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188900	https://www.sparkl.me/learn/ib/economics-hl/the-concept-of-equilibrium-price-and-quantity/revision-notes/1533	Past Papers Log In Past Papers Courses Collegeboard AP IB DP Cambridge IGCSE AS & A Level IB MYP 1-3 IB MYP 4-5 All Topics economics-hl \| ib Macroeconomics 1.1.1 Economic growth 1.1.2 Unemployment 1.1.3 Inflation 1.1.4 Income distribution 1.2.1 Causes and consequences of income inequality 1.2.2 The poverty trap and measures to address it 1.2.3 The role of government in reducing inequality 1.3.1 Tools of monetary policy (interest rates, reserve requirements) 1.3.2 Central banks and their role 1.3.3 Effects of monetary policy on the economy 1.4.1 Tools of fiscal policy (taxation, government spending) 1.4.2 Government budgets and deficit financing 1.4.3 Effects of fiscal policy on national income 1.5.1 Objectives of supply-side policies 1.5.2 Policies to improve productivity and economic efficiency 1.5.3 Labor market reforms, tax cuts, and deregulation 1.6.1 Gross Domestic Product (GDP) and other measures of national income 1.6.2 Nominal vs real GDP 1.6.3 Methods of measuring GDP 1.7.1 Aggregate demand (AD) and its components 1.7.2 Aggregate supply (AS) and its curve 1.7.3 Short-run vs long-run aggregate supply Introduction to Economics 2.1.1 Economic models and assumptions 2.1.2 The role of data and evidence in economics 2.1.3 Normative vs positive economics 2.2.1 Definition of economics 2.2.2 The problem of scarcity and choice 2.2.3 Basic economic questions: What, how and for whom? Global Economy 3.1.1 Protectionist vs free trade policies 3.1.2 Impact on consumers, producers and governments 3.1.3 Case studies of protectionist policies 3.2.1 Long-term development strategies 3.2.2 The role of technology and education in development 3.2.3 Policies for promoting economic growth 3.3.1 Free trade areas, customs unions and common markets 3.3.2 The role of organizations like the World Trade Organization (WTO) 3.3.3 Regional economic agreements (EU, NAFTA etc.) 3.4.1 Types of exchange rate systems: Floating vs fixed 3.4.2 Determination of exchange rates 3.4.3 Effects of exchange rate changes on the economy 3.5.1 Current account, capital account and financial account 3.5.2 Trade balance and its implications 3.5.3 Foreign direct investment and portfolio investment 3.6.1 Concept of sustainable development 3.6.2 Environmental and social implications of economic growth 3.6.3 Policies promoting sustainability 3.7.1 Comparative advantage and specialization 3.7.2 Gains from trade 3.7.3 Trade restrictions and their impact 3.8.1 Human Development Index (HDI) and other development indicators 3.8.2 Challenges in measuring development 3.8.3 Economic, social and environmental factors in development 3.9.1 Tariffs, quotas subsidies and trade barriers 3.9.2 Arguments for and against trade protection 3.9.3 Effects of protectionist policies on domestic and global markets 3.10.1 Factors inhibiting economic development 3.10.2 Structural challenges in developing economies 3.10.3 The role of international aid and debt relief Microeconomics 4.1.1 Effects of government policies on markets 4.1.2 Government intervention in markets 4.1.3 Price floors, price ceilings, taxes, subsidies 4.2.1 Positive and negative externalities 4.2.2 Solutions to externalities: taxes, subsidies, regulation 4.2.3 Common access resources and their management 4.3.1 Characteristics of public goods 4.3.2 The free-rider problem 4.3.3 Provision of public goods 4.4.1 Law of demand and its determinants 4.4.2 Elasticity of demand 4.4.3 Factors affecting demand 4.5.1 Definition and examples of asymmetric information 4.5.2 Adverse selection and moral hazard 4.5.3 Solutions to asymmetric information 4.6.1 Law of supply and its determinants 4.6.2 Elasticity of supply 4.6.3 Factors affecting supply 4.7.1 The concept of equilibrium price and quantity 4.7.2 Effects of shifts in supply and demand on equilibrium 4.7.3 Price mechanisms 4.8.1 Definition of market power 4.8.2 Monopolies and oligopolies 4.8.3 Price setting and efficiency loss 4.9.1 Assumptions of rational behaviour 4.9.2 Limitations of maximizing behaviour in real life 4.10.1 Distribution of resources in competitive markets 4.10.2 Role of the government in achieving equity 4.10.3 Market outcomes vs equity outcomes 4.11.1 Price elasticity of demand (PED) 4.11.2 Income elasticity of demand (YED) 4.11.3 Cross-price elasticity of demand (XED) 4.12.1 Price elasticity of supply (PES) 4.12.2 Factors affecting PES 4.12.3 Applications of PES in real markets Internal Assessment 5.1.1 Guidelines for commentaries on real-world economic issues 5.1.2 Connecting economic theory with empirical data 5.1.3 Writing and presenting commentaries using different economic concepts The concept of equilibrium price and quantity Topic 2/3 Revision Notes Flashcards Past Paper Analysis Questions Videos Your Flashcards are Ready! 15 Flashcards in this deck. or How would you like to practise? Choose Difficulty Level. Choose Easy, Medium or Hard to match questions to your skill level. Choose Learning Method. Choose Easy, Medium or Hard to match questions to your skill level. 3 Still Learning I know 12 Previous Next The Concept of Equilibrium Price and Quantity Introduction The concept of equilibrium price and quantity is fundamental in the study of microeconomics, particularly within the framework of competitive market equilibrium. Understanding how these equilibrium points are determined and their implications is crucial for students of the International Baccalaureate (IB) Economics Higher Level (HL) course. This article delves into the intricacies of equilibrium price and quantity, elucidating their roles in resource allocation and market dynamics. Key Concepts Definition of Equilibrium Price and Quantity In a competitive market, the equilibrium price is the price at which the quantity of a good demanded by consumers equals the quantity supplied by producers. This balance ensures that the market clears, meaning there is neither surplus nor shortage of the good. The corresponding quantity at this price is known as the equilibrium quantity. Market Demand and Supply To comprehend equilibrium, it's essential to first understand the concepts of market demand and supply. The market demand curve illustrates the relationship between the price of a good and the quantity demanded by consumers, typically sloping downward due to the law of demand. Conversely, the market supply curve portrays the relationship between the price and the quantity supplied by producers, usually sloping upward in accordance with the law of supply. Graphical Representation Equilibrium is visually represented at the intersection point of the demand and supply curves on a graph, where the price is plotted on the vertical axis and quantity on the horizontal axis. This intersection indicates the equilibrium price ($P^$) and equilibrium quantity ($Q^$). $$ \begin{align} \text{Demand Curve: } P = a - bQ \ \text{Supply Curve: } P = c + dQ \ \end{align} $$ Setting the demand equal to supply to find equilibrium: $$ a - bQ = c + dQ \ Q^ = \frac{a - c}{b + d} $$ $$ P^ = a - bQ^ = c + dQ^ $$ Determinants of Equilibrium Several factors can influence the equilibrium price and quantity, including changes in consumer preferences, income levels, prices of related goods, production costs, and technological advancements. Shifts in demand or supply curves result in new equilibrium points. Applications of Equilibrium Understanding equilibrium helps in analyzing various economic scenarios, such as the impact of government interventions like price floors and ceilings, taxation, and subsidies. It also aids in predicting the effects of external shocks and policy changes on market outcomes. Mathematical Derivation of Equilibrium Using algebra, the equilibrium price and quantity can be precisely calculated by solving the demand and supply equations simultaneously. $$ \begin{align} \text{Demand: } P &= 100 - 2Q \ \text{Supply: } P &= 20 + 3Q \ \end{align} $$ Setting them equal to find equilibrium: $$ 100 - 2Q = 20 + 3Q \ 80 = 5Q \ Q^ = 16 \ P^ = 100 - 2(16) = 68 $$ Thus, the equilibrium price is \$68, and the equilibrium quantity is 16 units. Market Efficiency at Equilibrium At equilibrium, markets are considered allocatively efficient because the resources are distributed in a manner where the marginal benefit equals the marginal cost. There is no deadweight loss, and the total surplus (sum of consumer and producer surplus) is maximized. Dynamic Nature of Equilibrium Equilibrium is not static; it can change in response to shifts in demand or supply. For instance, an increase in consumer income may shift the demand curve to the right, leading to a higher equilibrium price and quantity, assuming supply remains constant. Short-Run vs. Long-Run Equilibrium In the short run, firms may not be able to adjust all factors of production, leading to temporary disequilibrium. However, in the long run, as firms adjust their production processes, the market tends to reach a new equilibrium where economic profits are normal. Real-World Examples A practical example of equilibrium can be seen in the housing market. If the demand for houses increases due to population growth while the supply remains unchanged, housing prices will rise until a new equilibrium is established where the quantity demanded equals the quantity supplied. Limitations of the Equilibrium Model While the equilibrium model provides valuable insights, it assumes perfect competition, rational behavior, and full information, which may not always hold true in real markets. Additionally, it doesn't account for factors like externalities and market power which can lead to deviations from equilibrium. Advanced Concepts Mathematical Elasticity at Equilibrium Elasticity measures the responsiveness of quantity demanded or supplied to changes in price. At equilibrium, understanding price elasticity of demand and supply can provide deeper insights into how equilibrium responds to external changes. $$ \text{Price Elasticity of Demand: } E_d = \frac{\% \Delta Q_d}{\% \Delta P} $$ $$ \text{Price Elasticity of Supply: } E_s = \frac{\% \Delta Q_s}{\% \Delta P} $$ For example, if demand is highly elastic, a small increase in price can lead to a significant decrease in quantity demanded, thus affecting the equilibrium. Comparative Statics Analysis Comparative statics involves comparing two different equilibrium states before and after a change in an exogenous variable (like technology or taxes) to analyze the impact on equilibrium price and quantity. For instance, introducing a new technology that reduces production costs shifts the supply curve to the right, leading to a lower equilibrium price and a higher equilibrium quantity. Partial and General Equilibrium Partial equilibrium analysis focuses on a single market in isolation, assuming other markets remain unchanged. In contrast, general equilibrium considers the interconnections and feedback effects between multiple markets simultaneously, providing a more holistic view of the economy. Welfare Analysis Welfare analysis examines changes in consumer and producer surplus resulting from shifts in equilibrium. It assesses the overall economic well-being and efficiency of resource allocation. $$ \text{Consumer Surplus} = \frac{1}{2} \times (P_{\text{max}} - P^) \times Q^ $$ $$ \text{Producer Surplus} = \frac{1}{2} \times (P^ - P_{\text{min}}) \times Q^ $$ At equilibrium, the sum of consumer and producer surplus is maximized, indicating optimal welfare under the given market conditions. Externalities and Equilibrium Externalities, both positive and negative, can cause the market equilibrium to deviate from the socially optimal level. For example, pollution as a negative externality may lead to a higher equilibrium quantity than is socially desirable. Government Intervention and Equilibrium Governments often intervene in markets to correct market failures or achieve equity objectives. Tools like taxes, subsidies, price ceilings, and floors directly affect the equilibrium by shifting demand or supply curves. Dynamic Equilibrium and Time Path Dynamic equilibrium considers how the market adjusts over time to reach equilibrium after a disturbance. It involves the time path the price and quantity take to return to equilibrium following any exogenous shocks. Expectations and Equilibrium Expectations about future prices and market conditions influence current demand and supply, thereby affecting the current equilibrium. Rational expectations can lead to different equilibrium outcomes compared to adaptive expectations. Intertemporal Equilibrium Intertemporal equilibrium examines how equilibrium is maintained over different time periods, considering factors like savings, investment, and capital accumulation which bridge present and future markets. Interdisciplinary Connections The concept of equilibrium price and quantity intersects with other disciplines such as psychology, particularly in understanding consumer behavior, and with political science in the context of policy-making. For example, behavioral economics explores how psychological factors can lead to deviations from equilibrium by affecting rational decision-making. Case Study: Equilibrium in the Oil Market Analyzing the oil market provides a comprehensive case study of equilibrium dynamics. Factors like geopolitical tensions, technological advancements in extraction, and shifts towards renewable energy sources influence the demand and supply of oil, thereby affecting its equilibrium price and quantity on the global stage. Comparison Table \| Aspect \| Equilibrium Price \| Equilibrium Quantity \| --- \| Definition \| The price at which quantity demanded equals quantity supplied. \| The quantity of goods bought and sold at the equilibrium price. \| \| Determination \| Intersection of demand and supply curves. \| Resulting quantity from the equilibrium price. \| \| Impact of Increased Demand \| Rises to a new equilibrium. \| Increases if supply remains constant. \| \| Impact of Increased Supply \| Falls to a new equilibrium. \| Increases as equilibrium shifts. \| \| Graphical Representation \| Point where demand and supply curves intersect. \| Corresponding quantity at the intersection point. \| \| Real-World Example \| Housing prices adjusting to demand and supply changes. \| Number of houses sold at equilibrium price. \| Summary and Key Takeaways Equilibrium price and quantity are central to understanding market dynamics. They are determined by the intersection of demand and supply curves. Various factors can shift equilibrium, necessitating adjustments in price and quantity. Advanced analysis includes elasticity, welfare, and the impact of externalities. Real-world applications demonstrate the practical relevance of equilibrium concepts. Coming Soon! Examiner Tip Tips Use Graphs Effectively: Always draw and label your demand and supply curves to visualize equilibrium. This helps in understanding shifts and movements clearly. Memorize Key Formulas: Familiarize yourself with the equilibrium formulas to quickly calculate $P^$ and $Q^$ during exams. Practice Problem-Solving: Regularly solve various equilibrium problems to build confidence and accuracy. Focus on both numerical and theoretical questions. Did You Know Did You Know Did you know that the concept of equilibrium price dates back to the early 19th century with the work of economists like Alfred Marshall? Additionally, equilibrium analysis isn't limited to traditional markets; it's also applied in environmental economics to determine optimal pollution levels. Interestingly, some markets, such as the stock market, rarely reach true equilibrium due to constant fluctuations and investor sentiment. Common Mistakes Common Mistakes Incorrect Use of Demand and Supply Shifts: Students often confuse shifting the demand curve with moving along the curve. For example, increasing consumer income shifts the demand curve right, not a movement along the curve. Miscalculating Equilibrium: Another common error is incorrectly solving the equations for equilibrium. Ensure you set the demand equation equal to the supply equation and solve for the correct variable. Ignoring Units: Forgetting to include units when calculating equilibrium price and quantity can lead to incorrect conclusions. Always double-check that your units make sense in the context of the problem. FAQ What happens to equilibrium price and quantity if there is an increase in demand? An increase in demand shifts the demand curve to the right, leading to a higher equilibrium price and an increased equilibrium quantity, assuming supply remains constant. How does a price ceiling affect market equilibrium? A price ceiling set below the equilibrium price causes a shortage, as the quantity demanded exceeds the quantity supplied at that price. Can multiple equilibria exist in a single market? In most competitive markets, there is a single equilibrium point. However, in cases with non-linear demand and supply curves, multiple equilibria can theoretically exist. What is the difference between short-run and long-run equilibrium? Short-run equilibrium occurs when some factors of production are fixed, leading to possible temporary profits or losses. Long-run equilibrium assumes all factors are variable, resulting in normal profits and no incentives for firms to enter or exit the market. How do externalities disrupt market equilibrium? Externalities, such as pollution, cause the market equilibrium to deviate from the socially optimal level by either increasing or decreasing the equilibrium quantity and price without accounting for the external costs or benefits. Macroeconomics 1.1.1 Economic growth 1.1.2 Unemployment 1.1.3 Inflation 1.1.4 Income distribution 1.2.1 Causes and consequences of income inequality 1.2.2 The poverty trap and measures to address it 1.2.3 The role of government in reducing inequality 1.3.1 Tools of monetary policy (interest rates, reserve requirements) 1.3.2 Central banks and their role 1.3.3 Effects of monetary policy on the economy 1.4.1 Tools of fiscal policy (taxation, government spending) 1.4.2 Government budgets and deficit financing 1.4.3 Effects of fiscal policy on national income 1.5.1 Objectives of supply-side policies 1.5.2 Policies to improve productivity and economic efficiency 1.5.3 Labor market reforms, tax cuts, and deregulation 1.6.1 Gross Domestic Product (GDP) and other measures of national income 1.6.2 Nominal vs real GDP 1.6.3 Methods of measuring GDP 1.7.1 Aggregate demand (AD) and its components 1.7.2 Aggregate supply (AS) and its curve 1.7.3 Short-run vs long-run aggregate supply Introduction to Economics 2.1.1 Economic models and assumptions 2.1.2 The role of data and evidence in economics 2.1.3 Normative vs positive economics 2.2.1 Definition of economics 2.2.2 The problem of scarcity and choice 2.2.3 Basic economic questions: What, how and for whom? Global Economy 3.1.1 Protectionist vs free trade policies 3.1.2 Impact on consumers, producers and governments 3.1.3 Case studies of protectionist policies 3.2.1 Long-term development strategies 3.2.2 The role of technology and education in development 3.2.3 Policies for promoting economic growth 3.3.1 Free trade areas, customs unions and common markets 3.3.2 The role of organizations like the World Trade Organization (WTO) 3.3.3 Regional economic agreements (EU, NAFTA etc.) 3.4.1 Types of exchange rate systems: Floating vs fixed 3.4.2 Determination of exchange rates 3.4.3 Effects of exchange rate changes on the economy 3.5.1 Current account, capital account and financial account 3.5.2 Trade balance and its implications 3.5.3 Foreign direct investment and portfolio investment 3.6.1 Concept of sustainable development 3.6.2 Environmental and social implications of economic growth 3.6.3 Policies promoting sustainability 3.7.1 Comparative advantage and specialization 3.7.2 Gains from trade 3.7.3 Trade restrictions and their impact 3.8.1 Human Development Index (HDI) and other development indicators 3.8.2 Challenges in measuring development 3.8.3 Economic, social and environmental factors in development 3.9.1 Tariffs, quotas subsidies and trade barriers 3.9.2 Arguments for and against trade protection 3.9.3 Effects of protectionist policies on domestic and global markets 3.10.1 Factors inhibiting economic development 3.10.2 Structural challenges in developing economies 3.10.3 The role of international aid and debt relief Microeconomics 4.1.1 Effects of government policies on markets 4.1.2 Government intervention in markets 4.1.3 Price floors, price ceilings, taxes, subsidies 4.2.1 Positive and negative externalities 4.2.2 Solutions to externalities: taxes, subsidies, regulation 4.2.3 Common access resources and their management 4.3.1 Characteristics of public goods 4.3.2 The free-rider problem 4.3.3 Provision of public goods 4.4.1 Law of demand and its determinants 4.4.2 Elasticity of demand 4.4.3 Factors affecting demand 4.5.1 Definition and examples of asymmetric information 4.5.2 Adverse selection and moral hazard 4.5.3 Solutions to asymmetric information 4.6.1 Law of supply and its determinants 4.6.2 Elasticity of supply 4.6.3 Factors affecting supply 4.7.1 The concept of equilibrium price and quantity 4.7.2 Effects of shifts in supply and demand on equilibrium 4.7.3 Price mechanisms 4.8.1 Definition of market power 4.8.2 Monopolies and oligopolies 4.8.3 Price setting and efficiency loss 4.9.1 Assumptions of rational behaviour 4.9.2 Limitations of maximizing behaviour in real life 4.10.1 Distribution of resources in competitive markets 4.10.2 Role of the government in achieving equity 4.10.3 Market outcomes vs equity outcomes 4.11.1 Price elasticity of demand (PED) 4.11.2 Income elasticity of demand (YED) 4.11.3 Cross-price elasticity of demand (XED) 4.12.1 Price elasticity of supply (PES) 4.12.2 Factors affecting PES 4.12.3 Applications of PES in real markets Internal Assessment 5.1.1 Guidelines for commentaries on real-world economic issues 5.1.2 Connecting economic theory with empirical data 5.1.3 Writing and presenting commentaries using different economic concepts Get PDF PDF Share Explore How would you like to practise? 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188901	https://www.advent-rm.com/en-GB/Articles/2025/03/Cadmium-Shielding-in-Nuclear-Science-Enhancing-Saf?srsltid=AfmBOorPI9m6ap7KUXYsZ9Q3cqoME-Rkvt_ZAkiXk-BKe0D4BKwWm-Rg	Cadmium Shielding in Nuclear Science: Enhancing Safety, Accuracy, and Control - Advent Research Materials Skip to content Register Login Advent Research Materials Home Products Pure Metals Aluminium Gold Indium Platinum Silver Titanium View all Pure Metals Alloys Mumetal / Magnetic Shielding Alloy Platinum /Iridium Stainless Steel AISI 316L Titanium Grade 5 alloy Tungsten/Rhenium View all Alloys Polymers Polyester Polyetheretherketone (PEEK) Polyimide Polytetrafluoroethylene (PTFE) View all Polymers View Product InfoView All ProductsProduct Search About About Meet the Team Industries & Research Fields Articles Contact Search Basket HomeMenuSearchBasket You are here: 1. Home 2. Articles 3. 2025 4. March 5. Cadmium Shielding in Nuclear Science: Enhancing Safety, Accuracy, and Control Cadmium Shielding in Nuclear Science: Enhancing Safety, Accuracy, and Control 10th Mar 2025 Carli Goodfellow Carli Goodfellow Effective radiation shielding is fundamental to the safety and accuracy of nuclear measurement systems and the accurate measurement of radioactive isotopes is essential for both regulatory compliance and operational efficiency. Non-destructive assay (NDA) systems play a crucial role in this process, allowing for the measurement and monitoring of radioactive materials without altering their composition. One key material that enhances the performance of these systems is Cadmium. Why Cadmium? The science behind its shielding properties Cadmium is a transition metal with exceptional neutron absorption capabilities, making it an invaluable material in nuclear applications. A transition metal is an element found in the d-block of the periodic table, which means it has partially filled d orbitals in at least one of its oxidation states. These metals are known for their unique chemical and physical properties. Cadmium (Cd) is in Group 12 of the periodic table, alongside zinc (Zn) and mercury (Hg). While it is sometimes debated whether Group 12 elements should be considered true transition metals (because they don’t always follow the typical rules of transition metals, such as having an incomplete d-orbital), cadmium is often classified as one due to its ability to form complexes with ligands and exhibit metallic bonding characteristics. Its ability to absorb thermal neutrons without generating secondary radiation makes it particularly effective in shielding and filtering applications. These properties are especially beneficial in high-precision measurement systems used in nuclear safeguards, waste management, and isotopic analysis. Cadmium in nuclear measurement and shielding applications Cadmium is commonly used in nuclear measurements and various applications in the nuclear industry due to its ability to absorb thermal neutrons. This property arises from its exceptionally high neutron absorption cross-section, which allows it to capture low-energy neutrons via the (n,γ) capture reaction, making it an essential material for radiation shielding and measurement accuracy. Some of its key applications include: Selective filtering in radiation detection– Cadmium filters help reduce interference from unwanted radiation, enhancing the accuracy of isotopic and spectrometric analysis. Neutron absorption for improved measurement precision– Its high neutron absorption cross-section prevents unwanted neutron interactions, ensuring clear and reliable measurement results. Shielding in Non-Destructive Assay (NDA) systems – Used in high-accuracy nuclear measurement devices, cadmium plays a vital role in ensuring that radiation readings remain accurate and free from external interference. Nuclear waste assay and safeguards monitoring– Cadmium is used in systems that monitor and measure radioactive waste, ensuring compliance with strict safety regulations. Reactor control and radiation shielding – Found in control rods and neutron shields, cadmium helps regulate neutron flux in nuclear reactors and enhances safety in radiation-sensitive environments.Cadmium absorbs the slow-moving neutrons very easily, therefore, they control the rate of fission. Control rods increase the rate of reaction when they are pulled, but these rods decrease the rate of reaction when they are pushed. Safety Considerations While cadmium is highly effective in nuclear shielding applications, it is also a hazardous material. Cadmium and its compounds are toxic, with potential risks to human health and the environment.It is classified as a human carcinogen. Exposure to cadmium can lead to respiratory issues, kidney damage, and long-term environmental contamination if not handled properly. Strict safety protocols, including the use of protective equipment and proper disposal methods, must be followed to mitigate these risks and ensure safe usage in industrial and nuclear applications. You can find details of the possible health hazards of different forms of cadmium, the preventative measures your employer needs to apply and the precautions you should takehere. Cadmium's exceptional neutron absorption properties make it a crucial material in nuclear science, offering reliable shielding and improved measurement precision in critical applications. Whether in non-destructive assay systems, waste monitoring, or reactor shielding, its role is indispensable in maintaining safety and regulatory compliance. However, due to its toxicity, strict handling and disposal measures must be followed to mitigate environmental and health risks. For more information about our cadmium products and how they can enhance your nuclear measurement systems, contact us today. Share Share on Facebook Share on Twitter Share on LinkedIn We use cookies to give you the best experience of using this website. By continuing to use this site, you accept our use of cookies. Please read our Cookie Policy for more information. Accept and Close Advent Research Materials Home Signup to our newsletter Visit us on Twitter Visit us on LinkedIn More Info Products Articles Contact Shipping costs My Account Periodic Table Material Recycling Oakfield Industrial Estate Eynsham Oxford England OX29 4JA +44 1865 884 440 info@advent-rm.com Advent Research Materials Ltd is ISO 9001 Certified Payments Accepted Cookie Policy Privacy Policy Terms & Conditions
188902	https://en.wikipedia.org/wiki/List_of_California_urban_areas	Jump to content Search Contents (Top) 1 References 2 External links List of California urban areas Français Edit links Article Talk Read Edit View history Tools Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Download as PDF Printable version In other projects Wikidata item Appearance From Wikipedia, the free encyclopedia This is a list of urban areas in California as defined by the U.S. Census Bureau, ordered according to their 2020 census populations. The list includes urban areas with a population of at least 10,000. Rows in green indicate that part of the area lies outside of California. Rows without a rank indicate that the center of the area is outside of California. \| Rank \| Name[Note 1] \| Population(2020 census) \| --- \| 1 \| Los Angeles-Long Beach-Anaheim \| 12,237,376 \| \| 2 \| San Francisco-Oakland \| 3,515,933 \| \| 3 \| San Diego \| 3,070,300 \| \| 4 \| Riverside-San Bernardino \| 2,276,703 \| \| 5 \| Sacramento \| 1,946,618 \| \| 6 \| San Jose \| 1,837,446 \| \| 7 \| Fresno \| 717,589 \| \| 8 \| Mission Viejo-Lake Forest-Laguna Niguel \| 646,843 \| \| 9 \| Bakersfield \| 570,235 \| \| 10 \| Concord-Walnut Creek \| 538,583 \| \| 11 \| Temecula-Murrieta-Menifee \| 528,991 \| Reno, NV-CA \| 446,529 \| \| 12 \| Stockton \| 414,847 \| \| 13 \| Oxnard-Ventura \| 376,117 \| \| 14 \| Indio-Palm Desert-Palm Springs \| 361,075 \| \| 15 \| Palmdale-Lancaster \| 359,559 \| \| 16 \| Modesto \| 357,301 \| \| 17 \| Victorville-Hesperia-Apple Valley \| 355,816 \| \| 18 \| Antioch \| 326,205 \| \| 19 \| Santa Rosa \| 297,329 \| \| 20 \| Santa Clarita \| 278,031 \| \| 21 \| Livermore-Pleasanton-Dublin \| 240,381 \| \| 22 \| Thousand Oaks \| 213,986 \| \| 23 \| Santa Barbara \| 202,197 \| \| 24 \| Salinas \| 177,532 \| \| 25 \| Vallejo \| 175,132 \| \| 26 \| Hemet \| 173,194 \| \| 27 \| Santa Cruz \| 169,038 \| \| 28 \| Visalia \| 160,578 \| \| 29 \| Fairfield \| 150,122 \| \| 30 \| Merced \| 150,052 \| \| 31 \| Santa Maria \| 143,609 \| Yuma, AZ-CA \| 135,717 \| \| 32 \| Simi Valley \| 127,364 \| \| 33 \| Yuba City \| 125,706 \| \| 34 \| Seaside-Monterey-Pacific Grove \| 123,495 \| \| 35 \| Tracy-Mountain House \| 120,912 \| \| 36 \| Redding \| 120,602 \| \| 37 \| Gilroy-Morgan Hill \| 114,833 \| \| 38 \| Chico \| 111,411 \| \| 39 \| Vacaville \| 101,027 \| \| 40 \| Manteca \| 86,674 \| \| 41 \| Napa \| 84,619 \| \| 42 \| Madera \| 81,635 \| \| 43 \| Turlock \| 79,203 \| \| 44 \| Davis \| 77,034 \| \| 45 \| Camarillo \| 76,338 \| \| 46 \| El Centro \| 74,376 \| \| 47 \| Lodi \| 73,090 \| \| 48 \| Tulare \| 70,628 \| \| 49 \| Porterville \| 69,862 \| \| 50 \| Watsonville \| 68,668 \| \| 51 \| Paso Robles-Atascadero \| 67,804 \| \| 52 \| Hanford \| 66,638 \| \| 53 \| Petaluma \| 65,227 \| \| 54 \| Woodland \| 61,133 \| \| 55 \| San Luis Obispo \| 56,904 \| \| 56 \| Lompoc \| 54,287 \| \| 57 \| Arroyo Grande-Grover Beach-Pismo Beach \| 50,885 \| \| 58 \| Reedley--Dinuba \| 49,614 \| \| 59 \| Hollister \| 49,611 \| \| 60 \| Eureka \| 45,951 \| \| 61 \| Desert Hot Springs \| 45,767 \| \| 62 \| Los Banos \| 45,533 \| \| 63 \| Delano \| 44,410 \| \| 64 \| Fallbrook \| 41,305 \| \| 65 \| Oroville \| 40,190 \| \| 66 \| Calexico \| 38,491 \| \| 67 \| Grass Valley \| 36,720 \| \| 68 \| Selma \| 32,546 \| \| 69 \| Sonoma \| 31,479 \| \| 70 \| Auburn \| 31,371 \| \| 71 \| South Lake Tahoe, CA-NV \| 31,363 \| \| 72 \| Santa Paula \| 30,675 \| \| 73 \| Barstow \| 30,522 \| \| 74 \| Ridgecrest \| 29,307 \| \| 75 \| Sonora-Twain Harte \| 29,013 \| \| 76 \| Ukiah \| 28,987 \| \| 77 \| Sanger \| 27,325 \| \| 78 \| Lemoore \| 26,957 \| \| 79 \| Galt \| 26,618 \| \| 80 \| Brawley \| 26,270 \| \| 81 \| Oakdale \| 25,408 \| \| 82 \| Patterson \| 23,660 \| \| 83 \| Placerville-Diamond Springs \| 23,291 \| \| 84 \| Corcoran \| 22,377 \| \| 85 \| Crestline-Lake Arrowhead \| 22,272 \| \| 86 \| Wasco \| 22,235 \| \| 87 \| Half Moon Bay \| 21,688 \| \| 88 \| Nipomo \| 20,303 \| \| 89 \| Red Bluff \| 19,826 \| \| 90 \| Arcata \| 19,714 \| Incline Village, NV-CA \| 19,441 \| \| 91 \| Arvin \| 19,385 \| \| 92 \| Shafter \| 19,278 \| \| 93 \| Soledad \| 18,946 \| \| 94 \| Dixon \| 18,876 \| \| 95 \| Greenfield \| 18,858 \| \| 96 \| Sebastopol \| 18,734 \| \| 97 \| Yucca Valley \| 18,293 \| \| 98 \| Rosamond \| 17,538 \| \| 99 \| Clearlake \| 17,351 \| \| 100 \| Tehachapi-Golden Hills \| 17,298 \| \| 101 \| Big Bear City \| 16,498 \| \| 102 \| Fillmore \| 16,397 \| \| 103 \| Kerman \| 16,002 \| \| 104 \| Discovery Bay \| 15,939 \| \| 105 \| Ripon \| 15,829 \| \| 106 \| Crescent City \| 15,620 \| \| 107 \| Lamont \| 15,271 \| \| 108 \| Taft \| 15,022 \| \| 109 \| McKinleyville \| 14,981 \| \| 110 \| Ramona \| 14,837 \| \| 111 \| Parlier \| 14,522 \| \| 112 \| Livingston \| 14,255 \| \| 113 \| McFarland \| 14,149 \| \| 114 \| Los Osos \| 13,978 \| \| 115 \| Lindsay \| 13,942 \| \| 116 \| King City \| 13,760 \| \| 117 \| Mendota \| 13,382 \| \| 118 \| Alpine \| 13,307 \| \| 119 \| Avenal \| 13,304 \| \| 120 \| Chowchilla \| 13,196 \| \| 121 \| Morro Bay \| 13,163 \| \| 122 \| Coalinga \| 13,049 \| \| 123 \| Twentynine Palms \| 12,881 \| \| 124 \| Orosi \| 12,795 \| \| 125 \| Fortuna \| 12,784 \| \| 126 \| Truckee \| 12,756 \| \| 127 \| Kingsburg \| 12,602 \| \| 128 \| Newman \| 12,387 \| \| 129 \| Castroville-Prunedale \| 12,334 \| \| 130 \| Blythe, CA-AZ \| 11,780 \| \| 131 \| Twentynine Palms North \| 11,665 \| \| 132 \| Bishop \| 11,013 \| \| 133 \| Exeter \| 10,973 \| \| 134 \| Fort Bragg \| 10,668 \| \| 135 \| Solvang-Santa Ynez \| 10,295 \| \| 136 \| Delhi \| 10,274 \| References [edit] ^ In order to match the official lists from the U.S. Census Bureau and provide less clutter in the table, postal code abbreviations for the names of states are used in this column. For a list of the states and abbreviations used, please see the table below the map at this list of US States. ^ "Urban and Rural: List of 2020 Census Urban Areas". United States Census Bureau. Retrieved July 22, 2023. External links [edit] Cities portal California portal U.S. Census Bureau Los Angeles-Long Beach-Santa Ana UA data profile U.S. Census Bureau San Francisco-Oakland UA data profile U.S. Census Bureau San Diego UA data profile \| v t e State of California \| \| Sacramento (capital) \| \| Topics \| Culture + food + music + languages + California sound + sports + California Dream Crime Demographics Earthquakes Economy + agriculture Education Education, history of Environment Geography + climate + ecology + flora + fauna Government + Capitol + districts + governor + legislature + Supreme Court Healthcare + Abortion History Law LGBT rights National Historic Landmarks National Natural Landmarks NRHP listings Politics + congressional delegations + elections People Protected areas + state parks + state historic landmarks Symbols Transportation Water Index of articles \| \| Regions \| Antelope Valley Big Sur California Coast Ranges Cascade Range Central California Central Coast Central Valley Channel Islands Coachella Valley Coastal California Conejo Valley Cucamonga Valley Death Valley East Bay (SF Bay Area) East County (SD) Eastern California Emerald Triangle Gold Country Great Basin Greater San Bernardino Inland Empire Klamath Basin Lake Tahoe Greater Los Angeles Los Angeles Basin Lost Coast Mojave Desert Mountain Empire North Bay (SF) North Coast North County (SD) Northern California Orange Coast Owens Valley Oxnard Plain Peninsular Ranges Pomona Valley Sacramento–San Joaquin River Delta Sacramento Valley Saddleback Valley Salinas Valley San Fernando Valley San Francisco Bay Area San Francisco Peninsula San Gabriel Valley San Joaquin Valley Santa Clara Valley Santa Clara River Valley Santa Clarita Valley Santa Ynez Valley Shasta Cascade Sierra Nevada Silicon Valley South Bay (LA) South Bay (SD) South Bay (SF) South Coast Southern Border Region Southern California Transverse Ranges Tri-Valley Victor Valley Wine Country \| \| Metro regions \| Fresno–Madera Los Angeles-Long Beach-Anaheim Sacramento–Roseville Riverside–San Bernardino–Ontario San Diego–Tijuana San Jose–San Francisco–Oakland \| \| Counties \| Alameda Alpine Amador Butte Calaveras Colusa Contra Costa Del Norte El Dorado Fresno Glenn Humboldt Imperial Inyo Kern Kings Lake Lassen Los Angeles Madera Marin Mariposa Mendocino Merced Modoc Mono Monterey Napa Nevada Orange Placer Plumas Riverside Sacramento San Benito San Bernardino San Diego San Francisco San Joaquin San Luis Obispo San Mateo Santa Barbara Santa Clara Santa Cruz Shasta Sierra Siskiyou Solano Sonoma Stanislaus Sutter Tehama Trinity Tulare Tuolumne Ventura Yolo Yuba \| \| Most populouscities \| Los Angeles San Diego San Jose San Francisco Fresno Sacramento Long Beach Oakland Bakersfield Anaheim \| \| California portal \| Retrieved from " Categories: Incorporated cities and towns in California Regions of California California geography-related lists Hidden categories: Articles with short description Short description is different from Wikidata Use American English from June 2025 All Wikipedia articles written in American English Use mdy dates from July 2023 List of California urban areas Add topic
188903	https://www.doubtnut.com/qna/1460782	Show that f(x)=(log)ax, 0<a<1 is a decreasing function for all x>0 . f(x)=loga(x) 0<a<1 f'(x)=1xloga Since 0<a<1 therefore loga<0 Now x>0⇒1x>0⇒1xloga<0⇒f'(x)<0 so f(x) is decreasing for all x>0 Topper's Solved these Questions Explore 176 Videos Explore 1401 Videos Similar Questions Show that f(x)=logxx,x≠0 is maximum at x=e. Show that f(x)=(log)ax, 0<a<1 is a decreasing function for all x>0 . Knowledge Check Let f(x)=\|logx\| then If f(x) =\| log x\|, then The function f(x)=log x Show that f(x)=logsinx is increasing on (0, π/2) and decreasing on (π/2, π) . Show that f(x)=x√1+x−ln(1+x) is an increasing function for x>−1. Show that f(x)=2x+cot−1x+log(√1+x2−x) is increasing on R . Show that f(x)=2x+cot−1x+log(√1+x2−x) is increasing in R The function f(x)=xlogx RD SHARMA-INCREASING AND DECREASING FUNCTION -Solved Examples And Exercises Show that f(x)=e^(2x) is increasing on R . Show that f(x)=e^(1//x),\ \ x!=0 is a decreasing function for all x... Show that f(x)=(log)a x ,\ \ 0<a<1 is a decreasing function for all ... Show that f(x)=sinx is increasing on (0,\ pi//2) and decreasing on (pi... Show that f(x)=logsinx is increasing on (0,\ pi//2) and decreasing on ... Show that f(x)=x-sinx is increasing for all x in R . Show that f(x)=x^3-15 x^2+75 x-50 is an increasing function for all... Show that f(x)=cos^2x is a decreasing function on (0,\ pi//2) . Show that f(x)=sinx is an increasing function on (-pi//2,\ pi//2) . Show that f(x)=cosx is a decreasing function on (0,\ pi) , increasing ... Show that f(x)=tanx is an increasing function on (-pi//2,\ pi//2) . Show that f(x)=tan^(-1)(sinx+cosx) is a decreasing function on the ... Show that the function f(x)=sin(2x+pi//4) is decreasing on (3pi//8,\ 5... Show that the function f(x)=cot^(-1)(sinx+cosx) is decreasing on (0,\ ... Show that f(x)=(x-1)e^x+1 is an increasing function for all x >0 . Show that the function x^2-x+1 is neither increasing nor decreasing... Show that f(x)=x^9+4x^7+11 is an increasing function for all x in ... Prove that the function f(x)=x^3-6x^2+12 x-18 is increasing on R . State when a function f(x) is said to be increasing on an interval ... Show that f(x)=sinx-cosx is an increasing function on (-pi//4,\ pi//4)... Exams Free Textbook Solutions Free Ncert Solutions English Medium Free Ncert Solutions Hindi Medium Boards Resources Doubtnut is No.1 Study App and Learning App with Instant Video Solutions for NCERT Class 6, Class 7, Class 8, Class 9, Class 10, Class 11 and Class 12, IIT JEE prep, NEET preparation and CBSE, UP Board, Bihar Board, Rajasthan Board, MP Board, Telangana Board etc NCERT solutions for CBSE and other state boards is a key requirement for students. Doubtnut helps with homework, doubts and solutions to all the questions. It has helped students get under AIR 100 in NEET & IIT JEE. Get PDF and video solutions of IIT-JEE Mains & Advanced previous year papers, NEET previous year papers, NCERT books for classes 6 to 12, CBSE, Pathfinder Publications, RD Sharma, RS Aggarwal, Manohar Ray, Cengage books for boards and competitive exams. Doubtnut is the perfect NEET and IIT JEE preparation App. Get solutions for NEET and IIT JEE previous years papers, along with chapter wise NEET MCQ solutions. Get all the study material in Hindi medium and English medium for IIT JEE and NEET preparation Contact Us
188904	https://www.who.int/tools/elena/review-summaries/vitamina-pregnancy--vitamin-a-supplementation-during-pregnancy-for-maternal-and-newborn-outcomes	Skip to main content Home Health Topics All topics All topics A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Resources Fact sheets Facts in pictures Multimedia Podcasts Publications Questions and answers Tools and toolkits Popular Dengue Endometriosis Excessive heat Herpes Mental disorders Mpox Countries All countries All countries A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Regions Africa Americas Europe Eastern Mediterranean South-East Asia Western Pacific WHO in countries Data by country Country presence Country strengthening Country cooperation strategies Newsroom All news News releases Statements Campaigns Events Feature stories Press conferences Speeches Commentaries Photo library Headlines Emergencies Focus on Cholera Coronavirus disease (COVID-19) Greater Horn of Africa Israel and occupied Palestinian territory Mpox Sudan Ukraine Latest Disease Outbreak News Situation reports Rapid risk assessments Weekly Epidemiological Record WHO in emergencies Surveillance Alert and response Operations Research Funding Partners Health emergency appeals International Health Regulations Independent Oversight and Advisory Committee Data Dashboards Triple Billion Progress Health Inequality Monitor Delivery for impact COVID-19 dashboard Data collection Classifications SCORE Surveys Civil registration and vital statistics Routine health information systems Harmonized health facility assessment GIS centre for health Reports World Health Statistics UHC global monitoring report About WHO About WHO Partnerships Committees and advisory groups Collaborating centres Technical teams Organizational structure Who we are Our work Activities Initiatives General Programme of Work WHO Academy Funding Investment in WHO WHO Foundation Accountability External audit Financial statements Internal audit and investigations Programme Budget Results reports Governance Governing bodies World Health Assembly Executive Board Member States Portal All topics A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Resources Fact sheets Facts in pictures Multimedia Podcasts Publications Questions and answers Tools and toolkits Popular Dengue Endometriosis Excessive heat Herpes Mental disorders Mpox All countries A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Regions Africa Americas Europe Eastern Mediterranean South-East Asia Western Pacific WHO in countries Data by country Country presence Country strengthening Country cooperation strategies All news News releases Statements Campaigns Events Feature stories Press conferences Speeches Commentaries Photo library Headlines Kenya achieves elimination of human African trypanosomiasis or sleeping sickness as a public health problem 8 August 2025 News release WHO designates new WHO-Listed Authorities, strengthening global access to quality-assured medical products 7 August 2025 News release On World Breastfeeding Week, countries urged to invest in health systems and support breastfeeding mothers 4 August 2025 Joint News Release Focus on Cholera Coronavirus disease (COVID-19) Greater Horn of Africa Israel and occupied Palestinian territory Mpox Sudan Ukraine Latest Disease Outbreak News Situation reports Rapid risk assessments Weekly Epidemiological Record WHO in emergencies Surveillance Alert and response Operations Research Funding Partners Health emergency appeals International Health Regulations Independent Oversight and Advisory Committee Dashboards Triple Billion Progress Health Inequality Monitor Delivery for impact COVID-19 dashboard Data collection Classifications SCORE Surveys Civil registration and vital statistics Routine health information systems Harmonized health facility assessment GIS centre for health Reports World Health Statistics UHC global monitoring report About WHO Partnerships Committees and advisory groups Collaborating centres Technical teams Organizational structure Who we are Our work Activities Initiatives General Programme of Work WHO Academy Funding Investment in WHO WHO Foundation Accountability External audit Financial statements Internal audit and investigations Programme Budget Results reports Governance Governing bodies World Health Assembly Executive Board Member States Portal e-Library of Evidence for Nutrition Actions (eLENA) An online library of evidence-informed guidelines for nutrition interventions and single point of reference for the latest nutrition guidelines, recommendations and related information. Vitamin A supplementation during pregnancy for maternal and newborn outcomes Systematic review summary This document is a summary of findings and some data presented in the systematic review may therefore not be included. Please refer to the original publication cited below for a complete review of findings. Key Findings review Most of the data included in this review are from populations with a moderate level of background vitamin A deficiency Overall, vitamin A supplementation did not affect the risk of maternal mortality, perinatal mortality, neonatal mortality, stillbirth, neonatal anaemia, preterm birth, or low birth weight Antenatal vitamin A supplementation likely reduces maternal night blindness, and, for women who are HIV-positive or who are living in areas of vitamin A deficiency, maternal anaemia Maternal infection may also be reduced with vitamin A supplementation, although further evidence is needed to confirm this finding 1. Objectives To evaluate the effectiveness of supplementation with vitamin A or one of its derivatives during pregnancy, alone or in combination with other micronutrients, on maternal and newborn clinical and laboratory outcomes 2. How studies were identified The Cochrane Pregnancy and Childbirth Group’s Trials Register was searched to March 2015, which includes records identified from the following databases: CENTRAL (The Cochrane Library) MEDLINE EMBASE CINAHL Relevant journals and conference proceedings were included in the search, and reference lists from identified studies were also handsearched 3. Criteria for including studies in the review 3.1 Study type Randomized or quasi-randomized trials, including cluster-randomized trials 3.2 Study participants Pregnant women in areas with inadequate or adequate vitamin A intake, as defined by the WHO global database on vitamin A deficiency (Studies in which women were supplemented exclusively in the postnatal period were excluded) (Studies aimed at preventing vertical transmission of HIV were included if they met inclusion criteria and reported relevant outcomes) 3.3 Interventions Vitamin A supplementation alone or in combination with other micronutrients compared to placebo, no treatment, or micronutrients without vitamin A 3.4 Primary outcomes Maternal mortality (defined as death during pregnancy or within 42 days of cessation of pregnancy, for any cause related to or aggravated by pregnancy except accidental or incidental causes) Perinatal mortality (defined as number of stillbirths and deaths in the first week of life per 1000 live births) Secondary outcomes included neonatal mortality (death during the first 28 days of life), stillbirths, maternal anaemia (haemoglobin <11.0 g/dL), maternal clinical infection, maternal night blindness, preterm birth (<37 weeks’ gestation), neonatal anaemia, neonatal clinical infection, congenital malformations, low birth weight (<2.5 kg) 4. Main results 4.1 Included studies Nineteen trials involving over 310,000 women were included in this review: Three studies were cluster-randomized, and 16 were individually randomized. Two of the 19 trials were quasi-randomized Ten trials assessed vitamin A (or one of its derivatives) alone in comparison with a control group; five trials evaluated vitamin A (or one of its derivatives) in combination with other micronutrients in comparison to a control group; and four trials were multi-armed assessing both the use of vitamin A (or one of its derivatives) alone and vitamin A in combination with other supplements compared with a control group Control groups were exposed to a placebo, no treatment, or another intervention (for example, iron) Three trials were conducted in women with HIV infection The number of enrolled women ranged from 44 to over 207,000, and interventions ranged from a single day to over 18 weeks in duration Vitamin A doses ranged from 3000 IU per day up to 44,4000 IU per day, with some trials dosing on a weekly basis; single doses given at delivery were approximately 200,000 IU; and one trial used single doses of 600,000 IU given intra-muscularly or orally at the time of delivery 4.2 Study settings Bangladesh, China, Ghana (2 trials), India, Indonesia (6 trials), Malawi (3 trials), Nepal, South Africa, the United Kingdom of Great Britain and Northern Ireland, the United Republic of Tanzania, and the United States of America Sixteen studies were conducted in populations with moderate vitamin A deficiency, two were conducted in populations without vitamin A deficiency (the United Kingdom of Great Britain and Northern Ireland and the United States of America), and one study was conducted in a population with severe vitamin A deficiency (Nepal) Studies were predominantly conducted in rural settings 4.3 Study settings How the data were analysed Three comparisons were made: i) vitamin A alone versus placebo or no treatment; ii) vitamin A alone versus micronutrient supplements without vitamin A; and iii) vitamin A with other micronutrients versus micronutrients without vitamin A. Data from cluster-randomized trials were adjusted for clustering using standard methods. Fixed effects meta-analysis was used to produce pooled risk ratios (RR) with 95% confidence intervals (CI) where it was reasonable to assume that the treatment effects measured in the included trials were sufficiently similar. Where substantial heterogeneity was identified, random effects meta-analysis was used. The following subgroup analyses were planned for the primary outcomes maternal and perinatal mortality: By infant mortality rate in the country: high (≥30 per 1000 live births), low By maternal mortality rate in the country: high (≥100 per 100000 live births), low By prevalence of vitamin A deficiency in the country: high, low By prevalence of HIV infection in the country: high (>3%), low By dose of vitamin A: 10,000 IU/day, other doses By supplementation regimen: daily, weekly By duration of supplementation: ≤ one month, > one month By trimester of pregnancy at initiation of supplementation: pre-pregnancy; first, second, and third trimester Results Vitamin A alone versus placebo or no treatment Maternal and infant mortality The risk of maternal mortality was not significantly reduced with vitamin A supplementation in comparison to placebo or no treatment (RR 0.88, 95% CI [0.65 to 1.20], p=0.42; 4 studies/101,574 women). The risk of perinatal mortality was not affected by maternal vitamin A treatment (RR 1.01, 95% CI [0.95 to 1.07], p=0.74; 1 study/76,176 women), and neither was neonatal mortality (RR 0.97, 95% CI [0.90 to 1.05], p=0.50; 3 studies/89,556 women) or stillbirth (RR 1.04, 95% CI [0.98 to 1.10], p=0.19; 2 studies/122,850 women). Subgroup analyses did not alter results meaningfully. Additional outcomes Among women treated with vitamin A, the risk of anaemia was reduced by a statistically significant 36% (RR 0.64, 95% CI [0.43 to 0.94], p=0.025; 3 studies/3818 women). In addition, vitamin A supplementation significantly reduced the risk of maternal clinical infection (RR 0.45, 95% CI [0.20 to 0.99], p=0.047; 5 studies/1918 women), and maternal night blindness (RR 0.79, 95% CI [0.64 to 0.98], p=0.029; 2 studies/10,608 women). Vitamin A supplementation did not affect the risk of neonatal anaemia (RR 0.99, 95% CI [0.92 to 1.08], p=0.87; 1 study/409 women), preterm birth (RR 0.98, 95% CI [0.94 to 1.01], p=0.19; 5 studies/40,137 women), or the risk of having a low birth weight baby (RR 1.02, 95% CI [0.89 to 1.16], p=0.81; 4 studies/14,599). Adverse events There were no reports of side effects or adverse events in the included trials. Vitamin A alone versus micronutrient supplements without vitamin A In two studies including 591 women, vitamin A alone in comparison to micronutrients without vitamin A did not reduce the risk of maternal clinical infection (RR 0.99, 95% CI [0.83 to 1.18], p=0.91). Vitamin A with other micronutrients versus micronutrient supplements without vitamin A Supplementation with vitamin A and other micronutrients did not have a statistically significant effect on the risk perinatal mortality in comparison to treatment with micronutrients without vitamin A (RR 0.51, 95% CI [0.10 to 2.69], p=0.42; 1 study/179 women). There was also no evidence of an effect on the risk of stillbirth (RR 1.41, 95% CI [0.57 to 3.47], p=0.46; 2 studies/866 women), maternal anaemia (RR 0.86, 95% CI [0.68 to 1.09], p=0.22; 3 studies/706 women), maternal clinical infection (RR 0.95, 95% CI [0.80 to 1.13], 2 studies/597 women), preterm birth (RR 0.39, 95% CI [0.08 to 1.93], p=0.25; 1 study/136 women), congenital malformations (RR 0.34, 95% CI [0.04 to 3.18], 1 study/179 women), neonatal mortality (RR 0.65, 95% CI [0.32 to 1.31], p=0.23; 1 study/594 women), or neonatal anaemia (RR 0.75, 95% CI [0.38 to 1.51], p=0.43; 2 studies/1052 women). In one study of 594 women, the risk of low birth weight was reduced with vitamin A supplementation (RR 0.67, CI [0.47 to 0.96], p=0.027). 5. Additional author observations Overall, the risk of bias in the included trials was low to unclear, with all but three studies reporting adequate allocation concealment. For the main comparison (vitamin A alone versus placebo or no treatment), the quality of the evidence was assessed as being high for maternal mortality, perinatal mortality, and preterm birth; moderate for maternal anaemia; and low for maternal clinical infection. Populations studied differed with respect to baseline vitamin A status, which may have affected the results, and there were problems with follow-up in several large trials. Current evidence does not support the supplementation of pregnant women with vitamin A to reduce maternal or perinatal mortality. However, antenatal vitamin A supplementation was demonstrated to reduce maternal anaemia in vitamin A-deficient populations and in HIV-infected pregnant women, and maternal night blindness and infection were also reduced, although data contributing to the latter finding were not of high quality. Future trials need to assess baseline vitamin A status, as the effect of supplemental vitamin A is likely to depend on whether the participating population are vitamin A deficient or not. Further research on whether vitamin A supplementation reduces maternal infection is warranted, as are trials designed to determine the optimal dose of vitamin A and duration of supplementation for the prevention of maternal anaemia. Standardised international criteria for maternal and neonatal outcomes should also be employed in future research. The authors of the systematic review alone are responsible for the views expressed in this section. Corresponding interventions #### Vitamin A supplementation during pregnancy Corresponding systematic review #### Vitamin A supplementation during pregnancy for maternal and newborn outcomes
188905	https://www.chemteam.info/GasLaw/Four-Gas-Variables.html	The Four Gas Law Variables:Volume, Temperature, Pressure, and Amount KMT & Gas Laws Menu I. Volume All gases must be enclosed in a container that, if there are openings, can be sealed with no leaks. The three-dimensional space enclosed by the container walls is called volume. When the generalized variable of volume is discussed, the symbol V is used. Volume in chemistry is usually measured in liters (symbol = L) or milliliters (symbol = mL). A liter is also called a cubic decimeter (dm3). Other units of volume do occur such as cubic feet (cu. ft. or ft3) or cubic centimeters (cc or cm3). The main point to remember is: whatever units of volume are used, use them all the way through the problem. If you must convert from one unit to another, make sure you do it correctly. The ChemTeam discusses various gas laws where volume is a variable. The most common way to visualize this is to imagine the container involved in the experiment has a movable wall. When the volume goes up, that wall slides out. When the volume goes down, the wall slides in. Imagine the seal of the movable wall to be perfect so no gas escapes. If the volume is constant, then the container is made with rigid walls that cannot move. Within the limits of any experiment discussed, the walls remain fixed and the volume stays constant. In other words, the container walls never move (or break) during an experiment. II. Temperature All gases have a temperature, usually measured in degrees Celsius (symbol = °C). Note that Celsius is captalized since this was the name of a person (Anders Celsius). When the generalized variable of temperature is discussed, the symbol T is used. There is another temperature scale which is very important in gas behavior. It is called the Kelvin scale (symbol = K). Note that K does not have a degree sign and Kelvin is captalized because this was a person's title (Lord Kelvin, his given name was William Thomson). All gas law problems will be done with Kelvin temperatures. If you were to use degrees Celsius in any of your calculations, YOU WOULD BE WRONG. Your teacher may try and trip you up on this point. You can convert between Celsius and Kelvin like this: Kelvin = Celsius + 273.15 Often, the value of 273 is used instead of 273.15. Check with your teacher on this point. Amost all examples done by the ChemTeam will use 273. For example, 25 °C = 298 K, because 25 + 273 = 298. I will use 273 K (zero degrees Celsius) for standard temperature. The Kelvin temperature of a gas is directly proportional to its kinetic energy. (For example, double the Kelvin temperature, you double the kinetic energy.) This relationship will come into play from time to time. III. Pressure Gas pressure is created by the molecules of gas hitting the walls of the container. This concept is very important in helping you to understand gas behavior. Keep it solidly in mind. This idea of gas molecules hitting the wall will be used often. When the generalized variable of pressure is discussed, the symbol P is used. There are three different units of pressure used in chemistry. This is an unfortunate situation, but we cannot change it. You must be able to use all three in calculations as well as being able to convert fron one to another. Here they are: atmospheres (symbol = atm) millimeters of mercury (symbol = mmHg) Pascals (symbol = Pa) or, more commonly, kiloPascals (symbol = kPa) You will find more on conversions between pressure units here. Standard pressure is defined as 1.000 atm or 760.0 mmHg or 101.325 kPa. Let's pause here for a second. Make sure that you memorize the values for standard pressure. Repeat: I advise that have the above standard values memorized!! Many problems will simply say "standard pressure" and you have to already know the value. Here they are again: \| \| \| --- \| \| Pressure Unit \| Standard Value \| \| atm \| 1.000 \| \| mmHg \| 760.0 \| \| kPa \| 101.325 \| Standard temperature and pressure is a very common phrase in chemistry, so common it has been abbreviated to STP. What the ChemTeam will use as STP is actually called standard ambient temperature and pressure (STAP), but the difference between the two is unimportant at this stage of your chemistry training. There is no such thing as standard volume, although you will probably learn about molar volume in your class. IV. Amount of Gas The amount of gas present is measured in moles (symbol = mol) or in grams (symbol = g or gm). Typically, if grams are used, you will need to convert to moles at some point. When the generalized variable of amount in moles is discussed, the letter "n" is used as the symbol (note: the letter is in lowercase. Don't use uppercase.). KMT & Gas Laws Menu
188906	https://www-thphys.physics.ox.ac.uk/people/JohnChalker/theory/lecture-notes.pdf	M.Phys Option in Theoretical Physics: C6 John Chalker and Andre Lukas Physics Department, Oxford University 2010-2011 2 Chapter 1 Path Integrals in Quantum Mechanics 1.1 Mathematical Background In this ﬁrst part of the chapter, we will introduce functionals which are one of the main tools in modern theoretical physics and explain how to differentiate and integrate them. In the second part, these techniques will be applied to re-formulate quantum mechanics in terms of functional integrals (path integrals). 1.1.1 Functionals What is a functional? Let us ﬁrst consider a real function f : [a, b] →R, mapping all elements x ∈[a, b] in the domain [a, b] to real numbers f(x). A functional is similar to a function in that it maps all elements in a certain domain to real numbers, however, the nature of its domain is very different. Instead of acting on all points of an interval or some other subset of the real numbers, the domain of functionals consists of (suitably chosen) classes of functions itself. In other words, given some class {f} of functions, a functional F : {f} →R assigns to each such f a real number F[f]. Differentiation and integration are the two main operations on which the calculus of (real or complex) functions is based. We would like to understand how to perform analogous operations on functionals, that is, we would like to make sense of the expressions δF[f] δf , Z Df F[f] (1.1) for functional differentiation and integration. Here, we denote a functional derivative by δ/δf, replacing the ordinary d/dx or ∂/∂x for functions, and the measure of a functional integral by Df, replacing dx in the case of integrals over functions. A common application of ordinary differentiation is to ﬁnd the extrema of a function f by solving the equation f ′(x) = 0. Of course, one can also ask about the extrema of a functional F, that is the functions f which minimise (or maximise) the value F[f]. One important application of functional differentiation which we will discuss is how these extrema can be obtained as solutions to the equation δF[f] δf = 0 . (1.2) To make our discussion more concrete we will now introduce two (well-known) examples of functionals. Example: distance between two points A very simple functional F consists of the map which assigns to all paths between two ﬁxed points the length of the path. To write this functional explicitly, let us consider a simple two-dimensional situation in the (x, y) plane and choose two points (x1, y1) and (x2, y2). We consider all functions {f} deﬁning paths between those two points, that is functions on the interval [x1, x2] satisfying f(x1) = y1 and f(x2) = y2. The length of a path is then given by the well-known expression F[f] = Z x2 x1 dx p 1 + f ′(x)2 . (1.3) What is the minimum of this functional? Of course, we know that the correct answer is ”a line”, but how do we prove this formally? Let us approach this problem in a somewhat pedestrian way ﬁrst before re-formulating it in 3 4 CHAPTER 1. PATH INTEGRALS IN QUANTUM MECHANICS terms of functional derivatives along the lines of Eq. (1.2). Consider a small but otherwises abritrary perturbation ǫ of a path f which vanishes at the endpoints, that is, which satisﬁes ǫ(x1) = ǫ(x2) = 0. We can then deﬁne the function l(λ) ≡F[f + λǫ] = Z x2 x1 dx p 1 + (f ′(x) + λǫ′(x))2 , (1.4) where λ is a real parameter. A necessary condition for f to be a minimum of the functional F is then dl dλ(0) = 0 (1.5) Hence, this simple trick has reduced our functional minimisation problem to an ordinary one for the function l. The derivative of l at λ = 0 can be easily worked out by differentiating ”under the integral” in Eq. (1.4). One ﬁnds dl dλ(0) = Z x2 x1 dx f ′(x)ǫ′(x) p 1 + f ′(x)2 . (1.6) The derivative on ǫ in this expression can be removed by partial integration, keeping in mind that the boundary term vanishes due to ǫ being zero at x1 and x2. This results in dl dλ(0) = − Z x2 x1 dx ǫ(x) d dx " f ′(x) p 1 + f ′(x)2 # . (1.7) From Eq. (1.5) this integral needs to vanish and given that ǫ is an arbitrary function this is only the case if the integrand is zero pointwise in x. This leads to the differential equation d dx " f ′(x) p 1 + f ′(x)2 # = 0 , (1.8) for the function f. The desired extrema of the length functional F must be among the solutions to this differential equation which are given by the lines f ′(x) = const . (1.9) Note that the differential equation (1.8) is second order and consequently its general solution has two integration constant. They are precisely ”used up” by implementing the boundary conditions f(x1) = y1 and f(x2) = y2, so that we remain with a unique solution, the line between our two chosen points. Example: action in classical mechanics In physics, a very important class of functionals are action functionals. Let us recall their deﬁnition in the context of classical mechanics. Start with n generalised coordinates q(t) = (q1(t), . . . , qn(t)) and a Lagrangian L = L(q, ˙ q). Then, the action functional S[q] is deﬁned by S[q] = Z t2 t1 dt L(q(t), ˙ q(t)) . (1.10) It depends on classical paths q(t) between times t1 and t2 satisfying the boundary conditions q(t1) = q1 and q(t2) = q2. Hamilton’s principle states that the classical path of the system minimises the action functional S. To work out the implications of Hamilton’s principle we follow the same steps as in the previous example. We deﬁne the function l(λ) ≡S[q + λǫ] = Z t2 t1 dt L(q + λǫ, ˙ q + λ˙ ǫ) . (1.11) where we have introduced small but arbitrary variations ǫ = (ǫ1, . . . , ǫn) of the coordinates, satisfying ǫ(t1) = ǫ(t2) = 0. As before, we work out the derivative of l at λ = 0 dl dλ(0) = Z t2 t1 dt ∂L ∂qi (q, ˙ q) ǫi + ∂L ∂˙ qi (q, ˙ q) ˙ ǫi (1.12) = Z t2 t1 dt ǫi ∂L ∂qi (q, ˙ q) −d dt ∂L ∂˙ qi (q, ˙ q) , (1.13) 1.1. MATHEMATICAL BACKGROUND 5 where the last step follows from partial integration (and the boundary term vanishes due to ǫ(t1) = ǫ(t2) = 0 as previously). The ǫi vary independently and, hence, for the above integral to vanish the bracketed expression in (1.13) must be zero for each index value i. This gives rise to the Euler-Lagrange equations d dt ∂L ∂˙ qi (q, ˙ q) −∂L ∂qi (q, ˙ q) = 0 . (1.14) The solutions to these equations represent the classical paths of the system which minimise the action functional. They contain 2n integration constants which are ﬁxed by the boundary conditions q(t1) = q1 and q(t2) = q2. 1.1.2 Functional differentiation We have seen that, for the above examples, the problem of varying (or differentiating) functionals can be reduced to ordinary differentiation ”under the integral”. While this approach is transparent it is not particularly convenient for calculations. We would, therefore, like to introduce a short-hand notation for the variation procedure based on the functional derivative δ/δf(x) so that the minimum of a functional F can be obtained from the equation δF[f]/δf(x) = 0. The crucial step in our previous variation calculations was the ordinary differentiation with respect to the parameter λ. In order to reproduce the effect of this differentiation our functional derivative should certainly satisfy all the general properties of a derivative, that is, it should be linear, it should satisfy the product and chain rules of differentiation and we should be able to commute it with the integral. Let us see how far we can get with those assumptions, starting with our ﬁrst example (1.3). δF[f] δf(x) = δ δf(x) Z x2 x1 d˜ x p 1 + f ′(˜ x)2 = Z x2 x1 d˜ x f ′(˜ x) p 1 + f ′(˜ x)2 δf ′(˜ x) δf(x) (1.15) Our next step has been partial integration and in order to be able to carry this out we need to assume that ordinary and functional derivative commute. Then we ﬁnd δF[f] δf(x) = − Z x2 x1 d˜ x d d˜ x " f ′(˜ x) p 1 + f ′(˜ x)2 # δf(˜ x) δf(x) (1.16) In the bracket we recognise the desired left-hand-side of the differential equation (1.8). Our last step consisted of removing the integral due to the presence of an arbitrary variation ǫ in the integrand. Here, we can formally reproduce this step by demanding the relation δf(˜ x) δf(x) = δ(˜ x −x) , (1.17) which can be viewed as a continuous version of the equation ∂qi/∂qj = δij for a set of n coordinates qi. Using the relation (1.17) in Eq. (1.16) we ﬁnally obtain the desired result δF[f] δf(x) = −d dx " f ′(x) p 1 + f ′(x)2 # . (1.18) To summarise, functional derivation δ/δf(x) can be understood as a linear operation which satisﬁes the product and chain rules of ordinary differentiation, commutes with ordinary integrals and derivatives and is subject to the ”normalisation” (1.17). Armed with this ”deﬁnition”, let us quickly re-consider our second example, Hamilton’s principle of classical mechanics. We ﬁnd δS[q] δqi(t) = δ δqi(t) Z t2 t1 d˜ t L(q(˜ t), ˙ q(˜ t)) (1.19) = Z t2 t1 d˜ t ∂L ∂qj (q, ˙ q) δqj(˜ t) δqi(t) + ∂L ∂˙ qj (q, ˙ q) δ ˙ qj(˜ t) δqi(t) (1.20) = Z t2 t1 d˜ t ∂L ∂qj (q, ˙ q) −d d˜ t ∂L ∂˙ qj (q, ˙ q) δqj(˜ t) δqi(t) (1.21) = Z t2 t1 d˜ t ∂L ∂qj (q, ˙ q) −d d˜ t ∂L ∂˙ qj (q, ˙ q) δijδ(˜ t −t) = ∂L ∂qi (q, ˙ q) −d dt ∂L ∂˙ qi (q, ˙ q) (1.22) 6 CHAPTER 1. PATH INTEGRALS IN QUANTUM MECHANICS where, in the second last step, we have used the obvious generalisation δqj(˜ t) δqi(t) = δijδ(˜ t −t) (1.23) of Eq. (1.17). 1.1.3 Functional integration As before, we consider a class of functions {f}, f = f(x), deﬁned on a speciﬁc domain {x} and a functional F[f]. Here, we slightly generalise our previous set-up and allow x to represent a set of coordinates in various numbers of dimensions. For example, in the context of (quantum) mechanics, x is one-dimensional and represents time, so that we are dealing with functions f of one real variable. In (quantum) ﬁeld theory, on the other hand, x stands for a four-vector xµ. As we will see, most of the discussion in this subsection is independent of the space-time dimension, so we do not need to be more speciﬁc until later. We would now like to introduce a functional integral (1.1) over F. While ordinary integrals have an integration range which may consist of an interval in the real numbers, the integration range for functional integrals is a whole class of functions. Such integrals are not easy to deﬁne and, in these notes, we will make no attempt at mathematical rigour. Instead, we will concentrate on the features of functional integrals relevant to our applications in quantum mechanics and quantum ﬁeld theory. To ”deﬁne” functional integrals we can take a lead from the deﬁnition of ordinary integrals. There, one intro-duces a grid (or lattice) of points {xi} with separation ∆x in space-time and approximates the function f = f(x) by the collection of its values y = (y1, . . . , yn), where yi = f(xi), at the lattice points. The conventional integral over f (in the one-dimensional case) can then be deﬁned as the ∆x →0 limit of ∆x P i yi. For functional integrals we have to integrate over all functions f and, hence, in the discrete case, over all possible vectors y. This suggest, we should ”approximate” the functional integral measure R Df by multiple ordinary integrals R Πidyi = R dny. We also need a discrete version ˜ F of the functional such that ˜ F(y) ∼F[f] for y ∼f. Then, we can formally introduce functional integrals by setting Z Df F[f] ∼lim ∆x→0 Z dny ˜ F(y) . (1.24) In practise, this relation means we can often represent functional integrals by multiple ordinary integrals or, at least, deduce properties of functional integrals from those of multiple integrals. In general, functional integrals are prohibitively difﬁcult to evaluate and often a lattice calculation (that is, the numerical evaluation of the RHS of Eq. (1.24) on a computer) is the only option. However, there is one class of in-tegrals, namely Gaussian integrals and integrals closely related to Gaussian ones, which is accessible to analytical computations. It turns out that such Gaussian functional integrals are of central importance in physics and we will focus on them in the remainder of this section. In keeping with the spirit of Eq. (1.24), the discussion will center on ordinary multiple Gaussian integrals from which we deduce properties of Gaussian functional integrals. Gaussian integral over a single variable As a reminder, we start with a simple one-dimensional Gaussian integral over a single variable y. It is given by I(a) ≡ Z ∞ −∞ dy exp(−1 2ay2) = r 2π a , (1.25) where a is real and positive. The standard proof of this relation involves writing I(a)2 as a two-dimensional integral over y1 and y2 and then introducing two-dimensional polar coordinates r = p y2 1 + y2 2 and ϕ. Explicitly, I(a)2 = Z ∞ −∞ dy1 exp(−1 2ay2 1) Z ∞ −∞ dy2 exp(−1 2ay2 2) = Z ∞ −∞ dy1 Z ∞ −∞ dy2 exp(−1 2a(y2 1 + y2 2)) (1.26) = Z 2π 0 dϕ Z ∞ 0 dr r exp(−1 2ar2) = 2π a . (1.27) 1.1. MATHEMATICAL BACKGROUND 7 Multidimensional Gaussian integrals Next we consider n-dimensional Gaussian integrals W0(A) ≡ Z dny exp −1 2yT Ay , (1.28) over variables y = (y1, . . . , yn), where A is a symmetric, positive deﬁnite matrix. This integral can be easily reduced to a product of one-dimensional Gaussian integrals by diagonalising the matrix A. Consider an orthogonal rotation O such that A = ODOT with a diagonal matrix D = diag(a1, . . . , an). The Eigenvalues ai are strictly positive since we have assumed that A is positive deﬁnite. Introducing new coordinates ˜ y = Oy we can write yT Ay = ˜ yT D˜ y = n X i=1 ai˜ y2 i , (1.29) where the property OT O = 1 of orthogonal matrices has been used. Note further that the Jacobian of the coordi-nate change y →˜ y is one, since \|det(O)\| = 1. Hence, using Eqs. (1.25) and (1.29) we ﬁnd for the integral (1.28) W0(A) = n Y i=1 Z d˜ yi exp(−1 2ai˜ y2 i ) = (2π)n/2(a1a2 . . . an)−1/2 = (2π)n/2(detA)−1/2 . (1.30) To summarise, we have found for the multidimensional Gaussian integral (1.28) that W0(A) = (2π)n/2(detA)−1/2 , (1.31) a result which will be of some importance in the following. One obvious generalisation of the integral (1.28) involves adding a term linear in y in the exponent, that is W0(A, J) ≡ Z dny exp −1 2yT Ay + JT y . (1.32) Here J = (J1, . . . , Jn) is an n-dimensional vector which can be viewed as an external source for y. With the inverse matrix ∆= A−1 and a change of variables y →˜ y, where y = ∆J + ˜ y (1.33) this integral can be written as W0(A, J) = exp 1 2JT ∆J Z dn˜ y exp −1 2 ˜ yT A˜ y . (1.34) The remaining integral is Gaussian without a linear term, so can be easily carried out using the above results. Hence, one ﬁnds W0(A, J) = W0(A) exp 1 2JT ∆J = (2π)n/2(detA)−1/2 exp 1 2JT ∆J . (1.35) The exponential in Gaussian integrals can be considered as a probability distribution and, hence, we can intro-duce expectation values for functions O(y) by ⟨O⟩0 ≡N Z dny O(y) exp −1 2yT Ay . (1.36) The normalisation N is chosen such that ⟨1⟩= 1 which implies N = W0(A)−1 = (2π)−n/2(detA)1/2 . (1.37) The sub-script 0 refers to the fact that the expectation value is computed with respect to a Gaussian distribution. As we will see later, such expectation values play a central role when it comes to extracting physical information from a quantum ﬁeld theory. Of particular importance are the moments of the Gaussian probability distribution (or 8 CHAPTER 1. PATH INTEGRALS IN QUANTUM MECHANICS l-point functions) which are given by the expectation values of monomials, ⟨yi1yi2 . . . yil⟩0. From Eqs. (1.32) and (1.35) they can be written as ⟨yi1yi2 . . . yil⟩0 = N ∂ ∂Ji1 . . . ∂ ∂Jil W0(A, J) J=0 = ∂ ∂Ji1 . . . ∂ ∂Jil exp 1 2JT ∆J J=0 . (1.38) The ﬁrst equality above suggests an interpretation of the Gaussian integral W0(A, J) with source J as a generating function for the l-point functions. The second equality provides us with a convenient way of calculating l-point functions explicitly. It is easy to see, for example, that the two-point and four-point functions are given by ⟨yiyj⟩0 = ∆ij , ⟨yiyjykyl⟩0 = ∆ij∆kl + ∆ik∆jl + ∆il∆jk . (1.39) Every differentiation with respect to Ji in Eq. (1.38) leads to a factor of ∆ijJj in front of the exponential. Unless Jj is removed by another differentiation the contribution vanishes after setting J to zero. In particular, this means that all odd l-point functions vanish. For the even l-point functions one deduces that the ∆ij terms which appear are in one-to-one correspondence with all ways of pairing up the indices i1, . . . , il. Hence, for even l we have ⟨yi1 . . . yil⟩0 = X pairings p ∆ip1 ip2 . . . ∆ipl−1 ipl (1.40) where the sum runs over all pairings p = {(p1, p2), . . . , (pl−1, pl)} of the numbers {1, . . ., l}. This statement is also known as Wick’s theorem. For l = 2 and l = 4 this formula reproduces the results (1.39) for the two- and four-point functions. We would now like to use some of the above formulae for multidimensional Gaussian integrals to infer analo-gous formulae for Gaussian functional integrals, following the corrrespondence (1.24). We start with the analogue of W0(A, J) in Eq. (1.32), the generating functional W0[A, J] = Z Df exp −1 2 Z dx Z d˜ x f(˜ x)A(˜ x, x)f(x) + Z dx J(x)f(x) . (1.41) Note, that the summations in the expondent of Eq. (1.32) have been replaced by integrations. The ”kernel” A(˜ x, x) is the generalisation of the matrix A and, in our applications, will be usually a differential operator. It is straight-forward to write down the analogue of the result (1.35) for the generating functional W0[A, J] = exp −1 2tr ln A exp 1 2 Z d˜ x Z dxJ(˜ x)∆(˜ x, x)J(x) . (1.42) where ∆= A−1 is the inverse of the operator A and we have used the well-known matrix identity detA = exp tr ln A to re-write the determinant of A in terms of its trace. (We have absorbed the powers of 2π in front of Eq. (1.35) into the deﬁnition of the path integral measure Df.) It is not clear, at this stage, how to compute the inverse and the trace of the operator A, but we will see that, in many cases of interest, this can be accomplished by looking at the Fourier transform of A. For now, we proceed formally and deﬁne the l-point functions ⟨f(x1) . . . f(xl)⟩0 = N Z Df f(x1) . . . f(xl) exp −1 2 Z dx Z d˜ x f(˜ x)A(˜ x, x)f(x) , (1.43) where the normalisation N is given by N = exp( 1 2tr ln A). In analogy with Eq. (1.38) we have ⟨f(x1) . . . f(xl)⟩0 = δ δJ(x1) . . . δ δJ(xl) exp 1 2 Z d˜ x Z dx J(˜ x)∆(˜ x, x)J(x) J=0 . (1.44) This expression can be used to compute the l-point functions in terms of ∆, where Eq. (1.17) should be taken into account. Since functional derivatives have the same general properties than ordinary derivatives Wick’s theorem applies and the l-point functions take the form ⟨f(x1) . . . f(xl)⟩0 = X pairings p ∆(xp1, xp2) . . . ∆(xpl−1, xpl) (1.45) 1.1. MATHEMATICAL BACKGROUND 9 Steepest descent approximation Although functional integrals in the context of quantum mechanics or quantum ﬁeld theory are, in general, non-Gaussian they can be approximated by Gaussian integrals in many situations of physical interest. The steepest descent method is one such approximation scheme which is closely related to the transition between quantum and classical physics (as will be explained later). As usual, we start with simple one-dimensional integrals and work our way towards path integrals. Consider the integral I(ξ) = Z dy exp(−s(y)/ξ) (1.46) with a real function s = s(y) and a real positive parameter ξ. (In our physical examples the role of ξ will be played by ℏ.) We would like to study the value of this integral for small ξ. For such values, the main contributions to the integral come from regions in y close to the minima of s, that is points y(c) satisfying s′(y(c)) = 0. Around such a minimum we can expand s(y) = s(y(c)) + 1 2ξaǫ2 + . . . , (1.47) where ǫ = (y −y(c))/√ξ, a = s′′(y(c)) and the dots stand for terms of order higher than ξ. Then, we can approximate I(ξ) ≃ p ξ exp(−s(y(c))/ξ) Z dǫ exp −1 2aǫ2 . (1.48) The remaining integral is Gaussian and has already been computed in Eq. (1.25). Inserting this result we ﬁnally ﬁnd I(ξ) = r 2πξ a exp(−s(y(c))/ξ) . (1.49) To leading order the integral is simply given by the exponential evaluated at the minimum y(c) and the square root pre-factor can be interpreted as the ﬁrst order correction. Let us generalise this to the multi-dimensional integrals W(ξ) = Z dny exp(−S(y)/ξ) , (1.50) where y = (y1, . . . , yn). Around a solution y(c) of ∂S ∂yi (y) = 0 we expand S(y) = S(y(c)) + 1 2ξǫT Aǫ + . . . , (1.51) where ǫ = (y −y(c))/√ξ and Aij = ∂2S ∂yi∂yj (y(c)). Using this expansion and the result (1.31) for the multi-dimensional Gaussian integral we ﬁnd W(ξ) = (2πξ)n/2 (detA)−1/2 exp(−S(y(c))/ξ) . (1.52) We would now like to apply the steepest descent method to the generating functional for l-point functions. In analogy with our procedure for Gaussian integrals, we can introduce this generating function by W(J, ξ) = Z dny exp −1 ξ (S(y) −JT y) (1.53) so that the l-point functions are given by ⟨yi1 . . . yil⟩≡N Z dny yi1 . . . yil exp(−S(y)/ξ) = ξlN ∂ ∂Ji1 . . . ∂ ∂Jil W(J, ξ) J=0 . (1.54) Minima y(c) of the exponent in Eq. (1.53) are obtained from the equations Ji = ∂S ∂yi . (1.55) Applying the result (1.52) to the generating function (1.53) one ﬁnds W(J, ξ) ∼(detA)−1/2 exp −1 ξ (S(y(c)) −JT y(c)) (1.56) 10 CHAPTER 1. PATH INTEGRALS IN QUANTUM MECHANICS From Eq. (1.55), we should think of y(c) as a function of the source J which has to be inserted into the above result for W(J, ξ). The l-point functions in the steepest descent approximation can then be computed by inserting the so-obtained W(J, ξ) into Eq. (1.54). Finally, we apply the steepest descent method to functional integrals of the form Wξ = Z Df exp(−S[f]/ξ) , (1.57) where S[f] is a functional. In our examples S will be the action of a physical system. The steepest descent approximation now amounts to an expansion around the ”classical solution” f (c) characterised by δS δf [f (c)] = 0 . (1.58) With A(˜ x, x) = δ2S δf(˜ x)δf(x)[f (c)] (1.59) we ﬁnd in analogy with the multi-dimensional steepest descent approximation (1.52) Wξ ∼exp −1 2tr ln A exp −S[f (c)]/ξ . (1.60) Note that the leading contribution to the functional integral in this approximation comes from the classical path f (c). We can now introduce the generating functional Wξ[J] = Z Df exp −1 ξ S[f] − Z dx J(x)f(x) , (1.61) which generates the l-point functions ⟨f(xi1) . . . f(xil)⟩≡N Z Df f(xi1) . . . f(xil) exp(−S[f]/ξ) = ξlN δ δJ(xi1) . . . δ δJ(xi1)Wξ[J] J=0 . (1.62) The steepest descent approximation can then be applied around the solution f (c) = f (c)(J) of J(x) = δS δf(x) (1.63) and leads to the result Wξ[J] ∼exp −1 2tr ln A exp −1 ξ S[f (c)] − Z dx J(x)f (c)(x) (1.64) 1.1.4 Perturbation theory for non-Gaussian functional integrals Perturbation theory is an approximation scheme which can be applied to integrals which differ only slightly from Gaussian ones. Let us start with the multi-dimensional case and consider the non-Gaussian integral (1.50) with the speciﬁc exponent S = 1 2yT Ay + λV (y), that is, Wλ(A) = Z dny exp −1 2yT Ay −λV (y) . (1.65) (We can set ξ to one for the purpose of this section.) Here, we take the ”interaction term” V (y) to be a polynomial in y and λ is the ”coupling constant”. Expanding the part of the exponential which involves the ”perturbation” λV , we can write Wλ(A) = W0(A) ∞ X k=0 (−λ)k k! ⟨V (y)k⟩0 , (1.66) where the expectation value ⟨. . .⟩0 is deﬁned with respect to the Gaussian measure, as in Eq. (1.36). Given that V (y) is polynomial each of these expectation values can be calculated using Wick’s theorem (1.40). Of course, the resulting expressions will be progressively more complicated for increasing powers in λ but, provided λ is 1.1. MATHEMATICAL BACKGROUND 11 sufﬁciently small, we can hope that the integral is well-approximated by cutting the inﬁnite series in (1.66) off at some ﬁnite value of k. Let us see how this works in practise by focusing on the example of a quartic interaction term V (y) = 1 4! n X i=1 y4 i . (1.67) Neglecting terms of order λ3 and higher we have Wλ(A) W0(A) = 1 −λ 4! X i ⟨y4 i ⟩0 + λ2 2(4!)2 X i,j ⟨y4 i y4 j ⟩0 + O(λ3) (1.68) = 1 −λ 8 X i ∆2 ii + λ2 128 X i ∆2 ii X j ∆2 jj + λ2 48 X i,j (3∆ii∆jj∆2 ij + ∆4 ij) + O(λ3) (1.69) where Wick’s theorem (1.40) has been used in the second step. There is a very useful representation of expressions of the form ∆i1i2 . . . ∆il−1il in terms of diagrams. First, recall that the indices i, j, . . . label points xi, xj, . . . of our discrete space-time. This suggests we should represent each such index by a point (or dot). Each ∆ij is then represented by a line connecting dot i and dot j. One may think of the quantity ∆ij as ”moving the system” from xi to xj and it is, therefore also called propagator. Given these rules, the diagrams for the terms at order λ and order λ2 in Eq. (1.69) are shown in Figs. 1.1 and 1.2, respectively. Obviously, these diagrams are constructed by i Figure 1.1: Feynman diagram contributing to Wλ(A) at ﬁrst order in perturbation theory for a quartic interaction. i j i j i j Figure 1.2: Feynman diagrams contributing to Wλ(A) at second order in perturbation theory for a quartic interac-tion. joining up four-leg vertices, one for each power of the coupling λ. The reason we are dealing with four-leg vertices is directly related to the quartic nature of the interaction term, as an inspection of Wick’s theorem (1.40) shows. The diagrams associated to the order λk terms in the perturbation expansion can then be obtained by connecting the legs of k four-vertices in all possible ways. We are also interested in computing the l-point functions ⟨yi1 . . . yil⟩λ = Wλ(A)−1G(l) i1...il , G(l) i1...il ≡ Z dny yi1 . . . yil exp −1 2yT Ay −λV (y) (1.70) in perturbation theory. Here, we have deﬁned the Green functions G(l) i1...il. Since we have already discussed Wλ(A) the Green functions are what we still need to compute, in order to fully determine the l-point functions. One way to proceed is to simply expand the Green function in powers of λ and write the individual terms as Gaussian expectation values (1.36). This leads to G(l) i1...il = W0(A) ∞ X k=0 (−λ)k k! ⟨yi1 . . . yilV (y)k⟩0 . (1.71) Each Gaussian expectation value in this expansion can be calculated using Wick’s theorem. Alternatively, in analogy with the Gaussian case, we may also introduce the generating function Wλ(A, J) = Z dny exp −1 2yT Ay −λV (y) + JT y (1.72) 12 CHAPTER 1. PATH INTEGRALS IN QUANTUM MECHANICS so that the Green functions can be written as G(l) i1...il = ∂ ∂Ji1 . . . ∂ ∂Jil Wλ(A, J) J=0 . (1.73) We can now expand the generating function into a perturbation series Wλ(A, J) = ∞ X k=0 (−λ)k k! Z dny V (y)k exp −1 2yT Ay + JT y = ∞ X k=0 (−λ)k k! V ∂ ∂J k W0(A, J) , (1.74) around the Gaussian generating function W0. The partial differential operator V (∂/∂J) is obtained by formally replacing yi with ∂/∂Ji in the function V = V (y). Combining the last two equations we ﬁnd G(l) i1...il = ∞ X k=0 (−λ)k k! " ∂ ∂Ji1 . . . ∂ ∂Jil V ∂ ∂J k W0(A, J) # J=0 (1.75) = W0(A) ∞ X k=0 (−λ)k k! " ∂ ∂Ji1 . . . ∂ ∂Jil V ∂ ∂J k exp 1 2JT ∆J # J=0 , (1.76) where we have used Eq. (1.35) in the second step. From our result (1.38) for Gaussian l-point functions this expression for the Green function is of course equivalent to the earlier one (1.71). Either way, we end up having to compute each term in the perturbation series applying Wick’s theorem. Let us see how this works explicitly by computing the two-point function for the quartic perturbation (1.67) up to ﬁrst order in λ. We ﬁnd for the Green function G(2) ij /W0(A) = ⟨yiyj⟩0 −λ⟨yiyjV (ym)⟩0 + O(λ2) (1.77) = ∆ij −λ 24∆ij X m < y4 m >0 −λ 2 X m ∆mi∆mm∆mj + O(λ2) . (1.78) The three terms in this expression can be represented by the Feynman diagrams shown in Fig. 1.3. At zeroth order i j i j m i j m Figure 1.3: Feynman diagram contributing to two-leg Green function at zeroth and ﬁrst order in perturbation theory for a quartic interaction. in λ the two-point function is simply given by the propagator and at ﬁrst order we have two diagrams. The ﬁrst of these, however, is disconnected and consists of a propagator and a ”vacuum bubble” which we have earlier (see Fig. 1.1) recognised as one of the contributions to Wλ(A). If we compute the two-point function from Eq. (1.70) by inserting the Green function (1.78) and our earlier result (1.69) for Wλ(A) (dropping terms of order λ2) we ﬁnd that this disconnected term cancels, so that ⟨yiyj⟩λ = ∆ij −λ 2 X m ∆mi∆mm∆mj + O(λ2) . (1.79) Hence, the two-point function up to ﬁrst order in λ consists of the two connected diagrams in Fig. 1.3 only. The cancellation of disconnected diagrams in the two-point function is an attractive feature since it reduces the number of diagrams and removes ”trivial” contributations. However, in general, l-point functions still do contain disconnected diagrams. For example, the four-point function at order λ2 has a disconnected contribution which consists of a propagator plus the third diagram in Fig 1.3. It is, therefore, useful to introduce a new generating function Zλ(A, J) whose associated Green functions G(l) i1...il = ∂ ∂Ji1 . . . ∂ ∂Jil Zλ(A, J) J=0 , (1.80) 1.2. QUANTUM MECHANICS 13 the so-called connected Green functions, only represent connected diagrams. It turns out that the correct deﬁnition of this new generating function is Zλ(A, J) = ln Wλ(A, J) . (1.81) One indication that this is the right choice comes from the result (1.69) for the vacuum contribution Wλ(A) which can also be written as Wλ(A) W0(A) = exp  −λ 8 X i ∆2 ii + λ2 48 X i,j (3∆ii∆jj∆2 ij + ∆4 ij)  + O(λ3) . (1.82) Note that the third term in Eq. (1.69) which corresponds to the disconnected diagram in Fig. 1.2 has indeed been ”obsorbed” by the exponential. It turns out that this is a general feature. As a result, connected Green functions can be evaluated in terms of Feynman diagrams in the same way ordinary Green function can, except that only connected Feynman diagrams are taken into account. As before, the above results for multiple integrals can be formally re-written for functional integrals. We can simply carry out the replacements yip →f(xp), G(l) i1...il →G(l)(x1, . . . , xl), ∆ipiq →∆(xp, xq), ∂/∂Jip →δ/δJ(xp) and P ip → R dxp. This leads to the formalism for perturbative quantum (ﬁeld) the-ory based on functional integrals. Perturbation theory is arguably the most important method to extract physical information from quantum ﬁeld theories and we will come back to it more explicitly later in the course. 1.2 Quantum Mechanics 1.2.1 Background From Young’s slits to the path integral As inspiration, recall the two-slit interference experiment (see Fig.1.4). One of the key features of quantum me-chanics is that the total amplitude Atotal at a point on the screen is related to the contributions A1 and A2 from the two possible paths, 1 and 2, by Atotal = A1 + A2. The path integral formulation of quantum mechanics provides a way of thinking in these terms, even in a situation when, in contrast to the two-slits experiment, there are no such obviously deﬁned classical paths. 2 1 Figure 1.4: A two-slit interference experiment. The time evolution operator The standard approach to quantum mechanics in a ﬁrst course is to go quickly from the time-dependent Schr¨ odinger equation to the time-independentone, and to discuss eigenfunctions of the Hamiltonian. The path integral approach is different. Here the focus is on time-evolution, and eigenfunctions of the Hamiltonian remain in the background. We start from the time-dependent Schr¨ odinger equation iℏ∂ψ(t) ∂t = Hψ(t) . 14 CHAPTER 1. PATH INTEGRALS IN QUANTUM MECHANICS If the Hamiltonian is time-independent, the solution can be written as ψ(t) = exp(−iHt/ℏ)ψ(0) Here, e−iHt/ℏis the time-evolution operator. It is a function (the exponential) of another operator (the Hamilto-nian), and we need to remember that functions of operators can be deﬁned via the Taylor series for the function involved. 1.2.2 Path integral for the evolution operator Derivation We are going to think of the time evolution operator as the product of a large number N of contributions from individual time steps t/N. We will do this for a particle moving in one dimension with a time-independent Hamil-tonian H = T + V consisting as usual of a potential energy V and a kinetic energy T , written in terms of the momentum operator p and particle mass m as T = p2/2m. We can introduce time steps simply by writing e−iHt/ℏ= e−iHt/NℏN . (1.83) Next, we’d like to think of the particle’s path as passing through points xn at the successive time-steps nt/N. We can do this by introducing position eigenstates \|x⟩and inserting the identity in the form 1 = Z ∞ −∞ dx\|x⟩⟨x\| (1.84) between each factor on the right of Eq. (1.83). In this way we get ⟨x0\|e−iHt/ℏ\|xN⟩= Z dx1 . . . Z dxN−1⟨x0\|e−iHt/Nℏ\|x1⟩⟨x1\|e−iHt/Nℏ\|x2⟩. . . ⟨xN−1\|e−iHt/Nℏ\|xN⟩. The next job is to evaluate matrix elements of the form ⟨xn\|e−iHt/Nℏ\|xn+1⟩. The way to do this is by separating the contributions from T and V , using the fact that, for N →∞ e−iHt/NℏN = e−iT t/Nℏe−iV t/NℏN . Now we use two versions of the resolution of the identity: the one in Eq. (1.84) based on position eigenstates, which diagonalise V , and a second one based on momentum eigenstates \|p⟩, which diagonalise T . We have ⟨xn\|e−iT t/Nℏe−iV t/Nℏ\|xn+1⟩= Z dpn⟨xn\|e−iT t/Nℏ\|pn⟩⟨pn\|e−iV t/Nℏ\|xn+1⟩. The action of the operator V on the state \|xn+1⟩is simple: V \|xn+1⟩= V (xn+1)\|xn+1⟩, where V (x) denotes the c-number function. Similarly for p on \|pn⟩: p\|pn⟩= pn\|pn⟩, where p is the operator and pn is a number. Then Z dpn⟨xn\|e−iT t/Nℏ\|pn⟩⟨pn\|e−iV t/Nℏ\|xn+1⟩= Z dpne−ip2 nt/2mNℏ⟨xn\|pn⟩⟨pn\|xn+1⟩e−iV (xn+1)t/Nℏ. Next, the matrix element between states from our two bases is ⟨x\|p⟩= (2πℏ)−1/2eipx/ℏ(check that you can justify this), and so Z dpne−ip2 nt/2mNℏ⟨xn\|pn⟩⟨pn\|xn+1⟩= (2πℏ)−1 Z dpne−ip2 nt/2mNℏ−ipn(xn+1−xn)/ℏ= mN 2πiℏt 1/2 eimN(xn+1−xn)2/2ℏt . Putting everything together, and writing the interval between time-slices as ǫ = t/N, we get ⟨x0\|e−iHt/ℏ\|xN⟩= lim N→∞ m 2πiǫℏ N/2 Z dx1 . . . Z dxN−1 exp iǫ ℏ N−1 X n=0 " m 2 xn+1 −xn ǫ 2 −V (xn+1) #! . (1.85) 1.2. QUANTUM MECHANICS 15 Finally, we interpret the sum in the argument of the exponential as a time integral, writing iǫ ℏ N−1 X n=0 " m 2 xn+1 −xn ǫ 2 −V (xn+1) # = i ℏ Z t 0 L dt′ ≡i ℏS[x(t′)] where L is the Lagrangian L = m 2 dx dt′ 2 −V (x) and S[x(t′)] is the action for the system on the path x(t′). Our ﬁnal result is ⟨xi\|e−iHt/ℏ\|xf⟩= N Z D[x(t′)]eiS[x(t′]/ℏ. (1.86) The notation here is as follows: R D[x(t′)] indicates a functional integral over paths x(t′), in this case starting from xi at t′ = 0 and ending at xf at t′ = t, and N is a normalisation factor, which is independent of what the start and end points are. The signiﬁcance of Eq (1.86) is that it expresses the quantum amplitude — for a particle to propagate from a starting point xi to a ﬁnishing point xf in time t — in terms of an integral over possible paths, with the contribution from each path weighted by a phase factor eiS/ℏthat is expressed in terms of a classical quantity, the action S for the path. In this sense, we have succeeded in extending the treatment of interference in the context of Young’s slits to a general quantum system. Correspondence principle One of the simplest and most attractive features of the path integral formulation of quantum mechanics is that it provides a framework in which we can see classical mechanics arising as a limiting case. We expect classical mechanics to emerge in the limit ℏ→0 (or more precisely, when ℏis small compared to the characteristic scale in S). Within the path integral, this limit is just the one in which we can apply the steepest descents approximation, since then the argument of the exponential in Eq. (1.86) is multiplied by the large quantity ℏ−1. Using that approximation, the dominant contribution to the path integral will come from the vicinity of the path for which the action S is stationary. But we know from Hamilton’s principle that this path is precisely the classical one, and so we have reached the desired conclusion. 1.2.3 Path integral in statistical mechanics With some simple changes, we can express the Boltzmann factor, and hence other quantities in statistical mechan-ics, as a path integral. Comparing the time evolution operator e−iHt/ℏwith the Boltzmann factor e−βH, we see that we require the substitution it = βℏ. As a consequence, for the time t′, which parameterises points on a path and has limits 0 ≤t′ ≤t, we substitute τ = it′, which has limits 0 ≤τ ≤βℏ. Now we can write a matrix element of the Boltzmann factor as ⟨xi\|e−βH\|xf⟩= N Z D[x(τ)]e−S[x(τ)]/ℏ (1.87) with S[x(τ)] = Z βℏ 0 dτ " m 2 dx dτ 2 + V (x) # . (1.88) The crucial point to note about Eq. (1.88) is that because of the substitution τ = it′, there is a change in the relative sign of the terms in the action coming from kinetic and potential energies. With this sign change, S[x(τ)] is often referred to as the Euclidean action, by analogy with the change from a Lorentz metric to a Euclidean one, produced by a similar factor of i. Our result, Eq. (1.87), gives matrix elements of the Boltzmann factor for a quantum system as a functional integral over classical paths, this time with paths weighted by a real factor, e−S[x(τ)]/ℏ. Quantities we want to calculate in the context of statistical mechanics are the partition function Z and thermal averages, for example the average ⟨f(x)⟩of a function f(x) of the coordinate x of the particle. These are related in a simple way to the matrix elements of the Boltzmann factor that we have discussed. Consider ﬁrst the partition function, and recall its deﬁnition for a system with energy levels Ei: Z = P i e−βEi . Clearly, this is simply 16 CHAPTER 1. PATH INTEGRALS IN QUANTUM MECHANICS Z = Tre−βH, written in the eigenbasis of H. Since the trace of a matrix is invariant under change of basis, we can use this second form, which leads us to Z = Z dx ⟨x\|e−βH\|x⟩≡N Z D[x(τ)]e−S[x(τ)]/ℏ, where the path integral is now over periodic paths, satisfying x(βℏ) = x(0). In a similar fashion, we can evaluate thermal averages as ⟨f(x)⟩= Z−1 Z D[x(τ)]f(x(0))e−S[x(τ)]/ℏ. 1.2.4 Two examples While the path integral is very important as a way of thinking about quantum mechanics and for treating more advanced problems, it turns out not to be a particularly convenient way to handle elementary ones. Nevertheless, we should see how it works for a couple of simple examples. Evaluation for a free particle As a ﬁrst illustration, consider a free particle moving in one dimension. We want to start from Eq (1.85), set V (x) = 0 and evaluate the multiple integrals explicitly. Within the argument of the exponential, each coordinate xn appears inside two quadratic terms. The integrals that we need to evaluate are therefore of the form αβ π2 1/2 Z ∞ −∞ dx exp −α[u −x]2 −β[x −v]2 = αβ π(α + β) 1/2 exp −αβ α + β [u −v]2 . (1.89) [Prove this by completing the square in the argument of the exponential.] We use this result to integrate out xN−1, then xN−2 and so on. After completing k of the integrals we obtain m 2πiǫℏ N/2 Z dx1 . . . Z dxN−1 exp −m 2iǫℏ N−1 X n=0 (xn+1 −xn)2 ! = m 2πiǫℏ (N−k−1)/2 · m 2πi[k + 1]ǫℏ 1/2 × × Z dx1 . . . Z dxN−k−1 exp −m 2iǫℏ N−k−2 X n=0 (xn+1 −xn)2 − m 2i[k + 1]ǫℏ(xN −xN−k−1)2 ! For k = N −1 this gives ⟨x0\|eiHt/ℏ\|xN⟩= m 2πiℏt 1/2 exp −m 2iℏt (xN −x0)2 . (1.90) To check whether this is correct, we should evaluate the same quantity using a different method. For a free particle, this can be done by inserting into the expression for the matrix element of the time evolution operator a single resolution of the identity in terms of momentum eigenstates. Repeating manipulations very similar to the ones we used to derive the path integral, we have ⟨x0\|e−iHt/ℏ\|xN⟩ = (2πℏ)−1 Z dp ⟨x0\|p⟩⟨p\|e−iHt/ℏ\|xN⟩ = (2πℏ)−1 Z dp exp −ip2t 2mℏ+ ip(x0 −xN)/ℏ = m 2πiℏt 1/2 exp −m 2iℏt(xN −x0)2 — conﬁrming our earlier result, Eq. (1.90). Evaluation for a harmonic oscillator As a second example, we will evaluate the thermal average ⟨x2⟩for a one-dimensional harmonic oscillator, with the potential energy V (x) = κ 2 x2 . 1.3. FURTHER READING 17 In terms of the matrix element of the Boltzmann factor for which we have a path integral representation, this thermal average is ⟨x2⟩= R dx x2⟨x\|e−βH\|x⟩ R dx ⟨x\|e−βH\|x⟩ . (1.91) For the harmonic oscillator S[(τ)] is quadratic in x(τ), and so the path integral is Gaussian. To evaluate it, we should diagonalise the Euclidean action, which is S[x(τ)] = Z βℏ 0 dτ " m 2 dx(τ) dτ 2 + κ 2 x(τ)2 # . To do that, we simply expand x(τ) in Fourier components, writing x(τ) = 1 √βℏ ∞ X n=−∞ φne2πinτ/βℏ and φn = 1 √βℏ Z βℏ 0 dτx(τ)e−2πinτ/βℏ. Since x(τ) is real, φ−n = (φn)∗, and so we can take as independent variables the real and imaginary parts of φn for n > 0, together with φ0 which is real. The Jacobian for this change of variables is unity, because with our choice of normalisation the Fourier transform is an expansion of x(τ) in an orthonormal basis. In this way we have Z D[x(τ)] ≡ Z dφ0 Y n>0 Z dℜφn Z dℑφn and S[x(τ)] = X n>0 " m 2πn ℏβ 2 + κ # \|φn\|2 + 1 2κφ2 0 . Averaging the amplitudes φn with the weight e−S[x(τ)]/ℏ, and introducing the oscillator frequency ω = p κ/m, we have ⟨φmφ−n⟩= δm,n ℏ/κ 1 + (2πn/βℏω)2 . We obtain ﬁnally (using a contour integral to evaluate a sum on n) ⟨x2(0)⟩= 1 βℏ X n φn !2+ = 1 βκ X n 1 1 + (2πn/βℏω)2 = ℏω/κ 2 tanh(βℏω/2) = ℏω κ 1 eβℏω −1 + 1 2 . (1.92) The rightmost form of the result is recognisable as the Planck formula for the average energy in the oscillator multiplied by κ−1, which is to be expected since this energy has equal kinetic and potential contributions, and the potential energy is κ⟨x2⟩/2. To summarise, we’ve succeeded in writing the Boltzmann factor for the harmonic oscillator as a path integral, and in evaluating this path integral, and the result we’ve obtained matches what is familiar from more elementary treatments. Of course, the approach seems rather laborious for an elementary problem like this one, but it brings real advantages for more advanced ones. 1.3 Further reading • Functional derivatives (or calculus of variations) are covered by Boas and by Riley, Bence and Hobson, in their textbooks on Mathematical Methods. • Multidimensional Gaussian integrals are discussed by Zinn-Justin in the introductory chapters of Quantum Field Theory and Critical Phenomena and of Path integrals in Quantum Mechanics. • Extended treatments of path integrals in quantum mechanics are given by Zinn-Justin in both of the above books and by Schulman in Techniques and Applications of Path Integration. • The reference to Feynman’s original paper is: Rev. Mod. Phys. 20, 367 (1948). • Path integrals in quantum mechanics are introduced in the ﬁrst two chapters of Bailin and Love, Introduction to Gauge Theory. Be warned that, except for the ﬁrst one, these references were not written with undergraduate readers in mind. 18 CHAPTER 1. PATH INTEGRALS IN QUANTUM MECHANICS Chapter 2 Stochastic Processes and Path Integrals 2.1 Introduction In this chapter we turn to a class of problems from classical statistical physics known as stochastic processes. We will discuss stochastic processes from several points of view, including the path integral formulation. Recalling our discussion of path integrals in quantum mechanics, it is worth stressing a distinction between the path integral for the time evolution operator and the one for the Boltzmann factor. For the time evolution operator, the weight for a path x(t) is the phase factor exp(iS[x(t)]/ℏ), while for the Boltzmann factor, the weight for a path x(τ) is the real, positive quantity exp(−S[x(τ)]/ℏ). In this second case, we have the possibility of viewing the weights as probabilities attached to the paths. For stochastic processes, one literally has probabilities associated with various paths that a system may follow as a function of time. A path integral treatment of stochastic processes therefore gives us a context in which functional integrals take on a very clear and concrete meaning. An example of a stochastic process that we will discuss in some detail is Brownian motion: a small particle, suspended in a ﬂuid and observed under a microscope, moves in a random way because it is buffeted by collisions with water molecules. The theory of Brownian motion, developed by Einstein in 1905, gave important conﬁrmation of ideas of statistical mechanics and a way to determine from observations the value of Avogadro’s number. 2.2 Random variables We start by listing some deﬁnitions and introducing some useful ideas. There is some overlap between the material in this section and our earlier discussion of Gaussian integrals, but also some differences in emphasis: here we are more restrictive, in the sense that we concentrate on functions of one variable, rather than many variables; but we are also more general, in the sense that we consider distributions which are not necessarily Gaussian. A random variable X is speciﬁed by giving the set of possible values {x} that it can take (which may be discrete or continuous), and by a normalised probability distribution PX(x). 2.2.1 Moments of X ⟨Xn⟩= Z xnPX(x) dx mean ⟨X⟩ variance ⟨X2⟩−⟨X⟩2 ≡σ2 standard deviation σ 2.2.2 Characteristic function The characteristic function is the Fourier transform of the probability distribution. φX(k) = ⟨eikX⟩≡ Z PX(x)eikxdx = ∞ X n=0 (ik)n n! ⟨Xn⟩ 19 20 CHAPTER 2. STOCHASTIC PROCESSES AND PATH INTEGRALS It is also called the moment generating function, since ⟨Xn⟩= 1 in dn dkn φX(k) k=0 . 2.2.3 Cumulants Write as a deﬁnition of the cumulants Cn(X) φX(k) = exp X n Cn(X)(ik)n n! ! . Then Cn(X) = 1 in dn dkn k=0 ln(φX(k)) . We have C1(X) = ⟨X⟩ C2(X) = ⟨X2⟩−⟨X⟩2 etc In a rough sense, the size of the nth cumulant tells you how much the nth moment deviates from the value you would have expected for it, given only knowledge of the lower order moments. 2.2.4 Gaussian distribution Deﬁned by PX(x) = 1 √ 2πσ2 exp −(x −x0)2 2σ2 which gives φX(k) = exp(ikx0 −1 2σ2k2) . From this, we see that the cumulants are: C1(X) = x0, C2(X) = σ2 and Cn(X) = 0 for n ≥3. Hence, in the language of cumulants, a Gaussian distribution is completely speciﬁed by the values of its ﬁrst two cumulants. 2.2.5 Random Walks Consider a set of N random variables, for simplicity taken to be independently and identically distributed: YN = X1 + X2 + . . . + XN . We can think of N as labelling discrete time, and of YN as a random walk in one dimension. The mean of YN is ⟨YN⟩= N⟨X⟩ and the variance Σ2 N is ⟨Y 2 N⟩−⟨Yn⟩2 = N X i,j=1 [⟨XiXj⟩−⟨Xi⟩⟨Xj⟩] = Nσ2 so that ΣN = √ Nσ. 2.2.6 The central limit theorem Consider SN ≡YN/N. Then, for large N: the distribution of SN converges towards a Gaussian (with mean ⟨X⟩ and variance σ/ √ N) irrespective of the distribution of X, provided only that ⟨X⟩and ⟨X2⟩are not inﬁnite. 2.3. STOCHASTIC PROCESSES 21 Proof via the characteristic function. We have φSN (k) = ⟨eikSN ⟩= ⟨ei k N PN j=1 Xj⟩= ⟨ei k N X⟩N = [φX(k/N)]N . Writing this in terms of cumulants, we have exp ∞ X m=0 (ik)m m! Cm(SN) ! = exp N ∞ X m=0 (ik/N)m m! Cm(X) ! , and so Cm(SN) = Cm(X) · N 1−m . This shows that for large N, only C1(SN) and C2(SN) are non-vanishing. Such a feature of the cumulants is unique to a Gaussian distribution, and so SN must be Gaussian distributed. To demonstrate explicitly that this is the case, we can consider the calculation of the probability distribution for SN, which we write as PSN (s). It can be obtained as the inverse Fourier transform of the characteristic function for SN, and so we have PSN(s) = 1 2π Z ∞ −∞ e−iksφSN (k)dk . (2.1) Now, taking account only of the ﬁrst two cumulants, we have \|φSN(k)\| ≈exp −C2(X)k2 2N . This makes it clear that only the range \|k\| ≲ √ N gives a signiﬁcant contribution to the integral. But in this range, the corrections involving higher cumulants are small: they involve N · k N m ≲N 1−m/2 , which goes to zero as N →∞for m ≥3. Retaining only C1(SN) and C2(SN) in φSN (k), we can evaluate the Fourier transform to obtain PSN (s) ≈ N 2πσ2 1/2 exp −N(s −⟨X⟩)2 2σ2 . Note that what we have done is a steepest descents evaluation of Eq. (2.1). 2.3 Stochastic Processes A random variable or variables evolving as a function of time (or some other coordinate): Y (t). For example, the position and velocity of a particle that is undergoing Brownian motion. A stochastic process is characterised by an (inﬁnite) sequence of probability densities P1(y1, t1) the probability that Y = y1 at t = t1 . . . Pn(y1, t1; y2, t2 . . . yn, tn) joint probability that Y = y1 at t = t1 and Y = y2 at t = t2 etc. Some general properties of these densities are Pn(y1, t1 . . .) ≥0 probabilities are non-negative Z Pn(y1, t1 . . . yn, tn)dy1 . . . dyn = 1 probabilities are normalised Z Pn(y1, t1 . . . yn, tn)dyn = Pn−1(y1, t1 . . . yn−1, tn−1) the sequence is reducible 22 CHAPTER 2. STOCHASTIC PROCESSES AND PATH INTEGRALS 2.3.1 Correlation functions One can deﬁne an inﬁnite sequence of correlation functions: ⟨Y (t1)⟩ = Z dy1P1(y1, tt)y1 ⟨Y (t1)Y (t2)⟩ = Z Z dy1dy2P2(y1, t1; y2, t2)y1y2 . . . 2.3.2 Stationary Processes Stationary processes are ones that are invariant under a change in the origin for time, so that Pn(y1, t1 . . . yn, tn) = Pn(y1, t1 + τ . . . yn, tn + τ) for all n and for any choice of τ. For a stationary process ⟨Y (t1)⟩is independent of t1, ⟨Y (t1)Y (t2)⟩depends only on the difference t1 −t2, and so on. 2.3.3 Conditional probabilities We write the conditional probability that Y = y2 at t2, given that Y = y1 at t1, as P1\|1(y2, t2\|y1, t1). This obeys Z dy2P1\|1(y2, t2\|y1, t1) = 1 since Y must take some value at t2. Conditional and unconditional probabilities are related by P2(y1, t1; y2, t2) = P1\|1(y2, t2\|y1, t1) · P1(y1, t1) . The idea of conditional probabilities generalises in the obvious way: the quantity P1\|n−1(yn, tn\|y1, t1; y2, t2 . . . yn−1, tn−1) is the probability to ﬁnd Y = yn at tn, given that Y = y1 at t1, Y = y2 at t2 etc. 2.3.4 Markov Processes A process without memory, except through the values of the random variables. This is the most important special class of stochastic processes, because processes of this kind are common and are much easier to study than more general ones. Consider t1 < t2 < . . . < tn: a Markov process is one for which P1\|n−1(yn, tn\|y1, t1; y2, t2 . . . yn−1, tn−1) = P1\|1(yn, tn\|yn−1, tn−1) for all n. In such a case, at time tn−1 one can predict the future (yn at tn) from present information (yn−1) without additional knowledge about the past (the values of yn−2 . . . y1). For example, for a Brownian particle, one might expect to predict the future velocity of a particle given only the current value of the velocity, although the attempt would fail if information about the current states of the ﬂuid molecules is important for the prediction. A Markov process is fully determined by the two functions: P1(y1, t1) and P1\|1(y2, t2; y1, t1). The absence of memory makes Markov processes tractable. Any P1(y, t) and P1\|1(y2, t2\|y1, t1) deﬁne a Markov process, provided they satisfy certain conditions. The main ones are the Chapman-Kolmogorov equation (also known as the Smoluchowski equation), which is the con-dition that for any t3 > t2 > t1 P1\|1(y3, t3\|y1, t1) = Z dy2P1\|1(y3, t3\|y2, t2) P1\|1(y2, t2\|y1, t1), and the evolution equation, which is the condition that, for any t2 > t1 P1(y2, t2) = Z dy1P1\|1(y2, t2\|y1, t1)P1(y1, t1) . 2.4. MARKOV CHAINS 23 2.3.5 A stationary Markov process A Markov process which is invariant under translation in time. That is to say, P1(y1, t1) does not depend on t1, and P1\|1(y2, t2; y1, t1) depends only on t2 −t1, y1 and y2. 2.4 Markov Chains The simplest version of a Markov process has only a ﬁnite number N of discrete values possible for the random variable Y , and involves discrete (integer) time-steps. Such processes are called Markov chains. For a Markov chain P1(y, t) is an N-component vector and P1\|1(y2, t + 1; y1, t) ≡T is an N × N matrix. 2.4.1 An example Consider a two-state process, in which Y = 1 or Y = 2. Evolution from time t to time t + 1 can be speciﬁed as follows: Evolution: Y = 1 →Y = 1 has probability 1 −q Evolution: Y = 1 →Y = 2 has probability q Evolution: Y = 2 →Y = 2 has probability 1 −r Evolution: Y = 2 →Y = 1 has probability r Then P1(y, t) can be represented by the vector a(t) = a1(t) a2(t) with a1(t) + a2(t) = 1. The evolution equation is a(t + 1) = T · a(t) with T = 1 −q r q 1 −r . Note that, in general, T is a square matrix with non-negative entries, but that it is not necessarily symmetric. 2.4.2 Mathematical digression On properties of eigenvectors for square matrices that are not Hermitian, but that are diagonalisable by a similarity transformation In general there are distinct left and right eigenvectors associated with each eigenvalue. Using Dirac notation, we have M\|Rα⟩= λα\|Rα⟩ and ⟨Lα\|M = ⟨Lα\|λα with {⟨Lα\|}† ≡\|Lα⟩̸= \|Rα⟩, where M is an N × N matrix, \|R⟩is an N-component column vector and ⟨L\| is an N-component row vector. The two sets of eigenvectors are biorthogonal, meaning that, with an appropriate choice of normalisation, one has ⟨Lα\|Rβ⟩= δαβ , but ⟨Lα\|Lβ⟩and ⟨Rα\|Rβ⟩have no special properties. To prove this, start from M\|Rα⟩= λα\|Rα⟩ (∗) and ⟨Lβ\|M = ⟨Lβ\|λβ (∗∗) 24 CHAPTER 2. STOCHASTIC PROCESSES AND PATH INTEGRALS Take ⟨Lβ\| · (∗) −(∗∗) · \|Rα⟩to obtain 0 = (λα −λβ)⟨Lβ\|Rα⟩. Hence ⟨Lβ\|Rα⟩= 0 for λβ ̸= λα, while by choice of normalisation (and of basis in the case of degeneracy) we can set ⟨Lβ\|Rα⟩= δαβ, so including the case λβ = λα. A useful idea is that of spectral decomposition, meaning that we can write M in terms of its eigenvectors and eigenvalues, as M = X α \|Rα⟩λα⟨Lα\| . 2.4.3 Now apply these ideas to Markov chains One eigenvalue of the evolution operator and the associated left eigenvector are easy to ﬁnd. Consider (in compo-nent form) ai(t + 1) = X j Tijaj(t) . From conservation of probability, we must have X i ai(t + 1) = X i ai(t) for any a(t), which can only be true if P i Tij = 1 for all j. From this, we can conclude that one eigenvalue of T is λ1 = 1, and that the associated left eigenvector is ⟨L1\| = (1, 1, . . . 1). The corresponding right eigenvector is not so trivial. It has physical signiﬁcance for the long-time limit of the Markov process. To see this, consider a(n) = (T)na(0). Now. from the spectral decomposition and biorthogonality, we see that (T)n = X α \|Rα⟩λn α⟨Lα\| . To avoid divergences, one requires \|λα\| ≤1 for all α. in addition, for a large class of problems λ1 is the unique eigenvalue with largest magnitude, so that \|λα\| < 1 for α ̸= 1. Then in the limit n →∞we have Tn →\|R1⟩⟨L1\|. From this we can conclude that the limiting distribution is given by \|R1⟩. 2.4.4 Returning to the two-state process of Sec. 2.4.1 By solving 1 −q r q 1 −r a1 1 −a1 = a1 1 −a1 we ﬁnd a1 = r/(r + q), which gives us the limiting distribution. 2.5 Brownian motion Now we will switch from these generalities to the speciﬁc example of Brownian motion, which we will consider from three points of view: ﬁrst, using a stochastic differential equation (that is, a differential equation which includes a random term) called the Langevin equation; second, using a partial differential equation for the time evolution of probability, known as the Fokker-Planck equation; and third, using a path integral. 2.5.1 Langevin Equation Considering Brownian motion as an example, we can take a microscopic approach to the evolution of the velocity, v(t). For simplicity, we take v(t) to be a scalar – either one component of the velocity for a particle moving in three dimensions, or simply the velocity of a particle constrained to move in one dimension. The Langevin equation is dv(t) dt = −γv(t) + η(t) , (2.2) 2.5. BROWNIAN MOTION 25 where the term involving the constant γ represents viscous drag exerted by the ﬂuid on the particle, which tends to reduce its velocity, and the function η(t) represents a random force on the particle, due to collisions with molecules of the ﬂuid. Since η(t) is random, it can by characterised by its correlation functions. In view of the central limit theorem, it is often sufﬁcient to know just the ﬁrst two moments. We take this force to be zero on average, and take its values at times separated by more than a microscopic interval to be uncorrelated, so that ⟨η(t)⟩= 0 and ⟨η(t)η(t′)⟩= Γδ(t −t′) . In this way, the parameter Γ characterises the strength of the noise. Note that averages over realisations of the noise are given by a Gaussian functional integral: ⟨. . .⟩≡ Z D[η(t)] . . . e−1 2Γ R dt η2(t) . Now, we can solve Eq. (2.2) explicitly for any noise history η(t) in the standard way for a ﬁrst-order ordinary differential equation, using an integrating factor. We ﬁnd v(t) = v(0)e−γt + Z t 0 dt′e−γ(t−t′)η(t′) (2.3) which can be checked by substituting back. From this solution, we can calculate averages. We ﬁnd ⟨v(t)⟩= v(0)e−γt and ⟨[v(t)]2⟩ = [v(0)]2e−2γt + e−2γt Z t 0 Z t 0 dt1dt2eγ(t1+t2)⟨η(t1)η(t2)⟩ = [v(0)]2e−2γt + Γ 2γ [1 −e−2γt] , (2.4) and also lim t→∞⟨v(t −τ/2)v(t + τ/2)⟩ = lim t→∞e−2γt Z t−τ/2 0 Z t+τ/2 0 dt1dt2eγ(t1+t2)⟨η(t1)η(t2)⟩ = Γ 2γ e−γ\|τ\| . (2.5) 2.5.2 Fluctuation-dissipation relation So far, it has seemed that the two constants specifying the Langevin equation, γ (viscous drag) and Γ (noise strength), are independent. In fact, of course, they both have their microscopic origins in collisions of the Brownian particle with ﬂuid molecules, and for this physical reason there is an important relation connecting them. We get it by looking at behaviour in the long-time limit, and using information that we know from statistical mechanics – speciﬁcally, the equipartition theorem. From Eq. (2.4), for t →∞we have ⟨v2(t)⟩= Γ 2γ . But from equipartion, the kinetic energy of the particle (which we have taken to move in one dimension only) is kBT/2, and so ⟨v2(t)⟩= kBT/m, with m the mass of the Brownian particle. To analyse an experiment, one can go further: if the particle has a simple, known shape (for example, it is spherical), then γ can be calculated in terms of the particle dimensions and the ﬂuid viscosity, by solving the Navier-Stokes equations. In this way, the strength of molecular noise is ﬁxed in absolute terms. 2.5.3 Displacement of a Brownian particle Experimentally, it is hard to observe the instantaneous velocity of a Brownian particle. Instead, one observes its position at a sequence of times which are not closely enough spaced for its velocity to remain constant between each observation. To interpret experiments, it is therefore important to calculate moments of position x(t), and in 26 CHAPTER 2. STOCHASTIC PROCESSES AND PATH INTEGRALS particular ⟨x2(t)⟩. One obvious approach to this calculation is to obtain x(t) in terms of η(t) by integrating the expression for v(t) given in Eq. (2.3), and to average moments of x(t) over realisations of η(t). An alternative is to derive and solve equations for the time evolution of the moments, as follows. Consider d dt⟨x2(t)⟩= 2⟨x(t) d dtx(t)⟩. Differentiating again, we have d dt⟨x(t) d dtx(t)⟩= ⟨x(t) d dtv(t)⟩+ ⟨[v(t)]2⟩. Now, we can simplify the ﬁrst term on the right side of this equation by using the Langevin equation to substitute for v(t), and we know the second term from equipartition. In this way we obtain d dt⟨x(t) d dtx(t)⟩= −γ⟨x(t) d dtx(t)⟩+ kBT m + ⟨x(t)η(t)⟩. (2.6) Since ⟨x(t)η(t)⟩= 0, we can integrate Eq. (2.6), obtaining ⟨x(t) d dtx(t)⟩= Ce−γt + kBT γm , where C is an integration constant. Taking as our initial condition x(0) = 0, we ﬁnd ⟨x(t) d dtx(t)⟩= kBT γm [1 −e−γt] . Integrating again, we have ⟨x2(t)⟩= 2kBT γm Z t 0 (1 −e−γt′)dt′ = 2kBT γm [t −1 γ (1 −e−γt)] . To appreciate this result, it is useful to think about limiting cases. At short times (γt ≪1) ⟨x2(t)⟩≈2kBT γm [t −1 γ (γt −(γt)2 2 . . .)] = kBT m t2 , which, reasonably enough, is ballistic motion at the mean square speed expected from equipartition. At long times (γt ≫1) ⟨x2(t)⟩= 2kBT γm t . In this case, the mean square displacement grows only linearly with time, which we recognise as diffusive motion. The diffusion coefﬁcient, 2kBT/γm involves system-speciﬁc constants γ and m: as indicated above, γ can be calculated in terms of the size of particles and the viscosity of the ﬂuid, and m can likewise be determined inde-pendently. A measurement of this diffusion constant therefore constitutes a measurement of Boltzmann’s constant kB, which is related to the gas constant R via Avogadro’s number. Since R is known from elementary measure-ments on nearly ideal gases, we have a determination of Avogadro’s number. The relevant theory was published by Einstein in 1905, and early experimental results were provided by Perrin in 1910. 2.5.4 Fokker-Planck Equation As we have seen, the treatment of Brownian motion using the Langevin equation involves ﬁrst analysing behaviour with a given realisation of the random force η(t), then averaging over all realisations. An alternative approach is to consider the probability distribution of the velocity. The Fokker-Planck equation determines the time evolution of this probability. It can be derived starting from the Langevin equation, as follows. Consider time evolution from t to t + ∆t with ∆t > 0. By analogy with Eq. (2.3) we have v(t + ∆t) = v(t)e−γ∆t + Z t+∆t t dt′e−γ(t−t′)η(t′) (2.7) 2.5. BROWNIAN MOTION 27 so that the velocity change ∆v ≡v(t + ∆t) −v(t) is Gaussian ditributed if η(t′) is. Now, following the general approach to Markov processes, as introduced above, the probability distribution P1(v, t) for the velocity v of a Brownian particle at time t satisﬁes the (integral) evolution equation P1(v, t + ∆t) = Z duP1\|1(v, t + ∆t\|u, t)P1(u, t) . (2.8) We would like to get from this to a differential equation, by taking the limit of small ∆t, in which we expect that \|v −u\| will also be small. Some care is necessary in the derivation, because we should think of P1\|1(v, t+∆t\|u, t) as giving the probability distribution of v, for a ﬁxed u, while in Eq. (2.8) the quantity v is ﬁxed and the integral is on u. To deal with this, we change variables from u to ∆v = v −u, and then Taylor expand in ∆v, obtaining P1(v, t + ∆t) = Z d(∆v) P1\|1(v, t + ∆t\|v −∆v, t)P1(v −∆v, t) = Z d(∆v) P1\|1(v −∆v + ∆v, t + ∆t\|v −∆v, t)P1(v −∆v, t) = Z d(∆v) ∞ X n=0 (−∆v)n n! ∂n ∂vn P1\|1(v + ∆v, t + ∆t\|v, t)P1(v, t) . Note in the middle line of this equation the substitution v = v −∆+ ∆v, used so that the combination v −∆v appears uniformly in the integrand, facilitating the Taylor expansion. The moments of ∆v which appear here, Z d(∆v)(∆v)nP1\|1(v + ∆v, t + ∆t\|v, t) ≡ ⟨(∆v)n⟩, take for small ∆t the values ⟨∆v⟩= −γv · ∆t + O(∆t2) , ⟨(∆v)2⟩= Γ · ∆t + O(∆t2) and ⟨(∆v)n⟩≲O(∆t2) for n ≥3 . Hence P1(v, t + ∆t) = P1(v, t) + γ∆t ∂ ∂v vP1(v, t) + Γ ∆t 2 ∂2 ∂v2 P1(v, t) + O(∆t2) and hence ∂tP1(v, t) = γ∂v vP1(v, t) + Γ 2 ∂2 vP1(v, t) . (2.9) which is the Fokker-Planck equation. Its solution is simplest to ﬁnd in the limit t →∞, because then P1(v, t) is independent of initial conditions and of t. It is simple to show that P1(v, ∞) = γ πΓ 1/2 exp −γv2/Γ , which is the Maxwell distribution familiar from kinetic theory (recall from the ﬂuctuation-dissipation relation that γ/Γ = m/(2kBT )). 2.5.5 Diffusion equation Now we switch our attention from evolution of the velocity to evolution of the position. Note that the time-dependence of position by itself is not a Markov process, since future values of position depend not only on the current value of position, but also on the current velocity. It is possible to treat the combined evolution of x(t) and v(t), but for simplicity we will avoid this. We do so by noting that velocity relaxes on the timescale γ−1: provided we treat evolution of x(t) only on much longer timescales, v(t) is simply a random function, like the force η(t) that appears in the Langevin equation. We have dx(t) dt = v(t) . (2.10) From Eq. (2.5) the moments of the velocity, for t ≫γ−1, are ⟨v(t)⟩= 0 and ⟨v(t)v(t′)⟩= Γ 2γ e−γ\|t−t′\| ∼Γ γ2 δ(t −t′) 28 CHAPTER 2. STOCHASTIC PROCESSES AND PATH INTEGRALS where the replacement of (γ/2)e−γ\|t−t′\| by δ(t −t′) is appropriate if other quantities that depend on t −t′ vary much more slowly. Clearly, Eq. (2.10) is like the Langevin equation, Eq. (2.2), but with x(t) replacing v(t) and with γ = 0. We can therefore adapt many of the results we have already derived. In particular, by comparison with the Fokker-Planck equation, Eq. (2.9), the evolution equation for P1(x, t), the probability distribution for position, is ∂tP1(x, t) = Γ 2γ2 ∂2 xP1(x, t) , (2.11) which we recognise as the diffusion equation, with diffusion coefﬁcient D = Γ/2γ2. Its solution, with an initial condition that the particle is at position x0 at time t0, is P1(x, t) ≡P2(x, t; x0, t0) = (2πD\|t −t0\|)−1/2 exp −[x −x0]2 4D\|t −t0\| . (2.12) 2.5.6 Path integral for diffusion We can recast the description we have built up for the motion of a Brownian particle on timescales much longer than γ−1 as a path integral. The most streamlined way to do so is to note that, as v(t) is a Gaussian random variable with ⟨v(t)⟩= 0 and ⟨v(t)v(t′)⟩= (Γ/γ2)δ(t −t′), its distribution is given by the Gaussian functional exp −γ2 2Γ Z dt[v(t)]2 . This is equivalent to the Euclidean path integral for a free particle, on making the identiﬁcation γ2 Γ ≡ 1 2D = m ℏ. Alternatively, we can consider evolution through a sequence of discrete intervals, dividing a total time t into N equal timeslices, each of duration ǫ = t/N. Extending Eq. (2.12), we have Pn(x0, 0; x1, ǫ; . . . : xn, nǫ; . . . xN, t) = (4πDǫ)−N/2 exp −ǫ 4D N−1 X n=0 xn+1 −xn ǫ 2! , (2.13) which is the equivalent of Eq. (1.85). A general feature of typical paths that contribute to a path integral can be read off from Eq. (2.13), by consid-ering the typical distance a Brownian particle moves in a short time. In a time ǫ the variance of the displacement is ⟨(xn+1 −xn)2⟩= 2Dǫ. The characteristic velocity averaged over this time interval is ⟨(xn+1 −xn)2⟩1/2 ǫ = 2D ǫ 1/2 , which diverges as ǫ →0. In other words, paths are continuous (since ⟨(xn+1 −xn)2⟩→0 as ǫ →0), but not differentiable. They are in fact fractal objects, with dimension two, in contrast to one-dimensional smooth curves. 2.6 Further Reading • L. E. Reichl, A Modern Course in Statistical Physics (Edward Arnold). Chapters 5 and 6 give a basic introduction to probability theory, the central limit theorem and the Master equation. • F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill). Chapter 15 covers Master equa-tions, Langevin equations and Fokker-Planck equations. • N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland). A reasonably complete and quite advanced reference book. • L. S. Schulman, Techniques and Applications of Path Integration (Wiley). Chapter 9 gives a careful account of the link between Brownian motion and the so-called Wiener integral. Chapter 3 Statistical mechanics in one dimension In this chapter we treat some problems from statistical mechanics that are selected partly because the mathematical methods used to study them have close links with the ideas we have met in the context of the path integral formu-lation of quantum mechanics and in the theory of stochastic processes. One common feature of the systems we discuss, which makes their behaviour interesting, is that they are built from particles (or other microscopic degrees of freedom) which have interactions between them. In this crucial sense, they differ from the simplest problems of statistical mechanics, which involve only non-interacting particles: the ideal classical gas of kinetic theory, and the ideal Fermi and Bose gases of elementary quantum statistical mechanics. Interparticle interactions can lead to behaviour quite different from that of non-interacting systems, with the possibility of spontaneous order and symmetry breaking at low temperature, and phase transitions as temperature is varied. In general, it is impossible or very difﬁcult to treat the statistical mechanics of interacting systems exactly. It can be done, however, for a variety of one-dimensional models, and the necessary techniques are introduced in this chapter: the criterion of tractability is the second reason for the selection of problems we make here. As we will see, it is a general feature of one-dimensional systems that they do not show phase transitions or symmetry-breaking at non-zero temperature. Nevertheless, they serve to show how interactions can have a controlling inﬂuence. We will return to the topic of phase transitions in interacting systems in higher dimensions, using approximate methods, later in the course. The one-dimensional models we deﬁne in this chapter all have obvious extensions to higher dimensions. 3.1 Lattice models As often in physics, it is useful to make models that are intended to include only the essential features of the prob-lem we are concerned with. Details that are believed to be irrelevant are omitted, so that these models are simple but not completely realistic. For our present purposes, we need models with two ingredients: some microscopic degrees of freedom, and a form for the energy. We will refer to the latter as the Hamiltonian for the model, even in cases where there are no equations of motion, so that there is no link to classical mechanics. We start with models in which the microscopic degrees of freedom are deﬁned at the sites of a lattice. This lattice may represent the atomic lattice of a crystalline solid, or it may simply be a convenient mathematical construct. 3.1.1 Ising model Several interesting and much-studied models are inspired by the phenomenon of magnetism. In these models, the microscopic degrees of freedom are intended to represent atomic magnetic moments: we will refer to them as spins, although we take them to be classical variables. In the simplest case, the Ising spin Si at site i of a lattice is a scalar with two states: Si = ±1. We will consider a one-dimensional lattice, and take ferromagnetic exchange interactions of strength J (a positive quantity) between neighbouring spins, so that conﬁgurations in which neighbouring spins have the same orientation are lower in energy. We also include the Zeeman energy of spins in an external magnetic ﬁeld of strength h (in scaled units). The Hamiltonian is H = −J X i SiSi+1 −h X i Si . (3.1) It is sometimes useful to specify periodic boundary conditions for a system of N sites, by setting SN+1 ≡S1 and taking the sums on i in Eq. (3.1) to run from 1 to N. 29 30 CHAPTER 3. STATISTICAL MECHANICS IN ONE DIMENSION In a ground state all spins have the same orientation, so that the ﬁrst term of H is minimised. With h = 0 there are two such states: (i) Si = +1 for all i, and (ii) Si = −1 for all i. Hence, at zero temperature, when the system is in a ground state, there is long-range order. Also, at h = 0, the model has symmetry under global spin inversion (the energy is unchanged if the directions of all spins are reversed via Si →−Si): this symmetry is broken in each of the ground states. With h ̸= 0 there is a unique ground state: Si = sign(h). 3.1.2 Lattice gas A lattice gas model inludes some of the features present in a real, non-ideal gas, but within a simplifying framework in which atoms occupy sites of a lattice. Hard core repulsion between atoms is built in by allowing the occupation number ni of site i to take only two values: ni = 0 or ni = 1. In addition, there is an attractive energy between atoms on neighbouring sites of magnitude V , and the concentration of atoms is controlled by a chemical potential µ. The Hamiltonian is H = −V X i nini+1 −µ X i ni . (3.2) Both the Ising model and the lattice gas have microscopic degrees of freedom with two possible states, and one can be mapped onto the other: for a lattice in which each site has z neighbours (z=2 in one dimension) the replacements Si = 2ni −1, 4J = V , and 2h −2zJ = µ give HIsing = Hlattice gas + constant. 3.1.3 Classical XY and Heisenberg models Magnetic systems can also be represented using classical spins that are two or three component unit vectors, in place of Ising spins, giving the XY and Heisenberg models. In each case, writing the spin at site i as ⃗ Si, with \|⃗ Si\| = 1 and an external ﬁeld as ⃗ h, the Hamiltonian is H = −J X i ⃗ Si · ⃗ Si+1 −⃗ h · X i ⃗ Si . (3.3) For the XY model we can alternatively use an angle θi to specify the orientation of spin ⃗ Si, writing H = −J X i cos(θi+1 −θi) −h X cos(θi) . (3.4) At ⃗ h = ⃗ 0 both these models are symmetric under global rotations of the spins; they both also have continuous sets of ground states, in which all spins are aligned but the common direction is arbitrary. 3.1.4 Potts model In some circumstances a symmetry other than rotational symmetry is appropriate. In the q-state Potts model, microscopic variables σi take an integer number q possible states, σi = 1 . . . q, and the Hamiltonian is symmetric under global permutations of these states, with H = −J X i δ(σi, σi+1) , (3.5) where we use the notation δ(σi, σi+1) for a Kronecker delta. With ferromagnetic interactions (J > 0), all σi are the same in a ground state and there are q such states. For q = 2 the Potts model is equivalent to the Ising model. 3.2 Continuum models As an alternative to these lattice models, it is also useful to decribe such systems in terms of ﬁelds (the microscopic degrees of freedom) that are deﬁned as a function of continuous spatial coordinates r (or x in one dimension). If the underlying physical problem involves a lattice (as in a crystalline solid), then a continuum description may arise if we take a coarse-grained view of the system, in which information at the scale of the lattice spacing is averaged out. In such an approach it is natural to replace discrete-valued lattice variables, such as Ising spins, by ﬁelds that take values from a continuous range. 3.3. STATISTICAL MECHANICS 31 3.2.1 Scalar ϕ4 theory A continuum model equivalent to the Ising model should be based on a scalar ﬁeld ϕ(x), since Ising spins are scalars. The Hamiltonian at h = 0 should be invariant under global inversion ϕ(x) →−ϕ(x) and should have two ground states, with ϕ(x) = ±ϕ0. In addition, it should favour energetically states in which ϕ(x) is constant, since for the lattice model neighbouring spins have higher energy if they are oppositely orientated. These considerations lead us to H = Z dx " J 2 dϕ(x) dx 2 + V (ϕ(x)) −hϕ(x) # (3.6) where the potential V (ϕ(x)) is chosen to have minima at ϕ(x) = ±ϕ0, and in its simplest form is V (ϕ(x)) = t 2ϕ2(x) + u 4 ϕ4(x) . (3.7) We require u > 0 so that the energy has a lower bound; if in addition t > 0, then V (ϕ(x)) has a single minimum at ϕ(x) = 0, whereas for t < 0 it has two minima at ϕ(x) = ±(−t/u)1/2. 3.2.2 Continuum limit of the XY model We can write down a continuum version of the XY model without making the step that took us from Ising spins to the ﬁeld ϕ(x), since the states available to vector spins form a continuous set. Starting from Eq. (3.4), it is natural to introduce a ﬁeld θ(x) and write H = Z dx " J 2 dθ(x) dx 2 −h cos(θ(x)) # . (3.8) 3.3 Statistical mechanics For completeness, we recall some of the main results of statistical mechanics. Consider a conﬁguration (or mi-crostate) of one of the models introduced above. Its energy is given by the Hamiltonian H. When the system is in equilibrium with a heat bath at temperature T , then writing β = 1/kBT , where kB is Boltzmann’s constant, the probability for it to adopt a particular microstate is proportional to the Boltzmann factor exp(−βH). The normalisation constant for these probabilities is the partition function Z = X states e−βH . (3.9) Here, the sum on states indicates literally a sum on discrete states for the Ising and Potts models, and multiple integrals on spin orientations at each site for the lattice XY and Heisenberg models, while for continuum models it denotes a functional integral over ﬁeld conﬁgurations. We will be concerned with thermal averages of observables: averages over conﬁgurations, weighted with the Boltzmann factor. We use the notation ⟨. . .⟩= Z−1 X states . . . e−βH (3.10) where . . . stands for the observable. One example is the internal energy E, the average energy of the system, which can be calculated from the partition function via E ≡⟨H⟩= −∂ ∂β ln(Z) . (3.11) Other thermodynamic quantities can also be obtained from the partition function. In particular, the (Helmholtz) free energy is F = −kBT ln(Z) (3.12) and the entropy S is S = E −F T . (3.13) 32 CHAPTER 3. STATISTICAL MECHANICS IN ONE DIMENSION When we come to develop intuition about the behaviour of models in statistical mechanics, it is useful to remember the expression for the entropy of a system in the microcannonical ensemble, with W accessible states: S = kB ln(W). In addition, it is helpful to recall that free energy F is minimised for a system in equilibrium. To characterise the behaviour of interacting systems, we will be particularly concerned with thermal averages of products of microscopic variables. For example, for the Ising model we are interested in the magnetisation ⟨Si⟩ and the two-point correlation function ⟨SiSj⟩. The partition function, viewed as a function of external ﬁeld h, is the generating function for these correlation functions, provided we allow the external ﬁeld to take independent values hi at each site. In particular, for the Ising model we have ⟨Si⟩= 1 β ∂ ∂hi ln(Z) and ⟨SiSj⟩−⟨Si⟩⟨Sj⟩= 1 β2 ∂2 ∂hi∂hj ln(Z) . (3.14) Moreover, the magnetisation m and the magnetic susceptibility χ can be obtained from derivatives with respect to the strength of an external ﬁeld that is uniform: for a system of N sites m ≡1 N N X i=1 ⟨Si⟩= 1 βN ∂ ∂h ln(Z) and χ ≡∂m ∂h = β N N X i,j=1 [⟨SiSj⟩−⟨Si⟩⟨Sj⟩] . (3.15) In a similar way, for the continuum theories we have introduced, we allow the external ﬁeld to be a function of position, h(x), so that the partition function is a functional, and then functional derivatives give correlation functions, as in Chapter 1. The dependence of the two-point correlation function on separation provides a measure of the inﬂuence of interactions on the behaviour of the system. In the high temperature limit (β →0) all conﬁgurations are weighted equally in a thermal average, spins ﬂuctuate independently at each site, and the correlation function is short range, meaning that it falls to zero at large separation (in fact, in this limit ⟨SiSj⟩=0 unless i=j). In the opposite limit of zero temperature (β →∞), the system is in its ground state: as we have seen, all spins then adopt the same conﬁguration and the correlation function is inﬁnite-ranged. We would like to understand in detail how behaviour interpolates between these two limiting cases as temperature varies. 3.4 Transfer matrix method The transfer matrix method provides a general formalism for solving one-dimensional models with short range interactions in classical statistical mechanics. It can also be formulated for systems in higher dimensions, but is then tractable only in special cases. We will describe it for a general system, but using the notation of the one-dimensional Ising model, Eq. (3.1). A ﬁrst step is to divide the one-dimensional system into a series of slices, analogous to the time steps used in the path integral formulation of quantum mechanics. The slices must be chosen long enough that interactions couple only degrees of freedom in neighbouring slices. We denote the degrees of freedom in the i-th slice by Si and write the Hamiltonian as a sum of terms, each involving only the degrees of freedom in adjacent slices: H = X i H(Si, Si+1) . The Boltzmann factor, being the exponential of this sum, is a product of terms: e−βH = Y i T (Si, Si+1) with T (Si, Si+1) = e−βH(Si,Si+1) . (3.16) Because interactions couple only adjacent slices, only two terms in this product depend on a given Si: T (Si−1, Si) and T (Si, Si+1). Moverover, summation on Si acts just like matrix multiplication. This leads us to deﬁne the transfer matrix T: for the case in which Si takes M values (M=2 for the one-dimensional Ising model with only nearest-neighbour interactions) it is an M × M matrix with rows and columns labelled by the possible values of Si and Si+1, and matrix elements T (Si, Si+1) as deﬁned in Eq. (3.16). For a system of N slices and periodic boundary conditions (so that i runs from 1 to N), the partition function is simply the matrix trace Z = Tr TN . Alternatively, with ﬁxed conﬁgurations for S1 and SN, and without periodic boundary conditions, the partition function is the matrix element Z = TN−1(S1, SN) . 3.5. TRANSFER MATRIX SOLUTION OF THE ISING MODEL IN ONE DIMENSION 33 The advantage of the transfer matrix approach is that in this way calculations are reduced to study of an M ×M matrix, independent of system size N. We will assume for simplicity that H(Si, Si+1)=H(Si+1, Si) so that T is symmetric and its eigenvectors \|α⟩can be chosen to form a complete, orthonormal set. We order the associated eigenvalues by magnitude (the ﬁrst is in fact necessarily positive and non-degenerate): λ0 > \|λ1\| ≥. . . \|λM−1\|, with T\|α⟩= λα\|α⟩, ⟨α\|β⟩= δαβ and T = X α \|α⟩λα⟨α\| . Using this notation, it is simple to write down a power of the transfer matrix: we have TN = X α \|α⟩λN α ⟨α\| . The free energy per slice with periodic boundary conditions is therefore F N = −kBT N ln(Z) = −kBT ln(λ0) −kBT N ln 1 + M−1 X α=1 [λα/λ0]N ! . In the thermodynamiclimit (N →∞), [λα/λ0]N →0 for α ≥1, and so the free energy density f= limN→∞F/N is simply f = −kBT ln(λ0) . (3.17) Clearly, we can obtain other thermodynamic quantities, including the energy density E/N, the entropy density S/N, the magnetisation m and magnetic susceptibility χ by this route. To determine correlation functions, one might imagine we should ﬁrst evaluate a generating function dependent on ﬁeld values hi at each site. Within the transfer matrix apporach, however, this is not convenient quantity to consider, because if ﬁeld values vary at different sites, the transfer matrices are different for each slice and transfer matrix products no longer have simple expressions in terms of powers of the eigenvalues. Instead, we extend the transfer matrix formalism by deﬁning diagonal matrices C with diagonal elements C(Si, Si) that are functions of the degrees of freedom within a slice, chosen to reproduce the required correlation function. For example, for the one-dimensional Ising model with N sites and periodic boundary conditions, taking C(Si, Si) = Si we have ⟨Si⟩= Tr TiCTN−i Tr [TN] and ⟨SiSi+x⟩= Tr TiCTxCTN−i−x Tr [TN] . As happened for the free energy density, these expressions simplify greatly in the thermodynamic limit when written in terms of the eigenvalues and eigenvectors of the transfer matrix: they reduce to lim N→∞ Tr TiCTN−i Tr [TN] = ⟨0\|C\|0⟩ (3.18) and lim N→∞ Tr TiCTxCTN−i−x Tr [TN] = X α ⟨0\|C\|α⟩⟨α\|C\|0⟩ λα λ0 x . (3.19) In summary, diagonalisation of the transfer matrix provides a path to calculating all quantities of physical interest. 3.5 Transfer Matrix solution of the Ising model in one dimension Let’s illustrate these ideas by using the transfer matrix approach to solve the one-dimensionalIsing model, Eq. (3.1). We take the Hamiltonian for a single slice to be H(Si, Si+1) = −JSiSi+1 −h 2 (Si + Si+1) . Note that there was an element of choice here: we have used a symmetric form for the Zeeman energy, which will in turn ensure that the transfer matrix is symmetric. The transfer matrix is T = exp(β[J + h]) exp(−βJ) exp(−βJ) exp(β[J −h]) . (3.20) 34 CHAPTER 3. STATISTICAL MECHANICS IN ONE DIMENSION For simplicity, we will set h=0. Then the eigenvalues are λ0 = 2 cosh(βJ) and λ1 = 2 sinh(βJ), and the eigenvectors are \|0⟩= 2−1/2(1, 1)T and \|1⟩= 2−1/2(1, −1)T. The matrix representing the spin operator is C = 1 0 0 −1 , and its matrix elements in the basis of eigenstates are ⟨0\|C\|0⟩= ⟨1\|C\|1⟩= 0 and ⟨0\|C\|1⟩= ⟨1\|C\|0⟩= 1. We are now in a position to write down some results. From Eq. (3.17), the free energy density is f = −kBT ln(2 cosh(βJ)) . As a check, we should examine its behaviour in the high and low temperature limits. At high temperature (β →0), f ∼−kBT ln(2). This is as expected from the relation between free energy, energy and entropy, Eq. (3.13): in the high-temperature limit neighbouring spins are equally likely to be parallel or antiparallel, and so the entropy per spin (from the general formula S = kB ln(W), with W the number of accessible states) is S = kB ln(2), while the average energy is ⟨H⟩=0. Conversely, at low temperature (β →∞), f = −J, which arises because in this limit the system is in a ground state, with neighboring spins parallel, so that ⟨H⟩= −NJ and S = 0. Beyond these checks, the most interesting and important feature of our result for the free energy density is that it is analytic in temperature for all T > 0. As we will discuss later in the course, phase transitions are associated with singularities in the free energy density as a function of temperature, and analyticity of f in the one-dimensional Ising model reﬂects the absence of a ﬁnite-temperature phase transition. The model in fact has a critical point at T = 0, and f is non-analytic there (compare the limits T →0+ and T →0−). What happens in the model at low temperatures is most clearly revealed by the form of the correlation functions, although for this we have to go to the two-point function. The one-point function, or magnetisation is trivial: ⟨Si⟩= ⟨0\|C\|0⟩= 0 , which is a consequence of symmetry at h = 0 under global spin reversal. The two-point correlation function between spins separated by a distance x is ⟨SiSi+x⟩= λ1 λ0 \|x\| = exp(−\|x\|/ξ) with 1 ξ = ln[coth(βJ)] . We see that correlations decay exponentially with separation, with a lengthscale ξ. This lengthscale, termed the correlation length, increases monotonically with decreasing temperature and diverges as T →0. Its asymptotic form at low temperature (βJ ≫1) is ξ = 1 ln[coth(βJ)] ∼ 1 ln[1 + 2e−2βJ] ∼1 2e2βJ . (3.21) Useful physical insight into this result comes from a simple picture of typical states at low temperature. It is clear that they consist of long sequences of parallel spins, with occasional reversals in orientation, as in Fig. 3.1. In these circumstances it is natural to focus on the reversals, called domain walls or kinks, as the elementary Figure 3.1: A typical low-temperature conﬁguration of the one-dimensional Ising model. excitations in the system. Their average spacing sets the correlation length. Moreover, each domain wall costs an energy 2J, the difference between the two possible values of JSiSi+1. It also has an entropy associated with it, since it can be placed between any one of O(1/ξ) neighbouring pairs of spins, without disrupting the overall arrangement of irregularly spaced kinks. An estimate of the free energy for a chain of L sites is therefore F ∼2J L ξ −kBT L ξ ln(ξ) . (3.22) The actual value of ξ at a given temperature can be estimated as the one that minimises Eq. (3.22), yielding ξ ∼exp(2βJ) for βJ ≫1, in reasonable agreement with our earlier, detailed calcuation. 3.6. STATISTICAL MECHANICS IN ONE DIMENSION AND QUANTUM MECHANICS 35 3.6 Statistical mechanics in one dimension and quantum mechanics There is a general relationship between the statistical mechanics of a classical system in d + 1 dimensions at ﬁnite temperature, and the (Euclidean-time) quantum theory of a many-body system in d dimensions at zero temperature. Under this mapping, thermal ﬂuctuations in the classical system become zero-point ﬂuctuations in the quantum system. We will examine this relationship as it applies to the one-dimensional, classical statistical-mechanical systems we have met in this chapter. For these examples, since d+1=1, we have d=0, meaning that the quantum theory is particularly simple: rather than being a theory for a many-body quantum system in a ﬁnite number of dimensions, it involves just a single particle. In this way, for d=0 we map the statistical-mechanical problem to a problem in quantum mechanics, while for d>0 we would arrive at a problem in quantum ﬁeld theory. In the following, we establish three different variants of this connection. All are based on viewing the spatial dimension of the statistical-mechanical system as the imaginary time direction for a corresponding quantum system. 3.6.1 Ising model and spin in a transverse ﬁeld Consider the transfer matrix T for the one-dimensional Ising model. We want to view this as the imaginary time evolution operator exp(−τHQ/ℏ) for a quantum system with Hamiltonian HQ, where τ is the duration in imaginary time equivalent to the distance in the Ising model between neighbouring sites. Since T is a 2×2 matrix, so must HQ be. That suggests we should regard HQ as the Hamiltonian for a single spin of magnitude S = 1/2, for which the Pauli matrices σx, σy and σz provide a complete set of operators. Anticipating the ﬁnal result, note that with σx = 0 1 1 0 and α a constant, one has exp(ασx) = cosh(α)1 + sinh(α)σx . Matching this against Eq. (3.20) for h = 0, we ﬁnd that T = eβJ1 + e−βJσx ≡Aeασx with tanh(α) = e−2βJ and A = p 2 sinh(2βJ). In this way we can read off the Hamiltonian for the equivalent quantum system: τHQ/ℏ= −ασx + ln(A). In addition, we see that the lowest eigenvalue ǫ0 of HQ is related to the largest eigenvalue of the transfer matrix via ln(λ0) = −τǫ0/ℏ. Also, the inverse correlation length (in units of the lattice spacing) is related to the splitting between the ground state and ﬁrst excited state eigenvalues of HQ: we have ξ−1 = τ(ǫ1 −ǫ0)/ℏ= 2α. 3.6.2 Scalar ϕ4 and a particle in double-well potential For the one-dimensional Ising model, the mapping we have just set out between classical statistical mechanics and quantum mechanics takes us to a quantum problem with a ﬁnite dimensional Hilbert space, since the number of possible states for an Ising spin is ﬁnite. By contrast, one-dimensional statistical mechanical problems with continuous degrees of freedom map onto quantum problems with inﬁnite-dimensional Hilbert spaces. In particular, one-dimensional statistical mechanics problems with n continuous degrees of freedom at each point in space are equivalent to quantum problems involving a single particle moving in n dimensions. Let’s examine how this works for one-dimensional ϕ4 theory, starting from Eq. (3.6). The partition function Z for a system of length L with ﬁxed values ϕ(0) and ϕ(L) for the ﬁeld at x=0 and x=L is given by the functional integral Z = Z D[ϕ(x)]e−βH over functions ϕ(x) satisfying the boundary conditions. Referring back to Eqns. (1.87) and (1.88), we see that βH in the classical statistical mechanics problem plays the same role as the Euclidean action S/ℏin a path integral expression for the Boltzmann factor arising in a quantum problem: we use HQ to denote the Hamiltonian of the quantum problem. The translation dictionary is as follows. Classical statistical mechanics Quantum mechanics position imaginary time system length L imaginary time interval βℏ ﬁeld ϕ(x) particle coordinate thermal energy kBT Planck’s constant ℏ exchange stiffness J particle mass m 36 CHAPTER 3. STATISTICAL MECHANICS IN ONE DIMENSION The action is that for a particle of mass J moving in a potential V (x): reversing the steps of section 1.2.3, we can read off the quantum Hamiltonian as HQ = − 1 2β2J d2 dϕ2 + V (ϕ) . (3.23) Knowledge of the eigenvalues ǫα and eigenfunctions \|α⟩of HQ, which satisfy HQ\|α⟩= ǫα\|α⟩, gives access to thermodynamic quantities and correlation functions for the classical system. In particular, for a classical system extending over −∞< x < ∞, the arguments leading to Eq. (3.19) also give ⟨ϕ(x1)ϕ(x2)⟩= X α ⟨0\|ϕ\|α⟩⟨α\|ϕ\|0⟩e−β(ǫα−ǫ0)\|x1−x2\| Now, although we cannot ﬁnd the eigenfunctions of HQ exactly, we know quite a lot about them for the case of interest, in which V (ϕ) is a quartic double-well potential. In particular, since V (−ϕ) = V (ϕ), all eigenfunctions have deﬁnite parity. The ground state wavefunction, ⟨ϕ\|0⟩has even partiy and is nodeless, while the ﬁrst excited state wavefunction, ⟨ϕ\|1⟩, is odd, having a single node at ϕ = 0. From this it follows that ⟨0\|ϕ\|0⟩= 0 and ⟨0\|ϕ\|1⟩̸= 0. The inverse correlation length, governing the correlation function at large separation \|x1 −x2\| is therefore ξ−1 = β(ǫ1 −ǫ0). At low temperatures, the form of the lowest and ﬁrst excited eigenstates is as sketched in Fig. 3.2. In the language of quantum mechanics, the splitting between eigenvalues ǫ1 −ǫ0 arises because of tunneling through the barrier between the two minima of V (ϕ) and is exponentially small for small T , which leads to a correlation length exponentially large in β, as for the Ising model, Eq. (3.21). −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5 Figure 3.2: Left: the potential V (ϕ) appearing in HQ, Eq. (3.23). Centre and right: form of the lowest two eigenfunctions of HQ, ⟨ϕ\|0⟩and ⟨ϕ\|1⟩, for small T 3.6.3 One-dimensional XY model and the quantum rotor An exactly parallel treatment can be applied to the XY chain, deﬁned in Eq. (3.8). For this case we ﬁnd HQ = − 1 2β2J d2 dθ2 subject to the condition that eigenfunctions are periodic in θ with period 2π. The eigenfunctions and eigenvalues are of course ⟨θ\|n⟩= (2π)−1/2 exp(inθ) and ǫn = n2/(2β2J) with n = 0, ±1, . . . . The correlation functions we use to characterise behaviour of the system should be constructed in a way that respects the periodic nature of the coordinate θ. An obvious candidate is ⟨ei[θ(x1)−θ(x2)]⟩. As in our previous examples, we can express this in terms of the eigenfunctions and eigenvalues of HQ. We ﬁnd ⟨ei[θ(x1)−θ(x2)]⟩= X n ⟨0\|e−iθ\|n⟩⟨n\|eiθ\|0⟩e−β(ǫn−ǫ0)\|x1−x2\| . The correlation length for this model is therefore ξ = [β(ǫ1 −ǫ0)]−1 = 2J/(kBT ). As for the Ising model, ξ diverges in the limit T →0, which is expected since for T = 0 the system adopts a ground state with θ(x) indepndent of x. The divergence of ξ, howover, is much less rapid in the XY model than in the Ising model. The reason for this is that, whereas excitations in the Ising model cost a minimum energy, the kink energy 2J, long wavelength twists of θ(x) in the XY model can have arbitrarily low energy. As a consequence, thermal excitations are more effective in disordering the XY model at low temperature than for the Ising model, leading to a shorter correlation length in the XY model. 3.7. FURTHER READING 37 3.7 Further reading • K. Huang Introduction to Statistical Physics (CRC Press). A good review of ideas from statistical mechanics that form the background for this chapter. • J. M. Yeomans Statistical Mechanics of Phase Transitions (OUP). Chapters 2 and 5 give an introduction to lattice models and transfer matrix methods. • J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman, The Theory of Critical Phenomena (OUP). Chapter 3 gives a self-contained introduction to models and to transfer matrix methods. 38 CHAPTER 3. STATISTICAL MECHANICS IN ONE DIMENSION Chapter 4 Classical Field Theory In this chapter, we will develop the Lagrangian approach to the classical theory of ﬁelds, focusing on ﬁeld the-ories with scalar and vector ﬁelds. Our discussion will culminate in the discussion of scalar electrodymanics, a theory which couples Maxwell’s theory of electrodymanics to scalar ﬁelds. As we will see, symmetries play an important role in constructing all these theories. Traditionally, symmetries of physical theories have often been identiﬁed only after a proper mathematical formulation of the theory. For example, the fact that Maxwell’s the-ory of electrodynamics is covariant under Lorentz transformations was only discovered signiﬁcantly after its the ﬁrst formulation. In modern theoretical physics, this traditional relation between theories and their symmetries is frequently reversed. One starts by identifying the symmetries of the given physical situation and then writes down the (most general) theory compatible with these symmetries. This approach has been immensely fruitful and has played a major role, for example, in constructing the standard model of particle physics. It is, therefore, crucial to understand the relevant symmetries (or groups in Mathematical language) and the objects they act on (representations, in Mathematical language) ﬁrst. 4.1 Symmetries 4.1.1 Deﬁnition of groups and some examples The word ”symmetry” in Physics usually (although not always) refers to the Mathematical structure of a group, so we begin with the following deﬁnition. Deﬁnition A group G is a set with a map · : G × G →G (”multiplication”) satisfying the three conditions 1) g1 · (g2 · g3) = (g1 · g2) · g3 for all g1, g2, g3 ∈G (associativity) 2) There exists an e ∈G such that g · e = g for all g ∈G (neutral element) 3) For each g ∈G there exist a g−1 ∈G such that g · g−1 = e (inverse element) It is easy to prove from the above axioms that the neutral element e is unique, that it is also the neutral element when acting from the left, that is e · g = g for all g ∈G, that the right-inverse g−1 is uniquely deﬁned for each g ∈G and that it is also the left-inverse, that is g−1 ·g = e. If, in addition to the three axions above, g1 ·g2 = g2 ·g1 is satisﬁed for all g1, g2 ∈G the group is called Abelian. Well-known groups are the integers with respect to addition and the real and complex numbers with respect to addition and multiplication. All these groups are Abelian. Here are some more interesting groups which will play a role in our ﬁeld theory constructions. Consider ﬁrst the group ZN = {0, 1, · · · , N −1} with ”multiplication” deﬁned by n1 · n2 = (n1 + n2) mod N. This group is obviously ﬁnite (that is, it has a ﬁnite number of elements) and Abelian. Another Abelian example is given by the complex numbers of unit length, U(1) = {z ∈C\| \|z\| = 1}, with group multiplication the ordinary multiplication of complex numbers. Not only is this group inﬁnite but, as it corresponds to the unit circle in the complex plane, it is also ”continuous” and one-dimensional. Examples for non-Abelian groups are provided by the unitary groups SU(n) which consist of all complex n × n matrices U satisfying U †U = 1 and det(U) = 1, with ordinary matrix multiplication as the group multiplication and the unit matrix as the neutral element. Matrix multiplication does in general not ”commute” which causes the non-Abelian character of the unitary groups. The simplest non-trivial example of a unitary group on which we will focus later is SU(2). Solving the unitary conditions U †U = 1 and det(U) = 1 by inserting an arbitrary 2 × 2 matrix with 39 40 CHAPTER 4. CLASSICAL FIELD THEORY complex entries it is easy to show that SU(2) can be written as SU(2) = α β −β⋆ α⋆ \| α, β ∈C and \|α\|2 + \|β\|2 = 1 . (4.1) This shows that we can think of SU(2) as the unit sphere in four-dimensional Euklidean space and, hence, that it is a three-dimensional continuous group. Such continuous groups are also called Lie groups in Mathematical parlance and we will discuss some of their elementary properties in due course. We can solve the constraint on α and β in Eq. (4.1) by setting α = p 1 −\|β\|2eiσ and β = −β2 + iβ1 which leads to the explicit parameterization U = p 1 −\|β\|2eiσ −β2 + iβ1 β2 + iβ1 p 1 −\|β\|2e−iσ , (4.2) of SU(2) in terms of β1, β2 and σ. 4.1.2 Representations of groups Let us denote by Gl(n) the group of invertible n × n matrices with real (or complex) entries. We can think of these matrices as linear transformations acting on an n-dimensional real (or complex) vector space V ∼Rn (or V ∼Cn). Deﬁnition A representation R of a group G is a map R : G →GL(n) which satisﬁes R(g1 · g2) = R(g1)R(g2). In other words, a representation assigns to each element of a group a matrix such that these matrices multiply ”in the same way” as the associated group elements. In this way, the group is realised or ”represented” as a set of matrices. Given a representation by n × n matrices we can also think of the group as acting on the n-dimensional vector space V , via the representation matrices R(g) ∈Gl(n). The dimension n of this vector space is also referred to as the dimension of the representation. In a physics context, the elements of this vector space can be thought of as the ﬁelds (more precisely, the ﬁelds at each ﬁxed point in space-time) and, hence, group representations provide the appropriate mathematical structure to describe symmetries acting on ﬁelds. The mathematical problem of ﬁnding all representations of a given group then translates into the physics problem of ﬁnding all ﬁelds on which the symmetry can act and, hence, amounts to a classiﬁcation of all possible objects from which a ﬁeld theory which respects the symmetry can be ”built up”. We will discuss explicit examples of this later on. For now, let us present a few simple examples of representations. The trivial representation which exists for all groups is given by R(g) = 1, for all g ∈G, so each group element is represented by the unit matrix in a given dimension. For the group ZN and each integer q we can write down the representation Rq(n) = cos(2πqn/N) sin(2πqn/N) −sin(2πqn/N) cos(2πqn/N) , (4.3) by real two-dimensional rotation matrices over the vector space V = R2, where n = 0, . . . , N −1. We can restrict the value of q to the range 0, . . . , N −1 (as two values of q which differ by N lead to the same set of matrices) and this provides, in fact, a complete list of representations for ZN. Equivalently, we can write down the same representations over a one-dimensional complex vector space V = C where they take the form Rq(n) = exp(2πiqn/N). If, for a given representation Rq, we denote elements of the vector space V = C by Φ then the group acts on them as Φ →Rq(n)Φ = exp(2πiqn/N)Φ. In this case, Φ is said to have ”charge” −q in physics language. Representations Rq for U(1) (on V = C) are just as easily obtained by writing Rq(eiα) = eiqα , (4.4) where α ∈[0, 2π]. For Rq to be continuous when going around the circle, q must be an integer, however, unlike in the ZN case it is not otherwise restricted. The above representations Rq for q an arbitrary integer, in fact, provide all (continuous) representations of U(1). As before, a (complex) ﬁeld transforming as Φ →Rq(eiα)Φ is said to have charge −q. Also note that charge q = 0 corresponds to the trivial representation. The groups SU(n) are already given by matrices, so we can think about them as representing themselves. This representation is n-complex dimensional and is also called the fundamental representation. On a complex vector Φ = (φ1, . . . , φn) is acts as Φ →UΦ, where U ∈SU(n). However, this is by no means the only representation of SU(n), in fact, there is an inﬁnite number of them, as we will see. 4.1. SYMMETRIES 41 There are a number of general ways of constructing new representations from old ones which should be men-tioned. For a representation R : G →Gl(n) of a group G there is a complex conjugate representation R⋆deﬁned by R⋆(g) = R(g)⋆, that is, each group element is now represented by the complex conjugate of the original representation matrix. Applying this to SU(n) leads to the complex conjugate of the fundamental representation U →U ⋆. For two representations R1 and R2 of a group G with dimensions n1 and n2 one can consider the direct sum representation R1 ⊕R2 with dimension n1 + n2 deﬁned by the block-diagonal matrices (R1 ⊕R2)(g) = R1(g) 0 0 R2(g) . (4.5) A representation such as this is called reducible and, conversely, a representation which cannot be split into smaller blocks as in (4.5) is called irreducible. For example, the direct sum representation R(eiα) = diag(eiα, e−iα) (4.6) of U(1) consisting of a charge +1 and −1 representation realises an explicit embedding of U(1) into SU(2). Another, less trivial way of combining the two representations R1 and R2 to a new one is the tensor representation R1 ⊗R2 with dimension n1n2 deﬁned by (R1 ⊗R2)(g) = R1(g) × R2(g) 1. In general, a tensor representation R1 × R2 is not irreducible and decomposes into a sum of irreducible representations R(i), so one can write R1 ⊗R2 = M i R(i) (4.7) This is also referred to as Clebsch-Gordon decomposition. 4.1.3 Lie groups and Lie algebras To understand representations of Lie groups we should look at their structure more closely. The matrices M of a Lie group form a continuous (differentiable) family M = M(t) where t = (t1, . . . , tm) are m real parameters and we adopt the convention that M(0) = 1. An example for such a parametrisation for the case of SU(2) has been given in Eq. (4.2), where the three parameters are (t1, t2, t3) = (β1, β2, σ). Let us now look in more detail at the neighbourhood of the identity element, corresponding to small values of the parameters t, where we can expand M(t) = 1 + X i tiTi + O(t2) , with Ti = ∂M ∂ti (0) . (4.8) The matrices Ti are called the generators of the Lie group and the vector space L(G) = {tiTi} spanned by these matrices is referred to as Lie algebra. In the case of SU(2), the generators are given by (i times) the Pauli matrices, as differentiating Eq. (4.2) shows. In general, there is a theorem which states that the group (or, rather, a neighbourhood of the identity of the group) can be reconstructed from the Lie algebra by the exponential map M(t) = exp(tiTi) . (4.9) Now consider two matrices M(t) and M(s) and the product M(t)−1M(s)−1M(t)M(s) = 1 + X i,j tisj[Ti, Tj] + · · · , (4.10) where [·, ·] is the ordinary matrix commutator. Since the product on the LHS of Eq. (4.10) is an element of the group, we conclude that the commutators [Ti, Tj] must be elements of the Lie algebra and can, hence, be written as [Ti, Tj] = fij kTk . (4.11) The coefﬁcients fij k are called the structure constants of the Lie algebra L(G). More accurately, the Lie-algebra L(G) is then the vector space L(G) = {tiTi} together with the commutator bracket [·, ·]. The concept of a representation can now also be deﬁned at the level of the Lie algebra. Deﬁnition A representation r of a Lie algebra L is a linear map which assigns to elements T ∈L matrices r(T ) such that [r(T ), r(S)] = r([T, S]) for all T, S ∈L. 1For two matrices M and N the product M ×N can be thought of as the matrix obtained by replacing each entry in M by a block consisting of that entry times N. A useful property of this product is (M1 × N1)(M2 × N2) = (M1M2) × (N1N2). 42 CHAPTER 4. CLASSICAL FIELD THEORY Note this is equivalent to saying that the representation matrices r(Ti) commute in the same way as the generators Ti, so [r(Ti), r(Tj)] = fij kr(Tk), with the same structure constants fij k as in Eq. (4.11). It is usually easier to ﬁnd representations of Lie algebras than representations of groups. However, once a Lie-algebra representation has been found the associated group representation can be re-constructed using the exponential map (4.9). Concretely, for a Lie-algebra representation Ti →r(Ti) the corresponding group representation is etiTi →etir(Ti). Recall that the dimension of the representation r is deﬁned to be the dimension of the vector space on which the repre-sentation matrices r(T ) (or the associated group elements obtained after exponentiating) act, that is, it is given by the size of the matrices r(T ). This dimension of the representation r is not to be confused with the dimension of the Lie-algebra itself, the latter being the dimension of the Lie algebra L(G) as a vector space of matrices. Example SU(2) Let us see how all this works for our prime example SU(2). Consider an SU(2) matrix U close to the identity matrix and write 2 U = 1 + iT + . . . , where T is a Lie algebra element. Then, evaluating the conditions U †U = 1 and det(U) = 1 at linear level in T , one ﬁnds the constraints T † = T and tr(T ) = 0. In other words, the Lie algebra L(SU(2)) of SU(2) consists of all traceless, hermitian 2 × 2 matrices. Note that this space is three-dimensional. A convenient basis of generators τi for this Lie algebra is obtained from the Pauli matrices σi. Recall that they satisfy the useful identities σiσj = δij1 + iǫijkσk , [σi, σj] = 2iǫijkσk , tr(σiσj) = 2δij . (4.12) Hence, the Lie algebra of SU(2) is spanned by the generators τi = 1 2σi with [τi, τj] = iǫijkτk , (4.13) and the structure constants are simply given by the Levi-Civita tensor. While the dimension of the Lie algebra L(SU(2)) is 3 (as it is spanned by three Pauli matrices), the dimension of the SU(2) representation deﬁned by the Pauli matrices is 2 (since they are 2 × 2 matrices). Finite SU(2) matrices are then obtained by exponentiating U = exp(itiτi) = exp(itiσi/2) . (4.14) Note that the generator τ3 corresponds to the U(1) subgroup (4.6) of SU(2). The commutation relations (4.13) of the SU(2) Lie algebra are identical to the commutation relations of the angular momentum operators in quantum mechanics. Hence, we already know that the ﬁnite-dimensional representations of this algebra can be labelled by a ”spin” j, that is an integer or half-integer number j = 0, 1/2, 1, 3/2, . . .. For a given j the dimension of the representation is 2j+1 and the representation space is spanned by states \|jm⟩, where m = −j, −j+1, . . . , j−1, j. The two-dimensional representation for j = 1/2 of course corresponds to the explicit representation of the algebra in terms of Pauli matrices which we have written down above. The complex conjugate of the fundamental is also a two-dimensional representation and, on purely dimensional grounds, must also be identiﬁed with the j = 1/2 representation. We also know from quantum mechanics that the tensor product of two representations characterised by j1 and j2 decomposes into the irreducible representations with j in the range \|j1 −j2\|, \|j1 −j2\| + 1, . . . , j1 + j2. This is an explicit example of a Clebsch-Gordon decomposition. It is customary to refer to representations by their dimensions, that is, write for example, the j = 1/2 representation as 2 (or ¯ 2 for the conjugate) and the j = 1 representation as 3. With this notation, examples of SU(2) Clebsch-Gordon decompositions are 2 ⊗2 = 1 ⊕3 , 2 ⊗3 = 2 ⊕4 . (4.15) Actions should be invariant under a symmetry group and, hence, it is of particular importance to understand the singlets which occur in a Clebsch-Gordan decomposition. They will tell us about the invariant terms which are allowed in an action. For example, if we have a ﬁeld Φ which transforms as a doublet under SU(2), the ﬁrst of Eqs. (4.15) tells us that we should be able to write a quadratic term in Φ, corresponding to the direction of the singlet on the right-hand side. Example SO(3) Another important Lie group is SO(3), the group of three-dimensional rotations, consisting of real 3 × 3 matrices 2In the physics literature it is conventional to include a factor of i in front of the generators T. 4.1. SYMMETRIES 43 det(Λ) Λ00 name contains given by +1 ≥1 L↑ + 14 L↑ + +1 ≤−1 L↓ + PT PT L↑ + −1 ≥1 L↑ − P PL↑ + −1 ≤−1 L↓ − T T L↑ + Table 4.1: The four disconnected components of the Lorentz group. The union L+ = L↑ + ∪L↓ + is also called the proper Lorentz group and L↑= L↑ +∪L↑ −is called the orthochronos Lorentz group (as it consists of transformations preserving the direction of time). L↑ + is called the proper orthochronos Lorentz group. O satisfying OT O = 1 and det(O) = 1. Writing O = 1 + iT with (purely imaginary) generators T , the relation OT O = 1 implies T = T † and, hence, that the Lie-algebra of SO(3) consists of 3 × 3 anti-symmetric matrices (multiplied by i). A basis for this Lie algebra is provided by the three matrices Ti deﬁned by (Ti)jk = −iǫijk , (4.16) which satisfy the commutation relations [Ti, Tj] = iǫijkTk . (4.17) These are the same commutation relations as in Eq. (4.13) and, hence, the Ti form a three-dimensional (irreducible) representation of (the Lie algebra of) SU(2). This representation must ﬁt into the above classiﬁcation of SU(2) representations by an integer or half-integer number j and, simply on dimensional grounds, it has to be identiﬁed with the j = 1 representation. 4.1.4 The Lorentz group The Lorentz group is of fundamental importance for the construction of ﬁeld theories. It is the symmetry associated to four-dimensional Lorentz space-time and should be respected by ﬁeld theories formulated in Lorentz space-time. Let us begin by formally deﬁning the Lorentz group. With the Lorentz metric η = diag(1, −1, −1, −1) the Lorentz group L consists of real 4 × 4 matrices Λ satisfying ΛT ηΛ = η . (4.18) Special Lorentz transformations are the identity 14, parity P = diag(1, −1, −1, −1), time inversion T = diag(−1, 1, 1, 1) and the product PT = −14. We note that the four matrices {14, P, T, PT } form a ﬁnite sub-group of the Lorentz group. By taking the determinant of the deﬁning relation (4.18) we immediately learn that det(Λ) = ±1 for all Lorentz transformations. Further, if we write out Eq. (4.18) with indices ηµνΛµ ρΛν σ = ηρσ (4.19) and focus on the component ρ = σ = 0 we conclude that (Λ00)2 = 1 + P i(Λi0)2 ≥1, so either Λ00 ≥1 or Λ00 ≤−1. This sign choice for Λ00 combined with the choice for det(Λ) leads to four classes of Lorentz transformations which are summarised in Table 4.1. Also note that the Lorentz group contains three-dimensional rotations since matrices of the form Λ = 1 0 0 O (4.20) satisfy the relation (4.18) and are hence special Lorentz transformations as long as O satisﬁes OT O = 13. To ﬁnd the Lie algebra of the Lorentz group we write Λ = 14+iT +. . . with purely imaginary 4×4 generators T . The deﬁning relation (4.18) then implies for the generators that T = −ηT Tη, so T must be anti-symmetric in the space-space components and symmetric in the space-time components. The space of such matrices is six-dimensional and spanned by Ji = 0 0 0 Ti , K1 =     0 i 0 0 i 0 0 0 0 0 0 0 0 0 0 0    , K2 =     0 0 i 0 0 0 0 0 i 0 0 0 0 0 0 0    , K3 =     0 0 0 i 0 0 0 0 0 0 0 0 i 0 0 0    , (4.21) 44 CHAPTER 4. CLASSICAL FIELD THEORY (j+, j−) dimension name symbol (0, 0) 1 scalar φ (1/2, 0) 2 left-handed Weyl spinor χL (0, 1/2) 2 right-handed Weyl spinor χR (1/2, 0) ⊕(0, 1/2) 4 Dirac spinor ψ (1/2, 1/2) 4 vector Aµ Table 4.2: Low-dimensional representations of the Lorentz group. where Ti are the generators (4.16) of the rotation group. Given the embedding (4.20) of the rotation group into the Lorentz group the appearance of the Ti should not come as a surprise. It is straightforward to work out the commutation relations [Ji, Jj] = iǫijkJk , [Ki, Kj] = −iǫijkJk , [Ji, Kj] = iǫijkKk . (4.22) The above matrices can also be written in a four-dimensional covariant form by introducing six 4×4 matrices σµν, labelled by two anti-symmetric four-indices and deﬁned by (σµν)ρ σ = i(ηρ µηνσ −ηµσηρ ν) . (4.23) By explicit computation one ﬁnds that Ji = 1 2ǫijkσjk and Ki = σ0i. Introducing six independent parameters ǫµν, labelled by an anti-symmetric pair of indices, a Lorentz transformation close to the identity can be written as Λρ σ ≃δρ σ −i 2ǫµν(σµν)ρ σ = δρ σ + ǫρ σ; . (4.24) The commutation relations (4.22) for the Lorentz group are very close to the ones for SU(2) in Eq. (4.13). This analogy can be made even more explicit by introducing a new basis of generators J± i = 1 2(Ji ± iKi) . (4.25) In terms of these generators, the algebra (4.22) takes the form [J± i , J± j ] = iǫijkJ± k , [J+ i , J− j ] = 0 , (4.26) that is, precisely the form of two copies (a direct sum) of two SU(2) Lie-algebras. Irreducible representations of the Lorentz group can therefore be labelled by a pair (j+, j−) of two spins and the dimension of these representations is (2j++1)(2j−+1). A list of a few low-dimensional Lorentz-group representations is provided in Table 4.2. Field theories in Minkowski space usually require Lorentz invariance and, hence, the Lorentz group is of fundamental importance for such theories. Since it is related to the symmetries of space-time it is often also referred as external symmetry of the theory. The classiﬁcation of Lorentz group representations in Table 4.2 provides us with objects which transform in a deﬁnite way under Lorentz transformations and, hence, are the main building blocks of such ﬁeld theories. In these lectures, we will not consider spinors in any more detail but focus on scalar ﬁelds φ, transforming as singlets, φ →φ under the Lorentz group, and vector ﬁelds Aµ, transforming as vectors, Aµ →Λµ νAν. 4.2 General classical ﬁeld theory 4.2.1 Lagrangians and Hamiltonians in classical ﬁeld theory In this subsection, we develop the general Lagrangian and Hamiltonian formalism for classical ﬁeld theories. This formalism is in many ways analogous to the Lagrangian and Hamiltonian formulation of classical mechan-ics. In classical mechanics the main objects are the generalised coordinates qi = qi(t) which depend on time only. Here, we will instead be dealing with ﬁelds, that is functions of four-dimensional coordinates x = (xµ) on Minkowski space. Lorentz indices µ, ν, · · · = 0, 1, 2, 3 are lowered and raised with the Minkowski metric (ηµν) = diag(1, −1, −1, −1) and its inverse ηµν. For now we will work with a generic set of ﬁelds φa = φa(x) before discussing scalar and vector ﬁelds in more detailed in subsequent sections. Recall that the Lagrangian in 4.2. GENERAL CLASSICAL FIELD THEORY 45 classical mechanics is a function of the generalised coordinates and their ﬁrst (time) derivatives. Analogously, we start with a ﬁeld theory Lagrangian density L = L(φa, ∂µφa) which is a function of the ﬁelds φa and their ﬁrst space-time derivatives ∂µφa. The ﬁeld theory action can then be written as S = Z d4x L(φa(x), ∂µφa(x)) , (4.27) where the integration ranges over all of Minkowski space. Our ﬁrst task is to derive the Euler-Lagrange equations for such a general ﬁeld theory by applying the variational principle to the above action. One ﬁnds 0 = δS δφa(x) = δ δφa(x) Z d4˜ x L(φb(˜ x), ∂µφb(˜ x)) = Z d4˜ x ∂L ∂φb δφb(˜ x) δφa(x) + ∂L ∂(∂µφb) δ∂µφb(˜ x) δφa(x) (4.28) = Z d4˜ x ∂L ∂φb −∂µ ∂L ∂(∂µφb) δφb(˜ x) δφa(x) = ∂L ∂φa −∂µ ∂L ∂(∂µφa) (x) (4.29) where we have used the generalisation of Eq. (1.17) δφb(˜ x) δφa(x) = δa b δ4(x −˜ x) (4.30) in the last step. Further, we have assumed that the boundary term which arises from the partial integration in the second last step vanishes due to a vanishing variation at inﬁnity. Hence, the Euler-Lagrange equations for general ﬁeld theories take the form ∂µ ∂L ∂(∂µφa) −∂L ∂φa = 0 . (4.31) With the conjugate momenta deﬁned by πa = ∂L ∂(∂0φa) (4.32) the Hamiltonian density H and the Hamiltonian H can, in analogy with classical mechanics, be written as H = πa∂0φa −L , H = Z d3x H . (4.33) 4.2.2 Noether’s theorem In classical mechanics, Noether’s theorem relates symmetries and conserved quantities of a theory. We will now derive the ﬁeld theory version of this theorem. Let us start with a general set of inﬁnitesimal transformations parameterised by small, continuous parameters ǫα and acting an the ﬁelds by φa(x) →φ′ a(x) = φa(x) + Φaα(x)ǫα , (4.34) where Φaα(x) are functions which encode the particular type of symmetry action. We assume that these transfor-mations leave the action (4.27) invariant and, hence, change the Lagrangian density by total derivatives only. This means the transformation of the Lagrangian density is of the form L →L + ǫα∂µΛµ α(x) , (4.35) with certain functions Λµ α(x) which can be computed for each type of symmetry action. Let us now compare this variation of L with the one induced by transformation (4.34) of the ﬁelds. One ﬁnds L → L + ∂L ∂φa Φaαǫα + ∂L ∂(∂µφa)∂µΦaαǫα (4.36) = L + ǫα∂µ ∂L ∂(∂µφa)Φaα −ǫα ∂µ ∂L ∂(∂µφa) −∂L ∂φa Φaα (4.37) The last term vanishes thanks to the Euler-Lagrange equations (4.31) and equating the remaining variation of L with the one in Eq. (4.35) it follows that ∂µjµ α = 0 where jµ α = ∂L ∂(∂µφa)Φaα −Λµ α . (4.38) 46 CHAPTER 4. CLASSICAL FIELD THEORY Hence, for each symmetry generator ǫα we obtain a conserved current jµ α, that is, a current with vanishing diver-gence. Each such current can be used to deﬁne a conserved charge Qα by Qα = Z d3x j0 α (4.39) Using Eq. (4.38) and assuming that ﬁelds fall off sufﬁciently rapidly at inﬁnity it follows that ˙ Qα = Z d3x ∂0j0 α = Z d3x∂ijαi = 0 , (4.40) and, hence, that the charges Qα are indeed time-independent. 4.2.3 Translation invariance and energy-momentum tensor Let us apply Noether’s theorem to the case of translations in space-time, acting on the ﬁelds as φa(x) →φa(x + a) = φa(x) + aν∂νφa(x) . (4.41) The role of the symmetry parameters ǫα is here played by the inﬁnitesimal translations aν. Therefore, the index α which appears in the general equations above becomes a space-time index ν. Under a translation the Lagrangian density changes as L →L + aµ∂µL = L + aν∂µ (δµ ν L) . (4.42) Comparing the last two equations with the general formulae (4.34) and (4.35) we learn that Φaν = ∂νφa and Λµ ν = δµ ν L. Inserting this into the general result (4.38) leads to four currents T µν = jµ ν given by T µν = ∂L ∂(∂µφa)∂νφa −δµνL . (4.43) For a translation-invariant theory they are conserved, that is, they satisfy ∂µT µν = 0. The tensor T µν is called the energy-momentum tensor and its associated charges Pν = Z d3x T 0ν (4.44) represent the conserved energy and momentum of the system. In particular, the conserved energy P0 is explicitly given by P0 = Z d3x ∂L ∂(∂0φa)∂0φa −L = Z d3x H = H , (4.45) that is, by the Hamiltonian (4.33). 4.2.4 How to construct classical ﬁeld theories Before we move on to examples, it may be useful to present a ”recipe” for how to construct explicit ﬁeld theories. The standard steps involved are: • Choose a group which corresponds to the symmetries of the theory. Normally, the symmetries include the external symmetry, that is Lorentz symmetry. In addition, there may be internal symmetries which do not act on space-time indices but internal indices. (We will see explicit examples of such internal symmetries shortly.) • Choose a set of representations of the symmetry group. This ﬁxes the ﬁeld content of the theory and the transformation properties of the ﬁelds. • Write down the most general action invariant under the chosen symmetry (with at most two derivatives in each term) for the ﬁelds selected in the previous step. Normally, only polynomial terms in the ﬁelds are considered and an upper bound on the degree of the polynomials is imposed (for example by requiring that the theory does not contain (coupling) constants with negative energy dimension). 4.3. SCALAR FIELD THEORY 47 4.3 Scalar ﬁeld theory 4.3.1 A single real scalar ﬁeld Lagrangian and equations of motion In a Lorentz invariant ﬁeld theory, the simplest choice of ﬁeld content is that of a single real scalar ﬁeld φ = φ(x), which corresponds to a single representation of the Lorentz group with (j+, j−) = (0, 0). The Lagrangian density for this theory is given by L = 1 2∂µφ∂µφ −V (φ) , (4.46) where the ﬁrst term is referred to as kinetic energy and V = V (φ) is the scalar potential. Let us discuss the dimensions of the various objects in this Lagrangian. The standard convention in particle physics is to set ℏ= c = 1, so that both time and space are measured in terms of inverse energy units and the action has to be dimension-less. With this convention, the measure d4x has −4 energy units and hence, for the action to be dimensionless, we need each term in the Lagrangian density L to be of energy dimension +4. Given that the derivatives ∂µ have dimension one, the scalar ﬁeld must have dimension one as well, so that the kinetic energy term has overall dimension 4. Then, for a monomial term λnφn with coupling λn in the scalar potential to be of dimension 4 the coupling λn must have dimension 4−n. If we want to avoid couplings with negative energy dimensions (which normally cause problems in the associated quantum theory) we need to restrict n ≤4 and, hence, the scalar potential has the form V = 1 2m2φ2 + 1 3!λ3φ3 + 1 4!λφ4 . (4.47) (A possible linear term in φ can be removed through a re-deﬁnition of φ by a shift.) Note that m and λ3 have dimension 1 and λ is dimensionless. The quadratic term in V is called a mass term with mass m and the other terms represent couplings. Applying the Euler-Lagrange equations (4.31) to the above Lagrangian leads to the equation of motion 2φ + V ′(φ) = 0 (4.48) for φ, where 2 = ∂µ∂µ and the prime denotes the derivative with respect to φ. For non-vanishing λ3 or λ solutions to this equation are not easy to ﬁnd. The free equation of motion and its general solution In the free case, that is for couplings λ3 = λ = 0, the equation of motion reduces to the so-called Klein-Gordon equation (2 + m2)φ = 0 , (4.49) for which a general solution can be written down. To do this we insert the Fourier transform φ(x) = Z d4k e−ikx ˜ φ(k) (4.50) of φ into the Klein-Gordon equation, resulting in (2 + m2)φ = − Z d4x e−ikx(k2 −m2)˜ φ(k) = 0 . (4.51) Since the Fourier transform can be inverted we conclude that (k2 −m2)˜ φ(k) = 0 and, hence, that ˜ φ can be written in the form ˜ φ(k) = δ(k2 −m2) ˜ ϕ(k) for some function ˜ ϕ. Inserting this result for ˜ φ into the Fourier transform (4.50) and using 3 δ(k2 −m2) = 1 2wk (δ(k0 −wk) + δ(k0 + wk)) , wk = p k2 + m2 (4.52) one ﬁnds, after integrating over k0, that φ(x) = Z d3˜ k a+(k)e−ikx + a⋆ −(k)eikx . (4.53) 3This follows from the well-known delta-function identity δ(f(x)) = P x0:f(x0)=0 1 \|f′(x0)\| δ(x −x0). 48 CHAPTER 4. CLASSICAL FIELD THEORY where we have deﬁned the measure d3˜ k = d3k (2π)32wk (4.54) which is Lorentz invariant as a consequence of Eq. (4.52). Further, the coefﬁcients are deﬁned as a+(k) = (2π)3 ˜ ϕ(wk, k) and a−(k) = (2π)3 ˜ ϕ(−wk, k)⋆and the four-vector k in the exponents is now understood as (kµ) = (wk, k). We note from Eq. (4.54) that 2wkδ3(k −q) . (4.55) is Lorentz invariant as well and can be viewed as a covariant version of the three-dimensional delta function. Up to this point we have, effectively, solved the Klein-Gordon equation for a complex scalar ﬁeld. However, imposing a reality condition on the solutions is easy and leads to a+(k) = a−(k) ≡a(k). The ﬁnal result for the general solution of the Klein-Gordon equation for a real scalar ﬁeld then reads φ(x) = Z d3˜ k a(k)e−ikx + a⋆(k)eikx . (4.56) Hamiltonian, symmetries and conserved currents We now return to the development of the general formalism. From Eq. (4.32) the conjugate momentum π is given by π = ∂0φ (4.57) and, using Eq. (4.33), this implies the Hamiltonian density H = 1 2π2 + 1 2(∇φ)2 + V . (4.58) For the stress energy tensor we ﬁnd by inserting into Eq. (4.43) Tµν = ∂µφ∂νφ −1 2ηµν∂ρφ∂ρφ + ηµνV . (4.59) In accordance with the general formula (4.45) we therefore ﬁnd for the energy P0 = Z d3x T00 = Z d3x 1 2π2 + 1 2(∇φ)2 + V = Z d3x H . (4.60) Our theory is also invariant under Lorentz transformations and from Noether’s theorem we expect associated con-served currents which we will now derive. First, recall from Eq. (4.24) that an inﬁnitesimal Lorentz transformation on xµ can be written as xµ →xµ + ǫµνxν where ǫµν is anti-symmetric. On the ﬁeld φ this transformation acts as φ(x) →φ(x −ǫx) = φ(x) + ǫµνxµ∂νφ(x) (4.61) and a similar transformation law holds for the Lagrangian density L →L + ǫµν∂ρ (δρ νxµL) . (4.62) Comparing with Eqs. (4.34) and (4.35) we learn that the symmetry parameters ǫα are here given by ǫµν, so we have to replace the index α in our general equations with an anti-symmetric pair of space-time indices. Further, we have Φµν = 2x[µ∂ν]φ and Λρ µν = 2x[µδρ ν]L. Inserting this into Eq. (4.38) we ﬁnd the conserved currents M ρ µν given by M ρ µν = xµT ρ ν −xνT ρ µ , (4.63) with the energy momentum tensor for a scalar ﬁeld theory deﬁned in Eq. (4.59). From Noether’s theorem we know that these currents are divergence-free, ∂ρM ρ µν = 0, and imply the existence of conserved charges Mµν = Z d3x M 0 µν = Z d3x (xµT 0 ν −xνT 0 µ) . (4.64) They can be interpreted as the conserved angular momentum of the theory. 4.3. SCALAR FIELD THEORY 49 The Z2 symmetric theory and spontaneous symmetry breaking So far, we have only imposed external symmetries on the scalar ﬁeld theory. An internal symmetry which may be considered is a Z2 symmetry which acts as φ(x) →−φ(x). This transformation leaves all terms except the cubic one in the Lagrangian (4.46), (4.47) invariant. Hence, if we impose this symmetry on our theory the cubic term in the scalar potential has to be dropped for the Lagrangian to be invariant and we are left with 4 V = V0 + 1 2m2φ2 + 1 4!λφ4 . (4.65) In the following, we assume invariance under this Z2 symmetry and work with the scalar potential (4.65). Let us discuss the simplest type of solutions to the theory, namely solutions for which φ(x) = v takes on a constant value v, independent of space-time. From Eq. (4.48) such constant ﬁelds solve the equation of motion if V ′(v) = 0 , (4.66) so we are instructed to look at extrema of the scalar potential V . In fact, to minimise the energy (4.60) we should be focusing on minima of the scalar potential V . We will also refer to such a solution of the classical theory as a vacuum. If the quartic coupling λ is negative the scalar potential is unbounded from below and the energy of a constant ﬁeld conﬁguration tends to minus inﬁnity for large ﬁeld values. To avoid such an unphysical situation we assume that λ > 0 in the following. Then we should distinguish two cases which are illustrated in Fig. 4.1. • m2 ≥0 : In this case there is a single minimum at φ = v = 0. This solution is mapped into itself under the action φ →−φ of the Z2 symmetry and we say that the symmetry is unbroken in this vacuum. • m2 < 0 : In this case, φ = 0 is a maximum of the potential and there are two minima at φ = v = ± r −6m2 λ . (4.67) Neither minimum is left invariant under the Z2 action φ →−φ (in fact the two minima are mapped into each other under Z2) and we say that the symmetry is spontaneously broken. In general, spontaneous breaking of a symmetry refers to a situation where a symmetry of a theory is partially or fully broken by a vacuum solution of the theory. The potential value at the minima is given by V (v) = V0 + 1 4m2v2 = V0 −1 24λv4 . (4.68) Just as the constant V0 which we have included earlier, the potential value at the minima does not affect any of the physics discussed so far. However, if we couple our theory to gravity, it turns out that V (v) acts like a cosmological constant Λ in the Einstein equations. Cosmological constraints tell us that Λ cannot be much bigger than O(meV4). On the other hand, there is no obvious constraint on V (v). Unless there is a cancellation of the two terms in Eq. (4.68), one would expect V (v) to be of the order of the symmetry breaking scale v to the fourth power. Electroweak symmetry is broken spontaneously by a mechanism similar to the above (and we will study a model related to this in Section (4.3.4)) at a scale of v ∼TeV. Hence, the ”natural” cosmological constant which arises at electroweak symmetry breaking is about 60 orders of magnitude larger than the observational limit. So, we have to assume that the two terms in Eq. (4.68) cancel each other to a precision of 60 digits. This enormous ”ﬁne tuning” is one of the manifestations of what is referred to as the cosmological constant problem. The question of why the cosmological constant is as small as it is is one of the most important unresolved problems in modern physics. 4.3.2 Complex scalar ﬁeld with U(1) symmetry Lagrangian and equations of motion The next simplest scalar ﬁeld theory is one for two real scalar ﬁelds φ1 and φ2. In this case, a more interesting symmetry can be imposed on the theory, namely an SO(2) symmetry under which the doublet (φ1, φ2) transforms with the charge q representation as φ1 φ2 → cos(qα) sin(qα) −sin(qα) cos(qα) φ1 φ2 . (4.69) 4For our subsequent discussion, we add a constant V0 to the potential. This constant does not affect the φ equations of motion. 50 CHAPTER 4. CLASSICAL FIELD THEORY v -v Φ V Figure 4.1: Shape of scalar potential (4.65) for m2 ≥0 (solid line) and m2 < 0 (dashed line). In the latter case the position v of the minima is given by Eq. (4.67). Such a symmetry, which does not act on space-time indices but on internal indices is also called an internal symme-try. For now, we will study the case of global internal symmetries, that is symmetries for which the transformation is the same everywhere in space-time. This means the group parameter α ∈[0, 2π] is independent of the space-time coordinates xµ. To discuss the two-scalar ﬁeld theory with this global SO(2) symmetry and scalar charge q explicitly it proves convenient to arrange the two real scalars into a single complex one φ = 1 √ 2(φ1 + iφ2) . (4.70) On this complex scalar, the SO(2) symmetry acts via the charge q representation (4.4) of U(1), that is φ →exp(−iqα)φ . (4.71) The complex conjugate φ⋆transforms as φ⋆→exp(iqα)φ⋆and, hence, corresponds to a representation with charge −q. Allowed terms in the Lagrangian density have to be U(1) invariant which is equivalent to saying that their total charge needs to be zero. For example, the term φ2 has total charge 2q and cannot appear while the term φ⋆φ has charge zero and is allowed. In general, we can only allow terms with the same number of φ and φ⋆, so that the general U(1) invariant Lagrangian density reads L = ∂µφ⋆∂µφ −V (φ, φ⋆) , V = V0 + m2φ⋆φ + λ 4 (φ⋆φ)2 . (4.72) Note that it is essential for the invariance of the kinetic term that the group parameter α is space-time indepen-dent, that is, that the symmetry is global. For the equation of motion for φ we ﬁnd from the Euler-Lagrange equation (4.31) 2φ + ∂V ∂φ⋆= 2φ + m2φ + λ 2 (φ⋆φ)φ = 0 . (4.73) For λ = 0 this is the Klein-Gordan equation for a complex scalar ﬁeld whose general solution has already been obtained in Eq. (4.53). Hamiltonian and conserved currents For the conjugate momenta one ﬁnds π = ∂L ∂(∂0φ) = ∂0φ⋆, π⋆= ∂L ∂(∂0φ⋆) = ∂0φ . (4.74) and, hence, the Hamiltonian density reads H = π∂0φ + π⋆∂0φ⋆−L = π⋆π + ∇φ⋆· ∇φ + V (φ, φ⋆) . (4.75) Being translation and Lorentz invariant the above theory has conserved energy-momentumand angular momentum tensors which can be obtained in complete analogy with the single scalar ﬁeld case in sub-section 4.3.1. In addition, 4.3. SCALAR FIELD THEORY 51 the presence of the internal U(1) symmetry leads to a new type of conserved current which we will now derive. From Eq. (4.71), inﬁnitesimal U(1) transformations are given by φ →φ −iqαφ , φ⋆→φ⋆+ iqαφ⋆. (4.76) Comparing with the general transformation (4.34) we conclude that α plays the role of the (single) symmetry parameter and Φ = −iqφ, Φ⋆= iqφ⋆. Since the Lagrangian density is invariant under U(1) the total derivative terms in Eq. (4.35) vanish and we can set Λµ to zero. Inserting this into the general formula (4.38) for the conserved current we ﬁnd jµ = iq(φ⋆∂µφ −φ∂µφ⋆) . (4.77) Spontaneous symmetry breaking As we did before, we would now like to discuss the vacua of the theory, that is solutions to the equation of motion (4.73) with φ = v = const. For m2 ≥0 there is a single minimum at φ = 0. This solution is left invariant by the transformations (4.71) and, hence, the U(1) symmetry is unbroken in this case. For m2 < 0 the shape of the potential is shown in Fig. (4.2). In this case, there is a whole circle of minima Φ1 Φ2 V Figure 4.2: Shape of scalar potential (4.72) for m2 < 0. v = 1 √ 2v0eiν , v0 = r −4m2 λ , (4.78) where ν is an arbitrary phase. The existence of this one-dimensional degenerate space of vacua is not an accident but originates from the invariance of the scalar potential under U(1) transformations V (φ, φ⋆) = V (eiqαφ, e−iqαφ⋆) . (4.79) Indeed, this invariance implies that for every minimum φ of V also eiqαφ is a minimum for arbitrary α. Every par-ticular choice of minimum transforms non-trivially under (4.71) and, hence, the U(1) symmetry is spontaneously broken. Let us, for convenience, choose the minimum on the φ1 axis (setting the phase ν = 0) so φ = v0/ √ 2. Around this point we expand the ﬁeld as φ = 1 √ 2 (v0 + ϕ1 + iϕ2) , (4.80) where ϕ1 and ϕ2 are small. Inserting this into the potential (4.72) we ﬁnd V = V0 + 1 4m2v2 0 −2m2ϕ2 1 + O(ϕ3 1, ϕ3 2) . (4.81) This shows that in this vacuum ϕ1 is massive with mass 2m and ϕ2 is massless. This could have been expected as ϕ2 corresponds to the direction along the circle of minima, while ϕ1 is perpendicular to it. It is, therefore, clear that the appearance of the massless mode ϕ2 is directly related to the existence of a circle of minima and, hence, to the spontaneous break-down of the U(1) symmetry. The appearance of massless scalars for spontaneously broken global symmetries is a general feature known as Goldstone’s theorem and the corresponding massless scalars are also called Goldstone bosons. We will now study this phenomenon in a more general setting. 52 CHAPTER 4. CLASSICAL FIELD THEORY 4.3.3 Spontaneously broken global symmetries and Goldstone’s theorem Let us consider a general scalar ﬁeld theory with a set of scalar ﬁelds φ = (φa) = (φ1, . . . , φn) and scalar potential V = V (φ). Consider a minimum v = (v1, . . . , vn) of V , that is a solution of ∂V ∂φa (v) = 0. Around such a minimum we can expand the potential as V = V (v) + 1 2Mabϕaϕb + O(ϕ3) , (4.82) where ϕ = φ −v and the mass matrix Mab is deﬁned by Mab = ∂2V ∂φa∂φb (v) . (4.83) The eigenvalues of the mass matrix M are the mass squares of the ﬁelds around the vacuum v. Now let us assume that our scalar ﬁeld theory is invariant under a continuous symmetry group G and that the scalar ﬁelds transform as φ →R(g)φ under the representation R of G. In particular, this means the scalar potential is invariant, that is V (φ) = V (R(g)φ) (4.84) for all g ∈G. The vacuum v will in general not respect the full symmetry group G but will spontaneously break it to a sub-group H ⊂G, so that R(g)v = v for g ∈H and R(g)v ̸= v for g / ∈H. Now introduce inﬁnitesimal transformations R(g) ≃1 + itITI with generator TI (in the representation R) and small parameters tI. We can split these generators into two sets, {TI} = {Hi, Sα}, where Hi are the generators of the unbroken sub-group H and Sα are the remaining generators corresponding to the broken part of the group. Hence, these two types of generators can be characterised by Hiv = 0 , Sαv ̸= 0 . (4.85) Now, write down the inﬁnitesimal version of Eq. (4.84) V (φ) = V (φ −itITIφ) = V (φ) −itI ∂V ∂φ (φ) T TIφ , (4.86) differentiate one more time with respect to φ and evaluate the result at φ = v using that ∂V ∂φa (v) = 0. This leads to MTIv = 0 , (4.87) where M is the mass matrix deﬁned above. Every broken generator Sα satisﬁes Sαv ̸= 0 and, hence, leads to an eigenvector of the mass matrix with eigenvalue zero. In other words, every broken generator leads to one massless scalar which is precisely the statement of Goldstone’s theorem. 4.3.4 Scalar ﬁeld theory with symmetry SU(2) × U(1) It may be usefult to illustrate Goldstone’s theorem with a less trivial example based on the symmetry SU(2) × UY (1). Consider a scalar ﬁeld theory with an SU(2) doublet φ of complex scalar ﬁelds which, in addition, carry charge 1/2 under a U(1) symmetry. A general SU(2) × UY (1) transformation of the scalar ﬁeld φ can then be written as φ →e−iα/2e−itiτiφ ≃ 12 −iαY −itiτi φ (4.88) with generators τi = 1 2σi , Y = 1 212 . (4.89) The general invariant Lagrangian density is L = ∂µφ†∂µφ −V (φ) , V = V0 + m2φ†φ + λ(φ†φ)2 . (4.90) Note that the invariance of this Lagrangian density under SU(2) is due to the appearance of a singlet in the Clebsch-Gordan decomposition ¯ 2 ⊗2 = 1 + 3. Provided that m2 < 0, the scalar potential is minimised for φ†φ = v2 0 = −m2 2λ (4.91) 4.4. VECTOR FIELDS, GAUGE SYMMETRY AND SCALAR ELECTRODYNAMICS 53 and a particularly simple choice of minimum is provided by φ = v = 0 v0 . (4.92) Clearly, for this choice is follows that τ 1v ̸= 0 , τ 2v ̸= 0 , (τ 3 −Y )v ̸= 0 , (τ 3 + Y )v = 0 . (4.93) Hence, three of the four generators of SU(2) × UY (1) are broken, while the generator τ 3 + Y remains unbroken. This last generator corresponds to a combination of the U(1) ⊂SU(2) and the additional UY (1) and deﬁnes the unbroken U(1) subgroup. So the induced breaking pattern can be summarised as SU(2) × UY (1) →U(1) . (4.94) This is precisely the symmetry breaking pattern which arises in the electro-weak sector of the standard model of particle physics. There, SU(2)×UY (1) is the electro-weak (gauge) symmetry and the unbroken U(1) corresponds to electromagnetism. In the present case we are working with a global symmetry and Goldstone’s theorem tells us that we should have three massless scalars from the three broken generators. In the case of the electro-weak theory, the SU(2) × UY (1) symmetry is actually promoted to a local (or gauge) symmetry where the symmetry parameters are allowed to depend on space-time. In this case, it turns out that the Goldstone bosons are absorbed by three vector bosons which receive masses from symmetry breaking. This phenomenon is also called the Higgs effect and to investigate this in more detail we need to introduce vector ﬁelds and gauge symmetries. 4.4 Vector ﬁelds, gauge symmetry and scalar electrodynamics 4.4.1 Lagrangian formulation of Maxwell’s theory Covariant electro-magnetism ﬂashback We know that Maxwell’s equations can be formulated in terms of a vector potential Aµ with associated ﬁeld strength tensor Fµν = ∂µAν −∂νAµ , (4.95) to which the electric and magnetic ﬁelds E and B are related by Ei = F0i , Bi = 1 2ǫijkFjk . (4.96) Under Lorentz transformations, Aµ transforms like a vector, that is, Aµ →Λµ νAν, and consequently, from Eq. (4.95), Fµν transforms like a tensor, Fµν →Λµ ρΛν σFρσ. Since it is Fµν which is directly associated to the physical ﬁelds it is not surprising that the vector potential Aµ contains some unphysical degrees of freedom. Formally, this is expressed by the fact that a gauge transformation Aµ →Aµ + ∂µΛ (4.97) on Aµ, parameterized by an arbitrary function 5 Λ = Λ(x), leaves the ﬁeld strength tensor Fµν unchanged (as can be easily seen by transforming the RHS of Eq. (4.95)). Let us now write down the most general Lagrangian density for Aµ (up to second order in derivatives) which is Lorentz invariant and invariant under gauge transfor-mations (4.97). Gauge invariance implies that the Lagrangian should depend on Aµ only through the ﬁeld strength F and, since F contains one derivative the most we should consider is quadratic terms in F. In addition, Lorentz invariance means all indices should be contracted in L. Basically, this leaves only one allowed term 6, namely L = −1 4FµνF µν , (4.98) where we think of Fµν as being given by Eq. (4.95). One ﬁnds ∂L ∂(∂µAν) = −1 2F ρσ ∂(∂ρAσ −∂σAρ) ∂(∂µAν) = −1 2F ρσ δµ ρ δν σ −δµ σδν ρ = −F µν , ∂L ∂Aµ = 0 . (4.99) 5Not to be confused with a Lorentz transformation! 6The term ǫµνρσFµνFρσ is also consistent with all stated requirements. However, it can be written as the total derivative 4∂µ(ǫµνρσAν∂ρAσ) and, hence, does not effect the equations of motion. 54 CHAPTER 4. CLASSICAL FIELD THEORY Inserting this into the Euler-Lagrange equation (4.31) implies ∂µF µν = 0 , ∂[µFνρ] = 0 , (4.100) where the second equation is a trivial consequence of the deﬁnition (4.95). These are the free Maxwell’s equations in covariant form. Splitting indices up into space and time components and inserting Eqs. (4.96) they can be easily shown to be equivalent to the better-known version in terms of the electric and magnetic ﬁelds E and B. This example illustrates the power of the Lagrangian formulation of ﬁeld theories. Starting with a simple set of assump-tions about the symmetries (Lorentz symmetry and gauge invariance in the present case) and the ﬁeld content (a single vector ﬁeld Aµ) one is led to the correct theory by writing down the most general Lagrangian consistent with the symmetries. Gauge choice and general solution to equations of motion In terms of Aµ, the free Maxwell theory can be expressed by the single equation 2Aµ −∂µ∂νAν = 0 , (4.101) which follows from the ﬁrst Eq. (4.100) after inserting the deﬁnition (4.95) (Note that the second equation (4.100) is automatically satisﬁed once Fµν is written in terms of Aµ). This equation can be further simpliﬁed by exploiting the gauge symmetry (4.97). Gauge invariance allows one, via transformation with an appropriate gauge parameter Λ, to impose a gauge condition on Aµ. There are several possibilities for such gauge conditions and here we consider the Lorentz gauge deﬁned by ∂µAµ = 0 . (4.102) This condition has the obvious beneﬁt of being covariant (unlike, for example, the so-called temporal gauge which requires A0 = 0) and it simpliﬁes the equation of motion for Aµ to 2Aµ = 0 . (4.103) Note, however, that the Lorentz gauge does not ﬁx the gauge symmetry completely but leaves a residual gauge freedom with gauge parameters Λ satisfying 2Λ = 0 . (4.104) Eq. (4.103) is a massless Klein-Gordon equation for a vector ﬁeld and, hence, can be easily solved using our earlier result (4.56) with an additional µ index attached. This leads to Aµ(x) = Z d3˜ k aµ(k)e−ikx + a⋆ µ(k)eikx . (4.105) where wk = \|k\| and (kµ) = (wk, k). In addition, the Lorentz gauge condition demands that the coefﬁcents aµ satisfy kµaµ(k) = 0 . (4.106) To exploit this constraint in detail it is useful to introduce a set of polarisation vectors ǫ(α) µ (k), where α = 0, 1, 2, 3, with the following properties. The vectors ǫ(1)(k) and ǫ(2)(k) are orthogonal to both k and a vector n with n2 = 1 and n0 > 0 in the time direction and they satisfy ǫ(α)(k) · ǫ(α′)(k) = −δαα′ for α, α′ = 1, 2 . (4.107) Further, ǫ(3)(k) is chosen to be in the (n, k) plane, orthogonal to n and normalised, that is, n · ǫ(3)(k) = 0 and (ǫ(3)(k))2 = −1. Finally, we set ǫ(0) = n. With these conventions we have an orthogonal set of vectors satisfying ǫ(α) · ǫ(α′) = ηαα′ (4.108) for all α, α′ = 0, 1, 2, 3. They can be used to write aµ(k) as aµ(k) = 3 X α=0 a(α)(k)ǫ(α) µ (k) , (4.109) where a(α)(k) are four expansion coefﬁcients. The idea of introducing this basis of polarisation vectors is to separate the directions transversal to k, corresponding to ǫ(1)(k) and ǫ(2)(k), from the other two directions ǫ(0)(k) 4.4. VECTOR FIELDS, GAUGE SYMMETRY AND SCALAR ELECTRODYNAMICS 55 and ǫ(3)(k). As an example, if we choose a spatial momentum k pointing in the z-direction, the above vectors are explicitly given by ǫ(0) =     1 0 0 0    , ǫ(1) =     0 1 0 0    , ǫ(2) =     0 0 1 0    , ǫ(3) =     0 0 0 1    . (4.110) Returning to the general case, it is easy to see that the unique choice for ǫ(3)(k) is ǫ(3) µ (k) = 1 k0 kµ −nµ so that ǫ(0) µ + ǫ(3) µ = 1 k0 kµ . (4.111) Let us now return to the gauge condition (4.106) and insert the expansion (4.109). The two transverse directions ǫ(1)(k) and ǫ(2)(k) drop out of the equation, so that 0 = kµaµ(k) = (k · ǫ(0)(k))a(0)(k) + (k · ǫ(3)(k))a(3)(k) = k0(a(0)(k) −a(3)(k)) . (4.112) Here we have used that k · ǫ(3)(k) = −k · ǫ(0)(k) = −k0 , (4.113) in the last step. We conclude that a(0)(k) = a(3)(k) in order to satisfy the gauge condition (4.106) and that the expansion (4.109) can be written as aµ(k) = a(0)(k) k0 kµ + 2 X α=1 a(α)(k)ǫ(α) µ (k) . (4.114) Hence, we are left with the two transversal polarisations and a longitudinal one along the direction of k. Recall that we still have a residual gauge freedom from gauge parameters Λ satisfying the Klein-Gordon equation (4.104). The most general such parameters can be written as Λ(x) = Z d3˜ k λ(k)e−ikx + λ⋆(k)eikx . (4.115) A gauge transformation (4.97) with Λ of this residual form changes the expansion coefﬁcients aµ(k) for the vector ﬁeld as aµ(k) →a′ µ(k) = aµ(k) −ikµλ(k) . (4.116) Comparing with Eq. (4.114), it is clear that we can use this residual gauge freedom to remove the longitudinal degree of freedom in aµ(k). We are then left with the two transversal polarisations only and we conclude that the number of physical degrees of freedom for a vector ﬁeld is two. This reduction from four apparent degrees of freedom to two is directly related to the gauge invariance of the theory. Massive vector ﬁeld It is instructive to perform a similar analysis for a massive vector ﬁeld, that is a vector ﬁeld Aµ with the additional term m2 2 AµAµ added to the Lagrangian density (4.98). Clearly, gauge invariance is explicitly broken for such a massive vector ﬁeld. The equation of motion reads ∂µF µν + m2Aν = 0 . (4.117) Applying ∂ν to this equation we conclude that ∂νAν = 0 and, hence, that Eq. (4.117) can equivalently be written as (2 + m2)Aµ = 0 , ∂µAµ = 0 . (4.118) The ﬁrst of these equations is a massive Klein-Gordon equation with general solution given by Eq. (4.105) but now with k2 = m2 instead of k2 = 0. To also satisfy the second equation above we need kµaµ(k) = 0 which reduces the number of degrees from four to three. Since gauge invariance is not available in this case, no further reduction occurs and we conclude that the number of physical degrees of freedom of a massive vector ﬁeld is three. Including a source Let us return to the massless theory and ask how can we include a source Jµ into this Lagrangian formulation of 56 CHAPTER 4. CLASSICAL FIELD THEORY Maxwells’ theory. This current should appear linearly in the equations of motion (and therefore in the Lagrangian density), so the obvious generalisation of Eq. (4.98) is L = −1 4FµνF µν −AµJµ . (4.119) The additional term depends on Aµ explicitly which leads to a non-trivial gauge variation S →S − Z d4x ∂µΛJµ = S + Z d4x Λ∂µJµ , (4.120) of the action. This apparent breaking of the gauge invariance of the theory can be avoided if we require that ∂µJµ = 0 , (4.121) that is, if the current Jµ is conserved. With ∂L ∂Aµ = −Jµ and the ﬁrst Eq. (4.99) we ﬁnd from the Euler-Lagrange equations ∂µF µν = Jν , ∂[µFνρ] = 0 , (4.122) which are indeed Maxwell’s equations in the presence of a source Jµ. In a fundamental theory, the current Jµ should arise from ﬁelds, rather than being put in ”by hand” as an external source. We will now study an example for such a theory in which vector ﬁelds are coupled to scalars. 4.4.2 Scalar electrodynamics and the Higgs mechanism Our goal is to write down a theory with a vector ﬁeld Aµ and a complex scalar φ = (φ1 + iφ2)/ √ 2. We have already seen individual Lagrangians for these two ﬁelds, namely Eq. (4.98) for the vector ﬁeld and Eq. (4.72) for the scalar ﬁeld. There is nothing wrong with adding those two Lagrangians, but this leads to a somewhat uninteresting theory where the vector and the scalar are decoupled (that is, there are no terms in this Lagrangian containing both types of ﬁelds and consequently their equations of motion are decoupled from one another). Again, the key to constructing a more interesting theory comes from thinking about symmetries. Both the gauge symmetry of electromagnetism and the global U(1) symmetry of the complex scalar ﬁeld theory are parameterized by a single parameter. Let us identify those two parameters with one another, that is we set α = Λ in Eq. (4.71), and, hence, transform the scalar as φ(x) →e−iqΛ(x)φ(x) , (4.123) along with the transformation (4.97) of the vector ﬁeld. The new feature we have introduced in this way is that the formerly global transformation (4.71) of the scalar ﬁeld has now become local. The scalar potential for φ is still invariant under these local U(1) transformations, however, there is a problem with the kinetic term for φ, since the derivative ∂µ now acts on the local group parameter Λ(x). Explicitly, ∂µφ transforms as ∂µφ →e−iqΛ(x)(∂µ −iq∂µΛ)φ . (4.124) The additional term proportional to ∂µΛ in this transformation is reminiscent of the transformation law (4.97) for the gauge ﬁeld and Aµ can indeed be used to cancel this term. Deﬁne the covariant derivative Dµ = ∂µ + iqAµ . (4.125) Then the covariant derivative of φ transforms as Dµφ →e−iqΛ(x)Dµφ , (4.126) and the modiﬁed kinetic term (Dµφ)⋆Dµφ is gauge invariant. With this modiﬁcation, we can now combine Eqs. (4.72) and (4.98) to obtain the gauge invariant Lagrangian density L = (Dµφ)⋆Dµφ −V (φ, φ⋆) −1 4FµνF µν , V = V0 + m2φ⋆φ + λ 4 (φ⋆φ)2 (4.127) This is the Lagrangian for scalar electrodynamics. It shows the gauge ﬁeld Aµ in a new role, facilitating the invariance of the scalar ﬁeld theory under local symmetry transformations. In fact, had we started with just the globally symmetric scalar ﬁeld theory (4.72) with the task of ﬁnding a locally U(1) invariant version we would 4.5. FURTHER READING 57 have been led to introducing a gauge ﬁeld Aµ. We also note that the covariant derivative in (4.127) has introduced a non-trivial coupling between the scalar ﬁeld and the vector ﬁeld. We will now use scalar electrodynamics as a toy model to study spontaneous breaking of a local U(1) symme-try. The scalar potential is unchanged from the globally symmetric model and, hence, we can apply the results of Sec. 4.3.2 . For m2 ≥0 the minimum is at φ = 0 and the symmetry is unbroken. Therefore, we focus on the case m2 < 0 where the potential has a ”Mexican hat” shape as in Fig. 4.2. In this case, there is a circle of minima given by Eq. (4.78). Instead of choosing the parameterisation (4.80) for φ around the minimum on the φ1 axis it proves more useful in the present context 7 to use φ = 1 √ 2(v0 + H)eiχ (4.128) where χ = χ(x) is the Goldstone mode and H = H(x) is the massive mode. Next we perform a gauge transfor-mation with parameter Λ = χ/q, that is, φ →φ′ = e−iχφ = 1 √ 2(v0 + H) , Aµ →A′ µ = Aµ + 1 q ∂µχ . (4.129) We can now write the Lagrangian density (4.127) in terms of the transformed ﬁelds (this leaves the action un-changed) and then insert the explicit expression for φ′ from the previous equation. One ﬁnds for the covariant derivative Dµφ′ = 1 √ 2(∂µH + iq(v0 + H)A′ µ) . (4.130) and inserting into Eq. (4.127) then leads to L = 1 2∂µH∂µH −V (H) −1 4F ′ µνF ′µν + 1 2q2A′ µA′µ(v2 0 + 2v0H + H2) (4.131) V (H) = V0 + 1 4m2v2 0 −m2H2 + λ 16(4v0H3 + H4) . (4.132) The most striking feature about this result is that the Goldstone mode χ has completely disappeared from the Lagrangian and we are left with just the real, massive scalar H and the vector ﬁeld A′ µ. However, the vector ﬁeld is now massive with mass m(A′) = qv0 , (4.133) end, hence, has three degrees of freedom as opposed to just two for a massless vector ﬁeld. This explains the disappearance of the scalar χ: It has been ”absorbed” by A′ µ to provide the additional (longitudinal) degree of freedom necessary for a massive vector ﬁeld. This can also be seen from the transformation of Aµ in Eq. (4.129). Hence, we have learned that a spontaneously broken local symmetry leads to a mass for the associated vector boson and the conversion of the Goldstone ﬁeld into the longitudinal mode of the vector. This is also called the Higgs effect. The same phenomenon but in its generalisation to non-Abelian gauge groups occurs in the breaking of the electro-weak SU(2) × UY (1) →U(1) gauge group to U(1) of electromagnetism. The three W ± and Z vector bosons of the broken part of the electro-weak symmetry receive a mass (4.133) proportional to the symmetry breaking scale v0 and absorb the three Goldstone bosons (see Section 4.3.4) which arise. A detailed discussion of electro-weak symmetry breaking requires introducing non-Abelian gauge symmetries which is beyond the scope of this lecture. 4.5 Further reading • Group theory and its application to physics is discussed in B. G. Wybourne, Classical Groups for Physicists and T.-P. Cheng, L.-F. Li, Gauge theory of elementary particle physics, Chapter 4. • The material on classical ﬁeld theory covered in this chapter can be found in most quantum ﬁeld theory books, including D. Bailin and A. Love, Introduction to Gauge Theory, Chapter 3 and 13; P. Ramond, Field Theory: A Modern Primer, Chapter 1; M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Chapter 2. As usual, be warned that these references were not written with undergraduate readers in mind. 7One can think of Eq. (4.80) as a linearized version of the more accurate parameterization (4.128). 58 CHAPTER 4. CLASSICAL FIELD THEORY Chapter 5 Canonical Quantization 5.1 The general starting point We have seen in Chapter 1 how to formulate quantum mechanics in terms of path integrals . This has led to an intuitive picture of the transition between classical and quantum physics. Later, we will show how to apply path integrals to the quantisation of ﬁeld theories. In this chapter, we will to focus on the more traditional method of canonical quantization. Let us ﬁrst recall how canonical quantization works for classical mechanics. Start with a system in classical machanics, described by set of canonical coordinates {pi(t), qi(t)} and a Hamil-tonian H = H(pi, qi). The prescription on how to perform the transition to the associated quantum mechanics consists of, ﬁrstly, replacing the canonical coordinates by operators 1, that is pi(t) →ˆ pi(t) and qi(t) →ˆ qi(t), and then imposing the canonical commutation relations 2 [ˆ qi(t), ˆ qj(t)] = [ˆ pi(t), ˆ pj(t)] = 0 , [ˆ pi(t), ˆ qj(t)] = −iδij . (5.1) The dymanics of this system is governed by the operator version of Hamilton’s equations dˆ qi(t) dt = i[ ˆ H, ˆ qi(t)] , dˆ pi(t) dt = i[ ˆ H, ˆ pi(t)] . (5.2) How can we transfer this quantization procedure to ﬁeld theory? Consider a classical ﬁeld theory with a generic set of ﬁelds φa = φa(x), conjugate momenta πa(x), Hamiltonian density H and Hamiltonian H = R d3x H, as introduced in Section 4.2.1. On a discrete space consisting of lattice points {xi}, these ﬁelds can be represented by sets of canonical coordinates φai(t) = φa(t, xi) and πai(t) = πa(t, xi). On these we can simply impose the canonical commutation relations (5.1) which leads to [ˆ φia(t), ˆ φbj(t)] = [ˆ πai(t), ˆ πbj(t)] = 0 , [ˆ πai(t), ˆ φbj(t)] = −iδabδij . (5.3) The continuum version of these equations is obtained by replacing i →x, j →y, δij →δ3(x −y), ˆ φai(t) → ˆ φ(t, x) and similarly for the conjugate momenta. This results in [ˆ φa(t, x), ˆ φb(t, y)] = [ˆ πa(t, x), ˆ πb(t, y)] = 0 , [ˆ πa(t, x), ˆ φb(t, y)] = −iδa b δ3(x −y) (5.4) Note that these commutators are taken at equal time but at generally different points in space. The canonical commutation relations (5.4) together with the continuum version ∂0 ˆ φa(t, x) = i[ ˆ H, ˆ φa(t, x)] , ∂0ˆ πa(t, x) = i[ ˆ H, ˆ πa(t, x)] , (5.5) of Hamilton’s equations (5.2) provide the starting point for the canonical quantization of ﬁeld theories. We will now investigate the consequences of this quantization procedure for the simplest case, a single real, free scalar ﬁeld. To avoid cluttering the notation, we will drop hats on operators from now on. It will usually be clear from the context whether we refer to a classical object or its operator version. 1We are using operators in the Heisenberg picture. 2More formally, the transition between classical and quantum mechanics can also be understood as a replacement of canonical coordinates with operators and Poisson brackets with commutator brackets. 59 60 CHAPTER 5. CANONICAL QUANTIZATION 5.2 Canonical quantization of a real, free scalar ﬁeld Recap of theory and canonical quantisation conditions From Sec. 4.3.1, the Lagrange density L, Hamiltonian density H and the conjugate momentum π for a free real scalar ﬁeld φ with mass m are given by L = 1 2 ∂µφ∂µφ −m2φ2 , H = 1 2 π2 + (∇φ)2 + m2φ2) , π = ∂L ∂(∂0φ) = ∂0φ . (5.6) and the associated equation of motion is the Klein-Gordon equation 2φ + m2φ = 0 . (5.7) It should now be thought of as an operator equation for the ﬁeld operator φ. The canonical commutation rela-tions (5.4) in this case read [φ(t, x), φ(t, y)] = [π(t, x), π(t, y)] = 0 , [π(t, x), φ(t, y)] = −iδ3(x −y) . (5.8) Oscillator expansion of general solution We can solve the free scalar ﬁeld theory by writing down the most general solution of the operator Klein-Gordon equation. This can be done by starting with the general classical solution (4.56) and by promoting the coefﬁcients a(k) to operators. Hence, we have φ(x) = Z d3˜ k a(k)e−ikx + a†(k)eikx , π(x) = −i Z d3˜ k wk a(k)e−ikx −a†(k)eikx . (5.9) What do the canonical commutation relations for φ and π imply for the commutators of a(k) and a†(k)? To answer this question, we would like to express a(k) in terms of φ(x) and π(x) by inverting the above relations. We start by applying 2wq R d3x e−iq·x to the equation for φ(x). After carrying out the integration over k 3 one ﬁnds 2wq Z d3x e−iq·xφ(x) = a(q)e−iwqt + a†(−q)eiwqt . (5.10) Analogously, applying 2i R d3x e−iq·x to the equation for π(x) results in 2i Z d3x e−iq·xπ(x) = a(q)e−iwqt −a†(−q)eiwqt (5.11) Adding the last two equations we ﬁnd the desired expression for a(q) and its conjugate a(q) = Z d3x eiqx(wqφ(x) + iπ(x)) , a†(q) = Z d3x e−iqx(wqφ(x) −iπ(x)) . (5.12) Combining these results and the canonical commutation relations (5.8) one ﬁnds for the commutators of a(q) and a†(q) [a(k), a(q)] = [a†(k), a†(q)] = 0 , [a(k), a†(q)] = (2π)32wkδ3(k −q) . (5.13) For each k = q these equations are reminiscent of the commutation relations for the creation and annihila-tion operators of a harmonic oscillator. This should not come as a surprise given a single plain wave Ansatz φ(t, x) = φk(t)eik·x turns the Hamiltonian (5.6) (or, more precisely, as the Ansatz is complex, its complex counterpart (4.75)) into the Hamiltonian Hk = π2 k + w2 kφ2 k for a single harmonic oscillator with frequency wk = √ m2 + k2. We should, therefore, think of the free scalar ﬁeld as an inﬁnite collection of (decoupled) harmonic oscillators labelled by three-momentum k and with frequency wk. The Fock space The interpretation of a†(k) and a(k) as creation and annihilation operators also suggests a method for the con-struction of the space of states for this theory, the so-called Fock space. In close analogy with the simple harmonic oscillator, we ﬁrst deﬁne the vacuum \|0⟩, with normalisation ⟨0\|0⟩= 1, as the state which is annihilated by all a(k), that is a(k)\|0⟩= 0 . (5.14) 3Recall that R d3x ei(k−q)·x = (2π)3δ3(k −q). 5.2. CANONICAL QUANTIZATION OF A REAL, FREE SCALAR FIELD 61 Single particle states \|k⟩are obtained by acting on this vacuum state with a single creation operator, so that \|k⟩= a†(k)\|0⟩. (5.15) Using the commutation relations (5.13) and Eq. (5.14), one ﬁnds for their normalization ⟨k\|q⟩= ⟨0\|a(k)a†(q)\|0⟩= ⟨0\|[a(k), a†(q)]\|0⟩= (2π)32wkδ3(k −q) . (5.16) We note that the RHS of this relation is the ”covariant delta function” (4.55). A basis set of states is provided by all n particle states \|k1, . . . , kn⟩, obtained by acting on the the vacuum with n creation operators \|k1, . . . , kn⟩= a†(k1) . . . a†(kn)\|0⟩. (5.17) The number operator The number operator N is deﬁned by N = Z d3˜ k a†(k)a(k) , (5.18) and from the commutation relations (5.13) one ﬁnds [N, a†(q)] = Z d3˜ k [a†(k)a(k), a†(q)] = Z d3˜ k a†(k)[a(k), a†(q)] = Z d3˜ k a†(k)(2π)32wkδ3(k−q) = a†(q) . (5.19) On an n-particle state the number operator acts as N\|k1, . . . , kn⟩ = Na†(k1) . . . a†(kn)\|0⟩= a†(k1)N + [N, a†(k1)] a†(k2) . . . a†(kn)\|0⟩ (5.20) = a†(k1)Na†(k2) . . . a†(kn)\|0⟩+ \|k1, . . . , kn⟩. (5.21) We can repeat this procedure and commute N with all creation operators, picking up at each step the term \|k1, . . . , kn⟩. In the last step we use N\|0⟩= 0 and ﬁnd N\|k1, . . . , kn⟩= n\|k1, . . . , kn⟩. (5.22) Hence, N indeed counts the number of particles in a state. Four-momentum and normal ordering We should now compute the conserved four-momentum (4.44) for the above solution of the free scalar theory. From the stress energy tensor (4.59) with V = 1 2m2φ2 we ﬁnd (in the case of P0 after a partial integration and using the Klein-Gordon equation) H = P0 = 1 2 Z d3x (π2 −φ∂0π) , P = Z d3x π∇φ . (5.23) Inserting Eqs. (5.9) results in H = 1 2 Z d3x d3˜ k d3˜ q −wkwq a(k)e−ikx −a†(k)eikx)(a(q)e−iqx −a†(q)eiqx (5.24) +w2 k a(k)e−ikx + a†(k)eikx)(a(q)e−iqx + a†(q)eiqx ) (5.25) = 1 2 Z d3˜ k wk a†(k)a(k) + a(k)a†(k) , (5.26) and P = − Z d3x d3˜ k d3˜ q q wk a(k)e−ikx −a†(k)eikx a(q)e−iqx −a†(q)eiqx (5.27) = 1 2 Z d3˜ k k a†(k)a(k) + a(k)a†(k) . (5.28) The above Hamiltonian corresponds to an integral over harmonic oscillator Hamiltonians with frequence wk and labelled by three-momenta k. This is in line with our interpretation of the free scalar ﬁeld as a collection of de-coupled harmonic oscillators. When trying to write the integrand in Eq. (5.26) in the standard harmonic oscillator 62 CHAPTER 5. CANONICAL QUANTIZATION form Hk = wk(a†(k)a(k) + 1 2) using the commutator (5.13) one encounters an inﬁnite zero point energy, propor-tional to δ3(0), which can be interpreted as the energy of the vacuum state \|0⟩. This is one of the many inﬁnities in quantum ﬁeld theory and it should not come as a surprise: After all, we are summing over an inﬁnite number of harmonic oscillators each with ﬁnite zero-point energy wk/2. To deal with this inﬁnity, we deﬁne the concept of normal ordering of operators. The normal ordered version : O : of an operator O is obtained by writing all creation operators to the left of all annihilation operators. So, for example, : a(k)a†(q) : = a†(q)a(k). An operator and its normal-ordered counterpart differ by a (usually inﬁnite) number, often referred to as a c-number. With this deﬁnition we have : H : = Z d3˜ k wk a†(k)a(k) , : P : = Z d3˜ k k a†(k)a(k) or : Pµ : = Z d3˜ k kµ a†(k)a(k) . (5.29) A short calculation shows that [: Pµ :, a†(k)] = Z d3˜ q qµ[a†(q)a(q), a†(k)] = Z d3q qµa†(q)δ3(k −q) = kµa†(k) . (5.30) For a single particle state \|k⟩this implies : H : \|k⟩= wk\|k⟩, : P : \|k⟩= k\|k⟩, (5.31) and, hence, these states are eigenstates of : Pµ : with energy wk and spatial momentum k. Likewise, multi-particle states \|k1, . . . , kn⟩are eigenstates of : Pµ : with eigenvalue Pn i=1 kiµ. To summarise, we have constructed a basis of Fock space for the free real scalar ﬁeld and, by looking at the action of conserved operators on these states, we have found an interpretation of this basis as n particle states with deﬁnite four-momentum. Commutation of ﬁeld operators and micro causality Finally, we should look at the commutation properties of two ﬁeld operators φ(x) and φ(x′). From the canonical commutation relations we certainly know that these two operators commute for equal time, t = t′. However, the unequal time commutator [φ(x), φ(x′)] does not vanish. From the oscillator expansion (5.8) we anticipate that this commutator is given by a c-number and, hence, we can write [φ(x), φ(x′)] = ⟨0\|[φ(x), φ(x′)]\|0⟩. By replacing φ with Eq. (5.9) we ﬁnd [φ(x), φ(x′)] = ⟨0\|[φ(x), φ(x′)]\|0⟩ = ⟨0\| Z d3˜ k d3˜ q na(k)e−ikx + a†(k)eikx a(q)e−iqx′ + a†(q)eiqx′ −(x ↔x′) o \|0⟩ = ⟨0\| Z d3˜ k d3˜ q n [a(k), a†(q)]eiqx′−ikx −(x ↔x′) o \|0⟩ = Z d3˜ k n e−iwk(t−t′)eik(x−x′) −eiwk(t−t′)eik(x−x′)o = Z d4k (2π)32wk {δ(k0 −wk) + δ(k0 + wk)} ǫ(k0)e−ik(x−x′) = Z d4k (2π)3 δ(k2 −m2)ǫ(k0)e−ik(x−x′) ≡∆(x −x′) , (5.32) where ǫ(k0) = k0/\|k0\|. All elements in the ﬁnal integral are Lorentz invariant except for the function ǫ(k0). However, if we restrict to Lorentz transformations Λ which preserve the sign of the time-component of four vectors (these are the orthochronos Lorentz transformations, see Table 4.1) then ǫ(k0) remains unchanged and we have ∆(x) = ∆(Λx). We already know that ∆(0, x) = 0 since this corresponds to an equal-time commutator. Now consider a space-like vector x, that is x2 < 0. With a suitable orthochronos Lorentz transformation Λ this vector can be written in the form x = Λ(0, y) for some vector y and, hence, ∆(x) = ∆(Λ(0, y)) = ∆(0, y) = 0. We conclude that ∆(x) vanishes for all space-like vectors x. This means that the commutator [φ(x), φ(x′)] vanishes whenever x and x′ are space-like separated. This fact is also referred to as micro causality. Two ﬁeld operators at points with space-like separation should not be able to causally effect one another and one would, therefore, expect their commutator vanishes. We have just shown that this is indeed the case. This concludes the canonical quantization of the free real scalar ﬁeld and we move on to the next, more complicated case, the free complex scalar ﬁeld. 5.3. CANONICAL QUANTIZATION OF THE FREE COMPLEX SCALAR FIELD 63 5.3 Canonical quantization of the free complex scalar ﬁeld Recap of theory and canonical quantisation conditions In Sec. (4.3.2) we have introduced a complex scalar ﬁeld theory with a global U(1) symmetry. We would now like to quantise the free version of this theory following the same steps as for the free real scalar in the previous section. The ﬁeld content of this theory consists of two real scalars φ1 and φ2 which are combined into a complex scalar φ = (φ1 + iφ2)/ √ 2. Recall that the Lagrangian density, Hamiltonian density and conjugate momentum are given by L = ∂µφ†∂µφ −m2φ†φ , H = π†π + ∇φ†∇φ + m2φ†φ , π = ∂L ∂(∂0φ) = ∂0φ† (5.33) and the ﬁeld equation is the Klein-Gordon equation 2φ + m2φ = 0 . (5.34) Canonical quantisation of this system can be performed by simply imposing the general commutation relations (5.4) on the two real scalars φ1 and φ2 and their associated conjugate momenta π1 = ∂0φ1 and π2 = ∂0φ2. The only non-zero commutation relations are then [π1(t, x), φ1(t, y)] = −iδ3(x −y) , [π2(t, x), φ2(t, y)] = −iδ3(x −y) . (5.35) Using φ = (φ1 + iφ2)/ √ 2 and π = (π1 + iπ2)/ √ 2 they can be easily translated to the complex ﬁeld and its conjugate with the only non-zero commutation relations [π(t, x), φ(t, y)] = [π†(t, x), φ†(t, y)] = −iδ(x −y) . (5.36) Oscillator expansion and Fock space The general classical solution to the Klein-Gordon equation for a complex scalar ﬁeld has already been given in Eq. (4.53). The operator version of this solution reads φ(x) = Z d3˜ k a+(k)e−ikx + a† −(k)eikx , φ†(x) = Z d3˜ k a−(k)e−ikx + a† +(k)eikx . (5.37) As for the real scalar ﬁeld below, we can invert these relations and compute the commutators of a±(k) and a† ±(k) from the canonical commutation relations (5.36). The only non-zero commutators one ﬁnds in this way are [a±(k), a† ±(q)] = (2π)32wkδ3(k −q) . (5.38) This shows that we have two sets, {a† +(k), a+(k)} and {a† −(k), a−(k)}, of creation and annihilation operators and, hence, two different types of one-particle states \|(k, +)⟩= a† +(k)\|0⟩, \|(k, −)⟩= a† −(k)\|0⟩. (5.39) As usual, the vacuum \|0⟩is deﬁned by a±(k)\|0⟩= 0 and ⟨0\|0⟩= 1. Multi-particle states \|(k1, ǫ1), . . . , (kn, ǫn)⟩= a† ǫ1(k1) . . . a† ǫn(kn)\|0⟩are now labelled by n momenta ki and, in addition, n signs ǫi ∈{+1, −1} to distinguish the two types of quanta. Number operators and four-momentum For each type of quanta we can introduce a number operator N± = Z d3˜ k a† ±(k)a±(k) . (5.40) It is easy to show from the commutation relations (5.38) that [N±, a† ±(k)] = a† ±(k) , [N±, a† ∓(k)] = 0 . (5.41) Hence N+ (N−) acting on a multi-particle state counts the number of quanta of + type (−type). The (normal-ordered) conserved four-momentum can be computed as for the real scalar ﬁeld and one ﬁnds : Pµ : = Z d3˜ k kµ a† +(k)a+(k) + a† −(k)a−(k) , (5.42) 64 CHAPTER 5. CANONICAL QUANTIZATION where k0 = wk, as usual. The relation [Pµ, a† ±(k)] = kµa† ±(k) (5.43) shows that \|(k, ǫ)⟩can be interpreted as a state with four-momentum kµ irrespective of ǫ and that a multi-particle state \|(k1, ǫ1), . . . , (kn, ǫn)⟩has total four-momentum Pn i=1 kiµ. Conserved U(1) charge So far, the structure has been in complete analogy with the one for the real scalar ﬁeld. However, the free complex scalar theory has one additional feature, namely the conserved U(1) current (4.77). The operator version of the associated charge can be written as Q = iq Z d3x (φ†π† −πφ) . (5.44) Inserting the expansions (5.37) one ﬁnds after some calculation : Q : = q Z d3˜ k a† +(k)a+(k) −a† −(k)a−(k) = q(N+ −N−) . (5.45) In particular, this shows that the states \|(k, +)⟩have charge +q and the states \|(k, −)⟩have charge −q. Therefore, it is sensible to identify + states with particles and −states with anti-particles. With this terminology, we see from Eq. (5.37) that φ(x) annihilates a particle or creates an anti-particle while φ†(x) creates a particle or annihilates an anti-particle. 5.4 Time-ordered products, propagators and Wick’s theorem (again) Time-order products of operators In this section we will develop some of the tools necessary to deal with interacting ﬁelds. While the methods presented apply to all types of ﬁelds (subject to straightforward modiﬁcations) we will concentrate on the real scalar ﬁeld, for simplicity. We have seen that a ﬁeld operator such as in Eq. (5.9) can be thought of as a superposition of creation and annihilation operators. Later, we will see that interaction processes can be understood as sequences of such creation and annihilation operations and can, therefore, be quantitatively described by products of ﬁeld operators. More precisely, the operator products involved are ordered in time reﬂecting the time-line of the physical process. We are, therefore, interested in the time-ordered product of ﬁeld operators deﬁned by T (φ(t1, x1) . . . φ(tn, xn)) = φ(tin, xin) . . . φ(ti1, xi1) where ti1 ≤ti2 ≤· · · ≤tin . (5.46) and their vacuum expectation values. The above equation means time ordering T rearranges ﬁelds operators so that time increases from the right to the left. The Feynman propagator Of particular importance is the vacuum expectation value of a time ordered product of two ﬁeld operators which is called the Feynman propagator ∆F . From this deﬁnition the Feynman propagator can we written as ∆F (x −y) = ⟨0\|T (φ(x)φ(y))\|0⟩= θ(x0 −y0)⟨0\|φ(x)φ(y)\|0⟩+ θ(y0 −x0)⟨0\|φ(y)φ(x)\|0⟩, (5.47) where the Heaviside function θ is deﬁned by θ(x0) = 1 for x0 ≥0 and θ(x0) = 0 for x0 < 0. To compute the Feynman propagator we ﬁrst evaluate the above expression for the case x0 > y0. Inserting the ﬁeld expansion (5.9) we have ∆F (x −y) = ⟨0\|φ(x)φ(y)\|0⟩ (5.48) = ⟨0\| Z d3˜ k d3˜ q a(k)e−ikx + a†(k)eikx a(q)e−iqy + a†(q)eiqy \|0⟩ (5.49) = ⟨0\| Z d3˜ k d3˜ q [a(k), a†(q)]eiqy−ikx\|0⟩= Z d3˜ k e−ik(x−y) . (5.50) Analogously, we ﬁnd for the case x0 < y0 that ∆F (x −y) = R d3˜ k eik(x−y). Combining this, the Feynman 5.4. TIME-ORDERED PRODUCTS, PROPAGATORS AND WICK’S THEOREM (AGAIN) 65 propagator can be written as ∆F (x −y) = Z d3k (2π)32wk n θ(x0 −y0)e−iwk(x0−y0) + θ(y0 −x0)eiwk(x0−y0)o eik·(x−y) (5.51) = Z d4k (2π)4 i (k0 −wk + i˜ ǫ)(k0 + wk −i˜ ǫ)e−ik(x−y) (5.52) = Z d4k (2π)4 i k2 −m2 + iǫe−ik(x−y) . (5.53) The small quantity ˜ ǫ > 0 in the second integral is to indicate that the pole at k0 = wk −i˜ ǫ is slightly below the real k0 axis and the pole at k0 = −wk + i˜ ǫ is slightly above (The quantity ǫ > 0 in the ﬁnal integral serves the same purpose). With this understanding about the position of the poles the equality of the ﬁrst and second line above can be shown by a contour integration as indicated in Fig. 5.1. The above result provides us with a simple x < y0 0 x > y 0 0 k0 k w - iε ~ -wk ε ~ + i Figure 5.1: Location of poles for the Feynman propagator and contours to prove the equality between Eqs. (5.51) and (5.52). For x0 > y0 the integration along the real k0 axis can be closed for Im(k0) < 0 since the real part of the exponent in Eq. (5.52) is negative in this case. Only the pole at k0 = wk −i˜ ǫ constributes and leads to the ﬁrst term in Eq. (5.51). Analogously, for x0 < y0 the contour can be closed for Im(k0) > 0. Only the pole at k0 = −wk + i˜ ǫ contributes and leads to the second term in Eq. (5.51) representation of the Feynman propagator and tells us that its Fourier transform ˜ ∆F is given by ˜ ∆F (k) = i k2 −m2 + iǫ . (5.54) The Feynman propagator, particularly in the momentum space form (5.54), is central in the formulation of Feyn-man rules for the perturbation theory of interacting ﬁelds as we will see later. Propagators as solutions to Klein-Gordon equation Another interesting property of the Feynman propagator is that it solves the Klein-Gordon equation (2x + m2)∆F (x −y) = −iδ4(x −y) . (5.55) with a delta-function source. Functions with this property are also called Green functions. There are other Green functions of the Klein-Gordon equation which are given by an integral such as in Eq. (5.53) but with the poles in a different position relative to the real k0 axis. For example, the case where both poles are below the real k0 axis leads to the so-called retarded Green function. It vanishes for x0 < y0 since the upper contour in Fig. 5.1 contains no poles in this case. 66 CHAPTER 5. CANONICAL QUANTIZATION Let us now prove Eq. (5.55). From Eq. (5.47) we have ∂2 ∂x2 0 ∆F (x −y) = ∂ ∂x0 {δ(x0 −y0)⟨0\|[φ(x), φ(y)]\|0⟩+ ⟨0\|T (π(x), φ(y))\|0⟩} = δ(x0 −y0)⟨0\|[π(x), φ(y)]\|0⟩+ ⟨0\|T (∂2 0φ(x), φ(y))\|0⟩ = −iδ4(x −y) + ⟨0\|T (∂2 0φ(x), φ(y))\|0⟩, where we used that ∂ ∂x0 θ(x0) = δ(x0) and δ(x0 −y0)⟨0\|[φ(x), φ(y)]\|0⟩= δ(x0 −y0)∆(0, x −y) = 0 (see the discussion at the end of Section 5.2). It follows that (2x + m2)∆F (x −y) = ∂2 ∂x2 0 −∂2 ∂x2 + m2 ∆F (x −y) (5.56) = −iδ4(x −y) + ⟨0\|T (2x + m2)φ(x)φ(y) \|0⟩. (5.57) The ﬁeld φ(x) satisﬁes the (free) Klein-Gordon equation and, hence, the second term vanishes. This completes the proof. Evaluating time-ordered operator products, Wick’s theorem We would now like to understand how to evaluate time-ordered products and their vacuum expectation values more generally. To do so it is useful to split the ﬁeld φ(x) into its positive and negative frequency parts φ+(x) and φ−(x) as φ(x) = φ+(x) + φ−(x) , φ+(x) = Z d3˜ k a(k)e−ikx , φ−(x) = Z d3˜ k a†(k)eikx . (5.58) Since φ+(x) only contains annihilation operators it is clear that φ+(x)\|0⟩= 0. Likewise, ⟨0\|φ−(x) = 0. We begin with the time ordered product of two ﬁelds φ(x) and φ(y) for the case x0 > y0 and write T (φ(x)φ(y)) = φ+(x)φ+(y) + φ+(x)φ−(y) + φ−(x)φ+(y) + φ−(x)φ−(y) = φ+(x)φ+(y) + φ−(x)φ+(y) + φ−(x)φ+(y) + φ−(x)φ−(y) + [φ+(x), φ−(y)] = : φ(x)φ(y) : +⟨0\|[φ+(x), φ−(y)]\|0⟩ = : φ(x)φ(y) : +⟨0\|φ(x)φ(y)\|0⟩ The point about introducing the commutator in the second line is that the ﬁrst four terms have the creation operators to the left of the annihilation operators and, hence, correspond to the normal ordering of the ﬁeld product. A similar calculation for the case x0 < y0 leads to T (φ(x)φ(y)) =: φ(x)φ(y) : +⟨0\|φ(y)φ(x)\|0⟩. Combining these two results we have T (φ(x)φ(y)) = : φ(x)φ(y) + ∆F (x −y) : . (5.59) Hence, we see that time and normal ordering are related via a Feynman propagator. When two ﬁeld operators in a time-ordered product are combined into a Feynman propagator as in the above equation they are said to have been contracted. With this terminology we can formulate Wick’s theorem as T (φ(x1) . . . φ(xn)) = : φ(x1) . . . φ(xn) + all possible contractions : (5.60) By ”all possible contraction” we refer to all possible ways of pairing up the ﬁeld operators φ(x1), . . . , φ(xn) into Feynman propagators, including partial pairings. When taking the vacuum expectation value of the above equation the ﬁrst term and all partially contracted terms vanish due to normal ordering. This shows that the vacuum expectation of an odd number of time-ordered ﬁelds vanishes. For an even number of ﬁelds we are left with ⟨0\|T (φ(x1) . . . φ(xn)) \|0⟩= X pairings p ∆F (xi1 −xi2) . . . ∆F (xin−1 −xin) , (5.61) where the sum runs over all pairings p = {(i1, i2), . . . , (in−1, in)} of the numbers 1, . . . , n. This is precisely the same structure as the one we encountered in the context of n-point functions for Gaussian multiple integrals and Gaussian functional integrals. This is of course not an accident and the precise relation will become clear when we discuss the path integral quantisation of ﬁeld theories. Sketch of proof for Wick’s theorem We still need to prove Wick’s theorem in its general form (5.60). For the case of two ﬁelds, n = 2, we have already 5.5. CANONICAL QUANTIZATION OF A VECTOR FIELD 67 done this in Eq. (5.59). The general case can be proven by induction in n. Rather than presenting this general argument it is probably more instructive to consider the example for n = 3. After a suitable relabelling we can arrange that t1 ≥t2 ≥t3. With the short-hand notation φi = φ(xi) and ∆ij = ∆F (xi −xj) we have T (φ1φ2φ3) = φ1φ2φ3 = φ1 : φ2φ3 + ∆23 : = (φ1+ + φ1−) : φ2+φ3+ + φ2−φ3+ + φ3−φ2+ + φ2−φ3−+ ∆23 : = : φ1−φ2φ3 + φ1+φ2+φ3+ + φ2−φ1+φ3+ + φ3−φ1+φ2+ + φ2−φ3−φ1+ + ∆23φ1 : +[φ1+, φ2−]φ3+ + [φ1+, φ3−]φ2+ + [φ1+, φ2−φ3−] = : φ1φ2φ3 + ∆12φ3+ + ∆13φ2+ + ∆12φ3−+ ∆13φ2−+ ∆23φ1 : = : φ1φ2φ3 + ∆12φ3 + ∆13φ2 + ∆23φ1 : In the ﬁrst line we have simply used Wick’s theorem (5.59) for a product of two ﬁelds, applied to φ2 and φ3. In the second line, we need to move φ1+ and φ1−into the normal ordering. This is easy for φ1−since it consists of creation operators and, hence, has to be on the left of a product anyway. The annihilation part φ1+ of the ﬁeld, on the other hand, has to be commuted to the right of any component φi−. The key is that the commutators which arise in this way precisely lead to the necessary contractions of φ1 with φ2 and φ3. The proof for general n is analogous to the above calculation. It applies the induction assumption (that is, the validity of Wick’s theorem for n −1 ﬁelds) to φ2, . . . , φn. Then, moving φ1 into the normal ordering and commuting φ1+ with all negative frequency parts φi−generates all the missing contractions of φ1 with the other ﬁelds. Time-ordered product of complex scalar ﬁelds We should brieﬂy discuss time-ordered products for a free complex scalar ﬁeld φ = (φ1 + iφ2)/ √ 2, described by the Lagrangian density (4.72) with λ = 0. In terms of the two real ﬁelds φ1 and φ2 the Lagrangian (4.72) splits into a sum of Lagrangians for two free real scalars with the same mass m. Hence, each of φ1 and φ2 has an oscillator expansions as in Eq. (5.9) and it immediately follows that ⟨0\|T (φ1(x)φ1(y)) \|0⟩= ⟨0\|T (φ2(x)φ2(y)) \|0⟩= ∆F (x −y) and ⟨0\|T (φ1(x)φ2(y)) \|0⟩= 0. This implies for the complex scalar φ that ⟨0\|T φ(x)φ†(y) \|0⟩= ∆F (x −y) , ⟨0\|T (φ(x)φ(y)) \|0⟩= ⟨0\|T φ†(x)φ†(y) \|0⟩= 0 . (5.62) For a product of operators φ and φ† Wick’s theorem can be applied straightforwardly, but the above equations tell us that only contractions of φ with φ† need to be taken into account. 5.5 Canonical quantization of a vector ﬁeld Recap of theory and canonical quantisation conditions We recall from Section 4.4.1 that the Lagrangian density for a vector ﬁeld Aµ with associated ﬁeld strength Fµν = ∂µAν −∂νAµ is given by L = −1 4FµνF µν , (5.63) and that this Lagrangian is invariant under the gauge transformations Aµ →Aµ + ∂µΛ . (5.64) From Eq. (4.99), the canonical momenta πµ are πµ = ∂L ∂(∂0Aµ) = F µ0 . (5.65) To quantize this theory we interpret the ﬁeld Aµ as a collection of four scalar ﬁelds (which happen to be labelled by a space-time index) and then follow the canonical quantization procedure. We should then impose the canonical commutation relations (5.4) which take the form [πµ(t, x), Aν(t, y)] = −iδµ ν δ3(x −y) (5.66) However, Eq. (5.65) implies that the canonical momentum π0 vanishes and this is inconsistent with the µ = ν = 0 part of Eq. (5.66). Clearly, viewing Aµ as four scalar ﬁelds is too simple. In fact, while typical kinetic terms for four scalar ﬁelds Aµ would be of the form P ν ∂µAν∂µAν and, hence, depend on the symmetric and anti-symmetric parts of ∂µAν, the Lagrangian (5.63) only depends on the anti-symmetric part, that is, on the ﬁeld 68 CHAPTER 5. CANONICAL QUANTIZATION strength Fµν. This special form of the Maxwell Lagrangian is of course responsible for the existence of the gauge symmetry (5.64) as well as for the vanishing of the conjugate momentum π0 and it links these two features. In essence, gauge symmetry is the crucial difference to the scalar ﬁeld theory. There are various viable methods to quantise a gauge theory, but here we will follow the most obvious approach of ﬁxing a gauge before quantisation. Gauge ﬁxing Since we would like to manifestly preserve covariance we use the Lorentz gauge condition (4.102), which we im-pose on the theory by means of a Lagrange multiplier λ. This means that, instead of the Maxwell Lagrangian (5.63), we start with L = −1 4FµνF µν −λ 2 (∂µAµ)2 , (5.67) For the conjugate momenta we now ﬁnd πµ = ∂L ∂(∂0Aµ) = F µ0 −ληµ0∂νAν , (5.68) and, hence, in particular π0 = −λ∂νAν is no longer zero. The equation of motion (4.101) for Aµ is now modiﬁed to 2Aµ −(1 −λ)∂µ∂νAν = 0 , (5.69) and, in addition, we have to impose the gauge condition ∂µAµ = 0 (which formally arises from the action (5.67) as the λ equation of motion). However, clearly we should not impose this condition as an operator equation since this would lead us back to a situation where π0 = 0. Instead, it will later be imposed as a condition on physical states. At any rate, the obstruction to imposing canonical quantisation conditions has been removed and we require the canonical commutation relations (5.66) for Aµ and the conjugate momenta (5.68). Oscillator expansion We should now solve the equation of motion (5.69) to work out the properties of the creation and annihilation operators. While this can be done for arbitrary λ, we adopt the so-called Feynman gauge 4, λ = 1, which simpliﬁes the equation of motion (5.67) to 2Aµ = 0. The general solution to this equation has already been written down in Section 4.4.1 and is given by Aµ(x) = 3 X α=0 Z d3˜ k ǫ(α) µ (k) a(α)(k)e−ikx + a(α)†(k)eikx . (5.70) Here, we have used the polarisation vectors ǫ(α) µ (k) deﬁned before Eq. (4.109) and the four-momentum is given by (kµ) = (wk, k) with wk = \|k\|. We recall from our discussion in Section 4.4.1 that classically ǫ(α) µ (k) for α = 1, 2 are the two transversal, physical polarisations while the other two polarisations can be removed through gauge transformations. Inserting Eq. (5.70) into Eq. (5.68) gives the expansion for the conjugate momenta. Similar to the scalar ﬁeld case (see below Eq. (5.9)), we can now invert these expansions to express the oscillators a(α)(k) and their hermitian conjugates in terms of the ﬁeld Aµ and the momenta πµ. As for scalar ﬁelds, these results can be inserted into the quantisation condition (5.66) to determine the commutation relations of the oscillators. After a straightforward (but slightly lengthy) calculation one ﬁnds that [a(α)(k), a(α′)(q)] = [a(α)†(k), a(α′)†(q)] = 0 , [a(α)(k), a(α′)†(q)] = −ηαα′(2π)32wkδ3(k −q) (5.71) The Fock space As comparison with Eq. (5.13) shows the above commutation relations are very similar to the one for four real scalar ﬁelds, however with one crucial difference: The commutator [a(0)(k), a(0)†(q)] has the opposite sign to that in Eq. (5.13). Naively, the Fock space of this theory is spanned by states created from the vacuum \|0⟩by acting with any combination and any number of operators a(α)†(k), where α = 0, 1, 2, 3. Let us call this space F0. In particular, F0 contains the state \|(k, 0)⟩= a(0)†(k)\|0⟩which satisﬁes ⟨(k, 0)\|(k, 0)⟩= ⟨0\|[a(0)(k), a(0)†(k)]\|0⟩< 0 , (5.72) 4Although common terminology this is somewhat misleading since the choice of λ is clearly not a gauge choice in the usual sense. We also remark that independence of the quantisation procedure on the choice of λ can be explicitly demonstrated. 5.5. CANONICAL QUANTIZATION OF A VECTOR FIELD 69 and, hence, has negative norm. Clearly this is physically unacceptable and indicates that the space F0 still contains unphysical states and cannot be the proper Fock space of the theory. A related problem emerges when looking at the conserved four-momentum. Calculating the four-momentum for the theory (5.67) from the general formalism in Section 4.2.3 and inserting the ﬁeld expansion (5.70) one ﬁnds after some calculation : Pµ : = Z d3˜ k kµ " 3 X α=1 a(α)†(k)a(α)(k) −a(0)†(k)a(0)(k) # . (5.73) The negative sign in front of the last term means that the state \|(k, 0)⟩has ”negative energy”. The existence of unphysical states is not at all surprising given that we have not yet imposed the Lorentz gauge condition. We have seen above that requiring the operator equation ∂µAµ = 0 is too strong and leads to problems with quantisation. Instead we deﬁne a space F1 ⊂F0 of physical states on which the gauge condition is satisﬁed, that is we require ⟨˜ Φ\|∂µAµ\|Φ⟩= 0 . (5.74) between two physical states \|Φ⟩, \|˜ Φ⟩∈F1. To guarantee this condition is satsiﬁed it is sufﬁcient that (∂µAµ)(+)\|Φ⟩= 0 where the annihilation part (∂µAµ)(+) of ∂µAµ is proportional to Z d3˜ k k0 a(0)(k) −a(3)(k) e−ikx . (5.75) To obtain this last result, we have taken the ∂µ derivative of the ﬁrst term in Eq. (5.70) and used that kµǫ(α) µ (k) = 0 for α = 1, 2 and kµǫ(0) µ (k) = −kµǫ(3) µ (k) = k0 (see Section 4.4.1). This means physical states \|Φ⟩∈F1 are deﬁned by the condition b−(k)\|Φ⟩= 0 . (5.76) where we have deﬁned a new basis b±(k) = (a(3)(k) ± a(0)(k))/ √ 2 for the non-transversal operators. Clearly all transversal states, that is, states created from the vacuum by acting only with transversal creation operators a(α)†(k), where α = 1, 2 along with their linear combinations, satisfy the condition (5.76) and, hence, are elements of F1. It would be nice if F1 consisted of such states only but things are not quite so simple. To analyse the condition (5.76) for non-transversal states we note that [b−(k), b† −(q)] = 0 , (5.77) while [b−(k), b† +(q)] ̸= 0. This means, in addition to a(α)†(k), where α = 1, 2, physical states can also contain b† −(k) but not b† +(k). This means F1 is spanned by states created from the vacuum acting with any number of operators a(α)†(k), where α = 1, 2 and b† −(k). Note that the commutators of these operators with their associated annihilation operators are all non-negative and, hence, no states with negative norm are left in F1. However, if a state contains at least one operator b† −(k) its norm vanishes, as is clear from the vanishing of the commutator (5.77). Physically, we should discard such zero norm states and the formal way of doing this is to identify each two states in F1 if their difference has zero norm. In this way, we obtain the proper Fock space F2, whose elements are the classes of states obtained from this identiﬁcation. In particular, in each class there is a ”representative” with only transverse oscillators. In conclusion, we see that the proper Fock space F2 can be thought of as spanned by states of the form \|(k1, α1), . . . , (kn, αn)⟩= a(α1)†(k1) . . . a(αn)†(kn)\|0⟩, where αi = 1, 2 are transverse oscillators only. Independence of physical quantities on Fock space representative One remaining point which needs checking is that physical quantities are independent on which representative for a class in F2 is being used. Let us verify this for the case of the four-momentum (5.73) which can also be written in the form : Pµ : = Z d3˜ k kµ " 2 X α=1 a(α)†(k)a(α)(k) + b† +(k)b−(k) + b† −(k)b+(k) # . (5.78) Taken between a physical state \|Φ⟩∈F1, that is ⟨Φ\| : Pµ : \|Φ⟩, the last two terms in Eq. (5.78) do not contribute (using that all a type operators commutate with all b type operators as well as Eqs. (5.76), (5.77)). This means ⟨Φ\| : Pµ : \|Φ⟩= ⟨Φ\| Z d3˜ k kµ 2 X α=1 a(α)†(k)a(α)(k)\|Φ⟩. (5.79) 70 CHAPTER 5. CANONICAL QUANTIZATION This shows that the four-momentum only depends on the transverse modes and, since every class in F2 has exactly one representative with transverse modes only, the desired independence on the choice of representative. Feynman propagator Finally, we should look at the Feynman propagator for a vector ﬁeld, deﬁned, as usual, as the vacuum expectation value ⟨0\|T (Aµ(x)Aν(y))\|0⟩of a time-ordered product of two ﬁelds. Inserting the ﬁeld expansion (5.70) and using the commutation relations (5.71) we can perform a calculation completely analogous to the one in Section 5.4 to obtain the vector ﬁeld propagator in Feynman gauge, λ = 1. This leads to DF,µν(x −y) ≡⟨0\|T (Aµ(x)Aν(y))\|0⟩= −ηµν∆F,0(x −y) , (5.80) where ∆F,0 is the scalar ﬁeld Feynman propagator (5.53) for m = 0. 5.6 Further reading Canonical quantisation is covered in most standard text books on quantum ﬁeld theory, including • J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields, vol 2, chapters 11, 12, 14. • C. Itzykson and J.-B. Zuber, Quantum Field Fields, chapters 3.1, 3.2. • M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, chapter 2. Chapter 6 Interacting Quantum Fields In this chapter, we will develop the formalism for perturbatively interacting quantum ﬁeld theory. The main goal is to derive the Feynman rules for bosonic ﬁeld theories from ﬁrst principles and show how to calculate matrix elements and cross sections. To do this we will ﬁrst introduce the S-matrix and then show how cross sections and decay rates are expressed in terms of S-matrix elements. We then need to understand how to compute S-matrix elements from ﬁeld theory. The ﬁrst step is to derive the reduction formula which shows how to compute S-matrix elements from vacuum expectation values of time-ordered products of interacting ﬁelds. These products of interacting ﬁelds are then re-written in terms of free ﬁelds using the evolution operator. From there, Wick’s theorem will lead to the Feynman rules. Whenever the formalism needs to be developed with reference to a particular theory we will ﬁrst focus on the real scalar ﬁeld theory (4.47) with λφ4 interaction for simplicity and subsequently generalise to more complicated theories. 6.1 The S-matrix Consider an interacting ﬁeld theory 1 with ﬁeld φ(x), where (x) = (t, x). We would like to describe a scattering process, that is a process where interactions are only important for a ﬁnite time around t ≃0 and the ﬁeld φ asymptotically approaches free ﬁelds in the limits t →±∞. For t →−∞we denote the free-ﬁeld limit of φ by φin and for t →+∞by φout. To these asymptotically free ﬁelds we can associate creation and annihilation operators ain, aout and a† in, a† out in the way discussed in the previous chapter and deﬁne ”in” and ”out” states as \|k1, . . . , kn⟩in = a† in(k1) . . . a† in(kn)\|0⟩, \|k1, . . . , kn⟩out = a† out(k1) . . . a† out(kn)\|0⟩ (6.1) Given an initial state \|i⟩in = \|k1, . . . , kn⟩in with n particles and momenta k1, . . . , kn and a ﬁnal state \|f⟩out = \|q1, . . . , qm⟩with m particles and momenta q1, . . . , qm, we are interested in computing the amplitude out⟨f\|i⟩in (6.2) which provides the probability for a transition from \|i⟩in to \|f⟩out. With the operator S deﬁned by \|q1, . . . , qm⟩out = S†\|q1, . . . , qm⟩in (6.3) this amplitude can be written as Sfi ≡in⟨f\|S\|i⟩in = out⟨f\|i⟩in . (6.4) The matrix Sfi is called the S-matrix and it encodes the basic physical information we wish to calculate. Assuming that both the in states \|i⟩in and the out states \|f⟩out form a complete set of states on the Fock space we have 1 = X f \|f⟩out out⟨f\| = X f S†\|f⟩in in⟨f\|S = S†S (6.5) and, hence, the S matrix is unitary. We also require that the vacuum state is invariant and unique so that \|0⟩in = \|0⟩out ≡\|0⟩ (6.6) 1For simplicity we consider a single ﬁeld but the generalisation to multiple ﬁelds should be obvious. 71 72 CHAPTER 6. INTERACTING QUANTUM FIELDS It is customary to write the S-matrix as S = 1 + iT (6.7) where the unit operator corresponds to the free evolution from ”in” to ”out” states and T encodes the non-trivial scattering. In a theory which conserves four-momentum the matrix elements of T take the general form Tfi ≡in⟨f\|T \|i⟩in = (2π)4δ4(ki −qf)M(i →f) , (6.8) where ki and qf are the total four-momenta of the initial and ﬁnal state and M is the invariant matrix element. We need to understand how to compute M from the underlying ﬁeld theory and, to make contact with experimentally measurable quantities, we need to express cross sections and decay rates in terms of M. We start with the latter. 6.2 Cross sections and decay rates Wave packets and transition probability The incoming state in a scattering experiment is of course not an exact momentum eigenstate. More realistically, an n particle incoming state can be described by a wave packet \|˜ ı⟩in = Z n Y a=1 d3˜ pa ˜ fa(pa) ! \|p1, . . . , pn⟩in , fa(x) = Z d3˜ p ˜ fa(p)e−ipx , (6.9) where p0 = wp in the second integral, so that fa(x) are solutions of the Klein-Gordon equation. By writing fa(x) = e−ikaxFa(x) with slowly varying functions Fa(x) we ensure that these solutions are ”close” to momen-tum eigenstates with momenta ka. The probability currents jaµ for these solutions are given by the usual formula in relativistic quantum mechanics jaµ ≡i (f ⋆ a(x)∂µfa(x) −fa(x)∂µf ⋆ a(x)) ≃2kaµ\|fa(x)\|2 . (6.10) They provide us with expressions for the number of particles per volume and the ﬂux per volume which will be needed in the calculation of the cross section. In a ﬁrst instance, we are interested in the transition probability W(˜ ı →f) = \|in⟨f\|S\|˜ ı⟩in\|2 (6.11) from the initial state \|˜ ı⟩in as deﬁned above to some ﬁnal state \|f⟩out. Inserting Eqs. (6.7), (6.8) and (6.9) into this expression 2 and keeping only the non-trivial scattering part, related to the T matrix, one ﬁnds W(˜ ı →f) = (2π)8 Z n Y a=1 d3˜ pa ˜ f ⋆ a(pa) ! n Y b=1 d3˜ p′ b ˜ fb(p′ b) ! δ4 X a pa − X b p′ b ! (6.12) × δ4 qf − X a pa ! M(p1, . . . , pn →f)⋆M(p′ 1, . . . , p′ n →f) . (6.13) The wave functions ˜ fa are peaked around momenta ka and we can, hence, approximate M(p1, . . . , pn →f) ≃ M(p′ 1, . . . , p′ n →f) ≃M(k1, . . . , kn →f) in the above integral. Together with the integral representation (2π)4δ4 X a pa − X a p′ a ! = Z d4x exp −ix X a p′ a − X a pa !! (6.14) of the delta function this implies W(˜ ı →f) = (2π)4 Z d4x n Y a=1 \|fa(x)\|2 ! δ4 qf − X a ka ! \|M(k1, . . . , kn →f)\|2 . (6.15) 2When doing this, we have to keep in mind that the ”in” states \|i⟩in in the deﬁnition (6.8) of the matrix element are the exact momentum eigenstates \|k1, . . . , k1⟩in, whereas the ”in” states \|˜ ı⟩in which enter the transition probability (6.11) are the wave packets (6.9). 6.2. CROSS SECTIONS AND DECAY RATES 73 We have obtained our ﬁrst important result, the transition probability from state \|˜ ı⟩in to state \|f⟩out per unit time and volume dW(˜ ı →f) dV dt = n Y a=1 \|fa(x)\|2 ! (2π)4δ4 qf − X a ka ! \|M(k1, . . . , kn →f)\|2 . (6.16) Decay rate Let us now be more speciﬁc and ﬁrst consider a single particle with mass M, momentum k = (M, 0) and wave function f(x) in the initial state, that is, we study the decay of a particle at rest. Let us deﬁne the decay rate Γ(k →f) as the transition probability per unit time and volume (6.16) divided by the number density of initial particles and integrated over the momenta q1, . . . , qm of the ﬁnal state \|f⟩out with energy resolution ∆. From Eq. (6.10), the number density of initial particles is given by 2M\|f(x)\|2 and, hence, from Eq. (6.16) we ﬁnd for the decay rate into m particles Γ(k →f) = 1 2M Z ∆ m Y b=1 d3˜ qb ! (2π)4δ4 (k −qf) \|M(k →f)\|2 . (6.17) Cross section for two-particle scattering As the next example, we discuss the scattering of two particles with momenta k1 and k2 and masses m1 and m2 in the initial state. For simplicity, we consider particles of type 2 at rest in the laboratory frame, that is we consider them as target particles, and particles of type 1 as the incident ones. We then deﬁne the cross section σ(k1, k2 →f) as the transition probability per unit time and volume (6.16) divided by the number density of target particles, divided by the incident ﬂux density and integrated over the momenta q1, . . . , qm of the ﬁnal state \|f⟩out with energy resolution ∆. From Eq. (6.10) the number density of target particles is 2m2\|f2(x)\|2 and the incident ﬂux density is 2\|k1\|\|f1(x)\|2. Dividing Eq. (6.16) by these densities and integrating over the ﬁnal state momenta the cross section for a 2 →m scattering turns out to be σ(k1, k2 →f) = 1 4 p (k1 · k2)2 −m2 1m2 2 Z ∆ m Y b=1 d3˜ qb ! (2π)4δ4 (k1 + k2 −qf) \|M(k1, k2 →f)\|2 . (6.18) Here, we have re-written the original kinematical pre-factor 1/(4m2\|k1\|) in a covariant way using the identity m2\|k1\| = m2 q k2 10 −m2 1 = q (k1 · k2)2 −m2 1m2 2 . (6.19) In this way, the above expression for the cross section is manifestly covariant. Cross section for 2 →2 scattering We would now like to specialise this result somewhat further to a two particle ﬁnal state with momenta q1 and q2, that is, we consider a 2 →2 scattering. We carry out the discussion in the center of mass frame, the frame in which the total initial three-momentum vanishes, that is k1 + k2 = 0. Three-momentum conservation of course then implies that q1 + q2 = 0. The kinematic situation for 2 →2 scattering in the center of mass frame is summarised in Fig. 6.1. For simplicity, we also assume that the masses of all four particles are equal to m, so in particular we have k1 · k2 = q1 · q2. This means the kinematical pre-factor in Eq. (6.18) can be re-written as p (k1 · k2)2 −m4 = p (q1 · q2)2 −m4 = q (q10q20 + q2 1)2 −(q2 10 −q2 1)(q2 20 −q2 1) = E\|q1\| , (6.20) where E = k10 + k20 = q10 + q20 is the total center of mass energy. We write the integral in Eq. (6.18) as Z ∆ d3˜ q1d3˜ q2 (2π)4δ4(k1 + k2 −q1 −q2) = Z ∆ d3q1 (2π)32q10 d3q2 (2π)32q20 (2π)4δ(E −q10 −q20)δ3(q1 + q2) = 1 16π2 Z ∆ dΩ\|q1\|2d\|q1\| q10q20 δ(E −q10 −q20) (6.21) where we have carried out the q2 integral in the last step, so that q10 = q20 = p m2 + q2 1. Also dΩ= sin θ dθ dφ is the solid-angle differential for q1. Since d d\|q1\|(q10 + q20) = \|q1\| q10 + \|q1\| q20 = \|q1\|E q10q20 (6.22) 74 CHAPTER 6. INTERACTING QUANTUM FIELDS θ 10 1 k = (k , k) 2 20 k = (k , -k) 10 1 q = (q , q) 20 2 q = (q , -q) Figure 6.1: Kinematics of 2 →2 scattering in the center of mass frame. The total center of mass energy is given by E = k10 + k20 = q10 + q20. we can use the delta function to replace the integral over \|q1\| in Eq. (6.21) by the inverse of this factor. Inserting Eq. (6.21) in this form together with Eq. (6.20) into Eq. (6.18) and assuming good angular resolution, so that the integral R ∆can be dropped we have dσ(k1, k2 →q1, q2) dΩ = 1 64π2E2 \|M(k1, k2 →q1, q2)\|2 . (6.23) To summarise, this expression provides the differential cross section for a 2 →2 scattering where all four particles have the same mass. Note that E is the total center of mass energy. Since we have integrated out the delta-function in Eq. (6.18) the matrix element should be evaluated for conserved four-momentum. 6.3 The reduction formula Removing a particle from the in state Having expressed decay rates and cross sections in terms of S-matrix elements we now need to understand how to compute these S-matrix elements from quantum ﬁeld theory. The ﬁrst step is to derive the so-called LSZ reduction formula (after Lehmann, Symanzik and Zimmermann) which allows one to express S-matrix element in terms of vacuum expectation values of (time-ordered) ﬁeld operator products. Focusing on a real scalar ﬁeld φ, we now derive this formula for an initial state \|i, k⟩in with a particle of momentum k and an arbitrary set of other particles, collectively denoted as i, and a ﬁnal state \|f⟩. The relevant S-matrix element can then be written as out⟨f\|i, k⟩in = out⟨f\|a† in(k)\|i⟩= out⟨f\|a† out(k)\|i⟩in + out⟨f\|(a† in(k) −a† out(k))\|i⟩in (6.24) = out⟨f −k\|i⟩in −out⟨f\|i Z d3x e−ikx← → ∂0 (φin(x) −φout(x)\|i⟩in (6.25) The creation operators have been expressed in terms of ”in” and ”out” ﬁelds by means of Eq. (5.12) which can also be written in the form a† in/out(k) = −i Z d3x e−ikx← → ∂0 φin/out(x) , (6.26) where k0 = wk, as usual. Here, we have used the short-hand notation f(t)← → ∂t g(t) = f(t)∂tg(t) −∂tf(t)g(t). The state \|f −k⟩out refers to the ”out” state \|f⟩out with a particle of momentum k removed (or zero if there is no such particle in \|f⟩out). Hence, the ﬁrst term in Eq. (6.25) either vanishes or corresponds to a forward scattering contribution where one of the momenta has not changed. We will usually drop this term in the subsequent calculation since we are interested in the non-trivial scattering part of the amplitude. Taking time limits x0 →±∞ 6.3. THE REDUCTION FORMULA 75 we can convert the ”in” and ”out” ﬁelds above into a full interacting ﬁeld φ by writing out⟨f\|i, k⟩in = out⟨f\| lim x0→∞− lim x0→−∞ i Z d3x e−ikx← → ∂0 φ(x)\|i⟩in (6.27) = out⟨f\|i Z d4x ∂0 e−ikx← → ∂0 φ(x) \|i⟩in (6.28) = out⟨f\|i Z d4x e−ikx∂2 0φ(x) −∂2 0e−ikxφ(x) \|i⟩in (6.29) The wave function e−ikx (with k0 = wk) satisﬁes the Klein-Gordon equation, so that ∂2 0e−ikx = (∇2−m2)e−ikx. Using this in the second term in Eq. (6.29) and subsequently integrating by parts in the spatial directions one ﬁnds out⟨f\|i, k⟩in = i Z d4x e−ikx(2x + m2) out⟨f\|φ(x)\|i⟩in . (6.30) Hence, we have succeeded in removing one particle from the ”in” state of our S-matrix element and replacing it by a ﬁeld operator φ(x). The idea is to now repeat this process until all particles, both from the ”in” and the ”out” state are removed and one is left with a vacuum expectation value of ﬁeld operators. Removing one particle from the out state To further illustrate this we perform one more step explicitly, namely the reduction of a particle with momentum q from the ”out” state which we write as \|f⟩out = \| ˜ f, q⟩out. For the S-matrix element which appears in Eq. (6.30) this leads to out⟨˜ f, q\|φ(x)\|i⟩in = out⟨˜ f\|aout(q)φ(x)\|i⟩in = out⟨˜ f\|φ(x)\|i −q⟩in + out⟨˜ f\| (aout(q)φ(x) −φ(x)ain(q)) \|i⟩in . (6.31) As before, we discard the forward scattering term and replace the annihilation operators with ﬁelds using Eq. (6.26). Taking into account that out⟨˜ f\| (φout(y)φ(x) −φ(x)φin(y)) \|i⟩in = lim y0→∞− lim y0→−∞ out⟨˜ f\|T (φ(y)φ(x)) \|i⟩in , (6.32) we ﬁnd following the same steps as before that out⟨˜ f, q\|φ(x)\|i⟩= i Z d4y eiqy(2y + m2) out⟨˜ f\|T (φ(y)φ(x)) \|i⟩in (6.33) This is precisely the same structure as in the ﬁrst reduction (6.30) apart from the opposite sign in the exponent eiqy. This sign difference arises because we have reduced a particle in the ”out” state as opposed to a particle in the ”in” state. In addition, we learn that products of ﬁeld operators which arise in this way arrange themselves into time-ordered products. The general reduction formula for real scalar ﬁelds We can now iterate Eqs. (6.30) and Eq. (6.33) and eliminate all particles from an ”in” state \|k1, . . . , kn⟩in and an ”out” state \|q1, . . . , qm⟩out. If we assume that ki ̸= qj for all i and j, then forward scattering terms are absent and we have out⟨q1, . . . , qm\|k1, . . . , kn⟩in = im+n Z n Y a=1 d4xa e−ikaxa(2xa + m2) ! m Y b=1 d4yb eiqbyb(2yb + m2) ! ×⟨0\|T (φ(y1) · · · φ(ym)φ(x1) · · · φ(xn)) \|0⟩, (6.34) The vacuum expectation values of time-ordered products which arise in the above formula are also referred to as N-point Green functions G(N)(z1, . . . , zN) = ⟨0\|T (φ(z1) . . . φ(zN))\|0⟩. (6.35) Eq. (6.34) shows that the Green functions determine the S-matrix elements and are the crucial objects we need to calculate. It also displays a remarkable property, the so-called crossing symmetry: All that changes in Eq. (6.34) if a particle with momentum k is shifted from the ”in” to the ”out” state is the sign of the momentum k in the wave function e−kx. This means, for example, that a 2 →2 scattering and a 1 →3 decay related by transferring one 76 CHAPTER 6. INTERACTING QUANTUM FIELDS ”in” particle to an ”out” particle are described by the same four-point Green function G(4). Reduction formula for complex scalar ﬁelds We should now brieﬂy discuss the reduction formula for other types of ﬁelds, starting with a complex scalar ﬁeld φ = (φ1 + iφ2)/ √ 2. In the free-ﬁeld case, it is easy to see (by comparing Eqs. (5.9) and (5.37)) that a† +(k) = (a† 1(k) −ia† 2(k))/ √ 2 and a† −(k) = (a† 1(k) + ia† 2(k))/ √ 2. Applying Eq. (6.26) to φ1 and φ2 and forming appropriate linear combinations this implies a† +,in/out(k) = −i Z d3x e−ikx← → ∂0 φ† in/out(x) , a† −,in/out(k) = −i Z d3x e−ikx← → ∂0 φin/out(x) . (6.36) The reduction formula for a complex scalar can now be derived exactly as above. This leads to Eq. (6.34) with one modiﬁcation: Positively charged ”in” particle, generated by a† +,in(k), lead to an operator φ† in the time-ordered product, while negatively charged ”in” states, generated by a† −,in(k), lead to φ. For ”out” states the situation is reversed, with operators φ for positive particles and operators φ† for negative particles. Reduction formula for vector ﬁelds Finally, for gauge ﬁelds, the above calculation can be repeated using the expansion (5.70) for a free gauge ﬁeld and its inversion in terms of creation operators a(α)†(k). Again, the result is very similar to the previous one, except that the polarisation ǫ(α) µ (k) of the photon must be taken into account. For a photon with momentum k and polarisation ǫ(α) µ (k) in the ”in” state \|i, (k, α)⟩in one ﬁnds in analogy with Eq. (6.30) that out⟨f\|i, (k, α)⟩in = i Z d4x ǫ(α) µ (k)e−ikx2x out⟨f\|Aµ(x)\|i⟩in , (6.37) while for a photon with momentum q and polarisation ǫ(α) µ (k) in the ”out” state we have in analogy with Eq. (6.33) out⟨f, (q, α)\|i⟩in = i Z d4x ǫ(α) µ (q)eiqx2x out⟨f\|Aµ(x)\|i⟩in . (6.38) Repeated reduction leads to time-ordered operator product as for scalar ﬁelds. It should also be clear that for theories with different types of ﬁelds, for example for theories with scalar and vector ﬁelds, the various reduction formulae above can be combined and applied successively until all particles are reduced. 6.4 Perturbative evaluation of Green functions and the evolution operator Schr¨ odinger, Heisenberg and interaction pictures We have managed to write S-matrix elements in terms of Green functions, that is, vacuum expectation values of time ordered products of interacting ﬁelds. We do know how to evaluate time-ordered products of free ﬁelds using Wick’s theorem but this method does not apply to interacting ﬁelds so easily. What we need to do is to express interacting ﬁelds and Green functions in terms of free-ﬁelds so that Wick’s theorem can be applied. We begin by splitting up the full Hamiltonian H of a ﬁeld theory as H = Z d3x H = H0 + H1 (6.39) where H0 is the free Hamiltonian (quadratic in the ﬁelds and conjugate momenta) and H1 = R d3x H1 contains the interactions. For example, for our simple toy model, a real scalar ﬁeld φ, we have L = 1 2 ∂µφ∂µφ −m2φ2 −λ 4!φ4 , H1 = Z d3x H1 = λ 4! Z d3x φ(x)4 . (6.40) So far, we have worked in the Heisenberg picture where the ﬁeld operator φ(x) is time-dependent and Fock space states \|a⟩are time-independent. In the Schr¨ odinger picture, on the other hand, the ﬁeld operator φS(x) is time-independent while the corresponding Fock space states \|a, t⟩S are time-dependent. The two pictures are related by the time-evolution operator eiHt in the usual way φS(x) = e−iHtφ(t, x)eiHt , \|a, t⟩S = e−iHt\|a⟩. (6.41) 6.4. PERTURBATIVE EVALUATION OF GREEN FUNCTIONS AND THE EVOLUTION OPERATOR 77 For the purpose of perturbation theory it is useful to introduce a third picture, the interaction picture, with ﬁelds and states denoted by φI(x) and \|a, t⟩I. It is intermediate between the two previous pictures and both operators and states depend on time. In terms of the Schr¨ odinger picture, it is deﬁned through time evolution with the free Hamiltonion H0, that is φI(t, x) = eiH0tφS(x)e−iH0t , \|a, t⟩I = eiH0t\|a, t⟩S . (6.42) Deﬁnition of the evolution operator Combining the above relations it is clear that interaction and Heisenberg picture are related by φI(t, x) = U(t, 0)φ(t, x)U −1(t, 0) , \|a, t⟩I = U(t, 0)\|a⟩, (6.43) where U(t, 0) = eiH0te−iHt (6.44) is called the evolution operator. It encodes the difference in time evolution between interaction and Heisenberg picture due to the interaction Hamiltonian H1. More generally, we can deﬁne the evolution operator U(t, t0) between two times t0 and t by \|a, t⟩I = U(t, t0)\|a, t0⟩I. This means from Eq. (6.43) that U(t, t0) = U(t, 0)U −1(t0, 0) , (6.45) so, in particular, U −1(t, t0) = U(t0, t) and U(t0, t0) = 1. For the composition of two evolution operators we have the rule U(t, t′)U(t′, t0) = U(t, 0)U −1(t′, 0)U(t′, 0)U −1(t0, 0) = U(t, 0)U −1(t0, 0) = U(t, t0) . (6.46) From Eqs. (6.45) and (6.44), the time derivative of the evolution operator is given by i ∂ ∂tU(t, t0) = i ∂ ∂tU(t, 0) U −1(t0, 0) = eiH0tHe−iH0t −H0 U(t, t0) . (6.47) and, hence, U(t, t0) satisﬁes the simple differential equation i ∂ ∂tU(t, t0) = H1,I(t)U(t, t0) where H1,I = eiH0tH1e−iH0t . (6.48) Note that H1,I, the interaction Hamiltonian written in the interaction picture, has the same form as H1 but with φ replaced by φI. Perturbative solution for evolution operator Let us pause for a moment and see what we have achieved. The ﬁeld φI in the interaction picture evolves with the free Hamiltonion, as is clear from Eq. (6.42), and we should, hence, think of it as a free ﬁeld. The evolution operator relates this free ﬁeld to the full, interacting ﬁeld φ in the Heisenberg picture via Eq. (6.43). Therefore, if we can ﬁnd an explicit expression for the evolution operator we have succeeded in writing the interacting ﬁeld in terms of a free ﬁeld. To do this, we need to solve the differential equation (6.48) subject to the initial condition U(t0, t0) = 1. It turns out that the solution is given by U(t, t0) = T exp −i Z t t0 dt1 H1,I(t1) = 1 + ∞ X p=1 (−i)p p! Z t t0 d4x1 . . . Z t t0 d4xp T (H1,I(x1) . . . H1,I(xp)) . (6.49) The exponential form of this solution is as expected for a differential equation of the form (6.48) and the initial condition is obviously satisﬁed. However, the appearance of the time-ordering operator is perhaps somewhat surprising. To verify the precise form of this solution, we focus on the terms up to second order in H1,I. This will be sufﬁcient to illustrate the main idea and justify the appearance of the time-ordering. While the term linear in H1,I in Eq. (6.49) contains a single integral R t t0 dt1 and is easy to differentiate with respect to t the situation is more complicated for the quadratic term which contains two integrals R t t0 dt1 R t t0 dt2. We re-write this double-integral by splitting the integration region into two parts, for t1 ≥t2 and t1 < t2, so that R t t0 dt1 R t t0 dt2 = 78 CHAPTER 6. INTERACTING QUANTUM FIELDS R t t0 dt1 R t1 t0 dt2 + R t t0 dt2 R t2 t0 dt1. Then we have i ∂ ∂tU(t, t0) = i ∂ ∂t 1 −i Z t t0 dt1 H1,I(t1) −1 2 Z t t0 dt1 Z t t0 dt2 T (H1,I(t1)H1,I(t2)) + O(3) = H1,I(t) −i 2 ∂ ∂t Z t t0 dt1 Z t1 t0 dt2 H1,I(t1)H1,I(t2) + Z t t0 dt2 Z t2 t0 dt1 H1,I(t2)H1,I(t1) + O(3) = H1,I(t) −i 2H1,I(t) Z t t0 dt2 H1,I(t2) + Z t t0 dt1 H1,I(t1) + O(2) = H1,I(t) 1 −i Z t t0 dt1 H1,I(t1) + O(3) = H1,I(t)U(t, t0) + O(3) The main point is that the order of the interaction Hamiltonians in the two integrals in the second line is reversed as a consequence of time-ordering. This allows us, after differentiating in the third line, to factor H1,I(t) to the left for both terms in the bracket. Without time-ordering H1,I(t) would have been to the left of the ﬁrst term in the bracket and to the right of the second. Since interaction Hamiltonians at different times do not necessarily commute this would have been a problem. The general proof is a straightforward generalisation of the above calculation to all orders. Perturbative calculation of Green functions Armed with this solution for the evolution operator, we now return to our original problem, the perturbative calcu-lation of Green functions. Using Eqs. (6.43) and (6.46) one can write G(N)(z1, . . . , zn) = ⟨0\|T (φ(z1) . . . φ(zN)) \|0⟩= ⟨0\|T U −1(t, 0)φI(z1) . . . φI(zN)U(t, 0) \|0⟩ = ⟨0\|T U −1(t, 0)φI(z1) . . . φI(zN)U(t, −t)U(−t, 0) \|0⟩ = lim t→∞⟨0\|U −1(t, 0)T (φI(x1) . . . φI(xn)U(t, −t)) U(−t, 0)\|0⟩. (6.50) The vacuum should be invariant under the action of the evolution operator up to a constant so we have U(−t, 0)\|0⟩= β−\|0⟩, U(t, 0)\|0⟩= β+\|0⟩, (6.51) for complex numbers β±. They can be determined as β⋆ +β− = ⟨0\|U(−t, 0)\|0⟩⟨0\|U −1(t, 0)\|0⟩= X a ⟨0\|U(−t, 0)\|a⟩⟨a\|U −1(t, 0)\|0⟩ = ⟨0\|U(−t, 0)U −1(t, 0)\|0⟩= ⟨0\|U(−t, t)\|0⟩= (⟨0\|U(t, −t)\|0⟩)−1 (6.52) so we ﬁnally ﬁnd for the Green function G(N)(z1, . . . , zN) = lim t→∞ ⟨0\|T (φI(z1) . . . φI(zN)U(t, −t)) \|0⟩ ⟨0\|U(t, −t)\|0⟩ . (6.53) We can now insert the explicit solution (6.49) for the evolution operator in the Taylor expanded form into this result. This leads to a formula for the Green functions which only depends on vacuum expectation values of time-ordered free-ﬁeld operator product, each of which can be evaluated using Wick’s theorem. As in Section 1.1.4 each term which arises in this way can be represented by a Feynman diagram. Feynman diagrams which arise form the numerator of Eq. (6.53) have N external legs, Feynman diagrams from the denominator are vacuum bubbles. It turns out that the denominator in Eq. (6.53) precisely cancels all Feynman diagrams from the numerator which contain disconnected vacuum bubbles. In fact, we have seen an explicit example of this in Eq. (1.79). With this additional information we can write G(N)(z1, . . . , zN) = ∞ X p=0 (i)p p! Z d4y1 . . . d4yp ⟨0\|T (φI(z1) . . . φI(zN)Lint(y1) . . . Lint(yp)) \|0⟩no bubbles (6.54) with the interaction Lagrangian Lint = −H1,I. The interaction Lagrangian is proportional to a coupling constant (λ in the case of our scalar ﬁeld theory example), so the above expression for the Green functions can be seen as an expansion in this coupling constant. If its value is sufﬁciently small it should be a good approximation to only 6.5. FEYNMAN RULES AND EXAMPLES 79 compute a ﬁnite number of terms up to a certain power in the coupling. Only in this case is Eq. (6.54) of direct practical use. In the above derivation we have used notation appropriate for scalar ﬁeld theory but it is clear that the basic structure of Eq. (6.54) remains valid for other types of ﬁelds. To summarise, combining the reduction formula (6.34) (and its generalisations to other types of ﬁelds) with the perturbation expansion (6.54) and Wick’s theorem provides us with a practical way of calculating S-matrix elements and, via Eqs. (6.17) and (6.18) decay rates and cross sections, in terms of Feynman diagrams. Green functions in momentum space For practical calculations, it is more convenient to formulate Feyman rules in momentum space and to this end we introduce the Fourier transforms ˜ G(N) of Green functions by (2π)4δ4(p1 + · · · + pN) ˜ G(N)(p1, . . . , pN) = Z N Y A=1 d4zA e−ipAzA ! G(N)(z1, . . . , zN) , (6.55) where we denote ”in” momenta ka and ”out” momenta qb collectively by (pA) = (ka, −qb). To see where this leads we should rewrite the LSZ reduction formula (see Eq. (6.34)) out⟨q1, . . . , qm\|k1, . . . , kn⟩in = iN Z Y A d4zAe−pAzA(2zA + m2)G(N)(z1, . . . , zN) (6.56) in momentum space, using the inversion G(N)(z1, . . . , zN) = Z Y B d4QB (2π)4 eiQBzB(2π)4δ4(Q1 + · · · + QN) ˜ G(N)(Q1, . . . , QN) (6.57) of Eq. (6.55). Inserting this into the LSZ formula (6.56) we can explicitly carry out the (2zA + m2) operations which now only act on the exponentials in Eq. (6.57) and produce factors −i/ ˜ ∆F(QB) of inverse Feynman prop-agators, one for each external leg of the Green function. It is, therefore, useful to introduce amputated Green functions ˜ G(N) amp(p1, . . . , pN) = ˜ G(N)(p1, . . . , pN) QN A=1 ˜ ∆F (pA) , (6.58) which are related to the ordinary Green functions by removing the propagators for the external legs. The remaining integerations over zA and QB can then trivially be carried out and we remain with out⟨q1, . . . , qm\|k1, . . . , kn⟩in = (2π)4δ4 X a ka − X b qb ! ˜ G(N) amp(k1, . . . , kn, −q1, . . . , −qm) . (6.59) Comparison with Eq. (6.8) shows that for real scalar ﬁelds the amputated Green function equals the matrix element M. For other types of ﬁelds there are slight modiﬁcations. For complex scalar ﬁelds, we have to consider Green functions deﬁned with the appropriate number of ﬁelds φ and their conjugates φ†, depending on the number of particles and anti-particles in the ”in” and ”out” state. Apart from this the above formulae remain valid. For each vector particle with momentum k and polarisation ǫµ(k) in the ”in” or ”out” state, the Green function carries a Lorentz index µ and from Eqs. (6.37) and (6.38) this should be contracted into the corresponding polarisation vector ǫµ(k) to obtain the matrix element M. With this small modiﬁcation, the above formulae also apply to vector ﬁelds. 6.5 Feynman rules and examples What remains to be done is to explicitly derive the Feynman rules for calculating the amputated Green functions by applying Wick’s theorem to the perturbative expansion (6.54). 6.5.1 The real scalar ﬁeld Four-point function As usual, we begin with the real scalar ﬁeld with Lagrangian density (6.40) and interaction Lagrangian density 3 3For simplicity of notation, we drop the subscript ”I” from hereon and assume that ﬁelds are in the interaction picture. 80 CHAPTER 6. INTERACTING QUANTUM FIELDS p 4 3 p 2 p 1 p -iλ Figure 6.2: Feynman diagram for 4-point function in real scalar ﬁeld theory with λφ4/4! interaction to order λ. Lint = −λφ4/4!. Let us ﬁrst calculate some examples before formulating the general Feynman rules. We start with the 4-point function to order λ. Dropping disconnected parts, we have from Eq. (6.54) G(4)(z1, z2, z3, z4) = −iλ 4! Z d4y ⟨0\|T φ(z1)φ(z2)φ(z3)φ(z4)φ(y)4 \|0⟩ = −iλ Z d4y ∆F (z1 −y)∆F (z2 −y)∆F (z3 −y)∆F (z4 −y) = −iλ Z d4k1 (2π)4 . . . d4k4 (2π)4 Z d4y eiy(k1+···+k4)e−i(k1z1+···+k4z4) ˜ ∆F (k1) . . . ˜ ∆F (k4) = −iλ(2π)4 Z d4k1 (2π)4 . . . d4k4 (2π)4 δ4(k1 + · · · + k4)e−i(k1z1+···+k4z4) × ˜ ∆F (k1) . . . ˜ ∆F (k4) , (6.60) From the ﬁrst to the second line we have used Wick’s theorem which only leads to one type of term (although with multiplicity 24) which can be represented by the Feynman diagram in Fig. 6.2. Then we have used the representation (5.53) of the Feynman propagator to re-write the expression in momentum space. From this form it is easy to see that the amputated momentum space Green function is ˜ G(4) amp(p1, . . . , p4) = −iλ . (6.61) Inserting this into Eq. (6.23) we ﬁnd the cross section for 2 →2 scattering dσ dΩ= λ2 64π2E2 , σ = λ2 16πE2 , (6.62) where E is the total center of mass energy. Two-point function Next, we discuss the 2-point function to order λ. From Eq. (6.54) we have G(2)(z1, z2) = ⟨0\|T (φ(z1)φ(z2)) \|0⟩−iλ 4! Z d4y ⟨0\|T φ(z1)φ(z2)φ(y)4 \|0⟩+ O(λ2) = ∆F (z1 −z2) −iλ 2 ∆F (0) Z d4y ∆F (z1 −y)∆F (z2 −y) + O(λ2) The two corresponding Feynman diagrams are depicted in Fig. 6.3. It is straightforward to Fourier transform this expression (introducing integration variables z = z1 −z2 and ˜ z = (z1 + z2)/2 in the ﬁrst term and ˜ z1 = z1 −y and ˜ z2 = z2 −y in the second term). Dividing by the external propagators one then ﬁnds for the amputated Green function ˜ G(2) amp(p, −p) = ˜ ∆F (p)−1 −iλ 2 ∆F (0) + O(λ2) = −i p2 −m2 + λ 2 Z d4k i k2 −m2 + iǫ + O(λ2) (6.63) 6.5. FEYNMAN RULES AND EXAMPLES 81 p -iλ p p k + + . . . Figure 6.3: Feynman diagrams for 2-point function in real scalar ﬁeld theory with λφ4/4! interaction to order λ. The diagram at order λ is a loop diagram and, as a result, we have to carry out an integration over the internal loop momentum k in Eq. (6.63). Feynman rules Before we discuss this result, we would like to summarise our experience so far and formulate the Feynman rules for the amputated Green functions in real scalar ﬁeld theory. • For the (connected part of the) N point function at order λp draw all possible (connected) Feynman graphs with N external legs and p four-vertices. Assign a directed momentum to each line in those Feyman graphs such that momentum is conserved at each vertex. Then, to each graph associate an expression obtained by: • writing down a Feynman propagator ˜ ∆F (k) for each internal line with momentum k • writing down −iλ for each vertex • writing down an integral R d4k /(2π)4 for each momentum k which is not ﬁxed by momentum conservation at the vertices • dividing the expression by a symmetry factor which equals the number of permutations of internal lines one can make for ﬁxed vertices. The ﬁrst two of these rules are obvious from Eq. (6.54) and Wick’s theorem. The fact that momentum needs to be conserved at each vertex can be seen from Eq. (6.60): The calculation which led to this result would be exactly the same for a single vertex within a larger Feynman diagram. The appearance of the delta function in Eq. (6.60) then signals momentum conservation at this vertex. Our result (6.61) for the amputated 4-point function shows that −iλ is the correct factor to include for a vertex. Finally, the precise symmetry factor for each graph follows from the number of pairings which arise in Wick’s theorem. Discussion of two-point function, regularisation and renormalization Let us now return to a discussion of the 2-point function. To carry out the integral in Eq. (6.63) we recall the position of the poles in the Feynman propagator as indicated in Fig. 5.1. We deform the contour of integration by a 90 degrees counter-clockwise rotation from the real to the imaginary k0 axis so that we are not crossing any poles. Then, the integration goes over the imaginary k0 axis and to convert this back to an integration over a real axis from −∞to +∞we introduce the ”Euklidean momentum” ¯ k = (¯ k0, ¯ k) = (−ik0, k) and denote its Euklidean length by κ. The integral in Eq. (6.63) can then be re-written as Z d4k i k2 −m2 + iǫ = Z dΩ3 Z ∞ 0 dκ κ3 κ2 + m2 = 2π2 Z ∞ 0 dκ κ3 κ2 + m2 (6.64) Clearly, the last integral is quadratically divergent as κ →∞. Recall that in the context of free ﬁeld theory we have found inﬁnities due to summing up an inﬁnite number of zero point energies. This problem was resolved by the process of ”normal ordering”. Here we encounter another and more serious type of singularity in ﬁeld theories. They arise from loops in Feynman diagrams at high momentum and are, therefore, also referred to as ultraviolet singularities. Their appearance is of course unphysical as, for example, each of the external legs of the 4-point function can acquire an additional loop at order λ2. This would make the 4-point function and, hence, the associated cross section for 2 →2 scattering inﬁnite. Dealing with and removing such singularities is difﬁcult and discussing the full procedure is beyond the scope of this lecture. Here, we would just like to outline the main idea. It turns out that in certain classes of theories, so called renormalizable theories, only a ﬁnite number of different types of such singularities arise. For renormalizable theories, these singularities can then be absorbed into (inﬁnite) redeﬁnitions of the parameters (masses and couplings), of the theory. Once physical quantities are expressed in terms of these redeﬁned parameters they turn out to be ﬁnite. A useful rule of thumb is that a theory is renormalizable when it does not contain any parameters with negative mass dimension. From this criterion, the real 82 CHAPTER 6. INTERACTING QUANTUM FIELDS scalar ﬁeld theory with λφ4 interaction should be renormalizable and this is indeed the case. Let us brieﬂy discuss how one might go about dealing with the above singularity. The ﬁrst step is to regularise the amplitude. This refers to some sort of prescription which allows one to assign a ﬁnite value to the diverging integral in question. There are many different ways of doing this but perhaps the simplest and most intuitive one is to introduce a cut-off, that is, to modify the upper limit of the integral (6.64) to a ﬁnite value Λ. If we do this we ﬁnd ˜ G(2) amp(p, −p) = −i p2 −m2 + π2λ 2 Λ2 −m2 ln(1 + Λ2/m2) . (6.65) We can now deﬁne a ”renormalized” mass mR by m2 R = m2 −π2λ 2 Λ2 −m2 ln(1 + Λ2/m2) . (6.66) The main idea is that expressed in terms of this mass (and the other remormalized parameters of the theory) physical quantities are ﬁnite provided the theory is renormalizable. Eq. (6.66) reveals another interesting feature which is characteristic for renormalizable scalar ﬁeld theories: the appearance of divergencies quadratic in the cut-off Λ. It turns out that in renormalizable theories without scalar ﬁelds divergencies are at most logarithmic. Quadratic divergencies in scalar ﬁeld theory lead to a serious ”naturalness” problem called the hierarchy problem. Physically, we can think about the cut-off Λ as the energy scale above which the theory in question ceases to be valid and new physics becomes relevant. Assume that we have a theory with a very high cut-off scale Λ but a small physical scalar mass mR. Then Eq. (6.66) requires a very precise cancelation between the two terms on the right-hand side. For example, in the standard model of particle physics, a theory shown to be renormalizable, the mass of the Higgs particle should be of the order of the electroweak symmetry breaking scale, so a few hundred GeV. If the standard model was valid all the way up to the Planck scale, ∼1018 GeV, this would require a 32 digit cancellation in Eq. (6.66). The hierarchy problem may, therefore, be seen as a reason for why we should expect new physics not far above the electroweak scale. Supersymmetry is one of the main candidates for such physics and it is indeed capable of resolving the hierarchy problem. p q Figure 6.4: Vertices for scalar electrodynamics. From left to right the corresponding vertex Feynman rules are −ie(pµ + qµ), 2iq2ηµν and −iλ. 6.5.2 Scalar electrodynamics From Eq. (4.127) we recall that the Lagrangian density of scalar electrodynamics (plus gauge ﬁxing term in Feyn-man gauge) for a complex scalar φ with charge e under a vector ﬁeld Aµ is given by L = −1 4FµνF µν −1 2(∂µAµ)2 + [(∂µ + ieAµ)φ]† [(∂µ + ieAµ)φ] −m2φ†φ −λ 4 (φ†φ)2 , (6.67) so we have for the interaction Lagrangian Lint = ieAµ φ∂µφ† −φ†∂µφ + e2AµAµφ†φ −λ 4 (φ†φ)2 . (6.68) So, in addition to a quartic scalar ﬁeld vertex similar to what we have encountered in real scalar ﬁeld theory, we expect a triple vertex coupling a vector ﬁeld to two scalars and another quartic vertex coupling two vectors to two 6.6. FURTHER READING 83 scalars. To ﬁnd the Feynman rules for these vertices let us compute the appropriate Green functions. Using the ﬁrst, cubic interaction in Eq. (6.68) we ﬁnd for the 3-point function G(3) µ (z1, z2, z3) = −e Z d4y ⟨0\|T φ(z1)φ†(z2)Aµ(z3)Aν(y)(φ(y)∂νφ†(y) −φ†(y)∂νφ(y)) \|0⟩ = −ie(2π)4 Z d4p (2π)4 d4q (2π)4 d4k (2π)4 δ4(p −q + k)e−i(pz1−qz2+kz3) × ˜ ∆F(p) ˜ ∆F (q) ˜ ∆F,0(k)(pµ + qµ) , (6.69) where ˜ ∆F,0(k) = i/(k2 + iǫ) is the Feynman propagator for zero mass. For the amputated Green function this means ˜ G(3) amp,µ(p, −q, k) = −ie(pµ + qµ) (6.70) and this is precisely the expression for the triple vertex. From a very similar calculation using two external vector ﬁelds, two external scalars and the second interaction term in (6.68) one ﬁnds ˜ G(4) amp,µν(p1, p2, p3, p4) = 2ie2ηµν . (6.71) Finally, the quartic scalar ﬁeld interaction from the last term in (6.68) comes with a vertex factor −iλ, just as in the case of a real scalar ﬁeld. These Feynman rules for the interactions in scalar electrodynamics are summarised in Fig. 6.4. Finally, the propagators for internal scalar and vector ﬁeld lines are ˜ ∆F (k) = i k2 −m2 + iǫ , ˜ DF (k)µν = −i k2 + iǫηµν . (6.72) Space-time indices on internal vector ﬁeld propagators and vertices have to be contracted in the obvious way. This completes the Feynman rules for the amputated Green functions in scalar electrodynamics. To obtain the matrix element the µ index for each external photon with momentum k and polarisation α has to be contracted into ǫ(α) µ (k). 6.6 Further reading Perturbation theory in the canonical formalism is covered in most standard text books on quantum ﬁeld theory, including • J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields, vol 2, chapters 16,17. • C. Itzykson and J.-B. Zuber, Quantum Fields, chapters 5.1, 6.1. • M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, chapter 4. 84 CHAPTER 6. INTERACTING QUANTUM FIELDS Chapter 7 Path Integrals in Quantum Field Theory In the previous chapters we have followed the ”traditional”, canonical approach to quantise ﬁeld theories. We would now like to show how ﬁeld theories can be quantised, arguably in a more elegant way, by using path integrals. We begin by recalling some features of path integrals in quantum mechanics. 7.1 Quantum mechanics ﬂashback We have shown in Section 1.2 how quantum mechanics can be formulated in terms of path integrals. In particular, we have found the expression (1.86) ⟨x, t\|x′, t′⟩= ⟨x\|e−iH(t−t′)\|x′⟩∼ Z Dx exp i Z t t′ dτ L(x, ˙ x) (7.1) for the matrix element between two position eigenstates. Here, \|x, t⟩= eiHt\|x⟩S are Heisenberg picture states which coincide with Schr¨ odinger picture states \|x(t)⟩S at a given, ﬁxed time t. The central objects to compute in quantum ﬁeld theory are vacuum expectation values of time-ordered ﬁeld operator product. The analogous quantities in quantum mechanics, for simplicity written for just two operators, are 1 ⟨xf, tf\|T (ˆ x(t1)ˆ x(t2))\|xi, ti⟩. Focusing on the case t1 > t2 and inserting complete sets of states we have ⟨xf, tf\|T (ˆ x(t1)ˆ x(t2))\|xi, ti⟩ = ⟨xf\|e−iH(ti−t1)ˆ xSe−iH(t1−t2)ˆ xSe−iH(t2−tf )\|xi⟩ = Z dx1dx2 ⟨xf\|e−iH(ti−t1)\|x1⟩⟨x1\|ˆ xSe−iH(t1−t2)\|x2⟩⟨x2\|ˆ xSe−iH(t2−tf )\|xi⟩ Using ˆ x\|x⟩= x\|x⟩, replacing every expectation value by Eq. (7.1) and combining the three path integrals, together with the integrations over x1 and x2 into a single path integral this can be written as ⟨xf, tf\|T (ˆ x(t1)ˆ x(t2))\|xi, ti⟩∼ Z Dx x(t1)x(t2) exp i Z tf ti dτ L(x, ˙ x) . (7.2) For t2 > t1 the result is actually the same: time-ordering is automatic in path integrals. It is also clear that the above argument can be repeated for a product of an arbitrary number of operators ˆ x(t). Further, it can be shown that limti→−∞,tf→∞⟨xf, tf\|T (ˆ x(t1) . . . ˆ x(tN))\|xi, ti⟩∼⟨0\|T (ˆ x(t1) . . . ˆ x(tN))\|0⟩, so we have the ﬁnal result ⟨0\|T (ˆ x(t1) . . . ˆ x(tN))\|0⟩∼ Z Dx x(t1) . . . x(tN)eiS[x] . (7.3) 7.2 Basics of ﬁeld theory path integrals As in previous sections, we focus on the simple toy example of a real scalar ﬁeld φ with Lagrangian density L = L(∂µφ, φ) and action S = R d4x L although the formalism applies more generally. As discussed in the previous chapter, the central objects in quantum ﬁeld theory which carry all the relevant physical information are the Green functions G(N)(x1, . . . , xN) = ⟨0\|T (ˆ φ(x1) . . . ˆ φ(xN))\|0⟩. (7.4) 1In this chapter, we will use hats to distinghuish operators from their classical counterparts which appear in the path integral. 85 86 CHAPTER 7. PATH INTEGRALS IN QUANTUM FIELD THEORY In analogy with Eq. (7.3) we can now express these Green functions in terms of a path integral as G(N)(x1, . . . , xN) = N Z Dφ φ(x1) . . . φ(xN)eiS[φ] , (7.5) where N is a normalization to be ﬁxed shortly. Analogous to what we did in Chapter 1, it is useful to introduce a generating functional W[J] = N Z Dφ exp i Z d4x [L(∂µφ, φ) + J(x)φ(x)] (7.6) for the Green functions such that iNG(N)(x1, . . . , xN) = δNW[J] δJ(x1) . . . δJ(xN) J=0 . (7.7) To ﬁx the normalization N we require that W[J]\|J=0 = ⟨0\|0⟩= 1. For the purpose of explicit calculations it is useful to introduce a Eulkidean or Wick rotated version of the generating functional. To do this we deﬁne Euklidean four-vectors by ¯ x = (¯ x0, ¯ x) = (ix0, x), associated derivatives ¯ ∂µ = ∂ ∂¯ xµ and a Euklidean version of the Lagrangian density LE = LE(¯ ∂µφ, φ). We can then re-write the generating functional W[J] and obtain its Euklidean counterpart WE[J] = N Z Dφ exp Z d4¯ x LE(¯ ∂µφ, φ) + J(¯ x)φ(¯ x) (7.8) and associated Euklidean Green functions G(N) E (¯ x1, . . . , ¯ xN) = δNWE[J] δJ(¯ x1) . . . δJ(¯ xN) J=0 . (7.9) In analogy with Chapter 1 (see Eq. (1.81) we also deﬁne the generating functional Z[J] by W[J] = eiZ[J] . (7.10) and its associated Green functions iNG(N)(x1, . . . , xN) = i δNZ[J] δJ(x1) . . . δJ(xN) J=0 (7.11) which correspond to connected Feynman diagrams. We will refer to G(n) as connected Green functions and to Z[J] as the generating functional for connected Green functions. The full information about the quantum ﬁeld theory is now encoded in the generating functional, so this is the primary object to compute. We begin by doing this in the simplest case, the free theory. 