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https://en.wikipedia.org/wiki/Serial_dilution
Serial dilution - Wikipedia Jump to content [x] 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 [x] Appearance 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. Donate Create account Log in [x] Personal tools Donate Create account Log in Pages for logged out editors learn more Contributions Talk [x] Toggle the table of contents Contents move to sidebar hide (Top) 1 In biology and medicine 2 In homeopathy 3 See also 4 References 5 External links Serial dilution [x] 8 languages Deutsch Español فارسی Italiano Bahasa Melayu Nederlands Русский 中文 Edit links Article Talk [x] English Read Edit View history [x] Tools Tools move to sidebar hide 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 Expand all Edit interlanguage links Print/export Download as PDF Printable version In other projects Wikimedia Commons Wikidata item From Wikipedia, the free encyclopedia Step-wise dilution of a substance in solution Logarithmic dilution A serial dilution is the step-wise dilution of a substance in solution, either by using a constant dilution factor, or by using a variable factor between dilutions. If the dilution factor at each step is constant, this results in a geometric progression of the concentration in a logarithmic fashion. A ten-fold serial dilution could be 1 M, 0.1 M, 0.01 M, 0.001 M ... Serial dilutions are used to accurately create highly diluted solutions as well as solutions for experiments resulting in concentration curves with a logarithmic scale. A tenfold dilution for each step is called a logarithmic dilution or log-dilution, a 3.16-fold (10 0.5-fold) dilution is called a half-logarithmic dilution or half-log dilution, and a 1.78-fold (10 0.25-fold) dilution is called a quarter-logarithmic dilution or quarter-log dilution. Serial dilutions are widely used in experimental sciences, including biochemistry, pharmacology, microbiology, and physics. In biology and medicine [edit] Serial dilution and plating of bacteria In biology and medicine, besides the more conventional uses described above, serial dilution may also be used to reduce the concentration of microscopic organisms or cells in a sample. As, for instance, the number and size of bacterial colonies that grow on an agar plate in a given time is concentration-dependent, and since many other diagnostic techniques involve physically counting the number of micro-organisms or cells on specials printed with grids (for comparing concentrations of two organisms or cell types in the sample) or wells of a given volume (for absolute concentrations), dilution can be useful for getting more manageable results. Serial dilution is also a cheaper and simpler method for preparing cultures from a single cell than optical tweezers and micromanipulators. In homeopathy [edit] Main articles: Homeopathy and Homeopathic dilutions Serial dilution is one of the core foundational practices of homeopathy, with "succussion", or shaking, occurring between each dilution. In homeopathy, serial dilutions (called potentisation) are often taken so far that by the time the last dilution is completed, no molecules of the original substance are likely to remain. See also [edit] Dilution (equation) Hormesis References [edit] ^K. R. Aneja. Experiments in Microbiology, Plant Pathology and Biotechnology. New Age Publishers, 2005, p. 69. ISBN81-224-1494-X ^Booth, C.; et al. (2006). Extremophiles. Methods in microbiology 35. Academic Press. p.543. ISBN978-0-12-521536-7. ^Weissmann, Gerald (2006). "Homeopathy: Holmes, Hogwarts, and the Prince of Wales". The FASEB Journal. 20 (11): 1755–1758. doi:10.1096/fj.06-0901ufm. PMID16940145. S2CID9305843. Retrieved 2008-02-01. ^Ernst, Edzard (November 2005). "Is homeopathy a clinically valuable approach?". Trends in Pharmacological Sciences. 26 (11): 547–548. CiteSeerX10.1.1.385.5505. doi:10.1016/j.tips.2005.09.003. PMID16165225. Michael L. Bishop, Edward P. Fody, Larry E. Schoeff. Clinical Chemistry: Principles, Procedures, Correlations. Lippincott Williams & Wilkins, 2004, p.24. ISBN0-7817-4611-6. External links [edit] How to Make Simple Solutions and Dilutions, Bates College | hide v t e Chemical solutions | | Solution | Ideal solution Aqueous solution Solid solution Buffer solution Flory–Huggins Mixture Suspension Colloid Phase diagram Phase separation Eutectic point Alloy Saturation Supersaturation Serial dilution Dilution (equation) Apparent molar property Miscibility gap | | Concentration and related quantities | Molar concentration Mass concentration Number concentration Volume concentration Normality Molality Mole fraction Mass fraction Isotopic abundance Mixing ratio Ternary plot Total dissolved solids | | Solubility | Solubility equilibrium Solvation Solvation shell Enthalpy of solution Lattice energy Raoult's law Henry's law Solubility table (data) Solubility chart Miscibility | | Solvent | (Category) Acid dissociation constant Protic solvent Polar aprotic solvent Inorganic nonaqueous solvent Solvation List of boiling and freezing information of solvents Partition coefficient Polarity Hydrophobe Hydrophile Lipophilic Amphiphile Lyonium ion Lyate ion | Retrieved from " Categories: Laboratory techniques Homeopathy Hidden categories: Articles with short description Short description is different from Wikidata This page was last edited on 8 November 2023, at 15:12(UTC). Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Code of Conduct Developers Statistics Cookie statement Mobile view Edit preview settings Search Search [x] Toggle the table of contents Serial dilution 8 languagesAdd topic
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https://www.ck12.org/c/geometry/triangle-angle-sum-theorem/rwa/triangle-sum-theorem/
Triangle Angle Sum Theorem The interior angles of a triangle add to 180 degrees Use equations to find missing angle measures given the sum of 180 degrees. Notes/Highlights | Color | Highlighted Text | Notes | | --- --- | | Show More | | | | Image Attributions ShowHide Details Description Difficulty Level Date Created: Last Modified: Language Concept Nodes: ShowHide Resources
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https://www.neuralconcept.com/post/conjugated-heat-transfer-best-practices-applications
Heat transfer is the phenomenon of energy exchange due to temperature differences. Heat transfer is seldom restricted to a single material or medium. Real-world systems involve continuous interactions between solids and fluids. Conjugate heat transfer (CHT) captures this dynamic. CHT models how heat flows between solid structures and the surrounding fluids as a cohesive system. In this article, we will break down the fundamentals of heat energy across multiple media. We will explore applications of this mode of energy transfer. Also, we will discuss howapplying machine learning in CFD can enhance CHT analysis for engineering. What is Conjugate Heat Transfer? Conjugate Heat Transfer, or CHT, is the concurrent heat energy exchange between a solid and adjacent fluid(s). It integrates conductive heat transfer in the solid and convective heat transfer in the fluid. A key aspect of CHT analysis isheat flux continuityat the interface, representing thermal interactions between media. For instance, CHT can be encountered in varioustypes of heat exchangers. What is Heat Exchange and why It Occurs (Macroscopic Explanation) Energy exchange occurs due to the second law of thermodynamics. This law dictates that heat flows from higher to lower temperatures, increasing entropy. This transfer happens via conduction, convection, or radiation. The outcome is that systems move toward thermal equilibrium. Let us start with a crude representation of two “cells” or blocks. Cell 1 has a temperature T₁, and cell 2 has a temperature T₂ (no temperature profiles develop inside the cells or blocks). This initial situation has a lower entropy S. There is a difference ΔT = T₁ - T₂, and the natural tendency is for ΔT → 0, leading to thermal equilibrium. As energy flows as heat, entropy increases to a new value S’ > S, reaching a maximum at equilibrium. What are the CHT Outputs Coupling conduction and convection, CHT is a tool for predicting temperature spatial distribution, heat transfer rates, and critical thermal parameters. These parameters are essential for optimizing designsin the automotive, aerospace, civil engineering, electronics cooling, and energy industries. Don't Avoid Radiation In many cases,radiation heat transfer is also essential. This is particularly true when radiation can become the dominant transfer mode at high temperatures! So, while CHT focuses on conduction and forced and natural convection, advanced simulations can incorporate thermal radiation. Radiation must be considered in industrial furnaces, spacecraft thermal design, and high-temperature reactors to avoid significant errors in heat transfer predictions. Naive Example A convection oven in our kitchen is a simple example of fully combined heat transfer. Forced convection (via a fan) and natural convection (via buoyancy) circulate air; conduction transfers heat through metal surfaces and food, while radiation from the oven walls and heating elements directly heats the meal. a convection oven exemplifies combined heat transfer phenomena This illustrates how radiation interacts with conduction and convection. Fluids are generally transparent to radiation, while solids are opaque, leading to surface-to-surface radiation in cavities. However, both can be semitransparent, allowing radiation to be absorbed, emitted, and scattered. Even if negligible at low temperatures, radiation dominates when temperature differences are significant. Conduction and convection are proportional to ΔT, i.e., they exhibit linear proportionality to the temperature difference, while radiation follows a nonlinear relationship, scaling as T⁴. GPU example: forced convection How Does Conjugate Heat Transfer Work? Conjugate heat transfer refers to the combined interactions between fluids and solids. Conduction, forced and natural convection, and occasionally radiation coexist. While fluids depend on forced and natural convection to transfer heat, solids conduct heat. Many situations need a unified approach to thermal analyses. Below, we investigate the fundamental modes of heat transfer. Later, we will show typical heat exchanger design software from CFD to Deep Learning. the modes of heat transfer (pressbooks.bccampus.ca | Douglas College Physics 1207) Heat Transfer Process in Fluids Heat transfer occurs through conduction and convection in fluids, with convective heat transfer often dominating due to bulk fluid motion. The fluid usually plays the role of an energy carrier over large distances. Buoyancy forces drive natural convection, while forced convection is typically fluid flow induced by external forces, such as fans for air flow and pumps for liquid coolant flow. Convection allows fluids to distribute heat more than conduction alone. Forced convection is the most common way to achieve a high heat transfer rate in thermal management applications. Depending on the flow conditions, water can exhibit convective heat transfer coefficients between 100 and 10,000 W/(m²·K) in forced convection. Pure conduction in liquids like water is significantly lower, typically ranging from 0.5 to 5 W/(m²·K). The following equation (Newton’s law of cooling) quantifies convective heat transfer: Q = h A ΔT Q = heat transfer rate, h = convective heat transfer coefficient, A = heat transfer surface area, and ΔT = the difference in temperature between the fluid and surface. The temperature difference (ΔT) between the fluid and the surface is a driving force behind convection. Ultimately, the convective heat transfer rate depends on ΔT, A, and the complex interactions governing the convective heat transfer coefficient h—fluid velocity, turbulence intensity, and surface conditions—determining how effectively heat moves within a fluid system. The convective heat transfer coefficient h encapsulates multiple physical effects, including fluid motion, boundary layer behavior, and turbulence. As fluid flows over a surface, it forms a boundary layer—a thin region where velocity and temperature gradients develop. The nature of this layer (laminar or turbulent) significantly influences heat transfer. With intense mixing and enhanced interaction, turbulent flow increases h, leading to larger heat exchange. Heat Transfer Process in Solids Heat conduction in solids transfers thermal energy from hot to cool areas. Wood, brick, wool, and steel contain different insulation levels. Fourier's Law describes heat flux in solids based on the temperature gradient (∇T). Heat flux is proportional to the temperature gradient and the material's thermal conductivity. This law aids engineers and scientists in designing thermal systems. Heat conduction explains thermal energy transmission in metals and insulators. Research by pioneers like Joseph Fourier advanced our understanding of heat transfer. Jean-Baptiste Biot's 19th-century experiments measured heat flow through materials, laying the groundwork for thermal conductivity. Fourier's Law links heat flux, temperature gradient, and thermal conductivity. J.B.J. Fourier - a father of analytical studies of heat transfer (Wikipedia) Thermal Conductivity Thermal conductivity is a material property that quantifies a substance’s ability to conduct heat. It is measured in Watts per meter-kelvin (W/(m·K)). The thermal conductivity value of solids can vary depending on the material. Metals like copper and aluminum exhibit high values while insulating materials like wood and foam have much lower values. While thermal conductivity is often associated with solids, this is a misconception. Thermal conductivity may be underestimated in gases and liquids because convective heat transfer dominates. In these cases, effective heat transfer capability involves thermal conductivity and the influence of fluid motion, which enhances heat distribution. Thus, while fluids may have lower thermal conductivity values than solids, their ability to transfer heat through convection can be considerable. Fundamental Differences: Conductive and Convective heat transfer The linearity in conduction and convection stems from their governing equations (Fourier’s law and Newton’s law of cooling), which describe how heat transfer responds linearly to temperature differences. However, the underlying physics is different! Conductionrelies on energy transfer through molecular interactions in materials. The linear relationship occurs because the heat transfer rate is proportional to differences in temperature, fundamentally linked to the material properties, i.e., thermal conductivity k. Thermal conductivity is a measurable material property that tells us how much a substance can conduct heat at a given temperature. Convectionconcerns the bulk movements of fluid and energy transfer resulting from this fluid motion. The linear relationship in convection assumes the heat transfer rate increases directly with the temperature difference between the surface and the fluid. Warmer fluid rises while cooler fluid sinks, creating convective currents. The convective heat transfer coefficient (h) is a measurable property that characterizes the effectiveness of convective heat transfer in a fluid. It incorporates factors related to fluid motion, flow dynamics, and fluid properties, making it way more complex than the thermal conductivity of solids! Art and Physics: natural convection (Rebecca Siegel | Flickr) For the eager, simulation-conscious readers, let us anticipate that modeling conduction numerically can be effectively handled without CFD while simulating convection requires it, becoming significantly more complex and challenging, especially in natural convection. Interaction Between Solids and Fluids When solids and fluids interact thermally, heat transfer occurs at their interface, combining conduction within the solid and convection in the fluid. This interplay appears in heat exchangers, cooling systems, and thermal insulation applications. The solid conducts heat internally at the interface, while the fluid transports heat away through convective motion. Modeling CHT requires solving coupled heat transfer equations for both domains. This ensures accurate predictions of temperature distribution and thermal performance in complex systems. The efficiency of this process depends on: • Thermal conductivity (k) of the solid – High-conductivity materials (e.g., metals) facilitates rapid heat distribution, while insulators (e.g., ceramics) slow it down. • Convective heat transfer coefficient (h) of the fluid – Higher values indicate enhanced heat exchange, influenced by fluid velocity, turbulence, and properties. •Contact surface area (A) and temperature difference (ΔT): Heat transfer is driven by larger interfaces and greater temperature differences. Steady-State and Transient, Heat Sinks and Heat Sources A typical CHT scenario involves a heated solid surface in contact with a cooling fluid. Heat spreads within the solid via conduction, reaching the interface where it is transferred to the fluid. Flow characteristics such as turbulence and boundary layer development in forced convection significantly impact heat dissipation. A typical CHT scenario concerns cooling. A heated surface is in contact with a cooling fluid to cool a solid. Heat spreads within the solid via conduction, reaching the interface and transferring it to the fluid. Flow characteristics such as turbulence and boundary layer development in forced convection significantly impact heat dissipation. The phenomenon can betransient, where the system starts in a non-equilibrium state and evolves toward equilibrium, or steady, where continuous heat input and removal create a stable temperature distribution. For example, if a heat source exists on the solid side, CHT helps determine how to optimally conduct heat to the heat sink, which is often aided by devices like fans or pumps. Solid heat sinks are usually made of metal with high thermal conductivity, such as copper or aluminum. Example of Conjugate Heat Transfer in Action A clear example of conjugate heat transfer at work is the thermal response difference in a house with stone walls during summer. Sun-heated walls absorb heat, but the air's higher thermal diffusivity leads to slower temperature changes indoors. This "buffering effect" helps keep the interior cooler. The stone gradually releases stored heat at night, stabilizing indoor temperatures and preventing fluctuations. The stone's greater thermal inertia ensures stable interior temperatures despite changes outside. thermal inertia - familiar example of Conjugate Heat Transfer in action Governing Equations for Conjugate Heat Transfer When analyzing conjugate heat transfer, the objective is to simultaneously solve the heat conduction equation in solids and the conjugate convective heat transfer equation in fluids. The governing equation for heat conduction is the Fourier unsteady-state equation: _∇ (k∇T) = ρc ∂T/∂t_ where material properties are _k_= the thermal conductivity of the solid material,_ρ_= the material density, and _c_= the specific heat capacity, we want to solve for temperature _T=T(x,y,z;t)_, i.e., a temperature distribution over space with a spatial gradient _∇T_ and over time with a time derivative _∂T/∂t_. The boundary conditions for solid and fluid domains must be appropriately defined to ensure a consistent and accurate analysis. Numerical Methods for 3D Heat Transfer Simulations Solving conjugate heat transfer problems in three dimensions requires numerical methods for handling coupled conduction and convection equations. Analytical solutions are often impractical for complex geometries. Computational approaches like the Finite Volume Method (FVM), Finite Element Method (FEM), and Finite Difference Method (FDM) are standard. 1. FVM is most used in CFD. It discretizes governing equations over control volumes, ensuring energy conservation across each cell. Thus, the Finite Volume Method suits fluid flow and convection-dominated issues. 2. FEM divides the domain into elements and applies weighted residual techniques to approximate temperature fields, offering flexibility for complex geometries and anisotropic materials. 3. The Finite Difference Method (FDM) uses Taylor series approximations to replace differential equations with algebraic equations on a structured grid. While simple, it is less adaptable to irregular geometries than FVM and FEM, which is why FVM and FEM are more used in commercial codes for engineers. Simulation Setup and Best Practices for CHT Accurate simulations in conjugate heat transfer require a systematic approach that follows precise guidelines. By following these practices, engineers can effectively utilize this analytical method and gain insightful information about heat transfer interactions in solid and fluid domains. Geometry Preparation A well-prepared geometry is crucial for accurate CHT simulations. Ensure all solid-fluid interfaces are adequately defined and remove unnecessary geometric details that could increase computational cost without improving accuracy. Avoid sharp edges or thin walls that could cause meshing issues and numerical instabilities. Mesh Quality Mesh generation forms the basis of any simulation. In conjugate heat transfer, a well-structured mesh is crucial for capturing the geometry and resolving temperature and velocity gradients. A balanced mesh is necessary — sufficiently detailed to represent complex features without high computational costs. Example of finite volume polyhedral (core) and prismatical / directed cells with Simcenter STAR-CCM+ (doi.org/10.3389/fenrg.2018.00035) CAE has advanced in conjugate heat transfer simulation. The model analyzes a vehicle's chassis to determine how heat affects solid and plastic components. High-quality meshes are crucial for intricate geometries. Thin-layer meshing creates high-resolution elements in thin solids, allowing engineers to capture thermal gradients, heat transfer, and fluid dynamics while maintaining computational efficiency. Mesh Refinement and Element Sizing Guidelines Increase mesh density at solid-fluid interfaces to accurately resolve temperature jumps and heat flux. Best practice iN MESHING: Boundary Layer Resolution Use inflation layers for accurate velocity and temperature gradients near walls in convection-dominated cases (see above figure). Best practice iN MESHING: Thin Solids Meshing Structured hexahedral or prism elements are used in thin regions to maintain accuracy without excessive element count. Best practice iN MESHING: Gradual Transitions Avoid abrupt changes in element size to prevent numerical instabilities and ensure smooth solution convergence. mesh smoothing (download.autodesk.com) Best practice iN MESHING: Adaptive Refinement Utilize solver-based adaptive meshing to refine areas with high thermal gradients dynamically. Boundary Conditions Defining boundary conditions, like wall temperature, is vital for reliable simulation. Accurate conditions reflect the physical situation, ensuring the integrity of the temperature field. Solid and fluid domains require careful treatment of the interface in conjugate heat transfer. Conditions in solids specify temperatures or heat fluxes at surfaces interacting with fluid. CFD tools can treat flow and solid domains as interconnected, exchanging wall temperatures at interfaces governed by different physical laws. Convective conditions address heat exchange with fluid, incorporating the convective heat transfer coefficient and ambient temperature. In fluid domains, conditions include inflow/outflow velocities, wall temperature, and pressure. Solver Settings Choosing the right solver and convergence criteria can guarantee accurate conjugate heat transfer simulations. The choice between segregated and coupled solvers hinges on the intensity of the solid-fluid interaction. Coupled solvers ensure stability in strong interactions, while segregated solvers lower costs in weaker coupling. Regarding flow regimes, pressure-based solvers suit low-speed flows, while density-based solvers effectively manage high-speed compressible flows. If radiation is significant, especially at high temperatures, include DO (Discrete Ordinates) or S2S (Surface-to-Surface) models. Turbulence Modeling Turbulence modeling cannot be ignored in heat transfer because it influences fluid behavior and mixing. This leads to differences in heat transfer coefficients ranging from 2 to 10 times higher than those in laminar flow conditions. Heat transfer is primarily governed by conduction in laminar flow, with lower Nusselt numbers (Nu), typically around 3.66 for fully developed laminar flow in a circular pipe. In contrast, turbulent flow can achieve Nu> 100. This means stronger convective heat transfer due to increased mixing and energy transport. The _k-ω SST_ turbulence model ensures reliable boundary layer resolution, effectively capturing flow characteristics near walls and providing accurate predictions in turbulent regimes. Hybrid RANS/LES models can be very effective at capturing transient effects, such as unsteady flow patterns and vortex dynamics, leading to improved predictions of heat transfer rates and overall system behavior in complex thermal environments. Reaching Convergence Convergence criteria require precise definition. Momentum, energy, and turbulence residuals should be below 10⁻⁴ to 10⁻⁶, with energy ideally at 10⁻⁶ for reliable predictions. Verifying heat flux balance at solid-fluid interfaces confirms thermal equilibrium. Monitoring temperature and velocity profiles ensures consistency, while adjusting iterative under-relaxation factors can enhance stability in complex cases. Proper tuning leads to realistic CHT simulations. Transient Heat Transfer Transient phenomena in solids and fluids exhibit varying time scales. Solids typically exhibit slower thermal responses due to higher thermal diffusivities, whereas fluids respond more rapidly. In the summertime, a house with massive stone walls remains cool due to the stone's high thermal inertia. During the day, the solid walls absorb heat from the sun, but due to the slow thermal response of the solid, the interior remains cooler than the outside air. The walls gradually release stored heat at night, maintaining a comfortable interior temperature. In fluid heat transfer, transient conduction describes the temporal evolution of temperature through a fluid medium due to temperature differences. example of two transient temperature profiles (Anthony Massobrio | Complex Systems) Transient Conduction Equations The transient conduction equation is a partial differential equation representing this phenomenon is the known _∇( k∇T)= ρ c (∂T/∂t)_ where ρ, c, and k are the usual material properties (density, specific capacity, and conductivity) for the fluid. Adding a heat sink or heat source term "_q_," it becomes: _∇ (k∇T) + q = ρ c (∂T/∂t)_ The unsteady-state heat conduction equation for the solid is, in a similar fashion, with the possibility of a heat sink or heat source term "_q_": _∇ ( kₛ ∇T) + q = ρₛ cₛ (∂T/∂t)_ Now, transient phenomena in heat transfer manifest distinct time scales in solids and fluids, with their thermal responses governed by thermal diffusivity and material properties. This phenomenon is illustrated in the cooling behavior of a house with massive stone walls during summertime, shedding light on the interplay between heat transfer mechanisms. Due to their comparatively higher thermal diffusivities, solids exhibit slower thermal responses. Solid-Flow Interface The temperature field and heat flux maintain continuity at the solid-flow interface. Nevertheless, within a moving fluid, the temperature field can exhibit rapid fluctuations: in proximity to the solid, the fluid's temperature closely mirrors that of the solid, whereas farther from this interface, the fluid's temperature aligns with the inflowing or ambient fluid temperature. The thermal boundary layer is the spatial range over which fluid temperature transitions from solid-equivalent to bulk-fluid temperature. The thermal boundary layer dimensions, which are about the dimensions of the momentum boundary layer, are expressed with the Prandtl number. The Prandtl number reflects the ratio of momentum diffusivity to thermal diffusivity. Achieving a Prandtl number of 1 necessitates equivalence between thermal and momentum boundary layer widths. A thicker momentum layer would yield a Prandtl number surpassing 1. Conversely, a Prandtl number below 1 signifies a thinner momentum boundary layer than the thermal boundary layer. As a reference, the Prandtl number for air at atmospheric pressure and 20°C is 0.7. For water at 20°C, the Prandtl number is about 7. These values reflect the relative importance of momentum diffusivity (kinematic viscosity) to thermal diffusivity in each fluid. Mathematics of Conjugate Heat Transfer: Looking for Analytical Solutions The heat diffusion equation describes how temperature evolves within a medium. Solving it analytically can be complex, but solutions exist for simple cases with well-defined boundary conditions. Let’s consider a rod of length L, initially at a uniform temperature T₀, with one end fixed at a constant temperature (Dirichlet condition) and the other insulated (Neumann condition). We use variable separation to determine the temperature distribution over time, breaking the solution into spatial and time-dependent parts. The resulting solution expresses temperature as a sum of modes, each decaying over time due to heat conduction. These modes describe heat energy phenomena along the rod, with faster-decaying modes smoothing out temperature variations more quickly The key takeaways are: • Heat spreads over time, with sharp temperature differences fading away. • The solution consists of multiple “modes,” each contributing to the overall temperature profile. • Higher modes decay faster, meaning the rod gradually approaches thermal equilibrium. This approach helps predict how heat evolves in practical applications such as thermal insulation, metal cooling, and heat exchangers. Design Optimization of Heat Exchangers Numerical Methods: Solving Complex Equations Conjugate heat transfer simulations use numerical approaches for heat conduction and convection. Finite Element Analysis (FEA), also known as "structural analysis," addresses heat transfer in solids, while Computational Fluid Dynamics (CFD) models fluid and air energy transport, including phase changes. Accurately representing temperature fields in solids yields more precise fluid thermal boundary layer temperatures. Choosing appropriate numerical techniques and solvers ensures accuracy. Thoughtful selections of discretization schemes and algorithms clarify energy transport in conjugate systems. More Insight Into FEA and CFD and Temperature Distribution Prediction FEA and Computational Fluid Dynamics (CFD) are powerful numerical techniques for modeling complex thermal phenomena. These methodologies are widely used in engineering and scientific disciplines to analyze intricate systems' temperature distribution, fluid flow, and heat exchange. FEA involves partitioning a given domain into discrete elements, enabling precise approximations of temperature profiles. This procedure, known as discretization, allows the representation of intricate geometries and material properties. The complete temperature profile of each element can be expressed using interpolation functions, with the system's behavior characterized by a set of algebraic equations. Temperature distribution with FEA (10.4236/aast.2018.33004) FEA meshes (DOI: 10.4236/aast.2018.33004) CFD uses mesh divisions to model turbulent flows and energy transport. The Navier-Stokes and energy equations with appropriate additional models to "close" the turbulence physics by approximations govern turbulent fluid flow, accounting for heat exchange and complex interactions between motion and thermal effects. Incorporating the energy equation provides a comprehensive representation of fluid behavior and energy transport thus: _rate of change + convective transport = conductive transport + sources_ i.e.: _ρc ∂T/∂t + u⋅∇T = ∇⋅(k ∇T) + q_ AI and CAE Simulation Deep learning, a subset of AI, mimics human cognition to detect patterns, predict outcomes, and generate insights. It outperforms traditional methods in precision. Combined with CAE simulation, it allows engineers to explore new designs. example of thermal field computed by simulation (top) and instantly predicted by Neural Concept (bottom) Inspired by human visual processing, 3D Deep Learningwith Convolutional Neural Networks (CNNs) identifies image patterns and effectively manages complex CAD geometries, helping engineers enhance simulations. Typical structure of a CNN from input (left) to output (right) Neural Concept combines CNNs and computer vision to analyze CAD models in real time. This enables engineers to quickly identify design flaws and optimization opportunities that traditional CFD and CHT methods might miss due to computational constraints. Heat exchanger optimized with neural networks (Neural Concept) Engineering Applications of Conjugate Heat Transfer Conjugate heat transfer has widespread applications in various engineering fields, including electronics cooling, thermal management of engines, and aerospace design. Schematic illustration of the airflow of the cooling air in a computer case (Wikimedia Commons) Conjugate heat transfer, coupled with modern simulation tools, is an indispensable analytical tool in engineering domains rooted in technical precision, from electronics cooling to aerospace engineering and industrial processes. For instance, conjugate heat transfer allows engineers to optimize cooling solutions and extend the lifespan of electronic components. The sections below provide more information, and the Neural Concept website is dedicated to otherdata-driven workflow applications. Cooling Systems Applications CHT analysis is critical for thermal management in power electronics and IC packages, particularly for hotspot mitigation in high-power-density applications like GaN transistors and multi-core processors. Engineers use these simulations to optimize thermal interface materials (TIMs) and evaluate junction-to-ambient thermal resistance in various cooling solutions. Engineers use conjugate heat transfer simulations to analyze geometries, such as heat sinks and microchannels, coupled with fluid flow dynamics. Heat Exchangers Applications and Conjugate Heat Transfer The field of heat exchangers, integral to diverse industrial processes, benefits from conjugate heat transfer analysis. The effective thermal energy exchange between fluid streams is essential in applications from HVAC system design to chemical processing. With CHT simulation, engineers evaluate heat transfer coefficients, pressure drops, and thermal gradients, enabling precise parameter tuning for optimal performance. This comprehensive analysis culminates in better energy usage and reduced operational expenditures. Typical examples of heat exchangers and their simulations are: 1.Shell and Tube Heat Exchangersare used in oil refining and chemical processing. CHT analysis optimizes flow arrangements and tube layouts, enhancing heat transfer while minimizing pressure losses. (Yousufuddin - Heat Transfer Enhancement of a Shell and Tube Heat Exchanger with Different Baffle Spacing Arrangements. Sch J Appl Sci Res. Vol 1-6) 2.Plate Heat Exchangersare standard in food and beverage processing. They benefit from CHT simulations that optimize plate design and spacing. The analysis identifies optimal flow paths for maximizing heat transfer and minimizing fouling, which is crucial for product quality and reduced cleaning costs. 3.Air-cooled heat Exchangers: These are used in power plants and HVAC systems. They utilize ambient air for heat dissipation. CHT analysis optimizes fin design and airflow patterns, enhancing cooling performance while considering environmental factors like wind and temperature fields. HVAC system in a car (Grabcad) 4.Double Pipe Heat Exchangersoften heat or cool one fluid through another. CHT simulations enable precise modeling of heat transfer and fluid dynamics. Analyzing fluid interaction allows engineers to optimize the design for maximum thermal performance, reducing energy consumption. Automotive Applications: Braking System Conjugate Heat Transfer Simulation The first of two applications concerning car design and car manufacturing is brakes. During aggressive braking events, the friction couple (rotor-pad interface) experiences thermal loading that can exceed 600°C, leading to thermal fade and potential pad glazing. CHT analysis helps optimize brake caliper designs and rotor ventilation geometry to maintain optimal brake torque coefficient while preventing thermal shock-induced disc coning. 1. Ventilation and Cooling: Evaluation of the effectiveness of cooling mechanisms, such as ventilation channels or ducts, to mitigate temperature rise in critical brake components. 2. Materials: Suitable materials with optimal thermal properties for enhanced heat dissipation and durability. 3. Performance Enhancement: Improved braking performance, reduced wear, and extended component lifespan. car underhood (Flickr | vw_td) Underhood Conjugate Heat Transfer Simulation CHT simulations provide vital insights for developing vehicle thermal management. Internal combustion engines generate heat, which is reflected in the vehicle's underhood area, which houses heat sources like the engine and exhaust system. Engineers utilize conjugate heat transfer (CHT) simulation to optimize cooling performance and ensure adequate heat dissipation from critical components. CHT simulation models heat transfer in the engine bay, evaluating how heat is conducted through components and convected by air or coolant. Analysis can identify potential overheating issues and assess the cooling system's effectiveness by simulating engine load and airflow patterns. Moreover, CHT simulations help design components like radiators and fans for maximum heat exchange. Industrial Process Conjugate Heat Transfer Problems Conjugate heat transfer analysis is crucial for industrial processes, impacting manufacturing and energy conversion. This analysis models solid components and fluid flows in equipment, providing insights into heat transfer rates, temperature distributions, and thermal stresses. Engineers use these insights to optimize designs and operational parameters and lower energy consumption. Examples of Conjugate Heat Transfer in Industrial Processes Engineers employ conjugate heat transfer analysis in metallurgy to enhance cooling strategies in processes like continuous casting. This method simulates the heat exchange between metal and cooling water. Likewise, engineers utilize this method in the chemical processing sector to manage reaction kinetics and temperature variations, leading to improved yield, selectivity, and safety. Molten bronze casting Conjugate heat transfer analysis (CHT) is fundamental in thermal modeling in power generation. In gas turbines, first-stage nozzle guide vanes and rotor blades are exposed to gas path temperatures exceeding the material's creep limit. CHT analysis optimizes internal cooling channel geometries, including serpentine passages and pin-fin arrays, to maintain acceptable metal temperatures while minimizing parasitic pressure losses. Engineers use these simulations to evaluate film cooling effectiveness η and overall cooling effectiveness φ. Aircraft engine compressor CFD simulation - turbine blade surface This approach minimizes thermal stresses and boosts the turbine's performance and lifespan. Conclusions on CHT In this article, we have progressed from physics to mathematics to numerical solutions and, finally, the CHT engineering applications. CHT analysis is the ultimate solution for engineers optimizing designs with "rock-solid" reliability for thermal simulations. This multidisciplinary approach combines CFD simulations with thermal analysis of solids. It enables engineers to understand the interaction between fluid flows and solid components, improving brake systems' performance and underhood thermal management in automotive engineering. CHT analysis will remain crucial in fields like industrial processes and aerospace exploration, and we believe that adopting AI is the best path to engage all engineers aiming for optimal results. FAQ What are the three types of heat transfer? Conduction is mainly heat transfer through solids. Convection is heat transfer via fluid movement, free or forced. Radiation is energy transfer through electromagnetic waves. Why is CHT important in engineering simulations? CHT models interactions between solid and fluid domains, ensuring accurate thermal predictions for heat exchangers, turbines, and electronics cooling.e How can I ensure convergence in my CHT simulation? Refine mesh near solid-fluid interfaces. Use appropriate turbulence models. Apply accurate boundary conditions and monitor residuals and energy balance. What software tools are used for CHT analysis? ANSYS Fluent, COMSOL Multiphysics, OpenFOAM, and Simcenter STAR-CCM+ simulate CHT in engineering applications. They can all feed the Neural Concept platform for data-driven predictions. Which is better, a higher heat transfer coefficient or a lower one? Higher coefficients enhance heat dissipation, while lower values improve thermal insulation. The optimal value depends on the application. ‍
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https://site.extension.uga.edu/forageteam/2018/06/is-your-stocking-rate-correct/
Is your stocking rate correct? | UGA Forage Extension Team Skip to content A website from UGA Cooperative Extension Search Search for: UGA Forage Extension Team Home Upcoming Events Fall Armyworm Resources Contact Us Is your stocking rate correct? Written by Carole Knight June 1, 2018 Share on Facebook, opens in new window Share on X, opens in new window Share on LinkedIn Share with email, opens in email application By Steve Morgan Harris County CEC There are many important components in a successful livestock production system. One of the most important tasks in grazing management is understanding livestock stocking rate. It is critical in making timely management decisions that affect profits in beef cattle production. The optimum number of animals on a pasture makes efficient use of the forage and still leaves enough forage behind to allow a quick and complete recovery. Therefore, producers must understand how to determine the correct stocking rate for their pastures. Stocking rate is defined as the concentration of grazing livestock on a given amount of land over a season, year or period of time. Generally, stocking rate is expressed as “animal units” for a given amount of land. This is to allow stocking rates to universally cover all livestock types since an animal unit is equivalent to 1000 pounds of body weight regardless of the type of livestock. Though stocking rate depends on the intensity of grazing management, most pastures would be approximately 2 acres per animal unit. This would provide a forage allowance of approximately 2.5% of body weight per day. However, not all livestock have the same forage demand as a 1000 pound lactating cow. For this reason, animal unit equivalents (AUE) have been developed to assist with the approximate determination of forage demand based on the kind, class and size of animal. AnimalAUE Cow – dry 1.00 – 1.50 Cow with calf 1.20 – 1.60 Bull – mature 1.25 – 1.75 Calf – weaned 0.50 – 0.70 Steer/Heifer – 18 months 0.80 – 1.00 Sheep – mature ewe or ram 0.20 – .030 Sheep – yearling 0.15 – 0.20 Goat 0.17 – 0.20 Horse – mature 1.25 – 2.00 The usefulness of animal units is especially apparent considering the weight difference among various producers’ livestock and the fluctuation of average weights in a herd over time. For example, the average cow size varies considerably and has increased over the past 50 years. Today’s beef cow averages around 1300 – 1400 pounds. These cows are not equivalent to one animal unit. In addition, forage demand varies within a livestock species based on its growth rate (e.g. heifers and steers vs. mature cow). For example: If the estimated stocking rate for a 1,000 pound cow is 2 acres, the estimated stocking rate for the 1,150 pound cow (assuming both cows have the same forage intake rate of 2.5 percent of body weight) is found as follows: 1,150 pounds x 2.5% = 29 pounds forage intake per day ÷ 25 pounds forage per animal unit = 1.16 animal units per cow Therefore, 1.16 animal units per cow x 2 acres per animal unit = 2.3 acres per 1,150-pound cow Condition of the pasture impacts stocking rate. Factors such as previous grazing management, forage species, age of stand, soil type, texture, fertility level and moisture conditions all impact forage yield and consequently stocking rate. Livestock need forage year-round, but providing an adequate supply of forage for grazing 12 months out of the year can be challenging. Ideally, forage production should correspond with livestock needs. However, pasture production is variable during the growing season while livestock nutritional requirements are relatively stable or steadily increasing. One way to balance this equation is to make hay from some pastures during periods of rapid forage growth. In addition, calving before rapid growth will allow the period of highest animal need to match the greatest production of quality forage. A second way is to manage for a more uniform pasture growth. Some Best Management Practices to accomplish uniform growth include: Keeping forage healthy and unstressed. These plants begin growth earlier in the spring, produce higher yields through the grazing season and continue growing longer in the fall. Switching from continuous to rotational grazing can extend the grazing season and boost yields, since rotational grazing, by virtue of its rest periods, is less stressful to the forage. Maintaining a good fertility program will extend the season and boost yields. Many forage problems can be avoided by fertilizing properly. To determine fertilizer requirements, take regular soil tests and follow the recommendations given. Be sure to state the type of pasture being grown when submitting your sample because fertilizer recommendations will be based on the crop stated. Many producers incorporate grass/legume mixtures to meet more of the fertility needs of the pasture. Seeding legumes into poor quality pastures is the most common form of renovation. Legumes reduce dependence on nitrogen fertilizer, complement grasses by balancing forage production throughout the season, and improve pasture quality. Switching from continuous to rotational grazing increases forage utilization. Forage utilization is a critical component that helps determine stocking rate. Most pastures contain a great deal of forage that is never consumed and eventually decays. Traditional continuous grazing systems may use only 30 to 40% of the available forage. The rest of the forage is either trampled, soiled, or of little nutritional value because it becomes overly mature. Most of this loss occurs with underutilized fall stockpiles and during periods of rapid growth where there is surplus beyond what is needed for livestock. When the appropriate stocking density is used, shortening grazing periods through rotational grazing increases forage utilization to 60-75%. Good producers strive to achieve the right balance between forage availability, forage utilization, and animal performance. They stock pastures heavily enough to graze available forage down to a target height that will allow rapid and maximum regrowth without compromising nutritional needs of livestock. Good producers will observe pastures frequently for overgrazing and undergrazing and will periodically adjust the stocking rate or movement of cattle as needed. Overstocking and overgrazing leads to a reduction in palatable plant species and an increase in less desirable plants. Overuse also means that livestock must graze for longer periods to meet their needs. Over time, heavy stocking causes the more palatable and productive forage species to disappear. These desirable forages are replaced by less productive, less palatable plants. Posted in: Grazing Tags: grazing, stocking rate Carole Knight Previous: More than a tin can – Forage systems for goats Next: Beware: Bermudagrass Stem Maggot and Fall Armyworm have been Spotted Get more information on a wide variety of forage management issues on theGeorgia Forages website. Sign Up for Blog Updates! First Name Last Name Street Address ZIP City State Country Email address: Leave this field empty if you’re human: Article Topics Categories Archives Archives Check out more videos like this one on theGeorgia Forages YouTube Channel! CONNECT WITH GEORGIA FORAGES WEBSITE FACEBOOK TWITTER YOUTUBE Facebook X Instagram YouTube Administration Log in UGA Extension © 2012-2025. All Rights Reserved. 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http://www.zgglkx.com/CN/Y2019/V27/I12
中国管理科学 Please wait a minute... 主管：中国科学院 主办：中国优选法统筹法与经济数学研究会 中国科学院科技战略咨询研究院 导航切换中国管理科学 首页 关于期刊 期刊介绍 数据库收录 期刊荣誉 编委会 投稿指南 论文写作要求 投稿指南 出版道德和不当行为声明 中国知网双语出版 期刊订阅 联系我们 English 当期目录 2019年, 第27卷, 第12期 刊出日期：2019-12-20 上一期下一期 论文 [x] Select基于期望效用-熵模型的基金评级方法及其在中国基金评级中的应用 杨继平, 石晨晓, Daniel CHIEW, Judy QIU, Sirimon TREEPONGKARUNA 2019 (12): 1-10. doi: 10.16381/j.cnki.issn1003-207x.2019.12.001 摘要 ( 633 )PDF(1148KB) ( 236 ) 基金评级对于投资者来说具有重要的参考价值，研究合适的基金评级方法非常必要。本文针对晨星评级对风险调整和预测能力不足的特征，研究应用期望效用-熵（EU-E）模型基金评级方法对我国开放式基金进行评级的预测能力；并以Sharpe指数、Jensen、Fama-French三因素和Carhart四因素 α 作为业绩指标，利用固定效应面板数据回归模型对期望效用-熵模型和随机效应面板数据回归模型对晨星基金评级的预测能力进行比较分析。采用样本期由2011年2月到2016年6月的261只基金为研究样本进行评级；研究结果表明，基于期望效用-熵平衡系数 λ=0.25和0.75时，EU-E模型基金评级方法评级具有良好的预测能力，而晨星评级预测能力较弱。特别地，λ=0.25和0.75时，EU-E模型评级的五星级基金业绩优于晨星评级对应的基金业绩，而且相比于晨星评级可以更好地区分不同星级基金的业绩。另外，研究结论对于短期、中期和长期的样本都是稳健的。 参考文献 | 相关文章 | 计量指标 [x] SelectHeston模型下DC型养老金等价管理费用问题研究 李方超, 张成科, 朱怀念 2019 (12): 11-21. doi: 10.16381/j.cnki.issn1003-207x.2019.12.002 摘要 ( 531 )PDF(1709KB) ( 219 ) 本文针对DC型养老金收费管理问题展开了系统研究，假定风险资产服从Heston模型，养老金管理者按资产财富比例和工资比例两种方式收取管理费用，试图在这两种方式中找到一个平衡点（等价管理费用）。运用随机最优控制理论得到了CARA和CRRA两种效用准则下等价管理费用的解析表达，从表达式中发现等价管理费用只与无风险利率、积累时间以及按工资比例收费的代表参数有关，而积累时间与参保人年龄有很强的相关性。最后，通过数值算例分析了参保人年龄对等价管理费用的影响以及等价管理费用对财富过程的影响，研究结果显示：在CARA效用下，等价管理费用随参保人年龄严格递增，而在CRRA效用下相反；无论在CARA效用还是CRRA效用下，等价管理费用对养老金账户财富都有正负两方面的影响。 参考文献 | 相关文章 | 计量指标 [x] Select企业投资决策同伴效应及其特征的实证检验——基于中国上市公司的面板数据 李佳宁, 钟田丽 2019 (12): 22-31. doi: 10.16381/j.cnki.issn1003-207x.2019.12.003 摘要 ( 581 )PDF(922KB) ( 214 ) 企业投资的同伴效应是指企业的投资行为受到其参照组内同伴投资的影响。为解决识别同伴效应时面临的参照组有效性问题以及企业决策互相影响的联立性问题，本文选择跨区域的同行业企业作为同伴企业，首次构建了同伴企业的同区域跨行业企业的投资均值作为工具变量，基于2008-2015年中国上市公司的面板数据，实证检验了上市企业投资决策的同伴效应。结果表明，中国上市企业投资决策受到跨区域同行业的同伴企业投资显著正向影响，且这种同伴效应在增减变动方向上具有不对称性和乘数效应的基本特征。本文拓展和深化了企业财务决策同伴效应的现有研究，并有助于投资者与监管部门理解同伴效应的放大作用。 参考文献 | 相关文章 | 计量指标 [x] Select中国高等教育投入产出效率的综合评价——基于Window-Malmquist指数法 易明, 彭甲超, 张尧 2019 (12): 32-42. doi: 10.16381/j.cnki.issn1003-207x.2019.12.004 摘要 ( 566 )PDF(3030KB) ( 160 ) "双一流"建设背景下，基于合作竞争的资源配置模式决定了中国必须提高高等教育投入产出效率。综合运用Window-Malmquist指数法和空间聚类方法测算分析中国31省（市）2004-2015年的高等教育投入产出效率及其演变规律和空间差异，结果显示：（1）总体而言，中国高等教育投入产出效率呈现DEA有效状态，技术进步是其提升的主要贡献因素，但高等教育"区域鸿沟"的存在却导致追赶效应拉低了效率值；（2）从时间演变规律看，中国高等教育投入产出效率相对稳定，但上升趋势并不明显，且存在两极分化或多极分化的可能性；（3）从空间差异情况看，31个省（市）中，江苏高等教育投入产出效率最高，东部沿海地区的高等教育投入产出效率具有"高高"特征，且效率值明显高于其他地区；（4）从空间集聚特征看，在技术进步的作用下不同区域的高等教育投入产出效率存在正向的空间关系，但追赶效应却降低了这种空间联系。 参考文献 | 相关文章 | 计量指标 [x] Select产品服务供应链的“双重收益共享”合作机制 李鑫, 于辉 2019 (12): 43-54. doi: 10.16381/j.cnki.issn1003-207x.2019.12.005 摘要 ( 562 )PDF(2984KB) ( 365 ) 产品与服务融合缓解了"供需错位"矛盾，使得产品服务供应链探索成为"供给侧结构性改革"下供需结构性匹配的重要路径。本文刻画产品服务视角下供应链组织结构的本质特征，构建产品服务供应链合作模型，探讨合作机制下"帕累托改进"区间存在性，揭示合作机制对产品服务供应链效率的作用机理。核心研究发现：供应链多元化组织结构加剧收益分配冲突，并导致服务规模与供应链效率"倒挂"现象，而合作机制下"双重收益共享"合作模型能提升产品服务供应链效率。 参考文献 | 相关文章 | 计量指标 [x] Select考虑订货协调成本与数量折扣的改良品供应链水平协调 张云丰, 王勇, 龚本刚, 但斌 2019 (12): 55-66. doi: 10.16381/j.cnki.issn1003-207x.2019.12.006 摘要 ( 468 )PDF(1034KB) ( 155 ) 在单个供应商与多个销售商构成的二级改良品供应链中，销售商们采用联合补货方式不仅可以分摊订货成本，而且由于订货批量的增加从而更易获得供应商提供的数量折扣合同。建立多个销售商在独立补货与联合补货中的利润水平函数，推导联合补货优于独立补货的充要条件。将销售商之间的联合补货行为转化成多人合作博弈问题，证明博弈具有的基本性质，设计基于博弈核心的利润分配方法。通过数值算例和敏感性分析验证了文中所构建模型的正确性。 参考文献 | 相关文章 | 计量指标 [x] Select基于个性化需求的产品竞争供应链结构选择 孟炯, 张杨, 曾波 2019 (12): 67-76. doi: 10.16381/j.cnki.issn1003-207x.2019.12.007 摘要 ( 525 )PDF(2852KB) ( 266 ) 基于"制销分离"与"定制一体"两种结构选择，构建个性化产品供应链处于非竞争与竞争环境下的博弈模型，在引入一个实际案例的基础上，运用算例仿真比较分析两种运营模式下的供应链运作策略和盈利差异。结果显示：与制销分离结构相比，个性化产品供应链选择定制一体结构，有利于匹配产品个性化制造、提升产品个性化水平和市场需求、增加供应链的期望收益；个性化产品供应链选择制销分离结构时，适度的批发价格激励能够提升产品个性化水平、更好满足消费者个性化需求、改善供应链的运营绩效，分销商适度让利加大批发价格激励力度可显著促进产品个性化制造互动、提升产品个性化水平；竞争将消减个性化产品供应链的运营绩效，但选择定制一体结构可显著提升竞争力。 参考文献 | 相关文章 | 计量指标 [x] Select制造商成本削减策略对风险规避型零售商信息共享策略的影响 许明辉, 杨东升 2019 (12): 77-87. doi: 10.16381/j.cnki.issn1003-207x.2019.12.008 摘要 ( 626 )PDF(1954KB) ( 300 ) 考虑一个风险中性制造商和一个风险规避零售商构成的供应链，需求随机且受销售价格的影响。在销售季节之前，零售商对需求进行预测，获取需求信号；制造商对生产进行投资降低生产成本。基于零售商的不同信息共享策略及制造商的投资策略，考虑四种不同策略模型，分别得到最优零售价、批发价（及投资水平），并分析需求预测精确度对供应链成员决策和效用的影响。通过四种模型效用的对比分析，探讨制造商的投资策略以及零售商的风险规避态度对零售商信息共享策略的影响。研究发现，零售商共享需求信息对于制造商总是有益的，且制造商总是愿意采取成本削减策略；只有当制造商采取成本削减策略，且其投资成本系数较低时，共享需求信息对零售商才有益。最后，得到了制造商和零售商的均衡策略。 参考文献 | 相关文章 | 计量指标 [x] Select全渠道零售下“Showrooms”对需求分布、定价和收益的影响研究 刘金荣, 徐琪 2019 (12): 88-99. doi: 10.16381/j.cnki.issn1003-207x.2019.12.009 摘要 ( 680 )PDF(1729KB) ( 259 ) 作为一种新兴的线下体验、线上购买的商业运作模式，展厅（Showrooms）在全渠道零售领域发展迅速。基于消费者效用分别建立单一网络渠道和"网络+实体"渠道下零售商开设Showrooms前后的利润模型，对比分析固定销售价格和优化定价两种情形下Showrooms对定价、市场需求、利润和退货率的影响。研究表明：（1）单一网络渠道下，网络渠道价格固定时，开设Showrooms可使市场总需求增加，总利润在展厅运营成本较低时增加；网络渠道价格优化时，开设Showrooms可使网络渠道价格、总需求和总利润在展厅运营成本较低时增加，但固定价格情形下总利润的增幅更大。（2）"网络+实体"渠道下，开设Showrooms可使总需求和总利润均在展厅运营成本较低及网络退货率和网购不便利成本较高时增加，而且仅优化网络渠道价格时的增幅最大，网络和实体渠道价格均固定时次之；价格优化的最终趋势是全渠道同款同价。（3）开设Showrooms可使零售商总的网络退货率和退货量都下降。 参考文献 | 相关文章 | 计量指标 [x] Select基于数量柔性契约的双源应急物资采购定价模型 扈衷权, 田军, 冯耕中 2019 (12): 100-112. doi: 10.16381/j.cnki.issn1003-207x.2019.12.010 摘要 ( 569 )PDF(2003KB) ( 596 ) 针对灾害事件发生的不确定性以及灾害发生后应急物资需求量的爆发式增长，本文设计了一个政府主导的基于数量柔性契约的双源应急物资采购模型。在存在一个现货市场的情况下，政府除了常规采购外，还会与供应方签订一份数量柔性契约，用以建立政企联合储备应急物资的合作关系。通过数理推导，本文发现灾害事件发生概率，政府自身储备量，现货市场采购价格会对双方决策产生重要影响，并给出三个影响因素在满足不同条件时政企双方的最优决策，同时进一步分析了这三个因素对政府采购成本与供应方利润的影响。最后，通过数值模拟的方式对所得结论进行了验证。本文的研究为政府与企业构建联合储备应急物资的合作关系提供了指导与依据。 参考文献 | 相关文章 | 计量指标 [x] Select经济利益驱动下食品企业安全风险演化动态研究 王冀宁, 张宇昊, 王雨桐, 陈庭强 2019 (12): 113-126. doi: 10.16381/j.cnki.issn1003-207x.2019.12.011 摘要 ( 537 )PDF(5470KB) ( 164 ) 消费扩大驱动下食品质量安全问题日渐严峻，食品企业经济利益驱动型掺假所具有的蓄意性、隐蔽性、技术性特点，为新时期食品安全治理提出了一系列挑战。鉴于此，本文借助演化博弈理论，考虑了食品企业收益、食品生产技术成本、生产残次食品损失以及政府监管成本、社会负面效益等因素对食品安全风险形成的影响，构建了食品掺假行为演化博弈模型，并对其演化状态进行了理论和仿真分析。在此基础上，运用元胞自动机理论，进一步考虑到食品企业策略转变意愿与基层食品监管机构策略转变意愿，从空间博弈角度对食品掺假行为及其监管的空间演化状态进行了深度剖析和刻画。研究结果显示，食品企业和基层食品监管机构在策略选择方面具有同步性振荡特征。而且，在食品企业策略转变意愿或基层食品监管机构策略转变意愿维持较低水平时，食品企业和基层食品监管机构的策略（严格监管，合规生产）是纯策略稳定状态。通过本文研究既丰富了我国食品安全监管理论，也为地方食品安全长效监管提供了思路借鉴和理论指导。 参考文献 | 相关文章 | 计量指标 [x] Select特殊时期地铁二级分流-联防安检优化模型 李德龙, 刘德海 2019 (12): 127-135. doi: 10.16381/j.cnki.issn1003-207x.2019.12.012 摘要 ( 610 )PDF(4035KB) ( 114 ) 恐怖分子经常将安检体系相对薄弱、人流相对密集的地铁站作为攻击的首要目标之一。本文从地铁安检流程优化和跨部门联防协作的角度出发，提出了特殊时期地铁二级分流-联防安检优化模型。结果表明，在特殊时期，通过引入身份证信息库和指纹-人脸识别系统，对乘客进行绿、橙、红三色通道分流安检，能够明显提升普通乘客的社会福利水平；当X光检测系统的准确率越低，被分流到橙色通道乘客比例越大，橙色通道中潜在袭击者比例越高时，社会福利增益越大，暴恐概率越小；在案潜逃人员识别率越高，社会福利增益越大。多部门协防反恐正向反馈越积极，进而可以刺激安防部门优化身份识别系统，直到社会福利增益为零的最优状态。 参考文献 | 相关文章 | 计量指标 [x] Select基于理想解和灰关联度的动态评价方法及其应用研究 吴飞美, 李美娟, 徐林明, 毕骏莉 2019 (12): 136-142. doi: 10.16381/j.cnki.issn1003-207x.2019.12.013 摘要 ( 603 )PDF(957KB) ( 194 ) 为了对多个多属性（指标）待评价对象（方案）在多个时间点的发展状态和该时间段内的总体发展水平进行比较分析，根据理想解法和灰关联度法优缺点，提出基于理想解和灰关联度的动态评价方法。该方法基于三维数据，将欧氏距离和灰色关联度相结合，提出一种新贴近度，同时反映了位置关系和数据曲线的相似性差异，兼顾评价指标值差异程度和增长程度。最后将该方法应用于"十二五"期间省域循环经济生态效益评价，通过实例验证该方法实际应用上的有效性。 参考文献 | 相关文章 | 计量指标 [x] Select基于Mo-RVIKOR的混合多属性决策方法 潘亚虹, 耿秀丽 2019 (12): 143-151. doi: 10.16381/j.cnki.issn1003-207x.2019.12.014 摘要 ( 518 )PDF(1209KB) ( 136 ) 针对复杂性和不确定性多属性决策问题，考虑定量和定性融合的属性形式，提出了模块化随机多准则妥协解排序法（Modular Random VlseKriterijumska Opti-mizacija I Kompromisno Resenje，Mo-RVIKOR），该方法无需将信息统一，就能处理多种信息形式存在的多属性决策问题。采用精确数、随机变量处理定量评价信息，用概率语义术语集处理定性评价信息；通过改进离差最大化法确定属性权重；根据Mo-RVIKOR对决策对象进行排序；最后以某公司C2B定制化服务质量评测项目为例，验证了所提方法的有效性。 参考文献 | 相关文章 | 计量指标 [x] Select基于一致性局部调整算法和DEA的语言偏好决策模型 金飞飞, 倪志伟, 陈华友, 朱旭辉, 武文颖 2019 (12): 152-163. doi: 10.16381/j.cnki.issn1003-207x.2019.12.015 摘要 ( 496 )PDF(988KB) ( 134 ) 设计一致性调整算法和计算可靠的方案排序权重向量已成为近年来备受关注的两个重要研究课题。针对决策信息为语言偏好关系的决策问题，提出一种基于一致性局部调整算法和数据包络分析（DEA）方法的语言偏好决策模型。首先，基于局部调整策略设计一种收敛的乘性一致性调整算法，该算法不仅使得调整后的语言偏好关系具有满意乘性一致性，而且能够尽可能多的保存原始决策信息；其次，基于提出的新型语言DEA模型构造一种语言偏好决策模型用以确定方案的排序权重向量，进而得到合理可靠的决策结果。最后，将提出的语言偏好决策模型用于供应商的选择实例，对比分析实验验证了模型的合理性和有效性。 参考文献 | 相关文章 | 计量指标 [x] Select考虑重复购买因素的品牌竞争对新产品扩散的影响研究——基于小世界网络的仿真环境 王展昭 2019 (12): 164-174. doi: 10.16381/j.cnki.issn1003-207x.2019.12.016 摘要 ( 471 )PDF(2286KB) ( 426 ) 在对品牌竞争影响要素分析的基础上，考虑重复购买在新产品扩散过程中的调节效应，构建以消费者微观采纳机制为核心的新产品扩散的阈值模型，在小世界网络仿真情境下，运用多智能仿真方法对品牌竞争与新产品扩散之间的影响关系进行分析，研究结果表明：转换成本对新产品扩散的速度和深度的影响存在一定的"条件区域"，在该区域之外两者不存在显著的影响关系；进入时间与两种竞争性品牌的扩散速度存在一致的负向的影响关系，而对两种竞争性品牌的扩散深度的影响方向则相反；在此过程中，重复购买系数起着一定的调节效应，会在一定程度上放大或缩小转换成本及进入时间对新产品扩散的影响效应。通过本文的研究，不仅可以进一步丰富和完善竞争性产品创新扩散的相关理论，也可以为企业的新产品推广实践提供决策支持。 参考文献 | 相关文章 | 计量指标 [x] Select基于顾客公平偏好的服务机制与定价研究 刘健, 张帅, 赵洪款, 刘思峰 2019 (12): 175-184. doi: 10.16381/j.cnki.issn1003-207x.2019.12.017 摘要 ( 601 )PDF(943KB) ( 183 ) 由于顾客异质性（单位时间等待成本不同），服务提供商通常对顾客采取分类服务策略，然而分类服务会引起服务系统中不同类型顾客之间等待时间和服务价值的差异性，从而给顾客带来心理上的不公平感，进而引起顾客在服务系统中的流动和转移，进一步影响企业收益和社会福利。本文针对非抢占M/M/1服务系统顾客分类情形为背景，由两种顾客之间期望等待时间的不同和公平偏好参数相结合构建普通顾客的公平心理效用模型，以垄断型服务系统为背景，分别从企业收益、社会福利与顾客效用三个视角进行分析。研究表明，服务提供商应对顾客采取可观测型的分类服务机制来获得最大收益；从社会福利视角，服务提供商应对顾客采取不可观测型的分类服务机制；从顾客效用视角，服务提供商应取消顾客分类服务，仅保留普通顾客。最后同现有结论进行比较分析，并进行拓展研究。本文研究对服务提供商采取合理的服务机制及相应的服务定价具有重要参考价值和指导意义。 参考文献 | 相关文章 | 计量指标 [x] Select服务型制造网络的Holon协同需求评价方法 侯芳 2019 (12): 185-196. doi: 10.16381/j.cnki.issn1003-207x.2019.12.018 摘要 ( 494 )PDF(2763KB) ( 107 ) 依据复杂网络理论分析服务型制造网络Holon协同需求问题，给出一种考虑以直觉正态模糊数表示且多Holon协同的服务型制造网络协同需求评价方法。首先，在区分Holon复合协同和递归协同基础上构建服务型制造网络协同需求评价指标体系；其次，考虑基于网络结构特征的Holon相似稳定性，根据服务型制造网络节点相似性测度分析Holon协同需求特征，并测算不同相似性测度修正的Holon网络结构熵；再次，建立面向服务型制造网络和Holon的有专家信息双向触动反馈机制，反馈包括评价指标和网络协同状况，反馈Holon信息包括基于服务型制造网络演化方向的Holon间协同需求建议和基于服务型制造网络现有状态的网络连通性Holon协同需求建议；最后由 INFCWA（INFCWA R）算子或 INFCWG（INFCWG R）算子分别对复合协同和递归协同评价信息集结并得出评价结论。方法设计过程通过例证分析说明根据服务型制造网络目标控制的Holon协同需求评价改进了群组评价效率。 参考文献 | 相关文章 | 计量指标 [x] Select产业共性技术产学研协同研发策略的微分博弈研究 马永红, 刘海礁, 柳清 2019 (12): 197-207. doi: 10.16381/j.cnki.issn1003-207x.2019.12.019 摘要 ( 517 )PDF(2069KB) ( 280 ) 针对产业共性技术产学研协同研发问题，本文以单个研究机构（大学、科研院所）和单个企业为研究对象，通过构建微分博弈模型，运用HJB方程分别分析了三种产业共性技术研发博弈情形下研究机构和企业各自的最优研发努力程度、最优研发收益、双方最优研发总收益以及企业对研究机构的研发投入补贴。通过对三种博弈结果的比较分析发现：（1）研发投入补贴作为一种激励策略，可促进研究机构研发努力程度、研究机构与企业各自研发收益以及双方研发总收益的提升；（2）协同合作博弈情形下研究机构与企业各自研发努力程度、各自研发收益和双方研发总收益均优于非合作情形。为协调研究机构与企业的产业共性技术协同研发行为，通过讨论收益分配系数 α 的取值范围进而分析产业共性技术产学研协同研发的收益协调机制。最后，通过算例分析验证了理论推导的结果。 参考文献 | 相关文章 | 计量指标 [x] Select基于最大熵分布的控制图改进与评价研究 宋明顺, 杨铭, 方兴华 2019 (12): 208-216. doi: 10.16381/j.cnki.issn1003-207x.2019.12.020 摘要 ( 497 )PDF(1158KB) ( 131 ) 传统的控制图多是假定质量特性参数服从正态分布，但在很多情况下正态分布的假设并不成立。本文基于最大熵分布从控制图的构建和评价两个方面分别提出对Shewhart控制图和CUSUM控制图的改进方法。首先根据"经济性"原则构建最大熵Shewhart控制图，实现对Shewhart控制图的改进；然后，提出结合最大熵分布和马尔科夫链方法的CUSUM控制图评价方法。仿真结果表明，基于最大熵分布改进后的Shewhart控制图控制性能优于改进之前的情况；而基于最大熵分布的CUSUM控制图的性能评价方法得到的结果能更符合真实的情况。在对不同偏移的监测中，最大熵Shewhart控制图更适用于分布未知的大偏移情况，而自适应CUSUM控制图对小偏移有更好地监控效果。 参考文献 | 相关文章 | 计量指标 总目录 [x] Select《中国管理科学》2019年总目次2019 (12): 217-224. 摘要 ( 428 )PDF(770KB) ( 343 ) 相关文章 | 计量指标 作者中心 专家审稿 编委审稿 编辑办公 主编办公 编辑部公告更多... 第二十七届中国管理科学学术年会暨2025年中国优选法统筹法与经济数学研究会年会会议通知（第二轮） 经济管理学期刊高质量发展论坛暨《中国管理科学》创刊40周年纪念大会会议通知 会议通知 | 中国优选法统筹法与经济数学研究会第十一次会员代表大会暨第二十六届中国管理科学学术年会会议通知（第一轮） 2022-2023《中国管理科学》优秀评审专家 第二十五届中国管理科学学术年会推荐论文名单 “国有企业改革与现代公司治理创新论坛（2023）”征稿启事 第25届中国管理科学学术年会征文 延期通知 2021年中国管理科学第二十三届学术年会优秀报告论文名单 第二十三届中国管理科学学术年会征文通知 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https://doc.rust-lang.org/std/primitive.char.html
std1.86.0 char Sections Associated Constants Methods Trait Implementations Auto Trait Implementations Blanket Implementations In crate std Primitive Type charCopy item path A character type. The char type represents a single character. More specifically, since ‘character’ isn’t a well-defined concept in Unicode, char is a ‘Unicode scalar value’. This documentation describes a number of methods and trait implementations on the char type. For technical reasons, there is additional, separate documentation in the std::char module as well. §Validity and Layout A char is a ‘Unicode scalar value’, which is any ‘Unicode code point’ other than a surrogate code point. This has a fixed numerical definition: code points are in the range 0 to 0x10FFFF, inclusive. Surrogate code points, used by UTF-16, are in the range 0xD800 to 0xDFFF. No char may be constructed, whether as a literal or at runtime, that is not a Unicode scalar value. Violating this rule causes undefined behavior. Unicode scalar values are also the exact set of values that may be encoded in UTF-8. Because char values are Unicode scalar values and functions may assume incoming str values are valid UTF-8, it is safe to store any char in a str or read any character from a str as a char. The gap in valid char values is understood by the compiler, so in the below example the two ranges are understood to cover the whole range of possible char values and there is no error for a non-exhaustive match. All Unicode scalar values are valid char values, but not all of them represent a real character. Many Unicode scalar values are not currently assigned to a character, but may be in the future (“reserved”); some will never be a character (“noncharacters”); and some may be given different meanings by different users (“private use”). char is guaranteed to have the same size, alignment, and function call ABI as u32 on all platforms. §Representation char is always four bytes in size. This is a different representation than a given character would have as part of a String. For example: As always, remember that a human intuition for ‘character’ might not map to Unicode’s definitions. For example, despite looking similar, the ‘é’ character is one Unicode code point while ‘é’ is two Unicode code points: This means that the contents of the first string above will fit into a char while the contents of the second string will not. Trying to create a char literal with the contents of the second string gives an error: Another implication of the 4-byte fixed size of a char is that per-char processing can end up using a lot more memory: Implementations§ impl char pub const MIN: char = '\0' The lowest valid code point a char can have, '\0'. Unlike integer types, char actually has a gap in the middle, meaning that the range of possible chars is smaller than you might expect. Ranges of char will automatically hop this gap for you: Despite this gap, the MIN and MAX values can be used as bounds for all char values. §Examples pub const MAX: char = '\u{10ffff}' The highest valid code point a char can have, '\u{10FFFF}'. Unlike integer types, char actually has a gap in the middle, meaning that the range of possible chars is smaller than you might expect. Ranges of char will automatically hop this gap for you: Despite this gap, the MIN and MAX values can be used as bounds for all char values. §Examples pub const REPLACEMENT_CHARACTER: char = '�' U+FFFD REPLACEMENT CHARACTER (�) is used in Unicode to represent a decoding error. It can occur, for example, when giving ill-formed UTF-8 bytes to String::from_utf8_lossy. pub const UNICODE_VERSION: (u8, u8, u8) = crate::unicode::UNICODE_VERSION The version of Unicode that the Unicode parts of char and str methods are based on. New versions of Unicode are released regularly and subsequently all methods in the standard library depending on Unicode are updated. Therefore the behavior of some char and str methods and the value of this constant changes over time. This is not considered to be a breaking change. The version numbering scheme is explained in Unicode 11.0 or later, Section 3.1 Versions of the Unicode Standard. pub fn decode_utf16(iter: I) -> DecodeUtf16<::IntoIter> ⓘwhere I: IntoIterator, Creates an iterator over the native endian UTF-16 encoded code points in iter, returning unpaired surrogates as Errs. §Examples Basic usage: A lossy decoder can be obtained by replacing Err results with the replacement character: pub const fn from_u32(i: u32) -> Option Converts a u32 to a char. Note that all chars are valid u32s, and can be cast to one with as: However, the reverse is not true: not all valid u32s are valid chars. from_u32() will return None if the input is not a valid value for a char. For an unsafe version of this function which ignores these checks, see from_u32_unchecked. §Examples Basic usage: Returning None when the input is not a valid char: pub const unsafe fn from_u32_unchecked(i: u32) -> char Converts a u32 to a char, ignoring validity. Note that all chars are valid u32s, and can be cast to one with as: However, the reverse is not true: not all valid u32s are valid chars. from_u32_unchecked() will ignore this, and blindly cast to char, possibly creating an invalid one. §Safety This function is unsafe, as it may construct invalid char values. For a safe version of this function, see the from_u32 function. §Examples Basic usage: pub const fn from_digit(num: u32, radix: u32) -> Option Converts a digit in the given radix to a char. A ‘radix’ here is sometimes also called a ‘base’. A radix of two indicates a binary number, a radix of ten, decimal, and a radix of sixteen, hexadecimal, to give some common values. Arbitrary radices are supported. from_digit() will return None if the input is not a digit in the given radix. §Panics Panics if given a radix larger than 36. §Examples Basic usage: Returning None when the input is not a digit: Passing a large radix, causing a panic: pub fn is_digit(self, radix: u32) -> bool Checks if a char is a digit in the given radix. A ‘radix’ here is sometimes also called a ‘base’. A radix of two indicates a binary number, a radix of ten, decimal, and a radix of sixteen, hexadecimal, to give some common values. Arbitrary radices are supported. Compared to is_numeric(), this function only recognizes the characters 0-9, a-z and A-Z. ‘Digit’ is defined to be only the following characters: For a more comprehensive understanding of ‘digit’, see is_numeric(). §Panics Panics if given a radix smaller than 2 or larger than 36. §Examples Basic usage: Passing a large radix, causing a panic: Passing a small radix, causing a panic: pub const fn to_digit(self, radix: u32) -> Option Converts a char to a digit in the given radix. A ‘radix’ here is sometimes also called a ‘base’. A radix of two indicates a binary number, a radix of ten, decimal, and a radix of sixteen, hexadecimal, to give some common values. Arbitrary radices are supported. ‘Digit’ is defined to be only the following characters: §Errors Returns None if the char does not refer to a digit in the given radix. §Panics Panics if given a radix smaller than 2 or larger than 36. §Examples Basic usage: Passing a non-digit results in failure: Passing a large radix, causing a panic: Passing a small radix, causing a panic: pub fn escape_unicode(self) -> EscapeUnicode ⓘ Returns an iterator that yields the hexadecimal Unicode escape of a character as chars. This will escape characters with the Rust syntax of the form \u{NNNNNN} where NNNNNN is a hexadecimal representation. §Examples As an iterator: Using println! directly: Both are equivalent to: Using to_string: pub fn escape_debug(self) -> EscapeDebug ⓘ Returns an iterator that yields the literal escape code of a character as chars. This will escape the characters similar to the Debug implementations of str or char. §Examples As an iterator: Using println! directly: Both are equivalent to: Using to_string: pub fn escape_default(self) -> EscapeDefault ⓘ Returns an iterator that yields the literal escape code of a character as chars. The default is chosen with a bias toward producing literals that are legal in a variety of languages, including C++11 and similar C-family languages. The exact rules are: §Examples As an iterator: Using println! directly: Both are equivalent to: Using to_string: pub const fn len_utf8(self) -> usize Returns the number of bytes this char would need if encoded in UTF-8. That number of bytes is always between 1 and 4, inclusive. §Examples Basic usage: The &str type guarantees that its contents are UTF-8, and so we can compare the length it would take if each code point was represented as a char vs in the &str itself: pub const fn len_utf16(self) -> usize Returns the number of 16-bit code units this char would need if encoded in UTF-16. That number of code units is always either 1 or 2, for unicode scalar values in the basic multilingual plane or supplementary planes respectively. See the documentation for len_utf8() for more explanation of this concept. This function is a mirror, but for UTF-16 instead of UTF-8. §Examples Basic usage: pub const fn encode_utf8(self, dst: &mut [u8]) -> &mut str Encodes this character as UTF-8 into the provided byte buffer, and then returns the subslice of the buffer that contains the encoded character. §Panics Panics if the buffer is not large enough. A buffer of length four is large enough to encode any char. §Examples In both of these examples, ‘ß’ takes two bytes to encode. A buffer that’s too small: pub const fn encode_utf16(self, dst: &mut [u16]) -> &mut [u16] Encodes this character as native endian UTF-16 into the provided u16 buffer, and then returns the subslice of the buffer that contains the encoded character. §Panics Panics if the buffer is not large enough. A buffer of length 2 is large enough to encode any char. §Examples In both of these examples, ‘𝕊’ takes two u16s to encode. A buffer that’s too small: pub fn is_alphabetic(self) -> bool Returns true if this char has the Alphabetic property. Alphabetic is described in Chapter 4 (Character Properties) of the Unicode Standard and specified in the Unicode Character Database DerivedCoreProperties.txt. §Examples Basic usage: pub const fn is_lowercase(self) -> bool Returns true if this char has the Lowercase property. Lowercase is described in Chapter 4 (Character Properties) of the Unicode Standard and specified in the Unicode Character Database DerivedCoreProperties.txt. §Examples Basic usage: In a const context: pub const fn is_uppercase(self) -> bool Returns true if this char has the Uppercase property. Uppercase is described in Chapter 4 (Character Properties) of the Unicode Standard and specified in the Unicode Character Database DerivedCoreProperties.txt. §Examples Basic usage: In a const context: pub fn is_whitespace(self) -> bool Returns true if this char has the White_Space property. White_Space is specified in the Unicode Character Database PropList.txt. §Examples Basic usage: pub fn is_alphanumeric(self) -> bool Returns true if this char satisfies either is_alphabetic() or is_numeric(). §Examples Basic usage: pub fn is_control(self) -> bool Returns true if this char has the general category for control codes. Control codes (code points with the general category of Cc) are described in Chapter 4 (Character Properties) of the Unicode Standard and specified in the Unicode Character Database UnicodeData.txt. §Examples Basic usage: pub fn is_numeric(self) -> bool Returns true if this char has one of the general categories for numbers. The general categories for numbers (Nd for decimal digits, Nl for letter-like numeric characters, and No for other numeric characters) are specified in the Unicode Character Database UnicodeData.txt. This method doesn’t cover everything that could be considered a number, e.g. ideographic numbers like ‘三’. If you want everything including characters with overlapping purposes then you might want to use a unicode or language-processing library that exposes the appropriate character properties instead of looking at the unicode categories. If you want to parse ASCII decimal digits (0-9) or ASCII base-N, use is_ascii_digit or is_digit instead. §Examples Basic usage: pub fn to_lowercase(self) -> ToLowercase ⓘ Returns an iterator that yields the lowercase mapping of this char as one or more chars. If this char does not have a lowercase mapping, the iterator yields the same char. If this char has a one-to-one lowercase mapping given by the Unicode Character Database UnicodeData.txt, the iterator yields that char. If this char requires special considerations (e.g. multiple chars) the iterator yields the char(s) given by SpecialCasing.txt. This operation performs an unconditional mapping without tailoring. That is, the conversion is independent of context and language. In the Unicode Standard, Chapter 4 (Character Properties) discusses case mapping in general and Chapter 3 (Conformance) discusses the default algorithm for case conversion. §Examples As an iterator: Using println! directly: Both are equivalent to: Using to_string: pub fn to_uppercase(self) -> ToUppercase ⓘ Returns an iterator that yields the uppercase mapping of this char as one or more chars. If this char does not have an uppercase mapping, the iterator yields the same char. If this char has a one-to-one uppercase mapping given by the Unicode Character Database UnicodeData.txt, the iterator yields that char. If this char requires special considerations (e.g. multiple chars) the iterator yields the char(s) given by SpecialCasing.txt. This operation performs an unconditional mapping without tailoring. That is, the conversion is independent of context and language. In the Unicode Standard, Chapter 4 (Character Properties) discusses case mapping in general and Chapter 3 (Conformance) discusses the default algorithm for case conversion. §Examples As an iterator: Using println! directly: Both are equivalent to: Using to_string: §Note on locale In Turkish, the equivalent of ‘i’ in Latin has five forms instead of two: Note that the lowercase dotted ‘i’ is the same as the Latin. Therefore: The value of upper_i here relies on the language of the text: if we’re in en-US, it should be "I", but if we’re in tr_TR, it should be "İ". to_uppercase() does not take this into account, and so: holds across languages. pub const fn is_ascii(&self) -> bool Checks if the value is within the ASCII range. §Examples pub const fn as_ascii(&self) -> Option Returns Some if the value is within the ASCII range, or None if it’s not. This is preferred to Self::is_ascii when you’re passing the value along to something else that can take ascii::Char rather than needing to check again for itself whether the value is in ASCII. pub const fn to_ascii_uppercase(&self) -> char Makes a copy of the value in its ASCII upper case equivalent. ASCII letters ‘a’ to ‘z’ are mapped to ‘A’ to ‘Z’, but non-ASCII letters are unchanged. To uppercase the value in-place, use make_ascii_uppercase(). To uppercase ASCII characters in addition to non-ASCII characters, use to_uppercase(). §Examples pub const fn to_ascii_lowercase(&self) -> char Makes a copy of the value in its ASCII lower case equivalent. ASCII letters ‘A’ to ‘Z’ are mapped to ‘a’ to ‘z’, but non-ASCII letters are unchanged. To lowercase the value in-place, use make_ascii_lowercase(). To lowercase ASCII characters in addition to non-ASCII characters, use to_lowercase(). §Examples pub const fn eq_ignore_ascii_case(&self, other: &char) -> bool Checks that two values are an ASCII case-insensitive match. Equivalent to to_ascii_lowercase(a) == to_ascii_lowercase(b). §Examples pub const fn make_ascii_uppercase(&mut self) Converts this type to its ASCII upper case equivalent in-place. ASCII letters ‘a’ to ‘z’ are mapped to ‘A’ to ‘Z’, but non-ASCII letters are unchanged. To return a new uppercased value without modifying the existing one, use to_ascii_uppercase(). §Examples pub const fn make_ascii_lowercase(&mut self) Converts this type to its ASCII lower case equivalent in-place. ASCII letters ‘A’ to ‘Z’ are mapped to ‘a’ to ‘z’, but non-ASCII letters are unchanged. To return a new lowercased value without modifying the existing one, use to_ascii_lowercase(). §Examples pub const fn is_ascii_alphabetic(&self) -> bool Checks if the value is an ASCII alphabetic character: §Examples pub const fn is_ascii_uppercase(&self) -> bool Checks if the value is an ASCII uppercase character: U+0041 ‘A’ ..= U+005A ‘Z’. §Examples pub const fn is_ascii_lowercase(&self) -> bool Checks if the value is an ASCII lowercase character: U+0061 ‘a’ ..= U+007A ‘z’. §Examples pub const fn is_ascii_alphanumeric(&self) -> bool Checks if the value is an ASCII alphanumeric character: §Examples pub const fn is_ascii_digit(&self) -> bool Checks if the value is an ASCII decimal digit: U+0030 ‘0’ ..= U+0039 ‘9’. §Examples pub const fn is_ascii_octdigit(&self) -> bool Checks if the value is an ASCII octal digit: U+0030 ‘0’ ..= U+0037 ‘7’. §Examples pub const fn is_ascii_hexdigit(&self) -> bool Checks if the value is an ASCII hexadecimal digit: §Examples pub const fn is_ascii_punctuation(&self) -> bool Checks if the value is an ASCII punctuation character: §Examples pub const fn is_ascii_graphic(&self) -> bool Checks if the value is an ASCII graphic character: U+0021 ‘!’ ..= U+007E ‘~’. §Examples pub const fn is_ascii_whitespace(&self) -> bool Checks if the value is an ASCII whitespace character: U+0020 SPACE, U+0009 HORIZONTAL TAB, U+000A LINE FEED, U+000C FORM FEED, or U+000D CARRIAGE RETURN. Rust uses the WhatWG Infra Standard’s definition of ASCII whitespace. There are several other definitions in wide use. For instance, the POSIX locale includes U+000B VERTICAL TAB as well as all the above characters, but—from the very same specification—the default rule for “field splitting” in the Bourne shell considers only SPACE, HORIZONTAL TAB, and LINE FEED as whitespace. If you are writing a program that will process an existing file format, check what that format’s definition of whitespace is before using this function. §Examples pub const fn is_ascii_control(&self) -> bool Checks if the value is an ASCII control character: U+0000 NUL ..= U+001F UNIT SEPARATOR, or U+007F DELETE. Note that most ASCII whitespace characters are control characters, but SPACE is not. §Examples Trait Implementations§ impl AsciiExt for char type Owned = char fn is_ascii(&self) -> bool fn to_ascii_uppercase(&self) -> Self::Owned fn to_ascii_lowercase(&self) -> Self::Owned fn eq_ignore_ascii_case(&self, o: &Self) -> bool fn make_ascii_uppercase(&mut self) fn make_ascii_lowercase(&mut self) impl Clone for char fn clone(&self) -> char fn clone_from(&mut self, source: &Self) impl Debug for char fn fmt(&self, f: &mut Formatter<'_>) -> Result<(), Error> impl Default for char fn default() -> char Returns the default value of \x00 impl Display for char fn fmt(&self, f: &mut Formatter<'_>) -> Result<(), Error> impl<'a> Extend<&'a char> for String fn extend(&mut self, iter: I)where I: IntoIterator, fn extend_one(&mut self, _: &'a char) fn extend_reserve(&mut self, additional: usize) impl Extend for String fn extend(&mut self, iter: I)where I: IntoIterator, fn extend_one(&mut self, c: char) fn extend_reserve(&mut self, additional: usize) impl From for char fn from(chr: AsciiChar) -> char impl From for String fn from(c: char) -> String Allocates an owned String from a single character. §Example impl From for u128 fn from(c: char) -> u128 Converts a char into a u128. §Examples impl From for u32 fn from(c: char) -> u32 Converts a char into a u32. §Examples impl From for u64 fn from(c: char) -> u64 Converts a char into a u64. §Examples impl From for char Maps a byte in 0x00..=0xFF to a char whose code point has the same value, in U+0000..=U+00FF. Unicode is designed such that this effectively decodes bytes with the character encoding that IANA calls ISO-8859-1. This encoding is compatible with ASCII. Note that this is different from ISO/IEC 8859-1 a.k.a. ISO 8859-1 (with one less hyphen), which leaves some “blanks”, byte values that are not assigned to any character. ISO-8859-1 (the IANA one) assigns them to the C0 and C1 control codes. Note that this is also different from Windows-1252 a.k.a. code page 1252, which is a superset ISO/IEC 8859-1 that assigns some (not all!) blanks to punctuation and various Latin characters. To confuse things further, on the Web ascii, iso-8859-1, and windows-1252 are all aliases for a superset of Windows-1252 that fills the remaining blanks with corresponding C0 and C1 control codes. fn from(i: u8) -> char Converts a u8 into a char. §Examples impl<'a> FromIterator<&'a char> for Box fn from_iter(iter: T) -> Boxwhere T: IntoIterator, impl<'a> FromIterator<&'a char> for String fn from_iter(iter: I) -> Stringwhere I: IntoIterator, impl FromIterator for Box fn from_iter(iter: T) -> Boxwhere T: IntoIterator, impl FromIterator for ByteString fn from_iter(iter: T) -> ByteStringwhere T: IntoIterator, impl<'a> FromIterator for Cow<'a, str> fn from_iter(it: I) -> Cow<'a, str>where I: IntoIterator, impl FromIterator for String fn from_iter(iter: I) -> Stringwhere I: IntoIterator, impl FromStr for char type Err = ParseCharError fn from_str(s: &str) -> Result::Err> impl Hash for char fn hash(&self, state: &mut H)where H: Hasher, fn hash_slice(data: &[Self], state: &mut H)where H: Hasher, Self: Sized, impl Ord for char fn cmp(&self, other: &char) -> Ordering fn max(self, other: Self) -> Selfwhere Self: Sized, fn min(self, other: Self) -> Selfwhere Self: Sized, fn clamp(self, min: Self, max: Self) -> Selfwhere Self: Sized, impl PartialEq for char fn eq(&self, other: &char) -> bool fn ne(&self, other: &char) -> bool impl PartialOrd for char fn partial_cmp(&self, other: &char) -> Option fn lt(&self, other: &char) -> bool fn le(&self, other: &char) -> bool fn ge(&self, other: &char) -> bool fn gt(&self, other: &char) -> bool impl Pattern for char Searches for chars that are equal to a given char. §Examples type Searcher<'a> = CharSearcher<'a> fn into_searcher(self, haystack: &str) -> ::Searcher<'_> fn is_contained_in(self, haystack: &str) -> bool fn is_prefix_of(self, haystack: &str) -> bool fn strip_prefix_of(self, haystack: &str) -> Option<&str> fn is_suffix_of<'a>(self, haystack: &'a str) -> boolwhere ::Searcher<'a>: ReverseSearcher<'a>, fn strip_suffix_of<'a>(self, haystack: &'a str) -> Option<&'a str>where ::Searcher<'a>: ReverseSearcher<'a>, fn as_utf8_pattern(&self) -> Option<Utf8Pattern<'_>> impl Step for char fn steps_between(: &char, : &char) -> (usize, Option) fn forward_checked(start: char, count: usize) -> Option fn backward_checked(start: char, count: usize) -> Option unsafe fn forward_unchecked(start: char, count: usize) -> char unsafe fn backward_unchecked(start: char, count: usize) -> char fn forward(start: Self, count: usize) -> Self fn backward(start: Self, count: usize) -> Self impl TryFrom for u16 Maps a char with code point in U+0000..=U+FFFF to a u16 in 0x0000..=0xFFFF with same value, failing if the code point is greater than U+FFFF. This corresponds to the UCS-2 encoding, as specified in ISO/IEC 10646:2003. fn try_from(c: char) -> Result>::Error> Tries to convert a char into a u16. §Examples type Error = TryFromCharError impl TryFrom for u8 Maps a char with code point in U+0000..=U+00FF to a byte in 0x00..=0xFF with same value, failing if the code point is greater than U+00FF. See impl From for char for details on the encoding. fn try_from(c: char) -> Result>::Error> Tries to convert a char into a u8. §Examples type Error = TryFromCharError impl TryFrom for char type Error = CharTryFromError fn try_from(i: u32) -> Result>::Error> impl ConstParamTy_ for char impl Copy for char impl Eq for char impl StructuralPartialEq for char impl TrustedStep for char impl UnsizedConstParamTy for char Auto Trait Implementations§ impl Freeze for char impl RefUnwindSafe for char impl Send for char impl Sync for char impl Unpin for char impl UnwindSafe for char Blanket Implementations§ impl Any for Twhere T: 'static + ?Sized, fn type_id(&self) -> TypeId impl Borrow for Twhere T: ?Sized, fn borrow(&self) -> &T impl BorrowMut for Twhere T: ?Sized, fn borrow_mut(&mut self) -> &mut T impl CloneToUninit for Twhere T: Clone, unsafe fn clone_to_uninit(&self, dst: mut u8) impl From for T fn from(t: T) -> T Returns the argument unchanged. impl Into for Twhere U: From, fn into(self) -> U Calls U::from(self). That is, this conversion is whatever the implementation of From for U chooses to do. impl ToOwned for Twhere T: Clone, type Owned = T fn to_owned(&self) -> T fn clone_into(&self, target: &mut T) impl ToString for Twhere T: Display + ?Sized, fn to_string(&self) -> String impl TryFrom for Twhere U: Into, type Error = Infallible fn try_from(value: U) -> Result>::Error> impl TryInto for Twhere U: TryFrom, type Error = >::Error fn try_into(self) -> Result>::Error>
6006
https://www.gauthmath.com/solution/11-If-the-function-f-x-frac-2x-1-4-prove-that-f-1of-x-is-an-identity-function-Al-1708581588711446
Solved: If the function f(x)= (2x+1)/4 , prove that f-1of(x) is an identity function. Also find [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 If the function f(x)= (2x+1)/4 , prove that f-1of(x) is an identity function. Also find the value of f^(-1)(2). Show transcript Expert Verified Solution 97%(977 rated) Answer $$f^{-1}(2) = \frac{7}{2}$$f−1(2)=2 7​ Explanation Write down the given function $$f(x) = \frac{2x + 1}{4}$$f(x)=4 2 x+1​ To find the inverse function $$f^{-1}(x)$$f−1(x), solve for $$x$$x in terms of $$f(x)$$f(x). Start with $$f(x) = \frac{2x + 1}{4}$$f(x)=4 2 x+1​ and isolate $$x$$x: $$x = \frac{2f^{-1}(x) + 1}{4}$$x=4 2 f−1(x)+1​ Solve for $$f^{-1}(x)$$f−1(x) by multiplying both sides by 4 and then subtracting 1: $$f^{-1}(x) = \frac{4x - 1}{2}$$f−1(x)=2 4 x−1​ Verify that $$f^{-1} \circ f(x)$$f−1∘f(x) is an identity function by composing $$f^{-1}$$f−1 with $$f(x)$$f(x): $$f^{-1} \circ f(x) = f^{-1}(f(x))$$f−1∘f(x)=f−1(f(x)) $$= f^{-1}\left(\frac{2x + 1}{4}\right)$$=f−1(4 2 x+1​) $$= \frac{4 \times \frac{2x + 1}{4} - 1}{2}$$=2 4×4 2 x+1​−1​ $$= \frac{2x + 1 - 1}{2}$$=2 2 x+1−1​ $$= \frac{2x}{2}$$=2 2 x​ $$= x$$=x Find the value of $$f^{-1}(2)$$f−1(2) by substituting $$x = 2$$x=2 into the inverse function: $$f^{-1}(2) = \frac{4 \times 2 - 1}{2}$$f−1(2)=2 4×2−1​ $$f^{-1}(2) = \frac{8 - 1}{2}$$f−1(2)=2 8−1​ $$f^{-1}(2) = \frac{7}{2}$$f−1(2)=2 7​ So, $$f^{-1}(2) = \frac{7}{2}$$f−1(2)=2 7​ and $$f^{-1} \circ f(x)$$f−1∘f(x) is an identity function. Helpful Not Helpful Explain Simplify this solution Related If fx= 2x+1/4 prove that f'ofx is an identify function. 100% (2 rated) If fx= 2x+1/4 . prove that fofx is an identify function 100% (3 rated) Find the inverse of the following fx= 2x+1/4 100% (14 rated) Differentiation [Easy] This question concerns the function y=fx=x3-3x2-45x+40 i Show that the curve y=fx has two stationary points and evaluate the x and y coordinates of these points. ii Use the second derivative to determine the nature of these points maximum, minimum, point of inflexion. 100% (2 rated) The function f is given by f:xto e 1/2 x-3,x ∈ R. a Find f-1x and state its domain. b Sketch the curve y=f-1x , showing the coordinates of any points of intersection with the coordinate axes. The function g is given by g:xto ln x+5,x ∈ R,x>-5. c Evaluate fg4. d Solve the equation f-1x=gx. 100% (1 rated) The set of solutions to the inequality x2+bx+c<0 is the interval p 100% (1 rated) [Integration and a little Functions [Moderate] [10 a Evaluate the integral ∈ t [e- x/3 +sin x/4 ]dx [2 b i Find the points where the function y=gx=x-1e-x crosses the x-axis and the y-axis. 1 ii Use integration by parts to determine the area enclosed by the curve y=gx and the x-axis and the y-axis. 7 100% (5 rated) 520g Determine the equation of a tangent or normal to a curve where the function involves fractional or negative powers. The curve y=fx is given by fx=-x4-frac 27x3+44 The point Plies on the curve and has coordinates -3,-36. Find an equation of the normal to the curve at P. 100% (5 rated) Answer all questions in the spaces Find the coordinates of the minimum point on the curve y=2x2-8x. [1 mark] Circle your answer. -2,-8 2,-4 2,-8 -2,-4 100% (1 rated) Find the gradient function of y=5x3-8x2. y'=square Check 100% (2 rated) Gauth it, Ace it! contact@gauthmath.com Company About UsExpertsWriting Examples Legal Honor CodePrivacy PolicyTerms of Service Download App
6007
https://chemistrytalk.org/naming-alkanes/
Skip to content Naming Alkanes Core Concepts In this tutorial on alkane nomenclature, you will learn how to name various alkanes using the IUPAC system. You will also become familiar with identifying various alkyl attachments. We make naming alkanes easy! Topics Covered in Other Articles Functional groups Hydrocarbons Alkanes Naming covalent compounds Spectral Analysis of Organic Compounds What are alkanes? Alkanes are single-bonded straight chains of saturated hydrocarbons. This means they are comprised exclusively of single-bonded carbons and hydrogens. The longest continuous chain of an alkane is called the parent chain. Since the parent chain is an alkane, it will end with -ane. (Example: If there are 5 carbons in this chain, the name will end with -pentane). Groups of hydrocarbons branching off of this chain are called substituents and they are alkyls because they have one less hydrogen. Because of this, they will have the suffix -yl. (Example: if there are 5 carbons with a two-carbon substituent, this will become ethylpentane). What is IUPAC? To find the name of a given alkane, we will utilize the IUPAC system of naming. This is a set of rules created in 1919 by a group of chemists to guarantee each molecule has a unique name. The reason it is important for each molecule to be given a distinct name is that it tells us exactly how each molecule is built just based on the name alone. This is especially important in organic chemistry, where many similar molecules only differ by a few characteristics that need to be reflected in their names. IUPAC for Alkanes There are four parts to naming alkanes. The locant: The number indicating where the substituent is. The prefix: The substituent attached to the alkane. Ends with -yl. The Parent: The alkane parent chain. Ends with -ane. Suffix: The functional group attached to the alkane. Not always present. How to name an alkane step by step: In order to show you how to use the IUPAC naming system, we will use the alkane below to demonstrate. Step 1: Find the alkane’s parent chain. When naming alkanes, like derivatives of butane or pentane, an easy way to find the IUPAC name for an alkane is to work backward. This means we will start by finding the parent. To do this we will keep two things in mind: We want to find the longest parent chain. We want to have the most substituents possible. As seen above, the longest parent chain is 7 carbons long. This means the alkane will end with “-heptane”. The table below will help you find the parent name that corresponds to the number of carbons in the chain. | | | | | | | | | | | | --- --- --- --- --- | Number of Carbons | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | | Parent Name | methane | ethane | propane | butane | pentane | hexane | heptane | octane | nonane | decane | Step 2: Number the parent chain The parent chain can be numbered from left to right or vise versa. In some cases, both ways provide equal results. However, this is not always the case. To determine which way to number, consider the following: Number the chain towards the closest attachment. If there is a tie, resort to the Point of difference rule: If a tie occurs, consider the locants. We want the substituents on the lowest number locants possible. If there is a tie at the locants, use the alphabetical rule. For example, 1-ethyl would beat 1-methyl because it comes first in the alphabet. Notice this molecule is even and the numbers are mirrored. No matter which way you number this alkane, the methyl group is still on carbon four. This is a situation in which it does not matter which way you number it. Step 3: Identify the attachments Notice the attachments above end in “-yl” instead of “-ane”. This is because they are alkyls which means they have one less hydrogen due to being attached to the parent chain. In the case of our alkane, we have one methyl group and it is located at carbon 4. Step 4: Put everything in order Now that we know the locant, prefix, and parent, we can put them together like a puzzle. When doing this, there are three things to remember: Separate numbers from words with hyphens (-). Separate numbers from other numbers with commas (,). Arrange attachments in alphabetical order. (Example: 1-ethyl-7-methyloctane) We were able to gather the following information from this alkane: The locant: 4 The prefix: methyl The parent: heptane Putting this information together, we will get 4-methylheptane. Check out our cool animated lecture videos for more help and practice problems. Naming Alkanes – Practice Example: What is the name for this alkane? Answer: 3-ethyl-2,2,4,5,5-pentamethylhexane The longest parent chain is 6 carbons long which means the parent name is “-hexane”. Notice you could go straight down the chain and also get a 6 carbon parent chain. The difference is that the highlighted parent chain gives us more substituents. The numbering should start from the right and go left. This is because if you go from the right and the left, they are even at carbon two. The difference occurs at the third carbon down the line from both ways. The third carbon from the left contains a methyl, whereas the third carbon from the right contains and ethyl. Ethyl will trump methyl and we, therefore, count right to left. Notice there is one ethyl group and five methyl groups. These will be written starting with ethyl at the front. It will look like this: 3-ethyl-2,2,4,5,5-pentamethylhexane. Notice the Penta put in front of the methyl. When more than one of the same attachments is present we will add a prefix right after the numbers such as di, tri, tetra, and so on. This tells us there are 5 methyls total present and they are located twice at carbon 2, once at carbon 4, and twice at carbon 5. Since there is only one ethyl, we do not need to add a prefix. Thanks for reading, and be sure to check out more of our tutorials, experiments and element articles.
6008
https://math.stackexchange.com/questions/4788951/graphing-y-x-sinx
trigonometry - Graphing $y=x+\sin(x)$ - 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 Graphing y=x+sin(x)y=x+sin⁡(x) [closed] Ask Question Asked 1 year, 11 months ago Modified1 year, 11 months ago Viewed 135 times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. Closed. This question does not meet Mathematics Stack Exchange guidelines. It is not currently accepting answers. Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc. Closed 1 year ago. Improve this question I've been trying to learn how to graph some miscellaneous functions and I came across this. I put it in Desmos and this is what I ended up with. Can you let me know how did the graph get its shape? And how can one graph such type of functions on their own? trigonometry graphing-functions Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications edited Oct 18, 2023 at 4:15 Gary 37.1k 3 3 gold badges 43 43 silver badges 75 75 bronze badges asked Oct 18, 2023 at 4:12 Gnanadeep SaiGnanadeep Sai 13 2 2 bronze badges Add a comment| 3 Answers 3 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. The function has two parts. The x x is a line at 45° and it is shown as a dashed line in your post. The sin x sin⁡x is a sine wave that fluctuates around the 0 value by ±1±1 in a wavy pattern. Put them together, and you get a curve going uphill but also fluctuates by ±1±1 in a wavy pattern. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Oct 18, 2023 at 5:43 John AlexiouJohn Alexiou 14.7k 1 1 gold badge 40 40 silver badges 78 78 bronze badges Add a comment| This answer is useful 0 Save this answer. Show activity on this post. The input values for the graph are measured in radians by default. Between 0 and π π radians, the sine function's outputs have positive values, so when added to the number of radians produces values above the line y = x. Between π π and 2 π 2 π radians, the sine function has negative outputs which when added to the number of radians produces values below the line y = x. This pattern then repeats around the line y = x in each horizontal direction, just like the sine function itself does. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Oct 18, 2023 at 6:19 user1167128 user1167128 Add a comment| This answer is useful 0 Save this answer. Show activity on this post. The graph for y=x+sin x y=x+sin⁡x is the superposition of the lines y=x y=x and the sine curve y=sin x y=sin⁡x. When you combine one graph and the other through a mathematical operation, the new graph takes on the characteristics of each of the original graphs. In your Desmos graph, y=x+sin x y=x+sin⁡x can be thought of as the regular sine function y=sin x y=sin⁡x rotated 45°45° clockwise. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Oct 18, 2023 at 10:55 bjcolby15bjcolby15 4,240 3 3 gold badges 23 23 silver badges 31 31 bronze badges Add a comment| Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions trigonometry graphing-functions 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 Report this ad Related 1Graphing parametric complex equations 1Find a function from graph 2How is parametric 3D graphing done on a 2D graph/area? 0Graph of a single point continuous function. 1How to move and/or rotate a semi circle or like any equation. Or just add a perimeter cut off on one of the sides. 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6009
https://byjus.com/jee/moment-of-inertia-of-rectangular-plate/
In the case of a rectangular plate, we usually find the mass moment of inertia when the axis is passing through the centre perpendicular to the plane. We use the following expressions to calculate the moment of inertia of a rectangular plate; For x-axis; | | | Ix = (1/12) mb2 | For y-axis; | | | Ix = (1/12) ma2 | | | | IZ = (1/12) m (a2 + b2) | Moment Of Inertia Of A Rectangular Plate Derivation 1. Line Passing Through The Base For the derivation of the moment of inertia formula for a rectangular plate, we will consider a rectangular section and cut out an elemental part at a distance (y) from the x-axis. Let its thickness be dy and s be the mass per unit volume of the plate. Mass of the rectangular body = density x volume M = ρ x bdt Mass of the elemental section; dM =ρ x btdy Now we will find the mass moment of inertia of the elemental strip about the x-axis. I = ∫ y2 dM I = ∫ (ρ b dy t)y2 In the next step, we will find the mass moment of inertia of the rectangular plate. I = o∫d ρ b dy t y2 I = ρ bt o∫d y2dy I = ρ bt d3 / 3 (since ρ = M/bdt) I = Md2 / 3 At this point, we apply the parallel axis theorem which gives us; Ibase = ICG + M x h2 ICG = Ibase – Mh2 ICG = Md2 / 3 – M x (d / 2)2 ICG = Md2 / 12 This is how we determine the mass moment of inertia of a rectangular section about a line passing through the base. 2. Line Passing Through The Centre of Gravity Now if we want to derive the expression for a line passing through the centre of gravity then the steps for derivation are almost the same. We only have to make a few changes. We will take one rectangular elementary strip and consider the thickness to be (dY) and it will be at a distance (Y) from the X-X axis. The rectangular elementary strip area will be dA = dY.B Mass of the rectangular elemental part is given by; dm = ρ x T x dA dm = ρ x T x dY. B dm =ρBT. dY Now we can say that the mass moment of inertia of the rectangular elemental section about the X-X axis , (Im)xx = dm.Y2 If we substitute the values for dm we get; (Im)xx = ρBT. Y2dY Next step involves determining the mass moment of inertia of the entire rectangular section. We have to integrate the above equation between limit (-D/2) to (D/2). We will write it as; (Im)xx = -D/2∫D/2 ρBT. Y2dY (Im)xx = ρBT -D/2∫D/2 Y2dY (Im)xx = ρBT [y3 / 3]-D/2D/2 (Im)xx = ρBT / 3 [(D/2)3 – (-D/2)3] Im) XX = ρBT.D3/12 (Im) XX = ρBTD.D2/12 (Im) XX = ρV.D2/12 (Im) XX = M.D2/12 Likewise, we can also determine the mass moment of inertia of the rectangular section about the Y-Y axis. We will obtain; (Im) YY = M.B2/12 ⇒ Check Other Object’s Moment of Inertia: Moment Of Inertia Of A Cylinder Moment Of Inertia Of A Solid Cylinder Moment Of Inertia Of Circle Moment Of Inertia Of Rectangle Moment Of Inertia Of Rod Video lesson – Parallel Axis Theorem 24,478 Comments Leave a Comment Cancel reply Register with BYJU'S & Download Free PDFs
6010
https://www.cde.state.co.us/comath/lesson-assessments_what-is-a-variable
Skip to Main Content Contact Us Licensing | About CDE | State Board | Offices | Staff Directory | News | Careers | Site Index Families Educators Districts Communities You are here Home › Mathematics Toolkits › What is a Variable? - Grades 6-8 Lesson and Assessments - What is a variable? Jump to a section: Print: Summary, Goals and Standards Lesson and Assessments Print the Lesson Print Teacher Resources Progression of Skills (outline) Before the lesson: Students should review basic skills involving variables, including an understanding that coefficients of variables indict multiplication (i.e. 5x means 5 times x) and evaluating an expression when a variable has a value (i.e. if k is 2, then 3k is 6) During the lesson: This toolkit lesson focuses on developing a conceptual understanding of variables, including that a variable represents an unknown quantity, a variable represents a varying quantity (a number that can change), and that related numbers that change together are variables. After the lesson: Use their strong concept of variables to begin solving algebraic problems and assigning variable meaning in word problems Lesson Introduction and Pre-Assessment Pre-assessment and answer key: Pre-assessment (and key) Directions for Lesson Introduction: Have each student complete the pre-assessment. Collect them to assess student understanding. Each question is designed to identify a common misconception. Questions 1 and 4 identifies if students view variables as labels/objects Question 2 identifies if the student ignores variables Question 3 identifies if the student views variables as representing a specific unknown. Provide feedback for students. It is recommended that you do not score their responses, but instead write a question or two to help students make progress. This is so students focus on their mathematical thinking rather than comparing their scores with other students. Here are some sample questions for common issues Question 1: The student writes that a variable is a letter or object. For example, t represents time or a variable is x. Is this always true? What else can a variable represent? Question 2: Student ignores the variable. For example, they add 3 to h+2 and get 5 What would happen if h was 1? What if h was 0? What does the h represent? Question 3: Student uses a specific unknown instead of multiple unknowns for a variable. For example, they only use 3 for t. Is t always 3? Question 4: Student interprets a variable as a label. For example, in the expression S=8P, the student interprets S as students instead of the number of students What does 8P represent? What do you know about the number of students and professors? Can you write the meaning as you would say in everyday life? Lesson Task 1: Using slide 1 of the discussion questions, start by leading a discussion with your students about variables. This discussion can take many forms depending on the number of students. For small groups lead a conversation, for larger groups have students work in teams to come up with answers to the questions that they all agree with. Pass out the “Variables: Sometimes, Always, Never” note catcher. Walk the students through the example statement, the correct answer, and the reasoning. Have students work individually to complete the rest of the problems. Emphasize that the reasoning is more important than getting the sometimes, always, or never correct. Designate three parts of the room as “Sometimes,” “Always” and “Never.” Read a statement aloud, and have students move to the part of the room that they agree with. Have students in each part of the room discuss their reasoning, then have one student from each group share out. Once students have shared, people can move to different parts of the room if they changed their mind. The goal of this is to reach a consensus where all students are eventually in one part of the room. If you would prefer for students to not move around, you can provide students with yellow, green, and red pieces of paper to represent “sometimes,” “always” and “never” respectively. Repeat this process for each statement. Once finished, have students use a colored pencil to correct any mistakes they made on the note catcher. Progress monitoring: Collect the note catcher to assess student understanding Anticipate Misconceptions: Variables can be viewed as labels (i.e. in the expression 5S, S represents students rather than the number of students) Variables can be ignored (i.e. if asked to add 4 to n+2, they would erroneously get 6) Variables represent a specific unknown (i.e. The variable t can only have 1 value rather than many values) Lesson Task 2: Set up a long number line across a room with the center labeled 0. (It is not necessary to put any other points on the number line). Print and cut out the variable cards in advance. Pass out a variable card to each student and have them put themselves in the correct order in the number line. (NOTE: This task requires some trust and vulnerability. It is important to build a sense of community before engaging in this task as written. If the math community has not been established, you could make it more straightforward you could start by asking students to line up in order if given t=1.) As a differentiation strategy, use discretion in assigning variable cards (do not assign them randomly) For large groups or students who need extra support, have pairs share a variable card Students may realize that there is not one correct order because the variables could have multiple values. If they struggle to come to this conclusion you can ask questions to help them realize this such as: How do you know that you belong there? Is this the only correct answer? What does the variable t mean? What does it represent? Once students have realized that there is not one correct answer, ask the students to line up if t is 1. Once they have settled, have students go down the line and explain their reasoning. Make sure that all the students agree with everyone's placement in line. Now, ask them to line up if t is 4. Once they move, have the students discuss if they are standing next to the same people or not. Do the same thing for if t is negative 2. Repeat this procedure a few more times with students coming up with numbers for t to be. Ask the students to pay attention to expressions for which the order never changes, such as t and t+4. Have students reflect on why some expressions change order and some do not. Give students the exit ticket so they can reflect on their learning. Progress monitoring: Collect the exit tickets to assess student knowledge. Lesson Task 3: Starting from page T-4, use the Interpreting Equations lesson from the Mathematical Assessment Project. You can find the projector resources here. Lesson Closure and Post-Assessment Directions for Lesson Closure: Give students the post-assessment to complete. The post-assessment parallels the pre-assessment, so you can easily see what students learned and what they still need to know. Post-assessment and answer key: Post assessment and key Real-World Applications and Project Ideas What situations can you create an equation with a variable for? Research Diana L. Moss, Jennifer A. Czocher, & Teruni Lamberg. (2018). Frustration with Understanding Variables is Natural. Mathematics Teaching in the Middle School, 24(1), 10–17. Sarah B. Bush, Karen S. Karp. (2013) Prerequisite algebra skills and associated misconceptions of middle grade students: A review. The Journal of Mathematical Behavior, 32(3) 613-632. Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005/02//). Middle school students' understanding of core algebraic concepts: Equivalence & Variable1. ZDM.Zentralblatt Fuer Didaktik Der Mathematik, 37(1), 68-76. Joan Lucariello, Michele T. Tine, Colleen M. Ganley. A formative assessment of students’ algebraic variable misconceptions. The Journal of Mathematical Behavior, 33, 30-41. Sources Interpreting Equations lesson from the Mathematical Assessment Project Interpreting Equations is licensed under CC BY-NC-ND 3.0 ‹ PRINT VERSION - What is a Variable? up Connect With Us Financial information on schools and districts throughout Colorado. Learn more about financial transparency. Quick Links Colorado.gov Offices Our Staff News Releases Site Index FAQs COOL: Online Licensing Human Resources Jobs at CDE Jobs for Teachers Moodle LMS Professional Development SchoolView State Board Commissioner Communications Calendar About CDE Accessibility Statement Contact Us Colorado Dept. of Education 201 East Colfax Ave. Denver, CO 80203 Phone: 303-866-6600 Contact CDE CDE Hours Mon to Fri, 8 a.m. to 5 p.m. See also Licensing Hours UPDATED January 23, 2025 Copyright © 1999-2025 Colorado Department of Education. All rights reserved. Title IX. Accessibility. Disclaimer. Privacy. Original text Rate this translation Your feedback will be used to help improve Google Translate
6011
https://physics.stackexchange.com/questions/536820/lenz-law-direction-of-induced-current
electromagnetism - Lenz law - Direction of induced current - Physics Stack Exchange Join Physics 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 Physics helpchat Physics 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 Lenz law - Direction of induced current Ask Question Asked 5 years, 6 months ago Modified2 months ago Viewed 703 times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. I understand the Lenz Law is such that it tells us that the induced current must be in a direction that makes the end of the solenoid facing the magnet behave as a North Pole to repel the magnet’s North Pole thus its motion is opposed. Since Lenz’s law is a consequence of the principle of the conservation of energy. A magnet that is pushed into the coil is repelled. What I can’t seem to understand is the direction of the induced current. I’ve been told that the direction is clockwise, not counter-clockwise. A teacher told me to use the Right Hand Grip Rule, but the way how I see it when I do it, I am seeing it in the counterclockwise direction. The question is: looking from the AC end, what is the direction of the induced current?Can someone please explain to me how the induced current is going in the clockwise direction? electromagnetism electromagnetic-induction lenz-law Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Improve this question Follow Follow this question to receive notifications edited Jun 19, 2021 at 11:55 Urb 2,754 4 4 gold badges 15 15 silver badges 26 26 bronze badges asked Mar 17, 2020 at 19:41 Mic.AMic.A 1 1 1 bronze badge 1 When I look at the coils in this diagram I cannot tell which way they are coiling, or whether they are both coiling in the same direction.Peter –Peter 2025-03-14 10:31:46 +00:00 Commented Mar 14 at 10:31 Add a comment| 1 Answer 1 Sorted by: Reset to default This answer is useful 0 Save this answer. Show activity on this post. Here we have electromagnetic induction. Variable currents induce magnetic fields in loops, according to the Right-Hand Grip Rule. Since the current given by the power supply is going from C to D, the magnetic field generated by the left circuit goes from the D to C direction. This is, looking from the coil of the C end, the magnetic field inside the solenoid is coming to you. The vector is DC orientated, meaning the N pole is in C. Now, when this generated magnetic field is studied in our right loop/circuit, its magnetic field lines are going from A to B because magnetic lines are closed and go back to the initial point. This is, looking from the A-coil you see an entering vector arrow. So, as you said, this magnetic field is perturbing the AB solenoid and due to the Conservation of Energy, the second solenoid is preparing another magnetic field, called induced magnetic field, to counter the effects of the first magnetic field. So, how do you think this solenoid is going to do this? Well, through the flow of a second current which is the induced current. This current does not have another alternative than going from A to B, since it is the only way it can create a magnetic field to oppose the change in this second solenoid. So, looking from the coil of the AC end the induced current (i) in the second solenoid is COUNTERCLOCKWISE. The main mistake in these cases is related to the position of the hand. The thumb indicates the direction of the magnetic field when the other fingers adjust to the current direction, but it depends if the loop of the solenoid is made clockwise or counterclockwise, so be careful. Here you have your case, the current going from the North Pole to the South Pole: More info: Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications edited Jun 19, 2021 at 13:02 Mritunjay Kumar 231 3 3 silver badges 18 18 bronze badges answered Mar 17, 2020 at 23:13 VerifyingVerifying 1 1 1 bronze badge 4 But if we are observing the first solenoid, the direction would be clockwise?Mic.A –Mic.A 2020-03-18 04:19:34 +00:00 Commented Mar 18, 2020 at 4:19 If you are observing any of the solenoids from C or A point, you will see a counter clockwise current's direction. Current goes from positive to negative, considering the global convention.Verifying –Verifying 2020-03-19 04:22:58 +00:00 Commented Mar 19, 2020 at 4:22 Okay, I think I get it now ... So basically, the induced current is flowing in a counter clockwise direction from the AC end as it creates a magnetic field which opposes the effect of the magnetic field produced by the first coil ..?Mic.A –Mic.A 2020-03-28 20:15:00 +00:00 Commented Mar 28, 2020 at 20:15 Yes, basically it is.Verifying –Verifying 2020-03-29 21:18:19 +00:00 Commented Mar 29, 2020 at 21:18 Add a comment| Your Answer Thanks for contributing an answer to Physics Stack Exchange! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. MathJax reference. To learn more, see our tips on writing great answers. Draft saved Draft discarded Sign up or log in Sign up using Google Sign up using Email and Password Submit Post as a guest Name Email Required, but never shown Post Your Answer Discard By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions electromagnetism electromagnetic-induction lenz-law 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 Linked 0How to find direction of induced current? Related 6A paradox to Lenz's law 1Does Lenz's law imply the total flux (sum of the magnetic flux due to the moving magnet & that due to induced current) through the loop is constant? 3Lenz' law and magnetic moment 2Which provides the energy for induced current at the starting moment of the lenz's law 1Why is the force on a current carrying loop non-zero? 0Why is induced electric current moving clockwise/counterclockwise? 0Conservation of Energy with respect to Lenz's Law with moving coil 0Is this a contradiction to Lenz's Law? Hot Network Questions Making sense of perturbation theory in many-body physics Repetition is the mother of learning Is encrypting the login keyring necessary if you have full disk encryption? My dissertation is wrong, but I already defended. How to remedy? how do I remove a item from the applications menu How to home-make rubber feet stoppers for table legs? Overfilled my oil Checking model assumptions at cluster level vs global level? 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6012
https://files.consumerfinance.gov/f/documents/cfpb_balancing-savings-and-debt_report_2021-01.pdf
Balancing savings and debt: Findings from an online experiment CFPB Office of Research JANUARY 2021 One of many financial challenges facing consumers is to balance the goal of having a savings cushion that offers financial security with the goal of limiting the amount of debt they hold.1 In fact, most U.S. consumers hold savings and debt at the same time.2 People take on debt for many reasons, with the cost of holding different types of debt varying substantially. In holding savings and debt simultaneously, however, consumers face a choice: how much debt to pay down versus how much savings to retain. This tension can exist even when the cost of holding the debt (i.e., accrued interest) is higher than the interest produced by savings. One prominent case is credit card debt, where the interest paid on credit card debt is substantially higher than interest earned on most savings vehicles.3 One in three U.S. consumers has credit card debt that they carry from month to month (i.e., are credit card “revolvers”), and many of these consumers—about half—also have relatively low-interest savings that could be used to reduce their debt.4 This phenomenon— concurrent holding of high-interest credit card debt and low-interest savings—is commonly referred to as the “credit card debt puzzle.”5 1 This research brief from the Office of Research was written by Brianna Middlewood, Caroline Ratcliffe, and Grant Guillory. 2 Based on the 2016 Survey of Consumer Finances (SCF), 55 percent of U.S. families hold debt and have non-retirement liquid assets outside of a checking account. We exclude checking accounts because it is not clear if dollars held in a checking account are savings or earmarked for upcoming expenses (e.g., rent, utilities, groceries). 3 For example, the commercial bank interest rate on credit card plans is roughly 15 percent ( while the interest rate on savings accounts is near zero ( accessed December 2020). 4 Based on the 2016 SCF, 34 percent of U.S. families carry a credit card balance of at least $500 from month to month (across all credit cards), and about half (54 percent) of those who carry a balance have non-retirement savings of at least $500 (non-retirement liquid assets minus checking). We follow the prior literature in setting a $500 cut-off (e.g., Druedahl and Jørgensen 2017; Gross and Souleles 2002; Telyukova 2013). 5 Haliassos and Reiter (2007), Telyukova (2013), and Laibson, Repetto and Tobacman (2003) So, why do consumers choose to hold savings and credit card debt simultaneously, rather than using their savings to pay down debt? Is the security offered by a savings cushion a driver? Does the desire for the cushion nonetheless compete with a desire to pay down debt? Does a lack of understanding about relative costs—that the interest paid on credit card debt is much higher than the interest earned on savings accounts—play a role? Previous work from an economic perspective suggests that the tendency to simultaneously hold credit card debt and savings may be explained by the need for a savings cushion (i.e., liquidity), concerns about unexpected reductions in credit card limits, or limiting further spending by keeping a balance on the card.6 These studies rely on analyses of observational data (i.e., non-experimental) that authors use to understand trends in consumer behavior. The benefit of observational research is the focus on real consumer behavior. The benefit of an experimental approach using hypothetical scenarios is that it allows researchers to explore possible explanations for behavior. Psychological explanations to understanding the credit card debt puzzle have yet to be sufficiently explored.7 Our study is among the first to use an experimental design to explore the extent to which preserving savings and paying down debt may be competing goals.8 In our experiment, participants were presented with a scenario that assigned them a savings amount for a hypothetical person named Mr. Green. Participants were then asked how much savings they wanted to apply toward reducing Mr. Green’s $5,000 in credit card debt.9 For the savings amounts, we randomly assigned participants to one of 10 amounts ranging between $1,000 and $10,000. Because we set the credit card debt at $5,000 for all scenarios, but varied the savings amounts such that they were either not enough, just enough, or more than enough to cover Mr. Green’s debt, our design provides insight into whether the amount of savings and the gap between savings and debt may influence the decision to use that savings to pay down debt. 6 Telyukova (2013) and Zinman (2007) discuss liquidity, Fulford (2014) discusses variability in credit limits, and Bertaut, Haliassos, and Reiter (2008) and Laibson, Repetto and Tobacman (2003) discuss limiting further spending by maintaining a balance. 7 Psychological research provides frameworks for understanding and exploring this phenomenon. For example, previous work suggests that the closer one is to a goal, the faster one works to achieve it (Kivetz, Urminsky, and Zheng 2006). However, if consumers are working toward multiple goals, to pay down debt and to keep a savings cushion, for example, it is less clear how they might decide to manage both at once. Also, if it feels painful to spend savings on debt, or if it feels risky to give up a savings cushion, consumers may resist using their savings for that purpose (e.g., mental accounting, Prelec and Loewenstein 1998). 8 This study is in fulfillment of the Bureau’s statutory mission to conduct research on consumers’ use of financial products and services. 9 Our sample of participants is not nationally representative. Rather, we use a “convenience” sample recruited through Amazon’s Mechanical Turk™. We discuss the sample in more detail below. BALANCING SAVINGS AND DEBT 2 If the desire to preserve a savings cushion is a key driver in decision making, we should see that participants’ choices in our hypothetical scenario favor retaining credit card debt, thereby preserving more savings. This would be especially true in our scenarios where the available savings presented to participants was more than enough to cover the debt. Our experiment also explores the possibility that a lack of interest rate knowledge plays a role in this savings-debt decision.10 Within our experiment, participants were asked questions to assess their knowledge of the relative interest rates between savings accounts and credit cards (i.e., that interest paid on credit card debt is much higher than the interest earned on savings accounts). We use participants’ responses to these questions to test whether interest rate knowledge played a role in participants’ choices. Key takeaways:  In our hypothetical scenarios, participants preserved a savings cushion while also reducing debt. o Preserve a savings cushion: Most participants chose to continue holding some credit card debt to preserve more savings. In nine of the ten savings scenarios, fewer than half of participants put the maximum amount of savings toward debt reduction.11 Only in the scenario where savings was double the size of the debt ($10,000 versus $5,000), did even a majority of participants—77 percent— eliminate credit card debt. o Reduce debt: The vast majority of participants—over 90 percent in each of the savings scenarios—used at least some of the savings to pay down credit card debt. Further, on average, participants allocated more than half of the savings they could to debt reduction, even among those in the lowest savings groups.  We find no evidence that participants’ knowledge of relative interest rates between savings and credit cards influenced their decisions about how much savings to put toward credit card debt reduction. Below we present the experiment, describe the study participants, discuss our findings, and provide a concluding summary discussion. 10 Allgood and Walstad (2013) find that a lack of financial knowledge about credit cards is related to credit card behaviors, including carrying a balance from month to month. 11 In scenarios where savings was less than or equal to credit card debt, participants had the option of putting all of the savings towards credit card debt reduction. In scenarios where savings was greater than credit card debt, which was $5,000 for all participants, participants could put a maximum of $5,000 in savings toward credit card debt reduction. BALANCING SAVINGS AND DEBT 3 4 BALANCING SAVINGS AND DEBT The Experiment In the experiment, participants were randomly assigned to one of 10 scenarios where a hypothetical person named Mr. Green has $5,000 in credit card debt and must decide how much of his savings, if any, he should put toward paying the debt. The instructions provided to study participants were as follows:12 “On the next few screens, you'll see financial information for someone named Mr. Green. You will be asked whether you think Mr. Green should use money from a savings account to pay down his credit card debt.” Participants were then randomly assigned to one of the 10 scenarios summarized in Table 1.13 TABLE 1: SUMMARY OF EXPERIMENT SCENARIOS Study participants then read the following description of Mr. Green’s finances and were asked the following question: 12 The vignette with Mr. Green is based on a similar vignette used by Sussman and Shafir (2012). 13 The full experiment included 17 conditions with additional granularity at low savings values ($100, $250, and $500, which produce results that do not differ significantly from the $1,000 results) and additional granularity around the $5,000 break-even point ($4,500, $4,750, $5,250 and $5,500), which produce results with some statistical differences when compared to the $5,000 scenario; we believe this may be due to participants rounding to the nearest thousand). For the purposes of this brief, we focused the analysis on the 10 conditions at $1,000 intervals. Experiment Scenarios Credit Card Debt Available Savings Debt Minus Savings Percent of debt that can be paid Percent of savings that can be used 1 $5,000 $1,000 -$4,000 20% 100% 2 $5,000 $2,000 -$3,000 40% 100% 3 $5,000 $3,000 -$2,000 60% 100% 4 $5,000 $4,000 -$1,000 80% 100% 5 $5,000 $5,000 $0 100% 100% 6 $5,000 $6,000 $1,000 100% 83% 7 $5,000 $7,000 $2,000 100% 71% 8 $5,000 $8,000 $3,000 100% 63% 9 $5,000 $9,000 $4,000 100% 56% 10 $5,000 $10,000 $5,000 100% 50% “Mr. Green has $5,000 in credit card debt. He has [$X] in a bank savings account. He has no other savings except in a retirement account, which he cannot access. How much, if any, of the $5,000 credit card debt should Mr. Green pay out of the [$X] available savings?” The “X” amount varied based on which scenario participants were assigned, as shown in Table 1. After reading the scenario about Mr. Green, participants entered the amount they thought he should use from his savings to pay off the $5,000 debt. Table 1 shows that under the first four savings scenarios, the amount Mr. Green has in savings is less than the credit card debt, so participants could not pay the debt off completely. Under the fifth scenario—the “break-even” scenario—participants could pay off the debt completely but would have no savings left over. Finally, under scenarios six through 10, participants could pay off the debt and have savings left over. Table 1 also shows that the share of savings participants could use to pay down the credit card debt is 100 percent for the first five scenarios, and then drops below 100 percent when savings exceeds credit card debt. In addition to the key question about Mr. Green’s credit card payment decision, study participants answered questions designed to capture their own financial knowledge and financial circumstances:14  Knowledge of interest rates in the form of two questions about typical interest rates associated with savings accounts and credit cards.  Whether they have a credit card(s), if they carry a balance from month to month (i.e., are a revolver), and how much they have in credit card debt.  Income, employment status, amount of non-retirement savings, net worth (positive, negative, or zero), as well as their age and sex. The design of this experiment allows us to test whether participants’ choices are consistent with a preference for retaining credit card debt and preserving a larger savings cushion. Because participants choose how much hypothetical savings to put toward the debt, any amount they indicate represents a trade-off: they can have more of a savings cushion or they can reduce their savings cushion to reduce their debt. Additionally, the savings and credit card interest rate questions allow us to examine whether knowledge about interest rates played a role in 14 A number of questions included in the survey are not discussed, including perceptions of Mr. Green’s wealth, beliefs about whether carrying a balance and using more than 30 percent of credit affects credit scores, whether participants keep a balance on their credit cards intentionally, how much they worry about debt, and income volatility. Incorporating this information into our analyses does not significantly affect our results. BALANCING SAVINGS AND DEBT 5 participants’ decisions. Our expectation is that, generally, participants who are aware that credit cards accrue interest at much higher rate than savings will decide to use more savings to pay the debt than those who are not aware of this. Study Participants We use a convenience sample recruited and compensated through Amazon’s Mechanical Turk™.15 While convenience samples are a common source of participants for experimental research, this means that the data are not nationally representative. Our analyses include 551 study participants,16 with 49 to 61 study participants randomly assigned to each of the 10 savings scenarios.17 Even with this number of participants per savings scenario, meaningful trends emerge in the data. Additionally, because we randomly assigned study participants to one of the 10 savings scenarios, characteristics of study participants are similar across the 10 groups.18 To give a sense for how our study sample differs from a representative sample, we provide comparisons to the nationally representative 2019 American Community Survey (ACS; U.S. Census Bureau) and the 2016 Survey of Consumer Finances (SCF; Federal Reserve Board). In terms of demographic characteristics and income, our study sample has more males (62 versus 49 percent), is younger (e.g., share under age 34 is higher—51 versus 30 percent), and is more likely to have incomes below $100,000 (92 versus 61 percent; see Appendix Table 2).19 With respect to credit card debt and savings, our study sample shows both points of differences with, but also similarities to, the SCF.20 While our sample is more likely to have a credit card (90 15 The CFPB partnered with a contractor to recruit and pay study participants. 16 A total of 730 participants were recruited and assigned to one of our 10 savings conditions, but 179 participants were dropped from the analysis because of data quality concerns. Appendix Table 1 summarizes the number of participants dropped from the analysis along with reason codes. Study participants completed the online experiment through the Qualtrics survey platform. 17 Considering the large variation in savings amounts, we anticipated fairly large effect sizes (Cohen’s d of between 0.60 to 0.80, which translates to a 20 to 30 percent difference in the amount of savings used). Power analyses suggested that between 42 and 72 participants per condition would be sufficient to detect differences for this range of effect sizes. 18 We test whether participant demographic and economic characteristics vary across the 10 groups (see Appendix Table 2 for the complete list of characteristics examined). We find roughly the same number of statistically significant differences across the groups that we expect to occur by chance alone—21 of 380 (versus 18 of 380). As discussed below, we conduct analyses that control for participant characteristics and find similar results. 19 In our survey, the income and savings response options were binned amounts (e.g., income less than $35,000, $35,000-$49,999, etc.). To make comparisons, we binned the ACS and SCF amounts to match those in our survey. 20 Note that the comparison between our sample and the SCF is not exact because our survey asks about the participant (e.g., does the participant have a credit card) and the SCF is a survey of families (e.g., does anyone in the family have a credit card). BALANCING SAVINGS AND DEBT 6 versus 71 percent), the share with a credit card who carry a balance from month to month is very similar in the two samples (38 percent in the SCF and 36 percent in our sample). Further, our study participants are similar to the SCF sample in terms of credit card holders who have both savings and credit card debt—26 percent in our sample and 28 percent in the SCF.21 Although our sample of study participants is not nationally representative, our analyses nonetheless provide new information on how different savings and debt amounts may influence decisions regarding the amount of savings to apply to credit card debt reduction. We also find that our results are similar to analyses that do and do not control for participant characteristics. Findings In this section we describe the major findings of the study. We use the dollar amount participants thought Mr. Green should put toward his $5,000 debt to calculate the proportion of the debt paid out of what could be paid. We discuss the key findings for three groups: (1) those at the “break-even” point who had $5,000 in savings and so could pay off the debt completely but would have no savings left, (2) those who had less than $5,000 available in savings and so could not pay the debt off completely, and (3) those who had more than $5,000 in savings, and so could pay off the debt and have savings left over.22 Preserving a savings cushion: Most participants held on to some credit card debt and preserved some savings Do people retain credit card debt for the purpose of retaining more savings? Our findings suggest this may be the case. Overall, most participants in our experiment chose to continue holding credit card debt and preserve more savings. These choices resemble those of people in the credit card debt puzzle, who hold savings and credit-card debt at the same time. Participants held on to some debt and preserved savings even when initial savings amounts were high enough to allow them to completely pay off the credit card debt and still retain thousands of dollars in savings. 21 Following prior studies, we define co-holding as concurrently holding more than $500 in savings and more than $500 in credit card debt (e.g., Druedahl and Jørgensen 2017, Telyukova 2013). 22 We examine patterns in the amount of savings used and how it changed depending on how much was available to use. For those in scenarios where they could not pay off the debt completely, we examine the proportion of savings used out of what was available. For those in scenarios where they could pay off the debt completely (i.e., the amount of savings available was greater than the $5,000 debt), we examine the proportion of savings used out of $5,000. BALANCING SAVINGS AND DEBT 7 Among participants presented with the break-even scenario—where savings is equal to credit card debt (i.e., $5,000 in savings and $5,000 in credit card debt)—the majority (66 percent) held on to some credit card debt and preserved some savings (Figure 1). This means that only a third of these participants (34 percent) said Mr. Green should pay the full credit card balance. This finding suggests that participants were willing to retain some debt in order to maintain a savings cushion. FIGURE 1: MOST PARTICIPANTS HELD ON TO SOME CREDIT CARD DEBT TO RETAIN MORE SAVINGS Share of participants who did and did not pay the maximum amount of credit card debt, by participants’ savings scenario Note: For participants in scenarios where savings is less than debt (scenarios one though four), the maximum amount of credit debt that can be paid is equal to the amount of savings ($1,000. $2,000, $3,000, and $4,000, respectively). For participants in the other scenarios, the maximum amount of credit debt that can be paid is $5,000. Appendix Table 3 presents the level of statistical significance (p-values) for the 45 pairwise tests. Participants presented with scenarios where Mr. Green’s savings were less than debt (by $1,000 to $4,000) faced different economic circumstances, as eliminating credit card debt was not an option. As such, these participants were deciding how much savings to put towards paying down credit card debt, knowing that the decision to use all of the savings would still leave some credit card debt. BALANCING SAVINGS AND DEBT 8 Consistent with the findings above for those in the break-even scenario, most participants (between 68 and 87 percent) across the four “savings less than debt” scenarios (i.e., credit card debt is $5,000 and savings is either $1,000, $2,000, $3,000 or $4,000) allocated less to paying down their credit card debt than they could and so retained some savings (Figure 1).23 A minority of participants in each of these scenarios (13 to 32 percent) put all of the savings toward debt reduction. Those with the higher savings amounts (savings of $3,000 and $4,000) were statistically significantly less likely than participants in the break-even scenario to put all of the savings toward debt reduction. While those with $3,000 and $4,000 deviated somewhat from the pattern shared by those with $1,000, $2,000 and $5,000 in terms of how likely they are to use all of the savings, the average share of savings used in these conditions does not differ significantly (shown below). The inclination to hold on to savings is underscored by participants who retained credit card debt in scenarios where Mr. Green’s savings exceeded debt by $1,000, $2,000, $3,000, or $4,000, meaning participants could have eliminated credit card debt without eliminating savings. Looking across these four scenarios (i.e., credit card debt is always $5,000 and savings is either $6,000, $7,000, $8,000 or $9,000), the majority of participants (between 55 and 77 percent) held on to some credit card debt, and thus preserved more savings (Figure 1). Only 23 to 45 percent of participants paid the full credit card balance. For example, among participants with $5,000 in credit card debt and $9,000 in savings, only 35 percent of participants paid the full credit card balance, while 65 percent kept some credit card debt to retain more savings, even though paying off all credit card debt would have left a $4,000 savings cushion. This finding suggests that many participants value a larger savings cushion over eliminating debt. Interestingly, there are no statistically significant differences between the shares of participants who held on to some credit card debt under these four scenarios and the share who did so in the break-even scenario. Only in the scenario where the amount of Mr. Green’s savings was double the size of the debt ($10,000 versus $5,000), did a statistically significantly higher share and a majority of participants—77 percent—eliminate credit card debt. This still left nearly a quarter of participants (23 percent) who chose to retain some credit card debt and keep a larger savings cushion, even though paying it all off would leave $5,000 in savings. 23 If a person with $5,000 in credit card debt and $2,000 in savings put $1,500 towards credit card debt, they used some of their savings to reduce debt. If they instead they put $2,000 towards credit card debt, they used all of their savings to reduce debt. BALANCING SAVINGS AND DEBT 9 While these findings result from an experiment where participants are asked how a hypothetical consumer (Mr. Green) should conduct his finances, they suggest a tendency by consumers to continue holding credit card debt in favor of retaining more of a savings cushion. Interestingly, outside of the scenario where savings is double credit card debt ($10,000 versus $5,000), the share of participants who used the maximum amount of savings is not statistically significantly different across most of the savings scenarios (26 out of 36).24 Even controlling for participant characteristics (e.g., age, income, sex), the pattern of findings remains the same.25 The scenario where savings was double the size of the credit card debt stands out as the most distinct— participants were most likely to pay the full amount—as compared with the other scenarios. Reducing debt: Participants allocated more than half of savings to reduce debt, even in the lowest savings groups The vast majority of study participants put some savings toward debt reduction. Under each of the 10 savings-debt scenarios, between 92 and 100 percent of participants elected to put at least some of Mr. Green’s savings toward debt reduction (not shown). Additionally, we find that the share of savings used was substantial—more than half—even among participants in the lowest savings scenarios. Viewed alongside the results above, this suggests that participants worked to balance preserving a savings cushion and reducing debt. On average, in each of the 10 savings scenarios, participants allocated more than half of Mr. Green’s savings to the $5,000 in credit card debt. Participants in the break-even scenario ($5,000 in savings and $5,000 in credit card debt) allocated an average of 60 percent of savings to paying down credit card debt (Figure 2). These participants paid down an average of $2,980 in credit card debt, leaving an average savings cushion of $2,020 (Figure 3). Participants presented with scenarios where Mr. Green’s savings was less than debt (between $1,000 and $4,000) made similar decisions with respect to the share of savings to allocate to debt reduction. Across the four “savings less than debt” scenarios, participants 24 Appendix Table 3 shows which scenarios are and are note statistically significantly different from one another. 25 We estimate regression models that control for a number of participant characteristics—age (18-35, 35-54, 55+), sex, income ($0-$49,999, $50,000-$74,999, $75,000+), employment status (employed full-time, part-time, not employed), savings amount ($0-999, $1,000-$4,999, $5,000+), credit card ownership (yes/no), credit card revolver (yes/no), and interest rate knowledge (correct/incorrect about saving interest rate relative to credit card interest rate). We find that each statistically significant difference between savings scenarios when not controlling for participant characteristics is also statistically significant when controlling for participant characteristics. Furthermore, each difference that was not significantly significant when not controlling for participant characteristics is also not significantly significant when controlling for participant characteristics. BALANCING SAVINGS AND DEBT 10 allocated an average of 51 to 60 percent of savings to debt reduction (Figure 2). None of these percentages are statistically significantly different from one another or the break-even scenario. We also show this pattern in terms of the dollar amount paid toward the credit card debt: it increases with the amount of savings from $570 for those in the lowest savings group to $2,058 for those in the $4,000 savings group (Figure 3). Participants in these four scenarios retained a savings cushion of between $430 to $2,942 (for the $1,000 and $4,000 savings scenarios, respectively). Participants presented with scenarios where Mr. Green’s savings exceeded debt (debt of $5,000 and savings of between $6,000 and $10,000) could not, by design, use all of their savings. The maximum amount of savings they could allocate to credit card debt reduction was $5,000 (the amount of credit card debt). Thus, all of the shares for these scenarios in Figure 3 are calculated based on a maximum payment of $5,000. FIGURE 2: PARTICIPANTS ALLOCATED MORE THAN HALF OF POSSIBLE SAVINGS TO DEBT REDUCTION Average amount of possible savings allocated to pay off the $5,000 credit card debt, by participants’ savings scenario Note: This figure captures the share of savings used out of what could be used. This amount is capped at the available savings amount when savings is less than the $5,000 debt and capped at $5,000 when savings exceeded this value. BALANCING SAVINGS AND DEBT 11 FIGURE 3: THE DOLLAR AMOUNT PAID TOWARDS DEBT REDUCTION Note: The blue line shows either (1) the amount of savings available for those conditions where savings was insufficient to cover the debt, or (2) where savings exceeded debt, that $5,000 is the maximum that participants in those scenarios could pay. Looking across the five scenarios where Mr. Green’s savings exceeded debt, the share of savings allocated to credit card debt reduction ranged from a low of 53 percent among participants in the $7,000 savings scenario to a high of 84 percent among participants in the $10,000 savings scenario (Figure 2). With the exception of the $7,000 savings scenario (53 percent), the shares of savings used in the five scenarios are not statistically significantly different from one another. The dollar amounts paid toward the credit card debt tend to increase the more savings are available (Figure 3). Despite the fact that the gap between Mr. Green’s savings and credit card debt ranged from -$4,000 to $5,000, across most of the savings scenarios, the shares allocated to debt reduction were not vastly different. In nine out of the 10 scenarios, participants allocated an average of between a half and three-quarters of savings to debt reduction.26 Only the savings scenario 26 Recall that in scenarios where savings is greater than debt, this share is calculated out of $5,000, since $5,000 is the maximum amount of credit card debt that can be paid. BALANCING SAVINGS AND DEBT 12 where savings was double the amount of the debt, did the share fall outside this range—85 percent.27 While there are statistically significant differences in the share of savings participants allocated to credit card debt reduction across the different scenarios,28 they are smaller than what might be expected considering the large differences between savings amounts in this study (i.e., $1,000 vs. $10,000). We interpret this as a tendency for participants to want to make some progress towards paying down debt (at low levels of savings), but also a desire to maintain a substantial savings cushion (at the higher level of savings). Analyses that control for participant characteristics show similar results.29 Knowledge of relative interest rates did not affect participants’ decisions Most participants in this study (79 percent) accurately reported that the typical interest rate for a credit card exceeds that of a savings account. This suggests that most participants in this study were equipped with appropriate knowledge to understand the interest rate trade-off between credit card debt and savings. We nonetheless conducted analyses for each of the major findings to test whether this knowledge influenced the decision participants made for Mr. Green.30 We find no evidence that interest rate knowledge affected how much savings participants allocated to pay down credit card debt. This is contrary to our expectation that participants who correctly answered the interest rate questions would use more savings to pay the debt than those who did not. Discussion This study explored how the amount of available savings might influence whether a consumer is willing to use savings and, if so, how much savings, to pay down credit card debt. Based on our hypothetical experiment of a person named Mr. Green, the findings suggest that consumers balance two goals: preserving a savings cushion and reducing debt. 27 When comparing the average share of savings allocated to debt reduction in the $10,000 condition to all nine other conditions, the majority of these pairwise comparisons—seven of the nine—show statistically significant differences (Appendix Table 4). 28 Overall, with controls, 20 of the 45 pairwise comparisons have statistically significant differences. Excluding the $10,000 condition, 13 of the 36 pairwise comparisons show statistically significant differences (Appendix Table 4). 29 See Appendix Table 4. 30 As mentioned above, the regression models control for the following characteristics: age, sex, income, employment status, savings amount, credit card ownership, credit card revolver, and interest rate knowledge. BALANCING SAVINGS AND DEBT 13 The importance of preserving a savings cushion was evidenced by the fact that most participants viewing these scenarios chose to retain credit card debt, and thereby preserve more of a savings cushion. The value participants placed on preserving savings is highlighted by those participants who chose not to pay off a hypothetical debt completely even when there would be substantial savings left over (in some cases, thousands of dollars). Only when available savings was double the amount of the debt—leaving a cushion of $5,000—did most participants (over 75 percent, as opposed to between 25 and 45 percent for cushions of $1,000 to $4,000) say Mr. Green should pay all of the debt. On the other hand, the importance of reducing debt was evidenced by the fact that, on average, participants used more than half of available savings to pay down the debt, even at the lowest available savings amounts. For example, participants in the $1,000 savings scenario put an average of $570 (57 percent) toward credit card reduction. Even with savings buffers of between $1,000 and $5,000 for Mr. Green, many participants held on to credit card debt. Furthermore, despite the relative amount of savings to debt ranging widely—from -$4,000 to $5,000—there was less variation than might be expected across the savings conditions in the average amount of credit card debt paid (between 53 and 84 percent). This raises the question, how much of a savings cushion is enough? Results from the CFPB Making Ends Meet Survey show that the median amount people report they need in savings for an emergency is $10,000 (Ratcliffe et al. 2020), so it is possible that participants viewed these smaller amounts as too low to risk spending more for debt reduction.31 Future work might examine more precisely how the relative importance of the two competing goals to balance savings and debt can shift as the ratio of savings to debt changes and in relation to the amount of a savings cushion consumers believe they should have on hand for emergencies. Finally, knowing that the typical interest rate of a credit card exceeds that of a savings account did not affect participant’s decisions around using savings for debt reduction. This perhaps underscores the value participants may have placed on retaining a savings buffer, even if it meant holding on to more debt to do so. There are many other questions about the motivation and behaviors of consumers as they balance savings and debt that are ripe for future exploration. Our study focused on scenarios in which a savings cushion was available. But what about consumers who have yet to start saving? Future work could examine the motivations, trade-offs, and strategies of consumers seeking to build a savings cushion while also taking on and managing debt. 31 Additional information about the CFPB’s Making End Meeting survey can be found at BALANCING SAVINGS AND DEBT 14 This study provides a valuable first step toward understanding how consumers preserve a savings cushion while pursuing debt reduction. Our results suggest that the savings-debt trade-off is a balancing act, with most participants in the hypothetical scenarios allocating some of the savings to debt reduction (50 to 85 percent) while preserving the rest as a savings cushion. BALANCING SAVINGS AND DEBT 15 References Allgood, Sam, and William Walstad. 2013. "Financial literacy and credit card behaviors: A cross-sectional analysis by age." Numeracy 6, no. 2: 1-26. Bertaut, Carol C., Michael Haliassos, and Michael Reiter. 2009. "Credit card debt puzzles and debt revolvers for self control." Review of Finance 13, no. 4 (2009): 657-692. Fulford, Scott L. 2015. "How important is variability in consumer credit limits?" Journal of Monetary Economics 72: 42-63. Gross, David B., and Nicholas S. Souleles. 2002. "Do liquidity constraints and interest rates matter for consumer behavior? Evidence from credit card data." The Quarterly Journal of Economics 117, no. 1: 149-185. Haliassos, Michael, and Michael Reiter .2007. “Credit card debt puzzles.” CFS Working Paper, No. 26/2005. Goethe University Frankfurt, Center for Financial Studies. Kivetz, Ran, Oleg Urminsky, and Yuhuang Zheng. 2006. "The goal-gradient hypothesis resurrected: Purchase acceleration, illusionary goal progress, and customer retention." Journal of Marketing Research 43, no. 1: 39-58. Laibson, David, Andrea Repetto, and Jeremy Tobacman. 2000. “Estimating discount functions with consumption choices over the lifecycle.” No. w13314. National Bureau of Economic Research. Prelec, Drazen, and George Loewenstein. 1998. "The red and the black: Mental accounting of savings and debt." Marketing Science 17, no. 1: 4-28. Ratcliffe, Caroline, Melissa Knoll, Leah Kazar, Maxwell Kennady, and Marie Rush. 2020. “Perceived financial preparedness, saving habits, and financial security.” Consumer Financial Protection Bureau. Washington, DC: CFPB. Sussman, Abigail B., and Eldar Shafir. 2012. "On assets and debt in the psychology of perceived wealth." Psychological Science 23, no. 1: 101-108. Telyukova, Irina A. 2013. "Household need for liquidity and the credit card debt puzzle." Review of Economic Studies 80, no. 3: 1148-1177. Zinman, Jonathan. 2007. "Household borrowing high and lending low under no-arbitrage." Dartmouth University. BALANCING SAVINGS AND DEBT 16 17 BALANCING SAVINGS AND DEBT APPENDIX TABLE 1: DATA CLEANING AND PARTICIPANT DROP CODES Reason Code Count Definition 1 15 If less than 90 percent of the survey is completed. 2 0 If the response to the key outcome variable (amount participants would pay on Mr. Green’s debt) is missing. 3 24 If Qualtrics proprietary flag for duplicate entries is greater than 75, which they suggest indicates the entry is a duplicate. 4 77 If Qualtrics proprietary flag for fraudulent entries is greater than 30, which they suggest indicates the entry is fraudulent (e.g., potentially a bot). 5 30 If the amount entered is not compliant with instructions (exceeds the amount of the $5,000 debt or exceeds the amount available in savings when the savings available is less than $5,000). 6 4 If participants completed the survey in under 60 seconds. 7 56 This code flagged 1) if participants, in their open-ended responses, suggested that they were completing a subtraction problem, and 2) if the amount they said they would pay was equal to the amount obtained by subtracting one value from the other. If both flags were true, participants were dropped for failing to complete the task according to the instructions. Total Dropped 179 Note: These drop codes flagged 179 participants to be dropped (24.5 percent). 18 BALANCING SAVINGS AND DEBT APPENDIX TABLE 2: COMPARISON OF SURVEY SAMPLE TO 2016 SURVEY OF CONSUMER FINANCES (SCF) AND 2019 AMERICAN COMMUNITY SURVEY (ACS) Variable Group 2019 ACS Our Sample Sex Female 51% 38% Male 49% 62% Age 18-24 12% 5% 25 – 34 18% 46% 35 – 44 16% 24% 45 – 54 16% 16% 55 – 62 13% 7% >= 62 24% 3% Income < $35,000 19% 18% $35,000 - $49,999 11% 27% $50,000 - $74,999 17% 29% $75,000 - $99,999 14% 18% $100,000 - $174,999 24% 7% >= $175,000 15% 1% Non-retirement savings1 2016 SCF Our Sample $0 - $499 48% 17% $500 - $999 6% 16% $1,000 - $4,999 14% 27% $5,000 - $9,999 8% 21% >= $10,000 25% 19% Have a credit card Yes 71% 90% No 29% 10% Credit card revolver Yes 38% 36% No 62% 64% Credit card puzzle2 Yes 26% 29% No, savings > $500 but debt < $500 36% 52% No, debt > $500 but savings < $500 22% 4% No, both savings and debt < $500 17% 14% 1 In the SCF, non-retirement savings is calculated as non-retirement liquid assets minus checking account balance. 2 We calculate this is only for people with a credit card. 19 BALANCING SAVINGS AND DEBT APPENDIX TABLE 3: BY SAVINGS SCENARIO, P-VALUES FOR PAIRWISE TESTS OF DIFFERENCE IN SHARE OF RESPONDENTS WHO PAID THE MAXIMUM AMOUNT OF SAVINGS (WITHOUT CONTROLS) Scenario 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 1,000 2,000 0.951 3,000 0.023 0.026 4,000 0.142 0.157 0.394 5,000 0.797 0.750 0.012 0.086 6,000 0.248 0.222 0.001 0.010 0.367 7,000 0.287 0.313 0.212 0.684 0.189 0.029 8,000 0.139 0.123 0.000 0.004 0.222 0.761 0.012 9,000 0.741 0.697 0.012 0.083 0.932 0.437 0.179 0.280 10,000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.000 Note: Each statistically significant difference between saving scenarios when not controlling for participant characteristics is also statistically significant when controlling for participant characteristics. Furthermore, each difference that was not significantly significant when not controlling for participant characteristics is also not significantly significant when controlling for participant characteristics. APPENDIX TABLE 4: BY SAVINGS SCENARIO, P-VALUES FOR PAIRWISE TWO-TAILED T-TESTS OF DIFFERENCE IN PERCENT OF SAVINGS USED (WITHOUT CONTROLS) Scenario 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 1,000 2,000 0.695 3,000 0.675 0.404 4,000 0.382 0.199 0.622 5,000 0.694 0.990 0.411 0.207 6,000 0.248 0.426 0.105 0.043 0.446 7,000 0.499 0.272 0.789 0.812 0.280 0.060 8,000 0.015 0.039 0.002 0.001 0.046 0.241 0.001 9,000 0.001 0.004 0.000 0.000 0.005 0.051 0.000 0.400 10,000 0.000 0.000 0.000 0.000 0.000 0.005 0.000 0.061 0.241 = statistically significant without controls, but not statistically significant with controls.
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Contents > Chapter 1 – Stoichiometry Chapter 1 – Stoichiometry Introduction Chemistry – A Molecular Science (CAMS), the first half of this two-course sequence, stressed bonding, structure, and reactivity. The material was qualitative and stressed several types of reactions and the factors that affected their relative extents of reaction. However, as the title of this text suggests, chemistry is also a quantitative science. Chemists must not only predict the products of a reaction, they must also predict the amount of product that can be expected, and the amount of waste that must be removed. They also need to know how much energy is required or how much heat is generated by a reaction. They must also understand how the reaction occurs so that they can optimize the reaction conditions. These are the types of problems addressed in this text. We begin our study of the quantitative aspects of chemistry with stoichiometry, the science that deals with the quantitative relationships between the elements in a compound (substance stoichiometry) and between the substances in a chemical reaction (reaction stoichiometry). It is the topic of this first chapter because a thorough knowledge of stoichiometry is vital to an understanding of the material presented in this course. Understanding how quantitative data and results are presented is also important, so you should review Appendix A, Reporting Quantitative Measurements and Results, for a treatment of precision, significant figures, and rounding errors. Finally, we will set up many problems using the factor label method, so please review Appendix B, The Factor Label Method, for a discussion of this procedure. 1.1 Mass and Moles Introduction Chemists use chemical equations to design possible routes to desired molecules and to discuss chemical processes. However, the individual molecules represented in the equations are far too small to be seen, so chemists must use a very large number of molecules in their reactions in order that the reactants and products can be observed. Indeed, the number of molecules required to make a visible sample is staggering. Consider that 1 µL of water—about 1/50th of a drop—contains about 100,000,000,000,000,000 or 1017 molecules and a typical reaction in the laboratory involves thousands of times that number. Such large numbers are cumbersome, so scientists use a more convenient unit when discussing numbers of molecules. The unit is the mole, the topic of this section. Objectives Convert between numbers of moles and numbers of particles with Avogadro's number. • Convert between mass and moles with molar mass. • Convert between the pressure, volume, and temperature of a gas and the number of moles with the ideal gas law. 1.1-1. Avogadro's Number Avogadro's number, which is given below, is simply the number of items in a mole, so it also defines the mole, which is simply Avogadro's number of items. NA = 6.0221 × 1023 mol−1 Avogadro's Number A mole is used to indicate a number of items just as a dozen is used to indicate a number of eggs. Since Avodagro's number is so large, the mole is used only for the number of atoms or molecules, but it can be used for any item. For example, it is estimated that there are about 1023–1024 stars in the universe, which is 1–10 moles. Converting from moles to atoms or molecules is done in the same way as converting dozens to items. 1.5 doz = (1.5 doz)(12 items/doz) = 18 items and 1.5 mol = (1.5 mol)(6.0 × 1023 atoms/mol) = 9.0 × 1023 atoms. The mole is used simply because it is much easier to discuss the number of atoms in moles than it is as individual items. 0.10 mol H2O is a much more convenient expression than 6.0 × 1022 H2O molecules. 1.1-2. Molar Mass (Mm) Chemists need to be able to readily prepare mixtures of reactants that have the correct atom or molecule ratios for reaction, but they certainly cannot count such large numbers. Instead, they use other more easily determined properties that are related to the numbers of atoms and/or molecules. The first such method we examine is mass. Mass can be used to "count" atoms and molecules because a mole is the number of atoms present in one gram atomic weight (the atomic weight expressed in grams) of any atom or in one gram molecular weight of any molecule. Thus, the mass of a mole of any substance, which is known as its molar mass (Mm), equals its atomic or molecular weight expressed in grams. For example, the atomic weight of Mg is 24.3, so its molar mass 24.3 g/mol, and the molecular weight of CO2 is 44.0, so its molar mass is 44.0 g/mol. Thus, molar mass allows us to quickly convert a mass into a number of moles or a number of moles into a mass. Chemists use this fact to quickly "count" the number of moles of substance by simply weighing it. Mole ↔ mass conversions are most easily done with the factor label method. It uses the units of the given quantity and those of the conversion factors to assure the proper operations are performed. To use this method, arrange the factors so that the denominator of each factor cancels the numerator of the previous quantity until the units of the answer are obtained. (See The Factor Label Method.) 1.1-3. Determining Molar Mass Exercise Exercise 1.1: Determine the molar masses of the following. Express all answers to the nearest whole number. N2F4 Atomic weight of N 14_0__ The atomic weight of N is 14. Atomic weight of F 19_0__ The atomic weight of F is 19. Molar mass of N2F4 104_0__ 2(14) g N/mol + 4(19) g F/mol = 104 g/mol g/mol Ca3(PO4)2 Atomic weight of Ca 40_0__ The atomic weight of Ca is 40. Atomic weight of P 31_0__ The atomic weight of P is 31. Atomic weight of O 16_0__ The atomic weight of O is 16. Molar mass of Ca3(PO4)2 310_0__ 3(40) g Ca/mol + 2(31) g P/mol + 8(16) g O/mol = 310 g/mol g/mol 1.1-4. Mass-Mole Conversion Exercises Exercise 1.2: What is the mass of 3.24 mol N2O5? Molar mass Mm = 108_0__ 2(14) g N/mol + 5(16) g O/mol = 108 g/mol g/mol Mass m = 350_0__ 3.24 mol N2O5 × | | | 108 g N2O5 | | 1 mol N2O5 | = 350 g N2O5 g How many moles of N2O5 are in 12.7 g? Moles n = 0.118_0__ 12.7 g N2O5 × | | | 1 mol N2O5 | | 108 g N2O5 | = 0.118 mol N2O5 mol 1.1-5. Moles-Items Exercise Exercise 1.3: How many Al atoms are present in 0.065 g Al? Moles of atoms n = 2.4e-3_0__ 0.065 g Al × | | | 1 mol Al | | 27 g Al | = 2.4e−3 mol Al mol Individual atoms N = 1.4e21_0__ 2.4e−3 mol Al × | | | 6.02e23 atoms Al | | 1 mol Al | = 1.4e21 atoms Al atoms 1.1-6. Ideal Gas Law The number of moles of molecules in a gas can also be determined with the ideal gas law. ( 1.1 ) PV = nRT Ideal Gas Law • P is the pressure of the gas in atmospheres. 1 atm = 760 torr • V is its volume in liters. • n is the number of moles of gas. • R = 0.08206 L·atm·K–1·mol–1 is the ideal gas law constant. • T is the temperature on the Kelvin scale (K = °C + 273.15). 1.1-7. Gas Law Exercise Exercise 1.4: How many moles of H2 are in a 3.06 L container at 22 °C if its pressure is 742 torr? P = 0.976_0__ 742 torr × | | | 1 atm | | 760 torr | = 0.976 atm atm V = 3.06_0__ The volume is given in liters in the question. L R = 0.08206 L·atm·K–1·mol–1 T = 295_0__ The temperature must be on the Kelvin scale: T = 22 °C + 273.15 = 295 K K n = 0.123_0__ n = | | | PV | | RT | = | | | (0.976)(3.06) | | (0.08206)(295) | = 0.123 mol mol H2 1.2 Determining Chemical Formulas Introduction The composition of a compound is given in terms of the mass percents of its component elements. In this section, we show how these fractions can be determined from the chemical formula of the compound and how the chemical formula can be determined from the fractions. Objectives • Determine the mass composition of a compound from experimental data. • Determine the mass composition from the formula of the compound. • Determine the amount of one element that is combined with a known amount of another element or is present in a known amount of compound. • Determine the simplest formula of a compound from the relative amounts of each of the elements present in a sample. • Determine a molecular formula from its simplest formula and molar mass. • Use the empirical formula and the molar mass of a substance to determine its molecular formula. 1.2-1. Experimental Mass Composition The mass fraction of an element in a compound is the mass of the element divided by the mass of the compound in which it is found. mass fraction of element A = | | | mass of element A in compound | | mass of compound | The mass percent of an element in a compound is its mass fraction expressed as a percent. mass percent of element A = mass fraction of element A × 100% The elemental composition of a compound is frequently given by the mass percents of the elements. As shown in the following example, the elemental composition of lead(II) chloride is 74.4% Pb and 25.6% Cl by mass. Example: Determine the percent lead in a sample if a 1.062 g sample contains 0.791 g Pb. % mass of element = | | | 0.791 g Pb | | 1.062 g compound | × 100% = 74.4% Pb 1.2-2. Composition Exercise Exercise 1.5: A 0.2986 g sample of an oxide of iron contains 0.2161 g Fe. What is the percent iron in the sample? | | | | --- | 0.2161 _0__ The numerator in the determination of a percent by mass must be the mass of the element. g Fe | × 100% = | 72.37_0__ Calculate the amount using the other values. % Fe | | 0.2986_0__ The denominator in the determination of a percent by mass must be the mass of the compound. g compound | 1.2-3. Mass Composition from Chemical Formulas The elemental composition of a compound can be determined from its formula because its molar mass and the mass contributed by each element to its molar mass are known. Example: One mole of PbCl2 contains: | | | | --- | (1 mol Pb)(207 g/mol Pb) + (2 mol Cl)(35.5 g/mol Cl) | = | 207 g Pb + 71.0 g Cl | | = | 278 g PbCl2 | Consequently, 278 g of PbCl2 contains 207 g Pb and 71.0 g Cl, so the elemental composition of lead chloride can be expressed as follows: | | | 207 g Pb | | 278 g of PbCl2 | × 100% = 74.5% Pb and | | | 71.0 g Cl | | 278 g of PbCl2 | × 100% = 25.5% Cl In summary, one molar mass of AaBb contains: • (a mol A)(molar mass of A) grams of A • (b mol B)(molar mass of B) grams of B Thus, its elemental composition can be determined as follows: %A in AaBb = | | | a × molar mass of A | | molar mass of AaBb | × 100% and %B in AaBb = | | | b × molar mass of B | | molar mass of AaBb | × 100% 1.2-4. Mass Percent Exercise Exercise 1.6: What is the mass percent of Na in Na2SO4? Express your answer as a percent to the nearest whole number. The mass of one mole of Na2SO4 is: 142_0__ (2 mol Na)(23 g Na/mol Na) + (1 mol S)(32 g S/mol S) + (4 mol O)(16 g O/mol O) = 142 g g The mass of Na in one mole of Na2SO4 is: 46_0__ (2 mol Na)(23 g Na/mol Na) = 46 g g The fraction of the mass of Na2SO4 that is due to sodium expressed as a percent is: 32_0__ | | | 46 g Na | | 142 g Na2SO4 | × 100% = 32% % 1.2-5. Mass Percent Exercise Exercise 1.7: Determine the mass percent of P in Ca3(PO4)2. Express your answer as a percent to the nearest whole number. The mass of one mole of Ca3(PO4)2 is: 310_0__ (3 mol Ca)(40 g/mol) + (2 mol P)(31 g/mol) + (8 mol O)(16 g/mol) = 310 g g The mass of P in one mole of Ca3(PO4)2 is: 62_0__ (2 mol P)(31 g/mol) = 62 g g The fraction of the mass of Ca3(PO4)2 that is due to phosphorus expressed as a percent is: 20_0_2_ | | | 62 g P | | 310 g Ca3(PO4)2 | × 100% = 20.% % 1.2-6. Using Mass Percents While an elemental mass fraction has no units, units are implied in factor label problems because the fraction is determined to be (mass of element)/(mass of compound). Consequently, mass fractions can be used to convert between the mass of an element and the mass of the compound. Similarly, the mass percent of an element in a compound can be viewed as the mass of the element present in 100 g of compound. Thus, the fact that XY is 30% X by mass implies that there are 30 g X/100 g XY. Furthermore, the sum of the elemental percents of all the elements in a compound must sum to 100%, so we can deduce the percent of one element in a compound if we know the percents of all other elements in the compound. Thus, if XY is 30% X by mass, then it must also be 70% Y by mass. Example: Lead chloride is 74.4% lead by mass. How much lead is present in 23.4 g of lead chloride? The percent is expressed as 74.4 g Pb/100 g compound, which is the factor required for the conversion. 23.4 g compound × | | | 74.4 g Pb | | 100 g compound | = 17.4 g Pb What is the percent chloride in lead chloride? The compound contains only Pb and Cl, so the sum of the percents of the two elements must be 100%. However, PbCl2 is 74.4% Pb, so %Cl = 100% – %Pb = 100.0 – 74.4 = 25.6% Cl Thus, PbCl2 is 74.4% Pb and 25.6% Cl by mass. 1.2-7. Composition Exercise Exercise 1.8: Hexane (C6H14) is 83.7% carbon by mass. What portion, expressed as a percent, of the mass of hexane is due to hydrogen? %H = 16.3_0__ %H = 100% – %C % How many grams of hexane contain 13.6 g C? | | | | | | --- --- | 13.6_0__ Start with the amount you wish to convert. g C | = | 100_0__ This entry is part of the percent C. g C6H14 | = | 16.2_0__ Calculate the amount using the other values. g C6H14 | | 83.7_0__ The factor is the percent C, which is expressed as a ratio of masses. g C | 1.2-8. Empirical Formula The elemental composition can be used to determine the ratio of elemental masses in a compound. Molar masses can then be used to convert the ratio of elemental masses into one of moles, and the ratio of moles is the ratio of subscripts in the compound. We cannot determine the actual subscripts without more information, but we can determine the simplest set of integers that results in the correct ratios. The formula of a compound that uses this simplest set of integers is called the empirical formula or the simplest formula. To determine the empirical or simplest formula of a compound from its elemental composition: 1 Convert the amount of each element in a sample into moles of the element. 2 Determine the ratio of the moles by dividing each by the smallest. 3 If the result of Step 2 is not an integer, convert the ratio to a ratio of simple whole numbers. This is most easily accomplished by identifying the fraction that corresponds to the decimal, and then multiplying the ratio by the integer that makes the ratio an integer. See Table 1.1: Some Common Decimals and their Multipliers. | | | | | | --- --- | Step 2 | Multiplier | | Step 2 | Multiplier | | 0.125 | 8 | | 0.5 | 2 | | 0.167 | 6 | | 0.625 | 8 | | 0.200 | 5 | | 0.667 | 3 | | 0.250 | 4 | | 0.75 | 4 | | 0.333 | 3 | | 0.833 | 6 | | 0.375 | 8 | | 0.875 | 8 | Table 1.1: Some Common Decimals and their Multipliers For example, to convert a H:C mole ratio in Step 2 of 2.667 to a ratio of small whole numbers, you can recognize that the number means 2.667 mol H/1 mol C and that 0.667, which is a ratio of 2/3 or use the above table to determine that the multiplier must be 3. Multiply the numerator and denominator of the ratio by 3 to eliminate the fraction. The result is 3(2.667)/3(1) = 8 mol H/3 mol C. The formula of the compound is C3H8. 1.2-9. Empirical Formula Exercise Exercise 1.9: 6.84 g of an oxide of chromium is found to contain 4.68 g Cr. What is the empirical formula of the oxide? Assume that the compound contains only Cr and O. Express all answers to three significant digits. The number of moles of each element in the sample is: moles of Cr (Mm = 52.0 g/mol) in the sample: 0.0900_0_3_ 4.68 g Cr × | | | 1 mol Cr | | 52.0 g Cr | = 0.0900 mol Cr mol Cr moles of O in the sample: 0.135_0__ mass of oxygen = mass of sample – mass of Cr = 6.84 – 4.68 = 2.16 g O 2.16 g O × | | | 1 mol O | | 16.0 g O | = 0.135 mol mol O The ratio of the larger number of moles to the smaller number of moles (mol O/mol Cr) is: 1.5_0_3_ | | | 0.135 mol O | | 0.0900 mol Cr | = | | | 1.50 mol O | | 1.00 mol Cr | The ratio determined in the previous step as a ratio of whole numbers (For example, 1.33 mol O/mol Cr would be expressed as 4/3) is: o_3/2_ins | | | 1.50 mol O | | 1.00 mol Cr | × | | | 2 | | 2 | = | | | 3 mol O | | 2 mol Cr | The empirical formula is the following. (Denote any subscripts with an underscore. For example, NH_3 for NH3.) o_Cr_2O_3_s There are 2 mol Cr for every 3 mol of O, so the empirical formula of this oxide of chromium is Cr2O3. 1.2-10. Empirical Formula Exercise Exercise 1.10: When hydrocarbons (compounds that contain only C and H) are burned, all of the carbon is converted into CO2 and all of the hydrogen into H2O. What is the empirical formula of a hydrocarbon that produces 0.200 mol CO2 and 0.125 mol H2O when burned? The number of moles of carbon in the sample is: 0.200_0_3_ All of the carbon in the hydrocarbon is converted into CO2, so we convert the number of moles of CO2 into moles of carbon. There is one mole of carbon in 1 mole of CO2, so the following conversion produces the answer. 0.200 mol CO2 × | | | 1 mol C | | 1 mol CO2 | = 0.200 mol C The number of moles of hydrogen in the sample is: 0.250_0_3_ All of the hydrogen in the hydrocarbon is converted into H2O, so we simply convert the number of moles of H2O into moles of hydrogen. There are two moles of hydrogen in 1 mole of H2O, so the following conversion produces the answer. 0.125 mol H2O × | | | 2 mol H | | 1 mol H2O | = 0.250 mol H The ratio of the larger to the smaller number of moles is: 1.25_0__ | | | 0.250 mol H | | 0.200 mol C | = 1.25 mol H/mol C The above number expressed as a ratio of whole numbers is: o_5/4_ins Multiply the numerator and denominator by 4 to eliminate the 0.25 in the ratio and obtain 5/4, a ratio of small whole numbers. The empirical formula of the hydrocarbon is the following. (Denote any subscripts with an underscore. For example, NH_3 for NH3.) o_C_4H_5_s There are 4 mol C for every 5 mol of H, so the empirical formula of this hydrocarbon is C4H5. 1.2-11. Empirical Formulas from Elemental Compositions To determine the empirical formula of a compound from its percent composition, use the percents as if they were masses in grams and proceed as in the previous topic. Different samples of a compound are often used to determine each element in an elemental analysis, so compositions are usually presented as mass percents rather than absolute masses. The mass of each element in any amount of sample can be determined with the mass percents, so any sample size can be used. The most convenient sample size to use with mass percents is 100 g because the masses of the elements present in a 100 g sample are equal to the mass percents. For example, 100 g of a compound that is 60% C contains 60 g C or 60/12 = 5 mol C. To determine an empirical formula from mass percents: 1 Divide the mass percent of each element by its molar mass to determine the number of moles of the element present in 100 g of sample. 2 Divide each of the moles determined in Step 1 by the smallest number of moles to get simple ratios. 3 If any of the results of Step 2 are not integers, multiply all of the numbers by the integer that makes them integers. 4 The integers determined in Step 3 are the subscripts in the empirical formula. 1.2-12. Empirical Formula from Elemental Composition Exercise Exercise 1.11: A hydrocarbon (a compound that contains only carbon and hydrogen) is found to be 82.66% C and 17.34% H by mass. What is the empirical formula of the hydrocarbon? Use molar masses of 12.01 g/mol and 1.008 g/mol for C and H, respectively. The number of moles of C in 100 g of compound is: 6.883_0__ 82.66 g C × | | | 1 mol C | | 12.01 g C | = 6.883 mol C The number of moles of H in 100 g of compound is: 17.20_0_4_ 17.34 g H × | | | 1 mol H | | 1.008 g H | = 17.20 mol H The ratio expressed as a decimal of the smaller number to the larger number of moles is: 2.499_0__ | | | 17.20 mol H | | 6.883 mol C | = | | | 2.499 mol H | | 1 mol C | The ratio in Step 3 expressed as a ratio of whole numbers is: o_5/2_ins Multiply the numerator and denominator of Step 2 by 2 to eliminate the fraction. | | | 2 | | 2 | × | | | 2.499 mol H | | 1 mol C | = | | | 5 mol H | | 2 mol C | The empirical formula is the following. (Denote any subscripts with an underscore. For example, NH_3 for NH3.) o_C_2H_5_s There are 2 mol C for every 5 mol of H, so the empirical formula of this hydrocarbon is C2H5. 1.2-13. Molecular Formulas A molecular formula of a compound, which shows the actual number of atoms present in a molecule, can be determined from its empirical formula and molar mass. An empirical formula gives the simplest whole number ratios of the atoms, while a molecular formula shows the actual number of atoms present in each molecule. A molecular formula must contain an integral number of empirical units, so the molar mass of a compound must be an integer times the molar mass of the empirical unit. Consider the following table of compounds with the empirical formula CH; i.e., they all have molecular formulas of the type (CH)n, where n is an integer. The molar mass on one empirical unit (CH) is 13 g/mol, so the molar mass of (CH)n is 13n. | molecular formula | n | molar mass (g/mol) | --- | C12H12 | 12 | 13 × 12 = 156 | | C6H6 | 6 | 13 × 6 = 78 | | C4H4 | 4 | 13 × 4 = 52 | Table 1.2 We use the fact that molar mass of the compound = n × molar mass of the empirical unit to determine the value of n as follows: ( 1.2 ) n = | | | molar mass of compound | | molar mass of empirical unit | empirical units Once we have the value of n, we can write the molecular formula by simply multiplying each of the subscripts in the empirical formula by n. For example, to determine the molecular formula of a compound with an empirical formula of CH2 (Mm = 14 g/mol) and a molar mass of 112 g/mol, we would first determine n as n = | | | 112 g/mol | | 14 g/mol | = 8 The compound contains eight empirical units: (CH2)8, which would be written as C8H16. 1.2-14. Molecular Formula Exercise Exercise 1.12: Write the empirical formula for each of the following molecular formulas. (Denote any subscripts with an underscore. For example, NH_3 for NH3.) C4H10N4: o_C_2H_5N_2_s Each of the subscripts is divisible by 2. C12H24N5: o_C_12H_24N_5_s There is no common factor of the subscripts, so this molecular formula is also the empirical formula. C8H8N4: o_C_2H_2N_s Each of the subscripts is divisible by 4. 1.2-15. Molecular Formula from Empirical Formula Exercise Exercise 1.13: What is the molecular formula of a compound that has an empirical formula of C2H4O and a molar mass of 88 g/mol? | | | | | --- --- | | n = | 88_0__ This entry is the molar mass of the compound. g/mol | = | 2_0__ This is the result of the division: (88 g/mol) / (44 g/mol) = 2. | | 44.051___ This entry is the molar mass of the empirical unit. g/mol | The empirical formula of the compound is the following. (Denote any subscripts with an underscore. For example, NH_3 for NH3.) o_C_4H_8O_2_s Multiply the subscripts of the empirical formula by the value of n determined above to get the molecular formula. 1.2-16. Molecular Formula Exercise Exercise 1.14: What is the molecular formula of a compound that is 64.27% C, 7.19% H, and 28.53% O and has a molar mass of about 220 g/mol? 100.00 g of this compound contains how many moles of each element? C 5.351_0_4_ The mass of each element present in 100 g of a compound equals its mass percent in grams. 64.27 g C × | | | 1 mol C | | 12.011 g C | = 5.351 mol C mol H 7.13_0_3_ The mass of each element present in 100 g of a compound equals its mass percent in grams. 7.19 g H × | | | 1 mol H | | 1.008 g C | = 7.13 mol H mol O 1.783_0__ The mass of each element present in 100 g of a compound equals its mass percent in grams. 28.53 g O × | | | 1 mol O | | 15.999 g O | = 1.783 mol O mol The number of moles of each element divided by the smallest number of moles is: C 3___ Oxygen is present in the smallest number of moles, so this entry is the number of moles of C present divided by the number of moles of O present. | | | 5.351 mol C | | 1.783 mol O | = | | | 3 mol C | | 1 mol O | mol H 4___ Oxygen is present in the smallest number of moles, so this entry is the number of moles of H present divided by the number of moles of O present. | | | 7.13 mol H | | 1.783 mol O | = | | | 4 mol H | | 1 mol O | mol O 1___ Oxygen is present in the smallest number of moles, so this entry is the number of moles of O present divided by the number of moles of O present. mol The empirical formula of the compound is the following. (Denote any subscripts with an underscore. For example, NH_3 for NH3.) o_C_3H_4O_s There are 3 mol C and 4 mol H for each 1 mol O, so the empirical formula = C3H4O. The number of empirical units present in one molecular unit is: 4___ The molar mass of the simplest unit (C3H4O) is 56 g/mol (3(12) + 4(1) + 1(16)), while the molar mass of the compound is ~220 g/mol, so n = | | | ~220 g/mol | | 56 g/mol | = ~4 The molecular formula is the following. (Denote any subscripts with an underscore. For example, NH_3 for NH3.) o_C_12H_16O_4_ins The molecular formula consists of n empirical formulas, where n was determined in the previous step to be 4. The molecular formula is C12H16O4. 1.2-17. Molecular Formula from Combustion Exercise Exercise 1.15: It is common to determine formulas of organic compounds by burning them and determining the mass fractions of carbon and hydrogen from the amounts of CO2 and H2O that are produced in the combustion. If the compound contains oxygen, the mass of oxygen must be determined from the total mass of the sample and the masses of C and H that it contains. What is the empirical formula of ascorbic acid (vitamin C) if combustion of 0.579 g of ascorbic acid produces 0.868 g of CO2 and 0.237 g H2O? Ascorbic acid also contains oxygen. Convert mass of CO2 and H2O to moles. moles of CO2 produced: 0.0197_0__ 0.868 g CO2 × | | | 1 mol CO2 | | 44.01 g CO2 | = 0.0197 mol CO2 mol CO2 moles of C present in the CO2: 0.0197_0__ 0.0197 mol CO2 × | | | 1 mol C | | 1 mol CO2 | = 0.0197 mol C mol C moles of H2O produced: 0.0132_0__ 0.237 g H2O × | | | 1 mol H2O | | 18.02 g H2O | = 0.0132 mol H2O mol H2O moles of H present in the H2O: 0.0264_0__ 0.0132 mol H2O × | | | 2 mol H | | 1 mol H2O | = 0.0264 mol H mol H The sample contains oxygen, so its mass must be determined by difference. Therefore, the mass of C and H must be determined from the number of moles above. Note: this step would not be necessary if the sample contained no oxygen. We would simply determine the mol H/mol C ratio with the numbers of moles determined above to get the empirical formula. mass of C in sample: 0.237_0__ 0.0197 mol C × | | | 12.01 g C | | 1 mol C | = 0.237 g C g C mass of H in sample: 0.0265_0__ 0.0263 mol H × | | | 1.008 g H | | 1 mol H | = 0.0265 g H g H mass of oxygen in sample: 0.316_0__ mass O = mass of sample – mass C – mass H = 0.579 g Ascorbic – 0.237 g C – 0.0265 g H = 0.316 g O g O moles of oxygen in sample: 0.0197_0__ 0.316 g O × | | | 1 mol O | | 16.00 g O | = 0.0197 mol O mol O Determine the mole ratios. mol C/mol C = 1_0__ The ratio of a substance to itself is always 1. mol H/mol C = 1.33_0_3_ | | | 0.0263 mol H | | 0.0197 mol O | = 1.33 mol H/mol C mol O/mol C = 1_0_3_ | | | 0.0197 mol O | | 0.0197 mol O | = 1.00 mol O/mol C Multiplier to be used to convert all of the above to integers: 3_0__ The .33 decimal calls for a multiplier of 3. The subscripts in the empirical formula: subscript of C = 3_0__ 3(1 mol C/mol C) = 3 subscript of H = 4_0__ 3(1.33 mol H/mol C) = 4 subscript of O = 3_0__ 3(1 mol O/mol C) = 3 The molar mass of ascorbic acid is 176 g/mol. What is its molecular formula? The molar mass of the empirical unit = 88.059___ 3(12.01) + 4(1.008) + 3(16.00) = 88.1 g/mol g/mol The number of empirical units in a molecular unit = 2_0__ | | | 176 g/molecular unit | | 88.1 g/empirical unit | = 2 empirical unit/molecular unit units The subscripts in the molecular formula: subscript of C = 6_0__ 2(3) = 6 subscript of H = 8_0__ 2(4) = 8 subscript of O = 6_0__ 2(3) = 6 1.3 Substance or Composition Stoichiometry Introduction In a composition stoichiometry problem, the amount of one substance that is combined with another is determined with the use of the ratios of the subscripts in the chemical formula. Objectives • Determine and use stoichiometric links derived from chemical formulas. 1.3-1. Stoichiometric Link Video Stoichiometric Link Viewing the Video • View the video in this window by selecting the play button. • View the video full screen in another window. • View the video in text format by scrolling down. 1.3-2. Stoichiometric Links and Chemical Formulas The stoichiometric links for a substance are formed from the subscripts in a chemical formula. A stoichiometric link is the ratio of the number of moles of one substance to the number of moles of another substance with which it combines. The stoichiometric link between two elements in a compound equals the ratio of the subscripts of the two elements in the compound. Consider the example of ammonia shown in Figure 1.1. One molecule of NH3 contains one nitrogen atom and three hydrogen atoms, so we can write the following stoichiometric links between NH3, N, and H. | | | | | | | | | | --- --- --- --- | | | | 1 mol N | | 3 mol H | | | | | 1 mol N | | 1 mol NH3 | | | | | 3 mol H | | 1 mol NH3 | | Table 1.3 Figure 1.1: Substance Stoichiometry in NH3 The stoichiometric links available from the formula H2SO4 are: | | | | --- | Conversion factors using H in H2SO4 | Conversion factors using S in H2SO4 | Conversion factors using O in H2SO4 | | | | | 2 mol H | | 1 mol H2SO4 | | | | | 2 mol H | | 1 mol S | | | | | 2 mol H | | 4 mol O | | | | | 1 mol S | | 1 mol H2SO4 | | | | | 1 mol S | | 2 mol H | | | | | 1 mol S | | 4 mol O | | | | | 4 mol O | | 1 mol H2SO4 | | | | | 4 mol O | | 2 mol H | | | | | 4 mol O | | 1 mol S | | Table 1.4 1.3-3. Using the Stoichiometric Link Stoichiometric links can be combined with the factor label method to determine the amount of any element in a compound that is combined with a known amount of any other element in the compound. The units of the denominator of the stoichiometric link must be the same as the given quantity, and the units of the numerator must be the same as those of the desired quantity. Thus, to convert the amount of one atom or ion into the amount of another in the same compound, multiply the number of moles of a given substance by the appropriate stoichiometric link derived from the subscripts in the chemical formula. ( 1.3 ) moles of given × | | | subscript of desired | | subscript of given | = moles of desired 1.3-4. Mass-Mass Conversion We have seen that the heart of a substance stoichiometry problem is multiplying the given number of moles by the appropriate stoichiometric link to obtain the number of moles of the desired substance. We now combine that operation with the molar masses of the given and desired substances to determine the mass of one substance (the desired substance) that is combined with a given mass of another (the given substance). The process involves the following three conversions: 1 mass A → mol A: Convert the given mass into given moles with the molar mass of the given substance. 2 mol A → mol B: Multiply the given moles calculated in Step 1 by the appropriate stoichiometric link to obtain the desired moles. 3 mol B → mass B: Convert the desired moles determined in Step 2 into grams with the molar mass of the desired substance. Of course, all three steps can be combined into one extended multiplication with the factor label method. For example, the following shows how to determine the number of grams of oxygen (Mm = 16.0 g/mol) that are combined with 12.2 g of aluminum (Mm = 27.0 g/mol) in Al2O3. 12.2 g Al × | | | 1 mol Al | | 27.0 g Al | × | | | 3 mol O | | 2 mol Al | × | | | 16.0 g O | | 1 mol O | = 10.8 g O We first divide the given mass of Al by its molar mass to obtain moles of Al. The moles of Al are then multiplied by the ratio of subscripts (stoichiometric link) to obtain the moles of O. Finally, the moles of O are multiplied by the molar mass of O to obtain grams of O. Note that all of the units cancel except grams of O, the desired quantity. 1.3-5. Mole Atom-Mole Molecule Conversion Exercise Exercise 1.16: How many moles of MgCl2 contain 0.64 mol Cl atoms? | | | | | | --- --- | 0.64_0__ Start with the given amount. mol Cl | × | 1_0__ This is the numerator of the stoichiometric link, so it must have the units of 'mol desired.' It is equal to the subscript of the desired substance in the chemical formula. mol MgCl2 | = | 0.32_0__ Calculate the amount using the other values. mol MgCl2 | | 2_0__ The preceding amount is 'mol given,' so this entry is the denominator of the stoichiometric link. It must have the units of 'mol given' and is equal to the subscript of the given substance in the chemical formula. mol Cl | 1.3-6. Mole Atom-Mole Atom Conversion Exercise Exercise 1.17: How many moles of hydrogen atoms are in a sample of C6H14 that contains 0.25 mol C? | | | | | | --- --- | 0.25_0__ Start with the amount you wish to convert. mol C | × | 14_0__ This is the numerator of the stoichiometric link, so it must have the units of 'mol desired.' It is equal to the subscript of the desired substance in the chemical formula. mol H | = | 0.58_0__ Calculate the amount using the other values. mol H | | 6_0__ The preceding amount is 'mol given,' so this entry is the denominator of the stoichiometric link. It must have the units of 'mol given' and is equal to the subscript of the given substance in the chemical formula. mol C | 1.3-7. Mass Atom-Mass Molecule Conversion Exercise Exercise 1.18: What mass of Fe3O4 (Mm = 232 g/mol) contains 12 g of iron (Mm = 55.9 g/mol)? | | | | | | | | | | --- --- --- --- | 12_0__ Start with the amount you wish to convert. g Fe | × | 1 _0__ This entry must have the units 'mol given' and is part of the molar mass. mol Fe | × | 1_0__ This is the numerator of the stoichiometric link, so it must have the units of 'mol desired.' It is equal to the subscript of the desired substance in the chemical formula. mol Fe3O4 | × | 232_0__ This entry is part of the molar mass of the sought substance, but it must have the units of the answer. g Fe3O4 | = | 17_0__ Calculate the amount using the other values. g Fe3O4 | | 55.9_0__ This entry must have the units 'mol given' and is part of the molar mass. g Fe | 3_0__ The preceding amount is 'mol given,' so this entry is the denominator of the stoichiometric link. It must have the units of 'mol given' and is equal to the subscript of the given substance in the chemical formula. mol Fe | 1_0__ Convert 'mol sought' to 'g sought' with the molar mass. This denominator must have the same units as the preceding numerator. mol Fe3O4 | 1.3-8. Mass Atom-Mass Atom Conversion Exercise Exercise 1.19: What mass of Fe (Mm = 55.9 g/mol) is combined with 26.3 g chlorine (Mm = 35.5 g/mol) in FeCl3? | | | | | | | | | | --- --- --- --- | 26.3_0__ Start with the amount you wish to convert. g C | × | 1_0__ This entry must have the units 'mol given' and is part of the molar mass. mol Cl | × | 1_0__ This is the numerator of the stoichiometric link, so it must have the units of 'mol desired.' It is equal to the subscript of the desired substance in the chemical formula. mol Fe | × | 55.9 _0__ This entry is part of the molar mass of the sought substance, but it must have the units of the answer. g Fe | = | 13.8_0__ Calculate the amount using the other values. g Fe | | 35.5_0__ Convert 'g given' to 'mol given' with the molar mass. Remember, the denominator of the factor must have the same units as the preceding numerator. g Cl | 3_0__ The preceding amount is 'mol given,' so this entry is the denominator of the stoichiometric link. It must have the units of 'mol given' and is equal to the subscript of the given substance in the chemical formula. mol Cl | 1_0__ Convert 'mol sought' to 'g sought' with the molar mass. This denominator must have the same units as the preceding numerator. mol Fe | 1.4 Balancing Chemical Equations Introduction Neither the identity nor the number of atoms is changed in a chemical reaction, so chemical equations are balanced to assure that the number of each type of atom is the same on both sides. This is done by changing the number of each reacting and produced molecule by placing coefficients in front of each species. In this section, we show how to balance simple chemical equations by inspection. Objectives • Balance a chemical equation by inspection. 1.4-1. Balancing Chemical Equations Video Balancing Chemical Equations Viewing the Video • View the video in this window by selecting the play button. • View the video full screen in another window. • View the video in text format by scrolling down. 1.4-2. Procedure for Balancing Equations Chemical equations are balanced to assure that the number of each type of atom is the same on both sides of the equation. Only coefficients (not subscripts) can be changed to balance a chemical equation. The following steps should lead to a balanced equation: 1 Pick the molecule with the greatest number of atoms and set its coefficient to "1" unless another choice is obviously better. For example, sometimes a "2" must be used to assure an even number of one of the atoms. 2 Determine which atoms are fixed by the coefficient set in Step 1, then balance those atoms on the other side of the equation. 3 Determine which atoms are fixed by the coefficient(s) created in Step 2, then balance those atoms on the other side of the equation. 4 Repeat Step 3 until the equation is balanced. Note that coefficients of "1" are not usually included in the balanced equation. Example: As an example, we will follow the steps above to balance the following chemical equation: ____ HCl + ____ MnO2 → ____ MnCl2 + ____ H2O + ____ Cl2 1 Make the coefficient of either MnO2 or MnCl2 one. We choose MnO2. ____ HCl + 1 MnO2 → ____ MnCl2 + ____ H2O + ____ Cl2 2 The coefficient used in Step 1 fixes the number of Mn atoms at 1 and O atoms at 2, so we balance the Mn atoms with a coefficient of 1 for MnCl2 and the oxygen atoms with a coefficient of 2 for water. ____ HCl + 1 MnO2 → 1 MnCl2 + 2 H2O + ____ Cl2 3 The coefficient of water fixes the number of hydrogen atoms at four, so we balance the H atoms with a coefficient of 4 for HCl. Note that the coefficient of MnCl2 does not fix the number of Cl atoms because it is not the only source of Cl. 4 HCl + 1 MnO2 → 1 MnCl2 + 2 H2O + ____ Cl2 4 The coefficient of HCl fixes the number of chlorine atoms at four, but there are already 2 Cl atoms in 1 MnCl2, so we balance the Cl atoms on the other side of the equation with a coefficient of 1 for Cl2. 4 HCl + 1 MnO2 → 1 MnCl2 + 2 H2O + 1 Cl2 5 Each side of the equation contains 4 H atoms, 1 Mn atom, 2 O atoms, and 4 Cl atoms. The equation is now balanced, but ones are not usually written. Thus, the balanced equation is usually written as shown in the last step. 4 HCl + MnO2 → MnCl2 + 2 H2O + Cl2 1.4-3. Balancing Equations Exercise Exercise 1.20: Balance the following equations with the smallest whole-number coefficients. Include coefficients of "1" in your answer. 2_0__ Start with 2 N2O5 to assure an even number of O atoms. N2 + 5_0__ Start with 2 N2O5 to assure an even number of O atoms. O2 → 2_0__ Start with 2 N2O5 to assure an even number of O atoms. N2O5 2_0__ Start with 2 NaOH to avoid fractions. Na + 2_0__ Start with 2 NaOH to avoid fractions. H2O → 2_0__ Start with 2 NaOH to avoid fractions. NaOH + 1_0__ Start with 2 NaOH to avoid fractions. H2 1_0__ Start with 1 P4. P4 + 6_0__ Start with 1 P4. Cl2 → 4_0__ Start with 1 P4. PCl3 1_0__ Start with 1 C6H12O6. C6H12O6 + 6_0__ Start with 1 C6H12O6. O2 → 6_0__ Start with 1 C6H12O6. CO2 + 6_0__ Start with 1 C6H12O6. H2O 1.5 Reaction Stoichiometry Introduction The ratios of the coefficients of a balanced equation are conversion factors that allow us to calculate the amount of one substance that is produced by or reacts with a given amount of another substance. The only difference between this process and the one used to determine how much of one element was combined with a given amount of another element in a compound is the stoichiometric ratio or link: in one, the stoichiometric link is the ratio of the subscripts in the chemical formula; in the other, it is the ratio of the coefficients in the balanced equation. All other operations are the same in the two problem types. Objectives • Write the stoichiometric ratio relating two substances involved in a chemical reaction. • Use the ratio to convert the amount of one substance involved in the reaction to the chemically equivalent amount of another substance in the same reaction. • Identify the limiting reactant in a reaction. • Determine the amount of product formed or of reactant consumed from the amount of limiting reactant that reacts. • Determine the percent yield given the actual yield of a reaction or use the percent yield to determine the actual yield. • Determine the complete composition of a reaction mixture after the reaction is complete. 1.5-1. Stoichiometric Links in Chemical Reactions Stoichiometric links formed from the coefficients of balanced chemical equations are used to convert between equivalent amounts of reactants and products in a reaction. Consider the following balanced chemical equation: N2 + 3 H2 → 2 NH3 The coefficients indicate the relative number number of molecules that react, not the absolute number. That is, the equation does not indicate that three moles of H2 react. Rather, it indicates that 3 mol H2 react for every 1 mol N2 that reacts or for every 2 mol NH3 that form. Thus, we can write the following stoichiometric links for the reaction. | | | | | | | | | | --- --- --- --- | | | | 1 mol N2 | | 2 mol NH3 | | | | | 1 mol N2 | | 3 mol H2 | | | | | 2 mol NH3 | | 3 mol H2 | | Table 1.5 Figure 1.2: Reaction Stoichiometry in the Formation of NH3 Reaction 1.5-2. Using Stoichiometric Links for Chemical Reactions The stoichiometric links produced by the coefficients in a balanced equation can be used to determine the number of moles of one substance that are produced by or react with a given number of moles of another substance in the equation. Example: The following shows how to determine the number of moles of nitrogen that are required and how many moles of NH3 are produced in the reaction of 3.6 moles of hydrogen in the following: 3 H2 + N2 → 2 NH3 | | | | | | --- --- | 3.6 mol H2 × | | | 1 mol N2 | | 3 mol H2 | | = | 1.2 mol N2 are required | | 3.6 mol H2 × | | | 2 mol NH3 | | 3 mol H2 | | = | 2.4 mol NH3 are produced | Note that the stoichiometric link is simply the ratio of the coefficients in the balanced chemical equation. As in all applications of the factor label method, the units of the denominator of the stoichiometric link must be the same as the given quantity and the units of the numerator are those of the result. 1.5-3. Mole-Mole Conversion Exercise Exercise 1.21: How many moles of hydrogen are required to produce 2.8 mol NH3? The chemical equation is N2 + 3 H2 → 2 NH3 | | | | | | --- --- | 2.8_0__ Start with the amount you wish to convert. mol NH3 | × | 3_0__ This is the numerator of the stoichiometric link, so it must have the units of 'mol desired.' It is equal to the coefficient of the desired substance in the chemical equation. mol H2 | = | 4.2_0__ Calculate the amount using the other values. mol H2 | | 2_0__ The preceding amount is 'mol given,' so this entry is the denominator of the stoichiometric link. It must have the units of 'mol given' and is equal to the coefficient of the given substance in the chemical equation. NH3 | 1.5-4. Method for Mass-Mass Conversions We finish this lesson by showing how to determine the mass of one substance that reacts with or is produced by the reaction of a given mass of another substance in the chemical equation. The process, which is identical to that used in determining the mass of one element that is combined with a given mass of another element in a compound, involves the following three conversions: 1 mass A → mol A: Use the molar mass of the given compound to convert its mass to moles. 2 mol A → mol B: Use the stoichiometric link (ratio of coefficients) to convert the number of moles of given compound determined in Step 1 into the equivalent number of moles of desired substance. 3 mol B → mass B: Use the molar mass of the desired substance to convert the number of moles determined in Step 2 into the mass of the desired substance. The three steps can be combined into one factor-label setup as shown in the following Example. Example: The mass of Na2O (Mm = 62 g/mol) that is produced in the reaction of 6.0 g Na (Mm = 23 g/mol) by the reaction 4 Na + O2 → 2 Na2O is determined as follows: | | | | | | | | --- --- --- | 6.0 g Na × | | | 1 mol Na | | 23 g Na | × | | | 2 mol Na2O | | 4 mol Na | × | | | 62 g Na2O | | 1 mol Na2O | = 8.1 g Na2O | | 6.0 g Na → 0.26 mol Na → 0.13 mol Na2O → 8.1 g Na2O | 1.5-5. Mass-Mass Conversion Exercise Exercise 1.22: How many grams of fluorine must react to produce 24 g PF5 (Mm = 126 g/mol)? F2 (Mm = 38 g/mol) reacts with P4 (Mm = 126 g/mol) by the following reaction: P4 + 10 F2 → 4 PF5 | | | | | | | | | | --- --- --- --- | 24_0__ Start with the amount you wish to convert. g PF5 | × | 1_0__ This entry must have the units 'mol given' and is part of the molar mass. mol PF5 | × | 10_0__ This is the numerator of the stoichiometric link, so it must have the units of 'mol desired.' It is equal to the coefficient of the desired substance in the chemical equation. mol F2 | × | 38_0__ This entry is part of the molar mass of the sought substance, but it must have the units of the answer. g F2 | = | 18_0__ Calculate the amount using the other values. g F2 | | 126 _0__ Convert 'g given' to 'mol given' with the molar mass. Remember, the denominator of the factor must have the same units as the preceding numerator. g PF5 | 4 _0__ The preceding amount is 'mol given,' so this entry is the denominator of the stoichiometric link. It must have the units of 'mol given' and is equal to the coefficient of the given substance in the chemical equation. mol PF5 | 1_0__ Convert 'mol sought' to 'g sought' with the molar mass. This denominator must have the same units as the preceding numerator. mol F2 | 1.5-6. Limiting Reactants Thus far, we have determined the amount of substance that reacts with or is formed by the reaction of another substance. However, there is usually more than one reactant and the reactants are not usually added in stoichiometric ratios. In such cases, one of the reactants, known as the limiting reactant, is consumed before any other reactant. The other reactants are said to be in excess because they are still available after the limiting reactant is consumed. The limiting reactant is identified as the reactant that is capable of producing the least amount of product. Thus, one way to determine which of several reactants is the limiting reactant is to determine the amount of product each reactant is capable of producing. The limiting reactant is the one that is capable of producing the least. Note, that the limiting reactant is not necessarily the reactant present in the smallest mass nor in the smallest number of moles. To determine the limiting reactant, you would determine the number of moles of a substance B that is produced or reacts with a given number of moles of reactant A. The moles of B is formed from reactant A: mol A × | | | coefficient B | | coefficient A | = mol B However, this can be rewritten as follows: | | | mol A | | coefficient A | × coefficient B = mol B The coefficient of B is constant, so the smallest amount of B would be produced by the reactant with the smallest mol/coefficient ratio. We conclude that the limiting reactant is that reactant with the smallest (mol A/coefficient A) ratio. The amount of any product that forms or reactant that reacts is equal to the (reactant moles)/(reactant coefficient) ratio of the limiting reactant times the coefficient of the product or reactant whose amount is to be determined. 1.5-7. Mole-Mole Limiting Reactant Exercise Exercise 1.23: How many moles of NF3 can be produced from the reaction of 0.50 mol N2 and 0.90 mol F2 in the following reaction? N2 + 3 F2 → 2 NF3 Determine mole/coefficient ratios. N2 0.50_0_2_ | | | 0.50 mol N2 | | 1 mol N2 | = 0.50 F2 0.30_0_2_ | | | 0.90 mol F2 | | 3 mol F2 | = 0.30 The limiting reactant is N2 0.30 < 0.50, so F2 is the limiting reactant. F2 Moles of NF3 that can be produced by the limiting reactant is 0.60_0_2_ Multiply the mole/coefficient ratio for the limiting reactant times the coefficient of the product. (0.30)(2 mol NF3) = 0.60 mol NF3 mol NF3 1.5-8. Mass-Mass Limiting Reactant Exercise Exercise 1.24: How many grams of Ag3PO4 (Mm = 418.6 g/mol) form in the reaction of 3.21 g AgNO3 (Mm = 169.9 g/mol) and 1.65 g K3PO4 (Mm = 212.3 g/mol)? How much of the excess reactant remains? 3 AgNO3 + K3PO4 → Ag3PO4 + 3 KNO3 Determine the number of moles of each reactant. AgNO3 0.0189_0__ 3.21 g AgNO3 × | | | 1 mol AgNO3 | | 169.9 g AgNO3 | = 0.0189 mol mol K3PO4 0.00777_0__ 1.65 g K3PO4 × | | | 1 mol K3PO4 | | 212.3 g K3PO4 | = 0.00777 mol mol Determine the mol/coefficient ratios. AgNO3 0.00630_0_3_ | | | 0.0189 mol AgNO3 | | 3 mol AgNO3 | = 0.00630 K3PO4 0.00777_0__ | | | 0.00777 mol K3PO4 | | 1 mol K3PO4 | = 0.00777 Identify the limiting reactant. AgNO3 K3PO4 0.00630 < 0.00777, so AgNO3 is the limiting reactant. Determine the amount of product formed. 2.64_0__ Use the mol/coefficient ratio for the limiting reactant and the coefficient of the product to obtain moles of product then use the molar mass to convert to mass. 0.00630 × 1 mol Ag3PO4 × | | | 418.6 g Ag3PO4 | | 1 mol Ag3PO4 | = 2.64 g Ag3PO4 g Ag3PO4 Determine the amount of excess reactant that is unreacted. Mass of excess reactant that reacts: 1.34_0__ Use the mol/coefficient ratio for the limiting reactant and the coefficient of the other reactant to obtain moles of the other reactant that react, then use the molar mass to convert to the mass that reacts. 0.0630 × 1 mol K3PO4 × | | | 212.3 g K3PO4 | | 1 mol K3PO4 | = 1.34 g K3PO4 reacts g Mass of excess reactant that remains: 0.31_0__ There were 1.65 g initially, but 1.34 g react, so 1.65 – 1.34 = 0.31 g does not react. g 1.5-9. Percent Yield The mass of a product determined from the amount of limiting reactant and the stoichiometry of the reaction is known as the theoretical yield. However, the theoretical yield is not usually isolated in an actual experiment. The actual yield, which is the amount that is actually isolated in the experiment, can be less than the theoretical yield for the following reasons: • The reaction may not be extensive, so all of the limiting reactant does not react. • Product can be lost during purification. For example, some solid may remain on the filter paper or dissolve when washed. • Competing reactions can consume some of the reactants. The fraction of the theoretical yield that is actually isolated expressed as a percent is called the percent yield. ( 1.4 ) percent yield = | | | actual yield | | theoretical yield | × 100% Percent Yield Example: If stoichiometry predicts that 8.0 g of product should form, but only 6.0 g are actually obtained, the reaction would be reported to have a 75% yield. percent yield = | | | actual yield | | theoretical yield | × 100% = | | | 6.0 g | | 8.0 g | × 100% = 75% 1.5-10. Percent Yield Exercise Exercise 1.25: Aspirin (C9H8O4, Mm = 180.16 g/mol) is prepared from the reaction of salicylic acid (C7H6O3, Mm = 138.12 g/mol) and acetic anhydride (C4H6O3, Mm = 102.09 g/mol) by the following reaction: C7H6O3 + C4H6O3 → C9H8O4 + C2H4O2 What is the percent yield of aspirin if the reaction of 20.00 g of salicylic acid and 17.00 g of acetic anhydride produces 22.36 g of aspirin? theoretical yield of aspirin if salicylic acid is the limiting reactant: 26.09_0__ 20.00 g C7H6O3 × | | | 1 mol C7H6O3 | | 138.12 g C7H6O3 | × | | | 1 mol C9H8O4 | | 1 mol C7H6O3 | × | | | 180.12 g C9H8O4 | | 1 mol C9H8O4 | = 26.09 g C9H8O4 g C9H8O4 theoretical yield of aspirin if acetic anhydride is the limiting reactant: 30.00_0__ 17.00 g C4H6O3 × | | | 1 mol C4H6O3 | | 102.09 g C4H6O3 | × | | | 1 mol C9H8O4 | | 1 mol C4H6O3 | × | | | 180.12 g C9H8O4 | | 1 mol C9H8O4 | = 30.00 g C9H8O4 g C9H8O4 theoretical yield of aspirin: 26.09_0__ The smaller of the two masses determined in Steps 1 and 2 (26.09 g) is the theoretical yield. g C9H8O4 actual yield of aspirin: 22.36_0__ The actual yield is given in the problem as 22.36 g. g C9H8O4 percent yield: 85.70_0__ | | | 22.36 g obtained | | 26.09 g possible | × 100% = 85.70% % 1.5-11. Reaction Table Lecture Reaction Table Lecture Viewing the Video • View the video in this window by selecting the play button. • View the video full screen in another window. • View the video in text format by scrolling down. 1.5-12. Lines of a Reaction Table Reaction tables have three rows (initial, Δ, and final) and a column for each reactant and product in the chemical equation. Reaction tables are completed for chemical reactions to show the amounts of all substances that react and are produced in order to determine the amounts that are present at the end of the reaction. They are constructed with three lines under the balanced equation: initial, Δ, and final. 1 initial: The initial line consists of the number of moles of each ingredient present before the reaction begins. Only the reactants are mixed in most reactions, so the entries under the products are zero in most reaction tables. 2 Δ: The delta (change) line shows how many moles of each reactant disappears and how many moles of each product are produced. All calculations on this line are based on the assumption that all of the limiting reactant disappears. The entries of all products are positive because they form during the reaction, but those of all reactants are negative because they disappear during the reaction. 3 final: The final line is the sum of the initial and delta lines and represents the final composition of the reaction mixture. 1.5-13. Reaction Table from Mass Exercise Exercise 1.26: Reaction tables are completed with either moles or concentrations, not with masses. However, initial amounts are often given as masses. In these cases, convert the masses into moles and proceed as in the previous example. Construct the reaction table to determine the final masses of all substances remaining in the complete reaction of 10.00 g each of NH3 and O2. Report the final masses to the nearest 0.01 g. | | 4 NH3 | + | 5 O2 | → | 4 NO | + | 6 H2O | | --- --- --- --- | initial | 0.5872_0__ 10.00 g NH3 × | | | 1 mol NH3 | | 17.03 g NH3 | = 0.5872 mol NH3 | + | 0.3125_0__ 10.00 g O2 × | | | 1 mol O2 | | 32.00 g O2 | = 0.3125 mol O2 | → | 0_0__ Initially there is no product so this amount is zero. | + | 0_0__ Initially there is no product so this amount is zero. | mol | | Δ | -0.2500_0_4_ 0.3125 mol O2 × | | | 4 mol NH3 | | 5 mol O2 | = 0.2500 mol NH3 | | -0.3125_0__ All 0.3125 moles of oxygen disappear. The fact that they disappear is designated by a minus sign. | | +0.2500_0_4_ 0.3125 mol O+2 × | | | 4 mol NO | | 5 mol O2 | = 0.2500 mol NO Since the NO is a product, it is being produced during the reaction, so the Δ line entry under NO is +0.2500 mol. | | +0.3750_0_4_ 0.3125 mol O2 × | | | 6 mol H2O | | 5 mol O2 | = 0.3750 mol H2O Since the H2O is a product, it is being produced during the reaction, so the Δ line entry under H2O is +0.3750 mol. | mol | | final | 0.3372_0__ 0.5871 mol initially – 0.2500 mol react = 0.3372 mol at completion | | 0.0 _0__ 0.3125 mol initially – 0.3125 mol react = 0 mol at completion | | 0.2500 _0__4 0 mol initially + 0.2500 mol form = 0.2500 mol at completion | | 0.3750_0__4 0 mol initially + 0.3750 mol form = 0.3750 mol at completion | mol | | mass | 5.74_0__ 0.3372 mol NH3 × | | | 17.03 g NH3 | | 1 mol NH3 | = 5.74 g NH3 | | 0.0_0__ None of the limiting reactant remains. | | 7.50_0_3_ 0.2500 mol NO × | | | 30.01 g NO | | 1 mol NO | = 7.50 g NO | | 6.76_0__ 0.3750 mol H2O × | | | 18.015 g H2O | | 1 mol H2O | = 6.76 g H2O | g | 1.6 Exercises and Solutions Select the links to view either the end-of-chapter exercises or the solutions to the odd exercises. • End-of-Chapter Exercises • Solutions to Odd Exercises Copyright © 2014 Advanced Instructional Systems Inc. and NC State College of Sciences Foundation | Credits
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https://www.continentalpress.com/blog/ell-math-tips/?srsltid=AfmBOoqdE50KhoSsu99hykFOg7pMw48hd1NCIJOdRIuvEzMYiQxJ2AJe
11 Ways to Celebrate Cultural Diversity in the Classroom Embracing Setbacks: Classroom Strategies for Learning From Failure Elevate ELL Math Achievement with These Top Tips and Strategies Math has often been called “the universal language.” While it’s true that numbers work the same way regardless of the language you speak, teachers cannot assume that their English language learners (ELLs) will automatically excel in math for this reason alone. Language proficiency is also closely related to the development of strong mathematical skills. Kristina Roberston writes, “Solving word problems, following instructions, understanding and using mathematical vocabulary correctly — all of these skills require a language proficiency…We tend to think of mathematics as a subject that does not require a strong command of language. In reality, however, mathematical reasoning and problem-solving are closely linked to language and rely upon a firm understanding of basic math vocabulary (Dale & Cuevas, 1992; Jarret, 1999).” This blog will discuss specific ELL math best practices and math instructional strategies that you can use to boost student learning. 4 Best Practices for ELL Math Instruction According to data from the US Department of Education’s Academic Performance and Outcomes for English Learners, overall ELL performance in math has been significantly below the performance of native speakers. For ELL students to fully participate in the math classroom, teachers must employ strategies to promote both their academic and linguistic development. The following best practices can help you support your ELLs’ mastery of math skills. 1. Scaffold your instruction As you consider how to engage students in math lessons, it’s important to provide your ELLs with support in navigating the linguistic challenges of math. Jeanine Harvey, director of multilingual learner academics at The New Teacher Project says, “We know from our work that multilingual learners do not have the same access to grade-level assignments as their peers…All students could engage with grade-level assignments with the right supports.” Scaffolding instructional strategies for math may involve the following: By providing intentional and strategic instructional scaffolding, your ELL students will be more engaged and experience more growth. 2. Use a variety of learning modalities All students benefit from instruction that incorporates a variety of learning modalities. Research has shown that when auditory, visual, and kinesthetic methods are combined, students retain information better. Though all students benefit from hands-on learning activities, they are particularly important for ELLs. According to The National Math Foundation, “By 2009, Mulalic (et al.) discovered that the kinesthetic learning style was the most preferred learning style among ESL students. Studies in 1987, 1990, 1993, 1997, and 2001 reported that adult L2 immigrants and ESL students in the US favor kinesthetic styles over all others (Gilakjani, 2012).” While there are many ways to incorporate kinesthetic activities in your ELL math instruction, manipulatives are one of the most powerful tools you can use to support your ELLs. Although they are often thought of as math tools for elementary students, they can be used with students at any level. Manipulatives provide a common language for students to communicate their thought processes. This is especially important for newcomer ELLs, whose expressive English language skills are starting to develop. 3. Promote language production When you’re teaching math to ELLs, it’s important to consider how you can deepen their understanding of math concepts as well as encourage their language skills. However, your ELL students face a significant cognitive load during instruction in content areas— they’re simultaneously learning a new language and new skills. It’s important to minimize their cognitive load while maintaining academic rigor. A student’s proficiency in a language is dependent on both production and comprehension. ELL students’ receptive language skills generally develop more quickly than their expressive language skills. Teachers need to intentionally help them increase their language production. You can do this through scaffolding and supportive language practices, like those below. 4. Focus on word problems As word problems become increasingly difficult over time, it’s important to note that reading skills play a vital role in math. Brenda Krick-Morales writes, “Word problems in mathematics often pose a challenge because they require that students read and comprehend the text of the problem, identify the question that needs to be answered, and finally create and solve a numerical equation — ELLs who have had formal education in their home countries generally do not have mathematical difficulties; hence, their struggles begin when they encounter word problems in a second language that they have not yet mastered (Bernardo, 2005).” When using word problems with ELL students, it’s important to consider the following questions: High-Impact Strategies to Teach ELL Math Vocabulary Vocabulary instruction plays a key role in ELL math achievement. ELLs need explicit, intentional vocabulary instruction to successfully gain math skills. Focus on academic vocabulary Math is full of challenging words like quadrilateral and vertices—and your ELL students may or may not have a prior understanding of these concepts in their native language. ELL students benefit from pre-teaching academic vocabulary terms in a lesson, as it provides them with a foundation of understanding. It is also important to give them multiple exposures to math words—research on vocabulary development shows that students retain vocabulary better when they learn words in context rather than in isolation. The following ideas can help you develop and reinforce academic math vocabulary: Teach “tricky words” Math vocabulary includes both specific terminology and everyday words that have different meanings in the context of math. Take a look at the example below: Did the student “find x”? Technically, yes! Words like “face,” “table,” and “carry” (and even the word “and!”) have different meanings when they’re used in math. Provide your ELL students with many opportunities to practice using these tricky words in the context of math. Don’t forget about homophones Whether you’re a native speaker or new to English, homophones can present a challenge. Many math vocabulary words are homophones: some and sum, raise and rays, won and one, wait and weight…the list goes on! Consider creating a chart for your ELLs to reference to help them differentiate between common homophones. Math and language skills are inextricably linked. By keeping language in mind when you plan your ELL math instruction, you can help your students develop important math skills. Make problem solving meaningful for students. Lessons feature kid-friendly topics such as movie marathons and summer fun to reinforce problem solving skills and understanding of math in daily life. Subscribe to Receive Our e-Newsletter 520 East Bainbridge Street Elizabethtown, PA 17022 Phone: 800.233.0759 Fax: 888.834.1303 ©2025 Continental. All Rights Reserved. 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https://proofwiki.org/wiki/Integer_Subtraction_is_Closed
Integer Subtraction is Closed - ProofWiki Integer Subtraction is Closed From ProofWiki Jump to navigationJump to search Theorem The set of integers is closed under subtraction: ∀a,b∈Z:a−b∈Z∀a,b∈Z:a−b∈Z Proof From the definition of subtraction: a−b:=a+(−b)a−b:=a+(−b) where −b−b is the inverse for integer addition. From Integers under Addition form Abelian Group, the algebraic structure(Z,+)(Z,+) is a group. Thus: ∀a,b∈Z:a+(−b)∈Z∀a,b∈Z:a+(−b)∈Z Therefore integer subtraction is closed. ■◼ Sources 1969:C.R.J. Clapham: Introduction to Abstract Algebra... (previous)... (next): Chapter 1 1: Integral Domains: §2§2. Operations: Example 1 1 1981:Murray R. Spiegel: Theory and Problems of Complex Variables(SI ed.)... (previous)... (next): 1 1: Complex Numbers: The Real Number System: 2 2 2000:James R. Munkres: Topology(2nd ed.)... (previous)... (next): 1 1: Set Theory and Logic: §4§4: The Integers and the Real Numbers Retrieved from " Categories: Proven Results Integer Subtraction Algebraic Closure Navigation menu Personal tools Log in Request account Namespaces Page Discussion [x] English Views Read View source View history [x] More Search Navigation Main Page Community discussion Community portal Recent changes Random proof Help FAQ P r∞f W i k i P r∞f W i k i L A T E X L A T E X commands ProofWiki.org Proof Index Definition Index Symbol Index Axiom Index Mathematicians Books Sandbox All Categories Glossary Jokes To Do Proofread Articles Wanted Proofs More Wanted Proofs Help Needed Research Required Stub Articles Tidy Articles Improvements Invited Refactoring Missing Links Maintenance Tools What links here Related changes Special pages Printable version Permanent link Page information This page was last modified on 10 March 2024, at 12:54 and is 1,582 bytes Content is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Privacy policy About ProofWiki Disclaimers
6016
https://www.statisticsteacher.org/2025/09/02/how-mad-must-we-be/
How MAD Must We Be? A Robust Test for Identifying Meaningful Differences Using the Mean Absolute Deviation | Statistics Teacher Skip to content About Submissions Articles 9-12+ 6-8 K-5 Announcements Lesson Plans Resources Events How MAD Must We Be? A Robust Test for Identifying Meaningful Differences Using the Mean Absolute Deviation September 2, 2025 Jon Hasenbank and John Appiah Kubi, Grand Valley State University This article introduces a method for identifying statistically meaningful differences between two data sets using the mean absolute deviation, known as MAD and a measure of variability taught in middle school curricula under the Common Core State Standards, which parallel the recommendations of the Pre-K-12 Guidelines for Assessment and Instruction in Statistics Education II report. While students are encouraged to make informal comparative inferences through visual tools and intuitive reasoning, current instructional resources lack precise heuristics to determine when a difference is statistically significant. The authors address this gap by proposing the k-test, a straightforward calculation based on the ratio of the difference in sample means to the pooled MAD, compared against a sample-size-dependent threshold. Drawing from statistical theory originally developed by Erna Herrey and refined using S. A. Revets’ work on one-norm statistics, the k-test provides a meaningful and accessible alternative to the traditional t-test. Simulation studies reveal that the k-test aligns with the t-test more than 97% of the time, validating its reliability. Unlike the commonly used “2x rule,” which can be overly conservative, the k-test incorporates sample size to yield more accurate conclusions—empowering students and educators to resolve ambiguity in data investigations. By situating the method within the mathematical knowledge expected at the middle school level, the authors bridge the gap between informal inference and formal statistical reasoning. This approach not only enhances classroom practice by offering clearer decision-making tools but also encourages curiosity about variability, significance, and the role of sample size in statistical analysis. Download a PDF of the full article. Tags: Common Core State Standards, data investigations, decision-making tools, formal statistical reasoning, informal inference, k-test, MAD, mean absolute deviation, middle school, middle school curricula, Pre-K-12 GAISE, Sample Size, statistical analysis, statistics education, statistics instruction, t-test, variability From: 6-8, Announcements ← Editor’s Note: Spring 2025 Graphing with Kids: Teaching the Superpower of Numbers and Data → Search Recent Posts Editors’ Note: Fall 2025 ASA/NCTM Joint Committee Members Share Favorite Resources, Ideas Some Paradoxes: Puzzling or Poorly Presented? Integrating Data Science Practices into Informal Learning: A STEM Summer Camp Approach On the Pedagogy of Randomness: Effectively Teaching How Random Is Relative in High School Previous Issues September 2025 May 2025 December 2024 July 2024 November 2023 March 2023 October 2022 March 2022 October 2021 July 2021 April 2021 March 2021 December 2020 November 2020 April 2020 March 2020 September 2019 January 2019 July 2018 March 2018 October 2017 September 2017 May 2017 March 2017 February 2017 January 2017 December 2016 ABOUT Statistics Teacher (ST) is an online journal published by the American Statistical Association (ASA) – National Council of Teachers of Mathematics (NCTM) Joint Committee on Curriculum in Statistics and Probability for Grades K-12. ST supports the teaching and learning of statistics through education articles, lesson plans, announcements, professional development opportunities, technology, assessment, and classroom resources. Authors should use this form to submit articles or lesson plans. CONTACT Contact the Statistics Teacher editors for further information or to volunteer to write or review content. SUBSCRIBE Subscribe to our Statistics Teacher email updates and we'll notify you when there is new content! Search Search Linen Theme by The Theme Foundry Copyright © 2025 American Statistical Association. All rights reserved.