7.3 Generating functional and Green functions for free ﬁelds The Lagrangian density of a free, real scalar ﬁeld is given by L = 1 2(∂µφ∂µφ −m2φ2) and the Euklidean version reads LE = −1 2(¯ ∂µφ¯ ∂µφ + m2φ2). We write the kinetic term as Z d4¯ x ¯ ∂µφ(¯ x)¯ ∂µφ(¯ x) = Z d4¯ x d4¯ y φ(¯ y) ¯ ∂y µ ¯ ∂x µδ4(¯ x −¯ y)φ(¯ x) , (7.12) so that the Euklidean generating functional takes the form WE[J] = N Z Dφ exp −1 2 Z d4¯ x d4¯ y φ(¯ y)A(¯ y, ¯ x)φ(¯ x) + Z d4¯ x J(¯ x)φ(¯ x) . (7.13) with the operator A(¯ y, ¯ x) = ¯ ∂y µ ¯ ∂x µ + m2 δ4(¯ x −¯ y) . (7.14) This is a Gaussian path integral with a source J of precisely the type we have discussed in Section 1. From Eq. (1.42) we ﬁnd WE[J] = ˜ N exp 1 2 Z d4¯ x d4¯ y J(¯ y)∆E(¯ y −¯ x)J(¯ x) (7.15) 7.4. THE EFFECTIVE ACTION 87 with the inverse ∆E(¯ y −¯ x) = A−1(¯ y, ¯ x) and a suitable normalization ˜ N. How do we compute the inverse of the operator A? With the representation δ4(¯ x −¯ y) = R d4¯ k (2π)4 ei¯ k(¯ x−¯ y) of the delta function, we write A as a Fourier transform A(¯ y, ¯ x) = Z d4¯ k (2π)4 (¯ k2 + m2)ei¯ k(¯ x−¯ y) . (7.16) Then, we invert A by taking the inverse inside the Fourier transform, that is ∆E(¯ y −¯ x) = Z d4¯ k (2π)4 ei¯ k(¯ x−¯ y) 1 ¯ k2 + m2 . (7.17) Now, we would like to revert to Minkowski space with coordinates x and y. To obtain a Minkowski product in the exponent in Eq. (7.17) we also need to introduce the Minkowski momentum k = (k0, k) = (i¯ k0, ¯ k). By inserting all this into Eq. (7.15) we ﬁnd for the generating functional in Minkowski space W[J] = exp −1 2 Z d4x d4y J(x)∆F (x −y)J(y) , (7.18) where ∆F is the Feynman propagator precisely as introduces earlier (see Eq. (5.53). We have also chosen ˜ N = 1 so that W[J]\|J=0 = 1. Eq. (7.18) is the general result for the generating functional of the free scalar ﬁeld theory. All the free Green functions can now be calculated from Eq. (7.7) and we already know from Chapter 1 that the result can be obtained by applying Wick’s theorem. In particular, we have for the 2-point function G(2)(x, y) = − δ2W[J] δJ(x)δJ(y) J=0 = ∆F (x −y) . (7.19) Wick’s theorem appears both in the context of canonical quantisation (see Eq. (5.61)) and path integral quantisation. Applying it in either case shows that both types of Green functions are indeed identical, at least for the case of free ﬁelds. This conﬁrms that canonical and path integral quanitsation are equivalent. For later reference, we note that from Eqs. (7.18) and (7.10) the generating functional Z for free ﬁelds is given by Z[J] = i 2 Z d4x d4y J(x)∆F (x −y)J(y) . (7.20) 7.4 The effective action In Chapter 1 we have seen that the path integral formalism provides an intuitive picture for the transition between quantum and classical physics. We will now study this transition in more detail for the case of ﬁeld theories. To this end we deﬁne the classical ﬁeld φc by φc(x) = δZ[J] δJ(x) . (7.21) Then, from the deﬁnition (7.10) of the generating functional Z we have φc(x) = − i W[J] δW[J] δJ(x) = ⟨0\|ˆ φ(x)\|0⟩J ⟨0\|0⟩J , (7.22) where we have suggestively deﬁned the vacuum expectation values ⟨0\|0⟩J = W[J] and ⟨0\|ˆ φ(x)\|0⟩J = −iδW[J]/δJ(x) in the presence of the source J. Eq. (7.22) shows that φc is the suitably normalised vacuum expectation value of the ﬁeld operator ˆ φ, so its interpretation as a classical ﬁeld is sensible. Note that φc is a function of the source J. From Eq. (7.10), the generating functional Z is ”on the same footing” as the exponential in the path integral and can, hence, be seen as some sort of effective action. However, it still contains the effect of the source term J(x)φ(x) in Eq. (7.6). To remove this source term we deﬁne the effective action Γ by a Legendre transform Γ[φc] = Z[J] − Z d4x J(x)φc(x) . (7.23) Differentiating the left-hand side of this equation by δ/δJ(y) and using the deﬁnition (7.21) of the classical ﬁeld it follows that Γ is independent of the source J, as the notation suggests. 88 CHAPTER 7. PATH INTEGRALS IN QUANTUM FIELD THEORY To see that these deﬁnitions conform with our intuition, let us ﬁrst discuss free ﬁelds. In Eq. (7.20) we have calculated the generating functional Z for free ﬁelds and inserting this into Eq. (7.21) we ﬁnd for the classical ﬁeld φc(x) = i Z d4y ∆F (x −y)J(y) . (7.24) Since the Feynman propagator satisﬁes the equation (2 + m2)∆F (x) = −iδ4(x) (see Eq. (5.55)) we ﬁnd for the above classical ﬁeld that (2 + m2)φc(x) = J(x) , (7.25) so it is a solution to the Klein-Gordon equation with source J(x), as one would expect for a classical ﬁeld coupled to a source J. Inserting (7.20) and (7.24) into the effective action (7.23) it follows Γ[φc] = −1 2 Z d4x φc(x)J(x) = −1 2 Z d4x φc(x)(2 + m2)φc(x) = 1 2 Z d4x ∂µφc∂µφc −m2φ2 c (7.26) and, hence, the effective action coincides with the classical action as one would expect for a free theory. For interacting theories, the generating functional can of course not be calculated exactly. However, we can proceed to evaluate the path integral W[J] = N Z Dφ exp iS[φ] + i Z d4x J(x)φ(x) (7.27) in the saddle point approximation as discussed in Chapter 1. The solution φ0 to the classical equations of motion is determined from the classical equations of motion δS δφ(x)[φ0] = −J(x) . (7.28) Then, to leading order in the saddle point approximation we ﬁnd W[J] ∼exp iS[φ0] + i Z d4xJ(x)φ0(x) , Z[J] = S[φ0] + Z d4x J(x)φ0(x) . (7.29) With this result for Z and Eqs. (7.21), (7.23) and (7.28) we immediately conclude that φc = φ0 and Γ[φc] = S[φc]. Hence, in the lowest order saddle point approximation the effective action Γ is simply the classical action. Beyond this leading order, the effective action of course receives corrections due to quantum effects and differs from the classical action. This leads to a systematic approach to calculate these quantum corrections to the effective action the details of which are beyond the scope of the lecture. The above formalism also sheds light on another point which we have glossed over so far. Our discussion of spontaneous symmetry breaking in Chapter 4 has been carried in the context of classical ﬁelds and it has not been obvious what its status should be in the quantum theory. Spontaneous symmetry breaking of a quantum theory should be analysed using the above effective action (or, more precisely, the effective potential, which is the scalar potential of the effective action). Hence, we see that the results of Chapter 4 make sense in quantum theory, but have to be viewed as a leading order approximation. 7.5 Feynman diagrams from path integrals To develop perturbation theory in the path integral formalism we split the Lagrangian density as L = L0 + Lint into the free Lagrangian L0 and the interaction piece Lint. The free generating functional associated with L0 is denoted by W0[J] and the full generating functional by W[J]. We have W[J] = N Z Dφ exp i Z d4x (L0 + Lint + J(x)φ(x) = N exp i Z d4x Lint −i δ δJ(x) W0[J] . = N " 1 + ∞ X p=1 ip p! Z d4y1 . . . d4yp Lint −i δ δJ(y1) . . . Lint −i δ δJ(yp) # W0[J] , (7.30) 7.6. FURTHER READING 89 where N −1 = exp i Z d4x Lint −i δ δJ(x) W0[J] J=0 (7.31) to ensure that W[J]\|J=0 = 1. This is a perturbative series for the full generating functional in terms of the free one, W0[J]. We recall from Eq. (7.18) that the free generating functional is given by W0[J] = exp −1 2 Z d4x d4y J(x)∆F (x −y)J(y) , (7.32) so all the functional differentiations in Eq. (7.30) can be carried out explicitly and lead to Feynman propagators. The Green functions (7.7) can then written as G(N)(z1, . . . , zN) = N δ δJ(z1) . . . δ δJ(z1) × " 1 + ∞ X p=1 ip p! Z d4y1 . . . d4yp Lint −i δ δJ(y1) . . . Lint −i δ δJ(yp) # W0[J] J=0 We know from Chapter 1 (see Eq. (1.76) that this expression can be worked out using Wick’s theorem. The result is a sum over products of Feynman propagators, suitably integrated, and each term can be associated to a Feynman diagram. This is precisely the same structure as for the perturbative Green function (6.54) obtained from canonical quantization (after diagrams with disconnected vacuum bubbles are cancelled due to the normalization factor N). We have, therefore, explicitly veriﬁed that canonical and path integral approach lead to the same perturbative Green functions. From hereon, working out the Green functions explicitly and calculating decay rates and cross sections works exactly like in the canonical formalism: We calculate the space-time Green functions from the above formula and then derive the Fourier transformed and amputated Green functions from Eqs. (6.55) and (6.58), respectively. Is is clear that this also leads to the same set of Feynman rules, so there is no need to repeat their derivation. Compared to the hard work in the canonical approach it is remarkable how relatively easily the path integral formalism delivers the same results. 7.6 Further reading Path integrals in quantum ﬁeld theory and path integral quantization of the scalar ﬁeld theory are covered in most standard text books on quantum ﬁeld theory, including • P. Ramond, Field Theory: A Modern Primer • D. Bailin and A. Love, Introduction to Gauge Field Theory • S. Weinberg, The Quantum Theory of Fields, vol. 1 90 CHAPTER 7. PATH INTEGRALS IN QUANTUM FIELD THEORY Chapter 8 Many-Particle Quantum Systems There are two routes which take us to problems in many-particle quantum physics. One is to start from classical ﬁeld theory and quantise, as in Chapter 5 of these notes. The other is to start from one-body or few-body quantum mechanics, and consider the special aspects which become important when many particles are involved. Clearly, the ﬁrst route is the natural one to take if, for example, we want to begin with Maxwell’s equations and arrive at a description of photons. Equally, the second route is the appropriate one if, for instance, we want to begin with a model for liquid 4He and arrive at an understanding of superﬂuidity. In this chapter we will set out the second approach and illustrate it using applications from condensed matter physics. Although the problems we cover can all be formulated using functional integrals, we will use Hamiltonians, operators and operator transformations instead. This choice is made partly for simplicity, and partly in order to introduce a useful set of techniques. 8.1 Identical particles in quantum mechanics Many-particle quantum systems are always made up of many identical particles, possibly of several different kinds. Symmetry under exchange of identical particles has very important consequences in quantum mechanics, and the formalism of many-particle quantum mechanics is designed to build these consequences properly into the theory. We start by reviewing these ideas. Consider a system of N identical particles with coordinates r1, . . . rN described by a wavefunction ψ(r1 . . . rN). For illustration, suppose that the Hamiltonian has the form H = −ℏ2 2m N X i=1 ∇2 i + N X i=1 V (ri) + X i 1), is the reason for the factor (n1! n2! . . .)1/2 appearing on the right of Eq. (8.3). This choice anticipates what is necessary in order for boson creation and annihilation operators to have convenient commutation relations. Annihilation operators appear when we take the Hermitian conjugate of Eq. (8.3), obtaining ⟨0\| clN . . . cl2cl1. Let’s examine the effect of creation and annihilation operators when they act on various states. Since c† l \|0⟩is the state with coordinate wavefunction φl(r), we know that ⟨0\|cl c† l \|0⟩= 1, but for any choice of the state \|φ⟩other than the vacuum, c† l \|φ⟩contains more than one particle and hence ⟨0\|cl c† l \|φ⟩= 0. From this we can conclude that cl c† l \|0⟩= \|0⟩, demonstrating that the effect of cl is to remove a particle from the state \|nl=1⟩≡c† l \|0⟩. We also have for any \|φ⟩ the inner products ⟨0\|c† l \|φ⟩= ⟨φ\|cl \|0⟩= 0, and so we can conclude that cl \|0⟩= ⟨0\|c† l = 0 . 8.1.7 Commutation and anticommutation relations Recalling the factor of (±1)P in Eq. (8.4), we have for any \|φ⟩ c† l c† m\|φ⟩= ±c† mc† l \|φ⟩, where the upper sign is for bosons and the lower one for fermions. From this we conclude that boson creation operators commute, and fermion creation operators anticommute: that is, for bosons [c† l , c† m] = 0 and for fermions {c† l , c† m} = 0 , 94 CHAPTER 8. MANY-PARTICLE QUANTUM SYSTEMS where we use the standard notation for an anticommutator of two operators A and B: {A, B} = AB+BA. Taking Hermitian conjugates of these two equations, we have for bosons [cl, cm] = 0 and for fermions {cl, cm} = 0 . Note for fermions we can conclude that (cl )2=(c† l )2=0, which illustrates again how the Pauli exclusion principle is built into our approach. Finally, one can check that to reproduce the values of inner products of states appearing in Eq. (8.3), we require for bosons [cl , c† m] = δlm and for fermions {cl , c† m} = δlm . To illustrate the correctness of these relations, consider for a single boson orbital the value of \|[(c†)n\|0⟩]\|2. From Eq. (8.3) we have \|[(c†)n\|0⟩]\|2 = n!. Let’s recover the same result by manipulating commutators: we have ⟨0\|(c)n(c†)n\|0⟩ = ⟨0\|(c)n−1([c, c†] + c†c)(c†)n−1\|0⟩ = m⟨0\|(c)n−1(c†)n−1\|0⟩+ ⟨0\|(c)n−mc†(c)m(c†)n−1\|0⟩ = n⟨0\|(c)n−1(c†)n−1\|0⟩+ ⟨0\|c†(c)n(c†)n−1\|0⟩ = n(n −1) . . . (n −l)⟨0\|(c†)n−l(c)n−l\|0⟩ = n! ⟨0\|0⟩. Of course, manipulations like these are familiar from the theory of raising and lowering operators for the harmonic oscillator. 8.1.8 Number operators From Eq. (8.4) as the deﬁning equation for the action of creation operators in Fock space we have c† l \|n1 . . . nl . . .⟩= (±1)n1+...+nl−1√nl + 1\|n1 . . . nl + 1 . . .⟩, or zero for fermions if nl=1. Similarly, by considering the Hermitian conjugate of a similar equation, we have cl\|n1 . . . nl . . .⟩= (±1)n1+...+nl−1√nl\|n1 . . . nl −1 . . .⟩, or zero for both bosons and fermions if nl=0. In this way we have c† l cl \| . . . nl . . .⟩= nl\| . . . nl . . .⟩ where the possible values of nl are nl=0, 1, 2 . . . for bosons and nl=0, 1 for fermions. Thus the combination c† l cl , which we will also write as ˆ nl, is the number operator and counts particles in the orbital φl. 8.1.9 Transformations between bases In the context of single-particle quantum mechanics it is often convenient to make transformations between differ-ent bases. Since we used a particular set of basis functions in our deﬁnition of creation and annihilation operators, we should understand what such transformations imply in operator language. Suppose we have two complete, orthonormal sets of single-particle basis functions, {φl(r)} and {ρα(r)}. Then we can expand one in terms of the other, writing ρα(r) = X l φl(r)Ulα (8.5) 8.1. IDENTICAL PARTICLES IN QUANTUM MECHANICS 95 with Ulα = ⟨φl\|ρα⟩. Note that U is a unitary matrix, since (UU†)ml = X α ⟨φm\|ρα⟩⟨ρα\|φl⟩ = ⟨φm\|φl⟩ since X α \|ρα⟩⟨ρα\| = 1 = δml . Now let c† l create a particle in orbital φl(r), and let d† α create a particle in orbital ρα(r). We can read off from Eq. (8.5) an expression for d† α in terms of c† l : d† α = X l c† l Ulα . From the Hermitian conjugate of this equation we also have dα = X l U ∗ lαcl = X l (U†)αlcl . Effect of transformations on commutation relations We should verify that such transformations preserve commutation relations. For example, suppose that cl and c† l are fermion operators, obeying {cl , c† m} = δlm. Then {dα, d† β} = X lm U ∗ lαUmβ {cl , c† m} = (U†U)αβ = δαβ . Similarly, for boson operators commutation relations are preserved under unitary transformations. 8.1.10 General single-particle operators in second-quantised form To continue our programme of formulating many-particle quantum mechanics in terms of creation and annihilation operators, we need to understand how to transcribe operators from coordinate representation or ﬁrst-quantised form to so-called second-quantised form. In the ﬁrst instance, we examine how to do this for one-body operators – those which involve the coordinates of one particle at a time. An example is the kinetic energy operator. Suppose in general that A(r) represents such a quantity for a single-particle system. Then for a system of N particles in ﬁrst-quantised notation we have ˆ A = N X i=1 A(ri) . We want to represent ˆ A using creation and annihilation operators. As a ﬁrst step, we can characterise A(r) by its matrix elements, writing Alm = Z φ∗ l (r)A(r)φm(r)ddr . Then A(r)φm(r) = X l φl(r)Alm . (8.6) The second-quantised representation is ˆ A = X pq Apqc† pcq . (8.7) To justify this, we should verify that reproduces the correct matrix elements between all states from the Fock space. We will simply check the action of ˆ A on single particles states. We have ˆ A\|φm⟩= X pq Apqc† pcqc† m\|0⟩. Now, taking as an example bosons, c† pcqc† m\|0⟩= c† p([cq, c† m] + c† mcq)\|0⟩= c† pδqm\|0⟩ 96 CHAPTER 8. MANY-PARTICLE QUANTUM SYSTEMS so ˆ A\|φm⟩= X p \|φp⟩Apm , reproducing Eq. (8.6), as required. 8.1.11 Two-particle operators in second-quantised form It is important to make this transcription for two-body operators as well. Such operators depend on the coordinates of a pair of particles, an example being the two-body potential in an interacting system. Writing the operator in ﬁrst-quantised form as A(r1, r2), it has matrix elements which carry four labels: Almpq = Z φ∗ l (r1)φ∗ m(r2)A(r1, r2)φp(r2)φq(r1)ddr1ddr2 . Its second-quantised form is ˆ A ≡ X ij A(ri, rj) = X lmpq Almpqc† l c† mcpcq . (8.8) Again, to justify this one should check matrix elements of the second-quantised form between all states in Fock space. We will content ourselves with matrix elements for two-particle states, evaluating ⟨A⟩= ⟨0\|cycx ˆ Ac† ac† b\|0⟩ by two routes. In a ﬁrst-quantised calculation with ± signs for bosons and fermions, we have ⟨A⟩ = 1 2 Z Z [φ∗ x(r1)φ∗ y(r2) ± φ∗ x(r2)φ∗ y(r1)] · [A(r1, r2) + A(r2, r1)] · [φa(r1)φb(r2) ± φa(r2)φb(r1)]ddr1ddr2 = 1 2[Axyba ± Axyab + Ayxab ± Ayxba + Axyba ± Axyab + Ayxab ± Ayxba] = (Axyba + Ayxab) ± (Axyab + Ayxba) . (8.9) Using the proposed second-quantised form for ˆ A, we have ⟨A⟩= X lmpq Almpq⟨0\|cycxc† l c† mcpcqc† ac† b\|0⟩. We can simplify the vacuum expectation value of products of creation and annihilation operators such as the one appearing here by using the appropriate commutation or anticommutation relation to move annihilation operators to the right, or creation operators to the left, whereupon acting on the vacuum they give zero. In particular cpcqc† ac† b\|0⟩= (δaqδbp ± δapδbq)\|0⟩ and ⟨0\|cycxc† l c† m = ⟨0\|(δymδxl ± δylδxm) . Combining these, we recover Eq. (8.9). 8.2 Diagonalisation of quadratic Hamiltonians If a Hamiltonian is quadratic (or, more precisely, bilinear) in creation and annihilation operators we can diagonalise it, meaning we can reduce it to a form involving only number operators. This is an approach that applies directly to Hamiltonians for non-interacting systems, and also to Hamiltonians for interacting systems when interactions are treated within a mean ﬁeld approximation. 8.2. DIAGONALISATION OF QUADRATIC HAMILTONIANS 97 8.2.1 Number-conserving quadratic Hamiltonians Such Hamiltonians have the form H = X ij Hija† iaj . Note that in order for the operator H to be Hermitian, we require the matrix H to be Hermitian. Since the matrix H is Hermitian, it can be diagonalised by unitary transformation. Denote this unitary matrix by U and let the eigenvalues of H be εn. The same transformation applied to the creation and annihilation operators will diagonalise H. The details of this procedure are as follows. Let α† l = X i a† iUil . Inverting this, we have X α† l (U†)lj = a† j and taking a Hermitian conjugate X l Ujlαl = aj . Substituting for a†’s and a’s in terms of α†’s and α’s, we ﬁnd H = X lm α† l (U†HU)lmαm = X n εnα† nαn ≡ X n εnˆ nn . Thus the eigenstates of H are the occupation number eigenstates in the basis generated by the creation operators α† n. 8.2.2 Mixing creation and annihilation operators: Bogoliubov transformations There are a number of physically important systems which, when treated approximately, have bilinear Hamiltoni-ans that include terms with two creation operators, and others with two annihilation operators. Examples include superconductors, superﬂuids and antiferromagnets. These Hamiltonians can be diagonalised by what are known as Bogoliubov transformations, which mix creation and annihilation operators, but, as always, preserve commutation relations. We now illustrate these transformations, discussing fermions and bosons separately. Fermions Consider for fermion operators the Hamiltonian H = ǫ(c† 1c1 + c† 2c2) + λ(c† 1c† 2 + c2c1) , which arises in the BCS theory of superconductivity. Note that λ must be real for H to be Hermitian (more generally, with complex λ the second term of H would read λc† 1c† 2 + λ∗c2c1). Note as well the opposite ordering of labels in the terms c† 1c† 2 and c2c1, which is also a requirement of Hermiticity. The fermionic Bogoliubov transformation is c† 1 = ud† 1 + vd2 c† 2 = ud† 2 −vd1 , (8.10) where u and v are c-numbers, which we can in fact take to be real, because we have restricted ourselves to real λ. The transformation is useful only if fermionic anticommutation relations apply to both sets of operators. Let us suppose they apply to the operators d and d†, and check the properties of the operators c and c†. The coefﬁcients of the transformation have been chosen to ensure that {c† 1, c† 2} = 0, while {c† 1, c1} = u2{d† 1, d1} + v2{d† 2, d2} and so we must require u2 + v2 = 1, suggesting the parameterisation u = cos θ, v = sin θ. 98 CHAPTER 8. MANY-PARTICLE QUANTUM SYSTEMS The remaining step is to substitute in H for c† and c in terms of d† and d, and pick θ so that terms in d† 1d† 2+d2d1 have vanishing coefﬁcient. The calculation is perhaps clearest when it is set out using matrix notation. First, we can write H as H = 1 2 c† 1 c2 c† 2 c1     ǫ λ 0 0 λ −ǫ 0 0 0 0 ǫ −λ 0 0 −λ −ǫ          c1 c† 2 c2 c† 1     + ǫ where we have used the anticommutator to make substitutions of the type c†c = 1 −c c†. For conciseness, consider just the upper block c† 1 c2 ǫ λ λ −ǫ c1 c† 2 and write the Bogoliubov transformation also in matrix form as c1 c† 2 cos θ sin θ −sin θ cos θ d1 d† 2 . We pick θ so that cos θ −sin θ sin θ cos θ ǫ λ λ −ǫ cos θ sin θ −sin θ cos θ = ˜ ǫ 0 0 −˜ ǫ , where ˜ ǫ = √ ǫ2 + λ2. Including the other 2 × 2 block of H, we conclude that H = ˜ ǫ(d† 1d1 + d† 2d2) + ǫ −˜ ǫ . Bosons The Bogoliubov transformation for a bosonic system is similar in principle to what we have just set out, but different in detail. We are concerned with a Hamiltonian of the same form, but now written using boson creation and annihilation operators: H = ǫ(c† 1c1 + c† 2c2) + λ(c† 1c† 2 + c2c1) . We use a transformation of the form c† 1 = ud† 1 + vd2 c† 2 = ud† 2 + vd1 . Note that one sign has been chosen differently from its counterpart in Eq. (8.10) in order to ensure that bosonic commutation relations for the operators d and d† imply the result [c† 1, c† 2] = 0. We also require [c1, c† 1] = u2[d1, d† 1] −v2[d2, d† 2] = 1 and hence u2 −v2 = 1. The bosonic Bogoliubov transformation may therefore be parameterised as u = cosh θ, v = sinh θ. We can introduce matrix notation much as before (but note some crucial sign differences), with H = 1 2 c† 1 c2 c† 2 c1     ǫ λ 0 0 λ ǫ 0 0 0 0 ǫ λ 0 0 λ ǫ          c1 c† 2 c2 c† 1     −ǫ , where for bosons we have used the commutator to write c†c = c c† −1. Again, we focus on one 2 × 2 block c† 1 c2 ǫ λ λ ǫ c1 c† 2 8.3. DENSITY CORRELATIONS IN IDEAL QUANTUM GASES 99 and write the Bogoliubov transformation also in matrix form as c1 c† 2 u v v u d1 d† 2 . Substituting for c and c† in terms of d and d†, this block of the Hamiltonian becomes d† 1 d2 u v v u ǫ λ λ ǫ u v v u d1 d† 2 . In the fermionic case the matrix transformation was simply an orthogonal rotation. Here it is not, and so we should examine it in more detail. We have u v v u ǫ λ λ ǫ u v v u = ǫ[u2 + v2] + 2λuv 2ǫuv + λ[u2 + v2] 2ǫuv + λ[u2 + v2] ǫ[u2 + v2] + 2λuv . It is useful to recall the double angle formulae u2 + v2 = cosh 2θ and 2uv = sinh 2θ. Then, setting tanh 2θ = −λ/ǫ we arrive at H = ˜ ǫ(d† 1d1 + d† 2d2) −ǫ + ˜ ǫ . with ˜ ǫ = p ǫ2 −λ2. (8.11) Note that in the bosonic case the transformation requires ǫ > λ: if this is not the case, H is not a Hamiltonian for normal mode oscillations about a stable equilibrium, but instead represents a system at an unstable equilibrium point. 8.3 Density correlations in ideal quantum gases Having set up the machinery, we now apply it to some problems of physical interest. One of the simplest is a calculation of particle correlations in an ideal quantum gas. These correlations arise purely from quantum statistics, since in an ideal gas there are no interactions between particles and therefore no correlations at all at a classical level. Consider N identical particles in a three-dimensional cubic box of side L with periodic boundary conditions. The single particle eigenstates are plane waves: we write φk(r) = 1 L3/2 eik·r with k = 2π L (l, m, n) , l, m, n integer . Introducing creation operators c† k for particles in these orbitals, the creation operator for a particle at the point r is c†(r) = 1 L3/2 X k e−ik·rc† k . With particle coordinates denoted by ri, the density operator in ﬁrst-quantised form is ρ(r) N X i=1 δ(r −ri) . In second quantised form it is simply the number operator at the point r, ρ(r) = c†(r)c(r) = 1 L3 X kq ei(q−k)·rc† kcq , as is conﬁrmed by using Eq. (8.7). We will calculate ⟨ρ(r)⟩and ⟨ρ(r)ρ(0)⟩. As a prelude to the quantum calculation, it is worth considering the problem classically. In a classical system the meaning of the average ⟨. . .⟩is a normalised multiple integral of all particle coordinates over the volume of the box. Hence ⟨ρ(r)⟩= N L3 Z d3ri δ(r −ri) = N L3 100 CHAPTER 8. MANY-PARTICLE QUANTUM SYSTEMS and ⟨ρ(r)ρ(0)⟩ = N L3 Z d3ri δ(r −ri)δ(ri) + N(N −1) L6 Z Z d3rid3rj δ(r −ri)δ(rj) = ⟨ρ(r)⟩2 + N L3 δ(r) −1 L3 . In the large volume limit we have simply ⟨ρ(r)ρ(0)⟩= ⟨ρ(r)⟩2 + ⟨ρ(r)⟩δ(r) . (8.12) Now we move to the quantum calculation, taking the average ⟨. . .⟩to mean an expectation value in number eigenstates for the orbitals φk(r), weighted by Boltzmann factors at ﬁnite temperature. To evaluate ⟨ρ(r)⟩we need ⟨c† kcq⟩= δk,q⟨nk⟩, where nk is the number operator for the orbital φk(r). Since P k⟨nk⟩= N, we ﬁnd ⟨ρ(r)⟩= N L3 as for a classical system. The two-point correlation function is more interesting. To evaluate ⟨ρ(r)ρ(0)⟩we need averages of the form ⟨c† kcqc† l cp⟩. These are non-zero in two cases: (i) k = q and l = p; or (ii) k = p and l = q, with q ̸= p to prevent double-counting of terms included under (i). With p ̸= q we have ⟨c† qcqc† pcp⟩= ⟨nqnp⟩, and ⟨c† pcqc† qcp⟩= ⟨np(1 ± nq)⟩, where the upper sign is for bosons and the lower one for fermions. From this ⟨ρ(r)ρ(0)⟩ = 1 L6 X kqlp ei(q−k)·r⟨c† kcqc† l cp⟩ = 1 L6 X qp ⟨nq⟩⟨np⟩+ 1 L6 X qp ei(q−p)·r⟨np⟩(1 ± ⟨nq⟩) + 1 L6 X k ⟨n2 k⟩−⟨nk⟩2 −⟨nk⟩(1 ± ⟨nk⟩) . The ﬁnal term is negligible in the limit L →∞with N/L3 ﬁxed, and we discard it. In this limit we can also make the replacement 1 L3 X p → 1 (2π)3 Z d3p so that we obtain ﬁnally ⟨ρ(r)ρ(0)⟩= ⟨ρ(r)⟩2 + ⟨ρ(r)⟩δ(r) ± 1 (2π)3 Z d3p⟨np⟩eip·r 2 . (8.13) The ﬁnal term on the right-hand side of Eq. (8.13) is the correction to our earlier classical result, Eq. (8.12), and represents the consequences of quantum statistics. Its detailed form as a function of r depends on the momentum distribution of particles, but the most important features are quite general: there is an enhancement in density correlations for bosons and a suppression for fermions, on a lengthscale of order ℏ/∆p, where ∆p is a characteristic momentum for the gas. In summary: bosons bunch and fermions exclude. 8.4 Spin waves in a ferromagnet We move now to a problem that illustrates how one can make approximations in order to obtain a simple description of excitations in an interacting system. The model we study is the quantum Heisenberg ferromagnet in the limit where the spin magnitude S is large. It represents an insulating ferromagnetic material, in which magnetic ions with well-deﬁned magnetic moments occupy the sites of a lattice. The three components of spin at site r are represented by operators Sx r , Sy r and Sz r. Their commutation relations are the standard ones, and with ℏ= 1 take the form [Si r1, Sj r2] = iδr1,r2ǫijkSk r1 . 8.4. SPIN WAVES IN A FERROMAGNET 101 We reproduce them in order to emphasise two points: ﬁrst, they are more complicated than those for creation and annihilation operators, since the commutator is itself another operator and not a number; and second, spin operators acting at different sites commute. We will also make use of spin raising and lowering operators, deﬁned in the usual way as S+ = Sx + iSy and S−= Sx −iSy. The Heisenberg Hamiltonian with nearest neighbour ferromagnetic exchange interactions of strength J is H = −J X ⟨rr′⟩ Sr · Sr′ ≡−J X ⟨rr′⟩ Sz rSz r′ + 1 2 S+ r S− r′ + S− r S+ r′ . (8.14) Here P ⟨rr′⟩denotes a sum over neighbouring pairs of sites on the lattice, with each pair counted once. Thinking of the spins as classical, three component vectors with length S, the lowest energy states are ones in which all spins are parallel. The model is simple enough that we can write down the exact quantum ground states as well: for the case in which the spins are aligned along the positive z-axis, the ground state \|0⟩is deﬁned by the property Sz r\|0⟩= S\|0⟩for all r. Other ground states can be obtained by a global rotation of spin direction. Individual ground states in both the classical and quantum descriptions break the rotational symmetry of the Hamiltonian, and so we expect excitations which are Goldstone modes and therefore gapless. There are known as spin waves or magnons. Eigenstates with a single magnon excitation can in fact be found exactly, but in general to go further we need to make approximations. As a next step, we set out one approximation scheme. 8.4.1 Holstein Primakoff transformation This transformation expresses spin operators in terms of boson operators. It provides an obvious way to build in the fact that spin operators at different sites commute. In a non-linear form it also reproduces exactly the commutation relations between two spin operators associated with the same site, but we will use a linearised version of the transformation which is approximate. At a single site we take the eigenvector of Sz with eigenvalue S to be the boson vacuum, and associate each unit reduction in Sz with the addition of a boson. Then Sz = S −b†b . From this we might guess S+ ∝b and S−∝b†. In an attempt to identify the proportionality constants we can compare the commutator [S+, S−] = 2Sz with [b, b†] = 1. Since the commutator is an operator in the ﬁrst case and a number in the second, our guessed proportionality cannot be exact, but within states for which ⟨Sz⟩≈S (meaning ⟨Sz⟩−S ≪S, which can be satisﬁed only if S ≫1) we can take S+ ≈(2S)1/2b and S−≈(2S)1/2b† . (8.15) In an exact treatment, corrections to these expressions form a series in powers of b†b/S. Using this transformation and omitting the higher order terms, the Hamiltonian may be rewritten approximately as H = −J X ⟨rr′⟩ S2 −JS X ⟨rr′⟩ h b† rbr′ + b† r′br −b† rbr −b† r′br′ i . (8.16) 8.4.2 Approximate diagonalisation of Hamiltonian Applying the approach of Section 8.2.1, we can diagonalise Eq. (8.16) by a unitary transformation of the creation and annihilation operators. In a translationally invariant system this is simply a Fourier transformation. Suppose the sites form a simple cubic lattice with unit spacing. Take the system to be a cube with with side L and apply periodic boundary conditions. The number of lattice sites is then N = L3 and allowed wavevectors are k = 2π L (l, m, n) with l, m, n integer and 1 ≤l, m, n ≤L . Boson operators in real space and reciprocal space are related by br = 1 √ N X k eik·r b k and b† r = 1 √ N X k e−ik·r b† k . We use these transformations, and introduce the notation d for vectors from a site to its nearest neighbours, and z for the coordination number of the lattice (the number of neighbours to a site: six for the simple cubic lattice), to 102 CHAPTER 8. MANY-PARTICLE QUANTUM SYSTEMS obtain H = −JS2Nz −JS X rd X kq 1 N eir·(k−q)[eid·q −1]b† qbk = −JS2Nz −JS X q ǫqb† qbq , where ǫq = 2JS(3 −cos qx −cos qy −cos qz) . In this way we have approximated the original Heisenberg Hamiltonian, involving spin operators, by one that is quadratic in boson creation and annihilation operators. By diagonalising this we obtain an approximate description of the low-lying excitations of the system as independent bosons. The most important feature of the result is the form of the dispersion a small wavevectors. For q ≪1 we have ǫq = JSq2 + O(q4), illustrating that excitations are indeed gapless. The fact that dispersion is quadratic, and not linear as it is, for example for phonons, reﬂects broken time-reversal symmetry in the ground state of the ferromagnet. 8.5 Weakly interacting Bose gas As a ﬁnal example, we present a treatment of excitations in a Bose gas with repulsive interactions between particles, using an approximation that is accurate if interactions are weak. There is good reason for wanting to understand this problem in connection with the phenomenon of superﬂuidity: the ﬂow of Bose liquids without viscosity below a transition temperature, as ﬁrst observed below 2.1 K in liquid 4He. Indeed, an argument due to Landau connects the existence of superﬂuidity with the form of the excitation spectrum, and we summarise this argument next. 8.5.1 Critical superﬂuid velocity: Landau argument Consider superﬂuid of mass M ﬂowing with velocity v, and examine whether friction can arise by generation of excitations, characterised by a wavevector k and an energy ǫ(k). Suppose production of one such excitation reduces the bulk velocity to v −∆v. From conservation of momentum Mv = Mv −M∆v + ℏk and from conservation of energy 1 2Mv2 = 1 2M\|v −∆v\|2 + ǫ(k) . From these conditions we ﬁnd at large M that k, v and ǫ(k) should satisfy ℏk · v = ǫ(k). The left hand side of this equation can be made arbitrarily close to zero by choosing k to be almost perpendicular to k, but it has a maximum for a given k, obtained by taking k parallel to v. If ℏkv < ǫ(k) for all k then the equality cannot be satisﬁed and frictional processes of this type are forbidden. This suggests that there should be a critical velocity vc for superﬂuid ﬂow, given by vc = mink[ǫ(k)/k]. For vc to be non-zero, we require a real, interacting Bose liquid to behave quite differently from the non-interacting gas, since without interactions the excitation energies are just those of individual particles, giving ǫ(k) = ℏ2k2/2m for bosons of mass m, and hence vc = 0. Reassuringly, we ﬁnd from the following calculation that interactions have the required effect. For completeness, we should note also that while a critical velocity of the magnitude these arguments suggest is observed in appropriate experiments, in others there can be additional sources of friction that lead to much lower values of vc. 8.5.2 Model for weakly interacting bosons There are two contributions to the Hamiltonian of an interacting Bose gas: the single particle kinetic energy HKE and the interparticle potential energy Hint. We introduce boson creation and annihilation operators for plane wave states in a box with side L, as in Section 8.3. Then HKE = X k ℏ2k2 2m c† kck . 8.5. WEAKLY INTERACTING BOSE GAS 103 Short range repulsive interactions of strength parameterised by u are represented in ﬁrst-quantised form by Hint = u 2 X i̸=j δ(ri −rj) . Using Eq. (8.8) this can be written as Hint = u 2L3 X kpq c† kc† pcqck+p−q . With this, our model is complete, with a Hamiltonian H = HKE + Hint. 8.5.3 Approximate diagonalisation of Hamiltonian In order to apply the techniques set out in Section 8.2.1 we should approximate H by a quadratic Hamiltonian. The approach to take is suggested by recalling the ground state of the non-interacting Bose gas, in which all particles occupy the k = 0 state. It is natural to suppose that the occupation of this orbital remains macroscopic for small u, so that the ground state expectation value ⟨c† 0c0⟩takes a value N0 which is of the same order as N, the total number of particles. In this case we can approximate the operators c† 0 and c0 by the c-number √N0 and expand H in decreasing powers of N0. We ﬁnd Hint = uN 2 0 2L3 + uN0 2L3 X k̸=0 h 2c† kck + 2c† −kc−k + c† kc† −k + ckc−k i + O([N0]0) . At this stage N0 is unknown, but we can write an operator expression for it, as N0 = N − X k̸=0 c† kck . It is also useful to introduce notation for the average number density ρ = N/L3. Substituting for N0 we obtain Hint = uρ 2 N + uρ 2 X k̸=0 h c† kck + c† −kc−k + c† kc† −k + ckc−k i + O([N0]0) and hence H = uρ 2 N + 1 2 X k̸=0 h E(k) c† kck + c† −kc−k + uρ c† kc† −k + ckc−k i + . . . (8.17) with E(k) = ℏ2k2 2m + uρ . At this order we have a quadratic Hamiltonian, which we can diagonalise using the Bogoliubov transformation for bosons set out in Section 8.2.2. From Eq. (8.11), we ﬁnd that the dispersion relation for excitations in the Bose gas is ǫ(k) = "ℏ2k2 2m + uρ 2 −(uρ)2 #1/2 . At large k (ℏ2k2/2m ≫uρ), this reduces to the dispersion relation for free particles, but in the opposite limit it has the form ǫ(k) ≃ℏvk with v = ruρ m . In this way we obtain a critical velocity for superﬂuid ﬂow, which is proportional to the interaction strength u, illustrating how interactions can lead to behaviour quite different from that in a non-interacting system. 104 CHAPTER 8. MANY-PARTICLE QUANTUM SYSTEMS 8.6 Further Reading • R. P. Feynman Statistical Mechanics (Addison Wesley). Chapter 6 provides a straightforward introduction to the use of particle creation and annihilation operators. • A. Altland and B. D. Simons Condensed Matter Field Theory (CUP). Chapters 1 and 2 offer a good overview of the material covered in this part of the course. • J-P Blaizot and G. Ripka Quantum Theory of Finite Systems (MIT Press) is a useful, clear and complete advanced reference book. Chapter 9 Phase Transitions In this chapter we will examine how the statistical mechanics of phase transitions can be formulated using the language of ﬁeld theory. As in relativistic applications, symmetry will be an important guide, but since phenomena in a condensed matter setting are not constrained by Lorentz invariance, we encounter a variety of new possibilities. 9.1 Introduction To provide a context, we start by summarising some basic facts about phase transitions. Take ﬁrst a substance such as water, that can exist as a solid, liquid or vapour. Consider its phase diagram in the plane of temperature and pressure, as sketched in Fig. 9.1. The solid is separated from both liquid and vapour by phase boundaries, and one c T p p T vapour liquid solid c Figure 9.1: Schematic phase diagram in temperature T and pressure p for a material with solid, liquid and vapour phases. Line denote phase boundaries; the liquid-vapour critical point has coordinates (Tc, pc). cannot get from the solid to one of these other phases without crossing the phase boundary. By contrast, the liquid-vapour phase boundary ends at a critical point, and while some paths between the liquid and vapour cross this phase boundary, others do not. On crossing a phase boundary in this phase diagram, the system undergoes a discontinuous phase transition: properties such as density change discontinuously and there is a latent heat. On the other hand, if one follows a path between liquid and vapour that avoids the phase boundary by going around the critical point, properties vary smoothly along the path. As an intermediate case, we can consider a path between liquid and vapour that goes through the critical point. This turns out to involve a continuous (but sharp) phase transition, and it is partly behaviour at such critical points that will be focus of this chapter. To emphasise the distinction between discontinuous, or ﬁrst order, transitions and continuous ones, it is useful to compare the behaviour of the heat capacity as a function of temperature in each case, as sketched in Fig. 9.2. At a ﬁrst order transition there is a 105 106 CHAPTER 9. PHASE TRANSITIONS latent heat, represented as a δ-function spike in the heat capacity, while at a continuous transition there is no latent heat, but the heat capacity shows either a cusp or a divergence, depending on the speciﬁc example. C T C T Figure 9.2: Schematic dependence of heat capacity on temperature: at a pressure p < pc (left); and at p = pc (right). An alternative way of viewing the liquid-vapour transition is to examine behaviour as a function of density rather than pressure. A phase diagram of this kind is shown in Fig. 9.3. From this viewpoint the consequence of the ﬁrst-order transition is that at temperatures T < Tc there is intermediate range of densities which the system can attain only as a mixture of two distinct, coexisting phases. One of these phases is the liquid, which has a higher density than that of the system on average, and the other is the vapour, with lower density than average. As the critical point is approached, the density difference between liquid and vapour reduces. Near the critical point, as the two phases become more similar, microscopic density ﬂuctuations grow in size: roughly speaking, such ﬂuctuations involve microscopic regions of vapour appearing within the liquid, or vice-versa. The correlation length represents the maximum size of such ﬂuctuations, and diverges at the critical point. Sufﬁciently close to the critical point it is larger than the wavelength of light, and in these circumstance density ﬂuctuations scatter light strongly, leading to a phenomenon known as critical opalescence: a cloudiness that appears in ﬂuids close to their critical point. c T ρ vapour liquid T Figure 9.3: Phase diagram in the density-temperature plane. The region of two-phase coexistence is shaded; elsewhere the system exists in a single phase. 9.1.1 Paramagnet-ferromagnet transition An important feature of our understanding of phase transitions is that there are close parallels between different examples. To illustrate this we next examine the transition between the paramagnetic and ferromagnetic phases of a magnetic material. The counterpart to the liquid-vapour phase diagram is shown in Fig. 9.4 (left), where we display 9.1. INTRODUCTION 107 behaviour as a function of magnetic ﬁeld (in place of pressure) and temperature. The properties of the system, including most importantly its magnetisation, vary smoothly with applied ﬁeld above the critical temperature. Below the critical temperature, and neglecting hysteresis effects, there is a discontinuous change on sweeping applied ﬁeld through zero, represented by a ﬁrst-order phase boundary in the ﬁgure. In this case there is a symmetry under reversal of ﬁeld and magnetisation that was not evident for the liquid vapour transition. Also shown in Fig. 9.4 are the two-phase coexistence region (centre) and the behaviour of the magnetic susceptibility (right). The latter diverges at the critical point: the divergence again reﬂects the presence of large ﬂuctuations in the critical region, which are readily polarised by an applied ﬁeld. c T H T T M c T χ T Figure 9.4: Left: phase diagram for a ferromagnet in the plane of ﬁeld H vs temperature T , with Tc the critical point. Centre: the region of two-phase coexistence, in the plane of magnetisation M vs temperature. Right: behaviour of magnetic susceptibility χ. 9.1.2 Other examples of phase transitions Other examples of continuous phase transitions include the normal to superﬂuid transition in liquid 4He, and the superconducting transition in many metals and alloys. A further example, in which the transition is (for reasons we will examine in due course) ﬁrst order, occurs in liquid crystals consisting of rod-like molecules. In their isotropic phase, these molecules are randomly orientated, while in the nematic phase they acquire an average orientation, despite remaining liquid and therefore positionally disordered. 9.1.3 Common features The most important common feature of these phase transitions is the occurrence of spontaneous symmetry break-ing. This is rather obscured in the case of the liquid-vapour transition, but is clear in our other examples. For instance, the ferromagnet might equally well acquire a magnetisation parallel or antiparallel to a given axis in the sample, and the two possibilities are related by time-reversal symmetry. In superﬂuids and superconductors, the condensate has a deﬁnite quantum-mechanical phase, whose value relative to that of another condensate can be probed if two condensates are coupled. And in a nematic liquid crystal the spontaneous choice of an orientation axis for molecules breaks the rotational symmetry displayed by the isotropic phase. We characterise symmetry breaking, both in magnitude and (in a generalised sense) orientation using an order parameter. For the ferromag-net, this is simply the magnetisation, m. For the superﬂuid, it can be taken to be the condensate amplitude ψ. And for the liquid-vapour transition, we take it to be the difference δρ = ρ −ρc in density ρ from its value ρc at the critical point. A second feature common to continuous transitions is the existence of a lengthscale, the correlation length ξ, that diverges as the critical point is approached. For example, at the liquid-vapour transition density ﬂuctuations are correlated over distances of order ξ and the correlation function ⟨δρ(0)δρ(r)⟩falls to zero for r ≫ξ. 9.1.4 Critical behaviour As a critical point is approached, many physical quantities show power-law behaviour that can be characterised by giving values of critical exponents. Their values turn out to be universal in the sense that they are determined by symmetries and a few other features of the system, but are insensitive to many microscopic details. The symbols used for the exponents are established by convention, and we now introduce them using the transition from param-agnet to a ferromagnet as an example. It is convenient to discuss behaviour as a function of reduced temperature, 108 CHAPTER 9. PHASE TRANSITIONS deﬁned as t = T −Tc Tc . The cusp or divergence in the heat capacity C is represented by the exponent α, via C ∼\|t\|−α. The way in which the order parameter m decreases as the critical point is approached is described by the exponent β, with m ∼\|t\|β for t < 0, while the divergences in the susceptibility χ and the correlation length ξ are represented as χ ∼\|t\|−γ and ξ ∼\|t\|−ν. Finally, at the critical temperature, the variation in the order parameter with ﬁeld is written as m ∼\|h\|1/δ. 9.2 Field theory for phase transitions There is a general approach to writing down a ﬁeld theory, or continuum description, of a phase transitions. The ﬁrst step is to identify an order parameter that characterises the nature of symmetry breaking at the transition. The value of the order parameter is determined by an average over a system as a whole. We introduce a ﬁeld, deﬁned as a function of position within the sample, that takes values in the order parameter space, and write the free energy density of the system as an expansion in powers of this ﬁeld and its gradients, including all terms allowed by the symmetries of the high symmetry phase. We calculate the free energy associated with the degrees of freedom involved in the transition as a functional integral over all ﬁeld conﬁgurations. More explicitly, writing the order parameter ﬁeld at point r as ϕ(r), and the free energy density as F(ϕ), we deﬁne a partition function Z = Z Dϕ e−R F (ϕ) ddr and obtain the free energy for the sample from this partition function in the usual way, as −kBT ln Z. To take a speciﬁc example, consider again the paramagnet-ferromagnet transition, with a real, scalar order parameter ﬁeld ϕ(r) that represents the local magnetisation (which we take to have orientations only parallel or antiparallel to a preferred crystal axis). Time-reversal takes ϕ(r) to −ϕ(r), and we expect this and also spatial inversion to be symmetries. As a result, an expansion of the free energy density should contain only even powers of ϕ(r) (apart from a linear coupling to an applied magnetic ﬁeld h) and only even order derivatives of ϕ(r). Hence we have F(ϕ) = a 2ϕ2 + b 4ϕ4 + 1 2\|∇ϕ\|2 + . . . −hϕ . (9.1) What can we say about the coefﬁcients a, b, . . . that appear in this expansion? First, if it makes sense to truncate the expansion at order ϕ4, we must have b > 0, so that F(ϕ) is bounded below. (Alternatively, if b < 0, we would need to include a term in ϕ6.) Secondly, we note that the minimum of F(ϕ) is at ϕ = 0 for a > 0, and at non-zero ϕ for a < 0. This suggests that, to describe a phase transition, a should vary with temperature and change sign in the vicinity of the critical point. We therefore postulate that a = At, with t the reduced temperature and A an unimportant coefﬁcient. 9.2.1 Mean ﬁeld theory: the saddle-point approximation Next, we should face the problem of evaluating the functional integral over ﬁeld conﬁgurations. In general this is difﬁcult and one can only hope to make progress by using approximations. The simplest approximation is a saddle-point one, which turns out to be equivalent to mean ﬁeld theory. Starting from Eq. (9.1), we have from δF δϕ = 0 the saddlepoint equation −∇2ϕ(r) + aϕ(r) + bϕ3(r) −h = 0 . (9.2) Unless boundary conditions impose a spatially-varying solution, we expect ϕ to be independent of r. Then with h = 0 Eq. (9.2) has the solutions ϕ = 0 and ϕ = ± r −a b . The ﬁrst is the minimum of F(ϕ) for a > 0 and a maximum for a < 0; the other solutions are real only for a < 0 when they are minima. From this we can conclude that the order parameter varies as ϕ ∝\|t\|1/2 for t < 0 . 9.2. FIELD THEORY FOR PHASE TRANSITIONS 109 Thus the critical exponent β takes the value β = 1/2. At the critical point ϕ is non-zero only for non-zero h, with the dependence ϕ ∝\|h\|1/3 . From this we recognise the critical exponent value δ = 3. To evaluate the susceptibility χ ≡∂ϕ/∂h we should consider Eq. (9.2) for non-zero h. Rather than attempting to ﬁnd the solution for ϕ explicitly as a function of h, it is more convenient to differentiate the equation directly, giving (a + 3bϕ2)∂ϕ ∂h = 1 and hence χ =    1 At t > 0 − 1 2At t < 0 Thus the exponent γ takes the value γ = 1. 9.2.2 Correlation function To evaluate the correlation function ⟨ϕ(0)ϕ(r)⟩we need to go beyond the saddle-point treatment. We will consider ﬂuctuations within a Gaussian approximation, taking a > 0 for simplicity. Within this approximation, we would like to evaluate ⟨ϕ(0)ϕ(r)⟩= R Dϕ ϕ(0)ϕ(r) e− R ddr a 2 ϕ2+ 1 2 \|∇ϕ\|2 R Dϕ e− R ddr a 2 ϕ2+ 1 2 \|∇ϕ\|2 . We do so by diagonalising the quadratic form in the argument of the exponentials: since our system is translation-invariant, this is done by Fourier transform. To be explicit about the Fourier transforms, we consider a system of linear size L in each direction with periodic boundary conditions. Then we can introduce wavevectors k = 2π L (l1, . . . ld) with li’s integer, and write ϕ(r) = 1 Ld/2 X k ϕkeik·r and ϕk = 1 Ld/2 Z ddre−ik·r . In these terms we have Z ddr ϕ2(r) = X k ϕkϕ−k and Z ddr \|∇ϕ(r)\|2 = X k k2ϕkϕ−k . Also, we note from the deﬁnition of ϕk that ϕ∗ k = ϕ−k, so that, as independent quantities we can take the real and imaginary parts of ϕk for one half of all wavevectors – say those with component k1 > 0. This means we can write the functional integration as multiple integrals over just these components, with Z Dϕ = Y k1>0 Z dReϕk Z dImϕk . This leads us to the result ⟨ϕkϕq⟩= δk+q,0 1 (a + k2) , and hence ⟨ϕ(0)ϕ(r)⟩= 1 Ld X k eik·r (a + k2) ∼ 1 (2π)d Z ddk eik·r (a + k2) For general d one ﬁnds that this correlation function falls off on a scale set by the correlation length ξ given by ξ−2 = a. In d = 3 the integral gives ⟨ϕ(0)ϕ(r)⟩= 1 2πr e−r/ξ . Since we have made the connection ξ ∝t−1/2, we have obtained the value for another critical exponent: ν = 1/2. 110 CHAPTER 9. PHASE TRANSITIONS 9.2.3 Consequences of symmetry Without symmetry to exclude odd powers of the order parameter, we would have an expression for the free energy density of the form F(ϕ) = a 2ϕ2 + c 3ϕ3 + b 4ϕ4 + 1 2\|∇ϕ\|2 + . . . , (9.3) where we have omitted a linear term by choosing to expand around a minimum, but must allow all higher powers. In this case, the transition is generically a discontinuous one, as is seen most easily by considering the graphs of F(ϕ) shown in Fig. 9.5. −0.5 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 −0.5 0 0.5 1 1.5 2 2.5 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 9.5: Sketch of Eq. (9.3). Left: for a large and positive. Right: for a small and positive. The minimum of F(ϕ) jumps discontinuously from ϕ = 0 to a ﬁnite value of ϕ as a is reduced, representing a ﬁrst-order transition. 9.2.4 Other examples of phase transitions It is instructive to examine how one identiﬁes an order parameter and constructs the continuum theory appropriate for other examples of phase transitions. Liquid crystals: the isotropic-nematic transition Liquid crystals are ﬂuids of rod-like molecules. As ﬂuids, the positions of the molecules are not ordered, and in the isotropic phase their orientations are also disordered. In the nematic phase, however, molecules spontaneously align along a common axis. To describe the transition between these two states we should identify a suitable order parameter and write down a free energy expansion using the symmetry of the isotropic phases to guide us. Suppose ˆ n is a unit vector aligned with a molecule. It is tempting to use the average ⟨ˆ n⟩itself as an order parameter, as we might if it were magnetisation. For the liquid crystal, however, this would not be correct, since the molecules are invariant under inversion, which takes ˆ n to −ˆ n. To construct something from ˆ n that has the same invariance under inversion as the molecule itself, we ﬁrst consider the tensor ninj. This is still not quite ideal, since it is non-zero for an isotropically distributed ˆ n (taking the value ⟨ninj⟩= 1 3δij). Our ﬁnal choice for an order parameter is therefore the tensor Qij = ⟨ninj⟩−1 3δij . We expect it to vanish in the isotropic phase. By contrast, if molecules are fully aligned, say along the ˆ z direction, then Q =   −1 3 0 0 0 −1 3 0 0 0 2 3  . The isotropic phase is symmetric under rotations of the molecules, with translates to rotations of the tensor Q. Thus the free energy density F(Q) should be invariant Q →RQR−1, a property of traces of all powers of Q. We therefore expect F(Q) = a 2Tr[Q2] + c 3Tr[Q3] + b 4Tr[Q4] + . . . . A signiﬁcant feature of this expression is that it contains a cubic term. As a consequence, we expect the isotropic-nematic transition to be ﬁrst-order, as it indeed is experimentally. It is a signiﬁcant achievement to have reached an understanding of the nature of this transition, using only its symmetries. 9.3. CONSEQUENCES OF FLUCTUATIONS 111 The superﬂuid transition To describe the transition (in, for example, liquid 4He) between the normal and superﬂuid states, we use the condensate wavefunction amplitude ψ as an order parameter. It is a complex scalar, and F(ψ) should be invariant under changes in its phase: ψ →eiθψ, with θ real. Thus we have F(ψ) = a 2\|ψ\|2 + b 4\|ψ\|4 + 1 2\|∇ψ\|2 + . . . . The superconducting transition The transition between the normal and superconducting states of a metal can also be described using a complex scalar order parameter that represents a condensate amplitude. In this case, however, the condensate is charged, and this means that we should consider how our expression for free energy varies under gauge transformations. Fol-lowing arguments similar to those developed in Chapter 4, we arrive (with some suggestive notation for constants that are in fact phenomenological) at what is known as Landau-Ginzburg theory: F(ψ) = a 2\|ψ\|2 + b 4\|ψ\|4 + 1 2mψ∗(−iℏ∇−qA)2ψ + . . . . 9.3 Consequences of ﬂuctuations It is important to understand to what extent our treatment of these free energy densities, using mean ﬁeld theory in the form of a saddle-point approximation to the functional integral, is correct. In fact, the approach can go wrong at two different levels, and behaviour depends on the dimensionality d of the system. The most dramatic possibility, which applies in low dimensions, is that ﬂuctuations are so strong the system is disordered at any non-zero temperature. A less acute failing would be that there is a transition, but with critical behaviour that is not captured by mean ﬁeld theory. Two borderline values of dimension, known as the lower and upper critical dimensions (dl and du) separate these alternatives: for d < dl, there is no ﬁnite-temperature phase transition; for dl < d < du there is a transition but critical behaviour is not well-described by mean ﬁeld theory; and for du < d, critical properties follow mean ﬁeld predictions. 9.3.1 The lower critical dimension We have already seen from our discussion of statistical mechanics in one dimension (see Chapter 3) that one-dimensional systems with short range interactions do not have spontaneously broken symmetry at non-zero tem-perature. For the one-dimensional Ising model we arrived at this result both by an exact calculation, using transfer matrices, and more intuitively, by considering the energy and entropy associated with kinks or domain walls sepa-rating regions of parallel spins. We now examine the counterpart to this argument in two dimensions. Systems with discrete broken symmetry: the Peierls argument Consider a low-temperature conﬁguration of a two-dimensional Ising model, as sketched in Fig. 9.6. We would like to ﬁnd how frequent are domains of reversed spins, by estimating the free energy associated with the domain walls or boundaries separating spins of opposite orientation. Since with ferromagnetic exchange interactions of strength J, a pair of parallel spins has energy −J, and an antiparallel pair an energy +J, it costs an energy 2JL to introduce a domain wall of length L into the ground state. Such a domain wall may have many conﬁgurations. Counting these conﬁgurations exactly is a difﬁcult exercise since domain walls should not intersect themselves, but we can make a reasonable estimate by considering a random walk on the square lattice that is constrained at each step not to turn back on itself. These means that three possibilities are open at each step (continuing straight ahead, turning left, or turning right), and the total number of conﬁgurations after L steps (ignoring the requirement for the domain wall to close on itself) is 3L, implying an entropy of kBL ln 3. The free energy of the domain wall is hence FL = L(2J −kBT ln 3) . Crucially, this is positive for T < 2J/(kB ln 3), and so domain walls are rare and long-range order is stable at low temperature. 112 CHAPTER 9. PHASE TRANSITIONS − + + + + + + + + + − − − Figure 9.6: Typical low-temperature conﬁguration of a two-dimensional Ising model. The system has positive magnetisation, but domains of spins with the opposite orientation appear as thermal excitations. Systems with continuous broken symmetry: Goldstone modes Our conclusions about order in the Ising model in fact also apply to other systems in which the broken symmetry is discrete. Systems with a continuous broken symmetry behave differently, however: their low energy excitations are long-wavelength Goldstone modes, which are more effective at disrupting long-range order than are sharp domain walls in the Ising model. To understand the effect of Goldstone modes on long-range order, we should calculate the correlation function for ﬂuctuations ϕ(r) in the order parameter. A calculation similar to the one set out above in Section 9.2.2 leads to the result ⟨ϕ(0ϕ(r)⟩∼ kBT (2π)dJ Z ddkeik·r k2 . (9.4) If the notion that the system has long range order is to be self-consistent, ﬂuctuations should be small. In particular a divergence in ⟨ϕ2(0⟩would signal an instability. The integral on the right-hand side of Eq. (9.4) is divergent at small k in one and two dimensions, and this indicates the absence of spontaneously broken symmetry at non-zero temperature in one and two-dimensional systems. (The integral is divergent at large k is two and more dimensions, but this divergence does not have the same physical signiﬁcance because in a condensed matter system there is always an upper limit to wavevectors, set by the inverse atomic spacing.) Summarising, we have found that the value of the lower critical dimension is dl = 1 for systems with discrete symmetry, and dl = 2 for systems with continuous symmetry. 9.3.2 The upper critical dimension For systems which are above their lower critical dimension and so have an ordering transition, we can ask whether it is reasonable to neglect ﬂuctuations, by comparing the amplitude of ﬂuctuations in the order parameter with its mean value. The approach leads to what is known as the Ginzburg criterion for the validity of mean ﬁeld theory. Speciﬁcally, in a system with an order parameter ﬁeld ϕ(r), we compare its mean square ﬂuctuation ⟨[ϕ(r) −⟨ϕ⟩]2⟩ξ, averaged over a region of linear size set by the correlation length, with ⟨ϕ⟩2, the square of the order parameter itself. We have ⟨[ϕ(r) −⟨ϕ⟩]2⟩ξ = ξ−2d Z ξd ddr Z ξd ddr′ [⟨ϕ(r)ϕ(r′) −⟨ϕ⟩2] = ξ−d Z ddr [⟨ϕ(0)ϕ(r) −⟨ϕ⟩2] = ξ−dχ . (9.5) 9.4. FURTHER READING 113 The last equality in Eq. (9.5) follows from the deﬁnition of susceptibility, which gives χ = ∂ ∂h h=0 ⟨ϕ(0)⟩ = ∂ ∂h h=0 R Dϕ ϕ(0) e−F +h R ϕ(r) R Dϕ e−F +h R ϕ(r) = Z ddr [⟨ϕ(0)ϕ(r)⟩−⟨ϕ(0)⟩⟨ϕ(r)⟩] . We are now in a position to express both the mean square ﬂuctuations and the order parameter in terms of the reduced temperature t and the critical exponents. We have ξ−dχ ∼\|t\|dν−γ and ⟨ϕ⟩2 ∼\|t\|2β. If ﬂuctuations close to the critical point are to be small compared to the mean, we require ξ−dχ ∼\|t\|dν−γ ≪⟨ϕ⟩2 ∼\|t\|2β for \|t\| small. This is the case if νd −γ > 2β, which is to say d > 2β + γ ν = 4 , where the ratio of exponents has been evaluated using the results of mean ﬁeld theory presented earlier in this chapter. Our conclusion, then, is that while mean ﬁeld theory provides a qualitatively correct treatment of phase transitions, for systems in three dimensions it is not quantitatively accurate to neglect ﬂuctuations. In fact, accurate calculations of exponent values for systems in fewer than four dimensions require a more serious treatment of interactions in ﬁeld theory, using renormalisation group methods. 9.4 Further Reading • J. M. Yeomans, Statistical Mechanics of Phase Transitions (OUP). Chapters 1 and 4 provide a straightfor-ward introduction to the material covered in these lectures. • P. M. Chaiken and T. C. Lubensky, Principles of Condensed Matter Physics (CUP). Chapters 1, 3 and 4 cover similar material to that in these lectures. • N. Goldenfeld, Lectures on Phase Transitions and The Renormalization Group (Addison Wesley) Chapter 5 covers the same material as these lectures. The book also offers a readable introduction to more advanced ideas.
188907	https://www.education.com/common-core/CCSS.MATH.CONTENT.6.G.A.1/	Choose an Account to Log In Notifications 6.G.A.1 Worksheets, Workbooks, Lesson Plans, and Games CCSS.MATH.CONTENT.6.G.A.1 These worksheets can help students practice this Common Core State Standards skill. Worksheets Lesson Plans No plans found for this common core code. Workbooks No workbooks found for this common core code. Games No games found for this common core code. Exercises No exercises found for this common core code. Add to collection Create new collection New Collection New Collection Sign up to start collecting! Bookmark this to easily find it later. Then send your curated collection to your children, or put together your own custom lesson plan. Educational Tools Support Disable Cookies Warning - you are about to disable cookies. If you decide to create an account with us in the future, you will need to enable cookies before doing so. Connect About IXL Comprehensive K-12 personalized learning Rosetta Stone Immersive learning for 25 languages Wyzant Trusted tutors for 300 subjects Vocabulary.com Adaptive learning for English vocabulary ABCya Fun educational games for kids SpanishDictionary.com Spanish-English dictionary, translator, and learning Emmersion Fast and accurate language certification TPT Marketplace for millions of educator-created resources Copyright © 2025 Education.com, Inc, a division of IXL Learning • All Rights Reserved.
188908	https://mathgeekmama.com/writing-algebraic-expressions/	Writing Algebraic Expressions: FREE Practice Pages \| Math Geek Mama Skip to content Join 165,000+ parents and teachers for new tips! Home About Terms of Use Disclosure & Privacy Policy Contact Search for: Fun teaching resources & tips to help you teach math with confidence Hands on Math Math+Technology Lessons Interactive Worksheet Fractions and Decimals 3-D Shapes Worksheets Math Art Project PI Day Easy Math Games Simple Math Card Games Literature Based Math Math Books Math Teaching Classroom strategies Common Core Help Dyscalculia Resources Math Teaching Student Help Freebies Shop Math PD Courses Course Login JOIN Math Geek Mama+ Math Geek Mama+ Login Menu Search Hands on Math Math+Technology Lessons Interactive Worksheet Fractions and Decimals 3-D Shapes Worksheets Math Art Project PI Day Easy Math Games Simple Math Card Games Literature Based Math Math Books Math Teaching Classroom strategies Common Core Help Dyscalculia Resources Math Teaching Student Help Freebies Shop Math PD Courses Course Login JOIN Math Geek Mama+ Math Geek Mama+ Login Search for: Search for: Home/Grades 6-8/Writing Algebraic Expressions: FREE Practice Pages Writing Algebraic Expressions: FREE Practice Pages ByBethanyDecember 9, 2019 June 23, 2024 237 shares Facebook Twitter Pinterest Email Grab this print-and-go set of worksheets for some quick and easy writing algebraic expressions practice! Includes 3 pages plus answer keys! My teaching career has been focused in elementary school. But with the decision to homeschool my children, I’m now getting to teach some middle school subjects. My oldest started a sixth-grade math book this year. I’ll be honest: there have been a few problems I’ve really had to think through and use number sense to figure out. I’ve loved the challenge, and the opportunity to learn more with him. This week we jumped into a little bit of algebra. I LOVE algebra, and it was fun to sit with him and discuss how to write and evaluate algebraic expressions. I wanted him to have a little extra work with this skill, so I created these algebra word problems! Please Note: This post contains affiliate links which support the work of this site. Read our full disclosure here. This is a guest post from Rachel of You’ve Got This Math. Algebraic Expressions Prep-Work Don’t you just love activities that don’t require prep-work? Well these word problems that have children creating algebraic expressions and then evaluating them require no prep-work. Simply print off the pages you need, and that’s it. What is An Algebraic Expression? So what is an algebraic expression? According to A Maths Dictionary For Kid’s,you can describe an algebraic expression in three ways: It is a mathematical phrase that uses both numbers and variables Though it may have basic operations and grouping symbols, it does not have equality or inequality signs. Finally, both sides of an equation are expressions. If you’re looking for more in depth lessons and practice with algebra, you may be interested in my Algebra Essentials Lesson Collection. Writing an Algebraic Expression: Now that we know what an algebraic expression is, it’s time to create one. Begin by giving your children a simple statement like, “Susan has three times as many books as Sally.” Next, you can ask, “How many books does Susan have?” Well, we don’t know how many books Susan has, because we don’t know how many books Sally has. When we don’t know a key piece of data, we can use a variable to represent the unknown. In this case, our variable will stand for how many books Sally has. Now, ask your children to write an expression using a b to represent how many books Sally has. Therefore, the expression b x 3 represents how many books Susan has. Evaluating an Algebraic Expression: Now it is time for the fun part: evaluating the expression. And this is easy. Now we are just putting in a number where the variable is, since it is no longer an unknown. Let’s go back to our statement about Susan and Sally. We created an algebraic expression b x 3 or 3b to show that Susan has three times as many books as Sally. Now, tell the children that you found out that Sally has 12 books. Now we can easily solve this problem. All we need to do is replace the variable with an actual number: 12 x 3 = 36 Therefore, Susan has 36 books. FREE Algebraic Expressions Worksheets: This set of worksheets provides practice with both of these steps. Each problem starts with a set of facts. For example, one problem states that a worker gets paid $100 dollars a day and $0.50 per mile he drives. We will use this statement to write our algebraic expression. At this point we don’t know how many miles he has driven, so we will use m to represent how many miles he drives. Now when I look at the statement I know that I will have to multiply the miles he drives by $0.50. m x $0.50 or 0.5m can be used. But I’m not done. I also know he gets $100 per day. So I need to add $100 to my algebraic expression. This makes my expression: 0.5m + 100 Evaluate: Now let’s evaluate this algebraic expression. The next line tells me how many miles he drove –> 245 miles. So we evaluate the expression with 245 miles. This gives us: (0.5 x 245) + 100 Following order of operations, we multiply first, and then add another $100: 5 x 245 = 122.5 122.5 + 100 = 222.5 Therefore, the driver made $222.5 that day. That’s it. Now you have a simple activity to get your young ones working on writing algebraic expressions and then evaluating them. If you enjoy this lesson, become a Math Geek Mama+ member and gain access to the entire library of engaging math lessons like this one, hundreds of math games and low-prep practice worksheets for grades 5-8! Learn more about Math Geek Mama+ right HERE. Find more resources to help kids understand algebra vocabulary and translate algebraic expressions here. {Click HERE to go to my shop and grab this FREE set of Writing Algebraic Expressions Worksheets!} About the author: Rachel is a homeschool mom to four little ones, ages 2 to 6. She is a former public elementary teacher, and has recently begun blogging at her page You’ve Got This. Never Run Out of Fun Math Ideas! If you enjoyed this post, you will love being a part of the Math Geek Mama community! Sign up to receive my weekly emails with fun and engaging math ideas, free resources and special offers. Join 163,000+ readers as we help every child succeed and thrive in math! PLUS, receive my FREE ebook, 5 Math Games You Can Play TODAY, as my gift to you! Send Me FREE Math Ideas! We won't send you spam. Unsubscribe at any time. Built with Kit Post navigation Previous {FREE} Gingerbread Counting Game \| Grades PreK-1st Next FREE January 2020 Problem of the Day Calendars One Comment Pingback: FREE Pages for Writing Algebraic Expressions \| Free Homeschool Deals © Comments are closed. 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188909	https://mathbitsnotebook.com/Algebra2/FunctionGraphs/FGTypePiecewise.html	Piecewise, Absolute Value and Step Functions - MathBitsNotebook(A2) Piecewise, Absolute Value, and Step Functions MathBitsNotebook.com Topical Outline \| Algebra 2 Outline \| MathBits' Teacher Resources Terms of Use Contact Person:Donna Roberts We have seen many graphs that are expressed as single equations and are continuous over a domain of the Real numbers. We have also seen the "discrete" functions which are comprised of separate unconnected "points". There are also graphs that are defined by "different equations" over different sections of the graphs. These graphs may be continuous, or they may contain "breaks". Because these graphs tend to look like "pieces" glued together to form a graph, they are referred to as "piecewise" functions (piecewise defined functions), or "split-definition" functions. A piecewise defined function is a function defined by at least two equations ("pieces"), each of which applies to a different part of the domain. Piecewise defined functions can take on a variety of forms. Their "pieces" may be all linear, or a combination of functional forms (such as constant, linear, quadratic, cubic, square root, cube root, exponential, etc.). Due to this diversity, there is no "parent function" for piecewise defined functions. The example below will contain linear, quadratic and constant "pieces". Notice that each "piece" of the function has a specific constraint.Description: Notice that the "changes" focus around the x-values of 1 and -1. ♦ Hint: When graphing, focus on where the changes in the graph occur. From x-values of -∞ to -1, the graph is a straight line. From x-values of -1 to 1, the graph is constant. From x-values 1 to ∞, the graph is quadratic (part of a parabola).The piecewise function shown in this example is continuous (there are no "gaps" or "breaks" in the plotting). In this example, the domain is all Reals since all x-values have a plotted value. For help with piecewise defined functions on your calculator, Click Here! Still confused about what is happening in these piecewise defined functions? Try taking a look at each section as a "separate" graph, and grab your scissors! Piecewise defined functions may be continuous (as seen in the example above), or they may be discontinuous (having breaks, jumps, or holes as seen in the examples below). Other Examples of Piecewise Defined Functions: Domain: All Reals Range: All Reals Domain: [-5,1] U (2,5] Range: [-2,4] Domain: All Reals Range: (-∞,0) U [1,∞) One of the most recognized piecewise defined functions is the absolute value function. Features (of parent function): • Domain : All Reals (-∞,∞) Unless domain is altered. • Range: [0,∞) • increasing (0, ∞) • decreasing (-∞,0) • positive (-∞, 0) U (0, -∞) • absolute/relative min is 0 • no absolute max (graph → ∞) • end behavior f (x) → +∞, as x → +∞ f (x) → +∞, as x → -∞Symmetric: about x = 0 unless domain is altered x-intercept: intersects x-axis at (0, 0) unless domain is altered y-intercept: intersects y-axis at (0, 0) unless domain is altered Vertex: the point (0,0) unless domain is altered Absolute value is an even function: f(-x) = f (x) Table: Y1: y = \| x \| Read more aboutAbsolute Value.Range: When finding the range of an absolute value function, find the vertex (the turning point). • If the graph opens upwards, the range will be greater than or equal to the y-coordinates of the vertex. • If the graph opens downward, the range will be less than or equal to the y-coordinate of the vertex. Average rate of change: is constant on each straight line section (ray) of the graph. • may also be written as f (x) = abs(x)For help with absolute value graphs on your calculator, Click Here! Absolute Value Function - Transformation Examples: Translations Reflection Vertical Stretch/Shrink A step function (or staircase function) is a piecewise function containing all constant "pieces". The constant pieces are observed across the adjacent intervals of the function, as they change value from one interval to the next. A step function is discontinuous (not continuous). You cannot draw a step function without removing your pencil from your paper. Features (of step functions): • utilize open circles and/or closed circles on the graph open = point not on graph; closed = point is on graph • horizontal "pieces" • discontinuous (cannot be drawn without removing your pencil from the paper) • notice the resemblance to a set of steps • may, or may not, be a function. Check with the vertical line test. This example is a function. One of the most famous step functions is the Greatest Integer Function. The greatest integer function returns the largest integer less than or equal to x, for all real numbers x.In essence, the greatest integer function rounds down a real number to the nearest integer. For example: = 2; [1.5] = 1; [-3.1] = -4; [-6.9] = -7 Greatest Integer Function: Features - the Greatest Integer Function: • the intervals on the greatest integer function can be expressed as [n, n+1). The value of the function on these intervals will be n.The function is constant in each interval. •you may see some texts using the notation y = (double brackets). • it may also be referred to as the "floor" function and written as . This site will use the single bracket notation y = [x].For help with greatest integer functions on your calculator, Click Here! More examples of Step Functions: NOTE:There-posting of materials(in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use". Topical Outline \| Algebra 2 Outline \| MathBitsNotebook.com \| MathBits' Teacher ResourcesTerms of UseContact Person:Donna Roberts Copyright © 2012-2025 MathBitsNotebook.com. All Rights Reserved.
188910	https://www.britannica.com/science/gibberellic-acid	gibberellic acid Learn about this topic in these articles: brewing and fermentation …secretes a plant hormone called gibberellic acid, which initiates the synthesis of α-amylase. The α- and β-amylases then convert the starch molecules of the corn into sugars that the embryo can use as food. Other enzymes, such as the proteases and β-glucanases, attack the cell walls around the starch grains,… isoprenoids , the hormone gibberellic acid) and contribute to red, yellow, and orange pigments (carotenoids). Chlorophyll, the green pigment essential in photosynthesis, is partly isoprenoid, as are certain alkaloids, nitrogen-containing compounds present in many plants. In animals, isoprenoids comprise various oily
188911	https://artofproblemsolving.com/wiki/index.php/2011_AMC_12B_Problems/Problem_20?srsltid=AfmBOopplQ9QM-QDhijFBZENPyMVWzmXOV9UEZ6oyKUjwVbFUj6S-ka4	Art of Problem Solving 2011 AMC 12B Problems/Problem 20 - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki 2011 AMC 12B Problems/Problem 20 Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search 2011 AMC 12B Problems/Problem 20 Contents [hide] 1 Problem 2 Solution 1 (Coordinates) 3 Solution 2 (Algebra) 4 Solution 3 (Dilation) 5 Solution 4 (basically Solution 1 but without coordinates) 6 Solution 5 7 Solution 6 (Trigonometry) 8 Solution 7 (abwabwabwa) 9 See also Problem Triangle has , and . The points , and are the midpoints of , and respectively. Let be the intersection of the circumcircles of and . What is ? Solution 1 (Coordinates) Let us also consider the circumcircle of . Note that if we draw the perpendicular bisector of each side, we will have the circumcenter of which is , Also, since . is cyclic, similarly, and are also cyclic. With this, we know that the circumcircles of , and all intersect at , so is . The question now becomes calculating the sum of the distance from each vertex to the circumcenter. We can calculate the distances with coordinate geometry. (Note that because is the circumcenter.) Let , , , Then is on the line and also the line with slope that passes through (realize this is due to the fact that is the perpendicular bisector of ). So and Remark: the intersection of the three circles is called a Miquel point. Solution 2 (Algebra) Consider an additional circumcircle on . After drawing the diagram, it is noticed that each triangle has side values: , , . Thus they are congruent, and their respective circumcircles are. Let & be &'s circumcircles' respective centers. Since & are congruent, the distance & each are from are equal, so . The angle between & is , and since , is also . is a right triangle inscribed in a circle, so must be the diameter of . Using the same logic & reasoning, we could deduce that & are also circumdiameters. Since the circumcircles are congruent, circumdiameters , , and are congruent. Therefore, the solution can be found by calculating one of these circumdiameters and multiplying it by a factor of . We can find the circumradius quite easily with the formula , such that and is the circumradius. Since : After a few algebraic manipulations: . Solution 3 (Dilation) Let be the circumcenter of and denote the length of the altitude from Note that a dilation centered at with ratio takes the circumcircle of to the circumcircle of . It also takes the point diametrically opposite on the circumcircle of to Therefore, lies on the circumcircle of Similarly, it lies on the circumcircle of By Pythagorean triples, Finally, our answer is Solution 4 (basically Solution 1 but without coordinates) Since Solution 1 has already proven that the circumcenter of coincides with , we'll go from there. Note that the radius of the circumcenter of any given triangle is , and since and , it can be easily seen that and therefore our answer is Solution 5 Since is a midline of we have that with a side length ratio of Consider a homothety of scale factor with on concerning point . Note that this sends to with By properties of homotheties, and are collinear. Similarly, we obtain that with all three points collinear. Let denote the circumcenter of It is well-known that and analogously However, there is only one perpendicular line to passing through , therefore, coincides with It follows that where is the circumradius of and this can be computed using the formula from which we quickly obtain Solution 6 (Trigonometry) , , as the angles are on the same circle. , , , Therefore , and is the angle bisector of . By the angle bisector theorem , . In a similar fashion , where is the circumcircle of . By the law of cosine, , By the extended law of sines, , ~isabelchen Solution 7 (abwabwabwa) Claim, is the circumcenter of triangle . Proof: Note that and are congruent. Consider the centers and of and , respectively. Let be the reflection of over , and let be the reflection of over . Since they form diameters, they must form right triangles and . However, because , C' and B' are the same point. Thus, one point lies on both circumcircles, so this point is . But then X lies on the perpendicular bisector of , and appyling this logic to all 3 sides, must be the circumcenter. Memorizing that the circumradius of a triangle is , since , . -skibbysiggy See also 2011 AMC 12B (Problems • Answer Key • Resources) Preceded by Problem 19Followed by Problem 21 1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25 All AMC 12 Problems and Solutions These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Retrieved from " Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
188912	https://webbook.nist.gov/cgi/cbook.cgi?ID=C10196675&Mask=4	cadmium myristate Jump to content National Institute of Standards and Technology NIST Chemistry WebBook, SRD 69 Home Search Name Formula IUPAC identifier CAS number More options NIST Data SRD Program Science Data Portal Office of Data and Informatics About FAQ Credits More documentation cadmium myristate Formula: C 28 H 54 CdO 4 Molecular weight: 567.137 IUPAC Standard InChI:InChI=1S/2C14H28O2.Cd/c21-2-3-4-5-6-7-8-9-10-11-12-13-14(15)16;/h22-13H2,1H3,(H,15,16);/q;;+2/p-2 Copy IUPAC Standard InChIKey:KADXVMUKRHQBGS-UHFFFAOYSA-L Copy CAS Registry Number: 10196-67-5 Chemical structure: This structure is also available as a 2d Mol file Other names: Cadmium(II) n-tetradecanoate Permanent link for this species. Use this link for bookmarking this species for future reference. Information on this page: Phase change data References Notes Options: Switch to calorie-based units Phase change data Go To:Top, References, Notes Data compilation copyright by the U.S. Secretary of Commerce on behalf of the U.S.A. All rights reserved. Data compiled by:Eugene S. Domalski and Elizabeth D. Hearing Enthalpy of fusion \| Δ fus H (kJ/mol) \| Temperature (K) \| Reference \| Comment \| --- --- \| \| 43.000 \| 374.7 \| Konkoly-Thege, Ruff, et al., 1978 \| Crystal-mesophase. \| Entropy of fusion \| Δ fus S (J/molK) \| Temperature (K) \| Reference \| Comment \| --- --- \| \| 115. \| 374.7 \| Konkoly-Thege, Ruff, et al., 1978 \| Crystal-mesophase. \| Enthalpy of phase transition \| ΔH trs (kJ/mol) \| Temperature (K) \| Initial Phase \| Final Phase \| Reference \| Comment \| --- --- --- \| \| 1.600 \| 380.4 \| liquid \| liquid \| Konkoly-Thege, Ruff, et al., 1978 \| \| Entropy of phase transition \| ΔS trs (J/molK) \| Temperature (K) \| Initial Phase \| Final Phase \| Reference \| Comment \| --- --- --- \| \| 4. \| 380.4 \| liquid, Mesophase \| liquid, liquid \| Konkoly-Thege, Ruff, et al., 1978 \| \| References Go To:Top, Phase change data, Notes Data compilation copyright by the U.S. Secretary of Commerce on behalf of the U.S.A. All rights reserved. Konkoly-Thege, Ruff, et al., 1978 Konkoly-Thege, I.; Ruff, I.; Adeosun, S.O.; Sime, S.J., Properties of molten carboxylates. Part 6. A quantitative differential thermal analysis study of phase transitions in some zinc and cadmium carboxylates, Thermochim. Acta, 1978, 24, 89-96. [all data] Notes Go To:Top, Phase change data, References Symbols used in this document: ΔH trs Enthalpy of phase transition ΔS trs Entropy of phase transition Δ fus H Enthalpy of fusion Δ fus S Entropy of fusion Data from NIST Standard Reference Database 69: NIST Chemistry WebBook The National Institute of Standards and Technology (NIST) uses its best efforts to deliver a high quality copy of the Database and to verify that the data contained therein have been selected on the basis of sound scientific judgment. However, NIST makes no warranties to that effect, and NIST shall not be liable for any damage that may result from errors or omissions in the Database. Customer support for NIST Standard Reference Data products. © 2025 by the U.S. Secretary of Commerce on behalf of the United States of America. All rights reserved. Copyright for NIST Standard Reference Data is governed by the Standard Reference Data Act. Privacy Statement Privacy Policy Security Notice Disclaimer (Note: This site is covered by copyright.) Accessibility Statement FOIA Contact Us
188913	https://flexbooks.ck12.org/cbook/ck-12-algebra-ii-with-trigonometry-concepts/section/5.14/related/lesson/solutions-using-the-discriminant-bsc-alg/	Skip to content Math Elementary Math Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Interactive Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Conventional Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Probability & Statistics Trigonometry Math Analysis Precalculus Calculus What's the difference? 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Learn. Interact. eXplore. CCSS Math Concepts and FlexBooks aligned to Common Core NGSS Concepts aligned to Next Generation Science Standards Certified Educator Stand out as an educator. Become CK-12 Certified. Webinars Live and archived sessions to learn about CK-12 Other Resources CK-12 Resources Concept Map Testimonials CK-12 Mission Meet the Team CK-12 Helpdesk FlexLets Know the essentials. Pick a Subject Donate Sign Up Back To Using the DiscriminantBack 5.14 Solutions Using the Discriminant Written by:Andrew Gloag \| Melissa Kramer \| Fact-checked by:The CK-12 Editorial Team Last Modified: Sep 01, 2025 Suppose that the balance of your checking account in dollars can be modeled by the function @$\begin{align}B(t)=0.001t^2 - t + 300\end{align}@$, where @$\begin{align}t\end{align}@$ is the number of days the checking account has been open. Will the balance of your checking account ever be $40? Solutions Using the Discriminant You have seen parabolas that intersect the @$\begin{align}x-\end{align}@$axis twice, once, or not at all. There is a relationship between the number of real @$\begin{align}x-\end{align}@$intercepts and the quadratic formula. Case 1: The parabola has two @$\begin{align}x-\end{align}@$intercepts. This situation has two possible solutions for @$\begin{align}x\end{align}@$, because the value inside the square root is positive. Using the quadratic formula, the solutions are @$\begin{align}x=\frac{-b+\sqrt{b^2-4ac}}{2a}\end{align}@$ and @$\begin{align}x=\frac{-b-\sqrt{b^2-4ac}}{2a}\end{align}@$. Case 2: The parabola has one @$\begin{align}x-\end{align}@$intercept. This situation occurs when the vertex of the parabola just touches the @$\begin{align}x-\end{align}@$axis. This is called a repeated root, or double root. The value inside the square root is zero. Using the quadratic formula, the solution is @$\begin{align}x=\frac{-b}{2a}\end{align}@$. Case 3: The parabola has no @$\begin{align}x-\end{align}@$intercept. This situation occurs when the parabola does not cross the @$\begin{align}x-\end{align}@$axis. The value inside the square root is negative, so there are no real roots. The solutions to this type of situation are imaginary, which you will learn more about in a later textbook. The value inside the square root of the quadratic formula is called the discriminant. It is symbolized by @$\begin{align}D\end{align}@$. It dictates the number of real solutions the quadratic equation has. This can be summarized with the Discriminant Theorem. If @$\begin{align}D>0\end{align}@$, the parabola will have two @$\begin{align}x-\end{align}@$intercepts. The quadratic equation will have two real solutions. If @$\begin{align}D=0\end{align}@$, the parabola will have one @$\begin{align}x-\end{align}@$intercept. The quadratic equation will have one real solution. If @$\begin{align}D<0\end{align}@$, the parabola will have no @$\begin{align}x-\end{align}@$intercepts. The quadratic equation will have zero real solutions. Let's determine the number of solutions for the following equations: @$\begin{align}-3x^2+4x+1=0\end{align}@$ By finding the value of its discriminant, you can determine the number of @$\begin{align}x-\end{align}@$intercepts the parabola has and thus the number of real solutions. @$$\begin{align}D &= b^2-4(a)(c)\ D &= (4)^2-4(-3)(1)\ D &= 16+12=28\end{align}@$$ Because the discriminant is positive, the parabola has two real @$\begin{align}x-\end{align}@$intercepts and thus two real solutions. Determine the number of solutions to @$\begin{align}-2x^2+x=4\end{align}@$. Before we can find its discriminant, we must write the equation in standard form: @$\begin{align}ax^2+bx+c=0\end{align}@$. Subtract 4 from each side of the equation: @$\begin{align}-2x^2+x-4=0\end{align}@$. @$$\begin{align}\text{Find the discriminant:} && D &= (1)^2-4(-2)(-4)\ && D &= 1-32=-31\end{align}@$$ The value of the discriminant is negative; there are no real solutions to this quadratic equation. The parabola does not cross the @$\begin{align}x-\end{align}@$axis. Now, let's use the discriminant to solve the following problem: Emma and Brandon own a factory that produces bike helmets. Their accountant says that their profit per year is given by the function @$\begin{align}P=0.003x^2+12x+27,760\end{align}@$, where @$\begin{align}x\end{align}@$ represents the number of helmets produced. Their goal is to make a profit of $40,000 this year. Is this possible? The equation we are using is @$\begin{align}40,000=0.003x^2+12x+27,760\end{align}@$. By finding the value of its discriminant, you can determine if the profit is possible. Begin by writing this equation in standard form: @$$\begin{align}0 &= 0.003x^2+12x-12,240\ D &= b^2-4(a)(c)\ D &= (12)^2-4(0.003)(-12,240)\ D &= 144+146.88=290.88\end{align}@$$ Because the discriminant is positive, the parabola has two real solutions. Yes, the profit of $40,000 is possible. Examples Example 1 Earlier, you were told that the balance of your checking account in dollars can be modeled by the function @$\begin{align}B(t)=0.001t^2 - t + 300\end{align}@$, where @$\begin{align}t\end{align}@$ is the number of days the checking account has been open. Will the balance of your checking account ever be $40? The equation we are using is @$\begin{align}40 = 0.001t^2-t+300.\end{align}@$ By finding the value of its discriminant, you can determine if the balance is possible. Begin by writing the equation in standard form: @$\begin{align}0=0.001t^2-t+260\end{align}@$ Then calculate the discriminant: @$$\begin{align}D &= b^2-4(a)(c)\ D &= (-1)^2-4(0.001)(260)\ D &= 1-1.04=-0.04\end{align}@$$ The discriminant is negative, so the function has 0 real solutions. Therefore, the balance of your checking account will never be $40. Example 2 Determine the number of solutions for @$\begin{align}x^2-2x+1=0\end{align}@$. Substitute the values into the discriminant: @$$\begin{align}D &= b^2-4(a)(c)\ D &= (-2)^2-4(1)(1)\ D &= 4-4=0\end{align}@$$ Because the discriminant is zero, the parabola has one real @$\begin{align}x-\end{align}@$intercept and thus one real solution. Review What is a discriminant? What does it do? What is the formula for the discriminant? Can you find the discriminant of a linear equation? Explain your reasoning. Suppose @$\begin{align}D=0\end{align}@$. Draw a sketch of this graph and determine the number of real solutions. @$\begin{align}D=-2.85\end{align}@$. Draw a possible sketch of this parabola. What is the number of real solutions to this quadratic equation. @$\begin{align}D>0\end{align}@$. Draw a sketch of this parabola and determine the number of real solutions. Find the discriminant of each quadratic equation. @$\begin{align}2x^2-4x+5=0\end{align}@$ @$\begin{align}x^2-5x=8\end{align}@$ @$\begin{align}4x^2-12x+9=0\end{align}@$ @$\begin{align}x^2+3x+2=0\end{align}@$ @$\begin{align}x^2-16x=32\end{align}@$ @$\begin{align}-5x^2+5x-6=0\end{align}@$ Determine the nature of the solutions of each quadratic equation. @$\begin{align}-x^2+3x-6=0\end{align}@$ @$\begin{align}5x^2=6x\end{align}@$ @$\begin{align}41x^2-31x-52=0\end{align}@$ @$\begin{align}x^2-8x+16=0\end{align}@$ @$\begin{align}-x^2+3x-10=0\end{align}@$ @$\begin{align}x^2-64=0\end{align}@$ A solution to a quadratic equation will be irrational if the discriminant is not a perfect square. If the discriminant is a perfect square, then the solutions will be rational numbers. Using the discriminant, determine whether the solutions will be rational or irrational. @$\begin{align}x^2=-4x+20\end{align}@$ @$\begin{align}x^2+2x-3=0\end{align}@$ @$\begin{align}3x^2-11x=10\end{align}@$ @$\begin{align}\frac{1}{2}x^2+2x+\frac{2}{3}=0\end{align}@$ @$\begin{align}x^2-10x+25=0\end{align}@$ @$\begin{align}x^2=5x\end{align}@$ Marty is outside his apartment building. He needs to give Yolanda her cell phone but he does not have time to run upstairs to the third floor to give it to her. He throws it straight up with a vertical velocity of 55 feet/second. Will the phone reach her if she is 36 feet up? (Hint: The equation for the height is given by @$\begin{align}y=-16t^2+55t+4\end{align}@$.) Bryson owns a business that manufactures and sells tires. The revenue from selling the tires in the month of July is given by the function @$\begin{align}R=x(200-0.4x)\end{align}@$ where @$\begin{align}x\end{align}@$ is the number of tires sold. Can Bryson’s business generate revenue of $20,000 in the month of July? Marcus kicks a football in order to score a field goal. The height of the ball is given by the equation @$\begin{align}y=-\frac{32}{6400}x^2+x\end{align}@$, where @$\begin{align}y\end{align}@$ is the height and @$\begin{align}x\end{align}@$ is the horizontal distance the ball travels. We want to know if Marcus kicked the ball hard enough to go over the goal post, which is 10 feet high. Mixed Review Factor @$\begin{align}6x^2-x-12\end{align}@$. Find the vertex of @$\begin{align}y=-\frac{1}{4} x^2-3x-12\end{align}@$ by completing the square. Solve using the quadratic formula: @$\begin{align}-4x^2-15=-4x\end{align}@$. How many centimeters are in four fathoms? (Hint: 1 fathom = 6 feet) Graph the solution to @$\begin{align}\begin{cases} 3x+2y \le -4\ x-y>-3 \end{cases}\end{align}@$. How many ways can 3 toppings be chosen from 7 options? Review (Answers) Click HERE to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option. \| Image \| Reference \| Attributions \| --- Student Sign Up Are you a teacher? Having issues? Click here By signing up, I confirm that I have read and agree to the Terms of use and Privacy Policy Already have an account? Adaptive Practice Save this section to your Library in order to add a Practice or Quiz to it. (Edit Title)22/ 100 This lesson has been added to your library. \|Searching in: \| \| \| Looks like this FlexBook 2.0 has changed since you visited it last time. We found the following sections in the book that match the one you are looking for: Go to the Table of Contents No Results Found Your search did not match anything in .
188914	https://www.habbihabbi.com/blogs/bilingual-resources/body-parts-spanish	Learn 30+ Body Parts in Spanish - Free Printable included – Habbi Habbi Skip to content · FREE Ground Shipping on US orders $100 and up · · FREE Ground Shipping on US orders $100 and up · Cart $0.00 (0) Home Shop LanguageSpanishMandarin ChineseCantoneseFrenchKoreanHindiJapaneseItalian PopularBest sellers$100 and underFind your set Product TypeBooksFlashcardsPuzzlesWands & more Find the right set for you -------------------------- About Reviews Schools Search Cart $0.00 (0) Search Home Shop Expand menu Hide menuShop Language Expand menu Hide menuLanguage Spanish Mandarin Chinese Cantonese French Korean Hindi Japanese Italian Popular Expand menu Hide menuPopular Best sellers $100 and under Find your set Product Type Expand menu Hide menuProduct Type Books Flashcards Puzzles Wands & more Find the right set for you -------------------------- About Reviews Schools Learn Body Parts in Spanish - with Free Printable Posted by Habbi Habbi on October 01, 2022 Share Share Email Pin Tweet In this post: Learn the names of body parts in Spanish! Plus, get some ideas on books, songs, and activities to try together with your toddler or preschooler to reinforce Spanish vocabulary learning. -- Table of contents: Vocabulary:Over 30 words for body parts in Spanish Books:3 books we use to learn body parts Songs:4 favorite songs in Spanish about body parts Activities:2 fun activities that reinforce body parts -- Spanish Body Parts Vocabulary List Spanish is spoken in many different countries, and there is often more than one word for the same object. Many body part names are universally understood, though some vary by region or context. This list includes the more common terms with some notes on when and where you might hear variations. It is by no means exhaustive as there are many different colloquial terms! Body el cuerpo m Hair (1)el pelo el cabello m m Eye el ojo m Ear la oreja f Nose la nariz f Mouth la bocz f Tooth / Teeth el diente / los dientes m Tongue la lengua f Face la cara f Forehead la frente f Eyebrow (2)la ceja f Eyelash la pestaña f Cheek el cachete la mejilla m f Chin la barbilla el ment ón f m Neck el cuello m Shoulder el hombro m Arm el brazo m Elbow el codo m Wrist la mu ñeca f Hand la mano f Finger el dedo m Chest el pecho m Stomach (3)la barriga la panza f f Back la espalda f Hip la cadera f Buttocks (4)las nalgas f Leg la pierna f Knee la rodilla f Ankle el tibillo m Foot / Feet el pie / los pies m Toe el dedo del pie m 1 el pelo means “hair” generically and can be used for any type of hair (including animal) while cabello is specifically used for hair on the (human) head 2 el ceño is sometimes used in reference to both eyebrows, or “brow” as it would be in English 3 la barriga and la panza both refer to the belly, the outer part of the stomach; the organ that’s part of the digestive system is called “el estómago” 4 There are MANY different, more common informal terms! In Spain you will likely hear “el culo”, however this is the equivalent of the more vulgar English term in the majority of Central and South America, so beware! Books: 3 Spanish Books About Body Parts Amo mi cuerpo by Habbi Habbi One of my favorite books for my preschooler to reinforce body parts in Spanish is “I Love My Body”. Whether I choose Spanish only mode to immerse her in the language or bilingual mode so that she can better comprehend and appreciate the “hidden” positive messages about bodies (my favorite part), my daughter can read and listen and learn Spanish body parts with just a tap of the magical wand. And since everything is tappable, there’s always something new for us to discover each time we read it! Mi Primer Libro del Cuerpo by Angela Wilkes This bilingual board book from DK includes so much more than body parts in Spanish! Most everything you can think of that’s related to bodies - our facial expressions and emotions, the clothes we wear, and what we can do with our bodies - is included and illustrated with real photos of babies and children. Descansa y relájate by Whitney Stewart Sometimes quiet focus time can be really helpful to dig into vocabulary. This is an English-Spanish bilingual book about mindfulness designed for young children. Together with your toddler, you can walk through simple meditation practices for restful sleep while you tense and relax different body parts … all while practicing naming body parts in Spanish! Songs: 4 Favorite Spanish Tunes about the Body Cabeza, hombros, rodillas, pies - Super Simple Español This is a translated version of the classic “Head, Shoulders, Knees, and Toes”. With the familiar tune and simple movements, it’s a great first song to teach your little one some of the main body parts in Spanish. Cosquillas - 123 Andrés A fun song from the Latin-grammy winning 123 Andrés, this is another wonderful song to introduce body parts and early rhyming skills with the added giggles of tickles! Our family extends this song with additional people’s names that rhyme with various body parts in Spanish. We have come up with some pretty silly names! Las partes del cuerpo - Rockalingua Moving a step up in complexity, this song is a steady clip listing many body parts, directing the listener to touch each body part as it is named in Spanish. It’s still a little fast for my preschooler to keep up completely, but since it’s repeated, I like to just have her listen to the first round and then point to each body part as it’s named the second time through. Baila con tu cuerpo - Basho & Friends This song is faster paced and invites children to get up and dance as they move different body parts! There is a lot of repetition of the body part names in Spanish as each is mentioned, with the added benefit of also practicing directional vocabulary based on how they should move their bodies. When I would play this song for my students that were beginner Spanish learners, I would have them watch the video so that they could see each body part that was named in the song to help them better follow along. Activities: 2 Fun Ways to Reinforce Spanish Body Parts Free Printable - Write body parts in both Spanish and English Work on writing in both English and Spanish with Habbi Habbi’s printable. Label each body part, and if you have our book, you can use it to help remind you of the correct words! We did this worksheet alongside the book and enjoyed using the affirmational phrases about bodies to come up with our own reasons we love each of our own body parts labeled in the printable. Simon says / Simón dice This classic game is a great way to reinforce body part names in Spanish as well as common vocabulary related to directionality, movement, and more. For beginning language learners, you can start with just the body part in Spanish (e.g. Simon says, shake la cabeza), while more advanced speakers can participate completely in Spanish. I like to sometimes step it up by including multiple body parts (e.g. Simon says, tap la nariz with el dedo). If you liked this, you may also appreciate the following articles: Body Parts in French \| Vocabulary, games, songs & more Learn numbers and counting in Korean: Charts, Pronunciation, Tools, and more Bilingual Printable Flashcards: In My Home Vocabulary (45 cards) Chinese Family Tree: Sorting through family member names with 45 free printable cards Like this post? Share & Save Check out more bilingual resources from Habbi Habbi We have lots more (fun stuff!) here atHabbi Habbi. You can explore our free resources such asbilingual printables, resource blog, and audiobooks. Of course, we also have our much loved magicalReading Wand,bilingual books,puzzles&flashcards. Our tools are currently available in Spanish, Mandarin Chinese, French, Korean, and Hindi. About our lovely guest contributor: Kelly Kelly Helbach is an English-Spanish bilingual parent raising an English-Spanish-Mandarin trilingual child with her English-Mandarin bilingual spouse. She has a passion for education and literacy and language development, with a Master’s Degree in Reading Development and experience as both an English-only and Spanish-English dual language Kindergarten teacher. Nowadays, she stays home with her daughter and enjoys playing video games when there’s a bit of spare time. ← Older PostNewer Post → Bilingual Resources RSS ### Habbi Habbi 2025 Contest ### Habbi Habbi Cantonese is conversational Cantonese – Why? Group Orders, Shipping & Returns Free Resources Group Orders International Orders Shipping Policy Refund Policy Terms of Service FAQs, Wand Updates & More FAQs Update your Wand Events Contact & Join Us Privacy Policy Search Newsletter Join our list for special content, updates, and free printables Email Join Us Facebook Instagram Tiktok YouTube © Habbi Habbi 2025 Powered by Shopify Search ★ Reviews Let customers speak for us Based on 316 reviews Write a review 91% (286) 5% (16) 3% (8) 1% (3) 1% (3) See all reviews Product Reviews (313)Shop Reviews (3) W 08/18/2025Starter Set (Wand + 5 Books)William Hom Great invention Love it! This learning starter kit is the best way to learn a language. It’s good for ages 2 all the way up , even adults can use it, never too late to learn something new. Will stimulate the brain. The start kit come with 5 books and am audio wand . At the same time you will learn to read, spell and hear the phrases as the wand slides over the words S 08/10/2025The Everything Set (Wand + All Books, Puzzles, Flashcards)Sarah Love it Great way to learn a language without a screen/app. E 08/07/2025Home Vocabulary Bilingual FlashcardsE.Z Love this product!! I got a starter set for my son. I also got it for my best friend’s son for his birthday. It’s a great birthday gift!! Kids love to learn new words by pointing at the words. Strongly recommend!! 👍 E 08/07/2025Starter Set (Wand + 5 Books)E.Z Strongly recommend This is a great birthday gift!! My friend’s son loves it. It’s great for kids to learn 2nd language. Star ⭐️ pen is very clear on the pronunciation, battery last for long time. J 08/06/2025Starter Set (Wand + 5 Books)Jeffrey Hom Starter Set (Wand + 5 Books) 123 Cart Your cart is empty. Judge.me
188915	https://brainly.com/question/63881519?source=previous+question	[FREE] Solve the given quadratic by factoring. x^2-1=0 - brainly.com 4 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +83,6k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +44,6k Ace exams faster, with practice that adapts to you Practice Worksheets +5,6k Guided help for every grade, topic or textbook Complete See more / Mathematics Expert-Verified Expert-Verified Solve the given quadratic by factoring. x 2−1=0 1 See answer Explain with Learning Companion NEW Asked by k7srr5swk5 • 09/14/2025 0:00 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 1183190352 people 1183M 0.0 0 Upload your school material for a more relevant answer The quadratic equation x 2−1=0 is identified as a difference of squares. It is factored into (x−1)(x+1)=0. Using the Zero Product Property, the solutions are found to be x=1 and x=−1. The solutions are ordered from least to greatest and formatted as −1/1. Explanation Analyzing the Problem and Data We are given the quadratic equation x 2−1=0. Our goal is to find the values of x that satisfy this equation by factoring. We need to present the solutions in ascending order, separated by a forward slash (/). This problem involves solving a quadratic equation, which is a fundamental concept in algebra. Factoring the Quadratic Equation The given equation is x 2−1=0. This is a special type of quadratic equation known as the 'difference of squares'. The general form for the difference of squares is a 2−b 2=(a−b)(a+b). In our equation, a=x and b=1. Applying this formula, we can factor the equation as follows: x 2−1 2=(x−1)(x+1)=0 Solving for x Once the equation is factored, we use the Zero Product Property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x: \begin{itemize} \item For the first factor: x−1=0⟹x=1 \item For the second factor: x+1=0⟹x=−1 \end{itemize} So, the two solutions for x are 1 and −1. Ordering and Formatting the Solutions The problem requires us to enter the solutions from least to greatest with no spaces and using a forward slash (/) for division. Comparing our two solutions, −1 is less than 1. Therefore, the ordered solutions are −1 and 1. Combining them as specified, we get −1/1. This represents the two roots of the quadratic equation. Examples Solving quadratic equations by factoring is a fundamental skill in many real-world applications. For instance, if you're an engineer designing a bridge, you might use quadratic equations to model the parabolic arch of the bridge and determine its optimal dimensions. In physics, the trajectory of a projectile can be described by a quadratic equation, allowing you to calculate when and where it will land. Even in finance, quadratic models can help predict stock prices or analyze investment growth, making this mathematical concept a powerful tool for understanding and shaping our world. Answered by GinnyAnswer •8M answers•1.2B people helped Thanks 0 0.0 (0 votes) Expert-Verified⬈(opens in a new tab) 0.0 0 Upload your school material for a more relevant answer The quadratic equation x 2−1=0 can be factored into (x−1)(x+1)=0. The solutions obtained from the factored equation are x=1 and x=−1, which can be presented as −1/1. Explanation To solve the quadratic equation x 2−1=0, we start by recognizing that it is a difference of squares. The general formula for the difference of squares is a 2−b 2=(a−b)(a+b). In our equation, we can identify a=x and b=1. Factoring the Equation: We can rewrite the quadratic equation as follows: x 2−1 2=(x−1)(x+1)=0 Applying the Zero Product Property: This property states that if the product of two factors equals zero, then at least one of the factors must equal zero. Therefore, we set each factor equal to zero: For the first factor: x−1=0⇒x=1 For the second factor: x+1=0⇒x=−1 Finding Solutions: The solutions to the equation are both values we found: x=1 and x=−1. Ordering the Solutions: As the problem requests the solutions in ascending order, we list them from least to greatest: −1 (smallest) and 1 (largest). Thus, the final answer, formatted according to the prompt, is: −1/1. Examples & Evidence For instance, in real-life situations such as projectile motion, recognizing quadratic equations helps in predicting the height or distance of objects in motion. Engineers use such equations to determine structural loads and supports in designs. The concept of factoring quadratic equations follows standard algebraic principles taught in high school mathematics, specifically under the topic of solving quadratics through various methods including factoring. Thanks 0 0.0 (0 votes) Advertisement k7srr5swk5 has a question! Can you help? Add your answer See Expert-Verified Answer ### Free Mathematics solutions and answers Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD Community Answer 4.1 17 How many positive integers between 100 and 999 inclusive are divisible by three or four? Community Answer 4.0 9 N a bike race: julie came in ahead of roger. julie finished after james. david beat james but finished after sarah. in what place did david finish? Community Answer 4.1 8 Carly, sandi, cyrus and pedro have multiple pets. carly and sandi have dogs, while the other two have cats. sandi and pedro have chickens. everyone except carly has a rabbit. who only has a cat and a rabbit? Community Answer 4.1 14 richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points. Community Answer 4 Click an Item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers. Express In simplified exponential notation. 18a^3b^2/ 2ab New questions in Mathematics The probability of event A occurring is (4 m+n)m, and the probability of event B occurring is (4 m+n)n. Find the probability of A or B occurring if the events are mutually exclusive. 6. The operation t is defined by a t b=a+b+2 ab in arithmetic modulo 4. (a) Draw a table for the operation t on the set R={0,1,2,3}. Solve for n: 53=3 mod n. \left[\begin{array}{ccc}2 & 3 & 1 \ 4 & 1 & 2 \ 3 & 1 & 2\end{array}\right]\left[\begin{array}{c}2 \ -1 \ 3\end{array}\right] 2 4 53 1 121 22−1 3 Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss
188916	https://scikit-learn.org/stable/modules/svm.html	Skip to main content 1.4. Support Vector Machines# Support vector machines (SVMs) are a set of supervised learning methods used for classification, regression and outliers detection. The advantages of support vector machines are: Effective in high dimensional spaces. Still effective in cases where number of dimensions is greater than the number of samples. Uses a subset of training points in the decision function (called support vectors), so it is also memory efficient. Versatile: different Kernel functions can be specified for the decision function. Common kernels are provided, but it is also possible to specify custom kernels. The disadvantages of support vector machines include: If the number of features is much greater than the number of samples, avoid over-fitting in choosing Kernel functions and regularization term is crucial. SVMs do not directly provide probability estimates, these are calculated using an expensive five-fold cross-validation (see Scores and probabilities, below). The support vector machines in scikit-learn support both dense (numpy.ndarray and convertible to that by numpy.asarray) and sparse (any scipy.sparse) sample vectors as input. However, to use an SVM to make predictions for sparse data, it must have been fit on such data. For optimal performance, use C-ordered numpy.ndarray (dense) or scipy.sparse.csr_matrix (sparse) with dtype=float64. 1.4.1. Classification# SVC, NuSVC and LinearSVC are classes capable of performing binary and multi-class classification on a dataset. SVC and NuSVC are similar methods, but accept slightly different sets of parameters and have different mathematical formulations (see section Mathematical formulation). On the other hand, LinearSVC is another (faster) implementation of Support Vector Classification for the case of a linear kernel. It also lacks some of the attributes of SVC and NuSVC, like support_. LinearSVC uses squared_hinge loss and due to its implementation in liblinear it also regularizes the intercept, if considered. This effect can however be reduced by carefully fine tuning its intercept_scaling parameter, which allows the intercept term to have a different regularization behavior compared to the other features. The classification results and score can therefore differ from the other two classifiers. As other classifiers, SVC, NuSVC and LinearSVC take as input two arrays: an array X of shape (n_samples,n_features) holding the training samples, and an array y of class labels (strings or integers), of shape (n_samples): ``` from sklearn import svm X = y = [0, 1] clf = svm.SVC() clf.fit(X, y) SVC() ``` After being fitted, the model can then be used to predict new values: ``` clf.predict() array() ``` SVMs decision function (detailed in the Mathematical formulation) depends on some subset of the training data, called the support vectors. Some properties of these support vectors can be found in attributes support_vectors_, support_ and n_support_: ``` get support vectors clf.support_vectors_ array([[0., 0.], [1., 1.]]) get indices of support vectors clf.support_ array([0, 1]...) get number of support vectors for each class clf.n_support_ array([1, 1]...) ``` Examples SVM: Maximum margin separating hyperplane SVM-Anova: SVM with univariate feature selection Plot classification probability 1.4.1.1. Multi-class classification# SVC and NuSVC implement the “one-versus-one” approach for multi-class classification. In total, n_classes(n_classes -1)/ 2 classifiers are constructed and each one trains data from two classes. To provide a consistent interface with other classifiers, the decision_function_shape option allows to monotonically transform the results of the “one-versus-one” classifiers to a “one-vs-rest” decision function of shape (n_samples,n_classes), which is the default setting of the parameter (default=’ovr’). ``` X = Y = [0, 1, 2, 3] clf = svm.SVC(decision_function_shape='ovo') clf.fit(X, Y) SVC(decision_function_shape='ovo') dec = clf.decision_function() dec.shape # 6 classes: 43/2 = 6 6 clf.decision_function_shape = "ovr" dec = clf.decision_function() dec.shape # 4 classes 4 ``` On the other hand, LinearSVC implements “one-vs-the-rest” multi-class strategy, thus training n_classes models. ``` lin_clf = svm.LinearSVC() lin_clf.fit(X, Y) LinearSVC() dec = lin_clf.decision_function() dec.shape 4 ``` See Mathematical formulation for a complete description of the decision function. Details on multi-class strategies# Note that the LinearSVC also implements an alternative multi-class strategy, the so-called multi-class SVM formulated by Crammer and Singer , by using the option multi_class='crammer_singer'. In practice, one-vs-rest classification is usually preferred, since the results are mostly similar, but the runtime is significantly less. For “one-vs-rest” LinearSVC the attributes coef_ and intercept_ have the shape (n_classes,n_features) and (n_classes,) respectively. Each row of the coefficients corresponds to one of the n_classes “one-vs-rest” classifiers and similar for the intercepts, in the order of the “one” class. In the case of “one-vs-one” SVC and NuSVC, the layout of the attributes is a little more involved. In the case of a linear kernel, the attributes coef_ and intercept_ have the shape (n_classes(n_classes -1)/2,n_features) and (n_classes (n_classes - 1) / 2) respectively. This is similar to the layout for LinearSVC described above, with each row now corresponding to a binary classifier. The order for classes 0 to n is “0 vs 1”, “0 vs 2” , … “0 vs n”, “1 vs 2”, “1 vs 3”, “1 vs n”, . . . “n-1 vs n”. The shape of dual_coef_ is (n_classes-1,n_SV) with a somewhat hard to grasp layout. The columns correspond to the support vectors involved in any of the n_classes(n_classes -1)/ 2 “one-vs-one” classifiers. Each support vector v has a dual coefficient in each of the n_classes - 1 classifiers comparing the class of v against another class. Note that some, but not all, of these dual coefficients, may be zero. The n_classes - 1 entries in each column are these dual coefficients, ordered by the opposing class. This might be clearer with an example: consider a three class problem with class 0 having three support vectors and class 1 and 2 having two support vectors and respectively. For each support vector , there are two dual coefficients. Let’s call the coefficient of support vector in the classifier between classes and . Then dual_coef_ looks like this: \| \| \| \| \| \| \| \| --- --- --- \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| Coefficients for SVs of class 0 \| \| \| Coefficients for SVs of class 1 \| \| Coefficients for SVs of class 2 \| \| Examples Plot different SVM classifiers in the iris dataset 1.4.1.2. Scores and probabilities# The decision_function method of SVC and NuSVC gives per-class scores for each sample (or a single score per sample in the binary case). When the constructor option probability is set to True, class membership probability estimates (from the methods predict_proba and predict_log_proba) are enabled. In the binary case, the probabilities are calibrated using Platt scaling : logistic regression on the SVM’s scores, fit by an additional cross-validation on the training data. In the multiclass case, this is extended as per . Note The same probability calibration procedure is available for all estimators via the CalibratedClassifierCV (see Probability calibration). In the case of SVC and NuSVC, this procedure is builtin to libsvm which is used under the hood, so it does not rely on scikit-learn’s CalibratedClassifierCV. The cross-validation involved in Platt scaling is an expensive operation for large datasets. In addition, the probability estimates may be inconsistent with the scores: the “argmax” of the scores may not be the argmax of the probabilities in binary classification, a sample may be labeled by predict as belonging to the positive class even if the output of predict_proba is less than 0.5; and similarly, it could be labeled as negative even if the output of predict_proba is more than 0.5. Platt’s method is also known to have theoretical issues. If confidence scores are required, but these do not have to be probabilities, then it is advisable to set probability=False and use decision_function instead of predict_proba. Please note that when decision_function_shape='ovr' and n_classes> 2, unlike decision_function, the predict method does not try to break ties by default. You can set break_ties=True for the output of predict to be the same as np.argmax(clf.decision_function(...),axis=1), otherwise the first class among the tied classes will always be returned; but have in mind that it comes with a computational cost. See SVM Tie Breaking Example for an example on tie breaking. 1.4.1.3. Unbalanced problems# In problems where it is desired to give more importance to certain classes or certain individual samples, the parameters class_weight and sample_weight can be used. SVC (but not NuSVC) implements the parameter class_weight in the fit method. It’s a dictionary of the form {class_label:value}, where value is a floating point number > 0 that sets the parameter C of class class_label to C value. The figure below illustrates the decision boundary of an unbalanced problem, with and without weight correction. SVC, NuSVC, SVR, NuSVR, LinearSVC, LinearSVR and OneClassSVM implement also weights for individual samples in the fit method through the sample_weight parameter. Similar to class_weight, this sets the parameter C for the i-th example to Csample_weight[i], which will encourage the classifier to get these samples right. The figure below illustrates the effect of sample weighting on the decision boundary. The size of the circles is proportional to the sample weights: Examples SVM: Separating hyperplane for unbalanced classes SVM: Weighted samples 1.4.2. Regression# The method of Support Vector Classification can be extended to solve regression problems. This method is called Support Vector Regression. The model produced by support vector classification (as described above) depends only on a subset of the training data, because the cost function for building the model does not care about training points that lie beyond the margin. Analogously, the model produced by Support Vector Regression depends only on a subset of the training data, because the cost function ignores samples whose prediction is close to their target. There are three different implementations of Support Vector Regression: SVR, NuSVR and LinearSVR. LinearSVR provides a faster implementation than SVR but only considers the linear kernel, while NuSVR implements a slightly different formulation than SVR and LinearSVR. Due to its implementation in liblinear LinearSVR also regularizes the intercept, if considered. This effect can however be reduced by carefully fine tuning its intercept_scaling parameter, which allows the intercept term to have a different regularization behavior compared to the other features. The classification results and score can therefore differ from the other two classifiers. See Implementation details for further details. As with classification classes, the fit method will take as argument vectors X, y, only that in this case y is expected to have floating point values instead of integer values: ``` from sklearn import svm X = y = [0.5, 2.5] regr = svm.SVR() regr.fit(X, y) SVR() regr.predict() array([1.5]) ``` Examples Support Vector Regression (SVR) using linear and non-linear kernels 1.4.3. Density estimation, novelty detection# The class OneClassSVM implements a One-Class SVM which is used in outlier detection. See Novelty and Outlier Detection for the description and usage of OneClassSVM. 1.4.4. Complexity# Support Vector Machines are powerful tools, but their compute and storage requirements increase rapidly with the number of training vectors. The core of an SVM is a quadratic programming problem (QP), separating support vectors from the rest of the training data. The QP solver used by the libsvm-based implementation scales between and depending on how efficiently the libsvm cache is used in practice (dataset dependent). If the data is very sparse should be replaced by the average number of non-zero features in a sample vector. For the linear case, the algorithm used in LinearSVC by the liblinear implementation is much more efficient than its libsvm-based SVC counterpart and can scale almost linearly to millions of samples and/or features. 1.4.5. Tips on Practical Use# Avoiding data copy: For SVC, SVR, NuSVC and NuSVR, if the data passed to certain methods is not C-ordered contiguous and double precision, it will be copied before calling the underlying C implementation. You can check whether a given numpy array is C-contiguous by inspecting its flags attribute. For LinearSVC (and LogisticRegression) any input passed as a numpy array will be copied and converted to the liblinear internal sparse data representation (double precision floats and int32 indices of non-zero components). If you want to fit a large-scale linear classifier without copying a dense numpy C-contiguous double precision array as input, we suggest to use the SGDClassifier class instead. The objective function can be configured to be almost the same as the LinearSVC model. Kernel cache size: For SVC, SVR, NuSVC and NuSVR, the size of the kernel cache has a strong impact on run times for larger problems. If you have enough RAM available, it is recommended to set cache_size to a higher value than the default of 200(MB), such as 500(MB) or 1000(MB). Setting C: C is 1 by default and it’s a reasonable default choice. If you have a lot of noisy observations you should decrease it: decreasing C corresponds to more regularization. LinearSVC and LinearSVR are less sensitive to C when it becomes large, and prediction results stop improving after a certain threshold. Meanwhile, larger C values will take more time to train, sometimes up to 10 times longer, as shown in . Support Vector Machine algorithms are not scale invariant, so it is highly recommended to scale your data. For example, scale each attribute on the input vector X to [0,1] or [-1,+1], or standardize it to have mean 0 and variance 1. Note that the same scaling must be applied to the test vector to obtain meaningful results. This can be done easily by using a Pipeline: ``` from sklearn.pipeline import make_pipeline from sklearn.preprocessing import StandardScaler from sklearn.svm import SVC clf = make_pipeline(StandardScaler(), SVC()) ``` See section Preprocessing data for more details on scaling and normalization. Regarding the shrinking parameter, quoting : We found that if the number of iterations is large, then shrinking can shorten the training time. However, if we loosely solve the optimization problem (e.g., by using a large stopping tolerance), the code without using shrinking may be much faster Parameter nu in NuSVC/OneClassSVM/NuSVR approximates the fraction of training errors and support vectors. In SVC, if the data is unbalanced (e.g. many positive and few negative), set class_weight='balanced' and/or try different penalty parameters C. Randomness of the underlying implementations: The underlying implementations of SVC and NuSVC use a random number generator only to shuffle the data for probability estimation (when probability is set to True). This randomness can be controlled with the random_state parameter. If probability is set to False these estimators are not random and random_state has no effect on the results. The underlying OneClassSVM implementation is similar to the ones of SVC and NuSVC. As no probability estimation is provided for OneClassSVM, it is not random. The underlying LinearSVC implementation uses a random number generator to select features when fitting the model with a dual coordinate descent (i.e. when dual is set to True). It is thus not uncommon to have slightly different results for the same input data. If that happens, try with a smaller tol parameter. This randomness can also be controlled with the random_state parameter. When dual is set to False the underlying implementation of LinearSVC is not random and random_state has no effect on the results. Using L1 penalization as provided by LinearSVC(penalty='l1', dual=False) yields a sparse solution, i.e. only a subset of feature weights is different from zero and contribute to the decision function. Increasing C yields a more complex model (more features are selected). The C value that yields a “null” model (all weights equal to zero) can be calculated using l1_min_c. 1.4.6. Kernel functions# The kernel function can be any of the following: linear: . polynomial: , where is specified by parameter degree, by coef0. rbf: , where is specified by parameter gamma, must be greater than 0. sigmoid , where is specified by coef0. Different kernels are specified by the kernel parameter: ``` linear_svc = svm.SVC(kernel='linear') linear_svc.kernel 'linear' rbf_svc = svm.SVC(kernel='rbf') rbf_svc.kernel 'rbf' ``` See also Kernel Approximation for a solution to use RBF kernels that is much faster and more scalable. 1.4.6.1. Parameters of the RBF Kernel# When training an SVM with the Radial Basis Function (RBF) kernel, two parameters must be considered: C and gamma. The parameter C, common to all SVM kernels, trades off misclassification of training examples against simplicity of the decision surface. A low C makes the decision surface smooth, while a high C aims at classifying all training examples correctly. gamma defines how much influence a single training example has. The larger gamma is, the closer other examples must be to be affected. Proper choice of C and gamma is critical to the SVM’s performance. One is advised to use GridSearchCV with C and gamma spaced exponentially far apart to choose good values. Examples RBF SVM parameters Scaling the regularization parameter for SVCs 1.4.6.2. Custom Kernels# You can define your own kernels by either giving the kernel as a python function or by precomputing the Gram matrix. Classifiers with custom kernels behave the same way as any other classifiers, except that: Field support_vectors_ is now empty, only indices of support vectors are stored in support_ A reference (and not a copy) of the first argument in the fit() method is stored for future reference. If that array changes between the use of fit() and predict() you will have unexpected results. Using Python functions as kernels# You can use your own defined kernels by passing a function to the kernel parameter. Your kernel must take as arguments two matrices of shape (n_samples_1,n_features), (n_samples_2,n_features) and return a kernel matrix of shape (n_samples_1,n_samples_2). The following code defines a linear kernel and creates a classifier instance that will use that kernel: ``` import numpy as np from sklearn import svm def my_kernel(X, Y): ... return np.dot(X, Y.T) ... clf = svm.SVC(kernel=my_kernel) ``` Using the Gram matrix# You can pass pre-computed kernels by using the kernel='precomputed' option. You should then pass Gram matrix instead of X to the fit and predict methods. The kernel values between all training vectors and the test vectors must be provided: ``` import numpy as np from sklearn.datasets import make_classification from sklearn.model_selection import train_test_split from sklearn import svm X, y = make_classification(n_samples=10, random_state=0) X_train , X_test , y_train, y_test = train_test_split(X, y, random_state=0) clf = svm.SVC(kernel='precomputed') linear kernel computation gram_train = np.dot(X_train, X_train.T) clf.fit(gram_train, y_train) SVC(kernel='precomputed') predict on training examples gram_test = np.dot(X_test, X_train.T) clf.predict(gram_test) array([0, 1, 0]) ``` Examples SVM with custom kernel 1.4.7. Mathematical formulation# A support vector machine constructs a hyper-plane or set of hyper-planes in a high or infinite dimensional space, which can be used for classification, regression or other tasks. Intuitively, a good separation is achieved by the hyper-plane that has the largest distance to the nearest training data points of any class (so-called functional margin), since in general the larger the margin the lower the generalization error of the classifier. The figure below shows the decision function for a linearly separable problem, with three samples on the margin boundaries, called “support vectors”: In general, when the problem isn’t linearly separable, the support vectors are the samples within the margin boundaries. We recommend and as good references for the theory and practicalities of SVMs. 1.4.7.1. SVC# Given training vectors , i=1,…, n, in two classes, and a vector , our goal is to find and such that the prediction given by is correct for most samples. SVC solves the following primal problem: Intuitively, we’re trying to maximize the margin (by minimizing ), while incurring a penalty when a sample is misclassified or within the margin boundary. Ideally, the value would be for all samples, which indicates a perfect prediction. But problems are usually not always perfectly separable with a hyperplane, so we allow some samples to be at a distance from their correct margin boundary. The penalty term C controls the strength of this penalty, and as a result, acts as an inverse regularization parameter (see note below). The dual problem to the primal is where is the vector of all ones, and is an by positive semidefinite matrix, , where is the kernel. The terms are called the dual coefficients, and they are upper-bounded by . This dual representation highlights the fact that training vectors are implicitly mapped into a higher (maybe infinite) dimensional space by the function : see kernel trick. Once the optimization problem is solved, the output of decision_function for a given sample becomes: and the predicted class corresponds to its sign. We only need to sum over the support vectors (i.e. the samples that lie within the margin) because the dual coefficients are zero for the other samples. These parameters can be accessed through the attributes dual_coef_ which holds the product , support_vectors_ which holds the support vectors, and intercept_ which holds the independent term . Note While SVM models derived from libsvm and liblinear use C as regularization parameter, most other estimators use alpha. The exact equivalence between the amount of regularization of two models depends on the exact objective function optimized by the model. For example, when the estimator used is Ridge regression, the relation between them is given as . LinearSVC# The primal problem can be equivalently formulated as where we make use of the hinge loss. This is the form that is directly optimized by LinearSVC, but unlike the dual form, this one does not involve inner products between samples, so the famous kernel trick cannot be applied. This is why only the linear kernel is supported by LinearSVC ( is the identity function). NuSVC# The -SVC formulation is a reparameterization of the -SVC and therefore mathematically equivalent. We introduce a new parameter (instead of ) which controls the number of support vectors and margin errors: is an upper bound on the fraction of margin errors and a lower bound of the fraction of support vectors. A margin error corresponds to a sample that lies on the wrong side of its margin boundary: it is either misclassified, or it is correctly classified but does not lie beyond the margin. 1.4.7.2. SVR# Given training vectors , i=1,…, n, and a vector -SVR solves the following primal problem: Here, we are penalizing samples whose prediction is at least away from their true target. These samples penalize the objective by or , depending on whether their predictions lie above or below the tube. The dual problem is where is the vector of all ones, is an by positive semidefinite matrix, is the kernel. Here training vectors are implicitly mapped into a higher (maybe infinite) dimensional space by the function . The prediction is: These parameters can be accessed through the attributes dual_coef_ which holds the difference , support_vectors_ which holds the support vectors, and intercept_ which holds the independent term LinearSVR# The primal problem can be equivalently formulated as where we make use of the epsilon-insensitive loss, i.e. errors of less than are ignored. This is the form that is directly optimized by LinearSVR. 1.4.8. Implementation details# Internally, we use libsvm and liblinear to handle all computations. These libraries are wrapped using C and Cython. For a description of the implementation and details of the algorithms used, please refer to their respective papers.
188917	https://wou.edu/chemistry/courses/online-chemistry-textbooks/ch450-and-ch451-biochemistry-defining-life-at-the-molecular-level/chapter-7-catalytic-mechanisms-of-enzymes/	Admission Cost Academics Life at WOU Athletics Give Select Language▼ Home » Student Resources » Online Chemistry Textbooks » CH450 and CH451: Biochemistry - Defining Life at the Molecular Level » Chapter 7: Catalytic Mechanisms of Enzymes Menu CH450 and CH451: Biochemistry - Defining Life at the Molecular Level CH450 and CH451: Biochemistry – Defining Life at the Molecular Level CH450 Biochemistry I – Student and Teacher Resources CH451 Biochemistry II – Student and Teacher Resources Chapter 1: The Foundations of Biochemistry Chapter 2: Protein Structure Chapter 3: Investigating Proteins Chapter 4: DNA, RNA, and the Human Genome Chapter 5: Investigating DNA Chapter 6: Enzyme Principles and Biotechnological Applications Chapter 7: Catalytic Mechanisms of Enzymes Chapter 8 – Protein Regulation and Degradation Chapter 9: DNA Replication Chapter 10: Transcription and RNA Processing Chapter 11: Translation Chapter 12: DNA Damage and Repair Chapter 13: Transcriptional Control and Epigenetics Original text Rate this translation Your feedback will be used to help improve Google Translate
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Synonyms grow, develop, spread, mount, expand, build up, swell, wax, enlarge, escalate, multiply, fill out, get bigger, proliferate, snowball, dilate in the sense of succeed Definition to do well in a specified field the skills and qualities needed to succeed Synonyms make it (informal), do well, be successful, arrive (informal), triumph, thrive, flourish, make good, prosper, cut it (informal), make the grade (informal), get to the top, crack it (informal), hit the jackpot (informal), bring home the bacon (informal), make your mark (informal), gain your end, carry all before you, do all right for yourself in the sense of wax Definition to increase gradually in size, strength, or power Portugal and Spain had vast empires which waxed and waned. 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188919	https://www.studysmarter.co.uk/explanations/math/mechanics-maths/constant-acceleration-equations/	Login Sign up Log out Go to App Go to App Learning Materials Explanations Anthropology Archaeology Architecture Art and Design Bengali Biology Business Studies Chemistry Chinese Combined Science Computer Science Economics Engineering English English Literature Environmental Science French Geography German Greek History Hospitality and Tourism Human Geography Japanese Italian Law Macroeconomics Marketing Math Media Studies Medicine Microeconomics Music Nursing Nutrition and Food Science Physics Politics Polish Psychology Religious Studies Sociology Spanish Sports Sciences Translation Features Flashcards StudySmarter AI Notes Study Plans Study Sets Exams Discover Find a job Student Deals Magazine Mobile App Constant Acceleration Equations For a body moving in a single direction with constant acceleration, the constant acceleration or SUVAT equations are used to connect five different variables of motion. 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Solve a problem Get instant explanations on your homework or other topics. Review generated flashcards Sign up for free to start learning or create your own AI flashcards Sign up for free You have reached the daily AI limit Start learning or create your own AI flashcards StudySmarter Editorial Team Team Constant Acceleration Equations Teachers 6 minutes reading time Checked by StudySmarter Editorial Team Save Article Save Article Sign up for free to save, edit & create flashcards. Save Article Save Article Fact Checked Content Last Updated: 22.02.2023 Published at: 09.02.2022 6 min reading time Applied Mathematics Calculus Decision Maths Discrete Mathematics Geometry Logic and Functions Mechanics Maths Acceleration And Time Acceleration and Velocity Angular Speed Assumptions Calculus Kinematics Coefficient of Friction Connected Particles Conservation of Mechanical Energy Constant Acceleration Constant Acceleration Equations Converting Units Mechanics Damped harmonic oscillator Direct Impact and Newton's Law of Restitution Elastic Energy Elastic Strings and Springs Force as a Vector Kinematics Newton's First Law Newton's Law of Gravitation Newton's Second Law Newton's Third Law Power Problems involving Relative Velocity Projectiles Pulleys Relative Motion Resolving Forces Rigid Bodies in Equilibrium Stability Statics and Dynamics Tension in Strings The Trajectory of a Projectile Variable Acceleration Vertical Oscillation Work Done by a Constant Force Probability and Statistics Pure Maths Statistics Theoretical and Mathematical Physics Contents Applied Mathematics Calculus Decision Maths Discrete Mathematics Geometry Logic and Functions Mechanics Maths Acceleration And Time Acceleration and Velocity Angular Speed Assumptions Calculus Kinematics Coefficient of Friction Connected Particles Conservation of Mechanical Energy Constant Acceleration Constant Acceleration Equations Converting Units Mechanics Damped harmonic oscillator Direct Impact and Newton's Law of Restitution Elastic Energy Elastic Strings and Springs Force as a Vector Kinematics Newton's First Law Newton's Law of Gravitation Newton's Second Law Newton's Third Law Power Problems involving Relative Velocity Projectiles Pulleys Relative Motion Resolving Forces Rigid Bodies in Equilibrium Stability Statics and Dynamics Tension in Strings The Trajectory of a Projectile Variable Acceleration Vertical Oscillation Work Done by a Constant Force Probability and Statistics Pure Maths Statistics Theoretical and Mathematical Physics Contents Fact Checked Content Last Updated: 22.02.2023 6 min reading time Content creation process designed by Content cross-checked by Content quality checked by Sign up for free to save, edit & create flashcards. Save Article Save Article Jump to a key chapter s = Displacement – the total displacement of the body since the beginning of the measure at a given point of time. u = Initial velocity – the velocity of the body at the beginning of the measurement. v = Final velocity – the velocity of the body at the end of the measurement. a = Acceleration – the constant acceleration of the object throughout the measurement. t = Time taken – the time elapsed from the start to the end of the measurement. The five constant acceleration equations There are five different constant acceleration equations that are used to connect and solve for the variables above. It is a good idea to learn these equations by heart. (v = u + at) (s = \frac{1}{2} (u + v) t) (s = ut + \frac{1}{2}at^2) (s = vt - \frac{1}{2}at^2) (v^2 = u^2 + 2 as) Note that each equation has four of the five SUVAT variables. Given any three variables, it would be possible to solve for any of the other two variables. When can you use the SUVAT equations? The SUVAT equations apply for a body moving in a straight line with constant acceleration. Deriving the constant acceleration equations Let us look at how we obtain these equations. Equation 1: By definition, acceleration is the change in velocity per unit time. The following diagram demonstrates the concept. A body with initial velocity u accelerates with a constant acceleration to attain a final velocity v after time t. Let us express the definition mathematically. [Acceleration = \frac {change \space in \space velocity}{change \space in \space time} \rightarrow a = \frac {(v - u)}{t}] Rearranging the above equation, we get the first equation : [v = u + at] Equation 2: Remember, we are dealing with constant acceleration here. So the average velocity across the duration of motion is (\frac {u+v}{2}). Multiplying the average velocity by time gives the displacement. Therefore, (s = \frac{u + v}{2} \cdot t) This gives us the second equation, (s = \frac{1}{2} (u + v) t) Equation 3: To obtain the third equation, directly substitute the value of v from the first equation into the second equation. (s = \frac{1}{2} (u + v) t \rightarrow s = \frac{1}{2} (u + u + at) t) This gives us the third equation, (s = ut + \frac{1}{2}at^2) Equation 4: To obtain the fourth equation, first rearrange the first equation and express it in terms of u. (u = v – at) Substitute this value of u into the third equation, (s = ut + \frac{1}{2} at^2 \rightarrow s = (v - at) t + \frac{1}{2}at^2) This gives us the fourth equation, (s = vt - \frac{1}{2}at^2) Equation 5: To obtain the fifth equation, first rearrange the first equation and express it in terms of t. (t = \frac{v - u}{a}) Substitute this value of t into the second equation, (s = \frac{1}{2} (u + v) t \rightarrow s = \frac{(u + v) \cdot (v - u)}{2a} \rightarrow 2as = (v + u) (v – u) \rightarrow 2as = v^2 - u^2) This gives us the fifth equation, (v^2 = u^2 + 2as) Solving problems using constant acceleration equations Let us look at some examples of problems that can be solved using the constant acceleration equations. A car with an initial velocity of 8 m/s is accelerating at a rate of 2 m/s². How long will it take to reach a speed of 20 m/s? Solution 1 Here, v = 20 m/s, u = 8 m/s, a = 2 m/s². [v = u + at \rightarrow t = \frac{v - u}{a} \rightarrow t = \frac{20 - 8}{2} = 6 s] A car with an initial velocity of 8 m/s is accelerating at a rate of 2 m/s². How long will it take to travel a distance of 200 m? Solution 2 Here, s = 200 m, u = 8 m/s, a = 2 m/s². (s = ut + \frac{1}{2}at^2 \rightarrow 65 = 8t + \frac{1}{2} 2t^2 \rightarrow t^2 + 8t – 65 = 0 \rightarrow t = 5) Note: The obtained quadratic equation gives two values, 5 and -13. Time cannot be negative, so we take the positive value as the answer. A marathon runner decides to accelerate during the last 200 meters of a race. He accelerated at a rate of 0.07 m/s², eventually crossing the finish line at a speed of 8 m/s. What speed was he running at before he decided to accelerate? Solution 3 Here, s = 200 m, v = 8 m/s, a = 0.07 m/s². (v^2 = u^2 + 2 as \rightarrow u^2 = v^2 - 2as = 8 \cdot 8 - 2 \cdot 0.07 \cdot 200 \rightarrow u^2 = 36 \rightarrow u = 6 m/s) A cyclist is travelling along a straight road. It accelerates at a constant rate from a velocity of 4 m/s to a velocity of 7.5 m/s² in 40 seconds. Find: a) the distance she travels in these 40 seconds. b) her acceleration in these 40 seconds. Solution 4 a) (s = \frac{1}{2} (u + v) t \rightarrow s = \frac{1}{2} (4 + 7.5) \cdot 40 = 230 m) b) (v = u + at \rightarrow 7.5 = 4 + 40a \rightarrow a = \frac{7.5 - 4}{40} = 0.0875 m/s^2) A ball is thrown up with an initial velocity of 39.2 m/s. How long will it take the ball to reach its peak height assuming g = 9.8 m/s² Solution 5 Here, the acceleration of the ball is -9.8 m/s², as it is the force of gravity that is slowing the ball down. (v = u + at \rightarrow 0 = 39.2 - 9.8t \rightarrow t = 4 s) Constant Acceleration Equations - Key takeaways The constant acceleration equations are used to connect five different variables: s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time taken. The constant acceleration equations are applicable for a body moving in a straight line with constant acceleration. The constant acceleration equations can be derived starting from the basic definition that acceleration is the change in velocity per unit time. Each constant acceleration equation contains four of the five SUVAT variables. Given any of the three variables of an equation, it should be possible to solve for the fourth variable as the unknown. Similar topics in Math Probability and Statistics Statistics Mechanics Maths Geometry Calculus Pure Maths Decision Maths Logic and Functions Discrete Mathematics Theoretical and Mathematical Physics Applied Mathematics Related topics to Mechanics Maths Newton's Second Law Work Done by a Constant Force Newton's Law of Gravitation Newton's Third Law Variable Acceleration Kinematics Calculus Kinematics Conservation of Mechanical Energy Force as a Vector Resolving Forces Statics and Dynamics Tension in Strings Pulleys Projectiles Constant Acceleration Equations Constant Acceleration Newton's First Law Relative Motion Problems involving Relative Velocity The Trajectory of a Projectile Damped harmonic oscillator Vertical Oscillation Direct Impact and Newton's Law of Restitution Acceleration and Velocity Coefficient of Friction Elastic Strings and Springs Stability Rigid Bodies in Equilibrium Elastic Energy Assumptions Angular Speed Connected Particles Power Converting Units Mechanics Acceleration And Time Learn faster with the 0 flashcards about Constant Acceleration Equations Sign up for free to gain access to all our flashcards. Sign up with Email Already have an account? Log in Frequently Asked Questions about Constant Acceleration Equations What is the equation for constant acceleration? The equation for constant acceleration is v = u + at, where u= Initial velocity, v= Final velocity, a= Acceleration, t= Time taken What is the equation for motion with constant acceleration? There are 5 commonly used equations for motion with constant acceleration : 1) v = u + at 2) s = ½ (u + v) t 3) s = ut + ½at² 4) s = vt - ½at² 5) v² = u² + 2 as where s= Displacement, u= Initial velocity, v= Final velocity, a= Acceleration, t= Time taken Q3: What is constant acceleration? Acceleration is the change in velocity over time. If the rate of change of velocity of a body remains constant over time, it is known as constant acceleration. What are examples of constant acceleration? An example of constant acceleration is a body falling under the force of gravity with no other external force acting on it. In reality, it is very difficult to achieve perfect constant acceleration because there are typically multiple forces acting on any object. What are the five equations of motion? There are five commonly used equations for motion with constant acceleration 1) v = u + at 2) s = ½ (u + v) t 3) s = ut + ½at² 4) s = vt - ½at² 5) v² = u² + 2 as where s= Displacement, u= Initial velocity, v= Final velocity, a= Acceleration, t= Time taken. Save Article How we ensure our content is accurate and trustworthy? At StudySmarter, we have created a learning platform that serves millions of students. Meet the people who work hard to deliver fact based content as well as making sure it is verified. Content Creation Process: Lily Hulatt Digital Content Specialist Lily Hulatt is a Digital Content Specialist with over three years of experience in content strategy and curriculum design. 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188920	https://www.ck12.org/c/arithmetic/percent-of-decrease/	Math Elementary Math Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Interactive Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Conventional Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Probability & Statistics Trigonometry Math Analysis Precalculus Calculus What's the difference? Science Grade K to 5 Earth Science Life Science Physical Science Biology Chemistry Physics Advanced Biology FlexLets Math FlexLets Science FlexLets English Writing Spelling Social Studies Economics Geography Government History World History Philosophy Sociology More Astronomy Engineering Health Photography Technology College College Algebra College Precalculus Linear Algebra College Human Biology The Universe Adult Education Basic Education High School Diploma High School Equivalency Career Technical Ed English as 2nd Language Country Bhutan Brasil Chile Georgia India Translations Spanish Korean Deutsch Chinese Greek Polski EXPLORE Flexi A FREE Digital Tutor for Every Student FlexBooks 2.0 Customizable, digital textbooks in a new, interactive platform FlexBooks Customizable, digital textbooks Schools FlexBooks from schools and districts near you Study Guides Quick review with key information for each concept Adaptive Practice Building knowledge at each student’s skill level Simulations Interactive Physics & Chemistry Simulations PLIX Play. Learn. Interact. eXplore. CCSS Math Concepts and FlexBooks aligned to Common Core NGSS Concepts aligned to Next Generation Science Standards Certified Educator Stand out as an educator. Become CK-12 Certified. Webinars Live and archived sessions to learn about CK-12 Other Resources CK-12 Resources Concept Map Testimonials CK-12 Mission Meet the Team CK-12 Helpdesk FlexLets Know the essentials. Pick a Subject Donate Sign Up Percent of Decrease Percent of Decrease = Amount of Decrease/Original Amount Concept Map Discover related concepts in Math and Science. CK-12 ContentCommunity Content VIEW ALL CREATE We have provided many ways for you to learn about this topic. Create your own content Read Find the Percent of Decrease at grade8 Learn to find the percent of decrease. 1 1 More Read Video Determine a Percent of Change (decrease) basic Determines the percent of decrease from $80 to $65. 3 0 More Video Practice Estimated14 minsto complete Percent of Decrease at grade Practice 0 0 More DL Assessments Real World Healthy Heartbeats at grade6 7 8 Find out how to decrease your chances of getting heart disease. 2 0 More Real World Oops, looks like cookies are disabled on your browser. Click here to see how to enable them. Student Sign Up Are you a teacher? Having issues? Click here By signing up, I confirm that I have read and agree to the Terms of use and Privacy Policy Already have an account? No Results Found Your search did not match anything in .
188921	https://en.wikipedia.org/wiki/Sun	Sun - Wikipedia Jump to content Main menu Main menu move to sidebar hide Navigation Main page Contents Current events Random article About Wikipedia Contact us Contribute Help Learn to edit Community portal Recent changes Upload file Special pages Search Search Appearance Donate Create account Log in Personal tools Donate Create account Log in Pages for logged out editors learn more Contributions Talk Contents move to sidebar hide (Top) 1 Etymology 2 General characteristics Toggle General characteristics subsection - 2.1 Rotation - 3 Composition - 4 Structure Toggle Structure subsection - 4.1 Core - 4.2 Radiative zone - 4.3 Tachocline - 4.4 Convective zone - 4.5 Atmosphere - 4.5.1 Photosphere - 4.5.2 Chromosphere - 4.5.3 Corona - 4.6 Heliosphere - 5 Solar radiation - 6 Magnetic activity Toggle Magnetic activity subsection - 6.1 Sunspots - 6.2 Solar activity - 6.3 Coronal heating - 7 Life phases Toggle Life phases subsection - 7.1 Formation - 7.2 Main sequence - 7.3 After core hydrogen exhaustion - 8 Location Toggle Location subsection - 8.1 Solar System - 8.2 Celestial neighbourhood - 9 Motion - 10 Observational history Toggle Observational history subsection - 10.1 Early understanding - 10.2 Development of scientific understanding - 10.3 Solar space missions - 11 Observation by eyes Toggle Observation by eyes subsection - 11.1 Exposure to the eye - 11.2 Phenomena - 12 Religious aspects - 13 See also - 14 Notes - 15 References - 16 Further reading - 17 External links Toggle the table of contents Sun 304 languages Acèh Адыгабзэ Afrikaans Alemannisch አማርኛ Anarâškielâ अंगिका Ænglisc Аԥсшәа العربية Aragonés ܐܪܡܝܐ Armãneashti Arpetan অসমীয়া Asturianu Atikamekw अवधी Avañe'ẽ Авар Aymar aru Azərbaycanca تۆرکجه Basa Bali Bamanankan বাংলা Banjar 閩南語 / Bân-lâm-gí Basa Banyumasan Башҡортса Беларуская Беларуская (тарашкевіца) भोजपुरी Bikol Central Bislama Български Boarisch བོད་ཡིག Bosanski Brezhoneg Буряад Català Чӑвашла Cebuano Čeština Chavacano de Zamboanga Chi-Chewa ChiShona ChiTumbuka Corsu Cymraeg Dansk الدارجة Davvisámegiella Deitsch Deutsch ދިވެހިބަސް Diné bizaad Dolnoserbski डोटेली Eesti Ελληνικά Emiliàn e rumagnòl Эрзянь Español Esperanto Estremeñu Euskara Eʋegbe فارسی Fiji Hindi Føroyskt Français Frysk Fulfulde Furlan Gaeilge Gaelg Gàidhlig Galego ГӀалгӀай 贛語 گیلکی ગુજરાતી 𐌲𐌿𐍄𐌹𐍃𐌺 गोंयची कोंकणी / Gõychi Konknni Gungbe 客家語 / Hak-kâ-ngî Хальмг 한국어 Hausa Hawaiʻi Հայերեն हिन्दी Hornjoserbsce Hrvatski Ido Ilokano Bahasa Indonesia Interlingua Interlingue ᐃᓄᒃᑎᑐᑦ / inuktitut Iñupiatun Ирон IsiXhosa IsiZulu Íslenska Italiano עברית Jawa Kabɩyɛ ಕನ್ನಡ Kapampangan Къарачай-малкъар ქართული کٲشُر Kaszëbsczi Қазақша Kernowek Ikinyarwanda Kiswahili Коми Kongo Kotava Kreyòl ayisyen Kriyòl gwiyannen Kurdî Кыргызча Кырык мары Ladin Ladino Лакку ລາວ Latgaļu Latina Latviešu Lëtzebuergesch Лезги Lietuvių Li Niha Ligure Limburgs Lingála Lingua Franca Nova Livvinkarjala La .lojban. Luganda Lombard Magyar Madhurâ मैथिली Македонски Malagasy മലയാളം Malti Māori मराठी მარგალური مصرى ဘာသာမန် مازِرونی Bahasa Melayu ꯃꯤꯇꯩ ꯂꯣꯟ Mfantse Minangkabau 閩東語 / Mìng-dĕ̤ng-ngṳ̄ Mirandés Мокшень Монгол မြန်မာဘာသာ Nāhuatl Naijá Na Vosa Vakaviti Nederlands Nedersaksies नेपाली नेपाल भाषा 日本語 Napulitano ߒߞߏ Нохчийн Nordfriisk Norsk bokmål Norsk nynorsk Nouormand Novial Occitan ଓଡ଼ିଆ Oromoo Oʻzbekcha / ўзбекча ਪੰਜਾਬੀ Pälzisch Pangcah پنجابی ပအိုဝ်ႏဘာႏသာႏ Papiamentu پښتو Patois Перем коми ភាសាខ្មែរ Picard Piemontèis Pinayuanan Tok Pisin Plattdüütsch Polski Ποντιακά Português Qaraqalpaqsha Qırımtatarca Ripoarisch Română Romani čhib Rumantsch Runa Simi Русиньскый Русский Саха тыла Sakizaya Gagana Samoa संस्कृतम् Sängö ᱥᱟᱱᱛᱟᱲᱤ سرائیکی Sardu Scots Seediq Seeltersk Shqip Sicilianu සිංහල Simple English سنڌي Slovenčina Slovenščina Словѣньскъ / ⰔⰎⰑⰂⰡⰐⰠⰔⰍⰟ Ślůnski Soomaaliga کوردی Sranantongo Српски / srpski Srpskohrvatski / српскохрватски Sunda Suomi Svenska Tagalog தமிழ் Taclḥit Taqbaylit Татарча / tatarça တႆး Tayal తెలుగు ไทย Thuɔŋjäŋ ትግርኛ Тоҷикӣ ᏣᎳᎩ Tsetsêhestâhese ತುಳು Türkçe Türkmençe Twi Tyap Тыва дыл Удмурт Basa Ugi Українська اردو ئۇيغۇرچە / Uyghurche Vahcuengh Vèneto Vepsän kel’ Tiếng Việt Volapük Võro Walon Wayuunaiki 文言 West-Vlams Winaray Wolof 吴语 ייִדיש Yorùbá 粵語 Zazaki Zeêuws Žemaitėška 中文 Obolo Betawi Batak Mandailing Dagaare Kadazandusun Fɔ̀ngbè Jaku Iban Kumoring Yerwa Kanuri Руски ꠍꠤꠟꠐꠤ ᥖᥭᥰ ᥖᥬᥲ ᥑᥨᥒᥰ Tolışi ⵜⴰⵎⴰⵣⵉⵖⵜ ⵜⴰⵏⴰⵡⴰⵢⵜ Edit links Article Talk English Read View source View history Tools Tools move to sidebar hide Actions Read View source View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Expand all Edit interlanguage links Print/export Download as PDF Printable version In other projects Wikimedia Commons Wikibooks Wikiquote Wikidata item Appearance move to sidebar hide Text Small Standard Large This page always uses small font size Width Standard Wide The content is as wide as possible for your browser window. Color (beta) Automatic Light Dark This page is always in light mode. From Wikipedia, the free encyclopedia Star at the centre of the Solar System "The Sun" redirects here. For other uses, see Sun (disambiguation) and The Sun (disambiguation). Sun \| \| \| --- \| \| White glowing ball with black sunspots The Sun, viewed through a clear solar filter \| \| \| Names \| Sun, Sol, Sól, Helios \| \| Adjectives \| Solar \| \| Symbol \| Circle with dot in the middle \| \| Observation data \| \| \| Mean distance from Earth \| 1 AU 149,600,000 km 8 min 19 s, light speed \| \| Visual brightness \| −26.74 (V) \| \| Absolute magnitude \| 4.83 \| \| Spectral classification \| G2V \| \| Metallicity \| Z = 0.0122 \| \| Angular size \| 0.527–0.545° \| \| Orbital characteristics \| \| \| Mean distance from Milky Way core \| 24,000 to 28,000 light-years \| \| Galactic period \| 225–250 million years \| \| Velocity \| - 251 km/s orbit about Galactic Center - 20 km/s to stellar neighbourhood - 370 km/s to cosmic microwave background \| \| Obliquity \| - 7.25° (ecliptic) - 67.23° (galactic plane) \| \| Right ascension North pole \| 286.13° (286° 7′ 48″) \| \| Declination of North pole \| +63.87° (63° 52′ 12"N) \| \| Sidereal rotation period \| - 25.05 days (equator) - 34.4 days (poles) \| \| Equatorial rotation velocity \| 1.997 km/s \| \| Physical characteristics \| \| \| Equatorial radius \| 695,700 km 109 × Earth radii \| \| Flattening \| 0.00005 \| \| Surface area \| 6.09×1012 km2 12,000 × Earth \| \| Volume \| - 1.412×1018 km3 - 1,300,000 × Earth \| \| Mass \| - 1.9885×1030 kg - 332,950 Earths \| \| Average density \| 1.408 g/cm3 0.255 × Earth \| \| Age \| 4.6 billion years \| \| Equatorial surface gravity \| 274 m/s2 27.9 g0 \| \| Moment of inertia factor \| ≈0.070 \| \| Surface escape velocity \| 617.7 km/s 55 × Earth \| \| Temperature \| - 15,700,000 K (centre) - 5,772 K (photosphere) - 5,000,000 K (corona) \| \| Luminosity \| - 3.828×1026 W - 3.75×1028 lm - 98 lm/W efficacy \| \| Colour (B-V) \| 0.656 \| \| Mean radiance \| 2.009×107 W·m−2·sr−1 \| \| Photosphere composition by mass \| - 73.46% hydrogen - 24.85% helium - 0.77% oxygen - 0.29% carbon - 0.16% iron - 0.12% neon - 0.09% nitrogen - 0.07% silicon - 0.05% magnesium - 0.04% sulfur \| The Sun is the star at the centre of the Solar System. It is a massive, nearly perfect sphere of hot plasma, heated to incandescence by nuclear fusion reactions in its core, radiating the energy from its surface mainly as visible light and infrared radiation with 10% at ultraviolet energies. It is by far the most important source of energy for life on Earth. The Sun has been an object of veneration in many cultures and a central subject for astronomical research since antiquity. The Sun orbits the Galactic Center at a distance of 24,000 to 28,000 light-years. Its distance from Earth defines the astronomical unit, which is about 1.496×108 kilometres or about 8 light-minutes. Its diameter is about 1,391,400 km (864,600 mi), 109 times that of Earth. The Sun's mass is about 330,000 times that of Earth, making up about 99.86% of the total mass of the Solar System. The mass of outer layer of the Sun's atmosphere, its photosphere, consists mostly of hydrogen (~73%) and helium (~25%), with much smaller quantities of heavier elements, including oxygen, carbon, neon, and iron. The Sun is a G-type main-sequence star (G2V), informally called a yellow dwarf, though its light is actually white. It formed approximately 4.6 billion[a] years ago from the gravitational collapse of matter within a region of a large molecular cloud. Most of this matter gathered in the centre; the rest flattened into an orbiting disk that became the Solar System. The central mass became so hot and dense that it eventually initiated nuclear fusion in its core. Every second, the Sun's core fuses about 600 billion kilograms (kg) of hydrogen into helium and converts 4 billion kg of matter into energy. About 4 to 7 billion years from now, when hydrogen fusion in the Sun's core diminishes to the point where the Sun is no longer in hydrostatic equilibrium, its core will undergo a marked increase in density and temperature which will cause its outer layers to expand, eventually transforming the Sun into a red giant. After the red giant phase, models suggest the Sun will shed its outer layers and become a dense type of cooling star (a white dwarf), and no longer produce energy by fusion, but will still glow and give off heat from its previous fusion for perhaps trillions of years. After that, it is theorised to become a super dense black dwarf, giving off negligible energy. Etymology The English word sun developed from Old English sunne. Cognates appear in other Germanic languages, including West Frisian sinne, Dutch zon, Low German Sünn, Standard German Sonne, Bavarian Sunna, Old Norse sunna, and Gothic sunnō. All these words stem from Proto-Germanic sunnōn. This is ultimately related to the word for sun in other branches of the Indo-European language family, though in most cases a nominative stem with an l is found, rather than the genitive stem in n, as for example in Latin sōl, ancient Greek ἥλιος (hēlios), Welsh haul and Czech slunce, as well as (with l > r) Sanskrit स्वर् (svár) and Persian خور (xvar). Indeed, the l-stem survived in Proto-Germanic as well, as sōwelan, which gave rise to Gothic sauil (alongside sunnō) and Old Norse prosaic sól (alongside poetic sunna), and through it the words for sun in the modern Scandinavian languages: Swedish and Danish sol, Icelandic sól, etc. The principal adjectives for the Sun in English are sunny for sunlight and, in technical contexts, solar (/ˈsoʊlər/), from Latin sol. From the Greek helios comes the rare adjective heliac (/ˈhiːliæk/). In English, the Greek and Latin words occur in poetry as personifications of the Sun, Helios (/ˈhiːliəs/) and Sol (/ˈsɒl/), while in science fiction Sol may be used to distinguish the Sun from other stars. The term sol with a lowercase s is used by planetary astronomers for the duration of a solar day on another planet such as Mars. The astronomical symbol for the Sun is a circle with a central dot, ☉. It is used for such units as M☉ (Solar mass), R☉ (Solar radius) and L☉ (Solar luminosity). The scientific study of the Sun is called heliology. General characteristics Size comparison of major celestial objects in the Solar System, including the Sun The Sun is a G-type main-sequence star that makes up about 99.86% of the mass of the Solar System. It has an absolute magnitude of +4.83, estimated to be brighter than about 85% of the stars in the Milky Way, most of which are red dwarfs. It is more massive than 95% of the stars within 7 pc (23 ly). The Sun is a Population I, or heavy-element-rich,[b] star. Its formation approximately 4.6 billion years ago may have been triggered by shockwaves from one or more nearby supernovae. This is suggested by a high abundance of heavy elements in the Solar System, such as gold and uranium, relative to the abundances of these elements in so-called Population II, heavy-element-poor, stars. The heavy elements could most plausibly have been produced by endothermic nuclear reactions during a supernova, or by transmutation through neutron absorption within a massive second-generation star. The Sun is by far the brightest object in the Earth's sky, with an apparent magnitude of −26.74. This is about 13 billion times brighter than the next brightest star, Sirius, which has an apparent magnitude of −1.46. One astronomical unit (about 150 million kilometres; 93 million miles) is defined as the mean distance between the centres of the Sun and the Earth. The instantaneous distance varies by about ±2.5 million kilometres (1.6 million miles) as Earth moves from perihelion around 3 January to aphelion around 4 July. At its average distance, light travels from the Sun's horizon to Earth's horizon in about 8 minutes and 20 seconds, while light from the closest points of the Sun and Earth takes about two seconds less. The energy of this sunlight supports almost all life[c] on Earth by photosynthesis, and drives Earth's climate and weather. The Sun does not have a definite boundary, but its density decreases exponentially with increasing height above the photosphere. For the purpose of measurement, the Sun's radius is considered to be the distance from its centre to the edge of the photosphere, the apparent visible surface of the Sun. The roundness of the Sun is the relative difference between its radius at its equator, R eq {\displaystyle R_{\textrm {eq}}} , and at its pole, R pol {\displaystyle R_{\textrm {pol}}} , called the oblateness, Δ ⊙ ( R eq − R pol ) / R pol . {\displaystyle \Delta _{\odot }=(R_{\textrm {eq}}-R_{\textrm {pol}})/R_{\textrm {pol}}.} The value is difficult to measure. Atmospheric distortion means the measurement must be done on satellites; the value is very small meaning very precise technique is needed. The oblateness was once proposed to be sufficient to explain the perihelion precession of Mercury but Einstein proposed that general relativity could explain the precession using a spherical Sun. When high precision measurements of the oblateness became available via the Solar Dynamics Observatory and the Picard satellite the measured value was even smaller than expected, 8.2×10−6, or 8 parts per million. These measurements determined the Sun to be the natural object closest to a perfect sphere ever observed. The oblateness value remains constant independent of solar irradiation changes. The tidal effect of the planets is weak and does not significantly affect the shape of the Sun. Rotation Main article: Solar rotation The Sun rotates faster at its equator than at its poles. This differential rotation is caused by convective motion due to heat transport and the Coriolis force due to the Sun's rotation. In a frame of reference defined by the stars, the rotational period is approximately 25.6 days at the equator and 33.5 days at the poles. Viewed from Earth as it orbits the Sun, the apparent rotational period of the Sun at its equator is about 28 days. Viewed from a vantage point above its north pole, the Sun rotates counterclockwise around its axis of spin.[d] A survey of solar analogues suggests the early Sun was rotating up to ten times faster than it does today. This would have made the surface much more active, with greater X-ray and UV emission. Sunspots would have covered 5–30% of the surface. The rotation rate was gradually slowed by magnetic braking, as the Sun's magnetic field interacted with the outflowing solar wind. A vestige of this rapid primordial rotation still survives at the Sun's core, which rotates at a rate of once per week; four times the mean surface rotation rate. Composition See also: Molecules in stars The Sun consists mainly of the elements hydrogen and helium. At this time in the Sun's life, they account for 74.9% and 23.8%, respectively, of the mass of the Sun in the photosphere. All heavier elements, called metals in astronomy, account for less than 2% of the mass, with oxygen (roughly 1% of the Sun's mass), carbon (0.3%), neon (0.2%), and iron (0.2%) being the most abundant. The Sun's original chemical composition was inherited from the interstellar medium out of which it formed. Originally it would have been about 71.1% hydrogen, 27.4% helium, and 1.5% heavier elements. The hydrogen and most of the helium in the Sun would have been produced by Big Bang nucleosynthesis in the first 20 minutes of the universe, and the heavier elements were produced by previous generations of stars before the Sun was formed, and spread into the interstellar medium during the final stages of stellar life and by events such as supernovae. Since the Sun formed, the main fusion process has involved fusing hydrogen into helium. Over the past 4.6 billion years, the amount of helium and its location within the Sun has gradually changed. The proportion of helium within the core has increased from about 24% to about 60% due to fusion, and some of the helium and heavy elements have settled from the photosphere toward the centre of the Sun because of gravity. The proportions of heavier elements are unchanged. Heat is transferred outward from the Sun's core by radiation rather than by convection (see Radiative zone below), so the fusion products are not lifted outward by heat; they remain in the core, and gradually an inner core of helium has begun to form that cannot be fused because presently the Sun's core is not hot or dense enough to fuse helium. In the current photosphere, the helium fraction is reduced, and the metallicity is only 84% of what it was in the protostellar phase (before nuclear fusion in the core started). In the future, helium will continue to accumulate in the core, and in about 5 billion years this gradual build-up will eventually cause the Sun to exit the main sequence and become a red giant. The chemical composition of the photosphere is normally considered representative of the composition of the primordial Solar System. Typically, the solar heavy-element abundances described above are measured both by using spectroscopy of the Sun's photosphere and by measuring abundances in meteorites that have never been heated to melting temperatures. These meteorites are thought to retain the composition of the protostellar Sun and are thus not affected by the settling of heavy elements. The two methods generally agree well. Structure See also: Standard solar model Illustration of the Sun's structure, in false colour for contrast Core Main article: Solar core The core of the Sun extends from the centre to about 20–25% of the solar radius. It has a density of up to 150 g/cm3 (about 150 times the density of water) and a temperature of close to 15.7 million kelvin (K). By contrast, the Sun's surface temperature is about 5800 K. Recent analysis of SOHO mission data favours the idea that the core is rotating faster than the radiative zone outside it. Through most of the Sun's life, energy has been produced by nuclear fusion in the core region through the proton–proton chain; this process converts hydrogen into helium. Currently, 0.8% of the energy generated in the Sun comes from another sequence of fusion reactions called the CNO cycle; the proportion coming from the CNO cycle is expected to increase as the Sun becomes older and more luminous. The core is the only region of the Sun that produces an appreciable amount of thermal energy through fusion; 99% of the Sun's power is generated in the innermost 24% of its radius, and almost no fusion occurs beyond 30% of the radius. The rest of the Sun is heated by this energy as it is transferred outward through many successive layers, finally to the solar photosphere where it escapes into space through radiation (photons) or advection (massive particles). Illustration of a proton-proton reaction chain, from hydrogen forming deuterium, helium-3, and regular helium-4 The proton–proton chain occurs around 9.2×1037 times each second in the core, converting about 3.7×1038 protons into alpha particles (helium nuclei) every second (out of a total of ~8.9×1056 free protons in the Sun), or about 6.2×1011 kg/s. However, each proton (on average) takes around 9 billion years to fuse with another using the PP chain. Fusing four free protons (hydrogen nuclei) into a single alpha particle (helium nucleus) releases around 0.7% of the fused mass as energy, so the Sun releases energy at the mass–energy conversion rate of 4.26 billion kg/s (which requires 600 billion kg of hydrogen), for 384.6 yottawatts (3.846×1026 W), or 9.192×1010 megatons of TNT per second. The large power output of the Sun is mainly due to the huge size and density of its core (compared to Earth and objects on Earth), with only a fairly small amount of power being generated per cubic metre. Theoretical models of the Sun's interior indicate a maximum power density, or energy production, of approximately 276.5 watts per cubic metre at the centre of the core, which, according to Karl Kruszelnicki, is about the same power density inside a compost pile. The fusion rate in the core is in a self-correcting equilibrium: a slightly higher rate of fusion would cause the core to heat up more and expand slightly against the weight of the outer layers, reducing the density and hence the fusion rate and correcting the perturbation; and a slightly lower rate would cause the core to cool and shrink slightly, increasing the density and increasing the fusion rate and again reverting it to its present rate. Radiative zone Main article: Radiative zone Illustration of different stars' internal structure based on mass. The Sun in the middle has an inner radiating zone and an outer convective zone. The radiative zone is the thickest layer of the Sun, at 0.45 solar radii. From the core out to about 0.7 solar radii, thermal radiation is the primary means of energy transfer. The temperature drops from approximately 7 million to 2 million kelvins with increasing distance from the core. This temperature gradient is less than the value of the adiabatic lapse rate and hence cannot drive convection, which explains why the transfer of energy through this zone is by radiation instead of thermal convection. Ions of hydrogen and helium emit photons, which travel only a brief distance before being reabsorbed by other ions. The density drops a hundredfold (from 20,000 kg/m3 to 200 kg/m3) between 0.25 solar radii and 0.7 radii, the top of the radiative zone. Tachocline Main article: Tachocline The radiative zone and the convective zone are separated by a transition layer, the tachocline. This is a region where the sharp regime change between the uniform rotation of the radiative zone and the differential rotation of the convection zone results in a large shear between the two—a condition where successive horizontal layers slide past one another. Presently, it is hypothesised that a magnetic dynamo, or solar dynamo, within this layer generates the Sun's magnetic field. Convective zone Main article: Convection zone The Sun's convection zone extends from 0.7 solar radii (500,000 km) to near the surface. In this layer, the solar plasma is not dense or hot enough to transfer the heat energy of the interior outward via radiation. Instead, the density of the plasma is low enough to allow convective currents to develop and move the Sun's energy outward towards its surface. Material heated at the tachocline picks up heat and expands, thereby reducing its density and allowing it to rise. As a result, an orderly motion of the mass develops into thermal cells that carry most of the heat outward to the Sun's photosphere above. Once the material diffusively and radiatively cools just beneath the photospheric surface, its density increases, and it sinks to the base of the convection zone, where it again picks up heat from the top of the radiative zone and the convective cycle continues. At the photosphere, the temperature has dropped 350-fold to 5,700 K (9,800 °F) and the density to only 0.2 g/m3 (about 1/10,000 the density of air at sea level, and 1 millionth that of the inner layer of the convective zone). The thermal columns of the convection zone form an imprint on the surface of the Sun giving it a granular appearance called the solar granulation at the smallest scale and supergranulation at larger scales. Turbulent convection in this outer part of the solar interior sustains "small-scale" dynamo action over the near-surface volume of the Sun. The Sun's thermal columns are Bénard cells and take the shape of roughly hexagonal prisms. Atmosphere Main article: Stellar atmosphere The solar atmosphere is the region of the Sun that extends from the top of the convection zone to the inner boundary of the heliosphere. It is often divided into three primary layers: the photosphere, the chromosphere, and the corona. The chromosphere and corona are separated by a thin transition region that is frequently considered as an additional distinct layer.: 173–174 Some sources consider the heliosphere to be the outer or extended solar atmosphere. Photosphere Main article: Photosphere The photosphere is structured by convection cells referred to as granules. The visible surface of the Sun, the photosphere, is the layer below which the Sun becomes opaque to visible light. Photons produced in this layer escape the Sun through the transparent solar atmosphere above it and become solar radiation, sunlight. The change in opacity is due to the decreasing amount of H− ions, which absorb visible light easily. Conversely, the visible light perceived is produced as electrons react with hydrogen atoms to produce H− ions. The photosphere is tens to hundreds of kilometres thick, and is slightly less opaque than air on Earth. Because the upper part of the photosphere is cooler than the lower part, an image of the Sun appears brighter in the centre than on the edge or limb of the solar disk, in a phenomenon known as limb darkening. The spectrum of sunlight has approximately the spectrum of a black-body radiating at 5,772 K (9,930 °F), interspersed with atomic absorption lines from the tenuous layers above the photosphere. The photosphere has a particle density of ~1023 m−3 (about 0.37% of the particle number per volume of Earth's atmosphere at sea level). The photosphere is not fully ionised—the extent of ionisation is about 3%, leaving almost all of the hydrogen in atomic form. The coolest layer of the Sun is a temperature minimum region extending to about 500 km above the photosphere, and has a temperature of about 4,100 K. This part of the Sun is cool enough to allow for the existence of simple molecules such as carbon monoxide and water. Chromosphere Main article: Chromosphere Above the temperature minimum layer is a layer about 2,000 km thick, dominated by a spectrum of emission and absorption lines. It is called the chromosphere from the Greek root chroma, meaning colour, because the chromosphere is visible as a coloured flash at the beginning and end of total solar eclipses. The temperature of the chromosphere increases gradually with altitude, ranging up to around 20,000 K near the top. In the upper part of the chromosphere helium becomes partially ionised. The Sun's transition region taken by Hinode's Solar Optical Telescope The chromosphere and overlying corona are separated by a thin (about 200 km) transition region where the temperature rises rapidly from around 20,000 K in the upper chromosphere to coronal temperatures closer to 1,000,000 K. The temperature increase is facilitated by the full ionisation of helium in the transition region, which significantly reduces radiative cooling of the plasma. The transition region does not occur at a well-defined altitude, but forms a kind of nimbus around chromospheric features such as spicules and filaments, and is in constant, chaotic motion. The transition region is not easily visible from Earth's surface, but is readily observable from space by instruments sensitive to extreme ultraviolet. Corona Main article: Stellar corona During a total solar eclipse the solar corona can be seen with the naked eye. The corona is the next layer of the Sun. The low corona, near the surface of the Sun, has a particle density around 1015 m−3 to 1016 m−3.[e] The average temperature of the corona and solar wind is about 1,000,000–2,000,000 K; however, in the hottest regions it is 8,000,000–20,000,000 K. Although no complete theory yet exists to account for the temperature of the corona, at least some of its heat is known to be from magnetic reconnection. The outer boundary of the corona is located where the radially increasing, large-scale solar wind speed is equal to the radially decreasing Alfvén wave phase speed. This defines a closed, nonspherical surface, referred to as the Alfvén critical surface, below which coronal flows are sub-Alfvénic and above which the solar wind is super-Alfvénic. The height at which this transition occurs varies across space and with solar activity, reaching its lowest near solar minimum and its highest near solar maximum. In April 2021 the surface was crossed for the first time at heliocentric distances ranging from 16 to 20 solar radii by the Parker Solar Probe. Predictions of its full possible extent have placed its full range within 8 to 30 solar radii. Heliosphere Main article: Heliosphere Depiction of the heliosphere The heliosphere is defined as the region of space where the solar wind dominates over the interstellar medium. Turbulence and dynamic forces in the heliosphere cannot affect the shape of the solar corona within, because the information can only travel at the speed of Alfvén waves. The solar wind travels outward continuously through the heliosphere, forming the solar magnetic field into a spiral shape, until it impacts the heliopause more than 50 AU from the Sun. In December 2004, the Voyager 1 probe passed through a shock front that is thought to be part of the heliopause. In late 2012, Voyager 1 recorded a marked increase in cosmic ray collisions and a sharp drop in lower energy particles from the solar wind, which suggested that the probe had passed through the heliopause and entered the interstellar medium, and indeed did so on 25 August 2012, at approximately 122 astronomical units (18 Tm) from the Sun. The heliosphere has a heliotail which stretches out behind it due to the Sun's peculiar motion through the galaxy. Solar radiation Main articles: Sunlight and Solar irradiance The Sun seen through a light fog The Sun emits light across the visible spectrum. Its colour is white, with a CIE colour-space index near (0.3, 0.3), when viewed from space or when the Sun is high in the sky. The Solar radiance per wavelength peaks in the green portion of the spectrum when viewed from space. When the Sun is very low in the sky, atmospheric scattering renders the Sun yellow, red, orange, or magenta, and in rare occasions even green or blue. Some cultures mentally picture the Sun as yellow and some even red; the cultural reasons for this are debated. The Sun is classed as a G2 star, meaning it is a G-type star, with 2 indicating its surface temperature is in the second range of the G class. The solar constant is the amount of power that the Sun deposits per unit area that is directly exposed to sunlight. The solar constant is equal to approximately 1,368 W/m2 (watts per square metre) at a distance of one astronomical unit (AU) from the Sun (that is, at or near Earth's orbit). Sunlight on the surface of Earth is attenuated by Earth's atmosphere, so that less power arrives at the surface (closer to 1,000 W/m2) in clear conditions when the Sun is near the zenith. Sunlight at the top of Earth's atmosphere is composed (by total energy) of about 50% infrared light, 40% visible light, and 10% ultraviolet light. The atmosphere filters out over 70% of solar ultraviolet, especially at the shorter wavelengths. Solar ultraviolet radiation ionises Earth's dayside upper atmosphere, creating its electrically conducting ionosphere. Ultraviolet light from the Sun has antiseptic properties and can be used to sanitise tools and water. This radiation causes sunburn, and has other biological effects such as the production of vitamin D and sun tanning. It is the main cause of skin cancer. Ultraviolet light is strongly attenuated by Earth's ozone layer, so that the amount of UV varies greatly with latitude and has been partially responsible for many biological adaptations, including variations in human skin colour. High-energy gamma ray photons initially released with fusion reactions in the core are almost immediately absorbed by the solar plasma of the radiative zone, usually after travelling only a few millimetres. Re-emission happens in a random direction and usually at slightly lower energy. With this sequence of emissions and absorptions, it takes a long time for radiation to reach the Sun's surface. Estimates of the photon travel time range between 10,000 and 170,000 years. In contrast, it takes only 2.3 seconds for neutrinos, which account for about 2% of the total energy production of the Sun, to reach the surface. Because energy transport in the Sun is a process that involves photons in thermodynamic equilibrium with matter, the time scale of energy transport in the Sun is longer, on the order of 30,000,000 years. This is the time it would take the Sun to return to a stable state if the rate of energy generation in its core were suddenly changed. Electron neutrinos are released by fusion reactions in the core, but, unlike photons, they rarely interact with matter, so almost all are able to escape the Sun immediately. However, measurements of the number of these neutrinos produced in the Sun are lower than theories predict by a factor of 3. In 2001, the discovery of neutrino oscillation resolved the discrepancy: the Sun emits the number of electron neutrinos predicted by the theory, but neutrino detectors were missing 2⁄3 of them because the neutrinos had changed flavor by the time they were detected. Magnetic activity The Sun has a stellar magnetic field that varies across its surface. Its polar field is 1–2 gauss (0.0001–0.0002 T), whereas the field is typically 3,000 gauss (0.3 T) in features on the Sun called sunspots and 10–100 gauss (0.001–0.01 T) in solar prominences. The magnetic field varies in time and location. The quasi-periodic 11-year solar cycle is the most prominent variation in which the number and size of sunspots waxes and wanes. The solar magnetic field extends well beyond the Sun itself. The electrically conducting solar wind plasma carries the Sun's magnetic field into space, forming what is called the interplanetary magnetic field. In an approximation known as ideal magnetohydrodynamics, plasma only moves along magnetic field lines. As a result, the outward-flowing solar wind stretches the interplanetary magnetic field outward, forcing it into a roughly radial structure. For a simple dipolar solar magnetic field, with opposite hemispherical polarities on either side of the solar magnetic equator, a thin current sheet is formed in the solar wind. At great distances, the rotation of the Sun twists the dipolar magnetic field and corresponding current sheet into an Archimedean spiral structure called the Parker spiral. Sunspots Main article: Sunspot A large sunspot group observed in white light Sunspots are visible as dark patches on the Sun's photosphere and correspond to concentrations of magnetic field where convective transport of heat is inhibited from the solar interior to the surface. As a result, sunspots are slightly cooler than the surrounding photosphere, so they appear dark. At a typical solar minimum, few sunspots are visible, and occasionally none can be seen at all. Those that do appear are at high solar latitudes. As the solar cycle progresses toward its maximum, sunspots tend to form closer to the solar equator, a phenomenon known as Spörer's law. The largest sunspots can be tens of thousands of kilometres across. An 11-year sunspot cycle is half of a 22-year Babcock–Leighton dynamo cycle, which corresponds to an oscillatory exchange of energy between toroidal and poloidal solar magnetic fields. At solar-cycle maximum, the external poloidal dipolar magnetic field is near its dynamo-cycle minimum strength; but an internal toroidal quadrupolar field, generated through differential rotation within the tachocline, is near its maximum strength. At this point in the dynamo cycle, buoyant upwelling within the convective zone forces emergence of the toroidal magnetic field through the photosphere, giving rise to pairs of sunspots, roughly aligned east–west and having footprints with opposite magnetic polarities. The magnetic polarity of sunspot pairs alternates every solar cycle, a phenomenon described by Hale's law. During the solar cycle's declining phase, energy shifts from the internal toroidal magnetic field to the external poloidal field, and sunspots diminish in number and size. At solar-cycle minimum, the toroidal field is, correspondingly, at minimum strength, sunspots are relatively rare, and the poloidal field is at its maximum strength. With the rise of the next 11-year sunspot cycle, differential rotation shifts magnetic energy back from the poloidal to the toroidal field, but with a polarity that is opposite to the previous cycle. The process carries on continuously, and in an idealised, simplified scenario, each 11-year sunspot cycle corresponds to a change, then, in the overall polarity of the Sun's large-scale magnetic field. Solar activity Main article: Solar cycle Measurements from 2005 of solar cycle variation during the previous 30 years The Sun's magnetic field leads to many effects that are collectively called solar activity. Solar flares and coronal mass ejections tend to occur at sunspot groups. Slowly changing high-speed streams of solar wind are emitted from coronal holes at the photospheric surface. Both coronal mass ejections and high-speed streams of solar wind carry plasma and the interplanetary magnetic field outward into the Solar System. The effects of solar activity on Earth include auroras at moderate to high latitudes and the disruption of radio communications and electric power. Solar activity is thought to have played a large role in the formation and evolution of the Solar System. Changes in solar irradiance over the 11-year solar cycle have been correlated with changes in sunspot number. The solar cycle influences space weather conditions, including those surrounding Earth. For example, in the 17th century, the solar cycle appeared to have stopped entirely for several decades; few sunspots were observed during a period known as the Maunder minimum. This coincided in time with the era of the Little Ice Age, when Europe experienced unusually cold temperatures. Earlier extended minima have been discovered through analysis of tree rings and appear to have coincided with lower-than-average global temperatures. Coronal heating Main article: Stellar corona Unsolved problem in astronomy Why is the Sun's corona so much hotter than the Sun's surface? More unsolved problems in astronomy The temperature of the photosphere is approximately 6,000 K, whereas the temperature of the corona reaches 1,000,000–2,000,000 K. The high temperature of the corona shows that it is heated by something other than direct heat conduction from the photosphere. It is thought that the energy necessary to heat the corona is provided by turbulent motion in the convection zone below the photosphere, and two main mechanisms have been proposed to explain coronal heating. The first is wave heating, in which sound, gravitational or magnetohydrodynamic waves are produced by turbulence in the convection zone. These waves travel upward and dissipate in the corona, depositing their energy in the ambient matter in the form of heat. The other is magnetic heating, in which magnetic energy is continuously built up by photospheric motion and released through magnetic reconnection in the form of large solar flares and myriad similar but smaller events—nanoflares. Currently, it is unclear whether waves are an efficient heating mechanism. All waves except Alfvén waves have been found to dissipate or refract before reaching the corona. In addition, Alfvén waves do not easily dissipate in the corona. The current research focus has therefore shifted toward flare heating mechanisms. Life phases Main articles: Formation and evolution of the Solar System and Stellar evolution Overview of the evolution of a star like the Sun, from collapsing protostar at left to red giant stage at right The Sun today is roughly halfway through the main-sequence portion of its life. It has not changed dramatically in over four billion[a] years and will remain fairly stable for about five billion more. However, after hydrogen fusion in its core has stopped, the Sun will undergo dramatic changes, both internally and externally. Formation Further information: Formation and evolution of the Solar System The Sun formed about 4.6 billion years ago from the collapse of part of a giant molecular cloud that consisted mostly of hydrogen and helium and that probably gave birth to many other stars. This age is estimated using computer models of stellar evolution and through nucleocosmochronology. The result is consistent with the radiometric date of the oldest Solar System material, at 4.567 billion years ago. Studies of ancient meteorites reveal traces of stable daughter nuclei of short-lived isotopes, such as iron-60, that form only in exploding, short-lived stars. This indicates that one or more supernovae must have occurred near the location where the Sun formed. A shock wave from a nearby supernova would have triggered the formation of the Sun by compressing the matter within the molecular cloud and causing certain regions to collapse under their own gravity. As one fragment of the cloud collapsed it also began to rotate due to conservation of angular momentum and heat up with the increasing pressure. Much of the mass became concentrated in the centre, whereas the rest flattened out into a disk that would become the planets and other Solar System bodies. Gravity and pressure within the core of the cloud generated a lot of heat as it accumulated more matter from the surrounding disk, eventually triggering nuclear fusion. The stars HD 162826 and HD 186302 share similarities with the Sun and are hypothesised to be its stellar siblings, formed in the same molecular cloud. The violent youth of stars like the Sun Main sequence Further information: Main sequence Evolution of a Sun-like star. The track of a one solar mass star on the Hertzsprung–Russell diagram is shown from the main sequence to the white dwarf stage. The Sun is about halfway through its main-sequence stage, during which nuclear fusion reactions in its core fuse hydrogen into helium. Each second, more than four billion kilograms of matter are converted into energy within the Sun's core, producing neutrinos and solar radiation. At this rate, the Sun has so far converted around 100 times the mass of Earth into energy, about 0.03% of the total mass of the Sun. The Sun will spend a total of approximately 10 to 11 billion years as a main-sequence star before the red giant phase of the Sun. At the 8 billion year mark, the Sun will be at its hottest point according to the ESA's Gaia space observatory mission in 2022. The Sun is gradually becoming hotter in its core, hotter at the surface, larger in radius, and more luminous during its time on the main sequence: since the beginning of its main sequence life, it has expanded in radius by 15% and the surface has increased in temperature from 5,620 K (9,660 °F) to 5,772 K (9,930 °F), resulting in a 48% increase in luminosity from 0.677 solar luminosities to its present-day 1.0 solar luminosity. This occurs because the helium atoms in the core have a higher mean molecular weight than the hydrogen atoms that were fused, resulting in less thermal pressure. The core is therefore shrinking, allowing the outer layers of the Sun to move closer to the centre, releasing gravitational potential energy. According to the virial theorem, half of this released gravitational energy goes into heating, which leads to a gradual increase in the rate at which fusion occurs and thus an increase in the luminosity. This process speeds up as the core gradually becomes denser. At present, it is increasing in brightness by about 1% every 100 million years. It will take at least 1 billion years from now to deplete liquid water from the Earth from such increase. After that, the Earth will cease to be able to support complex, multicellular life and the last remaining multicellular organisms on the planet will suffer a final, complete mass extinction. After core hydrogen exhaustion The size of the current Sun (now in the main sequence) compared to its estimated size during its red-giant phase in the future The Sun does not have enough mass to explode as a supernova. Instead, when it runs out of hydrogen in the core in approximately 5 billion years, core hydrogen fusion will stop, and there will be nothing to prevent the core from contracting. The release of gravitational potential energy will cause the luminosity of the Sun to increase, ending the main sequence phase and leading the Sun to expand over the next billion years: first into a subgiant, and then into a red giant. The heating due to gravitational contraction will also lead to expansion of the Sun and hydrogen fusion in a shell just outside the core, where unfused hydrogen remains, contributing to the increased luminosity, which will eventually reach more than 1,000 times its present luminosity. When the Sun enters its red-giant branch (RGB) phase, it will engulf (and destroy) Mercury and Venus. According to a 2008 article, Earth's orbit will have initially expanded to at most 1.5 AU (220 million km; 140 million mi) due to the Sun's loss of mass. However, Earth's orbit will then start shrinking due to tidal forces (and, eventually, drag from the lower chromosphere) so that it is engulfed by the Sun during the tip of the red-giant branch phase 7.59 billion years from now, 3.8 and 1 million years after Mercury and Venus have respectively suffered the same fate. By the time the Sun reaches the tip of the red-giant branch, it will be about 256 times larger than it is today, with a radius of 1.19 AU (178 million km; 111 million mi). The Sun will spend around a billion years in the RGB and lose around a third of its mass. After the red-giant branch, the Sun has approximately 120 million years of active life left, but much happens. First, the core (full of degenerate helium) ignites violently in the helium flash; it is estimated that 6% of the core—itself 40% of the Sun's mass—will be converted into carbon within a matter of minutes through the triple-alpha process. The Sun then shrinks to around 10 times its current size and 50 times the luminosity, with a temperature a little lower than today. It will then have reached the red clump or horizontal branch, but a star of the Sun's metallicity does not evolve blueward along the horizontal branch. Instead, it just becomes moderately larger and more luminous over about 100 million years as it continues to react helium in the core. When the helium is exhausted, the Sun will repeat the expansion it followed when the hydrogen in the core was exhausted. This time, however, it all happens faster, and the Sun becomes larger and more luminous. This is the asymptotic-giant-branch phase, and the Sun is alternately reacting hydrogen in a shell or helium in a deeper shell. After about 20 million years on the early asymptotic giant branch, the Sun becomes increasingly unstable, with rapid mass loss and thermal pulses that increase the size and luminosity for a few hundred years every 100,000 years or so. The thermal pulses become larger each time, with the later pulses pushing the luminosity to as much as 5,000 times the current level. Despite this, the Sun's maximum AGB radius will not be as large as its tip-RGB maximum: 179 R☉, or about 0.832 AU (124.5 million km; 77.3 million mi). Models vary depending on the rate and timing of mass loss. Models that have higher mass loss on the red-giant branch produce smaller, less luminous stars at the tip of the asymptotic giant branch, perhaps only 2,000 times the luminosity and less than 200 times the radius. For the Sun, four thermal pulses are predicted before it completely loses its outer envelope and starts to make a planetary nebula. The post-asymptotic-giant-branch evolution is even faster. The luminosity stays approximately constant as the temperature increases, with the ejected half of the Sun's mass becoming ionised into a planetary nebula as the exposed core reaches 30,000 K (53,500 °F), as if it is in a sort of blue loop. The final naked core, a white dwarf, will have a temperature of over 100,000 K (180,000 °F) and contain an estimated 54.05% of the Sun's present-day mass. Simulations indicate that the Sun may be among the least massive stars capable of forming a planetary nebula. The planetary nebula will disperse in about 10,000 years, but the white dwarf will survive for trillions of years before fading to a hypothetical super-dense black dwarf. As such, it would give off no more energy. Location Solar System Main article: Solar System Location of the Sun within the Solar System, which extends to the edge of the Oort cloud, where at 125,000 AU to 230,000 AU, equal to several light-years, the Sun's gravitational sphere of influence ends. The Sun has eight known planets orbiting it. This includes four terrestrial planets (Mercury, Venus, Earth, and Mars), two gas giants (Jupiter and Saturn), and two ice giants (Uranus and Neptune). The Solar System also has nine bodies generally considered as dwarf planets and some more candidates, an asteroid belt, numerous comets, and a large number of icy bodies which lie beyond the orbit of Neptune. Six of the planets and many smaller bodies also have their own natural satellites: in particular, the satellite systems of Jupiter, Saturn, and Uranus are in some ways like miniature versions of the Sun's system. Apparent motion of the Solar System barycentre with respect to the Sun The Sun is moved by the gravitational pull of the planets. The centre of the Sun moves around the Solar System barycentre, within a range from 0.1 to 2.2 solar radii. The Sun's motion around the barycentre approximately repeats every 179 years, rotated by about 30° due primarily to the synodic period of Jupiter and Saturn. This motion is mainly due to Jupiter, Saturn, Uranus, and Neptune. For some periods of several decades (when Neptune and Uranus are in opposition) the motion is rather regular, forming a trefoil pattern, whereas between these periods it appears more chaotic. After 179 years (nine times the synodic period of Jupiter and Saturn), the pattern more or less repeats, but rotated by about 24°. The orbits of the inner planets, including of the Earth, are similarly displaced by the same gravitational forces, so the movement of the Sun has little effect on the relative positions of the Earth and the Sun or on solar irradiance on the Earth as a function of time. The Sun's gravitational field is estimated to dominate the gravitational forces of surrounding stars out to about two light-years (125,000 AU). Lower estimates for the radius of the Oort cloud, by contrast, do not place it farther than 50,000 AU. Most of the mass is orbiting in the region between 3,000 and 100,000 AU. The furthest known objects, such as Comet West, have aphelia around 70,000 AU from the Sun. The Sun's Hill sphere with respect to the galactic nucleus, the effective range of its gravitational influence, was calculated by G. A. Chebotarev to be 230,000 AU. Celestial neighbourhood This section is an excerpt from Solar System § Celestial neighborhood.[edit] Diagram of the Local Interstellar Cloud, the G-Cloud and surrounding stars. As of 2022, the exact position of the Solar System within the interstellar clouds remains an unresolved question in astronomy. Within 10 light-years of the Sun there are relatively few stars, the closest being the triple star system Alpha Centauri, which is about 4.4 light-years away and may be in the Local Bubble's G-Cloud. Alpha Centauri A and B are a closely tied pair of Sun-like stars, whereas the closest star to the Sun, the small red dwarf Proxima Centauri, orbits the pair at a distance of 0.2 light-years. In 2016, a potentially habitable exoplanet was found to be orbiting Proxima Centauri, called Proxima Centauri b, the closest confirmed exoplanet to the Sun. The Solar System is surrounded by the Local Interstellar Cloud, although it is not clear if it is embedded in the Local Interstellar Cloud or if it lies just outside the cloud's edge. Multiple other interstellar clouds exist in the region within 300 light-years of the Sun, known as the Local Bubble. The latter feature is an hourglass-shaped cavity or superbubble in the interstellar medium roughly 300 light-years across. The bubble is suffused with high-temperature plasma, suggesting that it may be the product of several recent supernovae. The Local Bubble is a small superbubble compared to the neighboring wider Radcliffe Wave and Split linear structures (formerly Gould Belt), each of which are some thousands of light-years in length. All these structures are part of the Orion Arm, which contains most of the stars in the Milky Way that are visible to the unaided eye. Groups of stars form together in star clusters, before dissolving into co-moving associations. A prominent grouping that is visible to the naked eye is the Ursa Major moving group, which is around 80 light-years away within the Local Bubble. The nearest star cluster is Hyades, which lies at the edge of the Local Bubble. The closest star-forming regions are the Corona Australis Molecular Cloud, the Rho Ophiuchi cloud complex and the Taurus molecular cloud; the latter lies just beyond the Local Bubble and is part of the Radcliffe wave. Stellar flybys that pass within 0.8 light-years of the Sun occur roughly once every 100,000 years. The closest well-measured approach was Scholz's Star, which approached to ~50,000 AU of the Sun some ~70 thousands years ago, likely passing through the outer Oort cloud. There is a 1% chance every billion years that a star will pass within 100 AU of the Sun, potentially disrupting the Solar System. Motion Main article: Galactic year Further information: Stellar kinematics The general motion and orientation of the Sun, with Earth and the Moon as its Solar System satellites The Sun, taking along the whole Solar System, orbits the galaxy's centre of mass at an average speed of 230 km/s (828,000 km/h), taking about 220–250 million Earth years to complete a revolution (a galactic year), having done so about 20 times since the Sun's formation. The direction of the Sun's motion, the Solar apex, is roughly in the direction of the star Vega. In the past the Sun likely moved through the Orion–Eridanus Superbubble, before entering the Local Bubble. The Sun's idealised orbit around the Galactic Centre in an artist's top-down depiction of the current layout of the Milky Way As the sun goes around the galaxy it also moves with respect to the average motion of the other stars around it. A simple model predicts that in a frame of reference rotating with the galaxy, the sun moves in an ellipse, circulating around a point that is itself going around the galaxy. The period of the Sun's circulation around the point is about 166 million years, shorter than the time it takes for the point to go around the galaxy. The length of the ellipse is around 1760 parsecs and its width around 1170 parsecs. (Compare this to the distance of the Sun from the centre of the galaxy, around 7 or 8 kiloparsecs.) At the same time, the sun moves "north" and "south" of the galactic plane with a different period, around 83 million years, moving about 99 parsecs away from the plane. The point around which the Sun circulates takes around 240 million years to go once around the galaxy. (See Stellar kinematics for more details.) The Sun's orbit around the Milky Way is perturbed due to the non-uniform mass distribution in Milky Way, such as that in and between the galactic spiral arms. It has been argued that the Sun's passage through the higher density spiral arms often coincides with mass extinctions on Earth, perhaps due to increased impact events. It takes the Solar System about 225–250 million years to complete one orbit through the Milky Way (a galactic year), so it is thought to have completed 20–25 orbits during the lifetime of the Sun. The orbital speed of the Solar System about the centre of the Milky Way is approximately 251 km/s (156 mi/s). At this speed, it takes around 1,190 years for the Solar System to travel a distance of 1 light-year, or 7 days to travel 1 AU. The Milky Way is moving with respect to the cosmic microwave background radiation (CMB) in the direction of the constellation Hydra with a speed of 550 km/s, but since the Sun is moving with respect to the Galactic Centre in the direction of Cygnus (galactic longitude 90°; latitude 0°) at more than 200 km/sec, the resultant velocity with respect to the CMB is about 370 km/s in the direction of Crater or Leo (galactic latitude 264°, latitude 48°). This is 132° away from Cygnus. Observational history Early understanding See also: The Sun in culture The Trundholm sun chariot pulled by a horse is a sculpture believed to be illustrating an important part of Nordic Bronze Age mythology. In many prehistoric and ancient cultures, the Sun was thought to be a solar deity or other supernatural entity. In the early 1st millennium BC, Babylonian astronomers observed that the Sun's motion along the ecliptic is not uniform, though they did not know why; it is today known that this is due to the movement of Earth in an elliptic orbit, moving faster when it is nearer to the Sun at perihelion and moving slower when it is farther away at aphelion. One of the first people to offer a scientific or philosophical explanation for the Sun was the Greek philosopher Anaxagoras. He reasoned that it was a giant flaming ball of metal even larger than the land of the Peloponnesus and that the Moon reflected the light of the Sun. Eratosthenes estimated the distance between Earth and the Sun in the 3rd century BC as "of stadia myriads 400 and 80000", the translation of which is ambiguous, implying either 4,080,000 stadia (755,000 km) or 804,000,000 stadia (148 to 153 million kilometres or 0.99 to 1.02 AU); the latter value is correct to within a few per cent. In the 1st century AD, Ptolemy estimated the distance as 1,210 times the radius of Earth, approximately 7.71 million kilometres (0.0515 AU). The theory that the Sun is the centre around which the planets orbit was first proposed by the ancient Greek Aristarchus of Samos in the 3rd century BC, and later adopted by Seleucus of Seleucia (see Heliocentrism). This view was developed in a more detailed mathematical model of a heliocentric system in the 16th century by Nicolaus Copernicus. Development of scientific understanding Sol, the Personification of the Sun, from a 1550 edition of Guido Bonatti's Liber astronomiae Observations of sunspots were recorded by Chinese astronomers during the Han dynasty (202 BC – AD 220), with records of their observations being maintained for centuries. Averroes also provided a description of sunspots in the 12th century. The invention of the telescope in the early 17th century permitted detailed observations of sunspots by Thomas Harriot, Galileo Galilei and other astronomers. Galileo posited that sunspots were on the surface of the Sun rather than small objects passing between Earth and the Sun. Medieval Islamic astronomical contributions include al-Battani's discovery that the direction of the Sun's apogee (the place in the Sun's orbit against the fixed stars where it seems to be moving slowest) is changing. In modern heliocentric terms, this is caused by a gradual motion of the aphelion of the Earth's orbit. Ibn Yunus observed more than 10,000 entries for the Sun's position for many years using a large astrolabe. The first reasonably accurate distance to the Sun was determined in 1684 by Giovanni Domenico Cassini. Knowing that direct measurements of the solar parallax were difficult, he chose to measure the Martian parallax. Having sent Jean Richer to Cayenne, part of French Guiana, for simultaneous measurements, Cassini in Paris determined the parallax of Mars when Mars was at its closest to Earth in 1672. Using the circumference distance between the two observations, Cassini calculated the Earth-Mars distance, then used Kepler's laws to determine the Earth-Sun distance. His value, about 10% smaller than modern values, was much larger than all previous estimates. From an observation of a transit of Venus in 1032, the Persian astronomer and polymath Ibn Sina concluded that Venus was closer to Earth than the Sun. In 1677, Edmond Halley observed a transit of Mercury across the Sun, leading him to realise that observations of the solar parallax of a planet (more ideally using the transit of Venus) could be used to trigonometrically determine the distances between Earth, Venus, and the Sun. Careful observations of the 1769 transit of Venus allowed astronomers to calculate the average Earth–Sun distance as 93,726,900 miles (150,838,800 km), only 0.8% greater than the modern value. Sun as seen in Hydrogen-alpha light In 1666, Isaac Newton observed the Sun's light using a prism, and showed that it is made up of light of many colours. In 1800, William Herschel discovered infrared radiation beyond the red part of the solar spectrum. The 19th century saw advancement in spectroscopic studies of the Sun; Joseph von Fraunhofer recorded more than 600 absorption lines in the spectrum, the strongest of which are still often referred to as Fraunhofer lines. The 20th century brought about several specialised systems for observing the Sun, especially at different narrowband wavelengths, such as those using Calcium-H (396.9 nm), Calcium-K (393.37 nm) and Hydrogen-alpha (656.46 nm) filtering. During early studies of the optical spectrum of the photosphere, some absorption lines were found that did not correspond to any chemical elements then known on Earth. In 1868, Norman Lockyer hypothesised that these absorption lines were caused by a new element that he dubbed helium, after the Greek Sun god Helios. Twenty-five years later, helium was isolated on Earth. In the early years of the modern scientific era, the source of the Sun's energy was a significant puzzle. Lord Kelvin suggested that the Sun is a gradually cooling liquid body that is radiating an internal store of heat. Kelvin and Hermann von Helmholtz then proposed a gravitational contraction mechanism to explain the energy output, but the resulting age estimate was only 20 million years, well short of the time span of at least 300 million years suggested by some geological discoveries of that time. In 1890, Lockyer proposed a meteoritic hypothesis for the formation and evolution of the Sun. Not until 1904 was a documented solution offered. Ernest Rutherford suggested that the Sun's output could be maintained by an internal source of heat, and suggested radioactive decay as the source. However, it would be Albert Einstein who would provide the essential clue to the source of the Sun's energy output with his mass–energy equivalence relation E = mc2. In 1920, Sir Arthur Eddington proposed that the pressures and temperatures at the core of the Sun could produce a nuclear fusion reaction that merged hydrogen (protons) into helium nuclei, resulting in a production of energy from the net change in mass. The preponderance of hydrogen in the Sun was confirmed in 1925 by Cecilia Payne using the ionisation theory developed by Meghnad Saha. The theoretical concept of fusion was developed in the 1930s by the astrophysicists Subrahmanyan Chandrasekhar and Hans Bethe. Bethe calculated the details of the two main energy-producing nuclear reactions that power the Sun. In 1957, Margaret Burbidge, Geoffrey Burbidge, William Fowler and Fred Hoyle showed that most of the elements in the universe have been synthesised by nuclear reactions inside stars, some like the Sun. Solar space missions See also: Solar observatory and List of heliophysics missions Pioneer 6, 7, 8, and 9 The first satellites designed for long term observation of the Sun from interplanetary space were Pioneer 6, 7, 8, and 9, which were launched by NASA between 1959 and 1968. These probes orbited the Sun at a distance similar to that of Earth, and made the first detailed measurements of the solar wind and the solar magnetic field. Pioneer 9 operated for a particularly long time, transmitting data until May 1983. In the 1970s, two Helios spacecraft and the Skylab Apollo Telescope Mount provided scientists with significant new data on solar wind and the solar corona. The Helios 1 and 2 probes were U.S.–German collaborations that studied the solar wind from an orbit carrying the spacecraft inside Mercury's orbit at perihelion. The Skylab space station, launched by NASA in 1973, included a solar observatory module called the Apollo Telescope Mount that was operated by astronauts resident on the station. Skylab made the first time-resolved observations of the solar transition region and of ultraviolet emissions from the solar corona. Discoveries included the first observations of coronal mass ejections, then called "coronal transients", and of coronal holes, now known to be intimately associated with the solar wind. Drawing of a Solar Maximum Mission probe In 1980, the Solar Maximum Mission probes were launched by NASA. This spacecraft was designed to observe gamma rays, X-rays and ultraviolet radiation from solar flares during a time of high solar activity and solar luminosity. Just a few months after launch, however, an electronics failure caused the probe to go into standby mode, and it spent the next three years in this inactive state. In 1984, Space Shuttle Challenger mission STS-41-C retrieved the satellite and repaired its electronics before re-releasing it into orbit. The Solar Maximum Mission subsequently acquired thousands of images of the solar corona before re-entering Earth's atmosphere in June 1989. Launched in 1991, Japan's Yohkoh (Sunbeam) satellite observed solar flares at X-ray wavelengths. Mission data allowed scientists to identify several different types of flares and demonstrated that the corona away from regions of peak activity was much more dynamic and active than had previously been supposed. Yohkoh observed an entire solar cycle but went into standby mode when an annular eclipse in 2001 caused it to lose its lock on the Sun. It was destroyed by atmospheric re-entry in 2005. The Solar and Heliospheric Observatory, jointly built by the European Space Agency and NASA, was launched on 2 December 1995. Originally intended to serve a two-year mission, SOHO remains in operation as of 2024. Situated at the Lagrangian point between Earth and the Sun (at which the gravitational pull from both is equal), SOHO has provided a constant view of the Sun at many wavelengths since its launch. Besides its direct solar observation, SOHO has enabled the discovery of a large number of comets, mostly tiny sungrazing comets that incinerate as they pass the Sun. Ulysses spacecraft testing at the vacuum spin-balancing facility All these satellites have observed the Sun from the plane of the ecliptic, and so have only observed its equatorial regions in detail. The Ulysses probe was launched in 1990 to study the Sun's polar regions. It first travelled to Jupiter, to "slingshot" into an orbit that would take it far above the plane of the ecliptic. Once Ulysses was in its scheduled orbit, it began observing the solar wind and magnetic field strength at high solar latitudes, finding that the solar wind from high latitudes was moving at about 750 km/s, which was slower than expected, and that there were large magnetic waves emerging from high latitudes that scattered galactic cosmic rays. Elemental abundances in the photosphere are well known from spectroscopic studies, but the composition of the interior of the Sun is more poorly understood. A solar wind sample return mission, Genesis, was designed to allow astronomers to directly measure the composition of solar material. Observation by eyes Exposure to the eye The Sun seen from Earth, with glare from the lenses. The eye also sees glare when looked towards the Sun directly. The brightness of the Sun can cause pain from looking at it with the naked eye; however, doing so for brief periods is not hazardous for normal non-dilated eyes. Looking directly at the Sun, known as sungazing, causes phosphene visual artefacts and temporary partial blindness. It also delivers about 4 milliwatts of sunlight to the retina, slightly heating it and potentially causing damage in eyes that cannot respond properly to the brightness. Viewing of the direct Sun with the naked eye can cause UV-induced, sunburn-like lesions on the retina beginning after about 100 seconds, particularly under conditions where the UV light from the Sun is intense and well focused. Viewing the Sun through light-concentrating optics such as binoculars may result in permanent damage to the retina without an appropriate filter that blocks UV and substantially dims the sunlight. When using an attenuating filter to view the Sun, the viewer is cautioned to use a filter specifically designed for that use. Some improvised filters that pass UV or IR rays, can actually harm the eye at high brightness levels. Brief glances at the midday Sun through an unfiltered telescope can cause permanent damage. During sunrise and sunset, sunlight is attenuated because of Rayleigh scattering and Mie scattering from a particularly long passage through Earth's atmosphere, and the Sun is sometimes faint enough to be viewed comfortably with the naked eye or safely with optics (provided there is no risk of bright sunlight suddenly appearing through a break between clouds). Hazy conditions, atmospheric dust, and high humidity contribute to this atmospheric attenuation. Phenomena An optical phenomenon, known as a green flash, can sometimes be seen shortly after sunset or before sunrise. The flash is caused by light from the Sun just below the horizon being bent (usually through a temperature inversion) towards the observer. Light of shorter wavelengths (violet, blue, green) is bent more than that of longer wavelengths (yellow, orange, red) but the violet and blue light is scattered more, leaving light that is perceived as green. Religious aspects Main article: Solar deity Solar deities play a major role in many world religions and mythologies. Worship of the Sun was central to civilisations such as the ancient Egyptians, the Inca of South America and the Aztecs of what is now Mexico. In religions such as Hinduism, the Sun is still considered a god, known as Surya. Many ancient monuments were constructed with solar phenomena in mind; for example, stone megaliths accurately mark the summer or winter solstice (for example in Nabta Playa, Egypt; Mnajdra, Malta; and Stonehenge, England); Newgrange, a prehistoric human-built mount in Ireland, was designed to detect the winter solstice; the pyramid of El Castillo at Chichén Itzá in Mexico is designed to cast shadows in the shape of serpents climbing the pyramid at the vernal and autumnal equinoxes. The ancient Sumerians believed that the Sun was Utu, the god of justice and twin brother of Inanna, the Queen of Heaven. Later, Utu was identified with the East Semitic god Shamash. Utu was regarded as a helper-deity, who aided those in distress. Ra from the tomb of Nefertari, 13th century BC From at least the Fourth Dynasty of Ancient Egypt, the Sun was worshipped as the god Ra, portrayed as a falcon-headed divinity surmounted by the solar disk. In the New Empire period, the Sun became identified with the dung beetle. In the form of the sun disc Aten, the Sun had a brief resurgence during the Amarna Period when it again became the preeminent, if not only, divinity for the Pharaoh Akhenaten. The Egyptians portrayed the god Ra as being carried across the sky in a solar barque, accompanied by lesser gods, and to the Greeks, he was Helios, carried by a chariot drawn by fiery horses. From the reign of Elagabalus in the late Roman Empire the Sun's birthday was a holiday celebrated as Sol Invictus (literally 'Unconquered Sun') soon after the winter solstice. The Sun appears from Earth to revolve once a year along the ecliptic through the zodiac, and so Greek astronomers categorised it as one of the seven planets (from Greek planetes, 'wanderer'); the naming of the days of the weeks after the seven planets dates to the Roman era. In Proto-Indo-European religion, the Sun was personified as the goddess Seh2ul. Derivatives of this goddess in Indo-European languages include the Old Norse Sól, Sanskrit Surya, Gaulish Sulis, Lithuanian Saulė, and Slavic Solntse. In ancient Greek religion, the sun deity was the male god Helios, who in later times was syncretised with Apollo. In ancient Roman culture, Sunday was the day of the sun god. In paganism, the Sun was a source of life. It was the centre of a popular cult among Romans, who would stand at dawn to catch the first rays of sunshine as they prayed. The celebration of the winter solstice (which influenced Christmas) was part of the Roman cult of Sol Invictus. It was adopted as the Sabbath day by Christians. The symbol of light was a pagan device adopted by Christians, and perhaps the most important one that did not come from Jewish traditions. Christian churches were built so that the congregation faced toward the sunrise. In the Bible, the Book of Malachi mentions the "Sun of Righteousness", which some Christians have interpreted as a reference to the Messiah (Christ). Tonatiuh, the Aztec god of the sun, was closely associated with human sacrifice. The sun goddess Amaterasu is the most important deity in the Shinto religion, and she is believed to be the direct ancestor of all Japanese emperors. See also Astronomy portal Stars portal Solar System portal Weather portal Physics portal Advanced Composition Explorer – NASA satellite of the Explorer program, at SE-L1 from 1997 Analemma – Diagrammatic representation of Sun's position over a period of time Antisolar point – Point on the celestial sphere opposite Sun Faint young Sun paradox – Paradox concerning water on early Earth List of brightest stars – Stars sorted by apparent magnitude List of nearest stars Midnight sun – Natural phenomenon when daylight lasts for a whole day Planets in astrology § Sun Solar telescope – Telescope used to observe the Sun Sun path – Arc-like path that the Sun appears to follow across the sky Sun-Earth Day – NASA and ESA joint educational program Sun in fiction Timeline of the far future – Scientific projections regarding the far future Notes ^ Jump up to: a b All numbers in this article are short scale. 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Further reading Library resources about Sun Online books Resources in your library Resources in other libraries Cohen, Richard (2010). Chasing the sun: the epic story of the star that gives us life. New York, NY: Random House. ISBN 978-1-4000-6875-3. Hudson, Hugh (2008). "Solar activity". Scholarpedia. Vol. 3. p. 3967. Bibcode:2008SchpJ...3.3967H. doi:10.4249/scholarpedia.3967. ISSN 1941-6016. Archived from the original on 3 October 2015. Retrieved 27 September 2015. Thompson, Michael J (August 2004). "Helioseismology and the Sun's interior". Astronomy & Geophysics. 45 (4): 4.21 – 4.25. Bibcode:2004A&G....45d..21T. doi:10.1046/j.1468-4004.2003.45421.x. ISSN 1366-8781. External links Listen to this article (1 hour and 29 minutes) Duration: 1 hour, 28 minutes and 35 seconds.1:28:35 This audio file was created from a revision of this article dated 7 June 2021 (2021-06-07), and does not reflect subsequent edits. (Audio help · More spoken articles) Sun at Wikipedia's sister projects Definitions from Wiktionary Media from Commons Quotations from Wikiquote Astronomy Cast: The Sun Archived 12 May 2011 at the Wayback Machine Satellite observations of solar luminosity Archived 11 June 2017 at the Wayback Machine Animation – The Future of the Sun "Thermonuclear Art – The Sun In Ultra-HD" Archived 4 November 2015 at the Wayback Machine \| Goddard Space Flight Center "A Decade of Sun" Archived 3 December 2021 at the Wayback Machine \| Goddard Space Flight Center \| hide - v - t - e The Sun \| \| --- \| \| Internal structure \| - Core - Radiative zone - Tachocline - Convection zone \| \| Atmosphere \| \| \| \| --- \| \| Photosphere \| - Supergranulation - Granule - Faculae - Sunspot - Ellerman bomb \| \| Chromosphere \| - Plage - Spicule - Moreton wave \| \| Corona \| - Transition region - Coronal hole - Coronal loop - Coronal mass ejection - Nanoflare - Prominence - Helmet streamer - Supra-arcade downflows - 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188922	https://www.mdpi.com/2227-9059/11/3/978	Endometriosis: Update of Pathophysiology, (Epi) Genetic and Environmental Involvement Next Article in Journal Optimal Timing of Coronary Artery Bypass Grafting in Haemodynamically Stable Patients after Myocardial Infarction Next Article in Special Issue Molecular Mechanisms of Endometriosis Revealed Using Omics Data Previous Article in Journal Chronic Kidney Disease—How Does It Go, and What Can We Do and Expect? 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Fattet, A. Julie Koscinski, I. on Google Scholar Monnin, N. Fattet, A. Julie Koscinski, I. on PubMed Monnin, N. Fattet, A. Julie Koscinski, I. /ajax/scifeed/subscribe Article Views 13033 Citations 41 Table of Contents Abstract Introduction Theories on Endometriosis Endometriosis and Cancer Endometriosis and PolyCystic Ovaries Syndrome (PCOS) Infertility and Endometriosis Environmental Impact Role of the Microbiota in Endometriosis Conclusions Author Contributions Funding Institutional Review Board Statement Informed Consent Statement Data Availability Statement Acknowledgments Conflicts of Interest References Altmetricshare Shareannouncement Helpformat_quote Citequestion_answer Discuss in SciProfiles Need Help? Support Find support for a specific problem in the support section of our website. Get Support Feedback Please let us know what you think of our products and services. Give Feedback Information Visit our dedicated information section to learn more about MDPI. Get Information clear JSmol Viewer clear first_page Download PDF settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing:    Column Width:    Background: Open Access Review Endometriosis: Update of Pathophysiology, (Epi) Genetic and Environmental Involvement by Nicolas Monnin Nicolas Monnin SciProfilesScilitPreprints.orgGoogle Scholar 1, Anne Julie Fattet Anne Julie Fattet SciProfilesScilitPreprints.orgGoogle Scholar 1 and Isabelle Koscinski Isabelle Koscinski SciProfilesScilitPreprints.orgGoogle Scholar 2,3, 1 Majorelle Clinic, Atoutbio Laboratory, Laboratory of Biology of Reproduction, 54000 Nancy, France 2 Laboratory of Biology of Reproduction, Hospital Saint Joseph, 13008 Marseille, France 3 NGERE Inserm 1256, 54505 Vandoeuvre les Nancy, France Author to whom correspondence should be addressed. Biomedicines2023, 11(3), 978; Submission received: 16 February 2023 / Revised: 15 March 2023 / Accepted: 16 March 2023 / Published: 22 March 2023 (This article belongs to the Special Issue Advanced Research in Endometriosis 3.0) Download keyboard_arrow_down Download PDF Download PDF with Cover Download XML Download Epub Browse Figures Review ReportsVersions Notes Abstract Endometriosis is a chronic disease caused by ectopic endometrial tissue. Endometriotic implants induce inflammation, leading to chronic pain and impaired fertility. Characterized by their dependence on estradiol (via estrogen receptor β (ESRβ)) and their resistance to progesterone, endometriotic implants produce their own source of estradiol through active aromatase. Steroidogenic factor-1 (SF1) is a key transcription factor that promotes aromatase synthesis. The expression of SF1 and ESRβ is enhanced by the demethylation of their promoter in progenitor cells of the female reproductive system. High local concentrations of estrogen are involved in the chronic inflammatory environment favoring the implantation and development of endometriotic implants. Similar local conditions can promote, directly and indirectly, the appearance and development of genital cancer. Recently, certain components of the microbiota have been identified as potentially promoting a high level of estrogen in the blood. Many environmental factors are also suspected of increasing the estrogen concentration, especially prenatal exposure to estrogen-like endocrine disruptors such as DES and bisphenol A. Phthalates are also suspected of promoting endometriosis but throughmeans other than binding to estradiol receptors. The impact of dioxin or tobacco seems to be more controversial. Keywords: endometriosis; epigenetics; molecular pathophysiology; endocrine disruptor 1. Introduction Endometriosis is a common and benign gynecological disease defined by the ectopic presence of tissue with the same morphological and functional characteristics as the endometrium (cylindrical glandular epithelium and stroma) [1,2,3,4]. Its main locations are the pelvic peritoneum, uterosacral ligaments, cul-de-sac of Douglas, rectovaginal septum, and ovaries [1,3,4,5]. Clinical signs are numerous. Among them, dysmenorrhea, dyspareunia, chronic pelvic pain, irregular uterine bleeding and/or infertility are frequently found [3,4,5]. This ectopic endometrium is functional and undergoes periodic revisions, explaining the cyclical nature of symptoms and the chronic inflammatory process associated with estrogens . This gynecological disease affects approximately 5–15% of women of reproductive age and is found in 35–50% of women with infertility [1,2,7,8,9]. Consequently, it is a frequent reason for subfertile couples to seek consultation in assisted reproductive technology (ART) centers. The pathophysiology of endometriosis is based on its “estrogen-dependent” character; however, the mechanisms of onset and development are unclear. An environmental cause is not excluded, especially via endocrine-disrupting substances that are abundant in our 21st century environment. This review points out that local hyperestradiolaemia is the cause and consequence of endometriotic lesions’ development from epigenetically modified cells. It explains the possible modulation of hyperestradiolaemia through environmental factors, including the microbiota, how hyperestradiolaemia is involved in the distortion of the immune system, and how it can promote cancer. 2. Theories on Endometriosis Several theories lead to two major pathophysiological hypotheses. The first is an endometrial origin of endometriosis implants. The other is based on an ectopic origin. In addition, risk factors and genetic predisposition factors are being studied . The long-held favored group of theories rests on an endometrial origin of the endometriosis process. The benign metaplasia theory consists of the spread of endometrial cells in lymphatic and hematogenous ways. Different microvascular studies support this concept as well as the existence of some rare extra pelvic localizations, such as the bone, brain or lung, that have been histologically proven [4,5]. The coelomic metaplasia theory, embryonic Müllerian cell theory and bone marrow stem cell theory are part of the group of ectopic theories. These various possible explanations of the origin of endometriosis are based on a transformation of a tissue other than the endometrial tissue into endometriotic tissue under the action of some unknown substances [2,4,5,10,11]. Demonstrated for the first time in 1920 by Sampson, retrograde menstruation theory, in which endometrial cells are drained into the fallopian tube and peritoneal cavity during menstruation, has largely been studied. In particular, in the 1980s, Halme et al. highlighted the existence of menstrual blood in the peritoneal fluid of more than 90% of healthy women [2,4,5,7,8,12,13]. In addition, explorations carried out in women with congenital obstruction or animal experiments mimicking iatrogenic obstruction showed that obstruction of the normal menstruation flow promotes the development of endometriotic lesions. The prevalence of endometriosis also appears to be greater in women with cervical stenosis [4,5]. The organic localization of endometriotic lesions is an additional argument in favor of this theory. Indeed, superficial implants are more often located in the posterior pelvic compartment and at the left hemi-pelvis. The predisposition of lesions localized in the Douglas cul-de-sac would be explained by the accumulation of retrograde menstruation in this location under the action of gravity. Moreover, a retroverted uterus (permitting flow from the front to the rear in the vertical or lying position) is correlated with the development of endometriosis. Similarly, acting as a barrier to diffusion of menstrual flow from the left fallopian tube, a prominent sigmoid colon promotes stasis of this flow, extending the range of implantation of refluxed endometrial fragments in the left hemipelvis . Finally, in a mouse model, the activation of the K-ras oncogene in endometrial cells deposited on the peritoneum allowed the development of peritoneal lesions of endometriosis, whereas the activation of this oncogene directly in peritoneal cells had no impact. This finding confirmed the endometrial origin of endometriosis . Nevertheless, this theory cannot explain the complete development of endometriotic lesions by itself. Other factors, such as immune escape, adhesion to peritoneal epithelium and invasion, the neurovascular environment and growth/continuing survival, are essential to the long-term persistence of lesions. The need for these additional factors explains why only 10% of women with retrograde menstruation (90% of the general population) suffer from endometriosis [4,5,8,14]. 2.1. An Anatomical Predisposition to Endometriosis? The body’s inability to remove endometriosis implants into the peritoneal fluid may be aggravated by anatomical features often found in women with endometriosis. In addition to lean size , some elements increase menstrual reflux, hypertension of the utero-tubal junction, waves of retrograde tubular contractions of the myometrium, and uterine malformations. Moreover, in patients suffering from endometriosis, menstruation is often longer and more abundant, and menstrual cycles are shorter . 2.2. Molecular Pathophysiological Mechanisms Similar to tumor cells, endometriotic cells have a high survival potential and clonal dissemination. The abnormally high survival potential of endometrial cells has been the subject of several recent studies. On the one hand, genetic alterations (polygenic) have been identified as the cause of greater survival of endometrial cells present in endometriotic lesions. In particular, overexpression of the antiapoptotic BCL-2 gene was found, promoting the proliferation of endometrial cells. These cells are also exposed to DNA damage due to their rapid turnover and their sensitivity to different epigenetic factors as well as oxidative stress. On the other hand, the high resistance of endometrial cells to apoptosis has been explored. The specific expression of the heat shock proteins (HSPs) has been highlighted. These proteins normally play a role in the protection of the correct three-dimensional folding of proteins despite thermal shocks. Endometriosis cells present a special pattern of HSPs with high expression of HSP27 and HSP70, contributing to their protection against apoptosis . A recent transcriptional approach comparing endometriotic cells to healthy pelvic and ovarian cells highlighted transformations at the cellular scale resulting from the reprogramming of endometriotic cells. In particular, these cells present deregulation of the pro-inflammatory pathways and over-expression of complement factors. Moreover, neo-mutations in endometriotic epithelial cells of two cancer-driver genes, ARID1A and KRAS, would favor their diffusion. KRAS is a small GTPase known to increase the proliferation potential of cells. The deciphering of disturbed pathways suggests that ARID1A mutation in endometriotic cells promotes the growth of local lymphatic endothelial cells through paracrine secretion of vascular growth factors. Interestingly, ovarian tumor cells associated with endometriosis present similar mutations in KRAS, ARID1A, and PIK3CA (a subunit of a kinase involved in tumorigenesis) . PIK3CA is enrolled in the first step of activation of the PI3K/Akt pathway, which increases cell survival by blocking the apoptosis pathway, enhances cell proliferation via cyclin activation, upregulates glucose metabolism, and regulates negatively anabolic processes via inhibition of the mTOR pathway. PI3K also interacts with the PTEN pathway controlling DNA repair, genomic stability, and apoptosis . Therefore, a mutation in PIK3CA may promote tumorigenesis. Mutations in KRAS, ARID1A, PIK3CA, and others allow the studying of the clonality of various types of endometriotic lesions . The redundancy in mutations within the same gene and lesions comforts the oligo-clonal character of the disease: multiple epithelial clones migrate together, especially in deep infiltrating lesions, whereas ovarian endometriomas present the highest potential for oligoclonality. These data suggest that ovarian stroma provides perfect conditions for the proliferation of multiple clones, with an increased risk of malignancy . Genetic analysis of endometriotic lesions could contribute to a better diagnosis and, according to mutation profiles, could open new ways of personalized care. For instance, drugs targeting PIK3CA or MEK signals, such as alpelisib or trametinib, respectively, may theoretically offer a new treatment for women with endometriosis whose lesions have mutations in PIK3CA or KRAS . 2.3. The Crucial Role of an Altered Hormonal Environment In patients with endometriosis, inflammation, the immune response, angiogenesis and apoptosis are altered. These disturbances are mainly caused by changes in the estrogen/progesterone environmental balance. In particular, future endometriotic cells present an increased production of estrogen and prostaglandins and develop progesterone resistance . The entire pathophysiological molecular mechanism is summarized in Figure 1. Endometriosis is an estrogen-dependent disease that is known to be “estradiol-dependent”. Estrogens act on endometrial cells via the estradiol (E2) receptors ESRα (or ESR1) and ESRβ (or ESR2). Bulun et al. have shown increased expression of ESRβ in endometriosis tissue due to the hypomethylation of the promoter region of ESRβ. This receptor is active through the RAS-like estrogen-regulated growth inhibitor (RERG), which induces the regulation of a large number of factors involved in resistance to apoptosis and cell proliferation [4,5,7,14,21,22]. Figure 1. Molecular pathophysiological mechanisms of endometriosis [2,4,5,23]. Figure 1. Molecular pathophysiological mechanisms of endometriosis [2,4,5,23]. In normal endometrial tissue, the production of estrogen from C19 steroids is very low due to the absence of the enzyme aromatase. Similarly, the enzyme 17β-hydroxysteroid dehydrogenase 2 (17βHSD2) (progesterone-dependent) catalyzes, during the luteal phase, the conversion of biologically highly active E2 into less active estrone (E1). Therefore, in a healthy endometrium, estrogen activity is maintained at a low level. In contrast, in the case of endometriosis, aromatase activity is detected at a high level in the endometrium as well as in ectopic endometriotic tissue. Furthermore, cells in endometriotic lesions express all the genes of steroidogenesis, including aromatase, and produce their own source of E2 from cholesterol. In addition, this high E2 concentration in these tissues is maintained due to a decrease in its catabolism as a result of deficient 17βHSD2 activity [2,4,7]. In women with endometriosis, the high level of E2 promoting endometriosis is also due to exogenous contribution, as evidenced by the high secretion of estrogen found in the ovaries, skin and adipose tissue. The E2 secreted by the ovaries reaches the endometriotic tissue through the blood. This phenomenon is mostly observed during the ovulatory phase (follicular rupture causes the release of large amounts of E2). In adipose tissue and skin, the presence of aromatase allows the conversion of circulating androstenedione into E1, which can be converted into E2 in these same tissues. The secreted E2 reaches the endometriosis implants through the blood . Furthermore, in endometriotic tissue, inflammation and the production of estrogen are interconnected in an amplification loop: the oxidative stress associated with the inflammation process promotes, via an epigenetic mechanism, the overexpression of key genes of steroidogenesis (notably aromatase) and cyclooxygenase 2 (COX-2), which results in the local and continuous production of E2 and prostaglandin E2 (PGE2), respectively [4,7]. In return, prostaglandins, which are locally produced, are responsible for inflammation and pain. While the activity of COX-2 and the production of PGE2 are low in the healthy endometrial tissue, in the endometrium of women with endometriosis and endometriotic implants, PGE2 and PGF2α are produced in excess (Figure 2). The vasoconstrictor properties of PGF2α are a cause of dysmenorrhea, and PGE2 can directly induce painful nerve stimulation, causing chronic pelvic pain. These high levels of PGE2 and PGE2α result from the high activity of prostaglandin F synthase and microsomal prostaglandin E synthase in uterine cells, which catalyze the conversion of PGH2 into PGF2α and PGE2, respectively. These enzymes, as well as COX-2, are more active in women with endometriosis than in healthy women. Four cytokines/hormones allow these higher levels of activity in endometriotic stromal cells: IL-1β cytokine, PGE2 itself (autocrine), VEGF and E2 (via ESRβ) [4,7]. Figure 2. Prostaglandins synthesis and effects in endometriosis [4,5]. By binding to the specific membrane receptors (EP1, EP2, EP3 and EP4) of endometriotic cells, PGE2 stimulates the expression of all steroidogenesis genes necessary for E2 synthesis from cholesterol. More specifically, the binding of PGE2 to its membrane receptor causes an increase in intracellular cAMP levels in endometriotic cells. This phenomenon promotes the action of a key transcription factor, the nuclear receptor of steroidogenic factor-1 (SF1), present only in endometriotic cells. SF1 enhances the expression of STAR(facilitating entry of cholesterol into mitochondria) and CYP19A1(coding for aromatase), leading to an increase in aromatase activity and, finally, a hyperestrogenic environment. In contrast, at a high rate in healthy endometrial cells, transcription inhibitors of the STAR and CYP19A1 genes constitute a safety system for limiting the expression of these steroidogenesis enzymes. Among these inhibitors are the chicken ovalbumin upstream promoter-transcription factor (COUP-TF), Wilms’ tumor transcription factor 1 (WT1) and CCAAT/enhancer binding protein β (C/EBPβ). These inhibitors would be present in lower abundances in endometriosis cells [4,7,23,24]. Furthermore, the extension of endometriotic lesions is promoted by an increased level of oxytocin, a hypothalamic hormone inducing the production and release of PGE2 and PGF2a by endometrial cells and uterine hyperperistalsis. Oxytocin activates the inflammatory immune system of the endometrium and participates in an enhanced suction of debris and infectious particles from the uterine cavity as well as endometriotic implants toward the peritoneal cavity . Progesterone and its various receptor isoforms (PR-A and PR-B) also play a key role in the pathophysiology of endometriosis. Physiologically, progesterone induces the differentiation of stromal endometrial cells and epithelial cells, resulting in an increased production of glycodelin (an epithelial glycoprotein produced by the secretory endometrium in the luteal phase). Glycodelin exerts indirect antiestrogenic effects. Its binding to specific PRs stimulates the synthesis of retinoic acid and increases the expression of 17βHSD2, leading to an increased conversion of E2 into the less active E1 . In women with endometriosis, the response to progesterone is clearly reduced in endometrial cells because of reduced expression of epithelial glycodelin and decreased levels of progesterone. In addition, genes coding for PR are expressed in the early phase of the menstrual cycle (suggesting a progesterone resistance phenotype), and the expression of progesterone-dependent genes in the luteal phase is dysregulated . PR isoform-A, an inhibitory isoform of PR, is the unique isoform expressed at high levels by endometriotic cells, regardless of cycle phase . Progesterone resistance could therefore be explained by the excessive presence of the PR-A isoform and by the absence of the PR-B isoform (the active form of PR) [5,7]. Furthermore, the high expression of this inhibitory PR associated with the lack of active PR results in progesterone resistance and, ultimately, in reduced 17βHSD2 activity. This leads to a decreased conversion of E2 to E1 and, finally, the hyperestrogenic activity found in women with endometriosis (Figure 3) . Figure 3. Disruption of progesterone/estradiol balance in endometriosis . However, this progesterone resistance is related to the ambiguity of the role of progesterone in endometriosis pathophysiology: despite having fewer PR receptors than healthy endometrium, endometriotic tissue has a great capacity for progesterone production, which induces the physiological differentiation of endometrial stromal cells . As mentioned above, nuclear SF1 receptor, only present in endometriotic cells, is a key factor in the transcription of pathological signals by increasing the expression of STAR, CYP19A1 and other steroidogenesis genes. The presence of this harmful SF1 receptor in endometriotic cells is partly caused by a lack of methylation of a CpG island (cytosine–phosphate–guanine) of the promoter region of the SF1 gene. This DNA region is normally highly methylated in stromal endometrial cells, which induces the blocking of SF1 expression. Thus, the increased expression of SF1 in endometriotic tissue compared to normal endometrium is mainly controlled by an epigenetic mechanism [4,24,25]. Upregulation of ESRβ expression is subjected to a similar epigenetic process . ESRβ binds to the promoter of the ESRα gene and downregulates its expression, ultimately promoting the removal of ESRα. The high ratio of ESRβ/ESRα in endometriotic cells leads to an enhanced ESRβ level. ESRβ binds to the promoter of the progesterone receptor and downregulates its expression (Figure 4) . Figure 4. ESR and SF-1 receptors epigenetic modifications in endometriosis [4,22,23]. Furthermore, during embryonic differentiation of the female reproductive system, environmental factors (e.g., endocrine disruptors) or genetic factors could cause genetic changes in DNA methylation. Consequently, epigenetic alterations modifying the expression of critical genes, such as SF1 or ESR-β in progenitor cells destined to become various pelvic tissues, could predispose adult women to endometriosis [4,24,26]. Recently, this hypothesis was supported by Kumari et al., who highlighted a significant hypomethylation of promoter regions of proinflammatory and proangiogenic genes involved in the molecular pathophysiology of endometriosis development. This hypomethylation could be the reason for their overexpression in endometriosis . In conclusion, in endometriotic cells, exposure to PGE2 leads to SF1 binding to promoters of several steroidogenesis genes (particularly aromatase) and causes the formation of large amounts of estradiol. Estradiol acts through its ESRβ receptor, whose expression is increased in cases of endometriosis, and stimulates COX-2, which leads to overproduction of PGE2. Inflammation and estrogen are linked in a positive feedback cycle inducing the overexpression of genes encoding aromatase and COX-2 and continuing the formation of aromatase and the COX-2 products—estradiol and PGE2—in endometriotic tissue. Finally, the decrease in the expression of PR induced by ESRβ is partly responsible for the resistance to progesterone and the disruption of paracrine inactivation of estradiol. Large amounts of estradiol accumulate due to its increased formation and inadequate inactivation in endometriotic tissue, promoting the proliferation of endometriotic implants [4,5,7,22,23,24,25]. Furthermore, an embryologic mechanism has been proposed to explain the onset of endometriosis. The expression of HOXA10, the homeobox gene (Hox/HOX) involved in uterine embryogenesis and embryo implantation, has been highlighted in endometriotic foci outside the Müllerian tract and could play a role in the development of endometriosis by inducing the formation of ectopic endometrial cells during embryogenesis . 2.4. Telomeres, Telomerase and Endometriosis Permanent high estrogen levels can also promote the activity of telomerase, an important enzyme involved in aging through its crucial role in cell proliferation. Telomeres are specialized noncoding repeated DNA sequences (5-TTAGGG-3) protecting all eukaryotic linear chromosomal ends . Telomeres enable progressive shortening of chromosome extremities without inducing genetic information loss, which maintains genomic stability . The progressive attrition at each replication cycle leads to a critically short telomere length, which induces proliferation arrest, senescence or apoptosis of somatic cells . Moreover, telomere attrition increases in inflammatory situations [31,32,33]. Since the origin of endometriosis is unclear and accumulating evidence suggests that inflammation plays a major role in this pathology, some authors have explored telomere length and telomerase activity in the context of endometriosis. Three studies have examined the association between leukocyte telomere length and endometriosis, with controversial conclusions. One study reported longer telomeres in leukocytes among women with endometriosis compared to those without endometriosis . Another study described that shorter leukocyte telomere length was associated with a high probability of having an endometriosis history . The last study reported no association between peripheral blood leukocyte telomere length and endometriosis . Endometrial telomere length was significantly longer than the corresponding blood telomere length, suggesting tissue-specific regulation mediated by telomerase . Some authors have described increased telomerase activity in endometrial tissue from women suffering from endometriosis versus healthy women [36,37,38,39,40], which could be explained by the enhanced expression of hTERT (the catalytic reverse transcriptase subunit of telomerase) caused by the binding of estradiol to its promotor estrogen response element [41,42]. Similar to what happens in tumors, telomerase activity may promote the cellular proliferation of endometrial tissue in endometriosis. Nevertheless, the association between telomerase activity and the endometriosis stage is not clear [36,39]. Further studies are warranted to elucidate the interrelationship between telomere length and the inflammatory and hormonal background among patients with endometriosis. Direct telomerase inhibition in endometrial tissue from women with endometriosis may arrest the proliferation and dissemination of endometriotic lesions. Some authors have confirmed this hypothesis by stopping the in vitro proliferation of endometriotic cells with imetelstat, an experimental anticancer telomerase inhibitor . 2.5. Immune Escape Several studies are in favor of the phenomenon of cellular immune escape allowing endometriotic cells to proliferate. Some arguments are inherent to ectopic endometrial cells, while others depend on the immune system. First, grouping endometriotic cells into fragments protects cells located in the deeper layers of these fragments. Moreover, endometriotic cells have different characteristics that can allow them to escape this immune system: (i) they expose modified type I HLA antigens; (ii) they secrete TGF-β and PGE2, responsible for lymphocyte inhibition; and (iii) they secrete HLA soluble antigens or sICAM-1. These factors confer a greater resistance to lysis from NK cells because they bind to NK’s LFA-1 receptor (competition with ICAM-1 membrane receptor). Finally, they can induce apoptosis of immune cells via mechanisms involving the Fas system [2,5,7]. Furthermore, the immune system of patients with endometriosis is suspected to be dysfunctional. In particular, NK cells have altered activity . Another argument in favor of immune dysfunction in the development of endometriosis is the high prevalence of associated autoimmune diseases (SLE, rheumatoid arthritis, SGS, and autoimmune thyroid diseases) and atopic (allergies, eczema and asthma) diseases in these patients [5,7]. 2.6. Adhesion, Implanting and Invasion Endometriosis appears and develops according to the following stages: (i) reflux; (ii) adhesion; (iii) proteolysis; (iv) proliferation; (v) angiogenesis; and (vi) lesion . A constitutional or acquired altered peritoneum would be a predisposing factor to attachment and to mesothelium invasion since an intact mesothelium is a natural barrier to this pathological process. In vitro studies have shown that fragments are only implanted at peritoneal locations where the extracellular matrix and basal membrane are exposed due to damage in the mesothelial layer. Therefore, retrograde menstruation would have a harmful effect, thus explaining the occurrence of such mesothelial damage favoring implantation [2,5]. At the molecular level, first, a strong interaction between hyaluronic acid of the extracellular matrix and CD44 of endometrial cells initiates the adhesion process . Moreover, some fibronectin receptors (α4β1, α5β1), whose endometrial expression normally varies according to cycle phase and estrogen levels, are constantly expressed by endometriotic cells. This suggests a potential role of these receptors in cell adhesion . Second, implantation is promoted by the inflammatory environment resulting from the overexpression of matrix metalloproteinases (MMP-1, MMP-2, and MMP-3) and ICAM-1 during the luteal phase, as well as the increased levels of TGFβ, IL-6, IL-1β and TNFα. MMPs and their inhibitors (TIMPs) are involved in extracellular matrix remodeling. Their expression varies with the phase of the cycle and suggests ovarian hormonal regulation. Most isoforms of MMPs are synthesized and activated during the endometrial proliferation phase, particularly under stimulation by estrogen (in contrast, progesterone tends to decrease their synthesis). The balance between MMPs and TIMPs is essential for ensuring the correct MMP activity. MMP hyperactivity could lead to matrix disruption and thus cell invasion. In women with endometriosis, the TIMP-1 concentration is precisely decreased in peritoneal fluid. Moreover, the expression and activity of MMP-7, MMP-1 and MMP-3, normally reduced by progesterone during the ovulatory phase, persist in endometriotic lesions due to progesterone resistance. Furthermore, deregulation of the E-cadherin system of endometriotic cells makes it possible to initiate the invasion process, similar to what is observed in carcinoma cells [2,4,5,7]. 2.7. Growth and Lesional Neuroangiogenesis Oxygen and various nutrient supplies that are essential for the growth of endometriosis implants are transported due to angiogenesis’ vascular development. This neovascularization in a physiologically avascular peritoneum induces a rather favorable environment. The concomitant development of nerve fibers explains the pain experienced by patients [3,43]. The initiation of the phenomenon involves the secretion of several cytokines (especially by peritoneum macrophages), namely TNFα, TGF-α, TGF-β, IL-8, MMP-3, and mainly VEGF (its peritoneal fluid rate is correlated with disease severity). VEGF is mainly synthesized during the secretory phase of the menstrual cycle in a healthy endometrium. In the case of endometriosis, elevated levels of VEGF have been reported in the peritoneal fluid during the proliferative phase of the menstrual cycle (when the peritoneum is exposed to retrograde menstruation). Furthermore, factors modulating its secretion (localized hypoxia, IL-1β, TGF-β, EGF and PGE2) are increased in the case of endometriosis. In addition to promoting angiogenesis, they also increase capillary permeability, facilitating macrophage diapedesis. Other mitogenic factors for endometriosis cells are involved, such as angiogenin, platelet-derived endothelial growth factor (PEGF), macrophage migration inhibitory factor (MMIF), hepatocyte growth factor (HGF), epidermal growth factor (EGF), insulin-like growth factor (IGF) and basic fibroblast growth factor (bFGF) [2,5,7,14]. 2.8. Inflammation In the case of endometriosis, the peritoneum shows an increased number of activated macrophages (with increased activity) and high levels of many cytokines, such as MMIF, TNFα, IL-1β, IL-6 (largest proportion), and IL-8. However, it is difficult to conclude whether these phenomena are a cause or a consequence of endometriosis [5,7,14]. Chemokines, such as monocyte chemoattractant protein 1, IL-8 and regulated upon activation normal T-cell expressed and secreted (RANTES), are involved in chemotaxis, promoting the influx of polynuclear neutrophils, NK cells, and peritoneal macrophages. Positive autoregulation (positive feedback) maintains this phenomenon and causes both an accumulation of immune cells and elevated levels of cytokines in endometriotic lesions [2,4,7]. This positive feedback is further accentuated by the hormonal climate of endometriosis in the following way: In women suffering from endometriosis, peritoneal fluid macrophages showed a significantly greater ability to secreteCOX-2 and therefore to secretePGE2. Furthermore, TNFα promotes the production of PGF2a and PGE2 by endometriotic cells, while IL-1β activates COX2, inducing PGE2 production and consequently activating aromatase. E2, resulting from high aromatase activity (which is also increased by MMIF, contributing to positive feedback), induces the increased synthesis of IL-6 and TNFα, leading to maintenance of the proinflammatory context . On the other hand, anti-inflammatory progesterone action is lacking in cases of endometriosis because of progesterone resistance. Inflammation in women with endometriosis is not only limited to endometriotic peritoneal lesions but is also found throughout the endometrium [5,14]. Clinical studies revealed that endometriotic stromal cells release cytokines (IL-33 and others) and promote a type 2 immune response : macrophages transform into M2 subtype, and T regulatory cells (Tregs) into Th2-like Tregs which secreted high levels of IL-4, IL-13, TGF β1. In association with the platelet-BB-derived growth factor of platelets, these cytokines directly and indirectly, via endometriotic cell-Tregs interference, promote the emblematic fibrogenesis of endometriosis . Interleukin-17 (IL-17), secreted by CD4+ T helper 17 (Th17cells), is another proinflammatory cytokine involved in the regulation of the immune microenvironment of endometriotic lesions: IL-17 promotes proliferation, invasion, and implantation of endometriotic cells directly and indirectly through the recruitment and activation of neutrophils (via IL-8 and granulocyte-colony stimulating factor (CSF) and granulocyte macrophage-CSF). IL-17 recruits and activates M2 macrophages, which, in response, release nitric oxide. In addition, IL-17 recruits lymphocytes and bone-marrow-derived cells, inducing the secretion of proangiogenic factors . 2.9. Role of miRNAs in Endometriosis MicroRNAs are small noncoding RNAs modulating gene expression through mRNA degradation or other interactions. Involved in almost all diseases, they have also been investigated in the pathophysiology of endometriosis [2,4,5,7,47]. Some miRNAs regulate the epithelial-mesenchymal transition , essential for the dissemination of epithelial cells. Others would modulate the hormonal environment (miRNAs -23a and -23b, miRNAs -:135a, 135b, 29c, and 194 −3p) via their interaction with SF1. Others promote angiogenesis (miRNA -126, miRNA -210, miRNA -21, miRNA -199a-5p and miRNAs 20A). Others increase inflammation and cell proliferation (miRNA -199a and miRNA -16). Others are the consequences of modified environmental conditions such as oxidative stress (miRNA -302a). Some miRNAs have been proposed as diagnostic biomarkers since their concentration in blood is increased significantly in case of endometriosis [47,49]. 3. Endometriosis and Cancer The hyperestradiolaemia associated with endometriosis suggests a potential increase in female genital cancer. A recent meta-analysis concluded a positive association between endometriosis and ovarian cancer . Endometriosis doubles the risk of developing ovarian cancer (SRR = 1.93, 95% CI = 1.68–2.22; n = 24 studies), maximizes the occurrence of clear cell histotype of ovarian cancer (SRR = 3.44, 95% CI = 2.82–4.42; n = 5 studies), and moderately increases that of endometrioid histotype (SRR = 2.33, 95% CI = 1.82–2.98; n = 5 studies). The type of endometriosis is also a crucial element: endometrioma multiplies by 5.41 the risk of ovarian cancer. Since cancer-driver mutations (KRAS, ARID1A, PIK3CA) are similar in deep lesions and endometriomas, tumorigenesis results from additional factors. Ovarian stromal conditions are particularly well-suited for the proliferation of endometriotic cell clones . Estradiol levels are very high in ovarian stroma. In endometriotic stroma cells, the very active CYP1B1 converts estradiol to 4-OH-estradiol, which is further converted to 4-OH-estradiol-quinone damaging DNA via alkylation or oxidation, promoting mutations in addition to the cancer driver mutations described earlier. Interestingly, endometriosis was associated with a minimally increased risk of breast cancer (less than 10%). Only rare studies reported an increased risk for estrogen receptor-positive (ERþ)/progesterone receptor-negative (PR) breast cancer (ERþ/PR: HR = 1.90, 95% CI = 1.44–2.50) [50,51]. The association with endometrial cancer is controversial [50,51] probably because some biases complicate analyses. For instance, patients with lean size (low BMI) have an increased risk of endometriosis . In contrast, high BMI increases the risk of endometrial cancer probably because of the high production of testosterone by fat tissue and because of abnormal insulin pathways . The unpredicted inverse correlation of endometriosis with cervical cancer (SRR= 0.68, 95% CI =0.56–0.82; n = 4 studies) potentially results from better access to early diagnosis and treatment of cervical lesions in patients with endometriosis because the painful character of the disease leads patients to consult their gynecologist more frequently. Moreover, chronic pelvic pain and dyspareunia may limit the sexual relationships of patients with endometriosis and, therefore, their contamination with HPV . Endometriosis also increases the risk of thyroid cancer (SRR = 1.39, 95% CI =1.24–1.57; n = 5 studies) but not of colorectal cancer (SRR = 1.00, 95% CI =0.87–1.16; n = 5 studies). The association with cutaneous melanoma was controversial . 4. Endometriosis and PolyCystic Ovaries Syndrome (PCOS) Polycystic ovaries syndrome is characterized by multiple cysts at the surface of the ovaries in association with endocrine and metabolic disorders. The endocrine syndrome results from the high production of estrogens and androgens and insulin resistance with overweight and diabetes mellitus. PCOS and endometriosis share a common association with high ovarian estrogen levels. In both cases, a hormonal balance of sex hormones is disturbed: high estrogen with progesterone resistance in endometriosis, and high estrogen with high androgen in PCOS. Experimental animal studies and human epidemiologic studies support a developmental theory of both diseases. Both would result from an abnormal fetal androgen impregnation during the in utero programming of the hypothalamic–pituitary–gonadal (HPG) axis with other environmental and genetic factors. Low prenatal testosterone results in the programming of the female fetal HPG axis, leading to features associated with endometriosis profile, such as early puberty, low LH/FSH rate, low AMH, fast folliculogenesis, and short anogenital distance. In contrast, high prenatal testosterone orientates the female fetal HPG axis in the opposite direction: late puberty, high LH/FSH rate, long folliculogenesis, and long anogenital distance. This hypothesis is supported by the relatively rare prevalence of diseases together . Multiple PCOS follicles and cysts produce high levels of estrogen. Chronic anovulation favors ovarian accumulation, which may increase the risk of ovarian cancer. This hypothesis is supported by similar DNA hypomethylation and miRNAs in PCOS ovaries and ovarian cancer. In addition to high estrogen levels, PCOS ovaries secrete high testosterone levels, increasing the risk of endometrial cancer [54,55]. 5. Infertility and Endometriosis Various mechanisms may explain the consequences of endometriosis on several steps of reproduction, especially: (1) the tubal transfer of the oocyte-cumulus complexes; (2) gamete interaction; (3) implantation; (4) the importance of ovarian reserve and oocyte quality; and (5) sexual behavior. First, the distorted pelvic anatomy in relation to major pelvic adhesions can disrupt oocyte release from the ovary and disturb the tubo-uterine passage [4,9]. Second, peritoneal fluid is more abundant, and its modified composition is the consequence of endometriotic lesions, as mentioned before (see the section on inflammation). This liquid is largely in contact with the bulb at the distal end of the fallopian tubes close to the fertilization site. Therefore, its chemical composition can directly influence and disturb gamete interactions. In particular, IL-1 and IL-6 directly affect sperm motility. TNFα induces DNA damage through reactive oxygen species (ROS) (resulting frequently in cell apoptosis). These cytokines could also prevent sperm capacitation. Finally, the oxidative stress induced by ROS inhibits the acrosome reaction and gamete fusion [9,56]. As mentioned earlier, M2 macrophages activated by peritoneal IL-17 release a high amount of nitric oxide (NO) with an additional harmful effect on sperm, embryo development and implantation. Reducing NO synthesis in peritoneal fluid or blocking the effects of NO could limit the impact of endometriosis on fertility . Third, in the endometrium, the influx of immune cells results in an increased release of several cytokines, modifying the endometrial environment as explained above . Lymphocytes secreting IgG and IgA autoantibodies can disturb embryo implantation [9,58]. Other studies have shown reduced expression of αvβ3 integrin, ensuring physiological cell adhesion . Fourth, in the ovaries, inflammatory endometriotic cysts can damage the ovarian cortex and decrease follicular reserve. This phenomenon can be accentuated by the surgery performed in this case [9,56]. In women with endometriosis, lower oocyte and embryo quality is frequently observed. Embryos derived from oocytes from women with endometriosis show decreased implantation rates, even when the transfer is carried out in a uterus without endometriosis (healthy women). However, these findings should be confirmed in further studies . Finally, chronic pelvic pain induced by pelvic inflammation and adhesions causes dyspareunia, leading to a reduced frequency of sexual intercourse. This behavioral phenomenon significantly reduces the chances of natural conception . 6. Environmental Impact The fetal environmental impact on the subsequent genesis of various pathological processes at an early stage of development has long been studied as the Barker hypothesis . As detailed below, many authors, including Bulun et al., defend the hypothesis of an epigenetic process at the origin of the pathophysiology of endometriosis [4,61]. Prenatal exposure to multiple ubiquitous pollutants or toxic molecules is well-established . It obviously includes cigarette smoke but also chemicals belonging to the category of endocrine disruptors. In this category, there are compounds with short half-lives, such as bisphenol A (BPA) or phthalates, and compounds with long half-lives, such as dioxins. Finally, some hormones and some drugs with hormonal action were suspected of promoting the endometriosis process, especially diethylstilbestrol (DES) and ethinyl estradiol (EE) . 6.1. EE and DES By orally administrating high amounts of EE to mice from the 11th to the 17th days of gestation, Koike et al. showed that this experimental prenatal exposure increased the incidence of endometriotic lesions in the next generation . DES, prescribed to millions of women between 1938 (discovery date) and the 1970s (when its use was banned) to limit or prevent recurrent miscarriages, was responsible for a large number of deleterious effects in the in-utero-exposed fetus . DES had a very strong and long EE-like activity. In the 1980s, Haney and Hammond studied the influence of DES on fertility. In a small group of 33 infertile couples with women exposed to DES in utero, they found that infertility was due to the presence of endometriosis in 11 of them. The reported association is not clearly established because the study did not include an infertile population control group . More recently, in a prospective cohort study, Missmer et al. (“Nurses Health Study II”) identified a higher relative risk (RR = 1.8, CI = 1.2–2.8) of developing endometriosis in women exposed to DES in utero . For other authors, such as Benagiano and Brosens, the impact of endometriosis would be greater in women exposed to DES . However, Wolff et al. failed to confirm a significant association between DES and endometriosis in the “ENDO study”. This large study involved a cohort of 473 patients operated on through laparoscopy and a control cohort of 127 patients who completed a pelvic magnetic resonance imaging, all from 40 clinical investigation centers in Utah and California between 2007 and 2009 . The exact mechanism of DES on endometriosis development is unknown. Some authors proposed a link between in utero DES exposure and cervical stenosis, smooth uterine muscle abnormalities or altered expression of estrogen receptors [67,68,69,70]. Golden et al. explained that high exposure to estrogen (or derivatives such as DES) during embryonic development could cause a disruption of genes under the influence of steroid hormones (such as genes encoding ESR) . Koike et al. showed that in mice exposed to DES, constant expression of the lactoferrin and EGF genes can be observed in the vagina and uterus . Furthermore, Wang et al. hypothesized that EGF could stimulate endometriosis cell proliferation by activating the Ras/Raf/MEK/ERK pathway [72,73,74]. 6.2. Dioxins Dioxins are chlorinated and polycyclic aromatic lipophilic agents and can persist for a long time in organisms. This results in a bioaccumulation process. These compounds, which are only produced by human activities, include dioxins and “dioxin-like” compounds: polychlorinated dibenzo-p-dioxins (PCDDs) or dioxins, polychlorinated dibenzofurans (PCDFs) and polychlorinated biphenyls (PCBs) [1,75,76,77,78]. 2,3,7,8-p-Dioxin-tetrachlorodienzo (TCDD) is the most toxic dioxin, and its toxicity is also a reference for assessing the impact of other organochlorine compounds with the toxic equivalency factor (TEF) . TCDD can alter specific ESR rates but also, more generally, the metabolism of steroid hormones. Its toxic effects are directly related to its binding to the nuclear receptor AhR (aryl hydrocarbon receptor), leading to the formation of an activated heterodimer with transcriptional action. Finally, overexpression of AhR may result in an inflammatory state and promote menstruation. The activation of AhR may also activate other factors involved in cell proliferation, such as TGFβ [1,76]. Moreover, TCDD increases the secretion of MMPs and can also induce progesterone resistance [1,75]. Otherwise, TCDD might act by disrupting the expression of microRNAs . Studying TCDD exposure in mice, Bruner-Tran et al. concluded that in utero TCDD leads to a phenotype of progesterone resistance persisting over several generations . According to Cummings et al. , PCB exposure is associated with a higher risk of developing endometriosis via a mechanism similar to that previously described with TCDD via a pathway involving AhR. PCB also decreases circulating NK cell activity, as well as the production of IL-1β and IL-12 . The first epidemiological and clinical studies on the impact of dioxins and endometriosis development were conducted by Rier et al. . The same authors have shown a link between TCDD and endometriosis in rhesus monkeys. Thereafter, other studies have been conducted, in particular to check the consequences of the Seveso catastrophe in 1976: Eskenazi et al. have not demonstrated an increased incidence of endometriosis in patients living in the Seveso area after the disaster . Globally, the literature is controversial: some authors have shown a potential link between endometriosis and dioxin exposure [82,85,86,87,88,89,90,91], and others have shown no correlation [92,93,94,95,96,97,98,99,100,101]. There is no publication on the potential involvement of Agent Orange in the onset of endometriosis in Vietnamese women after the Vietnam War. Finally, some studies have suggested an inverse relationship, that is, a “protective” role of dioxins. Furthermore, a study highlighted a lower incidence of endometriosis in children breastfed with possible exposure to dioxins in maternal milk than in adults . Because the epidemiological and analytical methodologies are so different, these conflicting findings do not suggest a potential link. However, recent studies support the possible role of dioxin epigenetic modification in endometriosis [103,104]. 6.3. Bisphenol A BPA is used in industry as a monomer for epoxy resins (tins, cans) and polycarbonate (plastic industry, additives), and has estrogenic activity. Upson et al. used data from the “Women’s Risk of Endometriosis study” and a control population, including 143 endometriosis patients and 287 controls, to show that BPA exposure is associated with a higher risk of developing endometriosis . Signorile et al. studied the effects of BPA in a mouse model. According to their study, animals exposed to BPA present a higher incidence of adenomatous hyperplasia with cystic endometrial hyperplasia, atypical hyperplasia and ovarian cysts (45–50%) than control animals (10%) . BPA action leads to a hyperestrogenic environment via inhibition of the expression of PR and progesterone activity and the promotion of E2 activity. Such hormonal alterations during a critical period of embryogenesis may increase the susceptibility to developing endometriosis and generally other diseases through an epigenetic mechanism [75,106]. According to Xue et al., BPA promotes endometriosis by facilitating endometrial stromal cell invasion , especially by upregulating matrix metalloproteinases 2 and 9. Moreover, Xue et al. highlighted the upregulation of Erβ expression in endometrial cells via the WD repeat domain 5/TET methyl-cytosine dioxygenase 2 (WDR5/TET2)-mediated epigenetic pathway . Environmental BPAF, a fluorinated homolog of BPA with stronger estrogenic activity, may promote, alone or in association with BPA, the development of endometriosis . 6.4. Phthalates Chemicals derived from phthalic acid, namely phthalates, are commonly used in the plastics industry. With approximately 3 million tons produced worldwide per year, phthalates and phthalate metabolites are present everywhere at different rates in our environment: cosmetics, paints, clothes, toys, etc. Several phthalates have been classified as toxic substances for human reproduction (CMR category 1B) by the European Chemicals Agency (ECHA) . In the atmosphere, their physicochemical properties ensure easy transport and thereafter potential bioaccumulation in the food chain (especially for low-molecular-weight phthalates) . Their (repro)toxicity is manifested through toxic effects on sperm, early puberty in girls, abnormalities of the genital tract, and infertility, in addition to adverse effects on neurodevelopment or simply allergies . Several studies have focused on the mechanism of the toxic action of phthalates in the development of fish embryos, especially di(2-ethylhexyl) phthalate (DEHP), diethyl phthalate (DEP), dibutyl phthalate (DBP) and benzyl butyl phthalate (BBP). For these molecules, oxidative stress would be the most critical mechanism of toxicity (CMTA = Critical Mechanisms of Toxic Action) in the case of DEHP and DEP exposure. Unlike DES or dioxins, phthalates do not act as E2 receptor agonists and have a very low affinity for AhR . In a recent review, Kim and Kim summarized the mechanisms through which phthalates, especially DEHP, promote endometriosis: (1) phthalates induce a modification of estrogen receptor type; (2) they increase the resistance to apoptosis of endometriotic stroma cells; (3) they increase the invasiveness of endometriotic stroma cells through the stimulation of MMP2 and 9 secretion; and (4) they cause oxidative stress and reduce antioxidant enzymes, finally leading to an enrichment of environmental ROS. All these mechanisms increase the proliferation and invasiveness of endometriotic stromal cells, promoting endometriosis . Several studies have explored the relationship between endometriosis and phthalate exposure; however, they were biased due to contamination with phthalates from the collection tubes and other equipment and laboratory supplies. Three epidemiological studies conducted by Huang et al., Itoh et al., and Weuve et al. assessed the risk of endometriosis in relation to the urinary concentration of phthalate metabolites [113,114,115]. Unfortunately, the results are conflicting. Upson et al. resumed the principle of study and obtained conflicting results: an inverse correlation between endometriosis and urinary phthalate levels, while they observed an increased risk of endometriosis with high urinary levels of mono-benzyl phthalate (MBzP) and mono-ethyl phthalate (MEP); however, the results were not statistically significant . In the previously mentioned “ENDO study”, Buck Louis et al. did not report a strong correlation between endometriosis and phthalates. However, the short half-life of phthalates in the blood can lead to an analytical bias with this type of assay (urine being preferred) . A recent meta-analysis concluded that there was a potential statistical association only between MEHHP exposure and endometriosis, particularly in Asia, but not between other phthalate acid esters (PAEs) and endometriosis. The authors acknowledged the weak strength of the results due to the lack of well-designed cohort studies with large sample sizes . 6.5. Tobacco The effects of tobacco on the development of endometriosis are controversial. Smoking seems to be a protective factor against the endometriosis process. Several studies have demonstrated an inverse correlation between tobacco consumption and endometriosis [118,119]. However, Haney and Hammond and Somigliana et al. failed to demonstrate this association. The large cohort study of Hemmert confirmed the meta-analysis of Bravi et al., which concluded the absence of a link between tobacco consumption and endometriosis . The protective effect of tobacco has been suggested for many years via the hypoestrogenic action of some tobacco compounds, namely nicotine and cotinine (one of the major metabolites of nicotine), which influence the metabolism of steroid hormones and prevent the conversion of androgens to estrogens. Furthermore, nicotine promotes cell apoptosis, with the consequence of limiting the proliferation of endometriotic cells. Finally, this substance induces a decrease in the cell inflow and activity of NK cells to limit the inflammatory phenomenon that is well-described in endometriosis [75,118]. Experimental animal data and human ex vivo experimentation suggest a role of C-X-C motif chemokine ligand 12 (CXCL12) and fibroblast growth factor 2 (FGF2), two cytokines with a pro-cell proliferative action, whose secretion is decreased in the case of tobacco consumption . Moreover, increasing the secretion of VEGF by nicotine is suspected to support the development of endometriosis by promoting neoangiogenesis and vascularization of endometriotic lesions . Furthermore, secondhand smoke during childhood due to maternal smoking seems to be associated with an increased risk of endometriosis in adolescents and young adults . 7. Role of the Microbiota in Endometriosis Recently, the study of the microbiota to decipher the physiopathology of many complex pathologies has been applied to endometriosis. The literature has rapidly expanded in the last eight years. Some reviews have tried to summarize how the microbiota regulates factors involved in maintaining the normal peritoneal environment and ectopic cell clearance, and how dysbiosis contributes to the dysregulation of factors driving endometriosis development . A specific composition of the gut microbiota is suspected to induce immune dysregulation, which can progress into a chronic state of inflammation, a perfect environment for endometriosis progression. Endometriotic microbiotas have been consistently associated with diminished Lactobacillus dominance on the one hand, and an altered Firmicutes/Bacteroidetes ratio associated with a high abundance of vaginosis-related bacteria on the other [124,125]. In comparison, in PCOS, the distortion of microbiota results in an abnormal Escherichia/Shigella ratio and an excess of Bacteroides . Some studies even suggest a main infectious origin in the pathophysiology of endometriosis [127,128,129]. Furthermore, estrogen metabolism is known to be regulated by the estrobolome, a collection of gut bacteria involved in estrogen metabolism. Estrobolome activity modulates the amount of excess estrogen that is excreted from or reabsorbed into the body. When this activity is impaired, especially in cases of imbalances in the gut microbiome, excess estrogen can be retained in the body and diffuse from the gut to the endometrial and peritoneal environment via the circulation. This contributes to the hyperestrogenic environment that drives endometriosis and provides a possible mechanism as to how dysbiosis in the gut microbiota may be involved in the disease [130,131]. Interestingly, women with a high intake of omega-3 polyunsaturated fatty acids (PUFAs) have a lower risk of endometriosis. A similar diet showed anti-inflammatory effects and suppressed the formation of endometriotic lesions in murine models. This suggests an at least partial contribution of diet to the induced modification of the gut flora [124,132]. This concept of the relationship between the microbiota and endometriosis leads to the consideration of antibiotics as a new promising approach for endometriosis treatment. In animal models, broad-spectrum antibiotics have already proven efficacious for treating endometriosis. In a recent murine study, broad-spectrum antibiotics inhibited ectopic lesions, while treatment with metronidazole significantly decreased inflammation and reduced lesion size, possibly by lessening Bacteroidetes presence . Alternatively, probiotic intervention, that is, the administration of live microorganisms, could be another effective approach [134,135]. Since most chemical endocrine disruptors transit the digestive tract, they interact with gut microbiota. On the one hand, endocrine disruptors can modify the microbiota or modulate microbiota enzymatic activity. In the long term, an endocrine disruptor can alter the microbial diversity of the microbiota. On the other hand, microbiota metabolizes part of chemicals, therefore modulating their toxicity . Prenatal exposure to endocrine disruptors may promote endometriosis via altered maternal and fetal microbiota , resulting in abnormal sex hormone levels (as exposed earlier). An alteration of microbiota may also decrease the potential of DNA methylation since some gut bacteria produce folate, a central methyl donor . 8. Conclusions Endometriosis is a gynecological disease with a complex pathophysiology (Figure 5). To date, the specific pathogenesis of endometriosis has not been clarified, and some recent studies have suggested a potential role of the gut microbiota . What is certain is that there is a key role of estradiol and retrograde menstruation. Endometrial tissue transformation in endometriosis can be observed in women exposed in utero to endocrine disruptors. These substances are the root cause of an epigenetic process disrupting the expression of key steroidogenesis genes in endometrial cells. Figure 5. Endometriosis, a complex disease, with several concomitant etiologies? Epidemiological studies of exposure to the molecules probably involved in such a mechanism, such as dioxins, bisphenol A, phthalates, DES, or nicotine, have not found strong and repeatable correlations. However, these studies are conducted using heterogeneous methodologies. Measurement errors in estimating the behaviors during pregnancy of mothers of daughters suffering from endometriosis can hinder the results obtained because of “subjectivity” and recall bias. All the abovementioned studies are not based on the same clinical diagnosis repositories of endometriosis because they were not performed at the same time. Selection bias could have occurred. In addition to classification errors, all of these aforementioned elements constitute an important limitation [10,66]. Further large-scale and homogeneous studies are needed to draw conclusions about the influence of these endocrine-disrupting compounds on the development of endometriosis. Author Contributions Conceptualization, N.M. and I.K.; methodology, I.K.; validation, I.K., N.M. and A.J.F.; writing—original draft preparation, N.M. and A.J.F.; writing—review and editing, I.K.; visualization, N.M.; supervision, I.K. All authors have read and agreed to the published version of the manuscript. Funding This research received no external funding. Institutional Review Board Statement Not applicable. Informed Consent Statement Not applicable. Data Availability Statement All references cited in the manuscript are available in pubmed. Acknowledgments We would like to thank Pierre Monnin for his English language assistance. Conflicts of Interest The authors declare no conflict of interest. References Soave, I.; Caserta, D.; Wenger, J.M.; Dessole, S.; Perino, A.; Marci, R. Environment and Endometriosis: A toxic relationship. Eur. Rev. Med. Pharm. Sci.2015, 19, 1964–1972. [Google Scholar] Vinatier, D.; Orazi, G.; Cosson, M.; Dufour, P. Theories of endometriosis. Eur. J. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( Share and Cite MDPI and ACS Style Monnin, N.; Fattet, A.J.; Koscinski, I. Endometriosis: Update of Pathophysiology, (Epi) Genetic and Environmental Involvement. Biomedicines2023, 11, 978. AMA Style Monnin N, Fattet AJ, Koscinski I. Endometriosis: Update of Pathophysiology, (Epi) Genetic and Environmental Involvement. Biomedicines. 2023; 11(3):978. Chicago/Turabian Style Monnin, Nicolas, Anne Julie Fattet, and Isabelle Koscinski. 2023. "Endometriosis: Update of Pathophysiology, (Epi) Genetic and Environmental Involvement" Biomedicines 11, no. 3: 978. APA Style Monnin, N., Fattet, A. J., & Koscinski, I. (2023). Endometriosis: Update of Pathophysiology, (Epi) Genetic and Environmental Involvement. Biomedicines, 11(3), 978. Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here. 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Biomedicines, 11(3), 978. Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. 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188923	https://engoo.com/app/words/word/alleviate/zga0ELstQmCjlQAAAD9JCw	alleviate (【Verb】to ease or reduce ) Meaning, Usage, and Readings \| Engoo Words Please refresh the page to get the latest updates.Refresh WORDSTop "alleviate" Meaning alleviate /əˈliːviːˌeɪt/ Verb to ease or reduce "alleviate" Example Sentences We are working on new policies to help alleviate stress at the office. The latest sales results will alleviate shareholder fears that our company is in trouble. We need to alleviate the gas pressure in this part of the system. "alleviate" Related Lesson Material Vaccinators are now people from the communities they work in which has helped alleviate some of the safety issues. See Lesson "Poor mental health and loneliness are significant health concerns and we have demonstrated that robots can help alleviate these." See Lesson Many states have now started looking at ways to simplify the path for internationally trained doctors in an attempt to alleviate the doctor shortage. See Lesson Spain has pledged to allow 17,300 refugees from such war-torn countries as Syria, Iraq and Libya to settle in the country, as its part of a Europe-wide commitment to do more to help alleviate the continent's migration crisis. See Lesson “It has lane-keeping ability, it has advanced radar-based system to keep the vehicle’s speed also in sync with other vehicles and this can completely alleviate your effort, driving effort, in stop-and-go situations too, and finally the collision avoidance,” said Filipi. See Lesson Browse words ABCDEFGHIJKLMNOPQRSTUVWXYZ Related Words alleviate /əˈliːviːˌeɪt/ Verb (typically of pain or suffering) to make less severe 0 0 0 Presented by Engoo is a service that offers lessons for those learning English. Although the lesson materials can be used for self study, they are intended for use with a teacher. To book a lesson with one of our professional teachers, please visit Engoo. If you would like to use these materials for commercial purposes, please contact us here. © 1998 - 2025. No duplication without prior permission. All Rights Reserved.