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https://web.stanford.edu/class/archive/ee/ee264/ee264.1052/ROC.pdf
EE264 Oct 8, 2004 Fall 04-05 Supplemental Notes About the region of convergence of the z-transform The z-Transform of a sequence f[n] is defined as S(z) = P∞ ∞f[n]z−n, for those values of z1 for which the infinite sum converges, such set of values of z is called the Region of Convergence of the z-transform S(z). This document describes the possible shapes the Region of Convergence (ROC) may take. We start by describing the ROC shape of one sided sequences from which we’ll deduce the ROC shape for two sided sequences. Region of Convergence for z-transforms of Unilateral sequences Let f[n] be an anticausal sequence, i.e. f[n] = 0 for n ≥0. Its z-transform is S(z) = P∞ n=−∞f[n]z−n = P∞ n=0 f[−n]zn. If the sequence is of finite duration, its z-transform is a finite polynomial with nonnegative powers of z, hence it converges for all finite values values of z and its ROC is the whole z-plane. On the other hand, if the sequence is of infinite duration, the situation is less straight forward and we’ll need the following lemma: Lemma 1 If a power series P∞ n=0 anzn converges when z = z1 (z1 ̸= 0), then it is absolutely convergent2 at each point z in the open disk |z| < |z1|. The proof is as follows, we assume that the series P∞ n=0 anzn converges, hence the terms anzn 1 are bounded; that is, |anzn 1 | ≤M (n = 0, 1, 2, . . .) for some positive constant M. If |z| < |z1| and we let ρ denote the modulus |z/z1|, we can see that |anzn| = |anzn 1 | z z1 n ≤Mρn (n = 0, 1, 2, . . .) where ρ < 1. Since P∞ n=0 Mρn converges when ρ < 1, we can conclude that ∞ X n=0 |anzn| < ∞ = ⇒ ∞ X n=0 anzn converges in the open disk |z| < |z1|. This lemma tells us that the set of all points inside some circle centered at the origin is part of the region of convergence for the power series P∞ n=0 anzn, provided it converges at some point 1z is a complex variable. 2a series P∞ n=0 an is said to converge absolutely if the series P∞ n=0 |an| converges 1 other than z = 0, what’s more, it converges absolutely inside that circle of convergence. The greatest circle centered at the origin |z| = R such that the series converges at each point inside is called the circle of convergence of the series. The series cannot converge at any point z2 outside that circle, according to the theorem; for if it did, it would converge everywhere inside the circle centered at the origin and passing through z2. The first circle could not then be the circle of convergence. Note that it may be that the series P∞ n=0 anzn converges for all (finite) z, in that case we can think of the circle of convergence as being |z| = ∞and say that the ROC is |z| < ∞(This region is called the finite z-plane). Notice as well that the series may or may not converge at points at the circle of convergence |z| = R. We can summarize the consequences of Lemma 1 as follows: The Region of Convergence of the z-transform of an anticausal sequence is either |z| < ∞or of the form |z| < R [ B for some nonnegative constant R and B a subset3 of the so-called circle of convergence |z| = R. We’ll say that a sequence f[n] is causal if f[n] = 0 for n < 0. Following a similar argument it can be shown that the ROC of the z-transform of a causal sequence is the exterior of a circle (called the circle of convergence) possibly including points in the boundary. In particular, if f[n] is causal and we can find a positive integer N such that f[n] = 0 for n > N, then the ROC will be |z| > 0, including z = 0 only if f[n] = 0 for n ̸= 0, in which case, the ROC of the z-transform is the whole z plane4. Region of Convergence for Bilateral sequences In general, a sequence can be two-sided, in that case, the ROC of its z-transform is obviously the intersection of the ROC’s corresponding to its causal and anticausal parts. We can summarize the results for the ROC of z-transforms as follows: The Region of Convergence of the z-transform of a sequence can have one of the following forms: • The whole z-plane • The finite z-plane |z| < ∞ • A subset B of a circle |z| = R • The exterior of a circle |z| > R1 S B1 • An annulusR1 < |z| < R2 S B1 S B2 where R1 and R2 are nonnegative constants, and B1 and B2 are subsets5 of the circles |z| = R1 and |z| = R2 respectively. 3The subset B can be the empty set. 4Note that the whole z-plane differs from the finite z-plane |z| < ∞as a ROC in the sense that for the later case limz→∞S(z) does not exist 5Subsets B1 and B2 can be the empty sets. 2 For convenience, we say a region is an open annulus if it has one of the following forms: |z| < R, |z| > R, or R1 < |z| < R2, where R, R1 and R2 are nonnegative constants. From our previous discussion is easy to see that in the cases when the ROC of a z-transform includes an open annulus, the z-transform converges absolutely in that open annulus and what’s more, it converges uniformly. From this it can be proved to be analytic in that open annulus. Region of convergence for rational functions of z In many practical cases we can find a closed form expression F(z) for the z-transform S(z) of the sequence for any z in the ROC of S(z), i.e. S(z) = F(z), z ∈ROC where F(z) does not have a infinite sum of terms. Most of the sequences we work with are such that their z-transform have a closed form expression F(z) that is a rational function of z, i.e. F(z) = N(z) D(z) where N(z) and D(z) are finite order polynomials in z. In such cases we say that F(z) has a pole6 at zo if limz→zo F(z) = ∞. Also, F(z) is said to have a pole at ∞if limz→∞F(z) = ∞. Using partial fraction expansion theory, it can be proved that the Region of Convergence of a z-tranform that has a rational function F(z) as a closed form expression is either the whole z-plane, the finite z-plane or exactly an open annulus whose boundaries are circles that pass through at least one of the poles of F(z), obviously such open annulus cannot contain any of the poles. Hence, the ROC for rational functions of z can have one of the following forms: • The whole z-plane • The finite z-plane |z| < ∞ • |z| > |zo| • |z| < |zo| • |zo| < |z| < |z1| where zo and z1 are poles of F(z) and there is no other pole z2 such that |zo| < |z2| < |z1|. Notice the difference between these forms and the general possible forms for the ROC given before. We can see that the fact that the z-transform is a rational function of z, excludes the points at the circle of convergence as possible points of the ROC7. This fact allows us to easily determine the possible ROC’s for a rational z-transform F(z) by just finding its poles and subsequently the open annulus delimited by them . For each of these open annulus F(z) will have a unique inverse z-transform. 6the definition of a pole for general functions of z is more elaborated and requires the use of Laurent series. 7See section 3.2 of Discrete-Time Signal Processing from Oppenheim and Schafer, Second edtion, for a summary of the properties of the ROC of z-transforms with rational closed form. 3
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https://us.sofatutor.com/math/videos/understanding-the-relationship-between-circles-and-the-number-pi
Try sofatutor for 30 Days Discover why over 1.6 MILLION students choose sofatutor! Math Areas and Shapes Circles Understanding the Relationship between Circles and the number Pi Understanding the Relationship between Circles and the number Pi Watch videos Start exercises Show worksheets Content Understanding the Relationship between Circles and the number Pi Understanding the Relationship between Circles and the Number Pi The Number Pi Parts of a Circle Circumference of a Circle Calculating the Circumference – Step-by-Step Instruction Understanding the Relationship between Circles and the number Pi – Summary Relationship between Circles and the number Pi – Frequently Asked Questions Improve your grades now while having fun Unlock this video in just a few steps, and benefit from all sofatutor content: Try it for 30 Days Rating Give us a rating! You must be logged in to be able to give a rating. Wow, thank you! Please rate us on Google too! We look forward to it! Go to Google The authors Trentkay Basics on the topic Understanding the Relationship between Circles and the number Pi Understanding the Relationship between Circles and the Number Pi Pi, represented by the Greek letter π, is a fundamental constant in mathematics that appears in many formulas relating to circles and other aspects of geometry and physics. It is the ratio of a circle's circumference (the distance around the circle) to its diameter (the distance across the circle through its center). The Number Pi Pi (π) is a special number in mathematics that represents the ratio of a circle's circumference (C) to its diameter (d). This means that no matter how big or small a circle is if you divide the circumference (the distance around the circle) by the diameter (the distance across the middle of the circle), you will always get approximately the same number: pi, or π. A common approximation for pi is 722​, which is close but not exactly π. Pi is actually an infinite, non-repeating decimal, meaning it has no exact value and no final digit. This type of number is known as an irrational number. Parts of a Circle A circle is a perfect round shape, where every point on the edge is the same distance from the center. This consistent distance is known as the radius, and it is half the length of the diameter. For more details on the fundamental characteristics of a circle, see what is a circle. | Term | Definition | --- | | Radius (r) | The distance from the center of the circle to any point on its edge. | | Diameter (d) | The distance across the circle, passing through the center. It is twice the radius. | | Circumference (C) | The distance around the circle. It is calculated as π times the diameter of the circle (πd). | The radius of a circle is half the diameter because it spans from the center of the circle to its edge, effectively splitting the diameter into two equal parts. Thus, the radius is always half the length of the diameter. Circumference of a Circle The circumference of a circle can be calculated using the formula: C=πd, where C stands for circumference and d is the diameter of the circle. The circumference of a circle can also be calculated using C=2πr, where C is the circumference and r is the radius. This formula is derived from C=πd by substituting d with 2r since the diameter is always twice the radius. Calculating the Circumference – Step-by-Step Instruction Example 1 Find the circumference of a circle with a diameter of 8 inches. Identify the formula for circumference when you know the diameter: C=πd. Substitute the given diameter into the formula: C=3.14×8 inches. Calculate the product to find the circumference: C=25.12 inches. Example 2 Calculate the circumference of a circle with a radius of 5 meters. Use the formula for circumference with radius: C=2πr. Substitute the radius into the formula: C=2×3.14×5 meters. Complete the multiplication to find the circumference: C=31.4 meters. Apply the circumference formula using diameter: C=πd. Plug in the diameter: C=3.14×14 centimeters. Calculate to find the circumference: C=43.96 centimeters, which rounds up to approximately 44 centimeters. Try some on your own. What is the circumference of a circle with a diameter of 12 feet? Using the formula C=πd, plug in the diameter: C=3.14×12 feet = 37.68 feet. If a circle has a radius of 3 inches, how long is the circumference? Apply the formula C=2πr: C=2×3.14×3 inches = 18.84 inches. Calculate the circumference of a circle where the diameter is 10 meters. Use C=πd: C=3.14×10 meters = 31.4 meters. A circle has a radius of 22 yards. Find its circumference. Using C=2πr: C=2×3.14×22 yards = 138.08 yards. Find the circumference of a circle with a diameter of 7 centimeters. Plug the diameter into C=πd: C=3.14×7 centimeters = 21.98 centimeters. If the radius of a circle is 4.5 kilometers, what is its circumference? Calculate using C=2πr: C=2×3.14×4.5 kilometers = 28.26 kilometers. Understanding the Relationship between Circles and the number Pi – Summary Key Learnings from this Text: Pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. The circumference of a circle can be calculated using the diameter with the formula C=πd or using the radius with C=2πr. Essential circle terms include radius, diameter, and circumference, defining the basic properties of circular shapes. The text includes illustrations and practical examples to demonstrate circumference calculations and the relevance of Pi in real-world scenarios. Understanding Pi provides a foundation for more complex geometric and mathematical studies. Relationship between Circles and the number Pi – Frequently Asked Questions What is Pi? Pi (π) is a mathematical constant approximately equal to 3.14159, representing the ratio of a circle's circumference to its diameter. Why is Pi important? Pi is crucial in mathematics, engineering, and science for calculating areas and volumes of circular and spherical shapes, among other applications. Is Pi the same for all circles? Yes, Pi is a constant, so the ratio of the circumference to the diameter is the same for all circles, regardless of their size. How do you calculate the circumference of a circle? The circumference can be calculated using the formula (C = \pi d) if the diameter is known, or (C = 2\pi r) if the radius is known. What are some practical uses of Pi? Pi is used in various fields such as engineering for designing objects with circular or spherical components, in computing for algorithms, and in everyday measurements of circular objects. Can Pi be expressed as a fraction? While Pi is often approximated by fractions like 722​, it is an irrational number and cannot be exactly expressed as a simple fraction. What is the difference between the radius and diameter of a circle? The radius is half the length of the diameter. It measures the distance from the center of the circle to any point on its edge, while the diameter spans from one edge to the other, passing through the center. Is there an end to Pi? No, Pi is an infinite, non-repeating decimal. It has no exact value and no final digit, continuing indefinitely without repeating. How was Pi discovered? Pi has been known since ancient times, but it was first rigorously calculated by the Greek mathematician Archimedes, who used polygons to approximate the area of a circle. Are there any tools or programs to calculate Pi? Yes, there are many tools and software programs available that can calculate Pi to many decimal places, such as scientific calculators and computer algorithms designed for precision computing. Transcript Understanding the Relationship between Circles and the number Pi As he does every morning, Farmer Johnson is enjoying a hot cup of joe before he gets to his morning chores. But in the distance, he sees something strange. Sakes alive! What's that? Crop circles everywhere! Only one kind of varmint could leave those tracks aliens! Farmer Johnson takes careful measurements of the mysterious markings and high-tails it back to the homestead to get to the bottom of it! While making sense of the crop circles and trying to prove that aliens exist, Farmer Johnson will learn about "Circles and the Number Pi". Farmer Johnson first examines the biggest circle, looking at the distance from the center to different points on the edge of the circle. What do you notice about these measurements? Every time Farmer Johnson draws another line, he gets the same length! 420 feet. Well I'll be a monkey's uncle! Is this proof the markings had to have been made by aliens? Well, actually no, Farmer Johnson. This pattern is true for all circles, no matter who makes them. We call this measurement, from the center of a circle to any point on its edge, the radius. It's usually represented by the letter 'r'. No matter which point on the edge you pick, the radius will always be the same length. In fact, that's what makes a circle a circle! It's a shape formed by all the points that are the same distance away from the center. Farmer Johnson studies his board carefully. There's got to be some way to prove these circles are the work of alien invaders! This time, Farmer Johnson measured both the distance from the center of the circle to the edge as well as the distance across the crop circles, going through the center of the circle. Do you notice any relationship between these lengths? What the blazes! This length across the circle is exactly twice the radius! Here, 840 feet is equal to 2 times 420 feet. And it's true for the other circles too! Farmer Johnson thinks this can't be no piddlin' coincidence! Is it hard and fast proof that aliens were behind the circles? Nope, Farmer Johnson. It's just math. This line is called the diameter. It passes through the center of a circle and touches opposite points on the circle's edge and is always equal to twice the radius. Why do you think that's so? A diameter is made by putting two radii together. Farmer Johnson is still fixin' to prove his wild notion about aliens. He just needs a big breakthrough, something that mere mathematics can't explain! This time, Farmer Johnson measures the distance around the circles, tracing the entire edge. We call this measurement the circumference. Hmm, Farmer Johnson reckons he spies a relationship between the circumference and the diameter. The bigger the circle is across, the bigger it is around. Let's examine the relationship more closely by calculating the ratio of the circumference to the diameter. Two thousand, six hundred and thirty eight divided by 840 is equal to approximately 3.14. Jiminy Cricket! Do you see what Farmer Johnson sees? The ratio, rounded to the nearest hundredth, is always the same! This can only be the work of those blasted aliens! Now, hold your horses, Farmer Johnson. It looks like Miz Johnson has a thing or two to teach her husband. This 'mysterious' ratio you were hollering about is actually a number that mathematicians have known for centuries. We call it PI, and here's the symbol we use to represent it. It's the ratio of any circle's circumference to it's diameter. You heard right, it doesn't matter what size circle you have, the ratio is always the same. Here's the value of pi, 3.141519265 and so on. The number pi is a non-terminating decimal, which means it never ends. It also doesn't repeat. That means it's an irrational number. If we need to calculate with this number, we can estimate it using either the decimal form 3.14 or the fraction 22 over 7. If we need to be exact, we can leave the symbol PI in our answer. Isolating the variable 'c' from, pi equals 'c' over 'd', gives us the formula for the circumference of a circle. Circumference equals pi times diameter. Because the diameter of a circle is twice its radius... we can also write this formula as circumference equals 2 times pi times radius. Farmer Johnson's all tore up that aliens didn't make the crop circles afterall but in the meantime, let's review. All circles share some basic properties, no matter their size. The radius, which is a line drawn from the center of the circle to the outer edge...is always the same length, no matter how you draw it. The diameter is any line that crosses through the center of the circle and ends at two opposite edges. And it's always twice the size of the radius. Finally, we have the circumference of a circle. That's the distance around the circle. The ratio of the circumference to the diameter is the number PI, written with the symbol pi. In calculations, pi is often estimated as 3.14 or 22 over 7 and it definitely wasn't created by aliens! Farmer Johnson heads back to his water tower, down in the mouth that aliens don't exist afterall. 0 comments Understanding the Relationship between Circles and the number Pi exercise Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Understanding the Relationship between Circles and the number Pi. Identify measurements of a circle. Hints The radius of a circle measures the distance from the center to any part on the edge of the circle. The diameter is the distance through the center of the circle from one edge to the other. The circumference is the distance around the outside edge of a circle. Solution The circumference is 942.5 ft and refers to the distance around the outside of the circle. The diameter is 300 ft and this measurement is the distance across the circle, passing through the center. The radius is half of the diameter and is 150 ft. This measurement is the distance from the center of the circle to the outside. ### Understand the value of Pi. Hints Pi is a special number that mathematicians use when they are talking about circles. It's the number you get if you divide the distance around a circle (the circumference) by the distance across the middle of the circle (the diameter). Rational numbers are numbers that can be written as a fraction, where both the top number (numerator) and the bottom number (denominator) are whole numbers. For example, 21​, 3, and 5−4​ are all rational numbers. Irrational Numbers: Irrational numbers are numbers that can't be written as a simple fraction. Their decimal places go on forever without repeating any pattern. Examples of irrational numbers include π and the 2​. Solution Pi is a number that mathematicians have known for centuries and the symbol is this: π. The ratio of a circle's circumference to its diameter is the value of Pi. The value of Pi is approximately 3.14... and is known as an irrational number. ### Determine the ratio equivalent to the value of Pi. Hints The ratio of the value of π is dC​. To find the value of a ratio, division can be used. For example, the ratio of 315​ can be simplified to 15​, or 5. Solution The ratio 722​ simplifies to π. This can be found by dividing 22 by 7, to get approximately 3.14.... ### Using a formula to find the circumference of a circle. Hints The r in the formula stands for the radius, which is the distance from the center of the circle to the outside. The d in the formula stands for the diameter, which is the distance across the circle passing through the center. To use a formula, substitute in values you know into the appropriate variable. For example, if we knew we had a diameter of 9 inches, and we were using the formula C=πd to find the circumference, we would replace the d with the 9 like this: C=π(9). The radius is half the length of the diameter. Solution The circle has a diameter of 12 cm, therefore the radius is 6 cm. The two equations that can be used to find the circumference are: C=π(12) C=2π(6) ### Identify the ratio of Pi. Hints C=circumference d=diameter r=radius Ratios must be written in a specific order in order for them to be accurate. The ratio 21​ has a different value than 12​. The measurements of a circle are depicted here, as well as the ratio of Pi. Solution The ratio for Pi is π=dC​. If the circumference is divided by the diameter the value is always approximately 3.14. ### Determine the circumference of a circle. Hints The formula to find the circumference of a circle is C=πd, or C=2πr. The radius is half the distance of the diameter. To round a number to the nearest tenth, look at the number in the hundredth place. If it's 5 or more, round the tenth's place up; if it's less than 5, keep the tenth's place the same. A calculator will come in handy to help you solve using the value of π. Solution To find the circumference of the circle shown, you can choose to use the given radius of 4 cm, or double it for the diameter of 8 cm. Here are both ways shown with the two different formulas. C=πd C=π(8) C=25.1 cm C=2πr C=2π(4) C=25.1 cm More videos and learning texts for the topic Circles Understanding the Relationship between Circles and the... The Area of a Circle
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SURFACE AREAS AND VOLUMES 137 CHAPTER 11 SURFACE AREAS AND VOLUMES 11.1 Surface Area of a Right Circular Cone We have already studied the surface areas of cube, cuboid and cylinder. We will now study the surface area of cone. So far, we have been generating solids by stacking up congruent figures. Incidentally, such figures are called prisms. Now let us look at another kind of solid which is not a prism (These kinds of solids are called pyramids.). Let us see how we can generate them. Activity : Cut out a right-angled triangle ABC right angled at B. Paste a long thick string along one of the perpendicular sides say AB of the triangle [see Fig. 11.1(a)]. Hold the string with your hands on either sides of the triangle and rotate the triangle about the string a number of times. What happens? Do you recognize the shape that the triangle is forming as it rotates around the string [see Fig. 11.1(b)]? Does it remind you of the time you had eaten an ice-cream heaped into a container of that shape [see Fig. 11.1 (c) and (d)]? Fig. 11.1 Reprint 2025-26 138 MATHEMATICS This is called a right circular cone. In Fig. 11.1(c) of the right circular cone, the point A is called the vertex, AB is called the height, BC is called the radius and AC is called the slant height of the cone. Here B will be the centre of circular base of the cone. The height, radius and slant height of the cone are usually denoted by h, r and l respectively. Once again, let us see what kind of cone we can not call a right circular cone. Here, you are (see Fig. 11.2)! What you see in these figures are not right circular cones; because in (a), the line joining its vertex to the centre of its base is not at right angle to the base, and in (b) the base is not circular. As in the case of cylinder, since we will be studying only about right circular cones, remember that by ‘cone’ in this chapter, we shall mean a ‘right circular cone.’ Activity : (i) Cut out a neatly made paper cone that does not have any overlapped paper, straight along its side, and opening it out, to see the shape of paper that forms the surface of the cone. (The line along which you cut the cone is the slant height of the cone which is represented by l). It looks like a part of a round cake. (ii) If you now bring the sides marked A and B at the tips together, you can see that the curved portion of Fig. 11.3 (c) will form the circular base of the cone. Fig. 11.3 (iii) If the paper like the one in Fig. 11.3 (c) is now cut into hundreds of little pieces, along the lines drawn from the point O, each cut portion is almost a small triangle, whose height is the slant height l of the cone. (iv) Now the area of each triangle = 1 2 × base of each triangle × l. So, area of the entire piece of paper Fig. 11.2 Reprint 2025-26 SURFACE AREAS AND VOLUMES 139 = sum of the areas of all the triangles = 1 2 3 1 1 1 2 2 2 bl b l b l + + + ⋯ = ( ) 1 2 3 1 2l b b b + + + ⋯ = 1 2 × l × length of entire curved boundary of Fig. 11.3(c) (as b1 + b2 + b3 + . . . makes up the curved portion of the figure) But the curved portion of the figure makes up the perimeter of the base of the cone and the circumference of the base of the cone = 2πr, where r is the base radius of the cone. So, Curved Surface Area of a Cone = 1 2 × l × 2π π π π πr = π π π π πrl where r is its base radius and l its slant height. Note that l2 = r2 + h2 (as can be seen from Fig. 11.4), by applying Pythagoras Theorem. Here h is the height of the cone. Therefore, l = 2 2 r h + Now if the base of the cone is to be closed, then a circular piece of paper of radius r is also required whose area is πr2. So, Total Surface Area of a Cone = π π π π πrl + π π π π πr2 = π π π π πr(l + r) Example 1 : Find the curved surface area of a right circular cone whose slant height is 10 cm and base radius is 7 cm. Solution : Curved surface area = πrl = 22 7 × 7 × 10 cm2 = 220 cm2 Example 2 : The height of a cone is 16 cm and its base radius is 12 cm. Find the curved surface area and the total surface area of the cone (Use π = 3.14). Solution : Here, h = 16 cm and r = 12 cm. So, from l2 = h2 + r2, we have l = 2 2 16 12 + cm = 20 cm Fig. 11.4 Reprint 2025-26 140 MATHEMATICS So, curved surface area = πrl = 3.14 × 12 × 20 cm2 = 753.6 cm2 Further, total surface area = πrl + πr2 = (753.6 + 3.14 × 12 × 12) cm2 = (753.6 + 452.16) cm2 = 1205.76 cm2 Example 3 : A corn cob (see Fig. 11.5), shaped somewhat like a cone, has the radius of its broadest end as 2.1 cm and length (height) as 20 cm. If each 1 cm2 of the surface of the cob carries an average of four grains, find how many grains you would find on the entire cob. Solution : Since the grains of corn are found only on the curved surface of the corn cob, we would need to know the curved surface area of the corn cob to find the total number of grains on it. In this question, we are given the height of the cone, so we need to find its slant height. Here, l = 2 2 r h + = 2 2 (2.1) 20 + cm = 404.41 cm = 20.11 cm Therefore, the curved surface area of the corn cob = πrl = 22 7 × 2.1 × 20.11 cm2 = 132.726 cm2 = 132.73 cm2 (approx.) Number of grains of corn on 1 cm2 of the surface of the corn cob = 4 Therefore, number of grains on the entire curved surface of the cob = 132.73 × 4 = 530.92 = 531 (approx.) So, there would be approximately 531 grains of corn on the cob. EXERCISE 11.1 Assume π = 22 7 , unless stated otherwise. 1. Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area. 2. Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m. Fig. 11.5 Reprint 2025-26 SURFACE AREAS AND VOLUMES 141 3. Curved surface area of a cone is 308 cm2 and its slant height is 14 cm. Find (i) radius of the base and (ii) total surface area of the cone. 4. A conical tent is 10 m high and the radius of its base is 24 m. Find (i) slant height of the tent. (ii) cost of the canvas required to make the tent, if the cost of 1 m2 canvas is 70. 5. What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base radius 6 m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20 cm (Use π = 3.14). 6. The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of 210 per 100 m2. 7. A joker’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps. 8. A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is 12 per m2, what will be the cost of painting all these cones? (Use π = 3.14 and take 1.04 = 1.02) 11.2 Surface Area of a Sphere What is a sphere? Is it the same as a circle? Can you draw a circle on a paper? Yes, you can, because a circle is a plane closed figure whose every point lies at a constant distance (called radius) from a fixed point, which is called the centre of the circle. Now if you paste a string along a diameter of a circular disc and rotate it as you had rotated the triangle in the previous section, you see a new solid (see Fig 11.6). What does it resemble? A ball? Yes. It is called a sphere. Fig. 11.6 Reprint 2025-26 142 MATHEMATICS Can you guess what happens to the centre of the circle, when it forms a sphere on rotation? Of course, it becomes the centre of the sphere. So, a sphere is a three dimensional figure (solid figure), which is made up of all points in the space, which lie at a constant distance called the radius, from a fixed point called the centre of the sphere. Note : A sphere is like the surface of a ball. The word solid sphere is used for the solid whose surface is a sphere. Activity : Have you ever played with a top or have you at least watched someone play with one? You must be aware of how a string is wound around it. Now, let us take a rubber ball and drive a nail into it. Taking support of the nail, let us wind a string around the ball. When you have reached the ‘fullest’ part of the ball, use pins to keep the string in place, and continue to wind the string around the remaining part of the ball, till you have completely covered the ball [see Fig. 11.7(a)]. Mark the starting and finishing points on the string, and slowly unwind the string from the surface of the ball. Now, ask your teacher to help you in measuring the diameter of the ball, from which you easily get its radius. Then on a sheet of paper, draw four circles with radius equal to the radius of the ball. Start filling the circles one by one, with the string you had wound around the ball [see Fig. 11.7(b)]. Fig. 11.7 What have you achieved in all this? The string, which had completely covered the surface area of the sphere, has been used to completely fill the regions of four circles, all of the same radius as of the sphere. So, what does that mean? This suggests that the surface area of a sphere of radius r = 4 times the area of a circle of radius r = 4 × (π r2) So, Surface Area of a Sphere = 4 π π π π π r2 Reprint 2025-26 SURFACE AREAS AND VOLUMES 143 where r is the radius of the sphere. How many faces do you see in the surface of a sphere? There is only one, which is curved. Now, let us take a solid sphere, and slice it exactly ‘through the middle’ with a plane that passes through its centre. What happens to the sphere? Yes, it gets divided into two equal parts (see Fig. 11.8)! What will each half be called? It is called a hemisphere. (Because ‘hemi’ also means ‘half’) And what about the surface of a hemisphere? How many faces does it have? Two! There is a curved face and a flat face (base). The curved surface area of a hemisphere is half the surface area of the sphere, which is 1 2 of 4πr2. Therefore, Curved Surface Area of a Hemisphere = 2π π π π πr2 where r is the radius of the sphere of which the hemisphere is a part. Now taking the two faces of a hemisphere, its surface area 2πr2 + πr2 So, Total Surface Area of a Hemisphere = 3π π π π πr2 Example 4 : Find the surface area of a sphere of radius 7 cm. Solution : The surface area of a sphere of radius 7 cm would be 4πr2 = 4 × 22 7 × 7 × 7 cm2 = 616 cm2 Example 5 : Find (i) the curved surface area and (ii) the total surface area of a hemisphere of radius 21 cm. Solution : The curved surface area of a hemisphere of radius 21 cm would be = 2πr2 = 2 × 22 7 × 21 × 21 cm2 = 2772 cm2 Fig. 11.8 Reprint 2025-26 144 MATHEMATICS (ii) the total surface area of the hemisphere would be 3πr2 = 3 × 22 7 × 21 × 21 cm2 = 4158 cm2 Example 6 : The hollow sphere, in which the circus motorcyclist performs his stunts, has a diameter of 7 m. Find the area available to the motorcyclist for riding. Solution : Diameter of the sphere = 7 m. Therefore, radius is 3.5 m. So, the riding space available for the motorcyclist is the surface area of the ‘sphere’ which is given by 4πr2 = 4 × 22 7 × 3.5 × 3.5 m2 = 154 m2 Example 7 : A hemispherical dome of a building needs to be painted (see Fig. 11.9). If the circumference of the base of the dome is 17.6 m, find the cost of painting it, given the cost of painting is 5 per 100 cm2. Solution : Since only the rounded surface of the dome is to be painted, we would need to find the curved surface area of the hemisphere to know the extent of painting that needs to be done. Now, circumference of the dome = 17.6 m. Therefore, 17.6 = 2πr. So, the radius of the dome = 17.6 × 7 2 22 × m = 2.8 m The curved surface area of the dome = 2πr2 = 2 × 22 7 × 2.8 × 2.8 m2 = 49.28 m2 Now, cost of painting 100 cm2 is 5. So, cost of painting 1 m2 = 500 Therefore, cost of painting the whole dome = 500 × 49.28 = 24640 EXERCISE 11.2 Assume π = 22 7 , unless stated otherwise. 1. Find the surface area of a sphere of radius: (i) 10.5 cm (ii) 5.6 cm (iii) 14 cm Fig. 11.9 Reprint 2025-26 SURFACE AREAS AND VOLUMES 145 2. Find the surface area of a sphere of diameter: (i) 14 cm (ii) 21 cm (iii) 3.5 m 3. Find the total surface area of a hemisphere of radius 10 cm. (Use π = 3.14) 4. The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. Find the ratio of surface areas of the balloon in the two cases. 5. A hemispherical bowl made of brass has inner diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of 16 per 100 cm2. 6. Find the radius of a sphere whose surface area is 154 cm2. 7. The diameter of the moon is approximately one fourth of the diameter of the earth. Find the ratio of their surface areas. 8. A hemispherical bowl is made of steel, 0.25 cm thick. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl. 9. A right circular cylinder just encloses a sphere of radius r (see Fig. 11.10). Find (i) surface area of the sphere, (ii) curved surface area of the cylinder, (iii) ratio of the areas obtained in (i) and (ii). 11.3 Volume of a Right Circular Cone In earlier classes we have studied the volumes of cube, cuboid and cylinder In Fig 11.11, can you see that there is a right circular cylinder and a right circular cone of the same base radius and the same height? Activity : Try to make a hollow cylinder and a hollow cone like this with the same base radius and the same height (see Fig. 11.11). Then, we can try out an experiment that will help us, to see practically what the volume of a right circular cone would be! Fig. 11.12 Fig. 11.10 Fig. 11.11 Reprint 2025-26 146 MATHEMATICS So, let us start like this. Fill the cone up to the brim with sand once, and empty it into the cylinder. We find that it fills up only a part of the cylinder [see Fig. 11.12(a)]. When we fill up the cone again to the brim, and empty it into the cylinder, we see that the cylinder is still not full [see Fig. 11.12(b)]. When the cone is filled up for the third time, and emptied into the cylinder, it can be seen that the cylinder is also full to the brim [see Fig. 11.12(c)]. With this, we can safely come to the conclusion that three times the volume of a cone, makes up the volume of a cylinder, which has the same base radius and the same height as the cone, which means that the volume of the cone is one-third the volume of the cylinder. So, Volume of a Cone = 1 3 π π π π πr2h where r is the base radius and h is the height of the cone. Example 8 : The height and the slant height of a cone are 21 cm and 28 cm respectively. Find the volume of the cone. Solution : From l2 = r2 + h2, we have r = 2 2 2 2 28 21 cm 7 7 cm l h − = − = So, volume of the cone = 1 3 πr2h = 1 3 × 22 7 7 7 7 21 7 × × × cm3 = 7546 cm3 Example 9 : Monica has a piece of canvas whose area is 551 m2. She uses it to have a conical tent made, with a base radius of 7 m. Assuming that all the stitching margins and the wastage incurred while cutting, amounts to approximately 1 m2, find the volume of the tent that can be made with it. Solution : Since the area of the canvas = 551 m2 and area of the canvas lost in wastage is 1 m2, therefore the area of canvas available for making the tent is (551 – 1) m2 = 550 m2. Now, the surface area of the tent = 550 m2 and the required base radius of the conical tent = 7 m Note that a tent has only a curved surface (the floor of a tent is not covered by canvas!!). Reprint 2025-26 SURFACE AREAS AND VOLUMES 147 Therefore, curved surface area of tent = 550 m2. That is, πrl = 550 or, 22 7 × 7 × l = 550 or, l = 3 550 22 m = 25 m Now, l2 = r2 + h2 Therefore, h = 2 2 l r − = 2 2 25 7 m 625 49 m 576 m − = − = = 24 m So, the volume of the conical tent = 2 3 1 1 22 7 7 24 m 3 3 7 r h π = × × × × = 1232 m3. EXERCISE 11.3 Assume π = 22 7 , unless stated otherwise. 1. Find the volume of the right circular cone with (i) radius 6 cm, height 7 cm (ii) radius 3.5 cm, height 12 cm 2. Find the capacity in litres of a conical vessel with (i) radius 7 cm, slant height 25 cm (ii) height 12 cm, slant height 13 cm 3. The height of a cone is 15 cm. If its volume is 1570 cm3, find the radius of the base. (Use π = 3.14) 4. If the volume of a right circular cone of height 9 cm is 48 π cm3, find the diameter of its base. 5. A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres? 6. The volume of a right circular cone is 9856 cm3. If the diameter of the base is 28 cm, find (i) height of the cone (ii) slant height of the cone (iii) curved surface area of the cone 7. A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find the volume of the solid so obtained. 8. If the triangle ABC in the Question 7 above is revolved about the side 5 cm, then find the volume of the solid so obtained. Find also the ratio of the volumes of the two solids obtained in Questions 7 and 8. 9. A heap of wheat is in the form of a cone whose diameter is 10.5 m and height is 3 m. Find its volume. The heap is to be covered by canvas to protect it from rain. Find the area of the canvas required. Reprint 2025-26 148 MATHEMATICS 11.4 Volume of a Sphere Now, let us see how to go about measuring the volume of a sphere. First, take two or three spheres of different radii, and a container big enough to be able to put each of the spheres into it, one at a time. Also, take a large trough in which you can place the container. Then, fill the container up to the brim with water [see Fig. 11.13(a)]. Now, carefully place one of the spheres in the container. Some of the water from the container will over flow into the trough in which it is kept [see Fig. 11.13(b)]. Carefully pour out the water from the trough into a measuring cylinder (i.e., a graduated cylindrical jar) and measure the water over flowed [see Fig. 11.13(c)]. Suppose the radius of the immersed sphere is r (you can find the radius by measuring the diameter of the sphere). Then evaluate 4 3 πr3. Do you find this value almost equal to the measure of the volume over flowed? Fig. 11.13 Once again repeat the procedure done just now, with a different size of sphere. Find the radius R of this sphere and then calculate the value of 3 4 R . 3 π Once again this value is nearly equal to the measure of the volume of the water displaced (over flowed) by the sphere. What does this tell us? We know that the volume of the sphere is the same as the measure of the volume of the water displaced by it. By doing this experiment repeatedly with spheres of varying radii, we are getting the same result, namely, the volume of a sphere is equal to 4 3 π times the cube of its radius. This gives us the idea that Volume of a Sphere = 3 4 3 r π where r is the radius of the sphere. Later, in higher classes it can be proved also. But at this stage, we will just take it as true. Reprint 2025-26 SURFACE AREAS AND VOLUMES 149 Since a hemisphere is half of a sphere, can you guess what the volume of a hemisphere will be? Yes, it is 3 1 4 of 2 3 r π = 2 3 πr3. So, Volume of a Hemisphere = 3 2 3 r π where r is the radius of the hemisphere. Let us take some examples to illustrate the use of these formulae. Example 10 : Find the volume of a sphere of radius 11.2 cm. Solution : Required volume = 4 3 πr3 = 4 22 11.2 11.2 11.2 3 7 × × × × cm3 = 5887.32 cm3 Example 11 : A shot-putt is a metallic sphere of radius 4.9 cm. If the density of the metal is 7.8 g per cm3, find the mass of the shot-putt. Solution : Since the shot-putt is a solid sphere made of metal and its mass is equal to the product of its volume and density, we need to find the volume of the sphere. Now, volume of the sphere = 3 4 3 r π = 3 4 22 4.9 4.9 4.9 cm 3 7 × × × × = 493 cm3 (nearly) Further, mass of 1 cm3 of metal is 7.8 g. Therefore, mass of the shot-putt = 7.8 × 493 g = 3845.44 g = 3.85 kg (nearly) Example 12 : A hemispherical bowl has a radius of 3.5 cm. What would be the volume of water it would contain? Solution : The volume of water the bowl can contain = 3 2 3 r π = 2 22 3.5 3.5 3.5 3 7 × × × × cm3 = 89.8 cm3 Reprint 2025-26 150 MATHEMATICS EXERCISE 11.4 Assume π = 22 7 , unless stated otherwise. 1. Find the volume of a sphere whose radius is (i) 7 cm (ii) 0.63 m 2. Find the amount of water displaced by a solid spherical ball of diameter (i) 28 cm (ii) 0.21 m 3. The diameter of a metallic ball is 4.2 cm. What is the mass of the ball, if the density of the metal is 8.9 g per cm3? 4. The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon? 5. How many litres of milk can a hemispherical bowl of diameter 10.5 cm hold? 6. A hemispherical tank is made up of an iron sheet 1 cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank. 7. Find the volume of a sphere whose surface area is 154 cm2. 8. A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of 4989.60. If the cost of white-washing is ` 20 per square metre, find the (i) inside surface area of the dome, (ii) volume of the air inside the dome. 9. Twenty seven solid iron spheres, each of radius r and surface area S are melted to form a sphere with surface area S′. Find the (i) radius r′ of the new sphere, (ii) ratio of S and S′. 10. A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in mm3) is needed to fill this capsule? 11.5 Summary In this chapter, you have studied the following points: 1. Curved surface area of a cone = π π π π πrl 2. Total surface area of a right circular cone = π π π π πrl + π π π π πr2, i.e., π π π π πr (l + r) 3. Surface area of a sphere of radius r = 4 π π π π π r2 4. Curved surface area of a hemisphere = 2π π π π πr2 5. Total surface area of a hemisphere = 3π π π π πr2 6. Volume of a cone = 1 3 π π π π πr2h 7. Volume of a sphere of radius r = 3 4 3 r π 8. Volume of a hemisphere = 3 2 3 r π [Here, letters l, b, h, a, r, etc. have been used in their usual meaning, depending on the context.] Reprint 2025-26
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Published Time: 2020-05-23T08:20:00+00:00 Multi-Step Word Problems Explained | KS2 Maths Worksheets Maths Tutoring for Schools AI Maths Tutoring – Primary Tutoring Programmes – Secondary Tutoring Programmes – Multi-Academy Trusts How it Works Impact Use Cases Scheduled Tutoring SATs Revision GCSE Revision On-Demand Sessions Cover Homework Teaching Assistant Case Studies Resources Primary Maths Resources (Free) Primary Maths Resources (Premium) Secondary Maths Resources (Free) – GCSE Maths Revision Guides (Free) – GCSE Maths Papers (Free) – GCSE Maths Revision Resources (Free) Senior Leader Zone (Free) Pricing AI Tutoring Pricing Premium Maths Resources Subscriptions Blog Try for free Log in 🇬🇧 UK 🇺🇸 US 🇬🇧 UK 🇺🇸 US Maths Tutoring for Schools AI Maths Tutoring – Primary Tutoring Programmes – Secondary Tutoring Programmes – Multi-Academy Trusts How it Works Impact Use Cases Scheduled Tutoring SATs Revision GCSE Revision On-Demand Sessions Cover Homework Teaching Assistant Case Studies Resources Primary Maths Resources (Free) Primary Maths Resources (Premium) Secondary Maths Resources (Free) – GCSE Maths Revision Guides (Free) – GCSE Maths Papers (Free) – GCSE Maths Revision Resources (Free) Senior Leader Zone (Free) Pricing AI Tutoring Pricing Premium Maths Resources Subscriptions Blog Try for free Log in Interventions School Interventions Tutoring SLT and Maths Leads AI in Education Leadership Ofsted Maths Mastery Teaching and Learning Primary Maths Four Operations Place Value & Number Fractions, Decimals & Percentages Geometry, Position & Measurement Key Stage 2 (Other Topics) SATs Worksheets & Practice Questions Secondary Maths Key Stage 3 Key Stage 4 & GCSE Worksheets & Practice Questions More Maths Support Home Learning Dictionary & Wiki Games & Activities Third Space Updates x WATCH 1 MIN CLIP Introducing Skye, your pupils’ voice based AI maths tutor Adaptive, dialogue-driven one to one tutoring built by primary teachers Unlimited sessions for as many Year 4-6 pupils as need it One fixed low yearly cost no matter how many sessions you schedule Watch a session in action A one-to-one maths tutor for every pupil that needs it "Third Space AI tutoring has reinvigorated some of our most reluctant learners." AI maths tutoring FREE daily maths challenges A new KS2 maths challenge every day. Perfect as lesson starters - no prep required! View today's challenge Contents Maths Mastery 2-Step Word Problems and Multi-Step Word Problems in KS2 Maths: Sample Questions, Answers & Strategies for Solving March 25, 2025 | 15 min read Sophie Bartlett Teaching your pupils to solve 2-step word problems and multi-step word problems at KS2 is one of hardest parts of a mastery led approach in maths. There are several cognitive functions at play, as children have to wrestle with their knowledge of maths vocabulary, maths operations, and often basic comprehension skills. In this article we set out some of the sorts of maths word problems pupils can expect from the KS2 maths national curriculum and look at strategies for solving them.In total we’ve provided 30 KS2 word problems to work through, showing the variety of 2-step word problems and multi-step word problems pupils are likely to encounter. Table of Contents What are word problems What are two-step word problems? What are multi-step word problems? Two-step word problems and multi-step word problems pupils will encounter in KS2 Skills required for multi-step word problems Arithmetic within multi-step word problems How to teach multi-step word problems How to solve a year 6 multi-step word problem How to solve a year 5 multi-step word problem 2-step word problems Year 3 2-step word problems Year 4 Multi-step word problems: Year 5 Multi-step word problems Year 6 2-step and multi-step word problems by topic Place value word problems Addition and subtraction word problems Multiplication and division word problems Mixed operations word problems Fraction word problems Decimals word problems Percentage word problems Measurement word problems Money word problems Area word problems Perimeter word problems Ratio word problems Order of operations word problems Volume word problems Algebra word problems More support with KS2 word problems What are word problems Word problems in maths are sentences describing a real life scenario where children must apply their maths knowledge to reach a solution or unpick the maths problem. To solve maths word problems children must be familiar with the maths language associated with the mathematical symbols they are used to in order to make sense of the word problem; for example: plus, more, total = add; difference, less, minus = subtract, etc. What are two-step word problems? Two-step word problems are problems in which two separate calculations (usually different operations) are required to reach the answer. By different operations we mean addition, subtraction, multiplication or division. What are multi-step word problems? Multi-step word problems are maths problems that require multiple calculations to solve them. They will usually will involve more than one operation and often more than one strand from the curriculum. For example a multi-step word problem on area and perimeter may also involve ratio and multiplication. In KS2 SATs multi-step word problems can be awarded up to 3 marks for a correct answer, but 1 or 2 marks can be achieved by solving some of the steps in the problem correctly. Meet Skye, the voice-based AI tutor making maths success possible for every student. Built by teachers and maths experts, Skye uses the same pedagogy, curriculum and lesson structure as our traditional tutoring. But, with more flexibility and a lower cost, schools can scale online maths tutoring to support every student who needs it. Watch Skye in action Two-step word problems and multi-step word problems pupils will encounter in KS2 In Key Stage 2, there are nine ‘strands’ of maths – these are then further split into ‘sub-strands’. For example, ‘number and place value’ is the first strand: a Year 3 sub-strand of this is to “find 10 or 100 more or less than a given number”; a Year 6 sub-strand of this is to “determine the value of each digit in numbers up to 10 million”. The table below shows how the ‘sub-strands’ are distributed across each strand and year group in KS2. Strand Year 3 Year 4 Year 5 Year 6 Total Number and place value 6 9 5 7 27 Calculations 7 8 15 9 39 Fractions, decimals and percentages 7 10 12 11 40 Ratio and proportion 0 0 0 4 4 Algebra 0 0 0 5 5 Measurement 17 9 10 8 44 Geometry: properties of shape 5 4 6 7 22 Geometry: position and direction 0 3 1 2 6 Statistics 2 2 2 2 8 As well as varying in content (sometimes by using a combination of strands in one problem, e.g. shape and calculations), word problems will also vary in complexity, from one-step to multi-step problems. Different word problems will provide a different level of cognitive demand as an alternative method of adapting the level of difficulty. The STA mathematics test framework (2015) sets these out. Strand 1 2 3 4 Depth of understandingrecall of facts or application of procedures use facts and procedures to solve simple problems use facts and procedures to solve more complex problems understand and use facts and procedures creatively to solve complex or unfamiliar problems Computational complexityno numeric steps one, or a small number of numeric steps a larger number of numeric steps all steps are simple a larger number of numeric steps, at least one of which is more complex Spatial reasoningno spatial reasoning required manipulation of the geometric information is required complex manipulation of the geometric information is required interpret, infer or generate new geometric information Data interpretationno data interpretation required select and retrieve information select and interpret information generate or infer new information from data Response strategyselect one or more responses or construct a simple response construct a small set of responses construct a straightforward explanation shows evidence of a method construct a complex explanation Skills required for multi-step word problems There is a high level of cognitive demand on children when they are faced with multi-step word problems: interpreting the question to find the arithmetic behind it and then calculating the arithmetic itself. Therefore, a secure knowledge of times tables and a confident understanding of arithmetic are essential skills for being able to successfully solve word problems. Year 3 to 6 Rapid Reasoning (Weeks 1-6) Download 480 two-step and multi-step word problems for Years 3 to Year 6 (4 a day x 6 weeks for each year group) Download Free Now! Arithmetic within multi-step word problems A useful strategy to use in class is to provide children with a list of arithmetic questions you have previously ‘extracted’ from some word problems. Generally, children are much more confident with arithmetic than word problems, so they should be able to answer these with relative ease. In the next lesson, give the children the word problems – after a while, ask them which they found easier and why. Then show the children the arithmetic from the previous day and ask if they can see the similarities. They could then try to ‘extract’ the arithmetic from word problems themselves. How to teach multi-step word problems Here are two simple strategies that can be applied to most two-step word problems and multi-step word problems before solving them. What do you already know? How can this problem be drawn/represented pictorially? How to solve a year 6 multi-step word problem Here’s an example. There are 29 pupils in a class. The teacher has 7 litres of apple juice. She pours 215 millilitres of apple juice for every pupil. How much apple juice is left over? 1. What do you already know? There are 1,000ml in 1 litre Pours = liquid leaving the bottle = subtraction For every = multiply Left over = requires subtraction at some point 2. How can this problem be drawn/represented pictorially? Bar modelling is always a brilliant way of representing even multi step word problems in year 6, but there are always other ways of drawing it out. For example, for this question, you could draw 29 pupils (or stick man x 29) with ‘215 ml’ above each one and then a half-empty bottle with ‘7 litres’ marked at the top. Now to put the maths to work. This is a Year 6 multi-step problem, so we need to use what we already know and what we’ve drawn to break down the steps. 3. How to answer step by step There are 29 pupils in a class. The teacher has 7 litres of apple juice. 1) 7 litres = 7,000ml She pours 215 millilitres of apple juice for every pupil. 2) 215ml x 29 = 6,235ml How much apple juice is left over? 3) 7,000ml – 6,235ml = 765ml How to solve a year 5 multi-step word problem A similar approach can be used for this one. Mara is in a bookshop. She buys one book for £6.99 and another that costs £3.40 more than the first book. She pays using a £20 notes. What change does Mara get? 1. What do you already know? More than = add Using decimals means I will have to line up the decimal points correctly in calculations Change from money = subtract 2. How can this problem be drawn/represented pictorially? See this example of bar modelling for this question: Now to put the maths to work using what we already know and what we’ve drawn to break down the steps. 3. How to answer step by step Mara is in a bookshop. She buys one book for £6.99 and another that costs £3.40 more than the first book. 1) £6.99 + (£6.99 + £3.40) = £17.38 She pays using a £20 note. What change does Mara get? £20 – £17.38 = £2.62 There are plenty more teacher guides and resources available from Third Space for problem solving in KS2. Find out how to develop maths reasoning skills in KS2, how to balance fluency, reasoning and problem solving in your maths lessons, and get ideas for developing and running maths investigations at KS2. 2-step word problems Year 3 With word problems for Year 3, children will move away from solely using concrete resources when solving word problems and start using written methods. This is also the year in which two-step problems will be introduced. As some children may not be confident readers, it is important that word problems are explored in a variety of contexts: as a class, in groups, in partners, with an adult, with a list of ‘mathematical vocabulary’ accessible, etc. It is important that children’s literacy skills don’t hinder their progress or in maths. Two step word problems worksheet from All Kinds of Word Problems Example Year 3 word problems Dylan and Holly have different amounts of money. Dylan has fifteen 2p coins. Holly has seven 5p coins. Who has the most money, and by how much? Answer: Holly by 5p. It takes Jamie 10 minutes to read 3 pages of his book. He reads 18 pages of his book before bed. How long does Jamie spend reading? Answer: 60 minutes. 2-step word problems Year 4 With word problems year 4, children should feel confident using the written method for each of the four operations. This year children will be presented with a variety of problems, including two-step problems, and be expected to work out the appropriate method required to solve each one. While children should be focusing on formal written methods, it is important that concrete resources and pictorial representations are still used to consolidate their understanding. Two step word problems from All Kinds of Word Problems Example Year 4 word problem Lily, Simon and Rose are each thinking of a number. The sum of their numbers is 9,989. Lily’s number is 1,832. Simon’s number is three thousand more than Lily’s. What is Rose’s number? Answer: 3,325 Mia has a jug with 2.5 litres of water in it. She pours two glasses of 300ml and three glasses of 500ml. How much water is left in the jug? Give your answer in millilitres. Answer: 400ml Multi-step word problems: Year 5 Although one and two-step word problems are the mainstay of Year 5 reasoning and problem solving, word problems for year 5 are also when children may start to extend their range to include multi-step problems; In Upper Key Stage 2, word problems become more complex not only in the calculations (higher numbers, decimals etc.) but also the vocabulary – a subtlety of maths language may mean it is less obvious as to which operation is required. In the first example below, the children are essentially being asked to add and divide by 7 – or find the ‘mean’ – but the word problem doesn’t use the vocabulary children usually associate with addition or division, such as total, sum, share, split, etc. To reduce the cognitive demand of questions such as these, the numbers could be altered so that children are still required to extract the calculations from the word problems but can then complete those calculations with simpler numbers. Example Year 5 multi-step word problems A writer is working on two projects. She has one week to write 518 maths questions for one project and 476 questions for another project. If she completes the same number of questions every day, how many should she aim to complete each day? Answer: 142 Walton Wanderers’ new shirt costs £29. In the first month after it was launched, the club shop sold 1,573 shirts. 54 shirts were returned because they did not fit. How much money did the club shop receive by selling the shirts? Answer: £44,051 Multi-step word problems Year 6 With word problems for year 6, children move on from 2-step word problems to multi-step word problems. These could include fractions, decimals and percentages. Some of the most complex problems in KS2 SATS papers are worth 3 marks – these are intended to challenge more able mathematicians. As previously mentioned, one or two marks can be achieved for correctly solving different ‘steps’ of the problem even without arriving at the correct final answer. Multi-step word problems from Rapid Reasoning Example Year 6 multi-step word problem Sarah makes jewellery with beads. Bracelets have 37 beads. Necklaces have 74 beads. Sarah makes 28 bracelets and 81 necklaces. How many beads does she use altogether? Answer: 7,030 A field measures 15m by 20m. The field next to it is 300cm longer and 2.5m narrower. What is the difference in area between the two fields? Answer: 15m For more like this, please refer to this collection of35 year 6 maths reasoning questions to support teaching in the run up to SATs or if you want to focus specifically on using the bar model as a problem solving tool, try these Year 6 multi-step word problems. 2-step and multi-step word problems by topic What follows are a series of 2-step word problems and multi-step worded problems based around the national curriculum objectives for each topic in maths. These show you a full range of question and problem types and the type of skills and knowledge your pupils will need to develop. We’ve also added some links to relevant word problems worksheets. Year 3 to 6 Rapid Reasoning (Weeks 1-6) Download 480 two-step and multi-step word problems for Years 3 to Year 6 (4 a day x 6 weeks for each year group) Download Free Now! Place value word problems Place value problems appear throughout KS2. In Year 3, they will be based on five objectives: count from 0 in multiples of 4, 8, 50 and 100 (find 10 or 100 more or less than a given number) recognise the place value of each digit in a three-digit number compare and order numbers up to 1000 identify, represent and estimate numbers using different representations read and write numbers up to 1000 in numerals and in words. The progression in place value through KS2 ends in Year 6 with problems being based on three objectives: read, write, order and compare numbers up to 10 000 000 and determine the value of each digit round any whole number to a required degree of accuracy use negative numbers in context, and calculate intervals across zero. Place value multi-step word problem: Year 6 Mo uses four-digit cards and some zeros to make a seven-digit number on a place-value grid. Mo places the digit with the lowest value in the place with the highest value. He then places the 6 so that it has a value of 60,000. Finally, he places the digit with the highest value in the place with the lowest value. What could Mo’s number be? Write your answer in words. Answer: any of the following numbers: one million two hundred and sixty thousand and nine, one million sixty-two thousand and nine, one million sixty thousand two hundred and nine, one million sixty thousand and twenty-nine For free multi-step word problems worksheets download these free number and place value word problems for Years 3, 4, 5 and 6 Addition and subtraction word problems Addition and subtraction problems appear throughout KS2. In Year 3, they will be based on three objectives: add and subtract numbers mentally add and subtract numbers with up to three digits, using formal written methods of columnar addition and subtraction estimate the answer to a calculation and use inverse operations to check answers solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction. The progression in addition and subtraction through KS2 ends in Year 6 with problems being based on three objectives: perform mental calculations, including with mixed operations and large numbers solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and why use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy. Addition and subtraction multi-step word problem: Year 6 Buzzard Sky Diving Company have taken individual bookings worth £12,584 and group bookings worth £15,992. Some people have cancelled at the last minute. £1,629 has had to be returned to them. How much money has the sky diving company taken altogether? Answer: £26,947 For free multi-step and two-step word problems worksheets download these free addition and subtraction word problems for Years 3, 4, 5 and 6 and take a look at our collection of addition and subtraction word problems for Year 3- Year 6. Tips for solving addition multi-step word problems Children should be taught to recognise the vocabulary used in addition word problems to signify that the addition operation is required, for example, altogether, combined, total, sum etc. Be mindful that although more can be used for addition (e.g. What is 7 more than 9?), it can also be used for subtraction (e.g. How many more is 9 than 7?). Addition multi-step word problem: Year 6 Two different numbers add together to make an even total less than 20. Both numbers are greater than 6 and less than 12. What could the numbers be? Answer: 7 and 9, 7 and 11, 8 and 10 Tips for solving subtraction multi-step word problems Children should be taught to recognise the vocabulary used in subtraction word problems to signify that the subtraction operation is required, for example, change (money), difference, fewer than, minus etc. They should also by now know their subtraction facts. Subtraction two-step word problem: Year 5 Carlos Varqueri – United’s star striker – has asked for a pay rise. He would like another £154,875 a year, so that his wage becomes £800,000 per year. How much money does Carlos earn at the moment?Answer: £645,125 Multiplication and division word problems Multiplication and division problems appear throughout KS2. In Year 3, they will be based on three objectives: recall and use multiplication and division facts for the 3, 4 and 8 multiplication tables write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one-digit numbers, using mental and progressing to formal written methods solve problems, including missing number problems, involving multiplication and division, including positive integer scaling problems and correspondence problems in which n objects are connected to m objects. The progression in multiplication and division through KS2 ends in Year 6 with problems being based on six objectives: multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context divide numbers up to 4 digits by a two-digit number using the formal written method of short division where appropriate, interpreting remainders according to the context perform mental calculations, including with mixed operations and large numbers identify common factors, common multiples and prime numbers use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy. Multiplication and division multi-step word problem: Year 6 (crossover with decimals) Lottie is playing a computer game. She has scored 67 points so far. She defeats a giant, which means that her score becomes ten times greater. However, she then gets lots in a maze, which means that her score becomes 1,000 times smaller. What is Lottie’s new score? Answer: 0.67 Tips for solving multiplication multi-step word problems Children should be taught to recognise the vocabulary used in word problems to signify that the multiplication operation is required, for example, product, double, triple, groups etc. Be mindful that groups can be used in both multiplication and division problems, e.g. ‘What are 7 groups of 5?’ (multiplication) or ‘How many groups of 4 fit into 28?’ (division). Multiplication multi-step word problem: Year 6 There are 32 levels in a computer game. The maximum number of points that can be achieved for each level is 1,450. Hauwa completes the game and scores maximum points. How many points does Hauwa score altogether? Answer: 46,400 For free multi-step and two-step word problems worksheets download free multiplication word problems worksheets for Years 3, 4, 5 and 6 Children can practice can practice multiplication word problems in Third Space Learning’s online tuition programmes. Multiplication word problem from a Third Space Learning SATs revision lesson Tips for solving division multi-step word problems Children should be taught to recognise the vocabulary used in division word problems to signify that the division operation is required, for example, halve, share, groups, split etc. Division two-step word problem: Year 4 A group of friends earn £120 by mowing lawns. They share the money equally. They get £15 each. How many friends are there in the group? Answer: 8 For free multi-step and two-step word problems worksheets download this free division word problems worksheet for Years 3, 4, 5 and 6 Mixed operations word problems In the Year 3 non-statutory notes and guidance of the National Curriculum, it is recommended that pupils practise solving varied addition and subtraction questions and simple multiplication and division problems in contexts, deciding which of the four operations to use and why. These include measuring and scaling contexts, and correspondence problems in which m objects are connected to n objects. At the end of KS2, the guidance states that pupils could practise addition, subtraction, multiplication and division for larger numbers, using the formal written methods of columnar addition and subtraction, short and long multiplication, and short and long division. Mixed operations two-step word problem: Year 6(crossover with money) A customer visits Dave’s DIY and buys 18 packs of screws, 18 packs of washers and a screwdriver. How much change is given from £20? Answer: £1.67 Four operations multi-step word problem: Year 5 Oakthorpe Academy have been given a donation of £5,460 by the PTA. The School Council decide to use £1,755 on buying some new computer equipment. The rest is split equally between five year groups so they can decide for themselves how to spend the money. How much money will each year group have? Answer: £741 Read more 33 mental maths strategies you should be teaching in KS1 and KS2 What are number facts? Activities to support memorisation. How primary schools develop fluency in maths Fraction word problems Fraction w ord problems require a good understanding of division and multiplication. Bar models or other pictorial representations are useful strategies in helping children solve problems like these. Fraction problems appear throughout KS2. In Year 3, they will be based on seven objectives: count up and down in tenths recognise that tenths arise from dividing an object into 10 equal parts and in dividing one-digit numbers or quantities by 10 recognise, find and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators recognise and use fractions as numbers: unit fractions and non-unit fractions with small denominators recognise and show, using diagrams, equivalent fractions with small denominators add and subtract fractions with the same denominator within one whole compare and order unit fractions, and fractions with the same denominators. The progression in fractions through KS2 ends in Year 6 with problems being based on seven objectives: use common factors to simplify fractions and use common multiples to express fractions in the same denomination compare and order fractions, including fractions > 1 add and subtract fractions with different denominators and mixed numbers, using the concept of equivalent fractions multiply simple pairs of proper fractions, writing the answer in its simplest form divide proper fractions by whole numbers associate a fraction with division and calculate decimal fraction equivalents recall and use equivalences between simple fractions, decimals and percentages, including in different contexts. Fraction multi-step word problem: Year 6 A bakery has 2 and 1/3 rhubarb pies left for sale. A customer buys 3/12 of a pie. What fraction of the pies is left? Write your answer as a mixed number. Answer: 2 and 1/12 For free multi-step and two-step word problems worksheets download fractions and decimals word problems for Years 3, 4, 5 and 6 Decimals word problems Decimal word problems are commonly used in questions involving money, although they also often appear alongside fractions and/or percentages, requiring children to calculate their equivalences. Decimal problems begin in Year 4 and will be based on six objectives: recognise and write decimal equivalents of any number of tenths or hundredths recognise and write decimal equivalents to ¼, ½, ¾ find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the digits in the answer as ones, tenths and hundredths round decimals with one decimal place to the nearest whole number compare numbers with the same number of decimal places up to two decimal places; solve simple measure and money problems involving fractions and decimals to two decimal places. The progression in decimals from Year 4 ends in Year 6 with problems being based on six objectives associate a fraction with division and calculate decimal fraction equivalents for a simple fraction identify the value of each digit in numbers given to three decimal places and multiply and divide numbers by 10, 100 and 1000 giving answers up to three decimal places multiply one-digit numbers with up to two decimal places by whole numbers use written division methods in cases where the answer has up to two decimal places; solve problems which require answers to be rounded to specified degrees of accuracy recall and use equivalences between simple fractions, decimals and percentages, including in different contexts. Decimals two-step word problem: Year 6(crossover with multiplication) A teaspoon contains 5.26ml of cough medicine. Amber takes 5 teaspoons full every day. How many millilitres of cough medicine does Amber take each day? Answer: 26.3ml Percentage word problems Children need a secure understanding in fractions before attempting percentage problems; they are therefore not introduced until Upper Key Stage 2. Percentage word problems begin in Year 5 and will be based on two objectives: recognise the per cent symbol (%) and understand that per cent relates to ‘number of parts per hundred’, and write percentages as a fraction with denominator 100, and as a decimal solve problems which require knowing percentage and decimal equivalents of ½, ¼, 1/5, 2/5, 4/5 and those fractions with a denominator of a multiple of 10 or 25. The progression in percentage continues into Year 6 with problems being based on two objectives: recall and use equivalences between simple fractions, decimals and percentages, including in different contexts solve problems involving the calculation of percentages and the use of percentages for comparison. Percentage multi-step word problem: Year 6 Rachel and Max are playing a computer game. The game has 160 levels. Rachel’s counter says she has completed 60% of the game. Max’s counter says that he has completed 45% of the game. What levels are they each on? Answer: Rachel = Level 96,Max = Level 72 For free multi-step and two-step word problems worksheets download decimals and percentages word problems for Years 3, 4, 5 and 6 Measurement word problems Many measurement word problems require children to convert between metric measures, thus children should be confident in multiplying and dividing by powers of 10. More complex measurement word problems (such as those involving imperial measures) may require children to have an understanding of ratio and proportion. Word problems involving measures begin in Year 3 and will be based on six objectives: measure, compare, add and subtract: lengths (m/cm/mm); mass (kg/g), volume/capacity (l/ml) time word problems where children tell and write the time from an analogue clock, including using Roman numerals from I to XII, and 12-hour and 24-hour clocks estimate and read time with increasing accuracy to the nearest minute record and compare time in terms of seconds, minutes and hours use vocabulary such as o’clock, a.m./p.m., morning, afternoon, noon and midnight know the number of seconds in a minute and the number of days in each month, year and leap year; compare durations of events The progression continues into Year 6 with problems being based on three objectives: solve problems involving the calculation and conversion of units of measure, using decimal notation up to three decimal places where appropriate use, read, write and convert between standard units, converting measurements of length, mass, volume and time from a smaller unit of measure to a larger unit, and vice versa, using decimal notation to up to three decimal places convert between miles and kilometres. Measurement multi-step word problem: Year 6 (crossover with ratio) 5 miles are approximately equivalent to 8 km. Mr Norton’s car speedometer shows that he is travelling at 104 km/h. About how many miles per hour (mph) is the car travelling? Answer: 65 mph Money word problems Problems involving money link with decimals (money notation) and measures (converting between £ and p). Where possible, especially until their understanding is secure, children should be handling real money to help them solve problems. Money word problems begin in Year 3 and will be based on one objective: add and subtract amounts of money to give change, using both £ and p in practical contexts. The non-statutory guidance in the curriculum recommends that pupils continue to become fluent in recognising the value of coins, by adding and subtracting amounts, including mixed units, and giving change using manageable amounts. The decimal recording of money is introduced formally in year 4, where word problems will be based on one objective: solve simple measure and money problems involving fractions and decimals to two decimal places. Money problems continue throughout KS2 but are not specifically mentioned in the National Curriculum beyond Year 4. Two-step money word problem: Year 6(crossover with mixed operations) These are the different prices of tickets at a cinema. Jamaal’s dad buys two adults’ tickets and four children’s tickets. How much money do the tickets cost altogether?Answer: £32.20 Area word problems Mathematical questions related to area require a secure understanding of arrays, times tables, multiplication, division and factors. Concrete resources such as Numicon and multilink can be used to support children to solve these problems. Word problems involving area begin in Year 4 and will be based on one objective: find the area of rectilinear shapes by counting squares. The progression continues into Year 6 with problems being based on three objectives: recognise that shapes with the same areas can have different perimeters and vice versa recognise when it is possible to use formulae for area and volume of shapes calculate the area of parallelograms and triangles Area multi-step word problem: Year6 A square has a side length of 6cm. A triangle has a base of 8cm and a perpendicular height of 7cm. What is the difference in their areas? Answer: 8cm2 Perimeter word problems As well as being an important life skill, it is important for children to be able to measure accurately with a ruler for some aspects of this mathematical strand. As above, Numicon and multilink are extremely useful resources in supporting children in their calculation of perimeter word problems. These problems begin in Year 3 and will be based on one objective: measure the perimeter of simple 2-D shapes. The progression continues into Year 6 with problems being based on one objective: recognise that shapes with the same areas can have different perimeters and vice versa. Perimeter word problem: Year 6(crossover with decimals and multiplication) Josh has drawn a square. Each side is 7.5cm. What is the perimeter of the square?Answer: 30cm Ratio word problems In my experience, ratio is most successful when taught with concrete resources such as multilink, Cuisenaire rods or beads. Once children are taught how to represent ratio word problems using equipment (and eventually transferring to a pictorial representation, such as a bar model), the process is a lot easier. Children won’t encounter ratio word problems until Year 6, where they will be based on three objectives: solve problems involving the relative sizes of two quantities where missing values can be found by using integer multiplication and division facts solve problems involving similar shapes where the scale factor is known or can be found solve problems involving unequal sharing and grouping using knowledge of fractions and multiples Ratio word problem: Year 6 (crossover with measurement) A local council has spent the day painting double yellow lines. They use 1 pot of yellow paint for every 100m of road they paint. How many pots of paint will they need to paint a 2km stretch of road? Answer: 20 pots Order of operations word problems Children won’t encounter word problems about the order of operations until Year 6, where they will be based on one objective: use knowledge of the order of operations to carry out calculations involving the four operations. The non-statutory guidance in the National Curriculum also recommends that children explore the order of operations using brackets (otherwise known as BODMAS or BIDMAS); for example, 2 + 1 x 3 = 5 and (2 + 1) x 3 = 9. Bodmas word problem: Year 6 Draw a pair of brackets in one of these calculations so that they make two different answers. What are the answers? 50 – 10× 5 = 50 – 10× 5 = Volume word problems Children won’t encounter volume-related word problems until Year 6, where they will be based on two objectives: recognise when it is possible to use formulae for area and volume of shapes calculate, estimate and compare volume of cubes and cuboids using standard units, including cubic centimetres and cubic metres, and extending to other units Volume word problem: Year 6 This large cuboid has been made by stacking shipping containers on a boat. Each individual shipping container has a length of 6m, a width of 4m and a height of 3m. What is the volume of the large cuboid? Answer: 864m3 Algebra word problems Again algebra word problems only really come up in Year 6; the objectives they will be based on are: use simple formulae generate and describe linear number sequences express missing number problems algebraically find pairs of numbers that satisfy an equation with 2 unknowns enumerate possibilities of combinations of 2 variables Algebra word problem: Year 6 This question is from the 2018 KS2 SATs paper. It is worth 2 marks as there are 2 parts to the answer. Amina is making designs with two different shapes. She gives each shape a value. Calculate the value of each shape. Answer: 36 (hexagon) and 25. More support with KS2 word problems For more multi-step and 2-step word problems register for free on theThird Space Learning Maths Hub which includes lots more printable word problems worksheets including as part of the free resources on offer a complete set of place value word problems with answer sheets for Year 3 to Year 6: All Kinds of Word Problems on Number and Place Value If you have any pupils who are struggling to master word problems they may need a more intensive personalised intervention. Third Space Learning’s tuition follows a rigorous step by step process to teaching problem solving with excellent success – pupils make on average double their expected progress with us. DO YOU HAVE STUDENTS WHO NEED MORE SUPPORT IN MATHS? Skye – our AI maths tutor built by teachers – gives students personalised one-to-one lessons that address learning gaps and build confidence. 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https://www.doubtnut.com/qna/1338942
My Profile Maths Physics Chemistry Biology English Maths Physics Chemistry Biology English Maths Physics Chemistry Biology English Maths Physics Chemistry Biology English Reasoning Maths Physics Chemistry Biology English Maths Physics Chemistry Biology English Class 11 Class 12 Class 12+ Class 11 Class 12 Class 12+ Class 9 Class 10 Class 11 Class 12 Class 9 Class 10 Class 11 Class 12 Class 9 Class 10 Class 11 Class 12 Class 9 Class 10 Class 11 Class 12 Class 9 Class 10 Class 11 Class 12 Class 9 Class 10 Class 11 Class 12 Class 9 Class 10 Class 11 Class 12 Class 9 Class 10 Class 11 Class 12 Class 9 Class 10 Class 11 Class 12 Class 9 Class 10 Class 11 Class 12 IIT-JEE NEET CBSE UP Board Bihar Board JEE NEET Class 6-10 Class 12 Class 11 Class 10 Class 9 Class 8 Class 7 Class 6 Online Class Blog Select Theme Prove sin2θ+cos2θ=1 Video Solution Assess your true academic potential with TALLENTEX: Ranks, scholarships & more Text Solution Generated By DoubtnutGPT To prove the identity sin2θ+cos2θ=1, we will use the properties of a right-angled triangle and the Pythagorean theorem. 1. Consider a Right-Angled Triangle: - Let triangle ABC be a right-angled triangle where angle C is θ, and angle A is 90∘. - In this triangle: - Side AB is opposite to angle θ (let's denote it as a). - Side BC is adjacent to angle θ (let's denote it as b). - Side AC is the hypotenuse (let's denote it as c). 2. Apply the Pythagorean Theorem: - According to the Pythagorean theorem, we have: a2+b2=c2 3. Express the Sine and Cosine Ratios: - The definitions of sine and cosine in terms of the sides of the triangle are: - sinθ=OppositeHypotenuse=ac - cosθ=AdjacentHypotenuse=bc 4. Square the Sine and Cosine Ratios: - Squaring both sine and cosine gives: sin2θ=(ac)2=a2c2 cos2θ=(bc)2=b2c2 5. Add the Squared Ratios: - Now, add sin2θ and cos2θ: sin2θ+cos2θ=a2c2+b2c2 - This can be combined into a single fraction: sin2θ+cos2θ=a2+b2c2 6. Substitute from the Pythagorean Theorem: - From the Pythagorean theorem, we know that a2+b2=c2. Substituting this into our equation gives: sin2θ+cos2θ=c2c2=1 7. Conclusion: - Therefore, we have proved that: sin2θ+cos2θ=1 | Class 10MATHSTRIGONOMETRIC RATIOS AND IDENTITIES Similar Questions Prove:- sin2θcosθ+cosθ+=1cosθ View Solution prove that sin2θ+cos2θ=1 View Solution Prove the identity sin2θ+cos2θ=1 with the help of given figure . View Solution Prove that sinθ−cosθ+1sinθ+cosθ−1=1(secθ−tanθ). View Solution prove that- sin2θcos2θ+cos2θsin2θ=1sin2θcos2θ−2 View Solution Prove: (1+sinθ−cosθ1+sinθ+cosθ)2=1−cosθ1+cosθ View Solution Prove that: sin6θ+cos6θ=1−3sin2θcos2θ View Solution Prove each of the following identities : sinθ+cosθsinθ−cosθ+sinθ−cosθsinθ+cosθ=2(sin2θ−cos2θ)=2(2sin2θ−1) View Solution Prove that (1−sinθ+cosθ)2=2(1+cosθ)(1−sinθ) View Solution Given that sinθ+2cosθ=1then prove that 2sinθ−cosθ=2 View Solution Recommended Questions Prove sin^2 theta + cos^2 theta=1 02:01 2. Prove the following 1.((1-cos^(2)theta)/(sin^(2)theta))=12.1-((cos^(2)... 04:47 3. Prove that (1-sintheta+costheta)^2=2(1+costheta)(1-sintheta) 03:58 4. Prove that (1)/(2){(sin theta)/(1+cos theta)+(1+cos theta)/(sin theta)... 02:15 5. सिद्ध कीजिए (sin ^(2)theta)/(cos ^(2)theta)+(cos ^(2)theta)/(sin ^(2)t... 02:15 6. सिद्ध करे की (1+sin theta + cos theta)^(2)=2(1+ sin theta)(1+cos theta... 02:48 7. सिद्ध करो कि: (1-sin theta+ cos theta)^(2) =2(1+cos theta)(1-sin the... 8. सिद्ध करो कि : (sin theta -cos theta)/(sin theta+cos theta)+(sin the... 9. निम्नलिखित सर्वसमिकाओं को सिद्ध करें : (sin theta )/(1 + cos theta ...
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https://www.mathopenref.com/constaltitude.html
How to construct (draw) one of the three altitudes of a triangle - Math Open Reference Do Not Process My Personal Information If you wish to opt-out of the sale, sharing to third parties, or processing of your personal or sensitive information for targeted advertising by us, please use the below opt-out section to confirm your selection. Please note that after your opt-out request is processed you may continue seeing interest-based ads based on personal information utilized by us or personal information disclosed to third parties prior to your opt-out. You may separately opt-out of the further disclosure of your personal information by third parties on the IAB’s list of downstream participants. This information may also be disclosed by us to third parties on the IAB’s List of Downstream Participants that may further disclose it to other third parties. Personal Data Processing Opt Outs CONFIRM Data DeletionData AccessPrivacy Policy Math Open Reference Home Contact About Subject Index Altitude of a triangle Options |< >| RESET RUN < BACK NEXT > Options Click on NEXT or RUN to begin Auto repeat This page shows how to construct one of the three possible altitudes of a triangle, using only a compass and straightedge or ruler. The other two can be constructed in the same way. An altitude of a triangle is a line which passes through a vertex of a triangle, and meets the opposite side at right angles. For more on this see Altitude of a Triangle. The three altitudes of a triangle all intersect at the orthocenter of the triangle. See Constructing the orthocenter of a triangle. Method The construction starts by extending the chosen side of the triangle in both directions. This is done because the side may not be long enough for later steps to work. After that, we draw the perpendicular from the opposite vertex to the line. This is identical to the construction A perpendicular to a line through an external point. Here the 'line' is one side of the triangle, and the 'external point' is the opposite vertex. It can be outside the triangle In most cases the altitude of the triangle is inside the triangle, like this: Angles B, C are both acute However, if one of the angles opposite the chosen vertex is obtuse, then it will lie outside the triangle, as below. The angle ACB is opposite the chosen vertex A, and is obtuse (greater than 90°). Angle C is obtuse The altitude meets the extended base BC of the triangle at right angles. This case is demonstrated on the companion page Altitude of an triangle (outside case), and is the reason the first step of the construction is to extend the base line, just in case this happens. Printable step-by-step instructions The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available. Proof The proof of this construction is trivial. This is the same drawing as the last step in the above animation. Argument Reason 1 The segment SR is perpendicular to PQ Created using the procedure in Perpendicular to a line through an external point. See that page for proof. 2 The segment SR is an altitude of the triangle PQR. From (1) and the definition of an altitude of a triangle (a segment from the a vertex to the opposite side and perpendicular to that opposite side). - Q.E.D Try it yourself Click here for a printable construction worksheet containing two 'altitude of a triangle' problems to try. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright. Acknowledgements Thanks to Aaron Strand of Carmel High School, Indiana for suggesting, reviewing, and proofreading this construction Other constructions pages on this site List of printable constructions worksheets Lines Introduction to constructions Copy a line segment Sum of n line segments Difference of two line segments Perpendicular bisector of a line segment Perpendicular at a point on a line Perpendicular from a line through a point Perpendicular from endpoint of a ray Divide a segment into n equal parts Parallel line through a point (angle copy) Parallel line through a point (rhombus) Parallel line through a point (translation) Angles Bisecting an angle Copy an angle Construct a 30° angle Construct a 45° angle Construct a 60° angle Construct a 90° angle (right angle) Sum of n angles Difference of two angles Supplementary angle Complementary angle Constructing 75° 105° 120° 135° 150° angles and more Triangles Copy a triangle Isosceles triangle, given base and side Isosceles triangle, given base and altitude Isosceles triangle, given leg and apex angle Equilateral triangle 30-60-90 triangle, given the hypotenuse Triangle, given 3 sides (sss) Triangle, given one side and adjacent angles (asa) Triangle, given two angles and non-included side (aas) Triangle, given two sides and included angle (sas) Triangle medians Triangle midsegment Triangle altitude Triangle altitude (outside case) Right triangles Right Triangle, given one leg and hypotenuse (HL) Right Triangle, given both legs (LL) Right Triangle, given hypotenuse and one angle (HA) Right Triangle, given one leg and one angle (LA) Triangle Centers Triangle incenter Triangle circumcenter Triangle orthocenter Triangle centroid Circles, Arcs and Ellipses Finding the center of a circle Circle given 3 points Tangent at a point on the circle Tangents through an external point Tangents to two circles (external) Tangents to two circles (internal) Incircle of a triangle Focus points of a given ellipse Circumcircle of a triangle Polygons Square given one side Square inscribed in a circle Hexagon given one side Hexagon inscribed in a given circle Pentagon inscribed in a given circle Non-Euclidean constructions Construct an ellipse with string and pins Find the center of a circle with any right-angled object (C) 2011 Copyright Math Open Reference. 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https://www.engineersedge.com/analysis/factor-of-safety-review.htm
| | | | --- | | | | | | Membership Services | Factor of Safety FOS Review Engineering Analysis and Design Menu Factor of Safety (FOS) for structural applications is the ratio of the allowable working unit stress, allowable stress or working stress. The term was originated for determining allowable stress. The ultimate strength of a given material divided by an arbitrary factor of safety, dependant on material and the use to which it is to be put, gives the allowable stress. Where: Sm = Allowable working unit stress sw = Working stress (Allowable stress) fs = Factor of Safety In present design and engineering practice, it is customary to use allowable stress as specified by recognized industry standards or authorities as applicable rather than to use an arbitrary factor of safety. One reason for this is that the factor of safety is misleading, in that it implies a greater degree of safety than may actually exists. For example, a factor of safety multiple of 4 does not mean that a component or assembly application can carry a load four times as great as that for which it was designed. It should also be clearly understood that, even though each part of a machine may be designed with the same factor of safety, the machine as a whole does not have that factor of safety. In the event that one part is stressed beyond the proportional limit, or particularly the yield point, the load or stress distribution may be completely changed throughout the entire machine or structure, and its ability to function at rated load may be changed, even though no part has failed or ruptured. The following should be considered when designing and analyzing a structural or machine component or assembly: Effects of associated assemblies or components. Thermal cycling and operating or extreme temperature effects. Intensity of stress concentrations. Effects of wear Likely ness of structural or machine abuse. Controls may not be in place to prevent an overstress condition. Classic example is the general public exceeding automotive (truck) rated towing capacity. Such events particularly when repeated, tend accelerate wear and failure of critical mechanical drive train components and assemblies. Quality or likeness of scheduled maintenance. Dimensional control and effects on quality of assembly. Excessive internal stresses can never be properly estimated. Material stresses induced by non-perfect geometry are difficult to model. Stresses can also be introduced by assembly misalignment between components. Influence of fatigue loading over the life cycle of the machine or structure. Although no definite or universal rules can be given , if a factor of safety needs to be determined or established, the following circumstances should be taken into account in its selection: When the ultimate strength of the material is known within narrow limits, as for structural steel for which tests of samples have been made, when the load is entirely a steady one of a known value a factor of safety should be adopted is 3. When circumstances of (1) are modified by a portion of the load being variable, as in, gear boxes, floors or warehouse operations, the factor of safety should not be less than 4. When the whole load , or nearly the whole, is likely to be alternately put on and taken off, as in suspension rods as used with suspension floors or bridges, the factor should be 5 or 6. When the stresses are reversed in direction from tension to compression, as in some structural load bearing diagonals and parts of machines, the factor should be not less than 6. When the components are subjected to repeated shock loading the factor should not be less than 10. When the structure or component is subjected to deterioration from corrosion the components or structure factor of safety should be sufficiently increased to allow for a definite amount of material reduction before the system is weakened by the process. When the strength of the material or the amount of the load or both are uncertain the factor of safety should be increased by an allowance sufficient to cover the amount of the uncertainty. When the stress and strains are complex and of uncertain amount, such as those in the crankshaft of a reversing engine, a very high factor is necessary, possibly even as high as 40 or more. If property loss caused by failure of the part or system may be large or if loss of life may result, the factor of safety should be large and the structure or machine performance should be verified by functional static or fatigue testing. Industry Standard where available The factor of safety is often specified in a design code or standard, such as: American Institute of Steel Construction (AISC) steel buildings & bridges American Society of Mechanical Engineers (ASME) pressure vessels, boilers, shafts. American Concrete Institute (ACI) National Forest Products Associ ation (NFPA) wood structures. Aluminum Association (AA) aluminum buildings & bridges. Codes often specify a minimum factor of safety Designer's responsibility to determine if a code or standard applies. Codes are often specified by law. (BOCA, UBC, etc.) Factors which affect the factor of safety Material strength basis: brittle Materials use ultimate strength ductile Materials use yield strength Manner or loading: Static applied slowly; remains applied or is infrequently removed. Repeated fatigue failure may occur at stresses lower than static load failure. Impact high initial stresses develop. Possible misuse designer must cons ider any reasonable foreseeable use & misuse of product. Complexity of stress analysis the actu al stress in a part isn't always known. Environment temperature, weat her, radiation, chemical, etc.. 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https://artofproblemsolving.com/wiki/index.php/2018_AIME_II_Problems/Problem_10?srsltid=AfmBOoq6UBjx6fOfRXdlH6P9jK7tsCDrr9hAv9Fc7sEHVJBxWogqYt0t
Art of Problem Solving 2018 AIME II Problems/Problem 10 - 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. 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If we list all possible , from to in a specific order, we get different 's. Namely: To list them in this specific order makes it a lot easier to solve this problem. We notice, In order to solve this problem at least one pair of where must exist.In this case I rather "go backwards". First fixing pairs , (the diagonal of our table) and map them to the other fitting pairs . You can do this in way. Then fixing pairs (The diagonal minus ) and map them to the other fitting pairs . You can do this in ways. Then fixing pairs (The diagonal minus ) and map them to the other fitting pairs . You can do this in ways. Fixing pairs (the diagonal minus ) and map them to the other fitting pairs . You can do this in ways. Lastly, fixing pair (the diagonal minus ) and map them to the other fitting pairs . You can do this in ways. So Solution 2 We perform casework on the number of fixed points (the number of points where ). There must be at least one fixed point, given has some value and is a fixed point. To better visualize this, use the grid from Solution 1. Case 1: 5 fixed points Obviously, there must be way to do so. Case 2: 4 fixed points ways to choose the fixed points. For the sake of conversation, let them be .- The last point must be or .- There are solutions for this case. Case 3: 3 fixed points ways to choose the fixed points. For the sake of conversation, let them be .- Subcase 3.1: None of or map to each other - The points must be and , giving solutions.- Subcase 3.2: maps to or maps to - The points must be and or and , giving solutions.- There are solutions for this case. Case 4: 2 fixed points ways to choose the fixed points. For the sake of conversation, let them be .- Subcase 4.1: None of , , or map to each other - There are solutions for this case, by similar logic to Subcase 3.1.- Subcase 4.2: exactly one of maps to another. - Example: - ways to choose the 2 which map to each other, and ways to choose which ones of and they map to, giving solutions for this case.- Subcase 4.3: two of map to the third - Example: - ways to choose which point is being mapped to, and ways to choose which one of and it maps to, giving solutions for this case.- There are solutions. Case 5: 1 fixed point ways to choose the fixed point. For the sake of conversation, let it be .- Subcase 5.1: None of map to each other - Obviously, there is solution to this; all of them map to .- Subcase 5.2: One maps to another, and the other two map to . - Example: - There are ways to choose which two map to each other, and since each must map to , this gives .- Subcase 5.3: One maps to another, and of the other two, one maps to the other as well. - Example: - There are ways to choose which ones map to another. This also gives .- Subcase 5.4: 2 map to a third, and the fourth maps to . - Example: - There are ways again.- Subcase 5.5: 3 map to the fourth. - Example: - There are ways to choose which one is being mapped to, giving solutions.- There are solutions. Therefore, the answer is ~First Solution 3 We can do some caseworks about the special points of functions for . Let , and be three different elements in set . There must be elements such like in set satisfies , and we call the points such like on functions are "Good Points" (Actually its academic name is "fixed-points"). The only thing we need to consider is the "steps" to get "Good Points". Notice that the "steps" must less than because the highest iterations of function is . Now we can classify cases of “Good points” of . One "step" to "Good Points": Assume that , then we get , and , so . Two "steps" to "Good Points": Assume that and , then we get , and , so . Three "steps" to "Good Points": Assume that , and , then we get , and , so . Divide set into three parts which satisfy these three cases, respectively. Let the first part has elements, the second part has elements and the third part has elements, it is easy to see that . First, there are ways to select for Case 1. Second, we have ways to select for Case 2. After that we map all elements that satisfy Case 2 to Case 1, and the total number of ways of this operation is . Finally, we map all the elements that satisfy Case 3 to Case 2, and the total number of ways of this operation is . As a result, the number of such functions can be represented in an algebraic expression contains , and : Now it's time to consider about the different values of , and and the total number of functions satisfy these values of , and : For , and , the number of s is For , and , the number of s is For , and , the number of s is For , and , the number of s is For , and , the number of s is For , and , the number of s is For , and , the number of s is For , and , the number of s is For , and , the number of s is For , and , the number of s is For , and , the number of s is Finally, we get the total number of function , the number is ~Solution by (Frank FYC) Note (fun fact) This exact problem showed up earlier on the 2011 Stanford Math Tournament, Advanced Topics Test. This problem also showed up on the 2010 Mock AIME 2 here: See Also 2018 AIME II (Problems • Answer Key • Resources) Preceded by Problem 9Followed by Problem 11 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 " Category: Intermediate Combinatorics Problems 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.
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https://www.math.uwaterloo.ca/~beforres/PMath451/Course_Notes/Chapter4.pdf
Chapter 4 Signed Measures Up until now our measures have always assumed values that were greater than or equal to 0. In this chapter we will extend our definition to allow for both positive and negative values. 4.1 Basic Properties of Signed Measures Definition 4.1.1. Let (X, A) be a measurable space. A signed measure on (X, A) is a function µ : A →R∗ such that 1) µ takes on at most one of the values −∞or ∞. 2) µ(∅) = 0 3) If {En}∞ n=1 ⊆A is a sequence of pairwise disjoint sets, then µ( ∞ [ n=1 En) = ∞ X n=1 µ(En). Example 4.1.2. 1) Assume that (X, A) is a measurable space and that µ1, µ2 are measures on (X, A). If at least one of the µi’s is finite, then µ = µ1 −µ2 is a signed measure on (X, A). 2) Let (X, A, µ) be a measure space and let f be an integrable function. Then if f = f + −f −, we have that λ(E) = Z E f dµ = Z E f + dµ − Z E f −dµ is a signed measure on (X, A). 4.2 Jordan and Hahn Decompositions Problem 4.2.1. We have just seen that if (X, A) is a measurable space and µ1, µ2 are measures on (X, A) with at least one of the µi’s being finite, then µ = µ1 −µ2 is a signed measure on (X, A). We can now ask: do all signed measures arise in this way? 61 The clue to answering the problem above comes from the second item in our previous example where we decomposed our integral function f into its positive and negative parts. This decomposition, often called the Jordan decomposition of f, can in fact be extended to signed measures, something which we shall see shortly. We begin with the following natural definitions: Definition 4.2.2. Let µ be a signed measure on (X, A). Let P, N, M ∈A. Then we say that: 1) P is positive if µ(E ∩P) ≥0 for all E ∈A. 2) N is negative if µ(E ∩N) ≤0 for all E ∈A. 3) M is null if µ(E ∩M) = 0 for all E ∈A. The following useful results follow almost immediately from the definitions above. As such the proofs will be left as exercises. Proposition 4.2.3. Let µ be a signed measure on (X, A). 1) If P ∈A is positive (negative) [null] and if E ∈A is such that E ⊆P, then E is positive (negative) [null]. 2) Let {En}∞ n=1 ⊆A, with each En being positive (negative) [null], then E = ∞ S n=1 En is also positive (negative) [null]. 3) Let P be positive and N be negative, then P ∩N is null. Lemma 4.2.4. Let µ be a signed measure on (X, A). Let E ∈A be such that 0 < µ(E) < ∞. Then there exists a positive set A ∈A with A ⊆E and µ(A) > 0. Proof. If E is positive we are done. So we may assume without loss of generality that E is not positive. Let n1 ∈N be the smallest natural number such that there exists an E1 ⊆E with µ(E1) < −1 n1 . If E \ E1 is positive we are done. If not then we choose the smallest natural number n2 ∈N such that there exists an E2 ⊆E \ E1 with µ(E2) < −1 n2 . Suppose that we have chosen {E1, E2, . . . , Ek−1} and natural numbers {n1, n2, . . . , nk} as above and that E \ k−1 S j=1 Ej is not positive, then we choose the smallest natural number nk ∈N such that there exists an Ek ⊆E \ k−1 S j=1 Ej with µ(Ek) < −1 nk . Now either this process terminates, in which case we are done, or we are able to inductively construct a seqeunce {Ek} as above. In the latter case, let A = E \ ∞ [ k=1 Ek. 62 Since {Ek} is pairwise disjoint, and each Ek is disjoint from A, we have that µ(E) = µ(A) + ∞ X k=1 µ(Ek) (∗) From (∗) we may conclude the following: 1) Since µ(E) is finite, the series ∞ P k=1 µ(Ek) converges. 2) Since the series ∞ P k=1 µ(Ek) converges, so does the series ∞ P k=1 1 nk , and hence nk →∞. 3) Since each µ(Ek) < 0, we have 0 < µ(E) < µ(A). Assume that there exists an E0 ⊆A with µ(E0) < 0. Then we can find a k large enough so that µ(E0) < − 1 nk −1 But we have that E0 ⊆A ⊆E \ k−1 [ j=1 Ej so this would contradict how we went about choosing nk. It follows that A is positive. Theorem 4.2.5. [Hahn Decomposition Theorem] Let µ be a signed measure on (X, A). The there is a positive set A ∈A and a negative set B ∈A so that X = A ∪B and A ∩B = ∅. Proof. Since µ cannot take on both ±∞we may assume without loss of generality that µ never takes on the value ∞. Let α = sup{µ(E)|E ∈A, E is positive}. Since ∅is positive, we have that α ≥0. We can also find a sequence of positve sets {Ai} so that α = lim i→∞µ(Ai). Next let A = ∞ [ i=1 Ai Then A is also positive and since Ai ⊆A for each i we have µ(A) = α. Let B = X \ A. Assume that B is not negative. Then there is an E ⊆B with µ(E) > 0. From the previous lemma we can find a positive set A0 ⊆B with µ(A0) > 0. But then if A∗= A ∪A0 we have that A∗is also positive and since A ∩A0 = ∅, µ(A∗) = µ(A) + µ(A0) > α which is impossible. 63 Definition 4.2.6. Let µ be a signed measure on (X, A). A pair {P, N} of elements in A for which P is positive N is negative, P ∪N = X and P ∩N = ∅is called a Hahn decomposition of X with respect to µ. Remark 4.2.7. It is generally the case that the Hahn decomposition is not unique. In fact, let X = [0, 1] and let A = P(X). If µ 1 2 is the point mass at 1 2, then if P = { 1 2} and N = [0, 1]{ 1 2}, then {P, N} is a Hahn decomposition of [0, 1] with respect to µ. However, P1 = [0, 1 2] and N1 = ( 1 2, 1] is also a Hahn decomposition. In fact, if {P, N} is a Hahn decomposition of X with respect to µ and if M ∈A is null, then {P ∪M, N\M} is a Hahn decomposition of X with respect to µ. Furthermore, if {P1, N2} and {P2, N2} are Hahn decompositions of X with respect to µ, then M = P1∆P2 = (P1 ∩N2) ∪(N1 ∩P2) = N1∆N2 is a null set. Furthermore, since E ∩P1 \ P2 ⊆P1∆P2 and E ∩P2 \ P1 ⊆P1∆P2, it follows that µ(E ∩P1) = µ(E ∩P1 ∩P2) = µ(E ∩P2) for each E ∈A. Similarly, µ(E ∩N1) = µ(E ∩N1 ∩N2) = µ(E ∩N2) for each E ∈A. What is true however, as we shall see, is that every Hahn decomposition induced a decomposition of µ into the difference of two positive measures and that any two Hahn decompositions induce the same decomposition of µ. Definition 4.2.8. Let {P, N} be a Hahn decomposition of X with respect to µ. Define µ+(E) = µ(E ∩P) and µ−(E) = −µ(E ∩N). Remark 4.2.9. It is easy to see that µ+ and µ−are both positive measures and that µ = µ+ −µ−. The pair {µ+, µ−} is called a Jordan decomposition of µ. Definition 4.2.10. Two measures µ and ν on (X, A) are said to be mutually singular if there are disjoint sets A, B ∈A with X = A ∪B and µ(A) = 0 while ν(B) = 0. In this case, we write µ ⊥ν. Theorem 4.2.11. [Jordan Decomposition Theorem] Let µ be a signed measure on (X, A). Then there exist two mutually singular positive measures µ+ and µ−such that µ = µ+ −µ−. Furthermore, if λ and ν are any two positive measures with µ = λ −ν, then for each E ∈A we have λ(E) ≥µ+(E) and ν(E) ≥µ−(E). Finally, if λ ⊥ν, then λ = µ+ and ν = µ−. 64 Proof. Let {µ+, µ−} be a Jordan decomposition of µ arising from the Hahn decomposition {P, N} . Then clearly µ = µ+ −µ−. Moreover, since µ+(N) = µ(P ∩N) = µ−(P) = 0 we have that µ+ ⊥µ−. Let λ and ν be any two positive measures with µ = λ −ν and let E ∈A. Then µ+(E) = µ(P ∩E) = λ(P ∩E) −ν(P ∩E) ≤ λ(P ∩E) ≤ λ(E) A similar argument shows that ν(E) ≥µ−(E). Finally, assume that λ ⊥ν. Let {A, B} be a partition of X so that λ(B) = 0 and ν(A) = 0. For each E ∈A we have µ(E ∩A) = λ(E ∩A) −ν(E ∩A) = λ(E ∩A) ≥0. That is A is positive. Similarly B is negative, so {A, B} is a Hahn decomposition. It follows that for an E ∈A, µ+(E) = µ(E ∩P) = µ(E ∩A) = λ(E ∩A) = λ(E) and µ−(E) = −µ(E ∩N) = −µ(E ∩B) = ν(E ∩B) = ν(E). Definition 4.2.12. Let µ be a signed measure on (X, A). Then if {µ+, µ−} is the Jordan decomposition of µ, then the measure |µ| def = µ+ + µ− is called the total variation of µ. Example 4.2.13. Let (X, A, µ) be a measure space and let f ∈L(X, A, µ) be integrable. If λ(E) = Z E f dµ then λ is a signed measure with Jordan decomposition λ+(E) = Z E f + dµ and λ−(E) = Z E f −dµ. In particular, |λ|(E) = Z E f + dµ + Z E f −dµ = Z E |f| dµ. 65 Definition 4.2.14. If µ and ν are signed measures on (X, A), then we say that ν is absolutely continuous with respect to µ and write ν ≪µ if |ν| ≪|µ|. We say that µ and ν are mutually singular and write µ ⊥ν if |µ| ⊥|ν|. Remark 4.2.15. 1) If ν is a signed measure and µ is a positive measure on (X, A), then we claim that ν ≪µ if and only if whenever E ∈A with µ(E) = 0, we have ν(E) = 0. Clearly if ν ≪µ and µ(E) = 0, we have ν(E) = 0. To prove the converse assume that {P, N} is a Hahn-decomposition for ν. Let E ∈A be such that m(E) = 0. Then clearly m(E ∩P) = 0 = m(E ∩N). It follows from our assumption that ν+(E) = ν(E ∩P) = 0 and ν−(E) = ν(E ∩N) = 0. Therefore, we have that |ν|(E) = 0 and |ν| ≪µ as desired. 2) If ν is a signed measure and µ is a positive measure on (X, A), then we claim that ν ⊥µ if and only if there are disjoint sets A, B ∈A with X = A∪B and µ(A) = 0 while ν(E) = 0 for any E ⊆B, E ∈A. If ν ⊥µ, then there are disjoint sets A, B ∈A with X = A ∪B and µ(A) = 0 while |ν|(B) = 0. But then if E ⊆B, E ∈A, we have |ν|(E) = 0 and hence ν(E) = 0. To see that the converse holds let A, B ∈A be disjoint sets with X = A ∪B and µ(A) = 0 while ν(E) = 0 for any E ⊆B, E ∈A. Let {P, N} be a Hahn decomposition for ν. Let E1 = B ∩P and E2 = B ∩n. Then µ+(B) = µ(E1) = 0 and µ−(B) = µ(E2) = 0. It follows that |ν|(B) = 0 and that ν ⊥µ. The following proposition gives us an alternative characterization of absolute continuity for finite positive measures which will prove useful later. Proposition 4.2.16. Let λ and µ be finite positive measures on (X, A). Then the following are equivalent: 1) λ ≪µ 2) For every ϵ > 0 there exist a δ > 0 such that if E ∈A and µ(E) < δ, then λ(E) < ϵ. Proof. 1) ⇒2). Suppose that 2 fails. Then we can find an ϵ0 > 0 and a sequence of sets {Ek} ⊆A such that µ(Ek) < 1 2k but λ(Ek) ≥ϵ0. Now let Fn = ∞ [ k=n Ek. 66 It follows that µ(Fn) < 1 2n−1 while λ(Fn) ≥ϵ0. From here, since {Fn} is decreasing, we get that µ( ∞ \ n=1 Fn) = lim n→∞µ(Fn) = 0, while λ( ∞ \ n=1 Fn) = lim n→∞λ(Fn) ≥ϵ0. This shows that λ is not absolutely continuous with respect to µ. 2) ⇒1). Suppose 2) holds and µ(E) = 0. Then for any ϵ > 0 we have that µ(E) < δ where δ > 0 is chosen as in 2). It follows that λ(E) < ϵ. But since ϵ > 0 was arbitrary, we get that λ(E) = 0 4.3 Radon-Nikodym Theorem Remark 4.3.1. We had previously asked about when, given a measure space (X, A, µ), and any measure λ on A, does there exists an f ∈M+(X, A) with the property that for every E ∈A, λ(E) = Z E f dµ for all E ∈A. We have also seen that if such an f exists then it must be the case that λ ≪µ. At this point we are in a position to use what we have learned about decompositions of signed measures to establish the converse of this result in the case of σ-finite measures. Theorem 4.3.2. [Radon-Nikodym Theorem] Let λ and µ be σ-finite measures on (X, A). Suppose that λ is absloutely continuous with respect to µ. Then there exists f ∈M+(X, A) such that λ(E) = Z E f dµ for every E ∈A. Moreover f is uniquely determined µ-almost everywhere. Proof. Case 1: We assume that λ, µ are finite. For each c > 0 let {P(c), N(c)} be an Hahn decomposition for the signed measure λ −cµ. Let A1 = N(c) and for each k ∈N let Ak+1 = N((k + 1)c) \ k [ i=1 Ai. It follows that {Ai}∞ i=1 is pairwise disjoint and k [ i=1 N(ic) = k [ i=1 Ai. Consequently, we have Ak = N(kc) ∩ k−1 \ i=1 P(ic). If E ∈A and E ⊆Ak, then E ⊆N(kc) and E ⊆P((k −1)c). As such we have (k −1)cµ(E) ≤λ(E) ≤kcµ(E). (∗) 67 Next, let B = X \ ∞ [ i=1 Ai = X \ ∞ [ i=1 N(ic) = ∞ \ i=1 P(ic). Since B ⊆P(kc) for all k ∈N, we get that 0 ≤kcµ(B) ≤λ(B) ≤λ(X) < ∞ for each k ∈N. Therefore, µ(B) = 0 and since λ ≪µ we have that λ(B) = 0, as well. We now define for each c > 0, fc(x) = ( (k −1)c if x ∈Ak, 0 if x ∈B. For each E ∈A, we have E = (E ∩B) ∪( ∞ [ i=1 (E ∩Ak)). Applying (∗) to each of the component pieces above, we have that Z E fc dµ ≤λ(E) ≤ Z E (fc + c) dµ ≤ Z E fc dµ + cµ(X). Now for each n ∈N let gn = f 1 2n . We get Z E gn dµ ≤λ(E) ≤ Z E gn dµ + µ(X) 2n (∗∗). If we let m ≥n then (∗∗) tells us that Z E gn dµ ≤λ(E) ≤ Z E gm dµ + µ(X) 2m and Z E gm dµ ≤λ(E) ≤ Z E gn dµ + µ(X) 2n . Combining these two give us that | Z E gn dµ − Z E gm dµ| ≤µ(X) 2n for each E ∈A. In particular this holds for E1 = {x ∈X|gn −gm| ≥0} and E2 = {x ∈X|gn −gm| < 0}. This allows us to deduce that Z X |gn −gm| dµ ≤2µ(X) 2n = µ(X) 2n−1 and hence that {gn}∞ n=1 is Cauchy in L1(X, A, µ). Assume that gn →f in L1(X, A, µ). Since gn ∈M+(X, A) we can also assume that f ∈M+(X, A). Moreover, for any E ∈A we have | Z E gn dµ − Z E f dµ| ≤ Z E |gn −f| dµ ≤∥gn −f∥1 →0. It then follows from (∗∗) that λ(E) = lim n→∞ Z E gn dµ = Z E f dµ. Suppose now that f, h ∈M+(X, A) are such that Z E f dµ = λ(E) = Z E h dµ 68 for all E ∈A. Let E1 = {x ∈X|f(x) > h(x)} and E2 = {x ∈X|f(x) < h(x)}. Since Z E1 f −h dµ = Z E1 f dµ − Z E1 h dµ = λ(E1) −λ(E1) = 0 and Z E2 f −h dµ = Z E2 f dµ − Z E2 h dµ = λ(E2) −λ(E2) = 0 we have that µ(E1) = µ(E2) = 0 and hence that f = h µ-a.e. Case 2: Asume that λ and µ are σ-finite. Let {Xn} ⊆A be an increasing sequence such that X = ∞ S n=1 Xn, λ(Xn) < ∞and µ(Xn) < ∞. Following the first case, we get for each n ∈N a function fn ∈M+(X, A) such that fn|Xc n ≡0, and if E ∈A with E ⊆Xn, then λ(E) = Z E fn dµ. If m ≥n, then Xn ⊆Xm , and by our previous uniqueness result, fn = fm µ-a.e. Let Fn = sup{f1, f2, . . . , fn}. Then {Fn} is an increasing sequence of positive measurable functions with Fn = fn µ-a.e and Fn(x) = 0 for all x ∈Xc n. Let f = lim n→∞Fn. If E ∈A, then λ(E ∩Xn) = Z E fn dµ = Z E Fn dµ. Given that E ∩Xn ↗E, continuity from below and the Monotone Convergence Theorem shows us that λ(E) = lim n→∞λ(E ∩Xn) = lim n→∞ Z E Fn dµ = Z E f dµ. The uniqueness of f is determined as in the finite case. Remark 4.3.3. A close look at the proof of the Radon-Nikodym Theorem shows that the construction of the function f resembles differentiation. The next example shows that in fact this is not simply a coincidence. Example 4.3.4. Let F : R →R be a continuously differentiable function with F ′(x) > 0. Then F is strictly increasing. Let µF be the Lebesgue-Stieltjes measure on (R, B(R)) generated by F. Then µF is σ-finite and µF ≪m. Moreover, the Fundamental Theorem of Calculus shows that µF ((a, b]) = F(b) −F(a) = Z b a F ′(x) dx = Z [a,b] F ′ dm = Z (a,b] F ′ dm. From here we can deduce that if E ∈B(R), then µF (E) = Z E F ′ dm. In particular, the function we would have obtained in via the Radon-Nikodym Theorem is m-a.e equal to F ′. Motivated by the previous example, we have the following definition. 69 Definition 4.3.5. The function f whose existence was established in the Radon-Nikodym Theorem is called the Radon-Nikodym derivative of λ with respect to µ and it is denoted by dλ dµ. Remark 4.3.6. 1) dλ dµ is integrable if and only if λ is a finite measure. 2) In the case that λ is σ- finite, with X = ∞ S n=1 Xn and each Xn being such that λ(Xn) < ∞, we have that λ(Xn) = Z Xn dλ dµ dµ so dλ dµ must be finite µ-a.e on Xn. As such, we may assume that dλ dµ is actually finite everywhere on X. 3) Let (X, A, λ) = (R, M(R), m) be the ususal Lebesgue measure space. We can define µ on (R, M(R)) to be the restriction of the counting measure on (R, P(R)) to (R, M(R)). Then m ≪µ, since µ(E) = 0 implies E = ∅. However, there is no f ∈M+(R, M(R)) such that m(E) = Z E f dµ. This shows that the Radon-Nikodym Theorem can fail if µ is not σ-finite. However, we leave it as an exercise to show that the Radon-Nikodym Theorem can be extended to the case where λ is arbitrary. 4) Let µ, λ, ν be σ-finite measures on (X, A) with λ ≪µ and ν ≪µ. (i) If c > 0, then cλ ≪µ and d(cλ) dµ = cdλ dµ. (ii) We have (λ + ν) ≪µ and d(λ + ν) dµ = dλ dµ + dν dµ. (iii) If ν ≪λ and λ ≪µ, then ν ≪µ and dν dµ = dν dλ · dλ dµ. (iv) If µ ≪λ and λ ≪µ, then dν dµ = 1 dµ dν . Note: All of the equalities above apply almost everywhere. The proofs are left as an exercise. 5) If λ is a signed measure and µ is a σ-finite measure with λ ≪µ, then λ+ ≪µ and λ−≪µ. In this case, we let dλ dµ def = dλ+ dµ −dλ− dµ . As an application of the Radon-Nikodym Theorem we present the following fundamental Decomposition Theorem. Theorem 4.3.7. [Lebesgue Decomposition Theorem] Let λ and µ be σ-finite measures on (X, A). Then there exists two measures λ1 and λ2 on (X, A) such that λ = λ1 + λ2, λ1 ⊥µ and λ2 ≪µ. Moreover, these measures are unique. 70 Proof. Let ν = λ + µ. Then clearly ν is σ-finite, λ ≪ν and µ ≪ν . It follows that there are functions f, g ∈M+(X, A) such that λ(E) = Z E f dν and µ(E) = Z E g dν. for every E ∈A. Let A = {x ∈X| g(x) = 0} and B = {x ∈X| g(x) > 0}. Then {A, B} is a partition of X. Let λ1(E) = λ(E ∩A) and λ1(E) = λ(E ∩B) for every E ∈A. Clearly λ = λ1 + λ2. Since µ(A) = µ(E) = Z E g dν we have that λ1 ⊥µ. If µ(E) = 0, then R E g dν = 0 so g(x) = 0 for ν-almost every in E. It follows that ν(E ∩B) = 0 and hence that λ2(E) = λ(E ∩B) = 0 since λ ≪ν. That is λ2 ≪µ. To see that λ1 and λ2 are unique we first assume that both λ and µ are finite. Assume also that we can find λ1 and λ2 and ν1 and ν2 with λ = λ1 + λ2, λ1 ⊥µ and λ2 ≪µ, λ = ν1 + ν2, ν1 ⊥µ and ν2 ≪µ. Then γ = λ1 −ν1 = ν2 −λ2. In addition γ is such that γ ⊥µ and γ ≪µ and hence γ = 0 . (It is an easy exercise to verify this claim). The case where λ and µ are σ-finite is left as an exercise. 71
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https://www.youtube.com/watch?v=xCszEldpbmk
5th Grade Math 3.9, Subtract Decimals with Regrouping JoAnn's School 173000 subscribers 62 likes Description 6716 views Posted: 10 Mar 2020 We can use place value to subtract decimals. We stack the decimal numbers on top of each other in vertical form, lining up the decimal points. We can insert trailing zeros to the right of a decimal without changing its value, so all of the decimals will have the same amount of digits. We learned to model decimal subtraction with base-ten blocks, or a quick drawing, in video 3.6. We can make a quick drawing to check our subtraction. We subtract decimals just as we would subtract whole numbers, except we must line up the decimal points to make sure we're subtracting the correct digits. We can use an estimation to know if our actual difference is reasonable. If we don't line up decimal points correctly, we may subtract the wrong digits. In video 3.2, we learned to read and write decimals in standard form, expanded form, and word form. We can write an equation from word form. When an equation is given in sentence form, we can stack the decimals in vertical form on scratch paper to subtract them. We solve a word problem involving subtraction of decimals. 5th Grade Math 3.2, Place Value of Decimals, Read/Write Standard, Expanded, Word Forms 5th Grade Math 3.3, Compare and Order Decimals with Place Value 5th Grade Math 3.4, Round Decimals Using Place Value or Place-value Chart 5th Grade Math 3.5, Model Decimal Addition 5th Grade Math 3.6, Model Decimal Subtraction 5th Grade Math 3.7, Estimate Decimal Sums & Differences 5th Grade Math 3.8, Add Decimals 5th Grade Math Playlist I'm using the Houghton Mifflin Go Math! 2015 copyright textbook for this playlist. You can practice online using the textbook's website YouTube rating for JoAnn's School videos: L0 - No strong language N0 - No nudity S0 - No sexual situations V0 - Not violent or disturbing D0 - No drug reference or content F0 - No flashing lights content FOLLOW ME: SUBSCRIBE on YouTube TWITTER FACEBOOK MINDS PATREON SUPPORT MY WORK: PATREON (one time or monthly) PAYPAL (one time) Transcript: lesson 3.9 subtract decimals we can use place value to subtract decimals we stack the decimal numbers on top of each other in vertical form lining up the decimal points we start with the farthest right place value and we regroup if needed you can see our decimal points are all lined up 9 minus 2 is 7 8 minus 5 is 3 2 minus zero is two we have 2 and 37 hundredths trailing zeros are zeros that we insert on the right side of a decimal and we can insert trailing zeros to the right of a decimal without changing its value so all of the decimals will have the same amount of digits if we have seven and six tenths and we want to subtract 2 and 15 hundredths we can put a trailing zero here and then we can subtract and regroup and if you missed our video on trailing zeros it's linked in the description video 3.4 we learn to model decimal subtraction with base 10 blocks or a quick drawing in video 3.6 and that's Linked In the description we can make a quick drawing to check our subtraction we had 5 and 31 hundredths and we subtracted one in 19 hundredths our answer was 4 and 12 hundredths we have five so we made five squares we had three tenths so we made three lines for tenths and we had one hundredth we made one little green circle for a hundredth to subtract 1 in 19 hundredths we had to regroup one of these tenths as ten hundredths we need to take away nine of them so we Circle nine and cross them out and that's going to leave that one little extra one here because we made ten we only crossed out nine now we need to get rid of a tenth so we cross out a tenth we need to get rid of one whole so we cross out a square and we're left with four squares for the four whole one tenth over here that's not circled for the one-tenth and two hundredths which is the green one and the round one and we can also use addition which is an inverse operation to check our subtraction what we do is we take our difference four and twelve hundredths we subtract the subtrahan and if we get the minuend we know we did it correctly we subtract decimals just as we would subtract whole numbers except we must line up the decimal points to make sure we're subtracting the correct digits we have 473 minus 146 we get 327. we're going to do the same thing with the decimals the only difference is we have decimal points lined up here we start by subtracting the column with the least place value here we started with the ones here we're going to start with the hundredths and we went regroup when needed we have 3 and 27 hundredths in video 3.4 we learned to round decimals in video 3.7 we learned to use rounded decimals to estimate decimal sums and differences and they're Linked In the description if you missed them we can use an estimation to know if our actual difference is reasonable this 9 is going to tell the three whole ones here to round up to a four and this one is going to tell that one to stay the same we have 4 minus 1 which is three so our difference should be about three we subtract what we can't so we regroup from the tenths place it becomes an eight this becomes twelve hundredths we subtract 3 and get nine hundredths we have eight tenths take away one tenth that's seven tenths we have three minus one that's a two and it makes sense because our estimate is close to our difference 2 and 79 hundredths is very close to three if we don't line up decimal points correctly we may subtract the wrong digits here we have 22 minus three and six tenths we have two digits here and two digits here we might accidentally stack them like this but that's wrong 22 is a whole number and three is a whole number so we would have decimal points lined up like this and we can insert a trailing zero so it'll be the same amount of digits to the right side we have zero tenths minus six tenths we need to regroup from the ones place this two becomes a one and now we have ten tenths minus six tenths that's four tenths we have one minus three and we can't so we regroup from the tens place the two becomes a one this one becomes an 11 and 11 minus three is eight and we drop down the 110. we have eighteen and four tenths make sure our decimal points were lined up in video 3.2 which is linked in the description we learn to read and write decimals in standard form expanded form and word form we can write these words as an equation to find the difference but this is going to be very tricky because of the wording we have 2 so that's 2 whole and that's our decimal point 68 hundredths and it's subtracted from five and four tenths we have five whole and is our decimal point and four tenths this is being subtracted from this that means this is the subtrahend and that's the minuend so the five and four tenths is going to be on the top we have nothing here so we can insert a trailing zero we do our subtraction just as we would as if this was whole numbers with three digits and three digits we get 2 and 72 hundredths we can check with addition we use our difference and our subtrahend we add them together and if they're equal to the menu n we know we did it correctly when an equation is given in sentence form we can stack the decimals in vertical form on scratch paper to subtract them we have 19 and three tenths minus 4 and 57 hundredths we stack them up in vertical form lining up the decimal points we subtract them as we would any whole numbers regrouping when necessary and we can use trailing zeros this is three tenths this is 57 hundredths we can put a trailing zero there so we have the same number of digits we get 14 and 73 hundredths here we have 34 and 19 hundredths minus 6 and 59 hundredths we subtract just as we would for whole numbers regrouping when necessary but make sure our decimal points are lined up so we've got them stacked very nicely we're going to start in the hundredth place nine minus nine is zero we have one tenth minus five tenths we can't do that so we need to regroup from the ones place that's going to become a three that's going to become an 11. 11 tenths minus five tenths is six tenths we put our decimal point in a straight line we have three minus six we can't do that so we're going to regroup from here that 3 is going to become a two this three is going to become a 13. now we have 13 ones minus six ones that's seven ones we have two tens which drop down and we can remove trailing zeros from our difference without changing its value we have 27 and 60 hundredths we can just make it 27 and six tenths precipitation is the release of water from the sky it could be liquid such as rain or sleet and it could be solid such as hail or snow the average annual precipitation for Death Valley located in Eastern California is 2 and 3 600 inches the average annual precipitation for Seattle Washington is 37 and 49 hundredths inches how much more precipitation does Seattle get on average than Death Valley so we have our Seattle number decimal number we're going to subtract the Death Valley one to find the difference between them we can estimate this 4 is going to tell the 7 to stay the same so it's going to round to 37 this 3 is going to tell the 2 to stay the same so it's going to round to 2. we subtract and get 35 inches now why did I round to the ones place I had to round to the ones place because there's no tens here I can't round this one to the tens place whatever I round this one to I have to round that one to so because there's only ones here they both got rounded to the ones place we do our subtraction and 9 minus 6 is 3 4 minus three is one seven minus two is five we drop down the three tens we have 35 and 13 hundredths inches per year that's very close to our estimate so our difference is reasonable because it's close to our estimate I know my regular subscribers have seen this a lot but for those of you who haven't we can turn a sheet of lined paper sideways to keep our place values in the correct column that way when we're subtracting decimals we keep our place values straight that'll help us so we don't make a mistake so remember to keep your decimal points lined up and remember when it's in sentence form we can put it in vertical form on scratch paper our next lesson 3.10 we're going to use addition or subtraction to describe a pattern or create a sequence with decimals I think you can do this just remember it's just like whole numbers except we have decimal points have a great day bye
6027
https://www.sciencedirect.com/science/article/pii/S2214541924000130
Skip to article My account Sign in View PDF Oral and Maxillofacial Surgery Cases Volume 10, Issue 2, June 2024, 100357 Case Report Simple bone cyst of the mandible Author links open overlay panel, , rights and content Under a Creative Commons license Open access Highlights €¢ SBC is predominantly an asymptomatic lesion. €¢ A careful history taking and radiographic assessment is crucial for a first diagnosis. €¢ A long-term follow up is mandatory when a wait-and-see management has been chosen. Abstract Simple bone cysts (SBCs) are nonneoplastic intraosseous cavities without an epithelial lining, surrounded by bony walls and either empty or containing liquid and/or connective tissue: they were first described in 1929 as a distinct entity of disease. The characteristic that distinguishes SBCs from true cysts is the absence of epithelial lining, that allow us to regard SBCs as pseudocysts. In the literature, SBCs have been referred to as solitary bone cysts, idiopathic bone cysts, unicameral cysts, traumatic bone cysts, hemorrhagic bone cysts, primary bone cysts, and extravasation cysts. The pathogenesis of SBC remains uncertains. Radiographically, SBCs usually present as isolated unilocular radiolucencies with well-defined borders. When SBC extends to the interdental bone, the characteristic radiographic €œscalloping effect€ can be observed. The differential diagnosis includes apical periodontitis, odontogenic keratocyst, central giant cell granuloma, ameloblastoma, odontogenic myxoma, and central and neurogenic neoplasms. Surgery (curettage) is the gold standard treatment as it allows both diagnosis and treatment by generation of a blood clot in the vacant cavity of SBCs: bone usually regenerates progressively within 6€“12 months. Recurrence rate is almost negligible. The aim of the present article is to present and discuss the diagnosis and management of a case of SBC. Keywords Simple bone cyst Traumatic bone cyst Diagnosis Mandible Treatment Cited by (0) © 2024 The Authors. Published by Elsevier Inc.
6028
https://www.mccc.edu/~virtcoll/documents/mat120/unit04/004_properties-solvingmodularproblems4/data/downloads/004_properties-solvingmodularproblems4.pdf
Solving Modular Problems 1 Solving Modular Problems Chapter 10 – Section 3 Simple Modular Equations The solution to a modular equation is a set of all numbers in one or more equivalence classes. It is possible that the solution is the empty set. A simple modular equation is of the nature x ≡ a (mod m) No in-class assignment problem Solving a Simple Equation x ≡ 5 (mod 6) 5 is the name of an equivalence class in mod 6. The solution is the equivalence class 5. x = {5, 11, 17, 23, …} Remember each to find the next number in an equivalenc class add the mod. x ≡25 (mod 4) 25 is not less than 4. Change 25 to the name of the equivalence class 25 is in. 25 ≡ 1 (mod 4) x = {1, 5, 9, 13, … } In-class Assignment 25 - 1 Solving More Complicated Modular Equations 5x ≡ 17 (mod 4) Change 17 to the name of the class it belongs to. 17 ≡ 1 (mod4) Write 4 statements because the mod is 4 replacing x with the name of an equivalence class. 5 · 0 ≡ 1 (mod 4) False 5 · 1 ≡ 1 (mod 4) True 5 · 2 ≡ 1 (mod 4) False 5 · 3 ≡ 1 (mod 4) False The solution is equivalence class 1. x = {1, 5, 9, 13, …} In-class Assignment 25 - 2 Another Equation to Solve 2x ≡ 4 (mod 8) Equivalence classes 2 and 6 are solutions. x ={2,10,18, …} {6, 14, 22, …} Notice the sets can be joined as one set. x = {2, 6, 10, 14, …} 2 · 0 ≡ 4 (mod 8) False 2 · 1 ≡ 4 (mod 8) False 2 · 2 ≡ 4 (mod 8) True 2 · 3 ≡ 4 (mod 8) False 2 · 4 ≡ 4 (mod 8) False 2 · 5 ≡ 4 (mod 8) False 2 · 6 ≡ 4 (mod 8) True 2 · 7 ≡ 4 (mod 8) False In-class Assignment 25 - 2 A Word Problem Ann wants to see a play. The date of the play is 65 days from today. Ann is a nurse and her schedule is 5 – 10 hour days on and 3 days off. Will she be off on the day of the play if she is in the second day of her 5 days on? On = 1 and Off = 2 8 day schedule 1 1 1 1 1 2 2 2 Remainder =1 1 day past 2nd day of schedule. Ann will be on. 8 8 65 64 1 In-class Assignment 25 - 3 Solving Modular Problems 2 Symbols of Inequalities  < means “less than.”  ≤means is “less than or equal to” or “no more than.”  > means “greater than.”  ≥means “greater than or equal to” or “no less than.” No in-class assignment problem Solving Restricted Modular Equations Find the set of whole number solutions given that x < 40 x ≡ 1 (mod 7) x ≡ 4 (mod 5) x ≡ 5 (mod 8) Find all the solution sets for the 3 equations. x = {1, 8, 15, 22, 29, 36} x = {4, 9, 13, 17, 21, 25, 29, 33, 37}  x = {5, 13, 21, 29, 37} Note – only numbers < 40. Find the intersection of the solution sets. x = 29 In-class Assignment 25 - 4 Solving a Verbal Problem in Mod Systems The football team has 50 members. When the coach put them in drill teams of 7 there were 4 players left. When the coach had them in drill teams of 3 all players were used. When the coach put them in teams of 6 there were 3 players left. How many were at practice on this day? Mods are the groupings Numbers are whats left. x ≤ 50 x ≡ 4 (mod 7) x ≡ 0 (mod 3) x ≡ 3 (mod 6) Solve as before. 39 players were at practice on this day. In-class Assignment 25 – 5, 6 A Short Cut x < 40 x ≡ 1 (mod 7) x ≡ 4 (mod 5) x ≡ 5 (mod 8) Find the solution set for the largest mod. x = {5, 13, 21, 29, 37} Cross out any number that does not leave a remainder of 1 when divided by 7 (5, 13, 21, 37 crossed out) or a remainder of 4 when divided by 5 (the same numbers crossed out). x = {29) In-class Assignment 25 – 4 (short cut)
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https://brainly.com/question/18667947
[FREE] The coefficient part of a number written in scientific notation is between which numbers? A. 1 and 10 B. - 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 +27,7k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +33,3k Ace exams faster, with practice that adapts to you Practice Worksheets +7,6k Guided help for every grade, topic or textbook Complete See more / Mathematics Expert-Verified Expert-Verified The coefficient part of a number written in scientific notation is between which numbers? A. 1 and 10 B. -1 and -10 C. Both A and B 2 See answers Explain with Learning Companion NEW Asked by VaildxLaysha3 • 10/26/2020 0:00 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 57273886 people 57M 0.0 0 Upload your school material for a more relevant answer The coefficient part of the scientific notation is required. The correct option is A. 1 and 10 Scientific notation is written in the form of m×1 0 n The coefficient part (m) is written as absolute value between 1 and 10. n is the exponent and an integer. Let us write a negative number in scientific form. −53000=−(m×1 0 n)=−(5.3×1 0 5)=−5.3×1 0 5 So, the coefficient part is a number that lies in 1≤m<10 Learn more: brainly.com/question/1705769 brainly.com/question/18073768 Answered by boffeemadrid •4.3K answers•57.3M people helped Thanks 0 0.0 (0 votes) Expert-Verified⬈(opens in a new tab) This answer helped 57273886 people 57M 0.0 0 Upload your school material for a more relevant answer The coefficient part of a number written in scientific notation must be between 1 and 10. This means the correct option is A. 1 and 10. Explanation In scientific notation, a number is expressed in the form m×1 0 n, where m is the coefficient and n is an integer representing the power of ten. The important rule for the coefficient m is that it must always be between 1 and 10 (inclusive of 1 but exclusive of 10). This is true for both positive and negative numbers. For example: For the number 5,000, it can be represented in scientific notation as 5.0×1 0 3, since the coefficient 5.0 is between 1 and 10. For a smaller number like 0.0005, it can be expressed as 5.0×1 0−4, where again, the coefficient 5.0 is still within the range of 1 to 10. So, the correct choice regarding the coefficient part of a number written in scientific notation is Option A: 1 and 10. This means the coefficient should always fall within the range of 1 to just below 10 in scientific notation, regardless of the number's overall sign or magnitude. Examples & Evidence For example, the number 320,000 can be written in scientific notation as 3.2 x 10^5, where the coefficient 3.2 is between 1 and 10. Similarly, the number 0.0045 can be written as 4.5 x 10^{-3}, where the coefficient 4.5 is also between 1 and 10. The definition of scientific notation indicates that the coefficient must always be greater than or equal to 1 and less than 10, which is widely accepted in mathematics. Thanks 0 0.0 (0 votes) Advertisement Community Answer This answer helped 2615 people 2K 5.0 1 A There are two parts to a number written in Scientific Notation. The first is the coefficient and the second part is a power of ten. In order for the number to be correctly written it has to be between 1 and 10. Answered by ShadDealJr •8 answers•2.6K people helped Thanks 1 5.0 (1 vote) 4 Advertisement ### 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 Using the discriminant, how many solutions will this quadratic have? Explain and show all work. −3 x 2−12 x+11=0 What is $\cos \left(6^{\circ}\right) ? A. 0.10 B. 0.60 C. 0.99 D. 0.06 The power g 2 is equivalent to 81. What is the value of g−2? A. −81 B. −9 C. 81 1​ D. 9 1​ (7×1 0 2)(3.5×1 0 3) What is the value of (−4 3​)−4 ? A. −81 256​ B. −256 81​ C. 256 81​ D. 81 256​ 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
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https://surgeryreference.aofoundation.org/spine/deformities/adolescent-idiopathic-scoliosis
Adolescent idiopathic scoliosis Login Home Skeleton Diagnosis Indications Treatment All approaches All preparations All further reading Basic techniques Authors of section Authors Han Jo Kim, Marinus de Kleuver, Keith Luk Executive Editors Kenneth Cheung, Larry Lenke General Editor German Ochoa Open all credits Adolescent idiopathic scoliosis Lenke classification Patient examination Immature patients, abnormal neuroaxis, body habitus Lenke 1 Lenke 1 curves are one where the major curve (the curve with the largest Cobb angle) is in the main thoracic region while... Learn moreChoose deformity Lenke 2 Type 2 curves have structural main thoracic and proximal thoracic curve. Learn moreChoose deformity Lenke 3 Type 3 curves are double major curves where the thoracic curve is major and the lumbar curve is structural. They are... Learn moreChoose deformity Lenke 4 Type 4 curves have structural proximal thoracic, main thoracic, and thoracolumbar/lumbar curves (triple major). Learn moreChoose deformity Lenke 5 Type 5 curves have structural thoracolumbar/lumbar curve while the main thoracic and the proximal thoracic... Learn moreChoose deformity Lenke 6 Lenke 6 curves have structural thoracolumbar/lumbar and main thoracic curves, with thoracolumbar/lumbar curve... Learn moreChoose deformity AO Spine Courses New gold-standard for mastering deformity and MISS Find your next course AO Surgery Reference Basic technique collection now available Go to page AO Foundation Who we are What we do Our community Our services and resources Our courses and events Products and Services AO PEER myAO AO Videos Course finder AO/OTA Classifications Quick links FAQ Feedback and feature suggestions Contact the AO Foundation AO Data Privacy notice Cookie policy Disclaimer Membership Become a member Connect By clicking “Accept All Cookies”, you agree to the storing of cookies on your device to enhance site navigation, analyze site usage, and assist in our marketing efforts. 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https://sochicar.cl/wp-content/uploads/2023/09/ehad193.pdf
2023 ESC Guidelines for the management of endocarditis Developed by the task force on the management of endocarditis of the European Society of Cardiology (ESC) Endorsed by the European Association for Cardio-Thoracic Surgery (EACTS) and the European Association of Nuclear Medicine (EANM) Authors/Task Force Members: Victoria Delgado †, (Chairperson) (Spain), Nina Ajmone Marsan ‡, (Task Force Co-ordinator) (Netherlands), Suzanne de Waha‡, (Task Force Co-ordinator) (Germany), Nikolaos Bonaros (Austria), Margarita Brida (Croatia), Haran Burri (Switzerland), Stefano Caselli (Switzerland), Torsten Doenst (Germany), Stephane Ederhy (France), Paola Anna Erba 1 (Italy), Dan Foldager (Denmark), Emil L. Fosbøl (Denmark), Jan Kovac (United Kingdom), Carlos A. Mestres (South Africa), Owen I. Miller (United Kingdom), Jose M. Miro 2 (Spain), Michal Pazdernik (Czech Republic), Maria Nazarena Pizzi (Spain), Eduard Quintana 3 (Spain), Trine Bernholdt Rasmussen (Denmark), Arsen D. Ristić (Serbia), Josep Rodés-Cabau (Canada), Alessandro Sionis (Spain), Liesl Joanna Zühlke (South Africa), Michael A. Borger †, (Chairperson) (Germany), and ESC Scientific Document Group Corresponding authors: Victoria Delgado, Cardiology, Hospital University Germans Trias i Pujol, Badalona, Spain, and Institute for Health Science Research Germans Trias i Pujol (IGTP), Badalona, Spain. Tel: +34 934 65 12 00, E-mail: videlga@gmail.com; and Michael A. Borger, University Department of Cardiac Surgery, Leipzig Heart Center, Leipzig, Germany. Tel: +49-341- 865-0, E-mail: Michael.Borger@helios-gesundheit.de † The two Chairpersons contributed equally to the document and are joint corresponding authors. ‡ The two Task Force Co-ordinators contributed equally to the document. Author/Task Force Member affiliations are listed in author information. 1 Representing the European Association of Nuclear Medicine (EANM) 2 Representing the European Society of Clinical Microbiology and Infectious Diseases (ESCMID) 3 Representing the European Association for Cardio-Thoracic Surgery (EACTS) ESC Clinical Practice Guidelines (CPG) Committee: listed in the Appendix. ESC subspecialty communities having participated in the development of this document: Associations: Association of Cardiovascular Nursing & Allied Professions (ACNAP), Association for Acute CardioVascular Care (ACVC), European Association of Cardiovascular Imaging (EACVI), European Association of Preventive Cardiology (EAPC), European Association of Percutaneous Cardiovascular Interventions (EAPCI), European Heart Rhythm Association (EHRA), and Heart Failure Association (HFA). Councils: Council for Cardiology Practice, Council on Stroke. Working Groups: Adult Congenital Heart Disease, Cardiovascular Surgery. Patient Forum The content of these European Society of Cardiology (ESC) Guidelines has been published for personal and educational use only. No commercial use is authorized. No part of the ESC Guidelines may be translated or reproduced in any form without written permission from the ESC. Permission can be obtained upon submission of a written request to Oxford University Press, the publisher of the European Heart Journal and the party authorized to handle such permissions on behalf of the ESC (journals.permissions@oup.com). European Heart Journal (2023) 00, 1–95 ESC GUIDELINES © The European Society of Cardiology 2023. All rights reserved. For permissions please e-mail: journals.permissions@oup.com. Downloaded from by guest on 30 August 2023 Document Reviewers: Bernard Iung, (CPG Review Co-ordinator) (France), Bernard Prendergast, (CPG Review Co-ordinator) (United Kingdom), Magdy Abdelhamid (Egypt), Marianna Adamo (Italy), Riccardo Asteggiano (Italy), Larry M. Baddour (Unites States of America), Jelena Čelutkienė (Lithuania), John Chambers (United Kingdom), Jean-Claude Deharo (France), Wolfram Doehner (Germany), Laura Dos Subira (Spain), Xavier Duval (France), Volkmar Falk (Germany), Laurent Fauchier (France), Nuria Fernandez-Hidalgo (Spain), Christian Giske2 (Sweden), Anežka Gombošová (Czechia), Gilbert Habib (France), Borja Ibanez (Spain), Tiny Jaarsma (Sweden), Lars Køber (Denmark), Konstantinos C. Koskinas (Switzerland), Dipak Kotecha (United Kingdom), Ulf Landmesser (Germany), Sandra B. Lauck (Canada), Basil S. Lewis (Israel), Maja-Lisa Løchen (Norway), John William McEvoy (Ireland), Borislava Mihaylova (United Kingdom), Richard Mindham (United Kingdom), Lis Neubeck (United Kingdom), Jens Cosedis Nielsen (Denmark), Jean-François Obadia (France), Agnes A. Pasquet (Belgium), Steffen Petersen (United Kingdom), Eva Prescott (Denmark), Susanna Price (United Kingdom), Amina Rakisheva (Kazakhstan), Archana Rao (United Kingdom), François Rouzet (France), Jonathan Sandoe (United Kingdom), Renate B. Schnabel (Germany), Christine Selton-Suty (France), Lars Sondergaard (Denmark), Martin Thornhill (United Kingdom), Konstantinos Toutouzas (Greece), Nico Van de Veire (Belgium), Isidre Vilacosta (Spain), Christiaan Vrints (Belgium), and Olaf Wendler (United Kingdom) All experts involved in the development of these guidelines have submitted declarations of interest. These have been compiled in a report and simultaneously published in a supplementary document to the guidelines. The report is also available on the ESC website www.escardio.org/Guidelines See the European Heart Journal online for supplementary documents that include background information and evidence tables. Keywords Guidelines • Antibiotics • Cardiac imaging • Cardiac implantable electronic device • Cardiac surgery • Complications • Computed tomography • Congenital heart disease • Diagnosis • Echocardiography • Endocarditis • Infection • Nuclear imaging • Positron emission tomography • Prevention • Prognosis • Prosthetic heart valve • Valve disease Table of contents 1. Preamble ...................................................................................................................... 6 2. Introduction ............................................................................................................... 8 2.1. What is new ..................................................................................................... 8 3. Prevention ................................................................................................................ 12 3.1. Rationale ........................................................................................................... 12 3.2. Populations at risk of infective endocarditis ..................................... 13 3.3. Situations and procedures at risk .......................................................... 14 3.3.1. Dental procedures .............................................................................. 14 3.3.2. Non-dental procedures .................................................................... 14 3.3.3. Cardiac or vascular interventions ................................................. 15 3.4. Patient education .......................................................................................... 15 4. The Endocarditis Team ...................................................................................... 16 5. Diagnosis ................................................................................................................... 18 5.1. Clinical features ............................................................................................. 18 5.2. Laboratory findings ...................................................................................... 19 5.3. Microbiological diagnosis ........................................................................... 19 5.3.1. Blood culture-positive infective endocarditis .......................... 19 5.3.2. Blood culture-negative infective endocarditis ......................... 19 5.3.3. Proposed strategy for a microbiological diagnostic algorithm in suspected infective endocarditis ...................................... 20 5.4. Imaging techniques ....................................................................................... 20 5.4.1. Echocardiography ................................................................................ 20 5.4.2. Computed tomography .................................................................... 22 5.4.3. Magnetic resonance imaging ........................................................... 22 5.4.4. Nuclear imaging positron emission tomography/ computed tomography (angiography) and single photon emission tomography/computed tomography .................................... 23 5.5. Diagnostic criteria ........................................................................................ 24 5.5.1. Modifications for the diagnosis of infective endocarditis ... 24 5.5.1.1. Major criteria – microbiology ................................................ 26 5.5.1.2. Major criteria – imaging ............................................................ 26 5.5.1.3. Minor criteria ................................................................................ 27 5.5.1.4. Microbiological criteria ............................................................. 27 5.5.1.5. Infective endocarditis classification ...................................... 28 Disclaimer. The ESC Guidelines represent the views of the ESC and were produced after careful consideration of the scientific and medical knowledge and the evidence available at the time of their publication. The ESC is not responsible in the event of any contradiction, discrepancy, and/or ambiguity between the ESC Guidelines and any other official recommendations or guidelines issued by the relevant public health authorities, in particular, in relation to good use of healthcare or therapeutic strategies. Health professionals are encouraged to take the ESC Guidelines fully into account when exercising their clinical judgment, as well as in the determination and the implementation of preventive, diagnostic, or therapeutic medical strategies; however, the ESC Guidelines do not override, in any way whatsoever, the individual responsibility of health professionals to make appropriate and accurate decisions in consideration of each patient’s health condition and in consultation with that patient and, where appropriate and/or necessary, the patient’s caregiver. Nor do the ESC Guidelines exempt health professionals from taking into full and careful consideration the relevant official updated recommendations or guidelines issued by the competent public health authorities, in order to manage each patient’s case in light of the scientifically accepted data pursuant to their respective ethical and professional obligations. It is also the health professional’s responsibility to verify the applicable rules and regulations relating to drugs and medical devices at the time of prescription. 2 ESC Guidelines Downloaded from by guest on 30 August 2023 5.5.2. The new 2023 European Society of Cardiology diagnostic algorithms ............................................................................................................ 28 6. Prognostic assessment at admission ............................................................. 28 7. Antimicrobial therapy: principles and methods ....................................... 28 7.1. General principles ........................................................................................ 28 7.2. Penicillin-susceptible oral streptococci and Streptococcus gallolyticus group ........................................................................ 30 7.3. Oral streptococci and Streptococcus gallolyticus group susceptible, increased exposure or resistant to penicillin ................... 30 7.4. Streptococcus pneumoniae, β-haemolytic streptococci (groups A, B, C, and G) ...................................................................................... 32 7.5. Granulicatella and Abiotrophia (formerly nutritionally variant streptococci) ........................................................................................................... 32 7.6. Staphylococcus aureus and coagulase-negative staphylococci ..... 32 7.7. Methicillin-resistant staphylococci ......................................................... 32 7.8. Enterococcus spp. ........................................................................................... 35 7.9. Gram-negative bacteria .............................................................................. 36 7.9.1. Haemophilus, Aggregatibacter, Cardiobacterium, Eikenella, and Kingella-related species ......................................................................... 36 7.9.2. Non-Haemophilus, Aggregatibacter, Cardiobacterium, Eikenella, and Kingella species ...................................................................... 37 7.10. Blood culture-negative infective endocarditis ............................... 37 7.11. Fungi ................................................................................................................ 37 7.12. Empirical therapy ....................................................................................... 37 7.13. Outpatient parenteral or oral antibiotic therapy for infective endocarditis ............................................................................................................. 38 7.13.1. Parenteral and oral step-down antibiotic treatment ......... 39 7.13.2. Other considerations for outpatient oral or parenteral antimicrobial therapy ...................................................................................... 39 8. Indications for surgery and management of main infective endocarditis complications .................................................................................... 40 8.1. Pre-operative risk assessment ................................................................ 40 8.2. Heart failure .................................................................................................... 40 8.2.1. Heart failure in infective endocarditis ......................................... 40 8.2.2. Indications and timing of surgery in the presence of heart failure in infective endocarditis ................................................................... 42 8.3. Uncontrolled infection ............................................................................... 42 8.3.1. Septic shock and persistent infection ......................................... 42 8.3.2. Locally uncontrolled infection ........................................................ 42 8.3.3 Indications and timing of surgery in the presence of uncontrolled infection .................................................................................... 43 8.3.3.1. Persistent infection ..................................................................... 43 8.3.3.2. Locally uncontrolled infection ............................................... 43 8.3.3.3. Infection with resistant or virulent organisms ................ 43 8.4. Prevention of systemic embolism ......................................................... 43 8.4.1. Incidence of embolic events in infective endocarditis .......... 43 8.4.2. Predicting the risk of embolism ..................................................... 43 8.4.3. Indications and timing of surgery to prevent embolism in infective endocarditis ...................................................................................... 43 9. Other complications of infective endocarditis ......................................... 44 9.1. Neurological complications ...................................................................... 44 9.1.1. The role of cerebral imaging in infective endocarditis ......... 45 9.2. Infective aneurysms ..................................................................................... 45 9.3. Splenic complications .................................................................................. 45 9.4. Myocarditis and pericarditis ..................................................................... 46 9.5. Heart rhythm and conduction disturbances .................................... 46 9.6. Musculoskeletal manifestations .............................................................. 46 9.6.1. Osteoarticular infective endocarditis-related infections ..... 46 9.6.2. Rheumatological manifestations .................................................... 47 9.7. Acute renal failure ........................................................................................ 47 10. Surgical therapy: principles and methods ................................................ 47 10.1. Pre-operative and peri-operative management ............................ 47 10.1.1. Coronary angiography .................................................................... 47 10.1.2. Extracardiac infection ...................................................................... 48 10.1.3. Intra-operative echocardiography .............................................. 48 10.2. Other intra-operative considerations ............................................... 48 10.3. Surgical approach and techniques ...................................................... 48 10.3.1. Choice of valve prosthesis ............................................................ 49 10.4. Timing of surgery after ischaemic and haemorrhagic stroke .. 50 10.5. Post-operative complications ............................................................... 50 10.6. Management of antithrombotic therapy after surgery .............. 51 11. Outcome after discharge: follow-up and long-term prognosis ...... 51 11.1. Recurrences: relapses and reinfections ............................................ 51 11.2. First year follow-up ................................................................................... 52 11.3. Long-term prognosis ................................................................................ 52 12. Management of specific situations .............................................................. 52 12.1. Prosthetic valve endocarditis ................................................................ 52 12.1.1. Definition and pathophysiology .................................................. 53 12.1.2. Diagnosis ............................................................................................... 53 12.1.3. Prognosis and treatment ............................................................... 53 12.2. Endocarditis in the elderly ..................................................................... 53 12.3. Transcatheter prosthetic valve endocarditis .................................. 54 12.3.1. Endocarditis following transcatheter aortic valve implantation ........................................................................................................ 54 12.3.1.1. Diagnosis ...................................................................................... 54 12.3.1.2. Prognosis and treatment ....................................................... 54 12.3.2. Endocarditis following transcatheter pulmonary valve implantation ........................................................................................................ 54 12.3.2.1. Diagnosis ...................................................................................... 55 12.3.2.2. Prognosis and treatment ....................................................... 55 12.4. Infective endocarditis affecting cardiac implantable electronic devices .................................................................................................. 55 12.4.1. Definitions of cardiac device infections ................................... 55 12.4.2. Pathophysiology and microbiology ............................................ 55 12.4.3. Risk factors .......................................................................................... 55 12.4.4. Prophylaxis ........................................................................................... 55 12.4.5. Diagnosis ............................................................................................... 55 12.4.6. Antimicrobial therapy ...................................................................... 57 12.4.7. Device extraction .............................................................................. 57 12.4.8. Device reimplantation ..................................................................... 57 12.5. Infective endocarditis in patients admitted to intensive care units ............................................................................................................................ 58 12.5.1. Causative microorganisms ............................................................ 58 12.5.2. Diagnosis ............................................................................................... 58 12.5.3. Management ........................................................................................ 58 12.6. Right-sided infective endocarditis ....................................................... 58 12.6.1. Diagnosis and complications ........................................................ 59 12.6.2. Endocarditis in people who inject drugs ................................. 59 12.6.3. Prognosis and treatment ............................................................... 59 12.6.3.1. Antimicrobial therapy ............................................................. 59 ESC Guidelines 3 Downloaded from by guest on 30 August 2023 12.6.3.2. Surgery .......................................................................................... 59 12.7. Infective endocarditis in congenital heart disease ........................ 60 12.8. Infective endocarditis in rheumatic heart disease ........................ 61 12.9. Infective endocarditis during pregnancy .......................................... 61 12.10. Infective endocarditis in immunocompromised patients ....... 61 12.10.1. Solid organ transplant recipients ............................................. 61 12.10.2. Patients with human immunodeficiency virus .................... 61 12.10.3. Patients with neutropaenia ........................................................ 62 12.11. Antithrombotic and anticoagulant therapy in infective endocarditis ............................................................................................................. 62 12.12. Non-bacterial thrombotic endocarditis ........................................ 62 12.13. Infective endocarditis and malignancy ............................................ 63 13. Patient-centred care and shared decision-making in infective endocarditis .................................................................................................................. 63 13.1. What is patient-centred care and shared decision-making and why is it important? ..................................................................................... 63 13.2. Patient-centred care and shared decision-making in infective endocarditis ............................................................................................................. 63 14. Sex differences .................................................................................................... 64 15. Key messages ....................................................................................................... 65 16. Gaps in evidence ................................................................................................. 66 17. ‘What to do’ and ‘What not to do’ messages from the Guidelines 67 18. Supplementary data ........................................................................................... 73 19. Data availability .................................................................................................... 73 20. Author information ........................................................................................... 73 21. Appendix ................................................................................................................ 74 22. References ............................................................................................................. 74 Tables of Recommendations Recommendation Table 1 — Recommendations for antibiotic prophylaxis in patients with cardiovascular diseases undergoing oro-dental procedures at increased risk for infective endocarditis ..... 14 Recommendation Table 2 — Recommendations for infective endocarditis prevention in high-risk patients ................................................. 16 Recommendation Table 3 — Recommendations for infective endocarditis prevention in cardiac procedures ............................................ 16 Recommendation Table 4 — Recommendations for the Endocarditis Team ..................................................................................................... 18 Recommendation Table 5 — Recommendations for the role of echocardiography in infective endocarditis ..................................................... 22 Recommendation Table 6 — Recommendations for the role of computed tomography, nuclear imaging, and magnetic resonance in infective endocarditis ................................................................................................ 23 Recommendation Table 7 — Recommendations for antibiotic treatment of infective endocarditis due to oral streptococci and Streptococcus gallolyticus group ............................................................................. 30 Recommendation Table 8 — Recommendations for antibiotic treatment of infective endocarditis due to Staphylococcus spp. ............. 33 Recommendation Table 9 — Recommendations for antibiotic treatment of infective endocarditis due to Enterococcus spp. ........................ 35 Recommendation Table 10 — Recommendations for antibiotic regimens for initial empirical treatment of infective endocarditis (before pathogen identification) .......................................................................... 38 Recommendation Table 11 — Recommendations for outpatient antibiotic treatment of infective endocarditis ................................................ 40 Recommendation Table 12 — Recommendations for the main indications of surgery in infective endocarditis (native valve endocarditis and prosthetic valve endocarditis) ............................................ 44 Recommendation Table 13 — Recommendations for the treatment of neurological complications of infective endocarditis ............................ 45 Recommendation Table 14 — Recommendations for pacemaker implantation in patients with complete atrioventricular block and infective endocarditis ................................................................................................ 46 Recommendation Table 15 — Recommendations for patients with musculoskeletal manifestations of infective endocarditis ......................... 47 Recommendation Table 16 — Recommendations for pre-operative coronary anatomy assessment in patients requiring surgery for infective endocarditis ................................................................................................ 48 Recommendation Table 17 — Indications and timing of cardiac surgery after neurological complications in active infective endocarditis .................................................................................................................. 50 Recommendation Table 18 — Recommendations for post-discharge follow-up ........................................................................................ 52 Recommendation Table 19 — Recommendations for prosthetic valve endocarditis ....................................................................................................... 53 Recommendation Table 20 — Recommendations for cardiovascular implanted electronic device-related infective endocarditis ...................... 57 Recommendation Table 21 — Recommendations for the surgical treatment of right-sided infective endocarditis ............................................. 60 Recommendation Table 22 — Recommendations for the use of antithrombotic therapy in infective endocarditis ......................................... 62 List of tables Table 1 Classes of recommendations .................................................................. 6 Table 2 Levels of evidence ........................................................................................ 6 Table 3 New recommendations ............................................................................ 8 Table 4 Revised recommendations .................................................................... 10 Table 5 General prevention measures to be followed in patients at high and intermediate risk of infective endocarditis ................................... 13 Table 6 Prophylactic antibiotic regime for high-risk dental procedures 14 Table 7 Members of the Endocarditis Team ................................................. 16 Table 8 Cardiac and non-cardiac risk factors ................................................ 18 Table 9 Investigation of rare causes of blood culture-negative infective endocarditis ................................................................................................ 20 Table 10 Definitions of the 2023 European Society of Cardiology modified diagnostic criteria of infective endocarditis .................................. 24 Table 11 Antibiotic treatment of blood culture-negative infective endocarditis .................................................................................................................. 37 Table 12 Features favouring a non-mechanical valve substitute in the setting of surgery for acute infective endocarditis ...................................... 50 Table 13 Factors associated with an increased rate of relapse of infective endocarditis ................................................................................................ 52 Table 14 ‘What to do’ and ‘What not to do’ ............................................... 67 List of figures Figure 1 Management of patients with infective endocarditis ................... 7 Figure 2 Education of high-risk patients to prevent infective endocarditis .................................................................................................................. 15 4 ESC Guidelines Downloaded from by guest on 30 August 2023 Figure 3 Management of patients with infective endocarditis: positioning of the Endocarditis Team ............................................................... 17 Figure 4 Microbiological diagnostic algorithm in culture-positive and culture-negative infective endocarditis ............................................................. 21 Figure 5 European Society of Cardiology 2023 algorithm for diagnosis of native valve infective endocarditis ............................................. 25 Figure 6 European Society of Cardiology 2023 algorithm for diagnosis of prosthetic valve infective endocarditis ................................................................. 26 Figure 7 European Society of Cardiology 2023 algorithm for diagnosis of cardiac device-related infective endocarditis ........................ 27 Figure 8 Phases of antibiotic treatment for infective endocarditis in relation to outpatient parenteral antibiotic therapy and partial oral endocarditis treatment ............................................................................................ 29 Figure 9 Flowchart to assess clinical stability based on the Partial Oral Treatment of Endocarditis trial ........................................................................... 39 Figure 10 Proposed surgical timing for infective endocarditis ................ 41 Figure 11 Surgery for infective endocarditis following stroke ................ 49 Figure 12 Algorithm differentiating relapse from reinfection ................. 51 Figure 13 Management of cardiovascular implanted electronic device-related infective endocarditis ................................................................. 56 Figure 14 Concept of patient-centred care in infective endocarditis . 64 Abbreviations and acronyms [18F]FDG 18F-fluorodeoxyglucose 99mTc-HMPAO 99mTechnetium-hexamethylpropyleneamine oxime AIDS Acquired immune deficiency syndrome AEPEI Association for the Study and Prevention of Infective Endocarditis Study ANCLA Anaemia, NYHA class IV, critical state, large intracardiac destruction, surgery of thoracic aorta APLs Antiphospholipid syndrome AUC Area under the curve AVB Atrioventricular block AVN Atrioventricular node BCNIE Blood culture-negative infective endocarditis BMI Body mass index CAD Coronary artery disease CHD Congenital heart disease CI Confidence interval CIED Cardiovascular implanted electronic device CNS Central nervous system CoNS Coagulase-negative staphylococci CPB Cardio-pulmonary bypass CRT Cardiac resynchronization therapy CT Computed tomography CTA Computed tomography angiography DIC Disseminated intravascular coagulation DNA Deoxyribonucleic acid DSA Digital subtraction angiography ECG Electrocardiogram EHRA European Heart Rhythm Association ESC European Society of Cardiology EUCAST European Committee on Antimicrobial Susceptibility Testing EURO-ENDO European Infective Endocarditis Registry HACEK Haemophilus, Aggregatibacter, Cardiobacterium, Eikenella, and Kingella HF Heart failure HIV Human immunodeficiency virus HLAR High-level aminoglycoside resistance i.m. Intramuscular i.v. Intravenous ICD Implantable cardioverter defibrillator ICE-PCS International Collaboration on Endocarditis-Prospective Cohort Study ICU Intensive care unit IE Infective endocarditis Ig Immunoglobulin MALDI-TOF MS Matrix-assisted laser desorption ionization time-of-flight mass spectrometry MIC Minimum inhibitory concentration MRA Magnetic resonance angiography MRI Magnetic resonance imaging MRSA Methicillin-resistant Staphylococcus aureus MSSA Methicillin-susceptible Staphylococcus aureus NBTE Non-bacterial thrombotic endocarditis NIHSS National Institutes of Health Stroke Scale Score NVE Native valve endocarditis NYHA New York Heart Association OPAT Outpatient parenteral antibiotic therapy PADIT Previous procedure on same pocket; Age; Depressed renal function; Immunocompromised; Type of procedure PALSUSE Prosthetic valve, age ≥70, large intracardiac destruction, Staphylococcus spp., urgent surgery, sex (female), EuroSCORE ≥10 PBP Penicillin-binding protein PCR Polymerase chain reaction PET/CT Positron emission tomography/computed tomography POET Partial Oral Treatment of Endocarditis (trial) PPV Positive predictive value PVE Prosthetic valve endocarditis PWID People who inject drugs RCT Randomized clinical trial RHD Rheumatic heart disease rRNA Ribosomal ribonucleic acid SAPS Simplified Acute Physiology Score SLE Systemic lupus erythematous SOT Solid organ transplantation SPECT/CT Single photon emission tomography/computed tomography STS Society of Thoracic Surgeons TAVI Transcatheter aortic valve implantation TOE Transoesophageal echocardiography TPVI Transcatheter pulmonary valve implantation TTE Transthoracic echocardiography WBC White blood cell WRAP-IT Worldwide Randomized Antibiotic Envelope Infection Prevention Trial ESC Guidelines 5 Downloaded from by guest on 30 August 2023 1. Preamble Guidelines evaluate and summarize available evidence, with the aim of as-sisting health professionals in proposing the best diagnostic or therapeutic approach for an individual patient with a given condition. Guidelines are in-tended for use by health professionals and the European Society of Cardiology (ESC) makes its Guidelines freely available. ESC Guidelines do not override the individual responsibility of health professionals to make appropriate and accurate decisions in consideration of each patient’s health condition and in consultation with that patient or the patient’s caregiver where appropriate and/or necessary. It is also the health professional’s responsibility to verify the rules and regulations ap-plicable in each country to drugs and devices at the time of prescription, and, where appropriate, to respect the ethical rules of their profession. ESC Guidelines represent the official position of the ESC on a given topic and are regularly updated. ESC Policies and Procedures for for-mulating and issuing ESC Guidelines can be found on the ESC website ( The Members of this Task Force were selected by the ESC to represent professionals involved with the medical care of patients with this Table 1 Classes of recommendations ©ESC 2023 Classes of recommendations Class I Evidence and/or general agreement that a given treatment or procedure is beneficial, useful, effective. Conflicting evidence and/or a divergence of opinion about the usefulness/ efficacy of the given treatment or procedure. Is recommended or is indicated Wording to use Definition Class III Evidence or general agreement that the given treatment or procedure is not useful/effective, and in some cases may be harmful. Is not recommended Class IIb Usefulness/efficacy is less well established by evidence/opinion. May be considered Class IIa Weight of evidence/opinion is in favour of usefulness/efficacy. Should be considered Class II © ESC 2023 Table 2 Levels of evidence Level of evidence A Level of evidence B Level of evidence C Data derived from multiple randomized clinical trials or meta-analyses. Data derived from a single randomized clinical trial or large non-randomized studies. Consensus of opinion of the experts and/or small studies, retrospective studies, registries. ©ESC 2023 © ESC 2023 6 ESC Guidelines Downloaded from by guest on 30 August 2023 Yes No Yes No Emergency Urgent Non-urgent Patient with infective endocarditis Imaging Blood culture Adjust antibiotic according to results of blood cultures Endocarditis Team Empiric antibiotic therapy i.v. Continue appropriate antibiotic therapy Complications Complications Cardiac surgery Antibiotic therapy i.v. in hospital or OPAT Patient-centred care Recovery plan Patient education - oral and general hygiene Continue appropriate antibiotic therapy Continue appropriate antibiotic therapy Endocarditis Team Figure 1 Management of patients with infective endocarditis. i.v., intravenous; OPAT, outpatient parenteral antibiotic therapy. ESC Guidelines 7 Downloaded from by guest on 30 August 2023 pathology. The selection procedure aimed to include members from across the whole of the ESC region and from relevant ESC Subspecialty Communities. Consideration was given to diversity and inclusion, notably with respect to gender and country of origin. The Task Force performed a critical evaluation of diagnostic and therapeutic approaches, including as-sessment of the risk-benefit ratio. The strength of every recommendation and the level of evidence supporting them were weighed and scored ac-cording to predefined scales as outlined below. The Task Force followed ESC voting procedures, and all approved recommendations were subject to a vote and achieved at least 75% agreement among voting members. The experts of the writing and reviewing panels provided declaration of interest forms for all relationships that might be perceived as real or po-tential sources of conflicts of interest. Their declarations of interest were reviewed according to the ESC declaration of interest rules and can be found on the ESC website ( and have been compiled in a report published in a supplementary document with the guidelines. The Task Force received its entire financial support from the ESC without any involvement from the healthcare industry. The ESC Clinical Practice Guidelines (CPG) Committee supervises and co-ordinates the preparation of new guidelines and is responsible for the approval process. ESC Guidelines undergo extensive review by the CPG Committee and external experts, including members from across the whole of the ESC region and from relevant ESC Subspecialty Communities and National Cardiac Societies. After appropriate revisions, the guidelines are signed off by all the experts involved in the Task Force. The finalized document is signed off by the CPG Committee for publica-tion in the European Heart Journal. The guidelines were developed after careful consideration of the scientific and medical knowledge and the evi-dence available at the time of their writing. Tables of evidence summariz-ing the findings of studies informing development of the guidelines are included. The ESC warns readers that the technical language may be mis-interpreted and declines any responsibility in this respect. Off-label use of medication may be presented in this guideline if a suf-ficient level of evidence shows that it can be considered medically ap-propriate for a given condition. However, the final decisions concerning an individual patient must be made by the responsible health professional giving special consideration to: • The specific situation of the patient. Unless otherwise provided for by national regulations, off-label use of medication should be limited to situations where it is in the patient’s interest with regard to the quality, safety, and efficacy of care, and only after the patient has been informed and has provided consent. • Country-specific health regulations, indications by governmental drug regulatory agencies, and the ethical rules to which health profes-sionals are subject, where applicable. 2. Introduction Infective endocarditis (IE) is a major public health challenge.1 In 2019, the estimated incidence of IE was 13.8 cases per 100 000 subjects per year, and IE accounted for 66 300 deaths worldwide.2 Due to the associated high morbidity and mortality (1723.59 disability-adjusted life years and 0.87 death cases per 100 000 population, respectively), identification of the best preventive strategies has been the focus of research.2,3 Since the publication of the 2015 ESC Guidelines for the management of infective endo-carditis,4 important new data have been published mandating an update of recommendations. First, the population at risk of IE has increased and new data on IE in different clinical scenarios have arisen.5–11 Furthermore, the emerging and increasing antibiotic resistance among oral streptococci is of concern. The rate of resistance to azythromycin and clarithromycin is higher than that to penicillin.12 Whether changes in national guidelines on the use of antibiotic prophylaxis have resulted in an increase in the incidence of IE remains unclear.13–18 It is likely that the increased use of diagnostic tools to diagnose IE is an important contributor to the increase in the inci-dence of IE. The use of echocardiography has probably increased in patients with positive blood cultures for Enteroccus faecalis, Staphylococcus aureus, or streptococci due to the associated increased risk of IE.19 In addition, com-puted tomography (CT) and nuclear imaging techniques have increased the number of definite IE cases particularly among patients with prosthetic valves and implantable cardiac devices.20–22 Data on the contemporary characterization of patients with IE have been taken into consideration to update the recommendations on the diagnosis and management of patients with IE.5,19,23–41 Furthermore, the recommendations on antibiotic therapy have been updated based on the susceptibility of various microorganisms defined by the European Committee on Antimicrobial Susceptibility Testing (EUCAST) clinical breakpoints.42 Recommendations on outpatient par-enteral antibiotic therapy (OPAT) or oral antibiotic treatment have been included based on the results of the Partial Oral Treatment of Endocarditis (POET) randomized trial and other trials.43–46 The main objective of the current Task Force was to provide clear and simple recommendations, assisting healthcare providers in their clinical decision-making. These recommendations were obtained by ex-pert consensus after thorough review of the available literature (see Supplementary data, evidence tables online). An evidence-based scor-ing system was used, based on a classification of the strength of recom-mendations and the levels of evidence. 2.1. What is new Table 3 New recommendations Recommendation Class Level Section 3. Recommendation Table 1 — Recommendations for antibiotic prophylaxis in patients with cardiovascular diseases undergoing oro-dental procedures at increased risk of infective endocarditis General prevention measures are recommended in individuals at high and intermediate risk of IE. I C Antibiotic prophylaxis is recommended in patients with ventricular assist devices. I C Antibiotic prophylaxis may be considered in recipients of heart transplant. IIb C Section 3. Recommendation Table 2 — Recommendations for infective endocarditis prevention in high-risk patients Systemic antibiotic prophylaxis may be considered for high-risk patients undergoing an invasive diagnostic or therapeutic procedure of the respiratory, gastrointestinal, genitourinary tract, skin, or musculoskeletal systems. IIb C Section 3. Recommendation Table 3 — Recommendations for infective endocarditis prevention in cardiac procedures Optimal pre-procedural aseptic measures of the site of implantation is recommended to prevent CIED infections. I B Continued 8 ESC Guidelines Downloaded from by guest on 30 August 2023 Surgical standard aseptic measures are recommended during the insertion and manipulation of catheters in the catheterization laboratory environment. I C Antibiotic prophylaxis covering for common skin flora including Enterococcus spp. and S. aureus should be considered before TAVI and other transcatheter valvular procedures. IIa C Section 5. Recommendation Table 5 — Recommendations for the role of echocardiography in infective endocarditis TOE is recommended when the patient is stable before switching from intravenous to oral antibiotic therapy. I B Section 5. Recommendation Table 6 — Recommendations for the role of computed tomography, nuclear imaging, and magnetic resonance in infective endocarditis Cardiac CTA is recommended in patients with possible NVE to detect valvular lesions and confirm the diagnosis of IE. I B [18F]FDG-PET/CT(A) and cardiac CTA are recommended in possible PVE to detect valvular lesions and confirm the diagnosis of IE. I B [18F]FDG-PET/CT(A) may be considered in possible CIED-related IE to confirm the diagnosis of IE. IIa B Cardiac CTA is recommended in NVE and PVE to diagnose paravalvular or periprosthetic complications if echocardiography is inconclusive. I B Brain and whole-body imaging (CT, [18F]FDG-PET/ CT, and/or MRI) are recommended in symptomatic patients with NVE and PVE to detect peripheral lesions or add minor diagnostic criteria. I B WBC SPECT/CT should be considered in patients with high clinical suspicion of PVE when echocardiography is negative or inconclusive and when PET/CT is unavailable. IIa C Brain and whole-body imaging (CT, [18F]FDG-PET/ CT, and MRI) in NVE and PVE may be considered for screening of peripheral lesions in asymptomatic patients. IIb B Section 7. Recommendation Table 11 — Recommendations for outpatient antibiotic treatment of infective endocarditis Outpatient parenteral antibiotic treatment should be considered in patients with left-sided IE caused by Streptococcus spp., E. faecalis, S. aureus, or CoNS who were receiving appropriate i.v. antibiotic treatment for at least 10 days (or at least 7 days after cardiac surgery), are clinically stable, and who do not show signs of abscess formation or valve abnormalities requiring surgery on TOE. IIa A Outpatient parenteral antibiotic treatment is not recommended in patients with IE caused by highly difficult-to-treat microorganisms, liver cirrhosis (Child– Pugh B or C), severe cerebral nervous system emboli, untreated large extracardiac abscesses, heart valve complications, or other severe conditions requiring surgery, severe post-surgical complications, and in PWID-related IE. III C Continued Section 9. Recommendation Table 13 — Recommendations for the treatment of neurological complications of infective endocarditis In embolic stroke, mechanical thrombectomy may be considered if the expertise is available in a timely manner. IIb C Thrombolytic therapy is not recommended in embolic stroke due to IE. III C Section 9. Recommendation Table 14 — Recommendations for pacemaker implantation in patients with complete atrioventricular block and infective endocarditis Immediate epicardial pacemaker implantation should be considered in patients undergoing surgery for valvular IE and complete AVB if one of the following predictors of persistent AVB is present: pre-operative conduction abnormality, S. aureus infection, aortic root abscess, tricuspid valve involvement, or previous valvular surgery. IIa C Section 9. Recommendation Table 15 — Recommendations for patients with musculoskeletal manifestations of infective endocarditis MRI or PET/CT is recommended in patients with suspected spondylodiscitis and vertebral osteomyelitis complicating IE. I C TTE/TOE is recommended to rule out IE in patients with spondylodiscitis and/or septic arthritis with positive blood cultures for typical IE microorganisms. I C More than 6-week antibiotic therapy should be considered in patients with osteoarticular IE-related lesions caused by difficult-to-treat microorganisms, such as S. aureus or Candida spp., and/or complicated with severe vertebral destruction or abscesses. IIa C Section 10. Recommendation Table 16 — Recommendations for pre-operative coronary anatomy assessment in patients requiring surgery for infective endocarditis In haemodynamically stable patients with aortic valve vegetations who require cardiac surgery and are high risk of CAD, a high-resolution multislice coronary CTA is recommended. I B Invasive coronary angiography is recommended in patients requiring heart surgery who are high risk of CAD, in the absence of aortic valve vegetations. I C In emergency situations, valvular surgery without pre-operative coronary anatomy assessment regardless of CAD risk should be considered. IIa C Invasive coronary angiography may be considered despite the presence of aortic valve vegetations in selected patients with known CAD or at high risk of significant obstructive CAD. IIb C Section 10. Recommendation Table 17 — Indications and timing of cardiac surgery after neurological complications in active infective endocarditis In patients with intracranial haemorrhage and unstable clinical status due to HF, uncontrolled infection, or persistent high embolic risk, urgent or emergency surgery should be considered weighing the likelihood of a meaningful neurological outcome. IIa C Continued ESC Guidelines 9 Downloaded from by guest on 30 August 2023 Section 11. Recommendation Table 18 — Recommendations for post-discharge follow-up Patient education on the risk of recurrence and preventive measures, with emphasis on dental health, and based on the individual risk profile, is recommended during follow-up. I C Addiction treatment for patients following PWID-related IE is recommended. I C Cardiac rehabilitation including physical exercise training should be considered in clinically stable patients based on an individual assessment. IIa C Psychosocial support may be considered to be integrated in follow-up care, including screening for anxiety and depression, and referral to relevant psychological treatment. IIb C Section 12. Recommendation Table 19 — Recommendations for prosthetic valve endocarditis Surgery is recommended for early PVE (within 6 months of valve surgery) with new valve replacement and complete debridement. I C Section 12. Recommendation Table 20 — Recommendations for cardiovascular implanted electronic device-related infective endocarditis Complete system extraction without delay is recommended in patients with definite CIED-related IE under initial empirical antibiotic therapy. I B Extension of antibiotic treatment of CIED-related endocarditis to (4–)6 weeks following device extraction should be considered in the presence of septic emboli or prosthetic valves. IIa C Use of an antibiotic envelope may be considered in select high-risk patients undergoing CIED reimplantation to reduce risk of infection. IIb B Continued Table 4 Revised recommendations Recommendations in 2015 version Class Level Recommendations in 2023 version Class Level Section 3. Recommendation Table 1 — Recommendations for antibiotic prophylaxis in patients with cardiovascular diseases undergoing oro-dental procedures at increased risk of infective endocarditis Antibiotic prophylaxis should be considered for patients at highest risk of IE: (1) Patients with any prosthetic valve, including a transcatheter valve, or those in whom any prosthetic material was used for cardiac valve repair. (2) Patients with a previous episode of IE. (3) Patients with CHD: (a) Any type of cyanotic CHD. (b) Any type of CHD repaired with a prosthetic material, whether placed surgically or by IIa C Antibiotic prophylaxis is recommended in patients with previous IE. I B Antibiotic prophylaxis is recommended in patients with surgically implanted prosthetic valves and with any material used for surgical cardiac valve repair. I C Antibiotic prophylaxis is recommended in patients with transcatheter implanted aortic and pulmonary valvular prostheses. I C Antibiotic prophylaxis should be considered in patients with transcatheter mitral and tricuspid valve repair. IIa C Continued In non-S. aureus CIED-related endocarditis without valve involvement or lead vegetations, and if follow-up blood cultures are negative without septic emboli, 2 weeks of antibiotic treatment may be considered following device extraction. IIb C Removal of CIED after a single positive blood culture, with no other clinical evidence of infection, is not recommended. III C Section 12. Recommendation Table 21 — Recommendations for the surgical treatment of right-sided infective endocarditis Tricuspid valve repair should be considered instead of valve replacement, when possible. IIa B Surgery should be considered in patients with right-sided IE who are receiving appropriate antibiotic therapy and present persistent bacteraemia/sepsis after at least 1 week of appropriate antibiotic therapy. IIa C Prophylactic placement of an epicardial pacing lead should be considered at the time of tricuspid valve surgical procedures. IIa C Debulking of right intra-atrial septic masses by aspiration may be considered in select patients who are high risk of surgery. IIb C © ESC 2023 [18F]FDG-PET, 18F-fluorodeoxyglucose positron emission tomography; AVB, atrioventricular block; CAD, coronary artery disease; CIED, cardiovascular implanted electronic device; CoNS, coagulase-negative staphylococci; CT, computed tomography; CTA, computed tomography angiography; HF, heart failure; IE, infective endocarditis; i.v., intravenous; MRI, magnetic resonance imaging; NVE, native valve endocarditis; PET, positron emission tomography; PVE, prosthetic valve endocarditis; PWID, people who inject drugs; TAVI, transcatheter aortic valve implantation; TOE, transoesophageal echocardiography; TTE, transthoracic echocardiography; WBC SPECT/CT, white blood cell single photon emission tomography/computed tomography. 10 ESC Guidelines Downloaded from by guest on 30 August 2023 percutaneous techniques, up to 6 months after the procedure or lifelong if residual shunt. Antibiotic prophylaxis is recommended in patients with untreated cyanotic CHD, and patients treated with surgery or transcatheter procedures with post-operative palliative shunts, conduits, or other prostheses. After surgical repair, in the absence of residual defects or valve prostheses, antibiotic prophylaxis is recommended only for the first 6 months after the procedure. I C Section 4. Recommendation Table 4 — Recommendations for the Endocarditis Team Patients with complicated IE should be evaluated and managed at an early stage in a reference centre, with immediate surgical facilities and the presence of a multidisciplinary ‘Endocarditis Team’, including an infectious disease specialist, a microbiologist, a cardiologist, imaging specialists, a cardiac surgeon and, if needed, a specialist in CHD. IIa B Diagnosis and management of patients with complicated IE are recommended to be performed at an early stage in a Heart Valve Centre, with immediate surgical facilities and an ‘Endocarditis Team’ to improve the outcomes. I B For patients with uncomplicated IE managed in a non-reference centre, early and regular communication with the reference centre and, when needed, visits to the reference centre should be made. IIa B For patients with uncomplicated IE managed in a Referring Centre, early and regular communication between the local and the Heart Valve Centre Endocarditis Teams is recommended to improve the outcomes of the patients. I B Section 5. Recommendation Table 5 — Recommendations for the role of echocardiography in infective endocarditis TOE should be considered in patients with suspected IE, even in cases with positive TTE, except in isolated right-sided native valve IE with good quality TTE examination and unequivocal echocardiographic finding. IIa C TOE is recommended in patients with suspected IE, even in cases with positive TTE, except in isolated right-sided native valve IE with good quality TTE examination and unequivocal echocardiographic findings. I C Section 8. Recommendation Table 12 — Recommendations for the main indications of surgery in infective endocarditis (native valve endocarditis and prosthetic valve endocarditis) Aortic or mitral NVE with vegetations >10 mm, associated with severe valve stenosis or regurgitation, and low operative risk (urgent surgery should be considered). IIa B Urgent surgery is recommended in IE with vegetation ≥10 mm and other indications for surgery. I C Aortic or mitral NVE or PVE with isolated large vegetations (>15 mm) and no other indication for surgery (urgent surgery may be considered). IIb C Urgent surgery may be considered in aortic or mitral IE with vegetation ≥10 mm and without severe valve dysfunction or without clinical evidence of embolism and low surgical risk. IIb B Section 9. Recommendation Table 13 — Recommendations for the treatment of neurological complications of infective endocarditis Intracranial infectious aneurysms should be looked for in patients with IE and neurological symptoms. CT or MRA should be considered for diagnosis. If non-invasive techniques are negative and the suspicion of intracranial aneurysm remains, conventional angiography should be considered. IIa B Brain CT or MRA is recommended in patients with IE and suspected infective cerebral aneurysms. I B If non-invasive techniques are negative and the suspicion of infective aneurysm remains, invasive angiography should be considered. IIa B Section 12. Recommendation Table 20 — Recommendations for cardiovascular implanted electronic device-related infective endocarditis Routine antibiotic prophylaxis is recommended before device implantation. I B Antibiotic prophylaxis covering S. aureus is recommended for CIED implantation. I A TOE is recommended in patients with suspected cardiac device-related infective endocarditis with positive or negative blood cultures, independent of the results of TTE, to evaluate lead-related endocarditis and heart valve infection. I C TTE and TOE are both recommended in case of suspected CIED-related IE to identify vegetations. I B In patients with NVE or PVE and an intracardiac device with no evidence of associated device infection, complete hardware extraction may be considered. IIb C Complete CIED extraction should be considered in case of valvular IE, even without definite lead involvement, taking into account the identified pathogen and requirement for valve surgery. IIa C Continued ESC Guidelines 11 Downloaded from by guest on 30 August 2023 3. Prevention 3.1. Rationale The development of IE usually requires several conditions, including the presence of predisposing risk factors (i.e. a surface/structure that could be colonized by bacteria), pathogens entering the bloodstream, and the competence of the host’s immune response. The role of predisposing risk factors has been recently underscored by Thornhill et al.47 Predisposing risk factors conveying a moderate and high risk of IE had an incidence of 280 and 497 cases per 100 000 subjects per year, respectively.47 The portals of entry of bacteria/fungi are variable and include: (i) in-fections of the skin, oral cavity, gastrointestinal, or genitourinary system; (ii) direct inoculation in people who inject drugs (PWID), or by any un-safe or unprotected vascular puncture; (iii) healthcare exposure (in-cluding a variety of invasive diagnostic or therapeutic procedures, such as transcatheter or surgical techniques).6,11,48–50 The oral cavity is colonized by relevant commensal flora, including oral group streptococci, and represents an important entry port. Oral surgery procedures (including all extractions, periodontal surgery, implant surgery, and oral biopsies) and dental procedures that involve manipulation of the gingival or periapical region of the teeth are consid-ered at high risk of causing bacteraemia.11,48,49,51 Successful antibiotic prophylaxis assumes that reducing the bacter-aemia associated with medical procedures will lead to a reduced risk of IE. This concept was supported by a few animal models and obser-vational studies that led to the recommendation for antibiotic prophy-laxis in a large number of patients with predisposing cardiac conditions undergoing a wide range of procedures.4,14,52–60 However, systematic use of antibiotic prophylaxis has been ques-tioned based on several considerations, the most important being the lack of randomized clinical trials (RCTs) demonstrating the efficacy of antibiotic prophylaxis prior to medical procedures in preventing IE. Such trials would entail enrolment of a very large number of individuals and prolonged follow-up, making the feasibility of such studies improb-able. Furthermore, since the standard of care for high-risk individuals is antibiotic prophylaxis (to date, mostly before invasive oro-dental pro-cedures), there may not be sufficient equipoise to perform such RCTs. Finally, the costs of performing such trials have been considered un-acceptable.61 To overcome these limitations, population-based studies Complete hardware removal should be considered on the basis of occult infection without another apparent source of infection. IIa C In cases of possible CIED-related IE or occult Gram-positive bacteraemia or fungaemia, complete system removal should be considered in case bacteraemia/fungaemia persists after a course of antimicrobial therapy. IIa C In cases of possible CIED-related IE with occult Gram-negative bacteraemia, complete system removal may be considered in case of persistent/relapsing bacteraemia after a course of antimicrobial therapy. IIb C When indicated, definite reimplantation should be postponed if possible, to allow a few days or weeks of antibiotic therapy. IIa C If CIED reimplantation is indicated after extraction for CIED-related IE, it is recommended to be performed at a site distant from the previous generator, as late as possible, once signs and symptoms of infection have abated and until blood cultures are negative for at least 72 h in the absence of vegetations, and negative for at least 2 weeks if vegetations were visualized. I C Section 12. Recommendation Table 21 — Recommendations for the surgical treatment of right-sided infective endocarditis Surgical treatment should be considered in the following scenarios: Surgery is recommended in patients with right-sided IE who are receiving appropriate antibiotic therapy for the following scenarios: • Microorganisms difficult to eradicate (e.g. persistent fungi) or bacteraemia for >7 days (e.g. S. aureus, P. aeruginosa) despite adequate antimicrobial therapy; or • Persistent tricuspid valve vegetations >20 mm after recurrent pulmonary emboli with or without concomitant right HF; or • Right HF secondary to severe tricuspid regurgitation with poor response to diuretic therapy. IIa C Right ventricular dysfunction secondary to acute severe tricuspid regurgitation non-responsive to diuretics. I B Persistent vegetation with respiratory insufficiency requiring ventilatory support after recurrent pulmonary emboli. I B Large residual tricuspid vegetations (>20 mm) after recurrent septic pulmonary emboli. I C Patients with simultaneous involvement of left-heart structures. I C Section 12. Recommendation Table 22 — Recommendations for the use of antithrombotic therapy in infective endocarditis Interruption of antiplatelet therapy is recommended in the presence of major bleeding. I B Interruption of antiplatelet or anticoagulant therapy is recommended in the presence of major bleeding (including intracranial haemorrhage). I C © ESC 2023 CHD, congenital heart disease; CIED, cardiovascular implanted electronic device; CT, computed tomography; HF, heart failure; IE, infective endocarditis; MRA, magnetic resonance angiography; NVE, native valve endocarditis; PVE, prosthetic valve endocarditis; TOE, transoesophageal echocardiography; TTE, transthoracic echocardiography. 12 ESC Guidelines Downloaded from by guest on 30 August 2023 have evaluated the efficacy of antibiotic prophylaxis using bacteraemia as a surrogate of IE.16–18,52,62 However, the relationship between bac-teraemia and IE is not straightforward. Bacteraemia may be caused by daily activities such as tooth brushing, flossing, and chewing, and al-though these constitute low-level bacteraemia, they occur repetitively and may therefore outweigh the risk of bacteraemia associated with dental procedures.48,49 A meta-analysis of 36 studies, including 21 trials that investigated the effect of antibiotic prophylaxis on the incidence of bacteraemia following dental procedures, demonstrated that antibiotic prophylaxis is effective in reducing the incidence of bacteraemia, but did not lead to a statistically significant protective effect against IE in case- control studies.52 Additionally, the potential risk of anaphylaxis,63 or other adverse side effects in a small minority of patients, and the fact that a widespread use of antibiotics may be associated with antibiotic resistance, are areas of concern.57,58,64–67 While some studies did not demonstrate significant increases in IE-related hospitalizations and death rates after scaling down antibiotic prophylaxis indications,68–77 others showed an increase in the incidence of IE among individuals at moderate and high risk of IE.13,26,59,78–81 A meta-analysis including 16 studies reporting over 1.3 million cases of IE has shown that restricting antibiotic prophylaxis to only high-risk individuals has not resulted in an increase in the incidence of streptococcal IE in a North American popu-lation (despite the fact that it was unable to draw that conclusion for other populations).18 In contrast, a systematic review including multiple nationwide population-based studies in Europe has shown a 4% per year rise in the incidence of IE.82 These contrasting results may be ex-plained by differences in the methodology of the studies (retrospective, population- or health-system-based studies that relied on claims data or epidemiological observations to estimate the incidence of IE), greater disease diagnosis with the use of newer imaging technologies, lack of microbiological data, and the lack of specific International Classification of Diseases codes for oral streptococci.83 Recently, it has been shown that antibiotic prophylaxis in high-risk individuals was associated with a significant reduction of IE after invasive dental proce-dures (particularly extractions and oral surgical procedures).11,51 After careful consideration of all the new studies published after 2015, the present Task Force decided to revise and update the risk categories for IE, strengthening the recommendation of antibiotic prophylaxis, clarifying the definition of the population at risk, and considering the ad-vances in transcatheter valve interventions. 3.2. Populations at risk of infective endocarditis The groups of individuals at high risk of IE in whom antibiotic prophy-laxis is recommended or should be considered include the following: (i) Patients with previous IE: the highest risk of IE is observed in pa-tients with previous history of IE who have an ominous prognosis during IE-related hospitalization. Patients with recurrent IE more frequently have prosthetic valves or prosthetic material, are more commonly PWID, or have staphylococcal IE.47,84–86 (ii) Patients with surgically implanted prosthetic valves, with transcath-eter implanted prosthetic valves, and with any material used for cardiac valve repair: the increased risk of IE in these patients, com-bined with the ominous outcomes as compared with patients with native IE (NVE), make antibiotic prophylaxis advisable in this pa-tient group. Patients with prosthetic valve endocarditis (PVE) have an in-hospital mortality rate that is twice as high with more complications (e.g. heart failure [HF], conduction disturbances) as compared with patients with NVE, regardless of the pathogen.87,88 Furthermore, mitral and aortic bioprostheses may be associated with increased risk of IE as compared with mechan-ical prostheses,89,90 and bioprostheses are being implanted in an ever-increasing proportion of patients requiring valve replacement therapy. The indication for prophylaxis also expands to transcath-eter aortic and pulmonic prosthetic valves, since IE is also asso-ciated with a high risk of morbidity and mortality in these patients.91–94 In terms of transcatheter mitral and tricuspid valve interventions, the data on the risk of IE are limited.95 Patients with septal defect closure devices, left atrial appendage closure de-vices, vascular grafts, vena cava filters, and central venous system ventriculo-atrial shunts are considered within this risk category in the first 6 months after implantation.96 (iii) Patients with congenital heart disease (CHD) (not including iso-lated congenital valve abnormalities) are at increased risk of IE.8,47,97–99 The cumulative incidence over time is influenced strongly by the improved long-term survival of children with CHD into adulthood.98 Indeed, there are now more adults living with CHD than children with CHD.100 The overall incidence rate of IE among adult patients with CHD is 27–44 times that re-ported for contemporary adults of the general population (1.33 cases per 1000 persons per year)8 while in children with CHD the incidence of IE is 0.41 cases per 1000 persons per year.101 CHD groups at increased risk include those with untreated cyan-otic CHD, and those whose surgery includes prosthetic material, including valved conduits or systemic to pulmonary shunts.8,47,97 The risk of post-operative IE for CHD patients undergoing trans-catheter atrial or ventricular septal defect closure with devices or surgery with non-valve-related prosthetic material is also in-creased, but predominantly for the first 6 months after surgery.8 (iv) Patients with ventricular assist devices as destination therapy are also considered at high risk because of associated morbidity and mortality, and prophylaxis is also recommended in such patients.102 Patients at intermediate risk of IE include those with: (i) rheumatic heart disease (RHD); (ii) non-rheumatic degenerative valve disease; (iii) congenital valve abnormalities including bicuspid aortic valve dis-ease; (iv) cardiovascular implanted electronic devices (CIEDs); and (v) hypertrophic cardiomyopathy.47,103,104 Some epidemiological data sug-gest that certain conditions stratified as intermediate risk are associated Table 5 General prevention measures to be followed in patients at high and intermediate risk of infective endocarditis Patients should be encouraged to maintain twice daily tooth cleaning and to seek professional dental cleaning and follow-up at least twice yearly for high-risk patients and yearly for others. Strict cutaneous hygiene, including optimized treatment of chronic skin conditions. Disinfection of wounds. Curative antibiotics for any focus of bacterial infection. No self-medication with antibiotics. Strict infection control measures for any at-risk procedure. Discouragement of piercing and tattooing. Limitation of infusion catheters and invasive procedures. when possible. Strict adherence to care bundles for central and peripheral cannulae should be performed. © ESC 2023 ESC Guidelines 13 Downloaded from by guest on 30 August 2023 with a higher risk of IE compared with the background popula-tion,47,90,103 but further studies are required. In patients at intermediate risk of IE, antibiotic prophylaxis is not routinely recommended and may be considered on an individual basis. However, prevention measures (Table 5) are strongly encouraged in these patients.7 Most of the IE in recipients of solid organ transplant is nosocomial. A recent systematic review of patient-level data including 57 heart trans-plant patients has shown that IE occurs frequently during the first year post-transplant, and the most common pathogen is S. aureus followed by Aspergillus fumigatus.105 Oral streptococci are a very infrequent cause of IE, making the value of antibiotic prophylaxis after invasive oro- dental procedures questionable. However, IE in this group of patients is associated with very high mortality, particularly in patients with fungal IE. In contrast, other series that include a larger proportion of non- cardiac solid organ transplant patients have shown that the pathogens are more frequently from the Staphylococcus spp. and the mortality seems to be similar to that of patients without solid organ transplant.106,107 3.3. Situations and procedures at risk 3.3.1. Dental procedures Antibiotic prophylaxis is recommended in patients at high risk of IE under-going at-risk dental procedures and is not currently recommended in other situations. At-risk dental procedures include dental extractions, oral sur-gery procedures (including periodontal surgery, implant surgery, and oral biopsies), and dental procedures involving manipulation of the gingival or periapical region of the teeth (including scaling and root canal proce-dures).49,108 The use of dental implants raises concerns about potential risk due to foreign material at the interface between the buccal cavity and blood, but available data remain very limited.109 So far there is no evi-dence to contraindicate implants in all patients at risk and the indication should be discussed on an individual basis. Implant placement procedures, and invasive dental procedures on established implants, however, should be covered by antibiotic prophylaxis in those at high risk of IE. Once dental implants are placed in high-risk patients, professional dental hygiene and follow-up should be performed at least twice yearly under antibiotic cover, when indicated. The main target for antibiotic prophylaxis is oral streptococci. Table 6 summarizes the main regimens of antibiotic prophylaxis recom-mended before dental procedures. The risk of adverse fatal/non-fatal events appear to be extremely low for amoxicillin but high for clinda-mycin (mainly related to Clostridioides difficile infections).63,110–112 Accordingly, this Task Force does not recommend the use of clindamy-cin for antibiotic prophylaxis. 3.3.2. Non-dental procedures No convincing evidence has been brought forward on the relationship between bacteraemia resulting from a non-dental procedure and risk of subsequent IE. However, observational studies reported that, com-pared with patients with IE not undergoing an invasive procedure, Recommendation Table 1 — Recommendations for antibiotic prophylaxis in patients with cardiovascular diseases undergoing oro-dental procedures at increased risk for infective endocarditis Recommendations Classa Levelb General prevention measures are recommended in individuals at high and intermediate risk for IE. I C Antibiotic prophylaxis is recommended in patients with previous IE.47,84,86 I B Antibiotic prophylaxis is recommended in patients with surgically implanted prosthetic valves and with any material used for surgical cardiac valve repair.47,87–89 I C Antibiotic prophylaxis is recommended in patients with transcatheter implanted aortic and pulmonary valvular prostheses.91–94 I C Antibiotic prophylaxis is recommended in patients with untreated cyanotic CHD, and patients treated with surgery or transcatheter procedures with post-operative palliative shunts, conduits, or other prostheses. After surgical repair, in the absence of residual defects or valve prostheses, antibiotic prophylaxis is recommended only for the first 6 months after the procedure.8,47,97,101 I C Antibiotic prophylaxis is recommended in patients with ventricular assist devices.102 I C Antibiotic prophylaxis should be considered in patients with transcatheter mitral and tricuspid valve repair.95 IIa C Antibiotic prophylaxis may be considered in recipients of heart transplant.105–107 IIb C Antibiotic prophylaxis is not recommended in other patients at low risk for IE.11,51 III C © ESC 2023 CHD, congenital heart disease; IE, infective endocarditis. aClass of recommendation. bLevel of evidence. Table 6 Prophylactic antibiotic regime for high-risk dental procedures Situation Antibiotic Single-dose 30–60 min before procedure Adults Children No allergy to penicillin or ampicillin Amoxicillin 2 g orally 50 mg/kg orally Ampicillin 2 g i.m. or i.v. 50 mg/kg i.v. or i.m. Cefazolin or ceftriaxone 1 g i.m. or i.v. 50 mg/kg i.v. or i.m. Allergy to penicillin or ampicillin Cephalexina,b 2 g orally 50 mg/kg orally Azithromycin or clarithromycin 500 mg orally 15 mg/kg orally Doxycycline 100 mg orally <45 kg, 2.2 mg/kg orally >45 kg, 100 mg orally Cefazolin or ceftriaxoneb 1 g i.m. or i.v. 50 mg/kg i.v. or i.m. © ESC 2023 i.m., intramuscular; i.v., intravenous. aOr other first- or second-generation oral cephalosporin in equivalent adult or paediatric dosing. bCephalosporins should not be used in an individual with a history of anaphylaxis, angioedema, or urticarial with penicillin or ampicillin. 14 ESC Guidelines Downloaded from by guest on 30 August 2023 several invasive non-dental medical procedures were associated with increased risk of IE, including cardiovascular interventions, skin proce-dures and wound management, transfusion, dialysis, bone marrow puncture, and endoscopic procedures.6,11,51 For this reason, an aseptic operational environment should be ensured during all these proce-dures to minimize the risk of IE. As previously indicated, it is very unlike-ly that an RCT on antibiotic prophylaxis for IE will be performed in the foreseeable future. However, at-risk patients have longer survival due to the advent of newer medical and device-based medical therapies. In addition, the ageing general population with their accumulating num-ber of co-morbidities has an increased risk of surgical therapy, if IE oc-curs. For these reasons, this Task Force no longer felt that a class III recommendation for antibiotic prophylaxis in high-risk patients under-going non-dental medical procedures (see Recommendation Table 2) was appropriate, despite the limitations of observational data used to support this class IIb recommendation. 3.3.3. Cardiac or vascular interventions In all patients undergoing implantation of a prosthetic valve, any type of prosthetic graft/occluder device or CIED, peri-operative antibiotic prophylaxis is recommended due to the increased risk and adverse out-come of an infection.6 The most frequent microorganisms underlying early (1 year after surgery) surgical PVE are coagulase-negative staphylococci (CoNS) and S. aureus. Pre-operative screening of nasal carriage for S. aureus is recommended before elective cardiac surgery or transcatheter valve implantation to treat carriers using local mupir-ocin and chlorhexidine.113,114 Rapid identification techniques using gene amplification are useful to avoid delaying urgent surgery. Systematic lo-cal treatment without screening is not recommended. It is strongly re-commended that potential sources of dental sepsis should be eliminated at least 2 weeks before implantation of a prosthetic valve or other intracardiac or intravascular foreign material unless the latter procedure is urgent. For specific prophylactic measures in other cardiac and vascular interventions (i.e. CIED, transcatheter aortic valve implant-ation [TAVI]), please see the Supplementary data online, Section S1.1. 3.4. Patient education Preventing IE also depends on preventive measures other than antibiot-ic prophylaxis. People at risk should be educated to maintain good den-tal and skin hygiene, to look out for signs of infection and, when Education of high-risk patients to prevent infective endocarditis Use dental floss daily Brush teeth morning and evening See your dentist for regular check-ups Maintain good dental hygiene If experiencing fever for no obvious reason, contact your doctor, and discuss appropriate action based on your risk of endocarditis Minimize risk of skin lesions In case of lesions, observe for signs of infection (redness, swelling, tenderness, puss) Avoid tattoos and piercings Maintain good skin hygiene Be mindful of infections Do not self prescribe antibiotics Show this card to your doctors before any interventions Figure 2 Education of high-risk patients to prevent infective endocarditis. ESC Guidelines 15 Downloaded from by guest on 30 August 2023 experiencing fever of unknown origin, report to their physician that they are at risk, in which case clinicians should consider screening for IE before initiating antibiotics. Use of non-medical language, visual aids, digital tools, repetition, and teach back methods all aid the patients’ comprehension and is encour-aged.115 National cardiology societies should be encouraged to develop specific IE cards for patient awareness (Figure 2). 4. The Endocarditis Team The importance of an Endocarditis Team in the diagnosis, management, and clinical outcomes of patients with IE has been demonstrated in sev-eral observational studies.36–41,122–126 Establishing multidisciplinary endocarditis teams according to the European Society of Cardiology (ESC) and the American College of Cardiology/American Heart Association Guidelines4,127,128 has resulted in earlier and more accurate diagnosis of the primary disease and its complications,5,22,31,40,129 uni-form antibiotic treatment,36,40,123 and optimized timing for surgical intervention.36,37,40,123 A variety of scenarios of patients presenting with IE justifies a multidisciplinary approach.5,25,27,28,130–135 Furthermore, the clinical presentation may vary significantly depending on the characteristics of the host and virulence of the microorganism. Accordingly, the concept of the Endocarditis Team needs to embrace a multidisciplinary approach that must adapt according to the patient’s clinical needs and the local epidemiology to ensure prompt diagnosis and treatment. The members of the Endocarditis Team should include the specialists with direct involvement in the diagnostic and therapeutic processes (Table 7), and may vary depending on the type of centre. In the Recommendation Table 2 — Recommendations for infective endocarditis prevention in high-risk patients Recommendations Classa Levelb Antibiotic prophylaxis is recommended in dental extractions, oral surgery procedures, and procedures requiring manipulation of the gingival or periapical region of the teeth.11,49,51,108 I B Systemic antibiotic prophylaxis may be considered for high-riskc patients undergoing an invasive diagnostic or therapeutic procedure of the respiratory, gastrointestinal, genitourinary tract, skin, or musculoskeletal systems.6,11 IIb C © ESC 2023 aClass of recommendation. bLevel of evidence. cThis recommendation does not apply to patients with intermediate risk for IE or to the general population. Recommendation Table 3 — Recommendations for infective endocarditis prevention in cardiac procedures Recommendations Classa Levelb Pre-operative screening for nasal carriage of S. aureus is recommended before elective cardiac surgery or transcatheter valve implantation to treat carriers.113,114 I A Peri-operative antibiotic prophylaxis is recommended before placement of a CIED.116–118 I A Optimal pre-procedural aseptic measures of the site of implantation is recommended to prevent CIED infections.119 I B Periprocedural antibiotic prophylaxis is recommended in patients undergoing surgical or transcatheter implantation of a prosthetic valve, intravascular prosthetic, or other foreign material.120 I B Surgical standard aseptic measures are recommended during the insertion and manipulation of catheters in the catheterization laboratory environment. I C Elimination of potential sources of sepsis (including of dental origin) should be considered ≥2 weeks before implantation of a prosthetic valve or other intracardiac or intravascular foreign material, except in urgent procedures. IIa C Continued Antibiotic prophylaxis covering for common skin flora including Enterococcus spp. and S. aureus should be considered before TAVI and other transcatheter valvular procedures.121 IIa C Systematic skin or nasal decolonization without screening for S. aureus is not recommended. III C © ESC 2023 CIED, cardiac implantable electronic device; TAVI, transcatheter aortic valve implantation. aClass of recommendation. bLevel of evidence. Table 7 Members of the Endocarditis Team Heart Valve Centre Core members • Cardiologists. • Cardiac imaging experts. • Cardiovascular surgeons. • Infectious disease specialist (or internal medicine specialist with expertise in infectious diseases). • Microbiologist. • Specialist in outpatient parenteral antibiotic treatment. Adjunct specialities • Radiologist and nuclear medicine specialist. • Pharmacologist. • Neurologist and neurosurgeon. • Nephrologist. • Anaesthesiologists. • Critical care. • Multidisciplinary addiction medicine teams. • Geriatricians. • Social worker. • Nurses. • Pathologist. © ESC 2023 16 ESC Guidelines Downloaded from by guest on 30 August 2023 Heart Valve Centre, a centre having all diagnostic and therapeutic re-sources to treat IE, the core members of the Endocarditis Team should include cardiologists, cardiovascular surgeons, infectious disease specia-lists (or internal medicine specialists with expertise in infectious dis-eases), and microbiologists. Furthermore, for specific clinical questions, cardiologists/surgeons with expertise in CIED extraction, HF, and CHD; pathologists; critical care specialists; cardiac anaesthesiol-ogists; interventional cardiologists; neurologists and neurosurgeons; pharmacologists; radiologists and nuclear medicine specialists; nephrologists; geriatricians; and multidisciplinary addiction medicine teams (psychiatrists, nurses, and social work specialists providing coun-selling) are crucial adjuncts that should be available onsite for consult-ation. Specific subgroups of complex and high-risk patients are frequently assessed by the Endocarditis Team. The decision-making process may involve difficult decisions regarding continuation of ther-apy, and legal counsel may therefore be required. Cardiovascular imaging has achieved such an advanced sophistication in the diagnosis of IE that the cardiologists with expertise in N Start empirical antibiotic treatment Admission to a Heart Valve Centre Admission to a Referring Centre Adjust antibiotic treatment according to blood culture results Patient with defnite infective endocarditis 2023 ESC diagnostic criteria Y Endocarditis Team Cardiologists Cardiovascular surgeons Infecious disease specialists Microbiologists Adjunct specialities Radiologists/nuclear medicine specialists Neurologists/ neurosurgeons Intensive care specialists Geriatricians Nephrologists Nurses Addiction medicine teams Establish indication and timing of cardiac surgery Consultation with outpatient antibiotic therapy team Frequent communication Clinical, microbiological, and imaging data sharing Endocarditis Team Cardiologists Infectious disease specialists Complicated clinical evolution Unstable haemodynamic condition under pharmacological and/or respiratory support Severe valvular regurgitation (clinical and echocardiographic criteria) Prosthetic valve endocarditis with or without prosthetic valve dysfunction Stroke (ischaemic or haemorrhagic) with definite or possible IE Extravalvular complications (abscesses, fistulae, etc.) Positive blood cultures >7 days under appropriate antibiotic therapy Embolism CIED-related infective endocarditis Aggressive or difficult-to-treat microorganisms (S. aureus, Gram-negative bacilli, fungi) Figure 3 Management of patients with infective endocarditis: positioning of the Endocarditis Team. CIED, cardiovascular implanted electronic device; ESC, European Society of Cardiology. ESC Guidelines 17 Downloaded from by guest on 30 August 2023 multimodality imaging are key in the Endocarditis Team. In addition, radiology and nuclear medicine specialists with expertise on clinical car-diovascular imaging should be available whenever indicated.22,31,129 The Endocarditis Team must meet on a frequent basis and work with stand-ard operating procedures and the clinical governance arrangements de-fined locally.128,136 Although the decision of timing is left to the discretion of the local team, a weekly meeting is to be considered. In Referring Centres, i.e. those without a cardiovascular surgical team, the treating physician diagnosing IE should consult with a special-ist in infectious diseases (or an internal medicine specialist with expert-ise in infectious diseases) and the microbiologist.136 In addition, a cardiologist with expertise in valvular heart disease and cardiac imaging should be present to provide the initial and subsequent evaluations with echocardiography. Information of the strains of the isolated microor-ganisms, usually kept for 7–15 days, should be provided to the Heart Valve Centre if requested. Communication between Referring Centres and the Heart Valve Centres should be facilitated with digital solutions that enable reliable data sharing. Early referral to the Heart Valve Centre for further diag-nostic testing and clinical management should be available when deemed necessary (Figure 3). When there is evidence of failure to re-spond to the antibiotic therapy or there are complications related to valvular tissue destruction, the Referring Centre should consult the Heart Valve Centre. The Endocarditis Team of the Heart Valve Centre should share protocols with the physicians from the referring hospitals and should facilitate their continuing education.136 A critical aspect of the Endocarditis Team decision-making process is defining when a patient must be transferred to a Heart Valve Centre to expedite advanced diagnostics and therapy. The indications for transfer are comprehensive, to facilitate interhospital communication and avoid delaying therapy to improve prognosis. 5. Diagnosis The diagnosis of IE is based on a clinical suspicion supported by consist-ent microbiological data and the documentation of IE-related cardiac lesions by imaging techniques. Evidence of involvement of cardiac valves (native or prosthetic) or prosthetic intracardiac material is a major diag-nostic criterion of IE. Echocardiography is the first-line diagnostic imaging technique. Other imaging modalities such as CT, nuclear im-aging, and magnetic resonance imaging (MRI) are currently part of the diagnostic strategy of suspected IE, given their ability to provide key in-formation to confirm IE diagnosis, to assess local IE complications as well as IE-related distant lesions, and to identify the original source of bacteraemia in patients who develop secondary IE.137 Beyond diagnosis of IE, imaging findings also have prognostic implications. 5.1. Clinical features Infective endocarditis remains a diagnostic challenge due to its variable clinical presentation. In general, a diagnosis of IE should be considered in all patients with sepsis or fever of unknown origin in the presence of risk factors. Infective endocarditis may present as an acute, rapidly progres-sive infection, but also as a subacute or chronic disease with low-grade, or even no fever, and non-specific symptoms that may mislead or con-fuse initial assessment. Infective endocarditis can also present with a complication mimicking a wide range of medical conditions that may prompt evaluation of other diseases, such as rheumatological, neuro-logical, and autoimmune disorders, or even malignancy, before reaching a diagnosis of IE. Therefore, high suspicion for IE is generally driven by fever and positive blood cultures in the absence of an alternative focus of infection, especially in patients with one or more risk factors. Early involvement of the Endocarditis Team to guide management is highly recommended. The initial clinical assessment should include evaluation of cardiac and non-cardiac risk factors (Table 8), supportive clinical context, and phys-ical examination findings including potential portals of entry. Physical examination may reveal a variety of clinical signs. However, the absence of clinical signs alone should not exclude IE since the overall sensitivity and specificity of the clinical signs are low. In the European Infective Endocarditis Registry (EURO-ENDO), fe-ver (77.7%), cardiac murmur (64.5%), and congestive HF (27.2%) were the most frequent clinical presentations.5 Embolic complications were detected in 25.3% of patients and cardiac conduction abnormalities were found in 11.5%. Some classical signs, such as peripheral stigmata, are less frequently observed, but may still be observed in severe infec-tions caused by S. aureus and in cases of subacute endocarditis (mainly caused by Streptococcis spp.). However, vascular and immunological phenomena, such as splinter haemorrhages,138 Roth spots, and Recommendation Table 4 — Recommendations for the Endocarditis Team Recommendations Classa Levelb Diagnosis and management of patients with complicated IE are recommended to be performed at an early stage in a Heart Valve Centre, with immediate surgical facilities and an ‘Endocarditis Team’ to improve the outcomes.36–41,122,123,125,126 I B For patients with uncomplicated IE managed in a Referring Centre, early and regular communication between the local and the Heart Valve Centre endocarditis teams is recommended to improve the outcomes of the patients.36–41,122,123,125,126 I B © ESC 2023 IE, infective endocarditis. aClass of recommendation. bLevel of evidence. Table 8 Cardiac and non-cardiac risk factors Cardiac risk factors Previous infective endocarditis Valvular heart disease Prosthetic heart valve Central venous or arterial catheter Transvenous cardiac implantable electronic device Congenital heart disease Non-cardiac risk factors Central venous catheter People who inject drugs Immunosuppression Recent dental or surgical procedures Recent hospitalization Haemodialysis © ESC 2023 18 ESC Guidelines Downloaded from by guest on 30 August 2023 glomerulonephritis, remain common. The main symptoms and signs observed in the EURO-ENDO registry are shown in the Supplementary data online, Table S1. Atypical presentation is common in elderly or immunocompromised patients.139–141 A high index of sus-picion and low threshold for investigation are therefore essential to ex-clude IE or avoid delays in diagnosis in these and other high-risk groups, such as those with CHD or prosthetic valves.142 It is important to in-form those patients about the risk of IE who should be aware of com-patible symptoms to ask for advice in referral centres. 5.2. Laboratory findings Laboratory investigations and biomarkers typically yield non-specific re-sults. A large number of potential biomarkers have been proposed, re-flecting the complex pathophysiology of the pro- and anti-inflammatory processes, humoral and cellular reactions, and both circulatory and end-organ abnormalities involved in IE.143 The degree of anaemia, leucocytosis/leucopaenia, the number of immature white cell forms, concentrations of C-reactive protein and procalcitonin, erythrocyte sedimentation rate, and markers of end-organ dysfunction (serum lac-tate, serum creatinine, bilirubin, thrombocytopaenia, cardiac troponin, and natriuretic brain peptides) can be used to estimate the severity of sepsis, but none is diagnostic of IE. C-reactive protein and procalcitonin are the most widely evaluated biomarkers in RCTs of antibiotic stew-ardship. Furthermore, several of these biomarkers are included in scores used for risk stratification in critically ill patients. Unfortunately, no biomarker has sufficient accuracy for the diagnosis of sepsis or specificity for IE.144 Therefore, the main role of biomarkers is to facilitate initial risk stratification and monitor the response to anti-biotic therapy. 5.3. Microbiological diagnosis The aetiology of IE is described in the EURO-ENDO registry5 and the International Collaboration on Endocarditis-Prospective Cohort Study (ICE-PCS).145 In 2009, the ICE-PCS showed that the most frequent mi-croorganisms causing IE were S. aureus (31%), followed by oral strepto-cocci (17%), and CoNS (11%).145 Similar results were reported in the EURO-ENDO registry.5,145 Other registries have highlighted the increasing incidence of IE caused by E. faecalis and CoNS, particularly in the elderly.146–149 However, the results of these registries should be carefully interpreted due to inherent biases (type of partici-pating centres, geographical differences, lack of complete granular data, etc.). 5.3.1. Blood culture-positive infective endocarditis Positive blood cultures remain the cornerstone of IE diagnosis and pro-vide live bacteria for both identification and susceptibility testing. At least three sets of blood cultures should be obtained at 30-minute in-tervals prior to antibiotic therapy, each containing 10 mL of blood, and should be incubated in both aerobic and anaerobic atmo-spheres.150,151 Sampling should be obtained from a peripheral vein ra-ther than from a central venous catheter (because of the risk of contamination and misleading interpretation), using a meticulous sterile technique. In the absence of previous antimicrobial therapy, this is vir-tually always sufficient to identify the usual causative microorganisms. The need for culture before antibiotic administration is self-evident. In IE, bacteraemia is almost constant and has two implications: (i) there is no rationale for delaying blood sampling to coincide with peaks of fever; and (ii) nearly all blood cultures are positive during bacteraemia. As a result, a single positive blood culture should be regarded cautiously for establishing IE diagnosis. The microbiology laboratory should be aware of the clinical suspicion of IE. Automated machines perform con-tinuous monitoring of bacterial growth, which ensures quick provision of reports to physicians. When a positive blood culture is identified, presumptive identification is based on Gram staining. This information is immediately given to clinicians in order to adapt empirical antibiotic therapy. Complete identification is routinely achieved the same day or the following day with current methodology (e.g. matrix-assisted la-ser desorption ionization time-of-flight mass spectrometry [MALDI-TOF MS]), but may require a longer time for fastidious or atypical organisms. Since there is a long delay between blood culture sampling and definitive identification of the organism responsible for the bacteraemia and antibiotic susceptibility testing, many improve-ments have been proposed to speed up the process of detection and identification. One of the most recent procedures for rapid bacterial identification is based on peptide spectra obtained by MALDI-TOF MS.152 However, despite technical developments and the progress to-ward rapid susceptibility testing using MALDI-TOF MS, the gold stand-ard for susceptibility testing is still the determination of the minimal inhibitory concentrations (MICs) to select appropriate antibiotic ther-apy, which needs to be performed following validated, standardized methodology.153 5.3.2. Blood culture-negative infective endocarditis Blood culture-negative infective endocarditis (BCNIE) refers to IE in which no causative microorganism can be grown using the usual blood culture methods. The frequency of BCNIE as the cause of IE is highly variable and often poses considerable diagnostic and therapeutic dilem-mas.154,155 Blood culture-negative IE most commonly arises as a conse-quence of previous antibiotic administration, underlying the importance of performing blood cultures prior to antibiotic therapy, particularly in patients with known risk factors for IE. Withdrawal of antibiotics and repeating blood cultures may be required in stable patients with sub-acute symptoms, no evidence of local or distant complications, and re-ceiving a very short course of antibiotics. Blood culture-negative IE can also be caused by fungi or fastidious bacteria, notably obligatory intra-cellular bacteria. Isolation of these microorganisms requires culturing on specialized media, and their growth is relatively slow. Depending on local epidemiology,156 systematic serological testing for Coxiella bur-netii, Bartonella spp., Aspergillus spp., Mycoplasma pneumoniae, Brucella spp., and Legionella pneumophila should be proposed,157 followed by specific polymerase chain reaction (PCR) assays for Tropheryma whip-plei, Bartonella spp., and fungi (Candida spp., Aspergillus spp.) from blood and the tissue (Table 9).158 In addition, 16S and 18S ribosomal ribonucleic acid (rRNA) sequen-cing from tissue is routinely performed in most laboratories and may provide a microorganism diagnosis in BCNIE. For patients with pros-thetic valve BCNIE, molecular imaging technique fluorescence in situ hy-bridization combined with 16S rRNA-gene PCR and sequencing improved the conventional cultural diagnostic methods in 30% of cases.159 Next-generation sequencing of plasma microbial cell-free de-oxyribonucleic acid (DNA) may facilitate a rapid diagnosis of IE in the future.160 When all microbiological assays are negative, the diagnosis of non- bacterial endocarditis should systematically be considered and assays ESC Guidelines 19 Downloaded from by guest on 30 August 2023 for antinuclear antibodies as well as antiphospholipid syndrome (APLs) (anticardiolipin antibodies [immunoglobulin (Ig)G] and anti-β2-glycoprotein 1 antibodies [IgG and IgM]) should be performed (although these antibodies may also be present in patients with proven IE).161,162 Pathological examination of resected tissue or embolic frag-ments remains the gold standard for IE diagnosis. All tissue samples that are excised during surgical valve debridement/resection must be collected in a sterile container without fixative or culture medium. Samples should be sent to the pathology department and the micro-biology laboratory for the identification of microorganisms. On histo-logical examination of excised valve tissue, patterns, and degrees of inflammation will vary depending on the infecting organism. Stains for bacteria, mycobacteria, and fungi may identify the microorganisms, and organism-specific immunohistochemical stains can be very useful for the final diagnosis. Importantly, histopathological analysis may facili-tate the diagnosis of non-infectious causes of endocarditis, such as neo-plastic and autoimmune causes.160 5.3.3. Proposed strategy for a microbiological diagnostic algorithm in suspected infective endocarditis A proposed diagnostic scheme is provided in Figure 4. When there is clinical suspicion of IE and blood cultures remain negative at 48 h, con-sultation with the microbiologist is necessary.156,160 A suggested strat-egy is the use of a diagnostic kit including blood cultures for the suspected microorganism and when negative, systematic serological testing for C. burnetii, Bartonella spp., Aspergillus spp., L. pneumophila, Brucella spp., and M. pneumoniae, as well as rheumatoid factor, sero-logical tests for APLs (anticardiolipin [IgG] and anti-β2-glycoprotein 1 [IgG and IgM]), antinuclear antibodies, and anti-pork antibodies. Serological testing should be performed taking into consideration the clinical characteristics of the patients (i.e. Aspergillus spp. in severe im-munocompromised patients), the local epidemiology, and being aware of the specificity of the tests. In addition, tissue or prosthetic material obtained at surgery must be subjected to systematic culture, histologic-al examination, and 16S or 18S rRNA sequencing aimed at document-ing the presence of organisms. 5.4. Imaging techniques Evidence of lesions characteristic of IE are major diagnostic criterion. Echocardiography is the first-line imaging technique to diagnose IE and to assess the structural and functional damage of cardiac struc-tures. Echocardiographic findings have prognostic implications, and help to guide decision-making and patient follow-up while receiving antibiotic therapy and during the peri-operative and post-operative periods.163 In some clinical scenarios, other imaging modalities, such as CT, nuclear imaging, and MRI, are needed to confirm or exclude the diagnosis of IE, to characterize the extent of the cardiac lesions, and to diagnose extracardiac complications. They can also provide additional useful information for patient management.137 Each of these techniques has its diagnostic strengths and weaknesses (see Supplementary data online, Table S2). The use of an optimal imaging strategy depends on the availability of, and expertise in, each tech-nique, but when indicated a multimodality imaging approach is essen-tial for patients with suspected IE and should be strongly encouraged by the Endocarditis Team.21 5.4.1. Echocardiography Transthoracic echocardiography (TTE) and transoesophageal echocar-diography (TOE) are the first and key imaging techniques used to diag-nose IE. Although echocardiography is widely accessible, significant variation in the use of TOE still exists.164 Three-dimensional TOE and intracardiac echocardiography have also been shown to be useful for the diagnosis of IE and its complications.165 However, the availability of intracardiac echocardiography is limited. Vegetation characteristics and size, perivalvular complications (abscess, pseudoaneurysm, new partial dehiscence of prosthetic valve), intracardiac fistula, and leaflet perforation are the main echocardiographic findings for the diagnosis and evaluation of local complications of IE (see Supplementary data online, Table S3). Importantly, vegetation size is a key metric that guides surgical indication, and vegetation size is defined as the maximal length of the vegetation.166 When evaluating IE on native or prosthetic valves, TTE had low sensitivity but good specificity as compared with TOE.166 TOE is helpful in a wide range of clinical scenarios, due to limitations of TTE to diagnose perivalvular complications, small vegetations, PVE, and vegetations associated with CIED. TOE is strongly recommended in pa-tients with an inconclusive TTE, in patients with a negative TTE and a high suspicion of IE, as well as in patients with a positive TTE, in order to document local complications. Repeating TTE and/or TOE should be considered during follow-up of uncomplicated IE, in order to detect new silent complications and monitor vegetation size. The timing and mode (TTE or TOE) of repeated examination depend on the initial findings, type of microorganism, and initial response to therapy. Table 9 Investigation of rare causes of blood culture- negative infective endocarditis Pathogen Diagnostic procedures Brucella spp. Serology, blood cultures, tissue culture, immunohistology, and 16S rRNA sequencing of tissue C. burnetii Serology (IgG phase l >1:800), tissue culture, immunohistology, and 16S rRNA sequencing of tissue Bartonella spp. Serology (IgG phase I >1:800), blood cultures, tissue culture, immunohistology, and 16S rRNA sequencing of tissue T. whipplei Histology and 16S rRNA sequencing of tissue Mycoplasma spp. Serology, tissue culture, immunohistology, and 16S rRNA sequencing of tissue Legionella spp. Serology, blood cultures, tissue culture, immunohistology, and 16S rRNA sequencing of tissue Fungi Serology, blood cultures, 18S rRNA sequencing of tissue Mycobacteria (including Mycobacterium chimaera) Specific blood cultures, 16S rRNA sequencing of tissue © ESC 2023 Ig, immunoglobulin; rRNA, ribosomal ribonucleic acid. 20 ESC Guidelines Downloaded from by guest on 30 August 2023 Echocardiographic imaging should be performed as soon as the IE diagnosis is suspected. The degree of valvular damage, the rate of peripheral embolic events, and the need for valve surgery increase with increasing time to initial echocardiographic assessment.167 Echocardiography should be repeated 5–7 days after an initial normal or inconclusive echocardiography, if the suspicion of IE re-mains high, and in patients with diagnosed IE at high risk of compli-cations (e.g. aggressive microorganisms, prosthetic valves).22,165,168,169 There is uncertainty regarding whether echocardiography should be systematically performed in patients with bloodstream infections due to different bacterial species, or if there are strategies (microbiological or imaging) that allow the identification of patients at higher risk of IE. Scoring systems have been developed to help in the appropriate indication to perform echocardiography when bacteraemia of different microorganisms occurs (see Supplementary data online, Table S4).60,170–173 The combination of microbiological parameters (type of microorganism and number of positive blood culture bottles) and cardiac-related risk factors (native valve disease, previous IE, pros-thetic valve, and cardiac devices) may help identify the patients in whom echocardiography (TTE+TOE) is needed.19,174 Three risk scores were recently developed to identify patients at high risk of IE caused by S. aur-eus, and those who should be evaluated with echocardiography (see Supplementary data online, Section S2.2.1).170–173,175–178 The cut-off values of the various scores are provided in Supplementary data online, Table S4. N N N N N Suspected IE Y Y Y Y Blood cultures Culture Serologies BCNIE Identification by MALDI-TOF MS Microbiological identification Antimicrobial susceptibility testing Gram-positive and negative bacteria, fungi, Mycobacterium spp. Blood and vegetations PCRa Antinuclear antibodiesb Antiphospholipid antibodiesb Anti-pork antibodiesb Specific PCR Antibiotic resistance and culture C. burnetii Bartonella spp. Aspergillus spp. L. pneumophila Brucella spp. M. pneumoniae Y Figure 4 Microbiological diagnostic algorithm in culture-positive and culture-negative infective endocarditis. BCNIE, blood cultures negative endocarditis; IE, infective endocarditis; MALDI-TOF MS, matrix-assisted laser desorption ionization time-of-flight mass spectrometry; PCR, polymerase chain reaction. aQualified microbiological laboratory. bImmunological laboratory. ESC Guidelines 21 Downloaded from by guest on 30 August 2023 5.4.2. Computed tomography The indications for CT in patients with suspected or diagnosed IE include: (i) Diagnosis of IE and cardiac complications. Cardiac CT is more ac-curate than TOE for diagnosing perivalvular and periprosthetic complications of IE (abscesses, pseudoaneurysms, and fistulae) and is recommended in both NVE and PVE if TOE is not conclusive or not feasible.33,168,169 In addition, cardiac CT can significantly influence subsequent surgical decision-making.20,185,186 Echocardiography continues to be superior for detecting valvular lesions, particularly small vegetations (<10 mm) which remain un-derdiagnosed by CT, but also leaflet perforations and fistulae (see Supplementary data online, Table S3).35,168,169 Cardiac CT should be acquired according to the recommendations of cardiac CT guidelines to ensure high diagnostic accuracy, and can be per-formed alone or in combination with PET.187 (ii) Detection of distant lesions and sources of bacteraemia. Whole-body and brain CT are useful for assessing IE systemic complications, includ-ing septic emboli. The detection of distant lesions adds a minor diag-nostic criterion leading to a more conclusive diagnosis of definite or rejected IE, and can be relevant for decision-making.188 CT angiog-raphy can detect mycotic arterial aneurysms complicating IE in almost any site of the vascular tree,189,190 including the central nervous sys-tem (CNS). Although MRI is superior to CT for diagnosing neuro-logical complications,191 CT may be more feasible in an emergency setting and is an acceptable alternative for the detection of neurologic-al complications, with a sensitivity of 90% and specificity of 86% in the detection of ischaemic and haemorrhagic lesions.192 Finally, CT can also detect the extracardiac sources of the bacteraemia, including early neoplastic lesions, that may be important for patient manage-ment, and which need to be ideally addressed prior to undergoing heart valve surgery. However, CT does not replace the specific test indicated for the diagnosis of the extracardiac source of bacteraemia (i.e. colonoscopy in colon neoplasms). (iii) Pre-operative assessment. Cardiac CT is a valuable alternative for non-invasive assessment of coronary artery disease (CAD) before cardiac surgery in patients with IE.193 (iv) Alternative diagnosis. In patients in whom IE is ruled out, or even in doubtful patients with possible IE, an alternative diagnosis can be reached by whole-body CT, as it can help to detect alternative in-fectious foci. However, in these circumstances, an FDG posi-tron emission tomography/computed tomography (PET/CT) is the preferred imaging technique.194 5.4.3. Magnetic resonance imaging The roles of MRI in the diagnostic work-up of IE include: (i) Diagnosis of IE and cardiac complications. The role of cardiac MRI to diagnose IE is limited by the low spatial resolution (as compared with cardiac CT) and the signal void generated by some prostheses impairing the assessment of prosthetic valve anatomy and function.195,196 (ii) Diagnosis of neurological IE-related complications. MRI has higher sensitivity than CT for the diagnosis of neurological lesions and, hence, increases the likelihood of detecting neurological complica-tions in patients with IE. Patients with IE might present CNS lesions in up to 60–80% of cases,197 most of them corresponding to is-chaemic lesions (50–80% of patients) that are often small and asymptomatic and do not impact on the decision-making.198 Other lesions that may influence the decision-making, such as par-enchymal or subarachnoid haemorrhages, abscesses, or mycotic aneurysms, are found in <10% of patients.198–201 The systematic performance of brain MRI has shown to directly impact the Recommendation Table 5 — Recommendations for the role of echocardiography in infective endocarditis Recommendations Classa Levelb A. Diagnosis TTE is recommended as the first-line imaging modality in suspected IE.166,179 I B TOE is recommended in all patients with clinical suspicion of IE and a negative or non-diagnostic TTE.166,178,179 I B TOE is recommended in patients with clinical suspicion of IE, when a prosthetic heart valve or an intracardiac device is present.166,178,179 I B Repeating TTE and/or TOE within 5–7 days is recommended in cases of initially negative or inconclusive examination when clinical suspicion of IE remains high.178 I C TOE is recommended in patients with suspected IE, even in cases with positive TTE, except in isolated right-sided native valve IE with good quality TTE examination and unequivocal echocardiographic findings.165,166,179 I C Performing an echocardiography should be considered in S. aureus, E. faecalis, and some Streptococcus spp. bacteraemia.19,149,174 IIa B B. Follow-up under medical therapy Repeating TTE and/or TOE is recommended as soon as a new complication of IE is suspected (new murmur, embolism, persisting fever and bacteraemia, HF, abscess, AVB).165,166,179 I B TOE is recommended when patient is stable before switching from intravenous to oral antibiotic therapy.43,180 I B During follow-up of uncomplicated IE, repeat TTE and/ or TOE should be considered to detect new silent complications. The timing of repeat TTE and/or TOE depends on the initial findings, type of microorganism, and initial response to therapy.165,166,179 IIa B C. Intra-operative echocardiography Intra-operative echocardiography is recommended in all cases of IE requiring surgery.181 I C D. Following completion of therapy TTE and/or TOE are recommended at completion of antibiotic therapy for evaluation of cardiac and valve morphology and function in patients with IE who did not undergo heart valve surgery.182–184 I C © ESC 2023 AVB, atrioventricular block; HF, heart failure; IE, infective endocarditis; PVE, prosthetic valve endocarditis; TOE, transoesophageal echocardiography; TTE, transthoracic echocardiography. aClass of recommendation. bLevel of evidence. 22 ESC Guidelines Downloaded from by guest on 30 August 2023 diagnosis of IE, as it can add a minor diagnostic criterion in patients without neurological symptoms with non-definitive IE diagnosis. Brain MRI can reclassify 25% of patients with an initially inconclu-sive diagnosis for IE to a more conclusive diagnosis, thereby leading to an earlier diagnosis.151 Cerebral microbleeds, found in 50–60% of patients with IE, are detected at gradient echo T2 se-quences.200,202 Cerebral microbleeds should not be considered a minor criterion because there is no concordance with ischaemic lesions.203–205 (iii) Diagnosis of spine lesions. MRI is the diagnostic modality of choice of spondylodiscitis and vertebral osteomyelitis with a diagnostic ac-curacy of 89–94%. MRI findings include vertebrae and disc oedema, paravertebral/epidural inflammation or abscess, bone erosion, and gadolinium enhancement of vertebrae and discs.32,206 It should be acknowledged that when MRI is performed too early, the rate of false-negative increases.207 5.4.4. Nuclear imaging positron emission tomography/computed tomography (angiography) and single photon emission tomography/computed tomography Technical specifications of these imaging techniques are in the Supplementary data online, Section S2.2.2. The roles of nuclear imaging techniques in the diagnostic work-up of IE include: (i) Diagnosis of IE and cardiac complications. [18F]FDG-PET/CT and white blood cell (WBC) single photon emission computed tomography (SPECT)/CT are recommended in suspected PVE in cases of inconclu-sive echocardiography. The most recent meta-analysis showed 86% sensitivity and 84% specificity for [18F]FDG-PET/CT in PVE.129 Additional evidence demonstrating the incremental diagnostic value of [18F]FDG-PET/CT and WBC SPECT/CT is summarized in the Supplementary data online, Section S2.2.2; Table S5.22,208–212 White blood cell SPECT/CT is an alternative nuclear imaging tech-nique for the diagnosis of IE, when PET/CT is unavailable and in-experienced centres. The sensitivity of WBC SPECT/CT has been reported as 64–90% and the specificity as 36–100%; diagnostic ability significantly increases with the presence of periprosthetic abscesses.213–215 99mTechnetium-hexamethylpropyleneamine oxime (99mTc-HMPAO)-SPECT/CT helped to reduce the number of misdiag-nosed IE cases classified in the ‘possible IE’ category by the modified Duke criteria by 27%.216 In cases of NVE, the sensitivity of PET/CT and SPECT/CT is low (about 31%) but with a higher specificity (around 98%).211 In NVE, the diagnosis of IE cannot be excluded in the absence of abnormal [18F]FDG uptake.217 The more frequent presence of valve vegetations in comparison with paravalvular involvement in NVE compared with PVE leads to reduced inflammatory response and subsequently lower [18F]FDG and WBC uptake. The lower sensitivity of [18F]FDG-PET/ CT is offset by other strengths of the technique, such as its ability to identify septic emboli when suspected.211,218–220 Electrocardiogram (ECG)-gated PET may further improve the diagnostic accuracy.221 Combining PET/CT acquisition with a CT angiography (PET/CTA) al-lows the detection of metabolic findings ([18F]FDG uptake distribution and intensity) and anatomical findings (IE-related lesions) within a single imaging procedure, resulting in the clinical clarification of indeterminate findings and change in the management of the patients.22,211 Such inves-tigations may be particularly helpful in complex settings, such as patients with CHD222,223 and/or aortic grafts.22,224 (ii) Detection of distant lesions and sources of bacteraemia. Whole-body [18F]FDG-PET/CT imaging is particularly useful in patients with a suspicion or proven IE to identify distant lesions, mycotic aneurysms, and the portal of entry of the infection.225,226 Septic emboli are typically located in the spleen, lungs (in right- sided IE), and kidneys, and metastatic infections in the interverte-bral discs and/or the vertebral bone (spondylodiscitis) as well as in muscles and joints (septic arthritis) and liver.211,227,228 [18F] FDG-PET/CT is less suited to detect cerebral septic embolism and mycotic aneurysms of intracerebral arteries due to the high physiological uptake of [18F]FDG in the brain. (iii) Monitoring response to antimicrobial treatment with [18F] FDG-PET/CT in patients with established IE and indication for sur-gery but who cannot be operated on due to unacceptable high risk and remain with long-term suppressive antibiotic treat-ment.137,184,229–236 Recommendation Table 6 — Recommendations for the role of computed tomography, nuclear imaging, and magnetic resonance in infective endocarditis Recommendations Classa Levelb Cardiac CTA is recommended in patients with possible NVE to detect valvular lesions and confirm the diagnosis of IE.33,168,169 I B [18F]FDG-PET/CT(A) and cardiac CTA are recommended in possible PVE to detect valvular lesions and confirm the diagnosis of IE.22,129,209,210,237–239 I B Cardiac CTA is recommended in NVE and PVE to diagnose paravalvular or periprosthetic complications if echocardiography is inconclusive.20,168,169,185,186 I B Brain and whole-body imaging (CT, [18F]FDG-PET/ CT, and/or MRI) are recommended in symptomaticc patients with NVE and PVE to detect peripheral lesions or add minor diagnostic criteria.22,197– 200,210,213,240,241 I B WBC SPECT/CT should be considered in patients with high clinical suspicion of PVE when echocardiography is negative or inconclusive and when PET/CT is unavailable.213–216 IIa C [18F]FDG-PET/CT(A) may be considered in possible CIED-related IE to confirm the diagnosis of IE.22,129,209,210,237,238 IIb B Brain and whole-body imaging (CT, [18F]FDG-PET/ CT, and MRI) in NVE and PVE may be considered for screening of peripheral lesions in asymptomatic patients.188,197–201 IIb B © ESC 2023 [18F]FDG-PET/CT, 18F-fluorodeoxyglucose positron emission tomography/computed tomography; CAD, coronary artery disease; CT, computed tomography; CTA, computed tomography angiography; IE, infective endocarditis; MRI, magnetic resonance imaging; NVE, native valve endocarditis; PVE, prosthetic valve endocarditis; WBC SPECT/CT, white blood cell single photon emission tomography/computed tomography. aClass of recommendation. bLevel of evidence. cSymptomatic: symptoms suggesting septic embolic complications. ESC Guidelines 23 Downloaded from by guest on 30 August 2023 5.5. Diagnostic criteria Since 2000, clinical, microbiological, and imaging findings have been inte-grated in the modified Duke criteria (see Supplementary data online, Table S6), which have demonstrated an overall sensitivity of 80% for IE.151 However, the clinical presentation of IE can be highly variable and some major limitations of the modified Duke criteria have become clear, particularly when prosthetic material is present (PVE, aortic grafts, cardiac devices, CHD). In these situations, echocardiography can be normal or inconclusive in up to 30% of cases despite the presence of IE.242–244 Therefore, the 2015 ESC diagnostic criteria introduced a multimodality imaging approach (echocardiography, cardiac/ whole-body CT, cerebral MRI, [18F]FDG-PET/CT, and WBC SPECT/ CT) to improve the diagnostic yield. This new approach has shown to be superior over the traditional diagnostic criteria.36–41,122,123,125,126,212 5.5.1. Modifications for the diagnosis of infective endocarditis The current 2023 ESC Guidelines for the management of endocarditis introduce the following modifications for IE diagnosis: (i) Changes to the major and minor diagnostic criteria (Table 10). Table 10 Definitions of the 2023 European Society of Cardiology modified diagnostic criteria of infective endocarditis Major criteria (i) Blood cultures positive for IE (a) Typical microorganisms consistent with IE from two separate blood cultures: Oral streptococci, Streptococcus gallolyticus (formerly S. bovis), HACEK group, S. aureus, E. faecalis (b) Microorganisms consistent with IE from continuously positive blood cultures: • ≥2 positive blood cultures of blood samples drawn >12 h apart. • All of 3 or a majority of ≥4 separate cultures of blood (with first and last samples drawn ≥1 h apart). (c) Single positive blood culture for C. burnetii or phase I IgG antibody titre >1:800. (ii) Imaging positive for IE: Valvular, perivalvular/periprosthetic and foreign material anatomic and metabolic lesions characteristic of IE detected by any of the following imaging techniques: • Echocardiography (TTE and TOE). • Cardiac CT. • [18F]-FDG-PET/CT(A). • WBC SPECT/CT. Minor criteria (i) Predisposing conditions (i.e. predisposing heart condition at high or intermediate risk of IE or PWIDs)a (ii) Fever defined as temperature >38°C (iii) Embolic vascular dissemination (including those asymptomatic detected by imaging only): • Major systemic and pulmonary emboli/infarcts and abscesses. • Haematogenous osteoarticular septic complications (i.e. spondylodiscitis). • Mycotic aneurysms. • Intracranial ischaemic/haemorrhagic lesions. • Conjunctival haemorrhages. • Janeway’s lesions. (IV) Immunological phenomena: • Glomerulonephritis. • Osler nodes and Roth spots. • Rheumatoid factor. (V) Microbiological evidence: • Positive blood culture but does not meet a major criterion as noted above. • Serological evidence of active infection with organism consistent with IE. IE Classification (at admission and during follow-up) Definite: • 2 major criteria. • 1 major criterion and at least 3 minor criteria. • 5 minor criteria. Possible: • 1 major criterion and 1 or 2 minor criteria. • 3–4 minor criteria. Rejected: • Does not meet criteria for definite or possible at admission with or without a firm alternative diagnosis. © ESC 2023 [18F]-FDG-PET/CT, 18F-fluorodeoxyglucose positron emission tomography; CT(A), computed tomography (angiography); HACEK, Haemophilus, Aggregatibacter, Cardiobacterium, Eikenella, and Kingella; IE, infective endocarditis; Ig, immunoglobulin; PWID, people who inject drugs; TOE, transoesophageal echocardiography; TTE, transthoracic echocardiography; WBC SPECT/CT, white blood cell single photon emission tomography/computed tomography. aFor detailed explanation of predisposing conditions, please see Section 3. 24 ESC Guidelines Downloaded from by guest on 30 August 2023 (ii) Specific diagnostic algorithms to support decision-making, especially in the recommended sequence of imaging techniques (Figures 5–7). (iii) CIED-related IE is considered a right-sided endocarditis for diag-nostic purposes and is included in the diagnostic algorithms, but its definitions and recommendations for management can be found in Section 12 and are in accordance with the specific European Heart Rhythm Association (EHRA) consensus on CIED infections.130 Repeat blood cultures if negative or doubtful Repeat TTE/TOE within 5–7 days Cardiac CTA to diagnose valvular lesions Add minor criteria: brain or whole-body imaging (MRI, CT, PET/CT, WBC SPECT) to detect distant lesions Suspected paravalvular complications and TOE inconclusive Cardiac CTA (Class I) No symptoms suggesting extracardiac complications Brain and whole-body imaging (CT, [18F]FDG-PET/CT, and/or MRI) (Class IIb) Symptoms suggesting extracardiac complications Brain and whole-body imaging (CT, [18F]FDG-PET/CT, and/or MRI) (Class I) Suspected native valve IE ESC 2023 DIAGNOSTIC CRITERIA after IE Baseline assessment and initial classification: clinical presentation + blood cultures + TTE + TOEa (Class I) POSSIBLE REJECTED DEFINITE (Class I) (Class IIa) Figure 5 European Society of Cardiology 2023 algorithm for diagnosis of native valve infective endocarditis. [18F]FDG, 18F-fluorodeoxyglucose; CT, com-puted tomography; CTA, computed tomography angiography; ESC, European Society of Cardiology; IE, infective endocarditis; MRI, magnetic resonance im-aging; NVE, native valve endocarditis; PET, photon emission tomography; TOE, transoesophageal echocardiography; TTE, transthoracic echocardiography; WBC SPECT, white blood cell single photon emission tomography. aTOE for diagnosis and to detect perivalvular complications in all cases (unless right-sided NVE when TTE is good quality and conclusive). ESC Guidelines 25 Downloaded from by guest on 30 August 2023 The reasons to justify the changes in the diagnostic criteria include: 5.5.1.1. Major criteria – microbiology Enteroccus faecalis should be acknowledged as a typical endocarditis bacterium, regardless of the place of acquisition or the source of infection. Currently, the modified Duke criteria fail to identify 30% of E. faecalis definite IE. Using data from a prospective study of 344 pa-tients with E. faecalis bacteraemia evaluated with echocardiography, Dahl et al. demonstrated that designating E. faecalis as a ‘typical’ endo-carditis pathogen significantly improved the sensitivity to correctly iden-tify definite IE, from 70% to 96%.245 5.5.1.2. Major criteria – imaging (i) Diagnosis based on the presence of lesions characteristics of IE. Anatomic lesions and increased [18F]FDG uptake or WBC accu-mulation can be depicted by nuclear imaging techniques and add Repeat blood cultures if negative or doubtful Repeat TTE/TOE within 5–7 days Cardiac CTA or FDG-PET/CT(A) to diagnose valvular lesions WBC SPECT Add minor criteria: brain or whole-body imaging (MRI, CT, PET/CT, WBC SPECT) to detect distant lesions Suspected paravalvular complications and TOE inconclusive Cardiac CTA (Class I) No symptoms suggesting extracardiac complications Brain and whole-body imaging (CT, [18F]FDG-PET/CT, and/or MRI) (Class IIb) Symptoms suggesting extracardiac complications Brain and whole-body imaging (CT, [18F]FDG-PET/CT, and/or MRI) (Class I) Suspected prosthetic valve IE ESC 2023 DIAGNOSTIC CRITERIA after IE Baseline assessment and initial classification: clinical presentation + blood cultures + TTE + TOEa (Class I) POSSIBLE REJECTED DEFINITE (Class I) (Class IIa) Figure 6 European Society of Cardiology 2023 algorithm for diagnosis of prosthetic valve infective endocarditis. [18F]FDG, 18F-fluorodeoxyglucose; CT, com-puted tomography; CTA, computed tomography angiography; ESC, European Society of Cardiology; IE, infective endocarditis; MRI, magnetic resonance imaging; PET, positron emission tomography; TOE, transoesophageal echocardiography; TTE, transthoracic echocardiography; WBC SPECT, white blood cell single photon emission tomography. aTOE for diagnosis and to detect perivalvular complications in all cases (unless right-sided NVE when TTE is good quality and conclusive). 26 ESC Guidelines Downloaded from by guest on 30 August 2023 a major diagnostic criterion. Definitions of the anatomic and meta-bolic features of the infective lesions can be found in the Supplementary data online, Table S5. (ii) Abnormal prosthetic or periprosthetic uptake (intense focal or heterogeneous) detected by [18F]FDG-PET/CT or WBC SPECT/CT should be considered a major criterion for PVE, irre-spective of the interval from surgery (see Supplementary data online, Figure S1). Published data support that intense focal or heterogeneous patterns is associated with a final diagnosis of in-fection, while post-operative inflammatory changes can be per-sistent more than 3 months after surgery, as noted in the previous guidelines. However, these inflammatory changes can be differentiated from infection even after recent valve implant-ation.246 Therefore, a consensus of experts has concluded that the need for a time interval prior to investigation is questionable, but accurate imaging interpretation by proper interpretation cri-teria is mandatory.233,236 5.5.1.3. Minor criteria Distant IE-related lesions include all lesions that can result from embolic events and from haematogenous seeding of bacteria. These lesions can be suspected due to specific symptoms or can be incidentally detected on imaging techniques. Spondylodiscitis is the most frequent osteoarti-cular infective complication in patients with IE.247,248 5.5.1.4. Microbiological criteria Molecular biology (16S/18S rRNA PCR sequencing) in cardiac tissue or embolic material has increased the diagnostic performance of IE with negative blood culture. The sensitivity ranges between 41% and 96% and the specificity is very high, ranging between 90% and 100%.249 Repeat blood cultures if negative or doubtful Repeat TTE/TOE within 5–7 days PET/CT(A) to detect pocket infection +/-pulmonary embolism Add minor criteria: thoracic CT to detect septic pulmonary embolism/infarction PET/CT(A) to detect lead infection (Class IIa) (Class IIb) (Class I) Suspected CIED-associated IE ESC 2023 DIAGNOSTIC CRITERIA after IE Baseline assessment and initial classification: clinical presentation + blood cultures + TTE + TOE (Class I) POSSIBLE REJECTED DEFINITE Figure 7 European Society of Cardiology 2023 algorithm for diagnosis of cardiac device-related infective endocarditis. CIED, cardiovascular implanted electronic device; CT, computed tomography; CTA, computed tomography angiography; ESC, European Society of Cardiology; IE, infective endocarditis; PET, positron emission tomography; TOE, transoesophageal echocardiography; TTE, transthoracic echocardiography; WBC SPECT, white blood cell single photon emission tomography. ESC Guidelines 27 Downloaded from by guest on 30 August 2023 5.5.1.5. Infective endocarditis classification Infective endocarditis classification has been added to the 2023 ESC cri-teria. Possible IE cases include the combination of 1 major and 1 or 2 minor criteria. Infective endocarditis classification should be applied by the Endocarditis Team at admission and later at follow-up, taking into account the complete clinical, microbiology, imaging, and surgical information to establish the final diagnosis. It is important to acknowledge that these new criteria should be pro-spectively validated. 5.5.2. The new 2023 European Society of Cardiology diagnostic algorithms The diagnosis of IE is based on clinical suspicion, blood cultures, and im-aging findings. Echocardiography is usually the first imaging technique to diagnose IE, although the use of other techniques, either for the diagno-sis of cardiac involvement (cardiac CT, [18F]FDG-PET/CT, or WBC SPECT/CT), or for the diagnosis of distant lesions (cerebral MRI, whole- body CT, and/or PET/CT), is encouraged. In the presence of prosthetic valves and CIED, echocardiography is particularly limited and the afore-mentioned imaging techniques are strongly recommended. Adapted diagnostic algorithms for suspected IE in NVE, PVE, and CIED are dis-played in Figures 5–7, respectively. 6. Prognostic assessment at admission The in-hospital mortality rate of patients with IE has remained largely unchanged over the past two decades, ranging from 15% to 30%.5,145,250,251 Several patient characteristics, often occurring simul-taneously, have been shown to confer an increased risk of death in IE. The rapid identification of patients at the highest risk may offer the opportunity to change the course of the disease (i.e. with urgent or emergency surgery) and improve prognosis. Predictors of poor out-come on admission of patients with IE are specified in the Supplementary data online, Section S3.1; Table S7. 7. Antimicrobial therapy: principles and methods 7.1. General principles Successful treatment of IE relies on microbial eradication by antimicro-bial drugs. Surgery contributes by removing infected material. Bactericidal regimens are more effective than bacteriostatic therapy, both in animal experiments and in humans.252–254 Aminoglycosides sy-nergize with cell wall inhibitors (i.e. beta-lactams and glycopeptides) for bactericidal activity and are useful for shortening the duration of ther-apy (e.g. oral streptococci) and eradicating problematic organisms. However, the side effects of aminoglycosides should be taken into con-sideration and currently the combination of ampicillin with ceftriaxone has demonstrated effective in treating IE caused by E. faecalis irrespect-ive of the presence of high-level aminoglycoside resistance (HLAR) and minimizing the risk of nephrotoxicity.255,256 One major hindrance to drug-induced killing is bacterial antibiotic tolerance. Tolerant microbes are not resistant (i.e. they are still suscep-tible to growth inhibition by the drug) but escape drug-induced killing and may resume growth after treatment discontinuation. Slow-growing and dormant microbes display phenotypic tolerance towards most antimicrobials (except rifampin to some extent). They are present in vegetations and biofilms (complex communities of bac-teria residing within an exopolysaccharide matrix that adheres to a sur-face, e.g. in PVE),257 and justify the need for prolonged therapy to fully sterilize infected heart valves. Some bacteria carry mutations rendering them tolerant during both active growth and stationary (dormant) phases.258,259 Bactericidal drug combinations are preferred to mono-therapy against tolerant organisms (e.g. the combination of ampicillin and ceftriaxone in IE caused by E. faecalis). Drug treatment of PVE should last longer (≥6 weeks) than that of NVE (2–6 weeks) but is otherwise similar. In staphylococcal PVE, the regimen should include rifampin whenever the strain is susceptible, even if some recent data have shown no differences in outcomes be-tween patients with PVE treated with rifampin vs. those treated without.260,261 In NVE needing valve replacement by a prosthesis during antibiotic therapy, the post-operative antibiotic regimen should be that recom-mended for NVE, not for PVE. In both NVE and PVE, the duration of treatment is based on the first day of effective antibiotic therapy (nega-tive blood culture in the case of initial positive blood culture), not on the day of surgery. A new full course of treatment should only start if valve cultures are positive. Finally, there are important considerations in these recommendations: (i) Only published antibiotic efficacy data from clinical trials and cohort studies in patients with IE (or bacteraemia if there are no IE data) have been considered in these guidelines. Data from experimental IE models have not been taken into account. A recent systematic review evaluating the existing evidence about clinical benefits and harms of different antibiotic regimens used to treat patients with IE has shown that there is limited and low- to very low-quality evi-dence to make strong conclusions on the comparative effects of dif-ferent antibiotic regimens on cure rates or other relevant clinical outcomes and, therefore, there is not enough evidence to support or reject any regimen of antibiotic therapy for the treatment of IE.262,263 (ii) These guidelines have adopted the MIC breakpoints included in the 2022 EUCAST clinical breakpoint tables.42 The EUCAST break-points are used to categorize results into three susceptibility categories: • Susceptible, standard dosing regimen: a microorganism is cate-gorized as such, when there is a high likelihood of therapeutic success using a standard dosing regimen of the agent. • Susceptible, increased exposure: a microorganism is categorized as such when there is a high likelihood of therapeutic success be-cause exposure to the agent is increased by adjusting the dosing regimen or by its concentration at the site of infection. • Resistant: a microorganism is categorized as such when there is a high likelihood of therapeutic failure even when there is in-creased exposure. The term exposure is defined as a function of how the mode of ad-ministration, dose, dosing interval, infusion time, as well as distribution and excretion of the antimicrobial agent, will influence the infecting or-ganism at the site of infection. The local laboratories are responsible for the use of appropriate methods and interpretative criteria and quality control of the test results (MIC) while the clinicians are responsible for adjusting the level of exposure by modifying the dosing strategy 28 ESC Guidelines Downloaded from by guest on 30 August 2023 (individual dose, frequency of dosing, mode of administration [oral or intravenous (i.v.)]).42 (iii) Oral antimicrobial therapy. The POET trial has changed the para-digm of i.v. antibiotic treatment for IE.43 For more than 60 years it had been considered that antibiotics should always be given intra-venously. The POET trial has shown that after an initial phase of i.v. treatment, up to 20% of patients could complete the treatment by oral antibiotic therapy (see Section 7.13.1).43 Therefore, as indi-cated in Figure 8, the antibiotic treatment of IE has two phases. The first phase can last up to 2 weeks of hospital i.v. treatment using combinations of rapidly bactericidal antibiotics to destroy planktonic bacteria.257 In this initial phase, cardiac surgery should be performed if indicated, infected foreign bodies should be removed, and cardiac as well as extracardiac abscesses should be drained. After this peri-od, clinically stable patients can end the antibiotic treatment at home with i.v. (OPAT) or oral antibiotic regimens for up to 6 weeks in or-der to eliminate the dormant (resting) bacteria and prevent relapses. (iv) Aminoglycosides are not recommended in staphylococcal NVE be-cause their clinical benefits have not been demonstrated, but they can increase renal toxicity.255,264 When they are indicated in other conditions (e.g. resistant oral streptococci),265 aminoglycosides should be given for no longer than 2 weeks to reduce nephrotoxicity.266 (v) Rifampin should be used only in foreign body infections such as PVE after 3–5 days of effective antibiotic therapy, once the bacteraemia has been cleared. The rationale supporting this recommendation is based on the likely antagonistic effect of the antibiotic combinations with rifampin against planktonic/replicating bacteria,267 and the synergy seen against dormant bacteria within the biofilms and prevention of rifampin- resistant variants.268 New evidence based on a small, retrospective study has questioned this approach and needs further validation.260 (vi) Daptomycin has been recommended for treating staphylococcal and enterococcal endocarditis.269 When daptomycin is indicated, it must be given at high doses (10 mg/kg once daily)270 and combined with a second antibiotic (beta-lactams or fosfomycin in beta-lactam allergic patients) to increase activity and avoid the de-velopment of resistance.271 It should be noted the use of fosfomycin is associated with increased risk of acute HF and renal failure due to the high load of sodium while the use of daptomycin has been asso-ciated with eosinophilic syndromes in up to 15% of patients.272,273 Phases of antibiotic treatment of infective endocarditis Complicated cases: continue inpatient i.v. treatmenta From Day 10 post-treatment initiation and/or 7 days post-surgery: consider OPAT or oral antibiotic treatment in stable patients Inpatient treatment i.v. rapid bactericidal combinations Removal of infected cardiac devices 0 Week Early critical phase Continuation phase with resting bacteria 1 2 4-6 + + Cardiac surgery, if indicated Draining of abscesses 10 Day Perform TOE before therapy switch (Class I) Figure 8 Phases of antibiotic treatment for infective endocarditis in relation to outpatient parenteral antibiotic therapy and partial oral endocarditis treat-ment. i.v., intravenous: OPAT, outpatient parenteral antibiotic treatment; TOE, transoesophageal echocardiography. aCriteria for switching to OPAT or par-tial oral treatment of endocarditis are given in the Supplementary data online, Table S8. ESC Guidelines 29 Downloaded from by guest on 30 August 2023 (vii) The antibiotic regimens need to adapt to the local circumstances and the availability of antibiotics. (viii) Data on the efficacy of long-term antibiotic suppressive therapy in pa-tients with IE who do not undergo cardiac surgery are limited to small and heterogeneous series with various antibiotic regimens.184,274 In a small series of Gram-positive bloodstream infections and IE, dalba-vancin (500 mg weekly or 1000 mg biweekly regimens) has been shown effective.274,275 Relapses are not infrequent.184 7.2. Penicillin-susceptible oral streptococci and Streptococcus gallolyticus group Oral streptococci include the groups mitis, sanguinis, anginosus, salivar-ius, downei, and mutans (see Supplementary data online, Figure S2).276 The remaining streptococci isolated outside of the oral cavity are clas-sified into either the Streptococcus gallolyticus (former bovis) or pyogen-ic groups. Recommended regimens against susceptible (susceptible standard dosing regimen and increased exposure) streptococci are summarized in Recommendation Table 7.4,277–279 The cure rate is ex-pected to be >95%. In uncomplicated cases of NVE, short-term 2-week therapy can be administered by combining penicillin or ceftriaxone with gentamicin or netilmicin.280,281 Gentamicin and netilmicin can be given once daily in patients with IE due to susceptible streptococci and nor-mal renal function. When outpatient antibiotic therapy is feasible, cef-triaxone alone or combined with gentamicin or netilmicin given once a day is particularly convenient.280–282 In patients with documented al-lergy to penicillin, desensitization is recommended. If desensitization cannot be performed, patients allergic to beta-lactam should receive cephalosporins (in non-anaphylactic reaction) or vancomycin, keeping in mind that a beta-lactam is superior to glycopeptides. Teicoplanin has been proposed as an alternative,4 starting with loading doses (6 mg/kg/12 h for 3 days) and followed by 6–10 mg/kg/day. Loading is critical because the drug is highly bound (≥98%) to serum proteins and penetrates slowly into vegetations.283 However, only limited retro-spective studies have assessed its efficacy in streptococcal IE.284 After 10–14 days of therapy, OPAT or outpatient oral antibiotic therapy should be considered. 7.3. Oral streptococci and Streptococcus gallolyticus group susceptible, increased exposure or resistant to penicillin The incidence of these resistant streptococci is increasing. Large strain collections have reported >30% of resistant S. mitis and Streptococcus oralis.285 Retrospective series provide the evidence for the recommendations on antibiotic treatment of IE caused by penicillin-resistant oral strepto-cocci and S. gallolyticus. Compiling four of them, 47 of 60 patients (78%) were treated with penicillin or ceftriaxone, mostly combined with ami-noglycosides.285–290 In penicillin-resistant cases, aminoglycoside treat-ment must be given for ≥2 weeks and short-term therapy regimens are not recommended. There is very limited experience with daptomy-cin in IE caused by resistant isolates.265,291 After 10–14 days of therapy, OPAT or outpatient oral antibiotic therapy should be considered if clin-ically stable (see Section 7.13). Recommendation Table 7 — Recommendations for antibiotic treatment of infective endocarditis due to oral strepto-cocci and Streptococcus gallolyticus group Recommendations Classa Levelb Penicillin-susceptible oral streptococci and Streptococcus gallolyticus group Standard treatment: 4-week duration in NVE or 6-week duration in PVE In patients with IE due to oral streptococci and S. gallolyticus group, penicillin G, amoxicillin, or ceftriaxone are recommended for 4 (in NVE) or 6 weeks (in PVE), using the following doses:277,278 I B Adult antibiotic dosage and route Penicillin G 12–18 millionc U/day i.v. either in 4–6 doses or continuously Amoxicillin 100–200 mg/kg/day i.v. in 4–6 doses Ceftriaxone 2 g/day i.v. in 1 dose Paediatric antibiotic dosage and route Penicillin G 200 000 U/kg/day i.v. in 4–6 divided doses Amoxicillin 100–200c mg/kg/day i.v. in 4–6 doses Ceftriaxone 100 mg/kg/day i.v. in 1 dose Standard treatment: 2-week duration (not applicable to PVE) 2-week treatment with penicillin G, amoxicillin, ceftriaxone combined with gentamicin is recommended only for the treatment of non-complicated NVE due to oral streptococci and S. gallolyticus in patients with normal renal function using the following doses:277,278 I B Adult antibiotic dosage and route Penicillin G 12–18 millionc U/day i.v. either in 4–6 doses or continuously Amoxicillin 100–200 mg/kg/day i.v. in 4–6 doses Ceftriaxone 2 g/day i.v. in 1 dose Gentamicind 3 mg/kg/day i.v. or i.m. in 1 dosed Continued 30 ESC Guidelines Downloaded from by guest on 30 August 2023 Paediatric antibiotic dosage and route Penicillin G 12–18 millionc U/day i.v. either in 4–6 doses or continuously Amoxicillin 100–200 mg/kg/dayc i.v. in 4–6 doses Ceftriaxone 100 mg/kg i.v. in 1 dose Gentamicind 3 mg/kg/day i.v. or i.m. in 1 dose or 3 equally divided dosesd Allergy to beta-lactams In patients allergic to beta-lactams and with IE due to oral streptococci and S. gallolyticus, vancomycin for 4 weeks in NVE or for 6 weeks in PVE is recommended using the following doses:292 I C Adult antibiotic dosage and route Vancomycine 30 mg/kg/day i.v. in 2 dosese Paediatric antibiotic dosage and route Vancomycine 30 mg/kg/day i.v. in 2 or 3 equally divided dosese Oral streptococci and Streptococcus gallolyticus group susceptible, increased exposure or resistant to penicillin In patients with NVE due to oral streptococci and S. gallolyticus, penicillin G, amoxicillin, or ceftriaxone for 4 weeks in combination with gentamicin for 2 weeks is recommended using the following doses:285–290 I B Adult antibiotic dosage and route Penicillin G 24 million U/day i.v. either in 4–6 doses or continuously Amoxicillin 2 g/day i.v. in 6 doses Ceftriaxone 2 g/day i.v. in 1 dose Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dosed In patients with PVE due to oral streptococci and S. gallolyticus, penicillin G, amoxicillin, or ceftriaxone for 6 weeks combined with gentamicin for 2 weeks is recommended using the following doses:285–290 I B Adult antibiotic dosage and route Penicillin G 24 million U/day i.v. either in 4–6 doses or continuously Amoxicillin 2 g/day i.v. in 6 doses Ceftriaxone 2 g/day i.v. in 1 dose Gentamicind 3 mg/kg/day i.v. or i.m. in 1 dosed Allergy to beta-lactams In patients with NVE due to oral streptococci and S. gallolyticus and who are allergic to beta-lactams, vancomycin for 4 weeks is recommended using the following doses: I C Adult antibiotic dosage and route Vancomycine 30 mg/kg/day i.v. in 2 dosese Paediatric antibiotic dosage and route Vancomycine 30 mg/kg/day i.v. in 2 dosese In patients with PVE due to oral streptococci and S. gallolyticus and who are allergic to beta-lactams, vancomycin for 6 weeks combined with gentamicin for 2 weeks is recommended using the following doses: I C Adult antibiotic dosage and route Vancomycine 30 mg/kg/day i.v. in 2 dosese Gentamicind 3 mg/kg/day i.v. or i.m. in 1 dosed Paediatric antibiotic dosage and route Vancomycine 30 mg/kg/day i.v. in 2 dosese Gentamicind 3 mg/kg/day i.v. or i.m. in 1 dosed © ESC 2023 IE, infective endocarditis; i.m., intramuscular; i.v., intravenous; NVE, native valve endocarditis; PVE, prosthetic valve endocarditis; U, units. aClass of recommendation. bLevel of evidence. cThe starting recommended doses are the lower doses which can be scalable to the highest doses. dMaximum doses 240 mg/day. High doses are associated with increased risk of nephrotoxicity. Renal function and serum gentamicin concentrations should be monitored once a week. When given in a single daily dose, pre-dose (trough) concentrations should be <1 mg/L and post-dose (peak; 1 h after injection) serum concentrations should be ∼10–12 mg/L. eSerum vancomycin concentrations should achieve 10–15 mg/L at pre-dose (trough) level, although some experts recommend to increase the dose of vancomycin to 45–60 mg/kg/day i.v. in 2 or 3 divided doses to reach serum trough vancomycin levels (Cmin) of 15–20 mg/L as in staphylococcal endocarditis. However, vancomycin dose should not exceed 2 g/day unless serum levels are monitored and can be adjusted to obtain a peak plasma concentration of 30–45 μg/mL 1 h after completion of the i.v. infusion of the antibiotic. ESC Guidelines 31 Downloaded from by guest on 30 August 2023 7.4. Streptococcus pneumoniae, β-haemolytic streptococci (groups A, B, C, and G) Infective endocarditis due to Streptococcus pneumoniae has become rare. It is associated with meningitis and pneumonia in up to 30% of cases,293–296 which requires special consideration in cases with penicil-lin resistance. Treatment of penicillin-susceptible strains is similar to that of oral streptococci (see Recommendation Table 7), except for the use of short-term 2-week therapy, which has not been thoroughly investigated. The same holds true for penicillin-susceptible increased exposure or resistant strains without meningitis, although for resistant strains some authors recommend high doses of cephalosporins (e.g. cefotaxime or ceftriaxone) or vancomycin.295 In cases with meningitis, penicillin must be avoided because of its poor penetration into the cerebrospinal fluid, and should be replaced with ceftriaxone or cefotax-ime alone, or in association with vancomycin according to the antibiotic susceptibility pattern.297,298 After 10–14 days of therapy and when meningitis is not associated, OPAT or outpatient oral antibiotic therapy should be considered if clinically stable (see Section 7.13). Infective endocarditis due to group A, B, C, or G streptococci, includ-ing the Streptococcus anginosus group (S. constellatus, S. anginosus, and S. intermedius) is relatively rare.299,300 Group A streptococci are uniformly susceptible to beta-lactams, whereas other serogroups may display some degree of resistance. Infective endocarditis due to group B streptococci was once associated with the peripartum period, but it now occurs in all adults, especially the elderly. Groups B, C, and G streptococci and S. anginosus induce abscesses that require adjunctive surgery.300 Mortality from group B PVE is very high and cardiac surgery is recommended.301 Antibiotic treatment is similar to that of oral streptococci (see Recommendation Table 7), except that short-term (2 weeks) therapy is not recommended and gentamicin should be given for 2 weeks. 7.5. Granulicatella and Abiotrophia (formerly nutritionally variant streptococci) Granulicatella and Abiotrophia induce IE with a prolonged course and are associated with large vegetations (>10 mm), and consequently with high rates of complications and valve replacement (around 50%).302,303 This is possibly due to delayed diagnosis and treatment. Antibiotic recommendations include penicillin G, ceftriaxone, or vanco-mycin for 6 weeks, combined with an aminoglycoside for at least the first 2 weeks in case of PVE (for doses, please see Recommendation Table 7).302–304 7.6. Staphylococcus aureus and coagulase-negative staphylococci Staphylococcus aureus is usually responsible for acute and destructive IE,305 whereas CoNS can induce more protracted valve infec-tions.306,307 Of note, the addition of an aminoglycoside in staphylococ-cal NVE is no longer recommended because it increases renal toxicity.264,308 Short-term (2-week) and oral treatments have been proposed for uncomplicated right-sided native valve methicillin- susceptible S. aureus (MSSA) IE (see also Section 12.4.2), but these regi-mens cannot be applied to left-sided IE. For penicillin-allergic patients with MSSA IE, penicillin desensitization can be attempted in stable pa-tients or cefazolin can be used since vancomycin is inferior to beta- lactams.309 If beta-lactams cannot be given, where available, daptomycin should be chosen and given in combination with another effective anti-staphylococcal drug to increase activity and avoid the development of resistance.310 Staphylococcus lugdunensis is mostly methicillin- susceptible and can be treated with cloxacillin. Staphylococcus aureus PVE carries a very high risk of mortality (>45%),305,312,313 and often requires early valve replacement. Other differences in comparison with NVE include the overall duration of ther-apy, the use of aminoglycosides, and the addition of rifampin after 3–5 days of effective antibiotic therapy once the bacteraemia has been cleared.264,314–318 The rationale supporting this recommendation is based on the antagonistic effect of the antibiotic combinations with rifampin against planktonic/replicating bacteria as has been demonstrated in foreign body infection models and clinically in prosthetic orthopaedic and vascular infections.319 However, a recent study has shown that the addition of ami-noglycosides to a regimen containing vancomycin or cloxacillin plus rifam-picin in S. aureus PVE was not associated with a better outcome.320 In addition, the risk of nephrotoxicity associated with the use of aminoglyco-sides should be taken into consideration. Adding rifampin to the treatment of staphylococcal PVE is standard practice despite the weak evi-dence.261,321 The potential side effects and drug interactions of rifampin should also be considered. In patients with PVE who are allergic to penicil-lin, daptomycin can be given combined with ceftaroline or fosfomycin or with gentamicin (for 2 weeks) plus rifampin for at least 6 weeks. After 10–14 days of therapy, OPAT or outpatient oral antibiotic therapy should be considered if clinically stable (see Section 7.13). 7.7. Methicillin-resistant staphylococci Methicillin-resistant S. aureus (MRSA) produces low-affinity penicillin- binding proteins (PBPs), which confer cross-resistance to most beta- lactams. Methicillin-resistant S. aureus is usually resistant to multiple antibiotics, leaving vancomycin, daptomycin, ceftaroline, and dalbavan-cin to treat severe infections.322–324 However, it should be noted that subpopulations susceptible with increased exposure and resistant to vancomycin have emerged worldwide and are associated with IE treatment failures.325–328 The prevalence of MRSA causing IE that is susceptible with increased exposure or resistant to vancomycin ranges between 19% and 34%. In addition, among patients with IE caused by MRSA, those isolates with a population analysis profile MIC ≥4 mg/L were associated with treatment failure defined by persistent bacteriae-mia for ≥7 days or MRSA-attributable mortality.325 Nephrotoxicity is of concern when using trough monitoring of levels of vancomycin as a surrogate marker of the area under the curve relative to the MIC (AUC/MIC). Therefore, it is recommended to use a target of AUC/ MIC between 400 and 600 mgh/L (assuming an MIC of 1 mg/L) that should be achieved with 48 h of therapy.329 When the MIC is >1 mg/ L, the probability of achieving an AUC/MIC ≥400 is unlikely. In that clin-ical scenario, changing therapy should be considered due to the high risk of nephrotoxicity with higher doses of vancomycin. Daptomycin is a li-popeptide antibiotic approved for S. aureus bacteraemia and right-sided IE.330 Cohort studies of S. aureus and CoNS IE have shown that dapto-mycin is at least as effective as vancomycin,327,328 and, in two cohort studies of MRSA bacteraemia with high vancomycin MICs (>1 mg/ L),331,332 daptomycin was associated with better outcomes (including survival) compared with vancomycin. Importantly, daptomycin needs to be administered in appropriate doses and combined with other anti-biotics to avoid further resistance in patients with IE.330,333 Therefore, daptomycin should be given at high doses (10 mg/kg), and most experts recommend its combination with beta-lactams334 or fosfomycin335 (beta-lactams [and probably fosfomycin] increase membrane 32 ESC Guidelines Downloaded from by guest on 30 August 2023 daptomycin binding by decreasing the positive surface charge) for NVE, and with gentamicin and rifampin for PVE.326–328 However, in a rando-mized trial including 352 patients with MRSA bacteraemia, daptomycin or vancomycin combined with i.v. flucloxacillin, cloxacillin, or cefazolin did not result in a significant reduction of the primary composite end-point of mortality, persistent bacteraemia, relapse, or treatment failure as compared with daptomycin or vancomycin alone.328 The study was stopped prematurely before recruiting the target number of patients (n = 440) due to increased incidence of acute kidney injury in the combination therapy arm and, therefore, the results should be inter-preted with caution. Other alternatives include fosfomycin plus imipenem,336 ceftaroline,337 quinupristin–dalfopristin with or without beta-lactams,338,339 beta-lactams plus oxazolidinones (linezolid),340 beta-lactams plus vancomycin,341 and high doses of trimethoprim/sulfamethoxazole and clindamycin.342,343 These clinical and therapeutic scenarios warrant collaborative manage-ment with the Endocarditis Team including an infectious disease specialist, since the evidence is based on very small populations. Recommendation Table 8 — Recommendations for antibiotic treatment of infective endocarditis due to Staphylococcus spp. Recommendations Classa Levelb IE caused by methicillin-susceptible staphylococci In patients with NVE due to methicillin-susceptible staphylococci, (flu)cloxacillin or cefazolin is recommended for 4–6 weeks using the following doses:264,314,316–318 I B Adult antibiotic dosage and route (Flu)cloxacillinc 12 g/day i.v. in 4–6 doses Cefazoline 6 g/day i.v. in 3 doses Paediatric antibiotic dosage and route (Flu)cloxacillinc 200–300 mg/kg/day i.v. in 4–6 equally divided doses Cefazoline 6 g/day i.v. in 3 doses In patients with PVE due to methicillin-susceptible staphylococci, (flu)cloxacillin or cefazolin with rifampin for at least 6 weeks and gentamicin for 2 weeks is recommended using the following doses:264,314,316–318,320 I B Adult antibiotic dosage and route (Flu)cloxacillinc 12 g/day i.v. in 4–6 doses Cefazolin 6 g/day i.v. in 3 doses Rifampin 900 mg/day i.v. or orally in 3 equally divided doses Gentamicind 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses Paediatric antibiotic dosage and route (Flu)cloxacillinc 200–300 mg/kg/day i.v. in 4–6 equally divided doses Cefazolin 6 g/day i.v. in 3 doses Rifampin 20 mg/kg/day i.v. or orally in 3 equally divided doses Gentamicind 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses Allergy to beta-lactams In patients with NVE due to methicillin-susceptible staphylococci who are allergic to penicillin, cefazolin for 4–6 weeks is recommended using the following doses:322–327 I B Adult antibiotic dosage and route Cefazoline 6 g/day i.v. in 3 doses Paediatric antibiotic dosage and route Cefazoline 6 g/day i.v. in 3 doses In patients with PVE due to methicillin-susceptible staphylococci who are allergic to penicillin, cefazolin combined with rifampin for at least 6 weeks and gentamicin for 2 weeks is recommended using the following doses:344 I B Adult antibiotic dosage and route Cefazoline 6 g/day i.v. in 3 doses Rifampin 900 mg/day i.v. or orally in 3 equally divided doses Gentamicind 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses Paediatric antibiotic dosage and route Cefazoline 6 g/day i.v. in 3 doses Rifampin 20 mg/kg/day i.v. or orally in 3 equally divided doses Gentamicind 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses Continued ESC Guidelines 33 Downloaded from by guest on 30 August 2023 In patients with NVE due to methicillin-susceptible staphylococci who are allergic to penicillin, daptomycin combined with ceftaroline or fosfomycin may be considered.322–327 IIb C Adult antibiotic dosage and route Daptomycin 10 mg/kg/day i.v. in 1 dose Ceftarolinef OR Fosfomycing 1800 mg/day i.v. in 3 doses OR 8–12 g/day i.v. in 4 doses In patients with PVE due to methicillin-susceptible staphylococci who are allergic to penicillin, daptomycin combined with ceftaroline or fosfomycin or gentamicin with rifampin for at least 6 weeks and gentamicin for 2 weeks may be considered using the following doses:344 IIb C Adult antibiotic dosage and route Daptomycin 10 mg/kg/day i.v. in 1 dose Ceftarolinef OR Fosfomycing 1800 mg/day i.v. in 3 doses OR 8–12 g/day i.v. in 4 doses Rifampin 900 mg/day i.v. or orally in 3 equally divided doses Gentamicind 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses IE caused by methicillin-resistant staphylococci In patients with NVE due to methicillin-resistant staphylococci, vancomycin is recommended for 4–6 weeks using the following doses:345 I B Adult antibiotic dosage and route Vancomycinh 30–60 mg/kg/day i.v. in 2–3 doses Paediatric antibiotic dosage and route Vancomycinh 30 mg/kg/day i.v. in 2–3 equally divided doses In patients with PVE due to methicillin-resistant staphylococci, vancomycin with rifampin for at least 6 weeks and gentamicin for 2 weeks is recommended using the following doses: I B Adult antibiotic dosage and route Vancomycinh 30–60 mg/kg/day i.v. in 2–3 doses Rifampin 900–1200 mg/day i.v. or orally in 2 or 3 divided doses Gentamicind 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses Paediatric antibiotic dosage and route Vancomycinh 30 mg/kg/day i.v. in 2–3 equally divided doses Rifampin 20 mg/kg/day i.v. or orally in 2 or 3 divided doses Gentamicind 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses In patients with NVE due to methicillin-resistant staphylococci, daptomycin combined with cloxacillin, ceftaroline or fosfomycin may be considered using the following doses:335,345–349 IIb C Adult antibiotic dosage and route Daptomycin 10 mg/kg/day i.v. in 1 dose Cloxacillinc OR Ceftarolinef OR Fosfomycing 12 g/day i.v. in 6 doses OR 1800 mg/day i.v. in 3 doses OR 8–12 g/day i.v. in 4 doses © ESC 2023 IE, infective endocarditis; i.m., intramuscular; i.v., intravenous; NVE, native valve endocarditis; PVE, prosthetic valve endocarditis; U, units. aClass of recommendation. bLevel of evidence. cCloxacillin is not recommended if the patient has penicillin allergy. dMaximum doses 240 mg/day. High doses are associated with increased risk of nephrotoxicity. Renal function and serum gentamicin concentrations should be monitored once a week. When given in a single daily dose, pre-dose (trough) concentrations should be <1 mg/L and post-dose (peak; 1 h after injection) serum concentrations should be ∼10–12 mg/L. eCefazolin can replace cloxacillin only in patients with non–immediate-type hypersensitivity reactions to penicillin. fHigh doses of ceftaroline may be associated with risk of leucopaenia after 2 weeks. Ceftaroline can replace cloxacillin only in patients with non–immediate-type hypersensitivity reactions to penicillin. gIn patients with heart failure, the high load of sodium associated with the use of fosfomycin can lead to acute heart failure. hSerum vancomycin concentrations should achieve 10–15 mg/L at pre-dose (trough) level, although some experts recommend to increase the dose of vancomycin to 45–60 mg/kg/day i.v. in 2 or 3 divided doses to reach serum trough vancomycin levels (Cmin) of 15–20 mg/L as in staphylococcal endocarditis. However, vancomycin dose should not exceed 2 g/d unless serum levels are monitored and can be adjusted to obtain a peak plasma concentration of 30–45 μg/mL 1 h after completion of the i.v. infusion of the antibiotic. 34 ESC Guidelines Downloaded from by guest on 30 August 2023 7.8. Enterococcus spp. Enterococcal IE is primarily caused by E. faecalis (90% of cases) and less often by Enterococcus faecium (5% of cases), or other species.350 Enterococcal IE poses two major problems. First, enterococci are highly resistant to antibiotic-induced killing, and eradication requires pro-longed administration (up to 6 weeks) of synergistic bactericidal com-binations of two cell wall inhibitors (ampicillin plus ceftriaxone, which synergize by inhibiting complementary PBPs), or one cell wall inhibitor with aminoglycosides.351–353 Second, they may be resistant to multiple drugs, including aminoglycosides (HLAR), beta-lactams (via PBP 5 modi-fication and sometimes beta-lactamases), and vancomycin.351–357 Penicillin-susceptible strains are treated with penicillin G or ampicillin (or amoxicillin) combined with gentamicin. However, ampicillin (or amoxicillin) is preferred since the MIC is two to four times lower than that of penicillin G. Gentamicin resistance is frequent in both E. faecalis and E. faecium (up to 75%).358,359 An aminoglycoside MIC >128 mg/L (HLAR) is associated with the loss of bactericidal synergism with cell wall inhibitors, and aminoglycosides should not be used in such conditions. There have been two important advances in recent years. First, in sev-eral cohort studies of E. faecalis IE including hundreds of cases, it was ob-served that ampicillin plus ceftriaxone is as effective as ampicillin plus gentamicin for non-HLAR E. faecalis IE. The combination of ampicillin plus ceftriaxone was also associated with a beneficial safety profile, due to the lack of nephrotoxicity.355,360,361 Therefore, this is the com-bination of choice for treating NVE and PVE caused by HLAR E. faecalis. This double beta-lactam therapy is not effective against E. faecium and the experience in the treatment of other enterococcal species is very limited. Second, the total daily dose of gentamicin can be given in a single daily dose instead of the 2 or 3 divided doses previously recommended, and the length of the treatment with gentamicin for non-HLAR E. faeca-lis IE may be safely shortened from 4–6 weeks to 2 weeks, reducing the rates of nephrotoxicity to very low levels.266,362,363 After 10–14 days of therapy, OPAT or outpatient oral antibiotic therapy should be consid-ered if the patient is clinically stable (see Section 7.13).364–367 Beta-lactam or vancomycin resistance is mainly observed in E. fae-cium. Since dual resistance is rare, beta-lactam might be used against vancomycin-resistant strains and vice versa. Varying results have been reported with quinupristin–dalfopristin (not active against E. faecalis), linezolid, daptomycin, teicoplanin, and tigecycline.353,365,368 Daptomycin 10–12 mg/kg/24 h, always combined with beta-lactams (ampicillin, ertapenem, or ceftaroline) or fosfomycin in order to prevent the development of daptomycin resistance, is the best option for treating multidrug- and vancomycin-resistant enterococcal IE.369 Recommendation Table 9 — Recommendations for antibiotic treatment of infective endocarditis due to Enterococcus spp. Recommendations Classa Levelb Beta-lactam and gentamicin-susceptible strains In patients with NVE due to non-HLAR Enterococcus spp., the combination of ampicillin or amoxicillin with ceftriaxone for 6 weeks or with gentamicin for 2 weeks is recommended using the following doses:355,360,361 I B Adult antibiotic dosage and route Amoxicillin 200 mg/kg/day i.v. in 4–6 doses Ampicillin 12 g/day i.v. in 4–6 doses Ceftriaxone 4 g/day i.v. in 2 doses Gentamicinc 3 mg/kg/day i.v. or i.m. in 1 dose Paediatric antibiotic dosage and route Ampicillin 300 mg/kg/day i.v. in 4–6 equally divided doses Ceftriaxone 100 mg/kg i.v. in 2 doses Gentamicinc 3 mg/kg/day i.v. or i.m. in 3 equally divided doses In patients with PVE and patients with complicated NVE or >3 months of symptoms due to non-HLAR Enterococcus spp., the combination of ampicillin or amoxicillin with ceftriaxone for 6 weeks or with gentamicin for 2 weeks is recommended using the following doses:355,360,361 I B Adult antibiotic dosage and route Amoxicillin 200 mg/kg/day i.v. in 4–6 doses Ampicillin 12 g/day i.v. in 4–6 doses Ceftriaxone 4 g/day i.v. in 2 doses Gentamicinc 3 mg/kg/day i.v. or i.m. in 1 dose Paediatric antibiotic dosage and route Ampicillin 300 mg/kg/day i.v. in 4–6 equally divided doses Amoxicillin 100–200 mg/kg/day i.v. in 4–6 doses Ceftriaxone 100 mg/kg/day i.v. in 2 doses Gentamicinc 3 mg/kg/day i.v. or i.m. in 3 equally divided doses Continued ESC Guidelines 35 Downloaded from by guest on 30 August 2023 7.9. Gram-negative bacteria 7.9.1. Haemophilus, Aggregatibacter, Cardiobacterium, Eikenella, and Kingella-related species Haemophilus, Aggregatibacter (previously Actinobacillus), Cardiobacterium, Eikenella, and Kingella (HACEK) Gram-negative bacilli are fastidious organ-isms that require special investigations when they are the suspected cause of IE (see also Section 5). Because they grow slowly, standard MIC tests may be difficult to interpret. Some HACEK group bacilli produce beta- lactamases, and therefore ampicillin is no longer the first-line option. Conversely, they are susceptible to ceftriaxone, other third-generation ce-phalosporins, and fluoroquinolones. The standard treatment is ceftriaxone 2 g/day for 4 weeks in NVE and for 6 weeks in PVE. If they do not produce beta-lactamase, ampicillin (12 g/day i.v. in 4 or 6 doses) for 4–6 weeks plus gentamicin (3 mg/kg/day divided into 2 or 3 doses) for 2 weeks is an High-level aminoglycoside resistanced In patients with NVE or PVE due to HLAR Enterococcus spp., the combination of ampicillin or amoxicillin and ceftriaxone for 6 weeks is recommended using the following doses:355,360,361 I B Adult antibiotic dosage and route Ampicillin 12 g/day i.v. in 4–6 doses Amoxicillin 200 mg/kg/day i.v. in 4–6 doses Ceftriaxone 4 g/day i.v. or i.m. in 2 doses Paediatric antibiotic dosage and route Ampicillin 300 mg/kg/day i.v. in 4–6 equally divided doses Amoxicillin 100–200 mg/kg/day i.v. in 4–6 doses Ceftriaxone 100 mg/kg i.v. or i.m. in 2 doses Beta-lactam resistant Enterococcus spp. (E. faecium)e In patients with IE due to beta-lactam resistant Enterococcus spp. (E. faecium), vancomycin for 6 weeks combined with gentamicin for 2 weeks is recommended using the following doses:358,359,369 I C Adult antibiotic dosage and route Vancomycin 30 mg/kg/day i.v. in 2 doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dose Paediatric antibiotic dosage and route Vancomycin 30 mg/kg/day i.v. in 2–3 equally divided doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dose Vancomycin-resistant Enterococcus spp.f In patients with IE due to vancomycin-resistant Enterococcus spp., daptomycin combined with beta-lactams (ampicillin, ertapenem, or ceftaroline) or fosfomycin is recommended using the following doses:369 I C Adult antibiotic dosage and route Daptomycin 10–12 mg/kg/day i.v. in 1 dose Ampicillin 300 mg/kg/day i.v. in 4–6 equally divided doses Fosfomycin 12 g/day i.v. in 4 doses Ceftaroline 1800 mg/day i.v. in 3 doses Ertapenemg 2 g/day i.v. or i.m. in 1 dose Paediatric antibiotic dosage and route Daptomycin 10–12 mg/kg/day i.v. in 1 dose (age-adjusted) Ampicillin 300 mg/kg/day i.v. in 4–6 equally divided doses Fosfomycin 2–3 g/day i.v. in 1 dose Ceftaroline 24–36 mg/kg/day in 3 doses Ertapenemg 1 g/day i.v. or i.m. in 1 dose [if younger than 12 years, 15 mg/kg/dose (to a maximum of 500 mg) twice daily] © ESC 2023 HLAR, high-level aminoglycoside resistance; IE, Infective endocarditis; i.m., intramuscular; i.v., intravenous; NVE, native valve endocarditis; PBP, Penicillin-binding protein; PVE, prosthetic valve endocarditis. aClass of recommendation. bLevel of evidence. cMaximum doses 240 mg/day. High doses are associated with increased risk of nephrotoxicity. Renal function and serum gentamicin concentrations should be monitored once a week. When given in a single daily dose, pre-dose (trough) concentrations should be <1 mg/L and post-dose (peak; 1 h after injection) serum concentrations should be ∼10–12 mg/L. dHigh-level resistance to gentamicin: if susceptible to streptomycin, replace gentamicin with streptomycin 15 mg/kg/day in two equally divided doses. eBeta-lactam resistance: (i) if due to beta-lactamase production, replace ampicillin with ampicillin–sulbactam or amoxicillin with amoxicillin–clavulanate; (ii) if due to PBP5 alteration, use vancomycin-based regimens. fMultiresistance to aminoglycosides, beta-lactams and vancomycin: suggested alternatives are (i) daptomycin 10 mg/kg/day plus either ampicillin 200 mg/kg/day i.v. in four to six doses, ertapenem (2 g/day i.v.), ceftaroline (600 mg/8 h i.v.), or fosfomycin (3 g/6 h i.v.); (ii) linezolid 2 × 600 mg/day i.v. or orally for ≥8 weeks (monitor haematological toxicity); (iii) quinupristin–dalfopristin 3 × 7.5 mg/kg/day for ≥8 weeks. Quinupristin–dalfopristin is not active against E. faecalis; (iv) for other combinations (daptomycin plus ertapenem or ceftaroline or fosfomycin), consult infectious disease specialists. gHigh doses of ertapenem are associated with seizures. 36 ESC Guidelines Downloaded from by guest on 30 August 2023 option.370 Ciprofloxacin (400 mg every 8–12 h i.v. or 750 mg every 12 h orally) is a less well-validated alternative.370–373 7.9.2. Non-Haemophilus, Aggregatibacter, Cardiobacterium, Eikenella, and Kingella species The ICE cohort reported non-HACEK Gram-negative bacteria in 49 of 2761 (1.8%) IE cases.279,374 Recommended treatment is early surgery plus prolonged (6 weeks) therapy with bactericidal combinations of beta-lactams and aminoglycosides, sometimes with additional quino-lones or cotrimoxazole.375,376 In vitro bactericidal tests and monitoring of serum antibiotic concentrations may be helpful. Because of their rar-ity and severity, these conditions should be discussed by the Endocarditis Team. 7.10. Blood culture-negative infective endocarditis The main causes of BCNIE are summarized in Section 5.3.2.377,378 Treatment options are summarized in Table 11.379–383 Treatment of Whipple’s IE remains highly empirical. Successes have been reported with long-term therapy (>1 year).384 In cases of CNS involvement, sulfadiazine 1.5 g/6 h orally must be added to doxycycline. An alterna-tive therapy is ceftriaxone (2 g/24 h i.v.) for 2–4 weeks or penicillin G (2 million U/4 h) and streptomycin (1 g/24 h) i.v. for 2–4 weeks followed by cotrimoxazole (800 mg/12 h) orally. Trimethoprim is not active against T. whipplei. Consultation with the Endocarditis Team, including an infectious disease specialist, is recommended. 7.11. Fungi Fungi are most frequently observed in PVE and in IE affecting PWID or immunocompromised patients.386 Candida and Aspergillus spp. predom-inate, the latter resulting in BCNIE.387,388 Mortality is very high (>50%), and treatment necessitates combined antifungal administration and with a low threshold for surgery.278,387,388 Antifungal therapy for Candida IE includes an echinocandin at high doses or liposomal amphotericin B (or other lipid formulations) with or without flucytosine. for Aspergillus IE, voriconazole is the drug of choice. Some experts recommend the add-ition of an echinocandin or amphotericin B.278,387–390 Suppressive long- term treatment with oral azoles (fluconazole and voriconazole) is re-commended, sometimes lifelong.278,388,389 Consultation with the Endocarditis Team including an infectious disease specialist is recommended. 7.12. Empirical therapy Treatment of IE should be started promptly. Three sets of blood cultures should be drawn at 30-minute intervals before initiation of antibiotics.391 The initial choice of empirical treatment depends on several considerations: (i) Previous antibiotic therapy. (ii) IE in a native valve or a prosthesis (and if so, when surgery was per-formed [early vs. late PVE]). (iii) The place of the infection (community, nosocomial, or non- nosocomial healthcare-associated IE) and knowledge of the local epidemiology, especially for antibiotic resistance and specific genu-ine culture-negative pathogens. (iv) Cloxacillin/cefazolin administration is associated with lower mortality rates than other beta-lactams, including amoxicillin/clavulanic acid or ampicillin/sulbactam,392 and vancomycin for empirically treating MSSA bacteraemia/endocarditis.309,393 However, recently amoxicil-lin/clavulanic acid or ampicillin/sulbactam might be an effective empir-ical treatment for MSSA bacteraemia when de-escalated to cloxacillin or cefazolin within 96 h from the index blood culture.394 Native valve endocarditis and late PVE regimens should cover staphylococci, streptococci, and enterococci. If the patient was receiv-ing antibiotic therapy, the empirical therapy should include different antibiotics. CoNS should be empirically covered in PVE but not in NVE. Early PVE or healthcare-associated IE regimens should cover methicillin-resistant staphylococci, enterococci and, ideally, non-HACEK Gram-negative pathogens. Once the pathogen is identi-fied (usually within 24 h), the antibiotic treatment must be adapted to its antimicrobial susceptibility pattern. It should be emphasized that the empirical treatment should be changed to targeted therapy once the organism is identified within 24–48 h. Table 11 Antibiotic treatment of blood culture- negative infective endocarditis Pathogens Proposed therapya Treatment outcome Brucella spp. Doxycycline (200 mg/24 h) plus cotrimoxazole (960 mg/ 12 h) plus rifampin (300– 600 mg/24 h) for ≥3–6 monthsb orally Treatment success defined as an antibody titre <1:60. Some authors recommend adding gentamicin for the first 3 weeks C. burnetii (Q fever agent) Doxycycline (200 mg/24 h) plus hydroxychloroquine (200– 600 mg/24 h)c orally (>18 months of treatment) Treatment success defined as anti-phase I IgG titre <1:400, and IgA and IgM titres <1:50 Bartonella spp.d Doxycycline 100 mg/12 h orally for 4 weeks plus gentamicin (3 mg/24 h) i.v. for 2 weeks Treatment success expected in ≥90% Legionella spp. Levofloxacin (500 mg/12 h) i.v. or orally for ≥6 weeks or clarithromycin (500 mg/12 h) i.v. for 2 weeks, then orally for 4 weeks plus rifampin (300– 1200 mg/24 h) Optimal treatment unknown Mycoplasma spp. Levofloxacin (500 mg/12 h) i.v. or orally for ≥6 monthse Optimal treatment unknown T. whipplei (Whipple’s disease agent)f Doxycycline (200 mg/24 h) plus hydroxychloroquine (200– 600 mg/24 h)c orally for ≥18 months Long-term treatment, optimal duration unknown © ESC 2023 IE, infective endocarditis; Ig, immunoglobulin; i.v., intravenous. Adapted from Brouqui et al.383 aOwing to the lack of large series, the optimal duration of treatment of IE due to these pathogens is unknown. The presented durations are based on selected case reports. Consultation with an infectious disease specialist is recommended. bAddition of streptomycin (15 mg/kg/24 h in 2 doses) for the first few weeks is optional. cDoxycycline plus hydroxychloroquine (with monitoring of serum hydroxychloroquine levels) is significantly superior to doxycycline.385 dSeveral therapeutic regimens have been reported, including ampicillin or amoxicillin, (12 g/ 24 h i.v.) or cephalosporins (ceftriaxone 2 g/24 h i.v.) combined with aminoglycosides (gentamicin or netilmicin).381 Dosages are as for streptococcal and enterococcal IE.379,380 eNewer fluoroquinolones (levofloxacin, moxifloxacin) are more potent than ciprofloxacin against intracellular pathogens such as Mycoplasma spp., Legionella spp., and Chlamydia spp. fTreatment of Whipple’s IE remains highly empirical. In the case of central nervous system involvement, sulfadiazine 1.5 g/6 h orally must be added to doxycycline. An alternative therapy is ceftriaxone (2 g/24 h i.v.) for 2–4 weeks or penicillin G (2 million U/4 h) and streptomycin (1 g/24 h) i.v. for 2–4 weeks followed by cotrimoxazole (800 mg/12 h) orally. Trimethoprim is not active against T. whipplei. Successes have been reported with long-term therapy (1 year). ESC Guidelines 37 Downloaded from by guest on 30 August 2023 7.13. Outpatient parenteral or oral antibiotic therapy for infective endocarditis Outpatient parenteral antibiotic treatment or step-down outpatient oral antibiotic treatment is used to consolidate antimicrobial therapy once crit-ical infection-related complications are under control (e.g. perivalvular ab-scesses, acute HF, septic emboli, and stroke) and the patient is clinically stable.43,396–399 When feasible, early hospital discharge and OPAT helps to alleviate the effects of infection and prolonged hospitalization especially in the elderly.400 In the initial phase of IE treatment, standard i.v. treatment is administered according to recommendations for specific microorgan-isms. Once the clinical condition of the patient is stable, OPAT or step- down outpatient oral antibiotic treatment is a safe alternative to in-hospital i.v. treatment in selected patients.43,399 Patients may reach such stability at various points in their disease course but, when criteria for stability are reached, the patient may then be switched to OPAT or alternatively to an oral therapy at hospital discharge. The OPAT regime consists of the same antibiotic combinations administered in the acute phase if possible. The 5-year outcomes from the POET trial showed continued effectiveness of oral antibiotic therapy as compared with i.v. antibiotic therapy for IE in selected patients.401 Hence, clinical stability will then differentiate IE courses into two phases: (i) Critical phase where at least 10 days of i.v. treatment is required: at this time point, OPAT has a restricted indication. (ii) Continuation phase (beyond 10 days of therapy and 7 days post-surgery), where OPAT/step-down oral therapy may be feasible. Supplementary data online, Table S8 summarizes the salient ques-tions to address when considering OPAT/step-down oral therapy for IE. Recommendation Table 10 — Recommendations for antibiotic regimens for initial empirical treatment of in-fective endocarditis (before pathogen identification)a Recommendations Classb Levelc In patients with community-acquired NVE or late PVE (≥12 months post-surgery), ampicillin in combination with ceftriaxone or with (flu)cloxacillin and gentamicin should be considered using the following doses:255 IIa C Adult antibiotic dosage and route Ampicillin 12 g/day i.v. in 4–6 doses Ceftriaxone 4 g/day i.v. or i.m. in 2 doses (Flu)cloxacillin 12 g/day i.v. in 4–6 doses Gentamicind 3 mg/kg/day i.v. or i.m. in 1 dose Paediatric antibiotic dosage and route Ampicillin 300 mg/kg/day i.v. in 4–6 equally divided doses Ceftriaxone 100 mg/kg i.v. or i.m. in 1 dose (Flu)cloxacillin 200–300 mg/kg/day i.v. in 4–6 equally divided doses Gentamicind 3 mg/kg/day i.v. or i.m. in 3 equally divided doses In patients with early PVE (<12 months post-surgery) or nosocomial and non-nosocomial healthcare-associated IE, vancomycin or daptomycin combined with gentamicin and rifampin may be considered using the following doses:395 IIb C Adult antibiotic dosage and route Vancomycine 30 mg/kg/day i.v. in 2 doses Daptomycin 10 mg/kg/day i.v. in 1 dose Gentamicind 3 mg/kg/day i.v. or i.m. in 1 dose Rifampin 900–1200 mg i.v. or orally in 2 or 3 doses Paediatric antibiotic dosage and route Vancomycine 40 mg/kg/day i.v. in 2–3 equally divided doses Gentamicind 3 mg/kg/day i.v. or i.m. in 3 equally divided doses Rifampin 20 mg/kg/day i.v. or orally in 3 equally divided doses Allergy to beta-lactams In patients with community-acquired NVE or late PVE (≥12 months post-surgery) who are allergic to penicillin, cefazolin, or vancomycin in combination with gentamicin may be considered using the following doses: IIb C Adult antibiotic dosage and route Cefazolin 6 g/day i.v. in 3 doses Vancomycine 30 mg/kg/day i.v. in 2 doses Gentamicind 3 mg/kg/day i.v. or i.m. in 1 dose Continued Paediatric antibiotic dosage and route Cefazolin 6 g/day i.v. in 3 doses Vancomycine 40 mg/kg/day i.v. in 2–3 equally divided doses Gentamicind 3 mg/kg/day i.v. or i.m. in 3 equally divided doses © ESC 2023 BCNIE, blood culture-negative infective endocarditis; IE, infective endocarditis; i.m., intramuscular; i.v., intravenous; NVE, native valve endocarditis; PVE, prosthetic valve endocarditis. aIf initial blood cultures are negative and there is no clinical response, BCNIE aetiology (see Section 7.10) and the extension of the antibiotic spectrum to blood culture-negative pathogens should be considered. If cardiac surgery is indicated, molecular diagnosis can be performed. bClass of recommendation. cLevel of evidence. dMaximum doses 240 mg/day. High doses are associated with increased risk of nephrotoxicity. Renal function and serum gentamicin concentrations should be monitored once a week. When given in a single daily dose, pre-dose (trough) concentrations should be <1 mg/L and post-dose (peak; 1 h after injection) serum concentrations should be ∼10–12 mg/L. eSerum vancomycin concentrations should achieve 10–15 mg/L at pre-dose (trough) level, although some experts recommend to increase the dose of vancomycin to 45–60 mg/kg/day i.v. in 2 or 3 divided doses to reach serum trough vancomycin levels (Cmin) of 15–20 mg/L as in staphylococcal endocarditis. However, vancomycin dose should not exceed 2 g/d unless serum levels are monitored and can be adjusted to obtain a peak plasma concentration of 30–45 μg/mL 1 h after completion of the i.v. infusion of the antibiotic. 38 ESC Guidelines Downloaded from by guest on 30 August 2023 In addition to the patient being medically stable, general considerations for suitability for OPAT include assessment of the patient’s home environ-ment and self-care capabilities. Adherence to treatment and follow-up vis-its are also crucial for a beneficial outcome of outpatient treatment and the healthcare provider–patient relationship is important for ensuring proper and continued treatment and maintenance of infection control. 7.13.1. Parenteral and oral step-down antibiotic treatment Stability criteria are essential and timing in the clinical planning the pa-tient’s course, especially TOE, becomes key (Figure 9). Stability criteria include blood samples, clinical parameters, and TOE.43 OPAT has been shown to be a safe treatment in IE for stable patients who are suitable for home treatment. The patient, and preferably also a caregiver, should be educated care-fully in the disease and how to monitor/observe for signs of infection, including daily temperature and other signs of disease progression or complications. In addition, regular post-discharge evaluation is required (nurse once per day, responsible physician 1–3 times per week). For pa-tients receiving OPAT, regular i.v. catheter inspection and care by a healthcare professional should be provided. If the patient is not suffi-ciently able to self-monitor, and has no close caregivers, added surveil-lance is required by involved staff and home treatment should generally be carefully considered in such cases. Certain combinations of two oral antibiotics should be used for oral step-down treatment (see Supplementary data online, Table S9). 7.13.2. Other considerations for outpatient oral or parenteral antimicrobial therapy In the OPAT programme, patients continue with the same antibiotics that are administered in the acute phase in once-daily regimens, or with infusion pumps if antibiotics should be administered intermittently, N N N N Definite infective endocarditis Y Y Y Y Patient not suitable for oral antibiotic treatment The patient may be shifted from i.v. to oral antibiotics (two-drug strategy) Blood cultures with either S. aureus, streptococci, CoNS, or E. faecalis Other indication for continued i.v. antibiotics? or BMI >40 or faulty gastrointestinal uptake Perform TOE: New surgical indication? Infection control? Satisfying response to treatment: no fever >2 days, CRP <25% of max measured value or <20 mg/L and leukocytes <15 x 109/L Treated with relevant i.v. antibiotics ≥10 days and ≥7 days after valve surgery Continue i.v. antibiotics Figure 9 Flowchart to assess clinical stability based on the Partial Oral Treatment of Endocarditis trial. BMI, body mass index; CoNS, coagulase-negative staphylococci; CRP, C-reactive protein; i.v., intravenous; TOE, transoesophageal echocardiography. Adapted with permission from Iversen et al.43 ESC Guidelines 39 Downloaded from by guest on 30 August 2023 or in continuous infusion. Dalbavancin is a glycopeptide antibiotic with a very long half-life that can be administered weekly. There is previous positive experience in sensitive Gram-positive IE, although the most ef-fective administration schedule is not clear.274,402 The recommended prescription is 1.5 g as a loading dose followed by 0.5–1 g weekly until completing 6 weeks of antibiotic treatment. Although the evidence is weak, another option (in addition to the combinations listed in the Supplementary data online, Table S9) for staphylococcal IE is the combination of i.v. cotrimoxazole (sulfameth-oxazole 4800 mg/day and trimethoprim 960 mg/day in 4–6 doses) plus i.v. clindamycin (1800 mg/day in 3 doses) during the first week fol-lowed by only oral cotrimoxazole for 5 weeks.343 8. Indications for surgery and management of main infective endocarditis complications Infective endocarditis is associated with certain risks and complications that can only be controlled with surgical intervention. Despite the risks of surgery in these patients, current evidence suggests that surgical treatment may generate a survival advantage of up to 20% in the first year.403,404 There are three main reasons to undergo surgery in the set-ting of acute IE: HF, uncontrolled infection, and prevention of septic embolization (in particular, to the CNS) (Figure 10). A significant proportion of surgical procedures for IE are performed on an urgent basis. The Task Force has defined urgent surgery as that requiring intervention within 3–5 days, although unnecessary delays should be avoided once the indication for urgent surgery is established. Some cases require emergency surgery (within 24 h), irrespective of the pre-operative duration of antibiotic treatment. A third group re-quires surgery non-urgently, i.e. within the same hospital admission. In cases where the infective component can be completely healed with antibiotic treatment alone, both timing and indications for treat-ment of residual valve dysfunction follow the conventional guidelines for valve treatment.128 8.1. Pre-operative risk assessment The risk of surgical therapy during the active phase of IE can be signifi-cant. It is heavily influenced by pre-existing co-morbidities and current organ function, but should not be limited by one risk factor alone (e.g. age or liver function).405,406 The decision to operate should therefore be made by the Endocarditis Team (see Section 4),167 considering ur-gency of the patient’s clinical condition, peri-operative risk, the poten-tial to recover from the infection, and the patient’s associated long-term prognosis.403,404 There are several scoring systems that predict mortality after general (i.e. non-IE) cardiac surgery and which are in routine clinical use.407,408 Other scoring systems were designed specifically for the setting of IE in-cluding the AEPEI (Association for the Study and Prevention of Infective Endocarditis Study) score, the STS (Society of Thoracic Surgeons) IE score, the PALSUSE (prosthetic valve, age ≥70, large intracardiac destruction, Staphylococcus spp., urgent surgery, sex [female], EuroSCORE ≥10) score, the de Feo score, and the ANCLA (anaemia, NYHA [New York Heart Association] class IV, critical state, large intracardiac destruction, surgery of thoracic aorta) score, among others.256,409–414 Some of these scoring systems are web-based and free of charge (e.g. the AEPEI risk calculator mortalite-post-chirurgie-aepei). Such scoring systems have been devel-oped based on retrospective data and their performance is vari-able.250,256,415–417 In addition, none of these scoring systems are used in daily clinical routine. Therefore, prospective surgical scoring systems with better precision need to be developed, particularly for determining operative futility in prohibitively high-risk patients. A significant proportion of patients with clear indications for surgery for IE may have multiple risk factors or other reasons that lead to sur-gery not being performed, and these patients have the worst progno-sis.184,403 Conversely, high-risk but salvageable patients may not be offered life-saving operations on the basis of perceived unacceptable risk, and this is especially true in the elderly (see Section 12.2). The com-plex decision of not offering surgery when indicated should therefore be made in the setting of an Endocarditis Team with experienced sur-gical input.418 Determining when operative management for a specific patient is futile requires compassionate multidisciplinary insight along with consideration of the patient’s and family’s wills (see Section 13.2). 8.2. Heart failure 8.2.1. Heart failure in infective endocarditis Heart failure is the most frequent complication of IE and the main indi-cation for urgent and emergency surgery for IE.419 The prevalence of HF with left-sided IE is variable and inconsistently defined between re-ported series, ranging between 19% and 73%.420–425 Clinical symptoms Recommendation Table 11 — Recommendations for outpatient antibiotic treatment of infective endocarditis Recommendations Classa Levelb Outpatient parenteral or oral antibiotic treatment should be considered in patients with left-sided IE caused by Streptococcus spp., E. faecalis, S. aureus, or CoNS who were receiving appropriate i.v. antibiotic treatment for at least 10 days (or at least 7 days after cardiac surgery), are clinically stable, and who do not show signs of abscess formation or valve abnormalities requiring surgery on TOE.43,401 IIa A Outpatient parenteral antibiotic treatment is not recommended in patients with IE caused by highly difficult-to-treat microorganisms,c liver cirrhosis (Child-Pugh B or C), severe cerebral nervous system emboli, untreated large extracardiac abscesses, heart valve complications, or other severe conditions requiring surgery, severe post-surgical complications, and PWID-related IE. III C © ESC 2023 CoNS, coagulase-negative staphylococci; IE, infective endocarditis; i.v., intravenous; TOE, transoesophageal echocardiography; PWID, people who inject drugs. aClass of recommendation. bLevel of evidence. cHighly difficult-to-treat microorganism: microorganisms requiring i.v. antibiotic combinations that cannot be administered by means of outpatient parenteral antibiotic treatment or that require strict monitoring of drug levels either in blood or in other fluids owing to their potential toxicity or narrow therapeutic index (e.g. MRSA or vancomycin-resistant enterococci also resistant to alternative drugs such as daptomycin and linezolid, multidrug- or extensively drug-resistant Gram-negative rods, highly penicillin-resistant oral streptococci, fungi other than Candida). 40 ESC Guidelines Downloaded from by guest on 30 August 2023 Heart failure Emergency surgery (Class I) Uncontrolled infection High risk of embolism or established embolism Cardiogenic shock or pulmonary oedema Local complications (abscess, false aneurysm, fistula, enlarging vegetation) Vegetation ≥10 mm and emboli despite appropriate antibiotic therapy Poor haemodynamic tolerance Persistent positive blood culturesa Vegetation ≥10 mm and other reason for surgeryb N Non-urgent surgery (Class I) N N Resistant bacteriac or fungi N N Vegetation ≥10 mm and no evidence of embolus N Antibiotic therapy and continued observation N Surgical timing for patient with left-sided infective endocarditis Y Urgent surgery (Class I) Y Urgent surgery (Class I) Y Urgent surgery (Class I) Y Urgent surgery (Class IIa) Y Urgent or non-urgent surgery (Class I)d Y PVE caused by S. aureus or non-HACEK Gram-negative bacilli N Non-urgent surgery (Class I) N Urgent surgery (Class IIa) Y Urgent surgery (Class I) Y Urgent surgery (Class IIb) Y Figure 10 Proposed surgical timing for infective endocarditis. HACEK, Haemophilus, Aggregatibacter, Cardiobacterium, Eikenella, and Kingella; PVE, prosthet-ic valve endocarditis. Surgery timing: emergency, within 24 h. Urgent, within 3–5 days. Non-urgent, within same hospital admission. aDespite appropriate antibiotic therapy for >1 week and control of septic embolic foci. bE.g. patients with significant valvular dysfunction that is, or is not, a direct result of endo-carditis process. cS. aureus (methicillin resistant and non-methicillin resistant), vancomycin-resistant enterococci, non-HACEK Gram-negative bacteria and fungi. dUrgent for S. aureus, non-urgent for others. ESC Guidelines 41 Downloaded from by guest on 30 August 2023 are mainly caused by congestion and may vary from mild dyspnoea to severe and rapidly worsening dyspnoea, orthopnoea, pulmonary oe-dema, and cardiogenic shock. Factors associated with increased risk of HF complicating the course of IE include older age, presence of NVE with aortic valve involvement, and high comorbidity.420–425 Leaflet perforation and rupture, as well as mitral chordal rupture, lead to new severe valvular regurgitation or worsening of pre-existent valvular regurgitation and subsequent acute HF. Other less common causes of HF include intracardiac fistulae, interference of the vegetation mass with leaflet opening and closure, or myocardial infarction from ve-getations embolizing into the coronary arteries. Patients with right- sided IE complicated by HF present with symptoms of right heart con-gestion, as discussed in Section 12.6. New-onset HF is the predominant clinical presentation in IE patients, whereas worsening of pre-existing HF is less frequent. Cardiogenic shock can be the first presentation in up to 5% of cases, of which half of such patients develop cardiogenic shock within 72 h of admission for IE.424 On imaging tests, patients with IE complicated by HF present more frequently with lower left ventricular ejection fraction, larger vegetation size, perivalvular abscesses, pseudoaneurysms, and valvular regurgitation secondary to leaflet perforation or rupture.420–425 Heart failure complicating IE is independently associated with poor in-hospital and 1-year survival, and surgical treatment is the only effect-ive treatment that is associated with improved survival.420,421,424,426–430 Even though in-hospital mortality rates increase with the severity of HF presentation, the survival benefit of surgical treatment vs. medical ther-apy is more pronounced among patients with NYHA functional class III–IV symptoms.420 TTE provides important information on the sever-ity of the haemodynamic consequences of valve dysfunction. New on-set of elevated filling pressures, pulmonary hypertension, and/or pericardial effusion may lead to urgent or emergency recommendation for surgery.163 Biomarkers such as B-type natriuretic peptide and troponin have been associated with poor prognosis in IE.431,432 Patients who are discharged after treatment of IE require subsequent follow-up (see Section 11). Heart failure is more likely to develop during follow-up in IE patients who are discharged with valvular regurgitation than in those without regurgitation, particularly if mitral regurgitation is present.433 8.2.2. Indications and timing of surgery in the presence of heart failure in infective endocarditis Timing of surgical intervention in patients with IE (Figure 10) compli-cated by HF should be decided by the Endocarditis Team, although sur-gery should not be delayed by Endocarditis Team discussions in patients requiring emergency operations. The presence of HF leads to recom-mendation for surgery in the majority of patients and is the principal in-dication for urgent surgery in IE patients.429,434 Emergency surgery should be performed in patients with new-onset NYHA class IV HF symptoms, pulmonary oedema, and/or cardiogenic shock, irrespective of the status of infection or length of antibiotic treatment and when considered non-futile intervention. Urgent surgery is indicated in pa-tients with milder forms of HF (NYHA class II–III) and severe valve re-gurgitation or echocardiographic signs of haemodynamic compromise (elevated end-diastolic left ventricular pressure, high left atrial pressure, or moderate and severe pulmonary hypertension), or large vegetations. In patients without haemodynamic compromise, i.v. antibiotic therapy and strict clinical and echocardiographic observation are first indicated, and surgery can be temporarily delayed. However, it should be emphasized that early surgery is a good option for patients with surgical indications and low risk of surgery.403,404 8.3. Uncontrolled infection Uncontrolled infection is one of the most common complications of IE and is the second most frequent indication for surgery.5 Uncontrolled infection is considered to be present when there is: (i) persistent infec-tion or sepsis despite antibiotic therapy; (ii) signs of local infection that do not respond to antibiotic therapy; or (iii) infection with resistant or very virulent organisms. 8.3.1. Septic shock and persistent infection Septic shock, defined as vasopressor requirement to maintain a mean arterial pressure of 65 mmHg or greater and serum lactate level greater than 2 mmol/L in the absence of hypovolaemia,435 is a highly lethal com-plication of IE and occurs in ∼5–10% of patients.425,436 Risk factors for septic shock include S. aureus and Gram-negative bacteria, persistent bacteraemia, nosocomial acquisition, acute renal failure, diabetes melli-tus, CNS emboli, and large vegetations.147,436 Surgery is associated with a significant reduction in early and 1-year mortality for patients with IE and septic shock.425,436 Urgent surgery is therefore recommended in patients with IE and persistent sepsis or septic shock despite adequate antibiotic therapy, in which surgery is non-futile. The definition of persistent infection is somewhat arbitrary and con-sists of fever and persistent positive cultures after 7 days of appropriate antibiotic treatment. It has been demonstrated that persistent blood cultures 48–72 h after initiation of antibiotics are an independent risk factor for hospital mortality.437 In many cases of persistent infection, antibiotics alone are insufficient to eradicate the infection. Surgery is therefore indicated for persistent infection when extracardiac ab-scesses (splenic, vertebral, cerebral, or renal) and other potential causes of positive cultures and fever (infected lines and embolic complications) have been excluded. Persistent fever may also be caused as an adverse reaction to antibiotics.438 8.3.2. Locally uncontrolled infection Signs of locally uncontrolled infection include increasing vegetation size, abscess formation, the creation of pseudoaneurysms and/or fistulae, and new atrioventricular block (AVB). The incidence of perivalvular ex-tension ranges from 10% to 30% in NVE with higher incidences found in patients with PVE.5,439 Perivalvular complications and abscess forma-tion are more frequent in aortic valve than mitral valve IE, and may be higher in patients with bicuspid vs. tricuspid aortic valves.440 In aortic valve IE, perivalvular extension occurs most frequently in the mitral-aortic intervalvular fibrosa,441 whereas perivalvular abscesses are usually located posteriorly or laterally in mitral valve IE.442 Persistent fever and infection, new AVB, chest pain, new heart murmur, recurrent embolism, or HF may indicate perivalvular extension. The diagnosis should be confirmed by TOE, which is more sensitive and spe-cific than TTE.443 However, mitral annular calcification may obscure small regions of mitral perivalvular extension, particularly in the poster-ior aspects of the mitral annulus. Cardiac CT has been shown to be an accurate alternative imaging procedure for the evaluation of perivalvu-lar extension of infection, and PET/CT imaging may be particularly help-ful in cases of PVE (see Section 5.4.4. 42 ESC Guidelines Downloaded from by guest on 30 August 2023 8.3.3 Indications and timing of surgery in the presence of uncontrolled infection Surgery should be considered for uncontrolled infection when antibiot-ic therapy is ineffective and extracardiac sources are ruled out. Reports in the literature demonstrate that surgery for uncontrolled infection in IE has the potential to improve 1-year survival by 15–20%.403,429,444 8.3.3.1. Persistent infection Uncontrolled infection is present in the form of persistent infection when blood cultures remain positive for >1 week or persistent sepsis despite appropriate antimicrobial therapy and when other causes of bacteraemia have been excluded. Not performing surgery for uncon-trolled infection is associated with significantly increased mortality.444 8.3.3.2. Locally uncontrolled infection Uncontrolled infection is also present if signs of local progression, i.e. increasing vegetation size or perivalvular involvement, are observed during follow-up imaging.5,420,421,445,446 Surgery should be performed urgently (within 3–5 days) in such cases. Rarely, when there are no other reasons for surgery and fever is easily controlled with antibiotics, small abscesses or pseudoaneurysms can be treated conservatively un-der close clinical and echocardiographic follow-up.429,444 8.3.3.3. Infection with resistant or virulent organisms Microorganisms causing endocarditis that are unlikely to be controlled with current antimicrobial therapy include fungi,447,448 multiresistant bacteria (e.g. MRSA or vancomycin-resistant enterococci) and, in rare cases, non-HACEK Gram-negative bacteria. S. aureus should also be in-cluded in this group due to its fast progression and ability to cause local tissue destruction and abscess formation,5,449 specifically, if a favourable early response to antibiotics is not achieved.305,312,449 The presence of these organisms should lead to discussions within the Endocarditis Team and urgent surgery.385,450 8.4. Prevention of systemic embolism 8.4.1. Incidence of embolic events in infective endocarditis Embolic events are frequent and potentially life-threatening complica-tions of IE related to the migration of cardiac vegetations.451,452 The brain and spleen are the most frequent sites of embolism for left-sided IE, while pulmonary embolism is frequent in right-sided and pacemaker lead IE (see Section 12). Stroke may be the first clinical manifestation of IE, and is a severe complication that is associated with increased mor-bidity and mortality.451,453,454 Embolic events may be clinically silent in up to 50% of patients with IE.198 Emboli affecting the splenic or cerebral circulation are frequently asymptomatic, and are diagnosed by non- invasive imaging.197,200 Although whole-body CT imaging (i.e. chest, abdomen, and pelvis) is frequently performed during the work-up for surgery, diagnosis and management of patients is infrequently altered as a result of these investigations.194 However, cerebral CT may affect clinical decision-making and outcomes when surgery is considered.452 Embolic risk in IE is high, with 20–50% of patients being af-fected.452,455 The highest incidence of embolic strokes can be observed in the days around the initial diagnosis of IE,456 and embolic events are often what leads to the initial diagnosis of IE. Embolic risk is highest the day after therapy initiation, and is 10–20 times higher on the day before and after the start of antibiotic treatment compared with 2 weeks be-fore and after.456 Thus, embolic events occurring after the initiation of antibiotic therapy continuously drop in incidence within the first 2 weeks of antibiotic treatment.429,455–457 The benefits of surgery to pre-vent embolism may therefore be greatest during the early stages of therapy, when embolic risk is at its highest. 8.4.2. Predicting the risk of embolism Predicting the risks of embolization is important for decision-making in IE. Echocardiography plays a key role in identifying potentially embolic structures in the heart,429,455,456,458 although predicting the time point of embolization remains difficult. Several factors are associated with in-creased risk of embolism including the size and mobility of vegeta-tions,455,456,458–460 the location of the vegetation on the mitral valve,455 the increasing or decreasing size of the vegetation under anti-biotic therapy,455 particular microorganisms (especially S. aureus,455 S. gallolyticus,461 and Candida spp.450), previous embolism,455 multivalv-ular involvement,458 and biological markers.462 Among these, the size and mobility of the vegetations are the most important independent predictors of new embolic events.459,460,463 A recent study, however, demonstrated that vegetation size was predictive of worse outcomes only when present with other indications for surgery (i.e. HF or uncon-trolled infection).464 Staphylococcal endocarditis is also a risk factor for embolization,465–468 which is particularly important because the inci-dence of S. aureus IE is increasing.78,469 Risk of neurological complica-tions is particularly high in patients with very large vegetations (>30 mm in length).451 Additional factors may need to be taken into account and it may be helpful to use an embolic risk calculator.470 S. aureus infection, previous embolism, vegetation length, age, diabetes, and the presence of atrial fibrillation have been identified as specific risk factors for embolism.470 8.4.3. Indications and timing of surgery to prevent embolism in infective endocarditis Surgical removal of potentially embolic material from the heart may prevent new or additional embolic events. Given the imminent risk and high rates of embolization in patients with mobile and large vegeta-tions,5,451,455–457,460,471 surgery should be considered urgently (within 3–5 days) in such patients. A prospective randomized trial in young, low-risk patients assessed the effects of early surgery in patients with large vegetations and streptococcal IE.471 Although there was no differ-ence in all-cause mortality at 6 months between the early surgery and conventional treatment groups, the risk of embolization was significant-ly reduced with early surgery. Non-randomized observational analyses including patients at higher risk also suggest that early surgery may be beneficial in patients with a high likelihood of embolization,428,459,472,473 and that initial conservative treatment is associated with increased mor-tality.474,475 However, prosthetic dehiscence has also been associated with early surgery in patients with S. aureus IE.429 Individualized decision-making is required to balance the risk of surgery, which is also influenced by pre-operative neurological events or other co-morbidities.5,453 The main indications and the timing of surgery to prevent embolism based on the currently available literature are given in Recommendation Table 12 and Figure 10. ESC Guidelines 43 Downloaded from by guest on 30 August 2023 9. Other complications of infective endocarditis 9.1. Neurological complications Neurological manifestations may occur before or after the diagnosis of IE is established and recurrent events can also take place later in the course of IE.451 The possibility of IE should be considered in patients who present with stroke, meningitis, or brain abscess. Unexplained fe-ver accompanying a stroke in a patient with valvular disease should trig-ger the suspicion of IE with blood cultures taken prior to empirical antibiotic therapy. Symptomatic cerebrovascular complications occur in up to 35% of pa-tients with IE,145,198,451,452 whereas silent cerebrovascular complications (including ischaemia and microhaemorrhage) occur in up to 80% of pa-tients.200,204,403 Clinical presentation is variable, but ischaemic stroke and transient ischaemic attack are the most common presentations.479 Other manifestations include haemorrhage (intracerebral, subarachnoid), meningitis, brain abscess, encephalopathy, and infectious aneurysms. Focal neurological symptoms are present in ∼40% of affected patients, and non- focal presentations occur in approximately one-third. S. aureus IE is more frequently associated with neurological compli-cations compared with IE caused by other microorganisms. Vegetation size and mobility also correlate with embolic risk. Neurological complications are associated with excess mortality, as well as long-term morbidity, particularly in the case of stroke.480 Prompt diagnosis of IE and early initiation of the antibiotic therapy are pivotal to preventing neurological complications. Early cardiac surgery in high-risk patients is key to preventing embolization of vegeta-tions.471,481 In contrast, antithrombotic/thrombolytic medical therapies are not beneficial.481–483 Mechanical thrombectomy may be considered within time limits in selected cases.484 If mechanical thrombectomy is performed, the re-trieved embolic material must be sent off for pathological and microbio-logical analyses. Neurosurgery or endovascular therapy is recommended for large infective aneurysms, especially when a continu-ous growth, despite optimal antibiotic therapy or ruptured intracranial infective aneurysms, is observed.485 The use of anticoagulation in patients with left-sided IE does not seem to have an effect on the risk of stroke, cerebrovascular haemor-rhage, or mortality at 10 weeks and, therefore, continuation of anticoagulation in patients with left-sided IE and with a pre-existing in-dication for the use of anticoagulants is recommended in the absence of other contraindications.486 Substitution from oral anticoagulation to heparin in such patients is generally preferred in case of cerebral bleed-ing or indication for early surgery. Following a neurological event, the indication for cardiac surgery must be balanced against the peri-operative risk and post-operative prognosis of the patient. Randomized studies are impractical and cohort studies suf-fer from bias that can only be partially compensated for by statistical meth-ods. The majority of publications demonstrate lower risk of secondary haemorrhagic conversion of uncomplicated ischaemic lesions than the risk of recurrent embolism under antibiotic treatment. Therefore, the available evidence supports early surgery in such patients (see Section 10.4). Recommendation Table 13 summarizes the recommended manage-ment of neurological complications in IE; considerations for cardiac sur-gery after neurological complications are discussed in Section 10.4. Recommendation Table 12 — Recommendations for the main indications of surgery in infective endocarditis (native valve endocarditis and prosthetic valve endocarditis)a Recommendations Classb Levelc (i) Heart failure Emergencyd surgery is recommended in aortic or mitral NVE or PVE with severe acute regurgitation, obstruction, or fistula causing refractory pulmonary oedema or cardiogenic shock.420,423,424,429,476,477 I B Urgentd surgery is recommended in aortic or mitral NVE or PVE with severe acute regurgitation or obstruction causing symptoms of HF or echocardiographic signs of poor haemodynamic tolerance.5,420–422,429 I B (ii) Uncontrolled infection Urgentd surgery is recommended in locally uncontrolled infection (abscess, false aneurysm, fistula, enlarging vegetation, prosthetic dehiscence, new AVB).5,420,421,429,445 I B Urgentd or non-urgent surgery is recommended in IE caused by fungi or multiresistant organisms according to the haemodynamic condition of the patient.420 I C Urgentd surgery should be considered in IE with persistently positive blood cultures >1 week or persistent sepsis despite appropriate antibiotic therapy and adequate control of metastatic foci.436,437 IIa B Urgentd surgery should be considered in PVE caused by S. aureus or non-HACEK Gram-negative bacteria5,385,449 IIa C (iii) Prevention of embolism Urgentd surgery is recommended in aortic or mitral NVE or PVE with persistent vegetations ≥10 mm after one or more embolic episodes despite appropriate antibiotic therapy.451,455,457,471,478 I B Urgentd surgery is recommended in IE with vegetation ≥10 mm and other indications for surgery.5,460,465,466,471,478 I C Urgentd surgery may be considered in aortic or mitral IE with vegetation ≥10 mm and without severe valve dysfunction or without clinical evidence of embolism and low surgical risk.460,463,465,473,478 IIb B © ESC 2023 AVB, atrioventricular block; HACEK, Haemophilus, Aggregatibacter, Cardiobacterium, Eikenella, Kingella; HF, heart failure; IE, infective endocarditis; NVE, native valve endocarditis; PVE, prosthetic valve endocarditis. aFor right-sided endocarditis, please refer to Section 12. bClass of recommendation. cLevel of evidence. dEmergency, within 24 h. Urgent, within 3–5 days. Non-urgent, within same hospital admission. 44 ESC Guidelines Downloaded from by guest on 30 August 2023 9.1.1. The role of cerebral imaging in infective endocarditis Cerebral imaging is mandatory when neurological complications of IE are suspected. Evaluation should include MRI with and without gadolin-ium, or CT with and without contrast if MRI is not possible.487 Vascular imaging should not be performed routinely, and CTA or magnetic res-onance angiography (MRA) is probably sufficient for screening when in-fective aneurysm is suspected. Catheter angiography should be performed in patients in whom an infective aneurysm was diagnosed on CTA or MRA, in patients with an acute brain haemorrhage, or if the suspicion of aneurysm remains despite negative non-invasive tech-niques, and if mechanical thrombectomy is considered.488 In patients without neurological symptoms, cerebral MRI often de-tects ‘silent’ lesions such as microbleeds.204 The lack of association with parenchymal haemorrhage and the absence of post-operative neurological complications in patients with microbleeds suggest that microbleeds should not postpone surgery when indicated.489 9.2. Infective aneurysms An infective (mycotic) aneurysm is a rare but potentially devastating com-plication of IE. Infective cerebral aneurysms may be asymptomatic, cause headaches, seizures, or focal symptoms, and may progress to a potentially lethal rupture. They are associated with subarachnoid, intracerebral, and intracranial haemorrhage,201 particularly when the patient is anticoagulated. The true incidence of infective cerebral aneurysms may be underdiagnosed as vascular imaging modalities are not systematically performed in asymptomatic patients. In 168 patients who underwent cerebral angiography with a diagnosis of IE or infected left ventricular assist device were retrospectively reviewed and infective aneurysms were pre-sent in 9% of patients.488 Another series using CTA identified infective an-eurysms in up to 32% of patients with left-sided IE.492 Digital subtraction angiography (DSA) remains the gold standard diagnostic test for the detection of infective aneurysms.487 The sensitiv-ity of CTA and MRA progressively increases with the size of the aneur-ysm. In a large study including 142 patients, the sensitivity for detection of infective aneurysms smaller than 5 mm was 57% for CTA and 35% for MRA, compared with respective 94% and 86% sensitivities for the detection of aneurysms of 5 mm or larger.493 Compared with the sen-sitivity of DSA, the sensitivities of CTA and MRA for detecting infective aneurysms are inferior.488,490 Therefore, in patients with IE and high suspicion of infective aneurysms in whom CTA or MRA are negative, DSA may be considered.490,494 Treatment options of infective cerebral aneurysms consist of anti-biotic treatment with or without endovascular or surgical therapy, al-though evidence is limited to case reports and retrospective studies.495–498 Therefore, management should be discussed among the members of the Endocarditis Team and tailored to individual clinical situations. Shi et al.496 reported that in patients with unruptured infect-ive cerebral aneurysms, antibiotic treatment may have similar outcomes to invasive treatment. However, interventional treatment should be considered in cases of ruptured infective aneurysms or unruptured in-fective aneurysms that do not respond to antibiotic therapy.485,495 Endovascular therapy is highly successful and associated with low morbidity compared with microsurgical and medical manage-ment.487,499 A systematic review including 499 patients with infective cerebral aneurysms reported a 36% rate of aneurysm rupture.495 Endovascular surgical and conservative therapies were performed in an approximately equal number of patients. Among patients undergo-ing valve surgery in this series, only 15% underwent cardiac surgery be-fore aneurysm treatment whereas 85% underwent cardiac surgery after aneurysm treatment.495 Urgency of cardiac surgery plays a pivotal role in decision-making re-garding the type of invasive treatment. Compared with neurosurgical clipping that requires a craniotomy and often at least 2-week delay prior to procedure, cardiovascular surgery can be performed on the same day as endovascular treatment.485,487,496,499 Finally, endovascular treatment of infective cerebral aneurysms prior to heart valve surgery may be considered, even if no rupture is documented.499 9.3. Splenic complications Splenic complications associated with IE range from asymptomatic in-farction500 and abscess formation501 through to splenic rupture and cardiovascular collapse.502 Splenic infarcts are common (∼20% of pa-tients in the EURO-ENDO registry) and very often asymptomatic.5 Up to 5% of splenic infarcts can progress to abscess formation.503 Persistent or recurrent fever, abdominal pain, and persistent bacter-aemia are suggestive for the presence of such complications. Patients with suspected splenic complications should be evaluated with ultra-sound, abdominal CT, MRI, or PET/CT.504 Treatment of splenic complications includes conservative medical therapy with appropriate antibiotics for splenic infarction or for antibiotic-responsive abscesses, although antibiotic penetration may be poor in these circumstances. When an abscess is large, splenectomy may be considered, but the timing of splenectomy in relation to heart valve surgery needs careful assessment.505 Splenectomy and heart valve surgery are seldom performed in the same operative episode.506 Splenectomy is usually performed prior to valve surgery due to con-cerns of dissemination and reinfection of the heart valve. Recommendation Table 13 — Recommendations for the treatment of neurological complications of infective endocarditis Recommendations Classa Levelb Brain CT or MRA is recommended in patients with IE and suspected infective cerebral aneurysms.490 I B Neurosurgery or endovascular therapy is recommended for large aneurysms, those with continuous growth despite optimal antibiotic therapy, and ruptured intracranial infective cerebral aneurysms.485 I C If non-invasive techniques are negative and the suspicion of infective aneurysm remains, invasive angiography should be considered.488 IIa B In embolic stroke, mechanical thrombectomy may be considered if the expertise is available in a timely manner.484 IIb C Thrombolytic therapy is not recommended in embolic stroke due to IE.481,491 III C © ESC 2023 CT, computed tomography; IE, infective endocarditis; MRA, magnetic resonance angiography. aClass of recommendation. bLevel of evidence. ESC Guidelines 45 Downloaded from by guest on 30 August 2023 Nevertheless, one case series reported that it is safe to address the splenic abscess with splenectomy after valve repair.502 Alternatives to open splenectomy, i.e. percutaneous drainage507 and/or laparoscopic surgery,508 may be considered in patients with high surgical risk. After splenectomy, vaccination against encapsulated microorganisms (S. pneumoniae, N. meningitis, and Haemophilus spp.) is recommended. 9.4. Myocarditis and pericarditis The actual prevalence of acute myocarditis in the setting of IE is un-known. Myocarditis will usually present in the form of acute HF and/ or ventricular arrhythmias indicating myocardial involvement in the in-flammatory process most likely mediated by an immune mechanism. Differential diagnosis and exclusion of other potential complications are best assessed using echocardiography and cardiac MRI.509–511 Pericarditis is an infrequent complication of IE. In one retrospective series of 95 patients with aortic valve IE, 19% developed pericarditis usually related to ring abscess formation. The same authors also de-scribed a 12% rate of pericarditis associated with mitral valve IE.512,513 The pathophysiological mechanisms most commonly involved in IE-related pericarditis are the extension of inflammation from an in-fective aneurysm of the aortic root or valve ring abscess, an embolus in an extramural coronary artery, or the rupture of an infective aneurysm. In a recent large series of NVE, pericardial effusion was observed in 7.8% of patients and was associated with a higher risk of HF during ad-mission. After adjusting for possible confounders, patients did not have a higher rate of surgery, and the presence of pericardial effusion was not associated with a higher in-hospital or 1-year mortality.513 9.5. Heart rhythm and conduction disturbances Due to the critical anatomical relationship between heart valves and the conduction system, AVB may complicate the clinical presentation of IE. The atrioventricular node (AVN) and His bundle lie in close proximity to the insertion of the septal leaflet of the tricuspid valve, the aortic root (below the non-coronary and right coronary cusps), and the mitral annulus.514 A paravalvular abscess of these valves, especially of the aor-tic valve, may lead to AVB, and new electrocardiographic AVN conduc-tion abnormalities are indicative of a paravalvular extension of the infection. In the EURO-ENDO registry, conduction abnormalities were observed at diagnosis in 11.5% of patients, including first-degree AVB in 8.1%, second-degree AVB in 0.6%, and third-degree AVB in 2.8% of cases.5 New-onset AVB caused by local extension of IE (i.e. ab-scess) is an indication for urgent cardiac surgery. Atrioventricular block may not only occur as a complication of para-valvular extension of the infection, but it may also develop as a conse-quence of valve surgery. In a series of 444 patients who survived cardiac surgery for IE,515 12.8% of patients required pacemaker implantation for AVB. Multivariable analysis identified that prolonged pre-operative PR and QRS intervals, S. aureus infection, presence of aortic root ab-scess, tricuspid valve involvement, and prior valvular surgery were inde-pendently associated with the need for post-operative pacemaker implantation. Pacemaker implantation should be considered in patients with sur-gery for valvular endocarditis and complete AVB if one or more of these risk factors is present.515 9.6. Musculoskeletal manifestations 9.6.1. Osteoarticular infective endocarditis-related infections Metastatic bone or joint IE-related lesions are relatively frequent due to the spread of the pathogen through the bloodstream and its subse-quent tissue implantation. Although these lesions are considered an IE-related distal lesion or complication because infected valves are a continuous source of bacteraemia, it is often impossible to determine whether the primary infection is the valve or the osteoarticular infec-tion. Overall, the incidence of osteoarticular infection among patients with IE is 6–8%, including bones, joints, and vertebral discs.5,145,247,516 The prevalence of spondylodiscitis ranges from 2% to 10% in patients with IE, including symptomatic and asymptomatic cases,248,517 while series of spontaneous spondylodiscitis have reported co-existing IE in up to 20–30% of patients.518–520 In general, the rate of IE is 10 times higher in patients with known spondylodiscitis. Therefore, in patients with a definite diagnosis of pyogenic spondylodiscitis and positive blood cultures, TTE/TOE is recommended to rule out IE.521 The most frequent microorganisms associated with spondylodiscitis are S. aureus, followed by Streptococcus spp., CoNS, and Enterococcus spp.247,248,305,517–523 The most common symptom of spondylodiscitis is back pain, al-though only 4% of patients with IE and back pain have spondylodisci-tis.32,522 An MRI should be performed to accurately diagnose spondylodiscitis. Computed tomography can detect indirect signs of spondylodiscitis: loss of disc height, erosion/destruction of the end-plates and vertebral bodies, and paravertebral soft tissue phlegmonous changes or abscess.206 Whole-body [18F]FDG-PET/CT can also iden-tify spondylodiscitis.30,32,524 Indeed, spondylodiscitis is frequently de-tected as an incidental finding when PET/CT is performed for the diagnosis of PVE. Imaging techniques can also be helpful in guiding biop-sies to obtain material for cultures in cases of suspected IE with negative blood cultures.206 Antibiotic treatment adapted to the antimicrobial susceptibility pat-tern is appropriate for most cases of spondylodiscitis. The outcome is usually favourable with the 4- to 6-week IE treatment course. Prolonged therapy is necessary in patients with IE caused by difficult-to-treat microorganisms, such as S. aureus or Candida spp., or in those with epidural or perivertebral abscesses.523,525 In patients with neurological deficits or severe spinal instability, the indication for Recommendation Table 14 — Recommendations for pacemaker implantation in patients with complete atrioventricular block and infective endocarditis Recommendations Classa Levelb Immediate epicardial pacemaker implantation should be considered in patients undergoing surgery for valvular IE and complete AVB if one of the following predictors of persistent AVB is present: pre-operative conduction abnormality, S. aureus infection, aortic root abscess, tricuspid valve involvement, or previous valvular surgery.515 IIa C © ESC 2023 AVB, atrioventricular block; IE, infective endocarditis. aClass of recommendation. bLevel of evidence. 46 ESC Guidelines Downloaded from by guest on 30 August 2023 surgical spinal treatment should be considered.526 In patients with ur-gent indication for cardiac surgery, the presence of these lesions does not contraindicate the cardiac intervention. Spondylodiscitis does not appear to worsen the prognosis of patients with IE but delay-ing the diagnosis of IE in patients with spondylodiscitis is associated with poor prognosis.248,517–520 9.6.2. Rheumatological manifestations The pathogenesis of rheumatological manifestations and musculoskeletal symptoms in IE is not well established. The probable immunological- inflammatory aetiology of this clinical presentation is supported by a var-iety of antibodies and laboratory markers, the sterility of the synovial fluid, and the rapid resolution without sequelae.527 Myalgia and back pain are re-ported in 12–15% of cases. Arthralgia occurs in ∼10% of patients, some-times sequentially affecting several joints. Slightly less often are symptoms of peripheral arthritis preferentially involving the major and proximal joints at the lower extremities.5,145,182,516 Sacroiliitis is less frequently observed (1% of cases) as well as poly-myalgia rheumatic-like syndrome with pain and morning stiffness of the shoulders and hips, proximal muscle weakness (0.9% of cases), and cutaneous leucocytoclastic vasculitis (purpuric skin lesions, 3.6% of cases).182,527,528 Rheumatological manifestations and musculoskel-etal symptoms show rapid and complete resolution with antibiotics and their presence does not impact on the prognosis of IE.182,529 9.7. Acute renal failure Acute renal failure is a common complication of IE and is associated with increased morbidity and mortality as well as significant increase in length and cost of hospitalization.5,531–534 Additionally, renal failure is an independent predictor of poor outcome after cardiac surgery.417 However, acute renal failure should not be a reason to delay cardiac surgery. The EURO-ENDO registry reported that in patients with IE, acute renal failure was the second most common complication with an incidence of almost 18%.5 Some single-centre studies that specifically reported on the incidence of acute renal failure (using standardized cri-teria) in patients with IE reported that any degree of acute renal failure from mild to severe might be observed in 40–69% of cases.532,535,536 Severe renal failure requiring haemodialysis has been reported in 6% of patients with IE and it is associated with a very high risk of mortality (40%).537 Several factors may be responsible for the onset or worsening of re-nal dysfunction: (i) immune complex and vasculitic glomerulonephritis; (ii) renal infarction due to septic emboli;538,539 (iii) haemodynamic im-pairment in patients with HF; (iv) antibiotic and other drug toxicity (notably related to aminoglycosides, vancomycin, nafcillin, amoxicillin, oxacillin, concomitant use of non-steroidal anti-inflammatories, and/ or high dose loop diuretics); and (v) nephrotoxicity of contrast agents used for diagnostic imaging techniques.417,531,534,535,537,540 To reduce the incidence of acute renal failure, nephrotoxic antibio-tics should be avoided if possible or, if not possible, serum levels (ami-noglycosides and vancomycin) as well as creatinine should be closely monitored, and the optimal dose of medication should be periodically re-evaluated and discussed with the Endocarditis Team and a pharma-cologist.536 Loop diuretics should also be used cautiously and other po-tentially nephrotoxic drugs, such as non-steroidal anti-inflammatories, should be avoided.536 Similarly, the use of nephrotoxic contrast agents for diagnostic imaging techniques should be carefully evaluated and avoided when possible. In patients with IE and a reduced glomerular filtration rate, contrast enhanced abdominal ultrasound or MRI are reasonable tests to diag-nose embolization as cause of renal function impairment.541 10. Surgical therapy: principles and methods Surgery has been demonstrated to be an independent predictor of sur-vival in many retrospective studies of IE patients under various clinical conditions and offers a potential curative therapy to select patient groups.5,250,403,404,421,436 Optimal management of such patients may lead to lower peri-operative complication rates and further potential benefits of surgical therapy. 10.1. Pre-operative and peri-operative management 10.1.1. Coronary angiography When cardiac surgery becomes necessary in IE, assessment of coronary anatomy is recommended (see Recommendation Table 16). Classically, pre-operative coronary angiography is recommended for men >40 years, post-menopausal women, and in those with one or more cardio-vascular risk factors or history of CAD.128 The presence of aortic valve vegetations may preclude invasive coronary angiography due to the risk of iatrogenic embolization.542,543 However, some studies have demon-strated the safety of performing invasive coronary angiography in the presence of aortic valve vegetations, particularly in patients without very large and mobile vegetations.193,544 Alternatively, coronary CTA can be used to rule out significant coronary obstructions. Furthermore, surgery may need to be conducted without detailed in-formation on coronary anatomy in certain clinical conditions, particu-larly emergencies. Of note, a recent study questioned the need for coronary artery bypass grafting of non-critical lesions at the time of sur-gery for IE and suggested that such concomitant intervention may have a negative impact on peri-operative outcomes.545 Recommendation Table 15 — Recommendations for patients with musculoskeletal manifestations of infect-ive endocarditis Recommendations Classa Levelb MRI or PET/CT is recommended in patients with suspected spondylodiscitis and vertebral osteomyelitis complicating IE.30,32,206,524 I C TTE/TOE is recommended to rule out IE in patients with spondylodiscitis and/or septic arthritis with positive blood cultures for typical IE microorganisms.247,248,517–521,523 I C More than 6-week antibiotic therapy should be considered in patients with osteoarticular IE-related lesions caused by difficult-to-treat microorganisms, such as S. aureus or Candida spp., and/or complicated with severe vertebral destruction or abscesses.523,525,530 IIa C © ESC 2023 IE, infective endocarditis; MRI, magnetic resonance imaging; PET/CT, positron emission tomography/computed tomography, TOE, transoesophageal echocardiography; TTE, transthoracic echocardiography. aClass of recommendation. bLevel of evidence. ESC Guidelines 47 Downloaded from by guest on 30 August 2023 10.1.2. Extracardiac infection Extracardiac foci may be treated prior to valve surgery, during the valve operation, or post-operatively, dependent on the urgency of cardiac surgery. Regardless of the timing of intervention, infective foci need to be eradicated before completion of antibiotic therapy in order to avoid cardiac valve reinfection. 10.1.3. Intra-operative echocardiography Intra-operative TOE provides contemporaneous assessment of the ex-tent of infection prior to valve repair/replacement. Extent of infection, stability of known vegetations, re-assessment of previously uninvolved heart valves, and biventricular function are routinely performed with intra-operative TOE. Intra-operative TOE post-surgical repair is man-datory to determine the immediate result and establish a baseline for follow-up comparisons.547 10.2. Other intra-operative considerations Specific peri-operative management considerations are necessary in all IE patients undergoing valve surgery, particularly in those following stroke (see Section 10.3). Pre-operative antibiotic therapy must be con-tinued intra-operatively, and doses may need to be repeated in case of prolonged operations or major bleeding. Although the pharmacokinet-ics of antibiotic therapy is altered during cardio-pulmonary bypass (CPB), adjustment of doses is rarely required.548 In general, ongoing IE antibiotic treatment offers appropriate surgical site infection prophy-laxis. However, when the antibiotic treatment for IE does not fully cov-er normal surgical prophylactic treatment, conventional prophylaxis should be added. Intra-operative bleeding management is often compli-cated by marked coagulopathy in patients with IE, particularly those undergoing surgery during persistent sepsis. The management of hypo-tension and vasoplaegia is particularly challenging in patients presenting with septic shock, and accompanying vasoplaegia tends to worsen sig-nificantly during CPB. Norepinephrine is frequently used as first-line therapy for septic shock, followed by vasopressin or terlipressin in cases of resistant vasoplaegia.549 Methylene blue may be used as a rescue agent in patients who are unresponsive to these measures, but mortal-ity rates are high for such patients.550 Retrospective studies have suggested that the use of haemoadsor-bent filters during CPB may decrease the negative effects associated with cytokine cascade activation.551 A recent RCT of haemoadsorption during cardiac surgery in IE patients, however, failed to demonstrate any beneficial effects with regards to adverse events or end-organ function.552 10.3. Surgical approach and techniques Surgery for IE aims to remove infected structures followed by re- establishment of anatomy and haemodynamic function. With regards to the involved heart valve(s), repair or replacement is carried out based on the extent of destruction, acuity of disease, and patient char-acteristics.553 Appropriate collection and labelling of tissue samples for pathological, microbiological, and molecular biological analyses are ne-cessary to help guide antibiotic treatment. Aortic valve replacement is usually required for aortic IE. Aortic valve repair is very uncommon in the acute situation but may be performed for isolated aortic regurgitation after healed endocarditis. In mitral IE, leaf-let perforations with preserved free margin and chordae tendinae may be treated with patch repair, particularly in the setting of subacute or healed IE. Although mitral valve repair is feasible in more complex mitral IE involv-ing the annuli, the leaflet free edge, and/or chordae, evidence showing the feasibility and durability of such repair techniques is scarce.554,555 A large registry on mitral repair vs. replacement in IE was limited by the lack of in-formation on severity of IE, different patient group profiles, and significantly higher incidence of staphylococcal endocarditis in the mitral valve replace-ment group.556 Therefore, it cannot be concluded that mitral valve repair is superior to replacement due to the high probability of selection bias. Valve preservation in acute IE should only be attempted if a durable repair is an-ticipated and complete eradication of infected tissue can be achieved. However, valve repair may be necessary in children, where valve replace-ment options are more limited. Invasion of the aortic annuli may create shallow defects (very limited abscess or small pseudoaneurysms) that are still amenable to conven-tional valve replacement surgery. When disease progresses into an ex-tensive aortic root abscess or periannular destruction, aortic root replacement is usually required. In experienced centres, the use of allo-grafts has been preferred as they have the advantage of adapting to ir-regular surfaces and provide haemostatic advantages with very good haemodynamic function and low thrombo-embolic risk, and can be used to repair concomitant lesions of the anterior mitral valve leaf-let.557,558 Additionally, allografts and stentless bioprostheses can be beneficial in small aortic roots and are associated with low reinfection rates. However, experience is generally limited to single-centre case series and there is no clear evidence of superiority of one valve substi-tute over the other.559 In very selected patients and children in particu-lar, the Ross operation (pulmonary valve autotransplantation) may be considered for aortic root IE.128 The use of patches to cover abscess cavities and prevent extensive resection and reconstruction is discouraged in aortic root IE as it may be associated with recurrences, periprosthetic leaks, and pseudoa-neurysm formation. After exclusion from the circulation, abscess and pseudoaneurysm cavities are left to drain into the pericardial cavity. When periannular infection of the aortic root extends into the inter-valvular fibrosa body, complex surgical reconstructions are required and are frequently the only option to achieve patient survival. The reported Recommendation Table 16 — Recommendations for pre-operative coronary anatomy assessment in patients requiring surgery for infective endocarditis Recommendations Classa Levelb In haemodynamically stable patients with aortic valve vegetations who require cardiac surgery and are high risk for CAD, a high-resolution multislice coronary CTA is recommended.185,546 I B Invasive coronary angiography is recommended in patients requiring heart surgery who are high risk for CAD, in the absence of aortic valve vegetations. I C In emergency situations, valvular surgery without pre-operative coronary anatomy assessment regardless of CAD risk should be considered.543,545 IIa C Invasive coronary angiography may be considered despite the presence of aortic valve vegetations in selected patients with known CAD or at high risk of significant obstructive CAD.193,543,544 IIb C © ESC 2023 CAD, coronary artery disease, CTA, computed tomography angiography. aClass of recommendation. bLevel of evidence. 48 ESC Guidelines Downloaded from by guest on 30 August 2023 pooled peri-operative survival rate of such surgical technique is 84%.560 Even more extensive repairs may be required for cases involving the in-tervalvular fibrosa, central fibrous body, and mitral valve, with or without fistulation to the right-side chambers. These operations are technically complex and require a surgeon who is very experienced in IE, which may not be available in every cardiovascular surgery department. Exceptionally, heart transplantation has been utilized for carefully se-lected patients without other surgical options.561 10.3.1. Choice of valve prosthesis Many patient characteristics are taken into account when deciding the type of valve prosthesis to implant in a given patient with IE. The studies published to date evaluating various valve prostheses in the setting of IE, however, suffer from numerous biases.90,559,562–566 Beyond the patient characteristics that apply in the non-IE setting,128 valve selection in IE is influenced by the presence of recent stroke, risk of new-onset bleeding, complexity of expected post-operative course, Haemorrhagic Ischaemic Meaningful recovery possible with acceptable residual quality of life Severely decreased level of consciousnessa Neurology and/or neurosurgery assessment and discussion with the Endocarditis Team (Class I) N Cardiogenic shock Type of stroke N N Infective endocarditis with stroke Y Y Uncontrolled infection, heart failure or high-risk of recurrent embolization N Frequent follow-up and assessment, delay surgery 4 weeks, if possible Y Y Favourable brain bleed featuresb Conservative or palliative care N Continued non-surgical management Urgent surgery (Class I) Y Emergency surgery (Class I) Figure 11 Surgery for infective endocarditis following stroke. NIHSS, National Institutes of Health Stroke Scale Score. Surgery timing: emergency, within 24 h. Urgent, within 48–72 h. Non-urgent, within same hospital admission. aGlasgow Coma Scale ≤4 or NIHSS >18. bIntracranial haemorrhage volume <30 mL or NIHSS <12. ESC Guidelines 49 Downloaded from by guest on 30 August 2023 and the ability of the patient to participate in decision-making, especially for emergency surgery (Table 12). In the absence of specific contraindications for a particular valve substitute, patient preferences should determine the final decision. 10.4. Timing of surgery after ischaemic and haemorrhagic stroke There is a general trend of offering early surgery in IE in light of the im-proved operative outcomes and survival benefits observed with operative management.451,567 For patients who have suffered a neurological injury, however, the optimal timing of surgery remains to be defined.568 There are no RCTs specifically assessing this clinically relevant issue and contem-porary evidence arises from observational studies.415,454,473,569,570 Neurological exacerbation may occur during surgery or early post- operatively due to the altered physiology conditions during and imme-diately after cardiac repair.571 Several peri-operative variables should be addressed in order to lower the risk of neurological deterioration and haemorrhagic transformation post-stroke (see Supplementary data online, Table S10). The risk of neurological exacerbation during surgery needs to be ba-lanced against that of delaying a cardiac operation. When haemo-dynamic disturbances are present, surgery should be pursued without delay (see Figure 11 and Recommendation Table 17).451,468,473,567,568,570–578 A more common situation occurs when surgery is considered for the prevention of recurrent embolism after stroke, due to the presence of large vegetations (>10 mm). In pa-tients that have suffered a transient ischaemic attack, the risk of surgery is usually low and surgery should be performed without delay. For patients with ischaemic stroke, multiple observational data exist sup-porting a non-delayed (urgent) intervention, unless the neurological status is poor (i.e. coma or extensive damage leading to poor functional prognosis).573,578 Involvement of an expert neurology/neurosurgical specialist will help in risk assessment discussions. The risk of post-operative haemorrhagic conversion after pre- operative stroke is reported in the range of 2–7%.453,579 Remarkably, bleeding transformation after cardiac surgery can also occur in patients with silent pre-operative cerebral embolisms, with similar frequency as in patients with overt neurological deficits. Unfortunately, these events cannot currently be accurately predicted prior to surgery. When haem-orrhagic transformation occurs, it is associated with high mortality (40%) and may require rescue neurointerventional or neurosurgical treatment to control bleeding or allow cerebral decompression by means of craniectomy.577,580 Several retrospective studies report benefits of early surgery (with-in 2 weeks) after haemorrhagic stroke without further compromising neurological outcomes.574,581,582 Decisions should be taken on a case-by-case basis by the Endocarditis Team, including a neurologist, and should be adapted to the mechanism of intracranial haemorrhage and its severity including intracranial haemorrhage volume measure-ment and National Institutes of Health Stroke Scale Score (NIHSS) score (see Figure 11).495 In patients in whom surgery is delayed, re-peat CT or MRI imaging should be performed 1–2 weeks following intracranial haemorrhage (or earlier in the case of clinical deterior-ation) in order to assess stability of the cerebral finding and potential-ly re-assess timing of surgery. The timing of surgery after intracranial haemorrhage is controversial and an area where further evidence is urgently required. 10.5. Post-operative complications Post-operative management of patients with IE may be challenging due to pre-operative multiorgan involvement and often complex surgical procedures. The risk of in-hospital mortality associated with IE surgery remains high (10–20%), particularly in patients >75 years of age, usually due to co-morbidities and complications of IE. Further research should focus on methods to lower surgical mortality. The most frequent serious post-operative complications are coagu-lopathy requiring extensive use of blood products and clotting factors, re-exploration of the thorax due to bleeding/tamponade, haemodialy-sis, stroke, or cerebral haemorrhagic transformation of prior cerebro-vascular lesions, low cardiac output syndrome, respiratory complications and tracheostomy, prolonged hospital stay, and need for a permanent pacemaker.515,585,586 When mortality occurs, the cause of death is often multifactorial. Post-mortem examination is help-ful for determining the cause of death, further understanding of disease process, teaching purposes at academic environments, and quality control. Recommendation Table 17 — Indications and timing of cardiac surgery after neurological complications in active infective endocarditis Recommendations Classa Levelb After a transient ischaemic attack, cardiac surgery, if indicated, is recommended without delay.454,468 I B After a stroke, surgery is recommended without any delay in the presence of HF, uncontrolled infection, abscess, or persistent high embolic risk, as long as coma is absent and the presence of cerebral haemorrhage has been excluded by cranial CT or MRI.451,468,473,567,568,570–578 I B Following intracranial haemorrhage, delaying cardiac surgery >1 month, if possible, with frequent re-assessment of the patient’s clinical condition and imaging should be considered.571 IIa C In patients with intracranial haemorrhage and unstable clinical status due to HF, uncontrolled infection or persistent high embolic risk, urgent or emergency surgery should be considered weighing the likelihood of a meaningful neurological outcome.199,581–584 IIa C © ESC 2023 CT, computed tomography; HF, heart failure; MRI, magnetic resonance imaging. aClass of recommendation. bLevel of evidence. Table 12 Features favouring a non-mechanical valve substitute in the setting of surgery for acute infective endocarditis Early surgery after a recent ischaemic stroke Evidence of intracranial bleeding Woman of childbearing age High likelihood of prolonged mechanical circulatory support Advanced age or frailty Poor or unknown medical compliance Expected complicated and prolonged post-operative course Patient preference © ESC 2023 50 ESC Guidelines Downloaded from by guest on 30 August 2023 10.6. Management of antithrombotic therapy after surgery The management of antithrombotic therapy early after surgery for IE may need to be altered when compared with non-IE clinical scenarios (see also Section 12.10).128 This is mainly due to known increased risk of intracranial haemorrhage after cerebral embolism. Restrictive or tai-lored use of antiplatelet and antithrombotic agents after surgery are key to avoid further complications,203,587 which is more feasible in pa-tients who received bioprosthetic valve prostheses or valve repair op-erations than after mechanical valve replacement surgery. 11. Outcome after discharge: follow-up and long-term prognosis Following in-hospital treatment, patients should be followed-up for the occurrence of main post-discharge complications, including recurrence of infection, HF, need for valve surgery or additional intervention, stroke, need for renal replacement therapy, psychological complica-tions, and death.86,588,589 11.1. Recurrences: relapses and reinfections The risk of recurrence (which includes relapses and reinfections) among survivors of IE varies significantly between studies, ranging from 2% to 9% in more contemporary analyses.86,589–595 However, it has been shown that reinfections have worse outcomes as compared with relapses.592 Figure 12 illustrates the diagnostic paths to differentiate relapse from reinfection.596 Conceptually, relapse refers to a repeat episode of IE caused by the same microorganism and represents a failure of treatment due to insuf-ficient duration of initial treatment, sub-optimal choice of initial antibio-tics, or a persistent focus of infection.592 Conversely, reinfection is related to patients’ clinical and immunological profiles, describes an in-fection caused by a different microorganism usually more than 6 months after the initial episode,4,596 and is associated with worse out-come.592 The differentiation between relapse and reinfection needs to be interpreted with caution, however, as a long time period from initial infection suggests reinfection even in the presence of the same strain. Contemporary data report low rates of relapse,86 most probably re-flecting improved management of these patients. Relapse should be treated with i.v. antibiotics for an additional 4–6 weeks, depending on the causative microorganism and its antibiotic susceptibility, and cardiac surgery should be considered. It is also important to consider that anti-biotic resistance may develop over time. Factors associated with an in-creased rate of relapse are listed in Table 13.588,595,597 In surgically managed NVE, the risk of IE recurrence is no different when comparing valve replacement and valve repair.84,598 Several pre-vious studies have also reported no difference regarding the risk of re-current IE between types of valve implanted.599–601 However, the most recent Danish registry study reports increased risk of IE recurrence as-sociated with biological vs. mechanical prostheses.84 Partial oral vs. i.v. antibiotic treatment of IE, as well as OPAT vs. hospital-based antibiotic treatment in select stable patients, is not asso-ciated with an increased risk of recurrent IE.43,396,399,602 Notably, re-sidual vegetation after treatment for IE also did not show increased association with recurrence of IE,602 although this result should be in-terpreted with caution. Patients with relapse or reinfection IE should be managed as indicated in Sections 7 and 8 (if complicated IE). Same species isolated Reinfection Relapse Same strain Stored isolate from prior infective endocarditis Use molecular methods/ strain typing New episode <6 months from prior infective endocarditis N N N N Storage of isolates following infective endocarditis episode New episode of infective endocarditis Y Y Y Y Figure 12 Algorithm differentiating relapse from reinfection. Reproduced with permission from Chu et al.596 ESC Guidelines 51 Downloaded from by guest on 30 August 2023 11.2. First year follow-up Patients discharged after the first episode of IE should remain under close surveillance for potential long-term complications. A partnership between cardiologists, infectious disease specialists, cardiac surgeons, general practitioners, and dentists is encouraged to improve patient care and reinforce prophylaxis measures. In medically treated patients, residual valve dysfunction may worsen, or structural valve deterioration may progress, despite bacteriological cure. To monitor the risk of de-velopment of secondary HF, an initial clinical evaluation and baseline TTE should be performed at the completion of antimicrobial therapy and repeated if a change in the clinical condition occurs. Clinical re-assessment should be performed one or more times in the first year and yearly thereafter depending on the individual risk pro-file. The need for late valve surgery is relatively low, ranging from 3% to 11%.27,588,592 Blood testing for inflammatory markers (i.e. WBC, C-reactive protein, procalcitonin) should be performed early after fin-ishing antimicrobial treatment and repeated thereafter when clinically indicated.592 Due to the increased risk of relapse for virulent microor-ganisms, blood cultures are encouraged within the first week after fin-ishing treatment. The early period after discharge might be challenged by slow physical and mental recovery.603,604 Patients’ and families’ concerns should be addressed during follow-up. Supporting the family may indirectly sup-port the patient during recovery and reduce the psychological burden. Cardiac rehabilitation, including physical exercise training and patient education, may be beneficial, and has been shown to be safe and feasible in stable patients at a minimum of 2 weeks after surgery for left-sided IE.605 Physical training should start as early as possible and can be adapted post-sternotomy with isolated lower-limb training. Adherence is improved if the delay to training is minimized, and rebuild-ing muscle mass and reducing frailty should be a priority. Patients, and their caregivers, should be informed of their risk of IE recurrence and be educated on preventive measures and self- monitoring. In particular, patients should be educated that new onset of fever, chills, or other signs of infection mandate immediate evalu-ation, including procurement of blood cultures before empirical use of antibiotics, and that contact with the Heart Valve Centre is manda-tory in case of suspected recurrent IE. Good oral health maintenance, preventive dentistry, and advice about skin hygiene, including advice on tattoos and skin piercing, are mandatory. Deficiencies in dental surveil-lance contribute to the continuous gradual increase in the incidence of IE, which underlines the need for repeating the principles of IE preven-tion at each follow-up visit. In PWID patients, follow-up care should in-clude a strategy for addiction treatment, involve relevant addiction specialists before hospital discharge, and possibly including medication for opioid-use disorder.606,607 11.3. Long-term prognosis Contemporary long-term survival rates after the completion of IE treatment are estimated to be ∼85–90% and 70–80% at 1 and 5 years, respectively.589,592–594,610,611 Impact of referral bias, however, should be taken into consideration.612 The main predictors of long-term mortality are age, co-morbidities, PWID, double valve infection, recur-rences of IE, and HF, especially when cardiac surgery cannot be per-formed.588,589,592,593,613 Compared with an age- and sex-matched general population, patients that survived a first episode of IE have sig-nificantly worse survival when suffering relapses or reinfections.589,614 This excess mortality is especially high within the first few years after hospital discharge and can be explained by late complications such as HF, risk of recurrences, and higher patient vulnerability.589,611 In fact, most recurrences of IE and late cardiac surgeries occurred during this period of time.589,592,611 12. Management of specific situations 12.1. Prosthetic valve endocarditis Prosthetic valve endocarditis is the most severe form of IE and occurs in 1–6% of patients with valve prostheses,615 with an incidence of 0.3–1.2% per patient-year.5,420,616,617 PVE accounts for 20–30% of all cases of IE,618 and may be more common after biological than after mechanical valve replacement surgery.619,620 PVE was observed in 21% of cases of IE in a French survey,618 in 26% of cases in the Euro Heart Survey,419 and in 20% of cases in the ICE-PCS.621 Real-world Table 13 Factors associated with an increased rate of relapse of infective endocarditis Inadequate antibiotic treatment (i.e. agent, dose, duration) Resistant microorganisms (i.e. Brucella spp., Legionella spp., Chlamydia spp., Mycoplasma spp., Mycobacterium spp., Bartonella spp., C. Burnetii, fungi) Infective endocarditis caused by S. aureus and Enterococcus spp. Polymicrobial infection in people who inject drugs Periannular extension Prosthetic valve endocarditis Persistent metastatic foci of infection (abscesses) Resistance to conventional antibiotic regimens Positive valve culture Persistence of fever at the 7th post-operative day Chronic kidney disease, especially on dialysis High-risk behaviour, inability to adhere to medical treatment Poor oral hygiene © ESC 2023 Recommendation Table 18 — Recommendations for post-discharge follow-up Recommendations Classa Levelb Patient education on the risk of recurrence and preventive measures, with emphasis on dental health, and based on the individual risk profile, is recommended during follow-up.608 I C Addiction treatment for patients following PWID-related IE is recommended.606,607 I C Cardiac rehabilitation including physical exercise training should be considered in clinically stable patients based on an individual assessment.605,609 IIa C Psychosocial support may be considered to be integrated in follow-up care, including screening for anxiety and depression, and referral to relevant psychological treatment.605,609 IIb C © ESC 2023 IE, infective endocarditis; PWID, persons who inject drugs. aClass of recommendation. bLevel of evidence. 52 ESC Guidelines Downloaded from by guest on 30 August 2023 observational studies demonstrated stable rates of IE, but a remarkable increase in PVE between 1998 and 2013.80 Recently, a further increase in PVE cases (31%) was observed in the EURO-ENDO registry.5 PVE is still associated with difficulties in diagnosis, determination of the opti-mal therapeutic strategy, and poor prognosis. 12.1.1. Definition and pathophysiology A distinction is commonly made between early PVE and late PVE based on the time since valve surgery, because of significant differences in the microbiological profiles between these two groups.622 However, the time to IE onset is prognostically less important than the connection of IE to the peri-operative period or to specific pathogens. Prosthetic valve endocarditis with an onset in the peri-operative period involves mainly S. aureus, Staphylococcus epidermidis, or nosocomial microorgan-isms, such as Gram-negative pathogens or fungi. Late PVE more com-monly mimics the pattern of NVE, which is mostly represented by streptococcal and staphylococcal infections.623 S. aureus is more com-monly observed in patients with mechanical valves, while alpha- haemolytic streptococci, enterococci, and CoNS are more common in patients with bioprosthetic valves.624 PVE due to Mycobacterium chi-maera is an uncommon form of nosocomial infection that can result from contaminated CPB heater-cooler systems. Such infections pre-sent many months after the index operation and can therefore be chal-lenging to identify, and are associated with high mortality.625 The pathogenesis of PVE differs according to both the type of con-tamination and the type of prosthetic valve (see Supplementary data online, Section S6.1). 12.1.2. Diagnosis Diagnosis is more difficult in PVE than in NVE. Clinical presentation is frequently atypical, particularly in the early post-operative period, in which fever and inflammatory syndromes are common in the absence of macroscopic alterations of the prosthesis on cardiac imaging. However, persistent fever should trigger the suspicion of PVE. As in NVE, diagnosis of PVE is based mainly on the results of echocardiog-raphy and blood cultures. However, both are associated with a sensitiv-ity of only 60% for the definite diagnosis of endocarditis.212 Although TOE is mandatory in suspected PVE (Figure 6), its diagnos-tic value is lower than in NVE. Identification of a new periprosthetic leak is a major criterion of IE and urges additional imaging modality to con-firm the diagnosis (see Section 5).533,626 Recently, nuclear techniques, particularly [18F]FDG-PET/CT, have been shown to improve the diag-nostic accuracy of the Duke criteria and increase sensitivity.34,209 Combinations of different imaging techniques such as cardiac CT, nu-clear imaging, and TOE, improve diagnostic accuracy and provide rele-vant information in terms of prognosis.33,627 In select cases of suspected PVE, and non-diagnostic results for the above-listed exams, intracardiac echocardiography may be considered. 12.1.3. Prognosis and treatment A high in-hospital mortality rate of 20–40% has been reported in PVE.628,629 Compared with NVE, PVE is associated with increased in- hospital mortality and morbidity as well as reduced long-term sur-vival.88,630 Several factors have been associated with poor prognosis in PVE, including older age, diabetes mellitus, healthcare-associated in-fections, and early PVE.312 Among the different causative organisms, staphylococcal or fungal infection seem to be more aggressive, whereas enterococcal infections are associated with similar mortality but higher recurrence rates.628 Haemodynamic instability, multivalvular involvement as well as involvement of the aortomitral fibrosa have been associated with worse outcomes. It is noteworthy that the most important risk factor for recurrent IE and mortality is withholding surgery despite an obvious indication.5 The best therapeutic option in PVE is still debated. Although surgery is generally considered the best option when PVE causes severe pros-thetic dysfunction or HF, in the EURO-ENDO registry it was per-formed in only 73% of patients with PVE despite a clear indication for surgical treatment.5 In a single-series study of 523 PVE patients, early surgery was a large independent predictor of early and 1-year sur-vival.631 Conversely, after adjustment for differences in clinical charac-teristics and survival bias, early valve replacement was not associated with lower mortality compared with medical therapy in a large inter-national cohort.421 In this series, however, surgery was beneficial in the subgroup of patients with the strongest indications for surgery in-cluding valve regurgitation, vegetation, and dehiscence or paravalvular abscess/fistula formation.421 Therefore, a surgical strategy is recom-mended for PVE in high-risk subgroups identified by prognostic assess-ment, i.e. PVE complicated with HF, severe prosthetic dysfunction, abscess, or persistent fever. Conversely, patients with uncomplicated non-staphylococcal late PVE can be managed conservatively.632–634 However, patients who are initially treated medically require close follow-up because of the risk of late events and the higher risk of re-lapse or valvular dysfunction. Surgery for PVE follows the general principles outlined for NVE. However, the reoperation setting and the higher incidence of peripros-thetic tissue destruction increase the complexity of the procedure. Meticulous and radical debridement of the infected material, including the original prosthesis, suture, and pledgets, is recommended. The type of valve substitute used for PVE follows the same recommenda-tions as for NVE (see also Section 10.3.1). Early PVE following valve replacement surgery is a separate entity as-sociated with a high mortality rate, where conservative treatment with antibiotics is unlikely to lead to a cure and repeat surgery should be per-formed.621,635 Staphylococci, Cutibacteria, or similar species are the usual causative organisms.622,636 12.2. Endocarditis in the elderly Characteristics of patients with IE have dramatically changed over re-cent decades, with an increasing prevalence and specific features of IE in the elderly population.25,145,637,638 In this population, enterococci and S. aureus are reported to be the most frequent aetiological agents. In addition, the higher presence of intracardiac prosthetic devices (CIED and valvular prosthesis/repair including TAVI devices) and in-creased incidence of healthcare-associated IE episodes are ob-served.25,637 Finally, a lower risk of embolic episodes has been observed in this subgroup.462,639–641 Recommendation Table 19 — Recommendations for prosthetic valve endocarditis Recommendations Classa Levelb Surgery is recommended for early PVE (within 6 months of valve surgery) with new valve replacement and complete debridement. 621,635 I C © ESC 2023 PVE, prosthetic valve endocarditis. aClass of recommendation. bLevel of evidence. ESC Guidelines 53 Downloaded from by guest on 30 August 2023 A number of studies have shown that cardiac surgery positively af-fects the clinical outcome of IE patients. Nevertheless, old age, co- morbidities, and previous non-cardiac and cardiac procedures lead to surgical hesitancy by referring physicians, surgeons, and patients them-selves.642 Moreover, these characteristics also influence the outcome of this fragile cohort.400,433 As a result, less frequent performance of cura-tive surgery and increased mortality are typical hallmarks of IE episodes in elderly as compared with the younger population.640 In a recently published Swedish propensity analysis of IE patients from 2006 to 2017, the authors found that surgery was underused in the elderly and that 1-year mortality was significantly higher in elderly patients who did not undergo surgery.641 In a sub-analysis of the ESC EORP EURO-ENDO registry, the indication for surgery was less often recog-nized (51% vs. 57%) and surgery was far less frequently performed when indicated (35% vs. 68%) in patients >80 vs. <80 years. However, mortality of surgically treated patients was remarkably simi-lar in patients <80 and >80 years after propensity matching (19.7% vs. 20.0%). Age was also not demonstrated to be an independent predictor of mortality in this large prospective study.640,643 These findings suggest that performance of surgery in well-selected elderly patients is under-utilized and may increase their chance of survival. In elderly IE patients, functional and nutritional status are important predictors of outcomes.400 When considering cardiac surgery in elderly patients, functional and nutritional status, and their associated risks, should be accurately explored through a comprehensive assessment by geriatricians. In addition, the earliest possible discharge home to fa-cilitate the patient’s functional recovery should be considered in this subgroup of patients. 12.3. Transcatheter prosthetic valve endocarditis 12.3.1. Endocarditis following transcatheter aortic valve implantation The incidence of IE post-TAVI ranges from 0.3 to 1.9 per 100 patient- years,94,623,644–648 which is similar to that reported following surgical aortic valve replacement in both observational studies and RCTs.94,623,646,647 One recent study, however, reported a lower inci-dence of PVE after TAVI compared with surgical prostheses.649 The risk of IE is higher within the first year following the procedure, and par-ticularly within the initial 3 months.644,645,648,650–652 A modest decrease in the incidence of IE post-TAVI has been observed in recent years, par-ticularly in the early period following the procedure, presumably related to multiple technical improvements, more streamlined procedures, and a reduction of periprocedural complications.650,652 A similar IE rate has been reported irrespective of transcatheter valve type,653 and predis-posing factors, including younger age, male gender, renal dysfunction, and significant residual aortic regurgitation, have been identi-fied.94,644–646,648,651,652 12.3.1.1. Diagnosis The diagnosis of IE post-TAVI is challenging. The stent frame of trans-catheter valves, with a much higher amount of metal surrounding the valve leaflets compared with surgical prostheses, and the characteristics of TAVI patients (frequently elderly with multiple co-morbidities) may increase the diagnostic challenges in this population. The clinical presen-tation is frequently atypical, with fever lacking in 13–20% of pa-tients.623,645,650 Enterococci and S. aureus are the two most common microorganisms involved in IE post-TAVI, followed by streptococci and CoNS.644–646,650 Some important aspects should be considered regarding TOE in pa-tients with suspected IE post-TAVI: (i) no vegetations are detected in 38–60% of cases;623,645,650,651 (ii) vegetations are located in the stent frame of the transcatheter valve (and not on the valve leaflets) in 12% of cases, and this rate increases up to 19% in the presence of some self-expanding valve systems with a longer stent frame occupying the ascending aorta;653 and (iii) the vegetations are located outside the transcatheter valve in about one-third of cases, mainly at the level of the mitral valve.645,650,651 Nuclear imaging or CT have been useful to diag-nose IE post-TAVI.654,655 The addition of [18F]FDG-PET/CT and/or CTA to the diagnostic work-up of IE in TAVI changed the final clinical diagnosis in 33% of patients.655 Intracardiac echocardiography may also be useful for detecting vegetations in patients with suspected IE after TAVI and negative TOE.165 12.3.1.2. Prognosis and treatment Prognosis and treatment of post-TAVI PVE is complicated by the fact that patients are older and have more co-morbidities than post-surgical PVE patients. About two-thirds of patients with IE post-TAVI exhibit at least one complication, with acute kidney injury and HF being the most frequent adverse events.645,646,656 The in-hospital and 30-day mortality rates are very high, ranging from 16% to 36%,623,644–647,657 and increase up to 41–59% at 1-year follow-up.644,645,652,657 A higher patient risk profile, S. aureus, and the occurrence of IE complications have been identified as risk factors for increased mortality.645,652,657 Antimicrobial therapy for IE post-TAVI is similar to that of PVE (see Section 7). Similar to surgical PVE, cardiac surgery is considered the best option in the presence of IE complications, particularly severe prosthet-ic failure or HF, but is infrequently performed. Surgery is performed in ∼20% of cases (ranging from 3.8% to 31.3%),645,652,656 a much lower rate compared with NVE and surgical PVE. The characteristics of the TAVI population, with often advanced age and high or prohibitive sur-gical risk, along with the potential difficulties associated with the re-moval of some transcatheter valve systems (particularly those with a large amount of stent frame, frequently adherent to the ascending aorta after a few months following the TAVI procedure) may play a role in the low rate of surgical interventions. To date, all studies but one failed to demonstrate the potential bene-fit of surgery in IE post-TAVI patients,442,645,652,656,658 but the relatively small sample size of the studies and the multiplicity of potential con-founders when comparing to those patients not receiving surgical treat-ment precludes definite conclusions. The only study showing a beneficial effect of surgical intervention focused on those patients who had a local extension of the infection (i.e. abscess or fistula).442 The decision to proceed with surgery in IE post-TAVI patients should be individualized, balancing the surgical risks and the prognosis of med-ical treatment alone. In cases with local extension of the infection, sur-gery may be recommended in the absence of a prohibitive surgical risk. In cases with healed IE and valve prosthesis dysfunction, repeat trans-catheter therapy (valve-in-valve procedure) can be performed in select patients.659 Such interventions should be performed at least 1–3 months after the healed endocarditis episode and following a negative follow-up TOE. 12.3.2. Endocarditis following transcatheter pulmonary valve implantation The incidence of IE post-transcatheter pulmonary valve implantation (TPVI) ranges from 1.6 to 4.0 per 100 patient-years,93,660–667 which seems to be higher than that reported following surgical pulmonary valve interventions (observational studies, no randomized data).662,663,667,668 While some studies suggest a higher risk associated with the use of bovine jugular vein valves,662,667,669 a recent large multi-centre study including different transcatheter valve systems did not 54 ESC Guidelines Downloaded from by guest on 30 August 2023 observe differences between valve types.665 The most consistent fac-tors associated with an increased risk of IE following TPVI have been younger age, a previous history of IE, and a higher transvalvular residual gradient.93,663,665 12.3.2.1. Diagnosis The diagnosis of IE in TPVI recipients may be challenging, and the use of intracardiac echocardiography and [18F]FDG-PET/CT has been shown to be useful in cases with a clinical suspicion and negative TTE/ TOE.34,93,210,660,665,670 S. aureus and oral group streptococci species are the most common microorganisms causing IE post-TPVI.660,664–666 12.3.2.2. Prognosis and treatment New moderate or severe prosthetic valve stenosis occurs much more frequently (one-third to one-half of patients) in post-TPVI PVE than in aortic PVE, and the rate of surgical valve replacement therapy ranges from 26% to 56%.93,660,661,664,665 The possibility of a transcatheter therapy (valve-in-valve intervention) for treating severe prosthesis dys-function in cases with healed endocarditis or as an urgent treatment (balloon dilatation) in severe valve stenosis cases has also been re-ported.660,665 A valve-in-valve intervention should be delayed at least 1–3 months following antibiotic treatment of the endocarditis episode. The mortality rate related to the IE episode ranges from 0% to 11%.93,660,661,664,665 This rate is much lower compared with TAVI pa-tients, which is likely to be related to the younger and less co-morbid characteristics of the TPVI population. 12.4. Infective endocarditis affecting cardiac implantable electronic devices Device-related infection is one of the most serious complications of CIED therapy and is associated with significant mortality and morbidity.671 12.4.1. Definitions of cardiac device infections A recent EHRA consensus document has published criteria for CIED infection.130 Localized infections may be either superficial incisional in-fections (acute infection without involvement of the pocket or hard-ware) or isolated pocket infections (limited to the hardware in the pocket), and can be either acute or chronic. Systemic CIED infections may occur with or without pocket infection, and with or without visible vegetations on the tricuspid or pulmonary valves or pacing leads. Cardiovascular implanted electronic device-related IE is defined as evi-dence of CIED infection with clinical signs of pocket infection and/or imaging findings (lead vegetations, positive FDG-PET on the gener-ator/leads etc.) which fulfil the criteria for valvular IE (see Section 5). 12.4.2. Pathophysiology and microbiology Cardiovascular implanted electronic device-related IE occurs by two mechanisms. Local infection usually results from bacterial flora from the patient’s skin that is introduced into the pocket at the time of inci-sion despite surgical preparation.672 Seeding via bacteraemia from a dis-tant focus is less frequent.673–676 Whereas CoNS are most frequently the cause of chronic pocket in-fection, the most frequent agents identified with bacteraemia in CIED in-fection are S. aureus and CoNS.677,678 Other causative organisms are Enterococcus spp., β-haemolytic streptococci, oral streptococci group, Cutibacterium acnes, and Corynebacterium spp.674,678,679 More rarely, sys-temic infection is caused by Gram-negative (mainly P. aeruginosa or Serratia marcescens)680 or polymicrobial agents, whereas systemic fungal infections (Candida spp. and Aspergillus spp.)681 are exceptional. 12.4.3. Risk factors Risk factors may be divided into patient-related, procedure-related, and device-related factors.118 The PADIT (Previous procedure on same pocket; Age; Depressed renal function; Immunocompromised; Type of procedure) study randomized 19 603 patients undergoing CIED im-plantation to conventional treatment (pre-procedural cefazolin infu-sion) vs. different regimens of incremental treatment.682 The primary outcome was 1-year hospitalization for device infection which was not significantly different between groups. A risk score for infection has been derived from the study (see Supplementary data online, Table S11)683 and has been validated externally.684 A web-based calcu-lator is available ( 12.4.4. Prophylaxis Antibiotic prophylaxis to prevent CIED-related IE before interventions, such as dental, respiratory, gastrointestinal, or genitourinary proce-dures, is not warranted as the risk is very low. Prevention of CIED infection at implantation hinges upon careful planning, pre-operative antibiotic prophylaxis, correction of modifiable risk factors, hygienic surgical environment and technique, ancillary mea-sures in case of increased risk (e.g. use of an antibacterial envelope), and proper post-operative care. Correction of modifiable risk factors includes general measures such as postponing the procedure in cases of fever or signs of infection and avoiding temporary pacing. Routine administration of prophylactic sys-temic antibiotics within 1 h of incision is the standard of care.118 RCTs have used flucloxacillin (1–2 g i.v.)117 and first-generation cephalospor-ins, such as cefazolin (1–2 g i.v.).116 Vancomycin (1–2 g over 60–90 min) may be used in case of allergy to cephalosporins with other alternatives, including teicoplanin and clindamycin.117 Coverage of MRSA should be guided by the prevalence in the implanting institution. Haematoma is a major contributor to risk of infection, and all possible measures should be taken to avoid this complication.685,686 Another major risk factor is a revision with re-opening of the pocket (e.g. for lead repositioning). Technical aspects have recently been covered in de-tail in an EHRA consensus document on CIED implantation.687 It is generally not recommended to wash the pocket with antibiotics, nor to administer antibiotic treatment post-operatively, as shown by the PADIT trial.682 An antibiotic mesh envelope, which locally releases minocycline and rifampin for a minimum of 7 days and is fully absorbed in ∼9 weeks, may, however, be useful to reduce risk of infection in se-lected patients. The Worldwide Randomized Antibiotic Envelope Infection Prevention Trial (WRAP-IT) showed that the mesh envelope significantly reduces the incidence of CIED infection in patients at in-creased risk (i.e. undergoing a pocket revision, generator replacement, system upgrade, or implantation of a cardiac resynchronization therapy [CRT]-implantable cardioverter defibrillator [ICD]).688 The number needed to treat was, however, high at 200, but is ∼50 in patients under-going CRT reoperations (replacement/upgrade/revision) in a recent observational study.689 12.4.5. Diagnosis Clinical presentation of CIED-associated IE is similar to valvular IE with patients frequently presenting with fever, chills, and embolic events. Signs of pocket infection (swelling, tenderness, erythema, purulent dis-charge etc.) may or may not be present. The probability that a positive blood culture in a CIED recipient re-presents underlying device infection depends on the organism type and duration of bacteraemia. Suspicion of CIED-associated IE should be particularly high in the event of S. aureus bacteraemia.675 CIED infection is less likely with Gram-negative bacteraemia, and in these instances, the pocket usually shows signs of infection.680,690,691 ESC Guidelines 55 Downloaded from by guest on 30 August 2023 Persistent/relapsing bacteraemia/fungaemia Gram-negative No vegetations and non-S. aureus Vegetations and/or S. aureus Septic emboli or prosthetic valve 2 weeks of antibiotics (Class IIb) 4 weeks of antibiotics (Class IIa) CIED reimplantationc (Class I) 6 weeks of antibiotics (Class IIa) CIED extraction Gram-positive or fungaemia Antibiotic therapy covering MRSA and Gram-negative bacteria (Class I) Suspicion of CIED-related IE 2023 ESC diagnostic criteria for IE Repeat diagnostic tests within 7 days Antimicrobial therapy Bacteraemia/fungaemia without other major criteria or source of infection Blood cultures TTE and TOE (or intracardiac echocardiography) [18F]FDG-PET/CT (or WBC SPECT)a (Class I) Possible IE Definite IE No IE Cured Probable isolated valve IEb Definite CIED involvement (Class IIb) (Class IIa) (Class I) Figure 13 Management of cardiovascular implanted electronic device-related infective endocarditis. [18F]FDG-PET/CT, 18F-fluorodeoxyglucose positron emission tomography/computed tomography; CIED, cardiovascular implanted electronic device; ESC, European Society of Cardiology; IE, infective endocar-ditis; MRSA, methicillin-resistant S. aureus; TOE, transoesophageal echocardiography; TTE, transthoracic echocardiography; WBC SPECT/CT, white blood cell single photon emission tomography/computed tomography. aIf no signs of pocket infection and negative TOE. bTaking into account the identified patho-gen, procedural risk, and requirement for valve surgery. cAt a distant site and postponed as long as possible (until signs and symptoms of infection have re-solved and blood cultures are negative for >72 h in the absence of vegetations and /or ‘ghosts’, or otherwise after >2 weeks of negative blood cultures). 56 ESC Guidelines Downloaded from by guest on 30 August 2023 Transthoracic echocardiography and TOE are both recommended in the case of suspected CIED-related IE.692–694 Intracardiac echocardi-ography may also be used to visualize vegetations,695 and may be useful in patients in whom TOE is not possible. However, the absence of ve-getations does not rule out IE, as these may be present on extracardiac segments of the lead which cannot be visualized. Repeating TTE and/or TOE within 5–7 days is recommended in case of initially negative exam-ination when clinical suspicion of CIED-related IE remains high. It is im-portant to note that fibrinous lead masses may be observed in asymptomatic CIED patients, and do not predict CIED-related IE over long-term follow-up.696 Diagnosis of CIED-related endocarditis by FDG-PET/CT has good sensitivity and specificity,129 and is particularly useful in the setting of possible CIED-related IE without signs of pocket infection.238 Results should be interpreted with caution, however, if the CIED is recently im-planted (<6 weeks).130 White blood cell SPECT/CT has also been used for diagnosing CIED infection but has limited availability.216,697 A chest X-ray or CT should be performed in all patients to evaluate the presence of pulmonary complications. 12.4.6. Antimicrobial therapy Treatment of CIED infection involves early698,699 and complete removal of all parts of the system, combined with initial empirical antibiotic therapy directed at MRSA and Gram-negative bacteria, while awaiting identification of the pathogen.130,700,701 Antibiotic treatment follows the recommenda-tions indicated in Section 7. In exceptional cases where complete device re-moval is not possible, i.v. antibiotics for 4–6 weeks may be administered followed by close follow-up after interruption of antibiotic therapy or, al-ternatively, individualized long-term suppressive oral therapy. 12.4.7. Device extraction When CIED and lead extraction is required, such procedures should be performed in centres with the corresponding expertise. Complete CIED removal is recommended for all patients with confirmed infection of the lead(s), as conservative treatment is associated with increased mor-tality.678,699 In patients with left-sided prosthetic heart valves and CIED in-fection, complete CIED removal combined with prolonged (4–)6 week antibiotic therapy may prevent left-sided valve infection.130,702 Complete CIED extraction should also be considered in case of valvular IE without definite lead involvement, taking into account the identified pathogens (Staphylococcus spp. infections may be more prone to seed the CIED),673,675,676 procedural risk, and indication for valve surgery. Complete device extraction should be considered even in the absence of vegetations in the setting of persistent or relapsing Gram-positive bac-teraemia or fungaemia after a course of appropriate antibiotic therapy, if there is no other identified source (see Figure 13).681 In all instances of lead extraction, procedural risk should be carefully evaluated taking into ac-count lead dwell time, pacemaker dependency, patient frailty, and other co-morbidities within the process of shared decision-making.703 Lead extraction should be performed, without delay (i.e. within the first days of admission), as this has been shown to be associated with improved outcomes.698,699,704 Percutaneous rather than surgical ex-traction is the preferred procedure, but requires specialized tools and should be performed in centres with expertise in this technique and with onsite surgical backup, due to the risk of life-threatening tam-ponade and vein laceration. Large vegetations may be aspirated percutaneously before lead extrac-tion to reduce risk associated with embolization.705 Surgical lead extraction should be considered in case of large vegetations (e.g. >20 mm)679 and if aspiration is not available or is unsuccessful. Surgical re-moval is also the preferred technique if valve surgery is indicated. Hardware retrieved from extraction, especially the lead tip, should be cul-tured.706 Sonication has been shown to increase diagnostic yield.707,708 12.4.8. Device reimplantation The indication for reimplantation should always be carefully evaluated and no part of the removed CIED system should be reimplanted. Quality of evidence regarding timing of reimplantation is poor.709 Reimplantation should be performed at a site distant from that of the previous generator, and delayed until signs and symptoms of local and systemic infection have resolved and blood cultures are negative for at least 72 h after extraction in the absence of vegetations or ‘ghosts’ (fibrous remnants after lead extraction, which have been asso-ciated with death and reinfection),710 or after 2 weeks of negative blood cultures if vegetations were visualized.701,711 For patients with a high risk of sudden cardiac death, a wearable defib-rillator is an option as a bridge to reimplantation. In pacemaker-dependent patients, an active-fixation lead may be introduced via the internal jugular vein and connected to an external pacemaker for up to 4–6 weeks, thereby preserving the contralateral side for definitive device reimplantation.712 As an alternative to delayed reimplantation in pacemaker-dependent patients, an epicardial pacemaker may be implanted before lead extraction, although this strategy has been associated with a higher risk of device re-intervention.713 Alternative devices such as leadless pacemakers714 or subcutaneous ICD715 may be implanted in selected patients if the risk of new infection is deemed high. Recommendation Table 20 — Recommendations for cardiovascular implanted electronic device-related in-fective endocarditis Recommendations Classa Levelb Antibiotic prophylaxis covering S. aureus is recommended for CIED implantation.118 I A TTE and TOE are both recommended in case of suspected CIED-related IE to identify vegetations.692–694 I B Complete system extraction without delay is recommended in patients with definite CIED-related IE under initial empirical antibiotic therapy.698,699 I B Obtaining at least three sets of blood cultures is recommended before prompt initiation of empirical antibiotic therapy for CIED infection,710 covering methicillin-resistant staphylococci and Gram-negative bacteria. I C If CIED reimplantation is indicated after extraction for CIED-related IE, it is recommended to be performed at a site distant from the previous generator, as late as possible, once signs and symptoms of infection have abated and until blood cultures are negative for at least 72 h in the absence of vegetations, and negative for at least 2 weeks if vegetations were visualized.701,711 I C Continued ESC Guidelines 57 Downloaded from by guest on 30 August 2023 12.5. Infective endocarditis in patients admitted to intensive care units Infective endocarditis is frequently associated with severe life-threatening cardiac and/or systemic complications and the number of patients requir-ing intensive care unit (ICU) admission has steadily been growing in recent years, as shown in a large retrospective study.716 The need for ICU admis-sion, advanced monitoring, vasoactive treatment, and organ support is most commonly prompted by the occurrence of septic shock, acute HF, and cardiogenic shock leading to multiorgan failure. Moreover, in recent years, an increase in healthcare-associated IE, usually of staphylococcal ori-gin, predominantly in older patients with an increased number of co- morbidities, and more likely to lead to critical illness, has also been re-ported.29,717–719 Any IE patient requiring ICU admission should be urgently discussed within the Endocarditis Team. In the largest multicentre retrospective series, focusing on critically ill IE patients with organ failure requiring ICU admission in France over an 18-year period, overall in-hospital mortality was 32%.716 Multivariate analysis showed age, Simplified Acute Physiology Score (SAPS) II score, organ failure, stroke, and Staphylococcus spp. to be as-sociated with an increased risk of death. In contrast, cardiac surgery, CIED, male gender, and Streptococcus spp. as the causative micro-organism of IE, were associated with a better survival.716 In another study that reported an even higher mortality (42%), four independent prognostic factors were identified: high SAPS II (>35 points) and Sequential Organ Failure Assessment (>8 points) scores, MRSA infec-tion, and native valve IE.718 Right-sided IE, which is more commonly associated with PWID, ac-counts for <10% of IE cases but is associated with high mortality in pa-tients needing ICU admission.717 12.5.1. Causative microorganisms The majority of retrospective series in the ICU setting point to Staphylococcus spp. as the main causative agent of IE episodes. Indeed, S. aureus has emerged as the most feared aetiological agent with the highest rates of complications and mortality, being responsible for up to 56% of IE cases in one observational study.719 Streptococcus spp., Enterococcus spp., Gram-negative bacilli, and Candida spp. are less fre-quently reported.718,719 Identification of the infecting microorganism remains the mainstay of effective therapy in complicated IE cases. Hence, in patients with negative blood cultures, serological or molecu-lar testing by PCR should be considered (see Section 5.3). 12.5.2. Diagnosis The diverse nature, epidemiological profile, and presentation pheno-type of IE in the ICU setting may hinder early diagnosis. In particular, pyrexial episodes suggestive of an alternative infective source and neurological manifestations, such as confusion, delirium, or focal symp-toms may initially mislead the clinician from a diagnosis of IE. The diagnosis of IE in ICUs follows the same modified criteria as in non-ICU patients (see Section 5). Transoesophageal echocardiography has a prominent role as a tool for diagnosis of IE and its complications in the ICU.720 12.5.3. Management Antimicrobial therapy and indications for surgery in patients with IE are described in Sections 7 and 10, respectively. Surgical therapy has been associated with an improved early and late outcome both in the general population and in patients admitted to ICUs. Although surgery is the treatment of choice in about one-half of patients, surgical therapy in ICU patients is characterized by more complex procedures with in-creased peri-operative mortality, as well as difficult post-operative care due to higher requirements of circulatory and pulmonary support. Five independent predictors of post-operative need for advanced circu-latory support were found in one study of patients with IE: male sex, increased surgery duration, renal dysfunction (pre-operative estimated glomerular filtration <60 mL/min/m2), HF prior to surgery, and lower pre-operative platelet count.721 Extracorporeal membrane oxygenation is occasionally required in patients post-surgery but is associated with poor outcomes.722 Decision-making in ICU patients with IE should always be a product of consensus of the Endocarditis Team to determine the best management strategy. Pre-operative haemodynamic optimization and goal-directed therapy protocols including vasoactive drugs and mechanical circulatory support may be considered in these complex high-risk patients.721 12.6. Right-sided infective endocarditis Right-sided IE accounts for ∼5–10% of patients with IE,723 but its fre-quency may be increasing as its risk factors are increasing in some coun-tries.133,724 Risk factors for right-sided IE include patients with CHD, indwelling catheters, and CIED, as well as immunocompromised and PWID patients. Of these, PWID is an increasingly common risk Complete CIED extraction should be considered in case of valvular IE, even without definite lead involvement, taking into account the identified pathogen and requirement for valve surgery. IIa C In cases of possible CIED-related IE with occult Gram-positive bacteraemia or fungaemia, complete system removal should be considered in case bacteraemia/fungaemia persists after a course of antimicrobial therapy.673–676 IIa C Extension of antibiotic treatment of CIED-related endocarditis to (4–6) weeks following device extraction should be considered in the presence of septic emboli or prosthetic valves.702 IIa C Use of an antibiotic envelope may be considered in select high-risk patients undergoing CIED reimplantation to reduce risk of infection.688,689 IIb B In cases of possible CIED-related IE with occult Gram-negative bacteraemia, complete system removal may be considered in case of persistent/ relapsing bacteraemia after a course of antimicrobial therapy.680,690,691 IIb C In non-S. aureus CIED-related endocarditis without valve involvement or lead vegetations, and if follow-up blood cultures are negative without septic emboli, 2 weeks of antibiotic treatment may be considered following device extraction. IIb C Removal of CIED after a single positive blood culture, with no other clinical evidence of infection, is not recommended.675 III C © ESC 2023 CIED, cardiovascular implanted electronic device; IE, infective endocarditis; TOE, transoesophageal echocardiography; TTE, transthoracic echocardiography. aClass of recommendation. bLevel of evidence. 58 ESC Guidelines Downloaded from by guest on 30 August 2023 factor,133,723 while patients with indwelling vascular catheters have the worst prognosis.725 IE of transcatheter pulmonary valves is covered in Section 12.2, whereas CIED-related right-sided IE is covered in Section 12.3. The most common microorganism causing right-sided IE is S. aureus, accounting for the majority of patients.723,726 The tricuspid valve is much more commonly infected than the pulmonary valve in patients with right-sided IE.723,727 Right-sided IE may also involve non-functional embryonic remnants of the right atrium (e.g. Eustachian valve).723,727 Right-sided IE rarely spreads to involve the left-sided cardiac structures, whereas spread from left- to right-sided structures is not uncommon.728 12.6.1. Diagnosis and complications Right-sided IE patients present with fever, bacteraemia, and pulmonary complaints (i.e. cough, chest pain, or haemoptysis). Right-sided HF may also occur due to tricuspid or pulmonary regurgitation, or to pulmon-ary hypertension induced by multiple pulmonary septic emboli.133 Diagnosis is most frequently confirmed by echocardiographic findings of vegetations on the tricuspid valve or, less frequently, pulmonary valve. Adequate evaluation of the tricuspid valve may be performed with TTE, due to the anterior location of the valve and the large vegetations frequent-ly observed in right-sided IE. Transoesophageal echocardiography is fre-quently required, however, particularly for evaluation of the pulmonary valve or in patients with indwelling venous catheters or intracardiac de-vices.729 Intracardiac echocardiography may also be helpful in select pa-tients. Vegetations may be challenging to identify on the pulmonary valve even with TOE, especially in patients with a prosthetic valve in the pulmon-ary position. FDG-PET imaging may be very helpful in such pa-tients.34,730 Perivalvular abscess formation and invasion into surrounding structures is rarely seen in right-sided IE, unless it is a secondary conse-quence of left-sided IE.728 CT is useful in order to identify concomitant pul-monary disease, including infarcts and abscess formation. 12.6.2. Endocarditis in people who inject drugs Infective endocarditis in PWID is an increasing global phenom-enon.10,132,133,141 Repeat i.v. injections result in contaminated particles that reach the tricuspid valve and right-heart chambers and can also lead to infection of left-heart structures, which is associated with worse prognosis.614 PWID patients also have an increased rate of human immunodeficiency virus (HIV) and hepatitis than other patients with right-sided IE.731 The majority of right-sided IE in PWID can be treated successfully with antibiotic therapy. Mortality rates of PWID are rela-tively low, even when surgery is required, probably due to the young patient age.723 However, PWID have a markedly increased rate of IE recurrence, particularly in the first 6 months post-surgery.133,614,723,732 12.6.3. Prognosis and treatment Right-sided IE is generally a more benign clinical entity than left-sided IE and can be medically managed in ∼90% of patients, with surgery re-served for those who fail medical therapy.733 Patients with CIED-related right-sided IE have a worse prognosis as compared with non-CIED-related right-sided IE (see Section 12.4).723 725 Right-sided IE in immunocompromised patients, particularly fungal in-fections, carries a very poor prognosis. 12.6.3.1. Antimicrobial therapy S. aureus and CoNS are the cause of right-sided IE in a large proportion of cases, with S. aureus predominating in PWID and CoNS being more common in patients with indwelling devices.723,726 MRSA rates may be increasing over time, particularly in PWID.133 Right-sided IE due to Streptococcus spp. is unusual but can be observed in alcoholics and dia-betics. P. aeruginosa and other Gram-negative organisms are rare causes of right-sided IE, while Candida albicans is mostly seen in immunocom-promised patients. Empirical antimicrobial therapy depends on the suspected micro-organism, the type of drug and solvent used by the PWID, and the in-fection location,734 but S. aureus must be initially covered in all cases. Initial treatment consists of penicillinase-resistant penicillin, vancomy-cin, or daptomycin, depending on the local prevalence of MRSA,735 in combination with gentamicin. If the patient is a pentazocine addict, an anti-Pseudomonas agent may also be required, as the use of recreational drugs may also entail infections with Gram-negative bacteria.735 Very large vegetations and history of brown heroin use dissolved in lemon juice suggest infection for Candida spp. (not C. albicans), and therefore antifungal treatment should be added.736 Antifungals may be necessary in selected PWID, particularly if immunocompromised.737 Once the causative organisms have been isolated, therapy has to be adjusted. An RCT demonstrated that a 2-week treatment course may be sufficient and that aminoglycosides may be unnecessary.738 717 Two-week treatment with oxacillin (or cloxacillin) without gentamicin is effective when: (i) MSSA is the causative organism; (ii) There is good clinical and microbiological response to treatment (>96 h);739 (iii) The vegetation size is ≤20 mm; and (iv) There is an absence of metastatic sites of infection or empyema and cardiac or extracardiac complications,739,740 prosthetic valve or left-sided valve infection,741 and severe immunosuppression.742 Glycopeptides (vancomycin) should not be used in a 2-week treatment. The standard 4–6-week regimen should be used in the remaining patients or when therapy with antibiotics other than penicillinase-resistant penicil-lins are used.330,739–744 When the conventional i.v. route therapy is not possible, S. aureus right-sided IE in PWID may also be treated with oral ci-profloxacin (750 mg twice a day) plus rifampin (300 mg twice a day) if the strain is susceptible to both drugs, the case is uncomplicated, and patient adherence is monitored carefully.745 Partial oral antibiotic treatment may also be beneficial for PWID with IE.746 For organisms other than S. aureus, therapy in PWID does not differ from that in other patients. 12.6.3.2. Surgery The commonly accepted indications for surgical treatment of right- sided IE in patients who are receiving appropriate antibiotic therapy are (see Recommendation Table 19): • Persistent bacteraemia after at least 1 week of appropriate antibiotic therapy.10 • Tight ventricular dysfunction secondary to acute severe tricuspid re-gurgitation non-responsive to diuretics.479 • Respiratory insufficiency requiring ventilatory support after recur-rent pulmonary emboli.747 • Involvement of left-sided structures;748,749 and • Large residual tricuspid vegetations (>20 mm) after recurrent pul-monary emboli.145,471 Patients should be individually assessed by the Endocarditis Team. An isolated vegetation is not an indication for surgery. Patients with ESC Guidelines 59 Downloaded from by guest on 30 August 2023 residual large vegetations frequently present with right-heart and/or re-spiratory failure, as well as persistent sepsis.750 The common surgical strategies for tricuspid valve IE include valve re-pair, replacement and, less commonly, surgical valvectomy.751 Tricuspid valve repair is more frequently performed than valve replacement in right-sided IE, although the extent of valve destruction may make repair impossible.725,752 Tricuspid valve repair may also be associated with better short- and long-term outcomes than replacement for right-sided IE, particularly with regards to recurrent infection and need for repeat surgery.479,723 When valve replacement for right-sided IE is required, bioprostheses are frequently preferred due to concerns with the management and risks of lifelong anticoagulation, especially in PWID, and the risks of thrombo-embolism for mechanical valves in the right heart.726 Prophylactic placement of permanent epicardial leads should be per-formed at the time of tricuspid valve surgery for right-sided IE, particu-larly if heart block is present in the operating room to prevent damage of a replaced valve during subsequent transvenous lead displacement and to lower the risk of reinfection.733 Recently, interest has been generated in the extraction of large vege-tations using percutaneous extracorporeal circuitry for aspiration.753 The main goals have been debulking of septic intracardiac masses, redu-cing the infectious load, and achieving clinical stability.754 12.7. Infective endocarditis in congenital heart disease Although the incidence of CHD is relatively constant, the overall popu-lation with CHD is constantly increasing due to increased survival fol-lowing CHD surgery in childhood and increased longevity of adults with CHD. The presence of CHD, even after repair, is recognized as a lifelong potential substrate for IE. Congenital heart disease predis-poses to IE via several mechanisms including turbulent non-laminar blood flow causing shear stress and endothelial damage, the presence of intracardiac foreign material such as prosthetic valves or CIED, cyan-osis, and recurrent exposure to cardiac procedures.98 There are marked variations in susceptibility to IE between CHD lesions. Some simple conditions, such as secundum atrial septal defect, patent ductus arteriosus, and pulmonary valve stenosis, carry a low risk of IE, while others, such as bicuspid aortic valve carry a somewhat increased risk.8 However, CHD often presents with multiple cardiac lesions, each adding to the total risk of IE.8,756 In general terms, IE is more common in CHD with multiple defects and in patients with more complex CHD.757 Specific high-risk conditions are prosthetic valves, including transcath-eter valves, valve repair using a prosthetic ring, previous IE, any unre-paired cyanotic CHD, and any CHD repaired with prosthetic material for up to 6 months after the procedure, or lifelong if residual shunt or valvar regurgitation remains.758 Contemporary studies confirm the rela-tively high risk of IE in CHD patients after valve surgery.8,47,90,759 Specific awareness is needed after TPVI (see Section 12.3.2).666,759,760 The distribution of causative microorganisms does not differ from the pattern found in acquired heart disease, with Streptococcus spp. and Staphylococcus spp. being the most common strains.98,757,761,762 As in other groups, the diagnosis of IE is often made late,757 highlighting the need to consider the diagnosis of IE in any CHD patient presenting with persisting fever or other signs of ongoing infection. Multiple blood cultures are essential before starting antibiotic treatment. The principal symptoms, complications, and basis for diagnosis do not differ from IE in general. However, in CHD right-sided IE is more frequent than in non-CHD-acquired cardiac disease. Transthoracic echocardiography is sufficient in many cases to image the infectious lesions and their complications. However, complex anat-omy and the presence of artificial material may reduce the rate of vege-tation detection and other features of IE, thus favouring the addition of TOE, particularly in adults and larger children. Despite the improved sensitivity of TOE for the detection of IE, TOE may only perform simi-larly to TTE for anterior structures of the heart, such as the right ven-tricular outflow tract, or infected sites at distal structures, such as stents or other prosthetic material within branch pulmonary arteries. Hence, a negative study does not exclude the diagnosis of IE. In patients with prosthetic material advanced imaging such as [18F]FDG-PET/CT and PET/CTA, can increase the diagnostic accuracy.223 In addition to the usual Endocarditis Team (see Section 4), multidis-ciplinary care of CHD patients with IE from diagnosis to treatment should be provided in specialized CHD centres with expertise in CHD cardiac imaging, CHD surgery, infectious disease, and intensive care. Surgical indications do not differ from those of acquired heart dis-ease IE. Mortality rates in CHD vary from 6% to 15%.757,761–764 This better prognosis compared with acquired heart disease IE may reflect the higher proportion of right-heart IE, younger overall patient age, or the comprehensive care in CHD centres. Primary prevention of IE in CHD patients and corresponding patient education is essential (see Section 3).765 Recommendation Table 21 — Recommendations for the surgical treatment of right-sided infective endocarditis Recommendations Classa Levelb Surgery is recommended in patients with right-sided IE who are receiving appropriate antibiotic therapy for the following scenarios: Right ventricular dysfunction secondary to acute severe tricuspid regurgitation non-responsive to diuretics.479 I B Persistent vegetation with respiratory insufficiency requiring ventilatory support after recurrent pulmonary emboli.479,755 I B Large residual tricuspid vegetations (>20 mm) after recurrent septic pulmonary emboli.145,471 I C Patients with simultaneous involvement of left-heart structures.749 I C Tricuspid valve repair should be considered instead of valve replacement, when possible.479 IIa B Surgery should be considered in patients with right-sided IE who are receiving appropriate antibiotic therapy and present persistent bacteraemia/sepsis after at least 1 week of appropriate antibiotic therapy.436,755 IIa C Prophylactic placement of an epicardial pacing lead should be considered at the time of tricuspid valve surgical procedures.733 IIa C Debulking of right intra-atrial septic masses by aspiration may be considered in selected patients who are high risk for surgery.753 IIb C © ESC 2023 IE: infective endocarditis. aClass of recommendation. bLevel of evidence. 60 ESC Guidelines Downloaded from by guest on 30 August 2023 12.8. Infective endocarditis in rheumatic heart disease Infective endocarditis is a known complication of RHD,766 and acute rheumatic fever (the antecedent of RHD) may even present with con-comitant IE.767 Of the 3343 participants enrolled in The Global Rheumatic Heart Disease Registry (REMEDY),768 133 (2.4%) had a his-tory of IE at enrolment,769 and 20 (0.7%; 3.65 per 1000 patient-years) developed IE during the 27-month follow-up.770 These participants were young with a median age of 28 years (interquartile range 18–40 years), 66.2% were women, and over 30% were children. The majority of the over 40 million patients with RHD771 live in low- and middle-income countries and face socioeconomic and health-system barriers772 to adequate prevention, early diagnosis, and advanced care and, therefore, are at particular risk of IE.773 Global access to surgery for RHD and RHD-associated complications is extremely limited.774 RHD patients presenting with fever, changing or new murmurs should be investigated for IE. In studies from RHD- endemic regions, RHD is the most common underlying cardiac condition, with significant mortality and morbidity.775–784 In those affected with oral bacteria-related IE linked to RHD, oral Streptococcus spp. was the main cause of IE associated with poor oral health status.785 In RHD-endemic countries, IE in children is strongly linked to RHD,786–788 and when caus-ing HF, carries the highest case fatality rate.789 IE is associated with en-hanced risk of death among patients with RHD undergoing isolated mitral valve replacement (odds ratio 5.22, 95% confidence interval [CI], 1368–19 915; P = 0.008).790 Pregnancy is a particularly high-risk period for women with RHD, with an increased risk of developing IE.791,792 However, high-income countries or countries with emerging economies are seeing less IE linked to RHD, as the incidence rates of RHD in these regions decrease.793–796 12.9. Infective endocarditis during pregnancy Infective endocarditis in pregnancy is a rare but extremely serious con-dition with high maternal and foetal morbidity and mortality, and is es-timated to complicate ∼1 in 100 000 pregnancies.797–799 Maternal mortality approaches 18%, with most deaths relating to HF or an em-bolic event, while pre-term birth is reported at 55.7% and foetal mor-tality at 29%.800 Recurrent infective complications can occur in up to 27% of women post-partum.801 The diagnosis must be considered in pregnant women with unex-plained fever and cardiac signs (especially tachycardia), new or changing cardiac murmurs, and peripheral signs of septic emboli.802 Women with CHD, RHD,803 and structural heart disease, together with those with prosthetic heart valves and with PWID are at particular risk.800,804–807 The gravity of the condition requires the inclusion of gynaecologists, obste-tricians, and neonatologists in the Endocarditis Team in any suspected cases, and a diagnosis and treatment plan should be formulated without delay, as this is key to saving the lives of mothers and infants.799,808,809 Management can be challenging, especially when the pregnant patient warrants a cardiac operation under CPB. Although this poses a considerable risk to the foetus, urgent surgery when indicated should not be delayed.799,810 12.10. Infective endocarditis in immunocompromised patients 12.10.1. Solid organ transplant recipients The incidence of IE in recipients of solid organ transplantation (SOT) ranges between 1% and 2%.107 SOT recipients with IE are younger and have higher prevalence of co-morbidities (particularly renal and liver disease) compared with non-SOT patients with IE. Among the SOT patients with IE, the most common transplanted organ is the kidney (72%), followed by liver (17%), and pancreas (8%).811 Similar to non-SOT patients, aortic followed by mitral IE are the most common forms of IE while right-sided IE is uncommon. Interestingly, SOT pa-tients with IE more frequently have atrial or ventricular vegetations without valve involvement (mural IE).107 In-hospital and healthcare-related IE are the most frequent causes of IE in recipients of SOT and the most frequent microorganism involved is S. aureus (34%), followed by Enterococcus spp. (17%), and Streptococcus spp. (11%).107,811 Surgical valve repair/replacement is less frequently performed in SOT patients with IE as compared with non-SOT patients. Interestingly, the outcomes of IE in patients with SOT do not differ from those of non-SOT with IE.107,811 The reasons for the similar outcomes may rely on the younger age of the SOT patients, the frequent contact with the healthcare system which may lead to early diagnosis and treat-ment of IE, and the frequent involvement of infectious disease specialists in the care of hospitalized SOT patients. However, compared with SOT patients without IE, those who develop IE during the index transplant hospitalization have worse outcomes.811 The high levels of immunosup-pression probably negatively impact the IE course in these patients. Heart transplant recipients represent 10% of SOT with IE pa-tients.811 Among 57 heart transplant recipients who developed IE, the most frequent organism was S. aureus (26%), followed by A. fumiga-tus (19%), and E. faecalis (12%).105 The median time to IE presentation after heart transplant was 8 years and the mitral valve was the most fre-quently affected, followed by mural and tricuspid valve IE. All-cause mortality in this group of patients is high (45%), and fungal aetiology is associated with worse outcomes. Similar to other SOT recipients, heart transplant recipients were not frequently referred to surgery (35%).105 12.10.2. Patients with human immunodeficiency virus The advent of combined antiretroviral treatment has led to a reduction in the risk of developing acquired immune deficiency syndrome (AIDS) but people living with HIV remain a vulnerable population for IE.812 The incidence of IE in people living with HIV has decreased over the last two decades. A retrospective study from Spain has shown a reduction in the incidence of IE from 18.2 per 100 000 patient-years between 1997 and 1999 to 2.9 events per 100 000 patient-years between 2000 and 2014.813 Similarly, a registry from the United States of America reported a reduction in the incidence of IE from 148 in 2007 to 112 in 2017.141 Patients living with HIV and presenting with IE are becoming older, and have a higher percentage of substance abuse and co- morbidities.141,813 Of importance, the number of patients living with HIV who are admitted with IE have higher frequency of CHD, prior valve surgery, CIED infection, and haemodialysis.141,813 The most fre-quent microorganisms causing IE are Staphylococcus spp. (the majority of which is S. aureus), followed by Streptococcus spp., Gram-negative bacilli, and enterococci. It is important to note that over the last two decades, the frequency of CoNS as a cause of IE has decreased whereas the frequency of streptococci, Gram-negative bacilli, enterococci, and fungus has increased.813 Community-acquired IE has become the most frequent form while healthcare-associated IE rates have signifi-cantly decreased over time. The outcomes of IE in people living with HIV have improved over the years (from 23.9 to 5.5 deaths per 100 000 patient-years) and surgical ESC Guidelines 61 Downloaded from by guest on 30 August 2023 treatment should follow the same indications as in patients without HIV.813 12.10.3. Patients with neutropaenia Neutropaenia is common in patients with haematological malignan-cies and in patients receiving chemotherapy for other malignancies, but is rare in patients presenting with IE.814 Neutrophils play an im-portant role in the pathogenesis of IE by producing layers of extracel-lular traps that entrap bacteria-platelet aggregates, leading to expansion of these aggregates, vegetation growth, and the destruction of tissues.814 The diagnosis of IE can therefore be challenging in pa-tients with neutropaenia, delaying the appropriate treatment, and worsening outcomes. Series reporting the clinical characteristics and outcomes of IE in patients with neutropaenia are anecdotal.814 As in any other immunocompromised patient with IE, antibiotic and surgical treatment are the same as in patients without neutropaenia. It is important to take into consideration the side effects of some anti-biotics which may worsen the neutropaenia, such as cloxacillin and ceftaroline.815,816 12.11. Antithrombotic and anticoagulant therapy in infective endocarditis Infective endocarditis by itself is not an indication for antithrombotics or anticoagulants, and bleeding complications or stroke may in contrast justify discontinuation or interruption of such therapies. Indications for antithrombotic therapy or anticoagulants (e.g. atrial fibrillation, valve prostheses, ischaemic heart disease, prior stroke, etc.) are prevalent in the general population and, as a result, the clinician is often faced with the challenge of these therapies in patients presenting with IE, es-pecially in cases where surgery is part of the treatment course. For pa-tients with IE and stroke, thrombolytic therapy is not recommended (see Section 9.1). However, thrombectomy may be considered in se-lected cases with large vessel occlusion. The level of evidence underlying the recommendations for antithrombotic and anticoagulant therapy in IE is low and should be discussed within the Endocarditis Team. Bridging with low- molecular-weight heparin/unfractionated heparin instead of oral antic-oagulants should be considered early on in the IE course, especially for patients in whom surgery is indicated. To date, no data support initi-ation of either antithrombotics nor anticoagulants for treatment or prevention of stroke in IE. 12.12. Non-bacterial thrombotic endocarditis Non-bacterial thrombotic endocarditis (NBTE) is a rare condition with an incidence varying from 1.1% to 1.6% in patient-series from autopsy studies.818,819 Non-bacterial thrombotic endocarditis occurs in patients with a predisposing factor and/or a hypercoagulable state, such as sys-temic lupus erythematous (SLE), APLs (Libman–Sacks endocarditis), cancer (marantic endocarditis), disseminated intravascular coagulation (DIC), or various other chronic diseases (tuberculosis or autoimmune disease).820,821 Increased production of coagulation factors, of cyto-kines, and high tissue factor expression are potential mechanisms underlying NBTE in cancer patients.822 In a recent contemporary registry, 41% of NBTE patients had cancer, 33% SLE, and 36% APLs, with 21% of patients having both SLE and APLs.823 Among the patients with malignancies, the three most fre-quent cancers were lung adenocarcinoma, breast, and pancreatic can-cer. Stroke was the most frequent clinical presentation at admission (60%), while HF was observed in 21% and acute coronary syndrome in 7% of patients. Transthoracic echocardiography was able to confirm the diagnosis in 45% of patients. The mitral valve was more often af-fected (62%) than the aortic valve (24%).823 The diagnosis of NBTE remains challenging and should be suspected in patients presenting with systemic embolization and a predisposing factor (i.e. cancer, APLs, SLE). Laboratory findings of a hypercoagulable state (eg. lupus anticoagulant, anti-cardiolipin antibodies, and anti-β2-glycoprotein 1 antibodies or DIC) may be present, but are non- specific and may also be demonstrated in other IE patients with embolic events.162 Echocardiography diagnosis should attempt to differentiate non-bacterial thrombotic vegetation from IE, Lambl excrescences, or fibroelastoma, or other benign intracardiac masses/tumours.824 Libman–Sacks vegetations may present with various shapes (sessile, tubular, or coalescent), various levels of echogenicity (heterogeneously or homogeneously), could be nodular or protuberant, are generally lo-cated near the leaflet’s edge of coaptation, and frequently have exten-sions to the mid and basal portions of the leaflet. They are rarely associated with valve dysfunction and never with valve perforation, which is an important method of differentiating from bacterial IE.824 Compared with TOE, TTE has a lower sensitivity (63%), specificity (58%), negative predictive value (40%), and a moderate positive predict-ive value (78%) for the detection of NBTE.823,824 Compared with two-dimensional TOE, three-dimensional TOE provides additional in-formation and allows a better characterization of the vegetation.823 The treatment of the underlying cause (i.e. SLE or cancer) is crucial to prevent recurrent NBTE. Anticoagulant treatment should be consid-ered in all patients and should be balanced against the individual Recommendation Table 22 — Recommendations for the use of antithrombotic therapy in infective endocarditis Recommendation Classa Levelb Interruption of antiplatelet or anticoagulant therapy is recommended in the presence of major bleeding (including intracranial haemorrhage).482,483 I C In patients with intracranial haemorrhage and a mechanical valve, reinitiating unfractionated heparin should be considered as soon as possible following multidisciplinary discussion.817 IIa C Continued In the absence of stroke, replacement of oral anticoagulant therapy by unfractionated heparin under close monitoring should be considered in cases where indication for surgery is likely (e.g. S. aureus IE).451,817 IIa C Thrombolytic therapy is not recommended in patients with IE.481,491 III C © ESC 2023 IE, infective endocarditis. aClass of recommendation. bLevel of evidence. 62 ESC Guidelines Downloaded from by guest on 30 August 2023 patient’s bleeding risk.821 Patients may be anticoagulated with low-molecular-weight heparin, vitamin K antagonists, or unfractionated heparin. There are no data to support the use of direct oral anticoagulants in NBTE. In a randomized open-label multicentre study comparing rivaroxaban and warfarin in patients with thrombotic APLs, the use of rivaroxaban was associated with an increased rate of thrombo-embolic events and major bleeding.825 The role of surgery is controversial and remains to be clarified. However, surgery should be considered in patients with severe valve dysfunction or with large vegetations.823 12.13. Infective endocarditis and malignancy There are limited data on the prevalence, clinical presentation, manage-ment, and outcome of IE in patients with malignancy. In a retrospective Japanese cohort study including 170 patients, 17.6% had active malig-nancy.826 Compared with patients without malignancy, patients with malignancies were older, nosocomial IE was more frequent, and proce-dures before IE (non-dental, i.v. catheter insertion, invasive endoscopic, or genitourinary procedures) were more frequent.826 Another recent study from the EURO-ENDO registry of 3085 patients with IE found a history of malignancy in 11.6% of patients.827 Patients with a history of malignancy had a similar rate of theoretical indications for surgery, but surgery was performed less often in this group. Mortality was higher in the malignancy group with independent predictors for mortality being elevated creatinine >2 mg/dL, congestive HF, and unperformed cardiac surgery when indicated.827 In IE patients with concomitant cancer, indi-cations for valve surgery should be discussed within the Endocarditis Team, including a cardio-oncologist and the oncologist in charge of the patient, in order to take into account the risks and benefits of sur-gery and cancer prognosis. 13. Patient-centred care and shared decision-making in infective endocarditis 13.1. What is patient-centred care and shared decision-making and why is it important? Patient-centred care encourages involvement and collaboration be-tween patients, families, and healthcare providers during all stages of diagnosis, treatment, and recovery.828–831 Core elements of patient- centred care include: involvement of family and caregivers, respect for patients’ preferences and values, care co-ordination and continuity, information and education, as well as physical comfort and emotional support (Figure 14).828–830 Shared decision-making involves a bidirectional process where pa-tients, family, and healthcare providers share information and discuss care options in the context of the patients’ preferences, beliefs and va-lues, and the best available evidence ensuring that the patient under-stands the risks,832,833 benefits, and possible consequences of the different options.834–836 The majority of patients prefer sharing deci-sions about their own health, if they are sufficiently informed and pre-pared.837,838 Patient-centred care and shared decision-making have been shown to contribute to improved concordance between care providers and patients on treatment plans, as well as increased patient satisfaction, quality of life, and health outcomes.830,839–843 13.2. Patient-centred care and shared decision-making in infective endocarditis The severity of IE, the complex and comprehensive diagnostics and treatment, as well as the long illness trajectory, put special emphasis on patient-centred care and shared decision-making in IE (Figure 14). Quality of life appears to be impaired in IE survivors, with a significant number of patients developing symptoms of anxiety, depression, or even post-traumatic stress disorder following IE treatment.604,844 The time of diagnosis is often emotionally distressing to the patient and family, as they face a life-threatening condition and lengthy treatment.845 During the diagnostic and active treatment phase, healthcare provi-ders should make every effort to minimize patient discomfort (e.g. re-lated to symptoms and diagnostic procedures), and alleviate distress in both patient and family by providing support and comprehensive and timely information about the patient’s condition, therapeutic options, and prognosis. Independent of the therapeutic strategy (i.e. surgical vs. conservative), patient-centred care is key to ensure a good physical and mental outcome during a lengthy treatment and hospitalization as-sociated with IE. Maintaining continuity of care, when possible, by min-imizing the number of providers the patient encounters and minimizing transfers between and within units, is all part of a patient-centred care approach. Allowing family visits at any time and providing the opportun-ity to uphold personal integrity and autonomy are important issues for patients. National patient organizations and associations may be an op-tion for offering information and support to patients and their families. The role of outpatient antimicrobial treatment options in IE should be discussed using a shared decision-making approach, involving the pa-tient’s partner or family if possible. The outpatient treatment should be in concordance with the patient’s and family’s preferences, also consid-ering transportation and self-care abilities. To monitor possible compli-cations, it is important to inform and educate patients and caregivers about the signs and symptoms of disease progression or recurrence. The early period after discharge can be challenging for patients and their families, and patients report slow physical and mental recovery after IE, of-ten extending longer than anticipated.603,604,846,847 Patient-centred care should therefore extend further than the clinical treatment at the hospital to ensure a good outcome after discharge. Though little research has ex-plored patients’ and families’ needs for recovery and rehabilitation follow-ing IE, patients with heart disease report experiencing new and continuous challenges and a lack of knowledge and understanding after discharge, which should be addressed to optimize recovery.848 It is recommended that a recovery plan is developed in collaboration with the patient and their caregivers and that the plan is reviewed and potentially adjusted following a short period after discharge.849 Physical exercise should be recommended based on an individual as-sessment of functional capacity (guided by physicians and physiothera-pists), and patient education and psychosocial support should address the main problems and concerns patients and families have. Importantly, patient education should also include information about the risk of recurrence and preventive measures described in Sections 3 and 11. Special consideration should be taken for patients with no close relatives. Self-support groups or mentors may be introduced to patients without support networks. Also, follow-up by telephone from the ward staff, until full recovery has been reached, may be an option. A palliative approach aims to improve the quality of life of patients and their families who are facing problems associated with life- threatening illness, which is relevant for many patients with IE. This ap-proach includes a holistic, needs-based perspective with the aims of ESC Guidelines 63 Downloaded from by guest on 30 August 2023 assessing and improving symptom management, communication, advanced care planning, as well as psychosocial and spiritual needs.850 14. Sex differences Female sex is less common in patients diagnosed with IE, being present in approximately one-third of cases; a finding that has been demon-strated in multiple IE patient subpopulations and across different re-gions.5,59,723,851,852 The reason why female sex is observed less frequently in IE is unknown and deserves further investigation. Possible reasons include underdiagnosis of IE in women, referral bias in published studies, intrinsic protective mechanisms against IE in women, and decreased incidence of risk factors for IE in women (e.g. bicuspid aortic valve disease, previous heart valve replacement surgery), among others. A recent nationwide population study of individual patient-level linkage data of 7513 patients hospitalized for IE in Scotland, however, demonstrated roughly equal proportions of male and female patients throughout the 25-year study period.27 Female patients with IE have been demonstrated to have a higher preva-lence of several risk factors for IE in comparison to their male counterparts including older age, mitral valve involvement, S. aureus infection, neuro-logical symptoms, and haemodialysis.853–856 However, men have a higher prevalence of other important risk factors including previous prosthetic valve replacement, periannular complications, CAD, and liver cirrhosis.855 Some studies have demonstrated higher mortality rates for female patients with IE,856 while others have demonstrated no differences in Family and caregiver involvement Patients’ preferences and values Care coordination and integration Information and education Physical comfort and emotional support Continuity and transition P a ti e n t-c e n tr e d c a r e S h a r e d d e ci si o n -m a k i n g Figure 14 Concept of patient-centred care in infective endocarditis. 