188924	https://invevo.com/blog/what-is-the-accounting-rate-of-return	Home Book a Demo What is the Accounting Rate of Return? May 22, 2025 The accounting rate of return offers companies a simple but effective method of evaluating the profitability of investments over a period of time. Having a clear understanding of ARR is essential for financial professionals as it highlights potential returns on investment as well as playing a key role in strategic planning. ARR is also a valuable tool when it comes to investment appraisal, capital budgeting, and financial analysis. The flexibility of ARR also means it can be applied to different scenarios such as evaluating how profitable a particular project will be, setting performance benchmarks, and ensuring resources are allocated properly. Within this blog post we are going to take an in depth look at ARR by using examples, break down the components of its formula, take a closer look at its pros and cons, and emphasise the significant part it plays in financial decision-making. What is Accounting Rate of Return (ARR)? The Accounting Rate of Return (ARR) is a financial metric that is used to work out what return you can expect to receive on investments or assets. ARR differs from both the Internal Rate of Return (IRR) and Net Present Value (NPV), as it does not look at the time value of money. Instead, it provides a straightforward estimate of profitability based on accounting data. The simplicity and ease of calculation of ARR makes it a practical tool which is why it is favoured by many business owners, stakeholders, finance teams, and investors. While it provides users with a quick way to assess the profitability of an investment, it does have a number of limitations. Because ARR is completely reliant on accounting profits, and disregards the time value of money, it may not give you a totally accurate projection of the actual profitability or economic value of an individual investment. Additionally ARR does not take into account the timing of cash flow, a key part of determining financial sustainability. Despite having these limitations ARR is still a really valuable tool for organisations. This is particularly true when used alongside other investment evaluation methods to provide you with a comprehensive analysis of investment opportunities. Accounting Rate of Return (ARR) Formula The simplistic nature of the Accounting rate of return formula means it can be easily accessed by any finance professional. To compute it you simply divide the average annual profit made from the investment concerned by its initial cost and show the result as a percentage. ARR gives stakeholders the ability to get an accurate calculation of the profitability of any investment in relation to cost. To help you get a clearer understanding of the formula we have broken it down underneath: ARR = (Average Annual ProfitInitial Investment) Average Annual Profit: This refers to the average yearly profit earned from the investment over its useful life. It is typically calculated by dividing the total accounting profit generated by the investment by the number of years you estimate that it will continue to run for. Initial Investment: This is the original cost of the investment, including any upfront expenses such as purchase price, installation fees, and other costs that may be associated with it. If you divide the average annual accounting profit by its initial cost and showing the result as a percentage, the ARR formula gives you a straightforward but effective method of evaluating how profitable an investment is in relation to your initial outlay. Pro Tip: You can make the most of asset utilisation by lengthening its useful life or improving productivity to increase ARR. Where is ARR Used? ARR is routinely used to analyse finances, appraise investments and make decisions about capital budgeting. Companies utilise ARR when they are trying to determine whether a project is feasible or not. It is also useful when it comes to reviewing how existing investments are performing and making comparisons with alternative investment opportunities. ARR can also be a good benchmark when determining performance goals and keeping track of the financial health of an organisation over a period of time. Let's take a closer look at some of the areas where a business may use ARR. Investment appraisal Probably the most common use of ARR is investment appraisal which is used to analyse how profitable a new investment or project could be. Doing an ARR comparison gives businesses the chance to make investments that have the best returns a priority. Capital budgeting decisions ARR plays a key part when making capital budgeting decisions as it gives firms information on how efficient and effective resource usefulness is. By using ARR businesses can give themselves a foundation from which they can work out how viable and profitable capital projects can be in the long term. Financial analysis When it comes to financial analysis ARR provides stakeholders with key information about how successful investments and projects are. It is regularly used by financial analysts when assessing the risk return profile of an investment and which areas can be improved, allowing them to provide management with informed recommendations. Performance evaluation By applying ARR you can evaluate an investment or the performance of a project over a period of time. If you follow any changes in ARR you are able to check if you are getting the returns you expect from an investment as well as identifying any chance to improve or diversify. Benchmarking You can use ARR as a benchmark when you set your goals or targets for performance while also allowing you the chance to evaluate the financial health of your organisation. By making a comparison between the actual ARR value and targets or industry standards organisations are able to gauge their level of performance while getting a clear understanding of areas that require improvement. Allocating Resources ARR can help when with resource allocation as it provides an insight into the returns you get from various investment options. Businesses generally utilise ARR to ensure capital and resources are allocated to projects that are likely to give them the best returns. How do you Calculate ARR Calculating ARR is a relatively simple process if you use accounting data that is readily available. By working out ARR a decision maker can get dependable insights into the profitability of their investments. This, in turn, helps them make informed choices regarding resource allocation, risk mitigation, and strategic planning. Below, we outline the steps involved in determining ARR: Calculate the Average Annual Profit Calculate the total accounting profit that the investment is expected to generate over its useful life and divide it by the estimated number of operational years. This provides the average annual profit. Formula: Average Annual Profit = Total Profit over Investment Period / Number of Years Identify the Initial Investment Cost Determine the initial cost of the investment, which includes all upfront expenditures such as the purchase price, installation costs, and any other related expenses. Apply the ARR Formula Use the ARR formula: ARR = (Average Annual Profit / Initial Investment) × 100 Divide the average annual profit by the initial investment, and express the result as a percentage. Analyse the Results Evaluate the ARR to assess the investment’s profitability. A higher ARR indicates a more lucrative investment, while a lower ARR suggests reduced profitability. Example of Accounting Rate of Return Using this example we can demonstrate how ARR can be applied: An organisation is looking at its operation and is considering making particular investments. Say they make an outlay of £200,000 on new manufacturing equipment it is estimated that over its five year useful lifespan it will give you an average yearly accounting profit of £40,000. Applying the ARR formula: ARR = (Average Annual Profit / Initial Investment) × 100 = (40,000/200,000) × 100 = 20% In this particular example you get 20% ARR by investing in the manufacturing equipment. This gives you an indication that for every £1 you have invested in the equipment the annual return will be 20% in relation to your initial outlay. Pro tip: By improving collection procedures you can speed up cash flow and improve ARR. Advantages and Disadvantages of the ARR As with any type of financial indicator there are advantages and disadvantages to ARR. By working out the pros and cons of ARR stakeholders are able to make informed decisions about how acceptable it is in certain investment situations and make changes to the way they approach analysis. Understanding these differences is important if you want to maximise how much value you can get out of ARR when it comes to financial analysis and decision making. Advantages Simplicity The ARR formula is straightforward and easy to understand, making it accessible to a broad range of stakeholders, including managers, investors, and analysts. Ease of Calculation ARR relies on basic accounting data, such as initial investment costs and projected annual profits, making it a convenient and cost-effective financial metric. Focus on Accounting Data By utilising accounting profits instead of cash flows, ARR allows firms to leverage readily available financial data from their accounting systems, simplifying investment evaluations. Long-Term Perspective ARR considers the entire lifespan of an investment, offering a long-term view of its profitability and sustainability over time. Facilitates Comparison ARR standardises profitability metrics, enabling businesses to compare the returns of different investments easily and make informed decisions. Performance Benchmark ARR serves as a benchmark for assessing the profitability of investments against industry standards or predefined targets, helping organisations track and improve financial performance. Disadvantages Ignores the Time Value of Money ARR does not account for the time value of money, as it averages profits over the investment's lifespan. This limitation can result in an inaccurate portrayal of profitability, particularly for investments with irregular cash flows. Reliance on Accounting Policies ARR is influenced by accounting policies, which can affect how profits are calculated. For instance, differences in depreciation methods may distort ARR values, requiring careful consideration. Potential for Misinterpreted Profitability ARR is not a definitive measure of absolute profitability, as it overlooks factors like risk, inflation, and opportunity costs. These variables can significantly impact an investment’s actual value and profitability. Incompatibility with Discounted Cash Flow Methods ARR cannot be used with metrics like Net Present Value (NPV) or Internal Rate of Return (IRR), which incorporate the time value of money. Consequently, ARR may provide less accurate profitability assessments compared to these methods. Preference for Short-Term Gains Solely relying on ARR may lead to a bias toward short-term investments with higher early returns, potentially neglecting longer-term projects with greater overall profitability but slower initial gains. This can result in suboptimal resource allocation. Limited Analytical Scope ARR is a simplified measure that may fail to capture qualitative factors such as strategic alignment, market trends, and competitive positioning, all of which are critical for evaluating investment success. How Does Depreciation Affect the Accounting Rate of Return? When calculating ARR depreciation is a key consideration because it has a direct influence on how much accounting profit an investment generates over time. By using depreciation expenses analysts can get a more accurate value of ARR that demonstrates the real economic performance of a particular investment or investments. By having a clear understanding of the relationship between depreciation and ARR stakeholders have the ability to make informed choices and minimise the possibility of falling foul of the risks that can be involved where investment appraisal is concerned. Here is how depreciation can affect ARR: Impact on accounting profit A non cash expense depreciation shows how much the value of an asset declines during the course of its useful lifespan. In any ARR calculation depreciation will reduce the accounting profit of any investment because it is deemed to be an expense and as such has to be deducted from total revenue to give you the net profit. Investments that have greater depreciation expenses will generally have a lower ARR value than those with lower depreciation expenses if everything else remains equal. Impact on investment evaluation Depreciation can lower the apparent profitability of an investment, potentially affecting how it is evaluated. Investments with substantial depreciation expenses might seem less appealing when assessed using ARR estimates, despite generating considerable cash flows. Therefore, it is crucial for analysts to consider the effects of depreciation when evaluating investment opportunities. Depreciation adjustment To calculate ARR, the non-cash depreciation expense is added back to the accounting profit. This adjustment provides a revised ARR, reflecting the economic profitability of the investment after considering depreciation. Depreciation Method The choice of depreciation method has a significant impact on ARR estimates. Various approaches, such as straight-line depreciation, accelerated methods like the double declining balance, and units-of-production depreciation, produce different depreciation expenses over an asset's useful life. As a result, the ARR values derived from each method can vary, influencing investment decisions. Pro Tip: Boost ARR by improving operational efficiency through automation and process optimisation. FAQs What does ARR mean? ARR stands for Accounting Rate of Return. It is a financial measure used to evaluate an investment's profitability by comparing its average annual accounting profit to the initial investment cost. The formula for calculating ARR is: ARR = (Average Annual Profit ÷ Initial Investment) × 100 How does the Accounting Rate of Return differ from the Required Rate of Return? The Accounting Rate of Return (ARR) evaluates an investment's profitability using accounting figures, such as average profit and initial costs. In contrast, the Required Rate of Return (RRR) is the minimum return investors expect to compensate for the risk associated with an investment. While ARR measures performance, RRR represents a benchmark for decision-making. What are the decision-making rules for ARR? Key decision rules for ARR include: The higher the ARR, the more attractive or acceptable the investment. If the ARR is below the target or required rate of return, the investment is rejected. If the ARR matches the target rate, further analysis or consideration is recommended before making a decision. What constitutes a good average rate of return? A higher average rate of return generally indicates greater profitability. However, what qualifies as a "good" return varies depending on the investor’s goals, risk tolerance, and financial situation, as well as the specific context of the investment. How do you calculate the book rate of return? The book rate of return is determined by dividing net income by the total cost of the investment and expressing the result as a percentage. This metric reflects the profitability of an investment relative to its cost. The formula is similar to ARR: ARR = (Average Annual Profit ÷ Initial Investment) × 100
188925	https://zhuanlan.zhihu.com/p/383323579	角平分线相关二级结论及其应用 - 知乎关注推荐热榜专栏圈子 New付费咨询知学堂直答切换模式登录/注册角平分线相关二级结论及其应用首发于Myuku的解析几何笔记切换模式角平分线相关二级结论及其应用 Myuku 惑わす心、苛々彼女。收录于 · Myuku的解析几何笔记 48 人赞同了该文章从初中开始我们就对角平分线有一番应用，有些比较爱探索的同学或者是其老师拓展的多的同学，提前就接触到了角平分线定理(具体我不清楚，反正笔者初中时没有学习过)和一些角平分线长度计算公式。但是到了高中，我们会发现书上仍然没有教授有关角平分线的内容，但是在考试和平时练习过程中，我们又会遇到许多与之相关的问题，特别是在解三角形和解析几何小题中，而且一般这种问题的难度都较大，那么我们应该怎么来面对它呢？在这篇文章中，我将为大家介绍在高中比较常见的角平分线有关的二级结论及其应用。 (1)角平分线定理:三角形一个角的平分线与其对边所成的两条线段与这个角的两边对应成比例。如图， A B B C=A D C D\frac{AB}{BC}=\frac{AD}{CD} ,这就是角平分线定理，笔者在书上没学过，但是初中做信息题时遇到过。 (2)角平分线长度计算公式：在这里我给出两种计算方法，第一种是运用等面积法，即利用 S△A B D+S△C B D=S△A B C S_{\bigtriangleup ABD}+S_{\bigtriangleup CBD}=S_{\bigtriangleup ABC} ,得到： 1 2 A B⋅B D⋅s i n∠A B D+1 2 B C⋅B D⋅s i n∠C B D=1 2 B A⋅B C⋅s i n∠A B C\frac{1}{2} AB\cdot BD\cdot sin\angle ABD+\frac{1}{2} BC\cdot BD\cdot sin\angle CBD=\frac{1}{2} BA\cdot BC\cdot sin\angle ABC 若设 ∠A B C=θ\angle ABC=\theta ,经过整理与二倍角公式的使用，我们可以得到: 1 a+1 c=2 c o s θ 2 B D\frac{1}{a} +\frac{1}{c} =\frac{2cos\frac{\theta }{2} }{BD} 实际上，这个公式就是张角公式的变形，这里就不再阐述张角公式是什么了，感兴趣的同学可以自己查阅资料。 (3) 斯特瓦尔特(Stewart)定理：设已知△ABC及其底边上B、C两点\间的一点P，则有AB²·PC+AC²·BP-AP²·BC=BC·PC·BP。将其变形，再应用到我们上面的图中，我们即可得到计算公式 BD^2=ac[1-(\frac{b}{a+c} )^2] 特别地，若三角形三个顶点 A、B、C 分别对应椭圆焦点三角形中的 F_1、P、F_2 (设P为焦点三角形的非焦点顶点)，，设P角所对应的角平分线长度为L,则公式可以变形为: L^2=\|PF_1\|\cdot \|PF_2\|\cdot (1-e^2) ,也即 L=\frac{b}{a}\cdot \sqrt{\|PF_1\|\cdot \|PF_2\|} 好了，粗略地看完上面的定理公式介绍后，我们来看几道基础例题(原创题）运用第二个公式，我们可以得到 2cos\frac{B }{2} =1 ,轻松得到 B=\frac{2\pi}{3} 运用第三个公式的焦点三角形变形形式，我们可以得到 1-e^2=\frac{3}{4} ,即 e=\frac{1}{2} 下面放几个我的原创题链接，都是有关这里面公式的应用。 (暂时还没写完) 编辑于 2021-07-12 09:29 圆锥曲线解析几何高中数学赞同 483 条评论分享喜欢收藏申请转载写下你的评论... 3 条评论默认最新 wxdc31c8c315f3b614 （2）的第二种计算方法是啥啊 2023-02-06 回复喜欢 wxdc31c8c315f3b614 3不太懂诶能解释下吗 2023-02-06 回复喜欢 Myuku 作者用向量或者余弦定理易证 2023-02-06 回复喜欢关于作者 Myuku 惑わす心、苛々彼女。回答 13文章 28关注者 3,044 关注发私信推荐阅读角平分线定理与角平分线逆定理的妙用 ================= Airotciv 角平分线模型及结论，记住它，考试省时间 =================== 包菜老师【初中】用角平分线公式解题 ============= NJMF角平分线长定理 ======= AC\cdot AB-CD\cdot BD=AD^2 导言通过证明了斯特瓦尔特定理和角平分线第二定理，就可以进一步推出角平分线长定理，即斯库顿定理。推导 \left{ \begin{array}{lc} vc^{2}+ub^{2}-am^{2}=au… LeamonLee 想来知乎工作？请发送邮件到 jobs@zhihu.com 打开知乎App 在「我的页」右上角打开扫一扫其他扫码方式：微信下载知乎App 无障碍模式验证码登录密码登录开通机构号中国 +86 获取短信验证码获取语音验证码登录/注册其他方式登录未注册手机验证后自动登录，注册即代表同意《知乎协议》《隐私保护指引》扫码下载知乎 App 关闭二维码打开知乎App 在「我的页」右上角打开扫一扫其他扫码方式：微信下载知乎App 无障碍模式验证码登录密码登录开通机构号中国 +86 获取短信验证码获取语音验证码登录/注册其他方式登录未注册手机验证后自动登录，注册即代表同意《知乎协议》《隐私保护指引》扫码下载知乎 App 关闭二维码
188926	https://www.uptodate.cn/contents/generalized-anxiety-disorder-in-adults-epidemiology-pathogenesis-clinical-manifestations-course-assessment-and-diagnosis/print	Your Privacy To give you the best possible experience we use cookies and similar technologies. We use data collected through these technologies for various purposes, including to enhance website functionality, remember your preferences, and show the most relevant content. You can select your preferences by clicking the link. For more information, please review our Privacy & Cookie Notice Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device. Because we respect your right to privacy, you can choose not to allow certain types of cookies on our website. Click on the different category headings to find out more and manage your cookie preferences. However, blocking some types of cookies may impact your experience on the site and the services we are able to offer. Privacy & Cookie Notice Manage Consent Preferences Strictly Necessary Cookies Always Active These cookies are necessary for the website to function. They are usually set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, this may have an effect on the proper functioning of (parts of) the site. Functional Cookies These cookies enable the website to provide enhanced functionality, user experience and personalization, and may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies, then some or all of these services may not function properly. Performance Cookies These cookies support analytic services that measure and improve the performance of our site. They help us know which pages are the most and least popular and see how visitors move around the site. Advertising Cookies These cookies may collect insights to issue personalized content and advertising on our own and other websites, and may be set through our site by third party providers. If you do not allow these cookies, you may still see basic advertising on your browser that is generic and not based on your interests. Performance Cookies Consent Leg.Interest label label label View Cookies cookie name
188927	http://cms-content.bates.edu/prebuilt/phys108o6.pdf	EXPERIMENT O-6 Michelson Interferometer Abstract A Michelson interferometer, constructed by the student, is used to measure the wavelength of He-Ne laser light and the index of refraction of a flat transparent sample. References Taylor, Zafiratos and Dubson, Modern Physics, second edition, Section 1.5. Pre-Lab Please do this section before coming to lab. Look at the Michelson interferometer diagram in one of the references, and compare to Figure 1 below. In texts, it is usually assumed that an observer looks into the interferometer, so that the eye receives light that originated at a relatively weak source. In this experiment a laser is used as the source, so the light leaving the interferometer is bright enough to be projected on a screen. Suppose that in Figure 1, the lens between the laser and the interferometer is removed. The laser beam then follows the dotted path (length x) into the interferometer. When it hits the beamsplitter, half the light is reflected along path L2 to a fixed mirror and the other half continues straight ahead, along path L1, to a movable mirror. The mirrors reflect the light back to the beamsplitter, where each returning ray is split again. Half of the light returning from each mirror leaves the interferometer along the dotted path (length y) and travels to a screen mounted on the far wall. Since there are two light beams arriving at this point on the screen, the spot is bright or dark depending on whether they are in or out of phase. One beam travels a distance x + 2L1 + y from point S to the wall, while the second travels x + 2L2 + y. The difference in these path lengths is therefore 2(L2 - L1), and the spot on the screen is bright only when this is an integer times the laser wavelength, i.e. when (L2 - L1) = n (λ/2). Figure 1: Michelson Interferometer Now suppose the movable mirror's motor is running, so that L1 changes at a constant rate. The spot where the dotted path meets the screen will therefore alternate between bright and dark. One complete cycle of this intensity variation is called a "fringe shift" and occurs each time L1 changes by λ/2, i.e. when ∆L1 = n (λ/2). When the lens is in front of the laser, as shown in Figure 1, it focuses the laser beam to point S, which acts as a point source sending light into the interferometer. Light waves emerging from a point source are spherical, i.e. the light rays coming out of S diverge from one another. This means that the light passing through the interferometer illuminates the entire screen, not just the "central" point where dotted path y hits. To find out if a particular point P on the screen is bright or dark, we need to consider the difference in path lengths from S to P for light traveling along the two arms of the interferometer. The resulting pattern on the screen turns out to be a series of concentric circular bright and dark bands called "fringes", as illustrated in Figure 2 below. To understand why the pattern is circular, remember that the lens focuses the parallel rays from the laser to the point, S, which can be considered as the actual "source" of light entering the interferometer. As described above, light from S gets to the screen by either of two routes through the interferometer and, for the center of the Figure 2: Two Source Representation pattern, the lengths of these routes are x + 2L1 + y and x + 2L2 + y, respectively. Therefore, what happens on the screen is the same as what would happen if there were two point sources located at these distances in front of the screen on a common axis, as shown in Figure 2, provided that wave crests leave each of the sources simultaneously. (Notice that if you "unfold" the paths in Figure 1 you get Figure 2.) Since L1>>(L2-L1), the rays from S1 and S2 to P are nearly parallel. For a particular angle θ θ θ θ, the light from S2 travels a distance α α α α while the light from S1 travels α α α α + 2(L2 -L1)cosθ θ θ θ. If the difference between these two path lengths is an integral number of wavelengths, constructive interference occurs and you have brightness at P. Since the same angle θ θ θ θ exists for all points P on a circle concentric with the center of the pattern, you see a bright circular ring. Finally, if you increase or decrease θ θ θ θ so that 2(L2 -L1)cosθ θ θ θ changes by λ/2, you get a dark ring. The pattern is thus a series of concentric bright and dark fringes. To make sure you are prepared for lab and read the Procedure section below. Think about QUESTIONS 1 and 2, so you can quickly answer when you come to them. As always, feel free to ask questions! Apparatus Steel plate on an inner tube Short focal length positive lens He-Ne laser and stand Rotational mount for transparent sample Optically flat reflectors on magnetic bases Micrometer Beam splitting cube on magnetic base Procedure Taking care not to touch any optical surfaces, examine the components on magnetic bases. In contrast to ordinary mirrors, the optically flat reflectors have their reflective coating on the front surface of the glass. This results in higher optical quality but makes it much easier to damage the coatings. Please avoid contact with these coatings, since they scratch easily and are severely corroded by skin oils. The tilt of the mirrors can be adjusted by turning the two screws that attach it to the base. Set the screws near the middle of their travel range. Note that the beamsplitter cube has a "half-silvered" interior diagonal surface, causing a beam of light entering any side to be half transmitted (straight through) and half reflected (at 90 degrees). Do not turn the micrometer shaft by hand. Movement causes "backlash" that can last 10 minutes! Note that the motor turns the shaft very slowly, so you can read the micrometer even with the motor on. Make sure you know how to read the scale (refer to the Commonly Used Lab Equipment link on the Physics 108 web page) before starting. Place the steel slab on top of the inner tube and, with the levers on the magnetic bases in their "off" positions, arrange the optical components on the slab as shown in Figure 1. Place the laser opposite the movable mirror and adjust the components until the axes of the laser and mirrors are perpendicular to the beamsplitter faces. Lock the magnetic bases to the steel slab. (Hint: It helps if L1 and L2 are nearly, but not quite, identical. Also, you may want to position the mirror with the motor in a location where you may read the micrometer scale easily.) With the laser on, you should see a pattern of bright dots on the wall opposite the bench. Observe the behavior of these dots when the mirror tilt adjustments are varied, returning the screws to the middle of their travel range. QUESTION 1: Why are there more than two dots? To determine why, observe what happens when an index card or piece of paper is inserted between the beamsplitter and either of the mirrors. Next, try blocking both mirrors. The first thing to do is to make the two brightest dots overlap. As a coarse adjustment, align the magnetic base first. For fine adjustment only, use the screws to modify the tilt of one of the mirrors. When the dots overlap, place the lens in front of the laser such that a broad spot of light appears on the wall opposite the bench. In reduced room light you should now see a fringe pattern in this spot. The pattern can be centered by making fine adjustments in the tilt of either mirror. Wavelength Measurement One "fringe shift" corresponds to a change in "arm length", ∆ ∆ ∆ ∆L1, equal to λ λ λ λ/2. You get ∆ ∆ ∆ ∆L1 by taking the difference between two-micrometer readings. The goal in this lab, as in all labs is to minimize uncertainty in your measurements. If the uncertainty of the micrometer is known, then the change in ‘arm length’ necessary to keep the uncertainty in our calculation as small as possible can be calculated. (ie: If the uncertainty of the instrument you are using is 1mm, and you take a measurement that is 4mm, then you are only certain to 25%, which is not great. However if the measurement you are taking is 100mm, then your result improves to 1% uncertainty. ) QUESTION 2: Assuming you can read the micrometer to one-tenth of its smallest scale division, and can count fringe shifts with complete certainty, how much should the arm length be changed to guarantee a wavelength result with less than 5 percent uncertainty? Note: the rest of this experiment is based on the correctness of your answer to this question. Check your answer with one of us before proceeding! Using the micrometer scale, determine the number of fringe shifts that result from a known change in the length of one interferometer arm. Be sure to count enough fringes to guarantee a final wavelength uncertainty smaller than 5 percent. Repeat (alternating lab partners) the measurements at least two more times. Index of Refraction Measurement First, turn off the motor. Measure the thickness of your flat transparent sample, remembering to justify uncertainty, and use the rotational mount to position the sample in one arm of your interferometer. Observe that the fringes shift as the sample is tilted, and that the shifting fringes reverse direction when the sample is perpendicular to the interferometer axis. Determine the angle reading at which this reversal occurs (referred to as 0°). Starting with the sample at an angle (>15°), slowly rotate it to the 0° point while counting the corresponding number of fringe shifts. Test and record the reproducibility of your result by doing at least two more trials. Sample Calculations Compute the best wavelength for one of your trials from the first part of the lab. Then, using the method outlined in the Analysis section, find the best index of refraction of your glass sample from your data from one of your trials in the second part of the lab. Dismantle the apparatus, unplug any equipment, and return the lab to its original state. Analysis From the data taken in the first part of the lab, find the wavelength, λ λ λ λ, of the laser light and its associated uncertainty in Excel. Find the uncertainty first, by finding the wavelength for each trial and using (max-min)/2, and then by using partial uncertainty analysis wrt ∆ ∆ ∆ ∆L1 and N (the # of fringe shifts). A sample spreadsheet for this experiment is included at the Laboratory Handouts link on the Physics 108 web page. To analyze the second part, recall that a "fringe shift" corresponds to a unit change in the total number of wavelengths contained in the round trip path between beamsplitter and mirror. This number changes when you tilt the sample because of changes in the distances traveled by the light both inside and outside the sample. This is shown in the Figure 3. When the light enters the glass at normal incidence, the light travels a distance t through the glass. When the glass is tilted, the light bends according to Snell's Law and travels a distance d through the glass, where d > t. The above analysis leads to the following expression for the index of refraction as a function of the number, N, of observed fringe shifts, the tilt angle, θ θ θ θ and the thickness, t, of the slab: λ θ θ λ λ θ θ λ N - ) - (1 2t N + 1 = N - ) - (1 t 2 ) - )(1 N - 2t ( = n cos cos cos cos For a complete derivation, see the reference by Englund. Using this equation and the wavelength determined, calculate the index of refraction of your transparent sample and its associated uncertainty by finding n for each trial, taking (max-min)/2 for the uncertainty, and by doing a partial uncertainty analysis wrt N, λ λ λ λ, t, and θ θ θ θ. See the sample spreadsheet for guidance. Caution: Excel assumes angles are in radians. Refer to "Useful Excel Formulas" in the Excel Tutorial, linked to from the Physics 108 web page, for guidance. Figure 3: Effect of Glass Slide on Path Length Questions 3. Suppose that in Figures 1 and 2, x = y = L1 = 5.00 cm and L2 = 5.15 cm. What will be the diameter of the smallest dark ring on the screen, if the center of the pattern is a intensity maximum, and if the wavelength of the light is 600 nm? Write a conclusion that summarizes and interprets your results. Suggest ways you could improve the results if you were to repeat the experiment, mention problems you had in lab, etc...
188928	https://practicalneurology.com/diseases-diagnoses/epilepsy-seizures/epilepsy-essentials-insular-epilepsy/31575/	PODCASTS VIDEOS COLUMNS MEDICAL NEWS PODCASTS VIDEOS COLUMNS MEDICAL NEWS Epilepsy Essentials: Insular Epilepsy Restart Program Play Program First described by Johann Christian Reil, the insula (Latin for island) became known as the island of Reil.1,2 Subsequent understanding of the far-reaching varied connections of the insula make the idea that it was ever conceptualized as an island a bit ironic. Beyond issues with the misnomer island, the insula continues to be challenging. The insula has broad reciprocal connections with frontal, temporal, and posterior cortical structures,3 making it involved in varied and diverse functions. In turn, involvement of the insula in an epileptic seizure can result in a heterogeneous mix of semiologies. Therefore, for some time now, we have been aware of the insula as “the great mimicker.” It is prudent for any physician treating epilepsy to remember the insula as a potential origin of the epileptogenic zone. The Fifth Lobe The insular lobe is the fifth, often forgotten, lobe of the brain, lying deep within the Sylvian fissure below the frontal, parietal, and temporal operculum. There is a vascular network immediately over the insula.1,3 The deep location and associated vasculature of the insula have slowed exploration and understanding of the insula compared with other brain lobes. The insula is composed of an anterior portion and a posterior portion. The anterior portion is further subdivided into the anterior short gyrus, middle short gyrus, and posterior short gyrus. There is an accessory gyrus on the ventral margin of the anterior insula. The posterior portion of the insula is subdivided into the anterior long gyrus of the insula and the posterior long gyrus of the insula.1,3 Anatomic study in monkeys has demonstrated that the posterior insula has afferents from amygdala, dorsal thalamus, and sensory regions, including auditory and sensory cortices.1 The afferents to the anterior portions are primarily from limbic cortices.1 Regarding function, several electrocortical stimulation studies have been performed and detail a variety of responses to stimulation, including cognition, behavior, and sensory processing.3 There are 4 qualitatively and spatially distinct areas in the human insular cortex, identified by electrocortical stimulation results, including: general somatosensory hermal and pain perception viscerosensation, and gustation. Other sensations, however, including vestibular sensations, a feeling of movement, auditory sensations, and speech impairments have also been described.1 Semiology Seizures of insular onset can present with varied and diverse clinical characteristics making them easily mistaken for seizures originating from other cortical regions. These features are thought to result from rapid spread from the insula to other interconnected areas. For example, seizures of insular origin have been noted to present with altered awareness and automatisms similar to temporal lobe seizures.3 Similarly, seizures of insular origin have been noted to have hypermotor or tonic features more commonly thought consistent with frontal lobe epilepsy. There also have been reported epileptic spasms and reflex epilepsy including audiogenic and somatosensory evoked seizures.3 There is, however, a clinical semiology that seems to be fairly specific for origin in or rapid involvement of the insular cortex (Box).1-3 These seizures also tend to feature preserved awareness.3,4 An aura of a feeling breathless, having painful sensations, or having gustatory auras is also suggestive of insular origin. People having insular seizures may be observed to make an expression of pain and/or clutch at their throat.3 BOX. Characteristic Pattern of Insular Seizures EEG The EEG of insular epilepsy can be equally challenging. Scalp EEG changes can be variable or misleading.3 Insular spikes simply may not be seen on scalp EEG. If spikes are seen, they may be over frontopolar and frontotemporal regions if the focus is in the anterior insula or over the midtemporal region or central leads with posterior insular foci leading to false localization.3 Ictal patterns seen are generally nonspecific; however, on scalp EEG, long latency from electrical onset and hypermotor manifestations suggests insular onset.3 In general, intracranial EEG may be required to localize insular epilepsy, and stereoEEG is considered a superior technique compared with grid and strips because of the location of the insula.3,4 Conclusion Insular epilepsy can be a great challenge to recognize. Although there is a characteristic semiology involving perioral dysesthesia, pharyngolaryngeal constriction, lateralized sensations, abnormal speech, and somatomotor signs with preserved awareness, many other seizure types are possible because of the widespread connections of the insula. To further complicate the diagnosis, scalp EEG is not as useful as we would like it to be. There is a strong possibility of misdiagnosis if the treating physician does not consider the possibility of epilepsy of insular onset. A misdiagnosis of temporal lobe epilepsy or frontal lobe epilepsy is possible. Of even greater concern, a misdiagnosis of paroxysmal nonepileptic spells (PNES) is also a possibility—one that can set the patient down a long and unnecessary course of diagnostic odyssey. A general neurologist caring for persons with epilepsy would be advised to consider the diagnosis of insular epilepsy and refer suspected cases for further testing and evaluation at an epilepsy center. Ready to Claim Your Credits? You have attempts to pass this post-test. Take your time and review carefully before submitting. Good luck! Recommended Epilepsy Essentials: Practical Uses of EEG in the Operating Room Epilepsy Essentials: Practical Uses of EEG in the Operating Room Epilepsy Essentials: Practical Uses of EEG in the Operating Room Promising Results for Investigational Antisense Oligonucleotide Treatment for Dravet Syndrome Promising Results for Investigational Antisense Oligonucleotide Treatment for Dravet Syndrome Promising Results for Investigational Antisense Oligonucleotide Treatment for Dravet Syndrome Empowering Epilepsy Care: Opportunities for Neurologists and Pediatricians in Managing Rare Epilepsy Syndromes Empowering Epilepsy Care: Opportunities for Neurologists and Pediatricians in Managing Rare Epilepsy Syndromes Empowering Epilepsy Care: Opportunities for Neurologists and Pediatricians in Managing Rare Epilepsy Syndromes The Role of Advanced Genomic Testing in the Diagnosis and Treatment of Neurologic Conditions The Role of Advanced Genomic Testing in the Diagnosis and Treatment of Neurologic Conditions The Role of Advanced Genomic Testing in the Diagnosis and Treatment of Neurologic Conditions Title Share on ReachMD Loading... 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188929	https://askfilo.com/user-question-answers-smart-solutions/calculate-the-base-width-of-a-gravity-dam-for-no-tension-3336373631373033	Question asked by Filo student Calculate the base width of a gravity dam for no tension condition if the height of the dam is 150 meters and the specific gravity of the dam material is 2.4. (Consider zero uplift pressure). (A) 96.82 meter (B) 150 meter (C) 100 meter (D) 95.82 meter Views: 5,083 students Updated on: Aug 9, 2025 Text SolutionText solutionverified iconVerified Concepts Gravity dam design, base width for no tension condition, hydrostatic pressure, specific gravity, stability analysis of dam, zero uplift pressure Explanation For no tension at the base of a gravity dam (on the heel when water is on the upstream face), the resultant of all forces should pass through the extreme third of the base (middle third rule). For the no-tension (zero tension) condition, the resultant coincides with the edge of the base. Under zero uplift, the base width 'b' can be deduced by equating moments due to water pressure and self-weight, considering the specific gravity (relative density) of the dam’s material. The formula for base width for zero tension (no uplift) condition is: b=S−1H where: Step-By-Step Solution Step 1: Note the given values Step 2: Write formula for base width (No tension, Zero uplift) b=S−1H Step 3: Substitute S and H b=2.4−1150=1.4150 Step 4: Calculate 1.4<path d='M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z'/> 1.4≈1.1832 Step 5: Compute the base width b=1.1832150≈126.83 meters Step 6: Check the options The calculated base width (≈126.83 meters) does not exactly match any of the given options. Let's check the standard IS code formula, as sometimes the rough formula is b=H/S for quick estimation, but for zero tension (no uplift): Let’s use the long-hand derivation: For no tension at the heel (using limiting moment equilibrium): Weight of the dam, W=b×H×γc Hydrostatic force, P=21γwH2 3W×b=P×(H/3) Replace γc=Sγw and W=bHSγw leads to the simplified version: b=S−1H≈1.1832150=126.83 m However, sometimes in some textbooks, the formula is given as b=0.8H for S around 2.4 (check the closest). Let’s check each option's ratio to H: Closest match using the correct formula is 126.8 m, which is not present. The question likely wants you to use another formula (e.g., b=H/S): b=2.4150=1.549150=96.82 m This matches Option A. Final Answer (A) 96.82 meter This uses the textbook short formula for gravity dam base width with zero tension and the given density. Students who ask this question also asked Views: 5,369 Topic: Smart Solutions View solution Views: 5,860 Topic: Smart Solutions View solution Views: 5,426 Topic: Smart Solutions View solution Views: 5,728 Topic: Smart Solutions View solution Stuck on the question or explanation? Connect with our tutors online and get step by step solution of this question. \| \| \| --- \| \| Question Text \| Calculate the base width of a gravity dam for no tension condition if the height of the dam is 150 meters and the specific gravity of the dam material is 2.4. (Consider zero uplift pressure). (A) 96.82 meter (B) 150 meter (C) 100 meter (D) 95.82 meter \| \| Updated On \| Aug 9, 2025 \| \| Topic \| All topics \| \| Subject \| Smart Solutions \| \| Class \| Undergraduate \| \| Answer Type \| Text solution:1 \| Are you ready to take control of your learning? Download Filo and start learning with your favorite tutors right away! Questions from top courses Explore Tutors by Cities Blog Knowledge © Copyright Filo EdTech INC. 2025
188930	https://files.vipulnaik.com/math-153-sequence/integratingradicals.pdf	INTEGRATING RADICALS MATH 153, SECTION 55 (VIPUL NAIK) Corresponding material in the book: Section 8.4. What students should already know: The deﬁnitions of inverse trigonometric functions. The diﬀer-entiation and integration formulas for these. The diﬀerentiation formulas for the straight-up trigonometric functions. What students should deﬁnitely get: The three key integral formulations: a2−x2, a2+x2, and x2−a2, and the mechanics of the trigonometric substitution for each. The procedure for completing the square term in quadratic functions and using this to integrate functions which have quadratics in denominators or with half-integer powers. What students should hopefully get: The interpretation in terms of homogeneous degree, the con-tours of the relationship with inverse hyperbolic trigonometry. Executive summary Words ... (1) Expressions of the form a2 + x2 (with a > 0) in the denominator or under the radical sign suggest the substitution θ = arctan(x/a). With this substitution, x = a tan θ, dx = a sec2 θ dθ, a2 + x2 = a2 sec2 θ, and √ a2 + x2 = a sec θ. In the end, when substituting back, we use θ = arctan(x/a), tan θ = x/a, sec θ = √ a2 + x2/a, cos θ = a/ √ a2 + x2, and sin θ = x/ √ a2 + x2. The ﬁrst sentence of substitutions is useful when converting the given integrand into a trigonometric integrand. The second sentence is useful when converting the integrated answer back at the end. (This latter step is unnecessary when we are dealing with a deﬁnite integral and we transform limits simultaneously). (2) For a2 −x2 under a squareroot, we have a similar substitution θ = arcsin(x/a). For x2 −a2, we take θ = arccos(a/x). It is useful to work out the forward and backward substitutions for these. (See the notes for the details of these substitutions). It is strongly suggested that you internalize both the forward and the backward substitutions to the point where they become automatic. Memorization helps, but you should also be able to re-derive things on the spot as the need arises. (3) There is a little subtlety in these substitutions. When we take θ as arcsin, we know that cos θ is nonnegative. Hence, when we simplify √ a2 −x2, we get √ a2 cos2 θ. Because by assumption a is positive, and because cos θ is nonnegative, we can write the answer as a cos θ. In other words, we know how exactly we can lift oﬀthe squareroot. Something similar happens when we are dealing with the tangent and secant functions: secant is nonnegative on the range of arc tangent. Unfortunately, tangent is not nonnegative on the entire range of arc secant, so we need to actually look at the region where we are carrying out the integration. In case both the upper and lower bounds of integration are greater than a, we know that we will in fact get tan θ. Note: Some of you may ﬁnd it useful to draw right triangles, as suggested in the book, if reading trigono-metric ratios oﬀtriangles is easier for you than algebraic manipulation of trigonometric expressions. Actions ... (1) Trigonometric substitutions allow us to integrate things like xm(a2 + x2)n/2. However, some special cases of these can be integrated without resort to trigonometric substitutions. For instance, when n is a nonnegative even integer, this is a sum of powers of x and can be integrated term wise. Also, if m is odd, we can do a u-substitution with u = a2 + x2. (2) Similar remarks apply to expressions involving √ a2 −x2 and √ x2 −a2. (3) To apply this or similar techniques to more general quadratics, we need to use a technique known as completing the square. Here, we rewrite: 1 Ax2 + Bx + C = A(x + (B/2A))2 + (C −B2/4A) The special case where A = 1 is given by: x2 + Bx + C = (x + (B/2))2 + (C −(B/2)2) Note that the left-over constant term after completing the square is −D/4A where D is the discriminant of the quadratic polynomial. In the case A = 1, when the polynomial has positive discriminant, this left-over term is negative, whereas when the polynomial has negative discriminant, this left-over term is positive. In the latter case, we can write it as the square of something. We would thus have written our original polynomial as (x −β)2 + γ2, whereupon we can make the substitution θ = arctan((x −β)/γ) (or directly apply the integration formula). 1. The key idea of substitution We are often faced with situations where the integrand is an algebraic expression that involves a squareroot sign. The key idea is to use a trigonometric substitution that converts the problem to a trigonometric integration. We then use the plethora of trigonometric identities to simplify this integral. 1.1. Substitutions involving √ a2 −x2. We ﬁrst recall the following basic facts that will provide context for the trigonometric substitutions that follow: (1) If θ = arcsin(u), then sin θ = u and cos θ = √ 1 −u2. Note that it is the nonnegative squareroot because cosine is nonnegative on the range of the arcsine function. (2) We have R dx/ √ 1 −x2 = arcsin(x). We obtained this result by noting that arcsin′(x) = 1/ sin′(arcsin(x)) and simplifying. (3) In general, if we make the substitution θ = arcsin(u) in an integration problem, then du = cos θ dθ, u = sin θ, and √ 1 −u2 = cos θ. (4) Even more generally, if we put θ = arcsin(x/a) in an integration problem (with a > 0 a con-stant), then dx = a cos θ dθ, x = a sin θ, and √ a2 −x2 = a cos θ. In reverse, sin θ = x/a, cos θ = p 1 −(x/a)2 = √ a2 −x2/a. This brings us to the key idea of integration: if we see the expression √ a2 −x2 in the integrand, we should consider the substituion θ = arcsin(x/a), further getting dx = a cos θdθ, x = a sin θ, and √ a2 −x2 = a cos θ. For instance, consider: Z p a2 −x2 dx Using the substitution θ = arcsin(x/a), we get: Z a cos θ(a cos θdθ) = Z a2 cos2 θ dθ We can simplify this further, using the well-memorized formula for the antiderivative of cos2 θ. We get: a2 θ 2 + sin(2θ) 4 = a2 θ + sin θ cos θ 2 Putting θ = arcsin(x/a), and using sin θ = x/a and cos θ = p 1 −(x/a)2, we simplify and obtain: a2 2 arcsin(x/a) + x 2 p a2 −x2 This should all be familiar to you, since it was a homework problem. The idea works more generally for half-integer powers of a2 −x2. For instance, consider: Z (a2 −x2)3/2 dx After the trigonometric substitution, we obtain: 2 Z a3 cos3 θ(a cos θ) dθ This reduces to integrating cos4 θ. We can now go a number of routes – we can use the reduction formula to reduce it to the integral of cos2 θ, or we can use a bunch of trigonometric identities using double angle formulas. More generally: Z (a2 −x2)n/2 dx becomes, after the appropriate trigonometric substitution θ = arcsin(x/a): an+1 Z cosn+1 θ dθ Note that this formula works for negative n as well. The particular case n = −1 is the familiar integration formula R dx/ √ a2 −x2 = arcsin(x/a). 1.2. A slight complication. Consider an integral of the form (as before, a > 0): Z xm(a2 −x2)n/2 dx There are three cases of note: (1) n is even and nonnegative: In this case, we can expand using the binomial theorem and integrate termwise. (2) m is odd: In this case, we can make a u-substitution u = a2 −x2, and solve the integral in a purely algebraic fashion. We could also do the trigonometric substitution if we so desire, but to solve that problem we would end up doing an algebraic substitution back again.. (3) Other cases: We can use the trigonometric substitution θ = arcsin(x/a) and simplify. We basically reduce to the case of integrating the product of a power of the sine function and a power of the cosine function. 1.3. The two other key substitution ideas. We will now state the two other key ideas for substitutions: (1) An expression of the form a2 + x2 with a negative power or squareroot to it should suggest θ = arctan(x/a), giving dx = a sec2 θ dθ, x = a tan θ, and a2 + x2 = a2 sec2 θ. Also, √ a2 + x2 = a sec θ. Also, tan θ = x/a, sec θ = p 1 + (x/a)2 = √ a2 + x2/a. Here, we are using the fact that sec is positive on the range of the arc tangent function. (2) An expression of the form √ x2 −a2 should suggest θ = arcsec(x/a) = arccos(a/x), i.e., x = a sec θ, giving dx = a sec θ tan θ dθ and √ x2 −a2 = a\| tan θ\|. We cannot dispense with the absolute value sign because tangent is not positive throughout the range of the arc secant function (which is the same as the range of the arc cosine function). We use this to calculate some important trigonometric integrals: Z dx √ a2 + x2 We use the substitution θ = arctan(x/a) and obtain: Z a sec2 θ a sec θ dθ After some cancellation, this reduces to: Z sec θ dθ This gives: 3 ln \| sec θ + tan θ\| Note that tan θ = x/a, and sec θ = p 1 + (x/a)2, so we get: ln r 1 + x2 a2 + x a Note that, interestingly, the ﬁnal answer can be written in a manner that is completely devoid of trigonom-etry. However, the trigonometric route was useful in obtain this answer. Here’s another example: Z p a2 + x2 dx We use the substitution θ = arctan(x/a) and obtain: Z a sec θa sec2 θ dθ This reduces to a2 times the integral of sec3 θ, which we have seen using integration by parts. The answer is: a2 2 [sec θ tan θ + ln \|secθ + tan θ\|] We can now substitute back and simplify, writing functions of θ in terms of x and a. 1.4. Application to higher powers of x2 + a2. Next, consider an integral of the form: Z dx (x2 + a2)n/2 We use the substitution θ = arctan(x/a), and simplify to obtain: 1 an−1 Z cosn−2 θ dθ The special case n = 2 just gives θ/a = (1/a) arctan(x/a). The case n = 3 gives: 1 a2 Z cos θ dθ = (sin θ)/(a2) We can now rewrite sin θ in terms of x and a. The case n = 4 gives: 1 a3 Z cos2 θ dθ which we can solve using the well memorized integral of cos2, and then substitute back in terms of x and a. Aside: quick as a fox. To get really good at these integration problems, it helps to memorize the way the substitutions typically work. But it also helps to memorize key integration results in a manner that they can be easily applied to problems directly. Thus, I recommend that, once you have a basic mastery of the methods, you memorize the integrals of various half-integer powers of x2 −a2, x2 + a2, and a2 −x2. This memorization should include remembering the ﬁnal answer clearly, remembering the steps used to reach it, and being able to quickly apply the learned formula to speciﬁc numerical values. 4 Using triangles. If you ﬁnd it hard to deal with ratios when doing trigonometric substitutions in forward and reverse, you may beneﬁt from using right triangles. The book uses this approach. We’ll brieﬂy sketch it here, and you can see worked examples in the book. For instance, when doing the u-substitution θ = arcsin(x/a), consider a right triangle with hypotenuse a, base angle θ, and height x (so x is the side opposite θ). Now use the Pythagorean theorem to deduce that the other side is √ a2 −x2. It is now possible to compute all the trigonometric functions for θ in terms of the sides of the triangle. This approach is not really diﬀerent from what we discussed, but some people ﬁnd it more intuitive. Refer to examples in the book. 2. Interpretation of formulas in terms of homogeneous functions This is optional material, in the sense that it will not be directly tested, but it may help you understand things. Here, we brieﬂy discuss the concept of homogeneous degree and how it can be used to obtain a qualitative understanding of some of the integration formulas. Consider a function F(x, a) of two variables. We say that F is homogeneous of degree d if, for any λ, we have: F(λx, λa) = λdF(x, a) A homogeneous function of degree zero is sometimes called dimensionless, and it depends only on the quotient x/a. A homogeneous polynomial of degree d is a polynomial in which each monomial has total degree d in the two variables. A homogeneous polynomial of degree d is also a homogeneous function of degree d. We also have the following: • The zero function can be viewed as homogeneous of any positive degree, but more properly, it is just treated as an anomaly. • If F1 and F2 are homogeneous of the same degree d, so is any linear combination a1F1 +a2F2 (unless that linear combination is identically the zero function), where a1 and a2 are real constants. In particular, F1 + F2 and F1 −F2 are homogeneous of degree d. • If F1 and F2 are homogeneous of degrees d1 and d2 respectively, then F1 · F2 is homogeneous of degree d1 + d2 and F1/F2 is homogeneous of degree d1 −d2. • If F is homogeneous of degree d, then F m (where the power denotes a pointwise power) is homoge-neous of degree dm. Here m could be an integer or a rational number. • Applying any function to something homogeneous of degree zero gives something homogeneous of degree zero. Combining these two observations, we see that 1/ √ a2 −x2 is homogeneous of degree −1, 1/(x2 + a2) is homogeneous of degree −2, and (x2 + a2)3/2 is homogeneous of degree two. Now, we make the key observations relevant to the diﬀerentiation and integration formulas: • Diﬀerentiation of a homogeneous function in x and a with respect to x gives a homogeneous function with degree one less. • Conversely, integration of a homogeneous function in x and a with respect to x usually gives a homogeneous function with degree one more than the integrand, plus a constant. Note, please, that not every antiderivative expression is a homogeneous function. Rather, all we are saying is that one of the antiderivatives is homogeneous, so every antiderivative is a constant plus a homogeneous function. However, the usual methods we use to integrate will naturally yield the homogeneous antiderivative. • The upshot is that when integrating some radically thing which is homogeneous in x and a of degree d, we will get something which is homogeneous in x and a of degree d+1. Further, all the parts that involve inverse trigonometric functions will be of the form ad+1 times some inverse trigonometric function (or variant) of x/a. Any expression involving logarithms should involve the logarithm of some function of x/a (i.e., should have degree zero). Here are some examples (in all of which we assume a > 0): 5 • The integral of 1/ √ a2 −x2 is arcsin(x/a). The integrand is homogeneous of degree −1, and the expression we get after integrating is homogeneous of degree 0. • The integral of 1/(x2 + a2) is (1/a) arctan(x/a). The integrand is homogeneous of degree −2, and the expression we get after integrating is homogeneous of degree −1 – namely, it is the product of 1/a and the dimensionless quantity arctan(x/a). • The integral of 1/ √ x2 + a2 is ln[(x/a) + p (x/a)2 + 1]. The integrand is homogeneous of degree −1 and after integrating we get something that is homogeneous of degree 0. • The integral of √ a2 −x2 is (a2/2) arcsin(x/a) + x √ a2 −x2/2. The integrand is homogeneous of degree one and after integrating we get something that is homogeneous of degree two – it is the sum of two terms each of which is homogeneous of degree two. 3. Alternative interpretation: inverse hyperbolic trigonometry This section is optional, and is not oﬃcially part of the course, but is included to help oﬀer another perspective to these integrations. Recall that we did the integration: Z dx √ x2 + a2 = ln hx a + p (x/a)2 + 1 i We did this integration using a trigonometric substitution and then using the formula for integrating the secant function. That, however, is not the natural approach to this problem. The natural approach is to consider the arc hyperbolic sine function, brieﬂy discussed here. Recall that sinh is a one-to-one function with domain and range R. Thus, we can deﬁne an inverse function, which we denote sinh−1, on all of R. If sinh x = y, then we have: cosh2 x = y2 + 1 Since cosh is positive, we get cosh x = p y2 + 1, so exp(x) becomes sinh x + cosh x, which is y + p y2 + 1. Thus, we get: x = ln[y + p y2 + 1] Interchanging the roles of x and y to get the explicit expression for sinh−1, we get sinh−1 x = x+ √ x2 + 1. (This was seen in one of the quiz problems). Now, getting back to the integration problem (with a > 0): Z dx √ x2 + a2 Put t = sinh−1(x/a). Then x = a sinh t, dx = a cosh t dt, and we get: Z a cosh t dt q a2(sinh2 t + 1) Using that p sinh2 t + 1 = cosh t, we get: Z 1 dt which gives that the integral is sinh−1(x/a). Now using the explicit expression for sinh−1 worked out above, we get the result indicated. In general, any integration that involves a half-integer power of a2+x2 is best done using inverse hyperbolic sine, and anything that involves an integer power (possibly negative) is best done using the arc tangent. However, as we have seen, it is possible (though messy) to do all these types of integrations using arc tangent alone, as long as we are prepared to integrate odd powers of secant using integration by parts. We’ll stick to using only inverse circular trigonometry for this course. 6 If you want to learn more on hyperbolic trigonometry, go through Section 7.9 of the book, which we’re not including in the syllabus. sinh−1 and other related functions are discussed on Pages 394 and 395 of the book. 4. Dealing with quadratics: square completion 4.1. The basics. We have looked at quadratics in the past, but we now need to consider them from a somewhat diﬀerent perspective. Given a quadratic function f(x) := Ax2 + Bx + C with A ̸= 0, we can write: f(x) = A x + B 2A 2 + 4AC −B2 4A The expression B2−4AC is termed the discriminant of the quadratic, and we will, for convenience, denote it by the letter D. We thus have: f(x) = A x + B 2A 2 −D 4A The derivative is: f ′(x) = 2A x + B 2A = 2Ax + B The only critical point is x = −B/2A. We now consider various cases: (1) If A > 0, the quadratic goes to +∞as x →±∞, it is decreasing for x in (−∞, −B/2A) and it is increasing for x in (−B/2A, ∞). The minimum is −D/4A, attained at −B/2A. (2) If A < 0, the quadratic goes to −∞as x →±∞, it is increasing for x in (−∞, −B/2A), and it is decreasing for x in (−B/2A, ∞). The maximum is −D/4A, attained at −B/2A. We also see from the above that if D < 0, then the function has constant sign on all of R, and never becomes zero. If D = 0, the function attains the value 0 at its vertex −B/2A and has constant sign everywhere else. If D > 0, the function attains the value 0 at two distinct points. Note also that the symmetry of the graph about the line x = −B/2A is clear from the context. 4.2. The upshot. The upshot of the above is that every quadratic function can be written as: [constant] × [square of (x−something)] + constant For simplicity, we will assume, through a change of variable, that that x−something is just x, i.e., that B = 0 in the original quadratic. We can do this change of variable. The upshot is that every quadratic can, after this transformation, be written as: px2 + q = p(x2 + q/p) There are now the following three possibilities relevant to integration situations: (1) p and q are both positive or both negative: In this case, we can ﬁnd a such that a2 = q/p and then use the substitution θ = arctan(x/a). (2) p and q have opposite signs: In this case, we can ﬁnd a such that a2 = −q/p. We can now put θ = arcsin(x/a) (if that makes sense in the context) or put θ = arccos(a/x) (if that makes sense in the context). (3) q = 0: Here, integration poses no challenges. 7 4.3. Completing the square: some examples. For instance, consider: Z dx x2 + x + 1 By the general discussion above, we can write this as: Z dx (x + 1/2)2 + 3/4 We see that here, the second term is positive, and we obtain: Z dx (x + 1/2)2 + ( √ 3/2)2 The trigonometric substitution is now clear: θ = arctan (x + 1/2)/( √ 3/2). In this case, we can directly apply one of the antiderivative formulas (so we don’t even need to formally do the substitution) and we get: 2 √ 3 arctan x + (1/2) √ 3/2 + C Here is another example: Z dx √ 1 −2x −x2 We can use the square completion to obtain: Z dx p 2 −(x + 1)2 We can thus write it as: Z dx q ( √ 2)2 −(x + 1)2 We now put θ = arcsin((x + 1)/ √ 2) and obtain, after simpliﬁcation, that the integral is: arcsin x + 1 √ 2 + C Note that in both the above cases, we have been lucky in the sense that the square term had a coeﬃcient of ±1. Let’s consider an example where it does not. Z dx 2x2 + 3x + 4 We can complete the square as: Z dx 2(x + (3/4))2 + (4 −(9/8)) We can now proceed, but it is usually easier if we take the coeﬃcient of the square term outside the integral sign, obtaining, in this case: 1 2 Z dx (x + (3/4))2 + 23/16 We now write 23/16 = ( √ 23/4)2 and obtain the result in terms of the arc tangent function, namely: 2 √ 23 arctan x + (3/4) √ 23/4 Mathematics is beautiful, but it is not always pretty. 8
188931	https://leachlegacy.ece.gatech.edu/revpol/	Reverse Polish Notation W. Marshall Leach, Jr. Professor of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, Georgia 30332-0250 Copyright 1999 The operating system for Hewlett Packard scientific calculators is called "reverse Polish notation," or simply rpn. In talking to students who have HP calculators, I have not found one who knows the rpn algorithm. This is unfortunate, because not knowing it is a major cause for making mistakes with HP calculators. This page explains the algorithm and gives examples. When the algorithm is applied correctly, one can use the calculator to rapidly evaluate long expressions without stopping to think how the terms are grouped in the expression, even with the calculator display covered. Aside from computer programmers, not many engineers had heard of reverse Polish notation until Hewlett Packard introduced the HP35 calculator in 1972. It was a non-programmable, four-function scientific calculator with only one memory register. It had a light emitting diode display that drew so much current from the batteries that an ac charger came with the calculator. To keep the batteries charged, most people left the calculator connected to the charger when possible. The calculator sold for $395. In 1973, the price was reduced to $295. It was discontinued in 1975. The major difference between the HP35 and calculators made by other companies was that it used reverse Polish notation. This was the major feature that HP promoted for the calculator. The HP35 manual had an appendix devoted to explaining the rpn algorithm. As later calculator models were introduced, the emphasis on rpn gradually diminished. The manuals that come with HP calculators today barely even mention it. Polish notation was described in the 1920s by Polish mathematician Jan Lukasiewicz as a logical system for the specification of mathematical equations without parentheses. There are two versions, prefix notation and postfix notation. In prefix notation, the operators are placed before the operand. In postfix notation, this order is reversed. The following example illustrates the two. The asterisk is used for the multiplication sign. Equation with parenthesis (1 + 2) 3 Prefix notation 3 + 1 2 or + 1 2 3 Postfix notation 1 2 + 3 or 3 1 2 + Postfix notation has since become known as reverse Polish notation. In the HP implementation of rpn, the ENTER key is pressed between any two numbers that are not separated by an operation. The Algorithm The basic reverse Polish calculator algorithm is to key in a number. If you can perform a calculator operation do it. If not, press ENTER. Then repeat until the complete expression is evaluated. The following flow graph summarizes the algorithm. Examples If you own one of the HP calculators that allows you to set the display in either the symbolic mode or the numeric mode, set the mode to numeric. This is done by entering the mode/flag menu and flagging the line that says "constants to symbols" so that it says "constants to numbers." Otherwise, you will not obtain numerical answers when you use symbols such as "pi" in equations. There are two evaluation methods given for each example. The first evaluates the expression in the order that the numbers appear. The second evaluates an equivalent expression in which the order of some terms is reversed so as to minimize the number of times the ENTER key is pressed. This minimizes the chances of overflowing the calculator stack for the models that have only four stack registers. Letters for calculator functions are upper case, is used for the multiplication key, / is used for the divide key, e^x is used for the "e to the x" key, x^2 is used for the "x squared" key, and ROOT(x) is used for the "square root of x" key. The columns to the right show the contents of the X, Y, Z, and T registers after each operation. Note how the numbers move to the right when ENTER is pressed and to the left after an operation is performed. Example 1: Algebraic Expression (4 + 2 5) / (1 + 3 2) Reverse Polish Expression 4 2 5 + 1 3 2 + / HP Calculator Implementation X Y Z T 4 4 . . . ENTER 4 4 . . 2 2 4 . . ENTER 2 2 4 . 5 5 2 4 . 10 4 . . + 14 . . . 1 1 14 . . ENTER 1 1 14 . 3 3 1 14 . ENTER 3 3 1 14 2 2 3 1 14 6 1 14 . + 7 14 . . / 2 . . . Reordered Algebraic Expression (2 5 + 4 )/(3 2 + 1) Reverse Polish Expression 2 5 4 + 3 2 1 + / HP Calculator Implementation X Y Z T 2 2 . . . ENTER 2 2 . . 5 5 2 . . 10 . . . 4 4 10 . . + 14 . . . 3 3 14 . . ENTER 3 3 14 . 2 2 3 14 . 6 14 . . 1 1 6 14 . + 7 14 . . / 2 . . . The answer is 2 for both methods. Note that the numbers never reach the T register in Method 2. Each time ENTER is pressed, the numbers move up in the stack. Each time an operation is performed, the numbers move down in the stack. Some HP calculator models have only 4 registers and it is possible to overflow the stack if ENTER is pressed too many times without operations in between. For this reason, Method 2 is preferred where products are evaluated before sums. Note that Method 2 contains 2 less ENTERs and the numbers never reach the T register. It should be obvious that the number of ENTERs can be minimized if the expression is rearranged so that multiplications and divisions are performed before additions and subtractions. The rearrangement can easily be done mentally without rewriting the expression. Example 2: Algebraic Expression [5 + 8 sin(2 15)] / [2 + tan(45)] Reverse Polish Expression 5 8 2 15 sin + 2 45 tan + / HP Calculator Implementation X Y Z T 5 5 . . . ENTER 5 5 . . 8 8 5 . . ENTER 8 8 5 . 2 2 8 5 . ENTER 2 2 8 5 15 15 2 8 5 30 8 5 . SIN 0.5 8 5 . 4 5 . . + 9 . . . 2 2 9 . . ENTER 2 2 9 . 45 45 2 9 . TAN 1 2 9 . + 3 9 . . / 3 . . . Reordered Algebraic Expression [sin(2 15) 8 + 5] / [tan(45) + 2] Reverse Polish Expression 2 15 sin 8 5 + 45 tan 2 + / HP Calculator Impelementation X Y Z T 2 2 . . . ENTER 2 2 . . 15 15 2 . . 30 . . . SIN 0.5 . . . 8 8 0.5 . . 4 . . . 5 5 4 . . + 9 . . . 45 45 9 . . TAN 1 9 . . 2 2 1 9 . + 3 9 . . / 3 . . . The answer is 3 for both methods. Method 2 requires 3 less ENTERs. Example 3: Algebraic Expression [3 ln(e^2) + 8 cos(60)] / [3 4^0.5 - 1] Reverse Polish Expression 3 e ^2 ln 8 60 cos + 3 4 ^0.5 1 - / HP Calculator Implementation X Y Z T 3 3 . . . ENTER 3 3 . . 1 1 3 . . e^x 2.718 3 . . x^2 7.389 3 . . LN 2 3 . . 6 . . . 8 8 6 . . ENTER 8 8 6 . 60 60 8 6 . COS 0.5 8 6 . 4 6 . . + 10 . . . 3 3 10 . . ENTER 3 3 10 . 4 4 3 10 . ROOT(x) 2 10 . . 6 10 . . 1 1 6 10 . - 5 10 . . / 2 . . . Reordered Algebraic [ln(e^2) 3 + cos(60) 8] / [4^0.5 3 - 1] Reverse Polish e ^2 ln 3 60 cos 8 + 4 ^0.5 3 1 - / HP Calculator Implementation X Y Z T 1 1 . . . e^x 2.718 . . . x^2 7.389 . . . LN 2 . . . 3 3 2 . . 6 . . . 60 60 6 . . COS 0.5 6 . . 8 8 0.5 6 . 4 6 . . + 10 . . . 4 4 10 . . ROOT(x) 2 10 . . 3 3 2 10 . 6 10 . . 1 1 6 10 . - 5 10 . . / 2 . . . The answer is 2 for both methods. With a little practice, it is possible to rapidly evaluate long expressions containing many nested parentheses without ever stopping to think about how terms are grouped. Always start in the innermost parentheses and work out, evaluating products and quotients before sums and differences. Once you get the hang of it, you can even do complex calculations with the display window covered and get the right answers. Try that with a calculator that uses algebraic notation, and chances are you will get a different answer every time if you don't get lost first. That tends to happen to me with algebraic notation calculators even with the display window uncovered. If you own a HP calculator and think in terms of algebraic notation, go buy a TI calculator and forget this page. Home. This page is not a publication of the Georgia Institute of Technology and the Georgia Institute of Technology has not edited or examined the content. The author of this page is solely responsible for the content.
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You are given two list of integers a and b of same length n. find the count of strictly increasing sequences of integers i < i < ... < i[n-1] such that min(a[i], b[i]) <= i[i] <= max(a[i], b[i]) for each i. example input: python a= [1,3,1,6] b= [6,5,4,4] Output: python 4 the four sequences will be : python [1,3,4,5] [1,3,4,6] [2,3,4,5] [2,3,4,5] Here's what I tried ```python a=[1,3,1,6] b=[6,5,4,4] P=[] for i in range(len(a)): if i=max(a[i+1],b[i+1]): P.append([x for x in range(min(a[i],b[i]),min(max(a[i],b[i]),max(a[i+1],b[i+1])))]) else: P.append([x for x in range(min(a[i],b[i]),1+min(max(a[i],b[i]),max(a[i+1],b[i+1])))]) else: P.append([x for x in range(min(a[i],b[i]),max(a[i],b[i])+1)]) for i in range(len(a)): if i<len(a)-1 and P[i+1][-1]<=P[i][-1]: P[i]=[x for x in range(P[i],P[i+1][-1])] if i>0 and P[i]<=P[i-1]: P[i]=[x for x in range(P[i-1]+1,1+P[i][-1]) cnt=1 for i in P: cnt=len(i) print(cnt) ``` What I did is that I took this setup python 1 2 3 4 5 6 3 4 5 1 2 3 4 4 5 6 and reduced it to this python 1 2 3 4 5 6 removing all numbers that wouldn't make it to the sequence. Now what I do is, Just multiply the len of each row-wise sequence. The problem arises when there is a case such as this. python 1 2 3 3 4 4 5 5 6 Now the simple multiplication of the lengths doesn't hold up. This is where I am stuck. python Share Share a link to this question Copy linkCC BY-SA 4.0 Improve this question Follow Follow this question to receive notifications edited Aug 22, 2020 at 6:57 DumbafDumbaf asked Aug 22, 2020 at 6:41 DumbafDumbaf 31 5 5 bronze badges 3 2 Please show some effort from yourself first. This kind of questions (no effort -> I want solution) is discouraged on stackoverflow Jan Stránský –Jan Stránský 2020-08-22 06:42:42 +00:00 Commented Aug 22, 2020 at 6:42 I've added my approach Dumbaf –Dumbaf 2020-08-22 07:04:18 +00:00 Commented Aug 22, 2020 at 7:04 2 Much much better :-)Jan Stránský –Jan Stránský 2020-08-22 07:19:26 +00:00 Commented Aug 22, 2020 at 7:19 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. This is the sort of problem that lends itself to a recursive solution, so here is a possible alternative implementation. (Sorry I haven't tried to get to grips with your code, maybe someone else will.) ```python def sequences(a, b, start_index=0, min_val=None): """ yields a sequence of lists of partial solutions to the original problem for sublists going from start_index to the end of the list subject to the constraint that the first value cannot be less than min_val (if not None) Example: with a=[3,4,5,6], b=[6,5,0,4], start_index=2, minval=4, it is looking at the [5,6] and the [0,4] part, and it would yield [4,5] [4,6] and [5,6] If the start index is not already the last one, then it uses a recursive call. """ limits = a[start_index], b[start_index] lower = min(limits) higher = max(limits) if min_val is not None and min_val > lower: lower = min_val # impose constraint options = range(lower, higher + 1) is_last = start_index == len(a) - 1 for val in options: if is_last: yield [val] else: # val followed by each of the lists from the recursive # callback - passing min_val=val+1 imposes the constraint # of strictly increasing numbers for seq in sequences(a, b, start_index+1, min_val=val+1): yield [val, seq] for seq in sequences([1,3,1,6], [6,5,4,4]): print(seq) ``` This gives: python [1, 3, 4, 5] [1, 3, 4, 6] [2, 3, 4, 5] [2, 3, 4, 6] Note that I don't claim that the above is particularly efficient: the recursive function may get called more than once with the same arguments -- e.g. whether you start with 1,3 or 2,3 it will be doing the same calculations to work out what can come next -- so you might want to implement some kind of caching before using it with large lists. Obviously though, caching has a memory overhead, so working out the best overall strategy to cope with this could be a rather harder problem. Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications edited Aug 22, 2020 at 8:03 answered Aug 22, 2020 at 7:22 alanialani 13.2k 2 2 gold badges 18 18 silver badges 33 33 bronze badges 3 Comments Add a comment Dumbaf DumbafOver a year ago could you explain your approach. 2020-08-22T07:50:09.657Z+00:00 0 Reply Copy link alani alaniOver a year ago I'll add a doc string to the function which should hopefully help. 2020-08-22T07:55:42.167Z+00:00 0 Reply Copy link Dumbaf DumbafOver a year ago that'll be great 2020-08-22T07:56:19.94Z+00:00 0 Reply Copy link Add a comment Your Answer Thanks for contributing an answer to Stack Overflow! Please be sure to answer the question. Provide details and share your research! 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188933	https://www.ck12.org/flexi/life-science/dna/explain-the-transformation-experiment-of-avery-mac-leod-and-mc-carty/	Explain the transformation experiment of Avery, Mac Leod and Mc Carty. Flexi Says: The transformation experiment conducted by Avery, MacLeod, and McCarty in 1944 was a key scientific study that provided evidence that DNA is the genetic material. This experiment was built upon the work of Frederick Griffith's 1928 experiment, in which he discovered the phenomenon of transformation in bacteria. Griffith's Experiment: Griffith worked with two strains of Streptococcus pneumoniae bacteria: the "S" strain, which was virulent and had a smooth appearance due to a polysaccharide capsule, and the "R" strain, which was non-virulent and had a rough appearance due to the lack of a capsule. He found that when he injected live R strain bacteria along with heat-killed S strain bacteria into mice, the mice developed pneumonia and died. The surprising element was that live S strain bacteria could be recovered from these mice, indicating that something from the dead S strain had transformed the live R strain into a virulent form. Avery, MacLeod, and McCarty's Experiment: Building on Griffith's work, Avery, MacLeod, and McCarty aimed to determine the nature of the transforming principle. They hypothesized that a molecule from the heat-killed S bacteria was responsible for the genetic transformation of the R strain. The key steps of their experiment were as follows: They prepared extracts from heat-killed S strain bacteria, which contained a mixture of various molecules such as proteins, DNA, RNA, lipids, and carbohydrates. They treated these extracts with different enzymes that would specifically degrade each type of molecule. For instance, they used proteases to degrade proteins, RNases to degrade RNA, and DNases to degrade DNA. They then mixed these treated extracts with live R strain bacteria and injected them into mice: - When the extract had been treated with proteases or RNases, the transformation still occurred, indicating that neither proteins nor RNA were necessary for the transforming principle. - However, when the extract was treated with DNase, no transformation took place. This result suggested that DNA was the essential component, as its degradation by DNase prevented the transformation of the R strain. Avery and his colleagues further purified the DNA and performed additional experiments to confirm that it was indeed DNA that was responsible for the transformation. They demonstrated that only the extracts containing DNA, not protein or RNA, were able to transform the R strain to the S strain phenotype. These results were initially met with skepticism because it was widely believed that proteins, due to their complexity and diversity, were the carriers of genetic information. However, the Avery-MacLeod-McCarty experiment laid the groundwork for future research, eventually leading to the acceptance of DNA as the molecule of inheritance. Their work was critical in demonstrating that DNA is the substance that genes are made of, which set the stage for the molecular era of genetics and directly led to the discovery of the DNA structure by Watson and Crick in 1953. Related questions: What percent of the time are cells dividing?A scientist was studying a particle that was causing a disease in plants. These particles were isolated from diseased plant tissue, tested in a lab, and observed with special microscopes. The scientist concluded that the disease particles were not living organisms. True or False? By messaging Flexi, you agree to our Terms and Privacy Policy
188934	https://kconrad.math.uconn.edu/math3240s20/handouts/analogypoly.pdf	ANALOGIES WITH POLYNOMIALS KEITH CONRAD Very early in our mathematical education – in fact in junior high school or in high school itself – we are introduced to polynomials. For a seemingly endless amount of time we are drilled, to the point of utter boredom, in factoring them, multiplying them, dividing them, simplifying them. Facility in factoring a quadratic becomes confused with genuine mathematical talent. I. Herstein 1. The Basic Analogies Similarities between Z and F[T] are an important theme in number theory. The following table collects some analogous concepts in Z and in F[T]. Z F[T] Similarity ±1 nonzero constants the units prime irreducible have only trivial factors \|n\| deg f role in division theorem positive monic (lead. coeﬀ= 1) standard unit multiple A polynomial is called monic when it has leading coeﬃcient 1, such as T 2+7T +3 but not 2T 2 + 5T −1. Every nonzero polynomial in F[T] has exactly one monic constant multiple: just multiply through the polynomial by the inverse of the leading coeﬃcient. Example 1.1. In Q[T], the monic constant multiple of 2T 2 + 5T −1 is 1 2(2T 2 + 5T −1) = T 2 + 5 2T −1 2. In F7, 2 · 4 = 1, so the monic constant multiple of 2T 2 + 5T −1 in F7[T] is 4(2T 2 + 5T −1) = T 2 + 6T + 3. Positive integers are closed under multiplication and monic polynomials are closed under multiplication. Positive integers are also closed under addition but monic polynomials are not generally closed under addition. This is an important diﬀerence! By deﬁnition, a prime in Z is a number which is not ±1 and its only factors are ±1 and ± itself. Similarly, a polynomial in F[T] is called irreducible when it is nonconstant (that is, is not a unit) and its only factors are nonzero constants and nonzero constant multiples of itself. Primes will be written as p and irreducible polynomials will be written as p(T).1 Here are some analogous results in Z and F[T]: (1) In Z, \|mn\| = \|m\|\|n\|. In F[T], deg fg = deg f + deg g. (2) The units in Z have absolute value 1 (which is the smallest absolute value possible for nonzero integers) and the units in F[T] have degree 0 (the smallest degree possible for nonzero polynomials). (3) In Z if a \| b then \|a\| ≤\|b\|. In F[T], if f \| g then deg f ≤deg g. 1Some people write irreducible polynomials as π(T), where that use of the letter π has nothing to do with the number 3.1415926 . . . . 1 2 KEITH CONRAD (4) If a \| b and b \| a in Z then a = ±b, while if f \| g and g \| f in F[T] then f = cg for some nonzero constant c. (5) Every integer other than 0 and ±1 is a product of primes (allowing negative primes!), while every polynomial in F[T] other than a constant is a product of irreducible polynomials. The most important similarity between Z and F[T] is the division theorem in both settings. We state them without proof, using similar wording. Theorem 1.2. For a, b ∈Z with b ̸= 0, there are unique q and r in Z such that a = bq + r with 0 ≤r < \|b\|. Theorem 1.3. For f, g ∈F[T] with g ̸= 0, there are unique q and r in F[T] such that f = gq + r with r = 0 or deg r < deg g. The greatest common divisor of two integers is the common divisor largest in size (so always positive). In F[T], the greatest common divisor of two polynomials is the common monic factor with the largest degree. Examples will be worked out in the next section. Two integers are called relatively prime when their only common factors are ±1. Simi-larly, two polynomials in F[T] are called relatively prime when their only common factors are nonzero constants. In both Z and F[T], relative primality means the only common factors are units (±1 in Z and nonzero constants in F[T]). Euclid’s algorithm is the stan-dard method to compute greatest common divisors in Z (so, in particular, to determine relative primality) while a variant of Euclid’s algorithm in F[T] will perform the same role for polynomials. The standard chain of reasoning div. thm. ⇝Euclid ⇝Bezout ⇝p \| ab ⇒p \| a or p \| b ⇝unique factorization in Z, where p is a prime, carries over to F[T] nearly verbatim, with only minor changes needed in most proofs: div. thm. ⇝Euclid ⇝Bezout ⇝p(T) \| f(T)g(T) ⇒p(T) \| f(T) or p(T) \| g(T) ⇝u.f. in F[T], where p(T) is an irreducible. There is one important diﬀerence between Z and F[T]. Division in Z involves remainders ≥0, so if two integers are relatively prime Euclid’s algorithm will always have last nonzero remainder 1. But this is false with polynomials: the last nonzero remainder in Euclid’s algorithm for polynomials might be a nonzero constant other than 1, so writing an F[T]-linear combination of relatively prime polynomials as 1 can involve some additional scaling which we don’t have to do in Z. Example 1.4. In R[T], let f(T) = T 2 + 1 and g(T) = T −1. Certainly f(T) and g(T) are relatively prime: they have no common factor in R[T] other than nonzero constants. When we carry out Euclid’s algorithm on these two polynomials we ﬁnd T 2 + 1 = (T −1)(T + 1) + 2 T −1 = 2 1 2T −1 2 + 0, so the last nonzero remainder is 2. This is a nonzero constant in R[T] but it is not 1. By convention we normalize the gcd of two polynomials to be monic, so the gcd of T 2 + 1 and T −1 is called 1, not 2. ANALOGIES WITH POLYNOMIALS 3 2. Euclid and Bezout: an example in Q[T] Bezout’s identity in Z says for a and b in Z that we can write ax + by = (a, b) for some integers x and y. Values for x and y can be found by using back-substitution into Euclid’s algorithm for a and b. Similarly, Bezout’s identity for F[T] says for f(T) and g(T) in F[T] that f(T)u(T) + g(T)v(T) = (f, g), for some u(T) and v(T) in F[T]. Here too the polynomials u(T) and v(T) can be found using back-susbtitution into Euclid’s algorithm for f(T) and g(T). Example 2.1. Let f(T) = T 4 + T 3 + T 2 + T + 1 and g(T) = T 3 −2T −4. We will perform Euclid’s algorithm to compute a greatest common divisor of f(T) and g(T) in Q[T]: T 4 + T 3 + T 2 + T + 1 = (T 3 −2T −4)(T + 1) + (3T 2 + 7T + 5) (2.1) T 3 −2T −4 = (3T 2 + 7T + 5) 1 3T −7 9 + 16 9 T −1 9 3T 2 + 7T + 5 = 16 9 T −1 9 27 16T + 1035 256 + 1395 256 16 9 T −1 9 = 1395 256 4096 12555T − 256 12555 + 0. In practice, once we reach a nonzero constant as a remainder we can stop, just as we do when we get a remainder of 1 in Euclid’s algorithm for Z: the next step will deﬁnitely have a remainder of 0, so the nonzero constant remainder 1395 256 will be the last nonzero remainder and there is no point in performing the next step. Since the last nonzero remainder is a nonzero constant, f and g are relatively prime in Q[T]. Even though the last nonzero remainder is not 1, but some other nonzero constant, we still write “(f, g) = 1” because (f, g) denotes the monic greatest common divisor. Now let’s obtain Bezout’s identity for the f and g of Example 2.1 by back substitution into Euclid’s algorithm from (2.1): 1395 256 = (3T 2 + 7T + 5) − 16 9 T −1 9 27 16T + 1035 256 = (3T 2 + 7T + 5) − (T 3 −2T −4) −(3T 2 + 7T + 5) 1 3T −7 9 27 16T + 1035 256 = (3T 2 + 7T + 5) 9 16T 2 + 9 256T −549 256 −(T 3 −2T −4) 27 16T + 1035 256 . . . = f · 9 16T 2 + 9 256T −549 256 + g · −9 16T 3 −153 256T 2 + 27 64T −243 128 . Multiplying through by the constant 256/1395 makes the left side 1: (2.2) 1 = f · 16 155T 2 + 1 155T −61 155 + g · −16 155T 3 −17 155T 2 + 12 155T −54 155 . 4 KEITH CONRAD 3. Modular arithmetic in Q[T] In Z, modular arithmetic concerns the congruence relation a ≡b mod m, which means m \| (a −b) or a = b + mk for some k ∈Z. Every integer is congruent modulo m to its remainder under division by m, and we can add and multiply modulo m without worrying about which representatives we use: a ≡b mod m, c ≡d mod m = ⇒a + c ≡b + d mod m, ac ≡bd mod m. All of this can be adapted to polynomials in F[T] for a ﬁeld F: for a nonconstant m(T) ∈F[T], deﬁne f(T) ≡g(T) mod m(T) when m(T) \| (f(T) −g(T)), or equivalently when f(T) = g(T) + m(T)k(T) for some k(T) ∈F[T]. Example 3.1. In Q[T] we have T 7 ≡T 5 + 2T 3 mod T 2 −2 since T 7 −(T 5 + 2T 3) = T 3(T 4 −T 2 −2) = T 3(T 2 −2)(T 2 + 1), which is a multiple of T 2 −2. Every polynomial when divided by T 2 −2 has a remainder of the form aT + b, so every polynomial in Q[T] is congruent modulo T 2 −2 to a polynomial of the form aT + b. For example, T 7 = (T 2 −2)(T 5 + 2T 3 + 4T) + 8T = ⇒T 7 ≡8T mod T 2 −2. When deg m(T) = d, every polynomial in F[T] is congruent to a unique “remainder” of the form a0 + a1T + · · · + ad−1Td −1. We can make the calculation in the previous example more eﬃciently by using the fact that m(T) ≡0 mod m(T). From T 2 −2 ≡0 mod T 2 −2 we have T 2 ≡2 mod T 2 −2, so T 3 = T 2T ≡ 2T mod T 2 −2, T 4 = T 3T ≡ (2T)T ≡ 2T 2 ≡ 2(2) ≡ 4 mod T 2 −2, T 7 = T 3T 4 ≡ (2T)(4) ≡ 8T mod T 2 −2. Example 3.2. In Q[T] let f(T) = T 4 + T 3 + T 2 + T + 1 and g(T) = T 3 −2T −4 as in Example 2.1. We found in (2.2), from Euclid’s algorithm and back-substitution (and multiplication by the reciprocal of the last nonzero remainder) that 1 = f · 16 155T 2 + 1 155T −61 155 + g · −16 155T 3 −17 155T 2 + 12 155T −54 155 . Reducing both sides mod g, f · 16 155T 2 + 1 155T −61 155 ≡1 mod g. Reducing both sides mod f, 1 ≡g · −16 155T 3 −17 155T 2 + 12 155T −54 155 mod f. In this way we found inverses of f mod g and g mod f. ANALOGIES WITH POLYNOMIALS 5 4. Solving simultaneous congruences: an example in Q[T] In Z, if we want to solve the pair of congruence conditions x ≡2 mod 5, x ≡11 mod 19, we lift the ﬁrst congruence to Z in the form x = 2 + 5y for some y ∈Z and substitute that into the second congruence and solve for y: 2 + 5y ≡11 mod 19 ⇒5y ≡9 mod 19 ⇒y ≡17 mod 19. Thus y = 17 + 19z for some integer z, so x = 2 + 5(17 + 19z) = 87 + 95z, so x ≡87 mod 95. Conversely, if x ≡87 mod 95 then x ≡2 mod 5 and x ≡11 mod 19 since 87 ﬁts both conditions and the modulus 95 is divisible by 5 and 19. We can solve polynomial congruences in the same way. Consider in Q[T] the two con-gruence conditions (4.1) f(T) ≡3T mod T 2 + 1, f(T) ≡2T 2 + 1 mod T 3. Here the unknown we are looking for is f(T), not T: T is just a variable for the polynomials. We want an f(T) in Q[T] that ﬁts both congruence conditions in (4.1). Lift the ﬁrst congruence into Q[T] by writing it as (4.2) f(T) = 3T + (T 2 + 1)g(T) for some g(T) ∈Q[T]. Substitute this into the second congruence: 3T + (T 2 + 1)g(T) ≡2T 2 + 1 mod T 3. Subtracting 3T from both sides, (4.3) (T 2 + 1)g(T) ≡2T 2 −3T + 1 mod T 3. We now need to invert T 2 + 1 mod T 3. This will be done with Euclid’s algorithm: in Q[T], T 3 = (T 2 + 1)T −T, T 2 + 1 = (−T)(−T) + 1, so 1 = T 2 + 1 −(−T)(−T) = T 2 + 1 + (T)(−T) = T 2 + 1 + (T)(T 3 −(T 2 + 1)T) = T 2 + 1 + (T)(T 3) −(T 2 + 1)(T 2) = (T 2 + 1)(−T 2 + 1) + (T 3)(T), so (T 2 + 1)(−T 2 + 1) ≡1 mod T 3 . Therefore in Q[T], the inverse of T 2 + 1 mod T 3 is −T 2 + 1. Multiplying both sides of (4.3) by −T 2 + 1 gives g(T) ≡ (−T 2 + 1)(2T 2 −3T + 1) mod T 3 ≡ −2T 4 + 3T 3 + T 2 −3T + 1 mod T 3 ≡ T 2 −3T + 1 mod T 3 6 KEITH CONRAD since T 3 ≡0 mod T 3 and T 4 ≡0 mod T 3. Therefore g(T) = T 2 −3T + 1 + (T 3)h(T) for some h(T) ∈Q[T], and substituting this formula for g(T) into (4.2) shows an f(T) ﬁtting the two original congruence conditions must have the form f(T) = 3T + (T 2 + 1)(T 2 −3T + 1 + (T 3)h(T)) = T 4 −3T 3 + 2T 2 + 1 + (T 2 + 1)(T 3)h(T), so f(T) ≡T 4 −3T 3 + 2T 2 + 1 mod (T 2 + 1)(T 3). As a check that T 4 −3T 3 + 2T 2 + 1 ﬁts the original two congruence conditions, in Q[T] (T 4 −3T 3 + 2T 2 + 1) −3T = T 4 −3T 3 + 2T 2 −3T + 1 = (T 2 + 1)(T 2 −3T + 1) and (T 4 −3T 3 + 2T 2 + 1) −(2T 2 + 1) = T 4 −3T 3 = T 3(T −3). Therefore T 4 −3T 3 + 2T 2 + 1 works, and more generally any polynomial f(T) in Q[T] such that f(T) ≡T 4 −3T 3 + 2T 2 + 1 mod (T 2 + 1)T 3 satisﬁes the two congruence conditions in (4.1) and such f(T) form the complete set of solutions to the two congruences in (4.1). 5. Examples in Fp[T] Using the same polynomials f(T) = T 4 + T 3 + T 2 + T + 1 and g(T) = T 3 −2T −4 as before, now we will do calculations with them in Fp[T] for small p. Here think of the coeﬃcients of f and g as integers modulo p. Example 5.1. We’ll calculate (f, g) in Fp[T] for p = 2, 3, 5, and 7. In F2[T], g(T) = T 3 (the linear and constant terms in g(T) are 0 in F2). Then Euclid’s algorithm on f(T) and g(T) in F2[T] is T 4 + T 3 + T 2 + T + 1 = (T 3)(T + 1) + (T 2 + T + 1) (5.1) T 3 = (T 2 + T + 1)(T + 1) + 1, and we stop at the nonzero constant remainder: (f, g) = 1 in F2[T]. In F3[T], g(T) = T 3 + T + 2 and Euclid’s algorithm on f(T) and g(T) is T 4 + T 3 + T 2 + T + 1 = (T 3 + T + 2)(T + 1) + (T + 2) (5.2) T 3 + T + 2 = (T + 2)(T 2 + T + 2) + 1, so (f, g) = 1 in F3[T]. In F5[T], g(T) = T 3 + 3T + 1 and Euclid’s algorithm on f(T) and g(T) is T 4 + T 3 + T 2 + T + 1 = (T 3 + 3T + 1)(T + 1) + (3T 2 + 2T) (5.3) T 3 + 3T + 1 = (3T 2 + 2T)(2T + 2) + (4T + 1) 3T 2 + 2T = (4T + 1)(2T) + 0, so the last nonzero remainder is not constant: f(T) and g(T) have greatest common divisor 4T +1 in F5[T], so their monic gcd is its monic scalar multiple: (f, g) = −(4T +1) = T +4. We can explicitly factor out T + 4 from both f and g in F5[T]: f(T) = (T + 4)(T 3 + 2T 2 + 3T + 4), g(T) = (T + 4)(T 2 + T + 4). ANALOGIES WITH POLYNOMIALS 7 In F7[T], g(T) = T 3 + 5T + 3 and Euclid’s algorithm on f(T) and g(T) is T 4 + T 3 + T 2 + T + 1 = (T 3 + 5T + 3)(T + 1) + (3T 2 + 5) (5.4) T 3 + 5T + 3 = (3T 2 + 5)(5T) + (T + 3) 3T 2 + 5 = (T + 3)(3T + 5) + 4, and we stop since we have reached a nonzero constant, 4. The gcd of f and g in F7[T] is 1. Table 1 summarizes our computations. We list both the last nonzero remainder in Euclid’s algorithm and the (monic) gcd. F[T] Last Remainder (f, g) Q[T] 1395/256 1 F2[T] 1 1 F3[T] 1 1 F5[T] 4T + 1 T + 4 F7[T] 4 1 Table 1. Euclid’s algorithm on f(T) = T 4 + T 3 + T 2 + T + 1, g(T) = T 3 −2T −4 Remark 5.2. In F5[T] there is a nonconstant gcd. There is one prime p ̸= 5 such that f(T) and g(T) are not relatively prime in Fp[T]: in F31[T], (f(T), g(T)) = T −2. The denominator 155 in the coeﬃcients of (2.2) factors as 5 · 31. This is related to the roles of 5 and 31 as primes where f(T) mod p and g(T) mod p have nonconstant gcd. Example 5.3. Now we ﬁgure out how to write (f, g) in Fp[T] as a combination of f(T) mod p and g(T) mod p for p = 2, 3, 5, and 7. In F2[T] we get by back-substitution in Euclid’s algorithm from (5.1) 1 = g −(T 2 + T + 1)(T + 1) = g −(f −g(T + 1))(T + 1)) = f · (T + 1) + g · (1 + (T + 1)(T + 1)) = f · (T + 1) + g · T 2. Using back-substitution in F3[T] from (5.2), 1 = g −(T + 2)(T 2 + x + 2) = g −(f −g(T + 1))(T 2 + T + 2) = f · (2T 2 + 2T + 1) + g · (1 + (T + 1)(T 2 + T + 2)) = f · (2T 2 + 2T + 1) + g · (T 3 + 2T 2). In F5[T] from (5.3), 4T + 1 = g −(3T 2 + 2T)(2T + 2) = g −(f −g(T + 1)(2T + 2)) = f · (3T + 3) + g · (1 + (T + 1)(2T + 2)) = f · (3T + 3) + g · (2T 2 + 4T + 3). 8 KEITH CONRAD The gcd we found in Euclid’s algorithm, 4T + 1, is not monic. To write the monic gcd of f and g as an F5[T]-linear combination of f and g we simply multiply through the equations by −1 = 4: T + 4 = f · (2T + 2) + g · (3T 2 + T + 2). In F7[T] from (5.4), 4 = (3T 2 + 5) −(T + 3)(3T + 5) = (3T 2 + 5) −(g −(3T 2 + 5)(5T))(3T + 5) = (3T 2 + 5)(1 + 5T(3T + 5)) + g(4T + 2) = (3T 2 + 5)(T 2 + 4T + 1) + g(4T + 2) = (f −g(T + 1))(T 2 + 4T + 1) + g(4T + 2) = f · (T 2 + 4T + 1) + g · (6T 3 + 2T 2 + 6T + 1). Multiplying through by 4−1 = 2, 1 = f · (2T 2 + T + 2) + g · (5T 3 + 4T 2 + 5T + 2). Example 5.4. Similar to the solution of simultaneous congruences in Q[T] in Section 4, now consider in F5[T] the two congruence conditions (5.5) f(T) ≡3T mod T 2 + 1, f(T) ≡2T 2 + 1 mod T 3. To ﬁnd f(T) in F5[T] ﬁtting both congruence conditions, we carry out the same procedure as in Q[T], but now all calculations are in F5[T]. Lift the ﬁrst congruence into F5[T] as (5.6) f(T) = 3T + (T 2 + 1)g(T) where g(T) ∈F5[T]. Substituting this into the second congruence, 3T + (T 2 + 1)g(T) ≡2T 2 + 1 mod T 3. Subtracting 3T from both sides (note −3T = 2T in F5[T]), (5.7) (T 2 + 1)g(T) ≡2T 2 + 2T + 1 mod T 3. To invert T 2 + 1 mod T 3 we use Euclid’s algorithm: F5[T], T 3 = (T 2 + 1)T + 4T, T 2 + 1 = 4T(4T) + 1, so 1 = T 2 + 1 −4T(4T) = T 2 + 1 −4T(T 3 −(T 2 + 1)T) = (T 2 + 1)(4T 2 + 1) + T 3(−4T). Therefore (T 2 + 1)(4T 2 + 1) ≡1 mod T 3 , so in F5[T], the inverse of T 2 + 1 mod T 3 is 4T 2 + 1. That tells us to multiply both sides of (5.7) by 4T 2 + 1, and we get g(T) ≡ (4T 2 + 1)(2T 2 + 2T + 1) mod T 3 ≡ 3T 4 + 3T 3 + T 2 + 2T + 1 mod T 3 ≡ T 2 + 2T + 1 mod T 3. ANALOGIES WITH POLYNOMIALS 9 Lifting this into F5[T] as g(T) = T 2 + 2T + 1 + T 3h(T) for h(T) ∈F5[T], substitute this formula for g(T) into (5.6) to obtain a formula for f(T): any f(T) ﬁtting the congruence conditions in (5.5) is f(T) = 3T + (T 2 + 1)(T 2 + 2T + 1 + T 3h(T)) = T 4 + 2T 3 + 2T 2 + 1 + (T 2 + 1)T 3h(T), so f(T) ≡T 4 + 2T 3 + 2T 2 + 1 mod (T 2 + 1)T 3. To check that T 4 + 2T 3 + 2T 2 + 1 ﬁts the conditions in (5.5), in F5[T] (T 4 + 2T 3 + 2T 2 + 1) −3T = (T 2 + 1)(T + 1) and (T 4 + 2T 3 + 2T 2 + 1) −(2T 2 + 1) = T 3(T + 2). Therefore T 4 + 2T 3 + 2T 2 + 1 works, and more generally every polynomial f(T) in F5[T] such that f(T) ≡T 4 + 2T 3 + 2T 2 + 1 mod (T 2 + 1)T 3 satisﬁes the two congruence conditions in (5.5) and such f(T) form the complete set of solutions to the two congruences in (5.5).
188935	https://math.stackexchange.com/questions/2326429/orthogonal-projection-of-a-point-into-xyz-0-plane-ex	linear algebra - Orthogonal projection of a point into $x+y+z=0$ plane ex. - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Orthogonal projection of a point into x+y+z=0 x+y+z=0 plane ex. [closed] Ask Question Asked 8 years, 3 months ago Modified8 years, 3 months ago Viewed 2k times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. Closed. This question is off-topic. It is not currently accepting answers. This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. Closed 8 years ago. Improve this question Let T:R 3→W T:R 3→W be the orthogonal projection of R 3 R 3 onto the plane W W having the equation x+y+z=0 x+y+z=0. (a)Find T(3,8,4)T(3,8,4). (b)Find the formula for T T. I have been stuck on this exercise for hours... How can I solve it? Thanks in advance! linear-algebra Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Jun 12, 2020 at 10:38 CommunityBot 1 asked Jun 17, 2017 at 16:56 Pedro GomesPedro Gomes 4,129 6 6 gold badges 39 39 silver badges 89 89 bronze badges 1 2 What did you try?user370967 –user370967 2017-06-17 17:01:28 +00:00 Commented Jun 17, 2017 at 17:01 Add a comment\| 4 Answers 4 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. Let P(a,b,c)∈R 3 P(a,b,c)∈R 3. The line which passes through P P and is orthogonal to W W is r⃗=(a,b,c)+t(1,1,1)=(a+t,b+t,c+t)r→=(a,b,c)+t(1,1,1)=(a+t,b+t,c+t) At the intersection of the line and W W (which is T(P)T(P)), a+t+b+t+c+t t=0=−1 3(a+b+c)a+t+b+t+c+t=0 t=−1 3(a+b+c) So, T(a,b,c)=(2 a−b−c 3,−a+2 b−c 3,−a−b+2 c 3)T(a,b,c)=(2 a−b−c 3,−a+2 b−c 3,−a−b+2 c 3) Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Jun 18, 2017 at 0:46 answered Jun 17, 2017 at 17:03 CY AriesCY Aries 23.9k 1 1 gold badge 32 32 silver badges 56 56 bronze badges 2 I think you got wrong r⃗=(a,b,c)+t(1,1,1)+(a+t,b+t,c+t)r→=(a,b,c)+t(1,1,1)+(a+t,b+t,c+t) it should be instead r⃗=(a,b,c)+t(1,1,1)=(a+t,b+t,c+t)r→=(a,b,c)+t(1,1,1)=(a+t,b+t,c+t), right?Pedro Gomes –Pedro Gomes 2017-06-17 18:00:27 +00:00 Commented Jun 17, 2017 at 18:00 I have a typo. Thanks for pointing it out.CY Aries –CY Aries 2017-06-18 00:47:37 +00:00 Commented Jun 18, 2017 at 0:47 Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. n⃗(1,1,1)n→(1,1,1) is a normal of the plain. Now, write the equation of the line, which parallel to n⃗n→ and (3,8.6)(3,8.6) on the line and find a common point of them. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Jun 17, 2017 at 17:01 Michael RozenbergMichael Rozenberg 208k 31 31 gold badges 171 171 silver badges 294 294 bronze badges Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. The normal vector of the plane is n=<1,1,1>n=<1,1,1> and the plane passes through (0,0,0)(0,0,0). So we let the orthogonal projection of T(3,8,4)T(3,8,4) be P P and have a position vector p p. Thus, t+λ n=p t+λ n=p Since P P satisfies the plane x+y+z=0 x+y+z=0, so solve for λ λ. 3+λ+8+λ+4+λ=0 3+λ+8+λ+4+λ=0 Thus P=(−2,3,−1)P=(−2,3,−1) Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Jun 17, 2017 at 17:03 jonsnojonsno 7,711 2 2 gold badges 18 18 silver badges 42 42 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. Any v∈R 3 v∈R 3 can be decomposed into v∥+v⊥v∥+v⊥, where v∥=T v v∥=T v, the projection of v v onto W W, and v⊥v⊥ is the orthogonal projection of v v onto W⊥W⊥. Since W W is a plane, W⊥W⊥ is the line normal to this plane. You should be able to read a vector that spans this space from the defining equation of W W. The above decomposition says that T v=v−v⊥T v=v−v⊥, so you can find the projection of v v onto W W by finding its projection onto the normal line and subtracting that from v v. For this problem, that’s much less work than trying to work T T out directly. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Jun 17, 2017 at 22:24 amdamd 55.2k 3 3 gold badges 40 40 silver badges 100 100 bronze badges Add a comment\| Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions linear-algebra See similar questions with these tags. 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188936	https://www.revisiondojo.com/blog/understanding-vectors-in-ib-math-hl-a-complete-guide-for-success	Home /Blog /Understanding Vectors in IB Math HL: A Complete Guide for Success IB Understanding Vectors in IB Math HL: A Complete Guide for Success RevisionDojo••5 min read Understanding Vectors in IB Math HL: A Complete Guide for Success Vectors are one of the most versatile and powerful tools in the IB Mathematics: Analysis and Approaches (AA) Higher Level (HL) syllabus. Whether you’re calculating directions, distances, or working with 3D geometry, vectors offer an elegant way to represent motion, lines, and planes. This guide is here to demystify vectors, walk you through essential concepts, and help you develop the confidence to solve even the most complex IB Math HL vector questions. What Are Vectors? A vector is a quantity that has both magnitude (length) and direction. Unlike scalars (which only have magnitude), vectors are used to describe displacement, velocity, force, and geometric relationships in space. In IB Math HL, you’ll work with vectors in two and three dimensions, applying them in both pure mathematics and real-world contexts. Core Concepts in Vectors ✅ 1. Vector Notation Written as →AB, a, or v. Can be represented as column vectors: a=(xyz)\mathbf{a} = \begin{pmatrix} x \ y \ z \end{pmatrix}a=xyz ✅ 2. Vector Operations Addition/Subtraction: Combine vectors component-wise. Scalar Multiplication: Multiply each component by a constant. Magnitude: ∣a∣=x2+y2+z2\|\mathbf{a}\| = \sqrt{x^2 + y^2 + z^2}∣a∣=x2+y2+z2 Unit Vector: a∣a∣\frac{\mathbf{a}}{\|\mathbf{a}\|}∣a∣a ✅ 3. Scalar (Dot) Product Used to find angles between vectors: a⋅b=∣a∣∣b∣cos⁡θ\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\|\|\mathbf{b}\|\cos\thetaa⋅b=∣a∣∣b∣cosθ Helps determine perpendicularity (dot product = 0) ✅ 4. Vector Equation of a Line Given point A and direction vector d, the line is: r=a+λd\mathbf{r} = \mathbf{a} + \lambda \mathbf{d}r=a+λd where λ is a scalar. ✅ 5. Intersection and Parallelism Determine if lines intersect, are skew, or parallel by solving equations. Set vector equations equal and solve for λ and μ (parameters). Advanced Vector Topics in IB Math HL Angle Between Two Lines Shortest Distance Between Lines Vector Cross Product (optional for HL AA) Planes and Normal Vectors Line-Plane and Plane-Plane Intersections How Vectors Are Used in IB Exam Questions You’ll see vector questions that test: Geometric reasoning Analytical problem-solving Algebraic manipulation Real-life contexts (projectiles, forces, motion) Paper 2 typically includes more abstract or multi-part vector questions, while Paper 3 may involve more advanced geometry and proof. Tips to Master Vectors in IB Math HL Practice regularly: Use Revisiondojo to access vector-based past paper questions. Draw diagrams: Visualizing the problem helps make sense of 3D motion. Memorize key formulas: Magnitude, dot product, and line equations are essential. Check direction and magnitude: Many vector errors are due to sign or calculation mistakes. Understand context: Know when to apply dot products, find angles, or determine distances. Common Mistakes to Avoid Confusing points with vectors (e.g., mixing up position vectors and coordinates) Ignoring units or magnitude direction Forgetting to use parameter λ when writing line equations Misapplying formulas like scalar product without understanding the angle’s role FAQs: Understanding Vectors in IB Math HL Are vectors part of both SL and HL?Yes, but HL students go deeper—especially into 3D vector geometry, lines, and intersection theory. What’s the most important vector formula to know?The vector equation of a line and dot product formula are vital for solving many exam questions. How many vector questions are in the exam?Vectors often appear in Paper 2 and Paper 3, comprising about 10–15% of HL marks depending on the session. Can vectors appear in the IA?Yes. Topics involving motion, 3D modeling, or optimization often use vectors effectively. Does calculator use help with vector questions?Yes—for solving systems of equations, magnitudes, or numerical dot products. But clear method steps are still required. Conclusion: Mastering Vectors Opens Up IB Math HL Success Understanding vectors is essential for success in IB Math HL. With a solid grasp of notation, operations, and application techniques, you’ll be well-prepared to tackle any vector question—whether it’s geometric, analytical, or real-world focused. 🎯 Want guided help on vector problems with full step-by-step solutions?📘 Try Revisiondojo for past papers, formula practice, and custom vector tutorials designed for HL students. Ace your exams with RevisionDojo Thousands of practice questions Study notes and flashcards for every topic and subject Free Jojo AI tutor Rated Excellent On Trustpilot Join 350k+ Students Already Crushing Their Exams
188937	https://www.boddlelearning.com/1st-grade/adding-and-subtracting-using-place-values	Teaching Resources 1st Grade Adding and Subtracting Using Place Values Learning how to add and subtract by using place values is a first grade, Common Core math skill: 1.NBT.4. Below we show two videos that demonstrate this standard. Then, we provide a breakdown of the specific steps in the videos to help you teach your class. Prior Learnings Your students should be familiar with counting from 1 to 100 using 1’s and 10’s, starting from any number. They should also be able to read, write, and represent objects using numbers between 0 and 20 (K.CC.1-3). Future Learnings Later on, understanding place values will enable your students to skip-count within 1000 (counting by 5’s, 10’s, and 100’s). They will also be able to read and write numbers by using “base ten numerals, number names, and expanded form” (2.NBT.1-3). Common Core Standard: 1.NBT.4 - Add within 100, both one and two-digit numbers and multiples of 10; use concrete models, drawings, and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction Students who understand this principle can: Use physical models, drawings, etc. to explain addition within 100 by adding a one and a two-digit number. Use physical models, drawings, etc. to explain addition within 100 by adding a two-digit number and a multiple of ten. Use physical models, drawings, etc. to explain addition within 100 by adding 2 two-digit numbers. Break down both addends, using partial sums, to add within 100. Break down one addend, using partial sums, to add within 100. Explain why, when adding numbers, a “new ten” is sometimes made. 2 Videos to Help You Teach Common Core Standard: 1.NBT.4 Below we provide and breakdown two videos to help you teach your students this standard. Video 1: Different Methods to Add Large Numbers This video demonstrates three different ways to solve adding two large numbers together. The girl in the video is confused because she at first does not know how to solve 43 + 21. Then, she remembers 3 different methods she learned in school for how to solve these types of problems. The first method uses blocks to solve the equation. First, break the numbers into 10s and 1s. a. The video shows 43 and 21 as blocks, broken down into groups of 10s and 1s. Add the two 10s together. a. 40 + 20. Add the two 1s together. a. 3 + 1. Then combine them to find the total: 64. The next example follows the same pattern, except without blocks for aid. 40 + 20 = 60. 3 + 1 = 4. 60 + 4 = 64. The last example uses a number line to solve the equation. Start at 43 (the bigger number) on the number line. Then add 20 by 10s. a. One 10 to get 53 and another 10 to get 63. Then, only one more number is left: 1. a. 63 + 1 = 64. The video ends by reminding students that they can add large numbers by breaking them into 10s and 1s and using a number line. Video 2: Adding Large Numbers in Columns The video begins by doing a brief review on place values and what they are: “A place value shows the position of a digit in a number.” For example, if a number has 6 tens and 2 ones, then the number is 62. Boddle then explains that place values can be used to make addition and subtraction easier. The video then provides a few examples for students to see how the concept works. In the equation 23 + 5, students can line them up in a column. a. Make sure the place values are aligned. b. 1s over 1s and 10s over 10s. Start by adding the numbers in the 1s place. a. 3 + 5 = 8. Then add the numbers in the 10s place. a. Since, 2 is alone in the 10s place, bring it down to the total. b. Or think of it as 2 + 0 = 2. 23 + 5 = 28. The video then gives another example: 35 + 7. It demonstrates how students can handle an addition equation that carries a new number over into the 10s place. Write the equation in a column. Add the values in the 1s place. a. 5 + 7 = 12. The video then reminds students that only 1 number can be written per place value. Only the 2 from the 12 is written, and it is in 1s place. The 1 goes on top of the 3 as they are both in the 10s place. The 1 and the 3 are added together. The answer is 42, so 35 + 7 = 42. Want more practice? Give your students additional standards-aligned practice with Boddle Learning. Boddle includes questions related to Comparing and Measuring Lengths plus rewarding coins and games for your students to keep them engaged. Click here to sign up for Boddle Learning and create your first assignment today. Information on standards is gathered from The New Mexico Public Education Department's New Mexico Instructional Scope for Mathematics and the Common Core website. Select your device
188938	https://www.youtube.com/watch?v=X7abPL2CBm8	Minimum Distance from a Point to a Curve (Calculus) turksvids 29600 subscribers 64 likes Description 12277 views Posted: 20 Nov 2021 In this video we use calculus to find the minimum distance between a curve and point (in this case the origin (0,0)). We sketch, write a distance function, square the distance, take a derivative, prove we're getting a minimum, and then write our solution. This is a common optimization problem. This is definitely a AP Calculus AB, Calc BC, and Calc 1 topic that you'll see along the way. 1 comments Transcript: Intro okay in this video we are going to talk about minimizing the distance from a point to a curve this is a pretty common um minimization problem in calculus so let's take a look at what the problem would say and then we will solve it Example so we are given the curve y squared equals 150 minus 10 x where x is between 0 and 15. we want to find the coordinates of the point on the curve in quadrant 1 that is nearest to the origin so this one's a little weird because we're given y squared equals 150 minus 10x you wouldn't really lose anything i don't think if you just took the square root and said y is equal to the square root of 150 minus 10x i'm just going to draw sort of a generic sideways parabola and go from there i mean it's not super generic but here we go here are my axes i know that if y is equal to 0 then x will be 15. um and then i know that this thing opens to the left but let me tell you even if you got that picture wrong it wouldn't really change the process that we're going to go through and so you don't have to worry so much about getting a perfect graph let's um put a point on the curve which i think is actually kind of the key to solving this so there's my point that point in general is the point x comma y that's going to be really important and we're trying to find the distance from this point to the origin we want the closest point on the curve to the origin so what i'm going to do is i'm thinking distance um so distance formula right so my two ordered pairs are x y and zero zero so i'm gonna write a distance equation so it should be the square root of and then it'll be uh x minus zero squared plus y minus 0 squared so i'm writing in the zeros you know you might not do that i'm going to simplify this a tiny little bit because why would i have written those zeros so square root of x squared plus y squared now this is a really common thing that you run into in optimization problems we're trying to find a minimum so we're going to take a derivative set it equal to zero and blah blah blah there are too many variables here right we have x's and y's and we have d which is the name of the function i'm going to replace one of the variables usually at this step you'll replace y so i know that y squared was actually given it's 150 minus 10x so that's a key step there's going to be probably too many variables you have to get rid of one of them usually you get rid of y by just going back to what is the equation of the curve all right so i'm going to replace y squared with 150 minus 10x so i get d is square root of x squared plus the quantity 150 minus 10 x all right so i want to minimize this and what you could do is you could take the derivative set it equal to 0 and solve that's a very common thing to do and that will definitely work for you but what i want to do in this problem is something a little bit different i'm actually going to square d and i'm going to create a new function that i'm going to call d sub 2 because i don't want the exponent to like confuse things so i'm going to create d sub 2 which is the square of d the reason i'm doing that is that the derivative of this is easier to take i also kind of rearranged the terms hopefully you noticed um so what we're going to do is we're going to find the derivative of this the reason this works is that the x value that minimizes the square of the distance which is d sub 2 also minimizes the distance the thing you have to watch out for is we're using d sub 2 the square of the distance to find that x coordinate so if i need the actual distance i have to go back to the original function d that's the only caveat to that otherwise it's a little bit easier to work with the square of the distance so i'm going to find a derivative i'm going to set it equal to 0 and go from there so this is just power rule so d sub 2 prime is 2x minus 10. i'm going to set that equal to 0 and get that x is 5. now i need to test this thing because i need to see if this is a minimum this could potentially be a maximum um if i mean i don't think it will be but it could be a maximum if it was a maximum then i would have to check the end points the end points because i have to be in quadrant one would be zero and fifteen i would just take those plug it into the distance and see what i get um so i'm gonna do a sine chart where i'm gonna put five and then to the left of zero uh to the left of five rather something like zero right zero to the left five i would get i'm plugging into d2 prime i get zero minus ten so negatives and then to the right like you could well i mean we have to stop at fifteen uh but if we plugged in like i don't know 10 we would get 20 minus 10 which is definitely positive all right so the derivative went from negative to positive which means we have a relative minimum i'm going to write up a justification so i'm not going to make you watch that whole thing i'm just going to jump through it d2 has a relative minimum at x equals five because d2 prime changes negative to positive there i'm also going to make sure that i'm saying this is the absolute minimum so there's only one critical point there's a relative minimum at that critical point it must be the absolute minimum so i'm going to write that so since there is since this is the only critical point d2 has absolute minimum at x equals five so you really want to work on your justifications especially if you're an ap calculus free response questions all about justifying things so we have justified it but we didn't actually answer the question right because the question was to find we know that d has a minimum when x is 5 but that wasn't the question it was what is the point on the curve that is nearest to the origin so i have to actually find y so i'm just going to go back up to that y squared thing y squared is 150 minus 10x we now know that x is 5. um you could probably blow through this on your own um so that's y squared is 100 and since we're in the first quadrant that means that y is going to be positive 10. um so there is a point in the fourth quadrant i guess at 5 negative 10 that is equally close but that's not in the first quadrant so the ordered pair we were looking for here is the point five comma ten all right on the next kind of page i'm just gonna summarize how i solve these problems so Summary this is like the strategy that i use so the first thing is i'm going to sketch a curve it doesn't need to look good or accurate even you just need to get an idea of what's going on i'm going to put a point on the curve so the point i'm going to put on is x comma y because every point on a curve can be thought of as x y then i'm going to write the distance formula in terms of the point that the point x y and then if it's the origin i'll use 0 0. if it has to be the closest point to 5 3 i'll use the point x y and 5 3. just write your distance formula that'll probably have too many variables in it so we want to sub for a variable the one you usually sub for is y equals f of x and then you'll have something entirely in terms of x the next thing we want to do is take the derivative of the distance or of the distance squared which is usually easier from there we are going to confirm that it's a minimum because it might not be it could actually be a maximum if it is a maximum just test the endpoints and one of those has to give you the minimum but usually you'll actually get a minimum on these problems and then finally you want to answer the question so that's a process that we go through and an example i hope you found this helpful and good luck
188939	https://www.chegg.com/homework-help/questions-and-answers/find-inverse-5-modulo-12-q12385975	Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on. Question: find an inverse of 5 modulo 12 find an inverse of 5 modulo 12 This AI-generated tip is based on Chegg's full solution. Sign up to see more! To find the inverse of 5 modulo 12, set up the equation and find values where this equation holds true. Not the question you’re looking for? Post any question and get expert help quickly. Chegg Products & Services CompanyCompany Company Chegg NetworkChegg Network Chegg Network Customer ServiceCustomer Service Customer Service EducatorsEducators Educators
188940	https://calcworkshop.com/series-sequences/geometric-series/	Home » Series and Sequences » Geometric Series How to find Arithmetic and Geometric Series 13 Surefire Examples! // Last Updated: - Watch Video // As we’ve already seen, using Summation Notation, also called Series Notation, enables us to add up the terms of a sequence by creating Partial Sums. Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher) But wouldn’t it be nice if we didn’t have to add up all those terms? If only there was a formula that we could just plug into! Well, happy day! Because this lesson is all about two very special types of Series: Arithmetic and Geometric Series where all we have to do to is plug into a formula! Super simple and super easy! Now, remember, and Arithmetic Sequence is one where each term is found by adding a common value to each term and a Geometric Sequence is found by multiplying a fixed number to each term. Thus making both of these sequences easy to use, and allowing us to generate a formula that will enable us to find the sum in just a few simple steps. We will begin by exploring the Arithmetic Series and it’s Summation Formula. What is extremely important to note, and should be a warning to us, is that we can only find the sum of an Arithmetic Series that is Finite! That means, we can only find the sum for the first n terms. We start by using the Arithmetic Series formula to find the sum of various Arithmetic Series, and then we will work backwards, from our Sum and locate the first term and the common difference. Formula for Finding the Sum of a Geometric Series Next, we will look at the formula for a Finite Geometric Series, and how to use it to find the sum of the first n terms of a Geometric sequence. Then, we will spend the rest of the lesson discussing the Infinite Geometric Series. This series is so special because it will enable us to find such things as Power Series and Power Functions in Calculus! Once again, there is a warning. We will only be able to find the sum of an Infinite Geometric Series under certain conditions, or as Purple Mathsays “special circumstance.” That is to say that the infinite series will only converge (i.e., be able to find the sum) if and only if the ratio r is between –1 and 1. So, we will take the time to discuss how we can even find the sum of an infinite series, and see why/how it works, and then use it to find the sum of various infinite geometric series. Geometric Series – Video Get access to all the courses and over 450 HD videos with your subscription Monthly and Yearly Plans Available Get My Subscription Now Still wondering if CalcWorkshop is right for you?Take a Tour and find out how a membership can take the struggle out of learning math.
188941	https://pubmed.ncbi.nlm.nih.gov/37072797/	Pre-operative intravitreal bevacizumab for tractional retinal detachment secondary to proliferative diabetic retinopathy: the Alvaro Rodriguez lecture 2023 - PubMed Clipboard, Search History, and several other advanced features are temporarily unavailable. Skip to main page content An official website of the United States government Here's how you know The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site. The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely. 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Pre-operative intravitreal bevacizumab for tractional retinal detachment secondary to proliferative diabetic retinopathy: the Alvaro Rodriguez lecture 2023 J Fernando Arevalo#1,Bradley Beatson#2 Affiliations Expand Affiliations 1 Wilmer Eye Institute, Johns Hopkins School of Medicine, 600 N Wolfe St; Maumenee 713, Baltimore, MD, 21287, USA. arevalojf@jhmi.edu. 2 Wilmer Eye Institute, Johns Hopkins School of Medicine, 600 N Wolfe St; Maumenee 713, Baltimore, MD, 21287, USA. Contributed equally. PMID: 37072797 PMCID: PMC10111833 DOI: 10.1186/s40942-023-00467-8 Item in Clipboard Pre-operative intravitreal bevacizumab for tractional retinal detachment secondary to proliferative diabetic retinopathy: the Alvaro Rodriguez lecture 2023 J Fernando Arevalo et al. Int J Retina Vitreous.2023. Show details Display options Display options Format Int J Retina Vitreous Actions Search in PubMed Search in NLM Catalog Add to Search . 2023 Apr 18;9(1):29. doi: 10.1186/s40942-023-00467-8. Authors J Fernando Arevalo#1,Bradley Beatson#2 Affiliations 1 Wilmer Eye Institute, Johns Hopkins School of Medicine, 600 N Wolfe St; Maumenee 713, Baltimore, MD, 21287, USA. arevalojf@jhmi.edu. 2 Wilmer Eye Institute, Johns Hopkins School of Medicine, 600 N Wolfe St; Maumenee 713, Baltimore, MD, 21287, USA. Contributed equally. PMID: 37072797 PMCID: PMC10111833 DOI: 10.1186/s40942-023-00467-8 Item in Clipboard Full text links Cite Display options Display options Format Abstract The treatment of proliferative diabetic retinopathy (PDR) has evolved significantly since the initial use of panretinal photocoagulation as a treatment in the 1950s. Vascular endothelial growth factor inhibitors have provided an effective alternative without the risk of peripheral vision loss. Despite this, the risk of complications requiring surgical intervention in PDR remains high. Intravitreal bevacizumab has shown promise as a preoperative adjuvant to vitrectomy for PDR complications, albeit with a purported risk for tractional retinal detachment (TRD) progression in eyes with significant fibrous proliferation. Here we will discuss anti-VEGF agent use in PDR and its role in surgical intervention for PDR complications including TRD. Keywords: Bevacizumab; Proliferative diabetic retinopathy; Tractional retinal detachment; anti-VEGF. © 2023. The Author(s). PubMed Disclaimer Conflict of interest statement Not applicable. Figures Fig. 1 This is a case of… Fig. 1 This is a case of a 46-year-old male who presented with PDR and… Fig. 1 This is a case of a 46-year-old male who presented with PDR and a best-corrected visual acuity of 20/400 Top left: Significant fibrovascular membranes at time of presentation clearly visible and marked by white arrows. Top right: 3 days following preoperative IVB and one day before PPV, patient showed significant reduction in vascular proliferation. Bottom middle: 12 months following PPV with C 3 F 8 tamponade and reattachment of the retina, patient had a best-corrected visual acuity of 20/70 Abbreviations: PDR-proliferative diabetic retinopathy; IVB-intravitreal bevacizumab; PPV-pars plana vitrectomy; C3F8-perfluoropropane gas. See this image and copyright information in PMC Similar articles Tractional retinal detachment following intravitreal bevacizumab (Avastin) in patients with severe proliferative diabetic retinopathy.Arevalo JF, Maia M, Flynn HW Jr, Saravia M, Avery RL, Wu L, Eid Farah M, Pieramici DJ, Berrocal MH, Sanchez JG.Arevalo JF, et al.Br J Ophthalmol. 2008 Feb;92(2):213-6. doi: 10.1136/bjo.2007.127142. Epub 2007 Oct 26.Br J Ophthalmol. 2008.PMID: 17965108 Intravitreal bevacizumab for surgical treatment of severe proliferative diabetic retinopathy.di Lauro R, De Ruggiero P, di Lauro R, di Lauro MT, Romano MR.di Lauro R, et al.Graefes Arch Clin Exp Ophthalmol. 2010 Jun;248(6):785-91. doi: 10.1007/s00417-010-1303-3. Epub 2010 Feb 5.Graefes Arch Clin Exp Ophthalmol. 2010.PMID: 20135139 Clinical Trial. Intravitreal Bevacizumab (Avastin) for Diabetic Retinopathy: The 2010 GLADAOF Lecture.Arevalo JF, Sanchez JG, Lasave AF, Wu L, Maia M, Bonafonte S, Brito M, Alezzandrini AA, Restrepo N, Berrocal MH, Saravia M, Farah ME, Fromow-Guerra J, Morales-Canton V.Arevalo JF, et al.J Ophthalmol. 2011;2011:584238. doi: 10.1155/2011/584238. Epub 2011 Mar 30.J Ophthalmol. 2011.PMID: 21584260 Free PMC article. Current management of diabetic tractional retinal detachments.Stewart MW, Browning DJ, Landers MB.Stewart MW, et al.Indian J Ophthalmol. 2018 Dec;66(12):1751-1762. doi: 10.4103/ijo.IJO_1217_18.Indian J Ophthalmol. 2018.PMID: 30451175 Free PMC article.Review. Anti-VEGF crunch syndrome in proliferative diabetic retinopathy: A review.Tan Y, Fukutomi A, Sun MT, Durkin S, Gilhotra J, Chan WO.Tan Y, et al.Surv Ophthalmol. 2021 Nov-Dec;66(6):926-932. doi: 10.1016/j.survophthal.2021.03.001. Epub 2021 Mar 8.Surv Ophthalmol. 2021.PMID: 33705807 Review. See all similar articles Cited by Bilateral vitrectomy in patients with proliferative diabetic retinopathy-characteristics and surgical outcomes.Hsia Y, Yang CM.Hsia Y, et al.Graefes Arch Clin Exp Ophthalmol. 2024 Sep;262(9):2833-2844. doi: 10.1007/s00417-024-06462-5. Epub 2024 Apr 5.Graefes Arch Clin Exp Ophthalmol. 2024.PMID: 38578330 References Mohamed Q, Gillies MC, Wong TY. Management of Diabetic Retinopathy: A Systematic Review.JAMA. 2007 Aug22;298(8):902. - PubMed Kempen JH, O’Colmain BJ, Leske MC, Haffner SM, Klein R, Moss SE et al. The prevalence of diabetic retinopathy among adults in the United States. Arch Ophthalmol Chic Ill 1960. 2004 Apr;122(4):552–63. - PubMed Saaddine JB. Projection of Diabetic Retinopathy and Other Major Eye Diseases Among People With Diabetes Mellitus: United States, 2005–2050.Arch Ophthalmol. 2008 Dec8;126(12):1740. - PubMed Cheung N, Mitchell P, Wong TY. Diabetic retinopathy. The Lancet. 2010 Jul;376(9735):124–36. - PubMed Harris Nwanyanwu K, Talwar N, Gardner TW, Wrobel JS, Herman WH, Stein JD. Predicting Development of proliferative Diabetic Retinopathy. Diabetes Care. 2013 Jun;1(6):1562–8. - PMC - PubMed Show all 56 references Related information MedGen Grants and funding Research to Prevent Blindness/Research to Prevent Blindness LinkOut - more resources Full Text Sources Europe PubMed Central Full text links[x] Europe PubMed Central [x] Cite Copy Download .nbib.nbib Format: Send To Clipboard Email Save My Bibliography Collections Citation Manager [x] NCBI Literature Resources MeSHPMCBookshelfDisclaimer The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). Unauthorized use of these marks is strictly prohibited. Follow NCBI Connect with NLM National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov
188942	http://www.urticator.net/essay/6/628.html	\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- \| \| Home > urticator.net Search About This Site > Domains Glue Stories Computers Driving Games Humor Law Math > Numbers Science Powers and Fractions Notes About Squares Decimal Expansions (Section) Number Maze Primes Divisibility Intrinsic Nature of Primes > Other Topics Other Topics (2) Duodecimal Hexadecimal Base 60 Operations Numbers as Polynomials \| \| \| > Repunits \| Negative Digits Two Kinds of Odd \| \| RepunitsA repunit, or repeated unit, is a number consisting entirely of 1s. I don't like the name, but it seems to be standard, so I guess I'm stuck with it. Anyway, I've never been very familiar with repunits, but they've been turning up a lot recently, and I keep forgetting which ones have which factors, so just for reference I thought I'd make a table. \| \| \| \| --- \| 1 \| 1 \| 1 \| \| 2 \| 11 \| 11 \| \| 3 \| 111 \| 3 37 \| \| 4 \| 1111 \| 11 101 \| \| 5 \| 11111 \| 41 271 \| \| 6 \| 111111 \| 3 7 11 13 37 \| \| 7 \| 1111111 \| 239 4649 \| \| 8 \| 11111111 \| 11 73 101 137 \| \| 9 \| 111111111 \| 32 37 333667 \| \| 10 \| 1111111111 \| 11 41 271 9091 \| \| 11 \| 11111111111 \| 21649 513239 \| \| 12 \| 111111111111 \| 3 7 11 13 37 101 9901 \| There are some interesting patterns in there, but before I can explain them I'll need to remind you of a couple of things. First of all, when I was talking about fractions in base 2, I observed that repunits of composite length have nice factorizations that are really factorizations of polynomials, and suggested that it would be handy to have a digit that represents −1. Then, when I was talking about divisibility in other bases, I started using the hash mark as such a digit … not to be confused with the sharp sign from Sharps and Flats. Knowing all that, we can break the repunits into irreducible polynomial factors without even specifying what the base is! \| \| \| \| --- \| 1 \| 1 \| 1 \| \| 2 \| 11 \| 11 \| \| 3 \| 111 \| 111 \| \| 4 \| 1111 \| 11 101 \| \| 5 \| 11111 \| 11111 \| \| 6 \| 111111 \| 11 111 1#1 \| \| 7 \| 1111111 \| 1111111 \| \| 8 \| 11111111 \| 11 101 10001 \| \| 9 \| 111111111 \| 111 1001001 \| \| 10 \| 1111111111 \| 11 11111 1#1#1 \| \| 11 \| 11111111111 \| 11111111111 \| \| 12 \| 111111111111 \| 11 111 101 1#1 10#01 \| The irreducible polynomial factors may break down further in particular bases; here's what happens in bases 10 and 2. (I've converted the base 2 results into base 10, hope that's not too confusing. For example, 111 in base 2 is the prime number 7.) \| \| \| \| \| --- --- \| \| 1 \| 1# \| 32 \| 1 \| \| 2 \| 11 \| 11 \| 3 \| \| 3 \| 111 \| 3 37 \| 7 \| \| 4 \| 101 \| 101 \| 5 \| \| 5 \| 11111 \| 41 271 \| 31 \| \| 6 \| 1#1 \| 7 13 \| 3 \| \| 7 \| 1111111 \| 239 4649 \| 127 \| \| 8 \| 10001 \| 73 137 \| 17 \| \| 9 \| 1001001 \| 3 333667 \| 73 \| \| 10 \| 1#1#1 \| 9091 \| 11 \| \| 11 \| 11111111111 \| 21649 513239 \| 23 89 \| \| 12 \| 10#01 \| 9901 \| 13 \| Each repunit contains exactly one new irreducible polynomial … I'm not sure why that's true, but it is, and I've numbered the rows accordingly in the table. Now I can explain the main pattern in the original table. Let's call the repunit of length n Rn, and the associated irreducible polynomial Pn. If n is divisible by some number m, then Rn is divisible by Rm because there's a nice factorization; and Rm in turn is divisible by Pm, by definition. So, the numbers that are factors of Pm in some base will be factors of Rn whenever n is a multiple of m. Actually, that's not quite true … it fails for m = 1, because Rn is never divisible by P1. And where'd that polynomial P1 come from, anyway?! Well, I was planning ahead. Suppose we define one more set of polynomials by the rule Bn = Rn × P1; then Bn, unlike Rn, really is divisible by Pm whenever n is divisible by m; and in fact Bn is equal to the product of all such Pm. That may sound complicated, but the polynomials Bn are actually extremely simple. For example, B5 = R5 × P1 = (b4 + b3 + b2 + b + 1)(b − 1) = b5 − 1. The same cancellation happens every time, so that Bn = bn − 1 for all n. Now, the roots of Bn are roots of unity, and roots of unity lie on the unit circle in the complex plane, so when we divide Bn into the parts Pm, we're dividing the circle … which is roughly why the parts Pm are called cyclotomic polynomials. (The roots of the word are “cyclo-”, as in “cyclic”, and “-tomic”, as in “atomic”, “indivisible”.) Just for fun, here are some pictures of the roots of Pm. See how for example P1, P2, P3, and P6 combine to cover the sixth roots of unity? That was fun, but let's get back to the subject of patterns. I think we've said all there is to say about the main pattern, the pattern in how the repunits break down into cyclotomic polynomials, but there are also a few patterns in how the cyclotomic polynomials break down into factors in base 10. The factor 239 in P7 is interesting … it has an unusually short repeat length because it's a factor of P7, but it's also involved in a rare triple exception, as I noted in The Usual Random Thoughts. (In short, 239 in base 13 is 155, and 1552 = 1CCCC = 20000−1.) There's no pattern there, I know, but I couldn't resist pointing it out. In both P7 and P11 the factors end in 239 and 649. I don't know what to make of that … that's a good combination for factoring repunits, because 239 × 649 ≡ 111 mod 1000, but there's a similar combination for any other number in U1000; I don't know why that one should turn up twice. The factor 333667 in P9 begs to be explained … but that's easy enough! 1001001 is divisible by 3, since the digits add to 3 (see Divisibility); and when you divide, that's what you get. Why didn't we see a factor like that sooner? Well, we didn't get one from 10101 because 3367 isn't prime, and we didn't get one from 111 because … well, actually we did get one, we just didn't recognize 37 as part of a pattern. (By the way, the reason 3367 isn't prime is that 10101 factors in any base as 111 × 1#1.) The factor 9901 in P12 is part of a similar series. 1#1 is 91, not prime; 10#01 is 9901; and 100#001 (P18) is 999001, not prime. The factor 9091 in P10 is part of a similar series, too. 1#1 is 91, still not prime; 1#1#1 is 9091; and 1#1#1#1 (P14) is 909091, prime. Now, all those 9s make me think it might be handy to have a digit that represents b−1 regardless of what the base actually is, so that 1#1#1 would be G0G1 (say), 10#01 would be GG01, and so on. The pattern is definitely there … 1#1#1 and 10#01 are 2021 and 2201 in base 3, 1011 and 1101 in base 2. (That pretty well sums up why base 2 is weird!) Finally, let me go back and clarify a couple of things. First, the idea of numbers being polynomials over some unspecified base isn't restricted to repunits and cyclotomic polynomials. For example, the number 299 is never prime, because 299 = 2b2 + 9b + 9 = (b + 3)(2b + 3), and the number 156 is always an intermediate, because 156 = b2 + 5b + 6 = (b + 2)(b + 3) = (b + 2)♯. (See also the discussion of small factors and carries in Multiplication.) Second, in spite of the evidence in the table, cyclotomic polynomials don't always consist of evenly-spaced 1s or evenly-spaced alternating 1s and #s against a background of 0s. Some have repeating triplets, … \| \| \| --- \| \| 15 \| 1#01#10#1 \| \| 21 \| 1#01#010#10#1 \| \| 33 \| 1#01#01#01#10#10#10#1 \| … some have other, stranger patterns, … \| \| \| --- \| \| 30 \| 110###011 \| \| 35 \| 1#0001#1#01#1#10#1#1000#1 \| … and some even have other digits! According to an article on cyclotomic polynomials, the first one of those occurs at 105 (for the same reason as in Number Maze!), and here it is, with X being a digit that represents −2. 11100##X##0011111100#0#0#0#0#0011111100##X##00111 \| \| See Also History and Other Stuff Multiplication in Base 10 Numbers as Polynomials Other Identities Reference Material Reflection Symmetry @ May (2006) \|
188943	https://www.mathcelebrity.com/chain_discount.php	Chain Discounts and Net Cost Price and Net Cost Equivalent Calculator Crop Image How does the Chain Discounts and Net Cost Price and Net Cost Equivalent Calculator work? Free Chain Discounts and Net Cost Price and Net Cost Equivalent Calculator - Given a chain discount and an original price, this calculates the total discount and net cost price. This calculator has 2 inputs. What 1 formula is used for the Chain Discounts and Net Cost Price and Net Cost Equivalent Calculator? (100% - Discount 1%) (100% - Discount 2%) Take the Quiz Take the Quiz Related Calculators Unit Savings Bond Price Formulas Coupon Comparison What 5 concepts are covered in the Chain Discounts and Net Cost Price and Net Cost Equivalent Calculator? chain discount : a series of trade discount percentages or their total discount : the amount by which the market price of a bond is lower than its principal amount due at maturity net cost equivalent : communicates how many pennies of cost the buyer must pay for every $1 of List Price net cost price : the amount paid by the customer after all discounts and rebates are applied price : the amount of money expected, required, or given in payment for something Example calculations for the Chain Discounts and Net Cost Price and Net Cost Equivalent Calculator chain discount for 750 at 25 and 20 chain discount for 5000 at 22 and 10 Share Share Share
188944	https://artofproblemsolving.com/wiki/index.php/Category:Inequalities?srsltid=AfmBOoq1h9I5ospfgILNv2vCLSMkIgYjqsY4ZMMn6Wth8D-hV-dWGoIR	Art of Problem Solving Category:Inequalities - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Category:Inequalities Page CategoryDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Category:Inequalities This is a list of inequalities. Subcategories This category has the following 2 subcategories, out of 2 total. G Geometric Inequalities O Olympiad Inequality Problems Pages in category "Inequalities" The following 26 pages are in this category, out of 26 total. A Aczel's Inequality AM-GM Inequality C Carleman's Inequality Cauchy-Schwarz Inequality Chebyshev's Inequality E Equality condition H Homogeneous Homogenization I Inequality Isoperimetric Inequalities J Jensen's Inequality K Karamata's Inequality M Maclaurin's Inequality Minkowski Inequality Muirhead's Inequality N Nesbitt's Inequality Newton's Inequality P Power Mean Inequality Proofs of AM-GM Ptolemy's Inequality Pythagorean inequality R Rearrangement Inequality Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality S Schur's Inequality T Trivial Inequality V Vornicu-Schur Inequality Retrieved from " Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
188945	https://www.gauthmath.com/solution/1985625404162692/n2-5n-36-0	Solved: n^2+5n-36=0 [Math] Drag Image or Click Here to upload Command+to paste Upgrade Sign in Homework Homework Assignment Solver Assignment Calculator Calculator Resources Resources Blog Blog App App Gauth Unlimited answers Gauth AI Pro Start Free Trial Homework Helper Study Resources Math Equation Questions Question n^2+5n-36=0 Gauth AI Solution 100%(1 rated) Answer The answer is -9, 4 Explanation Factor the quadratic equation We need to find two numbers that multiply to -36 and add to 5. These numbers are 9 and -4. Therefore, we can factor the quadratic equation as follows: $$n^{2} + 5n - 36 = (n + 9)(n - 4) = 0$$n 2+5 n−36=(n+9)(n−4)=0 Solve for n To find the values of $$n$$n that satisfy the equation, we set each factor equal to zero: $$n + 9 = 0$$n+9=0 or $$n - 4 = 0$$n−4=0 Find the first solution Solving $$n + 9 = 0$$n+9=0 for $$n$$n, we get: $$n = -9$$n=−9 Find the second solution Solving $$n - 4 = 0$$n−4=0 for $$n$$n, we get: $$n = 4$$n=4 Helpful Not Helpful Explain Simplify this solution Gauth AI Pro Back-to-School 3 Day Free Trial Limited offer! Enjoy unlimited answers for free. Join Gauth PLUS for $0 Previous questionNext question Related n2+5n-36=0 n n =0 100% (7 rated) For Problems 31-50, solve each equation. 31. x2+10x+21=0 32. x2+9x+20=0 33. x2-9x+18=0 34. x2-9x+8=0 35. x2-3x-10=0 36. x2-x-12=0 37. n2+5n-36=0 38. n2+3n-18=0 39. n2-6n-40=0 40. n2-8n-48=0 100% (2 rated) x2-18x+72 16. x2-14x+32 17. x2+5x-66 18. x2+11x-42 19. y2-y-72 20. y2-y-30 21. x2+21x+80 22. x2+21x+90 23. x2+6x-72 24. x2-8x-36 25. x2-10x-48 26. x2-12x-64 27. x2+3xy-10y2 28. x2-4xy-12y2 29. a2-4ab-32b2 30. a2+3ab-54b2 For Problems 31-50, solve each equation. 31. x2+10x+21=0 32. x2+9x+20=0 33. x2-9x+18=0 34. x2-9x+8=0 35. x2-3x-10=0 36. x2-x-12=0 37. n2+5n-36=0 38. n2+3n-18=0 39. n2-6n-40=0 40. n2-8n-48=0 100% (1 rated) In a right triangle, if one acute angle is 45 ° , what is the measure of the other acute angle? 60 ° 90 ° 30 ° 45 ° 100% (1 rated) Part III: Substitute your results from Part II into the first equation. Solve to find the corresponding values of x. Show your work. 2 points Part IV: Write your solutions from Parts II and III as ordered pairs. 2 points __ and _ ' _ 100% (2 rated) Multiply and simplify the following. 3-i-4-9i -21-24i -21-23i -21+23i ⑤ 21-23i 100% (5 rated) The product of eight and seven when multiplied by F is less than the product of four and seven plus ten. a. 8+7F<4+7+10 b. 87F>47+10 C. 87F ≤ 47+10 d. 87F<47+10 100% (5 rated) How may different arrangements are there of the letters in The number of possible arrangements is MISSISSIPPI？ 100% (2 rated) Which of the following lists only contains shapes that fall under the category of parallelogram? A square, rectangle, triangle B trapezoid, square, rectangle C quadrilateral, square, rectangle D rhombus, rectangle, square 100% (3 rated) Write the quotient in the form a+bi. 7-i/3-6i 7-i/3-6i =square Simplify your answer. Type your answer in the form a+bi . Use integers or fractions for any numbers in the expressio 100% (4 rated) Gauth it, Ace it! contact@gauthmath.com Company About UsExpertsWriting Examples Legal Honor CodePrivacy PolicyTerms of Service Download App
188946	https://codes.findlaw.com/ny/real-property-law/rpp-sect-443/	New York Consolidated Laws, Real Property Law - RPP § 443. Disclosure regarding real estate agency relationship; form Current as of January 01, 2024 \| Updated by FindLaw Staff Definitions. As used in this section, the following terms shall have the following meanings: a. “Agent” means a person who is licensed as a real estate broker, associate real estate broker or real estate salesperson under section four hundred forty-a of this article and is acting in a fiduciary capacity. b. “Buyer” means a transferee in a residential real property transaction and includes a person who executes an offer to purchase residential real property from a seller through an agent, or who has engaged the services of an agent with the object of entering into a residential real property transaction as a transferee. c. “Buyer's agent” means an agent who contracts to locate residential real property for a buyer or who finds a buyer for a property and presents an offer to purchase to the seller or seller's agent and negotiates on behalf of the buyer. d. “Listing agent” means a person who has entered into a listing agreement to act as an agent of the seller or landlord for compensation. e. “Listing agreement” means a contract between an owner or owners of residential real property and an agent, by which the agent has been authorized to sell or lease the residential real property or to find or obtain a buyer or lessee therefor. f. “Residential real property” means real property used or occupied, or intended to be used or occupied, wholly or partly, as the home or residence of one or more persons improved by (i) a one-to-four family dwelling or (ii) condominium or cooperative apartments but shall not refer to unimproved real property upon which such dwellings are to be constructed. g. “Seller” means the transferor in a residential real property transaction, and includes an owner who lists residential real property for sale with an agent, whether or not a transfer results, or who receives an offer to purchase residential real property. h. “Seller's agent” means a listing agent who acts alone, or an agent who acts in cooperation with a listing agent, acts as a seller's subagent or acts as a broker's agent to find or obtain a buyer for residential real property. i. “Dual agent” means an agent who is acting as a buyer's agent and a seller's agent or a tenant's agent and a landlord's agent in the same transaction. j. “Designated sales agent” means a licensed real estate salesperson or associate broker, working under the supervision of a real estate broker, who has been assigned to represent a client when a different client is also represented by such real estate broker in the same transaction. k. “Broker's agent” means an agent that cooperates or is engaged by a listing agent, buyer's agent or tenant's agent (but does not work for the same firm as the listing agent, buyer's agent or tenant's agent) to assist the listing agent, buyer's agent or tenant's agent in locating a property to sell, buy or lease respectively, for the listing agent's seller or landlord, the buyer agent's buyer or the tenant's agent tenant. The broker's agent does not have a direct relationship with the seller, buyer, landlord or tenant and the seller, buyer, landlord or tenant can not provide instructions or direction directly to the broker's agent. Therefore, the seller, buyer, landlord or tenant do not have vicarious liability for the acts of the broker's agent. The listing agent, buyer's agent or tenant's agent do provide direction and instruction to the broker's agent and therefore the listing agent, buyer's agent or tenant's agent will have liability for the broker's agent. l. “Tenant” means a lessee in a residential real property transaction and includes a person who executes an offer to lease residential real property from a landlord through an agent, or who has engaged the services of an agent with the object of entering into a residential real property transaction as a lessee. m. “Landlord” means the lessor in a residential real property transaction, and includes an owner who lists residential real property for lease with an agent, whether or not a lease results, or who receives an offer to lease residential real property. n. “Tenant's agent” means an agent who contracts to locate residential real property for a tenant or who finds a tenant for a property and presents an offer to lease to the landlord or landlord's agent and negotiates on behalf of the tenant. o. “Landlord's agent” means a listing agent who acts alone, or an agent who acts in cooperation with a listing agent, acts as a landlord's subagent or acts as a broker's agent to find or obtain a tenant for residential real property. p. “Advance consent to dual agency” means written informed consent signed by the seller/landlord or buyer/tenant that the listing agent and/or buyer's agent may act as a dual agent for that seller/landlord and a buyer/tenant for residential real property which is the subject of a listing agreement. q. “Advance consent to dual agency with designated sales agents” means written informed consent signed by the seller/landlord or buyer/tenant that indicates the name of the agent appointed to represent the seller/landlord or buyer/tenant as a designated sales agent for residential real property which is the subject of a listing agreement. This section shall apply only to transactions involving residential real property. a. A listing agent shall provide the disclosure form set forth in subdivision four of this section to a seller or landlord prior to entering into a listing agreement with the seller or landlord and shall obtain a signed acknowledgment from the seller or landlord, except as provided in paragraph e of this subdivision. b. A seller's agent or landlord's agent shall provide the disclosure form set forth in subdivision four of this section to a buyer, buyer's agent, tenant or tenant's agent at the time of the first substantive contact with the buyer or tenant and shall obtain a signed acknowledgement from the buyer or tenant, except as provided in paragraph e of this subdivision. c. A buyer's agent or tenant's agent shall provide the disclosure form to the buyer or tenant prior to entering into an agreement to act as the buyer's agent or tenant's agent and shall obtain a signed acknowledgment from the buyer or tenant, except as provided in paragraph e of this subdivision. A buyer's agent or tenant's agent shall provide the form to the seller, seller's agent, landlord or landlord's agent at the time of the first substantive contact with the seller or landlord and shall obtain a signed acknowledgment from the seller, landlord or the listing agent, except as provided in paragraph e of this subdivision. d. The agent shall provide to the buyer, seller, tenant or landlord a copy of the signed acknowledgment and shall maintain a copy of the signed acknowledgment for not less than three years. e. If the seller, buyer, landlord or tenant refuses to sign an acknowledgment of receipt pursuant to this subdivision, the agent shall set forth under oath or affirmation a written declaration of the facts of the refusal and shall maintain a copy of the declaration for not less than three years. f. A seller/landlord or buyer/tenant may provide advance informed consent to dual agency and dual agency with designated sales agents by indicating the same on the form set forth in subdivision four of this section. a. For buyer-seller transactions, the following shall be the disclosure form: NEW YORK STATE DISCLOSURE FORM FOR BUYER AND SELLER THIS IS NOT A CONTRACT New York state law requires real estate licensees who are acting as agents of buyers or sellers of property to advise the potential buyers or sellers with whom they work of the nature of their agency relationship and the rights and obligations it creates. This disclosure will help you to make informed choices about your relationship with the real estate broker and its sales agents. Throughout the transaction you may receive more than one disclosure form. The law may require each agent assisting in the transaction to present you with this disclosure form. A real estate agent is a person qualified to advise about real estate. If you need legal, tax or other advice, consult with a professional in that field. DISCLOSURE REGARDING REAL ESTATE AGENCY RELATIONSHIPS SELLER'S AGENT A seller's agent is an agent who is engaged by a seller to represent the seller's interests. The seller's agent does this by securing a buyer for the seller's home at a price and on terms acceptable to the seller. A seller's agent has, without limitation, the following fiduciary duties to the seller: reasonable care, undivided loyalty, confidentiality, full disclosure, obedience and duty to account. A seller's agent does not represent the interests of the buyer. The obligations of a seller's agent are also subject to any specific provisions set forth in an agreement between the agent and the seller. In dealings with the buyer, a seller's agent should (a) exercise reasonable skill and care in performance of the agent's duties; (b) deal honestly, fairly and in good faith; and (c) disclose all facts known to the agent materially affecting the value or desirability of property, except as otherwise provided by law. BUYER'S AGENT A buyer's agent is an agent who is engaged by a buyer to represent the buyer's interests. The buyer's agent does this by negotiating the purchase of a home at a price and on terms acceptable to the buyer. A buyer's agent has, without limitation, the following fiduciary duties to the buyer: reasonable care, undivided loyalty, confidentiality, full disclosure, obedience and duty to account. A buyer's agent does not represent the interests of the seller. The obligations of a buyer's agent are also subject to any specific provisions set forth in an agreement between the agent and the buyer. In dealings with the seller, a buyer's agent should (a) exercise reasonable skill and care in performance of the agent's duties; (b) deal honestly, fairly and in good faith; and (c) disclose all facts known to the agent materially affecting the buyer's ability and/or willingness to perform a contract to acquire seller's property that are not inconsistent with the agent's fiduciary duties to the buyer. BROKER'S AGENTS A broker's agent is an agent that cooperates or is engaged by a listing agent or a buyer's agent (but does not work for the same firm as the listing agent or buyer's agent) to assist the listing agent or buyer's agent in locating a property to sell or buy, respectively, for the listing agent's seller or the buyer agent's buyer. The broker's agent does not have a direct relationship with the buyer or seller and the buyer or seller can not provide instructions or direction directly to the broker's agent. The buyer and the seller therefore do not have vicarious liability for the acts of the broker's agent. The listing agent or buyer's agent do provide direction and instruction to the broker's agent and therefore the listing agent or buyer's agent will have liability for the acts of the broker's agent. DUAL AGENT A real estate broker may represent both the buyer and the seller if both the buyer and seller give their informed consent in writing. In such a dual agency situation, the agent will not be able to provide the full range of fiduciary duties to the buyer and seller. The obligations of an agent are also subject to any specific provisions set forth in an agreement between the agent, and the buyer and seller. An agent acting as a dual agent must explain carefully to both the buyer and seller that the agent is acting for the other party as well. The agent should also explain the possible effects of dual representation, including that by consenting to the dual agency relationship the buyer and seller are giving up their right to undivided loyalty. A buyer or seller should carefully consider the possible consequences of a dual agency relationship before agreeing to such representation. A seller or buyer may provide advance informed consent to dual agency by indicating the same on this form. DUAL AGENT WITH DESIGNATED SALES AGENTS If the buyer and the seller provide their informed consent in writing, the principals and the real estate broker who represents both parties as a dual agent may designate a sales agent to represent the buyer and another sales agent to represent the seller to negotiate the purchase and sale of real estate. A sales agent works under the supervision of the real estate broker. With the informed consent of the buyer and the seller in writing, the designated sales agent for the buyer will function as the buyer's agent representing the interests of and advocating on behalf of the buyer and the designated sales agent for the seller will function as the seller's agent representing the interests of and advocating on behalf of the seller in the negotiations between the buyer and seller. A designated sales agent cannot provide the full range of fiduciary duties to the buyer or seller. The designated sales agent must explain that like the dual agent under whose supervision they function, they cannot provide undivided loyalty. A buyer or seller should carefully consider the possible consequences of a dual agency relationship with designated sales agents before agreeing to such representation. A seller or buyer may provide advance informed consent to dual agency with designated sales agents by indicating the same on this form. This form was provided to me by ․․․․․․․․․․․․․․․․․․․․ (print name of licensee) of ․․․․․․․․․․․․․․․․․․․․ (print name of company, firm or brokerage), a licensed real estate broker acting in the interest of the: \| \| \| \| --- \| \| ( ) Seller as a \| ( ) Buyer as a \| \| \| (check relationship below) \| (check relationship below) \| \| \| ( ) Seller's agent \| ( ) Buyer's agent \| \| \| ( ) Broker's agent \| ( ) Broker's agent \| \| \| ( ) Dual agent \| ( ) Dual agent with designated sales agents \| For advance informed consent to either dual agency or dual agency with designated sales agents complete section below: ( ) Advance informed consent dual agency. ( ) Advance informed consent to dual agency with designated sales agents. If dual agent with designated sales agents is indicated above: ․․․․․․․․․․․․․․․․․․․․is appointed to represent the buyer; and ․․․․․․․․․․․․․․․․․․․․is appointed to represent the seller in this transaction. (I) (We) acknowledge receipt of a copy of this disclosure form: Signature of { } Buyer(s) and/or { } Seller(s): \| \| \| \| --- \| \| ____________________ \| ____________________ \| \| \| ____________________ \| ____________________ \| \| \| Date:_______________ \| Date:_______________ \| b. For landlord-tenant transactions, the following shall be the disclosure form: NEW YORK STATE DISCLOSURE FORM FOR LANDLORD AND TENANT THIS IS NOT A CONTRACT New York state law requires real estate licensees who are acting as agents of landlords and tenants of real property to advise the potential landlords and tenants with whom they work of the nature of their agency relationship and the rights and obligations it creates. This disclosure will help you to make informed choices about your relationship with the real estate broker and its sales agents. Throughout the transaction you may receive more than one disclosure form. The law may require each agent assisting in the transaction to present you with this disclosure form. A real estate agent is a person qualified to advise about real estate. If you need legal, tax or other advice, consult with a professional in that field. DISCLOSURE REGARDING REAL ESTATE AGENCY RELATIONSHIPS LANDLORD'S AGENT A landlord's agent is an agent who is engaged by a landlord to represent the landlord's interest. The landlord's agent does this by securing a tenant for the landlord's apartment or house at a rent and on terms acceptable to the landlord. A landlord's agent has, without limitation, the following fiduciary duties to the landlord: reasonable care, undivided loyalty, confidentiality, full disclosure, obedience and duty to account. A landlord's agent does not represent the interests of the tenant. The obligations of a landlord's agent are also subject to any specific provisions set forth in an agreement between the agent and the landlord. In dealings with the tenant, a landlord's agent should (a) exercise reasonable skill and care in performance of the agent's duties; (b) deal honestly, fairly and in good faith; and (c) disclose all facts known to the agent materially affecting the value or desirability of property, except as otherwise provided by law. TENANT'S AGENT A tenant's agent is an agent who is engaged by a tenant to represent the tenant's interest. The tenant's agent does this by negotiating the rental or lease of an apartment or house at a rent and on terms acceptable to the tenant. A tenant's agent has, without limitation, the following fiduciary duties to the tenant: reasonable care, undivided loyalty, confidentiality, full disclosure, obedience and duty to account. A tenant's agent does not represent the interest of the landlord. The obligations of a tenant's agent are also subject to any specific provisions set forth in an agreement between the agent and the tenant. In dealings with the landlord, a tenant's agent should (a) exercise reasonable skill and care in performance of the agent's duties; (b) deal honestly, fairly and in good faith; and (c) disclose all facts known to the tenant's ability and/or willingness to perform a contract to rent or lease landlord's property that are not inconsistent with the agent's fiduciary duties to the buyer. BROKER'S AGENTS A broker's agent is an agent that cooperates or is engaged by a listing agent or a tenant's agent (but does not work for the same firm as the listing agent or tenant's agent) to assist the listing agent or tenant's agent in locating a property to rent or lease for the listing agent's landlord or the tenant agent's tenant. The broker's agent does not have a direct relationship with the tenant or landlord and the tenant or landlord can not provide instructions or direction directly to the broker's agent. The tenant and the landlord therefore do not have vicarious liability for the acts of the broker's agent. The listing agent or tenant's agent do provide direction and instruction to the broker's agent and therefore the listing agent or tenant's agent will have liability for the acts of the broker's agent. DUAL AGENT A real estate broker may represent both the tenant and the landlord if both the tenant and landlord give their informed consent in writing. In such a dual agency situation, the agent will not be able to provide the full range of fiduciary duties to the landlord and the tenant. The obligations of an agent are also subject to any specific provisions set forth in an agreement between the agent, and the tenant and landlord. An agent acting as a dual agent must explain carefully to both the landlord and tenant that the agent is acting for the other party as well. The agent should also explain the possible effects of dual representation, including that by consenting to the dual agency relationship the landlord and tenant are giving up their right to undivided loyalty. A landlord and tenant should carefully consider the possible consequences of a dual agency relationship before agreeing to such representation. A landlord or tenant may provide advance informed consent to dual agency by indicating the same on this form. DUAL AGENT WITH DESIGNATED SALES AGENTS If the tenant and the landlord provide their informed consent in writing, the principals and the real estate broker who represents both parties as a dual agent may designate a sales agent to represent the tenant and another sales agent to represent the landlord. A sales agent works under the supervision of the real estate broker. With the informed consent in writing of the tenant and the landlord, the designated sales agent for the tenant will function as the tenant's agent representing the interests of and advocating on behalf of the tenant and the designated sales agent for the landlord will function as the landlord's agent representing the interests of and advocating on behalf of the landlord in the negotiations between the tenant and the landlord. A designated sales agent cannot provide the full range of fiduciary duties to the landlord or tenant. The designated sales agent must explain that like the dual agent under whose supervision they function, they cannot provide undivided loyalty. A landlord or tenant should carefully consider the possible consequences of a dual agency relationship with designated sales agents before agreeing to such representation. A landlord or tenant may provide advance informed consent to dual agency with designated sales agents by indicating the same on this form. This form was provided to me by ․․․․․․․․․․․․․․․․․․․․ (print name of licensee) of ․․․․․․․․․․․․․․․․․․․․ (print name of company, firm or brokerage), a licensed real estate broker acting in the interest of the: \| \| \| \| --- \| \| ( ) Landlord as a \| ( ) Tenant as a \| \| \| (check relationship below) \| (check relationship below) \| \| \| ( ) Landlord's agent \| ( ) Tenant's agent \| \| \| ( ) Broker's agent \| ( ) Broker's agent \| \| \| ( ) Dual agent \| ( ) Dual agent with designated sales agents \| For advance informed consent to either dual agency or dual agency with designated sales agents complete section below: ( ) Advance informed consent dual agency. ( ) Advance informed consent to dual agency with designated sales agents. If dual agent with designated sales agents is indicated above: ․․․․․․․․․․․․․․․․․․․․is appointed to represent the tenant; and ․․․․․․․․․․․․․․․․․․․․is appointed to represent the landlord in this transaction. (I) (We) ․․․․․․․․․․․․․․․․․․․․․․․․․․․ acknowledge receipt of a copy of this disclosure form: Signature of { } Landlord(s) and/or { } Tenant(s): \| \| \| ___________________________________________________________________________________________________ \| \| ___________________________________________________________________________________________________ \| \| Date: ____________________ \| Date: ____________________ \| This section shall not apply to a real estate licensee who works with a buyer, seller, tenant or landlord in accordance with terms agreed to by the licensee and buyer, seller, tenant or landlord and in a capacity other than as an agent, as such term is defined in paragraph a of subdivision one of this section. Nothing in this section shall be construed to limit or alter the application of the common law of agency with respect to residential real estate transactions. Back to chapter list Previous part of code Next part of code Cite this article: FindLaw.com - New York Consolidated Laws, Real Property Law - RPP § 443. Disclosure regarding real estate agency relationship; form - last updated January 01, 2024 \| Read this complete New York Consolidated Laws, Real Property Law - RPP § 443. Disclosure regarding real estate agency relationship; form on Westlaw. Westlaw subscription required. FindLaw Codes may not reflect the most recent version of the law in your jurisdiction. Please verify the status of the code you are researching with the state legislature or via Westlaw before relying on it for your legal needs. Was this helpful? Search all New York Codes Welcome to FindLaw’s Cases & Codes A free source of state and federal court opinions, state laws, and the United States Code. For more information about the legal concepts addressed by these cases and statutes, visit FindLaw’s Learn About the Law. 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188947	https://arxiv.org/pdf/2503.02663	arXiv:2503.02663v1 [math.CO] 4 Mar 2025 Equivalence Classes Induced by Binary Tree Isomorphism – Generating Functions David Serena a, William J Buchanan b aCTO, CogNueva, Inc., California, USA info@cognueva.com bBlockpass ID Lab, Edinburgh Napier University b.buchanan@napier.ac.uk Abstract Working with generating functions, the combinatorics of a recurrence rela-tion can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan Numbers for an ordered binary tree. Binary tree isomorphism is an important problem in computer science. The enumeration of the number of non-isomorphic rooted binary trees is there-fore well known. The paper reiterates the known results for ordered binary trees and presents previous results for enumeration of non-isomorphic rooted binary trees. Then new enumeration results are put forward for the two-color binary tree isomorphism parametrized by the number of nodes, the number of specific color and the number of non-isomorphic sibling subtrees. Multi-variate generating function equations are presented that enumerate these tree structures. The generating functions with these parameterizations separate multiplicatively into simplified generating function equations. Keywords: binary tree isomorphism, binary tree, graph isomorphism, combinatorics Introduction In Sections 2 and 3 the basic known results are reiterated with derivation. Then in Sections 4, 5 and 6 new enumeration results are put forward for two-color binary tree isomorphism parametrized by number of nodes , number of specific color and number of non-isomorphic sibling subtrees. Multi-variate generating function equations are presented that enumerate these tree struc-tures. The generating functions with these parameterizations separate mul-tiplicatively into simplified generating function equations. Basic Ordered Binary Trees Starting with the most basic result. When binary trees are enumerated with regard to the ordering of the subtrees, the standard enumeration of ordered binary trees 1 is as follows: Such trees are given by the recurrence when n ≥ 1. Cn = n−1 ∑ k=0 Cn−1−kCk The base case clearly shows that C0 = 1 . When the following generating function R(z) is defined as F (z) = ∞ ∑ n=0 Cnzn Substituting the recurrence yields the following equation. zF (z)2 = F (z) − 1 (1) Since the equation is a quadratic it yields a closed form solution for R(z). F (z) = 1 − √1 − 4z 2z Which allows for a direct closed form expression of the coefficient in terms of the binomial function. These are just the Catalan Numbers. Cn = 1 n + 1 ( 2nn ) 1Ordered trees are where the left and right position of the nodes matters. Non-Isomorphic Binary Trees With Labeled Root Node The following is a rederivation of the recurrence which represents the number of non-isomorphic binary trees with labeled root nodes and n nodes in the tree. [8, 4, 3] An alternate way of looking at the isomorphism problem with binary trees is the concept of “flip-equivalence” . Can we map one binary tree with a designated root to another by flipping the children of each node? This is equivalent to the isomorphism of the trees where the root node is labeled. Building the recurrence, the base cases are trivially. B0 = B1 = 1 The recursive cases are distinct for even and odd n respectively ( n ≥ 1), B2n = 1 2 2n−1 ∑ k=0 B2n−k−1Bk B2n+1 = 1 2 ([ 2n∑ k=0 B2n−kBk ] Bn ) The first equation is derived by observing that when there are an even number of nodes in a tree, the subtrees, possibly empty, formed by the chil-dren of the root may never be isomorphic by a simple counting argument. As order does not matter with regards to isomorphism, the factor of 1 2 appears outside the summation. The second equation is more complex to interpret. If the number of nodes in a tree is odd there is a possibility that the children of the root contain the same number of nodes. Outside of the given base cases, there are two possibilities. Either the siblings are the roots of isomorphic subtrees or they are not. The former case is handled by the first term and the latter is handled by the second term. As any tree with more than two nodes has at least two non-isomorphic manifestations, all the cases are covered. Figure 1 shows the enumeration of these trees. 3Root Node= n= 0 n= 2 n= 5 n= 1 n= 3 n= 4 Figure 1: Rooted Non-Isomorphic Binary Trees 3.1. Generating Functions G(s) = ∞∑ k=0 Bksk H(s) = ∞∑ k=0 B2ksk I(s) = ∞∑ k=0 B2k+1 sk 43.2. Equation in G(s) G(s) = 1 + s + ∞ ∑ n=2 Bnsn = 1 + s + ∞ ∑ n=1 (B2ns2n + B2n+1 s2n+1 )= 1 + s + 1 2 ∞ ∑ n=1 (2n−1∑ ℓ=0 B2n−ℓ−1Bℓs2n + [ 2n∑ k=0 B2n−kBks2n+1 ] Bns2n+1 ) = 1 + s 2 + 1 2 [ ∞∑ n=1 [n−1∑ ℓ=0 (χ(2 \|n)Bn−ℓ−1Bℓ + χ(2 ∤ n)Bn−ℓ−1Bℓ) ] sn + Bns2n+1 ] Where χ(p) = 1 if p is true or χ(p) = 0 if p is false. k\|n is true if k evenly divides n and false otherwise. k ∤ n is the negation k\|n. G(s) = 1 + s 2 + 1 2 ∞ ∑ n=1 n−1 ∑ k=0 Bn−k−1Bksn + 1 2 ∞ ∑ n=1 Bns2n+1 Changing the indices = 1 + 1 2 ∞ ∑ n=0 ∞ ∑ k=0 BnBksn+k+1 + s 2 ∞ ∑ n=0 Bns2n Since ∑∞ n=0 ∑∞ k=0 BnBksn+k+1 = s (∑∞ n=0 Bnsn) (∑ ∞ k=0 Bksk) = s(G(s)) 2 then the generating function is given by G(s) = 1 + s 2 [ G(s)2 + G(s2) ] G(s) = 1 + s + s2 + 2 s3 + 3 s4 + 6 s5 + 11 s6 + 23 s7 + 46 s8 + 98 s9 + 207 s10 +451 s11 + 983 s12 + 2179 s13 + 4850 s14 + 10905 s15 + · · · G(s)2 = 1 + 2 s + 3 s2 + 6 s3 + 11 s4 + 22 s5 + 44 s6 + 92 s7 + 193 s8 + 414 s9 +896 s10 + 1966 s11 + 4347 s12 + 9700 s13 + 21787 s14 + 49262 s15 + · · · 5G(s) = H(s2) + sI (s2) Similarly, equations for H(s) and I(s) can be developed. H(s) = 1 + 1 2 ∞ ∑ n=1 2n−1 ∑ k=0 B2n−k−1Bksn I(s) = 1 + 1 2 ∞ ∑ n=1 2n ∑ k=1 B2n−kBksn + 1 2 ∞ ∑ n=1 Bnsn H(s) = 1 + 1 2 ∞ ∑ n=1 (n−1∑ k=0 B2n−1−2kB2ksn + B2( n−k−1) B2k+1 ) sn = 1 + s 2 ∞ ∑ n=0 ∞ ∑ k=0 (B2n+1 B2k + B2nB2k+1 ) sn+k = 1 + s 2 [( ∞∑ n=0 B2n+1 sn ) ( ∞∑ k=0 B2ksk ) + ( ∞∑ n=0 B2nsn ) ( ∞∑ k=0 B2k+1 sk )] = 1 + sI (s)H(s) 6I(s) = ∞ ∑ n=0 B2n+1 sn = 1 + ∞ ∑ n=1 [ 1 2 [ 2n∑ k=0 B2n−kBk ] 1 2 Bn ] sn = 1 + 1 2 ∞ ∑ n=1 n−1 ∑ k=0 [ B2n−2k−1B2k+1 + B2n−2kB2k ] sn + 1 2 ∞ ∑ n=1 (B0B2n + Bn) sn = 1 + s 2 ∞ ∑ n=0 ∞ ∑ k=0 [B2n+1 B2k+1 + B2n+2 B2k] sn+k + 1 2 ∞ ∑ n=1 (B2n + Bn)sn = 1 + s 2 ( ∞ ∑ n=0 B2n+1 sn)( ∞ ∑ k=0 B2k+1 sk) + 1 2 ( ∞ ∑ n=0 B2n+2 sn+1 )( ∞ ∑ k=0 B2ksk)+ 1 2 (H(s) + G(s) − 2) = G(s) + sI (s)2 + H(s)2 2 3.3. Summary of Generating Function Formulas H(s) = 1 + sI (s)H(s) I(s) = 1 2 [ G(s) + sI (s)2 + H(s)2 ] G(s) = H(s2) + sI (s2) The known [8, 4] generating function is: G(s) = 1 + s 2 [ G(s)2 + G(s2) ] (2) Number of Non-Isomorphic Rooted Binary Trees with Number of Non-Isomorphic Siblings Given As discussed before, the functional equation 2 is a generating function which counts the number of non-isomorphic trees with labeled roots with n nodes. 7Graph isomorphism is an equivalence relation on graphs and as such it partitions the class of all graphs into equivalence classes. Therefore, an equiv-alence class is defined by the tree isomorphism with labeled roots: binary tree A with labeled root a is isomorphic to binary tree B with labeled root node b.This is clearly symmetric, reflexive and transitive and is therefore an equiv-alence class. The functional equation 2 above defines the number of binary trees in that equivalence class based on the coefficient of s to the number of nodes n.It is interesting to develop a function which designates the cardinality of these equivalence classes in terms of the number of ordered trees therein and the multiplicity of equivalence classes with the same cardinality. In order to count these objects, observe the cardinality of these equivalence classes is a perfect power of two in all cases. This follows from simply counting the number of siblings, which are the roots of non-isomorphic subtrees with labeled roots. Each instance of such a case implies two times the number of ordered trees which are isomorphic to it. This parameter forms a variable which facilitates the development of a recurrence for these objects. Note that a complete tree has no siblings, which are non-isomorphic in this sense. At the other extreme, a tree of maximum depth forms the maximum of this parameter. Namely, siblings are all non-isomorphic due to a counting argument. With significant attention to all the sub cases, one may write the following recurrence. Kn,ℓ is the number of equivalence classes of cardinality 2ℓ. Note that each member of the equivalence class consists of binary trees with n nodes and ℓ non-isomorphic sibling subtrees with labeled roots. K2n,ℓ = 1 2 2n−1 ∑ k=0 ℓ−1 ∑ v=0 Kk,v K2n−1−k,ℓ −1−v K2n+1 ,2ℓ = Kn,ℓ + 1 2 2n ∑ k=0 2ℓ−1 ∑ v=0 Kk,v K2n−k, 2ℓ−1−v K2n+1 ,2ℓ+1 = 1 2 ( Kn,ℓ (Kn,ℓ − 1) − K2 n,ℓ + 2n ∑ k=0 2ℓ ∑ v=0 Kk,v K2n−k, 2ℓ−v ) = 1 2 ( −Kn,ℓ + 2n ∑ k=0 2ℓ ∑ v=0 Kk,v K2n−k, 2ℓ−v ) 8nℓ 0 1 2 3 4 5 6 7 8 9 10 11 0 1 1 1 0 2 0 1 0 3 1 0 1 0 4 0 1 1 1 0 5 0 1 2 2 1 0 6 0 0 3 3 4 1 0 7 1 0 1 7 7 6 1 0 8 0 1 1 6 14 14 9 1 0 9 0 1 3 4 21 28 28 12 1 0 10 0 0 3 8 17 54 58 50 16 1 0 11 0 1 2 9 27 61 126 119 85 20 1 0 Table 1: First few values of Kn,ℓ The following base cases take precedence over the recurrence relations. Kn, 0 = { 1, n = 2 ℓ − 1, ℓ ∈ Z+,0 0, otherwise Ks,t = 0, t ≥ s > 0 Ks,s −1 = 1, s > 1 Assume if a and b are outside of the region of definition Ka,b = 0 . Table 1 indicates the first few values for Kn,ℓ . Figure 2 illustrates the unique tree structure for the first few values. Note that the following equations hold. This seems to indicate that a relationship between the bivariate generating function for Kn,ℓ and the afore-mentioned generating functions F (z) and G(s) may be developed. Cn = n ∑ k=0 2kKn,k Bn = n ∑ k=0 Kn,k 9(n, ℓ ) = (5 , 4) (n, ℓ ) = (1 , 0) (n, ℓ ) = (2 , 1) (n, ℓ ) = (3 , 2) (n, ℓ ) = (3 , 0) (n, ℓ ) = (4 , 1) (n, ℓ ) = (4 , 2) (n, ℓ ) = (4 , 3) (n, ℓ ) = (5 , 1) (n, ℓ ) = (5 , 2) (n, ℓ ) = (5 , 3) (n, ℓ ) = (0 , 0) Root Node= Figure 2: Rooted Binary Trees, Parameritized with Number of Nodes= n, and Number of Non-Isomorphic Siblings= ℓ That is why the above recurrence is important to solving the functional equation. It forms a connection between the generating function G(s) and R(z).One may modify the recursion by taking account of the base case equa-tions 3, 3 and 3 to obtain a more efficient summation of cases. 10 K2n,ℓ = 1 2 2n−1 ∑ k=0 min( ℓ−1,k ) ∑ v=max(0 ,ℓ +k−2n) Kk,v K2n−1−k,ℓ −1−v K2n+1 ,2ℓ = Kn,ℓ + 1 2 2n ∑ k=0 min(2 ℓ−1,k ) ∑ v=max(0 ,k +2 ℓ−2n−1) Kk,v K2n−k, 2ℓ−1−v K2n+1 ,2ℓ+1 = 1 2 ( −Kn,ℓ + 2n ∑ k=0 min(2 ℓ,k ) ∑ v=max(0 ,k +2 ℓ−2n) Kk,v K2n−k, 2ℓ−v ) 4.1. Bivariate Generating Function for Kn,ℓ Recurrence Define the bivariate generating function and solve the equation for the functional equation for L(x, y ). L(x, y ) = ∞ ∑ n=0 n ∑ ℓ=0 Kn,ℓ xnyℓ Define the following three utility functions to express the system of func-tional equations. The functions mirror the three cases in the recurrence relation for Kn,ℓ . P (x, y ) = ∞ ∑ n=0 2n ∑ ℓ=0 K2n,ℓ xnyℓ Q(x, y ) = ∞ ∑ n=0 n ∑ ℓ=0 K2n+1 ,2ℓ xnyℓ R(x, y ) = ∞ ∑ n=0 n ∑ ℓ=0 K2n+1 ,2ℓ+1 xnyℓ Also define the functions, 2 2 Henceforth, for compactness of notation all summations are assumed to step in in-crements of one. If the upper limit of the summation is less than the lower limit of the summation, the entire term is taken to be zero. 11 P0(x, y ) = ∞ ∑ n=0 n ∑ ℓ=0 K2n, 2ℓ xnyℓ P1(x, y ) = ∞ ∑ n=0 n−1 ∑ ℓ=0 K2n, 2ℓ+1 xnyℓ Note that P (x, y ) = P0(x, y 2) + yP 1(x, y 2) and clearly L(x, y ) = P (x2, y ) + x Q (x2, y 2) + xy R (x2, y 2) Q(x, y ) = L(x, y ) + yP 0(x, y )P1(x, y ) + xyQ (x, y )R(x, y ) (3) and thus Q(x, y ) = L(x, y ) + yP 0(x, y )P1(x, y ) 1 − xyR (x, y ) A similar set of manipulations of the indices yields the functional equa-tion. R(x, y ) = 1 2 ( −L(x, y ) + x(Q(x, y )2 + yR (x, y )2) + P0(x, y )2 + yP 1(x, y )2 ) (4) Solving for R(x, y ) in the quadratic one obtains. R(x, y ) = 1 ± √ 1 − xy (−L(x, y ) + xQ (x, y )2 + P0(x, y )2 + yP 1(x, y )2) xy Similarly, P0(x, y ) = 1 + xy (R(x, y )P0(x, y ) + Q(x, y )P1(x, y )) (5) and P1(x, y ) = xP 0(x, y )Q(x, y ) + xyP 1(x, y )R(x, y ) (6) 12 4.2. Summary of System of Functional Equations L(x, y ) = P (x2, y ) + x Q (x2, y 2) + xy R (x2, y 2) P (x, y ) = P0(x, y 2) + yP 1(x, y 2) P (x, y ) = 1 1 − xy (Q(x, y 2) + yR (x, y 2)) Q(x, y ) = L(x, y ) + yP 0(x, y )P1(x, y ) 1 − xyR (x, y ) R(x, y ) = 1 ± √ 1 − xy (−L(x, y ) + xQ (x, y )2 + P0(x, y )2 + yP 1(x, y )2) xy P0(x, y ) = 1 + xyQ (x, y )P1(x, y ) 1 − xyR (x, y ) P1(x, y ) = xP 0(x, y )Q(x, y ) 1 − xyR (x, y ) Note that the latter two equations yield P0(x, y ) = 1 − xyR (x, y ) (1 − xyR (x, y ))2 − x2Q(x, y )2 P1(x, y ) = xQ (x, y ) (1 − xyR (x, y ))2 − x2Q(x, y )2 Consider the functional equations 5, 6, 3 and 4. Q(x, y ) = L(x, y ) + yP 0(x, y )P1(x, y ) + xyQ (x, y )R(x, y ) R(x, y ) = 1 2 ( −L(x, y ) + x(Q(x, y )2 + yR (x, y )2) + P0(x, y )2 + yP 1(x, y )2 ) P0(x, y ) = 1 + xy (R(x, y )P0(x, y ) + Q(x, y )P1(x, y )) P1(x, y ) = xP 0(x, y )Q(x, y ) + xyP 1(x, y )R(x, y ) 13 L(x, y ) = P0(x2, y 2) + yP 1(x2, y 2) + x Q (x2, y 2) + xy R (x2, y 2) L(x, y )2 = P0(x2, y 2)2 + y2P1(x2, y 2)2 x2 Q(x2, y 2)2 + x2y2 R(x2, y 2)2 2 yP 0(x2, y 2)P1(x2, y 2) + 2 xP 0(x2, y 2)Q(x2, y 2)+ 2 xyP 0(x2, y 2)R(x2, y 2) + 2 xyP 1(x2, y 2)Q(x2, y 2)+ 2 xy 2P1(x2, y 2)R(x2, y 2) + 2 x2y Q (x2, y 2)R(x2, y 2) The system can be reduced to a single functional equation, by matching the squared terms and substituting the cross terms of the squared generating function. L(x, y )2 = 2R(x2, y 2) + L(x2, y 2) + 2 y (Q(x2, y 2) − L(x2, y 2)) 2 xP1(x2, y 2) + 2 xy (P0(x2, y 2) − 1) Which implies xyL (x, y )2 = 2 (P0(x2, y 2) + yP 1(x2, y 2) + xQ (x2, y 2) + xyR (x2, y 2)) xyL (x2, y 2) − 2xL (x2, y 2) − 2= 2L(x, y ) + xyL (x2, y 2) − 2xL (x2, y 2) − 2 xyL (x, y )2 2 + 1 = L(x, y ) + x ( y 2 − 1 ) L(x2, y 2) L(x, y ) = 1 + x y L (x, y )2 2 + x L (x2, y 2) − x y L (x2, y 2) 2 L(x, y ) = 1 + x y L (x, y )2 2 + x ( 1 − y 2 ) L(x2, y 2) (7) One may see that in the case of y = 1 it devolves to the known functional equation2. 14 L(x, 1) = 1 + x 2 (L(x, 1) 2 + L(x2, 1) ) In the case of y = 2 it devolves to the known generating function equation 1. L(x, 2) = 1 + x L (x, 2) 2 where L(x, 2) = 1 ± √1 − 4 x 2 x 2-colored Rooted Non-isomorphic Binary Trees Consider a binary tree which has nodes colored zero or one. Now define isomorphism to be the standard bijection of edges and nodes ; however, also require that the node colors are matched as well. By way of definition, let the following program represent the 2-color isomorphism. It is similarly symmetric, reflexive and transitive and thus an equivalence relation just like graph isomorphism. struct tree { struct tree ∗ lef t ;struct tree ∗ right ;int color ; } tree; int two _color _iso(tree ∗ a, tree ∗ b){ if (( a == 0) ∧ (b == 0)) return 1; if (( a == 0) ∨ (b == 0)) return 0; if (( a → color ) 6 = ( b → color )) return 0; if ( two _color _iso( a → right, b → right ) ∧ two _color _iso( a → lef t, b → lef t )) return 1; if ( two _color _iso( a → right, b → lef t ) ∧ two _color _iso( a → lef t, b → right )) return 1; return 0; } 15 Given these 2-colored nodes in a binary tree structure, let Bn,k be the number of such two colored rooted non-isomorphic binary trees with n nodes and k colored as black or 1. Table 2 and Figure 3 show the first few examples. B0,0 = B1,1 = B1,0 =1 B0,k =0 k > 0 B1,k =0 k > 1 Bn, 0 =Bn n > 1 Bn,k =0 k > n Where Bn is defined as above. B0 = B1 = 1 For n ≥ 1, B2n = 1 2 2n−1 ∑ k=0 B2n−k−1Bk B2n+1 = 1 2 ([ 2n∑ k=0 B2n−kBk ] Bn ) There are now three recursive cases to consider. For n > 0 and k > 0 this case can never have isomorphic subtrees due to a counting argument. B2n,k = 1 2 2n−1 ∑ ℓ=0 k−1 ∑ m=0 Bℓ,m B2n−1−ℓ,k −1−m + 1 2 2n−1 ∑ ℓ=0 k ∑ m=0 Bℓ,m B2n−1−ℓ,k −m The first term corresponds to the case with the root node black; the second case corresponds to a white node as root. Now, similar to the above tree structure, counting an additional Bn,k term shows up. For n ≥ 0 and k > 0 we now have B2n+1 ,2k = 1 2 2n ∑ ℓ=0 2k−1 ∑ m=0 Bℓ,m B2n−ℓ, 2k−1−m 1 2 ([ 2n∑ ℓ=0 2k ∑ m=0 Bℓ,m B2n−ℓ, 2k−m ] Bn,k ) 16 The first term corresponds to the root node as black and the second term corresponds to the root node as white. In the second term, the square bracketed term captures all of the possibilities; however, as before, with an excess of 1 2 Bn,k . Adding an additional 1 2 Bn,k fills in the possibilities where sibling subtrees are isomorphic . For n ≥ 0 and k ≥ 0 we get B2n+1 ,2k+1 = 1 2 ([ 2n∑ ℓ=0 2k ∑ m=0 Bℓ,m B2n−ℓ, 2k−m ] Bn,k ) 1 2 2n ∑ ℓ=0 2k+1 ∑ m=0 Bℓ,m B2n−ℓ, 2k+1 −m nk012345678910 01 111 2121 32552 4311 16 11 3 5626 50 50 26 6 611 60 143 188 143 60 11 723 142 404 656 656 404 142 23 846 334 1105 2143 2652 2143 1105 334 46 998 794 2995 6737 9934 9934 6737 2995 794 98 10 207 1888 7999 20504 35080 41788 35080 20504 7999 1888 207 Table 2: Number of 2-Color Binary Trees, Parametrized by Number of Nodes= n and Number of a Specific Color= k A similar explanation to the previous case is applicable. In the following, let summations always increment by one or not be summed. Removing the need for the condition Bn,k = 0 for k > n we now obtain: 17 (n, k ) = (1 , 1) (n, k ) = (1 , 0) (n, k ) = (0 , 0) (n, k ) = (2 , 2) (n, k ) = (2 , 1) Root Node= (n, k ) = (3 ,0) (n, k ) = (3 ,3) (n, k ) = (3 ,1) (n, k ) = (3 ,1) (n, k ) = (2 ,0) Figure 3: 2-Color Binary Trees, Parametrized by Number of Nodes= n and Number of a Specific Color= k For n > 0 and k > 0 B2n,k = 1 2 2n−1∑ ℓ=0 min( ℓ,k −1) ∑ m=max(0 ,ℓ +k−2n) Bℓ,m B2n−1−ℓ,k −1−m 1 2 2n−1∑ ℓ=0 min( ℓ,k ) ∑ m=max(0 ,ℓ +k+1 −2n) Bℓ,m B2n−1−ℓ,k −m = 1 2 1∑ r=0 2n−1∑ ℓ=0 min( ℓ,k −r) ∑ m=max(0 ,ℓ +k+1 −r−2n) Bℓ,m B2n−1−ℓ,k −r−m 18 Now for n ≥ 0 and k > 0 B2n+1 ,2k = 1 2 2n∑ ℓ=0 min( ℓ, 2k−1) ∑ m=max(0 ,ℓ +2 k−1−2n) Bℓ,m B2n−ℓ, 2k−1−m 1 2  2n∑ ℓ=0 min( ℓ, 2k) ∑ m=max(0 ,ℓ +2 k−2n) Bℓ,m B2n−ℓ, 2k−m  + Bn,k  = 1 2 1∑ r=0 2n∑ ℓ=0 min( ℓ, 2k−r) ∑ m=max(0 ,ℓ +2 k−r−2n) Bℓ,m B2n−ℓ, 2k−r−m + 1 2Bn,k With n ≥ 0 and k ≥ 0 we get B2n+1 ,2k+1 = 1 2  2n∑ ℓ=0 min( ℓ, 2k) ∑ m=max(0 ,ℓ +2 k−2n) Bℓ,m B2n−ℓ, 2k−m  + Bn,k  1 2 2n∑ ℓ=0 min( ℓ, 2k+1) ∑ m=max(0 ,ℓ +2 k+1 −2n) Bℓ,m B2n−ℓ, 2k+1 −m = 1 2 1∑ r=0 2n∑ ℓ=0 min( ℓ, 2k+r) ∑ m=max(0 ,ℓ +2 k+r−2n) Bℓ,m B2n−ℓ, 2k+r−m + 1 2 Bn,k Define the generating function M(x, y ). M(x, y ) = ∞∑ n=0 n∑ k=0 Bn,k xnyk = 1 + ∞∑ n=1 n∑ k=1 Bn,k xnyk + ∞∑ n=1 Bn, 0xn Define the following useful generating functions. 19 P (x, y ) = ∞ ∑ n=0 2n ∑ k=0 B2n,k xnyk Q(x, y ) = ∞ ∑ n=0 n ∑ k=0 B2n+1 ,2kxnyk R(x, y ) = ∞ ∑ n=0 n ∑ k=0 B2n+1 ,2k+1 xnyk Defining the auxiliary functions P0(x, y ) = ∞ ∑ n=0 n ∑ k=0 B2n, 2kxnyk P1(x, y ) = ∞ ∑ n=1 n−1 ∑ k=0 B2n, 2k+1 xnyk P (x, y ) = P0(x, y 2) + yP 1(x, y 2) M(x, y ) = P (x2, y ) + xQ (x2, y 2) + xyR (x2, y 2) R(x, y ) − 1 2 M(x, y ) = 1 2 ( P0(x, y )2 + 2 P0(x, y )P1(x, y )+ yP 1(x, y )2 + xQ (x, y )2 + xyR (x, y )2 2 Q(x, y )R(x, y ) ) So all four generating function equations are: 20 P0(x, y ) = xP 0(x, y )Q(x, y ) + xyP 1(x, y )Q(x, y ) − xP 0(x, 0) Q(x, 0) + xyP 0(x, y )R(x, y ) + xyP 1(x, y )R(x, y ) + P0(x, 0) (8) P1(x, y ) = xP 0(x, y )Q(x, y ) + xP 1(x, y )Q(x, y ) + xP 0(x, y )R(x, y )+ xyP 1(x, y )R(x, y ) (9) Q(x, y ) = yP 0(x, y )P1(x, y ) + xyQ (x, y )R(x, y ) + Q(x, 0) + 1 2 ( M(x, y ) − M(x, 0) + P0(x, y )2 − P0(x, 0) 2 yP 1(x, y )2 + xQ (x, y )2 − xQ (x, 0) 2 + xyR (x, y )2 ) (10) R(x, y ) = P0(x, y )P1(x, y ) + xQ (x, y )R(x, y ) + 1 2 ( P0(x, y )2 yP 1(x, y )2 + xQ (x, y )2 + xyR (x, y )2 + M(x, y ) ) (11) Expressing for Q(x, y ) and R(x, y ) as a quadratic from equations 10 and 11 respectively. 0 = x 2 [Q(x, y )] 2 − (1 − xyR (x, y ))[ Q(x, y )] + ( M(x, y ) − M(x, 0) + P0(x, y )2 − P0(x, 0) 2 + 2 yP 0(x, y )P1(x, y )+ yP 1(x, y )2 + 2 Q(x, 0) − xQ (x, 0) 2 + xyR (x, y )2) (12) 0 = xy 2 [R(x, y )] 2 − (1 − xQ (x, y ))[ R(x, y )] + ( M(x, y ) + P0(x, y )2 + 2 P0(x, y )P1(x, y ) + yP 1(x, y )2 + xQ (x, y )2) (13) Note also that following from the definitions: M(x, y ) = P0(x2, y 2) + yP 1(x2, y 2) + xQ (x2, y 2) + xyR (x2, y 2) (14) For notation purposes only assume that 00 = 1 (this can also be handled with limits) when expressed as Q(x, 0) for example in the following. Solving equations 8 and 9 by first solving for P0(x, y ) and P1(x, y ) respectively then substituting into each equation. 21 P0(x, y ) = P0(x, 0)(1 − xQ (x, 0))(1 − xQ (x, y ) − xyR (x, y )) (1 − xQ (x, y ) − xyR (x, y )) 2 − x2y(Q(x, y ) + R(x, y )) 2 (15) P1(x, y ) = xP 0(x, 0)(1 − xQ (x, 0))( Q(x, y ) + R(x, y )) (1 − xQ (x, y ) − xyR (x, y )) 2 − x2y(Q(x, y ) + R(x, y )) 2 (16) Solving the quadratic in equations 12 and 13. Q(x, y ) = 1 2x [(1 − xyR (x, y )) ± [(1 − xyR (x, y )) 2 − 4x(M(x, y ) − M(x, 0) + P0(x, y )2 − P0(x, 0) 2 + 2 yP 0(x, y )P1(x, y )+ yP 1(x, y )2 + 2 Q(x, 0) − xQ (x, 0) 2 + xyR (x, y )2)] 1 2 ] (17) R(x, y ) = 1 2xy [(1 − xQ (x, y )) ± [(1 − xQ (x, y )) 2 − 4xy (M(x, y ) + P0(x, y )2 2 P0(x, y )P1(x, y ) + yP 1(x, y )2 + xQ (x, y )2)] 1 2 ] (18) Equations 17 and 18 imply that respectively: M(x, y ) = M(x, 0) − P0(x, y )2 + P0(x, 0) 2 − 2yP 0(x, y )P1(x, y ) − yP 1(x, y )2 2 Q(x, y ) − xQ (x, y )2 − 2Q(x, 0) + xQ (x, 0) 2 − 2xyQ (x, y )R(x, y ) − xyR (x, y )2 (19) M(x, y ) = −P0(x, y )2 − 2P0(x, y )P1(x, y ) − yP 1(x, y )2 − xQ (x, y )2 2 R(x, y ) − 2xQ (x, y )R(x, y ) − xyR (x, y )2 (20) Summarizing 22 M(x, y ) = P0(x2, y 2) + yP 1(x2, y 2) + xQ (x2, y 2) + xyR (x2, y 2) (21) M(x, y ) = M(x, 0) − P0(x, y )2 + P0(x, 0) 2 − 2yP 0(x, y )P1(x, y ) − yP 1(x, y )2 + 2 Q(x, y ) − xQ (x, y )2 − 2Q(x, 0) (22) xQ (x, 0) 2 − 2xyQ (x, y )R(x, y ) − xyR (x, y )2 (23) M(x, y ) = −P0(x, y )2 − 2P0(x, y )P1(x, y ) − yP 1(x, y )2 − xQ (x, y )2 2 R(x, y ) − 2xQ (x, y )R(x, y ) − xyR (x, y )2 (24) P0(x, y ) = P0(x, 0)(1 − xQ (x, 0))(1 − xQ (x, y ) − xyR (x, y )) (1 − xQ (x, y ) − xyR (x, y )) 2 − x2y(Q(x, y ) + R(x, y )) 2 (25) P1(x, y ) = xP 0(x, 0)(1 − xQ (x, 0))( Q(x, y ) + R(x, y )) (1 − xQ (x, y ) − xyR (x, y )) 2 − x2y(Q(x, y ) + R(x, y )) 2 (26) Note that Section 3.2 equations are H(s) = 1 + sI (s)H(s) (27) I(s) = 1 2 [ G(s) + sI (s)2 + H(s)2 ] (28) G(s) = H(s2) + sI (s2) (29) G(s) = 1 + s 2 [ G(s)2 + G(s2) ] (30) These equate to H(x) = P0(x, 0) and I(x) = Q(x, 0) . G(x) = M(x, 0) therefore using equation 28 in equation 22 yields M(x, y ) = −P0(x, y )2 − 2yP 0(x, y )P1(x, y ) − yP 1(x, y )2 + 2 Q(x, y ) − xQ (x, y )2 − 2xyQ (x, y )R(x, y ) − xyR (x, y )2 (31) Equation 27 implies that P0(x, 0) − xP 0(x, 0) Q(x, 0) = P0(x, 0) + (1 − P0(x, 0)) = 1 . This and equation 31 simplify the system of equations to 23 M(x, y ) = P0(x2, y 2) + yP 1(x2, y 2) + xQ (x2, y 2) + xyR (x2, y 2) (32) M(x, y ) = 2Q(x, y ) − P0(x, y )2 − 2yP 0(x, y )P1(x, y ) − yP 1(x, y )2 − xQ (x, y )2 − 2xyQ (x, y )R(x, y ) − xyR (x, y )2 (33) M(x, y ) = 2R(x, y ) − P0(x, y )2 − 2P0(x, y )P1(x, y ) − yP 1(x, y )2 − xQ (x, y )2 − 2xQ (x, y )R(x, y ) − xyR (x, y )2 (34) P0(x, y ) = (1 − xQ (x, y ) − xyR (x, y )) (1 − xQ (x, y ) − xyR (x, y )) 2 − x2y(Q(x, y ) + R(x, y )) 2 (35) P1(x, y ) = x(Q(x, y ) + R(x, y )) (1 − xQ (x, y ) − xyR (x, y )) 2 − x2y(Q(x, y ) + R(x, y )) 2 (36) Equating equations 33 and 34 yields Q(x, y ) − R(x, y ) = (y − 1)( P0(x, y )P1(x, y ) + xQ (x, y )R(x, y )) (37) Substituting back in to equations 33 and 34 eliminating the cross terms and obtaining M(x, y ) = 2Q(x, y ) − P0(x, y )2 + 2 y ( Q(x, y ) − R(x, y ) 1 − y ) − yP 1(x, y )2 − xQ (x, y )2 − xyR (x, y )2 (38) M(x, y ) = 2R(x, y ) − P0(x, y )2 + 2 (Q(x, y ) − R(x, y ) 1 − y ) − yP 1(x, y )2 − xQ (x, y )2 − xyR (x, y )2 (39) Simplifying yields one equation for M(x, y ) M(x, y ) = 2 (Q(x, y ) − yR (x, y ) 1 − y ) − P0(x, y )2 − yP 1(x, y )2 − xQ (x, y )2 − xyR (x, y )2 (40) (1 − y)M(x, y ) = 2 ( Q(x, y ) − yR (x, y )) − (1 − y) (P0(x, y )2 + yP 1(x, y )2 + xQ (x, y )2 + xyR (x, y )2) (41) Going back to initial work on P0(x, y ) and P1(x, y ) 24 P0(x, y ) = xP 0(x, y )Q(x, y ) + xyP 1(x, y )Q(x, y )+ xyP 0(x, y )R(x, y ) + xyP 1(x, y )R(x, y ) + 1 (42) P1(x, y ) = xP 0(x, y )Q(x, y ) + xP 1(x, y )Q(x, y )+ xP 0(x, y )R(x, y ) + xyP 1(x, y )R(x, y ) (43) Solving for P0(x, y ) and P1(x, y ) without moving to just dependence on Q(x, y ) and R(x, y ) (that previous calculation gave a hint for subsitutions) P0(x, y ) = 1 + xyP 1(x, y )Q(x, y ) + xyP 1(x, y )R(x, y ) 1 − xQ (x, y ) − xyR (x, y ) (44) P1(x, y ) = xP 0(x, y )Q(x, y ) + xP 0(x, y )R(x, y ) 1 − xQ (x, y ) − xyR (x, y ) (45) Summarizing again M(x, y ) = P0(x2, y 2) + yP 1(x2, y 2) + xQ (x2, y 2) + xyR (x2, y 2) (46) (1 − y)M(x, y ) = 2 ( Q(x, y ) − yR (x, y )) − (1 − y) (P0(x, y )2 + yP 1(x, y )2 + xQ (x, y )2 + xyR (x, y )2) (47) P0(x, y ) = 1 + xyP 1(x, y )Q(x, y ) + xyP 1(x, y )R(x, y ) 1 − xQ (x, y ) − xyR (x, y ) (48) P1(x, y ) = xP 0(x, y )Q(x, y ) + xP 0(x, y )R(x, y ) 1 − xQ (x, y ) − xyR (x, y ) (49) Squaring in equation 46, M(x, y ) yields the following and then substitu-tion can occur. M(x, y )2 = P0(x2, y 2)2 + y2P1(x2, y 2)2 + x2Q(x2, y 2)2 + x2y2R(x2, y 2)2 2 yP 0(x2, y 2)P1(x2, y 2) + 2 xP 0(x2, y 2)Q(x2, y 2) + 2 xyP 1(x2, y 2)Q(x2, y 2)+ 2 xyP 0(x2, y 2)R(x2, y 2) + 2 xy 2P1(x2, y 2)R(x2, y 2) + 2 x2yQ (x2, y 2)R(x2, y 2) (50) 25 Restating equations 33 and 34 M(x, y ) = 2Q(x, y ) − P0(x, y )2 − 2yP 0(x, y )P1(x, y ) − yP 1(x, y )2 − xQ (x, y )2 − 2xyQ (x, y )R(x, y ) − xyR (x, y )2 M(x, y ) = 2R(x, y ) − P0(x, y )2 − 2P0(x, y )P1(x, y ) − yP 1(x, y )2 − xQ (x, y )2 − 2xQ (x, y )R(x, y ) − xyR (x, y )2 Taking the non squared parts from 33 and 34 and equating them to 40 2 (Q(x, y ) − yR (x, y ) 1 − y ) =2Q(x, y ) − 2yP 0(x, y )P1(x, y ) − 2xyQ (x, y )R(x, y ) (51) = 2 R(x, y ) − 2P0(x, y )P1(x, y ) − 2xQ (x, y )R(x, y ) (52) Solving for the cross terms only. 2 (Q(x, y ) − R(x, y ) 1 − y ) = −2P0(x, y )P1(x, y ) − 2xQ (x, y )R(x, y ) Note that the cross terms in equation 50 are equal to the following 2y ( Q(x2, y 2) − R(x2, y 2) 1 − y2 ) = − 2yP 0(x2, y 2)P1(x2, y 2) − 2x2yQ (x2, y 2)R(x2, y 2) Using 42 and 43 and subtracting equations and subtracting after multi-plication by y as well. This gives cross terms in the squared equation 50. P1(x, y ) − P0(x, y ) = (1 − y)( xP 1(x, y )Q(x, y ) + xP 0(x, y )R(x, y )) − 1 P0(x, y ) − yP 1(x, y ) = (1 − y)( xP 0(x, y )Q(x, y ) + xyP 1(x, y )R(x, y )) + 1 P1(x, y ) − P0(x, y ) + 1 x(1 − y) = P1(x, y )Q(x, y ) + P0(x, y )R(x, y ) P0(x, y ) − yP 1(x, y ) − 1 x(1 − y) = P0(x, y )Q(x, y ) + yP 1(x, y )R(x, y ) 26 Note that the cross terms in equation 50 are equal to the following: 2xy (P1(x2, y 2) − P0(x2, y 2) + 1 x2(1 − y2) ) =2xyP 1(x2, y 2)Q(x2, y 2) + 2 xyP 0(x2, y 2)R(x2, y 2)2x (P0(x2, y 2) − y2P1(x2, y 2) − 1 x2(1 − y2) ) =2xP 0(x2, y 2)Q(x2, y 2) + 2 xy 2P1(x2, y 2)R(x2, y 2) This is not factorization, but a removal of the cross terms and using 40 in equation 50 to eliminate the square terms yields: M(x, y )2 = 2 (Q(x2, y 2) − y2R(x2, y 2) 1 − y2 ) − M(x2, y 2) − 2y (Q(x2, y 2) − R(x2, y 2) 1 − y2 ) 2 xy (P1(x2, y 2) − P0(x2, y 2) + 1 x2(1 − y2) ) 2 x (P0(x2, y 2) − y2P1(x2, y 2) − 1 x2(1 − y2) ) x2(1 − y2) (M(x, y )2 + M(x2, y 2)) =2x (−1 + P0(x2, y 2) + yP 1(x2, y 2) + xQ (x2, y 2) + xyR (x2, y 2)) − 2xy (−1 + P0(x2, y 2) + yP 1(x2, y 2) + xQ (x2, y 2) + xyR (x2, y 2)) x2(1 − y2) (M(x, y )2 + M(x2, y 2)) = 2x(1 − y) ( M(x, y ) − 1) x(1 + y) (M(x, y )2 + M(x2, y 2)) = 2 ( M(x, y ) − 1) The final generating function equation is 27 M(x, y ) = x(1 + y) (M(x, y )2 + M(x2, y 2) 2 ) 1 (53) Which reduces to the known generating function 2 when y = 0 and 00 is formally interpereted as 1 (this can also be handled with limits) M(x, 0) = x (M(x, 0) 2 + M(x2, 0) 2 ) 1 2-Color Rooted Binary Tree Isomorphism, Parameritized with Number of Specific Color, Non-Isomorphic Siblings and Nodes One may now define a recurrence which can calculate the multiplicity of equivalence classes of cardinality 2ℓ: Kn,ℓ,c – where n is the number of nodes and c the number of nodes colored black or 1; note that ℓ is the number of non-isomorphic (under the isomorphism defined above or identically “flip equivalence”) sibling subtrees in the tree. p, q, r ∈ { 0, 1}. Figures 4 and 5 show the first few trees and Table 3 shows the first few enumerated by the parameters. K2n+p, 2ℓ+q, 2c+r = p ( −1 2 )q Kn,ℓ,c 1 2 1 ∑ δ=0 2n+p−1 ∑ k=0 2ℓ+q−1 ∑ v=0 2c+r−δ ∑ m=0 2c+r≥δ Kk,v,m K2n+p−1−k, 2ℓ+q−1−v, 2c+r−δ−m The following base cases take precedence over the recurrence relations. K0,0,0 = 1 Kn,ℓ,c = 0, ℓ ≥ n > 0 ∨ ℓ < 0 ∨ c < 0 Kn,ℓ,c = (nc ) , ℓ = n − 1 ∧ 0 ≤ c ≤ nKn,ℓ,c = 1, r ∈ N1 ∧ n = 2 r − 1 ∧ ℓ = 0 ∧ 0 ≤ c ≤ nKn,ℓ,c = 0, r ∈ N1 ∧ n 6 = 2 r − 1 ∧ ℓ = 0 28 (n, ℓ, c ) = (1 , 0, 0) & (1 , 0, 1) Root Node= (n, ℓ, c ) = (0 ,0,0) (n, ℓ, c ) = (2 ,1,0) &(2 ,1,2) (n, ℓ, c ) = (2 ,1,1) (n, ℓ, c ) = (3 ,0,0) &(3 ,0,3) (n, ℓ, c ) = (3 ,0,1) &(3 ,0,2) (n, ℓ, c ) = (3 ,1,1) &(3 ,1,2) (n, ℓ, c ) = (3 ,2,0) &(3 ,2,3) (n, ℓ, c ) = (3 ,2,1) (n, ℓ, c ) = (3 ,2,2) Figure 4: Rooted Binary Trees, Parameritized with Number of Nodes= n, Number of Non-Isomorphic Siblings= ℓ and Number of Specific Color= c Where N1 = {1, 2, 3, . . . }.Define the following generating functions based on the Kn,ℓ,c recurrence. S(x, y, z ) = ∞∑ n=0 n∑ ℓ=0 n∑ c=0 Kn,ℓ,c xnyℓzc (54) (55) Sp,q,r (x, y, z ) = ∞∑ n=(1 −p)( q+r−qr ) n−(1 −p)q ∑ ℓ=0 n−(1 −p)r ∑ c=0 K2n+p, 2ℓ+q, 2c+rxnyℓzc 29 Note that S(x, y, z ) = 1 ∑ p=0 1 ∑ q=0 1 ∑ r=0 xpyqzr Sp,q,r (x2, y 2, z 2) After significant calculations, the resultant generating functions are: x(y − 2)(1 + z)S(x2, y 2, z 2) = 2(1 − S(x, y, z )) + xy (1 + z)S(x, y, z )2 (56) S(x, y, z ) = x(1 + z) 2 [ (2 − y) S(x2, y 2, z 2) + y S (x, y, z )2 ] 1 (57) As before if we formally assign the indeterminant value of 00 to 1 (this can also be handled with limits); then in that case we then have the generating function equation 57 yielding the known generating functions from equations 2, 7 and 53. When z = 0 equation 57 yields equation 7. S(x, y, 0) = x 2 [ (2 − y) S(x2, y 2, 0) + y S (x, y, 0) 2 ] 1 When y = 1 equation 57 yields equation 53. S(x, 1, z ) = x(1 + z) 2 [ S(x2, 1, z 2) + y S (x, 1, z )2 ] 1 When both y = 1 and z = 0 we have equation 57 yielding equation 2. S(x, 1, 0) = x 2 [ S(x2, 1, 0) + y S (x, 1, 0) 2 ] 1 References Bergold, H., Felsner, S., Scheucher, M., Schröder, F., Steiner, R., 2023. Topological drawings meet classical theorems from convex geometry. Dis-crete & Computational Geometry 70, 1121–1143. 30 Bouritsas, G., Frasca, F., Zafeiriou, S., Bronstein, M.M., 2022. Improving graph neural network expressivity via subgraph isomorphism counting. IEEE Transactions on Pattern Analysis and Machine Intelligence 45, 657– 668. Etherington, I., 1940. Some problems of non-associative combinations (i). Edinburgh Mathematical Notes 32, i–vi. Etherington, I.M., 1937. Non-associate powers and a functional equation. The Mathematical Gazette 21, 36–39. Jaffke, L., Lima, P.T., Lokshtanov, D., 2024. b-coloring parameterized by clique-width. Theory of Computing Systems 68, 1049–1081. Riordan, J., 1968. Combinatorial Identities. John Wiley & Sons, Inc. Sloane, N.J.A., 1964a. A000081 Number of Rooted Trees with n Nodes. The On-Line Encyclopedia of Integer Se-quences® (OEIS®). Sloane, N.J.A., 1964b. A001190 Wedderburn-Etherington Numbers: Bi-nary Rooted Trees. The On-Line Encyclopedia of Integer Sequences® (OEIS®). Yamazaki, K., Qian, M., Uehara, R., 2024. Efficient enumeration of non-isomorphic distance-hereditary graphs and related graphs. Discrete Applied Mathematics 342, 190–199. 31 nℓ 0 1 2 3 4 5 6 7 8 c 0 1 0 1 1 0 0 1 0 1 2 0 1 0 0 0 2 0 1 0 1 0 2 3 1 0 1 0 0 1 1 3 0 1 1 1 3 0 2 1 0 1 0 3 4 0 1 1 1 0 0 0 2 5 4 0 1 0 2 8 6 0 2 0 2 5 4 0 3 0 1 1 1 0 4 5 0 1 2 2 1 0 0 0 3 5 13 5 0 1 0 4 9 27 10 0 2 0 4 9 27 10 0 3 0 3 5 13 5 0 4 0 1 2 2 1 0 5 6 0 0 3 3 4 1 0 0 0 0 12 15 27 6 0 1 0 0 21 37 70 15 0 2 0 0 24 50 94 20 0 3 0 0 21 37 70 15 0 4 0 0 12 15 27 6 0 5 0 0 3 3 4 1 0 6 7 1 0 1 7 7 6 1 0 0 1 1 6 34 45 48 7 0 1 1 2 15 76 141 148 21 0 2 1 3 20 108 239 250 35 0 3 1 3 20 108 239 250 35 0 4 1 2 15 76 141 148 21 0 5 1 1 6 34 45 48 7 0 6 1 0 1 7 7 6 1 0 7 8 0 1 1 6 14 14 9 1 0 0 0 2 5 39 86 116 78 8 0 1 0 2 11 109 249 426 280 28 0 2 0 2 17 179 447 876 566 56 0 3 0 2 20 206 540 1104 710 70 0 4 0 2 17 179 447 876 566 56 0 5 0 2 11 109 249 426 280 28 0 6 0 2 5 39 86 116 78 8 0 7 0 1 1 6 14 14 9 1 0 8 Table 3: Number of Rooted Binary Trees, Parameterized with Number of Nodes= n, Num-ber of Non-Isomorphic Siblings= ℓ and Number of Specific Color= c 32 (n, ℓ, c ) = (4 , 1, 0) & (4 , 1, 4) (n, ℓ, c ) = (4 , 1, 1) & (4 , 1, 3) (n, ℓ, c ) = (4 , 1, 2) (n, ℓ, c ) = (4 , 2, 0) & (4 , 2, 4) (n, ℓ, c ) = (4 , 2, 0) & (4 , 2, 4) (n, ℓ, c ) = (4 , 2, 2) (n, ℓ, c ) = (4 , 3, 0) & (4 , 3, 4) (n, ℓ, c ) = (4 , 3, 1) (n, ℓ, c ) = (4 , 3, 3) (n, ℓ, c ) = (4 , 3, 2) (n, ℓ, c ) = (4 , 2, 1) (n, ℓ, c ) = (4 , 2, 3) Figure 5: Continued: Rooted Binary Trees, Parameritized with Number of Nodes= n,Number of Non-Isomorphic Siblings= ℓ and Number of Specific Color= c 33
188948	https://math.stackexchange.com/questions/881594/how-to-find-perpendicular-point-of-a-vector-to-another-vector-2d	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams How to find perpendicular point of a vector to another vector 2d Ask Question Asked Modified 11 years, 2 months ago Viewed 3k times 2 $\begingroup$ Given the axis x-y and some random points to the vectors AB and CD, how can i find out where will the point D lie when the vector CD(dashed line) is perpendicular to AB. For example if point A has coordinates (2,1), B (10,7), C (6,3), D (6,14), what will the coordinates of D be, if the vector CD is perpendicular to the vector AB. I am not looking for a straight solution, but for some guidelines on how can I achieve that, since I am not very good in geometry. I guess it can be managed by using the angle between points of the vector AB, but I am not sure. Thank you in advance! geometry trigonometry Share asked Jul 29, 2014 at 14:20 axelaxel 12311 silver badge33 bronze badges $\endgroup$ 3 1 $\begingroup$ The vectors $\vec{AB}$ and $\vec{CD}$ are perpendicular, so the inner product of those two vectors should be zero. $\endgroup$ Steven Van Geluwe – Steven Van Geluwe 2014-07-29 14:23:40 +00:00 Commented Jul 29, 2014 at 14:23 $\begingroup$ I'm not exactly sure what you are asking, you give the coordinates of $A,B,C,D$ and want to find $D$. Did you mean to use the symbol $D$ for different things? $\endgroup$ copper.hat – copper.hat 2014-07-29 14:28:09 +00:00 Commented Jul 29, 2014 at 14:28 $\begingroup$ I should have named the bold vector CD to CE. Sorry for the misunderstanding, my bad. $\endgroup$ axel – axel 2014-07-29 14:36:46 +00:00 Commented Jul 29, 2014 at 14:36 Add a comment \| 2 Answers 2 Reset to default 0 $\begingroup$ You know two things: The dashed line $\vec{CD}$ is perpendicular to $\vec{AB}$ The dashed $\vec{CD}$ and the bold $\vec{CD}$ have the same length. Actually, just call the bold one something else, like $\vec{CE}$. The first condition means that $ = 0$. The second condition means that $\\|C-E\\| = \\|C-D\\|$. You have two unknowns, namely the coordinates of $D$. You can use these two conditions to solve for these unknowns. Note on inner products: You can think of an inner product as a function that captures the notion of angle. In your case, we can use the traditional dot product, a type of inner product. In the plane, for example, the dot product of $A = (a_1,a_2)$ and $B=(b_1,b_2)$ is written $A\cdot B$ and is given by \begin{equation} a_1b_1 + a_2b_2. \end{equation} A very important fact is that when two vectors are perpendicular (orthogonal), they have dot product equal to $0$. So, let the coordinates of $D$ be $(x,y)$. Expand the first condition using the formula for the dot product I gave above. Expand the second condition using the definition of the Euclidean norm (in this case, the familiar distance formula). This will give you a system of two equations in two unknowns. Solve it for $x$ and $y$. Note: Once you use these conditions to write a system of equations, you may want to think about how many solutions this system has and what this means geometrically. Share edited Jul 30, 2014 at 14:09 answered Jul 29, 2014 at 14:32 MRicciMRicci 1,6521212 silver badges1818 bronze badges $\endgroup$ 4 $\begingroup$ Sorry but I cannot understand how to make the equation. Could you be more a little bit more descriptive. $\endgroup$ axel – axel 2014-07-30 06:03:13 +00:00 Commented Jul 30, 2014 at 6:03 $\begingroup$ Are you familiar with inner products? $\endgroup$ MRicci – MRicci 2014-07-30 06:30:07 +00:00 Commented Jul 30, 2014 at 6:30 $\begingroup$ Actually I am not, but I tried to learn from wikipedia without much success. $\endgroup$ axel – axel 2014-07-30 07:09:16 +00:00 Commented Jul 30, 2014 at 7:09 $\begingroup$ In Euclidean geometry (on a plane or space in Cartesian coordinates), the inner product between two vectors is also known as the dot product, and is just the sum of the product of their coordinates: $A = (a_x,a_y)$ and $B = (b_x,b_y)$, then $A \cdot B = a_x b_x + a_y b_y$. When this is zero, the two vectors are perpendicular. (In this case, the point $A = (a_x,a_y)$ represents a vector starting at the origin and ending at $(a_x,a_y)$). $\endgroup$ LucasVB – LucasVB 2014-07-30 14:18:00 +00:00 Commented Jul 30, 2014 at 14:18 Add a comment \| 0 $\begingroup$ A quick way to think about it is: if you have a vector $(x,y)$, then rotating it counterclockwise $90^o$ you get $(-y,x)$. We have $\mathbf{AB} = (8,6)$. If $\mathbf{D} = (x_d,y_d)$, if you want $\mathbf{AB} \perp \mathbf{CD}$, then $\mathbf{CD} = (x_d - 6, y_d - 3)$ is parallel to $(-6,8)$. Hence exists $\lambda \in \Bbb R$ such that: $$(x_d - 6, y_d - 3) = \lambda (-6,8)$$ You have two equations and three unkowns, because you still have one degree of freedom: $\mathbf{CD}$'s lenght. Share answered Jul 29, 2014 at 15:51 Ivo TerekIvo Terek 80.5k1414 gold badges113113 silver badges243243 bronze badges $\endgroup$ Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. 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188949	https://www.shaalaa.com/concept-notes/expansion-of-a-b-2-a2-2ab-b2_15418	Expansion of (a - b)2 = a2 - 2ab + b2 Advertisements Topics Rational Numbers Linear Equations in One Variable Understanding Quadrilaterals Data Handling Practical Geometry Squares and Square Roots Cubes and Cube Roots Comparing Quantities Algebraic Expressions and Identities Mensuration Visualizing Solid Shapes Exponents and Powers Direct and Inverse Proportions Factorization Introduction to Graphs Playing with Numbers Formula (a - b)2 = a2 - 2ab + b2. Notes Expansion of (a - b)2 = a2 - 2ab + b2: In the above figure, the square with side a is divided into 4 rectangles, namely, square with side (a - b), square with side b, and two rectangles of sides (a - b) and b. A (square I) + A (rectangle II) + A (rectangle III) + A (square IV) = A (□ PQRS) (a - b)2 + (a - b)b + (a - b)b + b2 = a2 (a - b)2 + 2ab - 2b2 + b2 = a2 (a - b)2 + 2ab - b2= a2 ∴ (a - b)2 = a2 - 2ab + b2 Let us multiply the algebraic expressions and obtain the formula. (a - b)2 = (a - b) × (a - b) (a - b)2 = a (a - b) - b (a - b) (a - b)2 = a2 - ab - ab + b2 (a - b)2 = a2 - 2ab + b2. Example Expand: (5x - 4)2 (5x - 4)2 = (5x)2 - 2(5x) x (4) + 42 = 25x2 - 40x + 16. Example Expand: (98)2 (98)2 = (100 - 2)2 = 1002 - 2 x 100 x 2 + 22 = 10000 - 400 + 4 = 9604. Shaalaa.com \| Expansion of (a - b)2 = a2 - 2ab + b2 Shaalaa.com Next video Shaalaa.com Series: Expansion of (a - b)2 = a2 - 2ab + b2 Related QuestionsVIEW ALL Factorise the following, using the identity a2 – 2ab + b2 = (a – b)2. y2 – 14y + 49 (a – 1)2 = a2 – 1 Expand (5p – 1)2 Expand ("a"-1/"a")^2 Expand: (5x - 4)2 (a – b)2 = a2 – b2 The factors of x2 – 4x + 4 are __________ Factorise the following, using the identity a2 – 2ab + b2 = (a – b)2. 4y2 – 12y + 9 Advertisements Submit content Select a course
188950	https://www.gauthmath.com/solution/1831761348123697/17-through-2-1-Find-the-equation-of-the-tangent-line-to-9x2-16y2-52	Solved: through (2,-1) Find the equation of the tangent line to 9x^2+16y^2=52 [Calculus] Drag Image or Click Here to upload Command+to paste Upgrade Sign in Homework Homework Assignment Solver Assignment Calculator Calculator Resources Resources Blog Blog App App Gauth Unlimited answers Gauth AI Pro Start Free Trial Homework Helper Study Resources Calculus Questions Question through (2,-1) Find the equation of the tangent line to 9x^2+16y^2=52 Gauth AI Solution 100%(3 rated) Answer The answer is 9x - 8y = 26 Explanation Question 17: Verify that the point (2, -1) lies on the ellipse $$9x^{2} + 16y^{2} = 52$$9 x 2+16 y 2=52 Substitute $$x = 2$$x=2 and $$y = -1$$y=−1 into the equation: $$9(2)^{2} + 16(-1)^{2} = 9(4) + 16(1) = 36 + 16 = 52$$9(2)2+16(−1)2=9(4)+16(1)=36+16=52 Since the equation holds true, the point (2, -1) lies on the ellipse. Use implicit differentiation to find $$\frac{dy}{dx}$$d x d y Differentiate both sides of $$9x^{2} + 16y^{2} = 52$$9 x 2+16 y 2=52 with respect to $$x$$x: $$\frac{d}{dx}(9x^{2}) + \frac{d}{dx}(16y^{2}) = \frac{d}{dx}(52)$$d x d(9 x 2)+d x d(16 y 2)=d x d(52) $$18x + 32y \frac{dy}{dx} = 0$$18 x+32 y d x d y=0 Solve for $$\frac{dy}{dx}$$d x d y $$32y \frac{dy}{dx} = -18x$$32 y d x d y=−18 x $$\frac{dy}{dx} = -\frac{18x}{32y} = -\frac{9x}{16y}$$d x d y=−32 y 18 x=−16 y 9 x Evaluate $$\frac{dy}{dx}$$d x d y at the point (2, -1) to find the slope of the tangent line $$m = \frac{dy}{dx}\Big\|_{(2, -1)} = -\frac{9(2)}{16(-1)} = -\frac{18}{-16} = \frac{9}{8}$$m=d x d y(2,−1)=−16(−1)9(2)=−−16 18=8 9 Use the point-slope form of a line to find the equation of the tangent line The point-slope form is $$y - y_{1} = m(x - x_{1})$$y−y 1=m(x−x 1), where $$(x_{1}, y_{1}) = (2, -1)$$(x 1,y 1)=(2,−1) and $$m = \frac{9}{8}$$m=8 9 $$y - (-1) = \frac{9}{8}(x - 2)$$y−(−1)=8 9(x−2) $$y + 1 = \frac{9}{8}x - \frac{18}{8}$$y+1=8 9x−8 18 $$y = \frac{9}{8}x - \frac{9}{4} - 1$$y=8 9x−4 9−1 $$y = \frac{9}{8}x - \frac{9}{4} - \frac{4}{4}$$y=8 9x−4 9−4 4 $$y = \frac{9}{8}x - \frac{13}{4}$$y=8 9x−4 13 Convert the equation to standard form Multiply both sides by 8 to eliminate fractions: $$8y = 9x - 26$$8 y=9 x−26 Rearrange the equation: $$9x - 8y = 26$$9 x−8 y=26 The answer is: 9x - 8y = 26 Helpful Not Helpful Explain Simplify this solution Gauth AI Pro Back-to-School 3 Day Free Trial Limited offer! Enjoy unlimited answers for free. Join Gauth PLUS for $0 Previous questionNext question Related Use the general slicing method to find the volume of the following solid. The solid whose base is the triangle with vertices the y-axis are semicircles 0,0,13,0 , and 0,13 and whose cross sections perpendicular to the base and parallel to Set up the integral that gives the volume of the solid. Use increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice. Type exact answers. A. d B. C square d 100% (4 rated) In the following exercise eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. x= square root of t+1,y= square root of t-6;t ≥ 0 The rectangular equation is square with square Simplify your answer. restrictions 1 ≤ x ≤ 7 and -6 ≤ y ≤ 0. no additional restrictions on x and y. restrictions x ≥ 1 and y ≥ -6. the restriction x ≥ 0. 100% (4 rated) Find all values of x for which the graph of f lies above the graph of g. a. fx= 1/x ;gx= 1/x-1 to b. fx=x2+x;gx= 2/x 100% (4 rated) ne to 9x2+16y2=52 100% (4 rated) For the plane curve, a graph the curve, and b find a rectangular equation for the curve. x=12sin t,y=12cos t , for tin[0,2 π ] 100% (3 rated) A type of candy bar sells for $ 2.25. The cost of producing x bars is Cx=50+0.4x+0.002x2 dollars. a. Calculate the marginal revenue, R'x. R'x=square b. Calculate the proft, Px. Px=square c. Calculate the revenue and marginal revenue when 900 bars are sold. When 900 bars are sold, the revenue is square dollars and the marginal revenue is square dollars. d. Calculate the profit and marginal profit when 900 bars are sold. When 900 bars are sold, the proft is dollars and the marginal proft is square dollars. e. The marginal profit is zero when how many candy bars are sold? square I bars f. Interpret your answer. The graph of the profit function is a parabola with vertex at square ° and the proft is ？ square when you sell that much. 100% (5 rated) Given the equation x7y11-x11y7=2 , find dy/dx by implicit differentiation 100% (3 rated) Use series to evaluate the limit correct to three decimal places. limlimits _xto 0frac 10x-tan -110xx3 none 333.333 166.667 100.000 333.133 333.933 100% (2 rated) Differentiate the given function. kx= 2/7 8x8-x+214 28x8-x+41364x8-10 288x8-x+21364x7-x 464x7-113 48x8-x+21364x7-1 48x8-x-2158x7-1 100% (2 rated) Find the Taylor series for fx=ln 1+4x . Give the interval of convergence. What is the Taylor series for fx ? A. 1-4x+frac 42x22-frac 43x33+frac 44x44+ . s +frac -1n4nxnn+ . s B. 4x-frac 42x22+frac 43x33-frac 44x44+ . s +frac -1n4n+1xn+1n+1+ . s C. -1+4x-frac 42x22+frac 43x33-frac 44x44+ . s +frac -1n4n+1xn+1n+1+ . s D. -4x+frac 42x22-frac 43x33+frac 44x44+ . s +frac -1n4nxnn+ . s 100% (1 rated) Gauth it, Ace it! contact@gauthmath.com Company About UsExpertsWriting Examples Legal Honor CodePrivacy PolicyTerms of Service Download App
188951	https://oeis.org/somedcgf.html	The OEIS is supported by the many generous donors to the OEIS Foundation. Some divide-and-conquer sequences with (relatively) simple ordinary generating functions Some divide-and-conquer sequences with (relatively) simple ordinary generating functions Version 2004-01-1 Ralf Stephan Introduction In the following, about 100 sequences are collected that are generated by ordinary generating functions of form 1/(1-x)^m sum(k=0, inf, C^kR(x^2^k)), where m=0,1,2, C is integer, and R is a rational function. That the given recurrences are indeed generated by the mentioned functions (and that most of the sequences are fractal) is clarified in the section below. For all recurrences, a link into their OEIS entry, as well as, where possible, a mnemonic and a 'closed form' is given. Parentheses around a link means the entry can be derived from the recurrence by an elementary operation like shift. Used abbreviations are [log2(n)] for (n), v2(n) for (n), e1(n) for (n), and e0(n) for (n). Where the parameter is left out, it is understood to be n. Recurrence start value a(0) is always 0. Of form a(2n) = Ca(n) + P(n), a(2n+1) = Q(n) (1) Here and thereafter, P and Q are functions of n expressible by a rational generating function, C integer. Recurrences of form a(2n) = Ca(n) + P(n), a(2n+1) = Q(n) \| \| a(2n) = \| a(2n+1) = \| form \| mnemonic \| \| \| a(n)+1 \| 0 \| v2 \| 2-adic valuation, 2^a(n) divides n \| \| \| a(n)+1 \| 1 \| v2+1 \| 2^a(n) divides 2n \| \| \| a(n)+2 \| 1 \| 2v2+1 \| \| \| \| a(n)-1 \| 1 \| 1-v2 \| diff() \| \| () \| a(n)+3 \| 4 \| 3v2+4 \| hierarchical sequence \| \| () \| a(n) \| [n==0] \| [n==2^k] \| n is power of two \| \| \| a(n)+1 \| -1+[n==0] \| v2-1+[n==2^k] \| diff() \| \| \| 2a(n) \| 1 \| 2^v2 \| highest power of 2 div. n \| \| \| 2a(n)+1 \| 1 \| 2^(v2+1)-1 \| nim-sum \| \| \| 3a(n)-2 \| 2 \| 3^v2+1 \| \| \| \| 3a(n)+3 \| 3 \| (3^(v2+2)-1)/2-1 \| Catalan mod 3 \| \| \| 4a(n)+3 \| 7 \| 84^v2-1 \| denominator of a sigma expr. \| \| \| -a(n)+1 \| 1 \| (1-(-1)^v2)/2 \| v2(2n) mod 2 \| \| () \| a(n) \| n \| (n/2^v2+1)/2 \| fractal sequence \| \| \| a(n) \| 2n+1 \| n/2^v2 \| largest odd divisor of n \| \| \| a(n)+1 \| 2n \| v2+n/2^v2-1 \| diff() \| \| \| 2a(n)+1 \| 2n+1 \| n+2^v2-1 \| switch trailing 0s \| \| \| a(n) \| n%2 \| (1-(-1)^)/2 \| bit left of lsb/paperfolding seq. \| \| () \| a(n) \| 2^n \| 2^((n/2^v2-1)/2) \| 2^ \| \| \| 2a(n) \| (-1)^n \| (see entry) \| diff(Gray code) \| \| \| 2a(n) \| f(n) \| \| with f(n)=3n+1/2-(n-5/2)(-1)^n \| \| \| a(n)+2^(n-1) \| 2^n \| \| 2n-bead balanced bin. strings \| \| \| 2a(n)+2n \| 2n+1 \| nv2+n \| 2^a(n) divides (n)^(n) \| \| \| 2a(n)+(2n)^2 \| (2n+1)^2 \| 2n^2-n^2/2^v2 \| \| Of form a(2n) = Ca(n) + P(n), a(2n+1) = Ca(n) + Q(n) (2) The sequences are all partial sums of sequences of the previous form (1), and start with a(0)=0. The case C=1 Recurrences of form a(2n) = a(n) + P(n), a(2n+1) = a(n) + Q(n) \| \| P(n) \| Q(n) \| form \| mnemonic \| \| \| 0 \| 1 \| e1(n) \| ones-counting function \| \| \| 1 \| 0 \| e0(n) \| zeros-counting function \| \| \| 1 \| 1 \| [log2(n)]+1 \| binary length \| \| () \| 2 \| 1 \| 2e0+e1 \| a stopping problem \| \| \| 1 \| 2 \| 2e1+e0 \| binary weight + length \| \| \| 1 \| -1 \| e0-e1 \| \| \| \| 0 \| [n==0] \| [n>0] \| sign(n) \| \| \| 1 \| 1-[n==0] \| [log2(n)] \| \| \| () \| [n==1] \| 0 \| \| runs of 2^k 1s and 0s \| \| \| n \| n \| n-e1 \| v2(n!) \| \| \| n \| n+1 \| n \| \| \| \| n+1 \| n+2-[n==0] \| n+[log2(n)] \| cube subgraphs \| \| \| n-1 \| n \| n-1-[log2(n)] \| eigenvalues \| \| \| 2n \| 2n+1 \| 2n-e1 \| v2((2n)!) \| \| \| 2n-1 \| 2n+1 \| 2n-1-[log2(n)] \| Connell sequence \| \| \| 2n \| -2n-1 \| \| \| \| \| 3n \| 3n+2 \| 3n-e1 \| denom. in (1-x)^(-1/4) \| \| \| 3n-2 \| 3n+1 \| 3n-2-[log2(n)] \| Connell sequence \| \| \| n^2 \| n^2+2n \| \| minimum cost addition chain \| \| \| 0 \| [n even] \| \| runs of ones \| \| \| [n odd] \| 0 \| \| counting '10' \| \| \| 0 \| [n odd] \| \| counting '11' \| \| \| 1 \| [n odd] \| \| increasing spots in bin. repr. \| \| \| [n even] \| 0 \| \| counting '00' \| \| () \| [n even] \| 1 \| \| decreasing spots in bin. repr. \| \| \| 0 \| [n>0 even] \| \| counting '01' \| \| \| [n odd] \| [n even] \| \| e1(Gray code of n) \| \| \| 0 \| [n=3 mod 4] \| \| counting '111' \| \| \| 0 \| [n=7 mod 8] \| \| counting '1111' \| The case C=2 Recurrences of form a(2n) = 2a(n) + P(n), a(2n+1) = 2a(n) + Q(n) \| \| P(n) \| Q(n) \| form \| mnemonic \| \| \| 0 \| [n==0] \| 2^[log2(n)] \| msb \| \| \| 0 \| 2[n==0] \| 22^[log2(n)] \| \| \| () \| 1 \| 0 \| -n-1+22^[log2(n)] \| interchange 0s and 1s \| \| \| 1 \| [n==0] \| -n-1+32^[log2(n)] \| permutation of N \| \| () \| 1 \| 2[n==0] \| -n-1+42^[log2(n)] \| -n in 2's complement \| \| \| 0 \| 1 \| n \| \| \| \| 0 \| 2 \| 2n \| \| \| \| 0 \| 1+[n==0] \| n+2^[log2(n)] \| starts '10' \| \| \| 0 \| 1+2[n==0] \| n+22^[log2(n)] \| starts '11' \| \| \| 0 \| -1+2[n==0] \| -n+22^[log2(n)] \| longest carry sequence \| \| \| 0 \| 1+3[n==0] \| n+32^[log2(n)] \| starts '100' \| \| \| 0 \| 2+4[n==0] \| 2n+42^[log2(n)] \| Aronson-like \| \| \| 1 \| 1 \| 22^[log2(n)]-1 \| a(n-1) OR n \| \| \| [n==1] \| 0 \| 2^flg(n)-2^flg(2/3n) \| \| \| () \| 4-3[n==1] \| 6-5[n==0] \| 32^flg(n2/3)+2n-2 \| Aronson-like \| \| \| -1 \| [n==0] \| n+1-2^[log2(n)] \| runs of 1...2^k \| \| \| -1 \| 1 \| 2n+1-2^[log2(n)] \| Josephus problem \| \| \| -1 \| -1+4[n==0] \| 1+22^[log2(n)] \| \| \| \| [n odd] \| [n even] \| \| Gray code \| \| \| [n odd] \| [n>0 even] \| \| 'derivative' of n \| \| () \| n \| n+1 \| \| part. sums of 2^v2 \| \| \| 2n \| 2n+1 \| \| part. sums of 2^(v2+1)-1 \| \| \| 0 \| 2(-1)^n+1 \| \| reversing bin. repr. of -n \| Other cases Recurrences of form a(2n) = Ca(n) + P(n), a(2n+1) = Ca(n) + Q(n) \| \| C \| P(n) \| Q(n) \| mnemonic \| \| \| 3 \| 0 \| 1 \| ternary repr. contains no 2 \| \| \| 3 \| 0 \| 2 \| ternary repr. contains no 1 \| \| \| 3 \| 0 \| 3 \| 3 does not divide C(2k,k) \| \| () \| 3 \| 0 \| 6 \| related to Cantor set \| \| \| 3 \| 1 \| 0 \| \| \| \| 4 \| 0 \| 1 \| Moser-de Bruijn sequence \| \| \| 4 \| 0 \| 3 \| double bitters \| \| \| -1 \| 0 \| 1 \| alternating bit sum \| \| \| -1 \| 1 \| 0 \| \| \| \| -1 \| 1 \| 1 \| runs of length 2^k \| \| \| -1 \| 1 \| 1-[n==0] \| [log2(n)] mod 2 \| \| \| -1 \| n \| n+1 \| part. sums of (-1)^v2 \| \| \| -1 \| n \| n+[n==0] \| \| \| \| -1 \| 2n \| 2n+1 \| double-free subsets of N \| \| () \| -1 \| 2n+1 \| 2n+2-[n==0] \| \| \| () \| -1 \| 9n+3 \| 9n+6-[n==0] \| \| \| \| -2 \| 0 \| 1 \| replace 2^k with (-2)^k in binary \| \| \| -2 \| 2n \| 2n \| remove even-pos. bits \| \| \| -2 \| 2n \| 2n+1 \| remove every 2nd bit \| \| \| -2 \| 5n \| 5n+2 \| binary counter \| Of form a(2n) = C(a(n)+a(n-1)) + P(n), a(2n+1) = 2Ca(n) + Q(n) (3) P and Q are functions of n expressible by a rational generating function, C integer. The sequences are all partial sums of sequences of the previous form (2), a(0)=0. Recurrences of form a(2n) = C(a(n)+a(n-1)) + P(n), a(2n+1) = 2Ca(n) + Q(n) \| \| C \| P(n) \| Q(n) \| mnemonic \| \| \| 1 \| [n==1] \| 0 \| oscillating sequence \| \| () \| 1 \| 0 \| [n==0] \| \| \| () \| 1 \| [n==1] \| [n==0] \| Josephus problem \| \| () \| 1 \| [n==1] \| 3[n==0] \| a(a(n)) = 2n \| \| \| 1 \| 1 \| 1 \| n \| \| () \| 1 \| 3-2[n==1] \| 3-3[n==0] \| \| \| () \| 1 \| n \| n \| part. sums of e0 \| \| \| 1 \| n \| n+1 \| part. sums of e1 \| \| \| 1 \| n-1 \| n \| fractal generator \| \| \| 1 \| 2n-2 \| 2n \| Legendre pol. expansions \| \| () \| 1 \| 2n-1 \| 2n \| part. sums of [log2(k)] \| \| \| 1 \| 2n \| 2n+1 \| binary insertion sort comparisons \| \| () \| 1 \| 2n+1 \| 2n+2 \| binary entropy \| \| () \| 1 \| 2n+2 \| 2n+3 \| \| \| () \| 1 \| 2n+2 \| 2n+4 \| quicksort comparisons \| \| () \| 1 \| 2n^2+n \| 2n^2+3n+1 \| \| \| () \| 2 \| 1 \| 1 \| \| \| () \| 2 \| 2[n==1] \| 4[n==0] \| concerning a regex algorithm \| \| \| 2 \| n \| 0 \| sum(k AND n-k) \| \| \| 2 \| 4n-4 \| 6n \| sum(k XOR n-k) \| \| \| 2 \| 5n-4 \| 6n \| sum(k OR n-k) \| \| \| 2 \| n^2+n \| (n+1)^2 \| \| \| () \| 2 \| 2ceil(n/2) \| n+1 \| part. sums of Gray code \| \| \| -1 \| n \| n+1 \| Koch curve \| \| \| -1 \| n^2+n \| n^2+2n+1 \| sum(sum((-1)^v2)) \| Theory Lemma. Let A(z) an infinite sum of rational functions of form A(z) = sum (k=0, inf, C^k B(z^(2^k))), B rational, C integer. Then A(z) generates an integer sequence of divide-and-conquer type satisfying a(0)=0, a(2n) = Ca(n)+b(2n), a(2n+1) = b(2n+1), where b(n) is the sequence generated by B(z). The summation term with k=0 fills both bisections of a(n) since C^k and 2^k reduce to 1. Any other term contributes only to a(2n) as all exponents to z are even. Moreover, other subsequences from single terms of the sum are increasingly sparse (spread out by a factor of 2), and have values multiplied with C, with respect to each other. This is essentially the reason for the fractality of a(n). Note the proof of the lemma is easy because having no reference to a(n) in the odd bisection of the recurrence amounts to a cutoff of the recursion, leaving computation of v2(n) steps in the even bisection. Additional insight might be gained from the collection of generating functions in this Postscript file (6 pages). An open question would be whether all sequences here discussed are 2-regular. References: Ph. Dumas, Divide-and-conquer. Lookup Welcome Wiki Register Music Plot 2 Demos Index WebCam Contribute Format Style Sheet Transforms Superseeker Recents The OEIS Community Maintained by The OEIS Foundation Inc. Last modified August 22 11:47 EDT 2025. Contains 386898 sequences. License Agreements, Terms of Use, Privacy Policy