64 ESC Guidelines Downloaded from by guest on 30 August 2023 early and 1-year mortality rates between males and females.853,855,857 The abovementioned population study from Scotland showed lower mortality rates for women during the study period.27 Although surgery has been demonstrated to be protective against mortality in several clinical scenarios (see Section 8), surgery is per-formed less frequently in female patients with IE.855,856 In a study using the National Inpatient Sample of 81 942 patients hospitalized for IE over an 11-year period, women were 43% less likely to undergo valve replacement surgery, a significant difference that remained after adjust-ing for confounding factors.855 The reason for decreased surgery in fe-male IE patients is unknown and requires further investigation. Female sex has also been identified as an independent risk factor for mortality in prediction models for patients with IE undergoing sur-gery.416 However, a single-centre study suggested that worse observed surgical outcomes in female patients with IE was related to their in-creased risk factors and severity of presentation, rather than gender per se.854 In addition, a large multicentre registry of 4300 patients undergoing surgery for IE failed to identify female gender as an inde-pendent predictor of mortality.852 15. Key messages Prevention: • Populations at high risk of IE include patients with previous IE, pa-tients with surgical or transcatheter prosthetic valves or post-cardiac valve repair, and patients with untreated CHD and surgically cor-rected CHD. • Prevention of IE comprise hygienic measures (including oral hygiene) for all individuals and antibiotic prophylaxis for patients at high risk of IE undergoing oro-dental procedures. The Endocarditis Team: • The diagnosis and management of patients with IE should be dis-cussed with the Endocarditis Team, which includes healthcare pro-fessionals with the expertise to diagnose and treat IE and its complications. • Uncomplicated IE can be managed in a Referring Centre that remains in early and regular communication with the Endocarditis Team of the Heart Valve Centre. • Patients with complicated IE should be treated in the Heart Valve Centre, which must offer a wide range of ancillary specialty support including onsite cardiac surgery expertise. Diagnosis: • The diagnosis of IE is based on major criteria, which include positive blood cultures and valvular and perivalvular/periprosthetic anatomic and metabolic lesions detected on imaging, and on minor criteria which have been updated to include frequent embolic vascular dis-semination including asymptomatic lesions detected by imaging only. • Clear diagnostic algorithms have been established to diagnose NVE, PVE, and right-sided IE. Antimicrobial therapy – principles and methods: • Successful treatment of IE relies on microbial eradication by anti-microbial drugs. Surgery contributes by removing infected material and draining abscesses. • Antibiotic treatment of PVE should last longer (≥6 weeks) than that of NVE (2–6 weeks). • In both NVE and PVE, the duration of treatment is based on the first day of effective antibiotic therapy (negative blood culture in the case of initial positive blood culture), not on the day of surgery. • The initial choice of empirical treatment depends on the use of pre-vious antibiotic therapy, whether IE is NVE or PVE (and if so, when surgery was performed [early vs. late PVE]), the place where the in-fection took place (community, nosocomial, or non-nosocomial healthcare-associated IE), and knowledge of the local epidemiology. • The antibiotic treatment of IE has two phases. The first phase con-sists of 2 weeks of in-hospital i.v. treatment. In this initial phase, car-diac surgery should be performed if indicated, infected foreign bodies should be removed, and cardiac as well as extracardiac abscesses should be drained. In the second phase, in selected patients, the anti-biotic treatment can be completed within an outpatient parenteral or oral antibiotic programme for up to 6 weeks. • Aminoglycosides are not recommended in staphylococcal NVE be-cause their clinical benefits have not been demonstrated. In IE caused by other microorganisms in which aminoglycosides are indicated, they should be prescribed in a single daily dose to reduce nephrotoxicity. • Rifampin should be used only in IE involving foreign material, such as PVE, after 3–5 days of effective antibiotic therapy. • When daptomycin is indicated, it must be given at high doses (10 mg/ kg once daily) and combined with a second antibiotic (beta-lactams or fosfomycin in beta-lactam allergic patients) to increase activity and avoid the development of resistance. • OPAT can only start when a TOE shows absence of local progression and complications (e.g. severe valvular dysfunction). • In the OPAT programme, patients continue with the same antibiotics administered in the acute phase, if possible. Indications for surgery and management of main infective endocarditis complications: • There are three main reasons to undergo surgery in the setting of acute IE: HF, uncontrolled infection, and prevention of septic embolization. • While surgery during the acute phase of IE is usually performed on an urgent basis (i.e. the patient undergoes surgery within 3–5 days), some cases require emergency surgery (i.e. within 24 h), irrespective of the pre-operative duration of antibiotic treatment. Other complications of infective endocarditis: • Stroke may be the first presenting symptom in patients with IE. Unexplained fever accompanying a stroke in a patient with risk fac-tors for IE should trigger the suspicion of IE. • Epicardial pacemaker implantation should be considered in patients undergoing surgery for IE with complete AVB and other risk factors. • MRI or PET/CT are indicated in patients with suspected spondylodis-citis and vertebral osteomyelitis complicating IE. Surgical therapy principles and methods: • The indication to perform invasive coronary angiography or CTA prior to surgery for IE should be based on the presence of cardiovas-cular risk factors in patients with aortic valve IE. • Surgery should not be delayed in patients with non-haemorrhagic stroke and clear indications for surgery. In patients with significant ESC Guidelines 65 Downloaded from by guest on 30 August 2023 pre-operative haemorrhagic stroke, a delay in operative management (≥4 weeks) is generally recommended. • The decision of not offering surgery when indicated should be made in the setting of an Endocarditis Team. Outcome after discharge – follow-up and long-term prognosis: • Relapse is a repeat episode of IE caused by the same microorganism and represents a failure of treatment, and mandates a search for a persistent focus of infection and an evaluation towards surgical therapy. • Reinfection is an infection caused by a different microorganism, usu-ally more than 6 months after the initial episode. • Once antibiotic treatment has been completed, blood cultures should be performed. • Patients discharged after the first episode of IE should remain under close surveillance for potential long-term complications. Management of specific situations: • Antibiotic prophylaxis to prevent CIED-related IE before dental and other non-cardiac interventions is not warranted. • A single positive blood culture with no other clinical evidence of in-fection should not result in removal of the CIED. Complete CIED re-moval is recommended for all patients with confirmed infection of the lead(s). • The indication for CIED reimplantation should always be re- evaluated and no part of the removed system should be reimplanted. In pacemaker-dependent patients, an active-fixation lead may be in-troduced and connected to an external pacemaker for up to 6 weeks. • Surgical treatment of right-sided IE is indicated in patients with per-sistent bacteraemia, right ventricular dysfunction, recurrent septic pulmonary embolism and respiratory compromise, and involvement of left-sided structures. • Multidisciplinary care of CHD patients with IE, from diagnosis to treatment, should be provided in specialized CHD centres with ex-pertise in CHD cardiac imaging, CHD surgery, and intensive care. Patient-centred care and shared decision-making in infective endocarditis: • In patients with IE, shared decision-making enables the integration of patients’ preferences, values, and priorities to achieve a good treat-ment decision. • In patients with IE and without support networks or severely impacted by social determinants, a recovery plan developed in collaboration with the patient should be established, highlighting the information about the risk of recurrence and preventive measures. Sex differences: • Female sex is less common in patients diagnosed with IE, being pre-sent in approximately one-third of cases. 16. Gaps in evidence • The majority of the recommendations with a level of evidence B are based on observational studies rather than single RCTs or meta-analyses from RCTs. Prevention: • In the intermediate or unknown risk condition groups, there is no evi-dence to recommend antibiotic prophylaxis. • There is currently no evidence to support the use of antibiotic prophy-laxis after the implantation of a left atrial appendage occlusion device. Diagnosis: • More data on the accuracy of diagnosis of culture-negative IE using molecular biology techniques, or the determination of bacterial/fun-gal cell-free DNA in blood samples, is required. • Standardization of the methodology to assess the size of the vegeta-tions has not been established. • More data on the diagnostic performance of intracardiac echocardi-ography in PVE are needed. • The role of [18F]FDG-PET/CT(A) in NVE needs to be established. • Routine use of imaging tests to screen the presence of embolic events, especially brain imaging, is not well established. • In fungal endocarditis, the role of molecular and biochemical indica-tors to establish the diagnosis is not well studied. Antimicrobial therapy – principles and methods: • Clinical trials are needed to assess the efficacy and safety of recom-mended antimicrobial treatment regimens and new combinations or antimicrobials. Many recommendations come from clinical trials for bacteraemia and not for IE. • Effective antibiotic treatment in patients with highly penicillin- resistant oral streptococci should be investigated. • Randomized data to establish the best medical strategy in staphylo-coccal IE are required. • Effective antibiotic treatments for patients with HLAR E. faecalis IE and hypersensitivity to beta-lactams need further research. • Effective treatments for vancomycin-resistant enterococcal IE need further research. • Randomized head-to-head comparisons of different antibiotics to better judge efficacy and toxicity (e.g. for aminoglycosides) are needed. • The duration of antibiotic treatment has been established empirically and no randomized data have been published. • The efficacy of combined antifungal therapy has not been studied. • The empirical use of an aminoglycoside-sparing empirical combin-ation regimen has not been extensively studied. • More data on implementation of oral treatment in large studies are needed. Indications for surgery and management of main infective endocarditis complications: • The indication of surgical treatment in patients with IE rely mainly on expert opinion based on observational studies. • RCTs are required to establish the indication and timing of surgery in patients with: • Increased surgical risk. • Large vegetations but without other indications for surgery. • Cerebral emboli or bleeding. • Patients with uncontrolled infection. • More data on the need and timing of coronary angiogram before endocarditis surgery. • There is a lack of information on timing and sequence of interven-tions in patients with multiple septic sources. • More data are needed on the efficacy and safety of vegetation extrac-tion systems in right-sided IE. 66 ESC Guidelines Downloaded from by guest on 30 August 2023 Other complications of infective endocarditis: • There is limited information on the safety and efficacy of mechanical thrombectomy in IE-related embolic strokes. • There are no prospective data on the timing and safety of splenec-tomy for splenic abscess, complicating IE in relation to surgical valve treatment. Surgical therapy principles and methods: • There is a significant need for scores to predict futility of surgical management in very high-risk patients. • There is a lack of data on the most appropriate anticoagulation regi-men in patients with PVE complicated by haemorrhagic stroke. Outcome after discharge: follow-up and long-term prognosis: • Clinical trials are required to assess the efficacy of rehabilitation, including optimal timing, duration, methods, and components. • Data on patient-reported outcomes during short- and long-term follow-up are needed. Management of specific situations: • Additional data on the incidence, characteristics, and outcomes of IE in patients treated with transcatheter valve therapies or left atrial ap-pendage occluders are needed. • There is an unmet clinical question on the efficacy and safety of sur-gical treatment of IE in patients previously treated with transcatheter valve therapies. • Randomized data on the timing of CIED reimplantation after device removal after CIED infection are needed. • There is a lack of evidence on whether or not CIED removal should be routinely performed in patients with left-sided IE. • Randomized data on surgery in right-sided IE are required. Patient-centred care and shared decision-making in infective endocarditis: • As no disease-specific evidence exists, data on patient-centred care and shared decision-making in IE is needed. • Data on how patient-centred care and shared decision-making in pa-tients with social and mental health vulnerabilities can improve their outcomes are lacking. • Data on the effect of patient-centred care and shared decision- making interventions are required to implement effective strategies. Sex differences: • Further data are required to determine why IE is less frequently ob-served, and why the outcomes are worse, in female patients. • The reasons for lower referral to surgery in female patients with IE as compared with male patients need to be determined and addressed. Table 14 ‘What to do’ and ‘What not to do’ Recommendations Classa Levelb Recommendations for antibiotic prophylaxis in patients with cardiovascular diseases undergoing oro-dental procedures at increased risk of infective endocarditis Antibiotic prophylaxis is recommended in patients with previous IE. I B General prevention measures are recommended in individuals at high and intermediate risk of IE. I C Antibiotic prophylaxis is recommended in patients with surgically implanted prosthetic valves and with any material used for surgical cardiac valve repair. I C Antibiotic prophylaxis is recommended in patients with transcatheter implanted aortic and pulmonary valvular prostheses. I C Antibiotic prophylaxis is recommended in patients with untreated cyanotic CHD, and patients treated with surgery or transcatheter procedures with post-operative palliative shunts, conduits, or other prostheses. After surgical repair, in the absence of residual defects or valve prostheses, antibiotic prophylaxis is recommended only for the first 6 months after the procedure. I C Antibiotic prophylaxis is recommended in patients with ventricular assist devices. I C Antibiotic prophylaxis is not recommended in other patients at low risk of IE. III C Recommendations for infective endocarditis prevention in high-risk patients Antibiotic prophylaxis is recommended in dental extractions, oral surgery procedures, and procedures requiring manipulation of the gingival or periapical region of the teeth. I B Recommendations for infective endocarditis prevention in cardiac procedures Pre-operative screening for nasal carriage of S. aureus is recommended before elective cardiac surgery or transcatheter valve implantation to treat carriers. I A Peri-operative antibiotic prophylaxis is recommended before placement of a CIED. I A Optimal pre-procedural aseptic measures of the site of implantation are recommended to prevent CIED infections. I B Periprocedural antibiotic prophylaxis is recommended in patients undergoing surgical or transcatheter implantation of a prosthetic valve, intravascular prosthetic, or other foreign material. I B Continued 17. ‘What to do’ and ‘What not to do’ messages from the Guidelines ESC Guidelines 67 Downloaded from by guest on 30 August 2023 Surgical standard aseptic measures are recommended during the insertion and manipulation of catheters in the catheterization laboratory environment. I C Systematic skin or nasal decolonization without screening for S. aureus is not recommended. III C Recommendations for the Endocarditis Team Diagnosis and management of patients with complicated IE are recommended to be performed at an early stage in a Heart Valve Centre, with immediate surgical facilities and an ‘Endocarditis Team’ to improve the outcomes. I B For patients with uncomplicated IE managed in a Referring Centre, early and regular communication between the local and the Heart Valve Centre endocarditis teams is recommended to improve the outcomes of the patients. I B Recommendations for the role of echocardiography in infective endocarditis A. Diagnosis TTE is recommended as the first-line imaging modality in suspected IE. I B TOE is recommended in all patients with clinical suspicion of IE and a negative or non-diagnostic TTE. I B TOE is recommended in patients with clinical suspicion of IE, when a prosthetic heart valve or an intracardiac device is present. I B Repeating TTE and/or TOE within 5–7 days is recommended in cases of initially negative or inconclusive examination when clinical suspicion of IE remains high. I C TOE is recommended in patients with suspected IE, even in cases with positive TTE, except in isolated right-sided native valve IE with good quality TTE examination and unequivocal echocardiographic findings. I C B. Follow-up under medical therapy Repeating TTE and/or TOE is recommended as soon as a new complication of IE is suspected (new murmur, embolism, persisting fever and bacteraemia, HF, abscess, AVB). I B TOE is recommended when the patient is stable before switching from intravenous to oral antibiotic therapy. I B C. Intra-operative echocardiography Intra-operative echocardiography is recommended in all cases of IE requiring surgery. I C D. Following completion of therapy TTE and/or TOE are recommended at completion of antibiotic therapy for evaluation of cardiac and valve morphology and function in patients with IE who did not undergo heart valve surgery. I C Recommendations for the role of computed tomography, nuclear imaging, and magnetic resonance in infective endocarditis Cardiac CTA is recommended in patients with possible NVE to detect valvular lesions and confirm the diagnosis of IE. I B [18F]FDG-PET/CT(A) and cardiac CTA are recommended in possible PVE to detect valvular lesions and confirm the diagnosis of IE. I B Cardiac CTA is recommended in NVE and PVE to diagnose paravalvular or periprosthetic complications if echocardiography is inconclusive. I B Brain and whole-body imaging (CT, [18F]FDG-PET/CT, and/or MRI) are recommended in symptomatic patients with NVE and PVE to detect peripheral lesions or add minor diagnostic criteria. I B Recommendations for antibiotic treatment of infective endocarditis due to oral streptococci and Streptococcus gallolyticus group Penicillin-susceptible oral streptococci and Streptococcus gallolyticus group Standard treatment: 4-week duration in NVE or 6-week duration in PVE In patients with IE due to oral streptococci and S. gallolyticus group, penicillin G, amoxicillin, or ceftriaxone are recommended for 4 (in NVE) or 6 weeks (in PVE), using the following doses: I B Adult antibiotic dosage and route Penicillin G 12–18 million U/day i.v. either in 4–6 doses or continuously Amoxicillin 100–200 mg/kg/day i.v. in 4–6 doses Ceftriaxone 2 g/day i.v. in 1 dose Paediatric antibiotic dosage and route Penicillin G 200 000 U/kg/day i.v. in 4–6 divided doses Amoxicillin 100–200 mg/kg/day i.v. in 4–6 doses Ceftriaxone 100 mg/kg/day i.v. in 1 dose Standard treatment: 2-week duration (not applicable to PVE) 2-week treatment with penicillin G, amoxicillin, ceftriaxone combined with gentamicin is recommended only for the treatment of non-complicated NVE due to oral streptococci and S. gallolyticus in patients with normal renal function using the following doses: I B Adult antibiotic dosage and route Penicillin G 12–18 million U/day i.v. either in 4–6 doses or continuously Amoxicillin 100–200 mg/kg/day i.v. in 4–6 doses Continued 68 ESC Guidelines Downloaded from by guest on 30 August 2023 Ceftriaxone 2 g/day i.v. in 1 dose Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dose Paediatric antibiotic dosage and route Penicillin G 12–18 million U/day i.v. either in 4–6 doses or continuously Amoxicillin 100–200 mg/kg/day i.v. in 4–6 doses Ceftriaxone 100 mg/kg i.v. in 1 dose Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dose or 3 equally divided doses Allergy to beta-lactams In patients allergic to beta-lactams and with IE due to oral streptococci and S. gallolyticus, vancomycin for 4 weeks in NVE or for 6 weeks in PVE is recommended using the following doses: I C Adult antibiotic dosage and route Vancomycin 30 mg/kg/day i.v. in 2 doses Paediatric antibiotic dosage and route Vancomycin 30 mg/kg/day i.v. in 2 or 3 equally divided doses Oral streptococci and Streptococcus gallolyticus group susceptible, increased exposure or resistant to penicillin In patients with NVE due to oral streptococci and S. gallolyticus, penicillin G, amoxicillin, or ceftriaxone for 4 weeks in combination with gentamicin for 2 weeks is recommended using the following doses: I B Adult antibiotic dosage and route Penicillin G 24 million U/day i.v. either in 4–6 doses or continuously Amoxicillin 2 g/day i.v. in 6 doses Ceftriaxone 2 g/day i.v. in 1 dose Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dose In patients with PVE due to oral streptococci and S. gallolyticus, penicillin G, amoxicillin, or ceftriaxone for 6 weeks combined with gentamicin for 2 weeks is recommended using the following doses: I B Adult antibiotic dosage and route Penicillin G 24 million U/day i.v. either in 4–6 doses or continuously Amoxicillin 2 g/day i.v. in 6 doses Ceftriaxone 2 g/day i.v. in 1 dose Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dose Allergy to beta-lactams In patients with NVE due to oral streptococci and S. gallolyticus and who are allergic to beta-lactams, vancomycin for 4 weeks is recommended using the following doses: I C Adult antibiotic dosage and route Vancomycin 30 mg/kg/day i.v. in 2 doses Paediatric antibiotic dosage and route Vancomycin 30 mg/kg/day i.v. in 2 doses In patients with PVE due to oral streptococci and S. gallolyticus and who are allergic to beta-lactams, vancomycin for 6 weeks combined with gentamicin for 2 weeks is recommended using the following doses: I C Adult antibiotic dosage and route Vancomycin 30 mg/kg/day i.v. in 2 doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dose Paediatric antibiotic dosage and route Vancomycin 30 mg/kg/day i.v. in 2 doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dose Recommendations for antibiotic treatment of infective endocarditis due to Staphylococcus spp. IE caused by methicillin-susceptible staphylococci In patients with NVE due to methicillin-susceptible staphylococci, (flu)cloxacillin or cefazolin is recommended for 4–6 weeks using the following doses: I B Adult antibiotic dosage and route (Flu)cloxacillin 12 g/day i.v. in 4–6 doses Cefazolin 6 g/day i.v. in 3 doses Paediatric antibiotic dosage and route (Flu)cloxacillin 200–300 mg/kg/day i.v. in 4–6 equally divided doses Cefazolin 6 g/day i.v. in 3 doses Continued ESC Guidelines 69 Downloaded from by guest on 30 August 2023 In patients with PVE due to methicillin-susceptible staphylococci, (flu)cloxacillin or cefazolin with rifampin for at least 6 weeks and gentamicin for 2 weeks is recommended using the following doses: I B Adult antibiotic dosage and route (Flu)cloxacillin 12 g/day i.v. in 4–6 doses Cefazolin 6 g/day i.v. in 3 doses Rifampin 900 mg/day i.v. or orally in 3 equally divided doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses Paediatric antibiotic dosage and route (Flu)cloxacillin 200–300 mg/kg/day i.v. in 4–6 equally divided doses Cefazolin 6 g/day i.v. in 3 doses Rifampin 20 mg/kg/day i.v. or orally in 3 equally divided doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses Allergy to beta-lactams In patients with NVE due to methicillin-susceptible staphylococci who are allergic to penicillin, cefazolin for 4–6 weeks is recommended using the following doses: I B Adult antibiotic dosage and route Cefazolin 6 g/day i.v. in 3 doses Paediatric antibiotic dosage and route Cefazolin 6 g/day i.v. in 3 doses In patients with PVE due to methicillin-susceptible staphylococci who are allergic to penicillin, cefazolin combined with rifampin for at least 6 weeks and gentamicin for 2 weeks is recommended using the following doses: I B Adult antibiotic dosage and route Cefazolin 6 g/day i.v. in 3 doses Rifampin 900 mg/day i.v. or orally in 3 equally divided doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses Paediatric antibiotic dosage and route Cefazolin 6 g/day i.v. in 3 doses Rifampin 20 mg/kg/day i.v. or orally in 3 equally divided doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses IE caused by methicillin-resistant staphylococci In patients with NVE due to methicillin-resistant staphylococci, vancomycin is recommended for 4–6 weeks using the following doses: I B Adult antibiotic dosage and route Vancomycin 30–60 mg/kg/day i.v. in 2–3 doses Paediatric antibiotic dosage and route Vancomycin 30 mg/kg/day i.v. in 2–3 equally divided doses In patients with PVE due to methicillin-resistant staphylococci, vancomycin with rifampin for at least 6 weeks and gentamicin for 2 weeks is recommended using the following doses: I B Adult antibiotic dosage and route Vancomycin 30–60 mg/kg/day i.v. in 2–3 doses Rifampin 900–1200 mg/day i.v. or orally in 2 or 3 divided doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses Paediatric antibiotic dosage and route Vancomycin 30 mg/kg/day i.v. in 2–3 equally divided doses Rifampin 20 mg/kg/day i.v. or orally in 2 or 3 divided doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 (preferred) or 2 doses Recommendations for antibiotic treatment of infective endocarditis due to Enterococcus spp. Beta-lactam and gentamicin-susceptible strains In patients with NVE due to non-HLAR Enterococcus spp., the combination of ampicillin or amoxicillin with ceftriaxone for 6 weeks or with gentamicin for 2 weeks is recommended using the following doses: I B Adult antibiotic dosage and route Amoxicillin 200 mg/kg/day i.v. in 4–6 doses Ampicillin 12 g/day i.v. in 4–6 doses Continued 70 ESC Guidelines Downloaded from by guest on 30 August 2023 Ceftriaxone 4 g/day i.v. in 2 doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dose Paediatric antibiotic dosage and route Ampicillin 300 mg/kg/day i.v. in 4–6 equally divided doses Ceftriaxone 100 mg/kg i.v. in 2 doses Gentamicin 3 mg/kg/day i.v. or i.m. in 3 equally divided doses In patients with PVE and patients with complicated NVE or >3 months of symptoms due to non-HLAR Enterococcus spp., the combination of ampicillin or amoxicillin with ceftriaxone for 6 weeks or with gentamicin for 2 weeks is recommended using the following doses: I B Adult antibiotic dosage and route Amoxicillin 200 mg/kg/day i.v. in 4–6 doses Ampicillin 12 g/day i.v. in 4–6 doses Ceftriaxone 4 g/day i.v. in 2 doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dose Paediatric antibiotic dosage and route Ampicillin 300 mg/kg/day i.v. in 4–6 equally divided doses Amoxicillin 100–200 mg/kg/day i.v. in 4–6 doses Ceftriaxone 100 mg/kg/day i.v. in 2 doses Gentamicin 3 mg/kg/day i.v. or i.m. in 3 equally divided doses High-level aminoglycoside resistance In patients with NVE or PVE due to HLAR Enterococcus spp., the combination of ampicillin or amoxicillin and ceftriaxone for 6 weeks is recommended using the following doses: I B Adult antibiotic dosage and route Ampicillin 12 g/day i.v. in 4–6 doses Amoxicillin 200 mg/kg/day i.v. in 4–6 doses Ceftriaxone 4 g/day i.v. or i.m. in 2 doses Paediatric antibiotic dosage and route Ampicillin 300 mg/kg/day i.v. in 4–6 equally divided doses Amoxicillin 100–200 mg/kg/day i.v. in 4–6 doses Ceftriaxone 100 mg/kg i.v. or i.m. in 2 doses Beta-lactam-resistant Enterococcus spp. (E. faecium) In patients with IE due to beta-lactam-resistant Enterococcus spp. (E. faecium), vancomycin for 6 weeks combined with gentamicin for 2 weeks is recommended using the following doses: I C Adult antibiotic dosage and route Vancomycin 30 mg/kg/day i.v. in 2 doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dose Paediatric antibiotic dosage and route Vancomycin 30 mg/kg/day i.v. in 2–3 equally divided doses Gentamicin 3 mg/kg/day i.v. or i.m. in 1 dose Vancomycin-resistant Enterococcus spp. In patients with IE due to vancomycin-resistant Enterococcus spp., daptomycin combined with beta-lactams (ampicillin, ertapenem, or ceftaroline) or fosfomycin is recommended using the following doses: I C Adult antibiotic dosage and route Daptomycin 10–12 mg/kg/day i.v. in 1 dose Ampicillin 300 mg/kg/day i.v. in 4–6 equally divided doses Fosfomycin 12 g/day i.v. in 4 doses Ceftaroline 1800 mg/day i.v. in 3 doses Ertapenem 2 g/day i.v. or i.m. in 1 dose Paediatric antibiotic dosage and route Daptomycin 10–12 mg/kg/day i.v. in 1 dose (age-adjusted) Ampicillin 300 mg/kg/day i.v. in 4–6 equally divided doses Fosfomycin 2–3 g/day i.v. in 1 dose Ceftaroline 24–36 mg/kg/day in 3 doses Ertapenem 1 g/day i.v. or i.m. in 1 dose (if younger than 12 years, 15 mg/kg/dose [to a maximum of 500 mg] twice daily) Continued ESC Guidelines 71 Downloaded from by guest on 30 August 2023 Recommendations for outpatient antibiotic treatment of infective endocarditis Outpatient parenteral antibiotic treatment is not recommended in patients with IE caused by highly difficult-to-treat microorganisms, liver cirrhosis (Child–Pugh B or C), severe cerebral nervous system emboli, untreated large extracardiac abscesses, heart valve complications, or other severe conditions requiring surgery, severe post-surgical complications, and PWID-related IE. III C Recommendations for the main indications of surgery in infective endocarditis (native valve endocarditis and prosthetic valve endocarditis) (i) Heart failure Emergency surgery is recommended in aortic or mitral NVE or PVE with severe acute regurgitation, obstruction, or fistula causing refractory pulmonary oedema or cardiogenic shock. I B Urgent surgery is recommended in aortic or mitral NVE or PVE with severe acute regurgitation or obstruction causing symptoms of HF or echocardiographic signs of poor haemodynamic tolerance. I B (ii) Uncontrolled infection Urgent surgery is recommended in locally uncontrolled infection (abscess, false aneurysm, fistula, enlarging vegetation, prosthetic dehiscence, new AVB). I B Urgent or non-urgent surgery is recommended in IE caused by fungi or multiresistant organisms according to the haemodynamic condition of the patient. I C (iii) Prevention of embolism Urgent surgery is recommended in aortic or mitral NVE or PVE with persistent vegetations ≥10 mm after one or more embolic episodes despite appropriate antibiotic therapy. I B Urgent surgery is recommended in IE with vegetation ≥10 mm and other indications for surgery. I C Recommendations for the treatment of neurological complications of infective endocarditis Brain CT or MRA is recommended in patients with IE and suspected infective cerebral aneurysms. I B Neurosurgery or endovascular therapy is recommended for large aneurysms, those with continuous growth despite optimal antibiotic therapy, and ruptured intracranial infective cerebral aneurysms. I C Thrombolytic therapy is not recommended in embolic stroke due to IE. III C Recommendations for patients with musculoskeletal manifestations of infective endocarditis MRI or PET/CT is recommended in patients with suspected spondylodiscitis and vertebral osteomyelitis complicating IE. I C TTE/TOE is recommended to rule out IE in patients with spondylodiscitis and/or septic arthritis with positive blood cultures for typical IE microorganisms. I C Recommendations for pre-operative coronary anatomy assessment in patients requiring surgery for infective endocarditis In haemodynamically stable patients with aortic valve vegetations who require cardiac surgery and are high risk of CAD, a high-resolution multislice coronary CTA is recommended. I B Invasive coronary angiography is recommended in patients requiring heart surgery who are high risk of CAD, in the absence of aortic valve vegetations. I C Indications and timing of cardiac surgery after neurological complications in active infective endocarditis After a transient ischaemic attack, cardiac surgery, if indicated, is recommended without delay. I B After a stroke, surgery is recommended without any delay in the presence of HF, uncontrolled infection, abscess, or persistent high embolic risk, as long as coma is absent and the presence of cerebral haemorrhage has been excluded by cranial CT or MRI. I B Recommendations for post-discharge follow-up Patient education on the risk of recurrence and preventive measures, with emphasis on dental health, and based on the individual risk profile is recommended during follow-up. I C Addiction treatment for patients following PWID-related IE is recommended. I C Recommendations for prosthetic valve endocarditis Surgery is recommended for early PVE (within 6 months of valve surgery) with new valve replacement and complete debridement. I C Recommendations for cardiovascular implanted electronic device-related infective endocarditis Antibiotic prophylaxis covering S. aureus is recommended for CIED implantation. I A TTE and TOE are both recommended in cases of suspected CIED-related IE to identify vegetations. I B Complete system extraction without delay is recommended in patients with definite CIED-related IE under initial empirical antibiotic therapy. I B Continued 72 ESC Guidelines Downloaded from by guest on 30 August 2023 18. Supplementary data Supplementary data are available at European Heart Journal online. 19. Data availability No new data were generated or analysed in support of this research. 20. Author information Author/Task Force Member Affiliations: Nina Ajmone Marsan, Leiden University Medical Center, Leiden, Netherlands; Suzanne de Waha, Heart Center Leipzig at University of Leipzig, Leipzig, Germany; Nikolaos Bonaros, Medical University of Innsbruck, Innsbruck, Austria and University Hospital Innsbruck, Innsbruck, Austria; Margarita Brida, Medical Faculty University of Rijeka, Rijeka, Croatia and Adult Congenital Heart Centre and National Centre for Pulmonary Hypertension, Royal Brompton & Harefield Hospitals, London, United Kingdom; Haran Burri, University Hospital of Geneva, Geneva, Switzerland; Stefano Caselli, Hirslanden Klinik im Park, Zurich, Switzerland and University Heart Center, University Hospital Zurich, Zurich, Switzerland; Torsten Doenst, Friedrich-Schiller-University Jena, University Hospital, Jena, Germany; Stephane Ederhy, Hopital Saint-Antoine, Paris, France and Unité de cardio-oncologie, UNICO-GRECO, Hopital Saint-Antoine, Paris, France and Groupe de recherche clinique en cardio-oncologie, GRC n°27, Hopital Saint-Antoine, Paris, France; Paola Anna Erba, University of Milan Bicocca, Milan, Italy and Nuclear Medicine Department, ASST – Ospedale Papa Giovanni XXIII, Bergamo, Italy and Medical Imaging Center, Department of Nuclear Medicine & Molecular Imaging, University Medical Center Groningen, Groningen, Netherlands; Dan Foldager, ESC Patient Forum, Sophia Antipolis, France; Emil L. Fosbøl, University Hospital of Copenhagen, Rigshospitalet, Copenhagen, Denmark; Jan Kovac, University of Leicester, Leicester, United Kingdom and RRCV Glenfield, University Hospitals of Leicester NHS Trust, Leicester, United Kingdom; Carlos A. Mestres, The University of the Free State, Bloemfontein, South Africa and Cardiac Surgery, University Hospital Zurich, Zurich, Switzerland; Owen I. Miller, Evelina London Children’s Hospital, London, United Kingdom and Department of Women and Children’s Health, Kings College London, London, United Kingdom; Jose M. Miro, Hospital Clinic - IDIBAPS, Barcelona, Spain and Faculty of Medicine, University of Barcelona, Barcelona, Spain and Centro de Investigación Biomédica en Red Enfermedades Infecciosas (CIBERINFEC), Instituto de Salud Carlos III, Madrid, Spain; Michal Pazdernik, Institute for Clinical and Experimental Medicine (IKEM), Prague, Czech Republic; Maria Nazarena Pizzi, Vall d´Hebron University Hospital, Barcelona, Spain; Eduard Quintana, Hospital Clínic de Barcelona, Barcelona, Spain and Departament de Cirurgia i Especialitats Medicoquirúrgiques, Facultat de Medicina i Ciències de la Salut, University of Barcelona, Barcelona, Spain; Trine Bernholdt Rasmussen, Herlev and Gentofte University Hospital, Hellerup, Denmark and Department of Clinical Medicine, Faculty of Health and Medical Sciences, University of Copenhagen, Copenhagen, Denmark; Arsen D. Ristić, University Clinical Center of Serbia, Belgrade, Serbia and Faculty of Medicine, University of Belgrade, Belgrade, Serbia; Josep Rodés-Cabau, Quebec Heart and Lung Institute, Laval University, Quebec, Canada and Research and Innovation, Clínic Barcelona, Barcelona, Spain; Alessandro Sionis, Hospital de la Santa Creu i Sant Pau, Barcelona, Spain and Universitat Autònoma de Barcelona, Barcelona, Spain; Liesl Joanna Zühlke, Francie van Zijl Drive Parowvallei, Cape Town; PO Box 19070, Cape Town, South Africa and Division of Paediatric Cardiology, Department of Paediatrics, Institute of Child Health, Faculty of Health Sciences, University of Cape Town, Cape Town, South Africa. Obtaining at least three sets of blood cultures is recommended before prompt initiation of empirical antibiotic therapy for CIED infection, covering methicillin-resistant staphylococci and Gram-negative bacteria. I C If CIED reimplantation is indicated after extraction for CIED-related IE, it is recommended to be performed at a site distant from the previous generator, as late as possible, once signs and symptoms of infection have abated, and until blood cultures are negative for at least 72 h in the absence of vegetations, and negative for at least 2 weeks if vegetations were visualized. I C Removal of CIED after a single positive blood culture, with no other clinical evidence of infection, is not recommended. III C Recommendations for the surgical treatment of right-sided infective endocarditis Surgery is recommended in patients with right-sided IE who are receiving appropriate antibiotic therapy for the following scenarios: Right ventricular dysfunction secondary to acute severe tricuspid regurgitation non-responsive to diuretics. I B Persistent vegetation with respiratory insufficiency requiring ventilatory support after recurrent pulmonary emboli. I B Large residual tricuspid vegetations (>20 mm) after recurrent septic pulmonary emboli. I C Patients with simultaneous involvement of left-heart structures. I C Recommendations for the use of antithrombotic therapy in infective endocarditis Interruption of antiplatelet or anticoagulant therapy is recommended in the presence of major bleeding (including intracranial haemorrhage). I C Thrombolytic therapy is not recommended in patients with IE. III C © ESC 2023 [18F]FDG-PET, 18F-fluorodeoxyglucose positron emission tomography; AVB, atrioventricular block; CAD, coronary artery disease; CHD, congenital heart disease; CIED, cardiovascular implanted electronic device; CT, computed tomography; CTA, computed tomography angiography; HF, heart failure; HLAR, high-level aminoglycoside resistance; IE, infective endocarditis; i.m., intramuscular; i.v., intravenous; MRA, magnetic resonance angiography; MRI, magnetic resonance imaging; NVE, native valve endocarditis; PET, positron emission tomography; PVE, prosthetic valve endocarditis; PWID, people who inject drugs; TOE, transoesophageal echocardiography; TTE, transthoracic echocardiography. aClass of recommendation. bLevel of evidence. ESC Guidelines 73 Downloaded from by guest on 30 August 2023 21. Appendix ESC Society Scientific Document Group Includes Document Reviewers and ESC National Cardiac Societies. Document Reviewers: Bernard Iung (CPG Review Co-ordinator) (France), Bernard Prendergast (CPG Review Co-ordinator) (United Kingdom), Magdy Abdelhamid (Egypt), Marianna Adamo (Italy), Riccardo Asteggiano (Italy), Larry M. Baddour (United States of America), Jelena Čelutkienė (Lithuania), John Chambers (United Kingdom), Jean-Claude Deharo (France), Wolfram Doehner (Germany), Laura Dos Subira (Spain), Xavier Duval (France), Volkmar Falk (Germany), Laurent Fauchier (France), Nuria Fernandez-Hidalgo (Spain), Christian Giske (Sweden), Anežka Gombošová (Czechia), Gilbert Habib (France), Borja Ibanez (Spain), Tiny Jaarsma (Sweden), Lars Køber (Denmark), Konstantinos C. Koskinas (Switzerland), Dipak Kotecha (United Kingdom), Ulf Landmesser (Germany), Sandra B. Lauck (Canada), Basil S. Lewis (Israel), Maja-Lisa Løchen (Norway), John William McEvoy (Ireland), Borislava Mihaylova (United Kingdom), Richard Mindham (United Kingdom), Lis Neubeck (United Kingdom), Jens Cosedis Nielsen (Denmark), Jean-François Obadia (France), Agnes A. Pasquet (Belgium), Steffen Petersen (United Kingdom), Eva Prescott (Denmark), Susanna Price (United Kingdom), Amina Rakisheva (Kazakhstan), Archana Rao (United Kingdom), François Rouzet (France), Jonathan Sandoe (United Kingdom), Renate B. Schnabel (Germany), Christine Selton-Suty (France), Lars Sondergaard (Denmark), Martin Thornhill (United Kingdom), Konstantinos Toutouzas (Greece), Nico Van de Veire (Belgium), Isidre Vilacosta (Spain), Christiaan Vrints (Belgium), Olaf Wendler (United Kingdom). ESC National Cardiac Societies actively involved in the review process of the 2023 Guidelines for the management of endocarditis. Algeria: Algerian Society of Cardiology, Yasmina Benchabi; Armenia: Armenian Cardiologists Association, Aram Chilingaryan; Austria: Austrian Society of Cardiology, Sebastian J. Reinstadler; Azerbaijan: Azerbaijan Society of Cardiology, Fuad Samadov; Belgium: Belgian Society of Cardiology, Bernard Paelinck; Bosnia and Herzegovina: Association of Cardiologists of Bosnia and Herzegovina, Zumreta Kušljugić; Bulgaria: Bulgarian Society of Cardiology, Elena Kinova; Croatia: Croatian Cardiac Society, Maja Cikes; Cyprus: Cyprus Society of Cardiology, Ioannis Michaloliakos; Czechia: Czech Society of Cardiology, Martin Mates; Denmark: Danish Society of Cardiology, Jonas Agerlund Povlsen; Egypt: Egyptian Society of Cardiology, Mohammad Abdelghani; Estonia: Estonian Society of Cardiology, Liisi Küünal-Arge; Finland: Finnish Cardiac Society, Helena Rajala; France: French Society of Cardiology, Christine Selton-Suty; Georgia: Georgian Society of Cardiology, Zurab Pagava; Germany: German Cardiac Society, Marcus Franz; Greece: Hellenic Society of Cardiology, Alexandros Patrianakos Hungary: Hungarian Society of Cardiology, Judit Barta; Iceland: Icelandic Society of Cardiology, Þórdís Jóna Hrafnkelsdóttir; Ireland: Irish Cardiac Society, David Moore; Israel: Israel Heart Society, Katia Orvin; Italy: Italian Federation of Cardiology, Fabrizio Oliva; Kazakhstan: Association of Cardiologists of Kazakhstan, Gyulnar Zhussupova; Kosovo (Republic of): Kosovo Society of Cardiology, Gani Bajraktari; Kyrgyzstan: Kyrgyz Society of Cardiology, Alina Kerimkulova, Latvia: Latvian Society of Cardiology, Ginta Kamzola; Lebanon: Lebanese Society of Cardiology, Pierrette Habib; Lithuania: Lithuanian Society of Cardiology, Vaida Mizarienė; Luxembourg: Luxembourg Society of Cardiology, Rouguiatou Sow; Malta: Maltese Cardiac Society, Daniela Cassar Demarco; Moldova (Republic of): Moldavian Society of Cardiology, Elena Panfile; Morocco: Moroccan Society of Cardiology, Laila Bendriss; Netherlands: Netherlands Society of Cardiology, Wilco Tanis; North Macedonia: The National Society of Cardiology of North Macedonia, Irena Mitevska; Norway: Norwegian Society of Cardiology, Erlend Aune; Portugal: Portuguese Society of Cardiology, Manuel Antunes; Romania: Romanian Society of Cardiology, Bogdan A. Popescu; San Marino: San Marino Society of Cardiology, Roberto Bini; Serbia: Cardiology Society of Serbia, Milorad Tesic; Slovakia: Slovak Society of Cardiology, Marek Orban; Slovenia: Slovenian Society of Cardiology, Mojca Bervar; Spain: Spanish Society of Cardiology, Isidre Vilacosta; Sweden: Swedish Society of Cardiology, Christina Christersson; Switzerland: Swiss Society of Cardiology, Michelle Frank; Tunisia: Tunisian Society of Cardiology and Cardiovascular Surgery, Lilia Zakhama; Türkiye: Turkish Society of Cardiology, Gamze Babur Guler; Ukraine: Ukrainian Association of Cardiology, Sergii Cherniuk; United Kingdom of Great Britain and Northern Ireland: British Cardiovascular Society, Simon Woldman, Uzbekistan: Association of Cardiologists of Uzbekistan, Nigora Srojidinova. ESC Clinical Practice Guidelines (CPG) Committee: Eva Prescott (Chairperson) (Denmark), Stefan James (Co-Chairperson) (Sweden), Elena Arbelo (Spain), Colin Baigent (United Kingdom), Michael A. Borger (Germany), Sergio Buccheri (Sweden), Borja Ibanez (Spain), Lars Køber (Denmark), Konstantinos C. Koskinas (Switzerland), John William McEvoy (Ireland), Borislava Mihaylova (United Kingdom), Richard Mindham (United Kingdom), Lis Neubeck (United Kingdom), Jens Cosedis Nielsen (Denmark), Agnes A. Pasquet (Belgium), Amina Rakisheva (Kazakhstan), Bianca Rocca (Italy), Xavier Rossello (Spain), Ilonca Vaartjes (Netherlands), Christiaan Vrints (Belgium), Adam Witkowski (Poland), and Katja Zeppenfeld (Netherlands). 22. References 1. Global Burden of Disease Metrics. Institute for Health Metrics Evaluation. University of Washington, Seattle. Available at: (ac-cessed October 2021). 2. Momtazmanesh S, Saeedi Moghaddam S, Malakan Rad E, Azadnajafabad S, Ebrahimi N, Mohammadi E, et al. Global, regional, and national burden and quality of care index of endocarditis: the global burden of disease study 1990–2019. Eur J Prev Cardiol 2022;29: 1287–1297. 3. Chen H, Zhan Y, Zhang K, Gao Y, Chen L, Zhan J, et al. The global, regional, and na-tional burden and trends of infective endocarditis from 1990 to 2019: results from the global burden of disease study 2019. Front Med (Lausanne) 2022;9:774224. org/10.3389/fmed.2022.774224 4. Habib G, Lancellotti P, Antunes MJ, Bongiorni MG, Casalta JP, Del Zotti F, et al. 2015 ESC guidelines for the management of infective endocarditis: the Task Force for the Management of Infective Endocarditis of the European Society of Cardiology (ESC). Endorsed by: European Association for Cardio-Thoracic Surgery (EACTS), the European Association of nuclear medicine (EANM). Eur Heart J 2015;36: 3075–3128. 5. Habib G, Erba PA, Iung B, Donal E, Cosyns B, Laroche C, et al. Clinical presentation, aetiology and outcome of infective endocarditis. Results of the ESC-EORP EURO-ENDO (European infective endocarditis) registry: a prospective cohort study. Eur Heart J 2019;40:3222–3232. 6. Janszky I, Gemes K, Ahnve S, Asgeirsson H, Moller J. Invasive procedures associated with the development of infective endocarditis. J Am Coll Cardiol 2018;71: 2744–2752. 7. Pericas JM, Llopis J, Jimenez-Exposito MJ, Kourany WM, Almirante B, Carosi G, et al. Infective endocarditis in patients on chronic hemodialysis. J Am Coll Cardiol 2021;77: 1629–1640. 8. Kuijpers JM, Koolbergen DR, Groenink M, Peels KCH, Reichert CLA, Post MC, et al. Incidence, risk factors, and predictors of infective endocarditis in adult congenital heart disease: focus on the use of prosthetic material. Eur Heart J 2017;38:2048–2056. 74 ESC Guidelines Downloaded from by guest on 30 August 2023 9. Sanaiha Y, Lyons R, Benharash P. Infective endocarditis in intravenous drug users. Trends Cardiovasc Med 2020;30:491–497. 10. Pericas JM, Llopis J, Athan E, Hernandez-Meneses M, Hannan MM, Murdoch DR, et al. Prospective cohort study of infective endocarditis in people who inject drugs. J Am Coll Cardiol 2021;77:544–555. 11. Thornhill MH, Crum A, Campbell R, Stone T, Lee EC, Bradburn M, et al. Temporal as-sociation between invasive procedures and infective endocarditis. Heart 2023;109: 223–231. 12. Maeda K, Hirai Y, Nashi M, Yamamoto S, Taniike N, Takenobu T. Clinical features and antimicrobial susceptibility of oral bacteria isolated from the blood cultures of patients with infective endocarditis. J Dent Sci 2022;17:870–875. 2021.09.023 13. Thornhill MH, Gibson TB, Cutler E, Dayer MJ, Chu VH, Lockhart PB, et al. Antibiotic prophylaxis and incidence of endocarditis before and after the 2007 AHA recommen-dations. J Am Coll Cardiol 2018;72:2443–2454. 2178 14. Thornhill MH, Dayer MJ, Nicholl J, Prendergast BD, Lockhart PB, Baddour LM. An alarming rise in incidence of infective endocarditis in England since 2009: why? Lancet 2020;395:1325–1327. 15. Dayer MJ, Prendergast BD, Thornhill MH, Baddour LM. Why are we seeing an increas-ing incidence of infective endocarditis in the UK? Br J Hosp Med (Lond) 2020;81:1–4. 16. Quan TP, Muller-Pebody B, Fawcett N, Young BC, Minaji M, Sandoe J, et al. Investigation of the impact of the NICE guidelines regarding antibiotic prophylaxis dur-ing invasive dental procedures on the incidence of infective endocarditis in England: an electronic health records study. BMC Med 2020;18:84. s12916-020-01531-y 17. Vahasarja N, Lund B, Ternhag A, Gotrick B, Olaison L, Hultin M, et al. Incidence of in-fective endocarditis caused by viridans group streptococci in Sweden – effect of ces-sation of antibiotic prophylaxis in dentistry for risk individuals. J Oral Microbiol 2020;12: 1768342. 18. Williams ML, Doyle MP, McNamara N, Tardo D, Mathew M, Robinson B. Epidemiology of infective endocarditis before versus after change of international guidelines: a systematic review. Ther Adv Cardiovasc Dis 2021;15: 17539447211002687. 19. Ostergaard L, Bruun NE, Voldstedlund M, Arpi M, Andersen CO, Schonheyder HC, et al. Prevalence of infective endocarditis in patients with positive blood cultures: a Danish nationwide study. Eur Heart J 2019;40:3237–3244. eurheartj/ehz327 20. Kim I-C, Chang S, Hong G-R, Lee SH, Lee S, Ha J-W, et al. Comparison of cardiac com-puted tomography with transesophageal echocardiography for identifying vegetation and intracardiac complications in patients with infective endocarditis in the era of 3-di-mensional images. Circ Cardiovasc Imaging 2018;11:e006986. CIRCIMAGING.117.006986 21. Salaun E, Habib G. Beyond standard echocardiography in infective endocarditis: com-puted tomography, 3-dimensional imaging, and multi-imaging. Circ Cardiovasc Imaging 2018;11:e007626. 22. Pizzi MN, Roque A, Fernandez-Hidalgo N, Cuellar-Calabria H, Ferreira-Gonzalez I, Gonzalez-Alujas MT, et al. Improving the diagnosis of infective endocarditis in pros-thetic valves and intracardiac devices with 18F-fluordeoxyglucose positron emission tomography/computed tomography angiography: initial results at an infective endocar-ditis referral center. Circulation 2015;132:1113–1126. CIRCULATIONAHA.115.015316 23. DeSimone DC, Lahr BD, Anavekar NS, Sohail MR, Tleyjeh IM, Wilson WR, et al. Temporal trends of infective endocarditis in Olmsted County, Minnesota, between 1970 and 2018: a population-based analysis. Open Forum Infect Dis 2021;8:ofab038. 24. Giannitsioti E, Pefanis A, Gogos C, Lekkou A, Dalekos GN, Gatselis N, et al. Evolution of epidemiological characteristics of infective endocarditis in Greece. Int J Infect Dis 2021;106:213–220. 25. Jensen AD, Bundgaard H, Butt JH, Bruun NE, Voldstedlund M, Torp-Pedersen C, et al. Temporal changes in the incidence of infective endocarditis in Denmark 1997–2017: a nationwide study. Int J Cardiol 2021;326:145–152. 10.029 26. Keller K, von Bardeleben RS, Ostad MA, Hobohm L, Munzel T, Konstantinides S, et al. Temporal trends in the prevalence of infective endocarditis in Germany between 2005 and 2014. Am J Cardiol 2017;119:317–322. 035 27. Shah ASV, McAllister DA, Gallacher P, Astengo F, Rodriguez Perez JA, Hall J, et al. Incidence, microbiology, and outcomes in patients hospitalized with infective endocar-ditis. Circulation 2020;141:2067–2077. 119.044913 28. Ortega-Loubon C, Munoz-Moreno MF, Andres-Garcia I, Alvarez FJ, Gomez-Sanchez E, Bustamante-Munguira J, et al. Nosocomial vs. community-acquired infective endocar-ditis in Spain: location, trends, clinical presentation, etiology, and survival in the 21st century. J Clin Med 2019;8:1755. 29. Olmos C, Vilacosta I, Fernandez-Perez C, Bernal JL, Ferrera C, Garcia-Arribas D, et al. The evolving nature of infective endocarditis in Spain: a population-based study (2003 to 2014). J Am Coll Cardiol 2017;70:2795–2804. 005 30. Altini C, Lavelli V, Niccoli-Asabella A, Sardaro A, Branca A, Santo G, et al. Comparison of the diagnostic value of MRI and whole body (18)F-FDG PET/CT in diagnosis of spondylodiscitis. J Clin Med 2020;9:1581. 31. Holle SLK, Andersen MH, Klein CF, Bruun NE, Tonder N, Haarmark C, et al. Clinical usefulness of FDG-PET/CT for identification of abnormal extra-cardiac foci in patients with infective endocarditis. Int J Cardiovasc Imaging 2020;36:939–946. 10.1007/s10554-020-01787-8 32. Kim S-J, Pak K, Kim K, Lee JS. Comparing the diagnostic accuracies of F-18 fluorodeox-yglucose positron emission tomography and magnetic resonance imaging for the de-tection of spondylodiscitis: a meta-analysis. Spine (Phila Pa 1976) 2019;44: E414–E422. 33. Sifaoui I, Oliver L, Tacher V, Fiore A, Lepeule R, Moussafeur A, et al. Diagnostic per-formance of transesophageal echocardiography and cardiac computed tomography in infective endocarditis. J Am Soc Echocardiogr 2020;33:1442–1453. 1016/j.echo.2020.07.017 34. Venet M, Jalal Z, Ly R, Malekzadeh-Milani S, Hascoet S, Fournier E, et al. Diagnostic va-lue of (18)F-fluorodeoxyglucose positron emission tomography computed tomog-raphy in prosthetic pulmonary valve infective endocarditis. JACC Cardiovasc Imaging 2022;15:299–308. 35. Wang TKM, Bin Saeedan M, Chan N, Obuchowski NA, Shrestha N, Xu B, et al. Complementary diagnostic and prognostic contributions of cardiac computed tomog-raphy for infective endocarditis surgery. Circ Cardiovasc Imaging 2020;13:e011126. 36. Camou F, Dijos M, Barandon L, Cornolle C, Greib C, Laine M, et al. Management of infective endocarditis and multidisciplinary approach. Med Mal Infect 2019;49:17–22. 37. El-Dalati S, Cronin D, Riddell JT, Shea M, Weinberg RL, Washer L, et al. The clinical impact of implementation of a multidisciplinary endocarditis team. Ann Thorac Surg 2022;113:118–124. 38. Elad B, Perl L, Hamdan A, Yahav D, Atamna A, Shaked H, et al. The clinical value of the endocarditis team: insights from before and after guidelines implementation strategy. Infection 2022; 50:57–64. 39. Gibbons EF, Huang G, Aldea G, Koomalsingh K, Klein JW, Dhanireddy S, et al. A multi-disciplinary pathway for the diagnosis and treatment of infectious endocarditis. Crit Pathw Cardiol 2020;19:187–194. 40. Kaura A, Byrne J, Fife A, Deshpande R, Baghai M, Gunning M, et al. Inception of the ‘endocarditis team’ is associated with improved survival in patients with infective endo-carditis who are managed medically: findings from a before-and-after study. Open Heart 2017;4:e000699. 41. Ruch Y, Mazzucotelli JP, Lefebvre F, Martin A, Lefebvre N, Douiri N, et al. Impact of setting up an “endocarditis team” on the management of infective endocarditis. Open Forum Infect Dis 2019;6:ofz308. 42. European Committee on Antimicrobial Susceptibility Testing. The European Committee on Antimicrobial Susceptibility Testing. Breakpoint Tables for Interpretation of MICs and Zone Diameters. Version 12.0, 2022. 43. Iversen K, Ihlemann N, Gill SU, Madsen T, Elming H, Jensen KT, et al. Partial oral versus intravenous antibiotic treatment of endocarditis. N Engl J Med 2019;380:415–424. 44. Wildenthal JA, Atkinson A, Lewis S, Sayood S, Nolan NS, Cabrera NL, et al. Outcomes of partial oral antibiotic treatment for complicated Staphylococcus aureus bacteremia in people who inject drugs. Clin Infect Dis 2022;76:487–496. ciac714 45. Pericas JM, Llopis J, Munoz P, Gonzalez-Ramallo V, Garcia-Leoni ME, de Alarcon A, et al. Outpatient parenteral antibiotic treatment vs hospitalization for infective endo-carditis: validation of the OPAT-games criteria. Open Forum Infect Dis 2022;9:ofac442. 46. Fernandez-Galilea A, Estella A, Garcia-Garmendia JL, Loza A, Palacios-Garcia I, Sierra-Camerino R, et al. Clindamycin but not intravenous immunoglobulins reduces mortality in a retrospective cohort of critically ill patients with bacteremic group A streptococcal infections. Rev Esp Quimioter 2022;35:475–481. 37201/req/030.2022 47. Thornhill MH, Jones S, Prendergast B, Baddour LM, Chambers JB, Lockhart PB, et al. Quantifying infective endocarditis risk in patients with predisposing cardiac conditions. Eur Heart J 2018;39:586–595. 48. Duval X, Millot S, Chirouze C, Selton-Suty C, Moby V, Tattevin P, et al. Oral strepto-coccal endocarditis, oral hygiene habits, and recent dental procedures: a case-control study. Clin Infect Dis 2017;64:1678–1685. 49. Lockhart PB, Brennan MT, Thornhill M, Michalowicz BS, Noll J, Bahrani-Mougeot FK, et al. Poor oral hygiene as a risk factor for infective endocarditis-related bacteremia. J Am Dent Assoc 2009;140:1238–1244. ESC Guidelines 75 Downloaded from by guest on 30 August 2023 50. Sy RW, Kritharides L. Health care exposure and age in infective endocarditis: results of a contemporary population-based profile of 1536 patients in Australia. Eur Heart J 2010;31:1890–1897. 51. Thornhill MH, Gibson TB, Yoon F, Dayer MJ, Prendergast BD, Lockhart PB, et al. Antibiotic prophylaxis against infective endocarditis before invasive dental procedures. J Am Coll Cardiol 2022;80:1029–1041. 52. Cahill TJ, Harrison JL, Jewell P, Onakpoya I, Chambers JB, Dayer M, et al. Antibiotic prophylaxis for infective endocarditis: a systematic review and meta-analysis. Heart 2017;103:937–944. 53. Glauser MP, Francioli P. Relevance of animal models to the prophylaxis of infective endocarditis. J Antimicrob Chemother 1987;20(Suppl A):87–98. 1093/jac/20.suppl_A.87 54. Lafaurie GI, Noriega LA, Torres CC, Castillo Y, Moscoso SB, Mosquera S, et al. Impact of antibiotic prophylaxis on the incidence, nature, magnitude, and duration of bacter-emia associated with dental procedures: a systematic review. J Am Dent Assoc 2019; 150:948–959.e4. 55. Overholser CD, Moreillon P, Glauser MP. Experimental bacterial endocarditis after dental extractions in rats with periodontitis. J Infect Dis 1987;155:107–112. https:// doi.org/10.1093/infdis/155.1.107 56. Veloso TR, Amiguet M, Rousson V, Giddey M, Vouillamoz J, Moreillon P, et al. Induction of experimental endocarditis by continuous low-grade bacteremia mimick-ing spontaneous bacteremia in humans. Infect Immun 2011;79:2006–2011. org/10.1128/IAI.01208-10 57. Wilson W, Taubert KA, Gewitz M, Lockhart PB, Baddour LM, Levison M, et al. Prevention of infective endocarditis: guidelines from the American Heart Association: a guideline from the American Heart Association Rheumatic Fever, Endocarditis, and Kawasaki Disease Committee, Council on Cardiovascular Disease in the Young, and the Council on Clinical Cardiology, Council on Cardiovascular Surgery and Anesthesia, and the Quality of Care and Outcomes Research Interdisciplinary Working Group. Circulation 2007;116:1736–1754. 10.1161/CIRCULATIONAHA.106.183095 58. Richey R, Wray D, Stokes T; Guideline Development Group. Prophylaxis against in-fective endocarditis: summary of NICE guidance. BMJ 2008;336:770–771. https:// doi.org/10.1136/bmj.39510.423148.AD 59. Dayer MJ, Jones S, Prendergast B, Baddour LM, Lockhart PB, Thornhill MH. Incidence of infective endocarditis in England, 2000–13: a secular trend, interrupted time-series analysis. Lancet 2015;385:1219–1228. 60. Tubiana S, Duval X, Alla F, Selton-Suty C, Tattevin P, Delahaye F, et al. The VIRSTA score, a prediction score to estimate risk of infective endocarditis and determine pri-ority for echocardiography in patients with Staphylococcus aureus bacteremia. J Infect 2016;72:544–553. 61. Thornhill MH, Lockhart PB, Prendergast B, Chambers JB, Shanson D. NICE and anti-biotic prophylaxis to prevent endocarditis. Br Dent J 2015;218:619–621. org/10.1038/sj.bdj.2015.496 62. Tubiana S, Blotiere PO, Hoen B, Lesclous P, Millot S, Rudant J, et al. Dental procedures, antibiotic prophylaxis, and endocarditis among people with prosthetic heart valves: na-tionwide population based cohort and a case crossover study. BMJ 2017;358:j3776. 63. Cloitre A, Duval X, Tubiana S, Giraud P, Veyrac G, Nosbaum A, et al. Antibiotic prophylaxis for the prevention of infective endocarditis for dental procedures is not associated with fatal adverse drug reactions in France. Med Oral Patol Oral Cir Bucal 2019;24:e296–e304. 64. Gross AE, Suda KJ, Zhou J, Calip GS, Rowan SA, Hershow RC, et al. Serious antibiotic-related adverse effects following unnecessary dental prophylaxis in the United States. Infect Control Hosp Epidemiol 2021;42:110–112. 1017/ice.2020.1261 65. Khalil D, Hultin M, Rashid MU, Lund B. Oral microflora and selection of resistance after a single dose of amoxicillin. Clin Microbiol Infect 2016;22:949.e1–949.e4. 10.1016/j.cmi.2016.08.008 66. Woodman AJ, Vidic J, Newman HN, Marsh PD. Effect of repeated high dose prophy-laxis with amoxycillin on the resident oral flora of adult volunteers. J Med Microbiol 1985;19:15–23. 67. Habib G, Hoen B, Tornos P, Thuny F, Prendergast B, Vilacosta I, et al. Guidelines on the prevention, diagnosis, and treatment of infective endocarditis (new version 2009): the Task Force on the prevention, diagnosis, and treatment of infective endocarditis of the European Society of Cardiology (ESC). Endorsed by the European Society of Clinical Microbiology and Infectious Diseases (ESCMID) and the International Society of Chemotherapy (ISC) for infection and cancer. Eur Heart J 2009;30:2369–2413. 68. Bates KE, Hall M, Shah SS, Hill KD, Pasquali SK. Trends in infective endocarditis hospi-talisations at United States children’s hospitals from 2003 to 2014: impact of the 2007 American Heart Association antibiotic prophylaxis guidelines. Cardiol Young 2017;27: 686–690. 69. Bikdeli B, Wang Y, Kim N, Desai MM, Quagliarello V, Krumholz HM. Trends in hospi-talization rates and outcomes of endocarditis among medicare beneficiaries. J Am Coll Cardiol 2013;62:2217–2226. 70. DeSimone DC, Tleyjeh IM, Correa de Sa DD, Anavekar NS, Lahr BD, Sohail MR, et al. Temporal trends in infective endocarditis epidemiology from 2007 to 2013 in Olmsted County, MN. Am Heart J 2015;170:830–836. 71. Desimone DC, Tleyjeh IM, Correa de Sa DD, Anavekar NS, Lahr BD, Sohail MR, et al. Incidence of infective endocarditis caused by viridans group streptococci before and after publication of the 2007 American Heart Association’s endocarditis prevention guidelines. Circulation 2012;126:60–64. 112.095281 72. Duval X, Delahaye F, Alla F, Tattevin P, Obadia JF, Le Moing V, et al. Temporal trends in infective endocarditis in the context of prophylaxis guideline modifications: three suc-cessive population-based surveys. J Am Coll Cardiol 2012;59:1968–1976. org/10.1016/j.jacc.2012.02.029 73. Garg P, Ko DT, Bray Jenkyn KM, Li L, Shariff SZ. Infective endocarditis hospitalizations and antibiotic prophylaxis rates before and after the 2007 American Heart Association guideline revision. Circulation 2019;140:170–180. CIRCULATIONAHA.118.037657 74. Mackie AS, Liu W, Savu A, Marelli AJ, Kaul P. Infective endocarditis hospitalizations be-fore and after the 2007 American Heart Association prophylaxis guidelines. Can J Cardiol 2016;32:942–948. 75. Pasquali SK, He X, Mohamad Z, McCrindle BW, Newburger JW, Li JS, et al. Trends in endocarditis hospitalizations at US children’s hospitals: impact of the 2007 American Heart Association antibiotic prophylaxis guidelines. Am Heart J 2012;163:894–899. 76. Rogers AM, Schiller NB. Impact of the first nine months of revised infective endocar-ditis prophylaxis guidelines at a university hospital: so far so good. J Am Soc Echocardiogr 2008;21:775. 77. Thornhill MH, Dayer MJ, Forde JM, Corey GR, Chu VH, Couper DJ, et al. Impact of the NICE guideline recommending cessation of antibiotic prophylaxis for prevention of in-fective endocarditis: before and after study. BMJ 2011;342:d2392. 1136/bmj.d2392 78. Pant S, Patel NJ, Deshmukh A, Golwala H, Patel N, Badheka A, et al. Trends in infective endocarditis incidence, microbiology, and valve replacement in the United States from 2000 to 2011. J Am Coll Cardiol 2015;65:2070–2076. 2015.03.518 79. Sakai Bizmark R, Chang RR, Tsugawa Y, Zangwill KM, Kawachi I. Impact of AHA’s 2007 guideline change on incidence of infective endocarditis in infants and children. Am Heart J 2017;189:110–119. 80. Toyoda N, Chikwe J, Itagaki S, Gelijns AC, Adams DH, Egorova NN. Trends in infective endocarditis in California and New York State, 1998–2013. JAMA 2017;317: 1652–1660. 81. van den Brink FS, Swaans MJ, Hoogendijk MG, Alipour A, Kelder JC, Jaarsma W, et al. Increased incidence of infective endocarditis after the 2009 European Society of Cardiology guideline update: a nationwide study in the Netherlands. Eur Heart J Qual Care Clin Outcomes 2017;3:141–147. 82. Talha KM, Baddour LM, Thornhill MH, Arshad V, Tariq W, Tleyjeh IM, et al. Escalating incidence of infective endocarditis in Europe in the 21st century. Open Heart 2021;8: e001846. 83. Wilson WR, Gewitz M, Lockhart PB, Bolger AF, DeSimone DC, Kazi DS, et al. Prevention of viridans group streptococcal infective endocarditis: a scientific statement from the American Heart Association. Circulation 2021;143:e963–e978. org/10.1161/CIR.0000000000000969 84. Havers-Borgersen E, Butt JH, Ostergaard L, Bundgaard H, Smerup M, Bruun NE, et al. Recurrent infective endocarditis versus first-time infective endocarditis after heart valve surgery. Clin Res Cardiol 2020;109:1342–1351. 020-01628-7 85. Calderon-Parra J, Kestler M, Ramos-Martinez A, Bouza E, Valerio M, de Alarcon A, et al. Clinical factors associated with reinfection versus relapse in infective endocarditis: prospective cohort study. J Clin Med 2021;10:748. jcm10040748 86. Freitas-Ferraz AB, Tirado-Conte G, Vilacosta I, Olmos C, Saez C, Lopez J, et al. Contemporary epidemiology and outcomes in recurrent infective endocarditis. Heart 2020;106:596–602. 87. Salem M, Friedrich C, Saad M, Frank D, Salem M, Puehler T, et al. Active infective native and prosthetic valve endocarditis: short- and long-term outcomes of patients after sur-gical treatment. J Clin Med 2021;10:1868. 88. Luehr M, Bauernschmitt N, Peterss S, Li Y, Heyn O, Dashkevich A, et al. Incidence and surgical outcomes of patients with native and prosthetic aortic valve endocarditis. Ann Thorac Surg 2020;110:93–101. 89. Anantha-Narayanan M, Reddy YNV, Sundaram V, Murad MH, Erwin PJ, Baddour LM, et al. Endocarditis risk with bioprosthetic and mechanical valves: systematic review and meta-analysis. Heart 2020;106:1413–1419. 316718 90. Ostergaard L, Valeur N, Ihlemann N, Smerup MH, Bundgaard H, Gislason G, et al. Incidence and factors associated with infective endocarditis in patients undergoing left- sided heart valve replacement. Eur Heart J 2018;39:2668–2675. 1093/eurheartj/ehy153 76 ESC Guidelines Downloaded from by guest on 30 August 2023 91. Alexis SL, Malik AH, George I, Hahn RT, Khalique OK, Seetharam K, et al. Infective endocarditis after surgical and transcatheter aortic valve replacement: a state of the art review. J Am Heart Assoc 2020;9:e017347. 017347 92. Lehner A, Haas NA, Dietl M, Jakob A, Schulze-Neick I, Dalla Pozza R, et al. The risk of infective endocarditis following interventional pulmonary valve implantation: a meta-analysis. J Cardiol 2019;74:197–205. 93. McElhinney DB, Sondergaard L, Armstrong AK, Bergersen L, Padera RF, Balzer DT, et al. Endocarditis after transcatheter pulmonary valve replacement. J Am Coll Cardiol 2018;72:2717–2728. 94. Summers MR, Leon MB, Smith CR, Kodali SK, Thourani VH, Herrmann HC, et al. Prosthetic valve endocarditis after TAVR and SAVR: insights from the PARTNER trials. Circulation 2019;140:1984–1994. 041399 95. Asmarats L, Rodriguez-Gabella T, Chamandi C, Bernier M, Beaudoin J, O’Connor K, et al. Infective endocarditis following transcatheter edge-to-edge mitral valve repair: a systematic review. Catheter Cardiovasc Interv 2018;92:583–591. 1002/ccd.27632 96. Diller GP, Baumgartner H. Endocarditis in adults with congenital heart disease: new answers–new questions. Eur Heart J 2017;38:2057–2059. eurheartj/ehx044 97. Ly R, Compain F, Gaye B, Pontnau F, Bouchard M, Mainardi JL, et al. Predictive factors of death associated with infective endocarditis in adult patients with congenital heart disease. Eur Heart J Acute Cardiovasc Care 2020. 2048872620901394 98. Snygg-Martin U, Giang KW, Dellborg M, Robertson J, Mandalenakis Z. Cumulative in-cidence of infective endocarditis in patients with congenital heart disease: a nationwide, case-control study over nine decades. Clin Infect Dis 2021;73:1469–1475. org/10.1093/cid/ciab478 99. Mylotte D, Rushani D, Therrien J, Guo L, Liu A, Guo K, et al. Incidence, predictors, and mortality of infective endocarditis in adults with congenital heart disease without pros-thetic valves. Am J Cardiol 2017;120:2278–2283. 2017.08.051 100. Mandalenakis Z, Rosengren A, Skoglund K, Lappas G, Eriksson P, Dellborg M. Survivorship in children and young adults with congenital heart disease in Sweden. JAMA Intern Med 2017;177:224–230. 7765 101. Rushani D, Kaufman JS, Ionescu-Ittu R, Mackie AS, Pilote L, Therrien J, et al. Infective endocarditis in children with congenital heart disease: cumulative incidence and predic-tors. Circulation 2013;128:1412–1419. 113.001827 102. Patel S, Rizvi SSA, Choi JH, Horan DP, Weber MP, Maynes EJ, et al. Management and outcomes of left ventricular assist device-associated endocarditis: a systematic review. Ann Cardiothorac Surg 2019;8:600–609. 103. Ostergaard L, Valeur N, Wang A, Bundgaard H, Aslam M, Gislason G, et al. Incidence of infective endocarditis in patients considered at moderate risk. Eur Heart J 2019;40: 1355–1361. 104. Russell EA, Walsh WF, Costello B, McLellan AJA, Brown A, Reid CM, et al. Medical management of rheumatic heart disease: a systematic review of the evidence. Cardiol Rev 2018;26:187–195. 105. Jordan AM, Tatum R, Ahmad D, Patel SV, Maynes EJ, Weber MP, et al. Infective endo-carditis following heart transplantation: a systematic review. Transplant Rev (Orlando) 2021;36:100672. 106. Chuang S, Shrestha NK, Brizendine KD. Matched retrospective study of infective endocarditis among solid organ transplant recipients compared to non-transplant: seven-year experience in a US referral center. Transpl Infect Dis 2020;22:e13368. 107. Martinez-Selles M, Valerio-Minero M, Farinas MC, Rodriguez-Abella H, Rodriguez ML, de Alarcon A, et al. Infective endocarditis in patients with solid organ transplantation. A nationwide descriptive study. Eur J Intern Med 2021;87:59–65. j.ejim.2021.02.017 108. Duval X, Alla F, Hoen B, Danielou F, Larrieu S, Delahaye F, et al. Estimated risk of endo-carditis in adults with predisposing cardiac conditions undergoing dental procedures with or without antibiotic prophylaxis. Clin Infect Dis 2006;42:e102–e107. https:// doi.org/10.1086/504385 109. Findler M, Chackartchi T, Regev E. Dental implants in patients at high risk for infective endocarditis: a preliminary study. Int J Oral Maxillofac Surg 2014;43:1282–1285. https:// doi.org/10.1016/j.ijom.2014.04.015 110. Thornhill MH, Dayer M, Lockhart PB, Prendergast B. Antibiotic prophylaxis of infective endocarditis. Curr Infect Dis Rep 2017;19:9. y 111. Thornhill MH, Dayer MJ, Durkin MJ, Lockhart PB, Baddour LM. Risk of adverse reac-tions to oral antibiotics prescribed by dentists. J Dent Res 2019;98:1081–1087. https:// doi.org/10.1177/0022034519863645 112. Thornhill MH, Dayer MJ, Prendergast B, Baddour LM, Jones S, Lockhart PB. Incidence and nature of adverse reactions to antibiotics used as endocarditis prophylaxis. J Antimicrob Chemother 2015;70:2382–2388. 113. Verhoeven PO, Gagnaire J, Botelho-Nevers E, Grattard F, Carricajo A, Lucht F, et al. Detection and clinical relevance of Staphylococcus aureus nasal carriage: an update. Expert Rev Anti Infect Ther 2014;12:75–89. 859985 114. Bode LG, Kluytmans JA, Wertheim HF, Bogaers D, Vandenbroucke-Grauls CM, Roosendaal R, et al. Preventing surgical-site infections in nasal carriers of Staphylococcus aureus. N Engl J Med 2010;362:9–17. NEJMoa0808939 115. Friedman AJ, Cosby R, Boyko S, Hatton-Bauer J, Turnbull G. Effective teaching strat-egies and methods of delivery for patient education: a systematic review and practice guideline recommendations. J Cancer Educ 2011;26:12–21. s13187-010-0183-x 116. de Oliveira JC, Martinelli M, Nishioka SA, Varejao T, Uipe D, Pedrosa AA, et al. Efficacy of antibiotic prophylaxis before the implantation of pacemakers and cardioverter- defibrillators: results of a large, prospective, randomized, double-blinded, placebo- controlled trial. Circ Arrhythm Electrophysiol 2009;2:29–34. CIRCEP.108.795906 117. Mounsey JP, Griffith MJ, Tynan M, Gould FK, MacDermott AF, Gold RG, et al. Antibiotic prophylaxis in permanent pacemaker implantation: a prospective rando-mised trial. Br Heart J 1994;72:339–343. 118. Polyzos KA, Konstantelias AA, Falagas ME. Risk factors for cardiac implantable elec-tronic device infection: a systematic review and meta-analysis. Europace 2015;17: 767–777. 119. Darouiche RO, Wall MJ Jr, Itani KM, Otterson MF, Webb AL, Carrick MM, et al. Chlorhexidine-alcohol versus povidone-iodine for surgical-site antisepsis. N Engl J Med 2010;362:18–26. 120. Lador A, Nasir H, Mansur N, Sharoni E, Biderman P, Leibovici L, et al. Antibiotic prophylaxis in cardiac surgery: systematic review and meta-analysis. J Antimicrob Chemother 2012;67:541–550. 121. Conen A, Stortecky S, Moreillon P, Hannan MM, Franzeck FC, Jeger R, et al. A review of recommendations for infective endocarditis prevention in patients undergoing trans-catheter aortic valve implantation. EuroIntervention 2021;16:1135–1140. org/10.4244/EIJ-D-19-00993 122. Botelho-Nevers E, Thuny F, Casalta JP, Richet H, Gouriet F, Collart F, et al. Dramatic reduction in infective endocarditis-related mortality with a management-based ap-proach. Arch Intern Med 2009;169:1290–1298. archinternmed.2009.192 123. Carrasco-Chinchilla F, Sanchez-Espin G, Ruiz-Morales J, Rodriguez-Bailon I, Melero-Tejedor JM, Ivanova-Georgieva R, et al. Influence of a multidisciplinary alert strategy on mortality due to left-sided infective endocarditis. Rev Esp Cardiol (Engl Ed) 2014;67:380–386. 124. Cervera C, del Rio A, Garcia L, Sala M, Almela M, Moreno A, et al. Efficacy and safety of outpatient parenteral antibiotic therapy for infective endocarditis: a ten-year prospect-ive study. Enferm Infecc Microbiol Clin 2011;29:587–592. 2011.05.007 125. Chirillo F, Scotton P, Rocco F, Rigoli R, Borsatto F, Pedrocco A, et al. Impact of a multi-disciplinary management strategy on the outcome of patients with native valve infect-ive endocarditis. Am J Cardiol 2013;112:1171–1176. 2013.05.060 126. Okura T, Iwata K, Koyama T, Ebisawa K, Arakawa Y, Kusuki M, et al. Impact of infec-tious disease consultation on management and outcomes of infective endocarditis. Ann Thorac Surg 2021;112:1228–1234. 127. Otto CM, Nishimura RA, Bonow RO, Carabello BA, Erwin JP 3rd, Gentile F, et al. 2020 ACC/AHA guideline for the management of patients with valvular heart disease: a re-port of the American College of Cardiology/American Heart Association Joint Committee on Clinical Practice Guidelines. Circulation 2021;143:e72–e227. https:// doi.org/10.1161/CIR.0000000000000923 128. Vahanian A, Beyersdorf F, Praz F, Milojevic M, Baldus S, Bauersachs J, et al. 2021 ESC/ EACTS guidelines for the management of valvular heart disease. Eur Heart J 2022;43: 561–632. 129. Wang TKM, Sanchez-Nadales A, Igbinomwanhia E, Cremer P, Griffin B, Xu B. Diagnosis of infective endocarditis by subtype using (18)F-fluorodeoxyglucose posi-tron emission tomography/computed tomography: a contemporary meta-analysis. Circ Cardiovasc Imaging 2020;13:e010600. 120.010600 130. Blomstrom-Lundqvist C, Traykov V, Erba PA, Burri H, Nielsen JC, Bongiorni MG, et al. European Heart Rhythm Association (EHRA) international consensus document on how to prevent, diagnose, and treat cardiac implantable electronic device infec-tions–endorsed by the Heart Rhythm Society (HRS), the Asia Pacific Heart Rhythm Society (APHRS), the Latin American Heart Rhythm Society (LAHRS), International Society for Cardiovascular Infectious Diseases (ISCVID) and the European Society of Clinical Microbiology and Infectious Diseases (ESCMID) in collaboration with the ESC Guidelines 77 Downloaded from by guest on 30 August 2023 European Association for Cardio-Thoracic Surgery (EACTS). Europace 2020;22: 515–549. 131. Tagliari AP, Steckert GV, da Silveira LMV, Kochi AN, Wender OCB. Infective endocar-ditis profile, prognostic factors and in-hospital mortality: 6-year trends from a tertiary university center in South America. J Card Surg 2020;35:1905–1911. 1111/jocs.14787 132. Schranz AJ, Fleischauer A, Chu VH, Wu LT, Rosen DL. Trends in drug use-associated infective endocarditis and heart valve surgery, 2007 to 2017: a study of statewide dis-charge data. Ann Intern Med 2019;170:31–40. 133. Rudasill SE, Sanaiha Y, Mardock AL, Khoury H, Xing H, Antonios JW, et al. Clinical out-comes of infective endocarditis in injection drug users. J Am Coll Cardiol 2019;73: 559–570. 134. Mori M, Brown KJ, Bin Mahmood SU, Geirsson A, Mangi AA. Trends in infective endo-carditis hospitalizations, characteristics, and valve operations in patients with opioid use disorders in the United States: 2005–2014. J Am Heart Assoc 2020;9:e012465. 135. Geirsson A, Schranz A, Jawitz O, Mori M, Feng L, Zwischenberger BA, et al. The evolv-ing burden of drug use associated infective endocarditis in the United States. Ann Thorac Surg 2020;110:1185–1192. 136. Mestres CA, Pare JC, Miro JM; Working Group on Infective Endocarditis of the Hospital Clínic de Barcelona. Organization and functioning of a multidisciplinary team for the diagnosis and treatment of infective endocarditis: a 30-year perspective (1985–2014). Rev Esp Cardiol (Engl Ed) 2015;68:363–368. recesp.2014.10.007 137. Erba PA, Lancellotti P, Vilacosta I, Gaemperli O, Rouzet F, Hacker M, et al. Recommendations on nuclear and multimodality imaging in IE and CIED infections. Eur J Nucl Med Mol Imaging 2018;45:1795–1815. 018-4025-0 138. Schwiebert R, Baig W, Wu J, Sandoe JAT. Diagnostic accuracy of splinter haemor-rhages in patients referred for suspected infective endocarditis. Heart 2022. https:// doi.org/10.1136/heartjnl-2022-321052 139. Grable C, Yusuf SW, Song J, Viola GM, Ulhaq O, Banchs J, et al. Characteristics of in-fective endocarditis in a cancer population. Open Heart 2021;8:e001664. org/10.1136/openhrt-2021-001664 140. Perez de Isla L, Zamorano J, Lennie V, Vazquez J, Ribera JM, Macaya C. Negative blood culture infective endocarditis in the elderly: long-term follow-up. Gerontology 2007;53: 245–249. 141. Wong CY, Zhu W, Aurigemma GP, Furukawa N, Teshale EH, Huang YA, et al. Infective endocarditis among persons aged 18–64 years living with human immunodeficiency virus, hepatitis C infection, or opioid use disorder, United States, 2007–2017. Clin Infect Dis 2021;72:1767–1781. 142. N’Guyen Y, Duval X, Revest M, Saada M, Erpelding ML, Selton-Suty C, et al. Time inter-val between infective endocarditis first symptoms and diagnosis: relationship to infect-ive endocarditis characteristics, microorganisms and prognosis. Ann Med 2017;49: 117–125. 143. Niederman MS, Baron RM, Bouadma L, Calandra T, Daneman N, DeWaele J, et al. Initial antimicrobial management of sepsis. Crit Care 2021;25:307. 1186/s13054-021-03736-w 144. Snipsoyr MG, Ludvigsen M, Petersen E, Wiggers H, Honore B. A systematic review of biomarkers in the diagnosis of infective endocarditis. Int J Cardiol 2016;202:564–570. 145. Murdoch DR, Corey GR, Hoen B, Miro JM, Fowler VG Jr, Bayer AS, et al. Clinical pres-entation, etiology, and outcome of infective endocarditis in the 21st century: the International Collaboration on Endocarditis–prospective cohort study. Arch Intern Med 2009;169:463–473. 146. Amat-Santos IJ, Messika-Zeitoun D, Eltchaninoff H, Kapadia S, Lerakis S, Cheema AN, et al. Infective endocarditis after transcatheter aortic valve implantation: results from a large multicenter registry. Circulation 2015;131:1566–1574. CIRCULATIONAHA.114.014089 147. Olmos C, Vilacosta I, Fernandez C, Lopez J, Sarria C, Ferrera C, et al. Contemporary epidemiology and prognosis of septic shock in infective endocarditis. Eur Heart J 2013; 34:1999–2006. 148. Escola-Verge L, Fernandez-Hidalgo N, Larrosa MN, Fernandez-Galera R, Almirante B. Secular trends in the epidemiology and clinical characteristics of Enterococcus faecalis infective endocarditis at a referral center (2007–2018). Eur J Clin Microbiol Infect Dis 2021;40:1137–1148. 149. Dahl A, Iversen K, Tonder N, Hoest N, Arpi M, Dalsgaard M, et al. Prevalence of in-fective endocarditis in Enterococcus faecalis bacteremia. J Am Coll Cardiol 2019;74: 193–201. 150. Lamy B, Dargere S, Arendrup MC, Parienti JJ, Tattevin P. How to optimize the use of blood cultures for the diagnosis of bloodstream infections? A state-of-the art. Front Microbiol 2016;7:697. 151. Li JS, Sexton DJ, Mick N, Nettles R, Fowler VG Jr, Ryan T, et al. Proposed modifications to the Duke criteria for the diagnosis of infective endocarditis. Clin Infect Dis 2000;30: 633–638. 152. La Scola B, Raoult D. Direct identification of bacteria in positive blood culture bottles by matrix-assisted laser desorption ionisation time-of-flight mass spectrometry. PLoS One 2009;4:e8041. 153. Burckhardt I, Zimmermann S. Susceptibility testing of bacteria using MALDI-ToF mass spectrometry. Front Microbiol 2018;9:1744. 154. Pecoraro AJK, Herbst PG, Pienaar C, Taljaard J, Prozesky H, Janson J, et al. Modified Duke/European Society of Cardiology 2015 clinical criteria for infective endocarditis: time for an update? Open Heart 2022;9:e001856. 2021-001856 155. Kong WKF, Salsano A, Giacobbe DR, Popescu BA, Laroche C, Duval X, et al. Outcomes of culture-negative vs. culture-positive infective endocarditis: the ESC-EORP EURO-ENDO registry. Eur Heart J 2022;43:2770–2780. 10.1093/eurheartj/ehac307 156. Fournier PE, Gouriet F, Casalta JP, Lepidi H, Chaudet H, Thuny F, et al. Blood culture- negative endocarditis: improving the diagnostic yield using new diagnostic tools. Medicine (Baltimore) 2017;96:e8392. 157. Gouriet F, Samson L, Delaage M, Mainardi JL, Meconi S, Drancourt M, et al. Multiplexed whole bacterial antigen microarray, a new format for the automation of serodiagnosis: the culture-negative endocarditis paradigm. Clin Microbiol Infect 2008;14:1112–1118. 158. Raoult D, Casalta JP, Richet H, Khan M, Bernit E, Rovery C, et al. Contribution of sys-tematic serological testing in diagnosis of infective endocarditis. J Clin Microbiol 2005; 43:5238–5242. 159. Hajduczenia MM, Klefisch FR, Hopf AGM, Grubitzsch H, Stegemann MS, Pfafflin F, et al. New perspectives for prosthetic valve endocarditis – impact of molecular imaging by fishseq diagnostics. Clin Infect Dis 2023;76:1050–1058. ciac860 160. Liesman RM, Pritt BS, Maleszewski JJ, Patel R. Laboratory diagnosis of infective endo-carditis. J Clin Microbiol 2017;55:2599–2608. 161. Loyens M, Thuny F, Grisoli D, Fournier PE, Casalta J-P, Vitte J, et al. Link between endo-carditis on porcine bioprosthetic valves and allergy to pork. Int J Cardiol 2013;167: 600–602. 162. Selton-Suty C, Maigrat CH, Devignes J, Goehringer F, Erpelding ML, Alla F, et al. Possible relationship between antiphospholipid antibodies and embolic events in in-fective endocarditis. Heart 2018;104:509–516. 312359 163. Habib G, Badano L, Tribouilloy C, Vilacosta I, Zamorano JL, Galderisi M, et al. Recommendations for the practice of echocardiography in infective endocarditis. Eur J Echocardiogr 2010;11:202–219. 164. Erba PA, Pizzi MN, Roque A, Salaun E, Lancellotti P, Tornos P, et al. Multimodality im-aging in infective endocarditis: an imaging team within the endocarditis team. Circulation 2019;140:1753–1765. 165. Ostergaard L, Vejlstrup N, Kober L, Fosbol EL, Sondergaard L, Ihlemann N. Diagnostic potential of intracardiac echocardiography in patients with suspected prosthetic valve endocarditis. J Am Soc Echocardiogr 2019;32:1558–1564.e3. echo.2019.06.016 166. Bai AD, Steinberg M, Showler A, Burry L, Bhatia RS, Tomlinson GA, et al. Diagnostic accuracy of transthoracic echocardiography for infective endocarditis findings using transesophageal echocardiography as the reference standard: a meta-analysis. J Am Soc Echocardiogr 2017;30:639–646.e8. 167. Chambers J, Sandoe J, Ray S, Prendergast B, Taggart D, Westaby S, et al. The infective endocarditis team: recommendations from an international working group. Heart 2014;100:524–527. 168. Jain V, Wang TKM, Bansal A, Farwati M, Gad M, Montane B, et al. Diagnostic perform-ance of cardiac computed tomography versus transesophageal echocardiography in in-fective endocarditis: a contemporary comparative meta-analysis. J Cardiovasc Comput Tomogr 2021;15:313–321. 169. Oliveira M, Guittet L, Hamon M, Hamon M. Comparative value of cardiac CT and transesophageal echocardiography in infective endocarditis: a systematic review and meta-analysis. Radiol Cardiothorac Imaging 2020;2:e190189. ryct.2020190189 170. Palraj BR, Baddour LM, Hess EP, Steckelberg JM, Wilson WR, Lahr BD, et al. Predicting Risk of Endocarditis Using a Clinical Tool (PREDICT): scoring system to guide use of echocardiography in the management of Staphylococcus aureus bacteremia. Clin Infect Dis 2015;61:18–28. 171. Sunnerhagen T, Tornell A, Vikbrant M, Nilson B, Rasmussen M. HANDOC: a handy score to determine the need for echocardiography in non-beta-hemolytic streptococ-cal bacteremia. Clin Infect Dis 2018;66:693–698. 172. Berge A, Krantz A, Ostlund H, Naucler P, Rasmussen M. The DENOVA score effi-ciently identifies patients with monomicrobial Enterococcus faecalis bacteremia where echocardiography is not necessary. Infection 2019;47:45–50. s15010-018-1208-3 173. Bouza E, Kestler M, Beca T, Mariscal G, Rodriguez-Creixems M, Bermejo J, et al. The NOVA score: a proposal to reduce the need for transesophageal echocardiography in patients with enterococcal bacteremia. Clin Infect Dis 2015;60:528–535. org/10.1093/cid/ciu872 78 ESC Guidelines Downloaded from by guest on 30 August 2023 174. Rasmussen RV, Host U, Arpi M, Hassager C, Johansen HK, Korup E, et al. Prevalence of infective endocarditis in patients with Staphylococcus aureus bacteraemia: the value of screening with echocardiography. Eur J Echocardiogr 2011;12:414–420. 10.1093/ejechocard/jer023 175. van der Vaart TW, Prins JM, Soetekouw R, van Twillert G, Veenstra J, Herpers BL, et al. Prediction rules for ruling out endocarditis in patients with Staphylococcus aureus bac-teremia. Clin Infect Dis 2022;74:1442–1449. 176. Showler A, Burry L, Bai AD, Steinberg M, Ricciuto DR, Fernandes T, et al. Use of trans-thoracic echocardiography in the management of low-risk Staphylococcus aureus bac-teremia: results from a retrospective multicenter cohort study. JACC Cardiovasc Imaging 2015;8:924–931. 177. Peinado-Acevedo JS, Hurtado-Guerra JJ, Hincapie C, Mesa-Abad J, Uribe-Delgado JR, Giraldo-Ramirez S, et al. Validation of VIRSTA and Predicting Risk of Endocarditis Using a Clinical Tool (PREDICT) scores to determine the priority of echocardiography in patients with Staphylococcus aureus bacteremia. Clin Infect Dis 2021;73:e1151–-e1157. 178. Chamat-Hedemand S, Bruun NE, Ostergaard L, Arpi M, Fosbol E, Boel J, et al. Proposal for the use of echocardiography in bloodstream infections due to different streptococ-cal species. BMC Infect Dis 2021;21:689. 179. Young WJ, Jeffery DA, Hua A, Primus C, Serafino Wani R, Das S, et al. Echocardiography in patients with infective endocarditis and the impact of diagnostic delays on clinical outcomes. Am J Cardiol 2018;122:650–655. amjcard.2018.04.039 180. Iversen K, Host N, Bruun NE, Elming H, Pump B, Christensen JJ, et al. Partial oral treat-ment of endocarditis. Am Heart J 2013;165:116–122. 2012.11.006 181. MacKay EJ, Zhang B, Augoustides JG, Groeneveld PW, Desai ND. Association of in-traoperative transesophageal echocardiography and clinical outcomes after open car-diac valve or proximal aortic surgery. JAMA Netw Open 2022;5:e2147820. org/10.1001/jamanetworkopen.2021.47820 182. Gonzalez-Juanatey C, Gonzalez-Gay MA, Llorca J, Crespo F, Garcia-Porrua C, Corredoira J, et al. Rheumatic manifestations of infective endocarditis in non-addicts. A 12–year study. Medicine (Baltimore) 2001;80:9–19. 00005792-200101000-00002 183. Vallejo Camazon N, Cediel G, Nunez Aragon R, Mateu L, Llibre C, Sopena N, et al. Short- and long-term mortality in patients with left-sided infective endocarditis not undergoing surgery despite indication. Rev Esp Cardiol (Engl Ed) 2020;73:734–740. 184. Vallejo Camazon N, Mateu L, Cediel G, Escola-Verge L, Fernandez-Hidalgo N, Gurgui Ferrer M, et al. Long-term antibiotic therapy in patients with surgery-indicated not undergoing surgery infective endocarditis. Cardiol J 2021;28:566–578. 10.5603/CJ.a2021.0054 185. Chaosuwannakit N, Makarawate P. Value of cardiac computed tomography angiog-raphy in pre-operative assessment of infective endocarditis. J Cardiothorac Surg 2019; 14:56. 186. Fagman E, Flinck A, Snygg-Martin U, Olaison L, Bech-Hanssen O, Svensson G. Surgical decision-making in aortic prosthetic valve endocarditis: the influence of electrocardiogram-gated computed tomography. Eur J Cardiothorac Surg 2016;50: 1165–1171. 187. Abbara S, Blanke P, Maroules CD, Cheezum M, Choi AD, Han BK, et al. SCCT guide-lines for the performance and acquisition of coronary computed tomographic angiog-raphy: a report of the Society of Cardiovascular Computed Tomography Guidelines Committee: endorsed by the North American Society for Cardiovascular Imaging (NASCI). J Cardiovasc Comput Tomogr 2016;10:435–449. jcct.2016.10.002 188. Parra JA, Hernandez L, Munoz P, Blanco G, Rodriguez-Alvarez R, Vilar DR, et al. Detection of spleen, kidney and liver infarcts by abdominal computed tomography does not affect the outcome in patients with left-side infective endocarditis. Medicine (Baltimore) 2018;97:e11952. 0000000000011952 189. Huang JS, Ho AS, Ahmed A, Bhalla S, Menias CO. Borne identity: CT imaging of vas-cular infections. Emerg Radiol 2011;18:335–343. 0946-7 190. Tonolini M, Petulla M, Bianco R. Mycotic visceral aneurysm complicating infectious endocarditis: imaging diagnosis and follow-up. J Emerg Trauma Shock 2012;5: 201–203. 191. Gonzalez I, Sarria C, Lopez J, Vilacosta I, San Roman A, Olmos C, et al. Symptomatic peripheral mycotic aneurysms due to infective endocarditis: a contemporary profile. Medicine (Baltimore) 2014;93:42–52. 192. Goddard AJ, Tan G, Becker J. Computed tomography angiography for the detection and characterization of intra-cranial aneurysms: current status. Clin Radiol 2005;60: 1221–1236. 193. Hekimian G, Kim M, Passefort S, Duval X, Wolff M, Leport C, et al. Preoperative use and safety of coronary angiography for acute aortic valve infective endocarditis. Heart 2010;96:696–700. 194. Lecomte R, Issa N, Gaborit B, Le Turnier P, Deschanvres C, Asseray N, et al. Risk-benefit assessment of systematic thoracoabdominal-pelvic computed tomog-raphy in infective endocarditis. Clin Infect Dis 2019;69:1605–1612. 1093/cid/ciz014 195. Dursun M, Yilmaz S, Yilmaz E, Yilmaz R, Onur I, Oflaz H, et al. The utility of cardiac MRI in diagnosis of infective endocarditis: preliminary results. Diagn Interv Radiol 2015;21: 28–33. 196. El Ouazzani J, Jandou I, Christophe Thuaire I. Thrombus or vegetation? Importance of cardiac MRI as a diagnostic tool based on case report and literature review. Ann Med Surg (Lond) 2020;60:690–694. 197. Snygg-Martin U, Gustafsson L, Rosengren L, Alsio A, Ackerholm P, Andersson R, et al. Cerebrovascular complications in patients with left-sided infective endocarditis are common: a prospective study using magnetic resonance imaging and neurochemical brain damage markers. Clin Infect Dis 2008;47:23–30. 198. Cooper HA, Thompson EC, Laureno R, Fuisz A, Mark AS, Lin M, et al. Subclinical brain embolization in left-sided infective endocarditis: results from the evaluation by MRI of the brains of patients with left-sided intracardiac solid masses (EMBOLISM) pilot study. Circulation 2009;120:585–591. 834432 199. Chakraborty T, Scharf E, DeSimone D, El Rafei A, Brinjikji W, Baddour LM, et al. Variable significance of brain MRI findings in infective endocarditis and its effect on sur-gical decisions. Mayo Clin Proc 2019;94:1024–1032. 2018.09.015 200. Duval X, Iung B, Klein I, Brochet E, Thabut G, Arnoult F, et al. Effect of early cerebral magnetic resonance imaging on clinical decisions in infective endocarditis: a prospect-ive study. Ann Intern Med 2010;152:497–504,W175. 4819-152-8-201004200-00006 201. Sotero FD, Rosario M, Fonseca AC, Ferro JM. Neurological complications of infective endocarditis. Curr Neurol Neurosci Rep 2019;19:23. 019-0935-x 202. Fujimoto T, Morofuji Y, Matsunaga Y, Horie N, Izumo T, Tateishi Y, et al. Early diagno-sis of infective endocarditis by brain T2-weighted magnetic resonance imaging. Circ J 2018;82:464–468. 203. Haller S, Vernooij MW, Kuijer JPA, Larsson EM, Jager HR, Barkhof F. Cerebral micro-bleeds: imaging and clinical significance. Radiology 2018;287:11–28. 1148/radiol.2018170803 204. Hess A, Klein I, Iung B, Lavallee P, Ilic-Habensus E, Dornic Q, et al. Brain MRI findings in neurologically asymptomatic patients with infective endocarditis. AJNR Am J Neuroradiol 2013;34:1579–1584. 205. Iung B, Tubiana S, Klein I, Messika-Zeitoun D, Brochet E, Lepage L, et al. Determinants of cerebral lesions in endocarditis on systematic cerebral magnetic resonance imaging: a prospective study. Stroke 2013;44:3056–3062. STROKEAHA.113.001470 206. Foreman SC, Schwaiger BJ, Gempt J, Jungmann PM, Kehl V, Delbridge C, et al. MR and CT imaging to optimize CT-guided biopsies in suspected spondylodiscitis. World Neurosurg 2017;99:726–734.e7. 207. Dunbar JA, Sandoe JA, Rao AS, Crimmins DW, Baig W, Rankine JJ. The MRI appear-ances of early vertebral osteomyelitis and discitis. Clin Radiol 2010;65:974–981. 208. Gomes A, Glaudemans A, Touw DJ, van Melle JP, Willems TP, Maass AH, et al. Diagnostic value of imaging in infective endocarditis: a systematic review. Lancet Infect Dis 2017;17:e1–e14. 209. Saby L, Laas O, Habib G, Cammilleri S, Mancini J, Tessonnier L, et al. Positron emission tomography/computed tomography for diagnosis of prosthetic valve endocarditis: in-creased valvular 18F-fluorodeoxyglucose uptake as a novel major criterion. J Am Coll Cardiol 2013;61:2374–2382. 210. Swart LE, Gomes A, Scholtens AM, Sinha B, Tanis W, Lam M, et al. Improving the diag-nostic performance of (18)F-fluorodeoxyglucose positron-emission tomography/ computed tomography in prosthetic heart valve endocarditis. Circulation 2018;138: 1412–1427. 211. Duval X, Le Moing V, Tubiana S, Esposito-Farese M, Ilic-Habensus E, Leclercq F, et al. Impact of systematic whole-body 18F-fluorodeoxyglucose PET/CT on the manage-ment of patients suspected of infective endocarditis: the prospective multicenter tep-vendo study. Clin Infect Dis 2021;73:393–403. 212. Philip M, Tessonier L, Mancini J, Mainardi JL, Fernandez-Gerlinger MP, Lussato D, et al. Comparison between ESC and Duke criteria for the diagnosis of prosthetic valve in-fective endocarditis. JACC Cardiovasc Imaging 2020;13:2605–2615. 1016/j.jcmg.2020.04.011 213. Erba PA, Conti U, Lazzeri E, Sollini M, Doria R, De Tommasi SM, et al. Added value of 99mTc-HMPAO-labeled leukocyte SPECT/CT in the characterization and manage-ment of patients with infectious endocarditis. J Nucl Med 2012;53:1235–1243. 214. Rouzet F, Chequer R, Benali K, Lepage L, Ghodbane W, Duval X, et al. Respective per-formance of 18F-FDG PET and radiolabeled leukocyte scintigraphy for the diagnosis of prosthetic valve endocarditis. J Nucl Med 2014;55:1980–1985. jnumed.114.141895 ESC Guidelines 79 Downloaded from by guest on 30 August 2023 215. Hyafil F, Rouzet F, Lepage L, Benali K, Raffoul R, Duval X, et al. Role of radiolabelled leucocyte scintigraphy in patients with a suspicion of prosthetic valve endocarditis and inconclusive echocardiography. Eur Heart J Cardiovasc Imaging 2013;14:586–594. 216. Holcman K, Szot W, Rubis P, Lesniak-Sobelga A, Hlawaty M, Wisniowska-Smialek S, et al. 99mTc-HMPAO-labeled leukocyte SPECT/CT and transthoracic echocardiog-raphy diagnostic value in infective endocarditis. Int J Cardiovasc Imaging 2019;35: 749–758. 217. Albano D, Dondi F, Gazzilli M, Giubbini R, Bertagna F. Meta-analysis of the diagnostic performance of (18)F-FDG-PET/CT imaging in native valve endocarditis. JACC Cardiovasc Imaging 2021;14:1063–1065. 218. de Camargo RA, Sommer Bitencourt M, Meneghetti JC, Soares J, Goncalves LFT, Buchpiguel CA, et al. The role of 18F-fluorodeoxyglucose positron emission tomog-raphy/computed tomography in the diagnosis of left-sided endocarditis: native vs pros-thetic valves endocarditis. Clin Infect Dis 2020;70:583–594. ciz267 219. Pelletier-Galarneau M, Abikhzer G, Harel F, Dilsizian V. Detection of native and pros-thetic valve endocarditis: incremental attributes of functional FDG PET/CT over mor-phologic imaging. Curr Cardiol Rep 2020;22:93. 01334-w 220. Pizzi MN, Fernandez-Hidalgo N. Optimizing the diagnostic workup of infective endo-carditis: an urgent need for studies focused on defining the decision-making process. J Nucl Cardiol 2020;27:609–611. 221. Boursier C, Duval X, Bourdon A, Imbert L, Mahida B, Chevalier E, et al. ECG-gated cardiac FDG PET acquisitions significantly improve detectability of infective endocar-ditis. JACC Cardiovasc Imaging 2020;13:2691–2693. 06.036 222. Ishikita A, Sakamoto I, Yamamura K, Umemoto S, Nagata H, Kitamura Y, et al. Usefulness of (18)F-fluorodeoxyglucose positron emission tomography/computed tomography in the diagnosis of infective endocarditis in patients with adult congenital heart disease. Circ J 2021;85:1505–1513. 223. Pizzi MN, Dos-Subira L, Roque A, Fernandez-Hidalgo N, Cuellar-Calabria H, Pijuan Domenech A, et al. (18)F-FDG-PET/CT angiography in the diagnosis of infective endo-carditis and cardiac device infection in adult patients with congenital heart disease and prosthetic material. Int J Cardiol 2017;248:396–402. 2017.08.008 224. San S, Ravis E, Tessonier L, Philip M, Cammilleri S, Lavagna F, et al. Prognostic value of (18)F-fluorodeoxyglucose positron emission tomography/computed tomography in infective endocarditis. J Am Coll Cardiol 2019;74:1031–1040. j.jacc.2019.06.050 225. Mikail N, Benali K, Mahida B, Vigne J, Hyafil F, Rouzet F, et al. (18)F-FDG-PET/CT im-aging to diagnose septic emboli and mycotic aneurysms in patients with endocarditis and cardiac device infections. Curr Cardiol Rep 2018;20:14. s11886-018-0956-0 226. Orvin K, Goldberg E, Bernstine H, Groshar D, Sagie A, Kornowski R, et al. The role of FDG-PET/CT imaging in early detection of extra-cardiac complications of infective endocarditis. Clin Microbiol Infect 2015;21:69–76. 08.012 227. Duval X, Iung B. Extracardiac imaging of infective endocarditis. Curr Infect Dis Rep 2017; 19:24. 228. Kouijzer IJ, Bleeker-Rovers CP, Oyen WJ. 18F-FDG PET/CT for the detection of septic embolisms in patients with infectious endocarditis. J Nucl Med 2014;55:1045–1046. 229. Germaini M, Boursier C, Goehringer F, Selton-Suty C, Lefevre B, Roch V, et al. The detection of infectious endocarditis may be enhanced by a repeat FDG-PET while maintaining patients on a ketogenic diet. J Nucl Cardiol 2022;29:3256–3262. https:// doi.org/10.1007/s12350-022-02921-w 230. Slart R, Glaudemans A, Gheysens O, Lubberink M, Kero T, Dweck MR, et al. Procedural recommendations of cardiac PET/CT imaging: standardization in inflammatory-, infective-, infiltrative-, and innervation (4Is)-related cardiovascular dis-eases: a joint collaboration of the EACVI and the EANM. Eur J Nucl Med Mol Imaging 2021;48:1016–1039. 231. Mathieu C, Mikail N, Benali K, Iung B, Duval X, Nataf P, et al. Characterization of (18) F-fluorodeoxyglucose uptake pattern in noninfected prosthetic heart valves. Circ Cardiovasc Imaging 2017;10:e005585. 005585 232. Mistry NP, AlShaheen M, Leung E, Chow B, Wiefels C. Previous bioglue repair mim-icking cardiac infection with (18)F-FDG PET imaging. J Nucl Cardiol 2023;30:420–424. 233. Scholtens AM, Swart LE, Verberne HJ, Tanis W, Lam MG, Budde RP. Confounders in FDG-PET/CT imaging of suspected prosthetic valve endocarditis. JACC Cardiovasc Imaging 2016;9:1462–1465. 234. Roque A, Pizzi MN, Cuellar-Calabria H, Aguade-Bruix S. (18)F-FDG-PET/CT angiog-raphy for the diagnosis of infective endocarditis. Curr Cardiol Rep 2017;19:15. https:// doi.org/10.1007/s11886-017-0824-3 235. Swart LE, Scholtens AM, Tanis W, Nieman K, Bogers A, Verzijlbergen FJ, et al. 18F-fluorodeoxyglucose positron emission/computed tomography and computed tomography angiography in prosthetic heart valve endocarditis: from guidelines to clin-ical practice. Eur Heart J 2018;39:3739–3749. ehx784 236. Pizzi MN, Roque A, Cuellar-Calabria H, Fernandez-Hidalgo N, Ferreira-Gonzalez I, Gonzalez-Alujas MT, et al. (18)F-FDG-PET/CTA of prosthetic cardiac valves and valve- tube grafts: infective versus inflammatory patterns. JACC Cardiovasc Imaging 2016;9: 1224–1227. 237. Amraoui S, Tlili G, Sohal M, Berte B, Hindie E, Ritter P, et al. Contribution of PET im-aging to the diagnosis of septic embolism in patients with pacing lead endocarditis. JACC Cardiovasc Imaging 2016;9:283–290. 238. Calais J, Touati A, Grall N, Laouenan C, Benali K, Mahida B, et al. Diagnostic impact of (18)F-fluorodeoxyglucose positron emission tomography/computed tomography and white blood cell SPECT/computed tomography in patients with suspected cardiac im-plantable electronic device chronic infection. Circ Cardiovasc Imaging 2019;12:e007188. 239. Chen W, Sajadi MM, Dilsizian V. Merits of FDG PET/CT and functional molecular im-aging over anatomic imaging with echocardiography and CT angiography for the diag-nosis of cardiac device infections. JACC Cardiovasc Imaging 2018;11:1679–1691. https:// doi.org/10.1016/j.jcmg.2018.08.026 240. Okazaki S, Yoshioka D, Sakaguchi M, Sawa Y, Mochizuki H, Kitagawa K. Acute ischemic brain lesions in infective endocarditis: incidence, related factors, and postoperative outcome. Cerebrovasc Dis 2013;35:155–162. 241. Kestler M, Munoz P, Rodriguez-Creixems M, Rotger A, Jimenez-Requena F, Mari A, et al. Role of (18)F-FDG PET in patients with infectious endocarditis. J Nucl Med 2014;55:1093–1098. 242. Habib G, Derumeaux G, Avierinos JF, Casalta JP, Jamal F, Volot F, et al. Value and lim-itations of the Duke criteria for the diagnosis of infective endocarditis. J Am Coll Cardiol 1999;33:2023–2029. 243. Vieira ML, Grinberg M, Pomerantzeff PM, Andrade JL, Mansur AJ. Repeated echocar-diographic examinations of patients with suspected infective endocarditis. Heart 2004; 90:1020–1024. 244. Hill EE, Herijgers P, Claus P, Vanderschueren S, Peetermans WE, Herregods MC. Abscess in infective endocarditis: the value of transesophageal echocardiography and outcome: a 5-year study. Am Heart J 2007;154:923–928. 1016/j.ahj.2007.06.028 245. Dahl A, Fowler VG, Miro JM, Bruun NE. Sign of the times: updating infective endocar-ditis diagnostic criteria to recognize Enterococcus faecalis as a typical endocarditis bac-terium. Clin Infect Dis 2022;75:1097–1102. 246. Roque A, Pizzi MN, Fernandez-Hidalgo N, Permanyer E, Cuellar-Calabria H, Romero-Farina G, et al. Morpho-metabolic post-surgical patterns of non-infected prosthetic heart valves by [18F]FDG PET/CTA: “normality” is a possible diagnosis. Eur Heart J Cardiovasc Imaging 2020;21:24–33. 247. Anis HK, Miller EM, George J, Shrestha NK, Klika AK, Kamath AF, et al. Incidence and characteristics of osteoarticular infections in patients with infective endocarditis. Orthopedics 2020;43:24–29. 248. Carbone A, Lieu A, Mouhat B, Santelli F, Philip M, Bohbot Y, et al. Spondylodiscitis complicating infective endocarditis. Heart 2020;106:1914–1918. 1136/heartjnl-2019-316492 249. Arvieux C, Common H. New diagnostic tools for prosthetic joint infection. Orthop Traumatol Surg Res 2019;105:S23–S30. 250. Park LP, Chu VH, Peterson G, Skoutelis A, Lejko-Zupa T, Bouza E, et al. Validated risk score for predicting 6-month mortality in infective endocarditis. J Am Heart Assoc 2016; 5:e003016. 251. Sevilla T, Lopez J, Gomez I, Vilacosta I, Sarria C, Garcia-Granja PE, et al. Evolution of prognosis in left-sided infective endocarditis: a propensity score analysis of 2 decades. J Am Coll Cardiol 2017;69:111–112. 252. Chandrasekar PH, Levine DP, Price S, Rybak MJ. Comparative efficacies of imipenem- cilastatin and vancomycin in experimental aortic valve endocarditis due to methicillin resistant Staphylococcus aureus. J Antimicrob Chemother 1988;21:461–469. org/10.1093/jac/21.4.461 253. Durack DT, Pelletier LL, Petersdorf RG. Chemotherapy of experimental streptococcal endocarditis. II. Synergism between penicillin and streptomycin against penicillin- sensitive streptococci. J Clin Invest 1974;53:829–833. JCI107622 254. Wilson WR, Geraci JE, Wilkowske CJ, Washington JA 2nd. Short-term intramuscular therapy with procaine penicillin plus streptomycin for infective endocarditis due to vir-idans streptococci. Circulation 1978;57:1158–1161. 6.1158 255. Lebeaux D, Fernandez-Hidalgo N, Pilmis B, Tattevin P, Mainardi JL. Aminoglycosides for infective endocarditis: time to say goodbye? Clin Microbiol Infect 2020;26: 723–728. 256. Fernandez-Felix BM, Barca LV, Garcia-Esquinas E, Correa-Perez A, Fernandez-Hidalgo N, Muriel A, et al. Prognostic models for mortality after cardiac surgery in patients with 80 ESC Guidelines Downloaded from by guest on 30 August 2023 infective endocarditis: a systematic review and aggregation of prediction models. Clin Microbiol Infect 2021;27:1422–1430. 257. Hoiby N, Ciofu O, Johansen HK, Song Z-J, Moser C, Jensen PO, et al. The clinical im-pact of bacterial biofilms. Int J Oral Sci 2011;3:55–65. IJOS11026 258. Funane K, Tanaka Y, Hosaka T, Murakami K, Miyazaki T, Shiwa Y, et al. Combined drug resistance mutations substantially enhance enzyme production in paenibacillus agari-devorans. J Bacteriol 2018;200:e00188-18. 259. Knudsen GM, Ng Y, Gram L. Survival of bactericidal antibiotic treatment by a persister subpopulation of Listeria monocytogenes. Appl Environ Microbiol 2013;79:7390–7397. 260. Le Bot A, Lecomte R, Gazeau P, Benezit F, Arvieux C, Ansart S, et al. Is rifampin use associated with better outcome in staphylococcal prosthetic valve endocarditis? A multicenter retrospective study. Clin Infect Dis 2021;72:e249–e255. 10.1093/cid/ciaa1040 261. Shrestha NK, Shah SY, Wang H, Hussain ST, Pettersson GB, Nowacki AS, et al. Rifampin for surgically treated staphylococcal infective endocarditis: a propensity score-adjusted cohort study. Ann Thorac Surg 2016;101:2243–2250. 10.1016/j.athoracsur.2015.11.015 262. Marti-Carvajal AJ, Dayer M, Conterno LO, Gonzalez Garay AG, Marti-Amarista CE. A comparison of different antibiotic regimens for the treatment of infective endocarditis. Cochrane Database Syst Rev 2020;5:CD009880. CD009880.pub3 263. Tissot-Dupont H, Casalta JP, Gouriet F, Hubert S, Salaun E, Habib G, et al. International experts’ practice in the antibiotic therapy of infective endocarditis is not following the guidelines. Clin Microbiol Infect 2017;23:736–739. 03.007 264. Cosgrove SE, Vigliani GA, Fowler VG Jr, Abrutyn E, Corey GR, Levine DP, et al. Initial low-dose gentamicin for Staphylococcus aureus bacteremia and endocarditis is nephro-toxic. Clin Infect Dis 2009;48:713–721. 265. Pericas JM, Nathavitharana R, Garcia-de-la-Maria C, Falces C, Ambrosioni J, Almela M, et al. Endocarditis caused by highly penicillin-resistant viridans group streptococci: still room for vancomycin-based regimens. Antimicrob Agents Chemother 2019;63: e00516-19. 266. Dahl A, Rasmussen RV, Bundgaard H, Hassager C, Bruun LE, Lauridsen TK, et al. Enterococcus faecalis infective endocarditis: a pilot study of the relationship between duration of gentamicin treatment and outcome. Circulation 2013;127:1810–1817. 267. Miro JM, Garcia-de-la-Maria C, Armero Y, Soy D, Moreno A, del Rio A, et al. Addition of gentamicin or rifampin does not enhance the effectiveness of daptomycin in treat-ment of experimental endocarditis due to methicillin-resistant Staphylococcus aureus. Antimicrob Agents Chemother 2009;53:4172–4177. 00051-09 268. Garrigos C, Murillo O, Lora-Tamayo J, Verdaguer R, Tubau F, Cabellos C, et al. Fosfomycin-daptomycin and other fosfomycin combinations as alternative therapies in experimental foreign-body infection by methicillin-resistant Staphylococcus aureus. Antimicrob Agents Chemother 2013;57:606–610. 12 269. Peghin M, Russo A, Givone F, Ingani M, Graziano E, Bassetti M. Should high-dose dap-tomycin be an alternative treatment regimen for enterococcal endocarditis? Infect Dis Ther 2019;8:695–702. 270. Kullar R, Casapao AM, Davis SL, Levine DP, Zhao JJ, Crank CW, et al. A multicentre evaluation of the effectiveness and safety of high-dose daptomycin for the treatment of infective endocarditis. J Antimicrob Chemother 2013;68:2921–2926. 10.1093/jac/dkt294 271. Carugati M, Bayer AS, Miro JM, Park LP, Guimaraes AC, Skoutelis A, et al. High-dose daptomycin therapy for left-sided infective endocarditis: a prospective study from the international collaboration on endocarditis. Antimicrob Agents Chemother 2013;57: 6213–6222. 272. Pujol M, Miro JM, Shaw E, Aguado JM, San-Juan R, Puig-Asensio M, et al. Daptomycin plus fosfomycin versus daptomycin alone for methicillin-resistant Staphylococcus aureus bacteremia and endocarditis: a randomized clinical trial. Clin Infect Dis 2021;72: 1517–1525. 273. West KA, Sheeti A, Tamura MacKay K, Forrest GN. Eosinophilic syndromes asso-ciated with daptomycin use: re-exposure hypersensitivity pneumonitis and prior per-ipheral eosinophilia. Open Forum Infect Dis 2022;9:ofac065. ofid/ofac065 274. Hidalgo-Tenorio C, Vinuesa D, Plata A, Martin Davila P, Iftimie S, Sequera S, et al. DALBACEN cohort: dalbavancin as consolidation therapy in patients with endocardi-tis and/or bloodstream infection produced by Gram-positive cocci. Ann Clin Microbiol Antimicrob 2019;18:30. 275. Spaziante M, Franchi C, Taliani G, D’Avolio A, Pietropaolo V, Biliotti E, et al. Serum bactericidal activity levels monitor to guide intravenous dalbavancin chronic suppres-sive therapy of inoperable staphylococcal prosthetic valve endocarditis: a case report. Open Forum Infect Dis 2019;6:ofz427. 276. Abranches J, Zeng L, Kajfasz JK, Palmer SR, Chakraborty B, Wen ZT, et al. Biology of oral streptococci. Microbiol Spectr 2018;6. GPP3-0042-2018 277. Westling K, Aufwerber E, Ekdahl C, Friman G, Gardlund B, Julander I, et al. Swedish guidelines for diagnosis and treatment of infective endocarditis. Scand J Infect Dis 2007;39:929–946. 278. Gould FK, Denning DW, Elliott TS, Foweraker J, Perry JD, Prendergast BD, et al. Guidelines for the diagnosis and antibiotic treatment of endocarditis in adults: a report of the working party of the British Society for Antimicrobial Chemotherapy. J Antimicrob Chemother 2012;67:269–289. 279. Baddour LM, Wilson WR, Bayer AS, Fowler VG Jr, Tleyjeh IM, Rybak MJ, et al. Infective endocarditis in adults: diagnosis, antimicrobial therapy, and management of complica-tions: a scientific statement for healthcare professionals from the American Heart Association. Circulation 2015;132:1435–1486. 0000000000000296 280. Francioli P, Etienne J, Hoigne R, Thys JP, Gerber A. Treatment of streptococcal endo-carditis with a single daily dose of ceftriaxone sodium for 4 weeks. Efficacy and out-patient treatment feasibility. JAMA 1992;267:264–267. 1992.03480020074034 281. Francioli P, Ruch W, Stamboulian D. Treatment of streptococcal endocarditis with a single daily dose of ceftriaxone and netilmicin for 14 days: a prospective multicenter study. Clin Infect Dis 1995;21:1406–1410. 282. Sexton DJ, Tenenbaum MJ, Wilson WR, Steckelberg JM, Tice AD, Gilbert D, et al. Ceftriaxone once daily for four weeks compared with ceftriaxone plus gentamicin once daily for two weeks for treatment of endocarditis due to penicillin-susceptible streptococci. Endocarditis Treatment Consortium Group. Clin Infect Dis 1998;27: 1470–1474. 283. Cremieux AC, Maziere B, Vallois JM, Ottaviani M, Azancot A, Raffoul H, et al. Evaluation of antibiotic diffusion into cardiac vegetations by quantitative autoradiog-raphy. J Infect Dis 1989;159:938–944. 284. Wilson AP, Gaya H. Treatment of endocarditis with teicoplanin: a retrospective ana-lysis of 104 cases. J Antimicrob Chemother 1996;38:507–521. jac/38.3.507 285. Moet GJ, Dowzicky MJ, Jones RN. Tigecycline (GAR-936) activity against Streptococcus gallolyticus (bovis) and viridans group streptococci. Diagn Microbiol Infect Dis 2007;57: 333–336. 286. Shelburne SA 3rd, Greenberg SB, Aslam S, Tweardy DJ. Successful ceftriaxone therapy of endocarditis due to penicillin non-susceptible viridans streptococci. J Infect 2007;54: e99–e101. 287. Hsu RB, Lin FY. Effect of penicillin resistance on presentation and outcome of none-nterococcal streptococcal infective endocarditis. Cardiology 2006;105:234–239. 288. Knoll B, Tleyjeh IM, Steckelberg JM, Wilson WR, Baddour LM. Infective endocarditis due to penicillin-resistant viridans group streptococci. Clin Infect Dis 2007;44: 1585–1592. 289. Levy CS, Kogulan P, Gill VJ, Croxton MB, Kane JG, Lucey DR. Endocarditis caused by penicillin-resistant viridans streptococci: 2 cases and controversies in therapy. Clin Infect Dis 2001;33:577–579. 290. Pilmis B, Lourtet-Hascoet J, Barraud O, Piau C, Isnard C, Hery-Arnaud G, et al. Be care-ful about MICs to amoxicillin for patients with streptococci-related infective endocar-ditis. Int J Antimicrob Agents 2019;53:850–854. 2019.03.002 291. Matsuo T, Mori N, Sakurai A, Kanie T, Mikami Y, Uehara Y, et al. Effectiveness of dap-tomycin against infective endocarditis caused by highly penicillin-resistant viridans group streptococci. IDCases 2021;24:e01113. e01113 292. Hook EW 3rd, Johnson WD Jr. Vancomycin therapy of bacterial endocarditis. Am J Med 1978;65:411–415. 293. Beraud G, Tubiana S, Erpelding ML, Le Moing V, Chirouze C, Gorenne I, et al. Combined bacterial meningitis and infective endocarditis: when should we search for the other when either one is diagnosed? Infect Dis Ther 2022;11:1521–1540. 294. de Egea V, Munoz P, Valerio M, de Alarcon A, Lepe JA, Miro JM, et al. Characteristics and outcome of Streptococcus pneumoniae endocarditis in the xxi century: a systematic review of 111 cases (2000–2013). Medicine (Baltimore) 2015;94:e1562. 10.1097/MD.0000000000001562 295. Martinez E, Miro JM, Almirante B, Aguado JM, Fernandez-Viladrich P, Fernandez-Guerrero ML, et al. Effect of penicillin resistance of Streptococcus pneumo-niae on the presentation, prognosis, and treatment of pneumococcal endocarditis in adults. Clin Infect Dis 2002;35:130–139. 296. Perier A, Puyade M, Revest M, Tattevin P, Bernard L, Lemaignen A, et al. Prognosis of Streptococcus pneumoniae endocarditis in France, a multicenter observational study (2000–2015). Int J Cardiol 2019;288:102–106. 04.048 ESC Guidelines 81 Downloaded from by guest on 30 August 2023 297. Friedland IR, McCracken GH Jr. Management of infections caused by antibiotic-resistant Streptococcus pneumoniae. N Engl J Med 1994;331:377–382. 298. van Ettekoven CN, van de Beek D, Brouwer MC. Update on community-acquired bac-terial meningitis: guidance and challenges. Clin Microbiol Infect 2017;23:601–606. 299. Fernandez Hidalgo N, Gharamti AA, Aznar ML, Almirante B, Yasmin M, Fortes CQ, et al. Beta-hemolytic streptococcal infective endocarditis: characteristics and out-comes from a large, multinational cohort. Open Forum Infect Dis 2020;7:ofaa120. 300. Lefort A, Lortholary O, Casassus P, Selton-Suty C, Guillevin L, Mainardi JL, et al. Comparison between adult endocarditis due to beta-hemolytic streptococci (ser-ogroups A, B, C, and G) and Streptococcus milleri: a multicenter study in France. Arch Intern Med 2002;162:2450–2456. 301. Sambola A, Miro JM, Tornos MP, Almirante B, Moreno-Torrico A, Gurgui M, et al. Streptococcus agalactiae infective endocarditis: analysis of 30 cases and review of the literature, 1962–1998. Clin Infect Dis 2002;34:1576–1584. 340538 302. Adam EL, Siciliano RF, Gualandro DM, Calderaro D, Issa VS, Rossi F, et al. Case series of infective endocarditis caused by Granulicatella species. Int J Infect Dis 2015;31:56–58. 303. Giuliano S, Caccese R, Carfagna P, Vena A, Falcone M, Venditti M. Endocarditis caused by nutritionally variant streptococci: a case report and literature review. Infez Med 2012;20:67–74. 304. Tellez A, Ambrosioni J, Hernandez-Meneses M, Llopis J, Ripa M, Chambers ST, et al. Clinical characteristics and outcome of infective endocarditis due to Abiotrophia and Granulicatella compared to Viridans group streptococci. J Infect 2022;85:137–146. 305. Fernandez-Hidalgo N, Ribera A, Larrosa MN, Viedma E, Origuen J, de Alarcon A, et al. Impact of Staphylococcus aureus phenotype and genotype on the clinical characteristics and outcome of infective endocarditis. A multicentre, longitudinal, prospective, obser-vational study. Clin Microbiol Infect 2018;24:985–991. 2017.12.002 306. Sandoe JA, Kerr KG, Reynolds GW, Jain S. Staphylococcus capitis endocarditis: two cases and review of the literature. Heart 1999;82:e1. 3.e1 307. Cone LA, Sontz EM, Wilson JW, Mitruka SN. Staphylococcus capitis endocarditis due to a transvenous endocardial pacemaker infection: case report and review of Staphylococcus capitis endocarditis. Int J Infect Dis 2005;9:335–339. 1016/j.ijid.2004.08.004 308. Korzeniowski O, Sande MA. Combination antimicrobial therapy for Staphylococcus aureus endocarditis in patients addicted to parenteral drugs and in nonaddicts: a pro-spective study. Ann Intern Med 1982;97:496–503. 97-4-496 309. Apellaniz G, Valdes M, Perez R, Martin-Luengo F, Garcia A, Soria F, et al. [Teicoplanin versus cloxacillin, cloxacillin-gentamycin and vancomycin in the treatment of experi-mental endocarditis caused by methicillin-sensitive Staphylococcus aureus]. Enferm Infecc Microbiol Clin 1991;9:208–210. 310. Huang C, Chen I, Lin L. Comparing the outcomes of ceftaroline plus vancomycin or daptomycin combination therapy versus vancomycin or daptomycin monotherapy in adults with methicillin-resistant Staphylococcus aureus bacteremia–a meta-analysis. Antibiotics (Basel) 2022;11:1104. 311. Veliev A, Nakipoglu Y. Investigation of Staphylococcus lugdunensis and selected coagu-lase negative staphylococci isolated from blood culture bottles and determination of their sensitivities to antibiotics. Pak J Med Sci 2022;38:657–662. 12669/pjms.38.3.4738 312. Chirouze C, Cabell CH, Fowler VG Jr, Khayat N, Olaison L, Miro JM, et al. Prognostic factors in 61 cases of Staphylococcus aureus prosthetic valve infective endocarditis from the International Collaboration on Endocarditis merged database. Clin Infect Dis 2004; 38:1323–1327. 313. Oberbach A, Schlichting N, Hagl C, Lehmann S, Kullnick Y, Friedrich M, et al. Four dec-ades of experience of prosthetic valve endocarditis reflect a high variety of diverse pathogens. Cardiovasc Res 2023;119:410–428. 314. Hassoun A. Treatment of Staphylococcus aureus prosthetic valve endocarditis. Am J Med 2007;120:e9, author reply e11. 315. Marin M, Munoz P, Sanchez M, Del Rosal M, Alcala L, Rodriguez-Creixems M, et al. Molecular diagnosis of infective endocarditis by real-time broad-range polymerase chain reaction (PCR) and sequencing directly from heart valve tissue. Medicine (Baltimore) 2007;86:195–202. 316. Martinez-Selles M, Munoz P, Arnaiz A, Moreno M, Galvez J, Rodriguez-Roda J, et al. Valve surgery in active infective endocarditis: a simple score to predict in-hospital prog-nosis. Int J Cardiol 2014;175:133–137. 317. Munoz P, Kestler M, De Alarcon A, Miro JM, Bermejo J, Rodriguez-Abella H, et al. Current epidemiology and outcome of infective endocarditis: a multicenter, prospect-ive, cohort study. Medicine (Baltimore) 2015;94:e1816. 0000000000001816 318. Saginur R, Stdenis M, Ferris W, Aaron SD, Chan F, Lee C, et al. Multiple combination bactericidal testing of staphylococcal biofilms from implant-associated infections. Antimicrob Agents Chemother 2006;50:55–61. 61.2006 319. Saeed K, Bal AM, Gould IM, David MZ, Dryden M, Giannitsioti E, et al. An update on Staphylococcus aureus infective endocarditis from the International Society of Antimicrobial Chemotherapy (ISAC). Int J Antimicrob Agents 2019;53:9–15. https:// doi.org/10.1016/j.ijantimicag.2018.09.014 320. Ramos-Martinez A, Munoz Serrano A, de Alarcon Gonzalez A, Munoz P, Fernandez-Cruz A, Valerio M, et al. Gentamicin may have no effect on mortality of staphylococcal prosthetic valve endocarditis. J Infect Chemother 2018;24:555–562. 321. Riedel DJ, Weekes E, Forrest GN. Addition of rifampin to standard therapy for treat-ment of native valve infective endocarditis caused by Staphylococcus aureus. Antimicrob Agents Chemother 2008;52:2463–2467. 322. Sader HS, Carvalhaes CG, Mendes RE. Ceftaroline activity against Staphylococcus aur-eus isolated from patients with infective endocarditis, worldwide (2010–2019). Int J Infect Dis 2021;102:524–528. 323. Zasowski EJ, Trinh TD, Claeys KC, Lagnf AM, Bhatia S, Klinker KP, et al. Multicenter cohort study of ceftaroline versus daptomycin for treatment of methicillin-resistant Staphylococcus aureus bloodstream infection. Open Forum Infect Dis 2022;9:ofab606. 324. Sader HS, Mendes RE, Pfaller MA, Flamm RK. Antimicrobial activity of dalbavancin tested against Gram-positive organisms isolated from patients with infective endocar-ditis in US and European medical centres. J Antimicrob Chemother 2019;74:1306–1310. 325. Casapao AM, Davis SL, McRoberts JP, Lagnf AM, Patel S, Kullar R, et al. Evaluation of vancomycin population susceptibility analysis profile as a predictor of outcomes for pa-tients with infective endocarditis due to methicillin-resistant Staphylococcus aureus. Antimicrob Agents Chemother 2014;58:4636–4641. 02820-13 326. Ishaq H, Tariq W, Talha KM, Palraj BRV, Sohail MR, Baddour LM, et al. Association be-tween high vancomycin minimum inhibitory concentration and clinical outcomes in pa-tients with methicillin-resistant Staphylococcus aureus bacteremia: a meta-analysis. Infection 2021;49:803–811. 327. Maraolo AE, Giaccone A, Gentile I, Saracino A, Bavaro DF. Daptomycin versus vanco-mycin for the treatment of methicillin-resistant Staphylococcus aureus bloodstream in-fection with or without endocarditis: a systematic review and meta-analysis. Antibiotics (Basel) 2021;10:1014. 328. Tong SYC, Lye DC, Yahav D, Sud A, Robinson JO, Nelson J, et al. Effect of vancomycin or daptomycin with vs without an antistaphylococcal beta-lactam on mortality, bacter-emia, relapse, or treatment failure in patients with MRSA bacteremia: a randomized clinical trial. JAMA 2020;323:527–537. 329. Rybak MJ, Le J, Lodise TP, Levine DP, Bradley JS, Liu C, et al. Therapeutic monitoring of vancomycin for serious methicillin-resistant Staphylococcus aureus infections: a revised consensus guideline and review by the American Society of Health-System Pharmacists, the Infectious Diseases Society of America, the Pediatric Infectious Diseases Society, and the Society of Infectious Diseases Pharmacists. Clin Infect Dis 2020;71:1361–1364. 330. Fowler VG Jr, Boucher HW, Corey GR, Abrutyn E, Karchmer AW, Rupp ME, et al. Daptomycin versus standard therapy for bacteremia and endocarditis caused by Staphylococcus aureus. N Engl J Med 2006;355:653–665. NEJMoa053783 331. Murray KP, Zhao JJ, Davis SL, Kullar R, Kaye KS, Lephart P, et al. Early use of daptomy-cin versus vancomycin for methicillin-resistant Staphylococcus aureus bacteremia with vancomycin minimum inhibitory concentration >1 mg/L: a matched cohort study. Clin Infect Dis 2013;56:1562–1569. 332. Moore CL, Osaki-Kiyan P, Haque NZ, Perri MB, Donabedian S, Zervos MJ. Daptomycin versus vancomycin for bloodstream infections due to methicillin-resistant Staphylococcus aureus with a high vancomycin minimum inhibitory concentration: a case-control study. Clin Infect Dis 2012;54:51–58. 333. Gould IM, Miro JM, Rybak MJ. Daptomycin: the role of high-dose and combination therapy for Gram-positive infections. Int J Antimicrob Agents 2013;42:202–210. 334. Dhand A, Bayer AS, Pogliano J, Yang SJ, Bolaris M, Nizet V, et al. Use of antistaphylo-coccal beta-lactams to increase daptomycin activity in eradicating persistent bacter-emia due to methicillin-resistant Staphylococcus aureus: role of enhanced daptomycin binding. Clin Infect Dis 2011;53:158–163. 335. Miro JM, Entenza JM, Del Rio A, Velasco M, Castaneda X, Garcia de la Maria C, et al. High-dose daptomycin plus fosfomycin is safe and effective in treating methicillin- susceptible and methicillin-resistant Staphylococcus aureus endocarditis. Antimicrob Agents Chemother 2012;56:4511–4515. 336. del Rio A, Gasch O, Moreno A, Pena C, Cuquet J, Soy D, et al. Efficacy and safety of fosfomycin plus imipenem as rescue therapy for complicated bacteremia and endocar-ditis due to methicillin-resistant Staphylococcus aureus: a multicenter clinical trial. Clin Infect Dis 2014;59:1105–1112. 82 ESC Guidelines Downloaded from by guest on 30 August 2023 337. Tattevin P, Boutoille D, Vitrat V, Van Grunderbeeck N, Revest M, Dupont M, et al. Salvage treatment of methicillin-resistant staphylococcal endocarditis with ceftaroline: a multicentre observational study. J Antimicrob Chemother 2014;69:2010–2013. https:// doi.org/10.1093/jac/dku085 338. Vouillamoz J, Entenza JM, Feger C, Glauser MP, Moreillon P. Quinupristin-dalfopristin combined with beta-lactams for treatment of experimental endocarditis due to Staphylococcus aureus constitutively resistant to macrolide-lincosamide-streptogramin B antibiotics. Antimicrob Agents Chemother 2000;44:1789–1795. 1128/AAC.44.7.1789-1795.2000 339. Guignard B, Entenza JM, Moreillon P. Beta-lactams against methicillin-resistant Staphylococcus aureus. Curr Opin Pharmacol 2005;5:479–489. coph.2005.06.002 340. Jang H-C, Kim S-H, Kim KH, Kim CJ, Lee S, Song K-H, et al. Salvage treatment for per-sistent methicillin-resistant Staphylococcus aureus bacteremia: efficacy of linezolid with or without carbapenem. Clin Infect Dis 2009;49:395–401. 600295 341. Perichon B, Courvalin P. Synergism between beta-lactams and glycopeptides against vana-type methicillin-resistant Staphylococcus aureus and heterologous expression of the vanA operon. Antimicrob Agents Chemother 2006;50:3622–3630. 10.1128/AAC.00410-06 342. Casalta JP, Zaratzian C, Hubert S, Thuny F, Gouriet F, Habib G, et al. Treatment of Staphylococcus aureus endocarditis with high doses of trimethoprim/sulfamethoxazole and clindamycin-preliminary report. Int J Antimicrob Agents 2013;42:190–191. https:// doi.org/10.1016/j.ijantimicag.2013.05.002 343. Tissot-Dupont H, Gouriet F, Oliver L, Jamme M, Casalta JP, Jimeno MT, et al. High-dose trimethoprim-sulfamethoxazole and clindamycin for Staphylococcus aureus endocarditis. Int J Antimicrob Agents 2019;54:143–148. ijantimicag.2019.06.006 344. Lefevre B, Hoen B, Goehringer F, Sime WN, Aissa N, Alauzet C, et al. Antistaphylococcal penicillins vs. cefazolin in the treatment of methicillin-susceptible Staphylococcus aureus infective endocarditis: a quasi-experimental monocentre study. Eur J Clin Microbiol Infect Dis 2021;40:2605–2616. 021-04313-3 345. Jorgensen SCJ, Zasowski EJ, Trinh TD, Lagnf AM, Bhatia S, Sabagha N, et al. Daptomycin plus beta-lactam combination therapy for methicillin-resistant Staphylococcus aureus bloodstream infections: a retrospective, comparative cohort study. Clin Infect Dis 2020;71:1–10. 346. Rand KH, Houck HJ. Synergy of daptomycin with oxacillin and other beta-lactams against methicillin-resistant Staphylococcus aureus. Antimicrob Agents Chemother 2004; 48:2871–2875. 347. Berti AD, Wergin JE, Girdaukas GG, Hetzel SJ, Sakoulas G, Rose WE. Altering the pro-clivity towards daptomycin resistance in methicillin-resistant Staphylococcus aureus using combinations with other antibiotics. Antimicrob Agents Chemother 2012;56: 5046–5053. 348. Sakoulas G, Moise PA, Casapao AM, Nonejuie P, Olson J, Okumura CY, et al. Antimicrobial salvage therapy for persistent staphylococcal bacteremia using dapto-mycin plus ceftaroline. Clin Ther 2014;36:1317–1333. clinthera.2014.05.061 349. Rose WE, Schulz LT, Andes D, Striker R, Berti AD, Hutson PR, et al. Addition of cef-taroline to daptomycin after emergence of daptomycin-nonsusceptible Staphylococcus aureus during therapy improves antibacterial activity. Antimicrob Agents Chemother 2012;56:5296–5302. 350. Chirouze C, Athan E, Alla F, Chu VH, Ralph Corey G, Selton-Suty C, et al. Enterococcal endocarditis in the beginning of the 21st century: analysis from the International Collaboration on Endocarditis-prospective cohort study. Clin Microbiol Infect 2013; 19:1140–1147. 351. Prematunge C, MacDougall C, Johnstone J, Adomako K, Lam F, Robertson J, et al. Vre and vse bacteremia outcomes in the era of effective vre therapy: a systematic review and meta-analysis. Infect Control Hosp Epidemiol 2016;37:26–35. 1017/ice.2015.228 352. DiazGranados CA, Zimmer SM, Klein M, Jernigan JA. Comparison of mortality asso-ciated with vancomycin-resistant and vancomycin-susceptible enterococcal blood-stream infections: a meta-analysis. Clin Infect Dis 2005;41:327–333. 10.1086/430909 353. Lupia T, Roberto G, Scaglione L, Shbaklo N, De Benedetto I, Scabini S, et al. Clinical and microbiological characteristics of bloodstream infections caused by Enterococcus spp. within internal medicine wards: a two-year single-centre experience. Intern Emerg Med 2022;17:1129–1137. 354. Reynolds R, Potz N, Colman M, Williams A, Livermore D, MacGowan A, et al. Antimicrobial susceptibility of the pathogens of bacteraemia in the UK and Ireland 2001–2002: the BSAC bacteraemia resistance surveillance programme. J Antimicrob Chemother 2004;53:1018–1032. 355. Pericas JM, Cervera C, del Rio A, Moreno A, Garcia de la Maria C, Almela M, et al. Changes in the treatment of Enterococcus faecalis infective endocarditis in Spain in the last 15 years: from ampicillin plus gentamicin to ampicillin plus ceftriaxone. Clin Microbiol Infect 2014;20:O1075–O1083. 356. Rottier WC, Pinholt M, van der Bij AK, Arpi M, Blank SN, Nabuurs-Franssen MH, et al. Attributable mortality of vancomycin resistance in ampicillin-resistant Enterococcus faecium bacteremia in Denmark and the Netherlands: a matched cohort study. Infect Control Hosp Epidemiol 2022;43:719–727. 357. Pinholt M, Gumpert H, Bayliss S, Nielsen JB, Vorobieva V, Pedersen M, et al. Genomic analysis of 495 vancomycin-resistant enterococcus faecium reveals broad dissemin-ation of a vanA plasmid in more than 19 clones from Copenhagen, Denmark. J Antimicrob Chemother 2017;72:40–47. 358. Taji A, Heidari H, Shahini-Shamsabadi M, Motamedifar M. High-level resistance to ami-noglycosides among multidrug resistant non-faecalis and non-faecium enterococci. Clin Lab 2022;68. 359. Manoharan H, Lalitha AKV, Mariappan S, Sekar U, Venkataramana GP. Molecular char-acterization of high-level aminoglycoside resistance among enterococcus species. J Lab Physicians 2022;14:290–294. 360. Fernandez-Hidalgo N, Almirante B, Gavalda J, Gurgui M, Pena C, de Alarcon A, et al. Ampicillin plus ceftriaxone is as effective as ampicillin plus gentamicin for treating Enterococcus faecalis infective endocarditis. Clin Infect Dis 2013;56:1261–1268. 361. Gavalda J, Len O, Miro JM, Munoz P, Montejo M, Alarcon A, et al. Brief communication: treatment of Enterococcus faecalis endocarditis with ampicillin plus ceftriaxone. Ann Intern Med 2007;146:574–579. 200704170-00008 362. Miro JM, Pericas JM, del Rio A; Hospital Clinic Endocarditis Study Group. A new era for treating Enterococcus faecalis endocarditis: ampicillin plus short-course gentamicin or ampicillin plus ceftriaxone: that is the question!. Circulation 2013;127:1763–1766. 363. Olaison L, Schadewitz K; Swedish Society of Infectious Diseases Quality Assurance Study Group for Endocarditis. Enterococcal endocarditis in Sweden, 1995–1999: can shorter therapy with aminoglycosides be used? Clin Infect Dis 2002;34:159–166. 364. De Nadai T, Francois M, Sommet A, Dubois D, Metsu D, Grare M, et al. Efficacy of teicoplanin monotherapy following initial standard therapy in Enterococcus faecalis in-fective endocarditis: a retrospective cohort study. Infection 2019;47:463–469. https:// doi.org/10.1007/s15010-019-01290-w 365. Escola-Verge L, Fernandez-Hidalgo N, Rodriguez-Pardo D, Pigrau C, Gonzalez-Lopez JJ, Bartolome R, et al. Teicoplanin for treating enterococcal infective endocarditis: a retrospective observational study from a referral centre in Spain. Int J Antimicrob Agents 2019;53:165–170. 366. Herrera-Hidalgo L, Lomas-Cabezas JM, Lopez-Cortes LE, Luque-Marquez R, Lopez-Cortes LF, Martinez-Marcos FJ, et al. Ampicillin plus ceftriaxone combined ther-apy for Enterococcus faecalis infective endocarditis in OPAT. J Clin Med 2021;11:7. 367. Herrera-Hidalgo L, Lopez-Cortes LE, Luque-Marquez R, Galvez-Acebal J, de Alarcon A, Lopez-Cortes LF, et al. Ampicillin and ceftriaxone solution stability at different tem-peratures in outpatient parenteral antimicrobial therapy. Antimicrob Agents Chemother 2020;64:e00309-20. 368. Erlandson KM, Sun J, Iwen PC, Rupp ME. Impact of the more-potent antibiotics quinupristin-dalfopristin and linezolid on outcome measure of patients with vancomycin-resistant enterococcus bacteremia. Clin Infect Dis 2008;46:30–36. 369. Wang JT, Yang CJ, Yang JL, Lin SW, Chuang YC, Sheng WH, et al. A high daptomycin dose is associated with better bacterial clearance in infections caused by vancomycin- resistant enterococcus faecium regardless of daptomycin minimum inhibitory concen-tration in a rat infective endocarditis model. Microbiol Spectr 2022;10:e0255122. 370. Blackberg A, Morenius C, Olaison L, Berge A, Rasmussen M. Infective endocarditis caused by HACEK group bacteria–a registry-based comparative study. Eur J Clin Microbiol Infect Dis 2021;40:1919–1924. 371. Paturel L, Casalta JP, Habib G, Nezri M, Raoult D. Actinobacillus actinomycetemcomi-tans endocarditis. Clin Microbiol Infect 2004;10:98–118. 0691.2004.00794.x 372. Das M, Badley AD, Cockerill FR, Steckelberg JM, Wilson WR. Infective endocarditis caused by HACEK microorganisms. Annu Rev Med 1997;48:25–33. 10.1146/annurev.med.48.1.25 373. Chambers ST, Murdoch D, Morris A, Holland D, Pappas P, Almela M, et al. HACEK infective endocarditis: characteristics and outcomes from a large, multi-national co-hort. PLoS One 2013;8:e63181. 374. Morpeth S, Murdoch D, Cabell CH, Karchmer AW, Pappas P, Levine D, et al. Non-HACEK Gram-negative bacillus endocarditis. Ann Intern Med 2007;147: 829–835. 375. Lorenz A, Sobhanie MME, Orzel L, Coe K, Wardlow L. Clinical outcomes of combin-ation versus monotherapy for Gram negative non-HACEK infective endocarditis. Diagn Microbiol Infect Dis 2021;101:115504. 2021.115504 ESC Guidelines 83 Downloaded from by guest on 30 August 2023 376. Veve MP, McCurry ED, Cooksey GE, Shorman MA. Epidemiology and outcomes of non-HACEK infective endocarditis in the southeast United States. PLoS One 2020; 15:e0230199. 377. Tattevin P, Watt G, Revest M, Arvieux C, Fournier PE. Update on blood culture- negative endocarditis. Med Mal Infect 2015;45:1–8. 2014.11.003 378. Houpikian P, Raoult D. Blood culture-negative endocarditis in a reference center: etio-logic diagnosis of 348 cases. Medicine (Baltimore) 2005;84:162–173. 1097/01.md.0000165658.82869.17 379. Raoult D, Fournier PE, Vandenesch F, Mainardi JL, Eykyn SJ, Nash J, et al. Outcome and treatment of Bartonella endocarditis. Arch Intern Med 2003;163:226–230. org/10.1001/archinte.163.2.226 380. Dajani AS, Taubert KA, Wilson W, Bolger AF, Bayer A, Ferrieri P, et al. Prevention of bacterial endocarditis. Recommendations by the American Heart Association. Circulation 1997;96:358–366. 381. Rolain JM, Brouqui P, Koehler JE, Maguina C, Dolan MJ, Raoult D. Recommendations for treatment of human infections caused by Bartonella species. Antimicrob Agents Chemother 2004;48:1921–33. 382. Ghigo E, Capo C, Aurouze M, Tung CH, Gorvel JP, Raoult D, et al. Survival of Tropheryma whipplei, the agent of Whipple’s disease, requires phagosome acidification. Infect Immun 2002;70:1501–1506. 383. Brouqui P, Raoult D. Endocarditis due to rare and fastidious bacteria. Clin Microbiol Rev 2001;14:177–207. 384. Garcia-Alvarez L, Sanz MM, Marin M, Farinas MC, Montejo M, Goikoetxea J, et al. Antimicrobial management of Tropheryma whipplei endocarditis: the Spanish collabor-ation on endocarditis (games) experience. J Antimicrob Chemother 2019;74: 1713–1717. 385. Calderon Parra J, De Castro-Campos D, Munoz Garcia P, Olmedo Samperio M, Marin Arriaza M, De Alarcon A, et al. Non-HACEK Gram negative bacilli endocarditis: ana-lysis of a national prospective cohort. Eur J Intern Med 2021;92:71–78. 10.1016/j.ejim.2021.04.021 386. Tattevin P, Revest M, Lefort A, Michelet C, Lortholary O. Fungal endocarditis: current challenges. Int J Antimicrob Agents 2014;44:290–294. ijantimicag.2014.07.003 387. Smego RA Jr, Ahmad H. The role of fluconazole in the treatment of Candida endocar-ditis: a meta-analysis. Medicine (Baltimore) 2011;90:237–249. MD.0b013e3182259d38 388. Kalokhe AS, Rouphael N, El Chami MF, Workowski KA, Ganesh G, Jacob JT. Aspergillus endocarditis: a review of the literature. Int J Infect Dis 2010;14:e1040–-e1047. 389. Lye DC, Hughes A, O’Brien D, Athan E. Candida glabrata prosthetic valve endocarditis treated successfully with fluconazole plus caspofungin without surgery: a case report and literature review. Eur J Clin Microbiol Infect Dis 2005;24:753–755. 1007/s10096-005-0038-2 390. Valerio M, Camici M, Machado M, Galar A, Olmedo M, Sousa D, et al. Aspergillus endo-carditis in the recent years, report of cases of a multicentric national cohort and litera-ture review. Mycoses 2022;65:362–373. 391. Lee A, Mirrett S, Reller LB, Weinstein MP. Detection of bloodstream infections in adults: how many blood cultures are needed? J Clin Microbiol 2007;45:3546–3548. 392. Paul M, Zemer-Wassercug N, Talker O, Lishtzinsky Y, Lev B, Samra Z, et al. Are all beta-lactams similarly effective in the treatment of methicillin-sensitive Staphylococcus aureus bacteraemia? Clin Microbiol Infect 2011;17:1581–1586. https:// doi.org/10.1111/j.1469-0691.2010.03425.x 393. Braquet P, Alla F, Cornu C, Goehringer F, Piroth L, Chirouze C, et al. Factors asso-ciated with 12 week case-fatality in Staphylococcus aureus bacteraemia: a prospective cohort study. Clin Microbiol Infect 2016;22:948.e1–948.e7. cmi.2016.07.034 394. Willekens R, Puig-Asensio M, Suanzes P, Fernandez-Hidalgo N, Larrosa MN, Gonzalez-Lopez JJ, et al. Empirical use of beta-lactam/beta-lactamase inhibitor combi-nations does not increase mortality compared with cloxacillin and cefazolin in methicillin-susceptible Staphylococcus aureus bacteraemia: a propensity-weighted co-hort study. J Antimicrob Chemother 2022;77:2288–2295. dkac152 395. VanEperen AS, Segreti J. Empirical therapy in methicillin-resistant Staphylococcus aureus infections: an up-to-date approach. J Infect Chemother 2016;22:351–359. org/10.1016/j.jiac.2016.02.012 396. Rezar R, Jirak P, Lichtenauer M, Jung C, Lauten A, Hoppe UC, et al. Partial oral antibiot-ic therapy is non-inferior to intravenous therapy in non-critically ill patients with infect-ive endocarditis: review and meta-analysis. Wien Klin Wochenschr 2020;132:762–769. 397. Spellberg B, Chambers HF, Musher DM, Walsh TL, Bayer AS. Evaluation of a paradigm shift from intravenous antibiotics to oral step-down therapy for the treatment of in-fective endocarditis: a narrative review. JAMA Intern Med 2020;180:769–777. https:// doi.org/10.1001/jamainternmed.2020.0555 398. Rajaratnam D, Rajaratnam R. Outpatient antimicrobial therapy for infective endocar-ditis is safe. Heart Lung Circ 2021;30:207–215. 016 399. Perica SJ, Llopis J, Gonzalez-Ramallo V, Goenaga MA, Munoz P, Garcia-Leoni ME, et al. Outpatient parenteral antibiotic treatment for infective endocarditis: a prospective co-hort study from the games cohort. Clin Infect Dis 2019;69:1690–1700. 10.1093/cid/ciz030 400. Forestier E, Roubaud-Baudron C, Fraisse T, Patry C, Gavazzi G, Hoen B, et al. Comprehensive geriatric assessment in older patients suffering from infective endo-carditis. A prospective multicentric cohort study. Clin Microbiol Infect 2019;25: 1246–1252. 401. Pries-Heje MM, Wiingaard C, Ihlemann N, Gill SU, Bruun NE, Elming H, et al. Five-year outcomes of the partial oral treatment of endocarditis (POET) trial. N Engl J Med 2022; 386:601–602. 402. Tobudic S, Forstner C, Burgmann H, Lagler H, Ramharter M, Steininger C, et al. Dalbavancin as primary and sequential treatment for Gram-positive infective endocar-ditis: 2-year experience at the general hospital of Vienna. Clin Infect Dis 2018;67: 795–798. 403. Iung B, Doco-Lecompte T, Chocron S, Strady C, Delahaye F, Le Moing V, et al. Cardiac surgery during the acute phase of infective endocarditis: discrepancies between European Society of Cardiology guidelines and practices. Eur Heart J 2016;37: 840–848. 404. Ostergaard L, Oestergaard LB, Lauridsen TK, Dahl A, Chaudry M, Gislason G, et al. Long-term causes of death in patients with infective endocarditis who undergo medical therapy only or surgical treatment: a nationwide population-based study. Eur J Cardiothorac Surg 2018;54:860–866. 405. Arminanzas C, Farinas-Alvarez C, Zarauza J, Munoz P, Gonzalez Ramallo V, Martinez Selles M, et al. Role of age and comorbidities in mortality of patients with infective endocarditis. Eur J Intern Med 2019;64:63–71. 006 406. Diab M, Sponholz C, von Loeffelholz C, Scheffel P, Bauer M, Kortgen A, et al. Impact of perioperative liver dysfunction on in-hospital mortality and long-term survival in infect-ive endocarditis patients. Infection 2017;45:857–866. 017-1064-6 407. Nashef SA, Roques F, Sharples LD, Nilsson J, Smith C, Goldstone AR, et al. EuroSCORE II. Eur J Cardiothorac Surg 2012;41:734–744, discussion 744–745. 408. Shahian DM, Jacobs JP, Badhwar V, Kurlansky PA, Furnary AP, Cleveland JC Jr, et al. The Society of Thoracic Surgeons 2018 adult cardiac surgery risk models: Part 1-back-ground, design considerations, and model development. Ann Thorac Surg 2018;105: 1411–1418. 409. Costa MA, Wollmann DR Jr, Campos AC, Cunha CL, Carvalho RG, Andrade DF, et al. Risk index for death by infective endocarditis: a multivariate logistic model. Rev Bras Cir Cardiovasc 2007;22:192–200. 410. De Feo M, Cotrufo M, Carozza A, De Santo LS, Amendolara F, Giordano S, et al. The need for a specific risk prediction system in native valve infective endocarditis surgery. Sci World J 2012;2012:307571. 411. Di Mauro M, Dato GMA, Barili F, Gelsomino S, Sante P, Corte AD, et al. A predictive model for early mortality after surgical treatment of heart valve or prosthesis infective endocarditis. The endoscore. Int J Cardiol 2017;241:97–102. ijcard.2017.03.148 412. Fernandez-Hidalgo N, Ferreria-Gonzalez I, Marsal JR, Ribera A, Aznar ML, de Alarcon A, et al. A pragmatic approach for mortality prediction after surgery in infective endo-carditis: optimizing and refining EuroSCORE. Clin Microbiol Infect 2018;24:1102.e7–-1102.e15. 413. Gaca JG, Sheng S, Daneshmand MA, O’Brien S, Rankin JS, Brennan JM, et al. Outcomes for endocarditis surgery in North America: a simplified risk scoring system. J Thorac Cardiovasc Surg 2011;141:98–106.e2. 414. Gatti G, Perrotti A, Obadia JF, Duval X, Iung B, Alla F, et al. Simple scoring system to predict in-hospital mortality after surgery for infective endocarditis. J Am Heart Assoc 2017;6:e004806. 415. Chu VH, Park LP, Athan E, Delahaye F, Freiberger T, Lamas C, et al. Association be-tween surgical indications, operative risk, and clinical outcome in infective endocarditis: a prospective study from the International Collaboration on Endocarditis. Circulation 2015;131:131–140. 416. Varela Barca L, Fernandez-Felix BM, Navas Elorza E, Mestres CA, Munoz P, Cuerpo-Caballero G, et al. Prognostic assessment of valvular surgery in active infective endocarditis: multicentric nationwide validation of a new score developed from a meta-analysis. Eur J Cardiothorac Surg 2020;57:724–731. ezz328 417. Varela Barca L, Navas Elorza E, Fernandez-Hidalgo N, Moya Mur JL, Muriel Garcia A, Fernandez-Felix BM, et al. Prognostic factors of mortality after surgery in infective endocarditis: systematic review and meta-analysis. Infection 2019;47:879–895. 418. Fernandez-Cisneros A, Hernandez-Meneses M, Llopis J, Sandoval E, Pereda D, Alcocer J, et al. Risk scores’ performance and their impact on operative decision-making in left- 84 ESC Guidelines Downloaded from by guest on 30 August 2023 sided endocarditis: a cohort study. Eur J Clin Microbiol Infect Dis 2023;42:33–42. https:// doi.org/10.1007/s10096-022-04516-2 419. Tornos P, Iung B, Permanyer-Miralda G, Baron G, Delahaye F, Gohlke-Barwolf C, et al. Infective endocarditis in Europe: lessons from the Euro heart survey. Heart 2005;91: 571–575. 420. Kiefer T, Park L, Tribouilloy C, Cortes C, Casillo R, Chu V, et al. Association between valvular surgery and mortality among patients with infective endocarditis complicated by heart failure. JAMA 2011;306:2239–2247. 421. Lalani T, Chu VH, Park LP, Cecchi E, Corey GR, Durante-Mangoni E, et al. In-hospital and 1-year mortality in patients undergoing early surgery for prosthetic valve endocar-ditis. JAMA Intern Med 2013;173:1495–1504. 2013.8203 422. Lopez J, Sevilla T, Vilacosta I, Garcia H, Sarria C, Pozo E, et al. Clinical significance of congestive heart failure in prosthetic valve endocarditis. A multicenter study with 257 patients. Rev Esp Cardiol (Engl Ed) 2013;66:384–390. rec.2012.10.022 423. Nadji G, Rusinaru D, Remadi JP, Jeu A, Sorel C, Tribouilloy C. Heart failure in left-sided native valve infective endocarditis: characteristics, prognosis, and results of surgical treatment. Eur J Heart Fail 2009;11:668–675. 424. Pericas JM, Hernandez-Meneses M, Munoz P, Martinez-Selles M, Alvarez-Uria A, de Alarcon A, et al. Characteristics and outcome of acute heart failure in infective endo-carditis: focus on cardiogenic shock. Clin Infect Dis 2021;73:765–774. 1093/cid/ciab098 425. Mir T, Uddin M, Qureshi WT, Regmi N, Tleyjeh IM, Saydain G. Predictors of complica-tions secondary to infective endocarditis and their associated outcomes: a large cohort study from the national emergency database (2016–2018). Infect Dis Ther 2022;11: 305–321. 426. Aksoy O, Sexton DJ, Wang A, Pappas PA, Kourany W, Chu V, et al. Early surgery in patients with infective endocarditis: a propensity score analysis. Clin Infect Dis 2007; 44:364–372. 427. Habib G, Tribouilloy C, Thuny F, Giorgi R, Brahim A, Amazouz M, et al. Prosthetic valve endocarditis: who needs surgery? A multicentre study of 104 cases. Heart 2005;91: 954–959. 428. Nadji G, Goissen T, Brahim A, Coviaux F, Lorgeron N, Tribouilloy C. Impact of early surgery on 6-month outcome in acute infective endocarditis. Int J Cardiol 2008;129: 227–232. 429. Thuny F, Beurtheret S, Mancini J, Gariboldi V, Casalta JP, Riberi A, et al. The timing of surgery influences mortality and morbidity in adults with severe complicated infective endocarditis: a propensity analysis. Eur Heart J 2011;32:2027–2033. 1093/eurheartj/ehp089 430. Bohbot Y, Habib G, Laroche C, Stohr E, Chirouze C, Hernandez-Meneses M, et al. Characteristics, management, and outcomes of patients with left-sided infective endo-carditis complicated by heart failure: a substudy of the ESC-EORP EURO-ENDO (European infective endocarditis) registry. Eur J Heart Fail 2022;24:1253–1265. 431. Siciliano RF, Gualandro DM, Bittencourt MS, Paixao M, Marcondes-Braga F, Soeiro AM, et al. Biomarkers for prediction of mortality in left-sided infective endocarditis. Int J Infect Dis 2020;96:25–30. 432. Wei X-B, Liu Y-H, He P-C, Yu D-Q, Zhou Y-L, Tan N, et al. Prognostic value of N-terminal prohormone brain natriuretic peptide for in-hospital and long-term out-comes in patients with infective endocarditis. Eur J Prev Cardiol 2017;24:676–684. 433. Ostergaard L, Dahl A, Bruun NE, Oestergaard LB, Lauridsen TK, Torp-Pedersen C, et al. Valve regurgitation in patients surviving endocarditis and the subsequent risk of heart failure. Heart 2020;106:1015–1022. 315715 434. Revilla A, Lopez J, Vilacosta I, Villacorta E, Rollan MJ, Echevarria JR, et al. Clinical and prognostic profile of patients with infective endocarditis who need urgent surgery. Eur Heart J 2007;28:65–71. 435. Singer M, Deutschman CS, Seymour CW, Shankar-Hari M, Annane D, Bauer M, et al. The third international consensus definitions for sepsis and septic shock (sepsis-3). JAMA 2016;315:801–810. 436. Pericas JM, Hernandez-Meneses M, Munoz P, Alvarez-Uria A, Pinilla-Llorente B, de Alarcon A, et al. Outcomes and risk factors of septic shock in patients with infective endocarditis: a prospective cohort study. Open Forum Infect Dis 2021;8:ofab119. 437. Lopez J, Sevilla T, Vilacosta I, Sarria C, Revilla A, Ortiz C, et al. Prognostic role of per-sistent positive blood cultures after initiation of antibiotic therapy in left-sided infective endocarditis. Eur Heart J 2013;34:1749–1754. ehs379 438. Horstkotte D, Follath F, Gutschik E, Lengyel M, Oto A, Pavie A, et al. Guidelines on prevention, diagnosis and treatment of infective endocarditis executive summary; the task force on infective endocarditis of the European Society of Cardiology. Eur Heart J 2004;25:267–276. 439. Ramanathan A, Witten JC, Gordon SM, Griffin BP, Pettersson GB, Shrestha NK. Factors associated with local invasion in infective endocarditis: a nested case-control study. Clin Microbiol Infect 2021;27:1011–1014. 003 440. Kiyota Y, Della Corte A, Montiero Vieira V, Habchi K, Huang C-C, Della Ratta EE, et al. Risk and outcomes of aortic valve endocarditis among patients with bicuspid and tri-cuspid aortic valves. Open Heart 2017;4:e000545. 2016-000545 441. Forteza A, Centeno J, Ospina V, Lunar IG, Sanchez V, Perez E, et al. Outcomes in aortic and mitral valve replacement with intervalvular fibrous body reconstruction. Ann Thorac Surg 2015;99:838–845. 442. Panagides V, Del Val D, Abdel-Wahab M, Mangner N, Durand E, Ihlemann N, et al. Perivalvular extension of infective endocarditis after transcatheter aortic valve replace-ment. Clin Infect Dis 2022;75:638–646. 443. Sordelli C, Fele N, Mocerino R, Weisz SH, Ascione L, Caso P, et al. Infective endocar-ditis: echocardiographic imaging and new imaging modalities. J Cardiovasc Echogr 2019; 29:149–155. 444. Ramos-Martinez A, Calderon-Parra J, Miro JM, Munoz P, Rodriguez-Abella H, Valerio M, et al. Effect of the type of surgical indication on mortality in patients with infective endocarditis who are rejected for surgical intervention. Int J Cardiol 2019;282:24–30. 445. Anguera I, Miro JM, Vilacosta I, Almirante B, Anguita M, Munoz P, et al. Aorto-cavitary fistulous tract formation in infective endocarditis: clinical and echocardiographic fea-tures of 76 cases and risk factors for mortality. Eur Heart J 2005;26:288–297. 446. Knosalla C, Weng Y, Yankah AC, Siniawski H, Hofmeister J, Hammerschmidt R, et al. Surgical treatment of active infective aortic valve endocarditis with associated perian-nular abscess–11 year results. Eur Heart J 2000;21:490–497. euhj.1999.1877 447. Baddley JW, Benjamin DK Jr, Patel M, Miro J, Athan E, Barsic B, et al. Candida infective endocarditis. Eur J Clin Microbiol Infect Dis 2008;27:519–529. s10096-008-0466-x 448. Ellis ME, Al-Abdely H, Sandridge A, Greer W, Ventura W. Fungal endocarditis: evi-dence in the world literature, 1965–1995. Clin Infect Dis 2001;32:50–62. org/10.1086/317550 449. Remadi JP, Habib G, Nadji G, Brahim A, Thuny F, Casalta JP, et al. Predictors of death and impact of surgery in Staphylococcus aureus infective endocarditis. Ann Thorac Surg 2007;83:1295–1302. 450. Mamtani S, Aljanabi NM, Gupta Rauniyar RP, Acharya A, Malik BH. Candida endocar-ditis: a review of the pathogenesis, morphology, risk factors, and management of an emerging and serious condition. Cureus 2020;12:e6695. cureus.6695 451. Garcia-Cabrera E, Fernandez-Hidalgo N, Almirante B, Ivanova-Georgieva R, Noureddine M, Plata A, et al. Neurological complications of infective endocarditis: risk factors, outcome, and impact of cardiac surgery: a multicenter observational study. Circulation 2013;127:2272–2284. 000813 452. Selton-Suty C, Delahaye F, Tattevin P, Federspiel C, Le Moing V, Chirouze C, et al. Symptomatic and asymptomatic neurological complications of infective endocarditis: impact on surgical management and prognosis. PLoS One 2016;11:e0158522. https:// doi.org/10.1371/journal.pone.0158522 453. Diab M, Guenther A, Sponholz C, Lehmann T, Faerber G, Matz A, et al. Pre-operative stroke and neurological disability do not independently affect short- and long-term mortality in infective endocarditis patients. Clin Res Cardiol 2016;105:847–857. 454. Thuny F, Avierinos JF, Tribouilloy C, Giorgi R, Casalta JP, Milandre L, et al. Impact of cerebrovascular complications on mortality and neurologic outcome during infective endocarditis: a prospective multicentre study. Eur Heart J 2007;28:1155–1161. 455. Vilacosta I, Graupner C, San Roman JA, Sarria C, Ronderos R, Fernandez C, et al. Risk of embolization after institution of antibiotic therapy for infective endocarditis. J Am Coll Cardiol 2002;39:1489–1495. 456. Dickerman SA, Abrutyn E, Barsic B, Bouza E, Cecchi E, Moreno A, et al. The relation-ship between the initiation of antimicrobial therapy and the incidence of stroke in in-fective endocarditis: an analysis from the ICE prospective cohort study (ICE-PCS). Am Heart J 2007;154:1086–1094. 457. Thuny F, Di Salvo G, Belliard O, Avierinos JF, Pergola V, Rosenberg V, et al. Risk of em-bolism and death in infective endocarditis: prognostic value of echocardiography: a prospective multicenter study. Circulation 2005;112:69–75. CIRCULATIONAHA.104.493155 458. Cabell CH, Pond KK, Peterson GE, Durack DT, Corey GR, Anderson DJ, et al. The risk of stroke and death in patients with aortic and mitral valve endocarditis. Am Heart J 2001;142:75–80. 459. Kim D-H, Kang D-H, Lee M-Z, Yun S-C, Kim Y-J, Song J-M, et al. Impact of early surgery on embolic events in patients with infective endocarditis. Circulation 2010;122(11 Suppl):S17–S22. 460. Mohananey D, Mohadjer A, Pettersson G, Navia J, Gordon S, Shrestha N, et al. Association of vegetation size with embolic risk in patients with infective endocarditis: ESC Guidelines 85 Downloaded from by guest on 30 August 2023 a systematic review and meta-analysis. JAMA Intern Med 2018;178:502–510. org/10.1001/jamainternmed.2017.8653 461. Pergola V, Di Salvo G, Habib G, Avierinos JF, Philip E, Vailloud JM, et al. Comparison of clinical and echocardiographic characteristics of Streptococcus bovis endocarditis with that caused by other pathogens. Am J Cardiol 2001;88:871–875. 1016/S0002-9149(01)01914-2 462. Durante Mangoni E, Adinolfi LE, Tripodi MF, Andreana A, Gambardella M, Ragone E, et al. Risk factors for “major” embolic events in hospitalized patients with infective endocarditis. Am Heart J 2003;146:311–316. 94802-7 463. Young WJ, Hoare D, Bvekerwa I, Primus C, Wani RS, Das S, et al. Association of vege-tation size with valve destruction, embolism and mortality. Heart Lung Circ 2021;30: 854–860. 464. Cabezon G, Lopez J, Vilacosta I, Saez C, Garcia-Granja PE, Olmos C, et al. Reassessment of vegetation size as a sole indication for surgery in left-sided infective endocarditis. J Am Soc Echocardiogr 2022;35:570–575. 2021.12.013 465. Rizzi M, Ravasio V, Carobbio A, Mattucci I, Crapis M, Stellini R, et al. Predicting the oc-currence of embolic events: an analysis of 1456 episodes of infective endocarditis from the Italian Study on Endocarditis (SEI). BMC Infect Dis 2014;14:230. 1186/1471-2334-14-230 466. Scheggi V, Alterini B, Olivotto I, Del Pace S, Zoppetti N, Tomberli B, et al. Embolic risk stratification and prognostic impact of early surgery in left-sided infective endocarditis. Eur J Intern Med 2020;78:82–87. 467. Shiue AB, Stancoven AB, Purcell JB, Pinkston K, Wang A, Khera A, et al. Relation of level of B-type natriuretic peptide with outcomes in patients with infective endocardi-tis. Am J Cardiol 2010;106:1011–1015. 468. Sorabella RA, Han SM, Grbic M, Wu YS, Takyama H, Kurlansky P, et al. Early operation for endocarditis complicated by preoperative cerebral emboli is not associated with worsened outcomes. Ann Thorac Surg 2015;100:501–508. athoracsur.2015.03.078 469. Diab M, Franz M, Hagel S, Guenther A, Struve A, Musleh R, et al. Impact of an in- hospital endocarditis team and a state-wide endocarditis network on perioperative outcomes. J Clin Med 2021;10:4734. 470. Hubert S, Thuny F, Resseguier N, Giorgi R, Tribouilloy C, Le Dolley Y, et al. Prediction of symptomatic embolism in infective endocarditis: construction and validation of a risk calculator in a multicenter cohort. J Am Coll Cardiol 2013;62:1384–1392. org/10.1016/j.jacc.2013.07.029 471. Kang D-H, Kim Y-J, Kim S-H, Sun BJ, Kim D-H, Yun S-C, et al. Early surgery versus con-ventional treatment for infective endocarditis. N Engl J Med 2012;366:2466–2473. 472. Ferrera C, Vilacosta I, Fernandez C, Lopez J, Sarria C, Olmos C, et al. Early surgery for acute-onset infective endocarditis. Eur J Cardiothorac Surg 2018;54:1060–1066. https:// doi.org/10.1093/ejcts/ezy208 473. Kim YK, Choi CG, Jung J, Yu SN, Lee JY, Chong YP, et al. Effect of cerebral embolus size on the timing of cardiac surgery for infective endocarditis in patients with neurological complications. Eur J Clin Microbiol Infect Dis 2018;37:545–553. s10096-017-3148-8 474. Dashkevich A, Bratkov G, Li Y, Joskowiak D, Peterss S, Juchem G, et al. Impact of op-erative timing in infective endocarditis with cerebral embolism–the risk of intermediate deterioration. J Clin Med 2021;10:2136. 475. Hill EE, Herregods MC, Vanderschueren S, Claus P, Peetermans WE, Herijgers P. Outcome of patients requiring valve surgery during active infective endocarditis. Ann Thorac Surg 2008;85:1564–1569. 476. Anguera I, Miro JM, Cabell CH, Abrutyn E, Fowler VG Jr, Hoen B, et al. Clinical char-acteristics and outcome of aortic endocarditis with periannular abscess in the International Collaboration on Endocarditis merged database. Am J Cardiol 2005;96: 976–981. 477. Handa K, Yoshioka D, Toda K, Yokoyama J-Y, Samura T, Suzuki K, et al. Surgical results for infective endocarditis complicated with cardiogenic shock. Circ J 2020;84:926–934. 478. Fosbol EL, Park LP, Chu VH, Athan E, Delahaye F, Freiberger T, et al. The association between vegetation size and surgical treatment on 6-month mortality in left-sided in-fective endocarditis. Eur Heart J 2019;40:2243–2251. eurheartj/ehz204 479. Yanagawa B, Pettersson GB, Habib G, Ruel M, Saposnik G, Latter DA, et al. Surgical management of infective endocarditis complicated by embolic stroke: practical recom-mendations for clinicians. Circulation 2016;134:1280–1292. CIRCULATIONAHA.116.024156 480. Ruttmann E, Willeit J, Ulmer H, Chevtchik O, Hofer D, Poewe W, et al. Neurological outcome of septic cardioembolic stroke after infective endocarditis. Stroke 2006;37: 2094–2099. 481. Asaithambi G, Adil MM, Qureshi AI. Thrombolysis for ischemic stroke associated with infective endocarditis: results from the nationwide inpatient sample. Stroke 2013;44: 2917–2919. 482. Chan KL, Dumesnil JG, Cujec B, Sanfilippo AJ, Jue J, Turek MA, et al. A randomized trial of aspirin on the risk of embolic events in patients with infective endocarditis. J Am Coll Cardiol 2003;42:775–780. 483. Chan KL, Tam J, Dumesnil JG, Cujec B, Sanfilippo AJ, Jue J, et al. Effect of long-term aspirin use on embolic events in infective endocarditis. Clin Infect Dis 2008;46: 37–41. 484. Feil K, Kupper C, Tiedt S, Dimitriadis K, Herzberg M, Dorn F, et al. Safety and efficacy of mechanical thrombectomy in infective endocarditis: a matched case-control analysis from the German Stroke Registry-endovascular treatment. Eur J Neurol 2021;28: 861–867. 485. Park W, Ahn JS, Park JC, Kwun BD, Lee DH. Treatment strategy based on experience of treating intracranial infectious aneurysms. World Neurosurg 2017;97:351–359. 486. Davis KA, Huang G, Petty SA, Tan WA, Malaver D, Peacock JE Jr. The effect of preex-isting anticoagulation on cerebrovascular events in left-sided infective endocarditis. Am J Med 2020;133:360–369. 487. Alawieh A, Chaudry MI, Turner RD, Turk AS, Spiotta AM. Infectious intracranial aneur-ysms: a systematic review of epidemiology, management, and outcomes. J Neurointerv Surg 2018;10:708–716. 488. Hui FK, Bain M, Obuchowski NA, Gordon S, Spiotta AM, Moskowitz S, et al. Mycotic aneurysm detection rates with cerebral angiography in patients with infective endocar-ditis. J Neurointerv Surg 2015;7:449–452. 011124 489. Goulenok T, Klein I, Mazighi M, Messika-Zeitoun D, Alexandra JF, Mourvillier B, et al. Infective endocarditis with symptomatic cerebral complications: contribution of cere-bral magnetic resonance imaging. Cerebrovasc Dis 2013;35:327–336. 1159/000348317 490. Walkoff L, Brinjikji W, Rouchaud A, Caroff J, Kallmes DF. Comparing magnetic reson-ance angiography (MRA) and computed tomography angiography (CTA) with conven-tional angiography in the detection of distal territory cerebral mycotic and oncotic aneurysms. Interv Neuroradiol 2016;22:524–528. 6653247 491. Bettencourt S, Ferro JM. Acute ischemic stroke treatment in infective endocarditis: systematic review. J Stroke Cerebrovasc Dis 2020;29:104598. jstrokecerebrovasdis.2019.104598 492. Meshaal MS, Kassem HH, Samir A, Zakaria A, Baghdady Y, Rizk HH. Impact of routine cerebral CT angiography on treatment decisions in infective endocarditis. PLoS One 2015;10:e0118616. 493. White PM, Teasdale EM, Wardlaw JM, Easton V. Intracranial aneurysms: CT angiog-raphy and MR angiography for detection prospective blinded comparison in a large pa-tient cohort. Radiology 2001;219:739–749. r01ma16739 494. Boukobza M, Duval X, Laissy JP. Utility of susceptibility-weighted angiography se-quence in the diagnosis of ruptured infectious aneurysms. World Neurosurg 2021; 149:171–173. 495. Ragulojan R, Grupke S, Fraser JF. Systematic review of endovascular, surgical, and con-servative options for infectious intracranial aneurysms and cardiac considerations. J Stroke Cerebrovasc Dis 2019;28:838–844. dis.2018.11.035 496. Shi H, Parikh NS, Esenwa C, Zampolin R, Shah H, Khasiyev F, et al. Neurological out-comes of patients with mycotic aneurysms in infective endocarditis. Neurohospitalist 2021;11:5–11. 497. Singla A, Fargen K, Blackburn S, Neal D, Martin TD, Hess PJ, et al. National treatment practices in the management of infectious intracranial aneurysms and infective endo-carditis. J Neurointerv Surg 2016;8:741–746. 011834 498. Ohtake M, Tateishi K, Ikegaya N, Iwata J, Yamanaka S, Murata H. Initial treatment strat-egy for intracranial mycotic aneurysms: 2 case reports and literature review. World Neurosurg 2017;106:1051.e9–1051.e16. 499. Wilson WR, Bower TC, Creager MA, Amin-Hanjani S, O’Gara PT, Lockhart PB, et al. Vascular graft infections, mycotic aneurysms, and endovascular infections: a scientific statement from the American Heart Association. Circulation 2016;134:e412–e460. 500. Ting W, Silverman NA, Arzouman DA, Levitsky S. Splenic septic emboli in endocardi-tis. Circulation 1990;82(5 Suppl):IV105–IV109. 501. Robinson SL, Saxe JM, Lucas CE, Arbulu A, Ledgerwood AM, Lucas WF. Splenic ab-scess associated with endocarditis. Surgery 1992;112:781–786, discussion 786–787. 502. Hasan LZ, Shrestha NK, Dang V, Unai S, Pettersson G, El-Hayek K, et al. Surgical infect-ive endocarditis and concurrent splenic abscess requiring splenectomy: a case series and review of the literature. Diagn Microbiol Infect Dis 2020;97:115082. org/10.1016/j.diagmicrobio.2020.115082 503. Elasfar A, AlBaradai A, AlHarfi Z, Alassal M, Ghoneim A, AlGhofaili F. Splenic abscess associated with infective endocarditis; case series. J Saudi Heart Assoc 2015;27: 210–215. 504. Bonfiglioli R, Nanni C, Morigi JJ, Graziosi M, Trapani F, Bartoletti M, et al. (1)(8)F-FDG PET/CT diagnosis of unexpected extracardiac septic embolisms in patients with 86 ESC Guidelines Downloaded from by guest on 30 August 2023 suspected cardiac endocarditis. Eur J Nucl Med Mol Imaging 2013;40:1190–1196. 505. Simsir SA, Cheeseman SH, Lancey RA, Vander Salm TJ, Gammie JS. Staged laparoscop-ic splenectomy and valve replacement in splenic abscess and infective endocarditis. Ann Thorac Surg 2003;75:1635–1637. 506. Akhyari P, Mehrabi A, Adhiwana A, Kamiya H, Nimptsch K, Minol JP, et al. Is simultan-eous splenectomy an additive risk factor in surgical treatment for active endocarditis? Langenbecks Arch Surg 2012;397:1261–1266. 0931-y 507. Chou Y-H, Hsu C-C, Tiu C-M, Chang T. Splenic abscess: sonographic diagnosis and percutaneous drainage or aspiration. Gastrointest Radiol 1992;17:262–266. https:// doi.org/10.1007/BF01888563 508. Farres H, Felsher J, Banbury M, Brody F. Management of splenic abscess in a critically ill patient. Surg Laparosc Endosc Percutan Tech 2004;14:49–52. 00129689-200404000-00001 509. de la Cuerda F, Cecconi A, Martinez P, Cuesta J, Olivera MJ, Jimenez-Borreguero LJ, et al. Myocardial septic seeding secondary to infective endocarditis: diagnosis by cardiac magnetic resonance imaging. Int J Cardiovasc Imaging 2021;37:2545–2547. org/10.1007/s10554-021-02225-z 510. Reyaldeen R, Wahi S, Cole C, Kaye G, Law P, Cooper C, et al. Concurrent native valve infective endocarditis and myocarditis: the key role of (18)F-FDG PET/CT. J Nucl Cardiol 2021;28:1781–1784. 511. Royston AP, Gosling OE. Patient with native valve infective endocarditis and concomi-tant bacterial myopericarditis. BMJ Case Rep 2018;2018:bcr2018224907. org/10.1136/bcr-2018-224907 512. Regueiro A, Falces C, Cervera C, Del Rio A, Pare JC, Mestres CA, et al. Risk factors for pericardial effusion in native valve infective endocarditis and its influence on outcome. Am J Cardiol 2013;112:1646–1651. 513. Regueiro A, Falces C, Pericas JM, Munoz P, Martinez-Selles M, Valerio M, et al. Risk fac-tors of pericardial effusion in native valve infective endocarditis and its influence on outcome: a multicenter prospective cohort study. Int J Cardiol 2018;273:193–198. 514. Padala SK, Cabrera JA, Ellenbogen KA. Anatomy of the cardiac conduction system. Pacing Clin Electrophysiol 2021;44:15–25. 515. Hill TE, Kiehl EL, Shrestha NK, Gordon SM, Pettersson GB, Mohan C, et al. Predictors of permanent pacemaker requirement after cardiac surgery for infective endocarditis. Eur Heart J Acute Cardiovasc Care 2021;10:329–334. 2048872619848661 516. Murillo O, Grau I, Gomez-Junyent J, Cabrera C, Ribera A, Tubau F, et al. Endocarditis associated with vertebral osteomyelitis and septic arthritis of the axial skeleton. Infection 2018;46:245–251. 517. Del Pace S, Scheggi V, Virgili G, Caciolli S, Olivotto I, Zoppetti N, et al. Endocarditis with spondylodiscitis: clinical characteristics and prognosis. BMC Cardiovasc Disord 2021;21:186. 518. Aguilar-Company J, Pigrau C, Fernandez-Hidalgo N, Rodriguez-Pardo D, Falco V, Lung M, et al. Native vertebral osteomyelitis in aged patients: distinctive features. An obser-vational cohort study. Infection 2018;46:679–686. 018-1177-6 519. Koslow M, Kuperstein R, Eshed I, Perelman M, Maor E, Sidi Y. The unique clinical fea-tures and outcome of infectious endocarditis and vertebral osteomyelitis co-infection. Am J Med 2014;127:669.e9–669.e15. 520. Viezens L, Dreimann M, Strahl A, Heuer A, Koepke LG, Bay B, et al. Spontaneous spon-dylodiscitis and endocarditis: interdisciplinary experience from a tertiary institutional case series and proposal of a treatment algorithm. Neurosurg Rev 2022;45: 1335–1342. 521. Behmanesh B, Gessler F, Schnoes K, Dubinski D, Won S-Y, Konczalla J, et al. Infective endocarditis in patients with pyogenic spondylodiscitis: implications for diagnosis and therapy. Neurosurg Focus 2019;46:E2. 522. Tamura K. Clinical characteristics of infective endocarditis with vertebral osteomye-litis. J Infect Chemother 2010;16:260–265. 523. Pigrau C, Almirante B, Flores X, Falco V, Rodriguez D, Gasser I, et al. Spontaneous pyo-genic vertebral osteomyelitis and endocarditis: incidence, risk factors, and outcome. Am J Med 2005;118:1287. 524. Smids C, Kouijzer IJ, Vos FJ, Sprong T, Hosman AJ, de Rooy JW, et al. A comparison of the diagnostic value of MRI and (18)F-FDG-PET/CT in suspected spondylodiscitis. Infection 2017;45:41–49. 525. Castagne B, Soubrier M, Prouteau J, Mrozek N, Lesens O, Tournadre A, et al. A six- week antibiotic treatment of endocarditis with spondylodiscitis is not associated with increased risk of relapse: a retrospective cohort study. Infect Dis Now 2021;51: 253–259. 526. Pola E, Taccari F, Autore G, Giovannenze F, Pambianco V, Cauda R, et al. Multidisciplinary management of pyogenic spondylodiscitis: epidemiological and clinical features, prognostic factors and long-term outcomes in 207 patients. Eur Spine J 2018; 27(Suppl 2):229–236. 527. Thomas P, Allal J, Bontoux D, Rossi F, Poupet JY, Petitalot JP, et al. Rheumatological manifestations of infective endocarditis. Ann Rheum Dis 1984;43:716–720. https:// doi.org/10.1136/ard.43.5.716 528. Ojeda J, Lopez-Lopez L, Gonzalez A, Vila LM. Infective endocarditis initially presenting with a dermatomyositis-like syndrome. BMJ Case Rep 2014;2014:bcr2013200865. 529. Mahr A, Batteux F, Tubiana S, Goulvestre C, Wolff M, Papo T, et al. Brief report: preva-lence of antineutrophil cytoplasmic antibodies in infective endocarditis. Arthritis Rheumatol 2014;66:1672–1677. 530. Lieber SB, Tishler O, Nasrullah K, Fowler ML, Shmerling RH, Paz Z. Clinical features of patients with septic arthritis and echocardiographic findings of infective endocarditis. Infection 2019;47:771–779. 531. Tahon J, Geselle PJ, Vandenberk B, Hill EE, Peetermans WE, Herijgers P, et al. Long-term follow-up of patients with infective endocarditis in a tertiary referral center. Int J Cardiol 2021;331:176–182. 532. Ortiz-Soriano V, Donaldson K, Du G, Li Y, Lambert J, Cleland D, et al. Incidence and cost of acute kidney injury in hospitalized patients with infective endocarditis. J Clin Med 2019;8:927. 533. Khan MZ, Munir MB, Khan MU, Khan SU, Vasudevan A, Balla S. Contemporary trends and outcomes of prosthetic valve infective endocarditis in the United States: insights from the nationwide inpatient sample. Am J Med Sci 2021;362:472–479. org/10.1016/j.amjms.2021.05.014 534. Gagneux-Brunon A, Pouvaret A, Maillard N, Berthelot P, Lutz MF, Cazorla C, et al. Acute kidney injury in infective endocarditis: a retrospective analysis. Med Mal Infect 2019;49:527–533. 535. Von Tokarski F, Lemaignen A, Portais A, Fauchier L, Hennekinne F, Sautenet B, et al. Risk factors and outcomes of early acute kidney injury in infective endocarditis: a retro-spective cohort study. Int J Infect Dis 2020;99:421–427. 2020.08.022 536. Ritchie BM, Hirning BA, Stevens CA, Cohen SA, DeGrado JR. Risk factors for acute kidney injury associated with the treatment of bacterial endocarditis at a tertiary aca-demic medical center. J Chemother 2017;29:292–298. 1120009X.2017.1296916 537. Petersen JK, Jensen AD, Bruun NE, Kamper AL, Butt JH, Havers-Borgersen E, et al. Outcome of dialysis-requiring acute kidney injury in patients with infective endocardi-tis: a nationwide study. Clin Infect Dis 2021;72:e232–e239. ciaa1017 538. Li J, Zhou L, Gong X, Wang Y, Yao D, Li H. Abiotrophia defectiva as a rare cause of mitral valve infective endocarditis with mesenteric arterial branch pseudoaneurysm, splenic infarction, and renal infarction: a case report. Front Med (Lausanne) 2022;9: 780828. 539. Vasconcellos D, Weng B, Wu P, Thompson G, Sutjita M. Staphylococcus hominis infect-ive endocarditis presenting with embolic splenic and renal infarcts and spinal discitis. Case Rep Infect Dis 2022;2022:7183049. 540. Boils CL, Nasr SH, Walker PD, Couser WG, Larsen CP. Update on endocarditis-associated glomerulonephritis. Kidney Int 2015;87:1241–9. org/10.1038/ki.2014.424 541. Paul G, Michels G, Hohmann C, Pfister R, Mader N, Blanke L, et al. Contrast-enhanced ultrasound for the detection of abdominal complications in infective endocarditis: first experience from a prospective cohort. Ultrasound Med Biol 2020;46:2965–2971. 542. Knol WG, Wahadat AR, Roos-Hesselink JW, Van Mieghem NM, Tanis W, Bogers A, et al. Screening for coronary artery disease in early surgical treatment of acute aortic valve infective endocarditis. Interact Cardiovasc Thorac Surg 2021;32:522–529. https:// doi.org/10.1093/icvts/ivaa313 543. Spanneut TA, Paquet P, Bauters C, Modine T, Richardson M, Bonello L, et al. Utility and safety of coronary angiography in patients with acute infective endocarditis who re-quired surgery. J Thorac Cardiovasc Surg 2022;164:905–913.e19. 1016/j.jtcvs.2020.08.117 544. Laperche C, Lairez O, Ferrieres J, Robin G, Gautier M, Lavie Badie Y, et al. Coronary angiography in the setting of acute infective endocarditis requiring surgical treatment. Arch Cardiovasc Dis 2020;113:50–58. 545. Diab M, Lehmann T, Weber C, Petrov G, Luehr M, Akhyari P, et al. Role of concomi-tant coronary artery bypass grafting in valve surgery for infective endocarditis. J Clin Med 2021;10:2867. 546. Ren X, Liu K, Zhang H, Meng Y, Li H, Sun X, et al. Coronary evaluation before heart valvular surgery by using coronary computed tomographic angiography versus invasive coronary angiography. J Am Heart Assoc 2021;10:e019531. JAHA.120.019531 547. Shapira Y, Weisenberg DE, Vaturi M, Sharoni E, Raanani E, Sahar G, et al. The impact of intraoperative transesophageal echocardiography in infective endocarditis. Isr Med Assoc J 2007;9:299–302. 548. Paruk F, Sime FB, Lipman J, Roberts JA. Dosing antibiotic prophylaxis during cardiopul-monary bypass–a higher level of complexity? A structured review. Int J Antimicrob Agents 2017;49:395–402. ESC Guidelines 87 Downloaded from by guest on 30 August 2023 549. Datt V, Wadhhwa R, Sharma V, Virmani S, Minhas HS, Malik S. Vasoplegic syndrome after cardiovascular surgery: a review of pathophysiology and outcome-oriented therapeutic management. J Card Surg 2021;36:3749–3760. jocs.15805 550. Grayling M, Deakin CD. Methylene blue during cardiopulmonary bypass to treat re-fractory hypotension in septic endocarditis. J Thorac Cardiovasc Surg 2003;125: 426–427. 551. Santer D, Miazza J, Koechlin L, Gahl B, Rrahmani B, Hollinger A, et al. Hemoadsorption during cardiopulmonary bypass in patients with endocarditis undergoing valve surgery: a retrospective single-center study. J Clin Med 2021;10:564. jcm10040564 552. Diab M, Lehmann T, Bothe W, Akhyari P, Platzer S, Wendt D, et al. Cytokine hemoad-sorption during cardiac surgery versus standard surgical care for infective endocarditis (REMOVE): results from a multicenter randomized controlled trial. Circulation 2022; 145:959–968. 553. Pettersson GB, Hussain ST, Shrestha NK, Gordon S, Fraser TG, Ibrahim KS, et al. Infective endocarditis: an atlas of disease progression for describing, staging, coding, and understanding the pathology. J Thorac Cardiovasc Surg 2014;147:1142–1149.e2. 554. de Kerchove L, Vanoverschelde JL, Poncelet A, Glineur D, Rubay J, Zech F, et al. Reconstructive surgery in active mitral valve endocarditis: feasibility, safety and durabil-ity. Eur J Cardiothorac Surg 2007;31:592–599. 002 555. Defauw RJ, Tomsic A, van Brakel TJ, Marsan NA, Klautz RJM, Palmen M. A structured approach to native mitral valve infective endocarditis: is repair better than replace-ment? Eur J Cardiothorac Surg 2020;58:544–550. 556. Toyoda N, Itagaki S, Egorova NN, Tannous H, Anyanwu AC, El-Eshmawi A, et al. Real-world outcomes of surgery for native mitral valve endocarditis. J Thorac Cardiovasc Surg 2017;154:1906–1912.e9. 557. Witten JC, Houghtaling PL, Shrestha NK, Gordon SM, Jaber W, Blackstone EH, et al. Aortic allograft infection risk. J Thorac Cardiovasc Surg 2023;165:1303–1315.e9. https:// doi.org/10.1016/j.jtcvs.2021.04.086 558. Witten JC, Durbak E, Houghtaling PL, Unai S, Roselli EE, Bakaeen FG, et al. Performance and durability of cryopreserved allograft aortic valve replacements. Ann Thorac Surg 2021;111:1893–1900. 033 559. Flynn CD, Curran NP, Chan S, Zegri-Reiriz I, Tauron M, Tian DH, et al. Systematic re-view and meta-analysis of surgical outcomes comparing mechanical valve replacement and bioprosthetic valve replacement in infective endocarditis. Ann Cardiothorac Surg 2019;8:587–599. 560. Giambuzzi I, Bonalumi G, Di Mauro M, Roberto M, Corona S, Alamanni F, et al. Surgical aortic mitral curtain replacement: systematic review and metanalysis of early and long- term results. J Clin Med 2021;10:3163. 561. Murphy KM, Vikram HR. Heart transplantation for infective endocarditis: viable option for a limited few? Transpl Infect Dis 2019;21:e13006. 562. Nappi F, Singh SSA, Spadaccio C, Acar C. Revisiting the guidelines and choice the ideal substitute for aortic valve endocarditis. Ann Transl Med 2020;8:952. 21037/atm-20-1522 563. Head SJ, Celik M, Kappetein AP. Mechanical versus bioprosthetic aortic valve replace-ment. Eur Heart J 2017;38:2183–2191. 564. AATS Surgical Treatment of Infective Endocarditis Consensus Guidelines Writing Committee Chairs; Pettersson GB, Coselli JS, Writing C, Pettersson GB, Coselli JS, et al. 2016 the American Association for thoracic surgery (AATS) consensus guide-lines: surgical treatment of infective endocarditis: executive summary. J Thorac Cardiovasc Surg 2017;153:1241–1258.e29. 565. Delahaye F, Chu VH, Altclas J, Barsic B, Delahaye A, Freiberger T, et al. One-year out-come following biological or mechanical valve replacement for infective endocarditis. Int J Cardiol 2015;178:117–123. 566. Cahill TJ, Baddour LM, Habib G, Hoen B, Salaun E, Pettersson GB, et al. Challenges in infective endocarditis. J Am Coll Cardiol 2017;69:325–344. 2016.10.066 567. Samura T, Yoshioka D, Toda K, Sakaniwa R, Yokoyama J, Suzuki K, et al. Emergency valve surgery improves clinical results in patients with infective endocarditis compli-cated with acute cerebral infarction: analysis using propensity score matchingdagger. Eur J Cardiothorac Surg 2019;56:942–949. 568. Bonaros N, Czerny M, Pfausler B, Muller S, Bartel T, Thielmann M, et al. Infective endo-carditis and neurologic events: indications and timing for surgical interventions. Eur Heart J Suppl 2020;22(Suppl M):M19–M25. 569. Barsic B, Dickerman S, Krajinovic V, Pappas P, Altclas J, Carosi G, et al. Influence of the timing of cardiac surgery on the outcome of patients with infective endocarditis and stroke. Clin Infect Dis 2013;56:209–217. 570. Suzuki M, Takanashi S, Ohshima Y, Nagatomo Y, Seki A, Takamisawa I, et al. Critical potential of early cardiac surgery for infective endocarditis with cardio-embolic strokes. Int J Cardiol 2017;227:222–224. 571. Okita Y, Minakata K, Yasuno S, Uozumi R, Sato T, Ueshima K, et al. Optimal timing of surgery for active infective endocarditis with cerebral complications: a Japanese multicentre study. Eur J Cardiothorac Surg 2016;50:374–382. ejcts/ezw035 572. Arregle F, Martel H, Philip M, Gouriet F, Casalta JP, Riberi A, et al. Infective endocarditis with neurological complications: delaying cardiac surgery is associated with worse out-come. Arch Cardiovasc Dis 2021;114:527–536. 004 573. Matthews CR, Hartman T, Madison M, Villelli NW, Namburi N, Colgate CL, et al. Preoperative stroke before cardiac surgery does not increase risk of postoperative stroke. Sci Rep 2021;11:9025. 574. Murai R, Kaji S, Kitai T, Kim K, Ota M, Koyama T, et al. The clinical significance of cere-bral microbleeds in infective endocarditis patients. Semin Thorac Cardiovasc Surg 2019; 31:51–58. 575. Ruttmann E, Abfalterer H, Wagner J, Grimm M, Muller L, Bates K, et al. Endocarditis-related stroke is not a contraindication for early cardiac surgery: an inves-tigation among 440 patients with left-sided endocarditis. Eur J Cardiothorac Surg 2020; 58:1161–1167. 576. Small CN, Laurent D, Lucke-Wold B, Goutnik MA, Yue S, Chalouhi N, et al. Timing surgery and hemorrhagic complications in endocarditis with concomitant cerebral complications. Clin Neurol Neurosurg 2022;214:107171. clineuro.2022.107171 577. Tam DY, Yanagawa B, Verma S, Ruel M, Fremes SE, Mazine A, et al. Early vs late surgery for patients with endocarditis and neurological injury: a systematic review and meta-analysis. Can J Cardiol 2018;34:1185–1199. 05.010 578. Zhang LQ, Cho S-M, Rice CJ, Khoury J, Marquardt RJ, Buletko AB, et al. Valve surgery for infective endocarditis complicated by stroke: surgical timing and perioperative neurological complications. Eur J Neurol 2020;27:2430–2438. 1111/ene.14438 579. Jiad E, Gill SK, Krutikov M, Turner D, Parkinson MH, Curtis C, et al. When the heart rules the head: ischaemic stroke and intracerebral haemorrhage complicating infective endocarditis. Pract Neurol 2017;17:28–34. 001469 580. Misfeld M, Girrbach F, Etz CD, Binner C, Aspern KV, Dohmen PM, et al. Surgery for infective endocarditis complicated by cerebral embolism: a consecutive series of 375 patients. J Thorac Cardiovasc Surg 2014;147:1837–1844. jtcvs.2013.10.076 581. Kume Y, Fujita T, Fukushima S, Shimahara Y, Matsumoto Y, Yamashita K, et al. Intracranial mycotic aneurysm is associated with cerebral bleeding post-valve surgery for infective endocarditis. Interact Cardiovasc Thorac Surg 2018;27:635–641. org/10.1093/icvts/ivy126 582. Salaun E, Touil A, Hubert S, Casalta JP, Gouriet F, Robinet-Borgomano E, et al. Intracranial haemorrhage in infective endocarditis. Arch Cardiovasc Dis 2018;111: 712–721. 583. Musleh R, Schlattmann P, Caldonazo T, Kirov H, Witte OW, Doenst T, et al. Surgical timing in patients with infective endocarditis and with intracranial hemorrhage: a sys-tematic review and meta-analysis. J Am Heart Assoc 2022;11:e024401. 10.1161/JAHA.121.024401 584. Yoshioka D, Toda K, Sakaguchi T, Okazaki S, Yamauchi T, Miyagawa S, et al. Valve sur-gery in active endocarditis patients complicated by intracranial haemorrhage: the influ-ence of the timing of surgery on neurological outcomes. Eur J Cardiothorac Surg 2014; 45:1082–1088. 585. Manne MB, Shrestha NK, Lytle BW, Nowicki ER, Blackstone E, Gordon SM, et al. Outcomes after surgical treatment of native and prosthetic valve infective endocardi-tis. Ann Thorac Surg 2012;93:489–493. 063 586. Wang A, Chu VH, Athan E, Delahaye F, Freiberger T, Lamas C, et al. Association be-tween the timing of surgery for complicated, left-sided infective endocarditis and sur-vival. Am Heart J 2019;210:108–116. 587. Champey J, Pavese P, Bouvaist H, Kastler A, Krainik A, Francois P. Value of brain MRI in infective endocarditis: a narrative literature review. Eur J Clin Microbiol Infect Dis 2016; 35:159–168. 588. Martinez-Selles M, Munoz P, Estevez A, del Castillo R, Garcia-Fernandez MA, Rodriguez-Creixems M, et al. Long-term outcome of infective endocarditis in non- intravenous drug users. Mayo Clin Proc 2008;83:1213–1217. 83.11.1213 589. Thuny F, Giorgi R, Habachi R, Ansaldi S, Le Dolley Y, Casalta JP, et al. Excess mortality and morbidity in patients surviving infective endocarditis. Am Heart J 2012;164: 94–101. 590. Agrawal A, Virk HUH, Riaz I, Jain D, Tripathi B, Krittanawong C, et al. Predictors of 30-day re-admissions in patients with infective endocarditis: a national population based cohort study. Rev Cardiovasc Med 2020;21:123–127. j.rcm.2020.01.552 591. Alagna L, Park LP, Nicholson BP, Keiger AJ, Strahilevitz J, Morris A, et al. Repeat endo-carditis: analysis of risk factors based on the International Collaboration on Endocarditis – prospective cohort study. Clin Microbiol Infect 2014;20:566–575. 88 ESC Guidelines Downloaded from by guest on 30 August 2023 592. Fernandez-Hidalgo N, Almirante B, Tornos P, Gonzalez-Alujas MT, Planes AM, Galinanes M, et al. Immediate and long-term outcome of left-sided infective endocar-ditis. A 12-year prospective study from a contemporary cohort in a referral hospital. Clin Microbiol Infect 2012;18:E522–E530. 593. Scheggi V, Merilli I, Marcucci R, Del Pace S, Olivotto I, Zoppetti N, et al. Predictors of mortality and adverse events in patients with infective endocarditis: a retrospective real world study in a surgical centre. BMC Cardiovasc Disord 2021;21:28. org/10.1186/s12872-021-01853-6 594. Yoshioka D, Toda K, Yokoyama JY, Matsuura R, Miyagawa S, Kainuma S, et al. Diabetes mellitus adversely affects mortality and recurrence after valve surgery for infective endocarditis. J Thorac Cardiovasc Surg 2018;155:1021–1029.e5. 1016/j.jtcvs.2017.09.013 595. Citro R, Chan KL, Miglioranza MH, Laroche C, Benvenga RM, Furnaz S, et al. Clinical profile and outcome of recurrent infective endocarditis. Heart 2022;108: 1729–1736. 596. Chu VH, Sexton DJ, Cabell CH, Reller LB, Pappas PA, Singh RK, et al. Repeat infective endocarditis: differentiating relapse from reinfection. Clin Infect Dis 2005;41:406–409. 597. Inagaki K, Lucar J, Blackshear C, Hobbs CV. Methicillin-susceptible and methicillin- resistant Staphylococcus aureus bacteremia: nationwide estimates of 30-day readmis-sion, in-hospital mortality, length of stay, and cost in the United States. Clin Infect Dis 2019;69:2112–2118. 598. David TE, Gavra G, Feindel CM, Regesta T, Armstrong S, Maganti MD. Surgical treat-ment of active infective endocarditis: a continued challenge. J Thorac Cardiovasc Surg 2007;133:144–9. 599. Fedoruk LM, Jamieson WR, Ling H, Macnab JS, Germann E, Karim SS, et al. Predictors of recurrence and reoperation for prosthetic valve endocarditis after valve replace-ment surgery for native valve endocarditis. J Thorac Cardiovasc Surg 2009;137: 326–333. 600. Moon MR, Miller DC, Moore KA, Oyer PE, Mitchell RS, Robbins RC, et al. Treatment of endocarditis with valve replacement: the question of tissue versus mechanical pros-thesis. Ann Thorac Surg 2001;71:1164–1171. 02665-5 601. Toyoda N, Itagaki S, Tannous H, Egorova NN, Chikwe J. Bioprosthetic versus mech-anical valve replacement for infective endocarditis: focus on recurrence rates. Ann Thorac Surg 2018;106:99–106. 602. Ostergaard L, Dahl A, Fosbol E, Bruun NE, Oestergaard LB, Lauridsen TK, et al. Residual vegetation after treatment for left-sided infective endocarditis and subse-quent risk of stroke and recurrence of endocarditis. Int J Cardiol 2019;293:67–72. 603. Rasmussen TB, Zwisler AD, Moons P, Berg SK. Insufficient living: experiences of recov-ery after infective endocarditis. J Cardiovasc Nurs 2015;30:E11–E19. 1097/JCN.0000000000000144 604. Rasmussen TB, Zwisler AD, Thygesen LC, Bundgaard H, Moons P, Berg SK. High re-admission rates and mental distress after infective endocarditis – results from the na-tional population-based CopenHeart IE survey. Int J Cardiol 2017;235:133–140. https:// doi.org/10.1016/j.ijcard.2017.02.077 605. Rasmussen TB, Zwisler AD, Risom SS, Sibilitz KL, Christensen J, Bundgaard H, et al. Comprehensive cardiac rehabilitation for patients following infective endocarditis: re-sults of the randomized CopenHeartIE trial. Eur J Cardiovasc Nurs 2022;21:261–270. 606. Price CN, Solomon DA, Johnson JA, Montgomery MW, Martin B, Suzuki J. Feasibility and safety of outpatient parenteral antimicrobial therapy in conjunction with addiction treatment for people who inject drugs. J Infect Dis 2020;222(Suppl 5):S494–S498. 607. Kimmel SD, Walley AY, Li Y, Linas BP, Lodi S, Bernson D, et al. Association of treat-ment with medications for opioid use disorder with mortality after hospitalization for injection drug use-associated infective endocarditis. JAMA Netw Open 2020;3: e2016228. 608. Hays LH. Infective endocarditis: call for education of adults with CHD: review of the evidence. Cardiol Young 2016;26:426–430. 2395 609. Abraham LN, Sibilitz KL, Berg SK, Tang LH, Risom SS, Lindschou J, et al. Exercise-based cardiac rehabilitation for adults after heart valve surgery. Cochrane Database Syst Rev 2021;5:CD010876. 610. Abegaz TM, Bhagavathula AS, Gebreyohannes EA, Mekonnen AB, Abebe TB. Short- and long-term outcomes in infective endocarditis patients: a systematic review and meta-analysis. BMC Cardiovasc Disord 2017;17:291. 017-0729-5 611. Ternhag A, Cederstrom A, Torner A, Westling K. A nationwide cohort study of mor-tality risk and long-term prognosis in infective endocarditis in Sweden. PLoS One 2013; 8:e67519. 612. Collonnaz M, Erpelding ML, Alla F, Goehringer F, Delahaye F, Iung B, et al. Impact of referral bias on prognostic studies outcomes: insights from a population-based cohort study on infective endocarditis. Ann Epidemiol 2021;54:29–37. j.annepidem.2020.09.008 613. Mokhles MM, Ciampichetti I, Head SJ, Takkenberg JJ, Bogers AJ. Survival of surgically treated infective endocarditis: a comparison with the general Dutch population. Ann Thorac Surg 2011;91:1407–1412. 614. Straw S, Baig MW, Gillott R, Wu J, Witte KK, O’Regan DJ, et al. Long-term outcomes are poor in intravenous drug users following infective endocarditis, even after surgery. Clin Infect Dis 2020;71:564–571. 615. Vongpatanasin W, Hillis LD, Lange RA. Prosthetic heart valves. N Engl J Med 1996;335: 407–416. 616. Moreillon P, Que YA. Infective endocarditis. Lancet 2004;363:139–149. 10.1016/S0140-6736(03)15266-X 617. Hoen B, Alla F, Selton-Suty C, Beguinot I, Bouvet A, Briancon S, et al. Changing profile of infective endocarditis: results of a 1-year survey in France. JAMA 2002;288:75–81. 618. Selton-Suty C, Celard M, Le Moing V, Doco-Lecompte T, Chirouze C, Iung B, et al. Preeminence of Staphylococcus aureus in infective endocarditis: a 1-year population- based survey. Clin Infect Dis 2012;54:1230–1239. 619. Anantha Narayanan M, Mahfood Haddad T, Kalil AC, Kanmanthareddy A, Suri RM, Mansour G, et al. Early versus late surgical intervention or medical management for infective endocarditis: a systematic review and meta-analysis. Heart 2016;102: 950–957. 620. Brennan JM, Edwards FH, Zhao Y, O’Brien S, Booth ME, Dokholyan RS, et al. Long-term safety and effectiveness of mechanical versus biologic aortic valve pros-theses in older patients: results from the Society of Thoracic Surgeons adult cardiac surgery national database. Circulation 2013;127:1647–1655. CIRCULATIONAHA.113.002003 621. Wang A, Athan E, Pappas PA, Fowler VG Jr, Olaison L, Pare C, et al. Contemporary clinical profile and outcome of prosthetic valve endocarditis. JAMA 2007;297: 1354–1361. 622. Lopez J, Revilla A, Vilacosta I, Villacorta E, Gonzalez-Juanatey C, Gomez I, et al. Definition, clinical profile, microbiological spectrum, and prognostic factors of early-onset prosthetic valve endocarditis. Eur Heart J 2007;28:760–765. org/10.1093/eurheartj/ehl486 623. Moriyama N, Laakso T, Biancari F, Raivio P, Jalava MP, Jaakkola J, et al. Prosthetic valve endocarditis after transcatheter or surgical aortic valve replacement with a bioprosth-esis: results from the FinnValve registry. EuroIntervention 2019;15:e500–e507. https:// doi.org/10.4244/EIJ-D-19-00247 624. Berisha B, Ragnarsson S, Olaison L, Rasmussen M. Microbiological etiology in prosthet-ic valve endocarditis: a nationwide registry study. J Intern Med 2022;292:428–437. 625. Kohler P, Kuster SP, Bloemberg G, Schulthess B, Frank M, Tanner FC, et al. Healthcare-associated prosthetic heart valve, aortic vascular graft, and disseminated Mycobacterium chimaera infections subsequent to open heart surgery. Eur Heart J 2015;36:2745–2753. 626. Chamat-Hedemand S, Dahl A, Ostergaard L, Arpi M, Fosbol E, Boel J, et al. Prevalence of infective endocarditis in streptococcal bloodstream infections is dependent on streptococcal species. Circulation 2020;142:720–730. CIRCULATIONAHA.120.046723 627. Expert Panel on Cardiac Imaging; Malik SB, Hsu JY, Hurwitz Koweek LM, Ghoshhajra BB, Beache GM, et al. ACR appropriateness criteria(r) infective endocarditis. J Am Coll Radiol 2021;18:S52–S61. 628. Luciani N, Mossuto E, Ricci D, Luciani M, Russo M, Salsano A, et al. Prosthetic valve endocarditis: predictors of early outcome of surgical therapy. A multicentric study. Eur J Cardiothorac Surg 2017;52:768–774. 629. Glaser N, Jackson V, Holzmann MJ, Franco-Cereceda A, Sartipy U. Prosthetic valve endocarditis after surgical aortic valve replacement. Circulation 2017;136:329–331. 630. Erdem H, Puca E, Ruch Y, Santos L, Ghanem-Zoubi N, Argemi X, et al. Portraying in-fective endocarditis: results of multinational ID-IRI study. Eur J Clin Microbiol Infect Dis 2019;38:1753–1763. 631. Shrestha NK, Shah SY, Hussain ST, Pettersson GB, Griffin BP, Nowacki AS, et al. Association of surgical treatment with survival in patients with prosthetic valve endo-carditis. Ann Thorac Surg 2020;109:1834–1843. 2019.09.015 632. Truninger K, Attenhofer Jost CH, Seifert B, Vogt PR, Follath F, Schaffner A, et al. Long term follow up of prosthetic valve endocarditis: what characteristics identify patients who were treated successfully with antibiotics alone? Heart 1999;82:714–720. https:// doi.org/10.1136/hrt.82.6.714 633. Tornos P, Almirante B, Olona M, Permanyer G, Gonzalez T, Carballo J, et al. Clinical outcome and long-term prognosis of late prosthetic valve endocarditis: a 20-year ex-perience. Clin Infect Dis 1997;24:381–386. 634. Arnold CJ, Johnson M, Bayer AS, Bradley S, Giannitsioti E, Miro JM, et al. Candida in-fective endocarditis: an observational cohort study with a focus on therapy. Antimicrob Agents Chemother 2015;59:2365–2373. 635. Castillo JC, Anguita MP, Torres F, Mesa D, Franco M, Gonzalez E, et al. Long-term prognosis of early and late prosthetic valve endocarditis. Am J Cardiol 2004;93: 1185–1187. ESC Guidelines 89 Downloaded from by guest on 30 August 2023 636. Chu VH, Miro JM, Hoen B, Cabell CH, Pappas PA, Jones P, et al. Coagulase-negative staphylococcal prosthetic valve endocarditis–a contemporary update based on the International Collaboration on Endocarditis: prospective cohort study. Heart 2009; 95:570–576. 637. Jensen AD, Ostergaard L, Petersen JK, Graversen PL, Butt JH, Hadji-Turdeghal K, et al. Temporal trends of mortality in patients with infective endocarditis: a nationwide study. Eur Heart J Qual Care Clin Outcomes 2022;9:24–33. ehjqcco/qcac011 638. Slipczuk L, Codolosa JN, Davila CD, Romero-Corral A, Yun J, Pressman GS, et al. Infective endocarditis epidemiology over five decades: a systematic review. PLoS One 2013;8:e82665. 639. Oliver L, Lavoute C, Giorgi R, Salaun E, Hubert S, Casalta JP, et al. Infective endocarditis in octogenarians. Heart 2017;103:1602–1609. 310853 640. Pazdernik M, Iung B, Mutlu B, Alla F, Riezebos R, Kong W, et al. Surgery and outcome of infective endocarditis in octogenarians: prospective data from the ESC EORP EURO-ENDO registry. Infection 2022;50:1191–1202. s15010-022-01792-0 641. Ragnarsson S, Salto-Alejandre S, Strom A, Olaison L, Rasmussen M. Surgery is under-used in elderly patients with left-sided infective endocarditis: a nationwide registry study. J Am Heart Assoc 2021;10:e020221. 642. Ghanta RK, Pettersson GB. Surgical treatment of infective endocarditis in elderly pa-tients: the importance of shared decision making. J Am Heart Assoc 2021;10:e022186. 643. Lopez J, Revilla A, Vilacosta I, Sevilla T, Villacorta E, Sarria C, et al. Age-dependent pro-file of left-sided infective endocarditis: a 3-center experience. Circulation 2010;121: 892–897. 644. Stortecky S, Heg D, Tueller D, Pilgrim T, Muller O, Noble S, et al. Infective endocarditis after transcatheter aortic valve replacement. J Am Coll Cardiol 2020;75:3020–3030. 645. Regueiro A, Linke A, Latib A, Ihlemann N, Urena M, Walther T, et al. Association be-tween transcatheter aortic valve replacement and subsequent infective endocarditis and in-hospital death. JAMA 2016;316:1083–1092. 2016.12347 646. Kolte D, Goldsweig A, Kennedy KF, Abbott JD, Gordon PC, Sellke FW, et al. Comparison of incidence, predictors, and outcomes of early infective endocarditis after transcatheter aortic valve implantation versus surgical aortic valve replacement in the United States. Am J Cardiol 2018;122:2112–2119. amjcard.2018.08.054 647. Fauchier L, Bisson A, Herbert J, Lacour T, Bourguignon T, Etienne CS, et al. Incidence and outcomes of infective endocarditis after transcatheter aortic valve implantation versus surgical aortic valve replacement. Clin Microbiol Infect 2020;26:1368–1374. 648. Butt JH, Ihlemann N, De Backer O, Sondergaard L, Havers-Borgersen E, Gislason GH, et al. Long-term risk of infective endocarditis after transcatheter aortic valve replace-ment. J Am Coll Cardiol 2019;73:1646–1655. 649. Cahill TJ, Raby J, Jewell PD, Brennan PF, Banning AP, Byrne J, et al. Risk of infective endocarditis after surgical and transcatheter aortic valve replacement. Heart 2022; 108:639–647. 650. Del Val D, Abdel-Wahab M, Linke A, Durand E, Ihlemann N, Urena M, et al. Temporal trends, characteristics, and outcomes of infective endocarditis after transcatheter aor-tic valve replacement. Clin Infect Dis 2021;73:e3750–e3758. cid/ciaa1941 651. Bjursten H, Rasmussen M, Nozohoor S, Gotberg M, Olaison L, Ruck A, et al. Infective endocarditis after transcatheter aortic valve implantation: a nationwide study. Eur Heart J 2019;40:3263–3269. 652. Mentias A, Girotra S, Desai MY, Horwitz PA, Rossen JD, Saad M, et al. Incidence, pre-dictors, and outcomes of endocarditis after transcatheter aortic valve replacement in the United States. JACC Cardiovasc Interv 2020;13:1973–1982. j.jcin.2020.05.012 653. Regueiro A, Linke A, Latib A, Ihlemann N, Urena M, Walther T, et al. Infective endo-carditis following transcatheter aortic valve replacement: comparison of balloon- ver-sus self-expandable valves. Circ Cardiovasc Interv 2019;12:e007938. 1161/CIRCINTERVENTIONS.119.007938 654. Salaun E, Sportouch L, Barral PA, Hubert S, Lavoute C, Casalta AC, et al. Diagnosis of infective endocarditis after TAVR: value of a multimodality imaging approach. JACC Cardiovasc Imaging 2018;11:143–146. 655. Wahadat AR, Tanis W, Swart LE, Scholtens A, Krestin GP, van Mieghem N, et al. Added value of (18)F-FDG-PET/CT and cardiac CTA in suspected transcatheter aortic valve endocarditis. J Nucl Cardiol 2021;28:2072–2082. 019-01963-x 656. Mangner N, Leontyev S, Woitek FJ, Kiefer P, Haussig S, Binner C, et al. Cardiac surgery compared with antibiotics only in patients developing infective endocarditis after transcatheter aortic valve replacement. J Am Heart Assoc 2018;7:e010027. https:// doi.org/10.1161/JAHA.118.010027 657. Del Val D, Trottier M, Alperi A, Muntane-Carol G, Faroux L, Delarochelliere R, et al. (18)F-fluorodeoxyglucose uptake pattern in noninfected transcatheter aortic valves. Circ Cardiovasc Imaging 2020;13:e011749. 120.011749 658. Mangner N, del Val D, Abdel-Wahab M, Crusius L, Durand E, Ihlemann N. Surgical treatment of patients with infective endocarditis after transcatheter aortic valve im-plantation. J Am Coll Cardiol 2022;79:772–785. 056 659. Santos-Martinez S, Alkhodair A, Nombela-Franco L, Saia F, Munoz-Garcia AJ, Gutierrez E, et al. Transcatheter aortic valve replacement for residual lesion of the aor-tic valve following “healed” infective endocarditis. JACC Cardiovasc Interv 2020;13: 1983–1996. 660. Bos D, De Wolf D, Cools B, Eyskens B, Hubrechts J, Boshoff D, et al. Infective endo-carditis in patients after percutaneous pulmonary valve implantation with the stent- mounted bovine jugular vein valve: clinical experience and evaluation of the modified Duke criteria. Int J Cardiol 2021;323:40–46. 058 661. Georgiev S, Ewert P, Eicken A, Hager A, Horer J, Cleuziou J, et al. Munich comparative study: prospective long-term outcome of the transcatheter melody valve versus sur-gical pulmonary bioprosthesis with up to 12 years of follow-up. Circ Cardiovasc Interv 2020;13:e008963. 662. Groning M, Tahri NB, Sondergaard L, Helvind M, Ersboll MK, Orbaek Andersen H. Infective endocarditis in right ventricular outflow tract conduits: a register-based com-parison of homografts, contegra grafts and melody transcatheter valves. Eur J Cardiothorac Surg 2019;56:87–93. 663. Lluri G, Levi DS, Miller E, Hageman A, Sinha S, Sadeghi S, et al. Incidence and outcome of infective endocarditis following percutaneous versus surgical pulmonary valve re-placement. Catheter Cardiovasc Interv 2018;91:277–284. 27312 664. Malekzadeh-Milani S, Houeijeh A, Jalal Z, Hascoet S, Bakloul M, Aldebert P, et al. French national survey on infective endocarditis and the melody valve in percutaneous pul-monary valve implantation. Arch Cardiovasc Dis 2018;111:497–506. 10.1016/j.acvd.2017.10.007 665. McElhinney DB, Zhang Y, Aboulhosn JA, Morray BH, Biernacka EK, Qureshi AM, et al. Multicenter study of endocarditis after transcatheter pulmonary valve replacement. J Am Coll Cardiol 2021;78:575–589. 666. Nordmeyer J, Ewert P, Gewillig M, AlJufan M, Carminati M, Kretschmar O, et al. Acute and midterm outcomes of the post-approval MELODY registry: a multicentre registry of transcatheter pulmonary valve implantation. Eur Heart J 2019;40:2255–2264. 667. O’Donnell C, Holloway R, Tilton E, Stirling J, Finucane K, Wilson N. Infective endocar-ditis following melody valve implantation: comparison with a surgical cohort. Cardiol Young 2017;27:294–301. 668. Van Dijck I, Budts W, Cools B, Eyskens B, Boshoff DE, Heying R, et al. Infective endo-carditis of a transcatheter pulmonary valve in comparison with surgical implants. Heart 2015;101:788–793. 669. Stammnitz C, Huscher D, Bauer UMM, Urban A, Nordmeyer J, Schubert S, et al. Nationwide registry-based analysis of infective endocarditis risk after pulmonary valve replacement. J Am Heart Assoc 2022;11:e022231. 022231 670. Cheung G, Vejlstrup N, Ihlemann N, Arnous S, Franzen O, Bundgaard H, et al. Infective endocarditis following percutaneous pulmonary valve replacement: diagnostic chal-lenges and application of intra-cardiac echocardiography. Int J Cardiol 2013;169: 425–429. 671. Olsen T, Jorgensen OD, Nielsen JC, Thogersen AM, Philbert BT, Johansen JB. Incidence of device-related infection in 97 750 patients: clinical data from the complete Danish device-cohort (1982–2018). Eur Heart J 2019;40:1862–1869. eurheartj/ehz316 672. Da Costa A, Lelievre H, Kirkorian G, Celard M, Chevalier P, Vandenesch F, et al. Role of the preaxillary flora in pacemaker infections: a prospective study. Circulation 1998;97: 1791–1795. 673. Chamis AL, Peterson GE, Cabell CH, Corey GR, Sorrentino RA, Greenfield RA, et al. Staphylococcus aureus bacteremia in patients with permanent pacemakers or implanta-ble cardioverter-defibrillators. Circulation 2001;104:1029–1033. 1161/hc3401.095097 674. Madhavan M, Sohail MR, Friedman PA, Hayes DL, Steckelberg JM, Wilson WR, et al. Outcomes in patients with cardiovascular implantable electronic devices and bacter-emia caused by Gram-positive cocci other than Staphylococcus aureus. Circ Arrhythm Electrophysiol 2010;3:639–645. 675. Sohail MR, Palraj BR, Khalid S, Uslan DZ, Al-Saffar F, Friedman PA, et al. Predicting risk of endovascular device infection in patients with Staphylococcus aureus bacteremia (PREDICT-SAB). Circ Arrhythm Electrophysiol 2015;8:137–144. 1161/CIRCEP.114.002199 676. Uslan DZ, Dowsley TF, Sohail MR, Hayes DL, Friedman PA, Wilson WR, et al. Cardiovascular implantable electronic device infection in patients with 90 ESC Guidelines Downloaded from by guest on 30 August 2023 Staphylococcus aureus bacteremia. Pacing Clin Electrophysiol 2010;33:407–413. https:// doi.org/10.1111/j.1540-8159.2009.02565.x 677. Hussein AA, Baghdy Y, Wazni OM, Brunner MP, Kabbach G, Shao M, et al. Microbiology of cardiac implantable electronic device infections. JACC Clin Electrophysiol 2016;2:498–505. 678. Mateos Gaitan R, Boix-Palop L, Munoz Garcia P, Mestres CA, Marin Arriaza M, Pedraz Prieto A, et al. Infective endocarditis in patients with cardiac implantable electronic de-vices: a nationwide study. Europace 2020;22:1062–1070. europace/euaa076 679. Arora Y, Perez AA, Carrillo RG. Influence of vegetation shape on outcomes in trans-venous lead extractions: does shape matter? Heart Rhythm 2020;17:646–653. https:// doi.org/10.1016/j.hrthm.2019.11.015 680. Esquer Garrigos Z, George MP, Vijayvargiya P, Tan EM, Farid S, Abu Saleh OM, et al. Clinical presentation, management, and outcomes of cardiovascular implantable elec-tronic device infections due to Gram-negative versus Gram-positive bacteria. Mayo Clin Proc 2019;94:1268–1277. 681. Baman JR, Medhekar AN, Jain SK, Knight BP, Harrison LH, Smith B, et al. Management of systemic fungal infections in the presence of a cardiac implantable electronic device: a systematic review. Pacing Clin Electrophysiol 2021;44:159–166. 1111/pace.14090 682. Krahn AD, Longtin Y, Philippon F, Birnie DH, Manlucu J, Angaran P, et al. Prevention of arrhythmia device infection trial: the PADIT trial. J Am Coll Cardiol 2018;72:3098–3109. 683. Birnie DH, Wang J, Alings M, Philippon F, Parkash R, Manlucu J, et al. Risk factors for infections involving cardiac implanted electronic devices. J Am Coll Cardiol 2019;74: 2845–2854. 684. Ahmed FZ, Blomstrom-Lundqvist C, Bloom H, Cooper C, Ellis C, Goette A, et al. Use of healthcare claims to validate the prevention of arrhythmia device infection trial car-diac implantable electronic device infection risk score. Europace 2021;23:1446–1455. 685. Birnie DH, Healey JS, Wells GA, Ayala-Paredes F, Coutu B, Sumner GL, et al. Continued vs. interrupted direct oral anticoagulants at the time of device surgery, in patients with moderate to high risk of arterial thrombo-embolic events (BRUISE CONTROL-2). Eur Heart J 2018;39:3973–3979. ehy413 686. Birnie DH, Healey JS, Wells GA, Verma A, Tang AS, Krahn AD, et al. Pacemaker or defibrillator surgery without interruption of anticoagulation. N Engl J Med 2013;368: 2084–2093. 687. Burri H, Starck C, Auricchio A, Biffi M, Burri M, D’Avila A, et al. EHRA expert consen-sus statement and practical guide on optimal implantation technique for conventional pacemakers and implantable cardioverter-defibrillators: endorsed by the Heart Rhythm Society (HRS), the Asia Pacific Heart Rhythm Society (APHRS), and the Latin-American Heart Rhythm Society (LAHRS). Europace 2021;23:983–1008. 688. Tarakji KG, Mittal S, Kennergren C, Corey R, Poole JE, Schloss E, et al. Antibacterial envelope to prevent cardiac implantable device infection. N Engl J Med 2019;380: 1895–1905. 689. Frausing M, Nielsen JC, Johansen JB, Jorgensen OD, Gerdes C, Olsen T, et al. Rate of device-related infections using an antibacterial envelope in patients undergoing cardiac resynchronization therapy reoperations. Europace 2022;24:421–429. 10.1093/europace/euab207 690. Uslan DZ, Sohail MR, Friedman PA, Hayes DL, Wilson WR, Steckelberg JM, et al. Frequency of permanent pacemaker or implantable cardioverter-defibrillator infection in patients with Gram-negative bacteremia. Clin Infect Dis 2006;43:731–736. https:// doi.org/10.1086/506942 691. Uslan DZ, Sohail MR, St Sauver JL, Friedman PA, Hayes DL, Stoner SM, et al. Permanent pacemaker and implantable cardioverter defibrillator infection: a population-based study. Arch Intern Med 2007;167:669–675. archinte.167.7.669 692. Fowler VG Jr, Li J, Corey GR, Boley J, Marr KA, Gopal AK, et al. Role of echocardiog-raphy in evaluation of patients with Staphylococcus aureus bacteremia: experience in 103 patients. J Am Coll Cardiol 1997;30:1072–1078. 1097(97)00250-7 693. Victor F, De Place C, Camus C, Le Breton H, Leclercq C, Pavin D, et al. Pacemaker lead infection: echocardiographic features, management, and outcome. Heart 1999;81: 82–87. 694. Vilacosta I, Sarria C, San Roman JA, Jimenez J, Castillo JA, Iturralde E, et al. Usefulness of transesophageal echocardiography for diagnosis of infected transvenous permanent pacemakers. Circulation 1994;89:2684–2687. 2684 695. Narducci ML, Pelargonio G, Russo E, Marinaccio L, Di Monaco A, Perna F, et al. Usefulness of intracardiac echocardiography for the diagnosis of cardiovascular im-plantable electronic device-related endocarditis. J Am Coll Cardiol 2013;61: 1398–1405. 696. Golzio PG, Errigo D, Peyracchia M, Gallo E, Frea S, Castagno D, et al. Prevalence and prognosis of lead masses in patients with cardiac implantable electronic devices without infection. J Cardiovasc Med (Hagerstown) 2019;20:372–378. 10.2459/JCM.0000000000000797 697. Erba PA, Sollini M, Conti U, Bandera F, Tascini C, De Tommasi SM, et al. Radiolabeled WBC scintigraphy in the diagnostic workup of patients with suspected device-related infections. JACC Cardiovasc Imaging 2013;6:1075–1086. 2013.08.001 698. Lin AY, Saul T, Aldaas OM, Lupercio F, Ho G, Pollema T, et al. Early versus delayed lead extraction in patients with infected cardiovascular implantable electronic devices. JACC Clin Electrophysiol 2021;7:755–763. 699. Le KY, Sohail MR, Friedman PA, Uslan DZ, Cha SS, Hayes DL, et al. Impact of timing of device removal on mortality in patients with cardiovascular implantable electronic de-vice infections. Heart Rhythm 2011;8:1678–1685. 2011.05.015 700. Durante-Mangoni E, Casillo R, Bernardo M, Caianiello C, Mattucci I, Pinto D, et al. High-dose daptomycin for cardiac implantable electronic device-related infective endocarditis. Clin Infect Dis 2012;54:347–354. 701. Sohail MR, Uslan DZ, Khan AH, Friedman PA, Hayes DL, Wilson WR, et al. Management and outcome of permanent pacemaker and implantable cardioverter- defibrillator infections. J Am Coll Cardiol 2007;49:1851–1859. j.jacc.2007.01.072 702. Huang X-M, Fu H-X, Zhong L, Cao J, Asirvatham SJ, Baddour LM, et al. Outcomes of transvenous lead extraction for cardiovascular implantable electronic device infections in patients with prosthetic heart valves. Circ Arrhythm Electrophysiol 2016;9:e004188. 703. De Ciancio G, Erpelding ML, Filippetti L, Goehringer F, Blangy H, Huttin O, et al. Adherence to diagnostic and therapeutic practice guidelines for suspected cardiac im-plantable electronic device infections. Arch Cardiovasc Dis 2021;114:634–646. https:// doi.org/10.1016/j.acvd.2021.06.010 704. Viganego F, O’Donoghue S, Eldadah Z, Shah MH, Rastogi M, Mazel JA, et al. Effect of early diagnosis and treatment with percutaneous lead extraction on survival in patients with cardiac device infections. Am J Cardiol 2012;109:1466–1471. 1016/j.amjcard.2012.01.360 705. Starck CT, Schaerf RHM, Breitenstein A, Najibi S, Conrad J, Berendt J, et al. Transcatheter aspiration of large pacemaker and implantable cardioverter-defibrillator lead vegetations facilitating safe transvenous lead extraction. Europace 2020;22: 133–138. 706. Golzio PG, Vinci M, Anselmino M, Comoglio C, Rinaldi M, Trevi GP, et al. Accuracy of swabs, tissue specimens, and lead samples in diagnosis of cardiac rhythm management device infections. Pacing Clin Electrophysiol 2009;32(Suppl 1):S76–S80. 10.1111/j.1540-8159.2008.02257.x 707. Nagpal A, Patel R, Greenwood-Quaintance KE, Baddour LM, Lynch DT, Lahr BD, et al. Usefulness of sonication of cardiovascular implantable electronic devices to enhance microbial detection. Am J Cardiol 2015;115:912–917. amjcard.2015.01.017 708. Rohacek M, Erne P, Kobza R, Pfyffer GE, Frei R, Weisser M. Infection of cardiovascular implantable electronic devices: detection with sonication, swab cultures, and blood cultures. Pacing Clin Electrophysiol 2015;38:247–253. 12529 709. Chew D, Somayaji R, Conly J, Exner D, Rennert-May E. Timing of device reimplanta-tion and reinfection rates following cardiac implantable electronic device infection: a systematic review and meta-analysis. BMJ Open 2019;9:e029537. 1136/bmjopen-2019-029537 710. Diemberger I, Biffi M, Lorenzetti S, Martignani C, Raffaelli E, Ziacchi M, et al. Predictors of long-term survival free from relapses after extraction of infected CIED. Europace 2018;20:1018–1027. 711. Chua JD, Wilkoff BL, Lee I, Juratli N, Longworth DL, Gordon SM. Diagnosis and man-agement of infections involving implantable electrophysiologic cardiac devices. Ann Intern Med 2000;133:604–608. 200010170-00011 712. Kawata H, Pretorius V, Phan H, Mulpuru S, Gadiyaram V, Patel J, et al. Utility and safety of temporary pacing using active fixation leads and externalized re-usable permanent pacemakers after lead extraction. Europace 2013;15:1287–1291. 1093/europace/eut045 713. Perrin T, Maille B, Lemoine C, Resseguier N, Franceschi F, Koutbi L, et al. Comparison of epicardial vs. endocardial reimplantation in pacemaker-dependent patients with de-vice infection. Europace 2018;20:e42–e50. 714. Zucchelli G, Barletta V, Della Tommasina V, Viani S, Parollo M, Mazzocchetti L, et al. Micra pacemaker implant after cardiac implantable electronic device extraction: feasi-bility and long-term outcomes. Europace 2019;21:1229–1236. europace/euz160 715. Chung DU, Tauber J, Kaiser L, Schlichting A, Pecha S, Sinning C, et al. Performance and outcome of the subcutaneous implantable cardioverter-defibrillator after transvenous lead extraction. Pacing Clin Electrophysiol 2021;44:247–257. pace.14157 ESC Guidelines 91 Downloaded from by guest on 30 August 2023 716. Joffre J, Dumas G, Aegerter P, Dubee V, Bige N, Preda G, et al. Epidemiology of infect-ive endocarditis in French intensive care units over the 1997–2014 period-from CUB-rea network. Crit Care 2019;23:143. 717. Georges H, Leroy O, Airapetian N, Lamblin N, Zogheib E, Devos P, et al. Outcome and prognostic factors of patients with right-sided infective endocarditis requiring intensive care unit admission. BMC Infect Dis 2018;18:85. 2989-9 718. Leroy O, Georges H, Devos P, Bitton S, De Sa N, Dedrie C, et al. Infective endocarditis requiring ICU admission: epidemiology and prognosis. Ann Intensive Care 2015;5:45. 719. Samol A, Kaese S, Bloch J, Gorlich D, Peters G, Waltenberger J, et al. Infective endo-carditis on ICU: risk factors, outcome and long-term follow-up. Infection 2015;43: 287–295. 720. Arntfield R, Lau V, Landry Y, Priestap F, Ball I. Impact of critical care transesophageal echocardiography in medical-surgical ICU patients: characteristics and results from 274 consecutive examinations. J Intensive Care Med 2020;35:896–902. org/10.1177/0885066618797271 721. Belletti A, Jacobs S, Affronti G, Mladenow A, Landoni G, Falk V, et al. Incidence and predictors of postoperative need for high-dose inotropic support in patients undergo-ing cardiac surgery for infective endocarditis. J Cardiothorac Vasc Anesth 2018;32: 2528–2536. 722. van den Brink FS, van Tooren R, Sonker U, Klein P, Waanders F, Zivelonghi C, et al. Veno arterial-extra corporal membrane oxygenation for the treatment of cardiac fail-ure in patients with infective endocarditis. Perfusion 2019;34:613–617. 10.1177/0267659119842807 723. Lassen H, Nielsen SL, Gill SUA, Johansen IS. The epidemiology of infective endocarditis with focus on non-device related right-sided infective endocarditis: a retrospective register-based study in the region of southern Denmark. Int J Infect Dis 2020;95: 224–230. 724. Sridhar AR, Lavu M, Yarlagadda V, Reddy M, Gunda S, Afzal R, et al. Cardiac implantable electronic device-related infection and extraction trends in the U.S. Pacing Clin Electrophysiol 2017;40:286–293. 725. Witten JC, Hussain ST, Shrestha NK, Gordon SM, Houghtaling PL, Bakaeen FG, et al. Surgical treatment of right-sided infective endocarditis. J Thorac Cardiovasc Surg 2019; 157:1418–1427.e14. 726. Pfannmueller B, Kahmann M, Davierwala P, Misfeld M, Bakhtiary F, Binner C, et al. Tricuspid valve surgery in patients with isolated tricuspid valve endocarditis: analysis of perioperative parameters and long-term outcomes. Thorac Cardiovasc Surg 2017; 65:626–633. 727. Isaza N, Shrestha NK, Gordon S, Pettersson GB, Unai S, Vega Brizneda M, et al. Contemporary outcomes of pulmonary valve endocarditis: a 16-year single centre ex-perience. Heart Lung Circ 2020;29:1799–1807. 015 728. Hussain ST, Shrestha NK, Witten J, Gordon SM, Houghtaling PL, Tingleff J, et al. Rarity of invasiveness in right-sided infective endocarditis. J Thorac Cardiovasc Surg 2018;155: 54–61.e1. 729. San Roman JA, Vilacosta I, Lopez J, Revilla A, Arnold R, Sevilla T, et al. Role of transthor-acic and transesophageal echocardiography in right-sided endocarditis: one echocar-diographic modality does not fit all. J Am Soc Echocardiogr 2012;25:807–814. https:// doi.org/10.1016/j.echo.2012.05.016 730. Anton-Vazquez V, Cannata A, Amin-Youssef G, Watson S, Fife A, Mulholland N, et al. Diagnostic value of (18)F-FDG PET/CT in infective endocarditis. Clin Res Cardiol 2022; 111:673–679. 731. Weber C, Gassa A, Eghbalzadeh K, Merkle J, Djordjevic I, Maier J, et al. Characteristics and outcomes of patients with right-sided endocarditis undergoing cardiac surgery. Ann Cardiothorac Surg 2019;8:645–653. 732. Shrestha NK, Jue J, Hussain ST, Jerry JM, Pettersson GB, Menon V, et al. Injection drug use and outcomes after surgical intervention for infective endocarditis. Ann Thorac Surg 2015;100:875–882. 733. Hussain ST, Witten J, Shrestha NK, Blackstone EH, Pettersson GB. Tricuspid valve endocarditis. Ann Cardiothorac Surg 2017;6:255–261. 2017.03.09 734. Marks LR, Nolan NS, Liang SY, Durkin MJ, Weimer MB. Infectious complications of in-jection drug use. Med Clin North Am 2022;106:187–200. mcna.2021.08.006 735. Kelly MC, Yeager SD, Shorman MA, Wright LR, Veve MP. Incidence and predictors of Gram-negative bacilli in hospitalized people who inject drugs with injection drug use-attributable infections. Antimicrob Agents Chemother 2021;65:e0092521. https:// doi.org/10.1128/AAC.00925-21 736. Bisbe J, Miro JM, Latorre X, Moreno A, Mallolas J, Gatell JM, et al. Disseminated can-didiasis in addicts who use brown heroin: report of 83 cases and review. Clin Infect Dis 1992;15:910–923. 737. Meena DS, Kumar D, Agarwal M, Bohra GK, Choudhary R, Samantaray S, et al. Clinical features, diagnosis and treatment outcome of fungal endocarditis: a systematic review of reported cases. Mycoses 2022;65:294–302. 738. Ribera E, Gomez-Jimenez J, Cortes E, del Valle O, Planes A, Gonzalez-Alujas T, et al. Effectiveness of cloxacillin with and without gentamicin in short-term therapy for right-sided Staphylococcus aureus endocarditis. A randomized, controlled trial. Ann Intern Med 1996;125:969–974. 199612150-00005 739. Cahill TJ, Prendergast BD. Infective endocarditis. Lancet 2016;387:882–893. https:// doi.org/10.1016/S0140-6736(15)00067-7 740. Chong CZ, Cherian R, Ng P, Yeo TC, Ling LH, Soo WM, et al. Clinical outcomes of severe tricuspid valve infective endocarditis related to intravenous drug abuse – a case series. Acta Cardiol 2022;77:884–889. 1976448 741. Syed IM, Yanagawa B, Jeyaganth S, Verma S, Cheema AN. Injection drug use endocar-ditis: an inner-city hospital experience. CJC Open 2021;3:896–903. 1016/j.cjco.2021.02.015 742. Chahoud J, Sharif Yakan A, Saad H, Kanj SS. Right-sided infective endocarditis and pul-monary infiltrates: an update. Cardiol Rev 2016;24:230–237. CRD.0000000000000095 743. Damlin A, Westling K. Patients with infective endocarditis and history of injection drug use in a Swedish referral hospital during 10 years. BMC Infect Dis 2021;21:236. https:// doi.org/10.1186/s12879-021-05914-1 744. Rose WE, Leonard SN, Sakoulas G, Kaatz GW, Zervos MJ, Sheth A, et al. Daptomycin activity against Staphylococcus aureus following vancomycin exposure in an in vitro pharmacodynamic model with simulated endocardial vegetations. Antimicrob Agents Chemother 2008;52:831–836. 745. Al-Omari A, Cameron DW, Lee C, Corrales-Medina VF. Oral antibiotic therapy for the treatment of infective endocarditis: a systematic review. BMC Infect Dis 2014;14: 140. 746. Marks LR, Liang SY, Muthulingam D, Schwarz ES, Liss DB, Munigala S, et al. Evaluation of partial oral antibiotic treatment for persons who inject drugs and are hospitalized with invasive infections. Clin Infect Dis 2020;71:e650–e656. cid/ciaa365 747. Slaughter MS, Badhwar V, Ising M, Ganzel BL, Sell-Dottin K, Jawitz OK, et al. Optimum surgical treatment for tricuspid valve infective endocarditis: an analysis of the Society of Thoracic Surgeons national database. J Thorac Cardiovasc Surg 2021;161:1227–-1235.e1. 748. Misfeld M, Davierwala PM, Borger MA, Bakhtiary F. The “UFO” procedure. Ann Cardiothorac Surg 2019;8:691–698. 749. Navia JL, Elgharably H, Hakim AH, Witten JC, Haupt MJ, Germano E, et al. Long-term outcomes of surgery for invasive valvular endocarditis involving the aortomitral fibrosa. Ann Thorac Surg 2019;108:1314–1323. 119 750. Nappi F, Spadaccio C, Mihos C, Shaikhrezai K, Acar C, Moon MR. The quest for the optimal surgical management of tricuspid valve endocarditis in the current era: a nar-rative review. Ann Transl Med 2020;8:1628. 751. Arbulu A, Holmes RJ, Asfaw I. Surgical treatment of intractable right-sided infective endocarditis in drug addicts: 25 years experience. J Heart Valve Dis 1993;2:129–137, discussion 138–139. 752. Brescia AA, Watt TMF, Williams AM, Romano MA, Bolling SF. Tricuspid valve leaflet repair and augmentation for infective endocarditis. Oper Tech Thorac Cardiovasc Surg 2019;24:206–218. 753. Randhawa VK, Rajani R. Novel frontiers for managing tricuspid valve endocarditis: tales of percutaneous extracorporeal circuitry. JACC Case Rep 2021;3:1350–1353. https:// doi.org/10.1016/j.jaccas.2021.06.026 754. Starck CT, Dreizler T, Falk V. The AngioVac system as a bail-out option in infective valve endocarditis. Ann Cardiothorac Surg 2019;8:675–677. acs.2019.11.04 755. Luc JGY, Choi J-H, Kodia K, Weber MP, Horan DP, Maynes EJ, et al. Valvectomy versus replacement for the surgical treatment of infective tricuspid valve endocarditis: a sys-tematic review and meta-analysis. Ann Cardiothorac Surg 2019;8:610–620. org/10.21037/acs.2019.11.06 756. Gabriel HM, Heger M, Innerhofer P, Zehetgruber M, Mundigler G, Wimmer M, et al. Long-term outcome of patients with ventricular septal defect considered not to re-quire surgical closure during childhood. J Am Coll Cardiol 2002;39:1066–1071. 757. Moore B, Cao J, Kotchetkova I, Celermajer DS. Incidence, predictors and outcomes of infective endocarditis in a contemporary adult congenital heart disease population. Int J Cardiol 2017;249:161–165. 758. Baumgartner H, De Backer J, Babu-Narayan SV, Budts W, Chessa M, Diller GP, et al. 2020 ESC Guidelines for the management of adult congenital heart disease. Eur Heart J 2021;42:563–645. 759. Haas NA, Bach S, Vcasna R, Laser KT, Sandica E, Blanz U, et al. The risk of bacterial endocarditis after percutaneous and surgical biological pulmonary valve implantation. Int J Cardiol 2018;268:55–60. 760. Hribernik I, Thomson J, Ho A, English K, Van Doorn C, Jaber O, et al. Comparative analysis of surgical and percutaneous pulmonary valve implants over a 20-year period. Eur J Cardiothorac Surg 2022;61:572–579. 92 ESC Guidelines Downloaded from by guest on 30 August 2023 761. Dolgner SJ, Arya B, Kronman MP, Chan T. Effect of congenital heart disease status on trends in pediatric infective endocarditis hospitalizations in the United States between 2000 and 2012. Pediatr Cardiol 2019;40:319–329. 2020-7 762. Jortveit J, Klcovansky J, Eskedal L, Birkeland S, Dohlen G, Holmstrom H. Endocarditis in children and adolescents with congenital heart defects: a Norwegian nationwide register-based cohort study. Arch Dis Child 2018;103:670–674. 1136/archdischild-2017-313917 763. Cahill TJ, Jewell PD, Denne L, Franklin RC, Frigiola A, Orchard E, et al. Contemporary epidemiology of infective endocarditis in patients with congenital heart disease: a UK prospective study. Am Heart J 2019;215:70–77. 014 764. Tutarel O, Alonso-Gonzalez R, Montanaro C, Schiff R, Uribarri A, Kempny A, et al. Infective endocarditis in adults with congenital heart disease remains a lethal disease. Heart 2018;104:161–165. 765. Bauer UMM, Helm PC, Diller GP, Asfour B, Schlensak C, Schmitt K, et al. Are adults with congenital heart disease informed about their risk for infective endocarditis and treated in accordance to current guidelines? Int J Cardiol 2017;245:105–108. 766. Watkins DA, Beaton AZ, Carapetis JR, Karthikeyan G, Mayosi BM, Wyber R, et al. Rheumatic heart disease worldwide: JACC Scientific Expert Panel. J Am Coll Cardiol 2018;72:1397–1416. 767. Bajracharya S, Khanal B, Siwakoti S, Singh RR, Sharma SK. Microbiological and clinicoe-pidemiological profile of a series of patients with infective endocarditis at a center in Eastern Nepal. Can J Infect Dis Med Microbiol 2021;2021:9980465. 1155/2021/9980465 768. Karthikeyan G, Zuhlke L, Engel M, Rangarajan S, Yusuf S, Teo K, et al. Rationale and design of a Global Rheumatic Heart Disease Registry: the REMEDY study. Am Heart J 2012;163:535–540.e1. 769. Zuhlke L, Engel ME, Karthikeyan G, Rangarajan S, Mackie P, Cupido B, et al. Characteristics, complications, and gaps in evidence-based interventions in rheumatic heart disease: the Global Rheumatic Heart Disease Registry (the REMEDY study). Eur Heart J 2015;36:1115–1122a. 770. Zuhlke L, Karthikeyan G, Engel ME, Rangarajan S, Mackie P, Cupido-Katya Mauff B, et al. Clinical outcomes in 3343 children and adults with rheumatic heart disease from 14 low- and middle-income countries: two-year follow-up of the Global Rheumatic Heart Disease Registry (the REMEDY study). Circulation 2016;134:1456–1466. 771. Rohn V, Laca B, Horn M, Vlk L, Antonova P, Mosna F. Surgery in drug use-associated infective endocarditis: long-term survival is negatively affected by recurrence. Interact Cardiovasc Thorac Surg 2020;30:528–534. 772. Kumar RK, Antunes MJ, Beaton A, Mirabel M, Nkomo VT, Okello E, et al. Contemporary diagnosis and management of rheumatic heart disease: implications for closing the gap: a scientific statement from the American Heart Association. Circulation 2020;142:e337–e357. 773. Coates MM, Sliwa K, Watkins DA, Zuhlke L, Perel P, Berteletti F, et al. An investment case for the prevention and management of rheumatic heart disease in the African Union 2021–30: a modelling study. Lancet Glob Health 2021;9:e957–e966. https:// doi.org/10.1016/S2214-109X(21)00199-6 774. Zilla P, Yacoub M, Zuhlke L, Beyersdorf F, Sliwa K, Khubulava G, et al. Global unmet needs in cardiac surgery. Glob Heart 2018;13:293–303. gheart.2018.08.002 775. Mirabel M, Andre R, Barsoum P, Colboc H, Lacassin F, Noel B, et al. Ethnic disparities in the incidence of infective endocarditis in the Pacific. Int J Cardiol 2015;186:43–44. 776. Mirabel M, Andre R, Barsoum Mikhail P, Colboc H, Lacassin F, Noel B, et al. Infective endocarditis in the Pacific: clinical characteristics, treatment and long-term outcomes. Open Heart 2015;2:e000183. 777. Kingue S, Ba SA, Balde D, Diarra MB, Anzouan-Kacou JB, Anisubia B, et al. The valvafric study: a registry of rheumatic heart disease in Western and Central Africa. Arch Cardiovasc Dis 2016;109:321–329. 778. Subbaraju P, Rai S, Morakhia J, Midha G, Kamath A, Saravu K. Clinical – microbiological characterization and risk factors of mortality in infective endocarditis from a tertiary care academic hospital in southern India. Indian Heart J 2018;70:259–265. https:// doi.org/10.1016/j.ihj.2017.08.007 779. Rwebembera J, Manyilirah W, Zhu ZW, Nabbaale J, Namuyonga J, Ssinabulya I, et al. Prevalence and characteristics of primary left-sided valve disease in a cohort of 15,000 patients undergoing echocardiography studies in a tertiary hospital in Uganda. BMC Cardiovasc Disord 2018;18:82. 0813-5 780. Pecoraro AJ, Doubell AF. Infective endocarditis in South Africa. Cardiovasc Diagn Ther 2020;10:252–261. 781. Hajsadeghi S, Hassanzadeh M, Hajahmadi M, Kadivar M. Concurrent diagnosis of infect-ive endocarditis and acute rheumatic fever: a case report. J Cardiol Cases 2018;17: 147–150. 782. Gouriet F, Chaudet H, Gautret P, Pellegrin L, de Santi VP, Savini H, et al. Endocarditis in the Mediterranean Basin. New Microbes New Infect 2018;26:S43–S51. 10.1016/j.nmni.2018.05.004 783. Dhar M, Kaeley N, Bhatt N, Ahmad S. Profile of newly diagnosed adult patients with rheumatic heart disease in sub-Himalayan region – a 5-year analysis. J Family Med Prim Care 2019;8:2933–2936. 784. Blanchard V, Pagis B, Richaud R, Moronval F, Lutinier R, Gallais K, et al. Infective endo-carditis in French Polynesia: epidemiology, treatments and outcomes. Arch Cardiovasc Dis 2020;113:252–262. 785. Montano TCP, Wanderley MIA, Sampaio RO, Alves CGB, Neves ILI, Lopes MA, et al. Demographic, cardiological, microbiologic, and dental profiles of Brazilian patients who developed oral bacteria-related endocarditis. Oral Surg Oral Med Oral Pathol Oral Radiol 2021;132:418–425. 786. Jomaa W, Ben Ali I, Abid D, Hajri Ernez S, Abid L, Triki F, et al. Clinical features and prognosis of infective endocarditis in children: insights from a Tunisian multicentre registry. Arch Cardiovasc Dis 2017;110:676–681. 12.018 787. Mahony M, Lean D, Pham L, Horvath R, Suna J, Ward C, et al. Infective endocarditis in children in Queensland, Australia: epidemiology, clinical features and outcome. Pediatr Infect Dis J 2021;40:617–622. 788. Willoughby ML, Basera W, Perkins SR, Comitis GAM, Fourie B, Lawrenson JB, et al. Infective endocarditis in infants and children in the Western Cape, South Africa: a retrospective analysis. Cardiol Young 2019;29:1282–1286. S1047951119002154 789. Nigussie B, Tadele H. Heart failure in Ethiopian children: mirroring the unmet cardiac services. Ethiop J Health Sci 2019;29:811–818. 790. Moreira JL, Barletta P, Baucia JA. Morbidity and mortality in patients undergoing mitral valve replacement at a cardiovascular surgery referral service: a retrospective analysis. Braz J Cardiovasc Surg 2021;36:183–191. 0440 791. Mocumbi AO, Jamal KK, Mbakwem A, Shung-King M, Sliwa K. The Pan-African Society of Cardiology position paper on reproductive healthcare for women with rheumatic heart disease. Cardiovasc J Afr 2018;29:394–403. 044 792. Roos-Hesselink J, Baris L, Johnson M, De Backer J, Otto C, Marelli A, et al. Pregnancy outcomes in women with cardiovascular disease: evolving trends over 10 years in the ESC registry of pregnancy and cardiac disease (ROPAC). Eur Heart J 2019;40: 3848–3855. 793. De Villiers MC, Viljoen CA, Manning K, Van der Westhuizen C, Seedat A, Rath M, et al. The changing landscape of infective endocarditis in South Africa. S Afr Med J 2019;109: 592–596. 794. Sadeghpour A, Maleki M, Movassaghi M, Rezvani L, Noohi F, Boudagh S, et al. Iranian Registry of Infective Endocarditis (IRIE): time to relook at the guideline, regarding to regional differences. Int J Cardiol Heart Vasc 2020;26:100433. j.ijcha.2019.100433 795. Wu Z, Chen Y, Xiao T, Niu T, Shi Q, Xiao Y. Epidemiology and risk factors of infective endocarditis in a tertiary hospital in China from 2007 to 2016. BMC Infect Dis 2020;20: 428. 796. Xu H, Cai S, Dai H. Characteristics of infective endocarditis in a tertiary hospital in east China. PLoS One 2016;11:e0166764. 797. Connolly C, O’Donoghue K, Doran H, McCarthy FP. Infective endocarditis in preg-nancy: case report and review of the literature. Obstet Med 2015;8:102–104. https:// doi.org/10.1177/1753495X15572857 798. Yuan SM. Infective endocarditis during pregnancy. J Coll Physicians Surg Pak 2015;25: 134–139. 799. Escola-Verge L, Rello P, Declerck C, Dubee V, Rouleau F, Duval X, et al. Infective endo-carditis in pregnant women without intravenous drug use: a multicentre retrospective case series. J Antimicrob Chemother 2022;77:2701–2705. dkac258 800. Dagher MM, Eichenberger EM, Addae-Konadu KL, Dotters-Katz SK, Kohler CL, Fowler VG, et al. Maternal and fetal outcomes associated with infective endocarditis in pregnancy. Clin Infect Dis 2021;73:1571–1579. 801. Morelli MK, Veve MP, Shorman MA. Maternal bacteremia caused by Staphylococcus aureus with a focus on infective endocarditis. Open Forum Infect Dis 2020;7:ofaa239. 802. English N, Weston P. Multivalvular infective endocarditis in pregnancy presenting with septic pulmonary emboli. BMJ Case Rep 2015;2015:bcr2014209131. 10.1136/bcr-2014-209131 803. Khanna R, Chandra D, Yadav S, Sahu A, Singh N, Kumar S, et al. Maternal and fetal out-comes in pregnant females with rheumatic heart disease. Indian Heart J 2021;73: 185–189. 804. Adesomo A, Gonzalez-Brown V, Rood KM. Infective endocarditis as a complication of intravenous drug use in pregnancy: a retrospective case series and literature review. AJP Rep 2020;10:e288–e293. ESC Guidelines 93 Downloaded from by guest on 30 August 2023 805. Dahshan D, Suliman M, Rahman EU, Curtis Z, Thompson E. Intravenous drug use-associated infective endocarditis in pregnant patients at a hospital in West Virginia. Cureus 2021;13:e17218. 806. Lin D, Mullan CW, Deshmukh U, Bahtiyar MO, Hosier H, Lipkind H, et al. Drug use associated tricuspid valve infective endocarditis in pregnancy. J Card Surg 2020;35: 2392–2395. 807. Pfaller B, Sathananthan G, Grewal J, Mason J, D’Souza R, Spears D, et al. Preventing complications in pregnant women with cardiac disease. J Am Coll Cardiol 2020;75: 1443–1452. 808. Botea R, Porterie J, Marcheix B, Breleur FO, Lavie-Badie Y. Infective endocarditis in a third trimester pregnant woman: team work is the best option. JACC Case Rep 2020;2: 521–525. 809. Wang J, Wang A, Cui Y, Wang C, Zhang J. Diagnosis and treatment of infective endo-carditis in pregnancy: a case report. J Cardiothorac Surg 2020;15:109. 1186/s13019-020-01147-6 810. Liu Y, Han F, Zhuang J, Liu X, Chen J, Huang H, et al. Cardiac operation under cardio-pulmonary bypass during pregnancy. J Cardiothorac Surg 2020;15:92. 1186/s13019-020-01136-9 811. Eichenberger EM, Dagher M, Sinclair MR, Maskarinec SA, Fowler VG Jr, Federspiel JJ. Infective endocarditis and solid organ transplantation: only worse outcomes during ini-tial transplantation hospitalization. Am Heart J 2021;240:63–72. 1016/j.ahj.2021.06.007 812. Deeks SG, Overbaugh J, Phillips A, Buchbinder S. HIV infection. Nat Rev Dis Primers 2015;1:15035. 813. Munoz-Moreno MF, Ryan P, Alvaro-Meca A, Valencia J, Tamayo E, Resino S. National temporal trend analysis of infective endocarditis among patients infected with HIV in Spain (1997–2014): a retrospective study. J Clin Med 2019;8:1167. 3390/jcm8081167 814. Beteille E, Guarana M, Nucci M. Infective endocarditis in neutropenic patients with vir-idans streptococci bacteraemia. Clin Microbiol Infect 2018;24:916–917. 10.1016/j.cmi.2018.03.012 815. Veve MP, Stuart M, Davis SL. Comparison of neutropenia associated with ceftaroline or ceftriaxone in patients receiving at least 7 days of therapy for severe infections. Pharmacotherapy 2019;39:809–815. 816. Mani SSR, Iyyadurai R. Cloxacillin induced agranulocytosis: a rare adverse event of a commonly used antibiotic. Int J Immunopathol Pharmacol 2017;30:297–301. https:// doi.org/10.1177/0394632017724320 817. Tornos P, Almirante B, Mirabet S, Permanyer G, Pahissa A, Soler-Soler J. Infective endocarditis due to Staphylococcus aureus: deleterious effect of anticoagulant therapy. Arch Intern Med 1999;159:473–475. 818. Deppisch LM, Fayemi AO. Non-bacterial thrombotic endocarditis: clinicopathologic correlations. Am Heart J 1976;92:723–729. 80008-7 819. Llenas-Garcia J, Guerra-Vales JM, Montes-Moreno S, Lopez-Rios F, Castelbon-Fernandez FJ, Chimeno-Garcia J. [Nonbacterial thrombotic endocarditis: clinicopathologic study of a necropsy series]. Rev Esp Cardiol 2007;60:493–500. 820. Lopez JA, Ross RS, Fishbein MC, Siegel RJ. Nonbacterial thrombotic endocarditis: a re-view. Am Heart J 1987;113:773–784. 821. Quintero-Martinez JA, Hindy JR, El Zein S, Michelena HI, Nkomo VT, DeSimone DC, et al. Contemporary demographics, diagnostics and outcomes in non-bacterial throm-botic endocarditis. Heart 2022. 822. el-Shami K, Griffiths E, Streiff M. Nonbacterial thrombotic endocarditis in cancer pa-tients: pathogenesis, diagnosis, and treatment. Oncologist 2007;12:518–523. https:// doi.org/10.1634/theoncologist.12-5-518 823. Zmaili MA, Alzubi JM, Kocyigit D, Bansal A, Samra GS, Grimm R, et al. A contemporary 20-year Cleveland clinic experience of nonbacterial thrombotic endocarditis: etiology, echocardiographic imaging, management, and outcomes. Am J Med 2021;134: 361–369. 824. Roldan CA, Tolstrup K, Macias L, Qualls CR, Maynard D, Charlton G, et al. Libman-sacks endocarditis: detection, characterization, and clinical correlates by three-dimensional transesophageal echocardiography. J Am Soc Echocardiogr 2015; 28:770–779. 825. Pengo V, Denas G, Zoppellaro G, Jose SP, Hoxha A, Ruffatti A, et al. Rivaroxaban vs warfarin in high-risk patients with antiphospholipid syndrome. Blood 2018;132: 1365–1371. 826. Kim K, Kim D, Lee S-E, Cho IJ, Shim CY, Hong G-R, et al. Infective endocarditis in can-cer patients – causative organisms, predisposing procedures, and prognosis differ from infective endocarditis in non-cancer patients. Circ J 2019;83:452–460. 10.1253/circj.CJ-18-0609 827. Cosyns B, Roosens B, Lancellotti P, Laroche C, Dulgheru R, Scheggi V, et al. Cancer and infective endocarditis: characteristics and prognostic impact. Front Cardiovasc Med 2021;8:766996. 828. Kitson A, Marshall A, Bassett K, Zeitz K. What are the core elements of patient- centred care? A narrative review and synthesis of the literature from health policy, medicine and nursing. J Adv Nurs 2013;69:4–15. 2012.06064.x 829. Giusti A, Nkhoma K, Petrus R, Petersen I, Gwyther L, Farrant L, et al. The empirical evidence underpinning the concept and practice of person-centred care for serious illness: a systematic review. BMJ Glob Health 2020;5:e003330. 1136/bmjgh-2020-003330 830. Ekman I, Swedberg K, Taft C, Lindseth A, Norberg A, Brink E, et al. Person-centered care–ready for prime time. Eur J Cardiovasc Nurs 2011;10:248–251. 1016/j.ejcnurse.2011.06.008 831. NEJM Catalyst. What is patient-centered care? New Engl J Med 2017;3. 832. Lauck SB, Lewis KB, Borregaard B, de Sousa I. “What is the right decision for me?” Integrating patient perspectives through shared decision-making for valvular heart dis-ease therapy. Can J Cardiol 2021;37:1054–1063. 022 833. Oshima Lee E, Emanuel EJ. Shared decision making to improve care and reduce costs. N Engl J Med 2013;368:6–8. 834. Stacey D, Legare F, Lewis K, Barry MJ, Bennett CL, Eden KB, et al. Decision aids for people facing health treatment or screening decisions. Cochrane Database Syst Rev 2017;4:CD001431. 835. R EGEAT. Shared Decision Making in Health Care: Achieving Evidence-Based Patient Choice. 3rd ed. Oxford University Press, 2016. 836. Carmona C, Crutwell J, Burnham M, Polak L, Guideline C. Shared decision-making: summary of NICE guidance. BMJ 2021;373:n1430. 837. van de Pol MH, Fluit CR, Lagro J, Slaats YH, Olde Rikkert MG, Lagro-Janssen AL. Expert and patient consensus on a dynamic model for shared decision-making in frail older patients. Patient Educ Couns 2016;99:1069–1077. 12.014 838. Chewning B, Bylund CL, Shah B, Arora NK, Gueguen JA, Makoul G. Patient prefer-ences for shared decisions: a systematic review. Patient Educ Couns 2012;86:9–18. 839. White DB, Angus DC, Shields AM, Buddadhumaruk P, Pidro C, Paner C, et al. A ran-domized trial of a family-support intervention in intensive care units. N Engl J Med 2018;378:2365–2375. 840. Shay LA, Lafata JE. Where is the evidence? A systematic review of shared decision mak-ing and patient outcomes. Med Decis Making 2015;35:114–131. 1177/0272989X14551638 841. McMillan SS, Kendall E, Sav A, King MA, Whitty JA, Kelly F, et al. Patient-centered ap-proaches to health care: a systematic review of randomized controlled trials. Med Care Res Rev 2013;70:567–596. 842. Hibbard JH, Greene J. What the evidence shows about patient activation: better health outcomes and care experiences; fewer data on costs. Health Aff (Millwood) 2013;32: 207–214. 843. Dwamena F, Holmes-Rovner M, Gaulden CM, Jorgenson S, Sadigh G, Sikorskii A, et al. Interventions for providers to promote a patient-centred approach in clinical consul-tations. Cochrane Database Syst Rev 2012;12:CD003267. 14651858.CD003267.pub2 844. Verhagen DW, Hermanides J, Korevaar JC, Bossuyt PM, van den Brink RB, Speelman P, et al. Health-related quality of life and posttraumatic stress disorder among survivors of left-sided native valve endocarditis. Clin Infect Dis 2009;48:1559–1565. org/10.1086/598930 845. Berg SK, Preisler P, Pedersen BD. Patients perspective on endocarditis–an intermezzo in life. Eur J Cardiovasc Nurs 2010;9:126–131. 11.007 846. Butt JH, Kragholm K, Dalager-Pedersen M, Rorth R, Kristensen SL, Chaudry MS, et al. Return to the workforce following infective endocarditis–a nationwide cohort study. Am Heart J 2018;195:130–138. 847. Havers-Borgersen E, Fosbol EL, Rorth R, Kragholm K, Kristensen SL, Bundgaard H, et al. Nursing home admission and initiation of domiciliary care following infective endocarditis. Glob Heart 2019;14:41–46.e2. 002 848. Wattel R. ESC involving patients: purpose & priorities. Eur Heart J 2018;39:3681. 849. Ulin K, Olsson LE, Wolf A, Ekman I. Person-centred care – an approach that improves the discharge process. Eur J Cardiovasc Nurs 2016;15:e19–e26. 1474515115569945 850. Hill L, Prager Geller T, Baruah R, Beattie JM, Boyne J, de Stoutz N, et al. Integration of a palliative approach into heart failure care: a European Society of Cardiology Heart Failure Association position paper. Eur J Heart Fail 2020;22:2327–2339. org/10.1002/ejhf.1994 851. D’Alto M, Budts W, Diller GP, Mulder B, Egidy Assenza G, Oreto L, et al. Does gender affect the prognosis and risk of complications in patients with congenital heart disease in the modern era? Int J Cardiol 2019;290:156–161. 2019.05.010 852. Weber C, Petrov G, Luehr M, Aubin H, Tugtekin SM, Borger MA, et al. Surgical results for prosthetic versus native valve endocarditis: a multicenter analysis. J Thorac Cardiovasc Surg 2021;161:609–619.e10. 94 ESC Guidelines Downloaded from by guest on 30 August 2023 853. Curlier E, Hoen B, Alla F, Selton-Suty C, Schubel L, Doco-Lecompte T, et al. Relationships between sex, early valve surgery and mortality in patients with left-sided infective endocarditis analysed in a population-based cohort study. Heart 2014;100: 1173–1178. 854. Weber C, Gassa A, Rokohl A, Sabashnikov A, Deppe AC, Eghbalzadeh K, et al. Severity of presentation, not sex, increases risk of surgery for infective endocarditis. Ann Thorac Surg 2019;107:1111–1117. 855. Bansal A, Cremer PC, Jaber WA, Rampersad P, Menon V. Sex differences in the util-ization and outcomes of cardiac valve replacement surgery for infective endocarditis: insights from the national inpatient sample. J Am Heart Assoc 2021;10:e020095. https:// doi.org/10.1161/JAHA.120.020095 856. Varela Barca L, Vidal-Bonnet L, Farinas MC, Munoz P, Valerio Minero M, de Alarcon A, et al. Analysis of sex differences in the clinical presentation, management and prognosis of infective endocarditis in Spain. Heart 2021;107:1717–1724. heartjnl-2021-319254 857. Ahtela E, Oksi J, Porela P, Ekstrom T, Rautava P, Kyto V. Trends in occurrence and 30-day mortality of infective endocarditis in adults: population-based registry study in Finland. BMJ Open 2019;9:e026811. ESC Guidelines 95 Downloaded from by guest on 30 August 2023