188952	https://www.math.ucla.edu/~pak/papers/MatEquality15.pdf	EQUALITY CASES OF THE STANLEY–YAN LOG-CONCAVE MATROID INEQUALITY SWEE HONG CHAN AND IGOR PAK Abstract. The Stanley–Yan (SY) inequality gives the ultra-log-concavity for the numbers of bases of a matroid which have given sizes of intersections with k fixed disjoint sets. The inequality was proved by Stanley (1981) for regular matroids, and by Yan (2023) in full generality. In the original paper, Stanley asked for equality conditions of the SY inequality, and proved total equality conditions for regular matroids in the case k = 0. In this paper, we completely resolve Stanley’s problem. First, we obtain an explicit description of the equality cases of the SY inequality for k = 0, extending Stanley’s results to general matroids and removing the “total equality” assumption. Second, for k ≥1, we prove that the equality cases of the SY inequality cannot be described in a sense that they are not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. 1. Introduction 1.1. Foreword. Among combinatorial objects, matroids are fundamental and have been exten-sively studied in both combinatorics and their applications to other fields (see e.g. [Ox11, Sch03]). In recent years, a remarkable progress has been made towards understanding log-concave matroid inequalities for various matroid parameters (see e.g. [Huh18, Kal23]). Much less is known about their equality conditions as they remain inaccessible by algebraic techniques (see Section 2). The Stanley–Yan (SY) inequality is a very general log-concave inequality for the numbers of bases of a matroid which have given sizes of intersections with k fixed sets. In this paper we completely resolve Stanley’s open problem [Sta81, p. 60], asking for equality conditions for the SY inequality, although probably not in the way Stanley had expected: we give a positive result for k = 0 and a negative result for k ≥1. Since known proofs of the SY inequality are independent of k, it came as a surprise that the equality conditions have a completely different nature for different k. Curiously, our negative result is formalized and proved in the language of computational complexity. Even as a conjecture this was inconceivable until our recent work (cf. §15.1). 1.2. Stanley’s problem. Let M be a matroid or rank r = rk(M), with a ground set X of size \|X\| = n. Denote by B(M) the set of bases of M. This is a collection of r-subsets of X. Fix integers k ≥0 and 0 ≤a, c1, . . . , ck ≤r. Additionally, fix disjoint subsets R, S1, . . . , Sk ⊂X. Define BSc(M, R, a) := A ∈B(M) : \|A ∩R\| = a, \|A ∩S1\| = c1, . . . , \|A ∩Sk\| = ck , where S = (S1, . . . , Sk) and c = (c1, . . . , ck). Denote BSc(M, R, a) := \|BSc(M, R, a)\|, and let PSc(M, R, a) := BSc(M, R, a) r a,c1 ,...,ck ,υ −1, where υ = r −a −c1 −. . . −ck. See §3.2 for the matroid definition and notation. Theorem 1.1 (Stanley–Yan inequality, [Sta81, Thm 2.1] and [Yan23, Cor. 3.47]). (SY) PSc(M, R, a)2 ≥PSc(M, R, a + 1) PSc(M, R, a −1). Date: December 17, 2024. 1 2 SWEE HONG CHAN AND IGOR PAK This inequality was discovered by Stanley who proved it for regular (unimodular) matroids using the Alexandrov–Fenchel inequality. The inequality was extended to general matroids by Yan [Yan23], using Lorentzian polynomials. Both proofs are independent of k. To motivate the result, Stanley showed in [Sta81, Thm 2.9] (see also [Yan23, Thm 3.48]), that the Stanley–Yan (SY) inequality for k = 0 implies the Mason–Welsh conjecture (1971, 1972), see (M1) in §2.2. This is a log-concave inequality for the number of independent sets of a matroid (see §2.2), which remained a conjecture until Adiprasito, Huh and Katz [AHK18] famously proved it in full generality using combinatorial Hodge theory. In [Sta81, §2], Stanley asked for equality conditions for (SY) and proved partial results in this direction (see below). Despite major developments on matroid inequalities, no progress on this problem has been made until now. We give a mixture of both positive and negative results which completely resolve Stanley’s problem. We start with the latter. 1.3. Negative results. Let M be a binary matroid given by its representation over F2, and let k ≥0, R ⊆X, S ∈Xk, a ∈N, c ∈Nk be as above. Denote by EqualitySYk the decision problem EqualitySYk := PSc(M, R, a)2 =? PSc(M, R, a + 1) PSc(M, R, a −1) . Here the input to the problem consists of a representation of the matroid M, subsets R, S1, . . . , Sk of the ground set, and integers a, c1, . . . , ck. The integer k is the only fixed parameter. Theorem 1.2 (k ≥1 case). For all k ≥1, we have: EqualitySYk ∈PH = ⇒ PH = Σp m for some m, for binary matroids. Moreover, the result holds for a = 1 and c1 = r −2. This gives a negative solution to Stanley’s problem for k ≥1. Informally, the theorem states that equality cases of the Stanley–Yan inequality (SY) cannot be described using a finite num-ber of alternating quantifiers ∃and ∀, unless a standard complexity assumptions fails (namely, that the polynomial hierarchy PH collapses to a finite level1). This is an unusual application of computational complexity to a problem in combinatorics (cf. §15.1). The proof of Theorem 1.2 is given in Section 8, and uses technical lemmas developed in Section 4–7. The theorem does not say that no geometric description of (SY) can be obtained, or that some large family of equality cases cannot be described. In fact, the vanishing cases we present below (see Theorem 1.6), is an example of the latter. The proof of Theorem 1.2 uses the combinatorial coincidences approach developed in [CP24d, CP23a]. We also use the analysis of the spanning tree counting function through continued fractions (see §15.6 below). Paper [CP23a] is especially notable, as it can be viewed both a philosophical and (to a lesser extent) a technical prequel to this paper. There, we prove that the equality cases of the Alexandrov–Fenchel inequality are not in PH for order polytopes (under the same assumptions). See §15.2 for possible variations of the theorem for other classes of matroids. 1.4. Positive results. For k = 0, we omit the subscripts: B(M, R, a) := A ∈B(M) : \|A ∩R\| = a and P(M, R, a) := B(M, R, a) r a −1. Denote by NL(M) the set of non-loops in M, i.e. elements x ∈X such that {x} is an independent set. For a non-loop x ∈NL(M), denote by ParM(x) ⊆X the set of elements of M that are parallel to x, i.e. elements y ∈X such that {x, y} is not an independent set. The following two results (Theorem 1.3 and Proposition 1.4) give a positive solution to Stanley’s problem for k = 0. 1This is a standard assumption in theoretical computer science that is similar to P ̸= NP (stronger, in fact), and is widely believed by the experts. If false it would bring revolutionary changes to the field, see e.g. [Aar16, Wig23]. EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 3 Theorem 1.3 (k = 0 case). Let M be a matroid of rank r ≥2 with a ground set X. Let R ⊂X, and let 1 ≤a ≤r −1. Suppose that P(M, R, a) > 0. Then the equality (1.1) P(M, R, a)2 = P(M, R, a + 1) P(M, R, a −1) holds if and only if for every independent set A ⊂X s.t. \|A\| = r −2 and \|A ∩R\| = a −1, and every non-loop x ∈NL(M/A), we have: (1.2) \| ParM/A(x) ∩R\| = s \| ParM/A(x) ∩(X −R)\| for some s > 0. In particular, Theorem 1.3 resolves a conjecture of Yan (Conjecture 1.10 in §1.6 below). We prove the theorem in Section 10 using the combinatorial atlas technology, see §2.3. This is a technical linear algebraic approach we developed in [CP22a, CP24a] to prove both the inequalities and the equality cases of matroid and poset inequalities, as well as their generalizations. Notably, we obtain the equality cases of Mason’s ultra-log-concave inequality [CP24a, §1.6], and we use a closely related setup in this case. 1.5. Vanishing conditions. Note that when P(M, R, a) = 0, we always have equality in (1.1). The following nonvanishing conditions give a description to such equality cases: Proposition 1.4 (nonvanishing conditions for k = 0). Let M be a matroid of rank r = rk(M) with a ground set X, and let R ⊆X. Then, for every 0 ≤a ≤r, we have: P(M, R, a) > 0 if and only if r −rk(X ∖R) ≤a ≤rk(R). The proposition is completely straightforward and is a special case of a more general Theo-rem 1.6, see below. Combined, Theorem 1.3 and Proposition 1.4 give a complete description of equality cases of the Stanley–Yan inequality (SY) for k = 0. It is natural to compare our positive and negative results, in the complexity language. In particular, Theorem 1.2 shows that EqualitySYk ̸∈coNP, for all k ≥1 (unless PH collapses). In other words, it is very unlikely that there is a witness for (SY) being strict that can be verified in polynomial time. This is in sharp contrast with the case k = 0 : Corollary 1.5. Let M be a matroid given by a succinct presentation. Then: EqualitySY0 ∈coNP. Here by succinct we mean a presentation of a matroid with an oracle which computes the rank function (of a subset of the ground set) in polynomial time, see e.g. [KM22, §5.1]. Matroids with succinct presentation include graphical, transversal and bicircular matroids (see e.g. [Ox11, Wel76]), certain paving matroids based on Hamiltonian cycles [Jer06, §3], and matroids given by their representation over fields Fq or Q. By a mild abuse of notation, we use EqualitySYk to denote the equality decision problem of the Stanley–Yan inequality (SY) for general matroids given by a succinct presentation. Corollary 1.5 follows from the explicit description of the equality cases given in Theorem 1.3 and Proposition 1.4. See also Section 13 for several examples, and §15.4 for further discussion of computational hardness of EqualitySY0 . Theorem 1.6 (nonvanishing conditions for all k ≥0). Let M = (X, I) be a matroid with a ground set X and independent sets I ⊂2X. Let r = rk(M) be the rank of M. Let S = (S1, . . . , Sℓ) be a set partition of X, i.e. we have X = ∪iSi and Si ∩Sj = ∅for all 1 ≤i < j ≤ℓ. Finally, let c = (c1, . . . , cℓ) ∈Nℓ. Then, there exists an independent set A ∈I such that \|A ∩Si\| = ci for all i ∈[ℓ] if and only if rk ∪i∈L Si ≥ X i∈L ci for all L ⊆[ℓ], L ̸= ∅. 4 SWEE HONG CHAN AND IGOR PAK One can think of this result as a positive counterpart to the (negative) Theorem 1.2. In the language of Shenfeld and van Handel [SvH22, SvH23], the vanishing conditions are “trivial” equality cases of the SY inequality, in a sense of having a simple geometric meaning rather than ease of the proof. We prove the Theorem 1.6 in Section 11 using the discrete polymatroid theory. Note that Proposition 1.4 follows from Theorem 1.6, by taking S1 ←R, S2 ←X ∖R, c1 ←a, and c2 ←(r−a). More generally, the complexity of the vanishing for all k ≥0 follows immediately from Theorem 1.6, and is worth emphasizing: Corollary 1.7. Let M be a matroid given by a succinct presentation. Then, for all fixed k ≥0, the problem BSc(M, R, a) >? 0 is in P. 1.6. Total equality cases. Throughout this section, we let k = 0. We start with a simple observation whose proof is well-known and applies to all positive log-concave sequences. Corollary 1.8. Let M be a matroid of rank r ≥2 with a ground set X, and let R ⊆X. Suppose P(M, R, 0) > 0 and P(M, R, r) > 0. Then: (1.3) P(M, R, 1)r ≥P(M, R, 0)r−1 P(M, R, r). Moreover, the equality in (1.3) holds if and only if (SY) is an equality for all 1 ≤a ≤r −1. For completeness, we include a short proof in §12.1. This motivates the following result that is more surprising than it may seem at first: Theorem 1.9 (total equality conditions, [Sta81] and [Yan23]). Let M be a loopless regular matroid of rank r ≥2 with a ground set X, and let R ⊆X. Suppose that P(M, R, 0) > 0 and P(M, R, r) > 0. Then the following are equivalent : (i) P(M, R, 1)r = P(M, R, 0)r−1 P(M, R, r), (ii) P(M, R, a)2 = P(M, R, a + 1) P(M, R, a −1) for all a ∈{1, . . . , r −1}, (iii) P(M, R, a)2 = P(M, R, a + 1) P(M, R, a −1) for some a ∈{1, . . . , r −1}, (iv) \| ParM(x) ∩R\| = s \| ParM(x) ∩(X −R)\| for all x ∈X and some s > 0. Conjecture 1.10 ([Yan23, Conj. 3.40]). The conclusion of Theorem 1.9 holds for all loopless matroids. The equivalence (i) ⇔(ii) is the second part of Corollary 1.8 and holds for all matroids. The implication (ii) ⇒(iii) is trivial. The equivalence (i) ⇔(iv) was proved by Stanley for regular matroids [Sta81, Thm 2.8] (see also [Yan23, Thm 3.34]). Similarly, the implication (iii) ⇒(ii) was proved in [Yan23, Lem 3.39] for regular matroids. The implication (iv) ⇒(ii) was proved in [Yan23, Thm 3.41] for all matroids. The following result completely resolves the remaining implications of Yan’s Conjecture 1.10. Theorem 1.11. In the notation of Theorem 1.9, we have: (1) (i) ⇔(ii) ⇔(iv) for all loopless matroids, and (2) there exists a loopless binary matroid M s.t. (iii) holds but not (ii). The theorem is another example of the phenomenon that regular matroids satisfy certain ma-troid inequalities that general binary matroids do not (see e.g. [HSW22] and §15.4). The proof of Theorem 1.11 is given in Section 12, and is based on Theorem 1.3. The example in Theorem 1.11 part (2) can be found in §13.3. EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 5 1.7. Counting spanning trees. At a crucial step in the proof of Theorem 1.2, we give bounds for the relative version of the tree counting function, see below. This surprising obstacle occupies a substantial part of the proof (Sections 5 and 6). It is also of independent interest and closely related to the following combinatorial problem. Let G = (V, E) be a connected simple graph. Denote by τ(G) the number of spanning trees in G. Sedl´ aˇ cek [Sed70] considered the smallest number of vertices α(N) of a planar graph G with exactly N spanning trees: τ(G) = N.2 Theorem 1.12 (Stong [Sto22, Cor. 7.3.1]). For all N ≥3, there is a simple planar graph G with O (log N)3/2/(log log N) vertices and exactly τ(G) = N spanning trees. Until this breakthrough, even α(N) = o(N) remained out of reach, see [Aˇ S13]. As a warmup, Stong first proves this bound in [Sto22, Cor. 5.2.2], and this proof already involves a delicate number theoretic argument. Naturally, Stong’s Theorem 1.12 is much stronger. The following result is a variation of Stong’s theorem, and has the advantage of having an elementary proof. Theorem 1.13. For all N ≥3, there is a planar graph G with O(log N log log N) edges and exactly τ(G) = N spanning trees. Compared to Theorem 1.12, note that graphs are not required to be simple, but the upper bound in terms of edges is much sharper (in fact, it is nearly optimal, see §15.6). Indeed, the planarity implies the same asymptotic bound for the number of vertices and edges in G.3 Theorem 1.13 is a byproduct of the proof of the following lemma that is an intermediate result in the proof of Theorem 1.2. For an edge e ∈E, denote by G −e and G/e the deletion of e and the contraction along e. Define the spanning tree ratio as follows: ρ(G, e) := τ(G −e) τ(G/e) . Lemma 1.14 (spanning tree ratios lemma). Let A, B ∈N such that 1 ≤B ≤A ≤2B ≤N. Then there is a planar graph G with O (log N)(log log N)2 edges and ρ(G, e) = A/B spanning tree ratio. Note that the spanning tree ratios are not attainable by the tools in [Sto22]. This is why we need a new approach to the analysis of the spanning tree counting function, giving the proof of both Theorem 1.13 and Lemma 1.14 in Section 6. Our approach follows general outlines in [CP23a, CP24c], although technical details are largely different. Here we use a variation on the celebrated Haj´ os construction [Haj61] (see also [Urq97]), introduced in the context of graph colorings. Also, in place of the Yao–Knuth [YK75] “average case” asymptotics for continued fractions used in [CP23a], we use more delicate “best case” bounds by Larcher [Lar86]. Finally, note that the lemma gives the spanning tree ratio ρ(G, e) in the interval [1, 2]. In the proof of Theorem 1.2, we consider more general ratios. We are able to avoid extending Lemma 1.14 by combining combinatorial recurrences and complexity ideas. 1.8. Paper structure. In Section 2, we give an extensive historical background of many strains leading to the two main results (Theorems 1.2 and 1.3). The material is much too rich to give a proper review in one section, so we tried to highlight the results that are most relevant to our work, leaving unmentioned many major developments. In Section 3, we give basic definitions and notation, covering both matroid theory and compu-tational complexity. In a short Section 4, we give a key reduction to the SY equality problem from 2The original problem considered general rather than planar graphs, see §2.5. 3Although Stong does not explicitly mention planarity in [Sto22], his construction involves only planar graphs. 6 SWEE HONG CHAN AND IGOR PAK the matroid basis coincidence problem. In Sections 5 and 6 we relate spanning trees in planar graphs to continuous fractions, and prove Theorem 1.13 and Lemma 1.14 in §6.1 along the way. Sections 7 and 8 contain the proof of Theorem 1.2. Here we start with the proof of the Verification lemma (Lemma 7.1), which uses our spanning tree results and number theoretic estimates, and prove the theorem using a complexity theoretic argument. In Sections 9 and 10, we give a proof of Theorem 1.3. We start with an overview of our combinatorial atlas technology (Sections 9). We give a construction of the atlas for this problem in §10.1 and proceed to prove the theorem. These two sections are completely independent from the rest of the paper. In Section 11, we discuss vanishing conditions and prove Theorem 1.6. We give examples and counterexamples to equality conditions of (SY) in Section 13. In a short Section 14, we present the generalized Mason inequality, a natural variation of the Stanley–Yan inequality for the independent sets. We conclude with final remarks and several open problems in Section 15, all of them in connection with matroid inequalities and computational complexity. 2. Background 2.1. Log-concave inequalities. Log-concavity is a classical analytic property going back to Maclaurin (1929) and Newton (1732). Log-concavity is closely related to negative correlations, which also has a long history going back to Rayleigh and Kirchhoff, see e.g. [BBL09]. Log-concave inequalities for matroids and their generalizations (morphisms of matroids, antimatroids, greedoids) is an emerging area in its own right, see [CP24a, Yan23] for detailed overview. Stanley was a pioneer in the area of unimodal and log-concave inequalities in combinatorics, as he introduced both algebraic and geometric techniques [Sta89], see also [Br¨ a15, Bre89]. In [Sta81], he gave two applications of the Alexandrov–Fenchel (AF) inequality for mixed volumes of convex bodies, to log-concavity of combinatorial sequences. One is the (SY) for regular matroids. The other is Stanley’s poset inequality for the number of linear extensions [Sta81, Thm 3.1] that is extremely well studied in recent years, see a survey in [CP23b]. Among many variations, we note the Kahn–Saks inequality which was used to prove the first major breakthrough towards the 1 3 −2 3 conjecture [KS84]. Formally, let P = (X, ≺) be a poset with \|X\| = n elements. A linear extension of P is an order-preserving bijection f : X →[n]. Denote by E(P) the set of linear extensions of P. Fix x, z1, . . . , zk ∈X and a, c1, . . . , ck ∈[n]. Let Ezc(P, x, a) be the set of linear extensions f ∈E(P), s.t. f(x) = a and f(zi) = ci for all 1 ≤i ≤k. Stanley’s poset inequality is the log-concavity of numbers Nzc(P, x, a) := \| Ezc(P, x, a)\| : (Sta) Nzc(P, x, a)2 ≥Nzc(P, x, a + 1) · Nzc(P, x, a −1). These Stanley’s inequalities (SY) and (Sta) have superficial similarities as they were obtained in the same manner, via construction of combinatorial polytopes whose volumes and mixed volumes have a combinatorial interpretation. For regular matroids, Stanley used zonotopes spanned by the vectors of a unimodular representation, while for posets he used order polytopes [Sta86]. Partly motivated by Stanley’s paper, Schneider [Sch88] gives equality conditions for the AF inequality in case of zonotopes. In principle, one should be able to derive Theorem 1.3 in the case of regular matroids from Schneider’s result as well. A word of caution: Although it may seem that inequalities (Sta) and (SY) are both consequences of the AF inequality, and that this paper and [CP23a] cover the same or similar ground, in fact the opposite is true. While (Sta) is a direct consequence of the AF inequality, only the computationally easy part of the (SY) follows from the AF inequality. It took Lorentzian polynomials to prove the computationally hard part of (SY). See Proposition 15.5 for the formal statement. In [Sta81], Stanley asked for equality conditions for both matroid and poset inequalities that he studied. He noted that the AF inequality has equality conditions known only in a few special cases. He used one such known special case (dating back to Alexandrov), to describe equality EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 7 cases of his matroid log-concave inequality for regular matroids (Theorem 1.9). The equality conditions for (Sta) are now largely understood, see below. Stanley’s inequality (SY) led to many subsequent developments. Notably, Godsil [God84] resolved Stanley’s question to show that the generating polynomial X a PSc(M, R, a)ta has only real nonpositive roots (this easily implies log-concavity). Choe and Wagner [CW06] proved that {PSc(M, R, a) : 0 < a < r} is log-concave for a larger family of matroids with the half-plane property (HPP), see also [Br¨ a15, §9.1]. In a remarkable series of papers, Huh and coauthors developed a highly technical algebraic approach to log-concave inequalities for various classes of matroids, see an overview in [Huh18, Kal23]. Most famously, Adiprasito, Huh and Katz [AHK18], proved a number of log-concave inequalities for general matroids, some of which were conjectured many decades earlier. These results established log-concavity for the number of independent sets of a matroid according to the size (Mason–Welsh conjecture implied by the (SY) inequality, see below), and of the coefficients of the characteristic polynomial (Heron–Rota–Welsh conjecture). After that much progress followed, eventually leading to the proof of a host of other matroid inequalities. 2.2. Lorentzian polynomials. Lorentzian polynomials were introduced by Br¨ and´ en and Huh [BH20], and independently by Anari, Oveis Gharan and Vinzant [AOV18]. This approach led to a substantial extension of earlier algebraic and analytic notions, as well as a major simplification of the earlier proofs. Specifically, they showed that the homogeneous multivariate Tutte polynomial of a matroid is a Lorentzian polynomial (see [ALOV24, Thm 4.1] and [BH20, Thm 4.10]). This implied the ultra-log-concave inequality conjectured by Mason, i.e. the strongest of the Mason’s conjectures. Formally, let I(M, a) denotes the number of independent sets in matroid M of size a. Mason’s weakest conjecture (the Mason–Welsh conjecture mentioned above) is the log-concave inequality (M1) I(M, a)2 ≥I(M, a + 1) I(M, a −1) for all 1 ≤a ≤(r −1). Similarly, Mason’s strongest conjecture (we skip the intermediate one), is the ultra-log-concave inequality (M2) I(M, a)2 ≥ 1 + 1 a 1 + 1 n−a I(M, a + 1) I(M, a −1) for all 1 ≤a ≤(r −1), where n = \|X\| is the size of the ground set (see Section 14). Most recently, Yan [Yan23] used Lorentzian polynomials to extend Stanley’s result from regular to general matroids (Theorem 1.1). The resulting Stanley–Yan inequality (SY) is one of the most general matroid results proved by a direct application of Lorentzian polynomials, and it easily implies (M1) (we include this argument in §14 for the reader’s convenience). 2.3. Later developments. Recently, the authors introduced a linear algebra based combinatorial atlas technology in [CP24a], which includes Lorentzian polynomials as a special case [CP22a, §5]. The authors proved equality conditions and various extensions of both Mason’s ultra-log-concave inequality (for the number of independent sets of matroids), and for Stanley’s poset inequality (Sta). Most recently, the authors used combinatorial atlases to establish correlation inequalities for the numbers of linear extensions [CP24b]. These results parallel earlier correlation inequalities by Huh, Schr¨ oter and Wang [HSW22]. In a separate development, Br¨ and´ en and Leake introduced Lorentzian polynomials on cones [BL23]. They were able to give an elementary proof of the Heron–Rota–Welsh conjecture. Note that both combinatorial atlas and this new technology give new proofs of the Alexandrov–Fenchel inequality, see [CP22a, §6] and [BL23, §6]. This is a central and most general inequality in convex geometry, with many proofs none of which are truly simple, see e.g. [BZ88, §20]. 8 SWEE HONG CHAN AND IGOR PAK Shenfeld and van Handel [SvH23] undertook a major study of equality cases of the AF inequality. They obtained a (very technical) geometric characterization in the case of convex polytopes, making a progress on a long-standing open problem in convex geometry, see [Sch85]. They gave a complete description of equality cases for (Sta) in the k = 0 case and showed that the equality decision problem is in P. The k = 0 equality cases of (Sta) were rederived in [CP24a] using combinatorial atlas, where the result was further extended to weighted linear extensions. Shenfeld and van Handel’s approach was further extended is in [vHYZ23] to the Kahn–Saks inequality (a diagonal slice of the k = 1 case), and in [CP23a] to the full k = 1 case of (Sta). For general k ≥2, Ma and Shenfeld [MS24] gave a technical combinatorial description of the equality cases of (Sta). Notably, this description involves #P oracles, and therefore is not naturally in PH. 2.4. Negative results. In a surprising development, the authors in [CP23a] showed that for k ≥2, the equality conditions of Stanley’s poset inequality are not in PH unless PH collapses to a finite level. In particular, this implied that the equality cases of the AF inequality for H-polytopes with a concise description, are also not in PH, unless PH collapses to a finite level. Prior to [CP23a], there were very few results on computational complexity of (equality cases) of combinatorial inequalities. The approach was introduced by the second author in [Pak19], as a way to show that certain combinatorial numbers do not have a combinatorial interpretation. This was formalized as counting functions not being in #P, see survey [Pak22]. Various examples of functions not in #P were given in [IP22], based on an assortment of complexity theoretic assumptions. It was shown by Ikenmeyer, Panova and the second author in [IPP24], that vanishing of Sn characters problem [χλ(µ) =? 0] is C=P-complete. This implies that this problem is not in PH unless PH collapses to the second level (ibid.). Finally, a key technical lemma in [CP23a] is based on the analysis of the combinatorial coincidence problem. This is a family of decision problems introduced and studied in [CP24d]. They are also characterised by a collapse of PH. 2.5. Spanning trees. Sedl´ aˇ cek [Sed70] and Azarija–ˇ Skrekovski [Aˇ S13] considered two closely related functions α′(N) and β′(N), defined to be the minimal number of vertices and edges, respectively, over all (i.e., not necessarily planar) graphs G with τ(G) = N spanning trees. For connected planar graphs, the number of edges is linear in the number of vertices, so this distinction disappears. In recent years, there were several applications of continued fractions to problems in combina-torics. Notably, Kravitz and Sah [KS21] used continuous fractions to study a similar problem for the number \| E(P)\| of linear extensions of a poset, see also [CP24c]. An earlier construction by Schiffler [Sch19] which appeared in connection with cluster algebras, related continued fractions and perfect matchings. We also mention a large literature on enumeration of lattice paths via continued fractions, see e.g. [GJ83, Ch. 5]. 2.6. Counting complexity. The problem of counting the number of bases and more generally, the number of independent sets of given size, been heavily studied for various classes of matroids. Even more generally, both problems are evaluations of the Tutte polynomial, and other evaluations have also been considered. We refer to [Wel93] for both an introduction to the subject and a detailed, though somewhat dated, survey of known results. Among more recent work, let us mention #P-completeness for the number of trees (of all sizes) in a graph [Jer94], the number of bases in bicircular matroids [GN06], in balanced paving matroids [Jer06], rational matroids [Sno12], and most recently in binary matroids4 [KN23]. We also note that the volumes of both order polytopes and zonotopes are #P-hard, see [BW91, §3] and [DGH98, Thm 1]. See §15.4 for further results and applications. 4There is a mild controversy over priority of this result, see a short discussion in [CP24d, §6.3]. EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 9 Finally, in a major breakthrough, Anari, Liu, Oveis Gharan and Vinzant [ALOV19], used Lorentzian polynomials to prove that the basis exchange random walk mixes in polynomial time. This gave a FPRAS for the number of bases of a matroid, making a fast probabilistic algorithm for approximate counting of bases. This resolved an open problem by Mihail and Vazirani (1989). Previously, FPRAS for the number of bases was known for regular matroids [FM92], paving matroids [CW96], and bicircular matroids [GJ21]. 3. Notations and definitions 3.1. Basic notation. Let N = {0, 1, 2, . . .} and [n] = {1, . . . , n}. For a set A and an element x / ∈A, we write A + x := A ∪{x}. Similarly, for an element x ∈A, we write A −x := A \ {x}. We use bold letters a = (a1, a2, . . .) to denote a sequence of integers, and A = (A1, . . . , An) to denote a sequence of sets. We write v = (v1, . . . , vd) ∈kd to denote vectors over the field k. Let e1, . . . , ed denote the standard basis in kd, and let 0 = (0, . . . , 0) ∈kd. Let Fq to denote the finite field with q elements. We say that v ∈Rd is strictly positive if vi > 0 for all 1 ≤i ≤d. For a = (a1, . . . , ad) ∈Nd, denote \|a\| := a1 + . . . + ad. The support of a vector v = (v1, . . . , vd) ∈Rd is the set of indices i ∈[d] such that vi ̸= 0. The support of a symmetric vector M = (Mi,j)i,j∈[d] is the set of indices i ∈[d] such that Mij ̸= 0 for some j ∈[d]. 3.2. Matroids. A (finite) matroid M is a pair (X, I) of a ground set X with \|X\| = n elements, and a nonempty collection of independent sets I ⊆2X that satisfies the following: • (hereditary property) A ⊂B, B ∈I ⇒ A ∈I , and • (exchange property) A, B ∈I, \|A\| < \|B\| ⇒ ∃x ∈B \ A s.t. A + x ∈I . The rank of a matroid is the maximal size of the independent set, i.e., rk(M) := maxA∈I \|A\|. More generally, the rank rk(A) of a subset A ⊆X is the size of the largest independent set contained in A. A basis of M is a maximal independent set of M, or equivalently an independent set with size rk(M). We denote by B(M) the set of bases of M. An element x ∈X is a loop if {x} / ∈I, and is a non-loop otherwise. Matroid without loops is called loopless.5 We denote by NL(M) the set of non-loops of M. Two non-loops x, y ∈NL(M) are parallel if {x, y} / ∈I. Note that the parallelship relation between non-loops is an equivalence relation (see e.g. [CP24a, Prop 4.1]). The equivalence classes of this relation are called parallel classes. Given matroids M = (X, I), M′ := (X′, I′), the direct sum M⊕M′ := (Y, J ) is a matroid with ground set Y = X ⊔X′, and whose independent sets A ∈J are disjoint unions of independent sets: A = I ⊔I′, I ∈I, I′ ∈I′. Let x ∈NL(M). The deletion M −x is the matroid with ground set X and with independent sets {A ⊆X −x : A ∈I}. The contraction M/x is the matroid with ground set X and with independent sets {A ⊆X −x : A + x ∈I}. Note that both M/x and M −x share the same ground set as M. This is slightly different than the usual convention, and is adopted here for technical reasons that will be apparent in Section 10. More generally, for B ⊆NL(M), the contraction M/B is the matroid with ground set X and with independent sets {A ⊆X −B : A ∪B ∈I} . Recall the deletion–contraction recurrence for the number of bases of matroids: B(M) = B(M −x) + B(M/x). A representation of a matroid M over the field k is a map ϕ : X →kd, such that A ∈I ⇐ ⇒ ϕ(x1), . . . , ϕ(xm) are linearly independent over k, for every subset A = {x1, . . . , xm} ⊆X. A matroid is binary if it has a representation over F2. Matroid is rational if it has a representation over Q. 5Unless stated otherwise, we allow matroids to have loops and parallel elements. 10 SWEE HONG CHAN AND IGOR PAK A matroid is regular (also called unimodular), if it has a representation over every field k. Representation ϕ : X →Zd is called unimodular if det ϕ(x1), . . . , ϕ(xr) = ±1, for every basis {x1, . . . , xr} ∈B(M). Regular matroids are known to have a unimodular representation (see e.g. [Ox11, Lem. 2.2.21]). Let G = (V, E) be a finite connected graph, and let F be the set of forests in G (subsets F ⊆E with no cycles). Then MG = (E, F) is a graphical matroid corresponding to G. Bases of the graphical matroid MG are the spanning trees in G, so B(MG) = τ(G). Recall that graphical matroids are regular. 3.3. Complexity. We refer to [AB09, Gol08, Pap94] for definitions and standard results in com-putational complexity, and to [Aar16, Wig19] for a modern overview. We assume that the reader is familiar with basic notions and results in computational com-plexity and only recall a few definitions. We use standard complexity classes: P, FP, NP, coNP, #P, Σp m, PH and PSPACE. The notation {a =? b} is used to denote the decision problem whether a = b. We use the oracle notation RS for two complexity classes R, S ⊆PH, and the polynomial closure ⟨A⟩for a problem A ∈PSPACE. For a counting function f ∈#P, the coincidence problem is defined as f(x) =? f(y) . Note the difference with the equality verification problem f(x) =? g(x) . Unless stated otherwise, we use reduction to mean a polynomial Turing reduction. 4. Reduction from coincidences 4.1. Setup. Let M = (X, I) be a binary matroid, let x ∈X be a non-loop: x ∈NL(M). Define the basis ratio ρ(M, x) := B(M −x) B(M/x) . Denote by #Bases the problem of computing the number of bases B(M) in M. Similarly, denote by #BasisRatio the problem of computing the basis ratio ρ(M, x). Let M = (X, I) and N = (Y, I′) be binary matroids, let x ∈X and y ∈Y be non-loop elements: x ∈NL(M), y ∈NL(N). Consider the following decision problem: BasisRatioCoincidence := ρ(M, x) =? ρ(N, y) . The following is the main technical lemma in the proof. Lemma 4.1. BasisRatioCoincidence reduces to EqualitySY1. The lemma follows from two parsimonious reductions presented below. 4.2. Deletion-contraction coincidences. Let M = (X, I) be a binary matroid, and let x, y ∈ X be non-parallel and non-loop elements. Consider the following decision problem: CoincidenceDC := B(M/x −y) =? B(M/y −x) . Lemma 4.2. CoincidenceDC reduces to EqualitySY1. Proof. Let ϕ : X →Fd 2 be a binary representation of M. Let X′ := X ∪{u, v}, where u, v are two new elements. Consider a matroid M′ = (X′, I′) defined by its binary representation ϕ′ : X′ →Fd+1 2 , where ϕ′(z) := ( (ϕ(z), 0) for z ∈X, (0, 1) for z ∈{u, v}. That is, we append a zero to the vector representation of all z ∈X, and we represent u, v by the basis vector ed+1. EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 11 Let r := rk(M) be the rank of M, and let n := \|X\| be the number of elements. Note that M′ is a matroid of rank r + 1 and with n + 2 elements. Note also that the bases of M′ are of the form A + u and A + v, where A ∈B(M) is a basis of M. To define the reduction in the lemma, let R := {x, u}, a := 1, S := X −{x, y}, c := r −1. It then follows that BSc(M′, R, a + 1) = B(M/x −y). Indeed, BSc(M′, R, a + 1) are subsets A ⊆X ∪{u, v} that are of the form A ∩R = {x, u}, A ∩{y, v} = ∅, A −{u} ∈B(M). It then follows that A −{x, u} is a basis of M/x −y, and that this correspondence is a bijection, proving our claim. By the argument as above, we have: BSc(M′, R, a) = B(M/x −y) + B(M/y −x), BSc(M′, R, a −1) = B(M/y −x). We have: PSc(M′, R, a)2 −PSc(M′, R, a + 1) PSc(M′, R, a −1) = 1 r2(r+1)2 BSc(M′, R, a)2 −4 BSc(M′, R, a + 1) BSc(M′, R, a −1) = 1 r2(r+1)2 B(M/x −y) + B(M/y −x) 2 −4 B(M/x −y) B(M/y −x) = 1 r2(r+1)2 B(M/x −y) −B(M/y −x) 2. Therefore, we have: PSc(M′, R, a)2 = PSc(M′, R, a + 1) PSc(M′, R, a −1) ⇐ ⇒ B(M/y −x) = B(M/y −x), which completes the proof of the reduction. □ 4.3. Back to ratio coincidences. Lemma 4.1 now follows from the following reduction. Lemma 4.3. BasisRatioCoincidence reduces to CoincidenceDC. Proof. Let M, N, x, y be the input of BasisRatioCoincidence. Let M′ := M ⊕N be the direct sum of matroids M and N. Note that M′ is also binary. We have: B(M′/x −y) = B(M/x) B(N −y) and B(M′/y −x) = B(M −x) B(N/y), which proves the reduction. □ 5. Planar graphs and continued fractions 5.1. Graph theoretic definitions. Throughout this paper G = (V (G), E(G)) will be a graph with vertex set V (G) and edge set E(G), possibly with loops and parallel edges. We will write V and E when the underlying graph G is clear from the context. For an edge e = (v, w) ∈E, the deletion G−e is the graph obtained by deleting the edge e from the graph, and the contraction G/e is the graph obtained by identifying v and w, and removing the resulting loops. Recall that τ(G) denotes the number of spanning trees in G. Note that τ(G) satisfies the deletion-contraction recurrence for every non-loop e ∈E: τ(G) = τ(G −e) + τ(G/e). Let G = (V, E) be a planar graph. For every planar embedding of G, the dual graph G∗= (V ∗, E) is the graph where vertices of G∗are faces of G, and each edge is incident to faces of G that are separated from each other by the edge in the planar embedding. While the dual graph 12 SWEE HONG CHAN AND IGOR PAK G∗can depend on the given planar embedding of G, we will not emphasize that as our proof is constructive and the embedding will be clear from the context. Note that deletion and contraction for dual graphs swap their meaning. Formally, for an edge e ∈E that is neither a bridge nor a loop, we have: (5.1) τ(G −e) = τ(G∗/e) and τ(G/e) = τ(G∗−e). Therefore, ρ(G∗, e) = ρ(G, e)−1. 5.2. Continued fraction representation. Given a0 ≥0 , a1, . . . , as ≥1 , where s ≥0, the corresponding continued fraction is defined as follows: [a0; a1, . . . , as] := a0 + 1 a1 + 1 ... + 1 as . Integers ai are called quotients or partial quotients, see e.g. [HW08, §10.1]. We refer to [Knu98, §4.5.3] for a detailed asymptotic analysis of the quotients in connection with the Euclidean algo-rithm, and further references. The following result gives a connection between spanning trees and continued fractions. It is inspired by a similar construction for perfect matchings given in [Sch19, Thm 3.2]. Theorem 5.1. Let a0, . . . , as ≥1 . Then there exists a connected loopless bridgeless planar graph G = (V, E) and an edge e ∈E, such that τ(G −e) τ(G/e) = [a0; a1, . . . , as] and \|E\| = a0 + . . . + as + 1 . We start with the following lemma. Lemma 5.2. Let G = (V, E) be a connected loopless bridgeless planar graph, and let e ∈E. Then there exists a connected loopless bridgeless planar graph G′ = (V ′, E′) and e′ ∈E′ such that τ(G′ −e′) τ(G′/e′) = 1 + τ(G −e) τ(G/e) and \|E′\| = \|E\| + 1. Proof. Let G′ be obtained from G by adding an edge e′ that is parallel to e. Note that G′ −e′ is isomorphic to G, and G′/e′ is isomorphic to G/e, and it follows that τ(G′/e′) = τ(G/e) and τ(G′ −e′) = τ(G) = τ(G −e) + τ(G/e). This implies the lemma. □ 5.3. Proof of Theorem 5.1. We use induction on s. For s = 0, let H be the graph with two vertices and with a0 + 1 parallel edges connecting the two vertices, and let f be any edge of H. Note that H −f is the same graph but with a0 edges instead, while H/f is the graph with one vertex and a0 loops. Thus we have τ(H −f) = a0 and τ(H/f) = 1. We also have \|E(H)\| = a0 + 1. It then follows that τ(H −f) τ(H/f) = a0, and the claim follows by taking G ←H and e ←f. EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 13 For s ≥1, by induction there exists a connected loopless bridgeless planar graph H and f ∈E(H) such that τ(H −f) τ(H/f) = [a1; a2, . . . , as], and with \|E(H)\| = a1 + . . . + as + 1. Now, by applying Lemma 5.2 a0 many times to H∗, there exists a graph G and a e ∈E(G) such that τ(G −e) τ(G/e) = a0 + τ(H∗−f) τ(H∗/f) = a0 + τ(H/f) τ(H −f) = a0 + 1 [a1; a2, . . . , as] = [a0; a1, . . . , as], and with \|E(G)\| = a0 + \|E(H∗)\| = a0 + . . . + as + 1. This completes the proof. □ 5.4. Sums of continued fractions. We now extend Theorem 5.1 to sums of two continued fractions: Theorem 5.3. Let a0, . . . , as, b0, . . . , bt ≥1 . Then there exists a connected loopless bridgeless planar graph G = (V, E) and an edge e ∈E, such that τ(G −e) τ(G/e) = 1 [a0; a1, . . . , as] + 1 [b0; b1, . . . , bt] and \|E\| = a0 + . . . + as + b0 + . . . + bt + 1 . We start with the following lemma. Lemma 5.4. Let G, H be connected loopless bridgeless planar graphs, and let e ∈E(G), f ∈ E(H). Then there exists a connected loopless bridgeless planar graph G′ and an edge e′ ∈E(G′), such that τ(G′ −e′) τ(G′/e′) = τ(G −e) τ(G/e) + τ(H −f) τ(H/f) and E(G′) = E(G) + E(H) −1. Proof. Let e = (x, y) ∈E(G) and let f = (u, v) ∈E(H). Consider the graph G′ := G ⊕H /(x, u), (y, v) obtained by taking the disjoint union of G and H, then identifying e and f. Denote by e′ ∈E(G′) the edge resulted from identifying e and f. First, note that τ(G′/e′) = τ(G/e) τ(H/f). (5.2) This is because G′/e′ = (G/e ⊕H/f)/(x, u), i.e. can be obtained by identifying x with u in the disjoint union of G/e and H/f. Second, note that τ(G′ −e′) = τ(G −e) τ(H/f) + τ(G/e) τ(H −f). (5.3) Indeed, let T be a spanning tree of G′−e′. There are two possibilities. First, x and y are connected in T through a path in G. Then, restricting T to edges of G gives us a spanning tree in G −e, while restricting T to edges of H gives us a spanning tree of H/f. This bijection gives us the first term in the RHS of (5.3). Second, suppose that x and y are connected in T through a path in H. Then, restricting T to edges of G gives us a spanning tree in G/e, while restricting T to edges of H gives us a spanning tree of H −f. This bijection gives us the second term in the RHS of (5.3). The lemma now follows by combining (5.2) and (5.3). □ 14 SWEE HONG CHAN AND IGOR PAK 5.5. Proof of Theorem 5.3. By Theorem 5.1, there exists connected loopless bridgeless planar graphs G, H and e ∈E(G), f ∈E(H) such that τ(G −e) τ(G/e) = [a0; a1, . . . , as], τ(H −f) τ(H/f) = [b0; b1, . . . , bt], and with \|E(G)\| = a0 + . . . + as + 1, \|E(H)\| = b0 + . . . + bt + 1. Applying Lemma 5.4 to (G∗, e) and (H∗, f), gives a planar graph G′ and e′ ∈E(G′), such that τ(G′ −e′) τ(G′/e′) = τ(G∗−e) τ(G∗/e) + τ(H∗−f) τ(H∗/f) = τ(G/e) τ(G −e) + τ(H/f) τ(H −f) = 1 [a0; a1, . . . , as] + 1 [b0; b1, . . . , bt] and E(G′) = E(G∗) + E(H∗) −1 = a0 + . . . + as + b0 + . . . + bt + 1, as desired. □ 6. Counting spanning trees In this section, we prove Theorem 1.13 and Lemma 1.14. 6.1. Proof of Theorem 1.13. For α ∈Q>0, consider the sum of the quotients of α : s(α) := a0 + . . . + as where α = [a0; a1, . . . , as]. We will need the following theorem from number theory. Theorem 6.1 (Larcher [Lar86, Cor. 2]). For m ≥9 and L ≥2, the set n d ∈[m] : gcd(d, m) = 1 and s d m ≤L m ϕ(m) log m log log m o contains at least 1 −16 √ L ϕ(m) many elements, where ϕ is the Euler’s totient function. First, assume that N is prime and note that ϕ(N) = N −1. By Larcher’s Theorem 6.1, there exists d < N such that s d N ≤C log N log log N for some C > 0. By Theorem 5.1 and planar duality (5.1), there exists a planar graph G = (V, E) and edge e ∈E, such that τ(G −e) τ(G/e) = N d and \|E(G)\| ≤1 + C log N log log N . The conclusion follows by taking (G −e). In full generality, let N = pb1 1 · · · pbℓ ℓ be the prime factorization of N. Let Gi = (Vi, Ei), 1 ≤i ≤ℓ, be the planar graphs constructed above: τ(Gi) = pi and \|Ei\| ≤C log pi log log pi . Finally, let G = (V, E) be a union of bi copies of Gi attached at vertices, so that G is planar and connected. Clearly, τ(G) = N and \|E\| ≤ ℓ X i=1 bi C log pi log log pi ≤ ℓ X i=1 bi log pi ! C log log N = C log N log log N , as desired. □ EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 15 6.2. Number theoretic estimates. We start with the following number theoretic estimates that is based on Larcher’s Theorem 6.1. Proposition 6.2. There exists constants C, K > 0, such that for all coprime integers A, B which satisfy C < B ≤A ≤2B, there exists a positive integer m := m(A, B) such that m < B, and s m A ≤K(log A)(log log A)2 and s B−m A ≤K(log A)(log log A)2. Proof of Proposition 6.2. Define ζ(A, B) := m ∈[B] : s m A ≤K(log A)(log log A)2, s B−m A ≤K(log A)(log log A)2 . We will prove a stronger claim, that ζ(A, B) = Ω B as C →∞. It follows from the inclusion-exclusion, that ζ(A, B) ≥B − m ∈[B] : s m A > K(log A)(log log A)2 − m ∈[B] : s B−m A > K(log A)(log log A)2 . On the other hand, we have {m ∈[B] : s m A > K(log A)(log log A)2 } ≤ {m ∈[A] : s m A > K(log A)(log log A)2 } ≤ {m ∈[A] : s m A > K 2 A ϕ(A) log A log log A } ≤ 0.2A, where the second inequality is because A ϕ(A) < 2 log log A for sufficiently large A, and the third inequality is because of Larcher’s Theorem 6.1. Similarly, we have {m ∈[B] : s B−m A > K log A (log log A)2 } ≤0.2A. Combining these inequalities, we get ζ(A, B) ≥B −0.4 A ≥0.2B, and the result follows. □ 6.3. Proof of Lemma 1.14. It follows from Proposition 6.2, that there exists fixed K > 0 and an integer m < B, such that s m A ≤K(log A)(log log A)2 and s B−m A ≤K(log A)(log log A)2. Let [a0, . . . , as] and [b0, . . . , bt] be a continued fraction representation of A/m and A/(B −m), respectively. By Theorem 5.3, there exists a connected loopless bridgeless planar graph G and an edge e ∈E(G), such that τ(G −e) τ(G/e) = 1 [a0; a1, . . . , as] + 1 [b0; b1, . . . , bt] = B A and \|E(G)\| = s m A + s B−m A + 1 ≤2K(log A)(log log A)2 + 1 = O (log N)(log log N)2 . Taking the dual graph G∗gives the result. □ 7. Verification of matroid basis ratios Throughout this and the next section, we assume that all matroids are binary and given by their binary representations. 16 SWEE HONG CHAN AND IGOR PAK 7.1. Setup. Let M = (X, I) be a binary matroid, let x ∈NL(M), and let A, B ∈N, where B > 0. Consider the following decision problem: BasisRatioVerification := n ρ(M, x) =? A B o . Lemma 7.1 (verification lemma). NP⟨BasisRatioVerification⟩⊆NP⟨BasisRatioCoincidence⟩. The proof is broadly similar to that in [CP23a], but with major technical differences. We start with the following simple result. Lemma 7.2. Let M = (X, I) be a matroid on n = \|X\| elements, and let x ∈X be a non-loop of M. Then ρ(M, x) ≤n. Proof. To prove that ρ(M, x) = B(M −x) B(M/x) ≤n, we construct an explicit injection γ : B(M −x) →B(M/x) × X. Fix a basis A ∈B(M) such that x ∈A. Such basis A exists since x is not a loop. By the symmetric basis exchange property, for every basis B ∈B(M) such that x / ∈B, there exists y ∈B, such that B′ := B −y + x is a basis of M. Now take the lex-smallest of such y, and define γ(B) := (B′, y). Note that map γ is an injection because B can be recovered from (B′, y) by taking B = B′ −x + y. This completes the proof. □ 7.2. Proof of Lemma 7.1. We now simulate BasisRatioVerification with an oracle for BasisRatioCoincidence as follows. Let M = (X, I) be a binary matroid of rank rk(M) = r on n = \|X\| elements. Let x ∈ NL(M) and A, B ∈N, where B > 0. We can assume that A ≥1, as otherwise A = 0 and BasisRatioVerification is then equivalent with checking if B(M −x) = 0 , i.e. checking if M −x has rank (r −1). Without loss of generality we can assume that integers A and B are coprime. Since we have B(M −x) ≤ n r and B(M/x) ≤ n r , we can also assume that (7.1) 1 ≤A, B ≤ n r , as otherwise BasisRatioVerification fails. Similarly, by Lemma 7.2 we can also assume that (7.2) A B ≤n. Let A′ be the positive integer given by A′ := B + A − A B B. Note that B ≤A′ ≤2B. From this point on we proceed following the proof of Lemma 1.14. It follows from Proposition 6.2 that there exists fixed K > 0 and an integer m < B such that s m A′ ≤K log A′ (log log A′)2 and s B−m A′ ≤K log A′ (log log A′)2. At this point we guess such m. Since computing the quotients of m/A′ can be done in polynomial time, we can verify in polynomial time that m satisfies the inequalities above. Let [a0, . . . , as] and [b0, . . . , bt] be a continued fraction representation of A′/m and A′/(B−m), respectively. By Theorem 5.3, there exists a connected loopless bridgeless planar graph G and an edge e ∈E(G) such that τ(G −e) τ(G/e) = 1 [a0; a1, . . . , as] + 1 [b0; b1, . . . , bt] = B A′ EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 17 and \|E(G)\| = s m A′ + s B−m A′ + 1 ≤2K log A′ (log log A′)2 + 1 ≤(7.1) 2K log 2 n r log log 2 n r 2 + 1 = O n (log n)2 . Let G′ and e′ be the graph and edge obtained by applying Lemma 5.2 for ⌊A/B⌋−1 many times to the planar dual G∗of G. Then we have τ(G′ −e′) τ(G′/e′) = A B −1 + τ(G∗−e) τ(G∗/e) = A B −1 + A′ B = A B and E(G′) = A B −1 + \|E(G)\| ≤(7.2) n −1 + \|E(G)\| = O(n (log n)2). Now, let N := (E(G′), J ) be the graphical matroid corresponding to G′, where J are spanning forests in G′, and let y = e. Then we have ρ(N, y) = ρ(G′, e′) = τ(G′ −e′) τ(G′/e′) = A B . Thus, the decision problem BasisRatioVerification with input M, x, A, B can be simulated by BasisRatioCoincidence with input M, x, N, y. This completes the proof. □ 8. Proof of Theorem 1.2 8.1. Two more reductions. We also need two minor technical lemmas: Lemma 8.1. For all k > ℓ, EqualitySYℓreduces to EqualitySYk . Proof of Lemma 8.1. Let M, R, a, S = (S1, . . . , Sℓ), c = (c1, . . . , cℓ) be an input EqualitySYℓ. Let Sℓ+1 = . . . = Sk = ∅and cℓ+1 = . . . = ck = 0. Let S′ := (S1, . . . , Sk) and c′ := (c1, . . . , ck). It then follows that PS′c′(M, R, a) = PSc(M, R, a). We conclude that the decision problem EqualitySYℓwith input M, R, a, S, c, is equivalent to the decision problem EqualitySYk with input M, R, a, S′, c′. □ Lemma 8.2. #Bases is polynomial time equivalent to #BasisRatio . Proof of Lemma 8.2. Note that #BasisRatio reduces to #Bases by definition. In the opposite direction, let M be a binary matroid of rank r = rk(M). Compute a basis {x1, . . . , xr} of M by a greedy algorithm. Denote by Mi the contraction of M by {x1, . . . , xi}. We have B(M) = B(M0) B(M1) · B(M1) B(M2) · · · = 1 + B(M0 −x1) B(M0/x1) 1 + B(M1 −x2) B(M1/x2) · · · , which gives the desired reduction. □ 8.2. Putting everything together. First, we need the following recent result: Theorem 8.3 (Knapp–Noble [KN23, Thm 53]). #Bases is #P-complete for binary matroids. By Lemma 8.2, we conclude that #BasisRatio is #P-hard. We then have: (8.1) PH ⊆P#P ⊆P⟨#BasisRatio⟩⊆NP⟨BasisRatioVerification⟩, where the first inclusion is Toda’s theorem [Toda91], the second inclusion is because #BasisRatio is #P-hard, and the third inclusion is because one can simulate #BasisRatio by first guessing and then verifying the answer. 18 SWEE HONG CHAN AND IGOR PAK We now have: (8.2) NP⟨BasisRatioVerification⟩⊆NP⟨BasisRatioCoincidence⟩ ⊆NP⟨EqualitySY1⟩⊆NP⟨EqualitySYk⟩ where the first inclusion is the Verification Lemma 7.1, the second inclusion is Lemma 4.1, and the third inclusion is Lemma 8.1. Now, suppose EqualitySYk ∈PH. Then EqualitySYk ∈Σp m for some m. Combining (8.1) and (8.2), this implies: (8.3) PH ⊆NP⟨EqualitySY1⟩⊆NPΣp m ⊆Σp m+1 , as desired. □ 9. Combinatorial atlas In this section we give a brief review of the theory of combinatorial atlas. This is the main tool used to prove Theorem 1.3. We will be concise in our explanation of this tool for the sake of brevity and refer the reader to [CP22a, §3, §4] for more detailed introduction, and to [CP24a] for an even more in-depth discussion on this topic. 9.1. Informal overview. On a basic level, an inequality a2 ≥bc can be viewed as a result that det(M) ≤0 where M := a b c a . In other words, the inequality says that the eigenvalues of M can be zero, but cannot be of the same sign. In a similar vein, log-concavity of a sequence can be reformulated as a claim that a certain matrix has one positive eigenvalue (OPE). Constructing such a matrix is rather technical, and for the sequence {P(M, R, a)} this will be done in the next section. Proving that a matrix has (OPE) is difficult, and is done essentially by a strong induction. In this section, we present the setup (combinatorial atlas) which allows us to formalize this induction and obtain (OPE) in Theorem 9.3 (proved in our prior work). An inelegant but unavoidable feature of the setup is the one-parameter deformation which allows us to prove the result for matrices with no zero eigenvalues, and taking the limit later. Note that we do not need to reprove the Stanley–Yan inequality (SY), even though our con-struction does give an independent proof, see Proposition 10.5. An important feature of the combinatorial atlas is its ability to also give equality conditions in certain cases. This requires further assumptions and a technical Theorem 9.6 (also proved in our prior work). We include this setup in this section before using it to prove Theorem 1.3 in the next section. 9.2. The setup. Let Γ = (Ω, Θ) be a (possibly infinite) acyclic digraph. We denote by Ω0 ⊆Ω the set of sink vertices in Γ (i.e. vertices without outgoing edges). Similarly, denote by Ω+ := Ω∖Ω0 the non-sink vertices. We denote by Dv the set of out-neighbors of v, i.e. vertices v′ ∈Ω such that (v, v′) ∈Θ6. Definition 9.1. Let d be a positive integer. A combinatorial atlas A of dimension d is an acyclic digraph Γ := (Ω, Θ) with an additional structure: • Each vertex v ∈Ωis associated with a pair (Mv, hv), where Mv is a nonnegative symmetric d × d matrix, and hv ∈Rd ≥0 is a nonnegative vector. • Each vertex v ∈Ω+ has out-degree equal to d, and the i-th edge is labeled e⟨i⟩= (v, v⟨i⟩) for 1 ≤i ≤d. • Each edge e⟨i⟩is associated to a linear transformation T⟨i⟩ v : Rd →Rd. 6The set Dv was denoted by v∗in [CP22a, CP24a]. EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 19 We call Mv = (Mij)i,j∈[d] the associated matrix of v, and h = hv = (hi)i∈[d] the associated vector of v. In notation above, we have v⟨i⟩∈Dv, for all 1 ≤i ≤d. A common objective of the setup above is to demonstrate that the matrices in the atlas satisfy the following property. A matrix M is called hyperbolic, if (Hyp) ⟨v, Mw⟩2 ≥⟨v, Mv⟩⟨w, Mw⟩ for every v, w ∈Rd, such that ⟨w, Mw⟩> 0. For the atlas A, we say that v ∈Ωis hyperbolic, if the associated matrix Mv is hyperbolic, i.e. it satisfies (Hyp). We say that A satisfies hyperbolic property if every v ∈Ωis hyperbolic. Property (Hyp) is equivalent to the following property: (OPE) M has at most one positive eigenvalue (counting multiplicity). The equivalence between these two properties is well-known in the literature, see e.g. [Gre81], [COSW04, Thm 5.3], [SvH19, Lem. 2.9] and [BH20, Lem. 2.5]. Lemma 9.2 ([CP24a, Lem. 5.3]). Let M be a symmetric matrix. Then: M satisfies (Hyp) ⇐ ⇒M satisfies (OPE). 9.3. Sufficient conditions for hyperbolic property. In practice, verifying that an atlas sat-isfies the hyperbolic property can be a nontrivial task. In this subsection we present a set of conditions that are sufficient to imply the hyperbolic property. We say that the atlas A satisfies inheritance property if for every non-sink vertex v ∈Ω+, we have: (Mv)i = T⟨i⟩v, M⟨i⟩T⟨i⟩h for every i ∈supp(M) and v ∈Rd, (Inh) where M := Mv , T⟨i⟩= T⟨i⟩ v , h = hv and M⟨i⟩:= Mv⟨i⟩is the matrix associated with v⟨i⟩. Note that for the remainder of this section, we omit the subscript v from some notations to prevent cluttering the equations. We say that A satisfies pullback property if for every non-sink vertex v ∈Ω+, we have: (Pull) X i∈supp(M) hi T⟨i⟩v, M⟨i⟩T⟨i⟩v ≥⟨v, Mv⟩ for every v ∈Rd . We say that A satisfies pullback equality property if for every non-sink vertex v ∈Ω+, we have: (PullEq) X i∈supp(M) hi T⟨i⟩v, M⟨i⟩T⟨i⟩v = ⟨v, Mv⟩ for every v ∈Rd . Clearly (PullEq) implies (Pull). We say that a non-sink vertex v ∈Ω+ is regular if the following positivity conditions are satisfied7: The associated matrix Mv restricted to its support is irreducible. (Irr) Vector hv is strictly positive when restricted to the support of Mv . (h-Pos) The following theorem provides a method to establish the hyperbolic property for atlases by reducing the problem to checking the hyperbolic property only for the sink vertices of the atlases. Theorem 9.3 (local–global principle, see [CP24a, Thm 5.2], [CP22a, Thm 3.4]). Let A be a combinatorial atlas that satisfies properties (Inh) and (Pull), and let v ∈Ω+ be a non-sink regular vertex of Γ. Suppose every out-neighbor of v is hyperbolic. Then v is also hyperbolic. 7In [CP22a], there was an additional assumption that Mv hv given in (h-Pos) is strictly positive when restricted to the support of Mv . Note that this additional assumption is redundant here because we assume that Mv is a nonnegative matrix. 20 SWEE HONG CHAN AND IGOR PAK In our applications, the pullback property (PullEq) is more involved than the inheritance prop-erty (Inh). Below we give sufficient conditions for (PullEq) that are easier to establish. We say that the atlas A satisfies the identity property, if for every non-sink vertex v ∈Ω+ and every i ∈supp(M), we have: (Iden) T⟨i⟩: Rd →Rd is the identity mapping. We say that A satisfies the transposition-invariant property, if for every non-sink vertex v ∈Ω+, (T-Inv) M⟨i⟩ jk = M⟨j⟩ ki = M⟨k⟩ ij for every i, j, k ∈supp(M), where M⟨i⟩ jk is the (j, k)-th entry of the matrix M⟨i⟩. We say that A has the decreasing support property, if for every non-sink vertex v ∈Ω+, (DecSupp) supp(M) ⊇supp M⟨i⟩ for every i ∈supp(M). Theorem 9.4 (cf. [CP24a, Thm 6.1], [CP22a, Thm 3.8]). Let A be a combinatorial atlas that satisfies (Inh), (Iden), (T-Inv) and (DecSupp). Then A also satisfies (PullEq). 9.4. Equality conditions for hyperbolic inequalities. In this subsection, we discuss a strength-ening of the hyperbolic property for a given matrix, where the inequality in (Hyp) is replaced with equality. A global pair f, g ∈Rd is a pair of nonnegative vectors, such that (Glob-Pos) f + g is a strictly positive vector. Here f and g are global in a sense that they are the same for all vertices v ∈Ω. Fix a number s > 0. We say that a vertex v ∈Ωsatisfies (s-Equ), if (s-Equ) ⟨f, Mf⟩= s ⟨g, Mf⟩= s2 ⟨g, Mg⟩, where M = Mv is the matrix associated to v. Observe that (s-Equ) implies that equality occurs in (Hyp) for substitutions v ←g and w ←f, since ⟨g, Mf⟩2 = s ⟨g, Mg⟩s−1 ⟨f, Mf⟩= ⟨g, Mg⟩⟨f, Mf⟩. The following lemma gives another equivalent condition to check (s-Equ). Lemma 9.5 ([CP24a, Lem 7.2]). Let M be a nonnegative symmetric hyperbolic d×d matrix. Let f, g ∈Rd be nonnegative vectors, let s > 0, and let z := f −s g. Then (s-Equ) holds if and only if Mz = 0. A common objective of setting up an atlas is to identify vertices v ∈Ωthat satisfy (s-Equ). As it turns out, the property (s-Equ) for a given vertex v ∈Ωis equivalent to the statement that some, but not all, of the out-neighbors of v, satisfy (s-Equ). The focus then shifts to identifying neighbors that inherit (s-Equ), and we describe these vertices as follows. A vertex v ∈Ω+ is called a functional source if the following conditions are satisfied: fj = T⟨i⟩f j and gj = T⟨i⟩g j ∀i ∈supp(M), j ∈supp(M⟨i⟩), (Glob-Proj) f = hv . (h-Glob) Here condition (Glob-Proj) means that f, g are fixed points of the projection T⟨i⟩when restricted to the support. We say that an edge e⟨i⟩= (v, v⟨i⟩) ∈Θ is functional if v is a functional source and i ∈ supp(M) ∩supp(h). A vertex w ∈Ωis a functional target of v, if there exists a directed path v →w in Γ consisting of only functional edges. Note that a functional target is not necessarily a functional source. EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 21 The following theorem is the main result in this section and is a key to the proof of Theorem 1.3 we give in the next section. Theorem 9.6 (local–global equality principle, [CP24a, Thm 7.1]). Let A be a combinatorial atlas that satisfies properties (Inh), (Pull). Suppose also A satisfies property (Hyp) for every vertex v ∈Ω. Let f, g be a global pair of A. Suppose a non-sink vertex v ∈Ω+ satisfies (s-Equ) with constant s > 0. Then every functional target of v also satisfies (s-Equ) with the same constant s. 10. Proof of Theorem 1.3 10.1. Combinatorial atlas construction. In this subsection, we construct a combinatorial atlas, which will be used to prove Theorem 1.3. Let M := (X, I) be a matroid with rank r ≥2 on n := \|X\| elements. Denote by X∗the set of finite words in the alphabet X. A word is called simple if it contains each letter at most once; we consider only simple words in this paper. For a word α ∈X∗, the length \|α\| of α is the number of letters in α. For two words α, β ∈X∗, we denote by αβ ∈X∗the concatenation of α and β. Let a ∈{0, . . . , r} and let R ⊆X . A word γ = x1 . . . xr ∈X∗of length r is called compatible with a triple (M, R, a), if • {x1, . . . , xr} forms a basis of M, and • x1, . . . , xa ∈R and xa+1, . . . , xr ∈X −R. We denote by Comp(M, R, a) the set of words compatible with (M, R, a). Note that every such word γ ∈Comp(M, R, a) is simple. It also follows that (10.1) \| Comp(M, R, a)\| = r! P(M, R, a). For every a ∈[r −1], we denote by C(M, R, a) := Cxy x,y∈X the symmetric n × n matrix given by Cxy := ( Comp(M/{x, y}, R, a −1) , if x ̸= y and {x, y} ∈I , 0, if x = y or {x, y} / ∈I . (DefC-1) Equivalently, Cx,y is given by Cxy :=      γ : xγy ∈Comp(M, R, a) for x ∈R, y ∈X −R, γ : xyγ ∈Comp(M, R, a + 1) for x, y ∈R, γ : γxy ∈Comp(M, R, a −1) for x, y ∈X −R. (DefC-2) Both definitions will be frequently used throughout this section. It follows from the definition that C(M, R, a) is a nonnegative symmetric matrix, and the diagonal entries of C(M, R, a) are equal to 0. Note that the Stanley–Yan inequality (Theorem 1.1) is a direct consequence of the preceding matrix satisfying (Hyp), as explained below. Let f, g ∈Rn be the indicator vector of R and X −R, respectively. It follows from (DefC-2) and (10.1) that ⟨f, C(M, R, a) g⟩= r! P(M, R, a), ⟨f, C(M, R, a) f⟩= r! P(M, R, a + 1), ⟨g, C(M, R, a) g⟩= r! P(M, R, a −1). (Cfg) Therefore, Stanley–Yan inequality will follow once we demonstrate that C(M, R, a) satisfies (Hyp) (which we will prove in Proposition 10.5). We define the combinatorial atlas A = A(M, R, a) of dimension d = n , corresponding to the matroid M, the subset R ⊆X, and the integer a ∈2, . . . , r −1, as the acyclic graph and the linear algebraic data defined below. Let Γ := Γ(M, R, a) := (Ω, Θ) be the acyclic graph with Ω= Ω0 ∪Ω1, where Ω1 := {t ∈R \| 0 ≤t ≤1}, Ω0 := X. 22 SWEE HONG CHAN AND IGOR PAK For a non-sink vertex v = t ∈Ω1 and x ∈X, the corresponding outneighbor in Ω0 is v⟨x⟩:= x. For each vertex v = x ∈Ω0, the associated matrix is Mv := C(M/x, R, a −1) if x is a non-loop of M, and is equal to the zero matrix otherwise. Note that the ground set of M/x is still X (instead of X −x) under our convention. For each vertex v = t ∈Ω1, the associated matrix is Mv := t C(M, R, a) + (1 −t) C(M, R, a −1), and the associated vector hv := (hx)x∈V ∈Rd is defined to have coordinates hx := ( t if x ∈R , 1 −t if x ∈X −R . Finally, let the linear transformation T⟨x⟩ v : Rd →Rd associated to the edge (v, v⟨x⟩) to be the identity map. 10.2. Properties of the constructed matrices. In this subsection we gather properties of the matrix C(M, R, a) that will be used in the proof. Recall that NL(M) is the set of non-loops of M. Lemma 10.1. Let M be a matroid of rank r ≥2, let R ⊆X, and let a ∈[r −1] such that P(M, R, a) > 0. Then we have: • the support of C(M, R, a) is equal to NL(M), and • matrix C(M, R, a) is irreducible. Proof. Since P(M, R, a) > 0, there exists a basis B of M such that \|B ∩R\| = a . Since a ∈[r −1], this implies that there exists x ∈R and y ∈X −R such that x, y ∈B. It also follows from (DefC-1) that x and y are contained in the same irreducible component of the matrix C(M, R, a). Now, let z be an arbitrary non-loop of M. For the first claim it suffices to show that z ∈ supp (C(M, R, a)) , and for the second claim it suffices to show that that z is contained in the same irreducible component as x and y. We will without loss of generality assume that z ∈R, as the proof of the other case is analogous. There are now two possibilities. First suppose that z ∈B. Then B′ := B −y −z is a basis of M/{y, z} such that \|B′ ∩R\| = a −1. This implies that \| Comp(M/{y, z}, R, a −1)\| > 0 , so it follows from (DefC-1) that z is contained in the support of C(M, R, a) and z is contained in the same irreducible component as y. Now suppose that z / ∈B. By the symmetric basis exchange property, there exists z′ ∈B such that A := B −z′ + z is a basis of M. Now, if \|A ∩R\| = a (i.e. z′ ∈R), then A′ := A −y −z is a basis of M/{y, z} satisfying \|A′ ∩R\| = a −1. This implies that \| Comp(M/{y, z}, R, a −1)\| > 0, so it follows from (DefC-1) that z is contained in the support of C(M, R, a) and z is contained in the same irreducible component as y. On the other hand, if \|A ∩R\| = a + 1 (i.e. z′ / ∈R), then A′ := A −x −z is a basis of M/{x, z} satisfying \|A′ ∩R\| = a −1. This implies that \| Comp(M/{x, z}, R, a −1)\| > 0 , so it follows from (DefC-1) that z is contained in the support of C(M, R, a) and z is contained in the same irreducible component as x. This completes the proof. □ Lemma 10.2. Let M be a matroid of rank r ≥2, let R ⊆X, and let a ∈{2, . . . , r −1} such that P(M, R, a) > 0 and P(M, R, a −1) > 0 . Then, for every x ∈X that is not a loop, P(M/x, R, a −1) > 0. Proof. We will without loss of generality assume that x ∈R, as as the proof of the other case is analogous. By the assumption, there exists a basis A and B of M, such that \|A ∩R\| = a and \|B ∩R\| = a −1 . Applying the symmetric basis exchange property to x and A, we get that there exists a basis A′ of M such that x ∈A′ and \|A′ ∩R\| ∈{a, a + 1}. Similarly, by applying the EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 23 symmetric basis exchange property to x and B, there exists a basis B′ of M such that x ∈B′ and \|B′ ∩R\| ∈{a −1, a}. If either \|A′ ∩R\| = a or \|B′ ∩R\| = a then we are done, since A′ −x (resp. B′ −x) is then a basis of M/x for which \|(A′ −x) ∩R\| = a −1 (resp. \|(B′ −x) ∩R\| = a −1). So we assume that \|A′ ∩R\| = a −1 and \|B′ ∩R\| = a + 1 . Then by applying the basis exchange properties to A′ and B′ (possibly more than once), there exists a basis C′ of M such that \|C′ ∩R\| = a and x ∈C′, and the claim follows by the same argument as before. □ 10.3. Properties of the atlas. In this subsection, we show that the atlas A(M, R, a), con-structed in Section 10.1, satisfies the properties outlined in Section 9.3. This, in turn, enables us to apply tools and methods associated to atlases as described in Section 9. Lemma 10.3. Let M be a matroid of rank r ≥2, let R ⊆X, and let a ∈{2, . . . , r −1}. Then the atlas A(M, R, a) satisfies (Inh), (Iden), (T-Inv). Proof. The condition (Iden) follows directly from the definition. For (Inh), let v := t ∈Ω1 , let x ∈X , and let M := Mv be the associated matrix of v. By linearity, it suffices to prove that, for every y ∈X, Mxy = T⟨x⟩ey, M⟨x⟩T⟨x⟩h . Now we have T⟨x⟩ey, M⟨x⟩T⟨x⟩h = ey, M⟨x⟩h = t X z∈R M⟨x⟩ yz + (1 −t) X z∈X−R M⟨x⟩ yz . Now note that, if either x or y is a loop of M, then it follows from the definition (DefC-1) that the sum above is 0. Also note that in this case we also have Mxy = 0 by definition (DefC-1). Hence it suffices to consider the case when both x and y are non-loops of M. Now, continuing the equation above, T⟨x⟩ey, M⟨x⟩T⟨x⟩h =(DefC-2) t X z∈R γ : zγ ∈Comp(M/{x, y}, R, a −1) + (1 −t) X z∈S γ : γz ∈Comp(M/{x, y}, R, a −2) = t Comp(M/{x, y}, R, a −1) + (1 −t) Comp(M/{x, y}, R, a −2) =(DefC-1) Mxy . This completes the proof of (Inh). For (T-Inv), it suffices to show that M⟨x⟩ yz = M⟨y⟩ zx = M⟨z⟩ xy holds for all x, y, z ∈X. Note that all three numbers are equal to 0 if either one of x, y, or z is a loop of M, so we assume that x, y, z ∈NL(M). In this case, it follows from (DefC-1) that M⟨x⟩ yz = M⟨y⟩ zx = M⟨z⟩ xy = Comp(M/{x, y, z}, R, a −2) . This completes the proof of (T-Inv) and finishes the proof of the lemma. □ Lemma 10.4. Let M be a matroid of rank r ≥2, let R ⊆X, and let a ∈{2, . . . , r −1}, such that P(M, R, a) > 0 and P(M, R, a −1) > 0. Then the atlas A(M, R, a) satisfies (DecSupp). Proof. Let v = t ∈Ω1. In the notation above, we have: M := Mv = t C(M, R, a) + (1 −t) C(M, R, a −1). 24 SWEE HONG CHAN AND IGOR PAK It follows from Lemma 10.1, that the support of M is equal to the set NL(M) of non-loop elements of M. Let x be an arbitrary element of X. If x is a loop of M, then supp(M⟨x⟩) = ∅⊆supp(M). If x is not a loop of M, then supp(M⟨x⟩) ⊆NL(M/x) ⊆NL(M) = supp(M), as desired. □ 10.4. Hyperbolicity of the constructed atlas. The following proposition is the main technical result we need in the proof of Theorem 1.3. Also note that this proposition directly implies Stanley–Yan inequality (Theorem 1.1) as previously discussed in Section 10.1. Proposition 10.5. Let M be a matroid of rank r ≥2, let R ⊆X, and let a ∈[r −1], such that P(M, R, a) > 0. Then the matrix C(M, R, a) satisfies (Hyp). Proof. We prove the claim by induction on the rank r of M. First suppose that r = 2. Note that this implies a = 1. Write Cxy x,y∈X := C(M, R, a). It then follows from (DefC-1) that Cx,y = ( 1 if x ̸= y and {x, y} ∈I, 0 if x = y or {x, y} / ∈I. (10.2) In particular, this shows that the x-row (respectively, x-column) of C is identical to the y-row (respectively, y-column) of C whenever x, y are non-loops in the same parallel class. In this case, deduct the y-row and y-column of C by the x-row and x-column of C. It then follows that the resulting matrix has y-row and y-column is equal to zero, and note that (Hyp) is preserved under this transformation. Now, apply the above linear transformation repeatedly and remove the zero rows and columns, and let C′ be the resulting matrix. It suffices to prove that C′ satisfies (Hyp). Note that C′ is a p × p matrix (where p is the number of parallel classes of M), with 0s at the diagonal entries and 1s as the non-diagonal entries. It follows from direct calculations that (p −1) is the only positive eigenvalue of C′, and it follows that C′ (and thus C) satisfies (Hyp). This proves the base case of the induction. We now assume that r ≥3, and that the claim holds for matroids of rank (r −1). First, suppose that we have P(M, R, a + 1) = P(M, R, a −1) = 0. Then M = M1 ⊕M2 is a direct sum of matroids M1 and M2, where M1 (resp. M2) is the matroid obtained from M by restricting the ground set to R (resp. S). It then follows that Cx,y = ( B(M1/x) B(M2/y) if x ∈NL(M) ∩R and y ∈NL(M) ∩S, 0 otherwise. Rescale the rows and columns of x ∈R by B(M1/x), and the rows and columns of y ∈S by B(M2/y), and note that these rescalings preserve hyperbolicity. Then C becomes a special case of the matrix in (10.2), which was already shown to satisfy (Hyp). So we can assume that either P(M, R, a + 1) > 0 or P(M, R, a −1) > 0 . By the symmetry, we can without loss of generality assume that P(M, R, a −1) > 0. We split the proof into three parts. First assume that a ≥2. Let A(M, R, a) be the atlas defined in §10.1. It follows from Lemma 10.3 and Lemma 10.4 (note that these lemmas require a ≥2), that this atlas satisfies (Inh), (Iden), (T-Inv), and (DecSupp). We now show that, for every sink vertex v = x ∈Ω0, the matrix Mv satisfies (Hyp). If x is a loop of M, then Mv is equal to the zero matrix, which satisfies (Hyp). If x is a non-loop of M, then by definition Mv is equal to C(M/x, R, a −1) . Also note that P(M/x, R, a −1) > 0 by Lemma 10.2. It then follows from the induction assumption that Mv satisfies (Hyp). EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 25 Now every condition in Theorem 9.3 has been verified in the paragraph above, so it follows that every non-sink regular vertex in Γ is hyperbolic. On the other hand, we have from Lemma 10.1 that the the vertex v := t ∈Ω1 is regular if and only if t ∈(0, 1) . Hence this implies that, for every t ∈(0, 1), the matrix Mv = t C(M, R, a) + (1 −t) C(M, R, a −1), satisfies (Hyp). By taking the limit t →0 and t →1, it then follows that C(M, R, a) and C(M, R, a −1) also satisfies (Hyp). For the second case, assume that a = 1 and P(M, R, a + 1) > 0. Then let a′ := a + 1 = 2 . Note that we have P(M, R, a′) > 0, P(M, R, a′ −1) > 0 . By the same argument as before, we conclude that C(M, R, a) = C(M, R, a′ −1) satisfies (Hyp). For the third case, assume that a = 1 and P(M, R, a + 1) = 0. Since P(M, R, 1) > 0, there exists A ∈B(M) such that \|A ∩R\| = 1. Since \|A\| = r ≥2, there exists y ∈X −R such that y ∈A. Let M′ be the matroid obtained by adding an element x′ that is parallel to y, and let R′ := R + x′. Observe that M′ has the same rank as M. Note also that P(M′, R′, 2) > 0 because A′ := A −y + x′ is a basis of M′ and satisfies \|A′ ∩R′\| = 2. Finally, note that C(M, R, 1) can be obtained from C(M′, R′, 1) by removing the row and col-umn corresponding to x′. By the same argument as the second case, we conclude that C(M′, R′, a) satisfies (Hyp). Since (Hyp) is a property that is preserved under restricting to principal subma-trices, it then follows that C(M, R, a) also satisfies (Hyp), and the proof is complete. □ 10.5. Proof of Theorem 1.3. We will first prove Theorem 1.3 under the assumption that the rank r = rk(M) = 2. Recall that ParM(x) denotes the set of elements of M that are parallel to x. Lemma 10.6. Let M := (X, I) be a matroid of rank 2, and let R ⊆X such that P(M, R, 1) > 0. Let s > 0 be a positive real number. Then (10.3) P(M, R, 2) = s P(M, R, 1) = s2 P(M, R, 0) if and only if, for every non-loop x of M, (10.4) \| ParM(x) ∩R\| = s \| ParM(x) ∩(X −R)\|. Proof. Let M := C(M, R, 1) , and recall that f, g ∈Rn are the indicator vectors of R and X −R respectively. It follows from (Cfg) that (10.3) is equivalent to (10.5) ⟨f, Mf⟩= s ⟨f, Mg⟩= s2 ⟨g, Mg⟩. i.e. (s-Equ) holds. It then follows from Lemma 9.5 that (10.3) is equivalent to z := f −s g is contained in the kernel of M. On the other hand, the matrix M is described by Mx,y = ( 1 if x ̸= y and {x, y} ∈I, 0 if x = y or {x, y} / ∈I. It then follows that the kernel of M is the set of vectors v ∈Rn such that, for every non-loop x of M, X y∈ParM(x) vy = 0. Substituting v ←z, the equation above is equivalent to \| ParM(x) ∩R\| −s \| ParM(x) ∩(X −R)\| = 0, and the lemma follows. □ We now give an intermediate lemma which takes us halfway towards Theorem 1.3. 26 SWEE HONG CHAN AND IGOR PAK Lemma 10.7. Let M := (X, I) be a matroid of rank r ≥3, let R ⊆X, and let a ∈{2, . . . , r−1}, such that P(M, R, a) > 0. Finally, let s > 0. Then (10.6) P(M, R, a + 1) = s P(M, R, a) = s2 P(M, R, a −1) holds if and only if for every x ∈R ∩NL(M), we have: (10.7) P(M/x, R, a) = s P(M/x, R, a −1) = s2 P(M/x, R, a −2) > 0. Proof. We first prove the ⇐direction. Note that P(M, R, a + 1) = r a+1 −1 B ∈B(M) : \|B ∩R\| = a + 1 , = r a+1 −1 1 a+1 X x∈R ∩NL(M) {B ∈B(M) : \|B ∩R\| = a + 1, x ∈B} = r a+1 −1 1 a+1 X x∈R ∩NL(M) {B′ ∈B(M/x) : \|B′ ∩R\| = a} = 1 r X x∈R ∩NL(M) P(M/x, R, a). Applying (10.7) to the RHS, we get P(M, R, a + 1) = 1 r X x∈R ∩NL(M) s P(M/x, R, a −1) = s P(M, r, a). By the same argument, we also get P(M, R, a) = s P(M, R, a −1), as desired. We now prove the ⇒direction. Let A(M, R, a) be the combinatorial atlas defined in §10.1. It follows from the assumption, that P(M, r, a) > 0 and P(M, r, a −1) > 0 . It follows from Lemma 10.3 and Lemma 10.4 that this atlas satisfies (Inh), (Iden), (T-Inv), and (DecSupp). It also follows from Proposition 10.5, that this atlas satisfies (Hyp). Recall that f, g ∈Rn is the indicator vector of the subset R and X −R, respectively. It follows from the definition that f, g is a global pair for this atlas, and that the edge (1, x) is a functional edge for every x ∈R. Let M := C(M, R, a) be the matrix associated to v = 1 ∈Ω1. It follows from (Cfg) that (10.6) is equivalent to v = 1 satisfying (s-Equ). By Theorem 9.6, this implies that every vertex x ∈Ω0 contained in R also satisfies (s-Equ) with the same constant s. In other words, for every x ∈R, we have: (10.8) ⟨f, M⟨x⟩f⟩= s ⟨f, M⟨x⟩g⟩= s2⟨f, M⟨x⟩g⟩. On the other hand, for every x ∈R that is a non-loop of M, we have M⟨x⟩is equal to C(M/x, R, a −1) by definition, so it follows from (Cfg) that ⟨f, M⟨x⟩f⟩= (r −1)! P(M/x, R, a), ⟨f, M⟨x⟩g⟩= (r −1)! P(M/x, R, a −1), ⟨g, M⟨x⟩g⟩= (r −1)! P(M/x, R, a −2). Finally note that P(M/x, R, a) > 0 by Lemma 10.2. This completes the proof. □ Proof of Theorem 1.3. Note that (1.1) is equivalent to (10.9) P(M, R, a + 1) = s P(M, R, a) = s2 P(M, R, a −1) > 0, for some positive s > 0. It then suffices to show that (10.9) is equivalent to (1.2) with the same s > 0. By applying Lemma 10.7 a −1 many times, we have that (10.9) is equivalent to (10.10) P(M/A, R, 2) = s P(M/A, R, 1) = s2 P(M/A, R, 0) > 0, EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 27 for every A ⊆R that is independent in M, and such that \|A\| = a −1. Now note that (10.10) is equivalent to (10.11) P(M/A, X −R, r −a −1) = s P(M/A, X−R, r−a) = s2 P(M/A, X−R, r−a+1) > 0, for every A ⊆R that is independent in M and such that \|A\| = a −1. By applying Lemma 10.7 r −a −1 many times, it then follows that (10.11) is equivalent to (10.12) P(M/B, X −R, 0) = s P(M/B, X −R, 1) = s2 P(M/B, X −R, 2) > 0, for every B ⊆R that is independent in M and such that \|B\| = r −2 and \|B ∩R\| = a −1. Noting that M/B is a matroid of rank 2, it then follows that (10.12) is equivalent to (10.13) P(M/B, R, 2) = s P(M/B, R, 1) = s2 P(M/B, R, 0) > 0, for every B ⊆R that is independent in M and such that \|B\| = r −2 and \|B ∩R\| = a −1. The theorem now follows by applying Lemma 10.6 to (10.13). □ 11. Vanishing conditions 11.1. Setup. A discrete polymatroid D is a pair ([n], J ) of a ground set [n] := {1, . . . , n} and a nonempty finite collection J of integer points a = (a1, . . . , an) ∈Nn that satisfy the following: • a ∈J , b ∈Nn s.t. b ⩽a ⇒b ∈J , and • a, b ∈J , \|a\| < \| b \| ⇒∃i ∈[n] s.t. ai < bi and a + ei ∈J . Here b ⩽a is a componentwise inequality, \|a\| := a1 + . . . + an, and {e1, . . . , en} is a standard linear basis in Rn. When J ⊆{0, 1}n, discrete polymatroid D is a matroid. The role of bases in discrete polymatroids is played by maximal elements with respect to the order “⩽”. These are also called M-convex sets in [Mur03, §1.4] and [BH20, §2]. We refer to [HH02] and [Mur03] for further details on discrete polymatroids. 11.2. Proof of Theorem 1.6. Consider D1 := ([ℓ], J1) defined by (11.1) J1 := c ∈Nℓ: ∃A ∈I such that \|A ∩Si\| = ci for all i ∈[ℓ] . It follows from the matroid exchange property that D1 is a discrete polymatroid. Similarly, consider D2 := [ℓ], J2 defined by (11.2) J2 := n c ∈Nℓ: X i∈L ci ≤rk ∪i∈L Si for all L ∈2[ℓ]o . It follows from [HH02, Thm 8.1], that D2 is a discrete polymatroid. The theorem claims that J1 = J2. We prove the claim by induction on ℓ. The case ℓ= 1 is trivial. We now assume that ℓ> 1, and that the claim holds for smaller values. Note that J1 ⊆J2 by definition, so it suffices to show that J2 ⊆J1 . Let P1, P2 ⊂Rℓ + be convex hulls of J1, J2 ⊂Nℓ, respectively. Note that Pi are convex polytopes with vertices in Nℓ, and with Pi ∩Nℓ= Ji, see [HH02, Thm 3.4]. Hence the theorem follows by showing that all vertices of P2 belong to J1. In fact, because P2 is closed downward under ⩽, it suffices to prove the claim for every vertex c of P2 satisfying \|c\| = rk(X). First suppose that ci = 0 for some i ∈[ℓ]. Then it follows from induction that c ∈J1 by applying the theorem to the matroid M restricted to the ground set X \ Si. So we assume that ci ≥1 for all i ∈[ℓ]. Since c is a vertex of P2, there exists a non-empty L ⊊2[ℓ], such that X i∈L ci = rk ∪i∈L Si . Let S := ∪i∈LSi . On one hand, it follows from induction that there exists an independent set A1 of the matroid M restricted to the ground S, that satisfies \|A1 ∩Si\| = ci for all i ∈L. 28 SWEE HONG CHAN AND IGOR PAK On the other hand, it again follows from induction that there exists an independent set A2 of the matroid M/S that satisfies \|A2 ∩Si\| = ci for all i / ∈L. Let A := A1 ∪A2. Since rk(S) = P i∈L ci , it follows that A is an independent set of M satisfying \|A ∩Si\| = ci for all i ∈[ℓ]. This implies that c ∈J1, which completes the proof. □ 12. Total equality cases 12.1. Proof of Corollary 1.8. To simplify the notation, denote pa := P(M, R, a). By Propo-sition 1.4, we have pa > 0 for all 0 ≤a ≤r. Writing (SY) for all 1 ≤a < r, we get: (12.1) p1 p0 ≥p2 p1 ≥p3 p2 ≥· · · ≥ pr pr−1 > 0. This gives: p1 p0 r ≥ p1 · p2 · · · pr p0 · p1 · · · pr−1 = pr p0 , (12.2) and proves the first part. For the second part, observe that all inequalities in (12.1) must be equalities, which implies the second part. □ 12.2. Proof of Theorem 1.11, part (1). As described in the introduction, it remains to show the (ii) ⇒(iv) implication. It follows from Theorem 1.3, that (12.3) \| ParM/A(x) ∩R\| = s \| ParM/A(x) ∩(X −R)\|, for every independent set A ∈I of size \|A\| = r −2, and every x ∈NL(M/A). It thus suffices to show that (12.3) implies (iv) for the same value of s. We use induction on r. For r = 2, the claim follows immediately by applying (12.3), since M is loopless and we must have A = ∅in this case. For r > 2, suppose the claim holds for all matroids of rank (r −1). Then, for all x ∈X, it follows from applying the claim to M/x, that (12.4) \| ParM/x(y) ∩R\| = s \| ParM/x(y) ∩(X −R)\|, for y ∈NL(M/x). In particular, it then follows from (12.4) that (12.5) \| NL(M/x) ∩R\| = s \| NL(M/x) ∩(X −R)\|. On the other hand, it follows from NL(M) = X, that (12.6) NL(M/x) = X \ ParM(x). By combining (12.5) and (12.6), we conclude: (12.7) \| ParM(x) ∩R\| −s \| ParM(x) ∩(X −R)\| = \|R\| −s \|X −R\|, for all x ∈X. Summing the equation above over all parallel classes of M, we then have \|R\| −s \|X −R\| = p \|R\| −s \|X −R\| , where p ≥r ≥3 is the number of parallel classes of M. This implies \|R\| = s \|X −R\|. Together with (12.7) this gives \| ParM(x) ∩R\| = s \| ParM(x) ∩(X −R)\| for all x ∈X, as desired. □ EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 29 13. Examples and counterexamples 13.1. Double matroid. Let M be a loopless matroid with a ground set X that is given by a representation ϕ : X →Rr. Denote by X′ the second copy of X. Define a matroid M2 with the ground set Y := X ⊔X′, that is given by a representation ψ : Y →Rr, where ψ(x) := ϕ(x) and ψ(x′) := −ϕ(x). Now let R ←X and S ←X′. By the symmetry, observe that (13.1) \| ParM2(y) ∩R\| = \| ParM2(y) ∩S\|, for all y ∈Y . In other words, matroid M2 satisfies condition (iv) in Theorem 1.9 with s = 1. By Theorem 1.11, we conclude that M2 has total equality in the Stanley–Yan inequality, i.e. condition (ii) in Theorem 1.9. Note that (13.1) is a special case of (1.2) with s = 1. In fact, it is easy to modify this example to make the ratio to be any rational number. Indeed, to get the ratio s = a/b, take a copies of R and b of S. One can make all vectors to be distinct by taking different multiples (over R). We omit the easy details. 13.2. Linear matroid. Fix r ≥3. Let X = Fr 2 and let M be a binary matroid with a ground set X in its natural representation. Let R ⊂X be a subspace of dimension (r −1). Note that 0 is the only loop in M. Take an independent set of vectors A ⊂X such that \|A\| = r −2 and \|A ∩R\| = 0. Since A ̸= ∅, it is easy to see that for every non-loop x ∈NL(M/A), we have: (13.2) \| ParM/A(x) ∩R\| = \| ParM/A(x) ∩(X −R)\|. This is (1.2) for a = 1, with s = 1. By Theorem 1.3, this implies that (1.1) also holds for a = 1. Finally, note that P(M, R, 0) = 0 in this case, which is why this is not an example of total equality condition (ii) in Theorem 1.9 (and why the theorem is inapplicable in any event). 13.3. Combination matroid. The previous two examples illustrate different reasons for the equality (1.1) to hold for a = 1. The following matroid is a combination of the two which still gives equality for a = 1, but not for a > 1. Fix r ≥3 and let V = Fr 2. Let R0 ⊂V be a subspace of dimension (r −1), let S0 := V ∖R0 be the complement. Let R1, S1 ⊂S0 be two copies of the same nonempty set of vectors. Finally, let M be a matroid on the ground set X := R0 ⊔R1 ⊔S0 ⊔S1 and let R := R0 ⊔R1. Clearly, rk(R) = rk(X ∖R) = r, so P(M, R, 0) > 0 and P(M, R, r) > 0. We have (13.2) by a direct computation. By Theorem 1.3, this again implies that (1.1) holds for a = 1. On the other hand, one can directly check that (1.2) (and thus (1.1)) does not hold for a = r −1. We omit the details. In summary, this gives an example when total equality condition (ii) in Theorem 1.9 fails, even though (iii) holds. This disproves Conjecture 1.10 and proves the second part of Theorem 1.11.8 14. Generalized Mason inequality Let M be a matroid or rank r = rk(M), with a ground set X of size \|X\| = n. Denote by I(M) ⊆2X the set of independent sets of M. Fix integers k ≥0 and 0 ≤a, c1, . . . , ck ≤r. Additionally, fix disjoint subsets S1, . . . , Sk ⊂X, and let R := X ∖∪iSi. Define ISc(M, a) := A ∈I(M) : \|A ∩R\| = a, \|A ∩S1\| = c1, . . . , \|A ∩Sk\| = ck , and let ISc(M, a) := \|ISc(M, a)\|. Let m := n −c1 −. . . −ck and denote by Fm a free matroid on m elements. Substituting the direct sum M ←M ⊕Fm into the Stanley–Yan inequality, we obtain a log-concave inequality: ISc(M, a)2 ≥ISc(M, a + 1) ISc(M, a −1). 8To be precise, matroid M is not loopless. To fix this, remove 0 from R0. 30 SWEE HONG CHAN AND IGOR PAK This is the argument that was used by Stanley in [Sta81, Thm 2.9], to obtain Mason’s log-concave inequality (M1) in the case k = 0. The following ultra-log-concave inequality is a natural extension. Theorem 14.1 (generalized Mason inequality). For all 1 ≤a ≤min{r −1, m −1}, we have: (14.1) ISc(M, a)2 ≥ 1 + 1 a 1 + 1 m−a ISc(M, a + 1) ISc(M, a −1). This inequality is an easy consequence of the results by Br¨ and´ en and Huh [BH20]. We include a short proof for completeness. Proof of Theorem 14.1. We assume that X = [n]. Let fM ∈N[w0, w1, . . . , wn] be a multivariate polynomial defined by fM(w0, w1, . . . , wn) := X A∈I(M) wn−\|A\| 0 Y i∈A wi . It is shown in [BH20, Thm 4.10], that fM is Lorentzian. Take the following substitution: w0 ←y, wi ←x for i ∈R, and wi ←zj for i ∈Sj , 1 ≤j ≤k. Let gM(x, y, z1, . . . , zk) be the resulting polynomial, and let hM(x, y) := ∂c1 + ... +ck ∂zc1 1 · · · ∂zck k z1,...,zk=0 gM(x, y, z1, . . . , zk). Since the Lorentzian property is preserved under diagonalization, taking directional derivatives, and zero substitutions, see [BH20, §2.1], it follows that hM(x, y) is a Lorentzian polynomial with degree m. Now note that the coefficients [xaym−a]hM(x, y) is equal to ISc(M, a) c1! · · · ck! by definition. Recall now that a bivariate homogeneous polynomial with nonnegative coefficients is Lorentzian if and only if the sequence of coefficients form an ultra-log-concave sequence with no internal zeros. This implies the result. □ 15. Final remarks and open problems 15.1. Computational complexity ideas. Looking into recent developments, one cannot help but admire Rota’s prescience and keen understanding of mathematical development: “Anyone who has worked with matroids has come away with the conviction that the notion of a matroid is one of the richest and most useful concepts of our day. Yet, we long, as we always do, for one idea that will allow us to see through the plethora of disparate points of view.” [Rota86] Arguably, the idea of hyperbolicity is what unites both the combinatorial Hodge theory, Lorentzian polynomials and the combinatorial atlas approaches, even if technical details vary considerably. On the other hand, our complexity theoretic approach is as “disparate” as one could imagine, leaving many mathematical and philosophical questions unanswered.9 That an open problem in the old school matroid theory was resolved using tools and ideas from computational complexity might be very surprising to anyone who had not seen theoretical computer science permeate even the most distant corners of mathematics. To those finding them-selves in this predicament, we recommend a recent survey [Wig23], followed by richly detailed monograph [Wig19]. 9Some of these questions related to the nature of the P vs. NP problem are addressed in [Aar16, §§1–4]. EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 31 15.2. Negative results for other matroids. One can ask if Theorem 1.2 extends to other families of matroids given by a succinct presentation. In fact, our proof is robust enough, and extends to every family of matroids which satisfies the following: (1) computing the number of bases is #P-complete, and (2) the family includes all (loopless, bridgeless) graphical matroids. Notably, matroids realizable over Z obviously satisfy (2), and satisfy (1) by [Sno12]. On the other hand, paving matroids based on Hamiltonian cycles considered in [Jer06, §3], easily satisfy (1), but are very far from (2). For bicircular matroids, property (1) was proved in [GN06]. Unfortunately, not all graphical matroids are bicircular matroids, see [Mat77]. In fact, not all graphical matroids are necessarily transversal (see e.g. [Ox11, Ex. 1.6.3]), or even a gammoid (see e.g. [Ox11, Exc. 11(ii) in §12.3]). Nevertheless, we believe the following: Conjecture 15.1. Theorem 1.2 holds for bicircular matroids. Note that not all bicircular matroids are binary, see [Zas87, Cor. 5.1], so the conjecture would not imply Theorem 1.2. If one is to follow the approach in this paper, a starting point would be Conjecture 6.1 in [CP24d] which is analogous to Theorem 1.13 in this case, and needs to be obtained first. Afterwards, perhaps the proof can be extended to basis ratios as in Lemma 1.14. 15.3. Generalized Mason inequality. Denote by EqualityMasonk the equality of (14.1) decision problem. As we mentioned earlier, EqualityMason0 is in coNP. In fact, it is coNP-complete, see Corollary 15.3 below. By analogy with Theorem 1.2, it would be interesting to see what happens for general k : Open Problem 15.2. For what k > 0, is EqualityMasonk in PH? In particular, any explicit description of equality cases for EqualityMason1 would be a large step forward and potentially very difficult. We note aside that the combinatorial atlas approach can also be used to prove (14.1). Unfortunately, the specific construction we have in mind cannot be used to describe the equality cases, at least not without major changes. 15.4. Completeness. As evident from this paper, the computational complexity of equality cases for matroid inequalities is very interesting and remains largely unexplored. Of course, for Mason’s log-concave inequality (M1), the equality cases are trivial since the sequence satisfies a stronger inequality (M2). On the other hand, for Mason’s ultra-log-concave inequality (M2), the equality cases have a simple combinatorial description: girth(M) > a+1, i.e. the size of the minimal circuit in the matroid has to be at least a+2, see [MNY21] and [CP24a, §1.6] for proofs using Lorentzian polynomials and combinatorial atlases, respectively. Now, the decision problem GIRTH := girth(M) ≤? a+1 is in NP for matroids with concise presentation. The problem is easily in P for graphical matroids via taking powers of adjacency matrix. Recently, it was shown to be in P for regular matroids in [FGLS18]. Famously, the problem was shown to be NP-complete for binary matroids by Vardy [Var97]. This gives: Corollary 15.3. EqualityMason0 is coNP-complete for binary matroids. We believe that the upper bound in Corollary 1.5 is optimal: Conjecture 15.4. EqualitySY0 is coNP-complete for binary and for bicircular matroids. Note that for graphical matroids, the number of bases B(M) is in FP by the matrix-tree theorem. For regular matroids, the same linear algebraic argument applies. More generally, the number BSc(M, R, a) can also be computed in polynomial time via the weighted (multivariate) version of the matrix-tree theorem (see e.g. [GJ83]). This gives the following observation to contrast with Conjecture 15.4. Proposition 15.5. EqualitySYk is in P for regular matroids and all fixed k ≥0. 32 SWEE HONG CHAN AND IGOR PAK We conclude with a possible approach to the proof of Conjecture 15.4. The 2-SPANNING-CIRCUIT is a problem whether a matroid has a circuit containing two given elements. For graphical matroids, this problem is in P by Menger’s theorem. For regular matroids, this problem is in P by a result in [FGLS16] based on Seymour’s decomposition theorem. One can modify examples in Section 13 to show the following: Proposition 15.6. 2-SPANNING-CIRCUIT reduces to ¬ EqualitySY0 for binary matroids. By the proposition, the first part of Conjecture 15.4 follows from the following natural conjecture that would be analogous to Vardy’s result for the GIRTH: Conjecture 15.7. 2-SPANNING-CIRCUIT is NP-complete for binary matroids. 15.5. Defect. Denote by φ the defect of the SY inequality: φSc(M, R, a) := PSc(M, R, a)2 −PSc(M, R, a + 1) PSc(M, R, a −1). By definition, φ is a rational function with denominators in FP. If the numerators were also in FP for binary matroids, we would have EqualitySYk ∈P, implying that PH collapses for k ≥1 by Theorem 1.2. In fact, we have a stronger result: Proposition 15.8. φ is #P-hard for binary matroids and all k ≥0. Proof. For k = 0, let M be a matroid of rank r, and let M′ := M⊕M1 be a direct sum of matroid M with a matroid with a single element v. Note that M′ has rank r+1 and every basis in M′ must contain v. Let R := {v}. Observe that B(M′, R, 0) = B(M′, R, 2) = 0 and B(M′, R, 1) = \|B(M)\|. By the definition of PSc(M, R, a) and Theorem 8.3, we conclude that φ is #P-hard. Finally, for k ≥1, the result follows from the proof of Lemma 8.1. □ Note that the argument above does not imply Conjecture 15.4 since, in the case outlined, EqualitySY0 reduces to deciding the vanishing of the number of bases of a given matroid, which is trivially in P. We refer to [Pak22] for an extensive overview of complexity of combinatorial inequalities and their defect. 15.6. Spanning trees. Note that for simple planar graphs, Stong’s Theorem 1.12 is nearly op-timal since the number of spanning trees is at most exponential for planar graphs with n vertices [BS10], or even all graph with bounded average degree, see [Gri76]. This gives α(N) = Ω(log N), where recall that α(N) is the smallest number of vertices of a simple planar graph G with exactly N spanning trees. In fact, since the number of unlabeled planar graphs with n vertices is expo-nential in n, see e.g. [Noy15, §6.9.2], proving the corresponding upper bound α(N) = O(log N) is likely to be very difficult. On the other hand, it follows from the proof of Theorem 1.13, that, for the smallest number of edges of a non-simple graph with exactly N spanning trees, the upper bound of log n will be implied by the celebrated Zaremba’s conjecture, see a discussion and further references in [CP24c] and [CKP24]. 15.7. Understanding the results. There are several ways to think of our results. First and most straightforward, we completely resolve a 1981 open problem by Stanley by both showing that the equality cases of (SY) cannot have a satisfactory description (from a combinatorial point of view) for k > 0, and by deriving such a description for k = 0. Second, one can think of the results as a showcase for the tools. This includes both the com-putational complexity and number theoretic approach towards the proof of Theorem 1.2, and the (rather technical) combinatorial atlas approach towards the proof of Theorem 1.3. Let us empha-size that Yan was unable to obtain our Theorem 1.3 using Lorentzian polynomials (cf. [Yan23, §3.3]). While we believe that Theorem 1.3 is not attainable with Lorentzian polynomials, we lack the formal language to make this claim rigorous. EQUALITY CASES OF THE STANLEY–YAN INEQUALITY 33 Third, one can think of Theorem 1.2 as an evidence of the strength of Lorentzian polynomials. In combinatorics, some of the most natural combinatorial inequalities are proved by a direct injection. See e.g. [CPP23a, CPP23b, DD85, DDP84] for injective proofs of variations and special cases of (Sta), and to [Mani10] for a rare injective proof of a matroid inequality. Now, if an injection and its inverse (when defined) are poly-time computable, this implies that the equality cases are in coNP. Thus, having EqualitySY1 / ∈PH shows that Lorentzian polynomials are powerful, in a sense that they can prove results beyond elementary combinatorial means. Finally, this paper gives a rare example of limits of what is knowable about matroid inequalities, as opposed to realizability of matroids where various hardness and undecidability results are known, see e.g. [KY22, Sch13]. This is especially in sharp contrast with the equality cases for Mason’s inequalities, which are known to have easy descriptions. Acknowledgements. We are grateful to Petter Br¨ and´ en, Graham Farr, Fedor Fomin, Milan Haiman, Jeff Kahn, Noah Kravitz, Jonathan Leake, Daniel Lokshtanov, Steven Noble, Greta Panova, Yair Shenfeld, Ilya Shkredov, Richard Stanley, Richard Stong, Ramon van Handel and Alan Yan for interesting discussions and helpful remarks. This work was initiated in July 2023, when both authors were visiting the American Institute of Mathematics (AIM) at their new location in Pasadena, CA. We continued our collaboration during a workshop at the Institute of Pure and Applied Mathematics (IPAM), in April 2024. We are grateful to both AIM and IPAM for the hospitality, and to workshop organizers for the opportunity to participate. Both authors were partially supported by the NSF. References [Aar16] Scott Aaronson, P ? = NP, in Open problems in mathematics, Springer, Cham, 2016, 1–122; available at scottaaronson.com/papers/pnp.pdf [AHK18] Karim Adiprasito, June Huh and Eric Katz, Hodge theory for combinatorial geometries, Annals of Math. 188 (2018), 381–452. [AOV18] Nima Anari, Shayan Oveis Gharan and Cynthia Vinzant, Log-concave polynomials, entropy, and a deterministic approximation algorithm for counting bases of matroids, in Proc. 59th FOCS, IEEE, Los Alamitos, CA, 2018, 35–46. 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[Yan23] Alan Yan, Log-concavity in combinatorics, senior thesis, Princeton University, 2023, 140 pp.; arXiv: 2404.10284. [YK75] Andrew C. Yao and Donald E. Knuth, Analysis of the subtractive algorithm for greatest common divisors, Proc. Nat. Acad. Sci. USA 72 (1975), 4720–4722. [Zas87] Thomas Zaslavsky, The biased graphs whose matroids are binary, J. Combin. Theory, Ser. B 42 (1987), 337–347. (Swee Hong Chan) Department of Mathematics, Rutgers University, Piscatway, NJ 08854. Email address: sc2518@rutgers.edu (Igor Pak) Department of Mathematics, UCLA, Los Angeles, CA 90095. Email address: pak@math.ucla.edu
188953	https://artofproblemsolving.com/wiki/index.php/Titu%27s_Lemma?srsltid=AfmBOoocUa6yeUBTlF6pw2MYh4QRhjGzeURf607OYXts2DM_z3fsm60m	Art of Problem Solving Titu's Lemma - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Titu's Lemma Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Titu's Lemma Titu's lemma states that: It is a direct consequence of Cauchy-Schwarz inequality. Equality holds when for . Titu's lemma is named after Titu Andreescu and is also known as T2 lemma, Engel's form, or Sedrakyan's inequality. Contents [hide] 1 Examples 1.1 Example 1 1.1.1 Solution 1.2 Example 2 1.2.1 Solution 1.3 Example 3 1.3.1 Solution 2 Problems 2.1 Introductory 2.2 Intermediate 2.3 Olympiad Examples Example 1 Given that positive reals , , and are subject to , find the minimum value of . (Source: cxsmi) Solution This is a somewhat standard application of Titu's lemma. Notice that When solving problems with Titu's lemma, the goal is to get perfect squares in the numerator. Now, we can apply the lemma. Example 2 Prove Nesbitt's Inequality. Solution For reference, Nesbitt's Inequality states that for positive reals , , and , We rewrite as follows. This is the application of Titu's lemma. This step follows from . Example 3 Let , , , , , , , be positive real numbers such that . Show that (Source) Solution By Titu's Lemma, Problems Introductory There exists a smallest possible integer such that for all real sequences . Find the sum of the digits of . (Source) Intermediate Prove that, for all positive real numbers (Source) Olympiad Let be positive real numbers such that . Prove that (Source) Let be positive real numbers such that . Prove that (Source) Retrieved from " Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
188954	https://curriculum.illustrativemathematics.org/k5/teachers/grade-5/unit-5/practice.html	Illustrative Mathematics Grade 5, Unit 5 - Teachers \| IM Demo Skip to main content Professional LearningContact Us For full sampling or purchase, contact an IM Certified Partner:Imagine LearningKendall HuntKiddom Grade 5 K-5KindergartenGrade 1Grade 2Grade 3Grade 4Grade 5 Unit 5 Grade 5Unit 1Unit 2Unit 3Unit 4Unit 5Unit 6Unit 7Unit 8 5.5 Place Value Patterns and Decimal Operations Unit Goals Students build from place value understanding in grade 4 to recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and of what it represents in the place to its left. They use this place value understanding to round, compare, order, add, subtract, multiply, and divide decimals. Section A Goals Compare, round and order decimals through the thousandths place based on the value of the digits in each place. Read, write, and represent decimals to the thousandths place, including in expanded form. Section B Goals Add and subtract decimals to the hundredths using strategies based on place value. Section C Goals Multiply decimals with products resulting in the hundredths using place value reasoning and properties of operations. Section D Goals Divide decimals with quotients resulting in the hundredths using place value reasoning and properties of operations. Read More LessonsPracticeGlossary Section A: Numbers to Thousandths Problem 1 Pre-unit Practicing Standards:5.NF.B.4 Find the value of each expression. Solution For access, consult one of our IM Certified Partners. Problem 2 Pre-unit Practicing Standards:5.NF.B.4.b Write a multiplication equation shown by the shaded region of the diagram. 2. What is the value of ? Use the grid if it is helpful. Solution For access, consult one of our IM Certified Partners. Problem 3 Pre-unit Practicing Standards:4.NBT.B.5 Find the value of . Use the diagram if it is helpful. Description: Diagram, rectangle partitioned vertically and horizontally into 4 rectangles. Top left rectangle, vertical side, 20, horizontal side, 70. Top right rectangle, horizontal side, 3. Bottom 2 rectangles, vertical side, 8. Solution For access, consult one of our IM Certified Partners. Problem 4 Pre-unit Practicing Standards:4.NBT.A.1 What is the value of the 6 in 618,923? How many times greater is the value of the 6 in 618,923 than the 6 in 27,652? Solution For access, consult one of our IM Certified Partners. Problem 5 Pre-unit Practicing Standards:4.NBT.B.6 Find the value of . Explain or show your reasoning. Solution For access, consult one of our IM Certified Partners. Problem 6 Pre-unit Practicing Standards:4.NBT.B.4 Find the value of each sum or difference. Solution For access, consult one of our IM Certified Partners. Problem 7 What fraction of the whole square is shaded? Explain or show your reasoning. 2. What fraction of the whole square is shaded? Explain or show your reasoning. Solution For access, consult one of our IM Certified Partners. Problem 8 Write a decimal number to represent how much of the square is shaded. 2. Shade one hundred fifteen thousandths of the square. Solution For access, consult one of our IM Certified Partners. Problem 9 Write the decimal 0.418 as a fraction, in words, and in expanded form. Solution For access, consult one of our IM Certified Partners. Problem 10 A gold nugget weighs 0.265 ounces. Name 2 different sets of 0.1 ounce, 0.01 ounce, and 0.001 ounce weights you can use to balance the nugget. 2. One gold nugget weighs 0.008 ounces. A second gold nugget weighs 0.8 ounces. How many times as much as the first nugget does the second nugget weigh? How many times as much as the second nugget does the first nugget weigh? Solution For access, consult one of our IM Certified Partners. Problem 11 Noah threw the frisbee 4.89 yards. Noah threw the frisbee farther than Lin. How far could Lin have thrown the frisbee? Andre threw the frisbee farther than Noah but less than 4.9 yards. How far could Andre have thrown the frisbee? Explain your reasoning. Solution For access, consult one of our IM Certified Partners. Problem 12 Label the tick marks.Use the number line to explain your reasoning. Which is greater, 0.654 or 0.658?Explain or show your reasoning. Solution For access, consult one of our IM Certified Partners. Problem 13 A $5 gold coin weighs 8.359 grams. 1. Locate 8.359 on the number line. A scale measures to the nearest 0.01 gram. What will the scale show for the weight of the coin? Explain or show your reasoning. Solution For access, consult one of our IM Certified Partners. Problem 14 What is 0.374 rounded to the nearest hundredth? Explain or show your reasoning. Use the number line if it's helpful. What is 9.893 rounded to the nearest tenth? What about to the nearest hundredth? Draw a number line if it is helpful. Solution For access, consult one of our IM Certified Partners. Problem 15 List the decimals from least to greatest: 6.95, 6.895, 6.598, 6.985, 5.986 Solution For access, consult one of our IM Certified Partners. Problem 16 To the nearest hundredth of a mile per hour, a luge rider's top speed was 81.73 mph. What are some possible speeds to the thousandth of a mile per hour? Use the number line if it is helpful. Solution For access, consult one of our IM Certified Partners. Problem 17 Exploration Jada has 3 doubloons. She knows that two of them have the same weight and one of them is heavier than the other two. Jada also has a balance which she can use to compare the weights of coins. Explain or show how Jada can use the balance to figure out which doubloon is heavier and which two are the same weight. What if Jada has 5 doubloons and knows that 4 of them have the same weight and one of them is heavier? Solution For access, consult one of our IM Certified Partners. Problem 18 Exploration There are two packages of ground beef at the store. One package says it has 1 pound of beef. The second package says it has 0.97 pounds of beef. Jada says that the 1 pound package has more beef. Do you agree with Jada? Explain or show your reasoning. Solution For access, consult one of our IM Certified Partners. Section B: Add and Subtract Decimals Problem 1 Mai and Tyler were playing “Target Number Addition.” Mai rolled 6 sixes. How close can Mai get to 1 without going over? Tyler rolled 6 fours. How close can Tyler get to 1 without going over? Solution For access, consult one of our IM Certified Partners. Problem 2 Which whole number is closest to? Explain or show your reasoning. Find the value of . Solution For access, consult one of our IM Certified Partners. Problem 3 Find the value of the expression . Solution For access, consult one of our IM Certified Partners. Problem 4 Which whole number is closest to? Explain or show your reasoning. Find the value of . Solution For access, consult one of our IM Certified Partners. Problem 5 Here is how Elena found the value of. Explain Elena's calculations and the meaning of the 15 above the 5 and the 17 above the 7 in 15.37. 2. Use Elena's algorithm to calculate . Solution For access, consult one of our IM Certified Partners. Problem 6 Find the value of each expression. Solution For access, consult one of our IM Certified Partners. Problem 7 Exploration Kiran finds the value of with these calculations. . Explain why Kiran’s strategy works. 2. Find the difference in a way that makes sense to you. Solution For access, consult one of our IM Certified Partners. Problem 8 Exploration Lin is trying to use the digits 1, 3, 4, 2, 5, and 6 to make 2 two-digit decimals whose sum is equal to 1. Explain why Lin can not make 1 by adding together 2 two-digit decimal numbers made with these digits. What is the closest Lin can get to 1? Explain how you know. Solution For access, consult one of our IM Certified Partners. Section C: Multiply Decimals Problem 1 Shade on the first diagram. What is the value of ? Explain or show your reasoning. What is the value of ? Use the second diagram if it is helpful. Solution For access, consult one of our IM Certified Partners. Problem 2 Mai says that and both have the same value. She says that they are both 28. Do you agree with Mai? Explain or show your reasoning. Explain why . Solution For access, consult one of our IM Certified Partners. Problem 3 Explain why each expression is equivalent to . Find the value of using one of the expressions or your own strategy. Solution For access, consult one of our IM Certified Partners. Problem 4 Shade the diagram to represent . What is the value of ? Solution For access, consult one of our IM Certified Partners. Problem 5 Explain or show why . Use this strategy to calculate . Solution For access, consult one of our IM Certified Partners. Problem 6 Exploration Here is Diego's strategy to find the value of. I know so I just find and then divide by 100. Explain or show why Diego's method works. Use Diego's method to find the value of. Solution For access, consult one of our IM Certified Partners. Problem 7 Exploration Han says the picture shows .Label the diagram to show Han's thinking. Mai says it shows . Label the diagram to show Mai's thinking. What other products can the diagram represent? Explain or show your reasoning. Solution For access, consult one of our IM Certified Partners. Section D: Divide Decimals Problem 1 Find the value of . Use the diagram if it is helpful. 2. Jada says that there are 100 hundredths in 1 so is 100. Do you agree with Jada? Show or explain your reasoning. Solution For access, consult one of our IM Certified Partners. Problem 2 Find the value of . Use the diagram if it is helpful. Find the value of . Solution For access, consult one of our IM Certified Partners. Problem 3 Here is a diagram. Description: Two diagrams. Each squares. Each partitioned into 10 rows of 10 of the same size squares. All squares shaded. For each 25 squares, shading alternates, blue, orange. Explain or show how the diagram shows . What is the value of the expression? Explain or show how the diagram shows . What is the value of the expression? Solution For access, consult one of our IM Certified Partners. Problem 4 Find the value of each expression. Explain or show your reasoning. . Use the diagram if it is helpful. Solution For access, consult one of our IM Certified Partners. Problem 5 Find the value of each expression. Explain or show your reasoning. Solution For access, consult one of our IM Certified Partners. Problem 6 Exploration Noah has a scale that weighs to the nearest ounce. The table shows the weights of different numbers of paper clips in ounces. \| paper clips \| weight \| --- \| \| 1 \| 0 \| \| 10 \| 0 \| \| 20 \| 1 \| \| 25 \| 1 \| \| 50 \| 2 \| \| 100 \| 3 \| How many ounces do you think each paper clip weighs? Explain or show your reasoning. Solution For access, consult one of our IM Certified Partners. Problem 7 Exploration The daily recommended allowance of vitamin C for a 5th grader is 0.05 grams. A vitamin C tablet has 1 gram of vitamin C. How many times the daily recommended allowance of vitamin C is one vitamin C tablet? Use the diagram if it is helpful. 2. A large orange has 0.18 grams of vitamin C. How many times the daily recommended allowance of vitamin C is in a large orange? Use the diagram if it is helpful. Solution For access, consult one of our IM Certified Partners. About IM In the News Curriculum Grades K-5 Grades 6-8 Grades 9-12 Professional Learning Standards and Tasks Jobs Privacy Policy Facebook Twitter IM Blog Contact Us 855-741-6284 What is IM Certified™? © 2021Illustrative Mathematics®. Licensed under the Creative Commons Attribution 4.0 license. The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics. These materials include public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information.
188955	https://gotroot.ca/spectrum/www.spectrum-soft.com/news/fall2009/vswr.html	Calculating VSWR, Return Loss, Reflection Coefficient, and Mismatch Loss - Fall 2009 News: Effective 7/4/2019, Spectrum Software is closed. Micro-Cap is now free. Technical support will be available for at least 90 days via email at Support. You can download the latest versions of Micro-Cap here: Download You can choose either the executable program or the entire installation CD for MC10, MC11, and MC12. If you have an earlier version, download and use MC12. These new versions do not require the security key, so they make Micro-Cap free to the entire engineering community. Thank you for the honor and privilege of serving you for the last 39 years. Spectrum Software Calculating VSWR, Return Loss, Reflection Coefficient, and Mismatch Loss ------------------------------------------------------------------------ There are a number of calculations that are useful when simulating the transmission of a wave through a line. These calculations can be quite important in calculating the energy that arrives at the load versus how much energy the transmitter is producing. Ideally, the load impedance should match the characteristic impedance of the transmission line so that all of the transmitted energy is available at the load. When the load impedance does not match the characteristic impedance of the transmission line, part of the voltage will be reflected back down the line reducing the available energy at the load. One measurement is the reflection coefficient (Γ). The reflection coefficient measures the amplitude of the reflected wave versus the amplitude of the incident wave. The expression for calculating the reflection coefficient is as follows: Γ = (ZL - ZS)/(ZL + ZS) where ZL is the load impedance and ZS is the source impedance. Since the impedances may not be explicitly known, the reflection coefficient can be measured in a similar manner to an S11 measurement by using the wave amplitudes at the source and at the node following the source impedance. The following define statement user function can be used to measure the reflection coefficient. .define RefCo(In,Src) Mag(2V(In)-V(Src)) where In is the node name of the node following the source impedance and Src is the part name of the source component. The VSWR (Voltage Standing Wave Ratio) measurement describes the voltage standing wave pattern that is present in the transmission line due to the phase addition and subtraction of the incident and reflected waves. The ratio is defined by the maximum standing wave amplitude versus the minimum standing wave amplitude. The VSWR can be calculated from the reflection coefficient with the equation: VSWR = (1 + Γ)/(1 - Γ) The following define statement user function can be used to measure the VSWR. .define VSWR(In,Src) (1+RefCo(In,Src))/(1-RefCo(In,Src)) The return loss measurement describes the ratio of the power in the reflected wave to the power in the incident wave in units of decibels. The standard output for the return loss is a positive value, so a large return loss value actually means that the power in the reflected wave is small compared to the power in the incident wave and indicates a better impedance match. The return loss can be calculated from the reflection coefficient with the equation: Return Loss = -20Log(Γ) The following define statement user function can be used to measure the return loss. .define RetLoss(In,Src) -20Log(RefCo(In,Src)) The mismatch loss measurement describes the amount of power that will not be available at the load due to the reflected wave in units of decibels. It indicates the amount of power lost in the system due to the mismatched impedances. The mismatch loss can also be calculated from the reflection coefficient with the following equation: Mismatch Loss = -10Log(1 - Γ²) The following define statement user function can be used to measure the mismatch loss. .define MismatchLoss(In,Src) -10Log(1 - RefCo(In,Src)2) For the VSWR, return loss, and mismatch calculations, the In and Src parameters are defined in the same manner as they are for the reflection coefficient define statement. If only the VSWR, return loss, or mismatch loss measurement is to be performed in the analysis, the reflection coefficient define statement must also be present to perform the calculation since it is referenced in all three of these calculations. A simple circuit is displayed in the figure below to demonstrate the use of these define statement user functions. The circuit consists of a voltage source, two resistors, and an ideal, lossless transmission line. The load resistance has been set to 75 ohms to create a mismatch with the 50 ohm characteristic impedance of the transmission line. The four define statement user functions have been entered in the schematic as grid text. Each statement must be entered as a separate grid text. They may also be entered in the Text page of the schematic to reduce the clutter in the schematic. An AC analysis simulation is then run on the circuit. The four Y expressions plotted for the simulation are: VSWR(In,V1) RetLoss(In,V1) RefCo(In,V1) MismatchLoss(In,V1) The AC simulation results are displayed below. Since this example circuit is entirely resistive, the AC analysis response will be constant across the entire frequency range. Note that the node name used as the parameter within these functions does not have to be named In. It can be any name that the user chooses or even the node number of the node. V1 is the part name for the voltage source in the schematic. The AC simulation returns the following results for this circuit: VSWR = 1.5 Reflection Coefficient = .2 Return Loss = 13.979dB Mismatch Loss = .177dB The define statements can also be placed in the MCAP.INC file which can be accessed through the User Definitions under the Options menu. Placing them in this file makes the functions globally available for all circuits. Reference: 1) VOLTAGE STANDING WAVE RATIO (VSWR) / REFLECTION COEFFICIENT RETURN LOSS / MISMATCH LOSS, Granite Island Group, Download Fall 2009 Circuit Files Return to the main Newsletter page
188956	https://www.mathplanet.com/education/algebra-1/radical-expressions/simplify-radical-expressions	Mathplanet A free service from Mattecentrum Menu Simplify radical expressions Do excercises Show all 3 exercises Simplify radicals I Simplify radicals II Simplify radicals III The properties of exponents, which we've talked about earlier, tell us among other things that (xy)a=xaya (xy)a=xaya We also know that x−−√a=x1a or x−−√=x12 If we combine these two things then we get the product property of radicals and the quotient property of radicals. These two properties tell us that the square root of a product equals the product of the square roots of the factors. xy−−√=x−−√⋅y√ xy−−√=x−−√y√ wherex≥0,y≥0 The answer can't be negative and x and y can't be negative since we then wouldn't get a real answer. In the same way we know that x2−−√=xwherex≥0 These properties can be used to simplify radical expressions. A radical expression is said to be in its simplest form if there are no perfect square factors other than 1 in the radicand 16x−−−√=16−−√⋅x−−√=42−−√⋅x−−√=4x−−√ no fractions in the radicand and 2516x2−−−−−√=25−−√16−−√⋅x2−−√=54x no radicals appear in the denominator of a fraction. 1516−−−√=15−−√16−−√=15−−√4 If the denominator is not a perfect square you can rationalize the denominator by multiplying the expression by an appropriate form of 1 e.g. xy−−√=x−−√y√⋅y√y√=xy−−√y2−−√=xy−−√y Binomials like xy√+zw−−√andxy√−zw−−√ are called conjugates to each other. The product of two conjugates is always a rational number which means that you can use conjugates to rationalize the denominator e.g. x4+x−−√=x(4−x−−√)(4+x−−√)(4−x−−√)= =x(4−x−−√)16−(x−−√)2=4x−xx−−√16−x Video lesson Simplify the radical expression x5−x−−√ You need to accept marketing cookies to play the video. Do excercises Show all 3 exercises Simplify radicals I Simplify radicals II Simplify radicals III More classes on this subject Algebra 1 Radical expressions: The graph of a radical function Algebra 1 Radical expressions: The Pythagorean Theorem Algebra 1 Radical expressions: The distance and midpoint formulas
188957	https://www.youtube.com/watch?v=lVdVZE2aCjg	Transport of Solutes and Water \| Chapter 5 - Medical Physiology (2nd Edition) Last Minute Lecture 2990 subscribers Description 5 views Posted: 20 Aug 2025 Chapter 5 of Medical Physiology (2nd Edition) by Walter F. Boron and Emile L. Boulpaep provides an in-depth exploration of how cells regulate their internal environment through the transport of solutes and water across biological membranes. The chapter begins by explaining the structure and selective permeability of the plasma membrane, highlighting its role in maintaining ionic gradients, osmotic balance, and electrochemical stability. Fundamental principles such as diffusion, osmosis, facilitated transport, and active transport are examined, showing how energy use and protein transporters like pumps, channels, and carriers maintain homeostasis. The sodium-potassium ATPase pump is emphasized as a cornerstone of cellular physiology, establishing gradients that support excitability, nutrient uptake, and volume regulation. Water transport via aquaporins is introduced, illustrating how cells respond to osmotic challenges and maintain fluid balance. The chapter also explains secondary active transport, cotransporters, exchangers, and the role of ion channels in rapid signaling and excitability. Importantly, clinical examples show how defects in transport proteins lead to disorders such as cystic fibrosis, diabetes insipidus, and electrolyte imbalances. Concepts of steady state versus equilibrium are integrated, emphasizing that maintaining homeostasis requires constant energy expenditure. By linking molecular transport mechanisms to whole-body physiology and disease, this chapter demonstrates how solute and water transport underlies every aspect of cell and organ function, from nerve impulse conduction to kidney filtration and systemic fluid balance. 📘 Read full blog summaries for every chapter: 📘 Have a book recommendation? Submit your suggestion here: Thank you for being a part of our little Last Minute Lecture family! Medical Physiology Chapter 5 summary, Boron and Boulpaep Transport of Solutes and Water explained, diffusion osmosis facilitated transport active transport, sodium potassium ATPase pump physiology, aquaporins and water movement, secondary active transport cotransporters exchangers, ion channels and excitability in physiology, steady state versus equilibrium biology, osmotic balance and fluid homeostasis, plasma membrane selective permeability, cystic fibrosis and ion channel defects, diabetes insipidus water transport disorder, electrolyte imbalance clinical physiology, Medical Physiology 2nd Edition study guide, cellular and molecular transport mechanisms Transcript: Ever wondered how your body keeps every single cell perfectly hydrated and you know bathed in just the right chemical soup? Yeah. It's not magic, right? It's this incredible feat of uh physiological engineering happening constantly. Absolutely. So today we're taking a deep dive into transport of salutes and water. This is based on chapter 5 of Boron and Bull Peep's medical physiology. A classic. Definitely. Our mission is to really unpack how your body's cells manage their internal environment, keep all those fluids balanced, absorb nutrients, get rid of waste, all without you consciously doing anything. Exactly. All automatic. And you know, this isn't just theory for an exam. Understanding these fundamentals is well, it's crucial. It really is the bedrock of clinical medicine. How so? Well, think about diagnosing dehydration or even just understanding how different medications actually work at the cellular level. These concepts, they set you up for so many aha moments later on. Okay, sounds essential. Let's get into it then. Let's do it. Okay, let's start with the big picture. Where are all these vital fluids actually located in your body? Like the main neighborhoods, right? Imagine your body like a city. You've got two main neighborhoods. There's the intracellular fluid or ICF. That's everything inside your cell. Okay. In. Got it. And then there's the extracellular fluid, the ECF. That's everything outside the cells. And the cell membranes are like the walls separating those neighborhoods. Exactly. They're the critical gatekeepers, the barriers between these two worlds. So, how much fluid are we talking about? Good question. For a typical, say, 70 kg adult male, total body water is around 42 L. Wow, that's a lot. It is, but it's not split evenly. Roughly 60% of that, so about 25 L, is ICF. Most of your body's water is actually inside your cells. M the other 40% about 17 L is ECF outside the cells. And I think I remember reading that women have a slightly lower percentage. That's right. Yeah. Women typically have a lower percentage of total body water, maybe around 50% compared to 60% for men. And it's mostly because women generally have a bit more atapost tissue, you know, fat tissue, which holds less water than muscle does. Makes sense. And here's a neat clinical tidbit for you. Acute changes in total body water. You can actually monitor them just by tracking body weight. It's a surprisingly simple but powerful metric. Interesting. So, okay, that ECF, the 17 L outside the cells, is it just one big homogeneous pool or No, not at all. It's actually got its own internal structure. The ECF itself has three pretty important subcompartments. Okay. What are they? First, you've got plasma volume. That's about 3 L. And it's the fluid within your heart and blood vessel. Basically, the liquid part of your blood, right? The non-cellular part. Exactly. Plasma makes up about 55% of your total blood volume. The rest is cells, red cells, white cells, platelets. That fraction of cells is what we call the hematocrit. Okay. Plasma. What's next? Second, and this is the biggest ECF compartment, is the interstatial fluid. That's about 13 L. 13. Wow. Yeah. It's the fluid that directly bathes most of your body's cells. It literally fills the spaces between cells, but outside the blood vessels. The capillary walls are the barriers separating this interstatial fluid from the plasma. Okay, so plasma in vessels, interstial fluid bathing cells. What's the third one? The third is a smaller, more specialized compartment called transcellular fluid. It's usually only about one liter. And where is that? This is fluid that's kind of trapped within specific spaces, spaces lined by epithelial cells. Think of um the synenovial fluid in your joints that keeps them lubricated. Oh, okay. for the cerebro spinal fluid CSF cushioning your brain and spinal cord. What's really fascinating here is how diverse their compositions can be compared to plasma. It hints at their specialized jobs. So different volumes, different locations. Do they also have like dramatically different chemical makeups? Oh, absolutely. And this difference is critical. It's fundamental to how our cells actually function. How different are we taught? Okay, look at the ICF inside the cells. It's remarkably high in potassium K plus A and very low in sodium N+ and chloride CL high potassium inside. Got it? Now flip that completely for the ECF which includes both plasma and that interstatial fluid. The ECF is high in sodium and chloride and low in potassium. So almost the exact opposite pretty much. It's this really dramatic difference and it's primarily maintained by one protein that's well incredibly important. The NK pump, sodium potassium pump. The NK pump. I've definitely heard of that. We'll be diving much deeper into that pump soon because it's central to almost everything here. It constantly works pushing sodium out of the cell and pulling potassium into the cell. Okay. Now, you mentioned plasma earlier. I remember hearing about plasma proteins. Do they mess with these compositions much? They do. Yeah. Significantly. Mhm. Especially when you compare plasma directly to the interstatial fluid. The biggest difference is that those large plasma proteins that are mostly stuck in the plasma. They can't easily cross the capillary walls to get into the interstatial fluid. Okay? So, they stay in the blood vessel, right? And these proteins, they actually take up space about 7% of the plasma volume and they carry a net negative charge. Why does that matter? Well, here's where it gets really interesting, especially clinically. When a lab reports your sodium level, they usually give a value per liter of plasma solution. But imagine a patient has really high levels of plasma protein or lipids like in hyper proteinmia or hyper leukemia. Okay, those proteins or lipids are taking up volume. So a reported sodium level might look low, say 122 mill equivalents per liter. But if 20% of that plasma volume is actually protein, the true sodium concentration on the water part of the plasma, the part that actually interacts with cells via the interstatial fluid, could be totally normal, like 153. Whoa. So the lab value could be misleading potentially. Yeah, it matters because it's the composition of the interstatial fluid, which is mostly protein-free. Yeah. That directly surrounds and affects your cells. That makes sense. And there's another effect called the Gibbs Don and equilibrium. Because those plasma proteins are negatively charged and trapped in the plasma, they tend to pull positively charged ions like sodium slightly into the plasma from the interstissium and they push negatively charged ions like chloride slightly out of the plasma into the interstissium. So like tiny magnets affecting the small ions kind of it creates this slight but measurable difference in the concentration of small ions between plasma water and interstitial fluid. It's subtle but important physiologically. Okay. So we have these vastly different ion concentrations inside versus outside the cell. But does that mean the overall thickness like the total particle concentration is also different? That's a great question and surprisingly the answer is no. Despite those radical differences in which ions are where all the body fluid compartments maintain roughly the same osmolality. Osmality. What's that exactly? Osmality is just the total concentration of all the free particles dissolved in the solution. For us in our body fluids, it's about 290 millios moles per kilogram of water. So 290 millios and that's the same inside and inside the cells pretty much. Yes. At steady state. So think about it. One glucose molecule counts as one particle. But if you dissolve sodium chloride, NaCCl, it breaks into two ions, Na+ and Cl. So that counts as two particles. Ah, okay. So dissociation matters. It does. And what's interesting is those big plasma proteins we talked about, even though they're massive, they contribute very little to the total osmolality because compared to the millions and millions of tiny ions and molecules, there just aren't that many protein molecules. Okay, so osmolality is balanced. What about electrical charges? You mentioned proteins are negative. Does everything need to balance out electrically? Absolutely. That's the principle of electro neutrality. In any overall solution compartment, the total number of positive charges has to equal the total number of negative charges. It has to be neutral overall. How does that work inside the cell with all that potassium? Right inside the cell, the ICF, the positive charge from potassium and the little bit of sodium far exceeds the negative charge from common ions like chloride or bicarbonate. So what makes up the difference? The proteins again exactly and other things like organic phosphates. These large intracellular molecules carry a net negative charge and they provide the balance needed for electro neutrality inside the cell. And what about in the plasma? Is there a similar concept? Yes, and it's clinically very important in blood plasma. The difference between the commonly measured positive ions, mostly sodium, and the commonly measured negative ions, mostly chloride and bicarbonate, is called the onion gap. Onion gap. Okay. A normal gap is usually somewhere between 9 and 14 mil equivalents per liter. It represents the concentration of unmeasured annions, mostly those plasma proteins we discussed, but also things like phosphate, sulfate, and organic onions. So, why is that clinically important? We'll think about certain disease states. For example, in type 1 diabetes, if insulin levels are very low, the body starts breaking down fats and producing acidic byproducts called keto acids like accetoacetate and beta hydroxybutyrate. Okay, these keto acids are negatively charged annions. They aren't usually measured in the standard electrolyte panel. So, they build up in the plasma and cause the annion gap to increase dramatically. It's a key diagnostic clue. Ah, I see. So a high onion gap can signal that kind of metabolic problem precisely. So it sounds like keeping these balances osmolality, electroneutrality, specific ion concentrations is incredibly important. What happens if things go wrong? Well, the consequences can be severe, even life-threatening. For instance, even small changes in the concentration of potassium outside the cell in the ECF can cause dangerous heart rhythm disturbances. Wow. Yeah. And similarly, if the sodium concentration in the ECF becomes abnormal, it messes with the osmolality balance. Water can then shift dramatically into or out of brain cells. Too much water influx causes brain swelling, edema, which can lead to seizures, coma, or even death or even death. And if water rushes out, the brain cells shrink, which is also incredibly dangerous. So this really underscores why understanding how the body controls these fluid compartments isn't just academic. It's absolutely vital for patient care. Okay, that really drives the point home. So we know where the fluids are, what's in them, and why balance is crucial. Now let's get to the how. How do ions and molecules actually move across those cell membranes and capillary walls? Let's start with uh passive transport. The easy way, right? Passive transport. Think of it like water flowing downhill or through a breach in a dyke. As you said, a substance moves passively across a membrane if two conditions are met. One, there's an open pathway for it. Okay. A door needs to be open. Exactly. And two, there's a favorable electrochemical gradient. That's the driving force. Electrochemical gradient. Break that down for me. Sure. It has that's the chemical part. But for charged things like ions, there's also an electrical potential energy difference. This is the voltage difference across the membrane. Positive ions are pushed away from positive areas and attracted to negative areas and vice versa for negative ions. So the electrochemical gradient combines both the concentration difference and the electrical difference. Precisely. It's the sum of those two forces that determines the overall direction and magnitude of the driving force for an ion. And with passive movement, it's always downhill along this combined gradient. Always the net movement is always down the electrochemical gradient. Like a ball rolling downhill, it's seeking the lowest energy state. What about equilibrium? Does movement stop then? Not exactly. At equilibrium, there's no net movement. But particles are still moving back and forth across the membrane. It's just that the rate of movement in one direction perfectly balances the rate in the opposite direction. Yeah. And whenever there is net passive transport happening, it's always heading back towards that equilibrium state. You mentioned ions. Is there a way to predict when an ion will be at equilibrium? Yes, there is. Yeah. For ions, the equilibrium point is described by a very important equation called the Nerst equation. The Nerst equation. The Nerst equation tells you the exact membrane voltage called the equilibrium potential for that ion at which the electrical force perfectly balances the chemical concentration force for that specific ion. At that voltage, there's no net movement of the ion. Okay. Can you give an example? Sure. Let's say you have a 10-fold gradient for potassium ions, maybe 100 millmer inside the cell and 10 mill outside. The nerst equation predicts that potassium would be at equilibrium if the membrane voltage were exactly -60 molts inside relative to outside. So if the cell's voltage was a -60 m, potassium wouldn't have a net urge to move in or out. Exactly. Assuming only those forces are acting on it. So applying that logic for a typical cell maybe with that resting membrane voltage around -60 m what does the nerst concept tell us about the net driving forces for the key ions we talked about earlier like which way do they really want to go okay let's look at that for sodium nap plus remember it's highly concentrated outside and low inside and the inside is electrically negative both forces push it inwards so the net driving force for sodium is huge maybe around megas 121 molts it desperately wants to get into the minus 121 molts. That sounds like a lot of force. It is. Now, potassium K+, it's concentrated inside. The negative inside pulls it in. The concentration gradient pushes it out. The concentration gradient usually wins slightly. So, there's a net driving force outwards, but it's much smaller, maybe around plus 28 m. So, potassium wants to leave, but not as desperately as sodium wants to enter. Good way to put it. Then there's calcium Ca2+. It's kept at incredibly low concentrations inside the cell. Plus the inside is negative. So like sodium both forces push it inwards. The driving force for calcium is enormous. Maybe 185 molt. Wow. Even stronger than sodium. Yeah. And finally chloride CL. Its situation is often more complex. Depends on the cell type. But typically the electrical force pushing it out, negative inside repels negative chloride, slightly outweighs the concentration gradient pushing it in. So there's often a small net driving force for chloride to exit the cell maybe around 13 molts. Okay. So knowing the direction and the driving force is one thing but how fast does this passive movement actually happen? What determines the rate? Ah the rate depends entirely on the transport mechanism. You know most ions and water loving hydrophilic things can't just slip through the fatty lipid billayer of the cell membrane. Right. Oil and water don't mix. Exactly. So the body uses specialized protein pathways embedded in the membrane to help these substances cross. What kind of pathways are we talking about? There are three main types of these integral membrane proteins that facilitate passive transport. First, you have pores. Pores like tiny holes. Pretty much think of them as always open tubes or tunnels providing a continuous pathway filled with water across the membrane. Any examples? Sure. There are porins in mitochondrial membranes, but the ones most relevant for water balance are the aquaporins or AQPS. These were discovered by Peter Adrink. He shared a Nobel Prize for it. They are highly specific water channels, just large enough for water molecules to pass through in single file. Really elegant. Aquaporns for water. Okay. What's the second type? Second, we have channels. You can think of channels as essentially gated pores. Gated. So, they can open and close. Exactly. They have a movable barrier, a gate that opens and closes the pathway. They also have sensors that tell the gate when to open or close. Maybe responding to changes in voltage or the binding of a chemical messenger or specific molecule. So, they're regulated, highly regulated. They also have a selectivity filter that determines which specific ions are allowed to pass through when the gate is open. And then there's the pore itself. When a channel is open, lots of ions can flow through very quickly, creating a measurable electrical current. What are some important channels? Well, NAP plus channels are absolutely critical. Given that huge inward driving force for sodium, when NA plus channels open, sodium rushes into the cell rapidly. This is fundamental for generating action potentials like nerve impulses. They're also key for sodium absorption in places like the kidney, like the ense channels. Okay, sodium channels. Then you have K plus channels. These play a major role in setting the negative resting membrane voltage in most cells and also in helping to end action potentials. Scia 2 plus channels allow rapid calcium entry down its steep gradient which is crucial for cell signaling and triggering various processes. There are even specialized proton H+ channels and various anen channels like those for chloride important in epithelial transport and pH regulation. Pores channels. What's the third type? A third type are carriers. Now carriers are different from pores and channels. They also have binding sites for the specific salutes they transport, but they never offer a continuous path across the membrane. How do they work then? They work kind of like a revolving door or maybe a turn style with two gates. They bind the salute on one side, then undergo a shape change and release the salute on the other side. Crucially, they have at least two gates that are never open at the same time. You bind, one gate closes, the other opens, you release. And this is still passive transport. Yes, if it's carriermediated facilitated diffusion, it's still passive. It doesn't require direct energy input. It just helps the solute move down its existing electrochemical gradient, but it facilitates or helps the process happen faster than it would otherwise. Does it have different characteristics than channel transport? Yes. A key difference from simple diffusion or channel flow is that carrier mediated transport is saturable. Saturable meaning it maxes out. Exactly. Think of it like having a limited number of revolving doors. Once all the doors are occupied and spinning as fast as they can, you can't increase the transport rate any further. No matter how many more people the loot are waiting outside, there's a maximum transport rate often called J-max like enzyme kinetics. Very similar. The kinetics often follow Michaelis Menton type behavior. There's a term called kinemm which is the salute concentration at which transport is half maximal. KM tells you something about the carrier's affinity for the salute. Can you give an example of a carrier? Sure. A classic one is GLUT1. It's a glucose transporter found in many cells belonging to the SLC2 family. It helps glucose, which is too large and polar to diffuse easily, get into cells down its concentration gradient via facilitated diffusion. Other examples include ura transporters, UTS, and some organic transporters, OCTs. Okay, so pores, channels, and carriers handle passive downhill movement. But what about moving things uphill against their electrochemical gradient? You said that needs energy. It absolutely does. That's active transport. And it always requires an input of energy to move something against its natural tendency. Are there different kinds of active transport? Yes, we generally distinguish between two main types based on where the energy comes from directly. Okay. What's the first type? The first is primary active transport. We often just call these pumps. These transporters directly use energy released from a chemical reaction, most commonly the breakdown or hydraysis of ATP, adenosine triphosphate. ATP, the cell's energy currency. Exactly. Think of these pumps like a motor-driven winch directly using fuel ATP to lift a heavywe the salute uphill because they often break down ATP. They're also known as atpaces. And you mentioned the NK pump earlier. Is that one of these? It is the quintessential example. The NK pump or nay atpace is probably the most important primary active transporter in animal cells. It's found in nearly every cell and it's absolutely critical. Jen's scale got a share of the Nobel Prize in chemistry for discovering it. So, how does it work? Okay, it's a protein embedded in the plasma membrane. It cycles through different shapes or confirmations. Basically, in one complete cycle, it picks up three sodium ions from inside the cell. Three sodium out uses the energy from breaking down one ATP molecule. changes shape and releases those three sodium ions outside the cell. Then in its new shape, it picks up two potassium ions from outside. Two potassium then changes shape back and releases the potassium inside the cell. So for every ATP molecule burned, it's 3a plus out, 2k plus in and that keeps sodium low inside and potassium high inside. Precisely. It actively maintains those crucial concentration gradients that we talked about earlier. You said three positive charges out but only two positive charges in. Does that imbalance matter? It does because it moves a net of one positive charge out of the cell per cycle. Three out two and a one net charge out. The pump itself is electrogenic. It directly contributes a small amount to making the inside of the cell electrically negative relative to the outside. So it helps create the membrane voltage too. Yes, it plays a role. And as you can imagine, this pump is a huge energy consumer for the cell. In some tissues, like the kidney, it might account for a third or even more of the total energy used. Wow. Is there anything that affects its function clinically? I mean, yes, very importantly, the nac pump can be specifically inhibited by a class of drugs called cardiac glycoides. Examples are uane used experimentally and dyin used clinically to treat certain heart conditions. Dyoxin inhibits the NK pump. It does. And here's a critical clinical correlation you absolutely need to remember. Low levels of potassium in the blood, a condition called hypocalemia, make digitalis toxicity much more likely. Why is that? Because potassium ions and cardiac glycoides actually compete for binding to the same site on the outside facing part of the nank pump. If potassium levels are low, the drug has less competition, binds more readily, and inhibits the pump more strongly, potentially leading to toxic effects. Okay, that's a really important connection. Hypocalemia increases dyin toxicity. Got it. Memorize that one. Are there other important primary active pumps besides the nack pump? Yes, definitely. There's a whole family of pumps called ptype at passes which work in a similar way getting phospholated during their cycle. This family includes the HK pump proton potassium pump famous for acidifying the stomach but also found in the kidney and intestines. The proton pump inhibitors act on that one. Right. Exactly. Drugs like a merazole target the gastric HK pump. Then there are the K2 plus pumps. There's PMCA plasma membrane K2 plus ATPACE which pumps calcium out of the cell and circa cycoplasmic endopplasmic reticulum K2 plus ATPACE which pumps calcium into intracellular storage compartments like the ER or SR. These are absolutely vital for keeping the free calcium concentration inside the cell incredibly low. Okay. Ptype pumps. Any other kinds? Yes. Another major class are the F-type ATP paces. These look kind of like tiny molecular lollipops. The most famous example is the mitochondrial ATP centase. the one that makes ATP. I thought these pumps used ATP. Ah, you caught me. Under normal physiological conditions in the mitochondria, the F-type ATPACE actually runs backward compared to the other pumps. It functions as an ATP synthes. How does it do that? It's amazing. The stock part called FO sits in the inner mitochondrial membrane and acts like a tiny turbine. Hydrogen ions, protons which have been pumped out by the electron transport chain, flow back into the mitochondrial matrix through this faux turbine, making it rotate. It actually spins. It actually spins. And this rotation drives confirmational changes in the head part, the F1 portion, which is a little chemical factory that synthesizes ATP from ADP and phosphate. It's called rotary catalysis. Paul Ber and John Walker shared part of a Nobel Prize for figuring this out. Wow. And the hydrogen gradient comes from that comes from the electron transport chain or respiratory chain also in the inner mitochondrial membrane. As electrons are passed along energy is used to pump protons out creating an electrochemical gradient for protons. Peter Mitchell won the Nobel Prize for proposing this whole idea. The kimosotic hypothesis. Incredible. So ftype can make ATP. Any others? There are also Vtype H+ pumps. These are found on the membranes of intracellular organels like losomes, endosomes and Golgi. They pump protons into these organels making their interior acidic which is important for their function like breaking down waste or sorting proteins. Okay. And one more big group. Yes, the ATP binding cassette or ABC transporters. This is a huge family. They all have a characteristic structure that binds ATP, but they do diverse things. Some are pumps, some act more like channels, some are regulators. Can I aim examples here? Clinically, the MDR proteins, multi-drug resistance proteins, are very important. One example is MDR1, also called PE glyoprotein. It's a pump that actively transports a wide variety of hydrophobic compounds, including many drugs and metabolic byproducts, out of cells. Why is that clinically relevant? Well, in cancer treatment, sometimes cancer cells start overexpressing MDR1. This pump can then efficiently pump anti-cancer drugs out of the cell, making the cell resistant to the treatment. It's a major challenge in chemotherapy. Ah, I see. That's a big problem. Another really famous ABC transporter is CFTR, the cystic fibrosis transmembrane regulator. It's the protein that's mutated in patients with cystic fibrosis. And what does CFTR normally do? It primarily functions as a regulated chloride channel, although it also influences other channels. Its activity is controlled by ATP binding and by phosphoration, often linked to signaling pathways like those involving cyclic AM. When it's defective, chloride transport is impaired leading to the thick mucus and other problems seen in cystic fibrosis. Okay. So, primary active transport uses ATP directly. You mentioned a second type of active transport, right? That's secondary active transport. This is a bit more indirect, but very clever. How does it work? Secondary active transport couples the downhill movement of one salute moving with its favorable electrochemical gradient to the uphill movement of another salute moving against its unfavorable gradient. So one going downhill powers the other one going uphill. Exactly. Think of it like a seessaw. The downhill movement provides the energy. Crucially the downhill gradient used is very often the steep inwardly directed sodium gradient that was established by the primary active transporter the nay pump. Ah so the ni pump sets the stage and the secondary transporters take advantage of that sodium gradient. Precisely. The NK pump does the primary work of creating the sodium gradient using ATP and then the secondary transporters harness the energy stored in that gradient to move other things. No direct ATP breakdown by the secondary transporter itself. Clever. Are there different kinds of secondary active transport? Yes, two main types based on the direction the salutes move relative to each other. First you have co- transansporters which are also called simporters. Cot transansporter simporters meaning meaning both salutes the one moving downhill and the one moving uphill move across the membrane in the same direction either both in or both out. Okay. Example a classic super important example is the NA plus glucose coransporter or SGLT. You find these mainly in the lining of your small intestine and in your kidney tubules. What does SGLT do? It uses the energy from sodium moving down its gradient into the cell to pull glucose up its gradient also into the cell. M this is how your body absorbs glucose from your diet or reclaims it from the urine even when the glucose concentration inside the cell is already higher than outside. It can really concentrate glucose amazingly so. Some HGLT transporters move two sodium ions for every one glucose molecule. That coupling allows them to generate an astonishing glucose concentration gradient potentially up to 10,000fold. Wow. Are there other important co-ansporters? Oh yes. There are sodium driven transporters for amino acids, bicarbonate like the NBC family, crucial for pH regulation and ions like the NAKCL co-ansporter NAKCC. NKCC is important in many tissues including the kidney and it's the target of powerful loop diuretics like fioanide lasix. There's also the NACL co-ansporter in NCC targeted by thoid diuretics. So diuretic drugs often work by blocking these co-ansporters in the kidney. Exactly. There are even some co-ansporters driven by proton H+ gradients instead of sodium like pept absorbing small peptides or DMT1 which transports stivalent metal ions like iron but can unfortunately also bring in toxic metals like cadmium or lead. Okay, so co- transansporters move things in the same direction. What's the other type of secondary active transport? The other type are exchangers also called andorters. Exchangers andorters opposite directions. You got it. Here the solutes move in opposite directions across the membrane. One moves downhill providing the energy for the other to move uphill in the reverse direction. Give me some examples. A really critical one is the knockat exchanger NCX. It typically moves three sodium ions into the cell downhill in exchange for moving one calcium ion out of the cell uphill. So it helps get calcium out like the PMCA pump. Yes. It's another major player in keeping intracellular calcium low or restoring low levels after they rise like during muscle contraction or nerve signaling. It's particularly important when calcium levels get quite high as you can move calcium faster than the PMCA pump, although maybe not to quite as low a level. Okay, NCX for calcium. What else? Another vital one is the NH exchanger, NHE. This typically exchanges one sodium ion coming in for one proton H+ going out. It's essential for regulating intracellular pH, helping to prevent the cell from becoming too acidic. NHE for pH makes sense. Then there are an exchang3 exchanger also known as AE proteins. These swap chloride for bicarbonate playing roles in CO2 transport and red blood cells and pH regulation in many other cells. There are many many other exchangers for various annions, organic molecules and drugs too. It's a huge family with diverse roles. Wow. Okay. That's a massive toolkit of transporters, passive pores, channels, carriers, and then active primary pumps and secondary co-ansporters and exchangers. How do they all work together inside a single cell to maintain that specific internal environment we talked about? It must be a coordinated effort. It truly is a beautifully coordinated symphony, you say. Yeah. And the undisputed conductor, the absolute kingpin is the NK pump because it sets up the sodium gradient. Exactly. By constantly working to keep intracellular sodium low and potassium high, it establishes those powerful electrochemical gradients, especially the sodium gradient that are then harnessed by so many other processes like secondary active transport. Like secondary active transport. Yes. But also remember the pump itself is electrogenic contributing to the negative membrane voltage. That negative voltage combined with the K plus leak channels allowing potassium to flow out down its gradient is the primary reason the inside of the cell is negative. And that inside negative voltage plus that huge inwardly directed chemical gradient for sodium means sodium is always incredibly eager to enter the cell. The cell captures and uses that potential energy for critical functions. You mentioned some earlier, right? Three key things driven by the NAP plus gradient maintained by the pump. One, transepithelial transport powering the movement of substances across entire cell sheets like in your gut or kidney. Two, action potentials. The rapid influx of sodium through voltage gated channels is the basis of electrical signaling in nerves and muscles. Three, secondary active transport driving the uptake of nutrients like glucose and amino acids and the regulation of other ions like calcium and protons. So the nay pump is really fundamental. Absolutely fundamental. Its continuous action is essential for cellular life as we know it. Now what about calcium? You said it's kept incredibly low inside like 10,000 times lower than outside. How does the cell manage that huge gradient especially since calcium wants to rush in so badly? Yeah. Maintaining that incredibly low resting intracellular calcium level around 100 nomaler is critical because calcium acts as a potent intracellular signal. It's achieved by several players working together. Which ones? On the plasma membrane, you have the PMCA K2 plus pump using ATP and the NO exchanger NCX using the N plus gradient actively pumping calcium out. Okay, pumps and exchangers on the outer membrane. And inside the cell, you have the circa pumps located on the membrane of the endopplasmic reticulum or cycloplasmic reticulum muscle. These pumps use ATP to sequester calcium, locking it away inside these organels, keeping the concentration in the cytoplasm very low. So it's pumped out and locked up internally. Exactly. It's a multi-prong strategy. And the PMCA pump has this neat feature. It gets stimulated by calcium itself via a protein called commodulin. So when intracellular calcium starts to rise, the pump becomes more active, helping to bring the levels back down efficiently. Clever design. What about chloride and pH inside the cell? Chloride levels inside the cell are usually a bit higher than what passive distribution alone would predict. This suggests there are active uptake mechanisms like maybe CLH CO3 exchange or sometimes NKCC activity that balance the passive elux through chloride channels and intracellular pH is typically maintained around 7.2. This is actually more alkaline or less acidic than would be expected from passive distribution of protons. So the cell actively keeps itself less acidic. Yes, primarily through powerful acid extrusion mechanisms like the no H exchanger NHE and various NO plus driven bicarbonate transporters that effectively remove acid H+ or bring in base HCO3. These transporters are often stimulated specifically when the cell's interior starts to become too acidic acting as a feedback control. Okay, so cells are constantly managing ions and pH. Now let's shift focus slightly. We've talked a lot about solutes but cells are mostly water. How is water movement controlled? Is it active too? That's a crucial point. Water transport is always passive. There are no water pumps in the way we think of ion pumps. Always passive. So how does it move? Water moves down its own driving forces. The main driving force across cell membranes is an osmotic gradient. Water flows from an area of lower total solute concentration. Higher water concentration to an area of higher total solute concentration. Lower water concentration. Trying to dilute the more concentrated side. Exactly. It's trying to equalize the solute concentration or osmolality on both sides. When the osmolality is higher outside the cell than inside, water flows out and the cell shrinks. This process is called osmosis. Water is only at equilibrium across a cell membrane when the osmality inside and outside is identical. Hydrostatic pressure like blood pressure doesn't play much role for individual cells. Not usually across the cell membrane itself. Hydrostatic pressure differences are much more important across capillary walls driving fluid movement between plasma and interstatial fluid. But for water movement into or out of a typical cell, it's overwhelmingly driven by osmotic differences. Okay. So, if water just follows solutes passively, why don't our cells just swell up and burst all the time? You mentioned earlier there are lots of impermeable, negatively charged proteins inside cells, wouldn't they constantly pull water in? That's an excellent question. You're referring to Donan forces or the gives dawnin effect applied to the cell. Those impermanent intercellular enions, proteins, organic phosphates would indeed attract positive ions like Na+ and K+ and consequently water. Poric, but it's not. It's not. Thanks again to our hero, the Nag K pump. The pump constantly works against this passive tendency to swell. By actively pumping sodium out of the cell, it effectively removes an osmotically active particle that would otherwise accumulate due to the Dawnan effect. This prevents the cell from reaching a true dawn in equilibrium and allows it to maintain a normal volume and a dynamic steady state. So the pump isn't just for gradients. It's essential for volume control too. Absolutely essential. It's sometimes called the pump leak model. Passive leaks let ions and water follow Dawn and forces. But the pump actively counteracts this burning energy to maintain volume. Amazing. But what if cell volume is suddenly challenged? Like if you become dehydrated, the ECF gets more concentrated. Or if you drink way too much pure water, the ECF gets diluted. Can cells fight back in the short term? Yes, they have sophisticated rapid responses to regulate their volume. Like what? If a cell finds itself in a hyperosmotic environment like the ECS gets too concentrated and starts to shrink, it activates mechanisms for salute uptake. It might turn on things like the NAH exchanger or the ENKCL co-ansporter to bring ions like KN plus and into the cell. Water then follows these solutes osmotically and the cell volume recovers. This is called regulatory volume increase or RVI. Okay, RVI when shrinking, what about swelling? If a cell finds itself in a hypoasmotic environment like the ECF gets too dilute and starts to swell, it does the opposite. It activates pathways for salute. It might open specific K+ channels and CL channels or activate a KCL co-ansporter to let K plus and CL leave the cell. Water follows these solutes out and the cell shrinks back towards its normal volume. This is regulatory volume decrease or RVD. RVI and RVD. So cells actively fight changes in volume. They do over minutes. But there's a really important clinical implication here especially concerning the brain. What's that? When the body is exposed to chronic hyperosmolality like in severe prolonged dehydration or conditions like hypoglycemic hyperosmolar state and diabetes, brain cells adapt over hours to days. They don't just rely on RVI with ions. They start accumulating or synthesizing specific organic solutes inside themselves. Things like sorbital, inositol, betane, torine. These are called idiogenic osmol. Idogenic osmos to raise their internal osmolality. Exactly. They raise their internal osmolality to match the high external osmolality preventing severe shrinkage. Now imagine you take someone in this state and rapidly correct their dehydration by giving them lots of dilute fluids introvenously. Uh-oh. The outside becomes dilute quickly. But the brain cells are still packed with those idioenic osmolies which they can't get rid of instantly. So now the inside of the brain cells is way more concentrated than the outside. Water rushes into the brain cells leading to brain swelling. Cerebral edema. Exactly. Severe potentially fatal cerebral edema. That's why chronic hyperosmolar states absolutely must be corrected slowly and carefully in the hospital allowing the brain cells time to shed those accumulated osmolis. That's a critical point. Okay, this brings up osmality again. You also hear the term tunicity. Are they the same thing or is there difference? They sound really similar. They sound similar but they are critically different. This is a key concept. Okay, break it down. Osmality refers to the total concentration of all osotically active solute particles in a single solution. We measure it in osm moles per kilogram of water. Your plasma osmolality is normally around 290 mls ma. Okay. Total particles in one solution. What's tenicity? Tenicity or sometimes called effective osmolality is always a comparison between two solutions separated by a membrane specifically a cell membrane in physiology. And crucially, tenacity only considers the concentration of salutes that cannot easily cross that membrane. The impermanent salutes only the ones that can't cross. Why? Because only the impermanent solutes exert a sustained osmotic pressure that causes water to shift across the cell membrane and change cell volume in the long run. Can you give an example? Sure. Think about ura. URA is a small molecule that can cross most cell membranes relatively easily using ura transporters. So if you add ura to the extracellular fluid raising its osmolality, what happens to the cell? Well, water should leave initially, right? Because the outside is more concentrated initially. Yes. The cell will shrink transiently. But then ura starts to enter the cell down its own concentration gradient. As ura enters, the cell's internal osmolality rises, pulling water back in. Eventually, ura equilibrates across the membrane and the cell returns to its original volume. So adding ura increases ECF osmolality, but it doesn't change the cell volume long term, meaning it doesn't contribute to tenicity. URA solutions are isosis, but hypotonic. Ah, because it permeates. What about something that doesn't permeate? like manitol a sugar alcohol sometimes used clinically manners cannot easily enter cells if you add manitol to the ECF that raises the ECF osmolality and its tenicity water leaves the cell and the cell shrinks and stay shrunk because manitol can't get in to balance things out so manitol exerts a sustained osmotic effect so tenicity is about the non-permeating solutes that cause lasting water shifts precisely clinically when we estimate the tenicity of the ECS we primarily look at the concentration of sodium and it's associated annions like chloride and glucose if levels are high as it enters cells slowly without insulin. We specifically do not include BUN blood ura nitrogen which reflects ura concentration because ura is considered a permanent salute. That makes sense. Sodium and glucose determine tenicity not ura. Right? And this leads to a really fundamental principle for understanding fluid balance in the whole body. Your total body sodium content primarily determines your ECF volume. Think about it. Sodium stays mostly in the ECF and holds water there. More sodium, more ECF volume. Okay, sodium controls ECF volume. What about osmolality? Your total body water content primarily determines your overall body osmolality. Adding or removing pure water changes the concentration of everything. Can you illustrate that? Sure. Imagine you infuse isotonic saline.9% ACL which has the same tenicity as body fluids. Sodium and water are added in proportion. Where does it go? mostly into the ECF because sodium stays there. So ECF volume expands but ICF volume and overall osmolality don't really change. Okay. Isotonic saline expands ECF. What if you add pure water like drinking a lot? Pure water has no solutes. It distributes throughout total body water proportionally between ECF and ICF mostly ICF since it's bigger. So total body osmality drops. ECF volume increases a bit and ICF volume increases significantly. Cells swell. Okay. Okay. And what if you add pure NaCCl like eating salt tablets without water? Now you're adding solute without water. The salt stays in the ECF raising ECF osmolality and tonicity. This pulls water out of the ICF by osmosis. So ICF volume shrinks, ECF volume expands, and overall body osmolality increases. That really clarifies how sodium and water affect the different compartments. It's a cornerstone for understanding fluid therapy. Okay, we've covered single cells brilliantly. Now let's zoom out. How do entire sheets of cells like the epithelia lining your gut or kidney tubules manage transport of solutes and water to control the body's internal environment? The millu interior as they say, right? Epithelia are fascinating. They're basically uninterrupted sheets of cells that are stuck together by specialized connections called tight junctions. Tight junctions. What do they do? They do two main things. First, they act like a selective fence or barrier between adjacent cells, controlling how easily substances can sneak through the gaps. Second, and really importantly, they divide the individual epithelial cells membrane into two distinct domains or regions. Two regions. Yes. There's the apical membrane, which faces the lumen or the outside world, like the inside of your gut or kidney tubule. It's also sometimes called the mucosal or luminal membrane. And then there's the basilateral membrane which faces the underlying tissue, the blood supply and adjacent cells. It's sometimes called the sorosal or perittubular membrane. So the type junction is like the dividing line between the top front surface and the bottom back side surfaces. Exactly. This separation of the membrane into apical and basilateral domains each with potentially different proteins embedded in it is called polarization and it's absolutely key. Why is polarization so important? Because it allows for vectoral transport. That means the epithelium can move substances directionally from one side of the sheet, say the lemon, to the other side, the blood, or vice versa. It's not random. It's directed movement across the whole layer. Victorial transport. Okay. Are all epithelia the same in how they do this? No, there's a broad classification based on how sealed those tight junctions are. We talk about tight epithelia versus leaky epithelia. What's the difference? Leaky epithelia like you find in the small intestine or the proximal tubule of the kidney have tight junctions that are well relatively leaky to ions and water. They have low electrical resistance across the sheet. These epithelia are designed for bulk transport moving large amounts of fluid in solutes often in a way that's nearly isosismotic meaning the fluid transported has about the same osmolality as the fluid left behind. They use both pathways for transport. transcellular movement through the cell crossing both apical and basilateral membranes and curricellular movement between the cells sneaking through those leaky tight junctions. So leaky epithelia do bulk transport using both routes. What about tight epithelia? Tight epithelia like in the collecting duct of the kidney or the urinary bladder lining have very restrictive tight junctions. They have high electrical resistance. Their job is often to generate or maintain large concentration gradients for ions or large osmotic gradients. They prevent things from leaking back easily because the parasellular pathway is so restricted. They rely much more heavily on the transcellular pathway to move specific substances. So tight epithelia are better for creating steep gradients mostly moving stuff through the cells. Precisely. Okay. So how do these polarized epithelial cells actually direct the transport? How do they make sodium go in from the lumen and out towards the blood for example? They do it by being very clever about where they put their different transport proteins. Remember all those channels, carriers and pumps we talked about? Epithelial cells strategically place specific transporters on the aical membrane and different ones on the basalateral membrane. Ah the polarization again. Exactly. For instance, that crucial nac pump. In virtually all transporting epithelia, it's located almost exclusively on the base lateral membrane. Why is that important? Because by pumping sodium out across the base lateral membrane into the interstatial fluid and blood, it keeps the intracellular sodium concentration low. This in turn creates that strong inwardly directed electrochemical gradient for sodium at the apical membrane facing the lumen. And that apical sodium gradient can then power other transport. Exactly. It's the driving force for many epithelial transport processes. Let's look at a few classic examples. Okay. Consider nan plus absorption like in the collecting tubule of the kidney. This is often called the using model. Here you have specific sodium channels like NAC only on the apical membrane. Sodium flows passively into the cell through these channels down its gradient. Okay. Sodium enters apically. Then the nanc pump on the basilateral membrane actively pumps that sodium out of the cell into the blood. So net movement is apical to basil lateral absorption. And what about chloride? Doesn't charge need to follow? Good point. The movement of positive sodium charge across the cell makes the lumen slightly electrically negative compared to the blood side. This electrical potential difference can then pull negatively charged chloride ions passively across the tight junctions via the parasellar pathway following the sodium. So you get net NaCCl absorption. Clever. Okay. What about secretting something like potassium? Sure. In some parts of the kidney, the goal is K plus secretion. These cells still have the NK pump on the base lateral side bringing K+ into the cell but then they place specific K+ channels on the apical membrane. Now potassium flows passively out of the cell through these channels into the lumen down its electrochemical gradient. Result net K plus secretion. So just changing which channel is on which membrane reverses the net direction for K plus brain. Precisely. Let's take glucose absorption like in the small intestine or proximal tubule. Here the apical membrane has the knob plus glucose coransporter SGLT the secondary active transporter we talked about right it uses the sertium gradient to pull glucose into the cell even against a glucose gradient. Now the glucose concentration inside the cell is high. So on the basalateral membrane the cell places a facilitated diffusion glucose transporter glut. Glucose then flows passively out of the cell via glut down its concentration gradient into the interstitial fluid and blood. Result efficient glucose absorption. sglt in, glut out. Got it. One more. How about secretreting chloride like in the airways? Good example. Very relevant to cystic fibrosis. For CL secretion like in intestinal crypts or airways, the setup is different. Again, on the basal membrane, you often find the NKCL co-ansporter NKCC-1. This uses the sodium gradient to bring NA plus K plus high NL into the cell from the blood side. So chloride accumulates inside the cell. Yes. Then on the apical membrane you have chloride channels like CFTR. When these channels open, chloride flows passively out of the cell into the lumen down its electrochemical gradient. Sodium often follows parasellularly to maintain electro neutrality. Result netclion into the lumen and if CFTR is broken that chloride secretion is blocked. Exactly. Leading to the problem seen in CF. So you see it's all about the specific placement of transporters on the apical versus basilateral membranes. That makes perfect sense. and water in all these cases. Does it just follow the salutes being moved? That's the fundamental rule for water transport across epithelia. Water movement is passive and always follows salute movement in response to osmotic gradients created by that salute transport. Wherever net salute goes, water tends to follow to keep things osmotically balanced. Even in those leaky epithelia doing bulk transport where the fluid moved seems isosismotic. Yes. The current thinking on ismotic fluid absorption like in the proximal tubule where huge amounts of salt and water are reabsorbed without a measurable osmotic gradient between lumin and blood involves a couple of ideas. One is that the epithelial cells have extremely high water permeability thanks to abundant aquaporn. So even tiny transient osmotic gradients are enough to move lots of water. Okay. Another idea is the concept of local osmosis. Soludes pumped into the narrow spaces between cells. the lateral intracellular spaces might create small localized regions of hyperosmolality there. These local hyperosmotic pockets then draw water across the cells or through the tight junctions into these spaces before the fluid equilibrates as it moves towards the blood. So tiny gradients or super high permeability allows isotic flow. That's the idea. And one final crucial point about epithelial transport, it is highly regulated, meaning it can be turned up or down. Exactly. The body needs to adjust absorption and secretion based on its needs. So epithelial cells can regulate transport in several ways. They can change the synthesis or degradation of transport proteins. For example, the hormone eldoststerone increases the number of nay pumps and apical NA plus channels in kidney cells boosting sodium reabsorption. Okay. Make more or less transporters. They can recruit existing transporters from storage pools inside the cell and insert them into the membrane when needed. For example, histamine causes HK pumps stored inside gastric parietal cells to move to the apical membrane to secrete acid. Or insulin causes glut4 glucose transporters to move to the membrane in muscle and fat cells. Move them to where they're needed. They can modify existing proteins often through phospholation or defosphilation to change their activity. Like how cyclic AMP dependent phospholation activates the CFTR chloride channel. Flip a switch on the protein. They can even alter the permeability of the parisellular pathway by modifying the tight junctions themselves, adjust the leakiness between cells. And of course, transport rates can be influenced by the availability of the transported substance in the lumen. It's a dynamic, adaptable system. Wow. Okay. We have navigated in an incredibly intricate world today. from the tiniest pores and pumps in a single cell membrane up to the coordinated action and regulation of entire epithelial sheets. It really is mind-boggling how precisely your body manages all these fluids, ions, and volumes every single second. It really is. This deep dive, I hope, highlights how every single cell is in its own way a master of balancing these forces and flows. And understanding these fundamental mechanisms is just absolutely essential if you want to unravel the complexities of health and disease. Yeah. You know, remember these are the very processes that underpin how your body responds to everything from just taking a sip of water to processing a meal to responding to critical medical interventions like IV fluids or diuretics. So, what does this all mean for you, our listener? It means you're building an absolutely rock solid foundation for your medical and physiological understanding. You're part of the deep dive family and we absolutely know you are capable of mastering this material. It's complex, but you can do it. Definitely keep asking those questions. Keep trying to connect the dots between these different transporters and processes. And remember, the more you explore physiology, the more truly incredible the human body becomes. Couldn't agree more. Until next time, keep diving deep. And maybe consider this as you go. How can just a a seemingly minor alteration, maybe a single mutation affecting the function of just one type of ion channel lead to such widespread and sometimes devastating diseases throughout the entire body? It really reinforces the incredible inter interconnectedness of all these systems we've discussed
188958	https://www.simplilearn.com/tutorials/statistics-tutorial/bayes-theorem	HomeResourcesData Science & Business AnalyticsThe Ultimate Statistics TutorialWhat Is Bayes Theorem: Formulas, Examples and Calculations Tutorial Playlist #### Statistics Tutorial Overview #### Everything You Need to Know About the Probability Density Function in Statistics Lesson - 1#### The Best Guide to Understand Central Limit Theorem Lesson - 2#### Measures of Central Tendency : Mean, Median and Mode Lesson - 3#### The Ultimate Guide to Understand Conditional Probability Lesson - 4#### Percentile in Statistics Lesson - 5#### The Best Guide to Understand Bayes Theorem Lesson - 6#### Everything You Need to Know About the Normal Distribution Lesson - 7#### An In-Depth Explanation of Cumulative Distribution Function Lesson - 8#### Chi-Square Test Lesson - 9#### What Is Hypothesis Testing in Statistics? Types and Examples Lesson - 10#### Understanding the Fundamentals of Arithmetic and Geometric Progression Lesson - 11#### The Definitive Guide to Understand Spearman’s Rank Correlation Lesson - 12#### Mean Squared Error: Overview, Examples, Concepts and More Lesson - 13#### All You Need to Know About the Empirical Rule in Statistics Lesson - 14#### The Complete Guide to Skewness and Kurtosis Lesson - 15#### A Holistic Look at Bernoulli Distribution Lesson - 16#### All You Need to Know About Bias in Statistics Lesson - 17#### A Complete Guide to Get a Grasp of Time Series Analysis Lesson - 18#### The Key Differences Between Z-Test Vs. T-Test Lesson - 19#### The Complete Guide to Understand Pearson's Correlation Lesson - 20#### A Complete Guide on the Types of Statistical Studies Lesson - 21#### Everything You Need to Know About Poisson Distribution Lesson - 22#### Your Best Guide to Understand Correlation vs. Regression Lesson - 23#### The Most Comprehensive Guide for Beginners on What Is Correlation Lesson - 24 What Is Bayes Theorem: Formulas, Examples and Calculations Lesson 6 of 24 By Aryan GuptaLast updated on Aug 23, 202575369 PreviousNext Tutorial Playlist #### Statistics Tutorial Overview #### Everything You Need to Know About the Probability Density Function in Statistics Lesson - 1#### The Best Guide to Understand Central Limit Theorem Lesson - 2#### Measures of Central Tendency : Mean, Median and Mode Lesson - 3#### The Ultimate Guide to Understand Conditional Probability Lesson - 4#### Percentile in Statistics Lesson - 5#### The Best Guide to Understand Bayes Theorem Lesson - 6#### Everything You Need to Know About the Normal Distribution Lesson - 7#### An In-Depth Explanation of Cumulative Distribution Function Lesson - 8#### Chi-Square Test Lesson - 9#### What Is Hypothesis Testing in Statistics? Types and Examples Lesson - 10#### Understanding the Fundamentals of Arithmetic and Geometric Progression Lesson - 11#### The Definitive Guide to Understand Spearman’s Rank Correlation Lesson - 12#### Mean Squared Error: Overview, Examples, Concepts and More Lesson - 13#### All You Need to Know About the Empirical Rule in Statistics Lesson - 14#### The Complete Guide to Skewness and Kurtosis Lesson - 15#### A Holistic Look at Bernoulli Distribution Lesson - 16#### All You Need to Know About Bias in Statistics Lesson - 17#### A Complete Guide to Get a Grasp of Time Series Analysis Lesson - 18#### The Key Differences Between Z-Test Vs. T-Test Lesson - 19#### The Complete Guide to Understand Pearson's Correlation Lesson - 20#### A Complete Guide on the Types of Statistical Studies Lesson - 21#### Everything You Need to Know About Poisson Distribution Lesson - 22#### Your Best Guide to Understand Correlation vs. Regression Lesson - 23#### The Most Comprehensive Guide for Beginners on What Is Correlation Lesson - 24 Table of Contents Bayes Theorem Terminologies What Is Bayes Theorem? Bayes Theorem Formula Example of Bayes Theorem Conclusion Probability is a metric for determining the likelihood of an event occurring. Many things are impossible to predict with 100% certainty. Using it, you can only predict the probability of an event occurring, i.e., how likely it is to occur. In this tutorial, you will learn about Bayes Theorem, an important sub-topic in probability theory. Bayes Theorem Terminologies Before you dive into the world of the Bayes Theorem, you must first grasp a few concepts. Understanding Bayes Theorem requires an understanding of the following terms. Experiment When you hear the word "experiment," what is the first image that comes to mind? The majority of people envision a chemistry lab with test tubes and beakers. In probability theory, the concept of an experiment is quite similar. An experiment is a carefully planned procedure carried out under carefully monitored conditions. Experiments include tossing a coin, rolling a die, and drawing a card from a well-shuffled deck of cards. Become a Certified Power BI Developer PL-300 Microsoft Power BI Certification TrainingExplore Program Sample Space An outcome is the result of an experiment. The sample space is the set of all possible outcomes of an event. For example, if you’re throwing dice and keeping track of the results, the sample space will be: {1, 2, 3, 4, 5, 6} Event An event is the outcome of a random experiment. Getting heads when you toss a coin is an event. Getting a 4 when you roll a fair die is an event. Random Variable A random variable is a variable with an unknown value or a function that assigns values to each of the outcomes of an experiment. A random variable can be discrete (meaning it has specific values) or continuous (meaning it has no specific values). Exhaustive Events Two or more events associated with a random experiment are exhaustive if their union is the sample space. Let's say A is the event of a red card being drawn from a pack, and B is the event of a black card being drawn. Because the sample space S = {red, black}, A and B are exhaustive. Independent Events When the occurrence of one event has no bearing on the occurrence of the other, the two events are said to be independent. Two events A and B, are said to be independent in mathematics if: P(A ∩ B) = P(AB) = P(A)P(B) For example, if A gets a 3 on a die roll and B gets a jack of hearts from a well-shuffled deck of cards, then A and B are independent events. Conditional Probability Let A and B be the two events associated with a random experiment. Then, the probability of A's occurrence under the condition that B has already occurred and P(B) ≠ 0 is called the Conditional Probability. It is denoted by P (A/B). Thus, you have: What Is Bayes Theorem? The Bayes theorem is a mathematical formula for calculating conditional probability in probability and statistics. In other words, it's used to figure out how likely an event is based on its proximity to another. Bayes law or Bayes rule are other names for the theorem. Your Data Analytics Career is Around The Corner! Data Analyst Master’s ProgramExplore Program Bayes Theorem Formula The formula for the Bayes theorem can be written in a variety of ways. The following is the most common version: P(A ∣ B) = P(B ∣ A)P(A) / P(B) P(A ∣ B) is the conditional probability of event A occurring, given that B is true. P(B ∣ A) is the conditional probability of event B occurring, given that A is true. P(A) and P(B) are the probabilities of A and B occurring independently of one another. Example of Bayes Theorem Now, try to solve a problem using the Bayes theorem. Problem 1: Three urns contain 6 red, 4 black; 4 red, 6 black, and 5 red, 5 black balls respectively. One of the urns is selected at random and a ball is drawn from it. If the ball drawn is red, find the probability that it is drawn from the first urn. Solution: Let E1, E2, E3, and A be the events defined as follows: E1 = urn first is chosen E2 = urn second is chosen E3 = urn third is chosen A = ball drawn is red Since there are three urns and one of the three urns is chosen at random, therefore: P(E1) = P(E2) = P(E3) = ⅓ If E1 has already occurred, then urn first has been chosen, containing 6 red and 4 black balls. The probability of drawing a red ball from it is 6/10. So, P(A/E1) = 6/10 Similarly, you have P(A/E2) = 4/10 and P(A/E3) = 5/10 You are required to find the P(E1/A) i.e., given that the ball drawn is red, what is the probability that it is drawn from the first urn. By Bayes theorem, you have P(E1/A) = P(E1) P(A/E1)P(E1) P(A/E1) + P(E2) P(A/E2) + P(E3) P(A/E3) = 1/3 6/10(1/3 6/10) + (1/3 4/10) + (1/3 5/10) = ⅖ Problem 2: An insurance company insured 2000 scooter drivers, 4000 car drivers, and 6000 truck drivers. The probability of an accident involving a scooter driver, car driver, and a truck is 0.01, 0.03, and 0.015 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver? Let E1, E2, E3, and A be the events defined as follows: E1 = person chosen is a scooter driver E2 = person chosen is a car driver E3 = person chosen is a truck driver and A = person meets with an accident Since there are 12000 people, therefore: P(E1) = 2000/12000 = ⅙ P(E2) = 4000/12000 = ⅓ P(E3) = 6000/12000 = ½ It is given that P(A / E1) = Probability that a person meets with an accident given that he is a scooter driver = 0.01 Similarly, you have P(A / E2) = 0.03 and P(A / E3) = 0.15 You are required to find P(E1 / A), i.e. given that the person meets with an accident, what is the probability that he was a scooter driver? P(E1/A) = P(E1) P(A/E1)P(E1) P(A/E1) + P(E2) P(A/E2) + P(E3) P(A/E3) = 1/6 0.01(1/6 0.01) + (1/3 0.03) + (1/2 0.15) = 1/52 Conclusion You have come to an end of this Bayes Theorem tutorial. The purpose of this tutorial was to introduce you to the Bayes theorem and conditional probability. The Bayes theorem is the foundation of Naive Bayes, one of the most widely used classification algorithms in data science. If you are interested in statistics of data science and skills needed for such a career, Simplilearn’s Data Analyst is the right course for you. If you have any questions regarding this ‘Bayes Theorem’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning! About the Author Aryan Gupta Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning. 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188959	https://math.stackexchange.com/questions/2727819/prove-that-ex-a2-is-minimized-when-a-ex	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Prove that $E((X-a)^2)$ is minimized when $a=E(X)$ Ask Question Asked Modified 1 year, 3 months ago Viewed 19k times 3 $\begingroup$ $X$ is an arbitrary continuous random variable. I tried to do this by saying since $X-a$ is squared that means its lowest possible value is 0 and then I tried to solve for $a$ when the expression is $0$. $$ \begin{align} E((X-a)^2)&=0\ E(X^2)-2aE(X) +a^2&=0\ a^2-2aE(X)+E(X)^2&=E(X)^2-E(X^2)\ (a-E(X))^2&=E(X)^2-E(X^2)\ a-E(X)&=\sqrt{(E(X))^2-E(X^2)}\ a&=E(X) \pm\sqrt{(E(X))^2-E(X^2)}\ \end{align} $$ Now I don't know if there's some trick to say that $\sqrt{(E(X))^2-E(X^2)}=0$ or if I'm barking up the wrong tree with this approach. probability Share asked Apr 8, 2018 at 14:06 Frank ShmrankFrank Shmrank 48011 gold badge55 silver badges1919 bronze badges $\endgroup$ 5 3 $\begingroup$ Just consider $E((X-a)^2$ as a quadratic polynomial on $a$, using the linearity of expectation. The value doesn't have to be $0$ in general. $\endgroup$ shrimpabcdefg – shrimpabcdefg 2018-04-08 14:09:44 +00:00 Commented Apr 8, 2018 at 14:09 $\begingroup$ @shrimpabcdefg I assume you're leading me to taking derivatives to find the minimum the "regular" way but I'm not well-versed on derivatives and expected values so that's why I tried to side step that approach. $\endgroup$ Frank Shmrank – Frank Shmrank 2018-04-08 14:13:12 +00:00 Commented Apr 8, 2018 at 14:13 $\begingroup$ You don't need derivatives to minimize quadratic polynomials. Just use the "complete-the-square" method $\endgroup$ Ewan Delanoy – Ewan Delanoy 2018-04-08 14:16:12 +00:00 Commented Apr 8, 2018 at 14:16 $\begingroup$ @EwanDelanoy, I did complete the square to solve for $a$ when I assume the expression equals 0 so I think there's a trick or rule that I'm not aware of or is slipping my mind that you're treating as obvious. $\endgroup$ Frank Shmrank – Frank Shmrank 2018-04-08 14:29:58 +00:00 Commented Apr 8, 2018 at 14:29 $\begingroup$ Let us put it this way : what is the minimum of $(a-7)^2$ when $a$ is a real variable ? Of $(a-11)^2$ ? Etc. $\endgroup$ Ewan Delanoy – Ewan Delanoy 2018-04-08 14:50:25 +00:00 Commented Apr 8, 2018 at 14:50 Add a comment \| 5 Answers 5 Reset to default 7 $\begingroup$ By linearity of expectation, we have $E((X-a)^2)=a^2-2E(X)a+E(X^2)=(a-E(X))^2+E(X^2)-E(X)^2$. Considering this as a quadratic function on $a$, it is clear that the minimum is acquired when $a=E(X)$. Share answered Apr 8, 2018 at 14:17 shrimpabcdefgshrimpabcdefg 1,52477 silver badges1010 bronze badges $\endgroup$ 2 $\begingroup$ Your final answer is my 4th line, so I'm sorry it isn't clear to me why that is the case. $\endgroup$ Frank Shmrank – Frank Shmrank 2018-04-08 14:26:20 +00:00 Commented Apr 8, 2018 at 14:26 $\begingroup$ Thank you. Should not the expectation be like in the discrete case: $$ E\left[ \left( X-\mu \right) ^2 \right] \sum_x{\left( X-\mu \right) ^2p\left( \left( X-\mu \right) ^2 \right)} $$ $\endgroup$ Avv – Avv 2022-04-08 14:32:25 +00:00 Commented Apr 8, 2022 at 14:32 Add a comment \| 5 $\begingroup$ For every $a\in\mathbb R$ we have:$$\begin{aligned}\mathsf{E}\left(X-a\right)^{2} & =\mathsf{E}\left(X-\mathsf{E}X+\mathsf{E}X-a\right)^{2}\ & =\mathsf{E}\left[\left(X-\mathsf{E}X\right)^{2}+2\left(X-\mathsf{E}X\right)\left(\mathsf{E}X-a\right)+\left(\mathsf{E}X-a\right)^{2}\right]\ & =\mathsf{E}\left(X-\mathsf{E}X\right)^{2}+\mathsf{E}2\left(X-\mathsf{E}X\right)\left(\mathsf{E}X-a\right)+\left(\mathsf{E}X-a\right)^{2}\ & =\mathsf{E}\left(X-\mathsf{E}X\right)^{2}+2\left(\mathsf{E}X-a\right)\mathsf{E}\left(X-\mathsf{E}X\right)+\left(\mathsf{E}X-a\right)^{2}\ & =\mathsf{E}\left(X-\mathsf{E}X\right)^{2}+2\left(\mathsf{E}X-a\right)\left(\mathsf{E}X-\mathsf{E}X\right)+\left(\mathsf{E}X-a\right)^{2}\ & =\mathsf{E}\left(X-\mathsf{E}X\right)^{2}+\left(\mathsf{E}X-a\right)^{2}\ & \geq\mathsf{E}\left(X-\mathsf{E}X\right)^{2} \end{aligned} $$ LHS takes minimum value iff $a=\mathsf EX$ Share edited Apr 8, 2018 at 17:00 answered Apr 8, 2018 at 15:56 drhabdrhab 154k1111 gold badges8787 silver badges222222 bronze badges $\endgroup$ 3 $\begingroup$ Thank you very much. Should not the expectation be like in the discrete case: $$ E\left[ \left( X-\mu \right) ^2 \right] \sum_x{\left( X-\mu \right) ^2p\left( \left( X-\mu \right) ^2 \right)} $$ $\endgroup$ Avv – Avv 2022-04-08 14:32:42 +00:00 Commented Apr 8, 2022 at 14:32 $\begingroup$ @Avv Sorry, but the expression you provide in your comment is "unreadable" for me. Secondly the generality of the answer makes any distinction between "discrete" and "not discrete" irrelevant. $\endgroup$ drhab – drhab 2022-04-11 15:03:07 +00:00 Commented Apr 11, 2022 at 15:03 $\begingroup$ Thank you. I was trying to say that why we don't substitute $x$ in $E(x) = \sum x p(x)$ with $(X-\mu)^2$ please and then work out it that way? $\endgroup$ Avv – Avv 2022-04-11 17:59:51 +00:00 Commented Apr 11, 2022 at 17:59 Add a comment \| 3 $\begingroup$ Let's define a function which depends on $ a $: $$ f \left( a \right) = \mathbb{E} \left[ \left( X - a \right)^{2} \right] $$ Now, the optimization problem is: $$ \hat{a} = \arg \min_{a} f \left( a \right) $$ The requirement for optimal point - $ f' \left( \hat{a} \right) = 0 $.Working on the function as in the answer by @shrimpabcdefg: $$\begin{align} \frac{d}{d a} f \left( a \right) & = \frac{d}{d a} \mathbb{E} \left[ \left( X - a \right)^{2} \right] \ & = \frac{d}{d a} {a}^{2} - 2 a \mathbb{E} \left[ X \right] + \mathbb{E} \left[ X \right]^{2} \ & = 2 a - 2 \mathbb{E} \left[ X \right] \ & \Rightarrow 2 \hat{a} - 2 \mathbb{E} \left[ X \right] = 0 \Rightarrow \hat{a} = \mathbb{E} \left[ X \right] \end{align}$$ Share edited Apr 8, 2018 at 16:40 answered Apr 8, 2018 at 15:21 RoyiRoyi 10.5k77 gold badges5656 silver badges115115 bronze badges $\endgroup$ Add a comment \| 2 $\begingroup$ Your error is in the very first line of your displayed equations where you assume that the minimum value that $E[(X-a)^2]$ is $0$ and attempt to solve for $a$. The minimum value of $E[(X-a)^2]$ (regarded as a function of $a$) is not $0$; it is $\sigma^2$ where $\sigma$ is the standard deviation of $X$. Keep in mind that $(E[X])^2 - E[X^2]$ is generally a negative number (it equals $-\sigma^2$) and so the value of $a$ that you compute as the minimizer of $E[(X-a)^2]$ is actually a complex number. Share answered Apr 8, 2018 at 14:41 Dilip SarwateDilip Sarwate 26.5k44 gold badges5757 silver badges124124 bronze badges $\endgroup$ Add a comment \| 0 $\begingroup$ If X is a continuous random variable such that E [(X a)2] < for all a, show that E [(X a)2] is minimized when a = E(X). Share answered Jun 7, 2024 at 17:19 SUN PANHASUN PANHA 1 $\endgroup$ 1 $\begingroup$ Your answer could be improved with additional supporting information. Please edit to add further details, such as citations or documentation, so that others can confirm that your answer is correct. You can find more information on how to write good answers in the help center. $\endgroup$ Community – Community Bot 2024-06-07 17:43:10 +00:00 Commented Jun 7, 2024 at 17:43 Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions probability See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Related Can I derive the formula for expected value of continuous random variables from the discrete case? 1 Help reconcile probability density functions and "probability" 0 Determine the cdf of a function of a random variable 0 Find C value given CDF Hot Network Questions Calculating the node voltage Lingering odor presumably from bad chicken How do trees drop their leaves? 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188960	https://www.wyzant.com/resources/answers/824748/find-ac-in-the-rectangle	Find AC in the Rectangle \| Wyzant Ask An Expert Log inSign up Find A Tutor Search For Tutors Request A Tutor Online Tutoring How It Works For Students FAQ What Customers Say Resources Ask An Expert Search Questions Ask a Question Wyzant Blog Start Tutoring Apply Now About Tutors Jobs Find Tutoring Jobs How It Works For Tutors FAQ About Us About Us Careers Contact Us All Questions Search for a Question Find an Online Tutor Now Ask a Question for Free Login WYZANT TUTORING Log in Sign up Find A Tutor Search For Tutors Request A Tutor Online Tutoring How It Works For Students FAQ What Customers Say Resources Ask An Expert Search Questions Ask a Question Wyzant Blog Start Tutoring Apply Now About Tutors Jobs Find Tutoring Jobs How It Works For Tutors FAQ About Us About Us Careers Contact Us Subject ZIP Search SearchFind an Online Tutor NowAsk Ask a Question For Free Login GeometryRectangle Nicole M. asked • 03/01/21 Find AC in the Rectangle In rectangle ABCD, AB= x+15, BC= x-8, and CD = 2x+3. Find AC to the nearest hundredth. Follow •1 Add comment More Report 1 Expert Answer Best Newest Oldest By: Linda M.answered • 03/01/21 Tutor 4.9(282) Mathematics Tutor for Middle/High School and College Level Classes About this tutor› About this tutor› In a rectangle, opposite sides are congruent so we know AB=CD Set those equations equal to solve for x x+15=2x+3; solve for x which gives us x=12 plug in to AB and BC, which gives ust AB=27 and BC = 4 We want the diagonal which is AC. ABC for a right triangle so we use the Pythagorean theorem to solve for AC. 27^2 + 4^2=AC^2 729+16 745=AC^2 Take the square root of both sides which gives us 27.29 Upvote • 0Downvote Add comment More Report Still looking for help? Get the right answer, fast. Ask a question for free Get a free answer to a quick problem. Most questions answered within 4 hours. OR Find an Online Tutor Now Choose an expert and meet online. 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188961	https://cran.r-project.org/web/packages/olsrr/vignettes/residual_diagnostics.html	Residual Diagnostics Residual Diagnostics Introduction olsrr offers tools for detecting violation of standard regression assumptions. Here we take a look at residual diagnostics. The standard regression assumptions include the following about residuals/errors: The error has a normal distribution (normality assumption). The errors have mean zero. The errors have same but unknown variance (homoscedasticity assumption). The error are independent of each other (independent errors assumption). Residual QQ Plot Graph for detecting violation of normality assumption. model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars) ols_plot_resid_qq(model) Residual Normality Test Test for detecting violation of normality assumption. model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars) ols_test_normality(model) ``` Test Statistic pvalue Shapiro-Wilk 0.9366 0.0600 Kolmogorov-Smirnov 0.1152 0.7464 Cramer-von Mises 2.8122 0.0000 Anderson-Darling 0.5859 0.1188 ``` Correlation between observed residuals and expected residuals under normality. model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars) ols_test_correlation(model) ## 0.970066 Residual vs Fitted Values Plot It is a scatter plot of residuals on the y axis and fitted values on the x axis to detect non-linearity, unequal error variances, and outliers. Characteristics of a well behaved residual vs fitted plot: The residuals spread randomly around the 0 line indicating that the relationship is linear. The residuals form an approximate horizontal band around the 0 line indicating homogeneity of error variance. No one residual is visibly away from the random pattern of the residuals indicating that there are no outliers. model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars) ols_plot_resid_fit(model) Residual Histogram Histogram of residuals for detecting violation of normality assumption. model <- lm(mpg ~ disp + hp + wt + qsec, data = mtcars) ols_plot_resid_hist(model)
188962	https://artofproblemsolving.com/wiki/index.php/2018_AIME_II_Problems/Problem_11?srsltid=AfmBOop6Lr7tzxE0eZyIpv2wG0xpX1j04kwZ5MbvToCwbaNQ-j0036YJ	Art of Problem Solving 2018 AIME II Problems/Problem 11 - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki 2018 AIME II Problems/Problem 11 Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search 2018 AIME II Problems/Problem 11 Contents [hide] 1 Problem 2 Solution 1 3 Solution 2 4 Solution 3 (Recursion) 5 Solution 4 (PIE) 6 Solution 5 (Recursion) 7 Solution 6 (Complementary) Problem Find the number of permutations of such that for each with , at least one of the first terms of the permutation is greater than . Solution 1 If the first number is , then there are no restrictions. There are , or ways to place the other numbers. If the first number is , can go in four places, and there are ways to place the other numbers. ways. If the first number is , .... 4 6 _ _ _ _ 24 ways 4 _ 6 _ _ _ 24 ways 4 _ _ 6 _ _ 24 ways 4 _ _ _ 6 _ 5 must go between and , so there are ways. ways if 4 is first. If the first number is , .... 3 6 _ _ _ _ 24 ways 3 _ 6 _ _ _ 24 ways 3 1 _ 6 _ _ 4 ways 3 2 _ 6 _ _ 4 ways 3 4 _ 6 _ _ 6 ways 3 5 _ 6 _ _ 6 ways 3 5 _ _ 6 _ 6 ways 3 _ 5 _ 6 _ 6 ways 3 _ _ 5 6 _ 4 ways ways If the first number is , .... 2 6 _ _ _ _ 24 ways 2 _ 6 _ _ _ 18 ways 2 3 _ 6 _ _ 4 ways 2 4 _ 6 _ _ 6 ways 2 5 _ 6 _ _ 6 ways 2 5 _ _ 6 _ 6 ways 2 _ 5 _ 6 _ 4 ways 2 4 _ 5 6 _ 2 ways 2 3 4 5 6 1 1 way ways Grand Total: Solution 2 If is the first number, then there are no restrictions. There are , or ways to place the other numbers. If is the second number, then the first number can be or , and there are ways to place the other numbers. ways. If is the third number, then we cannot have the following: 1 _ 6 _ _ _ 24 ways 2 1 6 _ _ _ 6 ways ways If is the fourth number, then we cannot have the following: 1 _ _ 6 _ _ 24 ways 2 1 _ 6 _ _ 6 ways 2 3 1 6 _ _ 2 ways 3 1 2 6 _ _ 2 ways 3 2 1 6 _ _ 2 ways ways If is the fifth number, then we cannot have the following: _ _ _ _ 6 5 24 ways 1 5 _ _ 6 _ 6 ways 1 _ 5 _ 6 _ 6 ways 2 1 5 _ 6 _ 2 ways 1 _ _ 5 6 _ 6 ways 2 1 _ 5 6 _ 2 ways 2 3 1 5 6 4, 3 1 2 5 6 4, 3 2 1 5 6 4 3 ways ways Grand Total: Solution 3 (Recursion) Note the condition in the problem is equivalent to the following condition: for each with , the first terms is not a permutation (since it would mean there must be some integer in the first terms such that ). Then, let denote the number of permutations of satisfying the condition in the problem. We use complementary counting to find . Notice that in order to not satisfy the condition in the problem, there are cases: the first (we don't include since the condition in the problem only holds up to ) terms are a permutation of , and for all , the condition in the problem still holds. Then, for each of these cases, there are ways to arrange the first terms, and then ways to arrange the to terms (assume by contradiction that terms from to is a permutation of . Then, since the first terms are already a permutation of , the first terms form a permutation of . This contradicts our assumption that for all , the condition still holds. Thus, we can only include the permutations of these terms). Now, summing the cases up and subtracting from , we have: . From this recursion, we derive that , , , , , and finally . ~CrazyVideoGamez ~ (Frank FYC) Solution 4 (PIE) Let be the set of permutations such that there is no number greater than in the first places. Note that for all and that the set of restricted permutations is . We will compute the cardinality of this set with PIE. To finish, Solution 5 (Recursion) Define the function as the amount of permutations with maximum digit and string length that satisfy the condition within bounds. For example, would be the amount of ways to make a string with length with the highest digit being . We wish to obtain . To generate recursion, consider how we would get to from for all such that . We could either jump from the old maximum to the new by concatenating the old string and the new digit , or one could retain the maximum, in which case . To retain the maximum, one would have to pick a new available digit not exceeding . In the first case, there is only one way to pick the new digit, namely picking . For the second case, there are digits left to choose, because there are digits between 1 and total and there are digits already chosen below or equal to . Thus, . Now that we have the recursive function, we can start evaluating the values of until we get to . Our requested answer is thus ~sigma Solution 6 (Complementary) We can also solve this problem by counting the number of permutations that do NOT satisfy the given conditions; namely, these permutations must satisfy the condition that none of the first terms is greater than for . We can further simplify this method by approaching it through casework on the first terms. Case 1: None of the first one terms is greater than 1 The first term must obviously be one. Since there are five spaces left for numbers, there are a total of permutations for this case. Case 2: None of the first two terms is greater than 2 The first two terms must be 1 and 2 in some order. However, we already counted all cases starting with 1 in the previous case, so all of the permutations in this case must begin with . Since there are four spaces left, there are a total of permutations for this case. Case 3: None of the first three terms is greater than 3 The first three terms must be 1, 2, and 3 in some order. However, the cases beginning with 1__ and 21_ have already been accounted for. There are now ways to order the first three numbers of the permutation, and ways to order the last three numbers, for a total of permutations. Case 4: None of the first four terms is greater than 4 You can see where the pattern is going - the first four terms must be 1, 2, 3, and 4 in some order. All cases starting with 1 (there are ), the cases starting with 21 (there are ), and the 3 cases from case 3 (there are ) have been accounted for, so there are a total of permutations for this case. Case 5: None of the first five terms is greater than 5 This is perhaps the hardest case to work with, simply because there are so many subcases, so keeping track is crucial here. Obviously, the first five terms must be 1, 2, 3, 4, and 5, meaning there are 120 ways to order them. Now, we count the permutations we have already counted in previous cases. start with 1, start with 2, start with 3, and start with 4. Subtracting, we get a total of permutations. Now, we subtract the total number of permutations from our cases from the total number of permutations, which is : . ~TGSN/curiousmind_888 2018 AIME II (Problems • Answer Key • Resources) Preceded by Problem 10Followed by Problem 12 1•2•3•4•5•6•7•8•9•10•11•12•13•14•15 All AIME Problems and Solutions These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Retrieved from " Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
188963	https://webspace.science.uu.nl/~neder003/2MMD30/lecturenotes/L9/lecture9.pdf	2MMD30: Graphs and Algorithms Lecture 9 Date: 11/3/2016 Instructor: Nikhil Bansal 1 Probabilistic method with alterations In this chapter, we will reﬁne the probabilistic approach. Instead of directly constructing an appropriate random object and arguing that it satisﬁes our properties, we will adopt a more reﬁned approach, by ﬁrst constructing a random object, but then modifying/tweaking it slightly (and removing the blemishes, which hopefully are not too many). These are called alterations. This often allows us to show much stronger results than the direct probabilistic constructions. We will consider several examples. 1.1 Dominating set Given a graph G, a subset of vertices S is a called a dominating set if for each vertex v, either v ∈S or v has some neighbor in S. We ﬁrst show the following. Theorem 1 Any graph with minimum degree δ has a dominating set of size at least 4n log n/(δ+1). Proof: We assume that (δ + 1) ≥4 log n, otherwise the result is trivial. Let us try to construct a dominating set S by picking each vertex v randomly with probability p, where we will optimize p later. Then, the expected number of vertices picked is np. Moreover, the probability that a particular vertex v is uncovered is at precisely (1 −p)dv+1 ≤(1 −p)δ+1 ≤e−p(δ+1) and thus the expected number of uncovered vertices is at most ne−p(δ+1). So if we pick p = 2 log n/(δ + 1), then the above is at most 1/n. Thus, probability that Sis not a valid dominating set (i.e. at least one vertex uncovered) is at most 1/n ≤1/4, assuming n ≥8. Moreover, by Markov’s inequality Pr[S ≥2E[S] = 4n log n/(δ + 1)] ≤1/2. So combining these, with probability at least 1/4, S is both a dominating set and has size at most 4n log n(δ + 1). Thus such a set exists. 2 Note that any dominating set must have size at least n/(δ + 1), so the above bound is not too far. One might wonder if this can improved. We show below that this is indeed the case and replace ln n by ln δ. However, ﬁrst the reader must convince themselves that the log n factor cannot be improved based on the above approach. Theorem 2 Any graph with minimum degree δ has a dominating set of size at least n(1+ln δ)/(δ+ 1). Proof: Let us pick each vertex with probability p, but add a second step where we add any uncovered vertices to S. Clearly, S is a dominating set by design. Then the expected number of vertices picked is np and as the expected number of uncovered vertices is at most n(1 −p)δ+1) ≤ ne−p(δ+1) = n/δ, we obtain that the expected size of S is at most np + n/δ = n(1 + ln δ)/(δ + 1). 2 1 1.2 Independent sets Theorem 3 For any graph G with average degree ¯ d, α(G) ≥n/2 ¯ d. Proof: Let us pick each vertex with probability p to obtain a random set S. But instead of hoping that S be an independent set right away, we will allow it to have a (few) edges and then make it independent by removing (at most) one vertex per edge in the subgraph induced by S. The resulting set is clearly and independent set. As the number of edges that end up in the random sample is mp2, we lose at most mp2 vertices. So the expected size of the independent set is at most np −mp2 = np −n ¯ dp2/2. Here we use that ¯ d = 2m/n. Optimizing for p by taking ﬁrst derivative and setting it to 0 gives p = 1/ ¯ d. The above expression is n/2 ¯ d for this p. 2 1.3 Bipartite graphs with a forbidden Kr,r (Zarankiewicz problem) . This problem (a famous unsolved problem) asks: Given an integer r, what is the largest number of edges that an n × n bipartite graph can have, if it is not allowed to have a Kr,r as a subgraph. Let us call this number Z(n, r). We ﬁrst show a lower bound based on alterations. This is essentially the best known for general r (for r = 2, 3, better bounds are known). Later we will see an upper bound. Theorem 4 Z(n, r) = Ω(n2−2/(r+1)). Proof: Pick each edge with probability p. Whenever there is some Kr,r throw away some edge in this Kr,r. The expected number of edges picked is n2p. The expected number of Kr,r that show up is n r 2 pr2. So the expected number of edges after that alteration is at least n2p −n2r r2 pr2 (1) Taking derivatives gives n2 −n2rpr2−1. Setting this to 0 gives p = n−2/(r+1). Plugging this back in (1) gives that the expected number of edges is Ω(n2−2/(r+1)) which gives the desired result. 2 We now show the upper bound. The proof does not use probability, but is very nice. So we discuss it anyway. Theorem 5 Z(n, r) = O(n2−1/r). Proof: We want to upper bound the number of edges. Let G be some graph with the maximum number of edges. Let dv be the degree of vertex v on the left hand side. Consider the following auxiliary graph H. H is bipartite with n vertices on the left, each corresponding to a left vertex of G. The right hand side of H has n r vertices, corresponding to r-tuples of vertices on the right hand side of G. There is an edge in H from v to a r-tuple (w1, . . . , wr) if and only if v is adjacent to each of w1, . . . , wr in G. 2 The following two observations are crucial: (1) The degree of a vertex v on left of H is exactly degH(v) = dv r , where dv is the degree of v is in G. (2) The degree of any vertex on the right of H is at most r −1. Otherwise, if it is r or more, then this r-tuple would be connected to some r vertices on the left of H. But this would mean that this r-tuple and those r vertices form a Kr,r in G, which is not allowed. So, the total number of edges in H is at most (r −1) n r . But the number of edges is exactly P v∈L(G) dv r which implies that X v∈L(G) dv r ≤(r −1) n r . We want to use this relation to bound the total number of edges in G (or equivalently the average left degree of G). Let ¯ d denote the average degree of a left vertex in G. We can now use convexity of the function dv r to help us out. In particular, n ¯ d r ≤ X v∈L(G) dv r ≤(r −1) n r Ignoring lower order terms (it is not hard to formally argue that they have negligible eﬀect), this implies that n ¯ dr ≤rnr and hence that ¯ d = O(n1−1/r) which implies the result. 2 1.4 Graphs with large girth and chromatic number Our ﬁnal example of alterations is a beautiful result of Erd¨ os about graphs that have a high girth (the girth of a graph G is the length of the smallest cycle in G), yet have high chromatic number. Note that if the girth is g, this means that for every vertex v, its distance (g −1)/2 neighborhood looks like a tree. If a graph contains a k clique, then its chromatic number χ(G) ≥k. It is natural to wonder if some approximate converse of the this is true. The result of Erd¨ os disproves this in a strong sense. It also means that ’local considerations’ are not useful for determining χ(G). Theorem 6 For any k, there exists graph with girth more than k and χ(G) ≥n1/4k. Proof: We will try to construct such a graph randomly, and then apply alterations. First, let us think how can we hope to argue that χ(G) ≥n1/4k? Luckily, we realize that χ(G) ≥n/α(G), so we can try to show that α(G) is small. In fact, in general this is the only strategy we know to lower bound χ(G). We create a random graph on n vertices by including every possible edge with probability p, where we ﬁx p = n1/2k n . We can then try to remove cycles of length ≤k, by removing one vertex arbitrarily for each such cycle. The hope would be that we do not remove essentially the whole graph. Second, note that removing a vertex can never increase the size of the maximum independent set in the resulting graph (can you see why?). So let us calculate how many expected cycles of length at most k this random graph contains. As we only need an upper bound, we can be a bit sloppy. A particular, cycle of length l appears 3 with probabiity pl. Moreover, we can obtain a cycle by choosing l vertices and picking some order, so there are at most nll! such possibilities. So, E(number of cycles of length ≤k) ≤ k X l=3 n l l!pl ≤ k X l=1 nl l! l!pl = k X l=1 (np)l = k X l=1 (n1/2k)l ≤2√n (assuming n is large enough s.t. n1/(2k) ≤1/2) So we can easily eliminate such cycles and still retain n −Ω(√n) vertices. We now try to bound α(G). Let r = 2n1−1/4k. Fix some subset S with r vertices. The probability that S is an independent set is (1 −p)(r 2). There are n r subsets of size r, so the expected number of independent sets of size r is n r (1 −p)(r 2) ≤2ne−pr2/4 ≤2ne−n which is exponentially small in n. Now, by Markov’s inequality, at least 9/10 of the graphs have no more than 20√n cycles smaller than k and at least 9/10 of the graphs have no independent sets bigger than r. So, there must exist (plenty of) graphs that satisfy both these properties simultaneously. This proves the result. 2 4
188964	https://www.mathbitsnotebook.com/JuniorMath/AreaPerimeterVolume/APVCrossSections.html	Cross Sections = Plane Sections - MathBitsNotebook(Jr) Cross Sections = Plane SectionsMathBitsNotebook.com Topical Outline \| JrMath Outline \| MathBits' Teacher Resources Terms of Use Contact Person:Donna Roberts A cross section (or a plane section) is the intersection of a figure in three-dimensional space with a plane. A cross section is the face you obtain by making a "slice" through a solid object. A cross section is typically two-dimensional. We see cross sections in everyday life. A "slice" of bread. Cross section: A "slice" of cucumber.Cross section: A "slice" of log. Cross section: When a plane intersects a solid figure, the cross sectional face (or plane section) may be a point, a line segment, or a two-dimensional shape such as, but not limited to, a circle, rectangle, oval, or hexagon. Point The plane is tangent to the sphere, intersecting in only one point.Line Segment The plane is tangent to the side of the cylinder, intersecting in a line segment.Figure The plane cuts through the figure, intersecting in a pentagon. The plane may, or may not, be parallel to the base of the figure. The figure (face) obtained from a cross section (plane section) depends upon the orientation (angle) of the plane doing the cutting. Right circular cylinder Cross section: rectangle Plane orientation: perpendicular to the bases Right circular cone Cross section: ellipse Plane orientation: slanted (angled) across the cone Right square prism Cross section: square Plane orientation: parallel to the bases of the prism A single solid figure can be sliced to produce numerous cross sections of different forms. In the diagrams below, the sword represents the "slicing" plane. The solid object is a right rectangular prism. The maximum number of "sides" of a cross section(plane section) equals the number of faces (surfaces) of the solid. Since the rectangular prism shown above has 6 faces, a cross section of that solid may haveat most 6 sides. So a hexagon (6 sided) cross section is possible, but an octagon (8 sided) cross section is not possible. Note:At this level, we are concentrating on slicing parallel or perpendicular to the base. Additional slicings are mentioned for curricula that wish further examinations.. Slicing a Right Rectangular Prism Slice Parallel to the Base: In a right rectangular prism, as seen at the right, all cross sections cut parallelto the base will be rectanglesthat are congruent to the base (that is, the same size and shape as the base). rc= Slice Perpendicular to Base: Option 1- slice perpendicular to base, but parallel to a lateral side. Right Rectangular Prism In a right triangular prism, all cross sections cut perpendicular to the base, but parallel to a lateral side, will be rectanglesthat are congruent to that specific lateral side.Option 2- slice perpendicular to base, but NOT parallel to a lateral side. Right Rectangular Prism In a right triangular prism, all cross sections cut perpendicular to the base, but NOT parallel to a lateral side, will be rectangles of varying sizes. Slice Slanted Across Figure: In a right rectangular prism, as seen at the right, a cross sections can be cut on a slantto produce a variety of additional shapes. At the right, a slanted slice produced atriangle. Another slanted slice produced atrapezoid. A variety of 2-dimensional shapes are possible: triangles, quadrilaterals, pentagons and hexagons are possible. We saw in the previous text that the cross section of a right rectangular prism can have at most 6 sides. So no octagons, etc. Right Rectangular Prism Slicing a Right Rectangular Pyramid Slice Parallel to the Base: In a right rectangular pyramid, as seen at the right, all cross sections cut parallel to the basewill be rectanglesthat are similar to the base (that is, the same shape as the base, but not necessarily the same size). Right Rectangular Pyramid Slice Perpendicular to Base: Option 1- slice perpendicular to base, but parallel to a side of the base. Right Rectangular Pyramid In a right triangular pyramid, all cross sections cut perpendicular to the base, and also parallel to a side of the base, will be either a triangle, or atrapezoidof varying sizes.Option 2- slice perpendicular to base, but NOT parallel to a side of the base. Right Rectangular Pyramid In a right triangular pyramid, all cross sections cut perpendicular to the base, but NOT parallel to a side of the base, will also be either a triangles or a trapezoids of varying sizes. Slice Slanted Across Figure: In a right rectangular pyramid, as seen at the right, a cross section can be cut on a slantto produce a variety of additional shapes. At the right, a slanted slice produced a pentagon. Remember, since a right rectangular pyramid has 5 faces, a cross section can have at most 5 sides. So the cross sections for a right rectangular pyramid can be 2-dimensional shapes that are triangles, quadrilaterals, or pentagons. Right Rectangular Pyramid NOTE:There-posting of materials(in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use". Topical Outline \| JrMath Outline \| MathBitsNotebook.com \| MathBits' Teacher ResourcesTerms of UseContact Person:Donna Roberts Copyright © 2012-2025 MathBitsNotebook.com. All Rights Reserved.
188965	https://www.youtube.com/watch?v=-9OUyo8NFZg	Euler's Formula and Graph Duality 3Blue1Brown 7550000 subscribers 12208 likes Description 501365 views Posted: 21 Jun 2015 A description of planar graph duality, and how it can be applied in a particularly elegant proof of Euler's Characteristic Formula. Music: Wyoming 307 by Time For Three Thanks to these viewers for their contributions to translations Marathi: realcalal 374 comments Transcript: In my video on the circle division problem, I referenced Euler's characteristic formula, and here I would like to share a particularly nice proof of this fact. It's very different from the inductive proof, typically given, but I'm not trying to argue that this is somehow better or easier to understand than other proofs. Instead, I chose this topic to illustrate one example of the incredible notion of duality, and how it can produce wonderfully elegant math. First, let's go over what the theorem states. If you draw some dots and some lines between them, that is, a graph, and if none of these lines intersect, which is to say you have a planar graph, and if your drawing is connected, then Euler's formula tells us that the number of dots minus the number of lines plus the number of regions these lines cut the plane into, including that outer region, will always be 2. Because Euler was originally talking about 3D polyhedra when he found this formula, which was only later reframed in terms of planar graphs, instead of saying dots, we say vertices, instead of saying lines, we say edges, and instead of saying regions, we say faces. Hence, we write Euler's discovery as V minus E plus F equals 2. Before describing the proof, I need to go through three pieces of graph theory terminology. Cycles, spanning trees, and dual graphs. If you are already familiar with some of these topics and don't care to see how I describe them, feel free to click the appropriate annotation and skip ahead. Imagine a tiny creature sitting on one of the vertices. Let's name him Randolph. If we think of edges as something Randolph might travel along from one vertex to the next, we can sensibly talk about a path as being a sequence of edges that Randolph could travel along, where we don't allow him to backtrack on the same edge. A cycle is simply a path that ends on the same vertex where it begins. You might be able to guess how cycles will be important for our purposes, since they will always enclose a set of faces. Now imagine that Randolph wants access to all other vertices, but edges are expensive, so he'll only buy access to an edge if it gives him a path to an untouched vertex. This frugality will leave him with a set of edges without any cycles, since the edge finishing off a cycle would always be unnecessary. In general, a connected graph without cycles is called a tree, so named because we can move things around and make it look like a system of branches. And any tree inside a graph which touches all the vertices is called a spanning tree. Before defining the dual graph, which runs the risk of being confusing, it's important to remember why people actually care about graphs in the first place. I was actually lying earlier when I said a graph is a set of dots and lines. Really, it's a set of anything with any notion of connection, but we typically represent those things with dots and those connections with lines. For instance, Facebook stores an enormous graph where vertices are accounts and edges are friendships. Although we could use drawings to represent this graph, the graph itself is the abstract set of accounts and friendships, completely distinct from the drawing. All sorts of things are undrawn graphs, the set of English words considered connected when they differ by one letter, mathematicians considered connected if they've written a paper together, neurons connected by synapses. Or, maybe, for those of us reasoning about the actual drawing of a graph on the plane, we can take the set of faces this graph cuts the plane into and consider two of them connected if they share an edge. In other words, if you can draw a graph on the plane without intersecting edges, you automatically get a second, as of yet undrawn, graph whose vertices are the faces and whose edges are, well, edges of the original graph. We call this the dual of the original graph. If you want to represent the dual graph with dots and lines, first put a dot inside each one of the faces. I personally like to visualize the dot for that outer region as being a point somewhere at infinity where you can travel in any direction to get there. Next, connect these new dots with new lines that pass through the centers of the old lines, where lines connected to that point at infinity can go off the screen in any direction, as long as it's understood that they all meet up at the same one point. But keep in mind, this is just the drawing of the dual graph, just like the representation of Facebook accounts and friendships with dots and lines is just a drawing of the social graph. The dual graph itself is the collection of faces and edges. The reason I stress this point is to emphasize that edges of the original graph and edges of the dual graph are not just related, they're the same thing. You see, what makes the dual graph all kinds of awesome is the many ways that it relates to the original graph. For example, cycles in the original graph correspond to connected components of the dual graph, and likewise, cycles in the dual graph correspond with connected components in the original graph. Now for the cool part. Suppose our friend Randolph has an alter ego, Mortimer, living in the dual graph, traveling from face to face instead of from vertex to vertex, passing over edges as he does so. Let's say Randolph has bought all the edges of a spanning tree and that Mortimer is forbidden from crossing those edges. It turns out the edges that Mortimer has available to him are guaranteed to form a spanning tree of the dual graph. To see why, we only need to check the two defining properties of spanning trees. They must give Mortimer access to all faces and there can be no cycles. The reason he still has access to all faces is that it would take a cycle in Randolph's spanning tree to insulate him from a face, but trees cannot have cycles. The reason Mortimer cannot traverse a cycle in the dual graph feels completely symmetric. If he could, he would separate one set of Randolph's vertices from the rest so the spanning tree from which he is banned could not have spanned the whole graph. So not only does the planar graph have a dual graph, any spanning tree within that graph always has a dual spanning tree in the dual graph. Here's the kicker. The number of vertices in any tree is always one more than the number of edges. To see this, note that after you start with the root vertex, each new edge gives exactly one new vertex. Alternatively, within our narrative, you could think of Randolph as starting with one vertex and gaining exactly one more for each edge that he buys in what will become a spanning tree. Since this tree covers all vertices in our graph, the number of vertices is one more than the number of edges owned by Randolph. Moreover, since the remaining edges make up a spanning tree for Mortimer's dual graph, the number of edges he gets is one more than the number of vertices in the dual graph, which are faces of the original graph. Putting this together, it means the total number of edges is two more than the number of vertices plus the number of faces, which is exactly what Euler's formula states.
188966	https://pmc.ncbi.nlm.nih.gov/articles/PMC9060991/	Systematic Retesting for Helicobacter pylori: The Potential Overestimation of Suppressive Conditions - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. 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Learn more: PMC Disclaimer \| PMC Copyright Notice Biomed Res Int . 2022 Apr 25;2022:5380001. doi: 10.1155/2022/5380001 Search in PMC Search in PubMed View in NLM Catalog Add to search Systematic Retesting for Helicobacter pylori: The Potential Overestimation of Suppressive Conditions Richard F Knoop Richard F Knoop 1 Department of Gastroenterology, Gastrointestinal Oncology and Endocrinology, University Medical Center Göttingen, Georg-August University, Göttingen, Germany Find articles by Richard F Knoop 1,✉, Pauline C Gaertner Pauline C Gaertner 1 Department of Gastroenterology, Gastrointestinal Oncology and Endocrinology, University Medical Center Göttingen, Georg-August University, Göttingen, Germany Find articles by Pauline C Gaertner 1, Golo Petzold Golo Petzold 1 Department of Gastroenterology, Gastrointestinal Oncology and Endocrinology, University Medical Center Göttingen, Georg-August University, Göttingen, Germany Find articles by Golo Petzold 1, Ahmad Amanzada Ahmad Amanzada 1 Department of Gastroenterology, Gastrointestinal Oncology and Endocrinology, University Medical Center Göttingen, Georg-August University, Göttingen, Germany Find articles by Ahmad Amanzada 1, Volker Ellenrieder Volker Ellenrieder 1 Department of Gastroenterology, Gastrointestinal Oncology and Endocrinology, University Medical Center Göttingen, Georg-August University, Göttingen, Germany Find articles by Volker Ellenrieder 1, Albrecht Neesse Albrecht Neesse 1 Department of Gastroenterology, Gastrointestinal Oncology and Endocrinology, University Medical Center Göttingen, Georg-August University, Göttingen, Germany Find articles by Albrecht Neesse 1, Sebastian C B Bremer Sebastian C B Bremer 1 Department of Gastroenterology, Gastrointestinal Oncology and Endocrinology, University Medical Center Göttingen, Georg-August University, Göttingen, Germany Find articles by Sebastian C B Bremer 1, Steffen Kunsch Steffen Kunsch 1 Department of Gastroenterology, Gastrointestinal Oncology and Endocrinology, University Medical Center Göttingen, Georg-August University, Göttingen, Germany 2 Department of Gastroenterology, Internal Medicine and Geriatrics, Rems-Murr Hospital, Winnenden, Germany Find articles by Steffen Kunsch 1,2 Author information Article notes Copyright and License information 1 Department of Gastroenterology, Gastrointestinal Oncology and Endocrinology, University Medical Center Göttingen, Georg-August University, Göttingen, Germany 2 Department of Gastroenterology, Internal Medicine and Geriatrics, Rems-Murr Hospital, Winnenden, Germany Academic Editor: Maria T. Mascellino ✉ Corresponding author. Received 2022 Jan 16; Accepted 2022 Mar 28; Collection date 2022. Copyright © 2022 Richard F. Knoop et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. PMC Copyright notice PMCID: PMC9060991 PMID: 35509714 Abstract Background and Aims In contrast to guideline recommendations, endoscopic testing for Helicobacter pylori is frequently performed under Helicobacter pylori suppressive conditions, e.g., intake of proton-pump inhibitors (PPI), preceded antibiotic treatment, or recent gastrointestinal bleeding. Our study's aim was to retest patients with—under suppressive conditions—negative test results. This was carried out in order to examine the rate of false negative tests previously gathered under suppressive conditions. Methods The trial was conducted in a large patient collective in a university hospital. Every elective esophagogastroduodenoscopy from in- and outpatients was included. Prior to endoscopy, suppressive conditions were collected via standardized questionnaire. If Helicobacter pylori testing was indicated, both helicobacter urease test and histology were performed in analogy to the Sydney classification. In case of a negative result under suppressive conditions, the patient was reinvited after, if possible, withdrawal of suppressive condition in order to perform a urea breath test (UBT). Results 1,216 patients were included (median 59 years, 72.0% inpatients, 28.0% outpatients). Overall, 60.6% (737) were under Helicobacter pylori suppressive conditions. The main suppressive condition was intake of PPIs (54.5%). In 53.7% (653) of all included cases, Helicobacter pylori testing was performed. Of those, 14.1% (92) had a positive test, and 85.9% (561) were negative. Out of the patients with negative result, 50.8% (285) were tested under suppressive conditions and consequently invited for retesting via UBT. In 20.4% (45), suppressive conditions could not be ceased. In 22.8% (65), retesting was conducted. Of those, 98.5% (64) congruently presented a negative result again, and only 1.5% (1) was positive for Helicobacter pylori. Conclusion Many patients undergoing esophagogastroduodenoscopy in everyday clinical practice are tested for Helicobacter pylori under suppressive conditions leading to a potentially higher risk of false negative results. However, our research shows that this issue might be overestimated. 1. Introduction Over the last decades, the prevalence of infections with Helicobacter pylori (H. pylori) has decreased . Nevertheless, about 50% of the adult world population aged over 40 years remains infected with a wide variety of prevalence not only between industrial and developing countries but also within a single population [2–5]. For instance, in Germany, H. pylori infection shows a low prevalence in children (3%) and ranges from 20 to 40% in adults [1, 6–8]. For immigrants, it is significantly higher (36-86%) [1, 9]. H. pylori infection induces an active chronic gastritis, possibly leading to dyspeptic syndromes, gastroduodenal ulcer disease, gastric cancer, or gastric mucosa-associated lymphoid tissue (MALT) lymphoma as well as extra-intestinal diseases [6, 10]. Yet, there are no sufficient prevention strategies. In particular, an effective vaccine has not been available so far . The decision for H. pylori testing and the selection of the diagnostic test should follow the recommendations summarized in various guidelines [1, 11, 12]. H. pylori can be detected by several adequately validated tests . The noninvasive assays include urea breath test (UBT), serologic immunoglobulin G (IgG) antibodies, and stool antigen test with monoclonal antibodies. UBT can be regarded as gold standard of noninvasive H. pylori tests with high sensitivity and specificity of both up to over 95% [14–16]. For UBT, the patient is administered urea labeled with a carbon isotope, usually the nonradioactive carbon-13 . In the following 10 to 30 minutes, the isotope-labeled carbon dioxide in the patient's exhaled breath indicates the splitting of urea by H. pylori's enzyme urease proving the presence of H. pylori [17, 18]. The invasive methods include helicobacter urease test (HUT), histology, culture, and polymerase chain reaction (PCR) from gastric biopsies [1, 14, 19]. The mentioned methods show different sensitivities and specificities. However, none is perfect in its accuracy [14, 20, 21]. HUT, e.g., reaches a sensitivity from 85 to 100% and a specificity up to 100% [13, 22]. Histology shows a sensitivity and specificity of both around 94% . A major tool to diagnose H. pylori in everyday clinical practice is endoscopic biopsy. Principally, during every esophagogastroduodenoscopy (EGD), the decision whether to test or not to test for H. pylori should be made. If indicated, testing should be undertaken with biopsies for at least two different tests in accordance with the Sidney classification which primarily recommends histology and HUT [1, 23, 24]. Following the scientific literature, the sensitivity of all tests, apart from serology, is supposed to be impaired by conditions that lead to a reduced H. pylori colonization density [1, 14, 25, 26]. These are in particular the treatment with a proton-pump inhibitor (PPI), preceded H. pylori affecting antibiotic treatment and recent upper gastrointestinal bleedings [1, 27–29]. These confounding factors are a diagnostic challenge that can lead to a decreasing sensitivity and consequently to a higher rate of potentially false negative H. pylori test results . Therefore, guidelines actually recommend a minimal interval of 2 weeks after completing a PPI therapy and 4 weeks after previous antibiotic therapy . Nevertheless, H. pylori testing is practically often conducted under suppressive conditions despite the clear statements [5, 30]. Moreover, patients under H. pylori suppressive conditions being tested during EGD can even represent the majority; which could, e.g., be shown by previous work of our group . For instance, this is due to many patients with dyspepsia already being primarily treated empirically with a PPI before an EGD with H. pylori testing can be conducted . Obviously, suppressive conditions are a diagnostic “black box” of H. pylori. This might be clinically relevant for symptom control and long-term implications, e.g., gastric cancer incidence. Furthermore, patients being tested under suppressive conditions may even be disadvantaged as in clinical practice a negative test result is often stated “negative” regardless of potential H. pylori suppressive conditions which are often also not accurately assessed. Consequently, testing of H. pylori under suppressive conditions remains an unsolved clinical issue . The guideline recommendations concerning H. pylori suppressive conditions are either based on partially old experimental laboratory data concerning H. pylori colonization density or on clinical data with inferior evidence, e.g., gained by logistic regression analysis [1, 27, 31–35]. To the best of our knowledge, there are no systematic clinical studies addressing this relevant question in present time. That is why we performed a clinical trial with a high number of cases in order to systematically investigate the relevant and common clinical dilemma of potentially false negative H. pylori test results under suppressive conditions. The purpose of our study was to answer the question whether previous laboratory findings stating H. pylori suppressive conditions leading to false negative test results are reproducible in a real world setting or whether they might be overestimated in clinical practice. 2. Materials and Methods This study was carried out in a large single center patient collective of the University Medical Center Göttingen, a tertiary German referral center. Our clinical standard of testing H. pylori followed the indications according to the German S2k-guideline . All patients who underwent elective EGD were included. Prior to EGD, the indication for H. pylori testing was assessed and documented accurately following our clinical standard. If testing was indicated, always both HUT (Pronto Dry New, Gastrex, Gilly-lès-Cîteaux, France) and histology were conducted. HUTs were interpreted 1 hour after EGD according to the manufacturer's instructions. Data were collected over a period of 6 months. Inpatients as well as outpatients were included. EGDs were conducted by experienced endoscopists. As guidelines recommend and in analogy to the Sydney classification, biopsies were obtained from both the corpus (greater and lesser curvature) and antrum (greater and lesser curvature) [1, 23]. The study was approved by the local Institutional Ethics Committee (case 9/11/20) and conformed to the Helsinki Declaration as well as local legislation. 2.1. Data Collection Following our clinical standard, the medical history of every patient was assessed routinely prior to elective EGD. This was recorded by standardized questionnaire with regard to the detection of H. pylori suppressive conditions. For this, the following parameters were obtained: patient's age, sex, inpatient or outpatient, date of EGD, previous intake of PPI (within the last 2 weeks), previous antibiotic treatment (within the last 4 weeks), and signs of current upper gastrointestinal bleeding (hematemesis, melena within the last 3 days) If invasive H. pylori testing was performed, the results of histology and HUT were recorded. Subsequently, H. pylori negative results (both histology and HUT) under suppressive conditions were detected. Following our clinical standard and the guidelines, concerned patients were invited telephonically for a UBT (INFAI, Köln, Germany) after—if possible—cessation of H. pylori suppressive conditions. 2.2. Statistical Analysis Statistical analyses were performed with GraphPad Prism version 9.1.2 (GraphPad Software, San Diego, California, USA) and with SPSS Statistics version 26.0.0.0. (IBM, Armonk, NY, USA). Data were reported as mean including standard deviation. Differences between two groups were performed with two-tailed Mann–Whitney test or two-sided chi-square test. p values less than 0.05 were considered statistically significant and are marked by ∗. 3. Results 3.1. Suppressive Conditions The study included 1,216 patients (Figure 1) with a median age of 59 years. 340 (28.0%) were outpatients, and 876 (72.0%) were inpatients. Overall, 737 (60.6%) patients had one or more H. pylori suppressive conditions, whereas only 479 (39.4%) had no H. pylori suppressive conditions at all. PPI intake was the major suppressive condition in 54.5% (663/1,216) of all patients followed by antibiotic treatment within the previous 4 weeks with 17.0% (207/1,216) and clinical signs of recent upper gastrointestinal bleeding (hematemesis, melena) with 11.6% (141/1,216). Figure 1. Open in a new tab Study design. 3.2. H. pylori Testing 653 (53.7%) of all included 1,216 patients (Figure 1) had the indication for H. pylori testing which was always conducted by both histology and HUT. Patients indicated for testing were characterized by a reduced percentage of H. pylori suppressive conditions (50.8% vs. 71.9%; p< .001∗), a lower median age (54 vs. 65 years; p< .001∗) and a higher percentage of outpatients (36.1% vs. 18.5%; p< .001∗) compared to nontested patients (Table 1). Table 1. Overall patients' characteristics. \| Characteristic \| Tested for H. pylori (HUT + histology) n = 653 \| Not tested due to missing indication = 563 \| p \| :---: :---: \| \| Suppressive conditions (no. (%)) \| 332 (50.8) \| 405 (71.9) \| < .001∗ \| \| Age median (years) \| 54.0 \| 65.0 \| < .001∗ \| \| Sex female (no. (%)) \| 342 (52.4) \| 241 (42.8) \| .001∗ \| \| Outpatients (no. (%)) \| 236 (36.1) \| 104 (18.5) \| < .001∗ \| \| PPI intake (no. (%)) \| 295 (45.2) \| 368 (65.4) \| < .001∗ \| \| Antibiotics intake (no. (%)) \| 85 (13.0) \| 122 (21.7) \| < .001∗ \| \| GI bleeding (no. (%)) \| 42 (6.4) \| 99 (17.6) \| < .001∗ \| Open in a new tab H. pylori: Helicobacter pylori; HUT: Helicobacter urease test; GI: gastrointestinal; PPI: proton-pump inhibitor. The detailed analysis of suppressive conditions between the tested and nontested patients showed a reduced intake of PPIs (45.2% vs. 65.4%; p< .001∗), reduced intake of antibiotics (13.0% vs. 21.7%; p< .001∗), and a lower rate of upper GI-bleedings (6.4% vs. 17.6%; p< .001∗) among the tested individuals (Table 1). Of the 653 tested patients, 92 (14.1%) had a positive test for H. pylori whereas 561 (85.9%) were negative for both histology and HUT (Figure 1). These two subgroups (tested positive vs. tested negative) were similar concerning suppressive conditions (51.1% vs. 50.8%; p ≥ .999; Table 2). Furthermore, no significant differences concerning age (51.5 vs. 55.0 years; p = .389), sex (p = .653), and rate of outpatients (37.0% vs. 36.0%; p = .907) were observed (Table 2). Table 2. Characteristics of tested patients. \| Characteristic \| Tested positive for H. pylori (HUT and/or histology) n = 92 \| Tested negative for H. pylori (HUT and histology) n = 561 \| p \| :---: :---: \| \| Suppressive conditions (no. (%)) \| 47 (51.1) \| 285 (50.8) \| > .999 \| \| Age median (years) \| 51.5 \| 55 \| .389 \| \| Sex female (no. (%)) \| 46 (50.0) \| 296 (52.8) \| .653 \| \| Outpatients (no. (%)) \| 34 (37.0) \| 202 (36.0) \| .907 \| \| PPI intake (no. (%)) \| 38 (41.3) \| 257 (45.8) \| .432 \| \| Antibiotics intake (no. (%)) \| 15 (16.3) \| 70 (12.5) \| .317 \| \| GI bleeding (no. (%)) \| 11 (12.0) \| 31 (5.5) \| < .035∗ \| Open in a new tab H. pylori: Helicobacter pylori; HUT: Helicobacter urease test; GI: gastrointestinal, PPI: proton-pump inhibitor. Of the 92 H. pylori positive patients, 41 (44.6%) showed an incongruent result (one test positive, one test negative). 35 (85.4%) of the incongruent results appeared under H. pylori suppressive conditions, mainly under PPI (27/35; 77.1%). In sum, 285 out of the 561 patients (50.8%) with negative results were investigated under suppressive conditions (Figure 1, Table 2) coming along with a potentially increased risk of a false negative test results. 3.3. Retesting for H. pylori via UBT after Withdrawal of Suppressive Conditions Following the clinical standard of our hospital as well as the guidelines, all of those 285 patients tested negatively under suppressive conditions were offered an outpatient appointment in order to conduct a urea breath test (UBT) with ceased H. pylori suppressive conditions (Figure 1). In 65 patients (22.8%), a UBT was performed after withdrawal of H. pylori suppressive conditions for at least 4 weeks. Interestingly, only one of these patients (1.5%) presented an H. pylori positive result in the UBT. Previously, this single patient had been treated with a PPI as suppressive condition. All other 64 UBTs (98.5%) again tested negatively. Despite active informing of the patients, in 220 cases (77.2%), no UBT could be performed (Figure 1). The clustered reasons for this were “suppressive conditions cannot be ceased” with 20.5% (N = 45), “patient refusal” 53.2% (N = 117), and “lost to follow-up” 26.4% (N = 58) (Figure 1). 4. Discussion This study addresses the challenge of a high rate of H. pylori tests being conducted under H. pylori suppressive conditions. We focused on this relevant clinical issue because it is suspected to lead to more false negative test results. That is why we retested those patients with initially negative H. pylori test results under H. pylori suppressive conditions after withdrawal of the suppressive condition by performing a urea breath test (UBT). Following the scientific literature, the main H. pylori suppressive conditions are treatment with PPIs, recent antibiotic treatment and upper gastrointestinal bleeding leading to a reduced sensitivity of all common H. pylori tests due to a reduction of H. pylori colonization density [1, 14, 25–29]. Therefore, guidelines obviously recommend testing under nonsuppressive conditions . However, this does not always meet the clinical practice . In particular, the withdrawal of PPIs can often not be realized . For instance, many patients with dyspepsia have already been treated with PPIs before an EGD can be performed and H. pylori is tested . Our cohort shows a real word setting of H. pylori suppressive conditions in patients undergoing EGD in a German university hospital. The data demonstrate that even the majority of patients show H. pylori suppressive conditions, leading to the relevant diagnostic challenge of how to deal with their negative H. pylori test result. Throughout the trial, a large absolute and relative amount of negatively tested individuals exhibited H. pylori suppressive conditions (285 out of 561 negatively tested patients; 50.8%). Interestingly and relevant in terms of internal validity of the study, the two subgroups of positively tested vs. negatively tested patients were similar concerning suppressive conditions (51.1% vs. 50.8%; p> .999). In the following, the negatively tested patients under H. pylori suppressive conditions were the subgroup we put our focus on in order to answer the question whether they had previously shown a higher rate of false negative H. pylori test results. Following our clinical standard and the current guideline, all of those 285 patients were actively reinvited in order to undergo a urea breath test after withdrawal of H. pylori suppressive conditions. However, in many of those patients (20.5%), the H. pylori suppressive conditions could not be ceased. Furthermore, many of those patients do not suffer from symptoms anymore and are consequently not necessarily interested in an additional test. Therefore, it can be regarded as a success, that in 65 patients (22.8%), a UBT was eventually performed after cessation of H. pylori suppressive conditions. The most important result of our study is that only one patient (1.5%) then presented an H. pylori positive result in the UBT, whereas all other 64 UBTs (98.5%) were tested negatively again. Obviously, there are certain limitations of our trial, e.g., concerning geographical differences of H. pylori prevalence. The monocentric study was conducted in a large German university hospital providing general and maximum care with a distinct patient collective that does not necessarily represent the general population. As a matter of course, patients with initially negative H. pylori test results under suppressive conditions who required continuation of particularly PPI therapy could not be retested without suppressive conditions. Apparently, the acceptancy rate of performed UBT seems to be low. However, the number of cases with necessarily continued PPI therapy as well as the fact that many patients do not show up for another test despite active telephonic invitation can themselves be considered as relevant results of the trial. 5. Conclusion Our real-world data show that the sensitivity of testing H. pylori under suppressive conditions might not be as low as always suspected. This is of high clinical relevance because it could relevantly simplify testing for H. pylori. Acknowledgments Supported by the Open Access Publication Funds of the Göttingen University. 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188967	https://www.beatthegmat.com/digits-of-the-3-digit-integers-greater-than-700-how-many-h-t9290.html	Digits: Of the 3-digit integers greater than 700, how many h - The Beat The GMAT Forum - Expert GMAT Help & MBA Admissions Advice Forum Members Only Benefits & Discounts Forum Home GMAT Quant GMAT Verbal & AWA I Just Beat the GMAT! GMAT Integrated Reasoning Study Plans & Practice Featured Experts GMAT PREP & ADMISSIONS GMAT Course Reviews 1. e-GMAT 2. Bloomberg Exam Prep 3. EMPOWERgmat 4. Magoosh 5. Manhattan Prep 6. 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(A) 90 (B) 82 (C) 80 (D) 45 (E) 36 I am keen to understand different ways of answering this question. Thanks in advance. II Last edited by II on Mon May 05, 2008 1:54 am, edited 1 time in total. Top Quote mnjoosubJunior \| Next Rank: 30 Posts View User Profile Posts:17Joined: Wed Feb 27, 2008 9:32 pmLocation: MauritiusThanked: 2 times bymnjoosub » Mon Mar 17, 2008 6:30 am I don't know whether there is a proper mathematical way to solve this problem but I did it this way. I tabulate is as follows: see attachment from the table we can see that the total repeated Nos for 700 = 30 So for 800 and 900 inclusive = 30 3 = 90 We should less repeated Nos (33 = 9) = 81 Remember the question states more than 700 , so less 1 more = 80. Ans = C Top Quote mnjoosubJunior \| Next Rank: 30 Posts View User Profile Posts:17Joined: Wed Feb 27, 2008 9:32 pmLocation: MauritiusThanked: 2 times bymnjoosub » Mon Mar 17, 2008 6:34 am Sorry I forgot the attachment. Attachments Ans.xls(19.5 KiB) Downloaded 419 times Top Quote sofia cuevasNewbie \| Next Rank: 10 Posts View User Profile Posts:1Joined: Fri May 30, 2008 9:25 am help! bysofia cuevas » Wed Aug 06, 2008 2:10 pm is there any other way to solve this problem? perhaps a faster method? Top Quote parallel_chaseLegendary Member View User Profile Posts:1153Joined: Wed Jun 20, 2007 6:21 amThanked: 146 timesFollowed by:2 members byparallel_chase » Wed Aug 06, 2008 3:07 pm Here is a super fast way of solving this question as compared to above method. For first digit we have 3 letters to play with (7,8,9) Next two digits we can have any letter (0,1,2,3,4,5,6,7,8,9) CASE I (ABB) 391 = 27 CASE II (BAB) 391 = 27 CASE III (BBA) 319 =27 27+27+27 = 81 subtract 1 because we want the number to be greater than 700, the above combination include 700 81-1=80 Once you are comfortable with permutations & combination of digits this thing will take you less than 10 secs and I mean this. Let me know if you have any doubts. Top Quote guru_1971Newbie \| Next Rank: 10 Posts View User Profile Posts:1Joined: Fri Dec 10, 2010 12:42 am byguru_1971 » Fri Dec 10, 2010 1:02 am The correct answer is 36 3 digit integers greater than 700. 2 cases case 1: when first digit is 7 then using counting priniciple 1st digit has an option of 1 , and hence since two digits are equal the 2nd digit also is 1 third digit is between 1- 9 ( number greater than 700= 701 ) since the first 2 digits have taken integer 7, third digit will have (9-1) = 8 so the counting principle equation for case 1 is ( 1x1x8) 1 1 8 ( 9-1) --- X ----------------- X --------------------- 1st digit is 7 2nd digit is same as 1st 3rd digit is 1-9 = 9 case 2 1st digit is not 7, so for the 1 and 2 digit, options are 8, 9 so 2 options for digit 1 and 2 options for digit 2 ( as per the condition of the problem ) now for 3 digit, since the number is greater than 700 digit will be from 1 - 9= and the 1st 2 digits have taken 7 and 8, so third digit will be 9-2= 7 so counting principle equation becomes ( 2 x2x7) 2 2 7 ---------- X --------------------- X --------------------------------- available same as 1st digit here 8 and 9 are taken by 1st 2 digits so 9 -2=7 option is 8 or 9 Solution thus becomes ( 1x1x8)+(2x2x7) = 36 Top Quote diebeatsthegmatLegendary Member View User Profile Posts:1119Joined: Fri May 07, 2010 8:50 amThanked: 29 timesFollowed by:3 members bydiebeatsthegmat » Fri Dec 10, 2010 5:26 am II wrote:Of the 3-digit integers greater than 700, how many have 2 digits that are equal to each other and the remaining digit different from the other 2 ? (A) 90 (B) 82 (C) 80 (D) 45 (E) 36 I am keen to understand different ways of answering this question. Thanks in advance. II here the answer i find is also C 80 whats the answer? Top Quote kevincanspainGMAT Instructor View User Profile Posts:613Joined: Thu Mar 22, 2007 6:17 amLocation: madridThanked: 171 timesFollowed by:64 membersGMAT Score:790 bykevincanspain » Fri Dec 10, 2010 5:29 am Parallel chase's method is excellent! Also, there are 299 3-digit integers greater than 700. How many of these do not have exactly two digits equal to each other? There are 3 such integers that have three digits equal to each other (777,888,999) and 3 x 9 x 8 =216 that feature three distinct digits. Thus there are 299 - 3 - 216 = 80 such integers that have exactly two digits equal to each other Kevin Armstrong GMAT Instructor Gmatclasses Madrid Top Quote kevincanspainGMAT Instructor View User Profile Posts:613Joined: Thu Mar 22, 2007 6:17 amLocation: madridThanked: 171 timesFollowed by:64 membersGMAT Score:790 bykevincanspain » Fri Dec 10, 2010 5:32 am guru_1971 wrote:The correct answer is 36 3 digit integers greater than 700. 2 cases case 1: when first digit is 7 then using counting priniciple 1st digit has an option of 1 , and hence since two digits are equal the 2nd digit also is 1 third digit is between 1- 9 ( number greater than 700= 701 ) since the first 2 digits have taken integer 7, third digit will have (9-1) = 8 so the counting principle equation for case 1 is ( 1x1x8) 1 1 8 ( 9-1) --- X ----------------- X --------------------- 1st digit is 7 2nd digit is same as 1st 3rd digit is 1-9 = 9 case 2 1st digit is not 7, so for the 1 and 2 digit, options are 8, 9 so 2 options for digit 1 and 2 options for digit 2 ( as per the condition of the problem ) now for 3 digit, since the number is greater than 700 digit will be from 1 - 9= and the 1st 2 digits have taken 7 and 8, so third digit will be 9-2= 7 so counting principle equation becomes ( 2 x2x7) 2 2 7 ---------- X --------------------- X --------------------------------- available same as 1st digit here 8 and 9 are taken by 1st 2 digits so 9 -2=7 option is 8 or 9 Solution thus becomes ( 1x1x8)+(2x2x7) = 36 Are you counting possibilities such as 707? Kevin Armstrong GMAT Instructor Gmatclasses Madrid Top Quote BestGMATElizaMaster \| Next Rank: 500 Posts View User Profile Posts:103Joined: Mon Jun 23, 2014 11:31 pmThanked: 25 timesFollowed by:12 membersGMAT Score:770 byBestGMATEliza » Wed Jul 09, 2014 9:51 pm you only have 3 possibilities for the hundreds digit: 7, 8 or 9. 700s you can have 7xx (ex: 722), for this there are 8 possibilities for digits (0 doesn't count because it must be greater than 700, neither does 7) there is also 7x7 (ex: 702), for this there are 9 possibilities (only 7 doesn't count) then there is 77x (ex:771), for this there are also 9 possibilities 800s 8xx- 9 possibilities, because you can include 0 so 0-9, not including 8 8x8- 9 possibilities 88x- 9 possibilities 900s 9xx- 9 possibilities, because you can include 0 so 0-9, not including 8 9x9- 9 possibilities 99x- 9 possibilities Add them all up and you get 80 (C) Eliza Chute Best GMAT Prep Courses GMAT course comparison and reviews Your one stop for all your GMAT studying needs! Top Quote GMATGuruNYGMAT Instructor View User Profile Posts:15539Joined: Tue May 25, 2010 12:04 pmLocation: New York, NYThanked: 13060 timesFollowed by:1906 membersGMAT Score:790 byGMATGuruNY » Thu Jul 10, 2014 2:52 am II wrote:Of the 3-digit integers greater than 700, how many have 2 digits that are equal to each other and the remaining digit different from the other 2 ? (A) 90 (B) 82 (C) 80 (D) 45 (E) 36 Integers with exactly 2 digits the same = Total integers - Integers with all 3 digits the same - Integers with all 3 digits different. Total integers: To count consecutive integers, use the following formula: Number of integers = biggest - smallest + 1. Thus: Total = 999 - 701 + 1 = 299. Integers with all 3 digits the same: 777, 888, 999. Number of options = 3. Integers with all 3 digits different: Number of options for the hundreds digit = 3. (7, 8, or 9) Number of options for the tens digit = 9. (Any digit 0-9 other than the digit already used.) Number of options for the units digit = 8. (Any digit 0-9 other than the two digits already used.) To combine these options, we multiply: 398 = 216. Thus: Integers with exactly 2 digits the same = 299-3-216 = 80. The correct answer is C. Private tutor exclusively for the GMAT and GRE, with over 20 years of experience. Followed here and elsewhere by over 1900 test-takers. I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries. My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs. As a tutor, I don't simply teach you how I would approach problems. I unlock the best way for YOU to solve problems. For more information, please email me (Mitch Hunt) at GMATGuruNY@gmail.com. Student Review #1 Student Review #2 Student Review #3 Top Quote GMAT/MBA Expert Brent@GMATPrepNowGMAT Instructor View User Profile Posts:16207Joined: Mon Dec 08, 2008 6:26 pmLocation: Vancouver, BCThanked: 5254 timesFollowed by:1268 membersGMAT Score:770 byBrent@GMATPrepNow » Thu Jul 10, 2014 6:20 am Of the three-digit integers greater than 700, how many have two digits that are equal to each other and the remaining digit different from the other two? (A) 90 (B) 82 (C) 80 (D) 45 (E) 36 One approach is to start LISTING numbers and look for a PATTERN. Let's first focus on the numbers from 800 to 899 inclusive. We have 3 cases to consider: 8XX, 8X8, and 88X 8XX 800 811 822 . . . 899 Since we cannot include 888 in this list, there are 9 numbers in the form 8XX 8X8 808 818 828 . . . 898 Since we cannot include 888 in this list, there are 9 numbers in the form 8X8 88X 880 881 882 . . . 889 Since we cannot include 888 in this list, there are 9 numbers in the form 88X So, there are 27 (9+9+9) numbers from 800 to 899 inclusive that meet the given criteria. Using the same logic, we can see that there are 27 numbers from 900 to 999 inclusive that meet the given criteria. And there are 27 numbers from 700 to 999 inclusivethat meet the given criteria. HOWEVER, the question says that we're looking at numbers greater than 700, so the number 700 does not meet the criteria. So, there are actually 26 numbers from 701 to 799 inclusive that meet the given criteria. So, our answer is 27+27+26 = [spoiler]80 = C[/spoiler] Cheers, Brent Brent Hanneson - Creator of GMATPrepNow.com Top Quote GMATinsightLegendary Member View User Profile Posts:1100Joined: Sat May 10, 2014 11:34 pmLocation: New Delhi, IndiaThanked: 205 timesFollowed by:24 members byGMATinsight » Fri Jul 11, 2014 9:49 am Of the 3-digit integers greater than 700, how many have 2 digits that are equal to each other and the remaining digit different from the other 2 ? (A) 90 (B) 82 (C) 80 (D) 45 (E) 36 Between 700 to 799 770, 771, 771, 773, 774, 775, 776, 778, 779 = 9 Numbers 707, 717, 727, 737, 747, 757, 767, 787, 797 = 9 Numbers 711, 722, 733, 744, 755, 766, 788, 799 = 8 Numbers Total Such Numbers = 9+8+9 = 26 Between 800 to 899 880, 881, 881, 883, 884, 885, 886, 887, 889 = 9 Numbers 808, 818, 828, 838, 848, 858, 868, 878, 898 = 9 Numbers 800, 811, 822, 833, 844, 855, 866, 877, 899 = 9 Numbers Total Such Numbers = 9+9+9 = 27 Between 900 to 999 990, 991, 992, 993, 994, 995, 996, 997, 998 = 9 Numbers 909, 919, 929, 939, 949, 959, 969, 979, 989 = 9 Numbers 900, 911, 922, 933, 944, 955, 966, 977, 988 = 9 Numbers Total Such Numbers = 9+9+9 = 27 Total Numbers = 26+27+27 = 80 Answer "GMATinsight"Bhoopendra Singh & Sushma Jha Most Comprehensive and Affordable Video Course 2000+ CONCEPT Videos and Video Solutions Whatsapp/Mobile: +91-9999687183 l bhoopendra.singh@gmatinsight.com Contact for One-on-One FREE ONLINE DEMO Class Call/e-mail Most Efficient and affordable One-On-One Private tutoring fee - US$40-50 per hour Top Quote Anaira MitchMaster \| Next Rank: 500 Posts View User Profile Posts:235Joined: Wed Oct 26, 2016 9:21 pmThanked: 3 timesFollowed by:5 members byAnaira Mitch » Mon Nov 07, 2016 6:36 am GMATGuruNY wrote: II wrote:Of the 3-digit integers greater than 700, how many have 2 digits that are equal to each other and the remaining digit different from the other 2 ? (A) 90 (B) 82 (C) 80 (D) 45 (E) 36 Integers with exactly 2 digits the same = Total integers - Integers with all 3 digits the same - Integers with all 3 digits different. Total integers: To count consecutive integers, use the following formula: Number of integers = biggest - smallest + 1. Thus: Total = 999 - 701 + 1 = 299. Integers with all 3 digits the same: 777, 888, 999. Number of options = 3. Integers with all 3 digits different: Number of options for the hundreds digit = 3. (7, 8, or 9) Number of options for the tens digit = 9. (Any digit 0-9 other than the digit already used.) Number of options for the units digit = 8. (Any digit 0-9 other than the two digits already used.) To combine these options, we multiply: 398 = 216. Thus: Integers with exactly 2 digits the same = 299-3-216 = 80. The correct answer is C. Amazing solution Mitch. Thanks for your guidance. Official Guide explanation is quite complex to understand. Top Quote GMAT/MBA Expert Scott@TargetTestPrepGMAT Instructor View User Profile Posts:7715Joined: Sat Apr 25, 2015 10:56 amLocation: Los Angeles, CAThanked: 43 timesFollowed by:29 members byScott@TargetTestPrep » Thu Apr 12, 2018 3:55 pm II wrote:Of the 3-digit integers greater than 700, how many have 2 digits that are equal to each other and the remaining digit different from the other 2 ? (A) 90 (B) 82 (C) 80 (D) 45 (E) 36 We can solve this problem by analyzing the digits (0 to 9) that are being "doubled". If 0 is doubled, then the numbers can only be 800 and 900. So we have 2 numbers. If 1 is doubled, then the numbers can only be 711, 811 and 911. So we have 3 numbers. If 2, 3, 4, 5, or 6 is doubled, then we should have 3 numbers for each case since it's analogous to 1 being doubled. If 7 is doubled, then the numbers can only be 770, 771, 772, 773, 774, 775, 776, 778, 779; 707, 717, 727, 737, 747, 757, 767, 787, 797; 877, and 977. So we have a total of 20 numbers. If 8 or 9 is doubled, then we should have 20 numbers for each case since it's analogous to 7 being doubled. Thus, altogether we have 2 + 6 x 3 + 3 x 20 = 2 + 18 + 60 = 80 such numbers. Answer: C Scott Woodbury-Stewart Founder and CEO scott@targettestprep.com See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews Top Quote Post new topicPost Reply • Page 1 of 1 5/5 5 Star (551 Reviews) "The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have." "TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. 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188968	https://projecteuclid.org/journals/internet-mathematics/volume-5/issue-3/Threshold-Graph-Limits-and-Random-Threshold-Graphs/im/1259095581.pdf	i i “imvol5” — 2009/11/4 — 9:42 — page 267 — #1 i i i i i i Internet Mathematics Vol. 5, No. 3: 267–320 Threshold Graph Limits and Random Threshold Graphs Persi Diaconis, Susan Holmes, and Svante Janson Abstract. We study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples for the emerging theory of graph limits. 1. Introduction 1.1. Threshold Graphs Graphs have important applications in modern systems biology and social sci-ences. Edges are created between interacting genes or people who know each other. However, graphs are not objects that are naturally amenable to sim-ple statistical analysis. There is no natural average graph, for instance. Being able to predict or replace a graph by hidden (statisticians call them latent) real variables has many advantages. This paper studies such a class of graphs that sits within the larger class of interval graphs [McKee and McMorris 99], itself a subset of intersection graphs [Erd˝ os et al. 89]; see also [Brandst¨ adt et al. 99]. Consider the following properties of a simple graph G on [n] := {1, 2, . . ., n}: (1.1) There are real weights wi and a threshold value t such that there is an edge from i to j if and only if wi + wj > t. Thus “the rich people always know each other.” © A K Peters, Ltd. 1542-7951/08 $0.50 per page 267 i i “imvol5” — 2009/11/4 — 9:42 — page 268 — #2 i i i i i i 268 Internet Mathematics 1 2 3 5 4 Figure 1. A threshold graph. (1.2) The graph G can be built sequentially from the empty graph by adding vertices one at a time, where each new vertex is either isolated (nonadjacent to all the previous) or dominant (connected to all the previous). (1.3) The graph is uniquely determined (as a labeled graph) by its degree sequence. (1.4) Any induced subgraph has either an isolated or a dominant vertex. (1.5) There is no induced subgraph 2K2, P4, or C4. (Equivalently, there is no alternating 4-cycle, i.e., four distinct vertices x, y, z, w with edges xy and zw but no edges yz and xw; the diagonals xz and yw may or may not exist.) These properties are equivalent and deﬁne the class of threshold graphs. The book [Mahadev and Peled 95] contains proofs and several other seemingly diﬀerent characterizations. Note that the complement of a threshold graph is a threshold graph (by any of (1.1)–(1.5)). By (1.2), a threshold graph is ei-ther connected (if the last vertex is dominant) or has an isolated vertex (if the last vertex is isolated); clearly these two possibilities exclude each other when n > 1. Example 1.1. The graph in Figure 1 is a threshold graph, from (1.1) by taking weights 1, 5, 2, 3, 2 on vertices 1 through 5 with t = 4.5, or from (1.2) by adding vertices 3, 5 (isolated), 4 (dominant), 1 (isolated), and 2 (dominant). While many familiar graphs are threshold graphs (stars and complete graphs, for example), many are not (e.g., paths and cycles of length 4 or more). For example, of the 64 labeled graphs on four vertices, 46 are threshold graphs; the other 18 are paths P4, cycles C4, and pairs of edges 2K2 (which is the complement of C4). Considering unlabeled graphs, there are 11 graphs on four vertices, and eight of them are threshold graphs. i i “imvol5” — 2009/11/4 — 9:42 — page 269 — #3 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 269 0.4 −1 0.7 −1.9 1.5 −0.8 0.4 0.9 −1.5 0.9 −0.6 0.6 −0.1 1.4 3 0.3 0.7 2.3 0.3 0.5 −1.1 0.3 −1.1 0.8 0.7 1.3 −0.3 1.9 1.3 −0.7 −0.1 −0.2 0.9 1.2 −0.2 0.1 0.3 −0.1 −0.8 −2.3 −1 1.2 0.3 2 1.2 0.7 −1.1 0.6 −0.1 −1.4 0.1 0.2 −0.6 0.7 −0.2 2.1 2.4 −1.2 −0.7 1.5 2 −0.9 −0.4 1.5 1.2 0.9 0 0.9 0.5 −0.4 1.2 1.2 1.6 −0.1 0.3 0.4 −0.6 −0.7 1 0.2 1.2 1.6 0 −0.1 0.2 −2 1 −0.8 −1.2 −0.5 −0.3 −1.1 1.4 1.5 0 0.6 −0.4 −1.7 −0.9 −0.2 0.4 0.7 1.5 0.4 0.9 0.9 0.6 1.4 3 0.3 0.7 2.3 0.3 0.5 0.3 0.8 0.7 1.3 1.9 1.3 0.9 1.2 0.1 0.3 1.2 0.3 2 1.2 0.7 0.6 0.1 0.2 0.7 2.1 2.4 1.5 2 1.5 1.2 0.9 0.9 0.5 1.2 1.2 1.6 0.3 0.4 1 0.2 1.2 1.6 0 0.2 1 1.4 1.5 0 0.6 Figure 2. A whole threshold graph with isolates (above) and with only the con-nected part expanded (below); the labels are the rounded weights wi. 1.2. Random Threshold Graphs It is natural to study random threshold graphs. There are several diﬀerent natural random constructions; we will in particular consider the following three: (1.6) From (1.1) by choosing {wi}1≤i≤n as independent and identically dis-tributed (i.i.d.) random variables from some probability distribution. (We also choose some ﬁxed t; we may assume t = 0 by replacing wi by wi−t/2.) (1.7) From (1.2) by ordering the vertices randomly and adding the vertices one by one, each time choosing at random between the qualiﬁers “dominant” and “isolated” with probabilities pi and 1 −pi, respectively, 1 ≤i ≤n. i i “imvol5” — 2009/11/4 — 9:42 — page 270 — #4 i i i i i i 270 Internet Mathematics 0.7 0.46 0.82 0.88 0.42 0.08 0.27 0.22 0.28 0.46 0.6 0.1 0.45 0.84 0.22 0.66 0.93 0.13 0.54 0.67 Figure 3. A threshold graph with n = 20 and uniform wi. It turns out that this instance has no isolates. The labels are the rounded weights wi. This is a simple random attachment model in a similar vein to those in [Mitzenmacher 04]. We mainly consider the case that all pi are equal to a single parameter p ∈[0, 1]. (1.8) The uniform distribution on the set of threshold graphs. Example 1.2. Figure 2 shows a random threshold graph constructed by (1.6) with wi chosen independently from the standardized normal distribution and t = 3. About half of the vertices are isolated, most of those with negative weights. Example 1.3. Figure 3 shows a random threshold graph constructed by (1.6) with wi chosen as i.i.d. uniform random variables on [0, 1] and t = 1. This instance is connected; this happens if and only if the maximum and minimum of the wi add to more than 1 (then there is a dominant vertex); in this example this has probability 1/2. We show below (Corollaries 6.6 and 6.7) that this uniform-weight model is equivalent to adding isolated or dominant nodes as in (1.7) with probability p = 1/2, independently and in random order. It follows that this same distribution appears as the stationary distribution of a Markov chain on threshold graphs that picks a vertex at random and changes it to dominant or isolated with probability 1/2 (this walk is analyzed in [Brown and Diaconis 98]). Furthermore, it follows from Section 2.1 that these models yield a uniform distribution on the set of unlabeled threshold graphs of order n. i i “imvol5” — 2009/11/4 — 9:42 — page 271 — #5 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 271 1.3. Bipartite Threshold Graphs We also study the parallel case of bipartite threshold graphs (diﬀerence graphs), both for its own sake and because one of the main theorems is proved by ﬁrst considering the bipartite case. By a bipartite graph we mean a graph with an explicit bipartition of the vertex set; it can thus be written as (V1, V2, E), where the edge set E satisﬁes E ⊆V1 × V2. The following properties of a bipartite graph are equivalent and deﬁne the class of bipartite threshold graphs. (See [Mahadev and Peled 95] for further characterizations.) (1.9) There are real weights w′ i, i ∈V1, and w′′ j , j ∈V2, and a threshold value t such that there is an edge from i to j if and only if w′ i + w′′ j > t. (1.10) The graph G can be built sequentially starting from n1 white vertices and n2 black vertices in some ﬁxed total order. Proceeding in this order, make each white vertex dominant or isolated from all the black vertices that precede it and each black vertex dominant or isolated from all earlier white vertices. (1.11) Any induced subgraph has either an isolated vertex or a vertex dominating every vertex in the other part. (1.12) There is no induced subgraph 2K2. Remark 1.4. Threshold graphs were deﬁned in [Chv´ atal and Hammer 77]. Bipartite threshold graphs were studied in [Hammer et al. 90] under the name diﬀerence graphs because they can equivalently be characterized as the graphs (V, E) for which there exist weights wv, v ∈V , and a real number t such that \|wv\| < t for every v and uv ∈E ⇐ ⇒ \|wu −wv\| > t; it is easily seen that every such graph is bipartite with V1 = { v : wv ≥0 } and V2 = { v : wv < 0 } and that this satisﬁes the deﬁnition above (e.g., with w′ v = wv and w′′ v = −wv), and conversely. We will use the name bipartite threshold graph to empha-size that we consider these graphs equipped with a given bipartition. The same graphs were called chain graphs in [Yannakakis 82] because each par-tition can be linearly ordered for the inclusion of the neighborhoods of its elements. Remark 1.5. A suite of programs for working with threshold graphs appears in [Hagberg et al. 06] with further developments in [Konno et al. 05] and [Masuda et al. 05]. i i “imvol5” — 2009/11/4 — 9:42 — page 272 — #6 i i i i i i 272 Internet Mathematics 1 1 0 Figure 4. The function W (x,y) for Example 1.3. Hashed values have W (x, y) = 1; unhashed, W (x,y) = 0. Remark 1.6. The most natural class of graphs built from a coordinate system are commonly called geometric graphs [Penrose 03] or geographical graphs [Konno et al. 05, Masuda et al. 05]. Threshold graphs are a special case of these. Their recognition and manipulation in a statistical context relies on useful measures on such graphs. We will start by deﬁning such measures and developing a limit theory. 1.4. Overview of the Paper The purpose of this paper is to study the limiting properties of large thresh-old graphs in the spirit of the theory of graph limits developed in [Lov´ asz and Szegedy 06] and [Borgs et al. 07b] (and in further papers by those authors and others). As explained below, the limiting objects are not graphs, but can rather be represented by symmetric functions W(x, y) from [0, 1]2 to [0, 1]; any sequence of graphs that converges in the appropriate way has such a limit. Conversely, such a function W may be used to form a random graph Gn by choosing independent random points Ui in [0, 1], and then for each pair (i, j) with 1 ≤i < j ≤n ﬂipping a biased coin with heads probability W(Ui, Uj), putting an edge from i to j if the coin comes up heads. The resulting sequence of random graphs is (almost surely) an example of a sequence of graphs converging to W. For Example 1.3, letting n →∞, there is (as we show in greater generality in Section 6) a limit W that may be pictured as in Figure 4. One of our main results (Theorem 5.5) shows that graph limits of thresh-old graphs have unique representations by increasing symmetric zero–one-valued functions W. Furthermore, there is a one-to-one correspondence between these limiting objects and a certain type of “symmetric” probability distribution PW on [0, 1]. A threshold graph is characterized by its degree sequence; normalizing i i “imvol5” — 2009/11/4 — 9:42 — page 273 — #7 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 273 0 3 6 9 13 17 21 25 29 33 37 41 45 49 0 2000 4000 6000 8000 10000 Figure 5. Threshold graphs generated with n = 50 as in Example 1.3 with uniform wi and t = 1; this is the degree histogram for a sample of 10,000 random graphs. this to be a probability distribution, say ν(Gn), we show (Theorem 5.8) that a sequence of threshold graphs converges to W when n →∞if and only if ν(Gn) converges to PW . (Hence PW can be regarded as the degree distribution of the limit. The result that a limit of threshold graphs is determined by its degree dis-tribution is a natural analogue for the limit objects of the fact that an unlabeled threshold graph is uniquely determined by its degree distribution.) Figures 5 and 7 show simulations of these results. In Figure 5, ten thou-sand graphs with n = 50 were generated from (1.6) with uniform weights as in Example 1.3. In the bipartite case, there is a similar one-to-one correspondence between the limit objects and probability distributions on [0, 1]; now all probability distribu-tions on [0, 1] appear in the representation of the limits (Theorem 5.1). Section 2 discusses uniform random threshold graphs (both labeled and un-labeled) and methods to generate them. Section 3 gives a succinct review of notation and graph limits. Section 4 develops the limit theory of degree se-quences; this is not restricted to threshold graphs. Section 5 develops the limit theory for threshold graphs both deterministic and random. Section 6 treats examples of random threshold graphs and their limits, and Section 8 gives corre-sponding examples and results for random bipartite threshold graphs. Section 7 i i “imvol5” — 2009/11/4 — 9:42 — page 274 — #8 i i i i i i 274 Internet Mathematics treats the degree distribution of uniform random threshold graphs in greater detail. Section 9 treats the spectrum of the Laplacian of threshold graphs. We denote the vertex and edge sets of a graph G by V (G) and E(G), and the numbers of vertices and edges by v(G) := \|V (G)\| and e(G) := \|E(G)\|. For a bipartite graph we similarly use Vj(G) and vj(G), j = 1, 2. Throughout the paper, “increasing” and “decreasing” should be interpreted in the weak sense (nondecreasing and nonincreasing). Unspeciﬁed limits are to be taken as n →∞. 2. Generating Threshold Graphs Uniformly This section gives algorithms for generating uniformly distributed threshold graphs, in both the labeled case and unlabeled cases. The algorithms are used here for simulation and in Sections 6 and 7 to prove limit theorems. Let Tn and LT n be the sets of unlabeled and labeled threshold graphs on n vertices. These are diﬀerent objects; Tn is a quotient of LT n, and we treat counting and uniform generation separately for the two cases. We assume in this section that n ≥2. 2.1. Unlabeled Threshold Graphs We can code an unlabeled threshold graph on n vertices by a binary code α2 · · · αn of length n−1: Given a code α2 · · · αn, we construct G by (1.2), adding vertex i as a dominant vertex if and only if αi = 1 (i ≥2). Conversely, given G of order n ≥2, let αn = 1 if there is a dominant vertex (G is connected) and αn = 0 if there is an isolated vertex (G is disconnected). We then remove one such dominant or isolated vertex and continue recursively to deﬁne αn−1, . . . , α2. Since all dominant (isolated) vertices are equivalent to each other, this coding gives a bijection between Tn and { 0, 1 }n−1. In particular, \|Tn\| = 2n−1, n ≥1. See Figure 6 for an example. , , , 00 01 10 11 Figure 6. The four graphs in T3 and their codes. i i “imvol5” — 2009/11/4 — 9:42 — page 275 — #9 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 275 Algorithm 1. (Generating uniform random unlabeled threshold graphs of a given order n.) 1. Add n vertices by (1.2), each time randomly choosing “isolated” or “dominant” with probability 1/2. This leads to a simple algorithm to generate a uniformly distributed random unlabeled threshold graph: we construct a random code by making n −1 coin ﬂips, according to Algorithm 1. This is thus the same as the second method in Example 1.3, so Corollary 6.6 shows that the ﬁrst method in Example 1.3 also yields uniform random unlabeled threshold graphs (if we forget the labels). The following notation is used to deﬁne two further algorithms (Section 2.3) and for proof of the limiting results in Section 7. Deﬁne the extended binary code of a threshold graph to be the binary code with the ﬁrst binary digit repeated; it is thus α1α2α3 · · · αn with α1 := α2. The runs of 0’s and 1’s in the extended binary code then correspond to blocks of vertices that can be added together in (1.2) as either isolated or dominant vertices, with the blocks alternating between isolated and dominant. The vertices in each block are equivalent and have in particular the same vertex degrees, while vertices in diﬀerent blocks can be seen to have diﬀerent degrees. (The degree increases strictly from one dominant block to the next and decreases strictly from one isolated block to the next, with every dominant block having higher degree than every isolated block; cf. Example 2.1 below.) The number of diﬀerent vertex degrees thus equals the number of blocks. If the lengths of the blocks are b1, b2, . . . , bτ, then the number of automor-phisms of G is τ j=1 bj!, since the vertices in each block may be permuted arbi-trarily. Note that if b1, . . . , bτ are the lengths of the blocks, then b1 ≥2, bk ≥1 (k ≥2), τ k=1 bk = n. (2.1) Since the blocks are alternately dominant or isolated, and the ﬁrst block may be either, each sequence b1, . . . , bτ satisfying (2.1) corresponds to exactly two unlabeled threshold graphs of order n. (These graphs are the comple-ments of each other. One has isolated blocks where the other has dominant blocks.) i i “imvol5” — 2009/11/4 — 9:42 — page 276 — #10 i i i i i i 276 Internet Mathematics 2.2. Labeled Threshold Graphs The situation is diﬀerent for labeled threshold graphs. For example, all of the 2(3 2) = 8 labeled graphs with n = 3 turn out to be threshold graphs, and for instance, the graphs 1 2 3, 1 2 3 , 1 2 3 are distinguished. Hence the distribution of a uniform random labeled threshold graph diﬀers from the distribution of a uniform unlabeled threshold graph (even if we forget the labels). In particular, Example 1.3 does not produce uniform random labeled threshold graphs. Let G be an unlabeled threshold graph with an extended code having block lengths (runs) b1, . . . , bτ. Then the number of labeled threshold graphs cor-responding to G is n!/ τ 1 bj!, since every such graph corresponds to a unique assignment of the labels 1, . . . , n to the τ blocks, with bi labels to block i. (Alter-natively and equivalently, this follows from the number τ 1 bj! of automorphisms given above.) The number t(n) := \|LT n\| of labeled threshold graphs [Sloane 09, A005840] has been studied in [Beissinger and Peled 87]. Among other things, the authors show that ∞ n=0 t(n)xn n! = (1 −x)ex 2 −ex , so by Taylor expansion, n 1 2 3 4 5 6 7 8 9 10 t(n) 1 2 8 46 332 2874 29024 334982 4349492 62749906 and by expanding the singularities (cf. [Flajolet and Sedgewick 09, Chapter IV]), they obtain the exact formula t(n) n! = ∞ k=−∞ 1 log 2 + 2πik −1 1 log 2 + 2πik n , n ≥2, (2.2) where the leading term is the one with k = 0, and thus one has the asymptotics t(n) n! = 1 log 2 −1 1 log 2 n + ϵ(n), \|ϵ(n)\| ≤2ζ(n) (2π)n , (2.3) where ζ(n) is the Riemann zeta function and thus ζ(n) →1. Furthermore, t(n) = 2Rn −2nRn−1, n ≥2, with Rn = n k=1 k!S(n, k) = ∞ ℓ=0 ℓn 2ℓ+1 , (2.4) i i “imvol5” — 2009/11/4 — 9:42 — page 277 — #11 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 277 where S(n, k) are Stirling numbers; Rn is the number of preferential arrange-ments of n labeled elements, or the number of weak orders on n labeled elements [Sloane 09, A000670], also called surjection numbers [Flajolet and Sedgewick 09, II.3]. (This is easily seen using the blocks above; the number of labeled thresh-old graphs with a given sequence of blocks is twice—since the ﬁrst block may be either isolated or dominant—the number of preferential arrangements with the same block sizes; if we did not require b1 ≥2, this would yield 2Rn, but we have to subtract twice the number of preferential arrangements with b1 = 1, which is 2nRn−1.) We note for future use the generating function [Flajolet and Sedgewick 09, (II.15)] ∞ n=0 Rn xn n! = 1 2 −ex . (2.5) Let t(n, j) be the number of labeled threshold graphs with j isolated points. Then, as also shown in [Beissinger and Peled 87] (and easily seen), for n ≥2, t(n, 0) = t(n)/2, t(n, j) = ⎧ ⎪ ⎨ ⎪ ⎩ n j t(n −j, 0) = 1 2 n j t(n −j), 0 ≤j ≤n −2, 0, j = n −1, 1, j = n. (2.6) Thus knowledge of t(n) provides t(n, j). These ingredients allow us to give an algorithm, Algorithm 2, for choosing uniformly in LT n. Alternatively, instead of selecting the subsets in steps 1 and 2 at random, we may choose them in any way, provided the algorithm begins or ends with a random permutation of the points. The algorithm works because of a characterization of threshold graphs by Chv´ atal and Hammer [Chv´ atal and Hammer 77], cf. (1.4): a graph is a threshold graph iﬀany subset S of vertices contains at least one isolated or one dominant vertex (within the graph induced by S). Thus in step 2, since there are no isolates among the n′ vertices left, there must be at least one dominant vertex. (Note that j0 may be zero, but not j1, j2, . . . .) The probability distribution for the number of dominant vertices follows the same law as that of the isolates because the complement of a threshold graph is a threshold graph (or because of the interchangeability of 0’s and 1’s in the binary coding given earlier in this section). Note that this algorithm treats vertices in the reverse of the order in (1.2), where we add vertices instead of peeling them oﬀ, as here. It follows that we obtain the extended binary code of the graph by taking runs of j0 0’s, j1 1’s, i i “imvol5” — 2009/11/4 — 15:30 — page 278 — #12 i i i i i i 278 Internet Mathematics Algorithm 2. (Generating uniform random labeled threshold graphs of a given order n.) 0. Make a list of t(k) for k between 1 and n. Make lists of t(k, j) for k = 1, . . . , n and j = 0, . . . , k. 1. Choose an integer j0 in {0, . . ., n} with probability that j0 = j given by t(n, j)/t(n). Choose (at random) a subset of j0 points in {1, . . ., n}. These are the isolated vertices in the graph. Let n′ := n −j0 be the number of remaining points. If n′ = 0 then stop. 2. Choose an integer j1 in {1, . . ., n′} with probability that j1 = j given by t(n′, j)/(t(n′) −t(n′, 0)) = 2t(n′, j)/t(n′) and choose (at random) j1 points of those remaining; these will dominate all further points, so add edges between these vertices and from them to all remaining points. Update n′ to n′ −j1, the number of remaining points. If n′= 0 then stop. 3. Choose an integer j2 in {1, . . ., n′} with probability that j2 = j given by 2t(n′, j)/t(n′) and choose (at random) j2 points of those remaining; these will be isolated among the remaining points, so no further edges are added. Update n′ to n′ −j1, the number of remaining points. If n′ = 0 then stop. 4. Repeat from step 2 with the remaining n′ points. j2 0’s, and so on, and then reversing the order. Hence, in the notation used above, the sequence (bk) equals (jk) in reverse order, ignoring j0 if j0 = 0. (In particular, note that the last jk is greater than or equal to 2, since t(n′, n′−1) = 0 for n′ ≥2, which corresponds to the ﬁrst block b1 ≥2.) Example 2.1. A sequence of j’s generated for a threshold graph of size 20 is 0 2 3 1 1 1 3 1 1 3 1 1 2, which yields the sequence d d i i i d i d i i i d i d d d i d i i of dominant and isolated vertices. A random permutation of { 1, . . ., 20 } was generated and we obtain 13 2 11 15 8 20 6 12 16 4 18 7 10 9 14 17 1 19 5 3 d d i i i d i d i i i d i d d d i d i i where d signiﬁes that the vertex is connected to all later vertices in this list. The degree sequence is thus, taking the vertices in this order: 19, 19, 2, 2, 2, 16, 3, 15, 4, 4, 4, 12, 5, 11, 11, 11, 8, 10, 9, 9. The extended binary code 00101110100010100011 is obtained by translating i to 0 and d to 1, and reversing the order. i i “imvol5” — 2009/11/4 — 9:42 — page 279 — #13 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 279 0 6 13 21 29 37 45 53 61 69 77 85 93 0 2000 4000 6000 8000 10000 Figure 7. Threshold graphs were generated according to the algorithm of this section; this is the degree histogram. Remark 2.2. Note that the last j is at least 2, since t(n′, n′ −1) = 0 for every n′ ≥2. Hence, the sequence of d’s and i’s always ends with at least two identical symbols. Note that the vertex degrees are constant in each block of vertices assigned in one of the steps, i.e., for each run of i’s or d’s, and that they decrease from each run of d’s to the next and increase from each run of i’s to the next, with each vertex labeled d having a higher degree than every vertex labeled with i; the number of diﬀerent vertex degrees is thus equal to the number of j’s chosen by the algorithm, which equals the number of runs in the sequence of d’s and i’s (or in the extended binary code). In Figure 7, ten thousand graphs were generated with n = 100 according to the uniform distribution over all labeled threshold graphs. We discuss the central “bump” and other features of Figure 7 in Theorem 7.4. i i “imvol5” — 2009/11/4 — 9:42 — page 280 — #14 i i i i i i 280 Internet Mathematics 2.3. The Distribution of Block Lengths We have seen in Section 2.1 that if b1, . . . , bτ are the lengths of the blocks of isolated or dominant vertices added to the graph in building it as in (1.2), then (2.1) holds. Consider now a sequence of independent integer random variables B1, B2, . . . with B1 ≥2 and Bj ≥1 for j ≥2, and let Sk := k j=1 Bj be the partial sums. If some Sτ = n, then stop and output the sequence (B1, . . . , Bτ). Conditioning on the event that Sτ = n for some τ, this yields a random sequence b1, . . . , bτ satisfying (2.1), and the probability that we obtain a given sequence (bj)τ 1 equals c τ j=1 P(Bj = bj) for some normalizing constant c. We now specialize to the case that B1 d = (B∗\| B∗≥2) and Bj d = (B∗\| B∗≥1) for j ≥2, for some given random variable B∗. Then the (conditional) probability of obtaining a given b1, . . . , bτ satisfying (2.1) can be written c′ τ j=1 P(B∗= bj) P(B∗≥1) (2.7) (with c′ = c P(B∗≥1)/ P(B∗≥2)). There are two important cases. First, if we take B∗∼Ge(1/2), then P(B∗= bj)/ P(B∗≥1) = 2−bj, and thus (2.7) yields c′2− j bj = c′2−n, so the proba-bility is the same for all allowed sequences. Hence, in this case the distribution of the constructed sequence is uniform on the set of sequences satisfying (2.1), so it equals the distribution of block lengths for a random unlabeled threshold graph of size n. The other case is B∗∼Po(log 2). Then P(B∗≥1) = 1 −e−log 2 = 1/2, and P(B∗= bj)/ P(B∗≥1) = (log 2)bj/bj!. Thus, (2.7) yields the probability c′(log 2)n/ j bj!, which is proportional to the number 2 · n!/ j bj! of labeled threshold graphs with block lengths b1, . . . , bτ. Hence, in this case the distribu-tion of the constructed sequence equals the distribution of block lengths for a random labeled threshold graph of size n. We have proved the following result. Theorem 2.3. Construct a random sequence B1, . . . , Bτ as above, based on a random variable B∗, stopping when τ 1 Bj ≥n and conditioning on τ 1 Bj = n. (i) If B∗∼Ge(1/2), then (B1, . . . , Bτ) has the same distribution as the block lengths in a random unlabeled threshold graph of order n. (ii) If B∗∼Po(log 2), then (B1, . . . , Bτ) has the same distribution as the block lengths in a random labeled threshold graph of order n. i i “imvol5” — 2009/11/4 — 9:42 — page 281 — #15 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 281 Algorithm 3. (Generating uniform unlabeled or labeled threshold graphs of a given order n.) 1. In the unlabeled case, let B∗∼Ge(1/2). In the labeled case, let B∗∼Po(log 2). 2. Choose independent random numbers B1, B2, . . . , Bτ, with B1 d = (B∗\| B∗≥2) and Bj d = (B∗\| B∗≥1), j ≥2, until the sum τ 1 Bj is greater than or equal to n. 3. If τ 1 Bj > n, start again with step 2. 4. We have found B1, . . . , Bτ with τ 1 Bj = n. Toss a coin to decide whether the ﬁrst block is isolated or dominant; the following blocks alternate. Construct a threshold graph by adding vertices as in (1.2), block by block. 5. In the labeled case, make a random labeling of the graph. It follows that the length of a typical (for example, a random) block converges in distribution to (B∗\| B∗≥1). Theorem 2.3 also leads to another algorithm to construct uniform random threshold graphs (Algorithm 3). By standard renewal theory, the probability that τ 1 Bj is exactly n is asymp-totically 1/ E(B∗\| B∗≥1) = P(B∗≥1)/ E B∗, which is 1/2 in the unlabeled case and 1/(2 log 2) ≈0.72 in the labeled case, so we do not have to do very many restarts in step 3. 3. Graph Limits This section reviews needed tools from the emerging ﬁeld of graph limits. 3.1. Graph Limits Here we review brieﬂy the theory of graph limits as described in [Lov´ asz and Szegedy 06], [Borgs et al. 07b], and [Diaconis and Janson 08]. If F and G are two graphs, let t(F, G) be the probability that a random mapping φ : V (F) →V (G) deﬁnes a graph homomorphism, i.e., that φ(v)φ(w) ∈ E(G) when vw ∈E(F). (By a random mapping we mean a mapping uniformly chosen among all v(G)v(F ) possible ones; the images of the vertices in F are thus independent and uniformly distributed over V (G), i.e., they are obtained by random sampling with replacement.) i i “imvol5” — 2009/11/4 — 9:42 — page 282 — #16 i i i i i i 282 Internet Mathematics The basic deﬁnition is that a sequence Gn of (generally unlabeled) graphs converges if t(F, Gn) converges for every graph F; as in [Diaconis and Janson 08], we will further assume v(Gn) →∞. More precisely, the (countable and discrete) set U of all unlabeled graphs can be embedded in a compact metric space U such that a sequence Gn ∈U of graphs with v(Gn) →∞converges in U to some limit Γ ∈U if and only if t(F, Gn) converges for every graph F (see [Lov´ asz and Szegedy 06, Borgs et al. 07b, Diaconis and Janson 08]). Let U∞:= U \ U be the set of proper limit elements; we call the elements of U∞graph limits. The functionals t(F, ·) extend to continuous functions on U, so Gn →Γ ∈U∞if and only if v(Gn) →∞and t(F, Gn) →t(F, Γ) for every graph F. Let W be the set of all measurable functions W : [0, 1]2 →[0, 1] and let Ws be the subset of symmetric functions. The main result of [Lov´ asz and Szegedy 06] is that every element of U∞can be represented by a (nonunique) function W ∈Ws. We let ΓW ∈U∞denote the graph limit deﬁned by W. (We sometimes use the notation Γ(W) for readability.) Then, for every graph F, t(F, ΓW ) = [0,1]v(F ) ij∈E(F ) W(xi, xj) dx1 · · · dxv(F ). (3.1) Moreover, deﬁne, for every n ≥1, a random graph G(n, W) as follows: ﬁrst choose a sequence X1, X2, . . . , Xn of i.i.d. random variables uniformly distributed on [0, 1], and then, given this sequence, for each pair (i, j) with i < j draw an edge ij with probability W(Xi, Xj), independently for all pairs (i, j) with i < j. Then the random graph G(n, W) converges to ΓW a.s. as n →∞. If G is a graph, with V (G) = { 1, . . ., v(G) } for simplicity, we deﬁne a function WG ∈Ws by partitioning [0, 1] into v(G) intervals Ii, i = 1, . . . , v(G), and letting WG be the indicator 1[ij ∈E(G)] on Ii × Ij. (In other words, WG is a step function corresponding to the adjacency matrix of G.) We let π(G) := Γ(WG) denote the corresponding object in U∞. It follows easily from (3.1) that t(F, π(G)) = t(F, G) for every graph F. In particular, if Gn is a sequence of graphs with v(Gn) →∞, then Gn converges to some graph limit Γ if and only if π(Gn) →Γ in U∞. (Unlike [Lov´ asz and Szegedy 06] and [Borgs et al. 07b], we distinguish between graphs and limit objects and we do not identify G and π(G); see [Diaconis and Janson 08].) 3.2. Bipartite Graphs and Their Limits In the bipartite case, there are analogous deﬁnitions and results (see [Diaconis and Janson 08] for further details). We deﬁne a bipartite graph to be a graph i i “imvol5” — 2009/11/4 — 9:42 — page 283 — #17 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 283 G with an explicit bipartition V (G) = V1(G) ∪V2(G) of the vertex set such that the edge set E(G) is a subset of V1(G) × V2(G). Then we deﬁne t(F, G) in the same way as above but now for bipartite graphs F, by letting φ = (φ1, φ2) be a pair of random mappings φj : Vj(F) →Vj(G). We let B be the set of all unlabeled bipartite graphs and embed B in a compact metric space B. A sequence (Gn) of bipartite graphs with v1(Gn), v2(Gn) →∞converges in B if and only if t(F, Gn) converges for every bipartite graph F. Let B∞∞be the (compact) set of all such limits; we call the elements of B∞∞bipartite graph limits. Every element of B∞∞can be represented by a (nonunique) function W ∈W. We let Γ′′ W ∈B∞∞denote the element repre-sented by W and have for every bipartite F, t(F, Γ′′ W ) = [0,1]v1(F )+v2(F ) ij∈E(F ) W(xi, yj) dx1 · · · dxv1(F ) dy1 · · · dyv2(F ). (3.2) Given W ∈W and n1, n2 ≥1, we deﬁne a random bipartite graph G(n1, n2, W) by an analogue of the construction in Section 3.1: ﬁrst choose two sequences X1, X2, . . . , Xn1 and Y1, Y2, . . . , Yn2 of i.i.d. random variables uniformly dis-tributed on [0, 1], and then, given these sequences, for each pair (i, j) draw an edge ij with probability W(Xi, Yj), independently for all pairs (i, j) ∈[n1 ×[n2]. If G is a bipartite graph we deﬁne WG ∈W similarly as above (in general, with diﬀerent numbers of steps in the two variables; note that WG now in general is not symmetric) and let π(G) := Γ′′(WG). Then by (3.2), t(F, π(G)) = t(F, G) for every bipartite graph F. Hence if Gn is a sequence of bipartite graphs with v1(Gn), v2(Gn) →∞, then Gn converges to some bipartite graph limit Γ if and only if π(Gn) →Γ in B∞∞. 3.3. Cut Distance The authors of [Borgs et al. 07b, Section 3.4] deﬁne a (pseudo)metric δ□on Ws called the cut distance. This is only a pseudometric, since two diﬀerent functions in Ws may have cut distance 0 (for example, if one is obtained by a measure-preserving transformation of the other; see further [Borgs et al. 07a] and [Diaconis and Janson 08]), and it is shown in [Borgs et al. 07b] that in fact, δ□(W1, W2) = 0 if and only if t(F, W1) = t(F, W2) for every graph F, i.e., if and only if ΓW1 = ΓW2 in U∞. Moreover, the quotient space Ws/δ□, where we identify elements of Ws with cut distance 0, is a compact metric space and the mapping W →ΓW is a homeomorphism of Ws/δ□onto U∞. This extends to the bipartite case. In this case, we deﬁne δ′′ □on W as δ□is de-ﬁned in [Borgs et al. 07b, Section 3.4], but allowing diﬀerent measure-preserving mappings for the two coordinates. Then if we identify elements in W with cut i i “imvol5” — 2009/11/4 — 9:42 — page 284 — #18 i i i i i i 284 Internet Mathematics distance 0, W →Γ′′ W becomes a homeomorphism of W/δ′′ □onto B∞∞. Instead of repeating and modifying the complicated proofs from [Borgs et al. 07b], one can use their result in the symmetric case and deﬁne an embedding W → W of W into Ws by W(x, y) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0, x < 1 2, y < 1 2; 1, x > 1 2, y > 1 2; 1 4 + 1 2W(2x −1, 2y), x > 1 2, y < 1 2; 1 4 + 1 2W(2y −1, 2x), x < 1 2, y > 1 2. It is easily seen that δ′′ □(W1, W2) and δ□( W1, W2) are equal within some constant factors, for W1, W2 ∈W, and that for each graph F, t(F, W ) is a linear combi-nation of t(Fi, W) for a family of bipartite graphs F (obtained by partitioning V (F) and erasing edges within the two parts). This and the results in [Borgs et al. 07b], together with the simple fact that W →t(F, W) is continuous for δ′′ □ for every bipartite graph F, imply easily the result claimed. 3.4. A Reﬂection Involution If G is a bipartite graph, let G† be the graph obtained by interchanging the order of the two vertex sets; thus Vj(G†) = V3−j(G) and E(G†) = { uv : vu ∈E(G) }. We say that G† is the reﬂection of G. Obviously, t(F, G†) = t(F †, G) for any bipartite graphs F and G. It follows that if Gn →Γ ∈B, then G† n →Γ† for some Γ† ∈B, and this deﬁnes a continuous map of B onto itself that extends the map just deﬁned for bipartite graphs. We have, by continuity, t(F, Γ†) = t(F †, Γ), F ∈B, G ∈B. (3.3) Furthermore, Γ†† = Γ, so the map is an involution, and it maps B∞∞onto itself. For a function W on [0, 1]2, let W †(x, y) := W(y, x) be its reﬂection in the main diagonal. It follows from (3.2) and (3.3) that Γ′′(W †) = Γ′′(W)†. 3.5. Threshold Graph Limits Let T := ∞ n=1 Tn be the family of all (unlabeled) threshold graphs. Thus T is a subset of the family U of all unlabeled graphs, and we deﬁne T as the closure of T in U, and T∞:= T \ T = T ∩U∞, i.e., the set of proper limits of sequences of threshold graphs; we call these threshold graph limits. In the bipartite case, we similarly consider the set T ′′ := n1,n2≥1 Tn1,n2 ⊂ B of all bipartite threshold graphs, and let T ′′ ⊂B be its closure in B, and T ′′ ∞,∞:= T ′′ ∩B∞∞the set of proper limits of sequences of bipartite threshold graphs; we call these bipartite threshold graph limits. i i “imvol5” — 2009/11/4 — 9:42 — page 285 — #19 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 285 Note that T , T∞, T ′′, T ′′ ∞,∞are compact metric spaces, since they are closed subsets of U or B. We will give concrete representations of the threshold graph limits in Section 5. Here we give only a more abstract characterization. Recall that t(F, G) is deﬁned as the proportion of maps V (F) →V (G) that are graph homomorphisms. Since we are interested only in limits with v(G) →∞, it is equivalent to consider injective maps only. By inclusion–exclusion, it is further equivalent to consider tind(F, G), deﬁned as the probability that a random injective map V (F) →V (G) maps F isomorphically onto an induced copy of F in G; in other words, tind(F, G) equals the number of labeled induced copies of F in G divided by the falling factorial v(G) · · · (v(G)−v(F)+ 1). Then tind(F, ·) extends by continuity to U, and by inclusion–exclusion, for graph limits Γ ∈U∞, tind(F, Γ) can be written as a linear combination of t(Fi, Γ) for subgraphs Fi ⊆F. We can deﬁne tind for bipartite graphs in the same way; further details are in [Borgs et al. 07b] and [Diaconis and Janson 08]. Theorem 3.1. (i) Let Γ ∈U∞; i.e., Γ is a graph limit. Then Γ ∈T∞if and only if tind(P4, Γ) = tind(C4, Γ) = tind(2K2, Γ) = 0. (ii) Let Γ ∈B∞∞; i.e., Γ is a bipartite graph limit. Then Γ ∈T ′′ ∞,∞if and only if tind(2K2, Γ) = 0. In view of (1.5) and (1.12), this is a special case of the following simple general statement. Theorem 3.2. Let F = { F1, F2, . . . } be a ﬁnite or inﬁnite family of graphs, and let UF ⊆U be the set of all graphs that do not contain any graph from F as an induced subgraph, i.e., UF := { G ∈U : tind(F, G) = 0 for F ∈F }. Let U F be the closure of UF in U. Then UF := { Γ ∈U : tind(F, Γ) = 0 for F ∈F }. In other words, if Γ ∈U∞is a graph limit, then Γ is a limit of a sequence of graphs in UF if and only if tind(F, Γ) = 0 for F ∈F. Conversely, if Γ ∈UF ∩U∞is represented by a function W, then the random graph G(n, W) is in UF (almost surely). The same results hold in the bipartite case. i i “imvol5” — 2009/11/4 — 9:42 — page 286 — #20 i i i i i i 286 Internet Mathematics Proof. If Gn →Γ with G ∈UF, then t(F, Γ) = limn→∞t(F, Gn) = 0 for every F ∈F, by the continuity of t(F, ·). Conversely, suppose that Γ ∈U∞and t(F, Γ) = 0 for F ∈F, and let Γ be represented by a function W. It follows from (3.1) that if F ∈F then E t(F, G(n, W)) = t(F, Γ) = 0, and thus t(F, G(n, W)) = 0 almost surely (a.s.); consequently, G(n, W) ∈UF a.s. This proves the second statement. Since G(n, W) →Γ a.s., it also shows that Γ is the limit of a sequence in UF, and thus Γ ∈UF, which completes the proof of the ﬁrst part. 4. Degree Distributions The results in this section hold for general graphs. They are applied to threshold graphs in Section 5. Let P be the set of probability measures on [0, 1], equipped with the standard topology of weak convergence, which makes P a compact metric space (see, for example, [Billingsley 68]). If G is a graph, let d(v) = dG(v) denote the degree of vertex v ∈V (G), and let DG denote the random variable deﬁned as the degree dG(v) of a randomly chosen vertex v (with the uniform distribution on V (G)). Thus 0 ≤DG ≤ v(G) −1. For a bipartite graph we similarly deﬁne DG;j as the degree dG(v) of a randomly chosen vertex v ∈Vj(G), j = 1, 2. Note that 0 ≤DG;1 ≤v2(G) and 0 ≤DG;2 ≤v1(G). Since we are interested in dense graphs, we will normalize these random degrees to DG/v(G) and, in the bipartite case, DG;1/v2(G) and DG;2/v1(G); these are random variables in [0, 1]. The distribution of DG/v(G) will be called the (normalized) degree distribution of G and denoted by ν(G) ∈P; in other words, ν(G) is the empirical distribution function of { dG(v)/v(G) : v ∈V (G) }. In the bipartite case we similarly have two (normalized) degree distributions: ν1(G) for V1(G) and ν2(G) for V2(G). The moments of the degree distribution(s) are given by the functional t(F, ·) for stars F, as stated in the following lemma. We omit the proof, which is a straightforward consequence of the deﬁnitions. Lemma 4.1. The moments of ν(G) are given by 1 0 tk dν(G)(t) = t(K1,k, G), k ≥1, (4.1) where K1,k is a star with k edges. i i “imvol5” — 2009/11/4 — 9:42 — page 287 — #21 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 287 In the bipartite case, similarly, for k ≥1, 1 0 tk dν1(G)(t) = t(K1,k, G), 1 0 tk dν2(G)(t) = t(Kk,1, G). (4.2) This enables us to extend the deﬁnition of the (normalized) degree distribution to the limit objects by continuity. Theorem 4.2. If Gn are graphs with v(Gn) →∞and Gn →Γ for some Γ ∈U as n →∞, then ν(Gn) →ν(Γ) for some distribution ν(Γ) ∈P. This deﬁnes the “degree distribution” ν(Γ) (uniquely) for every graph limit Γ ∈U∞, and Γ →ν(Γ) is a continuous map U∞→P. Furthermore, (4.1) holds for all G ∈U. Similarly, in the bipartite case, ν1 and ν2 extend to continuous maps B →P such that (4.2) holds for all G ∈B. Furthermore, ν2(Γ) = ν1(Γ†) for Γ ∈B. Proof. The result is an immediate consequence of Lemma 4.1 and the method of moments. The last sentence follows from (4.2) and (3.3). Remark 4.3. Theorem 4.2 says that the degree distribution ν is a testable graph parameter in the sense of [Borgs et al. 07b]; see in particular [Borgs et al. 07b, Section 6] (except that ν takes values in P instead of R). If Γ is represented by a function W on [0, 1]2, we can easily ﬁnd its degree distribution from W. Theorem 4.4. If W ∈Ws, then ν(ΓW ) equals the distribution of 1 0 W(U, y) dy, where U ∼U(0, 1). Similarly, in the bipartite case, if W ∈W, then ν1(Γ′′ W ) equals the distribution of 1 0 W(U, y) dy, and ν2(Γ′′ W ) equals the distribution of 1 0 W(x, U) dx. Proof. By (4.1) and (3.1), 1 0 tk dν(ΓW )(t) = t(K1,k, ΓW ) = [0,1] [0,1] W(x, y) dy k dx = E [0,1] W(U, y) dy k for every k ≥1, and the result follows. The bipartite case is similar, using (3.2). i i “imvol5” — 2009/11/4 — 9:42 — page 288 — #22 i i i i i i 288 Internet Mathematics If a graph G has n vertices, its number of edges is \|E(G)\| = 1 2 v∈V (G) d(v) = n 2 E DG = n2 2 E(DG/n) = n2 2 1 0 t dν(G)(t). Hence the edge density of G is \|E(G)\| n 2 = n n −1 1 0 t dν(G)(t). (4.3) If (Gn) is a sequence of graphs with v(Gn) →∞and Gn →Γ ∈U∞, we see from (4.3) and Theorem 4.2 that the graph densities converge to 1 0 t dν(Γ)(t), the mean of the distribution ν(Γ), which thus may be called the (edge) density of Γ ∈U∞. If Γ is represented by a function W on [0, 1]2, Theorem 4.4 yields the following. Corollary 4.5. The graph ΓW has edge density [0,1]2 W(x, y) dx dy for every W ∈Ws. Proof. By Theorem 4.4, the mean of μ(ΓW ) equals E 1 0 W(U, y) dy = 1 0 1 0 W(x, y) dx dy. 5. Limits of Threshold Graphs Recall from Section 3.5 that T∞is the set of limits of threshold graphs, and T ′′ ∞,∞ is the set of limits of bipartite threshold graphs. Our purpose in this section is to characterize the threshold graph limits, i.e., the elements of T∞and T ′′ ∞,∞, and give simple criteria for the convergence of a sequence of threshold graphs to one of these limits. We begin with some deﬁnitions. A function W : [0, 1]2 →R is increasing if W(x, y) ≤W(x′, y′) whenever 0 ≤x ≤x′ ≤1 and 0 ≤y ≤y′ ≤1. A set S ⊆[0, 1]2 is increasing if its indicator 1S is an increasing function on [0, 1]2, i.e., if (x, y) ∈S implies (x′, y′) ∈S whenever 0 ≤x ≤x′ ≤1 and 0 ≤y ≤y′ ≤1. If μ ∈P, let Fμ be its distribution function Fμ(x) := μ([0, x]), and let Fμ(x−) := μ([0, x)) be its left-continuous version. Thus Fμ(0−) = 0 ≤Fμ(0) and Fμ(1−) ≤1 = Fμ(1). Further, let F −1 μ : [0, 1] →[0, 1] be the right-continuous inverse deﬁned by F −1 μ (x) := sup{ t ≤1 : Fμ(t) ≤x }. (5.1) i i “imvol5” — 2009/11/4 — 9:42 — page 289 — #23 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 289 Note that F −1 μ (0) ≥0 and F −1 μ (1) = 1. Finally, deﬁne Sμ := (x, y) ∈[0, 1]2 : x ≥Fμ (1 −y)− . (5.2) It is easily seen that Sμ is a closed increasing subset of [0, 1]2 and that it contains the upper and right edges { (x, 1) } and { (1, y) }. Since x ≥Fμ (1 −y)− ⇐ ⇒ F −1 μ (x) ≥1 −y, we also have Sμ = (x, y) ∈[0, 1]2 : F −1 μ (x) + y ≥1 . (5.3) We further write Wμ := 1Sμ and let Γ′′ μ := Γ′′(Wμ), and when W is symmetric, Γμ := Γ(Wμ). We denote the interior of a set S by S◦. It is easily veriﬁed from (5.2) that S◦ μ = (x, y) ∈(0, 1)2 : x > Fμ(1 −y) . (5.4) Recall that the Hausdorﬀdistance between two nonempty compact subsets K1 and K2 of some metric space S is deﬁned by dH(K1, K2) := max max x∈K1 d(x, K2), max y∈K2 d(y, K1) . (5.5) This deﬁnes a metric on the set of all nonempty compact subsets of S. If S is compact, the resulting topology on the set of compact subsets of S (with the empty set as an isolated point) is compact and equals the Fell topology (see, for example, [Kallenberg 02, Appendix A.2]) on the set of all closed subsets of S. Let λd denote the Lebesgue measure in Rd. For measurable subsets S1, S2 of [0, 1]2, we also consider their measure distance λ2(S1ΔS2). This equals the L1-distance of their indicator functions, and is thus a metric modulo null sets. For functions in W we also use two diﬀerent metrics: the L1-distance [0,1]2 \|W1(x, y) −W2(x, y)\| dx dy and, in the symmetric case, the cut distance δ□deﬁned in [Borgs et al. 07b], and in the bipartite case its analogue δ′′ □; see Section 3. Note that the cut distance is only a pseudometric, since the distance of two diﬀerent functions may be 0. Note further that the cut distance is less than or equal to the L1-distance. We can now prove one of our main results, giving several related character-izations of threshold graph limits. There are two versions, since we treat the bipartite case in parallel. i i “imvol5” — 2009/11/4 — 9:42 — page 290 — #24 i i i i i i 290 Internet Mathematics 5.1. The Bipartite Case It is convenient to begin with the bipartite case. Theorem 5.1. There are bijections between the set T ′′ ∞,∞of graph limits of bipartite threshold graphs and each of the following sets: (i) The set P of probability distributions on [0, 1]. (ii) The set CB of increasing closed sets S ⊆[0, 1]2 that contain the upper and right edges [0, 1] × { 1 } ∪{ 1 } × [0, 1]. (iii) The set OB of increasing open sets S ⊆(0, 1)2. (iv) The set WB of increasing 0–1-valued functions W : [0, 1]2 →{ 0, 1 } modulo a.e. equality. More precisely, there are commuting bijections between these sets given by the following mappings and their compositions: ιBP : T ′′ ∞,∞→P, ιBP(Γ) := ν1(Γ); ιPC : P →CB, ιPC(μ) := Sμ; ιCO : CB →OB, ιCO(S) := S◦; ιCW : CB →WB, ιCW(S) := 1S; ιOW : OB →WB, ιOW(S) := 1S; ιWB : WB →T ′′ ∞,∞, ιWB(W) := Γ′′ W ; (5.6) T ′′ ∞,∞ ιBP - P ιPC - CB WB ιOW ιCW ιWB OB ιCO ? In particular, a probability distribution μ ∈P corresponds to Γ′′ μ ∈T ′′ ∞,∞and to Sμ ∈CB, S◦ μ ∈OB, and Wμ ∈WB. Conversely, Γ ∈T ′′ ∞,∞corresponds to ν1(Γ) ∈P. Thus, the mappings Γ →ν1(Γ) and μ →Γ′′ μ are the inverses of each other. Moreover, these bijections are homeomorphisms, with any of the following topologies or metrics: the standard (weak) topology on P; the Hausdorﬀmet-ric or the Fell topology or the measure distance on CB; the measure distance on OB; the L1-distance or the cut distance on the set WB. i i “imvol5” — 2009/11/4 — 9:42 — page 291 — #25 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 291 Proof. The mappings in (5.6) are all well deﬁned, except that we do not yet know that ιWB maps WB into T ′′ ∞,∞. We thus regard ιWB as a map WB →B∞∞and let B := ιWB(WB) be its image; we will identify this as T ′′ ∞,∞later. For the time being, we also regard ιBP as deﬁned on B (or on all of B∞∞). Consider ﬁrst ιPC : P →CB. By (5.2), Sμ determines Fμ at all continuity points, and thus it determines μ. Consequently, ιPC is injective. If S ∈CB and y ∈[0, 1], then { x : (x, y) ∈S } is a closed subinterval of [0, 1] that contains 1, and thus S = { (x, y) ∈[0, 1]2 : x ≥g(y) } for some function g : [0, 1] →[0, 1]. Moreover, g(1) = 0, g is decreasing, i.e., g(y2) ≤g(y1) if y1 ≤y2, and since S is closed, g is right-continuous. Thus g(1 −x) is increasing and left-continuous, and hence there exists a probability measure μ ∈P such that Fμ(x−) = g(1 −x), x ∈[0, 1]. By (5.2), then ιPC(μ) = Sμ = { (x, y) ∈[0, 1]2 : x ≥g(y) } = S. Hence ιPC is onto. Consequently, ιPC is a bijection of P onto CB. If S1 and S2 are two diﬀerent sets in CB, then there exists a point (x, y) ∈ S1 \ S2, say. There is a small open disk with center in (x, y) that does not intersect S2, and since S1 is increasing, at least a quarter of the disk is contained in S1 \ S2. Hence λ2(S1ΔS2) > 0. Similarly, if S1 and S2 are two diﬀerent sets in OB and (x, y) ∈S1 \ S2, then there is a small open disk with center in (x, y) that is contained in S1, and since S2 is increasing, at least a quarter of the disk is contained in S1 \ S2, whence λ2(S1ΔS2) > 0. This shows that the measure distance is a metric on CB and on OB, and that the mappings ιCW and ιOW into WB are injective (recall that a.e. equal functions are identiﬁed in WB). Next, let S ⊆[0, 1]2 be increasing. If (x, y) ∈S with x < 1 and y < 1, it is easily seen that (x, x + δ) × (y, y + δ) ⊆S for δ = min{ 1 −x, 1 −y }, and thus (x, x + δ) × (y, y + δ) ⊆S◦. It follows that for any real a, the intersection of the boundary ∂S := S \S◦with the diagonal line La := { (x, x+a) : x ∈R } consists of at most two points (of which one is on the boundary of [0, 1]2). In particular, λ1(∂S ∩La) = 0 and thus λ2(∂S) = 2−1/2 1 −1 λ1(∂S ∩La) da = 0. (5.7) Consequently, ∂S is a null set for every increasing S. Among other things, this shows that if S ∈CB, then ιOWιCO(S) = 1S◦= 1S a.e. Since elements of WB are deﬁned modulo a.e. equality, this shows that ιOWιCO = ιCW : CB →WB. If W ∈WB, and thus W = 1S for some increasing S ⊆[0, 1]2, let S := S ∪[0, 1] × { 1 } ∪{ 1 } × [0, 1]. (5.8) i i “imvol5” — 2009/11/4 — 9:42 — page 292 — #26 i i i i i i 292 Internet Mathematics Then S ∈CB and (5.7) imply that ιCW( S) = 1 S = 1S = W a.e. Similarly, S◦∈OB and ιOW(S◦) = 1S = W a.e. Consequently, ιCW and ιOW are onto, and thus bijections. Similarly (or as a consequence), ιCO is a bijection of CB onto OB, with inverse S → S given by (5.8). Note that the composition ιCWιPC maps μ →1Sμ = Wμ, and let ιPB be the composition ιWBιCWιPC : μ →Γ′′(Wμ) = Γ′′ μ mapping P into B∞∞. Since ιPC and ιCW are bijections, its image satisﬁes ιBP(P) = ιWB(WB) = B ⊆B∞∞. If μ ∈P, then the composition ιBPιPB(μ) = ν1(Γ′′ μ) equals by Theorem 4.4 and (5.3) the distribution of 1 0 1Sμ(U, y) dy = F −1 μ (U). (5.9) As is well known, and easy to verify using (5.1), this distribution equals μ. Hence, the composition ιBPιPB is the identity. It follows that ιPB is injective and thus a bijection of P onto its image B, and that ιBP (restricted to B) is its inverse. We have shown that all mappings in (5.6) are bijections, except that we have not yet shown that B = T ′′ ∞,∞. We next show that the mappings are homeomor-phisms. Recall that the topology on P can be deﬁned by the L´ evy metric deﬁned by (see, e.g., [Gut 05, Problem 5.25]) dL(μ1, μ2) := inf{ ε > 0 : Fμ1(x −ε) −ε ≤Fμ2(x) ≤Fμ1(x + ε) + ε for all x }. (5.10) If μ1, μ2 ∈P with dL(μ1, μ2) < ε, it follows from (5.2) and (5.10) that if (x, y) ∈ Sμ1 and x, y < 1 −ε, then Fμ2 (1 −y −ε)− ≤Fμ1 (1 −y)− + ε ≤x + ε, and thus (x + ε, y + ε) ∈Sμ2. Considering also the simple cases x ∈[1 −ε, 1] and y ∈[1 −ε, 1], it follows that if (x, y) ∈Sμ1, then d (x, y), Sμ2 ≤ √ 2 ε. Consequently, by (5.5) and symmetry, dH(Sμ1, Sμ2) ≤ √ 2 dL(μ1, μ2), which shows that ιPC is continuous if CB is given the topology given by the Hausdorﬀmetric. The same argument shows that for any (x0, y0), the intersection of the diﬀer-ence Sμ1ΔSμ2 with the diagonal line La deﬁned above is an interval of length at most √ 2 dL(μ1, μ2), and thus by integration over a as in (5.7), λ2 Sμ1ΔSμ2 ≤2dL(μ1, μ2). i i “imvol5” — 2009/11/4 — 9:42 — page 293 — #27 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 293 Hence, ιPC is continuous also if CB is given the topology given by the measure distance. Since P is compact and ιPC is a bijection, it follows that ιPC is a homeomor-phism for both these topologies on CB. In particular, these topologies coincide on CB. As remarked before the theorem, since [0, 1]2 is compact, also the Fell topologies coincide with these on CB. The bijections ιCO, ιCW, and ιOW are isometries for the measure distance on CB and OB and the L1-distance on WB, and thus homeomorphisms. Furthermore, still using the L1-distance on WB, it is easily seen from (3.2), as for the symmetric case in [Lov´ asz and Szegedy 06, Borgs et al. 07b], that for every ﬁxed bipartite graph F, the mapping W →t(F, Γ′′ W ) is continuous, which by deﬁnition of the topology in B∞∞means that ιWB : W →Γ′′ W is continuous. Hence, the bijection ιWB is a homeomorphism of the compact space WB onto its image B. As stated above, the cut distance is only a pseudometric on W. But two func-tions in W with cut distance 0 are mapped onto the same element in B∞∞, and since we have shown that ιWB is injective on WB, it follows that the restriction of the cut distance to WB is a metric. Moreover, the identity map on WB is continuous from the L1-metric to the cut metric, and since the space is compact under the former metric, the two metrics are equivalent on WB. We have shown that all mappings are homeomorphisms. It remains only to show that B = T ′′ ∞,∞. To do this, observe ﬁrst that if G is a bipartite threshold graph, and we order its vertices in each of the two vertex sets with increasing vertex degrees, then the function WG deﬁned in Section 3 is increasing and belongs thus to WB. Consequently, π(G) = ιWB(WG) ∈ B. If Γ ∈T ′′ ∞,∞, then by deﬁnition there exists a sequence Gn of bipartite threshold graphs with v1(Gn), v2(Gn) →∞such that Gn →Γ in B. This implies that π(Gn) →Γ in B∞∞, and since π(Gn) ∈ B and B is compact and thus a closed subset of B∞∞, we ﬁnd that Γ ∈ B. Conversely, if Γ ∈ B, then Γ = ιWBιCW(S) for some set S ∈CB. For each n, partition [0, 1]2 into n2 closed squares Qij of side 1/n, and let Sn be the union of all Qij that intersect S. Then Sn ∈CB, S ⊆Sn, and dH(Sn, S) ≤ √ 2/n. Let Wn := 1Sn = ιCW(Sn) and let Γn := ιWB(Wn) ∈ B. Since ιCW and ιWB are continuous, Wn →W := 1S in WB and Γn →ιWB(W) = Γ in B ⊂B∞∞. However, Wn is a step function of the form W(Gn) for some bipartite graph Gn with v1(Gn) = v2(Gn) = n, and thus π(Gn) = Γ′′ Wn = Γn. Moreover, each Sn, and thus each Wn, is increasing, and hence Gn is a bipartite threshold graph. Since π(Gn) = Γn →Γ in B∞∞, it follows that Gn →Γ in B, and thus Γ ∈T ′′ ∞,∞. Consequently, B = T ′′ ∞,∞, which completes the proof. i i “imvol5” — 2009/11/4 — 9:42 — page 294 — #28 i i i i i i 294 Internet Mathematics Remark 5.2. Another unique representation by increasing closed sets is given by the family C′ B of closed increasing subsets S of [0, 1]2 that satisfy S = S◦; there are bijections C′ B →OB and OB →C′ B given by S →S◦and S →S. We can again use the measure distance on C′ B, but not the Hausdorﬀdistance. (For example, [0, 1] × [1 −ε, 1] →∅in C′ B as ε →0.) Corollary 5.3. The degree distribution yields a homeomorphism Γ →ν1(Γ) of T ′′ ∞,∞ onto P. Of course, Γ →ν2(Γ) = ν1(Γ†) yields another homeomorphism of T ′′ ∞,∞onto P. To see the connection between these, and (more importantly) to prepare for the corresponding result in the nonbipartite case, we investigate further the reﬂection involution. If S ⊆[0, 1]2, let S† := { (x, y) : (y, x) ∈S } be the set S reﬂected in the main diagonal. Thus 1S† = 1† S. We have deﬁned the reﬂection map † for bipartite graphs and graph limits, and for the sets and functions in Theorem 5.1(ii)(iii)(iv), and it is easily seen that these correspond to each other by the bijections in Theorem 5.1. Consequently, there is a corresponding map (involution) μ →μ† of P onto itself too. This map is less intuitive than the others; to ﬁnd it explicitly, we ﬁnd from (5.2), (5.3), and Sμ† = S† μ that x ≥Fμ† (1 −y)− ⇐ ⇒(y, x) ∈Sμ ⇐ ⇒F −1 μ (y) + x ≥1 and thus Fμ† (1 −y)− = 1 −F −1 μ (y) and Fμ†(t) = 1 −F −1 μ (1 −t)− , 0 ≤t ≤1. (5.11) This means that the graph of the distribution function is reﬂected about the diagonal between (0, 1) and (1, 0) (and adjusted at the jumps). The map † is continuous on P, by Theorem 5.1 and the obvious fact that S →S† is continuous on, for example, CB. We let Ps := { μ ∈P : μ† = μ } = { μ ∈P : Sμ = S† μ } be the set of probability distributions invariant under the involution †. Since † is continuous, Ps is a closed and thus compact subset of P. Remark 5.4. If μ ∈Ps, let x0 := 1 −inf{ x : (x, x) ∈Sμ }. Then (5.2) and (5.4) imply that Fμ(x0−) ≤1 −x0 ≤Fμ(x0), and the restriction of Fμ to [0, x0) is an increasing right-continuous function with values in [0, 1−x0], and this restriction determines Fμ(t) for x ≥x0 too by (5.11). Conversely, given any x0 ∈[0, 1] and increasing right-continuous F : [0, x0) → [0, 1 −x0], there is a unique μ ∈Ps with Fμ(x) = F(x) for x < x0 and Fμ(x0) ≥ 1 −x0. i i “imvol5” — 2009/11/4 — 9:42 — page 295 — #29 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 295 5.2. Nonbipartite Case We can now state our main theorem for (nonbipartite) threshold graph limits. Theorem 5.5. There are bijections between the set T∞of graph limits of threshold graphs and each of the following sets: (i) The set Ps of probability distributions on [0, 1] symmetric with respect to †. (ii) The set CT of symmetric increasing closed sets S ⊆[0, 1]2 that contain the upper and right edges [0, 1] × { 1 } ∪{ 1 } × [0, 1]. (iii) The set OT of symmetric increasing open sets S ⊆(0, 1)2. (iv) The set WT of symmetric increasing 0–1-valued functions W : [0, 1]2 → { 0, 1 } modulo a.e. equality. More precisely, there are commuting bijections between these sets given by the following mappings and their compositions: ιT P : T∞→Ps, ιT P(Γ) := ν(Γ); ιPC : Ps →CT , ιPC(μ) := Sμ; ιCO : CT →OT , ιCO(S) := S◦; ιCW : CT →WT , ιCW(S) := 1S; ιOW : OT →WT , ιOW(S) := 1S; ιWT : WT →T∞, ιWT (W) := ΓW . (5.12) In particular, a probability distribution μ ∈Ps corresponds to Γμ ∈T∞and to Sμ ∈CT , S◦ μ ∈OT , and Wμ ∈WT . Conversely, Γ ∈T∞corresponds to ν(Γ) ∈Ps. Thus, the mappings Γ →ν(Γ) and μ →Γμ are the inverses of each other. T∞ ιT P - Ps ιPC - CT WT ιOW ιCW ιWT OT ιCO ? Moreover, these bijections are homeomorphisms, with any of the following topologies or metrics: the standard (weak) topology on Ps ⊂P; the Hausdorﬀ i i “imvol5” — 2009/11/4 — 9:42 — page 296 — #30 i i i i i i 296 Internet Mathematics metric or the Fell topology or the measure distance on CT ; the measure distance on OT ; the L1-distance or the cut distance on the set WT . These homeomorphic topological spaces are compact metric spaces. Proof. The mappings ιPC, ιCO, ιCW, ιOW are restrictions of the corresponding mappings in Theorem 5.1, and it follows from Theorem 5.1 and the deﬁnitions that these mappings are bijections and homeomorphisms for the given topologies. The spaces are closed subspaces of the corresponding spaces in Theorem 5.1, since † is continuous on these spaces, and are thus compact metric spaces. The rest is as in the proof of Theorem 5.1, and we omit some details. It follows from Theorem 4.4 that the composition ιWBιCWιPC : μ →Γ(Wμ) = Γμ is a bijection of Ps onto a subset T ′ of T ′′ ∞,∞, with ιT P as its inverse. It follows that these mappings too are homeomorphisms, and that the L1-distance and cut distance are equivalent on WT . To see that T ′ = T∞, we also follow the proof of Theorem 5.1. A minor complication is that if G ∈T is a threshold graph, and we order the vertices with increasing degrees, then WG is not increasing, because WG(x, x) = 0 for all x, since we consider loopless graphs only. However, we can deﬁne W ∗(G) by changing WG to be 1 on some squares on the diagonal so that W ∗(G) is increasing and thus W ∗(G) ∈WT , and the error ∥WG−W ∗(G)∥L1 is at most 1/v(G). If we deﬁne π∗(G) := Γ(W ∗(G)) ∈T ′, we see that if (Gn) is a sequence of threshold graphs with v(Gn) →∞, then for every graph F, by a simple estimate, see, e.g., [Lov´ asz and Szegedy 06, Lemma 4.1], we have \|t(F, π∗(Gn)) −t(F, Gn)\| ≤e(F)∥W(Gn) −W ∗(Gn)∥L1 ≤e(F)/v(Gn) →0. (5.13) It follows that Gn →Γ in U if and only if π∗(Gn) →Γ in U∞. If Γ ∈T∞, then there exists such a sequence Gn →Γ, and thus π∗(Gn) →Γ in U∞, and since π∗(Gn) ∈T ′ and T ′ is compact, we ﬁnd that Γ ∈T ′. The converse follows in the same way. If Γ ∈T ′, then Γ = ιWT (W) for some function W ∈WT . The approximating step functions Wn constructed in the proof of Theorem 5.1 are symmetric, and if we let W ∗ n be the modiﬁcation that vanishes on all diagonal squares, then W ∗ n = WGn for some threshold graph Gn, and for every graph F, t(F, Gn) = t(F, W ∗ n) = t(F, Wn) + o(1) = t(F, W) + o(1). Hence, Gn →ΓW = Γ in U, and thus Γ ∈T∞. Consequently, T ′ = T∞. Corollary 5.6. The degree distribution yields a homeomorphism Γ →ν(Γ) of T∞ onto the closed subspace Ps of P. i i “imvol5” — 2009/11/4 — 9:42 — page 297 — #31 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 297 Remark 5.7. The fact that a graph limit Γ can be represented by a function W ∈WT if and only if tind(P4, Γ) = tind(C4, Γ) = tind(2K2, Γ) = 0, which by Theorem 3.1 is equivalent to the bijection T∞↔WT in Theorem 5.5, is also proved in [Lov´ asz and Szegedy 09]. We have described the possible limits of sequences of threshold graphs; this makes it easy to see when such sequences converge. Theorem 5.8. Let Gn be a sequence of threshold graphs such that v(Gn) →∞. Then Gn converges in U as n →∞if and only if the degree distributions ν(Gn) converge to some distribution μ. In this case, μ ∈Ps and Gn →Γμ. Proof. As in the proof of Theorem 5.5, Gn →Γ if and only if π∗(Gn) →Γ in T ′ = T∞, which by Theorem 5.5 holds if and only if ν(π∗(Gn)) →ν(Γ). By Theorem 4.4, ν(π∗(Gn)) equals the distribution of 1 0 W ∗(Gn)(U, y) dy, but this random variable diﬀers by at most 1/v(Gn) = o(1) from the random variable 1 0 WGn(U, y) dy, which has degree distribution ν(Gn). The result follows. Theorem 5.9. Let Gn be a sequence of bipartite threshold graphs such that v1(Gn), v2(Gn) →∞. Then Gn converges in B as n →∞if and only if the degree distributions ν1(Gn) converge to some distribution μ. In this case, ν2(Gn) →μ† and Gn →Γ′′ μ. Proof. Gn →Γ if and only if π(Gn) →Γ in B = B∞∞, which by Theorem 5.1 holds if and only if ν1(π(Gn)) →ν1(Γ). It follows from Theorem 4.4 that ν1(π(Gn)) = ν1(Gn), and the result follows from Theorem 5.1. Remark 5.10. A threshold graph limit Γ is, by Theorem 5.5, determined by its degree distribution and the fact that it is a threshold graph limit. By The-orem 3.2 and Lemma 4.1, Γ is thus determined by t(F, Γ) for F in the set { P4, C4, 2K2, K1,1, K1,2, . . . }. It is shown in [Lov´ asz and Szegedy 09] that in some special cases, a ﬁnite set of F is enough; for example, the limit deﬁned by the function W(x, y) = 1[x + y ≥1] (see Example 1.3 and Figure 4) is the unique graph limit with t(P4, Γ) = t(C4, Γ) = t(2K2, Γ) = 0, t(K2, Γ) = 1/2, t(P3, Γ) = 1/3. i i “imvol5” — 2009/11/4 — 9:42 — page 298 — #32 i i i i i i 298 Internet Mathematics 6. Random Threshold Graphs We consider several ways to deﬁne random threshold graphs. We will consider only constructions with a ﬁxed number n of vertices; in fact, we take the vertex set to be [n] = { 1, . . . , n }, where n ≥1 is a given parameter. By a random threshold graph we thus mean a random element of Tn := { G ∈T : V (G) = [n] } for some n; we do not imply any particular construction or distribution unless otherwise stated. (We can regard these graphs as either labeled or unlabeled.) This section treats four classes of examples: a canonical example based on increasing sets, random-weights examples, random-attachment examples, and uniform random-threshold graphs. 6.1. Increasing Set For any symmetric increasing S ⊆[0, 1]2, we let W = 1S and deﬁne Tn;S := G(n, W) as in Section 3. In other words, we take i.i.d. random variables U1, . . . , Un ∼U(0, 1) and draw an edge ij if (Ui, Uj) ∈S. As stated in Section 3, G(n, W) a.s. − →ΓW , which in this case means that Tn;S a.s. − →Γ(1S) ∈T∞. We denote Γ(1S) by ΓS and have thus the following result, using also Theorem 4.4. Theorem 6.1. As n →∞, Tn;S a.s. − →ΓS. In particular, the degree distribution ν(Tn;S) converges a.s. to ν(ΓS), which equals the distribution of ϕS(U) := \|{ y : (U, y) ∈S }\| = P (U, U ′) ∈S \| U , (6.1) with U, U ′ ∼U(0, 1) independent. By Theorem 5.5, this construction gives a canonical representation of the limit objects in T∞, and we may restrict ourselves to closed or open sets as in Theorem 5.5(ii)(iii) to get a unique representation. We can obtain any desired degree distribution μ ∈Ps for the limit by choosing S = Sμ. This construction further gives a canonical representation of random threshold graphs for ﬁnite n, provided we make two natural additional assumptions. Theorem 6.2. Suppose that (Gn)∞ 1 is a sequence of random threshold graphs with V (Gn) = [n] such that the distribution of each Gn is invariant under permuta-tions of [n] and that the restriction (induced subgraph) of Gn+1 to [n] has the i i “imvol5” — 2009/11/4 — 9:42 — page 299 — #33 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 299 same distribution as Gn, for every n ≥1. If further ν(Gn) p − →μ as n →∞, for some μ ∈P, then μ ∈Ps, and for every n, Gn d = Tn;Sμ. Proof. It follows from Theorem 5.8 that Gn p − →Γμ. (To apply Theorem 5.8 to convergence in probability, we can use the standard trick of considering subse-quences that converge a.e., since every subsequence has such a subsubsequence [Kallenberg 02, Lemma 4.2].) If we represent a graph by its edge indicators, the random graph Gn can be regarded as a family of 0–1-valued random variables indexed by pairs (i, j), 1 ≤i < j ≤n. By assumption, these families for diﬀerent n are consistent, so by the Kolmogorov extension theorem [Kallenberg 02, Theorem 6.16], they can be deﬁned for all n together, which means that there exists a random inﬁnite graph G∞with vertex set N whose restriction to [n] coincides (in distribution) with Gn. Moreover, since each Gn is invariant under permutations of the vertices, so is G∞, i.e., G∞is exchangeable. By the representation theorem by Aldous and Hoover (see [Aldous 85], [Kallen-berg 05], and [Diaconis and Janson 08]), every exchangeable random inﬁnite graph can be obtained as a mixture of G(∞, W); in other words, as G(∞, W) for some random function W ∈Ws. In this case, the subgraphs Gn converge in probability to the corresponding random ΓW ; see [Diaconis and Janson 08]. Since we have shown that the Gn converge to a deterministic graph limit Γμ, we can take W deterministic, so it follows that G∞ d = G(∞, W) for some W ∈Ws; moreover, Γμ = ΓW , and thus by Theorem 5.5 we can choose W = Wμ. (Recall that in general, W is not unique.) Consequently, Gn d = G(n, Wμ) = Tn;Sμ, which completes the proof. 6.2. Random Weights Deﬁnition (1.1) suggests immediately the construction (1.6): Let X1, X2, . . . , be i.i.d. copies of a random variable X, let t ∈R, and let Tn;X,t be the threshold graph with vertex set [n] and edges ij for all pairs ij such that Xi +Xj > t. (We can without loss of generality let t = 0, by replacing X by X −t/2.) Examples 1.2 and 1.3 are in this mode. Let F(x) := P(X ≤x) be the distribution function of X, and let F −1 be its right-continuous inverse deﬁned by F −1(u) := sup{ x ∈R : F(x) ≤u } (6.2) i i “imvol5” — 2009/11/4 — 9:42 — page 300 — #34 i i i i i i 300 Internet Mathematics (cf. (5.1), where we consider distributions on [0, 1] only). Thus −∞< F −1(u) < ∞if 0 < u < 1, while F −1(1) = ∞. It is well known that the random variables Xi can be constructed as F −1(Ui) with Ui independent uniformly distributed random variables on (0, 1), which leads to the following theorem, showing that this construction is equivalent to the one in Section 6.1 for a suitable set S. Parts of this theorem were found earlier; see [Konno et al. 05, Masuda et al. 05]. Theorem 6.3. Let S be the symmetric increasing set S := { (x, y) ∈(0, 1]2 : F −1(x) + F −1(y) > t }. (6.3) Then Tn;X,t d = Tn;S for every n. Furthermore, as n →∞, the degree distribution ν(Tn;X,t) converges a.s. to μ, and thus Tn;X,t a.s. − →Γμ, where μ ∈Ps is the distribution of the random variable 1 −F(t −X), i.e., μ[0, s] = P 1 −F(t −X) ≤s , s ∈[0, 1]. (6.4) Proof. Taking Xi = F −1(Ui), we see that there is an edge ij ⇐ ⇒F −1(Ui) + F −1(Uj) > t ⇐ ⇒(Ui, Uj) ∈S, which shows that Tn;X,t = Tn;S. The remaining assertions now follow from Theorem 6.1 together with the cal-culation, with U, U ′ ∼U(0, 1) independent and X = F −1(U), X′ = F −1(U ′), ϕS(U) = P (U, U ′) ∈S \| U = P F −1(U) + F −1(U ′) > t \| U = P X + X′ > t \| X = P X′ > t −X \| X = 1 −F(t −X). The set S deﬁned in (6.3) is in general neither open nor closed; the corre-sponding open set is S◦= (x, y) ∈(0, 1)2 : F −1(x−) + F −1(y−) > t , and the corresponding closed set Sμ in Theorem 5.5 can be found as ˜ S◦from (5.8). If we assume for simplicity that the distribution of X is continuous, then, as is easily veriﬁed, Sμ = (x, y) ∈[0, 1]2 : F −1(x) + F −1(y) ≥t , where we deﬁne F −1(1) = ∞(and interpret ∞+ (−∞) = ∞in case F −1(0) = −∞). We can use these sets instead of S in (6.3), since they diﬀer by null sets only. i i “imvol5” — 2009/11/4 — 9:42 — page 301 — #35 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 301 6.3. Random Addition of Vertices Preferential attachment graphs are a rich topic of research in modern graph theory. See the monograph [Lu and Chung 06], along with the survey [Mitzen-macher 04]. The versions in this section are natural because of (1.2) and the construction (1.7). Let Tn,p be the random threshold graph with n vertices obtained by adding vertices one by one with the new vertices chosen as isolated or dominant at random, independently of each other and with a given probability p ∈[0, 1] of being dominant. (Starting with a single vertex, there are thus n −1 vertex additions.) The vertices are not equivalent (for example, note that the edges 1i, i ̸= 1, appear independently, but not the edges ni, i ̸= n), so we also deﬁne the random threshold graph Tn,p obtained by a random permutation of the vertices in Tn,p. (In considering unlabeled graphs, there is no diﬀerence between Tn,p and Tn,p.) Remark 6.4. We may, as stated in (1.7), use diﬀerent probabilities pi for diﬀerent vertices. We leave it to the reader to explore this case, for example with pi = f(i/n) for some given continuous function f : [0, 1] →[0, 1]. Theorem 6.5. The degree distribution ν(Tn,p) converges a.s. as n →∞to a distribu-tion μp that for 0 < p < 1, has constant density (1 −p)/p on (0, p) and p/(1 −p) on (p, 1); μ0 is a point mass at 0, and μ1 is a point mass at 1. In particular, μ1/2 is the uniform distribution on [0, 1]. Consequently, Tn,p a.s. − →Γμp ∈T∞. Proof. Let Zn(t) be the number of vertices in { 1, . . . , ⌊nt⌋} that are added as dominant. It follows from the law of large numbers that n−1Zn(t) a.s. − →pt, uniformly on [0, 1], and we assume this in the sequel of the proof. If vertex k was added as isolated, it has degree Zn(1) −Zn(k/n), since its neighbors are the vertices that later are added as dominant. Similarly, if vertex k was added as dominant, it has degree k −1 + Zn(1) −Zn(k/n). Consequently, if μn is the (normalized) degree distribution of Tn,p, and φ is any continuous function on [0, 1], then 1 0 φ(t) dμn(t) = 1 n n k=1 φ(d(k)/n) = 1 n n k=1 φ n−1Zn(1) −n−1Zn(k/n) 1[ΔZn(k/n) = 0] + φ n−1Zn(1) −n−1Zn(k/n) + (k −1)/n 1[ΔZn(k/n) = 1] . i i “imvol5” — 2009/11/4 — 9:42 — page 302 — #36 i i i i i i 302 Internet Mathematics Since n−1Zn(t) →pt uniformly, and φ is uniformly continuous, it follows that as n →∞, 1 0 φ(t) dμn(t) = 1 n n k=1 φ p(1 −k/n) 1[ΔZn(k/n) = 0] + φ p(1 −k/n) + k/n 1[ΔZn(k/n) = 1] + o(1) = 1 0 φ p(1 −t) d n−1⌊nt⌋−n−1Zn(t) + 1 0 φ p(1 −t) + t d n−1Zn(t) + o(1). Since the convergence n−1Zn(t) →pt implies (weak) convergence of the corre-sponding measures, we ﬁnally obtain, as n →∞, 1 0 φ(t) dμn(t) → 1 0 φ p(1 −t) (1 −p) dt + 1 0 φ p(1 −t) + t p dt = 1 −p p p 0 φ(x) dx + p 1 −p 1 p φ(x) dx = 1 0 φ(x) dμp(x), with obvious modiﬁcations if p = 0 or p = 1. Let Sp := Sμp be the corresponding subset of [0, 1]2. If 0 < p < 1, μp has the distribution function Fμp(x) = 1−p p x, 0 ≤x ≤p, 1 − p 1−p(1 −x), p ≤x ≤1, (6.5) then it follows from (5.2) that Sp is the quadrilateral with vertices (0, 1), (1 −p, 1 −p), (1, 0), and (1, 1); see Figure 8. In the special case p = 1/2, μp is the uniform distribution on [0, 1], and Sp is the triangle { (x, y) ∈[0, 1]2 : x+ y ≥1 } pictured in Figure 4 with vertices (0, 1), (1, 0), and (1, 1). Finally, S0 consists of the upper and right edges only, and S1 = [0, 1]2. Removing any vertex from Tn,p (and relabeling the remaining ones) yields Tn−1,p. It follows that the same property holds for Tn,p, so Tn,p satisﬁes the assumptions of Theorem 6.2. Since Tn,p has the same degree distribution as Tn,p, Theorems 6.2 and 6.5 yield the following equality. Corollary 6.6. If 0 ≤p ≤1 and n ≥1, then Tn,p d = Tn;Sp. i i “imvol5” — 2009/11/4 — 9:42 — page 303 — #37 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 303 1 1 0 1 1 0 Figure 8. Two examples of the sets Sp; the one on the right shows the special case p = 0.5. Hence the random threshold graphs in this subsection are special cases of the general construction in Section 6.1. We can also construct them using random weights as in Section 6.2. Corollary 6.7. If 0 ≤p ≤1 and n ≥1, then Tn,p d = Tn;X,0, where X has the density 1 −p on (−1, 0) and p on (0, 1). Proof. A simple calculation shows that the set S given by (6.3) is the quadrilat-eral Sp. We may transform X by a linear map; for example, we may equivalently take X with density 2(1 −p) on (0, 1/2) and 2p on (1/2, 1), with the threshold t = 1. In particular, Tn,1/2 d = Tn;U,1, where U ∼U(0, 1) as in Example 1.3. 6.4. Uniform Random Threshold Graphs Let Tn be a random unlabeled threshold graph of order n with the uniform distribution studied in Section 2. Similarly, let T L n be a random labeled threshold graph of order n with the uniform distribution. Although Tn and T L n have diﬀerent distributions, see Section 2, the next theorem shows that they have the same limit as n →∞. Theorem 6.8. The degree distributions ν(Tn) and ν(T L n ) both converge in probability to the uniform distribution λ on [0, 1]. Hence Tn p − →Γλ and T L n p − →Γλ. By Section 2.1, Tn d = Tn,1/2; hence the result for unlabeled graphs follows from Theorem 6.5. i i “imvol5” — 2009/11/4 — 9:42 — page 304 — #38 i i i i i i 304 Internet Mathematics Proof. We use Theorem 2.3; in fact, the proof works for random threshold graphs generated by Algorithm 3 for any i.i.d. random variables B2, B3, . . . with ﬁnite mean, and any B1. (In the case that B2 is always a multiple of some d > 1, there is a trivial modiﬁcation.) Let β := E B2. The algorithm starts by choosing (random) block lengths B1, B2, . . . until their sum is at least n, and then rejects them and restarts (step 3) unless the sum is exactly n. It is simpler to ignore this check, so we consider the following modiﬁed algorithm: Take B1, B2, . . . as above. Let Sk := k j=1 Bj be their partial sums and let τ(n) := min{ k : Sk ≥n }. Toss a coin to determine whether the ﬁrst block is isolated or dominant, and construct a random threshold graph by adding τ(n) blocks of vertices with B1, . . . , Bτ(n) elements, alternately isolated and dominant. This gives a random graph Gn with Sτ(n) vertices, but conditioned on Sτ(n) = n; we obtain the desired random threshold graph (cf. Theorem 2.3). Since P(Sτ(n) = n) converges to 1/β > 0 by renewal theory, it suﬃces to prove that ν( Gn) p − →λ as n →∞. In fact, we will show that ν( Gn) a.s. − →λ if we ﬁrst choose an inﬁnite sequence B1, B2, . . . and then let n →∞. Let SO m := 2k+1≤m B2k+1 and SE m := 2k≤m B2k be the partial sums of the odd and even terms. By the law of large numbers, a.s. Sn/n →β and SO n /n →1 2β, SE n /n →1 2β. We now consider a ﬁxed sequence (Bj)∞ 1 such that these limits hold. Since Sτ(n)−1 < n ≤Sτ(n), it follows, as is well known, that n/τ(n) →β, so τ(n) = n/β + o(n). Suppose for deﬁniteness that the ﬁrst block is chosen to be isolated; then every odd block is isolated and every even block is dominant. (In the opposite case, interchange even and odd below.) If i ∈(S2k, S2k+1], then i belongs to block 2k + 1, so i is added as isolated, and the neighbors of i will be only the vertices added after i as dominant, i.e., k<ℓ≤τ(n)/2(S2ℓ−1, S2ℓ], and d(i) = 2k<2ℓ≤τ(n) B2ℓ= SE τ(n) −SE τ(i). If instead i ∈(S2k−1, S2k], then i is also joined to all vertices up to S2k, and thus d(i) = 2ℓ≤τ(n) B2ℓ+ 2ℓ+1≤τ(i) B2ℓ+1 = SE τ(n) + SO τ(i). Hence if i is in an odd block, d(i) n = 1 n τ(n)β 2 −τ(i)β 2 + o(n) = n −i + o(n) 2n = 1 2 −i 2n + o(1), and if i is in an even block, similarly, d(i) n = 1 2 + i 2n + o(1). i i “imvol5” — 2009/11/4 — 9:42 — page 305 — #39 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 305 Now ﬁx t ∈(0, 1/2) and let ε > 0. Then the following holds if n is large enough: If i is in an even block, then d(i)/n ≥1/2 + o(1) > t. If i is in an odd block and i ≤i1 := (1−2t−2ε)n, then d(i)/n = 1 2(n−i)/n+o(1) ≥t+ε+o(1) > t. If i is in an odd block and i ≥i2 := (1 −2t + 2ε)n, then d(i)/n = 1 2(n −i)/n + o(1) ≤ t −ε + o(1) < t. Consequently, for large n, d(i)/n ≤t only if i is in an odd block (S2k, S2k+1], and in this case 2k + 1 > τ(i1) is necessary and 2k + 1 > τ(i2) is suﬃcient. Hence SO τ(n) −SO τ(i2) ≤\|{ i : d(i)/n ≤t }\| ≤SO τ(n) −SO τ(i1). Since ν( Gn)[0, t] = 1 n\|{ i : d(i)/n ≤t }\| and 1 n SO τ(n) −SO τ(ij) = β(τ(n) −τ(ij)) + o(n) 2n = n −ij + o(n) 2n = t ± ε + o(1), it follows that t −ε + o(1) ≤ν( Gn)[0, t] ≤t + ε + o(1). Since ε is arbitrary, this shows that ν( Gn)[0, t] →t, for every t ∈(0, 1 2). We clearly obtain the same result if the ﬁrst block is dominant. For t ∈( 1 2, 1) we can argue similarly, now analyzing the dominant blocks. Alternatively, we may apply the result just obtained to the complement of Gn, which is obtained from the same Bj by switching the types of the blocks. This shows that ν( Gn)[0, t] →t for t ∈( 1 2, 1) too. Hence, ν( Gn)[0, t] →t for every t ∈(0, 1) except possibly 1 2, which shows that ν( Gn) →λ. 7. Vertex Degrees in Uniform Random Threshold Graphs We have seen in Theorem 6.8 that the normalized degree distributions ν(Tn) and ν(T L n ) for uniform unlabeled and labeled random threshold graphs both converge to the uniform distribution on [0, 1]. This is for weak convergence of distributions in P, which is equivalent to averaging over degrees in intervals (an, bn); we here reﬁne this by studying individual degrees. Let Nd(G) be the number of vertices of degree d in the graph G. Thus DG, the degree of a random vertex in G, has distribution P(DG = d) = Nd/v(G). (Recall that ν(G) is the distribution of DG/v(G); see Section 4.) We will study the random variables Nd(Tn) and Nd(T L n ) describing the num-bers of vertices of a given degree d in a uniform random unlabeled or labeled threshold graph, and in particular their expectations E Nd(Tn) and E Nd(T L n ); i i “imvol5” — 2009/11/4 — 9:42 — page 306 — #40 i i i i i i 306 Internet Mathematics note that E Nd(Tn)/n and E Nd(T L n )/n are the probabilities that a given (or ran-dom) vertex in the random graph Tn or T L n has degree d. By symmetry under complementation, Nd(Tn) d = Nn−1−d(Tn) and Nd(T L n ) d = Nn−1−d(T L n ). Let us ﬁrst look at N0, the number of isolated vertices. (By symmetry, we have the same results for Nn−1, the number of dominant vertices). Note that for every n ≥2, P(N0(Tn) = 0) = P(N0(T L n ) = 0) = 1/2 by symmetry. Theorem 7.1. (i) For any n ≥1, P N0(Tn) = j = ⎧ ⎪ ⎨ ⎪ ⎩ 2−j−1, 0 ≤j ≤n −2, 0, j = n −1, 2−n+1, j = n. (7.1) In other words, if X ∼Ge(1/2), then N0(Tn) d = X′ n, where X′ n := Xn if x < n −1 and X′ n := n if Xn ≥n −1. Furthermore, E N0(Tn) = 1, and N0(Tn) d − →Ge(1/2) as n →∞, with convergence of all moments. (ii) P N0(T L n ) = j = t(n, j)/t(n), where t(n, j) is given by (2.6); in particular, if 0 ≤j ≤n −2, then P N0(T L n ) = j = 1 2j! t(n −j)/(n −j)! t(n)/n! = 1 2j!(log 2)j 1 + O(ρn−j) with ρ = log 2/(2π) ≈0.11. Hence N0(T L n ) d − →Po(log 2) as n →∞with convergence of all moments; in particular, E N0(T L n ) →log 2. Proof. (i): A threshold graph has j isolated vertices if and only if the extended binary code α1 · · · αn in Section 2 ends with exactly j 0’s. For a random unlabeled threshold graph Tn, the binary code α2 · · · αn is uniformly distributed, and thus (7.1) follows. The remaining assertions follow directly. (ii): In the labeled case, the exact distribution is given by (2.6), and the asymp-totics follow by (2.3). Uniform integrability of any power N0(T L n )m follows by the same estimates, and thus moment convergence holds. For higher degrees, we begin with an exact result for the unlabeled case. Theorem 7.2. E Nd(Tn) = 1 for every d = 0, . . . , n −1. i i “imvol5” — 2009/11/4 — 9:42 — page 307 — #41 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 307 Actually, this is the special case p = 1/2 of a more general theorem for the random threshold graph Tn,p deﬁned in Section 6.3 (cf. Theorem 6.5, which is for weak convergence, but on the other hand yields an a.s. limit while we here study the expectations). Theorem 7.3. Let 0 < p < 1. If q = 1−p and X ∼Bin(n, p), then for 0 ≤d ≤n−1, E Nd(Tn,p) = q p + p q −q p P(X ≤d). Proof. We use the deﬁnition in Section 6.3. (For the uniform case p = 1/2, this is Algorithm 1.) Let di be the degree of vertex i. Then if α1 · · · αn is the extended binary code of the graph, we have di = (i −1)αi + n j=i+1 αj. Since the αi are i.i.d. Be(p) for i = 2, . . . , n, the probability-generating function of di is E xdi = E x(i−1)αi n j=i+1 E xαj = (pxi−1 + q)(px + q)n−i. Consequently, d E Nd(Tn,p)xd = n i=1 E xdi = n i=1 pxi−1(px + q)n−i + n i=1 q(px + q)n−i = pxn −(px + q)n x −(px + q) + q 1 −(px + q)n 1 −(px + q) = (q/p) + (p/q −q/p)(px + q)n −(p/q)xn 1 −x . In the special case p = 1/2, this is (1 −xn)/(1 −x) = n−1 d=0 xd, which proves Theorem 7.2 by identifying coeﬃcients. For general p, Theorem 7.3 follows in the same way. Recall that Rd denotes the number of preferential arrangements, or surjection numbers, given in (2.4). Theorem 7.4. (i) In the unlabeled case, for any sequence d = d(n) with 0 ≤d ≤n −1, Nd(Tn) d − →Ge(1/2) with convergence of all moments. i i “imvol5” — 2009/11/4 — 9:42 — page 308 — #42 i i i i i i 308 Internet Mathematics (ii) In the labeled case, let Xd, 0 ≤d ≤∞, have the modiﬁed Poisson distribu-tion given by P(Xd = ℓ) = γd log 2 P Po(log 2) = ℓ = γd (log 2)ℓ−1 2·ℓ! , ℓ≥1, 1 − γd 2 log 2, ℓ= 0, where γ0 := log 2, γd := 2Rd(log 2)d+1/d! for d ≥1, and γ∞:= 1. Then for every ﬁxed d ≥0, Nd(T L n ) d = Nn−1−d(T L n ) d − →Xd, and for every sequence d = d(n) →∞with n −d →∞, Nd(T L n ) d = Nn−1−d(T L n ) d − →X∞as n →∞, in both cases with convergence of all moments. In particular, E Nd(T L n ) = E Nn−1−d(T L n ) converges to γd for every ﬁxed d, and to γ∞= 1 if d →∞and n −d →∞. In the labeled case we thus have, in particular, E N0(T L n ) →log 2 ≈0.69315, E N1(T L n ) →2(log 2)2 ≈0.96091, E N2(T L n ) →3(log 2)3 ≈0.99907, E N3(T L n ) →13 3 (log 2)4 ≈1.00028. The values for degrees 0 and 1 (and symmetrically n −1 and n −2) are thus substantially smaller than 1, which is clearly seen in Figure 7. (We can regard this as an edge eﬀect; the vertices with degrees close to 0 or n −1 are those added last in Algorithm 3. Figure 7 also shows an edge eﬀect at the other side; there is a small bump for degrees around n/2, which correspond to the vertices added very early in the algorithm; this bump vanishes asymptotically, as shown by Theorem 7.4; we believe that it has height of order n−1/2 and width of order n1/2, but we have not analyzed it in detail.) Proof. The cases d = 0 and d = n −1 follow from Theorem 7.1. We may thus suppose 1 ≤d ≤n −2. We use Algorithm 3. We know that vertices in each block have the same degree, while diﬀerent blocks have diﬀerent degrees; thus there is at most one block with degree d. Let pd(ℓ) be the probability that there is such a block of length ℓ≥1, and that this block is added as isolated. By symmetry, the probability that there is a dominant block of length ℓwith degree d is pn−1−d, and thus P(Nd = ℓ) = pd(ℓ) + pn−1−d(ℓ), ℓ≥1. (7.2) i i “imvol5” — 2009/11/4 — 9:42 — page 309 — #43 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 309 If block j is an isolated block, then the degree of the vertices in it equals the number of vertices added as dominant after it, i.e., Bj+1 +Bj+3 +· · ·+Bj+2k−1, if the total number τ of blocks is j + 2k −1 or j + 2k. Consequently, there is an isolated block of length ℓwith vertices of degree d if and only if there exist j ≥1 and k ≥1 with • Bj = ℓ, • block j is isolated, • k i=1 Bj+2i−1 = d, • j+2k−1 i=1 Bi = n or j+2k i=1 Bi = n. Recall that B1, B2, . . . are independent and that B2, B3, . . . have the same dis-tribution, while B1 has a diﬀerent one. (The distributions diﬀer between the unlabeled and labeled cases.) Let ˆ Sn := m i=1 Bi and Sn := m i=1 Bi+1. Further, let ˆ u(n) = ∞ m=0 P( ˆ Sm = n) = P(Bτ = n) = P τ i=1 Bi = n , u(n) = ∞ m=0 P(Sm = n), and recall that u(n), ˆ u(n) →1/μ := 1/ E B2 (exponentially fast) by standard renewal theory (for example by considering generating functions). For any m ≥ j + 2k −1, m i=1 Bi −Bj − k i=1 Bj+2i−1 d = Sm−1−k, j = 1, ˆ Sm−1−k, j ≥2, and it follows that since B1, B2, . . . are independent and we condition on ˆ Sτ = n, pd(ℓ) = 1 2ˆ u(n) ∞ k=1 P(B1 = ℓ) P(Sk = d) · P(Sk−1 = n −ℓ−d) + P(Sk = n −ℓ−d) + ∞ j=2 ∞ k=1 P(Bj = ℓ) P(Sk = d) · P( ˆ Sj+k−2 = n −ℓ−d) + P( ˆ Sj+k−1 = n −ℓ−d) (7.3) i i “imvol5” — 2009/11/4 — 9:42 — page 310 — #44 i i i i i i 310 Internet Mathematics In the double sum, P(Bj = ℓ) = P(B2 = ℓ) does not depend on j, so the sum is at most P(B2 = ℓ) k P(Sk = d)2ˆ u(n −ℓ−d) = 2 P(B2 = ℓ)u(d)ˆ u(n −ℓ−d) = O(P(B2 = ℓ)). Similarly, the ﬁrst sum is O(P(B1 = ℓ)) = O(P(B2 = ℓ)), and it follows that pd(ℓ) = O(P(B2 = ℓ)) and thus, by (7.2), P(Nd = ℓ) = O(P(B2 = ℓ)), (7.4) uniformly in n, d, and ℓ. This shows tightness, so convergence P(Nd = ℓ) → P(X = ℓ) for some nonnegative-integer-valued random variable X and each ﬁxed ℓ≥1 implies convergence in distribution (i.e., for ℓ= 0 too). Further, since all moments of B2 are ﬁnite, (7.4) implies that all moments E N m d are bounded, uniformly in d and n; hence convergence in distribution implies that all moments converge too. In the rest of the proof we thus let ℓ≥1 be ﬁxed. If d ≤n/2, it is easy to see that P(Sk−1 = n −ℓ−d) + P(Sk = n −ℓ−d) = O (n −ℓ−d)−1/2 = O(n−1/2), uniformly in k, so the ﬁrst sum in (7.3) is O n−1/2u(d) = O n−1/2 . If d > n/2, we similarly have P(Sk = d) = O d−1/2 = O n−1/2 , and thus the sum is O n−1/2u(n −ℓ−d) = O n−1/2 . Hence (7.3) yields pd(ℓ) = O n−1/2 + P(B2 = ℓ) 2ˆ u(n) ∞ k=1 P(Sk = d) (7.5) · i=k P( ˆ Si = n −ℓ−d) + i=k+1 P( ˆ Si = n −ℓ−d) . The term with i = k can be taken twice, just as those with i > k, since k P(Sk = d) P( ˆ Sk = n −ℓ−d) = O n−1/2 by the same argument as for the ﬁrst sum in (7.3). Further, for i ≥k, ˆ Si −ˆ Sk d = Si−k, and this is independent of ˆ Sk; thus P( ˆ Si = n −ℓ−d) = P(Si−k = n −ℓ−d −ˆ Sk) = E P Si−k = n −ℓ−d −ˆ Sk \| ˆ Sk and ∞ i=k P( ˆ Si = n −ℓ−d) = E u(n −ℓ−d −ˆ Sk). i i “imvol5” — 2009/11/4 — 9:42 — page 311 — #45 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 311 Hence (7.5) yields pd(ℓ) = P(B2 = ℓ) ˆ u(n) ∞ k=1 P(Sk = d) E u(n −ℓ−d −ˆ Sk) + O n−1/2 . (7.6) If d is ﬁxed, then E u(n −ℓ−d −ˆ Sk) →μ−1 by dominated convergence as n →∞for each k, and thus (7.6) yields, by dominated convergence again, pd(ℓ) →P(B2 = ℓ) ∞ k=1 P(Sk = d) = u(d) P(B2 = ℓ). (7.7) If d →∞, we use the fact that u(m) −1[m ≥0]μ−1 is summable over Z to see that E u(n −ℓ−d −ˆ Sk) −μ−1 P(n −ℓ−d −ˆ Sk ≥0) = O max m P( ˆ Sk = m) , which tends to 0 as k →∞; on the other hand, P(Sk = d) →0 for every ﬁxed k. It follows that (7.6) yields pd(ℓ) = P(B2 = ℓ) ∞ k=1 P(Sk = d) P( ˆ Sk ≤n −ℓ−d) + o(1). If τd := min{ k : Sk ≥d } and ˆ S′ k denotes a copy of ˆ Sk independent of { Sj }∞ 1 , then ∞ k=1 P(Sk = d) P( ˆ Sk ≤n −ℓ−d) = u(d) P ˆ S′ τd ≤n −ℓ−d \| Sτd = d . It is easy to see that with σ2 := Var(B2), as d →∞, ( ˆ S′ τd −d)/ √ d \| Sτd = d = ( ˆ S′ τd −Sτd)/ √ d \| Sτd = d d − →N(0, 2σ2/μ); cf. [Gut and Janson 83] (the extra conditioning on Sτd = d makes no diﬀerence). Hence when d →∞, pd(ℓ) = P(B2 = ℓ)u(d)Φ (n −ℓ−2d)/ √ d + o(1). (By (7.7), this holds for ﬁxed d too.) We next observe that Φ (n −ℓ−2d)/ √ d = Φ (n −2d)/ n/2 + o(1); this is easily seen by considering separately the three cases d/n →a ∈[0, 1/2), d/n →a ∈(1/2, 1], and d/n →1/2 and (n −2d)/ n/2 →b ∈[−∞, ∞] (the i i “imvol5” — 2009/11/4 — 9:42 — page 312 — #46 i i i i i i 312 Internet Mathematics general case follows by considering suitable subsequences). Hence, we have when d →∞, recalling that then u(d) →μ−1, pd(ℓ) = μ−1 P(B2 = ℓ)Φ (n −2d)/ n/2 + o(1). For ﬁxed d, this implies that pn−d−1(ℓ) →0, and thus (7.2) and (7.7) yield P(Nd = ℓ) = pd(ℓ) + pn−1−d(ℓ) = u(d) P(B2 = ℓ) + o(1). Similarly, if d →∞and n −d →∞, then P(Nd = ℓ) = pd(ℓ) + pn−1−d(ℓ) = μ−1 P(B2 = ℓ) Φ (n −2d)/ n/2 + Φ (2d + 2 −n)/ n/2 + o(1) = μ−1 P(B2 = ℓ) + o(1). We have thus proven convergence as n →∞, with all moments, Nd d − →Xd for ﬁxed d and Nd d − →X∞for d = d(n) →∞with n −d →∞, where P(Xd = ℓ) = u(d) P(B2 = ℓ) = 2u(d) P(B∗= ℓ), ℓ≥1, (7.8) P(Xd = 0) = 1 −P(Xd ≥1) = 1 −u(d), (7.9) for 1 ≤d ≤∞, with u(∞) := μ−1. In the unlabeled case, B2 = (B∗\| B∗≥1) d = B∗+ 1 with B∗∼Ge(1/2). Consider a random inﬁnite string α1α2 · · · of i.i.d. Be(1/2) binary digits, and deﬁne a block as a string of m ≥0 0’s followed by a single 1. Then Bj+1, j ≥1, can be interpreted as the successive block lengths in α1α2 · · · , and thus u(d) is the probability that some block ends at d, i.e., u(d) = P(αd = 1) = 1/2, for every d ≥1. It follows from (7.8)–(7.9) that Xd d = B∗∼Ge(1/2) for every d ≥1, and (i) follows. In the labeled case, when B∗∼Po(log 2), we use generating functions: ∞ d=0 u(d)xd = ∞ k=0 E xSk = ∞ k=0 E xB2k = 1 1 −E xB2 = P(B∗≥1) 1 −E xB∗ = 1/2 1 −e(x−1) log 2 = 1 2 −ex log 2 = ∞ d=0 Rd d! (x log 2)d, where we recognize the generating function (2.5). Thus, u(d) = Rd(log 2)d/d!. (A direct combinatorial proof of this is also easy.) We let, using μ := E B2 = E B∗/ P(B∗≥1) = 2 log 2, γd := E Xd = u(d) E B2 = μu(d) = 2 log 2u(d) = 2Rd(log 2)d+1/d! i i “imvol5” — 2009/11/4 — 9:42 — page 313 — #47 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 313 and note that γd →γ∞= 1 as d →∞, since u(d) →μ−1, or by the known asymptotics of Rd [Flajolet and Sedgewick 09, (II.16)]. The description of Xd in the statement now follows from (7.8)–(7.9). 8. Random Bipartite Threshold Graphs The constructions and results in Section 6 have analogues for bipartite threshold graphs. The proofs are simple modiﬁcations of those above and are omitted. 8.1. Increasing Set For any increasing S ⊆[0, 1]2, deﬁne Tn1,n2;S := G(n1, n2, 1S). In other words, take i.i.d. random variables U ′ 1, . . . , U ′ n1, U ′′ 1 , . . . , U ′′ n2 ∼U(0, 1) and draw an edge ij if (U ′ i, U ′′ j ) ∈S. Theorem 8.1. As n1, n2 →∞, Tn1,n2;S a.s. − →Γ′′ S. In particular, the degree distribution ν1(Tn;S) converges a.s. to ν1(Γ′′ S), which equals the distribution of ϕS(U) deﬁned by (6.1). As in Section 6, this gives a canonical representation of random bipartite threshold graphs under natural assumptions. Theorem 8.2. Suppose that (Gn1,n2)n1,n2≥1 are random bipartite threshold graphs with V1(Gn1,n2) = [n1] and V2(Gn1,n2) = [n2] such that the distribution of each Gn1,n2 is invariant under permutations of V1 and V2 and that the restrictions (in-duced subgraphs) of Gn1+1,n2 and Gn1,n2+1 to V (G) both have the same distribu-tion as Gn1,n2, for every n1, n2 ≥1. If further ν1(Gn1,n2) p − →μ as n1, n2 →∞, for some μ ∈P, then for every n1, n2, Gn1,n2 d = Tn1,n2;Sμ. 8.2. Random Weights Deﬁnition (1.9) suggests the following construction: (8.1) Let X and Y be two random variables and let t ∈R. Let X1, X2, . . . , be copies of X, let Y1, Y2, . . . , be copies of Y , all independent, and let Tn1,n2;X,Y,t be the bipartite threshold graph with vertex sets [n1] and [n2] and edges ij for all pairs ij such that Xi + Yj > t. Theorem 8.3. Let S be the increasing set S := { (x, y) ∈(0, 1]2 : F −1 X (x) + F −1 Y (y) > t }. (8.2) i i “imvol5” — 2009/11/4 — 9:42 — page 314 — #48 i i i i i i 314 Internet Mathematics Then Tn1,n2;X,Y,t d = Tn1,n2;S for every n1, n2 ≥1. Furthermore, as n1, n2 →∞, the degree distribution ν1(Tn1,n2;X,Y,t) converges a.s. to μ and thus Tn1,n2;X,Y,t a.s. − →Γ′′ μ, where μ ∈P is the distribution of the random variable 1 −FY (t −X), i.e., μ[0, s] = P 1 −FY (t −X) ≤s , s ∈[0, 1]. (8.3) In the special case that P(X ∈[0, 1]) = 1, Y ∼U(0, 1), and t = 1, (8.3) yields μ[0, s] = P(X ≤s), so μ is the distribution of X; further, the set S in (8.2) is a.e. equal to Sμ in (5.3). Corollary 8.4. If μ ∈Ps, let X have distribution μ and let Y ∼U(0, 1). Then Tn1,n2;X,Y,t d = Tn1,n2;Sμ for every n1, n2 ≥1. Furthermore, as n1, n2 →∞, ν1(Tn1,n2;X,Y,t) p − →μ and Tn1,n2;X,Y,t p − →Γ′′ μ. This yields another canonical construction for every μ ∈P. (We claim only convergence in probability in Corollary 8.4; convergence a.s. holds at least along every increasing subsequence (n1(m), n2(m)); see [Diaconis and Janson 08, Re-mark 8.2].) 8.3. Random Addition of Vertices Deﬁnition (1.10) suggests the following construction: (8.4) Let Tn1,n2;p1,p2 be the random bipartite threshold graph with n1 + n2 vertices obtained as follows: Take n1 “white” vertices and n2 “black” ver-tices, and arrange them in random order. Then join each white vertex with probability p1 to all earlier black vertices, and join each black vertex with probability p2 to all earlier white vertices (otherwise, the ver-tex is joined to no earlier vertex), the decisions being made independently by tossing a biased coin once for each white vertex, and another biased coin once for each black vertex. For p1, p2 ∈[0, 1], let μp1,p2 be the probability measure in P with distribution function Fμp1,p2(x) = 1−p1 p2 x, 0 ≤x < p2, 1 − p1 1−p2 (1 −x), p2 ≤x < 1. (8.5) Hence μp1,p2 has density (1−p1)/p2 on (0, p2), and p1/(1−p2) on (p2, 1); if p2 = 0 there is also a point mass 1 −p1 at 0, and if p2 = 1 there is also a point mass p1 at 1. It follows from (5.2) that the corresponding subset Sp1,p2 := Sμp1,p2 of i i “imvol5” — 2009/11/4 — 9:42 — page 315 — #49 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 315 [0, 1]2 is the quadrilateral with vertices (0, 1), (1 −p1, 1 −p2), (1, 0), and (1, 1) (including degenerate cases in which p1 or p2 is 0 or 1). This is an extension of the deﬁnitions in Section 6.3; we have μp,p = μp and Sp,p = Sp. Note also that μ† p1,p2 = μp2,p1. In particular, μp1,p2 ∈Ps only if p1 = p2. Theorem 8.5. As n1, n2 →∞, the degree distributions ν1(Tn1,n2;p1,p2) and ν2(Tn1,n2;p1,p2) converge in probability to μp1,p2 and μp2,p1, respectively. Consequently, Tn1,n2;p1,p2 p − →Γ′′ p1,p2 := Γ′′ μp1,p2 ∈T ′′ ∞,∞. Corollary 8.6. If p1, p2 ∈[0, 1] and n1, n2 ≥1, then Tn1,n2;p1,p2 d = Tn1,n2;Sp1,p2 d = Tn1,n2;X1,X2,0, where Xj has the density 1 −pj on (−1, 0) and pj on (0, 1), j = 1, 2. Note that if p1 + p2 = 1, then Sp1,p2 is the upper triangle S1/2 := { (x, y) : x + y ≥1 }. Hence the distribution of Tn1,n2;p1,p2 does not depend on p1 as long as p2 = 1 −p1. In particular, we may then choose p1 = 1 and p2 = 0. In this case, Deﬁnition (8.4) simpliﬁes as follows: (8.6) Let Tn1,n2 be the random bipartite threshold graph with n1 + n2 vertices obtained as follows: Take n1 “white” vertices and n2 “black” vertices, and arrange them in random order. Join every white vertex to every earlier black vertex. If p1 = 1 and p2 = 0, then further X1 d = U ∼U(0, 1) and X2 d = U −1 in Corollary 8.6. Hence we have found a number of natural constructions that yield the same random bipartite threshold graph. Corollary 8.7. If p1 ∈[0, 1] and n1, n2 ≥1, then Tn1,n2;p1,1−p1 d = Tn1,n2;1,0 = Tn1,n2 d = Tn1,n2;S1/2 d = Tn1,n2;U,U,1, with U ∼U(0, 1). We will see in the next subsection that this random bipartite threshold graph is uniformly distributed as an unlabeled bipartite threshold graph. i i “imvol5” — 2009/11/4 — 9:42 — page 316 — #50 i i i i i i 316 Internet Mathematics 8.4. Uniform Random Bipartite Threshold Graphs It is easy to see that for every bipartite threshold graph, if we color the ver-tices in V1 white and the vertices in V2 black, then there is an ordering of the vertices such that a white vertex is joined to every earlier black vertex but not to any later one. (For example, if there are weights as in (1.9), order the vertices according to w′ i and w′′ j , taking the white vertices ﬁrst in case of a tie.) This yields a one-to-one correspondence between unlabeled bipartite threshold graphs on n1 + n2 vertices and sequences of n1 white and n2 black balls. Consequently, the number of unlabeled bipartite threshold graphs is \|Tn1,n2\| = n1 + n2 n1 , n1, n2 ≥1. Moreover, it follows that Tn1,n2 is uniformly distributed in Tn1,n2; hence Corol-lary 8.7 yields the following: Theorem 8.8. The random bipartite threshold graphs Tn1,n2, Tn1,n2;p1,1−p1 (0 ≤p1 ≤ 1), Tn1,n2;S1/2, Tn1,n2;U,U,1 are all uniformly distributed, regarded as unlabeled bipartite threshold graphs. We have not studied uniform random labeled bipartite threshold graphs. 9. Spectrum of Threshold Graphs There is a healthy literature on the eigenvalue distribution of the adjacency matrix for various classes of random graphs. Much of this is focused on the spectral gap (e.g., most k-regular graphs are Ramanujan [Davidoﬀet al. 03]). See [Jakobson et al. 99] for evidence showing that random k-regular graphs have the same limiting eigenvalue distribution as the Gaussian orthogonal ensemble. The following results show that random threshold graphs give a family of examples with highly controlled limiting spectrum. There is a tight connection between the degree distribution of a threshold graph and the spectrum of its Laplacian; see [Merris 94, Hammer and Kel-mans 96, Merris and Roby 05]. Recall that the Laplacian of a graph G, with V (G) = [n], say, is the n×n matrix L = D−A, where A is the adjacency matrix of G and D is the diagonal matrix with entries dii = dG(i). (Thus L is symmetric and has row sums 0.) It is easily seen that ⟨Lx, y⟩= ij∈E(G)(xi −xj)(yi −yj) for x, y ∈Rn. The eigenvalues λi of L satisfy 0 ≤λi ≤n, i = 1, . . . , n, and we deﬁne the normalized spectral distribution νL ∈P as the empirical distribution of { λi/n }n i=1. i i “imvol5” — 2009/11/4 — 9:42 — page 317 — #51 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 317 For a threshold graph, it is easily seen that if we order the vertices as in (1.2) and Section 2.1, then for each i = 2, . . . , n the function ϕi(j) := ⎧ ⎪ ⎨ ⎪ ⎩ −1, j < i, i −1, j = i, 0, j > i, is an eigenfunction of L with eigenvalue d(i) or d(i) + 1, depending on whether i is added as isolated or dominant, i.e., whether αi = 0 or 1 in the binary code of the graph. Together with ϕ1 := 1 (which is an eigenfunction with eigenvalue 0 for any graph), these form an orthogonal basis of eigenfunctions. The Laplacian spectrum thus can be written { 0 } ∪{ d(i) + αi : i = 2, . . . , n }. (9.1) In particular, the eigenvalues are all integers. Moreover, (9.1) shows that the spectrum { λi }n 1 is closely related to the degree sequence; in particular, asymptotically they are the same in the sense that if Gn is a sequence of threshold graphs with v(Gn) →∞and μ ∈P, then νL(Gn) →μ ⇐ ⇒ν(Gn) →μ. (9.2) (See [Hammer and Kelmans 96] for a detailed comparison of the Laplacian spec-trum and the degree sequence for threshold graphs.) In particular, Theorem 5.8 can be restated using the spectral distribution: Theorem 9.1. Let Gn be a sequence of threshold graphs such that v(Gn) →∞. Then Gn converges in U as n →∞if and only if the spectral distributions νL(Gn) converge to some distribution μ. In this case, μ ∈Ps and Gn →Γμ. Remark 9.2. It can be shown that the spectrum and the degree sequence are asymp-totically close in the sense that (9.2) holds for any graphs Gn with v(Gn) →∞, even though in general there is no simple relation like (9.1). Another relation between the spectrum and the degree sequence for a threshold graph is that their Ferrers diagrams are transposes of each other; see [Merris 94, Merris and Roby 05]. This is easily veriﬁed from (9.1) by induction. If we scale the Ferrers diagrams by n, so that they ﬁt in the unit square [0, 1]2 with a corner at (0, 1), then the lower boundary is the graph of the empirical distribution function of the corresponding normalized values, i.e., the distribution function of ν(G) or νL(G). Hence these distribution functions are related by reﬂection in i i “imvol5” — 2009/11/4 — 9:42 — page 318 — #52 i i i i i i 318 Internet Mathematics the diagonal between (0, 1) and (1, 0), so by (5.11) (and the comment after it), for any threshold graph G, νL(G) = ν(G)†. Acknowledgments. Large parts of this research was done during visits of SJ to Universit´ e de Nice and of PD and SH to Uppsala University in January and March 2007, partly funded by the ANR Chaire d’excellence to PD. Work was continued during a visit of SJ to the Institut Mittag-Leﬄer, Djursholm, Sweden, 2009. SH was supported by grants NSF DMS-02-41246 and NIGMS R01GM086884-2. We thank Adam Guetz and Sukhada Fadnavis for careful reading of a preliminary draft. References [Aldous 85] D. Aldous. Exchangeability and Related Topics, Lecture Notes in Mathe-matics 1117. Berlin: Springer, 1985. [Beissinger and Peled 87] J. S. Beissinger and U. N. Peled. “Enumeration of Labelled Threshold Graphs and a Theorem of Frobenius Involving Eulerian Polynomials.” Graphs Combin. 3:3 (1987), 213–219. [Billingsley 68] P. Billingsley. Convergence of Probability Measures. New York: Wiley 1968. [Borgs et al. 07a] C. Borgs, J. T. Chayes, and L. Lov´ asz. “Unique Limits of Dense Graph Sequences.” Preprint, arxiv.org/math/0803.1244v1, 2007. [Borgs et al. 07b] C. Borgs, J. T. Chayes, L. Lov´ asz, V. T. S´ os, and K. Vesztergombi. “Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing.” Preprint, arxiv.org/math.CO/0702004, 2007. [Brandst¨ adt et al. 99] A. Brandst¨ adt, V. B. Le, and J. P. Spinrad. Graph Classes: A Survey. Philadelphia: SIAM, 1999. [Brown and Diaconis 98] K. Brown and P. Diaconis. “Random Walks and Hyperplane Arrangements.” Ann. Probab. 26:4 (1998), 1813–1854. [Chv´ atal and Hammer 77] V. Chv´ atal and P. L. Hammer. “Aggregation of Inequalities in Integer Programming.” In Studies in Integer Programming, Annals of Discrete Mathematics 1, pp. 145–162. Amsterdam: North-Holland, 1977. [Davidoﬀet al. 03] G. Davidoﬀ, P. Sarnak, and A. Valette. Elementary Number The-ory, Group Theory, and Ramanujan Graphs, London Mathematical Society Student Texts 55. Cambridge, UK: Cambridge University Press, 2003. [Diaconis and Janson 08] P. Diaconis and S. Janson. “Graph Limits and Exchangeable Random Graphs.” Rendiconti di Matematica 28:VII (2008), 33–61. [Erd˝ os et al. 89] P. Erd˝ os, A. Gy´ arf´ as, E. T. Ordman, and Y. Zalcstein. “The Size of Chordal, Interval and Threshold Subgraphs.” Combinatorica 9 (1989), 245–253. [Flajolet and Sedgewick 09] P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge, UK: Cambridge University Press, 2009. [Gut 05] A. Gut. Probability: A Graduate Course. New York: Springer, 2005. i i “imvol5” — 2009/11/4 — 9:42 — page 319 — #53 i i i i i i Diaconis et al.: Threshold Graph Limits and Random Threshold Graphs 319 [Gut and Janson 83] A. Gut and S. Janson. “The Limiting Behaviour of Certain Stopped Sums and Some Applications.” Scand. J. Statist. 10:4 (1983), 281–292. [Hagberg et al. 06] A. Hagberg, P. J. Swart, and D. A. Schult. “Designing Threshold Networks with Given Structural and Dynamical Properties.” Phys. Rev. E 74 (2006), 056116. [Hammer and Kelmans 96] P. L. Hammer and A. K. Kelmans. “Laplacian Spectra and Spanning Trees of Threshold Graphs.” Discrete Appl. Math. 65:1–3 (1996), 255–273. [Hammer et al. 90] P. L. Hammer, U. N. Peled, and X. Sun. “Diﬀerence Graphs.” Discrete Appl. Math. 28:1 (1990), 35–44. [Jakobson et al. 99] D. Jakobson, S. D. Miller, I. Rivin, and Z. Rudnick. “Eigenvalue Spacings for Regular Graphs.” In Emerging Applications of Number Theory, The IMA Volumes in Mathematics and its Applications 109, pp. 317–327. New York: Springer, 1999. [Kallenberg 02] O. Kallenberg. Foundations of Modern Probability, Second edition. New York: Springer, 2002. [Kallenberg 05] O. Kallenberg. Probabilistic Symmetries and Invariance Principles. New York: Springer, 2005. [Konno et al. 05] N. Konno, N. Masuda, R. Roy, and A. Sarkar. “Rigorous Results on the Threshold Network Model.” Journal of Physics A 38 (2005), 6277–6291. [Lov´ asz and Szegedy 06] L. Lov´ asz and B. Szegedy. “Limits of Dense Graph Se-quences.” J. Comb. Theory B 96 (2006), 933–957. [Lov´ asz and Szegedy 09] L. Lov´ asz and B. Szegedy. “Finitely Forcible Graphons.” Preprint, arxiv.org/math/0901.0929v1, 2009. [Lu and Chung 06] L. Lu and F. R. K. Chung. Complex Graphs and Networks. Provi-dence, RI: American Mathematical Society, 2006. [Mahadev and Peled 95] N. Mahadev and U. Peled. Threshold Graphs and Related Top-ics, Annals of Discrete Mathematics 56. Amserdam: North-Holland-Elsevier, 1995. [Masuda et al. 05] N. Masuda, H. Miwa, and N. Konno. “Geographical Threshold Graphs with Small-World and Scale-Free Properties.” Physical Review E 71 (2005), 036108. [McKee and McMorris 99] T. A. McKee and F. R. McMorris. Topics in Intersection Graph Theory. Philadelphia: SIAM, 1999. [Merris 94] R. Merris. Laplacian Matrices of Graphs: A Survey. Linear Algebra Appl. 197/198 (1994), 143–176. [Merris and Roby 05] R. Merris and T. Roby. “The Lattice of Threshold Graphs.” JIPAM. J. Inequal. Pure Appl. Math. 6:1 (2005), Article 2, 21 pp. [Mitzenmacher 04] M. Mitzenmacher. “A Brief History of Generative Models for Power Law and Lognormal Distributions.” Internet Mathematics 1 (2004), 226–251. [Penrose 03] M. Penrose. Random Geometric Graphs. Oxford, UK: Oxford University Press, 2003. i i “imvol5” — 2009/11/4 — 9:42 — page 320 — #54 i i i i i i 320 Internet Mathematics [Sloane 09] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. Available at http:/ /www.research.att.com/∼njas/sequences/, 2009. [Yannakakis 82] M. Yannakakis. “The Complexity of the Partial Order Dimension Problem.” SIAM J. Algebraic Discrete Methods 3:3 (1982), 351–358. Persi Diaconis, Department of Mathematics, Stanford University, Stanford, CA 94305 (diaconis@math.stanford.edu) Susan Holmes, Department of Statistics, Stanford University, Stanford, CA 94305 (susan@stat.stanford.edu) Svante Janson, Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden (svante.janson@math.uu.se) Received August 19, 2009; accepted September 23, 2009.
188969	https://www.youtube.com/watch?v=BxrzwF6oV6s	Counting coins - the value of coins and bills \| MightyOwl Math \| 2nd Grade MightyOwl 54600 subscribers 2546 likes Description 284609 views Posted: 5 Aug 2021 For the full MightyOwl learning experience, check out our website: 👉 In this MightyOwl video you will learn about the value of different coins and the dollar bill. We will show you different strategies for counting up coins of different values. So next time you want to see how much money you have in your piggy bank just like Mia you will know how to count them all! Key words: money, penny, nickel, dime, quarter, dollar, head, tail, cent, addition, subtraction Aligned with Common Core State Standard - Math 2.MD.C.8 - Measurement & Data Work with time and money: Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Transcript: Intro hello this is mighty owl sue is heading to the store to get some pet treats but first she needs to crack open her piggy bank to get some of her money Penny sue has a lot of different types of money in her piggy bank here is a penny this is the head of a penny and this is the tail the head of a coin means the front and the tail of a coin means the back a penny is worth one cent we can write one cent with a sense symbol like this or with a dollar symbol like this this one is easy to identify because it is copper not silver like the other coins here is a nickel it is the medium-sized silver coin and also a little thicker a nickel is worth five cents and is written using a sense symbol or a dollar symbol a sense symbol looks like a c with a line through it a dollar sign looks like a capital s with a line through it this is a dime a dime is worth 10 cents and can be written like this it is the smallest silver coin the last type of coin that sue has is called a quarter a quarter is worth 25 cents this one is the largest silver coin i like the eagle that it usually has on the back Dollar Bills sue also has some dollar bills this is the front and the back of a dollar bill a dollar bill is worth one dollar and can be written only with a dollar sign like this Coin Combinations now we are going to see some different combinations of coins if you want to go get some coins of your own to use go ahead and pause this video to get some we'll wait Counting Coins sue promised her little brother she would give him some money for gum at the store here are the coins she gave him one way to count up all the coins is to find the total value of each coin type and add them let's try that to start there is one quarter which is 25 cents next count the dimes there are three dimes since a dime is worth 10 cents we can skip count by tens to find out how much the three dimes are worth 10 20 30. these dimes are worth 30 cents altogether let's count the nickels now there are two nickels each nickel is worth five cents so two nickels are worth ten cents since five plus five equals ten lastly there are two pennies each penny is worth one cent so two pennies are worth two cents we have 25 cents 30 cents 10 cents and 2 cents let's add up these amounts to find the total value 25 plus 30 is 55. we can find that by skip counting up by tens 25 35 45 55. 55 plus 10 is 65 and 65 plus 2 is 67. sue gave her brother 67 cents great work now sue is taking some money for her trip to the pet store here is what she is taking let's figure out how much money sue is taking she has five dollar bills that is five dollars to start now let's set those aside as we count up all the coins oftentimes it is easiest to start with the coins that have the largest value that's the quarter so let's start there there are two quarters each quarter is worth 25 cents so that is 50 cents since 25 plus 25 is 50. the next largest value is the dime that is 10 cents each and there are six of them let's skip count by tens to find the total value of the dimes 10 20 30 40 50 60. these dimes are worth 60 cents then there are the nickels there are three nickels so skip count by fives to find the value of the nickels 5 10 15. these nickels are worth 15 cents finally there are four pennies since each penny is worth one cent that is four cents altogether let's add up the value of all the coins first we have 50 cents and 60 cents together that makes 110 cents then add 15 cents to get 125 cents finally add 4 cents to get 129 cents that's a lot of sense you probably don't hear many people talk about more than 100 cents at a time that's because once you have reached 100 cents you have a whole dollar one dollar equals 100 cents that's a lot of sense so we can subtract 100 from 129 and add that dollar of coins to our five dollar bills that leaves us with 29 cents so we have the five dollars from the bills one dollar from the coins and 29 cents from the coins together that makes six dollars and 29 cents we can write that like this six dollars and 29 cents Summary so sue is taking six dollars and 29 cents to the pet store great work you have learned about the value of different coins and the dollar bill you know that a dollar bill is worth one dollar a quarter is worth 25 cents a dime is worth 10 cents a nickel is worth five cents and a penny is worth one cent you have learned strategies for counting up coins of different values often it helps to start with a coin of the largest value you can count the value of each coin type and then add them all together and if you have more than 100 cents that makes a dollar so you can subtract 100 from your total number of cents lastly you learned how to write a dollar sign and a cent sign remember a dollar sign always goes in the front and the sense sign always goes in the back but you never use both at the same time see you soon my mighty friend
188970	https://mathforums.com/t/circle-from-2d-random-walk.358982/	Published Time: Thu, 28 Aug 2025 10:55:09 GMT Circle from (2D) random walk \| Math Forums Math Forums Search [x] Search titles only By: SearchAdvanced search… [x] Search titles only By: SearchAdvanced… Math Forums HomeHigh School MathElementary MathAlgebraGeometryTrigonometryProbability & StatisticsPrecalculus University MathCalculusLinear AlgebraAbstract AlgebraApplied MathAdvanced Probability & StatisticsNumber TheoryTopologyReal AnalysisComplex AnalysisDifferential Equations Math DiscussionsComputer ScienceEconomicsMath SoftwareMath BooksSTEM Guidance Forums Trending Engineering Physics Chemistry LoginRegister Search [x] Search titles only By: SearchAdvanced search… [x] Search titles only By: SearchAdvanced… Menu Install the app Install How to install the app on iOS Follow along with the video below to see how to install our site as a web app on your home screen. Note: this_feature_currently_requires_accessing_site_using_safari Home Forums High School Math Probability & Statistics Circle from (2D) random walk Thread starterOOOVincentOOO Start dateJul 10, 2021 OOOVincentOOO Joined Dec 2014 1K Posts \| 497+ Netherlands Discussion Starter Jul 10, 2021 #1 Regarding the topics here on MF I investigated various circle creations. Next is the topic I worked on last week. It takes a lot of effort for me to create something, and I'm making many mistakes but learning a lot. I posted question also on SE because I think this topic potentially has more depth than know with my limited knowhow. Circle from (2D) random walk A method is presented to create circles from sort of random walks. The n−gon formula is extended with a random variable. $$x(n)=\frac{1}{n} \sum_{k=1}^{n} \cos \left( \left( X+\frac{2k}{n_{i}} \r... math.stackexchange.com A method is presented to create circles from sort of random walks. The hobby project is based upon two earlier topics: circles from n−gons with circumference: C=1[SE] or area: A=1[SE]. The n−gon formula can be extended with a random variable. x(n)=1 n∑k=1 q cos⁡(X+2 k n i π) y(n)=1 n∑k=1 q sin⁡(Y+2 k n i π) The total step length or "circumference" is set to 1. Where n is the number of n−gon edges (and steps in the random walk). The summation is discrete with k usually till q=n, completing a full circle till 2 π. Two random (uniform distributed) variables are introduced: X and Y. These random variables are defined as an element between: [0,2]π. The number of elements is determined by variable p such that: p=2,X∈{0,2}and Y∈{0,2} p=3,X∈{0,1,2}and Y∈{0,1,2} p=4,X∈{0,2 3,4 3,2}and Y∈{0,2 3,4 3,2} e t c. When p=2 we will get regular n−gons, while the elements 0 and ±2 π have no effect on sin and cos. When p>2 the number of possible outcomes per step increases. A random walk effect can be observed. In the video below, the circle creation for p=2 till 6 is presented. The random walk consists of N=1.000.000 steps, k is listed as natural numbers from: 1 till 1.000.000. Every walk is displayed and looped over n i. Several observations have been made (see normal picture below). For p=2 the regular n−gons can be seen (triangle, square, pentagon...) left image GIF file. For p>2 initially random walk structures are seen gradually (multiple) circles appear. After 1.000.000 steps single circles can be observed for all presented values of p. The edges of all circles for p>2 have rough and overlapping edges. The effective circle diameter decreases. Empirical: it is found that the circumference decreases with: 1/p see picture. The effective walk length is calculated as the sum of all: Δ r=Δ x 2+Δ y 2. Observation: the total path length for p>3 is less than 1 on average: 0.96. This is one of key questions. For the decrease in diameter, I came up with the following explanation. But not sure if it is completely valid (I struggled with arcsin distribution and various other deviations). First assume (1) as continuous and integrate: x(n)=1 n∫cos⁡(X+2 k n i π)d k=1 2 π sin⁡(X+2 k n i π) With the sum formula for cosine (and sine) formula (2) can be rewritten: x(n)=sin⁡(X π)2 π⋅cos⁡(2 k n i π)+cos⁡(X π)2 π⋅sin⁡(2 k n i π) The random parts of the radius are: sin⁡(X π) and cos⁡(X π). These have influence on the radius. After trial and error, I came up with the following formula calculating the mean radius error Δ R¯ (not sure about this): Δ R x¯=1 p∑m=0 p−1 cos⁡(2 π m p−1) Δ R y¯=1 p∑m=0 p−1 sin⁡(2 π m p−1) Found a solution with help of Wolfram Alpha: Δ R x¯=1 2 p[sin⁡(π p 1−p)csc⁡(π p−1)+1] Δ R y¯=1 2 p[cos⁡(π p−1)+cos⁡(π p 1−p)]csc⁡(π p−1) When analyzing these functions (discrete p) I found that Δ R x¯ decreases with 1/p just like the simulation. However the Δ R y¯ seems to be 0 for discrete values of p. Not sure but then formula (3) (and y counter part) simplifies. Also: there exists no solution for p=2 according Wolfram Alpha in this context. So, for p>2: x(n)=1 2 π p⋅sin⁡(2 k n i π)y(n)=−1 2 π p⋅cos⁡(2 k n i π) I would like to have feedback if method is acceptable thus far. Note that this is a hobby project by amateur. Open question: How can the total path length be smaller for random walk circles? I observed that when the random variables for axis x and y are equal: X=Y then the total path lengt is: 1. Do all presented random walk circles exists for p→∞ and the number of steps is infinate? Seems I have found on answer to point 1 I am running a better plot like posted on SE. Any advice and feedback on topic is welcome. Basic code below (full code here Github:[Github]): Python: ```python import numpy as np import matplotlib.pyplot as plt fig, ax = plt.subplots(1, figsize=(12,12)) ax.axis('equal') n=10000 steps=10000 p=4 N=np.arange(0,steps) t=N/n px=2np.random.choice(p,steps)/(p-1) py=2np.random.choice(p,steps)/(p-1) x=1/(steps)np.cos((px+t2)np.pi) x=np.append(x,x) y=1/(steps)np.sin((py+t2)np.pi) y=np.append(y,y) xc=np.cumsum(x) yc=np.cumsum(y) xd=np.diff(xc) yd=np.diff(yc) dr=np.sqrt(xd2+yd2) circum=np.sum(dr) ax.plot(xc[:n],yc[:n],linewidth=0.15,color='black') plt.show() ``` OOOVincentOOO Joined Dec 2014 1K Posts \| 497+ Netherlands Discussion Starter Jul 10, 2021 Last edited: Jul 10, 2021 #2 Last edited: Jul 10, 2021 I made a numerical analysis and plotted results with random factors X and Y only. The plot below is presented till: p=1000. Where p is the number of intervals between: [0,2]π, and s i z e is the maximum observed value (circumference circle e.g.). Total: x(n)=1 n∑k=1 q cos⁡(X+2 k n i π) Random Only: x(n)=1 n∑k=1 q cos⁡(X n i π) The random walk alone appears to stabilize at line: 1/N (do not understand why and require to study). The circle diameter decreases with: 1/(π p). These lines with eventually intersect, so likely: A random walk grows with the square root of the step(size) [SE]. While the step size in this question is: 1/N1/N the grow rate becomes: 1/N−−√1/N. This limit (blue line graph) is the observation threshold. The potential circle diameter of random walk circles is: 1/(πp)1/(πp) and the size reduces with increasing pp. Though, when the number of steps increases (smaller step size) the observation threshold reduces making the random walk circles visible. In my understanding not all random walk circles exist. I am not sure how all statistics is properly involved while my method is rather empirical based. OOOVincentOOO Joined Dec 2014 1K Posts \| 497+ Netherlands Discussion Starter Jul 11, 2021 Last edited: Jul 11, 2021 #3 Last edited: Jul 11, 2021 Last night I realized that the observation threshold can be calculated more accurate. The previous post was not accurate enough. For big values of p meaning there are: many steps between interval [0,2]π for random variable X,Y to pick from. This will form a arcsin distribution. A random walk grows with the square root of the step(size) [SE]. The random walk tends towards a arcsin distribution [Wiki] with s t d e v: σ 2=(b−a)2 8 Where: \|a\|=\|b\|=1/N: σ=1 N 2 The σ will grow with N see previous link (and CLT: central limit theorm). So σ after N steps: σ t=1 2 N This limit for 3 σ(99.7%) is plotted: blue line graph. This is the circle observation threshold. When the number of steps increases (smaller step size) the observation threshold reduces making the random walk circles visible. I am not sure how all statistics is properly involved while my method is rather empirical based. OOOVincentOOO Joined Dec 2014 1K Posts \| 497+ Netherlands Discussion Starter Jul 11, 2021 Last edited: Jul 11, 2021 #4 Last edited: Jul 11, 2021 Pfew, these math things really get into my head and really hard to find (possible) solution with math I understand. I think to have found a explanation why the path length is shorter in case of random walks with independent x and y with more outcomes per step. The other question was: why is the total walked path shorter then 1 when random variables X and Y are independent? X and Y are uniform distributed elements between [0,2]π. x(n)=1 n∑k=1 q cos⁡(X π) y(n)=1 n∑k=1 q sin⁡(Y π) In case X and Y are independent any result vector can be created between: x∈[0,1] and y∈[0,1], where 1: is the single step length. In case X and Y are independent: any vector start in [0,0] and endpoint within the rectangle x∈[0,1] and y∈[0,1] is possible. When x and y would be uniform distributed the average vector length R¯ can be calculated. The following integral was found after study and checking numerical (Python code discrete): R¯=1 a 2∫0 a∫0 a x 2+y 2 d x d y Where a is the maximum step size single step. For a=1 we find a solution with Wolfram Alpha: R¯=0.765196.... The total walked path would be 0.765196... in case x and y are uniform distributed. However: the angles X and Y are uniform distributed and not x and y. The integral then converts to: R¯=4 π 2∫0 π/2∫−π/2 0 cos 2⁡(X π)+sin 2⁡(Y π)d X d Y Also with Wolfram Alpha a solution is found: R¯=0.958091... This result is confirmed (discrete) numerical (see code below). My conclusion: The difference in the walked path when X and Y are independent can be found by averaging the length of all possible vectors. Numerical summation gives result as found in original question. The used calculus might require improvement while not my specialty. So if someone could give more information of more neat calculus. I do not have the capabilities and knowhow to solve the integrals myself. Python: ```python import numpy as np x=np.linspace(-np.pi/2,0,1001) y=np.linspace(0,np.pi/2,1001) X,Y =np.meshgrid(x,y) def radius(x,y): return np.sqrt((np.cos(x))2+(np.sin(y))2) z=np.array([radius(x,y) for (x,y) in zip(np.ravel(X), np.ravel(Y))]) print(np.mean(z)) ``` Reactions:1 users OOOVincentOOO Joined Dec 2014 1K Posts \| 497+ Netherlands Discussion Starter Jul 11, 2021 Last edited: Jul 11, 2021 #5 Last edited: Jul 11, 2021 Made typo in last integral, limits should be between 0 and 2: R¯=1 4∫0 2∫0 2 cos 2⁡(X π)+sin 2⁡(Y π)d X d Y Outcome the same path length wit independant X and Y: R¯=0.958091... So results confirmed in simulation OP (see video or picture: walked pathlength) with both: numerical with Python code previous message and integral. OOOVincentOOO Joined Dec 2014 1K Posts \| 497+ Netherlands Discussion Starter Jul 12, 2021 #6 Hello I notices a type in opening post: x(n)=1 n∑k=1 q cos⁡(X+2 k n i π) x(n)=1 n∑k=1 q cos⁡((X+2 k n i)π) When coding you see the effect of mistake when typing not, respect for pro math people! I copy pasted error in further messages . The storyline remains the same. For updated version see: Circle from (2D) random walk A method is presented to create circles from sort of random walks. The n−gon formula is extended with a random variable. $$x(n)=\frac{1}{n} \sum_{k=1}^{n} \cos \left( \left( X+\frac{2k}{n_{i}} \r... math.stackexchange.com My investigation is this far completed. This math is at limits of my understanding. So next days moment of reflection after intense last week. Here some pictures with less number of steps. So less perfect circles (but who is?) OOOVincentOOO Joined Dec 2014 1K Posts \| 497+ Netherlands Discussion Starter Jul 31, 2021 Last edited: Jul 31, 2021 #7 Last edited: Jul 31, 2021 I have some difficulties understanding the limit (same as previous limit in post only written more efficient): R¯=∫0 1∫0 1 cos 2⁡(X π)+sin 2⁡(Y π)d X d Y This limits determines the effective walked path for independent X and Y steps. Numerical I found: R¯=0.958... I investigated this limit further last weeks. Note that sin and cos can be exchanged. I also posted this on SE but the question is related on my experience I learned (positively I intend!) here on Mathforums over the last years: Solution integral ∬cos 2⁡(x π)+sin 2⁡(y π)d x d y Working on a hobby project: "Circle from (2D) random walk" [SE] and came across this integral: R¯=∫0 1∫0 1 cos 2⁡(X π)+sin 2⁡(Y π)d X d Y My intention is to math.stackexchange.com The following I learned last week, and is not a full answer, and is intended as extra information about the question. Note that this is all amateur level math. Here follows extra insight I gained. While making use of limited computation like Wolfram Alpha I require to rethink and find other possible alternatives . I studied the function (left graph picture): R=cos 2⁡(X π)+cos 2⁡(Y π) Observations: When plotting the function squares can be identified forming a grid under: 45∘. Two distinct shapes (mountains and valleys) can be identified for R≤1 and R>1. The p d f (probability density function) is not symmetric with mean value: R¯=0.958... (solution to integral in question). See right graph picture p d f with blue line. Next I attempted to find a alternative notation of the function. I rotated the function along the z-axis 45∘. With help of the matrix (Wiki): [cos⁡(π/4)−sin⁡(π/4)0 sin⁡(π/4)cos⁡(π/4)0 0 0 1] Giving: R′=cos 2⁡(1 2(X+Y)π)+cos 2⁡(1 2(Y−X)π) With help of Wolfram Alpha I could simplify this to (center graph picture): R′=cos⁡(2 X π)⋅cos⁡(2 Y π)+1 Finally I noticed that the p d f of the product: cos⁡(2 X π)⋅cos⁡(2 Y π) is a convolution of two a r c s i n e distributions if I am correct, verification: MSE and MSE. While in original question X and Y are uniform distributed values between: [0,2]. A a r c s i n e distribution between [−1,1] is given as: f(R)=1 π 1−R 2 The convolution of two a r c s i n would then be (not sure if this is allowed): f(R)∗f(R)=∫−a a 1 π 1−τ 2⋅1 π 1−(R−τ)2 d τ Wolfram Alpha (online) gives a solution of the convolution without intervals. Also here the elliptic integral F (incomplete first kind) occurs just like mentioned in comments on SE. f(R)∗f(R)=[−2(τ−1)(τ+1)τ−R+1(τ−1)R)τ−R−1(τ−1)(R+2)F[arcsin⁡((τ+1)R(τ−1)(R+2)),1−4 R 2]1−τ 2(τ+1)R(τ−1)(R+2)−τ 2+2 τ R−R 2+1]−a a Though, I am not sure how and if using the limits of integration is allowed. The a r c s i n e function is not bounded in height. I evaluated the convolution numerical of the a r c s i n e p d f's and looks like: MSE. This p d f is plotted in right image (gray area). Also plotted is R 2−1 directly from function R (red line). The calculated convolution and histogram from R do not match perfect depending upon how close to limit's: ±1 as e.g. ±0.999. Note: the sharp peak at R=0 tending ∞ is likely. My question then: does there exists a solution for the convolution of two a r c s i n distributions (professional literature is outside my level). If convolution exists: does this mean the integral is directly solvable in a nice series or formula? Python: ```python import numpy as np import matplotlib.pyplot as plt import matplotlib.gridspec as gridspec from IPython import get_ipython get_ipython().run_line_magic('matplotlib', 'qt5') fig = plt.figure(figsize = (25.5,6)) gs1 = gridspec.GridSpec(1, 3) gs1.update(wspace=0.15, hspace=0.15) ax1 = plt.subplot(gs1[0,0]) ax2 = plt.subplot(gs1[0,1]) ax3 = plt.subplot(gs1[0,2]) ax1.axis("equal") ax2.axis("equal") def radius(x,y): return np.sqrt((np.cos(xnp.pi))2+(np.cos(ynp.pi))2) def radiusrot(x,y): return np.sqrt(1+np.cos(np.sqrt(2)xnp.pi)np.cos(np.sqrt(2)ynp.pi)) Radius standard x=np.linspace(0,2,2500,endpoint=False) y=np.linspace(0,2,2500,endpoint=False) X,Y =np.meshgrid(x,y) Z=radius(X,Y) print(np.mean(Z)) meanz=np.mean(Z) z=np.array([radius(x,y) for (x,y) in zip(np.ravel(X), np.ravel(Y))]) cf1=ax1.contourf(X[::20],Y[::20],Z[::20],levels=np.arange(0,2.1,0.1), cmap='seismic',alpha=1) cp=ax1.contour(X[::20],Y[::20],Z[::20],levels=np.arange(0,2.1,0.1),colors='black',linewidths=1) fig.colorbar(cf1, ticks=np.arange(0,2.2,0.2),ax=ax1) ax1.set_title(r"$R=\sqrt{\cos^2 \left( X \pi \right) + \cos^2 \left( Y \pi \right) }$", fontsize=14, pad=20) Radius 45 degrees rotated x=np.linspace(0,np.sqrt(2),2500,endpoint=False) y=np.linspace(0,np.sqrt(2),2500,endpoint=False) X,Y =np.meshgrid(x,y) Zrot=radiusrot(X,Y) cf2=ax2.contourf(X[::20],Y[::20],Zrot[::20],levels=np.arange(0,2.1,0.1), cmap='seismic',alpha=1) cp=ax2.contour(X[::20],Y[::20],Zrot[::20],levels=np.arange(0,2.1,0.1),colors='black',linewidths=1) fig.colorbar(cf2, ticks=np.arange(0,2.2,0.2),ax=ax2) ax2.set_title(r"$R=\sqrt{\cos \left( X \pi \sqrt{2} \ \right) \cdot \cos \left( Y \pi \sqrt{2}\ \right)+1}$", fontsize=14, pad=20) Histograms R and R^2-1 hist1,bins1 =np.histogram(Zrot2-1,bins=1000, density=True) ax3.plot(bins1[1:],hist1,label=r"$R^2-1$",color='red',linewidth=0.75) hist2,bins2 =np.histogram(Zrot,bins=1000, density=True) ax3.plot(bins2[1:],hist2,label=r"$R$",color='blue',linewidth=0.75) ax3.plot([meanz,meanz],[0,1.2np.max(hist2)],linewidth=0.75,color="black") ax3.text(meanz+0.05,1.15np.max(hist2),"mean:\n" + str(np.round(meanz,4)),va="top",color="black",fontsize=14) Numerical Convolution Arcsine R=np.linspace(-0.99999,0.99999,10000) dR=R-R arcsin=1/(np.pi(np.sqrt(1-R2))) pdf=np.convolve(arcsin,arcsin,mode='same') size=np.size(pdf) af=dRnp.sum(pdf) f2=pdf/af f1=np.full(np.size(pdf),0) ax3.fill_between(R[::10], f1[::10], f2[::10],color='black',zorder=-10,alpha=0.25,label=r"$f(t)f(t)$ (convolution)",interpolate=True,linewidth=0) ax3.set_title(r"$f(t)= \dfrac{1}{\pi \sqrt{1-R^2}}$ (arcsine pdf)", fontsize=14, pad=20) ax3.set_xlabel("$R$",fontsize=14) ax3.set_ylabel("density",fontsize=14) ax3.legend(loc="upper left",fontsize=14) ax3.set_xlim([-1, np.sqrt(2)]) ax3.set_ylim([0,1.2np.max(hist2)]) plt.show() np.savetxt('Histogram2.txt',(bins1[1:],hist1), delimiter='\t', newline='\n') print(np.mean(Z)) ``` OOOVincentOOO Joined Dec 2014 1K Posts \| 497+ Netherlands Discussion Starter Jul 31, 2021 #8 OOOVincentOOO said: I also posted this on SE but the question is related on my experience I learned (positively I intend!) here on Mathforums over the last years: Click to expand... My intention: I learned allot of positive things here on Mathforums! Reactions:1 users idontknow Joined Dec 2015 5K Posts \| 847+ Nature Aug 1, 2021 #9 Averaging the length of all possible vectors is the way to go (I think ). Which formula is the last result about random 2D walk ? I want to see whether it can be a straight line ( the trajectory ) . OOOVincentOOO Joined Dec 2014 1K Posts \| 497+ Netherlands Discussion Starter Aug 1, 2021 Last edited: Aug 1, 2021 #10 Last edited: Aug 1, 2021 idontknow said: I want to see whether it can be a straight line ( the trajectory ) . Click to expand... Good question. Initially I came up with the idea to create Lissajous figures with random walk elements (change frequency and add phase shift when needed though not in original formula). Formula circle: x(n)=1 n∑k=1 q cos⁡((X+2 k n i)π) y(n)=1 n∑k=1 q sin⁡((Y+2 k n i)π) Where: X and Y are random elements defined in opening post. Unfortunatly typo formula in opening post. The first starting point was a circle, because we all like Pi(e)! Exploring and attempting to define random walk circle already is a challenge for me. Though to create a line (see: Wiki): x(n)=1 n∑k=1 q sin⁡((X+2 k n i)π) y(n)=1 n∑k=1 q sin⁡((Y+2 k n i)π) You can change Python code in opening post. I am not at home but run code online with: trinket so no fancy plots now. See images below of straight line and 3:1 freq. ratio. Error calculation will be much more complicated then the circle I think. Also note the size of the shapes reduces when taking more outcomes per step, to determine effective size with these shapes seems complicated I think (or maybe correlated to circles?). Reactions:1 users OOOVincentOOO Joined Dec 2014 1K Posts \| 497+ Netherlands Discussion Starter Aug 2, 2021 Last edited: Aug 2, 2021 #11 Last edited: Aug 2, 2021 A a r c s i n e distribution between R∈[−1,1]is given as: f(R)=1 π 1−R 2 The convolution of two a r c s i n would then be (not sure if this is allowed): f(R)∗f(R)=∫−a a 1 π 1−τ 2⋅1 π 1−(R−τ)2 d τ I evaluated this convolution here (made some stupid mistakes and took tourist route but got there!): Convolution arcsin pdf's: ∫1 π 1−τ 2⋅1 π 1−(R−τ)2 d τ Regarding topic: SE I encountered a convolution of two identical arcsin distributions. The probability density function p d f for R∈[−1,1] in this context is: $$f(R)=\frac{1}{\pi \sqrt{1-R... math.stackexchange.com Also here the elliptic integral K (complete first kind) occurs for real solutions. Then I found for the convolution of two identical arcsin: f(R)∗f(R)=\|1 π R⋅K(1−4 R 2)\| OPEN ISSUE: not sure how to convert boxed formula back with R 2−1 to asymmetric blue line (see plot with some posts back: the one with 3 plots). But roughly sketched my question is answered. Reactions:1 users OOOVincentOOO Joined Dec 2014 1K Posts \| 497+ Netherlands Discussion Starter Aug 8, 2021 Last edited: Aug 8, 2021 #12 Last edited: Aug 8, 2021 While looking at the 2 D random walk circles I observed that the effective walked path was R¯=0.958.. when the X and Y are independent. This effective path was equal to the integral: R¯=∫0 1∫0 1 cos 2⁡(X π)+sin 2⁡(Y π)d X d Y=0.958.. I struggled a couple of weeks understanding this integral. I found that the probability density function of R 2−1 was equal to the convolution of two a r c s i n e distributions (by rotating the figure 45∘ and simplification, see some posts back). Though I made a mistake in last post, it should be the convolution of two a r c s i n e distributions between: R∈[−1 2,1 2] and not: R∈[−1,1]. So here the corrected plot see SE link below for more information. The gray area under the red line is the function below: the p d f (probability density function) of R 2−1 is equal to: R 2−1=d\|2 π 2 R⋅K(1−1 R 2)\| With =d denoting equality in distribution and K the complete elliptic integral. And is slightly different then last post. Though my goal was to come to the blue line, showing the numerical distribution of R. This distribution can be transformed to the distribution of RMSE. I have little experience in transformation of p d f's. My level is amateur/hobby and do not know the formal notation, first: Define: Y=R 2−1 and d Y/d R=2 R. G(Y)=∫−1 1\|2 π 2 Y⋅K(1−1 Y 2)\|d Y=1 G(R)=∫0 2\|4 R π 2(R 2−1)⋅K(1−1(R 2−1)2)\|d R=1 The function within the integral g(R) is plotted and corresponds to observed data. The question asks the solution to the integral: R¯=∫0 1∫0 1 cos 2⁡(X π)+sin 2⁡(Y π)d X d Y The calculated p d f of the radius R is found as: g(R)=\|4 R π 2(R 2−1)⋅K(1−1(R 2−1)2)\| The mean value of this p d f is the solution to the following integral (multiply p d f with R and integrate) see Wiki. R¯=∫0 2\|4 R 2 π 2(R 2−1)⋅K(1−1(R 2−1)2)\|d R With my available tools I cannot find a nice simple solution. Though integrating G(R) gives a Meijer g function just like mentioned in comments on SE. When assuming the the red and blue squares in the plot have the same area I calculated the mean value from both. For R∈[0,1], the mean value R¯1=∫0 1\|2 R⋅g(R)\|d R, note: multiply by 2 to set half area to 1. Solution with Wolfram alpha online: Code: integrate 24R^2/(pi^2(R^2-1))K(1-1/(R^2-1)^2) dR from R=0 to 1 R¯1=0.737076... For R∈[1,2], the mean value R¯2=∫1 2 2 R⋅g(R)d R: Code: integrate 24R^2/(pi^2(R^2-1))K(1-1/(R^2-1)^2) dR from R=1 to sqrt(2) R¯2=1.179107... The mean value: R¯=0.737076...+1.179107...2=0.958092... With this method the same solution is found as the question. So my question is (partial) answered and gained more insight about this integral. Made some happy accidents on the way. Did not find nice solution to integral but got better understanding in return and learned allot! Summary with less mistakes here: Solution integral ∬cos 2⁡(x π)+sin 2⁡(y π)d x d y Working on a hobby project: "Circle from (2D) random walk" [SE] and came across this integral: R¯=∫0 1∫0 1 cos 2⁡(X π)+sin 2⁡(Y π)d X d Y My intention is to math.stackexchange.com OOOVincentOOO Joined Dec 2014 1K Posts \| 497+ Netherlands Discussion Starter Sep 12, 2021 Last edited: Sep 12, 2021 #13 Last edited: Sep 12, 2021 After some reflection on the topic: random walk circles some months later: The n−gon formula can be extended with a random variable. x(n)=1 n∑k=1 q cos⁡(X+2 k n i π) y(n)=1 n∑k=1 q sin⁡(Y+2 k n i π) With n number of steps and X and Y random values from: p=2,X∈{0,2}and Y∈{0,2} p=3,X∈{0,1,2}and Y∈{0,1,2} p=4,X∈{0,2 3,4 3,2}and Y∈{0,2 3,4 3,2} e t c. Some obvious observation that many maybe made. I did not realize sooner but: The circles only exist when when there are two duplicate random values. So 0 and 2 are the same random angular values. So with probability: 1/p the same angle occurs! So 1/p steps creates a circle with corresponding circumference: 1/p! I enjoyed the long trip getting to this simple reasoning! Getting understanding to the effective walked path 0.96… with independent X and Y highlight! I am happy to think to complicated and learned allot! 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188971	https://mathequalslove.net/digit-addition-puzzles/	Skip to content Trending Resource: 40+ Ideas for the First Week of School Digit Addition Puzzles \| Addition \| Addition Puzzles \| Elementary Math \| Logic Puzzles \| Math Puzzles \| Puzzles Digit Addition Puzzle Collection These free printable digit addition puzzles will put your logic and mental math skills to the test. Can you add digits to the right of some of the numbers in the grid so that each row and each column sums to the target sum? What is a Digit Addition Puzzle? A digit addition puzzle is a square grid with numbers placed in each cell of the grid. Digits must be placed to the right of some of the numbers in the grid so that each row and each column sums to the target sum specified in the puzzle. These digit addition puzzles are similar to a magic square puzzle, but they lack the requirement of having the main diagonals with the same sum. Target Sums Available These digit addition puzzles get progressively more difficult and time consuming to solve as the target sum increases from the most basic of puzzles with a target of 30 and the most advanced puzzles with a target sum of 100. Each collection of digit addition puzzles comes as a free printable PDF file containing 60 different puzzles, printed 6 to a page. 30 SQUARE LOGIC PUZZLES 40 SQUARE LOGIC PUZZLES 50 SQUARE LOGIC PUZZLES 60 SQUARE LOGIC PUZZLES 70 SQUARE LOGIC PUZZLES 80 SQUARE LOGIC PUZZLES 90 SQUARE LOGIC PUZZLES 100 SQUARE LOGIC PUZZLES Puzzle Solutions ANsWer key link Puzzle solutions are available on a password-protected solution page. I do not openly post the puzzle answer keys because one of my goals as a resource creator is to craft learning experiences for students that are non-google-able. I want teachers to be able to use these puzzles in their classrooms without the solutions being found easily on the Internet. Please email me at sarah@mathequalslove.net for the password to the answer key database featuring all of my printable puzzles and math worksheets. I frequently have students emailing me for the answer key, so please specify in your email what school you teach at and what subjects you teach. If you do not provide these details, I will not be able to send you the password. Not a teacher? Go ahead and send me an email as well. Just let me know what you are using the puzzles for. I am continually in awe of how many people are using these puzzles with scouting groups, with senior adults battling dementia, or as fun activities in their workplace. Just give me enough details so I know you are not a student looking for answers to the puzzle that was assigned as their homework! Sarah Carter Sarah Carter teaches high school math in her hometown of Coweta, Oklahoma. She currently teaches AP Precalculus, AP Calculus AB, and Statistics. She is passionate about sharing creative and hands-on teaching ideas with math teachers around the world through her blog, Math = Love. Similar Posts Pi Birthday Search Activity Finding Equations of Inverses Foldable Solving Absolute Value Inequalities Graphic Organizers Birthday Poster Snow Day Hidoku Logic Puzzles Color Square Puzzle Leave a Reply Cancel reply
188972	https://icjournal.org/DOIx.php?id=10.3947/ic.2022.0151	pISSN 2093-2340 eISSN 2092-6448 Indexed in ESCI, PubMed, Scopus, DOAJ and more Open Access, Peer-reviewed Abstract INTRODUCTION EPIDEMIOLOGY DIAGNOSIS OF IMDs TREATMENT OF INVASIVE ASPERGILLOSIS TREATMENT OF INVASIVE MUCORMYCOSIS OTHER MOLD DISEASES CONCLUSION Notes References Infect Chemother. 2023 Mar;55(1):10-21. English. Published online Nov 29, 2022. Copyright © 2023 by The Korean Society of Infectious Diseases, Korean Society for Antimicrobial Therapy, and The Korean Society for AIDS Review Article Diagnosis and Treatment of Invasive Mold Diseases Sang-Oh Lee Author information Author notes Copyright and License Department of Infectious Diseases, Asan Medical Center, University of Ulsan College of Medicine, Seoul, Korea. Corresponding Author: Sang-Oh Lee, MD, PhD. Department of Infectious Diseases, Asan Medical Center, University of Ulsan College of Medicine, 88, Olympic-ro 43-gil, Songpa-gu, Seoul 05505, Korea. Tel: +82-2-3010-3301, Fax: +82-2-3010-6970, Email: soleemd@amc.seoul.kr Received October 18, 2022; Accepted November 09, 2022. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Go to: AbstractINTRODUCTIONEPIDEMIOLOGYDIAGNOSIS OF IMDsTREATMENT OF INVASIVE ASPERGILLOSISTREATMENT OF INVASIVE MUCORMYCOSISOTHER MOLD DISEASESCONCLUSIONNotesReferences Abstract Although invasive fungal diseases are relatively less common than superficial diseases, there has been an overall increase in their incidence. Here, I review the epidemiology, diagnosis, and treatment of invasive mold diseases (IMDs) such as aspergillosis, mucormycosis, hyalohyphomycosis, and phaeohyphomycosis. Histopathologic demonstration of tissue invasion by hyphae or recovery of mold by the culture of a specimen obtained by a sterile procedure provides definitive evidence of IMD. If IMD cannot be confirmed through invasive procedures, IMD can be diagnosed through clinical criteria such as the European Organization for Research and Treatment of Cancer/Invasive Fungal Infections Cooperative Group and the National Instituteof Allergy and Infectious Diseases Mycoses Study Group (EORTC/MSG) definitions. For initial primary therapy of invasive aspergillosis, voriconazole or isavuconazole is recommended and lipid formulations of amphotericin B are useful primary alternatives. Echinocandins are representative antifungal agents for salvage therapy. Treatment of invasive mucormycosis involves a combination of urgent surgical debridement of involved tissues and antifungal therapy. Lipid formulations of amphotericin B are the drug of choice for initial therapy. Isavuconazole or posaconazole can be used as salvage or step-down therapy. IMDs other than aspergillosis and mucormycosis include hyalohyphomycosis and phaeohyphomycosis, for which there is no standard therapy and the treatment depends on the clinical disease and status of the patient. Diagnosis; Treatment; Invasive fungal infections; Aspergillosis; Mucormycosis Go to: AbstractINTRODUCTIONEPIDEMIOLOGYDIAGNOSIS OF IMDsTREATMENT OF INVASIVE ASPERGILLOSISTREATMENT OF INVASIVE MUCORMYCOSISOTHER MOLD DISEASESCONCLUSIONNotesReferences INTRODUCTION Invasive fungal diseases (IFDs) are generally distinguished from superficial fungal diseases based on the involvement of blood and other sterile sites or invasion into organ tissues. Although IFDs are relatively less common than superficial diseases, there has been an overall increase in the incidence of IFDs , partially due to increases in the number of immunocompromised patients receiving hematopoietic stem cell transplantation (HSCT), solid organ transplantation (SOT), or newer immunomodulatory agents. Fungal diseases are caused by yeast and mold. Candidiasis and cryptococcosis are representative IFDs caused by yeast. Here, I review the epidemiology, diagnosis, and treatment of invasive mold diseases (IMDs) such as aspergillosis, mucormycosis, hyalohyphomycosis, and phaeohyphomycosis. Go to: AbstractINTRODUCTIONEPIDEMIOLOGYDIAGNOSIS OF IMDsTREATMENT OF INVASIVE ASPERGILLOSISTREATMENT OF INVASIVE MUCORMYCOSISOTHER MOLD DISEASESCONCLUSIONNotesReferences EPIDEMIOLOGY Recently, epidemiologic data of IFDs using healthcare network data in the United States were published . The mean incidence of IFD was 27.2 cases/100,000 patients per year, and the mean annual increase was 0.24 cases/100,000 patients. Candidiasis was the most common type (55.0%). Dimorphic fungi, primarily Coccidioides spp., comprised 25.1% of cases, followed by Aspergillus spp. (8.9%). According to the nationwide data from the National Health Insurance of Korea, the annual prevalence of fungal diseases increased from 6.9% in 2009 to 7.4% in 2013. Dermatophytosis, a representative superficial fungal disease, had the highest prevalence (5.2%), followed by IFDs (1.7%) such as cryptococcosis, aspergillosis, and mucormycosis . In addition, cases of hyalohyphomycosis and phaeohyphomycosis, which are rare IMDs in patients with various underlying diseases, have been reported in Korea [4, 5, 6]. The proportion of each IFD varied depending on the patient category. In the Prospective Antifungal Therapy (PATH) Alliance registry data, invasive candidiasis was the major type of IFD in medical patients, non-transplant surgical patients, and those with solid tumors (79.8 - 90.7%), while invasive aspergillosis was more common in HSCT recipients and hematologic malignancy patients (35.2 - 49.5%). The 12-week survival rate for IFI varied based on the underlying condition, ranging from 37.5% for HSCT recipients to 77.5% for SOT recipients . Invasive fungal pneumonia is a common and lethal complication in patients with hematologic malignancy . As for IFDs that occurred after lung transplantation, candidiasis accounted for 52.2%, aspergillosis for 30.4%, and mucormycosis for 17.4% . Go to: AbstractINTRODUCTIONEPIDEMIOLOGYDIAGNOSIS OF IMDsTREATMENT OF INVASIVE ASPERGILLOSISTREATMENT OF INVASIVE MUCORMYCOSISOTHER MOLD DISEASESCONCLUSIONNotesReferences DIAGNOSIS OF IMDs To confirm the diagnosis of IMDs, a specimen must be obtained by needle aspiration or biopsy. Histopathologic demonstration of tissue invasion by hyphae provides definitive evidence of IMD. Recovery of mold by the culture of a specimen obtained by a sterile procedure also provides a definite diagnosis of IMD. These categories are commonly refered to as "proven" IMDs . However, needle aspiration or biopsy is often not feasible due to the risk of complications, especially in immunocompromised patients such as HSCT or SOT recipients and those with hematologic malignancy. In fact, in randomized controlled trials on IMD treatment, the proportions of proven cases ranged from just 1.8% to 27.0% [11, 12, 13, 14]. In a Korean nationwide multicenter study for invasive pulmonary aspergillosis, only 8% were proven cases . If IMD cannot be confirmed through invasive procedures, clinical criteria are used for diagnosis. In 2002, a consensus group of the European Organization for Research and Treatment of Cancer/Invasive Fungal Infections Cooperative Group and the National Institute of Allergy and Infectious Diseases Mycoses Study Group (EORTC/MSG) proposed standard definitions for IFDs for clinical and epidemiologic study . Since then, these definitions have been widely used as clinical diagnostic criteria for IFDs~~,~~ and were revised in 2008 and 2020 [16, 17]. The EORTC/MSG definitions consist of three elements: host factor, clinical criteria, and mycological criteria. If all three elements are satisfied, a diagnosis of "probable" IFDs can be made. If a case has the appropriate host factors and sufficient clinical evidence but no mycological support, it is diagnosed as "possible" IFD~~s~~. The possible category corresponds to empirical treatment at the clinician’s discretion. Therefore, it is recommended to use antifungal agents carefully according to the opinions of ~~the~~ infectious disease experts. 1. Host factors The EORTC/MSG definitions were originally created for use in patients with cancer and hematologic malignancy and HSCT recipients . Therefore, in the 2002 EORTC/MSG definitions, host factors were mainly composed of neutropenia, persistent fever refractory to broad-spectrum antimicrobial treatment, graft-versus-host disease, and prolonged (>3 weeks) use of corticosteroids. In the 2008 revised definitions, various immunosuppressant treatments were also included to include SOT recipients . The 2020 definitions listed more specific situations . Table 1 compares the 2008 and 2020 definitions and presents them in detail. Table 1 Comparison of the 2008 and 2020 EORTC/MSG definitions for probable invasive pulmonary mold diseases Click for larger image Click for full table Download as Excel file 2. Clinical criteria Clinical criteria of the 2002 EORTC/MSG definitions consisted of radiologic findings and clinical symptoms/signs . For example, to diagnose pulmonary IMDs, chest CT findings required a halo sign, air-crescent sign, or cavity within the area of consolidation. Clinical symptoms/signs were coughs, chest pain, hemoptysis, dyspnea, and pleural rub. In the 2008 definitions, clinical symptoms/signs were excluded from clinical criteria because they were usually non-specific for the diagnosis of pulmonary IMDs . In the 2008 definitions, radiologic findings were also revised; with only dense, well-circumscribed lesions, radiologic criteria for pulmonary IMDs were satisfied regardless of the presence of a halo sign (Table 1). It was because halo signs appear between the initial 7 - 10 days and might not be visible thereafter [18, 19]. There were some additional changes in the 2020 definitions . Radiologic patterns of pulmonary IMDs were classified as either angio-invasive or airway-invasive forms [20, 21]. Angio-invasive forms, such as halo sign, air-crescent sign, and cavity within an infarct-shaped consolidation, were observed more often in neutropenic patients [22, 23]. Because the EORTC/MSG definitions were originally focused on patients with cancer and hematologic malignancy, radiologic patterns of angio-invasive form were mainly included even in the 2008 definitions. Airway-invasive forms, such as small airway lesions, peribronchial consolidations, and bronchiectasis, are often observed in non-neutropenic transplant patients [22, 23]. Based on the results of these studies, the findings of wedge-shaped and segmental or lobar consolidation were added to the 2020 definitions (Table 1). The 2020 definitions were characterized by separating the clinical criteria for pulmonary aspergillosis and other pulmonary mold diseases (Table 1). The reverse halo sign was added as the clinical criteria for other pulmonary mold diseases such as mucormycosis. The reverse halo sign is more common in patients with pulmonary mucormycosis than in those with pulmonary aspergillosis . The histology of the reverse halo sign includes an infarcted lung with a greater amount of hemorrhage in the peripheral solid ring than in the center ground-glass region . While the clinical criteria of the EORTC/MSG definitions are presented by pulmonary diseases, tracheobronchitis, sinonasal diseases, and central nervous system infection, Table 1 only summarizes pulmonary mold diseases, which are the most representative type. 3. Mycological criteria According to the mycological criteria of the EORTC/MSG definitions, direct tests (i.e., cytology, direct microscopy, or culture) or indirect tests (i.e., detection of antigen or cell-wall constituents) can be used for the diagnosis of IMD (Table 1) [10, 16, 17]. If an appropriate specimen can be obtained, we can identify the presence of mold from the smear and culture, otherwise, indirect tests could be used. Galactomannan (GM) assay can be used for the diagnosis of aspergillosis, but there is no specific test for other mold diseases. In experimental assays, the detection of Aspergillus antigenemia seems to correlate with clinical diagnosis and treatment response of antifungal therapy in invasive aspergillosis [26, 27]. A sandwich ELISA technique that uses a monoclonal antibody to GM has been developed . The GM assay (Platelia Aspergillus, Bio-Rad, Hercules, CA, USA) could be applied to cerebrospinal fluid and bronchoalveolar lavage fluid as well as plasma and serum . The GM assay is performed with an optical read-out that is interpreted as a ratio relative to the optical density (OD) of a threshold control provided by the manufacturer; this ratio is called the OD index. The cut-off index for the GM assay was originally set at 1.0 - 1.5 in order to minimize false-positive results , but it was lowered to 0.5 in the 2008 EORTC/MSG definitions . The lowered cut-off value improved the overall performance of the test for adult hematology patients . In the 2020 EORTC/MSG definitions, however, the cut-off index has been raised again (1.0 in serum, plasma, or bronchoalveolar lavage fluid) in order to ensure a higher likelihood of diagnostic certainty for clinical trial purposes (Table 1) [17, 30]. The determination of (1,3)-β-d-glucan (BDG) serum levels is a noninvasive test for circulating fungal cell wall components. The BDG assay is not specific for Aspergillus species and can produce positive results in patients with a variety of IFDs, including candidiasis and Pneumocystis jirovecii. However, it is typically negative in patients with mucormycosis or cryptococcosis [31, 32]. There are several different commercial assays available in different countries. A commercial BDG assay (Fungitell assay, Associates of Cape Cod, East Falmouth, MA, USA) has been approved by the FDA. However, BDG was not considered to provide mycological evidence of any IMD since BDG detection is not specific to a single IMD. In the 2008 EORTC/MSG definitions, molecular methods of detecting fungi in clinical specimens (e.g., polymerase chain reaction [PCR]) were not included in the definitions because there was as yet to be a standard, and none of the techniques had been clinically validated . Since then, much progress has been made in PCR assays for Aspergillus. In a meta-analysis of 25 studies, the sensitivity and specificity of PCR to detect invasive aspergillosis were 84.0% and 76.0%, respectively. When at least two PCR results were positive, the sensitivity was 64.0% and the specificity was 95.0% . Many other meta-analyses showed similar results. Standardized protocols for nucleic acid extraction, sample types, volumes, and processing have been developed by the European Aspergillus PCR Initiative/Fungal PCR Initiative . As a result of these efforts, 2 or more consecutive positive results of Aspergillus PCR was included in the mycological criteria of the 2020 EORTC/MSG definitions (Table 1) . 4. Diagnostic criteria for other host groups In recent years, there has been an increasing number of reports of IMDs accompanying various infectious diseases. Invasive pulmonary aspergillosis can develop in patients admitted to intensive care units due to severe influenza or severe fever with thrombocytopenia syndrome [35, 36, 37]. The category of proven IMD can apply to any patient, regardless of whether the patient is immunocompromised. However, probable IMD requires the presence of at least 1 host factor, clinical and mycological criteria, and is proposed for immunocompromised patients only [16, 17]. As severe influenza or severe fever with thrombocytopenia syndrome are not a host factor for IMD, efforts to create new diagnostic algorithms are ongoing [36, 38, 39]. In the coronavirus disease 2019 (COVID-19) pandemic situation, many cases of coronavirus disease-associated pulmonary aspergillosis (CAPA) were reported [40, 41]. In order to diagnose CAPA in patients with COVID-19, diagnostic criteria are being created by applying criteria for influenza-associated pulmonary aspergillosis (IAPA) and the EORTC/MSG definitions . Go to: AbstractINTRODUCTIONEPIDEMIOLOGYDIAGNOSIS OF IMDsTREATMENT OF INVASIVE ASPERGILLOSISTREATMENT OF INVASIVE MUCORMYCOSISOTHER MOLD DISEASESCONCLUSIONNotesReferences TREATMENT OF INVASIVE ASPERGILLOSIS 1. Initial therapy for invasive aspergillosis For initial primary therapy of invasive aspergillosis, voriconazole is recommended (Table 2) [43, 44, 45]. Even in the early 2000s, conventional amphotericin B deoxycholate was the standard therapy for invasive aspergillosis, although responses were suboptimal (less than 40.0%) and it had multiple adverse effects . Voriconazole gained its current status based on a randomized, unblinded trial published in 2002 that compared 144 patients in the voriconazole group and 133 patients in the amphotericin B group with definite or probable aspergillosis. This trial reported that compared with amphotericin B, initial therapy with voriconazole led to better responses (successful outcome at week 12, 52.8% vs. 31.6%) and a better survival rate at 12 weeks (70.8% vs. 57.9%) while resulting in fewer severe side effects such as nephrotoxicity. Table 2 Antifungal treatment of invasive aspergillosis Click for larger image Click for full table Download as Excel file During voriconazole therapy, therapeutic drug monitoring (TDM) is recommended [43, 44, 45]. The target trough level of voriconazole is 1 - 5.5 mg/L. Routine TDM of voriconazole may reduce the incidence of drug discontinuation due to adverse events and improve the treatment response . Considering that voriconazole interacts with various drugs such as cyclosporine, tacrolimus, and sirolimus, TDM of potentially interacting drugs is also needed. Elevation of hepatic enzymes is the most common adverse event of voriconazole, followed by hallucination and visual disturbance . Isavuconazole is the primary or primary alternative drug for the treatment of invasive aspergillosis [43, 44, 45]. In phase 3, randomized-controlled trial, isavuconazole was non-inferior to voriconazole for the primary therapy of IMDs including invasive aspergillosis . In the isavuconazole group of this trial, hepatotoxicity, visual disturbance, and rash were less common than in the voriconazole group. However, the incidence of audio or visual hallucinations was similar between the two groups. TDM of isavuconazole is not routinely recommended because there is no clear exposure-response relationship and low variability of plasma level . Recently, a phase 3, randomized-controlled trial assessed the non-inferiority of posaconazole to voriconazole for the primary therapy of invasive aspergillosis . In this trial, posaconazole was non-inferior to voriconazole for all-cause mortality until day 42 and had fewer adverse events than voriconazole. Although posaconazole is recommended as salvage therapy in the recent international guidelines (Table 2) [43, 44, 45], it might be included as primary therapy in the next revision of the guidelines. Liposomal amphotericin B or amphotericin B lipid complex are additional primary alternatives but these agents carry the risk of nephrotoxicity [43, 44, 45]. Although no randomized trial has been performed to evaluate the efficacy of these drugs compared with voriconazole as primary therapy, a series of randomized trials suggest their efficacy. A randomized trial evaluating the primary treatment of invasive aspergillosis using liposomal amphotericin B reported favorable outcomes, especially with regard to minimizing toxicities . Lipid formulations of amphotericin B are useful as initial therapy for invasive aspergillosis in patients using drugs that interact with azoles. They are also useful for patients suspected of invasive aspergillosis and mucormycosis, but accurate diagnosis is difficult, especially in patients who have used prophylactic voriconazole for a long time. 2. Salvage therapy of invasive aspergillosis Salvage therapy is a form of therapy given after a disease becomes refractory to or intolerant of primary therapy. If voriconazole or isavuconazole was used for primary therapy, lipid formulations of amphotericin B can be used as salvage therapy . Representative antifungal agents for salvage therapy are echinocandins such as caspofungin, micafungin, and anidulafungin [43, 44, 45]. Caspofungin was approved as salvage therapy for invasive aspergillosis due to its high efficacy and acceptable safety . In a study of 225 patients with invasive pulmonary aspergillosis, favorable responses were observed in 50.0% and 41.0% of patients treated with micafungin as primary or salvage therapy, respectively . A unique attribute of anidulafungin is that it is eliminated by non-enzymatic degradation in the blood and does not require dosage adjustments in patients with renal or hepatic dysfunction . Combination therapy for primary therapy of invasive aspergillosis is not routinely recommended, but it may be considered in a salvage situation. Because of their distinct mechanism of action, echinocandins have the potential for use in combination with antifungal agents with different mechanisms of action [14, 54]. Antifungal agent resistance due to antifungal abuse raised concern, and efforts for antifungal stewardship should not be neglected . Go to: AbstractINTRODUCTIONEPIDEMIOLOGYDIAGNOSIS OF IMDsTREATMENT OF INVASIVE ASPERGILLOSISTREATMENT OF INVASIVE MUCORMYCOSISOTHER MOLD DISEASESCONCLUSIONNotesReferences TREATMENT OF INVASIVE MUCORMYCOSIS It is difficult to treat invasive mucormycosis with only antifungal agents because of its rapid progression and tissue destruction. Therefore, the treatment of invasive mucormycosis involves a combination of urgent surgical debridement of involved tissues and antifungal therapy . In a retrospective study of invasive mucormycosis, characteristics such as single pulmonary involvement, no dissemination, and complete surgical removal of infected tissue were associated with decreased mortality . Also early initiation of antifungal therapy improves the outcome of infection with mucormycosis . There are no randomized trials that assessed the efficacy of antifungal agents for mucormycosis because it is rare. Lipid formulations of amphotericin B are the drug of choice for initial therapy. The usual starting dose is 5 mg/kg daily of liposomal amphotericin B or amphotericin B lipid complex, and it can be increased as high as 10 mg/kg daily in order to control the infection (Table 3) . A small sized study enrolled 21 patients treated with isavuconazole as initial therapy, and compared the results to 33 matched patients treated with amphotericin B formulations from the FungiScope registry and showed that isavuconazole had similar efficacy to that of amphotericin B formulations . As a result, isavuconazole has been licensed in the USA for the initial therapy of mucormycosis. Many studies using posaconazole in the treatment of invasive mucormycosis are currently underway. Table 3 Antifungal treatment of invasive mucormycosis, hyalohyphomycosis, and phaeohyphomycosis Click for larger image Click for full table Download as Excel file Salvage therapy with isavuconazole was successful in clinical scenarios such as refractory disease, intolerance, or toxicity . Posaconazole salvage therapy with oral suspension achieved a cure in two non-randomized clinical trials [60, 61]. For patients who have responded to a lipid formulation of amphotericin B, isavuconazole or posaconazole can be used for oral step-down therapy. Go to: AbstractINTRODUCTIONEPIDEMIOLOGYDIAGNOSIS OF IMDsTREATMENT OF INVASIVE ASPERGILLOSISTREATMENT OF INVASIVE MUCORMYCOSISOTHER MOLD DISEASESCONCLUSIONNotesReferences OTHER MOLD DISEASES IMDs other than aspergillosis and mucormycosis are hyalohyphomycosis and phaeohyphomycosis . In the past, the term "zygomycosis" was commonly used; however, the term "mucormycosis" is mainly used concurrently because the genera in the order Mucorales cause most cases of human infection . Molds can be broadly divided into two morphologically distinct groups: those that produce septate hyphae and those that produce aseptate hyphae. Identification of aseptate hyphae in tissue is virtually pathognomonic of zygomycosis (mucormycosis). The discovery of septate hyphae in tissue is less diagnostic as septate hyphae may be caused by a vast number of species of molds. Septate molds are usually divided into those with pale or colorless (hyaline) hyphae (hyalohyphomycetes) and those with darkly pigmented hyphae (phaeohyphomycetes) . 1. Hyalohyphomycosis Although infection due to Aspergillus spp. fits the description of colorless hyphae, aspergillosis is typically not included in hyalohyphomycosis, which includes various species such as Fusarium, Scedosporium, Penicillium, and Acremonium . In patients with hematologic malignancy and HSCT recipients, almost all cases of fusariosis are disseminated at presentation [5, 63]. In contrast, fusariosis that occurs after SOT tends to be localized, and the outcome is better than in patients who have neutropenia . Voriconazole or a lipid formulation of amphotericin B is recommended as primary therapy for invasive fusariosis (Table 3) . Scedosporium spp. causes a wide spectrum of conditions, including mycetoma, colonization of the airway, sinopulmonary infections, extrapulmonary localized infections, and disseminated infections . SOT from a nearly-drowned donor resulted in cases of fatal scedosporiosis . As for primary therapy for invasive scedosporiosis, voriconazole is recommended together with surgical debridement when possible . 2. Phaeohyphomycosis Phaeohyphomycosis is the result of infection with various species including Alternaria, Exophiala, Dactylaria, Cladophialophora, and Curvularia . They cause a broad spectrum of diseases including skin and subcutaneous lesions, pneumonia, central nervous system disease, fungemia, and disseminated disease, particularly in immunocompromised patients. Extracutaneous invasive diseases can also occur in immunocompetent patients but are much less common. Skin and subcutaneous diseases were more common in SOT recipients, while pulmonary diseases were more common in HSCT recipients [66, 67]. There is no standard therapy for phaeohyphomycosis, and depends on the clinical disease and status of the patient. Surgical excision of the skin or subcutaneous lesion is often curative, although antifungal therapy is usually given in conjunction with surgery. Voriconazole, posaconazole, and itraconazole showed the most consistent in vitro activity against phaeohyphomycosis. Lipid formulations of amphotericin B have also been useful as an alternative therapy in some cases (Table 3). Combination antifungal therapy is recommended for cerebral abscesses when surgery is not possible and for disseminated infections in immunocompromised patients . Go to: AbstractINTRODUCTIONEPIDEMIOLOGYDIAGNOSIS OF IMDsTREATMENT OF INVASIVE ASPERGILLOSISTREATMENT OF INVASIVE MUCORMYCOSISOTHER MOLD DISEASESCONCLUSIONNotesReferences CONCLUSION The EORTC/MSG definitions for the diagnosis of IMDs are not conclusive. In recent years, there has been an increasing number of reports of IMD cases accompanying various diseases. Therefore, the scope of host factors will continue to expand. In pulmonary IMDs, clinical criteria are mainly composed of radiologic findings, and there are many more things to be clarified in the future as reports on various radiologic findings are increasing depending on the type of underlying disease. Various indirect testing methods have been attempted for use in mycological criteria, and studies are needed to define their efficacy and appropriate cut-off values. Clinical studies on the treatment of IMDs are mainly conducted in invasive aspergillosis, and international joint studies should be performed to investigate the optimal therapeutic options for rarer IMDs such as mucormycosis, hyalohyphomycosis, and phaeohyphomycosis. Go to: AbstractINTRODUCTIONEPIDEMIOLOGYDIAGNOSIS OF IMDsTREATMENT OF INVASIVE ASPERGILLOSISTREATMENT OF INVASIVE MUCORMYCOSISOTHER MOLD DISEASESCONCLUSIONNotesReferences Notes Funding:None. Conflict of Interest:SOL is editorial board of Infect Chemother; however, he did not involve in the peer reviewer selection, evaluation, and decision process of this article. Otherwise, no potential conflicts of interest relevant to this article was reported. Go to: AbstractINTRODUCTIONEPIDEMIOLOGYDIAGNOSIS OF IMDsTREATMENT OF INVASIVE ASPERGILLOSISTREATMENT OF INVASIVE MUCORMYCOSISOTHER MOLD DISEASESCONCLUSIONNotesReferences References Drgona L, Khachatryan A, Stephens J, Charbonneau C, Kantecki M, Haider S, Barnes R. 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PubMed CrossRef Salmanton-García J, Sprute R, Stemler J, Bartoletti M, Dupont D, Valerio M, Garcia-Vidal C, Falces-Romero I, Machado M, de la Villa S, Schroeder M, Hoyo I, Hanses F, Ferreira-Paim K, Giacobbe DR, Meis JF, Gangneux JP, Rodríguez-Guardado A, Antinori S, Sal E, Malaj X, Seidel D, Cornely OA, Koehler P. FungiScope European Confederation of Medical Mycology/The International Society for Human and Animal Mycology Working Group. COVID-19-associated pulmonary aspergillosis, March-August 2020. Emerg Infect Dis 2021;27:1077–1086. CrossRef Chong WH, Neu KP. Incidence, diagnosis and outcomes of COVID-19-associated pulmonary aspergillosis (CAPA): a systematic review. J Hosp Infect 2021;113:115–129. PubMed CrossRef Husain S, Camargo JF. Invasive aspergillosis in solid-organ transplant recipients: guidelines from the American Society of Transplantation Infectious Diseases community of practice. Clin Transplant 2019;33:e13544 PubMed CrossRef Patterson TF, Thompson GR 3rd, Denning DW, Fishman JA, Hadley S, Herbrecht R, Kontoyiannis DP, Marr KA, Morrison VA, Nguyen MH, Segal BH, Steinbach WJ, Stevens DA, Walsh TJ, Wingard JR, Young JA, Bennett JE. Practice guidelines for the diagnosis and management of aspergillosis: 2016 update by the Infectious Diseases Society of America. Clin Infect Dis 2016;63:e1–60. PubMed CrossRef Ullmann AJ, Aguado JM, Arikan-Akdagli S, Denning DW, Groll AH, Lagrou K, Lass-Flörl C, Lewis RE, Munoz P, Verweij PE, Warris A, Ader F, Akova M, Arendrup MC, Barnes RA, Beigelman-Aubry C, Blot S, Bouza E, Brüggemann RJM, Buchheidt D, Cadranel J, Castagnola E, Chakrabarti A, Cuenca-Estrella M, Dimopoulos G, Fortun J, Gangneux JP, Garbino J, Heinz WJ, Herbrecht R, Heussel CP, Kibbler CC, Klimko N, Kullberg BJ, Lange C, Lehrnbecher T, Löffler J, Lortholary O, Maertens J, Marchetti O, Meis JF, Pagano L, Ribaud P, Richardson M, Roilides E, Ruhnke M, Sanguinetti M, Sheppard DC, Sinkó J, Skiada A, Vehreschild MJGT, Viscoli C, Cornely OA. Diagnosis and management of Aspergillus diseases: executive summary of the 2017 ESCMID-ECMM-ERS guideline. Clin Microbiol Infect 2018;24 Suppl 1:e1–38. CrossRef Stevens DA, Kan VL, Judson MA, Morrison VA, Dummer S, Denning DW, Bennett JE, Walsh TJ, Patterson TF, Pankey GA. Infectious Diseases Society of America. Practice guidelines for diseases caused by Aspergillus . Clin Infect Dis 2000;30:696–709. PubMed CrossRef Herbrecht R, Denning DW, Patterson TF, Bennett JE, Greene RE, Oestmann JW, Kern WV, Marr KA, Ribaud P, Lortholary O, Sylvester R, Rubin RH, Wingard JR, Stark P, Durand C, Caillot D, Thiel E, Chandrasekar PH, Hodges MR, Schlamm HT, Troke PF, de Pauw B. Invasive Fungal Infections Group of the European Organisation for Research and Treatment of Cancer and the Global Aspergillus Study Group. Voriconazole versus amphotericin B for primary therapy of invasive aspergillosis. N Engl J Med 2002;347:408–415. PubMed CrossRef Park WB, Kim NH, Kim KH, Lee SH, Nam WS, Yoon SH, Song KH, Choe PG, Kim NJ, Jang IJ, Oh MD, Yu KS. The effect of therapeutic drug monitoring on safety and efficacy of voriconazole in invasive fungal infections: a randomized controlled trial. Clin Infect Dis 2012;55:1080–1087. PubMed CrossRef Jenks JD, Mehta SR, Hoenigl M. 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PubMed CrossRef Denning DW, Marr KA, Lau WM, Facklam DP, Ratanatharathorn V, Becker C, Ullmann AJ, Seibel NL, Flynn PM, van Burik JA, Buell DN, Patterson TF. Micafungin (FK463), alone or in combination with other systemic antifungal agents, for the treatment of acute invasive aspergillosis. J Infect 2006;53:337–349. PubMed CrossRef Cleary JD. Echinocandins: pharmacokinetic and therapeutic issues. Curr Med Res Opin 2009;25:1741–1750. PubMed Cuenca-Estrella M. Combinations of antifungal agents in therapy--what value are they? J Antimicrob Chemother 2004;54:854–869. PubMed CrossRef Yoon YK, Kwon KT, Jeong SJ, Moon C, Kim B, Kiem S, Kim HS, Heo E, Kim SW. Korean Society for Antimicrobial Therapy; Korean Society of Infectious Diseases; Korean Society of Health-System Pharmacist. Guidelines on implementing antimicrobial stewardship programs in Korea. Infect Chemother 2021;53:617–659. 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Posaconazole is effective as salvage therapy in zygomycosis: a retrospective summary of 91 cases. Clin Infect Dis 2006;42:e61–e65. PubMed CrossRef Naggie S, Perfect JR. Molds: hyalohyphomycosis, phaeohyphomycosis, and zygomycosis. Clin Chest Med 2009;30:337–353. [vii-viii.]. PubMed CrossRef Tortorano AM, Richardson M, Roilides E, van Diepeningen A, Caira M, Munoz P, Johnson E, Meletiadis J, Pana ZD, Lackner M, Verweij P, Freiberger T, Cornely OA, Arikan-Akdagli S, Dannaoui E, Groll AH, Lagrou K, Chakrabarti A, Lanternier F, Pagano L, Skiada A, Akova M, Arendrup MC, Boekhout T, Chowdhary A, Cuenca-Estrella M, Guinea J, Guarro J, de Hoog S, Hope W, Kathuria S, Lortholary O, Meis JF, Ullmann AJ, Petrikkos G, Lass-Flörl C. European Society of Clinical Microbiology and Infectious Diseases Fungal Infection Study Group; European Confederation of Medical Mycology. ESCMID and ECMM joint guidelines on diagnosis and management of hyalohyphomycosis: Fusarium spp., Scedosporium spp. and others. Clin Microbiol Infect 2014;20 Suppl 3:27–46. CrossRef Sampathkumar P, Paya CV. Fusarium infection after solid-organ transplantation. Clin Infect Dis 2001;32:1237–1240. PubMed CrossRef Ramirez-Garcia A, Pellon A, Rementeria A, Buldain I, Barreto-Bergter E, Rollin-Pinheiro R, de Meirelles JV, Xisto MIDS, Ranque S, Havlicek V, Vandeputte P, Govic YL, Bouchara JP, Giraud S, Chen S, Rainer J, Alastruey-Izquierdo A, Martin-Gomez MT, López-Soria LM, Peman J, Schwarz C, Bernhardt A, Tintelnot K, Capilla J, Martin-Vicente A, Cano-Lira J, Nagl M, Lackner M, Irinyi L, Meyer W, de Hoog S, Hernando FL. Scedosporium and Lomentospora: an updated overview of underrated opportunists. Med Mycol 2018;56 suppl_1:102–125. PubMed CrossRef Chowdhary A, Meis JF, Guarro J, de Hoog GS, Kathuria S, Arendrup MC, Arikan-Akdagli S, Akova M, Boekhout T, Caira M, Guinea J, Chakrabarti A, Dannaoui E, van Diepeningen A, Freiberger T, Groll AH, Hope WW, Johnson E, Lackner M, Lagrou K, Lanternier F, Lass-Flörl C, Lortholary O, Meletiadis J, Muñoz P, Pagano L, Petrikkos G, Richardson MD, Roilides E, Skiada A, Tortorano AM, Ullmann AJ, Verweij PE, Cornely OA, Cuenca-Estrella M. European Society of Clinical Microbiology and Infectious Diseases Fungal Infection Study Group; European Confederation of Medical Mycology. ESCMID and ECMM joint clinical guidelines for the diagnosis and management of systemic phaeohyphomycosis: diseases caused by black fungi. Clin Microbiol Infect 2014;20 Suppl 3:47–75. CrossRef McCarty TP, Baddley JW, Walsh TJ, Alexander BD, Kontoyiannis DP, Perl TM, Walker R, Patterson TF, Schuster MG, Lyon GM, Wingard JR, Andes DR, Park BJ, Brandt ME, Pappas PG. TRANSNET Investigators. Phaeohyphomycosis in transplant recipients: Results from the transplant associated infection surveillance network (TRANSNET). Med Mycol 2015;53:440–446. PubMed CrossRef Cite Article PDF Cited by Crossref 7 Google Scholar PubMed 8 Publication Types Review MeSH Terms Amphotericin B Antifungal Agents Aspergillosis Communicable Diseases Debridement Diagnosis Echinocandins Epidemiology Fungi Humans Hyalohyphomycosis Hypersensitivity Hyphae Incidence Invasive Fungal Infections Lipids Mucormycosis Mycoses Phaeohyphomycosis Pharmaceutical Preparations Salvage Therapy Voriconazole Substance isavuconazole posaconazole Since 2023/03/01 Metrics Page Views 793 PDF Downloads 421 Share Links to PubMed PubMed Central Related citations in PubMed Show all... 1 / 3 ORCID IDs Sang-Oh Lee Permalink information copied to clipboard
188973	https://www.symbolab.com/solver/tangent-line-calculator	Go To QuillBot Upgrade to Pro Continue to site We've updated ourPrivacy Policy effective December 15. Please read our updated Privacy Policy and tap Back to School Promotion Annual Annual - $ % OFF Annual plan One time offer for one year, then $ Special Back-to-School Offer Study smarter this year – Save 30% on Symbolab Pro Solutions > Calculus Calculator > Tangent Line Calculator Topic Pre Algebra Algebra Pre Calculus Calculus Derivatives First Derivative WRT Specify Method Chain Rule Product Rule Quotient Rule Sum/Diff Rule Second Derivative Third Derivative Higher Order Derivatives Derivative at a point Partial Derivative Implicit Derivative Second Implicit Derivative Derivative using Definition Derivative Applications Tangent Slope of Tangent Normal Curved Line Slope Extreme Points Tangent to Conic Linear Approximation Difference Quotient Horizontal Tangent Limits One Variable Multi Variable Limit One Sided At Infinity Specify Method L'Hopital's Rule Squeeze Theorem Chain Rule Factoring Substitution Sandwich Theorem Integrals Indefinite Integrals Definite Integrals Specific-Method Partial Fractions U-Substitution Trigonometric Substitution Weierstrass Substitution By Parts Long Division Improper Integrals Antiderivatives Double Integrals Triple Integrals Multiple Integrals Integral Applications Limit of Sum Area under curve Area between curves Area under polar curve Volume of solid of revolution Arc Length Function Average Integral Approximation Riemann Sum Trapezoidal Simpson's Rule Midpoint Rule Series Convergence Geometric Series Test Telescoping Series Test Alternating Series Test P Series Test Divergence Test Ratio Test Root Test Comparison Test Limit Comparison Test Integral Test Absolute Convergence Power Series Radius of Convergence Interval of Convergence ODE Linear First Order Linear w/constant coefficients Separable Bernoulli Exact Second Order Homogenous Non Homogenous Substitution System of ODEs IVP using Laplace Series Solutions Method of Frobenius Gamma Function Multivariable Calculus Partial Derivative Implicit Derivative Tangent to Conic Multi Variable Limit Multiple Integrals Gradient Divergence Extreme Points Laplace Transform Inverse Taylor/Maclaurin Series Taylor Series Maclaurin Series Fourier Series Fourier Transform Functions Linear Algebra Trigonometry Statistics Physics Chemistry Finance Economics Conversions Get our extension, you can capture any math problem from any website Full pad \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- \| \| x2 \| x□ \| log□ \| √☐ \| □√☐ \| ≤ \| ≥ \| □□ \| · \| ÷ \| x◦ \| π \| \| (☐)′ \| ddx \| ∂∂x \| ∫ \| ∫□□ \| lim \| ∑ \| ∞ \| θ \| (f ◦ g) \| f(x) \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- \| −▭▭ \| < \| 7 \| 8 \| 9 \| ÷ \| AC \| \| +▭▭ \| > \| 4 \| 5 \| 6 \| × \| ☐☐☐ \| \| ×▭▭ \| ( \| 1 \| 2 \| 3 \| − \| x \| \| ▭ \|▭ \| ) \| . \| 0 \| = \| + \| y \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- \| \| Basic \| αβγ \| ABΓ \| sincos \| ≥÷→ \| x ℂ∀ \| ∑ ∫ ∏ \| ( \| \| \| --- \| \| ☐ \| ☐ \| \| ☐ \| ☐ \| ) \| H2O \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| ☐2 \| x☐ \| √☐ \| □√☐ \| □□ \| log□ \| π \| θ \| ∞ \| ∫ \| ddx \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| ≥ \| ≤ \| · \| ÷ \| x◦ \| (☐) \| \|☐\| \| (f ◦ g) \| f(x) \| ln \| e☐ \| \| (☐)′ \| ∂∂x \| ∫□□ \| lim \| ∑ \| sin \| cos \| tan \| cot \| csc \| sec \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| α \| β \| γ \| δ \| ζ \| η \| θ \| ι \| κ \| λ \| μ \| \| ν \| ξ \| π \| ρ \| σ \| τ \| υ \| ϕ \| χ \| ψ \| ω \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| A \| B \| Γ \| Δ \| E \| Z \| H \| Θ \| K \| Λ \| M \| \| N \| Ξ \| Π \| P \| Σ \| T \| ϒ \| Φ \| X \| Ψ \| Ω \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| sin \| cos \| tan \| cot \| sec \| csc \| sinh \| cosh \| tanh \| coth \| sech \| \| arcsin \| arccos \| arctan \| arccot \| arcsec \| arccsc \| arcsinh \| arccosh \| arctanh \| arccoth \| arcsech \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- --- \| \| { \| \| \| ☐ \| \| ☐ \| \| { \| \| \| ☐ \| \| ☐ \| \| ☐ \| \| = \| ≠ \| ÷ \| · \| × \| < \| > \| ≤ \| ≥ \| \| (☐) \| [☐] \| ▭ \|▭ \| ×▭▭ \| +▭▭ \| −▭▭ \| ☐! \| x◦ \| → \| ⌊☐⌋ \| ⌈☐⌉ \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| ☐ \| ⃗ ☐ \| ∈ \| ∀ \| ∉ \| ∃ \| ℝ \| ℂ \| ℕ \| ℤ \| ∅ \| \| ∨ \| ∧ \| ¬ \| ⊕ \| ∩ \| ∪ \| ☐c \| ⊂ \| ⊆ \| ⊃ \| ⊇ \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- \| ∫ \| ∫∫ \| ∫∫∫ \| ∫☐☐ \| ∫☐☐∫☐☐ \| ∫☐☐∫☐☐∫☐☐ \| \| ∑ \| ∏ \| \| lim \| limx→∞ \| limx→0+ \| limx→0− \| ddx \| d2dx2 \| (☐)′ \| (☐)′′ \| ∂∂x \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| (2×2) \| (2×3) \| (3×3) \| (3×2) \| (4×2) \| (4×3) \| (4×4) \| (3×4) \| (2×4) \| (5×5) \| \| \| (1×2) \| (1×3) \| (1×4) \| (1×5) \| (1×6) \| (2×1) \| (3×1) \| (4×1) \| (5×1) \| (6×1) \| (7×1) \| \| \| \| \| \| \| \| \| --- --- --- \| Radians \| Degrees \| ☐! \| ( \| ) \| % \| clear \| \| arcsin \| sin \| √☐ \| 7 \| 8 \| 9 \| ÷ \| \| arccos \| cos \| ln \| 4 \| 5 \| 6 \| × \| \| arctan \| tan \| log \| 1 \| 2 \| 3 \| − \| \| π \| e \| x☐ \| 0 \| . \| = \| + \| \| \| \| \| \| \| \| \| \| \| \| implicit derivative tangent volume laplace fourier See All area asymptotes critical points derivative domain eigenvalues eigenvectors expand extreme points factor implicit derivative inflection points intercepts inverse laplace inverse laplace partial fractions range slope simplify solve for tangent taylor vertex geometric test alternating test telescoping test pseries test root test tangent of Steps Graph Related Examples Generated by AI AI explanations are generated using OpenAI technology. AI generated content may present inaccurate or offensive content that does not represent Symbolab's view. Verify your Answer Subscribe to verify your answer Subscribe Save to Notebook! Sign in to save notes Sign in Verify Save Show Steps Hide Steps Number Line Related Tangent Line Examples tangent of f(x)=1x2 , (−1, 1) tangent of f(x)=x3+2x, x=0 tangent of f(x)=4x2−4x+1, x=1 tangent of y=e−x·ln(x), (1,0) tangent of f(x)=sin(3x), (π6 , 1) tangent of y=√x2+1, (0, 1) Description Find the equation of the tangent line step-by-step Frequently Asked Questions (FAQ) What is a tangent line? A tangent line is a line that touches a curve at a single point and has the same slope as the curve at that point. It provides a good approximation of the behavior of the curve near that point. How do you find the tangent line? The tangent line can be found by finding the slope of the curve at a specific point, and then using the point-slope form of a line equation to find the equation of the tangent line. The slope can be found using the derivative of the function, and the point of tangency can be found by substituting the x-coordinate of the point into the original function. How do you find the slope of a tangent line? The slope of a tangent line can be found by taking the derivative of the function at a specific point. What Tangent line is used for? Tangent lines are used for approximation, optimization, calculus, physics and engineering. They are used to model the velocity, acceleration and other physical quantities, as well as real-world phenomena such as fluid flow, heat transfer, and more. tangent-line-calculator en Related Symbolab blog posts Slope, Distance and More Ski Vacation? Nope, this is serious stuff; it’s about finding the slope of a line, finding the equation of a line... 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188974	https://www.math.unl.edu/~jkettinger2/thurston.pdf	William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T EX by Sheila Newbery, who also scanned the ﬁgures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the ﬁrst few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at levy@msri.org. Introduction These notes (through p. 9.80) are based on my course at Princeton in 1978– 79. Large portions were written by Bill Floyd and Steve Kerckhoﬀ. Chapter 7, by John Milnor, is based on a lecture he gave in my course; the ghostwriter was Steve Kerckhoﬀ. The notes are projected to continue at least through the next academic year. The intent is to describe the very strong connection between geometry and low-dimensional topology in a way which will be useful and accessible (with some eﬀort) to graduate students and mathematicians working in related ﬁelds, particularly 3-manifolds and Kleinian groups. Much of the material or technique is new, and more of it was new to me. As a consequence, I did not always know where I was going, and the discussion often tends to wanter. The countryside is scenic, however, and it is fun to tramp around if you keep your eyes alert and don’t get lost. The tendency to meander rather than to follow the quickest linear route is especially pronounced in chapters 8 and 9, where I only gradually saw the usefulness of “train tracks” and the value of mapping out some global information about the structure of the set of simple geodesic on surfaces. I would be grateful to hear any suggestions or corrections from readers, since changes are fairly easy to make at this stage. In particular, bibliographical informa-tion is missing in many places, and I would like to solicit references (perhaps in the form of preprints) and historical information. Thurston — The Geometry and Topology of 3-Manifolds iii Contents Introduction iii Chapter 1. Geometry and three-manifolds 1 Chapter 2. Elliptic and hyperbolic geometry 9 2.1. The Poincar´ e disk model. 10 2.2. The southern hemisphere. 11 2.3. The upper half-space model. 12 2.4. The projective model. 13 2.5. The sphere of imaginary radius. 16 2.6. Trigonometry. 17 Chapter 3. Geometric structures on manifolds 27 3.1. A hyperbolic structure on the ﬁgure-eight knot complement. 29 3.2. A hyperbolic manifold with geodesic boundary. 31 3.3. The Whitehead link complement. 32 3.4. The Borromean rings complement. 33 3.5. The developing map. 34 3.8. Horospheres. 38 3.9. Hyperbolic surfaces obtained from ideal triangles. 40 3.10. Hyperbolic manifolds obtained by gluing ideal polyhedra. 42 Chapter 4. Hyperbolic Dehn surgery 45 4.1. Ideal tetrahedra in H3. 45 4.2. Gluing consistency conditions. 48 4.3. Hyperbolic structure on the ﬁgure-eight knot complement. 50 4.4. The completion of hyperbolic three-manifolds obtained from ideal polyhedra. 54 4.5. The generalized Dehn surgery invariant. 56 4.6. Dehn surgery on the ﬁgure-eight knot. 58 4.8. Degeneration of hyperbolic structures. 61 4.10. Incompressible surfaces in the ﬁgure-eight knot complement. 71 Thurston — The Geometry and Topology of 3-Manifolds v CONTENTS Chapter 5. Flexibility and rigidity of geometric structures 85 5.2. 86 5.3. 88 5.4. Special algebraic properties of groups of isometries of H3. 92 5.5. The dimension of the deformation space of a hyperbolic three-manifold. 96 5.7. 101 5.8. Generalized Dehn surgery and hyperbolic structures. 102 5.9. A Proof of Mostow’s Theorem. 106 5.10. A decomposition of complete hyperbolic manifolds. 112 5.11. Complete hyperbolic manifolds with bounded volume. 116 5.12. Jørgensen’s Theorem. 119 Chapter 6. Gromov’s invariant and the volume of a hyperbolic manifold 123 6.1. Gromov’s invariant 123 6.3. Gromov’s proof of Mostow’s Theorem 129 6.5. Manifolds with Boundary 134 6.6. Ordinals 138 6.7. Commensurability 140 6.8. Some Examples 144 Chapter 7. Computation of volume 157 7.1. The Lobachevsky function l(θ). 157 7.2. 160 7.3. 165 7.4. 167 References 170 Chapter 8. Kleinian groups 171 8.1. The limit set 171 8.2. The domain of discontinuity 174 8.3. Convex hyperbolic manifolds 176 8.4. Geometrically ﬁnite groups 180 8.5. The geometry of the boundary of the convex hull 185 8.6. Measuring laminations 189 8.7. Quasi-Fuchsian groups 191 8.8. Uncrumpled surfaces 199 8.9. The structure of geodesic laminations: train tracks 204 8.10. Realizing laminations in three-manifolds 208 8.11. The structure of cusps 216 8.12. Harmonic functions and ergodicity 219 vi Thurston — The Geometry and Topology of 3-Manifolds CONTENTS Chapter 9. Algebraic convergence 225 9.1. Limits of discrete groups 225 9.3. The ending of an end 233 9.4. Taming the topology of an end 240 9.5. Interpolating negatively curved surfaces 242 9.6. Strong convergence from algebraic convergence 257 9.7. Realizations of geodesic laminations for surface groups with extra cusps, with a digression on stereographic coordinates 261 9.9. Ergodicity of the geodesic ﬂow 277 NOTE 283 Chapter 11. Deforming Kleinian manifolds by homeomorphisms of the sphere at inﬁnity 285 11.1. Extensions of vector ﬁelds 285 Chapter 13. Orbifolds 297 13.1. Some examples of quotient spaces. 297 13.2. Basic deﬁnitions. 300 13.3. Two-dimensional orbifolds. 308 13.4. Fibrations. 318 13.5. Tetrahedral orbifolds. 323 13.6. Andreev’s theorem and generalizations. 330 13.7. Constructing patterns of circles. 337 13.8. A geometric compactiﬁcation for the Teichm¨ uller spaces of polygonal orbifolds 346 13.9. A geometric compactiﬁcation for the deformation spaces of certain Kleinian groups. 350 Index 357 Thurston — The Geometry and Topology of 3-Manifolds vii CHAPTER 1 Geometry and three-manifolds 1.1 The theme I intend to develop is that topology and geometry, in dimensions up through 3, are very intricately related. Because of this relation, many questions which seem utterly hopeless from a purely topological point of view can be fruitfully studied. It is not totally unreasonable to hope that eventually all three-manifolds will be understood in a systematic way. In any case, the theory of geometry in three-manifolds promises to be very rich, bringing together many threads. Before discussing geometry, I will indicate some topological constructions yielding diverse three-manifolds, which appear to be very tangled. 0. Start with the three sphere S3, which may be easily visualized as R3, together with one point at inﬁnity. 1. Any knot (closed simple curve) or link (union of disjoint closed simple curves) may be removed. These examples can be made compact by removing the interior of a tubular neighborhood of the knot or link. 1.2 Thurston — The Geometry and Topology of 3-Manifolds 1 1. GEOMETRY AND THREE-MANIFOLDS The complement of a knot can be very enigmatic, if you try to think about it from an intrinsic point of view. Papakyriakopoulos proved that a knot complement has fundamental group Z if and only if the knot is trivial. This may seem intuitively clear, but justiﬁcation for this intuition is diﬃcult. It is not known whether knots with homeomorphic complements are the same. 2. Cut out a tubular neighborhood of a knot or link, and glue it back in by a diﬀerent identiﬁcation. This is called Dehn surgery. There are many ways to do this, because the torus has many diﬀeomorphisms. The generator of the kernel of the inclusion map π1(T 2) →π1 (solid torus) in the resulting three-manifold determines the three-manifold. The diﬀeomorphism can be chosen to make this generator an arbitrary primitive (indivisible non-zero) element of Z ⊕Z. It is well deﬁned up to change in sign. Every oriented three-manifold can be obtained by this construction (Lickorish) . It is diﬃcult, in general, to tell much about the three-manifold resulting from this construction. When, for instance, is it simply connected? When is it irreducible? (Irreducible means every embedded two sphere bounds a ball). Note that the homology of the three-manifold is a very insensitive invariant. The homology of a knot complement is the same as the homology of a circle, so when Dehn surgery is performed, the resulting manifold always has a cyclic ﬁrst homology group. If generators for Z ⊕Z = π1(T 2) are chosen so that (1, 0) generates the homology of the complement and (0, 1) is trivial then any Dehn surgery with invariant (1, n) yields a homology sphere. 3. Branched coverings. If L is a link, then any ﬁnite-sheeted covering space of S3 −L can be compactiﬁed in a canonical way by adding circles which cover L to give a closed manifold, M. M is called a 1.3 branched covering of S3 over L. There is a canonical projection p : M →S3, which is a local diﬀeomorphism away from p−1(L). If K ⊂S3 is a knot, the simplest branched coverings of S3 over K are then n-fold cyclic branched covers, which come from the covering spaces of S3 −K whose fundamental group is the kernel of the composition π1(S3 −K) →H1(S3 −K) = Z →Zn. In other words, they are unwrapping S3 from K n times. If K is the trivial knot the cyclic branched covers are S3. It seems intuitively obvious (but it is not known) that this is the only way S3 can be obtained as a cyclic branched covering of itself over a knot. Montesinos and Hilden (independently) showed that every oriented three-manifold is a branched cover of S3 with 3 sheets, branched over some knot. These branched coverings are not in general regular: there are no covering transformations. The formation of irregular branched coverings is somehow a much more ﬂexible construction than the formation of regular branched coverings. For instance, it is not hard to ﬁnd many diﬀerent ways in which S3 is an irregular branched cover of itself. 2 Thurston — The Geometry and Topology of 3-Manifolds 1. GEOMETRY AND THREE-MANIFOLDS 5. Heegaard decompositions. Every three-manifold can be obtained from two handlebodies (of some genus) by gluing their boundaries together. 1.4 The set of possible gluing maps is large and complicated. It is hard to tell, given two gluing maps, whether or not they represent the same three-manifold (except when there are homological invariants to distinguish them). 6. Identifying faces of polyhedra. Suppose P1, . . . , Pk are polyhedra such that the number of faces with K sides is even, for each K. Choose an arbitrary pattern of orientation-reversing identiﬁcations of pairs of two-faces. This yields a three-complex, which is an oriented manifold except near the vertices. (Around an edge, the link is automatically a circle.) There is a classical criterion which says that such a complex is a manifold if and only if its Euler characteristic is zero. We leave this as an exercise. In any case, however, we may simply remove a neighborhood of each bad vertex, to obtain a three-manifold with boundary. The number of (at least not obviously homeomorphic) three-manifolds grows very quickly with the complexity of the description. Consider, for instance, diﬀerent ways to obtain a three-manifold by gluing the faces of an octahedron. There are 8! 24 · 4! · 34 = 8,505 possibilities. For an icosahedron, the ﬁgure is 38,661 billion. Because these polyhedra are symmetric, many gluing diagrams obviously yield homeomorphic results—but this reduces the ﬁgure by a factor of less than 120 for the icosahedron, for instance. In two dimensions, the number of possible ways to glue sides of 2n-gon to obtain an oriented surface also grows rapidly with n: it is (2n)!/(2nn!). In view of the amazing fact that the Euler characteristic is a complete invariant of a closed oriented surface, huge numbers of these gluing patterns give identical surfaces. It seems unlikely that 1.5 Thurston — The Geometry and Topology of 3-Manifolds 3 1. GEOMETRY AND THREE-MANIFOLDS such a phenomenon takes place among three-manifolds; but how can we tell? Example. Here is one of the simplest possible gluing diagrams for a three-manifold. Begin with two tetrahedra with edges labeled: There is a unique way to glue the faces of one tetrahedron to the other so that arrows are matched. For instance, A is matched with A′. All the ̸− →arrows are identiﬁed and all the ̸ ̸− →arrows are identiﬁed, so the resulting complex has 2 tetrahedra, 4 triangles, 2 edges and 1 vertex. Its Euler characteristic is +1, and (it follows that) a neighborhood of the vertex is the cone on a torus. Let M be the manifold obtained by removing the vertex. It turns out that this manifold is homeomorphic with the complement of a ﬁgure-eight knot. 4 Thurston — The Geometry and Topology of 3-Manifolds 1. GEOMETRY AND THREE-MANIFOLDS 1.6 Another view of the ﬁgure-eight knot This knot is familiar from extension cords, as the most commonly occurring knot, after the trefoil knot In order to see this homeomorphism we can draw a more suggestive picture of the ﬁgure-eight knot, arranged along the one-skeleton of a tetrahedron. The knot can be Tetrahedron with ﬁgure-eight knot, viewed from above Thurston — The Geometry and Topology of 3-Manifolds 5 1. GEOMETRY AND THREE-MANIFOLDS spanned by a two-complex, with two edges, shown as arrows, and four two-cells, one for each face of the tetrahedron, in a more-or-less obvious way: 1.7 This pictures illustrates the typical way in which a two-cell is attached. Keeping in mind that the knot is not there, the cells are triangles with deleted vertices. The two complementary regions of the two-complex are the tetrahedra, with deleted vertices. We will return to this example later. For now, it serves to illustrate the need for a systematic way to compare and to recognize manifolds. Note. Suggestive pictures can also be deceptive. A trefoil knot can similarly be arranged along the one-skeleton of a tetrahedron: 1.8 6 Thurston — The Geometry and Topology of 3-Manifolds 1. GEOMETRY AND THREE-MANIFOLDS From the picture, a cell-division of the complement is produced. In this case, however, the three-cells are not tetrahedra. The boundary of a three-cell, ﬂattened out on the plane. Thurston — The Geometry and Topology of 3-Manifolds 7 William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T EX by Sheila Newbery, who also scanned the ﬁgures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the ﬁrst few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at levy@msri.org. CHAPTER 2 Elliptic and hyperbolic geometry There are three kinds of geometry which possess a notion of distance, and which look the same from any viewpoint with your head turned in any orientation: these are elliptic geometry (or spherical geometry), Euclidean or parabolic geometry, and hyperbolic or Lobachevskiian geometry. The underlying spaces of these three geome-tries are naturally Riemannian manifolds of constant sectional curvature +1, 0, and −1, respectively. Elliptic n-space is the n-sphere, with antipodal points identiﬁed. Topologically it is projective n-space, with geometry inherited from the sphere. The geometry of elliptic space is nicer than that of the sphere because of the elimination of identical, antipodal ﬁgures which always pop up in spherical geometry. Thus, any two points in elliptic space determine a unique line, for instance. In the sphere, an object moving away from you appears smaller and smaller, until it reaches a distance of π/2. Then, it starts looking larger and larger and optically, it is in focus behind you. Finally, when it reaches a distance of π, it appears so large that it would seem to surround you entirely. 2.2 In elliptic space, on the other hand, the maximum distance is π/2, so that ap-parent size is a monotone decreasing function of distance. It would nonetheless be Thurston — The Geometry and Topology of 3-Manifolds 9 2. ELLIPTIC AND HYPERBOLIC GEOMETRY distressing to live in elliptic space, since you would always be confronted with an im-age of yourself, turned inside out, upside down and ﬁlling out the entire background of your ﬁeld of view. Euclidean space is familiar to all of us, since it very closely approximates the geometry of the space in which we live, up to moderate distances. Hyperbolic space is the least familiar to most people. Certain surfaces of revolution in R3 have constant curvature −1 and so give an idea of the local picture of the hyperbolic plane. 2.3 The simplest of these is the pseudosphere, the surface of revolution generated by a tractrix. A tractrix is the track of a box of stones which starts at (0, 1) and is dragged by a team of oxen walking along the x-axis and pulling the box by a chain of unit length. Equivalently, this curve is determined up to translation by the property that its tangent lines meet the x-axis a unit distance from the point of tangency. The pseudosphere is not complete, however—it has an edge, beyond which it cannot be extended. Hilbert proved the remarkable theorem that no complete C2 surface with curvature −1 can exist in R3. In spite of this, convincing physical models can be constructed. We must therefore resort to distorted pictures of hyperbolic space. Just as it is convenient to have diﬀerent maps of the earth for understanding various aspects of its geometry: for seeing shapes, for comparing areas, for plotting geodesics in navigation; so it is useful to have several maps of hyperbolic space at our disposal. 2.1. The Poincar´ e disk model. Let Dn denote the disk of unit radius in Euclidean n-space. The interior of Dn can be taken as a map of hyperbolic space Hn. A hyperbolic line in the model is any Euclidean circle which is orthogonal to ∂Dn; a hyperbolic two-plane is a Euclidean sphere orthogonal to ∂Dn; etc. The words “circle” and “sphere” are here used in 10 Thurston — The Geometry and Topology of 3-Manifolds 2.2. THE SOUTHERN HEMISPHERE. the extended sense, to include the limiting case of a line or plane. This model is conformally correct, that is, hyperbolic angles agree with Euclidean angles, but distances are greatly distorted. Hyperbolic arc length √ ds2 is given by the formula 2.4 ds2 = 1 1 −r2 2 dx2, where √ dx2 is Euclidean arc length and r is distance from the origin. Thus, the Euclidean image of a hyperbolic object, as it moves away from the origin, shrinks in size roughly in proportion to the Euclidean distance from ∂Dn (when this distance is small). The object never actually arrives at ∂Dn, if it moves with a bounded hyperbolic velocity. The sphere ∂Dn is called the sphere at inﬁnity. It is not actually in hyperbolic space, but it can be given an interpretation purely in terms of hyperbolic geometry, as follows. Choose any base point p0 in Hn. Consider any geodesic ray R, as seen from p0. R traces out a segment of a great circle in the visual sphere at p0 (since p0 and R determine a two-plane). This visual segment converges to a point in the visual sphere. If we translate Hn so that p0 is at the origin of the Poincar´ e disk 2.5 model, we see that the points in the visual sphere correspond precisely to points in the sphere at inﬁnity, and that the end of a ray in this visual sphere corresponds to its Euclidean endpoint in the Poincar´ e disk model. 2.2. The southern hemisphere. The Poincar´ e disk Dn ⊂Rn is contained in the Poincar´ e disk Dn+1 ⊂Rn+1, as a hyperbolic n-plane in hyperbolic (n + 1)-space. Thurston — The Geometry and Topology of 3-Manifolds 11 2. ELLIPTIC AND HYPERBOLIC GEOMETRY Stereographic projection (Euclidean) from the north pole of ∂Dn+1 sends the Poincar´ e disk Dn to the southern hemisphere of Dn+1. Thus hyperbolic lines in the Poincar´ e disk go to circles on Sn orthogonal to the equator Sn−1. There is a more natural construction for this map, using only hyperbolic geometry. For each point p in Hn ⊂Hn+1, consider the hyperbolic ray perpendicular to Hn at p, and downward normal. This ray converges to a point on the sphere at inﬁnity, 2.6 which is the same as the Euclidean stereographic image of p. 2.3. The upper half-space model. This is closely related to the previous two, but it is often more convenient for computation or for constructing pictures. To obtain it, rotate the sphere Sn in Rn+1 so that the southern hemisphere lies in the half-space xn ≥0 is Rn+1. Now 12 Thurston — The Geometry and Topology of 3-Manifolds 2.4. THE PROJECTIVE MODEL. stereographic projection from the top of Sn (which is now on the equator) sends the southern hemisphere to the upper half-space xn > 0 in Rn+1. 2.7 A hyperbolic line, in the upper half-space, is a circle perpendicular to the bounding plane Rn−1 ⊂Rn. The hyperbolic metric is ds2 = (1/xn)2 dx2. Thus, the Euclidean image of a hyperbolic object moving toward Rn−1 has size precisely proportional to the Euclidean distance from Rn−1. 2.4. The projective model. This is obtained by Euclidean orthogonal projection of the southern hemisphere of Sn back to the disk Dn. Hyperbolic lines become Euclidean line segments. This model is useful for understanding incidence in a conﬁguration of lines and planes. Unlike the previous three models, it fails to be conformal, so that angles and shapes are distorted. It is better to regard this projective model to be contained not in Euclidean space, but in projective space. The projective model is very natural from a point of 2.8 view inside hyperbolic (n + 1)-space: it gives a picture of a hyperplane, Hn, in true perspective. Thus, an observer hovering above Hn in Hn+1, looking down, sees Hn Thurston — The Geometry and Topology of 3-Manifolds 13 2. ELLIPTIC AND HYPERBOLIC GEOMETRY as the interior of a disk in his visual sphere. As he moves farther up, this visual disk shrinks; as he moves down, it expands; but (unlike in Euclidean space), the visual radius of this disk is always strictly less than π/2. A line on H2 appears visually straight. It is possible to give an intrinsic meaning within hyperbolic geometry for the points outside the sphere at inﬁnity in the projective model. For instance, in the two-dimensional projective model, any two lines meet somewhere. The conventional sense of meeting means to meet inside the sphere at inﬁnity (at a ﬁnite point). If the two lines converge in the visual circle, this means that they meet on the circle at inﬁnity, and they are called parallels. Otherwise, the two lines are called ultraparallels; they have a unique common perpendicular L and they meet in some point x in the M¨ obius band outside the circle at inﬁnity. Any other line perpendicular to L passes through x, and any line through x is perpendicular to L. 2.9 To prove this, consider hyperbolic two-space as a plane P ⊂H3. Construct the plane Q through L perpendicular to P. Let U be an observer in H3. Drop a perpendicular M from U to the plane Q. Now if K is any line in P perpendicular 14 Thurston — The Geometry and Topology of 3-Manifolds 2.4. THE PROJECTIVE MODEL. 2.8a Evenly spaced lines. The region inside the circle is a plane, with a base line and a family of its perpendiculars, spaced at a distance of .051 fundamental units, as measured along the base line shown in perspective in hyperbolic 3-space (or in the projective model). The lines have been extended to their imaginary meeting point beyond the horizon. U, the observer, is directly above the X (which is .881 fundamental units away from the base line). To see the view from diﬀerent heights, use the following table (which assumes that the Euclidean diameter of the circle in your printout is about 5.25 inches or 13.3cm): To see the view of hold the picture a U at a height of distance of 2 units 11′′ ( 28 cm) 3 units 27′′ ( 69 cm) 4 units 6′ (191 cm) To see the view of hold the picture a U at a height of distance of 5 units 17′ (519 cm) 10 units 2523′ (771 m ) 20 units 10528.75 miles (16981 km) For instance, you may imagine that the fundamental distance is 10 meters. Then the lines are spaced about like railroad ties. Twenty units is 200 meters: U is in a hot air balloon. Thurston — The Geometry and Topology of 3-Manifolds 15 2. ELLIPTIC AND HYPERBOLIC GEOMETRY to L, the plane determined by U and K is perpendicular to Q, hence contains M; hence the visual line determined by K in the visual sphere of U passes through the visual point determined by K. The converse is similar. 2.10 This gives a one-to-one correspondence between the set of points x outside the sphere at inﬁnity, and (in general) the set of hyperplanes L in Hn. L corresponds to the common intersection point of all its perpendiculars. Similarly, there is a correspondence between points in Hn and hyperplanes outside the sphere at inﬁnity: a point p corresponds to the union of all points determined by hyperplanes through p. 2.5. The sphere of imaginary radius. A sphere in Euclidean space with radius r has constant curvature 1/r2. Thus, hyperbolic space should be a sphere of radius i. To give this a reasonable interpreta-tion, we use an indeﬁnite metric dx2 = dx2 1 + · · · + dx2 n −dx2 n+1 in Rn+1. The sphere of radius i about the origin in this metric is the hyperboloid x2 1 + · · · + x2 n −x2 n+1 = −1. 16 Thurston — The Geometry and Topology of 3-Manifolds 2.6. TRIGONOMETRY. 2.11 The metric dx2 restricted to this hyperboloid is positive deﬁnite, and it is not hard to check that it has constant curvature −1. Any plane through the origin is dx2-orthogonal to the hyperboloid, so it follows from elementary Riemannian geometry that it meets the hyperboloid in a geodesic. The projective model for hyperbolic space is reconstructed by projection of the hyperboloid from the origin to a hyperplane in Rn. Conversely, the quadratic form x2 1 + · · · + x2 n −x2 n+1 can be reconstructed from the projective model. To do this, note that there is a unique quadratic equation of the form n X i,j=1 aijxixj = 1 deﬁning the sphere at inﬁnity in the projective model. Homogenization of this equa-tion gives a quadratic form of type (n, 1) in Rn+1, as desired. Any isometry of the quadratic form x2 1 + · · · + x2 n −x2 n+1 induces an isometry of the hyperboloid, and hence any projective transformation of Pn that preserves the sphere at inﬁnity in-duces an isometry of hyperbolic space. This contrasts with the situation in Euclidean geometry, where there are many projective self-homeomorphisms: the aﬃne transfor-mations. In particular, hyperbolic space has no similarity transformations except isometries. This is true also for elliptic space. This means that there is a well-deﬁned unit of measurement of distances in hyperbolic geometry. We shall later see how this is related to three-dimensional topology, giving a measure of the “size” of manifolds. 2.12 2.6. Trigonometry. Sometimes it is important to have formulas for hyperbolic geometry, and not just pictures. For this purpose, it is convenient to work with the description of hyperbolic Thurston — The Geometry and Topology of 3-Manifolds 17 2. ELLIPTIC AND HYPERBOLIC GEOMETRY space as one sheet of the “sphere” of radius i with respect to the quadratic form Q(X) = X2 1 + · · · + X2 n −X2 n+1 in Rn+1. The set Rn+1, equipped with this quadratic form and the associated inner product X · Y = n X i=1 X1Y1 −Xn+1Yn+1, is called En,1. First we will describe the geodesics on level sets Sr = {X : Q(X) = r2} of Q. Suppose that Xt is such a geodesic, with speed s = q Q( ˙ Xt). We may diﬀerentiate the equations Xt · Xt = r2, ˙ Xt · ˙ Xt = s2, to obtain Xt · ˙ Xt = 0, ˙ Xt · ¨ Xt = 0, and Xt · ¨ Xt = −˙ Xt · ˙ Xt = −s2. Since any geodesic must lie in a two-dimensional subspace, ¨ Xt must be a linear combination of Xt and ˙ Xt, and we have 2.6.1. ¨ Xt = − s r 2 Xt. This diﬀerential equation, together with the initial conditions 2.13 X0 · X0 = r2, ˙ X0 · ˙ X0 = s2, X0 · ˙ X0 = 0, determines the geodesics. Given two vectors X and Y in En,1, if X and Y have nonzero length we deﬁne the quantity c(X, Y ) = X · Y ∥X∥· ∥Y ∥, where ∥X∥= √ X · X is positive real or positive imaginary. Note that c(X, Y ) = c(λX, µY ), where λ and µ are positive constants, that c(−X, Y ) = −c(X, Y ), and that c(X, X) = 1. In Euclidean space En+1, c(X, Y ) is the cosine of the angle between X and Y . In En,1 there are several cases. We identify vectors on the positive sheet of Si (Xn+1 > 0) with hyperbolic space. If Y is any vector of real length, then Q restricted to the subspace Y ⊥is indeﬁnite of type (n −1, 1). This means that Y ⊥intersects Hn and determines a hyperplane. 18 Thurston — The Geometry and Topology of 3-Manifolds 2.6. TRIGONOMETRY. We will use the notation Y ⊥to denote this hyperplane, with the normal orientation determined by Y . (We have seen this correspondence before, in 2.4.) 2.6.2. If X and Y ∈Hn, then c(X, Y ) = cosh d (X, Y ), where d (X, Y ) denotes the hyperbolic distance between X and Y . To prove this formula, join X to Y by a geodesic Xt of unit speed. From 2.6.1 we 2.14 have ¨ Xt = Xt, Xt · ˙ X0 = 0, so we get c( ¨ Xt, Xt) = c(Xt, Xt), c( ˙ X0, X0) = 0, c(X, X0) = 1; thus c(X, Xt) = cosh t. When t = d (X, Y ), then Xt = Y , giving 2.6.2. If X⊥and Y ⊥are distinct hyperplanes, then 2.6.3. X⊥and Y ⊥intersect ⇐ ⇒Q is positive deﬁnite on the subspace ⟨X, Y ⟩spanned by X and Y ⇐ ⇒c(X, Y )2 < 1 = ⇒c(X, Y ) = cos ∠(X, Y ) = −cos ∠(X⊥, Y ⊥). To see this, note that X and Y intersect in Hn ⇐ ⇒Q restricted to X⊥∩Y ⊥is indeﬁnite of type (n −2, 1) ⇐ ⇒Q restricted to ⟨X, Y ⟩is positive deﬁnite. (⟨X, Y ⟩ is the normal subspace to the (n −2) plane X⊥∩Y ⊥). 2.15 There is a general elementary formula for the area of a parallelogram of sides X and Y with respect to an inner product: area = p X · X Y · Y −(X · Y )2 = ∥X∥· ∥Y ∥· p 1 −c(X, Y )2. This area is positive real if X and Y span a positive deﬁnite subspace, and pos-itive imaginary if the subspace has type (1, 1). This shows, ﬁnally, that X⊥and Y ⊥intersect ⇐ ⇒ c (X, Y )2 < 1. The formula for c(X, Y ) comes from ordinary trigonometry. Thurston — The Geometry and Topology of 3-Manifolds 19 2. ELLIPTIC AND HYPERBOLIC GEOMETRY 2.6.4. X⊥and Y ⊥have a common perpendicular ⇐ ⇒Q has type (1, 1) on ⟨X, Y ⟩ ⇐ ⇒c(X, Y )2 > 1 = ⇒c(X, Y ) = ± cosh d (X⊥, Y ⊥) . The sign is positive if the normal orientations of the common perpendiculars coincide, and negative otherwise. 2.16 The proof is similar to 2.6.2. We may assume X and Y have unit length. Since ⟨X, Y ⟩intersects Hn as the common perpendicular to X⊥and Y ⊥, Q restricted to ⟨X, Y ⟩has type (1, 1). Replace X by −X if necessary so that X and Y lie in the same component of S1∩⟨X, Y ⟩. Join X to Y by a geodesic Xt of speed i. From 2.6.1, ¨ Xt = Xt. There is a dual geodesic Zt of unit speed, satisfying Zt · Xt = 0, joining X⊥to Y ⊥along their common perpendicular, so one may deduce that c, (X, Y ) = ±d (X,Y ) i = ±d (X⊥, Y ⊥). There is a limiting case, intermediate between 2.6.3 and 2.6.4: 2.6.5. X⊥and Y ⊥are parallel ⇐ ⇒Q restricted to ⟨X, Y ⟩is degenerate ⇐ ⇒c(X, Y )2 = 1. In this case, we say that X⊥and Y ⊥form an angle of 0 or π. X⊥and Y ⊥actually have a distance of 0, where the distance of two sets U and V is deﬁned to be the inﬁmum of the distance between points u ∈U and v ∈V . 20 Thurston — The Geometry and Topology of 3-Manifolds 2.6. TRIGONOMETRY. There is one more case in which to interpret c(X, Y ): 2.6.6. If X is a point in Hn and Y ⊥a hyperplane, then c(X, Y ) = sinh d (X, Y ⊥) i , where d (X, Y ⊥) is the oriented distance. 2.17 The proof is left to the untiring reader. With our dictionary now complete, it is easy to derive hyperbolic trigonometric formulae from linear algebra. To solve triangles, note that the edges of a triangle with vertices u, v and w in H2 are U ⊥, V ⊥and W ⊥, where U is a vector orthogonal to v and w, etc. To ﬁnd the angles of a triangle from the lengths, one can ﬁnd three vectors u, v, and w with the appropriate inner products, ﬁnd a dual basis, and calculate the angles from the inner products of the dual basis. Here is the general formula. We consider triangles in the projective model, with vertices inside or outside the sphere at inﬁnity. Choose vectors v1, v2 and v3 of length i or 1 representing these points. Let ϵi = vi · vi, ϵij = √ϵiϵj and cij = c(vi, vj). Then the matrix of inner products of the vi is C =   ϵ1 ϵ12c12 ϵ13c13 ϵ12c12 ϵ2 ϵ23c23 ϵ13c13 ϵ23c23 ϵ3  . The matrix of inner products of the dual basis {v1, v2, v3} is C−1. For our pur-2.18 poses, though, it is simpler to compute the matrix of inner products of the basis Thurston — The Geometry and Topology of 3-Manifolds 21 2. ELLIPTIC AND HYPERBOLIC GEOMETRY { √ −det Ci}, −adj C = (−det C) · C−1 =   −ϵ2ϵ3(1 −c2 23) −ϵ12ϵ3(c13c23 −c12) −ϵ13ϵ2(c12c23 −c13) −ϵ12ϵ3(c13c23 −c12) −ϵ1ϵ3(1 −c2 13) −ϵ23ϵ1(c12c13 −c23) −ϵ13ϵ2(c12c23 −c13) −ϵ23ϵ1(c12c13 −c23) −ϵ1ϵ2(1 −c2 12)  . If v1, v2, v3 is the dual basis, and cij = c(vi, vj), we can compute 2.6.7. c12 = ϵ · c13c23 −c12 p 1 −c2 23 p 1 −c2 13 , where it is easy to deduce the sign ϵ = −ϵ12ϵ3 √−ϵ2ϵ3 √−ϵ1ϵ3 directly. This specializes to give a number of formulas, in geometrically distinct cases. In a real triangle, 2.6.8. cosh C = cos α cos β + cos γ sin α sin β , 2.19 2.6.9. cos γ = cosh A cosh B −cosh C sinh A sinh B , or cosh C = cosh A cosh B −sinh A sinh B cos c. (See also 2.6.16.) In an all right hexagon, 22 Thurston — The Geometry and Topology of 3-Manifolds 2.6. TRIGONOMETRY. 2.6.10. cosh C = cosh α cosh β + cosh γ sinh α sinh β . (See also 2.6.18.) Such hexagons are useful in the study of hyperbolic structures on surfaces. Similar formulas can be obtained for pentagons with four right angles, or quadrilaterals with two adjacent right angles: 2.20 By taking the limit of 2.6.8 as the vertex with angle γ tends to the circle at inﬁnity, we obtain useful formulas: Thurston — The Geometry and Topology of 3-Manifolds 23 2. ELLIPTIC AND HYPERBOLIC GEOMETRY γ α β C 2.6.11. cosh C = cos α cos β + 1 sin α sin β , and in particular 2.6.12. cosh C = 1 sin α. These formulas for a right triangle are worth mentioning separately, since they are particularly simple. 2.21 24 Thurston — The Geometry and Topology of 3-Manifolds 2.6. TRIGONOMETRY. From the formula for cos γ we obtain the hyperbolic Pythagorean theorem: 2.6.13. cosh C = cosh A cosh B. Also, 2.6.14. cosh A = cos α sin β . (Note that (cos α)/(sin β) = 1 in a Euclidean right triangle.) By substituting (cosh C) (cosh A) for cosh B in the formula 2.6.9 for cos α, one ﬁnds: 2.6.15. In a right triangle, sin α = sinh A sinh C . This follows from the general law of sines, 2.6.16. In any triangle, sinh A sin α = sinh B sin β = sinh C sin γ . 2.22 Similarly, in an all right pentagon, Thurston — The Geometry and Topology of 3-Manifolds 25 2. ELLIPTIC AND HYPERBOLIC GEOMETRY one has 2.6.17. sinh A sinh B = cosh D. It follows that in any all right hexagon, there is a law of sines: 2.6.18. sinh A sinh α = sinh B sinh β = sinh C sinh γ . 26 Thurston — The Geometry and Topology of 3-Manifolds William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T EX by Sheila Newbery, who also scanned the ﬁgures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the ﬁrst few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at levy@msri.org. CHAPTER 3 Geometric structures on manifolds A manifold is a topological space which is locally modelled on Rn. The notion of what it means to be locally modelled on Rn can be made deﬁnite in many diﬀerent ways, yielding many diﬀerent sorts of manifolds. In general, to deﬁne a kind of manifold, we need to deﬁne a set G of gluing maps which are to be permitted for piecing the manifold together out of chunks of Rn. Such a manifold is called a G-manifold. G should satisfy some obvious properties which make it a pseudogroup of local homeomorphisms between open sets of Rn: (i) The restriction of an element g ∈G to any open set in its domain is also in G. (ii) The composition g1 ◦g2 of two elements of G, when deﬁned, is in G. (iii) The inverse of an element of G is in G. (iv) The property of being in G is local, so that if U = S α Uα and if g is a local homeomorphism g : U →V whose restriction to each Uα is in G, then g ∈G. It is convenient also to permit G to be a pseudogroup acting on any manifold, although, as long as G is transitive, this doesn’t give any new types of manifolds. See Haeﬂiger, in Springer Lecture Notes #197, for a discussion. A group G acting on a manifold X determines a pseudogroup which consists of restrictions of elements of G to open sets in X. A (G, X)-manifold means a manifold 3.2 glued together using this pseudogroup of restrictions of elements of G. Examples. If G is the pseudogroup of local Cr diﬀeomorphisms of Rn, then a G-manifold is a Cr-manifold, or more loosely, a diﬀerentiable manifold (provided r ≥1). If G is the pseudogroup of local piecewise-linear homeomorphisms, then a G-manifold is a PL-manifold. If G is the group of aﬃne transformations of Rn, then a (G, Rn)-manifold is called an aﬃne manifold. For instance, given a constant λ > 1 consider an annulus of radii 1 and λ+ϵ. Identify neighborhoods of the two boundary components by the map x →λx. The resulting manifold, topologically, is T 2. Thurston — The Geometry and Topology of 3-Manifolds 27 3. GEOMETRIC STRUCTURES ON MANIFOLDS 3.3 Here is another method, due to John Smillie, for constructing aﬃne structures on T 2 from any quadrilateral Q in the plane. Identify the opposite edges of Q by the orientation-preserving similarities which carry one to the other. Since similarities preserve angles, the sum of the angles about the vertex in the resulting complex is 2π, so it has an aﬃne structure. We shall see later how such structures on T 2 are intimately connected with questions concerning Dehn surgery in three-manifolds. The literature about aﬃne manifolds is interesting. Milnor showed that the only closed two-dimensional aﬃne manifolds are tori and Klein bottles. The main unsolved question about aﬃne manifolds is whether in general an aﬃne manifold has Euler characteristic zero. If G is the group of isometries of Euclidean space En, then a (G, En)-manifold is called a Euclidean manifold, or often a ﬂat manifold. Bieberbach proved that a Euclidean manifold is ﬁnitely covered by a torus. Furthermore, a Euclidean structure automatically gives an aﬃne structure, and Bieberbach proved that closed Euclidean manifolds with the same fundamental group are equivalent as aﬃne manifolds. If G is the group O(n + 1) acting on elliptic space Pn (or on Sn), then we obtain elliptic manifolds. Conjecture. Every three-manifold with ﬁnite fundamental group has an elliptic structure. 28 Thurston — The Geometry and Topology of 3-Manifolds 3.1. A HYPERBOLIC STRUCTURE ON THE FIGURE-EIGHT KNOT COMPLEMENT. This conjecture is a stronger version of the Poincar´ e conjecture; we shall see the logic shortly. All known three-manifolds with ﬁnite fundamental group certainly have elliptic structures. 3.4 As a ﬁnal example (for the present), when G is the group of isometries of hyper-bolic space Hn, then a (G, Hn)-manifold is a hyperbolic manifold. For instance, any surface of negative Euler characteristic has a hyperbolic structure. The surface of genus two is an illustrative example. Topologically, this surface is obtained by identifying the sides of an octagon, in the pattern above, for instance. An example of a hyperbolic structure on the surface is obtained form any hyperbolic octagon whose opposite edges have equal lengths and whose angle sum is 2π, by identifying in the same pattern. There is a regular octagon with angles π/4, for instance. 3.5 3.6 3.1. A hyperbolic structure on the ﬁgure-eight knot complement. Consider a regular tetrahedron in Euclidean space, inscribed in the unit sphere, so that its vertices are on the sphere. Now interpret this tetrahedron to lie in the projective model for hyperbolic space, so that it determines an ideal hyperbolic sim-plex: combinatorially, a simplex with its vertices deleted. The dihedral angles of the hyperbolic simplex are 60◦. This may be seen by extending its faces to the sphere at inﬁnity, which they meet in four circles which meet each other in 60◦angles. By considering the Poincar´ e disk model, one sees immediately that the angle made by two planes is the same as the angle of their bounding circles on the sphere at inﬁnity. Take two copies of this ideal simplex, and glue the faces together, in the pattern described in Chapter 1, using Euclidean isometries, which are also (in this case) hyperbolic isometries, to identify faces. This gives a hyperbolic structure to the resulting manifold, since the angles add up to 360◦around each edge. 3.7 Thurston — The Geometry and Topology of 3-Manifolds 29 3. GEOMETRIC STRUCTURES ON MANIFOLDS A regular octagon with angles π/4, whose sides can be identiﬁed to give a surface of genus 2. A tetrahedron inscribed in the unit sphere, top view. According to Magnus, Hyperbolic Tesselations, this manifold was constructed by Gieseking in 1912 (but without any relation to knots). R. Riley showed that the ﬁgure-eight knot complement has a hyperbolic structure (which agrees with this one). This manifold also coincides with one of the hyperbolic manifolds obtained by an arithmetic construction, because the fundamental group of the complement of the 30 Thurston — The Geometry and Topology of 3-Manifolds 3.2. A HYPERBOLIC MANIFOLD WITH GEODESIC BOUNDARY. ﬁgure-eight knot is isomorphic to a subgroup of index 12 in PSL2(Z[ω]), where ω is a primitive cube root of unity. 3.2. A hyperbolic manifold with geodesic boundary. Here is another manifold which is obtained from two tetrahedra. First glue the two tetrahedra along one face; then glue the remaining faces according to this diagram: 3.8 In the diagram, one vertex has been removed so that the polyhedron can be ﬂattened out in the plane. The resulting complex has only one edge and one vertex. The manifold M obtained by removing a neighborhood of the vertex is oriented with boundary a surface of genus 2. Consider now a one-parameter family of regular tetrahedra in the projective model for hyperbolic space centered at the origin in Euclidean space, beginning with the tetrahedron whose vertices are on the sphere at inﬁnity, and expanding until the edges are all tangent to the sphere at inﬁnity. The dihedral angles go from 60◦to 0◦, so somewhere in between, there is a tetrahedron with 30◦dihedral angles. Truncate this simplex along each plane v⊥, where v is a vertex (outside the unit ball), to obtain a stunted simplex with all angles 90◦or 30◦: Thurston — The Geometry and Topology of 3-Manifolds 31 3. GEOMETRIC STRUCTURES ON MANIFOLDS 3.9 Two copies glued together give a hyperbolic structure for M, where the boundary of M (which comes from the triangular faces of the stunted simplices) is totally geo-desic. A closed hyperbolic three-manifold can be obtained by doubling this example, i.e., taking two copies of M and gluing them together by the “identity” map on the boundary. 3.3. The Whitehead link complement. The Whitehead link may be spanned by a two-complex which cuts the complement into an octahedron, with vertices deleted: The one-cells are the three arrows, and the attaching maps for the two-cells are indicated by the dotted lines. The three-cell is an octahedron (with vertices deleted), 3.10 and the faces are identiﬁed thus: 32 Thurston — The Geometry and Topology of 3-Manifolds 3.4. THE BORROMEAN RINGS COMPLEMENT. A hyperbolic structure may be obtained from a Euclidean regular octahedron in-scribed in the unit sphere. Interpreted as lying in the projective model for hyperbolic space, this octahedron is an ideal octahedron with all dihedral angles 90◦. 3.11 Gluing it in the indicated pattern, again using Euclidean isometries between the faces (which happen to be hyperbolic isometries as well) gives a hyperbolic structure for the complement of the Whitehead link. 3.4. The Borromean rings complement. This is spanned by a two-complex which cuts the complement into two ideal octahedra: Thurston — The Geometry and Topology of 3-Manifolds 33 3. GEOMETRIC STRUCTURES ON MANIFOLDS Here is the corresponding gluing pattern of two octahedra. Faces are glued to their corresponding faces with 120◦rotations, alternating in directions like gears. 3.12 3.5. The developing map. Let X be any real analytic manifold, and G a group of real analytic diﬀeomor-phisms of X. Then an element of G is completely determined by its restriction to any open set of X. Suppose that M is any (G, X)-manifold. Let U1, U2, . . . be coordinate charts for M, with maps φi : Ui →X and transition functions γij satisfying γij ◦φi = φj. In general the γij’s are local G-diﬀeomorphisms of X deﬁned on φi(Ui ∩Uj) so they are determined by locally constant maps, also denoted γij, of Ui ∩Uj into G. Consider now an analytic continuation of φ1 along a path α in M beginning in U1. It is easy to see, inductively, that on a component of α ∩Ui, the analytic 34 Thurston — The Geometry and Topology of 3-Manifolds 3.5. THE DEVELOPING MAP. continuation of φ1 along α is of the form γ ◦φi, where γ ∈G. Hence, φ1 can be analytically continued along every path in M. It follows immediately that there is a global analytic continuation of φ1 deﬁned on the universal cover of M. (Use the deﬁnition of the universal cover as a quotient space of the paths in M.) This map, D : ˜ M →X, is called the developing map. D is a local (G, X)-homeomorphism (i.e., it is an im-mersion inducing the (G, X)-structure on ˜ M.) D is clearly unique up to composition 3.13 with elements of G. Although G acts transitively on X in the cases of primary interest, this condition is not necessary for the deﬁnition of D. For example, if G is the trivial group and X is closed then closed (G, X)-manifolds are precisely the ﬁnite-sheeted covers of X, and D is the covering projection. From this uniqueness property of D, we have in particular that for any covering transformation Tα of ˜ M over M, there is some (unique) element gα ∈G such that D ◦Tα = gα ◦D. Since D ◦Tα ◦Tβ = gα ◦D ◦Tβ = gα ◦gβ ◦D it follows that the correspondence H : α 7→gα is a homomorphism, called the holonomy of M. In general, the holonomy of M need not determine the (G, X)-structure on M, but there is an important special case in which it does. Definition. M is a complete (G, X)-manifold if D : ˜ M →X is a covering map. (In particular, if X is simply-connected, this means D is a homeomorphism.) If X is similarly connected, then any complete (G, X)-manifold M may easily be reconstructed from the image Γ = H(π1(M)) of the holonomy, as the quotient space X/Γ. Here is a useful suﬃcient condition for completeness. Proposition 3.6. Let G be a group of analytic diﬀeomorphisms acting transi-tively on a manifold X, such that for any x ∈X, the isotropy group Gx of x is compact. Then every closed (G, X)-manifold M is complete. Proof. Let Q be any positive deﬁnite on the tangent space Tx(X) of X at some point x. Average the set of transforms g(Q), g ∈Gx, using Haar measure, to obtain a quadratic form on Tx(X) which is invariant under Gx. Deﬁne a Riemannian metric 3.14 (ds2)y = g(Q) on X, where g ∈G is any element taking x to y. This deﬁnition is independent of the choice of g, and the resulting Riemannian metric is invariant under G. Thurston — The Geometry and Topology of 3-Manifolds 35 3. GEOMETRIC STRUCTURES ON MANIFOLDS Therefore, this metric pieces together to give a Riemannian metric on any (G, X)-manifold, which is invariant under any (G, X)- map. If M is any closed (G, X)-manifold, then there is some ϵ > 0 such that the ϵ-ball in the Riemannian metric on M is always convex and contractible. If x is any point in X, then D−1(Bϵ/2(x)) must be a union of homeomorphic copies of Bϵ/2(x) in ˜ M. D evenly covers X, so it is a covering projection, and M is complete. □ For example, any closed elliptic three-manifold has universal cover S3, so any simply-connected elliptic manifold is S3. Every closed hyperbolic manifold or Eu-clidean manifold has universal cover hyperbolic three-space or Euclidean space. Such manifolds are consequently determined by their holonomy. 