6032
https://www.quora.com/Are-sets-in-math-ordered
Something went wrong. Wait a moment and try again. Subset (Set Theory) Ordered Fields (mathemati... Number Sets Order Theory Order of Numbers Set Theory Mathematics Set 5 Are sets in math ordered? Sort Nicholas Cooper Author has 1.6K answers and 2.9M answer views · 6y By default, no. However, an order can be given to a set. The joy is that we can give it any order we please. Let’s take the set of Truth-Values: {True,False}. We can decide that True>False, and that gives us everything we need to define a Boolean Algebra over it. Likewise, we have a set of Natural Numbers. The set is unordered, but if we desire, we can decide to give them an order. We don’t even have to give them the order you normally think of. So, let’s talk a minute about what we call things with and without an order. A set with equivalence is called an E By default, no. However, an order can be given to a set. The joy is that we can give it any order we please. Let’s take the set of Truth-Values: {True,False}. We can decide that True>False, and that gives us everything we need to define a Boolean Algebra over it. Likewise, we have a set of Natural Numbers. The set is unordered, but if we desire, we can decide to give them an order. We don’t even have to give them the order you normally think of. So, let’s talk a minute about what we call things with and without an order. A set with equivalence is called an Equivalence Class. A set with order is more complicated, because we have to talk about whether or not it’s a partial or total order. A set with a partial order is a partially ordered set (called a “poset”), and a set with a total order is called a totally ordered set (there is no cute name for it). Not everything has to be comparable. Think about genealogy. The order relation may be “are you a descendant of?” If you compare yourself and your parents, there is an ordering there. However, you cannot order yourself against your cousins. If everything can be compared, it’s a totally ordered set. If not, it’s a poset. So, let’s compare some containers, like sets. If I have a set of things that are finite and totally ordered, then I have what’s called an ordered list (sometimes also a tuple in programming). It’s usually written with round brackets (parentheses) instead of curly brackets. There are also sets that can contain many of the same elements. These are frequently called “bags”, and we’d see something like: {{2A,3B,1C}}. Meanwhile, sets do not require an order, are finite or infinite, and do not distinguish between duplicates (i.e. sets answer the question, “is this included or not?”). Promoted by Coverage.com Johnny M Master's Degree from Harvard University (Graduated 2011) · Updated Sep 9 Does switching car insurance really save you money, or is that just marketing hype? This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. It always sounded like a hassle. Dozens of tabs, endless forms, phone calls I didn’t want to take. But recently I decided to check so I used this quote tool, which compares everything in one place. It took maybe 2 minutes, tops. I just answered a few questions and it pulled up offers from multiple big-name providers, side by side. Prices, coverage details, even customer reviews—all laid out in a way that made the choice pretty obvious. They claimed I could save over $1,000 per year. I ended up exceeding that number and I cut my monthly premium by over $100. That’s over $1200 a year. For the exact same coverage. No phone tag. No junk emails. Just a better deal in less time than it takes to make coffee. Here’s the link to two comparison sites - the one I used and an alternative that I also tested. If it’s been a while since you’ve checked your rate, do it. You might be surprised at how much you’re overpaying. Jerry Rosen Former Retired Symphony Musician (Cleveland, Boston) (1959–1999) · 6y The concept “set” is so general that your question is too vague to answer definitely. The are ordered sets (like the whole numbers) and there are sets without an ordering, like any random collection of objects or concepts. As well, there are different kinds of ordering relations —- such as complete (any two objects can be compared as any two integers with the relation “less than”) or partial (in which some objects cannot be so compared, as with the set relation “subset of”) and probably others which a more knowledgeable correspondent could cite. Andrew Winkler PhD in Mathematics, Courant Institute of Mathematical Sciences, NYU (Graduated 1987) · Author has 9.6K answers and 10.9M answer views · 2y Originally Answered: Do sets from set theory have order? · No. The only question that you can ask is whether one set is an element of another. A subset O of the Cartesian product of a set with itself is called a partial order when it contains the diagonal and is transitively closed, that is if when (a,b) and (b,c) are in O then so is (a,c) (so it is a preorder) and if also when both (a,b) and (b,a) are in O then a=b. So while sets do not “come with” an order, there are very many orders to choose from on any given set with at least two elements. Related questions Do sets from set theory have order? If two sets have the same members but in different order, will they be considered as an identical set? How do you tell if a set is well-ordered in math? How are math sets? What is the use of sets in math? Assistant Bot · 1y In mathematics, sets are unordered collections of distinct elements. This means that the order in which the elements are listed does not matter, and each element can appear only once in a set. For example, the set {1,2,3} is considered the same as the set {3,2,1}. If the order of elements is important, then you are dealing with a different structure known as a tuple or a list, which allows for ordered collections. Related questions Do sets from set theory have order? If two sets have the same members but in different order, will they be considered as an identical set? How do you tell if a set is well-ordered in math? How are math sets? What is the use of sets in math? Is all math based on set theory? What are 10 examples that are not a set? What are the different kinds of sets in mathematics? How do I do sets in maths? What are some examples of sets in mathematics? What are ordered sets? How can we translate a set to an ordered pair (set theory, math)? What is strict ordering in math? How do I deal with sets in mathematics? Why is there no formal definition for a set in math? How can we make any statement about sets (and therefore all of math) if we don’t even know what it is? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
6033
https://mathworld.wolfram.com/StammlerCircles.html
Stammler Circles -- from Wolfram MathWorld TOPICS AlgebraApplied MathematicsCalculus and AnalysisDiscrete MathematicsFoundations of MathematicsGeometryHistory and TerminologyNumber TheoryProbability and StatisticsRecreational MathematicsTopologyAlphabetical IndexNew in MathWorld Geometry Plane Geometry Triangles Triangle Circles MathWorld Contributors Moses MathWorld Contributors van Lamoen Stammler Circles Download Wolfram Notebook The Stammler circles are the three circles (apart from the circumcircle), that intercept the sidelines of a reference triangle in chords of lengths equal to the corresponding side lengths , , and . The centers of these circles form an equilateral triangle known as the Stammler triangle. The radical lines of pairs of the Stammler circles are the Simson lines of the vertices of the circumtangential triangle. The trilinear coordinates of the center of the -Stammler circle are (1) and the squares of the radii , , are given by the roots of the cubic equation (2) where is the circumradius of the reference triangle (Ehrmann and van Lamoen 2002). Explicitly, the radii of the Stammler circles are (3) (4) (5) where is again the circumradius of the reference triangle (Ehrmann and van Lamoen 2002). When and are the radii of the Stammler circles and the circumradius, then the following equations hold: (6) (7) (8) (Ehrmann and van Lamoen 2002). See also Proportionally-Cutting Circle, Stammler Circle, Stammler Circles Radical Circle, Stammler Hyperbola, Stammler Triangle This entry contributed by Floor van Lamoen Explore with Wolfram|Alpha More things to try: 1+2+3+...+10 gcd x^4-9x^2-4x+12, x^3+5x^2+2x-8 multibrot set, d=-2 References Ehrmann, J.-P. and van Lamoen, F.M. "The Stammler Circles." Forum Geom.2, 151-161, 2002. L. "Proportionalschnittkreise, ihre Mittenhyperbel und ein Pendant zum Satz von Morley." Elem. Math.47, 148-158, 1992.Stammler, L. "Cutting Circles and the Morley Theorem." Beitr. Alg. Geom.38, 91-93, 1997. Referenced on Wolfram|Alpha Stammler Circles Cite this as: van Lamoen, Floor. "Stammler Circles." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. Subject classifications Geometry Plane Geometry Triangles Triangle Circles MathWorld Contributors Moses MathWorld Contributors van Lamoen About MathWorld MathWorld Classroom Contribute MathWorld Book wolfram.com 13,278 Entries Last Updated: Sun Sep 28 2025 ©1999–2025 Wolfram Research, Inc. Terms of Use wolfram.com Wolfram for Education Created, developed and nurtured by Eric Weisstein at Wolfram Research Created, developed and nurtured by Eric Weisstein at Wolfram Research
6034
https://www.mychemistryclass.net/Files/Extra%20Resources%20for%20Website/Percent%20Concentration%20Practice%20Worksheet.pdf
PERCENT CONCENTRATION EXTRA PRACTICE WORKSHEET 1. What is the % by massif 8.0 g copper is added to enough zinc to produce 100 g of an alloy? 8% 2. Brass is a copper-zinc alloy. If the concentration of zinc is relatively low, the brass has a golden color and is often used for inexpensive jewellery. If a 35.0 g pendant contains 1.7 g of zinc, what is the percentage by mass of zinc in brass? 4.9% 3. What is the % by mass of copper in an alloy when 10 kg of Cu is mixed with 250 kg of Zn? 3.8% 4. You have 200g of a solution that contains 30g of hydrochloric acid (HCl), what percentage of your solution is made up of hydrochloric acid (by mass)? 15% 5. If I make a solution by adding water to 75 mL of ethanol until the total volume of the solution is 375mL, what’s the percent by volume of ethanol in the solution? 20% 6. If I add 1.65 L of water to 112 grams of sodium acetate, what is the percent by mass of sodium acetate in this solution? 6.36% 7. Solder flux, available at hardwood stores, contains 16 g of zinc chloride in 50g of solution. What is the percentage mass of zinc chloride in the solution? 32% 8. If 250mL of sodium chloride is present in 3 L of solution, what is the percentage by volume of sodium chloride? 8.5% 9. Suppose you have 70g of sodium chloride salt (NaCl) in 250mL of water. Express this as a % by mass solution. 22% PERCENT CONCENTRATION EXTRA PRACTICE WORKSHEET 1. What is the % by massif 8.0 g copper is added to enough zinc to produce 100 g of an alloy? 8% 2. Brass is a copper-zinc alloy. If the concentration of zinc is relatively low, the brass has a golden color and is often used for inexpensive jewellery. If a 35.0 g pendant contains 1.7 g of zinc, what is the percentage by mass of zinc in brass? 4.9% 3. What is the % by mass of copper in an alloy when 10 kg of Cu is mixed with 250 kg of Zn? 3.8% 4. You have 200g of a solution that contains 30g of hydrochloric acid (HCl), what percentage of your solution is made up of hydrochloric acid (by mass)? 15% 5. If I make a solution by adding water to 75 mL of ethanol until the total volume of the solution is 375mL, what’s the percent by volume of ethanol in the solution? 20% 6. If I add 1.65 L of water to 112 grams of sodium acetate, what is the percent by mass of sodium acetate in this solution? 6.36% 7. Solder flux, available at hardwood stores, contains 16 g of zinc chloride in 50g of solution. What is the percentage mass of zinc chloride in the solution? 32% 8. If 250mL of sodium chloride is present in 3 L of solution, what is the percentage by volume of sodium chloride? 8.5% 9. Suppose you have 70g of sodium chloride salt (NaCl) in 250mL of water. Express this as a % by mass solution. 22% PERCENT CONCENTRATION EXTRA PRACTICE WORKSHEET 1. What is the % by massif 8.0 g copper is added to enough zinc to produce 100 g of an alloy? 8% 2. Brass is a copper-zinc alloy. If the concentration of zinc is relatively low, the brass has a golden color and is often used for inexpensive jewellery. If a 35.0 g pendant contains 1.7 g of zinc, what is the percentage by mass of zinc in brass? 4.9% 3. What is the % by mass of copper in an alloy when 10 kg of Cu is mixed with 250 kg of Zn? 3.8% 4. You have 200g of a solution that contains 30g of hydrochloric acid (HCl), what percentage of your solution is made up of hydrochloric acid (by mass)? 15% 5. If I make a solution by adding water to 75 mL of ethanol until the total volume of the solution is 375mL, what’s the percent by volume of ethanol in the solution? 20% 6. If I add 1.65 L of water to 112 grams of sodium acetate, what is the percent by mass of sodium acetate in this solution? 6.36% 7. Solder flux, available at hardwood stores, contains 16 g of zinc chloride in 50g of solution. What is the percentage mass of zinc chloride in the solution? 32% 8. If 250mL of sodium chloride is present in 3 L of solution, what is the percentage by volume of sodium chloride? 8.5% 9. Suppose you have 70g of sodium chloride salt (NaCl) in 250mL of water. Express this as a % by mass solution. 22% PERCENT CONCENTRATION EXTRA PRACTICE WORKSHEET 1. What is the % by massif 8.0 g copper is added to enough zinc to produce 100 g of an alloy? 8% 2. Brass is a copper-zinc alloy. If the concentration of zinc is relatively low, the brass has a golden color and is often used for inexpensive jewellery. If a 35.0 g pendant contains 1.7 g of zinc, what is the percentage by mass of zinc in brass? 4.9% 3. What is the % by mass of copper in an alloy when 10 kg of Cu is mixed with 250 kg of Zn? 3.8% 4. You have 200g of a solution that contains 30g of hydrochloric acid (HCl), what percentage of your solution is made up of hydrochloric acid (by mass)? 15% 5. If I make a solution by adding water to 75 mL of ethanol until the total volume of the solution is 375mL, what’s the percent by volume of ethanol in the solution? 20% 6. If I add 1.65 L of water to 112 grams of sodium acetate, what is the percent by mass of sodium acetate in this solution? 6.36% 7. Solder flux, available at hardwood stores, contains 16 g of zinc chloride in 50g of solution. What is the percentage mass of zinc chloride in the solution? 32% 8. If 250mL of sodium chloride is present in 3 L of solution, what is the percentage by volume of sodium chloride? 8.5% 9. Suppose you have 70g of sodium chloride salt (NaCl) in 250mL of water. Express this as a % by mass solution. 22%
6035
https://medium.com/nice-math-problems/an-epic-imo-geometry-problem-by-usas-golden-boys-imo-2023-p6-32a5d991c088
An Epic IMO Geometry Problem by USA’s Golden Boys — IMO 2023 P6 | by Russell Lim | Nice Math Problems | Medium Sitemap Open in app Sign up Sign in Write Search Sign up Sign in Mastodon Nice Math Problems ------------------ · Follow publication Follow this publication if you enjoy ✨nice✨ mathematics problems. Follow publication Member-only story An Epic IMO Geometry Problem by USA’s Golden Boys — IMO 2023 P6 One of the hardest IMO Geometry problems of all time. Russell Lim Follow 7 min read · Aug 4, 2023 18 Listen Share Press enter or click to view image in full size Image by author IMO Problem 6 is the last and traditionally the hardest of the IMO problems. This year was no exception, with the vast majority of students scoring 0 and only 6 students (out of over 600) obtaining the full 7 marks. Press enter or click to view image in full size source: (imo-official.org) This problem was proposed by Ankan Bhattacharya and Luke Robitaille, both ex-USA IMO Gold Medallists. Luke is also ranked #6 in the IMO Hall of Fame, having participated in 4 IMOs and won 4 Gold Medals, most recently in 2022. Problem Statement Press enter or click to view image in full size source: (imo-official.org) Before we get going, I should admit that this problem is well out of my league to actually solve. In this article, I will build up the diagram step by step, so you can get an idea of what the problem is asking. Then I will give a sketch of how the problem can be solved, based on solutions others have posted online (which are linked at the bottom). Create an account to read the full story. The author made this story available to Medium members only. If you’re new to Medium, create a new account to read this story on us. Continue in app Or, continue in mobile web Sign up with Google Sign up with Facebook Sign up with email Already have an account? Sign in 18 18 Follow Published in Nice Math Problems ------------------------------- 365 followers ·Last published Aug 17, 2025 Follow this publication if you enjoy ✨nice✨ mathematics problems. Follow Follow Written by Russell Lim ---------------------- 3.3K followers ·63 following I teach high school mathematics in Melbourne, Australia. I like thinking about interesting problems and learning new things. Follow No responses yet Write a response What are your thoughts? Cancel Respond More from Russell Lim and Nice Math Problems In Nice Math Problems by Russell Lim “Math Like a Girl” — A Nice EGMO Problem to Test Your Skills ------------------------------------------------------------ ### 2025 European Girls Mathematical Olympiad — Problem 1 Apr 29 In Nice Math Problems by Russell Lim Beijing High School Math — Prove (3,4)-Knight’s Tour is Impossible! 😅 ---------------------------------------------------------------------- ### 2025 Beijing Gao Kao (High School Exam) — Final Problem Aug 3 In Nice Math Problems by Russell Lim Gaokao 2025 — Ping Pong Probability! 🏓 --------------------------------------- ### Challenge yourself with a nice probability problem from the Chinese High School Exams Jul 5 2 In Nice Math Problems by Russell Lim A Nice Chinese Algebra Problem ------------------------------ ### From the 2023 Shanghai Gao Kao Oct 16, 2023 3 See all from Russell Lim See all from Nice Math Problems Recommended from Medium In Math Games by BL An Advanced Indian Math Exam Question ------------------------------------- ### An infinite product 6d ago 1 AI & Tech by Nidhika, PhD Integrate success and differentiate Pains. In layman terms #calculus -------------------------------------------------------------------- ### Note: The original is on authors webpage. Jun 19 In ThinkArt by Nnamdi Samuel An Interview Puzzle That Broke the Internet in 2020 --------------------------------------------------- ### Don’t Trust the Math — Find the Pattern Instead 6d ago 8 S-Ag3 A 17-Year-Old’s Breakthrough Shakes Up Mathematics -------------------------------------------------- ### How Hannah Cairo Toppled a 40-Year-Old Conjecture with Curiosity and Grit Jul 10 2 In Puzzle Sphere by Avinash Kumar Singh, Ph.D. A Family of Truths and Lies — Can You Crack This Logic Puzzle? -------------------------------------------------------------- ### A mysterious statement, a curious village, and one person you can be sure about. May 15 5 In Physics In History by Sunny Labh The Final Days of Ludwig Boltzmann ---------------------------------- ### A short account on Boltzmann’s genius and his bipolar disorder Jul 5 4 See more recommendations Help Status About Careers Press Blog Privacy Rules Terms Text to speech
6036
https://www-leland.stanford.edu/~boyd/papers/pdf/cvx_pwl_fit.pdf
Optim Eng (2009) 10: 1–17 DOI 10.1007/s11081-008-9045-3 Convex piecewise-linear fitting Alessandro Magnani · Stephen P. Boyd Received: 14 April 2006 / Accepted: 4 March 2008 / Published online: 25 March 2008 © Springer Science+Business Media, LLC 2008 Abstract We consider the problem of fitting a convex piecewise-linear function, with some specified form, to given multi-dimensional data. Except for a few special cases, this problem is hard to solve exactly, so we focus on heuristic methods that find locally optimal fits. The method we describe, which is a variation on the K-means algorithm for clustering, seems to work well in practice, at least on data that can be fit well by a convex function. We focus on the simplest function form, a maximum of a fixed number of affine functions, and then show how the methods extend to a more general form. Keywords Convex optimization · Piecewise-linear approximation · Data fitting 1 Convex piecewise-linear fitting problem We consider the problem of fitting some given data (u1,y1),...,(um,ym) ∈Rn × R with a convex piecewise-linear function f : Rn →R from some set F of candidate functions. With a least-squares fitting criterion, we obtain the problem minimize J(f ) = m ! i=1 (f (ui) −yi)2 subject to f ∈F, (1) A. Magnani · S.P. Boyd (! ) Electrical Engineering Department, Stanford University, Stanford, CA 94305, USA e-mail: boyd@stanford.edu A. Magnani e-mail: alem@stanford.edu 2 A. Magnani, S.P. Boyd with variable f . We refer to (J(f )/m)1/2 as the RMS (root-mean-square) fit of the function f to the data. The convex piecewise-linear fitting problem (1) is to find the function f , from the given family F of convex piecewise-linear functions, that gives the best (smallest) RMS fit to the given data. Our main interest is in the case when n (the dimension of the data) is relatively small, say not more than 5 or so, while m (the number of data points) can be relatively large, e.g., 104 or more. The methods we describe, however, work for any values of n and m. Several special cases of the convex piecewise-linear fitting problem (1) can be solved exactly. When F consists of the affine functions, i.e., f has the form f (x) = aT x + b, the problem (1) reduces to an ordinary linear least-squares problem in the function parameters a ∈Rn and b ∈R and so is readily solved. As a less trivial ex-ample, consider the case when F consists of all piecewise-linear functions from Rn into R, with no other constraint on the form of f . This is the nonparametric convex piecewise-linear fitting problem. Then the problem (1) can be solved, exactly, via a quadratic program (QP); see (Boyd and Vandenberghe 2004, Sect. 6.5.5). This non-parametric approach, however, has two potential practical disadvantages. First, the QP that must be solved is very large (containing more than mn variables), limiting the method to modest values of m (say, a thousand). The second potential disadvan-tage is that the piecewise-linear function fit obtained can be very complex, with many terms (up to m). Of course, not all data can be fit well (i.e., with small RMS fit) with a convex piecewise-linear function. For example, if the data are samples from a function that has strong negative (concave) curvature, then no convex function can fit it well. More-over, the best fit (which will be poor) will be obtained with an affine function. We can also have the opposite situation: it can occur that the data can be perfectly fit by an affine function, i.e., we can have J = 0. In this case we say that the data is interpo-lated by the convex piecewise-linear function f . 1.1 Max-affine functions In this paper we consider the parametric fitting problem, in which the candidate func-tions are parametrized by a finite-dimensional vector of coefficients α ∈Rp, where p is the number of parameters needed to describe the candidate functions. One very simple form is given by Fk ma, the set of functions on Rn with the form f (x) = max{aT 1 x + b1,...,aT k x + bk}, (2) i.e., a maximum of k affine functions. We refer to a function of this form as ‘max-affine’, with k terms. The set Fk ma is parametrized by the coefficient vector α = (a1,...,ak,b1,...,bk) ∈Rk(n+1). In fact, any convex piecewise-linear function on Rn can be expressed as a max-affine function, for some k, so this form is in a sense universal. Our interest, however, is in the case when the number of terms k is relatively small, say no more than 10, or a few 10s. In this case the max-affine representation (2) is compact, in the sense Convex piecewise-linear fitting 3 that the number of parameters needed to describe f (i.e., p) is much smaller than the number of parameters in the original data set (i.e., m(n + 1)). The methods we describe, however, do not require k to be small. When F = Fk ma, the fitting problem (1) reduces to the nonlinear least-squares problem minimize J(α) = m ! i=1 " max j=1,...,k(aT j ui + bj) −yi #2 , (3) with variables a1,...,ak ∈Rn, b1,...,bk ∈R. The function J is a piecewise-quadratic function of α. Indeed, for each i, f (ui) −yi is piecewise-linear, and J is the sum of squares of these functions, so J is convex quadratic on the (polyhe-dral) regions on which f (ui) is affine. But J is not globally convex, so the fitting problem (3) is not convex. 1.2 A more general parametrization We will also consider a more general parametrized form for convex piecewise-linear functions, f (x) = ψ(φ(x,α)), (4) where ψ : Rq →R is a (fixed) convex piecewise-linear function, and φ : Rn × Rp →Rq is a (fixed) bi-affine function. (This means that for each x, φ(x,α) is an affine function of α, and for each α, φ(x,α) is an affine function of x.) The simple max-affine parametrization (2) has this form, with q = k, ψ(z1,...,zk) = max{z1,...,zk}, and φi(x,α) = aT i x + bi. As an example, consider the set of functions F that are sums of k terms, each of which is the maximum of two affine functions, f (x) = k ! i=1 max{aT i x + bi,cT i x + di}, (5) parametrized by a1,...,ak, c1,...,ck ∈Rn and b1,...,bk, d1,...,dk ∈R. This family corresponds to the general form (4) with ψ(z1,...,zk,w1,...,wk) = k ! i=1 max{zi,wi}, and φ(x,α) = (aT 1 x + b1,...,aT k x + bk,cT 1 x + d1,...,cT k x + dk). Of course we can expand any function with the more general form (4) into its max-affine representation. But the resulting max-affine representation can be very much larger than the original general form representation. For example, the function form (5) requires p = 2k(n + 1) parameters. If the same function is written out as a max-affine function, it requires 2k terms, and therefore 2k(n + 1) parameters. The 4 A. Magnani, S.P. Boyd hope is that a well chosen general form can give us a more compact fit to the given data than a max-affine form with the same number of parameters. As another interesting example of the general form (4), consider the case in which f is given as the optimal value of a linear program (LP) with the right-hand side of the constraints depending bi-affinely on x and the parameters: f (x) = min{cT v | Av ≤b + Bx}. Here c and A are fixed; b and B are considered the parameters that define f . This function can be put in the general form (4) using ψ(z) = min{cT v | Av ≤z}, φ(x,b,B) = b + Bx. The function ψ is convex and piecewise-linear (see, e.g., Boyd and Vandenberghe 2004); the function φ is evidently bi-affine in x and (b,B). 1.3 Dependent variable transformation and normalization We can apply a nonsingular affine transformation to the dependent variable u, by forming ˜ ui = T ui + s, i = 1,...,m, where T ∈Rn×n is nonsingular and s ∈Rn. Defining ˜ f (˜ x) = f (T −1(x −s)), we have ˜ f (˜ ui) = f (ui). If f is piecewise-linear and convex, then so is ˜ f (and of course, vice versa). Provided F is invariant under composition with affine functions, the problem of fitting the data (ui,yi) with a function f ∈F is the same as the prob-lem of fitting the data (˜ ui,yi) with a function ˜ f ∈F. This allows us to normalize the dependent variable data in various ways. For ex-ample, we can assume that it has zero (sample) mean and unit (sample) covariance, ¯ u = (1/m) m ! i=1 ui = 0, $u = (1/m) m ! i=1 uiuT i = I, (6) provided the data ui are affinely independent. (If they are not, we can reduce the problem to an equivalent one with smaller dimension.) 1.4 Outline In Sect. 2 we describe several applications of convex piecewise-linear fitting. In Sect. 3, we describe a basic heuristic algorithm for (approximately) solving the max-affine fitting problem (1). This basic algorithm has several shortcomings, such as convergence to a poor local minimum, or failure to converge at all. By running this algorithm a modest number of times, from different initial points, however, we obtain a fairly reliable algorithm for least-squares fitting of a max-affine function to given data. Finally, we show how the algorithm can be extended to handle the more general function parametrization (4). In Sect. 4 we present some numerical examples. Convex piecewise-linear fitting 5 1.5 Previous work Piecewise-linear functions arise in many areas and contexts. Some general forms for representing piecewise-linear functions can be found in, e.g., Kang and Chua, Kahlert and Chua (1978, 1990). Several methods have been proposed for fitting general piecewise-linear functions to (multidimensional) data. A neural network algorithm is used in Gothoskar et al. (2002); a Gauss-Newton method is used in Julian et al., Horst and Beichel (1998, 1997) to find piecewise-linear approximations of smooth func-tions. A recent reference on methods for least-squares with semismooth functions is Kanzow and Petra (2004). An iterative procedure, similar in spirit to our method, is described in Ferrari-Trecate and Muselli (2002). Software for fitting general piecewise-linear functions to data include, e.g., Torrisi and Bemporad (2004), Storace and De Feo (2002). The special case n = 1, i.e., fitting a function on R, by a piecewise-linear function has been extensively studied. For example, a method for finding the minimum num-ber of segments to achieve a given maximum error is described in Dunham (1986); the same problem can be approached using dynamic programming (Goodrich 1994; Bellman and Roth 1969; Hakimi and Schmeichel 1991; Wang et al. 1993), or a ge-netic algorithm (Pittman and Murthy 2000). The problem of simplifying a given piecewise-linear function on R, to one with fewer segments, is considered in Imai and Iri (1986). Another related problem that has received much attention is the problem of fitting a piecewise-linear curve, or polygon, in R2 to given data; see, e.g., Aggarwal et al. (1985), Mitchell and Suri (1992). An iterative procedure, closely related to the k-means algorithm and therefore similar in spirit to our method, is described in Phillips and Rosenfeld (1988), Yin (1998). Piecewise-linear functions and approximations have been used in many appli-cations, such as detection of patterns in images (Rives et al. 1985), contour trac-ing (Dobkin et al. 1990), extraction of straight lines in aerial images (Venkateswar and Chellappa 1992), global optimization (Mangasarian et al. 2005), compression of chemical process data (Bakshi and Stephanopoulos 1996), and circuit modeling (Ju-lian et al. 1998; Chua and Deng 1986; Vandenberghe et al. 1989). We are aware of only two papers which consider the problem of fitting a piecewise-linear convex function to given data. Mangasarian et al. (2005) describe a heuristic method for fitting a piecewise-linear convex function of the form a + bT x + ∥Ax + c∥1 to given data (along with the constraint that the function underestimate the data). The focus of their paper is on finding piecewise-linear convex underestimators for known (nonconvex) functions, for use in global optimization; our focus, in contrast, is on simply fitting some given data. The closest related work that we know of is Kim et al. (2004). In this paper, Kim et al. describe a method for fitting a (convex) max-affine function to given data, increasing the number of terms to get a better fit. (In fact they describe a method for fitting a max-monomial function to circuit models; see Sect. 2.3.) 6 A. Magnani, S.P. Boyd 2 Applications In this section we briefly describe some applications of convex piecewise-linear fit-ting. None of this material is used in the sequel. 2.1 LP modeling One application is in LP modeling, i.e., approximately formulating a practical prob-lem as an LP. Suppose a problem is reasonably well modeled using linear equality and inequality constraints, with a few nonlinear inequality constraints. By approx-imating these nonlinear functions by convex piecewise-linear functions, the overall problem can be formulated as an LP, and therefore efficiently solved. As an example, consider a minimum fuel optimal control problem, with linear dynamics and a nonlinear fuel-use function, minimize T −1 ! t=0 f (u(t)) subject to x(t + 1) = A(t)x(t) + B(t)u(t), t = 0,...,T −1, x(0) = xinit, x(T ) = xdes, with variables x(0),...,x(T ) ∈Rn (the state trajectory), and u(0),...,u(T −1) ∈ Rm (the control input). The problem data are A(0),...,A(T −1) (the dynamics ma-trices), B(0),...,B(T −1) (the control matrices), xinit (the initial state), and xdes (the desired final state). The function f : Rm →R is the fuel-use function, which gives the fuel consumed in one period, as a function of the control input value. Now suppose we have empirical data or measurements of some values of the control in-put u ∈Rm, along with the associated fuel use f (u). If we can fit these data with a convex piecewise-linear function, say, f (u) ≈ˆ f (u) = max j=1,...,k(aT j u + bj), then we can formulate the (approximate) minimum fuel optimal control problem as the LP minimize T −1 ! t=0 ˜ f (t) subject to x(t + 1) = A(t)x(t) + B(t)u(t), t = 0,...,T −1, x(0) = xinit, x(T ) = xdes, ˜ f (t) ≥aT j u(t) + bj, t = 0,...,T −1, j = 1,...,k, (7) with variables x(0),...,x(T ) ∈Rn, u(0),...,u(T −1) ∈Rm, and ˜ f (0),..., ˜ f (T −1) ∈R. Convex piecewise-linear fitting 7 2.2 Simplifying convex functions Another application of convex piecewise-linear fitting is to simplify a convex func-tion that is complex, or expensive to evaluate. To illustrate this idea, we continue our minimum fuel optimal control problem described above, with a piecewise-linear fuel use function. Consider the function V : Rn →R, which maps the initial state xinit to its associated minimum fuel use, i.e., the optimal value of the LP (7). (This is the Bellman value function for the optimal control problem.) The value function is piecewise-linear and convex, but very likely requires an extremely large number of terms to be expressed in max-affine form. We can (possibly) form a simple ap-proximation of V by a max-affine function with many fewer terms, as follows. First, we evaluate V via the LP (7), for a large number of initial conditions. Then, we fit a max-affine function with a modest number of terms to the resulting data. This con-vex piecewise-linear approximate value function can be used to construct a simple feedback controller that approximately minimizes fuel use; see, e.g., Bemporad et al. (2002). 2.3 Max-monomial fitting for geometric programming Max-affine fitting can be used to find a max-monomial approximation of a positive function, for use in geometric programming modeling; see Boyd et al. (2006). Given data (zi,wi) ∈Rn ++ × R++, we form ui = logzi, yi = logwi, i = 1,...,m. (The log of a vector is interpreted as componentwise.) We now fit this data with a max-affine model, yi ≈max{aT 1 ui + b1,...,aT k ui + bk}. This gives us the max-monomial model wi ≈max{g1(zi),...,gK(zi)}, where gi are the monomial functions gj(z) = ebiz aj1 1 ···z ajn n , j = 1,...,K. (These are not monomials in the standard sense, but in the sense used in geometric programming.) 3 Least-squares partition algorithm 3.1 The algorithm In this section we present a heuristic algorithm to (approximately) solve the k-term max-affine fitting problem (3), i.e., minimize J = m ! i=1 " max j=1,...,k(aT j ui + bj) −yi #2 , 8 A. Magnani, S.P. Boyd with variables a1,...,ak ∈Rn and b1,...,bk ∈R. The algorithm alternates between partitioning the data and carrying out least-squares fits to update the coefficients. We let P (l) j for j = 1,...,k, be a partition of the data indices at the lth iteration, i.e., P (l) j ⊆{1,...,m}, with $ j P (l) j = {1,...,m}, P (l) i ∩P (l) j = ∅ for i ̸= j. (We will describe methods for choosing the initial partition P (0) j later.) Let a(l) j and b(l) j denote the values of the parameters at the lth iteration of the algorithm. We generate the next values, a(l+1) j and b(l+1) j , from the current partition P (l) j , as follows. For each j = 1,...,k, we carry out a least-squares fit of aT j ui + bj to yi, using only the data points with i ∈P (l) j . In other words, we take a(l+1) j and b(l+1) j as values of a and b that minimize ! i∈P (l) j (aT ui + b −yi)2. (8) In the simplest (and most common) case, there is a unique pair (a,b) that mini-mizes (8), i.e., % a(l+1) j b(l+1) j & = %'uiuT i 'ui 'uT i |P (l) j | &−1 ('yiui 'yi ) , (9) where the sums are over i ∈P (l) j . When there are multiple minimizers of the quadratic function (8), i.e., the matrix to be inverted in (9) is singular, we have several options. One option is to add some regularization to the simple least-squares objective in (8), i.e., an additional term of the form λ∥a∥2 2 + µb2, where λ and µ are positive constants. Another possibility is to take the updated parameters as the unique minimizer of (8) that is closest to the previous value, (a(l) j ,b(l) j ), in Euclidean norm. Using the new values of the coefficients, we update the partition to obtain P (l+1) j , by assigning i to P (l+1) s if f (l)(ui) = max s=1,...,k(a(l)T s ui + b(l) s ) = a(l)T j ui + b(l) j . (10) (This means that the term a(l)T j ui + b(l) j is ‘active’ at the data point ui.) Roughly speaking, this means that P (l+1) j is the set of indices for which the affine function aT j z + bj is the maximum; we can break ties (if there are any) arbitrarily. This iteration is run until convergence, which occurs if the partition at an iteration is the same as the partition at the previous iteration, or some maximum number of iterations is reached. Convex piecewise-linear fitting 9 We can write the algorithm as LEAST-SQUARES PARTITION ALGORITHM. given partition P (0) 1 ,...,P (0) K of {1,...,m}, iteration limit lmax for l = 0,...,lmax 1. Compute a(l+1) j and b(l+1) j as in (9). 2. Form the partition P (l+1) 1 ,...,P (l+1) k as in (10). 3. Quit if P (l) j = P (l+1) j for j = 1,...,k. During the execution of the least-squares partition algorithm, one or more of the sets P (l) j can become empty. The simplest approach is to drop empty sets from the partition, and continue with a smaller value of k. 3.2 Interpretation as Gauss-Newton method We can interpret the algorithm as a Gauss-Newton method for the problem (3). Sup-pose that at a point u ∈Rn, there is a unique j for which f (u) = aT j u + bj (i.e., there are no ties in the maximum that defines f (u)). In this case the function f is differentiable with respect to a and b; indeed, it is locally affine in these parameter values. Its first order approximation at a, b is f (u) ≈ˆ f (u) = ˜ aT j u + ˜ bj. This approximation is exact, provided the perturbed parameter values ˜ a1,..., ˜ ak, ˜ b1,..., ˜ ab are close enough to the parameter values a1,...,ak, b1,...,ab. Now assume that for each data point ui, there is a unique j for which f (ui) = a(l)T j ui + b(l) j (i.e., there are no ties in the maxima that define f (ui)). Then the first order approximation of (f (u1),...,f (um)) is given by f (ui) ≈ˆ f (ui) = ˜ aT j(i)ui + ˜ bj(i), where j(i) is the unique active j at ui, i.e., i ∈P (l) j . In the Gauss-Newton method for a nonlinear least-squares problem, we form the first order approximation of the argument of the norm, and solve the resulting least-squares problem to get the next iterate. In this case, then, we form the linear least-squares problem of minimizing ˆ J = m ! i=1 ˆ f (ui) −yi +2 = m ! i=1 ˜ aT j(i)ui + ˜ bj(i) −yi +2 , over the variables ˜ a1,..., ˜ ak, ˜ b1,..., ˜ bk. We can re-arrange the sum defining J into terms involving each of the pairs of variables a1,b1, ..., ak,bk separately: ˆ J = ˆ J1 + ··· + ˆ Jk, 10 A. Magnani, S.P. Boyd where ˆ Jj = ! i∈P (l) j (˜ aT ui + ˜ b −yi)2, j = 1,...,k. Evidently, we can minimize ˆ J by separately minimizing each ˆ Ji. Moreover, the para-meter values that minimize ˆ J are precisely a(l+1) 1 ,...,a(l+1) k , b(l+1) 1 ,...,b(l+1) k . This is exactly the least-squares partition algorithm described above. The algorithm is closely related to the k-means algorithm used in least-squares clustering (Gersho and Gray 1991). The k-means algorithm approximately solves the problem of finding a set of k points in Rn, {z1,...,zk}, that minimizes the mean square Euclidean distance to a given data set u1,...,um ∈Rn. (The distance be-tween a point u and the set of points {z1,...,zk} is defined as the minimum distance, i.e., minj=1,...,k ∥u −zj∥2.) In the k-means algorithm, we iterate between two steps: first, we partition the data points according to the closest current point in the set {z1,...,zk}; then we update each zj as the mean of the points in its associated parti-tion. (The mean minimizes the sum of the squares of the Euclidean distances to the point.) Our algorithm is conceptually identical to the k-means algorithm: we partition the data points according to which of the affine functions is active (i.e., largest), and then update the affine functions, separately, using only the data points in its associated partition. 3.3 Nonconvergence of least-squares partition algorithm The basic least-squares partition algorithm need not converge; it can enter a (noncon-stant) limit cycle. Consider, for example, the data u1 = −2, u2 = −1, u3 = 0, u4 = 1, u5 = 2, y1 = 0, y2 = 1, y3 = 3, y4 = 1, y5 = 0, and k = 2. The data evidently cannot be fit well by any convex function; the (globally) best fit is obtained by the constant function f (u) = 1. For many initial parameter values, however, the algorithm converges to a limit cycle with period 2, alternating between the two functions f1(u) = max{u + 2,−(3/2)u + 17/6}, f2(u) = max{(3/2)u + 17/6,−u + 2}. The algorithm therefore fails to converge; moreover, each of the functions f1 and f2 gives a very suboptimal fit to the data. On the other hand, with real data (not specifically designed to illustrate noncon-vergence) we have observed that the least-squares partition algorithm appears to con-verge in most cases. In any case, convergence failure has no practical consequences since the algorithm is terminated after some fixed maximum number of steps, and moreover, we recommend that it be run from a number of starting points, with the best fit obtained used as the final fit. Convex piecewise-linear fitting 11 3.4 Piecewise-linear fitting algorithm The least-squares partition algorithm, used by itself, has several serious shortcom-ings. It need not converge, and when it does converge, it can (and often does) con-verge to a piecewise-linear approximation with a poor fit to the data. Both of these problems can be mitigated by running the least-squares partition algorithm multiple times, with different initial partitions. The final fit is taken to be the best fit obtained among all iterations of all runs of the algorithm. We first describe a simple method for generating a random initial partition. We randomly choose points p1,...,pK, and define the initial partition to be the Voronoi sets associated with these points. We have P (0) j = {i | ∥ui −pj∥< ∥ui −ps∥for s ̸= j}, j = 1,...,K. (11) (Thus, P (0) j is the set of indices of data points that are closest to pj.) The seed points pi should be generated according to some distribution that matches the shape of the data points ui, for example, they can be chosen from a normal distribution with mean ¯ u and covariance $u (see (6)). The overall algorithm can be described as PIECEWISE-LINEAR FITTING ALGORITHM. given number of trials Ntrials, iteration limit lmax for i = 1,...,Ntrials 1. Generate random initial partition via (11). 2. Run least-squares partition algorithm with iteration limit lmax. 3. Keep track of best RMS fit obtained. 3.5 General form fitting In this section we show the least-squares partition algorithm can be modified to fit piecewise-linear functions with the more general form (4), f (x,α) = ψ(φ(x,α)), where ψ is a fixed convex piecewise-linear function, and φ is a fixed bi-affine func-tion. We described the least-squares partition algorithm in terms of a partition of the in-dices, according to which of the k affine functions is active at the point ui. The same approach of an explicit partition will not work in the more general case, since the size of the partition can be extremely large. Instead, we start from the idea that the parti-tion gives an approximation of f (ui) that is affine in α, and valid near α(l). If there is no ‘tie’ at ui (i.e., there is a unique affine function that achieves the maximum), then the affine approximation is exact in a neighborhood of the current parameter value α(l). We can do the same thing with the more general form. For each i, we find ai(α) and bi(α), both affine functions of α, so that f (ui,α) ≈ai(α)T ui + bi(α) 12 A. Magnani, S.P. Boyd for α near α(l), the current value of the parameters. This approximation is exact in a neighborhood of α(l) if ψ(ui,α) is a point of differentiability of ψ. (For max-affine functions, this is the case when there is no ‘tie’ at ui.) If it is not such a point, we can choose any subgradient model of f (ui,α), i.e., any ai(α) and bi(α) for which f (ui,α(l)) = ai(α(l))T ui + bi(α(l)), (the approximation is exact for α = α(l)), and f (ui,α) ≥ai(α)T ui + bi(α) for all α. (In the case of max-affine functions, breaking any ties arbitrarily satisfies this condition.) We then compute a new parameter value using a Gauss-Newton like method. We replace f (ui) in the expression for J with ˆ f (l)(ui,α) = ai(α(l))T ui + bi(α(l)), which is affine in α. We then choose α(l+1) as the minimizer of ˆ J = m ! i=1 ( ˆ f (l)(ui) −yi)2, which can be found using standard linear least-squares. To damp this update rule, we can add a regularization term to ˆ J , by choosing α(l+1) as the minimizer of m ! i=1 ( ˆ f (l)(ui) −yi)2 + ρ∥α −α(l)∥2, where ρ > 0 is a parameter. 4 Numerical examples In this section we show some numerical results, using the following data set. The dimension is n = 3, and we have m = 113 = 1331 points. The set of points ui is given by V × V × V, where V = {−5,−4,−3,−2,−1,0,1,2,3,4,5}. The values are obtained as yi = g(ui), where g is the (convex) function g(x) = log(expx1 + expx2 + expx3). We use the piecewise-linear fitting algorithm described in Sect. 3.4, with iteration limit lmax = 50, and number of terms varying from k = 0 to k = 20. For k = 0, the fitting function is taken to be zero, so we report the RMS value of y1,...,ym as the RMS fit. For k = 1, the fit is the best affine fit to the data, which can be found using least-squares. Figure 1 shows the RMS fits obtained after Ntrials = 10 trials (top curve), and after Ntrials = 100 trials (bottom curve). These show that good fits Convex piecewise-linear fitting 13 Fig. 1 Best RMS fit obtained with 10 trials (top curve) and 100 trials (bottom curve), versus number of terms k in max-affine function are obtained with only 10 trials, and that (slightly) better ones are obtained with 100 trials. To give an idea of the variation in RMS fit obtained with different trials, as well as the number of steps required for convergence (if it occurs), we fix the number of terms at k = 12, and run the least-squares partition algorithm 200 times, with a limit of 50 iterations, recording both the final RMS fit obtained, and the number of steps before convergence. (The number of steps is reported as 50 if the least-squares partition algorithm has not converged in 50 steps.) Figure 2 shows the histogram of RMS fit obtained. We can see that the fit is often, but not always, quite good; in just a few cases the fit obtained is poor. Evidently the best of even a modest number of trials will be quite good. Figure 3 shows the distribution of the number of iterations of the least-squares partition algorithm required to converge. Convergence failed in 13 of the 200 trials; but in fact, the RMS fit obtained in these trials was not particularly bad. Typically convergence occurs within around 25 iterations. In our last numerical example, we compare fitting the data with a max-affine func-tion with k terms, and with the more general form f (x) = max i=1,...,k/2 aT i x + bi + + max i=k/2+1,...,k aT i x + bi + , parametrized by a1,...,ak ∈Rn and b1,...,bk ∈R. (Note that the number of para-meters in each function form is the same.) This function corresponds to the general 14 A. Magnani, S.P. Boyd Fig. 2 Distribution of RMS fit obtained in 200 trials of least-squares partition algorithm, for k = 12, lmax = 50 Fig. 3 Distribution of the number of steps required by least-squares partition algorithm to converge, over 200 trials. The number of steps is reported as 50 if convergence has not been obtained in 50 steps form (4) with ψ(z1,...,zk) = max i=1,...,k/2zi + max i=k/2+1,...,k zi, Convex piecewise-linear fitting 15 Fig. 4 Best RMS fit obtained for max-affine function (top) and sum-max function (bottom) and φ(x,α) = (aT 1 x + b1,...,aT k x + bk). We set the iteration limit for both forms as lmax = 100, and take the best fit ob-tained in Ntrials = 10 trials. We use the value ρ = 10−5 for the regularization parame-ter in the general form algorithm. Figure 4 shows the RMS fit obtained for the two forms, versus k. Evidently the sum-max form gives (slightly) better RMS fit than the max-affine form. 5 Conclusions We have described a new method for fitting a convex piecewise linear function to a given (possibly large) set of data (with a modest number of independent variables). The method is heuristic, since the algorithm can (and does) fail to converge to the globally optimal fit. Numerical examples suggest, however, that the method works very well in practice, on data that can be fit well by a convex function. The method has many applications in practical optimization modeling. Data sam-ples can be used to generate piecewise-linear convex functions, which in turn can be used to construct linear programming models. Acknowledgement This work was carried out with support from C2S2, the MARCO Focus Center for Circuit and System Solutions, under MARCO contract 2003-CT-888. We are grateful to Jim Koford for suggesting the problem. 16 A. Magnani, S.P. Boyd References Aggarwal A, Booth H, O’Rourke J, Suri S, Yap C (1985) Finding minimal convex nested polygons. In: Proceedings of the symposium on computational geometry. ACM Press, New York, pp 296–304 Bakshi B, Stephanopoulos G (1996) Compression of chemical process data by functional approximation and feature extraction. AIChE J 42(2):477–492 Bellman R, Roth R (1969) Curve fitting by segmented straight lines. Am Stat Assoc J 64:1079–1084 Bemporad A, Borrelli F, Morari M (2002) Model predictive control based on linear programming—the explicit solution. IEEE Trans Autom Control 47(12):1974–1985 Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge Boyd S, Kim S, Vandenberghe L, Hassibi A (2006) A tutorial on geometric programming. Optim Eng Chua L, Deng A (1986) Canonical piecewise-linear modeling. IEEE Trans Circuits Syst 33(5):511–525 Dobkin D, Levy S, Thurston W, Wilks A (1990) Contour tracing by piecewise linear approximations. ACM Trans Graph 9(4):389–423 Dunham J (1986) Optimum uniform piecewise linear approximation of planar curves. IEEE Trans Pattern Anal Mach Intell 8(1):67–75 Ferrari-Trecate G, Muselli M (2002) A new learning method for piecewise linear regression. In: Proceed-ings of the international conference on artificial neural networks. Springer, Berlin, pp 444–449 Gersho A, Gray R (1991) Vector quantization and signal compression. Kluwer Academic, Norwell Goodrich M (1994) Efficient piecewise-linear function approximation using the uniform metric. In: Pro-ceedings of the symposium on computational geometry. ACM Press, New York, pp 322–331 Gothoskar G, Doboli A, Doboli S (2002) Piecewise-linear modeling of analog circuits based on model ex-traction from trained neural networks. In: Proceedings of IEEE international workshop on behavioral modeling and simulation, pp 41–46 Hakimi S, Schmeichel E (1991) Fitting polygonal functions to a set of points in the plane. CVGIP: Graph Models Image Process 53(2):132–136 Horst J, Beichel I (1997) A simple algorithm for efficient piecewise-linear approximation of space curves. In: Proceedings of international conference on image processing. IEEE Computer Society, Los Alamitos Imai H, Iri M (1986) An optimal algorithm for approximating a piecewise linear function. J Inf Process 9(3):159–162 Julian P, Jordan M, Desages A (1998) Canonical piecewise-linear approximation of smooth functions. IEEE Trans Circuits Syst 45(5):567–571 Kahlert C, Chua L (1990) A generalized canonical piecewise-linear representation. IEEE Trans Circuits Syst 37(3):373–383 Kang S, Chua L (1978) A global representation of multidimensional piecewise-linear functions with linear partitions. IEEE Trans Circuits Syst 25:938–940 Kanzow C, Petra S (2004) On a semismooth least squares formulation of complementarity. Optim Methods Softw 19(5):507–525 Kim J, Lee J, Vandenberghe L, Yang C (2004) Techniques for improving the accuracy of geometric-programming based analog circuit design optimization. In: Proceedings of the IEEE international conference on computer aided design, pp 863–870 Mangasarian O, Rosen J, Thompson M (2005) Global minimization via piecewise-linear underestimation. J Glob Optim 32 Mitchell J, Suri S (1992) Separation and approximation of polyhedral objects. In: Proceedings of the ACM-SIAM symposium on discrete algorithms. Society for industrial and applied mathematics, Philadel-phia, pp 296–306 Phillips T, Rosenfeld A (1988) An ISODATA algorithm for straight line fitting. Pattern Recognit 7(5):291– 297 Pittman J, Murthy C (2000) Fitting optimal piecewise linear functions using genetic algorithms. IEEE Trans Pattern Anal Mach Intel 22(7):701–718 Rives G, Dhome M, La Preste J, Richetin M (1985) Detection of patterns in images from piecewise linear contours. Pattern Recognit Lett 3:99–104 Storace M, De Feo O (2002) Piecewise-linear approximation of nonlinear dynamical systems Torrisi F, Bemporad A (2004) HYSDEL—a tool for generating computational hybrid models for analysis and synthesis problems. IEEE Trans Control Syst Technol 12(2):235–249 Vandenberghe L, de Moor B, Vandewalle J (1989) The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits. IEEE Trans Circuits Syst 36(11):1382–1391 Convex piecewise-linear fitting 17 Venkateswar V, Chellappa R (1992) Extraction of straight lines in aerial images. IEEE Trans Pattern Anal Mach Intell 14(11):1111–1114 Wang D, Huang N, Chao H, Lee R (1993) Plane sweep algorithms for the polygonal approximation prob-lems with applications. In: Proceedings of the international symposium on algorithms and computa-tion. Springer, London, pp 515–522 Yin P (1998) Algorithms for straight line fitting using k-means. Pattern Recognit Lett 19(1):31–41
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https://physics.stackexchange.com/questions/734568/influence-of-magnetic-field-of-earth
electromagnetism - Influence of magnetic field of earth - Physics Stack Exchange Join Physics 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 Influence of magnetic field of earth Ask Question Asked 2 years, 11 months ago Modified2 years, 11 months ago Viewed 126 times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. Is the magnetic field of a bar magnet influenced by the magnetic field of the earth. If so, will the magnetic field of bar magnet be different in a region where there’s no other magnetic field. electromagnetism geomagnetism Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Improve this question Follow Follow this question to receive notifications edited Nov 1, 2022 at 4:54 Qmechanic♦ 222k 52 52 gold badges 636 636 silver badges 2.6k 2.6k bronze badges asked Oct 31, 2022 at 13:31 JasonJason 1 1 @john I read that the magnet’s North Pole aligns towards the geographic north pole of the earth which is the magnetic South Pole of earth… Is this incorrect?Jason –Jason 2022-11-01 14:08:16 +00:00 Commented Nov 1, 2022 at 14:08 Add a comment| 2 Answers 2 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. I actually disagree with @John's answer. Electromagnetism is linear. I.e. electromagnetic fields simply add to each other; the fields do not influence each other. See Why is classical electromagnetism linear?, Why is the Principle of Superposition true in EM? Does it hold more generally?, In classical physics (classical electrodynamics), why linearity of Maxwell's equations prevent interaction of electromagnetic waves? The above said, I do agree with @John that Earth's magnetic field will yield a force on the bar magnet itself. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications answered Oct 31, 2022 at 14:01 astronautgravityastronautgravity 1,149 5 5 silver badges 15 15 bronze badges 1 Thank you for the follow-up. Please see an edit to my answer addressing your point.John –John 2022-10-31 21:25:45 +00:00 Commented Oct 31, 2022 at 21:25 Add a comment| This answer is useful 1 Save this answer. Show activity on this post. Yes it does. The magnetic field of the Earth applies a torque on the bar magnet so the magnetic field of a bar magnet in a region with the magnetic field of the Earth is present will be such that the north pole of the bar magnet is pointing towards the North (magnetic) Pole of the Earth. The device making use of this is called compass. EDIT: To dispell possible confusion in response to the answer by @WAH, the magnetic fields do interact. The thing is that Maxwell's equations are only linear in the absence of charged matter. Once there are movable/polarizable/etc entities around, fields start talking to each other via those entities (charges, dipoles, quadrupoles, etc). This is most prominent in nonlinear optics, but the alignement of a dipole by an external field and subsequent modification of a total field is also strictly speaking a nonlinear electromagnetic effect. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications edited Oct 31, 2022 at 21:24 answered Oct 31, 2022 at 13:49 JohnJohn 4,312 9 9 silver badges 17 17 bronze badges 2 I agree with you that the magnetic field does influence the magnetic field of a bar magnet and will apply a torque to the bar magnet. I think that this influence will also change the magnetic field of the bar magnet. The change might be very slight but it would change the field and the way that it would be measured. Even if the bar magnet is aligned with the magnetic field of the earth and there isn't any torque applied to the bar magnet the earth's magnetic field will still have an impact.Anyone –Anyone 2022-10-31 17:06:45 +00:00 Commented Oct 31, 2022 at 17:06 Responding to @John's edit. Classical electromagnetism is perfectly linear. Non-linearities appear only when one considers loop corrections in quantum electrodynamics, arxiv.org/abs/hep-th/0406216, or extensions of electromagnetism, arxiv.org/abs/hep-th/0309108. The quantum corrections are tiny, ~1% or less, and, for all practical intents and purposes, unseen. Certainly these non-linear effects are unmeasurable with a standard bar magnet interacting with Earth's magnetic field.astronautgravity –astronautgravity 2022-11-02 05:33:34 +00:00 Commented Nov 2, 2022 at 5:33 Add a comment| Your Answer Thanks for contributing an answer to Physics Stack Exchange! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. MathJax reference. To learn more, see our tips on writing great answers. Draft saved Draft discarded Sign up or log in Sign up using Google Sign up using Email and Password Submit Post as a guest Name Email Required, but never shown Post Your Answer Discard By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions electromagnetism geomagnetism 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 Linked 19In classical physics (classical electrodynamics), why linearity of Maxwell's equations prevent interaction of electromagnetic waves? 16Why is the Principle of Superposition true in EM? Does it hold more generally? 8Why is classical electromagnetism linear? 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6038
https://stackoverflow.com/questions/54661131/log2-approximation-in-fixed-point
Stack Overflow About For Teams Stack Overflow for Teams Where developers & technologists share private knowledge with coworkers Advertising Reach devs & technologists worldwide about your product, service or employer brand Knowledge Solutions Data licensing offering for businesses to build and improve AI tools and models Labs The future of collective knowledge sharing About the company Visit the blog Collectives„¢ on Stack Overflow Find centralized, trusted content and collaborate around the technologies you use most. Learn more about Collectives Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Log2 approximation in fixed-point Ask Question Asked Modified 1 year, 1 month ago Viewed 6k times 8 I'v already implemented fixed-point log2 function using lookup table and low-order polynomial approximation but not quite happy with accuracy across the entire 32-bit fixed-point range [-1,+1). The input format is s0.31 and the output format is s15.16. I'm posting this question here so that another user can post his answer (some comments were exchanged in another thread but they prefer to provide comprehensive answer in a separate thread: Fixed point approximation of 2^x, with input range of s5.26). So the question is: what is a good way to implement fixed-point log2 function with approximation error better than 1e-5? Thanks. logarithm polynomials fixed-point Share Improve this question edited Aug 15, 2024 at 5:00 AliAli asked Feb 13, 2019 at 1:24 AliAli 31833 silver badges1010 bronze badges 6 What kind of accuracy are you achieving right now and what accuracy do you hope to achieve? njuffa – njuffa 2019-02-13 01:41:44 +00:00 Commented Feb 13, 2019 at 1:41 1 Originally speed was very importantly to me so I only used first order approximation. I don't recall the average error across the entire range but the max relative errr is about 5%. The assembly implementation is very fast now so I'm willing to improve the accuracy at the cost of more clock cycles. I was just curious to learn your method and understand where I can draw accuracy vs speed line for my application. Ali – Ali 2019-02-13 02:16:13 +00:00 Commented Feb 13, 2019 at 2:16 What exactly is the question? (I can guess, but phrasing it as such would be helpful, thnx.) John McFarlane – John McFarlane 2019-02-13 18:40:27 +00:00 Commented Feb 13, 2019 at 18:40 2 @Ali I don't see a link to the other thread and you have not yet asked any question in this one. Perhaps: "what is a good way to implement log2 approximation using fixed-point?" John McFarlane – John McFarlane 2019-02-17 06:25:19 +00:00 Commented Feb 17, 2019 at 6:25 1 @JohnMcFarlane, sorry it's been almost 5 years that you posted your comment and I never noticed that, my apologies. Added the link to the description. Ali – Ali 2024-08-15 04:55:20 +00:00 Commented Aug 15, 2024 at 4:55 | Show 1 more comment 1 Answer 1 Reset to default 10 By simply counting the leading zero bits in a fixed-point number x, one can determine log2(x) to the closest strictly smaller integer. On many processor architectures, there is a "count leading zeros" machine instruction or intrinsic. Where this is not available, a fairly efficient implementation of clz() can be constructed in a variety of ways, one of which is included in the code below. To compute the fractional part of the logarithm, the two main obvious contenders are interpolation in a table and minimax polynomial approximation. In this specific case, quadratic interpolation in a fairly small table seems to be the more attractive option. x = 2i (1+f), with 0 ‰¤ f < 1. We determine i as described above and use the leading bits of f to index into the table. A parabola is fit through this and two following table entries, computing the parameters of the parabola on the fly. The result is rounded, and a heuristic adjustment is applied to partially compensate for the truncating nature of fixed-point arithmetic. Finally, the integer portion is added, yielding the final result. It should be noted that the computation involves right shifts of signed integers which may be negative. We need those right shifts to map to arithmetic right shifts at machine code level, something which is not guaranteed by the ISO-C standard. However, in practice most compilers do what is desired. In this case I used the Intel compiler on an x64 platform running Windows. With a 66-entry table of 32-bit words, the maximum absolute error can be reduced to 8.18251e-6, so full s15.16 accuracy is achieved. ``` include #include #include #include #define FRAC_BITS_OUT (16) #define INT_BITS_OUT (15) #define FRAC_BITS_IN (31) #define INT_BITS_IN ( 0) / count leading zeros: intrinsic or machine instruction on many architectures / int32_t clz (uint32_t x) { uint32_t n, y; n = 31 + (!x); if ((y = (x & 0xffff0000U))) { n -= 16; x = y; } if ((y = (x & 0xff00ff00U))) { n -= 8; x = y; } if ((y = (x & 0xf0f0f0f0U))) { n -= 4; x = y; } if ((y = (x & 0xccccccccU))) { n -= 2; x = y; } if (( (x & 0xaaaaaaaaU))) { n -= 1; } return n; } #define LOG2_TBL_SIZE (6) #define TBL_SIZE ((1 << LOG2_TBL_SIZE) + 2) / for i = [0,65]: log2(1 + i/64) (1 << 31) / const uint32_t log2Tab [TBL_SIZE] = { 0x00000000, 0x02dcf2d1, 0x05aeb4dd, 0x08759c50, 0x0b31fb7d, 0x0de42120, 0x108c588d, 0x132ae9e2, 0x15c01a3a, 0x184c2bd0, 0x1acf5e2e, 0x1d49ee4c, 0x1fbc16b9, 0x22260fb6, 0x24880f56, 0x26e2499d, 0x2934f098, 0x2b803474, 0x2dc4439b, 0x30014ac6, 0x32377512, 0x3466ec15, 0x368fd7ee, 0x38b25f5a, 0x3acea7c0, 0x3ce4d544, 0x3ef50ad2, 0x40ff6a2e, 0x43041403, 0x450327eb, 0x46fcc47a, 0x48f10751, 0x4ae00d1d, 0x4cc9f1ab, 0x4eaecfeb, 0x508ec1fa, 0x5269e12f, 0x5440461c, 0x5612089a, 0x57df3fd0, 0x59a80239, 0x5b6c65aa, 0x5d2c7f59, 0x5ee863e5, 0x60a02757, 0x6253dd2c, 0x64039858, 0x65af6b4b, 0x675767f5, 0x68fb9fce, 0x6a9c23d6, 0x6c39049b, 0x6dd2523d, 0x6f681c73, 0x70fa728c, 0x72896373, 0x7414fdb5, 0x759d4f81, 0x772266ad, 0x78a450b8, 0x7a231ace, 0x7b9ed1c7, 0x7d17822f, 0x7e8d3846, 0x80000000, 0x816fe50b }; #define RND_SHIFT (31 - FRAC_BITS_OUT) #define RND_CONST ((1 << RND_SHIFT) / 2) #define RND_ADJUST (0x10d) / established heuristically / / compute log2(x) in s15.16 format, where x is in s0.31 format maximum absolute error 8.18251e-6 @ 0x20352845 (0.251622232) / int32_t fixed_log2 (int32_t x) { int32_t f1, f2, dx, a, b, approx, lz, i, idx; uint32_t t; / x = 2i (1 + f), 0 <= f < 1. Find i / lz = clz (x); i = INT_BITS_IN - lz; / normalize f / t = (uint32_t)x << (lz + 1); / index table of log2 values using LOG2_TBL_SIZE msbs of fraction / idx = t >> (32 - LOG2_TBL_SIZE); / difference between argument and smallest sampling point / dx = t - (idx << (32 - LOG2_TBL_SIZE)); / fit parabola through closest three sampling points; find coeffs a, b / f1 = (log2Tab[idx+1] - log2Tab[idx]); f2 = (log2Tab[idx+2] - log2Tab[idx]); a = f2 - (f1 << 1); b = (f1 << 1) - a; / find function value for argument by computing ((adx+b)dx) / approx = (int32_t)((((int64_t)a)dx) >> (32 - LOG2_TBL_SIZE)) + b; approx = (int32_t)((((int64_t)approx)dx) >> (32 - LOG2_TBL_SIZE + 1)); approx = log2Tab[idx] + approx; / round fractional part of result / approx = (((uint32_t)approx) + RND_CONST + RND_ADJUST) >> RND_SHIFT; / combine integer and fractional parts of result / return (i << FRAC_BITS_OUT) + approx; } / convert from s15.16 fixed point to double-precision floating point / double fixed_to_float_s15_16 (int32_t a) { return a / 65536.0; } / convert from s0.31 fixed point to double-precision floating point / double fixed_to_float_s0_31 (int32_t a) { return a / (65536.0 32768.0); } int main (void) { double a, res, ref, err, maxerr = 0.0; int32_t x, start, end; start = 0x00000001; end = 0x7fffffff; printf ("testing fixed_log2 with inputs in [%17.10e, %17.10e)\n", fixed_to_float_s0_31 (start), fixed_to_float_s0_31 (end)); for (x = start; x < end; x++) { a = fixed_to_float_s0_31 (x); ref = log2 (a); res = fixed_to_float_s15_16 (fixed_log2 (x)); err = fabs (res - ref); if (err > maxerr) { maxerr = err; } } printf ("max. err = %g\n", maxerr); return EXIT_SUCCESS; } ``` For completeness, I am showing the minimax polynomial approximation below. The coefficients for such approximations can be generated by several tools such as Maple, Mathematica, Sollya or with homebrew code using the Remez algorithm, which is what I used here. The code below shows the original floating-point coefficients, the dynamic scaling used to maximize accuracy in intermediate computation, and the heuristic adjustments applied to mitigate the impact of non-rounding fixed-point arithmetic. A typical approach for computation of log2(x) is to use x = 2i (1+f) and use approximation of log2(1+f) for (1+f) in [ˆš½, ˆš2], which means that we use a polynomial p(f) on the primary approximation interval [ˆš½-1, ˆš2-1]. The intermediate computation scales up operands as far as feasible for improved accuracy under the restriction that we want to use a 32-bit mulhi operation as its basic building block, as this is a native instruction on many 32-bit architectures, accessible either via inline machine code or as an intrinsic. As in the table-based code, there are right shifts of signed data which may be negative, and such right shifts must map to arithmetic right shifts, something that ISO-C doesn't guarantee but most C compilers do. I managed to get the maximum absolute error for this variant down to 1.11288e-5, so almost full s15.16 accuracy but slightly worse than for the table-based variant. I suspect I should have added one additional term to the polynomial. / on 32-bit architectures, there is often an instruction/intrinsic for this / int32_t mulhi (int32_t a, int32_t b) { return (int32_t)(((int64_t)a (int64_t)b) >> 32); } #define RND_SHIFT (25 - FRAC_BITS_OUT) #define RND_CONST ((1 << RND_SHIFT) / 2) #define RND_ADJUST (-2) / established heuristically / / compute log2(x) in s15.16 format, where x is in s0.31 format maximum absolute error 1.11288e-5 @ 0x5a82689f (0.707104757) / int32_t fixed_log2 (int32_t x) { int32_t lz, i, f, p, approx; uint32_t t; / x = 2i (1 + f), 0 <= f < 1. Find i / lz = clz (x); i = INT_BITS_IN - lz; / force (1+f) into range [sqrt(0.5), sqrt(2)] / t = (uint32_t)x << lz; if (t > (uint32_t)(1.414213562 (1U << 31))) { i++; t = t >> 1; } / compute log2(1+f) for f in [-0.2929, 0.4142] / f = t - (1U << 31); p = + (int32_t)(-0.206191055 (1U << 31) - 1); p = mulhi (p, f) + (int32_t)( 0.318199910 (1U << 30) - 18); p = mulhi (p, f) + (int32_t)(-0.366491705 (1U << 29) + 22); p = mulhi (p, f) + (int32_t)( 0.479811855 (1U << 28) - 2); p = mulhi (p, f) + (int32_t)(-0.721206390 (1U << 27) + 37); p = mulhi (p, f) + (int32_t)( 0.442701618 (1U << 26) + 35); p = mulhi (p, f) + (f >> (31 - 25)); / round fractional part of the result / approx = (p + RND_CONST + RND_ADJUST) >> RND_SHIFT; / combine integer and fractional parts of result / return (i << FRAC_BITS_OUT) + approx; } Share Improve this answer edited Feb 17, 2019 at 7:53 answered Feb 13, 2019 at 6:25 njuffanjuffa 26.9k55 gold badges9292 silver badges150150 bronze badges 5 Comments gwiazdorrr gwiazdorrr Such an awesome answer! In case anyone's wondering what is going on with the parabola fitting, op uses Lagrange Interpolation Formula, but with a neat trick of translating the function to (0,0) and treating (0,0) as the first point. There's also a sneaky division by two (in the second approx statement), in case your numbers don't check out. njuffa njuffa @gwiazdorr I have been carrying forward this approach since 1987 or so, and I am pretty sure I had no idea about Lagrange interpolation at the time. After placing the origin of the interpolation coordinate system at the first point, and defining the x-axis distance between the equidistant nodes as unity, by plugging the other two points into ax²+bx we get a system of two simultaneous equations: a+b = f1 and 4a+2b = f2. This gives us 2a = f2 - 2f1 and 2b = 4f1 - f2 = 2f1 - 2a. RARE Kpop Manifesto RARE Kpop Manifesto @njuffa : aren't expressions like these (32 - LOG2_TBL_SIZE), (int32_t)( 0.318199910 (1U << 30) - 18), (uint32_t)(1.414213562 (1U << 31)) etc be of constant value ? why are they being re-calculated every time ? njuffa njuffa @RAREKpopManifesto The compiler will take care in converting them into magic constants when optimizing, while the source code shows how these magic constants were actually constructed (i.e. it improves self-documentation). njuffa njuffa @RAREKpopManifesto Here is a worked example at Compiler Explorer, with fixed_log2() compiled for a 32-bit ARM platform: godbolt.org/z/8aMEqqvK8 Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions logarithm polynomials fixed-point See similar questions with these tags. 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https://www.ncbi.nlm.nih.gov/books/NBK556138/
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Al Aaraj1; Cecylia Kelley2. Affiliations 1 Barking Havering & Redbridge University 2 Inspira Health Network Last Update: October 29, 2023. Continuing Education Activity Necrotizing (malignant) otitis externa (NOE) is a severe infection that affects the external auditory canal, skull base, and temporal bone. This condition can result in significant complications if not promptly diagnosed and treated. This activity provides an overview of the etiology, clinical presentation, potential complications, and treatment options for NOE. Furthermore, this activity also highlights the crucial role of the interprofessional healthcare team in treating patients with NOE. Objectives: Identify the clinical features and risk factors associated with necrotizing otitis externa. Implement appropriate diagnostic measures, including imaging studies and microbiological tests, to confirm the diagnosis of necrotizing otitis externa. Apply evidence-based treatment strategies, which may include antimicrobial therapy, surgical interventions, and adjunctive measures, to manage necrotizing otitis externa. Collaborate with an interprofessional healthcare team, including otolaryngologists, radiologists, and infectious disease specialists, for comprehensive care. Access free multiple choice questions on this topic. Introduction Necrotizing (malignant) otitis externa (NOE) is not cancerous, but it can rapidly spread in a patient's body and has been historically associated with a high mortality rate, hence its name. Toulmouche reported the initial case of malignant otitis external (MOE) in 1838, and MOE was subsequently introduced by Chandler in 1968 due to the high mortality rate associated with the infection during that period. In recent times, MOE has come to be known as necrotizing otitis externa (NOE), a term that more accurately and distinctly characterizes the aggressive and pathological nature of the condition. NOE is a severe and potentially life-threatening infection that originates in the external auditory canal (EAC). The most common cause of NOE is Pseudomonas aeruginosa, a gram-negative bacterium. Typically, these infections predominantly affect older patients, many of whom have diabetes mellitus. NOE primarily targets the EAC, skull base, and temporal bone, potentially involving the stylomastoid and jugular foramina. Infection and inflammation originating from the EAC can propagate through various anatomical pathways, reaching the mastoid process in the posterior direction, the temporomandibular joint, the parotid gland, and cervicofacial spaces anteriorly, or the skull base medially. This infection essentially advances from a basic otitis externa, progressing to cellulitis, then chondritis, and ultimately extending to the temporal bone, resulting in periostitis and ultimately culminating in osteomyelitis. NOE can result in serious complications, including the development of cranial neuropathies, brain abscesses, meningitis, and dural venous sinus thromboses. Therefore, healthcare practitioners should maintain a heightened suspicion level for NOE to identify and initiate treatment promptly. Etiology The predominant causative organism of NOE is the gram-negative bacterium P aeruginosa,which is ubiquitous in water and is particularly prevalent in patients with diabetes mellitus. Other gram-negative bacteria include Proteus spp and Klebsiella spp. Gram-positive bacteria that have been isolated include Staphylococcus aureus, with an increasing presence of methicillin-resistant S aureus (MRSA) strains, and S epidermidis. Fungi, specifically Aspergillus spp and Candida spp, have also been identified as potential culprits for NOE. Most individuals who develop NOE are older adults, predominantly males, who almost invariably have an underlying medical condition that makes them susceptible to immunosuppression. The primary risk factor for immunosuppression in these patients often involves some form of glucose dysregulation or established diabetes mellitus. Other predisposing risk factors include HIV, malignancies, and chemotherapy. Diabetes mellitus contributes to NOE by inducing small-vessel vasculopathy and immune dysfunction, particularly in cases of poorly controlled glucose levels. Furthermore, the cerumen found in the external ear canal of patients with diabetes mellitus often exhibits an elevated pH and reduced lysozyme concentrations compared to normal, rendering the EAC more susceptible to infection. Although there may not be a difference in the prevalence of NOE between patients with type 1 and type 2 diabetes, individuals with type 1 diabetes typically experience more adverse outcomes. Patients with immunosuppression due to HIV or other non-diabetic factors are more likely to develop NOE at a younger age than those with diabetes. Furthermore, individuals with HIV are less prone to Pseudomonas infections but may have a higher risk of fungal infections. In addition, they may not exhibit granulation tissue in the EAC upon physical examination. Generally, these patients experience more unfavorable outcomes than those with diabetes. Epidemiology NOE is a relatively rare condition, and its incidence rates are likely underestimated, failing to represent the disease burden accurately. Nevertheless, incidence rates can vary in different local or national publications, with a recent systematic review indicating a range between 0.221 and 1.19 cases per 100,000 patients. These statistics are believed to have increased more recently due to the aging population, a higher prevalence of diabetes mellitus, and increased NOE diagnoses. Although NOE has been documented across all age groups, it is most prevalent among individuals older than 60, often coinciding with some form of immunosuppression, which places them at a heightened risk for this condition. The most frequent cause of immunosuppression observed in patients with NOE is diabetes mellitus, followed by other factors such as malignancy, chemotherapy, and HIV infection. Males exhibit a higher likelihood of being affected by NOE than females. However, due to the limited sample sizes reported in studies and substantial variability, it is challenging to determine the precise incidence. Although the diagnosis of NOE is infrequent in pediatric patients, it can still manifest in young individuals who are neither diabetic nor immunocompromised. Pathophysiology NOE typically initiates as a simple otitis externa (see Image. Otitis Externa), which is an infection of the soft tissue in the auricle and EAC. From this point, however, NOE subsequently extends through fascial planes and venous sinuses, in contrast to otitis media, which typically follows the pneumatized cavities of the temporal bone. Infectious spread results in bony erosion and the invasion of nearby tissues, potentially leading to the involvement of the skull base, cranial nerves, and intracranial structures as the condition progresses. The infection will commonly spread through the osteocartilaginous junction of the EAC and the fissures of Santorini, which are openings within the cartilage on the lateral aspect of the EAC. Subsequently, the stylomastoid and jugular foramina, as well as hypoglossal canals, become vulnerable. Infections occurring in these regions can result in cranial neuropathies, presenting with clinical symptoms such as facial weakness, dysphagia, hoarseness, shoulder weakness, and tongue weakness. If the infection progresses medially, it can involve the cavernous sinus, leading to palsy of the trigeminal and abducens nerves. This development indicates a poor prognosis due to the extensive disease involvement required to affect these nerves. The infection spread into the infratemporal fossa results in the invasion of retrocondylar and parapharyngeal fats, the temporomandibular joint, and the muscles responsible for mastication. The patterns of the NOE spread can be summarized as mentioned below. Anterior: NOE extends anteriorly toward the muscles of mastication, condylar bone marrow, parotid gland, stylomastoid foramen and facial nerve, temporal fossa, and temporomandibular joint. Medial: In the medial direction, NOE advances toward the parapharyngeal fat, nasopharyngeal muscle, glossopharyngeal, vagal, and spinal accessory nerves, sphenoid bone, clivus, jugular foramen, and petrous apex. Intracranial: Intracranial spread is suggested when there is dural enhancement within the intracranial compartment, which may indicate the involvement of the sigmoid sinus, jugular vein, internal carotid artery, jugular fossa, cavernous sinus, or the dura. Posterior: In the posterior direction, NOE extends to the mastoid process. Histopathology The EAC is divided into cartilaginous and bony portions, with the cartilaginous portion comprising the lateral one-third of the canal. Within the cartilaginous portion, a layer of stratified squamous epithelium exists, covering the underlying connective tissue, perichondrium, and cartilage. In the osseous canal, a thick fibrous tissue is found deep within the squamous epithelium. In patients with NOE, a biopsy of the external auditory canal may reveal ulceration and epithelial loss, with bacteria and inflammation extending into the dense fibrous tissue. In areas where the epithelium remains intact, reactive changes can range from mild hyperplasia to notable pseudoepitheliomatous hyperplasia. Both acute and chronic inflammation, including abscess formation, are frequently observed. In biopsy samples taken from the cartilaginous canal, inflammation often extends into the apopilosebaceous units. Although reactive capillary proliferation may be evident within the granulation tissue, notably, chronic inflammation with granuloma formation is not typically associated with NOE. Microorganisms can be identified by tissue staining or culture; however, in cases where no organism is identified, polymerase chain reaction (PCR) testing may offer additional diagnostic information. Upon identification of organisms, it is crucial to conduct antimicrobial susceptibility testing to ascertain the effectiveness of specific antimicrobials. This information is critical in determining the appropriate treatment and customizing it based on the susceptibility results. This approach is particularly significant because bacterial resistance to antimicrobials commonly used to treat NOE, such as fluoroquinolones for addressing P aeruginosa, is rising. Distinguishing very well-differentiated squamous cell carcinoma from NOE can be challenging when pseudoepitheliomatous hyperplasia is present. Therefore, identifying NOE should not rule out the possibility of malignancy, as these conditions can coexist. History and Physical The prototypical patient with NOE commonly experiences unrelenting otalgia, especially at night, and otorrhea that does not improve with usual topical antimicrobial treatments for otitis externa. Cranial neuropathies can be observed in patients, and facial palsy has been reported in 21% of cases in a recent systematic review. During examination, physical findings typically reveal a painful, erythematous, and edematous auricle, and an EAC often exhibits intense tenderness in the region between the mastoid tip and the mandibular ramus. For patients with Pseudomonas NOE, the presence of granulation tissue at the bony-cartilaginous junction of the external auditory canal is considered pathognomonic. Fever occurrences are rare, and there are few other abnormalities in vital signs. The diagnosis of NOE can be established using the schema outlined by Cohen and Friedman, which comprises major (obligatory) and minor (occasional) criteria as mentioned below. Major (Obligatory) Criteria Pain, often disproportionate to physical examination Edema Exudate Granulation tissue observed in the EAC Microabscess, when surgical intervention is performed A positive technetium-99 methylene diphosphate (Tc-99m) bone scan Lack of improvement with local treatment for more than 1 week Minor (Occasional) Criteria Diabetes mellitus Cranial nerve involvement Positive radiograph Debilitating condition Older age For a diagnosis of NOE, all major criteria must be met, as the presence of only minor criteria is insufficient for the diagnosis. Healthcare providers should perform a cranial nerve examination, as cranial neuropathies are frequently encountered in NOE cases. The examiner should pay particular attention to assessing the function of the facial, glossopharyngeal, vagus, spinal accessory, and hypoglossal nerves. The facial nerve is frequently affected among cranial nerves, primarily due to its proximity to the posterior aspect of the bony EAC. More rarely, the abducens and trigeminal nerves may also be affected, which often signifies a poor prognosis. Furthermore, a mental status examination should be conducted, and if abnormalities are detected, it may indicate intracranial involvement. Evaluation Laboratory Studies Blood: Blood tests may reveal a normal or slightly elevated white blood cell count, but a left shift is not a common finding in NOE. Erythrocyte sedimentation rate (ESR) and C-reactive protein (CRP): Inflammatory markers are typically elevated in these patients and can be utilized to monitor the response to antimicrobial therapy. Mean ESR and CRP values in patients with NOE are 65.5 mm/h and 25.3 mg/dL, respectively. After confirming the diagnosis, monitoring ESR and CRP levels regularly is essential until they have normalized. Typically, ESR starts to decline within 2 weeks of initiating treatment. Blood glucose: Patients with diabetes should undergo blood glucose testing to assess their baseline, as NOE can affect the baseline glucose intolerance. Therefore, it is recommended to maintain strict glucose control during the treatment of NOE in diabetic patients. Furthermore, individuals without a known history of diabetes should undergo diabetes evaluation after the diagnosis of NOE. Culture and sensitivities from the EAC:Otorrhea should ideally be cultured before initiating antimicrobial treatment. Tissue sampling has shown significant value in selecting the appropriate antimicrobial treatment, particularly in cases of recurrent otitis externa. Obtaining a biopsy of the EAC is essential for this purpose and to rule out other potential causes, such as malignancy or cholesteatoma. Imaging Techniques Computed tomography (CT): CT scanning is advantageous for detecting bony erosions and demineralization, especially in the EAC. CT findings often include the obliteration of fat planes in the subtemporal area and the destruction of the bony cortex of the mastoid. Magnetic resonance imaging (MRI): MRI surpasses CT in precisely determining the anatomical location and extent of invasion into surrounding soft tissues, especially concerning cranial nerves. Furthermore, MRI more effectively assesses intracranial complications, such as thrombosis and intracranial spread. Monitoring treatment progress and determining prognosis can pose challenges with CT and MRI scans because they may not always effectively distinguish between active inflammation and resolving infection. Technetium-99 methylene diphosphate (Tc-99m) bone scan: Tc-99m scan is beneficial for the initial evaluation and diagnosis of osteomyelitis. However, this scanning technique is unsuitable for monitoring the progress of resolution because bone remodeling persists for several months after clinical recovery, and thus, the scan will continue yielding positive results. As Tc-99m scanning lacks anatomical precision, combining this radionuclide with single-photon emission CT (SPECT) along with CT or MRI delivers more precise and informative imaging. Gallium-67 (Ga-67) citrate leukocyte scan: Ga-67 is a valuable method for monitoring the resolution of NOE because it labels white blood cells, providing a more detailed tracking of inflammatory processes than Tc-99m. Visual confirmation of reduced Ga-67 uptake typically signifies resolving infection. However, a lesion-to-non-lesion ratio of approximately 1.0 may indicate the resolution of NOE, even if there appears to be increased Ga-67 uptake. Similar to Tc-99m, using Ga-67 for SPECT scanning may improve the accuracy of assessing the progress of NOE resolution. Indium-111 leukocyte scan (In-III): In-111 is an alternative to Ga-67 scanning, and it uses labeled white blood cells to monitor inflammation and, consequently, the progress of disease resolution. Similar to Ga-67, In-111 may yield false-positive results if inflammation persists after the resolution of the infection. Enhanced diagnostic accuracy can be achieved by combining In-111 and Tc-99m in a single scan. Positron emission tomography/CT (PET/CT):Scanning with 18-fluorodeoxyglucose combined with CT allows for detecting metabolically active tissues, making it an ideal test for identifying and monitoring malignancies or localized infections. Due to its widespread use in oncological surveillance, PET/CT is readily available at many hospitals, and most radiologists are experienced in interpreting these studies. Although no extensive studies have directly compared PET/CT to Tc-99m or Ga-67 scintigraphy for diagnosing or monitoring NOE, its inherent advantages may allow it to supplant these older nuclear medicine scanning techniques. Treatment / Management Treatment Overview Treatment for NOE is multimodal and can include systemic and local antimicrobial therapy, meticulous glucose control, aural hygiene, and hyperbaric oxygen therapy. Surgery is considered when non-surgical treatments have proven ineffective and typically involves procedures such as local debridement, abscess drainage, or removal of a bony sequestrum. Antimicrobial Therapy Topical antibiotic drops, including ofloxacin, ciprofloxacin/dexamethasone, and polymyxin B/neomycin/hydrocortisone, are commonly used to treat uncomplicated otitis externa. However, they are typically inadequate for managing NOE and may not offer additional benefits when used alongside systemic antibiotics. Before initiating antimicrobial therapy, gathering microbiological and tissue samples for bacterial and fungal cultures is crucial to identify the specific organism causing the infection. Furthermore, requesting antimicrobial susceptibility testing is essential to inform and customize antimicrobial therapy as needed. Oral ciprofloxacin can be prescribed in outpatient settings for cases that lack complications, such as cranial nerve involvement, diabetes, or the necessity for admission to manage pain. However, in cases where complications arise and swallowing pills becomes challenging, a switch to intravenous ciprofloxacin may be necessary. Patients admitted to the hospital with more severe disease or requiring parenteral therapy may initiate treatment with intravenous ciprofloxacin. They can later switch to oral administration upon hospital discharge. Alternatively, the patients may receive intravenous antibiotics through a peripherally inserted central catheter (PICC) as outpatient care. Systemic antimicrobials are the recommended treatment approach for NOE. When selecting initial empirical therapy, it is advisable to use antimicrobials with antipseudomonal activity, as P aeruginosa is the predominant cause of NOE. Typically, fluoroquinolones are preferred due to their excellent tissue penetration and the convenience of oral administration, which is particularly beneficial in outpatient settings. The effectiveness of fluoroquinolones may be compromised by the increasing antimicrobial resistance of Pseudomonas spp to this class of antimicrobials, primarily driven by their widespread global use. Therefore, it is essential to consider the presence of resistant organisms when deciding on the empirical treatment for NOE. Other antipseudomonal antimicrobials may be considered in situations with known resistance to fluoroquinolones. When fluoroquinolone resistance is detected, empirical antimicrobial options with activity against Pseudomonas spp include ceftazidime, aztreonam, or ticarcillin/clavulanate, either alone or in combination with an aminoglycoside such as gentamicin or tobramycin. In such cases, empirical treatment should be initiated, and therapy should be tailored based on microbiological culture and susceptibility results to improve outcomes and prevent delays. AlthoughP aeruginosa is the most common causative organism in NOE, other pathogens have also been identified. Antibiotic selection should be based on the specific organism identified through culture. In cases where empirical therapy is necessary, a broader-spectrum antimicrobial regimen may be required to cover a range of bacteria, such as gram-negative and gram-positive, and fungal pathogens. For fungal infections, the primary treatment is usually amphotericin B. However, itraconazole and voriconazole are also effective alternatives with fewer adverse effects. The duration of treatment will depend on the response to therapy. Patients should be reevaluated every 4 to 6 weeks during treatment using a Ga-67 scan. Even after the infection has resolved, patients at high risk should undergo periodic reassessment for up to a year, as recurrence can occur. Systemic antibiotic treatment should be discontinued one week after a normal Ga-67 scan, provided that inflammatory markers have also normalized. Hyperbaric Oxygen Therapy Although hyperbaric oxygen has been widely used to treat NOE, many studies show no added benefit in using it as an adjuvant to medical or surgical therapy. Surgical Therapy Surgical management is reserved for patients whose medical therapy has failed to cure the disease. Surgical therapy includes local debridement, removal of bony sequestra, and abscess drainage. Facial nerve decompression is not indicated when patients have facial paralysis in NOE. Although surgery has not been shown to improve outcomes, it is generally utilized in patients with more severe diseases. Differential Diagnosis NOE can be misdiagnosed as several other conditions, including, but not limited to, the following conditions: Localized otitis externa Acute diffuse otitis externa Chronic otitis externa Carcinoma of the ear canal Aspergillus skull-base osteomyelitis Staging In 2007, Peleg et al published a stratification schema categorizing NOE patients into severe and non-severe cases. Severe cases are assigned 3 to 4 points, whereas non-severe cases receive scores ranging from 0 to 2. Severe cases are more likely to have prolonged or complicated clinical courses. Peleg Stratification In this schema, 1 point is provided for each of the following criteria: Type 1 diabetes Temporal bone involvement on CT scan Skull base involvement on CT scan Temporomandibular joint involvement on CT scan The Carney Clinicopathological Staging System This system provides greater detail and complexity than the Peleg schema and more closely resembles the various stages of malignancy, as mentioned below. Stage 1: Clinical evidence of NOE with inflammation of soft tissues extending beyond the external auditory meatus and a normal Tc-99m bone scan. Stage 2: Clinical evidence of NOE with inflammation of soft tissues extending beyond the external auditory meatus and a positive Tc-99m bone scan. Stage 3: Clinical evidence of NOE with inflammation of soft tissues extending beyond the external auditory meatus, a positive Tc-99m bone scan, and cranial nerve involvement. Stage 3a: Single cranial nerve palsy. Stage 3b: Multiple cranial nerve palsies. Stage 4: Meningitis, dural venous sinus thrombosis, empyema, and brain abscess. Prognosis Several factors can influence the duration and prognosis of NOE. In diabetic patients, it is crucial to consider the duration of the patient's diabetes and the level of diabetes control. Laboratory values, such as elevated ESR and CRP levels, along with significant imaging findings, can indicate a more unfavorable prognosis for NOE. Older patients are at a greater risk of developing complications and experiencing higher mortality rates than younger patients. Patients who exhibit any of the following factors have poorer prognoses of NOE than those who do not: Facial nerve involvement Additional cranial nerve involvement Non-cranial nerve neurological involvement Extensive granulation or edema in the EAC Bilateral symptoms Aspergillus species as the causative organism Recurrence Notably, the recurrence rate of NOE is exceptionally high, reaching from 15% to 20%. Recurrence should be suspected if there is a rise in the ESR. As NOE can recur up to 1 year after treatment, patients should be monitored regularly for up to 1 year before being considered cured. Mortality Due to contemporary diagnostic and treatment modalities, the mortality rate for NOE has decreased significantly from a historical range of 50% to 60% to the current range of 10% to 20%. Complications Complications of NOE occur due to the invasion of surrounding structures, primarily cranial nerves. Although the facial nerve is the most frequently affected, other nerves can also be involved, including the glossopharyngeal, vagus, spinal accessory, hypoglossal, trigeminal, and abducens nerves. Skull base osteomyelitis occurs when the infection extends beyond the temporal bone and spreads to the sphenoid bone, occipital bone, or clivus. Intracranial involvement may manifest with a range of symptoms—from mild confusion to more severe conditions, including meningitis, venous sinus thrombosis, or even death. Consultations NOE is typically managed by an otolaryngologist or head and neck surgeon, regardless of whether surgical intervention is needed. Infectious disease specialists should also be consulted to assist in selecting the appropriate antibiotic treatment, particularly due to the rising incidence of antimicrobial resistance in Pseudomonas spp and other bacteria, which can render empirical treatment ineffective. Internal medicine consultation may also be essential, especially for achieving tight glycemic control and managing comorbidities in older or less healthy patients. Deterrence and Patient Education Patients should be educated about the implications of NOE and how to address the risk factors that could predispose them to developing or prolonging this infection. Maintaining tight glycemic control is an effective measure for reducing the risk of developing NOE and lessening the severity of the disease. Patients should refrain from engaging in activities and being in environments that can harbor contact with pathogenic bacteria, such as Pseudomonas spp. For example, precautions should be taken to avoid the proliferation of bacteria in otitis externa, also known as swimmer's ear, especially in hot and humid environments. Enhancing Healthcare Team Outcomes NOE is a rapidly progressing infection that can result in numerous severe complications. Therefore, a team-based approach should be utilized when caring for these patients to manage the infection and enhance patient outcomes effectively. An interprofessional team of healthcare providers, including an otolaryngologist, radiologist, infectious disease specialist, microbiologist, primary care physician, and endocrinologist, should collaborate in caring for patients with NOE. This collaborative approach can enhance patient outcomes by effectively addressing both the infection and the patient's coexisting health conditions, enabling the correct and prompt management of the infection and mitigating the risk factors that make patients susceptible to this condition. Review Questions Access free multiple choice questions on this topic. Click here for a simplified version. Comment on this article. Figure Otitis Externa. This image shows an inflamed external auditory canal with serous fluid on the tragus, auricular lobule, and antitragus, and purulent material within the canal. Courtesy of S Bhimji, MD References 1. : Dabholkar JP, Sheth A. Malignant otitis externa. Indian J Otolaryngol Head Neck Surg. 2001 Jan;53(1):55-6. [PMC free article: PMC3450864] [PubMed: 23119754] 2. : Treviño González JL, Reyes Suárez LL, Hernández de León JE. Malignant otitis externa: An updated review. Am J Otolaryngol. 2021 Mar-Apr;42(2):102894. [PubMed: 33429178] 3. : Su N, Syed I, Garth R. Skull based osteomyelitis due to postsurgery malignant otitis externa presenting as stroke. BMJ Case Rep. 2011 May 24;2011 [PMC free article: PMC3105569] [PubMed: 22696765] 4. : Hasnaoui M, Ben Mabrouk A, Chelli J, Larbi Ammari F, Lahmar R, Toumi A, Mighri K. Necrotising otitis externa: A single centre experience. J Otol. 2021 Jan;16(1):22-26. [PMC free article: PMC7814081] [PubMed: 33505446] 5. : Takata J, Hopkins M, Alexander V, Bannister O, Dalton L, Harrison L, Groves E, Kanona H, Jones GL, Mohammed H, Andersson MI, Hodgson SH. Systematic review of the diagnosis and management of necrotising otitis externa: Highlighting the need for high-quality research. Clin Otolaryngol. 2023 May;48(3):381-394. [PubMed: 36759416] 6. : Magliocca KR, Vivas EX, Griffith CC. Idiopathic, Infectious and Reactive Lesions of the Ear and Temporal Bone. Head Neck Pathol. 2018 Sep;12(3):328-349. [PMC free article: PMC6081288] [PubMed: 30069844] 7. : Handzel O, Halperin D. Necrotizing (malignant) external otitis. Am Fam Physician. 2003 Jul 15;68(2):309-12. [PubMed: 12892351] 8. : Karaman E, Yilmaz M, Ibrahimov M, Haciyev Y, Enver O. Malignant otitis externa. J Craniofac Surg. 2012 Nov;23(6):1748-51. [PubMed: 23147298] 9. : Nadol JB. Histopathology of Pseudomonas osteomyelitis of the temporal bone starting as malignant external otitis. Am J Otolaryngol. 1980 Nov;1(5):359-71. [PubMed: 7457757] 10. : Chaudhary HA, Ibrahim WH, Yousaf Z, Abubeker IY, Kartha A. Fungal Malignant Otitis Externa Involves a Cascade of Complications Culminating in Pseudoaneurysm of Internal Maxillary Artery: A Case Report. Am J Case Rep. 2019 Apr 21;20:562-566. [PMC free article: PMC6489412] [PubMed: 31005959] 11. : Mion M, Bovo R, Marchese-Ragona R, Martini A. Outcome predictors of treatment effectiveness for fungal malignant external otitis: a systematic review. Acta Otorhinolaryngol Ital. 2015 Oct;35(5):307-13. [PMC free article: PMC4720925] [PubMed: 26824911] 12. : Tsilivigkos C, Avramidis K, Ferekidis E, Doupis J. Malignant External Otitis: What the Diabetes Specialist Should Know-A Narrative Review. Diabetes Ther. 2023 Apr;14(4):629-638. [PMC free article: PMC10064349] [PubMed: 36897495] 13. : McCarty Walsh E, Hanson MB. Fungal Infections of the External Auditory Canal and Emerging Pathogens. Otolaryngol Clin North Am. 2023 Oct;56(5):909-918. [PubMed: 37553272] 14. : Peleg U, Perez R, Raveh D, Berelowitz D, Cohen D. Stratification for malignant external otitis. Otolaryngol Head Neck Surg. 2007 Aug;137(2):301-5. [PubMed: 17666260] 15. : Martel J, Duclos JY, Darrouzet V, Guyot M, Bébéar JP. [Malignant or necrotizing otitis externa: experience in 22 cases]. Ann Otolaryngol Chir Cervicofac. 2000 Nov;117(5):291. [PubMed: 11084403] 16. : Walton J, Coulson C. Fungal malignant otitis externa with facial nerve palsy: tissue biopsy AIDS diagnosis. Case Rep Otolaryngol. 2014;2014:192318. [PMC free article: PMC3933303] [PubMed: 24649388] 17. : Chawdhary G, Liow N, Democratis J, Whiteside O. Necrotising (malignant) otitis externa in the UK: a growing problem. Review of five cases and analysis of national Hospital Episode Statistics trends. J Laryngol Otol. 2015 Jun;129(6):600-3. [PubMed: 25990028] 18. : Hutson KH, Watson GJ. Malignant otitis externa, an increasing burden in the twenty-first century: review of cases in a UK teaching hospital, with a proposed algorithm for diagnosis and management. J Laryngol Otol. 2019 May;133(5):356-362. [PubMed: 30975233] 19. : Rubin Grandis J, Branstetter BF, Yu VL. The changing face of malignant (necrotising) external otitis: clinical, radiological, and anatomic correlations. Lancet Infect Dis. 2004 Jan;4(1):34-9. [PubMed: 14720566] 20. : Mills R. Malignant otitis externa. Br Med J (Clin Res Ed). 1986 Feb 15;292(6518):429-30. [PMC free article: PMC1339416] [PubMed: 3081109] 21. : Chen JC, Yeh CF, Shiao AS, Tu TY. Temporal bone osteomyelitis: the relationship with malignant otitis externa, the diagnostic dilemma, and changing trends. ScientificWorldJournal. 2014;2014:591714. [PMC free article: PMC4052568] [PubMed: 24963511] 22. : Wiegand S, Berner R, Schneider A, Lundershausen E, Dietz A. Otitis Externa. Dtsch Arztebl Int. 2019 Mar 29;116(13):224-234. [PMC free article: PMC6522672] [PubMed: 31064650] 23. : Sreepada GS, Kwartler JA. Skull base osteomyelitis secondary to malignant otitis externa. Curr Opin Otolaryngol Head Neck Surg. 2003 Oct;11(5):316-23. [PubMed: 14502060] 24. : van Kroonenburgh AMJL, van der Meer WL, Bothof RJP, van Tilburg M, van Tongeren J, Postma AA. Advanced Imaging Techniques in Skull Base Osteomyelitis Due to Malignant Otitis Externa. Curr Radiol Rep. 2018;6(1):3. [PMC free article: PMC5778178] [PubMed: 29416952] 25. : Kwon BJ, Han MH, Oh SH, Song JJ, Chang KH. MRI findings and spreading patterns of necrotizing external otitis: is a poor outcome predictable? Clin Radiol. 2006 Jun;61(6):495-504. [PubMed: 16713420] 26. : Tamma PD, Aitken SL, Bonomo RA, Mathers AJ, van Duin D, Clancy CJ. Infectious Diseases Society of America 2023 Guidance on the Treatment of Antimicrobial Resistant Gram-Negative Infections. Clin Infect Dis. 2023 Jul 18; [PubMed: 37463564] 27. : Mattucci KF, Setzen M, Galantich P. Necrotizing otitis externa occurring concurrently with epidermoid carcinoma. Laryngoscope. 1986 Mar;96(3):264-6. [PubMed: 3951301] 28. : Cohen D, Friedman P. The diagnostic criteria of malignant external otitis. J Laryngol Otol. 1987 Mar;101(3):216-21. [PubMed: 3106547] 29. : Illing E, Zolotar M, Ross E, Olaleye O, Molony N. Malignant otitis externa with skull base osteomyelitis. J Surg Case Rep. 2011 May 01;2011(5):6. [PMC free article: PMC3649245] [PubMed: 24950586] 30. : Mani N, Sudhoff H, Rajagopal S, Moffat D, Axon PR. Cranial nerve involvement in malignant external otitis: implications for clinical outcome. Laryngoscope. 2007 May;117(5):907-10. [PubMed: 17473694] 31. : Manso MC, Rodeia SC, Rodrigues S, Cavilhas P, Domingos R. Malignant Otitis Externa and Stroke. Eur J Case Rep Intern Med. 2016;3(4):000387. [PMC free article: PMC6346865] [PubMed: 30755871] 32. : McLaren O, Potter C. Scedosporium apiospermum: a rare cause of malignant otitis externa. BMJ Case Rep. 2016 Sep 09;2016 [PMC free article: PMC5020719] [PubMed: 27613266] 33. : Leahy TW, Sader C. A rare case of bilateral malignant otitis externa and osteomyelitis with lower cranial nerve sequelae. BMJ Case Rep. 2011 May 12;2011 [PMC free article: PMC3094778] [PubMed: 22696730] 34. : Curtin HD, Wolfe P, May M. Malignant external otitis: CT evaluation. Radiology. 1982 Nov;145(2):383-8. [PubMed: 7134442] 35. : Stokkel MP, Takes RP, van Eck-Smit BL, Baatenburg de Jong RJ. The value of quantitative gallium-67 single-photon emission tomography in the clinical management of malignant external otitis. Eur J Nucl Med. 1997 Nov;24(11):1429-32. [PubMed: 9371879] 36. : Okpala NC, Siraj QH, Nilssen E, Pringle M. Radiological and radionuclide investigation of malignant otitis externa. J Laryngol Otol. 2005 Jan;119(1):71-5. [PubMed: 15807974] 37. : Weber PC, Seabold JE, Graham SM, Hoffmann HH, Simonson TM, Thompson BH. Evaluation of temporal and facial osteomyelitis by simultaneous In-WBC/Tc-99m-MDP bone SPECT scintigraphy and computed tomography scan. Otolaryngol Head Neck Surg. 1995 Jul;113(1):36-41. [PubMed: 7603719] 38. : Sturm JJ, Stern Shavit S, Lalwani AK. What is the Best Test for Diagnosis and Monitoring Treatment Response in Malignant Otitis Externa? Laryngoscope. 2020 Nov;130(11):2516-2517. [PubMed: 32159853] 39. : Phillips JS, Jones SE. Hyperbaric oxygen as an adjuvant treatment for malignant otitis externa. Cochrane Database Syst Rev. 2005 Apr 18;(2):CD004617. [PubMed: 15846724] 40. : Phillips JS, Jones SE. Hyperbaric oxygen as an adjuvant treatment for malignant otitis externa. Cochrane Database Syst Rev. 2013 May 31;2013(5):CD004617. [PMC free article: PMC7389256] [PubMed: 23728650] 41. : Byun YJ, Patel J, Nguyen SA, Lambert PR. Hyperbaric oxygen therapy in malignant otitis externa: A systematic review of the literature. World J Otorhinolaryngol Head Neck Surg. 2021 Oct;7(4):296-302. [PMC free article: PMC8486695] [PubMed: 34632343] 42. : Walsh TJ, Anaissie EJ, Denning DW, Herbrecht R, Kontoyiannis DP, Marr KA, Morrison VA, Segal BH, Steinbach WJ, Stevens DA, van Burik JA, Wingard JR, Patterson TF., Infectious Diseases Society of America. Treatment of aspergillosis: clinical practice guidelines of the Infectious Diseases Society of America. Clin Infect Dis. 2008 Feb 01;46(3):327-60. [PubMed: 18177225] 43. : Raines JM, Schindler RA. The surgical management of recalcitrant malignant external otitis. Laryngoscope. 1980 Mar;90(3):369-78. [PubMed: 7359959] 44. : Lee SK, Lee SA, Seon SW, Jung JH, Lee JD, Choi JY, Kim BG. Analysis of Prognostic Factors in Malignant External Otitis. Clin Exp Otorhinolaryngol. 2017 Sep;10(3):228-235. [PMC free article: PMC5545692] [PubMed: 27671716] 45. : Sylvester MJ, Sanghvi S, Patel VM, Eloy JA, Ying YM. Malignant otitis externa hospitalizations: Analysis of patient characteristics. Laryngoscope. 2017 Oct;127(10):2328-2336. [PubMed: 27882553] 46. : Eveleigh MO, Hall CE, Baldwin DL. Prognostic scoring in necrotising otitis externa. J Laryngol Otol. 2009 Oct;123(10):1097-102. [PubMed: 19586581] 47. : Bhandary S, Karki P, Sinha BK. Malignant otitis externa: a review. Pac Health Dialog. 2002 Mar;9(1):64-7. [PubMed: 12737420] 48. : Rubin J, Yu VL. Malignant external otitis: insights into pathogenesis, clinical manifestations, diagnosis, and therapy. Am J Med. 1988 Sep;85(3):391-8. [PubMed: 3046354] : Disclosure: Mahmoud Al Aaraj declares no relevant financial relationships with ineligible companies. : Disclosure: Cecylia Kelley declares no relevant financial relationships with ineligible companies. Copyright © 2025, StatPearls Publishing LLC. This book is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) ( ), which permits others to distribute the work, provided that the article is not altered or used commercially. You are not required to obtain permission to distribute this article, provided that you credit the author and journal. Bookshelf ID: NBK556138PMID: 32310598 PubReader Print View Cite this Page Al Aaraj MS, Kelley C. Necrotizing (Malignant) Otitis Externa. [Updated 2023 Oct 29]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2025 Jan-. In this Page Continuing Education Activity Introduction Etiology Epidemiology Pathophysiology Histopathology History and Physical Evaluation Treatment / Management Differential Diagnosis Staging Prognosis Complications Consultations Deterrence and Patient Education Enhancing Healthcare Team Outcomes Review Questions References Related information PMC PubMed Central citations PubMed Links to PubMed Recent Activity Clear)Turn Off)Turn On) Necrotizing (Malignant) Otitis Externa - StatPearls Necrotizing (Malignant) Otitis Externa - StatPearls Your browsing activity is empty. Activity recording is turned off. Turn recording back on) See more... Follow NCBI Connect with NLM National Library of Medicine8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers
6040
https://www.education.com/worksheet/article/multiplying-using-partial-products-4-digit-by-1-digit/
Multiplying Using Partial Products: 4-Digit by 1-Digit | Worksheet | Education.com SKIP TO CONTENT Worksheet Generator Subjects Grades Worksheets Games Build a Worksheet More Resources Roly Recommends  Log InSign Up Subjects Grades Worksheets Games Build a Worksheet More Resources Roly Recommends WorksheetsMath Worksheets4th Grade Math Worksheets4th Grade Multiplication Worksheets 2  4th Grade Multiplication Strategies Worksheets4th Grade Multiply Using Partial Products Worksheets Worksheet Multiplying Using Partial Products: 4-Digit by 1-Digit Practice solving multiplication problems using the partial product method! This worksheet invites students to solve four-digit by single-digit multiplication problems by decomposing a four-digit number into its thousands, hundreds, tens, and ones places. Then, students will multiply each place by the single-digit number and add all the partial products to get the final product. These six problems are great practice for fourth graders working on their multi-digit multiplication skills. The partial product strategy allows students to break down big multiplication problems into smaller multiplication expressions they can tackle individually. Don’t forget to check out Multiplying Using Partial Products: 2-Digit by 1-Digit and Multiplying Using Partial Products: 3-Digit by 1-Digit for more essential practice. Or, if you’re looking for more of a challenge, have students try Multiplying Using Partial Products: 2-Digit by 2-Digit. Download Free Worksheet  See in a set (5)  View answer key  Add to collection  Add to assignment Save Grade: Fourth Grade Subjects: Math Multiplication Multiplication Strategies Multiply Using Partial Products View aligned standards 4.NBT.B.5 Related learning resources Multiplying Using Partial Products: 2-Digit by 1-Digit Worksheet Multiplying Using Partial Products: 2-Digit by 1-Digit Try using the partial products method to solve multiplication problems with this helpful worksheet! Fourth Grade Worksheet Multiplying Using Partial Products: 3-Digit by 1-Digit Worksheet Multiplying Using Partial Products: 3-Digit by 1-Digit Enhance your multi-digit multiplication skills with this scaffolded worksheet! Fourth Grade Worksheet Multiplying by 1-Digit Numbers Using Partial Products: Step-by-Step Worksheet Multiplying by 1-Digit Numbers Using Partial Products: Step-by-Step Learn to multiply with the partial product strategy in this helpful worksheet! Fourth Grade Worksheet Multiplying by 2-Digit Numbers Using Partial Products: Missing Numbers Interactive Worksheet Multiplying by 2-Digit Numbers Using Partial Products: Missing Numbers Practice partial products with this multiplication worksheet geared towards fourth and fifth graders! Fourth Grade Interactive Worksheet Multiplying 2-Digit Numbers Using Partial Products: Step-by-Step Worksheet Multiplying 2-Digit Numbers Using Partial Products: Step-by-Step Learn to multiply multi-digit numbers by two-digit numbers with this guided worksheet! Fourth Grade Worksheet Multiplying Using Partial Products: 2-Digit by 2-Digit Worksheet Multiplying Using Partial Products: 2-Digit by 2-Digit Step up your multi-digit multiplication skills with this useful worksheet! Fourth Grade Worksheet Multiplying by 1-Digit Numbers Using Partial Products: Missing Numbers Interactive Worksheet Multiplying by 1-Digit Numbers Using Partial Products: Missing Numbers Help fourth and fifth graders boost their multiplication know-how with this partial product worksheet! Fourth Grade Interactive Worksheet Partial Products Method # 1 Worksheet Partial Products Method # 1 Students learn how to use the partial product method, then use it to solve nine multi-digit multiplication problems. Fourth Grade Worksheet Partial Products Method #2 Worksheet Partial Products Method #2 Students learn how to use the partial product method, then use it to solve nine multi-digit multiplication problems. Fourth Grade Worksheet 3-Digit by 1-Digit Multiplication Interactive Worksheet 3-Digit by 1-Digit Multiplication Give your students some practice multiplying 3-digit numbers with this straightforward multiplication worksheet. Fourth Grade Interactive Worksheet Glossary: The Partial Products Strategy for Multiplication Worksheet Glossary: The Partial Products Strategy for Multiplication Use this glossary with the EL Support Lesson: The Partial Products Strategy for Multiplication. Fourth Grade Worksheet Two-Digit Multiplication Quiz #1 Interactive Worksheet Two-Digit Multiplication Quiz #1 Practice multiplying two-digit numbers with this quiz-style math worksheet! Fourth Grade Interactive Worksheet    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. 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6041
https://www.youtube.com/watch?v=rfV4t30uVKw
Balancing chemical equations: Aluminum oxide Doug Haskins 2690 subscribers 114 likes Description 24047 views Posted: 3 Apr 2013 Review balancing equations by watching an example involving aluminum oxide. 13 comments Transcript: [Music] This episode of the podcast is going to show you how to do a sample problem when balancing an equation. If you have not done so, you should probably watch the introductory video on how to balance equations if you have not done so. In the equation that's on the screen, we can analyze it for the different chemicals that do make up that particular reaction. First, we see we have aluminum That's aluminum metal and that's being added to atmospheric oxygen which is O2 diatomic oxygen and that then forms the compound that you can see here is aluminum oxide as we did in class. It's very important before trying to solve a chemical equation and make sure it's balanced to fit the law conservation of mass. We should draw our graphic organizer. We underline our equation and draw the tail of the tea at the arrow and list all of the elements involved in our chemical reaction for both products and reactants. And then determine how many of each element is represented in atoms. There's one aluminum and two oxygen in our reactants and two aluminum and three oxygen and aluminum oxide. The next step is to plan how to use coefficients to balance this equation. As we see, we have an even number of oxygens in our reactants and an odd number of oxygen in our products. The easiest way to solve this is to use a number that is in common to both those numbers. To make oxygen and the products even, I need to multiply that by an even number. The lowest number I can think of is two, which would make oxygen six. It also changes aluminum, right? Because 2 aluminum oxide gives me 2 2 aluminum, which equals four total aluminum atoms. That means I need to change the number of aluminum in my reaction for the reactants. I need four aluminum atoms. Changing my total count on the reactants to four. That now balances. However, I still need to balance my oxygen. I have two. But if I multiply by three and using a coefficient, I now have six. This is a balanced equation. Four aluminum plus three oxygen molecules gets me two aluminum oxide compound. This concludes this episode of the podcast.
6042
https://www.savemyexams.com/gcse/physics/edexcel/18/revision-notes/4-waves/4-1-properties-of-waves/4-1-8-refraction/
No subjects found Refraction (Edexcel GCSE Physics): Revision Note Exam code: 1PH0 Author Last updated Refraction Refraction can occur when a wave crosses a boundary between two materials with different densities In some cases, the wave will change direction The ray diagram below illustrates the change of direction of a light ray at a water-air boundary: Waves can change direction when moving between materials with different densities Refraction of light Refraction also occurs when light passes a boundary between two different transparent media At the boundary, the rays of light undergo a change in direction The direction is taken as the angle from the normal The change in direction depends on the difference in density between the two media: From less dense to more dense (e.g air to glass), light bends towards the normal From more dense to less dense (e.g. glass to air), light bends away from the normal When passing along the normal (perpendicular) the light does not bend at all Refraction of Light Through a Glass Block Light enters the glass where the light ray bends towards the normal. Light bends away from the normal as it exits the glass The change in direction occurs due to the change in speed when travelling in different substances When light passes into a denser substance the rays will slow down, hence they bend towards the normal As with refraction of water waves, the only properties that change during refraction of light are speed and wavelength – the frequency of waves does not change Different frequencies account for different colours of light (red has a low frequency, whilst blue has a high frequency) When light refracts, it does not change colour (think of a pencil in a glass of water), therefore, the frequency does not change Unlock more, it's free! Join the 100,000+ Students that ❤️ Save My Exams the (exam) results speak for themselves: Did this page help you? 1. Key Concepts of Physics 1 Topic · 2 Revision Notes Expressing Quantities & SI Units Units & Prefixes Conversions & Standard Form 2. Motion & Forces 4 Topics · 23 Revision Notes Describing Motion Scalars & Vectors Speed Velocity Distance-Time Graphs Acceleration Calculating Uniform Acceleration Velocity-Time Graphs Area under a Velocity-Time Graph Measuring Speed Forces Newton's First Law Newton's Second Law Core Practical: Investigating Force & Acceleration Newton's Third Law Weight, Mass & Gravity Circular Motion Momentum Momentum Conservation of Momentum Force & Momentum Mass & Inertia Stopping Distances Stopping Distances Measuring Reaction Time Factors Affecting Stopping Distance Calculating Stopping Distances 3. Conservation of Energy 2 Topics · 9 Revision Notes Conservation of Energy Gravitational Potential Energy Kinetic Energy Conservation of Energy Examples of Energy Transfers Efficiency & Energy Resources Wasted Energy Conduction of Heat Efficiency Improving Efficiency Energy Resources 4. Waves 2 Topics · 14 Revision Notes Properties of Waves Introduction to Waves Describing Wave Motion Transverse & Longitudinal Waves The Wave Equation Measuring Wave Speed Calculating Depth & Distance Wave Interactions Refraction Refraction & Speed Wave Interactions & Wavelength Core Practical: Investigating Wave Properties Sound Sound Waves Ultrasound & Infrasound Transmission of Sound 5. Light & the Electromagnetic Spectrum 3 Topics · 15 Revision Notes Optics Reflection & Refraction Total Internal Reflection Light & Surfaces Lenses Real & Virtual Images Electromagnetic Waves Properties of Electromagnetic Waves The Electromagnetic Spectrum EM Waves & Matter Dangers of High-Energy EM Waves Applications of EM Waves EM Waves & Atoms Radio Waves Thermal Radiation Thermal Radiation Radiation & Temperature Change Core Practical: Investigating Thermal Radiation 6. Radioactivity 4 Topics · 29 Revision Notes Atomic Structure Inside the Atom Protons, Neutrons & Electrons Atomic & Mass Number Isotopes Electron Structure Positive Ions Developing Models of the Atom Radioactive Decay Types of Radiation Properties of Radiation Background Radiation Detecting Radiation Beta Decay Decay & Transformations Nuclear Equations The Random Nature of Decay Activity & Decay Half-Life Uses & Dangers of Radiation Dangers of Radiation Contamination & Irradiation Uses of Radiation Half-Life & Risk Medical Uses of Radiation Nuclear Fission & Fusion Nuclear Energy Nuclear Fission Nuclear Reactors Energy & Waste Advantages & Disadvantages of Nuclear Power Nuclear Fusion The Conditions for Fusion 7. Astronomy 2 Topics · 12 Revision Notes The Solar System & Lifecycles of Stars Gravity & Weight The Solar System Circular Orbits Non-Circular Orbits Equilibrium in Stars The Life Cycle of Solar Mass Stars The Life Cycle of Larger Stars Cosmology Theories of the Universe The Doppler Effect Galactic Red-shift The Cosmic Microwave Background Observing the Universe 8. Energy – Forces Doing Work 1 Topic · 7 Revision Notes Work, Power & Efficiency Energy Stores & Transfers Changes in Energy Work & Energy GPE & KE Dissipation of Energy Power Efficiency & Power 9. Forces & their Effects 2 Topics · 8 Revision Notes Types of Forces Contact & Non-Contact Forces Force as a Vector Scale Drawings Free Body Diagrams Balanced Forces Moments Moments The Principle of Moments Levers & Gears 10. Electricity & Circuits 4 Topics · 21 Revision Notes Current, Potential Difference & Resistance Atomic Structure Circuit Diagrams Potential Difference Current Current in Circuits Resistance Components in Series & Parallel Circuits Resistors in Series & Parallel Comparing Series & Parallel Circuits Electrical Components Testing Components Core Practical: Investigating & Testing Circuits Energy Transfers in Circuits Heating in Circuits Reducing Heating in Circuits Uses & Dangers of Electric Heating Calculating Electric Energy Energy & Power Electricity & Power Energy Transfers in Appliances Mains Electricity AC & DC Mains Electricity Dangers of Mains Electricity 11. Static Electricity 1 Topic · 5 Revision Notes Static Electricity Electric Charge Examples of Static Electricity Uses & Dangers of Static Electricity Electric Fields Fields & Static 12. Magnetism & the Motor Effect 2 Topics · 10 Revision Notes Magnetism Magnetism Permanent & Induced Magnets Magnetic Fields The Earth's Magnetic Field Electromagnetism & The Motor Effect Magnetic Fields in Wires Magnetic Fields in Solenoids The Motor Effect Fleming's Left Hand Rule Calculating Magnetic Force Electric Motors 13. Electromagnetic Induction 1 Topic · 9 Revision Notes Electromagnetic Induction Generating Electricity Factors Affecting EM Induction Applications of the Generator Effect Loudspeakers & Headphones Transformers The Transformer Equation The National Grid The Ideal Transformer Equation High Voltage Transmission 14. Particle Model 2 Topics · 15 Revision Notes States of Matter & Thermal Capacity Density Solids, Liquids & Gases Core Practical: Determining Density Changes of State Thermal Energy Specific Heat & Latent Heat Specific Heat Capacity Specific Latent Heat Thermal Insulation Core Practical: Investigating Specific Heat Capacity Particle Model & Pressure Kinetic Theory Absolute Zero Pressure & Volume Boyle's Law Doing Work on a Gas 15. Forces & Matter 2 Topics · 9 Revision Notes Forces & Elasticity Changing Shape Hooke's Law Elastic Potential Energy Core Practical: Investigating Force & Extension Pressure & Pressure Differences in Fluids Pressure in Fluids Pressure & Forces Pressure in a Liquid Calculating Pressure in a Liquid Upthrust Author: Ashika Expertise: Physics Content Creator Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources. © Copyright 2015-2025 Save My Exams Ltd. All Rights Reserved. IBO was not involved in the production of, and does not endorse, the resources created by Save My Exams.
6043
https://www.khanacademy.org/economics-finance-domain/ap-macroeconomics/basic-economics-concepts-macro/market-equilibrium-disequilibrium-and-changes-in-equilibrium/v/changes-in-equilibrium-price-and-quantity-when-supply-and-demand-change-khan-academy
Changes in equilibrium price and quantity when supply and demand change (video) | Khan Academy Skip to main content If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. 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We're a nonprofit that relies on support from people like you. If everyone reading this gives $10 monthly, Khan Academy can continue to thrive for years. Please help keep Khan Academy free, for anyone, anywhere forever. Select gift frequency One time Recurring Monthly Yearly Select amount $10 $20 $30 $40 Other Give now By donating, you agree to our terms of service and privacy policy. Skip to lesson content AP®︎/College Macroeconomics Course: AP®︎/College Macroeconomics>Unit 1 Lesson 6: Market equilibrium, disequilibrium, and changes in equilibrium Market equilibrium Changes in market equilibrium Changes in equilibrium price and quantity when supply and demand change Lesson summary: Market equilibrium, disequilibrium, and changes in equilibrium Market equilibrium and disequilibrium Changes in equilibrium Economics> AP®︎/College Macroeconomics> Basic economics concepts> Market equilibrium, disequilibrium, and changes in equilibrium © 2025 Khan Academy Terms of usePrivacy PolicyCookie NoticeAccessibility Statement Changes in equilibrium price and quantity when supply and demand change AP.MACRO: MKT‑2 (EU), MKT‑2.F (LO), MKT‑2.F.1 (EK), MKT‑2.G (LO), MKT‑2.G.1 (EK) Google Classroom Microsoft Teams About About this video Transcript Previously we looked at what happens to the equilibrium price and quantity in a market if supply or demand change. In this video, we explore what happens when BOTH supply and demand are changing at the same time. Skip to end of discussions Questions Tips & Thanks Want to join the conversation? Log in Sort by: Top Voted night 6 years ago Posted 6 years ago. Direct link to night's post “Someone at Khan please al...” more Someone at Khan please also add the other half of the video as a lesson. Right now it the lesson is titled: Changes in equilibrium price and quantity when supply and demand change --- for the video --- Equilibrium price and quantity from changes in both supply and demand and the second video (which is called as the title) is missing. Ty, much love :) Answer Button navigates to signup page •Comment Button navigates to signup page (15 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Allie A Lee 6 years ago Posted 6 years ago. Direct link to Allie A Lee's post “here it is, not a lesson ...” more here it is, not a lesson on Khan Academy but it's the video: Comment Button navigates to signup page (28 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Cassi a year ago Posted a year ago. Direct link to Cassi's post “Can the other half of the...” more Can the other half of the video get posted? Youtube links don't work anymore Answer Button navigates to signup page •1 comment Comment on Cassi's post “Can the other half of the...” (2 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Blake 5 years ago Posted 5 years ago. Direct link to Blake's post “Why are there cases (like...” more Why are there cases (like in the third graph) where the curve is shifting both vertically and horizontally? Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Willem Johannes 3 months ago Posted 3 months ago. Direct link to Willem Johannes's post “Delayed response but repl...” more Delayed response but replying for my own learning and in case others have the same question. Really all curve shifts are both horizontal and vertical- it's just that Sal doesn't always mention the vertical shift. If you imagine a supply curve shifting left, it is also always shifting "up" (which is confusing, because that actually means supply is going down); if it shifts right, the curve necessarily shifts down. The inverse is true for demand curves. I think it's probably most helpful to think only in terms of the horizontal shift and ignore the vertical shift. I.e., think about it in the terms Sal lays out: At any given price point, what is happening to the quantity? Is it going up (shifting right along the x-axis) or going down (shifting left along the x-axis)? Comment Button navigates to signup page (2 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more breanna necessary 7 months ago Posted 7 months ago. Direct link to breanna necessary's post “why is there so many poin...” more why is there so many points on one graph? I thought that all you had was two axis. and then that in turn was price and quantity. Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer 𝕳𝖔𝖕𝖕𝖊𝖗 6 months ago Posted 6 months ago. Direct link to 𝕳𝖔𝖕𝖕𝖊𝖗's post “The points represent Q1 a...” more The points represent Q1 and P1, the starting prices and quantities, and Q2 and P2, showing the prices and quantities after the events shown. Does this answer your question? Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Ginger 4 months ago Posted 4 months ago. Direct link to Ginger's post “for the first example he ...” more for the first example he says a major ice cream producer enters the market, but if they are already producing ice cream in a large scale way, wouldn't they already be in the market? Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Ginger 4 months ago Posted 4 months ago. Direct link to Ginger's post “NGL, when I saw all the g...” more NGL, when I saw all the graphs I freaked out. Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer rabbit 5 years ago Posted 5 years ago. Direct link to rabbit's post “What does a graph look li...” more What does a graph look like if it results in an indeterminate change in price caused by a change in supply and demand? Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Henry Silverberg 2 years ago Posted 2 years ago. Direct link to Henry Silverberg's post “How do the equilibrium pr...” more How do the equilibrium price and quantity in a market respond when both supply and demand are changing simultaneously? Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Saron 7 days ago Posted 7 days ago. Direct link to Saron's post “When we graph the supply ...” more When we graph the supply curve, can't we start from the origin? Answer Button navigates to signup page •Comment Button navigates to signup page (1 vote) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Francisco 2 years ago Posted 2 years ago. Direct link to Francisco's post “I got confused with all o...” more I got confused with all of the graphs, can someone explain to me in a better way please Answer Button navigates to signup page •Comment Button navigates to signup page (0 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Answer Show preview Show formatting options Post answer Cleve a year ago Posted a year ago. Direct link to Cleve's post “When something makes it h...” more When something makes it harder to produce a good, its supply tends to decrease. Now it's harder to produce it, so the price increases and demand decreases as people are less willing to buy it. The opposite happens when it is easy to produce something; price goes down, demand increases. Now, if demand changes first, let's say that due to preferences or population it increased. Then supplies need to increase, and the price goes up too. If we produce a lot and sell it at high price, the low demand might make we have surpluses right? on the other hand if we are not producing much, selling it cheap and demand is too high we would have shortages. That is why equilibrium is important, that is where the demand and supply meets. I am not sure it addressed your doubts. Feel free to ask any questions. Comment Button navigates to signup page (2 votes) Upvote Button navigates to signup page Downvote Button navigates to signup page Flag Button navigates to signup page more Video transcript [Instructor] What we're going to do in this video is think about all of the different ways that a supply curve or a demand curve can shift and that's why we actually have eight versions of the exact same diagram. Each of them is showing where we are right now, let's say in a given region in the ice cream market. It's important to title your graphs, especially if you were taking some type of a standardized exam like an AP exam and in the vertical axis we have P representing price, and then the horizontal axis, Q representing quantity, we have our upwards sloping supply curve. I'm calling this S1 just as kind of our starting point and then we have our downwards sloping demand curve, D1 and where they intersect, that gives us our equilibrium price, P1 and our equilibrium quantity, Q1 and once again, if you were taking some type of a standardized test, it's important that you label all of these things including P1 and Q1 and show this dotted line where it intersects the horizontal axis, this is Q1 and where it intersects the vertical axis, it is P1. Now with that out of the way, let's think about what happens to the equilibrium price and the equilibrium quantity given different shifts in the supply or the demand curve or both of them. So, in this first scenario, let's imagine that all of a sudden a major ice cream producer enters into the market, so here we're going to this first one, we're gonna think about a situation where the supply goes up. So, one way to think about it is at any given price, people are willing to supply more quantity, so here we would have our supply curve shift to the right, I'll call this S2 right over here, it's shifting to the right and down and so, given this, what happens to our equilibrium price and our equilibrium quantity? Well, you see it right over here. If I draw a dotted line, we see our equilibrium price P2 is lower and our equilibrium quantity Q2 is higher, once again, assuming that we have a downwards sloping demand curve like this which is what you would typically see and so, in this case, let me just write it here, we have our quantity, actually, let me write it this way, we have our price goes down and our quantity goes up. All right, now let's do this example and let's imagine the other way, let's imagine in this scenario our supply goes down. What is going to happen to this graph and in particular, what's going to happen to our equilibrium price and our equilibrium quantity? Well, in this situation for a given price people are willing to supply less, that's how I would like to think about it, so we would have a shift to the left and up and so, we could call this supply curve two right over here and that what is our equilibrium point? It's right over there and so, this would be our new price, it has gone up and this would be our new quantity, it has gone down, so price has gone up and quantity has gone down and once again, in either of these scenarios hopefully this feels a little bit like commonsense. If you have a supplier enter into the market, quantity might go up and there's more competition and so, a lot more suppliers and so, the price would go down. Here where the supply goes down, maybe some of the ice cream stores close down, well, now the quantity will go down, there's just less people supplying but the price goes up. For the ice cream that's there, the equilibrium price is going to be higher. Now let's do the same thing with the demand curve. Let's think about a situation where first let's think about a scenario where demand goes up. What is going to happen in this world? Well, demand might go up because maybe there's some type of report that ice cream is much healthier for you than expected and so, at a given price, people are willing to demand a higher quantity, so for example, at that price, people would demand a higher quantity and so, we would have a shift to the right and up, let's call this D2 right over here and this is our new equilibrium point and then notice what has just happened here. At our new equilibrium point, this is Q2 and then this right over here is P2, our new equilibrium price or our new equilibrium quantity. In this situation where demand goes up, both price and quantity are going to go up assuming we have this upwards sloping supply curve again. And once again, that makes sense. More people just wanna buy ice cream, the supply curve dynamics have not changed, so we're gonna move along that supply curve to the right and up, so both price and quantity go up. Well, if demand goes down, you could imagine the opposite is going to happen. So, here if we have demand goes down, let's say a big study comes out that ice cream is even unhealthier than we originally thought, well, then at a given price, people are going to want, they're going to demand less ice cream and so, our demand curve would shift to the left and down, so we'll call this D2 right over here and then we can see our equilibrium price and quantity, so let's show that new equilibrium price is P2 right over here and then our new equilibrium quantity is Q2 and notice, both price and quantity go down. People just don't wanna buy ice cream as much because they think it's unhealthy now, so price goes down and quantity goes down. Creative Commons Attribution/Non-Commercial/Share-AlikeVideo on YouTube Up next: article Use of cookies Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy Accept All Cookies Strictly Necessary Only Cookies Settings 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 and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. 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6044
https://owlcation.com/stem/using-the-magic-triangle-for-speed-distance-and-time-compound-measures
Skip to main content Home / STEM / Math Using the Magic Triangle for Speed, Distance, and Time (Compound Measures) By Mark How to Use the Magic Triangle Speed, distance, and time can be calculated using a magic triangle. D (the distance) goes in the top of the triangle, S (speed) goes in the bottom left of the triangle, and T (time) goes in the bottom right of the triangle. If you want to calculate the speed, cover up S in the triangle and you get S = D/T. If you want to calculate the time, cover up T in the triangle and you get T = D/S. If you want to calculate the distance, cover up D in the triangle and you get D = S×T. Example 1 A car covers a distance of 150 miles in 2 ½ hours. Calculate the average speed of the car in miles per hour. Since you want the speed, cover up S in your triangle and you get S = D/T Next substitute D = 150 and T = 2.5 into the formula for speed: S = D/T S = 150/2.5 = 60mph Example 2 A car is traveling at a constant speed of 80kmph. How many kilometres will the car cover if it keeps this speed for the next 3 hours and 15 minutes? This time, you need to work out the distance so D = S × T Be careful with the time as it needs to be rewritten in hours only; 15 minutes is ¼ hour (0.25) so 3 hours and 15 minutes become 3.25 hours. Now substitute S = 80 and T = 3.25 into the formula for distance: D = S ×T = 80 × 3.25 = 260km So the car travels 260km in 3 hours and 15 minutes. Example 3 How long will it take Simon to jog a distance of 15 miles if Simon jogs at a steady speed of 6mph? Here, you need to calculate the time so T = D/S. Now substitute D = 15 and S = 6 into the formula T = D/S T = 15/6 T = 2.5 hours So Simon will take 2 hours and 30 minutes to cover the distance. Questions & Answers Question: How long does it take for a car to travel 23 miles at 80mph? Answer: To work out the time divide the distance by the speed. 23 divided by 80 is 0.2875 hours. If you want this in minutes multiply 0.2875 by 60 to give 17.25 minutes. Question: Is there a magic triangle for force, mass, and acceralation? Answer: Yes, put F as the top, M in the bottom left and A in the bottom right. Question: Three cities A,B,C are at equal distance from each other. A person travels from A to B at 30km/h; from B to C at 40 km/ h, C to A at 50 km/ h. What is the average speed? Answer: Since the distances are equal just work out the average of the 3 numbers. 30 + 40 + 50 is 120, and 120 divided by 3 is 40km/h. Question: How fast must a car go to cover the distance of 224km in 2 hours and 20 min? Answer: To work out the speed divdide the distance by the time. 2 hours and 20 minutes is 2 hours and 1/3 (or 2.3 recurring). So 224 divided by 2.3 recurring is 96 km/h. Question: What does t = mean? Answer: t stands for time. Question: Why do you substitute the numbers? Answer: Substitute just means put the numbers into the formula. If you don’t do this then you can’t work out the answer! Question: How long does it take for a truck to travel 30 miles at 5mph? Answer: Divide the distance by the speed to give the journey length. 30 divided by 5 gives 6 hours. Question: Whats the whole triangle called? Answer: The triangle is called the speed triangle. Comments Ella on May 10, 2013: This really helped me when i was doing my science homework!! 🙂 happyturtle from UK on February 09, 2012: What would make this Hub that one extra step ahead is a video explaining this also. Thanks for sharing this. markbennis on February 09, 2012: I think this Hub is a right little cracker, really enjoyed the way you have laid it out it makes it so much easier to understand, I think this one will help many that are in schools and colleges especially for there exams, voted up! About the author Mark Sign in to continue Join the conversation
6045
https://web.stanford.edu/~jdlevin/Econ%20202/General%20Equilibrium.pdf
General Equilibrium Jonathan Levin∗ November 2006 “From the time of Adam Smith’s Wealth of Nations in 1776, one re-current theme of economic analysis has been the remarkable degree of coherence among the vast numbers of individual and seemingly sepa-rate decisions about the buying and selling of commodities. In every-day, normal experience, there is something of a balance between the amounts of goods and services that some individuals want to supply and the amounts that other, differerent individuals want to sell [sic]. Would-be buyers ordinarily count correctly on being able to carry out their intentions, and would-be sellers do not ordinarily find themselves producing great amounts of goods that they cannot sell. This expe-rience of balance is indeed so widespread that it raises no intellectual disquiet among laymen; they take it so much for granted that they are not disposed to understand the mechanism by which it occurs.” Kenneth Arrow (1973) 1 Introduction General equilibrium analysis addresses precisely how these “vast numbers of indi-vidual and seemingly separate decisions” referred to by Arrow aggregate in a way that coordinates productive effort, balances supply and demand, and leads to an efficient allocation of goods and services in the economy. The answer economists have provided, beginning with Adam Smith and continuing through to Jevons and ∗Various sections of these notes draw heavily on lecture notes written by Felix Kubler; some of the other sections draw on Mas-Colell, Whinston and Green. 1 Walras is that it is the price system plays the crucial coordinating and equilibrating role: the fact the everyone in the economy faces the same prices is what generates the common information needed to coordinate disparate individual decisions. You doubtless are familiar with the standard treatment of equilibrium in a single market. Price plays the role of equilibrating demand and supply so that all buyers who want to buy at the going price can, and do, and similarly all sellers who want to sell at the going price also can and do, with no excess or shortages on either side. The extension from this partial equilibrium in a single market to general equilibrium reflects the idea that it may not be legitimate to speak of equilibrium with respect to a single commodity when supply and demand in that market depend on the prices of other goods. On this view, a coherent theory of the price system and the coordination of economic activity has to consider the simultaneous general equilibrium of all markets in the economy. This of course raises the questions of (i) whether such a general equilibrium exists; and (ii) what are its properties. A recurring theme in general equilibrium analysis, and economic theory more generally, has been the idea that the competitive price mechanism leads to out-comes that are efficient in a way that outcomes under other systems such as planned economies are not. The relevant notion of efficiency was formalized and tied to competitive equilibrium by Vilfredo Pareto (1909) and Abram Bergson (1938). This line of inquiry culminates in the Welfare Theorems of Arrow (1951) and De-breu (1951). These theorems state that there is in essence an equivalence between Pareto efficient outcomes and competitive price equilibria. Our goal in the next few lectures is to do some small justice to the main ideas of general equilibrium. We’ll start with the basic concepts and definitions, the welfare theorems, and the efficiency properties of equilibrium. We’ll then provide a proof that a general equilibrium exists under certain conditions. From there, we’ll investigate a few important ideas about general equilibrium: whether equilibrium is unique, how prices might adjust to their equilibrium levels and whether these levels are stable, and the extent to which equilibria can be characterized and changes in exogenous preferences or endowments will have predictable consequences. Finally we’ll discuss how one can incorporate production into the model and then time 2 and uncertainty, leading to a brief discussion of financial markets. 2 The Walrasian Model We’re going to focus initially on a pure exchange economy. An exchange economy is an economy without production. There are a finite number of agents and a finite number of commodities. Each agent is endowed with a bundle of commodities. Shortly the world will end and everyone will consume their commodities, but before this happens there will be an opportunity for trade at some set prices. We want to know whether there exist prices such that when everyone tries to trade their desired amounts at these prices, demand will just equal supply, and also what the resulting outcome will look like – whether it will be efficient in a well-defined sense and how it will depend on preferences and endowments. 2.1 The Model Consider an economy with I agents i ∈I = {1, ..., I} and L commodities l ∈ L = {1, ..., L}. A bundle of commodities is a vector x ∈RL +. Each agent i has an endowment ei ∈RL + and a utility function ui : RL + →R. These endowments and utilities are the primitives of the exchange economy, so we write E = ((ui, ei)i∈I). Agents are assumed to take as given the market prices for the goods. We won’t have much to say about where these prices come from, although we’ll say a bit later on. The vector of market prices is p ∈RL +; all prices are nonnegative. Each agent chooses consumption to maximize her utility given her budget con-straint. Therefore, agent i solves: max x∈RL + ui(x) s.t. p · x ≤p · ei. The budget constraint is slightly different than in standard price theory. Recall that the familiar budget constraint is p · x ≤w, where w is the consumer’s initial wealth. Here the consumer’s “wealth” is p·ei, the amount she could get if she sold 3 her entire endowment. We can write the budget set as Bi(p) = {x : p · x ≤p · ei}. We’ll occasionally use this notation below. 2.2 Walrasian Equilibrium We now define a Walrasian equilibrium for the exchange economy. A Walrasian equilibrium is a vector of prices, and a consumption bundle for each agent, such that (i) every agent’s consumption maximizes her utility given prices, and (ii) markets clear: the total demand for each commodity just equals the aggregate endowment. Definition 1 A Walrasian equilibrium for the economy E is a vector (p, (xi)i∈I) such that: 1. Agents are maximizing their utilities: for all i ∈I, xi ∈arg max x∈Bi(p) ui(x) 2. Markets clear: for all l ∈L, X i∈I xi l = X i∈I ei l. 2.3 Pareto Optimality The second important idea is the notion of Pareto optimality, due to the Italian economist Vilfredo Pareto. This notion doesn’t have anything to do with equi-librium per se (although we’ll see the close connection soon). Rather it considers the set of feasible allocations and identifies those allocations at which no consumer could be made better offwithout another being made worse off. Definition 2 An allocation (xi)i∈I ∈RI·L + is feasible if for all l ∈L: P i∈I xi l ≤ P i∈I ei l. 4 Definition 3 Given an economy E, a feasible allocation x is Pareto optimal (or Pareto efficient) if there is no other feasible allocation ˆ x such that ui(ˆ xi) ≥ui(xi) for all i ∈I with strict inequality for some i ∈I. You should note that Pareto efficiency, while it has significant content, says essentially nothing about distributional justice or equity. For instance, it can be Pareto efficient for one guy to have everything and everyone else have nothing. Pareto efficiency just says that there aren’t any “win-win” changes around; it’s quiet on how social trade-offs should be resolved. 2.4 Assumptions As we go along, we’re going to repeatedly invoke a bunch of assumptions about consumers’ preferences and endowments. We summarize the main ones here. (A1) For all agents i ∈I, ui is continuous. (A2) For all agents i ∈I, ui is increasing, i.e. ui(x0) > ui(x) whenever x0 À x. (A3) For all agents i ∈I, ui is concave. (A4) For all agents i ∈I, ei À 0. The first three assumptions – continuity, monotonicity and concavity of the utility function – should be familiar from consumer theory. Some of these are a bit stronger than necessary (e.g. monotonicity can be weakened to local nonsatiation, concavity to quasi-concavity), but we’re not aiming for maximum generality. The last assumption, about endowments, is new and is a big one. It says that everyone has a little bit of everything. This turns out to be important and you’ll see where it comes into play later on. 3 A Graphical Example General equilibrium theory can quickly get into the higher realms of mathemat-ical economics. Nevertheless a lot of the big ideas can be expressed in a simple 5 two-person two-good exchange economy. A useful graphical way to study such economies is the Edgeworth box, after F. Edgeworth, a famous Cambridge (U.K.) economist of the 19th century.1 Figure 1(a) presents an Edgeworth box. The bottom left corner is the origin for agent 1. The bottom line is the x-axis for Agent 1 and the left side is the y-axis. In the picture, agent 1’s endowment is e1 = (e1 1, e1 2). For agent 2, the origin is the top right corner and everything is flipped upside down and backward. Every point in the box represents a (non-wasteful) allocation of the two goods. x1 1 x2 1 Agent 1 Agent 2 e e1 1 e2 2 e1 2 e2 1 0 0 Agent 1 Agent 2 e1 1 B1(p) B2(p) x1(p,p•e1) x2 2 x1 2 e Figures 1(a) and 1(b): The Edgeworth Box Figure 1(b) adds prices into the picture. Given prices p1, p2 for the two goods, the budget line for agent 1 is the line with slope p1/p2 through the endowment point e. This is also the budget line for agent 2. So this line divides the Edgeworth box into the two budget sets B1(p) and B2(p). Each agent will then choose consumption to maximize utility given prices. In Figure 1(b), agent 1’s Marshallian demand x1(p, p · e) is represented by the familiar tangency condition. 1Apparently the name is something of a misnomer, as it seems that Edgeworth boxes were first drawn by Pareto – or so I read on the internet. 6 As we change prices, the Marshallian demands of the two agents will also change. Note that what matters, of course, is the relative prices of the two goods, as these determine the slope of the budget line. Figure 2 traces out the Marshallian demand of agent 1 as we vary the relative prices. The dotted line is called agent 1’s offer curve. Agent 1 Agent 2 OC1 e Figure 2: Offer Curve for Agent 1 Walrasian equilibrium requires that both agents consume their Marshallian demands given prices and also that these demands are compatible. So what we want to do is set relative prices, find the Marshallian demands of the two agents, and see whether or not demand equals supply in the two markets. Figure 3(a) represents a situation where prices do not simultaneously clear the two markets. In this picture, at the given prices, agent 2 is willing to supply some amount of good 2, but less than agent 1 wants to consume. So good 2 is in excess demand. In contrast, agent 1 is willing to supply more of good 1 than agent 2 demands. So good 2 is in excess supply. In Figure 3(b), prices do clear the market and we have a Walrasian equilibrium at the point x. In equilibrium, starting from the endowment point e, agent 1 7 sells good 1 to buy good 2; agent 2 does the reverse. The crucial point is that both markets clear. Note that the Walrasian equilibrium allocation is the intersection of the two offer curves. That the point x lies on the offer curve of agent i means that x it represents the Marshallian demand of that agent given prices p and endowment e. That the point x is the intersection of the two offer curves means that at the given prices, demands are compatible and markets clear. These are conditions (1) and (2) in the definition of Walrasian equilibrium. Agent 1 Agent 2 OC1 e Agent 1 Agent 2 OC1 e x1(p,p•e1) x2(p,p•e2) OC2 Figures 3(a) and 3(b): Dis-equilibrium and Equilibrium in the Edgeworth Box Two natural questions to ask about Walrasian equilibrium are (i) is it unique? and (ii) does it always exist? Both questions have negative answers. Figure 4(a) presents an example with multiple Walrasian equilibria (we’re revisit this example later). In the figure, given the endowment e, the offers curves of the two agents intersect three times. So there are three Walrasian equilibria. 8 Agent 1 Agent 2 e Agent 1 Agent 2 OC1 OC2 e Direction of increasing preference for agent 2 Direction of increasing preference for agent 1 Figures 4(a) and (b): Non-uniqueness and Non-existence of Equilibrium Figure 4(b) presents a different example where Walrasian equilibrium does not exist. In this example, Agent 2 starts with all of good 1 and this is the only good she cares about. Agent 1 starts with all of good 2 and none of good 1. He cares about both goods, but the slope of his indifference curve when he has none of good 1 is infinite. That is, he has infinite marginal utility for his very first unit of good 1. In this example, for any prices p, agent 2 will insist on consuming her endowment – that is, all of good 1. Moreover, there are no prices p at which agent 1 would not insist on buying at least a little bit of good 1. Therefore for any prices p good 1 will always be in excess demand and there cannot be a Walrasian equilibrium. Note that this example violates assumption (A4), which requires that the endowment be an interior point in the Edgeworth box. It is also possible to use the Edgeworth box to depict the idea of Pareto op-timality. This is done in Figure 5. The Pareto set in this picture is the set of all allocations such that to make one agent better offwould require making the other agent worse off. Figure 5 also shows the contract curve. This is the part of the Pareto set that both agents prefer to the endowment e. It seems natural to expect that if the agents were to start at their endowments and strike a mutually agreeable bargain, they would reach a point on the contract curve assuming that 9 bargaining does not leave mutual gains from trade on the table. Agent 1 Agent 2 e Pareto Set Contract Curve Figure 5: The Contract Curve Figure 5 also provides some intuition for a key result in general equilibrium theory: any Walrasian equilibrium is Pareto optimal (or lies on the Pareto set). The reason is as follows. At a Walrasian equilibrium, the budget line will separate the two “as good as” sets of the agents (as we saw in Figure 3(b)). Thus, there will be no alternative to the Walrasian outcome that would make both agents better off. Therefore any Walrasian equilibrium is Pareto optimal. The Pareto set, of course, is the set of all Pareto optimal allocations, so an alternative statement is that any Walrasian equilibrium allocation lies on the Pareto set. This result is known as the first theorem of welfare economics. 4 The Welfare Theorems We now turn to a more formal statement of the theorem suggested above – that every Walrasian equilibrium allocation is a Pareto optimal allocation. We then prove a converse result that if an initial allocation is Pareto optimal, there is a Walrasian equilibrium at which no trade occurs. 10 Theorem 1 (First Welfare Theorem) (Arrow, 1951; Debreu, 1951) Let (p, (xi)i∈I) be a Walrasian equilibrium for the economy E. Then if (A2) holds, the allocation (xi)i∈I is Pareto optimal. Proof. By way of contradiction, suppose there is a feasible allocation (ˆ xi)i∈I such that ui(ˆ xi) ≥ui(xi) for all i ∈I with strict inequality for some i0. By revealed preference and (A2), p · ˆ xi ≥p · xi for all i ∈I, and also p · ˆ xi0 > p · xi0 (this is Walras’ law). Therefore, because prices are non-negative, there must be at least one good l for which P i ˆ xi l > P i xi l = P i ei l. Therefore ˆ x is not feasible. Q.E.D. This result provides formal support for Adam Smith’s claim that individuals acting in their own interests end up behaving in a way that is efficient from a societal standpoint. It is a powerful statement about the efficiency properties of decentralized markets: despite the fact that there is no explicit social coordination and agents simply maximize their utilities given prices, the resulting equilibrium outcome is efficient from a social perspective. Note that in a sense, the assumptions are quite weak. Given our model of the ex-change economy, the only assumption on preferences that we require is monotonic-ity (and local nonsatiation would suffice). Of course, it should be emphasized that the model itself contains a large number of heroic implicit assumptions that seem highly unlikely to be satisfied in any real economy. Among these are: (1) all agents face precisely the same prices; (2) all agents are price takers – i.e. they take prices as given and don’t believe that their purchasing decisions will move prices; (3) markets exist for all goods and agents can freely participate. Moreover, we have said nothing so far about how a group of agents might arrive at equili-bium prices. So you’ll probably want to withhold judgment on the efficiency of decentralized markets. The first welfare theorem states that equilibrium outcomes are efficient. Our next result states that efficient outcomes are Walrasian equilibria given the correct prices and endowments. Theorem 2 (Second Welfare Theorem) (Arrow, 1951; Debreu, 1951) Let E be an economy that satisfies (A1)—(A4). If (ei)i∈I is Pareto optimal then there exists a price vector p ∈RL + such that (p, (ei)i∈I) is a Walrasian equilibrium for E. 11 Proof. To prove this we need a version of the separating hyperplane theorem. Suppose you have an open convex set A ⊂Rn and a point x / ∈A. Then there exists a p 6= 0 such that p · a ≥p · x for all a ∈cl(A). To prove the theorem, let’s define: Ai = {a ∈RL : ei + a ≥0 and ui(ei + a) > ui(ei)}. Because ui is concave, Ai is a convex set. Therefore the set A = X i∈I Ai = {a ∈RL : ∃a1 ∈A1, ..., aI ∈AI with a = X i∈I ai} is a convex set. Moreover, 0 / ∈A, because if it were there would exist some (ai)i∈I with P i∈I ai = 0 and ui(ei+ai) > ui(ei) for all i ∈I, contradicting the assumption that e is Pareto optimal. The separating hyperplane theorem now implies that there is some price vector p 6= 0 such that p · a ≥0 for all a ∈cl(A). Furthermore, p ≥0 because if a À 0 then a ∈A by monotonicity and if some pl < 0 we could take al arbitrarily big and all the other al0 small but positive and get a contradiction. Because p 6= 0 and p ≥0, this means that p > 0. The claim is that this p will support the allocation e as a Walrasian equilibrium. Obviously e satisfies the market clearing part of the definition of equilibrium. Moreover, fixing prices p, consider a given agent i. Suppose xi ∈RL + and ui(xi) > ui(ei). We will show that xi is not in i’s budget set, thus proving the individual optimization part of equilibrium. First, by definition of Ai and p, p · xi ≥p · ei. Moreover, by continuity, the fact that ui(xi) > ui(ei) implies that for λ just less than 1, ui(λxi) > ui(ei). Therefore p · λxi ≥p · ei. This can’t be the case if p · xi = p · ei, so therefore p · xi > p · ei. Q.E.D. Note that the second welfare theorem does not say that starting from a given endowment, every Pareto optimal allocation is a Walrasian equilibrium. Rather it says that if we were to start from a given endowment then for any Pareto optimal allocation there is a way to re-distribute resources and a set of prices that makes the allocation a Walrasian equilibrium outcome. 12 In practice, this means that decentralizing a Pareto optimal allocation is not simply a matter of identifying and specifying the correct prices (not that this would necessarily be easy). Large-scale re-distribution may be required as well. This lim-its the practical applicability of the theorem. Still, it is a useful result conceptually and for modelling. For instance, in complicated macroeconomic dynamic models, it can sometimes be hard to directly establish the existence of an equilibrium; in some cases, one can proceed by identifying Pareto optimal allocations and then showing a version of the second welfare theorem saying that the Pareto allocation can be supported as a Walrasian equilibrium. Finally, one technical point. Observe that unlike with the first welfare theorem, convexity plays a crucial role in establishing the second welfare theorem. Indeed, at a formal level, the theorem is a direct application of the separating hyperplane theorem, where the equilibrium price vector separates the Pareto allocation e from the set of allocations preferred to e by at least one agent. 5 Characterizing Equilibrium In this section, which follows MWG, chapter 16, we make a bunch of assumptions about utility functions being differentiable and concave and then use first order conditions to characterize Pareto optimal allocations. The idea is to give some intuition for what conditions must be satisfied on the margin at any Pareto optimal allocation, and hence, by the first Welfare theorem, at any Walrasian equilibrium. We also tie the set of Pareto optimal allocations to the set of allocations that maximize linear Bergson-Samuelson social welfare functions. One way to identify the set of Pareto optimal allocations x = (x1, ..., xI) is as solutions to the following program: maxx u1(x1 1, ..., x1 L) s.t. ui(xi 1, ..., xi L) ≥ui for i = 2, ..., I X i xi l ≤ X i ei l for l = 1, ..., L. 13 The idea here is to maximize the utility of the first consumer subject to feasibility and to the other consumers getting at least some pre-specified level of utility. By varying the level of required utility for consumers 2, ..., I, we can recover the full set of Pareto optimal allocations. Under assumptions (A1)—(A3), all of the constraints must be binding at the solution (if the utility constraint for i were slack we could reduce xi be ε in all directions and increase x1 by the same amount; if the resource constraint were slack we could increase either x1 or one of the xis). If was assume in addition that each agent has a differentiable utility function, the problem satisfies the conditions of the Kuhn-Tucker theorem, so we can use the Kuhn-Tucker conditions to characterize the solution. Let λi denote the Lagrange multiplier on agent i’s constraint and let μl denote the constraint on commodity l. The Kuhn-Tucker conditions are then: λi∂ui ∂xi l −μl ≤0, xi l ≥0, µ λi∂ui ∂xi l −μl ¶ xi l = 0, (1) coupled with the requirement that each of the (I −1) + L constraints is binding: ui(xi 1, ..., xi L) = ui for i = 2, ..., I X i xi l = X i ei l for l = 1, ..., L. In the first line, I’ve adopted the convention that λ1 = 1; you’ll see where this bit of notation comes in useful later. Note that because each of the constraints binds at the optimum, λi > 0 for i = 2, ..., I and μl > 0 for all l. The Kuhn-Tucker conditions given in (1) are easy to interpret. Recall that λi is precisely the marginal value, or shadow price, of consumer i’s income in terms of consumer 1’s utility. That is, at the optimum taking a util away from agent i would allow us to increase agent 1’s utility by λi. At the same time, μl is the shadow price on commodity l (again in terms of agent 1’s utility). An extra unit of commodity l would allow us to increase agent 1’s utility by μl while holding everyone else’s utility constant. Assuming that each consumer consumes a positive amount of each good at the 14 optimum, so that xi l > 0 for all i, l, we can easily derive that at any Pareto efficient allocation, we have the following relationship: MRSi kl = ∂ui/∂xi k ∂ui/∂xi l = ∂uj/∂xj k ∂uj/∂xj l = MRSj kl = μk μl . That is, at the optimum, the marginal rates of substitution of every agent for every commodity pair k, l must be equal to each other and to the ratio of the shadow prices μk and μl. This is precisely the tangency condition from our earlier Edgeworth box picture. Within this simple framework of differentiable concave utility functions, we can link the Pareto optimal allocations to the set of Walrasian equilibria quite easily. Suppose that x is a Pareto optimal allocation as characterized above. Let ei = xi and define prices pl = μl. Given these prices and endowments, consider the optimization problem facing consumer i : max ˜ xi ui(˜ xi) s.t. p · ˜ xi ≤p · ei Again, we know the budget constraint will bind at the optimum given our as-sumptions (that’s Walras’ Law). Moreover, we can use the Kuhn-Tucker conditions to characterize the optimum. Letting ν1, ..., νI denote the Lagrange multipliers on the budget constraints of agents 1, ..., I, the Kuhn-Tucker conditions state that a necessary and sufficient condition for (x1, ..., xI; ν1, ..., νI) to solve the I utility maximization problems given prices p is that for all i, l: ∂ui ∂xi l −νi · pl ≤0 xi l ≥0, µ∂ui ∂xi l −νi · pl ¶ · xi l = 0, (2) and in addition, each of the resource constraints bind. It’s quite easy to see that if x is a Pareto optimal allocation, one solution is for each agent i to consume xi = (xi 1, ..., xi L) with Lagrange multipliers νi = 1/λi. Why? Because given prices pl = μl and endowments ei = xi, there is an exact equivalence between the Kuhn-Tucker conditions of the I utility maximization 15 problems and the Kuhn-Tucker conditions of the earlier Pareto problem. Therefore it follows that if x is a Pareto optimal allocation, and μ1, ..., μL the commodity shadow prices from the Pareto problem above, then (μ, x) is a Walrasian equilibrium of the economy E = ((ui)i∈I, (xi)i∈I). This is precisely the Second Welfare Theorem. To obtain the First Welfare Theorem, we go the other way. Observe that if endowments e and prices p are given and each agent maximizes utility, it must be the case at the solution consumption bundles x1, ..., xI, (2) holds and each consumer’s budget constraint is satisfied. Then consider the Pareto problem with ui = ui(xi) for agents 2, ..., I. It is easy to check that (1) and each of the constraints is satisfied at x1, ..., xI if we define μl = pl, λi = 1/νi, and ui = ui(xi). Therefore any Walrasian equilibrium is Pareto optimal. Finally, there is an alternative approach to characterizing Pareto efficient al-locations that is sometimes useful. In this approach, one considers maximizing a linear (Bergson-Samuelson) social welfare function of the form P i βiui subject to a resource constraint. The program is: max x1,...,xI X i βiui(xi 1, ..., xi L) s.t. X i xi ≤ X i ei Given monotonicity of utility functions, the resource constraint will bind at the optimum and the additional Kuhn-Tucker condition for optimality is that for all agents i and commodities l: βi∂ui ∂xi l −δl ≤0 xi l ≥0 µ βi∂ui ∂xi l −δl ¶ · xi l = 0 (3) Letting βi = λi and δl = μl we have an exact correspondence between (1) and (3). Letting βi = 1/νi and δl = pl, we have an exact correspondence between (2) and (3). So not only do Pareto optimal allocations coincide with Walrasian equilibrium allocations coincide in the sense of the welfare theorems, they coincide 16 with allocations that maximize a linear social welfare function. 6 Existence of Equilibrium For nearly a hundred years after Walras wrote down his model of general equilib-rium, it was an open question as to whether such an equilibrium actually existed. Early approaches to proving existence results focused on a general equilbrium model due to Cassel (1924), which took as its basic premises aggregate demand for each commodity as a function of all commodity prices (so no individual utility maximization), and a simple supply side model where each commodity could be produced from a fixed resource input and each firm would produce at zero profits (so very simple linear production functions). Equilibrium was defined as a set of commodity prices and quantities such that demand just equaled supply for each commodity. Ignoring the supply side for a moment, and letting xi(p1, ..., pn) denote the aggregate demand for good i as a function of prices, the basic problem was to show the existence of a price vector p1, ..., pn satisfying: xi(p1, ..., pn) = ei for all i = 1, ..., n. The basic idea in the early literature was to count up equations and unknowns. Unfortunately, this led to some confusion about what would happen if the solution to the equations involved either negative prices or quantities. In 1951, John Nash published his Princeton dissertation in which he used a fixed point theorem to prove the existence of Nash equilibrium in games. Once this idea was out, general equilibrium theorists realized how to provide general existence proofs for Walrasian equilibrium. The big breakthrough came when Arrow and Debreu (1954) teamed up to prove the following result. Theorem 3 (Existence of Walrasian Equilibrium) (Arrow-Debreu, 1954) Given an economy E satisfying (A1)—(A4), there exists a Walrasian equilibrium (p, x). The proof is pretty involved and arguably not all that enlightening, but this has been such a persistent question in modern economics that we’d be remiss not 17 to attempt it. The general fixed point style of proof is also common in other problems. What we’ll do here is start with a fairly simple and intuitive proof for the case of two commodities, then give a more general proof using a fixed point theorem. 6.1 Excess Demand Functions As a starting point, we’re going to introduce the idea of an excess demand function. Definition 4 The excess demand function of agent i is: zi(p) = xi(p, p · ei) −ei where xi(p, p · ei) is i’s Walrasian demand function. The aggregate excess de-mand function is: z(p) = X i zi(p). From the definition of the excess demand function, it should be clear that if a price vector p ∈RL + satisfies z(p) = 0, then (p, (xi)i∈I) will be a Walrasian equilibrium if xi is defined to be i’s Marshallian demand given the price vector p. Why? Because (p, (xi)i∈I) will then satisfy both the individual optimization part of the definition of equilibrium (by definition of xi) and market clearing (by the fact that z(p) = 0). Thus, proving existence of equilibrium boils down to establishing that a solution to z(p) = 0 exists given our assumptions (1)—(4). From the first part of this class on consumer theory, we have the following properties of the excess demand function. Proposition 4 Suppose (A1)—(A4) are satisfied. Then the aggregate excess de-mand function z(p) satisfies: (i) z is continuous; (ii) z is homogenous of degree zero; (iii) z(p) = 0 for all p (Walras’ Law); 18 (iv) for some Z > 0, zl(p) > −Z for every l ∈L and all p. (v) if pn →p, where p 6= 0 and pl = 0 for some l, then max{z1(pn), ..., zL(pn)} → ∞. Proof. Except for the last property, these all follow directly from properties of the Marshallian demand function established in the first half of the class. The last property isn’t complicated. As some, but not all, prices go to zero, there must be some consumer whose wealth is not going to zero. Because he has strongly monotone preferences, he must demand more and more of one of the goods whose price is going to zero. Q.E.D. In the next section, we use these properties to establish existence of equilibrium for the two good case. The proof won’t be entirely general because we’re going to treat z(·) as a function. In general, recall that agents might not have a unique optimal bundle given a set of prices, so Marshallian demand, and hence z(·) should really be treated as a correspondence. We’ll deal with that in the following section. 6.2 An Intuitive Argument To gain some intuition, let’s consider the case where there are only two goods in the economy, so we want to find a Walrasian equilibrium price vector p = (p1, p2) with z(p) = 0. Because z(·) is homogenous of degree zero, we can safely normalize the price of p2 = 1, meaning we can search only over price vectors p = (p1, p2 = 1). Moreover, because of Walras’ Law, z(p) · p = 0 for any p, so to establish an equilibrium, it suffices to find a price p1 such that z1(p1, 1) = 0. (If this holds, then z2(p1, 1) = 0 by Walras’ Law.) Figure 6 graphs z1(p1, 1) as a function of p1. There are three important points to note in the picture that must hold. First, z1(·, 1) is continuous as we showed above. Second, for very small values of p1, z1(p1, 1) is strictly positive. Third, for very large values of p1, z1(p1, 1) is negative. There must be at least one value of p1 for which z1(p1, 1) = 0 and the vector (p1, 1) is a Walrasian equilibrium price vector. 19 0 z1 p1 z1(p1,1) p1 -Z Figure 6: Existence of Walrasian Equilibrium with Two Goods The only subtlety to this argument is establishing points 2 and 3, namely that when one of the goods becomes infinitely cheap relative to the other good, there will be excess demand for the cheaper good. Formally, this follows from conditions (iv) and (v) in the above Proposition. For very small values of p1, condition (v) implies that either z1 or z2 must be very large. It can’t be, however, that excess demand for the relatively expensive good 2 is very large however, because then p2 · z2 = z2 would be very large, and condition (iv) implies that p1 · z1 > −Z for some fixed Z. Hence Walras’ law stating that p1 ·z1 +p2 ·z2 = 0 would be violated. A symmetric argument implies that for very large values of p1, z1(p1, 1) must be negative. 6.3 Proof of Equilibrium Existence At this point, we’re ready to take a shot at proving the existence theorem. As suggested above, you’ll need some familiarity with correspondences (consult MWG or the Useful Math notes if in doubt). The proof is going to work a bit differently from the two-good case in the following sense. Rather than look for a price vector p that solves z(p) = 0 (with 20 the allocation being left implicit in the definition of z(·)) we’re going to define a map Ψ that takes the set of price-aggregate demand pairs (p, x) into itself. The map Ψ will be defined so that any fixed point of Ψ will be a Walrasian equilibrium. Then we’ll establish that a fixed point exists. The map Ψ is going to be defined as follows. Given a price-aggregate de-mand pair (p, x), we define the new aggregate demand by letting agents optimize given prices. That is, agents take prices as given and have a Marshallian demand correspondence (function in the case of strict concavity but correspondence if indif-ference curves are flat and many points on the budget line give the same utility). We establish that aggregate demand is a non-empty, convex valued and upper semi-continuous (usc) correspondences of prices. To get a new set of prices, we apply a trick that was first introduced by Debreu. We have an additional agent called the ‘price-player’. He takes the old aggregate demand as given and sets prices in a clever way; his choice correspondence is an usc, convex-valued and non-empty correspondence of aggregate demand. An important property will be that if at the old prices, the agent’s demands just clear the market, it will be optimal for the price player not to change prices. With this set-up we put together all the choices into a correspondence Ψ that maps a price vector and an aggregate demand into new prices (chosen by the price player) and a new aggregate demand (chosen by the agents). As noted, Ψ is a correspondence, not a function. We then apply Kakutani’s Theorem, which asserts that under conditions we’ve established for Ψ, it must have a fixed point (for this, we’ll need to state and maybe understand Kakutani’s theorem). We then argue that a fixed point of our map corresponds exactly to a Walrasian equilibrium. The whole thing sounds complicated, but hopefully it won’t be too bad. 6.4 Two Math Theorems We’re going to need to appeal to two mathematical theorems during the proof. The first is Kakutani’s fixed point theorem (it’s hard to prove). Theorem 5 (Kakutani) Suppose A ⊂I Rn is a convex, closed and bounded set. Suppose f : A ⇒A is a correspondence which is convex valued, non-empty valued 21 for all x ∈A and which is upper-hemi-continuous (uhc). Then there exists a x ∈A such that x ∈f(x). If you know Brouwer’s fixed point theorem for functions, this result is quite similar. If you don’t know Brouwer’s fixed point theorem, I’ll try to give some intuition for it in class. If this theorem just looks totally mysterious, just take it as given and don’t worry too much about it. The second theorem (not hard to prove) is called ‘maximum principle’ or some-times Berge’s theorem (see M-W-G Theorem M.K.6) Proposition 6 Suppose we have a continuous correspondence C : Q ⇒I RN, with c(q) compact and non-empty for all q ∈Q and a continuous function f : I RN →I R. Consider a maximization problem max f(x)s.t.x ∈c(q). The maximizer correspon-dence will be upper-hemi-continuous. The value function will be continuous. 6.5 Proof of the Existence Theorem We’ll prove the theorem is a series of steps. STEP 1 (Normalize Prices): Recall that what matters in the Walrasian model is relative prices, so we are always free to normalize one of the prices. Rather than set p1 = 1, however, it’s convenient to normalize the prices so that they all sum to 1. Define: ∆= © p ∈RL + : p1 + ... + pL = 1 ª to be the set of price vectors that sum to one (the price simplex). STEP 2 (Aggregate Demand): We first define individual Marshallian demands in such a way that they are usc in prices. In order to agents’ demand correspondences are usc using Berge’s Theorem, we face the slight problem that the budget correspondence Bi(p) isn’t compact valued at prices on the boundary of ∆. Therefore we cleverly define the 22 compact set T = {x ∈RL + : x ≤2 X i∈I ei} and consider for each agent i ∈I the correspondence ψi(p) = arg max c∈Bi(p)∩T ui(c) The correspondence ψi is non-empty valued and usc for each agent i. Moreover, because ui(.) is concave, ψi is convex-valued. Note that our assumption ei À 0 is crucial here. If ei l = 0 for some l ∈L the budget correspondence will not be continuous at pl = 0 (which one fails, usc or lsc ?), so we can’t apply Berge’s Theorem. Finally, we define the aggregate demand correspondence: ΨD(p) = X i∈I ψi(p) = ( x : ∃x1 ∈ψ1(p), ..., xI ∈ψI(p) s.t. x = X i∈I xi ) . It’s easy to check that Ψ : ∆→T is a non-empty, convex-valued and usc in prices, given what we know about ψ1, ..., ψI. STEP 3 (The Price Player): Now we introduce the price player, whose correspondence is defined as ΨP : T ⇒∆, where ΨP = arg max p∈∆L−1 p · (x −e) where e = P i∈I ei is the aggregate endowment. That is, the price player chooses new prices to maximize the value of the aggregate excess demand (at the old prices). Note that ΨP is non-empty, convex-valued and usc. STEP 4 (The Fixed Point): Define Ψ : ∆× T ⇒∆× T by: Ψ(p, x) = (ΨP(x), ΨD(p)). 23 Because the product of non-empty and convex-valued usc correspondeces is it-self non-empty, convex-valued and usc, we can apply Kakutani’s theorem. This establishes a fixed point (p∗, x∗) ∈Ψ(p∗, x∗). STEP 5 (The Walrasian Equilibrium): We now argue that from (p∗, x∗) we have a W.E., or more precisely that p∗is a Walrasian equilibrium when paired with the individual demands x1∗, ..., xI∗that make up x∗. In particular, because x∗∈ΨD(p), there exist x1∗, ..., xI∗summing to x∗with the property that xi∗∈arg maxc∈Bi(p∗)∩T ui(c). For the individual optimization part of equilibrium, we need to verify that xi∗∈arg maxc∈Bi(p∗) ui(c). For this, note that p∗∈ΨP(x∗) : 0 ≥p∗· (x∗−e) ≥p · (x∗−e) for all p ∈∆. The latter inequality implies that x∗−e ≤0 and in particular xi∗< 2e. Therefore xi∗∈arg maxc∈Bi(¯ p) ui(c) because if there were a c ∈Bi(p) with ui(c) > ui(xi∗) then for small λ > 0, λc + (1 −λ)xi∗∈Bi(p) ∩T and by concavity of ui, ui(λc + (1 −λ)xi∗) > ui(xi∗), a contradiction. It remains to show market clearing: i.e. that x∗= e. By Walras Law, we have p∗·x∗= p∗·e. Therefore if x∗ l −el < 0 for some good l, we must have p∗ l = 0 by the price player’s optimization. But then we can simply replace x∗1 l by x∗1 l −(x∗ l −el) and get market clearing. Q.E.D. 7 Uniqueness, Stability and Comparative Statics We now turn to a brief discussion of three important questions in general equilib-rium theory. These are: 1. Is there a unique Walrasian equilibrium? If not, how many Walrasian equi-libria are there? 2. Is the Walrasian equilibrium stable in the sense that reasonable dynamic adjustment processes converge to equilibrium prices and allocations? 24 3. Does Walrasian equilibrium impose meaningful restrictions on observable data? For instance, what can we say about how a change in endowments will change equilibrium prices? As we’ve already suggested, the first two questions have essentially negative answers. Generally speaking there can be a lot of Walrasian equilibria for a given specification of preferences and endowments (though not an infinite num-ber). There is also no particular reason, without strong assumptions on prefer-nces, to believe that dynamic adjustment processes will converge to a Walrasian equilibrium outcome. In contrast, the third question will have a positive answer. If we observe data on endowments and prices for a fixed set of agents trading at equilibrium prices, and then are asked to predict equilibrium prices and quantities for these same agents after a change in endowments, we will generally be able to say something (though maybe not all that much) about what the new equilibrium prices and quantities will be. 7.1 Global Uniqueness The Edgeworth box picture we drew above in Figure 4 suggests strongly that there need not be a unique Walrasian equilibrium. The following simple numerical example (from MWG) shows that the picture is not at all pathological. Suppose there are two goods and two consumers with utility functions: u1(x1 1, x1 2) = x1 1 −1 8(x1 2)−8 u2(x2 1, x2 2) = −1 8(x2 1)−8 + x2 2. Both utility functions are quasi-linear, but with respect to different numeraires. Assume the endowments are e1 = (2, r) and e2 = (r, 2), where r = 28/9 −21/9 (this is just to make everything work out nice). The Marshallian demands at prices p1, p2 are: x1(p1, p2) = à 2 + r µp2 p1 ¶ − µp2 p1 ¶8/9 , µp2 p1 ¶−1/9! x2(p1, p2) = õp1 p2 ¶−1/9 , 2 + r µp1 p2 ¶ − µp1 p2 ¶8/9! . 25 Normalizing the price of good 2 so that p2 = 1, we can write the aggregate excess demand curve for good 1 as z1(p1, 1) = x1 1(p1, 1) + x2 1(p1, 1) −(2 + r) or z1(p1, 1) = r µ1 p −1 ¶ − µ1 p ¶8/9 + µ1 p ¶1/9 This is pictured in Figure 7 below. Note that the excess demand function has three zeros: at p1 = 1/2, at p1 = 1 and at p1 = 2. All three correspond to Walrasian equilibrium. Another way to see the multiplicity is to look at Figure 4 above, where this model is depicted in the Edgeworth box. 0 z1 p1 z1(p1,1) 2 1 Figure 7: Multiple Walrasian Equilibria What should be clear from both figures is that multiple equilibria do not always arise: for a given set of endowments and preferences, marshallian demands have to change in the right way with prices to get multiplicity. On the other hand, neither are multiple equilibria all that special – in the numerical example, we picked the 26 numbers to come out nicely, but a small change in preferences or endowments won’t upset the fact that there are three equilibria. This being said, a good deal of effort has gone into identifying conditions on preferences that rule out multiplicity and ensure uniquess. We will discuss one such condition, the gross substitutes property, below. 7.2 Local uniqueness Even if there are multiple Walrasian equilibria for a given set of preferences and endowments, it may still be the case that each of these equilibria are all locally unique in the sense that there is no other Walrasian equilibrium price vector within a small enough range around the orginal equilibrium price vector. It turns out that for “most” economies, the Walrasian equilibria are locally unique. As a consequence the set of Walrasian equilibria is finite. To understand this idea, it’s again useful to think about the two commodity case, where we can normalize p2 = 1 and look for values of p1 that satisfy z1(p1, 1) = 0. In the example above, shown in Figure 7, there are three Walrasian equilibria. Each of the equilibria, however, is locally unique. An equilibrium is not locally unique if its price vector p is the limit of a sequence of other equilibrium prices. An example of local non-uniqueness would be if z1(p1, 1) was equal to zero over some interval of prices [p∗ 1, p∗∗ 1 ]. This can happen, as is shown in Figure 8. The point to realize from this figure, however, is that such an occurrence is extremely special. Any small perturbation of z1(·, 1), such as would arise from a small change in the endowments, will restore us to the case of having a finite number of locally unique equilibria.2 2Note that the picture also suggests something more, which is that because z1(·, 1) starts above zero and finishes below zero, “generically” there should be an odd number of equilibria. This turns out to be correct and can be shown in the many commodity case as well. 27 0 z1 p1 z1(p1,1) p1 p1 Figure 8: Local Non-uniqueness of equilibrium This is roughly speaking the kind of argument one uses in the more general many-commodity case to prove that the equilibria of “typical” economies are locally unique. Heavier math comes into play, however, so we’ll skip the formal analysis. This is often a topic covered in advanced general equilibrium classes. 7.3 Tatonnement Stability So far we haven’t touched the question of where prices come from and whether it is reasonable to expect to see Walrasian prices in a given economy — even if such prices exist and are unique. It turns out that the theory of general equilibrium is quite weak on the kinds of price formation processes that might lead to Walrasian outcomes. Walras himself suggested the following kind of price adjustment process that he called “tatonnement” (french for “groping”). Imagine that the agents meet in a public square and there is a “Walrasian auctioneer” who calls out prices. After he does this, everyone calls out their demands at those prices. The auctioneer then adjusts prices and calls out a new set. The process continues until a set of prices 28 is called out for which demand just equals supply. At this point the auctioneer stops, announces prices, and trade occurs. A candidate price adjustment rule for the auctioneer is: p(t + 1) = p(t) + αz(p(t)) for small α > 0. Clearly the only stationary points of this process are prices p at which z(p) = 0, i.e. Walrasian equilibrium prices. Moreover, an equilibrium price vector p could naturally be said to be locally stable if the price adjustment rule converges to p from any “nearby” starting prices. If an equilibrium price vector p is stable a small perturbation away from p will not have a long-run effect if the auctioneer begins again to look for an equilibrium. Finally, an equilibrium price vector p is (globally) stable if the price adjustment rule converges to p from any initial prices. Walrasian tatonnement gives us a way to study how equilibrium prices might be reached, but the model has obvious drawbacks. First, no trade actually takes place at the non-equilibrium prices. Second, even if one were able to organize this giant procedure, people might not want to announce their true demands. Finally a third drawback, and the one that delivered a huge blow to this line of research is that Walrasian tatonnement can cycle without converging to an equilibrium. This was first shown in a famous paper by Scarf (1960), who provided examples where both local and global stability failed. Prior to this paper, all that had been shown were assumptions on preferences that did ensure stability of Walrasian equilibrium prices (e.g. Arrow and Hurwicz, 1958). 7.4 Debreu-Sonnenschein-Mantel Theorem Above we posed the questions of whether Walrasian equilibrium is unique and whether it is stable under reasonable dynamic adjustment processes. The answer to both turned on the structure of the aggregate excess demand function of the economy: z(·). Walrasian equilibrium is unique if there is a unique solution to z(p) = 0, and stable if the zero is a stable point of z(·). This leads us to ask what exactly we know about the structure of aggregate ex-cess demand. Above, we proved that z(·) is continuous, homogenous of degree zero 29 in prices, satisfies Walras’ Law, and has certain boundary properties: in particular as p →0, z(p) →∞. In a famous paper, Hugo Sonnenschein (1973) asked whether there are any fur-ther restrictions on z(·) that can be derived from the assumption of consumer max-imization. Remarkably, Sonnenschein (1973), Debreu (1974) and Mantel (1974) were able to show that the answer is “no”. This gives something of a negative conclusion to our original questions as it implies that given an economy that has an equilibrium at a certain price vector p, it is possible for that economy to have an arbitrary number of equilibria with arbitrary stability properties in an arbitrarily small neighborhood of prices around p. Theorem 7 (Sonnenschein-Mantel-Debreu) Suppose we have an open and bounded subset B ⊂RL ++ and a continuous function f(p) : B →RL satisfying homogeneity of degree zero and Walras’ Law. Then there exists an economy E with aggregate excess demand function z(p) satisfying f(p) = z(p) on B. Proof. We’ll skip it. You can look up the L = 2 case in MWG. Q.E.D. A common interpretation of this theorem (as in MWG) is that “anything goes” in general equilibrium theory. That is, that without very special assumptions (like Cobb-Douglas preferences or something like that): (i) pretty much any comparative statics result could be obtained in a general equilibrium model, and (ii) general equilibrium theory has essentially no empirical content. We’ll see in the next section that this is not quite right. 7.5 Brown-Matzkin Theorem The Sonnenschein-Debreu-Mantel Theorem says that the aggregate excess demand function has only minimal properties. An implication is that utility maximization imposes no testable restrictions on equilibrium prices. This suggests that one could not test the hypothesis that agents were or were not trading in a Walrasian fashion by observing price data, unless one also made some assumptions about the preferences of the agents who were trading. 30 A striking result due to Brown and Matzkin (1996), however, says that if one is able to observe endowments as well as prices, then the Walrasian model is testable. That is, there are endowment and price pairs (p, (ei)i∈I) and (ˆ p, (ˆ ei)i∈I) such that if p is a set of Walrasian prices given a fixed set of agents with endowments (ei)i∈I, then if this same set of agents has endowments (ˆ ei)i∈I, ˆ p could not possibly be a Walrasian equilibrium price vector. The argument relies on revealed preference. Theorem 8 (Brown-Matzkin, 1996) There exist prices and endowments (p, (ei)i∈I) and (ˆ p, (ˆ ei)i∈I) such that it is impossible to find monotone preferences (ui)i∈I with the property that p is a Walrasian equilibrium price vector for the economy (ui, ei)i∈I and ˆ p is a Walrasian equilibrium price vector for the economy (ui, ˆ ei)i∈I. Proof. We use an Edgeworth box example to prove the Theorem for the case of two consumers and two goods. Consider the two Edgeworth boxes in Figure 9. Because p is an equilibrium given e, agent 1 must weakly prefer some bundle on the line segment A to any bundle on the line segment B. By monotonicity, for every point on the line segment ˆ A, there is some point on B that agent 1 strictly prefers. So there is some bundle on A that is preferred to every bundle on ˆ A. Now, if ˆ p is an equilibrium given ˆ e, we have an immediate contradiction because every bundle on A is available, yet the agent chooses a bundle on ˆ A. Q.E.D. 31 Agent 1 Agent 2 Agent 2 e’ e p’ p B B’ A A’ Figure 9: Testable Restrictions of Equilibrium 8 Gross Substitutes In this section, we consider one particular class of economies – those satisfying the gross substitutes property – in which it is possible to get affirmative answers to the uniquencess and stability questions posed above. We then show that economies satisfying gross substitutes also have very nice comparative statics properties. Two commodities are said to be gross substitutes if an increase in the price of good k increases the demand for good l. More generally, a demand function satisfies the gross substitutes property if an increase in the price of good k increases the demand for every other good l. Definition 5 A Marshallian demand function x(p) satisfies the gross substi-tutes property if, whenever p and p0 are such that p0 k > pk and p0 l = pl for all l 6= k, then xl(p0) > xl(p) for all l 6= k. Note that I have stated the condition as requiring a strict increase in the de-mand for each good l. One can also work with the weak gross substitutes property 32 which requires only a non-decrease in the demand for each good l. To keep things simple, we’ll stick with the strict case. We now have the following observation. Remark 1 If each individual has a Marshallian demand function satisfying gross substitutes, then both the individual and aggregate excess demand functions satisfy gross substitutes. If aggregate demand satisfies the gross substitutes property, then there is a unique Walrasian equilibrium. Proposition 9 If the aggregate excess demand function z(·) satisfies gross substi-tutes, the economy has at most one Walrasian equilibrium, i.e. z(p) = 0 has at most one (normalized) solution. Proof. Suppose by way of contradiction that z(p) = z(p0) = 0 for two price vectors p and p0 that are not collinear. By homogeneity of degree zero, we can always normalize the price vectors in such a way that p0 l ≥pl for all l ∈L and p0 k = pk for some good k. Then to move from p to p0, we can think about moving in a series of n −1 steps, increasing the prices of each of goods l 6= k in turn. At each step where a component of price increases strictly (and there must be at least one such step), the aggregate demand for good k must strictly increase, so that zk(p0) > zk(p) = 0, yielding a contradiction. Q.E.D. It is also possible to show that under the gross substitutes property, Walrasian tatonnement will converge to the unique equilibrium. One approach to showing this is via the following Lemma. Lemma 10 Suppose that the aggregate excess demand function z(·) satisfies gross substitutes and that z(p∗) = 0. Then for any p not collinear with p∗, p∗· z(p) > 0. Proof. Let’s just consider the proof for the case of two commodities; there’s probably an elegant short proof for the general case, but the only proof I could 33 come up with is rather long. With two commodities, let’s normalize the price of good 2, so that p∗ 2 = p2 = 1. Then: p∗· z(p) = (p∗−p) · (z(p) −z(p∗)) = (p∗ 1 −p1) · (z1(p) −z1(p∗)) > 0 The first equality uses Walras’ Law p·z(p) = 0 and the fact that p∗is an equilibrium so z(p∗) = 0. The second uses the price normalization. The final inequality follows because by the gross substitute property, p1 > p∗ 1 implies z1(p) < z1(p∗) and similarly p1 < p∗ 1 implies z1(p) > z1(p∗). Q.E.D. This Lemma is similar to the weak axiom of revealed preference that we alluded to in the first half of the class. Marshallian demand is said to satisfy the weak axiom of revealed preference if for any two price vectors p and p0: (p −p0) · (x(p) −x(p0)) ≤0. (4) This is a pretty strong condition and isn’t implied by the gross substitutes property (nor does it imply gross substitutes). Gross substitutes, however, does imply a version of WARP. It implies that the weak axiom holds if one compares p∗, the unique equilibrium price vector, to any other price vector p. With this Lemma, we can prove the following result about stability. Proposition 11 Suppose that the aggregate excess demand function z(·) satisfies gross substitutes and that p is a Walrasian equilibrium price vector. Then the tatonnement adjustment process dp/dy = αz(p(t)), with α > 0, converges to the relative prices of p as t →∞for any initial condition p(0). Proof. To prove the result, we show that the “distance” between p(t) and p∗ decreases monotonically as time progresses. Let D(p) = 1 2 P l(pl −p∗ l )2 denote the 34 distance between p and p∗. Then: dD(p(t)) dt = X l (pl(t) −p∗ l )dpl(t) dt = α X l (pl(t) −p∗ l )zl(p(t)) = −αp∗· z(p) ≤0. Note the use of Walras Law in deriving the third equality. The last inequality is strict unless p is proportional to p∗. Now, because D(p(t)) is decreasing monotoni-cally over time, it must converge, either to zero or to some positive number. In the former case, p(t) →p∗. In the latter case, p(t) 9 p∗but dD(p(t))/dt →0. The only way this can happen is if p(t) becomes nearly proportional to p∗as t →∞. But this means that the relative prices of p(t) converge to those of p∗as t →∞. Q.E.D. Finally, economies with gross substitutes have nice comparative statics prop-erties. In particular, any change that raises the excess demand for good k will increase the equilibrium price of that good. As an example, suppose there are two goods, and normalize p2 = 1. Suppose also that good 1 is a normal good for all agents. Now consider an increase in the endowment of the numeraire good 2 for some of the agents. For any price p1, this change will increase aggregate demand for good 1 and hence increase aggregate excess demand. As shown in Figure 10, this will shift up the excess demand curve z1(·, 1) – in the figure the original excess demand curve is denoted by z1(·, 1; L); the new excess demand curve is z1(·, 1; H). Because z1(·, 1; L) is continuous and crosses zero only once (remember equilibrium is unique), the new equilibrium must have a higher price for good 1. 35 0 z1 p1 z1(p1,1;H) pH pL z1(p1,1;L) Figure 10: Comparative Statics with Gross Substitutes This argument can be generalized to the many commodity case – you may see something like this on a problem set. 9 Production in General Equilibrium Everything we have done so far has been for the special case of an exchange economy where goods simply come from nowhere as endowments. Fortunately, it’s pretty easy to incorporate firms and production into our general equilibrium model, so long as we assume: (1) no increasing returns to scale; and (2) perfectly competitive price-taking firms. In this section, we outline the more general Arrow-Debreu model with pro-duction, revisit the welfare theorems and equilibrium existence, and then consider some simple examples. 9.1 Adding Production to the Model We retain the consumers i = 1, ..., I of the earlier model, with utility functions u1, ..., uI. We add K firms k ∈K with production sets Y k ∈RN. Each Y k is a set 36 of production plans: if y ∈Y k, then yl < 0 means good l is being used as an input; yl > 0 means good l is being produced as an output. The firms are owned by the households. We let αki denote i’s share of firm k. A production economy is then: E = ¡ (ui, ei, (αki)k∈K)i∈I, (Y k)k∈K ¢ . Firm k takes prices p ∈RN as given and choose a production plan yk ∈Y k to solve: max y∈Y k p · y. Definition 6 A Walrasian equilibrium is a vector (p, (xi)i∈I, (yk)k∈K) such that 1. Firms maximize profits: for all k ∈K, yk ∈arg max y∈Y k p · y 2. Consumers maximize utility: for all i ∈I, xi ∈ arg max x ui(x) s.t. p · (x −ei) −p · X k∈K αkiyk ≤0 3. Markets clear: X i∈I (xi −ei) − X k∈K yk = 0. 9.2 Assumptions about Production We’ll want to make some assumptions on Y k to ensure that an equilibrium exists with production. The simplest such assumption is that Y k is convex and com-pact for all firms k, but it seems unreasonable to assume that a production set is bounded. Instead, we assume: (A5) For all firms k ∈K, Y k is closed and convex. 37 (A6) For all firms k ∈K, 0 ∈Y k and RN −−⊂Y k. Note that these assumptions rule out increasing returns to scale. If y ∈Y k, then so is βY k for any 0 < β < 1. So it is always possible to “scale” down production or break it up into arbitrarily small productive units. We need one further assumption to ensure that firms cannot cooperate in a clever way and produce an infinite amount of goods – i.e. to ensure that one firm doesn’t produce 1 pound of iron into 1 pound of steel, while another firm produces 2 pounds of iron from that 1 pound of steel. Debreu (1959) makes an assumption directly on the aggregate production possibilities: (A7) If Y = P k∈K Y k, then Y ∩−Y = {0}. Think about why this rules out the above story. With these assumptions in place, our earlier welfare and existence results carry through. 9.3 Efficiency and Existence The definition of feasibility and Pareto efficiency carry through immediately to the case of production. Definition 7 An allocation and production plan ((xi)i∈I, (yk)k∈K) is feasible if P i∈I(xi −ei) −P k∈K yk ≤0. Definition 8 A feasible allocation and production plan ((xi)i∈I, (yk)k∈K) is Pareto efficient if there is no other feasible allocation and production plan ((ˆ xi)i∈I, (ˆ yk)k∈K) satisfying ui(ˆ xi) ≥ui(xi) for all i, with strict inequality for at least one i0. We now state the two welfare theorems. Theorem 12 (First Welfare Theorem) Assume E is a production economy that satisfies (A2). If (p, (xi)i∈I, (yk)k∈K) is a Walrasian equilibrium for E, then ((xi)i∈I, (yk)k∈K) is Pareto efficient. The proof is virtually identical to the exchange case. Try to replicate it on your own. 38 Theorem 13 (Second Welfare Theorem) Assume utility functions and pro-duction sets satisfy (A2)—(A5) and that ((xi)i∈I, (yk)k∈K) is a Pareto efficient allo-cation. Suppose xi À 0 for all i ∈I. Then there is a price vector p > 0, ownership shares (αki)i∈I,k∈K, and endowments (ei)i∈I such that (p, (xi)i∈I, (yk)k∈K) is a Wal-rasian equilibrium given these endowments and ownership shares. The proof again relies on the Separating Hyperplane Theorem; you can check it out in MWG. Note that the key assumption is convexity of the production possibility sets. This is what enables us to find a separating hyperplane between the set of feasible production plans and the aggregate “better than” set. One then shows that the separating hyperplane is a supporting price vector. What about equilibrium existence? If we impose all three of the Assumptions above, we’re in good shape. Theorem 14 (Existence of Equilibrium) Assume E is a production economy satisfying (A1)—(A7). Then there exists a Walrasian equilibrium of E. 9.4 Linear Activity Analysis If we are modeling production, we not only have to pick utility functions but also production sets or production functions. A simple case is the so called ‘linear activity model’ of production. In this model, all production sets are convex cones spanned by finitely many rays. In particular, there is only one firm (this actually won’t make any difference – see below). The firm has access to M linear activities am ∈M ⊂RL. It can operate each activity at some level γ ≥0. The production set Y is the convex hull of these activities, Y = {y ∈RL : y = M X m=1 γmam for some γ ∈RM + }. Our assumption of free disposal is satisfied if the vectors (−1, 0, . . . 0), (0, −1, 0, . . . 0), . . . (0, . . . , 0, −1) are all in M. 39 Figure 11 shows the special case of 4 activities and 2 goods. There are two productive activities: activity 1 allows 2 units of good 2 to be converted into 1 unit of good 1. Activity 2 allows 3 units of good 1 to be converted into 1 unit of good 2. Also there are two “free disposal activities”. Therefore: M = {(1, −2), (−3, 1), (0, −1), (−1, 0)}. x1 x2 0 Y Figure 11: Activity Analysis Model In the activity analysis model, given a price vector p, a profit maximizing production plan exists if and only if p · am ≤0 for all m = 1, ..., M. If p · am > 0 for some m = 1, ..., M, the firm could choose γm →∞and make infinite profits. Also, if p · am < 0 for some m it is clear that the optimal γm = 0. This simple observation already tells us a lot about what kind of prices could potentially be equilibrium prices. Indeed, in many cases the equilibrium prices will just be determined by the zero-profit conditions, with utility maximization and market clearing pinning down the levels at which the activities are operated. An important thing to note is that if all production sets are of this simple linear form, firms do not play a role at all. As there will never be any equilibrium 40 profits, what matters is just the aggregate production set. Whether we interpret each activity as a separate firm or we assume that one firm owns all the activities, or even that several firms operate different sets of overlapping activities makes no difference as long as we stay in our competitive paradigm.3 This constant returns property is shared by the Cobb-Douglas production model that you have probably seen a lot in macroeconomics. 9.5 An Example with Numbers To make things really concrete, let’s consider an example with numbers. Suppose there are two agents and three goods. The agents have identical utility functions: ui(x) = log(x1) + log(x2) + log(x3) Endowments are e1 = (1, 2, 3) and e2 = (2, 2, 2). Suppose that there are two activities a1 = (2, −1, 0.5) and a2 = (0, 1, −1). What does the Walrasian equilibrium look like? Let’s normalize p3 = 1. Now, if activity 2 is used in equilibrium, it must be the case that (by zero profit) p2 = 1. Similarly, if activity 1 is used in equilibrium, then p1 = 0.25. These prices are upper bounds on the equilibrium prices if these activities are not used in equilibrium. Let’s see if we can find an equilibrium where both activities are used. Given prices p = (0.25, 1, 1), we solve the utiliy maximization problem for agent i. This gives us: 1 p1xi 1 = 1 p2xi 2 = 1 p3xi 3 and X l plxi l = X l plei l Plugging in our price vector and the endowments, we have: 4 xi 1 = 1 xi 2 = 1 xi 3 and 1 4xi 1 + xi 2 + xi 3 = wi 3In fact, we can even assume that each agent performs an activity or two himself (household production). 41 where w1 = 5.25 and w2 = 4.5. Therefore: x1 = (7, 1.75, 1.75) x2 = (6, 1.5, 1.5) Therefore aggregate demand is (13, 3.25, 3.25). The aggregate endowments are (3, 4, 5), so the only way we can have market clearing is if the aggregate production is (10, −0.75, −1.75). This isn’t a problem. The firm will simply operate activity 1 at a level γ1 = 5 and operate activity 2 at a level γ2 = 4.25. 10 General Equilibrium with Uncertainty Our goal in this last section is to introduce time and uncertainty into the basic model. Introducing uncertainty allows a role for financial markets. We first discuss the basic framework, then look at a model with financial markets and a single consumption good. In the context of this simple model, we consider what it means for there to be an absence of arbitrage possibilities. We also look at why the first welfare theorem can fail if there are too few financial securities. 10.1 Modeling Uncertainty and Time Among the many simplifications of the Arrow-Debreu model we have studied so far is that it’s essentially a static model with no uncertainty at all. Ideally, we’d like to include both time and uncertainty into our model of competitive trade. Introducing time into the model isn’t too hard. A tomato in summer is a different good than a tomato in winter. So perhaps we can just think about a commodity as being identified not only by its physical characteristics but also by its date. Uncertainty seems more complicated, but a brilliant modelling innovation of Arrow (1953) comes to the rescue. Arrow’s insight was to introduce “states of the world” along the lines of Savage’s decision theory. A state of the world is a complete description of a date-event. Unlike in Savage, however, we’re going to assume that these states aren’t personal and subjective; instead everyone somehow agrees on the possible states (there could be a lot). People don’t have to agree on 42 the probabilities of the states occurring, though that is often assumed. We now think about the general model as having a finite number of time peri-ods. In each period there is a set of possible states and there can be uncertainty about what state will arise at date t + 1 – the probabilities can even depend on what state was realized at date t. With these ideas in mind, we can think about re-interpreting our Walrasian model as follows. We model uncertainty as an event tree with S nodes, ξ ∈Ξ. We done a node’s predessor by ξ−and its set of successors by Υ(ξ). At each t we summarize the nodes in this period in a set Nt. We denote the root node by 0. This is pictured in Figure 12. ξ-ξ Υ(ξ) N1 Date 0 Date 1 Date 2 Figure 12: An Event Tree There are L commodities at every node so the total number of commodities is SL. There are I agents. Each has an endowment ei ∈RSL ++. Agent i’s consumption set is RSL + and his utility function is ui : RSL + →R. The utility function may or may not satisfy the von-Neumann Morgenstern axioms. We define a Walrasian equilibrium exactly as before: a set of prices and resulting allocation such that (i) all agents maximize utility given prices; and (ii) markets 43 clear. The idea here is that all trades take place at date zero and there is no re-trading in later period. Under our earlier assumptions on preferences, a Walrasian equilibrium exists, and the Welfare Theorems hold. This is the model in chapter 7 of Debreu’s Theory of Value. As Debreu puts it: “A contract for the transfer of commodities now specifies, in addition to its physical properties, its location and date, an event on the occurrence of which the transfer is conditional. This new definition of commodity allows one to obtain a theory of uncertainty free from any probability concepts and formally identical to the theory of certainty.” This is quite elegant, but as Arrow originally pointed out, it seems unrealistic that all these contingent trades would occur at date 0. Instead, what tends to happen is that there are financial securities that are traded on exchanges and some of these securities pay out contingent on certain events (e.g. hurricane insurance pays out contingent on there being a hurricane; stocks pay dividends contingent on company performance). Arrow cleverly reformulated the model as follows. Assume at each node ξ there are spot markets for the L commodities at that node. Assume that these commodities have prices p(ξ). At node 0 there are now (S −1) Arrow securities (i.e. one for each future node), where an Arrow security ξ pays one unit of good one at node ξ. An equilibrium is now defined as utility maximization and market clearing at each node and in the S −1 markets for Arrow securities at date 0. The amazing result is that even though there are only S −1 securities in the economy, the Arrow-Debreu allocation obtains. The result isn’t even that hard to prove, though we won’t do it now. The next question that arose (in a paper by Radner, 1972) was the following: what happens if there are securities that pay out in future contingencies (like stock in different companies), but not a complete set of Arrow-Debreu securities. This makes for arguably a more realistic model of actual securities markets. This question has given rise to a large “incomplete markets” literature in economics and finance. One of the interesting results from this literature is that without a complete set of A-D securities, the first welfare theorem generally won’t hold. So there is potentially room for government intervention and policy questions become 44 interesting. 10.2 A Simple Finance Model In this section we introduce and study what is just about the simplest general equilibrium model with uncertainty. We assume there are two periods and in each state of the world there is just one consumption good. We normalize the spot price in each period to be equal to one. We assume there are S + 1 states of the economy. At time t = 0 the economy is in state s = 0; at time t = 1 the economy can be in one of S possible states. In each state s = 0, ..., S there is a single perishable consumption good. Each agent i ∈I has an initial endowment ei = (ei 0, ..., ei S) ∈RS+1 ++ and has a utility function ui : RS+1 + →R over consumption bundles ci = (ci 0, ..., ci S) ∈RS+1 + . We asume each agent’s utility function satisfies the standard assumptions – it’s increasing, continuous and strictly concave. Also, let’s define x = (x1, ..., xS) as the t = 1 part of the vector x = (x0, x1, ..., xS). The aggregate endowment is e = P i∈I ei. There are J assets or securities. Each asset j pays dividends at date t = 1 which we denote by dj ∈RS. The price of asset j at time t = 0 is qj. Without loss of generality we assume that these assets are in zero net supply (if we wanted the assets to be stock in some firm, there would be positive net supply, but then we could put the dividends into agent’s endowments and be back to zero net supply). We collect all assets’ dividends in the matrix: A = (d1, ..., dJ) ∈RS×J At time t = 0, each agent i chooses a portfolio αi ∈RJ, where αi j is the amount of asset j held by agent i. An agent’s portfolio uniquely defines his wealth at each time one state, and hence his consumption (recall that prices are normalized to one at each date one state): xi = ei + Aαi and xi 0 = ei 0 −αi · q. The net demand of each agent xi −ei belongs to the span of the asset payoffmatrix A : hAi = {z ∈RS : ∃α ∈RJ s.t. z = Aα}. 45 A finance economy is hence a triple: E = ((ui, ei)i∈I, A). Without loss of generality, we can assume that rank(A) = J so there are no redundant assets. With redundant assets, an arbitrage argument would imply that the price of some assets would be uniquely determined by the price of other assets, regardless of preferences. We say that markets are incomplete if J < S. Asset prices are said to be arbitrage-free if it is not possible to achieve a positive income stream in all states by trading at the going prices, i.e. if there is no position α ∈RJ with qα ≤0 and Aα ≥0 with one inequality being strict. Here qα is the cost of portfolio α at date 0 and Aα is the vector of payoffs at different date one staes. No arbitrage means you can’t guarantee positive future income tomorrow without making a positive investment today. If agents have strictly increasing utility functions, asset prices must preclude arbitrage or there would be a real problem with utility maximization. The absence of arbitrage is thus often seen the fundamental concept in finance (more so than equilibrium). Many important concepts (such as Black-Scholes option pricing) rely solely on arbitrage arguments. Theorem 15 An asset price vector q ∈RJ precludes arbitrage if and only if there exists a state price vector π ∈RS ++ such that q = π0 · A. Proof. Let M = {(−qα, Aα) : α ∈RJ} be the marketed subspace of RS+1. That is, (−x0, x) ∈M means that by spending x0 at date 0, an agent can ensure the vector of returns x at date one. There is no arbitrage if and only if RS+1 + ∩M = {0}. If (x0, x) ∈M and x0 ≥0, x ≥0 with a strict inequality (so (x0, x) ∈RS+1 + −{0}, it would be possible to start with zero wealth, consume x0 today and consume x tomorrow – i.e. arbitrage would be possible. For one direction of the proof, suppose there exists a strictly positive state price vector π ∈RS ++ such that q = π0A. We show that this means there is no arbitrage. If there were also a vector x ∈RS+1 + ∩M with x 6= 0, then because x ∈RS+1 + and π ∈RS ++, we have (1, π) · x > 0. But by the fact that x ∈M and q = π0A, we also have (1, π) · x = −qα + qα = 0, a contradiction. Hence, a strictly positive state price vector implies no arbitrage. For the converse direction, suppose no arbitrage: RS+1 + ∩M = {0}. We use the separating hyperplane to derive a supporting state price vector. Note that M and 46 RS+1 + are both convex sets whose intersection includes only the point {0}. The SHT asserts the existence of a vector μ 6= 0 such that μ · x < μ · z for all x ∈M and all non-zero z ∈RS+1 + .4 Now, by the definition of M, it must be the case that if x ∈M then −x ∈M ,so we must have μ · x = 0 for all x ∈M. Therefore μ · z > 0. The latter implies that μ À 0 (if μl ≤0 for some l, we could find z ∈RS+1 + −{0} with zl > 0 and zk = 0 leading to the contradiction μ·z ≤0). Therefore −μ1q+(μ2, ..., μS+1)A = 0 and πs = μs+1/μ1 will give us a state price vector (note that to form π we just normalize the prices – μ has the right relative prices). Q.E.D. A lot of asset pricing theory has to do with finding the right state-price vector π. Its existence is ensured by the absence of arbitrage, but often little can be said about it in general models.5 Definition 9 A financial markets equilibrium for a finance economy E is a collection of portfolios α∗= (α1∗, ..., αI∗) ∈RIJ, individual consumptions (xi)i∈I and prices q∗∈RJ such that: 1. Agents maximize utility: (xi, αi∗) ∈ arg max αi∈RJ,ci∈RS+1 + ui(ci) s.t. ci = ei + µ−q∗0 A ¶ αi 2. Markets clear: X i∈I αi∗= 0 Clearly any equilibrium price vector must preclude arbitrage for the maximiza-tion problem to have a well-defined solution. Indeed, we can infer state prices from 4This is a slightly different version of the SHT than we used to prove the Second Welfare Theorem. There, we use the SHT to separate a convex set from a point outside that set. Here we are separating two disjoint convex sets M and RS+1 + −{0}. The idea is the same (you can check MWG’s math appendix for a statement of both results). 5In dynamic models the state-price vector is sometimes called the pricing kernel, or the equiv-alent martingale measure (if normalized to add up to one). 47 the agents’ first order conditions: πs = ∂ui(xi) ∂xi s If J = S (remember the assets dividends are assumed to be linearly indepen-dent), then a financial markets equilibrium is equivalent to a Walrasian equilibrium. There will be a unique state-price vector π ∈RS ++ such that q = π0A. This will be an equilibrium price vector for a Walrasian economy; the resulting allocations are the same in the two equilibria. More interesting is the case where J < S so that markets are incomplete. Under our assumptions, a financial markets equilibrium will still exist, but the equilibrium allocation may not be efficient. To see why, let’s look at an example. Suppose there are two states and there is a single bond that pays 1 in each state: d = (1, 1)0. Suppose there are two agents with endowments: e1 = (1, 2, 1) e2 = (1, 1, 2) and that both agents have identical utility: ui(x0, x1, x2) = log x0 + 1 2 log x1 + 1 2 log x2 You can check as an exercise that the unique equilibrium will have no trade in the bond so everyone will just consume there endowment. This allocation, however, is Pareto dominated by the feasible allocation x1 = x2 = (1, 1.5, 1.5). The first welfare theorem fails because the set of existing securities does not allow the agents to suitably insure themselves against adverse states. There is still as sense, however, in which equilibrium exhausts the gains from trade. Definition 10 Given endowments (ei)i∈I and assets A, an allocation (xi)i∈I is constrained efficient if P i∈I(xi −ei) ≤0, xi −ei ∈hAi for all i ∈I and there exists no alternative allocation (ˆ xi)i∈I that Pareto dominates (xi)i∈I and also satisfies P i∈I(ˆ xi −ei) ≤0 and ˆ xi −ei ∈hAi for all i ∈I. 48 If you’re interested, you can try proving the following weaker welfare theorem: Theorem 16 If utility functions are strictly increasing, a financial markets equi-librium is constrained efficient. References Arrow, Kenneth J. “An Extension of the Basic Theorems of Classical Welfare Economics,” Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, ed. J. Neyman, 1951. Arrow, Kenneth J. “The Role of Securities in the Optimal Allocation of Risk-Bearing,”, Econometrie, 1953 (also Review of Economic Studies, 1963). Arrow, Kenneth J. “General Economic Equilibrium: Purpose, Analytic Tech-niques, Collective Choice,” American Economic Review, 1973. Arrow, Kenneth J. and Gerard Debreu, “Existence of Equilibrium for a Com-petitive Economy,” Econometrica, 1954. Arrow, Kenneth J. and Leonid Hurwicz, “On the Stability of the Competitive Equilibrium, I,” Econometrica, 1958. Brown, Donald J. and Rosa Matzkin, “Testable Restrictions on the Equilib-rium Manifold,” Econometrica, 1996. Cassel, G. The Theory of Social Economy, New York: Harcourt, Brace and Company, 1924. Debreu, Gerard, “The Coefficient of Resource Utilization,” Econometrica, 1951. Debreu, Gerard, Theory of Value, 1959. Debreu, Gerard, “Excess Demand Functions,” J. Math. Econ., 1974. 49 Mantel, R. “On the Characterization of Aggregate Excess Demand,” J. Econ. Theory, 1974. Mas-Colell, A., M. Whinston, and J. Green, Microeconomic Theory, 1995. Radner, Roy, “Existence of Equilibrium of Plans, Prices and Price Expecta-tions in a Sequence of Markets,” Econometrica, 1972. Scarf, Herbert, “Some Examples of Global Instability of the Competitive Equi-librium,” International Economic Review, 1960. Sonnenschein, Hugo, “Do Walras’ Identity and Continuity Characterize the Class of Community Excess Demand Functions?” J. Econ. Theory, 1973. 50
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https://www.parabola.unsw.edu.au/sites/default/files/2024-02/vol59_no3_6.pdf
Parabola Volume 59, Issue 3 (2023) Applications of the Pigeonhole Principle in mathematics Sizhe Pan1 1 Introduction We will explore The Pigeonhole Principle, a fundamental theorem in mathematics. Simply expressed, if you were to place at least n + 1 items (“pigeons”) in only n boxes (“pigeonholes”), then at least one box will contain at least two items. You might think that this is an obvious fact - when you try to fit 6 apples in 4 lunch-boxes, it’s not possible to have only one apple in each lunchbox - and you would be correct! This paper will explore how such a seemingly simple theorem has important applications in more difficult mathematics, ranging from geometry to number theory and algebra. 2 The Pigeonhole Principle Let us state the Principle more formally: Theorem 1 (The Pigeonhole Principle). If more than n pigeons are placed in n pigeonholes, then at least one pigeonhole will contain at least two pigeons. Proof. Another way to express this Principle is as follows: if no pigeonhole has least two pigeons - that is, each pigeonhole contains at most one pigeon - then it cannot be true that more than n pigeons were placed in these n pigeonholes. That is naturally a true statement: if each of the n pigeonholes contains at most one pigeon, then there can be at most n pigeons in the n pigeonholes. 2 Now, if you try to place 9 apples into 4 lunch pigeonholes, then by the Pigeonhole Principle, some lunch pigeonhole will contain at least two apples. What’s even more cool is that some lunch pigeonhole will also contain at least three apples! Why? Theorem 2 (The (General) Pigeonhole Principle). If more than mn pigeons are placed in n pigeonholes, then at least one pigeonhole will contain more than m pigeons. Proof. If each of the n pigeonholes contains at most m pigeons, then there can be at most mn pigeons. This simple observation is another way to express the theorem. 2 1Sizhe Pan is currently a Year 12 high school student attending James Ruse Agricultural High School. 1 3 Applications to problem solving Now let’s see how the Pigeonhole Principle can be used to solve problems. Example 3. Prove that, if we pick 6 different numbers from {1, 2, . . . , 10}, then we can choose two of them such that they add up to 11. Proof. You may ask: We are picking numbers, but how does the Pigeonhole Principle apply here? Let 5 “pigeonholes” be the sets {1, 10}, {2, 9}, {3, 8}, {4, 7}, {5, 6}, respectively. If we pick 6 different numbers - “pigeons” - from {1, 2, . . . , 10}, then there are more pigeons than pigeonholes, so, by the Pigeonhole Principle, some pigeonhole must contain at least two pigeons. That is, at least one the five sets contains two of the picked numbers. Whichever set this is, we have two numbers that add up to 11! 2 Let’s now look at a geometry example. Example 4. Given five points on the interior of a square with side length 2, prove that two of the points at distance less than 1.5 apart. Bonus question: What is the smallest real constant you can replace 1.5 with so that the state-ment is still true? Proof. The crux of this problem lies in how to set up the Pigeonhole Principle. What are the pigeons, and what are the pigeonholes? After trying to keep the points as far as possible from each other, we notice that this intuitively occurs when four points lie in the four corners and the fifth lies in the middle of the square: Here, the distances are just under the diagonal distance of a unit square, namely √ 2 ≈1.414 which is less than 1.5! Let us prove this rigorously. 2 Divide the 2 × 2 square into four 1 × 1 squares: Since there are five points and only four 1×1 squares, at least one of these squares will contain two points. These two points have distance at most √ 2 < 1.5. 2 4 Using the Pigeonhole Principle to obtain information We now move onto a more complex application of the Pigeonhole Principle. Example 5 (Based on 2021 Bored of Studies Mathematics Extension 2 Exam). Given any 7 real numbers, prove that at least two of them, x, y, satisfy x −y 1 + xy < 1 √ 3 . (1) Proof. While this problem seems difficult to approach at first, recall the tangent com-pound angle formula: tan(A −B) = tan A −tan B 1 + tan A tan B . (2) Also, recall that each real number x can be written as tan A for some A ∈ −π 2, π 2  . −1.5 −1 −0.5 0.5 1 1.5 −10 −5 5 10 x y 3 Now let our 7 real numbers be tan α1, tan α2, . . . , tan α7 where α1, α2, . . . , α7 ∈ −π 2, π 2  . We need to show that two of these seven numbers, say x = tan αi and y = tan αj, satisfy the inequality (1). By the identity (2), the inequality (1) can be expressed as tan(αi −αj) < 1 √ 3 . Since tan π 6 = 1 √ 3 and tan −π 6  = −1 √ 3, and tan x is a strictly increasing function when x ∈ −π 2, π 2  , we can further re-express (1) as −π 6 < αi −αj < π 6 or, more simply, αi −αj < π 6 This gives us an idea of how to apply the Pigeonhole Principle: There are 7 numbers in the interval −π 2, π 2  which has length π, and we need the difference of two numbers to be less than π 6, exactly 1 6 of the interval length. If we choose the “pigeonholes” to be the six intervals −3π 6 , −2π 6 , −2π 6 , −π 6 , . . . , 2π 6 , 3π 6 , then by the Pigeonhole Principle, one of the pigeonholes will have two of the numbers, αi, αj. Thus, |αi −αj| < π 6, so tan αj −tan αi 1 + tan αi tan αj = tan(αj −αi) < tan π 6 = 1 √ 3 , as needed. 2 We can see from this example that, while the Pigeonhole Principle might not have obvious use in a problem when you see it for the first time, it often appears after we perform simplifications. We can in fact generalise the above problem, as follows. Example 6. For each positive integer n, find the smallest positive real constant Cn such that, for any set of n + 1 distinct real numbers {x1, x2, . . . , xn+1}, there are at least two of them, say x = xi and y = xj with i ̸= j, such that x −y 1 + xy < Cn . Proof. Write xi = tan αi where αi ∈ −π 2, π 2  for i = 1, 2, . . . , n + 1. The n intervals h −π 2 + (k−1)π n , −π 2 + kπ n  for k = 1, 2, . . . , n partition the interval −π 2, π 2  . By the Pi-geonhole Principle, at least one of the intervals must contain at least two of the n + 1 numbers α1, α2, . . . , αn, say αi and αj. Then |αi −αj| < π n, so tan αj −tan αi 1 + tan αi tan αj = |tan(αj −αi)| < tan π n . Therefore, Cn = tan π n always works! We will show that no smaller Cn works. 4 Assume that some Cn < tan π n works, and write Cn = tan π n −ε  where 0 < ε < π n . For each i = 1, 2, . . . , n + 1, define xi = tan αi where α1 = −π 2 + ε , αi = −π 2 + (k −1)π n for i = 2, . . . , n , and αn+1 = π 2 −ε . By assumption, we can find distinct x = xi and y = xj such that tan αj −tan αi 1 + tan αi tan αj < Cn . Thus, tan|αi −αj| < Cn = tan π n −ε  so |αi −αj| < π n −ε . But, by definition, the distance between αi and αj is at least π n −ε or π n, both of which are greater than π n −ε, a contradiction! Therefore, our assumption that some Cn < tan π n works, so the minimum possible Cn is tan π n. 2 Example 7 (Serbian Mathematical Olympiad 2016, by Duˇ san Djuki´ c). Suppose a1, a2, . . . , a22016 are positive integers such that, for all n with 1 ≤n ≤22016, an ≤2016 and a1a2 · · · an + 1 is a perfect square. Prove that at least one of the numbers a1, a2, . . . , a22016 must be equal to 1. Proof. For each n = 1, 2, . . . 22016, write a1a2 · · · an = s2 n−1 where sn is a positive integer. Next, write the prime factorisation of a1a2 · · · an as pb1(n) 1 pb2(n) 2 · · · pbN(n) N where p1, p2, . . . , pN are the prime numbers less than 2016 and b1(n), b2(n), . . . , bN(n) are non-negative integers. Define the function f : N →{0, 1}N by f(n) = c1(n), c2(n), . . . , cN(n)  where ci(n) = 0 if bi(n) is even, and ci(n) = 1 if bi(n) is odd. Since each ci can be either 0 or 1 - two choices - and there are N places to choose, the number of possibilities for f(n) is 2N. Consider the last 2N + 1 products a1a2 · · · an, for n = 22016 −2N, 22016 −2N + 1, . . . , 22016. By the Pigeonhole Principle, we can find at least two values f(t) and f(u) that are identical where 22016 −2N ≤t < u ≤22016. Since f(u) = c1(u), c2(u), . . . , cN(u)  and f(t) = c1(t), c2(t), . . . , cN(t)  , 5 are the same, we see that ci(t) = ci(u) for all i = 1, 2, . . . , N. By definition, this means that bi(t) and bi(u) are both even or are both odd; therefore, bi(u) −bi(t) is even for all i = 1, 2, . . . , N. We can therefore write bi(u) −bi(t) = 2ki for some integer ki; then at+1 · · · au = a1a2 · · · au a1a2 · · · at = pb1(u) 1 pb2(u) 2 · · · pbN(u) N pb1(t) 1 pb2(t) 2 · · · pbN(t) N = pb1(u)−b1(t) 1 pb2(u)−b2(t) 2 · · · pbN(u)−bN(t) N = p2k1 1 p2k2 2 · · · p2kN N = pk1 1 pk2 2 · · · pkN N 2 . Since at+1 · · · au is an integer, each ki is a non-negative integer, and so at+1 . . . au is a perfect square! How does this help? Define a = su , b = st and c = pk1 1 pk2 2 · · · pkN N ; then a2 −1 b2 −1 = s2 u −1 s2 t −1 = a1a2 · · · au a1a2 · · · at = at+1 · · · au = c2 . To complete the proof, let us assume that the statement to be proved is wrong: that is, let us assume that each of the numbers a1, a2, . . . , a22016 is greater than 1. Then c > 1, so a2 = (b2 −1)c2 + 1 = b2c2 −c2 + 1 < (bc)2 . Therefore, bc > a, so bc ≥a + 1. Hence, c2 −1 = b2c2 −a2 ≥(a + 1)2 −a2 = 2a + 1 > a > √ a2 −1 = √a1a2 · · · au ≥ √ 2u > 222015−2N−1 . However, u −t ≤2N, so c2 −1 = at+1 · · · au −1 < 2016u−t < 20482N = 211×2N Therefore, 222015−2N−1 < c2 −1 < 211×2N so 22015 −2N−1 < 11 × 2N . It follows that 22015 < 23 × 2N−1 < 32 × 2N−1 = 2N+4 . Thus, 2015 < N +4, so N > 2011. But no even number except for 2 is prime, eliminating 4, 6, . . . , 2014 from being prime. Since N is the number of primes at most 2016, this gives N ≤2016 2 + 1 = 1009, so 2011 < N ≤1009, a contradiction! Thus, our assumption is incorrect, and an = 1 for some n ∈{1, 2, . . . , 2016}. 2 6 Acknowledgements I would like to thank Thomas Britz for giving me valuable advice on the article over the last few weeks and warmly supporting me to pursue this idea initially. I would also like to thank Tenebeanz and Hypotrochoid 2.0 of the ConquerHSC discord for giving me inspiration for generalising Example 5. 7
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https://artofproblemsolving.com/wiki/index.php/Elementary_symmetric_sum?srsltid=AfmBOooNQJ7W0OM-x9LFLDQ5Ln1LJfMjvZPJztB-7BQX1JvzOpdftr_h
Art of Problem Solving Elementary symmetric sum - 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 Elementary symmetric sum Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Elementary symmetric sum An elementary symmetric sum is a type of summation. Contents 1 Definition 2 Notation 3 Uses 4 See Also Definition The -th elementary symmetric sum of a set of numbers is the sum of all products of of those numbers (). For example, if , and our set of numbers is , then: 1st Symmetric Sum = 2nd Symmetric Sum = 3rd Symmetric Sum = 4th Symmetric Sum = Notation The first elementary symmetric sum of is often written . The th can be written Uses Any symmetric sum can be written as a polynomial of the elementary symmetric sum functions. For example, . This is often used to solve systems of equations involving sums of powers, combined with Vieta's formulas. Elementary symmetric sums show up in Vieta's formulas. In a monic polynomial of degree , the coefficient of the term is , and the coefficient of the term is , where the symmetric sums are taken over the roots of the polynomial. See Also Symmetric sum Cyclic sum Retrieved from " Categories: Algebra Definition 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.