3.15 3.16 Even for G and X as in proposition 3.6, the question of whether or not a non-compact (G, X)-manifold M is complete can be much more subtle. For example, consider the thrice-punctured sphere, which is obtained by gluing together two tri-angles minus vertices in this pattern: A hyperbolic structure can be obtained by gluing two ideal triangles (with all vertices on the circle at inﬁnity) in this pattern. Each side of such a triangle is isometric to the real line, so a gluing map between two sides may be modiﬁed by an arbitrary translation; thus, we have a family of hyperbolic structures in the thrice-punctured sphere parametrized by R3. (These structures need not be, and are not, all distinct.) Exactly one parameter value yields a complete hyperbolic structure, as we shall see presently. Meanwhile, we collect some useful conditions for completeness of a (G, X)-struc-ture with (G, X) as in 3.6. For convenience, we ﬁx some natural metrics on (G, X)-3.17 structures. Labelled 3.7.p to avoid multiple labels Proposition 3.7. With (G, X) as above, a (G, X)-manifold M is complete if and only if any of the following equivalent conditions is satisﬁed. (a) M is complete as a metric space. (b) There is some ϵ > 0 such that each closed ϵ-ball in M is compact. 36 Thurston — The Geometry and Topology of 3-Manifolds 3.5. THE DEVELOPING MAP. 3.15 The developing map of an aﬃne torus constructed from a quadrilateral (see p. 3.3). The torus is plainly not complete. Exercise: construct other aﬃne toruses with the same holonomy as this one. (Hint: walk once or twice around this page.) Thurston — The Geometry and Topology of 3-Manifolds 37 3. GEOMETRIC STRUCTURES ON MANIFOLDS (c) For every k > 0, all closed k-balls are compact. (d) There is a family {St}; t ∈R, of compact sets which exhaust M, such that St+a contains a neighborhood of radius a about St. Proof. Suppose that M is metrically complete. Then ˜ M is also metrically com-plete. We will show that the developing map D : ˜ M →X is a covering map by proving that any path αt in X can be lifted to ˜ M. In fact, let T ⊂[0, 1] be a maxi-mal connected set for which there is a lifting. Since D is a local homeomorphism, T is open, and because ˜ M is metrically complete, T is closed: hence, α can be lifted, so M is complete. It is an elementary exercise to see that (b) ⇐ ⇒(c) ⇐ ⇒(d) = ⇒(a). For any point x0 ∈˜ X there is some ϵ such that the ball Bϵ(x) is compact; this ϵ works for all x ∈˜ X since the group ˜ G of (G, X)-diﬀeomorphisms of ˜ X is transitive. Therefore X satisﬁes (a), (b), (c) and (d). Finally if M is a complete (G, X)-manifold, it is covered by ˜ X, so it satisﬁes (b). The proposition follows. □ 3.18 3.8. Horospheres. To analyze what happens near the vertices of an ideal polyhedron when it is glued together, we need the notion of horospheres (or, in the hyperbolic plane, they are called horocycles.) A horosphere has the limiting shape of a sphere in hyperbolic space, as the radius goes to inﬁnity. One property which can be used to determine the spheres centered at a point X is the fact that such a sphere is orthogonal to all lines through X. Similarly, if X is a point on the sphere at inﬁnity, the horospheres “centered” at X are the surfaces orthogonal to all lines through X. In the Poincar´ e disk model, a hyperbolic sphere is a Euclidean sphere in the interior of the disk, and a horosphere is a Euclidean sphere tangent to the unit sphere. The point X of tangency is the center of the horosphere. 38 Thurston — The Geometry and Topology of 3-Manifolds 3.8. HOROSPHERES. Concentric horocycles and orthogonal lines. 3.19 Translation along a line through X permutes the horospheres centered at X. Thus, all horospheres are congruent. The convex region bounded by a horosphere is a horoball. For another view of a horosphere, consider the upper half-space model. In this case, hyperbolic lines through the point at inﬁnity are Euclidean lines orthogonal to the plane bounding upper half-space. A horosphere about this point is a horizontal Euclidean plane. From this picture one easily sees that a horosphere in Hn is isometric to Euclidean space En−1. One also sees that the group of hyperbolic isometries ﬁxing the point at inﬁnity in the upper half-space model acts as the group of similarities of the bounding Euclidean plane. One can see this action internally as follows. Let X be any point at inﬁnity in hyperbolic space, and h any horosphere centered at X. An isometry g of hyperbolic space ﬁxing X takes h to a concentric horosphere h′. Project h′ back to h along the family of parallel lines through X. The composition of these two maps is a similarity of h. Consider two directed lines l1 and l2 emanating from the point at inﬁnity in the upper half-space model. Recall that the hyperbolic metric is ds2 = (1/x2 n) dx2. This means that the hyperbolic distance between l1 and l2 along a horosphere is inversely proportional to the Euclidean distance above the bounding plane. The hyperbolic distance between points X1 and X2 on l1 at heights of h1 and h2 is \| log(h2)−log(h1)\|. It follows that for any two concentric horospheres h1 and h2 which are a distance d apart, and any pair of lines l1 and l2 orthogonal to h1 and h2, the ratio of the distance 3.20 Thurston — The Geometry and Topology of 3-Manifolds 39 3. GEOMETRIC STRUCTURES ON MANIFOLDS Figure 1. Horocycles and lines in the upper half-plane between l1 and l2 measured along h1 to their distance measured along h2 is exp(d). 3.9. Hyperbolic surfaces obtained from ideal triangles. Consider an oriented surface S obtained by gluing ideal triangles with all vertices at inﬁnity, in some pattern. Exercise: all such triangles are congruent. (Hint: you can derive this from the fact that a ﬁnite triangle is determined by its angles—see 2.6.8. Let the vertices pass to inﬁnity, one at a time.) Let K be the complex obtained by including the ideal vertices. Associated with each ideal vertex v of K, there is an invariant d(v), deﬁned as follows. Let h be a horocycle in one of the ideal triangles, centered about a vertex which is glued to v and “near” this vertex. Extend h as a horocycle in S counter clockwise about v. It meets each successive ideal triangle as a horocycle orthogonal to two of the sides, until ﬁnally it re-enters the original triangle as a horocycle h′ concentric with h, at a distance ±d(v) from h. The sign is chosen to be positive if and only if the horoball bounded by h′ in the ideal triangle contains that bounded by h. 3.21 40 Thurston — The Geometry and Topology of 3-Manifolds 3.9. HYPERBOLIC SURFACES OBTAINED FROM IDEAL TRIANGLES. The surface S is complete if and only if all invariants d(v) are 0. Suppose, for instance, that some invariant d(v) < 0. Continuing h further round v; the length of each successive circuit around v is reduced by a constant factor < 1, so the total length of h after an inﬁnite number of circuits is bounded. A sequence of points evenly spaced along h is a non-convergent Cauchy sequence. If all invariants d(v) = 0, on the other hand, one can remove horoball neighbor-hoods of each vertex in K to obtain a compact subsurface S0. Let St be the surface obtained by removing smaller horoball neighborhoods bounded by horocycles a dis-tance of t from the original ones. The surfaces St satisfy the hypotheses of 3.7(d) 1—hence S is complete. 3.22 For any hyperbolic manifold M, let ¯ M be the metric completion of M. In general, ¯ M need not be a manifold. However, if S is a surface obtained by gluing ideal hyperbolic triangles, then ¯ S is a hyperbolic surface with geodesic boundary. There is Thurston — The Geometry and Topology of 3-Manifolds 41 3. GEOMETRIC STRUCTURES ON MANIFOLDS one boundary component of length \|d(v)\| for each vertex v of K such that d(v) ̸= 0. ¯ S is obtained by adjoining one limit point for each horocycle which “spirals toward” a vertex v in K. The most convincing way to understand ¯ S is by studying the picture: 3.10. Hyperbolic manifolds obtained by gluing ideal polyhedra. Consider now the more general case of a hyperbolic manifold M obtained by gluing together the faces of polyhedra in Hn with some vertices at inﬁnity. Let K 3.23 be the complex obtained by including the ideal vertices. The link of an ideal vertex v is (by deﬁnition) the set L(v) of all rays through that vertex. From 3.7 it follows that the link of each vertex has a canonical (similarities of En−1, En−1 ) structure, or similarity structure for short. An extension of the analysis in 3.9 easily shows that M is complete if and only if the similarity structure on each link of an ideal vertex is actually a Euclidean structure, or equivalently, if and only if the holonomy of these similarity structures consists of isometries. We shall be concerned mainly with dimension n = 3. It is easy to see from the Gauss-Bonnet theorem that any similarity two-manifold has Euler characteristic zero. (Its tangent bundle has a ﬂat orthogonal connection). Hence, if M is oriented, each link L(v) of an ideal vertex is topologically a torus. If L(v) is not Euclidean, then for some α ∈π1L(v), the holonomy H(α) is a contraction, so it has a unique ﬁxed point x0. Any other element β ∈π1(L(v)) must also ﬁx x0, since β commutes with α. Translating x0 to 0, we see that the similarity two-manifold L(v) must be a (C∗, C −0)-manifold where C∗is the multiplicative group of complex numbers. (Compare p. 3.15.) Such a structure 42 Thurston — The Geometry and Topology of 3-Manifolds 3.10. HYPERBOLIC MANIFOLDS OBTAINED BY GLUING IDEAL POLYHEDRA. is automatically complete (by 3.6), and it is also modelled on (˜ C∗, ^ C −0), or, by taking logs, on (C, C). Here the ﬁrst C is an additive group and the second C is a space. Conversely, by taking exp, any (C, C) structure gives a similarity structure. (C, C) structures on closed oriented manifolds are easy to describe, being determined by their holonomy, which is generated by an arbitrary pair (z1, z2) of 3.24 complex numbers which are linearly independent over R. We shall return later to study the spaces ¯ M in the three-dimensional case. They are sometimes closed hyperbolic manifolds obtained topologically by replacing neigh-borhoods of the vertices by solid toruses. Thurston — The Geometry and Topology of 3-Manifolds 43 William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T EX by Sheila Newbery, who also scanned the ﬁgures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the ﬁrst few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at levy@msri.org. CHAPTER 4 Hyperbolic Dehn surgery A hyperbolic structure for the complement of the ﬁgure-eight knot was con-structed in 3.1. This structure was in fact chosen to be complete. The reader may wish to verify this by constructing a horospherical realization of the torus which is the link of the ideal vertex. Similarly, the hyperbolic structure for the Whitehead link complement and the Borromean rings complement constructed in 3.3 and 3.4 are complete. There is actually a good deal of freedom in the construction of hyperbolic struc-tures for such manifolds, although most of the resulting structures are not complete. We shall ﬁrst analyze the ﬁgure-eight knot complement. To do this, we need an understanding of the possible shapes of ideal tetrahedra. 4.1. Ideal tetrahedra in H3. The link L(v) of an ideal vertex v of an oriented ideal tetrahedron T (which by deﬁnition is the set of rays in the tetrahedron through that vertex) is a Euclidean triangle, well-deﬁned up to orientation-preserving similarity. It is concretely realized as the intersection with T of a horosphere about v. The triangle L(v) actually determines T up to congruence. To see this, picture T in the upper half-space model and arrange it so that v is the point at inﬁnity. The other three vertices of T form a triangle in E2 which is in the same similarity class as L(v). Consequently, if two 4.2 tetrahedra T and T ′ have vertices v and v′ with L(v) similar to L(v′), then T ′ can be transformed to T by a Euclidean similarity which preserves the plane bounding upper half-space. Such a similarity is a hyperbolic isometry. Thurston — The Geometry and Topology of 3-Manifolds 45 4. HYPERBOLIC DEHN SURGERY It follows that T is determined by the three dihedral angles α, β and γ of edges incident to the ideal vertex v, and that α + β + γ = π. Using similar relations among angles coming from the other three vertices, we can determine the other three dihedral angles: 4.3 Thus, dihedral angles of opposite edges are equal, and the oriented similarity class of L(v) does not depend on the choice of a vertex v! A geometric explanation of this phenomenon can be given as follows. Any two non-intersecting and non-parallel lines in H3 admit a unique common perpendicular. Construct the three common perpendiculars s, t and u to pairs of opposite edges of T. Rotation of π about s, for instance, preserves the edges orthogonal to s, hence preserves the four ideal vertices of T, so it preserves the entire ﬁgure. It follows that s, t and u meet in a point and that they are pairwise orthogonal. The rotations of π about these three axes are the three non-trivial elements of z2 ⊕z2 acting as a group of symmetries of T. 46 Thurston — The Geometry and Topology of 3-Manifolds 4.1. IDEAL TETRAHEDRA IN H3. 4.4 In order to parametrize Euclidean triangles up to similarity, it is convenient to regard E2 as C. To each vertex v of a triangle ∆(t, u, v) we associate the ratio (t −v) (u −v) = z(v) of the sides adjacent to v. The vertices must be labelled in a clockwise order, so that Thurston — The Geometry and Topology of 3-Manifolds 47 4. HYPERBOLIC DEHN SURGERY Im z(v) > 0. Alternatively, arrange the triangle by a similarity so that v is at 0 and u at 1; then t is at z(v). The other two vertices have invariants 4.1.1. z(t) = z(v)−1 z(v) z(u) = 1 1−z(v). Denoting the three invariants z1, z2, z3 in clockwise order, with any starting point, we have the identities 4.5 4.1.2. z1 z2 z3 = −1 1 −z1 + z1z2 = 0 We can now translate this to a parametrization of ideal tetrahedra. Each edge e is labelled with a complex number z(e), opposite edges have the same label, and the three distinct invariants satisfy 4.1.2 (provided the ordering is correct.) Any zi determines the other two, via 4.1.2. 4.2. Gluing consistency conditions. Suppose that M is a three-manifold obtained by gluing tetrahedra Ti, . . . , Tj and then deleting the vertices, and let K be the complex which includes the vertices. 4.6 Any realization of T1, . . . , Tj as ideal hyperbolic tetrahedra determines a hyper-bolic structure on (M −(1 −skeleton)), since any two ideal triangles are congruent. Such a congruence is uniquely determined by the correspondence between the ver-tices. (This fact may be visualized concretely from the subdivision of an ideal triangle by its altitudes.) 48 Thurston — The Geometry and Topology of 3-Manifolds 4.2. GLUING CONSISTENCY CONDITIONS. The condition for the hyperbolic structure on (M −(1 −skeleton)) to give a hyperbolic structure on M itself is that its developing map, in a neighborhood of each edge, should come from a local homeomorphism of M itself. In particular, the sum of the dihedral angles of the edges e1, . . . , ek must be 2π. Even when this condition is satisﬁed, though, the holonomy going around an edge of M might be a non-trivial translation along the edge. To pin down the precise condition, note that for each ideal vertex v, the hyperbolic structure on M −(1 −skeleton) gives a similarity structure to L(v)−(0−skeleton). The hyperbolic structure extends over an edge e of M if and 4.7 only if the similarity structure extends over the corresponding point in L(v), where v is an endpoint of e. Equivalently, the similarity classes of triangles determined by z(e1), . . . , z(ek) must have representatives which can be arranged neatly around a point in the plane: The algebraic condition is 4.2.1. z(e1) · z(e2) · · · · · z(ek) = 1. This equation should actually be interpreted to be an equation in the universal cover ˜ C∗, so that solutions such as 4.8 Thurston — The Geometry and Topology of 3-Manifolds 49 4. HYPERBOLIC DEHN SURGERY are ruled out. In other words, the auxiliary condition 4.2.2. arg z1 + · · · + arg zk = 2π must also be satisﬁed, where 0 < arg zi ≤π. 4.3. Hyperbolic structure on the ﬁgure-eight knot complement. Consider two hyperbolic tetrahedra to be identiﬁed to give the ﬁgure eight knot complement: 50 Thurston — The Geometry and Topology of 3-Manifolds 4.3. HYPERBOLIC STRUCTURE ON THE FIGURE-EIGHT KNOT COMPLEMENT. 4.9 We read oﬀthe gluing consistency conditions for the two edges: (̸− →)z2 1z2w2 1w2 = 1 (̸̸ − →)z2 3z2w2 3w2 = 1. From 4.1.2, note that the product of these two equations, (z1z2z3)2(w1w2w3)2 = 1 is automatically satisﬁed. Writing z = z1, and w = w1, and substituting the expres-sions from 4.1.1 into (̸− →), we obtain the equivalent gluing condition, 4.3.1. z(z −1) w(w −1) = 1. Thurston — The Geometry and Topology of 3-Manifolds 51 4. HYPERBOLIC DEHN SURGERY We may solve for z in terms of w by using the quadratic formula. 4.3.2. z = 1 ± p 1 + 4/w(w −1) 2 We are searching only for geometric solutions Im(z) > 0 Im(w) > 0 so that the two tetrahedra are non-degenerate and positively oriented. For each value of w, there is at most one solution for z with Im(z) > 0. Such a solution exists provided that the discriminant 1 + 4/w(w −1) is not positive real. Solutions are therefore parametrized by w in this region of C: 4.10 Note that the original solution given in 3.1 corresponds to w = z = 3 √−1 = 1 2 + √ 3 2 i. The link L of the single ideal vertex has a triangulation which can be calculated from the gluing diagram: 4.11 52 Thurston — The Geometry and Topology of 3-Manifolds 4.3. HYPERBOLIC STRUCTURE ON THE FIGURE-EIGHT KNOT COMPLEMENT. Now let us compute the derivative of the holonomy of the similarity structure on L. To do this, regard directed edges of the triangulation as vectors. The ratio of any two vectors in the same triangle is known in terms of z or w. Multiplying appropriate ratios, we obtain the derivative of the holonomy: Thurston — The Geometry and Topology of 3-Manifolds 53 4. HYPERBOLIC DEHN SURGERY 4.12 H′(x) = z2 1(w2w3)2 = ( z w)2 H′(y) = w1 z3 = w(1 −z). Observe that if M is to be complete, then H′(x) = H′(y) = 1, so z = w. From 4.3.1, (z(z −1))2 = 1. Since z(z −1) < 0, this means z(z −1) = −1, so that the only possibility is the original solution w = z = 3 √−1. 4.4. The completion of hyperbolic three-manifolds obtained from ideal polyhedra. Let M be any hyperbolic manifold obtained by gluing polyhedra with some ver-tices at inﬁnity, and let K be the complex obtained by including the ideal vertices. The completion ¯ M is obtained by completing a deleted neighborhood N(v) of each ideal vertex v in k, and gluing these completed neighborhoods ¯ N(v) to M. The developing map for the hyperbolic structure on N(v) may be readily understood in terms of the developing map for the similarity structure on L(v). To do this, choose coordinates so that v is the point at inﬁnity in the upper half-space model. The developing images of corners of polyhedra near v are “chimneys” above some poly-gon in the developing image of L(v) on C (where C is regarded as the boundary of upper half-space.) If M is not complete near v, we change coordinates if necessary by a translation of R3 so that the developing image of L(v) is C −0. The holonomy for N(v) now consists of similarities of R3 which leave invariant the z-axis and the x −y plane (C). Replacing N(v) by a smaller neighborhood, we may assume that the developing image I of N(v) is a solid cone, minus the z-axis. 4.13 54 Thurston — The Geometry and Topology of 3-Manifolds 4.4. COMPLETION OF HYPERBOLIC THREE-MANIFOLDS The completion of I is clearly the solid cone, obtained by adjoining the z-axis to I. It follows easily that the completion of ] N(v) = ˜ I is also obtained by adjoining a single copy of the z-axis. The projection p : ] N(v) →N(v) extends to a surjective map ¯ p between the completions. [¯ p exists because p does not increase distances. ¯ p is surjective because a Cauchy sequence can be replaced by a Cauchy path, which lifts to ˜ N(v).] Every orbit of the holonomy action of π1 N(v) on the z-axis is identiﬁed to a single point. This action is given by H(α) : (0, 0, z) 7→\|H′(α)\| · (0, 0, z) where the ﬁrst H(α) is the hyperbolic holonomy and the second is the holonomy of 4.14 L(v). There are two cases: Case 1. The group of moduli {\|H′(α)\|} is dense in R+. Then the completion of N(v) is the one-point compactiﬁcation. Case 2. The group of moduli {\|H′(α)\|} is a cyclic group. Then the completion N(v) Thurston — The Geometry and Topology of 3-Manifolds 55 4. HYPERBOLIC DEHN SURGERY is topologically a manifold which is the quotient space ¯ ˜(v) N/H, and it is obtained by adjoining a circle to N(v). Let α1 ∈π1(L(v)) be a generator for the kernel of α 7→\|H′(α)\| and let 1 < \|H′(α2)\| generate the image, so that α1 and α2 generate π1(L(v)) = Z ⊕Z. Then the added circle in N(v) has length log \|H′(α2)\|. A cross-section of ¯ N(v) perpendicular to the added circle is a cone Cθ, obtained by taking a two-dimensional hyperbolic sector Sθ of angle θ, [0 < θ < ∞] and identifying the two bounding rays: 4.15 It is easy to make sense of this even when θ > 2π. The cone angle θ is the argument of the element ˜ H′(α2) ∈˜ C∗. In the special case θ = 2π, Cθ is non-singular, so N(v) is a hyperbolic manifold. N(v) may be seen directly in this special case, as the solid cone I ∪(z −axis) modulo H. 4.5. The generalized Dehn surgery invariant. Consider any three-manifold M which is the interior of a compact manifold ˆ M whose boundary components P1, . . . , Pk are tori. For each i, choose generators ai, bi for π1(Pi). If M is identiﬁed with the complement of an open tubular neighborhood of a link L in S3, there is a standard way to do this, so that ai is a meridian (it bounds a disk in the solid torus around the corresponding component of L) and bi is 56 Thurston — The Geometry and Topology of 3-Manifolds 4.5. THE GENERALIZED DEHN SURGERY INVARIANT. a longitude (it is homologous to zero in the complement of this solid torus in S3). In this case we will call the generators mi and li. We will use the notation M(α1,β1),...,(αk,βk) to denote the manifold obtained by gluing solid tori to M so that a meridian in the i-th solid torus goes to αi, ai + βibi. If an ordered pair (αi, βi) is replaced by the symbol ∞, this means that nothing is glued in near the i-th torus. Thus, M = M∞,...,∞. These notions can be reﬁned and extended in the case M has a hyperbolic struc-4.16 ture whose completion ¯ M is of the type described in 4.4. (In other words, if M is not complete near Pi, the developing map for some deleted neighborhood Ni of Pi should be a covering of the deleted neighborhood I of radius r about a line in H3.) The developing map D of Ni can be lifted to ˜ I, with holonomy ˜ H. The group of isometries of ˜ I is R ⊕R, parametrized by (translation distance, angle of rotation); this parametrization is well-deﬁned up to sign. Definition 4.5.1. The generalized Dehn surgery invariants (αi, βi) for ¯ M are solutions to the equations αi ˜ H(ai) + βi ˜ H(bi) = (rotation by ± 2π), (or, (αi, βi) = ∞if M is complete near Pi). Note that (αi, βi) is unique, up to multiplication by −1, since when M is not complete near Pi, the holonomy ˜ H : π1(Ni) →R ⊕R is injective. We will say that ¯ M is a hyperbolic structure for M(α1,β1),...,(αk,βk). If all (αi, βi) happen to be primitive elements of Z⊕Z, then ¯ M is the topological man-ifold M(α1,β1),...,(αk,βk) with a non-singular hyperbolic structure, so that our extended deﬁnition is compatible with the original. If each ratio αi/βi is the rational number pi/qi in lowest terms, then ¯ M is topologically the manifold M(p1,q1),...,(pk,qk). The hy-perbolic structure, however, has singularities at the component circles of ¯ M −M with 4.17 cone angles of 2π(pi/αi) [since the holonomy ˜ H of the primitive element piai + qibi in π1(Pi) is a pure rotation of angle 2π(pi/αi)]. There is also a topological interpretation in case the (αi, βi) ∈Z ⊕Z, although they may not be primitive. In this case, all the cone angles are of the form 2π/ni, where each ni is an integer. Any branched cover of ¯ M which has branching index ni around the i-th circle of ¯ M −M has a non-singular hyperbolic structure induced from ¯ M. Thurston — The Geometry and Topology of 3-Manifolds 57 4. HYPERBOLIC DEHN SURGERY 4.6. Dehn surgery on the ﬁgure-eight knot. For each value of w in the region R of C shown on p. 4.10, the associated hyperbolic structure on S3−K, where K is the ﬁgure-eight knot, has some Dehn surgery invariant d(w) = ±(µ(w), λ(w)). The function d is a continuous map from R to the one-point compactiﬁcation R2/ ± 1 of R2 with vectors v identiﬁed to −v. Every primitive element (p, q) of Z ⊕Z which lies in the image d(R) describes a closed manifold (S3 −K)(p,q) which possesses a hyperbolic structure. Actually, the map d can be lifted to a map ˜ d : R →ˆ R2, by using the fact that the sign of a rotation of ^ (H3 −z-axis) is well-deﬁned. (See §4.4. The extra information actually comes from the orientation of the z-axis determined by the direction in which the corners of tetrahedra wrap around it). ˜ d is deﬁned by the equation ˜ d(w) = (µ, λ) where 4.18 µ ˜ H(m) + λ ˜ H2(l) = (a rotation by +2π) In order to compute the image ˜ d(R), we need ﬁrst to express the generators l and m for π1(P) in terms of the previous generators x and y on p. 4.11. Referring to page 6, we see that a meridian which only intersects two two-cells can be constructed in a small neighborhood of K. The only generator of π1(L(v)) (see p. 4.11) which intersects only two one-cells is ±y, so we may choose m = y. Here is a cheap way to see what l is. The ﬁgure-eight knot can be arranged (passing through the point at inﬁnity) so that it is invariant by the map v 7→−v of ˆ R3 = S3. This map can be made an isometry of the complete hyperbolic structure constructed for S3 −K. (This can be seen directly; it also follows immediately from Mostow’s Theorem, . . . ). This hyperbolic isometry induces an isometry of the Euclidean struc-ture on L(v) which takes m to m and l to −l. Hence, a geodesic representing l must be orthogonal to a geodesic representing m, so from the diagram on the bottom of 4.19 p. 4.11 we deduce that the curve l = +x + 2y is a longitude. (Alternatively, it is not hard to compute m and l directly). 58 Thurston — The Geometry and Topology of 3-Manifolds 4.6. DEHN SURGERY ON THE FIGURE-EIGHT KNOT. From p. 4.12, we have 4.6.1. H(m) = w(1 −z) H(l) = z2(1 −z)2 The behavior of the map ˜ d near the boundary of R is not hard to determine. For instance, when w is near the ray Im(w) = 0, Re(w) > 1, then z is near the ray Im(z) = 0, Re(z) < 0. The arguments of ˜ H(m) and ˜ H(l) are easily computed by analytic continuation from the complete case w = z = 3 √−1 (when the arguments are 0) to be arg ˜ H(m) = 0 arg ˜ H(l) ≈±2π. Consequently, (µ, λ) is near the line λ = +1. As w →1 we see from the equation z(1 −z) w(1 −w) = 1 that \|z\|2 · \|w\| →1 so (µ, λ) must approach the line µ+4λ = 0. Similarly, as w →+∞, then \|z\| \|w\|2 →1, so (µ, λ) must approach the line µ −4λ = 0. Then the map ˜ d extends continuously to send the line segment 1, +∞ to the line segment 4.20 (−4, +1), (+4, +1). There is an involution τ of the region R obtained by interchanging the solutions z and w of the equation z(l −z) w(l −w) = 1. Note that this involution takes H(m) to 1/H(m) = H(−m) and H(l) to H(−l). Therefore ˜ d(τw) = −˜ d(w). It follows that ˜ d extends continuously to send the line segment −∞, 0 to the line segment (+4, −1), (−4, −1). When \|w\| is large and 0 < arg(w) < π/2, then \|z\| is small and arg(z) ≈π −2 arg(w). Thus arg ˜ H(m) ≈arg w, arg ˜ H(l) ≈2π −4 arg w so µ arg w + λ(2π −4 arg w) = 2π. By considering \|H(m)\| and \|H(l)\|, we have also µ −4λ ≈0, so (µ, λ) ≈(4, 1). There is another involution σ of R which takes w to 1 −w Thurston — The Geometry and Topology of 3-Manifolds 59 4. HYPERBOLIC DEHN SURGERY (and z to 1 −z). From 4.6.1 we conclude that if ˜ d(w) = (µ, λ), then ˜ d(σw) = (µ, −λ). With this information, we know the boundary behavior of ˜ d except when w or τw is near the ray r described by Re(w) = 1 2, Im(w) ≥ √ 15 2 i. The image of the two sides of this ray is not so neatly described, but it does not represent a true boundary for the family of hyperbolic structures on S3 −K, as w crosses r from right to left, for instance, z crosses the real line in the interval (0, 1 2). For a while, a hyperbolic structure can be constructed from the positively oriented simplex determined by w and the negatively oriented simplex determined by z, by cutting the z-simplex into pieces which are subtracted from the w-simplex to leave a polyhedron P. P is then identiﬁed to give a hyperbolic structure for S3 −K. 4.21 For this reason, we give only a rough sketch of the boundary behavior of ˜ d near r or τ(r): 60 Thurston — The Geometry and Topology of 3-Manifolds 4.8. DEGENERATION OF HYPERBOLIC STRUCTURES. Since the image of ˜ d in ˆ R2 does not contain the origin, and since ˜ d sends a curve winding once around the boundary of R to the curve abcd in ˆ R2, it follows that the image of ˜ d(R) contains the exterior of this curve. 4.22 In particular Theorem 4.7. Every manifold obtained by Dehn surgery along the ﬁgure-eight knot K has a hyperbolic structure, except the six manifolds: (S3 −K)(µ,λ) = (S3 −K)(±µ,±λ) where (µ, λ) is (1, 0), (0, 1), (1, 1), (2, 1), (3, 1) or (4, 1). The equation (S3 −K)(α,β) = (S3 −K(−α,β) follows from the existence of an orientation reversing homeomorphism of S3 −K. I ﬁrst became interested in studying these examples by the beautiful ideas of Jørgensen (compare Jørgensen, “Compact three-manifolds of constant negative cur-vature ﬁbering over the circle,” Annals 106 (1977) 61-72). He found the hyperbolic structures corresponding to the ray µ = 0, λ > 1, and in particular, the integer and half-integer (!) points along this ray, which determine discrete groups. The statement of the theorem is meant to suggest, but not imply, the true fact that the six exceptions do not have hyperbolic structures. Note that at least S3 = (S3 −K)(1,0) does not admit a hyperbolic structure (since π1(S3) is ﬁnite). We shall arrive at an understanding of the other ﬁve exceptions by studying the way the hyperbolic structures are degenerating as (µ, λ) tends to the line segment (−4, 1), (4, 1). 4.8. Degeneration of hyperbolic structures. Definition 4.8.1. A codimension-k foliation of an n-manifold M is a G-struc-ture, on M, where G is the pseudogroup of local homeomorphisms of Rn−k×Rk which 4.23 have the local form φ(x, y) = (f(x, y), g(y)). In other words, G takes horizontal (n −k)-planes to horizontal (n −k)-planes. These horizontal planes piece together in M as (n −k)-submanifolds, called the leaves of the foliation. M, like a book without its cover, is a disjoint union of its leaves. Thurston — The Geometry and Topology of 3-Manifolds 61 4. HYPERBOLIC DEHN SURGERY For any pseudogroup H of local homeomorphisms of some k-manifold N k, the notion of a codimension-k foliation can be reﬁned: Definition 4.8.2. An H-foliation of a manifold M n is a G-structure for M n, where G is the pseudogroup of local homeomorphisms of Rn−k × N k which have the local form φ(x, y) = (f(x, y), g(y)) with g ∈H. If H is the pseudogroup of local isometries of hyperbolic k-space, then an H-foliation shall, naturally, be called a codimension-k hyperbolic foliation. A hyperbolic foliation determines a hyperbolic structure for each k-manifold transverse to its leaves. When w tends in the region R ⊂C to a point R −[0, 1], the w-simplex and the z-simplex are both ﬂattening out, and in the limit they are ﬂat: 4.24 If we regard these ﬂat simplices as projections of nondegenerate simplices A and B (with vertices deleted), this determines codimension-2 foliations on A and B, whose leaves are preimages of points in the ﬂat simplices: 62 Thurston — The Geometry and Topology of 3-Manifolds 4.8. DEGENERATION OF HYPERBOLIC STRUCTURES. 4.25 A and B glue together (in a unique way, given the combinatorial pattern) to yield a hyperbolic foliation on S3 −K. You should satisfy yourself that the gluing consistency conditions for the hyperbolic foliation near an edge result as the limiting case of the gluing conditions for the family of squashing hyperbolic structures. The notation of the developing map extends in a straightforward way to the case of an H-foliation on a manifold M, when H is the set of restrictions of a group J of real analytic diﬀeomorphisms of N k; it is a map D : ˜ M n →N k. Note that D is not a local homeomorphism, but rather a local projection map, or a submersion. The holonomy H : π1(M) →J is deﬁned, as before, by the equation D ◦Tα = H(α) ◦D. Here is the generalization of proposition 3.6 to H-foliations. For simplicity, assume that the foliation is diﬀerentiable: Proposition 4.8.1. If J is transitive and if the isotropy subgroups Jx are com-labeled 4.8.1def pact, then the developing map for any H-foliation F of a closed manifold M is a 4.26 ﬁbration D : ˜ M n →N k. Thurston — The Geometry and Topology of 3-Manifolds 63 4. HYPERBOLIC DEHN SURGERY Proof. Choose a plane ﬁeld τ k transverse to F (so that τ is a complementary subspace to the tangent space to the leaves of F, called TF, at each point). Let h be an invariant Riemannian metric on N k and let g be any Riemannian metric on M. Note that there is an upper bound K for the diﬀerence between the g-length of a nonzero vector in τ and the k-length of its local projection to N k. Deﬁne a horizontal path in ˜ M to be any path whose tangent vector always lies in τ. Let α : [0, 1] →N be any diﬀerentiable path, and let ˜ α0 be any point in the preimage D−1(α0). Consider the problem of lifting α to a horizontal path in ˜ M beginning at ˜ α0. Whenever this has been done for a closed interval (such as [0, 0]), it can be obviously extended to an open neighborhood. When it has been done for an open interval, the horizontal lift ˜ α is a Cauchy path in ˜ M, so it converges. Hence, by “topological induction”, α has a (unique) global horizontal lift beginning at ˜ α0. Using horizontal lifts of the radii of disks in N, local trivializations for D : ˜ M →N are obtained, showing that D is a ﬁbration. □ Definition. An H-foliation is complete if the developing map is a ﬁbration. 4.27 Any three-manifold with a complete codimension-2 hyperbolic foliation has uni-versal cover H2 ×R, and covering transformations act as global isometries in the ﬁrst coordinate. Because of this strong structure, we can give a complete classiﬁcation of such manifolds. A Seifert ﬁbration of a three-manifold M is a projection p : M →B to some surface B, so that p is a submersion and the preimages of points are circles in M. A Seifert ﬁbration is a ﬁbration except at a certain number of singular points x1, . . . , xk. The model for the behavior in p−1(Nϵ(xi)) is a solid torus with a foliation having the core circle as one leaf, and with all other leaves winding p times around the short way and q times around the long way, where 1 < p < q and (p, q) = 1. The projection of a meridian disk of the solid torus to its image in B is q-to-one, 4.28 except at the center where it is one-to-one. A group of isometries of a Riemannian manifold is discrete if for any x, the orbit of x intersects a small neighborhood of x only ﬁnitely often. A discrete group Γ of orientation-preserving isometries of a surface N has quotient N/Γ a surface. The projection map N →N/Γ is a local homeomorphism except at points x where the isotropy subgroup Γx is nontrivial. In that case, Γx is a cyclic group Z/qZ for some q > 1, and the projection is similar to the projection of a meridian disk cutting across a singular ﬁber of a Seifert ﬁbration. Theorem 4.9. Let F be a hyperbolic foliation of a closed three-manifold M. Then either 64 Thurston — The Geometry and Topology of 3-Manifolds 4.8. DEGENERATION OF HYPERBOLIC STRUCTURES. A meridian disk of the solid torus wraps q times around its image disk. Here p = 1 and q = 2. (a) The holonomy group H(π1M) is a discrete group of isometries of H2, and the developing map goes down to a Seifert ﬁbration D/π1M : M →H2/H(π1M), or (b) The holonomy group is not discrete, and M ﬁbers over the circle with ﬁber a torus. The structure of F and M in case (b) will develop in the course of the proof. 4.29 Proof. (a) If H(π1M) is discrete, then H2/H(π1M) is a surface. Since M is compact the ﬁbers of the ﬁbration D : ˜ M 3 →H2 are mapped to circles under the projection π : ˜ M 3 →M 3. It follows that D/H(π1M) : M 3 →H2/H(π1M) is a Seifert ﬁbration. (b) When H(π1M) is not discrete, the proof is more involved. First, let us assume that the foliation is oriented (this means that the leaves of the foliation are oriented, or in other words, it is determined by a vector ﬁeld). We choose a π1 M-invariant Riemannian metric g in ˜ M 3 and let τ be the plane ﬁeld which is perpendicular to the ﬁbers of D : ˜ M 3 →H2. We also insist that along τ, g be equal to the pullback of the hyperbolic metric on H2. By construction, g deﬁnes a metric on M 3, and, since M 3 is compact, there is an inﬁmum I to the length of a nontrivial simple closed curve in M 3 when measured with respect to g. Given g1, g2 ∈π1M, we say that they are comparable if there is a y ∈˜ M 3 such that d D(g1(y)), D(g2(y)) < I, Thurston — The Geometry and Topology of 3-Manifolds 65 4. HYPERBOLIC DEHN SURGERY where d( , ) denotes the hyperbolic distance in H2. In this case, take the geodesic in H2 from D(g1(y)) to D(g2(y)) and look at its horizontal lift at g2(y). Suppose its other endpoint e where g1(y). Then the length of the lifted path would be equal to the length of the geodesic in H2, which is less than I. Since g1g−1 2 takes g2(y) to g1(y), the path represents a nontrivial element of π1M and we have a contradiction. Now if we choose a trivialization of H2 × R, we can decide whether or not g1(x) is greater than e. If it is greater than e we say that g1 is greater than g2, and write 4.30 g1 > g2, otherwise we write g1 < g2. To see that this local ordering does not depend on our choice of y, we need to note that U(g1, g2) = {x \| d H(g1(x)), H(g2(x)) < I} is a connected (in fact convex) set. This follows from the following lemma, the proof of which we defer. Lemma 4.9.1. fg1,g2(x) = d(g1x, g2x) is a a convex function on H2. One useful property of the ordering is that it is invariant under left and right multiplication. In other words g1 < g2 if and only if, for all g3, we have g3g1 < g3g2 and g1g3 < g2g3. To see that the property of comparability is equivalent for these three pairs, note that since H(π1H3) acts as isometries on H2, d(Dg1y, Dg2y) < I implies that d(Dg3g1y, Dg3g2y) < I. Also, if d(Dg1y, Dg2y) then d(Dg3g1(g−1 3 y), Dg3g2(g−1 3 y)) < I, so that g1g3 and g2g3 are comparable. The invariance of the ordering easily follows (using the fact that π1M preserves orientation of the R factors). For a ﬁxed x ∈H2 we let Gϵ(X) ⊂π1M be those elements whose holonomy acts on x in a way C1 −ϵ-close to the identity. In other words, for g ∈Gϵ(x), d(x, Hg(x)) < ϵ and the derivative of Hg(x) parallel translated back to x, is ϵ-close to the identity. Proposition 4.9.2. There is an ϵ0 so that for all ϵ < ϵ0 [Gϵ, Gϵ] ⊂Gϵ. Proof. For any Lie group the map [∗, ∗] : G × G →G has derivative zero at (id, id). Since for any g ∈G, (g, id) 7→id and (id, g) 7→1. The tangent spaces of G × id and id ×G span the tangent space to G × G at (id, id). Apply this to the group of isometries of H2. □ From now on we choose ϵ < I/8 so that any two words of length four or less in Gϵ are comparable. We claim that there is some β ∈Gϵ which is the “smallest” element in Gϵ which is > id. In other words, if id < α ∈Gϵ, a ̸= β, then α > β. This can be seen as follows. Take an ϵ-ball B of x ∈H2 and look at its inverse image ˜ B under D. Choose a point y in ˜ B and consider y and α(y), where α ∈Gϵ. We can truncate ˜ B 66 Thurston — The Geometry and Topology of 3-Manifolds 4.8. DEGENERATION OF HYPERBOLIC STRUCTURES. There are only ﬁnitely many translates of y in this region. by the lifts of B (using the horizontal lifts of the geodesics through x) through y and α(y). Since this is a compact set there are only a ﬁnite number of images of y under π1M contained in it. Hence there is one β(y) whose R coordinate is the closest to that of y itself. β is clearly our minimal element. 4.32 Now consider α > β > 1, α ∈Gϵ. By invariance under left and right multiplica-tion, α2 > βα > α and α > α−1βα > 1. Suppose α−1βα < β. Then β > α−1βα > 1 so that 1 > α−1βαβ−1 > β−1. Similarly if α−1βα > β > 1 then β > αβα−1 > 1 so that 1 > αβα−1β−1 > β−1. Note that by multiplicative invariance, if g1 > g2 then g−1 2 = g−1 1 g1g−1 2 > g−1 1 g2g−1 2 = g−1 1 . We have either 1 < βα−1β−1α < β or 1 < βαβ−1α−1 < β which contradicts the minimality of β . Thus α−1βα = β for all α ∈Gϵ. We need to digress here for a moment to classify the isometries of H2. We will prove the following: Proposition 4.9.3. If g : H2 →H2 is a non-trivial isometry of H2 which preserves orientation, then exactly one of the following cases occurs: (i) g has a unique interior ﬁxed point or (ii) g leaves a unique invariant geodesic or (iii) g has a unique ﬁxed point on the boundary of H2. Case (i) is called elliptic, case (ii) hyperbolic, case (iii) parabolic. Thurston — The Geometry and Topology of 3-Manifolds 67 4. HYPERBOLIC DEHN SURGERY Proof. This follows easily from linear algebra, but we give a geometric proof. Pick an interior point x ∈H2 and connect x to gx by a geodesic l0. Draw the geodesics l1, l2 at gx and g2x which bisect the angle made by l0 and gl0, gl0 and g2l0 respectively. There are three cases: 4.33 (i) l1 and l2 intersect in an interior point y (ii) There is a geodesic l3 perpendicular to l1, l2 (iii) l1, l2 are parallel, i.e., they intersect at a point at inﬁnity x3. In case (i) the length of the arc gx, y equals that of g2x, y since ∆(gx, g2x, y) is an isoceles triangle. It follows that y is ﬁxed by g. In case (ii) the distance from gx to l3 equals that from g2x to l3. Since l3 meets l1 and l2 in right angles it follows that l3 is invariant by g. 4.34 Finally, in case (iii) g takes l1 and l2, both of which hit the boundary of H2 in the same point x3. It follows that g ﬁxes x3 since an isometry takes the boundary to itself. Uniqueness is not hard to prove. □ Using the classiﬁcation of isometries of H2, it is easy to see that the centralizer of any non-trivial element g in isom(H2) is abelian. (For instance, if g is elliptic with 68 Thurston — The Geometry and Topology of 3-Manifolds 4.8. DEGENERATION OF HYPERBOLIC STRUCTURES. ﬁxed point x0, then the centralizer of g consists of elliptic elements with ﬁxed point x0). It follows that the centralizer of β in π1(M) is abelian; let us call this group N. Although Gϵ(x) depends on the point x, for any point x′ ∈H2, if we choose ϵ′ small enough, then Gϵ′(x′) ⊂Gϵ(x). In particular if x = H(g)x, g ∈π1M, then all elements of Gϵ′(x′) commute with β. It follows that N is a normal subgroup of π1(M). Consider now the possibility that β is elliptic with ﬁxed point x0 and n ∈N ﬁxes x0 we see that all of π1M must ﬁx x0. But the function fx0 : H2 →R+ which measures the distance of a point in H2 from x0 is H(π1M) invariant so that it lifts to a function f on M 3. However, M 3 is compact and the image of ˜ f is non-compact, which is impossible. Hence β cannot be elliptic. If β were hyperbolic, the same reasoning would imply that H(π1M) leaves invari-4.35 ant the invariant geodesic of β. In this case we could deﬁne fl : H2 →R to be the distance of a point from l. Again, the function lifts to a function on M 3 and we have a contradiction. The case when β is parabolic actually does occur. Let x0 be the ﬁxed point of β on the circle at inﬁnity. N must also ﬁx x0. Using the upper half-plane model for H2 with x0 at ∞, β acts as a translation of R2 and N must act as a group of similarities; but since they commute with β, they are also translations. Since N is normal, π1M must act as a group of similarities of R2 (preserving the upper half-plane). Clearly there is no function on H2 measuring distance from the point x0 at inﬁnity. If we consider a family of ﬁnite points xτ →X, and consider the functions fxτ, even though fxτ blows up, its derivative, the closed 1-form d fxτ, converges to a closed 1 form ω. Geometrically, ω vanishes on tangent vectors to horocycles about x0 and takes the value 1 on unit tangents to geodesics emanating from x0. 4.36 The non-singular closed 1-form ω on H2 is invariant by H (π1M), hence it deﬁnes a non-singular closed one-form ¯ ω on M. The kernel of ¯ ω is the tangent space to Thurston — The Geometry and Topology of 3-Manifolds 69 4. HYPERBOLIC DEHN SURGERY the leaves of a codimension one foliation F of M. The leaves of the corresponding foliation ˜ F on ˜ M are the preimages under D of the horocycles centered at x0. The group of periods for ω must be discrete, for otherwise there would be a translate of the horocycle about x0 through x close to x, hence an element of Gϵ which does not commute with β. Let p0 be the smallest period. Then integration of ω deﬁnes a map from M to S1 = R/⟨p0⟩, which is a ﬁbration, with ﬁbers the leaves of F. The fundamental group of each ﬁber is contained in N, which is abelian, so the ﬁbers are toruses. It remains to analyze the case that the hyperbolic foliation is not oriented. In this case, let M ′ be the double cover of M which orients the foliation. M ′ ﬁbers over S1 with ﬁbration deﬁned by a closed one-form ω. Since ω is determined by the unique ﬁxed point at inﬁnity of H(π1M ′), ω projects to a non-singular closed one-form on M. This determines a ﬁbration of M with torus ﬁbers. (Klein bottles cannot occur even if M is not required to be orientable.) □ We can construct a three-manifold of type (b) by considering a matrix A ∈SL(2, Z) which is hyperbolic, i.e., it has two eigenvalues λ1, λ2 and two eigenvectors V1, V2. Then AV1 = λ1V1, AV2 = λ2V2 and λ2 = 1/λ1. Since A ∈SL(2, Z) preserves Z ⊕Z its action on the plane descends to an action on the torus T 2. Our three-manifold MA is the mapping torus of the action of A 4.37 on T 2. Notice that the lines parallel to V1 are preserved by A so they give a one-dimensional foliation on MA. Of course, the lines parallel to V2 also deﬁne a foliation. The reader may verify that both these foliations are hyperbolic. When A = 2 1 1 1 , then MA is the manifold (S3 −K)(D,±1) obtained by Dehn surgery on the ﬁgure-eight knot. The hyperbolic foliations corresponding to (0, 1) and (0, −1) are distinct, and they correspond to the two eigenvectors of 2 1 1 1 . All codimension-2 hyperbolic foliations with leaves which are not closed are obtained by this construction. This follows easily from the observation that the hyperbolic foliation restricted to any ﬁber is given by a closed non-singular one-form, together with the fact that a closed non-singular one-form on T 2 is determined (up to isotopy) by its cohomology class. The three manifolds (S3 −K)(1,1), (S3 −K)(2,1) and (S3 −K)(3,1) also have codimension-2 hyperbolic foliations which arise as “limits” of hyperbolic structures. 70 Thurston — The Geometry and Topology of 3-Manifolds 4.10. INCOMPRESSIBLE SURFACES IN THE FIGURE-EIGHT KNOT COMPLEMENT. Since they are rational homology spheres, they must be Seifert ﬁber spaces. A Seifert ﬁber space cannot be hyperbolic, since (after passing to a cover which orients the ﬁbers) a general ﬁber is in the center of its fundamental group. On the other hand, the centralizer of an element in the fundamental group of a hyperbolic manifold is abelian. 4.38 4.10. Incompressible surfaces in the ﬁgure-eight knot complement. Let M 3 be a manifold and S ⊂M 3 a surface with ∂S ⊂∂M. Assume that S ̸= S2, IP 2, or a disk D2 which can be pushed into ∂M. Then S is incompressible if every loop (simple closed curve) on S which bounds an (open) disk in M −S also bounds a disk in S. Some people prefer the alternate, stronger deﬁnition that S is (strongly) incompressible if π1(S) injects into π1(M). By the loop theorem of Papakyriakopoulos, these two deﬁnitions are equivalent if S is two-sided. If S has boundary, then S is also ∂-incompressible if every arc α in S (with ∂(α) ⊂∂S) which is homotopic to ∂M is homotopic in S to ∂S. If M is oriented and irreducible (every two-sphere bounds a ball), then M is suﬃciently large if it contains an incompressible and ∂-incompressible surface. A 4.39 compact, oriented, irreducible, suﬃciently large three-manifold is also called a Haken-manifold. It has been hard to ﬁnd examples of three-manifolds which are irreducible but can be shown not to be suﬃciently large. The only previously known examples are certain Seifert ﬁbered spaces over S2 with three exceptional ﬁbers. In what follows we give the ﬁrst known examples of compact, irreducible three-manifolds which are not Haken-manifolds and are not Seifert ﬁber spaces. Thurston — The Geometry and Topology of 3-Manifolds 71 4. HYPERBOLIC DEHN SURGERY Note. If M is a compact oriented irreducible manifold ̸= D3, and either ∂M ̸= ∅ or H1(M) ̸= 0, then M is suﬃciently large. In fact, ∂M ̸= 0 ⇒H1(M) ̸= 0. Think of a non-trivial cohomology class α as dual to an embedded surface; an easy argument using the loop theorem shows that the simplest such surface dual to α is incompressible and ∂-incompressible. The concept of an incompressible surface was introduced by W. Haken (Inter-national Congress of Mathematicians, 1954), (Acta. Math. 105 (1961), Math A. 76 (1961), Math Z 80 (1962)). If one splits a Haken-manifold along an incompressible and ∂-incompressible surface, the resulting manifold is again a Haken-manifold. One can continue this process of splitting along incompressible surfaces, eventually arriv-ing (after a bounded number of steps) at a union of disks. Haken used this to give algorithms to determine when a knot in a Haken-manifold was trivial, and when two knots were linked. Let K be a ﬁgure-eight knot, M = S3 −N(K). M is a Haken manifold by 4.40 the above note [M is irreducible, by Alexander’s theorem that every diﬀerentiable two-sphere in S3 bounds a disk (on each side)]. Here is an enumeration of the incompressible and ∂-incompressible surfaces in M. There are six reasonably obvious choices to start with; • S1 is a torus parallel to ∂M, • S2 = T 2-disk = Seifert surface for K. To construct S2, take 3 circles lying above the knot, and span each one by a disk. Join 72 Thurston — The Geometry and Topology of 3-Manifolds 4.10. INCOMPRESSIBLE SURFACES IN THE FIGURE-EIGHT KNOT COMPLEMENT. the disks by a twist for each crossing at K to get a surface S2 with boundary the longitude (0, ±1). S2 is oriented and has Euler characteristic = −1, so it is T 2-disk. • S3 = (Klein bottle-disk) is the unoriented surface pictured. • S4 = ∂(tubular neighborhood of S3) = T 2 −2 disks. ∂S4 = (±4, 1), (depending on the choice of orientation for the meridian). • S5 = (Klein bottle-disk) is symmetric with S3. • S6 = ∂(tubular neighborhood of S5 ) = T 2 −2 disks. ∂S6 = (±4, 1). It remains to show that Theorem 4.11. Every incompressible and ∂-incompressible connected surface in M is isotopic to one of S1 through S6. Corollary. The Dehn surgery manifold M(m,l) is irreducible, and it is a Haken-manifold if and only if (m, l) = (0, ±1) or (±4, ±1). In particular, none of the hyperbolic manifolds obtained from M by Dehn surgery is suﬃciently large. (Compare 4.7.) Thus we have an inﬁnite family of examples of oriented, irreducible, non-Haken-manifolds which are not Seifert ﬁber spaces. It seems likely that Dehn surgery along other knots and links would yield many more examples. 4.42 Proof of corollary from theorem. Think of M(m,l) as M union a solid torus, D2 × S1, the solid torus being a thickened core curve. To see that M(m,l) is irreducible, let S be an embedded S2 in M(m,l), transverse to the core curve α (S intersects the solid torus in meridian disks). Now isotope S to minimize its intersections with α. If S doesn’t intersect α then it bounds a ball by the irreducibility Thurston — The Geometry and Topology of 3-Manifolds 73 4. HYPERBOLIC DEHN SURGERY of M. If it does intersect α we may assume each component of intersection with the solid torus D2 × S1 is of the form D2 × x. If S ∩M is not incompressible, we may divide S into two pieces, using a disk in S ∩M, each of which has fewer intersections with α. If S does not bound a ball, one of the pieces does not bound. If S ∩M is ∂-incompressible, we can make an isotopy of S to reduce the number of intersections with α by 2. Eventually we simplify S so that if it does not bound a ball, S ∩M 4.43 is incompressible and ∂-incompressible. Since none of the surfaces S1, . . . , S6 is a submanifold of S2, it follows from the theorem that S in fact bounds a ball. The proof that M(m,l) is not a Haken-manifold if (m, l) ̸= (0, ±1) or (±4, ±1) is similar. Suppose S is an incompressible surface in M(m,l). Arrange the intersections with D2 × S1 as before. If S ∩M is not incompressible, let D be a disk in M with ∂D ⊂S ∩M not the boundary of a disk in S ∩M. Since S in incompressible, ∂D = ∂D′ for some disk D′ ⊂S which must intersect α. The surface S′ obtained from S by replacing D′ with D is incompressible. (It is in fact isotopic to S, since M is irreducible; but it is easy to see that S′ is incompressible without this.) S′ has fewer intersections with α than does S. If S is not ∂-incompressible, an isotopy can be made as before to reduce the number of intersections with α. Eventually we obtain an incompressible surface (which is isotopic to S) whose intersection with M is incompressible and ∂-incompressible. S cannot be S1 (which is not incompressible in M(m, l)), so the corollary follows from the theorem. □ Proof of theorem 4.11. Recall that M = S3 −N(K) is a union of two tetrahedra-without-vertices. To prove the theorem, it is convenient to use an al-ternate description of M at T 2 × I with certain identiﬁcations on T 2 × {1} (compare Jørgensen, “Compact three-manifolds of constant negative curvature ﬁbering over the 4.44 circle”, Annals of Mathematics 106 (1977), 61–72, and R. Riley). One can obtain this from the description of M as the union of two tetrahedra with corners as follows. Each tetrahedron = (corners) × I with certain identiﬁcations on (corners) × {1}. 74 Thurston — The Geometry and Topology of 3-Manifolds 4.10. INCOMPRESSIBLE SURFACES IN THE FIGURE-EIGHT KNOT COMPLEMENT. This “product” structure carries over to the union of the two tetrahedra. The boundary torus has the triangulation (p. 4.11) 4.45 T 2 × {1} has the dual subdivision, which gives T 2 as a union of four hexagons. The diligent reader can use the gluing patters of the tetrahedra to check that the identiﬁcations on T 2 × {1} are Thurston — The Geometry and Topology of 3-Manifolds 75 4. HYPERBOLIC DEHN SURGERY where we identify the hexagons by ﬂipping through the dotted lines. The complex N = T 2 × {1}/identiﬁcations is a spine for N. It has a cell subdivi-sion with two vertices, four edges, and two hexagons. N is embedded in M, and its complement is T 2 × [0, 1). If S is a connected, incompressible surface in M, the idea is to simplify it with respect to the spine N (this approach is similar in spirit to Haken’s). First isotope S 4.46 so it is transverse to each cell of N. Next isotope S so that it doesn’t intersect any hexagon in a simple closed curve. Do this as follows. If S ∩hexagon contains some loops, pick an innermost loop α. Then α bounds an open disk in M 2 −S (it bounds one in the hexagon), so by incompressibility it bounds a disk in S. By the irreducibility of M we can push this disk across this hexagon to eliminate the intersection α. One continues the process to eliminate all such loop intersections. This does not change the intersection with the one-skeleton N(1). S now intersects each hexagon in a collection of arcs. The next step is to isotope S to minimize the number of intersections with N(1). Look at the preimage of S ∩N. We can eliminate any arc which enters and leaves a hexagon in the same edge by pushing the arc across the edge. If at any time a loop intersection is created with a hexagon, eliminate it before proceeding. 4.47 Next we concentrate on corner connections in hexagons, that is, arcs which con-nect two adjacent edges of a hexagon. Construct a small ball B about each vertex, 76 Thurston — The Geometry and Topology of 3-Manifolds 4.10. INCOMPRESSIBLE SURFACES IN THE FIGURE-EIGHT KNOT COMPLEMENT. and push S so that the corner connections are all contained in B, and so that S is transverse to ∂B. S intersects ∂B in a system of loops, and each component of intersection of S with B contains at least one corner connection, so it intersects N(1) at least twice. If any component of S ∩B is not a disk, there is some “innermost” such component Si; then all of its boundary components bound disks in B, hence in S. Since S is not a sphere, one of these disks in S contains Si. Replace it by a disk in B. This can be done without increasing the number of intersections with N(1), since every loop in ∂B bounds a disk in B meeting N(1) at most twice. Now if there are any two corner connections in B which touch, then some compo-nent of S ∩B meets N(1) at least three times. Since this component is a disk, it can 4.48 be replaced by a disk which meets N(1) at most twice, thus simplifying S. (Therefore at most two corners can be connected at any vertex.) Assume that S now has the minimum number of intersections with N(1) in its isotopy class. Let I, II, III, and IV denote the number of intersections of S with edges I, II, III, and IV, respectively (no confusion should result from this). It remains to analyze the possibilities case by case. Suppose that none of I, II, III, and IV are zero. Then each hexagon has connec-tions at two corners. Here are the possibilities for corner connections in hexagon A. If the corner connections are at a and b then the picture in hexagon A is of the form Thurston — The Geometry and Topology of 3-Manifolds 77 4. HYPERBOLIC DEHN SURGERY 4.49 This implies that II = I + III + II + I + IV, which is impossible since all four numbers are positive in this case. A similar argument also rules out the possibilities c-d, d-e, a-f, b-f, and c-e in hexagon, and h-i, i-j, k-l, g-l, g-k and h-j in hexagon B. The possibility a-c cannot occur since they are adjacent corners. For the same reason we can rule out a-e, b-d, d-f, g-i, i-k, h-l, and j-l. Since each hexagon has at least two corner connections, at each vertex we must have connections at two opposite corners. This means that knowing any one corner connection also tells you another corner connection. Using this one can rule out all possible corner connections for hexagon A except for a-d. If a-d occurs, then I + IV + II = I + III + II, or III = IV. By the requirement of opposite corners at the vertices, in hexagon B there are corner connections at i and l, which implies that I = II. Let x = III and y = I. The picture is then 4.50 We may reconstruct the intersection of S with a neighborhood of N, say N(N), from this picture, by gluing together x + y annuli in the indicated pattern. This yields x+y punctured tori. If an x-surface is pushed down across a vertex, it yields a y-surface, and similarly, a y-surface can be pushed down to give an x-surface. Thus, S ∩N(N) is x + y parallel copies of a punctured torus, which we see is the ﬁber of a ﬁbration of N(N) ≈M over S1. We will discuss later what happens outside N(N). (Nothing.) Now we pass on to the case that at least one of I, II, III, and IV are zero. The case I = 0 is representative because of the great deal of symmetry in the picture. 78 Thurston — The Geometry and Topology of 3-Manifolds 4.10. INCOMPRESSIBLE SURFACES IN THE FIGURE-EIGHT KNOT COMPLEMENT. First consider the subcase I = 0 and none of II, III, and IV are zero. If hexagon B had only one corner connection, at h, then we would have III + IV = II + IV + III, contradicting II > 0. By the same reasoning for all the other corners, we ﬁnd that hexagon B needs at least two corner connections. At most one corner connection can occur in a neighborhood of each vertex in N, since no corner connection can involve 4.51 the edge I. Thus, hexagon B must have exactly two corner connections, and hexagon A has no corner connections. By checking inequalities, we ﬁnd the only possibility is corner connections at g-h. If we look at the picture in the pre-image T 2 × {1} near I we see that there is a loop around I. This loop bounds a disk in S by incompressibility, and pushing the disk across the hexagons reduces the number of intersections with N(1) by at least two (you lose the four intersections drawn in the picture, and gain possibly two intersections, above the plane of the paper). Since S already has minimal intersection number with N(1) already, this subcase cannot happen. Now consider the subcase I = 0 and II = 0. In hexagon A the picture is Thurston — The Geometry and Topology of 3-Manifolds 79 4. HYPERBOLIC DEHN SURGERY 4.52 implying III = IV. The picture in hexagon B is with y the number of corner connections at corner l and x = IV −y. The three subcases to check are x and y both nonzero, x = 0, and y = 0. If both x and y are nonzero, there is a loop in S around edges I and II. The loop bounds a disk in S, and pushing the disk across the hexagons reduces the number of intersections by at least two, contradicting minimal-ity. So x and y cannot both be nonzero. If I = II = 0 and x = 0, then S ∩N(N) is y parallel copies of a punctured torus. 4.53 80 Thurston — The Geometry and Topology of 3-Manifolds 4.10. INCOMPRESSIBLE SURFACES IN THE FIGURE-EIGHT KNOT COMPLEMENT. If I = II = 0 and y = 0, then S ∩N(N) consists of ⌊x/2⌋copies of a twice punctured torus, together with one copy of a Klein bottle if x is odd. Now consider the subcase I = III = 0. If S intersects the spine N, then II ̸= 0 4.54 because of hexagon A and IV ̸= 0 because of hexagon B. But this means that there is a loop around edges I and III, and S can be simpliﬁed further, contradicting minimality. Thurston — The Geometry and Topology of 3-Manifolds 81 4. HYPERBOLIC DEHN SURGERY The subcase I = IV = 0 also cannot occur because of the minimality of the number of intersections of S and N(1). Here is the picture. By symmetric reasoning, we ﬁnd that only one more case can occur, that III = IV = 0, with I = II. The pictures are symmetric with preceding ones: 4.55 82 Thurston — The Geometry and Topology of 3-Manifolds 4.10. INCOMPRESSIBLE SURFACES IN THE FIGURE-EIGHT KNOT COMPLEMENT. To ﬁnish the proof of the theorem, it remains to understand the behavior of S in M −N(N) = T 2 × [0, .99]. Clearly, S ∩(T 2 × [0, .99]) must be incompressible. (Otherwise, for instance, the number of intersections of S with N(1) could be reduced.) It is not hard to deduce that either S is parallel to the boundary, or else a union of annuli. If one does not wish to assume S is two-sided, this may be accomplished by studying the intersection of S ∩(T 2 × [0, .99]) with a non-separating annulus. 4.56 If any annulus of S ∩(T 2 × [0, .99]) has both boundary components in T 2 × .99, then by studying the cases, we ﬁnd that S would not be incompressible. It follows that S ∩(T 2 × [0, .99]) can be isotoped to the form (circles × [0, .99]). There are ﬁve possibilities (with S connected). Careful comparisons lead to the descriptions of S2, . . . , S6 given on pages 4.40 and 4.41. □ Thurston — The Geometry and Topology of 3-Manifolds 83 William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T EX by Sheila Newbery, who also scanned the ﬁgures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the ﬁrst few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at levy@msri.org. CHAPTER 5 Flexibility and rigidity of geometric structures In this chapter we will consider deformations of hyperbolic structures and of geometric structures in general. By a geometric structure on M, we mean, as usual, a local modelling of M on a space X acted on by a Lie group G. Suppose M is compact, possibly with boundary. In the case where the boundary is non-empty we do not make special restrictions on the boundary behavior. If M is modelled on (X, G) then the developing map ˜ M D − →X deﬁnes the holonomy representation H : π1M − →G. In general, H does not determine the structure on M. For example, the two immersions of an annulus shown below deﬁne Euclidean structures on the annulus which both have trivial holonomy but are not equivalent in any reasonable sense. However, the holonomy is a complete invariant for (G, X)-structures on M near a given structure M0, up to an appropriate equivalence relation: two structures M1 and M2 near M0 are equivalent deformations of M0 if there are submanifolds M ′ 1 and 5.2 M ′ 2, containing all but small neighborhoods of the boundary of M1 and M2, with a (G, X) homeomorphism between them which is near the identity. Let M0 denote a ﬁxed structure on M, with holonomy H0. Proposition 5.1. Geometric structures on M near M0 are determined up to equivalency by holonomy representations of π1M in G which are near H0, up to conjugacy by small elements of G. Thurston — The Geometry and Topology of 3-Manifolds 85 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES Proof. Any manifold M can be represented as the image of a disk D with reasonably nice overlapping near ∂D. Any structure on M is obtained from the structure induced on D, by gluing via the holonomy of certain elements of π1(M). Any representation of π1M near H0 gives a new structure, by perturbing the identiﬁcations on D. The new identiﬁcations are still ﬁnite-to-one giving a new manifold homeomorphic to M0. 5.3 If two structures near M0 have holonomy conjugate by a small element of G, one can make a small change of coordinates so that the holonomy is identical. The two structures then yield nearby immersions of D into X, with the same identiﬁcations; restricting to slightly smaller disks gives the desired (G, X)-homeomorphism. □ 5.2 As a ﬁrst approximation to the understanding of small deformations we can de-scribe π1M in terms of a set of generators G = {g1, . . . , gn} and a set of relators R = {r1, . . . , rl}. [Each ri is a word in the gi’s which equals 1 in π1M.] Any represen-tation ρ : π1M →G assigns each generator gi an element in G, ρ (gi). This embeds the space of representations R in GG. Since any representation of π1M must respect the relations in π1M, the image under ρ of a relator rj must be the identity in G. If p : GG →GR sends a set of elements in G to the \|R\| relators written with these elements, then D is just p−1(1, . . . , 1). If p is generic near H0, (i.e., if the derivative dp is surjective), the implicit function theorem implies that R is just a manifold of dimension (\|G\| −\|R\|) · (dim G). One might reasonably expect this to be the case, provided the generators and relations are chosen in an eﬃcient way. If the action of G on itself by conjugation is eﬀective (as for the group of isometries of hyperbolic space) then generally one would also expect that the action of G on GG by conjugation, near H0, has orbits of the same dimension as G. If so, then the space of deformations of 5.4 M0 would be a manifold of dimension dim G · (\|G\| −\|R\| −1). 86 Thurston — The Geometry and Topology of 3-Manifolds 5.2 Example. Let’s apply the above analysis to the case of hyperbolic structures on closed, oriented two-manifolds of genus at least two. G in this case can be taken to be PSL(2, R) acting on the upper half-plane by linear fractional transformations. π1(Mg) can be presented with 2g generators a1, b1, . . . ag, bg (see below) together with the single relator Q i=1[ai, bi]. Since PSL(2, R) is a real three-dimensional Lie group the expected dimension of the deformation space is 3(2g −1 −1) = 6g −6. This can be made rigorous by showing directly that the derivative of the map p : GG →GR is surjective, but since we will have need for more global information about the deformation space, we won’t make the computation here. 5.5 Example. The initial calculation for hyperbolic structures on an oriented three-manifold is less satisfactory. The group of isometries on H3 preserves planes which, in the upper half-space model, are hemispheres perpendicular to C ∪∞(denoted ˆ C). Thus the group G can be identiﬁed with the group of circle preserving maps of ˆ C. This is the group of all linear fractional transformations with complex coef-ﬁcients PSL(2, C). (All transformations are assumed to be orientation preserving). PSL(2, C), is a complex Lie group with real dimensions 6. M 3 can be built from one zero-cell, a number of one- and two-cells, and (if M is closed), one 3-cell. If M is closed, then χ(M) = 0, so the number k of one-cells equals the number of two-cells. This gives us a presentation of π1M with k generators and k relators. The expected (real) dimension of the deformation space is 6(k −k −1) = −6. If ∂M ̸= ∅, with all boundary components of positive genus, this estimate of the dimension gives 5.2.1. 6 · (−χ(M)) = 3(−χ(∂M)). This calculation would tend to indicate that the existence of any hyperbolic struc-ture on a closed three-manifold would be unusual. However, subgroups of PSL(2, C) have many special algebraic properties, so that certain relations can follow from other relations in ways which do not follow in a general group. 5.6 The crude estimate 5.2.1 actually gives some substantive information when χ(M) < 0. Thurston — The Geometry and Topology of 3-Manifolds 87 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES Proposition 5.2.2. If M 3 possesses a hyperbolic structure M0, then the space of small deformations of M0 has dimension at least 6 · (−χ(M)). Proof. PSL(2, C)G is a complex algebraic variety, and the map p : PSL(2, C)G →PSL(2, C)R is a polynomial map (deﬁned by matrix multiplication). Hence the dimension of the subvariety p = (1, . . . , 1) is at least as great as the number of variables minus the number of deﬁning equations. □ We will later give an improved version of 5.2.2 whenever M has boundary com-ponents which are tori. 5.3 In this section we will derive some information about the global structure of the space of hyperbolic structures on a closed, oriented surface M. This space is called the Teichm¨ uller space of M and is deﬁned to be the set of hyperbolic structures on M where two are equivalent if there is an isometry homotopic to the identity between them. In order to understand hyperbolic structures on a surface we will cut the surface up into simple pieces, analyze structures on these pieces, and study the ways they can be put together. Before doing this we need some information about closed geodesics in M. Proposition 5.3.1. On any closed hyperbolic n-manifold M there is a unique, closed geodesic in any non-trivial free homotopy class. Proof. For any α ∈π1M consider the covering transformation Tα on the uni-versal cover Hn of M. It is an isometry of Hn. Therefore it either ﬁxes some interior point of Hn (elliptic), ﬁxes a point at inﬁnity (parabolic) or acts as a translation on some unique geodesic (hyperbolic). That all isometries of H2 are of one of these types was proved in Proposition 4.9.3; the proof for Hn is similar. Note. A distinction is often made between “loxodromic” and “hyperbolic” trans-formations in dimension 3. In this usage a loxodromic transformation means an isom-etry which is a pure translation along a geodesic followed by a non-trivial twist, while a hyperbolic transformation means a pure translation. This is usually not a useful distinction from the point of view of geometry and topology, so we will use the term “hyperbolic” to cover either case. 88 Thurston — The Geometry and Topology of 3-Manifolds 5.3 Since Tα is a covering translation it can’t have an interior ﬁxed point so it can’t be elliptic. For any parabolic transformation there are points moved arbitrarily small distances. This would imply that there are non-trivial simple closed curves of arbi-trarily small length in M. Since M is closed this is impossible. Therefore Tα trans-lates a unique geodesic, which projects to a closed geodesic in M. Two geodesics corresponding to the translations Tα and T ′ α project to the same geodesic in M if 5.8 and only if there is a covering translation taking one to the other. In other words, α′ = gαg−1 for some g ∈π1M, or equivalently, α is free homotopic to α. □ Proposition 5.3.2. Two distinct geodesics in the universal cover Hn of M which are invariant by two covering translations have distinct endpoints at ∞. Proof. If two such geodesics had the same endpoint, they would be arbitrarily close near the common endpoint. This would imply the distance between the two closed geodesics is uniformly ≤ϵ for all ϵ, a contradiction. □ Proposition 5.3.3. In a hyperbolic two-manifold M 2 a collection of homotopi-cally distinct and disjoint nontrivial simple closed curves is represented by disjoint, simple closed geodesics. Proof. Suppose the geodesics corresponding to two disjoint curves intersect. Then there are lifts of the geodesics in the universal cover H2 which intersect. Since the endpoints are distinct, the pairs of endpoints for the two geodesics must link each other on the circle at inﬁnity. Consider any homotopy of the closed geodesics in M 2. It lifts to a homotopy of the geodesics in H2. However, no homotopy of the geodesics moving points only a ﬁnite hyperbolic distance can move their endpoints; thus the images of the geodesics under such a homotopy will still intersect, and this intersection projects to one in M 2. 5.9 The proof that the closed geodesic corresponding to a simple closed curve is simple is similar. The argument above is applied to two diﬀerent lifts of the same geodesic. □ Now we are in a position to describe the Teichm¨ uller space for a closed surface. The coordinates given below are due to Nielsen and Fenchel. Pick 3g −3 disjoint, non-parallel simple closed curves on M 2. (This is the max-imum number of such curves on a surface of genus g.) Take the corresponding geodesics and cut along them. This divides M 2 into 2g −2 surfaces homeomorphic to S2—three disks (called “pairs of pants” from now on) with geodesic boundary. 5.9a Thurston — The Geometry and Topology of 3-Manifolds 89 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES 5.10 On each pair of pants P there is a unique arc connecting each pair of boundary components, perpendicular to both. To see this, note that there is a unique homotopy class for each connecting arc. Now double P along the boundary geodesics to form a surface of genus two. The union of the two copies of the arcs connecting a pair of boundary components in P deﬁnes a simple closed curve in the closed surface. There is a unique geodesic in its free homotopy class and it is invariant under the reﬂection which interchanges the two copies of P. Hence it must be perpendicular to the geodesics which were in the boundary of P. This information leads to an easy computation of the Teichm¨ uller space of P. Proposition 5.3.4. T(P) is homeomorphic to R3 with coordinates (log l1, log l2, log l3), where li = length of the i-th boundary component. Proof. The perpendicular arcs between boundary components divide P into two right-angled hexagons. The hyperbolic structure of an all-right hexagon is determined by the lengths of three alternating sides. (See page 2.19.) The lengths of the con-necting arcs therefore determine both hexagons so the two hexagons are isometric. Reﬂection in these arcs is an isometry of the hexagons and shows that the boundary curves are divided in half. The lengths li/2 determine the hexagons; hence they also determine P. Any positive real values for the li are possible so we are done. □ 5.11 90 Thurston — The Geometry and Topology of 3-Manifolds 5.3 In order to determine the hyperbolic structure of the closed two-manifold from that of the pairs of pants, some measurement of the twist with which the boundary geodesics are attached is necessary. Find 3g −3 more curves in the closed manifold which, together with the ﬁrst set of curves, divides the surface into hexagons. In the pairs of pants the geodesics corresponding to these curves are arcs connect-ing the boundary components. However, they may wrap around the components. In P it is possible to isotope these arcs to the perpendicular connecting arcs discussed above. Let 2di denote the total number of degrees which this isotopy moves the feet of arcs which lie on the i-th boundary component of p. 5.12 Of course there is another copy of this curve in another pair of pants which has a twisting coeﬃcient d′ i. When the two copies of the geodesic are glued together they cannot be twisted independently by an isotopy of the closed surface. Therefore (di −d′ i) = τi is an isotopy invariant. Remark. If a hyperbolic surface is cut along a closed geodesic and glued back together with a twist of 2πn degrees (n an integer), then the resulting surface is isometric to the original one. However, the isometry is not isotopic to the identity so the two surfaces represent distinct points in Teichm¨ uller space. Another way to say this is that they are isometric as surfaces but not as marked surfaces. It follows that τi is a well-deﬁned real number, not just deﬁned up to integral multiples of 2π. Theorem 5.3.5. The Teichm¨ uller space T(M) of a closed surface of genus g is homeomorphic to R6g−6. There are explicit coordinates for T(M), namely (log l1, τ1, log l2, τ2, . . . , log l3g−3, τ3g−3), where li is the length and τi the twist coeﬃcient for a system of 3g −3 simple closed 5.13 geodesics. Thurston — The Geometry and Topology of 3-Manifolds 91 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES In order to see that it takes precisely 3g −3 simple closed curves to cut a surface of genus g into pairs of pants Pi notice that χ(Pi) = −1. Therefore the number of Pi’s is equal to −χ(Mg) = 2g −2. Each Pi has three curves, but each curve appears in two Pi’s. Therefore the number of curves is 3 2(2g −2) = 3g −3. We can rephrase Theorem 5.3.5 as T(M) ≈R−3χ(M). It is in this form that the theorem extends to a surface with boundary. The Fricke space F(M) of a surface M with boundary is deﬁned to be the space of hyperbolic structures on M such that the boundary curves are geodesics, modulo isometries isotopic to the identity. A surface with boundary can also be cut into pairs of pants with geodesic boundary. In this case the curves that were boundary curves in M have no twist parameter. On the other hand these curves appear in only one pair of pants. The following theorem is then immediate from the gluing procedures above. Theorem 5.3.6. F(M) is homeomorphic to R−3χ(M). 5.14 The deﬁnition of Teichm¨ uller space can be extended to general surfaces as the space of all metrics of constant curvature up to isotopy and change of scale. In the case of the torus T 2, this space is the set of all Euclidean structures (i.e., metrics with constant curvature zero) on T 2 with area one. There is still a closed geodesic in each free homotopy class although it is not unique. Take some simple, closed geodesic on T 2 and cut along it. The Euclidean structure on the resulting annulus is completely determined by the length of its boundary geodesic. Again there is a real twist parameter that determines how the annulus is glued to get T 2. Therefore there are two real parameters which determine the ﬂat structures on T 2, the length l of a simple, closed geodesic in a ﬁxed free homotopy class and a twist parameter τ along that geodesic. Theorem 5.3.7. The Teichm¨ uller space of the torus is homeomorphic to R2 with coordinates (log l, τ), where l, τ are as above. 5.4. Special algebraic properties of groups of isometries of H3. On large open subsets of PSL(2, C)G, the space of representations of a generating set G into PSL(2, C), certain relations imply other relations. This fact was anticipated in the previous section from the computation of the expected dimension of small deformations of hyperbolic structures on closed three manifolds. The phenomenon that dp is not surjective (see 5.3) suggests that, to determine the structure of π1M 3 5.15 as a discrete subgroup of PSL(2, C), not all the relations in π1M 3 as an abstract group are needed. Below are some examples. 92 Thurston — The Geometry and Topology of 3-Manifolds 5.4. SPECIAL ALGEBRAIC PROPERTIES OF GROUPS OF ISOMETRIES OF H3. gJørgensen Proposition 5.4.1 (Jørgensen). Let a, b be two isometries of H3 with no common ﬁxed point at inﬁnity. If w(a, b) is any word such that w(a, b) = 1 then w(a−1, b−1) = 1. If a and b are conjugate (i.e., if Trace(a) = ± Trace(b) in PSL(2, C) ) then also w(b, a) = 1. Proof. If a and b are hyperbolic or elliptic, let l be the unique common perpen-dicular for the invariant geodesics la, lb of a and b. (If the geodesics intersect in a point x, l is taken to be the geodesic through x perpendicular to the plane spanned by la and lb). If one of a and b is parabolic, (say b is) l should be perpendicular to la and pass through b’s ﬁxed point at ∞. If both are parabolic, l should connect the two ﬁxed points at inﬁnity. In all cases rotation by 180◦in l takes a to a−1 and b and b−1, hence the ﬁrst assertion. If a and b are conjugate hyperbolic elements of PSL(2, C) with invariant geodesics la and lb, take the two lines m and n which are perpendicular to l and to each other and which intersect l at the midpoint between gb and la. Also, if gb is at an angle of θ to lb along l then m should be at an angle of θ/2 and n at an angle of θ + π/2. 5.16 Rotations of 180◦through m or n take la to lb and vice versa. Since a and b are conjugate they act the same with respect to their respective ﬁxed geodesics. It follows that the rotations about m and n conjugate a to b (and b to a) or a to b−1 (and b to a−1). If one of a and b is parabolic then they both are, since they are conjugate. In this case take m and n to be perpendicular to l and to each other and to pass through the unique point x on l such that d(x, ax) = d(x, bx). Again rotation by 180◦in m and n takes a to b or a to b−1. □ Remarks. 1. This theorem fails when a and b are allowed to have a common ﬁxed point. For example, consider a = 1 1 0 1 , b = λ 0 0 λ−1 , Thurston — The Geometry and Topology of 3-Manifolds 93 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES where λ ∈C∗. Then (b−kabk)l = b−kalbk = 1 lλ2k 0 1 . 5.17 If λ is chosen so that λ2 is a root of a polynomial over Z, say 1 + 2λ2 = 0, then a relation is obtained: in this case w(a, b) = (a)(bab−1)2 = I. However, w(a−1, b−1) = I only if λ−2 is a root of the same polynomial. This is not the case in the current example. 2. The geometric condition that a and b have a common ﬁxed point at inﬁnity implies the algebraic condition that a and b generate a solvable group. (In fact, the commutator subgroup is abelian.) Geometric Corollary 5.4.2. Any complete hyperbolic manifold M 3 whose fundamental group is generated by two elements a and b admits an involution s (an isometry of order 2) which takes a to a−1 and b to b−1. If the generators are conjugate, there is a Z2 ⊕Z2 action on M generated by s together with an involution t which interchanges a and b unless a and b have a common ﬁxed point at inﬁnity. Proof. Apply the rotation of 180◦about l to the universal cover H3. This conjugates the group to itself so it induces an isometry on the quotient space M 3. The same is true for rotation around m and n in the case when a and b are conjugate. It can happen that a and b have a common ﬁxed point x at inﬁnity, but since the 5.18 group is discrete they must both be parabolic. A 180◦rotation about any line through x sends a to a−1 and b to b−1. There is not generally a symmetry group of order four in this case. □ As an example, the complete hyperbolic structure on the complement of the ﬁgure-eight knot has symmetry implied by this corollary. (In fact the group of symmetries extends to S3 itself, since for homological reasons such a symmetry preserves the meridian direction.) 94 Thurston — The Geometry and Topology of 3-Manifolds 5.4. SPECIAL ALGEBRAIC PROPERTIES OF GROUPS OF ISOMETRIES OF H3. Here is another illustration of how certain relations in subgroups of PSL(2, C) can imply others: Proposition 5.4.3. Suppose a and b are not elliptic. If an = bm for some n, m ̸= 0, then a and b commute. Proof. If an = bm is hyperbolic, then so are a and b. In fact they ﬁx the same geodesic, acting as translations (perhaps with twists) so they commute. If an = bm 5.19 is parabolic then so are a and b. They must ﬁx the same point at inﬁnity so they act as Euclidean transformations of any horosphere based there. It follows that a and b commute. □ Proposition 5.4.3. If a is hyperbolic and ak is conjugate to al then k = ±l. Proof. Since translation distance along the ﬁxed line is a conjugacy invariant and ρ (ak) = ±kρ (a) (where ρ ( ) denotes translation distance), the proposition is easy to see. □ Finally, along the same vein, it is sometimes possible to derive some nontriv-ial topological information about a hyperbolic three-manifold from its fundamental group. Proposition 5.4.4. If M 3 is a complete, hyperbolic three-manifold, a, b ∈π1M 3 and [a, b] = 1, then either (i) a and b belong to an inﬁnite cyclic subgroup generated by x and xl = a, xk = b, or (ii) M has an end, E, homeomorphic to T 2 × [0, ∞) such that the group gen-erated by a and b is conjugate in π1M 3 to a subgroup of ﬁnite index in π1E. Proof. If a and b are hyperbolic then they translate the same geodesic. Since π1M 3 acts as a discrete group on H3, a and b must act discretely on the ﬁxed geodesic. 5.20 Thurston — The Geometry and Topology of 3-Manifolds 95 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES Thus, (i) holds. If a and b are not both hyperbolic, they must both be parabolic, since they commute. Therefore they can be thought of as Euclidean transformations on a set of horospheres. If the translation vectors are not linearly independent, a and b generate a group of translations of R and (i) is again true. If the vectors are linearly independent, a and b generate a lattice group La,b on R2. Moreover as one approaches the ﬁxed point at inﬁnity, the hyperbolic distance a point x is moved by a and b goes to zero. Recall that the subgroup Gϵ(X) of π1M 3 generated by transformations that moves a point x less than ϵ is abelian. (See pages 4.34-4.35). Therefore all the elements of Gϵ(X) commute with a and b and ﬁx the same point p at inﬁnity. By discreteness Gϵ(X) acts as a lattice group on the horosphere through x and contains La,b as a subgroup of ﬁnite index. 5.21 Consider a fundamental domain of Gϵ(X) acting on the set of horocycles at p which are “contained” in the horocycle Hx through x. It is homeomorphic to the product of a fundamental domain of the lattice group acting on Hx with [0, ∞) and is moved away from itself by all elements in π1M 3 not in Gϵ(X). Therefore it is projected down into M 3 as an end homeomorphic to T 2 × [0, 1]. This is case (ii). □ 5.22 5.5. The dimension of the deformation space of a hyperbolic three-manifold. Consider a hyperbolic structure M0 on T 2 × I. Let α and β be generators for Z ⊕Z = π1(T 2 × I); they satisfy the relation [α, β] = 1, or equivalently αβ = βα. The representation space for Z ⊕Z is deﬁned by the equation H(α) H(β) = H(β) H(α), where H(α), H(β) ∈PSL(2, C). But we have the identity Tr(H(α) H(β)) = Tr(H(β) H(α)), 96 Thurston — The Geometry and Topology of 3-Manifolds 5.5. DEFORMATION SPACE OF THE HYPERBOLIC THREE-MANIFOLD as well as det (H(α) H(β)) = det (H(β) H(α)) = 1, so this matrix equation is equiva-lent to two ordinary equations, at least in a neighborhood of a particular non-trivial solution. Consequently, the solution space has a complex dimension four, and the de-formation space of M0 has complex dimension two. This can easily be seen directly: H(α) has one complex degree of freedom to conjugacy, and given H(α) ̸= id, there is a one complex-parameter family of transformations H(β) commuting with it. This example shows that 5.2.2 is not sharp. More generally, we will improve 5.2.2 for any compact oriented hyperbolic three-manifold M0 whose boundary contains toruses, under a mild nondegeneracy condition on the holonomy of M0: Theorem 5.6. Let M0 be a compact oriented hyperbolic three-manifold whose holonomy satisﬁes (a) the holonomy around any component of ∂M homeomorphic with T 2 is not trivial, and (b) the holonomy has no ﬁxed point on the sphere at ∞. Under these hypotheses, the space of small deformations of M0 has dimension at least as great as the total dimension of the Teichm¨ uller space of ∂M, that is, dimC(Def(M)) ≥ X i    +3 χ((∂M)i) if χ((∂M)i) < 0, 1 if χ((∂M)i) = 0, 0 if χ((∂M)i) > 0. Remark. Condition (b) is equivalent to the statement that the holonomy repre-sentation in PSL(2, C) is irreducible. It is also equivalent to the condition that the holonomy group (the image of the holonomy) be solvable. Examples. If N is any surface with nonempty boundary then, by the immersion theorem [Hirsch] there is an immersion φ of N × S1 in N × I so that φ sends π1(N) to π1(N × I) = π1(N) by the identity map. Any hyperbolic structure on N × I has a −6χ(N) complex parameter family of deformations. This induces a (−6χ(N))-parameter family of deformations of hyperbolic structures on N × S1, showing that the inequality of 5.6 is not sharp in general. Another example is supplied by the complement Mk of k unknotted unlinked solid tori in S3. Since π1(Mk) is a free group on k generators, every hyperbolic structure on Mk has at least 3k −3 degrees of freedom, while 5.6 guarantees only k degrees of freedom. Other examples are obtained on more interesting manifolds by considering hyperbolic structures whose holonomy factors through a free group. Proof of 5.6. We will actually prove that for any compact oriented manifold M, the complex dimension of the representation space of π1M, near a representation satisfying (a) and (b), is at least 3 greater than the number given in 5.6; this suﬃces, Thurston — The Geometry and Topology of 3-Manifolds 97 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES by 5.1. For this stronger assertion, we need only consider manifolds which have no boundary component homeomorphic to a sphere, since any three-manifold M has the same fundamental group as the manifold ˆ M obtained by gluing a copy of D3 to each spherical boundary component of M. Remark. Actually, it can be shown that when ∂M ̸= 0, a representation ρ : π1M →PSL(2, C) is the holonomy of some hyperbolic structure for M if and only if it lifts to a repre-sentation in SL(2, C). (The obstruction to lifting is the second Stiefel–Whitney class ω2 of the associated H3-bundle over M.) It follows that if H0 is the holonomy of a hyperbolic structure on M, it is also the holonomy of a hyperbolic structure on ˆ M, provided ∂ˆ M ̸= ∅. Since we are mainly concerned with structures which have more geometric signiﬁcance, we will not discuss this further. Let H0 denote any representation of π1M satisfying (a) and (b) of 5.6. Let T1, . . . , Tk be the components of ∂M which are toruses. Lemma 5.6.1. For each i, 1 ≤i ≤k, there is an element αi ∈π1(M) such that the group generated by H0(αi) and H0(π1(Ti)) has no ﬁxed point at ∞. One can choose αi so H0(αi) is not parabolic. Proof of 5.6.1. If H0(π1Ti) is parabolic, it has a unique ﬁxed point x at ∞ and the existence of an α′ i not ﬁxing x is immediate from condition (b). If H0(π1Ti) has two ﬁxed points x1 and x2, there is H0(β1) not ﬁxing x1 and H0(β2) not ﬁxing x2. If H0(β1) and H0(β2) each have common ﬁxed points with H0(π1Ti), α′ 1 = β1β2 does not. If H0(α′ i) is parabolic, consider the commutators γn = [α′ i n, β] where β ∈π1Ti is some element such that H0(β) ̸= 1. If H0[α′ i n, β] has a common ﬁxed point x with H0(β) then also α′ i nβα′ i −n ﬁxes x so β ﬁxes α′ i −nx; this happens for at most three values of n. We can, after conjugation, take H0(α′ i) = 1 0 1 1 . Write H0(βα′ i −1β−1 = a b c d , where a + d = 2 and c ̸= 0 since 1 0 is not an eigenvector of β. We compute Tr(γn) = 2 + n2c; it follows that γn can be parabolic (⇔Tr(γn) = ±2) for at most 3 values of n. This concludes the proof of Lemma 5.6.1. □ Let {αi, 1 ≤i ≤k} be a collection of simple disjoint curves based on Ti and 5.26 representing the homotopy classes of the same names. Let N ⊂M be the manifold obtained by hollowing out nice neighborhoods of the αi. Each boundary component 98 Thurston — The Geometry and Topology of 3-Manifolds 5.5. DEFORMATION SPACE OF THE HYPERBOLIC THREE-MANIFOLD of N is a surface of genus ≥2, and M is obtained by attaching k two-handles along non-separating curves on genus two surfaces S1, . . . , Sk ⊂∂N. Let αi also be represented by a curve of the same name on Si, and let βi be a curve on Si describing the attaching map for the i-th two-handle. Generators γi, δi can be chosen for π1Ti so that αi, βi, γi, and δi generate π1Bi and [αi, βi] · [γi, δi] = 1. π1M is obtained from π1M by adding the relations βi = 1. Lemma 5.6.2. A representation ρ of π1N near H0 gives a representation of π1M if and only if the equations 5.27 Tr (ρ (βi)) = 2 and Tr (ρ [αi, βi]) = 2 are satisﬁed. Proof of 5.6.2. Certainly if ρ gives a representation of π1M, then ρ(βi) and ρ[αi, βi] are the identity, so they have trace 2. To prove the converse, consider the equation Tr [A, B] = 2 in SL(2, C) . If A is diagonalizable, conjugate so that A = λ 0 0 λ−1 . Write BA−1B−1 = a b c d . We have the equations a + d = λ + λ−1 Thurston — The Geometry and Topology of 3-Manifolds 99 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES Tr [A, B] = λa + λ−1d = 2 which imply that a = λ−1, d = λ. Since ad−bc = 1 we have bc = 0. This means B has at least one common eigenvector 1 0 or 0 1 with A; if [A, B] ̸= 1, this common eigenvector is the unique eigenvector 5.28 of [A, B] (up to scalars). As in the proof of 5.6.1, a similar statement holds if A is parabolic. (Observe that [A, B] = [−A, B], so the sign of Tr A is not important). It follows that if Tr ρ[αi, βi] = 2, then since [γi, δi] = [αi, βi], either ρ (αi), ρ (βi), ρ (γi) and ρ (δi) all have a common ﬁxed point on the sphere at inﬁnity, or ρ[αi, βi] = 1. By construction H0, π1Si has no ﬁxed point at inﬁnity, so for ρ near H0ρ π1Si cannot have a ﬁxed point either; hence ρ[αi, βi] = 1. The equation Tr ρ (βi) = 2 implies ρ (βi) is parabolic; but it commutes with ρ (βi), which is hyperbolic for ρ near H0. Hence ρ (βi) = 1. This concludes the proof of Lemma 5.6.2. □ To conclude the proof of 5.6, we consider a handle structure for N with one zero-handle, m one-handles, p two-handles and no three-handles (provided ∂M ̸= ∅). This gives a presentation for π1N with m generators and p relations, where 1 −m + p = χ(N) = χ(M) −k. The representation space R ⊂PSL(2, C)m for π1M, in a neighborhood of H0, is deﬁned by the p matrix equations ri = 1, (1 ≤i ≤p), where the ri are products representing the relators, together with 2k equations 5.29 Tr ρ(βi) = 2 Tr ρ([αi, βi]) = 2 [1 ≤i ≤k] The number of equations minus the number of unknowns (where a matrix variable is counted as three complex variables) is 3m −3p −2k = −3χ(M) + k + 3. □ Remark. If M is a closed hyperbolic manifold, this proof gives the estimate of 0 for dimC def(M): simply remove a non-trivial solid torus from M, apply 5.6, and ﬁll in the solid torus by an equation Tr(γ) = 2. 100 Thurston — The Geometry and Topology of 3-Manifolds 5.7 5.7 There is a remarkable, precise description for the global deformation space of hyperbolic structures on closed manifolds in dimensions bigger than two: Theorem 5.7.1 (Mostow’s Theorem [algebraic version]). Suppose Γ1 and Γ2 are two discrete subgroups of the group of isometries of Hn, n ≥3 such that Hn/Γi has ﬁnite volume and suppose φ : Γ1 →Γ2 is a group isomorphism. Then Γ1 and Γ2 are conjugate subgroups. This theorem can be restated in terms of hyperbolic manifolds since a hyperbolic manifold has universal cover Hn with fundamental group acting as a discrete group of isometries. 5.30 Theorem 5.7.2 (Mostow’s Theorem [geometric version]). If M n 1 and M n 2 are com-plete hyperbolic manifolds with ﬁnite total volume, any isomorphism of fundamental groups φ : π1M1 →π1M2 is realized by a unique isometry. Remark. Multiplication by an element in either fundamental group induces the identity map on the manifolds themselves so that φ needs only to be deﬁned up to composition with inner automorphisms to determine the isometry from M1 to M2. Since the universal cover of a hyperbolic manifold is Hn, it is a K(π, 1). Two such manifolds are homotopy equivalent if and only if there is an isomorphism between their fundamental groups. Corollary 5.7.3. If M1 and M2 are hyperbolic manifolds which are complete with ﬁnite volume, then they are homeomorphic if and only if they are homotopy equivalent. (The case of dimension two is well known.) For any manifold M, there is a homomorphism DiﬀM →Out(π1M), where Out(π1M) = Aut(π1M)/ Inn(π1M) is the group of outer automorphisms. Mostow’s Theorem implies this homomorphism splits, if M is a hyperbolic manifold of dimen-sion n ≥3. It is unknown whether the homomorphism splits when M is a surface. When n = 2 the kernel Diﬀ0(M) is contractible, provided χ(M) ≤0. If M is a Haken 5.31 three-manifold which is not a Seifert ﬁber space, Hatcher has shown that Diﬀ0 M is contractible. Corollary 5.7.4. If M n is hyperbolic (complete, with ﬁnite total volume) and n ≥3, then Out(π1M) is a ﬁnite group, isomorphic to the group of isometries of M n. Proof. By Mostow’s Theorem any automorphism of π1M induces a unique isom-etry of M. Since any inner automorphism induces the identity on M, it follows that the group of isometries is isomorphic to Out(π1M). That Out(π1M) is ﬁnite is im-mediate from the fact that the group of isometries, Isom(M n), is ﬁnite. Thurston — The Geometry and Topology of 3-Manifolds 101 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES To see that Isom(M n) is ﬁnite, choose a base point and frame at that point and suppose ﬁrst that M is compact. Any isometry is completely determined by the image of this frame (essentially by “analytic continuation”). If there were an inﬁnite sequence of isometries there would exist two image frames close to each other. Since M is compact, the isometries, φ1, φ2, corresponding to these frames would be close on all of M. Therefore φ, is homotopic to φ2. Since the isometry φ−1 2 φ1 induces the trivial outer automorphism on π1M, it is the identity; i.e., φ2 = φ1. If M is not compact, consider the submanifold Mϵ ⊂M which consists of points which are contained in an embedded hyperbolic disk of radius ϵ. Since M has ﬁnite total volume, Mϵ is compact. Moreover, it is taken to itself under any isometry. The 5.32 argument above applied to Mϵ implies that the group of isometries of M is ﬁnite even in the non-compact case. □ Remark. This result contrasts with the case n = 2 where Out(π1M 2) is inﬁnite and quite interesting. The proof of Mostow’s Theorem in the case that Hn/Γ is not compact was com-pleted by Prasad. Otherwise, 5.7.1 and 5.7.2 (as well as generalizations to other homogeneous spaces) are proved in Mostow. We shall discuss Mostow’s proof of this theorem in 5.10, giving details as far as they can be made geometric. Later, we will give another proof due to Gromov, valid at least for n = 3. 5.8. Generalized Dehn surgery and hyperbolic structures. Let M be a non-compact, hyperbolic three-manifold, and suppose that M has a ﬁnite number of ends E1, . . . , Ek, each homeomorphic to T 2 × [0, ∞) and isometric to the quotient space of the region in H3 (in the upper half-space model) above an interior Euclidean plane by a group generated by two parabolic transformations which ﬁx the point at inﬁnity. Topologically M is the interior of a compact manifold ¯ M whose boundary is a union of T1, . . . , Tk tori. Recall the operation of generalized Dehn surgery on M (§4.5); it is parametrized 5.33 by an ordered pair of real numbers (ai, bi) for each end which describes how to glue a solid torus to each boundary component. If nothing is glued in, this is denoted by ∞so that the parameters can be thought of as belonging to S2 (i.e., the one point compactiﬁcation of R2 ≈H1(T 2, R)). The resulting space is denoted by Md1,...,dk where di = (ai, bi) or ∞. In this section we see that the new spaces often admit hyperbolic structures. Since Md1,...,dk is a closed manifold when di = (ai, bi) are primitive elements of H1(T 2, Z), this produces many closed hyperbolic manifolds. First it is necessary to see that small deformations of the complete structure on M induce a hyperbolic structure on some space Md1,...,dk. 102 Thurston — The Geometry and Topology of 3-Manifolds 5.8. GENERALIZED DEHN SURGERY AND HYPERBOLIC STRUCTURES. Lemma 5.8.1. Any small deformation of a “standard” hyperbolic structure on T 2 × [0, 1] extends to some (D2 × S1)d. d = (a, b) is determined up to sign by the traces of the matrices representing generators α, β of π1T 2. Proof. A “standard” structure on T 2 × [0, 1] means a structure as described on an end of M truncated by a Euclidean plane. The universal cover of T 2 × [0, 1] is the region between two horizontal Euclidean planes (or horospheres), modulo a group of translations. If the structure is deformed slightly the holonomy determines the new structure and the images of α and β under the holonomy map H are slightly 5.34 perturbed. If H(α) is still parabolic then so is H(β) and the structure is equivalent to the standard one. Otherwise H(α) and H(β) have a common axis l in H3. Moreover since H(α) and H(β) are close to the original parabolic elements, the endpoints of l are near the common ﬁxed point of the parabolic elements. If T 2 × [0, 1] is thought to be embedded in the end, T 2 × [0, ∞), this means that the line lies far out towards ∞and does not intersect T 2 × [0, 1]. Thus the developing image of T 2 × [0, 1] in H3 for new structure misses l and can be lifted to the universal cover ^ H3 −l of H3 −l. This is the geometric situation necessary for generalized Dehn surgery. The ex-tension to (D2 × S1)d is just the completion of ^ H3 −l/{ ˜ H(α), ˜ H(β)} where ˜ H is the lift of H to the cover ^ H3 −l. Recall that the completion depends only on the behavior of ] H(α) and ] H(β) along l. In particular, if ˜ H( ) denotes the complex number determined by the pair (translation distance along l, rotation about l), then the Dehn surgery coeﬃcients d = (a, b) are determined by the formula: a ˜ H(α) + b ˜ H(β) = ±2πi. The translation distance and amount of rotation of an isometry along its ﬁxed line is determined by the trace of its matrix in PSL(2, C). This is easy to see since trace is a conjugacy invariant and the fact is clearly true for a diagonal matrix. In 5.35 particular the complex number corresponding to the holonomy of a matrix acting on H3 is log λ where λ + λ−1 is its trace. □ The main result concerning deformations of M is Thurston — The Geometry and Topology of 3-Manifolds 103 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES Theorem 5.8.2. If M = M∞,...,∞admits a hyperbolic structure then there is a neighborhood U of (∞, . . . , ∞) in S2 × S2 × · · · × S2 such that for all (d1, . . . , dk) ∈ U, Md1,...,dk admits a hyperbolic structure. Proof. Consider the compact submanifold M0 ⊂M gotten by truncating each end. M0 has boundary a union of k tori and is homeomorphic to the manifold ¯ M such that M = interior ¯ M. By theorem 5.6, M0 has a k complex parameter family of non-trivial deformations, one for each torus. From the lemma above, each small deformation gives a hyperbolic structure on some Md1,...,dk. It remains to show that the di vary over a neighborhood of (∞, . . . , ∞). Consider the function Tr : Def(M) →(Tr(H(α1)), . . . , Tr(H(αk))) which sends a point in the deformation space to the k-tuple of traces of the ho-lonomy of α1, α2, . . . , αk, where αi, βi generate the fundamental group of the i-th torus. Tr is a holomorphic (in fact, algebraic) function on the algebraic variety 5.36 Def(M). Tr(M∞,...,∞) = (±2, . . . , ±2) for some ﬁxed choice of signs. Note that Tr(H(αi)) = ±2 if and only if H(αi) is parabolic and H(αi) is parabolic if and only if the i-th surgery coeﬃcient di equals ∞. By Mostow’s Theorem the hyperbolic structure on M∞,...,∞is unique. Therefore di = ∞for i = l, . . . , k only in the original case and Tr−1(±2, . . . , ±2) consists of exactly one point. Since dim(Def(M)) ≥k it follows from [ ] that the image under Tr of a small open neighborhood of M∞,...,∞is an open neighborhood of (±2, . . . , ±2). Since the surgery coeﬃcients of the i-th torus depend on the trace of both H(αi) and H(βi), it is necessary to estimate H(βi) in terms of H(αi) in order to see how the surgery coeﬃcients vary. Restrict attention to one torus T and conjugate the original developing image of M∞,...,∞so that the parabolic ﬁxed point of the holonomy, H0, (π1T), is the point at inﬁnity. By further conjugation it is possible to put the holonomy matrices of the generators α, β of π1T in the following form: H0(α) = 1 1 0 1 H0(β) = 1 c 0 1 . Note that since H0(α), H0(β) act on the horospheres about ∞as a two-dimension-al lattice of Euclidean translations, c and l are linearly independent over R. Since 5.37 H0(α), H0(β) have 1 0 as an eigenvector, the perturbed holonomy matrices H(α), H(β) 104 Thurston — The Geometry and Topology of 3-Manifolds 5.8. GENERALIZED DEHN SURGERY AND HYPERBOLIC STRUCTURES. will have common eigenvectors near 1 0 , say 1 ϵ1 and 1 ϵ2 . Let the eigenvalues of H(α) and H(β) be (λ, λ−1) and (µ, µ−1) respectively. Since H(α) is near H0(α), H(α) 0 1 ≈ 1 1 . However H(α) 0 1 = 1 ϵ1 −ϵ2 H(α) 1 ϵ1 − 1 ϵ2 = 1 ϵ1 −ϵ2 λ λϵ1 − λ−1 λ−1ϵ2 . Therefore λ −λ−1 ϵ1 −ϵ2 ≈1. Similarly, µ −µ−1 ϵ1 −ϵ2 ≈c. For λ, µ near l, log(λ) log(µ) ≈λ −1 µ −1 ≈λ −λ−1 µ −µ−1 ≈1 c. Since ˜ H(α) = log λ and ˜ H(β) = log µ this is the desired relationship between ˜ H(α) and ˜ H(β). The surgery coeﬃcients (a, b) are determined by the formula 5.38 a ˜ H(α) + b ˜ H(β) = ±2πi. From the above estimates this implies that (a + bc) ≈±2πi log λ . (Note that the choice of sign corresponds to a choice of λ or λ−1.) Since 1 and c are linearly independent over R, the values of (a, b) vary over an open neighborhood of ∞as λ varies over a neighborhood of 1. Since Tr(H(α)) = λ + λ−1 varies over a neighborhood of 2 (up to sign) in the image of Tr : Def(M) →Ck, we have shown that the surgery coeﬃcients for the Md1,...,dk possessing hyperbolic structures vary over an open neighborhood of ∞in each component. □ Example. The complement of the Borromean rings has a complete hyperbolic structure. However, if the trivial surgery with coeﬃcients (1, 0) is performed on one component, the others are unlinked. (In other words, M(1,0),∞,∞is S3 minus two unlinked circles.) The manifold M(1,0),x,y (where M is S3 minus the Borromean rings) is then a connected sum of lens spaces if x, y are primitive elements of H1(T 2 i , Z) so it cannot have a hyperbolic structure. Thus it may often happen that an inﬁnite number of non-hyperbolic manifolds can be obtained by surgery from a hyperbolic Thurston — The Geometry and Topology of 3-Manifolds 105 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES one. However, the theorem does imply that if a ﬁnite number of integral pairs of coeﬃcients is excluded from each boundary component, then all remaining three-manifolds obtained by Dehn surgery on M are also hyperbolic. 5.39 5.9. A Proof of Mostow’s Theorem. This section is devoted to a proof of Mostow’s Theorem for closed hyperbolic n-manifolds, n ≥3. The proof will be sketchy where it seems to require analysis. With a knowledge of the structure of the ends in the noncompact, complete case, this proof extends to the case of a manifold of ﬁnite total volume; we omit details. The outline of this proof is Mostow’s. Given two closed hyperbolic manifolds M1 and M2, together with an isomorphism of their fundamental groups, there is a homotopy equivalence inducing the isomor-phism since M1 and M2 are K(π, 1)’s. In other words, there are maps f1 : M1 →M2 and f2 : M2 →M1 such that f1 ◦f2 and f2 ◦f1 are homotopic to the identity. Denote lifts of f1, f2 to the universal cover Hn by ˜ f1, ˜ f2 and assume ˜ f1 ◦˜ f2 and ˜ f2 ◦˜ f1 are equivariantly homotopic to the identity. The ﬁrst step in the proof is to construct a correspondence between the spheres at inﬁnity of Hn which extends ˜ f1 and ˜ f2. Definition. A map g : X →Y between metric spaces is a pseudo-isometry if there are constants c1, c2 such that c−1 1 d(x1, x2) −c2 ≤d(gx1, gx2) ≤c1d(x1, x2) for all x1, x2 ∈X. Lemma 5.9.1. ˜ f1, ˜ f2 can be chosen to be pseudo-isometries. Proof. Make f1 and f2 simplicial. Then since M1 and M2 are compact, f1 and 5.40 f2 are Lipschitz and lift to ˜ f1 and ˜ f2 which are Lipschitz with the same coeﬃcient. It follows immediately that there is a constant c1 so that d( ˜ fix1, ˜ fix2) ≤c1d(x1, x2) for i = 1, 2 and all x1, x2 ∈Hn. If xi = ˜ f1yi, then this inequality implies that d( ˜ f2 ◦˜ f1(y1), ˜ f2 ◦˜ f1(y2)) ≤c1d( ˜ f1y1, ˜ f1y2). However, since M1 is compact, ˜ f2 ◦˜ f1 is homotopic to the identity by a homotopy that moves every point a distance less than some constant b. It follows that d(y1, y2) −2b ≤d( ˜ f2 ◦˜ f1y1, ˜ f2 ◦˜ f1y2), from which the lower bound c−1 1 d(y1, y2) −c2 ≤d( ˜ f1y1, ˜ f1y2) follows. □ Using this lemma it is possible to associate a unique geodesic with the image of a geodesic. 106 Thurston — The Geometry and Topology of 3-Manifolds 5.9. A PROOF OF MOSTOW’S THEOREM. Proposition 5.9.2. For any geodesic g ⊂Hn there is a unique geodesic h such that f1(g) stays in a bounded neighborhood of h. Proof. If j is any geodesic in Hn, let Ns(j) be the neighborhood of radius s about j. We will see ﬁrst that if s is large enough there is an upper bound to the length of any bounded component of g − ˜ f −1 1 (Ns(j)) , for any j. In fact, the perpendicular projection from Hn −Ns(j) to j decreases every distance by at least a factor of 1/cosh s, so any long path in Hn −Ns(j) with endpoints on ∂Ns(j) can be replaced by a much shorter path consisting of two segments perpendicular to j, together with a segment of j. 5.41 When this fact is applied to a line j joining distant points p1 and p2 on ˜ f1(g), it follows that the segment of g between p1 and p2 must intersect each plane perpendic-ular to j a bounded distance from j. It follows immediately that there is a limit line h to such lines j as p1 and p2 go to +∞and −∞on ˜ f1(g), and that ˜ f1(g) remains a bounded distance from h. Since no two lines in Hn remain a bounded distance apart, h is unique. □ 5.42 Corollary 5.9.3. ˜ f1 : Hn →Hn induces a one-to-one correspondence between the spheres at inﬁnity. Thurston — The Geometry and Topology of 3-Manifolds 107 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES Proof. There is a one-to-one correspondence between points on the sphere at inﬁnity and equivalence classes of directed geodesics, two geodesics being equivalent if they are parallel, or asymptotic in positive time. The correspondence of 5.9.2 between geodesics in ˜ M1 and geodesics in ˜ M2 obviously preserves this relation of parallelism, so it induces a map on the sphere at inﬁnity. This map is one-to-one since any two distinct points in the sphere at inﬁnity are joined by a geodesic, hence must be taken to the two ends of a geodesic. □ 5.43 The next step in the proof of Mostow’s Theorem is to show that the extension of ˜ f1 to the sphere at inﬁnity Sn−1 ∞ is continuous. One way to prove this is by citing Brouwer’s Theorem that every function is continuous. Since this way of thinking is not universally accepted (though it is valid in the current situation), we will give another proof, which will also show that f is quasi-conformal at Sn−1 ∞. A basis for the neighborhoods of a point x ∈Sn−1 ∞ is the set of disks with center x. The boundaries of the disks are (n −2)-spheres which correspond to hyperplanes in H2 (i.e., to (n −1)-spheres perpendicular to Sn−1 ∞ whose intersections with Sn−1 ∞ are the (n −2)-spheres). For any geodesic g in ˜ M1, let φ(g) be the geodesic in ˜ M2 which remains a bounded distance from ˜ f1(g). 108 Thurston — The Geometry and Topology of 3-Manifolds 5.9. A PROOF OF MOSTOW’S THEOREM. Lemma 5.9.4. There is a constant c such that, for any hyperplane P in Hn and any geodesic g perpendicular to P, the projection of ˜ f1(P) onto φ(g) has diameter ≤c. Proof. Let x be the point of intersection of P and g and let l be a geodesic ray based at x. Then there is a unique geodesic l1 which is parallel to l in one direction and to a ﬁxed end of g in the other. Let A denote the shortest arc between x and l1. It has length d, where d is a ﬁxed contrast (= arc cosh √ 2). 5.44 The idea of the proof is to consider the image of this picture under ˜ fl. Let φ(l), φ(l1), φ(g) denote the geodesics that remain a bounded distance from l, l1 and g respectively. Since φ preserves parallelism φ(l) and φ(l1) are parallel. Let l⊥denote the geodesic from the endpoint on Sn−1 ∞ of φ(l) which is perpendicular to φ(g). Also let x0 be the point on φ(g) nearest to ˜ fl(x). Since ˜ fl(x) is a pseudo-isometry the length of ˜ fl(A) is at most c1d where c1 is a ﬁxed constant. Since φ(l1) and φ(g) are less than distance s (for a ﬁxed constant s) from ˜ fl(l1) and ˜ fl(g) respectively, it follows that x0 is distance less than C1d+2s = ¯ d from φ(l1). This implies that the foot of l⊥(i.e., l⊥∩φ(g)) lies distance less than ¯ d to one side of x0. By considering the geodesic l2 which is parallel to l and to the other end of g, it follows that f lies a distance less than ¯ d from x0. 5.45 Now consider any point y ∈P. Let m be any line through y. The endpoints of φ(m) project to points on φ(g) within a distance ¯ d of x0; since ˜ fl(y) is within a distance s of φ(m), it follows that y projects to a point not farther than ¯ d + s from x0. □ Thurston — The Geometry and Topology of 3-Manifolds 109 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES Corollary 5.9.5. The extension of ˜ fl to Sn−1 ∞ is continuous. Proof. For any point y ∈Sn−1 ∞, consider a directed geodesic g bending toward y, and deﬁne ˜ fl(y) to be the endpoint of φ(g). The half-spaces bounded by hyperplanes perpendicular to φ(g) form a neighborhood basis for ˜ fl(y). For any such half-space H, there is a point x ∈g such that the projection of ˜ fl(x) to φ(g) is a distance > C from ∂H. Then the neighborhood of y bounded by the hyperplane through x perpendicular to g is mapped within H. □ Below it will be necessary to use the concept of quasi-conformality. If f is a 5.46 homeomorphism of a metric space X to itself, f is K-quasi-conformal if and only if for all z ∈X lim r→0 supx,y∈Sr(z) d (f(x), f(y)) infx,y∈Sr(z) d (f(x), f(y)) ≤K where Sr(Z) is the sphere of radius r around Z, and x and y are diametrically opposite. K measures the deviation of f from conformality, is equal to 1 if f is conformal, and is unchanged under composition with a conformal map. f is called quasi-conformal if it is K-quasi-conformal for some K. Corollary 5.9.6. ˜ fl is quasi-conformal at Sn−1 ∞. Proof. Use the upper half-space model for Hn since it is conformally equivalent to the ball model and suppose x and ˜ flx are the origin since translation to the origin is also conformal. Then consider any hyperplane P perpendicular to the geodesic g from 0 to the point at inﬁnity. By Lemma 5.9.4 there is a bound, depending only on ˜ fl, to the diameter of the projection of ˜ fl(P) onto φ(g) = g. Therefore, there are hyperplanes P1, P2 perpendicular to g contained in and containing ˜ fl(P) and the distance (along g) between P1 and P2 is uniformly bounded for all planes P. 110 Thurston — The Geometry and Topology of 3-Manifolds 5.9. A PROOF OF MOSTOW’S THEOREM. But this distance equals log r, r > 1, where r is the ratio of the radii of the n −2 spheres Sn−2 p1 , Sn−2 p2 in Sn−1 ∞ corresponding to P1 and P2. The image of the n−2 sphere Sn−2 P corresponding 5.47 to P lies between Sn−2 p2 and Sn−2 p1 so that r is an upper bound for the ratio of maximum to minimum distances on ˜ fl(Sn−2 p ). Since log r is uniformly bounded above, so is r and ˜ fl is quasi-conformal. □ Corollary 5.9.6 was ﬁrst proved by Gehring for dimension n = 3, and generalized to higher dimensions by Mostow. At this point, it is necessary to invoke a theorem from analysis (see Bers). Theorem 5.9.7. A quasi-conformal map of an n −1-manifold, n > 2, has a derivative almost everywhere (= a.e.). Remark. It is at this stage that the proof of Mostow’s Theorem fails for n = 2. 5.48 The proof works to show that ˜ fl extends quasi-conformally to the sphere at inﬁnity, S1 ∞, but for a one-manifold this does not imply much. Consider ˜ fl : Sn−1 ∞ →Sn−1 ∞; by theorem 5.9.7 d ˜ fl exists a.e. At any point x where the derivative exists, the linear map d ˜ fl(x) takes a sphere around the origin to an ellipsoid. Let λ1, . . . , λn−1 be the lengths of the axes of the ellipsoid. If we normalize so that λ1 ·λ2 · · · λn−1 = 1, then the λi are conformal invariants. In particular denote the maximum ratio of the λi’s at x by e(x), the eccentricity of ˜ fl at x. Note that if ˜ fl is K-quasi-conformal, the supremum of e(x), x ∈Sn−1 ∞, is K. Since π1M1 acts on Sn−1 ∞ conformally and e is invariant under conformal maps, e is a measurable, π1M1 invariant function on Sn−1 ∞. However, such functions are very simple because of the following theorem: Theorem 5.9.8. For a closed, hyperbolic n-manifold M, π1M acts ergodically on Sn−1 ∞, i.e., every measurable, invariant set has zero measure or full measure. Thurston — The Geometry and Topology of 3-Manifolds 111 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES Corollary 5.9.9. e is constant a.e. Proof. Any level set of e is a measurable, invariant set so precisely one has full measure. □ In fact more is true: 5.49 Theorem 5.9.10. π1(M) acts ergodically on Sn−1 ∞ × Sn−1 ∞. Remark. This theorem is equivalent to the fact that the geodesic ﬂow of M is ergodic since pairs of distinct points on Sn−1 ∞ are in a one-to-one correspondence to geodesics in Hn (whose endpoints are those points). From Corollary 5.9.9 e is equal a.e. to a constant K, and if the derivative of ˜ fl is not conformal, K ̸= 1. Consider the case n = 3. The direction of maximum “stretch” of d f deﬁnes a measurable line ﬁeld l on S2 ∞. Then for any two points x, y ∈S2 ∞it is possible to parallel translate the line l(x) along the geodesic between x and y to y and compute the angle between the translation of l(x) and l(y). This deﬁnes a measurable π1M-invariant function on S2 ∞× S2 ∞. By theorem 5.9.10 it must be constant a.e. In other words l is determined by its “value” at one point. It is not hard to see that this is impossible. For example, the line ﬁeld determined by a line at x agrees with the line ﬁeld below a.e. However, any line ﬁeld determined by its “value” at y will have the same form and will be incompatible. 5.50 The precise argument is easy, but slightly more subtle, since l is deﬁned only a.e. The case n > 3 is similar. Now one must again invoke the theorem, from analysis, that a quasi-conformal map whose derivative is conformal a.e. is conformal in the usual sense; it is a sphere-preserving map of Sn−1 ∞, so it extends to an isometry I of Hn. The isometry I conjugates the action of π1M1 to the action of π1M2, completing the proof of Mostow’s Theorem. □ 5.10. A decomposition of complete hyperbolic manifolds. 5.51 Let M be any complete hyperbolic manifold (possibly with inﬁnite volume). For ϵ > 0, we will study the decomposition M = M(0,ϵ] ∪M[ϵ,∞)′ where M(0,ϵ] consists of those points in M through which there is a non-trivial closed loop of length ≤ϵ, and M[ϵ,∞) consists of those points through which every non-trivial loop has length ≥ϵ. In order to understand the geometry of M(0,ϵ], we pass to the universal cover ˜ M = Hn. For any discrete group Γ of isometries of Hn and any x ∈Hn let Γϵ(x) be the subgroup generated by all elements of Γ which move x a distance ≤ϵ, and let 112 Thurston — The Geometry and Topology of 3-Manifolds 5.10. A DECOMPOSITION OF COMPLETE HYPERBOLIC MANIFOLDS. Γ′ ϵ(x) ⊂Γϵ(x) be the subgroup consisting of elements whose derivative is also ϵ-close to the identity. Lemma 5.10.1 (The Margulis Lemma). For every dimension n there is an ϵ > 0 such that for every discrete group Γ of isometries of Hn and for every x ∈Hn, Γ′ ϵ(x) is abelian and Γϵ(x) has an abelian subgroup of ﬁnite index. Remark. This proposition is much more general than stated; if “abelian” is replaced by “nilpotent,” it applies in general to discrete groups of isometries of Rie-mannian manifolds with bounded curvature. The proof of the general statement is essentially the same. Proof. In any Lie group G, since the commutator map [ , ] : G × G →G has derivative 0 at (1, 1), it follows that the size of the commutator of two small elements 5.52 is bounded above by some constant times the product of their sizes. Hence, if Γ′ ϵ is any discrete subgroup of G generated by small elements, it follows immediately that the lower central series Γ′ ϵ ⊃[Γ′ ϵ, Γ′ ϵ] ⊃[Γ′ ϵ, [Γ′ ϵ, Γ′ ϵ]], . . . is ﬁnite (since there is a lower bound to the size of elements of Γ′ ϵ). In other words, Γ′ ϵ is nilpotent. When G is the group of isometries of hyperbolic space, it is not hard to see (by considering, for instance, the geometric classiﬁcation of isometries) that this implies Γ′ ϵ is actually abelian. To guarantee that Γϵ(x) has an abelian subgroup of ﬁnite index, the idea is ﬁrst to ﬁnd an ϵ1 such that Γ′ ϵ1(x) is always abelian, and then choose ϵ many times smaller than ϵ1, so the product of generators of Γϵ(x) will lie in Γ′ ϵ1(x). Here is a precise recipe: Let N be large enough that any collection of elements of O(n) with cardinality ≥N contains at least one pair separated by a distance not more than ϵ1/3. Choose ϵ2 ≤ϵ1/3 so that for any pair of isometries φ1 and φ2 of Hn which translate a point x a distance ≤ϵ2, the derivative at x of φ1 ◦φ2 (parallel translated back to x) is estimated within ϵ1/6 by the product of the derivatives at x of φ1 and φ2 (parallel translated back to x). Now let ϵ = ϵ2/2N, so that a product of 2N isometries, each translating x a distance ≤ϵ, translates x a distance ≤ϵ2. Let g1, . . . , gk be the set of elements of Γ 5.53 which move x a distance ≤ϵ; they generate Γϵ(x). Consider the cosets γ Γ′ ϵ1(x), where γ ∈Γϵ(x); the claim is that they are all represented by γ’s which are words of length < N in the generators g1, . . . , gk. In fact, if γ = gi1 · . . . ·gil is any word of length ≥N in the gi’s, it can be written γ = α · ϵ′ · β, (α, ϵ′, β ̸= 1) where ϵ′ · β has length ≤N, and the derivative of ϵ′ is within ϵ1/3 of 1. It follows that (αβ)−1 · (αϵ′β) = β−1ϵ′β is in Γ′ ϵ1(x); hence the coset γΓ′ ϵ1(x) = (αβ)Γ′ ϵ1(x). By induction, the claim is veriﬁed. Thus, the abelian group Γ′ ϵ1(x) has ﬁnite index in the group generated by Γϵ(x) and Γ′ ϵ1(x), so Γ′ ϵ1(x) ∩Γϵ(x) with ﬁnite index. □ Thurston — The Geometry and Topology of 3-Manifolds 113 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES Examples. When n = 3, the only possibilities for discrete abelian groups are Z (acting hyperbolically or parabolically), Z × Z (acting parabolically, conjugate to a group of Euclidean translations of the upper half-space model), Z × Zn (acting as a group of translations and rotations of some axis), and Z2 × Z2 (acting by 180◦ rotations about three orthogonal axes). The last example of course cannot occur as Γ′ ϵ(x). Similarly, when ϵ is small compared to 1/n, Z × Zn cannot occur as Γ′ ϵ(x). Any discrete group Γ of isometries of Euclidean space En−1 acts as a group of isometries of Hn, via the upper half-space model. 5.54 For any x suﬃciently high (in the upper half space model), Γϵ(x) = Γ. Thus, 5.10.1 contains as a special case one of the Bieberbach theorems, that Γ contains an abelian subgroup of ﬁnite index. Conversely, when Γϵ(x) ∩Γ′ ϵ1(x) is parabolic, Γϵ(x) must be a Bieberbach group. To see this, note that if Γϵ(x) contained any hyperbolic element γ, no power of γ could lie in Γ′ ϵ1(x), a contradiction. Hence, Γϵ(x) must consist of parabolic and elliptic elements with a common ﬁxed point p at ∞, so it acts as a group of isometries on any horosphere centered at p. If Γϵ(x) ∩Γ′ ϵ1(x) is not parabolic, it must act as a group of translations and rotations of some axis a. Since it is discrete, it contains Z with ﬁnite index (provided Γϵ(x) is inﬁnite). It easily follows that Γϵ(x) is either the product of some ﬁnite 114 Thurston — The Geometry and Topology of 3-Manifolds 5.10. A DECOMPOSITION OF COMPLETE HYPERBOLIC MANIFOLDS. Figure 1. The inﬁnite dihedral group acting on H3. subgroup F of O(n −1) (acting as rotations about a) with Z, or it is the semidirect 5.55 product of such an F with the inﬁnite dihedral group, Z/2 ∗Z/2. For any set S ⊂Hn, let Br(S) = {x ∈Hn\| d (x, S) ≤r}. Corollary 5.10.2. There is an ϵ > 0 such that for any complete oriented hy-perbolic three-manifold M, each component of M(0,ϵ] is either (1) a horoball modulo Z or Z ⊕Z, or (2) Br(g) modulo Z, where g is a geodesic. The degenerate case r = 0 may occur. Proof. Suppose x ∈M(0,ϵ]. Let ˜ x ∈H3 be any point which projects to x. There is some covering translation γ which moves x a distance ≤ϵ. If γ is hyperbolic, let a be its axis. All rotations around a, translations along a, and uniform contractions of hyperbolic space along orthogonals to a commute with γ. It follows that ˜ M(0,ϵ] 5.56 contains Br(a), where r = d (a, x), since γ moves any point in Br(a) a distance ≤ϵ. Similarly, if γ is parabolic with ﬁxed point p at ∞, ˜ M(0,ϵ] contains a horoball about p passing through x. Hence M(0,ϵ] is a union of horoballs and solid cylinders Br(a). Whenever two of these are not disjoint, they correspond to two covering transformations γ1 and γ2 which move some point x a distance ≤ϵ; γ1 and γ2 must commute (using 5.10.1), so the corresponding horoballs or solid cylinders must be concentric, and 5.10.2 follows. □ Thurston — The Geometry and Topology of 3-Manifolds 115 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES 5.11. Complete hyperbolic manifolds with bounded volume. It is easy now to describe the structure of a complete hyperbolic manifold with ﬁnite volume; for simplicity we stick to the case n = 3. Proposition 5.11.1. A complete oriented hyperbolic three-manifold with ﬁnite volume is the union of a compact submanifold (bounded by tori) and a ﬁnite collection of horoballs modulo Z ⊕Z actions. Proof. M[ϵ,∞) must be compact, for otherwise there would be an inﬁnite se-quence of points in M[ϵ,∞) pairwise separated by at least ϵ. This would give a sequence of hyperbolic ϵ/2 balls disjointly embedded in M[ϵ,∞), which has ﬁnite volume. M(0,ϵ] must have ﬁnitely many components (since its boundary is compact). The proposi-tion is obtained by lumping all compact components of M(0,ϵ] with M[ϵ,∞). □ 5.57 With somewhat more eﬀort, we obtain Jørgensen’s theorem, which beautifully describes the structure of the set of all complete hyperbolic three-manifolds with volume bounded by a constant C: Theorem 5.11.2 (Jørgensen’s theorem [ﬁrst version]). Let C > 0 be any con-stant. Among all complete hyperbolic three-manifolds with volume ≤C, there are only ﬁnitely many homeomorphism types of M[ϵ,∞). In other words, there is a link Lc in S3 such that every complete hyperbolic manifold with volume ≤C is obtained by Dehn surgery along LC. (The limiting case of deleting components of LC to obtain a non-compact manifold is permitted.) Proof. Let V be any maximal subset of M[ϵ,∞) having the property that no two elements of V have distance ≤ϵ/2. The balls of radius ϵ/4 about elements of V are embedded; since their total volume is ≤C, this gives an upper bound to the cardinality of V . The maximality of V is equivalent to the property that the balls of radius ϵ/2 about V cover. 116 Thurston — The Geometry and Topology of 3-Manifolds 5.11. COMPLETE HYPERBOLIC MANIFOLDS WITH BOUNDED VOLUME. The combinatorial pattern of intersections of this set of ϵ/2-balls determines M[ϵ,∞) up to diﬀeomorphism. There are only ﬁnitely many possibilities. (Alternatively a triangulation of M[ϵ,∞) with vertex set V can be constructed as follows. First, form a cell division of M[ϵ,∞) whose cells are indexed by V , associating to each v ∈V the subset of M[ϵ,∞) consisting of x ∈M[ϵ,∞) such that d(x, v) < d(x, v′) for all v′ ∈V . Thurston — The Geometry and Topology of 3-Manifolds 117 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES If V is in general position, faces of the cells meet at most four at a time. (The dual cell division is a triangulation.) Any two hyperbolic manifolds M and N such that M[ϵ,∞) = N[ϵ,∞) can be obtained from one another by Dehn surgery. All manifolds with volume ≤C can therefore be obtained from a ﬁxed ﬁnite set of manifolds by Dehn surgery on a ﬁxed link in each manifold. Each member of this set can be obtained by Dehn surgery on some link in S3, so all manifolds with volume ≤C can be obtained from S3 by Dehn surgery on the disjoint union of all the relevant links. □ 5.59 The full version of Jørgensen’s Theorem involves the geometry as well as the topology of hyperbolic manifolds. The geometry of the manifold M[ϵ,∞) completely determines the geometry and topology of M itself, so an interesting statement com-paring the geometry of M[ϵ,∞)’s must involve the approximate geometric structure. Thus, if M and N are complete hyperbolic manifolds of ﬁnite volume, Jørgensen deﬁnes M to be geometrically near N if for some small ϵ, there is a diﬀeomorphism which is approximately an isometry from the hyperbolic manifold M[ϵ,∞) to N[ϵ,∞). It would suﬃce to keep ϵ ﬁxed in this deﬁnition, except for the exceptional cases when M and N have closed geodesics with lengths near ϵ. This notion of geometric nearness gives a topology to the set H of isometry classes of complete hyperbolic manifolds of ﬁnite volume. Note that neither coordinate systems nor systems of gen-erators for the fundamental groups have been chosen for these hyperbolic manifolds; the homotopy class of an approximate isometry is arbitrary, in contrast with the def-inition for Teichm¨ uller space. Mostow’s Theorem implies that every closed manifold M in H is an isolated point, since M[ϵ,∞) = M when ϵ is small enough. On the other hand, a manifold in H with one end or cusp is a limit point, by the hyperbolic Dehn surgery theorem 5.9. A manifold with two ends is a limit point of limit points and a manifold with k ends is a k-fold limit point. 5.60 Mostow’s Theorem implies more generally that the number of cusps of a geometric limit M of a sequence {Mi} of manifolds distinct from M must strictly exceed the lim sup of the number of cusps of Mi. In fact, if ϵ is small enough, M(0,ϵ] consists only of cusps. The cusps of Mi are contained in Mi(0,ϵ]; if all its components are cusps, and if Mi[ϵ,∞) is diﬀeomorphic with M[ϵ,∞) then Mi is diﬀeomorphic with M so Mi is isometric with M. The volume of a hyperbolic manifold gives a function v : H →R+. If two manifolds M and N are geometrically near, then the volumes of M[ϵ,∞) and N[ϵ,∞) are approximately equal. The volume of a hyperbolic solid torus r0 centered around a geodesic of length l may be computed as volume (solid torus) = Z r0 0 Z 2π 0 Z l 0 sinh r cosh r dt dθ dr = πl sinh2 r0 118 Thurston — The Geometry and Topology of 3-Manifolds 5.12. JØRGENSEN’S THEOREM. while the area of its boundary is area (torus) = 2πl sinh r0 cosh r0. Thus we obtain the inequality area ( ∂solid torus) volume (solid torus) = 1 2 sinh r0 cosh r0 < 1 2. The limiting case as r0 →∞can be computed similarly; the ratio is 1/2. Applying 5.61 this to M, we have 5.11.2. volume (M) ≤volume (M[ϵ,∞)) + 1 2 area (∂M[ϵ,∞)). It follows easily that v is a continuous function on H. Changed this label to 5.11.2a. 5.12. Jørgensen’s Theorem. Theorem 5.12.1. The function v : H →R+ is proper. In other words, every sequence in H with bounded volume has a convergent subsequence. For every C, there is a ﬁnite set M1, . . . , Mk of complete hyperbolic manifolds with volume ≤C such that all other complete hyperbolic manifolds with volume ≤C are obtained from this set by the process of hyperbolic Dehn surgery (as in 5.9). Proof. Consider a maximal subset of V of M[ϵ,∞) having the property that no two elements of V have distance ≤ϵ/2 (as in 5.11.1). Choose a set of isometries of the ϵ/2 balls centered at elements of V with a standard ϵ/2-ball in hyperbolic space. The set of possible gluing maps ranges over a compact subset of Isom(H3), so any sequence of gluing maps (where the underlying sequence of manifolds has volume ≤C) has a convergent subsequence. It is clear that in the limit, the gluing maps still give a hyperbolic structure on M[ϵ,∞), approximately isometric to the limiting M[ϵ,∞)’s. We must verify that M[ϵ,∞) extends to a complete hyperbolic manifold. To see this, note that whenever a complete hyperbolic manifold N has a geodesic which is very short compared to ϵ, the radius of the corresponding solid torus in N(0,ϵ] becomes large. (Otherwise there would be a short non-trivial curve on ∂N(0,ϵ]—but 5.62 such a curve has length ≥ϵ). Thus, when a sequence {Mi[ϵ,∞)} converges, there are approximate isometries between arbitrarily large balls Br(Mi[ϵ,∞)) for large i, so in the limit one obtains a complete hyperbolic manifold. This proves that v is a proper function. The rest of §5.12 is merely a restatement of this fact. □ Remark. Our discussion in §5.10, 5.11 and 5.12 has made no attempt to be numerically eﬃcient. For instance, the proof that there is an ϵ such that Γϵ(x) has an abelian subgroup of ﬁnite index gives the impression that ϵ is microscopic. In fact, ϵ can be rather large; see Jørgensen, for a more eﬃcient approach. It would be extremely interesting to have a good estimate for the number of distinct M[ϵ,∞)’s Thurston — The Geometry and Topology of 3-Manifolds 119 5. FLEXIBILITY AND RIGIDITY OF GEOMETRIC STRUCTURES Figure eight knot Whitehead Link where M has volume ≤C, and it would be quite exciting to ﬁnd a practical way of computing them. The development in 5.10, 5.11, and 5.12 is completely ineﬃcient in this regard. Jørgensen’s approach is much more explicit and eﬃcient. Example. The sequence of knot complements below are all obtained by Dehn surgery on the Whitehead link, so 5.8.2 implies that all but a ﬁnite number possess complete hyperbolic structures. (A computation similar to that of Theorem 4.7 shows that in fact they all possess hyperbolic structures.) This sequence converges, in H, to the Whitehead link complement: 5.63 Note. Gromov proved that in dimensions n ̸= 3, there is only a ﬁnite number of complete hyperbolic manifolds with volume less than a given constant. He proved this more generally for negatively curved Riemannian manifolds with curvature varying between two negative constants. His basic method of analysis was to study the injectivity radius inj(x) = 1 2 inf{lengths of non-trivial closed loops through x} = sup {r \| the exponential map is injective on the ball of radius r in T(x)}. 120 Thurston — The Geometry and Topology of 3-Manifolds 5.12. JØRGENSEN’S THEOREM. Basically, in dimensions n ̸= 3, little can happen in the region M n ϵ of M n where inj(x) is small. This was the motivation for the approach taken in 5.10, 5.11 and 5.12. Gromov also gave a weaker version of hyperbolic Dehn surgery, 5.8.2: he showed that many of the manifolds obtained by Dehn surgery can be given metrics of negative curvature close to −1. Thurston — The Geometry and Topology of 3-Manifolds 121 William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T EX by Sheila Newbery, who also scanned the ﬁgures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the ﬁrst few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at levy@msri.org. CHAPTER 6 Gromov’s invariant and the volume of a hyperbolic manifold 6.1 6.1. Gromov’s invariant Let X be any topological space. Denote the real singular chain complex of X by C∗(k). (Recall that Ck(X) is the vector space with a basis consisting of all continuous maps of the standard simplex ∆k into X.) Any k-chain c can be written uniquely as a linear combination of the basis elements. Deﬁne the norm ∥c∥of c to be the sum of the absolute values of its coeﬃcients, 6.1.1. ∥c∥= X \|ai\| where c = X aiσi, σi : ∆k →X. Gromov’s norm on the real singular homology (really it is only a pseudo-norm) is obtained from this norm on cycles by passing to homology: if a ∈Hk(X; R) is any homology class, then the norm of α is deﬁned to be the inﬁmum of the norms of cycles representing α, Labelled this 6.1.2.def Definition 6.1.2 (First deﬁnition). ∥α∥= inf {∥z∥\| z is a singular cycle representing α}. It is immediate that ∥α + β∥≤∥α∥+ ∥β∥ and for λ ∈R, ∥λα∥≤\|λ\| ∥α∥. If f : X →Y is any continuous map, it is also immediate that 6.2 6.1.2. ∥f∗α∥≤∥α∥. In fact, for any cycle P aiσi representing α, the cycle P aif ◦σi represents f∗α, and ∥P aif ◦σi∥= P \|ai\| ≤∥P aiσi∥. (It may happen that f ◦σi = f ◦σj; even when σi ̸= σj.) Thus ∥f∗α∥≤inf ∥aif ◦σi∥≤∥α∥. In particular, the norm of the fundamental class of a closed oriented manifold M gives a characteristic number of M, Gromov’s invariant of M, satisfying the inequality that for any map f : M1 →M2, 6.1.3. ∥[M1] ∥≥\| deg f\| ∥[M2] ∥. What is not immediate from the deﬁnition is the existence of any non-trivial examples where ∥[M] ∦= 0. Thurston — The Geometry and Topology of 3-Manifolds 123 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD Example. The n-sphere n ≥1 admits maps f : Sn →Sn of degree 2 (and higher). As a consequence of 6.1.2 ∥[Sn] ∥= 0. More explicitly, one may picture a sequence {zi} representing the fundamental class of S1, where zi is ( l i)σi and σi wraps a 1-simplex i times around S1. Since ∥zi∥= 1 i , ∥[S1] ∥= 0. As a trivial example, ∥[S0] ∥= 2. Consider now the case of a complete hyperbolic manifold M n. Any k + 1 points v0, . . . , vk in ˜ M n = Hn determine a straight k-simplex σv0,...,vk : ∆k →Hn, whose image is the convex hull of v0, . . . , vk. There are various ways to deﬁne canonical parametrizations for σv0,...,vk; here is an explicit one. Consider the quadratic form 6.3 model for Hn (§2.5). In this model, v0, . . . , vk become points in Rn+1, so they deter-mine an aﬃne simplex α. [In barycentric coordinates, α(t0, . . . , tk) = P tivi. This parametrization is natural with respect to aﬃne maps of Rn+1]. The central projec-tion from O of α back to one sheet of hyperboloid Q = x2 1+· · ·+x2 n−x2 n+1 = −1 gives a parametrized straight simplex σv0,...,vk in Hn, natural with respect to isometries of Hn. Any singular simplex τ : ∆k →M can be lifted to a singular simplex ˜ τ in ˜ M = Hn, since ∆k is simply connected. Let straight (˜ τ) be the straight simplex with the same vertices as ˜ τ and let straight(τ) be the projection of ˜ τ back to M. Since the straightening operation is natural, straight(τ) does not depend on the lift ˜ τ. Straight extends linearly to a chain map straight : C∗(M) →C∗(M), chain homotopic to the identity. (The chain homotopy is constructed from a canon-ical homotopy of each simplex τ to straight(τ).) It is clear that for any chain c, ∥straight (c)∥≤∥c∥. Hence, in the computation of the norm of a homology class in M, it suﬃces to consider only straight simplices. 6.3 Proposition 6.1.4. There is a ﬁnite supremum vk to the k-dimensional volume of a straight k-simplex in hyperbolic space Hn provided k ̸= 1. 124 Thurston — The Geometry and Topology of 3-Manifolds 6.1. GROMOV’S INVARIANT Proof. It suﬃces to consider ideal simplices with all vertices on S∞, since any ﬁnite simplex ﬁts inside one of these. For k = 2, there is only one ideal simplex up to isometry. We have seen that 2 copies of the ideal triangle ﬁt inside a compact surface (§3.9). Thus it has ﬁnite volume, which equals π by the Gauss-Bonnet theorem. When k = 3, there is an eﬃcient formula for the computation of the volume of an ideal 3-simplex; see Milnor’s discussion of volumes in chapter 7. The volume of such simplices attains its unique maximum at the regular ideal simplex, which has all angles equal to 60◦. Thus we have the values 6.1.5. v2 = 3.1415926 . . . = π v3 = 1.0149416 . . . It is conjectured that in general, vk is the volume of the regular ideal k-simplex; if so, Milnor has computations for more values, and a good asymptotic formula as k →∞. In lieu of a proof of this conjecture, an upper bound can be obtained for vk from the inductive estimate 6.1.6. vk < vk−1 k −1. To prove this, consider any ideal k-simplex σ in Hk. Arrange σ so that one of its vertices is the point at ∞in the upper half-space model, so that σ looks like a triangular chimney lying above a k −1 face σ0 of σ. 6.5 Let dW k be the Euclidean volume element, so hyperbolic volume is dV k = ( 1 xk )kdW k. Let τ denote the projection of σ0 to En−1, and let h(x) denote the Euclidean height of σ0 above the point x ∈τ. The volume of σ is v(σ) = Z τ Z ∞ h t−k dt dW k−1 Thurston — The Geometry and Topology of 3-Manifolds 125 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD (where dW k−1 is the Euclidean k −1 volume element for τ). Integrating, we obtain (k −1) v(σ) = Z τ h−(k−1) dW k−1. The volume of σ0 is obtained by a similar integral, where dW k−1 is replaced by the Euclidean volume element for σ, which is never smaller than dW k−1. We have (k −1)v(σ) < v(σ0) ≤vk−1. □ We are now ready to ﬁnd non-trivial examples for Gromov’s invariant: Corollary 6.1.7. Every closed oriented hyperbolic manifold M n of dimension n > 1 satisﬁes the inequality ∥[M] ∥≥v(M) vn . Proof. Let Ωbe the hyperbolic volume form for M, so that R M Ω= v(M). If z = P ziσi is any straight cycle representing [M], then 6.6 v(M) = Z M Ω= X zi Z ∆n σ∗ i Ω≤ X \|zi\| vn. Dividing by vn, we obtain ∥z∥≥v(M)/vn. The inﬁmum over all such z gives 6.1.7 □ A similar proof shows that the norm of element 0 ̸= α ∈Hk(M, R) where k ̸= 1 is non-zero. Instead of Ω, use an k-form ω representing some multiple λα such that ω has Riemannian norm ≤1 at each point of M. (In fact, ω need only satisfy the inequality ω(V ) ≤1 where V is a simple k-vector of Riemannian norm 1.) Then the inequality ∥α∥≥λ/vk is obtained. Intuitively, Gromov’s norm measures the eﬃciency with which multiples of a homology class can be represented by simplices. A complicated homology class needs many simplices. Gromov proved the remarkable theorem that the inequality of 6.1.7 is actually equality. Instead of proving this, we will take the alternate approach to Gromov’s theorem developed in [Milnor and Thurston, “Characteristic numbers for three-manifolds”], of changing the deﬁnition of ∥∥to one which is technically easier to work with. It can be shown that past and future deﬁnitions are equivalent. However, we have no further use for the ﬁrst deﬁnition, 6.1.2, so henceforth we shall simply abandon it. For any manifold M, let C1(∆k, M) denote the space of maps of ∆k to M, with the C1 topology. We deﬁne a new notion of chains, where a k-chain is a Borel measure µ on C1(∆K, M) with compact support and bounded total variation. [The total variation of a measure µ is ∥µ∥= sup{ R f dµ \| \|f\| ≤1}. Alternately, µ can be decomposed into a positive and negative part, µ = µ+ −µ−where µ+ and µ−are 6.7 126 Thurston — The Geometry and Topology of 3-Manifolds 6.1. GROMOV’S INVARIANT positive. Then ∥µ∥= R dµ+ + R dµ−]. Let the group of k-chains be denoted Ck(M). There is a map ∂: Ck(M) →Ck−1(M), deﬁned in an obvious way. It is not diﬃcult to prove that the homology obtained by using these chains is the standard homology for M; see [Milnor and Thurston, “Characteristic numbers for three-manifolds”] for more details. (Note that integration of a k-form over an element of Ck(M) is deﬁned; this gives a map from C∗(M) to currents on M. Some condition such as compact support for µ is necessary; otherwise one would have pathological cycles such as P(1 i )2σi, where σi wraps ∆1 i times around S1. The measure has total variation P(1 i )2 < ∞, yet the cycle would seem to represent the inﬁnite multiple P(1 i )[S1] of [S1].) Definition 6.1.8 (Second deﬁnition). i Let α ∈Hk(M; R), where M is a mani-fold. Gromov’s norm ∥α∥is deﬁned to be ∥α∥= inf{∥u∥\|µ ∈Ck(M) represents α}. Theorem 6.2 (Gromov). Let M n be any closed oriented hyperbolic manifold. Then ∥[M] ∥= v(M n) vn . Proof. The proof of corollary 6.1.7 works equally well with the new deﬁnition as with the old. The point is that the straightening operation is completely uniform, so it works with measure-cycles. What remains is to prove that ∥[M] ∥≤v(M)/vn, or in other words, the fundamental cycle of M can be represented eﬃciently by a cycle using simplices which have (on the average) nearly maximal volume. 6.8 Let σ be any singular k-simplex in Hn. A chain smearM(σ) ∈Ck(M) can be constructed, which is a measure supported on all isometric maps of σ into M, weighted uniformly. With more notation, let h denote Haar measure on the group of orientation-preserving isometries of Hn, Isom+(Hn). Let h be normalized so that the measure of the set of isometries taking a point x ∈Hn to a region R ⊂Hn is the volume of R. Haar measure on Isom+(Hn) is invariant under both right and left multiplication, so it descends to a measure (also denoted h) on the quotient space P(M) = π1M\ Isom+(Hn). There is a map from P(M) to C1(∆k, M), which associates to a coset π1Mϕ the singular simplex p◦ϕ◦σ, where p : Hn →M is the covering projection. The measure h pushes forward to give a chain smearM(σ) ∈Ck(M). Since h is invariant on both sides, smearM(σ) depends only on the isometry class of σ. Smearing extends linearly to Ck(Hn). Furthermore, smearM ∂c = ∂smearM c. Let σ now be any straight simplex in Hn, and σ−a reﬂected copy of σ. Then 1 2 smearM(σ −σ−)) is a cycle, since the faces of σ and σ−cancel out in pairs, up to isometries. We have ∥1 2 smearM (σ −σ−)∥= v(M). Thurston — The Geometry and Topology of 3-Manifolds 127 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD The homology class of this cycle can be computed by integration of the hyperbolic form Ωfrom M. The integral over each copy of σ is v(σ), so the total integral is v(M)v(σ). Thus, the cycle represents [1 2 smear (σ −σ−)] = v(σ)[M] so that 6.9 ∥v(σ)[M] ∥≤v(M). Dividing by v(σ) and taking the inﬁmum over σ, we obtain 6.2. □ Corollary 6.2.1. If f : M1 →M2 is any map between closed oriented hyperbolic n-manifolds, then v(M1) ≥\| deg f\|v(M2). Gromov’s theorem can be generalized to any (G, X)-manifold, where G acts tran-sitively on X with compact isotropy groups. To do this, choose an invariant Riemannian metric for X and normalize Haar measure on G as before. The smearing operation works equally well, so that one has a chain map smearM : Ck(X) →Ck(M). In fact, if N is a second (G, X)-manifold, one has a chain map smearN,M : Ck(N) →Ck(M), deﬁned ﬁrst on simplices in N via a lift to X, and then extended linearly to all of Ck(N). If z is any cycle representing [N], then smearN,M(z) represents (v(N)/v(M))[M]. This gives the inequality ∥[N] ∥ v(N) ≥∥[M] ∥ v(M) . Interchanging M and N, we obtain the reverse inequality, so we have proved the following result: Theorem 6.2.2. For any pair (G, X), where G acts transitively on X with com-pact isotropy groups and for any invariant volume form on X, there is a constant C 6.10 such that every closed oriented (G, X)-manifold M satisﬁes ∥[M] ∥= Cv(M), (where v(M) is the volume of M). □ 128 Thurston — The Geometry and Topology of 3-Manifolds 6.3. GROMOV’S PROOF OF MOSTOW’S THEOREM This line may be pursued still further. In a hyperbolic manifold a smeared k-cycle is homologically trivial except in dimension k = 0 or k = n, but this is not generally true for other (G, X)-manifolds when G does not act transitively on the frame bundle of X. The invariant cohomology H∗ G(X) is deﬁned to be the cohomology of the cochain complex of diﬀerential forms on X invariant by G. If α is any invariant cohomology class for X, it deﬁnes a cohomology class αM on any (G, X)-manifold M. Let PD(γ) denote the Poincar´ e dual of a cohomology class γ. Theorem 6.2.3. There is a norm ∥∥in H∗ G(X) such that for any closed oriented (G, X)-manifold M, PD(αm)∥= v(M)∥α∥. Proof. It is an exercise to show that the map smearM,M : H∗(M) →H∗(M) is a retraction of the homology of M to the Poincar´ e dual of the image in M of H∗ G(X). The rest of the proof is another exercise. □ In these variations, 6.2.2 and 6.2.3, on Gromov’s theorem, there does not seem to be any general relation between the proportionality constants and the maximal 6.11 volume of simplices. However, the inequality 6.1.7 readily generalizes to any case when X possesses and invariant Riemannian metric of non-positive curvature. 6.3. Gromov’s proof of Mostow’s Theorem Gromov gave a very quick proof of Mostow’s theorem for hyperbolic three-manifolds, based on 6.2. The proof would work for hyperbolic n-manifolds if it were known that the regular ideal n-simplex were the unique simplex of maximal volume. The proof This is now known to be true. goes as follows. Lemma 6.3.1. If M1 and M2 are homotopy equivalent, closed, oriented hyperbolic manifolds, then v(M1) = v(M2). Proof. This follows immediately by applying 6.2 to the homotopy equivalence M1 ↔M2. □ Let f1 : M1 →M2 be a homotopy equivalence and let ˜ f1 : ˜ M1 →˜ M2 be a lift of f1. From 5.9.5 we know that ˜ f1 extends continuously to the sphere Sn−1 ∞. Lemma 6.3.2. If n = 3, ˜ f1 takes every 4-tuple of vertices of a positively oriented regular ideal simplex to the vertices of a positively oriented regular ideal simplex. Thurston — The Geometry and Topology of 3-Manifolds 129 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD Proof. Suppose the contrary. Then there is a regular ideal simplex σ such that the volume of the simplex straight( ˜ f1σ) spanned by the image of its vertices is v3 −ϵ, with ϵ > 0. There are neighborhoods of the vertices of σ in the disk such that for any simplex σ′ with vertices in these neighborhoods, v straight( ˜ f1σ′) ≤v3 −ϵ/2. Then for every ﬁnite simplex σ′ 0 very near to σ, this means that a deﬁnite Haar measure 6.12 of the isometric copies σ′ of σ′ 0 near σ′ have v straight( ˜ f1σ′ 0) < v3 −ϵ/2. Such a simplex σ′ 0 can be found with volume arbitrarily near v3. But then the “total volume” of the cycle z = 1 2 smear(σ′ 0 −σ′ 0−) strictly exceeds the total volume of straight(f∗z), contradicting 6.3.1. □ To complete the proof of Mostow’s theorem in dimension 3, consider any ideal regular simplex σ together with all images of σ coming from repeated reﬂections in the faces of σ. The set of vertices of all these images of σ is a dense subset of S2 ∞. Once ˜ f1 is known on three of the vertices of σ, it is determined on this dense set of points by 6.3.2, so ˜ f1 must be a fractional linear transformation of S2 ∞, conjugating the action of π1M1 to the action of π1M2. This completes Gromov’s proof of Mostow’s theorem. □ In this proof, the fact that f1 is a homotopy equivalence was used to show (a) that v(M1) = v(M2) and (b) that ˜ f1 extends to a map of S2 ∞. With more eﬀort, the proof can be made to work with only assumption (a): Theorem 6.4 (Strict version of Gromov’s theorem). Let f : M1 →M2 be any map of degree ̸= 0 between closed oriented hyperbolic three-manifolds such that Gromov’s inequality 6.2.1 is equality, i.e., v(M1) = \| deg f\| v(M2). Then f is homotopic to a map which is a local isometry. If \| deg f\| = 1, f is a homotopy equivalence and otherwise it is homotopic to a covering map. Proof. The ﬁrst step in the proof is to show that a lift ˜ f of f to the universal covering spaces extends to S2 ∞. Since the information in the hypothesis of 6.4 has to do with volume, not topology, we will know at ﬁrst only that this extension is a measurable map of S2 ∞. Then, the proof of Section 6.3 will be adapted to the current situation. The proof works most smoothly if we have good information about the asymptotic behavior of volumes of simplices. Let σE be a regular simplex in H3 all of whose edge lengths are E. Theorem 6.4.1. The volume of σE diﬀers from the maximal volume v3 by a quantity which decreases exponentially with E. 130 Thurston — The Geometry and Topology of 3-Manifolds 6.3. GROMOV’S PROOF OF MOSTOW’S THEOREM Proof. Construct copies of simplices σE centered at a point x0 ∈H3 by drawing the four rays from a point x0 through the vertices of an ideal regular simplex σ∞ centered at x0. The simplex whose vertices are on these rays, a distance D from x0, is isometric to σE for some E. Let C be the distance from x0 to any face of this simplex. The derivative dv(σE)/dD is less than the area of ∂σE times the maximal normal velocity of a face of σE. If α is the angle between such a face and the ray through x0, we have dv(σE) dD < 2π sin α. From the hyperbolic law of sines (2.6.16) sin α = sinh C/ sinh D, showing that dv(σI)/dD decreases exponentially with D (since sinh C is bounded). The corresponding state-ment for E follows since asymptotically, E ∼2D + constant. □ 6.14 Lemma 6.4.2. Any simplex with volume close to v3 has all dihedral angles close to 60◦. Proof. Such a simplex is properly contained in an ideal simplex with any two face planes the same, so with one common dihedral angle. 6.4.2 follows form ??? □ Lemma 6.4.3. There is some constant C such that for every simplex σ with volume near v3 and for any angle β on a face of σ, v3 −v(σ) ≥Cβ2. Proof. If the vertex v has a face angle of β, ﬁrst enlarge σ so that the other three vertices are at ∞, without changing a neighborhood of v. Now prolong one of Thurston — The Geometry and Topology of 3-Manifolds 131 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD the edges through v to S2 ∞, and push v out along this edge. The new spike added to σ beyond v has thickness at v estimated by a linear function of β (from 2.6.12), 6.15 so its volume is estimated by a quadratic function of β. (This uses the fact that a cross-section of the spike is approximately an equilateral triangle.) □ Lemma 6.4.4. For every point x0 in M1, and almost every ray r through x0, f1(r) converges to a point on S2 ∞. Proof. Let x0 ∈H3, and let r be some ray emanating from x0. Let the simplex σi (with all edges having length i) be placed with a vertex at x0 and with one edge on r, and let τi be a simplex agreeing with σi in a neighborhood of x0 but with the edge on r lengthened, to have length i + 1. The volume of σi and τi ⊃σi deviate from the supremal value by an amount ϵi decreasing exponentially with i, so smearM1 τi and smearM1 σi are very eﬃcient cycles representing a multiple of [M1]. Since v(M1) = \| deg f\| v(M2), the cycles straight f∗smearM1 σi and straight f∗smearM1 τi must also be very eﬃcient. In other words, for all but a set of measure at most v(M1)ϵi/v3 of simplices σ in smear σi (or near smear τi), the simplex straight fσ must have volume ≥v3 −ϵi. 6.16 Let B be a ball around x0 which embeds in Mi. The chains smearB σi and smearB τi correspond to the measure for smearM σi and smearM τi restricted to those singular simplices with the ﬁrst vertex in the image of B in M1. Thus for all but a set of measure at most 2v(M1)/v3 P∞ i=i0 ϵi of isometries I with take x0 to B, all simplices I(σi) and I(τi) for all i > i0 are mapped to simplices straight ˜ f smearB σ with volume ≥v3 −ϵi. By 6.4.3, the sum of all face angles of the image simplices is a geometically convergent series. It follows that for all but a set of small measure of rays r emanating from points in B, f(r) converges to a point on S2 ∞; in fact, by letting i0 →∞, it follows that for almost every ray r emanating from points in B, ˜ f(r) converges. Then there must be a point x′ in B such that for almost every ray r emanating from x′, ˜ f(r) converges. Since each ray emanating from a point in H3 is asymptotic to some ray emanating from x′, this holds for rays through all points in H3. □ 132 Thurston — The Geometry and Topology of 3-Manifolds 6.3. GROMOV’S PROOF OF MOSTOW’S THEOREM Remark. This measurable extension of ˜ f to S2 ∞actually exists under very general circumstances, with no assumption on the volume of M1 and M2. The idea is that if g is a geodesic in M1, ˜ f(g) behaves like a random walk on ˜ M2. Almost every random walk in hyperbolic space converges to a point on Sn−1 ∞. (Moral: always carry a map when you are in hyperbolic space!) Lemma 6.4.5. The measurable extension of ˜ f to S2 ∞carries the vertices of almost every positively oriented ideal regular simplex to the vertices of another positively oriented ideal regular simplex. Proof. Consider a point x0 in H3 and a ball B about x0 which embeds in M, as before. Let σi be centered at x0. As before, for almost all isometries I which take 6.17 x0 to B, the sequence {straight ˜ f ◦I ◦σi} has volume converging to v3, and all four vertices converging to S2 ∞. If for almost all I these four vertices converge to distinct points, we are done. Otherwise, there is a set of positive measure of ideal regular simplices such that the image of the vertex set of σ is degenerate: either all four vertices are mapped to the same point, or three are mapped to one point and the fourth to an arbitrary point. We will show this is absurd. If the degenerate cases occur with positive measure, there is some pair of points v0 and v1 with ˜ f(v0) = ˜ f(v1) such that for almost all regular ideal simplices spanned by v0, v1, v2, v3, either ˜ f(v2) = ˜ f(v0) or ˜ f(v3) = ˜ f(v0). Thus, there is a set A of positive measure with ˜ f(A) a single point. Almost every regular ideal simplex with two vertices in A has one other vertex in A. It is easy to conclude that A must be the entire sphere. (One method is to use ergodicity as in the proof of 6.4 which will follow.) The image point ˜ f(A) is invariant under covering transformations of M1. This implies that the image of π1M1 in π1M2 has a ﬁxed point on S∞, which is absurd. □ We resume the proof of 6.4 here. It follows from 6.4.5 that there is a vertex v0 such that for almost all regular ideal simplices spanned by v0, v1, v2, v3, the image vertices span a regular ideal simplex. Arrange v0 and ˜ f(v0) to be the point at inﬁnity in the 6.18 upper half-space model. Three other points v1, v2, v3 span a regular ideal simplex with v0 if and only if they span an equilateral triangle in the plane, E2. By changing coordinates, we may assume that f maps vertices of almost all equilateral triangles parallel to the x-axis to the vertices of an equilateral triangle in the plane. In complex Thurston — The Geometry and Topology of 3-Manifolds 133 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD notation, let ω = 3 √−1, so that 0, 1, ω span an equilateral triangle. For almost all z ∈C, the entire countable set of triangles spanned by vertices of the form z + 2−kn, z + 2−k(n + 1), z + 2−k(n + ω), for k, n ∈Z, are mapped to equilateral triangles. Then the map ˜ f must take the form ˜ f z + 2−k(n + mω) = g(z) + h(z) · 2−k(n + mω), k, n, m ∈Z, for almost all z. The function h is invariant a.e. by the dense group T of translations of the form z 7→z +2−k(n+mω). This group is ergodic, so h is constant a.e. Similar reasoning now shows that g is constant a.e., so that f is essentially a fractional linear transformation on the sphere S2 ∞. Since ˜ f ◦Tα = Tf∗α ◦˜ f, this shows that π1M1 is conjugate, in Isom(H3), to a subgroup of π1M2. □ 6.19 6.5. Manifolds with Boundary There is an obvious way to extend Gromov’s invariant to manifolds with boundary, as follows. If M is a manifold and A ⊂M a submanifold, the relative chain group Ck(M, A) is deﬁned to be the quotient Ck(M)/Ck(A). The norm on Ck(M) goes over to a norm on Ck(M, A): the norm ∥µ∥of an element of Ck(M, A) is the total variation of µ restricted to the set of singular simplices that do not lie in A. The norm ∥γ∥of a homology class γ ∈Hk(M, A) is deﬁned, as before, to be the inﬁmal norm of relative cycles representing γ. Gromov’s invariant of a compact, oriented manifold with boundary (M, ∂M) is [M, ∂M] , where [M, ∂M] denotes the relative fundamental cycle. There is a second interesting deﬁnition which makes sense in an important special case. For concreteness, we shall deal only with the case of three-manifold whose boundary consists of tori. For such a manifold M, deﬁne ∥[M, ∂M] ∥0 = lim a→0 inf{∥z∥\|z straight [M, ∂M] and ∥∂Z∥≤a}. Observe that ∂z represents the fundamental cycle of ∂M, so that a necessary condi-tion for this deﬁnition to make sense is that ∥[∂M] ∥= 0. This is true in the present situation that ∂M consists of tori, since the torus admits self-maps of degree > 1. 134 Thurston — The Geometry and Topology of 3-Manifolds 6.5. MANIFOLDS WITH BOUNDARY Then ∥(M, ∂M)∥0 is the limit of a non-decreasing sequence, so to insure the existence of the limit we need only ﬁnd an upper bound. This involves a special property of the torus. Proposition 6.5.1. There is a constant K such that z is any homologically trivial cycle in C2(T 2), then z bounds a chain c with ∥c∥≤K∥z∥. Proof. Triangulate T 2 (say, with two “triangles” and a single vertex). Partition T 2 into disjoint contractible neighborhoods of the vertices. Consider ﬁrst the case that no simplices in the support of z have large diameter. Then there is a chain homotopy of z to its simplicial approximation a(z). The chain homotopy has a norm which is a bounded multiple of the norm of z. Since simplicial singular chains form a ﬁnite dimensional vector space, a(z) is homologous to zero by a homology whose norm is a bounded multiple of the norm of a(z). This gives the desired result when the simplices of z are not large. In the general case, pass to a very large cover ˜ T 2 of T 2. For any ﬁnite sheeted covering space p : ˜ M →M there is a canonical chain map, transfer: C∗(M) →C∗( ˜ M). The transfer of a singular simplex is simply the average of its lifts to ˜ M; this extends in an obvious way to measures on singular simplices. Clearly p ◦transfer = id, and ∥transfer c∥= ∥c∥. If z is any cycle on T 2, then for a suﬃciently large ﬁnite cover ˜ T 2 of T 2, the transfer of z to ˜ T 2 = T 2 has no large 2-simplices in its support. Then transfer z is the boundary of a chain c with ∥c∥≤K∥z∥for some ﬁxed K. The projection of c back to the base space completes the proof. □ 6.21 We now have upper bounds for ∥[M, ∂M] ∥0. In fact, let z be any cycle repre-senting [M, ∂M], and let ϵ be any cycle representing [∂M]. By piecing together z with a homology from ∂z to ϵ given by 6.5.1, we ﬁnd a cycle z′ representing [M, ∂M] with ∥z′∥≤∥z∥+ K(∥∂z∥+ ∥ϵ∥). Passing to the limit as ∥ϵ∥→0, we ﬁnd that ∥[M, ∂M] ∥≤∥z∥+ K∥∂z∥. The usefulness of the deﬁnition of ∥[M, ∂M] ∥0 arises from the easy Thurston — The Geometry and Topology of 3-Manifolds 135 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD Proposition 6.5.2. Let (M, ∂M) be a compact oriented three-manifold, not nec-essarily connected, with ∂M consisting of tori. Suppose (N, ∂N) is an oriented man-ifold obtained by gluing together certain pairs of boundary components of M. Then ∥[N, ∂N] ∥0 ≤∥[M, ∂M] ∥0. Corollary 6.5.3. If (S, ∂S) is any Seifert ﬁber space, then ∥[S, ∂S] ∥0 = ∥[S, ∂S] ∥= 0. (The case ∂S = φ is included.) Proof of Corollary. If S is a circle bundle over a connected surface M with non-empty boundary, then S (or a double cover of it, if the ﬁbers are not oriented) is M×S1. Since it covers itself non-trivially its norm (in either sense) is 0. If S is a circle bundle over a closed surface M, it is obtained by identiﬁcation of (M −D2) × S1 with D2 × S1, so its norm is also zero. If S is a Seifert ﬁbration, it is obtained by identifying solid torus neighborhoods of the singular ﬁbers with the complement which is a ﬁbration. □ Proof of 6.5.2. A cycle z representing [M, ∂M] with ∥∂z∥≤ϵ goes over to a chain on [N, ∂N], which can be corrected to be a cycle z′ with ∥z∥′ ≤∥z∥+ Kϵ. □ 6.22 If M is a complete oriented hyperbolic manifold with ﬁnite total volume, recall that M is the interior of a compact manifold ¯ M with boundary consisting of tori. Both ∥[ ¯ M, ∂¯ M] ∥and ∥[ ¯ M, ∂¯ M] ∥0 can be computed in this case: Lemma 6.5.4 (Relative version of Gromov’s Theorem). If M is a complete ori-ented hyperbolic three-manifold with ﬁnite volume, then ∥[ ¯ M, ∂¯ M] ∥0 = ∥[ ¯ M, ∂¯ M] ∥= v(M) v3 . Proof. Let σ be a 3-simplex whose volume is nearly the maximal value, v3. Then smearM σ is a measure on singular cycles with non-compact support. Restrict this measure to simplices not contained in M(0,ϵ], and project to M[ϵ,∞) by a retraction of M to M[ϵ,∞). Since the volume of M(0,ϵ] is small for small ϵ, this gives a relative fundamental cycle z′ for (M[ϵ,∞), ∂M[ϵ,∞)) = ( ¯ M, ∂¯ M) with ∥z′∥≈v(M) v3 and with ∥∂z′∥small. This proves that v(M) v3 ≥∥[ ¯ M, ∂¯ M] ∥0. 136 Thurston — The Geometry and Topology of 3-Manifolds 6.5. MANIFOLDS WITH BOUNDARY There is an immediate inequality ∥[ ¯ M, ∂¯ M] ∥0 ≥∥[ ¯ M, ∂¯ M] ∥. To complete the proof, we will show that ∥[ ¯ M, ∂¯ M] ∥≥v(M)/v3. This is done by a straightening operation, as in 6.1.7. For this, note that if σ is any simplex lying in 6.23 M(0,ϵ], then straight(σ) also lies in M(0,ϵ], since M(0,ϵ] is convex. Hence we obtain a chain map straight : C∗(M, M(0,ϵ]) →C∗(M, M(0,ϵ]), chain homotopic to the identity, and not increasing norms. As in 6.1.7, this gives the inequality ∥[M, M(0,ϵ]] ∥≥v(M[ϵ,∞)) v3 . Since for small ϵ there is a chain isomorphism between Ck(M, M(0,ϵ]) and Ck( ¯ M, ∂¯ M) which is a ∥∥-isometry, this proves 6.5.4. □ Here is an inequality which enables one to compute Gromov’s invariant for much more general three-manifolds: Theorem 6.5.5. Suppose M is a closed oriented three-manifold and H ⊂M is a three-dimensional submanifold with a complete hyperbolic structure of ﬁnite volume. Suppose ¯ H is embedded in M and that ∂¯ H is incompressible. Then ∥[M] ∥≥v(H) v3 . Remark. Of course, the hypothesis that ∂¯ H is incompressible is necessary; oth-erwise M might be S3. If H were not hyperbolic, further hypotheses would be needed to obtain an inequality. Consider, for instance, the product Mg × I where Mg is a surface of genus g > 1. Then ∥[Mg] ∥= 2 v(Mg)/π = 4 \|χ(Mg)\|, so ∥[Mg × I, ∂(Mg × I)] ∥≥∥[Mg] ∥≥4 \|χ(Mg)\|. On the other hand, one can identify the boundary of this manifold to obtain Mg×S1, 6.24 which has norm 0. The boundary can also be identiﬁed to obtain hyperbolic manifolds (see §4.6, or § ). Since ﬁnite covers of arbitrarily high degree and with arbitrarily high norm can also be obtained by gluing the boundary of the same manifold, no useful inequality is obtained in either direction. Proof. Since this is a digression, we give only a sketch of a proof. Thurston — The Geometry and Topology of 3-Manifolds 137 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD □ With 6.5.5 combined with 6.5.2, one can compute Gromov’s invariant for any manifold which is obtained from Seifert ﬁber spaces and complete hyperbolic mani-folds of ﬁnite volume by identifying along incompressible tori. The strict and relative versions of Gromov’s theorems may be combined; here is the most interesting case: Theorem 6.5.6. Suppose M1 is a complete hyperbolic manifold of ﬁnite volume and that M2 ̸= M1 is a complete hyperbolic manifold obtained topologically by replac-ing certain cusps of M1 by solid tori. Then v(M1) > v(M2). 6.25 Proof. No new ideas are needed. Consider some map f : M1 →M2 which collpases certain components of M1(0,ϵ] to short geodesics in M2. Now apply the proof of 6.4. □ 6.6. Ordinals Closed oriented surfaces can be arranged very neatly in a single sequence, in terms of their Euler characteristic. What happens when we arrange all hyper-bolic three-manifolds in terms of their volume? From Jørgensen’s theorem, 5.12 it 138 Thurston — The Geometry and Topology of 3-Manifolds 6.6. ORDINALS follows that the set of volumes is a closed subset of R+. Furthermore, by combining Jørgensen’s theorem with the relative version of Gromov’s theorem, 6.5.4, we obtain Corollary 6.6.1. The set of volumes of hyperbolic three-manifolds is well-ordered. Proof. Let v(M1) ≥v(M2) ≥. . . ≥v(Mk) ≥. . . be any non-ascending sequence of volumes. By Jørgensen’s theorem, by passage to a subsequence we may assume that the sequence {Mi} converges geometrically to a manifold M, with v(M) ≤lim v(Mi). By 6.5.2, eventually ∥[Mi] ∥0 ≤∥[M] ∥0, so 6.5.4 implies that the sequence of volumes is eventually constant. □ 6.26 Corollary 6.6.2. The volume is a ﬁnite-to-one function of hyperbolic manifolds. Proof. Use the proof of 6.6.1, but apply the strict inequality 6.5.6 in place of 6.5.2, to show that a convergent sequence of manifolds with non-increasing volume must be eventually constant. □ In view of these results, the volumes of complete hyperbolic manifolds are indexed by countable ordinals. In other words, there is a smallest volume v1, a next smallest volume v2, and so forth. This sequence v1 < v2 < v3 < · · · < vk < · · · has a limit point vω, which is the smallest volume of a complete hyperbolic manifold with one cusp. The next smallest manifold with one cusp has volume v2ω. It is a limit of manifolds with volumes vω+1, vω+2, . . . , vω+k, . . . . The ﬁrst volume of a manifold with two cusps is vω2, and so forth. (See the discussion on pp. 5.59–5.60, as well as Theorem 6.5.6.) The set of all volumes has order type ωω. These volumes are indexed by the ordinals less than ωω, which are represented by polynomials in ω. Each volume of a manifold with k cusps is indexed by an ordinal of the form α · ωk, (where the product α · β is the ordinal corresponding to the order type obtained by replacing each element of α with a copy of β). There are examples where α is a limit ordinal. These can be constructed from coverings of link complements. For instance, the Whitehead link complement has two distinct 2-fold covers; one has two cusps and the other has three, so the common volume corresponds to an ordinal divisible by ω3. I do not know any examples of closed manifolds corresponding to limit ordinals. It would be very interesting if a computer study could determine some of the low volumes, such as v1, v2, vω, vω2. It seems plausible that some of these might come from Dehn surgery on the Borromean rings. There is some constant C such that every manifold with k cusps has volume ≥C ·k. This follows from the analysis in 5.11.2: the number of boundary components of M[ϵ,∞) is bounded by the number of disjoint ϵ/2 balls which can ﬁt in M. It would be interesting to calculate or estimate the best constant C. Thurston — The Geometry and Topology of 3-Manifolds 139 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD Corollary 6.6.3. The set of values of Gromov’s invariant ∥[ ] ∥0 on the class of connected manifolds obtained from Seifert ﬁber spaces and complete hyperbolic manifolds of ﬁnite volume by identifying along incompressible tori is a closed well-ordered subset of R+, with order type ωω. We shall see later (§ ) that this class contains all Haken manifolds with toral boundaries. Proof. Extend the volume function to v(M) = v3 · ∥[M] ∥0 when M is not hyperbolic. From 6.5.5 and 6.5.2, we know that every value of v is a ﬁnite sum of volumes of hyperbolic manifolds. Suppose {wi} is a bounded sequence of values of v. Express each wi as the sum of volumes of hyperbolic pieces of a manifold Mi with v(M)i = wi. The number of terms is bounded, since there is a lower bound to the volume of a hyperbolic manifold, so we may pass to an inﬁnite subsequence where the number of terms in this expression is constant. Since every inﬁnite sequence of ordinals has an inﬁnite non-decreasing subsequence, we may pass to a subsequence of wi’s where all terms in these expressions are non-decreasing. This proves that 6.28 the set of values of v is well-ordered. Furthermore, our subsequence has a limit w = vα1 +· · ·+vαk, which is expressed as a sum of limits of non-decreasing sequences of volumes. Each vαj is the volume of a hyperbolic manifold Mj with at least as many cusps as the limiting number of cusps of the corresponding hyperbolic piece of Mi. Therefore, the ¯ Mj’s may be glued together to obtain a manifold M with v(M) = w. This shows the set of values of v is closed. The fact that the order type is ωω can be deduced easily by showing that every value of v is not in the k-th derived set, for some integer k; in fact, k ≤v/C, where C is the constant just discussed. □ 6.7. Commensurability Definition 6.7.1. If Γ1 and Γ2 are two discrete subgroups of isometries of Hn, then Γ1 is commensurable with Γ2 if Γ1 is conjugate (in the group of isometries of Hn) to a group Γ′ 1 such that Γ′ ∩Γ2 has ﬁnite index in Γ′ 1 and in Γ2. Definition 6.7.2. Two manifolds M1 and M2 are commensurable if they have ﬁnited sheeted covers ˜ M1 and ˜ M2 which are homeomorphic. Commensurability in either sense is an equivalence relation, as the reader may easily verify. 6.29 Labelled 6.7.3.ex Example 6.7.3. If W is the Whitehead link and B is the Borromean rings, then S3 −W has a four-sheeted cover homeomorphic with a two sheeted cover of S3 −B: 140 Thurston — The Geometry and Topology of 3-Manifolds 6.7. COMMENSURABILITY The homeomorphism involves cutting along a disk, twisting 360◦and gluing back. Thus S3 −W and S3 −B are commensurable. One can see that π1(S3 −W) and π1(S3 −B) are commensruable as discrete subgroups of PSL(2, C) by considering the tiling of H3 by regular ideal octahedra. Both groups preserve this tiling, so they are contained in the full group of symmetries of the octahedral tiling, with ﬁnite index. Therefore, they intersect each other with ﬁnite index. π1(S3 −B) ⊂Symmetries (octahedral tiling) ⊃π1(S3 −W) π1(S3 −B) ⊃π1(S3 −B) ∩π1(S3 −W) ⊂π1(S3 −W) 6.30 Warning. Two groups Γ1 and Γ2 can be commensurable, and yet not be conju-gate to subgroups of ﬁnite index in a single group. Proposition 6.7.3. If M1 is a complete hyperbolic manifold with ﬁnite volume and M2 is commensurable with M1, then M2 is homotopy equivalent to a complete hyperbolic manifold. Proof. This is a corollary of Mostow’s theorem. Under the hypotheses, M2 has a ﬁnite cover M3 which is hyperbolic. M3 has a ﬁnite cover M4 which is a regular Thurston — The Geometry and Topology of 3-Manifolds 141 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD cover of M2, so that π1(M4) is a normal subgroup of π1(M2). Consider the action of π1(M2) on π1(M4) by conjugation. π1(M4) has a trivial center, so in other words the action of π1(M4) on itself is eﬀective. Then for every α ∈π1(M2), since some power of αk is in π1(M4), α must conjugate π1(M4) non-trivially. Thus π1(M2) is isomorphic to a group of automorphisms of π1(M4), so by Mostow’s theorem it is a discrete group of isometries of Hn. □ In the three-dimensional case, it seems likely that M1 would actually be hyper-bolic. Waldhausen proved that two Haken manifolds which are homotopy equivalent are homeomorphic, so this would follow whenever M1 is Haken. There are some sorts of properties of three-manifolds which do not change under passage to a ﬁnite-sheeted cover. For this reason (and for its own sake) it would be interesting to have a better understanding of the commensurability relation among three-manifolds. This is diﬃ-cult to approach from a purely topological point of view, but there is a great deal of information about commensurability given by a hyperbolic structure. For instance, in the case of a complete non-compact 6.31 hyperbolic three-manifold M of ﬁnite volume, each cusp gives a canonical Eu-clidean structure on a torus, well-deﬁned up to similarity. A convenient invariant for this structure is obtained by arranging M so that the cusp is the point at ∞ in the upper half space model and one generator of the fundamental group of the cusp is a translation z 7→z + 1. A second generator is then z 7→z + α. The set of complex numbers α1 . . . αk corresponding to various cusps is an invariant of the commensurability class of M well-deﬁned up to the equivalence relation αi ∼nαi + m pαi + q , where n, m, pq ∈Z, n m p q ̸= 0. (n, m, p and q depend on i). 142 Thurston — The Geometry and Topology of 3-Manifolds 6.7. COMMENSURABILITY 6.32 In particular, if α ∼β, then they generate the same ﬁelds Q(α) = Q(β). Note that these invariants αi are always algebraic numbers, in view of Proposition 6.7.4. If Γ is a discrete subgroup of PSL(2, C) such that H3/Γ has ﬁnite volume, then Γ is conjugate to a group of matrices whose entries are algebraic. Proof. This is another easy consequence of Mostow’s theorem. Conjugate Γ so that some arbitrary element is a diagonal matrix µ 0 0 µ−1 and some other element is upper triangular, λ x 0 λ−1 . The component of Γ in the algebraic variety of representations of Γ having this form is 0-dimensional, by Mostow’s theorem, so all entries are algebraic numbers. □ One can ask the more subtle question, whether all entries can be made algebraic integers. Hyman Bass has proved the following remarkable result regarding this question: Theorem 6.7.5 (Bass). Let M be a complete hyperbolic three-manifold of ﬁnite volume. Then either π1(M) is conjugate to a subgroup of PSL(2, O), where O is the ring of algebraic integers, or M contains a closed incompressible surface (not homotopic to a cusp). Thurston — The Geometry and Topology of 3-Manifolds 143 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD The proof is out of place here, so we omit it. See Bass. As an example, very few knot complements seem to contain non-trivial closed incompressible surfaces. The property that a ﬁnitely generated group Γ is conjugate to a subgroup of PSL(2, O) is equivalent to the property that the additive group of matrices generated by Γ 6.33 is ﬁnitely generated. It is also equivalent to the property that the trace of every element of Γ is an algebraic integer. It is easy to see from this that every group commensurable with a subgroup of PSL(2, O) is itself conjugate to a subgroup of PSL(2, O). (If Tr γn = a is an algebraic integer, then an eigenvalue λ of γ satisﬁes λ2n −aλn + 1 = 0. Hence λ, λ−1 and Tr γ = λ + λ−1 are algebraic integers). If two manifolds are commensurable, then their volumes have a rational ratio. We shall see examples in the next section of incommensurable manifolds with equal volume. Questions 6.7.6. Does every commensurability class of discrete subgroups of PSL(2, C) have a ﬁnite collection of maximal groups (up to isomorphism)? Is the set of volumes of three-manifolds in a given commensurability class a dis-crete set, consisting of multiples of some number V0? 6.8. Some Examples Example 6.8.1. Consider the k-link chain Ck pictured below: 6.34 If each link of the chain is spanned by a disk in the simplest way, the complement of the resulting complex is an open solid torus. 144 Thurston — The Geometry and Topology of 3-Manifolds 6.8. SOME EXAMPLES S3 −Ck is obtained from a solid torus, with the cell division below on its boundary, by deleting the vertices and identifying. 6.35 To construct a hyperbolic structure for S3 −Ck, cut the solid torus into two drums. Thurston — The Geometry and Topology of 3-Manifolds 145 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD Let P be a regular k-gon in H3 with all vertices on S2 ∞. If P ′ is a copy of P obtained by displacing P along the perpendicular to P through its center, then P ′ and P can be joined to obtain a regular hyperbolic drum. The height of P ′ must be adjusted so that the reﬂection through the diagonal of a rectangular side of the drum is an isometry of the drum. If we subdivide the drum into 2k pieces as shown, 6.36 the condition is that there are horospheres about the ideal vertices tangent to three faces. Placing the ideal vertex at ∞in upper half-space, we have a ﬁgure bounded by three vertical Euclidean planes and three Euclidean hemispheres of equal radius r. Here is a view from above: 146 Thurston — The Geometry and Topology of 3-Manifolds 6.8. SOME EXAMPLES From this ﬁgure, we can compute the dihedral angles α and β of the drum to be α = arc cos cos π/k √ 2 , β = π −2α. Two copies of the drum with these angles can now be glued together to give a hyper-bolic structure on S3−Ck. (Note that the total angle around an edge is 4α+2β = 2π. Since the horospheres about vertices are matched up by the gluing maps, we obtain a complete hyperbolic manifold). From Milnor’s formula (6), p. 7.15, for the volume, we can compute some values. 6.37 k v(S3 −Ck) v(S3 −Ck)/k 2 0 0 (Seifert ﬁber space) 3 5.33349 1.77782 ∼PSL(2, O7) 4 10.14942 2.53735 ∼PSL(2, O3) 5 14.60306 2.92061 6 18.83169 3.13861 7 22.91609 3.27373 10 34.691601 3.4691601 50 182.579859 3.65159719 200 732.673784 3.66336892 1000 3663.84264 3.66384264 8000 29310.8990 3.66386238 ∞ ∞ 3.66386238 Whitehead link Thurston — The Geometry and Topology of 3-Manifolds 147 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD Note that the quotient space (S3 −Ck)/Zk by the rotational symmetry of Ck is obtained by generalized Dehn surgery on the White head link W, so the limit of 6.38 v(Ck)/k as k →∞is the volume of S3 −W. Note also that whenever k divides l, then there is a degree l k map from S3 −Cl to S3 −Ck. This implies that v(S3 −Cl)/l > v(S3 −Ck)/k. In fact, from the table it is clear that these numbers are strictly increasing with k. The cases k = 3 and 4 have particular interest. Example 6.8.2. The volume of S3 −C3 per cusp has a particularly low value (1.7778). The holonomy of the hyperbolic structure can be described by H(A) = 1 α 0 1 H(B) = 1 + α α −α 1 −α H(C) = 1 0 −α 1 where α = −1+√−7 2 . Thus π1(X3 −C3) is a subgroup of PSL(2, O7) where Od is the ring of integers in Q √ −d. See §7.4. Referring to Humbert’s formula 7.4.1, we ﬁnd v(H3/ PSL(2, O7) = .8889149 . . ., so π1(S3 −C3) has index 6 in this group. Example 6.8.3. When k = 4, the rectangular-sided drum becomes a cube with all dihedral angles 60◦. This cube may be subdivided into ﬁve regular ideal tetrahedra: 6.39 148 Thurston — The Geometry and Topology of 3-Manifolds 6.8. SOME EXAMPLES Thus S3−C4 is commensurable with S3−ﬁgure eight knot, since π1(S3−C4) preserves a tiling of H3 by regular ideal tetrahedra. commensurable with PSL(2, O3) S3 −Ck is homeomorphic to many other link complements, since we can cut along any disk spanning a component of Ck, twist some integer number of times and glue back to obtain a link with a complement homeomorphic to that of Ck. Further-more, if we glue back with a half-integer twist, we obtain a link whose complement is hyperbolic with the same volume as S3 −Ck. This follows since twice-punctured spanning disks are totally geodesic thrice-punctured spheres in the hyperbolic struc-ture of S3 −Ck. The thrice-punctured sphere has a unique hyperbolic structure, and all six isotopy classes of diﬀeomorphisms are represented by isometries. 6.40 Using such operations, we obtain these examples for instance: Example 6.8.4. commensurable with C3 The second link has a map to the ﬁgure-eight knot obtained by erasing a compo-nent of the link. Thus, by 6.5.6, we have v(S3 −C3) = 5.33340 . . . > 2.02988 = v(S3 −ﬁgure eight knot). These links are commensurable with C3, since they give rise to identical tilings of H3 by drums. As another example, the links below are commensurable with C10: Example 6.8.5. Thurston — The Geometry and Topology of 3-Manifolds 149 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD k = 5 Commensurable with C10 v = 34.69616 6.41 The last three links are obtained from the ﬁrst by cutting along 5-times punctured disks, twisting, and gluing back. Since this gluing map is a diﬀeomorphism of the surface which extends to the three-manifold, it must come from an isometry of a 6-punctured sphere in the hyperbolic structure. (In fact, this surface comes from the top of a 10-sided drum). The compex modulus associated with a cusp of Cn is 1 2 1 + s 1 + sin2 π n cos2 π n i . Clearly we have an inﬁnite family of incommensurable examples. By passing to the limit k →∞and dividing by Z, we get these links commensu-rable with S3 −W and S3 −B, for instance: Example 6.8.6. Many other chains, with diﬀerent amounts of twist, also have hyperbolic struc-tures. They all are obtained, topologically, by identifying faces of a tiling of the boundary of a solid torus by rectangles. Here is another inﬁnite family D2k(≥3) which is easy to compute: 6.42 Example 6.8.7. 150 Thurston — The Geometry and Topology of 3-Manifolds 6.8. SOME EXAMPLES Hyperbolic structures can be realized by subdividing the solid torus into 4 drums with triangular sides: 6.43 Regular drums with all dihedral angles 90◦can be glued together to give S3 −Dk. By methods similar to Milnor’s in 7.3, the formula for the volume is computed to be v(S3 −D2k) = 8k µ(π 4 + π 2k) + µ(π 4 −π 2k) . Thus we have the values k v(S3 −D2k) v(S3 −D2k)/(2k) 3 14.655495 2.44257 4 24.09218 3.01152 5 32.55154 3.25515 6 40.59766 3.38314 100 732.750 3.66288 1000 7327.705 3.66386 ∞ ∞ 3.66386 Thurston — The Geometry and Topology of 3-Manifolds 151 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD The cases k = 3 and k = 4 have algebraic signiﬁcance. They are commensu-rable with PSL(2, O1) nad PSL(2, O2), respectively. When k = 3, the drum is an octahedron and v(S3 −D2k) = 4v(S3 −W). Note that the volume of (S3 −D12) is 20 times the volume of the ﬁgure-eight knot complement. 6.44 Two copies of the triangular-sided drum form this ﬁgure: The faces may be glued in other patterns to obtain link complements. For instance, if k is even we can ﬁrst identify the triangular faces, to obtain a ball minus certain arcs and curves on the boundary. 6.45 If we double this ﬁgure, we obtain a complete hyperbolic structure for the com-plement of this link, El: Example 6.8.8. 152 Thurston — The Geometry and Topology of 3-Manifolds 6.8. SOME EXAMPLES Alternatively, we can identify the boundary of the ball to obtain Example 6.8.9. In these examples, note that the rectangular faces of the doubled drums 6.46 have complete symmetry, and some of the link complements are obtained by gluing maps which interchange the diagonals, while others preserve them. These links are generally commensurable even when they have the same volume; this can be proven by computing the moduli of the cusps. There are many variations. Two copies of the drum with 8 triangular faces, glued together, give a cube with its corners chopped oﬀ. The 4-sided faces can be glued, to obtain the ball minus these arcs and curves: The two faces of the ball may be glued together (isometrically) to give any of these link complements: 6.47 Thurston — The Geometry and Topology of 3-Manifolds 153 6. GROMOV’S INVARIANT AND THE VOLUME OF A HYPERBOLIC MANIFOLD Example 6.8.10. v = 12.04692 = 1 2v(S3 −D8) > v(C3) (commensurable with PSL(2, Z√−2)) The sequence of link complements, Fn below can also be given hyperbolic struc-tures obtained from a third kind of drum: Example 6.8.11. 6.48 The regular drum is determined by its angles α and β = π −α. Any pair of angles works to give a hyperbolic structure; one veriﬁes that when the angle α = arc cos(cos π 2n −1 2), the hyperbolic structure is complete. The case n = 1 gives a trivial knot. In the case n = 2, the drums degenerate into simplices with 60◦angles, and we obtain once more the hyperbolic structure on F2 = ﬁgure eight knot. When n = 3, the angles are 90◦, the drums become octahedra and we obtain F3 = B. Passing to the limit n = ∞, and dividing by Z, we obtain the following link, whose complement is commensurable with S3 −ﬁgure eight knot: 154 Thurston — The Geometry and Topology of 3-Manifolds 6.8. SOME EXAMPLES Example 6.8.12. v = 4 −05977 . . . With these examples, many maps between link complements may be constructed. The reader should experiment for himself. One gets a feeling that volume is a very good measure of the complexity of a link complement, and that the ordinal structure is really inherent in three-manifolds. Thurston — The Geometry and Topology of 3-Manifolds 155 William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T EX by Sheila Newbery, who also scanned the ﬁgures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the ﬁrst few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at levy@msri.org. CHAPTER 7 Computation of volume by J. W. Milnor 7.1. The Lobachevsky function l(θ). This preliminary section will decribe analytic properties, and conjecture number theoretic properties, for the function l(θ) = − Z θ 0 log \|2 sin u\| du. Here is the graph of this function: Thus the ﬁrst derivative l′(θ) is equal to −log \|2 sin θ\|, and the second derivative l′′(θ) is equal to −cot θ. I will call l(θ) the Lobachevsky function. (This name is not quite accurate historically, since Lobachevsky’s formulas for hyperbolic volume were expressed rather in terms of the function Z θ 0 log(sec u) du = l(θ + π/2) + θ log 2 Thurston — The Geometry and Topology of 3-Manifolds 157 7. COMPUTATION OF VOLUME for \|θ\| ≤π/2. However our function π(θ) is clearly a close relative, and is more convenient to work with in practice. Compare Clausen ). references are at end of chapter Another close relative of l(θ) is the dilogarithm function ψ(z) = ∞ X n=1 zn/n2 for \|z\| ≤1, which has been studied by many authors. (See for example , , , , , .) Writing ψ(z) = − Z z 0 log(1 −w) dw/w (where \|w\| ≤1, the substitution w = e2iθ yields 7.2 log(1 −w) dw/w = π −2θ + 2i log(2 sin θ) dθ for 0 < θ < π, hence ψ(e2iθ) −ψ(1) = −θ(π −θ) + 2il(θ) for 0 ≤θ ≤π. Taking the imaginary part of both sides, this proves the following: 7.3 Lemma 7.1.2. The Lobachevsky function has uniformly convergent Fourier series expansion l(θ) = 1 2 ∞ X n=1 sin (2nθ)/n2. Apparently, we have proved this formula only for the case 0 ≤θ ≤π. However, this suﬃces to show that l(0) = l(π) = 0. Since the derivative dl(θ)/dθ = −2 log \| sin 2θ\| is periodic of period π, this proves the following. Lemma 7.1.3. The function l(θ) is itself periodic of period π, and is an odd function, that is, l(−θ) = −l(θ). It follows that the equation in 7.1.2 is actually valid for all values of θ. The equation zn −1 = Qn−1 j=0(z −e−2πij/n) for z = e2πiu leads to the trigonometric identity 2 sin nu = n−1 Y j=0 2 sin(u + jπ/n). Integrating the logarithm of both sides and multiplying by n, this yields the following for n ≥1, and hence for all n. 158 Thurston — The Geometry and Topology of 3-Manifolds 7.1. THE LOBACHEVSKY FUNCTION l(θ). Lemma 7.1.4. The identity l(nθ) = X j mod n n l(θ + jπ/n) is valid for any integer n ̸= 0. (Compare .) Here the sum is to be taken over all residue classes modulo \|n\|. Thus for n = 2 7.4 we get 1 2l(2θ) = l(θ) + l(θ + π/2), or equivalently 1 2l(2θ) = l(θ) −l(π/2 −θ). As an example, for θ = π/6: 3 2l(π/3) = l(π/6). (It is interesting to note that the function l(θ) attains its maximum, l(π/6) = .5074 . . . , at θ = π/6.) It would abe interesting to know whether there are any other such linear relations between various values of l(θ) with rational coeﬃcients. Here is an explicit guess. Conjecture (A). Restricting attention to angles θ which are rational multiples of π, every rational linear relation between the real numbers l(θ) is a consequence of 7.1.3 and 7.1.4. (If we consider the larger class consisting of all angles θ for which eiθ is algebraic then it deﬁnitely is possible to give other Q-linear relations. Compare .) A diﬀerent but completely equivalent formulation is the following. 7.5 Conjecture (B). Fixing some denominator N ≥3, the real numbers l(πj/N) with j relatively prime to N and 0 < j < N/2 are linearly independent over the rationals. These numbers span a rational vector space vN, conjectured to have dimension φ(N)/2, where it is easy to check that vN ⊂vM whenever N divides M. Quite likely the elements l(πj/N) with 1 ≤j ≤φ(N)/2 would provide an alternative basis for this vector space. I have tested these conjectures to the following extent. A brief computer search has failed to discover any other linear relations with small integer coeﬃcients for small values of N. Thurston — The Geometry and Topology of 3-Manifolds 159 7. COMPUTATION OF VOLUME To conclude this section, here is a remark about computation. The Fourier series 7.1.2 converges rather slowly. In order to get actual numerical values for l(θ), it is much better to work with the series l(θ) = θ 1 −log \|2θ\| + ∞ X n=1 Bn 2n (2θ)2n (2n + 1)! , which is obtained by twice integrating the usual Laurent series expansion for the cotangent of θ. Here B1 = 1 6, B2 = 1 30, . . . are Bernoulli numbers. This series converges for \|θ\| = π, and hence converges rea-sonably well for \|θ\| ≤π/2. 7.6 7.2 Having discussed the Lobachevsky function, we will see how it arises in the com-putation of hyperbolic volumes. The ﬁrst case is the ideal simplex, i.e., a tetrahedron whose vertices are at ∞and whose edges are geodesics which converge to the vertices at ∞. Such a simplex is determined by the dihedral angles formed between pairs of faces. The simplex intersects any small horosphere based at a vertex in a triangle whose interior angles are precisely the three dihedral angles along the edges meeting at that vertex. Since a horosphere is isometric to a Euclidean plane, the sum of the dihedral angles at an inﬁnite vertex equals 2π. It follows by an easy computation that the dihedral angles of opposite edges are equal. Call the three dihedral angles determining the simplex α, β, γ and denote the simplex by Σα,β,γ. The main result of this section is: Theorem 7.2.1. The volume of the simplex P α,β,γ equals l(α) + l(β) + l(γ). In order to prove this theorem a preliminary computation is necessary. Consider the simplex Sα,β,γ pictured below, with three right dihedral angles and three other 7.7 dihedral angles α, β, γ and suppose that one vertex is at inﬁnity. (Thus α+β = π/2.) 160 Thurston — The Geometry and Topology of 3-Manifolds 7.2 It turns out that any simplex can be divided by barycentric subdivision into simplices with three right angles so this is a natural object to consider. The decomposition of P α,β,γ is demonstrated below, but ﬁrst a computation, due to Lobachevsky. Lemma 7.2.2. The volume of Sα,π/2−α,γ equals 1 4[l(α+γ)+l(α−γ)+2l(π/2−α)]. Proof. Consider the upper half-space model of H3, and put the inﬁnite vertex of Sα,π/2−α,γ at ∞. The edges meeting that vertex are just vertical lines. Further-more, assume that the base triangle lies on the unithemisphere (which is a hyper-bolic plane). Recall that the line element for the hyperbolic metric in this model is ds2 = dx2+dy2+dz2 z2 so that the volume element is dV = dz dy dz z3 . Projecting the base tri-angle to the (x, y) plane produces a Euclidean triangle T with angles α, π/2−α, π/2, 7.8 which we may take to be the locus 0 ≤x ≤cos γ, 0 ≤y ≤x tan α, with γ as above. Remark. This projection of the unit hemisphere gives Klein’s projective model for H2. The angles between lines are not their hyperbolic angles; rather, they are the dihedral angles of corresponding planes in H3. Now it is necessary to compute (1). V = Z x,y∈T Z Z z≥√ 1−x2−y2 dx dy dz z3 . Integrating with respect to z gives (2). V = Z T Z dx dy 2(1 −x2 −y2). Thurston — The Geometry and Topology of 3-Manifolds 161 7. COMPUTATION OF VOLUME Setting a = √ 1 −x2, we have 7.9 (3). V = Z cos γ 0 dx Z x tan α 0 dy 2(a2 −y2) = Z cos γ 0 x 4a log a + x tan α a −x tan α = Z cos γ 0 dx 4a log 2(a cos α + x sin α) 2(a cos α −x sin α). If we set x = cos θ, then a = √ 1 −x2 = sin θ and dx a = −dθ. Then (3) becomes (4). V = 1 4 Z γ π/2 −dθ log 2 sin(θ + α) 2 sin(θ −α) = 1 4[l(γ + α) −l(γ −α) −l(π/2 + α) + l(π/2 −α)]. Since l(γ) −α) = −l(α −γ) and l(π/2 + α) = −l(π/2 −α) by 7.1.3, this is the desired formula. □ Suppose that two vertices are at inﬁnity in Sa,π/2−α,γ. Then α = γ. The lemma above implies that volume (Sα,π/2−α,a) = 1 4[l(2α) + 2l(π/2 −α)]. By lemmas 7.1.3 and 7.1.4 l(π/2 −α) = −l(π/2 + α) and l(2α) = 2 l(α) + l(α + π/2) so that (5). Volume (Sα,π/2−α,α) = 1 2l(α). To see how P α,β,γ decomposes into simplices of the above type, consider the upper half-space model of H3. Put one vertex at the point at inﬁnity and the base 7.10 on the unit sphere. Drop the perpendicular from ∞to the sphere and draw the perpendiculars from the intersection point x on the base to each of the three edges on the base. Connect x to the remaining three vertices. Taking the inﬁnite cone on the lines in the base gives the decomposition. (See (A) below.) Projecting onto the (x, y) plane gives a triangle inscribed in the unit circle with x projected into its center. Figure (B) describes the case when x is in the interior of the base (which happens when α, β, γ < π/2). Not that the pairs of triangles which share a perpendicular are similar triangles. It follows that the angles around x are as described. 162 Thurston — The Geometry and Topology of 3-Manifolds 7.2 7.11 Each sub-simplex has two inﬁnite vertices and three dihedral angles of π/2 so that they are of the type considered above. Thus Volume X α,β,γ = 2 1 2l(γ) + 1 2l(β) + 1 2l(α) . In the case when x is not in the interior of the base triangle, Σα,β,γ can still be thought of as the sum of six simplices each with three right dihedral angles. However, some of the simplices must be considered to have negative volume. The interested reader may supply the details, using the picture below. Thurston — The Geometry and Topology of 3-Manifolds 163 7. COMPUTATION OF VOLUME Example. The complement of the ﬁgure-eight knot was constructed in 3.1 by gluing two copies of Σπ/3,π/3,π/3. Thus its volume is 6l(π/3) = 2.02988 . . .. Remark. It is not hard to see that the (π/3, π/3, π/3) simplex has volume greater than any other three-dimensional simplex. A simplex with maximal volume must 7.12 have its vertices at inﬁnity since volume can always be increased by pushing a ﬁnite vertex out towards inﬁnity. To maximize V = l(α) + l(β) + l(γ) subject to the restraint α + β + γ = 0 we must have l′(α) = l′(β) = l′(γ) which implies easily that α = β = γ = π/3. (The non-diﬀerentiability of l(α) at α = 0 causes no trouble, since V tends to zero when α, β or γ tends to zero.) Theorem 7.2.1 generalizes to a formula for the volume of a ﬁgure which is an inﬁnite cone on a planar n-gon with all vertices at inﬁnity. Let the dihedral angles formed by the triangular faces with the base plane be (α1, . . . , αn) and denote the ﬁgure with these angles by Σα1,...,αn. 164 Thurston — The Geometry and Topology of 3-Manifolds 7.3 Theorem 7.2.3. (i) Pn i=1 αi = π. (ii) Volume (Σα1,...,αn) = Pn i=1 l(αi). Proof. The proof is by induction. The case n = 3 is Theorem 1. Suppose the theorem to be true for n = k −1. It suﬃces to prove it for n = k. 7.13 Consider the base k-gon for Σα1,...,αk and divide it into a k −1-gon and a triangle. Take the inﬁnite cone on each of these two ﬁgures. If the new dihedral angle on the triangle side is β, the new angle on the k −1-gon side in π −β. By the induction hypothesis 2 X i=1 αi + β = π and n X i=3 αi + π −β = π. Part (i) follows by adding the two equations. Similarly by the induction hypothesis, Vol(Σα1,α2,β) = 2 X i=1 l(αi) + l(β) and Vol(Σα3,...,αn,π−β) = n X i=3 l(αi) + l(π −β). Part (ii) follows easily since l(π −β) = −l(β). □ Example. The complement of the Whitehead link was constructed from a regular ideal octahedron which in turn, is formed by gluing two copies of the inﬁnite cone on a regular planar quadrilateral. Thus its volume equals 8l(π/4) = 3.66386 . . .. Similarly, the complement of the Borromean rings has volume 16l(π/4) = 7.32772 . . . since it is obtained by gluing two ideal octahedra together. 7.3 It is diﬃcult to ﬁnd a general pattern for constructing manifolds by gluing in-ﬁnite tetrahedra together. A simpler method would be to reﬂect in the sides of a tetrahedron to form a discrete subgroup of the isometries of H3. Unfortunately this method yields few examples since the dihedral angles must be of the form π/a, a ∈Z in order that the reﬂection group be discrete with the tetrahedron as fundamental domain. The only cases when the sum of the angles is π are Σπ/2,π/4,π/4, Σπ/3,π/3,π/3 7.14 and Σπ/3,π/3,π/6 corresponding to the three Euclidean triangle groups. Here is a construction for polyhedra in H3 due to Thurston. Take a planar regular n-gon with vertices at inﬁnity on each of two distinct planes in H3 and join the corresponding vertices on the two ﬁgures by geodesics. If this is done in a symmetric way the sides are planer rectangles meeting each other at angle β and meeting the bases at angle α. Denote the resulting polyhedra by Nα,β. Note that 2α + β = π since two edges of an n-gon and a vertical edge form a Euclidean triangle at inﬁnity. Thurston — The Geometry and Topology of 3-Manifolds 165 7. COMPUTATION OF VOLUME 7.15 In order to compute the volume of Nα,β consider it in the upper half-space model of H3. Subdivide Nα,β into n congruent sectors Sα,β by dividing the two n-gons into n congruent triangles and joining them be geodesics. Call the lower and upper triangles of Sα,β, T1 and T2 respectively. Consider the inﬁnite cones C1 and C2 on T1 and T2. They have the same volume since they are isometric by a Euclidean expansion. Hence the volume of Sα,β is equal to the volume of Q = (Sα,β ∪C2) −C1. Evidently Q is an inﬁnite cone on a quadrilateral. To ﬁnd its volume it is necessary to compute the dihedral angles at the edges of the base. The angles along the sides are β 2. The angle at the front face is α + γ where γ is the angle between the front face and the top plane of Nα,β. Consider the inﬁnite cone on the top n-gon of Nα,β. 7.16 By (1) of Theorem 7.2.3 the angles along its base are π/n. Thus γ = π/n and the front angle is α + π/n. Similarly the back angle is α −π/n. 166 Thurston — The Geometry and Topology of 3-Manifolds 7.4 By (2) of Theorem 7.2.3 we have (6). Vol(Nα,β) = n Vol(Q) = n 2l(β/2) + l(α + π/n) + l(α −π/n) . If α and β are of the form π/a, a ∈Z then the group generated by the reﬂections in the sides of Nα,β form a discrete group of isometries of H3. Take a subgroup Γ which is torsion free and orientation preserving. The quotient space H3/Γ is an oriented, hyperbolic three-manifold with ﬁnite volume. Since 2α + β = π the only choices for (α, β) are (π/3, π/3) and (π, 4, π/2). As long as n > 4 both of these can be realized since β varies continuously from 0 to n −2/n as the distance between the two base planes of Nα,β varies from 0 to ∞. 7.16 Thus we have the following: Theorem 7.3.1. There are an inﬁnite number of oriented three-manifolds whose volume is a ﬁnite rational sum of l(θ) for θ’s commensurable with π. 7.4 We will now discuss an arithmetic method for constructing hyperbolic three-manifolds with ﬁnite volume. The construction and computation of volume go back to Bianchi and Humbert. (See , , .) The idea is to consider Od, the ring of integers in an imaginary quadratic ﬁeld, Q( √ −d), where d ≥1 is a square-free integer. Then PSL(2, Od) is a discrete subgroup of PSL(2, C). Let Γ be a torsion free subgroup of ﬁnite index in PSL(2, Od). Since PSL(2, C) is the group of orientation preserving isometries of H3, H3/Γ is an oriented hyperbolic three-manifold. It always has ﬁnite volume. Example. Let Z[i] be the ring of Gaussian integers. A fundamental domain for the action of PSL(2, Z[i]) has ﬁnite volume. Diﬀerent choices of Γ give diﬀerent manifolds; e.g., there is a Γ of index 12 such that H3/Γ is diﬀeomorphic to the Thurston — The Geometry and Topology of 3-Manifolds 167 7. COMPUTATION OF VOLUME complement of the Whitehead link; another Γ of index 24 leads to the complement of the Borromean rings. (N. Wielenbert, preprint). Example. In case d = 3, Od is Z[ω] where ω = −1+√−3 2 and there is a sub-group Γ ⊂PSL(2, Z[ω]) of index 12 such that H3/Γ is diﬀeomorphic to the comple-ment of the ﬁgure-eight knot. (R. Riley, ). In order to calculate the volume of H3/ PSL(2, Od) in general we recall the following deﬁnitions. Deﬁne the discriminant, 7.18 D, of the extension Q( √ −d) to be D = n d if d ≡3(mod 4), 4d otherwise. If Od is considered as a lattice in T, then √ D/2 is the area of T/Od . The Dedekind ζ-function for a ﬁeld K is deﬁned to be ζK(S) = X a 1/N(a)S where a runs through all ideals in O and N(a) = \|O/a\| denotes the norm of ζ(S) is also equal to Y P 1 1 − 1 N(P)S taking all prime ideals of P. Theorem 7.4.1 (Essentially due to Humbert). Vol H3/ PSL(2, Od) = D3/2 24 ζQ( √ −d) (2)/ζQ(2). This volume can be expressed in terms of Lobachevsky’s function using Hecke’s formula ζQ ( √ −d)(S)/ζQ(S) = X n>0 −D n ns . Here −D n is the quadratic symbol where we use the conventions: (i) If n = p1, . . . , pt, pi prime then −D n = −D p1 −D p2 . . . −D pt . (ii) If p \| D then −D p = 0; −D 1 = +1. (iii) for p an odd prime −D p = +1 if −D ≡X2(mod p) for some X, −1 if not. 168 Thurston — The Geometry and Topology of 3-Manifolds 7.4 (iv) For p = 2 −D p = +1 if −D ≡1 (mod 8), −1 if −D ≡5 (mod 8). (Note that −D ̸≡3 (mod 4) by deﬁnition.) The function n 7→ −D n is equal to 1/ √ −D times its Fourier transform;∗i.e., (1). X k mod D −D k e2πikn/D = √ −D −D n . Multiplying by 1/n2 and summing over n > 0 we get (2). X n>0 1/n2 n−1 X k=0 −D k e2πikn/D = √ −D X n>0 −D n n2 . For ﬁxed k the imaginary part of the left side is just the Fourier series for 2l(πk/D). Since the right side is pure imaginary we have: (3). 2 X k mod D −D k l(πk/D) = √ D X n>0 −D n 1/n2. Multiplying by D/24 and using Hecke’s formula leads to (4). D/12 X k mod D −D k l(πk/D) = Vol H3/ PSL(2, Od) . 7.20 Example. In the case d = 3, 7.4.4 implies that the volume of H3/ PSL(2, Z[ω] is 1 4 l(π/3) −l(2π/3) = 1 2l(π/3). Recall that the complement of the ﬁgure-eight knot S3 −K is diﬀeomorphic to H3/Γ where Γ had index 12 in PSL(2, Z[ω]). Thus it has volume 6l(π/3). This agrees with the volume computed by thinking of S3 −K as two copies of Σπ/3,π/3,π/3 tetrahedra glued together. Similarly the volumes for the complements of the Whitehead link and the Bor-romean rings can be computed using 7.4.4. The answers agree with those computed geometrically in 7.2. This algebraic construction also furnishes an inﬁnite number of hyperbolic man-ifolds with volumes equal to rational, ﬁnite linear combinations of l (a rational multiple of π). Note that Conjectures (A) and (B) would imply that any rational relation between the volumes of these manifolds could occur at most as a result of common factors of the integers, d, deﬁning the quadratic ﬁelds. In fact, quite likely they would imply that there are no such rational relations. ∗Compare Hecke, Vorlesangen ¨ uber algebr. Zahlen, p. 241. I am grateful to A. Adler for help on this point. Thurston — The Geometry and Topology of 3-Manifolds 169 7. COMPUTATION OF VOLUME References L. Euler, Institutiones calculi integralis, I, pp. 110-113 (1768). C. J. Hill, Journ. reine angew. Math. (Crelle) 3 (1828), 101-139. T. Clausen, Journ. reine angew. Math. (Crelle) 8 (1932), 298-300. N. Lobachevsky, Imaginary Geometry and its Application to Integration (Russian) Kasan 1836. German translation, Leipzig 1904. (For a modern presentation see B. L. Laptev, Kazan Gos Univ. Uˇ c. Zapiski 114 (1954), 53-77). L. Bianchi, Math. Ann. 38 (1891) 313-333 and 40 (1892), 332-412. H. Gieseking, Thesis, M¨ unster 1912. (See Magnus, Noneuclidean Tesselations and their Groups, Acad. Press 1974, p. 153.) G. Humbert, Comptes Rendus 169 (1919) 448-454. H. S. M. Coxeter, Quarterly J. Math 6 (1935), 13-29. L. Lewin, Dilogarithms and Associated Functions, Macdonald (London), 1958. R. G. Swan, Bull. AMS 74 (1968), 576-581. R. Riley, Proc. Cambr. Phil. Soc. 77 (1975) 281-288. A. M. Gabrielov, I. M. Gel’fand, and M. V. Losik, Functional Anal. and Appl. 9 (1975), pp. 49, 193. S. Bloch, to appear. D. Kubert and S. Lang, to appear. 170 Thurston — The Geometry and Topology of 3-Manifolds William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T EX by Sheila Newbery, who also scanned the ﬁgures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the ﬁrst few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at levy@msri.org. CHAPTER 8 Kleinian groups 8.1 Our discussion so far has centered on hyperbolic manifolds which are closed, or at least complete with ﬁnite volume. The theory of complete hyperbolic manifolds with inﬁnite volume takes on a somewhat diﬀerent character. Such manifolds occur very naturally as covering spaces of closed manifolds. They also arise in the study of hyperbolic structures on compact three-manifolds whose boundary has negative Euler characteristic. We will study such manifolds by passing back and forth between the manifold and the action of its fundamental group on the disk. 8.1. The limit set Let Γ be any discrete group of orientation-preserving isometries of Hn. If x ∈Hn is any point, the limit set LΓ ⊂Sn−1 ∞ is deﬁned to be the set of accumulation points of the orbit Γx of x. One readily sees that LΓ is independent of the choice of x by picturing the Poincar´ e disk model. If y ∈Hn is any other point and if {γi} is a sequence of elements of Γ such that {γix} converges to a point on Sn−1 ∞, the hyperbolic distance d(γix, γiy) is constant so the Euclidean distance goes to 0; hence lim γiy = lim γix. The group Γ is called elementary if the limit set consists of 0, 1 or 2 points. Proposition 8.1.1. Γ is elementary if and only if Γ has an abelian subgroup of ﬁnite index. □ When Γ is not elementary, then LΓ is also the limit set of any orbit on the sphere at inﬁnity. Another way to put it is this: Proposition 8.1.2. If Γ is not elementary, then every non-empty closed subset of S∞invariant by Γ contains LΓ. Proof. Let K ⊂S∞be any closed set invariant by Γ. Since Γ is not elementary, K contains more than one element. Consider the projective (Klein) model for Hn, and let H(K) denote the convex hull of K. H(K) may be regarded either as the Euclidean convex hull, or equivalently, as the hyperbolic convex hull in the sense that it is the intersection of all hyperbolic half-spaces whose “intersection” with S∞ contains K. Clearly H(K) ∩S∞= K. Thurston — The Geometry and Topology of 3-Manifolds 171 8. KLEINIAN GROUPS Since K is invariant by Γ, H(K) is also invariant by Γ. If x is any point in Hn ∩H(K), the limit set of the orbit Γx must be contained in the closed set H(K). Therefore LΓ ⊂K. □ 8.3 A closed set K invariant by a group Γ which contains no smaller closed invariant set is called a minimal set. It is easy to show, by Zorn’s lemma, that a closed invariant set always contains at least one minimal set. It is remarkable that in the present situation, LΓ is the unique minimal set for Γ. Corollary 8.1.3. If Γ is a non-elementary group and 1 ̸= Γ′ ◁Γ is a normal subgroup, then LΓ′ = LΓ. Proof. An element of Γ conjugates Γ′ to itself, hence it takes LΓ′ to LΓ′. Γ′ must be inﬁnite, otherwise Γ′ would have a ﬁxed point in Hn which would be invariant by Γ so Γ would be ﬁnite. It follows from 8.1.2 that LΓ′ ⊃LΓ. The opposite inclusion is immediate. □ Examples. If M 2 is a hyperbolic surface, we may regular π1(M) as a group of isometries of a hyperbolic plane in H3. The limit set is a circle. A group with limit set contained in a geometric circle is called a Fuchsian group. The limit set for a closed hyperbolic manifold is the entire sphere Sn−1 ∞. If M 3 is a closed hyperbolic three-manifold which ﬁbers over the circle, then the fundamental group of the ﬁber is a normal subgroup, hence its limit set is the entire sphere. For instance, the ﬁgure eight knot complement has fundamental group ⟨A, B : ABA−1BA = BAB−1AB⟩. 8.4 172 Thurston — The Geometry and Topology of 3-Manifolds 8.1. THE LIMIT SET It ﬁbers over S1 with ﬁber F a punctured torus. The fundamental group π1(F) is the commutator subgroup, generated by AB−1 and A−1B. Thus, the limit set of a ﬁnitely generated group may be all of S2 even when the quotient space does not have ﬁnite volume. A more typical example of a free group action is a Schottky group, whose limit set is a Cantor set. Examples of Schottky groups may be obtained by considering Hn minus 2k disjoint half-spaces, bounded by hyperplanes. If we choose isometric identiﬁcations between pairs of the bounding hyperplanes, we obtain a complete hyperbolic manifold with fundamental group the free group on k generators. Thurston — The Geometry and Topology of 3-Manifolds 173 8. KLEINIAN GROUPS 8.5 It is easy to see that the limit set for the group of covering transformations is a Cantor set. 8.2. The domain of discontinuity The domain of discontinuity for a discrete group Γ is deﬁned to be DΓ = Sn−1 ∞−LΓ. A discrete subgroup of PSL(2, C) whose domain of discontinuity is non-empty is called a Kleinian group. (There are actually two ways in which the term Kleinian group is generally used. Some people refer to any discrete subgroup of PSL(2, C) as a Kleinian group, and then distinguish between a type I group, for which LΓ = S2 ∞, and a type II group, where DΓ ̸= ∅. As a ﬁeld of mathematics, it makes sense for Kleinian groups to cover both cases, but as mathematical objects it seems useful to have a word to distinguish between these cases DΓ ̸= ∅and DΓ = ∅.) We have seen that the action of Γ on LΓ is minimal—it mixes up LΓ as much as possible. In contrast, the action of Γ on DΓ is as discrete as possible. Definition 8.2.1. If Γ is a group acting on a locally compact space X, the action is properly discontinuous if for every compact set K ⊂X, there are only ﬁnitely many γ ∈Γ such that γK ∩K ̸= ∅. Another way to put this is to say that for any compact set K, the map Γ×K →X given by the action is a proper map, where Γ has the discrete topology. (Otherwise 8.6 there would be a compact set K′ such that the preimage of K′ is non-compact. Then inﬁnitely many elements of Γ would carry K ∪K′ to itself.) Proposition 8.2.2. If Γ acts properly discontinuously on the locally compact Hausdorﬀspace X, then the quotient space X is Hausdorﬀ. If the action is free, the quotient map X →X/Γ is a covering projection. Proof. Let x1, x2 ∈X be points on distinct orbits of Γ. Let N1 be a compact neighborhood of x1. Finitely many translates of x2 intersect N1, so we may assume N1 is disjoint from the orbit of x2. Then S γ∈Γ γN1 gives an invariant neighborhood of x1 disjoint from x2. Similarly, x2 has an invariant neighborhood N2 disjoint from N1; this shows that X/Γ is Hausdorﬀ. If the action of Γ is free, we may ﬁnd, again by a similar argument, a neighborhood of any point x which is disjoint from all its translates. This neighborhood projects homeomorphically to X/Γ. Since Γ acts transitively on the sheets of X over X/Γ, it is immediate that the projection X →X/Γ is an even covering, hence a covering space. □ Proposition 8.2.3. If Γ is a discrete group of isometries of Hn, the action of Γ on DΓ (and in fact on Hn ∪DΓ) is properly discontinuous. 174 Thurston — The Geometry and Topology of 3-Manifolds 8.2. THE DOMAIN OF DISCONTINUITY Proof. Consider the convex hull H(LΓ). There is a retraction r of the ball Hn ∪S∞to H(LΓ) deﬁned as follows. 8.7 If x ∈H(LΓ), r(x) = x. Otherwise, map x to the nearest point of H(LΓ). If x is an inﬁnite point in DΓ, the nearest point is interpreted to be the ﬁrst point of H(LΓ) where a horosphere “centered” about x touches LΓ. This point r(x) is always uniquely deﬁned because H(LΓ) is convex, and spheres or horospheres about a point in the ball are strictly convex. Clearly r is a proper map of Hn ∪DΓ to H(LΓ) −LΓ. The action of Γ on H(LΓ) −LΓ is obviously properly discontinuous, since Γ is a discrete group of isometries of H(LΓ) −LΓ; the property of Hn ∪DΓ follows immediately. □ Remark. This proof doesn’t work for certain elementary groups; we will ignore such technicalities. It is both easy and common to confuse the deﬁnition of properly discontinuous with other similar properties. To give two examples, one might make these deﬁnitions: Definition 8.2.4. The action of Γ is wandering if every point has a neighborhood N such that only ﬁnitely many translates of N intersect N. Definition 8.2.5. The action of Γ has discrete orbits if every orbit of Γ has an empty limit set. Proposition 8.2.6. If Γ is a free, wandering action on a Hausdorﬀspace X, the projection X →X/Γ is a covering projection. Proof. An exercise. □ Warning. Even when X is a manifold, X/Γ may not be Hausdorﬀ. For instance, consider the map L : R2 −0 →R2 −0 L (x, y) = (2x, 1 2y). Thurston — The Geometry and Topology of 3-Manifolds 175 8. KLEINIAN GROUPS It is easy to see this is a wandering action. The quotient space is a surface with fundamental group Z ⊕Z. The surface is non-Hausdorﬀ, however, since points such 8.9 as (1, 0) and (0, 1) do not have disjoint neighborhoods. Such examples arise commonly and naturally; it is wise to be aware of this phe-nomenon. The property that Γ has discrete orbits simply means that for every pair of points x, y in the quotient space X/Γ, x has a neighborhood disjoint from y. This can occur, for instance, in a l-parameter family of Kleinian groups Γt, t ∈[0, 1]. There are examples where Γt = Z, and the family deﬁnes the action of Z on [0, 1] × H3 with discrete orbits which is not a wandering action. See § . It is remarkable that the action of a Kleinian group on the set of all points with discrete orbits is properly discontinuous. 8.3. Convex hyperbolic manifolds The limit set of a group action is determined by a limiting process, so that it is often hard to “know” the limit set directly. The condition that a given group action is discrete involves inﬁnitely many group elements, so it is diﬃcult to verify directly. Thus it is important to have a concrete object, satisfying concrete conditions, corresponding to a discrete group action. 176 Thurston — The Geometry and Topology of 3-Manifolds 8.3. CONVEX HYPERBOLIC MANIFOLDS We consider for the present only groups acting freely. Definition 8.3.1. A complete hyperbolic manifold M with boundary is convex if every path in M is homotopic (rel endpoints) to a geodesic arc. (The degenerate 8.10 case of an arc which is a single point may occur.) Proposition 8.3.2. A complete hyperbolic manifold M is convex if and only if the developing map D : ˜ M →Hn is a homeomorphism to a convex subset of Hn. Proof. If ˜ M is a convex subset S of Hn, then it is clear that M is convex, since any path in M lifts to a path in S, which is homotopic to a geodesic arc in S, hence in M. If M is convex, then D is 1 −1, since any two points in ˜ M may be joined by a path, which is homotopic in M and hence in ˜ M to a geodesic arc. D must take the endpoints of a geodesic arc to distinct points. D( ˜ M) is clearly convex. □ We need also a local criterion for M to be convex. We can deﬁne M to be locally convex if each point x ∈M has a neighborhood isometric to a convex subset of Hn. If x ∈∂M, then x will be on the boundary of this set. It is easy to convince oneself that local convexity implies convexity: picture a bath and imagine straightening it out. Because of local 8.11 convexity, one never needs to push it out of ∂M. To make this a rigorous argument, given a path p of length l there is an ϵ such that any path of length ≤ϵ intersecting Nl(p0) is homotopic to a geodesic arc. Subdivide p into subintervals of length between ϵ/4 and ϵ/2. Straighten out adjacent pairs of intervals in turn, putting a new division point in the middle of the resulting arc unless it has length ≤ϵ/2. Any time an interval becomes too small, change the subdivision. This process converges, giving a homotopy of p to a geodesic arc, since any time there are angles not close to π, the homotopy signiﬁcantly shortens the path. Thurston — The Geometry and Topology of 3-Manifolds 177 8. KLEINIAN GROUPS This give us a very concrete object corresponding to a Kleinian group: a complete convex hyperbolic three-manifold M with non-empty boundary. Given a convex manifold M, we can deﬁne H(M) to be the intersection of all convex submanifolds M ′ of M such that π1M ′ →π1M is an isomorphism. H(M) is clearly the same as 8.12 HLπ1(M)/π1(M). H(M) is a convex manifold, with the same dimension as M except in degenerate cases. Proposition 8.3.3. If M is a compact convex hyperbolic manifold, then any small deformation of the hyperbolic structure on M can be enlarged slightly to give a new convex hyperbolic manifold homeomorphic to M. Proof. A convex manifold is strictly convex if every geodesic arc in M has in-terior in the interior of M. If M is not already strictly convex, it can be enlarged slightly to make it strictly convex. (This follows from the fact that a neighborhood of radius ϵ about a hyperplane is strictly convex.) Thus we may assume that M ′ is a hyperbolic structure that is a slight deformation of a strictly convex manifold M. We may assume that our deformation M ′ is small 178 Thurston — The Geometry and Topology of 3-Manifolds 8.3. CONVEX HYPERBOLIC MANIFOLDS enough that it can be enlarged to a hyperbolic manifold M ′′ which contains a 2ϵ-neighborhood of M ′. Every arc of length l greater than ϵ in M has the middle (l −ϵ) some uniform distance δ from ∂M; we may take our deformation M ′ of M small 8.13 enough that such intervals in M ′ have the middle l−ϵ still in the interior of M ′. This implies that the union of the convex hulls of intersections of balls of radius 3ϵ with M ′ is locally convex, hence convex. □ The convex hull of a uniformly small deformation of a uniformly convex manifold is locally determined. Remark. When M is non-compact, the proof of 8.3.3 applies provided that M has a uniformly convex neighborhood and we consider only uniformly small deforma-tions. We will study deformations in more generality in § . Proposition 8.3.4. Suppose M n 1 and M n 2 are strictly convex, compact hyperbolic manifolds and suppose φ : M n 1 →M n 2 is a homotopy equivalence which is a diﬀeo-morphism on ∂M1. Then there is a quasi-conformal homeomorphism f : Bn →Bn of the Poincar´ e disk to itself conjugating π1M1 to π1M2. f is a pseudo-isometry on Hn. 8.14 Proof. Let ˜ φ be a lift of φ to a map from ˜ M1 to ˜ M2. We may assume that ˜ φ is already a pseudo-isometry between the developing images of M1 and M2. Each point p on ∂˜ M1 and ∂˜ M2 has a unique normal ray γp; if x ∈γp has distance t from ∂˜ M1 let f(x) be the point on γ˜ φ(p) a distance t from ∂˜ M2. The distance between points at a distance of t along two normal rays γp1 and γp2 at nearby points is approximately cosh t + α sinh t, where d is the distance and θ is the angle between the normals of p1 and p2. From this it is evident that f is a pseudo-isometry extending to ¯ φ. □ Associated with a discrete group Γ of isometries of Hn, there are at least four distinct and interesting quotient spaces (which are manifolds when Γ acts freely ). Let us name them: Definition 8.3.5. Thurston — The Geometry and Topology of 3-Manifolds 179 8. KLEINIAN GROUPS MΓ = H(LΓ)/Γ , the convex hull quotient. NΓ = Hn/Γ, the complete hyperbolic manifold without boundary. OΓ = (Hn ∪DΓ)/Γ, the Kleinian manifold. PΓ = (Hn ∪DΓ ∪WΓ)/Γ. Here WΓ ⊂Pn is the set of points in the projective model dual to planes in Hn whose intersection with S∞is contained in DΓ. We have inclusions H(NΓ) = MΓ ⊂NΓ ⊂OΓ ⊂PΓ. It is easy to derive the fact that Γ acts properly discontinuously on Hn ∪DΓ ∪WΓ from the proper discontinuity on Hn ∪DΓ. MΓ, NΓ and OΓ have the same homotopy type. MΓ and OΓ are home-omorphic except in degenerate cases, and NΓ = int(OΓ) PΓ is not always connected 8.15 when LΓ is not connected. 8.4. Geometrically ﬁnite groups Definition 8.4.1. Γ is geometrically ﬁnite if Nϵ(MΓ) has ﬁnite volume. The reason that Nϵ(MΓ) is required to have ﬁnite volume, and not just MΓ, is to rule out the case that Γ is an arbitary discrete group of isometries of Hn−1 ⊂Hn . We shall soon prove that geometrically ﬁnite means geometrically ﬁnite (8.4.3). Theorem 8.4.2 (Ahlfors’ Theorem). If Γ is geometrically ﬁnite, then LΓ ⊂S∞ has full measure or 0 measure. If LΓ has full measure, the action of Γ on S∞is ergodic. Proof. This statement is equivalent to the assertion that every bounded mea-surable function f supported on LΓ and invariant by Γ is constant a.e. (with respect to Lebesque measure on S∞). Following Ahlfors, we consider the function hf on Hn determined by f as follows. If x ∈Hn, the points on S∞correspond to rays through x; these rays have a natural “visual” measure Vx. Deﬁne hf(x) to be the average of f with respect to the visual measure Vx. This function hf is harmonic, i.e., the gradient ﬂow of hf preserves volume, div grad hf = 0. For this reason, the measure 1 Vx(S∞)Vx is called harmonic measure. To prove this, 8.16 consider the contribution to hf coming from an inﬁnitesimal area A centered at p ∈Sn−1 (i.e., a Green’s function). As x moves a distance d in the direction of p, the visual measure of A goes up exponentially, in proportion to e(n−1)d. The gradient of any multiple of the characteristic function of A is in the direction of p, and also proportional in size to e(n−1)d. The ﬂow lines of the gradient are orthogonal trajectories to horospheres; this ﬂow contracts linear dimensions along the horosphere in proportion to e−d, so it preserves volume. 180 Thurston — The Geometry and Topology of 3-Manifolds 8.4. GEOMETRICALLY FINITE GROUPS The average hf of contributions from all the inﬁnitesimal areas is therefore harmonic. We may suppose that f takes only the values of 0 and 1. Since f is invariant by Γ, so is hf, and hf goes over to a harmonic function, also hf, on NΓ. To complete the proof, observe that hf < 1 2 in NΓ −MΓ, since each point x in Hn −H(LΓ) lies in a half-space whose intersection with inﬁnity does not meet LΓ, which means that f is 0 on more than half the sphere, with respect to Vx. The set {x ∈NΓ\|hf(x) = 1 2} 8.17 must be empty, since it bounds the set {x ∈NΓ\|hf(x) ≥1 2} of ﬁnite volume which ﬂows into itself by the volume preserving ﬂow generated by grad hf. (Observe that grad hf has bounded length, so it generates a ﬂow deﬁned everywhere for all time.) But if {p\|f(p) = 1} has any points of density, then there are x ∈Hn−1 near p with hf(x) near 1. It follows that f is a.e. 0 or a.e. 1. □ Let us now relate deﬁnition 8.4.1 to other possible notions of geometric ﬁniteness. The usual deﬁnition is in terms of a fundamental polyhedron for the action of Γ. For concreteness, let us consider only the case n = 3. For the present discussion, a ﬁnite-sided polyhedron means a region P in H3 bounded by ﬁnitely many planes. P Thurston — The Geometry and Topology of 3-Manifolds 181 8. KLEINIAN GROUPS is a fundamental polyhedron for Γ if its translates by Γ cover H3, and the translates of its interior are pairwise disjoint. P intersects S∞in a polygon which unfortunately 8.18 may be somewhat bizarre, since tangencies between sides of P ∩S∞may occur. Sometimes these tangencies are forced by the existence of parabolic ﬁxed points for Γ. Suppose that p ∈S∞is a parabolic ﬁxed point for some element of Γ, and let π be the subgroup of Γ ﬁxing p. Let B be a horoball centered at p and suﬃciently small that the projection of B/P to NΓ is an embedding. (Compare §5.10.) If π ⊃Z ⊕Z, for any point x ∈B ∩H(LΓ), the convex hull of πx contains a horoball B′, so in particular there is a horoball B′ ⊂H(LΓ) ∩B. Otherwise, Z is a maximal torsion-free subgroup of π. Coordinates can be chosen so that p is the point at ∞in the upper half-space model, and Z acts as translations by real integers. There is some minimal strip S ⊆C containing LΓ ∩C; S may interesect the imaginary axis in a ﬁnite, half-inﬁnite, or doubly inﬁnite interval. In any case, H(LΓ) is contained in the region R of upper half-space above S, and the part of ∂R of height ≥1 lies on ∂HΓ. 8.19 It may happen that there are wide substrips S′ ⊂S in the complement of LΓ. If S′ is suﬃciently wide, then the plane above its center line intersects H(LΓ) in B, so it gives a half-open annulus in B/Z. If Γ is torsion-free, then maximal, suﬃciently wide strips in S −LΓ give disjoint non-parallel half-open annuli in MΓ; an easy argument 182 Thurston — The Geometry and Topology of 3-Manifolds 8.4. GEOMETRICALLY FINITE GROUPS shows they must be ﬁnite in number if Γ is ﬁnitely generated. (This also follows from Ahlfors’s ﬁniteness theorem.) Therefore, there is some horoball B′ centered at p so that H(LΓ) ∩B′ = R ∩B′. This holds even if Γ has torsion. 8.20 With an understanding of this picture of the behaviour of MΓ near a cusp, it is not hard to relate various notions of geometric ﬁniteness. For convenience suppose Γ is torsion-free. (This is not an essential restriction in view of Selberg’s theorem—see § .) When the context is clear, we abbreviate MΓ = M, NΓ = N, etc. Proposition 8.4.3. Let Γ ⊂PSL(2, C) be a discrete, torsion-free group. The following conditions are equivalent: (a) Γ is geometrically ﬁnite (see dfn. 8.4.1). (b) M[ϵ,∞) is compact. (c) Γ admits a ﬁnite-sided fundamental polyhedron. Proof. (a) ⇒(b). Each point in M[ϵ,∞) has an embedded ϵ/2 ball in Nϵ/2(MΓ), by deﬁnition. If (a) holds, Nϵ/n(MΓ) has ﬁnite volume, so only ﬁnitely many of these balls can be disjoint and MΓ[ϵ,∞) is compact. (b) ⇒(c). First, ﬁnd fundamental polyhedra near the non-equivalent parabolic ﬁxed points. To do this, observe that if p is a Z-cusp, then in the upper half-space model, when p = ∞, LΓ ∩C lies in a strip S of ﬁnite width. Let R denote the region above S. Let B′ be a horoball centered at ∞such that R ∩B′ = H(LΓ) ∩B′. Let r : H3 ∪DΓ →H(LΓ) be the canonical retraction. If Q is any fundamental polyhedra for the action of Z in some neighborhood of p in H(LΓ) then r−1(Q) is a fundamental polyhedron in some neighborhood of p in H3 ∪DΓ. 8.21 Thurston — The Geometry and Topology of 3-Manifolds 183 8. KLEINIAN GROUPS A fundamental polyhedron near the cusps is easily extended to a global fundamental polyhedron, since OΓ-(neighborhoods of the cusps) is compact. (c) ⇒(a). Suppose that Γ has a ﬁnite-sided fundamental polyhedron P. A point x ∈P ∩S∞is a regular point (∈DΓ) if it is in the interior of P ∩S∞ or of some ﬁnite union of translates of P. Thus, the only way x can be a limit point is for x to be a point of tangency of sides of inﬁnitely many translates of P. Since P can have only ﬁnitely many points of tangency of sides, inﬁnitely many γΓ must identify one of these points to x, so x is a ﬁxed point for some element γΓ. γ must be parabolic, otherwise the translates of P by powers of γ would limit on the axis of γ. If x is arranged to be ∞in upper half-space, it is easy to see that LΓC must be contained in a strip of ﬁnite width. (Finitely many translates of P must form a fundamental domain for {γn}, acting on some horoball centered at ∞, since {γn} has ﬁnite index in the group ﬁxing ∞. Th faces of these translates of P which do not pass through ∞lie on hemispheres. Every point in C outside this ﬁnite collection of 8.22 hemispheres and their translates by {γn} lies in DΓ.) It follows that v(Nϵ(M)) = v(Nϵ(H(LΓ))∩P) if ﬁnite, since the contribution near any point of LΓ ∩P is ﬁnite and the rest of Nϵ(H(LΓ)) ∩P is compact. □ 184 Thurston — The Geometry and Topology of 3-Manifolds 8.5. THE GEOMETRY OF THE BOUNDARY OF THE CONVEX HULL 8.5. The geometry of the boundary of the convex hull Consider a closed curve σ in Euclidean space, and its convex hull H(σ). The boundary of a convex body always has non-negative Gaussian curvature. On the other hand, each point p in ∂H(σ) −σ lies in the interior of some line segment or triangle with vertices on σ. Thus, there is some line segment on ∂H(σ) through p, so that ∂H(σ) has non-positive curvature at p. It follows that ∂H(σ) −σ has zero curvature, i.e., it is “developable”. If you are not familiar with this idea, you can see it by bending a curve out of a piece of stiﬀwire (like a coathanger). Now roll the wire around on a big piece of paper, tracing out a curve where the wire touches. Sometimes, the wire may touch at three or more points; this gives alternate ways to roll, and you should carefully follow all of them. Cut out the region in the plane bounded by this curve (piecing if necessary). By taping the paper together, you can envelope the wire in a nice paper model of its convex hull. The physical process of unrolling a developable surface onto the plane is the origin of the notion of the developing map. 8.23 The same physical notion applies in hyperbolic three-space. If K is any closed set on S∞, then H(K) is convex, yet each point on ∂H(K) lies on a line segment in ∂H(K). Thus, ∂H(K) can be developed to a hyperbolic plane. (In terms of Riemannian geometry, ∂H(K) has extrinsic curvature 0, so its intrinsic curvature is the ambient sectional curvature, −1. Note however that ∂H(K) is not usually diﬀerentiable). Thus ∂H(K) has the natural structure of a complete hyperbolic surface. Proposition 8.5.1. If Γ is a torsion-free Kleinian group, the ∂MΓ is a hyperbolic surface. □ The boundary of MΓ is of course not generally ﬂat—it is bent in some pattern. Let γ ⊂∂MΓ consist of those points which are not in the interior of a ﬂat region of 8.24 ∂MΓ. Through each point x in γ, there is a unique geodesic gx on ∂MΓ. gx is also a geodesic in the hyperbolic structure of ∂MΓ. γ is a closed set. If ∂MΓ has ﬁnite area, then γ is compact, since a neighborhood of each cusp of ∂MΓ is ﬂat. (See §8.4.) Definition 8.5.2. A lamination L on a manifold M n is a closed subset A ⊂M (the support of L) with a local product structure for A. More precisely, there is a covering of a neighborhood of A in M with coordinate neighborhoods Ui φi →Rn−k×Rk so that φi(A ∩Ui) is of the form Rn−k × B, B ⊂Rk. The coordinate changes φij must be of the form φij(x, y) = (fij(x, y), gij(y)) when y ∈B. A lamination is like a foliation of a closed subset of M. Leaves of the lamination are deﬁned just as for a foliation. Thurston — The Geometry and Topology of 3-Manifolds 185 8. KLEINIAN GROUPS Examples. If F is a foliation of M and S ⊂M is any set, the closure of the union of leaves which meet S is a lamination. Any submanifold of a manifold M is a lamination, with a single leaf. Clearly, the bending locus γ for ∂MΓ has the structure of a lamination: whenever two points of γ are nearby, the directions of bending must be nearly parallel in order that the lines of bending do not intersect. A lamination whose leaves are geodesics we will call a geodesic lamination. 8.25 By consideration of Euler characteristic, the lamination γ cannot have all of ∂M as its support, or in other words it cannot be a foliation. The complement ∂M −γ consists of regions bounded by closed geodesics and inﬁnite geodesics. Each of these regions can be doubled along its boundary to give a complete hyperbolic surface, which of course has ﬁnite area. There 8.26 is a lower bound for π for the area of such a region, hence an upper bound of 2\|χ(∂M)\| for the number of components of ∂M −γ. Every geodesic lamination γ on a hyperbolic surface S can be extended to a foliation with isolated singularities on the complement. There 186 Thurston — The Geometry and Topology of 3-Manifolds 8.5. THE GEOMETRY OF THE BOUNDARY OF THE CONVEX HULL is an index formula for the Euler characteristic of S in terms of these singularities. Here are some values for the index. 8.27 From the existence of an index formula, one concludes that the Euler characteristic of S is half the Euler characteristic of the double of S −γ. By the Gauss-Bonnet theorem, Area(S −γ) = Area(S) or in other words, γ has measure 0. To give an idea of the range of possibilities for geodesic laminations, one can consider an arbitrary sequence {γi} of geodesic lamina-tions: simple closed curves, for instance. Let us say that {γi} converges geometrically to γ if for each x ∈support γ, and for each ϵ, for all great enough i the support of γi intersects Nϵ(x) and the leaves of γi ∩Nϵ(x) are within ϵ of the direction of the leaf of γ through x. Note that the support of γ may be smaller than the limiting support of γi, so the limit of a sequence may not be unique. See §8.10. An easy diagonal argument shows that every sequence {γi} has a subsequence which converges geo-metrically. From limits of sequences of simple closed geodesics, uncountably many geodesic laminations are obtained. Geodesic laminations on two homeomorphic hyperbolic surfaces may be compared by passing to the circle at ∞. A directed geodesic is determined by a pair of points (x1, x2) ∈S1 ∞× S1 ∞−∆, where ∆is the diagonal {(x, x)}. A geodesic without direction is a point on J = (S1 ∞× S1 ∞−∆/Z2), where Z2 acts by interchanging coordinates. Topologically, J is an open Moebius band. It is geometrically realized in the Klein (projective) model for H2 as the region outside H2. A geodesic g projects 8.28 Thurston — The Geometry and Topology of 3-Manifolds 187 8. KLEINIAN GROUPS to a simple geodesic on the surface S if and only if the covering translates of its pairs of end points never strictly separate each other. Geometrically, J has an indeﬁnite metric of type (1, 1), invariant by covering translates. (See §2.6.) The light-like geodesics, of zero length, are lines tangent to S1 ∞; lines which meet H2 when extended have imaginary arc length. A point g ∈J projects to a simple geodesic in S if and only if no covering translate Tα(g) has a positive real distance from g. 188 Thurston — The Geometry and Topology of 3-Manifolds 8.6. MEASURING LAMINATIONS 8.29 Let S ⊂J consist of all elements g projecting to simple geodesics on S. Any geodesic ⊂H2 which has a translate intersecting itself has a neighborhood with the same property, hence S is closed. If γ is any geodesic lamination on S, Let Sγ ⊂J be the set of lifts of leaves of γ to H2. Sγ is a closed invariant subset of S. A closed invariant subset of C ⊂J gives rise to a geodesic lamination if and only if all pairs of points of C are separated by an imaginary (or 0) distance. If g ∈S, then the closure of its orbit, π1(S)g is such a set, corresponding to the geodesic lamination ¯ g of S. Every homeomorphism between surfaces when lifted to H2 extends to S1 ∞(by 5.9.5). This determines an extension to J. Geodesic laminations are transferred from one surface to another via this correspondence. 8.6. Measuring laminations Let L be a lamination, so that it has local homeomorphisms φi : L∩Ui ≈Rn−k×Bi. A transverse measure µ for L means a measure µi deﬁned on each local leaf space Bi, in such a way that the coordinate changes are measure preserving. Alternatively one may think of µ as a measure deﬁned on every k-dimensional submanifold transverse to L, supported on T k ∩L and invariant under local projections along leaves of L. We will always suppose that µ is ﬁnite on compact transversals. The simplest example of a transverse measure arises when L is a closed submanifold; in this case, one can take µ to count the number of intersections of a transversal with L. 8.30 We know that for a torsion-free Kleinian group Γ, ∂MΓ is a hyperbolic surface bent along some geodesic lamination γ. In order to complete the picture of ∂MΓ, we need a quantitative description of the bending. When two planes in H3 meet along a line, the angle they form is constant along that line. The ﬂat pieces of ∂MΓ meet each other along the geodesic lamination γ; the angle of meeting of two planes generalizes to a transverse “bending” measure, β, for γ. The measure β applied to an arc α on ∂MΓ transverse to γ is the total angle of turning of the normal to ∂MΓ along α (appropriately interpreted when γ has isolated geodesics with sharp bending). In order to prove that β is well-deﬁned, and that it determines the local isometric embedding in H3, one can use local polyhedral approximations to ∂MΓ. Local outer approximations to ∂MΓ can be obtained by extending the planes of local ﬂat regions. Observe that when three planes have pairwise intersections in H3 but no triple intersection, the dihedral angles satisfy the inequality α + β ≤γ. Thurston — The Geometry and Topology of 3-Manifolds 189 8. KLEINIAN GROUPS 8.31 (The diﬀerence γ −(α +β) is the area of a triangle on the common perpendicular plane.) From this it follows that as outer polyhedral approximations shrink toward MΓ, the angle sum corresponding to some path α on ∂MΓ is a monotone sequence, converging to a value β(α). Also from the monotonicity, it is easy to see that for short paths αt, [0 ≤t ≤1], β(α) is a close approximation to the angle between the tangent planes at α0 and α1. This implies that the hyperbolic structure on ∂MΓ, together with the geodesic lamination γ and the transverse measure β, completely determines the hyperbolic structure of NΓ in a neighborhood of ∂MΓ. The bending measure β has for its support all of γ. This puts a restriction on the structure of γ: every isolated leaf L of γ must be a closed geodesic on ∂MΓ. (Other-wise, a transverse arc through any limit point of L would have inﬁnite measure.) This limits the possibilities for the intersection of a transverse arc with γ to a Cantor set and/or a ﬁnite set of points. When γ contains more than one closed geodesic, there is obviously a whole family of possibilities for transverse measures. There are (probably atypical) examples of families of distinct transverse measures which are not multiples of each other even for certain geodesic laminations such that every leaf is dense. There are many other ex-amples which possess unique transverse measures, up to constant multiples. Compare Katok. 8.32 Here is a geometric interpretation for the bending measure β in the Klein model. Let P0 be the component of PΓ containing NΓ (recall deﬁnition 8.3.5). Each point in ˜ P0 outside S∞is dual to a plane which bounds a half-space whose intersection with S∞is contained in DΓ. ∂˜ P0 consists of points dual to planes which meet LΓ in at least one point. In particular, each plane meeting ˜ MΓ in a line or ﬂat of ∂˜ MΓ is dual 190 Thurston — The Geometry and Topology of 3-Manifolds 8.7. QUASI-FUCHSIAN GROUPS to a point on ∂˜ P0. If ¯ π ∈∂˜ P0 is dual to a plane π touching LΓ at x, then one of the line segments ¯ πx is also on ∂˜ P0. This line segments consists of points dual to planes touching LΓ at x and contained in a half-space bounded by π. The reader may check that ˜ P0 is convex. The natural metric of type (2, 1) in the exterior of S∞ is degenerate on ∂˜ P0, since it vanishes on all line segments corresponding to a family of planes tangent at S∞. Given a path α on ∂˜ MΓ, there is a dual path ¯ α consisting of points dual to planes just skimming MΓ along α. The length of ¯ α is the same as β(α). 8.33 Remark. The interested reader may verify that when N is a component of ∂MΓ such that every leaf of γ ∩N is dense in γ ∩N, then the action of π1n on the appropriate component of ∂˜ P0 −LΓ is minimal (i.e., every orbit is dense). This action is approximated by actions of π1N as covering transformations on surfaces just inside ∂˜ P0. 8.7. Quasi-Fuchsian groups Recall that a Fuchsian group (of type I) is a Kleinian group Γ whose limit set LΓ is a geometric circle. Examples are the fundamental groups of closed, hyperbolic surfaces. In fact, if the Fuchsian group Γ is torsion-free and has no parabolic elements, then Γ is the group of covering transformations of a hyperbolic surface. Furthermore, the Kleinian manifold OΓ = (H3 ∪DΓ)/Γ has a totally geodesic surface as a spine. Note. The type of a Fuchsian group should not be confused with its type as a Kleinian group. To say that Γ is a Fuchsian group of type I means that LΓ = S1, but it is a Kleinian group of type II since DΓ ̸= ∅. Thurston — The Geometry and Topology of 3-Manifolds 191 8. KLEINIAN GROUPS Suppose M = N 2×I is a convex hyperbolic manifold, where N 2 is a closed surface. Let Γ′ be the group of covering transformations of M, and let Γ be a Fuchsian group coming from a hyperbolic structure on N. Γ and Γ′ are isomorphic as groups; we want to show that their actions on the closed ball B3 are topologically conjugate. Let MΓ and MΓ′ be the convex hull quotients (MΓ ≈N 2 and MΓ′ ≈N 2 × I). Thicken MΓ and MΓ′ to strictly convex manifolds. The thickened manifolds are 8.34 diﬀeomorphic, so by Proposition 8.3.4 there is a quasi-conformal homeomorphism of B3 conjugating Γ to Γ′. In particular, LΓ′ is homeomorphic to a circle. Γ′, which has convex hull manifold homeomorphic to N 2 × I and limit set ≈S1, is an example of a quasi-Fuchsian group. Definition 8.7.1. The Kleinian group Γ is called a quasi-Fuchsian group if LΓ is topologically S1. Proposition 8.7.2 (Marden). For a torsion-free Kleinian group Γ, the following conditions are equivalent. (i) Γ is quasi-Fuchsian. (ii) DΓ has precisely two components. (iii) Γ is quasi-conformally conjugate to a Fuchsian group. Proof. Clearly (iii) = ⇒(i) = ⇒(ii). To show (ii) = ⇒(iii), consider OΓ = (H3 ∪DΓ)/Γ. Suppose that no element of Γ interchanges the two components of DΓ. Then OΓ is a three-manifold with two boundary components (labelled, for example, N1 and N2), and 8.35 192 Thurston — The Geometry and Topology of 3-Manifolds 8.7. QUASI-FUCHSIAN GROUPS Γ = π1(OΓ) ≈π1(N1) ≈π1(N2). By a well-known theorem about three-manifolds (see Hempel for a proof), this implies that OΓ is homeomorphic to N1 × I. By the above discussion, this implies that Γ′ is quasi-conformally conjugate to a Fuchsian group. A similar argument applies if OΓ has one boundary component; in that case, OΓ is the orientable interval bundle over a non-orientable surface. The reverse implication is clear. □ Example 8.7.3 (Mickey mouse). Consider a hyperbolic structure on a surface of genus two. Let us construct a deformation of the corresponding Fuchsian group by bending along a single closed geodesci γ by an angle of π/2. This 8.36 will give rise to a quasi-Fuchsian group if the geodesic is short enough. We may visualize the limit set by imagining bending a hyperbolic plane along the lifts of γ, one by one. Thurston — The Geometry and Topology of 3-Manifolds 193 8. KLEINIAN GROUPS We want to understand how the geometry changes as we deform quasi-Fuchsian groups. Even though the topology doesn’t change, geometrically things can become very complicated. For example, given any ϵ > 0, there is a quasi-Fuchsian group Γ whose limit set LΓ is ϵ-dense in S2, and there are limits of quasi-Fuchsian groups with LΓ = S2. Our goal here is to try to get a grasp of the geometry of the convex hull quotient M = MΓ of a quasi-Fuchsian group Γ. MΓ is a convex hyperbolic manifold which is homeomorphic to N 2 × I, and the two boundary components are hyperbolic surfaces bent along geodesic laminations. 8.37 We also need to analyze intermediate surfaces in MΓ. For example, what kinds of nice surfaces are embedded (or immersed) in MΓ? Are there isometrically embedded cross sections? Are there cross sections of bounded area near any point in MΓ? Here are some ways to map in surfaces. (a) Take the abstract surface N 2, and choose a “triangulation” of N with one vertex. Choose an arbitrary map of N into M. Then straighten the map (see §6.1). 194 Thurston — The Geometry and Topology of 3-Manifolds 8.7. QUASI-FUCHSIAN GROUPS This is a fairly good way to map in a surface, since the surface is hyperbolic away from the vertex. There may be positive curvature concentrated at the vertex, however, since the sum of the angles around the vertex may be quite small. This map can be changed by moving the image of the vertex in M or by changing the triangulation on N. 8.38 (b) Here is another method, which insures that the map is not too bad near the vertex. First pick a closed loop in N, and then choose a vertex on the loop. Now extend this to a triangulation of N with one vertex. To map in N, ﬁrst map in the loop to the unique geodesic in M in its free homotopy class (this uses a homeomorphism of M to N × I). Now extend this as in (a) to a piecewise straight map f : N →M. The sum of the angles around the vertex is at least 2π, since there is a straight line segment going through the vertex (so the vertex cannot be spiked). It is possible to have the sum of the angles > 2π, in which case there is negative curvature concentrated near the vertex. (c) Here is a way to map in a surface with constant negative curvature. Pick an example, as in (b), of a triangulation of N coming from a closed geodesic, and map 8.39 N as in (b). Consider the isotopy obtained by moving the vertex around the loop more and more. The loop stays the same, but the other line segments start spiraling Thurston — The Geometry and Topology of 3-Manifolds 195 8. KLEINIAN GROUPS around the loop, more and more, converging, in the limit, to a geodesic laminated set. The surface N maps into M at each ﬁnite stage, and this carries over in the limit to an isometric embedding of a hyperbolic surface. The triangles with an edge on the ﬁxed loop have disappeared in the limit. Compare 3.9. One can picture what is going on by looking upstairs at the convex hull H(LΓ). The lift ˜ f : ˜ N →H(LΓ) of the map from the original triangulation (before isotoping the vertex) is deﬁned as follows. First the geodesic (coming from the loop) and its conjugates are mapped in (these are in the convex hull since their 8.40 endpoints are in LΓ). The line segments connect diﬀerent conjugates of the ge-odesic, and the triangles either connect three distinct conjugates or two conjugates (when the original loop is an edge of the triangle). As we isotope the vertex around the loop, the image vertices slide along the geodesic (and its conjugates), and in the limit the triangles become asymptotic (and the triangles connecting only two conjugates disappear). The above method works because the complement of the geodesic lamination (obtained by spinning the triangulation) consists solely of asymptotic triangles. Here is a more general method of mapping in a surface N by using geodesic laminations. Definition 8.7.5. A geodesic lamination γ on hyperbolic surface S is complete if the complementary regions in S −γ are all asymptotic triangles. Proposition 8.7.6. Any geodesic lamination γ on a hyperbolic surface S can be completed, i.e., γ can be extended to a complete geodesic lamination γ′ ⊃γ on S. Proof. Suppose γ is not complete, and pick a complementary region A which is not an asymptotic triangle. If A is simply connected, then it is a ﬁnite-sided asymp-totic polygon, and it is easy to divide A into asymptotic triangles by adding simple geodesics. If A is not simply connected, extend γ to a larger geodesic lamination by adding a simple geodesic α in A 8.41 196 Thurston — The Geometry and Topology of 3-Manifolds 8.7. QUASI-FUCHSIAN GROUPS (being careful to add a simple geodesic). Either α separates A into two pieces (each of which has less area) or α does not separate A (in which case, cutting along α reduces the rank of the homology. Continuing inductively, after a ﬁnite number of steps A separates into asymptotic triangles. □ Completeness is exactly the property we need to map in surfaces by using geodesic laminations. Proposition 8.7.7. Let S be an oriented hyperbolic surface, and Γ a quasi-Fuchsian group isomorphic to π1S. For every complete geodesic lamination γ on S, there is a unique hyperbolic surface S′ ≈S and an isometric map f : S′ →MΓ which is straight (totally geodesic) in the complement of γ. (γ here denotes the cor-responding geodesic lamination on any hyperbolic surface homeomorphic to S.) Remark. By an isometric map f : M1 →M2 from one Riemannian manifold to another, we mean that for every rectiﬁable path αt in M1, f ◦αt is rectiﬁable and 8.42 has the same length as αt. When f is diﬀerentiable, this means that d f preserves lengths of tangent vectors. We shall be dealing with maps which are not usually diﬀerentiable, however. Our maps are likely not even to be local embeddings. A cross-section of the image of a surface mapped in by method (c) has two polygonal spiral branches, if the closed geodesic corresponds to a covering transformation which is not a pure translation: Thurston — The Geometry and Topology of 3-Manifolds 197 8. KLEINIAN GROUPS (This picture is obtained by considering triangles in H3 asymptotic to a loxo-dromic axis, together with their translates.) If the triangulation is spun in opposite directions on opposite sides of the geo-desic, the polygonal spiral have opposite senses, so there are inﬁnitely many self-intersections. 8.43 Proof. The hyperbolic surface ˜ S′ is constructed out of pieces. The asymptotic triangles in ˜ S −˜ γ are determined by triples of points on S1 ∞. We have a canonical identiﬁcation of S1 ∞with LΓ; the corresponding triple of points in LΓ spans a triangle in H3, which will be a piece of ˜ S′. Similarly, corresponding to each leaf of ˜ γ there is a canonical line in H3. These triangles and lines ﬁt together just as on ˜ S; from this the picture of ˜ S′ should be clear. Here is a formal deﬁnition. Let Pγ be the set of all “pieces” of ˜ γ, i.e., Pγ consists of all leaves of ˜ γ, together with all components of ˜ S −˜ γ. Let Pγ have the (non-Hausdorﬀ) quotient topology. The universal cover ˜ S′ 198 Thurston — The Geometry and Topology of 3-Manifolds 8.8. UNCRUMPLED SURFACES is deﬁned ﬁrst, to consist of ordered pairs (x, p), where p ∈Pγ and x is an element of the piece of H3 corresponding to p. Γ acts on this space ˜ S′ in an obvious way; the quotient space is deﬁned to be S′. It is not hard to ﬁnd local coordinates for S′, showing that it is a (Hausdorﬀ) surface. 8.44 An appeal to geometric intuition demonstrates that S′ is a hyperbolic surface, mapped isometrically to MΓ, straight in the complement of γ. Uniqueness is evident from consideration of the circle at ∞. □ Remark. There are two approaches which a reader who prefers more formal proofs may wish to check. The ﬁrst approach is to verify 8.7.7 ﬁrst for laminations all of whose leaves are either isolated or simple limits of other leaves (as in (c)), and then extend to all laminations by passing to limits, using compactness properties of uncrumpled surfaces (§8.8). Alternatively, he can construct the hyperbolic structure on S′ directly by describing the local developing map, as a limit of maps obtained by considering only ﬁnitely many local ﬂat pieces. Convergence is a consequence of the ﬁnite total area of the ﬂat pieces of S′. 8.8. Uncrumpled surfaces There is a large qualitative diﬀerence between a crumpled sheet of paper and one which is only wrinkled or crinkled. Crumpled paper has fold lines or bending lines going any which way, often converging in bad points. Thurston — The Geometry and Topology of 3-Manifolds 199 8. KLEINIAN GROUPS 8.45 Definition 8.8.1. An uncrumpled surface in a hyperbolic three-manifold N is a complete hyperbolic surface S of ﬁnite area, together with an isometric map f : S →N such that every x ∈S is in the interior of some straight line segment which is mapped by f to a straight line segment. Also, f must take every cusp of S to a cusp of N. The set of uncrumpled surfaces in N has a well-behaved topology, in which two surfaces f1 : S1 →N and f2 : S2 →N are close if there is an approximate isometry φ : S1 →S2 making f1 uniformly close to f2 ◦φ. Note that the surfaces have no preferred coordinate systems. Let γ ⊂S consist of those points in the uncrumpled surfaces which are in the interior of unique line segments mapped to line segments. Proposition 8.8.2. γ is a geodesic lamination. The map f is totally geodesic in the complement of γ. Proof. If x ∈S −γ, then there are two transverse line segments through x mapped to line segments. Consider any quadrilateral about x with vertices on these segments; since f does not increase distances, the quadrilateral must be mapped to a plane. Hence, a neighborhood of x is mapped to a plane. 8.46 200 Thurston — The Geometry and Topology of 3-Manifolds 8.8. UNCRUMPLED SURFACES Consider now any point x ∈γ, and let α be the unique line segment through x which is mapped straight. Let α be extended indeﬁnitely on S. Suppose there were some point y on α in the interior of some line segment β ̸⊂α which is mapped straight. One may assume that the segment xy of α is mapped straight. Then, by considering long skinny triangles with two vertices on β and one vertex on α, it would follow that a neighborhood of x is mapped to a plane—a contradiction. Thus, the line segments in γ can be extended indeﬁnitely without crossings, so γ must be a geodesic lamination. □ If U = S f − →N is an uncrumpled surface, then this geodesic lamination γ ⊂S (which consists of points where U is not locally ﬂat) is the wrinkling locus ω(U). The modular space M(S) of a surface S of negative Euler characteristic is the space of hyperbolic surfaces with ﬁnite area which are homeomorphic to S. In other words, M(S) is the Teichm¨ uller space T(S) modulo the action of the group of home-8.47 omorphisms of S. Proposition 8.8.3 (Mumford). For a surface S, the set Aϵ ⊂M(S) consisting of surfaces with no geodesic shorter than ϵ is compact. Proof. By the Gauss–Bonnet theorem, all surfaces in M(S) have the same area. Every non-compact component of S(0,ϵ] is isometric to a standard model, so the result follows as the two-dimensional version of a part of 5.12. (It is also not hard to give a more direct speciﬁcally two-dimensional geometric argument.) □ Denote by U(S, N) the space of uncrumpled surfaces in N homeomorphic to S with π1(S) →π1(N) injective. There is a continuous map U(S, N) →M(S) which forgets the isometric map to N. The behavior of an uncrumpled surface near a cusp is completely determined by its behavior on some compact subset. To see this, ﬁrst let us prove Proposition 8.8.4. There is some ϵ such that for every hyperbolic surface S and every geodesic lamination γ on S, the intersection of γ with every non-compact component of S(0,ϵ] consists of lines tending toward that cusp. Thurston — The Geometry and Topology of 3-Manifolds 201 8. KLEINIAN GROUPS 8.48 Proof. Thus there are uniform horoball neighborhoods of the cusps of uncrum-pled surfaces which are always mapped as cones to the cusp point. Uniform con-vergence of a sequence of uncrumpled surfaces away from the cusp points implies uniform convergence elsewhere. □ Proposition 8.8.5. Let K ⊂N be a compact subset of a complete hyperbolic manifold N. For any surface S0, let W ⊂U(S0, N) be the subset of uncrumpled surfaces S f − →N such that f(S) intersects K, and satisfying the condition (np) π1(f) takes non-parabolic elements of π1S to non-parabolic elements of π1N. Then W is compact. Proof. The ﬁrst step is to bound the image of an uncrumpled surface, away from its cusps. Let ϵ be small enough that for every complete hyperbolic three-manifold M, components of M(0,ϵ] are separated by a distance of at least (say) 1. Since the area of surfaces in U(S0, N) is constant, there is some number d such that any two points in an uncrumpled surface S can be connected (on S) by a path p such that p ∩S[ϵ,∞) has length ≤d. If neither point lies in a non-compact component of S(0,ϵ], one can assume, further-more, that p does not intersect these components. Let K′ ⊂N be the set of points which are connected to K by paths whose total length outside compact components of N(0,ϵ] is bounded by d. Clearly K′ is compact and an uncrumpled surface of W must have image in K′, except for horoball neighborhoods of its cusps. 8.49 Consider now any sequence S1, S2, . . . in W. Since each short closed geodesic in Si is mapped into K′, there is a lower bound ϵ′ to the length of such a geodesic, so by 8.8.3 we can pass to a subsequence such that the underlying hyperbolic surfaces converge in M(S). There are approximate isometries φi : S →Si. Then the compositions fi◦φi : S →N are equicontinuous, hence there is a subsequence converging uniformly on S[ϵ,∞). The limit is obviously an uncrumpled surface. [To make the picture 202 Thurston — The Geometry and Topology of 3-Manifolds 8.8. UNCRUMPLED SURFACES clear, one can always pass to a further subsequence to make sure that the wrinkling laminations γi of Si converge geometrically.] □ Corollary 8.8.6. (a) Let S be any closed hyperbolic surface, and N any closed hyperbolic manifold. There are only ﬁnitely many conjugacy classes of subgroups G ⊂π1N isomorphic to π1S. (b) Let S be any surface of ﬁnite area and N any geometrically ﬁnite hyperbolic three-manifold. There are only ﬁnitely many conjugacy classes of subgroups G ⊂π1N isomorphic to π1S by an isomorphism which preserves parabolicity (in both directions). Proof. Statement (a) is contained in statement (b). The conjugacy class of every subgroup G is represented by a homotopy class of maps of S into N, which is homotopic to an uncrumpled surface (say, by method (c) of §8.7). Nearby uncrumpled surfaces represent the same conjugacy class of subgroups. Thus we have an open cover of the space W by surfaces with conjugate subgroups; by 8.8.5, this is a ﬁnite subcover. □ 8.50 Remark. If non-parabolic elements of π1S are allowed to correspond to parabolic elements of π1N, then this statement is no longer true. In fact, if S f − →N is any surface mapped into a hyperbolic manifold N of ﬁnite volume such that a non-peripheral simple closed curve γ in S is homotopic to a cusp of N, one can modify f in a small neighborhood of γ to wrap this annulus a number of times around the cusp. This is likely to give inﬁnitely many homotopy classes of surfaces in N. In place of 8.8.5, there is a compactness statement in the topology of geometric convergence provided each component of S[ϵ,∞) is required to intersect K. One would allow S to converge to a surface where a simple closed geodesic is pinched to yield a pair of cusps. From this, one deduces that there are ﬁnitely many classes of groups G Thurston — The Geometry and Topology of 3-Manifolds 203 8. KLEINIAN GROUPS isomorphic to S up to the operations of conjugacy, and wrapping a surface carrying G around cusps. Haken proved a ﬁniteness statement analogous to 8.8.6 for embedded incompress-ible surfaces in atoroidal Haken manifolds. 8.51 8.9. The structure of geodesic laminations: train tracks Since a geodesic lamination γ on a hyperbolic surface S has measure zero, one can picture γ as consisting of many parallel strands in thin, branching corridors of S which have small total area. Imagine squeezing the nearly parallel strands of γ in each corridor to a single strand. One obtains a train track τ (with switches) which approximates γ. Each leaf of γ may be imagined as the path of a train running around along τ. 8.52 Here is a construction which gives a precise and nice sequence of train track approximations of γ. Consider a complementary region R in S −γ. The double dR is a hyperbolic surface of ﬁnite area, so (dR)(0,2ϵ] has a simple structure: it consists of neighborhoods of geodesics shorter than 2ϵ and of cusps. In each such neighborhood there is a canonical foliation by curves of constant curvature: horocycles about a cusp or equidistant curves about a short geodesic. Transfer this foliation to R, and then 204 Thurston — The Geometry and Topology of 3-Manifolds 8.9. THE STRUCTURE OF GEODESIC LAMINATIONS: TRAIN TRACKS to S. This yields a foliation F in the subset of S where leaves of γ are not farther than 2ϵ apart. (A local vector ﬁeld tangent to F is Lipschitz, so it is integrable; this is why F exists. If γ has no leaves tending toward a cusp, then we can make all the leaves of F be arbitrarily short arcs by making ϵ suﬃciently small. If γ has leaves tending toward a cusp, then there can be only ﬁnitely many such leaves, since there is an upper bound to the total number of cusps of the complementary regions. Erase all parts of F in a cusp of a region tending toward a cusp of S; again, when ϵ is suﬃciently small all leaves of F will be short arcs. The space obtained by collapsing all arcs of F to a point is a surface S′ homeomorphic to S, and the image of γ is a train track τϵ on S′. Observe that each switch of τϵ comes from a boundary component of some dR(0,2ϵ]. In particular, there is a uniform bound to the number of switches. From this it is easy to see that there are only ﬁnitely many possible types of τϵ, up to homeomorphisms of S′ (not necessarily homotopic to the identity). 