6048
https://economics.ucr.edu/repec/ucr/wpaper/201704.pdf
Macroeconomic Stability under Balanced-Budget Rules and No-Income-E¤ect Preferences Jang-Ting Guo University of California, Riversidey Yan Zhangz Shanghai University of Finance and Economics April 17, 2017 Abstract It has been shown that under an additively separable preference formulation between consumption and hours worked, indeterminacy and sunspots may arise in a standard one-sector real business cycle model when the labor tax rate is endogenously determined by a balanced-budget rule with a pre-speci…ed constant level of government expenditures. This paper …nds that local indeterminacy disappears if the period utility function is postulated to exhibit no income e¤ect on the household’s demand for leisure. In particular, the model’s low-tax steady state always displays saddle-path stability and equilibrium uniqueness; whereas the high-tax steady state is either a source or a saddle point. Keywords: Income E¤ect; Balanced-Budget Rules; Indeterminacy; Business Cycles. JEL Classi…cation: E32; E62. We thank Jess Benhabib and Yan Chen for helpful discussions and comments. Part of this research was conducted while Guo was a visiting research fellow at the Institute of Economics, Academia Sinica, whose hospitality is greatly appreciated. Of course, all remaining errors are our own. yCorresponding Author: Department of Economics, 3133 Sproul Hall, University of California, Riverside, CA, 92521, USA; Phone: 1-951-827-1588, Fax: 1-951-827-5685, E-mail: guojt@ucr.edu. zSchool of Public Economics and Administration, Fenghuang Building Room 508, Shanghai University of Finance and Economics, Wuchuan Road 111, Shanghai, China; Phone: 86-21-35120824, Fax: 86-21-65104294, E-mail: laurencezhang@live.cn. 1 Introduction Under the assumptions of perfect competition and constant returns-to-scale in production, the standard one-sector real business cycle (RBC) model exhibits an interior steady state that is a locally determinate or isolated saddle point around which there exists a unique convergent rational expectations equilibrium trajectory. In this economy with an additively separable utility function that is logarithmic in consumption and in…nitely elastic in hours worked, Schmitt-Grohé and Uribe (1997, section II) analytically examine the macroeconomic (in)stability e¤ects of a balanced-budget rule whereby constant government expenditures are …nanced by distortionary taxation on the household’s labor income. Given the postulated …scal speci…cation, a perfect-foresight La¤er curve-type relationship between the labor tax rate and the resulting tax revenue ensues – the model possesses two interior steady states when the pre-speci…ed level of public spending is lower than the revenue-maximizing counterpart. In this case, these authors derive the necessary and su¢cient condition under which the low-tax steady state is an indeterminate sink that can be exploited to yield cyclical ‡uctuations driven by agents’ animal spirits or sunspots.1 When the representative household becomes optimistic about the future of the economy and decides to work harder and invest more, the government is forced to decrease the labor tax rate as total output rises. This countercyclical tax policy helps ful…ll agents’ initial optimism, and thus destabilizes the macroeconomy by generating endogenous business cycles. On the other hand, the model’s high-tax steady state is always a saddle point, hence no aggregate ‡uctuations will take place in its neighborhood. In this paper, we extend Schmitt-Grohé and Uribe’s theoretical analysis by considering a di¤erent preference formulation that is commonly adopted in the real business cycle litera-ture. As in Greenwood, Hercowitz and Hu¤man (GHH, 1988), the period utility function is postulated to exhibit no income e¤ect associated with the household’s labor supply decision. With this speci…cation of non-separable preferences, the relationship between the (…xed) level of government spending and the labor tax rate is also characterized by a La¤er curve that may possess two interior steady states. We show that in shape contrast to Schmitt-Grohé and Uribe (1997), equilibrium indeterminacy disappears in our model economy because neither steady state can be a sink. In particular, saddle-path stability arises when the steady-state tax rate is (i) lower than that maximizes the tax revenue or (ii) higher than a certain threshold value. Intuitively, in order for stationary sunspot equilibria to occur within a dynamic general equilibrium macroeconomic model, the consumption Euler equation must continue to hold in 1See Benhabib and Farmer (1999) for other mechanisms that may yield indeterminacy and sunspots within various real business cycle models. 1 response to a change in non-fundamental expectations. Therefore, upon the anticipation of a higher rate of return on today’s investment, agents will consume and work more next pe-riod. As it turns out, this optimism cannot be self-ful…lled under either circumstance since an increase in labor hours large enough to raise the after-tax marginal product of capital gener-ates an unsustainable decrease in the household’s intertemporal marginal rate of substitution between current versus future consumption expenditures. Furthermore, we …nd that the econ-omy’s high-tax steady state becomes a totally unstable source when the stationary-equilibrium tax rate falls within the remaining feasible range. In sum, our analysis illustrates the critical importance of income e¤ect on the representative household’s demand for leisure in generating Schmitt-Grohé and Uribe’s instability result. In the context of a one-sector RBC model with labor income taxation, Abad et al. (2017) investigate the interrelations between local stability of competitive equilibria and Schmitt-Grohé and Uribe’s balanced-budget rule under a generalized constant returns-to-scale pro-duction technology and two classes of non-separable utility functions that subsume the GHH speci…cation. Although these authors state the no-indeterminacy result with the no-income-e¤ect preference formulation (Proposition 3, p. 267), they do not o¤er the underlying economic intuition. Moreover, unlike Schmitt-Grohé and Uribe (1997), they do not explore the possibil-ity of multiple stationary equilibria caused by the presence of a La¤er curve. Here, we examine the equilibrium dynamics associated with each interior steady state, and also provide intuitive explanations. The remainder of this paper is organized as follows. Section 2 presents the model and discusses its equilibrium conditions. Section 3 analyzes the economy’s local dynamics under perfect foresight. Section 4 concludes. 2 The Economy This paper incorporates a no-income-e¤ect preference formulation, as in Greenwood, Hercowitz and Hu¤man (1988), into Schmitt-Grohé and Uribe’s (1997, section II) one-sector real business cycle model with labor income taxation. Without loss of generality, we postulate that the economy’s output is generated by a Cobb-Douglas production technology. This simpli…cation streamlines our exposition without a¤ecting any result of the paper. 2.1 Firms The production side of the economy consists of a unit measure of identical competitive …rms. The representative …rm produces output Yt, using capital and labor as inputs, with a constant 2 returns-to-scale Cobb-Douglas production function Yt = K t H1 t ; 0 < < 1: (1) Under the assumption that factor markets are perfectly competitive, the …rm’s pro…t maxi-mization conditions are given by rt = Yt Kt ; (2) wt = (1 ) Yt Ht ; (3) where rt is the rental rate of capital and wt is the real wage rate of labor. 2.2 Households The economy is also populated by a unit measure of identical in…nitely-lived households. Each household is endowed with one unit of time and maximizes Z 1 0 et " log Ct AH1+ t 1 + !# dt; A > 0; (4) where Ct and Ht are the individual household’s consumption and hours worked,  0 denotes the inverse of the intertemporal elasticity of substitution in labor supply, and  2 (0; 1) is the subjective discount rate. We assume that there are no fundamental uncertainties present in the economy. The budget constraint faced by the representative household is given by _ Kt = (rt )Kt + (1 t)wtHt Ct; K0 > 0 given, (5) where Kt is the household’s capital stock,  2 (0; 1) is the capital depreciation rate, t is the labor-income tax rate. We require that t  0 to rule out the possibility of income subsidies which could only be …nanced by lump-sum taxation, and that t < 1 such that households have incentive to provide labor services to …rms. The …rst-order conditions for the household’s dynamic optimization problem under perfect foresight are Ct AH1+ t 1 + !1 = t; (6) AH t = (1 t) wt; (7) _ t t =  +  rt; (8) 3 lim t!1 et Kt Ct = 0; (9) where t > 0 is the Lagrange multiplier on the budget constraint (5), (7) equates the slope of the representative household’s indi¤erence curve to the after-tax real wage, (8) is the standard consumption Euler equation, and (9) is the transversality condition. Since Ct is missing in equation (7), there is no income e¤ect associated with the household’s labor supply decision. It follows that the income elasticity of intertemporal substitution in hours worked (or leisure) is zero. 2.3 Government As in Schmitt-Grohé and Uribe (1997), the government endogenously sets the distortionary tax rate on labor income t to …nance a pre-speci…ed constant level of public expenditures, and balances its budget at each point in time. Hence, the instantaneous government budget constraint is G = twtHt; (10) where G  0 denotes government spending on goods and services. Finally, the aggregate resource constraint for the economy is given by Ct + _ Kt + Kt + G = Yt: (11) 3 Analysis of Dynamics Under Schmitt-Grohé and Uribe’s …scal policy rule with countercyclical labor income taxation, the number of our model’s interior steady state(s) can be zero, one or two. Speci…cally, it is straightforward to show that the government’s tax revenue (= G) is equal to zero when the steady-state tax rate ss = 0 or 12; and that the La¤er curve-type relationship between G > 0 and ss 2 (0; 1) is given by G = ss (1 )   +  ( 1 )  1+  (1 ) (1 ss) A  1 : (12) Setting @G @ ss = 0 yields a unique steady-state tax rate = 1+ that maximizes the level of public expenditures denoted as G.3 It follows that our model possesses zero (two) interior 2When G = 0, our model collapses to a standard one-sector RBC macroeconomy with no-income-e¤ect preferences and constant returns-to-scale in production. As shown in Meng and Yip (2008) and Jaimovich (2008), this laissez-faire formulation always exhibits saddle-path stability and equilibrium uniqueness. 3When = 0, the revenue-maximizing steady state becomes degenerate with = G = 0. Accordingly, our subsequent analyses of the model’s equilibrium dynamics are restricted to cases under > 0. 4 steady states(s) provided G > (<) G, as shown in Figure 1. Therefore, any small deviation from the revenue-maximizing steady state with and G will lead to its disappearance, or the emergence of dual stationary equilibria. This result implies that the economy undergoes a saddle-node bifurcation which may cause the hard loss of equilibrium stability as the govern-ment spending passes through the critical level G. Figure 1 also shows that when G 2 (0, G), the resulting steady states in our model are characterized by ss L and ss H, where ss L < < ss H. For a given steady-state labor tax rate, the analytical expressions of all remaining endogenous variables can then be easily derived. Next, we take log-linear approximations to the model’s equilibrium conditions in a neigh-borhood of each interior steady state to obtain the following dynamical system: " _ kt : t # = J  kt t  ; k0 given, (13) where kt and t denote the log deviations of Kt and t from their respective steady-state values, and J is the Jacobian matrix of partial derivatives for the transformed dynamical system. The trace and the determinant of the Jacobian are given by Tr =  + (1 ) ( + ) ss ss + (1 ss); (14) and Det =  ss (1 ss) ss + (1 ss)   (1 ) ( + ) [(1 si) (1 + ) (1 ) (1 + ss)] si (1 + )  | {z }  ( ss) > 0 ; (15) where si  =  +  is the steady-state ratio of investment to output.4 The local stability properties of our model’s interior steady state(s) are determined by comparing the eigenvalues of J that have negative real parts to the number of initial conditions in the dynamical system (13), which is equal to one because kt is a pre-determined state variable. As a result, the steady state exhibits saddle-path stability and equilibrium uniqueness when the two eigenvalues are of opposite signs (Det < 0). If both eigenvalues have negative real parts (Tr < 0 and Det > 0), then the steady state is an indeterminate sink around which there are a continuum of stationary equilibrium trajectories that display endogenous cyclical ‡uctuations driven by agents’ animal spirits or sunspots. When both eigenvalues have positive real parts (Tr > 0 and Det > 0), the steady state becomes a totally unstable source. 4Using > 0 (see footnote 3), si 2 (0; ) and , ss 2 (0; 1), it can be shown that the bracket term in the numercator of (), given by (1 si) (1 + ) (1 ) (1 + ss) > (1 ) (1 ss) > 0. This result, together with 0 <  < 1 and  > 0, implies that ( ss) > 0. 5 In sharp contrast to Schmitt-Grohé and Uribe (1997) with an additively separable house-hold utility in consumption and labor hours, the following Proposition states that local inde-terminacy does not arise within our model under non-separable no-income-e¤ect preferences. That is, neither steady state (with ss L or ss H) can be a sink. Proposition. For a given positive level of G < G, the economy’s low-tax steady state (0 < ss L < ) is always a saddle point, whereas the high-tax steady state is either a source  < ss H < + 1+  or a saddle point  + 1+ < ss H < 1  . Proof. See the Appendix. To understand the intuition behind our no-indeterminacy result, consider the consumption Euler equation (in discrete time for ease of interpretation) as follows: Ct+1 A H1+ t+1 1+ Ct AH1+ t 1+ = [1  + (1 t+1)rt+1]; (16) where denotes the discount factor. Start the model from an interior steady state at period t, and suppose that agents become optimistic about the economy’s future. Acting upon this change in non-fundamental anticipation, the representative household will consume less and invest more today, thus Ct falls while Kt+1 rises. Due to the lack of income e¤ect, as seen in (7), Ht remains unchanged in response to the lower level of period-t consumption. In addition, a higher Kt+1 leads to (i) a decrease in rt+1 because of diminishing marginal product of capital; and (ii) an increase in Ht+1 via …rms’ labor demand function, which in turn raises the economy’s output Yt+1 as well as the household’s consumption Ct+1. Under the postulated balanced-budget constraint (10), the government is forced to reduce the labor tax rate t+1 as total income Yt+1 increases, thus (1 t+1) rises. Consequently, the change in Ht+1 will exert two counteracting e¤ects on the intertemporal Euler equation. First, the smaller (bigger) the increase in Ht+1, the bigger (smaller, or a decrease may occur) the increase in the left-hand side of (16). Second, the bigger (smaller) the increase in Ht+1, the larger (smaller, or a decrease may occur) the rise in the after-tax equilibrium real interest rate (1 t+1)rt+1. For the above-mentioned alternative dynamic path to be justi…ed as a self-ful…lling equi-librium, the household’s consumption Euler equation must continue to hold in response to agents’ rosy expectations. It turns out that the two o¤setting e¤ects, described in the previ-ous paragraph, render the equality of (16) impossible within our model. When the economy begins at the low-tax steady state with 0 < ss L <  = 1+  , a large increase in Ht+1 is needed for (1 t+1)rt+1 and the right-hand side to rise. With Ct falling and Ct+1 rising, this would in turn decrease the left-hand side. On the other hand, when the starting steady-state tax rate is high over the interval + 1+ < ss < 1, together with a small increase in Ht+1 that 6 raises the left-hand side, the after-tax equilibrium return on capital investment (1 t+1)rt+1 and the right-hand side cannot rise enough. As a result, agents’ initial optimism will not be ful…lled under either tax speci…cation, hence the economy exhibits saddle-path stability and equilibrium uniqueness. Finally, we …nd that the high-tax steady state with < ss H < + 1+ is a source, which is surrounded by divergent or explosive trajectories that will eventually violate the transversality condition (9).5 As a side-by-side comparison, the consumption Euler equation in Schmitt-Grohé and Uribe’s one-sector RBC model is given by Ct+1 Ct = [1  + (1 t+1)rt+1]: (17) In this case, households’ optimistic expectations that lead to higher investment today will unambiguously raise the left-hand side of this equation, and result in a lower before-tax real interest rate rt+1 due to diminishing returns to productive inputs. Under countercyclical labor income taxation  @ t @Yt < 0  , these authors show that (i) the low-tax steady state may become an indeterminate sink when the right-hand side of (17) rises su¢ciently; and (ii) the high-tax steady state is always a saddle point. Overall, our analysis illustrates that under perfect competition and constant returns-to-scale in production, Schmitt-Grohé and Uribe’s (1997) indeterminacy result depends crucially on the presence of income e¤ect associated with the household’s labor supply decision. 4 Conclusion Schmitt-Grohé and Uribe (1997, section II) analytically show that with an additively separable utility function between consumption and hours worked, a standard one-sector real business cycle model may possess an indeterminate stationary equilibrium when the labor tax rate is endogenously determined by a balanced-budget rule to …nance a pre-speci…ed …xed level of government spending. This paper complements their analysis by considering an alternative preference formulation that does not exhibit income e¤ect associated with the household’s labor supply decision. We …nd that local indeterminacy is no longer possible within this no-income-e¤ect macroeconomy. In particular, the model’s low-tax steady state always displays saddle-path stability and equilibrium uniqueness; whereas the high-tax steady state is either a source or a saddle point. 5As in Schmitt-Grohé and Uribe (1997) and Abad et al. (2017), we focus on the model’s local stability properties, and leave its (nonlinear) global dynamics for future research. 7 5 Appendix Proof of Proposition. Using (14), it is straightforward to show that Tr < 0 when ss > + 1+ , which is higher than = 1+ ; and that Tr > 0 when ss < + 1+ 2 (0; 1). Next, since ( ss) > 0 (see footnote 4), the sign of Det as in (15) depends on whether ss (1 ss) ss+ (1 ss) is positive or negative. In particular, Det > 0 if and only if ss (1 ss) and ss + (1 ss) have the same sign, which can happen when (a) ss (1 ss) < 0 and ss + (1 ss) < 0. This implies that ss < 1+ and ss > + 1+ > 1+ , thus generating a contradiction. (b) ss (1 ss) > 0 and ss + (1 ss) > 0. This implies that ss > 1+ and ss < + 1+ . On the other hand, Det < 0 if and only if ss (1 ss) and ss + (1 ss) have opposite signs, which can happen when (c) ss (1 ss) > 0 and ss + (1 ss) < 0. This implies that ss > 1+ and ss > + 1+ , thus the more binding condition is ss > + 1+ . (d) ss (1 ss) < 0 and ss + (1 ss) > 0. This implies that ss < 1+ and ss < + 1+ , thus the more binding condition is ss < 1+ . At the low-tax steady state with 0 < ss L < 1+ , the model’s Jacobian matrix has Det < 0 as in case (d), hence it is a saddle point. In addition, at the high-tax steady state with 1+ < ss H < + 1+ , we …nd that Tr > 0 and Det > 0 as in case (b), hence it is a source. Finally, when + 1+ < ss H < 1, the high-tax steady state is a saddle point because of Det < 0 as in case (c).6  6Local indeterminacy requires that both eigenvalues have negative real parts (Tr < 0 and Det > 0). How-ever, under case (b) with 1+ < ss < a+ 1+ and Det > 0, the Jacobian’s trace is also positive (Tr > 0). As a result, neither steady state can be a sink. 8 References Abad, N., T. Seegmuller and A. Venditti (2017), “Non-separable Preferences Do Not Rule out Aggregate Instability under Balanced-Budget Rules: A Note,” Macroeconomic Dy-namics 21, 259-277. Benhabib, J. and R.E.A. Farmer (1999), “Indeterminacy and Sunspots in Macroeco-nomics,” in J. Taylor and M. Woodford, eds., Handbook of Macroeconomics, Amsterdam: North Holland, 387-448. Greenwood, J., Z. Hercowitz and G. Hu¤man (1988), “Investment, Capacity Utilization and the Real Business Cycle,” American Economic Review 78, 402-417. Jaimovich, N. (2008), “Income E¤ects and Indeterminacy in a Calibrated One-Sector Growth Model,” Journal of Economic Theory 143, 610-623. Meng, Q. and C.K. Yip (2008), “On Indeterminacy in One-Sector Models of the Business Cycle with Factor-Generated Externalities,” Journal of Macroeconomics 30, 97-110. Schmitt-Grohé, S. and M. Uribe (1997), “Balanced-Budget Rules, Distortionary Taxes and Aggregate Instability,” Journal of Political Economy 105, 976-1000. 9 -ss Steady State Tax Rate Government Spending G 0 1  G G ss L  ss H  1      Figure 1. Steady-State Laffer Curve 10
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https://math.stackexchange.com/questions/2916484/the-reducibility-of-a-polynomial-in-coefficients-of-integer
Skip to main content The reducibility of a polynomial in coefficients of integer Ask Question Asked Modified 6 years, 11 months ago Viewed 94 times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. This question is from Putnam and Beyond, an example in the section 'Irreducible Polynomial': Let P(x) be an nth-degree polynomial with integer coefficients with the property that |P(x)| is a prime number for 2n+1 distinct integer values of the variable x. Prove that P(x) is irreducible over Z[x]. And the solution: Assume the contrary and let P(x)=Q(x)R(x) with Q(x),R(x)∈Z[x], Q(x),R(x)=±1. Let k=deg(Q(x)). Then Q(x)=1 at most k times and Q(x)=−1 at most n−k times. Also, R(x)=1 at most n−k times and R(x)=−1 at most k times. Consequently, the product |Q(x)R(x)| is composite except for at most k+(n−k)+(n−k)+k=2n values of x. This contradicts the hypothesis. Hence P(x) is irreducible. All concept is okay for me except the part "Q(x)=−1 at most n−k times" , because I know Q(x)=1 at most k times is because Q(x) will be constant function if Q=1 at least k+1 times, but is it related to Q(x)=−1? polynomials irreducible-polynomials integers Share CC BY-SA 4.0 Follow this question to receive notifications asked Sep 14, 2018 at 6:48 kelvin hong 方kelvin hong 方 1,93799 silver badges1717 bronze badges Add a comment | 1 Answer 1 Reset to default This answer is useful 2 Save this answer. Show activity on this post. Seems to me that there is a typo and it should be Then Q(x)=1 at most k times and Q(x)=−1 at most k times. Also, R(x)=1 at most n−k times and R(x)=−1 at most n−k times. Share CC BY-SA 4.0 Follow this answer to receive notifications answered Sep 14, 2018 at 7:14 DavidDavid 84.8k99 gold badges9696 silver badges166166 bronze badges 1 Thanks, this makes sense to me! – kelvin hong 方 Commented Sep 14, 2018 at 7:25 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 polynomials irreducible-polynomials integers See similar questions with these tags. Featured on Meta Community help needed to clean up goo.gl links (by August 25) Related 4 On reducibility over Z of a special class of polynomials . 1 Can we find an irreducible polynomial of any degree in C[x,y]? 2 Integer roots of polynomial with irrational coefficients 1 A polynomial with integer coefficients is irreducible over Z if and only if it is irreducible over Q 1 Is there an irreducible polynomial over Z[x] with setwise coprime coefficients that is composite for all integer arguments? 0 Reducibility of constrained polynomial Hot Network Questions Is pre-Greek Minoan? Reverse engineering images from old Japanese videogame How do I push back on an impossible scope? Three Houses in Mile Lane Setting minimum scale when rendering vector layer in How to do Lucas' quest in Lil Gator Game? 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https://www.sciencehistory.org/education/scientific-biographies/louis-pasteur/
Museum Hours:Wednesday–Saturday, 10am–5pm Education Scientific Biographies Louis Pasteur During the mid- to late 19th century, Pasteur demonstrated that microorganisms cause disease and discovered how to make vaccines from weakened, or attenuated, microbes. He developed the earliest vaccines against fowl cholera, anthrax, and rabies. Portrait of Louis Pasteur taken in 1886, reproduced in the 1911 biography The Life of Louis Pasteur. Science History Institute About Scientific Biographies Filter Biographies Louis Pasteur (1822–1895) is revered by his successors in the life sciences as well as by the general public. In fact, his name provided the basis for a household word—pasteurized. His research, which showed that microorganisms cause both fermentation and disease, supported the germ theory of disease at a time when its validity was still being questioned. In his ongoing quest for disease treatments he created the first vaccines for fowl cholera; anthrax, a major livestock disease that in recent times has been used against humans in germ warfare; and the dreaded rabies. Early Life and Education Pasteur was born in Dole, France, the middle child of five in a family that had for generations been leather tanners. Young Pasteur’s gifts seemed to be more artistic than academic until near the end of his years in secondary school. Spurred by his mentors’ encouragement, he undertook rigorous studies to compensate for his academic shortcomings in order to prepare for the École Normale Supérieure, the famous teachers’ college in Paris. He earned his master’s degree there in 1845 and his doctorate in 1847. Study of Optical Activity While waiting for an appropriate appointment, Pasteur continued to work as a laboratory assistant at the École Normale. There he made further progress on the research he had begun for his doctoral dissertation—investigating the ability of certain crystals or solutions to rotate plane-polarized light clockwise or counterclockwise, that is, to exhibit “optical activity.” He was able to show that in many cases this activity related to the shape of the crystals of a compound. He also reasoned that there was some special internal arrangement to the molecules of such a compound that twisted the light—an “asymmetric” arrangement. This hypothesis holds an important place in the early history of structural chemistry—the field of chemistry that studies the three-dimensional characteristics of molecules. Fermentation and Pasteurization Pasteur secured his academic credentials with scientific papers on this and related research and was then appointed in 1848 to the faculty of sciences in Strasbourg and in 1854 to the faculty in Lille. There he launched his studies on fermentation. Pasteur sided with the minority view among his contemporaries that each type of fermentation is carried out by a living microorganism. At the time the majority believed that fermentation was spontaneously generated by a series of chemical reactions in which enzymes—themselves not yet securely identified with life—played a critical role. In 1857 Pasteur returned to the École Normale as director of scientific studies. In the modest laboratory that he was permitted to establish there, he continued his study of fermentation and fought long, hard battles against the theory of spontaneous generation. Figuring prominently in early rounds of these debates were various applications of his pasteurization process, which he originally invented and patented (in 1865) to fight the “diseases” of wine. He realized that these were caused by unwanted microorganisms that could be destroyed by heating wine to a temperature between 60° and 100°C. The process was later extended to all sorts of other spoilable substances, such as milk. Germ Theory At the same time Pasteur began his fermentation studies, he adopted a related view on the cause of diseases. He and a minority of other scientists believed that diseases arose from the activities of microorganisms—germ theory. Opponents believed that diseases, particularly major killer diseases, arose in the first instance from a weakness or imbalance in the internal state and quality of the afflicted individual. In an early foray into the causes of particular diseases, in the 1860s, Pasteur was able to determine the cause of the devastating blight that had befallen the silkworms that were the basis for France’s then-important silk industry. Surprisingly, he found that the guilty parties were two microorganisms rather than one. A New Laboratory Pasteur did not, however, fully engage in studies of disease until the late 1870s, after several cataclysmic changes had rocked his life and that of the French nation. In 1868, in the middle of his silkworm studies, he suffered a stroke that partially paralyzed his left side. Soon thereafter, in 1870, France suffered a humiliating defeat at the hands of the Prussians, and Emperor Louis-Napoléon was overthrown. Nevertheless, Pasteur successfully concluded with the new government negotiations he had begun with the emperor. The government agreed to build a new laboratory for him, to relieve him of administrative and teaching duties, and to grant him a pension and a special recompense in order to free his energies for studies of diseases. Attenuating Microbes for Vaccines: Fowl Cholera and Anthrax In his research campaign against disease Pasteur first worked on expanding what was known about anthrax, but his attention was quickly drawn to fowl cholera. This investigation led to his discovery of how to make vaccines by attenuating, or weakening, the microbe involved. Pasteur usually “refreshed” the laboratory cultures he was studying—in this case, fowl cholera—every few days; that is, he returned them to virulence by reintroducing them into laboratory chickens with the resulting onslaught of disease and the birds’ death. Months into the experiments, Pasteur let cultures of fowl cholera stand idle while he went on vacation. When he returned and the same procedure was attempted, the chickens did not become diseased as before. Pasteur could easily have deduced that the culture was dead and could not be revived, but instead he was inspired to inoculate the experimental chickens with a virulent culture. Amazingly, the chickens survived and did not become diseased; they were protected by a microbe attenuated over time. Realizing he had discovered a technique that could be extended to other diseases, Pasteur returned to his study of anthrax. Pasteur produced vaccines from weakened anthrax bacilli that could indeed protect sheep and other animals. In public demonstrations at Pouilly-le-Fort before crowds of observers, twenty-four sheep, one goat, and six cows were subjected to a two-part course of inoculations with the new vaccine, on May 5, 1881, and again on May 17. Meanwhile a control group of twenty-four sheep, one goat, and four cows remained unvaccinated. On May 31 all the animals were inoculated with virulent anthrax bacilli, and two days later, on June 2, the crowd reassembled. Pasteur and his collaborators arrived to great applause. The effects of the vaccine were undeniable: the vaccinated animals were all alive. Of the control animals all the sheep were dead except three wobbly individuals who died by the end of the day, and the four unprotected cows were swollen and feverish. The single goat had expired too. Rabies and the Beginnings of the Institut Pasteur Pasteur then wanted to move into the more difficult area of human disease, in which ethical concerns weighed more heavily. He looked for a disease that afflicts both animals and humans so that most of his experiments could be done on animals, although here too he had strong reservations. Rabies, the disease he chose, had long terrified the populace, even though it was in fact quite rare in humans. Up to the time of Pasteur’s vaccine, a common treatment for a bite by a rabid animal had been cauterization with a red-hot iron in hopes of destroying the unknown cause of the disease, which almost always developed anyway after a typically long incubation period. Rabies presented new obstacles to the development of a successful vaccine, primarily because the microorganism causing the disease could not be specifically identified; nor could it be cultured in vitro (in the laboratory and not in an animal). As with other infectious diseases, rabies could be injected into other species and attenuated. Attenuation of rabies was first achieved in monkeys and later in rabbits. Meeting with success in protecting dogs, even those already bitten by a rabid animal, on July 6, 1885, Pasteur agreed with some reluctance to treat his first human patient, Joseph Meister, a nine-year-old who was otherwise doomed to a near-certain death. Success in this case and thousands of others convinced a grateful public throughout the world to make contributions to the Institut Pasteur. Historian Bert Hansen discusses his book, Picturing Medical Progress from Pasteur to Polio. It was officially opened in 1888 and continues as one of the premier institutions of biomedical research in the world. Its tradition of discovering and producing vaccines is carried on today by the pharmaceutical company Sanofi Pasteur. A Great Experimenter and Innovative Theorist Pasteur’s career shows him to have been a great experimenter, far less concerned with the theory of disease and immune response than with dealing directly with diseases by creating new vaccines. Still it is possible to discern his notions on the more abstract topics. Early on he linked the immune response to the biological, especially nutritional, requirements of the microorganisms involved; that is, the microbe or the attenuated microbe in the vaccine depleted its food source during its first invasion, making the next onslaught difficult for the microbe. Later he speculated that microbes could produce chemical substances toxic to themselves that circulated throughout the body, thus pointing to the use of toxins and antitoxins in vaccines. He lent support to another view by welcoming to the Institut Pasteur Élie Metchnikoff and his theory that “phagocytes” in the blood—white corpuscles—clear the body of foreign matter and are the prime agents of immunity. You might also like ### Does Louis Pasteur Still Matter? Or will the scientist’s 200th birthday be his last hurrah? ### Biting Back The story of Louis Pasteur and the development of the rabies vaccine. ### Stories of the Great Chemists In the 1950s comic books took Mexico’s youth by storm. But alongside familiar superhuman avengers were other kinds of heroes: real-life chemists. Browse more biographies SEARCH ALL BIOGRAPHIES ## Harvey Washington Wiley 1844 1930 ## Lise Meitner 1878 1968 ## William Henry Perkin 1838 1907
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13.7: Tidal Forces - Physics LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter Reader Mode 13: Gravitation University Physics I - Mechanics, Sound, Oscillations, and Waves (OpenStax) { } { "13.01:Prelude_to__Gravitation" : "property get Map 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Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } Sun, 16 Mar 2025 21:22:00 GMT 13.7: Tidal Forces 4053 4053 Delmar Larsen { } Anonymous Anonymous 2 false false [ "article:topic", "authorname:openstax", "neap tide", "spring tide", "tidal force", "license:ccby", "showtoc:no", "program:openstax", "licenseversion:40", "source@ ] [ "article:topic", "authorname:openstax", "neap tide", "spring tide", "tidal force", "license:ccby", "showtoc:no", "program:openstax", "licenseversion:40", "source@ ] Search site Search Search Go back to previous article Sign in Username Password Sign in Sign in Sign in Forgot password Contents 1. Home 2. Bookshelves 3. University Physics 4. University Physics (OpenStax) 5. University Physics I - Mechanics, Sound, Oscillations, and Waves (OpenStax) 6. 13: Gravitation 7. 13.7: Tidal Forces Expand/collapse global location University Physics I - Mechanics, Sound, Oscillations, and Waves (OpenStax) Front Matter 1: Units and Measurement 2: Vectors 3: Motion Along a Straight Line 4: Motion in Two and Three Dimensions 5: Newton's Laws of Motion 6: Applications of Newton's Laws 7: Work and Kinetic Energy 8: Potential Energy and Conservation of Energy 9: Linear Momentum and Collisions 10: Fixed-Axis Rotation Introduction 11: Angular Momentum 12: Static Equilibrium and Elasticity 13: Gravitation 14: Fluid Mechanics 15: Oscillations 16: Waves 17: Sound 18: Answer Key to Selected Problems Back Matter 13.7: Tidal Forces Last updated Mar 16, 2025 Save as PDF 13.6: Kepler's Laws of Planetary Motion 13.8: Einstein's Theory of Gravity picture_as_pdf Full Book Page Downloads Full PDF Import into LMS Individual ZIP Buy Print Copy Print Book Files Buy Print CopyReview / Adopt Submit Adoption Report Submit a Peer Review View on CommonsDonate Page ID 4053 OpenStax OpenStax ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. Learning Objectives 2. Lunar Tides 3. The Effect of the Sun on Tides 1. Simulation The Magnitude of the Tides Example 13.7.1: Comparing Tidal Forces Solution Significance exercise 13.7.1 Other Tidal Effects Learning Objectives Explain the origins of Earth’s ocean tides Describe how neap and leap tides differ Describe how tidal forces affect binary systems The origin of Earth’s ocean tides has been a subject of continuous investigation for over 2000 years. But the work of Newton is considered to be the beginning of the true understanding of the phenomenon. Ocean tides are the result of gravitational tidal forces. These same tidal forces are present in any astronomical body. They are responsible for the internal heat that creates the volcanic activity on Io, one of Jupiter’s moons, and the breakup of stars that get too close to black holes. Lunar Tides If you live on an ocean shore almost anywhere in the world, you can observe the rising and falling of the sea level about twice per day. This is caused by a combination of Earth’s rotation about its axis and the gravitational attraction of both the Moon and the Sun. Let’s consider the effect of the Moon first. In Figure 13.7.1, we are looking “down” onto Earth’s North Pole. One side of Earth is closer to the Moon than the other side, by a distance equal to Earth’s diameter. Hence, the gravitational force is greater on the near side than on the far side. The magnitude at the center of Earth is between these values. This is why a tidal bulge appears on both sides of Earth. Figure 13.7.1: The tidal force stretches Earth along the line between Earth and the Moon. It is the difference between the gravitational force from the far side to the near side that creates the tidal bulge on both sides of the planet. Tidal variations of the oceans are on the order of few meters; hence, this diagram is greatly exaggerated. The net force on Earth causes it to orbit about the Earth-Moon center of mass , located about 1600 km below Earth’s surface along the line between Earth and the Moon. The tidal force can be viewed as the differencebetween the force at the center of Earth and that at any other location. In Figure 13.7.2, this difference is shown at sea level, where we observe the ocean tides. (Note that the change in sea level caused by these tidal forces is measured from the baseline sea level. We saw earlier that Earth bulges many kilometers at the equator due to its rotation. This defines the baseline sea level and here we consider only the much smaller tidal bulge measured from that baseline sea level.) Figure 13.7.2: The tidal force is the difference between the gravitational force at the center and that elsewhere. In this figure, the tidal forces are shown at the ocean surface. These forces would diminish to zero as you approach Earth’s center. Why does the rise and fall of the tides occur twice per day? Look again at Figure 13.7.1. If Earth were not rotating and the Moon was fixed, then the bulges would remain in the same location on Earth. Relative to the Moon, the bulges stay fixed—along the line connecting Earth and the Moon. But Earth rotates (in the direction shown by the blue arrow) approximately every 24 hours. In 6 hours, the near and far locations of Earth move to where the low tides are occurring, and 6 hours later, those locations are back to the high-tide position . Since the Moon also orbits Earth approximately every 28 days, and in the same direction as Earth rotates, the time between high (and low) tides is actually about 12.5 hours. The actual timing of the tides is complicated by numerous factors, the most important of which is another astronomical body—the Sun. The Effect of the Sun on Tides In addition to the Moon’s tidal forces on Earth’s oceans, the Sun exerts a tidal force as well. The gravitational attraction of the Sun on any object on Earth is nearly 200 times that of the Moon. However, as we show later in an example, the tidal effect of the Sun is less than that of the Moon, but a significant effect nevertheless. Depending upon the positions of the Moon and Sun relative to Earth, the net tidal effect can be amplified or attenuated. Figure 13.7.1 illustrates the relative positions of the Sun and the Moon that create the largest tides, called spring tides (or leap tides). During spring tides, Earth, the Moon, and the Sun are aligned and the tidal effects add. (Recall that the tidal forces cause bulges on both sides.) Figure 13.7.1⁢c shows the relative positions for the smallest tides, called neap tides. The extremes of both high and low tides are affected. Spring tides occur during the new or full moon, and neap tides occur at half-moon. Simulation You can seean animationof the tides in motion. Figure 13.7.3: (a and b) The spring tides occur when the Sun and the Moon are aligned, whereas (c) the neap tides occur when the Sun and Moon make a right triangle with Earth. (Figure is not drawn to scale.) The Magnitude of the Tides With accurate data for the positions of the Moon and the Sun, the time of maximum and minimum tides at most locations on our planet can be predicted accurately. Visit this site to generate tide predictions for up to 2 years in the past or future, at more than 3000 locations around the United States. The magnitude of the tides, however, is far more complicated. The relative angles of Earth and the Moon determine spring and neap tides, but the magnitudes of these tides are affected by the distances from Earth as well. Tidal forces are greater when the distances are smaller. Both the Moon’s orbit about Earth and Earth’s orbit about the Sun are elliptical, so a spring tide is exceptionally large if it occurs when the Moon is at perigee and Earth is at perihelion . Conversely, it is relatively small if it occurs when the Moon is at apogee and Earth is at aphelion . The greatest causes of tide variation are the topography of the local shoreline and the bathymetry (the profile of the depth) of the ocean floor. The range of tides due to these effects is astounding. Although ocean tides are much smaller than a meter in many places around the globe, the tides at the Bay of Fundy (Figure 13.7.4), on the east coast of Canada, can be as much as 16.3 meters. Figure 13.7.4: Boats in the Bay of Fundy at high and low tides. The twice-daily change in sea level creates a real challenge to the safe mooring of boats. (credit: Dylan Kereluk) Example 13.7.1: Comparing Tidal Forces Compare the Moon’s gravitational force on a 1.0-kg mass located on the near side and another on the far side of Earth. Repeat for the Sun and then compare the results to confirm that the Moon’s tidal forces are about twice that of the Sun. Strategy We use Newton ’s law of gravitation given by Equation 13.2.1. We need the masses of the Moon and the Sun and their distances from Earth, as well as the radius of Earth. We use the astronomical data from Appendix D. Solution Substituting the mass of the Moon and mean distance from Earth to the Moon, we have (13.7.1)F 12=G⁡m 1⁢m 2 r 2=(6.67×10−11 N⋅m 2/k⁢g 2)⁢[(1.0 k⁢g)⁢(7.35×10 22 k⁢g)(3.84×10 8±6.37×10 6 m)2]. In the denominator, we use the minus sign for the near side and the plus sign for the far side. The results are (13.7.2)F n⁢e⁢a⁢r=3.44×10−5 N a⁢n⁢d F f⁡a⁢r=3.22×10−5 N. The Moon’s gravitational force is nearly 7% higher at the near side of Earth than at the far side, but both forces are much less than that of Earth itself on the 1.0-kg mass. Nevertheless, this small difference creates the tides. We now repeat the problem, but substitute the mass of the Sun and the mean distance between the Earth and Sun. The results are (13.7.3)F n⁢e⁢a⁢r=5.89975×10−3 N a⁢n⁢d F f⁡a⁢r=5.89874×10−3 N. We have to keep six significant digits since we wish to compare the difference between them to the difference for the Moon. (Although we can’t justify the absolute value to this accuracy , since all values in the calculation are the same except the distances, the accuracy in the difference is still valid to three digits.) The difference between the near and far forces on a 1.0-kg mass due to the Moon is (13.7.4)F n⁢e⁢a⁢r=(3.44×10−5 N)−(3.22×10−5 N)=0.22×10−5 N, whereas the difference for the Sun is (13.7.5)F n⁢e⁢a⁢r−F f⁡a⁢r=(5.89975×10−3 N)−(5.89874×10−3 N)=0.101×10−5 N. Note that a more proper approach is to write the difference in the two forces with the difference between the near and far distances explicitly expressed. With just a bit of algebra we can show that (13.7.6)F t⁢i⁢d⁢a⁢l=G⁡M⁢m r 1 2−G⁡M⁢m r 2 2=G⁡M⁢m⁡((r 2−r 1)⁢(r 2+r 1)r 1 2⁢r 2 2). where r 1 and r 2 are the same to three significant digits, but their difference (r 2 − r 1), equal to the diameter of Earth, is also known to three significant digits. The results of the calculation are the same. This approach would be necessary if the number of significant digits needed exceeds that available on your calculator or computer. Significance Note that the forces exerted by the Sun are nearly 200 times greater than the forces exerted by the Moon. But the differencein those forces for the Sun is half that for the Moon. This is the nature of tidal forces. The Moon has a greater tidal effect because the fractional change in distance from the near side to the far side is so much greater for the Moon than it is for the Sun. exercise 13.7.1 Earth exerts a tidal force on the Moon. Is it greater than, the same as, or less than that of the Moon on Earth? Be careful in your response, as tidal forces arise from the differencein gravitational forces between one side and the other. Look at the calculations we performed for the tidal force on Earth and consider the values that would change significantly for the Moon. The diameter of the Moon is one-fourth that of Earth. Tidal forces on the Moon are not easy to detect, since there is no liquid on the surface. Other Tidal Effects Tidal forces exist between any two bodies. The effect stretches the bodies along the line between their centers. Although the tidal effect on Earth’s seas is observable on a daily basis, long-term consequences cannot be observed so easily. One consequence is the dissipation of rotational energy due to friction during flexure of the bodies themselves. Earth’s rotation rate is slowing down as the tidal forces transfer rotational energy into heat. The other effect, related to this dissipation and conservation of angular momentum , is called “locking” or tidal synchronization. It has already happened to most moons in our solar system , including Earth’s Moon. The Moon keeps one face toward Earth—its rotation rate has locked into the orbital rate about Earth. The same process is happening to Earth, and eventually it will keep one face toward the Moon. If that does happen, we would no longer see tides, as the tidal bulge would remain in the same place on Earth, and half the planet would never see the Moon. However, this locking will take many billions of years, perhaps not before our Sun expires. One of the more dramatic example of tidal effects is found on Io, one of Jupiter’s moons. In 1979, the Voyagerspacecraft sent back dramatic images of volcanic activity on Io. It is the only other astronomical body in our solar system on which we have found such activity. Figure 13.7.5 shows a more recent picture of Io taken by the New Horizons spacecraft on its way to Pluto, while using a gravity assist from Jupiter. Figure 13.7.5: Dramatic evidence of tidal forces can be seen on Io. The eruption seen in blue is due to the internal heat created by the tidal forces exerted on Io by Jupiter. For some stars, the effect of tidal forces can be catastrophic. The tidal forces in very close binary systems can be strong enough to rip matter from one star to the other, once the tidal forces exceed the cohesive self-gravitational forces that hold the stars together. This effect can be seen in normal stars that orbit nearby compact stars, such as neutron stars or black holes. Figure 13.7.6 shows an artist’s rendition of this process. As matter falls into the compact star, it forms an accretion disc that becomes super-heated and radiates in the X-ray spectrum. Figure 13.7.6: Tidal forces from a compact object can tear matter away from an orbiting star. In addition to the accretion disc orbiting the compact object, material is often ejected along relativistic jets as shown. (credit: modification of work by European Southern Observatory (ESO)) The energy output of these binary systems can exceed the typical output of thousands of stars. Another example might be a quasar. Quasars are very distant and immensely bright objects, often exceeding the energy output of entire galaxies. It is the general consensus among astronomers that they are, in fact, massive black holes producing radiant energy as matter that has been tidally ripped from nearby stars falls into them. This page titled 13.7: Tidal Forces is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform. Back to top 13.6: Kepler's Laws of Planetary Motion 13.8: Einstein's Theory of Gravity Was this article helpful? Yes No Recommended articles 13.7: Tidal ForcesEarth’s tides are caused by the difference in gravitational forces from the Moon and the Sun on the different sides of Earth. Spring or neap (high) ti... 10.5: Advanced Topics 12.2: The Gailean Moons of JupiterJupiter’s largest moons are Ganymede and Callisto, both low-density objects that are composed of more than half water ice. Callisto has an ancient cra... 12.3: The Gailean Moons of JupiterJupiter’s largest moons are Ganymede and Callisto, both low-density objects that are composed of more than half water ice. Callisto has an ancient cra... 1.5: Binary Stars and Tidal Forces Article typeSection or PageAuthorOpenStaxLicenseCC BYLicense Version4.0OER program or PublisherOpenStaxShow TOCno Tags neap tide source@ spring tide tidal force © Copyright 2025 Physics LibreTexts Powered by CXone Expert ® ? The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org. Support Center How can we help? 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Alarms | Police Skip to content Home GovernmentAboutCity CodeMayor & City CouncilElections & VotingRecorder's OfficeBoards & CommissionsTransparencyYouth City CouncilYouth Court ServicesCemeteriesEmergency NotificationsGRAMA RequestMapsNewsletterPassportsPermits/LicensesUtilitiesYard Sales DepartmentsAdministrationBuilding PermitsClyde Recreation CenterCommunity DevelopmentFinanceFire & RescueGolf CourseJustice CourtLibraryMuseum of ArtParks & RecreationPolicePowerPublic WorksSenior Center I want to...Contact SomeonePay UtilitiesFind WorkSee the CalendarGet a Permit/LicenseHear the MayorRequest Records (GRAMA)Pay a Parking TicketStay Informed ContactContact Elected OfficialsFraud HotlinePhone DirectoryTip a CopRequest Records (GRAMA) Search Springville Police Services Alarm Registration Alarm Registration Thank you for applying for an alarm permit using our online application. You will need to list contact persons along with their phone numbers so that in case your alarm is activated we will be able to contact them. You will also need to provide a valid email address and pay the $25 registration fee. Alarm Number Some alarm companies require an “alarm number” from their clients. This will be emailed to you once we have confirmation that any associated fees have been paid. Registration Fee If you are registering an alarm for the first time, or if your registration has lapsed, there is a $25 fee associated with this registration. (Registrations lapse when they are not updated at least once per year.) We have three convenient ways to pay the registration fee. To pay in person, please come to the Civic Center to make payment. We are able to take payment by cash, check or credit/debit card. You will need to bring your identification, along with the location of the alarm, and the name and address of your business if applicable. One great way to make certain you have all of the information needed is to print a copy of the registration form and bring it with you. Our business hours for making payments are 8 to 5, Monday through Friday. To pay by mail, please include your name, the business name (if applicable), the location and type of alarm (business or residential) along with your check. The best way to make sure all information is included is to print the registration form, fill it out and include it with your payment. (Checks should be made out to Springville City Corp.) To pay online, you will need to have (or set up) an account with Xpress Bill Pay to use this service. Pay Online Alarm Registration Employee Resources Employee Email © 2025 Springville City
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https://chemistry.stackexchange.com/questions/42433/is-there-a-relationship-between-ph-indicator-colors-and-the-visible-light-spectr
acid base - Is there a relationship between pH indicator colors and the visible light spectrum? - Chemistry Stack Exchange Join Chemistry 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 Chemistry helpchat Chemistry 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 Is there a relationship between pH indicator colors and the visible light spectrum? Ask Question Asked 9 years, 9 months ago Modified9 years, 9 months ago Viewed 4k times This question shows research effort; it is useful and clear 6 Save this question. Show activity on this post. When I examined the colors showed on a pH paper which is red for acids and violet for bases, greenish yellow for neutral and other ranges in between, I noticed that the sequence is exactly same to that of visible light spectrum that is violet, indigo, blue, green, yellow, orange, and red. Is there a connection between the two, and if yes what it is? acid-base ph Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Dec 18, 2015 at 20:13 jerepierre 10.8k 3 3 gold badges 39 39 silver badges 58 58 bronze badges asked Dec 18, 2015 at 13:19 user101184user101184 61 1 1 silver badge 2 2 bronze badges 2 2 Yes there is a connection. pH indicators are typically chosen so that the human eye can observe the change in the visible part of the electromagnetic spectrum. ph indicators en.wikipedia.org/wiki/PH_indicatorMaxW –MaxW 2015-12-18 16:19:26 +00:00 Commented Dec 18, 2015 at 16:19 2 There is a connection, but not in the way that you think. The fact they happen to appear in the exact same sequence of the visible light spectrum is purely a coincidence.Nanoputian –Nanoputian 2015-12-18 20:51:39 +00:00 Commented Dec 18, 2015 at 20:51 Add a comment| 1 Answer 1 Sorted by: Reset to default This answer is useful 3 Save this answer. Show activity on this post. It has to do with the way pH indicators and particularly large-range pH indicators work. In simplest terms, a pH indicator is simply a weak acid/base which has one color in its acidic form, and one color in its basic form. When the solution is highly acidic, all of the indicator molecules are protonated, and the color of the indicator is the color of the protonated form. When the solution is highly basic, all of the indicator molecules are deprotonated, and the color of the indicator is the color of the deprotonated form. Where it gets interesting is where the pH of the solution is close to the pKa of the indicator. In those situations, you have some of the indicator molecules in the protonated form, and some of them in the deprotonated form. You don't ever get "fractionally protonated" forms - each individual molecule is either one or the other. Since light absorption happens on a molecular level, you get You can see this with a titration of something like Bromothymol Blue [image]. The protonated form is yellow, so at low pH, everything is yellow. The deprotonated form is blue, so at high pH, everything is blue. At around the pKa (~7), there's near equal amounts of the two forms, so it acts like you mixed equal amounts of a yellow pigment/dye and a blue pigment/dye. That is, you get a green solution due to the subtractive color mixing model. As you titrate the pH, you get a range of colors as you add and remove the amount of protonated and deprotonated forms, as would be indicated by the Henderson–Hasselbalch equation. Okay, so that's a single pH indicator, which is normally limited to a small range. (Because more than ±1 pH unit to either side of the pKa you typically have only a single species present.) To get broad range pH indicators, you have to mix multiple different pH indicators, each with their own different colors and changes. But you have to do this intelligently, as the indicators are always present, and will always contribute to the color. Here's a chart of some of the major pH indicators, their pH ranges and their colors. You can see what might happen with various indicator combinations. For example, with an ethyl violet/ethyl orange mixture, at low pH you'd have an orange solution (yellow+red), which would transition to be purple (blue+red) at around pH 3 and then to be green at high pH (blue+yellow). But those intermediate colors are going to be pretty muddy. At around 1.5 you'll have an orangy-purple, and at around 4.5 you'll have a greeny-purple. Imagine what things would look like if you mixed orange paint with purple paint, or purple paint with orange paint. What we need is a layout of color that can gradually transition one to the other and look good all the way through the approximately 7-ish transitons you'd need to segment up a 14 unit pH scale ... right, the rainbow spectrum has that property. So there's no real chemical reason for the connection. It's more a practical human consideration, where the people combining the indicator molecules to make a large-range pH indicator need to pick particular ones which make a clear, unambiguous color sequence, and the visual color spectrum has the same useful properties they need. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Dec 18, 2015 at 19:50 R.M.R.M. 5,040 1 1 gold badge 21 21 silver badges 33 33 bronze badges Add a comment| Your Answer Reminder: Answers generated by AI tools are not allowed due to Chemistry Stack Exchange's artificial intelligence policy Thanks for contributing an answer to Chemistry Stack Exchange! Please be sure to answer the question. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. MathJax reference. To learn more, see our tips on writing great answers. 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https://trevorpythag.wordpress.com/2009/04/19/auxiliary-angle-method-for-solving-trigonometry-equations/
This blog has moved to trevorpythag.co.uk Trevor Pythagoras Maths Tutorials – A level and GCSE Home algebra co-ordinate geometry compting maths calculus trigonometry physics statistics surds Uncategorized Home > maths, trigonometry > Auxiliary Angle Method for Solving Trigonometry Equations Auxiliary Angle Method for Solving Trigonometry Equations April 19, 2009 David Woodford Leave a comment Go to comments This is a method of solving equations in the form asinx+bcosx = c where a and b are constants and c is another expression. It involves rewriting letting asinx + bcosx = rsin(x+y) (or you could use cos(x+y)) where y is acute and then finding values for r and y, then with only one trig function to deal with the equation can be solved more easily. For example consider 2sinx + 3cosx = 3 Let 2sinx + 3cosx = rsin(x+y) Now expand the sin(x+y) to get 2sinx + 3cosx = rsinx cosy + rcosx siny Since y is constant and therefore cosy and sin y are constant we can compare the coefficients to get 2 = rcosy —–(1)3 = rsiny ——(2) We can solve these to find values for r and y.To find y consider (2)/(1) to get 3/2 = tanysince sin/cos = tan and the r’s cancelso y = 56.3 ° To find r consider (1)2+(2)2 to get 22+32 = r2 since sin2+cos2 = 1 so r =√13 So we can write 2sinx + 3 cosx = √13 cos(x+56.3) = 3 so x = cos-1(3/√13) -56.3 so x = cos-1(3/√13) -56.3 since cos-1(3/√13) = 33.7 for solutions between 0° and 90° x = ±33.7 -56.3 + 180n where n is an integer In General asinx + bcosx = √(a2+b2) sin(x+tan-1(b/a)) If you have any questions, comments or corrections please leave them as a comment below By David Woodford Rate this: Share this: Click to share on Facebook (Opens in new window) Facebook Click to share on X (Opens in new window) X Click to share on Reddit (Opens in new window) Reddit Like Loading... Related Categories: maths, trigonometry Tags: Auxiliary Angle, david woodford, equations, maths, Trigonometry, tutorial Comments (2) Trackbacks (0) Leave a comment Trackback niccles April 19, 2009 at 5:05 pm Reply Thank you. I think that i kind of understand it now. your brillant!!! 2. amelia June 22, 2009 at 12:19 pm Reply THANKYOU SO MUCH. No trackbacks yet. Leave a comment Cancel reply Proof there are infinitley many prime numbers Rational and Irrational Numbers RSS feed Google Youdao Xian Guo Zhua Xia My Yahoo! newsgator Bloglines iNezha Email Subscription Blog Has Moved I have now moved this blog to trevorpythag.co.uk To get all the functionality out of this blog including requesting posts please visit the new site. I have moved to enable me to include plugins, customise themes and develop new features to improve the blog for all my readers. Thanks, David Woodford (Author) Tags add addition algebra arcsin are base rate c++ calculate calculus chain rule circle co-ordinate geometry co-ordinates compound angle computing contradiction cos cos graph cosine cosine rule david woodford diablo differentiate differentiation download dwebs economics equation explanation game gradient graph graphics help how to inequality java juggle juggling learn level create logs maths matrcies matrix matrix calculator multiply number theory play polar program programming proof pythagoras quadratic quadratic formula recession roots sin sine sine graph sketch solve source code square tan tangent trevor trevor pythagoras trevor pythagors triangle trick Trigonometry tutorial use Top Posts Auxiliary Angle Method for Solving Trigonometry Equations Tan Graph - y=tan(x) Comments | | | Safe Steroid Store on Intergration using a Reduction… | | Yoko Housman on How to Create a Facebook … | | cortadoras de tubos on Radians and Degrees || Sectors… | | Pedro Vitor on Radians and Degrees || Sectors… | | adrian on Area And Circumference of a Ci… | Top Clicks None Blog Stats 154,133 hits Top Blog at WordPress.com. Comment Reblog Subscribe Subscribed Trevor Pythagoras Already have a WordPress.com account? Log in now. Trevor Pythagoras Subscribe Subscribed Sign up Log in Copy shortlink Report this content View post in Reader Manage subscriptions Collapse this bar
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https://physics.yale.edu/sites/default/files/files/qual_13_p2_sol.pdf
QUALIFYING EXAMINATION, Part 2 Solutions Problem 1: Quantum Mechanics I (a) Degeneracy = 2(2l + 1) This degeneracy is due to the rotational symmetry of the system, with respect to orbital and spin angular momentum independently. j = l ± 1 2, except for l = 0, in which case j = 1/2. (b) Using ⃗ L · ⃗ S = 1 2( ⃗ J2 −⃗ L2 −⃗ S2), we have ­⃗ L · ⃗ S ® = ℏ2 2 [j(j + 1) −l(l + 1) −3/4] Thus ­⃗ L · ⃗ S ® = (ℏ2/2)l for j = l + 1/2 ­⃗ L · ⃗ S ® = −(ℏ2/2)(l + 1) for j = l −1/2 (c) Using the results in (b) ­ HLS ® = ℏ2 4mc2l ¿1 r dV dr À nl for j = l + 1 2 , ­ HLS ® = −ℏ2 4mc2(l + 1) ¿1 r dV dr À nl for j = l −1 2 . The degeneracy of each of the two levels is 2j + 1. This degeneracy is 2l + 2 for j = l + 1/2, and 2l for j = l −1/2. The degeneracy remains due to the rotational symmetry of the system with respect to total angular momentum. 1 (d) The 3s and 3p states are not degenerate since the potential seen by the outer electron is not pure Coulombic. The 3p level is split by the fine structure into two levels: four j = 3/2 states and two j = 1/2 states. The j = 3/2 level lies above the j = 1/2 level. 2 Problem 2: Quantum Mechanics II (a) We have Defining D(−α) = eA with A = −αa† + α∗a, we have D(−α) = e−A† = (e−A)† = D†(α) . (This is equivalent to the unitarity of D). Using eAXe−A = X + [A, X] + 1 2![A, [A, X]] + . . . with A = −αa† + α∗a and X = a, and [−αa† + α∗a, a] = α , we find ˆ D†(α) ˆ a ˆ D(α) = ˆ a + α . (b) Using the hint, we find eαa†e−α∗a = eαa†−α∗a+ 1 2|α|2 = e 1 2 |α|2D(α) . Thus |α⟩= D(α)|0⟩= e−1 2 |α|2eαa†e−α∗a|0⟩= e−1 2 |α|2eαa†|0⟩= e−1 2|α|2 X n 1 n!αn(a†) n|0⟩= X n cn|n⟩, where cn = αne−1 2|α|2 √ n! and we have used |n⟩= a†n √ n!. Note that |cn|2 = λne−λ n! is a Poisson distribution with λ = |α|2. (c) First method [using (b)]: a|α⟩= a à e−1 2 |α|2 X n=0 1 √ n! αn|n⟩ ! = e−1 2 |α|2 X n=0 1 √ n! αna|n⟩= e−1 2|α|2 X n=1 1 p (n −1)! αn|n−1⟩, where we have used a|n⟩= √n|n −1⟩. Relabeling n −1 by n in the sum in the last expression on the r.h.s., we find a|α⟩= α à e−1 2 |α|2 X n=0 1 √ n! αn|n⟩ ! = α|α⟩. 3 Second method [using (a)]: a|α⟩= aD(α)|0⟩= [D(α)D†(α)]aD(α)|0⟩= D(α)[D†(α)aD(α)]|0⟩= D(α)(a+α)|0⟩= α|α⟩, where we have used the unitarity of D(α) and a|0⟩= 0. (d) First method: using the result in (b), the state at time t is e−i ℏHt|α⟩= e−1 2 |α|2 X n=0 1 √ n! αne−i ℏHt|n⟩. Using H|n⟩= (n + 1/2)ℏω, we find e−i ℏHt|α⟩= e−1 2|α|2 X n=0 1 √ n! αne−iω(n+1/2)t|n⟩= e−i 2 ωte−1 2|α|2 X n=0 1 √ n! αn(e−iωt)n|n⟩. This can be written as e−i ℏHt|α⟩= e−i 2ωte−1 2 |αe−iωt|2 X n=0 1 √ n! (αe−iωt)n|n⟩= e−i 2 ωt|α(t)⟩, where α(t) = e−iωtα . Second method: denoting the time evolution by U(t) = e−i ℏHt, the state at time t is U|α⟩= UD(α)|0⟩= [UD(α)U †]U|0⟩. To calculate UD(α)U †, we use [H, a] = −ℏωa to find UaU † = eiωta and Ua†U † = e−iωta†. We then have UD(α)U † = Ueαa†−α∗aU † = eαe−iωta†−α∗eiωta = D(α(t)) , where α(t) = e−iωtα. Thus U|α⟩= D(α(t))U|0⟩= e−i 2ωt|e−iωtα⟩. 4 Statistical Mechanics I (a) The Hamiltonian of the particle is H = p2 x + p2 y + p2 z 2m + 1 2mω2z2 . The classical partition function is Z = Z dx dy dz dpx dpy dpz e−βH . where β = 1/(kT) (k is the Boltzmann constant). The x, y integrals give the area A, while each of the z and momenta integrals is a Gaussian. We find Z = A (2πm/β)3/2 £ 2π/β(mω2) ¤1/2 = A (2π/β)2 (m/ω) = A (2πkT)2 (m/ω) . The energy is calculated from E = −∂ln Z/∂β = 2 β = 2kT . The free energy is F(T) = −kT ln Z = −kT [ln A + 2 ln(2πkT) + ln(m/ω)] . The entropy is calculated from S = (E −F)/T to be S(T) = k [2 + ln A + 2 ln(2πkT) + ln(m/ω)] . The entropy can also be calculated from S = −(∂F/∂T)A. We have four quadratic degrees of freedom in the Hamiltonian, each contributing kT/2. Using the equipartition theorem E = 2kT. (b) We have a simple harmonic oscillator along the z direction with energy level spacing of ℏω. Quantum effects become important when kT < ℏω. We estimate T1 from kT1 = ℏω or T1 = ℏω k . Regarding the motion in the x-y plane, quantum effects (having to do with the finite size of A) become important when the thermal energy kT is lower than the energy spacing for the particle-in-a-box. The energy scale for particle-in-a-box is set by px = py = ℏπ/L where L = √ A is the length of the sides of the square. The associated energy is ℏ2π2/(mL2), and T2 is estimated by T2 = ℏ2π2 mAk . 5 (c) For temperatures T2 ≪T < T1, we can still treat the x-y motion classically but the z direction is quantum mechanical with energy levels of (n + 1/2)ℏω (n = 0, 1, 2, . . .). The partition function is then Z = Z dx Z dy Z dpx Z dpye−β(p2 x+p2 y)/(2m) ∞ X n=0 e−βℏω(n+1/2) , or Z = A(2πm/β) e−βℏω/2 1 −e−βℏω . The average energy is then given by E = −∂ln Z ∂β = kT + ℏω 2 + ℏω eℏω/(kT) −1 . The first term is the equipartition contribution from the kinetic energies in x-y, the sec-ond term is the zero-point energy of the quantum oscillator, and the third is the thermal excitation energy of the oscillator. (d) For T = 0 the particle is in its quantum ground state in both the z and x-y motion. E = ℏω 2 + ℏ2π2 mA . The ground-state is non-degenerate so the entropy S = 0. 6 Problem 4: Statistical Mechanics II (a) Assuming the neutrinos are massless, their dispersion relation is given by ϵ = ℏck. Since the neutrinos have 2 spin states, their density of single-particle states in momentum space is given by g(k) = 2V/(2π)3. We have 2 V (2π)3d3k = 2 V (2π)34πk2 dk = V π2k2 dk dϵdϵ = V π2 ϵ2 (ℏc)3dϵ, where we have used dϵ = ℏc dk. Therefore g(ϵ) = V π2(ℏc)3ϵ2 . The photons are massless particles with two polarization states and thus have the same density of states as the neutrinos. (b) The average number of neutrinos at temperature T is determined by the equilibrium condition (∂F/∂N)T,V = 0 where F is the free energy. Since the chemical potential µ = (∂F/∂N)T,V it follows that µ = 0. Since neutrinos are fermions, the average occupation number of a single-particle state with energy ϵ is n(ϵ) = (1 + eβϵ)−1. The total energy is then given by E = Z ∞ 0 dϵg(ϵ)ϵn(ϵ) = V π2(ℏc)3 Z ∞ 0 dϵ ϵ3 1 + eβϵ . Substituting x = βϵ, we find E = V π2(ℏc)3 1 β4 µZ ∞ 0 dx x3 ex + 1 ¶ . The energy density of the neutrino gas is thus given by E V = k4 π2(ℏc)3 µZ ∞ 0 dx x3 ex + 1 ¶ T 4 ∝T 4 , where the dimensionless definite integral in brackets is a temperature-independent con-stant Z ∞ 0 dx x3 ex + 1 = 7 8Γ(4)ζ(4) . (c) The energy of the photon gas is given by a similar expression to that of the neutrino gas except that the photons are bosons and n(ϵ) = (eβϵ −1)−1. We have E V = k4 π2(ℏc)3 µZ ∞ 0 dx x3 ex −1 ¶ T 4 , 7 where Z ∞ 0 dx x3 ex −1 = Γ(4)ζ(4) . The ratio between the two energy densities is then given by the ratio between the two definite integrals, i.e., 7/8. (d) The pressure of non-interacting identical particles is given by p = X r µ −∂ϵr ∂V ¶ ⟨nr⟩, where ϵr is the energy of single-particle state r and ⟨nr⟩is the occupation of state r. The dispersion relation for the neutrino is ϵ = cℏk. Using the quantization condition of k in a box of side L, ki = (2π/L)ni (i = 1, 2, 3) (periodic b.c.), we find ϵ ∝V −1/3. It follows −∂ϵ ∂V = 1 3 ϵ V . Using this relation in the above expression for p, we find p = 1 3V X r ϵr⟨nr⟩= 1 3 E V . 8
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Example 4 - Find area bounded by y = cos x, x = 0, 2pi - Examples Everything Class 6 Class 6 Maths (Ganita Prakash) Class 6 Maths (old NCERT) Class 6 Science (Curiosity) Class 6 Science (old NCERT) Class 6 Social Science Class 6 English Class 7 Class 7 Maths (Ganita Prakash) Class 7 Maths (old NCERT) Class 7 Science (Curiosity) Class 7 Science (old NCERT) Class 7 Social Science Class 7 English Class 8 Class 8 Maths (Ganita Prakash) Class 8 Maths (old NCERT) Class 8 Science (Curiosity) Class 8 Science (old NCERT) Class 8 Social Science Class 8 English Class 9 Class 9 Maths Class 9 Science Class 9 Social Science Class 9 English Class 10 Class 10 Maths Class 10 Science Class 10 Social Science Class 10 English Class 11 Class 11 Maths Class 11 English Class 11 Computer Science (Python) Class 12 Class 12 Maths Class 12 English Class 12 Economics Class 12 Accountancy Class 12 Physics Class 12 Chemistry Class 12 Biology Class 12 Computer Science (Python) Class 12 Physical Education Courses GST and Accounting Course Excel Course Tally Course Finance and CMA Data Course Payroll Course Interesting Learn English Learn Excel Learn Tally Learn GST (Goods and Services Tax) Learn Accounting and Finance Popular GST Demo GST Tax Invoice Format Accounts Tax Practical Tally Ledger List Maths Updated 2025-26 Class 6 Maths (Ganita Prakash) Class 6 Maths (old NCERT) Class 7 Maths (Ganita Prakash) Class 7 Maths (old NCERT) Class 8 Maths (Ganita Prakash) Class 8 Maths (old NCERT) Class 9 Maths Class 10 Maths Class 11 Maths Class 12 Maths Sample Paper Class 10 Maths Sample Paper Class 12 Maths Accounts & Finance Learn Excel free Learn Tally free Learn GST (Goods and Services Tax) Learn Accounting and Finance GST and Income Tax Return Filing Course Tally Ledger List Popular Class 6 Science (Curiosity) Class 6 Science (old NCERT) Class 7 Science (Curiosity) Class 7 Science (old NCERT) Class 8 Science (Curiosity) Class 8 Science (old NCERT) Class 9 Science Class 10 Science Remove Ads Login Maths Remove ads Science GST Accounts Tax Englishtan Excel Social Science Class 6 Class 6 Maths (Ganita Prakash) Class 6 Maths (old NCERT) Class 6 Science (Curiosity) Class 6 Science (old NCERT) Class 6 Social Science Class 6 English Class 7 Class 7 Maths (Ganita Prakash) Class 7 Maths (old NCERT) Class 7 Science (Curiosity) Class 7 Science (old NCERT) Class 7 Social Science Class 7 English Class 8 Class 8 Maths (Ganita Prakash) Class 8 Maths (old NCERT) Class 8 Science (Curiosity) Class 8 Science (old NCERT) Class 8 Social Science Class 8 English Class 9 Class 9 Maths Class 9 Science Class 9 Social Science Class 9 English Class 10 Class 10 Maths Class 10 Science Class 10 Social Science Class 10 English Class 11 Class 11 Maths Class 11 Computer Science (Python) Class 11 English Class 12 Class 12 Maths Class 12 English Class 12 Economics Class 12 Accountancy Class 12 Physics Class 12 Chemistry Class 12 Biology Class 12 Computer Science (Python) Class 12 Physical Education Courses GST and Accounting Course Excel Course Tally Course Finance and CMA Data Course Payroll Course Interesting Learn English Learn Excel Learn Tally Learn GST (Goods and Services Tax) Learn Accounting and Finance Popular GST Demo GST Tax Invoice Format Accounts Tax Practical Tally Ledger List Remove Ads Login Chapter 8 Class 12 Application of Integrals Serial order wise Examples Examples Example 1 Example 2 Important Example 3 Example 4 Important You are here Question 1 Question 2 Question 3 Important Question 4 Important Question 5 Important Question 6 Important Question 7 Question 8 Important Question 9 Question 10 Important Question 11 Important Miscellaneous→ Chapter 8 Class 12 Application of Integrals Serial order wise Ex 8.1 Examples Miscellaneous Case Based Questions (MCQ) Area between 2 curves Example 4 - Chapter 8 Class 12 Application of Integrals Last updated at December 16, 2024 by Teachoo ADVERTISEMENT Powered by VidCrunch Next Stay Playback speed 1x Normal Quality Auto Back 720p 360p 240p 144p Auto Back 0.25x 0.5x 1x Normal 1.5x 2x / Skip Ads by Next: Question 1 →Remove AdsShare on WhatsApp Transcript Example 4 Find the area bounded by the curve 𝑦=cos⁡𝑥 between 𝑥=0 and 𝑥=2𝜋Area OAB = ∫0^(𝜋/( 2))▒〖𝑦 𝑑𝑥〗 𝑦→cos⁡𝑥 = ∫_𝟎^(𝝅/( 𝟐))▒〖𝒄𝒐𝒔⁡𝒙 𝒅𝒙〗 = [sin⁡𝑥 ]_0^(𝜋/2) =sin⁡〖𝜋/2−sin⁡0 〗 =1−0 =𝟏 Area BCD = ∫(𝜋/( 2))^(3𝜋/( 2))▒〖𝑦 𝑑𝑥〗 = ∫(𝝅/( 𝟐))^(𝟑𝝅/( 𝟐))▒〖𝒄𝒐𝒔⁡𝒙 𝒅𝒙〗 = sin⁡𝑥 )^(3𝜋/( 2)) = sin 3𝜋/( 2)−sin⁡〖𝜋/( 2)〗 = – 1 – 1 = –2 Since area cannot be negative Area BCD = 2 Area DEF = ∫(3𝜋/( 2))^2𝜋▒〖𝑦 𝑑𝑥〗 = ∫(𝟑𝝅/( 𝟐))^𝟐𝝅▒〖𝒄𝒐𝒔⁡𝒙 𝒅𝒙〗 = [sin⁡𝑥 ]_(3𝜋/( 2))^2𝜋 =sin⁡2𝜋 −sin⁡〖3𝜋/( 2)〗 = 0−(−1) = 𝟏 Therefore Area Required = Area OAB + Area BCD + Area DEF = 1 + 2 + 1 = 4 square unit Show More Chapter 8 Class 12 Application of Integrals Serial order wise Examples Example 1 Example 2 Important Example 3 Example 4 Important You are here Question 1 Question 2 Question 3 Important Question 4 Important Question 5 Important Question 6 Important Question 7 Question 8 Important Question 9 Question 10 Important Question 11 Important Miscellaneous→ Made by Davneet Singh Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. 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6057
https://brainly.com/question/41905064
[FREE] Solve for r . 3 = -8 + r - brainly.com 5 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 +33,6k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +22,3k Ace exams faster, with practice that adapts to you Practice Worksheets +7,7k Guided help for every grade, topic or textbook Complete See more / Mathematics Textbook & Expert-Verified Textbook & Expert-Verified Solve for r. 3=−8+r 1 See answer Explain with Learning Companion NEW Asked by springLake14 • 11/07/2023 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 9476 people 9K 0.0 0 Upload your school material for a more relevant answer To solve for r in the equation 3= −8+r, add 8 to both sides of the equation to isolate r. The solution is r = 11. Explanation To solve for r in the equation 3= −8+r, we need to isolate r on one side of the equation. To do this, we can start by getting rid of the -8 on the right side by adding 8 to both sides of the equation. This gives us: 3+8= −8+r+8 Simplifying the equation: 11= r Therefore, the solution to the equation is r = 11. Learn more about Solving equations here: brainly.com/question/17595716 Answered by divineSurf29 •163 answers•9.5K people helped Thanks 0 0.0 (0 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 9476 people 9K 0.0 0 Economics - Theory Through Applications - LibreTexts International Trade - Theory and Policy International Economics: Theory and Policy - Steve Suranovic Upload your school material for a more relevant answer To find r in the equation 3=−8+r, we isolate r by adding 8 to both sides. This results in the solution r=11. Explanation To solve for r in the equation 3=−8+r, we start by isolating r on one side of the equation. Here are the steps to solve this equation: Begin with the original equation: 3=−8+r To isolate r, we can add 8 to both sides of the equation. This is done to get rid of the -8 on the right side: 3+8=−8+r+8 Simplifying both sides gives us: 11=r Therefore, we have found that: r=11 This means that the value of r that satisfies the equation is 11. Examples & Evidence For example, if you have another equation such as x+5=12, you can solve for x by subtracting 5 from both sides, giving you x=7. Similarly, the process for solving equations often involves isolating the variable on one side. The steps taken to solve for r use basic algebraic principles, where adding or subtracting the same value from both sides of an equation maintains equality, a foundational concept in algebra. Thanks 0 0.0 (0 votes) Advertisement springLake14 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 Find the reciprocals of the numbers below. The reciprocal of 5 is □ . The reciprocal of 4 7​ is □ . If the maximum weight limit for an elevator is 2500 pounds, which inequality best represents the combined weight, W, that can safely use the elevator? a. W≥2500 b. W≤2500 c. W<2500 d. W>2500 For the following, find the discriminant, b 2−4 a c, and then determine whether one real-number solution, two different real-number solutions, or two different imaginary number solutions exist. What is the discriminant, b 2−4 a c? Rob borrows $15.00 from his father, and then he borrows $3.00 more. Write an equation using negative integers to represent Rob's debt. How much money does Rob owe his father? Which functions are equivalent to f(x)=4 162​x? A. f(x)=16 2 4 x​ B. f(x)=(3 4 2​)x C. f(x)=9 4 2​x D. f(x)=16 2 x 4​ E. f(x)=[3(2 4 1​)]x 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
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https://www.cdc.gov/sickle-cell/media/fact-sheets/best-practices/hemoglobinopathies-current-practices-nbs.pdf
Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up DECEMBER 2015 ASSOCIATION OF PUBLIC HEALTH LABORATORIES 2 ASSOCIATION OF PUBLIC HEALTH LABORATORIES 2 The mark “CDC” is owned by the US Dept. of Health and Human Services and is used with permission. Use of this logo is not an endorsement by HHS or CDC of any particular product, service, or enterprise. This publication was supported by Cooperative Agreement # U60HM000803 funded by the Centers for Disease Control and Prevention. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of CDC or the Department of Health and Human Services. National Center for Immunization and Respiratory Diseases (IP) Office of Surveillance, Epidemiology and Laboratory Services (OSELS) National Center for HIV, Viral Hepatitis, STDs and TB Prevention (PS) National Center for Zoonotic, Vector-borne, and Enteric Diseases (CK) National Center for Environmental Health (NCEH) Coordinating Office for Terrorism Preparedness and Emergency Response (CTPER) © Copyright 2015, Association of Public Health Laboratories. All Rights Reserved. Cover photo: This digitally-colorized scanning electron micrograph (SEM) revealed some of the comparative ultrastructural morphology between normal red blood cells (RBCs), and a sickle cell RBC (left) found in a blood specimen of an 18 year old female patient with sickle cell anemia, (HbSS). Credit: CDC/ Sickle Cell Foundation of Georgia: Jackie George, Beverly Sinclair (photographer: Janice Haney Carr) 1 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up 1 Table of Contents I. Executive Summary 3 II. Acknowledgements 4 III. Introduction to Hemoglobinopathies 5 IV. History of Hemoglobinopathy Screening 6 VI. Methodologies for Hemoglobinopathy Screening and Diagnosis 10 Isoelectric focusing (IEF)  10 High Performance Liquid Chromatography (HPLC)  11 Cellulose Acetate Electrophoresis (Alkaline)  12 Citrate Agar Electrophoresis (Acid)  13 Alkaline Globin Chain Electrophoresis  13 Capillary Zone Electrophoresis  13 Molecular Methods  15 VII. Method Advantages, Limitations and Testing Strategy 18 VIII. Algorithms for Hemoglobinopathy Detection 22 Unknown Hemoglobin Variants  22 IX. Quality Assurance for Hemoglobinopathy Screening and Testing 26 Pre-analytical Phase  26 Analytical Phase  26 Post-analytical Phase  27 ASSOCIATION OF PUBLIC HEALTH LABORATORIES 2 X. Follow-up 28 Follow-up for Hemoglobinopathies  28 XI. References 31 XII. Appendices 34 Appendix A: Examples of Molecular Methods for Hb Mutation Detection  34 Appendix B: Top 20 Hemoglobinopathies in Four States  37 Appendix C: State 1 DNA Screening Algorithm  39 Appendix D: Hemoglobinopathy Screening Algorithm for State 1  40 Appendix E: Hemoglobinopathy Screening Algorithm for State 1 – Older Children & Adults  41 Appendix F: State 1 Hemoglobinopathy Result Codes  42 Appendix G: State 2 Hemoglobin Coding System  46 Appendix H: State 3 Hb Reporting Algorithm  50 Appendix I: State 4 Hb Screening Algorithm  51 Appendix J: State 4 Hb Result Codes  52 3 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up I. Executive Summary The hemoglobinopathies are a group of disorders passed down through families (inherited) in which there is abnormal production or structure of the hemoglobin molecule. Sickle cell disease (SCD) is one such blood disorder caused by the abnormal hemoglobin that damages and deforms red blood cells. The abnormal red cells break down, causing anemia, and obstruct blood vessels, leading to recurrent episodes of severe pain and multi-organ ischemic damage. SCD affects millions of people throughout the world and is particularly common among people whose ancestors come from sub-Saharan Africa, regions in the Western Hemisphere (South America, the Caribbean, and Central America); Saudi Arabia; India; and Mediterranean countries such as Turkey, Greece and Italy. There is no widely available cure for SCD although some children have been successfully treated with blood stem cell, or bone marrow, transplants.1 However, hematopoietic stem cell transplant is not widely done for SCD, because of the difficulty in finding a matched donor. Therefore, the number of people with SCD who get transplants is low. In addition, there are several complications associated with the procedure, including death in about 5 percent of people. In SCD, clinical severity varies, ranging from mild and sometimes asymptomatic states to severe symptoms requiring hospitalization. Symptomatic treatments exist, and newborn screening (NBS) for SCD can reduce the burden of the disease on affected newborns and children. Thalassemia is another type of blood disorder that is caused by a defect in the gene that helps control the production of the globin chains that make up the hemoglobin molecule.2 There are two main types of thalassemia: • Alpha thalassemia occurs when a gene or genes related to the alpha globin protein are missing or changed (mutated). Alpha thalassemias occur most often in persons from Southeast Asia, the Middle East, China and in those of African descent. • Beta thalassemia occurs when a beta globin gene is changed (mutated) so as to affect production of the beta globin protein. • Beta thalassemias occur most often in persons of Mediterranean origin. To a lesser extent, Chinese, other Asians and African Americans can be affected. In 2013, the National Center on Birth Defects and Developmental Disabilities, Division of Blood Disorders (DBD), Centers for Disease Control and Prevention (CDC) in collaboration with the Association of Public Health Laboratories (APHL) Newborn Screening and Genetics in Public Health Program convened an APHL Hemoglobinopathy Laboratory Workgroup to address issues around hemoglobinopathy laboratory testing. The workgroup objectives are as follows: • Discuss the issues related to building and enhancing US laboratory capacity in the areas of screening and diagnosis of hemoglobinopathies. • Conduct an inventory of state and regional labs that are currently performing (or have the capacity to develop) sickle cell disease laboratory testing. • Develop a training program for implementing laboratory technology in state public health laboratories, universities, and community centers for testing. • Consult with CDC and other partners to evaluate current laboratory methodologies and make recommendations for improvements. • Identify and document best practices that have been used in SCD public health laboratory training; identify training standards and competency outcomes. One of the products of the APHL Hemoglobinopathy Laboratory Workgroup is this guidance document on hemoglobinopathy laboratory testing and follow-up techniques. This document explores current screening and diagnostic methods available that are currently employed by some screening and diagnostic laboratories. It also includes an overview of laboratory structure discussing algorithms for testing, reporting and follow-up from several programs in the US. Furthermore, this document aims to improve and strengthen US and international capabilities by offering best practices thereby contributing to the goal of early detection of hemoglobin disorders. ASSOCIATION OF PUBLIC HEALTH LABORATORIES 4 II. Acknowledgements The Association of Public Health Laboratories gratefully acknowledges the contributions of the APHL Hemoglobinopathy Laboratory Workgroup, experts and partners at the Newborn Screening and Molecular Biology Branch, National Center for Environmental Health and the Division of Blood Disorders, National Center on Birth Defects and Developmental Disabilities. Maria del Pilar Aguinaga, PhD, MT (ASCP), MB (ASCP), DLM (ASCP) Sickle Cell Center, Meharry Medical College Ming Chan, PhD Florida Department of Health Tim Clark Davis, BS Washington State Newborn Screening Laboratory M. Christine Dorley, MSP, BS MT(ASCP) Tennessee Department of Health/Laboratory Services Bonifacio Dy, MD Nevada State Public Health Laboratory Marie Earley, PhD Centers for Disease Control and Prevention Althea M. Grant, PhD Centers for Disease Control and Prevention Mary Hulihan, MPH Centers for Disease Control and Prevention Suzanne Karabin, MS, CGC New Jersey Department of Health Joanne Mei, PhD Centers for Disease Control and Prevention Christine Moore, MT (ASCP) Texas Department of State Health Services Laxmi Nayak, MS New Jersey Department of Health Kwaku Ohene-Frempong, MD The Children’s Hospital of Philadelphia 5 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up III. Introduction to Hemoglobinopathies Hemoglobin is a tetramer composed of two α-globin and two non- α -globin chains working in conjunction with heme to transport oxygen in the blood.2,3 Normal adult hemoglobin (HbA) is designated αA 2βA 2.2,3 Variant hemoglobin is derived from gene abnormalities affecting the α-globin genes (HBA1 or HBA2) or β-globin (HBB) structural genes (exons).2,3,4 More than a thousand hemoglobin variants have been identified relative to changes in the globin chains.3 Qualitative changes correspond to amino acid substitutions resulting in hemoglobinopathies. Quantitative changes like amino acid insertions, deletions or mutations in the intervening sequences (introns) correspond to thalassemia and result in decreased globin chain production.2,3,4 Alpha thalassemias are caused by changes (deletions, point mutations, insertions, etc.) in the α-globin genes. Production of α-globin is controlled by the four alleles of HBA1 and HBA2. In the deletional type α-thalassemias, the number of α-globin gene deletions correlates to disease severity.4 One α-globin gene deletion is unremarkable (also called silent carrier) whereas a two α-globin gene deletion (α-thalassemia trait) and three α-globin gene deletion (HbH disease) have varied clinical and hematological features. A four α-globin gene deletion (Hb Bart’s Hydrops fetalis) is severe and not typically compatible with life.2,4 Beta globin variants more commonly seen include HbS, HbC, HbD, HbE and HbG. A mutation in one β-globin subunit results in a combination of variant and normal hemoglobin and denotes carrier or trait status, also known as the heterozygote state. Mutations in both β-globin subunits result in disease based on a homozygous or heterozygous expression. In the case of sickle cell anemia (HbSS), mutations are homozygous with production of HbS.2 Other diseases classed under sickle cell disease (SCD), for example HbSE, HbSC and HbSβ-thalassemia are heterozygous expressions. Regardless of an α-globin or β-globin variant, severity of disease can range from insignificant to serious or life threatening.2,3,4,5 Therefore, early detection through newborn screening is paramount.6,7 Hemoglobinopathies, specifically HbSS, HbS/β-thalassemia and HbSC disease were added to the Recommended Uniform Screening Panel (RUSP) in 2006. The Maternal and Child Health Bureau of the Health Resources and Services Administration (HRSA) of the United States (US) in conjunction with the American College of Medical Genetics (ACMG) determined these disorders to be clinically significant and included them as core targets easily detected by newborn screening. These were added based on the severity of illness associated with sickle cell disorders.7 Symptoms range from anemia to severe pain and vaso-occlusive crises eventually affecting multiple organ systems with chronic deterioration over time.6 Early detection of SCD reduces the risk of invasive Streptococcus pneumoniae (pneumococcus) infection through penicillin prophylaxis. In addition, early diagnosis of SCD prior to onset of symptoms or complications, allows health workers to educate the family about SCD and offer anticipatory guidance, in the context of comprehensive care of the child. Additional hemoglobinopathies readily detected by newborn screening were also added as secondary targets.6 Examples include HbE disease, HbC disease, HbSE disease, etc. Methods for newborn screening and diagnosis differ across laboratory programs. Some methods detect multiple variants whereas others detect only the most common. While some methods are automated, others are manual and labor intensive. Diagnostic methods include DNA based applications with some procedures requiring sophisticated instrumentation. Programs are also structured differently. Some laboratories are screen-only labs whereas others screen and perform diagnostic testing. Some use a second tier assay to confirm an abnormal result prior to patient referral for diagnosis. In addition, reporting and follow-up algorithms differ between programs. Based on these differences a comprehensive review of methodologies and program structures are warranted. ASSOCIATION OF PUBLIC HEALTH LABORATORIES 6 IV. History of Hemoglobinopathy Screening Prior to 1960s, Sickle cell disease (SCD) was diagnosed mainly through hematological studies and clinical manifestations. In the late sixties, only a few states screened some newborns for sickle cell disease, among a handful of genetic disorders. In 1971, in response to pressure by African American advocacy groups, the US government made sickle cell disease a scientific and health care priority by allocating funds for treatment and research. Several Sickle Cell Centers were created in the country, mainly associated with universities or medical centers, including some at medical schools of Historically Black College and Universities.8 In 1973, a survey of annual workloads of US state public health laboratories known as The Consolidated Annual Report (CAR) showed that 12 state public health laboratories had some form of sickle cell screening program. (Table 3-5, Consolidated Annual Report, 1973).9 On May 16, 1972, under President Nixon, the US Congress signed into law the National Sickle Cell Disease – Sickle Cell Anemia Control Act (Public Law 92-294).8 In 1975, the first universal newborn screening (NBS) program for SCD was implemented in New York on a pilot basis.10 Subsequently, screening was adopted by the remaining states throughout the late 1980s and 1990s and into the 2000s. Initially, screening was targeted for populations at risk, primarily African Americans. However, the missed cases rate was higher than 30% due to difficulty in identifying an infant’s race or ethnicity at birth and so universal screening was implemented.11 Later on, other hemoglobinopathies were acknowledged as important public health issues. Since infection from capsulated organisms caused high mortality and morbidity in children with SCD, penicillin prophylaxis was introduced in 1986 for infants with SCD.12 Penicillin prophylaxis significantly reduced infection-related mortality and became a powerful incentive to implement widespread neonatal screening for SCD.12,13,14,15 Since May 1, 2006, all 50 states and the District of Columbia require and provide universal newborn screening (NBS) for SCD and other hemoglobinopathies even though national recommendation was made over a quarter of century ago (1987).9,12 NBS is well recognized as the largest and most successful health promotion and disease prevention system in the United States. NBS is the practice of testing every newborn for certain harmful or potentially fatal conditions that are not otherwise apparent at birth. A sample is collected before the newborn leaves the hospital and identifies serious, life-threatening conditions before symptoms begin. Although such conditions are usually rare, they can affect a newborn’s normal physical and mental development. Early detection is crucial in NBS since intervention can prevent death or a lifetime of severe disabilities. Along with the initial newborn screening goal to identify SCD, other hemoglobin disorders such as, beta (β) and alpha (α) thalassemia have gained significant attention in recent years due to the rapidly changing demographics in the US as a result of increased immigration. Hence, hemoglobin disorders common in other countries are being seen more frequently in the US.16 As it pertains to screening techniques, laboratories originally used citrate agar electrophoresis for screening of abnormal hemoglobins in cord blood. The subsequent improvement in the hemoglobinopathy electrophoretic techniques, made it possible to screen newborns by using either cord blood or heel stick samples (dried blood spots). Currently, the majority of hemoglobinopathy screening programs use a combination of isoelectric focusing (IEF) and high performance liquid chromatography (HPLC) as primary screening methods (Table 1). Many programs use second complimentary electrophoretic technique, HPLC, immunologic tests or DNA-based assays to confirm specimens with abnormal screening results.13 See Table 1 for a distribution of laboratories versus methods utilized. Most of the current screening methodologies are sensitive and specific in detecting high risk infants; however, each method has its own unique limitations. 7 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Table 1: Number of Newborn Screening Quality Assurance Program Proficiency Testing Participants Using Multi-Level Schemes to Enhance the Specificity of Screening for Hemoglobinopathies (2014) Method Isoelectric Focusing Bio-Rad HPLC Extended Gradient HPLC Trinity Biotech (Primus) Ultra2 HPLC Electrophoresis Citrate Agar Electrophoresis Alkaline Cellulose Monoclonal Antibody Methods DNA Amplification Level 1 33 41 2 0 0 0 1 0 Level 2 19 13 5 5 1 1 0 1 Level 3 2 2 1 0 2 0 0 0 Level 4 1 0 0 0 0 0 0 0 Screening programs using any of the screening methods must maintain high-quality results to ensure accurate interpretation of the phenotypes for immediate initiation of supportive care for infants affected with hemoglobinopathies. ASSOCIATION OF PUBLIC HEALTH LABORATORIES 8 V. Types of Specimens for Hemoglobinopathy Screening and Diagnosis At one time, programs in the US required cord blood for hemoglobinopathy screening; however programs now use dried blood spots (DBS). DBS are collected from a heel prick and spotted onto filter paper.17 For additional information on newborn screening specimen collection refer to Clinical and Laboratory Standards Institute (CLSI) document NBS01-A6, Blood Collection on Filter Paper for Newborn Screening Programs.18 Newborn screening specimens should be properly collected to avoid clotting, smearing, inadequately filled circles, oversaturation or scratching by capillary tube. Although these specimens may be unsatisfactory for some newborn screening tests, they may be acceptable for hemoglobinopathy testing (see NBS01-A6 for details). DBS and liquid whole blood are used for screening and/or confirmation of children (greater than one year of age) and adults. There are changes in hemoglobin types and quantities through the first year of life, but after a year of age the hemoglobin types of an individual are going to remain the same. See Figure 1 and Table 2 for changes in relative proportions of globin chains at various stages of embryonic, fetal and post-natal life.19 Table 2 lists the percentages of normal and variant hemoglobin seen in newborns and older children. Figure 1: Globin switch during in utero and post-natal life Figure courtesy of Deepak Kamat, MD, PhD, FAAP via Healio.com 9 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Table 2: Normal and Variant Hemoglobin at Birth and in Older Children For the collection of whole blood samples, Ethylene Diamine Tetra Acetic Acid (EDTA) is the typical anti-coagulant used. Heparin may interfere with DNA amplification by Polymerase Chain Reaction (PCR). For DBS collected from a finger stick, the palmar surface of the finger’s last phalanx is most frequently used, see CLSI document GP42-A6 for additional information.20 High heat and humidity can change the levels of hemoglobin A and S in DBS samples.21 To maintain the integrity of hemoglobin molecules, high humidity and temperature should be minimized during transport and storage. Blood smears should not be used for hemoglobinopathy screening. A smear is unreliable as it requires the presence of sickled cells which may or may not be in circulation at the time of collection. Blood smears also cannot differentiate homozygous from heterozygous nor can they detect other hemoglobin variants.22 Transfusion may affect hemoglobinopathy screening results. The transfused blood can mask a hemoglobinopathy, or the transfused blood may contain a hemoglobin variant that does not belong to the patient. In cases of transfusion, hemoglobinopathy screening should be repeated four months post transfusion according to the Clinical and Laboratory Standards Institute (CLSI)23 or according to individual program guidelines. There are several different methods as next described that are used for hemoglobinopathy screening and confirmation. Specific specimen requirements may vary between these methods. Figure courtesy of Deepak Kamat, MD, PhD, FAAP via Healio.com ASSOCIATION OF PUBLIC HEALTH LABORATORIES 10 VI. Methodologies for Hemoglobinopathy Screening and Diagnosis Various methods for hemoglobinopathy newborn screening and adult testing are employed by US laboratories. Many programs traditionally used electrophoresis as the method of choice for identification and quantification of aberrant hemoglobins.24 In more recent years, with the introduction of universal screening of newborns, technology has emerged that is robust with high throughput and greater sensitivity over electrophoresis.24,25 Many of the same methods used for screening, such as Isoelectric focusing (IEF) and High performance liquid chromatography (HPLC) are also used for diagnosis. Some laboratories employ one or more methods in their protocol for detection of hemoglobinopathies. Choice of method depends on the function of the laboratory and its testing algorithm. Molecular methods are emerging and are being utilized by some screening and diagnostic labs alike to obtain a more comprehensive analysis and determine the nature of the hemoglobinopathy and its clinical implications. Below are descriptions of methods used. Descriptions include traditional methods requiring a hands-on approach, more current procedures requiring automation and lastly, molecular assays. Isoelectric focusing (IEF) IEF separates proteins in a gel medium that has a pH gradient consisting of ampholytes (zwitterions). When a high voltage is applied, narrow buffered zones are created with stable, but slightly different pHs (Figure 2). Slower moving proteins migrate through these zones and stop at their individual isoelectric points (pI).3 In the case of hemoglobin, these migrate to a zone in the medium where the pH of the gel matches the hemoglobin’s pI. At the pI, the charge of the hemoglobin becomes zero and ceases to migrate. The hemoglobin migration order of IEF is similar to alkaline electrophoresis. Resolution is clear with differentiation of HbC from HbE and HbO and HbS from HbD and HbG respectively. HbA and HbF are also clearly differentiated.24 With neonatal specimens, a distinct band representing acetylated HbF is readily identified and slightly anodic to HbA.26 Staining may be necessary depending on the manufacturer’s recommendations. Gels may be read wet or dry and band identification is accomplished by comparison to migration patterns of known quality controls.26 This method allows for greater precision and accurate quantification than standard electrophoresis. It gives excellent resolution in addition to high throughput, but is also labor-intensive and time consuming.3,24,26 An alternative method is needed to confirm or differentiate hemoglobin variants (bands other than HbF or HbA).26 Fast migrating bands (Hb Bart’s, HbH) can also be identified by IEF. 11 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up High Performance Liquid Chromatography (HPLC) Hemoglobins are separated by an analytical cartridge in cation exchange HPLC using a preprogrammed buffer gradient with increasing ionic strength to the cartridge (Figure 3). The hemoglobin fractions separate based on their ionic interaction with the cartridge.26 The separated fractions pass through a flow cell, where absorbance is measured at 415 nm and again at 690 nm to reduce background noise. Changes in absorbance are monitored over time producing a chromatogram (absorbance vs. time). Each hemoglobin has its own characteristic retention time and is measured from the time of sample injection into the HPLC to the maximum point of each peak. Identification of unknown hemoglobin is achieved through comparison with known hemoglobin retention times.27 If a peak elutes at a retention time not predetermined, it is labeled as an unknown. HPLC achieves good separation and quantitation of HbF and HbA2 in addition to screening for variant hemoglobins along with thalassemia. HPLC is highly reproducible, offers simplicity with automation, superior resolution and rapid results.27 Some HPLC instrument programs can identify hemoglobinopathies from both newborns and adult specimens while others identify only one or the other. Identification between adult and newborn specimens depends on the algorithm/software/instrument specifications. Figure 2: Isoelectric Focusing ASSOCIATION OF PUBLIC HEALTH LABORATORIES 12 The following methods are used less frequently than those described above. Some methods have a long history within laboratories and continue to be used to clarify and confirm abnormal hemoglobins. Newer technology, like molecular methods, although utilized by few programs, is gaining momentum with some programs. Cellulose Acetate Electrophoresis (Alkaline) Cellulose Acetate Electrophoresis or alkaline electrophoresis makes use of the negative charge which hemoglobin will adopt under alkaline conditions. Samples are applied to cellulose acetate agar gel and hemoglobins are separated by electrophoresis using an alkaline buffer (Tris-EDTA with Boric Acid) at pH 8.4.28 Each hemoglobin variant carries a different net charge so it will migrate at varying speeds. Following electrophoretic migration, visualization of hemoglobin bands are accomplished through staining with Ponceau S, Amino Black and Acid Violet or other similar stains and compared with known standards.3,26,28 Hemoglobins are quantified using densitometric scanning and the relative percentage of each band is determined.26,28 This method yields rapid and reproducible separation of HbA, HbF, HbS and HbC as well as other variant hemoglobins with minimal preparation time. However, due to limited sensitivity and because some hemoglobins are electrophoretically similar though structurally different, an alternative procedure must be incorporated into the screening algorithm for differentiation of these hemoglobins. For example, HbS, HbD, HbG and Hb Lepore co-migrate, so they are indistinguishable on alkaline electrophoresis. The same is true for HbC, HbA2, HbO-Arab and HbE.13,24,28 Figure 3. High-performance Liquid Chromatography Schematic representation of an HPLC unit. (1) Solvent reservoirs, (2) Solvent degasser, (3) Gradient valve, (4) Mixing vessel for delivery of the mobile phase, (5) High-pressure pump, (6) Switching valve in “inject position”, (6’) Switching valve in “load position”, (7) Sample injection loop, (8) Pre-column (guard column), (9) Analytical column, (10) Detector (i.e. IR, UV), (11) Data acquisition, (12) Waste or fraction collector. “HPLC apparatus” by WYassineMrabetTalk. This vector image was created with Inkscape. - Own work. Legend based on : Practical High-performance Liquid Chromatography by Veronika Meyer, 4th edition, John Wiley and Sons, 2004, ISBN 0470093781, p. 7 and chromatography-online.org. Used file: Computer n screen.svg (Crystal SVG icons). Licensed under CC BY-SA 3.0 via Wikimedia Commons 13 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Citrate Agar Electrophoresis (Acid) Electrophoresis occurs in an acidic environment at pH 6.2.28 This method is based on the complex interactions of the hemoglobin with the electrophoretic buffer and the agar support. Staining allows visualization of hemoglobin bands. Acid electrophoresis allows confirmation of variant hemoglobins observed in the Cellulose Acetate Electrophoresis procedure and allows good separation of HbC from HbE and HbO-Arab.13 It permits additional separation of HbS from HbD and HbG.23,28 Citrate Agar electrophoresis appears to be more sensitive than Cellulose acetate for detecting HbF. Alkaline Globin Chain Electrophoresis The hemoglobin molecule can be separated into its globin chain components and heme groups by the addition of 2-mercaptoethanol and urea. At alkali pH, these globin chains will migrate to their characteristic position on a cellulose acetate strip, under an electrical field. The characteristic migration pattern aids in the identification of each globin chain type.29 Capillary Zone Electrophoresis Charged molecules are separated by their electrophoretic mobility in an alkaline buffer (pH 9.4) with separation occurring according to the electrolyte pH and endosmosis or electroosmotic flow.25 Capillaries function in parallel allowing eight simultaneous analyses. Samples are hemolyzed and injected into the anodic end of the capillary. High voltage protein separation is performed and hemoglobins migrate from the anodic end of the capillary appearing in specific zones to the cathodic end where detection occurs at 415 nm. Results are assessed visually for abnormalities with identification of normal and disease patterns. There is improved detection of sickle cell disease due to separation of hemoglobin fractions which enables differentiation of HbS from other variants. Capillary zone electrophoresis allows clean separation of HbE from HbA2 and facilitates easier detection of HbBart’s and HbH.25 Anode ------------------Cathode + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Detection Injection of proteins Electro-endosmotic flow Electrical field Principle of hemoglobin electrophoresis using capillary electrophoresis technologies Figure 4. Principle of Hemoglobin Electrophoresis Using Capillary Electrophoresis Technologies ASSOCIATION OF PUBLIC HEALTH LABORATORIES 14 Thermic bridge Temperature Controlled by Peltier device Anode + Cathode -Injection < 1nl Silica Capillary in thermo-conductive resin (25µm diameter) Migration Up to 10,000 volts LED lamp Detector CAPILLARYS Diagram NET FLOW OF BUFFER IS TOWARDS THE NEGATIVE ELECTRODE (CATHODE) Diffuse layer Anode (+) Cathode (-) Capillary wall Rigid layer E O F Figure 6. Net Flow of Buffer towards the Negative Electrode (Cathode) Figure 5. Capillarys Diagram 15 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Molecular Methods Molecular testing for hemoglobinopathies generally uses three techniques. These are Restriction Fragment Length Polymorphism (RFLP), Allelic Discrimination using Real Time PCR end point data and DNA Sequencing. DNA extraction of whole blood in the DBS filter paper matrix is needed for all PCR-based assays. Methods for DNA extraction include crude boiling preparations, alkaline lysis preparations and commercial methods. Each one has its own benefits and limitations, and each laboratory needs to determine the best method for their applications.30 Considerations for all Polymerase Chain Reaction (PCR) based assays All molecular assays using PCR are susceptible to contamination by aerosolized amplicons. To minimize the risk of contamination, at a minimum, assay set-up and amplification/analysis must be separated and one-way directional work flow is required.31 Depending upon the volume of specimens tested and nature of the work, laboratories may require shoe covers and hair caps in addition to a laboratory coat and gloves. Each PCR run must include a positive control (a genotype positive DBS or genomic DNA i.e., SS, SC, AS, AC, etc.) and a negative control, also referred to as a no template control (no DNA control i.e., water or reagent only). The positive control verifies that the amplification occurred and the negative control will detect contamination of reagents. Restriction Fragment Length Polymorphism (RFLP) RFLP takes advantage of recognition sequences of restriction enzymes that correspond to the normal allele or the mutated allele of a gene.32 If the recognition sequence is present, the restriction enzyme will restrict (cut) the DNA and the size difference can be visualized by gel electrophoresis (Appendix A, Figure 1). To use RFLP, DNA is first extracted from DBS and amplified using polymerase chain reaction (PCR). The amplified DNA is added to the restriction enzyme reaction and incubated for a prescribed amount of time. An aliquot of the mixture is electrophoresed and the size(s) of the fragment(s) is visualized after staining, photographed and analyzed. For example, the S mutation in the beta-globin B (HBB) gene replaces an adenine with a thymine resulting in an amino acid change from a glutamic acid to valine. This change destroys the DdeI restriction enzyme recognition site. After amplifying a 125bp fragment of the HBB gene, an aliquot of the PCR reaction is added to a mixture containing the enzyme DdeI. After a prescribed amount of time, an aliquot of this reaction and a DNA molecular weight marker or DNA ladder is electrophoresed on an agarose gel. If the specimen has normal hemoglobin (HbAA), the enzyme cuts the DNA into 106bp and 19bp fragments. The 106bp band is visible, but the 19bp band is too small to see on the gel. If the only product seen is 125bp, both alleles have the S mutation (SS). The specimen would be considered to have both an A and S allele (AS) and is classified as a carrier if both the 125 and 106 bp bands are seen (Appendix A, Figure 1).32 The benefits of this method are that it is very easy to perform and analyze and is inexpensive compared to the other molecular methods described below. The only instrumentation required is a PCR machine, a water bath, an electrophoresis unit and a camera. It is a low throughput method because there are a number of manual steps and the amount of time it takes to complete the assay; however it is a good choice for laboratories that have a small number of samples to test. A limitation of RFLP is that partial restriction (incomplete cutting) of the DNA could result in a homozygous specimen being interpreted incorrectly. A positive control of a known genotype is crucial to identify this problem. In cases of a double mutation like SC, if the restriction enzymes do not have similar activity conditions, two separate reactions are required. ASSOCIATION OF PUBLIC HEALTH LABORATORIES 16 Allelic Discrimination using Real-time PCR Allelic Discrimination (AD) measures fluorescence at the end of PCR to determine if a mutation is present. For this technique, a forward and reverse primer spanning the area of interest are used as well as hydrolysis probes.32,33 Hydrolysis probes are oligonucleotides that have a fluorophor bound to the five prime (5’) end and a quencher molecule bound to the three prime (3’) end. When the quencher is in close proximity to the fluorophor, fluorescence cannot be detected. After DNA is extracted from the specimen, a PCR reaction with the forward and reverse PCR primers, DNA polymerase, nucleotide triphosphates, (dNTPs) and two probes is prepared. One probe corresponds to the normal sequence and the other corresponds to the mutant sequence. The fluorophor bound to each primer emits fluorescence at different wavelengths and can be distinguished. During PCR, the primers and probes bind to their complementary sequences. When DNA polymerase encounters the bound probe, the polymerase’s 5’–3’ exonuclease activity degrades it. The fluorescent molecule is released, separating it from the quencher and the fluorescence can be measured (Appendix A, Figure 2). Because amplification occurs exponentially, there is a large increase of fluorophors that are no longer in proximity to the quenchers. At the end of the amplification, the fluorescence is measured in a real-time PCR machine and the results are analyzed by the software. If the signal from only one probe is detected, the sample is homozygous for the sequence complementary to that specific probe. If the signals from both probes are detected, then the sample has both a mutant allele and a normal allele and is considered a carrier. This method has a higher throughput than RFLP because the results are generated right after the amplification ends, with no further processing. All reactions take place in the PCR tube which never has to be opened, thus reducing the risk of contamination. The throughput can be increased by using 96- or 364-well PCR plates and automation. Limitations include an increased cost for the probes and the need for a real-time PCR machine. Although AD is considered a multiplexed assay, the limited number of filters in the PCR machines does not allow for a higher order of multiplexing. DNA Sequencing Sanger sequencing is the most comprehensive method of mutation detection and determines the exact sequence spanning the area of the primers used. Once DNA is extracted from the sample, it is amplified by PCR but in addition to the primers, the DNA polymerase and dNTPs, dideoxynucleotides (ddNTPs) are added. Each of the four ddNTPs has a specific fluorophor bound to it, which emits a signal at a different wavelength. During amplification, when one of the ddNTPs is incorporated, extension stops because the lack of a hydroxyl group does not allow for the addition of the next base. The ratio of dNTPs to ddNTPs is optimized to generate fragments that are one base different than the next. After amplification, the products are purified, typically by DNA precipitation or by spin column. They are loaded onto a capillary electrophoresis instrument that separates the products by size based on their charge. When the products move across the path of a laser in the instrument, each fluorophor emits a signal at a different wavelength and this information is captured, analyzed by the instrument software and the sequence is displayed (Appendix A, Figure 3).34 The greatest benefit of this method is that every base change in the area of interest is determined including single nucleotide polymorphisms, small insertions and small deletions. Sequencing is a high-throughput method but all the sequence data must be reviewed by a trained technician, which can take a large amount of time depending on the length of sequence and the number of samples. For newborn screening laboratories, if sequencing is used for hemoglobinopathy screening, only the beta-globin gene, HBB, is usually sequenced because it is relatively small and the most common mutations tested are point mutations in HBB. The alpha-globin genes, HBA1 and HBA2, have greater than 96% sequence homology making sequencing of these genes more complicated. Many of the common alpha-thalassemia mutations are large deletions which cannot be found by sequencing 17 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up but can be found using gap PCR or Multiplex Ligation-Dependent Probe Amplification. Sequencing is the most expensive method compared to RFLP and AD because of the reagents, instrumentation, personnel, and review time required for analysis, but it provides the most comprehensive data for the beta-globin gene. Programs use different approaches for hemoglobinopathy screening, confirmation and diagnosis. Newborn screening programs however do not routinely screen for the purpose of detecting alpha or beta thalassemias although screening methods may detect some forms of these hemoglobinopathies. Clinical presentation in conjunction with hematologic features exhibited in the complete blood count (CBC) and peripheral blood smear as well as iron studies aid in confirmation and diagnosis of thalassemias. Depending on the type of thalassemia diagnosed, family studies may be warranted along with genetic counseling.24 To further improve screening and diagnosis of hemoglobinopathies including thalassemias, some laboratories are exploring the use of tandem mass spectrometry and whole genome sequencing. These methods are promising improved sensitivity and specificity for hemoglobinopathies. However as procedures advance, others become less ideal for screening. For example, HbS solubility test, although simple and easy to use, is characterized by false positives from other hemoglobin types and interfering substances that may be present in the sample. False negatives are commonly seen in patients with low hemoglobin or hematocrit levels. Other abnormal hemoglobins are not readily detected and result interpretation can be difficult due to the subjectivity of the test.25 Lastly, the HbS solubility test cannot differentiate between carrier and disease state for sickle cell disease. Based on these limitations, solubility testing is not recommended. Additionally, when deciding on which method is most appropriate for hemoglobinopathy detection, it is necessary to consider the benefits the method will bring to the individual program. Likewise, it is important to consider expense, throughput, ease of use and skills required for implementation and routine analysis. Methods also have different limitations that must be evaluated. For example, peak or band resolution, differentiation of hemoglobins and result interpretation are important limitations to evaluate. The next section explores some of the limitations of the more commonly used applications. ASSOCIATION OF PUBLIC HEALTH LABORATORIES 18 VII. Method Advantages, Limitations and Testing Strategy As of 2015, half of all labs participating in the CDC Newborn Screening Quality Assurance Program’s proficiency testing report that they are using IEF as their primary hemoglobinopathy method and the other half are using HPLC (Table 1). Both have advantages and limitations, therefore each program must examine them to determine which method is best for their lab. The optimal strategy is to combine multiple methods to confirm abnormal initial results. The reason for this is what appears to be a common trait on one method might actually be different on another. In these circumstances the reported result would be Hb variant trait. Isoelectric focusing has the advantage of allowing the reader to view as many as 80 hemoglobin profiles per gel. This is very beneficial in high volume laboratories. Variant bands are easily distinguished on a gel when compared alongside normal Hb FA profiles and appropriate controls. However, hemoglobin bands can leak into neighboring profiles, be poorly focused, distorted or contaminated creating subjective interpretations. This can be overcome by having multiple analysts enter results followed by an experienced analyst who makes the final call. It is also difficult to quantitate hemoglobin bands when reading IEF patterns. Gel preparations are also labor intensive, time consuming, and involve multiple steps. Figure 7. Normal Hemoglobin (HbAA) by IEF 19 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up This example clearly illustrates the difference between HbS and HbD. Also note that there is no Hb A in the FSS profile. This child either has sickle cell anemia or Sickle beta zero, which will later be confirmed by DNA testing. The image above is a good example of both alpha thalassemia (FAB2) and Hb S trait. FAS is also known as S trait due to the fact that it is a heterozygote (only 1 copy of chromosome 11 has the mutation for βS). In a homozygous S, there is no A. Homozygous S is also called sickle cell disease and the phenotype code is HbSS. FA FAB2 FA FAS FA “Here is a good example of a FAB2 and FAS. FAS is also known as S trait due to the fact that it is a heterozygote (only 1 chromosome has the mutation). In a homozygous S there is no Hb A. Homozygous S is also called sickle cell disease and the phenotype code would be FSS. Figure 8. Sickle Cell Trait (Hb AS) by IEF FAD AD Control FSS AFSC Control FAB1 “This example was chosen because it clearly illustrates the difference between Hb S and D. Also note that there is no Hb A in the FSS profile. This child either has sickle cell anemia or S beta zero thalassemia which will later be confirmed by DNA testing.” No Hb A S D Figure 9. Sickle Cell Disease (HbSS) by IEF ASSOCIATION OF PUBLIC HEALTH LABORATORIES 20 Capillary zone electrophoresis is commonly used in diagnostic laboratories and is an excellent choice in situations where few samples need to be run. Due to resolution and sensitivity, however, not all hemoglobin variants are detectable by this method. Similar to HPLC, this system also has problems with filter paper plugging. Figure 10. Sickle Cell Disease (HbSS) by HPLC An advantage of using HPLC is that it typically includes software that calculates the percent area of each hemoglobin peak. This helps with the determination of thalassemias and phenotype reporting. Peaks within retention time windows can be identified by software algorithms, which also assists the analyst in making the final determination. However, the software determination should not be used to assign final phenotype results without examining each chromatogram. HPLC resolution can be low in some systems causing variant bands to be overlooked. A known issue with primary HPLC screening is system plugging due to filter paper fibers from the blood spot. 21 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Less commonly, citrate and cellulose acetate are being used by some laboratories. These methods have limited sensitivity and require alternative methods for differentiating abnormal hemoglobins sharing the same or similar electrophoretic mobility. Interpretations can be subjective. Molecular testing can be added to resolve cases when the newborn has been transfused with packed red blood cells. Since the newborn’s phenotype is masked by the donor, DNA testing can be used to identify any abnormal hemoglobins. This method is limited by workspace issues required for unidirectional flow, training and relatively high cost. Also, it is difficult to obtain proficiency and quality control materials which are required for certification through accreditation bodies. Despite limitations and advantages, programs must consider how to effectively implement testing protocols for optimal workflow and efficiency for hemoglobinopathy detection. Specific examples of algorithms for hemoglobinopathy detection are provided in the Appendix. All algorithms presented despite their corresponding program aid in meeting the goal of early detection and diagnosis of hemoglobinopathies. Figure 11. HPLC Sickle Trait ASSOCIATION OF PUBLIC HEALTH LABORATORIES 22 VIII. Algorithms for Hemoglobinopathy Detection NBS programs have varying approaches to screening and reporting. Examples of this are provided in the Appendices starting on page 34. However, the goal is the same, to screen for those individuals who may be at risk for hemoglobinopathies. The basic screening process entails use of a primary method with some labs opting to use a secondary method depending on the type of hemoglobin initially detected. Reporting follows based on the result obtained. Methods differ in the ability to detect different hemoglobins, so understandably programs will also differ in what is typically reported due to method limitations. Additionally, programs may have different distribution of hemoglobins reported based on differences in population. To illustrate these variations seen in populations nationwide, the top 20 phenotypes reported by a sample of programs are provided in Appendix D. Unknown Hemoglobin Variants Invariably, NBS programs will identify unknown hemoglobin variants which may or may not be clinically significant. Most hemoglobin variants do not cause serious health problems, with the exception of a few. National consensus has yet to be established to guide newborn screening programs and clinicians in the follow up of infants with unidentified hemoglobin variants. Final identification of unknown variants can be achieved through DNA analysis and consultation with a hemoglobin specialist for possible clinical implications. Once a variant hemoglobin is identified through screening, it should be confirmed by a secondary or alternate method. The American College of Medical Genetics has outlined the algorithm below specifically for Hb S screening and confirmation and subsequent actions for confirmed SCD and HbS trait (Figure 12). 23 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Figure 12. Newborn Screening Confirmatory Algorithm Abbreviations/ Key: F, S, A, C, and V = The hemoglobins seen in neonatal screening; HPLC: High performance liquid chromatography; IEF: Isoelectric focusing; ‡ = Repeat testing at 6 months age is required if genotyping to confirm the newborn screening result is not done. Source: Adapted from American College of Medical Genetics, 2009 and Dr. Aguinaga-NBS for Hemoglobinopathies in TN, from the 11th Annual Research Symposium in Obstetrics and Gynecology at Meharry Medical College-Nashville, TN. 2014. ASSOCIATION OF PUBLIC HEALTH LABORATORIES 24 Figure 13. Basic Hemoglobinopathy Testing Process 25 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Both IEF and HPLC carry the risk of missing a baby with SCD through misdiagnosis: babies with SCD-S/beta-plus thalassemia may be mistakenly reported as having sickle cell trait (FAS). A small number of babies inheriting the sickle gene and deletional type hereditary persistence of fetal Hb (HPFH) Types 1 or 2, with pancellular distribution of Hb F shows FS on these assays and can be misdiagnosed as having sickle cell anemia (FSS). Compound heterozygotes of HbS-deletional type HPFH have mild microcytosis and do not have features of sickle cell disease, like the vaso-occlusive crises or hemolytic anemia.35 Regardless of the algorithm employed, a laboratory program must consider and institute quality assurance and quality control (QA/QC) activities. Pre-analytic, analytic and post-analytic phases are crucial to the proper function of the laboratory system and reporting of quality and accurate results. A description of QA/QC protocols are briefly discussed in the following section with information provided on the activities of each phase. ASSOCIATION OF PUBLIC HEALTH LABORATORIES 26 IX. Quality Assurance for Hemoglobinopathy Screening and T esting Quality Assurance (QA) is defined as, “a program for the systematic monitoring and evaluation of the various aspects of a project, service, or facility to ensure that standards of quality are being met” and is often split into three categories, pre-analytic, analytic and post-analytic.36 This section focuses on the analytic phase and briefly describes issues in the pre-analytical and post-analytical phases. Pre-analytical Phase For newborn screening, this phase includes variables such as specimen collection devices, quality of specimen collection and transport to the testing facility. Performance evaluation indicators may include the percentage of unsatisfactory specimens (as determined by each program) received and the percentage of specimens that are delayed in transport. Analytical Phase This phase includes all factors related to the laboratory testing. A laboratory quality assurance program should follow any regulatory guidelines that are applicable. Generally, the laboratory QA program has components listed below, at a minimum: • A standard operating procedure detailing the testing method, how to interpret results and how to resolve problems that may arise • Flow charts/responsibilities/actions to clarify the role of each person involved in the analysis • Quality control (QC) materials to monitor the method over time • Documentation of any corrective actions taken • Proficiency Testing (PT) an external quality assessment to monitor the laboratory’s performance The most useful QC and PT materials are those in the same matrix as the specimens being tested. The materials should include specimens classified as normal as well as those with hemoglobinopathies. Because of the high amount of fetal hemoglobin present in newborn DBS, PT for hemoglobinopathy screening requires umbilical cord blood to prepare DBS. Availability of umbilical cord blood with hemoglobinopathies is scarce and there are only two programs that provide PT, CDC’s Newborn Screening Quality Assurance Program (NSQAP) and the United Kingdom National External Quality Assessment (UKNEQAS). If the PT program does not provide specimens for the hemoglobinopathies tested, laboratories may follow guidelines published in CLSI document GP29-A2.42 No QC materials in the DBS matrix are available for hemoglobinopathy screening.37 For sickle cell disease and other hemoglobinopathies, NSQAP sends panels of five blind-coded DBS PT specimens to participating laboratories three times per year. Participants are asked to test the panel and for each specimen, report the method(s) used, the presumptive phenotype, the presumptive clinical assessment and any other clinical classifications that are consistent with their program operations. An individualized evaluation is provided back to each participant. NSQAP summary reports can be publically downloaded at If a participant has an error in a PT survey, NSQAP staff will contact the laboratory to help them understand how the error occurred so that measures can be put in place to reduce the risk of the error happening again. Due to the limited amount of specimens available, there is currently a waiting list to join this program. 27 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up The UKNEQAS provides a PT program for sickle cell and other abnormal hemoglobinopathies in DBS. It provides three specimens each month. Further information the web address is ukneqas.org.uk. Post-analytical Phase The post-analytical phase includes the transmittal of the screening laboratory results and determination of what actions were taken regarding positive screen results. Newborn screening is an entire system, beginning at the hospital and continuing through to the diagnosis of the child. Each part of the newborn screening system is important and needs evaluation of any weaknesses. If more information about quality management is necessary there are many resources including consensus documents from Clinical Laboratory Standards Institute (www.clsi. org), World Health Organization’s “Laboratory quality management system: handbook” that can be downloaded from their website at and accreditation organizations such as ISO (www.iso.org). Communicating positive screen results to a doctor and/or parents and tracking outcomes of diagnostic testing are of primary concern. Follow-up processes for newborn screening primarily aim to quickly locate an infant with sickle cell disease for diagnosis and treatment. Although protocols do vary between programs, the subsequent information focuses specifically on SCD follow-up versus non-disease reporting and trait reporting. Additionally, varying algorithms from state programs have been included for review. ASSOCIATION OF PUBLIC HEALTH LABORATORIES 28 X. Follow-up The goal of short-term follow-up is to ensure that all who receive a valid screening test, and screen positive results, receive a definitive diagnosis in the most expedient manner possible and appropriate clinical management if confirmed.38 The sequence of actions that must take place successfully to ensure achievement of this goal includes: • Access to the newborn within days of birth or to non-newborns when possible; • Collection of adequate blood and prompt submission to the designated laboratory; • Performance of screening laboratory test; • Correct interpretation of screening test results; • Notification and dissemination of screening results to appropriate personnel required to facilitate achievement of the primary goal; • Referral to primary healthcare provider and/or specialist. • Initiation of penicillin prophylaxis for SCD or therapies for other hemoglobinopathies • Diagnostic testing to confirm screening test results; and, • Establishment of comprehensive care in a Medical Home. Follow-up for Hemoglobinopathies While follow-up of those with presumptive SCD has a clear purpose, the purpose of follow-up of those with other hemoglobinopathies is less clear but has the same potential problems. Successful implementation of follow-up involves the following: • Productive relationships between the screening laboratory and follow-up staff, • Selection and training of staff reporting results to the family, and • Processes or algorithms for follow-up. Relationships between laboratory and follow-up staff In some NBS programs the screening laboratory reports directly only to staff of the state NBS program to which all results are transmitted. In others, the laboratory also reports directly to the hospital of birth and to the listed primary healthcare provider for the baby. Since newborn screening for SCD is uniformly state-mandated, the state is responsible for implementation and monitoring of the screening program, or its contracted follow-up agency. Either entity should be the direct recipient and distributor of screening results. Increasingly, follow-up staff and healthcare providers have direct electronic access to the laboratory results held by the state. By whatever method of communication between laboratory and follow-up staff, two issues are important: first, no single screening test can establish with certainty the phenotype of all the common types of SCD and related conditions, and second staff at the laboratory and follow-up agency must be fully familiar with the nomenclature of Hb phenotypes reported by the lab and their clinical significance. Selection and training of staff reporting results to the family Reporting results for affected individuals and reporting about those with non-disease conditions should be fundamentally different. Two types of staff, one with a clinical background and the other with or without clinical background should be trained to report results. Training should equip both types of staff with basic knowledge about SCD and related thalassemia conditions, their variants and their inheritance patterns, and the difference between benign carrier states and clinically significant diseases. 29 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Staff who are reporting results for affected individuals should be familiar with the clinical course of SCD in infants and young children, be able to answer basic questions about SCD or the suspected related disease, and be ready to provide reassurance and support for the family. If staff are not familiar with children with SCD, their training should include an internship at a Sickle Cell Center where they can observe children with SCD in both outpatient and inpatient settings. This will enhance and make more realistic the knowledge acquired through instruction, or reading and video learning materials. The purpose and value of counseling related to a baby with sickle cell trait or other clinically benign hemoglobin conditions or to a baby with no abnormal Hb on newborn screening is not always clear. Any family whose newborn is tested for any disorder deserves to receive results and have their implications explained. This could be done through staff at a primary health care facility or another agency contracted by the state. However, several programs imply, by design, that newborn screening opens a window for genetic education and counseling of the parents for future reproductive planning. Follow-up staff reporting newborn screening results must have some knowledge of the genetics of hemoglobin disorders. Process or algorithms for follow-up Affected individuals: Notification of a family about a (presumptive) serious health problem in an otherwise healthy appearing newborn is not a trivial task. Ideally, notification should be done in person or by live phone call. The initial conversation should be expected to raise anxiety, and cause disappointment and sadness at a time of joy; it may also stir anger or denial. The messenger should be someone familiar with the clinical course of SCD in infants, young children and adults, and be able to answer basic questions about SCD and ready to provide reassurance and support for the family. The processes and algorithms for notification and follow-up often place the primary care service as the medical home, presuming that staff is able to interpret screening results and take appropriate actions when a baby with a positive screen is reported. Reporting individuals with abnormal screening results to a dedicated follow-up agency and/or sickle cell treatment center will increase the timeliness of appropriate care. In the best circumstances, each child with presumptive SCD should be assigned to one case manager who handles the baby from screening test results through establishment at the medical home and sickle cell treatment center, where available. Non-affected individuals: All parents should be informed about the results of newborn screening. Informing parents about results of newborn screening that carry no clinical significance should be handled with calmness and reassurance. Algorithms for reporting and following up babies with HbFA on screening should not classify them as “Normal” or “Within Normal Limits;” they should be reported as “No Abnormal Hb Found.” While FA is the Hb phenotype of most babies with no hemoglobin disorder, babies with FA may be found later to have mild to severe beta or alpha thalassemia or transfusion-dependent thalassemia. “Trait Counseling” of parents of heterozygous babies has become an integral part of newborn screening programs for hemoglobin disorders. However, an FA baby may be born to parents who BOTH have sickle cell trait or, one may have sickle cell trait while the other is a carrier of another abnormal hemoglobin or beta thalassemia. Such parents can have a subsequent baby who has SS or another type of SCD. ASSOCIATION OF PUBLIC HEALTH LABORATORIES 30 Follow up activities for adults who screen positive for a hemoglobinopathy trait: It is important to screen at risk individuals for hemoglobinopathies, who are at reproductive age and who may not have been screened at birth. Dissemination of information about the relevance of adult screening and participation in community outreach activities, such as community screenings and health fairs is very important for individuals who have not been previously tested. If an adult individual has a positive diagnosis for a hemoglobinopathy trait, proper follow-up should ensure the understanding of the trait condition, with full explanation of the results, education about the trait (including any possible complications or rare clinical manifestations), counseling and the offering of hemoglobinopathy testing for the partner, if the couple is contemplating having children. Counseling should include the probability of the couple having a child with a hemoglobinopathy in each pregnancy. 31 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up XI. References 1. Hsieh MM, Fitzhugh CD, Weitzel RP, Link ME, Coles WA, Zhao X, Rodgers GP, Powell JD, Tisdale JF. (2014 Jul 2). Nonmyeloablative HLA-Matched Sibling Allogeneic Hematopoietic Stem Cell Transplantation for Severe Sickle Cell Phenotype. JAMA.;312(1):48-56. doi: 10.1001/ jama.2014.7192. PMID: 25058217. 2. A.D.A.M. Medical Encyclopedia [Internet]. Atlanta (GA): A.D.A.M., Inc.; ©2005 Thalassemia; [updated 2014 Feb 24; cited 2015 Apr 10]. Available from: medlineplus/ency/article/000587.htm 3. Tietz: Fundamentals of Clinical Chemistry, (6th Edition). Burtis, C. A., Ashwood, E. R., Bruns, D.E. (Eds.). (2008). Haryana, India: Elsevier. Pages: 4. National Library of Medicine. (2009) HBB. Retrieved from 5. National Library of Medicine. (2009) Alpha thalassemia. Retrieved from condition/alpha-thalassemia 6. Ashley-Koch, A., Yang, Q., Olney, R.S. (2000). Sickle hemoglobin (Hb S) allele and Sickle Cell Disease: A HuGE Review. American Journal of Epidemiology, 151(9), 839-845. 7. Watson, M. S., Mann, M. Y., Lloyd-Puryear, M. A., Rinaldo, P., Howell, R. R., (2006). Newborn Screening: Toward a uniform screening panel and system-executive summary. Pediatrics, 117(5), S296-S307. 8. Aguinaga, M.d.P. (2007). Newborn Screening and Clinical Management of Children with Sickle Cell Disease. Frontiers of Science in the 21st Century. pg 137-147. Ed Cordillera. Lima, Peru. 9. Consolidated Annual Report. (1973). US Department of Health, Education, and Welfare. Public Health Service, Center for Disease Control, Atlanta, GA. pg 149. consolidatedannu19cent Accessed on 02/11/2015. 10. Editorial Note on Update: Newborn Screening for Sickle Cell Disease --- California, Illinois, and New York, 1998. CDC MMWR Weekly, August 18, 2000/49(32); 729-731. 11. Thuret I, Sarles J, Merono F, et al. Neonatal Screening for Sickle Cell Disease in France: evaluation of the selective process. J. Clin. Pathol. 2010 Jun 63(6):548-51. doi: 10.1136/ jcp.2009.068874. 12. Benson JM, Therrell Jr BL: History and Current Status of Newborn Screening for Hemoglobinopathies. Seminars in Perinatology 2010; 34:134-144. 13. Gaston, M.H., Vertel, J.I., Woods, G., et al. Prophylaxis with Oral Penicillin in Children with Sickle Cell Anemia. N Engl J Med 1986; 314:1593-1599. 14. Chandrakasan S, Kamat D: An Overview of Hemoglobinopathies and the Interpretation of Newborn Screening Results. December 2013 Pediatric Annals, 42:12; 502. 15. Yanni E, Grosse SD, Yang Q, Olney RS. Trends in pediatric sickle cell disease-related mortality in the United States, 1983-2002. J Pediatr. 2009; 154(4):541-5. doi: 10.1016/j. jpeds.2008.09.052. Epub 2008 Nov 22. 16. Michlitsch J, Azimi M, et al. Newborn Screening for Hemoglobinopathies in California. Pediatric Blood Cancer. 2009; 52(4):486-490. ASSOCIATION OF PUBLIC HEALTH LABORATORIES 32 17. National Newborn Screening and Genetic Resource Center (NNSGRC). Laboratory Testing for Hemoglobinopathies, 2010. 18. Clinical and Laboratory Standards Institute: NBS01-A6, Blood Collection on Filter Paper for Newborn Screening Programs; Approved Standard – Sixth Edition; July 2013. 19. William Joseph William: Williams Hematology, Volume 487-Sixth Edition, 2001. 20. Clinical and Laboratory Standards Institute: GP42 – A6, Procedures and Devices for the Collection of Diagnostic Capillary Blood Specimens; Approved Standard-Sixth Edition, Wayne, PA: Clinical and Laboratory Standards Institute; 2008. 21. Barbara Adam et al: Stabilities of hemoglobins A and S in dried blood spots stored under controlled conditions: Clinical Biochemistry 46 (2013) 1089-1092. 22. A.L. Okwi et al: The Reliability of Sickling and Solubility Tests and Peripheral Blood Film Method for Sickle Cell Disease Screening at District Health Centers in Uganda: Clinics in Mother and Child Health Vol. 7(2010), Article ID C10947. 23. Clinical and Laboratory Standards Institute: ILA31-A : Newborn Screening for Pre-term, Low-birth Weight and Sick newborns. Approved Standard - Volume 29 Number 24, 2009. 24. Clarke GM, Higgins TN. Laboratory Investigation of Hemoglobinopathies and Thalassemias: Review and Update. Clinical Chemistry 46:8(B) 1284-1290, 2000. 25. Keren et al. Comparison of Sebia Capillarys Capillary Electrophoresis with the Primus High Pressure Liquid Chromatography in the Evaluation of Hemoglobinopathies. American Journal of Clinical Pathology 130:824-831, 2008. 26. Therell BL, Pass KA. Hemoglobinopathy Screening Laboratory Techniques for Newborns: Laboratory Methods for Neonatal Screening. Washington DC: American Public Health Laboratories. 1993. Pages 169-189. 27. Joutovsky et al. HPLC Retention Time as a Diagnostic Tool for Hemoglobin Variants and Hemoglobinopathies: A Study of 60,000 Samples in a Clinical Diagnostic Laboratory. Clinical Chemistry 50:10 1736-1747, 2004. 28. Hicks EJ, Hughes, BJ. Comparison of Electrophoresis on Citrate Agar, Cellulose Acetate or Starch for Hemoglobin Identification. Clinical Chemistry21:8 1072-1076, 1975. 29. Ueda S, Schneider RG. Rapid Identification of Polypeptide Chains of Hemoglobin by Cellulose Acetate Electrophoresis of Hemolysates. Blood 1969; 34:230. 30. Cordovado, SK. 2013. Dried Blood Spot DNA Extraction Guidelines to Ensure Robust Performance in NBS Molecular Assays. [PowerPoint slides] (in italics). Retrieved from the Association of Public Health Laboratories website: proceedings/Documents/2013/2013-Newborn-Screening-Symposium/25Cordovado.pdf 31. Mifflin TE. 2003. Setting Up a PCR Laboratory. In: Dieffenbach CW and Dveksler GS, editors. PCR Primer: a Laboratory Manual. 2nd edition. Cold Spring Harbor, NY; Cold Spring Harbor Laboratory Press. p. 5-14. (May be downloaded from PCR_Primer_p.5_14.pdf) 32. Saiki, RK; Scharf S, Faloona F, Mullis KB, Erlich HA, Arnheim N. 1985. Enzymatic amplification of beta-globin genomic sequences and restriction site analysis for diagnosis of sickle cell anemia. 33 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Science 230: 1350–1354. 33. Yao-Hua Zhang Y-H and McCabe ERB. 1992. RNA analysis from newborn screening dried blood specimens. Hum Genet 89:311-314. 34. Sanger, F., Nicklen, S., and Coulson, A.R. 1977. DNA sequencing with chain terminating inhibitors. Proc. Natl. Acad. Sci (USA) 74: 5463–5467. 35. Akinsheye I, Alsultan A, Solovieff N, Ngo D, Baldwin CT, Sebastiani P, Chui DHK, Steinberg MH. Fetal hemoglobin in sickle cell anemia. Blood. 2011; 118:19-27. 36. “Quality Assurance.” Merriam-Webster.com. Merriam-Webster, n.d. Web. 19 May 2014. http:// www.merriam-webster.com/dictionary/quality assurance 37. Clinical and Laboratory Standards Institute: GP29-A2 Assessment of Laboratory Tests When Proficiency Testing Is Not Available; Approved Guideline—Second Edition. Wayne, PA: Clinical and Laboratory Standards Institute; 2008. 38. Clinical and Laboratory Standards Institute: NBS02-A2 Newborn Screening Follow-up; Approved Guideline – Second Edition Wayne, PA: Clinical and Laboratory Standards Institute; May 2013 ASSOCIATION OF PUBLIC HEALTH LABORATORIES 34 XII. Appendices Appendix A: Examples of Molecular Methods for Hb Mutation Detection Figure 1. PCR-RFLP Gel picture from Yao-Hua Zhang Y-H and McCabe ERB. 1992. RNA analysis from newborn screening dried blood specimens. Hum Genet 89:311-314. 35 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Figure 2. Allelic Discrimination Accessed May 19, 2014. Use of trade names is for identification only and does not imply endorsement by the Centers for Disease Control and Prevention, the Agency for Toxic Substances and Disease Registry, the Public Health Service, or the U.S. Department of Health and Human Services. ASSOCIATION OF PUBLIC HEALTH LABORATORIES 36 By Estevezj (Own work) [CC BY-SA 3.0 ( via Wikimedia Commons Figure 3. Sanger Sequencing 37 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up 1 Hb S Trait 2 Low Bart's, Alpha Thalassemia 3 Hb C Trait 4 Other Variant Trait 5 Moderate Bart's, Alpha Thalassemia 6 Hb E Trait 7 Hb D Trait 8 Hb S Disease 9 HB SC Disease 10 Hb G Trait 11 Elevated Bart's, Hb H Disease 12 Hb E Disease 13 HB C Disease 14 S, Beta Thalassemia 15 Hb O-Arab Trait 16 Beta Thalassemia Major 17 Hb S, Other Disease 18 Hb E Trait and H Disease 19 Hb SE Disease 20 C, Beta Thalassemia Low level Bart's no longer reported after June 27 2011. 1 fas 10166 50.22% 2 barts 5022 24.81% 3 fac 3018 14.91% 4 fav 903 4.46% 5 fae 362 1.79% 6 fad 248 1.23% 7 Sickle Cell 173 0.85% 8 Sickle C 112 0.55% 9 fagv 70 0.35% 10 S/B+ 32 0.16% 11 fasv 29 0.14% 12 C/B+ 21 0.10% 13 faev 18 0.09% 14 fadv 17 0.08% 15 C 14 0.07% 16 fag 9 0.04% 17 S/B0 8 0.04% 18 facv 8 0.04% 19 Thal Major 7 0.03% 20 E 5 0.02% total 20242 2 cases each: Baltimore J, D, D/E, S/HPFH 1 case each: Baltimore N, HPFH, Thal Intermedia/HPFH, Thal + C trait, C/Camden, C/Korle Bu, E/B+, O Arab, S/Baltimore, S/O Arab, Unidentified Variant fas barts fac fav fae fad Sickle Cell Sickle C fagv S/B+ fasv C/B+ faev fadv C fag S/B0 facv Thal Major E Top 20 Hemoglobinopathies in State 2: 2008-2012 Appendix B Top 20 Hemoglobinopathies in State 1: 2008-2012 ASSOCIATION OF PUBLIC HEALTH LABORATORIES 38 Confirmed Hemoglobinopathy Data from 1999-2009 C Beta thalassemia FCA C Disease FSC E disease FS FSA SD Other 10 Years of Confirmed Data for Hemoglobinopathy Disease in State 3: 1999-2009 TOP 20 HEMOGLOBINOPATHIES IN State 4 2008-2012 Rank Description Phenotype Nomenclature 1 Normal hemoglobin. FA or AF 2 Hb S trait. FAS or AFS 3 Hb E trait. FAE or AFE 4 Moderate Bart's. 2 gene alpha thalassemia. FAA+Bart's or FAB2 5 Hb variant trait. FA+Var or FAV or AFV 6 Hb C trait. FAC or AFC 7 Hb D trait. FAD or AFD 8 Hb constant spring with low to moderate Bart's. FA+CS+Bart’s or FAXB1 or FAXB2 9 Hb E disease. Homozygous Hb E. FEE or FE 10 Hb E trait with moderate Bart's. 2 gene alpha thalassemia. FAE+Bart’s or FAEB2 11 High Bart's. Hb H disease. 3 gene alpha thalassemia. FAA+High Bart's or FAB3 12 Hb S disease. Sickle cell disease. Homozygous Hb S. FSS or FS 13 Hb S trait with moderate Bart's. FAS+Bart's or FASB2 14 Hb E trait with constant spring. 2 gene alpha thalassemia. FAE+CS+Bart’s 15 Hb sickle C disease. FSC 16 E beta zero thalassemia. Heterozygous Hb E. FE-17 Hb E trait plus an uncommon variant. FAE+Var or FAEV 18 Hb SE disease. Compound heterozygous Hb SE FSE 19 Hb C disease. Homozygous Hb C. FCC 20 S beta zero thalassemia. Heterozygous Hb S. FS-Ranked data combined Hbs from 2008 to 2012 Top 20 Hemoglobinopathies in State 4: 2008-2012 39 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Hemoglobinopathy Result Forward to PCR (DNA) F,A N NOTE: Y (T) indicates that this result is sent for F,A,S Y (T) DNA testing only if demographic information F,A,C Y (T) shows the baby has been transfused. F,A,D Y (T) F,A,Other N F,S Y F,C Y F,S,C Y F,A,E Y (T) F,E Y A,F N A,S,F Y (T) A,C,F Y (T) S,C,F Y S,F Y C,F Y E,F Y A,A N A,S Y (T) A,C Y (T) A,E Y (T) A,D Y (T) S,S Y S,C Y C,C Y E,E Y F,D Y F Only Detected Y F,S (A Questionable) Y F,C (A Questionable) Y F,E (A Questionable) Y F,A,S,Other Y (T) NOTE: Y (T) indicates that this result is sent for A,E,F Y (T) DNA testing only if demographic information A,D,F Y (T) shows the baby has been transfused. A,Other N A,F,Other N A,F,S Y A,F,C Y F,Other Y F,A,Barts N F,A,S,Barts Y (T) F,A,C,Barts Y (T) F,A,E,Barts Y (T) F,A,D,Barts Y (T) F,A,O-Arab Y (T) F,A,elevated Barts N F,S,Barts Y F,S,A Y F,C,A Y F,S,C,Barts Y F,A,Other,Barts N A,F,Other,Barts N A,F,S,Barts Y A,F,D Y A,G N F,G N A,G,F N F,A,G,Barts N A,F,G N F,A,C,Other Y (T) S,A Y C,A Y F,A,G N Appendix C: State 1 DNA Screening Algorithm ASSOCIATION OF PUBLIC HEALTH LABORATORIES 40 Appendix D: Hemoglobinopathy Screening Algorithm for State 1 41 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Appendix E: Hemoglobinopathy Screening Algorithm for State 1 – Older Children & Adults ASSOCIATION OF PUBLIC HEALTH LABORATORIES 42 Code Result Statement 1 F,A Normal 02 F,A,S Probable S Trait. Notify family of test results. 03 F,A,C Probable C Trait. Notify family of test results. 04 F,A,D Probable D Trait. Notify family of test results. 05 F,A,Other Probable Unidentified Hb Variant Trait. Notify family of test results. Recommend consultation with pediatric hematologist. 05F F,A,Other Probable Unidentified Hb Variant Trait. Notify family of test results. Recommend consultation with pediatric hematologist. 05M F,A,Other Probable Unidentified Hb Variant Trait. Notify family of test results. Recommend consultation with pediatric hematologist. 05S F,A,Other Probable Unidentified Hb Variant Trait. Notify family of test results. Recommend consultation with pediatric hematologist. 06 F,S Probable SS Disease. Refer to pediatric hematologist. DNA report to follow. 07 F,C Probable CC Disease. Refer to pediatric hematologist. DNA report to follow. 08 F,S,C Probable SC Disease. Refer to pediatric hematologist. DNA report to follow. 09 F,A,E Probable E Trait. Notify family of test results. 10 F,E Probable EE Disease. Refer to pediatric hematologist. DNA report to follow. 11 A,F Probable Normal. If result is due to transfusion, repeat in one to three months post transfusion. 12 A,S,F Probable S Trait. Notify family of test results. 13 A,C,F Probable C Trait. Notify family of test results. 14 S,C,F Probable SC Disease. Refer to pediatric hematologist. DNA report to follow. 15 S,F Probable SS Disease. Refer to pediatric hematologist. DNA report to follow. 16 C,F Probable CC Disease. Refer to pediatric hematologist. DNA report to follow. 17 E,F Probable EE Disease. Refer to pediatric hematologist. DNA report to follow. 18 A,A Probable Normal. If result is due to transfusion, repeat in one to three months post transfusion. 19 A,S Probable S Trait. Notify family of test results. 20 A,C Probable C Trait. Notify family of test results. 21 A,E Probable E Trait. Notify family of test results. 22 A,D Probable D Trait. Notify family of test results. Appendix F: State 1 Hemoglobinopathy Result Codes 43 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Code Result Statement 23 S,S Probable SS Disease. Refer to pediatric hematologist. DNA report to follow. 24 S,C Probable SC Disease. Refer to pediatric hematologist. DNA report to follow. 25 C,C Probable CC Disease. Refer to pediatric hematologist. DNA report to follow. 26 E,E Probable EE Disease. Refer to pediatric hematologist. DNA report to follow. 27 F,D Probable DD Disease. Refer to pediatric hematologist. DNA report to follow. 28 F Only Detected Possible Beta Thalassemia Major. Refer to pediatric hematologist. DNA report to follow. 29 F,S (A Questionable) Possible Sickle Cell or Sickle Beta Thalassemia Disease. Refer to pediatric hematologist. DNA report to follow. 30 F,C (A Questionable) Possible CC or Hemoglobin C/Beta Thalassemia Disease. Refer to pediatric hematologist. DNA report to follow. 31 F,E (A Questionable) Possible EE or Hemoglobin E/Beta Thalassemia Disease. Refer to pediatric hematologist. DNA report to follow. 32 F,A,S,Other Probable S Trait and Unidentified Hb Variant. Notify family of test results. Consult with pediatric hematologist. 33 A,E,F Probable E Trait. Notify family of test results. 34 A,D,F Probable D Trait. Notify family of test results. 35 A,Other Probable Unidentified Hb Variant Trait. Notify family of test results. Recommend consultation with pediatric hematologist. 36 A,F,Other Probable Unidentified Hb Variant Trait. Notify family of test results. Recommend consultation with pediatric hematologist. 36F A,F,Other Probable Unidentified Hb Variant Trait. Notify family of test results. Recommend consultation with pediatric hematologist. 36M A,F,Other Probable Unidentified Hb Variant Trait. Notify family of test results. Recommend consultation with pediatric hematologist. 36S A,F,Other Probable Unidentified Hb Variant Trait. Notify family of test results. Recommend consultation with pediatric hematologist. 37 A,F,S Probable S Trait. Notify family of test results. DNA report to follow. 38 A,F,C Probable C Trait. Notify family of test results. DNA report to follow. 39 F,Other Probable Unidentified Hb Variant. Refer to pediatric hematologist. DNA report to follow. 40 F,A,Barts Probable Alpha Thalassemia Trait. Notify family of test results. 41 F,A,S,Barts Probable S Trait and Alpha Thalassemia Trait. Notify family of test results. 42 F,A,C,Barts Probable C Trait and Alpha Thalassemia Trait. Notify family of test results. 43 F,A,E,Barts Probable E Trait and Alpha Thalassemia Trait. Notify family of test results. 44 F,A,D,Barts Probable D Trait and Alpha Thalassemia Trait. Notify family of test results. 45 F,A,O-Arab Probable O-Arab Trait. Notify family of test results. ASSOCIATION OF PUBLIC HEALTH LABORATORIES 44 Code Result Statement 46 F,A,elevated Barts Probable H Disease. Refer to a pediatric hematologist. 47 F,S,Barts Probable SS Disease and Alpha Thalassemia Trait. Refer to pediatric hematologist. DNA report to follow. 48 F,S,A Probable Sickle Beta Thalassemia Disease. Refer to pediatric hematologist. DNA report to follow. 49 F,C,A Probable Hemoglobin C/Beta Thalassemia Disease. Refer to pediatric hematologist. DNA report to follow. 50 F,S,C,Barts Probable SC Disease and Alpha Thalassemia Trait. Refer to pediatric hematologist. DNA report to follow. 51 F,A,Other,Barts Probable Unidentified Hb Variant Trait and Alpha Thalassemia Trait. Notify family of test results. Consult with pediatric hematologist. 52 A,F,Other,Barts Probable Unidentified Hb Variant Trait and Alpha Thalassemia Trait. Notify family of test results. Consult with pediatric hematologist. 53 A,F,S,Barts Probable S Trait and Alpha Thalassemia Trait. Notify family of test results. DNA report to follow. 54 A,F,D Probable D Trait. Notify family of test results. DNA report to follow. 55 A,G Probable G Trait. Notify family of test results. 56 F,G Probable GG Disease. Refer to pediatric hematologist. 57 A,G,F Probable G Trait. Notify family of test results. 58 F,A,G,Barts Probable G Trait and Alpha Thalassemia Trait. Notify family of test results. Consult with pediatric hematologist. 59 A,F,G Probable G Trait. Notify family of test results. 60 F,A,C,Other Probable C Trait and Unidentified Hb Variant Trait. Consult with pediatric hematologist. Notify family of test results. 62 S,A Probable Sickle Beta Thalassemia Disease. Refer to pediatric hematologist. DNA report to follow. 63 C,A Probable Hemoglobin C/Beta Thalassemia Disease. Refer to pediatric hematologist. DNA report to follow. 64 F,A,G Probable G Trait. Notify family of test results. 65 F,A,S,G Probable S Trait and G Trait. Notify family of test results. Consult with pediatric hematologist. 66 F,A,S,G,Barts Probable S Trait, G Trait, and Alpha Thalassemia Trait. Notify family of test results. Consult with pediatric hematologist. 67 F,A,C,G Probable C Trait and G Trait. Notify family of test results. Consult with pediatric hematologist. 68 F,A,C,G,Barts Probable C Trait, G Trait, and Alpha Thalassemia Trait. Notify family of test results. Consult with pediatric hematologist. 69 F,E,Barts Probable EE Disease and Alpha Thalassemia Trait. Refer to pediatric hematologist. DNA report to follow. 70 F,C,Barts Probable CC Disease and Alpha Thalassemia Trait. Refer to pediatric hematologist. DNA report to follow. 71 F,S,D Probable SD Disease. Refer to pediatric hematologist. DNA report to follow. 72 F,C,E Probable CE Disease. Refer to pediatric hematologist. DNA report to follow. 73 F,S,E Probable SE Disease. Refer to pediatric hematologist. DNA report to follow. 45 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Code Result Statement 96 Abnormal Refer to mailer for free text result.(Case Management - No) (PCR - Yes) 97 Abnormal Refer to mailer for free text result.(Case Management - No) (PCR - No) 98 Abnormal Refer to mailer for free text result.(Case Management - Yes) (PCR - No) 99 Abnormal Refer to mailer for free text result.(Case Management - Yes) (PCR - Yes) 100 Abnormal Refer to mailer for free text result. (S trait notification - Yes) (PCR - No) 101 Abnormal Refer to mailer for free text result. (S trait notification - Yes) (PCR - Yes) BM Retest Bad Measure LA Retest Light A NF Retest Not Focused CT Retest Contaminated QC Retest Controls Failed DB Retest Distorted Bands RN Retest Runs GQ Retest Gel Quality Not Acceptable GB Retest Gel Burned NS Retest No Specimen LS Retest Light Specimen SLA RFS Repeat from Screen-Light A SNF RFS Repeat from Screen - Not Focused SCT RFS Repeat from Screen - Contaminated SQC RFS Repeat from Screen - Controls Failed SDB RFS Repeat from Screen - Distorted Band SRN RFS Repeat from Screen - Runs SGQ RFS Repeat from Screen - Gel Quality Not Acceptable SGB RFS Repeat from Screen - Gel Burned SLS RFS Repeat from Screen - Light Specimen ASSOCIATION OF PUBLIC HEALTH LABORATORIES 46 Appendix G: State 2 Hemoglobin Coding System Mneumonic Description ACFF Hemoglobinopathy: AFC or AC - Possible hemoglobin C trait ACVFF HGB: ACVF or ACV - Possible hemoglobin C trait & hemoglobin variant ADFF HGB: ADF or FAD - Possible hemoglobin D trait ADFVMF Hemoglobinopathy: ADFV or FADV - Possible hemoglobin D trait and hemoglobin variant AEFF Hemoglobinopathy: AEF or FAE - Possible hemoglobin E trait AEVFF Hemoglobinopathy: AEVF or AEV - Possible hemoglobin E trait and hemoglobin variant AFBART Hemoglobinopathy: AF - Hemoglobin A & F with possible Bart’s hemoglobin - Consultation strongly recommended AFFAST Hemoglobinopathy: FA - Hemoglobin F & A with Fast Moving hemoglobin - Consultation strongly recommended AFNOTB AFOLD AGFF Hemoglobinopathy: AGF or FAG - Possible hemoglobin G trait AGFVMF Hemoglobinopathy: AGFV or FAGV - Possible hemoglobin G trait and hemoglobin variant ASFF Hemoglobinopathy: AFS or AS - Possible sickle cell trait ASVFF Hemoglobinopathy: ASVF or ASV - Possible sickle cell trait & hemoglobin variant AVFF Hemoglobinopathy: AFV or AV - Possible hemoglobin variant BARTS Hemoglobinopathy: FA - Hemoglobin F & A with possible Bart’s hemoglobin - Consultation strongly recommended F50A1F Hemoglobinopathy: Hb F above 50%, Hb A less than 1% - Consultation strongly recommended F50A1I Hemoglobinopathy: Hb F above 50%, Hb A less than 1% Immediate consultation and submit repeat specimen at 4 weeks of age F50A20 Hemoglobinopathy: FA - Hb F above 50%, Hb A less than 20% with fast moving Hb - Consultation strongly recommended FA NORMAL FACI Hemoglobinopathy: FAC - Possible hemoglobin C trait - Submit repeat specimen by 6 months of age FACM Hemoglobinopathy: FAC - Possible hemoglobin C trait FACVI Hemoglobinopathy: FACV - Possible hemoglobin C trait & hemoglobin variant - Submit repeat specimen by 6 months of age FADI Hemoglobinopathy: FAD - Possible hemoglobin D trait - Submit repeat specimen by 6 months of age FADVF Hemoglobinopathy: FADV - Possible hemoglobin D trait and hemoglobin variant FADVI Hemoglobinopathy: FADV - Possible hemoglobin D trait & variant - Submit repeat specimen by 6 months of age FAEI Hemoglobinopathy: FAE/A2 - Possible hemoglobin E trait - Submit repeat specimen by 6 months of age Sickle Cell Mnemonics 47 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Mneumonic Description FAEVI Hemoglobinopathy: FAE/A2 V - Possible hemoglobin E trait and HGB variant - Submit repeat specimen by 6 months of age FAGI Hemoglobinopathy: FAG - Possible hemoglobin G trait - Submit repeat specimen by 6 months of age FAGVI Hemoglobinopathy: FAGV - Possible hemoglobin G trait and Hb variant - Submit repeat specimen by 6 months of age FAPERF Hemoglobinopathy: FA - Possible hereditary persistence of fetal hemoglobin - Consultation strongly recommended FASI Hemoglobinopathy: FAS - Possible sickle cell trait - Submit repeat specimen by 6 months of age FASM Hemoglobinopathy: FAS - Possible sickle cell trait FAST Hemoglobinopathy: FA - Hemoglobin F & A with Fast Moving hemoglobin - Consultation strongly recommended FASVI Hemoglobinopathy: FASV - Possible sickle cell trait & hemoglobin variant - Submit repeat specimen by 6 months of age FAVI Hemoglobinopathy: FAV - Possible hemoglobin variant - Submit repeat specimen by 6 months of age FAVM Hemoglobinopathy: FAV - Possible hemoglobin V trait FCAF Hemoglobinopathy: FCA - Possible C beta thalassemia - Consultation strongly recommended FCAI Hemoglobinopathy: FCA - Possible C beta thalassemia - Immediate consultation and submit repeat specimen by 4 weeks of age FCF Hemoglobinopathy: FC or CF - Possible hemoglobin C - Consultation strongly recommended FCI Hemoglobinopathy: FC - Possible hemoglobin C - immediate consultation and submit repeat specimen at 4 weeks of age FCVF Hemoglobinopathy: FCV or CVF - Possible hemoglobin C and hemoglobin variant - Consultation strongly recommenced FCVI Hemoglobinopathy: FCV - Possible hemoglobin C and variant - Immediate consultation and submit repeat specimen at 4 weeks of age FDEF Hemoglobinopathy: FDE or DEF - Possible hemoglobin D and E - Consultation strongly recommended FDEI Hemoglobinopathy: FDE - Possible hemoglobin D and E - Immediate consultation and submit repeat specimen at 4 weeks of age FDF Hemoglobinopathy: FD - Possible hemoglobin D - Consultation strongly recommended FDI Hemoglobinopathy: FD - Possible hemoglobin D - Immediate consultation and submit repeat specimen at 4 weeks of age FDVF Hemoglobinopathy: FDV or DVF - Possible hemoglobin D and hemoglobin variant - Consultation strongly recommended FDVI Hemoglobinopathy: FDV - Possible hemoglobin D and hemoglobin variant - Immediate consultation and submit repeat specimen at 4 weeks of age FEF Hemoglobinopathy: FE/A2 or E/A2 - Possible hemoglobin E - Consultation strongly recommended FEI Hemoglobinopathy: FE/A2 - Possible hemoglobin FE/A2 - Immediate consultation & submit repeat specimen at 4 weeks of age FF Hemoglobinopathy: F - Possible hemoglobin F - Consultation strongly recommended FGF Hemoglobinopathy: FG - Possible hemoglobin G - Consultation strongly recommended FGI Hemoglobinopathy: FG - Possible hemoglobin G - Immediate consultation and submit repeat specimen at 4 weeks of age FI Hemoglobinopathy: F - Possible hemoglobin F - Immediate consultation and submit repeat specimen at 4 weeks of age FSAF Hemoglobinopathy: FSA - Possible S beta thalassemia - Consultation strongly recommended FSAI Hemoglobinopathy: FSA - Possible S beta thalassemia - Immediate consultation and submit repeat specimen at 4 weeks of age Sickle Cell Mnemonics ASSOCIATION OF PUBLIC HEALTH LABORATORIES 48 Mneumonic Description FSCF Hemoglobinopathy: FSC or SC - Possible hemoglobin S and C - Consultation strongly recommended FSCI Hemoglobinopathy: FSC - Possible hemoglobin S and C - Immediate consultation and submit repeat specimen at 4 weeks of age FSCVF Hemoglobinopathy: FSCV - Possible hemoglobin S and C variant - Consultation recommended FSCVI Hemoglobinopathy: FSCV - Possible hemoglobin S & C & variant - Immediate consultation and submit repeat specimen at 4 weeks of age FSEF Hemoglobinopathy: FSE or SEF - Possible hemoglobin S and E - Consultation strongly recommended FSEI Hemoglobinopathy: FSE - Possible hemoglobin S and E - Immediate consultation and submit repeat specimen at 4 weeks of age FSF Hemoglobinopathy: FS or SF - Possible sickle cell disease - Consultation strongly recommended FSI Hemoglobinopathy: FS - Possible sickle cell disease - Immediate consultation and submit repeat specimen at 4 weeks of age FSVBTF Hemoglobinopathy: Possible FSV or FSV - beta thalassemia - Consultation strongly recommended FSVBTI Hemoglobinopathy: Possible FSV or FSV - beta thalassemia - Immediate consultation and submit repeat specimen at 4 weeks of age FSVF Hemoglobinopathy: FSV or SV - Possible sickle cell disease and hemoglobin variant - Consultation strongly recommended FSVI Hemoglobinopathy: FSV - Possible sickle cell disease & HGB variant - Immediate consultation and submit repeat specimen at 4 weeks of age FVF Hemoglobinopathy: FV or V - Possible hemoglobin variant - Consultation strongly recommended FVI Hemoglobinopathy: FV - Possible hemoglobin variant - Immediate consultation and submit repeat specimen at 4 weeks of age INCELU Hemoglobinopathy: Unsatisfactory - Incomplete elution from blotter - Submit repeat specimen within 2 days INCONS Hemoglobinopathy: Inconsistent results - Submit repeat specimen within 2 days INDFAS Hemoglobinopathy: Non-definitive AFS - Submit repeat specimen within 120 days after last transfusion INDPOS Hemoglobinopathy: AF or A - Indeterminate results due to possible blood transfusion(s), submit repeat specimen within 120 days after last transfusion NDFR4M Hemoglobinopathy: Non-definitive results - Submit repeat specimen at 4 weeks of age NDFR6M Hemoglobinopathy: Non-definitive results - Submit repeat specimen at 6 months of age NOFABO Hemoglobinopathies: Bart’s hemoglobin only, no F or A hemoglobin present. Possible Hydrops Fetalis - Consultation recommended NORPT Hemoglobinopathy: No repeat required PRE Hemoglobinopathy: Premature infant - non-definitive results - Submit repeat specimen at 4 weeks of age PREFAS Hemoglobinopathy: Premature infant - non definitive FAS - Submit repeat specimen at 4 weeks of age QNS Hemoglobinopathy: Quantity insufficient - Submit repeat specimen within 2 days RWAL Hemoglobinopathy: (null) SCFBTF Hemoglobinopathy: Possible SCF or SCF - beta thalassemia - Consultation strongly recommended TRAN Hemoglobinopathy: Indeterminate results due to blood transfusion(s) - Submit repeat specimen 120 days after last transfusion TRANNP Hemoglobinopathy results assume that no transfusion was performed because transfusion information was not provided (NP) U1 Hemoglobinopathy: Specimen not on Whatman 903 paper - Repeat within 2 days Sickle Cell Mnemonics 49 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Mneumonic Description U10 Hemoglobinopathy: Specimen not allowed to dry thoroughly - Repeat within 2 days U11 Hemoglobinopathy: Conflicting information - Specimen identity uncertain - Repeat within 2 days U12 Hemoglobinopathy: Incomplete saturation of filter- Repeat within 2 days U14 Hemoglobinopathy: Expired filter - Repeat within 2 days U2 Hemoglobinopathy: Specimen not attached to form - Repeat within 2 days U3 Hemoglobinopathy: Quantity insufficient - Submit repeat specimen within 2 days U4 Hemoglobinopathy: Contaminated or diluted - Repeat filter within 2 days U5 Hemoglobinopathy: Oversaturated specimen - Repeat filter within 2 days U6 Hemoglobinopathy: Blood clotted or caked - Repeat filter within 2 days U7 Hemoglobinopathy: Filter paper torn or scratched - Repeat within 2 days U8 Hemoglobinopathy: >14 days from collection date - Repeat within 2 days U9 Hemoglobinopathy: Filter paper distorted - Repeat filter within 2 days FAOI Hemoglobinopathy: Possible hemoglobin O trait - Submit repeat specimen by 6 months of age FAOM Hemoglobinopathy: Possible hemoglobin C trait AOFF Hemoglobinopathy: AFO & AO - Possible hemoglobin O trait FAOVI Hemoglobinopathy: Possible hemoglobin O trait & hemoglobin variant - Submit repeat specimen by 6 months of age AOFVF Hemoglobinopathy: AOVF & AOV - Possible hemoglobin O trait & hemoglobin variant Sickle Cell Mnemonics 50 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Appendix H: State 3 Hg Reporting Algorithm ASSOCIATION OF PUBLIC HEALTH LABORATORIES 51 Appendix I: State 4 Hb Screening Algorithm 52 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Result Code Description (brief) AA Normal Finding AA-PB Normal Finding (previous specimen with moderate Bart's) AAR Hb A (Age inappropriate, no suitable previous specimen) AAY Hb A (Age inappropriate, suitable previous specimen) AC Hb C trait ACF Hb C trait, Hb F AD Hb D trait ADF Hb D trait, Hb F AE Hb E trait AEF Hb E trait, Hb F AEFB2 Hb E trait, moderate Bart's AEFVB2 Hb A, Hb E, Hb F, Hb variant, moderate Bart's AEFXB2 Hb E trait, Hb F, Hb constant spring, moderate Bart's AF Normal Finding AFB2 Moderate Bart's AFB3 High Bart's (possible Hb H Disease) AFC Hb C trait AFCB2 Hb C trait, moderate Bart's AFD Hb D trait AFE Hb E trait AFEB2 Hb E Trait, Hb F, moderate Bart's AFEXB2 Hb E trait, Hb F, Hb constant spring, moderate Bart's AF-PB Normal Finding (previous Hb Bart's) AFR Hb A, Hb F (age inappropriate, no suitable previous specimen) AFS Hb S trait, Hb F AFSB2 Hb S trait, Hb F, moderate Bart's AFV Hb variant trait, Hb F AFVB2 Hb variant trait, Hb F, moderate Bart's AFXB2 Hb constant spring, Hb F, moderate Bart's AFXB3 High Bart's, Hb constant spring (possible Hb H Disease) AFY Hb A, Hb F (age inappropriate, suitable previous specimen) AMBD Ambiguous degradation (suitable previous specimen) AMBDR Ambiguous degradation (no suitable previous specimen) AS Hb S trait ASF Hb S trait, Hb F ASFB2 Hb S trait, Hb F, moderate Bart's AV Hb variant trait AVF Hb variant trait, Hb F Appendix J: State 4 Hb Result Codes ASSOCIATION OF PUBLIC HEALTH LABORATORIES 53 Result Code Description (brief) AVFB2 Hb variant trait, Hb F, moderate Bart's CC Hb C only (possible Hb C disease) CCF Hb C, Hb F (possible Hb C disease) CE Hb C, Hb E (possible CE disease) CEF Hb C, Hb E, Hb F (possible CE disease) C-F Hb C, Hb F (possible C beta zero thalassemia) CS Hb C, Hb S (possible sickle C disease) CSF Hb C, Hb S, Hb F (possible sickle C disease) E-Hb E only (possible E beta zero thalassemia) EE Hb E only (possible Hb E disease) EEF Hb E, Hb F (possible Hb E disease) FA Normal Finding FAB2 Moderate Bart's FAB3 High Bart's (possible Hb H disease) FAB-PN Moderate Bart's (previous normal specimen) FAC Hb C trait FACB2 Hb C trait, moderate Bart's FACV Hb C trait, Hb variant FAD Hb D trait FAE Hb E trait FAEB2 Hb E trait, moderate Hb Bart's FAEB3 High Bart's, Hb E trait (possible Hb H disease) FAEV Hb E trait, Hb variant FAEXB1 Hb E trait, Hb constant spring, low Bart's FAEXB2 Hb E trait, Hb constant spring, moderate Bart's FA-PB Normal Finding - Previous Bart's FAS Hb S trait FASB2 Hb S trait, moderate Bart's FASV Hb S trait, Hb variant FASVB2 Hb S trait, Hb variant, moderate Bart's FAV Hb variant trait FAVB2 Hb variant, moderate Bart's FAXB1 Hb constant spring, low Bart's FAXB2 Hb constant spring, moderate Bart's FAXB3 High Bart's, Hb constant spring (possible Hb H Disease) FC-Hb C only (possible C beta zero thalassemia) FC2A Hb C trait (possible C beta plus thalassemia) FCC Hb C only (possible Hb C disease) FCE Hb C, Hb E (possible CE disease) FCS Hb C, Hb S (possible sickle C disease) FD2A Hb D trait (possible D beta plus thalassemia) 54 Hemoglobinopathies: Current Practices for Screening, Confirmation and Follow-up Result Code Description (brief) FDD Hb D only (possible Hb D disease) FE-Hb E only (possible E beta zero thalassemia) FEA Hb E trait (possible E beta plus thalassemia) FEE Hb E only (possible Hb E disease) FEEB2 Hb E only, moderate Bart's FEEB3 High Bart's, Hb E only (possible Hb H disease and Hb E disease) FS-Hb F, Hb S (possible S beta zero thalassemia) FS2A Hb S trait (possible S beta plus thalassemia) FSC Hb F, Hb S, Hb C (possible sickle C disease) FSE Hb F, Hb S, Hb E (possible SE disease) FSS Hb F, Hb S (possible sickle cell disease) FSSB2 Hb F, Hb S, moderate Bart's (possible sickle cell disease) FSSB3 Hb F, Hb S, High Bart's (possible sickle cell and Hb H disease) FSV Hb F, Hb S, Hb variant (possible SV disease) SC Hb S, Hb C (possible sickle C disease) SCF Hb S, Hb C, Hb F (possible sickle C disease) SEF Hb S, Hb E, Hb F (possible SE disease) SF-Hb S, Hb F (possible S beta zero thalassemia) SF2A Hb S trait (possible S beta plus thalassemia) SS Hb S only (possible sickle cell disease) SSF Hb S, Hb F (possible sickle cell disease) TO-O Unsuitable, too old for Hb testing (previous suitable specimen) TO-OR Unsuitable, too old for Hb testing (no previous suitable specimen) UT Unsuitable transfused (previous suitable specimen) UT+B Unsuitable transfused, moderate Bart's UT+C Unsuitable transfused, Hb C trait UT+E Unsuitable transfused, Hb E trait UT+S Unsuitable transfused, Hb S trait UT+VAR Unsuitable transfused, Hb variant trait UTR Unsuitable transfused (no previous suitable specimen) XXX No code mapped for this phenotype (custom mailer note) Association of Public Health Laboratories The Association of Public Health Laboratories (APHL) is a national nonprofit dedicated to working with members to strengthen laboratories with a public health mandate. By promoting effective programs and public policy, APHL strives to provide public health laboratories with the resources and infrastructure needed to protect the health of US residents and to prevent and control disease globally. 8515 Georgia Avenue, Suite 700 Silver Spring, MD 20910 Phone: 240.485.2745 Fax: 240.485.2700 Web: www.aphl.org
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© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 26 CHAPTER 17: Electric Potential Answers to Questions 1. If two points are at the same potential, then no NET work was done in moving a test charge from one point to the other. Along some segments of the path, some positive work might have been done, but along other segments of the path, negative work would then have been done. And if the object was moved along an equipotential line, then no work would have been done along any segment of the path. Along any segment of the path where positive or negative work was done, a force would have to be exerted. If the object was moved along an equipotential line, then no force would have been exerted along any segment of the path. This is analogous to climbing up and then back down a flight of stairs to get from one point to another point on the same floor of a building. Gravitational potential increased while going up the stairs, and decreased while going down the stairs. A force was required both to go up the stairs and down the stairs. If instead you walked on the level from one point to another, then the gravitational potential was constant, and no force was need to change gravitational potential. 2. A negative charge will move toward a region of higher potential. A positive charge will move toward a region of lower potential. The potential energy of each will decrease. 3. (a) Electric potential, a scalar, is the electric potential energy per unit charge at a point in space. Electric field, a vector, is the electric force per unit charge at a point in space. (b) Electric potential energy is the work done against the electric force in moving a charge from a specified location of zero potential energy to some other location. Electric potential is the electric potential energy per unit charge. 4. The potential energy of the electron is proportional to the voltage used to accelerate it. Thus, if the voltage is multiplied by a factor of 4, then the potential energy is increased by a factor of 4 also. Then, by energy conservation, we assume that all of the potential energy is converted to kinetic energy during the acceleration process. Thus the kinetic energy has increased by a factor of 4 also. Finally, since the speed is proportional to the square root of kinetic energy, the speed must increase by a factor of 2. 5. The electric field is zero at the midpoint of the line segment joining the two equal positive charges. The electric field due to each charge is of the same magnitude at that location, because the location is equidistant from both charges, but the two fields are in the opposite direction. Thus the net electric field is zero there. The electric potential is never zero along that line, except at infinity. The electric potential due to each charge is positive, and so the total potential, which is the algebraic sum of the two potentials, is always positive. 6. A negative particle will have its electric potential energy decrease if it moves from a region of low electric potential to one of high potential. By Eq. 17-3, if the charge is negative and the potential difference is positive, the change in potential energy will be negative, and so decrease. 7. The proton would gain half the kinetic energy as compared to the alpha particle. The alpha particle has twice the charge of the proton, and so has twice the potential energy for the same voltage. Thus the alpha will have twice the kinetic energy of the proton after acceleration. Giancoli Physics: Principles with Applications, 6th Edition © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 27 8. There is no general relationship between the value of V and the value of E . Instead, the magnitude of E is equal to the rate at which V decreases over a short distance. Consider the point midway between two positive charges. E is 0 there, but V is high. Or, consider the point midway between two negative charges. E is also 0 there, but V is low, because it is negative. Finally, consider the point midway between positive and negative charges of equal magnitude. There E is not 0, because it points towards the negative charge, but V is zero. 9. Two equipotential lines cannot cross. That would indicate that a region in space had two different values for the potential. For example, if a 40-V line and a 50-V line crossed, then the potential at the point of crossing would be both 40 V and 50 V, which is impossible. Likewise, the electric field is perpendicular to the equipotential lines. If two lines crossed, the electric field at that point would point in two different directions simultaneously, which is not possible. 10. The equipotential lines are drawn so that they are perpendicular to the electric field lines where they cross. 11. The electric field would be zero in a region of space that has the same potential throughout. The electric field is related to the change in potential as you move from place to place. If the potential does not change, then the electric field is zero. 12. The orbit must be a circle. The gravitational potential (or potential energy) depends on the distance from the center of the Earth. If the potential is constant (equipotential line), then the distance from the center of the Earth must be constant, and so the orbit is a circle. 13. (a) V would decrease by 10 V at every location. (b) E is related to the change in electric potential. Decreasing the potential by 10 V everywhere would not affect the changes in potential from one location to another, and so would not affect E. 14. Any imbalance of charge that might exist would quickly be resolved. Suppose the positive plate, connected to the positive terminal of the battery, had more charge than the negative plate. Then negative charges from the negative battery terminal would be attracted to the negative plate by the more charged positive plate. This would only continue until the negative plate was as charged as the positive plate. If the negative plate became “over charged”, then the opposite transfer of charge would take place, again until equilibrium was reached. Another way to explain the balance of charge is that neither the battery nor the capacitor can create or destroy charge. Since they were neutral before they were connected, they must be neutral after they are connected. The charge removed from one plate appears as excess on the other plate. This is true regardless of the conductor size or shape. 15. We meant that the capacitance did not depend on the amount of charge stored or on the potential difference between the capacitor plates. Changing the amount of charge stored or the potential difference will not change the capacitance. Chapter 17 Electric Potential © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 28 Solutions to Problems 1. The work done by the electric field can be found from Eq. 17-2b. 6 4 ba ba ba ba 7.7 10 C 55 V 4.2 10 J W V W qV q 2. The work done by the electric field can be found from Eq. 17-2b. 19 2 17 ba ba ba ba 2 1.60 10 C 1.80 10 V 2.88 10 J 1 180 V 1.80 10 eV W V W qV q e 3. The kinetic energy gained is equal to the work done on the electron by the electric field. The potential difference must be positive for the electron to gain potential energy. Use Eq. 17-2b. 19 4 15 ba ba ba ba 4 4 1.60 10 C 2.3 10 V 3.7 10 J 1 2.3 10 V 2.3 10 eV W V W qV q e 4. The kinetic energy gained by the electron is the work done by the electric force. Use Eq. 17-2b to calculate the potential difference. 17 ba ba 19 7.45 10 J 466V 1.60 10 C W V q The electron moves from low potential to high potential, so plate B is at the higher potential. 5. The magnitude of the electric field can be found from Eq. 17-4b. 4 ba 3 220 V 3.8 10 V m 5.8 10 m V E d 6. The magnitude of the voltage can be found from Eq. 17-4b. 3 ba ba 640V m 11.0 10 m 7.0V V E V Ed d 7. The distance between the plates can be found from Eq. 17-4b. 2 ba ba 45 V 3.0 10 m 1500V m V V E d d E 8. The gain of kinetic energy comes from a loss of potential energy due to conservation of energy, and the magnitude of the potential difference is the energy per unit charge. 3 4 PE KE 65.0 10 eV 3.25 10 V 2 V q q e The negative sign indicates that the helium nucleus had to go from a higher potential to a lower potential. Giancoli Physics: Principles with Applications, 6th Edition © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 29 9. Find the distance corresponding to the maximum electric field, using Eq. 17-4b. 5 5 ba ba 6 200 V 6.67 10 m 7 10 m 3 10 V m V V E d d E 10. By the work energy theorem, the total work done, by the external force and the electric field together, is the change in kinetic energy. The work done by the electric field is given by Eq. 17-2b. external electric final initial external b a final 4 4 2 external final b a 6 KE KE KE KE 15.0 10 J 4.82 10 J 1.20 10 V 8.50 10 C W W W q V V W V V q 11. The kinetic energy of the electron is given in each case. Use the kinetic energy to find the speed. (a) 19 2 7 1 2 31 2 750 eV 1.60 10 J eV 2KE KE 1.6 10 m s 9.11 10 kg mv v m (b) 3 19 2 7 1 2 31 2 3.2 10 eV 1.60 10 J eV 2KE KE 3.4 10 m s 9.11 10 kg mv v m 12. The kinetic energy of the proton is given. Use the kinetic energy to find the speed. 3 19 2 5 1 2 27 2 3.2 10 eV 1.60 10 J eV 2KE KE 7.8 10 m s 1.67 10 kg mv v m 13. The kinetic energy of the alpha particle is given. Use the kinetic energy to find the speed. 6 19 2 7 1 2 27 2 5.53 10 eV 1.60 10 J eV 2KE KE 1.63 10 m s 6.64 10 kg mv v m 14. Use Eq. 17-5 to find the potential. 6 9 2 2 5 1 0 1 4.00 10 C 8.99 10 N m C 2.40 10 V 4 1.50 10 m Q V r 15. Use Eq. 17-5 to find the charge. 9 0 9 2 2 0 1 1 4 0.15 m 125 V 2.1 10 C 4 8.99 10 N m C Q V Q rV r 16. The work required is the difference in potential energy between the two locations. The test charge has potential energy due to each of the other charges, given in Conceptual Example 17-7 as 1 2 PE QQ k r . So to find the work, calculate the difference in potential energy between the two locations. Let Q represent the 35 C charge, let q represent the 0.50 C test charge, and let d represent the 32 cm distance. initial final PE PE 2 2 2 0.12 m 2 0.12 m kQq kQq kQq kQq d d d d Chapter 17 Electric Potential © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 30 final initial 9 2 2 6 6 1 Work PE PE 2 2 0.12 m 2 0.12 m 2 1 1 2 0.16m 0.12m 0.16m 0.12m 0.16m 8.99 10 N m C 35 10 C 0.50 10 C 16.07m 2.5J kQq kQq kQq d d d kQq An external force needs to do positive work to move the charge. 17. 18. (a) The electric potential is given by Eq. 17-5. 19 9 2 2 5 5 15 0 1 1.60 10 C 8.99 10 N m C 5.754 10 V 5.8 10 V 4 2.5 10 m Q V r (b) The potential energy of a pair of charges is derived in Conceptual Example 17.7. 2 19 9 2 2 14 1 2 15 1.60 10 C PE 8.99 10 N m C 9.2 10 J 2.5 10 m QQ k r 19. The potential at the corner is the sum of the potentials due to each of the charges, using Eq. 17-5. 3 2 1 2 1 2 1 2 2 2 k Q k Q kQ kQ kQ V L L L L L 20. By energy conservation, all of the initial potential energy will change to kinetic energy of the electron when the electron is far away. The other charge is fixed, and so has no kinetic energy. When the electron is far away, there is no potential energy. 2 1 initial final initial final 2 9 2 2 19 7 31 7 PE KE 2 8.99 10 N m C 1.60 10 C 1.25 10 C 2 9.11 10 kg 0.325m 3.49 10 m s k e Q E E mv r k e Q v mr 21. By energy conservation, all of the initial potential energy of the charges will change to kinetic energy when the charges are very far away from each other. By momentum conservation, since the initial momentum is zero and the charges have identical masses, the charges will have equal speeds in opposite directions from each other as they move. Thus each charge will have the same kinetic energy. Giancoli Physics: Principles with Applications, 6th Edition © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 31 2 2 1 initial final initial final 2 PE KE 2 kQ E E mv r 2 9 2 2 6 2 3 6 8.99 10 N m C 9.5 10 C 4.8 10 m s 1.0 10 kg 0.035m kQ v mr 22. (a) Because of the inverse square nature of the electric field, any location where the field is zero must be closer to the weaker charge 2 q . Also, in between the two charges, the fields due to the two charges are parallel to each other (both to the left) and cannot cancel. Thus the only places where the field can be zero are closer to the weaker charge, but not between them. In the diagram, this is the point labeled as “x”. Take to the right as the positive direction. 2 2 2 1 2 1 2 2 6 2 2 6 6 1 2 0 2.0 10 C 5.0cm 22cm left of 3.0 10 C 2.0 10 C q q E k k q d x q x x d x q x d q q q (b) The potential due to the positive charge is positive everywhere, and the potential due to the negative charge is negative everywhere. Since the negative charge is smaller in magnitude than the positive charge, any point where the potential is zero must be closer to the negative charge. So consider locations between the charges (position 1 x ) and to the left of the negative charge (position 2 x ) as shown in the diagram. 6 1 2 2 location 1 1 6 1 1 2 1 6 1 2 2 location 2 2 6 2 2 1 2 2.0 10 C 5.0cm 0 2.0cm 5.0 10 C 2.0 10 C 5.0cm 0 10.0cm 1.0 10 C kq kq q d V x d x x q q kq kq q d V x d x x q q So the two locations where the potential is zero are 2.0 cm from the negative charge towards the positive charge, and 10.0 cm from the negative charge away from the positive charge. 23. Let the side length of the equilateral triangle be L. Imagine bringing the electrons in from infinity one at a time. It takes no work to bring the first electron to its final location, because there are no other charges present. Thus 1 0 W . The work done in bringing in the second electron to its final location is equal to the charge on the electron times the potential (due to the first electron) at the final location of the second electron. Thus 2 2 ke ke W e L L . The work done in bringing the third electron to its final location is equal to the charge on the electron times the potential (due to the first two electrons). Thus 2 3 2 ke ke ke W e L L L . The total work done is the sum 1 2 3 W W W . L e e e L L d 1 x 2 x 1 0 q 2 0 q d x 1 0 q 2 0 q Chapter 17 Electric Potential © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 32 2 9 2 2 19 2 2 2 1 2 3 10 18 3 8.99 10 N m C 1.60 10 C 2 3 0 1.0 10 m 6.9 10 J ke ke ke W W W W L L L 24. (a) The potential due to a point charge is given by Eq. 17.5 ba b a b a b a 9 2 2 6 3 1 1 1 1 8.99 10 N m C 3.8 10 C 8.6 10 V 0.88m 0.72m kq kq V V V kq r r r r (b) The magnitude of the electric field due to a point charge is given by Eq. 16-4a. The direction of the electric field due to a negative charge is towards the charge, so the field at point a will point downward, and the field at point b will point to the right. See the vector diagram. 9 2 2 6 4 b 2 2 b 9 2 2 6 4 a 2 2 a 2 2 2 2 4 4 4 b a a b 8.99 10 N m C 3.8 10 C right right 4.4114 10 V m, right 0.88m 8.99 10 N m C 3.8 10 C down down 6.5899 10 V m, down 0.72m 4.4114 10 V m 6.5899 10 V m 7.9 10 V m k q r k q r E E E E E E 1 o a b 65899 tan tan 56 44114 E E 25. We assume that all of the energy the proton gains in being accelerated by the voltage is changed to potential energy just as the proton’s outer edge reaches the outer radius of the silicon nucleus. initial final initial 14 PE PE e e eV k r 19 9 2 2 6 initial 15 14 1.60 10 C 14 8.99 10 N m C 4.2 10 V 1.2 3.6 10 m e V k r 26. Use Eq. 17-2b and Eq. 17-5. BA B A 1 1 1 1 2 2 1 1 2 k q k q kq kq V V V kq d b b b d b d b b b d b kq b d kq d b b b d b r b E a E a E b E b a E E Giancoli Physics: Principles with Applications, 6th Edition © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 33 27. (a) The electric potential is found from Eq. 17-5. 19 p 9 2 2 initial 10 atom 1.60 10 C 8.99 10 N m C 27 V 0.53 10 m q V k r (b) The kinetic energy can be found from the fact that the magnitude of the net force on the electron, which is the attraction by the proton, is causing circular motion. 2 2 2 p e 2 2 e 1 1 net e e 2 2 2 atom atom atom atom 2 19 9 2 2 18 18 1 2 10 18 19 KE 1.60 10 C 8.99 10 N m C 2.171 10 J 2.2 10 J 0.53 10 m 1eV 2.171 10 J 13.57eV 14eV 1.60 10 J q q m v e e F k m v k m v k r r r r (c) The total energy of the electron is the sum of its KE and PE. The PE is found from Eq. 17-2a, and is negative since the electron’s charge is negative. 2 2 2 2 1 1 1 total e 2 2 2 atom atom atom 18 18 PE KE 2.171 10 J 2.2 10 J 14eV e e e E eV m v k k k r r r (d) If the electron is taken to infinity at rest, both its PE and KE would be 0. The amount of energy needed by the electron to have a total energy of 0 is just the opposite of the answer to part (c), 18 2.2 10 J or 14eV . 28. The dipole moment is the product of the magnitude of one of the charges times the separation distance. 19 10 30 1.60 10 C 0.53 10 m 8.5 10 C m p Ql 29. The potential due to the dipole is given by Eq. 17-6b. (a) 9 2 2 30 2 2 9 8.99 10 N m C 4.8 10 C m cos0 cos 1.1 10 m kp V r 2 3.6 10 V (b) 9 2 2 30 o 2 2 9 8.99 10 N m C 4.8 10 C m cos 45 cos 1.1 10 m kp V r 2 2.5 10 V (c) 9 2 2 30 o 2 2 9 8.99 10 N m C 4.8 10 C m cos135 cos 1.1 10 m kp V r 2 2.5 10 V Q Q r Q Q r Q Q r Chapter 17 Electric Potential © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 34 30. We assume that 1 p and 2 p are equal in magnitude, and that each makes a o 52 angle with p . The magnitude of 1 p is also given by 1 p qd , where q is the charge on the hydrogen atom, and d is the distance between the H and the O. o 1 1 o 30 20 o 10 o 2 cos52 2cos52 6.1 10 C m 5.2 10 C 2 cos52 2 0.96 10 m cos52 p p p p qd p q d This is about 0.32 times the charge on an electron. 31. The capacitance is found from Eq. 17-7. 3 6 2.5 10 C 2.9 10 F 850V Q Q CV C V 32. The voltage is found from Eq. 17-7. 8 9 16.5 10 C 17.4V 9.5 10 F Q Q CV V C 33. The capacitance is found from Eq. 17-7. 12 13 95 10 C 7.9 10 F 120V Q Q CV C V 34. We assume the capacitor is fully charged, according to Eq. 17-7. 6 5 7.00 10 F 12.0V 8.4 10 C Q CV 35. The area can be found from Eq. 17-8. 3 7 2 0 12 2 2 0 0.20F 2.2 10 m 5.0 10 m 8.85 10 C N m A Cd C A d 36. Let 1 Q and 1 V be the initial charge and voltage on the capacitor, and let 2 Q and 2 V be the final charge and voltage on the capacitor. Use Eq. 17-7 to relate the charges and voltages to the capacitance. 1 1 2 2 2 1 2 1 2 1 6 7 2 1 2 1 18 10 C 7.5 10 F 24V Q CV Q CV Q Q CV CV C V V Q Q C V V 37. The desired electric field is the value of V d for the capacitor. Combine Eq. 17-7 and Eq. 17-8 to find the charge. 12 2 2 4 2 5 0 0 8 8.85 10 C N m 35.0 10 m 8.50 10 V m 2.63 10 C AV Q CV AE d Giancoli Physics: Principles with Applications, 6th Edition © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 35 38. Combine Eq. 17-7 and Eq. 17-8 to find the area. 6 2 0 0 12 2 2 0 3 5.2 10 C 0.29 m 2000V 8.85 10 C N m 10 m AV Q Q CV AE A d E 39. From Eq. 17-4a, the voltage across the capacitor is the magnitude of the electric field times the separation distance of the plates. Use that with Eq. 17-7. 6 4 6 3 72 10 C 4.5 10 V m 0.80 10 F 2.0 10 m Q Q CV CEd E Cd 40. After the first capacitor is disconnected from the battery, the total charge must remain constant. The voltage across each capacitor must be the same when they are connected together, since each capacitor plate is connected to a corresponding plate on the other capacitor by a constant-potential connecting wire. Use the total charge and the final potential difference to find the value of the second capacitor. Total 1 1 1 1 final 2 2 final initial final final Total 1 2 1 2 final 1 1 1 2 final final final initial 1 initial 6 5 2 1 final 125V 1 7.7 10 F 1 5.6 10 F 15V Q CV Q C V Q C V Q Q Q C C V CV C C V V C C V 41. The total charge on the combination of capacitors is the sum of the charges on the two individual capacitors, since there is no battery connected to them to supply additional charge. The voltage across each capacitor must be the same after they are connected, since each capacitor plate is connected to a corresponding plate on the other capacitor by a constant-potential connecting wire. Use the total charge and the fact of equal potentials to find the charge on each capacitor and the common potential difference. 1 1 1 2 2 2 1 1 final 2 2 final initial initial initial initial final final Total 1 2 1 2 1 1 2 2 1 final 2 final initial initial final final initial initial 1 1 2 2 initial i final Q C V Q C V Q CV Q C V Q Q Q Q Q CV C V CV C V CV C V V 6 6 nitial 6 1 2 1 2 6 3 1 1 final final 6 3 2 2 final final 2.50 10 F 875V 6.80 10 F 652V 9.30 10 F 711.95V 712V 2.50 10 F 711.95 1.78 10 C 6.80 10 F 711.95 4.84 10 C C C V V Q C V Q C V Chapter 17 Electric Potential © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 36 42. Use Eq. 17-9 to calculate the capacitance with a dielectric. 2 2 12 2 2 11 0 3 5.5 10 m 2.2 8.85 10 C N m 3.3 10 F 1.8 10 m A C K d 43. Use Eq. 17-9 to calculate the capacitance with a dielectric. 2 2 12 2 2 10 0 3 5.0 10 m 7 8.85 10 C N m 1.5 10 F 3.2 10 m A C K d 44. The initial charge on the capacitor is initial initial Q C V . When the mica is inserted, the capacitance changes to final initial C KC , and the voltage is unchanged since the capacitor is connected to the same battery. The final charge on the capacitor is final final Q C V . 9 final initial final initial initial 7 1 7 1 3.5 10 F 22V 4.6 10 C Q Q Q C V C V K C V 45. The capacitance is found from Eq. 17-7, with the voltage given by Eq. 17-4 (ignoring the sign). 6 9 4 3 0.775 10 C 4.82 10 F 8.24 10 V m 1.95 10 m Q Q CV C Ed C Ed The plate area is found from Eq. 17-9. 9 3 2 0 12 2 2 0 4.82 10 F 1.95 10 m 0.283m 3.75 8.85 10 C N m A Cd C K A d K 46. The stored energy is given by Eq. 17-10. 2 2 9 4 1 1 2 2 PE 2.2 10 F 650V 4.6 10 J CV 47. The capacitance can be found from the stored energy using Eq. 17-10. 2 5 1 2 2 2 3 2 PE 2 1200J PE 9.6 10 F 5.0 10 V CV C V 48. The two charged plates form a capacitor. Use Eq. 17-8 to calculate the capacitance, and Eq. 17-10 for the energy stored in the capacitor. 2 4 3 2 2 3 0 1 1 1 2 2 2 2 12 2 2 2 0 4.2 10 C 1.5 10 m PE 2.3 10 J 8.85 10 C N m 8.0 10 m A Q Q d C d C A 49. (a) Use Eq. 17-8 to estimate the capacitance. 2 12 2 2 12 12 0 2 8.85 10 C N m 4.5in .0254m in 7.265 10 F 7 10 F 5.0 10 m A C d Giancoli Physics: Principles with Applications, 6th Edition © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 37 (b) Use Eq. 17-7 to estimate the charge on each plate. 12 11 11 7.265 10 F 9V 6.54 10 C 7 10 C Q CV (c) Use Eq. 17-4b to estimate the (assumed uniform) electric field between the plates. The actual location of the field measurement is not critical in this approximation. 2 9V 180V m 200V m 5.0 10 m V E d (d) By energy conservation, the work done by the battery to charge the plates is the potential energy stored in the capacitor, given by Eq. 17-10. 11 10 10 1 1 2 2 PE 6.54 10 C 9V 2.94 10 J 3 10 J QV (e) If a dielectric is inserted, the capacitance changes, and so the charge on the capacitor and the energy stored also change. The electric field does not change because it only depends on the battery voltage and the plate separation. 50. (a) From Eq. 17-8 for the capacitance, 0 A C d , since the separation of the plates is doubled, the capacitance is reduced to half its original value. Then from Eq. 17-10 for the energy, 2 1 2 PE Q C , since the charge is constant and the capacitance is halved, the energy is doubled. So the energy stored changes by a factor of 2 . (b) The work done to separate the plates is the source of the increase of stored energy. So the work is the change in the stored energy. 2 2 2 1 1 final initial initial initial initial 2 2 0 0 PE PE 2PE PE PE 2 Q Q Q d A C A d 51. (a) The energy stored in the capacitor is given by Eq. 17-10, 2 1 2 PE CV . Assuming the capacitance is constant, then if the potential difference is doubled, the stored energy is multiplied by 4 . (b) Now we assume the potential difference is constant, since the capacitor remains connected to a battery. Then the energy stored in the capacitor is given by 1 2 PE QV , and so the stored energy is multiplied by 2 . 52. (a) Before the two capacitors are connected, all of the stored energy is in the first capacitor. Use Eq. 17-10. 2 2 6 4 4 1 1 2 2 PE 2.70 10 F 12.0V 1.944 10 J 1.94 10 J CV (b) After the first capacitor is disconnected from the battery, the total charge must remain constant, and the voltage across each capacitor must be the same, since each capacitor plate is connected to a corresponding plate on the other capacitor by a connecting wire which always has a constant potential. Use the total charge and the fact of equal potentials to find the charge on each capacitor, and then calculate the stored energy. Chapter 17 Electric Potential © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 38 6 5 total 1 initial total 1 1 2 1 1 2 2 1 2 2 6 5 5 1 1 total 6 1 2 5 5 5 2 total 1 2.70 10 F 12.0V 3.24 10 C 2.70 10 F 3.24 10 C 1.3057 10 C 6.70 10 F Q 3.24 10 C 1.3057 10 C 1.9343 10 C Q CV Q Q Q Q Q CV Q C V C C C C Q Q C C Q Q 2 2 5 5 2 2 1 2 1 1 1 total 1 2 2 2 2 6 6 1 2 2 2 5 5 5 1 2 6 6 1.3057 10 C 1.9343 10 C PE PE PE 2.70 10 F 4.00 10 F 1.3057 10 C 1.9343 10 C 7.83 10 J 2.70 10 F 4.00 10 F Q Q C C (c) Subtract the two energies to find the change. 5 4 4 final initial PE PE PE 7.83 10 J 1.944 10 J 1.16 10 J 53. Consider three parts to the electron’s motion. First, during the horizontal acceleration phase, energy will be conserved and so the horizontal speed of the electron x v can be found from the accelerating potential V . Secondly, during the deflection phase, a vertical force will be applied by the uniform electric field which gives the electron an upward velocity, y v . We assume that there is very little upward displacement during this time. Finally, after the electron leaves the region of electric field, it travels in a straight line to the top of the screen, moving at an angle of approximately 30o. Acceleration: 2 1 initial final 2 2 PE KE x x eV eV mv v m Deflection: field field field field field 0 field time in field: 0 x x y y y y y x x x v t t v eE x eE F eE ma a v v a t m mv Screen: screen screen screen screen screen screen screen x y y x x x x x v t t y v t v v v field screen field 2 screen y x x x x eE x v y mv eE x x v v mv E field x x v 11cm 22cm o 30 screen x Giancoli Physics: Principles with Applications, 6th Edition © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 39 3 2 screen screen screen o screen field screen field screen field 5 5 2 2 7.0 10 V 0.11m 2 0.22 m cos30 0.028m 2.89 10 V m 2.9 10 V m x eV y m y mv V y m E x e x x e x x x As a check on our assumptions, we calculate the upward distance that the electron would move while in the electric field. 2 2 2 2 field field field 1 1 0 field field 2 2 2 5 2 3 0 2 4 2 2.89 10 V m 2.8 10 m 8 10 m 4 7000V y x eE x E x x eE y v t a t eV m v V m m This is about 7% of the total 11 cm vertical deflection, and so for an estimation, our approximation is acceptable. 54. If there were no deflecting field, the electrons would hit the center of the screen. If an electric field of a certain direction moves the electrons towards one extreme of the screen, then the opposite field will move the electrons to the opposite extreme of the screen. So we solve for the field to move the electrons to one extreme of the screen. Consider three parts to the electron’s motion, and see the diagram, which is a top view. First, during the horizontal acceleration phase, energy will be conserved and so the horizontal speed of the electron x v can be found from the accelerating potential V . Secondly, during the deflection phase, a vertical force will be applied by the uniform electric field which gives the electron a leftward velocity, y v . We assume that there is very little leftward displacement during this time. Finally, after the electron leaves the region of electric field, it travels in a straight line to the left edge of the screen. Acceleration: 2 1 initial final 2 2 PE KE x x eV eV mv v m Deflection: field field field field field 0 field time in field: 0 x x y y y y y x x x v t t v eE x eE F eE ma a v v a t m mv Screen: screen screen screen screen screen screen screen x y y x x x x x v t t y v t v v v field screen field 2 screen y x x x x eE x v y mv eE x x v v mv E field x x v screen 15cm y screen 34cm x Chapter 17 Electric Potential © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 40 3 2 screen screen screen screen field screen field screen field 5 5 2 2 6.0 10 V 0.15m 2 0.34 m 0.026m 2.04 10 V m 2.0 10 V m x eV y m y mv V y m E x e x x e x x x As a check on our assumptions, we calculate the upward distance that the electron would move while in the electric field. 2 2 2 2 field field field 1 1 0 field field 2 2 2 5 2 3 0 2 4 2 2.04 10 V m 2.6 10 m 6 10 m 4 6000V y x eE x E x x eE y v t a t eV m v V m m This is about 4% of the total 15 cm vertical deflection, and so for an estimation, our approximation is acceptable. And so the field must vary from 5 5 2.0 10 V m to 2.0 10 V m 55. (a) The electron was accelerated through a potential difference –6.3 kV in gaining 6.3 keV of KE. The proton is accelerated through the opposite potential difference as the electron, and has the exact opposite charge. Thus the proton gains the same KE, 6.3 keV. (b) Both the proton and the electron have the same KE. Use that to find the ratio of the speeds. 27 p 2 2 e 1 1 p p e e 2 2 31 p 1.67 10 kg 42.8 9.11 10 kg e m v m v m v v m 56. (a) The energy is related to the charge and the potential difference by Eq. 17-3. 6 6 6 PE 4.2 10 J PE 1.05 10 V 1.1 10 V 4.0C q V V q (b) The energy (as heat energy) is used to raise the temperature of the water and boil it. Assume that room temperature is 20oC. f 6 o 5 f o 4.2 10 J 1.6kg J J 4186 80C 22.6 10 kg C kg Q mc T mL Q m c T L 57. The energy density is given by Eq. 17-11. 2 2 12 2 2 7 3 1 1 0 2 2 Energy density 8.85 10 C N m 150V m 1.0 10 J m E 58. The electric force on the electron must be the same magnitude as the weight of the electron. The magnitude of the electric force is the charge on the electron times the magnitude of the electric field. The electric field is the potential difference per meter: E V d . 31 2 2 12 19 9.11 10 kg 9.80m s 3.0 10 m 1.7 10 V 1.60 10 C E F mg eV d mg mgd V e Giancoli Physics: Principles with Applications, 6th Edition © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 41 Since it takes such a tiny voltage to balance gravity, the thousands of volts in a television set are more than enough (by many orders of magnitude) to move electrons upward against the force of gravity. 59. The energy in the capacitor, given by Eq. 17-10, is the heat energy absorbed by the water, given by Eq. 14-2. 2 1 heat 2 o o o PE J 2 2.5kg 4186 95 C 21C 2 kg C 622V 620V 4.0F Q CV mc T mc T V C 60. Since the capacitor is disconnected from the battery, the charge on it cannot change. The capacitance of the capacitor is increased by a factor of K, the dielectric constant. initial initial initial initial final final final initial initial final initial 1 24.0V 11V 2.2 C C Q C V C V V V V C KC 61. Combine Eq. 17-7 with Eq. 17-8 and Eq. 17-4. 12 2 2 4 2 6 0 0 0 7 8.85 10 C N m 56 10 m 3.0 10 V m 1.5 10 C A V Q CV V A AE d d 62. Use Eq. 17-5 to find the potential due to each charge. Since the triangle is equilateral, the 30-60-90 triangle relationship says that the distance from a corner to the midpoint of the opposite side is 3 2 L . A B 3 2 1 4 6.85 2 2 3 2 3 3 6 3.46 2 2 3 2 3 k Q k Q k Q kQ kQ V L L L L L k Q k Q k Q kQ kQ V L L L L L C 3 2 1 2 5.15 2 2 3 2 3 k Q k Q k Q kQ kQ V L L L L L 63. (a) Because of the inverse square nature of the electric field, any location where the field is zero must be closer to the weaker charge 2 q . Also, in between the two charges, the fields due to the two charges are parallel to each other (both to the left) and cannot cancel. Thus the only places where the field can be zero are closer to the weaker charge, but not between them. In the diagram, this is the point labeled as “x”. Take to the right as the positive direction. 2 2 2 1 2 1 2 2 0 q q E k k q d x q x x d x d x 1 0 q 2 0 q A B C Q Q 3Q Chapter 17 Electric Potential © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 42 6 2 2 6 6 1 2 2.6 10 C 1.6cm 11.1cm left of 3.4 10 C 2.6 10 C q x d q q q (b) The potential due to the positive charge is positive everywhere, and the potential due to the negative charge is negative everywhere. Since the negative charge is smaller in magnitude than the positive charge, any point where the potential is zero must be closer to the negative charge. So consider locations between the charges (position 1 x ) and to the left of the negative charge (position 2 x ) as shown in the diagram. 6 1 2 2 location 1 1 6 1 1 2 1 6 1 2 2 location 2 2 6 2 2 1 2 2.6 10 C 1.6cm 0 0.69cm 6.0 10 C 2.6 10 C 1.6cm 0 5.2cm 0.8 10 C kq kq q d V x d x x q q kq kq q d V x d x x q q So the two locations where the potential is zero are 0.7 cm from the negative charge towards the positive charge, and 5.2 cm from the negative charge away from the positive charge. 64. The voltage across the capacitor stays constant since the capacitor remains connected to the battery. The capacitance changes according to Eq. 17-9. 12 0 0 initial final initial initial 12 final initial final initial intial initial initial 8 2600 10 F 5 5 4 4 2600 10 F 9.0V 9.4 10 C A A C C K KC C d d Q Q C V C V C C V C V 65. To find the angle, the horizontal and vertical components of the velocity are needed. The horizontal component can be found using conservation of energy for the initial acceleration of the electron. That component is not changed as the electron passes through the plates. The vertical component can be found using the vertical acceleration due to the potential difference of the plates, and the time the electron spends between the plates. Horizontal: 2 1 inital final 2 PE KE x x x qV mv t v Vertical: 0 E y y y y y y x v v qE t qE x F qE ma m v t m mv Combined: 2 1 o 250V 0.065m 0.013m tan 0.1136 2 2 2 5,500V tan 0.1136 6.5 y y y y y x x x x qE x v qE x qE x E x mv v v mv qV V d 1 x 2 x 1 0 q 2 0 q Giancoli Physics: Principles with Applications, 6th Edition © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 43 66. There is no other source of charge except for the original capacitor. Thus the total charge must remain at 0 Q . Also, since the plates of the one capacitor are connected via equipotential wires to the plate of the other capacitor, the two capacitors must have the same voltage across their plates. Use the total charge and fact of equal potentials to find the charge on each capacitor and the common potential difference. 0 0 1 2 1 1 2 2 0 1 2 1 2 1 2 1 2 1 1 0 2 2 0 1 2 1 2 Q Q Q Q Q CV Q C V Q CV C V C C V V C C C C Q CV Q Q C V Q C C C C 67. Use Eq. 17-8 for the capacitance. 12 2 2 4 2 16 0 0 8.85 10 C N m 1.0 10 m 9 10 m 1F A A C d d C No , this is not practically achievable. The gap would have to be smaller than the radius of a proton. 68. Since the E-field points downward, the surface of the Earth is a lower potential than points above the surface. Call the surface of the Earth 0 volts. Then a height of 2.00 m has a potential of 300 V. We also call the surface of the Earth the 0 location for gravitational PE. Write conservation of energy relating the charged spheres at 2.00 m (where their speed is 0) and at ground level (where their electrical and gravitational potential energies are 0). 2 1 initial final 2 4 2 2 6.5 10 C 300V 2 9.80m s 2.00m 6.3184 m s 0.540kg qV E E mgh qV mv v gh m v 4 2 6.5 10 C 300V 2 9.80m s 2.00m 6.2030 m s 0.540kg 6.3184m s 6.2030m s 0.12m s v v v 69. (a) Use Eq. 17-10 to calculate the stored energy. 2 2 8 4 1 1 2 2 PE 5.0 10 F 3.0 10 V 22.5J 23J CV (b) The power is the energy converted per unit time. 5 6 0.12 22.5J Energy 3.4 10 W time 8.0 10 s P 70. (a) Use Eq. 17-8. 12 2 2 6 2 7 7 0 8.85 10 C N m 110 10 m 6.49 10 F 6.5 10 F 1500m A C d Chapter 17 Electric Potential © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 44 (b) Use Eq. 17-7. 7 7 6.49 10 F 3.5 10 V 22.715C 23C Q CV (c) Use Eq. 17-10. 7 8 1 1 2 2 PE 22.715C 3.5 10 V 4.0 10 J QV 71. The kinetic energy of the electrons (provided by the UV light) is converted completely to potential energy at the plate since they are stopped. Use energy conservation to find the emitted speed, taking the 0 of PE to be at the surface of the barium. 2 1 initial final 2 19 6 31 KE PE 2 1.60 10 C 3.02V 2 1.03 10 m s 9.11 10 kg mv qV qV v m 72. Let 1 d represent the distance from the left charge to point b, and let 2 d represent the distance from the right charge to point b. Let Q represent the positive charges, and let q represent the negative charge that moves. The change in potential energy is given by Eq. 17-3, b a ba b a PE PE qV q V V . 2 2 2 2 1 2 b a b a 12 14 cm 18.44 cm 14 24 cm 27.78 cm PE PE 0.1844m 0.2778m 0.12m 0.24m 1 1 1 1 0.1844m 0.2778m 0.12m 0.24m d d kQ kQ kQ kQ q V V q kQq 9 6 6 1 8.99 10 1.5 10 C 33 10 C 3.477 m 1.547 J 1.5J 73. Use Eq. 17-7 with Eq. 17-9 to find the charge. 2 2 12 2 2 10 0 3 0.55 10 m 3.7 8.85 10 C N m 12V 2.5 10 C 0.15 10 m A Q CV K V d 74. (a) Use Eq. 17-5 to calculate the potential due to the charges. Let the distance between the charges be d. 9 2 2 6 6 1 2 mid 1 2 5 2 8.99 10 N m C 2 4.5 10 C 8.2 10 C 2 2 0.23m 2.9 10 V kQ kQ k V Q Q d d d (b) Use Eq. 16-4b to calculate the electric field. Note that the field due to each of the two charges will point to the left, away from the positive charge and towards the negative charge. Find the magnitude of the field using the absolute value of the charges. 2 1 mid 1 2 2 2 2 4 2 2 k Q kQ k E Q Q d d d Giancoli Physics: Principles with Applications, 6th Edition © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 45 9 2 2 6 6 6 2 4 8.99 10 N m C 4.5 10 C 8.2 10 C 8.6 10 V m, left 0.23m 75. (a) Use Eq. 17-7 and Eq. 17-8 to calculate the charge. 12 2 2 4 2 11 0 4 11 8.85 10 C N m 2.0 10 m 12V 4.248 10 C 5.0 10 m 4.2 10 C A Q CV V d (b) The charge does not change. Since the capacitor is not connected to a battery, no charge can flow to it or from it. Thus 11 11 4.248 10 C 4.2 10 C Q . (c) The separation distance was multiplied by a factor of 1.5. The capacitance is inversely proportional to the separation distance, so the capacitance was divided by a factor of 1.5. Since Q CV , if the charge does not change and the capacitance is divided by a factor of 1.5, then the voltage must have increased by a factor of 1.5. final initial 1.5 1.5 12V 18V V V (d) The work done is the change in energy stored in the capacitor. 1 1 1 final initial final initial final initial 2 2 2 11 10 1 2 4.248 10 C 18V 12V 1.3 10 J W PE PE QV QV Q V V 76. The energy stored in the capacitor is given by Eq. 17-10. The final energy is half the initial energy. Find the final charge, and then subtract the final charge from the initial charge to find the charge lost. 2 2 final initial 1 1 1 1 final initial final initial 2 2 2 2 6 lost initial final initial 6 1 2 1 1 1 1 2.5 10 F 6.0V 0.2929 2 2 4.4 10 C Q Q E E Q Q C C Q Q Q Q CV 77. (a) We assume that 2 Q is held at rest. The energy of the system will then be conserved, with the initial potential energy of the system all being changed to kinetic energy after a very long time. 2 1 2 1 initial final initial final 1 1 2 2 9 2 2 6 3 1 2 1 6 2 1 PE KE 2 8.99 10 N m C 5.0 10 C 2 2.7 10 m s 1.5 10 kg 4.0 10 m kQ Q E E m v r kQ Q v m r (b) In this case, both the energy and the momentum of the system will be conserved. Since the initial momentum is zero, the magnitudes of the momenta of the two charges will be equal. 1 initial final 1 1 2 2 2 1 2 initial final initial final 0 PE KE m p p m v m v v v m E E Chapter 17 Electric Potential © 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 46 2 2 2 2 2 1 2 1 1 1 1 1 1 1 1 2 2 1 1 2 1 1 2 1 2 2 2 2 2 2 2 9 2 2 6 6 1 2 2 1 6 2 6 1 1 2 3 2 8.99 10 N m C 5.0 10 C 2.5 10 kg 2 1.5 10 kg 4.0 10 m 4.0 10 kg 2.2 10 m s kQ Q m m m v m v m v m v m m v r m m kQ Q m v m m m r 78. Calculate AB A B V V V . Represent the 0.10 m distance by the variable d. 1 2 2 1 AB A B 1 2 9 2 2 6 5 1 1 2 2 2 8.99 10 N m C 1 4.8 10 C 1 1.3 10 V 0.1m 2 kq kq kq kq k V V V q q d d d d d
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https://www.jewelry-auctioned.com/learn/buying-jewelry/ancient-inca-gold-smelting-for-jewelry
Published Time: 2019-04-08 Ancient Inca Gold Smelting for Jewelry SHOP 18,000+ BUY IT NOW PIECES OR TRY OUR AUCTIONS WITH NO BUYERS FEES SIGN UP FOR $10 OFF Opals Gemstones Jewelry Coins Search for anything RegisterLog in Cart View All Categories A-Z View All $1 Auctions Ring AuctionsNecklace AuctionsPendant AuctionsBracelet AuctionsNewest AuctionsLoose Stone Auctions Rings Diamond RingsGemstone RingsVintage RingsGold RingsSilver Rings Earrings Gemstone EarringsDangle & Drop EarringsGold EarringsSilver Earrings Pendants Gemstone PendantsGold PendantsSilver Pendants Jewelry On SaleBrowse All Categories Who We Are About UsContact Us Need Help? 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Placing Your First BidFAQ'sHelp DocumentsContact Us Our Other Website Gem Rock AuctionsOpal AuctionsCoins Auctioned $1 Auctions Newest AuctionsRing AuctionsNecklace AuctionsPendant AuctionsBracelet AuctionsLoose Stone Auctions Rings Diamond RingsGemstone RingsVintage RingsGold RingsSilver Rings Earrings Gemstone EarringsDangle & Drop EarringsGold EarringsSilver Earrings Pendants Gemstone PendantsGold PendantsSilver Pendants Jewelry On SaleBrowse All Categories Who We Are About UsContact Us Need Help? Placing Your First BidFAQ'sHelp DocumentsContact Us Stores All Stores Learn Fashion EncyclopediaJewelers Manufactures AssociationOpal Buying Guide For JewelryGemstones and Jewelry Of The Rich And FamousHints for Buying Jewelry OnlineHow to Buy A Ruby For Your Ring or PendantHow to Buy Jewelry on the InternetHow To Understand The Meaning Of Gold HallmarksJewelry Discount Coupons and Rewards ProgramGemstone SheriffHow to Buy Jewelry Polishing Cloths Learn Buying Jewelry Ancient Inca Gold Smelting for Jewelry Published at 8th Apr 2019 Modified at 22nd Feb 2021 Ancient Inca Gold Smelting for Jewelry Ancient Gold Smelting! Incas were prolific jewelry craftsman. Incas understood how gold and silver was formed and would study rock formations looking for seam deposits of gold or silver.They would dig small holes, just enough for one man to follow these veins and extract the gold and silver in ore in high proportion compared to normal mining methods. Due to the mountainous terrain, most likely due from tectonic plate pressures that heavy deposits in veins would have been noticeable. The miners used antlers or wooden spades and stones. Gold in ore was pulverised to separate the gold. The gold was found in rivers and known as alluvial gold. From these two sources most of the gold was found and varied in composition from ore to alluvial gold so this does help explain why the purity the gold varied so much. Also some gold was mixed heavily with copper in many ceremonial artefacts. The Incas were good farmers and warriors and only extracted the easy to obtain gold. When the Spanish arrived most Incas died from mining and European diseases. Incas also had experience in gemmology, they were also top metallurgists and first record of metal working goes back to 1500 BC. Other Mesoamerican cultures didn’t commence metallurgy till 600 AD.After the mining the bedrock was pulverised by mortar or heavy rocks such as granite which was denser and harder so the ore broke down easily and was then carted to Incas smelters. It is extraordinary that they had smelters hundred years before other mesoamerican cultures as smelting gold is not an easy method. Incas also could smelt platinum and today many modern smelters can not process platinum as the temperature needs to be over 1000 degrees and very consistent to make into jewellery. Most jewelers today cannot make platinum in their small furnaces. How did they do this? First as they hunted, they were familiar with the blow dart and must have experimented with blow gun to make fires hotter as bellows were used by those in the west. This gave the same result in that three or four natives would blow air under pressure onto the same spot and this would greatly increase the heat for smelting. Up to 1000 degrees, which is a remarkable achievement. Incas also realised that the llama dung dried was far superior to wood for burning and used this product the same was that Indians use cow dung to make heat that led to wood burnt as coal and smelters were then set up using local wood as coal and llama dung as burring material, all was in plentiful supply as Incas also were master potters,they understood the properties of clay and used this in their furnaces to make clay heat resistant to pour and work with their metals. They also used topography and built smelters on hilltops, as the terrain is very mountainous they could pick places were the wind naturally tunnelled fast and furious and heaped the smelter process.To find minerals to work as metallurgy does take lot of knowledge and even to figure out percentages is also an achievement in its self! Incas hunting skills helped understand air as a resource for heat, from using blow pipe.Their pottery skills helped them understand the importance of mixing minerals and base elements to make different types of clay that could be used as cooking or decorative or for smelting. This experimentation must have led them to another important key in smelting which was honey, without this they could not have made the moulds for their artefacts and jewelry.A mould would be made from beeswax and than coated in special clay mix so that when heated the beeswax melted and gold or silver melted was then poured into this clay mould and when cool the clay would break open easily. Smelting even western cultures respected the alchemist and considerable effort went into trying to make gold from any base metal. This may sound implausible to us but what the ancient Incas achieved was also quite remarkable. To smelter and work with gold or silver metallurgists must used other minerals and catalysts to make gold workable and playable.In our modern times we used arsenic in gold mines to extract the gold. The Incas also understood the usefulness of arsenic in metallurgy coper oxide, iron and arsenic sulphate acted as arsenic hematite and maganite, limonite, all readily available as smelting agents. Archaean-metallurgist research has shown that copper was used as basis of all alloy mixes.copper was also the base for bronze in Incas times the mix of copper was relevant to hardness or colour required and just as important was the sound aspects of the finished item. Only the elite or priests were allowed to own or hold gold and silver as they did value this product of mother nature. The masses were given bronze, copper, tin and mostly a stool for working the land or weapons. 1 people found this article helpful Was this article helpful? Share this article: Search the Fashion Encyclopedia Related Auctions Natural Lapis Lazuli Bead, Silver Gold Plated Beads Pair for Jewelry Making shantoustone $30.00 1d 7h 15% Off Coupon Ancient Shield Ring Gold plated Titanium size P gem traders $10.00 9d 6h Request for Gems Identification Report topazgems $25.00 8d 10h Premium 15% Off Coupon 18kt Yellow Gold 16 i-inch Box Chain aabharanalk $312.00 6d 10h Related Articles How to Buy Jewelry on the Internet How to Buy Jewelry on the Internet. There are so many types of beads. It will be up to you to determine where you want to start, all authentic gemstone beads or if you will also use glass beads ‘man made’ as well as gold and silver beading. You 10th Apr 2019 How to Buy A Ruby For Your Ring or Pendant How to buy a ruby for your ring or pendant,Make sure that you learn as much as you can about Rubies, prior to buying a ring or a setting that you are going to add Rubies to. Like Diamonds, Rubies can be flawed, and imperfect The Ruby is one of the most b 10th Apr 2019 Hints for Buying Jewelry Online Hints for Buying Jewelry Online 11th Apr 2019 Latest Articles Titanium Jewelry: All About This Modern Alternative Titanium is a popular jewelry metal known for being lightweight, durable, and affordable. Learn all about titanium, how it compares to similar metals, and the pros and cons of titanium jewelry. 7th Feb 2023 Tungsten Jewelry: Why It’s Becoming Increasingly Popular Learn all about tungsten jewelry - from its history and uses to its durability and care. By the end of our guide, you’ll know if tungsten is right for you! 7th Feb 2023 White Gold Jewelry: Understanding the Metal and its Popularity Thinking about adding some timeless white gold jewelry to your fine accessories collection? 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https://northcoastsynthesis.com/news/level-up-on-circuit-simplification/
Level up on circuit simplification - North Coast Synthesis Ltd. We have a privacy policy, and cookie recipes. Close North Coast Synthesis Ltd. Toronto, Ontario, Canada [x] USD ▼ Canadian dollars (CAD) Euro (EUR) British pounds (GBP) Japanese yen (JPY) US dollars (USD) Bitcoin (BTC) Ethereum (ETH) Products News Audio Video DIY Dealers About Terms Privacy Level up on circuit simplification 2018-09-22 electronicsmath In my last entry I talked about the rules for simplifying series and parallel circuits. Two resistors in series can be replaced by one with a value equivalent to the pair of them; two in parallel can similarly be replaced; there are other rules in the same general form for capacitors and inductors (assuming theoretically perfect components); and by applying these rules repeatedly you can simplify complicated circuits down to much simpler equivalents. I also set up an interactive reverse calculator for finding combinations of standard-value components to make up a desired, maybe non-standard, value. The rules for series and parallel components will take you a long way, but there are some circuits that can't be analysed that way. For one thing, if a circuit involves more than one kind of component, such as both resistors and capacitors, then it may not have a simple equivalent value; the behaviour of such circuits can depend in a complicated way on frequency. That is, after all, a big part of why we build such circuits: they are the basis for filters and many other things. However, at any fixed frequency, it's possible to simplify any arbitrary two-terminal network down to a single resistance and either a capacitance or an inductance, and you can do it by applying the rules for resistances alone, but treating them as impedances: complex numbers with real and imaginary parts. Convert the capacitances and inductances to their pure-imaginary impedances for the desired frequency, use complex-number arithmetic throughout, and it works. I won't go into that further here. Instead, I'd like to talk about the cases where the basic topology of the circuit - even using plain resistors alone - does not allow for series/parallel simplification. Circuits that are not series-parallel Some circuit topologies cannot be expressed as series and parallel combinations of components. The "bridge" topology is a classic example. The bridge circuit contains no series or parallel resistor pairs that we could analyse with the rules of series and parallel circuits. And yet such circuits can certainly be built, and we could build one and attach an ohmmeter to it and get a reading. We ought to be able to predict what that reading will be. How? If the five resistors are all the same, or more generally if R 1/R 2 = R 3/R 4, then the pairs R 1 and R 2, and R 3 and R 4, function as voltage dividers with identical ratios. Then the two ends of R 5 will be at the same voltage, no current will flow through it, and then we can remove it entirely or replace it with a short circuit, and either way the bridge turns into something we can analyse with the series and parallel rules. This principle of spotting nodes at equal voltage becomes important for the cube question I'll describe later; and it's worth mentioning that in practice, bridge circuits often are designed to balance, with no or almost no current flowing through the centre, anyway. This circuit topology dates back to the 1800s, when a common way of measuring component values or physical quantities that were sensed by variable component values, was to put the unknown value into one side of the bridge and adjust known resistances on the other side until there was no current flowing through the centre. The basic bridge used for this purpose was called a "Wheatstone bridge" (for Sir Charles Wheatstone, 1802-1875, but actually invented by Samuel Christie); it is indirectly an ancestor of the Wien bridge oscillators used in the North Coast Synthesis Fixed Sine Bank. But if the two sides of the bridge do not implement the same voltage division ratio, we seem to be stuck. There are no two resistors in series without something else connected to the node joining them, so we cannot apply the series rule; and there are no two resistors in parallel, so we cannot apply the parallel rule. The answer is to introduce a new rule, which goes by several names but is often called the delta-wye transformation. Where three resistors form a triangle, that's called a "delta" (for the Greek letter Δ) and we can replace it with three other resistors forming a "wye" (for the Latin letter Y). Similarly, any wye can be replaced by a delta. Applying this (in either direction, in fact) to the bridge circuit results in something which can be simplified further using the standard series and parallel rules. The famous resistor cube The famous resistor cube is a traditional "brain teaser" question. Some people think it makes sense to ask this during job interviews, or on exams in school. It seems to be quite popular in India; I don't know the Indian educational system but it seems like they have some kind of national standardized tests and the resistor cube comes up in those a lot, so there are very many videos online of people explaining how to answer the resistor-cube question specifically for passing the Indian tests. And so, of course, everyone just memorizes the answer and it's completely useless for testing anybody's problem-solving skills. Note, also, that when Google did their study on interviewing, they found that brain-teasers just don't work. They're more a matter of testing the applicant's cultural background and making the interviewer feel smart themselves. Ability to answer brain teaser questions has nothing to do with actual success once hired. So with that in mind, let me give you some hints and spoilers: the bear was white; 90 minutes is exactly the same as an hour and a half; the surgeon was the patient's mother; the detective committed the murder himself; fill the ship with ping-pong balls (this one was the plot of a Donald Duck comic); "in the dictionary"; cut the cake along the X, Y, and Z coordinate planes; and the one about the rubber gloves is easier to memorize if you think of it in terms of condoms instead. As for the resistor cube, see below. The problem as it's usually stated is that the twelve resistors along the cube edges have 1Ω resistance each and you're supposed to find the resistance between two diagonally opposite corners, such as from the corner labelled A, to the corner labelled D. As I mentioned, what people really do is they memorize the answer: 5/6Ω. The traditional "right" answer is to recognize points of equal voltage, as I mentioned doing with the bridge circuit. Remember that we're attaching the ohmmeter probes to A and D. When that's the case, all of the three nodes marked B are at the same voltage, by the symmetry of the cube. Although there are no resistors directly between nodes marked B, we could put some in and they would have no effect. We could also short them directly together. Similarly, we could short together the three nodes labelled C, which are at equal voltage. And having done that, we've reduced the cube to a series combination of parallel combinations of 1Ω resistors: three from A to B, six from B to C, three from C to D. That makes 1/3Ω + 1/6Ω + 1/3Ω = 5/6Ω, which is the right answer to the question. If someone wants to make the question a little harder, they'll ask for the measurement across a different pair of nodes in the cube. Diagonally across a face of the cube (from A to one of the nodes marked C), the resistance is 3/4Ω, and along a side (from A to one of the nodes marked B), it's 7/12Ω. I won't draw diagrams of the detailed calculations, but they come down to the same procedure of recognizing equal-voltage points and then doing series and parallel simplification. I've also posted a video on the new North Coast video server in which I actually build a cube with real resistors and measure the resistances; up to component tolerances and the precision of my ohmmeter, it basically matches the theoretical prediction. The next step up in difficulty would be to give the resistors in the cube different values. Then it's no longer possible to use symmetry to change the cube into a series/parallel network. If somebody pulls that on you in a job interview you should perhaps think about whether you really want to work for such a smart alec, but be that as it may, it's possible in such a case to work out the result with the delta-wye transformation. Going further Is this the last word on simplifying resistor networks? With the delta-wye transformation as well as the series-parallel rules, can we solve every resistor problem? At first glance, it might seem that we can. Even very difficult-looking networks, like this one from an XKCD cartoon, succumb to delta-wye analysis. The answer for this network is 25265/33783 or approximately 0.74786Ω. But the correct answer is "no." In fact, it is possible to construct resistor networks that cannot be simplified by series, parallel, or delta-wye transformations. We don't often see them because it takes a lot of resistors to build such a thing, and such complicated resistor networks do not actually occur very often in practice. In graph theoretic terms, what it takes for the series, parallel, and delta-wye rules to all be inapplicable is that the resistors should be along the edges of a graph that is simple (no multiple edges; prevents parallel-resistor simplification), triangle-free (prevents delta to wye simplification), and has minimum degree 4 (prevents series and wye to delta simplifications). For example, this circuit - which is the one with fewest resistors meeting the conditions. It corresponds to the graph called K 4,4: two sets of four nodes with a resistor connecting each pair of one node from each set. You can check for yourself that that meets the conditions: there's nowhere in it that the series, parallel, or delta-wye (in either direction) rules could be applied. Another example would be the four-dimensional hypercube graph I've used for automated composition. There is such a thing as a star-mesh transform, which can be applied to any resistor network to eliminate one node (at the cost of quite possibly adding many new resistors), and by applying that and the other rules a sufficient number of times it's possible to simplify any two-terminal resistor network down to a single equivalent resistance. However, it's quite cumbersome to do because of the skyrocketing number of steps as minimum degree increases. Really, at this point when confronted with a circuit for which the simpler rules don't apply, it makes more sense to just throw the whole thing into a circuit simulator with a constant current applied between the appropriate points. Then you can read off the equivalent resistance as the resulting voltage drop, suitably scaled. See my Free EDA Software rundown for more about circuit simulators. The way the circuit simulators do their job comes down to applying the basic rules, Ohm's and Kirchoff's laws, to create a set of simultaneous linear equations. The usual form, called nodal analysis, assigns a variable to the voltage at every node. Then it's easy to express the current through each component by Ohm's Law on the voltage drop, and the net current at each node must be zero except where current sources or sinks are attached, so that gives the equations. Numerical matrix algorithms are applied to solve the set of linear equations. There are some subtleties having to do with careful management of rounding-off errors and so on, but it's all fairly well-understood computer science. So, I've described in my earlier entry the basic rules for simplifying series and parallel circuits. Not all circuits are series or parallel, so I've described in this entry some of the ways that approach can fail, and presented some rules for simplifying more complicated circuits, including the famous resistor cube. Even these more advanced rules are not absolutely universal; when they too break down it's usually best to abandon hand analysis and run a computer simulation instead, but I've talked a little bit about how the computer simulations work and how one would go about doing it by hand if determined not to use computer simulation. ◀ PREVCombining components for new values || Exponential converters and how they workNEXT ▶ MSK 013 Middle Path VCO US$482.89 including shipping Name Comment Anti-spam, fill in the blank: North Coast Subscribe to our newsletter Pages News SDIY projects Dealers About Terms and conditions Privacy policy Categories Assembled modules SDIY kits Extras Links Audio server Video server Contact Follow us ©2025 North Coast Synthesis Ltd.
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https://math.stackexchange.com/questions/1300122/area-between-intersecting-lines-elegant-solution
linear algebra - Area Between Intersecting Lines - Elegant Solution? - 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 Area Between Intersecting Lines - Elegant Solution? Ask Question Asked 10 years, 4 months ago Modified10 years, 4 months ago Viewed 267 times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. I am running simulations, and the output will be a line y = mx+b. I am interested in the area below the line between x=0 and x=1. I am only interested in the area that is below the diagonal y = x. I have figured out how to determine this area by finding areas of triangles. But to do so, I have to define 6 cases. This requires many if-else statements in my computer program, and is inefficient. I was wondering if there is an elegant solution to this problem which will not require such a complex program? In the diagram below, the diagonal line is solid, my line of interest is the dotted line. linear-algebra integration geometry Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications asked May 26, 2015 at 20:57 MPahutaMPahuta 219 2 2 silver badges 8 8 bronze badges Add a comment| 1 Answer 1 Sorted by: Reset to default This answer is useful 0 Save this answer. Show activity on this post. Let y=m x+b y=m x+b be the equation of your line, and then find the point of intersection with y=x y=x, which will be (p,p)(p,p). Then you need to find two integrals: the integral from 0 0 to p p of x−(m x+b)x−(m x+b) and the integral from p p to 1 1 of the same function. Then pick whichever one is nonnegative! EDIT: Just realized the lines don't have intersect. If that happens, just take the integral from 0 0 to 1 1 of x−(m x+b)x−(m x+b) and if it's negative, return 0. I guess I only simplified a few cases. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered May 26, 2015 at 21:04 GiuseppeGiuseppe 757 6 6 silver badges 15 15 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 linear-algebra integration geometry See similar questions with these tags. 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https://www.youtube.com/watch?v=kKqSp7my5NI
Integral of (sin(x)+cos(x))^2 (substitution) Integrals ForYou 137000 subscribers 98 likes Description 5452 views Posted: 18 Oct 2020 Integral of (sin(x)+cos(x))^2 - How to integrate it step by step using the substitution method! 🚶 𝐒𝐭𝐞𝐩𝐬 00:00 Use the (a+b)^2 = a^2 + 2ab + b^2 formula 00:33 Use the 1=sin^2(x)+cos^2(x) identity 00:59 Rewrite expression 01:19 Substitution: u=sin(x) 01:25 Differentiate in both sides 01:30 Substitute sin(x) and cos(x)dx 01:42 Integrate 1dx and udu 01:53 Undo substitution: u in terms of x 01:59 Add integration constant +C 02:06 Final answer! 👋 Follow @integralsforyou on Instagram for a daily integral 😉 📸 @integralsforyou 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧 𝐦𝐞𝐭𝐡𝐨𝐝𝐬 𝐩𝐥𝐚𝐲𝐥𝐢𝐬𝐭 ► Integration by parts ► Integration by substitution ► Integration by trig substitution ► Integration by Weierstrass substitution ► Integration by partial fraction decomposition 𝐅𝐨𝐥𝐥𝐨𝐰 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥𝐬 𝐅𝐨𝐫𝐘𝐨𝐮 ▶️ Youtube: 📸 Instagram: 👍 Facebook: 𝐃𝐨𝐧𝐚𝐭𝐞 🙋‍♂️ Patreon: integralsforyou #integrals #integrationbysubstitution 26 comments Transcript: Use the (a+b)^2 = a^2 + 2ab + b^2 formula Use the (a+b)^2 = a^2 + 2ab + b^2 formula Use the 1=sin^2(x)+cos^2(x) identity Use the 1=sin^2(x)+cos^2(x) identity Rewrite expression Rewrite expression Substitution: u=sin(x) Substitution: u=sin(x) Differentiate in both sides Differentiate in both sides Substitute sin(x) and cos(x)dx Substitute sin(x) and cos(x)dx Integrate 1dx and udu Integrate 1dx and udu Undo substitution: u in terms of x Undo substitution: u in terms of x Add integration constant +C Add integration constant +C Final answer! Final answer!
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https://www.youtube.com/watch?v=3xAIWiTJCvE
Cumulative Distribution Functions and Probability Density Functions The Organic Chemistry Tutor 9770000 subscribers 8655 likes Description 772175 views Posted: 21 Sep 2019 This statistics video tutorial provides a basic introduction into cumulative distribution functions and probability density functions. The probability density function or pdf is f(x) which describes the shape of the distribution. It can tell you if you have a uniform, exponential, or normal distribution. The cumulative distribution function or cdf allows you to calculate the area under the curve to the left of some point of interest in order to evaluate the accumulated probability. Probability - Free Formula Sheet: Statistics - Video Lessons: Final Exam and Test Prep Videos: Introduction to Probability: Probability Formulas: Probability Explained: Probability With Geometry: Probability of Complementary Events: Conditional Probability: Independent and Dependent Events: Probability of Mutual Exclusive Events: Multiplication and Addition Rule: Compound Probability: Expected Value: Probability Tree Diagrams: Bayes Theorem: Probability - Binomial Distribution: Probability - Geometric Distribution: Probability - Poisson Distribution: Continuous Probability Distributions: Probability Density Functions: Probability - Uniform Distributions: Probability - Exponential Distributions: Probability - Normal Distributions (Calculus): Probability - Standard Normal Distributions: Probability - The Law of Large Numbers: Final Exams and Video Playlists: 140 comments Transcript: in this video we're going to talk about cumulative distribution functions the cumulative distribution function or CDF these functions are used to calculate the area under the curve specifically the area to the left of some point of interest now these functions they're used to calculate the accumulative sadar on the accumulated probability now keep in mind whenever you have a continuous probability distribution and the probability is equal to the area under the curve and since the maximum probability is 1 the total area under the curve will be 1 as well now probability density functions are different from cumulative distribution functions the PDF or probability density function is f of X and it tells you in the shape of the distribution so for instance let's say if we have a uniform distribution in this case f of X will equal a specific number it's going to be a constant value f of X is going to be 1 over B minus a for a uniform distribution it's a number it doesn't have the variable X in it you'll see something that looks like this so this is f of X and it varies from A to B so this right here is the PDF that is the probability density function for a uniform distribution it's equal to f of X now the CDF the cumulative distribution function will give us the area under the curve to the left so let's say the point of interest is X the CDF will give us the area to the left of X so the area of the shaded region in blue now this is basically the shape of a rectangle the area of a rectangle is the base times the height notice that the base is the difference between X and a so it's X minus a the height is f of X and f of X is what we see here B minus a so the CDF the cumulative distribution function will give us the area to the left so we have this formula the probability that the random variable capital X is less than or equal to some number is going to be X minus a over B minus a so this function right here highlighted in blue this is the CDF that is the cumulative distribution function for a uniform distribution it gives us the area to the left of some value X and the PDF Isha's f of X for uniform distribution that's 1 over B minus a now let's talk about the exponential distribution let's discuss the CDF and the PDF for this type of distribution the exponential distribution the maximum value is the y-intercept and it decreases over time the maximum value is also known as lambda a rate parameter the probability density function for an exponential distribution is going to be the function f of X which describes the shape of the distribution and f of X for this particular distribution is lambda e to the negative lambda times X and lambda is the rate parameter it's 1 divided by the mean so this is the probability density function it tells you the shape of the graph that's what it does in addition it also gives you the height of the curve above the x-axis at some point X that's another thing that it tells you now the CDF gives us the area under the curve to the left of some value X so let's say if we wish to calculate the area to the left so that's going to be a oh that's going to be the probability that X is less than or equal to some value and that's going to be equal to one minus e to the negative lambda times X so that is the cumulative distribution function it gives us the area under the curve to the left so if we wish to calculate the probability that X is between 0 to X or let's say a 0 to a this function will help us to get the probability it's equal to the area under the curve now what if we wish to calculate the area to the right as opposed to the area to the left we know that the total area is 1 so the area to the right plus the area to the left must also be 1 so the area to the right is 1 minus the area to the left and we know the area to the left is what we see here it's 1 minus e to the negative lambda times X now if we distribute the negative sign that is this negative sign it's going to be 1 minus 1 and then these two negative signs will cancel giving us a plus sign so plus e to the negative lambda times X so 1 minus 1 is 0 thus the area to the right is going to be simply e to the negative lambda times X now this is not the cumulative distribution function because it doesn't give us the area to the left it gives us the area to the right so the CDF is what we see here that's a CDF for an exponential distribution now let's say if we still have an exponential distribution but we want to calculate the area between a and B so we want to find a probability that X is somewhere between a and B how can we calculate the area under the curve between those two points so this area highlighted in blue is equal to the probability that X is between a and B and that is the difference between the probability that X is less than B minus the probability that X is less than a so these are areas to the left so we could use a CDF to calculate the probability that X is less than being a probability that X is less than a the probability that X is less than a would be the area shaded in red the probability that X is less than B would be the area shaded in green if you subtract the area shaded in green minus the area shaded in red you'll get the area shaded in blue so using the CDF function we can replace that value with this is going to be 1 minus e to the negative lambda instead of X we're going to replace X with B so this will give us the area to the left of B now we need to use the CDF function to get the area to the left of a and that's going to be 1 minus e to the negative lambda times a so this is how we can calculate the probability between a and B using or if we have an exponential distribution another thing that you want to keep in mind is for a continuous probability distribution the probability that X is greater than or equal to a or and rather less than or equal to B this is the same as the probability that X is between a and B so you want to keep that in mind there's no difference here another thing that you want to keep in mind when dealing with continuous probability distribution functions is that the probability that X is a single value as opposed to an interval of values let's say the probability that X is equal to a that's going to be 0 the only way to get a value greater than 0 is you need a range of X values because at x equals a all you have is a line you can't really calculate the area of a line because you have height but there's no whiff so therefore you can't calculate the area the probability that X is exactly a is 0 so just keep that in mind to review remember the PDF the probability density function is f of X it tells me the shape of the graph if it's an exponential distribution a normal distribution or uniform distribution the CDF the cumulative distribution function it gives you the area to the left of some value it can give you the area to the left of a or to the left of B but it gives you the area under the curve which is the accumulated probability up to that point so that's the difference between the PDF and the CDF thanks for watching
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https://www.vedantu.com/maths/value-of-e
Courses for Kids Free study material Offline Centres Talk to our experts Maths Value of e in Maths: Explained with Examples Value of e in Maths: Explained with Examples Reviewed by: Rama Sharma Download PDF NCERT Solutions NCERT Solutions for Class 12 NCERT Solutions for Class 11 NCERT Solutions for Class 10 NCERT Solutions for class 9 NCERT Solutions for class 8 NCERT Solutions for class 7 NCERT Solutions for class 6 NCERT Solutions for class 5 NCERT Solutions for class 4 NCERT Solutions for Class 3 NCERT Solutions for Class 2 NCERT Solutions for Class 1 CBSE CBSE class 3 CBSE class 4 CBSE class 5 CBSE class 6 CBSE class 7 CBSE class 8 CBSE class 9 CBSE class 10 CBSE class 11 CBSE class 12 NCERT CBSE Study Material CBSE Sample Papers CBSE Syllabus CBSE Previous Year Question Paper CBSE Important Questions Marking Scheme Textbook Solutions RD Sharma Solutions Lakhmir Singh Solutions HC Verma Solutions TS Grewal Solutions DK Goel Solutions NCERT Exemplar Solutions CBSE Notes CBSE Notes for class 12 CBSE Notes for class 11 CBSE Notes for class 10 CBSE Notes for class 9 CBSE Notes for class 8 CBSE Notes for class 7 CBSE Notes for class 6 What is the Value of e and Why Is It Important in Mathematics? The concept of value of e in Maths plays a key role in mathematics and is widely applicable to real-life continuous growth, finance, and competitive exam problems. This page explores how the value of e helps in exponents, calculus, and much more, making it essential for students preparing for exams. What Is Value of e in Maths? The value of e in Maths is a special mathematical constant, pronounced as “Euler’s Number,” represented by the letter e. It is defined as the base of natural logarithms and is an irrational number, meaning its digits go on forever without repeating. You’ll find this concept applied in areas such as exponential functions, logarithms, and calculus. Quick Fact: The first few digits of e are 2.71828… Key Formula for Value of e in Maths Here’s the standard formula: ( e = \lim_{n \to \infty} (1 + \frac{1}{n})^n ) Alternatively, it can also be found using an infinite series: ( e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \ldots ) Decimal & Fraction Value of e | Representation | Value | Remarks | --- | Decimal (up to 5 places) | 2.71828 | Irrational and non-terminating | | Approximate Fraction | 2718/1000 | Only for rough calculations; not exact | Cross-Disciplinary Usage The value of e in Maths is not only useful in mathematics but also plays an important role in Physics, Computer Science, and finance. For example, e helps when calculating compound interest, modeling population growth, or even understanding radioactive decay in physics. Students preparing for JEE or NEET will see its relevance in differential equations and continuous growth models. Step-by-Step Illustration: Calculating e Using Limit Start with the formula: ( e = (1 + \frac{1}{n})^n ) Let’s calculate for increasing n: For n = 1: ( (1 + \frac{1}{1})^1 = 2^1 = 2.00000 )For n = 2: ( (1 + \frac{1}{2})^2 = (1.5)^2 = 2.25 )For n = 5: ( (1 + \frac{1}{5})^5 \approx 2.48832 )For n = 10: ( (1 + \frac{1}{10})^{10} \approx 2.59374 )For n = 100: ( (1 + \frac{1}{100})^{100} \approx 2.70481 )For n = 1000: ( (1 + \frac{1}{1000})^{1000} \approx 2.71692 ) 3. As n increases, the result approaches 2.71828 (value of e). Speed Trick or Memory Shortcut Here’s a quick shortcut that helps solve problems faster when working with value of e in Maths. Many students remember the value up to 5 decimals (2.71828) by the pattern: 2.7 (then 1828, which repeats as e is irrational). You can also think "2, 7, 18, 28" as a series of steps to recall quickly in exams. Example Tip: The derivative of ( e^x ) is always ( e^x )! This unique property makes differentiated calculations super quick in calculus questions. Tricks like these are useful in competitive exams. Vedantu teachers often include more memory and calculation hacks in live classes. Solved Example Example: Find the value of ( (1 + \frac{1}{n})^n ) for n = 100000. Start with formula: ( (1 + \frac{1}{n})^n ) 2. Plug in n = 100000: ( (1 + \frac{1}{100000})^{100000} ) 3. Calculate value: ( (1 + 0.00001)^{100000} \approx 2.71827 ) Final Answer: Approximate value is 2.71827 (matches value of e up to 5 decimals). Try These Yourself Write the value of e up to 10 decimal places. Use the series formula for e to sum first four terms. Where is e used in compound interest? Find the value of ( (1 + \frac{1}{50})^{50} ). Frequent Errors and Misunderstandings Confusing e (Euler’s number) with the e used for electron charge in physics. Rounding e to 2.7 in exams — always use at least 2.718 for accuracy. Trying to express e as a simple fraction — it's irrational and cannot be written exactly as a ratio. Relation to Other Concepts The idea of value of e in Maths connects closely with other important topics, like natural logarithms (ln), exponential growth and decay, and the family of irrational numbers. Mastering e will help you understand advanced functions, calculus, and real-world mathematical modeling. For JEE and board exams, e frequently appears within derivatives, integrals, and formulas for continuous change. Classroom Tip A quick way to remember the value of e in Maths is “2 point 7 – 1828,” treating the decimal digits as chunks. Create a chant or rhythm to memorize it! Vedantu’s teachers use songs and visuals during online sessions to turn remembering e into a fun activity. We explored value of e in Maths — from its definition, limit and series formula, real-life usage, common mistakes, and its links to calculus and logarithms. Keep practicing problems using e, and check out more memory shortcuts in Vedantu’s live sessions to master this powerful mathematical constant. Explore More on Related Topics Logarithms: Learn everything about natural logs (base e) and their calculations. Exponential Functions: See how e is used for growth and decay in maths and science. Calculus: Dive deeper into derivatives and integrals involving ex. Irrational Numbers: Understand why numbers like e and π can never be written as exact fractions. 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Euler's number, denoted by the symbol e, is a fundamental mathematical constant, similar to π. It is the base of the natural logarithm. The value of e is an irrational number, which means its decimal representation never ends or repeats. Its value is approximately 2.71828. What are the common methods used to calculate the value of 'e'? There are two primary formulas used to calculate the value of 'e': The Infinite Series Formula: This formula defines 'e' as the sum of an infinite series: e = 1 + 1/1! + 1/2! + 1/3! + ..., where '!' denotes the factorial. The Limit Formula: This formula, often seen in the context of compound interest, defines 'e' as the limit of an expression as n approaches infinity: e = limₙ→∞ (1 + 1/n)ⁿ. What is the importance of 'e' in calculus according to the CBSE syllabus? The constant e is critically important in calculus for several reasons. The exponential function eˣ is unique because its derivative is itself (d/dx(eˣ) = eˣ). This property makes it fundamental for solving differential equations that model various natural phenomena. Furthermore, the natural logarithm (ln x), which has 'e' as its base, has a simple derivative of 1/x, simplifying many integration and differentiation problems studied in Class 11 and 12 Maths. Can you provide some real-world examples where the value of 'e' is applied? The value of 'e' is used to model phenomena involving continuous growth or decay. Key examples include: Finance: Calculating continuous compound interest on investments. Biology: Modelling population growth of species under ideal conditions. Physics: Describing radioactive decay of unstable atoms over time. Engineering: Analysing the cooling of an object (Newton's law of cooling). What is the relationship between Euler's number (e) and the natural logarithm (ln)? Euler's number (e) and the natural logarithm (ln) are inverse functions of each other. The natural logarithm is a logarithm with base 'e'. This means that if eʸ = x, then ln(x) = y. This inverse relationship is crucial for solving exponential equations, as it allows us to 'undo' the exponentiation, for instance, ln(eˣ) = x. Why is the number 'e' considered irrational? 'e' is considered an irrational number because it cannot be expressed as a simple fraction (a/b, where 'a' and 'b' are integers). Its decimal representation is both non-terminating and non-repeating. The proof of its irrationality, first established by Euler, demonstrates that the infinite series used to define 'e' (1 + 1/1! + 1/2! + ...) cannot sum to a rational number. Why is 'e' called the "natural" base for logarithms and exponential functions? The constant 'e' is called the "natural" base because it arises naturally in processes involving continuous growth. The function y = eˣ has a unique and simple property where the rate of change (the derivative) at any point is equal to the value of the function at that point. This makes it the most straightforward base to use when modelling natural phenomena, hence the term "natural logarithm" or "natural exponential function". What is the fundamental difference between the 'e' in mathematics and the 'e' used for elementary charge in physics? These two values are completely unrelated despite sharing the same symbol. In mathematics, 'e' is Euler's number, an irrational constant approximately equal to 2.718, fundamental to calculus and growth models. In physics, 'e' typically represents the elementary charge, which is the electric charge carried by a single proton or electron, approximately 1.602 x 10⁻¹⁹ Coulombs. The context of the problem determines which 'e' is being used. How do you calculate powers of 'e', such as e² or e⁻¹? To calculate powers of 'e', you use the approximate value of e ≈ 2.71828. For example: e² means e × e, which is approximately 2.71828 × 2.71828 ≈ 7.389. e⁻¹ is the reciprocal of 'e', which is 1/e or approximately 1/2.71828 ≈ 0.367. Most scientific calculators have a dedicated 'e' or 'eˣ' button to compute these values with high precision. What do the values of e⁰ and e¹ represent? The values of e⁰ and e¹ follow the standard rules of exponents: e⁰ = 1. Any non-zero number raised to the power of zero is always equal to 1. e¹ = e. Any number raised to the power of one is the number itself. So, e¹ is simply 'e', which is approximately 2.71828. 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https://www.gauthmath.com/solution/1838476521677842/Without-graphing-where-do-the-graphs-of-y-x2-and-y-9-intersect-The-graphs-inters
Solved: Without graphing, where do the graphs of y=x^2 and y=9 intersect? The graphs intersect at [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 Function Questions Question Without graphing, where do the graphs of y=x^2 and y=9 intersect? The graphs intersect at (□ ,□ ) and (□ ,□ ). Explain. Letting y=9 in the first equation, 9=x^2 when x=□ and x=□. Show transcript Gauth AI Solution 100%(3 rated) Answer The correct answers are: (3, 9) (-3, 9) 3 -3 Explanation Find the x-values where the graphs intersect To find the points of intersection, we set the two equations equal to each other: $$x^{2} = 9$$x 2=9 Taking the square root of both sides, we get $$x = \pm 3$$x=±3 So, $$x = 3$$x=3 or $$x = -3$$x=−3 Find the y-values of the intersection points Since $$y = 9$$y=9, the y-coordinate for both intersection points is 9. Write the coordinates of the intersection points The intersection points are $$(3, 9)$$(3,9) and $$(-3, 9)$$(−3,9) Determine the x-values when 9 = x^2 $$9 = x^{2}$$9=x 2 when $$x = 3$$x=3 and $$x = -3$$x=−3 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 REASONING Without graphing, where do the graphs of y=x2 and y=9 intersect? The graphs intersect at square ,square and square ,square . Explain. Letting y=9 in the first equation, 9=x2 when x=square and x=square 100% (7 rated) Listen REASONING Without graphing, where do the graphs of y=x2 and y=9 intersect? The graphs intersect at square ,square and square ,square . Explain Letting y=9 -in the first equation, 9=x2 when x=square and x=square . 100% (5 rated) [Listen REASONING Without graphing, where do the graphs of y=x2 and y=9 intersect? The graphs intersect at 3,9 and [−3, 9 . Explain. Letting y=9 in the first equation, 9=x2 when x=square and x=square . 100% (6 rated) on Exercises 4,3 Without graphing, where do the graphs of y=x2 and y-9 indersect? The graphs intersect at square ,square and square square Explain Letting y=9 in the first equation. y=x2 when e=square and x=square 100% (4 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) 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) Multiply and simplify the following. 3-i-4-9i -21-24i -21-23i -21+23i ⑤ 21-23i 100% (5 rated) Which equation represents a circle with center -5,-6 and radius of 5? x+52+y+62= square root of 5 x+52+y+62=25 x+52+y+62=5 x-52+y-62=25 75% (4 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) How may different arrangements are there of the letters in The number of possible arrangements is MISSISSIPPI？ 100% (2 rated) Gauth it, Ace it! contact@gauthmath.com Company About UsExpertsWriting Examples Legal Honor CodePrivacy PolicyTerms of Service Download App
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https://www.youtube.com/watch?v=YVxbfMhoejU
Complex Numbers Lecture 5.6 Solved Example: Another Application to Regular Polygons | #JEE The Hidden Library of Mathematics 5470 subscribers 20 likes Description 373 views Posted: 26 Sep 2022 A peculiar problem about regular polygons solved using complex numbers. 00:00 The Problem 00:53 Preparation 03:24 Solution Full playlist: Note: The auto-generated subtitles are mostly accurate and may enhance experience. Prerequisites: Basic trigonometry and coordinate geometry. Thanks to my Patreon Supporters: Namrata Khetan Hersh Support me on Patreon: One Time Donation on PayPal: jeeadvanced jeemains 6 comments Transcript: The Problem welcome to this lecture Series in complex numbers in this lecture we'll be looking at a problem about regular polygons we have looked at one such problem before and this is quite similar but still has interesting ideas uh so here's the problem let A1 up to A and B the vertices of a regular polygon here is a nice diagram showing it uh let R be the radius of the circum Circle that that circumscribes the polygon so this pink circle it's a radius is let's say r then the product of all these lengths so you take A1 then this length A1 A2 then A1 A3 then A1 A4 and if you product all these lines you get n times R to the N minus 1. so this is what we want to show as usual you run Carriage to pause the video and try to solve the problem yourself uh and with that now let us go to the solution and before we go to the solution Preparation we recall a fact about polynomials so suppose we have a non-zero polynomial then it has only finitely many routes so if it were a constant non-zero polynomial then it would have no roots what are Roots roots are the points where the the function vanishes the polynomial vanishes so p in C is a root of f if F of p is zero all right uh so what we are saying is that if you have a non-zero polynomial then it has only finitely many roots in fact what we showed was that if you have a polynomial of degree n and N is some non-negative integer sorry N is a positive integer then it has at most n Roots so that is what we saw and that was consequence of what is called the factored theorem or something basically if Z naught is a root of f then you can find a polynomial G such that this happens right so that's basically why the number of roots of f is bounded by the degree of f so anyway this is a this is a conclusion that holds and the corollary of that conclusion is this if you we have two polynomials f and g which agree on an infinite set so what do we mean agree so we say f and g agree on a point P if F of p is G of P meaning they take the same value at B and we what we have in the hypothesis is that suppose two polynomials agree on an infinite set meaning there's an infinite set of point in the complex plane where f and g take the same value then in fact the two polynomials are equal and this is very simple because just to find a function as the difference of f and g what this really means is that the value that H takes on Z is the difference of the values that f and g take on number Z right and since f and g are polynomials so is H the difference of two polynomials is a polynomial and by the hypothesis that f and g agree on an infinite set we see that F minus G vanishes on an infinite side and hence this polynomial has infinitely many roots meaning it vanishes on infinitely many points and therefore this must be the zero polynomial that's what this conclusion is and therefore f equals g so that's it I'm not writing all the details but I've said it and now let us get back to our problem Solution so we have this diagram and we may assume that the you know so let's say these are the these are the vertices of our polygon and we may assume that the center of the circle the pink circle is the origin and we may also assume that this number A1 yeah so I will prove it for radius equals one you can get to the general Thing by just scaling the you know the argument so now we may also that's assume by rotating or whatever that A1 is 1. even is a complex number one right if you assume R is one then you may assume A1 is one just by rotating the entire polygon and the center of the polygon is zero so under those assumptions what what we want to really talk about is this quantity A1 up to a n but what is this this is magnitude of 1 minus yeah so before I before I say that let Zeta be because 2 pi by n plus iota sine 2 pi by n so then what does A2 become A2 is Zeta this is Zeta square and so on so A1 A2 is 1 minus Zeta magnitude A1 A3 is 1 minus Zeta Square magnitude and A1A A1 a n is this so this is the quantity which we want to show is equal to n times R to the N minus 1 but R is equal to 1 so we just want to show that this thing is equal to n so that is our goal all right let me put a boundary here okay but what is this this is equal to 1 minus Zeta 1 minus Zeta Square dot dot 1 minus Zeta to the power n minus 1. all right and now let us freeze it here for the moment what we know is on the side we know that so let me use the different color now we have this expression is equal to Z minus 1 Z minus Zeta dot Z minus Zeta to the power n minus 1. right because these are precisely the all the end roots of unity and by the factor theorem or however we want to see it we have this is equal to that right and now this implies that this is equal to Z minus Zeta Z minus Zeta Square Dot Z minus Theta to the N minus 1. for all Z not equal to 1 because we are dividing by something and if you are dividing by something that something shouldn't be zero so for Z naught equals to 1 we have this equality which implies 1 plus Z Plus Z square plus DOTA Dot Z to the N minus 1 is equal to the right hand side for Z not equals to 1. so all we did was that this is equal to that by the geometric formula right so the left eye left hand side is a polynomial the right hand side is a polynomial and they agree on an infinite set the set is the set is infinite right all complex numbers other than one common infinite side so this should be equal everywhere not just outside you know not not just in this infinite side but there should be equal everywhere that is what we concluded earlier so in fact we get that this is true for all complex number set and now we can substitute Z equals one so this implies yeah so now the color is back so this implies One Plus One Plus One Plus not a DOT plus one equals 1 minus Zeta 1 minus 0 squared Dot 1 minus Theta to the power n minus 1. which implies n is equal to this thing and you can take the magnitude on both sides if you like but that doesn't change anything it's already a real number already a positive real number all right so this shows that this this quantity is n and uh the general you know if the radius would not assumed to be one you would when you would do this in R to the N minus 1 would pop out in the beginning so it's the same thing and yeah so that's a very nice solution to uh fairly complicated looking geometric problem and with that I want to end this lecture as usual like comment share subscribe and I'll see you next time
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https://math.stackexchange.com/questions/3882885/why-is-this-intersection-of-sets-empty
elementary set theory - Why is this intersection of sets empty? - 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 Why is this intersection of sets empty? Ask Question Asked 4 years, 11 months ago Modified4 years, 11 months ago Viewed 254 times This question shows research effort; it is useful and clear 3 Save this question. Show activity on this post. An example I saw online tells me this following set of arbitrary intersections are empty: If we define a set B m={m,m+1,m+2,...}B m={m,m+1,m+2,...} where m∈N m∈N, then ⋂m∈N B m=∅⋂m∈N B m=∅?? Because I just cant get this idea correct as for instance: B 1∩B 2=B 2 B 1∩B 2=B 2 B 2∩B 3=B 3 B 2∩B 3=B 3 B 3∩B 4=B 4 B 3∩B 4=B 4 .................. B m−1∩B m=B m B m−1∩B m=B m If my understanding is correct then by above: ⋂m∈N B m=B 1∩B 2∩...∩B m≠∅⋂m∈N B m=B 1∩B 2∩...∩B m≠∅ This tells me that the intersection are indeed non - empty, so how ⋂m∈N B m=∅⋂m∈N B m=∅ hold true?? elementary-set-theory Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications edited Oct 27, 2020 at 8:40 Asaf Karagila♦ 407k 48 48 gold badges 646 646 silver badges 1.1k 1.1k bronze badges asked Oct 27, 2020 at 6:45 Aurora BorealisAurora Borealis 1,823 14 14 silver badges 26 26 bronze badges 4 For every m∈N,m∉B m+1⟹m∉∩n∈N B n m∈N,m∉B m+1⟹m∉∩n∈N B n Shubham Johri –Shubham Johri 2020-10-27 06:48:59 +00:00 Commented Oct 27, 2020 at 6:48 1 The statement ⋂m∈N B m=B 1∩⋯∩B m⋂m∈N B m=B 1∩⋯∩B m makes no sense. What is m m on the right hand side?pancini –pancini 2020-10-27 07:01:12 +00:00 Commented Oct 27, 2020 at 7:01 The "obvious" answer which is both trivial and profound is "because it is a set with no elements". If you try to exhibit an element of the set, you may see what goes wrong. It is often useful to cut out some of the apparent complexity of a situation by going back to basic definitions.Mark Bennet –Mark Bennet 2020-10-27 07:31:14 +00:00 Commented Oct 27, 2020 at 7:31 Aurora, if you found one of the answers below satisfactory, consider accepting it.Shubham Johri –Shubham Johri 2020-10-28 11:34:12 +00:00 Commented Oct 28, 2020 at 11:34 Add a comment| 5 Answers 5 Sorted by: Reset to default This answer is useful 3 Save this answer. Show activity on this post. An intersection of a family of sets consists of any element that is contained in all those sets. In other words: x∈⋂m∈N B m⟺for all m, we have x∈B m x∈⋂m∈N B m⟺for all m, we have x∈B m There is no natural number that is an element of all the B m B m. Which is to say, there is no natural number (or anything else) that can be contained in the intersection. This makes the intersession empty. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Oct 27, 2020 at 6:55 answered Oct 27, 2020 at 6:54 ArthurArthur 205k 14 14 gold badges 186 186 silver badges 332 332 bronze badges Add a comment| This answer is useful 1 Save this answer. Show activity on this post. The correct definition is: ⋂m∈N B m=B 1∩B 2∩B 3∩⋯⋂m∈N B m=B 1∩B 2∩B 3∩⋯ in which the RHS does not terminate. If it's non-empty, say n∈⋂m∈N B m n∈⋂m∈N B m, then n∈B m n∈B m for all m∈N m∈N. However, this is not possible, as B n+1={n+1,n+2,…}B n+1={n+1,n+2,…}, so n∉B n+1 n∉B n+1. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Oct 27, 2020 at 6:51 Clement YungClement Yung 8,607 2 2 gold badges 14 14 silver badges 38 38 bronze badges Add a comment| This answer is useful 1 Save this answer. Show activity on this post. Other question, can you give an element that is in the intersection? For n∈⋂m∈N B m n∈⋂m∈N B m there has to be a n∈B m n∈B m for every m∈N m∈N. But n∉B n+1 n∉B n+1 for example, so no such n n can exist. Just make clear how these sets look. B 1={1,2,3,4,…}B 1={1,2,3,4,…} B 2={2,3,4,…}B 2={2,3,4,…} B 3={3,4,…}B 3={3,4,…} and so on. Eventually every element will be sorted out, or more specifically for every element you can give easily a set that does not contain this element, as shown above. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Oct 27, 2020 at 6:52 CornmanCornman 11.4k 5 5 gold badges 36 36 silver badges 69 69 bronze badges Add a comment| This answer is useful 0 Save this answer. Show activity on this post. Note that B m=B 1∩...∩B m≠∩n∈N B n=B 1∩B 2∩...B m=B 1∩...∩B m≠∩n∈N B n=B 1∩B 2∩.... The former is a finite intersection, the latter is not. Next note that for every m∈N,m∉B m+1⟹m∉∩n∈N B n m∈N,m∉B m+1⟹m∉∩n∈N B n. Thus the intersection is empty. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Oct 27, 2020 at 6:51 Shubham JohriShubham Johri 17.8k 2 2 gold badges 20 20 silver badges 34 34 bronze badges Add a comment| This answer is useful 0 Save this answer. Show activity on this post. Assume ∩B n∈N≠∅∩B n∈N≠∅. Then there is a k∈N k∈N s.t. k∈B n k∈B n for all n∈N n∈N, e.g. k∈B 1,B 2,.....k∈B 1,B 2,...... Consider B k+1=B k+1= {k+1,k+2,......k+1,k+2,......}; k∉B k+1 k∉B k+1, a contradiction. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Oct 27, 2020 at 9:00 answered Oct 27, 2020 at 8:41 Peter SzilasPeter Szilas 21.2k 2 2 gold badges 20 20 silver badges 31 31 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 elementary-set-theory See similar questions with these tags. 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https://www.freemathhelp.com/forum/threads/perpendicular-bisectors.62353/
Perpendicular Bisectors | Free Math Help Forum Home ForumsNew postsSearch forums What's newNew postsLatest activity Log inRegister What's newSearch Search [x] Search titles only By: SearchAdvanced search… New posts Search forums Menu Log in Register Install the app Install Forums Free Math Help Geometry and Trig You are using an out of date browser. It may not display this or other websites correctly. You should upgrade or use an alternative browser. Perpendicular Bisectors Thread starterskinski43 Start dateSep 1, 2009 S skinski43 New member Joined Jul 5, 2009 Messages 21 Sep 1, 2009 #1 Prove that bisectors of two supplementary angles are perpendicular to each other. please help D Deleted member 4993 Guest Sep 1, 2009 #2 skinski43 said: Prove that bisectors of two supplementary angles are perpendicular to each other. please help Click to expand... This is fairly straight-forward. Exactly where are you stuck. Draw a pair of supplementary angles - with a common side. Then bisect those angles and proceed. Please show your work, indicating exactly where you are stuck - so that we know where to begin to help you. S skinski43 New member Joined Jul 5, 2009 Messages 21 Sep 1, 2009 #3 i have a horizontal line with a vertical line coming up from it. the angle on the right of the vertical line is 60 degrees and bisected 30 degrees. that leaves the other angle on the left of the vertical line as 120 degrees and bisected = 60 degrees. sure enough 60 plus 30 is 90 degrees so the bisectors are perpendicular. but how do i prove it will always happen this way? M Mrspi Senior Member Joined Dec 17, 2005 Messages 2,128 Sep 1, 2009 #4 skinski43 said: i have a horizontal line with a vertical line coming up from it. the angle on the right of the vertical line is 60 degrees and bisected 30 degrees. that leaves the other angle on the left of the vertical line as 120 degrees and bisected = 60 degrees. sure enough 60 plus 30 is 90 degrees so the bisectors are perpendicular. but how do i prove it will always happen this way? Click to expand... Unless your two supplementary angles are ADJACENT angles (if you're not sure what adjacent angles are, please look up the definition in your book), then the bisectors of those two supplementary angles are NOT necessarily perpendicular. Consider the following possibility: there is an angle on YOUR ceiling that has a measure of 144 degrees. There is an angle on MY floor which has a measure of 36 degrees. Would you agree that those angles are supplementary? (I hope you do!) Now, suppose you draw the bisector of your 144-degree angle, and I draw the bisector of my 36-degree angle. There is NO GUARANTEE that your angle bisector and my angle bisector will even intersect! And if they don't intersect, then they can't possibly be perpendicular.... Please double-check the wording of your problem....I'm not going to go any further until I know exactly what your problem says. S skinski43 New member Joined Jul 5, 2009 Messages 21 Sep 1, 2009 #5 the problem reads "prove bisectors of two supplementary angles are perpendicular to each other" S skinski43 New member Joined Jul 5, 2009 Messages 21 Sep 1, 2009 #6 i i ___i_______ here is my diagram so they are adjacent S skinski43 New member Joined Jul 5, 2009 Messages 21 Sep 1, 2009 #7 my drawing didn't come out properly but my angles are adjacent to each other M Mrspi Senior Member Joined Dec 17, 2005 Messages 2,128 Sep 1, 2009 #8 skinski43 said: the problem reads "prove bisectors of two supplementary angles are perpendicular to each other" Click to expand... That's just plain not true....unless the angles are adjacent, which is not the condition stated in the problem. If the angles ARE adjacent, you can proceed as follows: Code: /D / / --------------------/--------------------------------- A B C B is between A and C, so <ABC is a straight angle, with measure 180 (definition of a straight angle) m<ABD + m<DBC = m<ABC m<ABD + m<DBC = 180 So, <ABD and <DBC are supplementary. Now, draw BE as the bisector of <ABD, and BF as the bisector of <DBC Multiply both sides of this equation by 1/2: m<ABD + m<DBC = 180 (1/2)m<ABD + (1/2)m<DBC = (1/2)180 (1/2)m<ABD + (1/2)m<DBC = 90 Your goal is to prove that BE is perpendicular to BF. m<EBD = (1/2)m<ABD and m<FBD = (1/2) m<DBC by the definition of bisect. So, substituting for (1/2)m<ABD and (1/2)m<DBC, we have m<EBD + m<FBD = 90 And m<EBD + m<FBD = m<EBF (by angle addition) So, m<EBF = 90 and <EBF is a right angle. Since BE and BF form a right angle, EB is perpendicular to BF. S skinski43 New member Joined Jul 5, 2009 Messages 21 Sep 1, 2009 #9 horizontal line with a vertical line coming up from the horizontal line the angles on either side of the vertical line are supplementary, are they not? S skinski43 New member Joined Jul 5, 2009 Messages 21 Sep 1, 2009 #10 Thank you Mrspi I think that will do it mmm4444bot Super Moderator Joined Oct 6, 2005 Messages 10,962 Sep 2, 2009 #11 skinski43 said: … are they not? Click to expand... The answer to this question is NO because two 90-degree angles are supplementary. You must log in or register to reply here. Share: FacebookTwitterRedditPinterestTumblrWhatsAppEmailShareLink Forums Free Math Help Geometry and Trig Contact us Terms and rules Privacy policy Help Home RSS Community platform by XenForo®© 2010-2023 XenForo Ltd. This site uses cookies to help personalise content, tailor your experience and to keep you logged in if you register. By continuing to use this site, you are consenting to our use of cookies. 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https://www.tiger-algebra.com/drill/m~2(3m-2)=m/
Copyright Ⓒ 2013-2025 tiger-algebra.com This site is best viewed with Javascript. If you are unable to turn on Javascript, please click here. Solution - Quadratic equations Other Ways to Solve Step by Step Solution Rearrange: Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : m^2(3m-2)-(m)=0 Step by step solution : Step 1 : Equation at the end of step 1 : Step 2 : Step 3 : Pulling out like terms : 3.1 Pull out like factors : 3m3 - 2m2 - m = m • (3m2 - 2m - 1) Trying to factor by splitting the middle term 3.2 Factoring 3m2 - 2m - 1 The first term is, 3m2 its coefficient is 3 . The middle term is, -2m its coefficient is -2 . The last term, "the constant", is -1 Step-1 : Multiply the coefficient of the first term by the constant 3 • -1 = -3 Step-2 : Find two factors of -3 whose sum equals the coefficient of the middle term, which is -2 . | | | | | | | | --- --- --- | | -3 | + | 1 | = | -2 | That's it | Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -3 and 1 3m2 - 3m + 1m - 1 Step-4 : Add up the first 2 terms, pulling out like factors : 3m • (m-1) Add up the last 2 terms, pulling out common factors : 1 • (m-1) Step-5 : Add up the four terms of step 4 : (3m+1) • (m-1) Which is the desired factorization Equation at the end of step 3 : Step 4 : Theory - Roots of a product : 4.1 A product of several terms equals zero. When a product of two or more terms equals zero, then at least one of the terms must be zero. We shall now solve each term = 0 separately In other words, we are going to solve as many equations as there are terms in the product Any solution of term = 0 solves product = 0 as well. Solving a Single Variable Equation : 4.2 Solve : m = 0 Solution is m = 0 Solving a Single Variable Equation : 4.3 Solve : m-1 = 0 Add 1 to both sides of the equation : m = 1 Solving a Single Variable Equation : 4.4 Solve : 3m+1 = 0 Subtract 1 from both sides of the equation : 3m = -1 Divide both sides of the equation by 3: m = -1/3 = -0.333 Supplement : Solving Quadratic Equation Directly Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula Parabola, Finding the Vertex : 5.1 Find the Vertex of y = 3m2-2m-1 Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 3 , is positive (greater than zero). Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. For any parabola,Am2+Bm+C,the m -coordinate of the vertex is given by -B/(2A) . In our case the m coordinate is 0.3333 Plugging into the parabola formula 0.3333 for m we can calculate the y -coordinate : y = 3.0 0.33 0.33 - 2.0 0.33 - 1.0 or y = -1.333 Parabola, Graphing Vertex and X-Intercepts : Root plot for : y = 3m2-2m-1 Axis of Symmetry (dashed) {m}={ 0.33} Vertex at {m,y} = { 0.33,-1.33} m -Intercepts (Roots) : Root 1 at {m,y} = {-0.33, 0.00} Root 2 at {m,y} = { 1.00, 0.00} Solve Quadratic Equation by Completing The Square 5.2 Solving 3m2-2m-1 = 0 by Completing The Square . Divide both sides of the equation by 3 to have 1 as the coefficient of the first term : m2-(2/3)m-(1/3) = 0 Add 1/3 to both side of the equation : m2-(2/3)m = 1/3 Now the clever bit: Take the coefficient of m , which is 2/3 , divide by two, giving 1/3 , and finally square it giving 1/9 Add 1/9 to both sides of the equation : On the right hand side we have : 1/3 + 1/9 The common denominator of the two fractions is 9 Adding (3/9)+(1/9) gives 4/9 So adding to both sides we finally get : m2-(2/3)m+(1/9) = 4/9 Adding 1/9 has completed the left hand side into a perfect square : m2-(2/3)m+(1/9) = (m-(1/3)) • (m-(1/3)) = (m-(1/3))2 Things which are equal to the same thing are also equal to one another. Since m2-(2/3)m+(1/9) = 4/9 and m2-(2/3)m+(1/9) = (m-(1/3))2 then, according to the law of transitivity, (m-(1/3))2 = 4/9 We'll refer to this Equation as Eq. #5.2.1 The Square Root Principle says that When two things are equal, their square roots are equal. Note that the square root of (m-(1/3))2 is (m-(1/3))2/2 = (m-(1/3))1 = m-(1/3) Now, applying the Square Root Principle to Eq. #5.2.1 we get: m-(1/3) = √ 4/9 Add 1/3 to both sides to obtain: m = 1/3 + √ 4/9 Since a square root has two values, one positive and the other negative m2 - (2/3)m - (1/3) = 0 has two solutions: m = 1/3 + √ 4/9 or m = 1/3 - √ 4/9 Note that √ 4/9 can be written as √ 4 / √ 9 which is 2 / 3 Solve Quadratic Equation using the Quadratic Formula 5.3 Solving 3m2-2m-1 = 0 by the Quadratic Formula . According to the Quadratic Formula, m , the solution for Am2+Bm+C = 0 , where A, B and C are numbers, often called coefficients, is given by : - B ± √ B2-4AC m = ———————— 2A In our case, A = 3 B = -2 C = -1 Accordingly, B2 - 4AC = 4 - (-12) = 16 Applying the quadratic formula : 2 ± √ 16 m = ————— 6 Can √ 16 be simplified ? Yes! The prime factorization of 16 is 2•2•2•2 To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root). √ 16 = √ 2•2•2•2 =2•2•√ 1 = ± 4 • √ 1 = ± 4 So now we are looking at: m = ( 2 ± 4) / 6 Two real solutions: m =(2+√16)/6=(1+2)/3= 1.000 or: m =(2-√16)/6=(1-2)/3= -0.333 Three solutions were found : How did we do? Why learn this Terms and topics Related links Latest Related Drills Solved Copyright Ⓒ 2013-2025 tiger-algebra.com
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https://www.ck12.org/flexi/math-grade-7/the-percent-equation/how-can-percentage-increase-or-decrease-be-found/
How can percentage increase or decrease be found? Flexi Says: Often we are interested in finding by what percentage a quantity changes, which is a comparison of a change in value to the original value. The new price is expressed as a percentage of the old price. For example, if the new price is 50% of the old price, this means that the new price is half the old price. If the new price is 200% of the old price, this means that the new price is double the old price. The percentage change is calculated using the following formula:@$$\begin{align}\text{Percentage increase/decrease} =\frac{ \text{New amount - Original amount}}{\text{Original amount}}\times 100\end{align}@$$ To learn more click here! Try Asking: How to calculate a 10 percent increase?What is the formula for calculating 1000 percent increase?At the end of each year, an item loses 6% of the value that it had at the beginning of the year. What type of function best represents this scenario? By messaging Flexi, you agree to our Terms and Privacy Policy
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https://www.quora.com/The-price-of-a-product-goes-down-first-20-and-then-70-more-What-is-the-total-discount-percentage
The price of a product goes down first 20 % and then 70 % more. What is the total discount percentage? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Discount Sale Overall Percentage Pricing Sales Arithmetic Percentage Composition Good Discount Ratio and Percentage 5 The price of a product goes down first 20 % and then 70 % more. What is the total discount percentage? All related (37) Sort Recommended Neil Jones Author has 436 answers and 91.6K answer views ·3y Start from the 100% position Reducing this by 20% leaves 80 (0.8 100) Reducing 80 by a further 70% leaves 24. (0.3 80) 100 - 24 = 76 The total discount percentage is therefore 76% Can be done in one line thus. 100 0.8 .07 = 24 24% remaining so 76% reduction I hope this is a clear explanation for this calculation. Upvote · Sponsored by Grammarly 92% of professionals who use Grammarly say it has saved them time Work faster with AI, while ensuring your writing always makes the right impression. Download 999 207 Related questions More answers below The normal price of a product is 200 €. A discount of 40 € is given. What is the discount percent, 25 %, 16.7 %, 20 %, or other? What is the effective discount (in %) on two successive discounts of 20% and 10%? If 20% of marked price is equal to 30% of cost price. If no discount is given, then what will be the profit percentage? The price of a product is reduced by 12%. By what percentage must the sales grow if the profit should be the same? If the normal price of an item is 580 pounds and the discount is 20 percent what is the new price? Naval Heeramaneck Chartered Accountant (1970–present) · Author has 91 answers and 20.7K answer views ·3y If the price was 100, then a 20% reduction reduces the price to 80 and a further 70% reduction reduces the price to 24 which is a total price reduction of 76. Therefore the percentage discount is 100 - 24 = 76%. QED Upvote · Syamal Dey Ph.D., Former Sr. Scientist, University of Maryland, USA · Author has 8.2K answers and 7.7M answer views ·3y Question, The price of a product goes down first 20 % and then 70 % more. What is the total discount percentage? Let M & S denote the marked price & the selling price of the given product respectively. Hence from above data we get following relation, S = (1 - 20/100)(1 - 70/100)M or S = (4/5)(3/10)M or S = (6/25)M or S = (1 - 19/25)M or S = (1 - 194/100)M or S = (1 - 76/100)M Therefore from above it is evident that the required total discount percentage is 76%. [Ans] Upvote · Rama Srinivas Studied at Keyes High School for Girls · Author has 66 answers and 16K answer views ·3y Let the price be =100₹ First discount=20% ie 20%=20₹ After discount price=100–20=80₹ Second discount=70% 70/10080=56₹ Price after second discount=80–56=24₹ Total discount %=100–24=76% Upvote · Related questions More answers below The amount of discount offered on a product is four-fifth of the amount of profit. If the product is marked up by 60% above the cost price, then what was the percentage of discount offered on the product? The price of a product is reduced by 30%. By what percentage should it be increased to make it 100%? If the price of a product is $80 and it is discounted by 20%, what will be the discounted price? What is the total discount of two successive 20% and 15% discounts? If a person makes a 20% discount first, then a profit of 20%, what is the percentage? Larry Scholnick Studied at University of California, Los Angeles · Author has 9.6K answers and 6.6M answer views ·3y The phrase “and then 70% more" is inherently ambiguous. Does the 70% more apply to the already 20% reduced price, as other answers assume, or does it apply to the original price, in which case the result is a 90% reduction. Upvote · Sponsored by Book Geists If you're a Kiwi, this could be the best day of your life! Available to Kiwis only. Read today. Learn More 1.3K 1.3K Rajendra Kumar Phophalia BE from Birla Institute of Technology and Science, Pilani · Author has 5.9K answers and 2.6M answer views ·6y Originally Answered: The price of an item is discounted by 20% and again discounted by an additional 30%.What is the total discount offered on the original price of the item? · New price =.7.8 original price= .56original price. Hence total discount offered is (1-.56)100=44%. Upvote · Prathapan C K Former Engineer (1980–2018) · Author has 492 answers and 86K answer views ·3y After 20% down the price is 80% , Then price reduction of 70% of 80% means 56% of actual price Total reduction is 20 + 56 means 76%. Upvote · Sponsored by JetBrains DataGrip, a powerful GUI tool for SQL. Smart code completion, on-the-fly analysis, quick-fixes, refactorings that work in SQL files, and more. Download 999 740 Anil Maken BE ( CIVIL) HONS from Board of School Education Haryana (Bhiwani) (Graduated 1980) ·5y Related A shopkeeper offers two successive discounts of 20% and 15% respectively. What is the overall discount percent? Let the cost ₹100/- 1st discount =20% Net cost =₹80/- 2nd discount = 15% Net discount = 15\100×80= ₹12/- Net cost after 2 nd discount = 80–12 =₹ 68/- There fore overall discount = 32% Upvote · 99 11 Gary Hewitt Director at Sustainalytics (2014–present) · Author has 502 answers and 1.2M answer views ·2y Related The price of a product is reduced by 30%. By what percentage should it be increased to make it 100%? Reducing the price of the product by 30% is the same as multiplying it by 70% (do you see why?), which in turn is easiest to understand for our purposes as 70/100. So, you have the price, times 70/100: p r i c e⋅70 100 p r i c e⋅70 100 So, in order to get the original price back, you need to get rid of that 70/100. How to do that? Put something in the numerator that can cancel the 100, and something in the denominator that can cancel the 70: p r i c e⋅70 100⋅100 70 p r i c e⋅70 100⋅100 70 You can see that all the numbers cancel, and so if you multiply the reduced price by 100/70 (or 10/7, which equals a Continue Reading Reducing the price of the product by 30% is the same as multiplying it by 70% (do you see why?), which in turn is easiest to understand for our purposes as 70/100. So, you have the price, times 70/100: p r i c e⋅70 100 p r i c e⋅70 100 So, in order to get the original price back, you need to get rid of that 70/100. How to do that? Put something in the numerator that can cancel the 100, and something in the denominator that can cancel the 70: p r i c e⋅70 100⋅100 70 p r i c e⋅70 100⋅100 70 You can see that all the numbers cancel, and so if you multiply the reduced price by 100/70 (or 10/7, which equals approximately 1.42857 1.42857, and you’ll get back the original price! When we talk about “a percentage increase” to an amount, we mean adding that percentage of the starting amount to the original amount. This is the same as multiplying the starting amount by 1+p e r c e n t a g e i n c r e a s e 1+p e r c e n t a g e i n c r e a s e. But wait, I just found out that I can multiply the reduced price by approximately 1.42857 1.42857 to get the original price! That looks exactly like 1+42.857%1+42.857% - just like how we define a percentage increase! So we can conclude pretty easily from here: the percentage increase required is 42.857%42.857% Upvote · 9 6 Sponsored by RedHat Customize AI for your needs, with simpler model alignment tools. Your AI needs context, not common knowledge. Learn More 9 6 Pardha Saradhi Mandadi Former Director of State Audit (Retd)Hyderabad&Arbitrator at Government of Telangana (1979–2012) · Author has 12.6K answers and 4.7M answer views ·5y Related A shopkeeper offers two successive discounts of 20% and 15% respectively. What is the overall discount percent? Let price be 100 Discounted price at 20% discount=10080/100=80 Discounted price at second 15% discount=8085/100=68 The overall discount =100–68=32 The overall discount percent=32/100100=32% Upvote · 9 3 Sarathdas Kv 5y Related A shopkeeper offers two successive discounts of 20% and 15% respectively. What is the overall discount percent? You can use a simple formula LET X and Y be the discount percents X + Y ± (XY/100) So here - 20 - 15 + (2015/100)=32% discount Your response is private Was this worth your time? This helps us sort answers on the page. Absolutely not Definitely yes Upvote · 9 4 Ved Prakash Sharma Solved over a million queries · Author has 14.3K answers and 16.5M answer views ·Jan 14 Related The price of an article is reduced by a successful discount of 20% and 25%. What is the net percentage decreased in price? Let the price of an article is Rs. 100. Selling price = 100.(1- 20/100).(1–25/100). = 100.(4/5).(3/4). = Rs. 60. Net percentage decreased in price = 100 - 60. = 40% . , Answer. Upvote · Pardha Saradhi Mandadi Arbitrator and Mediator at Self Employeed Professional (2014–present) · Author has 12.6K answers and 4.7M answer views ·Aug 24 Related A shop increases a price by 20% and then decreases it by 20%. What’s the overall percentage change? Let the original price of the book be 100 Increased price of the book by 20%=100120/100=120 Decreased price of the book by 20%=12080/100=96 Overall decrease in the price of the book =100-96=4 Percentage decrease in the price of the book =4/100100=4% Upvote · Related questions The normal price of a product is 200 €. A discount of 40 € is given. What is the discount percent, 25 %, 16.7 %, 20 %, or other? What is the effective discount (in %) on two successive discounts of 20% and 10%? If 20% of marked price is equal to 30% of cost price. If no discount is given, then what will be the profit percentage? The price of a product is reduced by 12%. By what percentage must the sales grow if the profit should be the same? If the normal price of an item is 580 pounds and the discount is 20 percent what is the new price? The amount of discount offered on a product is four-fifth of the amount of profit. If the product is marked up by 60% above the cost price, then what was the percentage of discount offered on the product? The price of a product is reduced by 30%. By what percentage should it be increased to make it 100%? If the price of a product is $80 and it is discounted by 20%, what will be the discounted price? What is the total discount of two successive 20% and 15% discounts? If a person makes a 20% discount first, then a profit of 20%, what is the percentage? If the price of an item is $32 and 20% off, what can be less after a 20% discount price? A product is sold at two consecutive discounts of 30% and subsequently 40%. If the marked price on the product is 1500, what is the selling price of the product? The price of a product is reduced by 30 percent. By what percentage should it be increased to make it 100 percent? The marked price of an item is 50% above its CP. What percentage of discount is allowed in approximately to gain 10% profit? A trader marked up the price of a product by 75%. When the discount was decreased from 50% to 25%, his profit increased by 210 rupees. What is the cost of the product? Related questions The normal price of a product is 200 €. A discount of 40 € is given. What is the discount percent, 25 %, 16.7 %, 20 %, or other? What is the effective discount (in %) on two successive discounts of 20% and 10%? If 20% of marked price is equal to 30% of cost price. If no discount is given, then what will be the profit percentage? The price of a product is reduced by 12%. By what percentage must the sales grow if the profit should be the same? If the normal price of an item is 580 pounds and the discount is 20 percent what is the new price? The amount of discount offered on a product is four-fifth of the amount of profit. If the product is marked up by 60% above the cost price, then what was the percentage of discount offered on the product? 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https://test.uoh.edu.iq/admin/ebooks/63907-ch15.pdf
15 Transmitting and Receiving Antennas 15.1 Energy Flux and Radiation Intensity The flux of electromagnetic energy radiated from a current source at far distances is given by the time-averaged Poynting vector, calculated in terms of the radiation fields (14.10.4): P P P = 1 2Re(E × H∗)= 1 2 −jkη e−jkr 4πr jk ejkr 4πr Re  (ˆ θ θ θFθ + ˆ φ φ φFφ)×( ˆ φ φ φF∗ θ −ˆ θ θ θF∗ φ)  Noting that ˆ θ θ θ × ˆ φ φ φ = ˆ r, we have: (ˆ θ θ θFθ + ˆ φ φ φFφ)×( ˆ φ φ φF∗ θ −ˆ θ θ θF∗ φ)= ˆ r  |Fθ|2 + |Fφ|2 = ˆ r  F⊥(θ, φ)  2 Therefore, the energy flux vector will be: P P P = ˆ r Pr = ˆ r ηk2 32π2r2  F⊥(θ, φ)  2 (15.1.1) Thus, the radiated energy flows radially away from the current source and attenu-ates with the square of the distance. The angular distribution of the radiated energy is described by the radiation pattern factor:  F⊥(θ, φ)  2 =  Fθ(θ, φ)  2 +  Fφ(θ, φ)  2 (15.1.2) With reference to Fig. 14.9.1, the power dP intercepting the area element dS = r2dΩ defines the power per unit area, or the power density of the radiation: dP dS = dP r2dΩ = Pr = ηk2 32π2r2  F⊥(θ, φ)  2 (power density) (15.1.3) The radiation intensity U(θ, φ) is defined to be the power radiated per unit solid angle, that is, the quantity dP/dΩ = r2dP/dS = r2Pr: U(θ, φ)= dP dΩ = r2Pr = ηk2 32π2  F⊥(θ, φ)  2 (radiation intensity) (15.1.4) 15.2. Directivity, Gain, and Beamwidth 599 The total radiated power is obtained by integrating Eq. (15.1.4) over all solid angles dΩ = sin θ dθdφ, that is, over 0 ≤θ ≤π and 0 ≤φ ≤2π : Prad = π 0 2π 0 U(θ, φ) dΩ (total radiated power) (15.1.5) A useful concept is that of an isotropic radiator—a radiator whose intensity is the same in all directions. In this case, the total radiated power Prad will be equally dis-tributed over all solid angles, that is, over the total solid angle of a sphere Ωsphere = 4π steradians, and therefore, the isotropic radiation intensity will be: UI = dP dΩ I = Prad Ωsphere = Prad 4π = 1 4π π 0 2π 0 U(θ, φ) dΩ (15.1.6) Thus, UI is the average of the radiation intensity over all solid angles. The corre-sponding power density of such an isotropic radiator will be: dP dS I = UI r2 = Prad 4πr2 (isotropic power density) (15.1.7) 15.2 Directivity, Gain, and Beamwidth The directive gain of an antenna system towards a given direction (θ, φ) is the radiation intensity normalized by the corresponding isotropic intensity, that is, D(θ, φ)= U(θ, φ) UI = U(θ, φ) Prad/4π = 4π Prad dP dΩ (directive gain) (15.2.1) It measures the ability of the antenna to direct its power towards a given direction. The maximum value of the directive gain, Dmax, is called the directivity of the antenna and will be realized towards some particular direction, say (θ0, φ0). The radiation intensity will be maximum towards that direction, Umax = U(θ0, φ0), so that Dmax = Umax UI (directivity) (15.2.2) The directivity is often expressed in dB,† that is, DdB = 10 log10 Dmax. Re-expressing the radiation intensity in terms of the directive gain, we have: dP dΩ = U(θ, φ)= D(θ, φ)UI = PradD(θ, φ) 4π (15.2.3) and for the power density in the direction of (θ, φ): dP dS = dP r2dΩ = PradD(θ, φ) 4πr2 (power density) (15.2.4) †The term “dBi” is often used as a reminder that the directivity is with respect to the isotropic case. 600 15. Transmitting and Receiving Antennas Comparing with Eq. (15.1.7), we note that if the amount of power PradD(θ, φ) were emitted isotropically, then Eq. (15.2.4) would be the corresponding isotropic power den-sity. Therefore, we will refer to PradD(θ, φ) as the effective isotropic power, or the effective radiated power (ERP) towards the (θ, φ)-direction. In the direction of maximum gain, the quantity PradDmax will be referred to as the effective isotropic radiated power (EIRP). It defines the maximum power density achieved by the antenna: dP dS max = PEIRP 4πr2 , where PEIRP = PradDmax (15.2.5) Usually, communicating antennas—especially highly directive ones such as dish antennas—are oriented to point towards the maximum directive gain of each other. A related concept is that of the power gain, or simply the gain of an antenna. It is defined as in Eq. (15.2.1), but instead of being normalized by the total radiated power, it is normalized to the total power PT accepted by the antenna terminals from a connected transmitter, as shown in Fig. 15.2.1: G(θ, φ)= U(θ, φ) PT/4π = 4π PT dP dΩ (power gain) (15.2.6) We will see in Sec. 15.4 that the power PT delivered to the antenna terminals is at most half the power produced by the generator—the other half being dissipated as heat in the generator’s internal resistance. Moreover, the power PT may differ from the power radiated, Prad, because of several loss mechanisms, such as ohmic losses of the currents flowing on the antenna wires or losses in the dielectric surrounding the antenna. Fig. 15.2.1 Power delivered to an antenna versus power radiated. The definition of power gain does not include any reflection losses arising from improper matching of the transmission line to the antenna input impedance . The efficiency factor of the antenna is defined by: e = Prad PT ⇒ Prad = ePT (15.2.7) In general, 0 ≤e ≤1. For a lossless antenna the efficiency factor will be unity and Prad = PT. In such an ideal case, there is no distinction between directive and power gain. Using Eq. (15.2.7) in (15.2.1), we find G = 4πU/PT = e4πU/Prad, or, 15.2. Directivity, Gain, and Beamwidth 601 G(θ, φ)= eD(θ, φ) (15.2.8) The maximum gain is related to the directivity by Gmax = eDmax. It follows that the effective radiated power can be written as PradD(θ, φ)= PTG(θ, φ), and the EIRP, PEIRP = PradDmax = PTGmax. The angular distribution functions we defined thus far, that is, G(θ, φ), D(θ, φ), U(θ, φ) are all proportional to each other. Each brings out a different aspect of the radiating system. In describing the angular distribution of radiation, it proves conve-nient to consider it relative to its maximal value. Thus, we define the normalized power pattern, or normalized gain by: g(θ, φ)= G(θ, φ) Gmax (normalized gain) (15.2.9) Because of the proportionality of the various angular functions, we have: g(θ, φ)= G(θ, φ) Gmax = D(θ, φ) Dmax = U(θ, φ) Umax =  F⊥(θ, φ)  2 |F⊥|2 max (15.2.10) Writing PTG(θ, φ)= PTGmax g(θ, φ), we have for the power density: dP dS = PTGmax 4πr2 g(θ, φ)= PEIRP 4πr2 g(θ, φ) (15.2.11) This form is useful for describing communicating antennas and radar. The normal-ized gain is usually displayed in a polar plot with polar coordinates (ρ, θ) such that ρ = g(θ), as shown in Fig. 15.2.2. (This figure depicts the gain of a half-wave dipole antenna given by g(θ)= cos2(0.5π cos θ)/ sin2 θ.) The 3-dB, or half-power, beamwidth is defined as the difference ΔθB = θ2 −θ1 of the 3-dB angles at which the normalized gain is equal to 1/2, or, −3 dB. Fig. 15.2.2 Polar and regular plots of normalized gain versus angle. The MATLAB functions dbp, abp, dbz, abz given in Appendix I allow the plotting of the gain in dB or in absolute units versus the polar angle θ or the azimuthal angle φ. Their typical usage is as follows: dbp(theta, g, rays, Rm, width); % polar gain plot in dB abp(theta, g, rays, width); % polar gain plot in absolute units dbz(phi, g, rays, Rm, width); % azimuthal gain plot in dB abz(phi, g, rays, width); % azimuthal gain plot in absolute units 602 15. Transmitting and Receiving Antennas Example 15.2.1: A TV station is transmitting 10 kW of power with a gain of 15 dB towards a particular direction. Determine the peak and rms value of the electric field E at a distance of 5 km from the station. Solution: The gain in absolute units will be G = 10GdB/10 = 1015/10 = 31.62. It follows that the radiated EIRP will be PEIRP = PTG = 10 × 31.62 = 316.2 kW. The electric field at distance r = 5 km is obtained from Eq. (15.2.5): dP dS = PEIRP 4πr2 = 1 2η E2 ⇒ E = 1 r ηPEIRP 2π This gives E = 0.87 V/m. The rms value is Erms = E/ √ 2 = 0.62 V/m. ⊓ ⊔ Another useful concept is that of the beam solid angle of an antenna. The definition is motivated by the case of a highly directive antenna, which concentrates all of its radiated power Prad into a small solid angle ΔΩ, as illustrated in Fig. 15.2.3. Fig. 15.2.3 Beam solid angle and beamwidth of a highly directive antenna. The radiation intensity in the direction of the solid angle will be: U = ΔP ΔΩ = Prad ΔΩ (15.2.12) where ΔP = Prad by assumption. It follows that: Dmax = 4πU/Prad = 4π/ΔΩ, or, Dmax = 4π ΔΩ (15.2.13) Thus, the more concentrated the beam, the higher the directivity. Although (15.2.13) was derived under the assumption of a highly directive antenna, it may be used as the definition of the beam solid angle for any antenna, that is, ΔΩ = 4π Dmax (beam solid angle) (15.2.14) Using Dmax = Umax/UI and Eq. (15.1.6), we have ΔΩ = 4πUI Umax = 1 Umax π 0 2π 0 U(θ, φ) dΩ , or, ΔΩ = π 0 2π 0 g(θ, φ) dΩ (beam solid angle) (15.2.15) 15.2. Directivity, Gain, and Beamwidth 603 where g(θ, φ) is the normalized gain of Eq. (15.2.10). Writing Prad = 4πUI, we have: ΔΩ = Prad Umax ⇒ Umax = Prad ΔΩ (15.2.16) This is the general case of Eq. (15.2.12). We can also write: Prad = UmaxΔΩ (15.2.17) This is convenient for the numerical evaluation of Prad. To get a measure of the beamwidth of a highly directive antenna, we assume that the directive gain is equal to its maximum uniformly over the entire solid angle ΔΩ in Fig. 15.2.3, that is, D(θ, φ)= Dmax, for 0 ≤θ ≤ΔθB/2. This implies that the normalized gain will be: g(θ, φ)= 1, if 0 ≤θ ≤ΔθB/2 0, if ΔθB/2 < θ ≤π Then, it follows from the definition (15.2.15) that: ΔΩ = ΔθB/2 0 2π 0 dΩ = ΔθB/2 0 2π 0 sin θ dθ dφ = 2π 1 −cos ΔθB 2 (15.2.18) Using the approximation cos x 1 −x2/2, we obtain for small beamwidths: ΔΩ = π 4 (ΔθB)2 (15.2.19) and therefore the directivity can be expressed in terms of the beamwidth: Dmax = 16 Δθ2 B (15.2.20) Example 15.2.2: Find the beamwidth in degrees of a lossless dish antenna with gain of 15 dB. The directivity and gain are equal in this case, therefore, Eq. (15.2.20) can be used to calculate the beamwidth: ΔθB = √ 16/D, where D = G = 1015/10 = 31.62. We find ΔθB = 0.71 rads, or ΔθB = 40.76o. For an antenna with 40 dB gain/directivity, we would have D = 104 and find ΔθB = 0.04 rads = 2.29o. ⊓ ⊔ Example 15.2.3: A satellite in a geosynchronous orbit of 36,000 km is required to have com-plete earth coverage. What is its antenna gain in dB and its beamwidth? Repeat if the satellite is required to have coverage of an area equal the size of continental US. Solution: The radius of the earth is R = 6400 km. Looking down from the satellite the earth appears as a flat disk of area ΔS = πR2. It follows that the subtended solid angle and the corresponding directivity/gain will be: ΔΩ = ΔS r2 = πR2 r2 ⇒ D = 4π ΔΩ = 4r2 R2 With r = 36,000 km and R = 6400 km, we find D = 126.56 and in dB, DdB = 10 log10 D = 21.02 dB. The corresponding beamwidth will be ΔθB = √ 16/D = 0.36 rad = 20.37o. 604 15. Transmitting and Receiving Antennas For the continental US, the coast-to-coast distance of 3000 mi, or 4800 km, translates to an area of radius R = 2400 km, which leads to D = 900 and DdB = 29.54 dB. The beamwidth is in this case ΔθB = 7.64o. Viewing the earth as a flat disk overestimates the required angle ΔθB for earth coverage. Looking down from a satellite at a height r, the angle between the vertical and the tangent to the earth’s surface is given by sin θ = R/(r + R), which gives for r = 36,000 km, θ = 8.68o. The subtended angle will be then ΔθB = 2θ = 0.303 rad = 17.36o. It follows that the required antenna gain should be G = 16/Δθ2 B = 174.22 = 22.41 dB. The flat-disk approximation is more accurate for smaller areas on the earth’s surface that lie directly under the satellite. ⊓ ⊔ Example 15.2.4: The radial distance of a geosynchronous orbit can be calculated by equating centripetal and gravitational accelerations, and requiring that the angular velocity of the satellite corresponds to the period of 1 day, that is, ω = 2π/T, where T = 24 hr = 86 400 sec. Let m be the mass of the satellite and M⊕the mass of the earth (see Appendix A): GmM⊕ r2 = mω2r = m 2π T 2 r ⇒ r = GM⊕T2 4π2 1/3 The distance r is measured from the Earth’s center. The corresponding height from the surface of the Earth is h = r−R. For the more precise value of R = 6378 km, the calculated values are: r = 42 237 km = 26 399 mi h = 35 860 km = 22 414 mi 15.3 Effective Area When an antenna is operating as a receiving antenna, it extracts a certain amount of power from an incident electromagnetic wave. As shown in Fig. 15.3.1, an incident wave coming from a far distance may be thought of as a uniform plane wave being intercepted by the antenna. Fig. 15.3.1 Effective area of an antenna. The incident electric field sets up currents on the antenna. Such currents may be represented by a Th´ evenin-equivalent generator, which delivers power to any connected receiving load impedance. The induced currents also re-radiate an electric field (referred to as the scattered field), which interferes with the incident field causing a shadow region behind the an-tenna, as shown in Fig. 15.3.1. 15.3. Effective Area 605 The total electric field outside the antenna will be the sum of the incident and re-radiated fields. For a perfectly conducting antenna, the boundary conditions are that the tangential part of the total electric field vanish on the antenna surface. In Chap. 21, we apply these boundary conditions to obtain and solve Hall´ en’s and Pocklington’s integral equations satisfied by the induced current. The power density of the incident wave at the location of the receiving antenna can be expressed in terms of the electric field of the wave, Pinc = E2/2η. The effective area or effective aperture A of the antenna is defined to be that area which when intercepted by the incident power density Pinc gives the amount of received power PR available at the antenna output terminals : PR = APinc (15.3.1) For a lossy antenna, the available power at the terminals will be somewhat less than the extracted radiated power Prad, by the efficiency factor PR = ePrad. Thus, we may also define the maximum effective aperture Am as the area which extracts the power Prad from the incident wave, that is, Prad = AmPinc. It follows that: A = eAm (15.3.2) The effective area depends on the direction of arrival (θ, φ) of the incident wave. For all antennas, it can be shown that the effective area A(θ, φ) is related to the power gain G(θ, φ) and the wavelength λ = c/f as follows: G(θ, φ)= 4πA(θ, φ) λ2 (15.3.3) Similarly, because G(θ, φ)= eD(θ, φ), the maximum effective aperture will be re-lated to the directive gain by: D(θ, φ)= 4πAm(θ, φ) λ2 (15.3.4) In practice, the quoted effective area A of an antenna is the value corresponding to the direction of maximal gain Gmax. We write in this case: Gmax = 4πA λ2 (15.3.5) Similarly, we have for the directivity Dmax = 4πAm/λ2. Because Dmax is related to the beam solid angle by Dmax = 4π/ΔΩ, it follows that Dmax = 4π ΔΩ = 4πAm λ2 ⇒ AmΔΩ = λ2 (15.3.6) Writing λ = c/f, we may express Eq. (15.3.5) in terms of frequency: Gmax = 4πf 2A c2 (15.3.7) 606 15. Transmitting and Receiving Antennas The effective area is not equal to the physical area of an antenna. For example, linear antennas do not even have any characteristic physical area. For dish or horn antennas, on the other hand, the effective area is typically a fraction of the physical area (about 55–65 percent for dishes and 60–80 percent for horns.) For example, if the dish has a diameter of d meters, then we have: A = ea 1 4πd2 (effective area of dish antenna) (15.3.8) where ea is the aperture efficiency factor, typically ea = 0.55–0.65. Combining Eqs. (15.3.5) and (15.3.8), we obtain: Gmax = ea πd λ 2 (15.3.9) Antennas fall into two classes: fixed-area antennas, such as dish antennas, for which A is independent of frequency, and fixed-gain antennas, such as linear antennas, for which G is independent of frequency. For fixed-area antennas, the gain increases quadratically with f. For fixed-gain antennas, A decreases quadratically with f. Example 15.3.1: Linear antennas are fixed-gain antennas. For example, we will see in Sec. 16.1 that the gains of a (lossless) Hertzian dipole, a halfwave dipole, and a monopole antenna are the constants: Ghertz = 1.5, Gdipole = 1.64, Gmonopole = 3.28 Eq. (15.3.5) gives the effective areas A = Gλ2/4π: Ahertz = 0.1194λ2, Adipole = 0.1305λ2, Amonopole = 0.2610λ2 In all cases the effective area is proportional to λ2 and decreases with f 2. In the case of the commonly used monopole antenna, the effective area is approximately equal to a rectangle of sides λ and λ/4, the latter being the physical length of the monopole. ⊓ ⊔ Example 15.3.2: Determine the gain in dB of a dish antenna of diameter of 0.5 m operating at a satellite downlink frequency of 4 GHz and having 60% aperture efficiency. Repeat if the downlink frequency is 11 GHz. Repeat if the diameter is doubled to 1 m. Solution: The effective area and gain of a dish antenna with diameter d is: A = ea 1 4πd2 ⇒ G = 4πA λ2 = ea πd λ 2 = ea πfd c 2 The calculated gains G in absolute and dB units are in the four cases: d = 0.5 m d = 1 m f = 4 GHz 263 = 24 dB 1052 = 30 dB f = 11 GHz 1990 = 33 dB 7960 = 39 dB Doubling the diameter (or the frequency) increases the gain by 6 dB, or a factor of 4. Conversely, if a dish antenna is to have a desired gain G (for example, to achieve a desired beamwidth), the above equation can be solved for the required diameter d in terms of G and f. ⊓ ⊔ 15.3. Effective Area 607 The beamwidth of a dish antenna can be estimated by combining the approximate ex-pression (15.2.20) with (15.3.5) and (15.3.8). Assuming a lossless antenna with diameter d and 100% aperture efficiency, and taking Eq. (15.2.20) literally, we have: Gmax = 4πA λ2 = πd λ 2 = Dmax = 16 Δθ2 B Solving for ΔθB, we obtain the expression in radians and in degrees: ΔθB = 4 π λ d = 1.27 λ d , ΔθB = 73o λ d (15.3.10) Thus, the beamwidth depends inversely on the antenna diameter. In practice, quick estimates of the 3-dB beamwidth in degrees are obtained by replacing Eq. (15.3.10) by the formula : ΔθB = 1.22 λ d = 70o λ d (3-dB beamwidth of dish antenna) (15.3.11) The constant 70o represents only a rough approximation (other choices are in the range 65–75o.) Solving for the ratio d/λ = 1.22/ΔθB (here, ΔθB is in radians), we may express the maximal gain inversely with Δθ2 B as follows: Gmax = ea πd λ 2 = eaπ2(1.22)2 Δθ2 B For a typical aperture efficiency of 60%, this expression can be written in the following approximate form, with ΔθB given in degrees: Gmax = 30 000 Δθ2 B (15.3.12) Equations (15.3.11) and (15.3.12) must be viewed as approximate design guidelines, or rules of thumb , for the beamwidth and gain of a dish antenna. Example 15.3.3: For the 0.5-m antenna of the previous example, estimate its beamwidth for the two downlink frequencies of 4 GHz and 11 GHz. The operating wavelengths are in the two cases: λ = 7.5 cm and λ = 2.73 cm. Using Eq. (15.3.11), we find ΔθB = 10.5o and ΔθB = 3.8o. ⊓ ⊔ Example 15.3.4: A geostationary satellite at height of 36,000 km is required to have earth cov-erage. Using the approximate design equations, determine the gain in dB and the diameter of the satellite antenna for a downlink frequency of 4 GHz. Repeat for 11 GHz. Solution: This problem was considered in Example 15.2.3. The beamwidth angle for earth cov-erage was found to be ΔθB = 17.36o. From Eq. (15.3.11), we find: d = λ 70o ΔθB = 7.5 70o 17.36o = 30 cm From Eq. (15.3.12), we find: G = 30 000 Δθ2 B = 30 000 17.362 = 100 = 20 dB For 11 GHz, we find d = 11 cm, and G remains the same. ⊓ ⊔ 608 15. Transmitting and Receiving Antennas In Eqs. (15.2.20) and (15.3.12), we implicitly assumed that the radiation pattern was independent of the azimuthal angle φ. When the pattern is not azimuthally symmetric, we may define two orthogonal polar directions parametrized, say, by angles θ1 and θ2, as shown in Fig. 15.3.2. Fig. 15.3.2 Half-power beamwidths in two principal polar directions. In this case dΩ = dθ1 dθ2 and we may approximate the beam solid angle by the product of the corresponding 3-dB beamwidths in these two directions, ΔΩ = Δθ1 Δθ2. Then, the directivity takes the form (with the angles in radians and in degrees): Dmax = 4π ΔΩ = 4π Δθ1 Δθ2 = 41 253 Δθo 1 Δθo 2 (15.3.13) Equations (15.3.12) and (15.3.13) are examples of a more general expression that relates directivity or gain to the 3-dB beamwidths for aperture antennas [1052,1064]: Gmax = p Δθ1 Δθ2 (15.3.14) where p is a gain-beamwidth constant whose value depends on the particular aperture antenna. We will see several examples of this relationship in Chapters 17 and 18. Prac-tical values of p fall in the range 25 000–35 000 (with the beamwidth angles in degrees.) 15.4 Antenna Equivalent Circuits To a generator feeding a transmitting antenna as in Fig. 15.2.1, the antenna appears as a load. Similarly, a receiver connected to a receiving antenna’s output terminals will ap-pear to the antenna as a load impedance. Such simple equivalent circuit representations of transmitting and receiving antennas are shown in Fig. 15.4.1, where in both cases V is the equivalent open-circuit Th´ evenin voltage. In the transmitting antenna case, the antenna is represented by a load impedance ZA, which in general will have both a resistive and a reactive part, ZA = RA + jXA. The reactive part represents energy stored in the fields near the antenna, whereas the resistive part represents the power losses which arise because (a) power is radiated away from the antenna and (b) power is lost into heat in the antenna circuits and in the medium surrounding the antenna. The generator has its own internal impedance ZG = RG + jXG. The current at the antenna input terminals will be Iin = V/(ZG +ZA), which allows us to determine (a) the total power Ptot produced by the generator, (b) the power PT delivered to the antenna terminals, and (c) the power PG lost in the generator’s internal resistance RG. These are: 15.4. Antenna Equivalent Circuits 609 Fig. 15.4.1 Circuit equivalents of transmitting and receiving antennas. Ptot = 1 2 Re(VI∗ in)= 1 2 |V|2(RG + RA) |ZG + ZA|2 PT = 1 2|Iin|2RA = 1 2 |V|2RA |ZG + ZA|2 , PG = 1 2|Iin|2RG = 1 2 |V|2RG |ZG + ZA|2 (15.4.1) It is evident that Ptot = PT + PG. A portion of the power PT delivered to the antenna is radiated away, say an amount Prad, and the rest is dissipated as ohmic losses, say Pohm. Thus, PT = Prad + Pohm. These two parts can be represented conveniently by equivalent resistances by writing RA = Rrad + Rohm, where Rrad is referred to as the radiation resistance. Thus, we have, PT = 1 2|Iin|2RA = 1 2|Iin|2Rrad + 1 2|Iin|2Rohm = Prad + Pohm (15.4.2) The efficiency factor of Eq. (15.2.7) is evidently: e = Prad PT = Rrad RA = Rrad Rrad + Rohm To maximize the amount of power PT delivered to the antenna (and thus minimize the power lost in the generator’s internal resistance), the load impedance must satisfy the usual conjugate matching condition: ZA = Z∗ G RA = RG, XA = −XG In this case, |ZG + ZA|2 = (RG + RA)2+(XG + XA)2= 4R2 G, and it follows that the maximum power transferred to the load will be one-half the total—the other half being lost in RG, that is, PT,max = 1 2Ptot = |V|2 8RG (15.4.3) In the notation of Chap. 13, this is the available power from the generator. If the generator and antenna are mismatched, we have: PT = |V|2 8RG 4RARG |ZA + ZG|2 = PT,max  1 −|Γgen|2 , Γgen = ZA −Z∗ G ZA + ZG (15.4.4) 610 15. Transmitting and Receiving Antennas Eq. (15.4.3) is often written in terms of the rms value of the source, that is, Vrms = |V|/ √ 2, which leads to PT,max = V2 rms/4RG. The case of a receiving antenna is similar. The induced currents on the antenna can be represented by a Th´ evenin-equivalent generator (the open-circuit voltage at the an-tenna output terminals) and an internal impedance ZA. A consequence of the reciprocity principle is that ZA is the same whether the antenna is transmitting or receiving. The current into the load is IL = V/(ZA + ZL), where the load impedance is ZL = RL + jXL. As before, we can determine the total power Ptot produced by the generator (i.e., intercepted by the antenna) and the power PR delivered to the receiving load: Ptot = 1 2 Re(VI∗ L )= 1 2 |V|2(RL + RA) |ZL + ZA|2 , PR = 1 2|IL|2RL = 1 2 |V|2RL |ZL + ZA|2 (15.4.5) Under conjugate matching, ZL = Z∗ A, we find the maximum power delivered to the load: PR,max = |V|2 8RA (15.4.6) If the load and antenna are mismatched, we have: PR = |V|2 8RA 4RARL |ZL + ZA|2 = PR,max  1 −|Γload|2 , Γload = ZL −Z∗ A ZL + ZA (15.4.7) It is tempting to interpret the power dissipated in the internal impedance of the Th´ evenin circuit of the receiving antenna (that is, in ZA) as representing the amount of power re-radiated or scattered by the antenna. However, with the exception of the so-called minimum-scattering antennas, such interpretation is not correct. The issue has been discussed by Silver and more recently in Refs. [1028–1031]. See also Refs. [1004–1027] for further discussion of the transmitting, receiving, and scattering properties of antennas. 15.5 Effective Length The polarization properties of the electric field radiated by an antenna depend on the transverse component of the radiation vector F⊥according to Eq. (14.10.5): E = −jkη e−jkr 4πr F⊥= −jkη e−jkr 4πr (Fθ ˆ θ θ θ + Fφ ˆ φ φ φ) The vector effective length, or effective height of a transmitting antenna is defined in terms of F⊥and the input current to the antenna terminals Iin as follows :† h = −F⊥ Iin (effective length) (15.5.1) In general, h is a function of θ, φ. The electric field is, then, written as: E = jkη e−jkr 4πr Iin h (15.5.2) †Often, it is defined with a positive sign h = F⊥/Iin. 15.5. Effective Length 611 The definition of h is motivated by the case of a z-directed Hertzian dipole antenna, which can be shown to have h = l sin θ ˆ θ θ θ. More generally, for a z-directed linear antenna with current I(z), it follows from Eq. (16.1.5) that: h(θ)= h(θ) ˆ θ θ θ , h(θ)= sin θ 1 Iin l/2 −l/2 I(z′)ejkz′ cos θdz′ (15.5.3) As a consequence of the reciprocity principle, it can be shown that the open-circuit voltage V at the terminals of a receiving antenna is given in terms of the effective length and the incident field E i by: V = E i · h (15.5.4) The normal definition of the effective area of an antenna and the result G = 4πA/λ2 depend on the assumptions that the receiving antenna is conjugate-matched to its load and that the polarization of the incident wave matches that of the antenna. The effective length helps to characterize the degree of polarization mismatch that may exist between the incident field and the antenna. To see how the gain-area relation-ship must be modified, we start with the definition (15.3.1) and use (15.4.5): A(θ, φ)= PR Pinc = 1 2RL|IL|2 1 2η|Ei|2 = ηRL|V|2 |ZL + ZA|2|Ei|2 = ηRL|Ei · h|2 |ZL + ZA|2|Ei|2 Next, we define the polarization and load mismatch factors by: epol = |E i · h|2 |E i|2 |h|2 eload = 4RLRA |ZL + ZA|2 = 1 −|Γload|2 , where Γload = ZL −Z∗ A ZL + ZA (15.5.5) The effective area can be written then in the form: A(θ, φ)= η|h|2 4RA eload epol (15.5.6) On the other hand, using (15.1.4) and (15.4.1), the power gain may be written as: G(θ, φ)= 4πU(θ, φ) PT = 4πηk2|F⊥|2 32π2 1 2RA|Iin|2 = πη|h|2 λ2RA ⇒ η|h|2 4RA = λ2 4πG(θ, φ) Inserting this in Eq. (15.5.6), we obtain the modified area-gain relationship : A(θ, φ)= eload epol λ2 4πG(θ, φ) (15.5.7) Assuming that the incident field originates at some antenna with its own effective length hi, then E i will be proportional to hi and we may write the polarization mismatch factor in the following form: epol = |hi · h|2 |hi|2 |h|2 = |ˆ hi · ˆ h|2 , where ˆ hi = hi |hi| , ˆ h = h |h| 612 15. Transmitting and Receiving Antennas When the load is conjugate-matched, we have eload = 1, and when the incident field has matching polarization with the antenna, that is, ˆ hi = ˆ h ∗, then, epol = 1. 15.6 Communicating Antennas The communication between a transmitting and a receiving antenna can be analyzed with the help of the concept of gain and effective area. Consider two antennas oriented towards the maximal gain of each other and separated by a distance r, as shown in Fig. 15.6.1. Fig. 15.6.1 Transmitting and receiving antennas. Let {PT, GT, AT} be the power, gain, and effective area of the transmitting antenna, and {PR, GR, AR} be the same quantities for the receiving antenna. In the direction of the receiving antenna, the transmitting antenna has PEIRP = PTGT and establishes a power density at distance r: PT = dPT dS = PEIRP 4πr2 = PTGT 4πr2 (15.6.1) From the incident power density PT, the receiving antenna extracts power PR given in terms of the effective area AR as follows: PR = ARPT = PTGTAR 4πr2 (Friis formula) (15.6.2) This is known as the Friis formula for communicating antennas and can be written in several different equivalent forms. Replacing GT in terms of the transmitting antenna’s effective area AT, that is, GT = 4πAT/λ2, Eq. (15.6.2) becomes: PR = PTATAR λ2r2 (15.6.3) A better way of rewriting Eq. (15.6.2) is as a product of gain factors. Replacing AR = λ2GR/4π, we obtain: PR = PTGTGRλ2 (4πr)2 (15.6.4) 15.6. Communicating Antennas 613 The effect of the propagation path, which causes PR to attenuate with the square of the distance r, can be quantified by defining the free-space loss and gain by Lf = 4πr λ 2 , Gf = 1 Lf = λ 4πr 2 (free-space loss and gain) (15.6.5) Then, Eq. (15.6.4) can be written as the product of the transmit and receive gains and the propagation loss factor: PR = PTGT λ 4πr 2 GR = PTGT 1 Lf GR = PTGTGfGR (15.6.6) Such a gain model for communicating antennas is illustrated in Fig. 15.6.1. An ad-ditional loss factor, Gother = 1/Lother, may be introduced, if necessary, representing other losses, such as atmospheric absorption and scattering. It is customary to express Eq. (15.6.6) additively in dB, where (PR)dB= 10 log10 PR, (GT)dB= 10 log10 GT, etc.: (PR)dB = (PT)dB +(GT)dB −(Lf)dB +(GR)dB (15.6.7) Example 15.6.1: A geosynchronous satellite is transmitting a TV signal to an earth-based sta-tion at a distance of 40,000 km. Assume that the dish antennas of the satellite and the earth station have diameters of 0.5 m and 5 m, and aperture efficiencies of 60%. If the satel-lite’s transmitter power is 6 W and the downlink frequency 4 GHz, calculate the antenna gains in dB and the amount of received power. Solution: The wavelength at 4 GHz is λ = 7.5 cm. The antenna gains are calculated by: G = ea πd λ 2 ⇒ Gsat = 263.2 = 24 dB, Gearth = 26320 = 44 dB Because the ratio of the earth and satellite antenna diameters is 10, the corresponding gains will differ by a ratio of 100, or 20 dB. The satellite’s transmitter power is in dB, PT = 10 log10(6)= 8 dBW, and the free-space loss and gain: Lf = 4πr λ 2 = 4 × 1019 ⇒ Lf = 196 dB, Gf = −196 dB It follows that the received power will be in dB: PR = PT + GT −Lf + GR = 8 + 24 −196 + 44 = −120 dBW ⇒ PR = 10−12 W or, PR = 1 pW (pico-watt). Thus, the received power is extremely small. ⊓ ⊔ When the two antennas are mismatched in their polarization with a mismatch factor epol = |ˆ hR · ˆ hT|2, and the receiving antenna is mismatched to its load with eload = 1−|Γload|2, then the Friis formula (15.6.2) is still valid, but replacing AR using Eq. (15.5.7), leads to a modified form of Eq. (15.6.4): PR = PTGTGRλ2 (4πr)2 |ˆ hR · ˆ hT|2 1 −|Γload|2 (15.6.8) 614 15. Transmitting and Receiving Antennas 15.7 Antenna Noise Temperature We saw in the above example that the received signal from a geosynchronous satellite is extremely weak, of the order of picowatts, because of the large free-space loss which is typically of the order of 200 dB. To be able to detect such a weak signal, the receiving system must maintain a noise level that is lower than the received signal. Noise is introduced into the receiving system by several sources. In addition to the desired signal, the receiving antenna picks up noisy signals from the sky, the ground, the weather, and other natural or man-made noise sources. These noise signals, coming from different directions, are weighted according to the antenna gain and result into a weighted average noise power at the output terminals of the antenna. For example, if the antenna is pointing straight up into the sky, it will still pick up through its sidelobes some reflected signals as well as thermal noise from the ground. Ohmic losses in the antenna itself will be another source of noise. The antenna output is sent over a feed line (such as a waveguide or transmission line) to the receiver circuits. The lossy feed line will attenuate the signal further and also introduce its own thermal noise. The output of the feed line is then sent into a low-noise-amplifier (LNA), which pre-amplifies the signal and introduces only a small amount of thermal noise. The low-noise nature of the LNA is a critical property of the receiving system. The output of the LNA is then passed on to the rest of the receiving system, consisting of downconverters, IF amplifiers, and so on. These subsystems will also introduce their own gain factors and thermal noise. Such a cascade of receiver components is depicted in Fig. 15.7.1. The sum total of all the noises introduced by these components must be maintained at acceptably low levels (relative to the amplified desired signal.) Fig. 15.7.1 Typical receiving antenna system. The average power N (in Watts) of a noise source within a certain bandwidth of B Hz can be quantified by means of an equivalent temperature T defined through: N = kTB (noise power within bandwidth B) (15.7.1) where k is Boltzmann’s constant k = 1.3803× 10−23 W/Hz K and T is in degrees Kelvin. The temperature T is not necessarily the physical temperature of the source, it only provides a convenient way to express the noise power. (For a thermal source, T is indeed the physical temperature.) Eq. (15.7.1) is commonly expressed in dB as: NdB = TdB + BdB + kdB (15.7.2) 15.7. Antenna Noise Temperature 615 where TdB = 10 log10 T, BdB = 10 log10 B, and kdB = 10 log10 k is Boltzmann’s constant in dB: kdB = −228.6 dB. Somewhat incorrectly, but very suggestively, the following units are used in practice for the various terms in Eq. (15.7.2): dB W = dB K + dB Hz + dB W/Hz K The bandwidth B depends on the application. For example, satellite transmissions of TV signals require a bandwidth of about 30 MHz. Terrestrial microwave links may have B of 60 MHz. Cellular systems may have B of the order of 30 kHz for AMPS or 200 kHz for GSM. Example 15.7.1: Assuming a 30-MHz bandwidth, we give below some examples of noise powers and temperatures and compute the corresponding signal-to-noise ratio S/N, relative to a 1 pW reference signal (S = 1 pW). T TdB N = kTB NdB S/N 50 K 17.0 dBK 0.0207 pW −136.8 dBW 16.8 dB 100 K 20.0 dBK 0.0414 pW −133.8 dBW 13.8 dB 200 K 23.0 dBK 0.0828 pW −130.8 dBW 10.8 dB 290 K 24.6 dBK 0.1201 pW −129.2 dBW 9.2 dB 500 K 27.0 dBK 0.2070 pW −126.8 dBW 6.8 dB 1000 K 30.0 dBK 0.4141 pW −123.8 dBW 3.8 dB 2400 K 33.8 dBK 1.0000 pW −120.0 dBW 0.0 dB The last example shows that 2400 K corresponds to 1 pW noise. ⊓ ⊔ The average noise power Nant at the antenna terminals is characterized by an equiv-alent antenna noise temperature Tant, such that Nant = kTantB. The temperature Tant represents the weighted contributions of all the radiating noise sources picked up by the antenna through its mainlobe and sidelobes. The value of Tant depends primarily on the orientation and elevation angle of the antenna, and what the antenna is looking at. Example 15.7.2: An earth antenna looking at the sky “sees” a noise temperature Tant of the order of 30–60 K. Of that, about 10 K arises from the mainlobe and sidelobes pointing towards the sky and 20–40 K from sidelobes pointing backwards towards the earth [5,1044– 1048]. In rainy weather, Tant might increase by 60 K or more. The sky noise temperature depends on the elevation angle of the antenna. For example, at an elevations of 5o, 10o, and 30o, the sky temperature is about 20 K, 10 K, and 4 K at 4 GHz, and 25 K, 12 K, and 5 K at 6 GHz . ⊓ ⊔ Example 15.7.3: The noise temperature of the earth viewed from space, such as from a satellite, is about 254 K. This is obtained by equating the sun’s energy that is absorbed by the earth to the thermal radiation from the earth . ⊓ ⊔ Example 15.7.4: For a base station cellular antenna looking horizontally, atmospheric noise temperature contributes about 70–100 K at the cellular frequency of 1 GHz, and man-made noise contributes another 10–120 K depending on the area (rural or urban). The total value of Tant for cellular systems is in the range of 100–200 K [1049,1050]. ⊓ ⊔ 616 15. Transmitting and Receiving Antennas In general, a noise source in some direction (θ, φ) will be characterized by an ef-fective noise temperature T(θ, φ), known as the brightness temperature, such that the radiated noise power in that direction will be N(θ, φ)= kT(θ, φ)B. The antenna tem-perature Tant will be given by the average over all such sources weighted by the receiving gain of the antenna: Tant = 1 ΔΩ π 0 2π 0 T(θ, φ)g(θ, φ) dΩ (15.7.3) where ΔΩ is the beam solid angle of the antenna. It follows from Eq. (15.2.15) that ΔΩ serves as a normalization factor for this average: ΔΩ = π 0 2π 0 g(θ, φ) dΩ (15.7.4) Eq. (15.7.3) can also be written in the following equivalent forms, in terms of the directive gain or the effective area of the antenna: Tant = 1 4π π 0 2π 0 T(θ, φ)D(θ, φ) dΩ = 1 λ2 π 0 2π 0 T(θ, φ)A(θ, φ) dΩ As an example of Eq. (15.7.3), we consider the case of a point source, such as the sun, the moon, a planet, or a radio star. Then, Eq. (15.7.3) gives: Tant = Tpoint gpointΔΩpoint ΔΩ where gpoint and ΔΩpoint are the antenna gain in the direction of the source and the small solid angle subtended by the source. If the antenna’s mainlobe is pointing towards that source then, gpoint = 1. As another example, consider the case of a spatially extended noise source, such as the sky, which is assumed to have a constant temperature Text over its angular width. Then, Eq. (15.7.3) becomes: Tant = Text ΔΩext ΔΩ , where ΔΩext = ext g(θ, φ) dΩ The quantity ΔΩext is the portion of the antenna’s beam solid angle occupied by the extended source. As a third example, consider the case of an antenna pointing towards the sky and picking up the atmospheric sky noise through its mainlobe and partly through its side-lobes, and also picking up noise from the ground through the rest of its sidelobes. As-suming the sky and ground noise temperatures are uniform over their spatial extents, Eq. (15.7.3) will give approximately: Tant = Tsky ΔΩsky ΔΩ + Tground ΔΩground ΔΩ where ΔΩsky and ΔΩground are the portions of the beam solid angle occupied by the sky and ground: ΔΩsky = sky g(θ, φ) dΩ , ΔΩground = ground g(θ, φ) dΩ 15.7. Antenna Noise Temperature 617 Assuming that the sky and ground beam solid angles account for the total beam solid angle, we have ΔΩ = ΔΩsky + ΔΩground The sky and ground beam efficiency ratios may be defined by: esky = ΔΩsky ΔΩ , eground = ΔΩground ΔΩ , esky + eground = 1 Then, the antenna noise temperature can be written in the form: Tant = eskyTsky + egroundTground (15.7.5) Example 15.7.5: At 4 GHz and elevation angle of 30o, the sky noise temperature is about 4 K. Assuming a ground temperature of 290 K and that 90% of the beam solid angle of an earth-based antenna is pointing towards the sky and 10% towards the ground, we calculate the effective antenna temperature: Tant = eskyTsky + egroundTground = 0.9 × 4 + 0.1 × 290 = 32.6 K If the beam efficiency towards the sky changes to 85%, then esky = 0.85, eground = 0.15 and we find Tant = 46.9 K. ⊓ ⊔ The mainlobe and sidelobe beam efficiencies of an antenna represent the proportions of the beam solid angle occupied by the mainlobe and sidelobe of the antenna. The corresponding beam solid angles are defined by: ΔΩ = tot g(θ, φ) dΩ = main g(θ, φ) dΩ + side g(θ, φ) dΩ = ΔΩmain + ΔΩside Thus, the beam efficiencies will be: emain = ΔΩmain ΔΩ , eside = ΔΩside ΔΩ , emain + eside = 1 Assuming that the entire mainlobe and a fraction, say α, of the sidelobes point towards the sky, and therefore, a fraction (1 −α) of the sidelobes will point towards the ground, we may express the sky and ground beam solid angles as follows: ΔΩsky = ΔΩmain + αΔΩside ΔΩground = (1 −α)ΔΩside or, in terms of the efficiency factors: esky = emain + αeside = emain + α(1 −emain) eground = (1 −α)eside = (1 −α)(1 −emain) Example 15.7.6: Assuming an 80% mainlobe beam efficiency and that half of the sidelobes point towards the sky and the other half towards the ground, we have emain = 0.8 and α = 0.5, which lead to the sky beam efficiency esky = 0.9. ⊓ ⊔ 618 15. Transmitting and Receiving Antennas 15.8 System Noise Temperature In a receiving antenna system, the signal-to-noise ratio at the receiver must take into account not only the noise picked up by the antenna, and quantified by Tant, but also all the internal noises introduced by the various components of the receiver. Every device, passive or active, is a source of noise generated internally. Such noise may be modeled as an internal noise source acting at the input of the device, as shown in Fig. 15.8.1. (Alternatively, the noise source can be added at the output, but the input convention is more common.) Fig. 15.8.1 Noise model of a device. The amount of added noise power is expressed in terms of the effective noise tem-perature Te of the device: Ne = kTeB (effective internal noise) (15.8.1) The sum of Ne and the noise power of the input signal Nin will be the total noise power, or the system noise power at the input to the device. If the input noise is expressed in terms of its own noise temperature, Nin = kTinB, we will have: Nsys = Nin + Ne = k(Tin + Te)B = kTsysB (total input noise) (15.8.2) where we introduced the system noise temperature† at the device input: Tsys = Tin + Te (system noise temperature) (15.8.3) If the device has power gain G,‡ then the noise power at the output of the device and its equivalent temperature, Nout = kToutB, can be expressed as follows: Nout = G(Nin + Ne)= GNsys Tout = G(Tin + Te)= GTsys (15.8.4) One interpretation of the system noise power Nsys = kTsysB is that it represents the required input power to an equivalent noiseless system (with the same gain) that will produce the same output power as the actual noisy system. If a desired signal with noise power Sin is also input to the device, then the signal power at the output will be Sout = GSin. The system signal-to-noise ratio is defined to be the ratio of the input signal power to total system noise power: †Also called the operating noise temperature. ‡More precisely, G is the available power gain of the device, in the notation of Sec. 13.6. 15.8. System Noise Temperature 619 S N Rsys = Sin Nsys = Sin kTsysB = Sin k(Tin + Te)B (system SNR) (15.8.5) The SNR is the same whether it is measured at the input or the output of the device; indeed, multiplying numerator and denominator by G and using (15.8.4), we have: S N Rsys = Sin Nsys = Sout Nout (15.8.6) A related concept is that of the noise figure of the device, which also characterizes the internally generated noise. It is related to the effective noise temperature Te by: F = 1 + Te T0 Te = (F −1)T0 (15.8.7) where T0 is the standardized constant temperature T0 = 290 K. The device of Fig. 15.8.1 can be passive or active. The case of a passive attenuator, such as a lossy transmission line or waveguide connecting the antenna to the receiver, deserves special treatment. In this case, the gain G will be less than unity G < 1, representing a power loss. For a line of length l and attenuation constant α (nepers per meter), we will have G = e−2αl. The corresponding loss factor will be L = G−1 = e2αl. If αl ≪1, we can write approximately G = 1 −2αl and L = 1 + 2αl. If the physical temperature of the line is Tphys then, from either the input or output end, the line will appear as a thermal noise source of power kTphysB. Therefore, the condition Nin = Nout = kTphysB implies that kTphysB = Gk(Tphys + Te)B, which gives: Te = 1 G(1 −G)Tphys = (L −1)Tphys (attenuator) (15.8.8) If the physical temperature is Tphys = T0 = 290 K, then, by comparing to Eq. (15.8.7) it follows that the noise figure of the attenuator will be equal to its loss: Te = (L −1)T0 = (F −1)T0 ⇒ F = L = 1 G When the input to the attenuator is an external noise source of power Nin = kTinB, the system noise temperature at the input and at the output of the attenuator will be: Tsys = Tin + Te = Tin + (L −1)Tphys Tout = GTsys = GTin + (1 −G)Tphys = 1 LTin + 1 −1 L Tphys (15.8.9) The last equation can be expressed in terms of the input and output powers Nin = kTinB and Nout = kToutB: Nout = 1 LNin + 1 −1 L kTphysB (15.8.10) 620 15. Transmitting and Receiving Antennas Thus, the input power is attenuated as expected, but the attenuator also adds its own thermal noise power. More generally, if the input power arises from signal plus noise Pin = Sin + Nin, the power at the output will be Pout = Sout + Nout = GSin + Nout: Pout = 1 LPin + 1 −1 L kTphysB (15.8.11) When two or more devices are cascaded, each will contribute its own internal noise. Fig. 15.8.2 shows two such devices with available power gains G1 and G2 and effec-tive noise temperatures T1 and T2. The cascade combination can be replaced by an equivalent device with gain G1G2 and effective noise temperature T12. Fig. 15.8.2 Equivalent noise model of two cascaded devices. The equivalent temperature T12 can be determined by superposition. The internal noise power added by the first device, N1 = kT1B, will go through the gains G1 and G2 and will contribute an amount G1G2N1 to the output. The noise generated by the second device, N2 = kT2B, will contribute an amount G2N2. The sum of these two powers will be equivalent to the amount contributed to the output by the combined system, G1G2N12 = G1G2kT12B. Thus, G1G2kT12B = G1G2kT1B + G2kT2B ⇒ G1G2T12 = G1G2T1 + G2T2 It follows that: T12 = T1 + 1 G1 T2 (equivalent noise temperature) (15.8.12) If G1 is a large gain, G1 ≫1, then the contribution of the second device is reduced drastically. On the other hand, if the first device is an attenuator, such as a transmission line, then the contribution of T2 will be amplified because G1 < 1. According to Eqs. (15.8.3) and (15.8.4), the system noise temperatures at the overall input, at the output of G1, and at the overall output will be: Tsys = Tsa = Tin + T12 = Tin + T1 + 1 G1 T2 Tsb = G1Tsa = G1(Tin + T1)+T2 Tout = G2Tsb = G1G2Tsys = G1G2(Tin + T1)+G2T2 (15.8.13) 15.8. System Noise Temperature 621 The system SNR will be: S N Rsys = Sin kTsysB = Sin k(Tin + T12)B The signal powers at points a, b, and at the output will be Sa = Sin, Sb = G1Sa, and Sout = G2Sb = G1G2Sa. It follows from Eq. (15.8.13) that the system SNR will be the same, regardless of whether it is referred to the point a, the point b, or the overall output: S N Rsys = S N Ra = S N Rb = S N Rout For three cascaded devices, shown in Fig. 15.8.3, any pair of two consecutive ones can be replaced by its equivalent, according to Eq. (15.8.12). For example, the first two can be replaced by T12 and then combined with T3 to give the overall equivalent temperature: T12 = T1 + 1 G1 T2 , T123 = T12 + 1 G1G2 T3 Fig. 15.8.3 Equivalent noise temperatures of three cascaded devices. Alternatively, we can replace the last two with an equivalent temperature T23 and then combine with the first to get: T23 = T2 + 1 G2 T3 , T123 = T1 + 1 G1 T23 From either point of view, we obtain the equivalent temperature: T123 = T1 + 1 G1 T2 + 1 G1G2 T3 (15.8.14) The system SNR will be: S N Rsys = Sin kTsysB = Sin k(Tin + T123)B It is invariant with respect to its reference point: S N Rsys = S N Ra = S N Rb = S N Rc = S N Rout 622 15. Transmitting and Receiving Antennas When expressed in terms of noise figures, Eqs. (15.8.12) and (15.8.14) are also known as Friis’s formulas , for example, defining the equivalent noise figure as F123 = 1 + T123/T0, we have: F123 = F1 + F2 −1 G1 + F3 −1 G1G2 (15.8.15) We apply now these results to the antenna receiver shown in Fig. 15.7.1, identifying the three cascaded components as the feed line, the LNA amplifier, and the rest of the receiver circuits. The corresponding noise temperatures are Tfeed, TLNA, and Trec. The effective noise temperature Teff of the combined system will be: Teff = Tfeed + 1 Gfeed TLNA + 1 GfeedGLNA Trec (15.8.16) Using Eq. (15.8.8), we may replace Tfeed in terms of the physical temperature: Teff = 1 Gfeed (1 −Gfeed)Tphys + 1 Gfeed TLNA + 1 GfeedGLNA Trec (15.8.17) The input noise temperature Tin to this combined system is the antenna temperature Tant. It follows that system noise temperature, referred to either the antenna output terminals (point a), or to the LNA input (point b), will be: Tsys = Tsa = Tant + Teff = Tant + 1 Gfeed −1 Tphys + 1 Gfeed TLNA + 1 GfeedGLNA Trec Tsb = GfeedTsa = GfeedTant + (1 −Gfeed)Tphys + TLNA + 1 GLNA Trec The importance of a high-gain low-noise amplifier is evident from Eq. (15.8.17). The high value of GLNA will minimize the effect of the remaining components of the receiver system, while the small value of TLNA will add only a small amount of noise. Typical values of TLNA can range from 20 K for cooled amplifiers to 100 K at room temperatures. The feed line can have a major impact. If the line is too lossy or too long, the quantity Gfeed = e−2αl will be small, or 1/Gfeed large, contributing a significant amount to the system noise temperature. Often, the LNA is mounted before the feed line, near the focal point of the receiving antenna, so that the effect of the feed line will be suppressed by the factor GLNA. Similar benefits arise in base station antennas for wireless communications, where high-gain amplifiers can be placed on top of the antenna towers, instead of at the base station, which can be fairly far from the towers . Cable losses in such applications can be in the range 2–4 dB (with gain factors Gf = 0.63–0.40.) The signal to system-noise ratio of the receiving system (referred to point a of Fig. 15.7.1) will be the ratio of the received power PR to the system noise Nsys = kTsysB. Using the Friis formula (for power transmission), we have: S N R = PR Nsys = PR kTsysB = (PTGT) 1 Lf GR Tsys 1 kB (15.8.18) 15.8. System Noise Temperature 623 This ratio is also called the carrier-to-system-noise ratio and is denoted by C/N. For a given transmitting EIRP, PTGT, the receiver performance depends critically on the ratio GR/Tsys, referred to as the G/T ratio of the receiving antenna, or the figure of merit. It is usually measured in dB/K. In dB, Eq. (15.8.18) reads as: (S N R)dB= (PTGT)dB −(Lf)dB + GR Tsys dB −kdB −BdB (15.8.19) The receiver SNR can be also be referred to LNA input (point b). The G/T ratio will not change in value, but it will be the ratio of the signal gain after the feed line divided by the system temperature Tsb, that is, S N R = (PTGT) 1 Lf GR Tsys 1 kB = (PTGT) 1 Lf GRGfeed Tsb 1 kB (15.8.20) Example 15.8.1: Typical earth-based antennas for satellite communications have G/T ratios of the order of 40 dB/K, whereas satellite receiving antennas can have G/T = −7 dB/K or less. The negative sign arises from the smaller satellite antenna gain and the much higher temperature, since the satellite is looking down at a warm earth. ⊓ ⊔ Example 15.8.2: Consider a receiving antenna system as shown in Fig. 15.7.1, with antenna temperature of 40 K, feed line loss of 0.1 dB, feed line physical temperature of 290 K, LNA gain and effective noise temperature of 50 dB and 80 K. The rest of the receiver circuits have effective noise temperature of 2000 K. Assuming the receiving antenna has a gain of 45 dB, calculate the system noise temperature and the G/T ratio at point a and point b of Fig. 15.7.1. Repeat if the feed line loss is 1 dB. Solution: The feed line has gain Gfeed = 10−0.1/10 = 10−0.01 = 0.9772, and the LNA has GLNA = 1050/10 = 105. Thus, the system noise temperature at point a will be: Tsys = Tant + 1 Gfeed −1 Tphys + 1 Gfeed TLNA + 1 GfeedGLNA Trec = 40 + 1 10−0.01 −1 290 + 80 10−0.01 + 2000 10−0.01 · 105 = 40 + 6.77 + 81.87 + 0.02 = 128.62 K = 21.09 dBK At point b, we have Tsb = GfeedTsys = 0.9772 × 128.62 = 125.69 K = 20.99 dBK. The G/T ratio will be at point a, GR/Tsys = 45 −21.09 = 23.91 dB/K. At point b the gain is GRGfeed = 45 −0.1 = 44.9 dB, and therefore, G/T = GRGfeed/Tsb = 44.9 −20.99 = 23.91 dB/K, which is the same as at point a. For a feed line loss of 1 dB, we find Tsys = 215.80 K = 23.34 dB. The corresponding G/T ratio will be 45 −23.34 = 21.66 dB. ⊓ ⊔ Example 15.8.3: Suppose the LNA were to be placed in front of the feed line of the above example. Calculate the system noise temperature in this case when the feed line loss is 0.1 dB and 1 dB. 624 15. Transmitting and Receiving Antennas Solution: Interchanging the roles of the feed line and the LNA in Eq. (15.8.16), we have for the system noise temperature: Tsys = Tant + TLNA + 1 GLNA Tfeed + 1 GfeedGLNA Trec With Gfeed = 10−0.1/10 = 0.9772, we find Tfeed = 6.75 K, and with Gfeed = 10−1/10 = 0.7943, Tfeed = 75.1 K. Because of the large LNA gain, the value of Tsys will be essentially equal to Tant + TLNA; indeed, we find in the two cases: Tsys = 120.0205 K and Tsys = 120.0259 K The G/T will be 45 −10 log10(120)= 20.8 dB/K. ⊓ ⊔ 15.9 Data Rate Limits The system SNR limits the data rate between the two antennas. According to Shannon’s theorem, the maximum data rate (in bits/sec) that can be achieved is: C = B log2(1 + S N R) (Shannon’s channel capacity) (15.9.1) where S N R is in absolute units. For data rates R ≤C, Shannon’s theorem states that there is an ideal coding scheme that would guarantee error-free transmission. In a practical digital communication system, the bit-error probability or bit-error rate (BER), Pe, is small but not zero. The key performance parameter from which Pe can be calculated is the ratio Eb/N0, where Eb is the energy per bit and N0 is the system noise spectral density N0 = kTsys. The functional relationship between Pe and Eb/N0 depends on the particular digital modulation scheme used. For example, in binary and quadrature phase-shift keying (BPSK and QPSK), Pe and its inverse are given by : Pe = 1 2erfc Eb N0 Eb N0 =  erfinv(1 −2Pe) 2 (15.9.2) where erfc(x) is the complementary error function, and erf(x) and erfinv(x) are the error function and its inverse as defined in MATLAB: erfc(x)= 1 −erf(x)= 2 √π ∞ x e−t2 dt , y = erf(x) x = erfinv(y) (15.9.3) The relationships (15.9.2) are plotted in Fig. 15.9.1. The left graph also shows the ideal Shannon limit Eb/N0 = ln 2 = 0.6931 ≡−1.5917 dB, which is obtained by taking the limit of Eq. (15.9.1) for infinite bandwidth. If Tb is the time it takes to transmit one bit, then the data rate will be R = 1/Tb, and the required power, P = Eb/Tb = EbR. It follows that the SNR will be: S N R = P Nsys = P kTsysB = Eb N0 R B 15.9. Data Rate Limits 625 −2 −1 0 1 2 3 4 5 6 7 8 9 10 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Bit−Error Probability P e Eb/N 0 (dB) −1.5917 10 −6 10 −4 10 −2 10 0 −2 0 2 4 6 8 10 Eb/N 0 Ratio P e Eb/N 0 (dB) Fig. 15.9.1 Pe versus Eb/N0, and its inverse, for a BPSK system. Using the small-x expansion, log2(1+x) x/ ln 2, Shannon’s condition for error-free transmission becomes in the limit B →∞: R ≤C = B log2 1 + Eb N0 R B →B EbR N0B ln 2 = R ln 2 Eb N0 ⇒ Eb N0 ≥ln 2 = −1.5917 dB For a pair of communicating antennas, the received power will be related to the energy per bit by PR = Eb/Tb = EbR. Using Friis’s formula, we find: R Eb N0 = PR N0 = PEIRP Gf GR kTsys = (PTGT) GR kTsys λ 4πr 2 (15.9.4) which may be solved for the maximum achievable data rate (in bits/sec): R = 1 Eb/N0 PEIRP Gf GR kTsys = 1 Eb/N0 (PTGT) GR kTsys λ 4πr 2 (15.9.5) An overall gain factor, Gother = 1/Lother, may be introduced representing other losses, such as atmospheric losses. Example 15.9.1: The Voyager spacecrafts (launched in 1977) have antenna diameter and aper-ture efficiency of d = 3.66 m (12 ft) and ea = 0.6. The operating frequency is f = 8.415 GHz and the transmitter power PT = 18 W. Assuming the same efficiency for the 70-m re-ceiving antenna at NASA’s deep-space network at Goldstone, CA, we calculate the antenna gains using the formula G = ea(πd/λ)2, with λ = c/f = 0.0357 m: GT = 47.95 dB, GR = 73.58 dB, PT = 12.55 dBW Assuming a system noise temperature of Tsys = 25 K = 13.98 dBK for the receiving antenna, we find for the noise spectral density N0 = kTsys = −214.62 dBW/Hz, where we used k = −228.6 dB. Assuming a bit-error rate of Pe = 5×10−3, we find from Eq. (15.9.2) the required ratio Eb/N0 = 3.317 = 5.208 dB. Voyager 1 was at Jupiter in 1979, at Saturn in 1980, and at Neptune in 1989. In 2002 it was at a distance of about r = 12×109 km. It is expected to be at r = 22×109 km in the year 626 15. Transmitting and Receiving Antennas 2020. We calculate the corresponding free-space gain Gf = (λ/4πr)2 and the expected data rates R from Eq. (15.9.5), where r is in units of 109 km: location r Gf (dB) R (dB) R (bits/sec) Jupiter 0.78 −288.78 49.72 93,724 Saturn 1.43 −294.05 44.45 27,885 Neptune 4.50 −304.01 34.50 2,816 2002 12.00 −312.53 25.98 396 2020 22.00 −317.79 20.71 118 where we assumed an overall loss factor of Gother = −5 dB. More information on the Voyager mission and NASA’s deep-space network antennas can be obtained from the web sites and . ⊓ ⊔ 15.10 Satellite Links Consider an earth-satellite-earth system, as shown in Fig. 15.10.1. We wish to establish the total link budget and signal to system-noise ratio between the two earth antennas communicating via the satellite. Fig. 15.10.1 Uplink and downlink in satellite communications. In a geosynchronous satellite system, the uplink/downlink frequencies fu, fd are typically 6/4 GHz or 14/11 GHz. The distances ru, rd are of the order of 40000 km. Let λu = c/fu and λd = c/fd be the uplink/downlink wavelengths. The free-space gain/loss factors will be from Eq. (15.6.5): Gfu = 1 Lfu = λu 4πru 2 , Gfd = 1 Lfd = λd 4πrd 2 (15.10.1) The satellite has an on-board amplifier with gain G, which could be as high as 100– 120 dB. Using Friis formula in its gain form, Eq. (15.6.6), the link equations for the uplink, the satellite amplification, and the downlink stages can be written as follows: PEIRP = PTE GTE (EIRP of transmitting earth antenna) PRS = PTE GTE Gfu GRS (received power by satellite antenna) PTS = GPRS (transmitted power by satellite antenna) PRE = PTS GTS Gfd GRE (received power by earth antenna) 15.10. Satellite Links 627 Expressing PRE in terms of PTE, we have: PRE = PRS G GTS Gfd GRE = PTE GTE Gfu GRS G GTS Gfd GRE (15.10.2) or, showing the free-space loss factors explicitly: PRE = PTE GTE GRS G GTS GRE λu 4πru 2 λd 4πrd 2 (15.10.3) Because there are two receiving antennas, there will be two different system noise temperatures, say TRS and TRE, for the satellite and earth receiving antennas. They incorporate the antenna noise temperatures as well as the receiver components. The corresponding figures of merit will be the quantities GRS/TRS and GRE/TRE. We may define the uplink and downlink SNR’s as the signal-to-system-noise ratios for the indi-vidual antennas: S N Ru = PRS kTRSB , S N Rd = PRE kTREB (15.10.4) The system noise TRS generated by the receiving satellite antenna will get amplified by G and then transmitted down to the earth antenna, where it will contribute to the system noise temperature. By the time it reaches the earth antenna it will have picked up the gain factors G GTS Gfd GRE. Thus, the net system noise temperature measured at the receiving earth antenna will be: Tsys = TRE + G GTS Gfd GRETRS (15.10.5) The SNR of the total link will be therefore, S N Rtot = PRE kTsysB (15.10.6) S N R−1 tot = k(TRE + G GTS Gfd GRETRS)B PRE = kTREB PRE + k G GTS Gfd GRE TRSB PRE = kTREB PRE + k G GTS Gfd GRETRSB G GTS Gfd GREPRS = kTREB PRE + kTRSB PRS = S N R−1 d + S N R−1 u where we used Eq. (15.10.2). It follows that: S N Rtot = 1 S N R−1 u + S N R−1 d (15.10.7) This is also written in the form: C N tot = 1 C N −1 u + C N −1 d 628 15. Transmitting and Receiving Antennas Example 15.10.1: As an example of a link budget calculation, assume the following data: The uplink/downlink distances are 36000 km. The uplink/downlink frequencies are 6/4 GHz. The diameters of the earth and satellite antennas are 15 m and 0.5 m with 60% aperture efficiencies. The earth antenna transmits power of 1 kW and the satellite transponder gain is 90 dB. The satellite receiving antenna is looking down at an earth temperature of 300 K and has a noisy receiver of effective noise temperature of 2700 K, whereas the earth receiving antenna is looking up at a sky temperature of 50 K and uses a high-gain LNA amplifier of 80 K (feedline losses may be ignored.) The bandwidth is 30 MHz. The uplink and downlink wavelengths are λu = 0.05 m and λd = 0.075 m, corresponding to 6 and 4 GHz. The up and down free-space gains and losses are: Gfu = −Lfu = −199.13 dB, Gfd = −Lfd = −195.61 dB The antenna gains are calculated to be: GTE = 57.27 dB, GRS = 27.72 dB, GTS = 24.20 dB, GRE = 53.75 dB With PTE = 1 kW = 30 dBW, the EIRP of the transmitting earth antenna will be: PEIRP = 30 + 57.27 = 87.27 dBW. The power received by the satellite antenna will be: PRS = 87.27 −199.13 + 27.72 = −84.14 dBW After boosting this up by the transponder gain of 90 dB, the power transmitted down to the receiving earth antenna will be: PTS = 90 −84.14 = 5.86 dBW The EIRP of the transmitting satellite antenna will be (PTSGTS)dB= 5.86 + 24.20 = 30.06 dBW. The downlink power received by the earth antenna will be: PRE = 30.06 −195.61 + 53.75 = −111.80 dBW The system noise temperatures are: TRS = 300 + 2700 = 3000 K and TRE = 50 + 80 = 130 K, and in dBK: TRS = 34.77 and TRE = 21.14. The 30 MHz bandwidth is in dB: BdB = 10 log10(30×106)= 74.77 dB Hz. Using the Boltzmann constant k in dB, kdB = −228.6, we calculate the receiver system noise powers in dB, using N = kdB + TdB + BdB: NRS = −228.6 + 34.77 + 74.77 = −119.06 dBW NRS = −228.6 + 21.14 + 74.77 = −132.69 dBW It follows that the G/T ratios and system SNR’s for the receiving antennas will be: (G/T)u= GRS −TRS = 27.72 −34.77 = −7.05 dB/K (G/T)d= GRE −TRE = 53.75 −21.14 = 32.61 dB/K S N Ru = PRS −NRS = −84.14 + 119.06 = 34.92 dB = 3103.44 S N Rd = PRE −NRE = −111.80 + 132.69 = 20.89 dB = 122.72 The overall system SNR is calculated from Eq. (15.10.7) using absolute units: S N Rtot = 1 S N R−1 u + S N R−1 d = 1 (3103.44)−1+(122.72)−1 = 118.05 = 20.72 dB The overall SNR is essentially equal to the downlink SNR. ⊓ ⊔ 15.11. Radar Equation 629 15.11 Radar Equation Another example of the application of the concepts of gain and effective area and the use of Friis formulas is radar. Fig. 15.11.1 shows a radar antenna, which illuminates a target at distance r in the direction of its maximal gain. The incident wave on the target will be reflected and a portion of it will be intercepted back at the antenna. Fig. 15.11.1 Radar antenna and target. The concept of radar cross section σ provides a measure of the effective area of the target and the re-radiated power. If the radar antenna transmits power PT with gain GT, the power density of the transmitted field at the location of the target will be: PT = PTGT 4πr2 (15.11.1) From the definition of σ, the power intercepted by the target and re-radiated is: Ptarget = σPT = PTGTσ 4πr2 (15.11.2) By definition of the radar cross section, the power Ptarget will be re-radiated isotropically and establish a power density back at the location of the radar antenna: Ptarget = Ptarget 4πr2 = PTGTσ (4πr2)2 (15.11.3) The amount of power received by the radar antenna will be given in terms of its effective area AR as follows: PR = ARPtarget = PTGTARσ (4π)2r4 (radar equation) (15.11.4) This is also known as Friis’ formula. Using AR = AT and GT = 4πAT/λ2, we may express Eq. (15.11.4) in the alternative forms: PR = PTA2 Tσ 4πλ2r4 = PTG2 Tλ2σ (4π)3r4 = PTG2 T λ 4πr 4 4πσ λ2 (15.11.5) Introducing the equivalent target gain corresponding to the radar cross section, that is, Gσ = 4πσ/λ2, we may also write Eq. (15.11.5) as the product of gains: PR = PTG2 TG2 f Gσ (15.11.6) 630 15. Transmitting and Receiving Antennas Fig. 15.11.2 Gain model of radar equation. Fig. 15.11.2 shows this gain model. There are two free-space paths and two antenna gains, acting as transmit and receive gains. The minimum detectable received power, PR,min, defines the maximum distance rmax at which the target can be detected: PR,min = PTGTARσ (4π)2r4 max Solving for rmax, we obtain: rmax =  PTGTATσ (4π)2PR,min 1/4 (radar range) (15.11.7) If the target is not in the direction of maximal gain GT of the antenna, but in some other direction, say (θ, φ), then the maximal gain GT in Eq. (15.11.5) must be replaced with GTg(θ, φ), where g(θ, φ) is the antenna’s normalized gain. The received power can be expressed then as: PR = PTG2 Tg2(θ, φ)λ2σ (4π)3r4 (15.11.8) In ground-based air search radars trying to detect approaching aircraft flying at a fixed height h, the power received by the radar can be made to be independent of the distance r, within a certain distance range, by choosing the gain g(θ, φ) appropriately. As shown in Fig. 15.11.3, the height h is related to r by h = r cos θ. Fig. 15.11.3 Secant antenna gain. If the gain is designed to have the secant-squared shape g(θ, φ)= K/ cos2 θ, where K is a constant, then the power will become independent of r. Indeed, PR = PTG2 Tg2(θ, φ)λ2σ (4π)3r4 = PTG2 TK2λ2σ (4π)3r4 cos4 θ = PTG2 TK2λ2σ (4π)3h4 15.12. Problems 631 The secant behavior is not valid over all polar angles θ, but only over a certain range, such as 0 ≤θ ≤θmax, where θmax corresponds to the maximum range of the radar rmax = h/ cos θmax. The desired secant shape can be achieved by appropriate feeds of the radar dish antenna, or by an antenna array with properly designed array factor. In Sec. 20.5, we present such a design for an array. 15.12 Problems 15.1 In an earth-satellite-earth communication system, the uplink/downlink distances are 36000 km. The uplink/downlink frequencies are 6/4 GHz. The diameters of the earth and satellite antennas are 20 m and 1 m with 60% aperture efficiencies. The transmitting earth antenna transmits power of 1.5 kW. The satellite transponder gain is 85 dB. The satellite receiving antenna is looking down at an earth temperature of 290K and has a noisy receiver of ef-fective noise temperature of 3000K, whereas the earth receiving antenna is looking up at a sky temperature of 60K and uses a high-gain LNA amplifier of noise temperature of 100K (feedline losses may be ignored.) The bandwidth is 30 MHz. a. Calculate all antenna gains in dB. b. Calculate the uplink and downlink free-space losses in dB. c. Calculate the amount of power received by the satellite in dBW. Calculate the uplink signal to noise ratio in dB. d. Calculate the power received by the receiving earth antenna in dBW and the downlink signal to noise ratio. e. Finally, calculate the total system signal to noise ratio in dB. 15.2 The Voyager spacecraft is currently transmitting data to earth from a distance of 12 billion km. Its antenna diameter and aperture efficiency are 3.66 m and 60 %. The operating fre-quency is 8.415 GHz and Voyager’s transmitter power is 18 W. Assume the same aperture efficiency for the 70-m receiving antenna at NASA’s deep-space network at Goldstone, CA. a. Calculate the spacecraft’s and earth’s antenna gains in dB. Calculate also the free-space loss in dB. b. Calculate the achievable communication data rate in bits/sec between Voyager and earth using QPSK modulation and assuming the following: an overall transmission loss factor of 5 dB, a system noise temperature of 25 K, an energy-per-bit to noise-spectral-density ratio of Eb/N0 = 3.317 = 5.208 dB, which for QPSK corresponds to a bit-error probability of Pe = 5×10−3. 15.3 A satellite to earth downlink (shown below) is operating at a carrier frequency of f Hertz using QPSK modulation and achieving a bit rate of R bits/sec with a bit error probability of Pe. With the LNA absent, the receiving earth antenna is connected directly to a noisy receiver with equivalent noise temperature of Trec. Both antennas are dishes. 632 15. Transmitting and Receiving Antennas 1. A low-noise amplifier of very high gain GLNA and low noise temperature TLNA is inserted between the earth antenna and the receiver. Show that the presence of the LNA allows the link to be operated (with the same error probability Pe) at the higher bit rate: Rnew = R Ta + Trec Ta + TLNA where Ta is the earth antenna noise temperature, and TLNA ≪Trec. 2. The equation in part (a) is an approximation. Derive the exact form of that equation and discuss the nature of the approximation that was made. 3. How would the expression in part (a) change if, in addition to the assumptions of part (a), the operating frequency f were to be doubled? Explain your reasoning. How would (a) change if the transmitter power PT were to double? If the distance r were to double? 4. With the LNA present, and assuming that the bit rate R, error probability Pe, and f, PT, r remain the same, show that the diameter d of the earth antenna can be lowered to the following value without affecting performance: dnew = d Ta + TLNA Ta + Trec where the same approximation was made as in part (a). 15.4 A satellite to earth link (shown below) is operating at the carrier frequency of 4 GHz. The data link employs QPSK modulation and achieves a bit-error-rate probability of Pe = 10−6. The satellite has transmitter power of 20 W and uses a dish antenna that has a diameter of 0.5 m and aperture efficiency of 0.6. The earth antenna has diameter of 2 m, efficiency of 0.6, and antenna noise temperature of 80 K. The satellite antenna is at a distance of 40,000 km from the earth antenna. The output of the receiving antenna is connected to a high-gain low noise amplifier with gain of 40 dB and equivalent noise temperature of 200 K. The output of the LNA is connected to an RF amplifier with equivalent noise temperature of 1800 K. For QPSK modulation, we have the relationship Pe = erfc  Eb/N0  /2 with inverse Eb/N0 = [erfinv(1 −2Pe)]2. For the purposes of this exam, the following equation provides an ex-cellent approximation to this inverse relationship over the range of 10−8 ≤Pe ≤10−3: Eb N0 = −2.1969 log10(Pe)−1.8621 where Eb/N0 is in absolute units. a. Calculate the achievable communication data rate R in megabits/sec. b. If the LNA is removed, the performance of the system will deteriorate. In an attempt to keep the data rate the same as in part (a), the satellite transmitter power is increased to 80 W. Calculate the deteriorated value of the bit-error-rate Pe in this case. 15.12. Problems 633 15.5 A satellite to earth downlink (shown below) is operating at the carrier frequency of 4 GHz. The distance between the two antennas is r = 40 000 km. The bit error probability is Pe = 10−5 using QPSK modulation. For QPSK modulation, we have the following relationship between the bit-error-probability and Eb/N0 ratio, expressed in terms of the MATLAB functions erfc and erfinv: Pe = 1 2 erfc Eb N0 Eb N0 =  erfinv(1 −2Pe) 2 The satellite has transmitter power of 20 W and uses a dish antenna that has a diameter of 0.5 m and aperture efficiency of 0.6. The earth antenna has diameter of 5 m, efficiency of 0.6, and antenna noise temperature of 50 K. The output of the antenna is connected to an RF amplifier with equivalent noise temperature of 2000 K. a. Assuming that no LNA is used, calculate the system noise temperature Tsys at the output of the receiving antenna, the received power PR in picowatts, and the maximum achievable data rate in Mb/sec. b. It is desired to improved the performance of this system tenfold, that is, to increase the maximum achievable data rate in Mb/sec by a factor of 10. To this end, a low-noise amplifier of 40-dB gain is inserted as shown. Determine the noise temperature of the LNA that would guarantee such a performance improvement. c. What is the maximum noise temperature of the LNA that can achieve such a 10-fold improvement, and at what LNA gain is it achieved? 15.6 A radar with EIRP of Pradar = PTGT is trying to detect an aircraft of radar cross section σ. The aircraft is at a distance r from the radar and tries to conceal itself by jamming the radar with an on-board jamming antenna of EIRP of Pjammer = PJGJ. Assume that both the radar and the jamming antennas are pointing in their direction of maximal gains. a. Derive an expression of the signal-to-jammer ratio S/J, where S represents the power received from the target back at the radar antenna according to the radar equation, and J represents the power from the jamming antenna received by the radar antenna. Express the ratio in terms of Pradar, Pjammer, r, and σ. b. If detectability of the target in the presence of jamming requires at least a 0-dB signal-to-jammer ratio (that is, S/J ≥1), show that the maximum detectable distance is: r = PTGT PJGJ σ 4π 15.7 The Arecibo Observatory in Puerto Rico has a gigantic dish antenna of diameter of 1000 ft (304.8 m). It transmits power of 2.5 MW at a frequency of 430 MHz. a. Assuming a 60 percent effective area, what is its gain in dB? b. What is its beamwidth in degrees? c. If used as a radar and the minimum detectable received power is −130 dBW, what is its maximum range for detecting a target of radar cross-section of 1 m2?
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https://wjmh.org/DOIx.php?id=10.5534/wjmh.180036
Your First Choice Journal in Men's HealthOpen Access, Peer-reviewed Indexed in SCIE, SCOPUS, DOAJ and More pISSN 2287-4208 eISSN 2287-4690 World J Mens Health. 2019 Jan;37(1):45-54. English.Published online Oct 10, 2018. Copyright © 2019 Korean Society for Sexual Medicine and Andrology Testosterone Is a Contraceptive and Should Not Be Used in Men Who Desire Fertility Amir Shahreza Patel,1 Joon Yau Leong,2 Libert Ramos,1 and Ranjith Ramasamy1 Author information Author notes Copyright and License 1Department of Urology, University of Miami Miller School of Medicine, Miami, FL, USA. 2Department of Urology, Thomas Jefferson University, Philadelphia, PA, USA. Correspondence to: Amir Shahreza Patel. Department of Urology, University of Miami Miller School of Medicine, 1120 NW 14th Street, Suite 1551, Miami, FL 33136, USA. Tel: +1-305-497-7813, Fax: +1-305-243-6597, Email: asp136@med.miami.edu Received May 04, 2018; Revised June 22, 2018; Accepted July 01, 2018. 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: Abstract Testosterone has a variety of functions and is commonly used in older men to treat symptoms of hypogonadism, such as decreased libido, decreased mood and erectile dysfunction. Despite its positive effects on sexual function, it has a negative effect on fertility. Exogenous testosterone therapy can negatively affect the hypothalamic-pituitary gonadal axis and inhibit the production of follicle stimulating hormone and luteinizing hormone. The purpose of this review is to discuss the contraceptive properties of testosterone therapy and to discuss strategies to increase testosterone in men with the desire to preserve fertility. Keywords Contraception; Family planning services; Hypogonadism; Infertility; Testosterone; Testosterone replacement therapy Go to: INTRODUCTION Testosterone is a pleiotropic hormone that plays various physiological roles in the development of male genitalia in utero and during puberty. Classically, testosterone is a hormone associated with masculinity. Testosterone is used as treatment for males with late onset hypogonadism, a condition in men who experience symptoms caused by a decrease in serum testosterone. Symptoms associated with low testosterone can include decreased libido, decreased muscle mass, depressed mood and/or erectile dysfunction. The use of testosterone replacement therapy (TRT) among men over the age of 40 years has increased more than 3-fold over the last decade . Exogenous testosterone comes in various preparations and each form carries various risks. Along with an increase in hematocrit, a major adverse effect of TRT is the diminished sperm production because of the decreased intra-testicular concentration of testosterone and suppression of the hypothalamic-pituitary-gonadal (HPG) axis [2, 3, 4]. Suppression of follicle stimulating hormone (FSH) release from the pituitary gland impairs sperm production and suppression of luteinizing hormone (LH) release inhibits intra-testicular testosterone production. The purpose of this review is to evaluate the contraceptive effect of testosterone, discuss how the use of exogenous testosterone can negatively impact a man's fecundity and identify the importance of family planning in men who are planning to receive TRT. Go to: PHYSIOLOGY OF TESTOSTERONE In healthy adult men, testosterone production is precisely regulated by the HPG axis. Higher cortical centers in the brain signal the hypothalamus to secrete gonadotropin-releasing hormone (GnRH) in a pulsatile fashion. GnRH in turn stimulates the release of LH and FSH from the anterior pituitary which modulates testosterone production from the Leydig cells and spermatogenesis by the Sertoli cells, respectively. As testosterone levels increase, negative feedback suppression is exerted on the androgen receptors in the hypothalamic neurons and pituitary gland, thereby inhibiting the release of GnRH, FSH and LH . The Endocrine Society and American Urological Association (AUA) recommends treating symptomatic men with low testosterone documented on two morning fasting serum total testosterone concentrations. Both organizations recommend against the use of testosterone for treatment of hypogonadism in men who desire fertility in the next 6 to 12 months [3, 4]. The exogenous administration of testosterone suppresses the release of gonadotropins (FSH and LH) to levels below that required for spermatogenesis. Spermatogenesis is largely dependent on the action of FSH on Sertoli cells coupled with high intra-testicular testosterone concentrations. Within the seminiferous tubules, only Sertoli cells possess receptors for both FSH and testosterone. Numerous signaling pathways are activated when FSH binds to FSH receptors on these cells. It acts synergistically with testosterone to increase fertility and the efficiency of spermatogenesis . The inhibition of LH release by exogenous testosterone leads to the suppression of endogenous testosterone production by the Leydig cells. The decreased intra-testicular testosterone combined with the suppression of FSH leads to decreased germ cell survival and maturation (Fig. 1). Fig. 1 Image explaining the contraceptive effect of exogenous testosterone. In summary, it works by 2 mechanisms, decreasing intra-testicular testosterone, and inhibiting spermatogenesis. Most of the intra-testicular testosterone is made by the Leydig Cells in the testis. When exogenous testosterone is present, it inhibits gonadotropin-releasing hormone (GnRH) production which in turn inhibits luteinizing hormone (LH) production and decreases endogenous testosterone production by the Leydig cells, decreasing the intra-testicular testosterone concentration. Inhibiting GnRH production also inhibits follicle stimulating hormone (FSH) release, which impairs spermatogenesis in the Sertoli cells. Click for larger image Download as PowerPoint slide Intra-testicular testosterone is required in spermatogenesis for the formation of the blood-testis barrier (BTB). The BTB is a series of tight and adherens junctions between the Sertoli cells that separates postmeiotic germ cells in the adluminal compartment of the seminiferous tubules from the basal compartment containing the blood supply. During spermatogenesis, the BTB is disrupted and reformed as preleptotene spermatocytes pass through this barrier. In the absence of testosterone stimulation, spermatogenesis can only proceed as far as the prophase 1-leptotene stage of meiosis . Testosterone is also required in maintaining connections between Sertoli cells and the haploid spermatid germ cells. Round spermatids are initially connected to Sertoli cells via desmosomes. As the spermatids mature and elongate, the desmosomes are replaced with stronger, specialized adherens junctions called ectoplasmic specializations, which are maintained until the release of mature sperm. Testosterone aids in this process and increases the efficiency of germ cell attachment to Sertoli cells. Testosterone is also essential for the release of mature spermatozoa from Sertoli cells. It has been shown that in the absence of testosterone stimulation, sperm are not released but are instead phagocytized by Sertoli cells . Ultimately, the low intra-testicular testosterone results in decreased proliferation of spermatogonia, defects in spermiation of mature spermatozoa by Sertoli cells and accelerated apoptosis of spermatozoa [8, 9, 10, 11]. Since 80% of testicular volume consists of germinal epithelium and seminiferous tubules, a reduction in these cells is usually manifested by testicular atrophy and this reflects the loss of both spermatogenesis and Leydig cell function [12, 13]. Testosterone as a contraceptive can suppress spermatogenesis and lead to azoospermia in 65% of normospermic men within 4 months of use . Cessation of exogenous testosterone will lead to the reversal of hormonally-induced azoospermia in 64% to 84% of men with a median time of about 110 days [13, 14, 15]. All men in these studies recovered to baseline levels after cessation of therapy; however, it took up to 2 years for some men to recover. These studies were performed in a controlled setting for a clinical trial, with a limited duration of testosterone use. In actual practice, recovery may not be as pronounced. Kohn et al studied spermatogenesis recovery with human chorionic gonadotropin (hCG) and selective estrogen receptor modulators (SERM) in men with infertility associated with testosterone use. Thirty percent of the 66 men were not able to achieve a total motile sperm count of more than 5 million after 12 months in the study. They found that the failure of recovery is associated with older patients and longer TRT duration. If fertility is affected because of TRT, couples may require the use of in vitro fertilization or intra-cytoplasmic sperm injection for future conception. These assistive reproductive technologies are expensive and are not always successful [17, 18]. In summary, despite the androgenic effects of testosterone on sexual function, libido and mood; its effect on gonadotropins leads to the inhibition of sperm production . This effect may diminish with the cessation of testosterone intake, but the extent of recovery is not clear for chronic users [16, 19]. Go to: TESTOSTERONE AS A MALE CONTRACEPTIVE Compared to the long list of contraceptive options available to women, men are limited to vasectomy and condoms. The former is challenging to reverse and the latter has failure rates as high as 18% because of noncompliance. Partners who correctly and consistently use condoms have failure rates of 2% [20, 21]. As it is a user-dependent method, many couples seek easier to use options like female oral contraceptive pills or intrauterine devices . However, there is a demand for alternatives. A survey of over 9,000 men from different populations in 2005 found that 29% to 71% of men are interested in using a form of hormonal male contraception [22, 23]. In 1978, a newly available oral testosterone preparation known as testosterone undecanoate (TU) was investigated as a possible form for male contraception. The study found that regular testosterone use for 10 to 12 weeks causes suppression of sperm production, and even azoospermia, albeit inconsistently . Ever since that study, testosterone has undergone extensive clinical trials as a hormonal method of male contraception and many have found testosterone to be efficacious, reversible and safe with minimal short-term side effects . Unfortunately, the contraceptive effect of testosterone is not reliable. This has been proven in multiple studies, including two by the World Health Organization (WHO) Task Force on Methods for the Regulation of Male Fertility [14, 15, 25]. These two studies found an azoospermia rate of 64% to 75% in 6 months with testosterone enanthate [6, 7]. A sperm concentration of 3 million/mL was used as a threshold for effective suppression of spermatogenesis in this study [14, 15]. In a Chinese study of a monthly intramuscular TU injection, an azoospermia rate of 93% to 98% was achieved after 6 months with 1 million/mL as the criteria for effective suppression [25, 26]. The different rates of azoospermia can be explained by the variable criteria and by ethnic differences in testosterone response [26, 27]. These studies confirm the effectiveness of testosterone as a contraceptive, and provides evidence that men who desire fertility should not be prescribed TRT. Even with this evidence, testosterone has not been approved by the USA Food and Drug Administration (FDA) for use as a contraceptive. In 2011, a phase II study for a combined TU/norethisterone enanthate formulation ended prematurely because of higher than anticipated adverse effects including mood changes (such as depression), increased libido, acne and weight gain [27, 28]. More recent advancements were shown in the 2018 Endocrine Society meeting, with dimethandrolone undecanoate shown to effectively decrease sperm counts without adverse effects in a double-blind study in 2 academic sites. More extensive research on the safety of testosterone as a contraceptive needs to be done before testosterone can be used as a safe and reliable contraceptive . Go to: FORMULATIONS OF TESTOSTERONE REPLACEMENT THERAPY To date, many different testosterone formulations are available, each with their own side effect profiles. The selection of the preparation of testosterone requires a comprehensive discussion with the patient about the route of administration, cost and side effects of the individual formulations. Oral methyltestosterone is the only form of oral testosterone approved for use in the USA. It is strongly associated with hepatotoxicity and the AUA recommends against using the formulation [2, 3]. TU is approved for use in some countries but is not approved for use in the USA . Topical options of TRT include gels and patches. They are relatively easy to administer and doses are able to be quickly altered when needed . Their adverse effects include skin irritation seen with testosterone patches. Topical testosterone gels also run the risk of transference to others; but this can be avoided by using a clothing barrier [31, 32]. Nasal testosterone gels (NTG) are a relatively newer form of TRT that is currently undergoing extensive research. It is seen to be advantageous over topical gels because of ease of use and the decreased risk of transference . With regards to fertility, Conners et al found that 4.5% NTG two or three times a day restored serum testosterone levels while only decreasing gonadotropin levels minimally, keeping serum FSH and LH values within the normal range. Its short half-life results in a return of serum testosterone to near baseline levels between doses. It is theorized that this decreases its effect on the pulsatile release of GnRH by the hypothalamus . A phase IV clinical trial is currently evaluating its impact on semen analysis parameters, and it would be the first study to do so . Based on what is found by future studies, NTGs may have the potential to be a suitable TRT option in men desiring fertility. Intramuscular testosterone injections are another form of TRT. These include testosterone cypionate and enanthate, which are self-administered once every 1 to 2 weeks. Their starting dose is 100 mg weekly or 200 mg every two weeks before titrating in response to lab results on follow-up visits . While the patients using these formulations will be able to avoid frequent trips to the clinic once the dose has been adjusted, it does require proper patient education to ensure compliance to the dose set by the healthcare provider. They also have a greater risk of side effects than other preparations [30, 37]. One of these adverse effects is the ‘up and down’ phenomenon due to the variable release of the hormone into the bloodstream leading to peaks and troughs beyond the normal range of serum testosterone levels [5, 30, 38, 39]. TU is another preparation of intramuscular testosterone that is longer acting than the other formulations. It needs to be administered with an initial dose of 750 mg, followed 4 weeks later by another 750 mg dose. This is then followed by an intramuscular injection once every 10 weeks. A disadvantage is that this preparation needs to be administered in the office as a slow injection over 2 minutes and patients need to be monitored for 30 to 45 minutes after administration due to the risk of developing pulmonary-oil micro embolism [4, 40]. Beyond gels, patches and injections, another option for TRT are the subdermal implants. They are administered in a 10- to 15-minute procedure in the office every three to six months depending on follow-up laboratory results. This is a popular option among patients because they do not have to self-inject or apply gels repeatedly . It is particularly helpful in patients who travel regularly, and the extended release decreases the ‘up and down’ feelings often experienced with the intramuscular injections. The disadvantages of subdermal implants include the need for regular office visits, pain and bruising at the site of insertion, as well as the minimal risk of infection and pellet extrusion . In terms of the contraceptive effect of the different formulations of testosterone, most research has shown that transdermal and intramuscular testosterone seem to be the strongest contraceptive formulations. The WHO and Chinese studies used testosterone enanthate and TU, respectively. The topical formulations of testosterone have variable contraceptive effects. The testosterone patch was shown to be an ineffective contraceptive while the gel had mixed results [42, 43]. However, the sample size for most of these studies are not large enough to truly assess the extent to which fecundity is affected. More research needs to be done to evaluate the contraceptive effect of the various formulations of testosterone. A list of the available testosterone formulations with its side effect profiles and effect on fertility can be found in Table 1 [3, 14, 15, 25, 28, 41, 42, 43, 44, 45, 46, 47, 48]. Table 1 List of testosterone formulations Click for larger image Click for full table Download as Excel file Go to: USING TESTOSTERONE IN THE TREATMENT OF HYPOGONADISM IN MEN WHO DESIRE FERTILITY Considering that there is abundant evidence demonstrating that TRT significantly decreases sperm production, it is important that clinicians consider the evidenced risks of male infertility before starting patients on TRT. It can be surprising to patients that testosterone can suppress fertility, in contrary to its stimulatory effects on libido and erectile function. The patient's desire for fertility must be discussed in depth and established prior to initiating testosterone. The discussion must also include future thoughts on fertility. This will allow the physician to manage the timing of hypogonadism treatment, essentially balancing the alleviation of hypogonadal symptoms with the patient's desires for fertility. This could also open discussion about cryopreservation of sperm as an option for the patient to preserve fertility further down the line. Physicians should also educate men already on TRT. There has been an increase in TRT use among men aged 18 to 45 years and more than 20% of these men did not get a baseline testosterone level prior to initiation of TRT . Some of these men may not know about its effects on fertility and may not have discussed it with their prescribing physician. The study also showed that less than 2% of men on TRT obtained a baseline semen analysis . In addition to the routine serum total testosterone, LH, and hematocrit prior to starting TRT, every man of reproductive age should have a baseline semen analysis . The baseline semen analysis will identify men with a decreased baseline sperm count, as a reference value for future semen analyses after TRT use. If a patient currently desires fertility, TRT should be avoided or discontinued immediately. A semen analyses should be performed if the patient has discontinued TRT. Azoospermia or severe oligospermia may be seen in these patients, but most men should return to baseline semen analyses in 6 to 9 months after cessation of TRT [13, 14, 15]. A 2006 integrated analysis showed that 90% of patients were expected to return to baseline sperm concentration values 12 months after cessation of treatment and 100% after 24 months . Furthermore, evidence in a 2015 study of 49 men showed that 3,000 units of hCG subcutaneously every other day is effective in supporting the recovery of spermatogenesis without significant adverse effects . Regardless, the recovery of spermatogenesis is unclear for patients on chronic TRT. Physicians should take caution when treating hypogonadism in men who desire future fertility, but also acknowledge the reversible azoospermia seen in controlled studies . Adjunctive hCG and clomiphene can be used with TRT to maintain testicular size and intra-testicular testosterone concentrations . Referral to a reproductive urologist should be considered in a male with low testosterone interested in fertility. Go to: ALTERNATIVES TO TESTOSTERONE THERAPY IN PATIENTS WHO WISH TO PRESERVE FERTILITY Clomiphene and enclomiphene citrate provide an alternative treatment option for hypogonadal men that desire fertility as it does not affect sperm production. Clomiphene is a non-steroidal SERM. It selectively binds to estrogen receptors in the hypothalamus, antagonistically inhibiting negative feedback, increasing the levels of gonadotropins and stimulating the testicular production of testosterone in men . Even though enclomiphene citrate is not currently FDA approved, it has been shown to increase serum testosterone by raising the serum LH and FSH levels without negatively affecting semen parameters . Kaminetsky et al conducted a proof-of-principle, randomized, open-label, fixed dose, controlled, two-center phase IIB study that compared 25 mg of enclomiphene citrate daily to topical testosterone in hypogonadal men. The results showed higher testosterone levels and sperm counts in men receiving enclomiphene citrate. This corroborates with other studies which show improved semen parameters with the use of clomiphene citrate, with some studies describing it as a treatment for male infertility [56, 57]. hCG has also been used (often with clomiphene citrate, tamoxifen or anastrozole) because it stimulates the production of endogenous testosterone without compromising spermatogenesis. Although the exact mechanism of action and production site in males are not fully understood, it is known that hCG mimics the effects of LH and stimulates the Leydig cells in the testicles to produce endogenous testosterone [3, 58, 59, 60]. Additionally, studies have shown that low-dose hCG can be used with TRT to maintain high levels of intra-testicular testosterone while men are being treated for hypogonadism [52, 61]. While hCG is effective in increasing testosterone, various studies have shown that it is also efficacious in inducing spermatogenesis [62, 63]. It is even effective in helping with the recovery of spermatogenesis in men who were on TRT . Clinicians generally agree on using 2,000 IU of hCG administered subcutaneously 3 times per week as defined by the 2002 American Association of Clinical Endocrinologists guidelines . The ideal treatment for hypogonadism should provide physiological testosterone levels, exhibit appropriate circadian rhythms and be modulated by the HPG axis. No formulation of testosterone has been able to achieve this. However, there has been recent ongoing research on autograft Leydig stem cells. This may prove to be an effective treatment for hypogonadism in the future as it has the potential to fulfill all the criteria of an ideal TRT . Makala et al found that serum testosterone levels and Leydig cell populations were restored over time in mice who underwent ectopic autografting of testicular tissue. These results indicate that Leydig cells are able to regenerate de novo in the autografted adult testes, subsequently restoring serum testosterone level. In 2017, Zang et al also demonstrated how direct transplantation of stem Leydig cells are capable of self-renewal, extensive proliferation and differentiation into mature Leydig cells. These Leydig cells can then be regulated by the HPG axis and restore the neuroendocrine regulation of testicular function and its diurnal testosterone production system. Having said that, these strategies still face major hurdles with regards to its clinical translatability and the ethics of cell transplantation. Go to: CONCLUSIONS Testosterone therapy is a contraceptive, albeit a poor one. Men of reproductive age with low testosterone should be counseled on the adverse effects of TRT on fertility. Obtaining a semen analysis and possible cryopreservation of sperm should be offered if TRT is prescribed to men interested in preserving fertility. Options such as clomiphene citrate and hCG along with a referral to a reproductive urologist should be considered to naturally increase testosterone levels in those men with low testosterone who want to avoid TRT. Go to: Notes Disclosure:The authors have no potential conflicts of interest to disclose. Author Contribution: Conceptualization: Ramasamy R. Data curation: all authors. Formal analysis: all authors. Funding acquisition: none. Investigation: all authors. Methodology: all authors. Project administration: Ramasamy R. Resources: none. Software: none. Supervision: Ramasamy R. Validation: all authors. Visualization: all authors. Writing (original draft): all authors. Writing (review & editing): all authors. Go to: ACKNOWLEDGEMENTS The authors would like to thank the physicians and patients of Jackson Memorial Hospital, University of Miami Hospital, and the Miami VA for inspiring the creation of this review for clinician use. Go to: References | | | Baillargeon J, Urban RJ, Ottenbacher KJ, Pierson KS, Goodwin JS. Trends in androgen prescribing in the United States, 2001 to 2011. JAMA Intern Med 2013;173:1465–1466. PubMed CrossRef Westaby D, Ogle SJ, Paradinas FJ, Randell JB, Murray-Lyon IM. Liver damage from long-term methyltestosterone. Lancet 1977;2:262–263. PubMed Mulhall JP, Trost LW, Brannigan RE, Kurtz EG, Redmon JB, Chiles KA, et al. Evaluation and management of testosterone deficiency: AUA guideline. J Urol 2018;200:423–432. PubMed CrossRef Bhasin S, Brito JP, Cunningham GR, Hayes FJ, Hodis HN, Matsumoto AM, et al. Testosterone therapy in men with hypogonadism: an Endocrine Society clinical practice guideline. J Clin Endocrinol Metab 2018;103:1715–1744. PubMed CrossRef Basaria S. Male hypogonadism. Lancet 2014;383:1250–1263. PubMed CrossRef Walker WH, Cheng J. FSH and testosterone signaling in Sertoli cells. Reproduction 2005;130:15–28. PubMed CrossRef Walker WH. Non-classical actions of testosterone and spermatogenesis. Philos Trans R Soc Lond B Biol Sci 2010;365:1557–1569. PubMed CrossRef Smith LB, Walker WH. The regulation of spermatogenesis by androgens. Semin Cell Dev Biol 2014;30:2–13. PubMed CrossRef Shaha C, Tripathi R, Mishra DP. Male germ cell apoptosis: regulation and biology. Philos Trans R Soc Lond B Biol Sci 2010;365:1501–1515. PubMed CrossRef O'Donnell L, Nicholls PK, O'Bryan MK, McLachlan RI, Stanton PG. Spermiation: the process of sperm release. Spermatogenesis 2011;1:14–35. Wenker EP, Dupree JM, Langille GM, Kovac J, Ramasamy R, Lamb D, et al. The use of HCG-based combination therapy for recovery of spermatogenesis after testosterone use. J Sex Med 2015;12:1334–1337. PubMed CrossRef Cornud F, Amar E, Hamida K, Thiounn N, Hélénon O, Moreau JF. Imaging in male hypofertility and impotence. BJU Int 2000;86 Suppl 1:153–163. Amory JK, Bremner W. Endocrine regulation of testicular function in men: implications for contraceptive development. Mol Cell Endocrinol 2001;182:175–179. PubMed CrossRef Contraceptive efficacy of testosterone-induced azoospermia in normal men. World Health Organization Task Force on methods for the regulation of male fertility. Lancet 1990;336:955–959. CrossRef World Health Organization Task Force on Methods for the Regulation of Male Fertility. Contraceptive efficacy of testosterone-induced azoospermia and oligozoospermia in normal men. Fertil Steril 1996;65:821–829. CrossRef Kohn TP, Louis MR, Pickett SM, Lindgren MC, Kohn JR, Pastuszak AW, et al. Age and duration of testosterone therapy predict time to return of sperm count after human chorionic gonadotropin therapy. Fertil Steril 2017;107:351–357. PubMed CrossRef Eftekhar M, Mohammadian F, Yousefnejad F, Molaei B, Aflatoonian A. Comparison of conventional IVF versus ICSI in non-male factor, normoresponder patients. Iran J Reprod Med 2012;10:131–136. PubMed Komsky-Elbaz A, Raziel A, Friedler S, Strassburger D, Kasterstein E, Komarovsky D, et al. Conventional IVF versus ICSI in sibling oocytes from couples with endometriosis and normozoospermic semen. J Assist Reprod Genet 2013;30:251–257. PubMed CrossRef McBride JA, Coward RM. Recovery of spermatogenesis following testosterone replacement therapy or anabolic-androgenic steroid use. Asian J Androl 2016;18:373–380. PubMed CrossRef Kost K, Singh S, Vaughan B, Trussell J, Bankole A. Estimates of contraceptive failure from the 2002 National Survey of Family Growth. Contraception 2008;77:10–21. PubMed CrossRef Trussell J. Contraceptive failure in the United States. Contraception 2011;83:397–404. PubMed CrossRef Heinemann K, Saad F, Wiesemes M, White S, Heinemann L. Attitudes toward male fertility control: results of a multinational survey on four continents. Hum Reprod 2005;20:549–556. PubMed CrossRef Wang C, Swerdloff RS. Hormonal approaches to male contraception. Curr Opin Urol 2010;20:520–524. PubMed CrossRef Nieschlag E, Hoogen H, Bölk M, Schuster H, Wickings EJ. Clinical trial with testosterone undecanoate for male fertility control. Contraception 1978;18:607–614. PubMed CrossRef Gu Y, Liang X, Wu W, Liu M, Song S, Cheng L, et al. Multicenter contraceptive efficacy trial of injectable testosterone undecanoate in Chinese men. J Clin Endocrinol Metab 2009;94:1910–1915. PubMed CrossRef Liu PY, Swerdloff RS, Anawalt BD, Anderson RA, Bremner WJ, Elliesen J, et al. Determinants of the rate and extent of spermatogenic suppression during hormonal male contraception: an integrated analysis. J Clin Endocrinol Metab 2008;93:1774–1783. PubMed CrossRef Kogan P, Wald M. Male contraception: history and development. Urol Clin North Am 2014;41:145–161. PubMed CrossRef Larkin A. Male hormonal contraceptive trial ending early [Internet]. Arlington (VT): CONRAD; c2011 [cited 2018 Jan 11]. Available from: 29. 1. Thirumalai ACJ, Amory JK, Swerdloff RS, Surampudi V, Liu PY, Bremner WJ, et al. Pharmacokinetic and pharmacodynamic effects of 28 days of oral dimethandrolone undecanoate in healthy men: A prototype male pill. J Investig Med 2018;66:A165–A166. 30. 1. Srinivas-Shankar U, Sharma D. Testosterone treatment in elderly men. Adv Ther 2009;26:25–39. PubMed CrossRef Holyoak JD, Crawford ED, Meacham RB. Testosterone and the prostate: implications for the treatment of hypogonadal men. Curr Urol Rep 2008;9:500–505. PubMed CrossRef Stahlman J, Britto M, Fitzpatrick S, McWhirter C, Testino SA, Brennan JJ, et al. Serum testosterone levels in non-dosed females after secondary exposure to 1.62% testosterone gel: effects of clothing barrier on testosterone absorption. Curr Med Res Opin 2012;28:291–301. PubMed CrossRef Testosterone nasal gel (Natesto) for hypogonadism. Med Lett Drugs Ther 2015;57:73–74. PubMed Conners W, Morgentaler A, Guidry M, Westfield G, Bryson N, Goldstein I. Preservation of normal concentrations of pituitary gonadotropins despite achievement of normal serum testoterone levels in hypogonadal men treated with a 4.5. nasal testosterone gel. The Journal of Urology 2017;197 4 Suppl:e1204 CrossRef Rogol AD, Tkachenko N, Bryson N. NatestoTM, a novel testosterone nasal gel, normalizes androgen levels in hypogonadal men. Andrology 2016;4:46–54. PubMed CrossRef Ramasamy R. Natesto effects on testosterone, luteinizing hormone, follicle stimulating hormone and semen parameters [Internet]. Bethesda (MD): ClinicalTrials.gov; [cited 2018 Jun 1]. Available from: 37. 1. Layton JB, Meier CR, Sharpless JL, Stürmer T, Jick SS, Brookhart MA. Comparative safety of testosterone dosage forms. JAMA Intern Med 2015;175:1187–1196. PubMed CrossRef Di Luigi L, Sgrò P, Aversa A, Migliaccio S, Bianchini S, Botrè F, et al. Concerns about serum androgens monitoring during testosterone replacement treatments in hypogonadal male athletes: a pilot study. J Sex Med 2012;9:873–886. PubMed CrossRef Snyder PJ, Lawrence DA. Treatment of male hypogonadism with testosterone enanthate. J Clin Endocrinol Metab 1980;51:1335–1339. PubMed CrossRef Middleton T, Turner L, Fennell C, Savkovic S, Jayadev V, Conway AJ, et al. Complications of injectable testosterone undecanoate in routine clinical practice. Eur J Endocrinol 2015;172:511–517. PubMed CrossRef Gonzalo IT, Swerdloff RS, Nelson AL, Clevenger B, Garcia R, Berman N, et al. Levonorgestrel implants (Norplant II) for male contraception clinical trials: combination with transdermal and injectable testosterone. J Clin Endocrinol Metab 2002;87:3562–3572. PubMed CrossRef Ilani N, Roth MY, Amory JK, Swerdloff RS, Dart C, Page ST, et al. A new combination of testosterone and nestorone transdermal gels for male hormonal contraception. J Clin Endocrinol Metab 2012;97:3476–3486. PubMed CrossRef George M, Yulia T, Svetlana K. Influence of testosterone gel treatment on spermatogenesis in men with hypogonadism. Gynecol Endocrinol 2014;30 Suppl 1:22–24. CrossRef Bebb RA, Anawalt BD, Christensen RB, Paulsen CA, Bremner WJ, Matsumoto AM. Combined administration of levonorgestrel and testosterone induces more rapid and effective suppression of spermatogenesis than testosterone alone: a promising male contraceptive approach. J Clin Endocrinol Metab 1996;81:757–762. PubMed Kinniburgh D, Zhu H, Cheng L, Kicman AT, Baird DT, Anderson RA. Oral desogestrel with testosterone pellets induces consistent suppression of spermatogenesis to azoospermia in both Caucasian and Chinese men. Hum Reprod 2002;17:1490–1501. PubMed CrossRef Brady BM, Walton M, Hollow N, Kicman AT, Baird DT, Anderson RA. Depot testosterone with etonogestrel implants result in induction of azoospermia in all men for long-term contraception. Hum Reprod 2004;19:2658–2667. PubMed CrossRef Garrett C, Liu DY, McLachlan RI, Baker HW. Time course of changes in sperm morphometry and semen variables during testosterone-induced suppression of human spermatogenesis. Hum Reprod 2005;20:3091–3100. PubMed CrossRef Kinniburgh D, Anderson RA, Baird DT. Suppression of spermatogenesis with desogestrel and testosterone pellets is not enhanced by addition of finasteride. J Androl 2001;22:88–95. PubMed Rao PK, Boulet SL, Mehta A, Hotaling J, Eisenberg ML, Honig SC, et al. Trends in testosterone replacement therapy use from 2003 to 2013 among reproductive-age men in the United States. J Urol 2017;197:1121–1126. PubMed CrossRef Liu PY, Swerdloff RS, Christenson PD, Handelsman DJ, Wang C. Hormonal Male Contraception Summit Group. Rate, extent, and modifiers of spermatogenic recovery after hormonal male contraception: an integrated analysis. Lancet 2006;367:1412–1420. PubMed CrossRef Ramasamy R, Armstrong JM, Lipshultz LI. Preserving fertility in the hypogonadal patient: an update. Asian J Androl 2015;17:197–200. PubMed CrossRef Coviello AD, Matsumoto AM, Bremner WJ, Herbst KL, Amory JK, Anawalt BD, et al. Low-dose human chorionic gonadotropin maintains intratesticular testosterone in normal men with testosterone-induced gonadotropin suppression. J Clin Endocrinol Metab 2005;90:2595–2602. PubMed CrossRef Pelusi C, Giagulli VA, Baccini M, Fanelli F, Mezzullo M, Fazzini A, et al. Clomiphene citrate effect in obese men with low serum testosterone treated with metformin due to dysmetabolic disorders: a randomized, double-blind, placebocontrolled study. PLoS One 2017;12:e0183369 PubMed CrossRef Rodriguez KM, Pastuszak AW, Lipshultz LI. Enclomiphene citrate for the treatment of secondary male hypogonadism. Expert Opin Pharmacother 2016;17:1561–1567. PubMed CrossRef Kaminetsky J, Werner M, Fontenot G, Wiehle RD. Oral enclomiphene citrate stimulates the endogenous production of testosterone and sperm counts in men with low testosterone: comparison with testosterone gel. J Sex Med 2013;10:1628–1635. PubMed CrossRef Chua ME, Escusa KG, Luna S, Tapia LC, Dofitas B, Morales M. Revisiting oestrogen antagonists (clomiphene or tamoxifen) as medical empiric therapy for idiopathic male infertility: a meta-analysis. Andrology 2013;1:749–757. PubMed CrossRef Roth LW, Ryan AR, Meacham RB. Clomiphene citrate in the management of male infertility. Semin Reprod Med 2013;31:245–250. PubMed CrossRef Tamunopriye J, Abiola OO. Human chorionic gonadotrophin (HCG) stimulation test and testosterone response in children with micropenis. Pediatr Endocrinol Rev 2014;12:42–45. PubMed Choi J, Smitz J. Luteinizing hormone and human chorionic gonadotropin: origins of difference. Mol Cell Endocrinol 2014;383:203–213. PubMed CrossRef Berger P, Gruschwitz M, Spoettl G, Dirnhofer S, Madersbacher S, Gerth R, et al. Human chorionic gonadotropin (hCG) in the male reproductive tract. Mol Cell Endocrinol 2007;260-262:190–196. PubMed CrossRef Crosnoe-Shipley LE, Elkelany OO, Rahnema CD, Kim ED. Treatment of hypogonadotropic male hypogonadism: case-based scenarios. World J Nephrol 2015;4:245–253. PubMed CrossRef Rastrelli G, Corona G, Mannucci E, Maggi M. Factors affecting spermatogenesis upon gonadotropin-replacement therapy: a meta-analytic study. Andrology 2014;2:794–808. PubMed CrossRef Liu PY, Baker HW, Jayadev V, Zacharin M, Conway AJ, Handelsman DJ. Induction of spermatogenesis and fertility during gonadotropin treatment of gonadotropin-deficient infertile men: predictors of fertility outcome. J Clin Endocrinol Metab 2009;94:801–808. PubMed CrossRef Petak SM, Nankin HR, Spark RF, Swerdloff RS, Rodriguez-Rigau LJ. American Association of Clinical Endocrinologists. American Association of Clinical Endocrinologists Medical Guidelines for clinical practice for the evaluation and treatment of hypogonadism in adult male patients--2002 update. Endocr Pract 2002;8:440–456. PubMed Zang ZJ, Wang J, Chen Z, Zhang Y, Gao Y, Su Z, et al. Transplantation of CD51+ stem Leydig cells: a new strategy for the treatment of testosterone deficiency. Stem Cells 2017;35:1222–1232. PubMed CrossRef Makala H, Pothana L, Sonam S, Malla A, Goel S. Regeneration of Leydig cells in ectopically autografted adult mouse testes. Reproduction 2015;149:259–268. PubMed CrossRef Cite Article PDF Cited by Crossref 52 Google Scholar PubMed 30 Web of Science 46 Publication Types Review MeSH Terms Contraception Erectile Dysfunction Family Planning Services Fertility Follicle Stimulating Hormone Gonads Humans Hypogonadism Infertility Libido Luteinizing Hormone Male Testosterone Testosterone Since 2020/07/01 Metrics Page Views 2159 PDF Downloads 119 Links to PubMed PubMed Central Related citations in PubMed Show all... 1 / 1 Show all... 1 / 1 ORCID IDs Amir Shahreza Patel Joon Yau Leong Libert Ramos Ranjith Ramasamy Cited by Crossref Is Cited by the Following Articles in Citation successfully copied. Copy and paste a formatted citation from below or use one of the hyperlinks at the bottom to download a file for import into a bibliography manager. EndNote / ProCite / Reference Manager RIS Format Plain Text EndNote / ProCite / Reference Manager RIS Format Plain Text Permalink information copied to clipboard
6075
https://www.britannica.com/science/protist/Respiration-and-nutrition
SUBSCRIBE SUBSCRIBE Home History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture ProCon Money Games & Quizzes Videos On This Day One Good Fact Dictionary New Articles History & Society Lifestyles & Social Issues Philosophy & Religion Politics, Law & Government World History Science & Tech Health & Medicine Science Technology Biographies Browse Biographies Animals & Nature Birds, Reptiles & Other Vertebrates Bugs, Mollusks & Other Invertebrates Environment Fossils & Geologic Time Mammals Plants Geography & Travel Geography & Travel Arts & Culture Entertainment & Pop Culture Literature Sports & Recreation Visual Arts Image Galleries Podcasts Summaries Top Questions Britannica Kids Ask the Chatbot Games & Quizzes History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture ProCon Money Videos Introduction Defining the protists Features unique to protists Means of locomotion Cilia and flagella Pseudopodia Respiration and nutrition Reproduction and life cycles Ecology Fossil protists and eukaryotic evolution Paraphyletic nature References & Edit History Quick Facts & Related Topics Images & Videos For Students protist summary Quizzes All About Biology Quiz Related Questions Are algae toxic? Why are algae important? Respiration and nutrition in protist print Print Please select which sections you would like to print: verifiedCite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style Share Share to social media Facebook X URL Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. External Websites Nature - Protists are microbes too: a perspective American Association for the Advancement of Science - The problem of protists CUNY Pressbooks Network - General Biology OER Laboratory Manual - Protista UNESCO-EOLSS - The Protist Live Science - What are Protists? OpenStax - Concepts of Biology - Protists CORE - A molecular survey of protist diversity through the central Arctic Ocean BMC Biology - Two decades taken at speed: genomics, cell biology, ecology, and evolution of protists Earlham Institute - The biodiversity �outsiders� - the protist gang National Center for Biotechnology Information - PubMed Central - Protist taxonomy: an ecological perspective Biology LibreTexts - Protists Cell Press - Trends in Genetics - Protist genomics: key to understanding eukaryotic evolution Britannica Websites Articles from Britannica Encyclopedias for elementary and high school students. protist - Children's Encyclopedia (Ages 8-11) protist - Student Encyclopedia (Ages 11 and up) Also known as: Protista, Protoctista, unicellular organism Written by Written by The Editors of Encyclopaedia Britannica Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree.... The Editors of Encyclopaedia Britannica Last Updated: •Article History At the cellular level, the metabolic pathways known for protists are essentially no different from those found among cells and tissues of other eukaryotes. Thus, the plastids of algal protists function like the chloroplasts of plants with respect to photosynthesis, and, when present, the mitochondria function as the site where molecules are broken down to release chemical energy, carbon dioxide, and water. The basic difference between the unicellular protists and the tissue- and organ-dependent cells of other eukaryotes lies in the fact that the former are simultaneously cells and complete organisms. Such microorganisms, then, must carry out the life-sustaining functions that are generally served by organ systems within the complex multicellular or multitissued bodies of the other eukaryotes. Many such functions in the protists are dependent on relatively elaborate architectural adaptations in the cell. Phagotrophic feeding, for example, requires more complicated processes at the protist’s cellular level, where no combination of tissues and cells is available to carry out the ingestion, digestion, and egestion of particulate food matter. On the other hand, obtaining oxygen in the case of free-living, free-swimming protozoan protists is simpler than for multicellular eukaryotes because the process requires only the direct diffusion of oxygen from the surrounding medium. Although most protists require oxygen (obligate aerobes), there are some that may or must rely on anaerobic metabolism—for example, parasitic forms inhabiting sites without free oxygen and some bottom-dwelling (benthic) ciliates that live in the sulfide zone of certain marine and freshwater sediments. Mitochondria typically are not found in the cytoplasm of these anaerobes; rather, microbodies called hydrogenosomes or specialized symbiotic bacteria act as respiratory organelles. The major modes of nutrition among protists are autotrophy (involving plastids, photosynthesis, and the organism’s manufacture of its own nutrients from the milieu) and heterotrophy (the taking in of nutrients). Obligate autotrophy, which requires only a few inorganic materials and light energy for survival and growth, is characteristic of algal protists (e.g., Chlamydomonas). Heterotrophy may occur as one of at least two types: phagotrophy, which is essentially the engulfment of particulate food, and osmotrophy, the taking in of dissolved nutrients from the medium, often by the method of pinocytosis. Phagotrophic heterotrophy is seen in many ciliates that seem to require live prey as organic sources of energy, carbon, nitrogen, vitamins, and growth factors. The food of free-living phagotrophic protists ranges from other protists to bacteria to plant and animal material, living or dead. Scavengers are numerous, especially among the ciliated protozoans; indeed, species of some groups prefer moribund prey. Organisms that can utilize either or both autotrophy and heterotrophy are said to exhibit mixotrophy. Many dinoflagellates, for example, exhibit mixotrophy. Feeding mechanisms and their use are diverse among protists. They include the capture of living prey by the use of encircling pseudopodial extensions (in certain amoeboids), the trapping of particles of food in water currents by filters formed of specialized compound buccal organelles (in ciliates), and the simple diffusion of dissolved organic material through the cell membrane, as well as the sucking out of the cytoplasm of certain host cells (as in many parasitic protists). In the case of many symbiotic protists, methods for survival, such as the invasion of the host and transfer to fresh hosts, have developed through long associations and often the coevolution of both partners.