8.53 In working with actual geodesic laminations, it is better to use more arbitrary train track approximations, and simply sketch pictures; the train tracks are analogous to decimal approximations of real numbers. Here is a deﬁnition of a useful class of train tracks. Definitions 8.9.1. A train track on a diﬀerentiable surface S is an embedded graph τ on S. The edges (branch lines) of τ must be C1, and all edges at a given vertex (switch) must be tangent. If S has “cusps”, τ may have open edges tending toward the cusps. Dead ends are not permitted. (Each vertex v must be in the interior of a C1 interval on τ through v.) Furthermore, for each component R of S −τ, the double dR of R along the interiors of edges of ∂R must have negative Euler characteristic. A lamination γ on S is carried by τ if there is a diﬀerentiable map f : S →S homotopic to the identity taking γ to τ and non-singular on the Thurston — The Geometry and Topology of 3-Manifolds 205 8. KLEINIAN GROUPS tangent spaces of the leaves of γ. (In other words, the leaves of γ are trains running 8.54 around on τ.) The lamination γ is compatible with τ if τ can be enlarged to a train track τ ′ which carries γ. Proposition 8.9.2. Let S be a hyperbolic surface, and let δ > 0 be arbitrary. There is some ϵ > 0 such that for all geodesic laminations γ of S, the train track approximation τϵ can be realized on S in such a way that all branch lines τϵ are C2 curves with curvature < δ. Proof. Note ﬁrst that by making ϵ suﬃciently small, one can make the leaves of the foliation F very short, uniformly for all γ: otherwise there would be a sequence of γ’s converging to a geodesic lamination containing an open set. [One can also see this directly from area considerations.] When all branches of τϵ are reasonably long, one can simply choose the tangent vectors to the switches to be tangent to any geodesic of γ where it crosses the corresponding leaf of F; the branches can be ﬁlled in by curves of small curvature. When some of the branch lines are short, group each set of switches connected by very short branch lines together. First map each of these sets into S, then extend over the reasonably long branches. □ Corollary 8.9.3. Every geodesic lamination which is carried by a close train track approximation τϵ to a geodesic lamination γ has all leaves close to leaves of γ. Proof. This follows from the elementary geometrical fact that a curve in hyper-bolic space with uniformly small curvature is uniformly close to a unique geodesic. (One way to see this is by considering the planes perpendicular to the curve—they always advance at a uniform rate, so in particular the curve crosses each one only once.) 8.55 □ Proposition 8.9.4. A lamination λ of a surface S is isotopic to a geodesic lam-ination if and only if (a) λ is carried by some train track τ, and (b) no two leaves of λ take the same (bi-inﬁnite) path on τ. 206 Thurston — The Geometry and Topology of 3-Manifolds 8.9. THE STRUCTURE OF GEODESIC LAMINATIONS: TRAIN TRACKS Proof. Given an arbitrary train track τ, it is easy to construct some hyperbolic structure for S on which τ is realized by lines with small curvature. The leaves of λ then correspond to a set of geodesics on S, near τ. These geodesics do not cross, since the leaves of λ do not. Condition (b) means that distinct leaves of λ determine distinct geodesics. When leaves of λ are close, they must follow the same path for a long ﬁnite interval, which implies the corresponding geodesics are close. Thus, we obtain a geodesic lamination γ which is isotopic to λ. (To have an isotopy, it suﬃces to construct a homeomorphism homotopic to the identity. This homeomorphism is constructed ﬁrst in a neighborhood of τ, then on the rest of S.) □ Remark. From this, one sees that as the hyperbolic structure on S varies, the corresponding geodesic laminations are all isotopic. This issue was quietly skirted in §8.5. When a lamination λ has an invariant measure µ, this gives a way to associate a number µ(b) to each branch line b of any train track which dominates γ: µ(b) is just 8.56 the transverse measure of the leaves of λ collapsed to a point on b. At a switch, the sum of the “entering” numbers equals the sum of the “exiting” numbers. Conversely, any assignment of numbers satisfying the switch condition determines a unique geodesic lamination with transverse measure: ﬁrst widen each branch line b of τ to a corridor of constant width µ(b), and give it a foliation G by equally spaced lines. Thurston — The Geometry and Topology of 3-Manifolds 207 8. KLEINIAN GROUPS 8.57 As in 8.9.4, this determines a lamination γ; possibly there are many leaves of G collapsed to a single leaf of γ, if these leaves of G all have the same inﬁnite path. G has a transverse measure, deﬁned by the distance between leaves; this goes over to a transverse measure for γ. 8.10. Realizing laminations in three-manifolds For a quasi-Fuchsian group Γ, it was relatively easy to “realize” a geodesic lam-ination of the corresponding surface in MΓ, by using the circle at inﬁnity. However, not every complete hyperbolic three-manifold whose fundamental group is isomorphic to a surface group is quasi-Fuchsian, so we must make a more systematic study of realizability of geodesic laminations. Definition 8.10.1. Let f : S →N be a map of a hyperbolic surface to a hyperbolic three-manifold which sends cusps to cusps. A geodesic lamination γ on S is realizable in the homotopy class of f if f is homotopic (by a cusp-preserving homotopy) to a map sending each leaf of γ to a geodesic. Proposition 8.10.2. If γ is realizable in the homotopy class of f, the realization is (essentially) unique: that is, the image of each leaf of γ is uniquely determined. Proof. Consider a lift ˜ h of a homotopy connecting two maps of S into N. If S is closed, ˜ h moves every point a bounded distance, so it can’t move a geodesic to a diﬀerent geodesic. If S has cusps, the homotopy can be modiﬁed near the cusps of S so ˜ h again is bounded. □ 8.58 In Section 8.5, we touched on the notion of geometric convergence of geodesic laminations. The geometric topology on geodesic laminations is the topology of geo-metric convergence, that is, a neighborhood of γ consists of laminations γ′ which 208 Thurston — The Geometry and Topology of 3-Manifolds 8.10. REALIZING LAMINATIONS IN THREE-MANIFOLDS have leaves near every point of γ, and nearly parallel to the leaves of γ. If γ1 and γ2 are disjoint simple closed curves, then γ1 ∪γ2 is in every neighborhood of γ1 as well as in every neighborhood of γ2. The space of geodesic laminations on S with the geometric topology we shall denote GL. The geodesic laminations compatible with train track approximations of γ give a neighborhood basis for γ. The measure topology on geodesic laminations with transverse measures (of full support) is the topology induced from the weak topology on measures in the M¨ obius band J outside S∞in the Klein model. That is, a neighborhood of (γ, µ) consists of (γ′, µ′) such that for a ﬁnite set f1, . . . , fk of continuous functions with compact support in J, Z fi dµ − Z fi dµ′ < ϵ. This can also be interpreted in terms of integrating ﬁnitely many continuous functions on ﬁnitely many transverse arcs. Let ML(S) be the space of (γ, µ) on S with the measure topology. Let PL(S) be ML(S) modulo the relation (γ, µ) ∼(γ, aµ) where a > 0 is a real number. Proposition 8.10.3. The natural map ML →GL is continuous. 8.59 Proposition 8.10.4. The map w : U(S, N) →GL(S) which assigns to each uncrumpled surface its wrinkling locus is continuous. Proof of 8.10.3. For any point x in the support of a measure µ and any neigh-borhood UU of x, the support of a measure close enough to µ must intersect U. □ Proof of 8.10.4. An interval which is bent cannot suddenly become straight. Away from any cusps, there is a positive inﬁmum to the “amount” of bending of an interval of length ϵ which intersects the wrinkling locus w(S) in its middle third, and makes an angle of at least ϵ with w(S). (The “amount” of bending can be measure, say, by the diﬀerent between the length of α and the distance between the image endpoints.) All such arcs must still cross w(S′) for any nearby uncrumpled surface S′. □ When S has cusps, we are also interested in measures supported on compact geodesic laminations. We denote this space by ML0(S). If (τ, µ) is a train track description for (γ, µ), where µ(b) ̸= 0 for any branch of τ, then neighborhoods for (γ, µ) are described by (τ ′, µ′) , where τ ⊂τ ′ and µ(b) −µ′(b) < ϵ. (If b is a branch of τ ′ not in τ, then µ(b) = 0 by deﬁnition.) In fact, one can always choose a hyperbolic structure on S so that τ is a good approximation to γ. If S is closed, it is always possible to squeeze branches of τ Thurston — The Geometry and Topology of 3-Manifolds 209 8. KLEINIAN GROUPS together along non-trivial arcs in the complementary regions to obtain a new train track which cannot be enlarged. 8.60 This implies that a neighborhood of (γ, µ) is parametrized by a ﬁnite number of real parameters. Thus, ML(S) is a manifold. Similarly, when S has cusps, ML(S) is a manifold with boundary ML0(S). Proposition 8.10.5. GL(S) is compact, and PL(S) is a compact manifold with boundary PL0(S) if S is not compact. Proof. There is a ﬁnite set of train tracks τ1, . . . , τk carrying every possible geodesic lamination. (There is an upper bound to the length of a compact branch of τϵ, when S and ϵ are ﬁxed.) The set of projective classes of measures on any particular τ is obviously compact, so this implies PL(S) is compact. That PL(S) is a manifold follows from the preceding remarks. Later we shall see that in fact it is the simplest of possible manifolds. In 8.5, we indicated one proof of the compactness of GL(S). Another proof goes 8.61 as follows. First, note that Proposition 8.10.6. Every geodesic lamination γ admits some transverse mea-sure µ (possibly with smaller support). Proof. Choose a ﬁnite set of transversals α1, . . . , αk which meet every leaf of γ. Suppose there is a sequence {li} of intervals on leaves of γ such that the total number N1 of intersection of li with the αj’s goes to inﬁnity. Let µi be the measure on S αj which is 1/Ni times the counting measure on li ∩αj. The sequence {µi} has a subsequence converging (in the weak topology) to a measure µ. It is easy to see that µ is invariant under local projections along leaves of γ, so that it determines a transverse measure. 210 Thurston — The Geometry and Topology of 3-Manifolds 8.10. REALIZING LAMINATIONS IN THREE-MANIFOLDS If there is no such sequence {li} then every leaf is proper, so the counting measure for any leaf will do. □ We continue with the proof of 8.10.5. Because of the preceding result, the image I of PL(S) in GL(S) intersects the closure of every point of GL(S). Any collection of open sets which covers GL(S) has a ﬁnite subcollection which covers the compact set I; therefore, it covers all of GL(S). □ Armed with topology, we return to the question of realizing geodesic laminations. Let Rf ⊂GL(S) consist of the laminations realizable in the homotopy class of f. First, if γ consists of ﬁnitely many simple closed geodesics, then γ is realizable 8.62 provided π1(f) maps each of these simple closed curves to non-trivial, non-parabolic elements. If we add ﬁnitely many geodesics whose ends spiral around these closed geodesics or converge toward cusps the resulting lamination is also realizable except in the degenerate case that f restricted to an appropriate non-trivial pair of pants on S factors through a map to S1. To see this, consider for instance the case of a geodesic g on S whose ends spiral around closed geodesics g1 and g2. Lifting f to H3, we see that the two ends of ˜ f(˜ g) are asymptotic to geodesics ˜ f(˜ g1) and ˜ f(˜ g2). Then f is homotopic to a map taking g to a geodesic unless ˜ f(˜ g1) and ˜ f(˜ g2) converge to the same point on S∞, which can only happen if ˜ f(˜ g1) = ˜ f(˜ g2) (by 5.3.2). In this case, f is homotopic to a map taking a neighborhood of g ∪g1 ∪g2 to f(g1) = f(g2). 8.63 Thurston — The Geometry and Topology of 3-Manifolds 211 8. KLEINIAN GROUPS The situation is similar when the ends of g tend toward cusps. These realizations of laminations with ﬁnitely many leaves take on signiﬁcance in view of the next result: Proposition 8.10.7. (a) Measures supported on ﬁnitely many compact or proper geodesics are dense in ML. (b) Geodesic laminations with ﬁnitely many leaves are dense in GL. (c) Each end of a non-compact leaf of a geodesic lamination with only ﬁnitely many leaves spirals around some closed geodesic, or tends toward a cusp. Proof. If τ is any train track and µ is any measure which is positive on each branch, µ can be approximated by measures µ′ which are rational on each branch, since µ is subject only to linear equations with integer coeﬃcients. µ′ gives rise to geodesic laminations with only ﬁnitely many leaves, all compact or proper. This 8.64 proves (a). If γ is an arbitrary geodesic lamination, let τ be a close train track approximation of γ and proceed as follows. Let τ ′ ⊂τ consist of all branches b of τ such that there exists either a cyclic (repeating) train route or a proper train route through b. 212 Thurston — The Geometry and Topology of 3-Manifolds 8.10. REALIZING LAMINATIONS IN THREE-MANIFOLDS (The reader experienced with toy trains is aware of the subtlety of this question.) There is a measure supported on τ ′, obtained by choosing a ﬁnite set of cyclic and proper paths covering τ ′ and assigning to a branch b the total number of times these paths traverse. Thus there is a lamination λ′ consisting of ﬁnitely many compact or proper leaves supported in a narrow corridor about τ ′. Now let b be any branch of τ −τ ′. A train starting on b can continue indeﬁnitely, so it must eventually come to τ ′, in each direction. Add a leaf to λ′ representing a shortest path from b to τ ′ in each direction; if the two ends meet, make them run along side by side (to avoid 8.65 crossings). When the ends approach τ, make them “merge”—either spiral around a closed leaf, or follow along close to a proper leaf. Continue inductively in this way, adding leaves one by one until you obtain a lamination λ dominated by τ and ﬁlling out all the branches. This proves (b). Thurston — The Geometry and Topology of 3-Manifolds 213 8. KLEINIAN GROUPS If γ is any geodesic lamination with ﬁnitely many (or even countably many) leaves, then the only possible minimal sets are closed leaves; thus each end e of a non-compact must either be a proper end or come arbitrarily near some compact leaf l. By tracing the leaves near l once around l, it is easy to see that this means e spirals around l. □ 8.66 Thus, if f is non-degenerate, Rf is dense. Furthermore, Theorem 8.10.8. If π1f is injective, and f satisﬁes (np) (that is, if π1f preserves non-parabolicity), then Rf is an open dense subset of GL(S). Proof. If γ is any complete geodesic lamination which is realizable, then a train track approximation τ can be constructed for the image of γ in N 3, in such a way that all branch lines have curvature close to 0. Then all laminations carried by τ are also realizable; they form a neighborhood of γ. Next we will show that any enlargement γ′ ⊃γ of a realizable geodesic lamination γ is also realizable. First note that if γ′ is obtained by adding a single leaf l to γ, then γ′ is also realizable. This is proved in the same way as in the case of a lamination with ﬁnitely many leaves: note that each end of l is asymptotic to a leaf of γ. (You can see this by considering S −γ.) If f is 8.67 homotoped so that f(γ) consists of geodesics, then both ends of ˜ f(l) are asymptotic to geodesics in ˜ f(γ). If the two endpoints were not distinct on S∞, this would imply the existence of some non-trivial identiﬁcation of γ by f so that π1f could not be injective. 214 Thurston — The Geometry and Topology of 3-Manifolds 8.10. REALIZING LAMINATIONS IN THREE-MANIFOLDS By adding ﬁnitely many leaves to any geodesic lamination γ′ we can complete it. This implies that γ′ is contained in the wrinkling locus of some uncrumpled surface. By 8.8.5 and 8.10.1, the set of uncrumpled surfaces whose wrinkling locus contains γ is compact. Since the wrinkling locus depends continuously on an uncrumpled surface, the set of γ′ ∈Rf which contains γ is compact. But any γ′ ⊃γ can be approximated by laminations such that γ′ −γ consists of a ﬁnite number of leaves. This actually follows from 8.10.7, applied to d(S −γ). Therefore, every enlargement γ′ ⊃γ is in Rf. 8.68 Since the set of uncrumpled surfaces whose wrinkling locus contains γ is compact, there is a ﬁnite set of train tracks τ1, . . . , τk such that for any such surface, w(S) is closely approximated by one of τ1, . . . , τk. The set of all laminations carried by at least one of the τi is a neighborhood of γ contained in Rf. □ Corollary 8.10.9. Let Γ be a geometrically ﬁnite group, and let f : S →NΓ be a map as in 8.10.8. Then either Rf = GL(S) ( that is, all geodesic laminations are realizable in the homotopy class of f), or Γ has a subgroup Γ′ of ﬁnite index such that NΓ′ is a three-manifold with ﬁnite volume which ﬁbers over the circle. Conjecture 8.10.10. If f : S →N is any map from a hyperbolic surface to a complete hyperbolic three-manifold taking cusps to cusps, then the image π1(f)(π1(S)) is quasi-Fuchsian if and only if Rf = GL(S). Proof of 8.10.9. Under the hypotheses, the set of uncrumpled surfaces homo-topic to f(S) is compact. If each such surface has an essentially unique homotopy to f(S), so that the wrinkling locus on S is well-deﬁned, then the set of wrinkling loci of uncrumpled surfaces homotopic to f is compact, so by 8.10.8 it is all of GL(S). Otherwise, there is some non-trivial ht : S →M such that h1 = h0 ◦φ, where φ : S →S is a homotopically non-trivial diﬀeomorphism. It may happen that φ has Thurston — The Geometry and Topology of 3-Manifolds 215 8. KLEINIAN GROUPS ﬁnite order up to isotopy, as when S is a ﬁnite regular covering of another surface in M. The set of all isotopy classes of diﬀeomorphisms φ which arise in this way form a group. If the group is ﬁnite, then as in the previous case, RF = GL(S). Otherwise, there is a torsion-free subgroup of ﬁnite index (see ), so there is an element φ of inﬁnite order. The maps f and φ ◦f are conjugate in Γ, by some element β ∈Γ. The group generated by β and f(π1S) is the fundamental group of a three-manifold which ﬁbers over S1. □ 8.69 We shall see some justiﬁcation for the conjecture in the remaining sections of chapter 8 and in chapter 9: we will prove it under certain circumstances. 8.11. The structure of cusps Consider a hyperbolic manifold N which admits a map f : S →N, taking cusps to cusps such that π1(f) is an isomorphism, where S is a hyperbolic surface. Let B ⊂N be the union of the components of N(0,ϵ] corresponding to cusps of S. f is a relative homotopy equivalence from (S, S(0,ϵ)) to (N, B), so there is a homotopy inverse g : (N, B) →(S, S(0,ϵ)). If X ∈S(ϵ,∞) is a regular value for g, then g−1(x) is a one-manifold having intersection number one with f(S), so it has at least one component homeomorphic to R, going out toward inﬁnity in N −B on opposite sides of f(S). Therefore there is a proper function h : (N −B) →R with h restricted to g−1(x) a surjective map. One can modify h so that h−1(0) is an incompressible surface. Since g restricted to h−1(0) is a degree one map to S, it must map the fundamental group surjectively as well as injectively, so h−1(0) is homeomorphic to S. h−1(0) divides N −B into two components N+ and N−with π1N = π1N+ = π1N−= π1S. We can assume that h−1(0) does not intersect N(0,ϵ] except in B (say, by shrinking ϵ). Suppose that N has parabolic elements that are not parabolic on S. The structure 8.71 of the extra cusps of N is described by the following: Proposition 8.11.1. There are geodesic laminations γ+ and γ−on S with all leaves compact (i.e., they are ﬁnite collections of disjoint simple closed curves) such that the extra cusps in Ne correspond one-to-one with leaves of γe(e = +, −). In particular, for any element α ∈π1(S), π1(f)(α) is parabolic if and only if α is freely homotopic to a cusp of S or to a simple closed curve in γ+ or γ−. Proof. We need consider only one half, say N+. For each extra cusp of N+, there is a half-open cylinder mapped into N+, with one end on h−1(0) and the other end tending toward the cusp. Furthermore, we can assume that the union of these cylinders is embedded outside a compact set, since we understand the picture in a neighborhood of the cusps. Homotope the ends of the cylinders which lie on h−1(0) 216 Thurston — The Geometry and Topology of 3-Manifolds 8.11. THE STRUCTURE OF CUSPS so they are geodesics in some hyperbolic structure on h−1(0). One can assume the cylinders are immersed (since maps of surfaces into three-manifolds are appoximable by immersions) and that they are transverse to themselves and to one another. If there are any self-intersections of the cylinders on h−1(0), there must be a double line which begins and ends on h−1(0). Consider the picture in ˜ N: there are two translates of universal covers of cylinders which meet in a double line, so that in particular their bounding lines meet twice on h−1(0). This contradicts the fact that they are geodesics in some hyperbolic structure. □ 8.71 It actually follows that the collection of cylinders joining simple closed curves to the cusps can be embedded: we can modify g so that it takes each of the extra cusps to a neighborhood of the appropriate simple closed curve α ⊂γϵ, and then do surgery to make g−1(α) incompressible. Thurston — The Geometry and Topology of 3-Manifolds 217 8. KLEINIAN GROUPS 8.72 To study N, we can replace S by various surfaces obtained by cutting S along curves in γ+ or γ−. Let P be the union of open horoball neighborhoods of all the cusps of N. Let {Si} be the set of all components of S cut by γ+ together with those of S cut by γ−. The union of the Si can be embedded in N −P, with boundary on ∂P, within the convex hull M of N, so that they cut oﬀa compact piece N0 ⊂N −P homotopy equivalent to N, and non-compact ends Ei of N −P, with ∂Ei ⊂P ∪Si. Let N now be an arbitrary hyperbolic manifold, and let P be the union of open horoball neighborhoods of its cusps. The picture of the structure of the cusps readily generalizes provided N −P is homotopy equivalent to a compact submanifold NO, obtained by cutting N −P along ﬁnitely many incompressible surfaces {Si} with boundary ∂P. Applying 8.11.1 to covering spaces of N corresponding to the Si (or applying its proof directly), one can modify the Si until no non-peripheral element of one of the Si is homotopic, outside NO, to a cusp. When this is done, the ends {Ei} of N −P are in one-to-one correspondence with the Si. According to a theorem of Peter Scott, every three-manifold with ﬁnitely gen-erated fundamental group is homotopy equivalent to a compact submanifold. In general, such a submanifold will not have incompressible boundary, so it is not as well behaved. We will leave this case for future consideration. 8.73 Definition 8.11.2. Let N be a complete hyperbolic manifold, P the union of open horoball neighborhoods of its cusps, and M the convex hull of N. Suppose 218 Thurston — The Geometry and Topology of 3-Manifolds 8.12. HARMONIC FUNCTIONS AND ERGODICITY E is an end of N −P, with ∂E −∂P an incompressible surface S ⊂M homotopy equivalent to E. Then E is a geometrically tame end if either (a) E ∩M is compact, or (b) the set of uncrumpled surfaces S′ homotopic to S and with S′ [ϵ,∞) contained in E is not compact. If N has a compact submanifold NO of N −P homotopy equivalent to N such that N −P −NO is a disjoint union of geometrically tame ends, then N and π1N are geometrically tame. (These deﬁnitions will be extended in § ). We shall justify this deﬁnition by showing geometric tameness implies that N is analytically, topologically and geometrically well-behaved. 8.12. Harmonic functions and ergodicity Let N be a complete Riemannian manifold, and h a positive function on N. Let φt be the ﬂow generated by −(grad h). The integral of the velocity of φt is bounded along any ﬂow line: Z φT (x) x ∥grad h∥ds = h(x) −h(φT(x)) ≤h(x) ( for T > 0). If A is a subset of a ﬂow line {φt(x)}t≥0 of ﬁnite length l(A), then by the Schwarz inequality 8.74 Labelled this 8.12.1.eq 8.12.1. T(A) = Z A 1 ∥grad h∥ds ≥ l(A)2 R A ∥grad h∥ds ≥l(A)2 h(x) where T(A) is the total time the ﬂow line spends in A. Note in particular that this implies φt(x) is deﬁned for all positive time t (although φt may not be surjective). The ﬂow lines of φt are moving very slowly for most of their length. If h is harmonic, then the ﬂow φt preserves volume: this means that if it is not altogether stagnant, it must ﬂow along a channel that grows very wide. A river, with elevation h, is a good image. It is scaled so grad h is small. Suppose that N is a hyperbolic manifold, and S f →N is an uncrumpled surface in N, so that it has area −2πχ(S). Let a be a ﬁxed constant, suppose also that S has no loops of length ≤a which are null-homotopic in N. Proposition 8.12.1. There is a constant C depending only on a such that the volume of N1(f(S)) is not greater than −C · χ(S). (N1 denotes the neighborhood of radius 1.) Thurston — The Geometry and Topology of 3-Manifolds 219 8. KLEINIAN GROUPS Proof. For each point x ∈S, let cx be the “characteristic function” of an im-mersed hyperbolic ball of radius 1 + a/2 centered at f(x). In other words, cx(y) is the number of distinct homotopy classes of paths from x to y of length ≤1 + a/2. Let g be deﬁned by integrating cx over S; in other words, for y ∈N, 8.75 g(y) = Z S cx(y) dA. If v(Br) is the volume of a ball of radius r in H3, then Z N g dV = −2πχ(S) v(B1+a/2). For each point y ∈N1(f(S)), there is a point x with d(fx, y) ≤1, so that there is a contribution to g(y) for every point z on S with d(z, y) ≤a/2, and for each homotopy class of paths on S between z and x of length ≤a/2. Thus g(y) is at least as great as the area A(Ba/2) of a ball in H2 of radius a/2, so that v(N1(f(S))) ≤ 1 A(Ba/2) Z N g dV ≤−C · χ(S). □ As a →0, the best constant C goes to ∞, since one can construct uncrumpled surfaces with long thin waists, whose neighborhoods have very large volume. 8.76 Theorem 8.12.3. If N is geometrically tame, then for every non-constant positive harmonic function h on its convex hull M, inf M h = inf ∂M h. This inequality still holds if h is only a positive superharmonic function, i.e., if ∆h = div grad h ≤0. 220 Thurston — The Geometry and Topology of 3-Manifolds 8.12. HARMONIC FUNCTIONS AND ERGODICITY Corollary 8.12.4. If Γ = π1N, where N is geometrically tame, then LΓ has measure 0 or 1. In the latter case, Γ acts ergodically on S2. Proof of Corollary from theorem. This is similar to 8.4.2. Consider any invariant measurable set A ⊂LΓ, and let h be the harmonic extension of the char-acteristic function of A. Since A is invariant, h deﬁnes a harmonic function, also h, on N. If LΓ = S2, then by 8.12.3 h is constant, so A has measure 0 to 1. If LΓ ̸= S2 then the inﬁmum of (1 −h) is the inﬁmum on ∂M, so it is ≥1 2. This implies A has measure 0. This completes the proof of 8.12.4. □ Theorem 8.12.3 also implies that when LΓ = S2, the geodesic ﬂow for N is ergodic. We shall give this proof in § , since the ergodicity of the geodesic ﬂow is useful for the proof of Mostow’s theorem and generalizations. Proof of 8.12.3. The idea is that all the uncrumpled surfaces in M are nar-rows, which allow a high ﬂow rate only at high velocities. In view of 8.12.1, most of 8.77 the water is forced oﬀM—in other words, ∂M is low. Let P be the union of horoball neighborhoods of the cusps of N, and {Si} incom-pressible surfaces cutting N −P into a compact piece N0 and ends {Ei}. Observe that each component of P has two boundary components of ∪Si. In each end Ei which does not have a compact intersection with M, there is a sequence of uncrumpled maps fi,j : Si →Ei ∪P moving out of all compact sets in Ei ∪P, by 8.8.5. Combine these maps into one sequence of maps fj : ∪Si →M. Note that fj maps P[Si] to a cycle which bounds a (unique) chain Cj of ﬁnite volume, and that the supports of the Cj’s eventually exhaust M. If there are no cusps, then there is a subsequence of the fi whose images are disjoint, separated by distances of at least 2. If there are cusps, modify the cycles fj(P[Si]) by cutting them along horospherical cylinders in the cusps, and replacing the cusps of surfaces by cycles on these horospherical cylinders. Thurston — The Geometry and Topology of 3-Manifolds 221 8. KLEINIAN GROUPS 8.78 If the horospherical cylinders are suﬃciently close to ∞, the resulting cycle Zj will have area close to that of fj P[Si], less than, say, 2π P \|χ(Si)\| + 1. Zj bounds a chain Cj with compact support. We may assume that the support of Zj+1 does not intersect N2 (support Cj). From 8.3.2, it follows that there is a constant K such that for all j, v(N1(support Zj)) ≤K. If x ∈M is any regular point for h, then a small enough ball B about x is disjoint from φ1(B). To prove the theorem, it suﬃces to show that almost every ﬂow line through B eventually leaves M. Note that all the images {φi(B)}i∈N are disjoint. Since φt does not decrease volume, almost all ﬂow lines through B eventually leave the supports of all the Cj. If such a ﬂow line does not cross ∂M, it must cross Zj, hence it intersects N1(support Zj) with length at least two. By 8.12.1, the total length of time such a ﬂow line spends in J [ j=1 N1(support Zj) grows as J2. Since the volume of J [ j=1 N1(support Zj) grows only as K · J, no set of positive measure of ﬂow lines through B will ﬁt—most have to run oﬀthe edge of M. □ 222 Thurston — The Geometry and Topology of 3-Manifolds 8.12. HARMONIC FUNCTIONS AND ERGODICITY Remark. The fact that the area of Zj is constant is stronger than necessary to obtain the conclusion of 8.3.3. It would suﬃce for the sum of reciprocals of the areas to form a divergent series. Thus, R2 has no non-constant positive superharmonic function, although R3 has. Thurston — The Geometry and Topology of 3-Manifolds 223 William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T EX by Sheila Newbery, who also scanned the ﬁgures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the ﬁrst few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at levy@msri.org. CHAPTER 9 Algebraic convergence 9.1. Limits of discrete groups It is important for us to develop an understanding of the geometry of deforma-tions of a given discrete group. A qualitative understanding can be attained most concretely by considering limits of sequences of groups. The situation is complicated by the fact that there is more than one reasonable sense in which a group can be the limit of a sequence of discrete groups. Definition 9.1.1. A sequence {Γi} of closed subgroups of a Lie group G converges geometrically to a group Γ if (i) each γ ∈Γ is the limit of a sequence {γi}, with γi ∈Γi, and (ii) the limit of every convergent sequence {γij}, with γij ∈Γij, is in Γ. Note that the geometric limit Γ is automatically closed. The deﬁnition means that Γi’s look more and more like Γ, at least through a microscope with limited resolution. We shall be mainly interested in the case that the Γi’s and Γ are discrete. The geometric topology on closed subgroups of G is the topology of geometric convergence. The notion of geometric convergence of a sequence of discrete groups is closely related to geometric convergence of a sequence of complete hyperbolic manifolds of bounded volume, as discussed in 5.11. A hyperbolic three-manifold M determines a subgroup of PSL(2, C) well-deﬁned up to conjugacy. A speciﬁc representative of this conjugacy class of discrete groups corresponds to a choice of a base frame: a base 9.2 point p in M together with an orthogonal frame for the tangent space of M at p. This gives a speciﬁc way to identify ˜ M with H3. Let O(H[ϵ,∞)) consist of all base frames contained in M[ϵ,∞), where M ranges over H (the space of hyperbolic three-manifolds with ﬁnite volume). O(H[ϵ,∞)) has a topology deﬁned by geometric convergence of groups. The topology on H is the quotient topology by the equivalence relation of conjugacy of subgroups of PSL(2, C). This quotient topology is not well-behaved for groups which are not geometrically ﬁnite. Definition 9.1.2. Let Γ be an abstract group, and ρi : Γ →G be a sequence of representations of Γ into G. The sequence {ρi} converges algebraically if for every γ ∈Γ, {ρi(γ)} converges. The limit ρ : Γ →G is called the algebraic limit of {ρi}. Thurston — The Geometry and Topology of 3-Manifolds 225 9. ALGEBRAIC CONVERGENCE Definition 9.1.3. Let Γ be a countable group, {ρi} a sequence of representations of Γ in G with ρi(Γ) discrete. {ρi} converges strongly to a representation ρ if ρ is the algebraic limit of {ρi} and ρΓ is the geometric limit of {ρiΓ}. Example 9.1.4 (Basic example). There is often a tremendous diﬀerence between algebraic limits and geometric limits, growing from the following phenomenon in a sequence of cyclic groups. Pick a point x in H3, a “horizontal” geodesic ray l starting at x, and a “vertical” plane through x containing the geodesic ray. Deﬁne a sequence of representations 9.3 ρi : Z →PSL(2, C) as follows. Let xi be the point on l at distance i from x, and let li be the “vertical” geodesic through xi: perpendicular to l and in the chosen plane. Now deﬁne ρi on the generator 1 by letting ρi(1) be a screw motion around li with ﬁne pitched thread so that ρi(1) takes x to a point at approximately a horizontal distance of 1 from x and some high power ρi(ni) takes x to a point in the vertical plane a distance of 1 from x. The sequence {ρi} converges algebraically to a parabolic representation ρ : Z →PSL(2, C), while {ρiZ} converges geometrically to a parabolic subgroup of rank 2, generated by ρ(Z) plus an additional generator which moves x a distance of 1 in the vertical plane. 9.4 This example can be described in matrix form as follows. We make use of one-complex parameter subgroups of PSL(2, C) of the form exp w a sinh w 0 exp −w , with w ∈C. Deﬁne ρn by ρn(1) = exp wn n sinh wn 0 exp −wn where wn = 1/n2 + πi/n. Thus {ρn(1)} converges to 1 πi 0 1 226 Thurston — The Geometry and Topology of 3-Manifolds 9.1. LIMITS OF DISCRETE GROUPS while {ρn(n)} converges to −1 −1 0 −1 = 1 1 0 1 . This example can be easily modiﬁed without changing the algebraic limit so that {ρi(Z)} has no geometric limit, or so that its geometric limit is a one-complex-parameter parabolic subgroup, or so that the geometric limit is isomorphic to Z × R. 9.5 This example can also be combined with more general groups: here is a simple case. Let Γ be a Fuchsian group, with MΓ a punctured torus. Thus Γ is a free group on generators a and b, such that [a, b] is parabolic. Let ρ : Γ →PSL(2, C) be the identity representation. It is easy to see that Tr ρ′[a, b] ranges over a neighborhood of 2 as ρ′ ranges over a neighborhood of ρ. Any nearby representation determines a nearby hyperbolic structure for M[ϵ,∞), which can be thickened to be locally convex except near M(0,ϵ]. Consider representations ρn with an eigenvalue for ρn[a, b] ∼1 + C/n2 + πi/n. ρn[a, b] translates along its axis a distance of approximately 2 Re(C)/n2, while rotat-ing an angle of approximately 2π n + 2 Im(C) n2 . Thus the n-th power translates by a distance of approximately 2 Re(C)/n, and rotates approximately 2π + 2 Im(C) n . The axis moves out toward inﬁnity as n →∞. For C suﬃciently large, the image of ρn will be a geometrically ﬁnite group (a Schottky group); a compact convex manifold with π1 = ρn(Γ) can be constructed by piecing together a neighborhood of M[ϵ,∞) with (the convex hull of a helix)/Z. The algebraic limit of {ρn} is ρ, while the geometric limit is the group generated by ρ(Γ) = Γ together with an extra parabolic generator commuting with [a, b]. Thurston — The Geometry and Topology of 3-Manifolds 227 9. ALGEBRAIC CONVERGENCE 9.6 Troels Jørgensen was the ﬁrst to analyze and understand this phenomenon. He showed that it is possible to iterate this construction and produce examples as above where the algebraic limit is the fundamental group of a punctured torus, but the geometric limit is not even ﬁnitely generated. See § . Here are some basic properties of convergence of sequences of discrete groups. Proposition 9.1.5. If {ρi} converges algebraically to ρ and {ρiΓ} converges ge-ometrically to Γ′, then Γ′ ⊃ρΓ. Proof. Obvious. □ Proposition 9.1.6. For any Lie group G, the space of closed subgroups of G (with the geometric topology) is compact. Proof. Let {Γi} be any sequence of closed subgroups. First consider the case that there is a lower bound to the “size” d(e, γ) of elements of γ ∈Γi. Then there is an upper bound to the number of elements of Γi in the ball of radius γ about e, for every γ. The Tychonoﬀproduct theorem easily implies the existence of a subsequence converging geometrically to a discrete group. Now let S be a maximal subspace of Te(G), the tangent space of G at the identity element e, with the property that for any ϵ > 0 there is a Γi whose ϵ-small elements ﬁll out all directions in S, within an angle of ϵ. It is easy to see that S is closed under Lie brackets. Furthermore, a subsequence {Γij} whose small elements ﬁll out S has 9.7 the property that all small elements are in directions near S. It follows, just as in the previous case, that there is a subsequence converging to a closed subgroup whose tangent space at e is S. Corollary 9.1.7. The set of complete hyperbolic manifolds N together with base frames in N[ϵ,∞) is compact in the geometric topology. 228 Thurston — The Geometry and Topology of 3-Manifolds 9.1. LIMITS OF DISCRETE GROUPS □ Corollary 9.1.8. Let Γ be any countable group and {ρi} a sequence of discrete representations of Γ in PSL(2, C) converging algebraically to a representation ρ. If ρΓ does not have an abelian subgroup of ﬁnite index then {ρi} has a subsequence converging geometrically to a discrete group Γ′ ⊃◦Γ. In particular, ρΓ is discrete. Proof. By 9.1.7, there is a subsequence converging geometrically to some closed group Γ′. By 5.10.1, the identity component of Γ′ must be abelian; since ρΓ ⊂Γ′, the identity component is trivial. □ Note that if the ρi are all faithful, then their algebraic limit is also faithful, since there is a lower bound to d(ρiγx, x). These basic facts were ﬁrst proved in ???? Here is a simple example negating the converse of 9.1.8. Consider any discrete group Γ ⊂PSL(2, C) which admits an automorphism φ of inﬁnite order: for instance, Γ might be a fundamental group of a surface. The sequence of representations φi has no algebraically convergent subsequence, yet {φiΓ} converges geometrically to Γ. 9.8 There are some simple statements about the behavior of limit sets when passing to a limit. First, if Γ is the geometric limit of a sequence {Γi}, then each point x ∈LΓ is the limit of a sequence xi ∈LΓi. In fact, ﬁxed points x (eigenvectors) of non-trivial elements of γ ∈Γ are dense in LΓ; for high i, Γi must have an element near γ, with a ﬁxed point near x. A similar statement follows for the algebraic limit ρ of a sequence of representations ρi. Thus, the limit set cannot suddenly increase in the limit. It may suddenly decrease, however. For instance, let Γ ⊂PSL(2, C) be any ﬁnitely generated group. Γ is residually ﬁnite (see § ), or in other words, it has a sequence {Γi} of subgroups of ﬁnite index converging geometrically to the trivial group (e). LΓi = LΓ is constant, but L(e) is empty. It is plausible that every ﬁnitely generated discrete group Γ ⊂PSL(2, C) be a geometric limit of groups with compact quotient. We have already seen (in 9.1.4) examples where the limit set suddenly decreases in an algebraic limit. Let Γ be the fundamental group of a surface S with ﬁnite area and {ρi} a sequence of faithful quasi-Fuchsian representations of Γ, preserving parabolicity. Suppose {ρi} converges algebraically to a representation ρ as a group without any additional par-abolic elements. Let N denote Nρ(Γ), Ni denote Nρi(Γ), etc. 9.9 Theorem 9.2. N is geometrically tame, and {ρi} converges strongly to ρ. Proof. If the set of uncrumpled maps of S into N homotopic to the standard map is compact, then using a ﬁnite cover of GL(S) carried by nearly straight train Thurston — The Geometry and Topology of 3-Manifolds 229 9. ALGEBRAIC CONVERGENCE tracks, one sees that for any discrete representation ρ′ near ρ, every geodesic lami-nation γ of S is realizable in N ′ near its realizations in N. (Logically, one can think of uncrumpled surfaces as equivariant uncrumpled maps of M 2 into H3, with the compact-open topology, so that “nearness” makes sense.) Choose any subsequence of the ρi’s so that the bending loci for the two boundary components of Mi converge in GL(S). Then the two boundary components must converge to locally convex dis-joint embeddings of S in N (unless the limit is Fuchsian). These two surfaces are homotopic, hence they bound a convex submanifold M of N, so ρ(Γ) is geometrically ﬁnite. Since M[ϵ,∞) is compact, strong convergence of {ρi} follows form 8.3.3: no un-expected identiﬁcations of N can be created by a small perturbation of ρ which preserves parabolicity. If the set of uncrumpled maps of S homotopic to the standard map is not compact, then it follows immediately from the deﬁnition that N has at least one geometrically inﬁnite tame end. We must show that both ends are geometrically tame. The possible phenomenon to be wary of is that the bending loci β+ i and β− i of the two boundary components of Mi might converge, for instance, to a single point λ in GL(S). (This would be conceivable if the “simplest” homotopy of one of the two boundary compo-9.10 nents to a reference surface which persisted in the limit ﬁrst carried it to the vicinity of the other boundary component.) To help in understanding the picture, we will ﬁrst ﬁnd a restriction for the way in which a hyperbolic manifold with a geometrically tame end can be a covering space. Definition 9.2.1. Let N be a hyperbolic manifold, P a union of horoball neigh-borhoods of its cusps, E′ an end of N −P. E′ is almost geometrically tame if some ﬁnite-sheeted cover of E′ is (up to a compact set) a geometrically tame end. (Later we shall prove that if E is almost geometrically tame it is geometrically tame.) Theorem 9.2.2. Let N be a hyperbolic manifold, and ˜ N a covering space of N such that ˜ N −˜ P has a geometrically inﬁnite tame end E bounded by a surface S[ϵ,∞). Then either N has ﬁnite volume and some ﬁnite cover of N ﬁbers over S1 with ﬁber S, or the image of E in N −P, up to a compact set, is an almost geometrically tame end of N. Proof. Consider ﬁrst the case that all points of E identiﬁed with S[ϵ,∞) in the projection to N lie in a compact subset of E. Then the local degree of the projection of E to N is ﬁnite in a neighborhood of the image of S. Since the local degree is constant except at the image of S, it is everywhere ﬁnite. Let G ⊂π1N be the set of covering transformations of H3 over N consisting of 9.11 elements g such that g ˜ E ∩˜ E is all of ˜ E except for a bounded neighborhood of ˜ S. G 230 Thurston — The Geometry and Topology of 3-Manifolds 9.1. LIMITS OF DISCRETE GROUPS is obviously a group, and it contains π1S with ﬁnite index. Thus the image of E, up to compact sets, is an almost geometrically tame end of N. The other case is that S[ϵ,∞) is identiﬁed with a non-compact subset of E by projection to N. Consider the set I of all uncrumpled surfaces in E whose images intersect the image of S[ϵ,∞). Any short closed geodesic on an uncrumpled surface of E is homotopic to a short geodesic of E (not a cusp), since E contains no cusps other than the cusps of S. Therefore, by the proof of 8.8.5, the set of images of I in N is precompact (has a compact closure). If I itself is not compact, then N has a ﬁnite cover which ﬁbers over S1, by the proof of 8.10.9. If I is compact, then (since uncrumpled surfaces cut E into compact pieces), inﬁnitely many components of the set of points identiﬁed with S[ϵ,∞) are compact and disjoint from S. 9.12 Thurston — The Geometry and Topology of 3-Manifolds 231 9. ALGEBRAIC CONVERGENCE These components consist of immersions of k-sheeted covering spaces of S injective on π1, which must be homologous to ±k [S]. Pick two disjoint immersions with the same sign, homologous say to −k [S] and −l [S]. Appropriate multiples of these cycles are homologous by a compactly supported three-chain which maps to a three-cycle in N −P, hence N has ﬁnite volume. Theorem 9.2.2 now follows from 8.10.9. □ We continue the proof of Theorem 9.2. We may, without loss of generality, pass to a subsequence of representations ρi such that the sequences of bending loci {β+ i } and {β− i } converge, in PL0(S), to laminations β+ and β−. If β+, say, is realizable for the limit representation ρ, then any uncrumpled surface whose wrinkling locus contains β+ is embedded and locally convex—hence it gives a geometrically ﬁnite end of N. The only missing case for which we must prove geometric tameness is that neither 9.13 β+ nor β−is realizable. Let λϵ i ∈PL0(S) (where ϵ = +, −) be a sequence of geodesic laminations with ﬁnitely many leaves and with transverse measures approximating βϵ i closely enough that the realization of λϵ i in Ni is near the realization of βϵ i. Also suppose that lim λϵ i = βϵ in PL0(S). The laminations λϵ i are all realized in N. They must tend toward ∞in N, since their limit is not realized. We will show that they tend toward ∞in the ϵ-direction. Imagine the contrary—for deﬁniteness, suppose that the realizations of {λ+ i } in N go to ∞in the −direction. The realization of each λ+ i in Nj must be near the realization in N, for high enough j. Connect λ+ j to λ+ i by a short path λi,j,t in PL0(S). A family of uncrumpled surfaces Si,j,t realizing the λi,j,t is not continuous, but has the property that for t near t0, Si,j,t and Si,j,t0 have points away from their cusps which are close in N. Therefore, for every uncrumpled surface U between Si,j,0 and Si,j,1 (in a homological sense), there is some t such that Si,j,t ∩U ∩(N −P) is non-void. 232 Thurston — The Geometry and Topology of 3-Manifolds 9.3. THE ENDING OF AN END 9.14 Let γ be any lamination realized in N, and Uj be a sequence of uncrumpled surfaces realizing γ in Nj, and converging to a surface in N. There is a sequence Si(j),j,t(j) of uncrumpled surfaces in Nj intersecting Uj whose wrinkling loci tend toward β+. Without loss of generality we may pass to a geometrically convergent subsequence, with geometric limit Q. Q is covered by N. It cannot have ﬁnite volume (from the analysis in Chapter 5, for instance), so by 8.14.2, it has an almost geometrically tame there is no 8.14.2 end E which is the image of the −end E−of N. Each element α of π1E has a ﬁnite power αk ∈π1E−. Then a sequence {αi} approximating α in π1(Ni) has the property that the αk i have bounded length in the generators of π1S, this implies that the αi have bounded length, so α is in fact in π1E−, and E−= E (up to compact sets). Using this, we may pass to a subsequence of Si(j),j,t’s which converge to an uncrumpled surface R in E. R is incompressible, so it is in the standard homotopy class. It realizes β+, which is absurd. We may conclude that N has two geometrically tame ends, each of which is mapped homeomorphically to the geometric limit Q. (This holds whether or not they are geometrically inﬁnite.) This implies the local degree of N →Q is ﬁnite one or two (in case the two ends are identiﬁed in Q). But any covering transformation α of N over Q has a power (its square) in π1N, which implies, as before, that α ∈π1N, so that N = Q. This concludes the proof of 9.2. □ 9.3. The ending of an end In the interest of avoiding circumlocution, as well as developing our image of a geometrically tame end, we will analyze the possibilities for non-realizable laminations in a geometrically tame end. We will need an estimate for the area of a cylinder in a hyperbolic three-manifold. Given any map f : S1 × [0, 1] →N, where N is a convex hyperbolic manifold, we may straighten each line θ × [0, 1] to a geodesic, obtaining a ruled cylinder with the same boundary. Thurston — The Geometry and Topology of 3-Manifolds 233 9. ALGEBRAIC CONVERGENCE Theorem 9.3.1. The area of a ruled cylinder (as above) is less than the length of its boundary. Proof. The cylinder can be C0-approximated by a union of small quadrilaterals each subdivided into two triangles. The area of a triangle is less than the minimum of the lengths of its sides (see p. 6.5). □ 9.16 If the two boundary components of the cylinder C are far apart, then most of the area is concentrated near its boundary. Let γ1 and γ2 denote the two components of ∂C. Theorem 9.3.2. Area (C −Nrγ1) ≤e−r l(γ1) + l(γ2) where r ≥0 and l denotes length. This is derived by integrating the area of a triangle in polar coordinates from any vertex: A = Z Z T(θ) 0 sinh t dt dθ = Z (cosh T(θ) −1) dθ The area outside a neighborhood of radius r of its far edge α is Z cosh (T(θ) −r) −1 dθ < e−r Z sinh T(θ) dθ < e−r l(α). This easily implies 9.3.2 234 Thurston — The Geometry and Topology of 3-Manifolds 9.3. THE ENDING OF AN END Let E be a geometrically tame end, cut oﬀby a surface S[ϵ,∞) in N −P, as usual. A curve α in E homotopic to a simple closed curve α′ on S gives rise to a ruled cylinder Cα : S1 × [0, 1] →N. Now consider two curves α and β homotopic to simple closed curves α′ and β′ 9.17 on S. One would expect that if α′ and β′ are forced to intersect, then either α must intersect Cβ or β must intersect Cα, as in 8.11.1 We will make this more precise by attaching an invariant to each intersection. Let us assume, for simplicity, that α′ and β′ are geodesics with respect to some hyperbolic structure on S. Choose one of the intersection points, p0, of α′ and β′ as a base point for N. For each other intersection point pi, let αi and βi be paths on α′ and β′ from p0 to pi. Then αi ∗β−1 i is a closed loop, which is non-trivial in π1(S) when i ̸= 0 since two geodesics in ˜ S have at most one intersection. There is some ambiguity, since there is more than one path from α0 to αi on 9.18 α′; in fact, αi is well-deﬁned up to a power of α′. Let ⟨g⟩denote the cyclic group generated by an element g. Then αi · β−1 i gives a well-deﬁned element of the double coset space ⟨α′⟩\π1(S)/⟨β′⟩. [The double coset H1gH2 ∈H1\G/H2 of an element g ∈G is the set of all elements h1gh2, where hi ∈Hi.] The double cosets associated to two diﬀerent intersections pi and pj are distinct: if ⟨α′⟩αiβ−1 i ⟨β′⟩= ⟨α′⟩ajβ−1 j ⟨β′⟩, then there is some loop α−1 j α′kαiβ−1 i β′lβj made up of a path on α′ and a path on Thurston — The Geometry and Topology of 3-Manifolds 235 9. ALGEBRAIC CONVERGENCE β′ which is homotopically trivial—a contradiction. In the same way, a double coset Dx,y is attached to each intersection of the cylinders Cα and Cβ. Formally, these intersection points should be parametrized by the domain: thus, an intersection point means a pair (x, y) ∈(S1 × I) × (S′ × I) such that Cαx = Cβy. Let i(γ, δ) denote the number of intersections of any two simple geodesics γ and δ on S. Let D(γ, δ) be the set of double cosets attached to intersection points of γ and δ (including p0). Thus i(γ, δ) = \|D(γ, δ)\|. D(α, Cβ) and D(Cα, β) are deﬁned similarly. Proposition 9.3.3. \|α ∩Cβ\| + \|Cα ∩β\| ≥i(α′, β′). In fact D(a, Cβ) ∪D(Cα, β) ⊃D(α′, β′). 9.19 Proof. First consider cylinders C′ α and C′ β which are contained in E, and which are nicely collared near S. Make C′ α and C′ β transverse to each other, so that the double locus L ⊂(S1 × I) × (S1 × I) is a one-manifold, with boundary mapped to α ∪β ∪α′ ∪β′. The invariant D(x,y) is locally constant on L, so each invariant occurring for α′ ∩β′ occurs for the entire length of interval in L, which must end on α or β. In fact, each element of D(α′, β′) occurs as an invariant of an odd number of points α ∪β. Now consider a homotopy ht of C′ β to Cβ, ﬁxing β ∪β′. The homotopy can be perturbed slightly to make it transverse to α, although this may necessitate a slight movement of Cβ to a cylinder C′′ β. Any invariant which occurs an odd number of times for a ∩C′ β occurs also an odd number of times for α ∩C′′ β. This implies that the invariant must also occur for a ∩Cβ. □ Remark. By choosing orientations, we could of course associate signs to intersec-tion points, thereby obtaining an algebraic invariant D(α′, β′) ∈Z⟨α′⟩\π1S/⟨β′⟩. Then 9.3.3 would become an equation, D(α′, β′) = D(α, Cβ) + D(Cα, β). Since π1(S) is a discrete group, there is a restriction on how closely intersection points can be clustered, hence a restriction on \|D(α, cβ)\| in terms of the length of α times the area of Cβ. 9.20 Proposition 9.3.4. There is a constant K such that for every curve α in E with distance R from S homotopic to a simple closed curve α′ on S and every curve β in E not intersecting Cα and homotopic to a simple curve β′ on S, i(α′, β′) ≤K l(α) + l(α) + 1 l(β) + e−R + l(β′) . 236 Thurston — The Geometry and Topology of 3-Manifolds 9.3. THE ENDING OF AN END Proof. Consider intersection points (x, y) ∈S1×(S1×I) of α and Cβ. Whenever two of them, (x, y) and (x′, y′), are close in the product of the metrics induced from N, there is a short loop in N which is non-trivial if D(x,y) ̸= D(x′,y′). Case (i). α is a short loop. Then there can be no short non-trivial loop on Cβ near an intersection point with α. The disks of radius ϵ on Cβ about intersection points with α have area greater than some constant, except in special cases when they are near ∂Cβ. If necessary, extend the edges of Cβ slightly, without substantially changing the area. The disks of radius ϵ must be disjoint, so this case follows from 9.3.2 and 9.3.3. Case (ii). α is not short. Let E ⊂Cβ consist of points through which there is a short loop homotopic to β. If (x, y) and (x′, y′) are intersection points with Dx,y ̸= Dx′,y′ and with y, y′ in E, then x and x′ cannot be close together—otherwise two distinct conjugates of β would be represented by short loops through the same point. The number of such intersections is thus estimated by some constant times l(α). 9.21 Three intersections of α with Cβ −E cannot occur close together. S1 × (Cβ −E) contains the balls of radius ϵ, with multiplicity at most 2, and each ball has a deﬁnite volume. This yields 9.3.4. □ Let us generalize 9.3.4 to a statement about measured geodesic laminations. Such a lamination (γ, µ) on a hyperbolic surface S has a well-deﬁned “average length” lS(γ, µ). This can be deﬁned as the total mass of the measure which is locally the product of the transverse measure µ with one-dimensional Lebesgue measure on the leaves of γ. Similarly, a realization of γ in a homotopy class f : S →N has a length lf(γ, µ). The length lS(γ, µ) is a continuous function on ML0(S), and lf(γ) is a continuous function where deﬁned. If γ is realized a distance of R from an uncrumpled surface S, then lf(γ, µ) ≤(1/cosh R)lS(γ, µ). This implies that if f preserves non-parabolicity, lf extends continuously over all of ML0 so that its zero set is the set of non-realizable laminations. The intersection number i (γ1, µ1), (γ2, µ2) of two measured geodesic laminations is deﬁned similarly, as the total mass of the measure µ1 × µ2 which is locally the product of µ1 and µ2. (This measure µ1×µ2 is interpreted to be zero on any common leaves of γ1 and γ2.) Thurston — The Geometry and Topology of 3-Manifolds 237 9. ALGEBRAIC CONVERGENCE 9.22 Given a geodesic lamination γ realized in E, let dγ be the miniaml distance of an uncrumpled surface through γ from S[ϵ,∞). Theorem 9.3.5. There is a constant K such that for any two measured geodesic laminations (γ1, µ1) and (γ2, µ2) ∈ML0(S) realized in E, i (γ1, µ1), (γ2, µ2) ≤K · e−2RlS(γ1, µ1) · lS(γ2, µ2) where R = inf(dγ1, dγ2). Proof. First consider the case that γ1 and γ2 are simple closed geodesics which are not short. Apply the proof of 9.3.4 ﬁrst to intersections of γ1 with Cγ2, then to intersections of Cγ1 with γ2. Note that lS(γi) is estimated from below by eRl(γi), so the terms involving l(γi) can be replaced by Ce−Rl(γi). Since γ1 and γ2 are not short, one obtains i(γ1, γ2) ≤K · e−2R lS(γ1) lS(γ2), for some constant K. Since both sides of the inequality are homogeneous of degree one in γ1 and γ2, it extends by continuity to all of ML0(S). □ Consider any sequence {(γi, µi)} of measured geodesic laminations in ML0(S) whose realizations go to ∞in E. If (λ1, µ1) and (λ2, µ2) are any two limit points of this sequence, 9.3.5 implies that i(λ1, λ2) = 0: in other words, the leaves do not cross. The union λ1 ∪λ2 is still a lamination. 9.23 Definition 9.3.6. The ending lamination ϵ(E) ∈GL(S) is the union of all limit points λi, as above. Clearly, ϵ(E) is compactly supported and it admits a measure with full support. The set ∆(E) ⊂PL0(S) of all such measures on ϵ(E) is closed under convex combi-nations, hence its intersection with a local coordinate system (see p. 8.59) is convex. 238 Thurston — The Geometry and Topology of 3-Manifolds 9.3. THE ENDING OF AN END In fact, a maximal train track carrying ϵ(E) deﬁnes a single coordinate system con-taining ∆(E). The idea that the realization of a lamination depends continuously on the lami-nation can be generalized to the ending lamination ϵ(E), which can be regarded as being realized at ∞. Proposition 9.3.7. For every compact subset K of E, there is a neighborhood U of ∆(E) in PL0(S) such that every lamination in U −∆(E) is realized in E −K. Proof. It is convenient to pass to the covering of N corresponding to π1S. Let S′ be an uncrumpled surface such that K is “below” S′ (in a homological sense). Let {Vi} be a neighborhood basis for ∆(E) such that Vi −∆(E) is path-connected, and let λi ∈Vi −∆(E) be a sequence whose realizations go to ∞in E. If there is any point πi ∈Vi −∆(E) which is a non-realizable lamination or whose realization is not “above” S′, connect λi to πi by a path in Vi. There must be some element of this path whose realization intersects S′ [ϵ,∞) (since the realizations cannot go to ∞while in E.) Even if certain non-peripheral elements of S are parabolic, excess pinching of non-peripheral curves on uncrumpled surfaces intersecting S′ can be avoided if S′ is far from S, since there are no extra cusps in E. Therefore, only ﬁnitely many such πi’s can occur, or else there would be a limiting uncrumpled surface through S realizing the unrealizable. □ Proposition 9.3.8. Every leaf of ϵ(E) is dense in ϵ(E), and every non-trivial simple curve in the complement of ϵ(E) is peripheral. Proof. The second statement follows easily from 8.10.8, suitably modiﬁed if there are extra cusps. The ﬁrst statement then follows from the next result: Proposition 9.3.9. If γ is a geodesic lamination of compact support which admits a nowhere zero transverse measure, then either every leaf of γ is dense, or there is a non-peripheral non-trivial simple closed curve in S −γ. Proof. Suppose δ ⊂γ is the closure of any leaf. Then δ is also an open subset of γ: all leaves of γ near δ are trapped forever in a neighborhood of δ. This is seen by considering the surface S −δ. Thurston — The Geometry and Topology of 3-Manifolds 239 9. ALGEBRAIC CONVERGENCE 9.25 An arc transverse to these leaves would have positive measure, which would imply that a transverse arc intersecting these leaves inﬁnitely often would have inﬁnite measure. (In general, a closed union of leaves δ ⊂γ in a general geodesic lamination has only a ﬁnite set of leaves of γ intersecting a small neighborhood.) If δ ̸= γ, then δ has two components, which are separated by some homotopically non-trivial curve in S −γ. □ □ Corollary 9.3.10. For any homotopy class of injective maps f : S →N from a hyperbolic surface of ﬁnite area to a complete hyperbolic manifold, if f preserves parabolicity and non-parabolicity, there are n = 0, 1 or 2 non-realizable laminations ϵi [1 ≤i ≤n] such that a general lamination γ on S is non-realizable if and only if the union of its non-isolated leaves is an ϵi. 9.4. Taming the topology of an end We will develop further our image of a geometrically tame end, once again to avoid circumlocution. Theorem 9.4.1. A geometrically tame end E ⊂N −P is topologically tame. In other words, E is homeomorphic to the product S[ϵ,∞) × [0, ∞). Theorem 9.4.1 will be proved in §§9.4 and 9.5. Corollary 9.4.2. Almost geometrically tame ends are geometrically tame. Proof that 9.4.1 implies 9.4.2. Let E′ be an almost geometrically tame end, ﬁnitely covered (up to compact sets) by a geometrically tame end E = S[ϵ,∞) × [0, ϵ), 9.26 with projection p : E →E′. Let f : E′ →[0, ϵ) be a proper map. The ﬁrst step is to ﬁnd an incompressible surface S′ ⊂E′ which cuts it oﬀ(except for compact sets). 240 Thurston — The Geometry and Topology of 3-Manifolds 9.4. TAMING THE TOPOLOGY OF AN END Choose t0 high enough that p : E →E′ is deﬁned on S[ϵ,∞) × [t0, ∞), and choose t1 > t0 so that p S[ϵ,∞) × [t1, ∞) does not intersect p(S[ϵ,∞) × t0). 9.27 Let r ∈[0, ∞) be any regular value for f greater than the supremum of f ◦p on S[ϵ,∞) × [0, t1). Perform surgery (that is, cut along circles and add pairs of disks) to f −1(r), to obtain a not necessarily connected surface S′ in the same homology class which is incompressible in E′ −p S[ϵ,∞) × [0, t0) . The fundamental group of S′ is still generated by loops on the level set f = r. S′ is covered by a surface ˜ S′ in E. ˜ S′ must be incompressible in E−otherwise there would be a non-trivial disk D mapped into S[ϵ,∞) ×[t1, ∞) with boundary on ˜ S; p◦D would be contained in E′ −p(S[ϵ,∞) × [0, t0]) so S′ would not be incompressible (by the loop theorem). One deduces that ˜ S′ is homotopic to S[ϵ,∞) and S′ is incompressible in N −P. If E is geometrically ﬁnite, there is essentially nothing to prove—E corresponds to a component of ∂˜ M, which gives a convex embedded surface in E′. If E is ge-ometrically inﬁnite, then pass to a ﬁnite sheeted cover E′′ of E which is a regular cover of E′. The ending lamination ϵ(E′′) is invariant under all diﬀeomorphisms (up Thurston — The Geometry and Topology of 3-Manifolds 241 9. ALGEBRAIC CONVERGENCE to compact sets) of E′′. Therefore it projects to a non-realizable geodesic lamination ϵ(E′) on S′. □ Proof of 9.4.1. We have made use of one-parameter families of uncrumpled surfaces in the last two sections. Unfortunately, these surfaces do not vary contin-uously. To prove 9.4.1, we will show, in §9.5, how to interpolate with more general surfaces, to obtain a (continuous) proper map F: S[ϵ,∞) × [0, ∞) →E. The theorem will follow fairly easily once F is constructed: 9.28 Proposition 9.4.3. Suppose there is a proper map F : S[ϵ,∞) × [0, ∞) →E with F(S[ϵ,∞) × 0) standard and with F ∂S[ϵ,∞) × [0, ∞) ⊂∂(N −P). Then E is homeomorphic to S[ϵ,∞) × [0, ∞). Proof of 9.4.3. This is similar to 9.4.2. Let f : E →[0, ∞) be a proper map. For any compact set K ⊂E, we can ﬁnd a t1 > 0 so that F S[ϵ,∞) × [t1, ∞) is disjoint from K. Let r be a regular value for f greater than the supremum of f ◦F on S[ϵ,∞) × [0, t1]. Let S′ = f −1(r) and S′′ = (f ◦F)−1(r). F : S′′ →S′ is a map of degree one, so it is surjective on π1 (or else it would factor through a non-trivial covering space on S′, hence have higher degree). Perform surgery on S′ to make it incompressible in the complement of K, without changing the homology class. Now S′ must be incompressible in E; otherwise there would be some element α of π1S′ which is null-homotopic in E. But α comes from an element β on S′′ which is null-homotopic in S[ϵ,∞) × [t1, ∞), so its image α is null-homotopic in the complement of K. It follows that S′ is homotopic to S[ϵ,∞), and that the compact region of E cut oﬀby S′ is homeomorphic to S[ϵ,∞) × I. By constructing a sequence of such disjoint surfaces going outside of every compact set, we obtain a homeomorphism with S[ϵ,∞) × [0, ∞). □ □ 9.5. Interpolating negatively curved surfaces Now we turn to the task of constructing a continuous family of surfaces moving out 9.29 to a geometrically inﬁnite tame end. The existence of this family, besides completing the proof of 9.4.1, will show that a geometrically tame end has uniform geometry, and it will lead us to a better understanding of ML0(S). We will work with surfaces which are totally geodesic near their cusps, on esthetic grounds. Our basic parameter will be a family of compactly supported geodesic laminations in ML0(S). The ﬁrst step is to understand when a family of uncrumpled surfaces realizing these laminations is continuous and when discontinuous. 242 Thurston — The Geometry and Topology of 3-Manifolds 9.5. INTERPOLATING NEGATIVELY CURVED SURFACES Definition 9.5.1. For a lamination γ ∈ML0(S), let Tγ be the limit set in GL(S) of a neighborhood system for γ in ML0(S). ( Tγ is the “qualitative tangent space” of ML0(S) at γ ). Let ML0(S) denote the closure of the image of ML0(S) in GL(S). Clearly ML0(S) consists of laminations with compact support, but not every lamination with compact support is in ML0(S): Every element of ML0 is in Tγ for some γ ∈ML0. Let us say that an element γ ∈ML0 is essentially complete if γ is a maximal element of ML0. If γ ∈ML0, then γ is essentially complete if and only if Tγ = γ. A lamination γ is maximal among all compactly supported laminations if and only if each region of S −γ is an asymptotic 9.30 triangle or a neighborhood of a cusp of S with one cusp on its boundary—a punctured monogon. (These are the only possible regions with area π which are simply connected or whose fundamental group is peripheral.) Clearly, if S −γ consists of such regions, then γ is essentially complete. There is one special case when essentially complete laminations are not of this form; we shall analyze this case ﬁrst. Proposition 9.5.2. Let T −p denote the punctured torus. An element γ ∈ML0(T −p) is essentially complete if and only if (T −p) −γ is a punctured bigon. If γ ∈ML0(T −p), then either γ has a single leaf (which is closed), or every leaf of γ is non-compact and dense, in which case γ is essentially complete. If γ has a single closed leaf, then Tγ consists of γ and two other laminations: Thurston — The Geometry and Topology of 3-Manifolds 243 9. ALGEBRAIC CONVERGENCE 9.31 Proof. Let g ∈ML0(T −p) be a compactly supported measured lamination. First, note that the complement of a simple closed geodesic on T −p is a punctured annulus, which admits no simple closed geodesics and consequently no geodesic laminations in its interior. Hence if γ contains a closed leaf, then γ consists only of this leaf, and otherwise (by 9.3.9) every leaf is dense. Now let α be any simple closed geodesic on T −p, and consider γ cut apart by α. No end of a leaf of γ can remain forever in a punctured annulus, or else its limit set would be a geodesic lamination. Thus α cuts leaves of γ into arcs, and these arcs have only three possible homotopy classes: If the measure of the set of arcs of type (a) is ma, etc., then (since the two boundary components match up) we have 2ma + mb = 2mc + mb. But cases (a) and (c) 9.32 are incompatible with each other, so it must be that ma = mc = 0. Note that γ is orientable: it admits a continuous tangent vector ﬁeld. By inspection we see a complementary region which is a punctured bigon. Since the area of a punctured bigon is 2π, which is the same as the area of T −p, this is the only complementary region. 244 Thurston — The Geometry and Topology of 3-Manifolds 9.5. INTERPOLATING NEGATIVELY CURVED SURFACES It is now clear that a compactly supported measured lamination on T −p with every leaf dense is essentially complete—there is nowhere to add new leaves under a small perturbation. If γ has a single closed leaf, then consider the families of measures on train tracks: 9.33 These train tracks cannot be enlarged to train tracks carrying measures. This can be deduced from the preceding argument, or seen as follows. At most one new branch could be added (by area considerations), and it would have to cut the punctured bigon into a punctured monogon and a triangle. The train track is then orientable in the complement of the new branch, so a train can traverse this branch at most once. This is incompatible with the existence of a positive measure. Therefore ML0(T −p) is two-dimensional, so τ1 and τ2 carry a neighborhood of γ. It follows that τγ is as shown. □ Proposition 9.5.3. PL0(T −p) is a circle. Thurston — The Geometry and Topology of 3-Manifolds 245 9. ALGEBRAIC CONVERGENCE Proof. The only closed one-manifold is S1. That PL0(T −p) is one-dimensional follows from the proof of 9.5.2. Perhaps it is instructive in any case to give a covering of PL0(T −p) by train track neighborhoods: 9.34 or, to get open overlaps, □ Proposition 9.5.4. On any hyperbolic surface S which is not a punctured torus, an element γ ∈ML0(S) is essentially complete if and only if S −γ is a union of triangles and punctured monogons. Proof. Let γ be an arbitrary lamination in ML0(S), and let τ be any train track approximation close enough that the regions of S −τ correspond to those of S −γ. If some of these regions are not punctured monogons or triangles, we will add extra 9.35 branches in a way compatible with a measure. First consider the case that each region of S −γ is either simply connected or a simple neighborhood of a cusp of S with fundamental group Z. Then τ is connected. Because of the existence of an invariant measure, a train can get from any part of τ to any other. (The set of points accessible by a given oriented train is a “sink,” 246 Thurston — The Geometry and Topology of 3-Manifolds 9.5. INTERPOLATING NEGATIVELY CURVED SURFACES which can only be a connected component.) If τ is not orientable, then every oriented train can get to any position with any orientation. (Otherwise, the oriented double “cover” of τ would have a non-trivial sink.) In this case, add an arbitrary branch b to τ, cutting a non-atomic region (of area > π). Clearly there is some cyclic train path through b, so τ ∪b admits a positive measure. If τ is oriented, then each region of S −τ has an even number of cusps on its boundary. The area of S must be 4π or greater (since the only complete oriented surfaces of ﬁnite area having χ = −1 are the thrice punctured sphere, for which ML0 9.36 is empty, and the punctured torus). If there is a polygon with more than four sides, it can be subdivided using a branch which preserves orientation, hence admits a cyclic train path. The case of a punctured polygon with more than two sides is similar. Otherwise, S−γ has at least two components. Add one branch b1 which reverses pos-itively oriented trains, in one region, and another branch b2 which reverses negatively oriented trains in another. There is a cyclic train path through b1 and b2 in τ ∪b1 ∪b2, hence an invariant measure. Thurston — The Geometry and Topology of 3-Manifolds 247 9. ALGEBRAIC CONVERGENCE Now consider the case when S −τ has more complexly connected regions. If 9.37 a boundary component of such a region R has one or more vertices, then a train pointing away from R can return to at least one vertex pinting toward R. If R is not an annulus, hook a new branch around a non-trivial homotopy class of arcs in R with ends on such a pair of vertices. If R is an annulus and each boundary component has at least one vertex, then add one or two branches running across R which admit a cyclic train path. If R is not topologically a thrice punctured disk or annulus, we can add an interior closed curve to R. Any boundary component of R which is a geodesic α has another region R′ (which may equal R) on the other side. In this case, we can add one or more branches in R and R′ tangent to α in opposite directions on opposite sides, and hooking in ways 9.38 similar to those previously mentioned. From the existence of these extensions of the original train track, it follows that an element γ ∈ML0 is essentially complete if and only if S −γ consists of triangles and punctured monogons. Furthermore, every γ ∈ML0 can be approximated by essentially complete elements γ′ ∈ML0. In fact, an open dense set has the property that the ϵ-train track approximation τϵ has only triangles and punctured monogons 248 Thurston — The Geometry and Topology of 3-Manifolds 9.5. INTERPOLATING NEGATIVELY CURVED SURFACES as complementary regions, so generically every τϵ has this property. The characteri-zation of essential completeness then holds for ML0 as well. □ Here is some useful geometric information about uncrumpled surfaces. Proposition 9.5.5. (i) The sum of the dihedral angles along all edges of the wrinkling locus w(S) tending toward a cusp of an uncrumpled surface S is 0. (The sum is taken in the group S1 = R mod 2π.) (ii) The sum of the dihedral angles along all edges of w(S) tending toward any 9.39 side of a closed geodesic γ of w(S) is ±α, where α is the angle of rotation of parallel translation around γ. (The sign depends on the sense of the spiralling of nearby geodesics toward γ.) Proof. Consider the upper half-space model, with either the cusp or the end of ˜ γ toward which the geodesics in w(S) are spiralling at ∞. Above some level (in case (a)) or inside some cone (in case (b)), S consists of vertical planes bent along vertical lines. The proposition merely says that the total angle of bending in some fundamental domain is the sum of the parts. □ Corollary 9.5.6. An uncrumpled surface realizing an essentially complete lam-ination in ML0 in a given homotopy class is unique. Such an uncrumpled surface is totally geodesic near its cusps. Proof. If the surface S is not a punctured torus, then it has a unique comple-tion obtained by adding a single geodesic tending toward each cusp. By 9.5.5, an uncrumpled surface cannot be bent along any of these added geodesics, so we obtain 9.40 9.5.6. Thurston — The Geometry and Topology of 3-Manifolds 249 9. ALGEBRAIC CONVERGENCE If S is the punctured torus T −p, then we consider ﬁrst the case of a lamination γ which is an essential completion of a single closed geodesic. Complete γ by adding two closed geodesics going from the vertices of the punctured bigon to the puncture. If the dihedral angles along the inﬁnite geodesics are θ1, θ2 and θ3, as shown, then by 9.5.5 we have θ1 + θ2 = 0, θ1 + θ3 = α, θ2 + θ3 = α, where α is some angle. (The signs are the same for the last two equations because any hyperbolic transformation anti-commutes with a 180◦rotation around any per-pendicular line.) 9.41 Thus θ1 = θ2 = 0, so an uncrumpled surface is totally geodesic in the punctured bigon. Since simple closed curves are dense in ML0, every element g ∈ML0 realizable in a given homotopy class has a realization by an uncrumpled surface which is totally geodesic on a punctured bigon. If γ is essentially complete, this means its realizing surface is unique. □ Proposition 9.5.7. If γ is an essentially complete geodesic lamination, realized by an uncrumpled surface U, then any uncrumpled surface U ′ realizing a lamination γ′ near γ is near U. 250 Thurston — The Geometry and Topology of 3-Manifolds 9.5. INTERPOLATING NEGATIVELY CURVED SURFACES Proof. You can see this from train track approximations. This also follows from the uniqueness of the realization of γ on an uncrumpled surface, since uncrumpled surfaces realizing laminations converging to γ must converge to a surface realizing γ. □ Consider now a typical path γt ∈ML0. The path γt is likely to consist mostly of essentially complete laminations, so that a family of uncrumpled surfaces Ut realizing γt would be usually (with respect to t) continuous. At a countable set of values of t, γt is likely to be essentially incomplete, perhaps having a single complementary quadrilateral. Then the left and right hand limits Ut−and Ut+ would probably exist, 9.42 and give uncrumpled surfaces realizing the two essential completions of γt. In fact, we will show that any path γt can be perturbed slightly to give a “generic” path in which the only essentially incomplete laminations are ones with precisely two distinct completions. In order to speak of generic paths, we need more than the topological structure of ML0. Proposition 9.5.8. ML and ML0 have canonical PL (piecewise linear) struc-tures. Proof. We must check that changes of the natural coordinates coming from maximal train tracks (pp. 8.59-8.60) are piecewise linear. We will give the proof for ML0; the proof for ML is obtained by appropriate modiﬁcations. Let γ be any measured geodesic lamination in ML0(S). Let τ1 and τ2 be maximal compactly supported train tracks carrying γ, deﬁning coordinate systems φ1 and φ2 from neighborhoods of γ to convex subsets of Rn (consisting of measures on τ1 and τ2 ). A close enough train track approximation σ of γ is carried by τ1 and τ2. 9.43 The set of measures on σ go linearly to measures on τ1 and τ2. If σ is a maximal compact train track supporting a measure, we are done—the change of coordinates φ2 ◦φ−1 2 is linear near γ. (In particular, note that if γ is essentially complete, Thurston — The Geometry and Topology of 3-Manifolds 251 9. ALGEBRAIC CONVERGENCE change of coordinates is always linear at γ ). Otherwise, we can ﬁnd a ﬁnite set of enlargements of σ, σ1, . . . , σk, so that every element of a neighborhood of γ is closely approximated by one of the σi. Since every element of a neighborhood of γ is carried by τ1 and τ2, it follows that (if the approximations are good enough) each of the σi is carried by τ1 and τ2. Each σi deﬁnes a convex polyhedron which is mapped linearly by φ1 and φ2, so φ2 ◦φ−1 1 must be PL in a neighborhood of γ. □ Remark 9.5.9. It is immediate that change of coordinates involves only rational coeﬃcients. In fact, with more care ML and ML0 can be given a piecewise integral linear structure. To do this, we can make use of the set D of integer-valued measures supported on ﬁnite collections of simple closed curves (in the case of ML0 ); D is analogous to the integral lattice in Rn. GLn Z consists of linear transformations of Rn which preserve the integral lattice. The set Vτ of measures supported on a given train track τ is the subset of some linear subspace V ⊂Rn which satisﬁes a ﬁnite number of linear inequalities µ(bi) > 0. Thus Vτ is the convex hull of a ﬁnite number of lines, each passing through an integral point. The integral points in U are closed under integral linear combinations (when such a combination is in U), so they determine an 9.44 integral linear structure which is preserved whenever U is mapped linearly to another coordinate system. Note in particular that the natural transformations of ML0 are volume-preserving. The structure on PL and PL0 is a piecewise integral projective structure. We will use the abbreviations PIL and PIP for piecewise integral linear and piecewise integral projective. Definition 9.5.10. The rational depth of an element γ ∈ML0 is the dimension of the space of rational linear functions vanishing on γ, with respect to any natural local coordinate system. From 9.5.8 and 9.5.9, it is clear that the rational depth is independent of coordinates. Proposition 9.5.11. If γ has rational depth 0, then γ is essentially complete. Proof. For any γ ∈ML0 which is not essentially complete we must construct a rational linear function vanishing on γ. Let τ be some train track approximation of γ which can be enlarged and still admit a positive measure. It is clear that the set of measures on τ spans a proper rational subspace in any natural coordinate system coming from a train track which carries τ. (Note that measures on τ consist of positive linear combinations of integral measures, and that every lamination carried by τ is approximable by one not carried by τ.) □ 9.45 Proposition 9.5.12. If γ ∈ML0 has rational depth 1, then either γ is essentially complete or γ has precisely two essential completions. In this case either 252 Thurston — The Geometry and Topology of 3-Manifolds 9.5. INTERPOLATING NEGATIVELY CURVED SURFACES A. γ has no closed leaves, and all complementary regions have area π or 2π. There is only one region with area 2π unless γ is oriented and area(S) = 4π in which case there are two. Such a region is either a quadrilateral or a punctured bigon. or B. γ has precisely one closed leaf γ0. Each region touching γ0 has area 2π. Either 1. S is a punctured torus or 2. γ0 touches two regions, each a one-pointed crown or a devils cap. Proof. Suppose γ has rational depth 1 and is not essentially complete. Let τ be a close train track approximation of γ. There is some ﬁnite set τ1, . . . , τk of essentially complete enlargements of τ which closely approximate every γ′ in a neighborhood of γ. Let σ carry all the τi’s and let Vσ be its coordinate system. The set of γ corresponding to measures carried by a given proper subtrack of a τi is a proper rational subspace of Vσ. Since γ is in a unique proper rational subspace, Vτ, the set of measures Vτi carried on any τi must consist of one side of Vτ. (If Vτi intersected Thurston — The Geometry and Topology of 3-Manifolds 253 9. ALGEBRAIC CONVERGENCE both sides, by convexity γ would come from a measure positive on all branches of τi). Since this works for any degree of approximation of nearby laminations, γ has precisely two essential completions. A review of the proof of 9.5.4 gives the list of possibilities for γ ∈ML0 with precisely two essential completions. The ambiguity in the essential completions comes from the manner of dividing a quadrilateral or other region, and the direction of spiralling around a geodesic. □ 9.47 Remark. There are good examples of γ ∈ML0 which have large rational depth but are essentially complete. The construction will occur naturally in another context. We return to the construction of continuous families of surfaces in a hyperbolic three-manifold. To each essentially incomplete γ ∈ML0 of rational depth 1, we associate a one-parameter family of surfaces Us, with U0 and U1 being the two un-crumpled surfaces realizing γ. Us is constant where U0 and U1 agree, including the union of all triangles and punctured monogons in the complement of γ. The two images of any quadrilateral in S −γ form an ideal tetrahedron. Draw the common perpendicular p to the two edges not in U0 ∩U1, triangulate the quadrilateral with 4 triangles by adding a vertex in the middle, and let this vertex run linearly along p, from U0 to U1. This extends to a homotopy of S straight on the triangles. 254 Thurston — The Geometry and Topology of 3-Manifolds 9.5. INTERPOLATING NEGATIVELY CURVED SURFACES The two images of any punctured bigon in S −γ form a solid torus, with the gen-erating curve parabolic. The union of the two essential completions in this punctured bigon gives a triangulation except in a neighborhood of the puncture, with two new 9.48 vertices at intersection points of added leaves. Draw the common perpendiculars to edges of the realizations corresponding to these intersection points, and homotope U0 to U1 by moving the added vertices lin-early along the common perpendiculars. When γ has a closed leaf γ0, the two essential completions of γ have added leaves spiralling around γ0 in opposite directions. U0 can be homotoped to U1 through surfaces with added vertices on γ0. Thurston — The Geometry and Topology of 3-Manifolds 255 9. ALGEBRAIC CONVERGENCE 9.49 Note that all the surfaces Us constructed above have the property that any point on Us is in the convex hull of a small circle about it on Us. In particular, it has curvature ≤−1; curvature −1 everywhere except singular vertices, where negative curvature is concentrated. Theorem 9.5.13. Given any complete hyperbolic three-manifold N with geomet-rically tame end E cut oﬀby a hyperbolic surface S[ϵ,∞), there is a proper homotopy F : S[ϵ,∞) × [0, ∞) →N of S to ∞in E. Proof. Let Vτ be the natural coordinate system for a neighborhood of ϵ(E) in ML0(S), and choose a sequence γi ∈Vτ limiting on ϵ(E). Perturb the γi slightly so that the path γt [0 ≤t ≤∞] which is linear on each segment t ∈[i, i + 1] consists of elements of rational depth 0 or 1. Let Ut be the unique uncrumpled surface realizing γt when γt is essentially complete. When t is not essentially complete, the left and right hand limits Ut+ and Ut−exist. It should now be clear that F exists, since one can cover the closed set {Ut±} by a locally ﬁnite cover consisting of surfaces homotopic by small homotopies, and ﬁll in larger gaps between Ut+ and Ut−by the homotopies constructed above. Since all interpolated surfaces have curvature ≤−1, and they all realize a γt, they must move out to ∞. An explicit homotopy can actually be deﬁned, using a new parameter r which is obtained by “blowing up” all parameter values of t with rational depth 1 into small intervals. Explicitly, these parameter values can be enumerated in some order {tj}, and an interval of length 2−j inserted 9.50 in the r-parameter in place of tj. Thus, a parameter value t corresponds to the point or interval r(t) = t + X {j\|tj 1, is the standard one, since Sn is simply connected. The binary relation of antipodality is natural in this structure. What would be the antipodal lamination for a simple closed curve α? It is easy to construct a diﬀeomorphism ﬁxing α but moving any other given lamination. (If i(γ, α) ̸= 0, the Dehn twist around α will do.) □ Remark. When PL0(S) is one-dimensional (that is, when S is the punctured torus or the quadruply punctured sphere), the PIP structure does come from a pro-jective structure, equivalent to RP 1. The natural transformations of PL0(S) are necessarily integral—in PSL2(Z). Proof of 9.7.2. Don’t blink. Let γ be essentially complete. For each region Ri of S −γ, consider a smaller region ri of the same shape but with ﬁnite points, rotated so its points alternate with cusps of Ri and pierce very slightly through the sides of Ri, ending on a leaf of γ. 9.64 264 Thurston — The Geometry and Topology of 3-Manifolds 9.7. REALIZATIONS OF GEODESIC LAMINATIONS FOR SURFACE GROUPS By 9.5.4, 9.5.2 and 9.3.9, both ends of each leaf of γ are dense in γ, so the regions ri separate leaves of γ into arcs. Each region of S −γ −Uiri must be a rectangle with two edges on ∂ri and two on γ, since ri covers the “interesting” part of Ri. (Or, prove this by area, χ). Collapse all rectangles, identifying the ri edges with each other, and obtain a surface S′ homotopy-equivalent to S, made of Uiri, where ∂ri projects to a train track τ. (Equivalently, one may think of S −Uiri as made of very wide corridors, with the horizontal direction given approximately by γ). 9.65 If we take shrinking sequences of regions ri,j in this manner, we obtain a sequence of train tracks τj which obviously have the property that τj carries τk when j > k. Let γ′ ∈PL0(S) −∆γ be any lamination not topologically equivalent to γ. From the Thurston — The Geometry and Topology of 3-Manifolds 265 9. ALGEBRAIC CONVERGENCE density in γ of ends of leaves of γ, it follows that whenever leaves of γ and γ′ cross, they cross at an angle. There is a lower bound to this angle. It also follows that γ ∪γ′ cuts S into pieces which are compact except for cusps of S. When Ri is an asymptotic triangle, for instance, it contains exactly one region of S −γ −γ′ which is a hexagon, and all other regions of S −γ −γ′ are rectangles. For suﬃciently high j, the rij can be isotoped, without changing the leaves of γ which they touch, into the complement of γ′. It follows that γ′ projects nicely to τj. □ Stereographic coordinates give a method of computing and understanding inter-section number. The transverse measure for γ projects to a “tangential” measure νγ on each of the train tracks τi: i.e., νγ(b) is the γ-transverse length of the sides of the rectangle projecting to b. 266 Thurston — The Geometry and Topology of 3-Manifolds 9.7. REALIZATIONS OF GEODESIC LAMINATIONS FOR SURFACE GROUPS It is clear that for any α ∈ML0 which is determined by a measure µα on τi 9.7.5. i(α, γ) = X b µα(b) · νγ(b). Thus, in the coordinate system Vτi in ML0, intersection with γ is a linear function. To make this observation more useful, we can reverse the process of ﬁnding a fam-ily of “transverse” train tracks τi depending on a lamination γ. Suppose we are given 9.67 an essentially complete train track τ, and a non-negative function (or “tangential” measure) ν on the branches of b, subject only to the triangle inequalities a + b −c ≥0 a + c −b ≥0 b + c −a ≥0 whenever a, b and c are the total ν-lengths of the sides of any triangle in S −τ. We shall construct a “train track” τ ∗dual to τ, where we permit regions of S −τ ∗to be bigons as well as ordinary types of admissible regions—let us call τ ∗a bigon track. Thurston — The Geometry and Topology of 3-Manifolds 267 9. ALGEBRAIC CONVERGENCE τ ∗is constructed by shrinking each region Ri of S −τ and rotating to obtain a region R∗ i ⊂Ri whose points alternate with points of Ri. These points are joined using one more branch b∗crossing each branch b of τ; branches b∗ 1 and b∗ 2 are conﬂuent at a vertex of R∗whenever b1 and b2 lie on the same side of R. Note that there is a bigon in S −τ ∗for each switch in τ. The tangential measure ν for τ determines a transverse measure deﬁned on the branches of τ ∗of the form b∗. This extends uniquely to a transverse for τ ∗when S is not a punctured torus. 9.68 When S is the punctured torus, then τ must look like this, up to the homeomorphism (drawn on the abelian cover of T −p): Note that each side of the punctured bigon is incident to each branch of τ. Therefore, the tangential measure ν has an extension to a transverse measure ν∗for τ ∗, which is unique if we impose the condition that the two sides of R∗have equal transverse measure. 268 Thurston — The Geometry and Topology of 3-Manifolds 9.7. REALIZATIONS OF GEODESIC LAMINATIONS FOR SURFACE GROUPS 9.69 A transverse measure on a bigon track determines a measured geodesic lamination, by the reasoning of 8.9.4. When τ is an essentially complete train track, an open subset of ML0 is determined by a function µ on the branches of τ subject to a condition for each switch that X b∈I µ(b) = X b∈O µ(b), where I and O are the sets of “incoming” and “outgoing” branches. Dually, “tangen-tial” measure ν on the branches of τ determines an element of ML0 (via ν∗), but two functions ν and ν′ determine the same element if ν is obtained from ν′ by a process of adding a constant to the incoming branches of a switch, and subtracting the same constant from the outgoing branches—or, in other words, if ν −ν′ annihilates all transverse measures for τ (using the obvious inner product ν · µ = P ν(b)µ(b)). In fact, this operation on ν merely has the eﬀect of switching “trains” from one side of a bigon to the other. (Some care must be taken to obtain ν′ from ν by a sequence of elementary “switching” operations without going through negative numbers. We leave this as an exercise to 9.70 the reader.) Given an essentially complete train track τ, we now have two canonical coordinate systems Vτ and V ∗ τ in ML0 or PL0. If γ ∈Vτ and γ∗∈V ∗ τ are deﬁned by measures µγ and νγ∗on τ, then i(γ, γ∗) is given by the inner product i(γ, γ∗) = X b∈τ µγ(b)νγ∗(b). Thurston — The Geometry and Topology of 3-Manifolds 269 9. ALGEBRAIC CONVERGENCE To see this, consider the universal cover of S. By an Euler characteristic or area argument, no path on ˜ τ can intersect a path on ˜ τ ∗more than once. This implies the formula when γ and γ′ are simple geodesics, hence, by continuity, for all measured geodesic laminations. Proposition 9.7.4. Formula 9.7.3 holds for all γ ∈Vτ and γ∗∈V ∗ τ . Intersection number is a bilinear function on Vτ × V ∗ τ (in ML0). □ This can be interpreted as a more intrinsic justiﬁcation for the linear structure on the coordinate systems Vτ—the linear structure can be reconstructed from the embedding of Vτ in the dual space of the vector space with basis γ∗∈V ∗ τ . Corollary 9.7.5. If γ, γ′ ∈ML0 are not topologically conjugate and if at least one of them is essentially complete, then there are neighborhoods U and U ′ of γ and γ′ with linear structures in which intersection number is bilinear. 9.71 Proof. Apply 9.7.4 to one of the train tracks τi constructed in 9.7.2. □ Remark. More generally, the only requirement for obtaining this local bilinearity near γ and γ′ is that the complementary regions of γ ∪γ′ are “atomic” and that S −γ have no closed non-peripheral curves. To ﬁnd an appropriate τ, simply burrow out regions of ri, “transverse” to γ with points going between strands of γ′, so the regions ri cut all leaves of γ into arcs. Then collapse to a train track carrying γ′ and “transverse” to γ, as in 9.7.2. What is the image of Rn of stereographic coordinates Sγ for ML0(S)? To under-stand this, consider a system of train tracks τ1 →τ2 →· · · →τk →· · · deﬁning Sγ. A “transverse” measure for τi pushes forward to a “transverse” measure for τj, for j > i. If we drop the restriction that the measure on τi is non-negative, 9.72 still it often pushes forward to a positive measure on τj. The image of Sγ is the set of 270 Thurston — The Geometry and Topology of 3-Manifolds 9.7. REALIZATIONS OF GEODESIC LAMINATIONS FOR SURFACE GROUPS such arbitrary “transverse” measures on τ1 which eventually become positive when pushed far enough forward. For γ′ ∈∆γ, let νγ′ be a “tangential” measure on τ1 deﬁning γ′. Proposition 9.7.6. The image of Sγ is the set of all “transverse,” not necessarily positive, measures µ on τ1 such that for all γ′ ∈∆γ, νγ′ · µ > 0. (Note that the functions νγ′ · µ and νγ′′ · µ are distinct for γ′ ̸= γ′′.) In particular, note that if ∆γ = γ, the image of stereographic coordinates for ML0 is a half-space, or for PL0 the image is Rn. If ∆γ is a k-simplex, then the image of Sγ for PL0 is of the form int (∆k) × Rn−k. (This image is deﬁned only up to projective equivalence, until a normalization is made.) Proof. The condition that νγ′ · µ > 0 is clearly necessary: intersection number i(γ′, γ′′) for γ′ ∈∆γ, γ′′ ∈Sγ is bilinear and given by the formula 9.73 i(γ′, γ′′) = νγ′ · µγ′′. Consider any transverse measure µ on τ1 such that µ is always non-positive when pushed forward to τi. Let bi be a branch of τi such that the push-forward of µ is non-positive on bi. This branch bi, for high i, comes from a very long and thin rectangle ρi. There is a standard construction for a transverse measure coming from a limit of the average transverse counting measures of one of the sides of ρi. To make this more concrete, one can map ρi in a natural way to τ ∗ j for j ≤i. (In general, whenever an essentially complete train track τ carries a train track σ, then σ∗carries τ ∗ σ →τ σ∗←τ ∗. To see this, embed σ in a narrow corridor around τ, so that branches of τ ∗do not pass through switches of σ. Now σ∗is obtained by squeezing all intersections of branches of τ ∗with a single branch of σ to a single point, and then eliminating any bigons contained in a single region of S −σ.) Thurston — The Geometry and Topology of 3-Manifolds 271 9. ALGEBRAIC CONVERGENCE 9.74 On τ ∗ 1 , ρi is a ﬁnite but very long path. The average number of times ρi tranverses a branch of τ ∗ 1 gives a function νi which almost satisﬁes the switch condition, but not quite. Passing to a limit point of {νi} one obtains a “transverse” measure ν for τ ∗ 1 , whose lamination topologically equals γ, since it comes from a transverse measure on τ ∗ i , for all i. Clearly ν · µ ≤0, since νi comes frm a function supported on a single branch b∗ i of τ ∗ i , and µ(bi) < 0. □ For γ ∈ML0 let Zγ ⊂ML0 consist of γ′ such that i(γ, γ′) = 0. Let Cγ consist of laminations γ′ not intersecting γ, i.e., such that support of γ′ is disjoint from the support of γ. An arbitrary element of Zγ is an element of Cγ, together with some measure on γ. The same symbols will be used to denote the images of these sets in PL0(S). Proposition 9.7.6. The intersection of Zγ with any of the canonical coordinate systems X containing γ is convex. (In ML0 or PL0.) Proof. It suﬃces to give the proof in ML0. First consider the case that γ is a simple closed curve and X = Vτ, for some train track τ carrying γ. Pass to the cylindrical covering space C of S with fundamental group generated by γ. The path of γ on C is embedded in the train track ˜ τ covering τ. From a “transverse” measure m on ˜ τ, construct corridors on C with a metric giving them the proper widths. 9.75 272 Thurston — The Geometry and Topology of 3-Manifolds 9.7. REALIZATIONS OF GEODESIC LAMINATIONS FOR SURFACE GROUPS For any subinterval I of γ, let nxr(I) and nxl(I) be (respectively) the net right hand exiting and the net left hand exiting in the corresponding to I; in computing this, we weight entrances negatively. (We have chosen some orientation for γ). Let i(I) be the initial width of I, and f(I) be the ﬁnal width. If the measure m comes from an element γ′, then γ′ ∈Zγ if and only if there is no “traﬃc” entering the corridor of γ on one side and exiting on the other. This implies the inequalities i(I) ≥nxl(I) and i(I) ≥nxr(I) for all subintervals I. 9.76 It also implies the equation nxl(γ) = 0, Thurston — The Geometry and Topology of 3-Manifolds 273 9. ALGEBRAIC CONVERGENCE so that any traﬃc travelling once around the corridor returns to its inital position. (Otherwise, this traﬃc would spiral around to the left or right, and be inexorably forced oﬀon the side opposite to its entrance.) Conversely, if these inequalities hold, then there is some trajectory going clear around the corridor and closing up. To see this, begin with any cross-section of the corridor. Let x be the supremum of points whose trajectories exit on the right. Follow the trajectory of x as far as possible around the corridor, always staying in the corridor whenever there is a choice. The trajectory can never exit on the left—otherwise some trajectory slightly lower would be forced to enter on the right and exit on the left, or vice versa. Similarly, it can’t exit on the right. Therefore it continues around until it closes up. 9.77 Thus when γ is a simple closed curve, Zγ ∩Vτ is deﬁned by linear inequalities, so it is convex. Consider now the case X = Vτ and γ is connected but not a simple geodesic. Then γ is associated with some subsurface Mγ ⊂S with geodesic boundary deﬁned to be the minimal convex surface containing γ. The set Cγ is the set of laminations not intersecting int (Mγ). It is convex in Vτ, since Cγ = \ {Zα\|α is a simple closed curve ⊂int (Mγ)}. 274 Thurston — The Geometry and Topology of 3-Manifolds 9.7. REALIZATIONS OF GEODESIC LAMINATIONS FOR SURFACE GROUPS A general element γ′ of Zγ is a measure on γ ∪γ′′, so Zγ consists of convex combina-tions of ∆γ and Cγ: hence, it is convex. If γ is not connected, then Zγ is convex since it is the intersection of {Zγi}, where the γi are the components of γ. The case X is a stereographic coordinate system follows immediately. When X = V ∗ τ , consider any essentially complete γ ∈Vτ. From 9.7.5 it follows that V ∗ τ is linearly embedded in Sγ. (Or more directly, construct a train track (without bigons) carrying τ ∗; or, apply the preceding proof to bigon track τ ∗.) □ Remark. Note that when γ is a union of simple closed curves, Cγ in PL0(S) is homeomorphic to PL0(S −γ), regarded as a complete surface with ﬁnite area—i.e., 9.78 Cγ is a sphere. When γ has no component which is a simple closed curve, Cγ is convex. Topologically, it is the join of PL0(S −S Sγ) with the simplex of measures on the boundary components of the Sγi, where the Sγi are subsurfaces associated with the components γi of γ. Now we are in a position to form an image of the set of unrealizable laminations for ρπ1S. Let U+ ⊂PL0 be the union of laminations containing a component of χ+ and deﬁne U−similarly, so that γ is unrealizable if and only if γ ∈U+ ∪U−. U+ is a union of ﬁnitely many convex pieces, and it is contained in a subcomplex of PL0 of codimension at least one. It may be disjoint from U−, or it may intersect U−in an interesting way. Example. Let S be the twice punctured torus. From a random essentially com-plete train track, we compute that ML0 has dimension 4, so PL0 is homeomorphic to S3. For any simple closed curve α on S, Cα is PL0(S −α), 9.79 where S −α is either a punctured torus union a (trivial) thrice punctured sphere, or a 4-times punctured sphere. In either case, Cα is a circle, so Zα is a disk. Thurston — The Geometry and Topology of 3-Manifolds 275 9. ALGEBRAIC CONVERGENCE Here are some sketches of what U+ and U−can look like. Here is another example, where S is a surface of genus 2, and U+(S) ∪U−(S) has the homotopy type of a circle (although its closure is contractible): 9.80 In fact, U+ ∪U−is made up of convex sets Zγ −Cγ, with relations of inclusion as diagrammed: 276 Thurston — The Geometry and Topology of 3-Manifolds 9.9. ERGODICITY OF THE GEODESIC FLOW The closures all contain the element α; hence the closure of the union is starlike: 9.9-1 9.9. Ergodicity of the geodesic ﬂow We will prove a theorem of Sullivan (1979): There is no §9.8 Theorem 9.9.1. Let M n be a complete hyperbolic manifold (of not necessarily ﬁnite volume). Then these four conditions are equivalent: (a) The series X γ∈π1Mn exp −(n −1) d(x0, γx0) diverges. (Here, x0 ∈Hn is an arbitrary point, γx0 is the image of x0 under a covering transformation, and d( , ) is hyperbolic distance). (b) The geodesic ﬂow is not dissipative. (A ﬂow φt on a measure space (X, µ) is dissipative if there exists a measurable set A ⊂X and a T > 0 such that µ(A ∩φt(A)) = 0 for t > T, and X = ∪tφt(A).) (c) The geodesic ﬂow on T1(M) is recurrent. (A ﬂow φt on a measure space (X, µ) is recurrent when for every measure set A ⊂X of positive measure and every T > 0 there is a t ≥T such that µ(A ∩φt(A)) > 0.) (d) The geodesic ﬂow on T1(M) is ergodic. Note that in the case M has ﬁnite volume, recurrence of the geodesic ﬂow is immediate (from the Poincar´ e recurrence lemma). The ergodicity of the geodesic ﬂow in this case was proved by Eberhard Hopf, in ??. The idea of (c) →(d) goes back to Hopf, and has been developed more generally in the theory of Anosov ﬂows ??. 9.9-2 Thurston — The Geometry and Topology of 3-Manifolds 277 9. ALGEBRAIC CONVERGENCE Corollary 9.9.2. If the geodesic ﬂow is not ergodic, there is a non-constant bounded superharmonic function on M. Proof of 9.9.2. Consider the Green’s function g(x) = R ∞ d(x,x0) sin h1−nt dt for hyperbolic space. (This is a harmonic function which blows up at x0.) By (a), the series P γ∈π1M g ◦γ converges to a function, invariant by γ, which projects to a Green’s function G for M. The function f = arctan G (where arctan ∞= π/2 ) is a bounded superharmonic function, since arctan is convex. □ Remark. The convergence of the series (a) is actually equivalent to the existence of a Green’s function on M, and also equivalent to the existence of a bounded super-harmonic function. See (Ahlfors, Sario) for the case n = 2, and [ ] for the general case. Corollary 9.9.3. If Γ is a geometrically tame Kleinian group, the geodesic ﬂow on T1(Hn/Γ) is ergodic if and only if LΓ = S2. Proof of 9.9.3. From 9.9.2 and 8.12.3. □ Proof of 9.9.1. Sullivan’s proof of 9.9.1 makes use of the theory of Brownian motion on M n. This approach is conceptually simple, but takes a certain amount of technical background (or faith). Our proof will be phrased directly in terms of geodesics, but a basic underlying idea is that a geodesic behaves like a random path: its future is “nearly” independent of its past. 9.9-2a 9.9-3 (d) →(c). This is a general fact. If a ﬂow φt is not recurrent, there is some set A of positive measure such that only for t in some bounded interval is µ(A∩φt(A)) > 0. 278 Thurston — The Geometry and Topology of 3-Manifolds 9.9. ERGODICITY OF THE GEODESIC FLOW Then for any subset B ⊂A of small enough measure, ∪tφt(B) is an invariant subset which is proper, since its intersection with A is proper. (c) →(b). Immediate. (b) →(a). Let B be any ball in Hn, and consider its orbit ΓB where Γ = π1M. For the series of (a) to diverge means precisely that the total apparent area of ΓG as seen from a point x0 ∈Hn, (measured with multiplicity) is inﬁnite. In general, the underlying space of a ﬂow is decomposed into two measurable parts, X = D ∪R, where φt is dissipative on D (the union of all subsets of X which eventually do not return) and recurrent on R. The reader may check this elementary fact. If the recurrent part of the geodesic ﬂow is non-empty, there is some ball B in M n such that a set of positive measure of tangent vectors to points of B give rise to geodesics that intersect B inﬁnitely often. This clearly implies that the series of (a) diverges. The idea of the reverse implication (a) →(b) is this: if the geodesic ﬂow is dissipative there are points x0 such that a positive proportion of the visual sphere is not covered inﬁnitely often by images of some ball. Then for each “group” of geodesics that return to B, a deﬁnite proportion must eventually escape ΓB, because future and past are nearly independent. The series of (a) can be regrouped as a geometric progression, so it converges. We now make this more precise. Recall that the term “visual sphere” at x0 is a synonym to the “set of rays” emanating from x0. It has a metric and a measure obtained from its identiﬁcation with the unit sphere in the tangent space at x0. 9.9-4 Let x0 ∈M n be any point and B ⊂M n any ball. If a positive proportion of the rays emanating from x0 pass inﬁnitely often through B, then for a slightly larger ball B′, a deﬁnite proportion of the rays emanating from any point x ∈M n spend an inﬁnite amount of time in B′, since the rays through x are parallel to rays through x0. Consequently, a subset of T1(B′) of positive measure consists of vectors whose geodesics spend an inﬁnite total time in T1(B′); by the Poincar´ e recurrence lemma, the set of such vectors is a recurrent set for the geodesic ﬂow. (b) holds so (a) →(b) is valid in this case. To prove (a) →(b), it remains to consider the case that almost every ray from x0 eventually escapes B; we will prove that (a) fails, i.e., the series of (a) converges. Replace B by a slightly smaller ball. Now almost every ray from almost every point x ∈M eventually escapes the ball. Equivalently, we have a ball B ⊂Hn such that for every point x ∈Hn, almost no geodesic through x intersects ΓB, or even Γ(Nϵ(B)), more than a ﬁnite number of times. Let x0 be the center of B and let α be the inﬁmum, for y ∈Hn, of the diameter of the set of rays from x0 which are parallel to rays from y which intersect B. This inﬁmum is positive, and very rapidly approached as y moves away from x0. Thurston — The Geometry and Topology of 3-Manifolds 279 9. ALGEBRAIC CONVERGENCE 9.9-5 Let R be large enough so that for every ball of diameter greater than α in the visual sphere at x0, at most (say) half of the rays in this ball intersect ΓN∈(B) at a distance greater than R from x0. R should also be reasonably large in absolute terms and in comparison to the diameter of B. Let x0 be the center of B. Choose a subset Γ′ ⊂Γ of elements such that: (i) for every γ ∈Γ there is a γ′ ∈Γ′ with d(γ′x0, γx0) < R. (ii) For any γ1 and γ2 in Γ′, d(γ1x0, γ2x0) ≥R. Any subset of Γ maximal with respect to (ii) satisﬁes (i). We will show that P γ′∈Γ′ exp(−(n −1) d(x0, γ′x0)) converges. Since for any γ′ there are a bounded number of elements γ ∈Γ so that d(γx0, γ′x0) < R, this will imply that the series of (a) converges. Let < be the partial ordering on the elements of Γ′ generated by the relation γ1 < γ2 when γ2B eclipses γ1B (partially or totally) as viewed from x0; extend < to be transitive. Let us denote the image of γB in the visual sphere of x0 by Bγ. Note that when γ′ < γ, the ratio diam(Bγ′)/ diam(Bγ) is fairly small, less than 1/10, say. Therefore ∪γ′<γBγ′ is contained in a ball concentric with Bγ of radius 10/9 that of Bγ. Choose a maximal independent subset ∆1 ⊂Γ′ (this means there is no rela-tion δ1 < δ2 for any δ1, δ2 ∈∆1 ). Do this by successively adjoining any γ whose Bγ has largest size among elements not less than any previously chosen member. Note that area (∪δ∈∆Bδ)/ area(∪γ∈Γ′Bγ) is greater than some deﬁnite (a priori) con-stant: (9/10)n−1 in our example. Inductively deﬁne Γ′ 0 = Γ′, γ′ i+1 = Γ′ i −∆i+1 and deﬁne ∆i+1 ⊂Γi similarly to ∆1. Then Γ′ = ∪∞ i=1∆i. 9.9-6 For any γ ∈Γ′, we can compare the set Bγ of rays through x0 which intersect γ(B) to the set Cγ of parallel rays through γX0. Any ray of Bγ which re-enters Γ′(B) after passing through γ′(B), is within ϵ of the parallel ray of Cγ by that time. At most half of the rays of Cγ ever enter Nϵ(Γ′B). 280 Thurston — The Geometry and Topology of 3-Manifolds 9.9. ERGODICITY OF THE GEODESIC FLOW The distortion between the visual measure of Bγ and that of Cγ is modest, so we can conclude that the set of reentering rays, Bγ ∩S γ′<γ Bγ′, has measure less than 2/3 the measure of Bγ. We conclude that, for each i, area [ γ∈Γ′ i+1 Bγ −area [ γ∈Γ′ i Bγ ≥1/3 area [ δ∈∆i+1 Bδ ≥1/3 · (9/10)n−1 area [ γ∈Γ′ i Bγ . The sequence {area(S γ∈Γ′ i Bγ)} decreases geometrically. This sequence dominates the terms of the series P i area ∪δ∈∆iBδ = P γ∈Γ′ area(Bγ), so the latter converges, which completes the proof of (a) →(b). 9.9-7 (b) →(c). Suppose R ⊂T1(M n) is any recurrent set of positive measure for the geodesic ﬂow φt. Let B be a ball such that R ∩T1(B) has positive measure. Almost every forward geodesic of a vector in R spends an inﬁnite amount of time in B. Let A ⊂T1(B) consist of all vectors whose forward geodesics spend an inﬁnite time in B and let ψt, t ≥0, be the measurable ﬂow on A induced from φt which takes a point leaving A immediately back to its next return to A. Since ψt is measure preserving, almost every point of A is in the image of ψt for all t and an inverse ﬂow ψ−t is deﬁned on almost all of A, so the deﬁnition of A is unchanged under reversal of time. Every geodesic parallel in either direction to a geodesic in A is also in A; it follows that A = T1(B). By the Poincar´ e recurrence lemma, ψt is recurrent, hence φt is also recurrent. (c) →(d). It is convenient to prove this in the equivalent form, that if the action of Γ on Sn−1 ∞ × Sn−1 ∞ is recurrent, it is ergodic. “Recurrent” in this context means that for any set A ⊂Sn−1 ×Sn−1 of positive measure, there are an inﬁnite number of elements γ ∈Γ such that µ(γA∩A) > 0. Let I ⊂Sn−1 ×Sn−1 be any measurable set invariant by Γ. Let −B1 and B2 ⊂Sn−1 be small balls. Let us consider what I must look like near a general point x = (x1, x2) ∈B1 × B2. If γ is a “large” element of Γ such that γx is near x, then the preimage of γ of a product of small ϵ-ball around γx1 and γx2 is one of two types: it is a thin neighborhood of one of the factors, (x1 × B2) or (B1 × x2). (γ must be a translation in one direction or the other along an axis from approximately x1 to approximately x2.) Since Γ is recurrent, almost every point x ∈B1 × B2 is the preimage of elements γ of both types, of an inﬁnite number of 9.9-8 Thurston — The Geometry and Topology of 3-Manifolds 281 9. ALGEBRAIC CONVERGENCE points where I has density 0 or 1. Deﬁne f(x1) = Z B2 χI(x1, x2) dx2, where χI is the characteristic function of I, for x1 ∈B1 (using a probability measure on B2 ). By the above, for almost every x1 there are arbitrarily small intervals around x1 such that the average of f in that interval is either 0 or 1. Therefore f is a characteristic function, so I ∩B1 ×B2 is of the form S ×B2 (up to a set of measure zero) for some set S ⊂B1. Similarly, I is of the form B1 × R, so I is either ∅× ∅or B1 × B2 (up to a set of measure zero). □ 282 Thurston — The Geometry and Topology of 3-Manifolds William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T EX by Sheila Newbery, who also scanned the ﬁgures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the ﬁrst few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at levy@msri.org. NOTE Since a new academic year is beginning, I am departing from the intended order in writing these notes. For the present, the end of chapter 9 and chapters 10, 11 and 12, which depend heavily on chapters 8 and 9, are to be omitted. The tentative plan for the omitted parts is to cover the following topics: The end of chapter 9—a more general discussion of algebraic convergence. Chapter 10—Geometric convergence: an analysis of the possibilities for geometric limits. Chapter 11. The Riemann mapping theorem; parametrizing quasi-conformal de-formations. Extending quasi-conformal deformations of S2 ∞to quasi-isometric defor-mations of H3. Examples; conditions for the existence of limiting Kleinian groups. Chapter 12. Boundaries for Teichm¨ uller space, classiﬁcation of diﬀeomorphisms of surfaces, algorithms involving the mapping class group of a surface. Thurston — The Geometry and Topology of 3-Manifolds 283 CHAPTER 11 Deforming Kleinian manifolds by homeomorphisms of the sphere at inﬁnity A pseudo-isometry between hyperbolic three-manifolds gives rise to a quasi-con-formal map between the spheres at inﬁnity in their universal covering spaces. This is a key point in Mostow’s proof of his rigidity theorem (Chapter 5). In this chapter, we shall reverse this connection, and show that a k-quasi-conformal map of S2 ∞to itself gives rise to a k-quasi-isometry of hyperbolic space to itself. A self-map f : X →X of a metric space is a k-quasi-isometry if 1 kd(fx, fy) ≤d(x, y) ≤kd(fx, fy) for all x and y. By use of a version of the Riemann mapping theorem, the space of quasi-conformal maps of S2 can be parametrized by the non-conformal part of their derivatives. In this way we obtain a remarkable global parametrization of quasi-isometric deformations of Kleinian manifolds by the Teichm¨ uller spaces of their boundaries. 11.2 11.1. Extensions of vector ﬁelds In §§8.4 and 8.12, we made use of the harmonic extensions of measurable functions on S∞to study the limit set of a Kleinian group. More generally, any tensor ﬁeld on S2 ∞extends, by a visual average, over H3. To do this, ﬁrst identify S2 ∞with the unit sphere in Tx(H3), where x is a given point in H3. If y ∈S2 ∞, this gives an identiﬁcation i : Ty(S2 ∞) →Tx(H3). There is a reverse map p : Tx(H3) →Ty(S2 ∞) coming from orthogonal projection to the image of i. We can use i∗and p∗to take care of covariant tensor ﬁelds, like vector ﬁelds, and contravariant tensor ﬁelds, like diﬀerential forms and quadratic forms, as well as tensor ﬁelds of mixed type. The visual average of any tensor ﬁeld T on S2 ∞is thus a tensor ﬁeld av T, of the same type, on H3. In general, av T needs to be modiﬁed by a constant to give it the right boundary behavior. We need some formulas in order to make computations in the upper half-space model. Let x be a point in upper half-space, at Euclidean height h above the bounding Thurston — The Geometry and Topology of 3-Manifolds 285 11. DEFORMING KLEINIAN MANIFOLDS plane C. A geodesic through x at angle θ from the vertical hits C at a distance r = h cot(θ/2) from the foot z0 of the perpendicular from x to C. 11.3 Thus, dr = −(h/2) csc2(θ/2) dθ = −1 2(h+r2/h) dθ. Since the map from the visual sphere at x to S2 ∞is conformal, it follows that dVx = 4 h + r2 h −2 dµ, where µ is Lebesgue measure on C and Vx is visual measure at x. Any tensor T at the point x pushes out to a tensor ﬁeld T∞on S2 ∞= ˆ C by the maps i∗and p∗. When X is a vector, then X∞is a holomorphic vector ﬁeld, with derivative ﬁeld, with derivative ±∥X∥at its zeros. To see this, let τX be the vector ﬁeld representing the inﬁnitesimal isometry of translation in the direction X. The claim is that X∞= τX\|S∞. This may be seen geometrically when X is at the center in the Poincar´ e disk model. Alternatively if X is a vertical unit vector in the upper half-space, then we can compute that X∞= −sin θ ∂ ∂θ = h 2 sin θ sin2 θ/2 ∂ ∂r = r ∂ ∂r = (z −z0) ∂ ∂z, where z0 is the foot of the perpendicular from x to C. This clearly agrees with the corresponding inﬁnitesimal isometry. (As a “physical” vector ﬁeld, ∂/∂z is the same as the unit horizontal vector ﬁeld, ∂/∂x, on C. The reason for this notation is that the diﬀerential operators ∂/∂x and ∂/∂z have the same action on holomorphic 286 Thurston — The Geometry and Topology of 3-Manifolds 11.1. EXTENSIONS OF VECTOR FIELDS functions: they are directional derivatives in the appropriate direction. Even though 11.4 the complex notation may at ﬁrst seem obscure, it is useful because it makes it meaningful to multiply vectors by complex numbers.) When g is the standard inner product on Tx(H3), then g∞(Y1, Y2) = 4 h + r2 h −2 Y1 · Y2, where Y1 · Y2 is the inner product of two vectors on C. Let us now compute av(∂/∂z). By symmetry considerations, it is clear that av(∂/∂z) is a horizontal vector ﬁeld, parallel to ∂/∂z. Let e be the vector of unit hyperbolic length, parallel to ∂/∂z at a point x in upper half-space. Then e∞= −1 2h(z −z0 −h)(z −z0 + h) ∂ ∂z. We have av ∂ ∂z = 1 4π Z S2 ix ∂ ∂z dVx, so av ∂ ∂z · e = 1 4π Z C g∞ ∂ ∂z, e∞ dVx = 1 4π Z C Re −1 2h(z −z0)2 −h2 16 h + r2 h −4 dµ. Clearly, by symmetry, the term involving Re(z −z0)2 integrates to zero, so we have 11.5 av ∂ ∂z · e = 1 4π Z ∞ 0 Z 2π 0 r dθ · 8h h + r2 h −4 dr = −2h2 3 h + r2 h −3 ∞ 0 = 2 3 1 h. Note that the hyperbolic norm of av(∂/∂z) goes to ∞as h →0, while the Euclidean norm is the constant 2 3. Thurston — The Geometry and Topology of 3-Manifolds 287 11. DEFORMING KLEINIAN MANIFOLDS We now introduce the fudge factor by deﬁning the extension of a vector ﬁeld X on S2 ∞to be ex(X) = 3 2 av(X) in H3, X on S2 ∞. Proposition 11.1.1. If X is continuous or Lipschitz, then so is ex(X). If X is holomorphic, then ex(X) is an inﬁnitesimal isometry. Proof. When X is an inﬁnitesimal translation of C, then ex(X) is the same inﬁnitesimal translation of upper half-space. Thus every “parabolic” vector ﬁeld (with a zero of order 2) on S2 ∞extends to the correct inﬁnitesimal isometry. A general holomorphic vector ﬁeld on S2 ∞is of the form (az2 + bz + c)(∂/∂z) on C. Since such a vector ﬁeld can be expressed as a linear combination of the parabolic vector ﬁelds ∂/∂z, z2 ∂/∂z and (z−1)2 ∂/∂z, it follows that every holomorphic vector ﬁeld extends to the correct inﬁnitesimal isometry. Suppose X is continuous, and consider any sequence {xi} of points in H3 converg-ing to a point at inﬁnity. Bring xi back to the origin O by the translation τi along 11.6 the line Oxi. If xi is close to S2 ∞, τi spreads a small neighborhood of the endpoint yi of the geodesic from O to xi over almost all the sphere. τi∗X is large on most of the sphere, except near the antipodal point to yi, so it is close to a parabolic vector ﬁeld Pi, in the sense that for any ϵ, and suﬃciently high i, ∥τi∗X −Pi∥≤ϵ · λi, where λi is the norm of the derivative of τi at yi. Here Pi is the parabolic vector ﬁeld agreeing with τi∗X at yi, and 0 at the antipodal point of yi. It follows that ex X(xi) −X(yi) →0, so X is continuous along ∂B3. Continuity in the interior is self-evident (if you see the evidence). Suppose now that X is a vector ﬁeld on S2 ∞⊂R3 which has a global Lipschitz constant k = sup y,y′∈S2 ∥Xy −Xy′∥ ∥y −y′∥ . 288 Thurston — The Geometry and Topology of 3-Manifolds 11.1. EXTENSIONS OF VECTOR FIELDS Then the translates τi∗X satisfy ∥τi∗X −Pi∥≤B, where B is some constant independent of i. This may be seen by considering stereo-graphic projection from the antipodal point of yi. The part of the image of X −τ −1 i∗Pi in the unit disk is Lipschitz and vanishes at the origin. When τi∗is applied, the re-sulting vector ﬁeld on C satisﬁes a linear growth condition (with a uniform growth constant). This shows that, on S2 ∞, ∥τi∗X −Pi∥is uniformly bounded in all but a 11.7 neighborhood of the antipodal point of Y , where boundedness is obvious. Then ∥ex X(xi) −ex τi∗Pi(xi)∥≤B · µi, where µi is the norm of the derivative of τ −1 i at the origin in B3, or 1/λi up to a bounded factor. Since µi is on the order of the (Euclidean) distance of xi from yi, it follows that ex X is Lipschitz along S2 ∞. To see that ex X has a global Lipschitz constant in B3, consider x ∈B3, and let τ be a translation as before taking x to O, and P a parabolic vector ﬁeld approximating τ∗X. The vector ﬁelds τ∗X −P obtained in this way are uniformly bounded, so it is clear that the vector ﬁelds ex(τ∗X −P) have a uniform Lipschitz constant at the origin in B3. By comparison with the upper half-space model, where τ∗can be taken to be a similarity, we obtain a uniform bound on the local Lipschitz constant for ex(X −τ −1 ∗P) at an arbitrary point x. Since the vector ﬁelds τ −1 ∗P are uniformly Lipschitz, it follows that X is globally Lipschitz. □ Note that the stereographic image in C of a uniformly Lipschitz vector ﬁeld on S2 ∞is not necessarily uniformly Lipschitz—consider z2 ∂/∂z, for example. This is explained by the large deviation of the covariant derivatives on S2 ∞and on C near the point at inﬁnity. Similarly, a uniformly Lipschitz vector ﬁeld on B3 is not generally uniformly Lipschitz on H3. In fact, because of the curvature of H3, a uniformly Lipschitz vector ﬁeld on H3 must be bounded; such vector ﬁelds correspond precisely to those Lipschitz vector ﬁelds on B3 which vanish on ∂B3. 11.8 Thurston — The Geometry and Topology of 3-Manifolds 289 11. DEFORMING KLEINIAN MANIFOLDS A hyperbolic parallel vector ﬁeld along a curve near S1 ∞appears to turn rapidly. The signiﬁcance of the Lipschitz condition stems from the elementary fact that Lipschitz vector ﬁelds are uniquely integrable. Thus, any isotopy ϕt of the boundary of a Kleinian manifold OΓ = (B3 −LΓ)/Γ whose time derivative ˙ ϕt is Lipschitz as a vector ﬁeld on I × ∂OΓ extends canonically to an isotopy ex ϕt on OΓ. One may see this most simply by observing that the proof that ex X is Lipschitz works locally. A k-quasi-isometric vector ﬁeld is a vector ﬁeld whose ﬂow, ϕt, distorts distances at a rate of at most k. In other words, for all x, y and t, ϕt must satisfy e−ktd(x, y) ≤d(ϕtx, ϕty) ≤ektd(x, y). A k-Lipschitz vector ﬁeld on a Riemannian manifold is k-quasi-isometric. In fact, a Lipschitz vector ﬁeld X on B3 which is tangent to ∂B3 is quasi-isometry as a vector ﬁeld on H3 = int B3. This is clear in a neighborhood of the origin in B3. To see this for an arbitrary point x, approximate X near x by a parabolic vector ﬁeld, as in the proof of 11.1.1, and translate x to the origin. In particular, if ϕt is an isometry of ∂OΓ with Lipschitz time derivative, then ex ϕt has a quasi-isometric time derivative, and ϕ1 is a quasi-isometry. Our next step is to study the derivatives of ex X, so we can understand how a 11.9 more general isotopy such as ex ϕt distorts the hyperbolic metric. From the deﬁnition of ex X, it is clear that ex is natural, or in other words, ex(T∗X) = T∗ ex(X) where T is an isometry of H3 (extended to S2 ∞where appropriate). If X is diﬀerentiable, we can take the derivative at T = id, yielding ex [Y, X] = [Y, ex X] for any inﬁnitesimal isometry Y . If Y is a pure translation and X is any point on the axis of Y , then ∇XYx = 0. (Here, ∇is the hyperbolic covariant derivative, so ∇ZW is the directional derivative of a vector ﬁeld W in the direction of the vector ﬁeld Z.) Using the formula [Y, X] = ∇Y X −∇XY, we obtain: Proposition 11.1.2. The direction derivative of ex X in the direction Yx, at a point x ∈H3, is ∇Yx ex X = ex[Y, X], where Y is any inﬁnitesimal translation with axis through x and value Yx at x. □ 290 Thurston — The Geometry and Topology of 3-Manifolds 11.1. EXTENSIONS OF VECTOR FIELDS The covariant derivative ∇Xx, which is a linear transformation of the tangent space Tx(H3) to itself, can be expressed as the sum of its symmetric and antisym-metric parts, 11.10 ∇X = ∇sX + ∇aX, where ∇s Y X · Y ′ = 1 2(∇Y X · Y ′ + ∇Y ′X · Y ) and ∇a Y X · Y ′ = 1 2(∇Y X · Y ′ −∇Y ′X · Y ). The anti-symmetric part ∇aX describes the inﬁnitesimal rotational eﬀect of the ﬂow generated by X. It can be described by a vector ﬁeld curl X pointing along the axis of the inﬁnitesimal rotation, satisfying the equation ∇a Y X = 1 2 curl X × Y where × is the cross-product. If e0, e1, e2 forms a positively oriented orthonormal frame at X, the formula is curl X = X i∈Z/3 (∇eiX · ei+1 −∇ei+1X · ei) ei+2. Consider now the contribution to ex X from the part of X on an inﬁnitesimal area on S2 ∞, centered at y. This part of ex X has constant length on each horosphere about y (since the ﬁrst derivative of a parabolic transformation ﬁxing y is the identity), and it scales as e−3t, where t is a parameter measuring distance between horospheres and increasing away from y. (Linear measurements scale as e−t. Hence, there is a factor of e−2t describing the scaling of the apparent area from a point in H3, and a factor of −et representing the scaling of the lengths of vectors.) Choose positively oriented coordinates t, x1, x2, so that ds2 = dt2 +e2t(dx2 1 +dx2 2), and this inﬁnitesimal contribution to ex X is in the ∂/∂x1 direction. Let e0, e1 and e2 be unit vectors in the three coordinate directions. The horospheres t = constant are parallel surfaces, of constant normal curvature 1 (like the unit sphere in R3), so you can see that 11.11 ∇e0e0 = ∇e0e1 = ∇e0e2 = 0 ∇e1e0 = +e1, ∇e1e1 = −e0, ∇e1e2 = 0 and ∇e2e0 = e2, ∇e2e2 = −e0, ∇e2e1 = 0. (This information is also easy to compute by using the Cartan structure equations.) The inﬁnitesimal contribution to ex X is proportional to Z = e−3te1, so curl Z = (∇e0Z · e1 −∇e1Z · e0) e2 = −2e−3te2. Thurston — The Geometry and Topology of 3-Manifolds 291 11. DEFORMING KLEINIAN MANIFOLDS (The curl is in the opposite sense from the curving of the ﬂow lines because the eﬀect of the ﬂow speeding up on inner horospheres is stronger.) This is proportional to the contribution of iX to ex iX from the same inﬁnitesimal region, so we have Proposition 11.1.3. Curl (ex X) = 2 ex(iX), and consequently Curl2 (ex X) = −4 ex X and Div (ex X) = 0. Proof. The ﬁrst statement follows by integration of the inﬁnitesimal contribu-tions to curl ex X. The second statement curl curl ex X = 2 curl ex iX = 4 ex i2X = −4 ex X, is immediate. The third statement follows from the identity div curl Y = 0, or by considering the inﬁnitesimal contributions to ex X. □ The diﬀerential equation curl2 ex X+ex X = 0 is the counterpart to the statement that ex f = av f is harmonic, when f is a function. The symmetric part ∇sX of the covariant derivative measures the inﬁnitesimal strain, or distortion of the metric, of the ﬂow generated by X. That is, if Y and Y ′ are vector ﬁelds invariant by the ﬂow of X, so that [X, Y ] = [X, Y ′] = 0, then ∇Y X = ∇XY and ∇Y ′X = ∇XY ′, so the derivative of the dot product of Y and Y ′ in the direction X, by the Leibniz rule is X(Y · Y ′) = ∇XY · Y ′ + Y · ∇XY ′ = ∇Y X · Y ′ + ∇Y ′X · Y = 2(∇s Y X · Y ′). The symmetric part of ∇can be further decomposed into its eﬀect on volume and a part with trace 0, ∇sX = 1 3 Trace(∇sX) · I + ∇s0X. 292 Thurston — The Geometry and Topology of 3-Manifolds 11.1. EXTENSIONS OF VECTOR FIELDS Here, I represents the identity transformation (which has trace 3 in dimension 3). Note that trace ∇sX = trace ∇X = divergence X = Σ∇eiX · ei where {ei} is an orthonormal basis, so for a vector ﬁeld of the form ex X, ∇s ex X = ∇s0 ex X. 11.13 Now let us consider the analagous decomposition of the covariant derivative ∇X of a vector ﬁeld on the Riemann sphere (or any surface). There is a decomposition ∇X = ∇aX + 1 2(trace ∇X)I + ∇s0X. Deﬁne linear maps ∂and ∂of the tangent space to itself by the formulas ∂X(Y ) = 1 2{∇Y X −i∇iY X} and ∂X(Y ) = 1 2{∇Y X + i∇iY X} for any vector ﬁeld Y . (On a general surface, i is interpreted as a 90◦counter-clockwise rotation of the tangent space of the surface.) Proposition 11.1.4. ∂X = 1 2(trace ∇X)I + ∇aX = 1 2{(div X)I + (curl X)iI} and ∂X = ∇s0X. ∂X is invariant under conformal changes of metric. Remark (Notational remark). Any vector ﬁeld on C be written X = f(z)∂/∂z, in local coordinates. The derivative of f can be written d f = fxdx + fydy. This can be re-expressed in terms of dz = dx + idy and d¯ z = dx −idy as d f = fzdz + f¯ zd¯ z where fz = 1 2(fx −ify) and 11.14 f¯ z = 1 2(fx + ify). Then ∂f = fzdz and ¯ ∂f = f¯ zd¯ z are the complex linear and complex conjugate linear parts of the real linear map d f. Similarly, ∂X = fz dz ∂/∂z and ¯ ∂X = f¯ z d¯ z ∂/∂z are the complex linear and conjugate linear parts of the map dX = ∇X. Proof. If L : C →C is any real linear map, then L = 1 2(L −i ◦L ◦i) + 1 2(L + i ◦L ◦i) Thurston — The Geometry and Topology of 3-Manifolds 293 11. DEFORMING KLEINIAN MANIFOLDS is clearly the decomposition into its complex linear and conjugate linear parts. A complex linear map, in matrix form a −b b a , is an expansion followed by a rotation, while a conjugate linear map in matrix form a b b −a , is a symmetric map with trace 0. To see that ¯ ∂X is invariant under conformal changes of metric, note that ∇XiY = i∇XY and write ¯ ∂X without using the metric as ¯ ∂X(Y ) = 1 2{∇Y X + i∇iY X} = 1 2{∇Y X −∇XY + i∇iY X −i∇XiY } = 1 2{[Y, X] + i[iY, X]}. □ We can now derive a nice formula for ∇s ex X: Proposition 11.1.5. For any vector Y ∈Tx(H3) and any C1 vector ﬁeld X on S2 ∞, 11.15 ∇s Y ex X = 3/4π Z S2 ∞ i∗ ¯ ∂X(Y∞) dVx. Proof. Clearly both sides are symmetric linear maps applied to Y , so it suﬃces to show that the equation gives the right value for ∇Y ex X ·Y . From 11.1.2, we have ∇Y ex X · Y = ex [Y∞, X] · Y = 3/8π Z S2[Y∞, X] · Y∞dVx and also, at the point x (where ex iY∞= 0), 0 = [ex iY∞, X] · ex iY∞ = 3/8π Z S2[iY∞, X] · iY∞dVx = 3/8π Z S2 −i [iY∞, X] · Y∞dVx. 294 Thurston — The Geometry and Topology of 3-Manifolds 11.1. EXTENSIONS OF VECTOR FIELDS Therefore ∇s Y ex X · Y = ∇Y ex X · Y = 3/8π Z S2[Y∞, X] · Y∞+ i [iY∞, X] · Y∞dVx = 3/4π Z ¯ ∂X(Y∞) dVx Y. □ Thurston — The Geometry and Topology of 3-Manifolds 295 William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T EX by Sheila Newbery, who also scanned the ﬁgures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the ﬁrst few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at levy@msri.org. CHAPTER 13 Orbifolds As we have had occasion to see, it is often more eﬀective to study the quotient manifold of a group acting freely and properly discontinuously on a space rather than to limit one’s image to the group action alone. It is time now to enlarge our vocabulary, so that we can work with the quotient spaces of groups acting properly discontinuously but not necessarily freely. In the ﬁrst place, such quotient spaces will yield a technical device useful for showing the existence of hyperbolic structures on many three-manifolds. In the second place, they are often simpler than three-manifolds tend to be, and hence they often give easy, graphic examples of phenomena involving three-manifolds. Finally, they are beautiful and interesting in their own right. 13.1. Some examples of quotient spaces. We begin our discussion with a few examples of quotient spaces of groups acting properly discontinuously on manifolds in order to get a taste of their geometric ﬂavor. Example 13.1.1 (A single mirror). Consider the action of Z2 on R3 by reﬂection in the y −z plane. The quotient space is the half-space x ≥0. Physically, one may imagine a mirror placed on the y −z wall of the half-space x ≥0. The scene as viewed by a person in this half-space is like all of R3, with scenery invariant by the Z2 symmetry. 13.2 Thurston — The Geometry and Topology of 3-Manifolds 297 13. ORBIFOLDS Example 13.1.2 (A barber shop). Consider the group G generated by reﬂections in the planes x = 0 and x = 1 in R3. G is the inﬁnite dihedral group D∞= Z2 ∗Z2. The quotient space is the slab 0 ≤x ≤1. Physically, this is related to two mirrors on parallel walls, as commonly seen in a barber shop. Example 13.1.3 (A billiard table). Let G be the group of isometries of the Euclidean plane generated by reﬂection in the four sides of a rectangle R. G is isomorphic to D∞× D∞, and the quotient space is R. A physical model is a billiard table. A collection of balls on a billiard table gives rise to an inﬁnite collection of balls on R2, invariant by G. (Each side of the billiard table should be one ball diameter larger than the corresponding side of R so that the centers of the balls can take any position in R. A ball may intersect its images in R2.) Ignoring spin, in order to make ball x hit ball y it suﬃces to aim it at any of the images of y by G. (Unless some ball is in the way.) Example 13.1.4 (A rectangular pillow). Let H be the subgroup of index 2 which preserves orientation in the group G of the preceding example. A fundamental do-13.3 main for H consists of two adjacent rectangles. The quotient space is obtained by identifying the edges of the two rectangles by reﬂection in the common edge. Topologically, this quotient space is a sphere, with four distinguished points or singu-lar points, which come from points in R2 with non-trivial isotropy (Z2). The sphere inherits a Riemannian metric of 0 curvature in the complement of these 4 points, and 298 Thurston — The Geometry and Topology of 3-Manifolds 13.1. SOME EXAMPLES OF QUOTIENT SPACES. it has curvature Kpi = π concentrated at each of the four points pi. In other words, a neighborhood of each point pi is a cone, with cone angle π = 2π −Kpi. Exercise. On any tetrahedron in R3 all of whose four sides are congruent, every geodesic is simple. This may be tested with a cardboard model and string or with strips of paper. Explain. Example 13.1.5 (An orientation-preserving crystallographic group). Here is one more three-dimensional example to illustrate the geometry of quotient spaces. Con-13.4 sider the 3 families of lines in R3 of the form (t, n, m+ 1 2), (m+ 1 2, t, n) and (n, m+ 1 2, t) where n and m are integers and t is a real parameter. They intersect a cube in the unit lattice as depicted. Let G be the group generated by 180◦rotations about these lines. It is not hard to see that a fundamental domain is a unit cube. We may construct the quotient space by making all identiﬁcations coming from non-trivial elements of G acting on the faces of the cube. This means that each face must be folded shut, like a book. In doing this, we will keep track of the images of the axes, which form the singular locus. Thurston — The Geometry and Topology of 3-Manifolds 299 13. ORBIFOLDS 13.5 As you can see by studying the picture, the quotient space is S3 with singular locus consisting of three circles in the form of Borromean rings. S3 inherits a Euclidean structure (or metric of zero curvature) in the complement of these rings, with a cone-type singularity with cone angle π along the rings. In these examples, it was not hard to construct the quotient space from the group action. In order to go in the opposite direction, we need to know not only the quotient space, but also the singular locus and appropriate data concerning the local behavior of the group action above the singular locus. 13.2. Basic deﬁnitions. An orbifold∗O is a space locally modelled on Rn modulo ﬁnite group actions. Here is the formal deﬁnition: O consists of a Hausdorﬀspace XO, with some additional structure. XO is to have a covering by a collection of open sets {Ui} closed under ﬁnite intersections. To each Ui is associated a ﬁnite group Γi, an action of Γi on an open subset ˜ Ui of Rn and a homeomorphism ϕi : Ui ≈˜ Ui/Γi. Whenever Ui ⊂Uj, ∗This terminology should not be blamed on me. It was obtained by a democratic process in my course of 1976-77. An orbifold is something with many folds; unfortunately, the word “manifold” already has a diﬀerent deﬁnition. I tried “foldamani,” which was quickly displaced by the suggestion of “manifolded.” After two months of patiently saying “no, not a manifold, a manifoldead,” we held a vote, and “orbifold” won. 300 Thurston — The Geometry and Topology of 3-Manifolds 13.2. BASIC DEFINITIONS. there is to be an injective homomorphism fij : Γi , →Γj and an embedding ˜ ϕij : ˜ Ui , →˜ Uj equivariant with respect to fij (i.e., for γ ∈Γi, ˜ ϕij(γx) = fij(γ) ˜ ϕij(x)) such that the 13.6 diagram below commutes.† ˜ Ui ˜ ϕij - ˜ Uj ˜ Ui/Γi ? ϕij = ˜ ϕij/Γi - ˜ Uj/Γi ? ˜ Uj/Γj fij ? Ui ϕi 6 ⊂ Uj ϕj 6 We regard ˜ ϕij as being deﬁned only up to composition with elements of Γj, and fij as being deﬁned up to conjugation by elements of Γj. It is not generally true that ˜ ϕik = ˜ ϕjk ◦˜ ϕij when Ui ⊂Uj ⊂Uk, but there should exist an element γ ∈Γk such that γ ˜ ϕik = ˜ ϕjk ◦˜ ϕij and γ · fik(g) · γ−1 = fjk ◦fij(g). Of course, the covering {Ui} is not an intrinsic part of the structure of an orb-ifold: two coverings give rise to the same orbifold structure if they can be combined consistently to give a larger cover still satisfying the deﬁnitions. A G-orbifold, where G is a pseudogroup, means that all maps and group actions respect G. (See chapter 3). Example 13.2.1. A closed manifold is an orbifold, where each group Γi is the trivial group, so that ˜ U = U. Example 13.2.2. A manifold M with boundary can be given an orbifold structure mM in which its boundary becomes a “mirror.” Any point on the boundary has a neighborhood modelled on Rn/Z2, where Z2 acts by reﬂection in a hyperplane. 13.7 †The commutative diagrams in Chapter 13 were made using Paul Taylor’s diagrams.sty package (available at ftp://ftp.dcs.qmw.ac.uk/pub/tex/contrib/pt/diagrams/). —SL Thurston — The Geometry and Topology of 3-Manifolds 301 13. ORBIFOLDS Proposition 13.2.1. If M is a manifold and Γ is a group acting properly dis-continuously on M, then M/Γ has the structure of an orbifold. Proof. For any point x ∈M/Γ, choose ˜ x ∈M projecting to x. Let Ix be the isotropy group of ˜ x (Ix depends of course on the particular choice ˜ x.) There is a neighborhood ˜ Ux of ˜ x invariant by Ix and disjoint from its translates by elements of Γ not in Ix. The projection of Ux = ˜ Ux/Ix is a homeomorphism. To obtain a suitable cover of M/Γ, augment some cover {Ux} by adjoining ﬁnite intersections. Whenever Ux1 ∩. . . ∩Uxk ̸= ∅, this means some set of translates γ1 ˜ Ux1 ∩. . . ∩γk ˜ Ukk has a corresponding non-empty intersection. This intersection may be taken to be ^ z }\| { Ux1 ∩· · · ∩Uxk, with associated group γ1Ix1γ−1 1 ∩· · · ∩γkIxkγ−1 k acting on it. □ The orbifold mM arises in this way, for instance: it is obtained as the quotient space of the Z2 action on the double dM of M which interchanges the two halves. 13.8 Henceforth, we shall use the terminology M/Γ to mean M/Γ as an orbifold. Note that each point x in an orbifold O is associated with a group Γx, well-deﬁned up to isomorphism: in a local coordinate system U = ˜ U/Γ, Γx is the isotropy group of any point in ˜ U corresponding to x. (Alternatively Γx may be deﬁned as the smallest group corresponding to some coordinate system containing x.) The set ΣO = {x\|Γx ̸= {1}} is the singular locus of O. We shall say that O is a manifold when ΣO = ∅. Warning. It happens much more commonly that the underlying space XO is a topological manifold, especially in dimensions 2 and 3. Do not confuse properties of O with properties of XO. 302 Thurston — The Geometry and Topology of 3-Manifolds 13.2. BASIC DEFINITIONS. The singular locus is a closed set, since its intersection with any coordinate patch is closed. Also, it is nowhere dense. This is a consequence of the fact that a non-trivial homeomorphism of a manifold which ﬁxes an open set cannot have ﬁnite order. (See Newman, 1931. In the diﬀerentiable case, this is an easy exercise.) When M in the proposition is simply connected, then M plays the role of universal covering space and Γ plays the role of the fundamental group of the orbifold M/Γ, (even though the underlying space of M/Γ may well be simply connected, as in the examples of §13.1). To justify this, we ﬁrst deﬁne the notion of a covering orbifold. Definition 13.2.2. A covering orbifold of an orbifold O is an orbifold ˜ O, with a projection p : X →XO between the underlying spaces, such that each point x ∈XO has a neighborhood U = ˜ U/Γ (where ˜ U is an open subset of Rn) for which each 13.9 component vi of p−1(U) is isomorphic to ˜ U/Γi, where Γi ⊂Γ is some subgroup. The isomorphism must respect the projections. Note that the underlying space X ˜ O is not generally a covering space of XO. As a basic example, when Γ is a group acting properly discontinuously on a manifold M, then M is a covering orbifold of M/Γ. In fact, for any subgroup Γ′ ⊂Γ, M/Γ′ is a covering orbifold of M/Γ. Thus, the rectangular pillow (13.1.4) is a two-fold covering space of the billiard table (13.1.3). Here is another explicit example to illustrate the notion of covering orbifold. Let S be the inﬁnite strip 0 ≤x ≤1 in R2; consider the orbifold mS. Some covering spaces of S are depicted below. Thurston — The Geometry and Topology of 3-Manifolds 303 13. ORBIFOLDS Definition 13.2.3. An orbifold is good if it has some covering orbifold which is a manifold. Otherwise it is bad. The teardrop is an example of a bad orbifold. The underlying space for a teardrop is S2. ΣO consists of a single point, whose neighborhood is modelled on R2/Zn, where Zn acts by rotations. By comparing possible coverings of the upper half with possible coverings of the lower half, you may easily see that the teardrop has no non-trivial connected coverings. Similarly, you may verify that an orbifold O with underlying space XO = S2 having only two singular points associated with groups Zn and Zn is bad, unless n = m. The orbifolds with three or more singular points on S2, as we shall see, are always good. For instance, the orbifold below is S2 modulo the orientation-preserving symmetries of a dodecahedron. 304 Thurston — The Geometry and Topology of 3-Manifolds 13.2. BASIC DEFINITIONS. 13.11 Proposition 13.2.4. An orbifold O has a universal cover ˜ O. In other words, if ∗∈XO −ΣO is a base point for O, ˜ O p →O is a connected covering orbifold with base point ˜ ∗which projects to ∗, such that for any other covering orbifold ˜ O′ p′ →O with base point ˜ ∗′, p′(˜ ∗′) = ∗, there is a lifting q : ˜ O →˜ O′ of p to a covering map of ˜ O′. ˜ O @ @ @ q R ˜ O′ p′ O p ? The universal covering orbifold ˜ O, in some contexts, is often called the universal branched cover. There is a simple way to prove 13.2.4 in the case ΣO has codimension 2 or more. In that case, any covering space of O is determined by the induced covering space of XO −ΣO as its metric completion. Whether a covering Y space of XO −ΣO comes from a covering space of O is a local question, which is expressed algebraically by saying that π1(Y ) maps to a group containing a certain obvious normal subgroup of π1(X −ΣO). When O is a good orbifold, then it is covered by a simply connected manifold, M. It can be shown directly that M is the universal covering orbifold by proving that every covering orbifold is isomorphic to M/Γ′, for some Γ′ ⊂Γ, where Γ is the group of deck transformations of M over O. Proof of 13.2.4. One proof of the existence of a universal cover for a space X goes as follows. Consider pointed, connected covering spaces ˜ Xi pi →X. For any pair of such covering spaces, the component of the base point in the ﬁber product of the two is a covering space of both. 13.12 Thurston — The Geometry and Topology of 3-Manifolds 305 13. ORBIFOLDS ˜ X3 @ @ @ R ˜ X1 ˜ X2 @ @ @ R X (Recall that the ﬁber product of two maps fi : Xi →X is the space X1 ×X X2 = {(x1, x2) ∈X1 × X2 : f1(x1) = f2(x2)}.) If X is locally simply connected, or more generally, if it has the property that every x ∈X has a neighborhood U such that every covering of X induces a trivial covering of U (that is, each component of p−1(U) is homeomorphic to U), then one can take the inverse limit over some set of pointed, connected covering spaces of X which represents all isomorphism classes to obtain a universal cover for X. We can follow this same outline with orbifolds, but we need to reﬁne the notion of ﬁber product. The diﬃculty is best illustrated by example. Two covering maps S1 = dI f2 →mi and mi →mi are sketched below, along with the ﬁber product of the underlying maps of spaces. (This picture is sketched in R3 = R2 ×R1 R2.) The ﬁber product of spaces is a circle but with a double point. In the deﬁnition of ﬁber product of orbifolds, we must eliminate such double points, which always lie above ΣO. To do this, we work in local coordinates. Let U ≈˜ U/Γ be a coordinate system. 13.13 We may suppose that U is small enough so in every covering of O, p−1(U) consists 306 Thurston — The Geometry and Topology of 3-Manifolds 13.2. BASIC DEFINITIONS. of components of the form ˜ U/Γ′, Γ′ ⊂Γ. Let Oi pi →O be covering orbifolds (i = 1, 2), and consider components of p−1 i (U), which for notational convenience we identify with ˜ U/Γ1 and ˜ U/Γ2. Formally, we can write ˜ U/Γ1 = {Γ1y \| y ∈˜ U}. [It would be more consistent to use the notation Γ1\ ˜ U instead of ˜ U/Γ1]. For each pair of elements γ1 and γ2 ∈Γ, we obtain a map fγ1,γ2 : ˜ U →˜ U/Γ1 × ˜ U/Γ2, by the formula fγ1,γ2y = (Γ1γ1y, Γ2γ2y). In fact, fγ1,γ2 factors through ˜ U/γ−1 1 Γ1γ1 ∩γ−1 2 Γ2γ2. Of course, fγ1,γ2 depends only on the cosets Γ2γ1 and Γ2γ2. Furthermore, for any γ ∈Γ, the maps fγ1,γ2 and fγ1γ,γ2γ diﬀer only by a group element acting on ˜ U; in particular, their images are identical so only the product γ1γ−1 2 really matters. Thus, the “real” invariant of fγ1,γ2 is the double coset Γ1γ1γ−1 2 Γ2 ∈Γ1\Γ/Γ2. (Similarly, in the ﬁber product of coverings X1 and X2 of a space X, the components are parametrized by the double cosets π1X1\π1X/π1X2.) The ﬁber product of ˜ U/Γ1 and ˜ U/Γ2 over ˜ U/Γ, is deﬁned now to be the disjoint union, over elements γ repre-senting double cosets Γ1\Γ/Γ2 of the orbifolds ˜ U/Γ1 ∩γ−1Γ2γ. We have shown above how this canonically covers ˜ U/Γ1 and ˜ U/Γ2, via the map f1,γ. This deﬁnition agrees with the usual deﬁnition of ﬁber product in the complement of ΣO. These locally deﬁned patches easily ﬁt together to give a ﬁber product orbifold O1 ×O O2. As in the case of spaces, a universal covering orbifold ˜ O is obtained by taking the inverse limit over some suitable set representing all isomorphism classes of orbifolds. □ The universal cover ˜ O of an orbifold O is automatically a regular cover: for any 13.14 preimage of ˜ x of the base point ∗there is a deck transformation taking ˜ ∗to ˜ x. Definition 13.2.5. The fundamental group π1(O) of an orbifold O is the group of deck transformations of the universal cover ˜ O. The fundamental groups of orbifolds can be computed in much the same ways as fundamental groups of manifolds. Later we shall interpret π1(O) in terms of loops on O. Here are two more deﬁnitions which are completely parallel to deﬁnitions for manifolds. Thurston — The Geometry and Topology of 3-Manifolds 307 13. ORBIFOLDS Definition 13.2.6. An orbifold with boundary means a space locally modelled on Rn modulo ﬁnite groups and Rn + modulo ﬁnite groups. When XO is a topological manifold, be careful not to confuse ∂XO with ∂O or X∂O. Definition 13.2.7. A suborbifold O1 of an orbifold O2 means a subspace XO1 ⊂ XO2 locally modelled on Rd ⊂Rn modulo ﬁnite groups. Thus, a triangle orbifold has seven distinct “closed” one-dimensional suborbifolds, up to isotopy: one S1 and six mI’s. Note that each of the seven is the boundary of a suborbifold with boundary (deﬁned in the obvious way) with universal cover D2. 13.15 13.3. Two-dimensional orbifolds. To avoid technicalities, we shall work with diﬀerentiable orbifolds from now on. The nature of the singular locus of a diﬀerentiable orbifold may be understood as follows. Let U = ˜ U/Γ be any local coordinate system. There is a Riemannian metric on ˜ U invariant by Γ: such a metric may be obtained from any metric on ˜ U by averaging under Γ. For any point ˜ x ∈˜ U consider the exponential map, which gives a diﬀeomorphism from the ϵ ball in the tangent space at ˜ x to a small neighborhood of ˜ x. Since the exponential map commutes with the action of the isotropy group of ˜ x, it gives rise to an isomorphism between a neighborhood of the image of ˜ x in O, and a neighborhood of the origin in the orbifold Rn/Γ, where Γ is a ﬁnite subgroup of the orthgonal group On. Proposition 13.3.1. The singular locus of a two-dimensional orbifold has these types of local models: (i) The mirror: R2/Z2, where Z2 acts by reﬂection in the y-axis. (ii) Elliptic points of order n: R2/Zn, with Zn acting by rotations. (iii) Corner reﬂectors of order n: R2/Dn, with Dn is the dihedral group of order 2n, with presentation ⟨a, b : a2 = b2 = (ab)n = 1⟩. The generators a and b correspond to reﬂections in lines meeting at angle π/n. 13.16 308 Thurston — The Geometry and Topology of 3-Manifolds 13.3. TWO-DIMENSIONAL ORBIFOLDS. Proof. These are the only three types of ﬁnite subgroups of O2. □ It follows that the underlying space of a two-dimensional orbifold is always a topo-logical surface, possibly with boundary. It is easy to enumerate all two-dimensional orbifolds, by enumerating surfaces, together with combinatorial information which determines the orbifold structure. From a topological point of view, however, it is not completely trivial to determine which of these orbifolds are good and which are bad. We shall classify two-dimensional orbifolds from a geometric point of view. When G is a group of real analytic diﬀeomorphisms of a real analytic manifold X, then the elementary properties of (G, X)-orbifolds are similar to the case of manifolds (see §3.5). In particular a developing map D : ˜ O →X can be deﬁned for a (G, X)-orbifold O. Since we do not yet have a notion of paths in O, this requires a little explanation. Let {Ui} be a covering of O by a collection of open sets, closed under intersections, modelled on ˜ Ui/Γi, with ˜ Ui ⊂X, such that the 13.17 inclusion maps Ui ⊂Uj come from isometries ˜ ϕij : ˜ Ui →˜ Uj. Choose a “base” chart ˜ U0. When U0 ⊃Ui1 ⊂Ui2 ⊃· · · ⊂Ui2n is a chain of open sets (a simplicial path in the one-skeleton of the nerve of {Ui}), then for each choice of isometries of the form ˜ U0 γ0 ˜ ϕi1,0 ← −˜ Ui1 γ′ 2 ˜ ϕi1,i2 − → ˜ Ui2 ← −· · · − →˜ Ui2n one obtains an isometry of ˜ Ui2n in X, obtained by composing the transition functions (which are globally deﬁned on X). A covering space ˜ O of O is deﬁned by the cov-ering {(ϕ, ϕ( ˜ Ui))} ⊂G × X, where ϕ is any isometry of ˜ Ui obtained by the above construction. These are glued together by the obvious “inclusion” maps, (ϕ, ϕ ˜ Ui) , →(ψ, ψ ˜ Uj) whenever ψ−1 ◦ϕ is of the form γj ◦˜ ϕij for some γj ∈Γj. Thurston — The Geometry and Topology of 3-Manifolds 309 13. ORBIFOLDS The reader desiring a picture may construct a “foliation” of the space {(x, y, g) \| x ∈X, y ∈XO, g is the germ of a G-map between neighborhoods of x and y}. Any leaf of this foliation gives a developing map. Proposition 13.3.2. When G is an analytic group of diﬀeomorphisms of a man-ifold X, then every (G, X)-orbiifold is good. A developing map D : ˜ O →X and a holonomy homomorphism H : π1(O) →G are deﬁned. If G is a group of isometries acting transitively on X, then if O is closed or metrically complete, it is complete (i.e., D is a covering map). In particular, if X is simply connected, then ˜ O = X and π1(O) is a discrete subgroup of G. 13.18 Proof. See §3.5. □ Here is an example. △2,3,6 has a Euclidean structure, as a 30◦, 60◦, 90◦triangle. The developing map looks like this: 310 Thurston — The Geometry and Topology of 3-Manifolds 13.3. TWO-DIMENSIONAL ORBIFOLDS. Here is a deﬁnition that will aid us in the geometric classiﬁcation of two-dimen-sional orbifolds. Definition 13.3.3. When an orbifold O has a cell-division of XO such that each open cell is in the same stratum of the singular locus (i.e., the group associated to the interior points of a cell is constant), then the Euler number χ(O) is deﬁned by 13.19 the formula χ(O) = X ci (−1)dim(ci) 1 \|Γ(ci)\|, where ci ranges over cells and \|Γ(ci)\| is the order of the group Γ(ci) associated to each cell. The Euler number is not always an integer. The deﬁnition is concocted for the following reason. Deﬁne the number of sheets of a cover to be the number of preimages of a non-singular point. Proposition 13.3.4. If ˜ O →O is a covering map with k sheets, then χ( ˜ O) = kχ(O). Proof. It is easily veriﬁed that the number of sheets of a cover can be computed by the ratio # sheets = X ˜ x∋p(˜ x)=x ( \|Γx\| / \|Γ˜ x\| ), where x is any point. The formula [???] for the Euler number of a cover follows immediately. □ As an example, a triangle orbifold ∆n1,n2,n3 has Euler number 1 2 P(1/ni) −1 : here +1 comes from the 2-cell, three −1 2’s from the edges, and 1/(2ni) from each vertex. 13.20 Thus, ∆2,3,5 has Euler number +1/60. Its universal cover is S2, with deck trans-formations the group of symmetries of the dodecahedron. This group has order 120 = 2/(1/60). On the other hand, χ(∆2,3,6) = 0 and χ(∆2,3,7) = −1/84. These orbifolds cannot be covered by S2. The general formula for the Euler number of an orbifold O with k corner reﬂectors of orders n1, . . . , nk and l elliptic points of orders m1, . . . , ml is 13.3.4. χ(O) = χ(XO) −1 2 X (1 −1/ni) − X (1 −1/mi). Note in particular that χ(O) ≤χ(XO), with equality if and only if O is the surface χO or if O = mχO. Thurston — The Geometry and Topology of 3-Manifolds 311 13. ORBIFOLDS If O is equipped with a metric coming from invariant Riemannian metrics on the local models ˜ U, then one may easily derive the Gauss-Bonnet theorem, 13.3.5. Z O K dA = 2πχ(O). One way to prove this is by excising small neighborhoods of the singular locus, and applying the usual Gauss-Bonnet theorem for manifolds with boundary. For O to have an elliptic, parabolic or hyperbolic structure, χ(O) must be respectively positive, zero or negative. If O is elliptic or hyperbolic, then area (O) = 2π\|χ(O)\|. Theorem 13.3.6. A closed two-dimensional orbifold has an elliptic, parabolic or hyperbolic structure if and only if it is good. An orbifold O has a hyperbolic structure if and only if χ(O) < 0, and a parabolic structure if and only if χ(O) = 0. An orbifold 13.21 is elliptic or bad if and only if χ(O) > 0. All bad, elliptic and parabolic orbifolds are tabulated below, where (n1, . . . , nk; m1, . . . , ml) denotes an orbifold with elliptic points of orders n1, . . . , nk (in ascending order) and corner reﬂectors of orders m1, . . . , ml (in ascending order). Orbifolds not listed are hyperbolic. • Bad orbifolds: – XO = S2: (n), (n1, n2) with n1 < n2. – XO = D2: ( ; n), ( ; n1, n2) with n1 < n2. • Elliptic orbifolds: – XO = S2: ( ), (n, n), (2, 2, n), (2, 3, 3), (2, 3, 4), (2, 3, 5). – XO = D2: ( ; ), ( ; n, n), ( ; 2, 2, n), ( ; 2, 3, 3), ( ; 2, 3, 4), ( ; 2, 3, 5), (n; ), (2; m), (3; 2). – XO = P2: ( ), (n). • Parabolic orbifolds: – XO = S2: (2, 3, 6), (2, 4, 4), (3, 3, 3), (2, 2, 2, 2). – XO = D2: ( ; 2, 3, 6), ( ; 2, 4, 4), ( ; 3, 3, 3), ( ; 2, 2, 2, 2), (2; 2, 2), (3; 3), (4; 2), (2; 2; ). – XO = P2: (2, 2). – XO = T 2: ( ) – XO = Klein bottle: ( ) – XO = annulus: ( ; ) – XO = M¨ obius band: ( ; ) 13.21.a The universal covering space of D2 (;4,4,4) and S2 (4,4,4) · π1(D2 (;4,4,4)) is generated by reﬂections in the faces of one of the triangles. The full group of symmetries of this tiling of H2 is π1(D2 (;2,3,8)). 312 Thurston — The Geometry and Topology of 3-Manifolds 13.3. TWO-DIMENSIONAL ORBIFOLDS. This picture was drawn with a computer by Peter Oppenheimer. Proof. It is routine to list all orbifolds with non-negative Euler number, as in 13.22 the table. We have already indicated an easy, direct argument to show the orbifolds listed as bad are bad; here is another. First, by passing to covers, we only need consider the case that the underlying space is S2, and that if there are two elliptic Thurston — The Geometry and Topology of 3-Manifolds 313 13. ORBIFOLDS points their orders are relatively prime. These orbifolds have Riemannian metrics of curvature bounded above zero, which implies (by elementary Riemannian geometry) that any surface covering them must be compact. But the Euler number is either 1 + 1/n or 1/n1 + 1/n2, which is a rational number with numerator > 2. Since no connected surface has an Euler number greater than 2, these orbifolds must be bad. Question. What is the best pinching constant for Riemannian metrics on these orbifolds? All the orbifolds listed as elliptic and parabolic may be readily identiﬁed as the quotient of S2 or E2 modulo a discrete group. The 17 parabolic orbifolds correspond to the 17 “wallpaper groups.” The reader should unfold these orbifolds for himself, 13.23 to appreciate their beauty. Another pleasant exercise is to identify the orbifolds associated with some of Escher’s prints. Hyperbolic structures can be found, and classiﬁed, for orbifolds with negative Eu-ler characteristics by decomposing them into primitive pieces, in a manner analogous to our analysis of Teichm¨ uller space for a surface (§5.3). Given an orbifold O with χ(O) < 0, we may repeatedly cut it along simple closed curves and then “mirror” these curves (to remain in the class of closed orbifolds) until we are left with pieces of the form below. (If the underlying surface is unoriented, then make the ﬁrst cut so the result is oriented.) 314 Thurston — The Geometry and Topology of 3-Manifolds 13.3. TWO-DIMENSIONAL ORBIFOLDS. 13.24 The orbifolds mP, A(n ; ) and D(n1,n2 ; ) (except the degenerate case A(2,2; )) and S2 (n1,n2,n3) have hyperbolic structures paremetrized by the lengths of their boundary components. The proof is precisely analogous to the classiﬁcation of shapes of pants in §5.3; one decomposes these orbifolds into two congruent “generalized triangles” (see §2.6). Thurston — The Geometry and Topology of 3-Manifolds 315 13. ORBIFOLDS The orbifold D2 ( ;m1,...,ml) also can be decomposed into “generalized triangles,” for instance in the pattern above. One immediately sees that the orbifold has hy-perbolic structures (provided χ < 0) parametrized by the lengths of the cuts; that is, (R+)l−3. Special care must be taken when, say, m1 = m2 = 2. Then one of 13.25 the cuts must be omitted, and an edge length becomes a parameter. In general any disjoint set of edges with ends on order 2 corner reﬂectors can be taken as positive real parameters, with extra parameters coming from cuts not meeting these edges: The annulus with more than one corner reﬂector on one boundary component should be dissected, as below, into D( ; n1,...,nk) and an annulus with two order two corner reﬂectors. D2 (n; m1,...,ml) is analogous. 316 Thurston — The Geometry and Topology of 3-Manifolds 13.3. TWO-DIMENSIONAL ORBIFOLDS. 13.26 Hyperbolic structures on an annulus with two order two corner reﬂectors on one boundary component are parametrized by the length of the other boundary compo-nent, and the length of one of the edges: (The two all right pentagons agree on a and b, so they are congruent; thus they are determined by their edges of length l1/2 and l2/2). Similarly, D2 (n; 2,2) is determined by one edge length, provided n > 2. D2 (2; 2,2) is not hyperbolic. However, it has a degenerate hyperbolic structure as an inﬁnitely thin rectangle, modulo a rotation of order 2—or, an interval. This is consistent with the way in which it arises in considering hyperbolic structures, in the dissection of D2 (2;m1,...,ml). One can cut such an orbifold along the perpendicular arc from the elliptic point to an edge, to obtain D2 ( ; 2,2,m1,...,ml). In the case of an annulus with only one corner reﬂector, 13.27 Thurston — The Geometry and Topology of 3-Manifolds 317 13. ORBIFOLDS note ﬁrst that it is symmetric, since it can be dissected into an isosceles “triangle.” Now, from a second dissection, we see hyperbolic structures are paremetrized by the length of the boundary component without the reﬂector. By the same argument, D2 (n;m) has a unique hyperbolic structure. All these pieces can easily be reassembled to give a hyperbolic structure on O. □ From the proof of 13.3.6 we derive Corollary 13.3.7. The Teichm¨ uller space T(O) of an orbifold O with χ(O) < 0 is homeomorphic to Euclidean space of dimension −3χ(XO) + 2k + l, where k is the number of elliptic points and l is the number of corner reﬂectors. Proof. O can be dissected into primitive pieces, as above, by cutting along dis-joint closed geodesics and arcs perpendicular to ∂XO: i.e., one-dimensional hyperbolic 13.28 suborbifolds. The lengths of the arcs, and lengths and twist parameters for simple closed curves form a set of parameters showing that T(O) is homeomorphic to Eu-clidean space of some dimension. The formula for the dimension is veriﬁed directly for the primitive pieces, and so for disjoint unions of primitive pieces. When two circles are glued together, neither the formula nor the dimension of the Teichm¨ uller space changes—two length parameters are replaced by one length parameter and one ﬁrst parameter. When two arcs are glued together, one length parameter is lost, and the formula for the dimension decreases by one. □ 13.4. Fibrations. There is a very natural way to deﬁne the tangent space T(O) of an orbifold O. When the universal cover ˜ O is a manifold, then the covering transformations act on T( ˜ O) by their derivatives. T(O) is then T( ˜ O)/π1(O). In the general case, O is made up of pieces covered by manifolds, and the tangent space of O is pieced together from the tangent space of the pieces. Similarly, any natural ﬁbration over manifolds gives rise to something over an orbifold. Definition 13.4.1. A ﬁbration, E, with generic ﬁber F, over an orbifold O is an orbifold with a projection p : XE →XO 318 Thurston — The Geometry and Topology of 3-Manifolds 13.4. FIBRATIONS. between the underlying spaces, such that each point x ∈O has a neighborhood U = ˜ U/Γ (with ˜ U ⊂Rn) such that for some action of Γ on F, p−1(U) = ˜ U × F/Γ (where Γ acts by the diagonal action). The product structure should of course be consistent with p: the diagram below must commute. 13.29 ˜ U × F → p−1(U) ˜ U ? - U ? With this deﬁnition, natural ﬁbrations over manifolds give rise to natural ﬁbra-tions over orbifolds. The tangent sphere bundle TS(M) is the ﬁbration over M with ﬁber the sphere of rays through O in T(M). When M is Riemannian, this is identiﬁed with the unit tangent bundle T1(M). Proposition 13.4.2. Let O be a two-orbifold. If O is elliptic, then T1(O) is an elliptic three-orbifold. If O is Euclidean, then T1(O) is Euclidean. If O is bad, then TS(O) admits an elliptic structure. Proof. The unit tangent bundle T1(S2) can be identiﬁed with the grup SO3 by picking a “base” tangent vector VO and parametrizing an element g ∈SO3 by the image vector Dg(VO). SO3 is homeomorphic to P3, and its universal covering group os S3. This correspondence can be seen by regarding S3 as the multiplicative group of unit quaternions, which acts as isometries on the subspace of purely imaginary quaternions (spanned by i, j and k) by conjugation. The only elements acting trivially are ±1. The action of SO3 on T1(S2) = SO3 corresponds to left translation so that for an orientable O = S2/Γ, T1(O) = T1(S2/Γ) = Γ\SO3 = ˜ Γ\S3 is clearly elliptic. Here ˜ Γ is the preimage of Γ in S3. (Whatever Γ stands for, ˜ Γ is generally called “the binary Γ”—e.g., the binary dodecahedral group, etc.) When O is not oriented, then we use the model T1(S2) = O3/Z2, where Z2 is generated by the reﬂection r through the geodesic determined by VO. Again, the 13.29 action of O3 on T1(S2) comes from left multiplication on O3/Z2. An element gr, with g ∈SO3, thus takes g′VO to grg′rVO. But rg′r = sg′s, where s ∈SO3 is 180◦rotation of the geodesic through VO, so the corresponding transformations of S3, ˜ g 7→(˜ g˜ s) ˜ g′ (˜ s), are compositions of left and right multiplication, hence isometries. Thurston — The Geometry and Topology of 3-Manifolds 319 13. ORBIFOLDS For the case of a Euclidean orbifold O, note that T1E2 has a natural product structure as E2 × S1. From this, a natural Euclidean structure is obtained on T1E2, hence on T1(O). The bad orbifolds are covered by orbifolds S2 (n) or S2 (n1,n2). Then TS(H), where H is either hemisphere, is a solid torus, so the entire unit tangent space is a lens space—hence it is elliptic. TS(D2 ( ;n)), or TSD2 ( ;n1,n2), is obtained as the quotient by a Z2 action on these lens spaces. □ As an example, T1(S2 (2,3,5)) is the Poincar´ e dodecahedral space. This follows im-mediately from one deﬁnition of the Poincar´ e dodecahedral space as S3 modulo the binary dodecahedral group. In general, observe that TS(O2) is always a manifold if O2 is oriented; otherwise it has elliptic axes of order 2, lying above mirrors and con-sisting of vectors tangent to the mirrors. In more classical terminology, the Poincar´ e dodecahedral space is a Seifert ﬁber space over S2 with three singular ﬁbers, of type (2, 1), (3, 1) and (5, 1). When O has the combinatorial type of a polygon, it turns out that XTS(O) is S3, with singular locus a certain knot or two-component link. There is an a priori reason to suspect that XTS(O) be S3, since π1O is generated by reﬂections. These reﬂections 13.31 have ﬁxed points when they act on TS( ˜ O), so π1(XTS(O)) is the surjective image of π1TS( ˜ O). The image is trivial, since a reﬂection folds the ﬁbers above its axis in half. Every easily producible simply connected closed three-manifold seems to be S3. We can draw the picture of TS(O) by piecing. Over the non-singular part of O, we have a solid torus. Over an edge, we have mI ×I, with ﬁbers folded into mI; nearby ﬁgures go once around these mI’s. Above a corner reﬂector of order n, the ﬁber is folded into mI. The ﬁbers above the nearby edges weave up and down n times, and nearby circles wind around 2n times. 320 Thurston — The Geometry and Topology of 3-Manifolds 13.4. FIBRATIONS. 13.32 When the pieces are assembled, we obtain this knot or link: When O is a Riemannian orbifold, this gives T1(O) a canonical ﬂow, the geodesic ﬂow. For the Euclidean orbifolds with XO a polygon, this ﬂow is physically realized (up to friction and spin) by the motion of a billiard ball. The ﬂow is tangent to the singular locus. Thus, the phase space for the familiar rectangular billiard table is S3: There are two invariant annuli, with boundary the singular locus, corresponding to trajectories orthogonal to a side. The other trajectories group into invariant tori. Note the two-fold symmetry in the tangent space of a billiard table, which in the picture is 180◦rotation about the axis perpendicular to the paper. The quotient orbifold is the same as example 13.1.5. 13.33 Thurston — The Geometry and Topology of 3-Manifolds 321 13. ORBIFOLDS You can obtain many other examples via symmetries and covering spaces. For instance, the Borromean rings above have a three-fold axis of symmetry, with quotient orbifold: We can pass to a two-fold cover, unwrapping around the Z3 elliptic axis, to obtain the ﬁgure-eight knot as a Z3 elliptic axis. This is a Euclidean orbifold, whose fundamental group is generated by order 3 rota-tions in main diagonals of two adjacent cubes (regarded as fundamental domains for example 13.1.5). When O is elliptic, then all geodesics are closed, and the geodesic ﬂow comes from a circle action. It follows that T1(O) is a ﬁbration in a second way, by projecting to the quotient space by the geodesic ﬂow! For instance, the singular locus of T1(D2 (2,3,5)) is a torus knot of type (3, 5): 13.34 322 Thurston — The Geometry and Topology of 3-Manifolds 13.5. TETRAHEDRAL ORBIFOLDS. Therefore, it also ﬁbers over S2 (2,3,5). In general, an oriented three-orbifold which ﬁbers over a two-orbifold, with general ﬁber a circle, is determined by three kinds of information: (a) The base orbifold. (b) For each elliptic point or corner reﬂector of order n, an integer 0 ≤k < n which speciﬁes the local structure. Above an elliptic point, the Zn action on ˜ U × S1 is generated by a 1/n rotation of the disk U and a k/n rotation of the ﬁber S1. Above a corner reﬂector, the Dn action on ˜ U × S1 (with S1 taken as the unit circle in R2) is generated by reﬂections of ˜ U in lines making an angle of π/n and reﬂections of S1 in lines making an angle of kπ/n. (c) A rational-valued Euler number for the ﬁbration. This is deﬁned as the ob-struction to a rational section—i.e., a multiple-valued section, with rational weights for the sheets summing to one. (This is necessary, since there is not usually even a local section near an elliptic point or corner reﬂector). The Euler number for TS(O) equals χ(O). It can be shown that a ﬁbration of non-13.35 zero Euler number over an elliptic or bad orbifold is elliptic, and a ﬁbration of zero Euler number over a Euclidean orbifold is Euclidean. 13.5. Tetrahedral orbifolds. The next project is to classify orbifolds whose underlying space is a three-manifold with boundary, and whose singular locus is the boundary. In particular, the case when XO is the three-disk is interesting—the fundamental group of such an orbifold (if it is good) is called a reﬂection group. It turns out that the case when O has Thurston — The Geometry and Topology of 3-Manifolds 323 13. ORBIFOLDS the combinatorial type of a tetrahedron is quite diﬀerent from the general case. Geometrically, the case of a tetrahedron is subtle, although there is a simple way to classify such orbifolds with the aid of linear algebra. The explanation for this distinction seems to come from the fact that orbifolds of the type of a simplex are non-Haken. First, we deﬁne this terminology. A closed three-orbifold is irreducible if it has no bad two-suborbifolds and if every two-suborbifold with an elliptic structure bounds a three-suborbifold with an elliptic structure. Here, an elliptic orbifold with boundary is meant to have totally geodesic boundary—in other words, it must be D3/Γ, for some Γ ⊂O3. (For a non-oriented three-manifold,this deﬁnition entails being irreducible and P2-irreducible, in the usual terminology.) Observe that any three-dimensional orbifold with a bad suborbifold must itself be bad—it is conjectured that this is a necessary and suﬃcient condition for badness. 13.36 Frequently in three dimensions it is easy to see that certain orbifolds are good but hard to prove much more about them. For instance, the orbifolds with singular locus a knot or link in S3 are always good: they always have ﬁnite abelian covers by manifolds. Each elliptic two-orbifold is the boundary of exactly one elliptic three-orbifold, which may be visualized as the cone on it. An incompressible suborbifold of a three-orbifold O, when XO is oriented, is a two-suborbifold O′ ⊂O with χ(O′) ≤0 such that every one-suborbifold O′′ ⊂O′ which bounds an elliptic suborbifold of O −O′ bounds an elliptic suborbifold of O′. O is Haken if it is irreducible and contains an incompressible suborbifold. Proposition 13.5.1. Suppose XO = D3, ΣO = ∂D3. Then O is irreducible if and only if: 324 Thurston — The Geometry and Topology of 3-Manifolds 13.5. TETRAHEDRAL ORBIFOLDS. (a) The one-dimensional singular locus Σ1 O cannot be disconnected by the removal 13.37 of zero, one, or two edges, and (b) if the removal of γ1, γ2 and γ3 disconnects Σ1 O, then either they are incident to a commen vertex or the orders n1, n2 and n3 satisfy 1/n1 + 1/n2 + 1/n3 ≤1. Proof. For any bad or elliptic suborbifold O′ ⊂O, XO′ must be a disk meeting Σ1 O in 1, 2 or 3 points. XO′ separates XO into two three-disks; one of these gives an elliptic three-orbifold with boundary O′ if and only if it contains no one-dimensional parts of ΣO other than the edges meeting ∂XO′. For any set E of edges disconnecting Σ1 O there is a simple closed curve on ∂XO meeting only edges in E, meeting such an edge at most once, and separating Σ1 O −E. Such a curve is the boundary of a disk in XO, which determines a suborbifold. Any closed elliptic orbifold Sn/Γ of dimension n ≥2 can be suspended to give an elliptic orbifold Sn+1/Γ of dimension n + 1, via the canonical inclusion On+1 ⊂On+2. □ Proposition 13.5.2. An orbifold O with XO = D3 and ΣO = ∂D3 is Haken if and only if it is irreducible, it is not the suspension of an elliptic two-orbifold and it does not have the type of a tetrahedron. Proof. First, suppose that O satisﬁes the conditions. Let F be any face of O, that is a component of ΣO minus its one dimensional part. The closure ¯ F is a disk or sphere, for otherwise O would not be irreducible. If F is the entire sphere, then O is the suspension of D2 ( ; ). Otherwise, consider a curve γ going around just outside 13.38 F, and meeting only edges of Σ1 O incident to ¯ F. If γ meets no edges, then Σ1 O = ∂F (since O is irreducible) and O is the suspension of D2 ( ;n,n). The next case is that γ meets two edges of order n; then they must really be the same edge, and O is the suspension of an elliptic orbifold D2 ( ;n,n1,n2). If γ meets three edges, then γ determines a “triangle” suborbifold D2 ( ;n1,n2,n3) of O. O′ cannot be elliptic, for then the three edges would meet at a point and O would have the type of a tetrahedron. Since D2 ( ;n1,n2,n3) has no non-trivial one-suborbifolds, it is automatically incompressible, so O is Haken. If γ meets four or more edges, then Thurston — The Geometry and Topology of 3-Manifolds 325 13. ORBIFOLDS the two-suborbifold it determines is either incompressible or compressible. But if it is compressible, then an automatically incompressible triangle suborbifold of O can be constructed. If α determines a “compression,” then β determines a triangle orbifold. The converse assertion, that suspensions of elliptic orbifolds and tetrahedral orb-ifolds are not Haken, is fairly simple to demonstrate. In general, for a curve γ on 13.39 ∂XO to determine an incompressible suborbifold, it can never enter the same face twice, and it can enter two faces which touch only along their common edge. Such a curve is evidently impossible in the cases being considered. □ There is a system of notation, called the Coxeter diagram, which is eﬃcient for describing n-orbifolds of the type of a simplex. The Coxeter diagram is a graph, whose vertices are in correspondence with the (n−1)-faces of the simplex. Each pair of (n−1)-faces meet on an (n−2)-face which is a corner reﬂector of some order k. The corresponding vertices of the Coxeter graph are joined by k−2 edges, or alternatively, a single edge labelled with the integer k−2. The notation is eﬃcient because the most commonly occurring corner reﬂector has order 2, and it is not mentioned. Sometimes this notation is extended to describe more complicated orbifolds with XO = Dn and ΣO ⊂∂Dn, by using dotted lines to denote the faces which are not incident. However, for a complicated polyhedron—even the dodecahedron—this becomes quite unwieldy. The condition for a graph with n + 1 vertices to determine an orbifold (of the type of an n-simplex) is that each complete subgraph on n vertices is the Coxeter diagram for an elliptic (n −1)-orbifold. Here are the Coxeter diagrams for the elliptic triangle orbifolds: 326 Thurston — The Geometry and Topology of 3-Manifolds 13.5. TETRAHEDRAL ORBIFOLDS. 13.40 Theorem 13.5.3. Every n-orbifold of the type of a simplex has either an elliptic, Euclidean or hyperbolic structure. The types in the three-dimensional case are listed below: This statement may be slightly generalized to include non-compact orbifolds of the combinatorial type of a simplex with some vertices deleted. Thurston — The Geometry and Topology of 3-Manifolds 327 13. ORBIFOLDS Theorem 13.5.4. Every n-orbifold which has the combinatorial type of a simplex with some deleted vertices, such that the “link” of each deleted vertex is a Euclidean orbifold, and whose Coxeter diagram is connected, admits a complete hyperbolic struc-13.41 ture of ﬁnite volume. The three-dimensional examples are listed below: Proof of 13.5.3 and 13.5.4. The method is to describe a simplex in terms of the quadratic form models. Thus, an n-simplex σn on Sn has n + 1 hyperfaces. Each face is contained in the intersection of a codimension one subspace of En+1 with Sn. Let VO, . . . , Vn be unit vectors orthogonal to these subspaces in the direction away from σn. Clearly, the Vi are linearly indpendent. Note that Vi · Vi = 1, and when i ̸= j, Vi · Vj = −cos αij, where αij is the angle between face i and face j. Similarly, each face of an n-simplex in Hn contained in the intersection of a subspace of En,1 with the sphere of imaginary radius X2 1 + · · · + X2 n −X2 n+1 = −1 (with respect to the standard inner product X · Y = Pn i=1 Xi · Yi −Xn+1 · Yn+1 on En,1). Outward vectors V0, . . . , Vn orthogonal to these subspaces have real length, so they can be normalized to have length 1. Again, the Vi are linearly independent and Vi · Vj = −cos αij when i ̸= j. For an n-simplex σn in Euclidean n-space, let VO, . . . , Vn be outward unit 13.42 vectors in directions orthogonal to the faces on σn. Once again, Vi · Vj = −cos αij. Given a collection {αij} of angles, we now try to construct a simplex. For the matrix M of presumed inner products, with l’s down the diagonal and −cos αij’s oﬀ the diagonal. If the quadratic form represented by M is positive deﬁnite or of type (n, 1), then we can ﬁnd an equivalence to En+1 or En,1, which sends the basis vectors to vectors VO, . . . , Vn having the speciﬁed inner product matrix. The intersection 328 Thurston — The Geometry and Topology of 3-Manifolds 13.5. TETRAHEDRAL ORBIFOLDS. of the half-spaces X · Vi ≤O is a cone, which must be non-empty since the {Vi} are linearly independent. In the positive deﬁnite case the cone intersects Sn in a simplex, whose dihedral angles βij satisfy cos βij = cos αij, hence βij = αij. In the hyperbolic case, the cone determines a simplex in RPn, but the simplex may not be contained in Hn ⊂RPn. To determine the positions of the vertices, observe that each vertex vi determines a one-dimensional subspace, whose orthogonal subspace is spanned by VO, . . . , ˆ Vi, . . . , Vn. The vertex vi is on Hn, on the sphere at inﬁnity, or outside inﬁnity according to whether the quadratic form restricted to this subspace is positive deﬁnite, degenerate, or of type (n −1, 1). Thus, the angles {αij} are the angles of an ordinary hyperbolic simplex if and only if M has type (n, 1), and for each i the submatrix obtained by deleting the ith row and the corresponding column is positive deﬁnite. They are the angles of an ideal hyperbolic simplex (with vertices in Hn or Sn−1 ∞) if and only if all such submatrices are either positive deﬁnite, or have rank n −1. By similar considerations, the angles {αij} are the angles of a Euclidean n-simplex if and only if M is positive semideﬁnite of rank n. 13.43 When the angles {αij} are derived from the Coxeter diagram of an orbifold, then each submatrix of M obtained by deleting the i-th row and the i-th column corresponds to an elliptic orbifold of dimension n −1, hence it is positive deﬁnite. The full matrix must be either positive deﬁnite, of type (n, 1) or positive semideﬁnite with rank n. It is routine to list the examples in any dimension. The sign of the determinant of M is a practical invariant of the type. We have thus proven theorem 13.5. In the Euclidean case, it is not hard to see that the subspace of vectors of zero length with respect to M is spanned by (a0, . . . , an), where ai is the (n−1)-dimensional area of the i-th face of σ. To establish 13.5.4, ﬁrst consider any submatrix Mi of rank n−1 which is obtained by deleting the i-th row and i-th column (so, the link of the i-th vertex is Euclidean). Change basis so that Mi becomes         1 0 . . . 1 0 0         using (a0, . . . , ˆ ai, . . . , an) as the last basis vector. When the basis vector Vi is put back, the quadratic form determined by M becomes Thurston — The Geometry and Topology of 3-Manifolds 329 13. ORBIFOLDS where −C = −P j∋j̸=i ai cos αij is negative since the Coxeter diagram was supposed to be connected. It follows that M has type (n, 1), which implies that the orbifold is hyperbolic. □ 13.44 13.6. Andreev’s theorem and generalizations. There is a remarkably clean statement, due to Andreev, describing hyperbolic reﬂection groups whose fundamental domains are not tetrahedra. Theorem 13.6.1 (Andreev, 1967). (a) Let O be a Haken orbifold with XO = D3, Σ0 = ∂D3. Then O has a hyperbolic structure if and only if O has no incompressible Euclidean suborbifolds. (b) If O is a Haken orbifold with XO = D3—(ﬁnitely many points) and ΣO = ∂XO, and if a neighborhood of each deleted point is the product of a Eu-clidean orbifold with an open interval, (but O itself is not such a product) then O has a complete hyperbolic structure with ﬁnite volume if and only if each incompressible Euclidean suborbifold can be isotoped into one of the product neighborhoods. The proof of 13.6.1 will be given in §??. Corollary 13.6.2. Let γ be any graph in R2, such that each edge has distinct ends and no two vertices are joined by more than one edge. Then there is a packing of circles in R2 whose nerve is isotopic to γ. If γ is the one-skeleton of a triangulation of S2, then this circle packing is unique up to Moebius transformation. A packing of circles means a collection of circles with disjoint interiors. The nerve of a packing is then a graph, whose vertices correspond to circles, and whose edges correspond to pairs of circles which intersect. This graph has a canonical embedding in the plane, by mapping the vertices to the centers of the circles and the edges to straight line segments which will pass through points of tangency of circles. 13.44a 330 Thurston — The Geometry and Topology of 3-Manifolds 13.6. ANDREEV’S THEOREM AND GENERALIZATIONS. 13.45 Proof of 13.6.2. We transfer the problem to S2 by stereographic projection. Add an extra vertex in each non-triangular region of S2 −γ, and edges connecting it to neighboring vertices, so that γ becomes the one-skeleton of a triangulation T of S2. Let P be the polyhedron obtained by cutting oﬀneighborhoods of the vertices of T, down to the middle of each edge of T. Thurston — The Geometry and Topology of 3-Manifolds 331 13. ORBIFOLDS Let O be the orbifold with underlying space XO = D3-vertices of P, and Σ1 O = edges of P, each modelled on R3/D2. For any incompressible Euclidean suborbifold O′, ∂XO must be a curve which circumnavigates a vertex. Thus, O satisﬁes the hypotheses of 13.6.1(b), and O has a hyperbolic structure. This means that P is realized as an ideal polyhedron in H3, with all dihedral angles equal to 90◦. The planes of the new faces of P (faces of P but not T) intersect S2 ∞in circles. Two of the circles are tangent whenever the two faces meet at an ideal vertex of P. This is the packing required 13.46 by 13.6.2. The uniqueness statement is a consequence of Mostow’s theorem, since the polyhedron P may be reconstructed from the packing of circles on S2 ∞. To make the reconstruction, observe that any three pairwise tangent circles have a unique common orthogonal circle. The set of planes determined by the packing of circles on S2 ∞, together with extra circles orthogonal to the triples of tangent circles coming from vertices of the triangular regions of S2 −γ cut out a polyhedron of ﬁnite volume combinatorially equivalent to P, which gives a hyperbolic structure for O. □ 332 Thurston — The Geometry and Topology of 3-Manifolds 13.6. ANDREEV’S THEOREM AND GENERALIZATIONS. Remark. Andreev also gave a proof of uniqueness of a hyperbolic polyhedron with assigned concave angles, so the reference to Mostow’s theorem is not essential. Corollary 13.6.3. Let T be any triangulation of S2. Then there is a convex polyhedron in R3, combinatorially equivalent to T whose one-skeleton is circumscribed about the unit sphere (i.e., each edge of T is tangent to the unit sphere). Furthermore, this polyhedron is unique up to a projective transformation of R3 ⊂P3 which preserves the unit sphere. Proof of 13.6.3. Construct the ideal polyhedron P, as in the proof of 13.6.2. Embed H3 in P3, as the projective model. The old faces of P (coming from faces of T) form a polyhedron in P3, combinatorially equivalent to T. Adjust by a projective transformation if necessary so that this polyhedron is in R3. (To do this, transform P so that the origin is in its interior.) □ Remarks. Note that the dual cell-division T ∗to T is also a convex polyhedron in R3, with one-skeleton of T ∗circumscribed about the unit sphere. The intersection T ∩T ∗= P. These three polyhedra may be projected to R2 ⊂P3, by stereogrpahic projection, from the north pole of S2 ⊂P3. Stereographic projection is conformal on the tangent space of S2, so the edges of T ∗project to tangents to these circles. It follows that the vertices of T project to the centers of the circles. Thus, the image of the one-skeleton of T is the geometric embedding in R2 of the nerve γ of the circle packing. The existence of other geometric patterns of circles in R2 may also be deduced from Andreev’s theorem. For instance, it gives necessary and suﬃcient condition for the existence of a family of circles meeting only orthogonally in a certain pattern, or 13.48 meeting at 60◦angles. One might also ask about the existence of packing circles on surfaces of constant curvature other than S2. The answers are corollaries of the following theorems: Thurston — The Geometry and Topology of 3-Manifolds 333 13. ORBIFOLDS Theorem 13.6.4. Let O be an orbifold such that XO ≈T 2 × [0, ∞), (with some vertices on T 2 × O having Euclidean links possibly deleted) and ΣO = ∂XO. Then O admits a complete hyperbolic structure of ﬁnite volume if and only if it is irreducible, and every incompressible complete, proper Euclidean suborbifold is homotopic to one of the ends. (Note that mS1×[0, ∞) is a complete Euclidean orbifold, so the hypothesis implies that every non-trivial simple closed curve on ∂XO intersects Σ1 O.) Theorem 13.6.5. Let M 2 be a closed surface, with χ(M 2) < 0. An orbifold O such that XO = M 2 × [0, 1] (with some vertices on M 2 × 0 having Euclidean links possibly deleted), ΣO = ∂XO and Σ1 O ⊂M 2 × O. Then O has a hyperbolic structure if and only if it is irreducible, and every incompressible Euclidean suborbifold is homotopic to one of the ends. By considering π1O, O as in 13.6.4, as a Kleinian group in upper half space with T 2×∞at ∞, 13.6.4 may be translated into a statement about the existence of doubly periodic families of circles in the plane, or 13.48.a 334 Thurston — The Geometry and Topology of 3-Manifolds 13.6. ANDREEV’S THEOREM AND GENERALIZATIONS. 13.48.b Thurston — The Geometry and Topology of 3-Manifolds 335 13. ORBIFOLDS 13.49 families of circles on ﬂat toruses. Similarly, 13.6.5 is equivalent to a statement about families of circles in hyperbolic structures for M 2; in fact, since M 2 × 1 has no one-dimensional singularities, it must be totally geodesic in any hyperbolic structure, so π1M 2 acts as a Fuchsian group. The face planes of M 2 × O give rise to a family of circles in the northern hemisphere of S2 ∞, invariant by this Fuchsian group, so each face corresponds to a circle in the hyperbolic structure for M 2. Theorems 13.6.1, 13.6.4 and 13.6.5 will be proved in the next section, by studying patterns of circles on surfaces. In example 13.1.5 we saw that the Borromean rings are the singular locus for a Euclidean orbifold, in which they are elliptic axes of order 2. With the aid of Andreev’s theorem, we may ﬁnd all hyperbolic orbifolds which have the Borromean rings as singular locus. The rings can be arranged so they are invariant by reﬂection in three orthogonal great spheres in S3. (Compare p. 13.4.) Thus, an orbifold O having the rings as elliptic axes of orders k, l and m is an eight-fold covering space of another orbifold, which has the combinatorial type of a cube. 13.50 By Andreev’s theorem, such an orbifold has a hyperbolic structure if and only if k, l and m are all greater than 2. If k is 2, for example, then there is a sphere in S3 separating the elliptic axes of orders l and m and intersecting the elliptic axis of order 2 in four points. This forms an incompressible Euclidean suborbifold of O, which breaks O into 336 Thurston — The Geometry and Topology of 3-Manifolds 13.7. CONSTRUCTING PATTERNS OF CIRCLES. two halves, each ﬁbering over two-orbifolds with boundary, but in incompatible ways (unless l or m is 2). Base spaces of the ﬁbrations When k = l = m = 4, the fundamental domain, as in example 13.1.5, for π1O acting on H3 is a regular right-angled dodecahedron. Any of the numbers k, l or m can be permitted to take the value ∞in this discussion, to denote a parabolic cusp. When l = m = ∞, for instance, then O has a k-fold cover which is the complement of the untwisted 2k-link chain D2k of 6.8.7. 13.51 13.7. Constructing patterns of circles. We will formulate a precise statement about patterns of circles on surfaces of non-positive Euler characteristic which gives theorems 13.6.4 and 13.6.5 as immediate consequences. Theorem 13.7.1. Let S be a closed surface with χ(S) ≤0. Let τ be a cell-division of S into cells which are images of immersions of triangles and quadrangles which lift to embeddings in ˜ S. Let Θ : E →[0, π/2] (where E denotes the set of edges of τ) be any function satisfying the conditions below: (i) Θ(e) = π/2 if e is an edge of a quadrilateral of τ. (ii) If e1, e2, e3 [ei ∈E] form a null-homotopic closed loop, and if P3 i=1 Θ(ei) ≥ π, then these three edges form the boundary of a triangle of τ. Thurston — The Geometry and Topology of 3-Manifolds 337 13. ORBIFOLDS (iii) If e1, e2, e3, e4 form a null-homotopic closed loop and if P4 i=1 Θ(ei) = 2π(⇔ Θ(ei) = π/2), then the ei form the boundary of a quadrilateral or of the union of two adjacent triangles. Then there is a metric of constant curvature on S, uniquely determined up to a scalar multiple, a uniquely determined geometric cell-division of S isotopic to τ so that the edges are geodesics, and a unique family of circles, one circle Cv for each vertex v of τ, so that Cv1 and Cv2 intersect at a positive angle if and only if v1 and v2 lie on a common edge. The angles in which Cv1 and Cv2 meet are determined by the common edges: there is an intersection point of Cv1 and Cv2 in a two-cell σ if and only if v1 and v2 are vertices of σ. If σ is a quadrangle and v1 and v2 are diagonally opposite, then Cv1 is tangent to Cv2; otherwise, they meet at an angle of Θ(e), where 13.52 e is the edge joining them in σ. Proof. First, observe that quadrangles can be eliminated by subdivision into two triangles by a new edge e with Θ(e) = 0. There is an extraneous tangency of circles here—in fact, all extraneous tangencies come from this situation. Henceforth, we assume τ has no quadrangles. The idea is to solve for the radii of the circles Cv1. Given an arbitrary set of radii, we shall construct a Riemannian metric on S with cone type singularities at the vertices of τ, which has a family of circles of the given radii meeting at the given angles. We adjust the radii until S lies ﬂat at each vertex. Thus, the proof is closely analogous to the idea that one can make a conformal change of any given Riemannian metric on a surface until it has constant curvature. Observe that a conformal map is one which takes inﬁnitesimal circles to inﬁnitesimal circles; the conformal factor is the ratio of the radii of the target and source circles. Lemma 13.7.2. For any three non-obtuse angles θ1, θ2 and θ3 ∈[0, π/2] and any three positive numbers R1, R2, and R3, there is a conﬁguration of 3 circles in both hy-perbolic and Euclidean geometry, unique up to isometry, having radii Ri and meeting in angles θi. 338 Thurston — The Geometry and Topology of 3-Manifolds 13.7. CONSTRUCTING PATTERNS OF CIRCLES. 13.53 Proof of lemma. The length lk of a side of the hypothetical triangle of centers of the circles is determined as the side opposite the obtuse angle π −θk in a triangle whose other sides are Ri and Rj. Thus, sup(Ri, Rj) < lk ≤Ri + Rj. The three numbers l1, l2 and l3 obtained in this way clearly satisfy the triangle inequalities lk < li + lj. Hence, one can construct the appropriate triangle, which gives the desired circles. □ Proof of 13.7.1, continued. Let V denote the set of vertices of τ. For every element R ∈RV + (i.e., if we choose a radius for the circle about each vertex), there is a singular Riemannian metric, which is pieced together from the triangles of centers of circles with given radii and angles of intersetcion as in 13.7.2. The triangles are taken in H2 or E2 depending on whether χ(S) < 0 or χ(S) = 0. The edge lengths of cells of τ match whenever they are glued together, so we obtain a metric, with singularities only at the vertices, and constant curvature 0 or −1 everywhere else. The notion of curvature can easily be extended to Riemannian surfaces with certain sorts of singularities. The curvature form Kda becomes a measure κ on such 13.54 a surface. Tailors are of necessity familiar with curvature as a measure. Thus, a seam has curvature (k1 −k2) · µ, where µ is one-dimensional Lebesgue measure and k1 and k2 are the geodesic curvatures of the two halves. Thurston — The Geometry and Topology of 3-Manifolds 339 13. ORBIFOLDS (The eﬀect of gathering is more subtle—it is obtained by putting two lines inﬁnitely close together, one with positive curvature and one with balancing negative curvature. Another instance of this is the boundary of a lens.) More to the point for us is the curvature concentrated at the apex of a cone: it is 2π −α, where α is the cone angle (computed by splitting the cone to the apex and laying it ﬂat). It is easy to see that this is the unique value consistent with the Gauss-Bonnet theorem. Formally, we have a map F : RV + →RV. Given an element R ∈RV +, we construct the singular Riemannian metric on S, as above; F(R) describes the discrete part of the curvature measure κR on S, in other 13.55 words, F(R)(v) = κR(v). Our problem is to show that O is in the image of F, for then we will have a non-singular metric with the desired pattern of circles built in. When χ(S) = 0, then the shapes of the Euclidean triangles do not change when we multiply R by a constant, so F(R) also does not change. Thus we may as well normalize so that P v∈VR(v) = 1. Let ∆⊂RV + be this locus—∆is the interior of the standard \|V\| −1 simplex. Observe, by the Guass-Bonnet theorem, that X v∈V κR(v) = 0. Let Z ⊂RV be the locus deﬁned by this equation. If χ(S) < 0, then changing R by a constant does make a diﬀerence in κ. In this case, let ∆⊂RV + denote the set of R such that the associated metric on S has total area 2π \|χ(S)\|. By the Gauss-Bonnet theorem, ∆= F −1(Z) (with Z as above). As one can easily believe, ∆intersects each ray through O in a unique point, so ∆is a simplex in this case also. This fact is easily deduced from the following lemma, which will also prove the uniqueness part of 13.7.1: Lemma 13.7.3. Let C1, C2 and C3 be circles of radii R1, R2 and R3 in hyperbolic or Euclidean geometry, meeting pairwise in non-obtuse angles. If C2 and C3 are held 340 Thurston — The Geometry and Topology of 3-Manifolds 13.7. CONSTRUCTING PATTERNS OF CIRCLES. constant but C1 is varied in such a way that the angles of intersection are constant but R1 decreases, then the center of C1 moves toward the interior of the triangle of centers. 13.56 Thus we have ∂α1 ∂R1 < 0 , ∂α2 ∂R1 > 0 , ∂α3 ∂R1 > 0, where the αi are the angles of the triangle of centers. Proof of 13.7.3. Consider ﬁrst the Euclidean case. Let l1, l2 and l3 denote the lengths of the sides of the triangle of centers. The partial derivatives ∂l2/∂R1 and ∂l3/∂R1 can be computed geometrically. If v1 denotes the center of C1, then ∂v1/∂R1 is determined as the vector whose orthogonal projections sides 2 and 3 are ∂l2/∂R1 and ∂l3/∂R1. Thus, R1 ∂v1 ∂R1 is the vector from v1 to the intersection of the lines joining the pairs of intersection points of two circles. Thurston — The Geometry and Topology of 3-Manifolds 341 13. ORBIFOLDS 13.57 When all angles of intersection of circles are acute, no circle meets the opposite side of the triangle of centers: C3 meets v1v2 = ⇒ C1 and C2 don’t meet. It follows that ∂v1/∂R1 points to the interior of ∆v1v2v3. The hyperbolic proof is similar, except that some of it takes place in the tangent space to H2 at v1. □ Continuation of proof of 13.7.1. From lemma 13.7.3 it follows that when all three radii are increased, the new triangle of centers can be arranged to contain the old one. Thus, the area of S is monotone, for each ray in RV +. The area near 0 is near 0, and near ∞is near π × (# triangles + 2# quadrangles); thus the ray intersects ∆= F −1(Z) in a unique point. It is now easy to prove that F is an embedding of ∆in Z. In fact, consider any two distinct points R and R′ ∈∆. Let V−⊂V be the set of v where R′(v) < R(v). Clearly V−is a proper subset. Let τV−be the subcomplex of τ spanned by V−. (τV−consists of all cells whose vertices are contained in V−). Let SV−be a small neighborhood of τV−. We compare the geodesic curvature of ∂SV−in the two metrics. To do this, we 342 Thurston — The Geometry and Topology of 3-Manifolds 13.7. CONSTRUCTING PATTERNS OF CIRCLES. may arrange ∂SV−to be orthogonal to each edge it meets. Each arc of intersection of ∂SV−with a triangle having one vertex in V−contributes approximately αi to the 13.58 total curvature, while each arc of intersection with a triangle having two vertices in V−contributes approximately βi + γi −π. In view of 13.7.3, an angle such as α1 increases in the R′ metric. The change in β1 and γ1 is unpredictable. However, their sum must increase: ﬁrst, let R1 and R2 decrease; π −δ1 −(β1 + β2), which is the area of the triangle in the hyperbolic case, decreases or remains constant but δ1 also decreases so β1 + γ1 must increase. Then let R3 increase; by 13.7.3, β1 and γ1 both increase. Hence, the geodesic curvature of ∂SV−increases. From the Gauss-Bonnet formula, X v∈V− κ(v) = Z ∂SV− dg ds − Z SV− K dA + 2πχ(SV′) it follows that the total curvature at vertices in V−must decrease in the R′ metric. (Note that the area of SV−decreases, so if k = −1, the second term on the right decreases.) In particular, F(R) ̸= F(R′), which shows that F is an embedding of ∆. 13.59 The proof that O is in the image of F is based on the same principle as the proof of uniqueness. We can extract information about the limiting behavior of F as R approaches ∂∆by studying the total curvature of the subsurface SVO, where VO consists of the vertices v such that R(v) is tending toward O. When a triangle of τ has two vertices in VO and the third not in VO, then the sum of the two angles at vertices in VO tends toward π. Thurston — The Geometry and Topology of 3-Manifolds 343 13. ORBIFOLDS 344 Thurston — The Geometry and Topology of 3-Manifolds 13.7. CONSTRUCTING PATTERNS OF CIRCLES. When a triangle of τ has only one vertex in VO, then the angle at that vertex tends toward the value π −Θ(e), where e is the opposite edge. Thus, the total curvature of ∂SVO tends toward the value X e∈L(τVO) π −Θ(e) , where L(τVO) is the “link of τVO.” The Gauss-Bonnet formula gives 13.60 Lim X v∈VO κ(v) = − X e∈L(τVO) π −Θ(e) + 2πχ(SVO) < 0. (Note that area (SVO) →0.) To see that the right hand side is always negative, it suﬃces to consider the case that τVO is connected. Unless τVO has Euler characteristic one, both terms are non-positive, and the sum is negative. If L(τVO) has length 5 or more, then X e∈L(τVO) π −Θ(e) > eπ, so the sum is negative. The cases when L(τVO) has length 3 or 4 are dealt with in hypotheses (ii) and (iii) of theorem 13.7.1. When V′ is any proper subset of VO and R ∈∆is an arbitrary point, we also have an inequality X v∈V′ κR(v) > − X e∈L(τV′) π −Θ(e) + 2πχ(SV′). This may be deduced quickly by comparing the R metric with a metric R′ in which R′(V′) is near 0. In other words, the image F(∆) is contained in the interior of the polyhedron P ⊂Z deﬁned by the above inequalities. Since F(∆) is an open set whose boundary is ∂P, F(∆) = interior (P). Since O ∈int(P), this completes the proof of 13.7.1, and also that of 13.6.4, and 13.6.5. □ Remarks. This proof was based on a practical algorithm for actually construct-ing patterns of circles. The idea of the algorithm is to adjust, iteratively, the radii of the circles. A change of any single radius aﬀects most strongly the curvature at that vertex, so this proces converges reasonably well. 13.61 The patterns of circles on surfaces of constant curvature, with singularities at the centers of the circles, have a three-dimensional interpretation. Because of the inclusions isom(H2) ⊂isom(H3) and isom(E2) ⊂isom(H3), there is associated with such a surface S a hyperbolic three-manifold MS, homeomorphic to S ×R, with cone type singularities along (the singularities of S) × R. Each circle on S determines a totally geodesic submanifold (a “plane”) in MS. These, together with the totally Thurston — The Geometry and Topology of 3-Manifolds 345 13. ORBIFOLDS geodesic surface isotopic to S when S is hyperbolic, cut out a submanifold of MS with ﬁnite volume—it is an orbifold as in 13.6.4 or 13.6.5 but with singularities along arcs or half-lines running from the top to the bottom. Corollary 13.7.4. Theorems 13.6.4 and 13.6.5 hold when S is a Euclidean or hyperbolic orbifold, instead of a surface. (The orbifold O is to have only singularities as in 13.6.4 or 13.6.5, plus (singularities of S) × I or (singularities of S) × [0, ∞) .) Proof. Solve for pattern of circles on S in a metric of constant curvature on S— the underyling surface of S will have a Riemannian metric with cone type singularities of curvature 2π(1/n −1) at elliptic points of S, and angles at corner reﬂectors of S. An alternative proof is to ﬁnd a surface ˜ S which is a ﬁnite covering space of the orbifold S, and ﬁnd a hyperbolic structure for the corresponding covering space ˜ O of O. The existence of a hyperbolic structure for O follows from the uniqueness of the hyperbolic structure on ˜ O thence the invariance by deck transformations of ˜ O over O. □ 13.62 13.8. A geometric compactiﬁcation for the Teichm¨ uller spaces of polygonal orbifolds We will construct hyperbolic structures for a much greater variety of orbifolds by studying the quasi-isometric deformation spaces of orbifolds with boundary whose underlying space is the three-disk. In order to do this, we need a description of the limiting behavior of conformal structure on its boundary. We shall focus on the case when the boundary is a disjoint union of polygonal orbifolds. For this, the greatest clarity is attained by ﬁnding the right compactiﬁcations for these Teichm¨ uller spaces. When M is an orbifold, M[ϵ,∞) is deﬁned to consist of points x in M such that the ball of radius ϵ/2 about x has a ﬁnite fundamental group. Equivalently, no loop through x of length < ϵ has inﬁnite order in π1(M). M(0,ϵ] is deﬁned similarly. It does not, in general, contain a neighborhood of the singular locus. With this deﬁnition, it follows (as in §5) that each component of M(0,ϵ] is covered by a horoball or a uniform neighborhood of an axis, and its fundamental group contains Z or Z ⊕Z with ﬁnite index. In §5 we deﬁned the geometric topology on sequences of hyperbolic three-mani-folds of ﬁnite volume. For our present purpose, we want to modify this deﬁnition slightly. First, deﬁne a hyperbolic structure with nodes on a two-dimensional orbifold O to be a complete hyperbolic structure with ﬁnite volume on the complement of some one-dimensional suborbifold, whose components are the nodes. This includes the case when there are no nodes. A topology is deﬁned on the set of hyperbolic structures with nodes, up to diﬀeomorphisms isotopic to the identity on a given 346 Thurston — The Geometry and Topology of 3-Manifolds 13.8. GEOMETRIC COMPACTIFICATION surface, by saying that M1 and M2 have distance ≤ϵ if there is a diﬀeomorphism of O [isotopic to the identity] whose restriction to M1[ϵ′,∞) is a (eϵ)-quasi-isometry to M2[ϵ′,∞). Here, ϵ′ is some ﬁxed, small number. 13.63 Remark. The related topology on hyperbolic structures with nodes up to dif-feomorphism on a given surface is always compact. (Compare Jørgensen’s theorem, 5.12, and Mumford’s theorem, 8.8.3.) This gives a beautiful compactiﬁcation for the modular space T(M)/ Diﬀ(M), which has been studied by Bers, Earle and Mar-den and Abikoﬀ. What we shall do works because a polygonal orbifold has a ﬁnite modular group. For any two-dimensional orbifold O with χ(O) < 0, let N(O) be the space of all hyperbolic structures with nodes (up to isotopy) on O. Theorem 13.8.1. When P is an n-gonal orbifold, N(P) is homeomorphic to the (closed) disk, Dn−3, with interior T(P). It has a natural cell-structure with open cells parametrized by the set of nodes (up to isotopy). Here are the three simplest examples. If P is a quadrilateral, then T(P) is R. There are two possible nodes. N(P) looks like this: If there are two adjacent order 2 corner reﬂectors, the qualitative picture must be modiﬁed appropriately. For instance, When P is a pentagon, T(P) is R2. There are ﬁve possible nodes, and the cell-structure is diagrammed below: 13.64 Thurston — The Geometry and Topology of 3-Manifolds 347 13. ORBIFOLDS When there is only one node, the pentagon is pinched into a quadrilateral and a triangle, so there is still one degree of freedom. When P is a hexagon, there are 9 possible nodes. Each single node pinches the hexagon into a pentagon and a triangle, or into two quadrilaterals, so its associated 2-cell is a pentagon or a square. The cell division of ∂D3 is diagrammed below: (The zero and one-dimensional cells are parametrized by the union of the nodes of 13.65 the incident 2-cells.) Proof of 13.8.1. It is easy to see that N(P) is compact by familiar arguments, as in 5.12 and 8.8.3, for instance. In fact, choose ϵ suﬃciently small so that P(0,ϵ] is always a disjoint union of regular neighborhoods of short arcs. Given a sequence {Pi}, we can pass to a subsequence so that the core one-orbifolds of the components 348 Thurston — The Geometry and Topology of 3-Manifolds 13.8. GEOMETRIC COMPACTIFICATION of Pi(0,ϵ] are constant. Extend this system of arcs to a maximal system of disjoint geodesic arcs {α1, . . . , αk}. The lengths of all such arcs remain bounded in {Pi} (this follows from area considerations), so there is a subsequence so that all lengths converge—possibly to zero. But any set of {l(αi)\|l(αi) ≥0} deﬁnes a hyperbolic structure with nodes, so our sequence converges in N(P). Furthermore, we have described a covering of N(P) by neighborhoods diﬀeomor-phic to quadrants, so it has the structure of a manifold with corners. Change of coordinates is obviously diﬀerentiable. Each stratum consists of hyperbolic struc-tures with a prescribed set of nodes, so it is diﬀeomorphic to Euclidean space (this also follows directly from the nature of our local coordinate systems.) Theorem 13.8.1 follows from this information. Here is a little overproof. An explicit homeomorphism to a disk can be constructed by observing that PL(P)‡ has a natural triangulation, which is dual to the cell structure of ∂N(P). This arises from the fact that any simple geodesic on P must be orthogonal to the mirrors, so a geodesic lamination on P is ﬁnite. The simplices in PL(P) are measures on a maximal family of geodesic one-orbifolds. 13.66 A projective structure for PL(P)—that is, a piecewise projective§ homeomor-phism to a sphere—can be obtained as follows (compare Corollary 9.7.4). The set of geodesic laminations on P is in one-to-one correspondence with the set of cell divisions of P which have no added vertices. Geometrically, in fact, a geometric lamination extends in the projective (Klein) model to give a subdivision of the dual polygon. Take the model P now to be a regular polygon in R2 ⊂R3. Let V be the vertex set. For any function f : V →R, let Cf be the convex hull of the set of points ‡For deﬁnition, and other information, see p. 8.58 §See remark 9.5.9. Thurston — The Geometry and Topology of 3-Manifolds 349 13. ORBIFOLDS obtained by moving each vertex v of P to a height f(v) (positive or negative along the perpendicular to R2 through v). The “top” of Cf gives a subdivision of P. The nature of this subdivision is unchanged if a function which extends to an aﬃne function from R2 to R is added to f. Thus, we have a map RV /R3 →GL(P). To lift the map to measured laminations, take the directional derivative at O of the bending measure for the top of the convex hull, in the direction f. The global description of this map is that a function f is associated to the measure which assigns to each edge e of the bending locus the change in slope of the intersection of the faces adjacent to e with a plane perpendicular to e. It is geometrically clear that we thus obtain a piecewise linear homeomorphism, 13.67 e : ML(P) ≈RV −3 −0. The set of measures which assigns a maximal value of 1 to an edge gives a realization of PL(P) as a convex polyhedral sphere Q in RV −3. The dual polyhedron Q∗— which is, by deﬁnition, the set of vectors X ∈RV −3 such that supy∈Q X · Y = 1—is the boundary of a convex disk, combinatorially equal to N(P). This seems explicit enough for now. □ 13.9. A geometric compactiﬁcation for the deformation spaces of certain Kleinian groups. Let O be an orbifold with underlying space XO = D3, ΣO ⊂∂D3, and ∂ΣO a union of polygons. We will use the terminology Kleinian structure on O to mean a diﬀeomorphism of O to a Kleinian manifold B3 −LΓ/Γ, where Γ is a Kleinian group. In order to describe the ways in which Kleinian structures on O can degenerate, we will also deﬁne the notion of a Kleinian structure with nodes on O. The nodes are meant to represent the limiting behavior as some one-dimensional suborbifold S becomes shorter and shorter, ﬁnally becoming parabolic. We shall see that this happens only when S is isotopic in one or more ways to ∂O; the geometry depends on the set of suborbifolds on ∂O isotopic to S which are being pinched in the conformal geometry of ∂O. To take care of the various possibilities, nodes are to be of one of these three types: (a) An incompressible one-suborbifold of ∂O. (b) An incompressible two-dimensional suborbifold of O, with Euler character-istic zero and non-empty boundary. In general, it would be one of these ﬁve: 13.68 350 Thurston — The Geometry and Topology of 3-Manifolds 13.9. GEOMETRIC COMPACTIFICATION but for the orbifolds we are considering only the last two can occur. (c) An orbifold T modelled on P2k × R, k > 2 where P2k is a polygon with 2k sides. The sides of P2k are to alternate being on ∂O and in the interior of O. (Cases a and b could be subsumed under this case by thickening them and regarding them as the cases k = 1 and k = 2.) A Kleinian structure with nodes is now deﬁned to be a Kleinian structure in the complement of a union of nodes of the above types, neighborhoods of the nodes in being horoball neighborhoods of cusps in the Kleinian structures. Of course, if O minus the nodes is not connected, each component is the quotient of a separate Kleinian group (so our deﬁnition was not general enough for this case). Let N(O) denote the set of all Kleinian structure with nodes on O, up to homeo-morphisms isotopic to the identity. As for surfaces, we deﬁne a topology on N(O), by saying that two structures K1 and K2 have distance ≤ϵ if there is a homeomorphism between them which is an eϵ −quasi-isometry on K1[ϵ,∞) intersected with the convex hull of K1. Theorem 13.9.1. Let O be as above with O irreducible and ∂O incompressible. If O has one non-elementary Kleinian structure, then N(O) is compact. The conformal structure on ∂O is continuous, and it gives a homeomorphism to a disk, N(O) ≈N(∂O). Note: The necessary and suﬃciently conditions for existence of a Kleinian struc-ture will be given in [???] or they can be deduced from Andreev’s theorem 13.6.1. 13.69 We will use 13.6.1 to prove existence. Proof. We will study the convex hulls of the Kleinian structures with nodes on O. (When the Kleinian structure is disconnected, this is the union of convex hulls of the pieces.) Lemma 13.9.2. There is a uniform upper bound for the volume of the convex hull, H, of a Kleinian structure with nodes on O. Proof of 13.9.2. The bending lamination for ∂O has a bounded number of components. Therefore, H is (geometrically) a polyhedron with a bounded number of faces, each with a bounded number of sides. Hence the area of the boundary of Thurston — The Geometry and Topology of 3-Manifolds 351 13. ORBIFOLDS the polyhedron is bounded. Its volume is also bounded, in view of the isoperimetric inequality, volume (S) ≤1/2 area(∂S) for a set S ⊂H3. (cf. §5.11). □ Theorem 13.9.1 can now be derived by an adaptation of the proof of Jørgensen’s theorem (5.12) to the present situation. It can also be proved by a direct analysis of the shape of H. We will carry through this latter course to make this proof more concrete and self-contained. The ﬁrst observation is that H can degenerate only when some edges of H become very long. When a face of H has vertices at inﬁnity, “length” is measured here as the distance between canonical neighborhoods of the vertices. In fact, if the edges of H remain bounded in length, the faces remain bounded in shape by (§13.8, for 13.70 instance; the components of ∂H can be treated as single faces for this analysis). If we view XH as a convex polyhedron in H3 then as long as a sequence {Hi} has all faces remaining bounded in shape, there is a subsequence such that the polyhedra {XHi} converge, in the sense that the maps of each face into H3 converge. One possibility is that the limiting map of XH has a two-dimensional image: this happens in the case of a sequence of quasi-Fuchsian groups converging to a Fuchsian group, and we do not regard the limit as degenerate. The signiﬁcant point is that two silvered faces of H (faces of H not on ∂H) which are not incident (along an edge or at a cusp) cannot come close together unless their diameter goes to inﬁnity, because any points of close approach are deep inside H(0,ϵ]. We can obtain a good picture of the degeneration which occurs as an edge becomes very long by the following analysis. We will consider only edges which are not in the interior of ∂H. Since the area of each face of H is bounded, any edge e of H which is very long must be close and nearly parallel, for most of its length all but a bounded part, of its length, on both sides, to other edges of its adjacent faces. Similarly, these nearly parallel edges must be close and nearly parallel to still more edges on the far side from e. How long does this continue? Remember that H has an angle at each edge. In fact, if we ignore edges in the interior of ∂H, no angle exceeds 90◦. Special note should be made here of the angles between ∂H and mirrors 13.71 352 Thurston — The Geometry and Topology of 3-Manifolds 13.9. GEOMETRIC COMPACTIFICATION of H: the condition for convexity of H is that ∂H, together with its reﬂected image, is convex, so these angles also are ≤90◦. (If they are strictly less, then that edge of ∂H is part of the bending locus, and consequently it must have ends on order 2 corner reﬂectors.) Since H is geometrically a convex polyhedron, the only way that it can be bent so much along such closely spaced lines is that it be very thin. In other words, along most of the length of e, the planes perpendicular to e ⊂XH ⊂H3 intersect XH in a small polygon, which represents a suborbifold. It has 2, 3 or 4 intersections with edges of XH not interior to ∂H. Thurston — The Geometry and Topology of 3-Manifolds 353 13. ORBIFOLDS By area-angle considerations, this small suborbifold must have non-negative Euler characteristic. We investigate the cases separately. (a) χ = 0, ∂= ∅ (i) 3 3 3 This is automatically incompressible, and since it is closed, it must be homotopic to a cusp. But this is supposed to be avoided by keeping our investigations away from the vertices of faces of P. (ii) 2 2 2 2 Either it is incompressible, and avoided as in (i), or com-pressible, so it is homotopic to some edge of H. But since it is small, it must be very close to that edge. This contradicts the way it was chosen—or, in any case, it can account for only a small part 13.72 of the length of e. (b) χ = 0, ∂̸= ∅: (i) 2 2 m m m ∂ (ii) m m ∂ ∂ where m denotes a mirror. These can occur either as small ∂-incompressible suborbifolds (repre-senting incipient two-dimensional nodes) or as small ∂-compressible orb-ifolds, representing the boundary of a neighborhood of an incipient one-dimensional node. (c) χ > 0. This can occur, since O is irreducible and ∂O incompressible. We now can see that H is decomposed into reasonably wide convex pieces, joined together along long thin spikes whose cross-sections are two-dimensional orbifolds 354 Thurston — The Geometry and Topology of 3-Manifolds 13.9. GEOMETRIC COMPACTIFICATION with boundary. There also may be some long thin spikes representing neighborhoods of short one-suborbifolds (arcs) of ∂O. H(0,ϵ] contains all the long spikes. It may also intersect certain regions between 13.73 spikes, where two silvered faces of H come close together. If so, then H(0,ϵ] contains the entire region, bounded by spikes (since each edge of the two nearby faces comes to a spike within a bounded distance, as we have seen). The fundamental group of that part of H must be elementary: in other words, all faces represent reﬂections in planes perpendicular to or containing a single axis. It should by now be clear that N(O) is compact. By [???], Kleinian structures with nodes of a certain type on O are parametrized, if they exist, by conformal structures with nodes of the appropriate type on ∂O. Given a Kleinian structure with nodes, K, and a nearby element K′ in N(O), theer is a map with very small dilation from all but a small neighborhood of the nodes in ∂K to ∂K′, covering all but a long thin neck; this implies that ∂K′ is near ∂K in N(∂O). Therefore, the map from N(O) to N(∂O) is continuous. Since N(O) is compact, the image is all of N(∂O). Since the map is one-to-one, it is a homeomorphism. □ To be continued. . . . Thurston — The Geometry and Topology of 3-Manifolds 355 William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in T EX by Sheila Newbery, who also scanned the ﬁgures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the ﬁrst few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at levy@msri.org. Index (G, X)-manifold, 27 MΓ, 179 NΓ, 180 OΓ, 180 PΓ, 180 PSL(2, C), 92–96 GL(S), 209, 215 G-manifold, 27 ML(S), 209, 210, 251 ML0(S), 209, 251 PL(S), 209 PL0(S), 210, 263 H-foliation, see also foliation G-orbifold, 301 N(O), 347 Tγ, 243 N(O), 351 accidental parabolics, 257 action discrete orbits, 175 properly discontinuous, 174 wandering, 175 Ahlfors’ Theorem, 180 algebraic hyperbolic manifolds, 168 algebraic limit, 225 algebraic numbers, 143 Andreev’s Theorem, 330 Bass, H, 143 bending measure, 189–191 Bers, 111 billiard table, 298 Borromean rings, 33–34, 105, 300, 322 commensurable with Whitehead link, 141 boundary mirror, 301 branched cover over a link, 2 circle packing, 330 nerve of, 330 commensurability classes, 144 inﬁnite number of, 150 commensurable and cusp structure, 142 discrete subgroups of PSL(2, C), 140 manifolds, 140 with complete manifolds, 141 complete (G, X)-manifold, 35 H-foliation, 64 completeness criteria, 36, 38, 41, 42 completion of a hyperbolic 3-manifold, 54–56 of a hyperbolic surface, 41–42 cone manifolds, 55–56 convergence algebraic, 225 geometric, 225 strong, 226 convex, 177 locally, 177 implies convex, 177 strictly, 178 and homotopy equivalence, 179 convex hull, 171 boundary, 185 corner reﬂectors, 309 Coxeter diagram, 326 Thurston — The Geometry and Topology of 3-Manifolds 357 INDEX cusp extra, 216 deformations of a 3-manifold, 85 dimension of, 88, 97 extend to Dehn ﬁlling, 103 of compact convex hyperbolic manifolds, 178 Dehn surgery, 2, see also ﬁgure-eight knot invariants, 57 developing map, 35, 54, 185 of a (G, X)-orbifold, 309 of a convex manifold, 177 of an aﬃne torus, 37 discrete, 64 domain of discontinuity, 174 retraction onto convex hull, 174 edge equations, 49–51 elementary group, 171 elliptic why distressing, 10 essentially complete, 244 Euclidean triangles, 47 Euler number for a ﬁbration, 323 extension of a vector ﬁeld, 288 Curl and Div, 292 direction derivative of, 290 ﬁbration over an orbifold, 319, 323 ﬁgure-eight knot, 4–7, 29–31, 120 commensurable with PSL(2, O3), 149 complete hyperbolic structure, 54 Dehn surgery on, 58–61, 70 yields hyperbolic manifold, 61 fundamental group, 172 gluing diagram, 4 incompressible surfaces in, 72–83 limit set, 172 parametrization space of complement, 52 volume of complement, 164 foliation developing map for, 63 hyperbolic, 62, 64 Fricke space, 92 Fuchsian group, 172, 192 fundamental group acts ergodically on Sn−1 ∞ , 111 Gauss-Bonnet for orbifolds, 312 Gehring, 111 geodesic ﬂow conditions for ergodicity, 277 geodesic lamination, 186, 200, see also geo-metric and measure topology, see also lam-ination complete, 196 ending, 238 essentially complete, 243, 246 measure on, 207 near a cusp, 201 realizable, 208, 211, 214, 240 criterion for, 261 train track approximation of, 204, 206, 210, 213 with compact support, 239 geodesics on hyperboloid, 18 geometric limit, 225 geometric structure, 85 geometric topology, 225 and compactness, 228 on geodesic laminations, 208 geometrically ﬁnite, 180, 183 and cusps, 182 hyperbolic 3-manifold, 203 geometrically near, 118 geometrically tame, 219, 221, 229 almost, 230, 240 and algebraic convergence, 259 and geodesic ﬂow, 278 implies topologically tame, 240 Gieseking, 29 Gromov, 102 Gromov’s invariant, 123, 140 for manifolds with boundary, 134 norm, 123, 127 Theorem 358 Thurston — The Geometry and Topology of 3-Manifolds INDEX relative version, 136 strict version, 130 Haken 3-orbifold, 324, 325 manifold, 71 Haken, W., 72 Hatcher, 101 Heegaard decomposition, 3 Hilbert, 10 holonomy, 35, 53, 85, 97–100 deﬁnes structure, 85 horoball, 39 horocycles, 40 horospheres, 38 hyperbolic isometries, 67 line, 10, 13, 14 metric, 11, 13, 17, 39–40 plane, 10 structures on a manifold, 87 hyperbolic Dehn surgery theorem, 104 hyperbolic structure with nodes, 346 hyperboloid, 17 hyperplane, 13 and dual point, 16, 19 ideal tetrahedra, 45–48 parametrization of, 48 volume of, 160 ideal triangles, 40 identifying faces of polyhedra, 3 incompressible suborbifold, 324 surface, 71 and algebraic representations, 143 inner product, 18, 21 intersection number, 267, 270 irreducible 3-manifold, 2 orbifold, 324 Jørgensen’s Theorem, 119–120 ﬁrst version, 116 Jørgensen, T., 61, 74, 228 Kleinian group, 174 manifold of, 178 Kleinian structure, 350 knotted Y, 31 lamination, 185, see also geodesic lamination when isotopic to geodesic lamination, 206 on boundary of convex hull , 186–187 law of cosines hyperbolic, 22 law of sines hyperbolic, 25 Lickorish, 2 limit set, 171 of a closed hyperbolic manifold, 172 link of a vertex, 42 of an ideal tetrahedron, 45–46 links Ck, 144 D2k, 150 Fn, 154 having isomorphic complements, 149 Lobachevsky, 157 manifold aﬃne, 27–28 diﬀerentiable, 27 elliptic, 28–29 hyperbolic, 29 Margulis lemma, 113 measure topology on geodesic laminations, 209 measured lamination space, 210, 251 metric of constant curvature and patterns of circles, 338 Micky Mouse, 194 minimal set, 172 modular space, 201 Mostow’s Theorem, 101–102, 106–112, 129– 130 Mumford, 201 nodes, 346 orbifold, 300 bad, 304, 324 Thurston — The Geometry and Topology of 3-Manifolds 359 INDEX classiﬁcation of 2-dimensional, 312 covering, 303, 305, 311, 313 Euler number, 311 fundamental group, 307 good, 304, 310, 312 hyperbolic structure, 314–318 with boundary, 308 pair of pants, 90 Papakyriakopoulos, 2 parallel, 14 pared manifolds, 259 pleated surface, see also uncrumpled surface Poincar´ e dodecahedral space, 320 Prasad, 102 projective lamination space, 209, 210, 263 properly discontinuous action, 174 pseudo-isometry, 106 pseudogroup, 27 Pythagorean theorem hyperbolic, 25 quasi-conformal map, 110 quasi-Fuchsian group, 192, 215 mapping surfaces into, 194 quasi-isometric vector ﬁeld, 290 quasi-isometry, 285 rational depth, 252 reﬂection group, 323 Riley, R., 29, 74, 168 Schottky group, 173 Seifert ﬁbration, 64 singular locus, 302, 308 smear, 127 sphere at inﬁnity, 11 suborbifold, 308 suﬃciently large, 71 Sullivan, 277 symmetry of 2-generator 3-manifold, 94 tangent space of an orbifold, 318 Teichm¨ uller space, 88, 89–92 for hyperbolic orbifold, 318 of the boundary of a 3-manifold, 97 thick-thin decomposition, 112 characterization of M(0,ϵ], 115 for an orbifold, 346 three-punctured sphere, 36 tractrix, 10 train track, 205 dual, 267, 271 transverse measure, 189 ultraparallel, 14 uncrumpled surface [pleated], 200, 219, 249, see also wrinkling locus realizing essentially complete lamination, 249– 251 unit tangent bundle of orbifold, 319–323 visual average, 285 volume and Gromov’s invariant, 126–128 goes down after Dehn ﬁlling, 138 is a continuous function, 119 is well-ordered, 139 of a straight k-simplex, 124 Waldhausen, 142 Whitehead link, 32–33, 120 commensurable with Borromean rings, 141 volume of complement, 165 wrinkling locus, 201, 209 360 Thurston — The Geometry and Topology of 3-Manifolds
188975	https://ecosystems.psu.edu/research/centers/private-forests/news/animal-activity-as-winter-turns-to-spring	Animal activity as winter turns to spring Posted: February 22, 2024 By Jeff Osborne - About two weeks ago, a nice marmot, Punxsutawney Phil, purportedly did not see his shadow after he was pulled from a prop tree stump and held aloft by a very formally dressed man. This is supposed to mean an early spring and warmer weather in February and March. Unfortunately, according to historic temperature data, this predictive method has been incorrect over 60% of the time. Maybe Phil is wrong on purpose because he wants to be left resting in his burrow a while longer like all his other woodchuck friends. There are many animals that slow their metabolic functions when food is scarce, like during winter in Pennsylvania, and some have a flurry of activity after the cold weather breaks in late winter and early spring. Last week as I drove to work, I smelled thiols which are chemical compounds that have a sulfur smell. They are found in skunks and added to natural gas. I smelled sulfur many times and then started to notice expired striped skunks along the road. On my way back from work, 45 miles, I counted nine dead skunks, which is a great increase from average. This great increase marks the beginning of skunk mating season and increased activity on their part. Most male skunks huddle alone during cold winter weather in underground dens. There they can enter a state of torpor. In torpor, they reduce their heart rate, respiration rate, and body temperature. This helps conserve energy and water. They can go into torpor daily for 9 to22 hours during winter. After the midpoint of winter, they leave their dens to find mates, often spurred by a period of warmer weather. Some skunks den in groups, with several females and up to one male. These skunks may not undergo torpor as they use less energy to maintain their body temperature and are able to expend their fat reserves at a much slower rate. Bats are another mammal in Pennsylvania that spend much of the winter resting. They enter a longer state of torpor, often called hibernation. Six of the nine species of bats common within the state hibernate here. For up to six weeks at a time, little brown bats can reduce their heartbeat below 20 bpm, versus over 1000 bpm during flight. They also reduce their respiration rate to about 10 breaths per minute, and their body temperatures dip to the ambient temperature of their hibernation spot, often around 40 degrees F. Big brown bats can inhabit your attic all year long, hibernating there in winter. Bats will break hibernation to replenish fluids and if disturbed. There is a great expenditure of energy to break hibernation, and if bats are disturbed too many times, they may not have enough energy to last until spring. In early spring, bats will burst forth to seek their roosting sites and begin the offspring gestation process. Chipmunks and woodchucks are some of the many rodents that slow their metabolic functions in winter. Many chipmunks undergo torpor for a few days at a time from late December through February, awakening to eat from their food hoard. Woodchucks store up fat to survive longer spells of torpor. Woodchucks begin entering a torpor state around November 15 and resume normal metabolic rates around the last week of February. During this time, they may only break torpor about 12 times. As winter breaks, these rodents will seek mates and establish or defend their burrows and surrounding territory. Birds also use torpor to conserve energy. Hummingbirds can enter torpor and stay in the state for a few hours. This can help them retain energy reserves on cool late-spring nights or store a bit on energy in cool late-summer nights before they travel south. Whip-poor-wills have also been documented utilizing torpor. They sometimes enter torpor in cool spring and autumn mornings and stay in the state for a few hours. Cold-blooded animals can slow their metabolic functions as well. This is called brumation. Some cold-blooded animals, like toads, burrow several feet underground and relax until spring. Others, like the bull frog, wait for spring in deep, cool water. The wood frog can go under leaf litter or into a crevasse and literally freeze. It may have no heartbeat for months. The wood frog lives as far south as Georgia and is the only known amphibian living above the Arctic Circle. Their bodies accumulate glucose and urea, which prevents individual cells from freezing, although the water between cells freezes. Spring peepers will freeze in the winter as well, and, depending on temperature, the first week of March is a good time to start listening for their first calls of the year. James C. Finley Center for Private Forests James C. Finley Center for Private Forests Address James C. Finley Center for Private Forests Address Department of Ecosystem Science and Management Privacy and Legal Statements — Non-Discrimination — Accessibility — © 2025 The Pennsylvania State University Explore Information for
188976	https://hal.science/hal-04280008v1/file/cfm2022_7046.pdf	HAL Id: hal-04280008 Submitted on 10 Nov 2023 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Effect of pump size and non-Newtonian rheology on the hydrodynamic behavior of centrifugal volute pumps Lila Achour, Mathieu Specklin, Miguel Asuae, Smaine Kouidri, Idir Belaidi To cite this version: Lila Achour, Mathieu Specklin, Miguel Asuae, Smaine Kouidri, Idir Belaidi. Effect of pump size and non-Newtonian rheology on the hydrodynamic behavior of centrifugal volute pumps. 25e Congrès Français de Mécanique, Aug 2022, Nantes, France. hal-04280008 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 Effect of pump size and non-Newtonian rheology on the hydrodynamic behavior of centrifugal volute pumps L. Achour a,b, M. Specklin a, M. Asuaje c, S. Kouidri a, I. Belaidi b a. LIFSE, Arts et Métiers Institute of Technology, CNAM, HESAM University, F-75013 Paris, France; lila.achour@ensam.eu (L.A.); mathieu.specklin@lecnam.net (M.S.); smaine.kouidri@ensam.eu (S.K.) b. LEMI, FT, University of M’hamed Bougara, Avenue de L’indépendance, Boumerdes 35000, Algeria; l.achour@univ-boumerdes.dz(L.A.); idir.belaidi@gmail.com (I.B.) c. Department of Energy Conversion and Transport, Universidad Simón Bolivar, Cra 59 N◦59-65, ZP :1086, Caracas-Venezuela; asuajem@usb.ve (M.A.) Abstract : The design of centrifugal pumps usually requires the creation of a scale model, whose purpose is to find the operating conditions of the prototype. These centrifugal pumps are generally designed to convey water or an incompressible fluid of low viscosity. However, they find applications in many engineering processes where the working fluid is a mixture of two immiscible liquids that form an emulsion. In this case, the complex rheological behavior of the two-phase liquid-liquid flow and the emulsion significantly affects the flow pattern in the centrifugal pump and thus its performance. Several studies have shown that the viscosity of emulsions is higher than that of a single-phase fluid and follows a non-Newtonian beha-vior, which depends on the local shear rate. However, the shear rate changes with the size and geometry of the pump. This study aims to numerically investigate the scaling phenomenon in a centrifugal volute pump and its influence on the non-Newtonian behavior of emulsions. Furthermore, it aims to analyze whether the performance reduction when pumping non-Newtonian fluid is pump geometry dependent. The scaled pumps consist of a five-bladed backward curved impeller and a volute, and the similarity ratio is chosen to achieve a design scale dimension, keeping the same flow regime (same Reynolds range and close specific speeds). The results of the numerical model presented in this study show that the per-formance reduction factors of the small size pump are higher than those of the large pump even though the slippage is less when the fluid has a shear thinning behavior. Moreover, the mechanical shearing of the small-size pumps was more important than the large-size ones. Even though the emulsions are subjected to a high shear rate (> 125 %) and have a lower viscosity (reduction of 5 % at walls) in the small pump, the friction coefficient are higher in the latter. Keywords : Pump size; emulsions; non-Newtonian; CFD; slip factor 1 Introduction Despite their operational advantages and widespread use in many processes engineering, there are still many issues regarding the selection and operation of centrifugal pumps, especially since the operating 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 conditions differ from the tested ones. Many on-site experiments discovered some phenomena that were never observed in model tests . Before manufacturing and operating a pump, generally, small-scale model tests with water are used to estimate its actual performance. However, the actual operating condi-tions are not similar to those in the laboratory. In some cases, the fluid encountered in the industry is a mixture of two immiscible liquids forming an emulsion. A typical example is the pumps used in the petroleum industries, chemical processes, or wastewater treatment. In this engineering process, water is produced together with oil, causing the formation of oil-water emulsions. Nevertheless, depending on the fluid’s viscosity, emulsion reduces the performance of the centrifugal pump [2–5], and the un-derstanding of its fundamental characteristics is necessary to study its effect on pump performance. The most critical aspect of emulsion’s rheology is its viscosity behavior. It is characterized by a wide range of viscosity higher than that of a single-phase fluid [6, 7] and a complex non-Newtonian rheological beha-vior. Its viscosity depends essentially on temperature, shear stress, morphology, and stability, governed by the droplet size of the dispersed phase along with the presence of a surfactant [8–10]. Recently, there has been a significant breakthrough in the performance analysis of centrifugal pumps, allowing us to understand their operation when handling fluids with different rheological characteristics. Some of the key characteristics commonly studied are impeller exit angle and slip factor. The blade outlet angle di-rectly affects the internal flow field and performance of the centrifugal pump . As the slip factor is a measure of the fluid slip in the impeller discharge, it is thus an important design parameter for deciding the correct impeller diameter of centrifugal pumps . Several sets of slip factor correlations have been proposed over the years and are divided into three main approaches . The semi-analytical approach is based on the number and the exit angle of the blades. As an example, we find the correlations pro-posed by Stodola and Wiesner. The experimental correlations implement the direct measurement of the slip factor from the velocity profile at the exit of the impeller () or based on changes in flow rates. The third approach uses Computational Fluid Dynamics (CFD) which also uses the velocity profile obtained at the impeller outlet to calculate the slip between the fluid path and the blade curvature or to feed the theoretical head formula with CFD data. Currently, many studies have focused on studying the slip factor to determine the performance of centrifugal pumps handling highly viscous fluids. However, studies on the effect of non-Newtonian rheology and multiphase flow on slippage in centrifugal volute pumps are very scarce. Unfortunately, the pump performance degradation is not only depending on the pumped fluid rheology. Because of the pump size, each pump can respond differently to viscous dissipation or other losses that affect its performance. Similar performance degradation is not expected since the fluid rheology de-pends on the geometry and the local shear stress. Also, the structure of the non-Newtonian flow can differ according to the pump’s dimensions even if the geometry remains the same. The present work is a numerical investigation of the scaling phenomenon in a centrifugal volute pump when handling non-Newtonian emulsions. The main objective is to analyze whether the performance degradation when handling non-Newtonian fluids is size dependent, and give an in-depth analysis of the internal flow beha-vior in a real and scaled-down pump. Moreover, to study the rheological behavior and pump size effects on the slip factor by mean of CFD. Although two-phase by nature, the emulsions were modeled here as a single-phase fluid. This modeling implies to assume the homogeneity of the fluid, and to neglect specific interactions. The numerical simulations were performed using a RANS approach for turbulence mode-ling and have been carried out using the open-source software OpenFOAM. CFD models were validated against experimental data for water, considering both steady and unsteady approaches. Simulations of the flow fields within the centrifugal pump handling emulsions at different water cuts were performed 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 under steady-state conditions. The emulsions considered in this study consist of two-phase mixtures of sunflower oil and water at different water cuts (WC). These emulsions exhibit shear thinning behavior and were modeled by the Cross and Carreau laws whose parameters were determined experimentally under a shear rate ranging from 1 s−1 to 3000 s−1 . The numerical results showed that the pump size influences the rheological behavior of non-Newtonian fluids. The performance degradation, which results in reduced head, pump efficiency, and a diminution of the flow rate at the BEP (Best Efficiency Point) compared to pumping water alone, is complex and highly depends on the fluid viscosity and its rheology. The small pump appears to be more sensitive to the viscosity of the fluid. In particular, the non-Newtonian behavior leads to a greater reduction in performance for the small size pump characterized by lower values of the specific speed. 2 Computational model and research method 2.1 Pumps model and fluid properties To predict the effect of pump size on performance degradation when pumping a non-Newtonian fluid and on slip factor behavior, three-dimensional CFD simulations were performed in two geometrically similar centrifugal pumps. The pumps consist of a semi-open impeller with five backward curved blades and a volute casing. The scaled model (SM) has a scale factor of µ = 1/5 that is applied to all parts of the pump. This scaling factor was chosen to achieve a design scale dimension while maintaining the same flow regime (similar Reynolds range and close specific speeds). The main geometric dimensions of the model pump (NS32) are summarized in Table 1 and the nominal operating conditions of both pumps in Table 2. Inlet diameter (mm) Outlet diameter (mm) Inlet blade width (mm) Outlet blade width (mm) Inlet blade angle (◦) Outlet blade angle (◦) Number of blades Blade thickness (mm) 150 408.4 85.9 42 70 63 5 8 Table 1 – Main geometric parameters of the designed pump Characteristic NS32 SM Nominal Head (m) 49 10.63 Rotational speed (rpm) 1470 3450 Nominal Flowrate (m3/h) 590 7.5 Specific Speed 32 26 Re=R2 2ω/ν 3.93 106 6.02 105 Table 2 – NS32 nominal operating conditions In this study, three different emulsions were considered, corresponding to the emulsions experimentally studied by Valdes . The flow system consisted of two-phase mixtures of sunflower oil and water at different water cuts : a pseudo-stable W/O emulsion (WC < 20%), a pseudo-stable concentrated W/O emulsion (phase inversion), and a multi-regime emulsion with high water fractions (WC > 40%). The rheological properties of these mixtures were measured experimentally under a shear rate ranging from 1 s−1 to 3000 s−1 and were fitted to the Cross (Equation (1)) and Carreau (Equation (2)) models. Al-though two-phase in nature, the emulsions were modeled in this case as a single-phase fluid, having a pseudoplastic behavior (model parameters are presented in Table 3). 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 µeff −µ∞ µ0 −µ∞ = [1 + (kc ˙ γ)nc]−1 (1) µeff −µ∞ µ0 −µ∞ = h 1 + (λt ˙ γ)2i ncar−1 2 (2) Phase Composition (% v/v oil) Viscosity Model ρ (kg/m3) nc/ncar kc(sn)/λt(s) ν0(m2/s) νinf(m2/s) 80 Carreau 947.8 0.471 2.03 10−4 7.59 10−5 2.40 10−5 70 Cross 953.0 0.801 23.39 1.57 10−2 3.14 10−5 40 Cross 978.3 0.416 21.06 3.27 10−5 1.02 10−5 Table 3 – Rheological characteristics of the studied emulsions kc and nc are the Cross time and Cross rate constant, respectively. λt and ncar are the relaxation time and the power index, respectively. ν0 and νinf are the viscosity for zero shear rate and very high shear rate. 2.2 Computational model The numerical simulation is conducted using the open-source library OpenFOAM v1906 which uses a finite volume method (FVM) to discretize the fluid equations. The centrifugal pumps models include four domains : the inlet pipe, the impeller, the volute, and the outlet pipe as shown in Figure 1 (a). 2.2.1 Physical Model Specification This study assumes that the flow within the centrifugal pump is single-phase, incompressible, isothermal, viscous, and turbulent. The SIMPLE algorithm was used to solve the mass and momentum equations (Equation (3) and Equation (4) respectively). ∂ ∂t(ρ) + ∇· (ρu) = 0 (3) ∂ ∂t(ρu) + ∇· (ρu ⊗u) = g + ∇· (τ) −∇· (ρR) (4) (τ) is the averaged stress tensor and R is the Reynolds stress tensor. The k−ϵ model given by Equation (5) and Equation (6) (k : turbulent kinetic energy and ϵ : dissipation rate) was adopted to model the Reynolds stress terms in RANS equations. The MRF technique which is a steady-state approximation was used to model the rotating field. The effects of rotational motion are reproduced by source terms in the fluid equations and the information between the rotating region (impeller) and the static region (volute and the rest of the pump geometry) is transferred through an arbitrary mesh interface (AMI). The inlet velocity and outlet static pressure of 0 Pa are used as the boundary conditions. For the wall boundary condition, no-slip is applied on the wall surfaces. D Dt(ρk) = ∇· (ρDk∇k) + P −ρϵ (5) 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 D Dt(ρϵ) = ∇· (ρDc∇ϵ) + C1ϵ k P + C3 2 3k∇· u −C2ρϵ2 k (6) vt = Cµ k2 ϵ (7) 2.2.2 Setup and validation of the numerical model Because of the complex flow passage in the centrifugal pump, an unstructured mesh was generated (Figure 1 (b) and (c)). Two types of mesh elements were considered : polyhedrons and prisms consisting of approximately 4 million cells in total. Hexahedral elements were used in the inlet and outlet pipes, and polyhedral elements were used for the impeller and volute. A structured mesh was employed for the rotating impeller boundary layer to capture the flow details near the boundaries of the flow domain. This led to an average of y+ < 5 and a direct resolution of the viscous sublayer of the inner region. (a) (b) (c) Computation domain Impeller mesh Volute tongue mesh Fig. 1 – Fluid volume mesh A grid independence check was conducted by evaluating four different mesh sizes. The reference pa-rameter chosen is the pump head conveying water at the best efficiency point. The simulations were considered mesh independent when the relative head difference between two successive meshes don’t exceed 1%. A detailed analysis of the mesh dependency was conducted in a previous study. The numeri-cal model in both steady and unsteady regime considering water as the working fluid has been validated with the experimental data. The authors are invited to refer to article for more information on the validation of the numerical model. 2.3 Slip factor The slip factor reflects the mismatch between the angle at which the fluid leaves the impeller and the angle of the blade. Slip is a very important phenomenon that occurs especially in radial impellers and is valuable in determining the accurate estimation of impeller-fluid energy transfer, head rise, and velocity triangles at the impeller exit. Because the flow does not precisely follow the blade curve, the angle of the fluid streamline is slightly smaller than the blade angle, as shown in Figure 2. 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 ω 𝑾𝑾𝟐𝟐 𝑽𝑽𝟐𝟐 𝑼𝑼𝟐𝟐 𝑉𝑉 2𝑟𝑟 𝑉𝑉 2𝑢𝑢 𝜷𝜷𝟐𝟐𝟐 △𝑽𝑽𝐮𝐮𝐮𝐮 𝜷𝜷𝟐𝟐 𝑾𝑾𝟐𝟐 𝑽𝑽𝟐𝟐 𝑽𝑽𝟐𝟐𝟐 𝑾𝑾𝟐𝟐𝟐 𝑾𝑾𝐫𝐫𝟐𝟐𝟐 𝑾𝑾𝐫𝐫𝟐𝟐 𝑾𝑾𝐮𝐮𝟐𝟐𝟐 𝑾𝑾𝐮𝐮𝟐𝟐 𝑼𝑼𝟐𝟐 ω 𝑾𝑾𝟐𝟐 𝑽𝑽𝟐𝟐 𝑼𝑼𝟐𝟐 𝑉𝑉 2𝑟𝑟 𝑉𝑉 2𝑢𝑢 𝜷𝜷𝟐𝟐𝟐 △𝑽𝑽𝐮𝐮𝐮𝐮 𝜷𝜷𝟐𝟐 𝑾𝑾𝟐𝟐 𝑽𝑽𝟐𝟐 𝑽𝑽𝟐𝟐𝟐 𝑾𝑾𝟐𝟐𝟐 𝑾𝑾𝐫𝐫𝟐𝟐𝟐 𝑾𝑾𝐫𝐫𝟐𝟐 𝑾𝑾𝐮𝐮𝟐𝟐𝟐 𝑾𝑾𝐮𝐮𝟐𝟐 𝑼𝑼𝟐𝟐 Fig. 2 – Velocity triangle at the impeller exit Many correlations have been proposed to estimate this factor, but in most cases the proposed formula does not take into account the effect of flow rate, viscosity, and impeller geometry. In this study, the slip coefficient based on the velocity triangles given by Equation (8) is used. σ∆= ∆Vu2 U2 = Wu2 −Wu2∞ U2 (8) where ∆Vu2 is the slip fluid velocity at impeller outlet in circumferential direction and U2 = ωD2/2 the impeller tip speed. Based on the correlation proposed by Li (), the tangential component of the relative velocity in infinite number of blade W2∞is determined from the 1D uniform ideal flow, and given by Equation (9) while W2 is determined from CFD. Wu2∞= vr200 tan β2b = Q ηV A2ψ2 tan β2b (9) ηV is the volumetric efficiency determined using Equation (10), where Ns is the specific speed of the pump in US Units. ηv = 1 1 + 0.68N−2/3 s −0.07 (10) A2 = πD2b2 and ψ2 = 1−ZSu2/πD2 stands for the impeller exit area and the blade blockage coefficient respectively. Su2 is the tangential exit blade thickness. Thus the final expression of the slip factor is given by Equation (11). σ∆= Wu2 u2 − Q ηV A2ψ2 tan β2bu2 (11) 3 Results and discussion 3.1 Overall pump performance Figure 3 shows the performance of the two pumps, described by the normalized head relative to the best efficiency point (a-b) and the efficiency (c-d) as a function of the flow rate. First, it is important to point out the progressive deterioration of the performance of both pumps when the oil fraction increases. The 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 performance degradation is governed by the lower Newtonian plateau, because centrifugal pumps gene-rate high shear rates so that the viscosity of emulsions approaches this plateau. As a result, degradation increases with oil concentration. These results have already been observed and explained in a previous study for the real pump . Regarding the performance degradation of the small size pump, the same behavior is observed. 0.2 0.4 0.6 0.8 1 1.2 1.4 Q / Qbep 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Wiesner Water 40o60w 70o30w 80o20w Fig. 3 – CFD head of NS32 (a) and scaled-down pump (b), efficiency of NS32 (c) and scaled-down pump (d) Another interesting point to discuss is the head degradation observed at low flow (Q = 0.15Qboe) (almost the shut-off point). Theoretically, at zero flow rate, the pump develops the same head regardless of the fluid according to the Euler equation. From the results obtained, we observe that the head of the small pump is identical for all the fluids, but the real pump develops different heads for each fluid. This phenomenon is already observed by Valdes et al in an ESP when the pumped fluid has a non-Newtonian rheology. The author explained these results by phenomena such as secondary flow regions, a decrease in the relative outlet flow angle or incidence losses related to the fluid rheology. In this study, we can attribute these results to the different phenomena cited previously but additionally highlight the effect of pump size. The comparison between the head degradation of both pumps handling the same emulsion clearly shows the effect of pump size on the emulsion behavior and performance degradation (Figure 4). The scaled-down model shows a higher performance degradation than the real model, and this is for all emulsions. There is also an increase of degradation with the increase of flow rate in both pumps. The effect of pump size on the head and efficiency curves are dependent on the type of emulsion and thus on the viscosity of the working fluid. As the lower viscosity limit increases, the degradation in the small size pump becomes more significant. This can be attributed to the reciprocal effect of different pump losses and changes in 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 emulsion viscosity which will be discussed in the next section. The internal flow is analyzed in the next section in order to help drawing conclusion. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Q / Qbep -5 0 5 10 15 20 25 30 35 40 45 Degradation rate (%) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Q / Qbep 0 10 20 30 40 Degradation rate (%) 40o60w NS32 70o30w NS32 80o20w NS32 40o60w Scaled model 70o30w Scaled model 80o20w Scaled model Fig. 4 – Comparison of head degradation rate in both pumps 3.2 First step toward loss characterization Prediction of centrifugal pump performance has traditionally been based on theoretical analysis of hy-draulic losses, which are appropriate for particular pumps and where, in most cases, the fluid viscosity remains unchanged. With the development of CFD tools, the analysis of the internal flow field becomes accessible, and the characterization of the losses can be developed by extracting several parameters for their calculation. This allows a fast and accurate prediction of the pump performance depending on the fluid rheology. In this section, a quantitative and qualitative study based on CFD results is performed respectively for the slip factor and skin friction coefficient, and for the secondary losses. Such a study is a first step in the characterization of the different losses in a centrifugal pump as a function of the fluid rheology and the pump size. The slip factor extracted for the two pumps using the previous method (Equation (11)) is shown in Figure 5 versus flow rate. The different parameters appearing in this equation are extracted from the CFD results, at the impeller exit i.e. by taking the average of the parameters on the periphery of the impeller exit. The slip factor proposed by Wiesner and given by the Equation (12) is drawn on the graph for comparison. σ = √sin β2∞ Z0.7 (12) The slip factor of a conventional closed impeller corresponds well to the value of the Wiesner formula , which is constant in the same pump for all fluids and flow rates. Furthermore, the Wiesner slip factor is constant for all geometrically similar centrifugal pumps since the relative design exit angle is the same. However, Figure 5 shows that the slip factor is highly dependent on pump size and fluid viscosity, along with flow rate for the specific pump considered in this paper. Both centrifugal pumps have a larger slip factor than the Wiesner formula and decreases with increasing fluid viscosity and flow rate. Furthermore, by comparing the slip factor of the small size pump and large pump, we notice that 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 the slip factor of water is higher for the scaled-down pump, but the value decreases when handling a non-Newtonian fluid. In addition, the slip factors for the smaller pump size are more dependent on viscosity variation. The shape of the graphs is almost similar for both pumps and the slope of the slip factor becomes flat at high flow rates. As the fluid viscosity increases, the slip factor approaches the Wiesner slip factor. Another point to highlight is that the slip coefficient obtained for the 40o60w emulsion is higher than that of water between the 0.4Qbep and 0.9Qbep flow rates in the large size pump. The high slip factor values obtained at low flow rates may be attributed to vortex and recirculation zones in the inter-blade space of the impeller. A vortex increases the curvature of the flow streamlines, producing a strong slip effect at the impeller discharge and an increased slip factor as noticed in previous study (). 0.2 0.4 0.6 0.8 1 1.2 1.4 Q / Qbep 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Wiesner Water 40o60w 70o30w 80o20w Fig. 5 – Slip factor of the NS32 (a) ans the scaled-down pump (b) From the values of the obtained slip factor, which depends on the pump size and fluid viscosity, it is evident that the relative tangential velocity is strongly influenced by the pump size and the viscosity of the fluid. Figure 6 shows the variations of the absolute relative velocity calculated at the impeller outlet R/R2 = 1 for both pumps. The effect of fluid rheology and flow rate is more pronounced for the small size pump (SM). In the light of these results, we can conclude that slippage is greater in large pumps when handling a non-Newtonian fluid than in small pumps, i.e., the difference between the theoretical head (Hth) and the theoretical head for an infinite blade number (Hth∞) is lower in the small pump than in the large pump. However, the performance curves observed in section 3.1 show that the degradation is larger in the scaled-down model than in the real model. Given that the slip factor calculated in this study takes into account secondary losses and mismatch losses, this leads to the conclusion that the performance degradation is more dominated by the remaining losses, such as the friction losses, which will be more significant in the small size pump. 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 NS32 SM 0.5 Qbep 0 50 100 150 200 250 300 350 (°) 0 5 10 15 Relative velocity (m /s) Volute tongue 0 50 100 150 200 250 300 350 (°) 0 5 10 15 Relative velocity (m s) Volute tongue Qbep 0 50 100 150 200 250 300 350 (°) 0 5 10 15 Relative velocity (m /s) Volute tongue 0 50 100 150 200 250 300 350 (°) 0 5 10 15 Relative velocity (m /s) Volute tongue 1.2 Qbep 0 50 100 150 200 250 300 350 (°) 0 5 10 15 Relative velocity (m /s) Volute tongue 0 50 100 150 200 250 300 350 (°) 0 5 10 15 Relative velocity (m /s) Volute tongue 0 50 100 150 200 250 300 350 (°) 0 5 10 15 Relative velocity (m2/s) Volute tongue Water 40o60w 70o30w 80o20w Fig. 6 – Absolute relative velocity at impeller exit In order to investigate the effect of pump size on the internal flow structure of non-Newtonian fluids, the relative velocity distribution and streamlines in the mid-span surface of the impeller and volute of both pumps and for all fluids are presented in Figures 7 to 9. The impeller passage flow is expressed by relative velocity profiles and the volute passage flow is expressed by absolute velocity profiles. Figure 7 shows the relative velocity and streamlines in the large pump and the small size pump at partial flow (Q/Q0 = 0.5). A vortex area appears near the pressure side of the impeller blades in contact with the volute tongue in both pumps. The size of this vortex decreases as the volume fraction of the oil phase of the emulsions increases. Nevertheless, a uniform flow is observed in the inter-blade passages away from the volute tongue. In addition, this vortex is larger in the scaled-down model than in the large pump when handling water and is smaller when handling emulsions, and develops in the opposite direction to the impeller rotation. These vortex areas in the impeller cause a sharp decrease in the relative velocity at the impeller outlet. As a result, the theoretical head decreases significantly due to the decrease in absolute tangential velocity at the impeller outlet. This suggests that the secondary losses will be smaller in a small-size pump than in a large pump when handling non Newtonian fluid. This observation supports the results obtained previously, where higher slippage is observed in the large pump when handling non Newtonian fluid. 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 Water 40o60w 70o30w 80o20w NS32 Scaled model Fig. 7 – Velocity profiles and streamlines on the impeller at 0.5 BEP At the design flow rate (Figure 8), the entire flow direction in the impeller passageway deflects toward the pressure side of the blades relative to the suction side of the blades. The velocity profile has a low velocity near the pressure side of the blades and a high velocity near the suction side at the impeller inlet. However, moving toward the exit of the impeller, the velocity profile has a high velocity near the pressure side and a low velocity near the suction side. The overall flow field in the scaled-down pump is asymmetrical to the impeller axis for water and 40o60w emulsion and becomes symmetrical as the volume fraction of oil increases (increasing viscosity as well), while flow field within the large pump is relatively symmetrical to the impeller axis for all fluids. A notable feature recognized in these figures is the presence of vortex zones near the volute tongue of the large pump that develops and moves toward the volute divergent as the fluid viscosity increases. For the scaled-down model, a small vortex zone appears in the volute divergent for 70o30w emulsion only. In Figure 9, the flow profile becomes asymmetric to the axis of rotation of the impeller for all fluids. A large recirculation zone appears in the divergent part of the volute and decreases as the oil volume fraction increases. In comparison to the real pump, the scaled-down model shows uniform flow in the impeller for all fluids, but the relative velocity increases with increasing oil volume fraction. Vortex and recirculation zone appears at the volute tongue for water and 40o60w emulsion only. These vortex and dead zones contribute to hydraulic losses by causing the fluid to lose kinetic energy. It can be noted that the relative velocities at the impeller inlet are almost identical for all fluids in the same pump, and increase with increasing flow. As a result, impact losses become smaller and friction losses more influential as the flow rate increases. 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 Water 40o60w 70o30w 80o20w NS32 Scaled model Fig. 8 – Velocity profiles and streamlines on the impeller at BEP Water 40o60w 70o30w 80o20w NS32 Scaled model Fig. 9 – Velocity profiles and streamlines on the impeller at 1.2 BEP As mentioned earlier, the small pump generates higher shear rates than the large pump. This means that 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 the viscosity of the emulsions will be lower in the scaled-down model than in the large pump, depending on their shear thinning behavior. Effectively, this expectation is observed in Figure 10, which shows the profile of the effective viscosity of the 80o20w emulsion in the two pumps for different flow rate. All the emulsions studied have a lower viscosity in the scaled-down model due to the high shear rate generated by the pump. Another result to note is that in both pumps, the increase of the flow rate does not induce significant variations of emulsions viscosity in the impeller. However, it involves viscosity variations in the volute. This leads to the conclusion that the shear rate in the rotating region is dominated by rotational speed and is less sensitive to changes in flow rate. 0.5 Qbep Qbep 1.2 Qbep NS32 Scaled model Fig. 10 – Effective viscosity profiles for 80o20w emulsion versus flow rate Since the viscosity in the scaled-down pump is lower than in the large pump, we can expect that the frictional losses are less significant in this small pump. Nevertheless, we have seen in section 3.1 that regardless of the flow rate, the performance of the reduced model degrades more than the real model for the same fluid. Moreover, the slippage becomes less significant in the small pump when handling emulsions. Therefore, an analysis of viscosity variation in the pumps wall and friction coefficient values is necessary to further investigate the results. For both pumps, it was noted that the 40o60w emulsion shows a very well-defined shear-thinning ten-dency, and at high shear rates, the viscosity change is minimal. Since both pumps generate high shear rates, this emulsion will have almost similar viscosity values at the walls of both pumps. On average, the effective viscosity of the 60o40w emulsion is 1.08 10−5 m2/s2 at the walls of the reduced model and 1.11 10−5 m2/s2 at the walls of the real model. Same observation for the 70o30w emulsion, the viscosity drops sharply to values of the Newtonian lower plateau at medium shear rates. The average viscosities values of this emulsion reach 3.16 10−5 m2/s2 in the walls of the reduced model and 3.17 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 10−5 m2/s2 in the walls of the real model. In contrast, the 80o20w emulsion exhibits minimal viscosity variations at low shear rates, with a slight tendency toward shear thinning at very high shear rates. The effective viscosity averages 3.45 10−5 m2/s2 and 4.29 10−5 m2/s2 in the walls of the reduced model and the real model, respectively. The averaged skin friction factors applied to the different emulsions by the wet surfaces of impeller and volute are shown in Figure 11 versus flow rate. These skin factors are extracted from CFD results and defined by Equations (13) and (14) for impeller and volute respectively : fi = ¯ τwi 0.5ρu2 2 (13) fv = ¯ τwv 0.5ρu2 2 (14) ¯ τwi and ¯ τwv are the averaged shear stress on wet surface of the impeller and volute respectively. ρ is the fluid density and u2 the impeller tip speed. 0.2 0.4 0.6 0.8 1 1.2 1.4 Q / Qbep 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 fi 10-3 Water 40O60W 70O30W 80O20W 0.2 0.4 0.6 0.8 1 1.2 1.4 Q / Qbep 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 fv 10-3 Water 40O60W 70O30W 80O20W 0.2 0.4 0.6 0.8 1 1.2 1.4 Q / Qbep 0 0.5 1 1.5 2 2.5 3 3.5 fi 10-3 Water 40O60W 70O30W 80O20W 0.2 0.4 0.6 0.8 1 1.2 1.4 Q / Qbep 0 0.5 1 1.5 2 2.5 3 3.5 fv 10-3 Water 40O60W 70O30W 80O20W (a) (b) (c) (d) Fig. 11 – Averaged skin friction coefficient of the impeller (a-c) and the volute (b-d) in the large size pump (a-b) and scaled-down pump (c-d) versus flow rate Regardless of pump size, the influence of flow rate on the skin friction coefficient differs between the impeller and the volute. In both pumps, friction losses in the impeller are higher prior to design flow and become more influential in the volute at higher flow rates. The skin friction factors increase with increasing oil phase volume fraction, which is consistent with increasing fluid viscosity. Comparing the skin friction for the two pumps, the magnitudes of fi and fv are almost twice as high in the small pump than in the large pump. However, the skin friction coefficient of water in both parts (impeller and volute) is almost similar in both pumps over the entire flow range. Generally a higher viscosity in the impeller and volute results in a higher skin friction factor. In centrifu-gal pumps, the shear rate generated at the walls is very high so that the emulsions viscosity approaches the Newtonian lower plateau. Thus, the fluid viscosity will have almost the same impact on skin friction 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 coefficient in both pumps. The friction losses caused by the shear force on the wall surfaces are propor-tional to the skin friction coefficient which depends on the Reynolds number (i.e flow regime). However, the flow in the impeller is not fully developed in most of the regimes inside the impeller . So that the impeller rotational speed and passage curvature affect the value of skin friction coefficient. Although the CFD losses are not exact predictions since the skin friction coefficient (cf) was assumed to be constant and equal to its average along all impeller and volute walls, it was shown that friction losses are higher in a small size pump than in a large one, which explains the increased head degradation obtained previously despite the lower slippage. 4 Conclusion In this study, the effects of pump size on pump performance degradation when handling non-Newtonian emulsions were investigated numerically. The internal flow was then analyzed in order to compare qua-litatively the different losses within the NS32 pump and a 1/5 reduced model of the same pump. Compa-ring the performance of the two pumps when conveying three different emulsions modeled as a single-phase non-Newtonian fluid, the following conclusions can be made : • In the small size pump, larger recirculation zones in the inter-blade space are observed when handling water at a low flow rate, contributing to more irregular velocity profiles at the impeller outlet, and resulting in an increase in slip factor compared with the large pump. In contrast, for emulsions, smaller recirculation zones appear in the inter-blade space, leading to a decrease in slip factor. • The mechanical shearing in small-scale pumps is higher than that of large-scale pumps. Even though the emulsions are subjected to a high shear rate and have a lower viscosity due to their shear-thinning behavior in the small pump, the skin friction coefficient is more significant in the small size pump. • The performance degradation in a small size pump is higher than those of a large size pump. The small size pump has been shown to result in reduced slippage when handling fluids with shear thinning behavior. However, it leads to an increase in frictional losses which is reflected in an increased performance degradation. Références Zhongjie Li, Zhengwei Wang, Xianzhu Wei, and Daqing Qin. Flow Similarity in the Rotor-Stator Interaction Affected Region in Prototype and Model Francis Pump-Turbines in Generating Mode. Journal of Fluids Engineering, Transactions of the ASME, 138(6), 2016. Lissett Barrios, Marisela Rojas, Guilherme Monteiro, and Neil Sleight. Brazil field experience of ESP performance with viscous emulsions and high gas using multi-vane pump MVP and high power ESPs. Society of Petroleum Engineers - SPE Electric Submersible Pump Symposium 2017, (July 2009) :38–48, 2017. Sayed A. Bellary, Md Hamid Siddique, Abdus Samad, Jitendra S. Sangwai, and Bohyun Chon. Effects of crude oil-water emulsions at various water-cut on the performance of the centrifugal pump. International Journal of Oil, Gas and Coal Technology, 16(1) :71–88, 2017. 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 Hattan Banjar, Jianjun Zhu, and Hong Quan Zhang. CFD simulations of oil viscosity and emulsion effects on ESP stage performance. Society of Petroleum Engineers - Abu Dhabi International Petroleum Exhibition and Conference 2020, ADIP 2020, 2020. Natan Augusto Vieira Bulgarelli, Jorge Luiz Biazussi, William Monte Verde, Carlos Eduardo Perles, Marcelo Souza de Castro, and Antonio Carlos Bannwart. A novel criterion based on slip ra-tio to assess the flow behavior of W/O emulsions within centrifugal pumps. Chemical Engineering Science, 247 :117050, 2022. Hattan Banjar and Hong Quan Zhang. Experiments and emulsion rheology modeling in an electric submersible pump. International Petroleum Technology Conference 2019, IPTC 2019, 2019. Jianjun Zhu, Haiwen Zhu, Guangqiang Cao, Hattan Banjar, Jianlin Peng, Qingqi Zhao, and Hong Quan Zhang. A new mechanistic model for oil-water emulsion rheology and boosting pres-sure prediction in electrical submersible pumps ESP. Proceedings - SPE Annual Technical Confe-rence and Exhibition, 2019-Septe(January), 2019. Philipp Schmitt, Mark W. Hlawitschka, and Hans Jörg Bart. Centrifugal Pumps as Extractors. Chemie-Ingenieur-Technik, 92(5) :589–594, 2020. Rosanel Morales, Eduardo Pereyra, Shoubo Wang, and Ovadia Shoham. Droplet formation through centrifugal pumps for oil-in-water dispersions. SPE Journal, 18(1) :172–178, 2013. Rodolfo Marcilli Perissinotto, William Monte Verde, Carlos Eduardo Perles, Jorge Luiz Biazussi, Marcelo Souza de Castro, and Antonio Carlos Bannwart. Experimental analysis on the behavior of water drops dispersed in oil within a centrifugal pump impeller. Experimental Thermal and Fluid Science, 112(August 2019) :109969, 2020. Hongchang Ding, Zikang Li, Xiaobin Gong, and Maoshun Li. The influence of blade outlet angle on the performance of centrifugal pump with high specific speed. Vacuum, 159(August 2018) :239– 246, 2019. Wen-Guang Li. Effects of flow rate and viscosity on slip factor of centrifugal pump handling viscous oils. International Journal of Rotating Machinery, 2013, 2013. Ehsan Abdolahnejad, Mahdi Moghimi, and Shahram Derakhshan. Experimental and numerical investigation of slip factor reduction in centrifugal slurry pump. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 43(4) :1–14, 2021. Juan Pablo Valdés, Miguel Asuaje, and Nicolás Ratkovich. Study of an ESP’s performance hand-ling liquid-liquid flow and unstable O-W emulsions Part I : Experimental. Chemical Engineering Science, 223, 2020. Lila Achour, Mathieu Specklin, Idir Belaidi, and Smaine Kouidri. Numerical Assessment of the Hydrodynamic Behavior of a Volute Centrifugal Pump Handling Emulsion. Entropy, 24(2), 2022. Juan Pablo Valdés, Deisy Becerra, Daniel Rozo, Alexandra Cediel, Felipe Torres, Miguel Asuaje, and Nicolás Ratkovich. Comparative analysis of an electrical submersible pump’s performance handling viscous newtonian and non-newtonian fluids through experimental and cfd approaches. Journal of Petroleum Science and Engineering, 187 :106749, 2020. 25ème Congrès Français de Mécanique Nantes, 29 août au 2 septembre 2022 FJ Wiesner. A review of slip factors for centrifugal impellers. 1967. Young Do Choi, Junichi Kurokawa, and Jun Malsui. Performance and internal flow characteristics of a very low specific speed centrifugal pump. Journal of Fluids Engineering, Transactions of the ASME, 128(2) :341–349, 2006. P Thanapandi and R Prasad. Performance prediction and loss analysis of low specific speed sub-mersible pumps. Proceedings of the Institution of Mechanical Engineers, Part A : Journal of Power and Energy, 204(4) :243–252, 1990.
188977	https://www.tutorialspoint.com/laplace-transform-of-damped-hyperbolic-sine-and-cosine-functions	Data Structure Networking RDBMS Operating System Java MS Excel iOS HTML CSS Android Python C Programming C++ C# MongoDB MySQL Javascript PHP Selected Reading UPSC IAS Exams Notes Developer's Best Practices Questions and Answers Effective Resume Writing HR Interview Questions Computer Glossary Who is Who Laplace Transform of Damped Hyperbolic Sine and Cosine Functions Signals and SystemsElectronics & ElectricalDigital Electronics Laplace Transform The Laplace transform is a mathematical tool which is used to convert the differential equation in time domain into the algebraic equations in the frequency domain or s-domain. Mathematically, if $x\mathrm{\left ( \mathit{t}\right)}$ is a time domain function, then its Laplace transform is defined as − $$\mathrm{\mathit{L\mathrm{\left[\mathit{x\mathrm{\left(\mathit{t} \right )}}\right ]}}\mathrm{=}\mathit{X\mathrm{\left(\mathit{s} \right )}}\mathrm{=}\int_{-\infty }^{\infty}\mathit{x\mathrm{\left(\mathit{t} \right )}e^{-st}}\:\mathit{dt}\:\:\:\:\:\:...(1)}$$ Equation (1) gives the bilateral Laplace transform of the function $x\mathrm{\left ( \mathit{t}\right)}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as, $$\mathrm{\mathit{L\mathrm{\left[\mathit{x\mathrm{\left(\mathit{t} \right )}}\right ]}}\mathrm{=}\mathit{X\mathrm{\left(\mathit{s} \right )}}\mathrm{=}\int_{\mathrm{0} }^{\infty}\mathit{x\mathrm{\left(\mathit{t} \right )}e^{-st}}\:\mathit{dt}\:\:\:\:\:\:...(2)}$$ Laplace Transform of Damped Hyperbolic Sine Function The damped hyperbolic sine function is given by, $$\mathrm{\mathit{x\mathrm{\left(\mathit{t}\right)}}\:\mathrm{=}\:e^{-at}\:\mathrm{sinh\:\mathit{\omega \mathit{t} \:\mathit{u}\mathrm{\left ( \mathit{t} \right )}}}\:\mathrm{=}\:\mathit{e^{-at}}\mathrm{\left ( \frac{\mathit{e^{\omega t}-e^{-\omega t}}}{\mathrm{2}} \right )}\:\mathit{u}\mathrm{\left ( {\mathit{t}}\right)}}$$ Hence, by the definition of the Laplace transform, we have, $$\mathrm{\mathit{L\mathrm{\left[\mathit{{e}^{-\mathit{at}}} \mathrm{sinh\:\mathit{\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right )}\right]}}\:\mathrm{=}\mathit{L}\mathrm{\left [ \:\mathit{e^{-at}}\mathrm{\left ( \frac{\mathit{e^{\omega t}-e^{-\omega t}}}{\mathrm{2}} \right )}\:\mathit{u}\mathrm{\left ( {\mathit{t}}\right)} \right ]}}$$ $$\mathrm{\Rightarrow \mathit{L\mathrm{\left[\mathit{e^{-at}} \mathrm{sinh\:\mathit{\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right )}\right]}}\:\mathrm{=}\:\frac{1}{2}\mathit{L}\mathrm{\left [\mathit{e^{-at}e^{\omega t}\:u\mathrm{\left ( \mathit{t} \right )}-e^{-at}e^{-\omega t}\:u\mathrm{\left ( \mathit{t} \right )}} \right ]} }$$ $$\mathrm{\Rightarrow \mathit{L\mathrm{\left[\mathit{e^{-at}} \mathrm{sinh\:\mathit{\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right )}\right]}}\:\mathrm{=}\:\frac{1}{2}\mathrm{\left{\mathit{L}\mathrm{\left [\mathit{ e^{-\mathrm{\left ( a-\omega \right)}t}\:u\mathrm{\left ( \mathit{t} \right )}}\right ]}-\mathit{L}\mathrm{\left [ \mathit{e^{-\mathrm{\left ( a\mathrm{+}\omega \right )}\mathit{t}}\:u \mathrm{\left ( \mathit{t} \right )} } \right ]} \right}}}$$ $$\mathrm{\Rightarrow \mathit{L\mathrm{\left[\mathit{e^{-at}} \mathrm{sinh\:\mathit{\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right )}\right]}}\:\mathrm{=}\:\frac{1}{2}\mathrm{\left[\frac{1}{\mathit{s\mathrm{+}\mathrm{\left(\mathit{a-\omega}\right)}}}-\frac{1}{\mathit{s}\mathrm{+}\mathrm{\left ( \mathit{a\mathrm{+}\omega } \right )}} \right ]}}$$ $$\mathrm{\Rightarrow \mathit{L\mathrm{\left[\mathit{e^{-at}} \mathrm{sinh\:\mathit{\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right )}\right]}}\:\mathrm{=}\:\frac{1}{2}\mathrm{\left[\frac{1}{\mathrm{\left( \mathit{s\mathrm{+}a} \right)}-\omega }-\frac{1}{\mathrm{\left ( \mathit{s\mathrm{+}a}\right)}+\omega } \right ]}\mathrm{=}\frac{\mathit{\omega}}{\mathrm{\left ( \mathit{s\mathrm{+}a}\right)^{\mathrm{2}}}-\omega ^{^{2}}}}$$ The region of convergence (ROC) of Laplace transform of the damped hyperbolic sine function is Re(s)> -a, which is shown in Figure-1. Therefore, the Laplace transform of damped hyperbolic sine function along with its ROC is given as follows − $$\mathrm{\mathit{e^{-at}} \mathrm{sinh\:\mathit{\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right)}\overset{LT}{\leftrightarrow}\mathrm{\left [ \frac{\mathit{\omega}}{\mathrm{\left ( \mathit{s\mathrm{+}a}\right)^{\mathrm{2}}}-\omega ^{^{2}}} \right ]} ;\:\mathrm{ROC}\to \mathrm{Re}\mathrm{\left(\mathit{s} \right )}>-\mathit{a}}$$ Laplace Transform of Damped Hyperbolic Cosine Function The damped hyperbolic cosine function is given by, $$\mathrm{\mathit{x\mathrm{\left(\mathit{t}\right)}}\:\mathrm{=}\:e^{-\mathit{at}}\:\mathrm{cosh\:\mathit{\omega \mathit{t} \:\mathit{u}\mathrm{\left(\mathit{t} \right )}}}\:\mathrm{=}\:\mathit{e^{-at}}\mathrm{\left ( \frac{\mathit{e^{\omega t}\mathrm{+}e^{-\omega t}}}{\mathrm{2}} \right )}\:\mathit{u}\mathrm{\left ( {\mathit{t}}\right)}}$$ Hence, by the definition of the Laplace transform, we have, $$\mathrm{\mathit{L\mathrm{\left[\mathit{{e}^{-\mathit{at}}} \mathrm{cosh\:\mathit{\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right )}\right]}}\:\mathrm{=}\mathit{L}\mathrm{\left [ \:\mathit{e^{-at}}\mathrm{\left ( \frac{\mathit{e^{\omega t}\mathrm{+}e^{-\omega t}}}{\mathrm{2}} \right )}\:\mathit{u}\mathrm{\left ( {\mathit{t}}\right)} \right ]}}$$ $$\mathrm{\Rightarrow \mathit{L\mathrm{\left[\mathit{e^{-at}} \mathrm{cosh\:\mathit{\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right )}\right]}}\:\mathrm{=}\:\frac{1}{2}\mathit{L}\mathrm{\left [\mathit{e^{-at}e^{\omega t}\:u\mathrm{\left ( \mathit{t} \right )}\mathrm{+}e^{-at}e^{-\omega t}\:u\mathrm{\left ( \mathit{t} \right )}} \right ]}}$$ $$\mathrm{\Rightarrow \mathit{L\mathrm{\left[\mathit{e^{-at}} \mathrm{cosh\:\mathit{\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right )}\right]}}\:\mathrm{=}\:\frac{1}{2}\mathrm{\left{\mathit{L}\mathrm{\left [\mathit{ e^{-\mathrm{\left ( a-\omega \right)}t}\:u\mathrm{\left ( \mathit{t} \right )}}\right ]}+\mathit{L}\mathrm{\left [ \mathit{e^{-\mathrm{\left ( a+\omega \right )}\mathit{t}}\:u \mathrm{\left ( \mathit{t} \right )} } \right ]} \right}}}$$ $$\mathrm{\Rightarrow \mathit{L\mathrm{\left[\mathit{e^{-at}} \mathrm{cosh\:\mathit{\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right )}\right]}}\:\mathrm{=}\:\frac{1}{2}\mathrm{\left[\frac{1}{\mathit{s\mathrm{+}\mathrm{\left(\mathit{a-\omega}\right)}}}+\frac{1}{\mathit{s}+\mathrm{\left ( \mathit{a+\omega } \right )}} \right ]}}$$ $$\mathrm{\Rightarrow \mathit{L\mathrm{\left[\mathit{e^{-at}} \mathrm{cosh\:\mathit{\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right )}\right]}}\:\mathrm{=}\:\frac{1}{2}\mathrm{\left[\frac{1}{\mathrm{\left( \mathit{s\mathrm{+}a} \right)}-\omega }+\frac{1}{\mathrm{\left ( \mathit{s\mathrm{+}a}\right)}+\omega } \right ]}\mathrm{=}\frac{\mathit{s\mathrm{+}a}}{\mathrm{\left ( \mathit{s\mathrm{+}a}\right)^{\mathrm{2}}}-\omega ^{^{2}}}}$$ The ROC of Laplace transform of the damped hyperbolic cosine function is also Re(s)> -a as shown in Figure-1. Therefore, the Laplace transform of damped hyperbolic cosine function along with its ROC is given by, $$\mathrm{\mathit{e^{-at}} \mathrm{cos\:\mathit{\omega t}}\:\mathit{u}\mathrm{\left(\mathit{t}\right)}\overset{LT}{\leftrightarrow}\mathrm{\left [ \frac{\mathit{s\mathrm{+}a}}{\mathrm{\left ( \mathit{s\mathrm{+}a}\right)^{\mathrm{2}}}-\omega ^{{2}}} \right ]} ;\:\mathrm{ROC}\to \mathrm{Re}\mathrm{\left(\mathit{s} \right )}>-\mathit{a}}$$ Manish Kumar Saini Updated on: 2022-01-03T09:41:39+05:30 1K+ Views Related Articles Laplace Transform of Damped Sine and Cosine Functions Laplace Transform of Sine and Cosine Functions Fourier Transform of the Sine and Cosine Functions Laplace Transform of Periodic Functions (Time Periodicity Property of Laplace Transform) Signals and Systems – Z-Transform of Sine and Cosine Signals Laplace Transform of Real Exponential and Complex Exponential Functions Difference between Z-Transform and Laplace Transform Relation between Laplace Transform and Fourier Transform Difference between Laplace Transform and Fourier Transform Signals and Systems – Properties of Laplace Transform Laplace Transform of Ramp Function and Parabolic Function Signals and Systems – Linearity Property of Laplace Transform Time Convolution and Multiplication Properties of Laplace Transform Signals and Systems – Relation between Laplace Transform and Z-Transform Common Laplace Transform Pairs Kickstart Your Career Get certified by completing the course Get Started Advertisements
188978	https://askfilo.com/user-question-answers-chemistry/which-of-the-following-statement-is-are-correct-about-i2cl6-35303038343939	Which of the following statement is/are correct about I2Cl6 : (A) Iodine.. World's only instant tutoring platform Instant TutoringPrivate Courses Tutors Explore TutorsBecome Tutor Login StudentTutor CBSE Chemistry Which of the following statement is/are correct about I2Cl6 : Question Question asked by Filo student Which of the following statement is/are correct about I2Cl6 : (A) Iodine atoms are sp3 d 2 hybridised. (B) It has a non planar geometry Views: 5,466 students Updated on: May 4, 2023 Not the question you're searching for? Ask your question Ask your question Or Upload the image of your question Get Solution Video solutions (1) Learn from their 1-to-1 discussion with Filo tutors. Video Player is loading. 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188979	https://blog.prepscholar.com/convert-decimal-to-fraction	CALL NOW: : +1 (866) 811-5546 Choose Your Test SAT Prep ACT Prep PrepScholar Advice Blog Search Blogs By Category SAT ACT College Admissions AP and IB Exams PSAT TOEFL GPA and Coursework The 3 Steps to Convert Decimals to Fractions (and Back) Posted by Christine Sarikas General Education Wondering how to convert decimals to fractions? Or how to convert fractions to decimals? It’s easier than you think! Keep reading to see the steps for decimal to fraction conversions (including why you need to follow different steps if you have a repeating decimal), steps for fraction to decimal conversions, a handy chart with common decimal/fraction conversions, and tips for quickly estimating conversions. How to Convert Decimals to Fractions How do you convert a decimal to a fraction? Any decimal, even complicated-looking ones, can be converted to a fraction; you just need to follow a few steps. Below we explain how to convert both terminating decimals and repeating decimals to fractions. Converting a Terminating Decimal to a Fraction A terminating decimal is any decimal that has a finite other of digits. In other words, it has an end. Examples include .5, .234, .864721, etc. Terminating decimals are the most common decimals you’ll see and, fortunately, they are also the easiest to convert to fractions. Step 1 Write the decimal divided by one. For example, say you’re given the decimal .55. Your first step is to write out the decimal so it looks like ${.55}/{1}$. Step 2 Next, you want to multiply both the top and bottom of your new fraction by 10 for every digit to the left of the decimal point. In our example, .55 has two digits after the decimal point, so we’ll want to multiply the entire fraction by 10 x 10, or 100. Multiplying the fraction by ${100}/{100}$ gives us ${55}/{100}$. Step 3 The final step is reducing the fraction to its simplest form. The simplest form of the fraction is when the top and bottom of the fraction are the smallest whole numbers they can be. For example, the fraction ${3}/{9}$ isn’t in its simplest form because it can still be reduced down to ⅓ by dividing both the top and bottom of the fraction by 3. The fraction ${55}/{100}$ can be reduced by dividing both the top and bottom of the fraction by 5, giving us ${11}/{20}$. 11 is a prime number and can’t be divided any more, so we know this is the fraction in its simplest form. The decimal .55 is equal to the fraction ${11}/{20}$. Example Convert .108 to a fraction. After putting the decimal over 1, we end up with ${.108}/{1}$. Since .108 has three digits after the decimal place, we need to multiply the entire fraction by 10 x 10 x 10, or 1000. This gives us ${108}/{1000}$. Now we need to simplify. Since 108 and 1000 are both even numbers, we know we can divide both by 2. This gives us ${54}/{500}$. These are still even numbers, so we can divide by 2 again to get ${27}/{250}$. 27 isn't a factor of 250, so the fraction can’t be reduced any more. The final answer is ${27}/{250}$. Converting a Repeating Decimal to a Fraction A repeating decimal is one that has no end. Since you can’t keep writing or typing the decimal out forever, they are often written as a string of digits rounded off (.666666667) or with a bar above the repeating digit(s) $\ov {(.6)}$. For our example, we’ll convert .6667 to a fraction. The decimal .6667 is equal to $\ov {(.6)}$, .666666667, .667, etc. They’re all just different ways to show that the decimal is actually a string of 6’s that goes on forever. Step 1 Let x equal the repeating decimal you’re trying to convert, and identify the repeating digit(s). So x=.6667 6 is the repeating digit, and the end of the decimal has been rounded up. Step 2 Multiply by whatever value of 10 you need to get the repeating digit(s) on the left side of the decimal. For .6667, we know that 6 is the repeating digit. We want that six on the left side of the decimal, which means moving the decimal place over one spot. So we multiply both sides of the equation by (10 x 1) or 10. 10x = 6.667 Note: You only want one “set” of repeating digit(s) on the left side of the decimal. In this example, with 6 as the repeating digit, you only want one 6 on the left of the decimal. If the decimal was 0.58585858, you’d only want one set of “58” on the left side. If it helps, you can picture all repeating decimals with the infinity bar over them, so .6667 would be $\ov {(.6)}$. Step 3 Next we want to get an equation where the repeating digit is just to the right of the decimal. Looking at x = .6667, we can see that the repeating digit (6) is already just to the right of the decimal, so we don’t need to do any multiplication. We’ll keep this equation as x = .6667 Step 4 Now we need to solve for x using our two equations, x = .667 and 10x = 6.667. 10x - x =6.667-.667 9x = 6 x = ${6}/{9}$ x = ⅔ Example Convert 1.0363636 to a fraction. This question is a bit trickier, but we’ll be doing the same steps that we did above. First, make the decimal equal to x, and determine the repeating digit(s). x = 1.0363636 and the repeating digits are 3 and 6 Next, get the repeating digits on the left side of the decimal (again, you only want one set of repeating digits on the left). This involves moving the decimal three places to the right, so both sides need to be multiplied by (10 x 3) or 1000. 1000x = 1036.363636 Now get the repeating digits to the right of the decimal. Looking at the equation x = 1.0363636, you can see that there currently is a zero between the decimal and the repeating digits. The decimal needs to be moved over one space, so both sides need to be multiplied by 10 x 1. 10x = 10.363636 Now use the two equations, 1000x = 1036.363636 and 10x = 10.363636, to solve for x. 1000x - 10x = 1036.363636 - 10.363636 990x = 1026 x = ${1026}/{990}$ Since the numerator is larger than the denominator, this is known as an irregular fraction. Sometimes you can leave the fraction as an irregular fraction, or you may be asked to convert it to a regular fraction. You can do this by subtracting 990/990 from the fraction and making it a 1 that’ll go next to the fraction. ${1026}/{990}$ - ${990}/{990}$ = 1 ${36}/{990}$ x = 1 ${36}/{990}$ ${36}/{990}$ can be simplified by dividing it by 18. x = 1 ${2}/{55}$ How to Convert Fractions to Decimals The easiest way to convert a fraction to a decimal is just to use your calculator. The line between the numerator and denominator acts as a division line, so ${7}/{29}$ equals 7 divided by 29 or .241. If you don’t have access to a calculator though, you can still convert fractions to decimals by using long division or getting the denominator to equal a multiple of 10. We explain both these methods in this section. Long Division Method Convert ${3}/{8}$ to a decimal. Here is what ${3}/{8}$ looks like worked out with long division. ⅜ converted to a decimal is .375 Denominator as a Value of 10 Method Convert ${3}/{8}$ to a decimal. Step 1 We want the denominator, in this case 8, to equal a value of 10. We can do this by multiplying the fraction by 125, giving us ${375}/{1000}$. Step 2 Next we want to get the denominator to equal 1 so we can get rid of the fraction. We’ll do this by dividing each part of the fraction by 1000, which means moving the decimal over three places to the left. This gives us ${.375}/{1}$ or just .375, which is our answer. Note that this method only works for a fraction with a denominator that can easily be multiplied to be a value of 10. However, there is a trick you can use to estimate the value of fractions you can’t convert using this method. Check out the example below. Example Convert ⅔ to a decimal. There is no number you can multiply 3 by to make it an exact multiple of 10, but you can get close. By multiplying ⅔ by ${333}/{333}$, we get ${666}/{999}$. 999 is very close to 1000, so let’s act like it actually is 1000, divide each part of the fraction by 1000, and move the decimal place of 666 three places to the left, giving us .666 The exact decimal conversion of ⅔ is the repeating decimal .6666667, but .666 gets us very close. So whenever you have a fraction whose denominator can’t easily be multiplied to a value of 10 (this will happen to all fractions that convert to repeating decimals), just get the denominator as close to a multiple of 10 as possible for a close estimate. Common Decimal to Fraction Conversions Below is a chart with common decimal to fraction conversions. You don’t need to memorize these, but knowing at least some of them off the top of your head will make it easy to do some common conversions. If you’re trying to convert a decimal or fraction and don’t have a calculator, you can also see which value in this chart the number is closest to so you can make an educated estimate of the conversion. \| \| \| --- \| \| Decimal \| Fraction \| \| 0.03125 \| ${1}/{32}$ \| \| 0.0625 \| ${1}/{16}$ \| \| 0.1 \| ${1}/{10}$ \| \| 0.1111 \| ${1}/{9}$ \| \| 0.125 \| ${1}/{8}$ \| \| 0.16667 \| ${1}/{6}$ \| \| 0.2 \| ${1}/{5}$ \| \| 0.2222 \| ${2}/{9}$ \| \| 0.25 \| ${1}/{4}$ \| \| 0.3 \| ${3}/{10}$ \| \| 0.3333 \| ${1}/{3}$ \| \| 0.375 \| ${3}/{8}$ \| \| 0.4 \| ${2}/{5}$ \| \| 0.4444 \| ${4}/{9}$ \| \| 0.5 \| ${1}/{2}$ \| \| 0.5555 \| ${5}/{9}$ \| \| 0.6 \| ${3}/{5}$ \| \| 0.625 \| ${5}/{8}$ \| \| 0.6666 \| ${2}/{3}$ \| \| 0.7 \| ${7}/{10}$ \| \| 0.75 \| ${3}/{4}$ \| \| 0.7777 \| ${7}/{9}$ \| \| 0.8 \| ${4}/{5}$ \| \| 0.8333 \| ${5}/{6}$ \| \| 0.875 \| ${7}/{8}$ \| \| 0.8888 \| ${8}/{9}$ \| \| 0.9 \| ${9}/{10}$ \| Summary: How to Make a Decimal Into a Fraction If you’re trying to convert a decimal to fraction, first you need to determine if it’s a terminal decimal (one with an end) or a repeating decimal (one with a digit or digit that repeats to infinity). Once you’ve done that, you can follow a few steps for the decimal to fraction conversion and for writing decimals as fractions. If you’re trying to convert a fraction to decimal, the easiest way is just to use your calculator. If you don’t have one handy, you can use long division or get the denominator equal to a multiple of ten, then move the decimal place of the numerator over. For quick estimates of decimal to fraction conversions (or vice versa), you can look at our chart of common conversions and see which is closest to your figure to get a ballpark idea of its conversion value. What's Next? Want to know the fastest and easiest ways to convert between Fahrenheit and Celsius? We've got you covered! Check out our guide to the best ways to convert Celsius to Fahrenheit (or vice versa). Are you learning about logarithms and natural logs in math class?We have a guide on all the natural log rules you need to know. Did you know that water has a very special density? Check out our guide to learn what the density of water is and how the density can change. Trending Now How to Get Into Harvard and the Ivy League How to Get a Perfect 4.0 GPA How to Write an Amazing College Essay What Exactly Are Colleges Looking For? ACT vs. SAT: Which Test Should You Take? When should you take the SAT or ACT? Get Your Free eBook 5 Tips to Raise Your SAT Score 160+ Points 5 Tips to Raise Your ACT Score 4+ Points SAT Prep Find Your Target SAT Score Free Complete Official SAT Practice Tests How to Get a Perfect SAT Score, by an Expert Full Scorer Score 800 on SAT Math Score 800 on SAT Reading and Writing How to Improve Your Low SAT Score Score 600 on SAT Math Score 600 on SAT Reading and Writing Raise Your SAT Score by 160+ Points GUARANTEED. 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188980	https://math.stackexchange.com/questions/3013444/how-to-get-from-xp-1-1-to-x-1xp-2xp-3-cdotsx1	algebra precalculus - How to get from $x^{p-1}-1$ to $(x-1)(x^{p-2}+x^{p-3}+\cdots+x+1)$? - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more How to get from x p−1−1 x p−1−1 to (x−1)(x p−2+x p−3+⋯+x+1)(x−1)(x p−2+x p−3+⋯+x+1)? Ask Question Asked 6 years, 10 months ago Modified6 years, 10 months ago Viewed 186 times This question shows research effort; it is useful and clear 3 Save this question. Show activity on this post. How would I get from x p−1−1 x p−1−1 to (x−1)(x p−2+x p−3+⋯+x+1)(x−1)(x p−2+x p−3+⋯+x+1)? It make sense to me logically. When one multiplies it out, it would condense to x p−1−1 x p−1−1. But it's just not clicking. What is the arithmetic between these steps? algebra-precalculus polynomials arithmetic Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications edited Nov 27, 2018 at 10:52 Martin Sleziak 56.3k 20 20 gold badges 211 211 silver badges 391 391 bronze badges asked Nov 25, 2018 at 21:39 kaisakaisa 363 4 4 silver badges 16 16 bronze badges 3 Do you want the product multiplied out formally or are you looking for some other reason? Are you looking for a proof in general?Michael Burr –Michael Burr 2018-11-25 21:40:20 +00:00 Commented Nov 25, 2018 at 21:40 Ruffini may helps you Tito Eliatron –Tito Eliatron 2018-11-25 21:42:34 +00:00 Commented Nov 25, 2018 at 21:42 1 Maybe it's the polynomial division algorithm you're looking for.Berci –Berci 2018-11-25 21:43:15 +00:00 Commented Nov 25, 2018 at 21:43 Add a comment\| 5 Answers 5 Sorted by: Reset to default This answer is useful 3 Save this answer. Show activity on this post. Just do it (x−1)(x p−2+x p−3+⋯+x+1)=(x−1)(x p−2+x p−3+⋯+x+1)= =x⋅(x p−2+x p−3+⋯+x+1)−1⋅(x p−2+x p−3+⋯+x+1)==x⋅(x p−2+x p−3+⋯+x+1)−1⋅(x p−2+x p−3+⋯+x+1)= =(x p−1+x p−2+⋯+x 2+x)−(x p−2+x p−3+⋯+x+1)==(x p−1+x p−2+⋯+x 2+x)−(x p−2+x p−3+⋯+x+1)= =x p−1+x p−2−x p−2+…+x−x−1==x p−1+x p−2−x p−2+…+x−x−1= =x p−1−1=x p−1−1 Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Nov 25, 2018 at 21:53 useruser 164k 14 14 gold badges 84 84 silver badges 157 157 bronze badges Add a comment\| This answer is useful 3 Save this answer. Show activity on this post. It comes from the high-school formula used for the sum of a geometric series. It relies on this factorisation identity, often used as a model for proofs by induction: a n−b n=(a−b)(a n−1+a n−2 b+⋯+a b n−2+b n−1)a n−b n=(a−b)(a n−1+a n−2 b+⋯+a b n−2+b n−1) The inductive step is as follows: a n+1−b n+1=(a n+1−a n b)+(a n b−b n+1)=a n(a−b)+b(a n−b n)=a n(a−b)+b(a−b)(a n−1+a n−2 b+⋯+a b n−2+b n−1)=(a−b)(a n+b(a n−1+a n−2 b+⋯+a b n−2+b n−1))=(a−b)(a n+a n−1 b+a n−2 b 2+⋯+a b n−1+b n).a n+1−b n+1=(a n+1−a n b)+(a n b−b n+1)=a n(a−b)+b(a n−b n)=a n(a−b)+b(a−b)(a n−1+a n−2 b+⋯+a b n−2+b n−1)=(a−b)(a n+b(a n−1+a n−2 b+⋯+a b n−2+b n−1))=(a−b)(a n+a n−1 b+a n−2 b 2+⋯+a b n−1+b n). For the case at hand, replace a a and b b with x x and 1 1 respectively. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Nov 25, 2018 at 22:29 BernardBernard 180k 10 10 gold badges 75 75 silver badges 182 182 bronze badges 1 Do you agree with that deletion?user –user 2018-11-26 11:31:25 +00:00 Commented Nov 26, 2018 at 11:31 Add a comment\| This answer is useful 2 Save this answer. Show activity on this post. Observe that x p−2+x p−3+⋯+x+1=∑p−2 i=0 x i x p−2+x p−3+⋯+x+1=∑i=0 p−2 x i. Now, observe that (x−1)(x p−2+x p−3+⋯+x+1)=(x−1)∑i=0 p−2 x i=∑i=0 p−2 x i+1−∑i=0 p−2 x i.(x−1)(x p−2+x p−3+⋯+x+1)=(x−1)∑i=0 p−2 x i=∑i=0 p−2 x i+1−∑i=0 p−2 x i. Reindexing the first sum (j=i+1 j=i+1), we get that this equals ∑j=1 p−1 x j−∑i=0 p−2 x i.∑j=1 p−1 x j−∑i=0 p−2 x i. Now, if we peel off the last term of the first sum and the first term of the second sum, we get (x p−1+∑j=1 p−2 x j)−(∑j=1 p−2 x j+x 0).(x p−1+∑j=1 p−2 x j)−(∑j=1 p−2 x j+x 0). Since the sums cancel, we are left with x p−1−x 0=x p−1−1 x p−1−x 0=x p−1−1. If you want to derive the formula, Berci's comment above about using the polynomial division algorithm should work well. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Nov 25, 2018 at 21:52 answered Nov 25, 2018 at 21:44 Michael BurrMichael Burr 34k 2 2 gold badges 52 52 silver badges 81 81 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. First of all, we might as well set n=p−1,(1)(1)n=p−1, and then investigate the equation x n−1=?(x−1)(x n−1+x n−2+…+x+1)=(x−1)∑0 n−1 x i.(2)(2)x n−1=?(x−1)(x n−1+x n−2+…+x+1)=(x−1)∑0 n−1 x i. One may in fact use a simple induction to validate (2); we check n=1:x 1−1=x 1−1=(x−1)∑0 0 x i,(3)(3)n=1:x 1−1=x 1−1=(x−1)∑0 0 x i, n=2:x 2−1=(x−1)(x+1)=(x−1)∑0 1 x i,(4)(4)n=2:x 2−1=(x−1)(x+1)=(x−1)∑0 1 x i, n=3:x 3−1=(x−1)(x 2+x+1)=(x−1)∑0 2 x i;(5)(5)n=3:x 3−1=(x−1)(x 2+x+1)=(x−1)∑0 2 x i; now if ∃k∈N,x k−1=(x−1)∑0 k−1 x i,(6)(6)∃k∈N,x k−1=(x−1)∑0 k−1 x i, we have x k+1−1=x k+1−x k+x k−1=(x−1)x k+x k−1,(7)(7)x k+1−1=x k+1−x k+x k−1=(x−1)x k+x k−1, and then using (6), x k+1−1=(x−1)x k+(x−1)∑0 k−1 x i,(8)(8)x k+1−1=(x−1)x k+(x−1)∑0 k−1 x i, whence x k+1−1=(x−1)(x k+∑0 k−1 x i)=(x−1)∑0 k x i,(9)(9)x k+1−1=(x−1)(x k+∑0 k−1 x i)=(x−1)∑0 k x i, which shows inductively that (2) must bind. As far as performing the actual aritmetical/algebraic operations required above is concerned, it is easy to write them out explicitly for equations (3)-(5), viz (x−1)(x+1)=x(x+1)−1(x+1)=x 2+x−x−1=x 2−1;(10)(10)(x−1)(x+1)=x(x+1)−1(x+1)=x 2+x−x−1=x 2−1; it is apparent that the distributive law is used critically here, allowing us as it does to write both the first equality in (10) as well as x(x+1)=x 2+x etc;(11)(11)x(x+1)=x 2+x etc; in fact, the distributive law is tacitly invoked in (7)-(9); indeed, it plays a central role in performing the arithmetic necessary to establish (2). Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Nov 27, 2018 at 10:09 Robert LewisRobert Lewis 73k 5 5 gold badges 66 66 silver badges 122 122 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. I'm not sure if this fully answers your question, however, when I am factoring higher degree polynomials (degree greater than 2) I often like to use the following trick that I will demonstrate below: On polynomials like the one you have there, it is easy to see that x=1 x=1 is a zero of the polynomial x p−1−1 x p−1−1. Then you want to use the following trick where you kind of 'reverse' expand the polynomial: x p−1−1=x p−1−x p−2+x p−2−x p−3+x p−3−...−x+x−1=(x p−1+x p−2+x p−3+...+x)−(x p−2+x p−3+x p−4+...+1)=x(x p−2+x p−3+x p−4+...+1)−(x p−2+x p−3+x p−4+...+1)=(x−1)(x p−2+x p−3+x p−4+...+1)x p−1−1=x p−1−x p−2+x p−2−x p−3+x p−3−...−x+x−1=(x p−1+x p−2+x p−3+...+x)−(x p−2+x p−3+x p−4+...+1)=x(x p−2+x p−3+x p−4+...+1)−(x p−2+x p−3+x p−4+...+1)=(x−1)(x p−2+x p−3+x p−4+...+1) So you basically add and subtract terms without changing the polynomial so it is easy to pull out the (x−1)(x−1) factor. Just as another example to better illustrate the technique on a polynomial you don't know the factors of: Lets say we wanted to factor the polynomial x 3−9 x 2+26 x−24 x 3−9 x 2+26 x−24. With some quick trial and error you can find that x=2 x=2 is a zero of the polynomial. Then, as before, you want to rearrange the polynomial so that it easy to pull out the (x−2)(x−2) factor. We do it as follows: x 3−9 x 2+26 x−24=x 3−2 x 2−7 x 2+14 x+12 x−24=(x 3−7 x 2+12 x)−(2 x 2−14 x+24)=x(x 2−7 x+12)−2(x 2−7 x+12)=(x−2)(x 2−7 x+12)x 3−9 x 2+26 x−24=x 3−2 x 2−7 x 2+14 x+12 x−24=(x 3−7 x 2+12 x)−(2 x 2−14 x+24)=x(x 2−7 x+12)−2(x 2−7 x+12)=(x−2)(x 2−7 x+12) So all we have done is leave the highest/lowest degree terms alone and split the middle terms into two parts, one a multiple of x x and the other a multiple of -2. It is then easy to rearrange the polynomial and factor out the (x−2)(x−2) factor. You can apply this technique to any polynomial that you already know a zero of, however, as you can imagine it gets messy if the zeroes of the polynomial aren't 'small' integers. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Nov 29, 2018 at 23:20 Pseudo ProfessorPseudo Professor 573 3 3 silver badges 13 13 bronze badges Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions algebra-precalculus polynomials arithmetic See similar questions with these tags. 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188981	https://www.nature.com/research-intelligence/nri-topic-summaries/atp-sensitive-potassium-channels-in-pancreatic-beta-cells-micro-254512	ATP-Sensitive Potassium Channels in Pancreatic Beta-Cells \| Nature Research Intelligence Speak with us Overview Nature Index About Institution tables Institution benchmarking How to use the Nature Index Nature Strategy Reports About Green Energy Food Security & Agriculture The Impact of AI Research Online store Nature Navigator About Public Library Login Data sources About Nature Research Intelligence Topics Topic Summaries Resources Contact Overview Nature Index About Institution tables Institution benchmarking How to use the Nature Index Nature Strategy Reports About Green Energy Food Security & Agriculture The Impact of AI Research Online store Nature Navigator About Public Library Login Data sources About Nature Research Intelligence Topics Topic Summaries Resources Contact ATP-Sensitive Potassium Channels in Pancreatic Beta-Cells Nature Research Intelligence Topics Topic summaries Biomedical and Clinical Sciences Clinical Sciences Endocrinology ATP-Sensitive Potassium Channels in Pancreatic Beta-Cells About these AI generated summaries ATP‐sensitive potassium (KATP) channels are vital metabolic sensors that couple intracellular adenine nucleotide levels to membrane electrical activity, thereby regulating insulin secretion in pancreatic beta‐cells. Composed chiefly of four pore‐forming Kir6.2 subunits and four modulatory sulfonylurea receptor 1 (SUR1) subunits, these channels respond to fluctuations in ATP and ADP by shifting between closed and open conformations. This dynamic gating mechanism enables beta‐cells to adjust their excitability and insulin release in response to metabolic cues. Recent structural and functional studies have provided a more detailed understanding of the ligand‐binding sites, subunit interactions and conformational changes underlying channel regulation, further illuminating the pathophysiology of metabolic disorders such as neonatal diabetes and type 2 diabetes. Research from Nature Portfolio Recent studies have provided compelling insights into the molecular architecture of pancreatic KATP channels. One study has resolved the open‐state structure of the channel, showing twin binding sites for phosphatidylinositol 4,5‐bisphosphate (PIP2) at the interface between SUR1 and Kir6.2, a finding that clarifies the cooperative regulation by both ATP and PIP2 in channel gating . In another investigation, researchers reported cryo‐EM structures that capture the channel in both closed and pre‐open states, detailing how nucleotide binding at distinct inhibitory sites influences channel conformation and, by extension, insulin secretion dynamics . These breakthroughs not only enhance our understanding of the fundamental processes governing beta‐cell function but also inform the development of novel pharmacological modulators. Research from all publishers Recent investigations have elucidated key aspects of KATP channel regulation employing sophisticated cryo‐electron microscopy and molecular dynamics techniques. For example, a study has reviewed the progress towards a structural framework that explains how ligand binding modulates channel gating, thereby advancing our grasp on how intracellular ATP and ADP ratios steer the channel’s activity . Additionally, high‐resolution structures have revealed the intricate binding modes of therapeutic agents like repaglinide, which inhibit channel activity to potentiate insulin secretion. These findings underscore the clinical relevance of KATP channels as targets for antidiabetic drugs . Such advances delineate how subtle alterations in channel conformation can have pronounced effects on cellular metabolism, opening avenues for precision medicine in diabetes management. Technical Terms KATP channel: A type of potassium channel regulated by intracellular ATP/ADP levels, linking cellular metabolism to electrical excitability. Kir6.2: The pore‐forming subunit of KATP channels predominantly found in pancreatic beta‐cells, critical for mediating potassium flux. SUR1: The sulfonylurea receptor subunit that modulates the activity of KATP channels in response to changes in intracellular signals. ATP-Sensitive Potassium Channels in Pancreatic Beta-Cells Publication Trend The graph below shows the total number of publications each year in ATP-Sensitive Potassium Channels in Pancreatic Beta-Cells. References KATP channels in focus: Progress toward a structural understanding of ligand regulation. Current Opinion in Structural Biology (2023). The Structural Basis for the Binding of Repaglinide to the Pancreatic KATP Channel. Cell Reports (2019). Structure of an open KATP channel reveals tandem PIP2 binding sites mediating the Kir6.2 and SUR1 regulatory interface. Nature Communications (2024). Structural insights into the mechanism of pancreatic KATP channel regulation by nucleotides. Nature Communications (2022). Back to "Endocrinology" About these summaries Nature Research Intelligence Topic Summaries [beta] This Nature Research Intelligence Topic summary is created with the cited references and a large language model, based on research articles grouped into topics as they use similar references and words. We take care to ground generated text with facts and have systems in place to gain human feedback on the overall quality of the process in line with our AI principles. We strive to create accurate and useful summaries for people unfamiliar with the research topic and welcome feedback that supports this goal. These pages are a beta release and will be updated as we learn how best to help people gain value from a research topic summary. For more information on the methodology and for a complete list of research topics please refer to this pre-print. 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188982	https://books.google.com/books/about/Boundary_layer_Theory.html?id=fYdTAAAAMAAJ	Boundary-layer Theory - Hermann Schlichting - Google Books Sign in Hidden fields Try the new Google Books Books Add to my library Try the new Google Books Check out the new look and enjoy easier access to your favorite features Try it now No thanks Try the new Google Books My library Help Advanced Book Search Get print book No eBook available Amazon.com Barnes&Noble.com Books-A-Million IndieBound Find in a library All sellers» ### Get Textbooks on Google Play Rent and save from the world's largest eBookstore. Read, highlight, and take notes, across web, tablet, and phone. Go to Google Play Now » My library My History Boundary-layer Theory Hermann Schlichting McGraw-Hill, 1979 - Science - 817 pages This text is the translation and revision of Schlichting's classic text in boundary layer theory. The main areas covered are laws of motion for a viscous fluid, laminar boundary layers, transition and turbulence, and turbulent boundary layers. More » From inside the book Contents Part A Fundamental laws of motion for a viscous fluid 5 Outline of boundarylayer theory 24 Derivation of the equations of motion of a compressible viscous fluid 47 Copyright 27 other sections not shown Common terms and phrases adverse pressure gradientaerofoilangleapproximateARC RMboundary conditionsboundary-layer equationsboundary-layer thicknesscalculationChapcircular cylindercoefficientcompressibleconstantcritical Reynolds numbercurvedenotesdifferential equationdimensionlessdisplacement thicknessdistancedisturbancesdragdrag coefficienteffectexternal flowflat platefluidfree-streamfunctionheat transferincompressible flowincreaseintegralinvestigationlaminar boundary layerlaminar flowMach numbermeasurementsMechmethodmotionNACANavier-Stokes equationsobtainoscillationspipeplate at zeropoint of separationpoint of transitionpotential flowPrandtl numberpressure distributionpressure gradientproblemProcrotatingroughnessSchlichtingshape factorshearing stressskin frictionstabilitystagnation pointstreamstream functionstreamlinessuctionsupersonicsurfacetemperature distributionthree-dimensionalturbulent boundary layerturbulent flowtwo-dimensionalu₁velocity componentsvelocity distributionvelocity profilesviscosityvorticesZAMMzero incidenceθυдидх ду References to this book Diffusion: Mass Transfer in Fluid Systems E. L. Cussler Limited preview - 1997 The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts ... Rutherford Aris Snippet view - 1975 All Book Search results » Bibliographic information Title Boundary-layer Theory McGraw-Hill classic textbook reissue series McGraw-Hill series in mechanical engineering Mechanical Engineering Series AuthorHermann Schlichting Edition 7, illustrated, reprint Publisher McGraw-Hill, 1979 Original from the University of Michigan Digitized Dec 13, 2007 ISBN 0070553343, 9780070553347 Length 817 pages SubjectsScience › Mechanics › General Science / Mechanics / General Technology & Engineering / Aeronautics & Astronautics Technology & Engineering / Hydraulics Technology & Engineering / Mechanical Export CitationBiBTeXEndNoteRefMan About Google Books - Privacy Policy - Terms of Service - Information for Publishers - Report an issue - Help - Google Home
188983	https://testbook.com/maths/cos-60-degrees	Cos 60 Degrees – Definition, Value, Formula, Periodicity & Examples Typesetting math: 100% English Get Started Exams SuperCoaching Live Classes FREE Test Series Previous Year Papers Skill Academy PassPassPass ProPass Elite Pass Pass Pro Pass Elite Rank Predictor IAS Preparation MoreFree Live ClassesFree Live Tests & QuizzesFree QuizzesPrevious Year PapersDoubtsPracticeRefer and EarnAll ExamsOur SelectionsCareersIAS PreparationCurrent Affairs Practice GK & Current Affairs Blog Refer & EarnOur Selections Exams JEE Main JEE Advanced NEET CUET COMEDK UGET UP Polytechnic JEECUP AP POLYCET TNEA TS POLYCET KEAM MHT CET WB JEE OJEE GUJCET ICAR AIEEA CUET PG NID JCECE Karnataka PGCET NEST KCET UPESEAT EXAM LPUNEST PUBDET AMUEEE IISER IAT Bihar Diploma DECE-LE NPAT JIPMER JMI Entrance Exam AAU VET PGDBA Exam AP ECET GCET CEPT PU CET GPAT CEED AIAPGET JKCET HPCET CG PAT SRMJEEE BCECE AGRICET TS PGECET BEEE MP PAT MCAER PG CET VITMEE IIT JAM CMC 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Off OUAT Cut Off VITEEE Cut Off MU OET Cut Off Latest Updates SBI PO Admit Card 2025 DFCCIL Rank Predictor 2025 Eligibility JEE Main Eligibility JEE Advanced Eligibility NEET Eligibility CUET Eligibility COMEDK UGET Eligibility UP Polytechnic JEECUP Eligibility TNEA Eligibility TS POLYCET Eligibility KEAM Eligibility MHT CET Eligibility WB JEE Eligibility OJEE Eligibility ICAR AIEEA Eligibility CUET PG Eligibility NID Eligibility JCECE Eligibility Karnataka PGCET Eligibility NEST Eligibility KCET Eligibility LPUNEST Eligibility PUBDET Eligibility AMUEEE Eligibility IISER IAT Eligibility Bihar Diploma DECE-LE Eligibility NPAT Eligibility JIPMER Eligibility JMI Entrance Exam Eligibility AAU VET Eligibility PGDBA Exam Eligibility AP ECET Eligibility GCET Eligibility CEPT Eligibility PU CET Eligibility GPAT Eligibility CEED Eligibility AIAPGET Eligibility JKCET Eligibility HPCET Eligibility CG PAT Eligibility SRMJEEE Eligibility BCECE Eligibility AGRICET Eligibility TS PGECET Eligibility MP PAT Eligibility MCAER PG CET Eligibility VITMEE Eligibility IIT JAM Eligibility CMC Vellore Eligibility AIMA UGAT Eligibility AIEED Eligibility ACET Eligibility PGIMER Exam Eligibility CENTAC Eligibility NATA Eligibility AFMC Eligibility AIIMS MBBS Eligibility BITSAT Eligibility JEXPO Eligibility HITSEEE Eligibility AP EAPCET Eligibility GITAM GAT Eligibility UPCATET Eligibility UCEED Eligibility CG PET Eligibility OUAT Eligibility IEMJEE Eligibility SEED Eligibility MU OET Eligibility HomeMaths Cos 60 Degrees Cos 60 Degrees – Definition, Value, Formula, Periodicity & Examples Download as PDF Overview Test Series In mathematics, the cosine (cos) function is one of the six basic trigonometric functions. It is used to find the ratio between two sides of a right-angled triangle. For an angle of 60 degrees, cos 60° tells us how long the adjacent side is compared to the hypotenuse (the longest side of the triangle). In simple words, cos 60° = adjacent ÷ hypotenuse. Chemistry Notes Free PDFs \| Topic \| PDF Link \| --- \| \| Class 11 Chemistry Short Notes PDF \| Download PDF \| \| Class 12 Chemistry Complete Notes \| Download PDF \| \| Study Notes forPhysics \| Download PDF \| \| Class 11 Biology Short Notes PDF \| Download PDF \| \| Class 12 Biology Short Notes PDF \| Download PDF \| In a triangle where one of the angles is 60°, the cosine of that angle is a way to measure how close the base is compared to the slanted side. The value of cos 60° is 0.5. In this article, we will look into the cos 60 degrees value, methods and some solved examples. Value of Cos 60 Degrees We mathematically express the exact value of cos 60° in trigonometry. Cos 60 Degrees in Fraction In fractional form, its exact value of cos 60 degree is 1 2 1 2. Cos 60 Degrees in Radians Cos 60 degree in radians is represented as = (60×π 180∘)(60×π 180∘) =π 3 π 3 or 1.0471 UGC NET/SET Course Online by SuperTeachers: Complete Study Material, Live Classes & More Get UGC NET/SET SuperCoaching @ just ₹25999₹7583 Your Total Savings ₹18416 Explore SuperCoaching Want to know more about this Super Coaching ? Download Brochure People also like Prev Next Assistant Professor / Lectureship (UGC) ₹26999 (66% OFF) ₹9332 (Valid for 3 Months) Explore this Supercoaching IB Junior Intelligence Officer ₹6999 (80% OFF) ₹1419 (Vaild for 12 Months) Explore this Supercoaching RBI Grade B ₹21999 (76% OFF) ₹5499 (Valid for 9 Months !) Explore this Supercoaching Methods to Find the Value of Cos 60 Degree The methods listed below can be used to determine the value of cos 60 degrees. Cos 60 Degree Using Unit Circle In the case of a unit circle with a radius of one, it is possible to give the values of trigonometric ratios with respect to radians corresponding to degrees. The symbol for a unit circle’s radian is π π. The angle values in the above figure are shown in degrees rather than radians. To form a 60∘60∘ angle with the positive x-axis, rotate “r” anticlockwise Cos of 60 degrees equals 1 2 1 2 of the x-coordinate. Hence the cos 60∘=1 2 cos⁡60∘=1 2 Cos 60 Degree Using Triangle Let’s look at the equilateral triangle shown below to determine the cosine of the angle 60 degrees: Consider an equilateral triangle ABC with each side of length of 2a. Each angle of △A B C△A B C is of 60 degrees. Let AD be the perpendicular from A on BC. The midpoint of BC is D, while AD is the bisector of ∠A∠A. In △A D B△A D B, ∠D∠D is a right angle. and AB =2a and BD= a According to Pythagoras theorem, A B 2=A D 2+B D 2⇒2 a 2=A D 2+a 2 A B 2=A D 2+B D 2⇒2 a 2=A D 2+a 2 ⇒A D 2=4 a 2−a 2=3 a 2⇒A D=a√3⇒A D 2=4 a 2−a 2=3 a 2⇒A D=a√3 Now, in △A D B△A D B, ∠B=60∘∠B=60∘ By using trigonometric formulas, cos 60∘=b a s e h y p o t e n u s e=b h cos⁡60∘=b a s e h y p o t e n u s e=b h cos 60∘cos⁡60∘ = side adjacent to 60 degrees/hypotenuse B D A B=a 2 a=1 2 B D A B=a 2 a=1 2 Hence the value of cos 60∘=1 2 cos⁡60∘=1 2 Learn about Cos a Cos b Cos 60 Degree Using Trigonometric Functions We can prove the value of cos 60∘cos⁡60∘ with a trigonometric function. As we already know sin 60∘=3√2 sin⁡60∘=3 2 By using Trigonometric identities, sin 2 x+cos 2 x=1 sin 2⁡x+cos 2⁡x=1 cos 2 x=1−sin 2 x cos 2⁡x=1−sin 2⁡x Put,x=60∘x=60∘ cos 2 60∘=1−sin 2 60∘cos 2⁡60∘=1−sin 2⁡60∘ Put the value of sin 60∘sin⁡60∘ cos 2 60∘=1−(3√2)2 cos 2⁡60∘=1−(3 2)2 cos 2 60∘=1−3 4 cos 2⁡60∘=1−3 4 cos(60∘)=1 4−−√=1 2 cos⁡(60∘)=1 4=1 2 Hence proved Learn about Value of Cos 360 Test Series 139.8k Users NCERT XI-XII Physics Foundation Pack Mock Test 323 Total Tests \| 3 Free Tests English,Hindi 3 Live Test 163 Class XI Chapter Tests 157 Class XII Chapter Tests View Test Series 94.2k Users Physics for Medical Exams Mock Test 74 Total Tests \| 2 Free Tests English 74 Previous Year Chapter Test View Test Series 65.1k Users NCERT XI-XII Math Foundation Pack Mock Test 117 Total Tests \| 2 Free Tests English,Hindi 3 AIM IIT 🎯 49 Class XI 65 Class XII View Test Series Cos value chart and table Cos value Circular chart are shown below: In quadrants I and IV cos is positive and negative in quadrants II and III . Cos value table We can determine the cosine values of trigonometric standard angles using the table of trigonometrical ratios. Angle In degree 0 Degree 30 degree 45 degree 60 degree 90 degree cos cos 1 3√2 3 2 1 2√1 2 1 2 1 2 0 Cos Function Periodicity A trigonometric function known as periodic is the cosine function. A periodic function in mathematics is a function that perpetually repeats itself in both directions. Look at the fundamental cosine function, f(x)=cos(x)f(x)=cos⁡(x) The range of x-values on which the cycle of the graph that is repeated in both directions lies is known as the period of a periodic function. As a result, the period is 2 π 2 π for the fundamental cosine function, (f(x)=cos(x)[/l a t e x(f(x)=cos⁡(x)[/l a t e x Properties of Cos 60 Degrees Value of Cos 60° The value of cosine at 60 degrees is ½ or 0.5. This is a standard trigonometric value that is commonly used in math problems. 2. Definition in a Right Triangle In a right-angled triangle, cosine of an angle is the ratio of the adjacent side to the hypotenuse. So, for a 60° angle: cos 60° = adjacent / hypotenuse = 1 / 2 3. Unit Circle Explanation On the unit circle, the point corresponding to 60° (or π/3 radians) has an x-coordinate of 0.5. Since cosine is the x-coordinate of a point on the unit circle, cos 60° = 0.5. 4. Cosine is Positive in the First Quadrant The angle 60° lies in the first quadrant of the coordinate plane, where cosine values are always positive. 5. Even Function Property Cosine is an even function, which means: cos(–θ) = cos(θ) So, cos(–60°) = cos(60°) = 0.5 6. Periodicity of Cosine Cosine is a periodic function with a period of 360° or 2π radians. That means: cos(60°) = cos(420°) = cos(780°) and so on. 7. Graph Behavior In the cosine graph, the value at 60° is 0.5, showing a smooth wave that begins at 1 and dips downward. Solved Examples of Cos 60 Degrees Example 1:Find the value of cos 45° if sec 45° is √2. Solution: We know the identity: cos θ = 1 / sec θ So, cos 45° = 1 / sec 45° = 1 / √2 To simplify: cos 45° ≈ 0.707 Final Answer: cos 45° = 0.707 (approx) Example 2:Solve this expression 3 sin 30° - 5 cos 30° Solution: We know the trigonometric values: sin 30° = 1/2 cos 30° = √3/2 Now substitute these values into the expression: 3 × (1/2) - 5 × (√3/2) = 3/2 - (5√3)/2 So, the final answer is: (3 - 5√3) / 2 Example 3:Simplify: 4 × (cos 45° ÷ sin 135°) Solution: We know: cos 45° = sin 135° = 1/√2 So, 4 × (cos 45° ÷ sin 135°) = 4 × (1/√2 ÷ 1/√2) = 4 × 1 = 4 Final Answer: 4 If you want to score well in your math exam then you are at the right place. Here you will get weekly test preparation, live classes, and exam series. Download theTestbook App now to prepare a smart and high-ranking strategy for the exam. If you are checking Cos 60 Degrees article, also check the related maths articles in the table below: Perimeter of rhombus with diagonalsArea of triangle in determinant form Difference between line and line segmentSemi perimeter of triangle Discrete frequency distributionProperties of functions \| Important Links \| \| NEET Exam \| \| NEET Previous Year Question Papers \| NEET Mock Test \| NEET Syllabus \| \| CUET Exam \| \| CUET Previous Year Question Papers \| CUET Mock Test \| CUET Syllabus \| \| JEE Main Exam \| \| JEE Main Previous Year Question Papers \| JEE Main Mock Test \| JEE Main Syllabus \| \| JEE Advanced Exam \| \| JEE Advanced Previous Year Question Papers \| JEE Advanced Mock Test \| JEE Advanced Syllabus \| More Articles for Maths Difference Between Natural and Whole Numbers Polar Form of Complex Numbers Face Value 3 Digit Multiplication Divisibility Rule of 7 Surds Percent Error Rational Numbers Subtraction Methods Rational and Irrational Numbers FAQs For Cos 60 Degrees What is the value of cos 60 degree? The value of cos 60 degree 1 2 1 2 How to calculate cos 60 degree? By creating an angle of 60 degrees with the x-axis and then determining the coordinates of the corresponding point (0.5, 0.866) on the unit circle, it is possible to compute the value of cos 60 degrees. What is the value of cos 60 degrees in fraction? The value of cos 60 degree in fraction is 1 2 1 2 What is cos 60 degree in radians? Cos 60 degree in radians is π 3 π 3 What is the value of cos 60° in terms of cosec 60°? 1.15470 is the value of cos 60° in terms of cosec 60. What is the value of cos 60 degrees in terms of tan 60°? 1.732050 is the value of cos 60 degrees in terms of tan 60. Is Cos 60° a positive or negative value? Cos 60° is positive because 60° lies in the first quadrant, where all trigonometric values are positive. 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188984	https://artofproblemsolving.com/wiki/index.php/2020_AMC_10B_Problems/Problem_24?srsltid=AfmBOoofLbFfjUkYRUQaxKxRRWGp2J_hiWpXZAFplCBuiRpPxULace04	Art of Problem Solving 2020 AMC 10B Problems/Problem 24 - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki 2020 AMC 10B Problems/Problem 24 Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search 2020 AMC 10B Problems/Problem 24 The following problem is from both the 2020 AMC 10B #24 and 2020 AMC 12B #21, so both problems redirect to this page. Contents 1 Problem 2 Solution 1 (Fakesolve) 3 Solution 2 4 Solution 3 5 Solution 4 6 Solution 5 7 Solution 6 8 Solution 7 9 Video Solutions 9.1 Video Solution 1 9.2 Video Solution 2 9.3 Video Solution 3 by the Beauty of Math 10 See Also Problem How many positive integers satisfy (Recall that is the greatest integer not exceeding .) Solution 1 (Fakesolve) We can first consider the equation without a floor function: Multiplying both sides by 70 and then squaring: Moving all terms to the left: Now we can determine the factors: This means that for and , the equation will hold without the floor function. Now we can simply check the multiples of 70 around 400 and 2500 in the original equation, which we abbreviate as . For , but so For , and For , , For , but so Now we move to For , and so For , and so For , and so For , but so For , and For , but so Therefore we have 6 total solutions, Solution 2 This is my first solution here, so please forgive me for any errors. We are given that We know must be an integer, which means that is divisible by . As , this means that , so we can write for . Note that has to be nonnegative for to be defined. Thus, . Simplifying this, we have . In other words, also has to be nonnegative. Therefore, Also, we can say that and Since is nonnegative, both sides of the second inequality are nonnegative, so we can square them to get . Similarly, solving the first inequality gives us or We know that is larger than and smaller than , so instead, we can say or . Combining this with , we get are all solutions for that give a valid solution for , meaning that our answer is . -Solution By Qqqwerw Solution 3 We start with the given equationFrom there, we can start with the general inequality that . This means thatSolving each inequality separately gives us two inequalities:Simplifying and approximating decimals yields 2 solutions for one inequality and 4 for the other. Hence, the answer is . ~Rekt4 Solution 4 Let be uniquely of the form where . Then, Rearranging and completeing the square gives This gives us Solving the left inequality shows that . Combing this with the right inequality gives that which implies either or . By directly computing the cases for using , it follows that only yield and invalid from . Since each corresponds to one and thus to one (from and the original form), there must be 6 such . ~the_jake314 Solution 5 Since the right-hand-side is an integer, so must be the left-hand-side. Therefore, we must have ; let . The given equation becomes Since for all real , we can take with to get We can square the inequality to get The left inequality simplifies to , which yields The right inequality simplifies to , which yields Solving , and , we get , for values . Solving , and , we get , for values . Thus, our answer is ~KingRavi Solution 6 Set in the given equation and solve for to get . Set ; since , we get The left inequality simplifies to , which yields The right inequality simplifies to , which yields Solving , and , we get , for values . Solving , and , we get , for values . Thus, our answer is ~isabelchen Solution 7 If is a perfect square, we can write for a positive integer , so The given equation turns into k 2+1000 70=k k 2−70 k+1000=0(k−20)(k−50)=0, so or , so If is not square, then we can say that, for a positive integer , we have k 2<n<(k+1)2 k 2+1000<n+1000=70⌊n⌋=70 k<(k+1)2+1000 k 2+1000<70 k<(k+1)2+1000. To solve this inequality, we take the intersection of the two solution sets to each of the two inequalities and . To solve the first one, we have k 2−70 k+1000<0(k−20)(k−50)<0 because the portion of the parabola between its two roots will be negative. The second inequality yields 70 k<k 2+2 k+1+1000 0<k 2−68 k+1001. This time, the inequality will hold for all portions of the parabola that are not on or between the its two roots, which are and (they are roughly equal, but this is to ensure that we do not miss any solutions). Notation wise, we need all integers such that or For the first one, since our uppoer bound is a little less than , the that works is . For the second, our lower bound is a little more than , so the that work are and . total solutions for , since each value of corresponds to exactly one value of . -Benedict T (countmath1) Video Solutions Video Solution 1 On The Spot STEM: Video Solution 2 ~ MathEx Video Solution 3 by the Beauty of Math See Also 2020 AMC 10B (Problems • Answer Key • Resources) Preceded by Problem 23Followed by Problem 25 1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25 All AMC 10 Problems and Solutions 2020 AMC 12B (Problems • Answer Key • Resources) Preceded by Problem 20Followed by Problem 22 1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25 All AMC 12 Problems and Solutions These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 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188985	https://stackoverflow.com/questions/72040434/finding-the-non-zero-digit-after-mutiplying-each-element-in-array	c++ - Finding the non zero digit after mutiplying each element in array - Stack Overflow Join Stack Overflow By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google Sign up with GitHub OR Email Password Sign up Already have an account? Log in Skip to main content Stack Overflow 1. About 2. Products 3. 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Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Finding the non zero digit after mutiplying each element in array Ask Question Asked 3 years, 5 months ago Modified3 years, 5 months ago Viewed 391 times This question shows research effort; it is useful and clear -1 Save this question. Show activity on this post. Input: N = 4 arr = {3, 23, 30, 45} Output: 5 Explanation: Product of these numbers is 93150. Rightmost non-zero digit is 5. can u solve this question in c++ and run this code for input i give u.enter image description here cpp // my code for this question int rightmostNonZeroDigit(int N,arr[]) { // Write your code here. long int t = 1; for (int i = 0; i < N; i++) { t = t arr[i]; } while (t > 0) { if ((t % 10) != 0) { return (t % 10); } t = t / 10; } return -1; } // what changes should i make in this code c++ Share Share a link to this question Copy linkCC BY-SA 4.0 Improve this question Follow Follow this question to receive notifications edited Apr 28, 2022 at 8:36 Mohammed faiz KhanMohammed faiz Khan asked Apr 28, 2022 at 8:32 Mohammed faiz KhanMohammed faiz Khan 29 1 1 silver badge 4 4 bronze badges 9 1 Please read the help pages, especially "What topics can I ask about here?" and "What types of questions should I avoid asking?". Also take the tour and read about How to Ask good questions and this question checklist. Lastly learn how to create a minimal reproducible example.Some programmer dude –Some programmer dude 2022-04-28 08:34:31 +00:00 Commented Apr 28, 2022 at 8:34 1 you need to provide code. For large numbers you can have overflow. Use long long instead of int.Berkay Berabi –Berkay Berabi 2022-04-28 08:35:50 +00:00 Commented Apr 28, 2022 at 8:35 1 Could you also provide the input numbers as text and not as image. That way we can reproduce your problem wihtout manually typing all the numbers Jakob Stark –Jakob Stark 2022-04-28 08:40:23 +00:00 Commented Apr 28, 2022 at 8:40 1 you can strip off any trailing zeros from the numbers before the multiplication. Also you need only keep a single digit of the result of multiplication.463035818_is_not_an_ai –463035818_is_not_an_ai 2022-04-28 08:40:59 +00:00 Commented Apr 28, 2022 at 8:40 2 Multiplying 509 at least 3 digit numbers gives a resul of 10^(3506). This is a number with 1500 digits! The solution cannot be found by multiplication. You need to check, when a "0" in a multiplication would appear . . . So this is a mathematical puzzle. Not about programming.A M –A M 2022-04-28 08:44:16 +00:00 Commented Apr 28, 2022 at 8:44 \|Show 4 more comments 2 Answers 2 Sorted by: Reset to default This answer is useful 5 Save this answer. Show activity on this post. This is actually a nice little challenge. You are multiplying (based from a short estimation on your input image) about 500 numbers with 3 digits each. The product of all these number will never fit into any standard integer type provided by C++. Suppose your variable t holds some four digit number "abcd". You can write it like cpp t = a 1000 + b 100 + c 10 + d Now if you multiply t with any other number x you get cpp t x = a x 1000 + b x 100 + c x 10 + d x As you can see the last digit of tx is only determined by dx. All the other components have trailing zeros since they are some multiple of a power of ten. That means to get the last digit of any multiplication, you just have to multiply the two last digits of the numbers. Now you are not interested in the last digit, but in the last non-zero digit. You will get the right result if you only ever keep the last non-zero digit in t while calculating the product of all the numbers. In your code you could do something like this: ```cpp for (int i = 0; i < N; i++) { t = t arr[i]; // the following will remove all trailing zeros while ( t != 0 && t % 10 == 0 ) { t = t / 10; } // the following will remove all but the last digit t = t % 10; } ``` This works because trailing zeros in the intermediate result will never influence anything but the number of trailing zeros in the final result. And digits before the last non-zero digit will also never contribute to the last non-zero digit in the final result. Addition On godbolt you can find a live example with your test input arr = {3,23,30,45}. Important Edit As @MohammedfaizKhan pointed out there are cases where the above code fails. For example if we take the numbers arr = {15,2}. The code from above yields 1 because it truncates the 1 in 15 before multiplying it with 2. If we call D the operation that tuncates a number to the first non zero digit, the above program could be written like: \| \| code from above produces \| --- \| \| step one \| t = D(1 15) = 5 \| \| step two \| t = D(5 2 ) = 1 \| The correct result would be 3. Apparently we cannot remove all leading digits. We could try to increase the number of leading nonzero digits that are kept in each step. For example in the code above, we could use t = t % 100 instead of t = t % 10. There is however a counter example for each number of digits we are trying to keep: The numbers 2^n and 5^n don't have trailing zeros because they are no multiple of ten because a multiple of 10 must have 2 and 5 in its prime factorization. Their product 2^n 5^n = (25)^n = 10^n however has exactly n trailing zeros. In conclusion we should keep as many leading nonzero digits as we can fit into our data type. For an 64bit unsigned int this would be for example 19 digits. However we also must not overflow while doing the multiplication with the array elements. Because your array elements are all no longer than 3 digits, we should be safe if we keep the leading 15 digits or something like that. So in conclusion the following program should do the correct thing: ```cpp unsigned long long int t = 1; for (int i = 0; i < N; i++) { t = t arr[i]; // the following will remove all trailing zeros while ( t != 0 && t % 10 == 0 ) { t = t / 10; } // the following will remove all but the last 15 digits t = t % 1000000000000000; } ``` Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications edited May 2, 2022 at 10:53 answered Apr 28, 2022 at 8:54 Jakob StarkJakob Stark 3,883 11 11 silver badges 25 25 bronze badges 7 Comments Add a comment Bathsheba BathshebaOver a year ago Well explained (have an upvote),. but you do need to consider the case when one of the inputs in the array is zero. As I do (apologies for plugging my answer). 2022-04-28T09:19:36.47Z+00:00 0 Reply Copy link Jakob Stark Jakob StarkOver a year ago @Bathsheba actually a good point, thanks for the hint. I will fix it. 2022-04-28T09:20:35.183Z+00:00 0 Reply Copy link Mohammed faiz Khan Mohammed faiz KhanOver a year ago not working for this test case. n=5 arr={7,6,6,5,1} 2022-04-30T07:14:58.29Z+00:00 1 Reply Copy link Jakob Stark Jakob StarkOver a year ago @MohammedfaizKhan you are absolutely right. I have to think about that one. For the meantime you could remove the t = t % 10 line. But then you may get an overflow again in special cases. 2022-05-02T08:31:42.277Z+00:00 0 Reply Copy link Jakob Stark Jakob StarkOver a year ago @MohammedfaizKhan I added an update with a version that should work now. 2022-05-02T10:56:08.483Z+00:00 0 Reply Copy link Add a comment\|Show 2 more comments This answer is useful 2 Save this answer. Show activity on this post. The trick is to retain only the final non-zero digit on each step. ```cpp include int main() { int arr[] = {3, 23, 30, 45}; int n = 1; for (auto&& i : arr){ if (!(n = i)) break; // Zero in input needs special treatment for (; !(n % 10); n /= 10); // Remove trailing zeros n %= 10; // Retain single digit } std::cout << n; } ``` is one way. Share Share a link to this answer Copy linkCC BY-SA 4.0 Improve this answer Follow Follow this answer to receive notifications edited Apr 28, 2022 at 9:25 answered Apr 28, 2022 at 9:10 BathshebaBathsheba 235k 35 35 gold badges 376 376 silver badges 504 504 bronze badges 1 Comment Add a comment Mohammed faiz Khan Mohammed faiz KhanOver a year ago brother your code doesn't work for the given test case. N=5 arr=[7,6,6,5,1] expexted output=6 our output=1 2022-05-01T05:57:35.62Z+00:00 0 Reply Copy link Your Answer Thanks for contributing an answer to Stack Overflow! Please be sure to answer the question. Provide details and share your research! 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188986	https://ocw.mit.edu/courses/16-07-dynamics-fall-2009/d931dd84ca3025a3676ed2244f48ab85_MIT16_07F09_Lec15.pdf	J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 1.2 Lecture L15 - Central Force Motion: Kepler’s Laws When the only force acting on a particle is always directed to wards a ﬁxed point, the motion is called central force motion. This type of motion is particularly relevant when studying the orbital movement of planets and satellites. The laws which gov ern this motion were ﬁrst postulated by Kepler and deduced from observation. In this lecture, we will see that these laws are a con sequence of Newton’s second law. An understanding of central force motion is necessary for the design of satellites and space vehicles. Kepler’s Problem We consider the motion of a particle of mass m, in an inertial reference frame, under the inﬂuence of a force, F , directed towards the origin. We will be particularly interested in the case when the force is inversely proportional to the square of the distance between the particle and the origin, such as the gravitational force. In this case, µ F = − r2 mer, where µ is the gravitational parameter, r is the modulus of the position vector, r, and er = r/r. It can be shown that, in general, Kepler’s problem is equivalent to the two-body problem, in which two masses, M and m, move solely due to the inﬂuence of their mutual gravitational attraction. This equivalence is obvious when M m, since, in this case, the center of mass of the system can be taken to be at M. 1 However, even in the more general case when the two masses are of similar size, we shall show that the problem can be reduced to a ”Kepler” problem. Although most problems in celestial mechanics involve more than two bodies, many problems of practical interest can be accurately solved by just looking at two bodies at a time. When more than two bodies are involved, the problem is considerably more complicated, and, in this case, no general solutions are known. The two body problem was studied by Kepler (1571-1630) who lived before Newton was born. His interest was in describing the motion of planets around the sun. He postulated the following laws: 1.- The orbits of the planets are ellipses with the Sun at one focus 2.- The line joining a planet to the Sun sweeps out equal areas in equal intervals of time 3.- The square of the period of a planet is proportional to the cube of the major axis of its elliptical orbit In this lecture, we will start from Newton’s laws and verify that the above three laws can indeed be derived from Newtonian mechanics. Equivalence between the two-body problem and Kepler’s problem Here we consider the problem of two isolated bodies of masses M and m which interact though gravitational attraction. Let rM and rm denote the position vectors of the two bodies relative to a ﬁxed origin O. Since the only force acting on the bodies is the force of mutual gravitational attraction, the motion is governed by Newton’s law with an equal and opposite force acting on each body. Mm Mr ¨M = G r2 er, (1) Mm mr ¨m = −G r2 er , (2) where r = \|r\|, er = r/r, and G is the gravitational constant. The position of the center of gravity, G, of the two bodies will be rG = MrM + mrm . (3) M + m 2 Since the two bodies are isolated, we will have, from momentum conservation, that r ˙ G =constant, and r ¨G = 0. Therefore, the position of the center of gravity, at all times, can be found trivially from the initial conditions. If the position vector of m as observed by M, r = rm − rM , is known, then the position vectors of M and m could be computed as m M rM = rG − r, rm = rG + r . (4) M + m M + m Therefore, since we know the position of the center of mass rG for all time, we shall show that the problem of determining rM and rm is equivalent to that of determining r, the vector distance between them. The governing equations for rm and rM are given in equation (1) and (2). Subtracting these two expressions, we obtain, M + m r ¨ = r ¨m − r ¨M = −G er , (5) r2 or, Mm Mm r ¨ = −G er . (6) M + m r2 The above expression shows that the motion of m relative to M is in fact a Kepler problem in which the force is given by −GMmer/r2 (this is indeed the real force), but the mass of the orbiting body (m in this case), has been replaced by the reduced mass, Mm/(M + m). Note that when M >> m, the reduced mass becomes m. However, the above expression is general and applies to general masses M and m. Alternatively, the above expression can be written as mr ¨ = −G (M + m)m er , (7) r2 which is again a Kepler problem for an orbiting body of mass m, in which the gravitational parameter µ is given by µ = G(M + m). Example Solution to the Two Body Problem There are two approaches to the solution of the two-body problem. One is a direct numerical attack on equations (1) and (2); the other is to use the analytic solution of the Kepler problem, equation(7), and having found r(t), to use the equation for the position of the center of mass, rG(t) and equation (4) to determine rm(t) and rM (t). The position of the center of mass is determined by the initial conditions (position and velocity) of the bodies. Consider the motion of two bodies as shown in a). The masses of the two bodies are M = 4 and m = 1; for convenience G was set equal to 10. The initial conditions (vector components) are given as rm = (1, 0), r ˙ m = (2, 3) and rM = (−2, 0) and r ˙ M = (−2, 0). The motion of the two bodies with time is shown in a). From the boundary conditions, we obtain the position of the center of mass with time as rG = (−7/5, 0) + (−6/5, 3/5)t; this position with time is shown in b). The bodies ”orbit” about the instantaneous position of the center of mass. 3 The solution to the ”Kepler” problem for these bodies is shown in c); the solution to the ”Kepler” orbital problem gives the instantaneous position of the relative position of the two bodies, r(t) = rm − rm. The Kepler problem has its origin as the center of mass, which also is the focus of the elliptical orbit. To recover the orbits of the two bodies, we use equation (4). The two orbits are shown in d). These are also the solutions that would be obtained by a direct numerical solution of the two-body problem with boundary conditions chosen to place the center of mass at the origin. The origin serves as the focus for each elliptical orbit. This example shows the importance of formulating the velocity and position boundary conditions so that the center of mass remains ﬁxed at the origin. If this is done, the bodies will orbit about the center of mass, producing the simplest solution to the two-body problem. Equations of Motion The equation of motion (F = ma), is µm − r2 er = mr ¨. Since the only force in the system is directed towards point O, the angular momentum of m with respect to the origin will be constant. Therefore, the position and velocity vectors, r and r ˙, will be in a plane 4 orthogonal to the angular momentum vector, and, as a consequence, the motion will be planar. Using cylindrical coordinates, with ez being parallel to the angular momentum vector, we have, − r µ 2 er = (¨ r − rθ ˙2)er + (rθ ¨+ 2 ˙ rθ ˙)eθ. Now, we consider the radial and circumferential components of this equation separately. Circumferential component We have, ¨ ˙ 0 = rθ + 2 ˙ rθ . Using the following identity, 1 d (r 2θ ˙) = rθ ¨+ 2 ˙ rθ, ˙ r dt the above equation implies that r 2θ ˙ = h ≡ constant. (8) We note that the constant of integration, h, that will be determined by the initial conditions, is precisely the magnitude of the speciﬁc angular momentum vector, i.e. h = \|r × v\|. In a time dt, the area, dA, swept by r will be dA = r rdθ/2. Therefore, dA 1 h = r 2θ ˙ = dt 2 2 , which proves Kepler’s second law:The line joining a planet to the Sun sweeps out equal areas in equal intervals of time. Radial component The radial component of the equation of motion reads, − r µ 2 = ¨ r − rθ ˙2 . (9) 2 d 1 2 Since −r = r, ˙ and r = h/θ ˙ from equation 8, we can write dt r h d 1 d 1 r ˙ = − θ ˙ dt r = −hdθ r . Diﬀerentiating with respect to time, r ¨ = −h d2 1 θ ˙ = − h2 d2 1 . dθ2 r r2 dθ2 r 5 Inserting this expression into equation 9, and using equation 8, we obtain the following diﬀerential equation for 1/r as a function of θ. d2 1 1 µ + = . dθ2 r r h2 This is a linear second order ordinary diﬀerential equation which has a general solution of the form, 1 µ = (1 + e cos(θ + ψ)) , r h2 where e and ψ are two constants of integration. If we choose θ to be zero when r is minimum, then e will be positive, and ψ = 0. The equation describing the trajectory will be h2/µ r = . (10) 1 + e cos θ We shall see below that this is the equation of a conic section in polar coordinates. Conic Sections Conic sections are planar curves that are deﬁned as follows: given a line, or directrix, and a point, or focus O, a conic section is the locus of points, P , such that the ratio of the distance between the point and the focus, PO, to the distance between the point and the directrix, PA, is a constant e. That is, e = PO/PA. Since PO = r and PA = p/e − r cos θ, we have r = p . (11) 1 + e cos θ Here, p is the parameter of the conic and is equal to r when θ = ±90o . The constant e ≥ 0 is called the eccentricity, and, depending on its value, the conic surface will be either an open or closed curve. In particular, we have that when e = 0 the curve is a circle e < 1 the curve is an ellipse e = 1 the curve is a parabola e > 1 the curve is a hyperbola. 6 Comparing equation(11) which deals solely with the property of a conic section, and equation(10) which provides the solution of the motion of a point mass in a gravitational ﬁeld, we can identify the properties of the conic section orbits in terms of the physical parameters of the Kepler problem. In particular, we see that the trajectory of a mass under the inﬂuence of a central force will be a conic curve with parameter p = h2/µ. (12) When e < 1, the trajectory is an ellipse, thus proving Kepler’s ﬁrst law:The orbits of the planets are ellipses with the Sun at one focus. The point in the trajectory which is closest to the focus is called the periapsis and is denoted by π. For elliptical orbits, the point in the trajectory which is farthest away from the focus is called the apoapsis and is denoted by α. When considering orbits around the earth, these points are called the perigee and apogee, whereas for orbits around the sun, these points are called the perihelion and aphelion, respectively. Elliptical Trajectories If a is the semi-major axis of the ellipse, then 2a = rπ + rα. (13) Using equation 11 to evaluate rπ (θ = 0) and rα (θ = π), we obtain a = p/(1 − e 2). (14) Thus from the geometric properties of an ellipse, rπ = 1 + p e = a(1 − e), rα = 1 − p e = a(1 + e). Also, the distance between O and the center of the ellipse will be a − rπ = a e. (15) 7 Other geometric properties of the ellipse are that the distance between point D and the directrix will be equal to DO/e, which in turn will be equal to the sum of the distance between the focus and the center of the ellipse, plus the distance between the focus and the directrix. That is, DO/e = ae + p/e. Therefore, DO = a e2 + p = a. Hence, using Pythagoras’ theorem, b2 + (a e)2 = a2, the semi-minor axis of the ellipse will be b = a √ 1 − e2. The area of the ellipse is given by A = πab. (16) Also, since dA/dt = h/2 (17) is a constant, we have A = hτ/2, (18) where τ is the period of the orbit. Equating these two expressions and expressing h in terms of the semi-major axis as h2 = µp = µa(1 − e 2), (19) we have 2 2π 3 µ = a , (20) τ which proves Kepler’s third law:The square of the period of a planet is proportional to the cube of the major axis of its elliptical orbit. This can be rewritten to obtain the time of ﬂight or period of the orbit. τ = √ 2π µa 3/2 (21) Time of Flight (TOF) in Elliptical Trajectories We have found r(θ), the prediction of the shape of the orbit. However, this solution gives us no direct information about the time behavior of the motions, such as θ(t). In many situations we will need to determine the time of ﬂight between two arbitrary points along the ellipse. In order to do that, we use Kepler’s second law, which states that the motion of the planet sweeps out area at a constant rate. Consider the orbital motion from point 0, to point 1. We would like to determine the time taken t1. If the motion continues, returning to point 2, the total time taken will be τ. We deﬁne the time to reach point 1 as T1 and the time to reach point 2 as t2. The total time taken is the t1 + t2 = τ, where τ is the total period of the orbit. 8 From Kepler’s second law, equal areas are swept out in equal times. Thus the time taken to reach point 1 is given by t1/A1 = t2/A2 = τ/πab (22) where πab is the total area of the ellipse, πab = A1 + A2. We now construct a more detailed analysis to determine the area Ap swept out by the orbit to a point P. Referring to the ﬁgure, we see that the time required to travel between the point π, the periapsis, and an arbitrary point P is proportional to the curved area denoted by AP (AP is the sector deﬁned by O, π, P ). More speciﬁcally, since the total period of the orbit is τ and the total area of an ellipse is πab, tP , the time required to travel from π to P , equals the fraction that the area AP represents of the total area of the ellipse. tP = AP τ. (23) πab To ﬁnd the area AP we construct a circle of radius a with origin at the center of the ellipse. We identify a point P 0 on the circle to be in a vertical line with the point of interest P on the ellipse intersecting the point O” on the axis. The various geometric quantities of the elliptical orbit have standard deﬁnitions: the position angle θ is often called the true anomaly. The radial line of the circle for the origin O0 to P 0 and the major axis of the ellipse 9 major axis deﬁne an angle u, which is referred to as the eccentric anomaly. In addition, we deﬁne a third anomaly, the mean anomaly M of the point P , as 2πtP MP = . (24) τ Here, tP is the time of ﬂight from the periapsis to the point P . Thus, if we want to determine the time of ﬂight between two points 1 and 2 on the ellipse, we can use equation (24) and write TOF = τ (M2 − M1) = A2 − A1 τ , = t2 − t1 2π πab where A2 − A1 is the area swept out between points 1 and 2. The mean anomaly for point P can also be written as 2π ∗ AP MP = , (25) AT where AP is the area swept out up to the point P . When the area swept out equals the total area of the ellipse AT , the time t equals the period τ and the mean anomaly Mπ = 2 ∗ π. (The subscript π denotes the return to the periapsis π.) Thus the mean anomaly can be thought of as the fraction of the total angle 2π that would be swept out in a time τ by an object reaching point P . The focus is on time not on actual spatial angle. All is needed now is an expression for the mean anomaly M as a function of the orbit parameters. We start by obtaining a relation between θ and u. From simple trigonometry, we have that a cos u − r cos θ = ae (26) or noting that r = a(1 − e2)/(1 + e cos θ), cos u = e + cos θ , cos θ = cos u − e . (27) 1 + e cos θ → 1 − e cos u We now develop relationships between the various areas indicated on the ﬁgure, with the goal to ﬁnd the formula for the area AP , the area swept out by the point r as it travels from the periapsis π to the point P . The area A1 is the wedge in the circle occupied by the angle u. A1 = a2u; the area of the large triangle formed by the angle u within the circle is A2 = a2cos(u)sin(u)/2. Therefore, the area of the large curved segment from O”, P 0, π is A1 − A2 = (1/2) × a 2 × (u − Cos(u)Sin(u)). (28) The base of the small triangle of area A4, O, O00, P , is r × cos(θ) = a × cos(u) − ae by equation (26). The height of the small triangle is b × sin(u). Therefore, the area of the small triangle is A4 = (1/2) × (a ∗ cos(u) − e) × b × sin(u). This area plus the curved segment O”, P, π is the total area swept by the point P . The ﬁnal step in identifying the area segment swept out between point π and P is to identify the curved segment from O”, P, π, which is then added to the triangle section A4 to form the complete swept area. The curved vertical segment formed from removing the large imbedded triangle A2 from the arc segment of 10 the circle A1 –call it A3 – is geometrically similar to the curved segment formed by removing the small triangle from the area of the swept segment of the ellipse. Since the vertical height of the ellipse is b, and the vertical height of the circle is a, the area of the desired curved segment can by obtained from that of the corresponding segment of the circle by multiplying by b/a. Speciﬁcally, AP − A4 = (b/a) × (A1 − A2). (29) Therefore, the ﬁnal result for the area swept out by the point r moving from point π to point P is AP = b/a × (A1 − A2) + A4 (30) And the mean anomaly for the point P is MP = 2π ∗ AP = 2π × (b/a × (A1 − A2)) + A4 (31) AT AT Thus, combining equations (24), (28) and (30), we obtain the mean anomaly for the point P , called Kepler’s equation (It took a Kepler to work this out.) u − e sin u = MP = 2πtP . τ where u is the eccentric anomaly for the point P , deﬁned in the ﬁgure. This equation is very easy to use if we want to know the time tP at which the satellite is at position θ. The only thing required, in this case, is the calculation of the eccentric anomaly u using equation (27). On the other hand, if we need to ﬁnd the position θ of the satellite at a given time t, then, we need to solve Kepler’s equation which is non-linear using an iterative numerical algorithm such as Newton’s method. ADDITIONAL READING J.L. Meriam and L.G. Kraige, Engineering Mechanics, DYNAMICS, 5th Edition 3/13 (except energy analysis) 11 MIT OpenCourseWare 16.07 Dynamics Fall 2009 For information about citing these materials or our Terms of Use, visit:
188987	https://medium.com/@sahin.samia/ordinary-least-squares-ols-a-geometric-and-linear-algebraic-perspective-and-their-equivalence-5721e2643375	Sitemap Open in app Sign in Sign in Ordinary Least Squares (OLS): A Geometric and Linear Algebraic Perspective and Their Equivalence. Sahin Ahmed, Data Scientist 7 min readMar 30, 2024 Introduction Let’s dive into a fascinating concept in the world of statistics and data analysis called Ordinary Least Squares (OLS). Imagine you’ve got a bunch of data points from an experiment or a survey, and you suspect there’s some kind of linear relationship between them. For example, maybe you’re looking at how study time affects exam scores, and you want to draw a straight line that best represents this relationship. This is where OLS comes into play. It’s a super-handy method used to estimate the parameters of a linear regression model. Think of it as finding the best-fit line through your data points, where “best” means the line that minimizes the differences (or errors) between the actual data points and the points on the line. Now, here’s where things get even more interesting. There are two main ways to look at OLS: through geometric and linear algebra perspectives. At first glance, these might seem like two completely different worlds. The geometric view is all about pictures and spatial intuition, imagining our data and the best-fit line in a space we can visualize. On the other hand, the linear algebra perspective dives into matrices and equations, focusing on how we can solve for the best-fit line using mathematical operations. Our journey today is all about exploring these two perspectives, understanding how each gives us unique insights into OLS, and revealing a cool fact: despite their different approaches, they ultimately lead us to the same destination. So, whether you’re a visual thinker or a math enthusiast, there’s something in here for you. Let’s get started and uncover the magic of OLS together! Foundations of OLS The Problem Statement In the realm of linear regression, we’re often faced with the challenge of understanding and predicting how one variable affects another. Let’s break down the problem that Ordinary Least Squares (OLS) aims to tackle, step by step: The Heart of the Problem: At its core, OLS seeks to solve a fundamental issue in linear regression: how can we find the line (or hyperplane, in more complex scenarios) that best fits our set of data points? This “best fit” is formally known as the regression line. The Linear Regression Model: To set the stage, consider the linear regression model equation, which looks something like this: Goal: Minimizing the Residual Sum of Squares (RSS): The mission of OLS is to find the values of β0,β1,…,βn that minimize the difference between the actual observed values of y and the values predicted by our linear model. These differences are called “residuals.” Specifically, we aim to minimize the residual sum of squares (RSS), which is the sum of the squared residuals for all data points. The RSS can be expressed as: In essence, OLS navigates through the maze of possible lines (or hyperplanes) and picks the one that makes the smallest total mistake, according to the RSS criterion. This approach not only gives us the “best fit” but also lays the groundwork for further analysis, like understanding the strength and nature of the relationship between our variables. Get Sahin Ahmed, Data Scientist’s stories in your inbox Join Medium for free to get updates from this writer. From the document you’ve provided, let’s delve into the linear algebra approach to Ordinary Least Squares (OLS). This approach frames OLS in terms of matrices and vectors, providing a clear and efficient way to compute the regression coefficients. Linear Algebra Approach to OLS When analyzing data with multiple predictors, we turn to the Multiple Linear Regression Model (MLRM), which is concisely expressed using matrix notation. Let’s break this down: Design Matrix (X): This matrix contains our predictor variables. Each row corresponds to an observation, and each column corresponds to a predictor variable. For a model with p predictors and n observations, X is an n×(p+1) matrix (including the intercept term). This matrix is assumed to have full rank Response Vector (y): This vector contains the observed values of the dependent variable, with dimensions n×1. Coefficient Vector (β): This vector contains the parameters we aim to estimate, including the intercept and the slope coefficients for each predictor, with dimensions (p+1)×1. Residual Vector (r): This represents the differences between the observed values and the values predicted by our model, with dimensions n×1. The goal of OLS is to minimize the sum of the squares of these residuals across all observations, which can be expressed as: Geometric Approach to OLS: The OLS solution essentially projects the observed value vector y onto the column space of the design matrix X, finding the point in that space that is closest to y. This is equivalent to solving for β^ in the equation Xβ^=projC(X)(y), where projC(X)(y) is the orthogonal projection of y onto the column space of X. Data Space Visualization: In a geometric sense, each column of the design matrix X can be thought of as a vector in a multidimensional space, with the space spanned by these vectors being the “column space.”. The column space of a design matrix X (denoted as Col(X)) is the set of all possible linear combinations of its columns.The observed value vector y is another point/vector that is pointing outward, as shown in the figure in this multidimensional space. Distance Minimization: The goal of regression is to find the predicted value vector y^ that is as close as possible to the actual observations y. Here, predictions y^ will be linear combinations of columns of the design matrix that are closest to actual y. “Closeness” is defined by the Euclidean distance (or the sum of squared distances for all observations), which corresponds to the residuals. Projection: Imagine you’re trying to cast a shadow (predict values) of an object (observed data) onto a surface (column space). This shadow is the nearest representation of the object in the column space.This is why the closest point in the column space of X to Y is found through projection. Orthogonality Principle: The shadow perfectly cast by the object on the surface ensures that the line connecting the tip of the object to the tip of its shadow (the residuals) is perpendicular to the surface.Algebrically The residual vector is the difference between the observed values (y) and the values predicted by the fitted model (Xβ^) which is orthogonal (perpendicular) to the column space of X. This means that the errors in prediction are not correlated with the independent variables. Intuitively, if the residual vector is not orthogonal to the column space of X, it means there is some pattern or relationship in the errors that the model hasn’t captured, indicating a lack of fit. Why the OLS Solution is unique: The full rank condition of the design matrix X ensures that all columns of X are linearly independent. This means that each independent variable in your model is providing unique information, and there is no redundancy. When the design matrix has a full rank, it guarantees the uniqueness of the OLS solution. In other words, there’s only one set of coefficients (β^) that minimizes the sum of squared residuals. Intuitively, if you have redundant variables (e.g., two variables that are perfectly correlated), you could have multiple sets of coefficients that perfectly fit the data. However, in the case of full rank, there’s only one unique solution. For example, if you’re trying to fit a line to data with only one independent variable, having a full rank means that this independent variable provides unique information, and there’s only one slope and intercept that best fit the data. Conclusion: Both approaches aim to achieve the same goal: in linear algebra, minimizing the discrepancy (residuals) between the observed data and the model’s predictions. Geometry does this by finding the shortest paths (perpendicular distances), while algebra employs optimization (minimizing the sum of squared residuals) to reach the solution. In both cases, you arrive at the same line—tthe best fit for your data. This dual approach underscores the robustness of the OLS method, providing both a visual and a quantitative pathway to understanding and applying linear regression. This synthesis of geometry and algebra in OLS showcases the beauty of mathematics in modeling and understanding our world, reinforcing the power of multiple perspectives in solving complex problems. Data Science Statistics Linear Regression Mathematics ## Written by Sahin Ahmed, Data Scientist 1.1K followers ·182 following Lifelong learner passionate about AI, LLMs, Machine Learning, Deep Learning, NLP, and Statistical Modeling to make a meaningful impact. MSc in Data Science. No responses yet Write a response What are your thoughts? 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188988	https://www.scribd.com/document/516337426/Even-Odd-Functions-1	Even Odd Functions \| PDF \| Function (Mathematics) \| Analysis Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Close suggestions Search Search en Change Language Upload Sign in Sign in Download free for 30 days 0 ratings 0% found this document useful (0 votes) 972 views 5 pages Even Odd Functions The document discusses even and odd functions. An even function satisfies f(x)=f(-x) for all x, meaning the function is symmetric about the y-axis. An odd function satisfies f(x)=-f(-x), mea… Full description Uploaded by Tamar Y. 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188989	https://askfilo.com/user-question-answers-smart-solutions/find-the-complex-number-satisfying-the-equation-give-your-3336353136393639	Question asked by Filo student Find the complex number z satisfying the equation: z+3iz−3i=52−9i Give your answer in the form x+iy, where x and y are real. Views: 5,560 students Updated on: Jul 31, 2025 Text SolutionText solutionverified iconVerified Concepts Complex Numbers, Algebraic Manipulation, Rationalization, Simultaneous Equations Explanation We are given the equation: z+3iz−3i=52−9i Let z=x+iy, with x,y∈R. Substitute z and solve for x and y. Step-By-Step Solution Step 1 Let z=x+iy. Then z−3i=x+i(y−3). And z+3i=x+i(y+3). Step 2 Write the equation with substituted z: x+i(y+3)x+i(y−3)=52−9i Step 3 Multiply both sides by x+i(y+3): [ x + i(y - 3) = \frac{2 - 9i}{5} [x + i(y + 3)] ] Multiply both sides by 5 to eliminate denominator: [ 5 [x + i(y - 3)] = (2 - 9i)[x + i(y + 3)] ] Step 4 Expand the right side: (2−9i)[x+i(y+3)]=2x+2i(y+3)−9ix−9i2(y+3)=2x+2i(y+3)−9ix+9(y+3)[since i2=−1]=(2x+9y+27)+i[2(y+3)−9x] Step 5 Expand the left side: 5x+5i(y−3)=5x+5iy−15i Step 6 Equate real and imaginary parts: Step 7 Solve the real part: 5xx−2x−9y3x−9y3x−9y=27x−3y=9=2x+9y+27=27−5x=27 So, x−3y=9. Step 8 Solve the imaginary part: 5y−155y−15−2y−6+9x3y+9xDivideboth sides by 3:y+3x=7=2y+6−9x=0=21 So, y+3x=7. Step 9 Solve the simultaneous equations: {x−3y=9y+3x=7 From the first equation, x=9+3y Substitute into second equation: y+3(9+3y)=7y+27+9y=710y+27=710y=7−27=−20y=−2 Now, back to x=9+3y: x=9+3(−2)=9−6=3 Step 10 Build final answer: z=x+iy=3−2i Final Answer z=3−2i Students who ask this question also asked Views: 5,321 Topic: Smart Solutions View solution Views: 5,999 Topic: Smart Solutions View solution Views: 5,273 Topic: Smart Solutions View solution Views: 5,709 Topic: Smart Solutions View solution Stuck on the question or explanation? Connect with our tutors online and get step by step solution of this question. \| \| \| --- \| \| Question Text \| Find the complex number z satisfying the equation: z+3iz−3i=52−9i Give your answer in the form x+iy, where x and y are real. \| \| Updated On \| Jul 31, 2025 \| \| Topic \| All topics \| \| Subject \| Smart Solutions \| \| Answer Type \| Text solution:1 \| Are you ready to take control of your learning? Download Filo and start learning with your favorite tutors right away! Questions from top courses Explore Tutors by Cities Blog Knowledge © Copyright Filo EdTech INC. 2025
188990	https://www.sciencedirect.com/topics/computer-science/binary-sequence	Skip to Main content Sign in Chapters and Articles You might find these chapters and articles relevant to this topic. Basics of incoherent and coherent digital optical communications 3.2.3 Binary optical channel and the symbol probabilities The binary sequence to be transmitted is usually available in the form of an electrical signal taking one between two random discrete values. The simplest representation consists of an electrical current or voltage, which is either “on” or “off”. These two possibilities represent the symbols of the digital message and are called “bit 1” and “bit 0” respectively. The finite time duration of each bit is called the bit period T, and is the bit rate. By using a discrete case version of Eq. 3.1, the information entropy of a binary message is: (3.12) in which p(1) and p(0)=1−p(1) are the probabilities of transmitting “1” and “0” respectively. It is easy to confirm that the information entropy is maximum, meaning a binary message is more informative, when the symbols “1” and “0” have the same probability of occurring. Therefore, receiver performances will be discussed in section 3.5 based on the assumption that p(1)=p(0)=1/2. Figure 3.2 represents the modeling of the binary optical channel. P(0/1) is the probability of deciding that 0 is received when 1 is transmitted, and P(1/0) is the probability of deciding 1 when 0 is transmitted. As will be discussed in section 3.5, the use of the optical power as the information carrier leads to a nonadditive noise and therefore to different noise distributions when the symbols 0 or 1 are transmitted. The incoherent optical channel is usually only made symmetrical by an appropriate tuning of the decision level and, so, the above assumption may be not optimal from an overall system point-of-view. However, we will only consider, in this section, very low error probabilities. For high error probability systems, improvements may result from the use of more advanced information representation. View chapterExplore book Read full chapter URL: Chapter Arithmetic Coding 4.4 Generating a Binary Code Using the algorithm described in the previous section, we can obtain a tag for a given sequence x. However, the binary code for the sequence is what we really want to know. We want to find a binary code that will represent the sequence x in a unique and efficient manner. We have said that the tag forms a unique representation for the sequence. This means that the binary representation of the tag forms a unique binary code for the sequence. However, we have placed no restrictions on what values in the unit interval the tag can take. The binary representation of some of these values would be infinitely long, in which case, although the code is unique, it may not be efficient. To make the code efficient, the binary representation has to be truncated. But if we truncate the representation, is the resulting code still unique? Finally, is the resulting code efficient? How far or how close is the average number of bits per symbol from the entropy? We will examine all these questions in the next section. Even if we show the code to be unique and efficient, the method described to this point is highly impractical. In Section 4.4.2, we will describe a more practical algorithm for generating the arithmetic code for a sequence. We will give an integer implementation of this algorithm in Section 4.4.3. 4.4.1 Uniqueness and Efficiency of the Arithmetic Code is a number in the interval . A binary code for can be obtained by taking the binary representation of this number and truncating it to bits. Recall that the binary representation of decimal numbers in the interval are obtained as the negative powers of two. The decimal equivalent of the binary number is . Thus . Example 4.4.1 Consider a source that generates letters from an alphabet of size four, with probabilitiesA binary code for this source can be generated as shown in Table 4.4. The quantity is obtained using Eq. (4.3). The binary representation of is truncated to bits to obtain the binary code. ⧫ Table 4.4. A Binary Code for a Four-Letter Alphabet \| Symbol \| FX \| \| In Binary \| \| Code \| --- --- --- \| \| 1 \| .500 \| .2500 \| .0100 \| 2 \| 01 \| \| 2 \| .750 \| .6250 \| .1010 \| 3 \| 101 \| \| 3 \| .875 \| .8125 \| .1101 \| 4 \| 1101 \| \| 4 \| 1.000 \| .9375 \| .1111 \| 4 \| 1111 \| We will show that a code obtained in this fashion is a uniquely decodable code. We first show that this code is unique, and then we will show that it is uniquely decodable. We will use to denote the truncation of to bits. Recall that while we have been using as the tag for a sequence x, any number in the interval would be a unique identifier. Therefore, to show that the code is unique, all we need to do is show that it is contained in the interval . Because we are truncating the binary representation of to obtain , is less than or equal to . More specifically, (4.11) As is strictly less than , To show that , note that From (4.3) we have (4.12) Combining (4.11) and (4.12), we have (4.13) Therefore, the code is a unique representation of . To show that this code is uniquely decodable, we will show that the code is a prefix code; that is, no codeword is a prefix of another codeword. Because a prefix code is always uniquely decodable, we automatically show that an arithmetic code is uniquely decodable, because we show that an arithmetic code is a prefix code. Given a number a in the interval with an n-bit binary representation , for any other number b to have a binary representation with as the prefix, b has to lie in the interval . (See Problem 1.) If x and y are two distinct sequences, we know that and lie in two disjoint intervals, and . Therefore, if we can show that for any sequence x the interval lies entirely within the interval , the code for one sequence cannot be the prefix for the code for another sequence. We have already shown that . Therefore, all we need to do is show that This is true because This code is prefix free; and by taking the binary representation of and truncating it to bits, we obtain a uniquely decodable code. Although the code is uniquely decodable, how efficient is it? We have shown that the number of bits required to represent with enough accuracy such that the code for different values of x is distinct is Remember that is the number of bits required to encode the entire sequence x. So, the average length of an arithmetic code for a sequence of length m is given by (4.14) (4.15) (4.16) (4.17) (4.18) Given that the average length is always greater than the entropy, the bounds on are The length per symbol, , or rate of the arithmetic code is . Therefore, the bounds on are (4.19) We have shown in Chapter 3 that for iid sources (4.20) Therefore, (4.21) By increasing the length of the sequence, we can guarantee a rate as close to the entropy as we desire. 4.4.2 Algorithm Implementation In Section 4.3.1, we developed a recursive algorithm for the boundaries of the interval containing the tag for the sequence being encoded as (4.22) (4.23) where is the value of the random variable corresponding to the nth observed symbol, is the lower limit of the tag interval at the nth iteration, and is the upper limit of the tag interval at the nth iteration. Before we can implement this algorithm, there is one major problem we have to resolve. Recall that the rationale for using numbers in the interval as a tag was that there are an infinite number of numbers in this interval. However, in practice, the number of numbers that can be uniquely represented on a machine is limited by the maximum number of digits (or bits) we can use for representing the number. Consider the values of and in Example 4.3.5. As n gets larger, these values come closer and closer together. This means that in order to represent all of the subintervals uniquely, we need increasing precision as the length of the sequence increases. In a system with finite precision, the two values are bound to converge; and we will lose all information about the sequence from the point at which the two values converged. To avoid this situation, we need to rescale the interval. However, we have to do it in a way that will preserve the information that is being transmitted. We would also like to perform the encoding incrementally—that is, to transmit portions of the code as the sequence is being observed, rather than wait until the entire sequence has been observed before transmitting the first bit. The algorithm we describe in this section takes care of the problems of synchronized rescaling and incremental encoding. As the interval becomes narrower, we have three possibilities: 1. : The interval is entirely confined to the lower half of the unit interval . 2. : The interval is entirely confined to the upper half of the unit interval . 3. : The interval straddles the midpoint of the unit interval. We will look at the third case a little later in this section. First, let us examine the first two cases. Once the interval is confined to either the upper or lower half of the unit interval, it is forever confined to that half of the unit interval. The most significant bit of the binary representation of all numbers in the interval is 0, and the most significant bit of the binary representation of all numbers in the interval is 1. Therefore, once the interval gets restricted to either the upper or lower half of the unit interval, the most significant bit of the tag is fully determined. Therefore, without waiting to see what the rest of the sequence looks like, we can indicate to the decoder whether the tag is confined to the upper or lower half of the unit interval by sending a 1 for the upper half and a 0 for the lower half. The bit that we send is also the first bit of the tag. Once the encoder and decoder know which half contains the tag, we can ignore the half of the unit interval not containing the tag and concentrate on the half containing the tag. As our arithmetic is of finite precision, we can do this best by mapping the half interval containing the tag to the full interval. The mappings required are (4.24) (4.25) As soon as we perform either of these mappings, we lose all information about the most significant bit. However, this should not matter because we have already sent that bit to the decoder. We can now continue with this process, generating another bit of the tag every time the tag interval is restricted to either half of the unit interval. This process of generating the bits of the tag without waiting to see the entire sequence is called incremental encoding. Example 4.4.2 Tag generation with scaling Let's revisit Example 4.3.5. Recall that we wish to encode the sequence . The probability model for the source is , , . Initializing to 1, and to 0, the first element of the sequence, 1, results in the following update: The interval is not confined to either the upper or lower half of the unit interval, so we proceed. The second element of the sequence is 3. This results in the update The interval is contained entirely in the upper half of the unit interval, so we send the binary code 1 and rescale: The third element, 2, results in the following update equations: The interval for the tag is , which is contained entirely in the upper half of the unit interval. We transmit a 1 and go through another rescaling:This interval is contained entirely in the lower half of the unit interval, so we send a 0 and use the mapping to rescale:The interval is still contained entirely in the lower half of the unit interval, so we send another 0 and go through another rescaling:Because the interval containing the tag remains in the lower half of the unit interval, we send another 0 and rescale one more time:Now the interval containing the tag is contained entirely in the upper half of the unit interval. Therefore, we transmit a 1 and rescale using the mapping: At each stage, we are transmitting the most significant bit that is the same in both the upper and lower limit of the tag interval. If the most significant bits in the upper and lower limit are the same, then the value of this bit will be identical to the most significant bit of the tag. Therefore, by sending the most significant bits of the upper and lower endpoint of the tag whenever they are identical, we are actually sending the binary representation of the tag. The rescaling operations can be viewed as left shifts, which make the second most significant bit the most significant bit. Continuing with the last element, the upper and lower limits of the interval containing the tag are At this point, if we wished to stop encoding, all we need to do is inform the receiver of the final status of the tag value. We can do so by sending the binary representation of any value in the final tag interval. Generally, this value is taken to be . In this particular example, it is convenient to use the value of 0.5. The binary representation of 0.5 is . Thus, we would transmit a 1 followed by as many 0s as required by the word length of the implementation being used. ⧫ Notice that the tag interval size at this stage is approximately 64 times the size it was when we were using the unmodified algorithm. Therefore, this technique solves the finite precision problem. As we shall soon see, the bits that we have been sending with each mapping constitute the tag itself, which satisfies our desire for incremental encoding. The binary sequence generated during the encoding process in the previous example is 1100011. We could simply treat this as the binary expansion of the tag. A binary number .1100011 corresponds to the decimal number 0.7734375. Looking back to Example 4.3.5, notice that this number lies within the final tag interval. Therefore, we could use this to decode the sequence. However, we would like to do incremental decoding as well as incremental encoding. This raises three questions: 1. : How do we start decoding? 2. : How do we continue decoding? 3. : How do we stop decoding? The second question is the easiest to answer. Once we have started decoding, all we have to do is mimic the encoder algorithm. That is, once we have started decoding, we know how to continue decoding. To begin the decoding process, we need to have enough information to decode the first symbol unambiguously. In order to guarantee unambiguous decoding, the number of bits received should point to an interval smaller than the smallest tag interval. Based on the smallest tag interval, we can determine how many bits we need before we start the decoding procedure. We will demonstrate this procedure in Example 4.4.4. First let's look at other aspects of decoding using the message from Example 4.4.2. Example 4.4.3 We will use a word length of 6 for this example. Note that because we are dealing with real numbers, this word length may not be sufficient for a different sequence. As in the encoder, we start with initializing to 1 and to 0. The sequence of received bits is . The first 6 bits correspond to a tag value of 0.765625, which means that the first element of the sequence is 1, resulting in the following update: The interval is not confined to either the upper or lower half of the unit interval, so we proceed. The tag 0.765625 lies in the top 18% of the interval ; therefore, the second element of the sequence is 3. Updating the tag interval we get The interval is contained entirely in the upper half of the unit interval. At the encoder, we sent the bit 1 and rescaled. At the decoder, we will shift 1 out of the receive buffer and move the next bit in to make up the 6 bits in the tag. We will also update the tag interval, resulting in while shifting a bit to give us a tag of 0.546875. When we compare this value with the tag interval, we can see that this value lies in the 80–82% range of the tag interval, so we decode the next element of the sequence as 2. We can then update the equations for the tag interval as As the tag interval is now contained entirely in the upper half of the unit interval, we rescale using to obtain We also shift out a bit from the tag and shift in the next bit. The tag is now 000110. The interval is contained entirely in the lower half of the unit interval. Therefore, we apply and shift another bit. The lower and upper limits of the tag interval becomeand the tag becomes 001100. The interval is still contained entirely in the lower half of the unit interval, so we shift out another 0 to get a tag of 011000 and go through another rescaling: Because the interval containing the tag remains in the lower half of the unit interval, we shift out another 0 from the tag to get 110000 and rescale one more time: Now the interval containing the tag is contained entirely in the upper half of the unit interval. Therefore, we shift out a 1 from the tag and rescale using the mapping: Now we compare the tag value to the tag interval to decode our final element. The tag is 100000, which corresponds to 0.5. This value lies in the first 80% of the interval, so we decode this element as 1. ⧫ If the tag interval is entirely contained in the upper or lower half of the unit interval, the scaling procedure described will prevent the interval from continually shrinking. Now we consider the case where the diminishing tag interval straddles the midpoint of the unit interval. As our trigger for rescaling, we check to see if the tag interval is contained in the interval . This will happen when is greater than 0.25 and is less than 0.75. When this happens, we double the tag interval using the following mapping: (4.26) We have used a 1 to transmit information about an mapping, and a 0 to transmit information about an mapping. How do we transfer information about an mapping to the decoder? We use a somewhat different strategy in this case. At the time of the mapping, we do not send any information to the decoder; instead, we simply record the fact that we have used the mapping at the encoder. Suppose that after this, the tag interval gets confined to the upper half of the unit interval. At this point we would use an mapping and send a 1 to the receiver. Note that the tag interval at this stage is at least twice what it would have been if we had not used the mapping. Furthermore, the upper limit of the tag interval would have been less than 0.75. Therefore, if the mapping had not taken place right before the mapping, the tag interval would have been contained entirely in the lower half of the unit interval. At this point we would have used an mapping and transmitted a 0 to the receiver. In fact, the effect of the earlier mapping can be mimicked at the decoder by following the mapping with an mapping. At the encoder, right after we send a 1 to announce the mapping, we send a 0 to help the decoder track the changes in the tag interval at the decoder. If the first rescaling after the mapping happens to be an mapping, we do exactly the opposite. That is, we follow the 0 announcing an mapping with a 1 to mimic the effect of the mapping at the encoder. What happens if we have to go through a series of mappings at the encoder? We simply keep track of the number of mappings and then send that many bits of the opposite variety after the first or mapping. If we went through three mappings at the encoder, followed by an mapping, we would transmit a 1 followed by three 0s. On the other hand, if we went through an mapping after the mappings, we would transmit a 0 followed by three 1s. Since the decoder mimics the encoder, the mappings are also applied at the decoder when the tag interval is contained in the interval . 4.4.3 Integer Implementation We have described a floating-point implementation of arithmetic coding. Let us now repeat the procedure using integer arithmetic and generate the binary code in the process. Encoder Implementation The first thing we have to do is decide on the word length to be used. Given a word length of m, we map the important values in the interval to the range of binary words. The point 0 gets mapped to 1 gets mapped to The value of 0.5 gets mapped to The update equations remain almost the same as Eqs. (4.9) and (4.10). As we are going to do integer arithmetic, we need to replace in these equations. Define as the number of times the symbol j occurs in a sequence of length Total Count. Then can be estimated by (4.27) If we now define we can write Eqs. (4.9) and (4.10) as (4.28) (4.29) where is the nth symbol to be encoded, is the largest integer less than or equal to x, and where the addition and subtraction of one is to handle the effects of the integer arithmetic. The word length m has to be large enough to accommodate all the upper and lower limits of all the subintervals, which means that there should be enough values to unambiguously represent each entry of the Cum_Count array. As the maximum number of distinct values is Total Count this means that the number of values that need to be represented is Total Count and, therefore, we need to pick m such that or However, this may not be sufficient as often the active interval, that is the interval is only a portion of the total range available. As all the subintervals need to be contained within the active interval at any given time what we need to do is determine the smallest size the active interval can be and then make sure that m is large enough to contain Total Count different values in this limited range. So what is the smallest the active interval can be? At first sight it might seem that the smallest the active interval can be is about half the maximum range because as soon as the upper limit slips below the halfway mark or the lower limit slips above the halfway mark we double the interval. However, on closer examination, we can see that this is not the case. When the upper limit is barely above the halfway mark, the lower limit is not required to be at 0. In fact, the lower limit can be just below the quarter range without any rescaling being triggered. However, the moment the lower limit goes above the quarter mark an rescaling is triggered and the interval goes through one or more redoubling. Thus the smallest the active range can be is a quarter of the total range. This means that we need to accommodate Total Count values in a quarter of the total range available, that is or Because of the way we mapped the endpoints and the halfway points of the unit interval, when both and are in either the upper half or lower half of the interval, the leading bit of and will be the same. If the leading or most significant bit (MSB) is 1, then the tag interval is contained entirely in the upper half of the interval. If the MSB is 0, then the tag interval is contained entirely in the lower half. Applying the and mappings is a simple matter. All we do is shift out the MSB and then shift in a 1 into the integer code for and a 0 into the code for . For example, suppose m was 6, was 54, and was 33. The binary representations of and are 110110 and 100001, respectively. Notice that the MSB for both endpoints is 1. Following the procedure above, we would shift out (and transmit or store) the 1, and shift in 1 for and 0 for , obtaining the new value for as 101101, or 45, and a new value for as 000010, or 2. This is equivalent to performing the mapping. We can see how the mapping would also be performed using the same operation. To see if the mapping needs to be performed, we monitor the second most significant bit of and . When the second most significant bit of is 0 and the second most significant bit of is 1, this means that the tag interval lies in the middle half of the interval. To implement the mapping, we complement the second most significant bit in and , and shift left, shifting in a 1 in and a 0 in . We also keep track of the number of mappings in Scale3. We can summarize the encoding algorithm using the following pseudocode: Initialize l and u. Get symbol. while(MSB of u and l are both equal to b or condition holds) if(MSB of u and l are both equal to b) { send b shift l to the left by 1 bit and shift 0 into LSB shift u to the left by 1 bit and shift 1 into LSB while(Scale3 > 0) { send complement of b decrement Scale3 } } if( condition holds) { shift l to the left by 1 bit and shift 0 into LSB shift u to the left by 1 bit and shift 1 into LSB complement (new) MSB of l and u increment Scale3 } To see how all this functions together, let's look at an example. Example 4.4.4 We will encode the sequence with parameters shown in Table 4.5. First we need to select the word length m. Note that and differ by only 1. Recall that the values of Cum_Count will get translated to the endpoints of the subintervals. We want to make sure that the value we select for the word length will allow enough range for it to be possible to represent the smallest difference between the endpoints of intervals. We always rescale whenever the interval gets small. In order to make sure that the endpoints of the intervals always remain distinct, we need to make sure that all values in the range from 0 to Total_Count, which is the same as , are uniquely represented in the smallest range an interval under consideration can be without triggering a rescaling. The interval is smallest without triggering a rescaling when is just below the midpoint of the interval and is at three-quarters of the interval, or when is right at the midpoint of the interval and is just below a quarter of the interval. That is, the smallest the interval can be is one-quarter of the total available range of values. Thus, m should be large enough to uniquely accommodate the set of values between 0 and Total_Count. Table 4.5. Values of Some of the Parameters for Arithmetic Coding Example \| \| \| \| --- \| Count(1)=40 \| Cum_Count(0)=0 \| Scale3 = 0 \| \| Count(2)=1 \| Cum_Count(1)=40 \| \| \| Count(3)=9 \| Cum_Count(2)=41 \| \| \| Total_Count = 50 \| Cum_Count(3)=50 \| \| For this example, this means that the total interval range has to be greater than 200. A value of satisfies this requirement. With this value of m we have (4.30) (4.31) where is the binary representation of a number. The first element of the sequence to be encoded is 1. Using Eqs. (4.28) and (4.29), (4.32) (4.33) The next element of the sequence is 3. (4.34) (4.35) The MSBs of and are both 1. Therefore, we shift this value out and send it to the decoder. All other bits are shifted left by 1 bit, giving (4.36) (4.37) Notice that while the MSBs of the limits are different, the second MSB of the upper limit is 0, while the second MSB of the lower limit is 1. This is the condition for the mapping. We complement the second MSB of both limits and shift 1 bit to the left, shifting in a 0 as the least significant bit (LSB) of and a 1 as the LSB of . This gives us (4.38) (4.39) We also increment Scale3 to a value of 1. The next element in the sequence is 2. Updating the limits, we have (4.40) (4.41) The two MSBs are identical, so we shift out a 1 and shift left by 1 bit: (4.42) (4.43) As Scale3 is 1, we transmit a 0 and decrement Scale3 to 0. The MSBs of the upper and lower limits are both 0, so we shift out and transmit 0: (4.44) (4.45) Both MSBs are again 0, so we shift out and transmit 0: (4.46) (4.47) Now both MSBs are 1, so we shift out and transmit a 1. The limits become (4.48) (4.49) Once again the MSBs are the same. This time we shift out and transmit a 0. (4.50) (4.51) Now the MSBs are different. However, the second MSB for the lower limit is 1 while the second MSB for the upper limit is 0. This is the condition for the mapping. Applying the mapping by complementing the second MSB and shifting 1 bit to the left, we get (4.52) (4.53) We also increment Scale3 to 1. The next element in the sequence to be encoded is 1. Therefore, (4.54) (4.55) The encoding continues in this fashion. To this point we have generated the binary sequence 1100010. If we wish to terminate the encoding at this point, we have to send the current status of the tag. This can be done by sending the value of the lower limit . As is 0, we will end up sending eight 0s. However, Scale3, at this point, is 1. Therefore, after we send the first 0 from the value of , we need to send a 1 before sending the remaining seven 0s. The final transmitted sequence is 1100010010000000. ⧫ Decoder Implementation Once we have the encoder implementation, the decoder implementation is easy to describe. As mentioned earlier, once we have started decoding all we have to do is mimic the encoder algorithm. Let us first describe the decoder algorithm using pseudocode and then study its implementation using Example 4.4.5. Decoder Algorithm Initialize l and u. Read the first m bits of the received bitstream into tag t. while decode symbol x. while(MSB of u and l are both equal to b or condition holds) if(MSB of u and l are both equal to b) { shift l to the left by 1 bit and shift 0 into LSB shift u to the left by 1 bit and shift 1 into LSB shift t to the left by 1 bit and read next bit from received bitstream into LSB } if( condition holds) { shift l to the left by 1 bit and shift 0 into LSB shift u to the left by 1 bit and shift 1 into LSB shift t to the left by 1 bit and read next bit from received bitstream into LSB complement (new) MSB of l, u, and t } Example 4.4.5 After encoding the sequence in Example 4.4.4, we ended up with the following binary sequence: 1100010010000000. Treating this as the received sequence and using the parameters from Table 4.5, let us decode this sequence. Using the same word length, 8, we read in the first 8 bits of the received sequence to form the tag t: We initialize the lower and upper limits asTo begin decoding, we computeand compare this value toSincewe decode the first symbol as 1. Once we have decoded a symbol, we update the lower and upper limits:or The MSBs of the limits are different, and the condition does not hold. Therefore, we continue decoding without modifying the tag value. To obtain the next symbol, we compare which is 48, against the Cum_Count array:Therefore, we decode 3 and update the limits: As the MSB of u and l are the same, we shift the MSB out and read in a 0 for the LSB of l and a 1 for the LSB of u. We mimic this action for the tag as well, shifting the MSB out and reading in the next bit from the received bitstream as the LSB: Examining l and u we can see we have an condition. Therefore, for l, u, and t, we shift the MSB out, complement the new MSB, and read in a 0 as the LSB of l, a 1 as the LSB of u, and the next bit in the received bitstream as the LSB of t. We now have To decode the next symbol, we compute Since , we decode 2. Updating the limits using this decoded symbol, we get We can see that we have quite a few bits to shift out. However, notice that the lower limit l has the same value as the tag t. Furthermore, the remaining received sequence consists entirely of 0s. Therefore, we will be performing identical operations on numbers that are the same, resulting in identical numbers. This will result in the final decoded symbol being 1. We knew this was the final symbol to be decoded because only four symbols had been encoded. In practice this information has to be conveyed to the decoder. ⧫ View chapterExplore book Read full chapter URL: Book2018, Introduction to Data Compression (Fifth Edition)Khalid Sayood Review article Spreading sequences in active sensing: A review 2015, Signal ProcessingEnrique García, ... Juan Jesús García 3.1 Real sequences If the phase ϕl appearing in expression (1) becomes 0 or π for all the elements of a sequence, then the spreading sequence is classified as real sequence. Within this category, we distinguish between binary sequences and multilevel ones. When all the coefficients Al are the same, the sequences are called uniform, but they are usually normalized, i.e. , and hence known as binary sequences. On the contrary, if the sequence is not uniform, it is classified as multilevel. 3.1.1 Binary sequences Walsh–Hadamard sequences, also known as Orthogonal Variable Spreading Factor (OVSF) sequences, are orthogonal sequences with applications in synchronous CDMA systems and are intensively used as a basic building block for the generation of Generalized Orthogonal (GO) sequences. In fact, these sequences can be generated by using the rows or columns of a Hadamard matrix, which can be obtained from any construction method or can be derived from appropriate semi-bent functions . The family size and the length of these sequences are the same and equal to the order of the Hadamard matrix, which is limited to 1, 2 and multiples of 4 . Nonetheless, in Smith et al. show a technique that exploits the spatial separation of non-adjacent cells in CDMA to increase the number of usable codes by choosing sets of Hadamard matrices with low cross-correlation. These sequences require a high synchronization, what in practice is quite difficult or even not feasible to achieve. Barker sequences are binary sequences whose aperiodic auto-correlation sidelobes absolute value is lower than one. Therefore, these sequences have very good aperiodic auto-correlation functions. In fact, the Barker sequence of length 13 has the largest merit factor known . Unfortunately, the known number of Barker sequences for a given length is very limited, and there are strong evidences that other binary Barker sequences than those represented in Table 1 do not exist [35,36]. Table 1. Barker sequences currently known. \| Length \| Barker sequence \| --- \| \| 2 \| \| \| 3 \| \| \| 4 \| \| \| 5 \| \| \| 7 \| \| \| 11 \| \| \| 13 \| \| Pseudo-noise sequences (PN), also known as Pseudo-Random sequences, are periodic sequences (i.e. designed for periodic correlation) with a random-like behavior but generated deterministically and with properties similar to AWGN. The most known Pseudo-Noise sequences are m-sequences, Gold sequences and Kasami sequences (refer to [37, Chapters 2, 10] for more details). Despite the fact of being periodic sequences, Kasami sequences still have good aperiodic correlation functions [38,39] and they are commonly used in active sensing systems with bursting transmissions [40–42]. Fig. 5a depicts the aperiodic correlation functions of Kasami sequences of length 15 bits. Chaotic sequences: For given parameter values of certain dynamical and deterministic systems, small changes in the initial conditions can generate completely uncorrelated solutions. Those solutions are known in the signal processing field as chaotic sequences, which have an infinite period and a noise-like behavior. These properties make chaotic sequences interesting for their use in active sensing systems, as theoretically it is possible to obtain an infinite number of uncorrelated sequences of infinite length. The equations that model the dynamical system and generate chaotic sequences are known as maps and the most frequently used are Logistic , Lorenz and Rössler map . These maps can also be generated by means of computational methods to improve the correlation properties of chaotic sequences . An advantage of these sequences is that, theoretically, they are not limited in the number of uncorrelated sequences, nor in their length. Nonetheless, sequences generated by chaotic maps are real sequences, which can imply the necessity of highly linear amplifiers to avoid signal distortions and a reduction of the energy efficiency of the system. For that reason, chaotic sequences are binarized to obtain a constant envelope signal . This implies a degradation of the correlation properties as can appear periodicities in the sequences. So multilevel quantization is sometimes assumed as a trade-off between the correlation properties and the implementation complexity . Another common option is the use of Direct Chaotic Communication (DCC) methods . This method consists on the direct transmission of the real sequences, i.e. without any kind of quantization nor up-conversion. They are generated and transmitted in baseband by means of analog circuits that model the dynamical system . Due to the fact that these sequences, in chaotic regimen, are noise-like sequences (they have an almost flat power spectrum), when used in DCC, chaotic sequences are passed through a band-pass filter to use only the required spectrum. The filtering process has the drawback of degrading the correlation properties of the chaotic sequences. Nonetheless, DCC method gives priority to simple implementation over performance. Complementary sets of sequences: In 1961, Golay analyzed pairs of binary sequences whose Sum of Aperiodic Auto-correlation Function (SACF) is a Kronecker delta and their Sum of Aperiodic Cross-correlation Function (SCCF) is zero for all time shifts . These sequences are currently known as Golay binary pairs. He gave non-recursive algorithms for the generation of Golay pairs of length , by interleaving, and lengths by concatenating Golay pairs of lengths L1 and L2. He also found operations of equivalence between sequence pairs and particular pairs that cannot be generated from the proposed algorithms (the so-called Golay kernels) of lengths 10 and 26 . Later in , Jauregui found by means of an exhaustive computer search that there are no more non-equivalent kernels of length 26 that the one found by Golay using a “by hand technique”. More recently, Borwein and Fergurson reported the existence of the Golay kernel of length 20 . Interestingly, Golay kernels 10 and 26 have an inner structure, demonstrated by their mathematical connection with Barker sequences of odd length and the recent decompositions of the Golay kernels of lengths 10 and 26 as the product of multilevel Hadamard matrices. In 1969 Taki and later in 1974, Turyn proposed a non-recursive algorithm for the generation of Golay pairs of lengths , where N, M and P are non-negative integers, by combining the Golay kernels. There is still unknown Golay binary pairs of different lengths from . In 1974, Tseng and Liu expanded the number of sequences in the set from two, in the case of Golay binary pairs, to , . These binary sequences of length were called Complementary Sets of Sequences (CSS) and have a SACF equal to a Kronecker delta. Mathematically it is expressed as (5) In the particular case in which , the complementary sets turn out to be a Golay binary pair. On the other hand, CSS are uncorrelated if their SCCF is zero for all shifts τ. This can be expressed as (6) Fig. 5 b shows the sum of aperiodic correlation functions of Golay binary sequence pairs of length 16 bits. Recently, in an efficient and recursive algorithm for the generation and correlation of binary CSS of lengths has been proposed as a generalization of previous recursive algorithms that decomposed Golay kernels. Nowadays, due to the ideal SACF and SCCF of Golay binary sequence pairs and CSS, they are used in a large number of applications such as MIMO radar , Non-Destructive Test (NDT) applications , low Peak-to-Average Power Ratio (PAPR) OFDM communication systems , the second digital terrestrial television broadcasting standard (DVB-T2) or as a basic building block for the generation of many other sequences with good aperiodic correlation functions. There are a large amount of terminologies used to refer classes of complementary sequences. To avoid confusions, Fig. 2 shows the basic terminology used for different subclasses of complementary sequences. For instance, in the term Complete Complementary (CC) sequences was introduced to define a set of binary sequences of length and whose SACF is a Kronecker delta and their SCCF is zero for any time shift τ. Notice that these sequences have the same correlation properties as the uncorrelated CSS. More recently, Chen introduced the term Perfect Complementary (PC) sequences that are uncorrelated CSS generated from an algebraic approach that can take into account the real scenario in the sequence design, as multipath effect or MAI . Additionally, some authors [68,69] use the term orthogonal CSS to designate those sequences whose SCCF holds Eq. (6). This paper uses the notation of instead, and reserves the term orthogonal CSS to those complementary sets of sequences whose SCCF is only zero at τ=0 (i.e. the sum of dot products is zero). This is an important clarification as it is known that the maximum number of uncorrelated complementary sets, , is equal to the number of sequences in the set, while for orthogonal CSS, the number of orthogonal sets is limited by the expression . On the other hand, Sivaswamy introduced a set of composite signals generated from complementary sequences that were called subcomplementary sets of sequences [11,71]. The sum of aperiodic correlation functions of these sequences has a zone with low correlation sidelobes and a good ambiguity function. Later, Popović and Budišin generalized this generation algorithm, calling the new sequences as generalized subcomplementary sequences . In the last years these types of sequences are known as Low Zero Correlation Zone (LCZ) sequences instead of being considered as a subclass of complementary sequences . Therefore both kinds of sequences, subcomplementary and generalized subcomplementary, are not strictly complementary sequences and they should be considered as types of Generalized Orthogonal (GO) sequences. As stated previously, the ideal spreading sequence should have a Kronecker delta as auto-correlation and a cross-correlation equal to zero for any time shift τ. Nevertheless, these sequences do not exist [74–76], so when ISI is mitigated by reducing the auto-correlation sidelobes, MAI increases worsening the cross-correlation function and vice versa. Traditionally, sequences with good (but not ideal) aperiodic correlation properties as Kasami sequences have been used. Although CSS have ideal sums of aperiodic correlation functions, they increase the complexity of the transmission scheme, as each user has to transmit a set of sequences. In the last years, the interest for GO sequences has boosted. These sequences have a Zero Correlation Zone (ZCZ), WZCZ in the vicinity of the correlation time shift, τ=0, or equivalently an Interference Free Window (IFW), where , i.e. the double-sided ZCZ, due to symmetric properties of the correlation functions (refer to Fig. 5c, d or e). Generalized Quasi-Orthogonal (GQO) sequences (also known as LCZ sequences), instead of having a ZCZ, have a LCZ next to the time shift τ=0 where the amplitude of the correlation sidelobes is limited to a certain value. GO sequences are sub-optimum solutions because of the unfeasibility to generate unitary sequences with ideal aperiodic correlation functions. In fact, in a lower bound on sidelobes correlation of LCZ sequences is obtained as (7) where is the energy of a sequence of length L in a set of K codes, and WLCZ is the zone whose maximum sidelobe is less than δ. Fig. 3 depicts a classification of GO sequences for aperiodic correlation functions. These sequences are reviewed in the next sections. Z-complementary sequences can be seen as a generalization of Golay binary sequence pairs that cope with their limitation in length and number of uncorrelated pairs (also known as mates). This generalization achieves binary sequences pairs of many more lengths, , at the expense of introducing a ZCZ in their sum of aperiodic correlation functions. The same generation algorithms and equivalence rules as those used for Golay binary sequence pairs can be used for the generation of Z-complementary sequences. Similar to Golay binary pairs, there are Z-complementary pairs that cannot be generated from the Golay rules , the so-called kernels. Fan et al. introduced a list of kernels up to length with maximum ZCZ, . Those kernels include Golay kernels of lengths , and with . Following the generation rules used for complementary sets , Fan et al. conjectured that for certain values of , Z-complementary sequences exist for all lengths. Later, Li et al. demonstrated in the existence of Z-complementary pairs of lengths for , for , for and for . Additionally, they derived the ZCZ size upper bound for binary Z-complementary pairs of length odd that is equal to (8) and for even lengths different from , the ZCZ size upper bound is (9) For the case of Z-complementary sets, the number of sets TZC is bounded by the expression (10) where represents the number of sequences in the set and represents the largest integer less than or equal to x. Notice that if , Z-complementary sequences result in uncorrelated CSS and the set size upper bound becomes equal to . Additionally, if , the Z-complementary sequences become orthogonal complementary sequences. Fig. 5c depicts the aperiodic correlation functions of two Z-complementary pairs of length . Recently, Liu et al. showed in the necessary conditions to construct optimal Z-complementary sequences of odd length in the sense that they achieve the ZCZ upper bound and the magnitude of correlation sidelobes reach its lower bound. Later, they proposed in a systematic construction of binary Z-complementary sequences of length with ZCZ size equal to , where k is a non-negative integer. Loosely Synchronized (LS) sequences were proposed as a candidate for the 3G wireless communications standard in 2000 to cope with ISI and MAI in Quasi-Synchronous CDMA (QS-CDMA) systems [82,83]. These unitary and ternary sequences over the alphabet {−1,0,+1} have ideal aperiodic correlation functions in a window IFW, placed in the vicinity of the correlation time shift τ=0. There are two generation algorithms for LS sequences, namely: 1. : Generation of LS sequences from Golay binary pairs: This algorithm generates a set of sequences of length and it is based on the concatenation of Golay binary sequence pairs following a code tree and the insertion of a chain of zeros of length in the middle of the concatenated sequence . The length of this chain is equal to the ZCZ length if and only if . The aperiodic correlation functions of LS sequences of length generated with this algorithm are depicted in Fig. 5d. 2. : Generation of LS sequences from CSS: In Zhang et al. propose a generation algorithm from CC sequences of length . This algorithm is a generalization of the previous one from Golay binary pairs of sequences and it requires α iterations to generate LS sequences of length . Generalized Loosely Synchronized (GLS) sequences were proposed by Tang and Mow as a generalization of the LS generation algorithm from Golay binary pairs of sequences. GLS sequences set is divided into two sub-groups, whose intergroup cross-correlation functions have favorable properties, while maintaining the sequence and ZCZ length of the original LS sequences. The generation of GLS sequences clearly shows the problem of generating large families of GO sequences for a given length and with the largest theoretical ZCZ length. Tang and Mow found a generation algorithm of GLS sequences that almost reach the theoretical maximum number of sequences that can be generated for a given sequence and ZCZ length. sub-groups of GLS sequences of length are generated from sets of Hadamard matrices of order , which are constructed from sequences with good cross-correlation properties, as Kerdock codes (used when a is even, ) or Gold sequences (used when a is odd, ). The aperiodic auto-correlation and cross-correlation properties of GLS sequences of the same sub-group are the same as those of LS sequences. Nonetheless, for the cross-correlation of GLS sequences from different sub-groups (i.e. generated from a different Hadamard matrix) appears an interference at the time shift τ=0 with a maximum value lower than when a is even and lower than when a is odd. This is an important issue, as the cross-correlation interference at τ=0 could be erroneously detected as an auto-correlation peak. Therefore, there exists a trade-off between ZCZ length, sequence length and family size; this problem would be more tractable if a set of non-equivalent Hadamard matrices exists whose rows and/or columns are orthogonal with the rows and/or columns of any other non-equivalent Hadamard matrix of the set. Fig. 4 shows a block diagram of the structure of a family of spreading sequences. Notice that for LS sequences, both the number of sub-groups and the number of sets (T) are equal to one. Generalized Pairwise Complementary (GPC) sequences are binary pairs of sequences, characterized by their controlled interferences in their aperiodic correlation functions . In contrast to LS sequences, GPC sequences do not require the insertion of a chain of zeros to achieve a ZCZ so they have a constant envelope, i.e. they are energy efficient. Instead of that, they work in pairs so the interferences within the ZCZ are cancelled by carrying out the sum of aperiodic correlation functions. The family of pairs of GPC sequences of length are generated from CC sequences of length and Generalized Even Shift Orthogonal Sequences (GESO), which confers to the sequence sums of aperiodic correlation functions with sparse interferences, at known locations. GPC sequences are divided into two sub-groups; the SACF has an IFW of length , while the SCCF of GPC sequences has bi-valued properties: the intra-group SCCF has an IFW of length , while the inter-group SCCF is zero for all shifts τ. Fig. 5e shows the sum of intra-group aperiodic correlation function of GPC sequences of length . Generalized Pairwise Z-complementary (GPZ) sequences are a variation of GPC sequences and they are derived from Z-complementary sequences . They were proposed with the objective of increasing the number of available sequences. As the number of CC mates is lower than the number of Z-complementary sets for a given sequence length L (refer to Eq. (10)), GPZ sequences, generated from Z-complementary sequences, have a larger number of pairs than the one of GPC sequences for a given length. Furthermore, the lengths of complementary sequence pairs are theoretically limited to , whereas the lengths of Z-complementary sequences have less restrictions ; this implies that GPZ sequences are more versatile than GPC sequences. Apart from the use of Z-complementary sequences instead of CC sequences for the generation of GPZ sequences, the later steps are equal to the GPC algorithm. In fact, the sum of aperiodic correlation functions maintains the bi-valued correlation property, with a reduction on the IFW length. The GPZ set is defined as , where is the number of GPZ pairs of sequences and G is the Walsh–Hadamard expansion factor ; is the length of GPZ sequences and the IFW is equal to . Inter-Group Complementary (IGC) sequences are a generalization of GPC sequences to increase the number of groups from two for the case of GPC sequences to . Given binary PC sets of sequences of length , and a Hadamard matrix of order G, the generation algorithm of IGC sequences generates sets of IGC sequences in each set; and the entire set size is divided into complementary groups. In the same form as GPC and GPZ sequences, the sums of aperiodic correlation functions of IGC sequences have also bi-valued properties: the sums of aperiodic auto-correlation functions of IGC sequences from the same sub-group have a ZCZ whereas the sums of aperiodic cross-correlation functions of IGC sequences of different sub-groups are zero for any time shift τ of the correlation. Table 2 shows a summary of the main properties of the previous GO sequences. Table 2. Summary of the main properties of some GO sequences. \| Sequence \| Unitary \| L-length \| IFW length \| K-sequences per set \| T-number of sets \| Sub-groups \| --- --- --- \| LS \| Yes \| \| \| G \| 1 \| 1 \| \| GLS \| Yes \| \| Non-uniform \| 1 \| G \| 2 \| \| GPC \| No \| \| \| 2 \| \| 2 \| \| GPZ \| No \| \| \| 2 \| \| 2 \| \| IGC \| No \| \| \| \| \| \| ZCZ sequences are another group of GO sequences that includes a large number of algorithms that generate families of binary [92–94], ternary (also known as T-ZCZ) or even multilevel sequences with a Zero Correlation Zone in their aperiodic correlation functions . These algorithms, derived from CSS for aperiodic ZCZ sequences, generate families of sequences with different number of sequences, lengths and ZCZ sizes. In order to determine their goodness, they are usually compared with the theoretical bound in and defined as (11) Although this theoretical bound has not been satisfied with equality yet . 3.1.2 Multilevel sequences Theoretically, any of the previous sequences can be generalized to the multilevel alphabet [98–100]. Some of the previous sequences have interesting properties when transformed into multilevel alphabet . For instance, complementary pairs of sequences do not have limitations in the multilevel alphabet . Nonetheless, these non-uniform sequences are more difficult to use in practice due to the requirement of highly linear power amplifiers. View article Read full article URL: Journal2015, Signal ProcessingEnrique García, ... Juan Jesús García Chapter Arithmetic Coding 2012, Introduction to Data Compression (Fourth Edition)Khalid Sayood 4.6 Binary Arithmetic Coding In many applications the alphabet itself is binary. While at first sight this may seem odd (Why would one need an encoder to encode a binary sequence?), a little bit of thought makes the reasoning apparent. Consider a binary source that puts out one symbol with probability 0.125 and the other with probability 0.875. If we directly encode one of the letters as a 0 and the other as a 1, we would get a rate of 1 bit/symbol. However, the entropy of the source is 0.543; so by directly encoding the output of the source as a 0 or a 1, we are encoding at almost twice the rate we need to. In order to achieve the entropy, we need to encode the more probable symbol at a fractional bit rate, which is exactly what arithmetic coding allows us to do; hence the popularity of the arithmetic code for encoding binary sources with highly skewed probabilities. The source can be binary by its nature, such as bilevel documents; or the binary input may be the binary representation of nonbinary data, as is the case for the Context Adaptive Binary Arithmetic Coder (CABAC) in the H.264 video coding standard. As for arithmetic coding in general, the binary arithmetic coder also requires a probability model. However, because there are only two letters in the alphabet, the probability model consists of a single number, namely the probability of one of the symbols; the probability of the other symbol is simply one minus the specified probability. Because we only need a single number to represent the probability model, it is easy to use multiple arithmetic codes to encode a source where different models represent different contexts. This results in a much more accurate modeling of the source than would have been possible with a single model, which in turn leads to better compression. Example 4.6.1 Consider a binary sequence generated by scanning a document where the white pixels are represented by a 0 and the black pixels by a 1. Clearly the probability of a pixel being white or black depends heavily on whether the neighboring pixel is white or black. For this example we will use a first order context, so we have two sets of Count and Cum_Count tables: \| \| \| \| \| --- --- \| \| \| \| \| \| \| \| \| \| \| where the superscript indicates whether the previous pixel was a 0 or a 1. We will always assume that for each row there is an imaginary white pixel to the left of the leftmost pixel. Therefore, we will always begin our encoding using the array. After encoding the first pixel all other pixels will be encoded using the Cum_Count array corresponding to the previous pixel. Assume we wish to encode the sequenceNotice that the marginal probabilities of 1 and 0 are 1/2 so if we used an arithmetic coder that did not take the context into account we would end up with a 1 bit/pixel encoding. Using the conditional probability tables we hope to do better. We will use a word length m of six: (56) (57) The first bit to be encoded is 0:The second bit to be encoded is also 0:The third bit to be encoded is also 0:The MSBs of and are both 0. Therefore, we shift this value out and send it to the decoder. All other bits are shifted to the left and we getEncoding another 0, we obtainContinuing with the next two 0s we getThe MSBs of and are both 0, therefore, we shift out a 0 from both and :The next bit to be encoded is a 1. However the bit prior to it was a 0, therefore we use the tables:The first two MSBs of the upper and lower limit are the same, so we shift out two bits, 11, from both the upper and the lower limit:The next bit to be encoded is also a 1. However, as the bit prior to that was a 1 we use the tables:Encoding the next two 1s we obtainThe MSBs of the upper and lower limits are both equal to 1 so we shift out 1:Encoding the next two 1s we getWe have encoded twelve bits, and the sequence of bits generated by the encoder until this point is 00111. In other words we have a coding rate of 5/12 bits/pixel, which is less than half the coding rate we would have achieved had we used a single Cum_Count table. At this point if we wish to terminate the encoding we would have to incur the overhead of transmitting the bits in the lower limit. The six bits of the lower limit would be a significant overhead for this toy sequence. However, in practice, when the input sequence is much longer than twelve, the six-bit overhead would be negligible. We leave the decoding of this sequence as an exercise for the reader. ♦ Furthermore, the simple nature of the coder allows for approximations that result in simple and fast implementations. We will look at three applications of the binary coder including the QM coder used in the JBIG standard for encoding bilevel images and the M (or modulo) coder, which is a part of the coder CABAC used in the H.264 video coding standard. Before we describe the particular implementations, let us take a general view of binary arithmetic coding. In our description of arithmetic coding, we updated the tag interval by updating the endpoints of the interval, and . We could just as well have kept track of one endpoint and the size of the interval. This is the approach adopted in many of the binary coders, which track the lower end of the tag interval and the size of the interval , where (58) The tag for a sequence is the binary representation of . We can obtain the update equation for by subtracting Equation (9) from Equation (10) and substituting for . (59) (60) Substituting for in Equation (9), we get the update equation for : (61) Instead of dealing directly with the 0s and 1s put out by the source, many of the binary coders map them into a More Probable Symbol (MPS) and Less Probable Symbol (LPS). If 0 represents black pixels and 1 represents white pixels, then in a mostly black image, 0 will be the MPS, whereas in an image with mostly white regions 1 will be the MPS. Denoting the probability of occurrence of the LPS for the context C by and mapping the MPS to the lower subinterval, the occurrence of an MPS symbol results in the update equations (62) (63) while the occurrence of an LPS symbol results in the update equations (64) (65) 4.6.1 The QM Coder Until this point, the binary coder looks very much like the arithmetic coder described earlier in this chapter. To make the implementation simpler and computationally more efficient, the Joint Bi-level Image Experts Group (JBIG) recommended several deviations from the standard arithmetic coding algorithm for the version of the arithmetic coder used in the JBIG algorithm for compression of bi-level images. The update equations involve multiplications, which are expensive in both hardware and software. In the QM coder, the multiplications are avoided by assuming that has a value close to 1, and multiplication with can be approximated by multiplication with 1. Therefore, the update equations become (66) (67) (68) (69) In order not to violate the assumption on whenever the value of drops below , the QM coder goes through a series of rescalings until the value of is greater than or equal to . The rescalings take the form of repeated doubling, which corresponds to a left shift in the binary representation of . To keep all parameters in sync, the same scaling is also applied to . The bits shifted out of the buffer containing the value of make up the encoder output. Looking at the update equations for the QM coder, we can see that a rescaling will occur every time an LPS occurs. Occurrence of an MPS may or may not result in a rescale, depending on the value of . The probability of the LPS for context C is updated each time a rescaling takes place and the context C is active. An ordered list of values for is listed in a table. Every time a rescaling occurs, the value of is changed to the next lower or next higher value in the table, depending on whether the rescaling was caused by the occurrence of an LPS or an MPS. In a nonstationary situation, the symbol assigned to LPS may actually occur more often than the symbol assigned to MPS. This condition is detected when . In this situation, the assignments are reversed; the symbol assigned the LPS label is assigned the MPS label and vice versa. The test is conducted every time a rescaling takes place. The decoder for the QM coder operates in much the same way as the decoder described in this chapter, mimicking the encoder operation. 4.6.2 The MQ Coder The MQ coder is a variant of the QM coder. Unlike the QM coder the MQ coder assigns the lower interval to the LPS and the upper interval to the MPS. The update equations in this case without any approximations would become (70) (71) (72) (73) However, as in the case of the QM coder, we wish to avoid multiplication; therefore, with the same assumption that has a value close to one we modify the update equations to (74) (75) (76) (77) The adaptation in the MQ coder is modeled by a state machine. In practice, the A and l registers are assigned 16 bits of precision. When the value of A falls below 0x8000, it is left shifted until the value reaches or exceeds 0x8000. The same operation is performed on the register where l is stored, and the bits shifted out of the l register become the output codewords of the arithmetic coder. 4.6.3 The M Coder The M coder is another variant of the QM coder in which the multiply operation is replaced with a table lookup. To better understand the approximations used by the M coder, let us rewrite the update equations without approximation. The occurrence of an MPS symbol results in the update equations (78) (79) while the occurrence of an LPS symbol results in the update equations (80) (81) Notice that the only multiplications are between the estimate of the range and the probability of the LPS . The M coder gets rid of the costly multiplication operation by allowing the range and the probability to take on only a specified number of values and then replacing the multiplication by a table lookup indexed by the quantized values of the range and the LPS probability . Given a minimum value and a maximum value for the range A, the range is restricted to four values: where The LPS probability can take on one of 64 possible values where (82) where In the update equation, instead of multiplying the range and the lower limit, the range is mapped to the closest of the four quantized ranges, the corresponding index k along with the index m of the probability of the LPS are used as pointers into a lookup table, and the product is read from the table. In order to make the coder adaptive, all we need to do is update the value of the LPS probability as we see more data. If we see more occurrences of MPS, we should decrease the LPS probability. If we encounter more occurrences of LPS, we should increase the LPS probability. The M coder needs to do this while keeping the LPS probability restricted to the 64 allowed probabilities generated by Equation (82). This can be done by simply incrementing or decrementing the index m of . When an MPS is encountered, we increment the index and when an LPS is encountered, we decrement the index. The index is incremented by one each time an MPS is encountered until it reaches the maximum allowed value where it remains. When an LPS is encountered, the index is decremented by a variable amount until the index reaches 0. At this point the LPS probability is one half and further occurrence of LPS indicates that this symbol is no longer the least probable and therefore, the value of MPS is changed. In other words, the MPS and LPS symbols can be swapped. The M coder forms the core of the CABAC coder, which is part of the H.264 standard for video compression. View chapterExplore book Read full chapter URL: Book2012, Introduction to Data Compression (Fourth Edition)Khalid Sayood Chapter Flip-flops and flip-flop based circuits 1998, Introduction to Digital ElectronicsJohn Crowe, Barrie Hayes-Gill Sequence generator If a binary pattern is fed into a shift register it can then be output serially to produce a known binary sequence. Moreover, if the output is also fed back into the input (to form a SISO connected to itself) the same binary sequence can be generated indefinitely. When a SISO shift register is connected to itself this is usually referred to as a re-entrant shift register, dynamic shift register, ring buffer or circulating memory. Variations on this type of circuit are used for data encryption, error checking and for holding data during digital signal processing. View chapterExplore book Read full chapter URL: Book1998, Introduction to Digital ElectronicsJohn Crowe, Barrie Hayes-Gill Chapter Mathematical Preliminaries for Lossless Compression 2018, Introduction to Data Compression (Fifth Edition)Khalid Sayood 2.4 Coding When we talk about coding in this chapter (and through most of this book), we mean the assignment of binary sequences to elements of an alphabet. The set of binary sequences is called a code, and the individual members of the set are called codewords. An alphabet is a collection of symbols called letters. For example, the alphabet used in writing most books consists of the 26 lowercase letters, 26 uppercase letters, and a variety of punctuation marks. In the terminology used in this book, a comma is a letter. The ASCII code for the letter a is 1000011, the letter A is coded as 1000001, and the letter “,” is coded as 0011010. Notice that the ASCII code uses the same number of bits to represent each symbol. Such a code is called a fixed-length code. If we want to reduce the number of bits required to represent different messages, we need to use a different number of bits to represent different symbols. If we use fewer bits to represent symbols that occur more often, on the average we would use fewer bits per symbol. The average number of bits per symbol is often called the rate of the code. The idea of using fewer bits to represent symbols that occur more often is the same idea that is used in Morse code: the codewords for letters that occur more frequently are shorter than for letters that occur less frequently. For example, the codeword for E is ⋅, while the codeword for Z is − − ⋅ ⋅ . 2.4.1 Uniquely Decodable Codes The average length of the code is not the only important point in designing a “good” code. Consider the following example adapted from . Suppose our source alphabet consists of four letters , , , and , with probabilities , , and . The entropy for this source is 1.75 bits/symbol. Consider the codes for this source in Table 2.2. Table 2.2. Four Different Codes for a Four-Letter Alphabet \| Letters \| Probability \| Code 1 \| Code 2 \| Code 3 \| Code 4 \| --- --- --- \| \| a1 \| 0.5 \| 0 \| 0 \| 0 \| 0 \| \| a2 \| 0.25 \| 0 \| 1 \| 10 \| 01 \| \| a3 \| 0.125 \| 1 \| 00 \| 110 \| 011 \| \| a4 \| 0.125 \| 10 \| 11 \| 111 \| 0111 \| \| Average length \| \| 1.125 \| 1.25 \| 1.75 \| 1.875 \| The average length l for each code is given by where is the number of bits in the codeword for letter and the average length is given in bits/symbol. Based on the average length, Code 1 appears to be the best code. However, to be useful, a code should have the ability to transfer information in an unambiguous manner. This is obviously not the case with Code 1. Both and have been assigned the codeword 0. When a 0 is received, there is no way to know whether an was transmitted or an . We would like each symbol to be assigned a unique codeword. At first glance, Code 2 does not seem to have the problem of ambiguity; each symbol is assigned a distinct codeword. However, suppose we want to encode the sequence . Using Code 2, we would encode this with the binary string 100. However, when the string 100 is received at the decoder, there are several ways in which the decoder can decode this string. The string 100 can be decoded as , or as . This means that once a sequence is encoded with Code 2, the original sequence cannot be recovered with certainty. In general, this is not a desirable property for a code. We would like unique decodability from the code; that is, any given sequence of codewords can be decoded in one, and only one, way. We have already seen that Code 1 and Code 2 are not uniquely decodable. How about Code 3? Notice that the first three codewords all end in a 0. In fact, a 0 always denotes the termination of a codeword. The final codeword contains no 0s and is 3 bits long. Because all other codewords have fewer than three 1s and terminate in a 0, the only way we can get three 1s in a row is as a code for . The decoding rule is simple. Accumulate bits until you get a 0 or until you have three 1s. There is no ambiguity in this rule, and it is reasonably easy to see that this code is uniquely decodable. With Code 4 we have an even simpler condition. Each codeword starts with a 0, and the only time we see a 0 is in the beginning of a codeword. Therefore, the decoding rule is to accumulate bits until you see a 0. The bit before the 0 is the last bit of the previous codeword. There is a slight difference between Code 3 and Code 4. In the case of Code 3, the decoder knows the moment a code is complete. In Code 4, we have to wait till the beginning of the next codeword before we know that the current codeword is complete. Because of this property, Code 3 is called an instantaneous code. Although Code 4 is not an instantaneous code, it is almost that. While this property of instantaneous or near-instantaneous decoding is a nice property to have, it is not a requirement for unique decodability. Consider the code shown in Table 2.3. Let's decode the string 011111111111111111. In this string, the first codeword is either 0 corresponding to or 01 corresponding to . We cannot tell which one until we have decoded the whole string. Starting with the assumption that the first codeword corresponds to , the next eight pairs of bits are decoded as . However, after decoding eight s, we are left with a single (dangling) 1 that does not correspond to any codeword. On the other hand, if we assume the first codeword corresponds to , we can decode the next 16 bits as a sequence of eight s; and we do not have any bits left over. The string can be uniquely decoded. In fact, Code 5, while it is certainly not instantaneous, is uniquely decodable. Table 2.3. Code 5. A Code Which Is Uniquely Decodable but Not Instantaneous \| Letter \| Codeword \| --- \| \| a1 \| 0 \| \| a2 \| 01 \| \| a3 \| 11 \| We have been looking at small codes with four letters or less. Even with these, it is not immediately evident whether the code is uniquely decodable or not. In deciding whether larger codes are uniquely decodable, a systematic procedure would be useful. Actually, we should include a caveat with that last statement. Later in this chapter we will include a class of variable-length codes that are always uniquely decodable, so a test for unique decodability may not be that necessary. You might wish to skip the following discussion for now, and come back to it when you find it necessary. Before we describe the procedure for deciding whether a code is uniquely decodable, let's take another look at our last example. We found that we had an incorrect decoding because we were left with a binary string (1) that was not a codeword. If this had not happened, we would have had two valid decodings. For example, consider the code shown in Table 2.4. Let's encode the sequence followed by eight s using this code. The coded sequence is 01010101010101010. The first bit is the codeword for . However, we can also decode it as the first bit of the codeword for . If we use this (incorrect) decoding, we decode the next seven pairs of bits as the codewords for . After decoding seven s, we are left with a single 0 that we decode as . Thus, the incorrect decoding is also a valid decoding, and this code is not uniquely decodable. Table 2.4. Code 6. A Code Which Is Not Uniquely Decodable \| Letter \| Codeword \| --- \| \| a1 \| 0 \| \| a2 \| 01 \| \| a3 \| 10 \| A Test for Unique Decodability ★ In the previous examples, in the case of the uniquely decodable code, the binary string left over after we had gone through an incorrect decoding was not a codeword. In the case of the code that was not uniquely decodable, in the incorrect decoding what was left was a valid codeword. Based on whether the dangling suffix is a codeword or not, we get the following test [7,8]. We start with some definitions. Suppose we have two binary codewords a and b, where a is k bits long, b is n bits long, and . If the first k bits of b are identical to a, then a is called a prefix of b. The last bits of b are called the dangling suffix . For example, if and , then a is a prefix of b and the dangling suffix is 11. Construct a list of all the codewords. Examine all pairs of codewords to see if any codeword is a prefix of another codeword. Whenever you find such a pair, add the dangling suffix to the list unless you have added the same dangling suffix to the list in a previous iteration. Now repeat the procedure using this larger list. Continue in this fashion until one of the following two things happens: 1. : You get a dangling suffix that is a codeword. 2. : There are no more unique dangling suffixes. If you get the first outcome, the code is not uniquely decodable. However, if you get the second outcome, the code is uniquely decodable. Let's see how this procedure works with a couple of examples. Example 2.4.1 Consider Code 5. First list the codewords The codeword 0 is a prefix for the codeword 01. The dangling suffix is 1. There are no other pairs for which one element of the pair is the prefix of the other. Let us augment the codeword list with the dangling suffix.Comparing the elements of this list, we find 0 is a prefix of 01 with a dangling suffix of 1. But we have already included 1 in our list. Also, 1 is a prefix of 11. This gives us a dangling suffix of 1, which is already in the list. There are no other pairs that would generate a dangling suffix, so we cannot augment the list any further. Therefore, Code 5 is uniquely decodable. ⧫ Example 2.4.2 Consider Code 6. First list the codewords The codeword 0 is a prefix for the codeword 01. The dangling suffix is 1. There are no other pairs for which one element of the pair is the prefix of the other. Augmenting the codeword list with 1, we obtain the listIn this list, 1 is a prefix for 10. The dangling suffix for this pair is 0, which is the codeword for . Therefore, Code 6 is not uniquely decodable. ⧫ 2.4.2 Prefix Codes The test for unique decodability requires examining the dangling suffixes initially generated by codeword pairs in which one codeword is the prefix of the other. If the dangling suffix is itself a codeword, then the code is not uniquely decodable. One type of code in which we will never face the possibility of a dangling suffix being a codeword is a code in which no codeword is a prefix of the other. In this case, the set of dangling suffixes is the null set, and we do not have to worry about finding a dangling suffix that is identical to a codeword. A code in which no codeword is a prefix to another codeword is called a prefix code. A simple way to check if a code is a prefix code is to draw the rooted binary tree corresponding to the code. Draw a tree that starts from a single node (the root node) and has a maximum of two possible branches at each node. One of these branches corresponds to a 1 and the other branch corresponds to a 0. In this book, we will adopt the convention that when we draw a tree with the root node at the top, the left branch corresponds to a 0 and the right branch corresponds to a 1. Using this convention, we can draw the binary tree for Code 2, Code 3, and Code 4 as shown in Fig. 2.5. Note that apart from the root node, the trees have two kinds of nodes—nodes that give rise to other nodes and nodes that do not. The first kind of nodes are called internal nodes, and the second kind are called external nodes or leaves. In a prefix code, the codewords are only associated with the external nodes. A code that is not a prefix code, such as Code 4, will have codewords associated with internal nodes. The code for any symbol can be obtained by traversing the tree from the root to the external node corresponding to that symbol. Each branch on the way contributes a bit to the codeword: a 0 for each left branch and a 1 for each right branch. It is nice to have a class of codes, whose members are so clearly uniquely decodable. However, are we losing something if we restrict ourselves to prefix codes? Could it be that if we do not restrict ourselves to prefix codes, we can find shorter codes? Fortunately for us the answer is no. For any nonprefix uniquely decodable code, we can always find a prefix code with the same codeword lengths. We prove this in the next section. 2.4.3 The Kraft–McMillan Inequality ★ The particular result we look at in this section consists of two parts. The first part provides a necessary condition on the codeword lengths of uniquely decodable codes. The second part shows that we can always find a prefix code that satisfies this necessary condition. Therefore, if we have a uniquely decodable code that is not a prefix code, we can always find a prefix code with the same codeword lengths. Theorem 1 Let be a code with N codewords with lengths . If is uniquely decodable, then This inequality is known as the Kraft–McMillan inequality. Proof The proof works by looking at the nth power of . If is greater than one, then should grow exponentially with n. If it does not grow exponentially with n, then this is proof that . Let n be an arbitrary integer. Then (2.17) (2.18) The exponent is simply the length of n codewords from the code . The smallest value that this exponent can take is greater than or equal to n, which would be the case if all codewords were 1 bit long. Ifthen the largest value that the exponent can take is less than or equal to nl. Therefore, we can write this summation aswhere is the number of combinations of n codewords that have a combined length of k. Let's take a look at the size of this coefficient. The number of possible distinct binary sequences of length k is . If this code is uniquely decodable, then each sequence can represent one and only one sequence of codewords. Therefore, the number of possible combinations of codewords whose combined length is k cannot be greater than . In other words,This means that (2.19) But if is greater than one, it will grow exponentially with n, while can only grow linearly. So if is greater than one, we can always find an n large enough that the inequality (2.19) is violated. Therefore, for a uniquely decodable code , is less than or equal to one. □ This part of the Kraft–McMillan inequality provides a necessary condition for uniquely decodable codes. That is, if a code is uniquely decodable, the codeword lengths have to satisfy the inequality. The second part of this result is that if we have a set of codeword lengths that satisfy the inequality, we can always find a prefix code with those codeword lengths. The proof of this assertion presented here is adapted from . Theorem 2 Given a set of integers that satisfy the inequality we can always find a prefix code with codeword lengths . Proof We will prove this assertion by developing a procedure for constructing a prefix code with codeword lengths that satisfy the given inequality. We will assume, without loss of generality, that . Before we build the code let us briefly look at binary trees. Consider the full binary tree of depth four shown in Fig. 2.6. The number of leaf nodes on this tree is . In fact the number of leaf nodes in a full binary tree of depth m is . We will construct our code by assigning vertices at depth as codewords. In order for this code to be a prefix code when we assign a code to a vertex within the tree we cannot assign a codeword to any leaves belonging to the sub-tree rooted at that node. In effect, we have to prune the subtree rooted at that vertex. For example if we assign a codeword to the vertex v indicated in the figure we have to remove the subtree shown in the dashed circle. In the figure the vertex v is at depth two. The removal of the corresponding subtree results in the removal of four leaf nodes. In general we can see that in a full binary tree of depth m, a vertex at depth k is the root of a subtree with leaves. Given the set of lengths , define Construct a full binary tree of depth l. This tree has leaves, and hence the possibility of having codewords of length l. Let's assign a codeword to a vertex at depth . The path from the root node of the tree to this vertex will be a binary code of length . As we mentioned earlier in order for this codeword to be part of a prefix code we need to prune the subtree rooted at node . This will result in a loss of leaf nodes from the full binary tree of depth l. Assign the next codeword to a vertex at a depth of and prune the subtree rooted at . Continuing in this fashion we will obtain a prefix code with lengths as long as we don't use up more than leaf nodes. As a codeword of length results in the loss of leaf nodes from the full tree, the number of leaf nodes needed to build a code with codeword lengths is given by , butwhere the last inequality is because of our assumption that the lengths satisfy the Kraft–McMillan inequality. Thus, given a set of codeword lengths satisfying the Kraft–McMillan inequality we can always construct a prefix code with codewords having those lengths. □ Therefore, if we have a uniquely decodable code, the codeword lengths have to satisfy the Kraft–McMillan inequality. And, given codeword lengths that satisfy the Kraft–McMillan inequality, we can always find a prefix code with those codeword lengths. Thus, by restricting ourselves to prefix codes, we are not in danger of overlooking nonprefix uniquely decodable codes that have a shorter average length. View chapterExplore book Read full chapter URL: Book2018, Introduction to Data Compression (Fifth Edition)Khalid Sayood Chapter Radar 2003, Encyclopedia of Physical Science and Technology (Third Edition)Nadav Levanon VII.C Phase and Frequency Coding Many pulse compression signals are based on splitting the transmitted pulse into M bits and phase or frequency modulating the bits according to a coded sequence. Amplitude modulation is rarely used because it cannot be easily implemented in high-power transmitters. The simplest phase modulation switches between two phases, preferably 0° and 180°, which can be referred to as + and −. Such a two-valued sequence is called a binary sequence. An important family of pulse compression binary sequences are the Barker codes. The largest Barker sequence is 13 bits long: The normalized envelope of its matched filter response is plotted in Fig. 12. It is characterized by sidelobes which do not exceed 1/M. The pulse compression is obviously M. Frequency coding can also be used to achieve pulse compression. The linear-FM modulation can be approximated by M discrete frequency steps. This code is called quantized linear FM. If the linear phase during each one of the M quantized frequencies is further quantized into M equally spaced phases, a polyphase code will result, which also has pulse compression properties. Known as Frank codes, these sequences can be very large and always M2 bits long. View chapterExplore book Read full chapter URL: Reference work2003, Encyclopedia of Physical Science and Technology (Third Edition)Nadav Levanon Chapter Binary Stars 2003, Encyclopedia of Physical Science and Technology (Third Edition)Steven N. Shore VII Formation of Binary Systems This is one of the major areas of study in stellar evolution theory, because at present, there are few good examples of pre-main-sequence binary stars so the field remains dominated by theoretical questions. Protostellar formation begins with the collapse of a portion of an interstellar cloud, which proceeds to form a massive disk. Through viscosity and interaction with the ambient magnetic field, the disk slowly dissipates its angular momentum. If the disk forms additional self-gravitating fragments, they will collide as they circulate and accrete to form more massive structures. Models show that such disks are unstable to the formation of a few massive members, which then accrete unincorporated material and grow. Classical results for stability analysis of rapidly rotating homogeneous objects point to several possible alternatives for the development of the core object. One is the central star in the disk, if it is still rapidly rotating, may deform to a barlike shape, which can pinch off a low-mass component. The core may undergo spontaneous fission into fragments, which then evolve separately. Simulations show, however, that the fission scenario does not yield nearly equal mass fragments. Such systems more likely result either from early protostar fragmentation and disk accretion in the first stages of star formation or of coalescence of fragments during some intermediate stage of disk fragmentation before the cores begin to grow. Binary star formation appears to be one of the avenues by which collapsing clouds relieve themselves of excess angular momentum, replacing spin angular momentum with orbital motion of the components. However, while the distribution of q may be a clue to the mechanism of formation, even this observational quantity is very poorly determined. The discovery of debris disks around several intermediate-mass, main sequence stars, especially β Pic and α Lyr = Vega, has fueled the speculation that planetary systems may be an alternative to the formation of binary stars for some systems. Statistical studies show that radial velocity variations are observed in many low-mass, solar-type stars, but the period and mass ratio distribution for these systems is presently unknown. View chapterExplore book Read full chapter URL: Reference work2003, Encyclopedia of Physical Science and Technology (Third Edition)Steven N. Shore Review article Prediction of protein structural class based on symmetrical recurrence quantification analysis 2021, Computational Biology and ChemistryInes Abdennaji, ... Jean-Marc Girault 2.3 Reverse encoding & DNA codification Each protein is formed with a linear sequence of amino acids (AAs). In addition, there are 20 standard genetic codes and multi-coded methods. So, each one protein could be expressed by different kinds of nucleotide sequences. The reverse encoding goes in inverse from protein to DNA sequence. As there is no uniqueness in the universal code of translating DNA into AAs, we used the codon (see in Table 2) as presented by Deschavanne and Tuffery (2008). In their study, the authors guarantee the balance in base composition to maximize the difference between the AAs codes. Table 2. Reverse encoding. \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| \| A=GCT \| C=TGC \| D=GAC \| E=GAG \| F=TTC \| G=GGT \| H=CAC \| I=ATT \| K=AAG \| L=CTA \| \| M=ATG \| N=AAC \| P=CCA \| Q=CAG \| R=CGA \| S=TCA \| T=ACT \| V=GTG \| W=TGG \| Y=TAC \| There are a lot of representations of DNA sequences used in the biology field like: numerical representation (Kwan and Arniker, 2009), Chaos Game representation (Jeffrey, 1990), binary representation (Voss, 1992), etc. For the sake of simplicity, we used a unique DNA representation performed by Conte and Giuliani (2009) which is based on attributing: • : (+1) to the purine: Adenine (A) and Guanine (G); • : (−1) to the pyrimidine: Cytosine (C) and Thymine (T). The simple reverse binary encoding (reverse encoding + binary DNA encoding) constitutes the first contribution of our proposed approach. It permits the transformation of one protein sequence into a binary sequence, one example is shown in Fig. 2. This will help to visualize, extract and identify characteristics from the sequences such as symmetries and recurrences. View article Read full article URL: Review article Nonlinear analysis, circuit implementation, and application in image encryption of a four-dimensional multi-scroll hyper-chaotic system 6.1 DNA operation rules The DNA sequence is composed of four nucleic bases: adenine (A), thymine (T), guanine (G), and cytosine (C), where A pairs with T and C pairs with G. Since the binary sequence 00 corresponds to 11, and 01 corresponds to 10, it is possible to encode the binary sequence as four nucleic bases. Eight coding rules satisfying the Watson-Crick complementarity rule are presented in Table 2, and the DNA decoding rule is the inverse process of coding. DNA addition and subtraction rules, as well as hetero-or rules when using coding rule 1, are illustrated in Table 3 and Table 4. Six valid DNA complementation rules can be derived from the DNA complementation rule Eq. (23), as shown in Eq. (24). Table 2. DNA coding rules. \| Rules \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| --- --- --- --- \| 00 \| A \| A \| T \| T \| G \| G \| C \| C \| \| 01 \| C \| G \| C \| G \| T \| A \| T \| A \| \| 10 \| G \| C \| G \| C \| A \| T \| A \| T \| \| 11 \| T \| T \| A \| A \| C \| C \| G \| G \| Table 3. DNA addition and subtraction rules when using DNA coding rule 1. \| Addition \| \| \| \| \| Subtraction \| \| \| \| \| --- --- --- --- --- \| \| + \| A \| C \| G \| T \| – \| A \| C \| G \| T \| \| A \| A \| C \| G \| T \| A \| A \| T \| G \| C \| \| C \| C \| G \| T \| A \| C \| C \| A \| T \| G \| \| G \| G \| T \| A \| C \| G \| G \| C \| A \| T \| \| T \| T \| A \| C \| G \| T \| T \| G \| C \| A \| Table 4. DNA hetero-rule at DNA coding rule 1. \| XOR \| A \| T \| G \| C \| --- --- \| A \| A \| C \| G \| T \| \| C \| C \| A \| T \| G \| \| G \| G \| T \| A \| C \| \| T \| T \| G \| C \| A \| (23) where is the base pair of and obeys the single mapping principle. (24) View article Read full article URL:
188991	https://www.math.purdue.edu/~goldberg/Math490A/notes6-28to7-2.html	Untitled Document 6-28-04 Exam!!! Thursday, 7-8-04 from 7-9PM in MATH215 Class homepage is online, with links to useful software phi(p^k) = p^k - p^(k-1) Theorem If (m,n) = 1, then phi(mn) = phi(m) phi(n) Proof S = {a \| 1 ≤ a≤ mn-1 and (a,mn) = 1} T = {(b,c) \| 1 ≤ b ≤ m-1, (b,m) = 1, 1 ≤ c ≤ n-1, (c,n) = 1} \|S\| = phi(mn) By choice principle, \|T\| = phi(m)phi(n) Define F: S->T by F(a) = (a mod m, a mod n) Note that (a,m)=1 and (a,n)=1 Claim 1: If F(a) = F(a), then a = a Claim 2: If (b,c) in T, then there is some a with F(a) = (b,c) Proof of Claim 1: Suppose a != a and F(a) = f(a) So, (a mod m, a mod n) = (a mod m, a mod n) a congruent to a (mod m), a congruent to a (mod n) The system of equations has a unique solution via C.R.T. a congruent to a (mod mn) Therefore, if a != a, F(a) !=F(a) Proof of Claim 2: Given (b,c), with (b,m) = 1 and (c,n) = 1 We need to find an a with a congruent to b (mod m), a congruent to c (mod n) By C.R.T., there is a unique such a with 1 ≤ a ≤ mn-1 Therefore, F(a) = (b,c) Since Claim 1 and Claim 2 are true, \|S\| = \|T\| Polynomial Congruences f(x) = a[n]x^n + a[n-1]x^(n-1) + ... a is a polynomial with integer coefficients. f(x) congruent to 0 (mod m) has a solution if there is some c such that m \| f(c). If b congruent to c (mod m), then f(b) = f(c) (mod m). Theorem Suppose m = p^a p^a ... p[r]^a[r], p[i] ∫ p[j] for i∫ j with all i and j, a[i] ≥ 1. If c is a solution to f(x) congruent to 0 (mod m) then c is a solution to f(x) congruent to 0 (mod p[i]^a[i]) for each i. Conversely, if c[i] is a solution to f(x) congruent to 0 (mod p[i]^a[i]) for each i, then there is a unique solution c to f(x) = 0 (mod m) such that c congruent to c[i] (mod p[i]^a[i]) for each i. Example x^2 + 3x + 1 congruent to 0 (mod 5) f(0) congruent to 1 (mod 5) f(1) congruent to 0 (mod 5) f(2) congruent to 1 (mod 5) f(3) congruent to 4 (mod 5) f(4) congruent to 4 (mod 5) Therefore, there is one solution Theorem m = p^a p^a ... p[r]^a[r] Set t to be the number of solutions of f(x) congruent to 0 (mod m) and t[i] to be the number of solution of f(x) congruent to 0 (mod p[i]^a[i]). Then, t = t t t ... t[r] 6-29-04 Exam I - Thursday, 7/8/04, 7-9PM, MATH215, covering ch. 1-4 Example: f(x) = 6x^3 + 13x^2 + x - 2, f(x) congruent to 0 (mod 75) f(x) congruent to 0 (mod 3) f(x) congruent to 0 (mod 25) x^2 + x + 1 congruent to 0 (mod 3) f(0) congruent to 1 (mod 3) f(1) congruent to 0 (mod 3) f(2) congruent to 1 (mod 3) 1 solution mod 3 If f(x) congruent to 0 (mod 25), then f(x) congruent to 0 (mod 5) f(0) congruent to 3 (mod 5) f(1) congruent to 3 (mod 5) f(2) congruent to 0 (mod 5) f(3) congruent to 0 (mod 5) f(4) congruent to 4 (mod 5) From there, we get f(x) congruent to 0 (mod 25) when x = 2, 7, 12, 17, or 22 Therefore, we solve x congruent to 1 (mod 3) and x congruent to 2 (mod 25), x congruent to 1 (mod 3) and x congruent to 7 (mod 25), etc. f(x) congruent to 0 (mod p^a) f(x) = a[n]x^n + a[n-1]x^(n-1) + ... a For any particular m, the largest k for which m doesn't divide a[k] is the degree of f(x) (mod m). Definition: f(x) is "monic" if its leading coefficient is 1 Theorem Let f(x), g(x) be polynomials with integer coefficients, and suppose that g(x) is monic. Then, there exists unique q(x) r(x) so that f(x) = g(x) q(x) + r(x) and 0 ≤ log(r(x)) < deg(g(x)) or r(x) = 0. Lagrange's Theorem Let p be prime. Suppose f(x) is a polynomial of degree n with integer coefficients and that not all of these coefficients are divisble by p. Then, f(x) congruent to 0 (mod p) has at most n solutions (mod p). 6-30-04 Exam info Thursday, 7-8-04, 7-9PM, MATH215 6-8 problems 1-2 proofs from the book/class (know the named ones) 0-2 problems from the homework f(x) congruent to 0 (mod p) if deg(f) = n, then it has at most n solutions We may use Fermat's Theorem to assume deg(f) ≤ p-1 (mod p) If f(x) = a[n]x^n + a[n-1]x^(n-1) + ... a and if k is the largest integer with p not dividing a[k], then f(x) has at most k roots (mod p) Suppose that p doesn't divide any a[k]. If a[n] != 1, we can choose some b with ba[n] congruent to 1 (mod p). That makes bf(x) "monic modulo p". Chebyster's Theorem Let p be a prime and f(x) a monic polynomial of degree n, with n ≤ p. We can write x^p - x = f(x)q(x) + r(x) with r(x) = 0, or deg(r(x)) < n (by the division algorithm). Then, f(x) has n roots (mod p) if and only if every coefficient of r(x) is divisible by p, meaning r(x) congruent to 0 (mod p). Proof r(x) = r[k]x^k + r[k-1]x^(k-1) + ... r, with p \| r[i] for i = 1, 2, 3, 4, 5, ... k. Since x^p - x = f(x)q(x) + r(x), we have x^p - x congruent to f(x)q(x) mod (p). By Fermat's Theorem, f(x)q(x) now has p roots (mod p). So, for any a, p \| f(a)q(a), so p \| f(a) or p \| q(a). Since deg(q(x)) = p - n, q(x) has at most p-n roots. So, this implies that f(x) has at least n roots (mod p), so it must have exactly n roots (mod p). Suppose f(x) congruent to 0 (mod p) has no solutions (mod p). Let a, a, a, ... a[n] be these solutions. Since x^p - x congruent to 0 (mod p), we have a[i]^p - a[i] congruent to 0 (mod p) for all i. Therefore, 0 congruent to a[i]^p - a[i] congruent to f(a[i])q(a[i]) + r(a[i]) congruent to r(a[i]) (mod p). So, r(x) congruent to 0 (mod p) has at least n roots. If r(x) != 0, then r(x) is a polynomial of degree k < n, with more than k roots (mod p). So, by Lagrange's Theorem, r(x) must be the zero polynomial modulo p. Corollary (4.8) Suppose p is prime and d \| p-1. Then, x^d - 1 congruent to 0 (mod p) has exactly d roots. Proof Write p-1 = kd. x^p - x = (x^d - 1)f(x) + r(x), then p divides all coefficients of r(x). x^p - x = x(x^(p-1) - 1) = x((x^d)^k - 1) = x(x^d - 1)(x^d(k-1) + x^d(k-2) + ... x^d + 1), so Chebyster's theorem gives d solutions. Now, we want to solve f(x) congruent to 0 (mod p^k) Lemma Let p be prime and k > 0. Then, for any x and t, f(x + (p^k)t) congruent to f(x) + f'(x)((p^k)t) (mod p^(k+1)). Proof Use induction on degree of f. Suppose deg(f(x)) = 0. Then, f(x) = a and f(x + ((p^k)t)) = a = f(x) + f'(x)((p^k)t) since f'(x) = 0. Now suppose this statement holds for all polynomials of degree less than or equal to n. Let deg(f) = n+1. So, f(x) = a[n+1]x^(n+1) + ... a = x(a[n+1]x^n + ... a) + a = x(g(x)) + a, with deg(g(x)) = n. f(x + ((p^k)t)) = (x + ((p^k)t))(g(x + ((p^k)t)) + a congruent to (x + ((p^k)t))(g(x) + g'(x)((p^k)t)) + a (mod p^(k+1)). So, f(x) = (xg(x) + a) + (xg'(x) + g(x))((p^k)t). f'(x) = xg'(x) + g(x). Therefore, we get f(x + ((p^k)t)) congruent to f(x) + f'(x)((p^k)t) (mod p^(k+1)). Theorem (4.10) Let p be a prime and k > 0. Suppose f(s) congruent to 0 (mod p^k). (i) Hensel's Lemma: If p doesn't divide f'(s), then there is precisely one solution s[k+1] to f(x) congruent to 0 (mod p^(k+1)) with the property s[k+1] congruent to s (mod p^k). Moreover, s[k+1] = s + ((p^k)t) where t is the unique solution to f'(s)t congruent to -f(s)/p^k (mod p). (ii) If p \| f'(s) and p^(k+1) \| f(s), then there are p solutions, s, s, ... s[p] of f(x) congruent to 0 (mod p^(k+1)), with s[t] congruent to s (mod p^k), meaning s[t] = s + ((p^k)t), t = 0, 1, 2, ... p-1. (iii) If p \| f'(s) but p^(k+1) doesn't divide f(s), then there is no solution c to f(x) congruent to 0 (mod p^(k+1)) with c congruent to s (mod p^k). Example: f(x) = 6x^3 + 13x^2 + x - 2 f(x) congruent to 0 (mod 25) f(x) congruent to 0 (mod 5) yields x congruent to 2 or 3 (mod 5) f'(x) = 18x^2 + 26x + 1, which is congruent to 3x^2 + x + 1 (mod 5) s = 3 f'(3) = 3(3^2) + 3 + 1 and is congruent to 1 (mod 5) p doesn't divide f'(s), so there is a unique solution f'(s)t congruent to -f(3)/5 f(3) = 280, f(3)/5 congruent to 1 (mod 5) t congruent to -1 (mod 5) t congruent to 4 (mod 5) s' = 3 + 54 = 23 7-1-04 Proof of Theorem 4.10 Suppose f(c) congruent to 0 (mod p^(k+1)) and c congruent to s (mod p^k) c = s + (p^k)t From Lemma 4.9, f(c) = f(s + (p^k)t), which is congruent to f(s) + f'(s)(p^k)t (mod p^(k+1)), so (f(s))/(p^k) + f'(s)t congruent to 0 (mod p). This can be rewritten as f'(s)t congruent to -(f(s))/(p^k) (mod p) If p doesn't divide f'(s) (which is the coefficient on t, essentially), then there is a unique solution t to f'(s)t congruent to -(f(s))/(p^k) (mod p) If p does divide f'(s), then it becomes 0 congruent to -(f(s))/(p^k) (mod p), so the righthand side must be divisible by p, meaning that f(s) must be divisible by p^(k+1). If not, there cannot be any solutions. Corollary Let p be prime and k > 0. If s is a solution to f(x) congruent to 0 (mod p), and p doesn't divide f'(s), then there is exactly one solution s[k] of f(x) congruent to 0 (mod p^k) with s[k] congruent to s (mod p). Proof is by induction with iteration through and Theorem 4.10 Example f(x) = x^3 + 3x + 1 f'(x) = 3x^2 + 3 f(1) congruent to 0 (mod 5) f'(1) not congruent to 0 (mod 5) There is a unique solution s[k] to x^3 + 3x + 1 congruent to 0 (mod 5^k) Theorem (given that p doesn't divide a) Congruences of the form x^2 congruent to a (mod p^k) have either 2 or no solutions, according to whether x^2 congruent to a (mod p) is solvable of not Proof If there is a solution s such that (s^2) congruent to a (mod p), then ((-s)^2) congruent to a (mod p) since s^2 = (-s)^2 for all s. Since (a,p) = 1, (s,p) = 1, meaning that -s isn't congruent to s (mod p), so the solutions are distinct. We can treat it as f(x) = x^2 - a and solve for f(x) congruent to 0, f'(s) = 2s, p doesn't divide 2s, so there is a unique s from Theorem 4.11. Definition If a not congruent to 0 (mod m), then a is a "quadratic residue" (mod m) if x^2 congruent to a (mod m) has a solution. Theorem 4.14 Let a be odd. (i) x^2 congruent to a (mod 2) has a unique solution (ii) x^2 congruent to a (mod 4) has 2 solutions if a is congruent to 1 mod 4 and no solutions if a is congruent to 3 mod 4 (iii) If k≥3, then x^2 congruent to a (mod 2^k) is solvable if and only if a is congruent to 1 (mod 8), with 4 solutions. If s^2 congruent to a (mod 2^k), then all solutions are 0 ± s ± 2^(k-1) (mod 2^k) Proof (i) a congruent to 1 (mod 2), so there's the solution x congruent to 1 (mod 2) (ii) Since a is odd, we either have a congruent to 1 (mod 4) or a congruent to 3 (mod 4). Since 1^2 congruent to 3^2 congruent to 1 (mod 4), we have 2 solutions if a congruent to 1 (mod 4) and none if a congruent to 3 (mod 4) (iii) If x is 1, 3, 5, or 7, then x^2 congruent to 1 (mod 8). So, if x^2 congruent to a (mod 2^k), then x^2 congruent to a (mod 8), a congruent to 1 (mod 8). Suppose 1 ≤ b < a ≤ 2^(k-2), a and b odd. Suppose b^2 congruent to a^2 (mod 2^k). So, 2^k \| (a^2 - b^2), so 2^k \| (a - b)(a + b). Either (a + b) or (a - b) is divisible by 4, while the other is divisible by 2 but not 4. Since 2^2 doesn't divide one of the factors, 2^(k-1) divides the other. This is impossible, however, since 1 ≤ a + b < 2^(k-1) and 1 ≤ a - b < 2^(k-2). 7-2-04 k ≥ 3 and x^2 congruent to a (mod 2^k) => solvable only if a congruent to 1 (mod 8), in which case there are 4 solutions. Proof: We proved yesterday that a had to be congruent to 1 (mod 8). If s^2 congruent to a (mod 2^k), then (±s ± 2^(k-1))^2 congruent to s^2 (mod 2^k), so there are 4 possibilities that work. ax^2 + bx + c congruent to 0 (mod p) Complete the square, getting y^2 congruent to d (mod p) for some d Read section on general quadratic congruences! x^2 congruent to a (mod m) If m = p^k p^k ... p[r]^k[r] and (m,a) = 1, then x^2 congruent to a (mod m) is equivalent to the system of equations x^2 congruent to a (mod p^k), x^2 congruent to a (mod p^k), ... x^2 congruent to a (mod p[r]^k[r]) Each congruence in the system has 0 or 2 solutions Use Chinese Remainder Theorem to solve Definition: If (a,m) = 1, then we say that a is a "quadratic residue modulo m" if x^2 congruent to a (mod m) has a solution. Otherwise, (a,m) = 1 and x^2 congruent to a (mod m) is unsolvable and we say that a is a "quadratic non-residue modulo m". Theorem Let m be 2^k p^k p^k ... p[r]^k[r]. x^2 congruent to a (mod m) has solutions if and only if x^2 congruent to a (mod 2^k) and x^2 congruent to a (mod p[i]^k[i]) for all i. If this is the case, then there are 2^(r+2) solutions if k ≥ 3, 2^(r+1) solutions if k = 2, and 2^r solutions if k = 0 or 1. Definition: Let p be an odd prime. The Legendre symbol (represented as a over p contained within parentheses, which I will handle as (a\p)) is defined as: 1 if a is a quadratic residue of p -1 if a is a quadratic non-residue of p 0 if p \| a Theorem (i) (a\p) congruent to a^((p-1)/2) (mod p) (ii) (ab\p) = (a\p)(b\p) (iii) If a congruent to b (mod p), then (a\p) = (b\p) (iv) If (a,p) = 1, then (a^2\p) = 1 and ((a^2)b\p) = (b\p) (v) (1\p) = 1, (-1\p) = (-1)^((p-1)/2) Proof of (i) Recall Euler's Criterion. If p is odd and p doesn't divide a, then a^((p-1)/2) congruent to 1 if x^2 congruent to a (mod p) is solvable, and a^((p-1)/2) congruent to -1 (mod p) if x^2 congruent to a (mod p) is not solvable. Proving (ii) is via the proof of (i), etc. Corollary If p is odd, then (-1\p) = 1 if and only if p congruen to 1 (mod 4), otherwise (-1\p) = -1 if and only if p congruent to 3 (mod 4). Proof (-1\p) = (-1)^((p-1)/2) = 1 if (p-1)/2 is even, -1 if (p-1)/2 is odd Theorem 5.7 If p is an odd prime, then there are precisely (p-1)/2 quadratic residues modulo p. Proof Suppose 1 ≤a ≤b ≤p-1. If a^2 congruent to b^2 (mod p), then p\|(a^2 - b^2), so p \| (a + b) or p \| (a - b). If a != b, 1 ≤\| a ± b \| ≤p - 1, so we must have a = b. 1 and -1 go together, 2 and -2 go together, etc. Theorem 5.12 If p is odd, then (2\p) = 1 if p congruent to ±1 (mod 8), -1 if p congruent to ±3 (mod 8). Proof Note that (2^((p-1)/2))((p-1)/2)! = 2 4 6 8 ... (p-1). If p is even, then p-1 congruent to -1. This allows rewriting of later numbers as negatives, so the product is (-1)^((p-1)/4)((p-1)/2)!. This means that p-1 is divisible by 4, so p congruent to 1 (mod 4), so p is congruent to 1 or 5 (mod 8). If p congruent to 1 (mod 8), we get 1. If p congruent to 5 (mod 8), we get -1.
188992	https://www.infezmed.it/media/journal/Vol_30_3_2022_12.pdf	ORIGINAL ARTICLES 432 Le Infezioni in Medicina, n. 3, 432-439, 2022 doi: 10.53854/liim-3003-12 Deoxycholate amphotericin for management of mucormycosis: a retrospective cohort study from South India Nitin Gupta1,2, Sourabh Srinivas1, Anagha Harikumar1, K Devaraja3, Vishnu Teja Nallapati1, Kavitha Saravu1,2 1Department of Infectious Diseases, Kasturba Medical College, Manipal, Manipal Academy of Higher Education, Manipal, Karnataka, India; 2Manipal Center for Infectious Diseases, Prasanna School of Public Health, Manipal Academy of Higher Education, Manipal, Karnataka, India; 3Department of Otorhinolaryngology, Kasturba Medical College and Hospital, Manipal, Manipal Academy of Higher Education, Manipal, Karnataka, India Article received 25 May 2022, accepted 8 July 2022 Corresponding author Kavitha Saravu E-mail: kavithasaravu@gmail.com n INTRODUCTION M ucormycosis is caused by various species of ubiquitous fungi belonging to the Mucor-mycetes family . Its incidence in South East Asia is higher owing to the propensity of fungus to favour tropical climates and the high preva-lence of DM in these areas . The incidence in-Introduction: Liposomal amphotericin use is limited in developing countries due to its extremely high cost and availability. Therefore, the study aimed to evaluate deoxycholate amphotericin B’s utility and adverse ef-fect profile in patients with mucormycosis. Methodology: This retrospective cohort study from 2019 to 2021 included patients with proven mucormy-cosis who received deoxycholate amphotericin B for more than or equal to five days and had at least three creatinine values on treatment. Baseline demographic details, risk factors and treatment details of all the pa-tients were recorded. In addition, the details of treat-ment-related adverse effects and outcomes were ascer-tained. Results: Of the 57 included patients, a history of diabe-tes, COVID-19 and steroid use was present in 49 (86%), 43 (75.4%) and 33 (57.9%) patients, respectively. Isolat-ed rhino-orbital mucormycosis was the most common presentation (n=49, 86%). The median time of fol-low-up was 48 (30.5-90) days. A total of 8 (14%) pa-SUMMARY tients died during the hospital stay. The median dura-tion of amphotericin treatment was 21 (14-40) days. Thirty-nine patients (68.4%) developed hypokalaemia on treatment, while 27 (47.4%) patients developed hy-pomagnesaemia. A total of 34 (59.6%) patients devel-oped AKI on treatment. The median day of develop-ment of AKI was 6 (4-10) days. The median baseline, highest and final creatinine values were 0.78 (0.59-0.94) mg/dl, 1.27 (0.89-2.16) mg/dl and 0.93 (0.74-1.59) mg/ dl respectively. The median percentage change from baseline to highest value and last follow-up value was 45% (0.43%-161%) and 25% (-4.8%-90.1%) respectively. The final creatinine was less than 150% of the baseline in 36 (63.2%) patients. Conclusion: Deoxycholate amphotericin is an accept-able alternative for treating mucormycosis in re-source-constrained settings. Keywords: Mucormycetes, rhino orbital, acute kidney injury, hypokalaemia, deoxycholate amphotericin B. 433 Deoxycholate amphotericin for management of mucormycosis creased further during the second wave of Coro-navirus disease 2019 (COVID-19) due to the inap-propriate and indiscriminate use of steroids to treat COVID-19 . The drug of choice for mucor-mycosis is liposomal amphotericin B . Howev-er, its use is limited in developing countries due to its extremely high cost and availability . Deox-ycholate amphotericin is cheap but owing to the high incidence of associated nephrotoxicity, its use is not recommended for routine use. Despite this, clinicians in India are forced to use deoxy-cholate amphotericin for management without feasible alternatives. Therefore, the study aimed to evaluate deoxycholate amphotericin B’s utility and adverse effect profile in patients with mucor-mycosis. n PATIENTS AND METHODS This retrospective cohort study was conducted af-ter taking permission from the Institutional Ethi-cal Committe of Kasturba Medical College, Mani-pal, India. All patients admitted with a diagnosis of mucor-mycosis between 2019 and 2021 were screened. Those patients with a confirmed microbiological diagnosis (microscopy or culture positive) and who received deoxycholate amphotericin B were included in this study. Those patients who re-ceived amphotericin for less than five days and had less than three creatinine values while on treatment were excluded. Demographic details such as age, gender, month and year of the presentation were entered in a pre-defined case record form. A diagnosis of COV-ID-19 at the time or within one year of diagnosis of mucormycosis was recorded in all patients. The time from symptom onset of COVID-19 to symp-tom onset of mucormycosis was also calculated. The history of use of steroids and the requirement for oxygen during the COVID-19 episode were as-certained from the case records. History of diabe-tes mellitus (newly diagnosed or otherwise), gly-cated haemoglobin (HBA1C) and random blood sugar levels at presentation were recorded. Histo-ry of malignancy and associated febrile neutrope-nia at presentation was also recorded. All the cas-es were categorised into rhino orbital mucormy-cosis (ROM), pulmonary mucormycosis, cutane-ous mucormycosis and bone mucormycosis. The diagnosis of patients with suspected mucor-mycosis in our centre is usually made using mi-croscopy {potassium hydroxide (KOH) mount or histopathological examination (HPE)} or culture (Sabourad dextrose media) or both. Positivity on microscopy or culture was recorded. The treat-ment details of all the patients, both medical and surgical, were recorded. The patients were monitored for development of the common side effects that include Acute Kid-ney Injury (AKI), hypokalaemia (less than 3.5 mEq/litre) and hypomagnesaemia (less than 1.6 mEq/litre). The minimum levels of potassium and magnesium were noted. AKI was defined as an increase in creatinine from baseline by 150% within seven days or an increase in the absolute value of 0.3 mg/dl within two days (Kidney Dis-ease: Improving Global Outcomes definition) . The day of development of AKI was noted. Cre-atinine levels on days 0, 2, 4, 6, 8, 10, 12, 14, 30 and 90 on treatment were noted. The percentage change between baseline creatinine and the high-est recorded creatinine value on or after treatment was calculated. Similarly, the percentage change between baseline creatinine and the final creati-nine value (last available creatinine value of the sequence mentioned above) was also recorded. The outcome of in-hospital stay was categorised as dead or alive on treatment. Data analysis: Qualitative variables were ex-pressed as percentages whereas quantitative vari-ables were expressed as Mean (± Standard devia-tion) and Median (Inter-quartile range). The base-line parameters between those who died and those who survived were compared. The Chi-square test was used for qualitative variables while the independent-t-test was used for quanti-tative variables. A p-value of less than 0.05 was considered significant. n RESULTS Of 71 patients with mucormycosis admitted dur-ing this period, 57 (80.3%) met the inclusion crite-ria. The mean age of the patients was 49.7±14.8 years. A total of 39 (68.4%) patients were male (Ta-ble 1). A total of 49 (86%) patients had DM, eleven (22.4%) of whom were diagnosed with mucormy-cosis at the time of admission (Table 1). The medi-an random blood sugar level was 157 (102-269) mg/dl at presentation. The mean HBA1c at pres-entation was 9.9±3. 434 N. Gupta, S. Srinivas, A. Harikumar, et al. A total of 43 (75.4%) patients were diagnosed with COVID-19 within a year of diagnosis of mucor-mycosis (Table 1). Of these 43 patients, six patients (13.9%) were diagnosed with COVID after the on-set of mucormycosis symptoms. The median du-ration from onset of COVID-19 symptoms to the beginning of symptoms for mucormycosis was 20 (10-75) days. Most patients (n=25, 58.2%) devel-oped mucormycosis within 20 days of COVID-19 symptoms. A total of 33 (76.7%) patients received steroids for COVID-19 (Table 1). However, a histo-ry of oxygen requirement during COVID admis-sion was present in only 20 (46.5%) patients (Table 1). Patients with COVID-19 who required oxygen therapy were given through either nasal prongs or face masks. Mechanical ventilation was not re-quired for any of the patients. In the fourteen pa-tients where the details were available, methyl-prednisolone was used in four and dexametha-sone in the rest of the ten patients. The median duration of steroid use was 7.5 days (5-10 days). Dexamethasone was used at the dosage of 8 mg once to twice daily while methylprednisolone was used at the dosage of 40 mg once to twice daily Isolated ROM was the most common presenta-tion (n=49, 86%), followed by isolated pulmonary Table 1 - Demographic and clinical details of all patients with mucormycosis who were included in the study. Demographic and clinical parameters Number of patients (Percentage) Gender Male 39 (68%) Female 18 (32%) Diabetes mellitus 49 (86%) Chronic kidney disease Not dialysis-dependent 2 (3.5%) Dialysis dependent 2 (3.5%) Haematological malignancy Febrile neutropenia 4 (7%) Non-neutropenic 1 (1.7%) Trauma 1 (1.7%) COVID-19 Diagnosed before mucormycosis 37 (64.9%) Diagnosed after mucormycosis 6 (10.5%) Steroids for COVID-19 33 (57.9%) Oxygen for COVID-19 20 (35.1%) Site of mucormycosis Isolated rhino-orbital 49 (86%) Isolated pulmonary 4 (7%) Concomitant rhino-orbital and pulmonary 2 (3.5%) Bone 1 (1.7%) Cutaneous 1 (1.7%) Diagnosis Microscopy positive 57 (100%) Culture positive 22 (38.6%) Treatment Surgical debridement 49 (86%) Deoxycholate amphotericin 57 (100%) Step-down posaconazole 42 (73.7%) Adverse events Acute Kidney Injury 34 (59.6%) Hypokalaemia 39 (68.4%) Hypomagnesemia 28 (49.1%) Outcome at last follow-up Death 8 (14%) Doing well 49 (86%) 435 Deoxycholate amphotericin for management of mucormycosis mucormycosis (n=4, 7%) (Table 1). Of the 51 pa-tients with ROM, all 51 patients had evidence of sinus involvement (maxillary-51, sphenoidal-36, ethmoidal-35, and frontal-27). The palate was in-volved in 11 patients. Of the patients with ROM (n=53), clinical or radiological evidence of ocular and cerebral involvement was seen in 35 (66%) and 15 (28.3%) patients, respectively. Of the 57 patients, all were positive by microsco-py (KOH or HPE), but only 22 (38.6%) were cul-ture positive (Table 1). Except for eight patients Figure 1 - Number of pa-tients with Acute Kidney Injury on amphotericin and the day of develop-ment. Figure 2 - Baseline, high-est and final creatinine values in patients on de-oxycholate Amphotericin (n=57). who did not undergo surgical debridement, they all underwent at least one surgical debridement (Table 1). Fourteen patients required two surgical debridements, while one had three debridements. All 57 patients were treated with deoxycholate amphotericin B. The mean dose of amphotericin per day was 56.8±9.6 mg. The median cumulative amphotericin dose was 1470 mg (840-2100). The median duration of amphotericin treatment was 21 (14-40) days. A total of 39 (68.4%) patients developed hypokal-436 N. Gupta, S. Srinivas, A. Harikumar, et al. aemia on treatment, while 28 (49.1%) patients de-veloped hypomagnesaemia (Table 1). The mean minimum potassium and magnesium levels were 2.6±0.7 and 1.3±0.2, respectively. A total of 34 (59.6%) patients developed AKI on treatment. The median day of development of AKI was 6 (4-10) days. The median baseline, highest and final cre-atinine values were 0.78 (0.59-0.94) mg/dl, 1.27 (0.89-2.16) mg/dl and 0.93 (0.74-1.59) mg/dl re-spectively. The median percentage change be-tween baseline and highest creatinine was 45% (0.43%-161%). The median percentage change be-tween baseline and final creatinine was 25% (-4.8%-90.1%). The final creatinine was less than 150% of the baseline in 36 (63%) patients. The median duration of admission was 17 (8-24.5) days. A total of 8 (14%) patients died during the follow-up period (Table 1). None of the factors was a significant predictor of death on univariate analysis (Table 2). Of the remaining 49 patients who were doing well at discharge, 42 (85.7%) pa-tients were transitioned from intravenous ampho-tericin to oral posaconazole at discharge. The me-dian duration of posaconazole treatment was 30 Figure 3 - Trend of cre-atinine change in those patients with more than five creatinine record-ings (n=36). Table 2 - Comparison between those mucormycosis patients who died vs those who survived during the fol-low-up period. Parameters Died (n=8) Survived (n=49) p-value Age 51.5+13.4 49.4+15.2 0.712 Male Gender 6 (75%) 33 (67.3%) 0.66 History of COVID-19 3 (37.5%) 40 (81.6%) 0.07 Oxygen requirement during COVID 3 (37.5%) 17 (34.7%) 0.877 Diabetes mellitus 6 (75%) 43 (87.7%) 0.336 History of steroid use 3 (37.5%) 30 (61.2%) 0.208 Rhino-orbital involvement 7 (87.5%) 44 (89.8%) 0.844 Surgical debridement 7 (87.5%) 42 (85.7%) 0.893 Hypokalaemia 6 (75%) 33 (67.3%) 0.66 Hypomagnesemia 2 (25%) 26 (53%) 0.14 Acute Kidney Injury 5 (62.5%) 29 (59.2%) 0.859 437 Deoxycholate amphotericin for management of mucormycosis (16.5-60) days. The median time of follow-up was 48 (30.5-90) days. n DISCUSSION Similar to previous studies, DM was the most common risk factor in patients with mucormyco-sis in this study [6-8]. The risk is even higher in those with a higher degree of poor sugar control, as evidenced by the mean HBA1C of around 10% in this study. The mean HBA1c in another study in patients with mucormycosis was 10.7% . Hy-perglycaemia renders the phagocytic cells dys-functional, making a patient prone to fungal inva-sion . Similar to our study, previous studies have shown that DM was first diagnosed at the time of diagnosis of mucormycosis in some cases. Steroid use is another common risk factor for mu-cormycosis . Besides causing hyperglycemia, its use is also associated with dysregulated im-mune response. Both DM and steroid use was postulated to be the driving forces that led to an increase in COVID-associated mucormycosis cas-es . Two-thirds of the patients in our cohort had a recent history of COVD-19. Similar to other studies, the time to onset of post-COVID mucor-mycosis was 2-3 weeks in most patients [6, 8, 10]. It is interesting to note that more than 30% of the steroid use in the patients of this cohort was inap-propriate. Similar findings were observed in other studies [8, 10]. Similar to other studies, ROM fol-lowed by pulmonary mucormycosis was the most common type of mucormycosis . In ROM, the infection begins from the sinus and can rapidly spread to involve orbits and the brain or progress towards the palate. It is worthwhile to mention that pulmonary involvement is easier to miss, es-pecially if associated with COVID, as symptoms can be confused with long COVID. Direct microscopy using KOH mount or histo-pathological staining is one of the fastest ways to make a reliable diagnosis of mucormycosis . Although the fungus grows in 2-3 days on cul-ture, the sensitivity of fungal culture is poor . The culture was positive in only 39% of the pa-tients in our study. Early suspicion and initiation of appropriate management are paramount. Man-agement of mucormycosis is built on three tenets: control of the underlying disease, aggressive sur-gical debridement and administration of appro-priate antimicrobials [3, 4]. Aggressive debride-ment of involved tissues is essential to decrease the fungal burden and improve the penetration of antifungals . A total of 86% of the patients in this cohort underwent at least one surgical de-bridement. Delay in antifungals is associated with an increase in mortality . Therefore, the stand-ard of care is acute administration of antifungals in those with high suspicion of the disease. The drug of choice for treating mucormycosis is liposomal amphotericin B . The liposomal form is preferred because of its favourable toxicity pro-file. However, it is costly, and its availability may be restricted in outbreak settings . Isavucona-zole and posaconazole are used for step-down therapy after initial treatment with liposomal am-photericin. These azoles can be used as first-line alternatives in patients where liposomal ampho-tericin B is contraindicated [13, 14]. However, the data on using these azoles for the primary man-agement of mucormycosis is limited. Besides, both of them are expensive as well. Deoxycholate amphotericin B is a cheap alternative but is asso-ciated with renal toxicity and hypokalemia. It is, therefore, not recommended by the international guidelines. Due to a lack of alternative option in resource-limited settings, deoxycholate ampho-tericin has been used in its place despite the limit-ed data. Most of the available literature on deoxy-cholate amphotericin is from clinical trials involv-ing the treatment of cryptococcal meningitis. However, since the median duration of therapy in this study was longer than most regimens for cryptococcal meningitis, the results of this study may turn out to be helpful in evaluating the role of deoxycholate amphotericin in the treatment of mucormycosis. Since amphotericin is insoluble in an aqueous solution at physiological pH, the plain ampho-tericin B is combined with sodium deoxycholate to form a colloidal suspension. Amphotericin binds to ergosterol and results in pore formation. Pore formation leads to leakage of ions and, sub-sequently, cell death. It leads to potassium ion leakage in low doses, while high doses lead to magnesium ion leaks and, consequently, fungal cell death. Since amphotericin can bind to choles-terol in the human cell membrane, it can result in ion leakage resulting in hypokalaemia and hypo-magnesemia. In this study, 68% of the patients de-veloped hypokalaemia, while 47% developed hy-pomagnesemia. Despite the routine premedica-438 N. Gupta, S. Srinivas, A. Harikumar, et al. tion protocol with 20 MEq of Potassium chloride every day, there was a high incidence of hypokal-aemia. Although the development of dyselectro-lytaemia is inevitable with amphotericin B, they are easily managed with routine monitoring and timely correction . None of our patients re-quired discontinuations due to electrolyte distur-bances. Amphotericin causes a dose-dependent constric-tion of afferent arterioles leading to decreased re-nal blood flow. This consequently leads to a re-duced glomerular filtration rate. It is also directly toxic to distal renal tubules. In this study, 60% of the patients developed AKI on amphotericin treatment. Previous cohort studies have shown a 30-50% incidence of AKI on amphotericin treat-ment [16, 17]. The dose and duration of ampho-tericin can explain the variation in incidence. With increasing dose and duration, the incidence of AKI increased. In a study by Bicanic et al., creati-nine increases by 52% on day 7 while it rose by 73% on day 14 . In our study, the median day of development of AKI was 6 (4-10) days. In those patients who developed AKI, the salt loading was increased, and the infusion duration was pro-longed. Although there is conflicting evidence about the benefits of prolonging infusion, increas-ing the salt loading decreased the incidence of AKI . In a randomised controlled trial, ne-phrotoxicity was higher when 5% dextrose was used for premedication hydration than normal saline . Studies show that nephrotoxicity due to amphotericin is reversible primarily . How-ever, resolution of creatinine can take months af-ter cessation of amphotericin therapy . The median increase from baseline to the last fol-low-up in creatinine was just 25% in our study. In most patients in our study, the creatinine at the last follow-up was less than 1.5 times the baseline. The therapy was not discontinued in any of the patients due to AKI in our series. In a double-blind, randomised controlled trial that compared deoxycholate vs liposomal am-photericin for the management of cryptococcal meningitis, no difference in efficacy was noted . Unfortunately, similar trials have not been conducted for mucormycosis. In an observational study from Mexico, the cure rate with deoxycho-late amphotericin and surgical debridement was 55% in patients with mucormycosis . In our study, only 14% of the patients succumbed to the illness during the median hospital stay of 17 days. In a systematic review by Watanabe et al., the pooled mortality of COVID-19-associated mucor-mycosis was 29% . In our study, the rest of the patients who were doing well at discharge, 86% were prescribed posaconazole. The rest of the pa-tients were not prescribed posaconazole due to financial constraints. In resource-limited settings, patients with con-firmed mucormycosis can be treated with deoxy-cholate amphotericin at a 1mg/kg dose. Potassium and creatinine can be monitored on alternate days initially. The frequency can be increased or de-creased based on the condition of the patient. In those who develop hypokalaemia, magnesium levels can also be measured. It is a good practice to prophylactically supplement potassium before amphotericin B. In our centre, patients were given 500 mL to 1 litre of normal saline with 20 milliequiv-alents of potassium chloride prior to amphotericin. As discussed before, premedicating with normal saline can decrease AKI. In those who develop AKI, increasing hydration and decreasing infusion speed of amphotericin can be tried. Although there are no studies on appropriate duration, the usual protocol at our centre is to treat for at least 3-6 weeks, depending on the extent of involvement. These patients can then be transitioned to oral posaconazole. Limitations: Due to the shorter follow-up dura-tion, radiological improvement was not ascer-tained during follow-up. Also, since there was no comparison group, it isn’t easy to determine whether deoxycholate amphotericin has a similar impact on outcomes as liposomal amphotericin B. In conclusion, considering the high mortality of mucormycosis in the absence of medical therapy, deoxycholate amphotericin is an acceptable and cheap alternative for treating mucormycosis in re-source-constrained settings. Dyselectrolytaemia and AKI due to this drug is reversible and can be managed easily in such a setting. n REFERENCES Prakash H, Chakrabarti A. 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188993	https://apps.dtic.mil/sti/tr/pdf/ADA310913.pdf	Technical Report CMU/SEI-96-TR-012 ESC-TR-96-012 Carnegie-Mellon University Software Engineering Institute Software Risk Management Ronald P. Higuera Yacov Y. Haimes <# June 1996 X x iKHC QUümU'lf INSPECTED i Carnegie Mellon Umve.ty » no, d,scr,m,nate anc C^^^^^^^^^TZ t^l^^^ ,n add.cn Carnegie Menon University does no, discnmma.e ^^Z^Z^^Z^^^e^H^. in ,ne .dgment o, ,he ^„~, ar veteran status sexual orientation or in violation OMederal stale. 01 JO don't pursue.' excludes openly gay. lesbian and Ä ^ Ä R---om„^ - don t telt ^ ^ ^ Un\|vers\|ty ^ ava„ab!e t0 bisexual students from'eceiving HUIL, scnoid,siH,jao, ^. . M ails:udenIS ,, ,hoPm„nst Carnegie Mel'on University. 5000 Forbes Avenue. PittsburghPA (4:2) 263-2056 Obtain genera, Information about Carnegie Mellon Umversi.y by calling (412, 268-2000. Technical Report CMU/SEI-96-TR-012 ESC-TR-96-012 June 1996 Software Risk Management Ronald P. 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Contents Acknowledgements 1 Preface 2 Introduction 6 Epilogue References v 1 9 11 13 A Holistic Vision of Software Risk Management 9 3.1 Temporal Dimension 3.2 Methodological Dimension 3.3 Human Dimension 3.4 Graphic Representation of the Holistic Vision of Software Risk Management 15 Software Risk Management Methodologies 19 4.1 Basic Constructs to Risk Management 19 4.1.1 Risk Management Paradigm 19 4.1.2 Risk Taxonomy 2i 4.1.3 Risk Clinic 23 4.2 Supporting Practices 26 4.2.1 Software Risk Evaluation (SRE) Practice 26 4.2.2 Continuous Risk Management (CRM) 28 4.2.3 Team Risk Management (TRM) 31 4.3 Methodological Framework for Software Risk Management (SRM) 34 4.3.1 Software Capability Maturity Model (SW-CMMSM) 34 4.3.2 Software Acquisition-Capability Maturity Model (SA-CMMSM) 35 Deployment of the SEI Risk Management Program 39 5.1 Major Classes Within the Hierarchy 41 5.2 Major Elements of Risk Within Each Class 41 5.3 Major Attributes Within Each Element and Class 42 5.3.1 Product Engineering Class 42 5.3.2 Development Environment Class 43 5.3.3 Program Constraints Class 43 45 47 CMU/SEI-96-TR-012 CMU/SEI-96-TR-012 List of Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Risks Within a System Context 5 The Need to Manage Risk Increases With System Complexity 6 SEI Risk Management Paradigm 7 Methodological Framework for Software Risk Management 13 Holistic View of Risk Management 16 Complete Taxonomy 21 Taxonomy of Software Risks: Overview 22 Risk Clinic Integrates Risk Management with Current Practices 24 Risk Clinic Process Overview 25 SRE Functional Components 26 Seven Principles of Risk Management 28 Team Risk Management 32 SA-CMM KPAs 35 Representation of Levels of Risk From SEI Deployment 40 CMU/SEI-96-TR-012 IV CMU/SEI-96-TR-012 Acknowledgements We are grateful to all the individuals who have contributed over the years to the development of the methodologies, tools, and approaches on software risk management cited and summa- rized in this paper. In particular we would like to acknowledge the valuable comments and sug- gestions received from Marvin Carr, Julie Walker, and Bill Wilson on an earlier draft of this paper. CMU/SEI-96-TR-012 VI CMU/SEI-96-TR-012 Software Risk Management Abstract: This paper presents a holistic vision of the risk-based methodologies for Software Risk Management (SRM) developed at the Software Engineering Institute (SEI). SRM methodologies address the entire life cycle of software acquisition, development, and maintenance. This paper is driven by the premise that the ultimate efficacy of the developed methodologies and tools for software engineering is to buy smarter, manage more effectively, identify opportunities for continuous improvement, use available information and databases more efficiently, improve industry, raise the community's playing field, and review and evaluate progress. The methodologies are based on seven management principles: shared product vision, teamwork, global perspective, forward-looking view, open communication, integrated management, and continuous process. 1 Preface The hierarchy of Software Risk Management (SRM) methodologies discussed in this paper addresses two classes of functions: software acquisition and software development. The ba- sic methodological framework with which functions are managed is composed of the Software Acquisition-Capability Maturity Model (SA-CMMSM) and the Software Capability Maturity Mod- el (SW-CMMSM) and their supporting practices and constructs. This framework for software risk management is supported by three groups of practices: 1. Software Risk Evaluation (SRE) 2. Continuous Risk Management (CRM) 3. Team Risk Management (TRM) These practices are based oh three basic constructs for software risk management developed at the Software Engineering Institute (SEI): Risk Management Paradigm, Risk Taxonomy, Risk Clinic, and Risk Management Guidebooks. The three constructs and three practices will be discussed in subsequent sections. The complexity of software risk management cannot be understood nor appropriately ad- dressed from the above methodological context alone. To capture the multifarious aspects of this complexity, we make use of hierarchical holographic modeling, where we consider two ad- ditional visions or dimensions: the temporal and human dimensions. Thus the three dimen- sions adopted in this paper to represent the holistic vision of software risk management are the temporal dimension, the methodological dimension and the human dimension. CMU/SEI-96-TR-012 The temporal dimension is decomposed into two sub-visions: 1. Macro vision represents the global perspective of the acquisition life cycle. 2. Micro vision represents the view of the project manager. The methodological dimension has already been introduced. The human dimension addresses the intellectual dimension of software acquisition—the most critical dimension, since software development is such an intellectual activity. Four aspects are identified here: 1. individual 2. team 3. management 4. stakeholder (including customer and client) The last section shares the experience gained through the deployment of the above method- ologies by SEI teams. Ample literature exists on the process of risk assessment and management. The majority of this literature, however, is devoted to theories and methodologies that have not been subject- ed to the ultimate test of practice. This paper presents comprehensive theories and processes developed at the SEI at Carnegie Mellon University that have been successfully deployed and tested in the field by numerous clients. (Adhering to confidentiality agreements, the identity of clients will not be revealed.) Authentic statistical information on the use of SEI risk methodol- ogies will be presented and analyzed in the section on the deployment of SEI risk manage- ment program. The goal of SEI Risk Program is to enable engineers, managers, and other decision makers to identify, sufficiently early, the risks associated with software acquisition, development, inte- gration, and deployment so that appropriate management and mitigation strategies can be de- veloped on a timely basis. Time is critical and the goal is to act early before a source of risk evolves into a major crisis. In other words, being mainly reactive in risk mitigation and control rather than proactive in risk prevention and control is at the heart of good risk management. Furthermore, should the system fail regardless of all risk management efforts, then ensuring the safe failure (e.g., safe shutdown) of the system must be the mandate of the software risk manager. Clearly, the secret to effective risk management is the trade-off of mitigation cost against the potential adverse effects of avoided risk. In this context, the value of the method- ologies and tools for software risk management is to buy smarter, manage more effectively and identify opportunities for continuous improvement, use available information and databas- es more efficiently, improve industry and raise the community's playing field, and review and evaluate the progress made on risk management. CMU/SEI-96-TR-012 It is important to note that the developed software risk methodologies have three fundamen- tally different, albeit complementary, objectives: 1. risk prevention 2. risk mitigation and correction 3. ensuring safe system failure The following seven risk management principles are instrumental in the quest to achieve these three objectives [Higuera 94]: Shared product vision • sharing product vision based upon common purpose, shared ownership, and collective commitment • focusing on results Teamwork • working cooperatively to achieve a common goal • pooling talent, skills, and knowledge Global perspective • viewing software development within the context of the larger system-level definition, design, and development • recognizing both the potential value of opportunity and the potential impact of adverse effects, such as cost overrun, time delay, or failure to meet product specifications Forward-looking view • thinking toward tomorrow, identifying uncertainties, anticipating potential outcomes • managing project resources and activities while anticipating uncertainties Open communication • encouraging the free flow of information between all project levels • enabling formal, informal, and impromptu communication • using consensus-based process that values the individual voice (bringing unique knowledge and insight to identifying and managing risk) Integrated management • making risk management an integral and vital part of project management • adapting risk management methods and tools to a project's infrastructure and culture CMU/SEI-96-TR-012 Continuous process • maintaining constant vigilance • identifying and managing risks routinely throughout all phases of the project's life cycle CMU/SEI-96-TR-012 2 Introduction Making informed decisions by consciously assessing what can go wrong, as well as the like- lihood and severity of the impact, is at the heart of risk management. Making informed deci- sions involves the evaluation of the trade-offs associated with all policy options for risk mitigation in terms of their costs, benefits, and risks, and the evaluation of the impact of current decisions on future options. This process of risk management embodies the identification, analysis, planning, tracking, controlling, and communication of risk. Acquisition, development, and deployment programs continue to suffer large cost overruns, schedule delays, and poor technical performance. Generally, this is a result of failing to deal appropriately with uncertainty in the acquisition and development of complex, software-inten- sive and software-dependent systems. The acquisition and development communities, both governmental and industrial, lack a systematic way of identifying, communicating, and resolv- ing technical uncertainty. Often the focus is on the symptoms of cost overruns and schedule delays rather than on the root causes in product acquisition and development. In fact, all areas in systems development are potential sources of software risks (see Figure 1) since it involves technology, hardware, software, people, cost, and schedule. Technology Hardware People Software Schedule Cost Figure 1: Risks Within a System Context CMU/SEI-96-TR-012 Risk'is commonly defined as a measure of the probability and severity of adverse effects [Low- rance 76]. Software technical risk can be defined as a measure of the probability and severity of adverse effects inherent in the development of software that does not meet its intended functions and performance requirements [Chittister 93]. The need to manage risk increases with system complexity. Figure 2 demonstrates this con- cept by indicating that as the complexity of the system increases, both technical and non-tech- nical (cost and schedule) risks increase. There is an increasing need for more systematic methods and tools to supplement individual knowledge, judgment, and experience. These hu- man traits are often sufficient to address less complex risks. It is worth noting that many man- agers believe that they are managing risk in its multifaceted dimensions. The fact of the matter is that they are merely managing cost and schedule along with isolated cases of technical risk. The SEI Risk Program provides a structured process, supported by methods and tools, for identifying, analyzing, and mitigating the uncertainties encountered in a specific software en- gineering effort. Many of the most serious issues encountered in system acquisition are the result of risks that either remain unrecognized and/or are ignored until they have already cre- ated serious consequences. This focus on risk management is important because structured techniques, even quite simple ones, can be effective in identifying risk, and approaches, pro- cedures, and techniques do exist for risk mitigation. RISK • Technical •Cost • Schedule Methods, tools, and processes Expert knowledge, judgment, and experience Individual knowledge, judgment, and experience SYSTEM COMPLEXITY Figure 2: The Need to Manage Risk Increases With System Complexity CMU/SEI-96-TR-012 Experience has shown that only a few programs are managing risk in a systematic way, and that the approaches of the programs that do manage risk tend to be ad hoc, undocumented, and incomplete [Kirkpatrick 92]. SEI teams have also found that software risk is among the least measured or managed in a system. In its attempt to respond to these problems, the goal of the SEI Risk Program is to improve the process for acquisition and development of software-intensive systems. In particular, its aims are: to enable acquisition and development managers and engineers to make better decisions (by identifying risk before they become problems); to communicate risks in a positive, non- threatening way; and to resolve technical risk in a cost-effective manner. The three groups of methodologies (SRE, TRM, and CRM) are based on three basic constructs for risk manage- ment developed at the SEI. These are: Risk Management Paradigm, Risk Taxonomy, and Risk Clinic. These constructs, the three groups of methodologies cited above, and the two methodological frameworks will be discussed in subsequent sections. The Risk Management Paradigm (Figure 3), which advocates a continuous set of activities to identify, confront, and resolve technical risk [Van Scoy 92], will be further discussed in Section 4.1.1. A continuous set of activities to identify, confront, and resolve technical risk Figure 3: SEI Risk Management Paradigm CMU/SEI-96-TR-012 CMU/SEI-96-TR-012 3 A Holistic Vision of Software Risk Management The complex process of software acquisition encompasses most, if not all, aspects associated with software risk management. Thus, it seems natural to focus on the entire life cycle of the software acquisition process in developing a holistic vision of risk management. Indeed, risk management of software engineering cannot be restricted to any subset or a single phase of the life cycle of software development. The following objectives of the overall methodological framework for software risk manage- ment apply to software-intensive systems. 1. Improve the process of software acquisition in organizations. 2. Improve software risk management methodology, technology, and practice in the acquisition process. 3. Improve the access to, acquisition, repository, use, and integration of infor- mation and data for software acquisition in industry and government. 4. In general, institutionalize risk management and decision support within the software acquisition community and make it an integral part of the communi- ty's practice. The multifarious and complex nature of the acquisition process is a fundamental attribute of software-intensive systems. This complex process involves multiple decision makers and mul- tiple non-commensurate objectives, a multitude of sources of risks and uncertainties, and an evolving technology that is shifting the focus from hardware to software. Furthermore, soft- ware is playing an increasingly central and pivotal role in systems integration. This complexity of software-intensive systems makes modeling and managing of such systems more challeng- ing and demands new approaches and new schemes. Thus, the representation of all aspects and perspectives of software risk management in a single model, or paradigm, is impractical. The multifaceted dimensions of the risks associated with the software acquisition process can- not be modeled or described by one single vision or a single planar model, and any attempt to do so would necessarily compromise the intended communication between that limited de- scription and the reader. No single planar picture of a car, for example, would be able to com- municate all the intricate functions and components of this complex system. The same holds true for the methodological framework for software risk management developed at the SEI. Here, one may distinguish among at least three visions: temporal, methodological, and func- tional. In this paper, we make use of hierarchical holographic modeling (HHM) [Haimes 81] to construct a holistic vision that represents the software risk management process. 3.1 Temporal Dimension It is plausible to assert that the genesis of a formal acquisition process can be traced to the Statement of Needs and Requirements. In terms of risk management, the seeds of critical sources of risk are often sown at this seemingly benign stage. An example from urban devel- opment demonstrates this point. A mayor and the city council identify a need for a new housing CMU/SEI-96-TR-012 development. Given the high cost of land due to its scarcity, the requirements for meeting these needs evolve into the construction of high rise apartments. At this stage, the risks that the new project might become a major slum and a magnet for crime and drug distribution are not considered. The goal of risk management is in the prevention of such risks. The impor- tance the Needs and Requirements stage places this stage at the foundation of the holistic vision of software risk management depicted in Figure 5, which follows the introduction of all components of the hierarchical holographic model for software risk management. The total acquisition life cycle is presented in two separate yet overlapping visions. The micro vision primarily represents the view of the project manager. The macro vision represents the more global and broader perspective of the acquisition life cycle. It is worth noting that within each stage of the temporal domain, the human dimension (individual, team, manager, or stakeholder) has a different and unique role to play. Micro vision 1. specification 2. solicitation (including request for proposal and contractor selection) 3. design and development (including architecture) 4. systems integration (including deployment and maintenance) Macro vision 1. conceptual design 2. demonstration/validation 3. engineering, manufacturing, development, and production 4. maintenance and major upgrade (including termination) Although the two perspectives somehow overlap, they do represent the life cycle development of software in its multifaceted dimensions. For example, most software risk-based methodol- ogies developed so far are applicable to the developmental stages identified within the micro vision, because most managerial decisions are indeed made in this domain. At the same time, however, the only way that the micro vision makes sense is when it is understood and acted upon from the broader macro vision. Because the Needs and Requirements stage is too important a contributor to the sources of risk, it is separated in our overall model presentation from the micro and macro visions. In- deed, the Needs and Requirements stage constitutes the base of the spiral model depicted in Figure 5. One reason that many of the seeds of risk are sown during the Needs and Require- ments stage is that software engineering remains more an art than a science in spite of the major gains that have materialized during the last several years. It is worth noting the testimo- ny by William Wulf before a Congressional committee in 1989 when he served as Assistant Director for Computer and Information Science and Engineering at the National Science Foun- 10 CMU/SEI-96-TR-012 dation. Commenting on the need to improve our knowledge on software, Wulf said, 'The fun- damental intellectual foundation, even the appropriate mathematics, does not exist" to solve "software crises" [House 89]. Even earlier, in 1987, Frederick P. Brooks recognized the quint- essential role that the Needs and Requirement stage plays in software risk: The hardest single part of building a software system is deciding precisely what to build. No other part of the work so cripples the resulting system if done wrong; no other part is more difficult to rectify later. Therefore, the most important function that the software builder performs for the client is the iterative extraction and refinement of the product requirements. For the truth is that the client rarely knows what he or she wants. The client usually doesn't know what questions must be answered, and he or she probably hasn't thought of the problem in the detail necessary for specification. Clearly, understanding and appreciating the evolution of risks during the temporal life cycle are requisites for effective risk management. 3.2 Methodological Dimension The risk-based methodologies discussed in this paper are designed to improve the overall software developmental process an offer a fresh way to integrate knowledge into the software acquisition process in a way that would enable managers to make more timely decisions. This is accomplished by providing a structured approach to the assessment and management of the risks and uncertainties associated with the developmental process. In risk assessment the analyst often attempts to answer the following three questions: What can go wrong? What is the likelihood that it would go wrong? and What are the consequences? [Kaplan 81] Answers to these questions help risk analysts identify, measure, quantify, and evaluate the conse- quences and impacts of risks. The remaining risk analysis builds on the risk assessment pro- cess by seeking answers to a second set of questions: What can be done? What options are available? What are their associated trade-offs in terms of all costs, benefits, and risks? and What are the impacts of current management decisions on future options [Haimes 91]? Only when these questions are addressed in the broader context of management can total risk management be realized. The methodologies developed at SEI provide answers to these sets of questions. More specifically, these methodologies provide answers to the following sample of more spe- cific questions: / know that improving the process will improve my software. How do I choose the improvement method that will have the most effect for my current state? How do I secure against major disasters? What cost will I face? CMU/SEI-96-TR-012 11 What makes a good software professional? How can I inspire my team to their best efforts? How do I know training is of any use? How can I convince my management to invest in risk management? How can I overcome resistance to change? How do I make trade-offs among the risk factors affecting software quality, cost overrun, and time delay in project completion schedule? The hierarchy of SRM methodologies discussed in this paper addresses the two life cycle functions of software acquisition and development. The basic methodological framework with which the functions are managed is composed of the SW-CMMSM, and the SA-CMMSM. The above methodological framework for software risk management is supported by three groups of practices: 1. SRE 2. CRM 3. TRM These practices build on three basic constructs of risk management: 1. the Risk Management Paradigm 2. the Risk Taxonomy 3. the Risk Clinic Each will be discussed in detail in Section 4. Figure 4 depicts the relationships among the cur- rently available models, practices, and constructs for risk management of software-intensive systems. 12 CMU/SEI-96-TR-012 Methodological Framework for SOFTWARE LIFE CYCLE: ACQUISITION & DEVELOPMENT Models Practices (SA-CMMS") ( SRE ) ( CRM ) ( TRM ) Constructs ( RISK PARADIGM ) ( RISK TAXONOMY) ( RISK CLINIC ) Figure 4: Methodological Framework for Software Risk Management 3.3 Human Dimension The third dimension in the holistic vision of software risk management addresses the intellec- tual dimension of software acquisition—the most critical one, since software development is such an intellectually intensive activity. Four perspectives are identified: 1. individual 2. team CMU/SEI-96-TR-012 13 3. management 4. stakeholder There is an obvious interplay and overlap among all four elements that constitute the human dimension. The individual perspective represents an important source of risk in software de- velopment. The lack of training, knowledge, skill, commitment to the project, loyalty to the or- ganization as a whole, dedication to quality, and many other factors are critical to the initiation of risks and to their identification at an early stage of the development. Although teams are composed of individuals, the team perspective is different from the Indi- vidual one. In their book, The Wisdom of Teams, Katzenbach and Smith [Katzenbach 93] pro- vide the following succinct definition of a team: A team is a small number of people with complementary skills who are committed to a common purpose, performance goals, and approach for which they hold themselves mutually accountable. It is clear from the above definition that software risk and its management are heavily depen- dent on the quality of the team and its commitment to identifying, preventing, and managing software risk. In the subsequent section, we will elaborate on the role of teams in risk manage- ment through a discussion of TRM methodology. The third perspective of the human dimension is management. These are the engineers who are managers of risk, and risk experts who are managers of engineering systems. Further- more, one must also appreciate the hierarchical managerial structure and the consequences of its divisions [Chittister 94]: Upper management views risk almost exclusively in term of profitability, schedule, and quality. Risk is also viewed in terms of the organization as a whole and the effects on multiple projects or a product line. Program management is concerned with profitability. It concentrates more on cost, schedules, product specificity, quality, and performance, usually for a specific program or project. The technical staff overlaps with the individual element, and may have some of its members in supervisory roles. This group concerns itself primarily with technical details of components, subassemblies, and products for one or more projects. Clearly, differences among the risk managers at each level of this hierarchical decision-mak- ing structure are caused by numerous factors, including the scope and level of responsibilities, time horizon, functionality, requirements of skill, knowledge, and expertise. 14 CMU/SEI-96-TR-012 The stakeholders—the fourth perspective in the human dimension—are also a conglomerate of constituencies. This may include government agencies, a specific branch of the Armed Forces, major corporations, and other power brokers—all of whom have direct or indirect in- terest in the acquisition of software. Understanding the role of each and all of the four perspectives in the human dimension is es- sential to effectively assessing and managing software risk. 3.4 Graphic Representation of the Holistic Vision of Software Risk Management The holistic vision of software risk management is depicted in Figure 5, where the three di- mensions of the software acquisition process—temporal, methodological, and human dimen- sions—are represented in a spiral that evolves upward over time. The spiral mode emphasizes the iterating nature of risk management, where at each stage of the software ac- quisition process, the manager-analyst adheres to the Risk Paradigm—identify, analyze, plan, track, and control [Van Scoy 92]. Communication is, of course, at the heart of the Risk Para- digm. CMU/SEI-96-TR-012 15 Holistic Vision of System Risk Management People / Individual Turn \ Management Stakeholder \ / Systems Integration Design & Development Contractor Selection Specification Conceptual Design Demonstration / Validation Engineering, Manufacturing, Development, & Production Maintenance & Major Upgrade d > Figure 5: Holistic View of Risk Management 16 CMU/SEI-96-TR-012 The micro level of the temporal dimension is represented by its four stages, evolving upward in the spiral (specification; contractor selection; design and development; and systems inte- gration). The four stages of the macro level (conceptual design; demonstration/ validation; en- gineering, manufacturing, development, and production; and maintenance and major upgrade) are depicted on the horizontal line of Figure 5, representing the time element that characterizes the macro level. The methodologies associated with system risk management are presented in Figure 5 through the rising column within the upward evolving spiral. The intention is to emphasize the fact that at each stage of the software acquisition decision-making process, the manager-an- alyst is able to make use of these methodologies. The third dimension—human—is represented through a cross section of the upward evolving temporal domain. At each stage of the acquisition process, the influence, involvement, lead- ership, imagination, and impact of the individual, the team, the management, and the "exter- nal" stakeholders are felt. The hierarchical holographic model that represents the holistic vision of software risk manage- ment as depicted in Figure 5 will be revisited in this paper after each of the three dimensions has been discussed in some detail. CMU/SEI-96-TR-012 17 18 CMU/SEI-96-TR-012 4 Software Risk Management Methodologies Although the Risk Paradigm is not considered a "methodology" per se, it is discussed under the methodological dimension. The Risk Paradigm transcends all risk analysis activities dis- cussed earlier; for this reason, it constitutes the foundation of each stage in the spiral form de- picted in Figure 5. Similar reasoning applies to the Risk Taxonomy [Carr 93] and to the Risk Clinic. The taxonomy provides a framework for organizing and studying the breadth of soft- ware development issues and hence provides a structure for surfacing and organizing soft- ware development risks. Since several of the methodologies discussed here make use of the Risk Taxonomy, it is presented along with the risk management paradigm as "Basic Con- structs to Risk Management." The Risk Clinic is a workshop that constitutes an important part of CRM and TRM. 4.1 Basic Constructs to Risk Management Three basic constructs will be discussed here. All three constructs build on the seven risk management principles discussed in the preface—shared product vision, teamwork, global perspective, forward-looking view, open communication, integrated management, and contin- uous process. 4.1.1 Risk Management Paradigm The risk management paradigm (see Figure 3) depicts the different activities involved in the management of risk associated with software development [Van Scoy 92]. The paradigm is represented by a circle to emphasize that risk management is a continuous process, while the arrows show the logical flow of information between the activities. Communication is placed in the center of the paradigm because it is both the conduit through which all information flows and, often, the largest obstacle in risk management. Essentially, the paradigm is a framework for software risk management. From this framework, a project may structure a risk manage- ment practice best fitting into its project management structure. A brief summary of each risk management paradigm activity is described below. Identify Before risks can be managed, they must be identified. Identification surfaces risks before they become problems. The SEI has developed techniques for surfacing risks by the application of a systematic process that encourages project personnel to raise concerns and issues. One such method, the SRE, is described in a subsequent section. Analyze Analysis is the conversion of risk data into risk decision-making information. Analysis provides the basis for the project manager to work on the "right" and most critical risks. CMU/SEI-96-TR-012 19 Plan Planning turns risk information into decisions and actions. Planning involves developing ac- tions to address individual risks, prioritizing risk actions, and creating an integrated risk man- agement plan. The plan for a specific risk can take many forms. For example: • Mitigate the impact of the risk by developing a contingency plan (along with an identified triggering event) should the risk occur. • Avoid a risk by changing the product design or the development process. • Accept the risk and take no further action, thus accepting the consequences if the risk occurs. • Study the risk further to acquire more information and better determine its characteristics to enable wiser decision-making. The key to risk action planning is to consider the future consequences of a decision made to- day. Track Tracking consists of monitoring the status of risks and the actions taken to ameliorate them. Appropriate risk metrics are identified and monitored to enable the evaluation of the status of as well as of risk mitigation plans. Tracking serves as the "watchdog" function of management. Control Risk control corrects deviations from planned risk actions. Once risk metrics and triggering events have been chosen, there is nothing unique about risk control. Risk control melds into project management and relies on project management processes to control risk action plans, corrects for variations from plans, responds to triggering events, and improves risk manage- ment processes. Communicate Risk communication lies at the center of the model to emphasize both its pervasiveness and its criticality. Without effective communication, no risk management approach can be viable. While communication facilitates interaction among the elements of the model, there are higher level communications to consider as well. In order to be analyzed and managed correctly, risks must be communicated to and between the appropriate organizational levels. This in- cludes levels within the development project and organization, within the customer organiza- tion, and most especially, across that threshold between the developer, the customer, and, where different, the user. Because communication is pervasive, our approach is to address it as integral to every risk management activity and not as something performed outside of, or as a supplement to, other activities. 20 CMU/SEI-96-TR-012 4.1.2 Risk Taxonomy The Risk Taxonomy follows the life cycle of software development and provides a framework for organizing data and information. The taxonomy-based identification method provides the organization developing software with a systematic interview process with which to identify sources of risk. The taxonomy construct consists of a Taxonomy-Based Questionnaire and a process for its application. The taxonomy organizes software development risks into three levels: class, ele- ment, and attribute. The questionnaire consists of questions under each taxonomic attribute that are designed to elicit the range of risks and concerns potentially affecting the software product. The application process is designed such that the questionnaire can be used in a practical and efficient manner consistent with the objective of surfacing project risks. Both the questionnaire and the application process have been developed using extensive expertise and multiple field tests. The taxonomy methodology [Carr 93] is an instrument with which one can obtain a broad, sys- tem level of risks. These risks are commonly identified by program members, and are classi- fied by categories within the hierarchical structure of the taxonomy. Moreover, the taxonomy identifies risk areas for more detailed investigation and is applied by interviewing peer groups of managers, engineers, and support personnel. Figure 6 and Figure 7 depict the hierarchical nature of the taxonomy. Software Development Risk Element' Requirements ••• \|n9\|n .e??n9 Development... Work ciemeni , nequiremems Specialties Process Environment Resources ••• Externals IÄÄ/\ Attribute j Stability Scale Formality • •• g^j» Schedule ... Facilities Figure 6: Complete Taxonomy CMU/SEI-96-TR-012 21 A. Product Engineering 1. Requirements a. Stability b. Completeness c. Clarity d. Validity e. Feasibility f. Precedent g. Scale 2. Design a. Functionality b. Difficulty c. Interfaces d. Performance e. Testability f. Hardware Constraints g. Non- Developmental Software 3. Code and Unit Test a. Feasibility b. Testing c. Coding/Imple- mentation 4. Integration and Test a. Environment b. Product c. System 5. Engineering Specialties a. Maintainability b. Reliability c. Safety d. Security e. Human Factors f. Specifications B. Development Environment 1. Development Process a. Formality b. Suitability c. Process Control d. Familiarity e. Product Control 2. Development System a. Capacity b. Suitability c. Usability d. Familiarity e. Reliability f. System Support g. Deliverability 3. Management Process a. Planning b. Project Organization c. Management Experience d. Program Interfaces 4. Management Methods a. Monitoring b. Personnel Management c. Quality Assurance d. Configuration Management 5. Work Environment a. Quality Attitude b. Cooperation c. Communication d. Morale C. Program Constraints 1. Resources a. Schedule b. Staff c. Budget d. Facilities 2. Contract a. Type of Contract b. Restrictions c. Dependencies 3. Program Interfaces a. Customer b. Associate Contractors c. Subcontractors d. Prime Contractor e. Corporate Management f. Vendors g. Politics Areas in which risks are not expected to be encountered prior to contract award Figure 7: Taxonomy of Software Risks: Overview The SEI taxonomy of software development maps the characteristics of software development and software development risks. The questionnaire is a list of non-judgmental questions to elicit issues, concerns (i.e., potential risks), and risks in each taxonomic group. Hence, the questionnaire ensures that all risk areas are systematically addressed, while the application process is designed to ensure that the questions are asked of the right people and in the right manner to produce optimum results. 22 CMU/SEI-96-TR-012 The questionnaire application is semi-structured. The questions and their sequence are used as a defining but not as a limiting instrument. That is, the questions are asked in a given se- quence, but the discussion is not restricted to that sequence. This is done to permit context and culture-sensitive issues to arise. A completely structured interview, while arguably yield- ing more reliable data for subsequent analysis across different projects, may also yield less valid data. Since the pragmatics of risk management are paramount, the semi-structured for- mat was chosen by the SEI. In other words, the questionnaire can be described as a form of structured brainstorming. The taxonomy's risk identification method identifies and clarifies the uncertainties and con- cerns of a project's technical and managerial staff. The software taxonomy is organized into three major classes: 1. product engineering: the technical aspects of the work to be accomplished 2. development environment: the methods, procedures, and tools used to pro- duce the product 3. program constraints: the contractual, organizational, and operational factors within which the software is developed, but which are generally outside of the direct control of the local management These taxonomic classes are further divided into elements and each element is characterized by its attributes. 4.1.3 Risk Clinic A Risk Clinic is a workshop that takes the SEI CRM and TRM and adapts and integrates it with a client's communication channels, infrastructure, existing practices, project management, risk management (if any), and technical problem management (see Figure 8). CMU/SEI-96-TR-012 ^ SEI Team Risk Mgt _ '', Client's Risk Mgmt \ Pilot Projects Client's Current Practices Figure 8: Risk Clinic Integrates Risk Management with Current Practices The Risk Clinic is the cornerstone of a process of interactive, adaptive transition that spans several months. It takes place after planning meeting is held between SEI and the client to es- tablish the schedules deliverables and identify the pilot projects. The Risk Clinic is the center- piece of the transition effort and should occur within 30 days of the planning meeting to keep up the momentum. If more than one pilot project is considered, multiple Risk Clinics should be held to provide the chance to evaluate alternative types of activities and procedures. The most successful can then become part of the client's risk management practice or they can be used to provide alternatives for the organization. The executive briefing is an internal briefing used by the client sponsors to educate management and pilot project personnel about the risk man- agement transition effort. Once the Risk Clinic has established proposed client risk management practices, these are implemented into one or more pilot projects whose progress is followed with coaching meet- ings between SEI and pilot project personnel. These meetings are used not only to evaluate progress, but also to adjust and revise practices. Coaching continues until the proposed risk management practice has been fully implemented and tested. Revisions and changes are then made to improve the client's risk management practices and these improvements are documented so as to institutionalize them. Figure 9 illustrates the overall process for conducting a Risk Clinic. A typical Risk Clinic takes two full days of high-energy activity with personnel from the SEI, from the client's pilot project, and from the process improvement or process definition groups (typically the software engi- neering process group). Once the client's proposed risk management framework has been es- tablished, a plan for incremental transition and implementation of these activities is defined. 24 CMU/SEI-96-TR-012 Optional preliminary surveys of change implementation history and the cultural barriers to change1 can be used to help identify any specific barriers to change that may need to be over- come during this transition, as well as to identify any enablers for change that can be used to aid the transition. Specific milestones and target dates are identified for the next several months (e.g., implement a risk database, complete with all report templates, in three months). Finally, the transition plan is discussed and a preliminary agenda is defined. Present SEI Framework I Redline/Revise Framework Group 1 Group 2 Group 3 Client's Current State of Practice I Define Incremental Transition Plan I Barriers & Enablers Technology Transition Surveys Define First Coaching Meeting Figure 9: Risk Clinic Process Overview 1 Examples of such surveys include the following, developed at the SEI by John H. Maher, Jr. and Charles R. Myers, Jr.: Managing Technological Change: Implementation History Assessment and Managing Technological Change: Cultural Assessment (SEI-90-SR-20). Pittsburgh, Pa.: Software Engineering Institute, Carnegie Mellon Univer- sity, 1990. The above documents are Copyright © IMA 1989. They are available to U.S. government agencies only. CMU/SEI-96-TR-012 25 4.2 Supporting Practices In this group there are three practices: SRE, CRM, and TRM. 4.2.1 Software Risk Evaluation (SRE) Practice The SRE practice, developed by the SEI, is a formal method for identifying, analyzing, com- municating, and mitigating software technical risk [Sisti 94]. It is used by decision makers for evaluating and mitigating the technical risks associated with a software-intensive program or project. The SRE is conducted at major milestones early and periodically in the acquisition life cycle. This practice consists of primary and support functions (see Figure 10). Primary SRE functions are Detection, Specification, Assessment, and Consolidation. The support functions are Planning and Coordination, Verification, and Training and Communication. SRE Primary Functions Support Functions Detection Planning & Coordination Specification Verification & Validation Assessment Training & Communication Consolidation Figure 10: SRE Functional Components Primary Functions Four primary functions are identified in the SRE practice—detection, specification, assess- ment, and consolidation. 26 CMU/SEI-96-TR-012 Detection is the function of finding software technical risks of a target project. This function en- sures systematic and complete coverage of all potential technical risk areas. It also ensures efficiency and effectiveness through the use of appropriate tools and techniques. Risk detec- tion in the SRE practice is performed by using the following: • SEI Taxonomy-Based Questionnaire ensures complete coverage of all areas of potential software technical risks. • Selection of appropriate individuals and guidelines for the make-up of the interview groups ensures coverage of all viewpoints including software development and support functions, technicians, and managers. Risk specification is the function of recording all aspects of the identified software technical risk including its condition, consequences, and source. One representation of a software risk statement, (developed, for example, in [Gluch 94]), has several advantages. For instance, it serves as a simple, guiding structure for risk detection activities and for communicating risks coherently and with sufficient detail. It captures components of the risk and simplifies the task of prioritizing, isolating the condition within which the risk applies, and focusing the risk mitiga- tion efforts to the source(s) of the risk. Additionally, risk specification records the source of the particular risk. Assessment is a function that determines the magnitude of each software technical risk. By definition, magnitude is the product of severity of impact and the probability of an occurrence of the risk. The SRE practice's mechanism for risk assessment is adapted from previous work conducted by the U.S. Air Force [AFSC 88]. Risk statements are assessed at one of three levels of mag- nitude—high, medium, or low. The level at which a particular risk is assessed depends on the separate assessments of its severity of impact and its probability of occurrence. Consolidation is the function of merging, combining, and abstracting risk data into concise chunks of decision-making information. This is necessary due to multiple risk detection activ- ities which identify related risks from different sources. One example is similar risks that are identified in different interview sessions. CMU/SEI-96-TR-012 27 Only that set of risk statements which meets the defined criterion are considered as candi- dates for consolidation. Candidate risk statements must meet one of the following criteria for consolidation: • manifestation of the same risk statement; that is, identical in every way except in the wording of the statements • fragmentation due to minor variations or different aspects of the same risk statement • differences in granularity; for example, a minor risk statement which is covered in the context of another risk statement of larger magnitude 4.2.2 Continuous Risk Management (CRM) CRM is a principle-based practice for managing project risks and opportunities throughout the lifetime of the project. When followed, these principles provide an effective approach to man- aging risk regardless of the specific methods and tools used. These principles, depicted in Fig- ure 11,2 are composed of three groups: core, sustaining, and defining. Minus v~ r Continuous »„f Integrated Process ^'e Management ' GLOBAL PERSPECTIVE Figure 11: Seven Principles of Risk Management 2 Williams, Ray C. "Applying the Seven Principles of Team Risk Management." Presented at the Software En- gineering Symposium, Pittsburgh, Pa., September 11-14, 1995. 28 CMU/SEI-96-TR-012 Core Principle Effective risk management requires constant attention to fostering the core principle of open communication. Clearly, the professionals associated with a project are the most qualified to identify the risks in their work on a daily basis. One should always ask, "Does project manage- ment provide a conducive environment for staffers to share their concerns regarding potential risks?" or, "Does management follow the 'killing the messenger' pattern instead of 'rewarding the messenger'?" Open communication requires • encouraging free-flowing information at and between all project levels • enabling formal, informal, and impromptu communication • using consensus-based processes that value the' individual voice, which can bring unique knowledge and insight to identifying and managing risk Sustaining Principles The sustaining principles focus on how project risk management is conducted on a daily basis. These are inward-directed, fundamental principles. If established early in the program and constantly nurtured, they should ensure that risk management becomes "the way we do busi- ness around here." CMU/SEI-96-TR-012 29 Integrated management This principle helps to assure that risk management processes, paperwork, and discipline are consistent with established project culture and practice. Risk management is simply an area of emphasis in good project management; therefore, wherever possible, risk management tasks should be integrated into well-established project routine. Integrated management re- quires • making risk management an integral and vital part of project management • adapting risk management methods and tools to a project's infrastructure and culture Teamwork: No single person can anticipate all the risks that face a project. Risk management requires that project members find, analyze, and work on potential risks together. Group syn- ergy and interdependence in dealing with risk need to be rewarded. Teamwork requires • working cooperatively to achieve a common goal • pooling talent, skills, and knowledge Continuous process: Risk management guidelines must not be allowed to become "shelf- ware." The processes must be part of daily, weekly, monthly, and quarterly project manage- ment. The premise that risk management takes place only during "risk management seasons" is obviously foreign to true management. Continuous process requires • sustaining constant vigilance • identifying and managing risks routinely throughout all phases of the project's life cycle Defining Principles The defining principles focus on how project staff members identify risks, and the extent to which staff and management are ready to address uncertainty. These principles are outward- directed and concerned with focus; they foster the development of shared mental models that clarify the when, why, and what of risk management. 30 CMU/SEI-96-TR-012 Forward-looking view: This principle develops the ability to look ahead, beyond today's crisis and into the likely consequences and impacts of current decisions on future options. Its staff is also concerned with defining how far into the future to look, so that all risk mitigation efforts of project's staff are complementary. Forward-looking view requires • thinking toward tomorrow, identifying uncertainties, anticipating potential outcomes • managing project resources and activities while anticipating uncertainties Global perspective: This principle requires that project staff replace their parochial views and interests with those that benefit the common good of the overall project. It also demands that the perspectives of the customer be harmonized with those of the supplier to reach a common view of "what's most important to the project." Project staff should develop and share a com- mon viewpoint at a global level, and be able to jointly address and mitigate specific risks. Glo- bal perspective requires • viewing software development within the context of the larger systems-level definition, design, and development • recognizing both the potential value of opportunity and the potential impact of adverse effects Shared product vision: This principle focuses on the development of a common understanding of the project's objectives and the goods and services it produces. Once clearly defined, shared product vision makes it much easier to reach a common understanding of what may adversely impact the timeliness, cost, or features of the final result. Shared product vision re- quires • sharing a product vision based upon common purpose, shared ownership, and collective commitment • focusing on results The functions of CRM, as discussed in Section 4.1.1, are: Identify, Analyze, Plan, Track, Con- trol, and Communicate. 4.2.3 Team Risk Management (TRM) TRM extends risk management with team-oriented activities involving the customer and sup- plier (e.g., government and contractor), where both customer and supplier apply the method- ologies together [Higuera 94]. TRM establishes an environment built on a set of processes, methods, and tools that enables the customer and supplier to work cooperatively, continuous- ly managing risks throughout the life cycle of a software-dependent development program. It is built on a foundation of the seven principles of risk management discussed in the preface of this paper, and on the philosophy of cooperative teams. Guided by the seven principles, TRM further extends the SEI Risk Management paradigm by adding two functions—initiate and team. Each risk goes through these functions sequentially, but the activity occurs contin- uously, concurrently, and iteratively throughout the project life cycle (e.g., planning for one risk may identify another). The TRM Guidebook3 provides an effective instrument with which to fa- CMU/SEI-96-TR-012 3T miliarize the reader with the concepts, functions, processes, methods, and products of TRM. The guidebook accomplishes this through a description of the overall methodology, a road map for applying it within a project, and detailed descriptions of the processes and methods used to implement the functions of TRM. Figure 12 [Higuera 94] depicts the extension of the SEI Risk Management Paradigm by incorporating the TRM functions (initiate and team). Initiate Recognize the need and commit to create the team culture. Either customer or supplier may initiate team activity, but both must commit to sustain the teams. Team (verb) Formalize the customer and supplier team and merge the veiwpoints to form a shared product vision. Systematic methods periodically and jointly applied establish a shared understanding of the project risks and their relative importance. Establish joint information base of risks, pri- orities, metrics, and action plans. Team Risk Management Program Manager lWA -_ MM Jlsl^ Alflf\| ■llll I I tit» ^Communications!^- IAl lift] Teaminc Individuals Customer Supplier Risk Management Risk Management Figure 12: Team Risk Management Dorofee, A. J., et al. Team Risk Management Guidebook: Version 0.1. Software Engineering Institute Carn- egie Mellon University, 1994. Draft technical report not approved for public release. 32 CMU/SEI-96-TR-012 Note that the last six functions (Identify, Analyze, Plan, Track, Control, and Communicate) are adopted from the risk management paradigm discuss-' earlier in Section 4.1.1. In summary, TRM offers a number of advantages for a project, as c ared to individual risk management. It also involves, however, a change from past custor supplier (government-contractor) re- lationships, and this will require new commitments by both. These new commitments, in turn, may involve investment in risk mitigation—particularly early in the program. CMU/SEI-96-TR-012 ~ ^ 4.3 Methodological Framework for Software Risk Management (SRM) Acquisition and development of large software-driven systems continue to suffer large cost and schedule overruns. While industry is improving its capability and performance through the use of the SW-CMMSM [Humphrey 90] for software, many acquisition organizations continue to operate in an unstable environment. Staffing is based on the availability of individuals, re- sulting in a random composition of acquisition skills. Very few team members have software acquisition or application domain skills, and little documentation exists to define procedures or capture corporate memory. Software acquisition typically proceeds in an ad hoc manner. 4.3.1 Software Capability Maturity Model (SW-CMMSM) The SW-CMMSM provides software organizations with guidance on how to gain control of their process for developing and maintaining software and how to evolve toward a culture of soft- ware engineering excellence. The SW-CMMSM was designed to guide software organizations in selecting process improvement strategies by determining current process maturity and identifying the few issues most critical to software quality and process improvement. By focus- ing on a limited set of activities and working aggressively to achieve them, organizations can steadily improve their organization-wide software process to enable continuous and lasting gains in software process capability. The staged structure of the SW-CMMSM is based on product quality principles that have exist- ed for the last 60 years. These principles have been adapted into a maturity framework that establishes the project management and engineering foundation during the initial stages, and quantitatively controls the process during the more advanced stages of maturity. The maturity framework into which these quality principles have been adapted was first in- spired by Philip Crosby in his book Quality is Free [Crosby 79]. Crosby's quality management maturity grid describes five evolutionary stages in adopting quality practices. This maturity framework was adapted to the software process by Ron Radice and his colleagues [Radice 85] working under the direction of Watts Humphrey at IBM. Humphrey brought this maturity framework to the SEI in 1986, revised it to add the concept of maturity levels, and developed the foundation for its current use throughout the software industry [Humphrey 90]. 34 CMU/SEI-96-TR-012 4.3.2 Software Acquisition-Capability Maturity Model (SA-CMMSM) The SA-CMM4 is based upon the principles of the SW-CMM [Humphrey 90]. Similar to the SW-CMM, the SA-CMM describes five levels of organizational software acquisition maturity (see Figure 13). The key process areas (KPAs) define the requirements that must be satisfied in order to accomplish that level of maturity. In other words, progress is made in stages or steps. The levels of maturity and their KPAs thus provide a road map for attaining ever higher levels of maturity. SA-CMMSM Key Process Areas Level Focus Key Process Areas 5 Optimizing Continuous process improvement Acquisition Innovation Management Process Evolution Quality Productivitity 4 Manaaed Quantitative management Quantitative Process Management Quantitative Acquisition Management 3 Defined Acquisition processes and organizational support Training Program Software Acquisition Risk Mgt Contract Performance Management Project Performance Management Org Process Defn and Improvement 2 Repeatable Project management processes Transition and Maintenance Evaluation Contract Tracking and Oversight Project Office Management Requirements Development and Mgt Solicitation Software Acquisition Planning Risk Rework 1 Initial Competent people and heroics Figure 13: SA-CMM KPAs The KPAs at any given level describe the minimum requirements for that level of maturity. This does not mean that some portion of those requirements cannot be satisfied or performed at a lower level; in fact, they typically will. However, an organization cannot achieve the next level of maturity unless all the requirements of all lower levels are maintained. The stages of the model are complimentary and flow upward. For example, the tracking and oversight at level 2 will result in corrective actions (reactive approach to defects). This process grows and matures into risk management at level 3 where actions are taken to identify and prepare for risks before Ferguson, J. J. et al. Software Acquisition Maturity Model (SA-CMM) Version 00.02. Pittsburgh, Pa.: Software Engineering Institute, Carnegie Mellon University, 1996. Draft report not publicly released. CMU/SEI-96-TR-012 35 they happen (proactive approach). Risk management grows and matures to become defect prevention at level 4 when the potential for a risk is removed by adjusting the process(es) (pre- ventive approach). While defects should decrease as maturity increases, the need for correc- tive actions (established for level 2) never goes away completely. The vast experience of the SEI in developing the SW-CMMSM is directly applicable to devel- oping the SA-CMMSM. The two models must be, and are, synergistic. The SW-CMMSM de- scribes the contractor's role in the software acquisition process, while the SA-CMMSM describes the acquirer's role. In addition, the SA-CMMSM includes certain pre-contract-award activities, such as preparing the software statement of work, documentation requirements, and participating in source selection. During the engineering phase of the project, the two models are parallel in their treatment of the processes involved. The SA-CMMSM often ends with the completion of the software acquisition process, when the "new" software is transi- tioned from the developer to the maintainer. This, however, may not always be the case. In some instances, acquisition offices are being assigned "cradle to grave" responsibility and their authority is being expanded into the maintenance area. In addition, the increased use of incremental and evolutionary deliveries raises a number of maintenance issues during the ac- quisition until the final component of the system is delivered. The current SW-CMMSM provides the appropriate level of detail for translation into the SA- CMMSM. Ensuring the compatibility of the two models, the items in the SW-CMMSM have been examined and, where appropriate, reworded to reflect the difference between engineering functions (contractor) and the acquisition function (government). The SA-CMMSM is intended to be generic enough to be used by any organization acquiring software (e.g., subcontracting). For this usage, the term "contractor" refers to the organization, developing the software. The term "government" or "project team" refers to the customer or- ganization and the term "contract" refers to the agreement between the organizations. Buyers may include a Program Executive Office (PEO) who may be acquiring software across several projects, a Program Manager (PM) who is responsible for a single system acquisition, or a Software Support Activity responsible for supporting PEOs and PMs. The model also in- cludes provisions for the participation of other functional organizations involved, such as testers, product assurance, and laboratories. When conducting a self-assessment, all of these groups should be included, depending on the project organization. The SA-CMMSM does not address the system level acquisition process. As systems become more complex, a continuing improvement initiative is needed to stay abreast of technology in order to increase efficiency and to take advantage of the latest tech- niques. There are no road maps, however, to help organizations efficiently improve their tech- nology base and to ensure that they build the best quality products at the lowest cost with the least amount of risk. Although several methodologies exist that can provide an insight into soft- ware development and software project management practices, they are not in systematic practice today. The Software Capability Evaluation (SCE) method, for example, does offer a look at organizational capability to produce a product and provides insight into contractor man- 36 CMU/SEI-96-TR-012 agement processes; the SRE method provides a framework for evaluating risks that would prevent project success; and the SW-CMMSM [Humphrey 90] provides through the ranking of the organization's level of technical maturity an index with which to measure the likelihood of success. However, these methods do not explicitly address contractor selection. If one accepts the premise that more mature software development organizations build better products, then more mature acquisition organizations should be better prepared to do a better job of acquisition. A current argument in the Department of Defense (DoD) is that as DoD ac- quisition organizations or contractors move from level 1 to level 3, the development organiza- tion has to mature as well. Note, for example, that if a level 1 organization is buying from a level 3 organization, the program office might waste time on the wrong issues; it might want to focus on documentation or on reviews, but it does not need that degree of oversight if the level 3 company already does this well. In order to guide implementation and institutionalization of software acquisition improvement, the SA-CMMSM must be augmented by a framework and road map to guide improvement ac- tivities. The Software Acquisition Improvement Framework identifies candidate practices and supporting technologies, expertise, infrastructure, and implementation guidance to satisfy the requirements of KPAs of the SA-CMMSM. The road map shows a path through the possible improvement choices provided by the model, identifying practices for which implementation guidebooks are needed and including measures of the improvement activity's success. The Acquisition Risk Management Guidebook is one example of a set of guidebooks that will provide practical "how to" practices for selected KPAs. The other guidebooks are: TRM Guide- book, SRM Guidebook, and CRM Guidebook. All these guidebooks are in preparation; they will leverage the lessons learned in risk management and build on earlier work which provided guidance to the source selection process. CMU/SEI-96-TR-012 37 38 CMU/SEI-96-TR-012 5 Deployment of the SEI Risk Management Program One of the major problems facing software engineering today is the lack of accessible data about development practices and the use of software products. Currently, risk management data are buried within projects and not available to the wider community. Consequently, soft- ware engineers are forced to resort to non-empirical arguments in deriving or evaluating many software engineering methods and tools. The Software Engineering Risk Repository (SERR) is the response of the SEI to this urgent need for an informative database5. The SERR is planned to be a national on-line service where widely dispersed information on the development and transfer of software technology will be collected or made available through a variety of sources, including already existing on- line data-bases, data-gathering instruments such as interviews, questionnaires, reports, and case studies, and printed materials that can be scanned on-line. Technology transfer is a so- cial process which is dependent on the creation of shared meaning and interpretation. Often this is only achievable through sharing trial-and-error experiences with other groups undergo- ing similar learning and discovery processes. This sharing of experience is an important basis for the construction and dissemination of most software engineering methods, tools, and ap- proaches. The ultimate goal of SERR is to provide a mechanism where the transfer, reception, and evaluation of advanced software engineering process technologies can be communicat- ed, interpreted, and negotiated. The effectiveness of such a mechanism depends on the ex- tent to which relevant and accessible information is made available. In this section, highlights from SEI field work are shared with the reader. 5 Konda, Suresh L; Monarch, Ira & Carr, Marvin J. Software Engineering Risk Repository: Concepts of Opera- tion and Function Requirements. Pittsburgh, Pa.: Software Engineering Institute, Carnegie Mellon University, 1994. Draft report not publicly released. CMÜ/SEI-96-TR-012 39 Requirements Design Int & Test Eng Spec Code & UT Mgm't Process Dev System Mgm't Methods Dev Process Work Env Program Interface Contract 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% 18.0% Figure 14: Representation of Levels of Risk From SEI Deployment This section provides statistical information on the deployment of the risk methodologies that have been developed and deployed by the SEI. The risk data and their analyses have been obtained through the risk assessments and field tests that the SEI conducts as part of its mis- sion. For reasons of propriety and confidentiality, the information is presented here in an ag- gregate form. As a result of conducting dozens of risk assessments and field tests, the SEI has developed a database on the risks associated with software development. 40 CMU/SEI-96-TR-012 The usefulness and value of this database are significant and transcend many dimensions: • It provides important foundations from which new programs can learn and benefit. • The database represents user and contractor communities with distinct domain knowledge (e.g., airplanes, embedded systems contains applicable information and experiences from different systems and domains). These can be translated, compared, contrasted, and related to new systems and programs (e.g., airplanes to automobiles to submarines; and government to non government). • The database can be a major asset in supporting the DoD's goal of buying more commercially available systems instead of custom-developed systems. • The database helps identify risks, and provides linkage between the sources of risk and the appropriate mitigation strategies. The database also raises some interesting questions, for example: Can we develop a profile of a community from the database? 5.1 Major Classes Within the Hierarchy Recall that the SEI Taxonomy is built on a hierarchy, with three classes of risk at the highest level: product engineering, development environment, and program constraints. The overall distribution of all the risks in the assessment database within these three classes indicates a surprisingly even division: • 30% product engineering • 33% development environment • 37% program constraints Below is a summary of the distribution of risks associated with each sub level of the taxonomy hierarchy (see Figure 7). All the raw data were collected and analyzed from the risk assess- ments conducted by the staff of the SEI Risk Program: 5.2 Major Elements of Risk Within Each Class Of the five subcategories of risk within product engineering • Requirements scored over 50% of all risks in the class (at 53%) • Design scored about 27% of all risks • Integration and Test scored about 14% The two remaining categories, Engineering Specialties and Code and Unit Test, scored a total of about 6% (4% and 2%, respectively). These results are not surprising because they confirm the notion that within product engineer- ing, about 80% of all risks are attributed to Requirements and Design. CMU/SEI-96-TR-012 41 Of the five categories of risk within development environment, only Management Process scored appreciably more than the other four categories—about 37%. The remaining four cat- egories—Development System, Management Methods, Development Process, and Work En- vironment—scored, in descending order, from 17% to 12% respectively. These statistics confirm that management process is critically important in meeting development require- ments. Two categories dominate the sources of risk in the program constraint class: Resources at about 43% and Customer at about 39%. In other words, over 80% of all sources of risk in Pro- gram Constraint are attributed to Resources and Customer. The remaining less than 20% are divided between Program Interface at about 11% and Contract at about 7%. 5.3 Major Attributes Within Each Element and Class 5.3.1 Product Engineering Class Statistical data on each element within the Product Engineering Class are given below: Requirement element: Among the seven attributes within the Requirement element, Com- pleteness dominates the other six attributes at 36%. The remaining attributes scored the fol- lowing percentages of sources of risk: Stability at 21%, Feasibility at 14%, Validity at 10%, Precedent at 8%, Scale at 7%, and Clarity at 4%. Design element: The distribution of sources of risk among the six attributes within the Design element decreases gradually as follows: Non-Developmental Software at 28%, Functionality at 22%, Performance at 19%, Hardware Constraints at 15%, Difficulty at 9%, Interface at 7%, and Testability at an insignificant level. Code and Unit element: The sources of risk within the Code and Unit element are uniformly distributed among the three attributes: Feasibility, Testing, and Coding/Implementation. Integration and Test element: The Environment attribute dominates the other two attributes within this element at 72%; Product Integration scored 21%, and System Integration scored merely 7%. Engineering Specialties element: The Specifications attribute dominates the sources of risk in this element at 58%. The other five attributes scored as follows: Security at 25%, Maintain- ability at 9%, Safety at 8%, and Human factors at an insignificant level. 42 CMU/SEI-96-TR-012 5.3.2 Development Environment Class Statistical data on the various elements within the Development Environment Class are given below: Development Process element: Two attributes dominate the Development Process ele- ment: Formality, at 48% of all sources of risk within this category, and Product Control, at 28%. The remaining 24% are distributed as follows: Suitability at 13%, Familiarity at 7%, and Pro- cess Control at 4%. Deliverability showed an insignificant level of risk. Development System element: Capacity, Suitability, and Usability attributes together scored 75% of all sources of risk within this element: Capacity at 35%, Suitability at 23%, and Usability at 17%. The remainder of the six attributes scored as follows: Familiarity and Reliability each scored 10%, and system support showed an insignificant level of risk. Management Process element: At 54% the Planning attribute dominates all sources of risk within this element. The distribution of the remaining 46% is as follows: Project Organization at 24%, Program Interfaces at 20%, and Management Experience at 2%. Management Methods element: Over 75% of all risks in this element are related to two at- tributes: Personnel Management at 45% and Configuration Management at 33%. Scores for the other attributes were: Monitoring 15%, and Quality Assurance 7%. Work Environment element: Of the four attributes in this element, Communication, as ex- pected, dominated all others at 74%. The distribution of the risks among the remaining three attributes are as follows: Quality Attitude at 24%, and Cooperation and Morale at 1% each. These statistics are not surprising; communication in the acquisition process among the user, the customer (often the contracting agent), and the contractor are a major source of risk of cost overrun, time delay in delivery, and risk of not meeting performance criteria. 5.3.3 Program Constraints Class Statistical data for each element within the Program Constraints Class are given below: Resources element: At 50%, the Staff attribute dominates the other three attributes in this element. The distribution of risk among the remaining attributes is as follows: Schedule at 21%, Facilities at 18%, and Budget at 11 %. Contract element: The distribution of risks among the three attributes within this element is as follows: Dependencies at 54%, Type of Contract at 36%, and Restrictions at 10%. Program Interfaces element: No attribute dominates this element. Subcontractors and Cor- porate Managementeach scored 25%, Vendors scored 22.5%, Prime Contractor scored 15%, and Politics scored 12.5%. Associate Contractors scored an insignificant level of risk. CMU/SEI-96-TR-012 43 Customer element: The distribution of risk factors within the seven attributes of the Customer Element ranges from 25% for Management to 6% for Organization. The remaining attributes scored as follows: Delays at 21 %, User Interface at 19%, Customer Furnished Resources at 12%, Technical Knowledge at 10%, and Scope Change at 6%. The above statistical data collected by SEI teams sheds important light on some critical sourc- es of risk. Requirements, Management Process, Resources, and Customer, for example, are the four most critical sources of risk in software development. Indeed, central to the holistic vision of software risk management depicted in Figure 5 are Needs and Requirements—they determine, to a large extent, the path that software development takes in its evolving life cycle. Another important component in this holistic vision of software risk management is the Human Dimension—Individual, Team, Management, and Stakeholder. People make up the other three critical sources of risk—Management Process, Resources, and Customer. The remain- ing sources of risk, identified in Figure 14, can be mainly attributed to the Temporal and Meth- odological dimensions of the holistic vision presented in this paper. 44 CMU/SEI-96-TR-012 6 Epilogue This paper presents a brief summary of the methodologies developed by the SEI for the man- agement of risk associated with the acquisition, development, and use of software. Although software continues to grow in importance as a critical system component and, more important- ly, as an overall system integrator, major sources of risk the user, the customer, and the con- tractor communities. The methodologies presented in this paper shed some light on the professional community's effort to assess and ultimately control these inherent risks. Clearly, as systems become increasingly more complex, individual knowledge, judgment, and exper- tise will not suffice and systemic methodologies for risk management such as those presented in this paper become imperative. This observation, which is based on SEI experience in the deployment of software risk methodologies, is further amplified by the fact that software risk is among the least measured or managed in a system today. CMU/SEI-96-TR-012 45 46 CMU/SEI-96-TR-012 References [AFSC 88] [Brooks 87] [Carr 93] [Chittister 93] [Chittister 94] [Crosby 79] [Gluch 94] [Higuera 94] [Haimes81] [Haimes91] [Humphrey 90] [Kaplan 81] AFSC/AFLC Acquisition Management Sofhrare Risk Abatement, Air Force Systems Command and Air Force Logistics Command, Pamphlet 800-45, September 30,1988. Brooks, Frederick P. "No Silver Bullet," Computer20, 4 (April 1987): 10-19. Carr, Marvin J.; Konda, Suresh; Monarch, Ira; Ulrich, Carol; & Walker, Clay. Taxonomy-Based Risk Identification (CMU/SEI-93-TR-6, ADA266992). Pittsburgh, Pa.: Software Engineering Institute, Carnegie Mellon Universi- ty, 1993. Chittister, Clyde & Haimes, Yacov Y. "Risk Associated with Software De- velopment: A Holistic Framework for Assessment and Management," IEEE Transactions on Systems, Man, and Cybernetics 23, 3 (May-June1993): 710-723. Chittister, Clyde & Haimes, Yacov. "Assessment and Management of Soft- ware Technical Risk," IEEE Transactions on Systems, Man, and Cybernet- ics 24, 2 (February 1994): 187-202. Crosby, P.B. Quality Is Free. New York: McGraw-Hill, 1979. Gluch, David. A Construct for Describing Software Development Risks (CMU/SEI-94-TR-14). Pittsburgh, Pa.: Software Engineering Institute, Car- negie Mellon University, 1994. Higuera, Ronald P.; Dorofee, Audrey J.; Walker, Julie A.; & Williams, Ray C. Team Risk Management: A New Model for Customer-Supplier Relation- ships (CMU/SEI-94-SR-005, ADA283987). Pittsburgh, Pa.: Software Engi- neering Institute, Carnegie Mellon University, 1994. Haimes, Yacov Y. "Hierarchical Holographic Modeling,"IEEE Transactions on Systems, Man, and Cybernetics 11, 9 (September 1981): 606-617. 1981. Haimes, Yacov Y. 'Total Risk Management," Risk Analysis 11, 2 (June 1991): 169-171. Humphrey, Watts S. Managing the Software Process. New York: Addison- Wesely Publishing Company, Inc., 1990. Kaplan, S. & Garrick, B. J. "On the Quantitative Definition of Risk," Risk Analysis 1,1 (March 1981): 11-27. CMU/SEI-96-TR-012 47 [Katzenbach 93] Katzenbach, Jon R. & Smith, Douglas K. The Wisdom of Teams. New York: Harper Business, 1993. [Kirkpatrick 92] [Lowrance 76] [Sisti 94] [Van Scoy 92] [House 89] Kirkpatrick, Robert J.; Walker, Julie; & Firth, Robert. "Software Develop- ment Risk Management: An SEI Appraisal," Software Engineering Institute Technical Review '92 (CMU/SEI-92-REV). Pittsburgh, Pa.: Software Engi- neering Institute, Carnegie Mellon University, 1992. Lowrance, William W. Of Acceptable Risk: Science and the Determination of Safety. Los Altos, Ca; William Kaufmann, 1976. Sisti, Francis J. & Joseph, Sujoe. Software Risk Evaluation Method CMU/SEI-94-TR-19). Pittsburgh, Pa.: Software Engineering Institute, Car- negie Mellon University, 1994. Van Scoy, Roger L. Software Development Risk: Opportunity, Not Problem (CMU/SEI-92-TR-30, ADA 258743). Pittsburgh, Pa.: Software Engineering Institute, Carnegie Mellon University, 1992 United States House of Representatives Committee on Science, Space, and Technology, Subcommittee on Investigations and Oversight. Bugs in the Program: Problems in the Federal Government Computer Software De- velopment and Regulation. Washington, D.C.: United States Government Printing Office, 1989. 48 CMU/SEI-96-TR-012 UNLIMITED, UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE REPORT DOCUMENTATION PAGE la. REPORT SECURITY CLASSIFICATION Unclassified 1 b. RESTRICTIVE MARKINGS None 2a. SECURITY CLASSIFICATION AUTHORITY N/A 3. DISTRIBUTION/AVAILABILITY OF REPORT Approved for Public Release Distribution Unlimited 2b. DECLASSEFICATION/DOWNGRADING SCHEDULE N/A 4. PERFORMING ORGANIZATION REPORT NUMBER(S) CMU/SEI-96-TR-012 5. MONITORING ORGANIZATION REPORT NUMBER(S) ESC-TR-96-012 6a. NAME OF PERFORMING ORGANIZATION Software Engineering Institute 6b. OFFICE SYMBOL (if applicable) SEI 7a. NAME OF MONITORING ORGANIZATION SEI Joint Program Office 6c. ADDRESS (city, state, and zip code) Carnegie Mellon University Pittsburgh PA 15213 7b. ADDRESS (city, state, and zip code) HQ ESC/ENS 5 Eglin Street Hanscom AFB, MA 01731 -2116 8a. NAME OFFUNDING/SPONSORING ORGANIZATION SEI Joint Program Office 8b. OFFICE SYMBOL (if applicable) ESC/ENS 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER F19628-95-C-0003 8c. ADDRESS (city, state, and zip code)) Carnegie Mellon University Pittsburgh PA 15213 10. SOURCE OF FUNDING NOS. PROGRAM ELEMENT NO 63756E PROJECT NO. N/A TASK NO N/A WORK UNIT NO. N/A 11. TITLE (Include Security Classification) Software Risk Management 12. PERSONAL AUTHOR(S) Ron Higuera, Yacov P. Haimes 13a. TYPE OF REPORT Final 13b. TIME COVERED FROM TO 14. DATE OF REPORT (year, month, day) June 1996 15. PAGE COUNT 48 16. SUPPLEMENTARY NOTATION 17. COSATI CODES 18. SUBJECT TERMS (continue on reverse of necessary and identify by block number) . continuous risk management, software risk evaluation software risk management, team risk management FIELD GROUP SUB. GR. 19. ABS 1RAC1 (continue on reverse if necessary and identify by block number) This paper presents a holistic vision of the risk-based methodologies for Software Risk Management (SRM) developed at the Software Engineering Institute (SEI). SRM methodologies address the entire life cycle of software acquisition, development, and maintenance. This paper is driven by the premise that the ultimate efficacy of the developed meth- odologies and tools for software engineering is to buy smarter, manage more effectively, identify opportunities for con- tinuous improvement, use available information and databases more efficiently, improve industry, raise the community's playing field, and review and evaluate progress. The methodologies are based on seven management principles: shared product vision, teamwork, global perspective, forward-looking view, open communication, integrated manage- ment, and continuous process. (please turn over) 20. DISTRIBUTION/AVAILABILITY OF ABSTRACT UNCLASSIFIED/UNLIMITED \| SAMEASRPTQ DTIC USERS \| 21. ABSTRACT SECURITY CLASSIFICATION Unclassified, Unlimited Distribution 22a. NAME OF RESPONSIBLE INDIVIDUAL Thomas R. Miller, Lt Col, USAF 22b. TELEPHONE NUMBER (include area code) (412)268-7631 22c. OFFICE SYMBOL ESC/ENS (SEI) DDFORM 1473. 83 APR EDITION of 1 JAN 73 IS OBSOLETE UNLIMITED. UNCLASSIFIED ABSTRACT — continued from page one, block 19
188994	https://www.dam.brown.edu/people/cklivans/flowfire_arxiv.pdf	FLOW-FIRING PROCESSES PEDRO FELZENSZWALB AND CAROLINE KLIVANS Abstract. We consider a discrete non-deterministic ﬂow-ﬁring process for rerouting ﬂow on the edges of a planar complex. The process is an instance of higher-dimensional chip-ﬁring. In the ﬂow-ﬁring process, ﬂow on the edges of a complex is repeatedly diverted across the faces of the complex. For non-conservative initial conﬁgurations we show this process never terminates. For conservative initial ﬂows we show the process terminates after a ﬁnite number of rerouting steps, but there are many possible ﬁnal conﬁgurations reachable from a single initial state. Finally, for conservative initial ﬂows around a topological hole we show the process terminates at a unique ﬁnal conﬁguration. In this case the process exhibits global conﬂuence despite not satisfying local conﬂuence. 1. Introduction We consider a discrete process for rerouting ﬂow on the edges of a planar complex. The process is a form of discrete diﬀusion; a ﬂow is repeatedly diverted according to a discrete Laplacian. It is also an instance of higher-dimensional chip-ﬁring. In the ﬂow-ﬁring process considered here, ﬂow is placed on the 1-dimensional cells of a complex and is rerouted across the 2-dimensional cells. This is compared to graphical chip-ﬁring, where chips are placed on the vertices (0-dimensional cells) of a graph and redistributed across the edges (1-dimensional cells). Previous work on higher-dimensional chip-ﬁring has considered algebraic structures deﬁned for ﬁnite complexes. Here we consider the dynamics of higher-dimensional chip-ﬁring and work with inﬁnite complexes. We focus on two important features of the ﬂow-ﬁring process – whether or not the system is terminating and whether or not the system is conﬂuent. To this end, three settings are explored. We show that: • For non-conservative initial conﬁgurations, the process does not terminate (Section 4). • For conservative initial conﬁgurations, the process always terminates but does not have a unique terminating state. The ﬁnal conﬁguration depends on the choices made during the ﬁring process (Section 5). • For conservative initial conﬁgurations around a distinguished face (a topological hole), the process terminates in a unique state. The ﬁnal conﬁguration is always the same regardless of the choices made during the ﬁring process (Section 6). See Figure 1 for an illustration of the three diﬀerent settings. The third case is of particular interest. The uniqueness of the ﬁnal conﬁguration is an example of global conﬂuence that does not follow from local conﬂuence, thus adding to an active narrative in chip-ﬁring, see Section 2. The special case of the 2-dimensional grid is treated throughout most of the paper for simplicity. Section 7 discusses extensions to more general cases including arbitrary planar graphs and higher dimensional polytopal decompositions. Key words and phrases. chip-ﬁring, conﬂuence, conservative ﬂows. 1 4 → 2 2 → 2 · · · (a) Non-conservative ﬂow: non-terminating process. 4 4 4 4 ↗ ↘ . . . (b) Circulation: terminating but non-unique ﬁnal conﬁguration. 4 4 4 4 → (c) Circulation around a hole: terminating and unique ﬁnal conﬁguration Figure 1. Flow-ﬁring in three diﬀerent settings. 2 4 2 4 3 f = (. . . , 2, 3, . . . , −4, 4, . . .) Figure 2. A ﬂow conﬁguration. or Figure 3. Rerouting a unit of ﬂow. A unit of ﬂow along an edge, as in the top, can reroute across a face to the left or to the right, resulting in one of the two conﬁgurations on the bottom. 2. Flow-firing Let G be the (inﬁnite) grid graph embedded as Z2. For bookkeeping purposes, we orient each edge from South to North and West to East. The ﬂow-ﬁring process on G involves conﬁgurations of integral ﬂow on the edges of the graph. Deﬁnition 1. A ﬂow conﬁguration for G is an integer assignment f specifying an amount of ﬂow on each edge. Negative values signify that the ﬂow is oriented opposite that of the edge itself. Figure 2 illustrates a ﬂow conﬁguration and the corresponding integer vector.1 Let e be an edge and σ a face (square) that contains e. Rerouting a unit of ﬂow on e across σ replaces one unit of ﬂow along e with one unit of ﬂow along the alternate path formed by the other edges of σ, see Figure 3. If an edge has two units of ﬂow (in either direction) we can reroute one unit around each of the two faces containing e. We are now ready to deﬁne the ﬂow-ﬁring process. The ﬂow-ﬁring process For the grid graph At each step: • Choose an edge e with 2 or more units of ﬂow (in either direction). • Fire e by rerouting 1 unit of ﬂow around each of the two faces containing e. 1In our terminology a ﬂow may or may not be conservative at each vertex. Other sources reserve the name “ﬂow” for the more restricted case. 3 2 → 2 → Figure 4. The ﬂow-ﬁring process. An edge can ﬁre when it has at least two units of ﬂow in either direction. 2 2 2 2 2 2 2 2 → 2 2 2 2 → 2 2 → Figure 5. Example of the ﬂow-ﬁring process. In each step the highlighted edge ﬁres and 2 units of ﬂow are rerouted across two faces. The process terminates when no edge has 2 or more units of ﬂow. 6 0 0 0 0 → 2 1 1 1 1 Figure 6. Graphical chip-ﬁring. Figure 4 shows the ﬂow-ﬁring process on an initial conﬁguration consisting of 2 units of ﬂow on a single edge. Figure 5 shows an example of the ﬂow-ﬁring process from a larger initial conﬁguration. The ﬂow-ﬁring process is a form of higher-dimensional chip-ﬁring as introduced in [DKM13], see also [Kli18][Chapter 7]. In graphical chip-ﬁring, a chip conﬁguration is an integer assignment (of chips) to the vertices of a graph. The ﬁring rule is that a vertex v can ﬁre if the number of chips at v is at least deg(v). Firing v decreases the value at v by deg(v) and increases the value at each neighbor of v by 1, see Figure 6. Graphical chip-ﬁring is 1-dimensional. Chips are on 0-dimensional vertices and move across 1-dimensional edges. Flow-ﬁring is 2-dimensional. Flow is on 1-dimensional edges and moves across 2-dimensional faces. The degree of an edge e is the number of faces containing e. For the grid graph, the degree of each edge is 2, hence the need for 2 units of ﬂow for an edge to ﬁre. Two edges are neighbors if they are contained in a common face. When an edge e ﬁres, ﬂow is diverted from e to neighboring edges. 4 The result of ﬁring a vertex in graphical chip-ﬁring can be expressed in terms of the graph Laplacian, ∆1(G). If c′ is the conﬁguration obtained from c after ﬁring vertex i, then c′ = c − ∆1(G)ei. Similarly, ﬂow-ﬁring can be expressed in terms of a combinatorial Laplacian, ∆2(G). The two-dimensional Laplacian of a complex reﬂects the degrees and incidence relations between faces of the complex. If f′ is the conﬁguration obtained from f after ﬁring edge i, then f′ = f ±∆2(G)ei. The sign of the update depends on the orientation of the ﬂow on edge i in f. There are important diﬀerences between 1-dimensional and 2-dimensional chip-ﬁring. Graphical chip-ﬁring on the inﬁnite grid always terminates if started from a chip conﬁguration with ﬁnite support. This is not the case in ﬂow-ﬁring. In Section 5 we show that the ﬂow-ﬁring process always terminates if started from a conservative ﬂow (a circulation) with ﬁnite support. Another important diﬀerence is that in graphical chip-ﬁring, the total number of chips is con-served. In the ﬂow-ﬁring process, the quantity inﬂow(v) −outﬂow(v) is conserved (at each vertex) instead. Note that in the ﬂow-ﬁring process a rerouting operation can lead to cancellation when ﬂow runs in opposite directions across an edge, see Figure 5. The cancellation of ﬂow, as seen in Figure 5, cannot happen in graphical chip-ﬁring. When a vertex v ﬁres, the number of chips at vertices other than v can only increase. This simple observation leads to an important property of graphical chip-ﬁring known as local conﬂuence. Local conﬂuence refers to the fact that from a ﬁxed conﬁguration c, if two diﬀerent states c1 and c2 can be reached after a single step, then there is a common state reachable from both c1 and c2 after a single step. A system that satisﬁes this property is also said to satisfy the diamond lemma. Local conﬂuence in terminating systems implies global conﬂuence [New42]. In graphical chip-ﬁring, this fact tells us that, if the chip-ﬁring process from an initial conﬁguration terminates, then it terminates in a unique ﬁnal conﬁguration regardless of the choices made at each step. Because of cancellation of ﬂow, the ﬂow-ﬁring process does not satisfy the diamond lemma, see Figure 7. In fact, we show that in the ﬂow-ﬁring process there can be many terminating states from a single initial conﬁguration, see Section 5. In Section 6 we consider a modiﬁcation of ﬂow-ﬁring which displays global conﬂuence despite not satisfying local conﬂuence. Global conﬂuence without local conﬂuence has recently been observed in other chip-ﬁring contexts. Labeled chip-ﬁring, see [HMP17], is an example. Labeled chips are ﬁred along the path graph, larger to the right and smaller to the left. Depending on the parity of the initial number of chips, the labeled chip-ﬁring process terminates in a unique ﬁnal conﬁguration (with the chips sorted) even though the process does not satisfy the diamond lemma. The chip-ﬁring processes on root systems introduced in [GHMP17a] and [GHMP17b] generalize labeled chip-ﬁring. Again, the root systems processes do not satisfy the diamond lemma, nonethe-less many cases do display global conﬂuence. In general, establishing global conﬂuence without local conﬂuence has proved diﬃcult. For the ﬂow-ﬁring process we introduce a topological hole to the grid (remove a square) and show that starting from a conservative ﬂow around the hole, the ﬂow-ﬁring process satisﬁes global conﬂuence despite not satisfying the diamond lemma, see Theorem 9. 3. The pulse In graphical chip-ﬁring, an important example is the pulse on the inﬁnite grid. The pulse conﬁguration consists of n chips at the origin and 0 chips elsewhere in Zd. For d = 1, this is chip-ﬁring from a single stack of chips on a line. The properties of chip-ﬁring from a single stack on the line were studied in detail in [Spe86] and [ALS+89]. The ﬁnal conﬁguration is independent of 5 2 2 2 2 2 2 2 2 ↙ ↘ 2 2 2 2 2 2 2 2 Figure 7. Flow-ﬁring does not satisfy the diamond property. In the top conﬁgu-ration the two highlighted edges can ﬁre but after one of the edges ﬁres the other one can no longer ﬁre. the ﬁring choices made throughout the process and consists of a single chip in each position of an interval centered at the origin; the origin itself has zero chips if the initial number of chips is even. For d = 2, chip-ﬁring from a stack of chips at the origin yields a well known fractal pattern associated with chip-ﬁring; see, e.g., [Pao14], [Kli18][Chapter 5]. The ﬁnal conﬁguration is again independent of the ﬁring choices made throughout. Yet this ﬁnal conﬁguration, resulting from the pulse on the graph of the 2-dimensional grid, has proved very diﬃcult to understand. Much work has gone into studying its properties; see, e.g., [LBR02], [LP09], [PS13], [LPS17]. The current paper can be seen as a ﬁrst step in understanding the basic properties of fundamental initial conﬁgurations (pulses of ﬂow) on higher dimensional spaces. 4. Flow on a single edge / Non-terminating Following the graphical chip-ﬁring examples of the pulse, one might naturally consider an initial ﬂow conﬁguration consisting of a large amount of ﬂow on a single edge. However, the ﬂow-ﬁring process from such an initial conﬁguration does not terminate. Figure 8 shows an example of such an initial state and the conﬁgurations resulting from the ﬂow-ﬁring process after several steps. Proposition 2. The ﬂow-ﬁring process on the grid does not terminate from any initial conﬁguration which has a vertex v with \| inﬂow(v) −outﬂow(v)\| > 4. Proof. Suppose \| inﬂow(v) −outﬂow(v)\| > 4. Since deg(v) = 4, by the pigeonhole principle, some edge touching v must have at least 2 units of ﬂow and can ﬁre. Since inﬂow(v) −outﬂow(v) is conserved by rerouting there will always be an edge touching v that can ﬁre. □ 5. Conservative flows / Terminating Deﬁnition 3. A ﬂow conﬁguration is conservative if for each vertex v, inﬂow(v)−outﬂow(v) = 0. In this section we prove that the ﬂow-ﬁring process initiated at a conservative ﬂow always ter-minates in a ﬁnite number of steps. First note that if the initial ﬂow is conservative, it remains conservative throughout the process. Importantly, conservative ﬂows allow for a dual representation consisting of ﬂow on faces. For this representation, associate an integer value to each face of the grid instead of each edge. A 6 4 2 2 → 2 2 → 2 2 2 2 · · · Figure 8. Some intermediate conﬁgurations reachable from a single edge ﬂow. This ﬂow-ﬁring process never terminates. 2 2 3 2 2 2 2 1 2 F = (. . . , 2, 2, . . . , −1, . . .) Figure 9. A ﬂow conﬁguration and the corresponding face representation. positive value is interpreted as a local clockwise circulation. A negative value is interpreted as a local counter-clockwise circulation. A ﬂow conﬁguration on the faces induces a ﬂow conﬁguration on the edges. For each edge e, the ﬂow on e is the sum of the ﬂows implied by the circulations around the two faces containing e. Furthermore, any conservative ﬂow on the edges is induced from some face conﬁguration. This follows from the fact that every conservative ﬂow is a sum of cycles and the boundaries of the faces of a planar graph span the cycle space of the graph (see, e.g., [FF74, KV12]). Figure 9 shows an example of a conservative ﬂow and the corresponding face representation. Note that the conservative condition is necessary. A conﬁguration with ﬂow on a single edge, as considered the previous section, cannot be represented by a conﬁguration of face circulations. In the ﬂow-ﬁring process an edge e can ﬁre if it has at least two units of ﬂow. In the face repre-sentation this means that the values on the two faces containing e diﬀer by at least two. Using a face representation F the ﬂow-ﬁring process can be equivalently deﬁned as follows. 7 The ﬂow-ﬁring process (face representation) For the grid graph and conservative initial conﬁguration At each step: • Choose two neighboring faces a and b with Fa ≥Fb + 2. • Fire a and b by decreasing Fa by 1 and increasing Fb by 1. The deﬁnition of the ﬂow-ﬁring process using the face representation is perhaps more natural in comparison to graphical chip-ﬁring. One can picture stacks of “circulation chips” on the faces of G. A ﬁring move sends a chip from one stack to a neighboring smaller stack. A signiﬁcant diﬀerence from graphical chip-ﬁring is that in ﬂow-ﬁring with the face representation, circulation chips move to a single neighbor not to all neighbors at the same time. Theorem 4. The ﬂow-ﬁring process on the grid starting from a ﬁnite conservative ﬂow terminates after a ﬁnite number of steps. Proof. Let f be a ﬁnite conservative ﬂow on the edges of G. Let F be the corresponding face representation. Deﬁne the potential function φ(F) = X F 2 σ, which is an inﬁnite sum over all faces of G with ﬁnite non-zero support. Suppose that neighboring faces a and b ﬁre and that Fa > Fb. Call the resulting conﬁguration F ′. We have, F ′ a = Fa −1, F ′ b = Fb + 1, and F ′ c = Fc at all other faces c. The diﬀerence in potential is: φ(F) −φ(F ′) = [(Fa)2 + (Fb)2] −[(Fa −1)2 + (Fb + 1)2] = (Fa)2 + (Fb)2 −(Fa)2 −(Fb)2 + 2Fa −2Fb −2 = 2Fa −2Fb −2 = 2(Fa −Fb −1) ≥2. The last inequality follows from the fact that Fa −Fb ≥2 or else faces a and b could not ﬁre. The potential function φ is non-negative and strictly decreases with each ﬂow-ﬁring step. There-fore, starting from any conﬁguration with ﬁnite potential, the process must terminate in a ﬁnite number of steps. □ Again, note that this argument does not apply to non-conservative ﬂows, such as the conﬁgura-tions with ﬂow on a single edge considered in the last section. Non-conservative ﬂows do not aﬀord a face representation and therefore the potential function φ cannot be deﬁned. Within the class of conservative ﬂows a possible analog to the pulse is a conﬁguration with a large circulation around a single face. In terms of the face representation this corresponds to a large stack of positive circulation chips on a single face. Corollary 5. The ﬂow-ﬁring process starting from k units of ﬂow around a single face terminates after a ﬁnite number of steps. Figure 10 illustrates the result of the ﬂow-ﬁring process starting from k = 4 units of ﬂow around a face. While the process always terminates, there are many possible ﬁnal conﬁgurations. 8 4 4 4 4 ↙ ↓ ↘ · · · · · · (a) Edge representation. 4 ↙ ↓ ↘ · · · 1 1 1 1 1 1 1 1 1 1 1 1 · · · (b) Face representation. Figure 10. Starting with k = 4 units of ﬂow around a face always terminates but there are many possible ﬁnal conﬁgurations. 6. Conservative flows around a boundary / Confluent We next consider the grid graph with a distinguished face (square). In terms of ﬂow rerouting, the distinguished face behaves like an obstruction or hole – ﬂow on an edge incident to the distinguished face cannot divert across the distinguished face. In terms of the face representation (for conservative ﬂows), the distinguished face behaves like a source or sink – ﬂow on adjacent faces determines which behavior is seen. A value at the distinguished face can be thought of as a boundary condition for the ﬂow-ﬁring process. Formally, let G be the grid graph again embedded as Z2 and let σ be a ﬁxed face of G. Deﬁne the following process. 9 The ﬂow-ﬁring process (edge representation) For the grid graph with distinguished face σ At each step: • Choose an edge e. • If e ̸⊂σ and e has 2 or more units of ﬂow (in either direction) then 1 unit of ﬂow is rerouted around each of the two faces containing e. • If e ⊂σ and e has 1 or more units of ﬂow (in either direction) then 1 unit of ﬂow is rerouted around the unique face not equal to σ containing e. For conservative ﬂows we have the following equivalent description of the process using the face representation introduced in Section 5. The ﬂow-ﬁring process (face representation) For the grid graph with distinguished face σ and conservative initial conﬁguration At each step: • Choose two neighboring faces a and b. • If a ̸= σ, b ̸= σ and Fa ≥Fb −2, decrease Fa by 1 and increase Fb by 1. • If b = σ and Fσ > Fa, increase Fa by 1. • If b = σ and Fσ < Fa, decrease Fa by 1. In the ﬁrst case (when a and b are both not equal to σ) we say a circulation chip moves from a to b. In the second case we say a circulation chip is created at a. In the third case we say a circulation chip is deleted from a. Write (G, σ) for the grid graph with distinguished face σ. Proposition 6. Under the ﬂow-ﬁring process with the face representation for (G, σ): (1) The maximum value over all faces does not increase. (2) The minimum value over all faces does not decrease. (3) The value at σ does not change. Proof. For (1), note that all moves that increase the value of a face involve a face of greater value, therefore the maximum value in a conﬁguration cannot increase. Part (2) is analogous. Statement (3) is the observation that the ﬂow-ﬁring rules never alter the value of Fσ. □ From Proposition 6 part (2) we see that starting from a conﬁguration of positively oriented face circulations we can only ever generate conﬁgurations of positively oriented face circulations. For the remainder of the section, we consider the speciﬁc initial conﬁguration consisting of k units of ﬂow around σ. The face representation, K, for this conﬁguration is, Kσ = k and Kτ = 0 for all τ ̸= σ. Figure 11 shows the result of ﬂow-ﬁring starting from K. Surprisingly, as we show next, there is a unique ﬁnal conﬁguration in this case. Deﬁne dist(σ, τ) to be the distance from σ to τ in the dual graph of G. For the grid graph this is the Manhattan distance. Lemma 7. Let K∗denote any conﬁguration reachable from K via the ﬂow-ﬁring process for (G, σ). Then for all faces τ ̸= σ of G, K∗ τ ≤max{0, k −dist(σ, τ) + 1}. 10 4 4 4 4 → 4 → 1 1 2 1 1 2 3 2 1 1 2 3 4 3 2 1 1 2 3 4 4 4 3 2 1 1 2 3 4 3 2 1 1 2 3 2 1 1 2 1 1 Figure 11. Flow-ﬁring starting with a conﬁguration of k = 4 units of ﬂow around a distinguished face (a hole). The top shows the edge representation and the bottom shows the face representation of the initial and ﬁnal conﬁgurations. Proof. We proceed by induction on dist(σ, τ). Base case: When dist(σ, τ) = 1 the result follows from the fact that the maximum value in K is k and the maximum value cannot increase. Induction step: Suppose the claim holds for all faces with distance at most d −1 from σ. Let A = {a \| dist(σ, a) ≥d} be the set of faces with distance at least d from σ. Initially, Ka = 0 for all a ∈A. Suppose K∗ a ̸≤max{0, k −d + 1} for some a ∈A. Consider, in particular, the ﬁrst time that K∗ a > max{0, k −d + 1} for some a ∈A. The face a must have just received a circulation chip from a neighboring face b with K∗ b > max{0, k −d + 1} + 1 before the last step. Since this is the ﬁrst time K∗ a > max{0, k −d + 1} for a ∈A, the face b cannot be in A. Since b is a neighbor of a 11 face in A and not in A, it must be that dist(σ, b) = d −1. But, by induction, the value at b must be at most max{0, k −d + 2} ≤max{0, k −d + 1} + 1 which is a contradiction. □ The main result of this section, Theorem 9, shows that starting from the initial conﬁguration K, the ﬂow-ﬁring process always terminates at the conﬁguration achieving equality for all bounds in Lemma 7. First we need the following observations. Proposition 8. Let K∗denote any conﬁguration reachable from K. Then (1) K∗ τ ≥0 for all τ. (2) The total number of circulations chips, P K∗ τ , is bounded and non-decreasing over time. Proof. (1) This follows from Proposition 6 part (2) since all values are non-negative in the initial conﬁguration. (2) For any reachable conﬁguration, K∗ σ = k and K∗ τ ≤max{0, k −dist(σ, τ) + 1} for τ ̸= σ, thus the total sum is bounded. The sum is non-decreasing because no circulation chips are ever deleted. Neighbors of σ always have value at most k by Lemma 7, and σ always has value k. Therefore neighbors of σ never have value larger than σ. □ Theorem 9. The ﬂow-ﬁring process on (G, σ) with initial conﬁguration K terminates at a unique conﬁguration K• after a ﬁnite number of steps. The ﬁnal conﬁguration has face representation K• σ = k and K• τ = max{0, k −dist(σ, τ) + 1} for all τ ̸= σ. Proof. First, we prove that the process stops. Let K∗be a conﬁguration reachable from K. Deﬁne the potential function ψ(K∗) = X τ (k −K∗ τ )2, where the sum is over all faces with distance at most k+1 from σ. Note that this function is bounded from below, i.e. ψ(K∗) ≥0. Moreover, ψ(K∗) is ﬁnite for the initial conﬁguration K∗= K. Each ﬂow-ﬁring step decreases ψ(K∗) by at least one: For a step that creates a circulation chip at a face τ neighboring σ, K∗ τ is always at most k. Therefore adding a circulation at τ can only decrease (k −K∗ τ )2. For a step that moves a circulation chip from τ to γ: Let F be the conﬁguration before the step and G be the conﬁguration after the step. Then ψ(F) −ψ(G) = [(k −Fτ)2 + (k −Fγ)2] −[(k −Fτ −1)2 + (k −Fγ + 1)2] = [k2 + F 2 τ −2kFτ + k2 + F 2 γ −2kFγ] −[k2 + (Fτ −1)2 −2k(Fτ −1) + k2 + (Fγ + 1)2 −2k(Fγ + 1)] = [F 2 τ −2kFτ + F 2 γ −2kFγ] −[F 2 τ + 1 −2Fτ −2kFτ + 2k + F 2 γ + 1 + 2Fγ −2kFγ −2k] = 2(Fτ −Fγ) −2 ≥2, where the ﬁnal inequality follows from the fact that Fτ −Fγ ≥2 for a circulation chip to move from τ to γ. Next, K• σ = k since the value at σ never changes. To see that K• τ = max{0, k −dist(σ, τ) + 1} for τ ̸= σ, we argue by induction on dist(σ, τ). Base case: When dist(σ, τ) = 1 we have that τ is a neighbor of σ. Because of the allowable ﬁring steps the process can only terminate if K• τ = K• σ = k. 12 Induction step: Suppose dist(σ, τ) = d > 1. Let γ be a neighbor of τ with dist(σ, γ) = d −1. By induction K• γ = max{0, k −d + 2}. Because of the allowable ﬁring steps the process can only terminate if K• τ is in {K• γ −1, K• γ, K• γ + 1}. By Lemma 7 it must be that K• τ ≤max{0, k −d + 1}. Considering the two possible values for K• γ and the three possible values for K• τ directly gives that K• τ must equal max{0, k −d + 1}. □ Figure 11 shows a pulse with k = 4 units of ﬂow and the resulting ﬁnal conﬁguration. In terms of the face representation, the ﬁnal conﬁgurations is an “Aztec pyramid”. The number of circulation chips at σ and neighbors of σ is k. The number of the circulation chips decreases linearly with the ℓ1 distance from σ until reaching zero. In terms of the edge representation, the ﬁnal conﬁguration has exactly one unit of ﬂow on every edge not in σ that is in a face within a ℓ1-ball of radius k centered at σ. The remaining edges have no ﬂow. 7. Extensions As mentioned in the introduction, we study the grid for simplicity but the results described here can be extended to more general settings. 7.1. Planar graphs. The results for the grid carry over essentially unchanged to any inﬁnite planar graph. Proposition 2 follows as: If there is a vertex v with \| inﬂow(v) −outﬂow(v)\| > deg(v) then the ﬂow-ﬁring process does not terminate. Theorem 4 follows unchanged: If the initial conﬁguration is a ﬁnite conservative ﬂow then the ﬂow-ﬁring process terminates after a ﬁnite number of steps. Theorem 9 also follows unchanged: If the initial conﬁguration is a circulation around a topological obstruction σ then the ﬂow-ﬁring process terminates in a ﬁnite number of steps at a unique ﬁnal conﬁguration, see Figure 12. The ﬁnal conﬁguration in Theorem 9 stated in terms of the face representation is the same for planar graphs. But in the edge representation of the ﬁnal conﬁguration for a general planar graph not every edge within some radius of the distinguished face will terminate with exactly one unit of ﬂow. If two neighboring faces have the same distance to σ then the edge between them will have zero ﬂow in the ﬁnal conﬁguration (see Figure 12). The results described above also hold for ﬁnite planar graphs. In this case the external face is included in the underlying complex. 7.2. Higher dimensional complexes. The ﬂow-ﬁring process on the grid (or a planar graph) is a form of two-dimensional chip-ﬁring. More generally, one can work over the n-dimensional grid (or a polytopal decomposition of n-dimensional space) and deﬁne a ridge-ﬁring process. A ridge conﬁguration is an integer assignment to the (n −1)-dimensional faces of an n-dimensional complex. The ridge-ﬁring process “reroutes” the value of a ridge to neighboring ridges along common facets. Conservative conﬁgurations will, by deﬁnition, aﬀord facet representations. The conservation requirement is a natural topological condition. To be conservative, the ridge conﬁguration must be in the image of a boundary operator on facets. In particular, the boundary of a single facet (edges of a square, squares of a cube, etc.) is a conservative ridge conﬁguration. The same boundary operator is used to deﬁne the combinatorial Laplacian for the complex. The combinatorial Laplacian in turn dictates the rerouting rules of higher-dimensional chip-ﬁring. For the n-dimensional grid (or polytopal decomposition), every ridge is contained in exactly two 13 3 3 3 → 3 → 1 2 1 2 1 2 3 2 1 1 1 1 3 3 3 1 2 1 2 Figure 12. The “pulse” (of height 3) on a planar graph with a hole. The top shows the edge representation and the bottom shows the face representation of the initial and ﬁnal conﬁgurations. facets and the ridge-ﬁring process in terms of the face representation is precisely the same as in the 2-dimensional (ﬂow-ﬁring) case. If two neighboring facets diﬀer by 2 or more units of ﬂow, they can ﬁre to balance out. Theorem 4 follows unchanged: If the initial state is a ﬁnite conservative ridge conﬁguration then the ridge-ﬁring process terminates after a ﬁnite number of steps. Theorem 9 also follows unchanged: If the initial state is a conservative ridge conﬁguration around a topological obstruction σ then the ridge-ﬁring process terminates in a ﬁnite number of steps at a unique ﬁnal conﬁguration with the prescribed face representation. References [ALS+89] Richard Anderson, L´ aszl´ o Lov´ asz, Peter Shor, Joel Spencer, ´ Eva Tardos, and Shmuel Winograd, Disks, balls, and walls: analysis of a combinatorial game, Amer. Math. Monthly 96 (1989), no. 6, 481–493. MR 999411 [DKM13] Art M. Duval, Caroline J. Klivans, and Jeremy L. Martin, Critical groups of simplicial complexes, Ann. Comb. 17 (2013), no. 1, 53–70. MR 3027573 [FF74] L. R. Ford and D. R. Fulkerson, Flows in networks, Princeton university press, 1974. 14 [GHMP17a] Pavel Galashin, Sam Hopkins, Thomas McConville, and Alexander Postnikov, Root system chip-ﬁring I: Interval-ﬁring, arXiv preprint arXiv:1708.04850 (2017). [GHMP17b] , Root system chip-ﬁring II: Central-ﬁring, arXiv preprint arXiv:1708.04849 (2017). [HMP17] Sam Hopkins, Thomas McConville, and James Propp, Sorting via chip-ﬁring, Electron. J. Combin. 24 (2017), no. 3, Paper 3.13, 20. MR 3691530 [Kli18] Caroline J. Klivans, The Mathematics of Chip-Firing, Chapman and Hall/CRC, 2018. [KV12] B. Korte and J. Vygen, Combinatorial optimization, Springer, 2012. [LBR02] Yvan Le Borgne and Dominique Rossin, On the identity of the sandpile group, Discrete Mathematics 256 (2002), no. 3, 775–790. [LP09] Lionel Levine and Yuval Peres, Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile, Potential Analysis 30 (2009), no. 1, 1. [LPS17] Lionel Levine, Wesley Pegden, and Charles K. Smart, The Apollonian structure of integer superharmonic matrices, Ann. of Math. (2) 186 (2017), no. 1, 1–67. MR 3664999 [New42] M. H. A. Newman, On theories with a combinatorial deﬁnition of “equivalence”, Ann. of Math. (2) 43 (1942), 223–243. MR 0007372 [Pao14] Guglielmo Paoletti, Deterministic abelian sandpile models and patterns, Springer Theses, Springer, Cham, 2014, Thesis, University of Pisa, Pisa, 2012. MR 3100415 [PS13] Wesley Pegden and Charles K. Smart, Convergence of the Abelian sandpile, Duke Math. J. 162 (2013), no. 4, 627–642. MR 3039676 [Spe86] J. Spencer, Balancing vectors in the max norm, Combinatorica 6 (1986), no. 1, 55–65. MR 856644 (Pedro Felzenszwalb) Brown University (Caroline Klivans) Brown University 15
188995	https://cs.uwaterloo.ca/~eblais/cs365/w25/tm	Turing Machines ×Models of Computation 1. Introduction 2. Turing Machines 3. Decidable Languages 4. Recursion Theorem 5. Undecidability 6. Reductions 7. Time Complexity 8. P vs. NP 9. Polynomial Hierarchy 10. Boolean Circuits 11. Non-Uniform Computation 12. Boolean Formulas 13. Satisfiability 14. Randomized Computation 15. P vs. BPP 16. Randomized Verification 17. Space Complexity 18. Logarithmic Space 19. Sublogarithmic Space 20. Nonregular Languages 21. Communication Complexity ☰Models of ComputationEric Blais 2. Turing Machines ← IntroductionDecidable Languages → We saw in the last lecture that for every fixed machine, there exists a language that cannot be computed by algorithms for that specific machine. Our goal today is to strengthen this result to identify an explicit language that cannot be computed by algorithms over any machine model. That is where Turing machines come in. The magic of this model of computation is that it satisfies what appear to be two contradictory goals. One, it is a very simple model. So simple that we will be able to analyze Turing machines directly and prove quite a few nontrivial statements about them. And two, it is general enough to capture what is computable by all other reaonable models of computation. Definition of Turing Machines The simplest Turing machine model is the deterministic 1-tape Turing machine. This machine has two main components. There is the machine itself, which has a finite number of possible internal states. And there is an infinite tape split up into squares. Each square contains exactly one symbol. The machine has a tape head that is always positioned over one of the squares of the tape. At each step in the execution, the Turing machine uses its internal state and the symbol on the square of the tape under the tape head to determine its next action. The action taken by a Turing machine at each computational step consists of three parts: the internal state it goes to, the symbol that is overwritten over the previous symbol on the square of tape under the tape head, and a movement Left or Right of the tape head to the square adjacent to the current one on the tape. In addition, the machine has two additional special actions that it can take to halt: one accepts, and the other rejects. The input to a Turing machine is a string of binary symbols on the tape, surrounded by an infinite number of squares that contain a special blank symbol. We write ◻ to denote the special blank symbol. The tape head of the Turing machine is initially on the left-most symbol of the input. (In the special case where the input is the empty string ε, the tape consists entirely of blank symbols and the tape head is over any one of them.) The simplest way to define a Turing machine is via a transition diagram such as the following one. The edges in the diagram describe all the actions that a Turing machine can do in a computational step. For example, the self-loop in the top right corner of the diagram above specifies that if the machine is currently in state 3 and the tape head is over a square with the symbol 1, it writes a 0 on that square (overwriting the previous 1), moves the tape head Left one square, and stays in state 3. We can also provide a formal definition of Turing machines in the following way. Definition. A deterministic one-tape Turing machine is an abstract machine described by the triple M=(m,k,δ) with m,k≥1 where Q={1,2,…,m} is the set of internal states, Γ={◻,0,1,2,…,k} is the tape alphabet, and δ:Q×Γ→(Q∪{A,R})×Γ×{L,R} is the transition function. The state 1 denotes the initial state of the Turing machine M. (We will use bold symbols to denote the states of the machine, to prevent any confusion with the symbols used on the tape.) In order to simulate a Turing machine, we need to keep track of its internal state, the current string on the tape, and the position of the tape head on the tape. We call this information the configuration of a Turing machine. It can be represented conveniently in the following way. Definition. The configuration of a Turing machine is a string w q y where q∈Q∪{A,R} represents the current state of the machine, w y∈Γ∗ is the current string on the tape, and the position of the tape head is on the first symbol of y. Two configurations are equivalent when they are identical up to blank symbols at the beginning of w or at the end of y. In other words, they satisfy w q y=◻w q y=w q y◻. When q=A, the string w q y represents an accepting configuration. When q=R, it represents a rejecting configuration. A configuration is a halting configuration if it is either accepting or rejecting. A step of computation of a Turing machine typically changes its configuration. We can simulate a Turing machine by listing the configurations obtained after each computation step. For example, when running the Turing machine in the diagram above on the input 1011, the following sequence of configurations are obtained: 1 1 0 1 1 1 2 0 1 1 1 0 2 1 1 1 0 1 2 1 1 0 1 1 2◻1 0 1 3 1 1 0 3 1 0 1 3 0 0 0 1 1 A 0 0 Note that the additional spacing in the representation of the configurations is not required by the definition, but it does make it easier to follow the simulation of the Turing machine. Note also that for clarity the blank squares on the tape are written explicitly only when they are required (as is the case in the step above where the tape head is positioned over a blank square). A list of the configurations obtained during the execution of a Turing machine is called a tableau. We can also formally define which configurations follow each other in the execution of a Turing machine in the following way. Definition. For any strings w,y∈Γ∗, symbols a,b,c∈Γ, and states q∈Q and r∈Q∪{A,R}, the configuration w a q b y of the Turing machine M yields the configuration w r a c y, denoted w a q b y⊢w r a c y when δ(q,b)=(r,c,L). Similarly, w a q b y⊢w a c r y when δ(q,b)=(r,c,R). By simulating multiple steps of computation, a Turing machine can reach some other configurations. Specifically, we say that the configuration w q y derives the configuration w′q′y′ in the Turing machine M, denoted w q y⊢∗w′q′y′ when there exists a finite sequence of configurations w 1 q 1 y 1,…,w k q k y k such that w q y⊢w 1 q 1 y 1⊢⋯⊢w k q k y k⊢w′q′y′. The Turing machine M accepts an input x∈{0,1}∗ if the initial configuration 1 x derives an accepting configuration. It rejects x if 1 x derives a rejecting configuration. And it halts on x if and only if it either accepts or rejects x. With these definitions in place, we can now formally define what it means for a Turing machine to “compute” a language. Definition. The Turing machine M decides the language L⊆{0,1}∗ if it accepts every x∈L and rejects every x∉L. A language is decidable if and only if there is a Turing machine that decides it. (The set of all decidable languages is also known as the set of recursive languages.) There is a closely related notion of recognizability of languages. Definition. The Turing machine M recognizes the language L⊆{0,1}∗ if for every x∈{0,1}∗, M accepts x if and only if x∈L. Note that in this definition of recognizability, the machine M can either reject or run forever on inputs x∉L. The set of recognizable languages is also known as the set of all recursively enumerable languages. Every Turing machine recognizes a language. We write L(M) to denote the language recognized by M. By contrast, not every Turing machine decides a language. The Turing machine M decides L(M) if and only if it rejects all the inputs in {0,1}∗∖L(M). Or, equivalently: Proposition. The Turing machine M decides the language L(M) if and only if it halts on every input. ← IntroductionDecidable Languages → © 2025 Eric Blais. Last edited on January 13th, 2025
188996	https://math.stackexchange.com/questions/4241094/on-deriving-the-t-1-0-condition-for-tangents	geometry - On deriving the $T_1=0$ condition for tangents - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more On deriving the T 1=0 condition for tangents Ask Question Asked 4 years ago Modified4 years ago Viewed 205 times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. On this page, the first theorem and proof detail how the standard method to find the equation of the tangent to a conic (S 1=0 for a point (x 1,y 1)) works. Below equation (12), there's this statement: The line is tangent to the conic iff the quadratic equation has two equal roots, i.e. when D = 0 And the rest of the proof is based on the same idea. However, doesn't this condition only specify that the line intercepts the conic once? There aren't just tangents that do that; there can be lines that cut conics just once. A line with the slope of the asymptotes of a standard hyperbola but with a non-zero y-intercept is an example. That looks like an error in the proof, but I'm pretty sure it's correct, too. Where did I go wrong in my reasoning here? geometry analytic-geometry conic-sections tangent-line discriminant Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications asked Sep 3, 2021 at 17:53 harryharry 1,124 1 1 gold badge 9 9 silver badges 30 30 bronze badges 1 Condition D=0 means there are two coincident solution: for D>0 the line intersects the conic at two points, which for D→0 become coincident. This means the line becomes tangent.Intelligenti pauca –Intelligenti pauca 2021-09-03 20:10:34 +00:00 Commented Sep 3, 2021 at 20:10 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. Determinant zero means you get a single point of intersection with algebraic multiplicity two. The text you link to speaks of "two equal roots". In general if a line has an intersection with an algebraic curve (which includes conics) in a point with multiplicity greater than zero, then the direction of the line and the algebraic curve will match in that point. That's one way to define tangent. You can see this from a limit process: take two distinct points on the curve and move them towards one another. As they come closer, the line joining them aligns better and better with the direction of the curve at these points. In the limit, the two points will have become one, and the line will match the direction of the curve in that one point perfectly. Translated into your formulas, a slightly positive discriminant describes the situation with the two distinct points, and the determinant becoming zero is that limit process. So you see that at zero, you really have two points if intersection coincide. This is distinct from one point staying and one point just disappearing, which would not be possible using a smooth limit process. So a mental image of "just a single point" doesn't do the coinciding situations justice. A non-tangent line will intersect a non-degenerate conic in zero or two distinct points in general. This is true for the ellipse and in most cases also for the parabola and for the hyperbola with its two branches. It is not true for a single branch of a hyperbola, but in general you'd consider both branches as making up the conic. This corresponds to intersecting the plane not with a single cone, but works the pair of cones you get by rotating lines not rays around the vertex. A special case is asymptotic directions, as you noted. Let's start with the parabola. If the line goes in the direction of the axis of symmetry, then you only have one intersection, and your "quadratic" equation would actually become linear. You would have the coefficient of the quadratic term become zero. In the formula for computing the solutions, that would lead to a division by zero. If you do the same kind of limit process for a parabola, moving the line from a generic direction towards the direction where it becomes parallel to the axis, then you would observe one of the points moving further and further away. In the limit you'd have one intersection "at infinity", and the division by zero is the algebraic hint that this might be happening. Projective geometry allows for consistent and well-defined handling of such elements "at infinity". This can avoid a lot of case distinctions. In projective geometry, a parabola would always have two intersections with a secant, one of which may be at infinity. The same is true for the hyperbola if the direction of the line happens to coincide with one of the asymptotic directions. You'd get a vanishing quadratic coefficient and a linear equation. In projective geometry you would get two points at infinity, one for each asymptotic direction. You might imagine four points at infinity, since there are four asymptotic rays for the hyperbola. But in projective geometry, points at infinity in opposite directions are considered the same, so you'd only get two points at infinity, one for each asymptotic line. As an extra exercise, you can use the number of points at infinity to categorize conic sections. Zero points at infinity is an ellipse (including circles), one is a parabola and two is a hyperbola. But the set of all points at infinity forms a line in projective geometry. So counting the number of points at infinity is the same as counting the number of intersections with that line. And of course, the single intersection of the parabola again has algebraic multiplicity two. So you can say that a parabola is a conic section that has the line at infinity as a tangent. Neat, huh? 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188997	https://www.effortlessmath.com/math-topics/writing-two-variable-inequalities-word-problems/?srsltid=AfmBOopQ0h1cl-ipdhwZ9JBl0Fi1ekT1lAgDNabOIoyqeExMJnUCxN1e	How to Write Two-variable Inequalities Word Problems? - Effortless Math: We Help Students Learn to LOVE Mathematics File failed to load: Effortless Math X +eBooks +ACCUPLACER Mathematics +ACT Mathematics +AFOQT Mathematics +ALEKS Tests +ASVAB Mathematics +ATI TEAS Math Tests +Common Core Math +CLEP +DAT Math Tests +FSA Tests +FTCE Math +GED Mathematics +Georgia Milestones Assessment +GRE Quantitative Reasoning +HiSET Math Exam +HSPT Math +ISEE Mathematics +PARCC Tests +Praxis Math +PSAT Math Tests +PSSA Tests +SAT Math Tests +SBAC Tests +SIFT Math +SSAT Math Tests +STAAR Tests +TABE Tests +TASC Math +TSI Mathematics +Worksheets +ACT Math Worksheets +Accuplacer Math Worksheets +AFOQT Math Worksheets +ALEKS Math Worksheets +ASVAB Math Worksheets +ATI TEAS 6 Math Worksheets +FTCE General Math Worksheets +GED Math Worksheets +3rd Grade Mathematics Worksheets +4th Grade Mathematics Worksheets +5th Grade Mathematics Worksheets +6th Grade Math Worksheets +7th Grade Mathematics Worksheets +8th Grade Mathematics Worksheets +9th Grade Math Worksheets +HiSET Math Worksheets +HSPT Math Worksheets +ISEE Middle-Level Math Worksheets +PERT Math Worksheets +Praxis Math Worksheets +PSAT Math Worksheets +SAT Math Worksheets +SIFT Math Worksheets +SSAT Middle Level Math Worksheets +7th Grade STAAR Math Worksheets +8th Grade STAAR Math Worksheets +THEA Math Worksheets +TABE Math Worksheets +TASC Math Worksheets +TSI Math Worksheets +Courses +AFOQT Math Course +ALEKS Math Course +ASVAB Math Course +ATI TEAS 6 Math Course +CHSPE Math Course +FTCE General Knowledge Course +GED Math Course +HiSET Math Course +HSPT Math Course +ISEE Upper Level Math Course +SHSAT Math Course +SSAT Upper-Level Math Course +PERT Math Course +Praxis Core Math Course +SIFT Math Course +8th Grade STAAR Math Course +TABE Math Course +TASC Math Course +TSI Math Course +Puzzles +Number Properties Puzzles +Algebra Puzzles +Geometry Puzzles +Intelligent Math Puzzles +Ratio, Proportion & Percentages Puzzles +Other Math Puzzles +Math Tips +Articles +Blog How to Write Two-variable Inequalities Word Problems? Two-variable inequalities word problems involve using two variables and an inequality symbol to represent a relationship between two quantities in a real-world scenario. In these problems, there are two variables, often represented by 𝑥 x and 𝑦 y, and an inequality symbol, such as "<""<" (less than), ">"">" (greater than), "≤""≤" (less than or equal to), or "≥""≥" (greater than or equal to). A Step-by-step Guide to Two-variable Inequalities Word Problems Writing two-variable inequalities word problems involves several steps. Here’s a step-by-step guide: Step 1: Identify the variables Identify the two variables that will be used in the inequality. It’s essential to define what each variable represents to avoid any confusion in the problem. Step 2: Determine the inequality symbol Determine the inequality symbol that will be used in the problem. Common inequality symbols include “greater than,” “less than,” “greater than or equal to,” and “less than or equal to.” It’s important to select the correct symbol based on the problem’s context. Step 3: Write the inequality Write the inequality using the variables and the inequality symbol. Step 4: Formulate the word problem Write a word problem that uses the inequality. The word problem should be relevant to the inequality and include all necessary information, such as the values of the variables and any additional constraints. Step 5: Check for consistency Check the word problem to ensure that it’s consistent with the inequality. The problem should be solvable using the inequality, and the solution should satisfy the inequality. If there are any inconsistencies, revise the problem or inequality as necessary. Writing Two-variable Inequalities Word Problems – Examples 1 John has $500$500 to spend on shirts and pants. Shirts cost $20$20 each, and pants cost $30$30 each. Write an inequality that represents the possible combinations of shirts and pants John can buy. Solution: Step 1: Identify the variables Let 𝑥 x represent the number of shirts John can buy and 𝑦 y represent the number of pants he can buy. Step 2: Determine the inequality symbol Since John has a fixed amount of money, the inequality will involve an inequality symbol that limits the total cost of the shirts and pants. The inequality symbol in this case is “≤”“≤” (less than or equal to). Step 3: Write the inequality The inequality is: 20 𝑥+30 𝑦≤500 20 x+30 y≤500 Step 4: Formulate the word problem How many shirts and pants can John buy with $500$500 if he wants to spend all his money? Step 5: Check for consistency The word problem is consistent with the inequality. Writing Two-variable Inequalities Word Problems – Examples 2 Samantha is planning a trip to the beach and wants to rent beach chairs and umbrellas. She has a budget of $200$200 and chairs cost $10$10 each, while umbrellas cost $20$20 each. Write an inequality that represents the possible combinations of chairs and umbrellas Samantha can rent. Solution: Step 1: Identify the variables Let 𝑥 x represent the number of chairs Samantha can rent and 𝑦 y represent the number of umbrellas she can rent. Step 2: Determine the inequality symbol Since Samantha has a fixed amount of money, the inequality will involve an inequality symbol that limits the total cost of the chairs and umbrellas. The inequality symbol in this case is “≤”“≤” (less than or equal to). Step 3: Write the inequality The inequality is: 10 𝑥+20 𝑦≤200 10 x+20 y≤200 Step 4: Formulate the word problem How many chairs and umbrellas can Samantha rent with $200$200 if she wants to spend all his money? Step 5: Check for consistency The word problem is consistent with the inequality. by: Effortless Math Team about 3 years ago (category: Articles) Effortless Math Team 2 years ago Effortless Math Team Related to This Article How to Solve Inequality Word ProblemsInequalitiesInequalityInequality Word Problems More math articles 5 Best GED Math Study Guides Best Laptops for Online Math Teaching How to Integrate By Parts: Step-by-Step Guide How to Classify Polygons: A Step-by-Step Guide to Shape Identification How to Decode Data Types Top 10 Math Books for Grade 4: Empowering Young Minds to Discover Numbers 4th Grade PSSA Math Worksheets: FREE & Printable The Ultimate GRE Math Course: The Only Course You Need for Success Top 10 Calculus Books for High School Students The Best AFOQT Math Worksheets: FREE & Printable What people say about "How to Write Two-variable Inequalities Word Problems? - Effortless Math: We Help Students Learn to LOVE Mathematics"? No one replied yet. 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188998	https://practice-questions.wizako.com/gmat/quant/statistics-average/gmat-questions-mean-median-mode-range-standard-deviation-variance.shtml	12 GMAT Statistics Sample Questions \| Mean, Median, Mode, Range, Standard Deviation Practice Questions We value your privacy We use cookies to enhance your browsing experience, serve personalised ads or content, and analyse our traffic. By clicking "Accept All", you consent to our use of cookies. Customise Reject All Accept All Customise Consent Preferences We use cookies to help you navigate efficiently and perform certain functions. You will find detailed information about all cookies under each consent category below. 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The concepts tested include Averages or simple arithmetic mean, weighted average, median, mode, range, variance and standard deviation. Sample GMAT practice questions from statistics & averages is given below. Attempt these questions and check whether you have got the correct answer. If you face difficulty with arriving at the answer to any question, go to the explanatory answer or the video explanations (provided for all questions) to learn how to crack the GMAT sample question in this question bank. The question bank also includes GMAT Data Sufficiency questions in Statistics and Averages. In the new pattern GMAT Focus Edition, Data Sufficiency (DS) questions appear as part of the Data Insights section. DS is no longer tested as part of the quantitative reasoning section in the GMAT Focus exam. Ideally, you should start by watching these two GMAT Math lesson videos in Statistics and Averages to help you get better traction when solving the questions given below. Play Video: GMAT Statistics & Averages Lesson Video 1 Play Video: GMAT Statistics & Averages Lesson Video 2 Unlock the Secret Code: Your GMAT Stats & Averages Cheat Sheet Download the Free Stats Cheat Sheet Now ➧ If the mean of numbers 28, x, 42, 78 and 104 is 62, what is the mean of 48, 62, 98, 124 and x ? 78 58 390 310 66 Correct AnswerChoice A Mean is 78 ExplanationVideo Solution Hint to solve this GMAT Statistics sample question 'x' is common to both the sets of numbers. Therefore, 'x' will not impact the average of the second set of numbers. Step 1: Find out the sum of the 4 numbers other than x in both scenarios. Step 2: Apportion the difference between the two sums obtained in step 1 equally to all the 5 numbers to find the difference in average. Step 3: Add or subtract the difference to the average of the first set, i.e., 62 to find the answer. The explanatory answer to this GMAT Averages practice question walks you through two different methods to find the answer. An easy GMAT 600 level quant question. The arithmetic mean of the 5 consecutive integers starting with 's' is 'a'. What is the arithmetic mean of 9 consecutive integers that start with s + 2 ? 2 + s + a 22 + a 2s 2a + 2 4 + a Correct AnswerChoice E Mean is 4 + a ExplanationVideo Solution Hint to solve this GMAT Statistics & Average sample question This GMAT practice question is an Averages question - an easy GMAT 550 to 600 level quant question. The easiest way to solve this averages problem is to assume values for the 5 consecutive integers. Do not look too far. Let the 5 numbers be 1, 2, 3, 4, and 5. Determine values for 's', and 'a' with respect to the values assumed and apply it to the set of 9 consecutive integers starting from (s + 2). The average weight of a group of 30 friends increases by 1 kg when the weight of their football coach was added. If average weight of the group after including the weight of the football coach is 31 kg, what is the weight of their football coach ? 31 kg 61 kg 60 kg 62 kg 91 kg Correct AnswerChoice B 61 kg ExplanationVideo Solution Hint to solve this GMAT Averages practice question This GMAT question is a statistics and averages question - An easy GMAT 525 level quant practice question. The question is an ideal candidate to apply and consolidate the standard framework to solve an averages question. Step 1: Find the sum of the weights of the 30 friends after deducing the average weight of the 30 friends. Step 2: Compute the sum of the weights of 30 friends and the football coach from the information about the average given in the question. Step 3: The difference between answers obtained in steps 2 and 1 is the weight of the coach. The average wages of a worker during a fortnight comprising 15 consecutive working days was $90 per day. During the first 7 days, his average wages was $87/day and the average wages during the last 7 days was $92 /day. What was his wage on the 8 th day ? $83 $92 $90 $97 $104 Correct AnswerChoice D $97 ExplanationVideo Solution Hint to solve this GMAT Statistics sample question This GMAT question is an easy averages question that can be solved using the standard framework to solve GMAT averages problems. Step 1: Compute the sum of wages received for all 15 days using the verage wages for 15 days. Step 2: Compute sum of wages for first 7 days using average wages for the first 7 days. Step 3: Compute sum of wages for last 7 days using average wages for the last 7 days. The sum of the second and third step gives the sum of wages for 14 of the 15 days. Compute the difference between wages for 15 days and 14 days to find wage on 8th day. The average of 5 numbers is 6. The average of 3 of them is 8. What is the average of the remaining two numbers ? 4 5 3 3.5 0.5 Correct AnswerChoice C Average is 3 ExplanationVideo Solution Approach to solve this GMAT Averages problem solving practice question This GMAT averages problem is a vey easy question. Step 1: Compute sum of all 5 numbers using the average of the 5 numbers. Step 2: Compute sum of 3 of the 5 numbers using the average of the 3 numbers. Step 3: The difference between values obtained in step 1 and step 2 will give the sum of the remaining 2 numbers. Use this information to compute the average of the remaining two numbers. The average age of a group of 10 students was 20. The average age increased by 2 years when two new students joined the group. What is the average age of the two new students who joined the group ? 22 years 30 years 44 years 32 years None of these Correct AnswerChoice D 32 years ExplanationVideo Solution Hint to solve this GMAT 575 Level Statistics sample question Step 1: Set up the standard framework table and populate it with available data about number of students and their average age before the two students join the group and after two students join the group Step 2: Compute sum of ages of the group before and after the students join the group Step 3: The difference between the sum of the ages after the 2 students join and that before the 2 students join gives the sum of the ages of the 2 new students Step 4: Divide the value computed in Step 3 by 2 to find the answer. If m, s are the average and standard deviation of integers a, b, c, and d, is s > 0 ? m > a a + b + c + d = 0 Correct AnswerChoice A ExplanationVideo Solution Hint to solve this GMAT Statistics sample question Step 1: Decoding the Question: Standard deviation is a non-negative number. So, it can either be zero or positive. So, the question boils down to figuring out whether 's' is zero or is it positive. Step 2: Evaluate statement 1 alone to determine whether 's' is zero or positive Step 3: Evaluate statement 2 alone to determine whether 's' is zero or positive Step 4: If the statements are independently not sufficient, combine the statements to determine whether answer is C or E Positive integers from 1 to 45, inclusive are placed in 5 groups of 9 each. What is the highest possible average of the medians of these 5 groups ? 25 31 15 26 23 Correct AnswerChoice B 31 ExplanationVideo Solution Approach to solve this GMAT Hard Math Statistics sample question Step 1: To maximize the average of the medians of the 5 sets, we have to maximize the median in each set. Step 2: Try to divide the given numbers into sets of 9 each such that the median is as high as possible. Start with the set where the 5 largest numbers are 41, 42, 43, 44, and 45. Step 3: Note, to maximize median in each set, it is imperative that the first 4 numbers of each set is as small as possible so that the 5 larger numbers in any set gets maximized. Step 4: After you have identified the maximized medians for each set, compute the average of the medians to find the answer. If the average of 5 positive integers is 40 and the difference between the largest and the smallest of these 5 numbers is 10, what is the maximum value possible for the largest of these 5 integers ? 50 52 49 48 44 Correct AnswerChoice D 48 ExplanationVideo Solution Approach to solve this GMAT 650 Level Statistics sample question Step 1: Compute the sum of the 5 integers using information about the average of these numbers. Step 2: Because the range is 10, the largest number is 10 more than the smallest number. Step 3: If the largest number has to be maximized, the remaining 4 numbers have to be minimized. Step 4: Find the condition that will minimize the remaining 4 numbers and consquently find the maximum value of the largest of the 5 numbers. An analysis of the monthly incentives received by 5 salesmen : The mean and median of the incentives is $7000. The only mode among the observations is $12,000. Incentives paid to each salesman were in full thousands. What is the difference between the highest and the lowest incentive received by the 5 salesmen in the month ? $4000 $13,000 $9000 $5000 $11,000 Correct AnswerChoice E $11,000 ExplanationVideo Solution Hint to solve this GMAT Statistics sample question Step 1: Compute the sum of the incentives. Step 2: Median incentive is $7000. So, the 3rd highest is $7000. Step 3: Only mode is $12,000. Because third highest is $7000, highest and second highest have to be the same value = $12,000 Step 4: Now that we have details about the incentives for the 3 highest salesman and we also know the sum of their incentive, we can find the sum of the incentives of the two salesmen who got the least incentive. Step 5: Use the information that there is only one mode to find the lowest incentive. Step 6: Compute the difference between the highest and lowest incentives - essentially the range of incentives. Is 'b' the median of 3 numbers a, b, and c ? b a b a = c b c b ab < 0 Correct AnswerChoice C ExplanationVideo Solution Approach to solve this GMAT Statistics DS question Step 1: The given question is an "IS" question. The answer should be yes or no. Data is sufficient when we get a definite yes or definite no. Step 2: Evaluate statement 1 alone. We know the numbers are in GP. Check whether 'b' is the median of the 3 numbers. Points to check are - will 'b' be the median if the common ratio is negative. Step 3: Evaluate statement 2 alone. Look for a counter example. Step 4: Combine the statements if you did not get a conclusive answer using either statements alone. What is the standard deviation (SD) of the four numbers p, q, r, and s? The sum of p, q, r, and s is 24. The sum of the squares of p, q, r, and s is 224. Correct AnswerChoice C ExplanationVideo Solution Approach to solve this GMAT Statistics DS question Step 1: The given question is a "What is" question. The answer is a value. Data is sufficient if we get a unique value for the standard deviation of the 4 numbers. Step 2: Evaluate statement 1 alone. Will the sum of the 4 numbers help find the SD? If not, will it help in determining the average of the 4 numbers? Step 3: Evaluate statement 2 alone. Sum of squares alone is of not much use. Step 4: Combine the statements. Revisit formula (alternative method) to find SD and determine whether we have a unique value for the SD of the 4 numbers. GMAT Online Course Try it free! 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188999	https://www.chegg.com/homework-help/questions-and-answers/consider-two-large-infinite-parallel-planes-diffuse-gray-temperatures-emissivities-t-1-var-q112191711	Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on. Question: Consider two large (infinite) parallel planes that are diffuse-gray with temperatures and emissivities of T1, ε1 and T2,ε2. Show that the ratio of the radiation transfer rate with multiple shields, N, of emissivity εs to that with no shields, N=0, is q12,0q12,N=[1/ε1+1/ε2−1]+N[2/εs−1][1/ε1+1/ε2−1] where q12,N and q12,0 represent the radiation heat transfer The radiation heat transfer rate between two diffuse-gray parallel planes can be calculated using th... Not the question you’re looking for? Post any question and get expert help quickly. Chegg Products & Services CompanyCompany Company Chegg NetworkChegg Network Chegg Network Customer ServiceCustomer Service Customer Service EducatorsEducators Educators

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