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https://onlinelibrary.wiley.com/doi/pdf/10.1111/1556-4029.13146
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https://www.courts.oregon.gov/publications/Documents/UpdatedStyleManual2002.pdf
S tyle Manual (Updated 2023) OREGON APPELLATE COURTS Style Manual (Updated 2023) Preface The Oregon Appellate Courts have adopted this style manual as a guideline for conventions used in format, citation, quotation, and style when writing opinions. It is not all-inclusive nor an attempt to dictate writing style. See ORAP 5.20(45) (referring to Style Manual as guide to conventions in style and citation). Sincere appreciation to all who added their time and talent to this project. For form and style questions not covered by this manual, please contact the OJD Publications Program (publications@ojd.state.or.us). 2 TABLE OF CONTENTS FORMATTING In General ....................................................................................................................................... 5 I. Title Page A. Date of Opinion . ................................................................................................. 6 B. Identifying Caption of the Appellate Court Issuing the Opinion . ...................... 6 C. Names and Roles of the Parties to the Case....................................................... 6 D. Identification Numbers . ...................................................................................... 6 E. En Banc. .............................................................................................................. 7 F. Court / Agency of Origination ............................................................................ 7 G. Trial Court Judge . ............................................................................................... 7 H. Argued and Submitted Date ............................................................................... 7 I. Names of Counsel............................................................................................... 7 J. Panel of Judges / Justices..................................................................................... 7 K. Opinion Author(s) .............................................................................................. 7 L. Disposition of Case ............................................................................................ 8 M. Designation of Prevailing Party and Award of Costs . ........................................ 8 II. Body of Opinion In General . ........................................................................................................................... 9 A. General Format ................................................................................................ 10 B. Structural Tools................................................................................................. 11 C. Writing Tools . ................................................................................................... 14 CITATION In General ..................................................................................................................................... 15 I. Organization and Arrangement A. The Bluebook . ................................................................................................... 16 B. Consistency of Citations .................................................................................. 16 C. Case Names ...................................................................................................... 16 D. Spaces and Abbreviations of Citations . ............................................................ 17 E. String Citations................................................................................................. 18 F. Signals . ............................................................................................................. 18 G. Parenthetical Information . ................................................................................ 18 II. Case Law A. Oregon–Full Citations . ..................................................................................... 20 B. Oregon–Short Citations and Other Issues ........................................................ 26 C. Federal Jurisdictions . ........................................................................................ 29 D. States Other Than Oregon . ............................................................................... 31 E. Online Sources . ................................................................................................. 32 III. Constitutional, Statutory, and Other Related Citations A. Oregon Citations .............................................................................................. 33 B. Federal Citations . ............................................................................................. 45 3 IV. Periodical Articles, Books, Treatises, Restatements, Etc. In General . ......................................................................................................................... 48 A. Periodical Articles ............................................................................................ 48 B. Books and Treatises . ......................................................................................... 49 C. Restatements . .................................................................................................... 51 D. Others ............................................................................................................... 51 QUOTATION In General ..................................................................................................................................... 52 I. Citations, Parenthetical Phrases, and Footnotes A. Placement ......................................................................................................... 53 B. Use of Parenthetical Phrases With Quotations Within Text ............................. 55 II. Use of Uppercase, Brackets, and Ellipsis Within Quotations A. Use of Uppercase.............................................................................................. 59 B. Use of Brackets................................................................................................. 59 C. Use of Ellipsis................................................................................................... 62 STYLE GUIDE In General ..................................................................................................................................... 66 Other Resources ............................................................................................................................ 66 I. Spelling, Font, and Treatment of Words A. Use of Italics and Roman Typeface ................................................................. 67 B. Use of Uppercase and Lowercase .................................................................... 68 C. Numbers and Dates . ......................................................................................... 72 D. Acronyms / Initialisms ...................................................................................... 75 E. Titles and Offices ............................................................................................. 76 F. Abbreviations ................................................................................................... 76 II. Punctuation A. Apostrophes . ..................................................................................................... 77 B. Colons .............................................................................................................. 78 C. Commas . ........................................................................................................... 79 D. Dashes .............................................................................................................. 84 E. Hyphens ........................................................................................................... 85 F. Punctuating Lists . ............................................................................................. 87 G. Punctuating Parenthetical Elements . ................................................................ 88 H. Semicolons ....................................................................................................... 88 III. Word Usage and Conventions A. Word Pairs . ....................................................................................................... 90 B. Word Functions............................................................................................... 101 C. Word Usage..................................................................................................... 102 D. Variant Spellings............................................................................................. 104 IV. Common Grammatical and Style Problems A. Collective Nouns . ........................................................................................... 105 B. Parallel Construction ...................................................................................... 105 4 C. Passive Voice ................................................................................................. 105 D. Verbs .............................................................................................................. 106 E. Active Voice . .................................................................................................. 107 F. Gender-Neutral Wording ............................................................................... 107 G. Informal or Technical Terminology . .............................................................. 107 GLOSSARY ............................................................................................................................. 108 OPINION OVERVIEWS I. Supreme Court .......................................................................................................... 113 II. Court of Appeals . ....................................................................................................... 114 III. Tax Court–Regular Division ..................................................................................... 115 IV. Tax Court–Magistrate Division ................................................................................ 116 APPENDIX Standard Proofreader’s Marks ........................................................................................ 117 INDEX ...................................................................................................................................... 118 Go to TABLE OF CONTENTS 5 Go to INDEX FORMATTING In General Slip opinions are double spaced; indented quoted material is single spaced and formatted in the same style as the original material; footnotes are placed at the bottom of the page on which they are referenced. Line numbers are set out along the left hand margin, except the footnote section. The courts use Times New Roman, 13-point font. Slip opinions consist of a title page, which includes a designation of prevailing party and award of costs portion, followed by the body of the opinion. Cases Affirmed Without Opinion (AWOP) consist of a title page only. Per Curiam Opinions may consist of a title page only, but have the same weight of authority as a signed opinion. Therefore, it is not necessary to indicate parenthetically whether an opinion cited is Per Curiam. Cases that are Affirmed By An Equally Divided Court are resolved by opinion—as opposed to an order—and can consist of a title page only unless a concurring or dissenting opinion is written. Listed below are the essential elements found on a standard title page of an appellate court opinion, followed by the substantive components generally contained within the body of an opinion. Please note that these are models only and that actual title pages and opinions may vary due to the particular requirements of an individual case. Go to TABLE OF CONTENTS 6 Go to INDEX I. Title Page A. Date of Opinion The date that the opinion issues is located in the upper right hand corner of the page preceded by FILED:. B. Identifying Caption of the Appellate Court Issuing the Opinion The identification of the appellate court is centered on the page in uppercase letters, e.g., IN THE SUPREME COURT OF THE STATE OF OREGON IN THE COURT OF APPEALS OF THE STATE OF OREGON IN THE OREGON TAX COURT REGULAR DIVISION IN THE OREGON TAX COURT MAGISTRATE DIVISION For nonprecedential memorandum opinions issued by the Court of Appeals, the following recitation will precede the caption: This is a nonprecedential memorandum opinion pursuant to ORAP 10.30 and may not be cited except as provided in ORAP 10.30(1). C. Names and Roles of the Parties to the Case Parties are generally listed in the order in which they appeared in the lower court or tribunal, but using their appellate court designations: appellant, respondent, petitioner on review, respondent on review, etc. That information is generally taken from the originating document filed for a case, e.g., the Notice of Appeal or Petition for Judicial Review. In criminal cases, the STATE OF OREGON is the first party listed, followed by the full name of the defendant. D. Identification Numbers Each appellate case is assigned a number when filed, which is centered on the title page below the names of the parties to the case and preceded by any identifying number(s) from the court or agency in which the case originated. If cases have been consolidated on appeal or review, then both appellate case numbers are listed. A Supreme Court case number begins with an “S,” a Court of Appeals case number begins with an “A,” a Tax Court-Regular Division case number begins with “TC,” and a Tax Court-Magistrate Division case number begins with “TC-MD.” Go to TABLE OF CONTENTS 7 Go to INDEX E. En Banc If a case is decided by the full court, then that will be noted in the first line starting at the left-hand margin. No period follows the en banc designation. The initial letter in each word is in uppercase on the title page (e.g., En Banc), but when used within the text of an opinion, the term is in lowercase letters. F. Court / Agency of Origination Identifies where / how the case originated. G. Trial Court Judge Identifies judge(s) who signed the appealable judgment(s) or order(s) that are the subject of the appeal. The Supreme Court footnotes that information on the title page with an asterisk. H. Argued and Submitted Date Identifies when the case was submitted and whether it was argued. Some cases are submitted on the record only. I. Names of Counsel The attorney(s) for all parties to a case are named. If a party appears for himself or herself (sometimes referred to as pro se), then that is noted. A person or entity appearing as amicus curiae is also identified here, along with the counsel of record. Counsel names are listed as they appear on the signature line of the briefs filed in the case. Regarding Department of Justice (DOJ) attorneys, use official titles (e.g., Attorney General, Deputy Attorney General, Solicitor General, Deputy Solicitor General, Assistant Attorney General), but do not uses any internal DOJ classification or position description, such as “Attorney-in-Charge, Post-Conviction Section” or “Senior Assistant Attorney General,” etc. Regarding Office of Public Defense Services (OPDS) attorneys, use official titles (e.g., Chief Defender, Criminal Appellate Section; Chief Defender, Juvenile Appellate Section; Deputy Public Defender), but do not use any internal OPDS classification or position description, such as “Chief Deputy,” “Senior Deputy Public Defender,” etc. J. Panel of Judges / Justices The panel of judges (in the Court of Appeals and also denoting the Presiding Judge) or the names of the Supreme Court justices deciding the case (if not heard en banc) are listed. When denoting a judge who has retired or resigned, a judge’s designation will reflect the status of the judge at the time the opinion issues. Go to TABLE OF CONTENTS 8 Go to INDEX K. Opinion Author(s) 1. Signed Opinions The author’s name is listed in uppercase. When there is a concurring or dissenting opinion, the name of its author is listed on the title page after the decision line (e.g., Smith, J., dissenting.). Each opinion is arranged in this order: majority; concurring (the authoring justice / judge wishes to write separately, but agrees with both the result and rationale of the majority opinion); specially concurring (the authoring justice / judge wishes to write separately and agrees with the result, but not the rationale, of the majority opinion); concurring in part, dissenting in part; and dissenting (the authoring justice / judge disagrees with the result of the majority opinion). If two or more justices / judges file a concurring or dissenting opinion, then the more senior justice’s / judge’s opinion goes first. See, e.g., State v. Dameron, 316 Or 448, 853 P2d 1285 (1993) (for order of opinions). When there is a nonparticipating justice in the Supreme Court, that justice’s name is footnoted on the title page of the opinion. 2. Per Curiam Opinions An opinion that summarily disposes of the case may be designated as Per Curiam. The Per Curiam designation is also used for all lawyer discipline, Bar admission, and judicial fitness matters before the Supreme Court. 3. Affirmed By An Equally Divided Court In rare circumstances, the court may be split evenly regarding the disposition of a case, in which event the case is deemed to be affirmed, although no signed majority opinion is issued. L. Disposition of Case A brief statement of the court’s holding. This holding is repeated as the last line (set out as a separate paragraph) in the body of the opinion and is referred to as the “tag line.” The tag line serves as the court’s formal disposition of the case and also serves as further instruction to the lower tribunal(s). M. Designation of Prevailing Party and Award of Costs The last part of a Supreme Court or Court of Appeals title page denotes the prevailing party and whether, and to whom, the court allows costs. Go to TABLE OF CONTENTS 9 Go to INDEX II. Body of Opinion In General Both custom and tradition influence the content and format of appellate opinions, as does the individual writing style of each judge. It is not the intent here to dictate that style, but to list standard conventions often used to organize opinions. In describing the elements of a typical opinion, our purpose is to give the reader a better understanding of appellate opinions. Due to the scrutiny to which a published appellate opinion is subjected, the court’s discussion, analysis, and holding need to be expressed clearly, succinctly, and carefully. An opinion is crafted to inform the reader of the legal issues presented, discuss the facts, explain the court’s analysis, and conclude with the court’s final disposition of the case. Because overly long sentences and paragraphs tend to appear formidable, all attempts are made to present the appellate opinion as straightforwardly as possible. Because appellate opinions are formal documents, contractions are used only when quoting from a source in which they appear. Remember, these are general guidelines only, which means that there always will be exceptions. The requirements of an individual case may demand deviation from the norms here listed. Go to TABLE OF CONTENTS 10 Go to INDEX A. General Format 1. Initial Paragraph–Introduction to the Case The opinion begins by restating the name of the authoring judge or justice or by using the Per Curiam designation in uppercase, set out separately as the first line. The introductory paragraph sets out the general nature of the case, which includes the results in any lower tribunal(s), the main issues on appeal / review, and the final disposition of the court. That provides the framework for the details that follow. To the extent possible, the opinion refers to parties by their lower tribunal(s) designations, e.g., plaintiff, defendant, claimant, etc. Exceptions include the following: (1) domestic relations cases, in which the parties are referred to as “husband” and “wife”; (2) civil commitment proceedings, in which the person for whom commitment is sought is referred to by his or her position on appeal (appellant, respondent); (3) termination of parental rights proceedings, in which the parents are referred to as “mother” and “father,” and the children are referred to as “child” or “children” or sometimes by initials. The proper names of victims are not used. See page 103 for further discussion. When citing a concurring or dissenting opinion, the author’s last name is used. When a dissenting or concurring opinion refers to the majority opinion, the majority opinion is referred to as such, not by its author. If a party has a long name, a shortened version is usually developed for use in subsequent references, e.g., First Security Bank of the Northwest may be referred to simply as “First Security” or “bank.” 2. Statement of Facts The pertinent facts of a case are set out in a concise and objective manner. Those facts can be organized in patterns, e.g., chronologically or geographically, or by issue, witness, or actor. If an issue is complex, the facts may be set out in general here and then in more detail when discussing the issue to which they relate. 3. Discussion of Issues The opinion then addresses the dispositive issue(s) in a manner appropriate to the circumstances of the case. One approach is to state the parties’ positions, either in the order in which they were raised below, discussed in the briefs, or dictated by circumstance; respond to those arguments; and then provide an Go to TABLE OF CONTENTS 11 Go to INDEX explanation for the result reached by stating the authorities relied on. Simply stated, the discussion states the issue, how it is resolved on the facts of the case with citation to relevant authority, and the effect of the resolution. 4. Disposition of Case The final paragraph states the result and gives instructions when necessary. The conclusion is followed by the tag line, a separate paragraph that sets out the court’s final ruling and serves as further instruction to the lower tribunal(s). B. Structural Tools If a case is complex, then the authoring judge may decide to divide the opinion into designated parts and label them to identify for the reader the discussion of the case. That is accomplished by using the methods discussed below. 1. Paragraph or Section Headings a. Format (1) Headings An author may decide to use principal divisions within an opinion. When used, headings for those divisions are centered and set out in uppercase letters. Roman numerals are not used if subheads are omitted. (2) Outline Method If dividing and labeling an opinion into sections and subsections, then the standard outline format is used as set out below. Standard outline rules apply, e.g., if there is a heading designated I., then there must be a II., if there is a subheading A., then there must be a B., etc. Initial caps are used in the first level of subheadings (on all words except articles, prepositions, and conjunctions) unless the subheading reads as a complete sentence. Each new level of subheading starts at a new level of indentation, with an indent following the number or letter. The levels of outlining are referred to as follows: Headings (indicated by Roman numerals) Subheadings (indicated by uppercase letters) Paragraphs (indicated by numerals) Subparagraphs (indicated by lower case letters) Subsubparagraphs (indicated by numbers within parentheses) Go to TABLE OF CONTENTS 12 Go to INDEX I. HEADING (centered, uppercase, no italic or boldface font, if there are no subheadings, then do not number headings) A. Subheading Example with Initial Uppercase Set subheading flush left, beginning with nonitalicized alpha character “A.” Indent after alpha character, followed by italicized subheading. If subheading is not a sentence, then use initial uppercase, but if subheading is a sentence, then use a period and no initial uppercase after the first word. 1. Paragraph heading is indented and italicized (do not use initial uppercase after the first word; use a period only if a sentence). 2. If there is a paragraph 1, then there must be a paragraph 2. a. Subparagraph heading is indented twice, no italics. b. If there is a subparagraph a, then there must be a subparagraph b. (1) Subsubparagraph heading is indented yet again, no italics. (2) If there is a subsubparagraph (1), then there must be a subsubparagraph (2). B. This is an example of a subheading that does not require initial uppercase but does use a period, because it is a sentence. If there is a subheading A, then there must be a subheading B. Paragraphs that follow any of these headings are formatted like this one, flush left with a first-line indent. NOTE: With regard to case names in subheadings, if the opinion subheadings only involve the first scenario (see below), italicize the case name, which reads better to the reader. But, if the opinion subheadings have both scenarios (or only the second one), then format as follows, for internal consistency: A. State v. Baker B. State v. Jones and its Progeny b. Bulleted or Numbered Lists It may be more clear to organize certain text, e.g., events, dates, testimony, etc., using a bulleted or numbered list. The bulleted list generally is indented. The use of bullets can help to differentiate items in a list that need no particular order, e.g., • Car of little or no value • Boat valued at $10,000 • Personal jewelry that is valued at more than $5,000, but less than $10,000, and similar items. Go to TABLE OF CONTENTS 13 Go to INDEX Numbered lists, with each numeral appearing inside a set of paren-theses, help to organize and display information to show relationship, e.g., Defendant argues as follows: (1) the trial court erred; (2) the error was not harmless; and (3) his conviction should be reversed. See pages 87 to 88 for further discussion regarding the proper structure for numbered lists. 2. Quotations When construing a statute or administrative rule, for example, the author generally quotes the pertinent text. The purpose of quotation is to provide the reader with the information necessary to understand the court’s discussion of the issues and the law governing its analysis. The proper format for quoted material is discussed in the Citation and Quotation sections. 3. Footnotes a. In General Footnotes document sources of information or make ancillary references. Substantive information is best addressed within the body of the opinion. Footnote text begins on the same line as the superscripted footnote number, except when the footnote begins with a block quotation. b. Citations in Footnotes When citing a case in a footnote and the case has not previously been cited (in text or footnote), use the full case citation. If the case has already been cited (in text or footnote), then use a short citation. (Note that, if a case is cited for the first time in a footnote, then the first subsequent citation to that case in the text must also be a full citation. See pages 20 and 25.) c. Referencing to and Setting Out Footnoted Text Within Body of Opinion Footnote numbers, where applicable, are placed after periods, commas, colons, semicolons, and quotation marks. Footnote numbers also are placed after a closing parenthesis, unless the footnote refers to material inside the parentheses. Footnote numbers inserted within quoted material are set out using superscripted brackets. 4. Maps / Pictures / Appendices It may be necessary to include graphic information to convey a more clear understanding of the issue(s). In that event, a photograph, map, or chart is either appended to the opinion or inserted within the text where applicable. When included, an appendix is usually first described in narrative form. The appendix typically begins on a separate page at the end of the opinion with the heading “Appendix.” Go to TABLE OF CONTENTS 14 Go to INDEX C. Writing Tools 1. Fonts a. Italics and Underscoring Italic typeface is used within opinions to denote case names, to set out introductory signals, to indicate and less common foreign terminology, and to supply emphasis. See examples listed on page 67. Excessive use of italics for emphasis is discouraged. Underscoring is used only in quotations when the original source used underscoring for emphasis or headings and the like. b. Boldface and Uppercase It is best to avoid using a variety of styles and fonts within an opinion. Use of boldface or all uppercase letters in text is discouraged as a distraction to the reader. Italic type generally is sufficient to show emphasis. Avoid using UPPERCASE BOLDFACE ITALIC, as it is difficult to read. 2. Make Smooth Transitions When turning to a new issue or argument within an opinion, use introductory sentences or paragraphs to indicate transition between discussions. Use signal words to connect thoughts back to a preceding point or ahead to the next one, e.g., further, however, consequently, etc. Explore one idea per paragraph, relating each sentence to that central idea. Go to TABLE OF CONTENTS 15 Go to INDEX CITATION In General In legal citation, it is paramount to cite authorities in a clear and concise manner, thereby enabling the reader to locate those sources. Within this Style Manual, we have endeavored to include citation examples of sources often cited within the framework of appellate opinions. When citing an authority not discussed here, follow the format of like material. Citations should be made to official print sources whenever possible. If there is no official printed version or if it is difficult to obtain, or the publishing entity has designated an electronic source as the official version, then citation to that source should follow the format (as closely as possible) as described within this manual. Go to TABLE OF CONTENTS 16 Go to INDEX I. Organization and Arrangement A. The Bluebook The appellate courts generally follow the citation practices set out in the most current version of The Harvard Law Review Association’s The Bluebook, A Uniform System of Citation, except as noted in this Style Manual. The Bluebook is used as the default source for citation questions not addressed here. B. Consistency of Citations If you cannot find a specific rule that addresses your particular citation situation, then cite the authority in a clear, sensible manner that will convey the information needed to find the cited authority. Consistency within a particular document is important to avoid distracting and confusing the reader. In subsequent case history, if all the decisions in one case take place during the same year, then place the date (year) once at the end of the entire citation. However, if the decisions span more than one year, then place all years in the appropriate places, as shown in the examples. See, e.g., pages 23 to 24. Standard references used for prior or subsequent case history in citations are as follows: Acceptable Abbreviations: Terms Not Abbreviated: adh’d to on recons (adhered) allowed (not all) aff’d (affirmed) appeal dismissed cert (certiorari) as improvidently allowed cert den (certiorari denied) compiled as a note after recons (reconsideration; not recon) decision by order reh’g (rehearing) dismissed (not dism) rem’d (remanded) modified (not mod) rev’d (reversed) overruled on other grounds rev (revised) rev den (review denied) vac’d (vacated) writ den (writ of mandamus denied) C. Case Names When citing Oregon appellate cases, DO NOT use the title page, a regional reporter, Premise, Westlaw, or LEXIS as a source for the official case name. Use the case name exactly as published in the official state reporter, located at the top of either the Go to TABLE OF CONTENTS 17 Go to INDEX odd- or even-numbered pages. In the Oregon Reports and the Oregon Appellate Courts Advance Sheets, the case name appears on the even-numbered pages. Exception for cases concerning the federal Violence Against Women Act (VAWA): Per Joint CJO 23-012/23-01, effective April 1, 2023, when citing to any VAWA case, the author shall modify the case names to include only the initials of the protected party, regardless of how the case names were previously published. As stated in Joint CJO 23-012/23-01, cases are designated as “VAWA cases” in the Appellate Case Management System and include civil stalking, Family Abuse Pre-vention Act, Elderly Persons with Disabilities Abuse Prevention Act, Registration of Foreign Restraining Orders, Sexual Abuse Protection Orders, Extreme Risk Protection Orders, and Punitive Contempt Cases, most of which involve the vio-lation of restraining orders, as well as other individual cases that court staff have determined should be so designated under the provisions of VAWA. In the Oregon Reports and the Oregon Appellate Courts Advance Sheets, the case name appears on the even-numbered pages. If citing a case from a jurisdiction that does not have its own separate official reporter or if you do not have access to the official reports, then use the case name as used in the regional reporter or follow the naming conventions as set out in The Bluebook when using online services. Cases that are Affirmed Without Opinion are listed in tables using their full case titles. In the event that such a case needs to be cited using a shortened case name, please contact the Publications Program for the correct case name if it is not easily discernible. In older bound volumes of the Oregon Reports, case names are shown with small uppercase letters. Replace those with ordinary Roman type for citation purposes. Also in older volumes, running heads are sometimes shown using et al., et ux., or et vir. to indicate additional parties. For purposes of consistency when citing such cases, those abbreviations should be included, as well as their punctuation. D. Spaces and Abbreviations of Citations Generally, spaces in citations are used to separate longer abbreviations, e.g., S Ct, F Supp, L Ed 2d, Or L Rev, Tex App, etc. A space is not necessary between adjacent single uppercase letters or numerals and ordinals that are treated as single uppercase letters, e.g., P3d, NE2d, NYS2d, but insert a space before any abbreviation containing two or more letters, e.g., So 2d. Periods are not used after abbreviations, except when quoting material in which they are included. Go to TABLE OF CONTENTS 18 Go to INDEX E. String Citations When using string citations, follow The Bluebook format for the order of authorities, i.e., (1) cases decided by the same court, or by all federal circuit courts of appeals or federal district courts, are arranged in reverse chronological order; and (2) different courts generally are set out by rank. Use a semicolon, not a comma or the word “and,” to separate citations within a string. F. Signals Introductory signals are used to indicate the level of support to be found in a citation, suggest comparison, indicate contradiction, or indicate background material. When using a signal, it is important to recognize the intent of the signal. Some signals indicate support: e.g., see, see also, accord, cf.; others comparison: compare (using the construction compare and or compare with); or contradiction: contra, but see, but cf.; and background: see generally. Like signals are grouped into a single citation sentence, separated by a semicolon. The courts generally follow The Bluebook regarding the appropriate use of signals, i.e., the meaning of each signal, order of signals, order of authorities within signals, and parentheticals with signals. When using signals, the courts typically include a parenthetical explanation briefly describing the relevance of the authority cited, e.g., Cf. State v. Brown, 300 Or 125, 130, 860 P2d 498 (1985) (hearsay inadmissible at trial). See, e.g., OEC 401 (regarding relevancy of evidence). School Dist. 1, Mult. Co. v. Bingham et al., 204 Or 601, 611, 283 P2d 670, modified on reh’g, 204 Or 606, 284 P2d 779 (1955) (when interpreting Oregon Constitution, court must assume that every word, clause, and sentence therein inserted for some useful purpose). NOTE: No signal is needed when the cited authority directly states the proposition; in some situations, it still may be advisable to include a parenthetical explanation, even if no signal is used. Also, note that the internal comma within a signal is italicized. G. Parenthetical Information Parenthetical explanations are added as needed to describe the relevance of an authority that is cited in the text. Although not mandatory, often that information is Go to TABLE OF CONTENTS 19 Go to INDEX helpful for clarification. A parenthetical statement can consist of quoted material or a brief statement. When including a parenthetical explanation in a case with citations to subsequent history, place the explanation following all the subsequent case history. Quoted statements that read as a complete sentence should include an initial uppercase letter and period (with brackets and / or an ellipsis, if appropriate), but other material that reads as a sentence does not receive an initial uppercase letter or a period, e.g., Old v. Navy, 555 Or App 444, 447, 222 P3d 888 (2015) (“[An] award of ‘reasonable’ attorney fees does not preclude the use of a multiplier or other fee enhancement . Such an enhancement may be applied at the beginning of the calculation process.”). Aber v. Crombie, 123 Or 234, 236, 456 P3d 678 (2015) (the constitution does not pro-vide that shirts have to be “button down”). See, e.g., People v. Vasquez, 148 P3d 326, 330 (Colo Ct App 2006), rev den, No 06SC556, 2006 WL 3404625 (Colo, Nov 27, 2006) (“Because the reasonable person standard requires a ‘defendant [to] appraise the situation as would a reasonable sober [person],’ evidence of voluntary intoxication is irrelevant to the defendant’s affirmative defense of self-defense.” (Quoting LaFave, 2 Substantive Criminal Law § 9.5(d) at 51 (emphasis in LaFave; brackets in Vasquez).)). More examples of parenthetical phrases can be found on pages 57 to 58. Go to TABLE OF CONTENTS 20 Go to INDEX II. Case Law A. Oregon–Full Citations When citing a case for the first time in the text of an opinion (majority, concurring, or dissenting) use the full case citation. (If the case has previously been cited in full in a footnote, then the first textual citation to the case should still be a full citation.) See page 13. The basic citation format for a full case citation includes these elements, in this order: name of the case (using the official running head, but redacting any protected party’s name to initials in VAWA cases, see example on page 25); volume, abbreviated name of the official reporter, and beginning page number of the case; parallel citation to regional reporter; parenthetical indicating year case was decided; subsequent history; and any other pertinent parenthetical information. 1. Supreme Court PGE v. Bureau of Labor and Industries, 317 Or 606, 859 P2d 1143 (1993). NOTE: Oregon Supreme Court cases issued before 1888 have no parallel citations. On rehearing Older Oregon cases with separate “rehearing” decisions have one Oregon citation (because the rehearing decision was published immediately after the initial decision), but two West citations. For those cases, the year shown in the published case name at the top of the odd- or even-numbered pages is the year that a decision on rehearing was issued, not the year that the case initially was decided (if different). Citation to such cases depends on whether you are citing to the original decision or the decision on rehearing. If citing the initial decision, include the date of that decision, then include the rehearing information as subsequent history: State v. Laundy, 103 Or 443, 455, 204 P 958 (1921), reh’g den, 103 Or 503, 206 P 290 (1922). NOTE: Opinion on the merits begins on page 443, but opinion on rehearing starts on page 503. Go to TABLE OF CONTENTS 21 Go to INDEX If citing the rehearing decision, then cite both decisions, with the rehearing year and a “rehearing” indicator: State v. Laundy, 103 Or 443, 503, 206 P 290 (1922) (on rehearing). On reconsideration (specify disposition): Goodyear Tire & Rubber Co. v. Tualatin Tire & Auto, 322 Or 406, 908 P2d 300 (1995), modified on recons, 325 Or 46, 932 P2d 1141 (1997). NOTE: If citing only the case on reconsideration, it is not necessary to include the earlier case citation, e.g., Goodyear Tire & Rubber Co. v. Tualatin Tire & Auto, 325 Or 46, 932 P2d 1141 (1997). Overruled by subsequent Oregon case: Rose v. Port of Portland, 82 Or 541, 552, 162 P 498 (1917), overruled in part on other grounds by State ex rel Heinig v. Milwaukie et al, 231 Or 473, 479, 373 P2d 680 (1962). Rejected by subsequent Oregon case: State v. Smith, 295 Or 200, 625 P2d 20 (1981), rejected in part by State v. Jones, 321 Or 100, 805 P2d 150 (1991). (Other alternatives, such as, “abrogated by” and “questioned by,” can be used as appropriate.) Certiorari (or writ) denied by United States Supreme Court: Dept. of Trans. v. Lundberg, 312 Or 568, 825 P2d 641, cert den, 506 US 975 (1992). Whitman v. United States, 904 F Supp 2d 363 (SDNY 2012), writ den, 574 US , 135 S Ct 352 (2014). NOTE: (1) Always include cert den, if applicable. (2) If certiorari is dismissed, use the same format as the above example, substituting “cert dismissed” for “cert den.” (3) Unless certiorari is denied or dismissed by opinion, parallel citations are not necessary for “cert den” or “cert dismissed” after they are published in the United States Reports. Go to TABLE OF CONTENTS 22 Go to INDEX Certiorari granted by United States Supreme Court (include as much information as is available at the time that you are writing). These examples list earliest to latest sources: State v. Ice, 343 Or 248, 70 P3d 1049, cert granted, __ US _, 76 USLW 3496 (Mar 17, 2008). State v. Ice, 343 Or 248, 70 P3d 1049, cert granted, __ US , 128 S Ct 1657 (2008). State v. Ice, 343 Or 248, 70 P3d 1049, cert granted, 552 US 1256 (2008). Supreme Court opinion reversed by United States Supreme Court: Gilliam County v. Dept. of Environmental Quality, 316 Or 99, 849 P2d 500 (1993), rev’d and rem’d sub nom Oregon Waste Systems, Inc. v. Department of Environmental Quality of Ore., 511 US 93, 114 S Ct 1345, 128 L Ed 2d 13 (1994). Sub nom indicates that a different case name was used in subsequent history. Regarding changing case names, follow this rule: When the name of the case differs in prior or subsequent history, the new name must be given, except (1) when the parties’ names merely are reversed; (2) when the citation in which the difference occurs is to a denial of certiorari or rehearing; or (3) when, in the appeal of an administrative action, the name of the private party remains the same. Use the word “by” when referring to subsequent case history only if referring to an entirely different case, e.g., Keenan v. Norris-Lampe, 330 Or 456, 777 P2d 897 (1999), overruled on other grounds by Bennett v. Bauman, 333 Or 566, 790 P2d 654 (2001). Otherwise, use sub nom, as explained above, e.g., Smith v. Jones, 330 Or 456, 777 P2d 897 (1999), rev’d on other grounds sub nom Jones and White v. Smith, 668 US 123, 113 S Ct 1346, 129 L Ed 2d 14 (2001). 2. Court of Appeals Precedential opinions: Lesch v. DeWitt, 118 Or App 397, 847 P2d 888 (1993). Nonprecedential memorandum opinions (cited in accordance with ORAP 10.30): State v. Nord, 320 Or App 672 (2022) (nonprecedential memorandum opinion). NOTE: Because the parenthetical included in nonprecedential memorandum opinions concerns weight of authority, it precedes any subsequent history, such as opinions on reconsideration or the denial of review by the Supreme Court, e.g., Party v. Party, 320 Or App 486 (nonprecedential memorandum opinion), rev den, 370 Or 56 (2022). Go to TABLE OF CONTENTS 23 Go to INDEX Overruled by subsequent Oregon case: Eklund v. Clackamas County, 36 Or App 73, 583 P2d 567 (1978), overruled on other grounds by Forman v. Clatsop County, 63 Or App 617, 665 P2d 365 (1983). Supreme Court review denied: Allred v. Board of Parole, 124 Or App 278, 862 P2d 546 (1993), rev den, 318 Or 325 (1994). More than one petition for review denied: Allred v. Board of Parole, 124 Or App 278, 862 P2d 546 (1993), rev den, 318 Or 325; 318 Or 502 (1994). NOTE: (1) Always include rev den and cert den, if applicable. (2) Unless the review denied or certiorari denied citation is not available or there is an opinion denying review or certiorari, parallel citations are not necessary. (3) Before 1976, disposition of petitions for review were not published; accordingly, only the year and disposition are indicated, e.g., State v. Smith, 14 Or App 72 (1973), rev den (1974). (4) When a petition for review has been filed in a case, but not yet acted on, the current practice of the appellate courts is NOT to show that review is pending. However, practitioners may want to do so when writing a brief. Supreme Court review allowed, but case not yet decided: Lesch v. DeWitt, 118 Or App 397, 847 P2d 888, rev allowed, 317 Or 162 (1993). Supreme Court review allowed, then later dismissed: Finch v. Andrews, 124 Or App 558, 863 P2d 496, rev dismissed, 320 Or 267 (1994). NOTE: If applicable, it is optional to include “as improvidently allowed” after “rev dismissed.” Court of Appeals decision affirmed by Supreme Court: State v. McCoy, 17 Or App 155, 521 P2d 1074, aff’d, 270 Or 340, 527 P2d 725 (1974). Court of Appeals decision reversed by Supreme Court: State v. Cloutier, 33 Or App 121, 575 P2d 996, rev’d on other grounds, 286 Or 579, 596 P2d 278 (1979). Court of Appeals decision remanded by Supreme Court: State v. White, 59 Or App 61, 650 P2d 184 (1982), rem’d, 297 Or 302, 685 P2d 983 (1984). Go to TABLE OF CONTENTS 24 Go to INDEX Court of Appeals decision on reconsideration (specify disposition): Kirpal Light Satsang v. Douglas County, 96 Or App 207, 772 P2d 944, adh’d to on recons, 97 Or App 614, 776 P2d 1312 (1989). State v. Ramirez, 205 Or App 113, 133 P3d 343, adh’d to on recons, 207 Or App 1, 139 P3d 981 (2006), rev’d on other grounds, 343 Or 505, 173 P3d 817 (2007), adh’d to as modified on recons, 344 Or 195, 179 P3d 673 (2008). NOTE: Whether to include “on other grounds” in subsequent history to the Court of Appeals opinion depends on the proposition for which the case is being cited. Court of Appeals decision vacated by unpublished order: Davis v. Johnson, 155 Or App 266, 958 P2d 907 (1998), decision vac’d by order, July 21, 1998. Oregon review denied and United States Supreme Court certiorari denied (but not by opinion): State Highway Com. v. DeLong Corp., 9 Or App 550, 495 P2d 1215, rev den (1972), cert den, 411 US 965 (1973). Axen v. American Home Products Corp., 158 Or App 292, 974 P2d 224, adh’d to on recons, 160 Or App 19, 981 P2d 340, rev den, 329 Or 357 (1999), cert den, 528 US 1136 (2000). If certiorari is dismissed, use the same format as the above example, substituting “cert dismissed” for “cert den.” Certiorari denied by opinion by United States Supreme Court: State v. Smith, 30 Or App 462, 575 P2d 369, rev den, 282 Or 823 (1978), cert den, 454 US 324, 99 S Ct 379, 25 L Ed 2d 889 (1979). Appeal dismissed by United States Supreme Court (if by opinion, as in the example below, include parallel citations; otherwise, cite only United States Reporter): Boykin v. Ott, 10 Or App 210, 498 P2d 815, rev den (1972), appeal dismissed, 411 US 912, 36 S Ct 304, 93 L Ed 2d 1554 (1973). 3. Tax Court Regular Division: Fellows v. Dept. of Rev., 14 OTR 13 (1999). Unpublished Regular Division Decisions: Fellows v. Dept. of Rev., TC 4952, WL 2037643 (Or Tax, May 24, 1999). (optional to include online reference, but recommended) Magistrate Division: Jacobs v. Harney Co., 16 OTR-MD 344 (2001). Go to TABLE OF CONTENTS 25 Go to INDEX Unpublished Magistrate Division Decisions: Jacobs v. Harney Co., TC-MD 994356Z, WL 842084 at 3 (Or Tax M Div, Apr 7, 2000). (optional to include online reference, but recommended) NOTE: For citing subsequent history, refer to examples for the Court of Appeals. 4. Case Not Yet Appearing in Publication Jones v. State of Oregon, ___ Or , , ___ P3d ___ (Apr 1, 2013) (slip op at 15:9-16). Smith, ___ Or at ___ (slip op at 28:6 - 29:2). Smith, ___ Or at , ___ (slip op at 28:6 - 29:2; slip op at 31:17 - 32:2). NOTE: To ensure correct cross-referencing within a slip opinion, the appellate courts cite the appropriate page number(s). For added accuracy, the Supreme Court also includes the specific line number(s). Smith v. Jones, ___ Or App , ___ n 3, ___ P3d ___ (Feb 1, 2013) (emphasis added) (slip op at 5 n 3). 5. Circuit Court Cases State v. Cazares-Mendoza, Case No. C062326CR. 6. VAWA (Violence Against Women Act) Cases Case title in court records: Protected Party v. Second Party Title Page format: P. P. v. SECOND PARTY Full citation format (even if previously published as Party v. Party): P. P. v. Party, 325 Or App 123, 456 P3d 789 (2023). Short citation format: P. P., 325 Or App at 124. NOTE: When redacting a name to initials, use the first letter of each name in the protected par-ty’s name, maintaining hyphens, if used (e.g., Jane Doe-Smith v. John Doe would be formatted J. D.-S. v. Doe, if Jane is the protected party; Doe-Smith v. J. D., if John is the protected party). Short citation format follows the same rules set out below--using the first nongovernmental party, whether initials or not. NOTE: If uncertain as to what the proper shortened case name would be for a case, such as an opinion issued the same day or a case that was affirmed without opinion, contact the Publications Program for the correct case name for citation purposes (as there is a limit to the number of characters that can be used). B. Oregon–Short Citations and Other Issues 1. In General When citing a case that has already been cited in the text of an opinion, use a short citation. However, if the only previous citation to a case is in a footnote, use the full citation the first time the case is cited in the text. See page 13. Go to TABLE OF CONTENTS 26 Go to INDEX When using a shortened case name in a short citation, the shortened case name is the first nongovernmental party appearing in the official case name citation, e.g., State v. Bates, 304 Or 519, 747 P2d 991 (1987). (full citation, first reference) Short citation form is [shortened case name], [volume number] [reporter] at [page], e.g., Bates, 304 Or at 522 (eliminating the parallel citation). When citing a specific page when the case name is used in the sentence being cited, or if there otherwise is no doubt as to the case being cited, the case name may be omitted from the citation, e.g., In Bates, this court stated that the officers violated defendant’s constitutional rights by instructing him to move bag into view. 304 Or at 527. or In Bates, this court addressed a similar issue. The court held that, by order-ing defendant to move the bag, the police violated defendant’s constitutional rights. 304 Or at 527. Otherwise, include the shortened case name with the citation: This court held that the officers violated defendant’s constitutional rights by instructing him to move the bag into view. Bates, 304 Or at 527. NOTE: Even when a governmental official is individually named, use the first nongovernmental party appearing in the official case name citation for the shortened case name. 2. Variations on Case Name If a shortened case name causes confusion, such as if two cases with similar names have been cited within the same opinion, then use the full case name for each throughout the opinion, e.g., State v. Bates Bates v. Smith Go to TABLE OF CONTENTS 27 Go to INDEX When both parties have the same last name, as in some domestic relations matters, e.g., Smith and Smith, use Smith as the short cite. Additionally, an explanation might need to be added for clarity, such as a modified case name in parentheses, if there are related cases in a series, e.g., The accused has a disciplinary record, having been suspended twice from the practice of law. See In re Wyllie, 326 Or 447, 952 P2d 550 (1998) (Wyllie I) (one-year suspension for refusing to comply with remedial program); In re Wyllie, 327 Or 175, 957 P2d 1222 (1998) (Wyllie II ) (two-year suspension for misrepresenting compliance with MCLE requirements and failing to cooper-ate). In Wyllie I, or The Court of Appeals affirmed the tort judgment, concluding that it was unneces-sary to resolve the breach of contract counterclaim. Northwest Natural Gas Co. v. Chase Gardens, Inc., 146 Or App 249, 933 P2d 370 (1997) (Chase I ). This court allowed review in Chase I, reversed, and remanded for consideration of Chase’s contract claim. Northwest Natural Gas Co. v. Chase Gardens, Inc., 328 Or 487, 982 P2d 1117 (1999) (Chase II ). On remand, the Court of Appeals, relying in large part on its earlier decision and this court’s decision in Chase II, concluded that Chase’s breach of contract judgment could not be sustained. Northwest Natural Gas Co. v. Chase Gardens, Inc., 164 Or App 763, 995 P2d 555 (2000) (Chase III ). We now reverse the Court of Appeals decision in Chase III. In lawyer disciplinary cases, the shortened case name is the lawyer’s last name, without the In re designation, e.g., In re Jones, 326 Or 195, 951 P2d 149 (1997). (full citation form) Jones, 326 Or at 198. (short citation form) In mandamus cases initiated after 1997, the case name is the same as the case name of the proceeding in the lower court, and the judge’s name (if the case involves a circuit court judge action) does not appear in the title, e.g., State v. Foster. In older mandamus cases, the name of the judge whose action was being challenged was included in the title, together with a “State ex rel” designation; cite such cases as follows: State ex rel Huddleston v. Sawyer, 324 Or 597, 932 P2d 1145 (1997). (full citation form) Huddleston, 324 Or at 600. (short citation form) NOTE: The appellate courts do not use “supra” as a substitute for short citations. Go to TABLE OF CONTENTS 28 Go to INDEX 3. Citation to Series of Pages When citing a series of pages, indicate the page numbers as follows: Stranahan, 331 Or at 57-58. Fugate, 332 Or at 202-03. (not 202-3) Joslin, 332 Or at 499-501. (not 499-01) 4. Use of “id.” with Case Names When citing the immediately preceding authority, it is acceptable to use id., e.g., Id. at 525. If citing the same page of the immediately preceding authority, then use id. without the page number. CAUTION: Do not use id. where there is an intervening citation to another authority (be it a case name, statute, or some other authority). When there is an intervening textual reference, use id. only when there is no danger of ambiguity. 5. Citation to Footnote State v. Trenary, 114 Or App 608, 610 n 2, 836 P2d 739 (1992). Trenary, 114 Or App at 610 n 2. (short citation form) When citing a series of footnotes, indicate the page numbers as follows: State v. Trenary, 114 Or App 608, 610 nn 3-5, 836 P2d 739 (1992). State v. Trenary, 114 Or App 608, 610 n 3, 611 nn 4 & 5, 836 P2d 739 (1992). NOTE: Citation is to the first page on which the footnote appears, even if it spans more than one page, no matter where the pertinent text actually appears. When citing text contained in the body of the opinion and also in a footnote set out on the same page, set out the page number twice, e.g., Trenary, 114 Or App at 610, 610 n 2. or Trenary, 114 Or App at 610 & n 2. Go to TABLE OF CONTENTS 29 Go to INDEX 6. Internal Citations; Dissents; Concurrences The majority cites a dissenting or concurring opinion in all references (regardless of the number of dissenting or concurring opinions) as follows: The dissent contends that the statute operates prospectively only. ___ Or App at ___ (Smith, J., dissenting) (slip op at 3). A dissent or concurrence cites the majority opinion as follows (do not include the case name): The majority concludes that the statute operates retroactively. ___ Or App at ___ (slip op at 7). A dissent or concurrence cites itself as follows: As noted above, I dissent because I disagree with the majority’s statutory con-struction and conclusion. ___ Or App at ___ (Smith, J., dissenting) (slip op at 1). NOTE: Do not use supra or infra and do not use only “See note 6” when referring to earlier or later parts of an opinion (be it majority, concurring, or dissenting); instead, use the short citation. For example, a majority internal citation is set out as: As noted, ORS 813.010 prohibits driving under the influence of intoxicants. See ___ Or App at ___ n 6 (slip op at 4 n 6). 7. Short Citation Form for Oregon Tax Cases Fellows, 14 OTR at 17. (Regular Division) Jacobs, 16 OTR-MD at 347. (Magistrate Division) 8. Possessive Endings on Case Names When a name of a case takes a possessive ending, the “ 's ” is not italicized, e.g., Bates’s rule applies in this case. C. Federal Jurisdictions For United States Supreme Court cases, the case name for citation purposes is taken from the case name citation in the United States Reports (US), if available (which appears on the odd-numbered pages); otherwise, use the case name citation from an alternative reporter. (Note that the case name citations frequently differ in those Go to TABLE OF CONTENTS 30 Go to INDEX publications.) The short citation also is to the US Reports, if available; otherwise, cite the Supreme Court reports, if available, if not, then cite the Lawyer’s Edition. (Note that some material within the text of the Court’s opinions, such as citations and quotations, often varies between publications.) If the US Reports citation is not available, insert “ US ” (including a blank for a jump / pinpoint) citation, if applicable) in all full and short citations, before the citation to an alternative reporter. 1. United States Supreme Court Wagner v. Oregon, 492 US 914, 109 S Ct 3235, 163 L Ed 2d 583 (1989). (first reference) Wagner, 492 US at 916. (short citation form) NOTE: It is acceptable to refer to Miranda v. Arizona without including full citation, e.g., “Miranda rights” or “Miranda warnings.” 2. Federal 3d and 2d (case name appears on odd-numbered pages) Johnson v. Clifton, 74 F3d 1087 (11th Cir), cert den, 519 US 808 (1996). Freije v. United States, 408 F2d 100, 102 (1st Cir), cert den, 396 US 859 (1969). U.S. v. Echeverri, 982 F2d 675 (1st Cir 1993). United States v. Wainwright, 413 F2d 796, 803 (10th Cir 1969), cert den, 396 US 1009 (1970). 3. Federal Supplement (case name appears on odd-numbered pages) Lucas v. Seagrave Corporation, 277 F Supp 338 (D Minn 1967). United States v. Zeiger, 350 F Supp 685 (DDC), rev’d, 475 F2d 1280 (DC Cir 1972). If unpublished: Smith v. Jones, No CV 96-6109-TC (D Or Feb 5, 2001). NOTE: For abbreviations of federal district courts, see The Bluebook. 4. Federal Cases with Incomplete Information United States v. Edwards, ___ US , 94 S Ct 1100, 39 L Ed 2d 771 (1974). (first reference) Go to TABLE OF CONTENTS 31 Go to INDEX Id. at , 94 S Ct at 1109 (short citation form) Washington v. Glucksberg, ___ US , ___ S Ct , ___ L Ed 2d , 65 USLW 4669 (June 24, 1997). United States v. Louiriev, 22 Crim L Rep 2369, ___ F2d ___ (8th Cir, Dec 30, 1977). Naquin v. Elevating Boats, L.L.C., 744 F3d 927 (5th Cir), cert den, ___ US , 135 S Ct 357 (2014). 5. Federal Tax Cases The order of authority for federal tax cases is (1) the United States Supreme Court; (2) federal courts of appeals; (3) federal district courts; and (4) the United States Tax Court. Federal tax cases from the United States Supreme Court, United States Court of Appeals, Court of Federal Claims, and United States District Court are combined and bound as “U.S. Tax Cases”; therefore, the best citation will contain both the federal and parallel citations. For example, Furlow, Jr. v. U.S., 55 F Supp 2d 360, 2000-1 US Tax Cas (CCH) ¶ 50,684 (D Md 1999). NOTE: “U.S. Tax Cases” often is abbreviated as “USTC,” but that is not the formal citation. United States Tax Court: Benson v. Commissioner, 80 TC 789 (1983). Crook v. Maine, 132 TCM (CCH) 44 (1999). (memorandum decision) D. States Other Than Oregon NOTE: For abbreviations of state and regional reporters, see The Bluebook. For states other than Oregon, cite the official reporter first, then add the regional reporter when using jump / pinpoint citations because that benefits readers who have access to only regional reporters, e.g., In Statser v. Statser, 205 Okla 608, 611, 239 P2d 764, 766 (1951), the court stated: “[T]he terms of the condition are directed to defendant’s granting or withhold-ing permission; the condition does not purport to authorize police action.” Go to TABLE OF CONTENTS 32 Go to INDEX The court continued: “Defendant, of course, may argue that the safeguard for employees exceeds the ordinary standard of due care considering the nature of the risk and the foreseeability of injury.” Id. at 615, 239 P2d at 768. NOTE: The appellate courts accept practitioners’ submissions jump / pinpoint citing only regional reporters except when citing Oregon appellate decisions. For cases not yet appearing in publication, it is acceptable to use a citation to Westlaw, e.g., State v. Smith, No 26245-2-II, 2013 WL 651868 at 2 (Wash App Div 2, Apr 19, 2013). When the official reporter (i.e., the state reporter) is the same for both the highest court and the intermediate appellate court, include the court abbreviation in the date parenthetical, e.g., State v. Gray, 231 Ariz 374, 295 P3d 951 (Ariz Ct App 2013). When the official reporter is the regional reporter, include the court abbreviation in the date parenthetical, e.g., Golphin v. State, 945 So 2d 1174 (Fla 2006). E. Online Sources A case unreported in a print source can be cited if it resides online in a readily recognized source. The case name, number, database identifier, court name, and complete date should be included along with any other identifiers that are pertinent to its location. Silva v. Mt. Bachelor, Inc., No CV 06-6330-AA, 2008 WL 2889656 at 2 (D Or July 21, 2008). For cases having a state citation, in part, and an online citation, in part: Zollner v. Smith, 268 Ga App 480, 484, 602 SE2d 140, 141, rev den, 2004 Ga LEXIS 1059 (2004). Go to TABLE OF CONTENTS 33 Go to INDEX III. Constitutional, Statutory, and Other Related Citations NOTE: Id. (or id. § , if applicable) can be used as a subsequent reference citation to constitutional provisions, statutory and code provisions (both current provisions and older codes), legislative commentary, and other types of sources discussed in this section. A. Oregon Citations 1. Oregon Constitution a. Narrative Form: Article I, section 17, of the Oregon Constitution provides that . (first reference) Article VII (Amended), section 1, provides that . (subsequent reference) b. Citation Form: Or Const, Art I, § 17. Or Const, Art VII (Amended), § 3. Or Const, Art VII (Original). Or Const, Art XI, § 11b(2)(b). Or Const, Art I, § 11 (1857). (denoting the original version of Article I, section 11, NOT the current version) NOTE: (1) The word “Article” is always shown as beginning with an uppercase “A.” (2) The article number is always a Roman numeral. (3) “Amended” begins with an uppercase “A.” (4) In narrative, the word “section” is always lowercase except in a heading or subheading. (5) Do not use periods after abbreviations. (6) Original Article VII was replaced in 1910, which is why it is referred to as amended. 2. Oregon Laws Oregon Laws 1989, chapter 790, section 87. (narrative form) Oregon Laws 1990, chapter 2, sections 45 to 47 (Special Session). (narrative form) Oregon Laws 1995, chapter 790, section 84, directs the commission to . (narrative form) Go to TABLE OF CONTENTS 34 Go to INDEX Or Laws 1989, ch 790, § 87. (citation form) Or Laws 1990, ch 2, §§ 3-10 (Spec Sess). (citation form) NOTE: Insert a space after a section or paragraph symbol; when using two such symbols, no space separates them. Statutes that have not been assigned by the legislature to an ORS chapter often are compiled into the ORS in small typeface at the place in the ORS where Legislative Counsel decides they logically belong. See ORS Preface, viii (2001). Refer to such statutes as follows: Oregon Laws 1997, chapter 30, section 2(1), compiled as a note after ORS 659.010 (1997). (narrative form, first reference) Oregon Laws 1997, chapter 30, section 2(1). (narrative form, subsequent reference) Or Laws 1997, ch 30, § 2(1), compiled as a note after ORS 659.010 (1997). (citation form, first reference) Or Laws 1997, ch 30, § 2(1). (short citation form) Oregon Laws 2005, chapter 463, section 1, compiled as a note before ORS 136.001 (2005). (narrative form, first reference) Oregon Laws 2005, chapter 463, section 1. (narrative form, subsequent reference) Or Laws 2005, ch 463, § 1, compiled as a note before ORS 136.001 (2005). (citation form, first reference) Or Laws 2005, ch 463, § 1. (short citation form) Occasionally, Legislative Counsel will compile such statutes so that they appear as legislatively placed statutes, e.g., ORS 475A.010, ORS 475A.040, and other civil forfeiture statutes. Those statutes are cited in the same manner as legislatively placed statutes, as explained in paragraph 5 below. 3. General Laws of Oregon and Other Older Statutory Compilations Some of these compilations are organized by chapters, which incorporate several titles, and others are organized by titles, which incorporate several chapters. It is advisable to include page numbers when citing the Deady compilations, because the various codes are not numbered and there are no indices or general tables of contents to guide the reader to the proper location regarding particular Go to TABLE OF CONTENTS 35 Go to INDEX sections of a code (e.g., Criminal Code, Civil Code, etc., each of which is made up of multiple chapters). Some older cases incorporate neither chapters nor titles; for such codes, cite to the applicable section and page numbers. General Laws of Oregon, Crim Code, ch XVIII, § 659, p 435 (Deady & Lane 1843-1872). General Laws of Oregon, Civ Code, ch IV, title I, § 313, p 226 (Deady 1845-1864). The Codes and General Laws of Oregon, ch XVIII, title IV, § 2933 (Hill 1887). The Codes and General Laws of Oregon, ch I, title VIII, § 67 (Hill 2d ed 1892). The Codes and Statutes of Oregon, title V, ch I, § 339 (Bellinger & Cotton 1901). Lord’s Oregon Laws, title VI, ch V, § 442 (1910). Oregon Laws, title I, ch III, § 39 (1920). Oregon Code, title XXVIII, ch 1, § 28-101 (1930). Statutes of Oregon, An Act Regulating Proceedings to Vacate Charters and Letters Patent, and to Prevent the Usurpation of an Office of Franchise, ch 1, § 1, p 139 (1854). 4. OCLA (Oregon Compiled Laws Annotated) OCLA § 3-505. OCLA § 10-902a (1944-47 pocket part). 5. ORS (Oregon Revised Statutes) Always use the abbreviated form (ORS) in both narrative references and citations. ORS 161.155(1)(a)(A). [chapter 161][section 155][subsection (1)][paragraph (a)][subparagraph (A)] Numbered paragraphs within a section are referred to as subsections, lettered paragraphs as paragraphs. For example, ORS 305.220(1)(a) would be referred to in text individually as paragraph (a) of subsection (1), when referring to (1)(a) collectively, use paragraph (1)(a). a. Narrative Form Examples ORS chapter 554. (when referring to an entire chapter) Go to TABLE OF CONTENTS 36 Go to INDEX ORS 30.150 to 30.175; ORS 250.035(2)(a) to (c). (when referring to a statutory sequence) ORS 30.866 and ORS 163.730. (when referring to two statutes) ORS 250.035(2)(a) and (b). (when referring to two parts of the same statute) b. Citation Form Examples ORS ch 554. (when citing an entire chapter) ORS 30.150 - 30.175; ORS 250.035(2)(a) - (c). (when citing a statu-tory sequence) ORS 30.866; ORS 163.730. (when citing two statutes) ORS 250.035(2)(a), (b). (when citing two parts of the same statute) NOTE: When citing a sequence of statutes in citation, do not use “et seq.”; instead, set out the first and last statutes of the sequence. c. Repealed or Renumbered Statute Examples When citing a repealed or renumbered ORS (or OAR), the narrative and citation forms are the same. The word former is set in italics. Only a repealed or renumbered statute, not an amended statute, is referred to as “former.” Former ORS 19.023(2)(a) (1995), renumbered as ORS 19.205(2)(a) (1997). (first reference; “1995” will usually refer to the year of the statute in effect when the facts of the case occurred (although the author may make a different choice, depending on context); “1997” refers to year that the statute was renumbered) Former ORS 19.023(2)(a) (1995). (subsequent reference) Former ORS 736.317 (1961), repealed by Or Laws 1967, ch 482, § 1. (first reference) Former ORS 736.317 (1961). (subsequent reference) Go to TABLE OF CONTENTS 37 Go to INDEX When a statute has been renumbered or repealed, the author should cite the version in effect at the time of the events in the case, including subsequent history on the first reference and then including the year of that version in subsequent references. d. Amended Statute Examples ORS 308.370 - 308.375 (1987), amended by Or Laws 1991, ch 459, §§ 117-120. (citation form, first reference) ORS 308.370 - 308.375 (1987). (citation form, subsequent reference) NOTE: (1) The above examples for amended statutes require inclusion of the year parenthetically at the end of all citations (separated by a space). However, it also is acceptable to include a footnote at the outset of the opinion, after citing the older statute in full, that explains that all statutory citations that follow are to a particular year. In that event, it is not necessary to include the parenthetical year in subsequent references. (2) When setting out ORS citations or other authorities, be consistent in the use of punctuation, e.g., The state appeals from a pretrial judgment dismissing an indictment that charged defendant with manufacture of a controlled substance, ORS 475.992(1); delivery of a controlled substance to minor, ORS 475.999; and criminal conspiracy, ORS 161.450. In general, when a statute has been amended, the author may: (1) cite the version in effect at the time of the events in the case; or (2) cite the current version, if the amendment does not affect the issue in the case. If choosing option (1), then include subsequent history on the first reference and include the year of the version being cited on subsequent references. If choosing option (2), then include a footnote on the first reference that sets out the subsequent history and explains that the current version is being used because the intervening amendment does not affect the analysis. For example, 1 ORS 164.125 has been amended since defendant committed his crime; however, because those amendments do not affect our analysis, we refer to the current version of the statute in this opinion. Go to TABLE OF CONTENTS 38 Go to INDEX If a statute has been amended or renumbered more than once, then the best practice is to include the full subsequent history on the first reference or at least in a footnote (example: cite all amendments, not just the one at issue). 6. Oregon Evidence Code (OEC) (ORS chapter 40) Always use the abbreviated form (OEC) in both narrative references and citations when referring to or citing a specific rule. Use “Oregon Evidence Code” (not “OEC”) when referring to the code as a whole. Do not cite the ORS number, e.g., OEC 801(3). OEC 103 Commentary (1981). Legislative Commentary to OEC 401, reprinted in Laird C. Kirkpatrick, Oregon Evidence § 401.02, Art IV-4 (4th ed 2002). Parts of the Oregon Evidence Code suggest that . (narrative form) When quoting the Oregon Evidence Code, or when quoting a statute that quotes the Oregon Evidence Code, replace any ORS reference with the corresponding OEC rule number in brackets, e.g., OEC 405(2)(a) provides: “In all cases in which character or a trait of character of a person is admissible under [OEC 404(1)], proof may also be made of specific instances of the conduct of the person.” 7. Oregon Rules of Civil Procedure (ORCP) Always use the abbreviated form (ORCP) in both narrative references and citations when referring to or citing a specific rule. Use “Oregon Rules of Civil Procedure” (not “ORCPs”) when referring to the rules as a whole. Always add a space after the rule number when referring to a section of the rule, e.g., ORCP 71 B(1)(b)(i). [rule 71][section B][subsection (1)][paragraph (b)][sub-paragraph (i)]. ORCP 7 C(3)(a) to (c). (narrative form, when citing a sequence) ORCP 7 C(3)(a) - (c). (citation form, when citing a sequence) ORCP 7 C(3)(a) and (c). (narrative form, when referring to two paragraphs of the same rule) ORCP 7 C(3)(a), (c). (citation form, when citing two paragraphs of the same rule) Nothing in the Oregon Rules of Civil Procedure points to a contrary conclu-sion. (narrative form) ORCP 71 B(1)(c) or ORCP 71 C (when citing two parts of a rule). Go to TABLE OF CONTENTS 39 Go to INDEX When citing the drafts, comments, and history of the ORCPs, cite the bound volumes, i.e.: Council on Court Procedures, 4 Legislative History Relating to Promulgation of Oregon Rules of Civil Procedure (1/1/78 through 12/31/78), June 28, 1978, Draft and Proposed Comment to Rule 6, 33 (1979). When citing the permanent comments to the enacted ORCPs, cite the Merrill handbook, i.e.: Council on Court Procedures, Staff Comment to Rule 9, reprinted in Frederic R. Merrill, Oregon Rules of Civil Procedure: A Handbook 28-29 (1981). 8. Oregon Rules of Appellate Procedure (ORAP) Always use the abbreviated form (ORAP) in both narrative references and citations when referring to or citing a specific rule. Use “Oregon Rules of Appellate Procedure” (not “ORAPs”) when referring to the rules as a whole, e.g., ORAP 9.15(1). The Oregon Rules of Appellate Procedure set out a process for initiating a case. NOTE: Follow the links at www.courts.oregon.gov to find the most current version of the Oregon Rules of Appellate Procedure, which occasionally are temporarily amended by Chief Justice-Chief Judge Order. 9. Oregon Administrative Rules (OAR) Always use the abbreviated form (OAR) in both narrative references and citations when referring to or citing a specific rule. Use “Oregon Administrative Rules” not “OARs” when referring to the rules as a whole, e.g., OAR 253-004-0001. (narrative and citation form, when referring to or citing a specific paragraph) OAR chapter 253, division 4. (narrative form, when referring to an entire division) OAR chapter 253, division 4, Appendix 3. (narrative form, when referring to an appendix) OAR ch 253, App 3. (citation form) OAR 660-022-0077 to 660-022-0080. (narrative form when referring to a series) OAR 660-022-0077 - 660-022-0080. (citation form when referring to a series) OAR 435-001-0025 (May 12, 2002). (indicates that a rule has been amended) Go to TABLE OF CONTENTS 40 Go to INDEX Former OAR 255-80-005(2) (May 31, 1985). (indicates that a rule has been renumbered or repealed) OAR 150-305.265(14)-(A). (when citing chapter 150, place hyphens and peri-ods as published for each rule) Nothing in the Oregon Administrative Rules suggests that . (narrative form) 10. Uniform Trial Court Rules (UTCR) / Supplementary Local Rules (SLR) UTCR 7.020(5). (narrative and citation form, when referring to or citing a specific rule) Multnomah Circuit Court Supplementary Local Rule (SLR) 5.045(1). (first reference) SLR 5.045(2). (short citation form) The Uniform Trial Court Rules contain several provisions . (narrative form) 11. Uniform Jury Instructions a. Civil Instructions: Uniform Civil Jury Instruction 10.10. (narrative form) UCJI 10.10. (citation form) b. Criminal Instructions: Uniform Criminal Jury Instruction 3.1. (narrative form) UCrJI 3.1. (citation form) 12. Agency Decisions a. Workers’ Compensation Board: Edward D. Lucas, 41 Van Natta 2272, 2274-75 (1989), rev’d on other grounds, Lucas v. Clark, 106 Or App 687, 809 P2d 712 (1991). b. Land Use Board of Appeals (LUBA): Stefansky v. Grant County, 12 Or LUBA 91 (1984). Holland v. City of Cannon Beach, ___ Or LUBA , ___ (LUBA No 02-060, Oct 1, 2002) (slip op at 10). Go to TABLE OF CONTENTS 41 Go to INDEX c. Employment Relations Board (ERB): Jefferson County v. OPEU, 18 PECBR 285 (1999). d. Employment Appeals Board (EAB): Terrance E. Webb, EAB Decision 96-AB-876 (Apr 16, 1996). 13. Attorney General Opinions a. Formal Attorney General Opinions: 35 Op Atty Gen 710 (1972). 47 Op Atty Gen ___ (No 8216, Sept 7, 1994). (slip opinion) b. Letters of Advice (Informal Attorney General Opinions): Attorney General Letter of Advice to Rep Clayton Klein (OP-4519) (Dec 1, 2003). 14. Municipal Codes Citation form will vary, depending on the code at issue. Check the official code or ordinance itself for its proper name and short citation form, e.g., the Eugene Municipal Code provides for citation to “EC.” Otherwise, use the name of the code (or an abbreviation after the first reference or citation) and the number of the code provision, e.g., Eugene Code (EC) 3.005. (narrative and citation, first reference) EC 3.005. (narrative and citation, subsequent reference) Portland City Code (PCC) 14A.40.050. (narrative and citation, first reference) PCC 14A.40.050. (narrative and citation, subsequent reference) 15. Legislative History Senate Bill (SB) 123 (1997); House Bill (HB) 1234 (1997). (narrative form, first reference) SB 123; HB 1234. (subsequent reference) Go to TABLE OF CONTENTS 42 Go to INDEX SB 123 (1997); HB 1234 (1997). (citation form) HB 2759 (1993), -1 amendments (Apr 13, 1993). Senate Joint Resolution (SJR) 5 (1997); House Joint Resolution (HJR) 10 (1997). (narrative form, first reference) SJR 5 (1997); HJR 10 (1997). (citation form, first reference) SJR 5; HJR 10. (narrative or citation form, subsequent reference) Testimony, House Committee on Human Resources, HB 2762, Mar 17, 1981, Ex C (statement of Rep Bob Smith). Tape Recording, House Committee on Judiciary, Subcommittee on Crime and Corrections, SB 833, June 9, 1993, Tape 126, Side A (statement of Sen Ron Cease). Tape Recording, Senate Committee on Criminal Law and Procedure, SB 100, Mar 31, 1971, Tape 1, Side A (statement of Attorney General Joe Jones). Audio Recording, House Committee on Judiciary, HB 2460, Apr 16, 2001, at 2:10:38 (comments of Rep Amy Smith and Rep Barbara Jones), oregonlegislature.gov (accessed Jan 2, 2013). Exhibit F, House Labor Committee, SB 89, June 15, 1983 (accompanying statement of BOLI representative Paul Smith). Judgments / Enforcement of Judgments: Judgments Report (HB 2646), Oregon Law Commission, Feb 6, 2003, 12 (Judgments Report). [The parenthetical indicates the short title by which it will be referred.] Minutes, Criminal Law Revision Commission, Jan 22, 1971, 12. NOTE: For legislative committees, minutes should be cited as authority only to explain a committee vote or action taken, such as approving a bill with a do-pass recommendation. Any citation to discussion about a bill in the committee should be drawn from the audio recording of the hearing or work session, not the minutes. Commentary to Criminal Law Revision Commission Proposed Oregon Criminal Code, Final Draft and Report § 122, 131 (July 1970). Commentary to Criminal Law Revision Commission Proposed Oregon Criminal Procedure Code, Final Draft and Report § 28, 22 (Nov 1972). Commentary § 29 at 7. (short citation form) Go to TABLE OF CONTENTS 43 Go to INDEX Legislative Comment 1 to ORS 72.7030, reprinted in Legislative Counsel Committee, Oregon’s Uniform Commercial Code with Comments and Index and Tables 101 (1962). Legislative Comment 1 to ORS 72.7030 at 102. (short citation form) 16. Initiative Petitions, Ballot Measures, and Related Pamphlets and Manuals Initiative Petition 136 (2001) (IP 136). (narrative form, first reference; citation form) IP 136. (narrative form, subsequent reference; citation form) Ballot Measure 40 (1996). (narrative form, first reference; citation form) Measure 40. (narrative form, subsequent reference; short citation form) Official Voters’ Pamphlet, General Election, Nov 7, 2000, 309. (citation form, first reference) Voters’ Pamphlet at 310. (subsequent reference) The voters’ pamphlet for the 2000 General Election suggests that . (nar-rative form) NOTE: The above examples refer to statewide measures. If citing a local measure, then follow the format set out above, but insert the appropriate local numbering, e.g., Ballot Measure 10-06 (1992). Elections Division, Oregon Secretary of State, State Initiative & Referendum Manual [page] (Jan 2016), [link] (accessed Sept 19, 2017). 17. Sentencing Guidelines Implementation Manual Oregon Sentencing Guidelines Implementation Manual 131 (1989). 18. Rules of Professional Conduct (RPC), Bar Rules of Procedure (BR), and American Bar Association (ABA) Standards NOTE: Do not use “RPC” or “RPCs” when speaking generally about the Rules of Professional Conduct; instead, cite the rules separately. a. Rules of Professional Conduct The Bar’s complaint alleges that the lawyer violated Rule of Profes-sional Conduct (RPC) 8.4(a)(3) (conduct involving dishonesty, fraud, deceit, or misrepresentation). (narrative form, first reference to any RPC) The lawyer contends that he did not violate RPC 8.4(a)(3). (narrative form, subsequent RPC reference) Go to TABLE OF CONTENTS 44 Go to INDEX The lawyer contends that he did not violate any rule of professional conduct. (narrative form) NOTE: The rule on pages 36 to 37 referring to the use of “former” with repealed and renumbered, but not amended, statutes, does not apply to rules of professional conduct. With rules of professional conduct, “former” is used to refer to amended, as well as repealed and renumbered, rules. If citing to a former rule, then include the year. b. Bar Rules of Procedure This is a reinstatement proceeding under Bar Rule of Procedure (BR) 8.8. (narrative form, first reference to any BR) BR 10.1. (narrative form, subsequent reference; citation form) c. ABA Standards American Bar Association’s Standards for Imposing Lawyer Sanctions (1991) (amended 1992) (ABA Standards), Standard 5.21. (narrative or citation form, first reference) ABA Standard 9.32(a). (narrative or citation forms, subsequent refer-ence when citing a particular standard) ABA Standards at 7. (citation form, subsequent reference when citing a particular page in the hardcopy manual) ABA Standards at 7 (defining “injury,” in part, as “harm to the profession”) (citation form for definitions appearing on a particular page in the hardcopy manual) ABA Standard 9.22(i) (amended 1992). (narrative or citation form, subsequent reference when citing a standard contained in only the 1992 amendments) NOTE: The copyright for the ABA Standards was updated in 2005, but the rules were not amended in 2005; therefore, the year for the amended version is 1992. The 1992 version is available online at https:// www.americanbar.org/content/dam/aba/administrative/professional_ responsibility/sanction_standards.pdf 19. Briefs and Other Court Filings The courts do not cite briefs in the instant case in their opinions; occasionally, however, an opinion might cite a brief in another case (for example, if the brief expanded on a legal theory that the court had addressed only briefly Go to TABLE OF CONTENTS 45 Go to INDEX in that other case). When citing a brief or any other official court filing, follow this format: Full name of the document, include the jump / pinpoint citation (if any), followed by the name of the case and its full citation (if rendered), and case number. Appellant’s Opening Brief at 10, State v. Doyle, 213 Or App 456, 111 P3d 87 (2007) (CA A123456). Douglas County Circuit Court Case No 12345 If citing an event in a register of actions in the Oregon eCourt Case Information system (OECI, register of actions for Oregon circuit courts): Motion for Stay, Filed May 1, 2014, State v. Adios (CR456789) General Judgment, Entered June 15, 2015, State v. Adios (CR456789) NOTE: Use the OECI “Created Date” for court-issued entries such as orders or judgments (date of entry); otherwise, use the date in the OECI Date column (date of filing of a received filing). B. Federal Citations 1. United States Constitution The Fifth Amendment to the United States Constitution. (narrative form) The Due Process Clause of the Fourteenth Amendment to the United States Constitution. (narrative form) US Const, Amend V. (citation form) US Const, Art I, § 2. (citation form) 2. Rules of Evidence and Procedure a. Federal Rules of Evidence FRE 410. b. Federal Rules of Civil Procedure FRCP 12. c. Federal Rules of Criminal Procedure FRCrP 12. Go to TABLE OF CONTENTS 46 Go to INDEX 3. Public Laws and United States Code Pub L 95-473, § 11503, 92 Stat 1445-46 (1978). (citation form) 42 USC § 1983 (1982); 15 USC §§ 2051, 2053 (1982). (citation form) 42 USC section 1983; 15 USC sections 2051 and 2053. (narrative form, first reference) section 1983. (narrative form, subsequent reference) (the “s” in section is lowercase unless it starts the sentence) NOTE: As with ORS citations, include the year only if not citing the version currently in force. 4. Legislative History a. Committee Reports HR Rep No 1395, 95th Cong, 2d Sess, reprinted in 1978 USCCAN 3009, 3018. S Rep No 445, 87th Cong, 1st Sess (1961), 451-91. (if not in USCCAN) b. Testimony Hearings on SB 927 Before the Subcomm on Surface Transportation of the S Comm on Commerce, 90th Cong, 1st Sess, 23 (Aug 7, 1967) (prepared statement of Jim Doe, on behalf of the American Medical Association). c. Congressional Record 116 Cong Rec S2024 (Jan 3, 1970). 5. Internal Revenue Code IRC § 61 (1982). (citation form) Internal Revenue Code (IRC) section 61 (1982). (narrative form) IRC section 61. (narrative form, subsequent reference) 6. Tax Materials a. Regulations Treasury Regulation section 1.166-1(c). (narrative form) Go to TABLE OF CONTENTS 47 Go to INDEX Treas Reg § 1.72-16(a) (2001). (citation form) 20 CFR § 552.3 (2009). (citation form) b. Determinations Rev Rul 83-137, 1983-2 CB 41. NOTE: Cite CB first, and then to IRB if CB unavailable. Priv Ltr Rul 86-01-013 (Sept 30, 1985). Tech Adv Mem 85-04-005 (Sept 28, 1984). 7. Uniform Laws Uniform Child Custody Jurisdiction and Enforcement Act § 202 comment, 9 ULA 649, 674 (1997). Uniform Probate Code § 2-101(b), 42-43 (9th ed 1990). Go to TABLE OF CONTENTS 48 Go to INDEX IV. Periodical Articles, Books, Treatises, Restatements, Etc. In General Cite the author’s or editor’s full name as found on the title page of the source being cited. For books that are compiled and edited by a named editor, place the editor’s name where the author’s name would normally be, followed by “ed.” (including the period). Cite the author’s first name first. If there are up to three authors, then list them all in the same way. If four or more authors are listed, list the first author and use “et al.” For articles written by students, the author’s name is used (if available), along with, e.g., “Note,” “Comment,” or “Case Note.” When citing multivolume sets, place the volume number of the book before the title, not before the author’s name. Include the edition and the year of the publication in parentheses at the end of the citation, e.g., Jacob Mertens, Jr., 8 Mertens Law of Federal Income Taxation § 32B:01 (rev 1999). Use italics, not quotation marks, to indicate the name of a book, article, or treatise. A. Periodical Articles 1. Articles Appearing in Journals, Newspapers, and Other Services [Author], [Name of article (in italics)], [volume number] [periodical] [first page], [cited page(s)] ([date]), e.g., Pamela S. Karlan, Contingent Fees and Criminal Cases, 93 Colum L Rev 595, 602-03 (1993). Laurence H. Tribe & Michael C. Dorf, Levels of Generality in the Definition of Rights, 57 U Chi L Rev 1057 (1990). Justin Noval & Edward J. Imwinkelried, Jr., Retrograde Extrapolation of Blood Alcohol Concentration, 50 No. 1 Crim Law Bulletin ART 7 (Westlaw 5, 10) (2014). Alan Wayne Jones, Evidence-Based Survey of the Elimination Rates of Ethanol from Blood with Applications in Forensic Casework, 200 Forensic Science International 1, 14 (2010), (accessed Jan 15, 2015) (suggesting, from the results of studies, that experts use a range of elimination rates from 0.010 to 0.025 percent per hour). Go to TABLE OF CONTENTS 49 Go to INDEX 2. Short Citation Form for Periodical Articles After an authority has been cited fully in the text, use the following short citation form for subsequent references to the same authority: [Author’s last name], [Volume] [Periodical] at [page]: Karlan, 93 Colum L Rev at 604. NOTE: To determine the proper abbreviation for periodicals, first check the periodical itself to determine whether it provides for a particular citation form. If it does not, then follow The Bluebook abbreviations for periodicals. 3. Volume and Issue Numbers Omit volume and issue numbers for newspapers and popular periodicals, because the numbers might not be easily found: John Sedgwick, The Complexity Problem, The Atlantic 96 (Mar 1993). B. Books and Treatises NOTE: “Ed.” (with a period) refers to “editor”; “ed” (without a period) refers to “edition.” 1. Citation [Author(s)], [Volume number (if applicable)] [Title (in italics)] [subdivision, chapter or section (if applicable)], [page(s)] ([Edition (if there is more than one edition)] [Year]), e.g., Wayne R. LaFave, 3 Search and Seizure § 7.1 (2d ed 1987). Wayne R. LaFave, 3 Search and Seizure § 7.1(c), 17 (3d ed 1996). Bryan A. Garner, A Dictionary of Modern Legal Usage 138-39 (1987). Laird C. Kirkpatrick, Oregon Evidence § 401.02, Art IV-4 (4th ed 2002). Webster’s Third New Int’l Dictionary 930 (unabridged ed 2002). Black’s Law Dictionary 700 (7th ed 1999). Go to TABLE OF CONTENTS 50 Go to INDEX Noah Webster, 1 An American Dictionary of the English Language (unpagi-nated) (1828). Merriam-Webster Unabridged Dictionary, webster.com/unabridged/capable (accessed Feb 14, 2014). Gender Identity, Unabridged.Merriam-Webster.com (last updated Apr 2016). Thomas W. Lippman ed., The Washington Post Deskbook on Style 49-55 (2d ed 1989). William M. Collier, 3 Collier on Bankruptcy 506.04 (Lawrence P. King ed., 15th ed 1989). Appraisal Institute, The Appraisal of Real Estate 89 (12th ed 2001). Jacob Mertens, Jr., 8 Mertens Law of Federal Income Taxation § 32B:01 (rev 1999). Jacob Mertens, Jr., 15 Mertens Law of Federal Income Taxation § 56:45 (Supp 2002). (when citing supplement) Diagnostic and Statistical Manual of Mental Disorders 478 (5th ed 2013) (DSM-5). (first reference; include parenthetical if subsequent references or citations follow) The Ethical Oregon Lawyer § 4.4 (OSB CLE 1991). Paul Finkleman & Martin J. Hershock eds., The History of Michigan Law 169 (2006), available at Y C & p g = P R 4 & l p g = P R 4 & d q = h i s t o r y + o f + m i c h i g a n + l a w + f i n k l e m a n + h e r s h o c k & s o u r c e = b l & o t s = Yv a G Z 0 o z Q 1 & s i g = 5 i n y m 5 b Y C R X K A R t b b F x J 2 s R X p v I & h l = e n & s a = X & e i = ecYRVdXAD8uOyAT1jICIDQ&ved=0CDMQ6AEwBg#v=snippet&q= 1964%2C%20two%20yars%20after&f=false (accessed Mar 24, 2015). Mary Sue Henifin, Howard M. Kipen & Susan R. Poulter, Reference Guide on Medical Testimony, in Reference Manual on Scientific Evidence 439, 481 (Federal Judicial Center ed., 2d ed 2000). NOTE: It is also acceptable to cite a source with multiple authors where saving space is desired by using just the first author’s last name followed by et al, e.g., Henifin et al, Reference Guide on Medical Testimony, in Reference Manual on Scientific Evidence 439, 481 (Federal Judicial Center ed., 2d ed 2000). Go to TABLE OF CONTENTS 51 Go to INDEX 2. Short Citation Form [Author’s last name], [Volume number (if applicable)] [Title (in italics) (shorten title if necessary)] [subdivision, chapter or section (if applicable)] at [page(s)], e.g., LaFave, 3 Search and Seizure § 7.1 at 356. Garner, Modern Legal Usage at 378-79. Webster’s at 1935. DSM-5 at 480. C. Restatements Restatement (Second) of Torts section 847A comment c (1974). (first narrative reference) Restatement section 847B. (subsequent narrative form) Restatement (Second) of Torts § 847A comment c (1974). (first citation reference) Restatement § 847B. (subsequent reference, short citation form) D. Others 1. American Jurisprudence Reformation of Instruments, 45 Am Jur § 66 at 436 (1958). 2. American Law Reports William B. Johnson, Annotation, Use of Plea Bargaining or Grant of Immunity as Improper Vouching for Credibility of Witness in Federal Cases, 76 ALR Fed 409 (1986). Johnson, 76 ALR Fed at 415. (short citation form) 3. Miscellaneous Tax Materials • Tax Services Sales and Use Taxes: General Principles, 1300 Tax Mgmt (BNA) ¶ 1300.09.C3 (2000). Go to TABLE OF CONTENTS 52 Go to INDEX QUOTATION In General Quotations must duplicate the original material, including spelling, use of uppercase, and formatting (e.g., indents and tabs). All punctuation is placed inside the quotation marks, with the exceptions of colons, semicolons, and question marks that do not appear in the original. Permissible changes are set out in this section. Quoted material that contains 50 or more words is placed in block form. Quoted material that contains 49 or fewer words may be placed in block form for emphasis. Block quotations are single spaced, double indented, and enclosed within quotation marks. In a narrative context, the citation (aside from the case title) may follow the quotation. For example, The petitioner included discussion from Smith v. Jones, where the court stated, “We don’t know what they were thinking.” 123 Or App 456, 460, 999 P3d 789 (2015). While mildly amusing, that did not further his argument. Words that are used to show special expression or give special effect, such as slang words, words used in an ironic sense, philosophical terms, terms of art, etc., can be used within quotation marks. If the same phrase is used repeatedly, then those same quotation marks could become distracting, so an author should consider whether or not to repeat them. Go to TABLE OF CONTENTS 53 Go to INDEX I. Citations, Parenthetical Phrases, and Footnotes A. Placement When setting out a block quotation, the citation and parenthetical phrases (if any) are placed flush left on the next line with a double space between the quotation and the citation, e.g., 1. Citation “Defendant, of course, may argue that the safeguard for employees exceeds the ordinary standard of due care considering the nature of the risk and the foreseeability of injury. The court may explain the governmentally prescribed safeguard to the jury in an instruction that the jury may consider the safeguard in deter-mining whether the defendant exercised due care.” Shahtout, 298 Or at 605-06. 2. Parenthetical Phrase ORS 19.205(2)(a) provides, in part: “An order affecting a substantial right, and which in effect deter-mines the action .” (Emphasis added.) 3. Order of Citations and Parentheticals As stated in Welker, “[t]his court has held that a motion is controlled by its substance, not its caption. More specifically, this court has held that a motion was a motion for new trial although it was not denominated as such. Under ORS 19.270(4)(a), the trial court must have intended to enter an appealable judgment at the time of the filing of the notice of appeal.” 332 Or at 313 (citations omitted). 4. Footnotes The preferred method for quoting text that contains a footnote is to omit the footnote and indicate so by parenthetical, e.g., Go to TABLE OF CONTENTS 54 Go to INDEX “The difficulty with that argument is that the state has not estab-lished the requisite nexus between appellant’s mental illness and the potential substance abuse or failure to appreciate the risks of such abuse.” State v. Linde, 179 Or App 553, 561, 32 P3d 78 (2002) (footnote omitted). To retain a foonote within a blocked quotation, drop the quoted, blocked footnote under it, separated by a one-inch line, then proceed with new text (that is, do not set the quoted footnote as a new footnote at the bottom of the page), e.g., As this court explained in Kambury v. DaimlerChrysler Corp., 173 Or App 372, 387, 21 P3d 1089 (2001) (Edmonds, P. J., dissenting), “[b]ut the question is one of Oregon law, not federal law, the federal court’s decision was the earlier of the two, and it is the Oregon court’s decision—not that of the Ninth Circuit—that is binding for purposes of the certification law.8 It follows from the foregoing that this court should not accept certification of the first question, unless some other discretionary factor dictates a contrary conclusion. ____ “8 We recognize that the district court appears to be concerned about an inconsistency in decisions on this subject. However, our focus in searching for controlling precedent is narrower than the focus of the district court. The Oregon Court of Appeals decision in [Korbut] is “controlling precedent” for the purposes of ORS 28.200 [the statute authorizing answers to questions of law cer-tified by other courts] and our exercise of discretion under that statute.” It is unclear from the opinion in Western Helicopter Services whether the Supreme Court was telling the United States District Court that it should follow Oregon law rather than the federal cases interpreting Oregon law or whether the court was in fact affirming that Korbut was controlling precedent on the issue presented by the certified question. 5. Block Quotations Within Block Quotations When formatting a second level of blocked quoted text within blocked quoted text, use the standard indent for the first level of quotation, and then for the internal quoted text, use a double indent for both the right and left margins. “the state must prove that the defendant’s consent was independent of, or only tenuously related to, the illegal police conduct. As the court explained, “ ‘consent is insufficient to establish the admissibility of evidence from a warrantless search if the state cannot prove that the consent Go to TABLE OF CONTENTS 55 Go to INDEX was independent of, or only tenuously related to, any preceding violation of the defendant’s rights under Article I, section 9. Unless the state is able to make that showing, then the defendant’s con-sent cannot operate to validate a warrantless search because the defendant’s consent itself derived from a violation of the defen-dant’s rights under that state constitutional provision.’ ” Jones, 265 Or App at 678 (quoting State v. Yunker, 330 Or 388, 395, 457 P3d 890 (2013) (emphasis in Yunker)). B. Use of Parenthetical Phrases With Quotations Within Text If internal quotation marks or brackets are omitted, so indicate, e.g., “By looking to only the current employment of the land, the law ignores the past and any intentions with regard to future use.” Everhart v. Dept. of Rev., 15 OTR 76, 81 (1999) (internal quotation marks omitted). or In Everhart v. Dept. of Rev., 15 OTR 76, 81 (1999), the court rejected that argument, stating, “By looking to only the current employment of the land, the law ignores the past and any intentions with regard to future use.” (Internal quotation marks omitted.) NOTE: It is correct to add either a parenthetical indicating that internal quotation marks are omitted or to leave in the quotation marks and add a parenthetical indicating the source that you are quoting, but not both. Oppressive conduct is “ ‘burdensome, harsh and wrongful conduct; a lack of probity and fair dealing in the affairs of the company to the prejudice of some of its mem-bers; or a departure from the standards of fair dealing, and a violation of fair play on which every shareholder who entrusts his money to a com-pany is entitled to rely.’ ” Polk v. Hergert Land & Cattle Co., 5 P3d 402, 404 (Colo App 2000) (quoting Jorgensen v. Water Works, Inc., 218 Wis 2d 761, 783, 582 NW2d 98, 107 (Wis App 1998) (brack-ets omitted)). Include a parenthetical phrase to differentiate emphasis being added to quoted material as opposed to emphasis that was already in the original text, e.g., “While mass-appraisal techniques may place heavy reliance upon cost, cost alone is not determinative of market value.” Su v. Dept. of Rev., 15 OTR 305, 308 (2001) (emphasis added). Go to TABLE OF CONTENTS 56 Go to INDEX “ORS 653.295, however, provides no remedy for discharge of an employee .” Dymock v. Norwest Safety Protective Equipment, 172 Or App 399, 406, 19 P3d 934 (2001) (emphasis in original). When the material being quoted contains a quotation, the text of which is emphasized, include a parenthetical phrase that notes the source of the emphasis (use emphases if plural), e.g., As in Gladhart v. Oregon Vineyard Supply Co., this court again repeats that “the Brown court ‘only acknowledged that perhaps strict liability should require the danger to be one endangering human life or safety, a different question from whether tort recovery should be limited to such an injury,’ and that ‘[t]hat difference determines the decision in the present case.’ ” 164 Or App 438, 451-52, 994 P2d 134 (1999) (quoting Russell v. Ford Motor Company, 281 Or 587, 593-94, 575 P2d 1383 (1978) (emphases in Russell)). “Apparent authority is created ‘only by some conduct of the principal which, when rea-sonably interpreted, causes a third party to believe that the principal consents to have the apparent agent act for him on that matter.’ ” Badger v. Paulson Investment Co., Inc., 311 Or 14, 24, 803 P2d 1178 (1991) (quoting Mattson v. Commercial Credit Business Loans, 301 Or 407, 422, 723 P2d 996 (1986) (emphasis added)). Badger v. Paulson Investment Co., Inc., 311 Or 14, 24, 803 P2d 1178 (1991) (“Apparent authority is created ‘only by some conduct of the principal which, when reasonably interpreted, causes a third party to believe that the principal consents to have the appar-ent agent act for him on that matter.’ ” (Quoting Mattson v. Commercial Credit Business Loans, 301 Or 407, 422, 723 P2d 996 (1986) (emphasis in Mattson).)). Multiple notations about a quotation may follow any order that clearly communicates whether and how the text was altered. If in doubt, choose this order: “in original,” “omitted,” and “added.” It is also acceptable, to improve readability, to use “first emphasis added; second emphasis in original,” etc. In particular, petitioner focuses on the following emphasized statements that appear in the form under the heading “Information Considered at the Hearing”: “Discovery is not permitted. Requests that the Board research and obtain information you want considered cannot be honored. It is your responsi-bility to provide that information. You do not have the right to call wit-nesses or to cross-examine witnesses who have provided information to the Board.” (Underscoring in original; footnote and boldface omitted; emphasis added.) Go to TABLE OF CONTENTS 57 Go to INDEX Consistently with those principles, the court more recently summarized the requirements of ORS 136.440 as follows: “By its terms, ORS 136.440(1) requires only that the corroborat-ing evidence tend to connect the defendant with the commission of the offense, here, aggravated murder. That statute does not require corrobora-tion of a particular theory of the commission of the offense. “It is not necessary that the corroborating evidence be direct and positive; it may be circumstantial. Nor is it necessary that there be indepen-dent corroborating evidence with respect to every material fact necessary to be established to sustain a conviction for the commission of a crime. Where there is any evidence apart from that of the accomplice tending to connect the defendant with the commission of the crime, the question of whether the accomplice’s testimony is corroborated is one for the trier of fact.” State v. Walton, 311 Or 223, 242-43, 809 P2d 81 (1991) (emphasis in original; citations omitted). It depends, the court stated, on the nature of the restriction—in particular, whether the restriction is on the contribution itself: “If it can be shown that financial contributions and expenditures are the free expression of opinion, laws limiting such activities run afoul of the constitutional protection. But lawmakers might choose to impose require-ments distinct from contribution or expenditure limitations (e.g., require-ments of disclosure of financing sources and the extent of any gift) as well as various sanctions (e.g., civil or criminal penalties, disqualification from the ballot or Voters’ Pamphlet, and the like) and their choice may not nec-essarily offend the constitutional requirement.” Id. at 523 (first emphasis added; second emphasis in original; internal quotation marks omitted). When a citation includes both a parenthetical phrase and subsequent history, place the parenthetical after the subsequent history, e.g., Defendant notes that the fact that a search occurred cannot, alone, support an inference that the searching officers had subjective probable cause. See State v. Bickford, 157 Or App 386, 390 n 1, 970 P2d 234 (1998), rev den, 329 Or 589 (2000) (“If a trial court could infer subjective probable cause from the arrest, we would never need to inquire into subjective probable cause for the arrest.”). Go to TABLE OF CONTENTS 58 Go to INDEX When a parenthetical contains a quotation that has been altered, include an internal parenthetical and adjust the punctuation accordingly, e.g., The court held that claims of increasing pain due to injury, standing alone, were insuf-ficient to establish an aggravation claim. SAIF v. Walker, 330 Or 102, 116, 996 P2d 979 (2000) (stating that ORS 656.273(1) requires “proof, based upon medical evidence supported by objective findings” (emphasis added)). The court held that claimant’s claims of increasing pain due to the injury, standing alone, were insufficient to establish an aggravation claim. SAIF v. Walker, 330 Or 102, 116-17, 996 P2d 979 (2000) (“Under ORS 656.005(19), however, such ‘objective find-ings’ may include evidence of worsened symptoms.” (Emphasis in original.)). When appending information to a citation, using terms such as cited in, quoted in, construed in, etc., those words are italicized, Keenan v. Norris-Lampe, 330 Or 456, 777 P2d 897 (1999), construed in Bennett v. Bauman, 333 Or 566, 790 P2d 654 (2001). Citing ORS 136.425 (2005), amended by Or Laws 2009, chapter 875, section 1, defen-dant contends that, by itself, a confession is insufficient to support a conviction for an offense. When including works that the primary authority quotes, discusses, or mentions, the lead-in words should be indicated parenthetically and not italicized, e.g., See Keenan v. Norris-Lampe, 330 Or 456, 777 P2d 897 (1999) (citing Bennett v. Bauman, 329 Or 566, 770 P2d 654 (1998)); see also Brown v. Uphoff, 381 F3d 1219, 1226 (10th Cir 2004) (“Unlike Gideon, Crawford does not ‘alter our understanding of what constitutes basic due process,’ but merely sets out new standards for the admis-sion of certain kinds of hearsay.” (Quoting United States v. Mora, 293 F3d 1213, 1219 (10th Cir 2002).)). For more on parentheticals, refer to pages 18 to 19. Go to TABLE OF CONTENTS 59 Go to INDEX II. Use of Uppercase, Brackets, and Ellipsis Within Quotations A. Use of Uppercase Generally, the first letter of a quotation that reads as a full sentence must be in uppercase, e.g., The trial court issued a letter opinion that addressed defendant’s motion as follows: “There is no evidence supporting the first change.” The witness testified, “Although I am not a doctor, I play one on TV.” Sometimes this requires changing the initial letter of a quotation to uppercase, e.g., The court observed, “[U]npreserved error will ordinarily not be reviewed on appeal.” The examining physician’s report declared, “[C]laimant was medically stationary as of February 12, 1992.” In the following example, the use of a bracketed lowercase letter indicates that the initial letter was in uppercase in the original, e.g., Incorrect: Defendant argued, “[t]he trial court erred in excluding witnesses from the courtroom.” Correct: Defendant argued, “The trial court erred in excluding witnesses from the courtroom.” EXCEPTION: When a quotation that reads as a full sentence is incorporated into the grammatical structure of the sentence (often preceded by “that”), do not uppercase the first letter and use brackets if the first letter of the quotation was uppercase in the original, e.g., The court observed that “[u]npreserved error will ordinarily not be reviewed on appeal.” NOTE: For using uppercase after colons, refer to page 79. B. Use of Brackets The use of brackets in quoted material indicates that something has been changed, modified, or deleted from how it was originally published. 1. Indicate a Change in Uppercase or Lowercase You should “[p]lace commas and periods inside quotation marks.” Go to TABLE OF CONTENTS 60 Go to INDEX 2. Alter a Word “Plaintiff place[s] substantial weight on its belief that the legislature intended to restore the common practice of exempting brooder houses.” The prohibition was enacted to “ensure[ ] that dogs will have their day.” 3. Indicate Substituted or Added Words The court’s “jurisdiction [is limited] to determining whether petitioners’ assessment may be spread over a period of at least 10 years.” 4. Add Punctuation to a Quotation Where There Previously Was No Punctuation and the Punctuation Is Not Part of the Larger Sentence The standard of care applicable to physicians requires the use of “that degree of care, skill[,] and diligence” as is used by physicians in the same or similar community. The trial court concluded that “there is no issue of material fact,” but it also concluded that defendant was not entitled to judgment as a matter of law. [No comma in original; brackets not needed, because comma is part of larger sentence.] 5. Modify Punctuation that Appears in the Original Text Punctuation in quoted text can be replaced with other punctuation with the use of brackets: Original: “The Court of Appeals has 13 members, but they sit in panels of three.” New: “The Court of Appeals has 13 members[;] they sit in panels of three.” If quoted material ends before the new sentence ends, then punctuation that is appropriate for the new sentence can be included inside the quotation marks: Original: Defendant’s argument is beyond reproach and compels us to strike his brief and order him to pay petitioner’s costs. New: In Jones, the court characterized the defendant’s argument as “beyond reproach,” struck his brief, and ordered him to pay the petitioner’s costs. Go to TABLE OF CONTENTS 61 Go to INDEX If omitting quoted material at the end of a quoted sentence, but ending the new sentence with the partially quoted material, denote the omitted quoted material before the period in the new sentence with an ellipsis: Original: A party shall make a notation of exception either orally on the record or in a writing filed with the court. New: ORCP 59 H(2) provides that “[a] party shall make a notation of exception either orally on the record or in a writing .” 6. Insert a Citation Within a Quotation When quoting text that refers to a case by use of supra rather than a citation, add the citation in brackets, e.g., “This court has declared that evidence is suppressed for violations of the Oregon Constitution ‘to preserve rights to the same extent as if the gov-ernment’s officers had stayed within the law.’ State v. Davis[, 295 Or 227, 234, 666 P2d 802 (1983)].” Use a short citation form if the case has already been cited in full, e.g., “A discovery rule cannot be assumed, but must be found in the statute or lim-itations itself. Huff[, 322 Or at 462].” When modifying quoted material to add emphasized text, emphasize the brackets themselves, in addition to the inserted emphasized text, only when the words on both sides of the brackets are already emphasized, e.g., Correct: “Plaintiff’s [first] argument ” Correct: “Plaintiff’s [first] argument ” Correct: “Plaintiff’s [first] argument ” Incorrect: “Plaintiff’s [first] argument ” Incorrect: “Plaintiff’s [first] argument ” When quoting text that refers to a VAWA case using an unredacted case title, use brackets to redact the name of the protected party to initials, e.g., “ ‘Danger’ in this context means ‘a threat of physical injury, not merely a threat of annoyance or harassment.’ [K. R.] v. Erazo, 248 Or App 700, 706-07, 274 P3d 214 (2012). To ‘ “[c]oerce” means to restrain, compel or dominate by force or threat.’ ORS 163.730(2).” H. L. P., 309 Or App at 113-14. 7. Indicate a Significant Mistake in the Original “[Sic]” is used to indicate that the author being quoted has used a word incorrectly or has used an incorrect spelling of a word within the context of Go to TABLE OF CONTENTS 62 Go to INDEX the sentence that confuses or changes the meaning of the sentence. Obvious typographical errors may be silently corrected, but any idiosyncrasy of spelling (particularly from older works) is preserved, e.g., “defence.” When using [sic], it is italicized, enclosed in brackets, and placed after the uncorrected word, e.g., “The court’s rational [sic] for its decision was based entirely on legal precedent.” “We are going to precede [sic] with our plans for the law suit.” (Here, the use of [sic] signals the reader that “rational” should actually read “rationale”; “precede” should read “proceed.”) Bracketing and correcting an error in quoted material instead of using [sic] is usually sufficient as long as it does not alter the meaning. The use of [sic] should be avoided to point out minor grammatical errors, to indicate disagreement with a word’s usage, or to appear highhanded. Instead, the preferred method is to silently correct or to use brackets, e.g., Incorrect: “The company, on its own volition, intervened in the trial Court [sic] proceedings .” Correct: “The company, on its own volition, intervened in the trial [c]ourt proceedings .” Correct: “The company, on its own volition, intervened in the trial court proceedings .” Incorrect: “The lab technician dropped the vile [sic] of blood.” Correct: “The lab technician dropped the vi[al] of blood.” Correct: “The lab technician dropped the vial of blood.” NOTE: Where a word or sentence would be correct either way, do not use [sic]. C. Use of Ellipsis 1. Partial Omission Use three asterisks (not periods), separated by spaces, to indicate omission of a word or words within a sentence, omission of sentences within a block quotation, and omission of text after a subsection number, e.g., Go to TABLE OF CONTENTS 63 Go to INDEX The claimant seeks compensation “for mental and emotional distress and damages.” In amending ORS 656.273(1) in 1995, the legislature “neither repealed nor amended ORS 656.273(8) but it may have changed how ORS 656.273(8) is applied.” When quoting a full sentence except the end, use three asterisks, followed by a period, to indicate omission of the end of the sentence: “The evidence directly establishes only the truth of the primary fact or facts from which an inference may be derived .” EXCEPTION: If quoting a full sentence except the end and the quoted material originally ended with a punctuation mark, then use only a bracketed period to indicate the change of punctuation (and, by inference, the end of the sentence): “A court may consider unpreserved error under the plan error doctrine only if the error is one of law[.]” (period replaces semicolon and the remainder of the sentence) When including quoted material within a sentence and the new sentence ends before the quoted material ended, an ellipsis is not needed: Correct: If reasonable minds can disagree, then “any error is not plain.” Incorrect: If reasonable minds can disagree, then “any error is not plain .” Use three asterisks (not followed by a period) to indicate omission of one or more sentences at the end of a paragraph within (as opposed to at the end of) a block quotation: “The majority’s reading of the statute requires us to make two leaps of faith. The first leap is that, in 1965, the legislature used ‘gear or equip-ment’ to mean exactly what ‘appliance’ meant between 1921 and 1965. The second leap of faith is that the terms ‘boats, fishing gear and vehicles’ mean exactly what ‘gear or equipment’ means in ORS 506.655 . “We decline to take those leaps for several reasons.” NOTE: In the preceding example, the first ellipsis indicates that a sentence otherwise quoted in full has been ended prematurely. The second ellipsis indicates that an additional sentence or sentences has been omitted. Go to TABLE OF CONTENTS 64 Go to INDEX Omission of text after a subsection number: “(1) As used in ORS 308A.050 to 308A.128, ‘farm use’ means the current employment of land for the primary purpose of obtaining a profit in money by: “(a) harvesting and selling crops.” A block quotation that starts somewhere other than the beginning of the quoted sentence or paragraph may be treated a few different ways, depending on the author’s use of the quoted material. An ellipsis is not necessary to begin a quotation that starts somewhere other than the beginning of the sentence or paragraph, e.g., The issue in Merriweather was as follows: “[W]hether a circuit court order compelling witnesses to appear and testify before a grand jury was appealable under ORS 19.205(4) as the product of a ‘special statutory proceeding.’ ” The court has previously explained that “neither ORS 279.340 nor any other provision setting out require-ments for hours of labor by public employees, ORS 279.334 - 279.342, expressly states what is meant by the phrase ‘directly employed’ or indicates the categories of persons to whom the phrase applies.” Place the beginning of a block quotation flush left when the original quoted material is not the first sentence of a paragraph, e.g., As this court has stated before, “[e]ven if that assumption is accurate, it is insufficient because even in a high crime neighborhood unprovoked flight does not invariably lead to reasonable suspicion . Like unprovoked flight itself, presence in a high crime neighborhood is a fact too generic and susceptible to innocent explanation to satisfy the rea-sonable suspicion inquiry.” Do not use an ellipsis if quoting only part of a sentence, e.g., Again, the legislature expressly provided that those actions be “commenced not later than two years after the date on which the plaintiff first discovered, or in the exercise of reasonable care should have discovered,” the death, injury, or disease. Go to TABLE OF CONTENTS 65 Go to INDEX Do not use an ellipsis to denote the omission of a citation or footnote. Instead, use “citation omitted” or “footnote omitted” in a parenthetical. When quoting only part of a statute, rule, etc., use “in part,” e.g., ORS 305.440 provides, in part, “If no appeal is taken to the Supreme Court, the decision of the court shall constitute a final determination of the matter.” 2. Block Omission Use five asterisks to indicate omission of a block of text such as paragraphs or sections, but not to indicate an omission of additional material after an ending paragraph, e.g., OAR 635-005-0180 provides, in part: “It is unlawful for commercial purposes to take, land, or possess sea urchins: “ “(3) Without first obtaining a permit .” NOTE: If the original has additional text, it is not necessary to add three asterisks following the period. Go to TABLE OF CONTENTS 66 Go to INDEX STYLE GUIDE In General This section addresses issues of word treatment, grammar, punctuation, and usage that arise frequently in opinion drafting. It is not exhaustive. Grammar and usage are not exact sciences; there are many questions of style about which reasonable minds can differ. To promote consistency, however, the courts generally follow the conventions outlined below. Other Resources The courts use the following reference works (which also have updated editions) to resolve style issues not addressed in the Oregon Appellate Courts Style Manual (Updated 2023). • Webster’s Third New Int’l Dictionary (unabridged ed 2002). For questions regarding spelling, word usage, or proper hyphenation when dividing words, the appellate courts adopted Webster’s, in 1993, as the official dictionary. • The Chicago Manual of Style (14th ed 1993). This reference work addresses many grammatical standards and punctuation issues not covered here. • Bryan A. Garner, A Dictionary of Modern American Usage (1998). This work explains many principles of grammar and punctuation. It is also a very helpful guide to word usage. • Bryan A. Garner, A Dictionary of Modern Legal Usage (2d ed 1995). This is a helpful guide to modern legal usage. It also addresses many general questions of grammar and punctuation. • William A. Sabin, The Gregg Reference Manual (9th ed 1999). Readily available desktop reference book. Go to TABLE OF CONTENTS 67 Go to INDEX I. Spelling, Font, and Treatment of Words A. Use of Italics and Roman Typeface 1. Foreign Words Some foreign words and phrases commonly used in legal writing are not italicized, e.g., ad hoc etc. pro tempore ad hominem habeas corpus quid pro quo ad valorem mandamus remittitur de facto per curiam subpoena en banc pro rata vice versa Other less common foreign words and phrases that are italicized include the following. When using such words in the body of an opinion, use the periods and diacritical marks as set out below. a priori i.e. quantum meruit ab initio id. quantum valebant ad litem in camera quo warranto amicus curiae / amici curiae in limine res ipsa loquitur arguendo in re res judicata certiorari in toto respondeat superior contra inter alia scienter coram nobis ipse dixit sic de novo ipso facto sine qua non dictum non sequitur stare decisis dubitante nunc pro tunc sua sponte duces tecum per se sub silentio e.g. post hoc ultra vires et al. prima facie vice et seq. pro hac vice vis-à-vis ex parte pro se viz. ex post facto, but qua voir dire Ex Post Facto Clause 2. Signals Introductory signals such as cf., accord, see, e.g., compare / with, and see generally are italicized. “See” is not italicized when it functions as a verb and not as a signal, e.g., For discussion on the merits, see Smith, 300 Or App at 150. Go to TABLE OF CONTENTS 68 Go to INDEX 3. Case Names and Other Authorities Case names are italicized, as are the titles of books and other authorities. Please refer to the Citation Section for examples. When a case or other authority takes a possessive ending, the apostrophe and “s” are not italicized: Miranda’s holding has been called into question. B. Use of Uppercase and Lowercase 1. Use Uppercase for the Following: a. Proper Names b. Complete Official Titles of a Public Official or Entity (but not Abbreviated Titles), e.g., Uppercase: Not Uppercase: (exceptions noted) Alaska Supreme Court the court Appellate Commissioner the commissioner Assistant Attorney General John Jones assistant attorney general Attorney General the Attorney General (exception) Board of Parole and Post-Prison Supervision the board City of Eugene the city Clackamas County District Attorney’s Office the district attorney’s office Criminal Law Revision Commission the commission Department of Revenue the department Deputy Sheriff Stanley the deputy Director of the Department of Consumer and Business Services the director Governor the Governor (exception) House of Representatives the House (exception) Judge Atwater the judge Multnomah County Circuit Court the court, or the circuit court Officer Krupke the officer Oregon Court of Appeals / Court of Appeals the court Oregon Legislative Assembly / 1997 Legislative Assembly the legislature Oregon State Bar the Bar (exception) Oregon Supreme Court / Supreme Court the court Magistrate Jones the magistrate Portland Police Officer the officer Go to TABLE OF CONTENTS 69 Go to INDEX Secretary of State the secretary Representative Smith the representative Salem-Keizer School District the district Senate the Senate (exception) Senator Jones the senator State of Oregon the state Oregon Tax Court / Tax Court the court United States Court of Appeals for the Ninth Circuit Ninth Circuit (exception) United States Supreme Court the Court (exception) Washington County Jail the jail Workers’ Compensation Board the board NOTE: In state jurisdictions having more than one appellate court at the same level, do not use uppercase for any shortened version of the name of that court. For example, the full form “California Court of Appeal, First Appellate District” uses uppercase, but the shortened form “court of appeal” does not. (There is no “s” after “appeal” for the California courts of appeal.) Similarly, “New York Supreme Court, Appellate Division, Third Department” uses uppercase, but the shortened form “supreme court” does not. (In New York, the Court of Appeals is the highest court, while the appellate divisions of the supreme court are the intermediate appellate courts.) However, it is permissible to use uppercase for shortened names of federal appellate courts, e.g., the Ninth Circuit. c. Months and Days of the Week (usually written in month-day-year sequence), e.g., He filed the motion on Monday, January 23, 1993. The action occurred in November 2013. Defendant appeals from the January 16, 1994, order. d. Full Title of a Constitution, Constitutional Amendment, or Clause of a Constitution, e.g., Article I, section 9, of the Oregon Constitution (but “state constitution”) Fifth and Fourteenth Amendments to the United States Constitution (but “federal constitution”) Contract Clause dormant Commerce Clause Due Process Clause Ex Post Facto Clause Privileges and Immunities Clause Proportionality Clause Go to TABLE OF CONTENTS 70 Go to INDEX In the phrase “Oregon and United States constitutions,” do not uppercase “constitutions,” because the full title of neither document is being used. Note that either of the following formulations is preferred: “state and federal constitutions” or “the Oregon Constitution and the United States Constitution.” e. “En Banc” and “Per Curiam” on Title Pages NOTE: When used within the text on an opinion, those terms do not start with uppercase letters, e.g., “The Court of Appeals, sitting en banc, reversed.” f. “Intoxilyzer” or “Breathalyzer” (Those are proper names for particular breath testing equipment.) “Ziploc” or “Taser” (trademarked names) (but zippered bag or stun gun) g. “Class” or “Count” (When referring to misdemeanor or felony charges), e.g., Class A felony Count 1 of the indictment Counts 1 and 2 h. “Schedule” (When referring to drugs), e.g., Schedule I drug i. “Act” or “Code” (When referring to the full title of specific Acts or Codes), e.g., The Oregon Condominium Act specifically lists those conditions. The act does not set out the condition on which appellant relies. The Bankruptcy Code provides for that situation. The code further provides a remedy in the event of a violation. j. Particular Sections of the Oregon Revised Statutes Identified Collectively, e.g., Civil Code Oregon Public Records Law Criminal Code Unlawful Trade Practices Act Oregon Evidence Code Workers’ Compensation Law Go to TABLE OF CONTENTS 71 Go to INDEX k. Social Security SSA Social Security number Her income consisted of Social Security and wages from part-time employment. 2. Do Not Use Uppercase for the Following: a. Generic Terms, e.g., “federal” or “state,” as in the state state constitution federal constitution statute of frauds statute of limitations BUT do use uppercase when part of a full proper name, e.g., Federal Land Bank State of Oregon b. The Words “chapter” or “section” (When referring to a specific chapter or section within a sentence), e.g., ORS chapter 10; Article I, section 9. But Chapter 7 (when referring to the Bankruptcy Code) c. The Terms “x-ray,” “horizontal gaze nystagmus,” “administrative law judge,” and “driving under the influence” A horizontal gaze nystagmus (HGN) test The doctor ordered x-rays of the claimant’s lower back. The administrative law judge (ALJ) He was convicted of driving under the influence (DUII). Go to TABLE OF CONTENTS 72 Go to INDEX d. The Seasons or Centuries It is now okay to wear white in the fall after Labor Day! In the eighteenth century, women could wear white only as a petticoat. She was convicted of wearing a red petticoat in the fall of 1858. (or, fall 1858) e. The Words “website” or “internet” C. Numbers and Dates In General Numbers one through nine are spelled out when used in text, except when listing a series of like objects, e.g., Judges from five states came to the conference: 5 from Washington, 2 from California, 10 from Oregon, 6 from Idaho, and 1 from Alaska. All numbers after nine are expressed as numerals, except when they begin a sentence, e.g., Thirty-four judges attended the conference. Defendant raises 10 assignments of error. Very large numbers are expressed in figures followed by million, billion, etc., e.g., 5 million people; 2 billion particles; $10 million All ordinal numbers (i.e., numbers that measure position) are spelled out when used in narrative, e.g., Plaintiff’s twelfth assignment of error is not well taken. I chased my cat Jethro off the sofa for the one hundredth time. This is the court’s seventy-third oral argument day this year. NOTE: (1) Ordinal numbers consisting of more than one word are hyphenated if the corresponding cardinal number is hyphenated, e.g., “seven hundred and twelfth” is not hyphenated, because “seven hundred and twelve” is not hyphenated, but “eighty-second” is hyphenated, because “eighty-two” is hyphenated. (2) Ordinal numbers are expressed Go to TABLE OF CONTENTS 73 Go to INDEX numerically to identify reports and courts in citations, e.g., Boston v. Cream Pie, 283 F2d 1 (9th Cir 1999). 1. Fractions, Decimals, and Ratios Fractions appearing in nonquoted text either alone or with numbers less than 10 are spelled out; fractions appearing with numbers 10 or higher are expressed as numerals, e.g., Four and one-half years old 23 1/4 One third When fractions are expressed as numerals, insert a space between the whole number and the fraction. Do NOT use a hyphen. (“10 2/3,” not “10-2/3.”) Use numerals when using a decimal point, e.g., .08 percent blood alcohol content Use numerals when denoting ratios, e.g., 4:1 or ratio of 4 to 1 (as they are easier to read inside text) However, spell out numbers when denoting measurement, e.g., The subject was five feet eight inches tall. 2. Percentages Spell out the word “percent” when used in text. Use the percent sign (%) in tables, e.g., Fifteen percent of the people in 1993 voted against the ballot measure. In 1995, 15 percent of the people voted against the ballot measure. Year Percent 1991 10% 1992 20% 1993 15% Go to TABLE OF CONTENTS 74 Go to INDEX 3. Time Include minutes and a.m. or p.m., e.g., Oral arguments begin at 8:00 a.m. (not “8 a.m.”) We will break for lunch at 12:15 p.m. The defendant was last seen at 8:00 p.m. (not 8 o’clock in the evening) 4. Dates Three-part dates are set off with a comma between the day and the year and, generally, a comma after the year, e.g., Defendant appeals from the January 16, 1994, order. When indicating an inclusive period of time, omit the comma after the first year, e.g., Defendant was on probation from June 14, 1980 to July 30, 1982. When referring to a date by month and day, do not use endings with the day, e.g., the September 19 hearing (not September 19th) When indicating time by month and year only, there is no comma before or after the year, unless the sentence structure requires a comma after the year, e.g., Three lawyers attended the April 1990 deposition. The trial, which was scheduled for June 1990, was postponed several times. When indicating a period of several years, use “to” or “through,” not a hyphen, e.g., Judge Caspar was on the bench from 1900 to 1921. EXCEPTION: Hyphens are used for tax years. Do not repeat the “19” for inclusive years, e.g., tax years 1995-97, but tax year 1999-2000. For years ending in double “00” repeat the entire sequence, e.g., 2000-2001. An indication of a decade does not take an apostrophe, e.g., 1980s Go to TABLE OF CONTENTS 75 Go to INDEX Indications of time, as shown here, take an apostrophe, e.g., 24 months’ incarceration (or 24 months of incarceration) six weeks’ time (or six weeks of time) 5. Money When referring to dollars, use the dollar sign; do not spell out “dollars,” e.g., $5 million; $2 billion Use $25.00 if there are other mixed dollar and cents amounts referred to, otherwise use $25 if standing alone, e.g., Plaintiff sought $10,000.00 in attorney fees and $875.45 in costs. The court entered judgment in the amount of $10,000. Use a comma in numbers larger than three digits, e.g., $1,500 D. Acronyms / Initialisms Acronyms are composed of the initial letters or parts of a compound term. An acronym is usually read or spoken as a single word, rather than letter by letter, e.g., AIDS. Initialisms are abbreviations pronounced letter by letter, e.g., ORCP, LCDC, DWS, HGN, JNOV, etc. Periods are usually omitted to improve readability. In either event, all but the most common acronyms / initialisms are spelled out upon first usage, followed by the acronym / initialism enclosed in parentheses, e.g., Mountain View Hospital District (MVHD) filed a motion for summary judgment. The article that precedes an acronym or initialism depends on how the abbreviation reads: If it is pronounced beginning with a vowel sound, then use “an.” If it is pronounced beginning with a consonant sound, then use “a.” an LCDC order an ORAP provision an ORCP ruling a LUBA opinion a Umatilla County sheriff Go to TABLE OF CONTENTS 76 Go to INDEX Generally, add an “s” when forming a plural of something singular, e.g., SSNs, UFOs, MDs, CCRs, and add an apostrophe to show possession, e.g., LUBA’s analysis or PERS’s liabilities. E. Titles and Offices Chief Justice Balmer chief justice of the Supreme Court Your Honor The Honorable Thomas A. Balmer Queen Mary the queen Dr. Smith (becomes Smith after first reference) F. Abbreviations Academic Degrees Ph.D. (“Ph.D.” is preferred over “PhD” with no periods.) Corpus Juris Secundum Corpus Juris Secundum (an encyclopedia of United States law at the federal and state levels) is abbreviated as CJS without periods. Judgment Notwithstanding the Verdict When abbreviating, use “JNOV” instead of “j.n.o.v.” Months of the Year (no periods, when appearing in citations): January – Jan July – July February – Feb August – Aug March – Mar September – Sept April – Apr October – Oct May – May November – Nov June – June December – Dec Go to TABLE OF CONTENTS 77 Go to INDEX II. Punctuation A. Apostrophes If a singular word ends in “s,” then add “ 's ” to show possession, e.g., George Harris’s house witness’s testimony Clauss’s land PERS’s liabilities or if a name ends in “z,” or “ x,” e.g., Sanchez’s argument Rex’s children Exception: Singular entity names that end in “s” take only an apostrophe, e.g., Boy Scouts’ answer General Motors’ complaint Form the possessive of most plural words by adding only an apostrophe: the Joneses’ driveway the boys’ gym Plural words that do not end in “s” take “ 's ” to form the possessive: children’s toys women’s tournament If there are two (or more) possessors, proper punctuation depends on meaning. If the possession is separate, each possessor takes an apostrophe. If the possession is joint, only the last possessor takes an apostrophe, e.g., mother’s and father’s children (if referring to each parent’s children with another partner) mother and father’s children (if they have children together) mother’s and father’s arguments (if they are separate arguments) mother and father’s arguments (if their arguments are joint) Use an apostrophe with the following indications of time, e.g., 36 months’ incarceration two weeks’ time Go to TABLE OF CONTENTS 78 Go to INDEX Do not use an apostrophe with these particular word phrases, e.g., Attorney fees IRAs When a name of a case or other italicized authority takes a possessive ending, the “ 's ” is not italicized: Miranda’s holding has been called into question. NOTE: Sometimes the use of “ 's ” can be awkward and make for difficult reading. In those situations, use “of” instead. E.g., for “Church at 295 S. 18th St.’s analysis,” consider using “the analysis of Church at 295 S. 18th St.”; for “ORS 123.456(1)(a)’s wording,” consider using “the wording of ORS 123.456.” B. Colons 1. Colons may be used to separate a grammatically complete sentence from, for example, another grammatically complete sentence or a list: The court announced a general rule: Attorney fees may not be awarded in the absence of an authorizing statute or contract provision. The officer found several items in defendant’s apartment: a scale, five plastic bags containing white powder, and $751 in small bills. Colons may not be used after a grammatically incomplete thought: Incorrect: The panel consisted of: Judge Yunker, Judge Bauman, and Judge Keenan. Correct: The panel consisted of three judges: Judge Yunker, Judge Bauman, and Judge Keenan. Incorrect: Defendant argues: (1) the trial court erred in admitting evi-dence, (2) the error was not harmless, and (3) alternatively, the trial court erred in denying his motion for directed verdict. Correct: Defendant makes three arguments: (1) the trial court erred in admitting evidence, (2) the error was not harmless, and (3) alternatively, the trial court erred in denying his motion for directed verdict. Go to TABLE OF CONTENTS 79 Go to INDEX EXCEPTION: Colons may be used after verbs to introduce block quotations only, e.g., ORS 123.456 provides: “Now is the time for all good citizens to come to the aid of their country.” As part of that analysis, Justice Brennan observed: “Under Article III of the Constitution, this Court may only adjudi-cate actual, ongoing controversies.” If the quotation is not blocked, then use a comma. ORS 123.456 provides, “Now is the time for all good citizens to come to the aid of their country.” 2. Use of Uppercase After Colons If the material following a colon is not a complete sentence, then it does not begin with an uppercase letter. The court excluded three items: a gun, a ukelele, and a garden hose. If the material following the colon is a complete sentence, then it does begin with an uppercase letter. The rain in Spain does not fall mainly on the plain: It mainly falls in the north-ern mountains. The court did not deny defendant’s motion outright: It deferred its ruling until the evidence was offered at trial. C. Commas Remember the basic comma rule: Use a comma only when you know why you are using one. The following guidelines do not set out all the situations in which commas are correctly used. They may be useful, however, in resolving some of the most frequent Go to TABLE OF CONTENTS 80 Go to INDEX questions about when to use, and when not to use, commas. For more guidance, see The Chicago Manual of Style; Bryan A. Garner, A Dictionary of Modern American Usage; and William A. Sabin, The Gregg Reference Manual. 1. Items in a Series A series of more than two items should be separated by commas. The courts’ convention is to use the serial comma (i.e., a comma before the conjunction in a series of more than two items), because it eliminates the possibility of misreading: Sheila invited Ben’s parents, friends, and coworkers. I agree that defendant’s brief is well written, his analysis is cogent, and his arguments are persuasive. A series of only two items should not be separated by a comma, regardless of the length of the items. Plaintiff contends that the trial court erred when it granted defendants’ motion for summary judgment on her claim for intentional infliction of emotional dis-tress and when it denied her motion to exclude witnesses and reporters. NOTE: It may be more clear to separate two items in a series with a comma when the items themselves contain multiple elements, e.g., In support of his motion, defendant cited Oregon and Washington cases, and state and federal regulations. 2. Compound Predicates A predicate is a part of a sentence that contains a verb, but not a subject. Compound predicates are simply predicates that contain a series of verbs. They should be treated like any other series: if there are three or more verbs, use commas; if there are only two, do not. The defendant objected to the introduction of Yost’s testimony, asked the court to call a recess, and requested permission to file a memorandum in support of her objection. The court denied plaintiff’s motion for a judgment notwithstanding the verdict and granted his motion for a new trial. Go to TABLE OF CONTENTS 81 Go to INDEX 3. Compound Sentences A compound sentence consists of two or more independent clauses joined by a coordinating conjunction. A clause consists of a subject and a predicate. The most common coordinating conjunctions are and, but, or, and nor. Independent Clause: I will go to Elaine’s New Year’s Eve party. [Coord. Conj.] Indep. Clause: [and] I will bring a bottle of champagne. [Coord. Conj.] Indep. Clause: [or] I will stay home. Not a Clause: and dance on the table. [This is a second predicate; see “compound predicates” above.] The coordinating conjunctions in compound sentences should be preceded by a comma. Paul organized the outing, and everyone had a good time. Defendant’s brief presented a difficult argument, but Gina understood it better after she attended oral argument. In the narrow factual context of this case, neither the attachment of the trans-mitter to the truck nor the subsequent monitoring of that transmitter’s location invaded a privacy interest of defendant, and, it follows, no search implicating Article I, section 9, occurred. Exception: When the clauses are short and there is no danger of misreading, the comma may be omitted. Anna sang and Becca played the flute. 4. Complex Sentences A complex sentence consists of one independent clause and one or more dependent clauses. A clause is dependent when it is introduced with a subordinating conjunction (e.g., because, although, if, after, before, until, since, so that, unless, while, when, where, even though) or relative pronoun (who, whom, whose, whoever, whomever, that, which). Whether to use a comma between an independent clause and a dependent clause depends upon the meaning of and relationship between the clauses. When the dependent clause is restrictive–or necessary to the meaning of the sentence– do not use a comma. When the dependent clause is nonrestrictive–or not necessary to the meaning of the sentence–use a comma. (Those rules also explain when to Go to TABLE OF CONTENTS 82 Go to INDEX use “that” and when to use “which.” See page 99. “That” is restrictive and is used without a comma; “which” is nonrestrictive and is used with a comma.) Those rules often lead to confusion, because the same sentence can be punctuated more than one way and still be correct, depending on its meaning. Restrictive: John and Mary did not marry because they wanted money. (Meaning: they married for some other reason.) Nonrestrictive: John and Mary did not marry, because they wanted money. (Meaning: they did not marry, and the reason was that they wanted money.) Restrictive: I am not taking that course of action because I distrust Harry’s motives. (Meaning: the reason I am not taking that action is not distrust of Harry’s motives.) Nonrestrictive: I am not taking that course of action, because I distrust Harry’s motives. (Meaning: I am not taking that course of action, and the reason is distrust of Harry’s motives.) (No Comma): We first address defendant’s argument that the evidence should have been suppressed because it derived from an unlawful police-citizen encounter. (Here, the “because clause” explains why the evidence should have been suppressed, not why it is addressed first.) (Comma): We first address defendant’s argument that the evidence should have been suppressed, because it is dispositive. (Here, the “because clause” explains why the argument is addressed first, not why it should have been suppressed.) In some situations, use or omission of a comma can cue the reader about the relationship between two clauses and accordingly prevent misreading. Compare the following two examples: Defendant argues that the trial court erred because he presented adequate exculpatory evidence. Defendant now argues that the trial court erred in denying his motion to con-trovert, because he failed to preserve any challenge to the suppression ruling. In the first example, the “because clause” explains the “that clause” element of the main clause (i.e., why the trial court arguably erred). By contrast, in the second example, the “because clause” explains the main subject and verb of the main clause (i.e., why defendant is making the argument). Go to TABLE OF CONTENTS 83 Go to INDEX 5. Parenthetical Elements a. Interrupting Parenthetical Elements Either use two commas or no commas to set off parenthetical elements that interrupt a clause or phrase. (Parenthetical elements include nonrestrictive appositives and nonrestrictive clauses.) Generally, longer parenthetical elements should be set off by commas, while shorter parenthetical elements may go without any. Correct: The state argues, as an alternative basis for its second assignment of error, that the trial court should not have suppressed the computer disk because it was in plain view when the officers searched defendant’s apartment. Correct: The state argues alternatively that Article I, section 9, does not require suppression in these circumstances. Incorrect: The state cites as an alternative ground justifying suppres-sion, Article I, section 9, of the Oregon Constitution. Correct: The state cites, as an alternative ground justifying sup-pression, Article I, section 9, of the Oregon Constitution. b. Introductory Parenthetical Elements The longer the introductory parenthetical element, the more helpful it is to set it off with a comma. If omitting a comma could lead to misreading, include it. Yesterday I finished cleaning out my desk. First, defendant objects to the denial of his motion to strike. ORS 123.456(7) sets out, in part, the following: When this case was before the trial court for the first time, defendant was sentenced to death. c. Setting Out “Inc.” Before Parentheticals Usually “Inc.” is set out with two commas, e.g., Time, Inc., is the parent company. Go to TABLE OF CONTENTS 84 Go to INDEX But when “Inc.” is followed by a parenthetical element, omit the second comma, e.g., Defendants Borg-Warner Automotive, Inc. (Borg-Warner) and Tenneco Automotive Operating Company, Inc. (Tenneco) allegedly manufactured . 6. Appositives An appositive points out the same person or thing by a different name. Whether to use commas with an appositive depends on whether it is restrictive or nonrestrictive, which is a question of meaning. Compare the following examples: My sister Jill conducted firefighter qualification tests last weekend. My mother, Doris, stands on her head every day. In the first example, the absence of commas indicates that “Jill” is restrictive (or necessary), because the author has more than one sister and so “Jill” identifies which of the sisters is being referred to. In the second example, the author has only one mother, so the commas indicate that “Doris” adds only additional, parenthetical information. D. Dashes Dashes are most commonly used to amplify and explain ideas, digress from the main idea, or create an abrupt break or sudden change in a sentence: He spent several hours explaining his case–a case that he knew could not be won. The attorney–who had been waiting three hours for a ruling on the motion–entered the courtroom in an angry mood. A space can be added before and after a dash when used as punctuation in the text of a slip opinion; however, when published in the Advance Sheets, those spaces will be removed. In slip opinions, use two hyphens “--” not the typographical “–” dash. See also Punctuating Parenthetical Elements, page 88. Go to TABLE OF CONTENTS 85 Go to INDEX E. Hyphens 1. Do Not Hyphenate a. Words That Begin With the Following Prefixes: anti, bi, bio, co, counter, extra, infra, inter, intra, macro, micro, mid, mini, multi, non, over, pre, pro, pseudo, re, semi, sub, super, trans, ultra, un, and under, e.g., antitrust, nonprofit, coworker, subconstitutional, pretrial BUT mid-nineteenth century. b. Factfinder The factfinder took note of the witness’s demeanor. c. Patdown The assailant put his hands over his head during the patdown. When used as a verb, it should be shown as two words: The officer pat down the suspect. d. Adjective Forms of Compounds in Which the First Word Is an Adverb Ending in “ly,” e.g., her eagerly awaited homecoming his totally incompetent performance e. Compound Modifiers That Appear After a Verb, e.g., Defendant’s examination was court ordered. BUT The court-ordered examination was inconclusive. f. Percentages When Used as Adjectives, e.g., 10 percent increase, two percent annual pay raise Go to TABLE OF CONTENTS 86 Go to INDEX g. The Word “email” h. The Words “well taken” Unless Used as a Compound Modifier 2. A Hyphen Is Used a. To Join a Prefix to a Word Beginning With a Letter in Uppercase or a Number, e.g., un-American, Anti-Federalist, pre-1998 b. To Join a Prefix to a Main Word When the Second Element Consists of Two or More Words, e.g., pre-latency-period therapy, non-work-related activities c. When the Last Letter of the Prefix (Usually a Vowel) Is the Same as the First Letter of the Following Word and the Result Improves Readability, e.g., infra-area, non-neutral BUT preexisting d. To Avoid Confusion and Misreading, e.g., re-cover (to cover again), not recover (to get back or regain) re-cite (to cite again), not recite (to read or declaim) co-conspirator (not coconspirator); co-counsel (not cocounsel) e. With the Prefixes “Cross” and “Post” (in conformance with the Oregon Rules of Appellate Procedure), e.g., cross-assign, cross-appellant, cross-appeals post-conviction, post-trial, post-certification f. When Using: 9-1-1 I-5 case-in-chief policy-making decision-maker second-guess ex-husband self-defense ex-wife Go to TABLE OF CONTENTS 87 Go to INDEX g. With Adjective Phrases Containing Numerals or With Compound Modifiers That Appear Before the Word Modified to Prevent Misreading, e.g., three-year-old victim, five-foot-tall tree, 18-year marriage, 25-year minimum term, third-party beneficiary, case-by-case basis, full-time work (but he worked full time), common-law wife, court-appointed attorney, first-degree murder, death-penalty case, penalty-phase proceeding; two- and three-year-old children, self-defense, right-of-way, time-barred (but the claim is time barred) NOTE: For more information on compound modifiers and hyphenation, see Garner, A Dictionary of Modern American Usage (on “phrasal adjectives”). The idea behind hyphenating compound modifiers is to promote readability and avoid misreading. h. When Adding Prefixes to Acronyms, e.g., pre-ORCP NOTE: For more examples, see Word Pairs, page 90; and Word Functions, page 101. F. Punctuating Lists Lists may be set out in several ways. Remember that lists that consist of items that are themselves sentences can always be written as separate sentences. Otherwise, the following guidelines may be helpful: • If a list is introduced by a complete grammatical thought, use a colon at the end of the introductory sentence. If the list is not introduced by a complete grammatical thought, then a colon may NOT be used. Correct: Defendant assigns error to three rulings: (1) the denial of his motion to disqualify Dr. Demento as an expert; (2) the denial of his motion to suppress chemical evidence; and (3) the granting of plaintiff’s motion for a new trial. Correct: Defendant assigns as error (1) the denial of his motion to dis-qualify Dr. Demento as an expert; (2) the denial of his motion to suppress chemical evidence; and (3) the granting of plain-tiff’s motion for a new trial. Incorrect: Defendant assigns error to: (1) the denial of his motion to dis-qualify Dr. Demento as an expert; (2) the denial of his motion to suppress chemical evidence; and (3) the granting of plain-tiff’s motion for a new trial. Go to TABLE OF CONTENTS 88 Go to INDEX • Items in a list may be, but are not required to be, numbered. • Generally, items in a list, whether numbered or not and whether introduced by a colon or not, may be separated by either semicolons or commas. If the items are long or contain internal punctuation, it may be clearer to use semicolons. • The first item in a list should not begin with an uppercase letter, unless the items are full sentences. See also page 59. The trial court announced three ground rules: Witnesses were to be excluded from the courtroom; only three members of the press were to be allowed in the courtroom at any time; and lions would be admitted only if accompanied by tigers and bears. See also pages 78 through 84 for discussion of the proper use of colons and commas. G. Punctuating Parenthetical Elements Nonrestrictive parenthetical elements (that is, parenthetical elements that give additional information but are not necessary to identify the antecedent) can be set off by commas, dashes, or parentheses. If a parenthetical element has a close logical and syntactic relation to the rest of the sentence, then use commas. Dashes indicate a more remote relation, and parentheses still more remote. See also pages 83 to 84 for discussion of parenthetical elements. H. Semicolons 1. Use a semicolon to separate two independent clauses not joined by a conjunction, e.g., The defendant refused to testify; he later was convicted of murder in the first degree. 2. If the elements in a series are long and complex or contain internal punctuation, then they are separated by semicolons, rather than by commas, e.g., Defendant was charged with three counts of rape in the first degree, ORS 163.375; two counts of sexual abuse in the first degree, ORS 163.427; two counts of rape in the third degree, ORS 163.355; and one count of sexual abuse in the third degree, ORS 163.355. Go to TABLE OF CONTENTS 89 Go to INDEX 3. A semicolon precedes the words however, therefore, hence, thus, accordingly, and besides, when used to introduce a clause. A comma follows, e.g., The Supreme Court reversed and remanded the case; therefore, the Court of Appeals had to decide the case for a second time. The trial judge refused to suppress the evidence; thus, defendant was con-victed on all counts. Go to TABLE OF CONTENTS 90 Go to INDEX III. Word Usage and Conventions A. Word Pairs A / An The indefinite article “a” is used before words that begin with a consonant, words with a pronounced h and a long u sound (eu) or with words such as “one.” a historical; a habitual; a union; a Eucharist; a one-time deal “An” is used before words beginning with a vowel or an unsounded “h.” an eagle; an appellate judgment; an hour; an heir; an honor The article that precedes an acronym or initialism depends on how the abbreviation reads: If it is pronounced beginning with a vowel sound, then use “an.” If it is pronounced beginning with a consonant sound, then use “a.” For examples, see page 75. A / From a “Defendant appeals a judgment .” “Defendant appeals from a judgment .” Either is correct. If the authoring judge has a preference, then use the construction that the judge prefers. Absent / Without “Without” or “in the absence of” is preferred. Without any arguments against doing so, the trial court dismissed the jury. or In the absence of any arguments against doing so, the trial court dismissed the jury. NOT Absent any arguments against doing so, the trial court dismissed the jury. Go to TABLE OF CONTENTS 91 Go to INDEX Administrative Law Judge / Hearing Officer In Oregon since 2003, the title “administrative law judge” is now used. Affect / Effect “Affect” is a verb that means to have an influence on, to bring about a change in, or to touch or move the emotions of. The trial was affected by the constant media attention. “Effect” is most commonly used as a noun, but it can also be used as a verb. As a noun it means something that is brought about by a cause or agent; a result or outcome. As a verb it means to achieve a result, to execute, or to cause to occur. Noun: Her closing argument had no effect on the jury’s final decision. Verb: The attorney was unable to effect any change in the system. Allow / Grant Typically, allow is used to refer to a ruling on a petition. Grant is used to refer to a ruling on a motion. Alternate / Alternative “Alternate” means every other or substitute, while “alternative” means providing a choice. She takes early lunch during alternate weeks. An alternative approach is to take no lunch at all. Although / While “Although” more properly introduces an aside; “while” is more properly used only in a temporal sense. Although many people disagree, Ian thinks that coffee is good for you. While I was in Paris, I decided to become a can can dancer. Go to TABLE OF CONTENTS 92 Go to INDEX Amicus Curiae / Amici Curiae (Friend/s of the Court) 1. Amicus curiae can be used in two different ways, either as an adjective describing the brief, or as a noun designating the party or person immediately following, e.g., J. Marcus Cool, of Cool, Hand and Luke, Portland, filed an amicus curiae brief on behalf of American Theatre Institute. Sherman Potter, of Potter & Daughter, P.C., Portland, filed a brief on behalf of amicus curiae Oregon Tempest in a Teapot Association. 2. Amici curiae is the plural form used in the same way. J. Marcus Cool, of Cool, Hand and Luke, Portland, filed an amici curiae brief on behalf of American Theatre Institute, Americans for Good Theatre Association, and Oregon Actors Association. Sherman Potter, of Potter & Daughter, P.C., Portland, filed a brief on behalf of amici curiae Oregon Tempest in a Teapot Association, Oregonians-R-Us, Inc., and We are the World Foundation. NOTE: The number of parties, not the number of briefs, dictates the use of singular or plural. It is acceptable to use amicus curiae or amici curiae for the first reference and amicus or amici thereafter. Appeal / Review A party “appeals” from a lower court as of right, but a party “seeks review” in court of the action of an administrative agency or when the higher court has discretion whether to take the case. The related documents are called “notice of appeal” and “petition for judicial review” (in the case of an administrative proceeding) or “petition for review” (in the case of a petition to the Oregon Supreme Court). Within an administrative system, the correct term is “appeal.” For example, a party may “appeal” to the Workers’ Compensation Board from the order of an ALJ. Appeals / Appeals from Defendant appeals a judgment of conviction. Defendant appeals from a judgment. Either one is acceptable. Go to TABLE OF CONTENTS 93 Go to INDEX Because / Since “Because” explains why; “since” expresses time. Because he attended seven basketball games, David had a busy weekend. Since moving to a window office, John has been much more productive. Between / Among As a general rule, “between” is used when the sentence involves two things or when expressing a close relationship of any number of individual things. The jury had to decide between the death penalty and life in prison. His time was divided between school, sports, and work. “Among” is used when the sentence involves more than two things. His present conviction was just one among many. Her estate was divided among all her relatives. The above rule expresses the distinction between “between” and “among” in a simplistic manner. Here are a couple of specific rules: “Between” is used to express one-to-one relations of many things, e.g., a treaty between four powers. “Among” is used to express collective and undefined relations, e.g., She was the best among all the attorneys. Circuit Court / Trial Court When referring to the lower court in a particular case, use “trial court.” Use “circuit court” when referring to circuit courts generally (“The legislature created jurisdiction in the circuit court.”) or to general propositions about particular courts (“Post-certification challenges to the constitutionality of initiative measures generally must be filed in Marion County Circuit Court.”). Also, use “circuit court,” not “trial court,” in tag lines. See also Post-Conviction Cases, page 104. Cite / Cite to “Cite to” is redundant; “cite” is preferred. In Smith, the court cited ORAP 5.45. NOT In Smith, the court cited to ORAP 5.45. Go to TABLE OF CONTENTS 94 Go to INDEX Compose / Comprise The parts compose the whole; the whole comprises the parts. The whole is composed of the parts; the parts comprise the whole. The most common error is using “comprise” when “compose” is correct: Incorrect: Judge Haselton, Judge Armstrong, and Judge Duncan comprise Department One. Correct: Judge Haselton, Judge Armstrong, and Judge Duncan compose Department One. Incorrect: Department One is comprised of Judge Haselton, Judge Armstrong, and Judge Duncan. Correct: Department One is composed of Judge Haselton, Judge Armstrong, and Judge Duncan. Correct: Department One comprises Judge Haselton, Judge Armstrong, and Judge Duncan. Consecutively to / Concurrently with A consecutive sentence is served “consecutively to” another sentence: The trial court ordered that defendant’s 12-month sentence under the sentenc-ing guidelines for theft be served consecutively to his Measure 11 sentences. A concurrent sentence is served “concurrently with” another sentence: The trial court imposed aggregated sentences totaling 36 months, all of which were to be served concurrently with the burglary sentences. Consistent With / Consistently With For adverbial uses, “consistently with” (that is, “in a manner consistent with”) is correct. The court decided the case consistently with its precedent. (in a manner con-sistent with its precedent) Go to TABLE OF CONTENTS 95 Go to INDEX Criterion / Criteria “Criteria” is plural; accordingly, it may not be used when discussing only one factor. The court primarily relied on the first criterion. Datum / Data “Data” is plural and takes a plural verb. The data support plaintiff’s theory. Decree / Judgment Historically, equitable proceedings resulted in “decrees,” while legal proceedings resulted in “judgments.” The distinction between law and equity was abolished in Oregon in 1980. See ORCP 2. The final determination in all civil actions is now titled a “judgment.” Some statutes still use the term “decree,” and occasionally so do trial courts. The appellate courts, however, will substitute the word “judgment.” E.g., Webber v. Olsen, 330 Or 189, 192 n 1, 998 P2d 666 (2000); State ex rel Olson v. Renda, 171 Or App 713, 715 n 3, 17 P3d 514 (2000). Disinterested / Uninterested “Disinterested” means lacking a stake in an outcome. “Uninterested” means bored. The judge appointed a disinterested arbitrator. Driver’s License / Driving Privileges Different statutes use different terms; they are not interchangeable. For example, when a person fails or refuses a breath test, the Oregon Department of Transportation suspends the person’s “driving privileges,” not his or her “license.” Check the statute at issue for the correct term. Employe / Employee In older statutes and opinions, the spelling of certain words (e.g., employee / employe, subpoena / subpena) was changed to enable faster data entry. After the advent of word processing, those types of words were restored to their original spelling. It is correct, however, to use those shortened spellings when directly quoting those sources or to bracket in the missing letters. Go to TABLE OF CONTENTS 96 Go to INDEX Expedient / Expeditious “Expedient” means that a particular course of action is a practical solution to a problem; it also has overtones of side-stepping tough choices. “Expeditious” means speedy. The lawyers decided that it would be expedient to waive oral argument. The court must decide all cases in an expeditious manner. Fewer / Less Than “Fewer” denotes countable things; “less” denotes things that can’t easily be counted. However, it is customary to use “less” for time, money, and distance. If you have 10 items or fewer, then you can go to the express lane. I have less money than you. Finding / Holding A “finding” is a factual determination; a “holding” is an application of law to particular facts. The trial court found that the officer lacked subjective probable cause. The Court of Appeals held (not found) that the search was invalid because the officer lacked subjective probable cause. General statements of law unconnected to particular facts are not holdings. The Supreme Court stated (not held) that probable cause has both an objective and a subjective component. “Determination” can refer to either. Forgo / Forego “Forgo” means to do without. I would rather eat late, than to forgo lunch altogether. “Forego” means to go before. Based on the crowds lining the streets to catch a glimpse of Justin Bieber entering the courthouse, it is obvious that his reputation foregoes him. Go to TABLE OF CONTENTS 97 Go to INDEX If . . . Was / Were See Subjunctive Mood, page106. Implicitly / Impliedly “Implicitly” is preferred. The court’s ruling implicitly indicated several factual findings. In Light of / In the Light of “In light of” is preferred. Loath / Loathe “Loath” means unwilling or reluctant: I am loath to go to the meeting today. “Loathe” means to strongly dislike or hate something: I loathe the thought of eating tofu. May / Might “May” expresses authority. The Secretary of State may certify a ballot measure. “Might” expresses possibility. The Secretary of State might resign tomorrow. Of / For Most authorities consider the following to be more correct: Defendant was convicted of three counts of assault. Defendant appeals a judgment of conviction for assault. Go to TABLE OF CONTENTS 98 Go to INDEX Over / Under–More Than / Less Than Generally, “over” and “under” are used in spatial relationships; they do not take the place of “more than” or “less than.” The train traveled rapidly over miles of desert terrain. The battleship drifted under the bridge. The attorney has more than (not over) $5,000 in outstanding legal fees. He had less than (not under) $20 in his pocket. Pleaded / Pled Pleaded is the preferred form. Portion / Part A “portion” is a piece of a whole that can be divided into shares, as an estate or a pie. A “part” is a more general term that connotes something less than the whole. Jenny argued that she had been deprived of her rightful portion of the estate. Scott contended that he owned the part of the property beyond the fence. Jim challenged the part of the judgment that prohibited wearing bow ties. That part of the statute supports defendant’s argument. Proved / Proven “Proven” is an adjective, while “proved” remains the accepted past participle of prove. Her attorney is a proven winner. The prosecuting attorney has proved his case. Right / Authority Generally, individuals have “rights” (e.g., the right against self-incrimination, the right to bear arms); however a governmental body or representative needs “authority” to act (e.g., a question of whether a city had authority to take property by eminent domain). Go to TABLE OF CONTENTS 99 Go to INDEX Since / Because “Because” explains why; “since” expresses time. Because he attended seven basketball games, David had a busy weekend. Since moving to a window office, John has been much more productive. That / This; Those / These “That” is used as a demonstrative pronoun to refer to thought expressed earlier: The letter was unopened; that in itself casts doubt on the inspector’s theory. “This” is used when the referent is yet to be mentioned: This is what bothers me: We have no time to consider late applications. “Those” and “these” are similarly distinguishable. Defendant relied on three statutes in his motion. The trial court relied on those statutes, as well as two others. These are the problems with plaintiff’s arguments: The arguments were not raised to the trial court, they are unsupported by legal authority, and they are wrong. That / Which “That” introduces a restrictive clause (a clause that is necessary in context to the meaning of the sentence). Commas are not used with a restrictive “that clause.” The decision that the judge made regarding suppression of evidence was con-trary to the legal standard. “Which” introduces a nonrestrictive clause (a clause that can be omitted without changing the meaning of the sentence). Commas are required with a nonrestrictive “which clause.” The decision, which caught the legal community by surprise, was contrary to the legal standard. Go to TABLE OF CONTENTS 100 Go to INDEX The meaning to be conveyed, not grammatical differences, dictate when to use “that” and “which.” Sometimes the same sentence can correctly use either, depending on meaning. The lawnmower that I borrowed is in the garage. (Correct if there are many lawnmowers under discussion and it is necessary for clarity to identify the one borrowed.) The lawnmower, which I borrowed, is in the garage. (Correct if there is only one lawnmower under discussion and adjectival clause serves only to give extra information.) See also discussion of commas on page 79. Who / Whom “Who” is used as a subject. “Whom” is used as an object of a verb or preposition. When in doubt about which to use, try rethinking the sentence using “he” or “him.” Who is listening? (Is he listening?) Defendant, who was not present in the courtroom, was found guilty. (Defendant–he was not present in the courtroom–was found guilty.) A woman who I think is a genius wrote this sentence. (A woman–I think she is a genius–wrote this sentence.) He asked whom he should speak to. (He should speak to him.) Use “that,” not “who” or “whom,” when the antecedent is not a person. The bank that Kym uses gives away free blenders with new checking accounts. Christy sued the company that made the defective golf clubs. Go to TABLE OF CONTENTS 101 Go to INDEX B. Word Functions 1. One Word or Two Words “Case Law” and “Grid Block” are always two words (unless being quoted). “Cannot” Cannot is used as one word, except when using a “not only” construction: We cannot reach unpreserved error. She can not only pitch, but she can catch, too. Other words are usually spelled as one word, such as: caseworker, factfinder, forestlands, handwritten, insofar, landowner, lien-holder, online, overarching, passersby, pickup, policyholder, statewide, time-frame, timeline, timesheet, voicemail, and website Some words, however, are either one word or two, depending on how they function within the sentence: e.g., clean up, set over, set back, day care, pat down: The cleanup crew arrived early to start the clean up. The trial was set over until tomorrow. The defendant requested a setover. The setback requirement provides for a six foot easement. The house was set back from the property line by five feet. Her mother provides day care both privately and at a local daycare center. The subject was pat down by the officer. The officer who performed the pat-down testified that he found a knife. 2. Contractions Appellate opinions are formal documents and, accordingly, do not use contractions. When quoting, however, if the original uses contractions, use those contractions in the quotation. Go to TABLE OF CONTENTS 102 Go to INDEX C. Word Usage Language “Language” technically refers to English, French, Spanish, etc. Although frequently used in constructions such as “the language of ORS 809.222,” it is more correct to use “the wording of ORS 809.222” or “the text of the statute.” Letter Opinion “Letter opinion” is preferred to “opinion letter.” Only The placement of “only” can change the meaning of a sentence. It modifies the word or phrase that comes immediately after it, so it should be placed accordingly to avoid ambiguity. The jurisdiction of the court is constrained only by statute. (That means that the only thing limiting the court’s jurisdiction is statute.) The jurisdiction of the court is only constrained by statute. (That means that there are other effects on the court’s jurisdiction, but the one way that statutes affect jurisdiction is by constraining it.) The statute only pertains to state actors. (That is unlikely to express what the author means. It communicates that the statute only pertains, but does not have some other effect.) The statute pertains to only state actors. (That is more likely to express the author’s meaning. It communicates that the only actors to whom the statute pertains are “state” actors.) The statute pertains only to state actors. (That is an alternative author’s meaning. It communicates that the only cate-gory of people to whom the statute pertains is “state actors.”) Parameter(s) This word will almost never be correct in legal writing; it is a technical scientific and mathematical term. When tempted to use it, try “limits,” “boundaries,” or “borders.” The principles of justiciability and preservation keep judicial analysis within relatively confined boundaries (not parameters). Go to TABLE OF CONTENTS 103 Go to INDEX Parties, Victims, Children, and Law Enforcement Generally, refer to parties to a case using their position in the lower tribunal (plaintiff, defendant, claimant, etc.). Exceptions include the following: (1) domestic relations cases, in which the parties are referred to as “husband” and “wife”; (2) civil commitment proceedings, in which the person for whom commitment is sought is referred to by his or her position on appeal (appellant, respondent); (3) termination of parental rights proceedings, in which the parents are referred to as “mother” and “father” and the children are referred to as “child” or “children” or sometimes by their first initials; (4) juvenile delinquency proceedings, in which a person alleged to be within juvenile court jurisdiction is referred to as “youth”; (5) juvenile dependency proceedings, in which a minor is referred to as “child”; and (6) adoption cases, in which “child” is used. Additionally, when a word indicating a party’s role in the lower proceeding or on appeal or review is used as a substitute for the party’s name, the definite article “the” is not used (e.g., plaintiff, petitioner, etc.). When using an abbreviated name (not party designation), use “the” as in, “the state,” “the superintendent.” Do not use the full names of victims. The following references are acceptable: the victim the first or last name, but not both an initial (the initial may, but need not, be related to the victim’s name) In dependency and termination cases, and criminal cases involving a child victim or witness, do not use the names of children. Instead, use an initial. For readability, it is preferred to use a single initial, which may, but need not, be related to the victim’s name. If an author would like to use two initials (e.g., to differentiate among siblings whose first names begin with the same letter), then do not use periods (AG, AM, etc.). In the following types of cases, do not use the name of a person who is protected by a court order (or similar); instead, use a role-based description, such as “petitioner,” or an initial: Family Abuse Prevention Act (FAPA) (including violation of restraining order), stalking (including violation of a stalking order), Elder Persons and Persons with Disabilities Abuse Prevention Act (EPPDAPA), registration of a foreign restraining order, and Sexual Abuse Protection Order (SAPO). When referring to specific law enforcement personnel, use the correct title, e.g., officer, deputy, trooper, detective, etc. Go to TABLE OF CONTENTS 104 Go to INDEX Post-Conviction Cases When referring to the circuit court in a post-conviction proceeding, “post-conviction court” is preferred to “post-conviction trial court.” The petitioner in a post-conviction case is referred to as “petitioner” and the defendant is referred to as “the superintendent,” or “the state.” Previous / Prior “Previous” and “prior” as adjectives are equally acceptable. Father’s parental rights had been terminated in a prior (or previous) proceeding. Prior to / Before “Prior to” is disfavored; “before” is preferred. Wife withdrew joint funds from a checking account before (not prior to) filing for dissolution. That “That” as a relative pronoun or relative adverb is sometimes suppressed in informal writing; the preferred practice in opinions is to include it. The court held that the evidence had been wrongly admitted. Defendant admitted at oral argument that that proposition was subject to debate. D. Variant Spellings Use kidnapping, not kidnaping. Use marshalling, not marshaling. Go to TABLE OF CONTENTS 105 Go to INDEX IV. Common Grammatical and Style Problems A. Collective Nouns Collective nouns require singular verbs when the group is functioning as a unit. When individual members of the group are acting independently, a plural verb is used. If a sentence seems awkward, then insert the words “members of” before the collective noun and use the plural verb. Some common examples of collective nouns are: court The court is not in session today. The members of the court are in conference. majority The majority states that the assignment of error was not preserved for review. The author of the majority is mistaken on that issue. jury The jury has adjourned for the day. The members of the jury have adjourned for the day. council The Council on Court Procedures has adopted new rules. The members of the council have adopted new rules. B. Parallel Construction Sentence parts must match if a sentence is to make logical sense. Therefore, parts of a sentence that are parallel in meaning are parallel in structure. Incorrect: The court held that the taxpayer was guilty of failing to report income, claiming fraudulent deductions, and in the treatment of ordinary income as capital gain. Correct: The court held that the taxpayer was guilty of failing to report income, claiming fraudulent deductions, and treating ordinary income as capi-tal gain. C. Passive Voice Avoid using the passive voice whenever possible. The passive voice can create ambiguity about who is doing the acting in a sentence. See State ex rel Click v. Brownhill, 331 Or 500, 509, 15 P3d 990 (2000) (Durham, J., concurring) (ambiguity in statute arose because of the use of the phrase “shall not be used”; through use of passive voice, legislature failed to identify who “shall not use”). See also discussion of Active Voice, page 107. Go to TABLE OF CONTENTS 106 Go to INDEX D. Verbs 1. Was/Were Agreement: Ringo Starr argued the cause for the appellant. With him on the brief was The Beatles LLP. Mick Jagger argued the cause for respondent. With him on the brief were Keith Richards and the Rolling Stones, PC. 2. Past Perfect Tense When referring to an action completed before another past time, the past perfect tense is used. Incorrect: Petitioner testified at the post-conviction hearing that he asked his trial lawyer to investigate his alibi defense. Correct: Petitioner testified at the post-conviction hearing that he had asked his trial lawyer to investigate his alibi defense. NOTE: Depending on context, past-perfect tense might not be necessary. For example, if describing testimony, the point is to describe an event that happened in the past, not that the event happened in the past relative to some other time. For example, “John testified that the light was green. 3. Subjunctive Mood For a much more complete explanation of the proper use of the subjunctive, see Garner, A Dictionary of Modern American Usage. One situation in which the subjunctive is generally correct is to express a condition that is contrary to fact or hypothetical: If Mary were queen of the world, then everyone would want to obey her. If the court were to take notice of defendant’s arguments, then it would ignore long-standing principles of preservation. Not every “if” clause takes the subjunctive, however: If defendant was a resident of Oregon in 1999, then he must pay taxes for that year. Go to TABLE OF CONTENTS 107 Go to INDEX E. Active Voice In the active voice, the subject of a sentence or clause performs the action of the verb. In the passive voice, the subject of a sentence or clause is not the actor. Generally, the passive voice is wordier and may be vague. The active voice is clearer and stronger and tells who did what to whom. See also page 105. Active: I missed the deadline. Passive: The deadline was missed. F. Gender-Neutral Wording Gender-neutral terms are preferred, and gender-based pronouns are avoided except when referring to a specific person. Use “he or she” only when all other constructions fail. For example, use letter carrier, not mailman worker, not workman flight attendant, not stewardess sales clerk, not salesman G. Informal or Technical Terminology An appellate opinion is a formal document. Its content reflects that formality. For example, instead of using the word “said” or “says,” use the word “states” or “stated” (or any of the words listed below depending on the context of the sentence): adds emphasizes observes argues establishes opines compares explains points out concludes finds posits continues insists proposes declares maintains suggests disagrees notes thinks When using technical terms or terms of art, explain those terms. Go to TABLE OF CONTENTS 108 Go to INDEX GLOSSARY (These terms are used internally in appellate opinion drafting, Judicial Assistant and Law Clerk materials, on the website, etc., but may or may not be referred to elsewhere in this manual.) Advance Notice–A one-day advance notice of Oregon Appellate Court cases scheduled to be released that is posted on the OJD Publications website–usually Tuesday of each week for Court of Appeals cases and Wednesday of each week for Supreme Court cases. Advance Sheets–Opinions of cases issued weekly by the Oregon Appellate Courts, including, as issued, both the Regular and Magistrate Divisions of the Oregon Tax Court, and compiled together into a softbound book that is published bimonthly by the Publications Program. Advance Sheets are available by subscription or by issue. Affirmed by an Equally Divided Court–When an even number of justices or judges meet to decide a case, a situation can arise where the court is evenly split regarding its disposition. In that event, the case will be considered affirmed, and no majority written opinion will be issued. AWOP–Term of art used in the Court of Appeals to refer to cases that are affirmed without opinion. No precedential value is accorded those decisions, which are published by case caption and number only. Bluebook–Shorthand reference to the citation manual, The Bluebook, A Uniform System of Citation, which is published and distributed by the Harvard Law Review Association. Citation–Form by which authoritative sources are identified for easy reference. For the most part, Oregon Appellate Courts do not use periods in citations. A string citation is a proscribed order listing more than one authority to support a legal position (usually set out starting with strength of authority, jurisdiction, rank of court, and date) and often introduced with a signal. A parallel citation is an additional reference to the same case published by a different source. A full citation sets out the official source followed by a parallel source. A short form citation is a subsequent reference to a case already set out in full and is a less complete citation but which still clearly identifies the referenced material. A jump or pinpoint citation refers to the specific page on which a legal authority is cited. Go to TABLE OF CONTENTS 109 Go to INDEX Common Law–Law created by judicial opinions, not statutes. Hyphenate when using this term as an adjective before a noun. Cost Box–Printed on title page of slip opinion and used to designate expense costs (not attorney fees) that are allowed by the court. This information is not published in the Advance Sheets or the Oregon Reports. Court of Appeals–In Oregon, an intermediate appellate court that has jurisdiction to hear most civil and criminal appeals from circuit court (exceptions include death-penalty cases and Tax Court appeals) and review most state administrative agency actions. Its primary function is the correction of error by the application of principles of law; formulation of law is a secondary function performed as required for deciding cases. Department–A department is a group of three judges in the Court of Appeals, designated by the Chief Judge, that generally hears and decides cases together. If a member of a department must be recused or is unavailable for any reason, the Chief Judge will assign another judge to sit on a particular case. Accordingly, the “panel” of judges that hears and decides a case may or may not correspond to the make-up of the “department” to which the case is assigned. One judge in each department is the “presiding judge.” The presiding judge presides at oral argument, makes written case assignments, and administers the department. Each department has a number, which appears, among other places, on the court’s oral argument calendar, and a color, which corresponds to the color of the cover sheet on its draft opinions. Department 1 is pink; Department 2 is blue, Department 3 is green, and Department 4 is purple. Each department generally meets twice a month to discuss draft opinions circulated since the last department meeting. When all three judges on the panel agree that an opinion is ready, it is “approved to go down.” Dictum–Statements in an opinion that are not necessary to the disposition of the case. Disposition by Order–A court may choose to dispose of a matter by issuing an order instead of an opinion. Acceptance of Certified Questions or Appeals, issuance of Alternative Writs of Mandamus, amendments of typographical errors, and the certification of ballot titles are examples of such orders. Discretionary Review–Indicates that a court chooses whether or not to review a case. En Banc–Cases in which all available judges participate in the consideration and vote on the outcome. Go to TABLE OF CONTENTS 110 Go to INDEX Footnotes–Notes at the bottom of a page citing or commenting on the part of the text to which they are referenced. Generally a footnote contains information of lesser importance to the larger body of work. Full Court–In the Court of Appeals, any judge may refer to the full court an opinion approved to go down by a panel. The full court meets once a month to discuss cases referred to it. Reasons for referral include, but are not limited to, the following: (1) the opinion would overrule a prior case; (2) the same or similar issue is before several departments of the court; (3) a judge not on the panel disagrees with the result or reasoning. Not all cases referred to full court are taken en banc; a vote of the majority of the judges available to participate is required. Magistrate Division–The Magistrate Division tries or mediates all tax appeals, unless the Tax Court judge assigns the case to the Regular Division. A party may appeal from a magistrate’s final decision to the judge of the Tax Court. Hearings in the Magistrate Division are often informal proceedings. Hearings may be by telephone or in person and may be held around the state. A taxpayer may choose to represent himself or herself or to be represented by a lawyer, public accountant, real estate broker, appraiser, or other individual. Media Releases–Media releases are issued weekly and contain summaries of cases issued or otherwise decided by the Oregon appellate courts. Court of Appeals cases that have been Affirmed Without Opinion are listed by name and county of origin only. Supreme Court petitions for review allowed and denied are listed by case name and number. Other miscellaneous Supreme Court matters, such as public meetings, are included on the media releases. Miranda–Refers to the United States Supreme Court decision Miranda v. Arizona, 384 US 436, 86 S Ct 1602, 16 L Ed 2d 694 (1966), from which the rules regarding the right to remain silent and the right to an attorney are taken. On Appeal / Review–A party “appeals” from a lower court as of right, but a party “seeks review” in court of the action of an administrative agency or when the higher court has discretion whether to take the case. The related documents are called “notice of appeal” and “petition for judicial review” (in the case of an administrative proceeding) or “petition for review” (in the case of a petition to the Oregon Supreme Court). Within an administrative system, the correct term is “appeal.” For example, a party may “appeal” to the Workers’ Compensation Board from the order of an administrative law judge. Go to TABLE OF CONTENTS 111 Go to INDEX ORAP–Acronym for Oregon Rules of Appellate Procedure. Those rules are applicable to proceedings in the Oregon Supreme Court and Court of Appeals and supplement the statutory requirements. The most recent version can be accessed on the Oregon Judicial Department’s website, on the Rules page. Oregon Reports–Hardbound volumes of the opinions issued by the Oregon Appellate Courts, published separately by individual court. The Oregon Reports is the official reporter for Oregon case law, and published and distributed by the Office of the State Court Administrator, Publications Program. Parenthetical Information–When used in relation to citation, text within parentheses that usually indicates alterations to text or provides explanatory statements. Per Curiam–By the court. Precedent–Except as provided in ORAP 10.30, all written Oregon appellate court opinions have the same precedential weight, whether signed or issued per curiam. Publications–The program of the State of Oregon Law Library that is responsible for publishing the official version of the Oregon Appellate Courts opinions (and miscellaneous other official materials) in hardbound volumes as the Oregon Reports, in softbound biweekly Advance Sheets, and online. Publications Website–The Publications website, managed by the Publications Program, includes links to Supreme Court, Court of Appeals, and Tax Court opinions since 1998, currently in publication format (not slip opinion format). The page numbers in opinions on the Publications website can slightly vary from the page numbers ultimately used in the Oregon Reports, due to minor edits that might be made after an opinion is issued. The Publications website also includes links to all media releases issued by the appellate courts, and printable and online versions of this Style Manual. Release Date–Date that opinions are released to the public via the Appellate Courts Records Section. Currently the Court of Appeals releases opinions to the public on Wednesdays, and the Supreme Court releases opinions to the public on Thursdays or other days as necessary. Running Head–Term of art used when referring to the official case name used for citation purposes that is printed as part of the header information when a case gets published in the official reports. Go to TABLE OF CONTENTS 112 Go to INDEX Signals–Introductory words indicating support, comparison, or contradiction of a stated proposition. Those words are italicized, and generally are followed by a parenthetical statement explaining the relevance of the citation. Slip Opinion–Appellate court decisions in an 8½″ x 11″ format using the court’s initial pagination, and filed in the electronic case file as a document image linked to the case register in the Appellate Case Management System. Once linked, the slip opinion can be viewed electronically by the public in the Records Office public viewing station, unless the case is confidential. Paper copies of slip opinions also are mailed to the parties. Summary Disposition–A summary disposition is a disposition of an appeal, a judicial review, or other proceeding pending in an appellate court without full briefing and submission to the courts on the merits of the case. Summary dispositions include dismissals, summary affirmances, and summary reversals. By various statutes governing particular kinds of cases, the Court of Appeals has authority to summarily affirm trial court decisions in particular cases if the court determines that the appeal does not present a substantial question of law. Supreme Court–The highest court in Oregon and the final arbiter of Oregon law. Its primary function is as law-announcing forum. The Supreme Court has discretionary review, e.g., cases on review from the Court of Appeals; direct review, e.g., death-penalty cases, tax cases, and disciplinary matters; and original discretionary jurisdiction, e.g., mandamus, quo warranto, habeas corpus, and certified questions. Tag Line–The court’s formal disposition of a case. The tag line is listed both on the title page and as the last line of an opinion. Tax Court–The Tax Court is a special court that has exclusive, statewide jurisdiction to hear only cases that involve Oregon’s tax laws, including income taxes, corporate excise taxes, property taxes, timber taxes, cigarette taxes, local budget laws, and property tax limitations. There are no jury trials, and appeals go directly to the Supreme Court. The Oregon Tax Court has two divisions–the Regular Division and the Magistrate Division. TCR / TCR-MD–Acronyms for the Tax Court rules and Tax Court Rules-Magistrate Division. Those rules are applicable to proceedings in the Oregon Tax Court and supplement statutory requirements. To the degree the wording of a Tax Court Rule mirrors that of an Oregon Rule of Civil Procedure, case interpretations of the Oregon Rules of Civil Procedure are authoritative for applying the Tax Court Rules. For circumstances not addressed by the Tax Court Rules-Magistrate Division, proceedings in the Magistrate Division defer to the guidance of the Tax Court Rules. Go to TABLE OF CONTENTS 113 Go to INDEX Opinion Overview I Petition filed Petition assigned to chambers Law clerk reads petition, summarizes issue/argument, and writes memo recommending whether to allow, deny, or hold Justice reads clerk's memo and makes own recommendation Petition and clerk memo with Justice recommendation are circulated to full court and placed on conference agenda At conference, justices vote to allow, deny, or hold ("Discuss" agenda items are discussed individually) Deny Allow Hold If denied, an order denying review issues nine days after conference; petitioner may petition for reconsideration. Dismissed as improvidently allowed At post-argument conference, justices discuss decision; if issue was different than originally thought or if preservation impediment is clear, case may be dismissed Oral argument If petition is allowed, an "allow" order issues, scheduling the case for argument. Occasionally, a staff person first prepares an allow memo to confirm that the case has no jurisdictional or preservation problems. Petition may be held for other case with a similar issue that is currently being considered by the SC Case assigned to chambers Justice drafts opinion and circulates to full court Draft opinion is discussed at conference and may be voted down or sent back for rewrites/edits Draft rewritten/edited and recirculated Court denies petition as improvidently allowed Problems Down (Occasionally, if the court is in agreement on the result but several suggestions are to be incorporated, a "final" draft goes back to conference for final court review before the court votes the opinion "down") Opinion is down-drafted; summary and media release are drafted. Final opinion sent to Records; final opinion and summary sent to Publications SUPREME COURT OPINIONS: A STEP-BY-STEP OVERVIEW Opinion posted to the case register and online at 8:00 a.m. (usually on Thursday) Go to TABLE OF CONTENTS 114 Go to INDEX Opinion Overview II POST-ARGUMENT CONFERENCE • Straw vote • Some cases may be AWOPed at this time • PJ assigns each case not AWOPed to a judge Judge edits / rewrites draft opinion New majority or dissent / concurrence drafted and distributed; original opinion may be revised Circulation to all judges – Referral deadline is 4pm the Tuesday after circulation date Case taken "en banc" • Straw vote • Assignment of judge to write new majority or dissent /concurrence COURT OF APPEALS OPINIONS: A STEP-BY-STEP OVERVIEW Cases assigned to departments Judges read briefs Pre-argument panel meeting ORAL ARGUMENT Research and writing of draft opinion or AWOP memo Opinion goes "into department” DEPARTMENT CONFERENCE Approval to “go down” Approval to “go down” FULL COURT CONFERENCE Opinion “pulled back” to dept. Pass Reassignment to another judge Circulation to all staff attorneys – SA conference Monday after circulation date Opinion prepared for publication – Must be “on the cart” by Friday after referral Opinion released to the public – Wednesday at 8am Publication in Advance Sheets Proofreading of Advance Sheets Publication in hard bound volume of Or App Referral to full court; referral memo written and distributed Referral to full court Go to TABLE OF CONTENTS 115 Go to INDEX Opinion Overview III Case Management Conference Case Filed REGULAR DIVISION Publish in Advance Sheets Approve Final Draft Research and Write Draft Opinion Trial or Oral Argument Prepare Opinion for Publication Edit/rewrite draft opinion Prepare title page Edit/rewrite draft opinion, summaries Write case and index summary Release opinion to public Set briefing schedule and oral argument Set trial Read Briefs/Memoranda TAX COURT - REGULAR DIVISION OPINIONS: A STEP-BY-STEP OVERVIEW Go to TABLE OF CONTENTS 116 Go to INDEX Opinion Overview IV Case Management Conference Case Filed MAGISTRATE DIVISION Publish in Advance Sheets Approve Final Draft Research and Write Draft Decision Trial or Oral Argument Prepare Decision for Publication Edit/rewrite draft decision Prepare title page Edit/rewrite summaries Write case and index summary Release decision to public Submit for publication Council approval Finalize Set briefing schedule and oral argument Set trial Read Briefs/Memoranda TAX COURT - MAGISTRATE DIVISION OPINIONS: A STEP-BY-STEP OVERVIEW Go to TABLE OF CONTENTS 117 Go to INDEX APPENDIX (Standard Proofreader’s Marks) Go to TABLE OF CONTENTS 118 Go to INDEX INDEX A A / an, word pairs .............................................................................................................. 90 A / from a, word pairs ....................................................................................................... 90 Abbreviations . ...................... 16, 17, 30, 31, 33, 35, 38, 39, 41, 49, 68-69, 75, 76, 90, 103 Absent / without, word pairs . ............................................................................................ 90 Academic degrees ............................................................................................................ 76 Acronyms . ............................................................................................ 75, 87, 90, 111, 112 Active voice ................................................................................................................... 107 Administrative Law Judge ....................................................................................... 71, 110 Administrative Law Judge / Hearing Officer, word pairs ................................................. 91 Administrative Rules, Oregon ......................................................................................... 39 Affect / effect, word pairs . ................................................................................................. 91 Affirmed by an equally divided court, generally ........................................................... 108 Agency Decisions, citation to .......................................................................................... 40 Allow / grant, word pairs .................................................................................................. 91 Alternate / alternative, word pairs . .................................................................................... 91 Although / while, word pairs ............................................................................................ 91 Amended Oregon Constitution . ............................................................................................. 33 Statutes . ........................................................................................................... 37-38 American Bar Association Standards, citation to ...................................................... 43-44 American Jurisprudence . .................................................................................................. 51 American Law Reports .................................................................................................... 51 Amicus curiae . .............................................................................................................. 7, 67 Amicus curiae / amici curiae, word pairs ......................................................................... 92 Among / between, word pairs ........................................................................................... 93 Apostrophes ......................................................................................................... 68, 74, 77 Appeal / review, on ........................................................................................... 10, 103, 110 Appeal / review, word pairs .............................................................................................. 92 Appeals / appeals from, word pairs . .................................................................................. 92 Appellate Procedure, Oregon Rules of .................................................................... 39, 111 Appendices, generally . ..................................................................................................... 13 Appositives ................................................................................................................ 83, 84 Attorney General Opinions, citation to . ........................................................................... 41 Authority, on Summary Disposition .............................................................................. 112 Authority / right, word pairs ............................................................................................. 98 Go to TABLE OF CONTENTS 119 Go to INDEX B Ballot measures, citation to . ............................................................................................. 43 Bar Rules of Procedure, citation to ............................................................................ 43-44 Because / since, word pairs . ........................................................................................ 93, 99 Before / prior to, word usage .......................................................................................... 104 Between / among, word pairs . ........................................................................................... 93 Block quotations .......................................................................... 13, 52, 53, 54, 62-64, 79 Bluebook, The .................................................................................. 16, 18, 30, 31, 49, 108 Body of opinion, generally ............................................................................................ 5, 9 Boldface ..................................................................................................................... 12, 14 Books ................................................................................................................... 48, 50, 68 Brackets, use of . ......................................................................................................... 59-62 To correct spelling ................................................................................................ 62 With periods . ......................................................................................................... 63 Briefs, citation to . ....................................................................................................... 44-45 C Cannot, word functions . ................................................................................................. 101 Case law Case names ............................................................................. 14, 16-17, 28, 29, 68 Cases not yet appearing in publication ................................................................. 32 Federal case law . ................................................................................................... 29 Oregon case law, citation of generally . ........................................................... 20-29 Certiorari denied by United States Supreme Court ...................... 21-22, 24 Court of Appeals decision affirmed / reversed by Supreme Court ............. 23 Court of Appeals decision remanded by Supreme Court . .......................... 23 Court of Appeals decision vacated by unpublished order ......................... 24 On reconsideration . .............................................................................. 21, 24 On rehearing ........................................................................................ 20-21 Oregon review denied and US Supreme Court certiorari denied ............. 24 Oregon Supreme Court opinion reversed by United States Supreme Court ...................................................................................................... 22 Oregon Supreme Court review allowed .................................................... 23 Oregon Supreme Court review allowed, then dismissed . .......................... 23 Oregon Supreme Court review denied ...................................................... 23 Oregon Tax Court case citations .......................................................... 24-25 Short form, Oregon .............................................................................. 25-29 United States Supreme Court appeal dismissed ........................................ 24 United States Supreme Court certiorari denied by opinion ...................... 24 Other state case law ........................................................................................ 31-32 Go to TABLE OF CONTENTS 120 Go to INDEX Certiorari ................................................................................................................... 16, 67 Certiorari denied by United States Supreme Court . ....................................... 21-22, 23-24 Children, using names of, word usage ........................................................................... 103 Circuit court / trial court, word pairs ................................................................................ 93 Citations Agency decisions ............................................................................................ 40-41 Attorney General Opinions . .................................................................................. 41 Ballot measures . .................................................................................................... 43 Case law Federal case law . .................................................................................. 29-31 Oregon Court of Appeals ..................................................................... 22-24 Oregon Supreme Court ........................................................................ 20-22 Oregon Tax Court ................................................................................ 24-25 Other state or court citation ................................................................. 31-32 Case names ....................................................................... 14, 16-17, 22, 28, 29, 68 Concurrence or dissent . ............................................................................. 10, 20, 29 Consistency of . ................................................................................................ 16, 17 Constitution, citation to Oregon ................................................................................................. 33, 69 United States ........................................................................................ 45, 69 Dictionaries . .................................................................................................... 49-51 Federal citations Internal Revenue Code .............................................................................. 46 Legislative history ..................................................................................... 46 Public Laws and United States Code . ........................................................ 46 Rules of Evidence and Procedure .............................................................. 45 Tax materials . ....................................................................................... 46-47 Uniform Laws ............................................................................................ 47 United States Constitution ................................................................... 45, 69 Footnotes . .............................................................................................................. 28 Citation within a footnote .......................................................................... 13 Form of ........................................................................................................... 16-19 Harvard Bluebook, cross references to ................................................................. 16 Id., use of ............................................................................................ 28, 33, 35, 67 Internet, citation to . ................................................................................... 15, 17, 50 Legislative history . .................................................................................... 41-43, 46 Municipal Codes ................................................................................................... 41 Oregon Administrative Rules . ......................................................................... 39-40 Oregon Evidence Code ......................................................................................... 38 Oregon Rules of Appellate Procedure .................................................................. 39 Go to TABLE OF CONTENTS 121 Go to INDEX Citations (continued) Oregon Rules of Civil Procedure . ................................................................... 38-39 Periodical articles, books, treatises, restatements, etc. ................................... 48-51 Rules of Professional Conduct . ....................................................................... 43-44 Short form Books, articles, periodicals ........................................................................ 51 Oregon Court of Appeals and Supreme Court cases ................................. 25 Oregon Tax Court cases ............................................................................. 29 Signals . .................................................................................................... 18, 67, 112 Statutory citation General Laws of Oregon . ........................................................................... 34 Oregon Compiled Laws Annotated ........................................................... 35 Oregon Laws . ....................................................................................... 33-34 Oregon Revised Statutes . ..................................................................... 35-38 Repealed or renumbered / use of former .................................. 36-37, 40, 44 String citations .............................................................................................. 18, 108 Supra, use of ............................................................................................. 27, 29, 61 Uniform Jury Instructions . .................................................................................... 40 Uniform Trial Court Rules / Supplementary Local Rules .................................... 40 Cite / cite to, word pairs . ................................................................................................... 93 Collective nouns, grammar ............................................................................................ 105 Colons, in general .......................................................................................... 13, 52, 78-79 Using uppercase following colons . ....................................................................... 79 Commas ............................................................................................ 13, 79-84, 88, 99-100 Comments .................................................................................... 33, 38, 42, 43, 47, 48, 51 Complex sentences . .................................................................................................... 81-82 Compose / comprise, word pairs . ...................................................................................... 94 Compound Predicates .............................................................................................................. 80 Sentences .............................................................................................................. 81 Comprise / compose, word pairs . ...................................................................................... 94 Concurrence, citation to ....................................................................................... 10, 20, 29 Concurrently with / consecutively to, word pairs ............................................................. 94 Concurring Opinion, in general ..................................................................................... 5, 8 Consecutively to / concurrently with, word pairs ............................................................. 94 Consistent with / consistently with, word pairs ................................................................ 94 Constitutional citations Oregon Constitution . ....................................................................................... 33, 69 United States Constitution .............................................................................. 45, 69 Consolidated cases ............................................................................................................. 6 Go to TABLE OF CONTENTS 122 Go to INDEX Contractions, word functions ..................................................................................... 9, 101 Corpus Juris Secundum (CJS), abbreviations . ................................................................. 76 Criterion / criteria, word pairs . .......................................................................................... 95 D Dashes ........................................................................................................................ 84, 88 Data / datum, word pairs . .................................................................................................. 95 Dates .................................................................................................................... 12, 72, 74 Decimals .......................................................................................................................... 73 Decree / judgment, word pairs .......................................................................................... 95 Disciplinary cases, lawyer ....................................................................................... 27, 112 Dictionary, citation to . ................................................................................................ 49-51 Disinterested / uninterested, word pairs . ........................................................................... 95 Dissent, citation to ............................................................................................... 10, 20, 29 Dissenting opinion, in general ....................................................................................... 5, 8 Driver’s license / driving privileges, word pairs . .............................................................. 95 E Editor ......................................................................................................................... 48, 49 Effect / affect, word pairs .................................................................................................. 91 Electronic (Online) sources . ........................................................................... 15, 17, 32, 50 Ellipsis, use of . ............................................................................................... 18, 61, 62-65 Employe / employee, word pairs ...................................................................................... 95 Evidence Code (Rules of Evidence) Federal................................................................................................................... 45 Oregon............................................................................................................. 38, 70 Expedient / expeditious, word pairs . ................................................................................. 96 F Factfinder Without hyphen . .................................................................................................... 85 Word functions . ................................................................................................... 101 Federal citations, in general .......................................................... 17, 29-31, 45-47, 69, 71 Case law . ............................................................................................................... 29 Federal Rules Of civil procedure ...................................................................................... 45 Of criminal procedure . ............................................................................... 45 Of evidence ................................................................................................ 45 Federal tax materials Regulations and Determinations . ......................................................... 46-47 Internal Revenue Code (IRC) ............................................................................... 46 Go to TABLE OF CONTENTS 123 Go to INDEX Fewer / less than, word pairs . ............................................................................................ 96 Finding / holding, word pairs . ..................................................................................... 96, 97 Fonts . .......................................................................................................................... 14, 67 Footnotes Citation in ................................................................................................. 13, 20, 25 Citation to ............................................................................................................. 28 Generally . .......................................................................................................... 5, 13 Omitted ................................................................................................................. 65 On title page . ....................................................................................................... 7-8 Placement in block quotations ........................................................................ 53-54 For / of, word pairs . ........................................................................................................... 97 Foreign words and phrases, italics ................................................................................... 67 Forgo / forego, word pairs ................................................................................................ 96 Formatting Body of opinion, generally ............................................................................... 9-14 Generally . ................................................................................................................ 5 Quotations . ............................................................................................................ 52 Title page ............................................................................................................ 6-8 Former ................................................................................................................. 36-37, 44 Fractions . .......................................................................................................................... 73 G Gender-neutral wording ................................................................................................. 107 Glossary ......................................................................................................................... 108 Grammar, references ........................................................................................................ 66 Grant / allow, word pairs . .................................................................................................. 91 H Headings, use of . ........................................................................................................ 11-12 Hearing Officer / Administrative Law Judge, word pairs ................................................. 91 Holding / finding, word pairs ............................................................................................ 96 Hyphens / hyphenation ................................................................ 40, 66, 72, 73, 74, 85-87 I Id., use of . ....................................................................................................... 28, 33, 35, 67 If . . . was / were, word pairs . ............................................................................................ 97 Implicitly / impliedly, word pairs . ..................................................................................... 97 Improvidently allowed ............................................................................................... 16, 23 In light of / in the light of, word pairs . .............................................................................. 97 Informal terminology ..................................................................................................... 107 Go to TABLE OF CONTENTS 124 Go to INDEX Infra, use of . ..................................................................................................................... 29 Initialisms . .................................................................................................................. 75-76 Initiative Petitions ............................................................................................................ 43 Internal Revenue Code . .................................................................................................... 46 Internet (Online) sources Citation . ..................................................................................................... 15, 17, 50 In general .............................................................................................................. 32 Italics Book, article, or treatise names . ................................................................ 48-51, 68 Case names ............................................................................................... 29, 68, 78 Foreign words and phrases ................................................................................... 67 Generally . .................................................................................................. 12, 14, 62 In citations . ...................................................................................................... 36, 58 Signals . .......................................................................................................... 18, 112 J Judgment / decree, word pairs .......................................................................................... 95 Judgment notwithstanding the verdict Abbreviations . ....................................................................................................... 76 Initialisms . ............................................................................................................. 75 Jump citations (see pinpoint citations) . .................................................. 30, 31, 32, 45, 108 Jury Instructions, Uniform, citation to . ............................................................................ 40 L Land Use Board of Appeals, citation to ........................................................................... 40 Language, word usage ................................................................................................... 102 Law Enforcement, names for, word usage . .................................................................... 103 Latin words, italics . .......................................................................................................... 67 Legislative history Federal, citation to ................................................................................................ 46 Oregon, citation to .......................................................................................... 41-43 Less than / fewer, word pairs ............................................................................................ 96 Less than / more than, word pairs ..................................................................................... 98 Letter opinion, word usage ............................................................................................ 102 License, driver’s / driving privileges, word pairs ............................................................. 95 Lists, bulleted or numbered . ....................................................................................... 12-13 Punctuating ..................................................................................................... 87-88 Loath / loathe, word pairs ................................................................................................. 97 Go to TABLE OF CONTENTS 125 Go to INDEX M May / might, word pairs . ................................................................................................... 97 Minutes, citation to .......................................................................................................... 42 Money .............................................................................................................................. 75 Months, abbreviations . ..................................................................................................... 76 More than / less than, word pairs ...................................................................................... 98 Municipal codes, citation to . ............................................................................................ 41 N Narrative citation form Initiative Petitions, Ballot Measures, and Voters’ Pamphlets ............................... 43 Legislative History . ............................................................................................... 41 Oregon Administrative Rules . ......................................................................... 39-40 Oregon Constitution . ............................................................................................. 33 Oregon Evidence Code ......................................................................................... 38 Oregon Laws . .................................................................................................. 33-34 Oregon Revised Statutes . ................................................................................ 35-38 Oregon Rules Civil Procedure ........................................................................ 38-39 Rules of Professional Conduct (RPC) ............................................................ 43-44 Uniform Jury Instructions . .................................................................................... 40 United States Code . ............................................................................................... 46 United States Constitution .................................................................................... 45 United States Rules of Evidence and Procedure . .................................................. 45 Nonprecedential memorandum opinions...................................................................... 6, 22 Nouns, collective, grammar ........................................................................................... 105 Numbers, in general ................................................................................................... 72-75 Case . ........................................................................................................................ 6 Citation page ................................................................................................... 28, 34 Footnote ................................................................................................................ 13 Slip opinion line . ..................................................................................................... 5 Subsubparagraph . .................................................................................................. 11 Volume and issue .................................................................................................. 49 O Of / for, word pairs . ........................................................................................................... 97 Offices .............................................................................................................................. 76 Online sources . ................................................................................................................. 32 Only, word usage ........................................................................................................... 102 Opinion Body of, in general . ................................................................................................. 9 Overviews ................................................................................................... 113-116 Slip opinions ................................................................................................... 5, 112 Go to TABLE OF CONTENTS 126 Go to INDEX Oregon Administrative Rules . .......................................................................................... 39 Oregon Compiled Laws Annotated . ................................................................................. 35 Oregon Constitution . ........................................................................................................ 33 Oregon Court of Appeals Defined . ............................................................................................................... 109 Oregon Evidence Code .................................................................................................... 38 Oregon Laws . ............................................................................................................. 33-34 Oregon Revised Statutes ............................................................................................ 35-38 Oregon Rules of Appellate Procedure . ..................................................................... 39, 111 Oregon Rules Civil Procedure ................................................................................... 38-39 Oregon Supreme Court Defined . ............................................................................................................... 112 Outlining .......................................................................................................................... 11 Sample illustration ................................................................................................ 12 Over / under–more than / less than, word pairs ................................................................. 98 Overruled case law . .................................................................................................... 21, 22 P Parallel citation .......................................................................... 20, 21, 23, 24, 26, 31, 108 Parallel construction, grammar ...................................................................................... 105 Parameter, word usage ................................................................................................... 102 Parenthetical elements ............................................................................................... 83-84 Punctuating ........................................................................................................... 88 Parenthetical information . ............................................. 18-19, 20, 32, 37, 42, 65, 111, 112 With block quotations ............................................................................... 53, 55-58 Part / portion, word pairs .................................................................................................. 98 Parties to cases Generally . ........................................................................................ 6, 10, 22, 27, 92 Word usage . ......................................................................................................... 103 Passive voice . ......................................................................................................... 105, 107 Past perfect tense . ........................................................................................................... 106 Patdown Without hyphen . .................................................................................................... 85 Word functions . ................................................................................................... 101 Pending cases ................................................................................................................... 23 Per curiam ............................................................................................ 5, 8, 10, 67, 70, 111 Percentages ................................................................................................................ 73, 85 Periodical articles . ...................................................................................................... 48-49 Ph.D., abbreviations . ........................................................................................................ 76 Pleaded / pled, word pairs . ................................................................................................ 98 Pinpoint citations ................................................................................... 30, 31, 32, 45, 108 Go to TABLE OF CONTENTS 127 Go to INDEX Portion / part, word pairs .................................................................................................. 98 Possessives, generally ................................................................................................ 77-78 Possessive endings on case names . ................................................................. 29, 68 Post-conviction cases, word usage . ................................................................................ 104 Previous / prior, word usage............................................................................................ 104 Prior / previous, word usage ........................................................................................... 104 Prior to / before, word usage . .......................................................................................... 104 Proofreader’s Marks . ...................................................................................................... 117 Proved / proven, word pairs .............................................................................................. 98 Punctuation .............................................................................................. 52, 60, 63, 77-89 Of elements ..................................................................................................... 83, 88 Of lists . ............................................................................................................ 87-88 Resources .............................................................................................................. 66 Q Quotations Block quotations ................................................................................. 52, 62-65, 79 Of footnotes ............................................................................................... 13 Footnotes within ........................................................................................ 54 Generally . ........................................................................................................ 13, 52 Order of citations and parenthetical phrases with block quotes ........................... 53 Parenthetical phrases within text .......................................................................... 55 Placement of citation ...................................................................................... 53-54 Use of brackets . ............................................................................................... 59-62 Use of ellipses . ................................................................................................ 62-65 Use of uppercase . .................................................................................................. 59 R Ratios ............................................................................................................................... 73 Reconsideration . ......................................................................................................... 21, 24 Reference guides . ....................................................................................................... 50, 66 Rehearing, Oregon cases . ..................................................................................... 16, 20-21 Repealed or renumbered statutes ................................................................... 36-37, 40, 44 Restatements .............................................................................................................. 48, 51 Review / appeal, on . .................................................................................. 10, 103, 110, 112 Review / appeal, word pairs . ............................................................................................. 92 Right / authority, word pairs ............................................................................................. 98 Rules of Appellate Procedure, Oregon, citation to .......................................................... 39 Rules of Civil Procedure, citation to Federal .................................................................................................................. 45 Oregon ............................................................................................................ 38-39 Rules of Professional Conduct, citation to . ................................................................ 43-44 Go to TABLE OF CONTENTS 128 Go to INDEX S Seasons . ............................................................................................................................ 72 Semicolons . ........................................................................................ 13, 17, 18, 52, 63, 88 Sentencing Guidelines, citation to ................................................................................... 43 Series, items in a series ................................................................ 27, 28, 39, 40, 72, 80, 88 Short citation form, generally .................................................................................... 10, 13 Books and Treatises .............................................................................................. 51 Federal Jurisdictions ....................................................................................... 29-31 Oregon Court of Appeals and Supreme Court cases . ............................................ 25 Oregon Tax Court ................................................................................................. 29 Periodical Articles . ................................................................................................ 49 States Other Than Oregon . .............................................................................. 31-32 Sic . .............................................................................................................................. 61-62 Signals, generally . .................................................................................................... 18, 112 Italics . .............................................................................................................. 14, 67 Since / because, word pairs . ........................................................................................ 93, 99 Spacing Around dashes . ...................................................................................................... 84 With abbreviations ................................................................................................ 17 With asterisks (ellipsis) . ........................................................................................ 62 Spelling, variant ............................................................................................................. 104 Statement of facts, generally . ........................................................................................... 10 Statutes, Oregon, citation to . ................................................................................ 35-38, 70 String citations ......................................................................................................... 18, 108 Sub nom . ........................................................................................................................... 22 Subjunctive mood .......................................................................................................... 106 Subsequent history ................................................................. 18, 20, 22, 24, 25, 37-38, 57 Supplementary Local Rules, citation to ........................................................................... 40 Supra, use of ........................................................................................................ 27, 29, 61 T Tag lines ................................................................................................. 8, 11, 93, 100, 112 Tax Court, Oregon, defined . ........................................................................................... 112 Case citation . ................................................................................................... 24-25 Short form .................................................................................................. 29 Magistrate Division, defined . .............................................................................. 110 Miscellaneous tax materials . ........................................................................... 46-47 Tax Court, United States Case citation . ......................................................................................................... 31 Federal tax cases ................................................................................................... 31 Go to TABLE OF CONTENTS 129 Go to INDEX Technical terminology ................................................................................................... 107 Tenses, verb, grammar ................................................................................................... 106 That, word usage . ........................................................................................................... 104 That / this; those / these, word pairs .................................................................................. 99 That / which, word pairs ................................................................................................... 99 Time ................................................................................................................................. 74 Title page, formatting . .................................................................................................... 6-8 Titles . .................................................................................................................... 68-69, 76 Transitions, between discussions ..................................................................................... 14 Treatises ..................................................................................................................... 48-51 Trial court / circuit court, word pairs ................................................................................ 93 Trial Court Rules, Uniform, citation to . ........................................................................... 40 Typeface ..................................................................................................................... 14, 67 Foreign words ................................................................................................. 14, 67 Italic ...................................................................................................................... 14 Of case names ................................................................................................. 14, 68 Roman . ............................................................................................................ 17, 67 Signals . .................................................................................................................. 67 To show emphasis ................................................................................................. 14 U Under / over, word pairs ................................................................................................... 98 Underscoring, use of ........................................................................................................ 14 Uniform Jury Instructions, citation to . ............................................................................. 40 Uniform Laws, citation to ................................................................................................ 47 Uniform Trial Court Rules, citation to . ............................................................................ 40 Uninterested / disinterested, word pairs . ........................................................................... 95 United States Code, citation to . ........................................................................................ 46 Unpublished decisions, citation to ............................................................................. 25, 30 Orders, citation to ................................................................................................. 24 Uppercase, use of ..................................... 6, 7, 8, 10, 11, 12, 14, 17, 18, 33, 59, 68-72, 88 After colons . .......................................................................................................... 79 With brackets ........................................................................................................ 59 With prefixes ......................................................................................................... 86 V VAWA (Violence Against Women Act). ............................................................17, 20, 25, 61 Verbs Active voice ........................................................................................................ 107 Before colons ........................................................................................................ 79 In complex sentences ............................................................................................ 82 Go to TABLE OF CONTENTS 130 Go to INDEX Verbs (continued) “See” ..................................................................................................................... 67 Tenses, grammar ................................................................................................. 106 Was / were, agreement ......................................................................................... 106 With affect / effect . ................................................................................................. 91 With collective nouns ................................................................................... 95, 105 With compound modifiers . .................................................................................... 85 With compound predicates ................................................................................... 80 With “whom” ...................................................................................................... 100 Victims, names of, word usage ................................................................................ 10, 103 Violence Against Women Act (VAWA)............................................................17, 20, 25, 61 Voters’ Pamphlets . ............................................................................................................ 43 W Which / that, word pairs . ................................................................................................... 99 While / although, word pairs . ............................................................................................ 91 Who / whom, word pairs . ................................................................................................ 100 Without / absent, word pairs ............................................................................................. 90 Word pairs ........................................................................................................................ 90 Other resources ..................................................................................................... 66 Workers’ Compensation Board .......................................................................... 69, 92, 110 Citation to ............................................................................................................. 40
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https://www.quora.com/ABC-is-an-isosceles-triangle-where-AB-AC-The-angle-bisector-of-B-angel-B-intersects-AC-to-D-where-BC-BD-AD-What-is-the-value-of-A-angle-A
∆ABC is an isosceles triangle where AB=AC. The angle bisector of <B (angel B) intersects AC to D, where BC=BD+AD. What is the value of <A (angle A)? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Angle One Bisectors Isosceles Triangles PLANE GEOMETRY Angles Construction of Triangles Angle Bisector Theorem Concept of Geometry 5 ∆ABC is an isosceles triangle where AB=AC. The angle bisector of <B (angel B) intersects AC to D, where BC=BD+AD. What is the value of <A (angle A)? All related (34) Sort Recommended Simon Tsai Lives in Taiwan · Author has 4.9K answers and 2M answer views ·Updated 4y I vaguely remember that months ago, I wrote an answer on Quora showing how to find the length of the bisector of an angle in a triangle, but I cannot find it, and I am not writing it again. Instead I am giving you a link to a Web page where you will find a proof of the results which follow: [math]\quad \displaystyle a \, \overline {AQ} = c \, \overline {CQ}[/math] [math]\quad \displaystyle \overline {BQ} = \sqrt {a c \left[ 1 - {\left( \frac {b}{a + c} \right)}^{2} \right]}[/math] Note the conventions [math]\overline {AB} = c[/math], [math]\overline {AC} = b[/math] and [math]\overline {BC} = a[/math]. As specified in this question, [math]a = y[/math] and [math]b = c = x[/math]: [math]\quad \di[/math] Continue Reading I vaguely remember that months ago, I wrote an answer on Quora showing how to find the length of the bisector of an angle in a triangle, but I cannot find it, and I am not writing it again. Instead I am giving you a link to a Web page where you will find a proof of the results which follow: [math]\quad \displaystyle a \, \overline {AQ} = c \, \overline {CQ}[/math] [math]\quad \displaystyle \overline {BQ} = \sqrt {a c \left[ 1 - {\left( \frac {b}{a + c} \right)}^{2} \right]}[/math] Note the conventions [math]\overline {AB} = c[/math], [math]\overline {AC} = b[/math] and [math]\overline {BC} = a[/math]. As specified in this question, [math]a = y[/math] and [math]b = c = x[/math]: [math]\quad \displaystyle \overline {AQ} = \frac {{x}^{2}}{x + y}[/math] [math]\quad \displaystyle \overline {CQ} = \frac {{y}^{2}}{x + y}[/math] [math]\quad \displaystyle \overline {BQ} = \frac {x y}{x + y} \sqrt {2 + \frac {y}{x}}[/math] If [math]\overline {BQ} = \overline {BC} - \overline {AQ}[/math], then [math]\quad \displaystyle \left( x + y \right) \left( {x}^{3} - 3 {x}^{2} y + {y}^{3} \right) = 0[/math] This can be derived without any ingenuity. Just do it, and you will agree. Set [math]x = 1[/math] for the sake of simplicity, but without loss of generality. Since [math]x + y \neq 0[/math], [math]\quad \displaystyle {y}^{3} - 3 y + 1 = 0[/math] Solving such a cubic equation is not very easy. You can directly apply Cardano’s formula or follow his approach. I personally prefer the latter, but I am not going to explain everything because I am lazy. Set [math]y = \alpha + \beta[/math] with [math]\alpha \beta = 1[/math] and [math]{\alpha}^{3} + {\beta}^{3} + 1 = 0[/math]. Solve the following equation, which appears to be sextic but is actually quadratic: [math]\quad \displaystyle {\lambda}^{6} + {\lambda}^{3} + 1 = 0[/math] [math]\quad \displaystyle {{\lambda}^{3}}_{+} = {\alpha}^{3} = \frac {- 1 + i \sqrt {3}}{2}[/math] [math]\quad \displaystyle {{\lambda}^{3}}_{-} = {\beta}^{3} = \frac {- 1 - i \sqrt {3}}{2}[/math] I have assumed that you have already known how to solve a quadratic equation and am going to assume that you have learnt complex numbers and Euler’s formula, with which we can rewrite the results above and obtain [math]\quad \displaystyle {\alpha}{1} = {e}^{- \frac {4}{9} i \, \pi} \quad\quad {\alpha}{2} = {e}^{+ \frac {2}{9} i \, \pi} \quad\quad {\alpha}_{3} = {e}^{+ \frac {8}{9} i \, \pi}[/math] [math]\quad \displaystyle {\beta}{1} = {e}^{+ \frac {4}{9} i \, \pi} \quad\quad {\beta}{2} = {e}^{- \frac {2}{9} i \, \pi} \quad\quad {\beta}_{3} = {e}^{- \frac {8}{9} i \, \pi}[/math] [math]\quad \displaystyle {y}{1} = {\alpha}{2} + {\beta}_{2} = 2 \cos \left( 2 \pi \, / \, 9 \right)[/math] [math]\quad \displaystyle {y}{2} = {\alpha}{1} + {\beta}_{1} = 2 \cos \left( 4 \pi \, / \, 9 \right)[/math] [math]\quad \displaystyle {y}{3} = {\alpha}{3} + {\beta}_{3} = 2 \cos \left( 8 \pi \, / \, 9 \right)[/math] [math]{y}_{3} < 0[/math], so we rule it out. We are very close to the answer now. What we are ultimately interested in is the the vertex angle, which can be calculated as follows: [math]\quad \displaystyle \angle {BAC} = 2 \arcsin \left( y \, / \, 2 \right)[/math] If [math]y = {y}{2}[/math], then [math]\angle {BAC} = 20°[/math]. This is obviously impossible; simply draw a graph, and you will see. If [math]y = {y}{1}[/math], then [math]\angle {BAC} = 100°[/math]. This is exactly the answer that we have been looking for. Upvote · 9 1 Promoted by Coverage.com Johnny M Master's Degree from Harvard University (Graduated 2011) ·Updated Sep 9 Does switching car insurance really save you money, or is that just marketing hype? This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. Continue Reading This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. It always sounded like a hassle. Dozens of tabs, endless forms, phone calls I didn’t want to take. But recently I decided to check so I used this quote tool, which compares everything in one place. It took maybe 2 minutes, tops. I just answered a few questions and it pulled up offers from multiple big-name providers, side by side. Prices, coverage details, even customer reviews—all laid out in a way that made the choice pretty obvious. They claimed I could save over $1,000 per year. I ended up exceeding that number and I cut my monthly premium by over $100. That’s over $1200 a year. For the exact same coverage. No phone tag. No junk emails. Just a better deal in less time than it takes to make coffee. Here’s the link to two comparison sites - the one I used and an alternative that I also tested. If it’s been a while since you’ve checked your rate, do it. You might be surprised at how much you’re overpaying. Upvote · 999 485 999 103 99 17 Gary Ward MaEd in Education&Mathematics, Austin Peay State University (Graduated 1997) · Author has 4.9K answers and 7.6M answer views ·4y ∆ABC is an isosceles triangle where AB=AC. The angle bisector of <B (angel B) intersects AC to D, where BC=BD+AD. What is the value of <A (angle A)? [math]\angle A = 100^\circ \,[/math]and the base angles are [math]\, 40^\circ [/math] When it was boxed in between A(0, 0.9) making AD + BD = 2.029 with apex angle [math]\, \approx 96^\circ \,[/math] and A(0, 0.8) making AD + BD = 1.9727 with apex angle [math]\, \approx 103^\circ \,[/math] , I took a guess that the apex angle was [math]\, 100^\circ \,[/math] with base angles equal to [math]\, 40^\circ \, [/math]and that gave me AD + BD = 2.0000579 and I am okay with that. Line equations I used were: [math]\, y = m(x+1); \, y = -m(x[/math] Continue Reading ∆ABC is an isosceles triangle where AB=AC. The angle bisector of <B (angel B) intersects AC to D, where BC=BD+AD. What is the value of <A (angle A)? [math]\angle A = 100^\circ \,[/math]and the base angles are [math]\, 40^\circ [/math] When it was boxed in between A(0, 0.9) making AD + BD = 2.029 with apex angle [math]\, \approx 96^\circ \,[/math] and A(0, 0.8) making AD + BD = 1.9727 with apex angle [math]\, \approx 103^\circ \,[/math] , I took a guess that the apex angle was [math]\, 100^\circ \,[/math] with base angles equal to [math]\, 40^\circ \, [/math]and that gave me AD + BD = 2.0000579 and I am okay with that. Line equations I used were: [math]\, y = m(x+1); \, y = -m(x - 1); \, y =\tan \left( \frac{\tan^{-1} m}{2} \right) (x + 1)[/math] Upvote · 9 1 Simon Bridge Scientist · Author has 86.2K answers and 38.8M answer views ·4y Hint: cosine rule ie. because angle B is bisected, then point D is half way between A and C. therefore: AB = AC = 2AD = 2DC and you are given: BC = BD + AD You can do the cosine rule for ABC and for ABD with A inside the cosine. Upvote · 9 1 Promoted by Webflow Metis Chan Works at Webflow ·Feb 4 Which HTML editor is the best for web development? With the various HTML editors available today there are several considerations to keep in mind when deciding on which is the right fit for your company including ease of use, SEO controls, high performance hosting, flexible content management tools and scalability. Webflow offers all those features and allows you to experience the power of code – without writing it. You can take control of HTML5, CSS3, and JavaScript in a completely visual canvas — and let Webflow translate your design into clean, semantic code that’s ready to publish to the web, or hand off to developers. If you prefer more cus Continue Reading With the various HTML editors available today there are several considerations to keep in mind when deciding on which is the right fit for your company including ease of use, SEO controls, high performance hosting, flexible content management tools and scalability. Webflow offers all those features and allows you to experience the power of code – without writing it. You can take control of HTML5, CSS3, and JavaScript in a completely visual canvas — and let Webflow translate your design into clean, semantic code that’s ready to publish to the web, or hand off to developers. If you prefer more customization you can also expand the power of Webflow by adding custom code on the page, in the , or before the of any page. Trusted by 200,000+ leading organizations – Get started for free today! Get started for free today! Upvote · 99 37 9 6 Related questions More answers below If I have the triangle ABC with AB=AC and the angle bisector of B is drawn so it intersects AC in D, how can I calculate the angle BAC? Such that BD+AD=BC. The triangle ABC's AB> AC and the isosceles AD of angle A intersects the BC arm at point D. How will I prove that angle ADB is an obtuse angle? ABC is a right triangle such that AB = AC and the bisector of angle C intersects the side AB at D. How can I prove that AC + AD = BC? ABC is an isosceles triangle with AB=AC. Angle ABC=80 degree. D is a point on AC such that AD=BC. What is the measure of angle BDC? There is a triangle ABC, where B is 90 degrees. If the internal bisector of angle B meets AC at D, how do I find the value of AD when the sides AB = 3cm and BC = 4cm? Assistant Bot · 1y To solve for the angle [math]\angle A[/math] in the isosceles triangle [math]\triangle ABC[/math] where [math]AB = AC[/math], we will use the properties of angle bisectors and the given condition [math]BC = BD + AD[/math]. Identify the triangle properties: Let [math]\angle A = \alpha[/math]. Since [math]\triangle ABC[/math] is isosceles, [math]\angle B = \angle C = \beta[/math]. The sum of angles in a triangle gives us: [math]\alpha + 2\beta = 180^\circ \quad \text{(1)}[/math] 2. Using the angle bisector: The angle bisector of [math]\angle B[/math] divides it into two equal angles: [math]\angle ABD = \angle DBC = \frac{\beta}{2}[/math] 3. Apply the Law of Sines: In [math]\triangle BDC[/math]: [math]\frac{BC}{[/math] Continue Reading To solve for the angle [math]\angle A[/math] in the isosceles triangle [math]\triangle ABC[/math] where [math]AB = AC[/math], we will use the properties of angle bisectors and the given condition [math]BC = BD + AD[/math]. Identify the triangle properties: Let [math]\angle A = \alpha[/math]. Since [math]\triangle ABC[/math] is isosceles, [math]\angle B = \angle C = \beta[/math]. The sum of angles in a triangle gives us: [math]\alpha + 2\beta = 180^\circ \quad \text{(1)}[/math] 2. Using the angle bisector: The angle bisector of [math]\angle B[/math] divides it into two equal angles: [math]\angle ABD = \angle DBC = \frac{\beta}{2}[/math] 3. Apply the Law of Sines: In [math]\triangle BDC[/math]: [math]\frac{BC}{\sin(\angle BDC)} = \frac{BD}{\sin(\angle DBC)}[/math] [math]\frac{BC}{\sin(\angle DCA)} = \frac{AD}{\sin(\angle DAC)}[/math] 4. Using the condition [math]BC = BD + AD[/math]: From the condition [math]BC = BD + AD[/math], we can express [math]BC[/math] in terms of [math]BD[/math] and [math]AD[/math]: Using the angles: Since [math]\angle BDC = 180^\circ - \angle B - \angle DBC = 180^\circ - \beta - \frac{\beta}{2} = 180^\circ - \frac{3\beta}{2}[/math]. Equating the two expressions: We can derive relationships between the angles and side lengths using the above properties, and ultimately find: [math]2\beta = 180^\circ - \alpha \implies \beta = 90^\circ - \frac{\alpha}{2}[/math] 7. Substituting back into equation (1): [math]\alpha + 2(90^\circ - \frac{\alpha}{2}) = 180^\circ[/math] Simplifying: [math]\alpha + 180^\circ - \alpha = 180^\circ[/math] 8. Conclude the value of [math]\angle A[/math]: We find that [math]\alpha + 2\beta = 180^\circ[/math] holds true, and since [math]\beta[/math] can be expressed in terms of [math]\alpha[/math], we can solve: [math]\alpha = 60^\circ \quad \text{and} \quad \beta = 60^\circ[/math] Thus, the value of [math]\angle A[/math] is [math]\boxed{60^\circ}[/math]. Upvote · Nicolai Dimitrov Updated 5y Related If I have the triangle ABC with AB=AC and the angle bisector of B is drawn so it intersects AC in D, how can I calculate the angle BAC? Such that BD+AD=BC. Draw the point [math]E[/math] on the edge [math]BC[/math] so that [math]BE=BD[/math]. Then triangle [math]BDE[/math] is isosceles. Since [math]BD+AD = BC[/math] and [math]BE=BD[/math], we conclude that [math]CE = BC - BE = BC - BD = AD[/math] Furthermore, since [math]BD[/math] is the angle bisector of angle [math]\angle \, ABC [/math]and [math]AB = CA[/math],[math] [/math] [math]\frac{CA}{CB} = \frac{AB}{CB} = \frac{AD}{CD} = \frac{CE}{CD}[/math] Hence, triangles [math]\Delta \, DEC[/math] and [math]\Delta\, ABC[/math] are similar, because they also share a common angle. However, triangle [math]\Delta \, ABC[/math] is isosceles, so triangle [math]\Delta \, DEC[/math] is also isosceles. Consequently, [math]\angle\, CDE = \angle \, DCE = \angle \, BCA = 2\,\alpha.[/math] Thus, angle [math]\angle \, DEC = 180 - (\angle \, CDE[/math] Continue Reading Draw the point [math]E[/math] on the edge [math]BC[/math] so that [math]BE=BD[/math]. Then triangle [math]BDE[/math] is isosceles. Since [math]BD+AD = BC[/math] and [math]BE=BD[/math], we conclude that [math]CE = BC - BE = BC - BD = AD[/math] Furthermore, since [math]BD[/math] is the angle bisector of angle [math]\angle \, ABC [/math]and [math]AB = CA[/math],[math] [/math] [math]\frac{CA}{CB} = \frac{AB}{CB} = \frac{AD}{CD} = \frac{CE}{CD}[/math] Hence, triangles [math]\Delta \, DEC[/math] and [math]\Delta\, ABC[/math] are similar, because they also share a common angle. However, triangle [math]\Delta \, ABC[/math] is isosceles, so triangle [math]\Delta \, DEC[/math] is also isosceles. Consequently, [math]\angle\, CDE = \angle \, DCE = \angle \, BCA = 2\,\alpha.[/math] Thus, angle [math]\angle \, DEC = 180 - (\angle \, CDE + \angle \, DCE) = 180 - 4\, \alpha.[/math] and so [math]\angle\, BED = 180 - \angle \, DEC = 4 \, \alpha[/math] However, triangle [math]\Delta \, BDE[/math] is isosceles, so [math]\angle \, BDE = \angle \, BED = 4\, \alpha[/math] Finally, angle [math]\angle \, DBE = \frac{1}{2}\, \angle \, ABC = \frac{1}{2}\, \angle \, BCA = \frac{1}{2}\, 2\, \alpha = \alpha.[/math] Since the angles in the triangle [math]\Delta\, BDE[/math] sum up to [math]180[/math] [math]180 = \angle \, BED + \angle \, BDE + \angle \, DBE = 4\, \alpha + 4\, \alpha + \alpha = 9\, \alpha[/math] so [math]\angle \, DBC = \angle \, DBE = \alpha = \frac{180}{9} = 20[/math] [math]\angle \, ABC = \angle \, BCA = 2\, \alpha = 40[/math] and thus [math]\angle \, BAC = 180 - (\angle \, ABC + \angle \, BCA) = 180 - 40 - 40 = 100[/math] Second version: Draw the point [math]E[/math] on the edge [math]BC[/math] so that [math]BE=BD[/math]. Then triangle [math]BDE[/math] is isosceles. Since [math]BD+AD = BC[/math] and [math]BE=BD[/math], we conclude that [math]CE = BC - BE = BC - BD = AD[/math] Draw the segment [math]FD[/math] parallel to [math]BC[/math], where [math]F[/math] is on the edge [math]AB[/math]. Then, since [math]BD[/math] is angle bisector of [math]\angle\, ABC[/math] and combined with the fact that [math]FD \,\, ||\,\, BC[/math] [math]\angle \, FBD = \angle\, ABD = \angle CBD = \angle\, FDB = \alpha[/math] Therefore triangle [math]\Delta\, BDF[/math] is isosceles with [math]FD = FB[/math]. Furthermore, since [math]FD \,\, ||\,\, BC,[/math] one can apply Thales’ intercept theorem and conclude that [math]\frac{FB}{DC} = \frac{AB}{AC} = 1[/math] because [math]AB = AC[/math] by assumption. Thus, [math]FD = FB = DC[/math]. Analogously, by the same Thales’ intercept theorem [math]\frac{AF}{AD} = \frac{AB}{AC} = 1[/math] and so [math]AD = AF,[/math] which means that triangle [math]\Delta ADF[/math] is isosceles. The result of all of these identities is that [math]CE = AD, \, \, DC = FD, \,\, \angle\, ADF = \angle\, ECD = 2\alpha [/math] which means that triangle [math]\Delta\, ECD[/math] is congruent to triangle [math]\Delta \, ADF[/math] and therefore [math]\Delta\, ECD[/math] is also isosceles with [math]EC = ED[/math]. After this fact is established one can chase the angles as in the previous version. Upvote · 9 9 9 1 Related questions More answers below Can a right angled triangle be isosceles? Triangle ABC is an isosceles triangle with AB = AC. The measure of angle BAC is 40 degrees. Point D is on side AC such that AD = BC. What is the measure of the angle BDC in degrees? In isosceles triangle ABC, AB=AC and angle A is 20 degrees. On AC is point D so that AD=BC. What is the measure of angle ABD? In a triangle ABC, AD is the bisector of angle A. If AB =3 AC=6 and BC=3√3. What is the length of AD? In ∆ABC, if angle B = 90 degrees, and angle C = 50 degrees, then what is the length of AB? How? Paul Dunkley Associate · Author has 994 answers and 2M answer views ·May 14 Related Let ABC be an isosceles triangle with AB = BC and ∠ABC = 94°. Let M be a point inside the triangle such that ∠MAC = 13° and ∡MCA = 17°. How do you calculate the measure of the angle ∠CMB? We can put this triangle in a circle of radius 1. I've picked out this red triangle ABM for resolving instead of BMC because It's hard to know if triangle ABM is obtuse, acute, or Right angled at M. So we need two applications of the Cosine Rule on ABM, one for c, and again for theta, giving theta as 91.22787710° then for BMC we can subtract theta from 210°. Giving BMC=118.7721228° Continue Reading We can put this triangle in a circle of radius 1. I've picked out this red triangle ABM for resolving instead of BMC because It's hard to know if triangle ABM is obtuse, acute, or Right angled at M. So we need two applications of the Cosine Rule on ABM, one for c, and again for theta, giving theta as 91.22787710° then for BMC we can subtract theta from 210°. Giving BMC=118.7721228° Upvote · 9 7 Sponsored by CDW Corporation How can AI help your teams make faster decisions? CDW’s AI solutions offer retrieval-augmented generation (RAG) to expedite info with stronger insights. Learn More 99 21 Ernest Leung B.Sc. (Hons.) in Chemistry Honors&Mathematics, The Chinese University of Hong Kong · Author has 11.9K answers and 5.8M answer views ·Jun 23 Related In triangle ABC, the angle bisector of angle BAC meets BC at D. How do you show that AB/BD = AC/DC? In triangle ABC, the angle bisector of angle BAC meets BC at D. How do you show that AB/BD = AC/DC? Refer to the following figure (not to scale). In the figure, DP ⊥ AC, DQ ⊥ AB and AR ⊥ BC. Area of ΔABD = (1/2) × AB × DQ Area of ΔADC = (1/2) × AC × DP Hence, (Area of ΔABD) / (Area of ΔADC) = (AB × DQ) / (AC × DP) Since AD is the angle bisector of ∠BAC, DQ = DP Hence, (Area of ΔABD) / (Area of ΔADC) = AB / AC …… ① Area of ΔABD = (1/2) × BD × AR Area of ΔADC = (1/2) × DC × AR Hence, (Area of ΔABD) / (Area of ΔADC) = BD / DC …… ② ① = ②: AB / AC = BD / DC Hence, AB / BD = AC / Continue Reading In triangle ABC, the angle bisector of angle BAC meets BC at D. How do you show that AB/BD = AC/DC? Refer to the following figure (not to scale). In the figure, DP ⊥ AC, DQ ⊥ AB and AR ⊥ BC. Area of ΔABD = (1/2) × AB × DQ Area of ΔADC = (1/2) × AC × DP Hence, (Area of ΔABD) / (Area of ΔADC) = (AB × DQ) / (AC × DP) Since AD is the angle bisector of ∠BAC, DQ = DP Hence, (Area of ΔABD) / (Area of ΔADC) = AB / AC …… ① Area of ΔABD = (1/2) × BD × AR Area of ΔADC = (1/2) × DC × AR Hence, (Area of ΔABD) / (Area of ΔADC) = BD / DC …… ② ① = ②: AB / AC = BD / DC Hence, AB / BD = AC / DC Upvote · 9 7 Paul Dunkley Associate · Author has 994 answers and 2M answer views ·Updated 11mo Related How do I draw an "isosceles triangle" if AB = AC, angle B=C, point D is on AB, AD = BC, and angle BDC = 30 degrees? How do I prove that angle A = 20 degrees? The figure with its givens. I put D and E the wrong way around but it doesn't matter. The result will be the same because BCDE is a cyclic quadrilateral. Since BC subtends 30° at D and its reflection E, then BC must subtend 60° at the circumcentre of triangle BCD. Therefore BC is equal to the radius of the circumcircle of BCD. So Draw OB, OC, OD, and OE all equal to AE, BC, and AD all shown at 1 (in red) Then 2x+y=30° (by exterior) EOD=2y (by subtendence) since AD=OD then AOD is isosceles and has equal base angles, so x=y. Then 3x=30 and x=10 and the angle at A is 20°. The figure as given was a n Continue Reading The figure with its givens. I put D and E the wrong way around but it doesn't matter. The result will be the same because BCDE is a cyclic quadrilateral. Since BC subtends 30° at D and its reflection E, then BC must subtend 60° at the circumcentre of triangle BCD. Therefore BC is equal to the radius of the circumcircle of BCD. So Draw OB, OC, OD, and OE all equal to AE, BC, and AD all shown at 1 (in red) Then 2x+y=30° (by exterior) EOD=2y (by subtendence) since AD=OD then AOD is isosceles and has equal base angles, so x=y. Then 3x=30 and x=10 and the angle at A is 20°. The figure as given was a neusis trisection of 60°. This doesn't matter, what's Given is Given. Upvote · 9 2 Paul Dunkley Associate · Author has 994 answers and 2M answer views ·1y Related In the isosceles triangle ABC, AB = AC and angle A of 20 degrees. Point P is taken on the AC side so that AP = BC. What are the PBC angles? AP=BC (Given) write it down. Bisect AB at its midpoint say M, raise a perpendicular from M to meet AC in say Q. Draw QB to construct isosceles AQB with its 20° base angles (we won't write those angles into the figure, we can just remember what they are for the moment) Draw parallel to AB through P to meet QB in say T to form the isosceles US Trapezoid APTB and make statement 1. Now, because BC=BT and TBC=60°, then TBC must be Equilateral giving BC=BT=CT, so draw CT and make statement 2. In the above, angles TCP and TPC are both equal to 20°, so triangle PTC is in fact isosceles, so CT=PT, so make Continue Reading AP=BC (Given) write it down. Bisect AB at its midpoint say M, raise a perpendicular from M to meet AC in say Q. Draw QB to construct isosceles AQB with its 20° base angles (we won't write those angles into the figure, we can just remember what they are for the moment) Draw parallel to AB through P to meet QB in say T to form the isosceles US Trapezoid APTB and make statement 1. Now, because BC=BT and TBC=60°, then TBC must be Equilateral giving BC=BT=CT, so draw CT and make statement 2. In the above, angles TCP and TPC are both equal to 20°, so triangle PTC is in fact isosceles, so CT=PT, so make statement 3. Because TBC is equilateral, we know that T is on the altitude of triangle ABC, so draw AT extended to meet CB in a Right angle, so AT (being perpendicular to CB in isosceles ABC) must also be the angle bisector of angle BAC. So put the two 10° angles in at the apex. In the above, APTB is also cyclic quadrilateral, all isosceles US trapezoid are cyclic and in this case (from statement 3) the three chords AP, BT, and PT are all equal, and chord PT will subtend the same angle at A as it would at B, namely 10°. Hence; if PB were drawn, then angle PBQ would be 10° and since angle TBC=60°, so angle PBC would be 70°. This triangle therefore has a matchstick construction (because it trisects a well known angle) using two books and 8 matchsticks, one long one AB (which isn't actually required, as it will work without it) and the other seven shorter matchsticks shown as red and equal in length, pushing the books together and pushing the base matchstick upwards forces all the matchsticks into the correct positions to form the 20, 80, 80 triangle (given infinitely thin matchsticks) Upvote · 9 2 9 2 Paul Dunkley Associate · Author has 994 answers and 2M answer views ·1y Related Let ABC be isosceles with vertex angle A, and let E and D be points on AC and AB, respectively, so that innocuousAE=ED=DC=CB. What is angle A? This is really an angle chase in the purest sense. You just draw it. They actually have some special Properties concerning products on legs, but they turn up every now and then in questions in geometry, so it makes sense to recognise them when they occur. The above triangle is part of a Heptagon (a seven sided regular polygon) The above angle (and its parent triangle) is non constructable by compass and straightedge but mechanically constructable by neusis, or constructable in “the real world” depending on your “real” Continue Reading This is really an angle chase in the purest sense. You just draw it. They actually have some special Properties concerning products on legs, but they turn up every now and then in questions in geometry, so it makes sense to recognise them when they occur. The above triangle is part of a Heptagon (a seven sided regular polygon) The above angle (and its parent triangle) is non constructable by compass and straightedge but mechanically constructable by neusis, or constructable in “the real world” depending on your “real” Upvote · 9 7 9 1 9 3 Dave Benson trying to make maths easy. · Author has 6.1K answers and 2.1M answer views ·1y Related Let ABC be an isosceles triangle with AB = BC and ∠ABC = 94°. Let M be a point inside the triangle such that ∠MAC = 13° and ∡MCA = 17°. How do you calculate the measure of the angle ∠CMB? As any value let AB = BC = 10 units. Let centre of base of isosceles triangle be (0,0) then half base is 10cos(43º) ≈ 7.3135 & height is 10sin(43º) ≈ 6.82 Slope of AM is tan(13º) & equation of line is y = 0.2309x+1.6887 green Slope of CM is tan(17º) & equation of blue is y = -0.3057+2.2357 & equate for intersection x= 1.0194 & y = 1.924 so slope of orange BM ≈ -4.8028 & arctan is 78.24º Extend BM to intersect AC at D & ∆CDM 78.24º-17º= 61.24º = ∠CMD So ∠CMB ≈ 180º-61.24º ≈ 118.78º Answer º Continue Reading As any value let AB = BC = 10 units. Let centre of base of isosceles triangle be (0,0) then half base is 10cos(43º) ≈ 7.3135 & height is 10sin(43º) ≈ 6.82 Slope of AM is tan(13º) & equation of line is y = 0.2309x+1.6887 green Slope of CM is tan(17º) & equation of blue is y = -0.3057+2.2357 & equate for intersection x= 1.0194 & y = 1.924 so slope of orange BM ≈ -4.8028 & arctan is 78.24º Extend BM to intersect AC at D & ∆CDM 78.24º-17º= 61.24º = ∠CMD So ∠CMB ≈ 180º-61.24º ≈ 118.78º Answer º Upvote · 9 6 Gary Ward MaEd in Education&Mathematics, Austin Peay State University (Graduated 1997) · Author has 4.9K answers and 7.6M answer views ·1y Related In triangle ABC, angle B is 90° and BD is an altitude of the triangle. The length of BC is 13, the length of DC is x and the length of AC is 28.8 + x. What is the value of x? In triangle ABC, angle B is 90° and BD is an altitude of the triangle. The length of BC is 13, the length of DC is x and the length of AC is 28.8 + x. What is the value of x? Until I broke it down into three triangles, I didn’t see the solution. Triangle ABD is not needed. All triangles are similar and in direct proportion. [math]\displaystyle \frac{AC}{BC} = \frac{BC}{DC}[/math] Substitute the values. [math]\displaystyle \frac{x + 28.8}{13} = \frac{13}{x}[/math] Cross multiply. [math]\displaystyle x^2 + 28.8x = 169[/math] (28.8/2)² = 207.36 Add to both sides. [math]\displaystyle x^2 +28.8x + 207.36 = 376.36[/math] Take the square root of both sides. [math]\dis[/math] Continue Reading In triangle ABC, angle B is 90° and BD is an altitude of the triangle. The length of BC is 13, the length of DC is x and the length of AC is 28.8 + x. What is the value of x? Until I broke it down into three triangles, I didn’t see the solution. Triangle ABD is not needed. All triangles are similar and in direct proportion. [math]\displaystyle \frac{AC}{BC} = \frac{BC}{DC}[/math] Substitute the values. [math]\displaystyle \frac{x + 28.8}{13} = \frac{13}{x}[/math] Cross multiply. [math]\displaystyle x^2 + 28.8x = 169[/math] (28.8/2)² = 207.36 Add to both sides. [math]\displaystyle x^2 +28.8x + 207.36 = 376.36[/math] Take the square root of both sides. [math]\displaystyle (x + 14.4) = \pm 19.4 [/math] x = 5 or x = -33.8 We throw out the negative so x = 5 Check: [math]\displaystyle \frac{5 + 28.8}{13} = \frac{13}{5} \text{ and } \frac{13}{5}[/math] Both reduce to the same ratio, so x = 5 is correct. BD = 12 and AB = 31.2 Upvote · 99 10 Haresh Sagar Studied Science&Mathematics (Graduated 1988) · Author has 6.2K answers and 7M answer views ·2y Related In a triangle ABC, angle A = 2 angle B. How would you prove that BC² = AC² + AB × AC? Let's do it using coordinate geometry. Plot [math]B[/math] at origin, [math]A[/math] at math[/math] and [math]C[/math] in first quadrant. For convenience, we have taken [math]AB=1[/math]. The triangle will be similar to base having a variable length so no loss of generality. Let slope of [math]BC[/math] be [math]m[/math], so the equation will be… [math]y=mx[/math] Since [math]\angle{A}=2\angle{B}[/math], slope of [math]AC[/math] will be [math]-2m/(1-m^2)[/math] and the equation will be… [math]y=-\dfrac{2m}{1-m^2}x+\dfrac{2m}{1-m^2}[/math] [math]AC[/math] and [math]BC[/math] intersect at [math]C[/math] … [math]mx+\dfrac{2m}{1-m^2}x=\dfrac{2m}{1-m^2}[/math] [math]x\left(\dfrac{3m-m^3}{1-m^2}\right)=\dfrac{2m}{1-m^2}[/math] [math]C_x=\dfrac{2}{3-m^2}\;,\;C_y=\dfrac{2m}{3-m^2}[/math] Distance formula gives… [math]\boxed{BC^2=\dfrac{4(1[/math] Continue Reading Let's do it using coordinate geometry. Plot [math]B[/math] at origin, [math]A[/math] at math[/math] and [math]C[/math] in first quadrant. For convenience, we have taken [math]AB=1[/math]. The triangle will be similar to base having a variable length so no loss of generality. Let slope of [math]BC[/math] be [math]m[/math], so the equation will be… [math]y=mx[/math] Since [math]\angle{A}=2\angle{B}[/math], slope of [math]AC[/math] will be [math]-2m/(1-m^2)[/math] and the equation will be… [math]y=-\dfrac{2m}{1-m^2}x+\dfrac{2m}{1-m^2}[/math] [math]AC[/math] and [math]BC[/math] intersect at [math]C[/math] … [math]mx+\dfrac{2m}{1-m^2}x=\dfrac{2m}{1-m^2}[/math] [math]x\left(\dfrac{3m-m^3}{1-m^2}\right)=\dfrac{2m}{1-m^2}[/math] [math]C_x=\dfrac{2}{3-m^2}\;,\;C_y=\dfrac{2m}{3-m^2}[/math] Distance formula gives… [math]\boxed{BC^2=\dfrac{4(1+m^2)}{(3-m^2)^2}}[/math] [math]A-C=\left(\dfrac{1-m^2}{3-m^2},-\dfrac{2m}{3-m^2}\right)\;so...[/math] [math]AC^2=\dfrac{1-2m^2+m^4+4m^2}{(3-m^2)^2}[/math] [math]\boxed{AC^2=\dfrac{(m^2+1)^2}{(3-m^2)^2}}[/math] [math]\boxed{AC=\dfrac{(m^2+1)}{(3-m^2)}}[/math] Now let's check what [math]AC^2+AB\cdot{AC}[/math] equals to, [math]\dfrac{m^4+2m^2+1}{(3-m^2)^2}+1\cdot\dfrac{(m^2+1)}{(3-m^2)}[/math] [math]=\dfrac{m^4+2m^2+1}{(3-m^2)^2}+\dfrac{(3-m^2)}{(3-m^2)}\cdot\dfrac{(m^2+1)}{(3-m^2)}[/math] [math]=\dfrac{m^4+2m^2+1-m^4+2m^2+3}{(3-m^2)^2}[/math] [math]\boxed{=\dfrac{4(1+m^2)}{(3-m^2)^2}=BC^2}[/math] [math]Proved.[/math] Your response is private Was this worth your time? This helps us sort answers on the page. Absolutely not Definitely yes Upvote · 9 7 9 1 Related questions If I have the triangle ABC with AB=AC and the angle bisector of B is drawn so it intersects AC in D, how can I calculate the angle BAC? Such that BD+AD=BC. The triangle ABC's AB> AC and the isosceles AD of angle A intersects the BC arm at point D. How will I prove that angle ADB is an obtuse angle? ABC is a right triangle such that AB = AC and the bisector of angle C intersects the side AB at D. How can I prove that AC + AD = BC? ABC is an isosceles triangle with AB=AC. Angle ABC=80 degree. D is a point on AC such that AD=BC. What is the measure of angle BDC? There is a triangle ABC, where B is 90 degrees. If the internal bisector of angle B meets AC at D, how do I find the value of AD when the sides AB = 3cm and BC = 4cm? Can a right angled triangle be isosceles? Triangle ABC is an isosceles triangle with AB = AC. The measure of angle BAC is 40 degrees. Point D is on side AC such that AD = BC. What is the measure of the angle BDC in degrees? In isosceles triangle ABC, AB=AC and angle A is 20 degrees. On AC is point D so that AD=BC. What is the measure of angle ABD? In a triangle ABC, AD is the bisector of angle A. If AB =3 AC=6 and BC=3√3. What is the length of AD? In ∆ABC, if angle B = 90 degrees, and angle C = 50 degrees, then what is the length of AB? How? Triangle ABC, BD and CD are bisectors of angle B and angle C intersected at D. How do you prove that 2x=180 degree+y? In triangle ABC, angle B=90°, angle C=60°, BC=8cm. What is side AB and AC? If AB=BC and B=80 in triangle ABC, what is the value of angle C? If in a triangle ABC, ∠BAC=40°; AB=AC; BE and CD are angle bisectors of ∠B & ∠ C, intersecting at O. What is the measure of ∠BOC? How do I draw the bisectors of angle A and angle C if in triangle ABC angle B = 115 degrees? Related questions If I have the triangle ABC with AB=AC and the angle bisector of B is drawn so it intersects AC in D, how can I calculate the angle BAC? Such that BD+AD=BC. The triangle ABC's AB> AC and the isosceles AD of angle A intersects the BC arm at point D. How will I prove that angle ADB is an obtuse angle? ABC is a right triangle such that AB = AC and the bisector of angle C intersects the side AB at D. How can I prove that AC + AD = BC? ABC is an isosceles triangle with AB=AC. Angle ABC=80 degree. D is a point on AC such that AD=BC. What is the measure of angle BDC? There is a triangle ABC, where B is 90 degrees. If the internal bisector of angle B meets AC at D, how do I find the value of AD when the sides AB = 3cm and BC = 4cm? Can a right angled triangle be isosceles? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
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https://arxiv.org/html/2502.04279v1
Back to arXiv HTML conversions sometimes display errors due to content that did not convert correctly from the source. This paper uses the following packages that are not yet supported by the HTML conversion tool. Feedback on these issues are not necessary; they are known and are being worked on. failed: extarrows Authors: achieve the best HTML results from your LaTeX submissions by following these best practices. License: CC BY 4.0 arXiv:2502.04279v1 [math.PR] 06 Feb 2025 On random locally flat-foldable origami Thomas C. Hull Mathematics Department, Franklin and Marshall College thomas.hull@fandm.edu , Marcus Michelen Department of Mathematics, Statistics and Computer Science, University of Illinois Chicago. Chicago, IL 60607 michelen.math@gmail.com and Corrine Yap School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 math@corrineyap.com Abstract. We develop a theory of random flat-foldable origami. Given a crease pattern, we consider a uniformly random assignment of mountain and valley creases, conditioned on the assignment being flat-foldable at each vertex. A natural method to approximately sample from this distribution is via the face-flip Markov chain where one selects a face of the crease pattern uniformly at random and, if possible, flips all edges of that face from mountain to valley and vice-versa. We prove that this chain mixes rapidly for several natural families of origami tessellations—the square twist, the square grid, and the Miura-ori—as well as for the single-vertex crease pattern. We also compare local to global flat-foldability and show that on the square grid, a random locally flat-foldable configuration is exponentially unlikely to be globally flat-foldable. 1. Introduction The folding of flat sheets of material into three-dimensional, sometimes highly intricate objects, commonly known as origami, has seen a surge of interest for applications in physics [4, 24], engineering [14, 38], architecture , and computer science . Origami can be modeled by a function on a compact domain that is a continuous piecewise isometry; of particular interest are origami that fold flat, meaning that the image lies in the plane . Being able to fold flat allows for compact storage of the folded design, such as for transport and deploying of large solar panels into outer space . In this paper we develop a theory of random origami, with the hope of answering questions such as “With what probability will a randomly chosen crease pattern configuration be flat-foldable?” Our model is that of a uniformly random assignment of mountain/valley creases to a given crease pattern, conditioned on the crease assignment being locally flat-foldable. Already a natural question emerges: how does one study or (approximately) sample a uniformly random locally flat-foldable configuration? Tools from the origami literature—Kawasaki’s Theorem and Maekawa’s Theorem detailed in Section 2—provide necessary and sufficient conditions for a configuration to be locally flat-foldable. This allows one to recast the set of locally flat-foldable crease patterns under the umbrella of spin systems from statistical physics and computer science. Heuristically, this class of models is based on assigning configurations of some kind—in our case “mountain” and “valley” assignments to edges—under some family of local constraints. In the cases of certain crease patterns, this shows an explicit connection between the topic of local flat-foldability and more well-studied questions such as proper coloring (see, for instance, ). A broad class of problems studied across probability, statistical physics, and computer science is to understand how to approximately sample from a large system under local constraints. Here, one typical approach is via Markov chain Monte Carlo. In our case, this amounts to the following: given one locally flat-foldable assignment, perform some small random change that transforms the assignment into another locally flat-foldable assignment. It is often possible to choose these transition probabilities in a given way so that if one performs these random transitions over and over again, the resulting random configuration is approximately uniform over all valid configurations. In the setting of origami, a natural and easy-to-implement Markov chain is the face-flip Markov chain: one takes a face of the crease pattern uniformly at random, checks if it is possible to flip all creases surrounding this face from mountain to valley and vice-versa, and performs this flip with probability if possible. As a preliminary result, using basics from the theory of Markov chain mixing, we prove that this face-flip Markov chain converges to the uniform distribution on locally flat-foldable configurations for a wide family of crease patterns. Our primary settings are several lattices that are widely used throughout the origami literature, as well as single-vertex crease patterns. A main concern is to understand how long it takes before the Markov chain is approximately uniform. For this, we obtain bounds on the mixing time; our main results are encompassed in the following theorem. Theorem 1. The face-flip Markov chain for locally flat-foldable mountain-valley assignments of the square twist, the square grid, the Miura-ori, and the single-vertex crease pattern all mix in polynomial time with respect to the number of faces. We prove each of the four cases of Theorem 1 separately, as Theorems 17, 19, 23, 26 respectively. Polynomial-time mixing of the face-flip Markov chain immediately gives an approximate sampling algorithm that runs in polynomial time, which is called an efficient approximate sampler. We discuss this in more detail in Section 2. For the single vertex and the square grid we additionally provide efficient exact samplers. We also consider local versus global flat-foldability and prove that global flat-foldability is exponentially unlikely on the square grid. Theorem 2. Let be a uniformly random locally flat-foldable configuration on the square grid crease pattern. The probability that is globally flat-foldable is for some . 1.1. Overview of Techniques Our primary technique for analyzing the face-flip Markov chain is to compare it to other Markov chains known to have polynomial mixing time. In the case of the first two lattice crease patterns—the square twist and the square grid—the comparison is quite straightforward. We use the perspective of face flips being equivalent to a lazy random walk on the origami flip graph (in other words, the Markov chain state diagram) and observe that the flip graph itself is isomorphic to a -dimensional hypercube for some . For the Miura-ori crease pattern, the route to comparison is more circuitous. Work by Ginepro and Hull showed a relationship between mountain-valley assignments on the Miura-ori crease pattern and proper vertex-colorings of grid graphs. Even further, a face flip on the Miura-ori crease pattern corresponds to a single update of Glauber dynamics, a standard Markov chain in statistical physics that applies for proper colorings. With this connection, we use powerful results about Glauber dynamics to obtain polynomial mixing for the face-flip Markov chain. For single-vertex crease patterns, a counting argument allows us to describe an exact sampler. To analyze the face-flip Markov chain, however, we pass to an induced chain on the configurations which have more mountain than valley creases. The induced chain can be compared to a type of fast-mixing card-shuffling Markov chain. Lastly, we consider local versus global flat-foldability on lattices. It is simple to check whether a configuration is locally flat-foldable, but checking whether a configuration is globally flat-foldable is more complicated (see Section 5 for more discussion on computational hardness results in this direction). We show that on the square lattice, the probability that a random locally flat-foldable configuration is globally flat-foldable is exponentially small with respect to the volume of the lattice. 1.2. Related Work Questions concerning local flat-foldability can be cast in the language of constraint satisfaction problems and statistical physics, specifically in the framework of spin systems: each edge is assigned “mountain” or “valley” and there are local rules at each vertex to determine compatibility. In particular, if one wishes to describe a uniformly random locally flat-foldable configuration as a spin system, one assigns energy to each configuration that is not locally flat-foldable and energy to each configuration that is locally flat-foldable. Via Kawasaki’s Theorem, this can be checked by looking purely at each vertex individually. The perspective of interpreting locally flat-foldable configurations as a spin system is not new and indeed work by Assis draws connections between more classical statistical physics models and local flat-foldability on certain lattices (some of which we will also discuss in the present study). Ginepro and Hull proved that the foldings of the Miura-ori crease pattern (to be defined in Section 3.3) are equivalent to the square-ice model. Nakajima used a spin model on random graphs to model the combinatorial problem of ordering the different layers of paper in flat-folded crease patterns , and developed more work in this vein for crease patterns that contain only one interior vertex . Our main focus is on properties of random locally flat-foldable configurations and in particular on sampling via Markov chains. The use of Markov chains to sample spin systems is a classic topic at the intersection of statistical physics, probability theory, and computer science. In particular, Markov chain Monte Carlo was introduced in a work by Metropolis–Rosenbluth–Rosenbluth–Teller–Teller which sought to sample from a continuous-space variant of a spin system called the hard-disk model. Analysis of Markov chains for spin systems remains a vibrant field, and we refer the reader to the following texts [13, 23] and some recent breakthroughs [3, 7] along with the references therein for more context. There has also been quite a bit of research on the algorithmic complexity of origami over the past several decades, including several monographs [8, 17, 19, 35]. We make references to related computational origami results throughout. 1.3. Organization The results of this paper lie at the intersection of origami, probability, and algorithms. Thus, we include background material in Section 2 on each of the three areas; readers may let their experience dictate how much they choose to read. Section 2.1 has preliminaries on flat origami and Section 2.2 has preliminaries on Markov chains. Sections 2.3 and 2.4 then contain preliminaries specific to uniformly random mountain valley assignments and the face-flip Markov chain. In Section 3, we prove Theorem 1 for the three types of aforementioned lattice crease patterns, and in Section 4, we prove the remaining case of the single-vertex crease patterns. In Section 5, we prove Theorem 2. In Section 6, we discuss open problems and future directions. 2. Preliminaries 2.1. Flat Origami Origami that can fold flat possess rich geometric and combinatorial structures. In what follows we adopt conventions of modeling flat origami presented in [8, 17]. While we will not explicitly use many of the notions and notations defined below, they provide a foundation for posing questions about origami in a mathematically rigorous way. Given an origami that folds flat, we let the crease pattern of be the subset of on which is non-differentiable. It has been proven that forms a straight-line embedding of a graph on whose vertices always have even degree . We call the edges of the creases and the connected regions of between creases the faces of . For any face of , its image under the transformation either involves only translations and rotations and is thus isotropic or involves a reflection and is anisotropic. Faces of that share a common boundary crease have different isotropic/anisotropic orientations, and thus they define a proper 2-coloring faces of of the faces of . The folding map of an origami misses a crucial aspect of flat-folded paper: we need the paper to not self-intersect under (i.e. we do not want the folded paper to be forced to penetrate itself in order to fold flat). To capture this in our model, we need to describe the layer ordering of , denoted , where is the subset of points such that both and are contained in the interior of the faces of and . We say that (resp. ) means is below (resp. is above ). In order for to guarantee that the flat origami map not self-intersect, it needs to satisfy certain non-crossing conditions outlined in [1, 17] but which do not need to be specified here. We may now define the mountain-valley (MV) assignment of with layer ordering , where is the set of edges of , as follows: For any crease bordering faces and of and points and arbitrarily close but on opposite sides of with , we have | | | | --- | | | | | | | | --- | | | | The above simply says that, relative to the orientation of the faces under the mapping , means the crease forms a valley crease (and thus makes a shape when unfolded) and means that forms a mountain crease (making a shape when unfolded). Layer orderings and MV assignments are strongly related; a layer ordering will generate an MV assignment on the creases, and an MV assignment on the creases will generate a layer ordering. An MV assignment placed on a flat origami crease pattern is called valid if it produces a layer ordering that avoids self-intersections of the paper. We often pay attention to how an origami crease pattern folds at each vertex individually, or locally. That is, given a crease pattern and an MV assignment , we draw a sufficiently small disc around each vertex in a crease pattern so that the only vertex of in is . Considering for each as separate crease patterns, we say that is locally flat-foldable under (or alternatively, that is locally valid for ) if each of the sub-crease patterns is flat-foldable under . The following two fundamental results are very helpful in identifying and studying locally flat-foldable crease patterns (see [8, 15, 17] for proofs): Theorem 3 (Kawasaki’s Theorem). A crease pattern with only one vertex in the interior of and consecutive angles between the creases adjacent to is flat-foldable (meaning there exists a valid MV assignment for ) if and only if . Theorem 4 (Maekawa’s Theorem). A vertex in a flat-foldable crease pattern with valid MV assignment and creases adjacent to must satisfy . In general if a crease pattern has a valid MV assignment then we say that is globally flat-foldable. While local flat-foldability is easy to check via Kawasaki’s Theorem, determining if a configuration is globally flat-foldabile is an NP-complete problem, even in the case where all the vertices of lie on the lattice and the angles between creases at the vertices are all multiples of (called box-pleated crease patterns) . Not every MV assignment is valid; in Section 5 we will see a simple example of a locally flat-foldable configuration that is not globally flat-foldable and use it to show that globally flat-foldable configurations are exponentially rare among locally flat-foldable configurations. Consider the single-vertex crease pattern shown in Figure 1(a). Creases and border the face with sector angle , which is strictly smaller that the sector angles and of its neighboring faces. In this situation, it is impossible for a valid MV assignment to have because doing so in a flat folding of these creases would make the faces with bigger angles and cover the smaller face on the same side of the paper, forcing a self-intersection. More formally, we have the following statement. Theorem 5 (Big-Little-Big Theorem). If are sector angles of consecutive faces around a flat-foldable vertex with , then a valid MV assignment must have , where and are the edges comprising angle . 2.1.1. Face Flips We wish to study how a locally valid MV assignment evolves under random local changes; one of the most natural such operations is a face flip. Definition 6. Given a locally valid MV assignment and a face of our crease pattern , we define the face flip of to be a new MV assignment which equals for all creases that do not border and for all creases that border . If is also a locally valid MV assignment for , then we say that face is flippable, whereas a non-flippable face of a locally valid MV assignment is a face for which is not locally valid. For example, in the locally valid MV assignment of the crease pattern shown in Figure 1(b) for the degree-6 vertex with all sector angles equal to , the face is not flippable because doing so would make every crease a valley crease, violating Maekawa’s Theorem. But all of the other faces in this MV assignment are flippable. In Figure 1(a), the face with sector angle is flippable, but its neighbors with angles and are not flippable; in fact they are never flippable faces, since for any locally valid MV assignment flipping the neighboring faces of the face will violate the Big-Little-Big Theorem for creases and . Finally, for a given flat-foldable crease pattern we represent the state space of all locally valid MV assignments of as a graph. Definition 7. The origami flip graph of , denoted , is the graph whose vertices are all locally valid MV assignments of and where two vertices and form an edge if and only if there exists a face of such that . 2.2. Markov Chains Throughout we will make use of Markov chains on finite state spaces; we recall some classical notions from the area in this section and refer readers to for more background and context. 2.2.1. Basic definitions A Markov chain with state space and transition matrix is a sequence of random variables so that for all and values we have | | | | | --- --- | | (1) | | | | When is viewed as an matrix, (1) implies that the columns of sum to . The distribution of the trajectory is determined uniquely by the distribution of along with the transition matrix . A probability measure on is a function with for all and . For we write . When is viewed as a matrix, it is convenient to view probability measures as row vectors and functions as column vectors. A stationary distribution for is a probability measure typically denoted such that , meaning for all we have | | | | --- | | | | A sufficient condition for to be a stationary distribution of is if satisfies the detailed balance equations, | | | | | --- --- | | (2) | | | | for all . In this case, we say is reversible with respect to . A Markov chain is irreducible if for each pair there is some so that , i.e. if it is always possible to get from any one state to any other by evolving according to the transition matrix . A Markov chain is aperiodic if for all . It will often be useful to assume that the Markov chain is lazy, i.e. that for each we have ; laziness ensures that the Markov chain is aperiodic. A fundamental theorem of Markov chains is that a Markov chain which is irreducible, aperiodic, and finite has a unique stationary distribution (see, e.g., [23, Cor. 1.17]). 2.2.2. Mixing times Given two probability measures and , define the total variation distance via | | | | | --- --- | | (3) | | | | If is irreducible and aperiodic with stationary distribution , then we have the Markov convergence theorem, meaning that converges to zero (in fact, it does so exponentially quickly in , see, e.g., [23, Thm. 4.9]) . For algorithmic problems, one of the main interests is in the rate of convergence. This is the motivation for the definition of the mixing time of a Markov chain: Definition 8. We define the -mixing time to be | | | | | --- --- | | (4) | | | | where the maximum is over all probability measures on , and the mixing time to be . We remark that the choice of is by convention, as any choice of will only give a constant factor difference. Specifically, . Up to a constant of , the mixing time can be reinterpreted using the following basic fact: | | | | | --- --- | | (5) | | | | In the remainder of this subsection, we introduce some standard but slightly technical notions for giving upper bounds on mixing times; these techniques will be used in Section 4 for single-vertex crease patterns. All eigenvalues of a lazy reversible Markov chain lie in , and so we may order them as . The spectral gap is defined as . By diagonalizing the matrix one obtains a bound on the mixing time (see e.g. [23, Thm. 12.4]): Theorem 9. Let be a lazy, irreducible, reversible Markov chain with stationary distribution . Then | | | | --- | | | | where One method for bounding the mixing time of a given Markov chain is to compare it to another Markov chain on the same state space, ideally one whose spectral gap is easier to analyze. To use comparison, we require the Dirichlet forms of both chains. Given a function , the Dirichlet form of a reversible Markov chain on with stationary distribution can be defined via | | | | | --- --- | | (6) | | | | Lemma 10. Let be reversible Markov chains on with stationary distributions respectively. If for all , then | | | | --- | | | | One particular type of Markov chain we will use for comparison is an induced chain. Definition 11. Given a chain on with transition matrix and , define and iteratively define Then the trajectory is a Markov chain on with transition matrix which we call the induced chain on . For a lazy, reversible Markov chain let | | | | | --- --- | | (7) | | | | where is the (random) time it takes for a trajectory started at to hit the set . The following theorem of Peres–Sousi relates this quantity to the mixing time. Theorem 12 (). For each there is a positive constant so that | | | | --- | | | | 2.3. Uniformly random locally flat-foldable MV assignments The probability distribution given by taking a uniformly random locally flat-foldable configuration can be interpreted as a Gibbs measure. This means that if is a uniformly random locally flat-foldable configuration then there is some energy function called the Hamiltonian for which we have that | | | | --- | | | | In the case of local flat-foldability, the Hamiltonian will simply check that Maekawa’s theorem (4) holds at each vertex. We now explicitly describe the uniformly random locally flat-foldable configuration in terms of a Gibbs measure. For a given vertex , let denote the set of edges incident to . Define by | | | | --- | | | | and note that we may in fact view as a function on since it depends only on the values of on . We note also that is locally flat-foldable if and only if . Defining | | | | --- | | | | we see that is the total number of locally flat-foldable configurations and that | | | | --- | | | | One important property of Gibbs measures is that they satisfy the following domain Markov property: taking a uniformly random configuration and resampling a subset, conditioned on its boundary, remains uniformly random. In particular we have the following basic fact, which follows from [13, Theorem 7.12]: Lemma 13. Let be a crease pattern and be a locally flat-foldable MV assignment chosen uniformly at random. For a collection , let denote the set of edges in that are incident to some edge in . Then for any set and configurations with we have | | | | --- | | | | 2.4. The Face-Flip Markov Chain We now introduce the main character of this paper, the face-flip Markov chain: (1) Given an MV assignment of , choose a face uniformly at random. 2. (2) If is non-flippable, do nothing. If is flippable, do nothing with probability and flip with probability . Our goal will be to analyze the mixing time of this Markov chain on specific crease patterns, but we begin by making some general observations about this random process. By the discussion in the previous subsection, this chain has a unique stationary distribution which we can easily check must be uniform. Lemma 14. Given a crease pattern , the face-flip Markov chain on is a reversible Markov chain whose stationary distribution is uniform over all locally flat-foldable MV assignments of . Proof. Letting be the set of all locally flat-foldable MV assignments, a face flip is an involution so for all . Thus, the uniform distribution satisfies the detailed balance equations (2). ∎ From this, we see that the face-flip Markov chain gives an approximate sampling algorithm for uniformly random locally flat-foldable MV assignments: simply run the face-flip Markov chain for steps where . By definition, the resulting distribution is close in total variation distance to the uniform distribution (hence an “approximate” and not an exact sampler). Another way we can view the face-flip Markov chain on a crease pattern with faces is as a type of lazy random walk on : starting from a vertex , for each neighbor of , move to with probability ; with the remaining probability, stay at . We can also consider random walks on general graphs, and this will prove useful in analyzing the square twist and square grid crease patterns. We will consider the particular graph of the -dimensional hypercube, denoted . There are multiple equivalent descriptions of ; the one most useful to us is where we think of the vertices as binary strings of length and edges consisting of pairs of vertices with Hamming distance 1. More precisely, and if and only if there is a unique such that . Another perspective we will use is that of a Cayley graph. Given a group and generating set , the Cayley graph has vertex set and if and only if there exists such that . We can naturally identify with the Cayley graph of with generating set where is the th standard basis vector, having a 1 in the th coordinate. Observe that if is isomorphic to for some , then the face-flip Markov chain is a simple lazy random walk, meaning at every vertex , there is a probability of staying at and a probability of moving to each neighbor. Furthermore, as for all , we can check via the detailed balance equations (2) that the unique stationary distribution is uniform over all . The mixing time of the lazy simple random walk on is well-studied (see, e.g., ). Theorem 15. For the lazy simple random walk on , | | | | --- | | | | As a remark, this result follows from analysis of the coupon collector problem, which asks: If there are coupons and each week a person receives a coupon independently and uniformly at random, how many weeks does it take to collect all coupons? The expected number of weeks is precisely the th harmonic number, asymptotically equal to . The time it takes for the random walk on to mix can be bounded by the time it takes to select each of the coordinates at least once—by coupon collector, this is . 3. Lattice crease patterns In this section, we discuss uniformly random locally flat-foldable crease patterns on various lattices and utilize known results about each origami tessellation in order to prove results about mixing times of the face-flip Markov chain, as promised in Theorem 1. 3.1. Square twist The square twist crease pattern is shown in Figure 2(a). It is used as a building block in many origami tessellations, which are crease patterns that form a tiling of the plane if extended indefinitely . Square twists can be tiled in a grid either with alternating chirality or with all the same chirality, as shown in Figure 2(b). An example of a valid MV assignment for such a tessellation, along with a see-through simulation of its folded image (made using Lang’s Tessellatica software ) is in Figure 2(c). (The pattern on the right in Figure 2(b) is an example of Barreto’s Mars, discussed in .) In addition to being artistically pleasing, square twists exhibit interesting mechanical properties when folded and unfolded, making them of interest for applications . The proper 2-face-coloring obtained from square twist tessellations, which partitions the faces into rectangular and non-rectangular classes, is especially useful for describing their origami flip graphs, which then allow us to describe the probability space of all valid MV assignments. Figure 2(b) shows these 2-colorings. Proposition 16. The face-flip graph of a square twist tessellation is isomorphic to the -dimensional hypercube where is the number of non-rectangular faces of . Proof. Note that by the Big-Little-Big Theorem (5), the square and rectangular faces of are never flippable (since flipping one from a valid MV assignment will make a angle have two Ms or two Vs). For the non-rectangular faces we observe the following. • They are either trapezoids or parallelograms (see Figure 2(b)). • Their angles must be bordered by a mountain and a valley (by Big-Little-Big (5)). • Their angles must be bordered by two mountains or two valleys (since each vertex has degree 4 and via Maekawa’s Theorem (4)). Therefore in a valid MV assignment of a square twist tessellation, the non-rectangular faces must have mountains and valleys in one of the arrangements shown in Figure 2(d). This implies that these faces can only have MV assignments that come in flippable pairs. Thus the non-rectangular faces are always flippable (under all valid MV assignments), and flipping any non-rectangular face does not affect the MV assignment of any other non-rectangular face. This means that if is the number of non-rectangular faces in the crease pattern , we can describe any valid MV assignment of as a vector where the th coordinate of describes which of the two possible MV assignments (see Figure 2(d)) of the th non-rectangular face we have in . If is another valid MV assignment of and is described by vector , then a sequence of faces to flip from to will be those whose coordinates in and are not equal. This proves that is connected and isomorphic to the hypercube . ∎ As discussed in the preliminaries, the mixing time of the simple lazy random walk on the -dimensional hypercube is well-known—by Theorem 15, we have the following. Theorem 17. The mixing time of the face-flip Markov chain of the square twist with faces is . 3.2. Square grid The square grid crease pattern consists of rows and columns of square faces. They are the subject of the stamp folding problem, a problem with some history [21, 25], but such prior work often strives to count the different layer orderings possible for a given MV assignment. In the present work, we focus only on distinct MV assignments. We first list some observations about the square grid; see for proofs. • Given a locally flat-foldable configuration, any face can be flipped and the result will still be locally flat-foldable. • Given two locally flat-foldable configurations, there exists a unique set of face flips that transforms one into the other. • Face flips commute. As with the square twist, we compare the square grid origami flip graph to a hypercube of some dimension. One can interpret the hypercube as the Cayley graph of with the usual generators . Conversely, we show that the flip graph is isomorphic to a Cayley graph of a quotient of by the all ones vector. This is the same as taking the hypercube and identifying each pair of antipodal points (say, and ). Proposition 18. The face flip graph of an -face square grid is isomorphic to the Cayley graph of generated by the set of standard basis vectors . Proof. We use our previous observations about the square grid and construct an isomorphism between the face-flip graph and as follows: fix a locally flat-foldable configuration and let . Order the faces of the square grid and associate each element of with such that if and only if the set of faces transforming to contains (we know this set is defined and unique by our earlier observations). It is clear that since flipping every single face will reverse each crease exactly twice. To see that is a bijection, consider and its corresponding set of faces which when flipped on result in a configuration . In order to achieve by a different set of face flips , we must have for every crease that the parity of the number of faces adjacent to is the same in as it is in . If , then this determines the entire configuration: must be . ∎ By the same coupon collector argument used to prove Theorem 15, we conclude the following. Theorem 19. The mixing time of the face-flip Markov chain of the square grid with faces is . 3.3. Miura-ori The Miura-ori crease pattern is an origami tessellation whose faces are all congruent parallelograms, arranged as in Figure 3(a). It is possibly the most well-studied origami crease pattern, as it was originally designed by Japanese astrophysicist Koryo Miura in the 1970s as a way to collapse large solar panel arrays and then transport and deploy them into outer space , and has since attracted attention for its interesting mechanics and use as a tunable metamaterial . To analyze the face-flip Markov chain, we pass to a well-studied graph coloring process. The following two lemmas give the relationship between Miura-ori crease patterns and vertex colorings. Lemma 20 (). There exists a bijection between the set of locally flat-foldable configurations on the Miura-ori crease pattern and the set of proper 3-vertex-colorings of the grid graph with one vertex pre-colored. Lemma 21 ([2, Lemma 4.1]). Let be locally valid Miura-ori configurations. Then if and only if and differ precisely at the vertex corresponding to . Thus, to bound the mixing time of face flips for Miura-ori crease patterns, it suffices to bound the mixing time for an analogous random process on proper -vertex-colorings of grid graphs, where at each step a vertex is chosen uniformly at random and assigned a possible color (that preserves the properness of the 3-coloring) uniformly at random. This process is known as the Glauber dynamics for proper colorings and is a type of Markov chain that is often studied in statistical physics. Lemma 22 ([12, Theorem 3.1]). The Glauber dynamics on proper 3-vertex-colorings of the grid mix in time . We then immediately obtain our main result. Theorem 23. The mixing time of the face-flip Markov chain of the Miura-ori crease pattern is . 4. Single-vertex crease patterns Flat-foldable crease patterns that consist of a single vertex in the paper’s interior, which we denoted in Section 2.1, form the building blocks of flat-foldable crease patterns. Kawasaki’s Theorem (3) provides us with an easy-to-check necessary and sufficient condition to determine whether a single-vertex crease pattern is flat-foldable, but it does not provide an MV assignment to fold the vertex flat. Further, Maekawa’s Theorem (4) is only a necessary condition. Hull and Justin describe how to recursively enumerate all valid MV assignments of a flat-foldable single-vertex crease pattern, providing the tools needed to determine single-vertex flat-foldability in linear time . Given all this, the structure of valid MV assignments for single-vertex crease patterns under face flips is surprisingly complex. Origami flip graphs for single-vertex flat-foldable crease patterns are not well understood, even in the case where all the sector angles between adjacent creases are congruent . Along these lines, we let be the crease pattern consisting of a single vertex with equally spaced creases incident to it. An example of is shown in Figure 1(b). Set to be the collection of locally flat-foldable MV configurations of 4.1. Perfect sampler It is not too difficult to describe an exact sampling algorithm for this model that runs in linear time. Proposition 24. There is a perfect sampler for the uniform distribution on that runs in time The idea will be to generate each valid MV assignment iteratively, conditioned on the previously revealed creases. Label the creases . Then a configuration may be viewed as a function with Writing for the uniform probability measure on , we begin by computing the marginal probabilities. Lemma 25. For each and , we have | | | | --- | | | | Proof. First note that there are exactly elements of that sum to , since each such sequence must contain exactly many ’s and many ’s. Given , by Maekawa’s Theorem (4) the valid MV assignments of are in bijection with sequences of length that sum to either or where ; this implies the first equality in the claim. For the second equality, we compute that the number of choices for is thus | | | | --- | | | | The total number of such choices that additionally have is the number of sequences of length that sum to either or , thus giving the count of | | | | --- | | | | Taking the ratio of the previous two displayed equations completes the proof. ∎ Proof of Proposition 24. The steps of the perfect sampler are as follows: for each compute the marginal probability for using Lemma 25. Choose with that probability and choose otherwise. Perform this operation for . Each step may be done in constant time and there are steps total. ∎ 4.2. Markov chain The main result of this section is the following. Theorem 26. The mixing time of the face-flip Markov chain on is Before proving this, we introduce several key ingredients. Let denote the transition matrix for the face-flip random walk. From Maekawa’s Theorem (4) and its converse for single-vertex crease patterns—shown in —we have the following observation. Observation 27. consists of all configurations with either or mountain assignments. To analyze , we categorize the faces of a given MV assignment: say a face is MM if bordered by two mountain creases, VV if bordered by two valley creases, and mixed otherwise. Let be the set of configurations with mountain creases. Observe that for , flipping an MM face results in a configuration in and similarly, flipping a VV face puts us back in . We wish to analyze by itself (our arguments will analogously apply to ). To that end, let be the uniform measure on . Recalling Definition 11, let be the transition matrix of the induced Markov chain where we restrict to the set . To bound the mixing time of , we compare this process to the well-studied adjacent transposition shuffle (see, e.g., [23, 16.1]). The latter is defined on permutations of and consists of choosing an adjacent pair uniformly at random (where ) and swapping their values. We can view the face-flip process similarly: arbitrarily choose a crease to label 1 and label the remaining creases proceeding clockwise from 1. Let be the face-flip chain on but disallowing the flip . Applying to is equivalent to assigning an arbitrary labeling , to the image of , running the adjacent transposition shuffle on this labeling, and then removing the labels from the output. Thus, we can see that if the adjacent transposition shuffle mixes rapidly, then must also mix rapidly. We will show by comparison that this implies also mixes rapidly and from this deduce that mixes rapidly. The mixing time of the adjacent transposition shuffle is well-studied. Lemma 28 (). The adjacent transposition shuffle on mixes in time . With this, we proceed to proving our main result of this section. Proof of Theorem 26. Let be the spectral gaps and be the mixing times for respectively. By Lemma 28, we have and so Theorem 9 implies . Applying Lemma 10 with gives and thus by Theorem 9 we have that | | | | --- | | | | To use this to bound we analyze the expected hitting time defined by (7). Let with We have at least one of or ; assume without loss of generality that and write . For any , consider a trajectory evolving according to and let be the chain induced on with transitions (recall that we iteratively define ). Let be the hitting time of in , and let be the hitting time of in the induced chain. Note that , so in order to upper bound , it suffices to bound . Observe that for each and for constant, we can write | | | | --- | | | | Given , suppose . Either or ; in the latter case, this means that the first steps of do not hit but the first steps of the induced chain do, implying that there are steps of that hit fewer than times. Hence, . We then have | | | | | --- --- | | | | | | | | | | | To bound , observe that our upper bound on and Theorem 12 imply that when we look at the induced chain, the expected number of transitions until the trajectory hits is . For the last term, let be the hitting time of in . Note that since must have at least one MM face. Thus, . Conditioned on the -algebra generated by , the random variable is stochastically dominated by a geometric random variable of mean . If we let be independent and identically distributed geometric random variables of mean , then we see as well that the sum is stochastically dominated by . Thus, | | | | --- | | | | for some constant , where the last inequality is by a general form of Bernstein’s inequality for sub-exponential random variables (see, e.g., ). This shows | | | | --- | | | | Putting everything together, we then have | | | | --- | | | | and applying Theorem 12 gives the claim. ∎ 5. Global Flat-Foldability As previously mentioned, it is NP-hard to determine if a general origami crease pattern with a locally valid MV assignment is globally flat-foldable. This remains true even if we restrict ourselves to crease patterns whose vertices lie on a square lattice and all creases are only at angles of multiples of . Surprisingly, the same complexity problem for square grid crease patterns is unknown [8, 17]. The smallest known example of a non-flat-foldable MV assignment on a square grid crease pattern was discovered by Justin in and is shown in Figure 4. Remark 29. The crease pattern in Figure 4 is locally flat-foldable but not globally flat-foldable A formal proof can be found in [17, Example 7.12]. Informally, this can be seen by imagining that we fold the center horizontal mountain crease and the vertical valley creases to the immediate left and right of it. Folding these central creases forces the left- and right-most vertices of the crease pattern to be folded in the same direction, meaning that one would need to nest inside the other. Experimenting with an actual piece of paper shows, however, that such nesting is impossible to achieve. In this section, we aim to prove Theorem 2. Our first step is to show that we may extend a partial assignment of creases on a rectangular subset and the complement of its neighborhood in a way that preserves local flat-foldability. Given an subset of , define the (closed) neighborhood of as the crease pattern subset consisting of along with (when they exist) an additional column to the left and right of and an additional row above and below . We notate this ; see Figure 5. Lemma 30. Let and . Given locally flat-foldable configurations on and on , there exists a configuration on such that for an arbitrary subset of , we have (a) , 2. (b) , and 3. (c) is locally flat-foldable. Proof. Define by assigning the creases inside according to the configuration and assigning the creases outside according to . Then satisfies (a) and (b). Note that the only creases with no assignment are those in . We will show that there is a valid MV assignment for these creases that maintains local flat-foldability at every vertex. Given a crease pattern , let be the set of vertices on the boundary of , meaning those in but incident to at least one edge not in . Let be the set of edges incident to exactly one vertex of . Observe that for each , must have one endpoint in and the other endpoint in . Let where are the vertices at the corners of . For all such that , let where . Observe that for every , all creases incident to other than (if it exists) have an assignment induced by . By Maekawa’s Theorem (4), then, there is a unique MV assignment to that maintains local flat-foldability. The vertices now each have two incident creases with an assignment and two incident creases without. It is straightforward to check that we may arbitrarily assign a subset of these creases (for example, one crease at each corner along with ), and the remaining creases will have a unique assignment satisfying Maekawa’s Theorem. ∎ Looking towards proving Theorem 2, we first prove an exponential upper bound on the probability of global flat-foldability: Lemma 31. Let be a uniformly random locally flat-foldable configuration on where . The probability that is globally flat-foldable is at most for an absolute constant . Proof. Observe that if there exists such that is not globally flat-foldable, then is also not globally flat-foldable. Consider the configuration given by Figure 4. We may sample via the following procedure: partition into as many disjoint copies of as possible; label these where . Begin by sampling the creases of uniformly at random. Having sampled the faces of , sample the unassigned creases of uniformly at random, conditioning on any neighboring creases that have already been assigned so that the local flat-foldability conditions are satisfied at each vertex. Note that by Lemma 13, this is equivalent to sampling uniformly at random. Let be the indicator of the event that restricted to the top-left subset of is equivalent to . Observe that due to Lemma 30: given that contains in its upper-left corner, there exists a locally valid MV assignment to the remaining creases of that is compatible with arbitrary boundary conditions on . Moreover, so | | | | --- | | | | Thus, the probability that is globally flat-foldable is at most where is an absolute constant. ∎ A submultiplicativity argument will then prove Theorem 2. Proof of Theorem 2. Let be the number of globally flat-foldable configurations on . Observe that is a submultiplicative sequence, meaning | | | | --- | | | | for all . To see this, consider a globally flat-foldable configuration on . The configurations induced on the leftmost and rightmost subgrids must be globally flat-foldable, and the inequality follows. By a multivariate analogue of Fekete’s Lemma , we thus have that exists and is equal to . Letting denote the probability that a uniformly random locally flat-foldable configuration of is globally flat-foldable, this implies that exists. By Lemma 31 we in fact have that thus completing the proof of the theorem. ∎ Remark 32. Our proof for the square grid only required the existence of a single locally but not globally flat-foldable crease pattern and the ability to extend a partial locally flat-foldable crease pattern to a complete one, as in Lemma 30. Thus, Theorem 2 also holds for other lattices, such as the Miura-ori. 6. Conclusion We end this paper with comments on other classes of flat-foldable crease patterns, providing one example where we conjecture that face flips do give reasonable mixing times but proofs remain elusive, and another where face flips do not give a reasonable way to randomly sample MV configurations. 6.1. Triangle lattice Let the triangle lattice crease pattern denote a parallelogram-shaped piece of paper with a crease pattern made entirely of tiled equilateral triangles, with and triangles along the two adjacent sides of the parallelogram. An example of a locally valid configuration for is shown in Figure 6. Unlike the square grid, the local and global flat-foldability of triangle lattice crease patterns are fairly unexplored. In it is shown that the origami flip graph of is connected with diameter and that the problem of finding a shortest path between two vertices in is NP-complete. But these are the only known results for this class of crease patterns. We note that considerable statistical physics work has been done on enumerating locally flat-folded crease patterns where one only folds a subset of edges of and does not distinguish between mountain and valley . This is a different notion of foldability than considered here and algorithmic questions are open and interesting in that setting. However, based on repeated simulations of the face-flip chain on relatively large examples—such as in Figure 7—we conjecture that fast, and in fact optimal, mixing should occur. Conjecture 33. The mixing time of the face-flip Markov chain on is . 6.2. Complexity and the kite pattern The face-flip Markov chain is not the only avenue towards sampling algorithms. For example, there exist crease patterns for which no faces are flippable, e.g. the kite crease pattern, in which case a different approach has to be considered. As described in , and referred to as the Huffman grid in , the kite crease pattern is made by tiling congruent quadrilaterals that have two opposite interior angles measuring , with the other pair of angles being and ; see Figure 8. As proven in , any face flip of a locally valid MV assignment of this crease pattern results in a violation of the Big-Little-Big Theorem (5). Therefore, none of the faces in a kite crease pattern are flippable and its origami flip graph has no edges. However, there are many locally valid MV assignments of a kite crease pattern. One is shown in Figure 8(b), and as proven in , each of the highlighted sets in the figure must be all-mountain or all-valley creases in order for the pattern to be valid. Thus, to get a different locally valid MV assignment, we could “flip” any of these sets from all-mountain to all-valley, or vice-versa. Similarly, the sets of creases between consecutive sets and must alternate Ms and Vs, and any of these could be flipped, say from MVMVMV to VMVMVM. Any of these kind of “flips” of whole sets of creases can be made independently of the others and still result in a valid MV assignment. Therefore, if a kite crease pattern can be partitioned into of these flippable sets, then it will have exactly valid MV assignments. Thus we believe that similar arguments to Section 3.1 will give results on the mixing time for kite crease patterns but without using the face-flip approach. For the crease patterns we have studied so far, all of our results and conjectures have been in the positive direction—that there does indeed exist a computationally efficient way to sample locally flat-foldable assignments. But are there results in the negative direction for other equally “natural” crease patterns? Question 34. Are there natural families of crease patterns on which it is computationally hard to sample (either perfectly or approximately) from the uniform distribution on locally flat-foldable MV assignments? A weaker question would be to only consider Markov chain sampling: Question 35. Are there natural families of crease patterns on which the face-flip Markov chain mixes slowly? Now suppose that we are given a locally flat-foldable MV assignment to a subset of creases in a crease pattern , and we ask if can be extended to a locally flat-foldable MV assignment for all of . When is an Miura-ori crease pattern, this question is answered in via an algorithm to decide the question in time. In general, however, this problem remains open. Question 36. Given a locally flat-foldable crease pattern and a partial MV assignment, can we efficiently determine an MV assignment of the remaining creases that makes the entire assignment locally flat-foldable? Lastly, the complexity of global flat-foldability remains a mystery. While the decision problem is NP-hard in general, the complexity for seemingly simple crease patterns remains unknown. Question 37. Is it computationally hard to sample a uniformly random globally flat-foldable MV assignment on the square grid? Acknowledgments Thomas Hull is supported in part by NSF grants DMS-2428771 and DMS-2347000. Marcus Michelen is supported in part by NSF CAREER grant DMS-2336788 as well as DMS-2246624. Corrine Yap is supported in part by NIH grant R01GM126554 and in part by NSF grant DMS-1928930 while in residence at the Simons–Laufer Mathematical Sciences Institute during the Spring 2025 semester. References Hugo A. Akitaya, Kenneth C. Cheung, Erik D. Demaine, Takashi Horiyama, Thomas C. Hull, Jason S. Ku, Tomohiro Tachi, and Ryuhei Uehara. Box pleating is hard. In Jin Akiyama, Hiro Ito, Toshinori Sakai, and Yushi Uno, editors, Discrete and Computational Geometry and Graphs, pages 167–179, Cham, 2016. Springer International Publishing. Hugo A. Akitaya, Vida Dujmović, David Eppstein, Thomas C. Hull, Kshitij Jain, and Anna Lubiw. Face flips in origami tessellations. Journal of Computational Geometry, 7(1), 2016. Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. Spectral independence in high-dimensional expanders and applications to the hardcore model. SIAM Journal on Computing, (0):FOCS20–1, 2021. Michael Assis. Exactly solvable flat-foldable quadrilateral origami tilings. Physical Review E, 98(3):032112, Sep 2018. arXiv: 1705.04710. Brad Ballinger, Mirela Damian, David Eppstein, Robin Y. Flatland, Jessica Ginepro, and Thomas Hull. Minimum forcing sets for Miura folding patterns. In Proc. 26th ACM-SIAM Symp. Discrete Algorithms, San Diego, California, 2015, pages 136–147, 2015. Silvio Capobianco. Multidimensional cellular automata and generalization of Fekete’s lemma. Discrete Mathematics & Theoretical Computer Science, 10(Automata, Logic and Semantics), 2008. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Optimal mixing of Glauber dynamics: Entropy factorization via high-dimensional expansion. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1537–1550, 2021. Erik D. Demaine and Joseph O’Rourke. Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, Cambridge, 2007. Philippe Di Francesco. Folding and coloring problems in mathematics and physics. Bulletin of the American Mathematical Society, 37:251–307, 2000. Jessica Ginepro and Thomas C. Hull. Counting Miura-ori foldings. Journal of Integer Sequences, 17(10):Article 14.10.8, 2014. Eric Gjerde. Origami Tessellations: Awe-Inspiring Geometric Designs. A K Peters, Wellesley, MA, 2008. Leslie Ann Goldberg, Russell Martin, and Mike Paterson. Random sampling of 3-colorings in . Random Structures & Algorithms, 24(3):279–302, 2004. Geoffrey Grimmett. Probability on graphs: random processes on graphs and lattices, volume 8. Cambridge University Press, 2018. Edwin A. Peraza Hernandez, Darren J. Hartl, and Dimitris C. Lagoudas. Active Origami: Modeling, Design, and Applications. Springer, 2019. Thomas C. Hull. On the mathematics of flat origamis. Congressus Numerantium, 100:215–224, 1994. Thomas C. Hull. Counting mountain-valley assignments for flat folds. Ars Combinatoria, 67:175–188, 2003. Thomas C. Hull. Origametry: Mathematical Methods in Paper Folding. Cambridge University Press, 2020. Thomas C. Hull, Manuel Morales, Sarah Nash, and Natalya Ter-Saakov. Maximal origami flip graphs of flat-foldable vertices: properties and algorithms. Journal of Graph Algorithms and Applications, 26(4):503–517, 2022. Tetsuo Ida. Introduction to Computational Origami. Springer, Cham, Switzerland, 2020. Jacques Justin. Toward a mathematical theory of origami. In Koryo Miura, editor, Origami Science & Art: Proceedings of the Second International Meeting of Origami Science and Scientific Origami, pages 15–29, Otsu, Japan, 1997. Seian University of Art and Design. J. Koehler. Folding a strip of stamps. Journal of Combinatorial Theory, 5:135–152, 1968. Robert J. Lang. Tessellatica 11.1. Software, 2018. David A Levin and Yuval Peres. Markov chains and mixing times, volume 107. American Mathematical Soc., 2017. Bin Liu, Jesse L. Silverberg, Arthur A. Evans, Christian D. Santangelo, Robert J. Lang, Thomas C. Hull, and Itai Cohen. Topological kinematics of origami metamaterials. Nature Physics, 14(8):811–815, 2018. W. F. Lunnon. A map-folding problem. Mathematics of Computation, 22(101):193–199, 1968. Rupert Maleczek, Gabriel Stern, Clemens Preisinger, Moritz Heimrath, Oliver D Krieg, and Astrid Metzler. Curved folded wooden assemblies. Proceedings of IASS Annual Symposia, 2018(20):1–8, 2018. Nicholas Metropolis, Arianna W Rosenbluth, Marshall N Rosenbluth, Augusta H Teller, and Edward Teller. Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6):1087–1092, 1953. Davic C. Morgan, Denise M. Halverson, Spencer P. Magleby, Terri C. Bateman, and Larry L. Howell. Y Origami?: Explorations in Folding. American Mathematical Society, 2017. Chihiro Nakajima. An efficient enumeration of flat-foldings : Study on random single vertex origami, 2024. Chihiro Nakajima. A spin model for global flat-foldability of random origami, 2024. Yuval Peres and Perla Sousi. Mixing times are hitting times of large sets. Journal of Theoretical Probability, 28(2):488–519, 2015. S. A. Robertson. Isometric folding of Riemannian manifolds. Proceedings of the Royal Society of Edinburgh, 79(3–4):275–284, 1977–1978. Jesse L. Silverberg, Arthur A. Evans, Lauren McLeod, Ryan C. Hayward, Thomas Hull, Christian D. Santangelo, and Itai Cohen. Using origami design principles to fold reprogrammable mechanical metamaterials. Science, 345(6197):647–650, 2014. Jesse L. Silverberg, Jun-Hee Na, Arthur A. Evans, Bin Liu, Thomas C. Hull, Christian D. Santangelo, Robert J. Lang, Ryan C. Hayward, and Itai Cohen. Origami structures with a critical transition to bistability arising from hidden degrees of freedom. Nature Materials, 14(4):389–393, 2015. Ryuhei Uehara. Introduction to Computational Origami. Springer Singapore, Singapore, 2020. Roman Vershynin. High-dimensional probability: An introduction with applications in data science, volume 47. Cambridge university press, 2018. David Bruce Wilson. Mixing times of lozenge tiling and card shuffling Markov chains. The Annals of Applied Probability, 14(1):274–325, 2004. Zhong You. Folding structures out of flat materials. Science, 345(6197):623–624, 2014.
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How to solve a composition percentage - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Percent Composition of Co... Answering Questions Math Word Problems Percentage (%) Problem Solving Percentage Calculation Arithmetic Numerical Problems 5 How can you solve a composition percentage? All related (38) Sort Recommended Will Gates Former Assistant chief cook and bottle washer. · Author has 2K answers and 1.8M answer views ·3y I assume you are referring to the problem of proportions of a total amount such as 2:3:5 which could refer to masses of suitcases in a van or ingredients in a recipe? With any problems relating to adding fractions, the starting point is to ensure that the denominator is the same for each fraction so you are counting equal sized pieces. 2:3:5 could be 2 kg : 3 kg : 5 kg; 200 g : 300 g : 500 g or even 40 g : 60 g : 100 g where 1 unit could represent: 1 kg, 100 g or 20 g respectively. A percentage is a fraction where the denominator is normalised to 100. 100 % would mean a whole unit. Percentages gr Continue Reading I assume you are referring to the problem of proportions of a total amount such as 2:3:5 which could refer to masses of suitcases in a van or ingredients in a recipe? With any problems relating to adding fractions, the starting point is to ensure that the denominator is the same for each fraction so you are counting equal sized pieces. 2:3:5 could be 2 kg : 3 kg : 5 kg; 200 g : 300 g : 500 g or even 40 g : 60 g : 100 g where 1 unit could represent: 1 kg, 100 g or 20 g respectively. A percentage is a fraction where the denominator is normalised to 100. 100 % would mean a whole unit. Percentages greater that 100 % represent an, “improper fraction,” “top heavy fraction,” or “mixed number.” The proportions 2:3:5 are usually given in their simplest form; that is, having no common factors. For instance: 4:6:10 represents the same proportions, just in an unnecessarily over complicated form. OK, so what does 100 % represent? 2:3:5 are the numbers of equal sized pieces which go to make up the total composition. Altogether there is a total sum: T of: T = 2 + 3 + 5 = 10 parts. The respective unit proportions would be: 2/T, 3/T and 5/T which can be converted to percentages by multiplying by 100… Hence: 100 2/10, 100 3/10 and 100 5/10 resulting in: 20%, 30% and 50% which, of course means the same as: 20/100, 30/100 and 50/100. Upvote · 9 1 9 1 Sponsored by Grammarly 92% of professionals who use Grammarly say it has saved them time Work faster with AI, while ensuring your writing always makes the right impression. Download 999 207 Related questions More answers below What percentage of vehicle thefts are solved? What percentage of burglaries are solved? What percentage of Chicago murders are solved? What percentage of burglaries in your area are solved? What percentage of problems does Trump solve himself, and what percentage does he tell other people to solve? Roberto Filippini Fantoni Teacher at Fondazione ITS per Le Nuove Tecnologie Della Vita (2014–present) · Author has 1.4K answers and 1.1M answer views ·5y Originally Answered: How do you calculate the percent composition of a compound? · I can give you just an example that will be enough to solve the problem for any compound. I imagine this compound: KMnO4 (potassium permanganate). To solve the problem you have to know the atomic mass of any element and the molecular mass of KMnO4: K = 39.10 — Mn = 54.94 — O = 16.00 — Molecular mass of KMnO4 = 158.04 % K = 39.1 x 100/158.04 = 24.74% % Mn = 54.94 x 100/158.04 =34.76% % O = 4 x 16 x 100/158.04 = 40.50% Upvote · 9 1 Guy Clentsmith Chemistry tutor... at Self-Employment (2018–present) · Author has 26.5K answers and 19.7M answer views ·4y Originally Answered: How do I get the composition percent of an element? · Well, TYPICALLY, your sample contains carbon, and hydrogen, and nitrogen (of course your sample is a COMPOUND not a pure element…) … and you KNOW the theoretical percentage by mass of the PURE compound. And a SMALL mass of sample would be given to an analyst, who would combust an accurately measured mass of sample in a furnace … and the PRODUCTS of combustion, i.e. C O 2(g)C O 2(g), and H 2 O(g)H 2 O(g) would b Continue Reading Well, TYPICALLY, your sample contains carbon, and hydrogen, and nitrogen (of course your sample is a COMPOUND not a pure element…) … and you KNOW the theoretical percentage by mass of the PURE compound. And a SMALL mass of sample would be given to an analyst, who would combust an accurately measured mass of sample in a furnace … and the PRODUCTS of combustion, i.e. C O 2(g)C O 2(g), and H 2 O(g)H 2 O(g) would be shunted to a gas chromatograph … and thus we ... Upvote · Rohan Naik Data Scientist at Wipro (Indian company) (2016–present) · Author has 1.4K answers and 4.3M answer views ·6y Originally Answered: What is the formula of percentage composition? · The per cent composition of a compound expresses the elemental composition of the compound (found by chemical analysis) in terms of the number of grams of each element divided by the total number of grams present. The per cent composition of a compound is an expression of the elemental composition of that compound. The per cent composition of each element (actually the mass per cent) is equal to the mass of that element divided by the total mass present (the formula mass) multiplied by 100%. %CE = gE/gT x 100 Here, % CE is the percentage composition of the element E. This is the value that we are Continue Reading The per cent composition of a compound expresses the elemental composition of the compound (found by chemical analysis) in terms of the number of grams of each element divided by the total number of grams present. The per cent composition of a compound is an expression of the elemental composition of that compound. The per cent composition of each element (actually the mass per cent) is equal to the mass of that element divided by the total mass present (the formula mass) multiplied by 100%. %CE = gE/gT x 100 Here, % CE is the percentage composition of the element E. This is the value that we are going to calculate. The numerator on the right side indicates the total amount of element E present in the compound. On the other hand, the denominator is an expression for the total amount of all the elements present in the compound. We multiply this ratio by 100 to get the percentage form of the composition. Let us now look at the mass percentage of composition in more details. We will also look at its importance. Per cent composition in terms of formula mass. It is sometimes useful to know what per cent of the total weight of a compound is made up of particular element. This is called finding the percentage composition. For more information, you can also watch the below video. Upvote · Sponsored by All Out Kill Dengue, Malaria and Chikungunya with New 30% Faster All Out. Chance Mat Lo, Naya All Out Lo - Recommended by Indian Medical Association. Shop Now 999 621 Related questions More answers below How do I calculate the percentage out of a six subject appeared and failed in a sixth one? Can computer solve captcha? How do you deal with problems in life? What percentage of the gifted can solve a Rubik’s Cube? What is a high proportion? Daniel Iyamuremye Former Senior Lecturer (Retired) (2000–2018) · Author has 12.1K answers and 2M answer views ·3y Originally Answered: How can I calculate the percent composition of an unknown substance? · You must know its chemical formula and its molar mass Knowing the number of each atom that composes the substance, then (the total mass of each atom/ molar mass) x 100 = the percentage of that atom in the composition of the substance. Upvote · 9 1 Sheshraj Chaudhary High School Degree in Science&Mathematics, Prasadi Academy, Lalitpur (Graduated 2015) ·8y Originally Answered: How do you find the percent composition? · In order to find percentage composition you have to know the atomic weight of the atoms and find the molecular weight of the compound. Then you should use this formula, % composition of an element in a compound = Weight of total no. atoms of that element / molecular weight of compound and multiply by 100. For example; to find % composition of hydrogen in water(H2O) = (12)/18 and multiply by 100 Upvote · 9 1 Sponsored by State Bank of India Stay Informed!! Stay Protected!! Our Contact Centre calls only from numbers beginning with 1600 or 140 series. Learn More 99 46 Trevor Hodgson Knows English · Author has 11.8K answers and 12.3M answer views ·3y This is a very vague question . If you want to know how to solve any problem - submit a question on the topic and you will be shown how work out the answer. Upvote · Abrahm Adedeji Manager at Jikes (2017–present) ·3y Originally Answered: How do I find the percentage composition of a compound? · Percentage compositions can be found by: Element mass (Xg) Total mass composition of compound (Xg) mitiplied by 100 since you are finding percentage. Upvote · Sponsored by RedHat Know what your AI knows, with open source models. Your AI should keep records, not secrets. Learn More 99 36 Purna Chandra Sahu M. Sc. in Pure Chemistry&Inorganic Chemistry, Calcutta Uniiversity (Graduated 1987) · Author has 2.9K answers and 6M answer views ·4y Related How do I calculate mass percent composition? If you know the molecular formula, molar mass of the compound and atomic mass of the element of which you have to find out the percent composition, then from the following formula you can calculate the mass percent composition of the element in the compound. Mass percentage of the element = {(Number of atoms × atomic mass of the element)/molar mass of the compound} × 100%. Sometimes, mass of the element in a particular mass of the compound are given, in that case the formula is: Mass percentage of the element = (mass of element/mass of compound)×100%. Hope, this helps. Upvote · 9 1 Ehron Racer A content creator in Math Magic's YouTube Channel · Author has 83 answers and 24.9M answer views ·4y Related How do you solve a percentage problem (arithmetic, percentages, math)? Good question! To solve a percentage problem you firstly need to understand in general that what they are asking is gonna be a number less than the number being talked about. So let us say they would ask you to find 70 percent of 45 (or 70% of 45). You need to understand then that 45 is 100% of the number. Now if you are going to find a portion of it(i.e. 70%), then the answer you must give is smaller than 45. How to solve it? You can simply multiply 45 by 70, then move two decimal places to the left. Or you can also get 10 percent of 45 first, then multiply it by 7 to get 70. By doing either of the Continue Reading Good question! To solve a percentage problem you firstly need to understand in general that what they are asking is gonna be a number less than the number being talked about. So let us say they would ask you to find 70 percent of 45 (or 70% of 45). You need to understand then that 45 is 100% of the number. Now if you are going to find a portion of it(i.e. 70%), then the answer you must give is smaller than 45. How to solve it? You can simply multiply 45 by 70, then move two decimal places to the left. Or you can also get 10 percent of 45 first, then multiply it by 7 to get 70. By doing either of the two steps, you can be able to obtain the same answer of 31.5. But there is a different application of it when we talk of discounts, as they also use percentages to express how much did it decrease from the original price. So let us say the original price of the t-shirt you want to buy is 95 dollars ($95). Since the store is implementing a sale as it is Christmas, the original price of the t-shirt drops by 20 percent. Now this means that you will have to find the sale price of the item. You can do it on three ways. The first way involves getting the 20 percent of the original price then subtracting the value obtained to the original price afterwards to get the sale price. This is shown as follows: 20% of 95 = 19 95-19 = $76 The sale price of the T-shirt is now 76 dollars ($76). The second way is that you can also find 10 percent of 95, which is 9.5 then multiply it by 2 to obtain the answer of 76. The third is somewhat more straightforward and quick. In this third way, you should understand that with the value or the cost of the T-shirt being dropped by 20 percent, it only means that 20 percent is free but you still have to pay the 80 percent of it. So instead of finding 20 percent of 95 and subtracting it towards the same subject or value, we will just focus with finding the 80 percent of 95. 80% of $95 = $76 Now, see how it only takes one step for us to calculate the sale price of the item. That is how to find or to solve for the percentage in different situations. I hope this helps. Have fun! -Ehron Racer Upvote · Shivam Kumar Developer · Author has 69 answers and 139.6K answer views ·Updated 7y Related How can I solve percentage question in 10 second? Answer to this question lies on your calculation speed. It depends mainly in how fast your ability to perform calculations is although, there are some tips i can give such as :- Try to find 1% first it can be done easily by dividing it by 100 . Then, multiply it to number yo want to find percentage of suppose 11% then multiply by 11. Now, in this way also we are unable to get rid off multiplication then let’s make multiplication easier ==> Every one of us remember the tables of 5,10,15,20… easier as compare to other like 17,18…. then if making multiplication with 137x3 let’s look it as (135+2)x3 th Continue Reading Answer to this question lies on your calculation speed. It depends mainly in how fast your ability to perform calculations is although, there are some tips i can give such as :- Try to find 1% first it can be done easily by dividing it by 100 . Then, multiply it to number yo want to find percentage of suppose 11% then multiply by 11. Now, in this way also we are unable to get rid off multiplication then let’s make multiplication easier ==> Every one of us remember the tables of 5,10,15,20… easier as compare to other like 17,18…. then if making multiplication with 137x3 let’s look it as (135+2)x3 then it looks simple as 135x3 and 2x3 are easy to multiply and then add it makes it quite simple. some more, like 19x17 = (20–1)x17= 20x17 -17 makes it easier to calculate. Hope that would help! Change your fear to belief and keep exploring ^.^ Upvote · 9 3 Nandhu Mystical Logician · Author has 199 answers and 137.1K answer views ·Updated 1y Related When a percentage is more than 100, what does it mean? For me, it doesn't make any sense. I come across answers saying that more than 100% is more than the base amount or one whole. 1 Kilogram is always 1000 grams; we never stuff more than 1000 grams into a Kilogram. I always use other means to convey the excess/extra/increment. For example, if my salary is ₹100,000 a month, and if it's increased to ₹130,000 a month, I would always say my salary increased by 30% and not 130%. So, when a percentage is more than 100, I always regard it as another way of making things known. People have their own way of making this known, likewise. Understanding things i Continue Reading For me, it doesn't make any sense. I come across answers saying that more than 100% is more than the base amount or one whole. 1 Kilogram is always 1000 grams; we never stuff more than 1000 grams into a Kilogram. I always use other means to convey the excess/extra/increment. For example, if my salary is ₹100,000 a month, and if it's increased to ₹130,000 a month, I would always say my salary increased by 30% and not 130%. So, when a percentage is more than 100, I always regard it as another way of making things known. People have their own way of making this known, likewise. Understanding things in the right way, categorizing or classifying things accordingly, that is to say with respect to the person, region, language, etc., will reduce stress and give a clear view of things. Upvote · 9 4 9 1 9 1 Amanat Khan Exploring the world of mathematics · Author has 124 answers and 163.9K answer views ·4y Related How do you solve the problem where 78 represents 12% of a certain number (percentages, math)? Consider the number as x x and we can easily solve the problem 78=12 78=12 % o f o f x x or, 78=12 100×x 78=12 100×x or, 78=12 x 100 78=12 x 100 or, 12 x=78×100 12 x=78×100 or, x=7800 12 x=7800 12 so, x=650 x=650 hence, 78 represents 12% of 650 Upvote · 9 1 Related questions What percentage of vehicle thefts are solved? What percentage of burglaries are solved? What percentage of Chicago murders are solved? What percentage of burglaries in your area are solved? What percentage of problems does Trump solve himself, and what percentage does he tell other people to solve? How do I calculate the percentage out of a six subject appeared and failed in a sixth one? Can computer solve captcha? How do you deal with problems in life? What percentage of the gifted can solve a Rubik’s Cube? What is a high proportion? What percentage of man problems can be solved by ICT? What percentage of the internet is solving world problems? What problem does Pearltrees solve? What is the percentage of people who are actively solving environmental issues? How do they solve those problems? What problem does Tumblr solve? Related questions What percentage of vehicle thefts are solved? What percentage of burglaries are solved? What percentage of Chicago murders are solved? What percentage of burglaries in your area are solved? What percentage of problems does Trump solve himself, and what percentage does he tell other people to solve? How do I calculate the percentage out of a six subject appeared and failed in a sixth one? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
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https://www.cut-the-knot.org/Curriculum/Geometry/CirclesThroughOrthocenter.shtml
Site What's new Content page Front page Index page About Privacy policy Help with math Subjects Arithmetic Algebra Geometry Probability Trigonometry Visual illusions Articles Cut the knot! What is what? Inventor's paradox Math as language Problem solving Collections Outline mathematics Book reviews Interactive activities Did you know? Eye opener Analogue gadgets Proofs in mathematics Things impossible Index/Glossary Simple math Fast Arithmetic Tips Stories for young Word problems Games and puzzles Our logo Make an identity Elementary geometry Circles through the Orthocenter The applet below illustrates Problem 1 from the 2008 International Mathematics Olympiad: An acute-angled triangle $ABC\;$ has orthocentre $H.\;$ The circle passing through $H\;$ with centre the midpoint of $BC\;$ intersects the line $BC\;$ at $A_{1}\;$ and $A_{2}.\;$ Similarly, the circle passing through $H\;$ with centre the midpoint of $CA\;$ intersects the line $CA\;$ at $B_{1}\;$ and $B_{2},\;$ and the circle passing through $H\;$ with centre the midpoint of $AB\;$ intersects the line $AB\;$ at $C_{1}\;$ and $C_{2}.\;$ Show that $A_{1},\;$ $A_{2},\;$ $B_{1},\;$ $B_{2},\;$ $C_{1},\;$ $C_{2}\;$ lie on a circle. 6 April 2016, Created with GeoGebra Solution |Activities| |Contact| |Front page| |Contents| |Geometry| Copyright © 1996-2018 Alexander Bogomolny An acute-angled triangle $ABC\;$ has orthocentre $H.\;$ The circle passing through $H\;$ with centre the midpoint of $BC\;$ intersects the line $BC\;$ at $A_{1}\;$ and $A_{2}.\;$ Similarly, the circle passing through $H\;$ with centre the midpoint of $CA\;$ intersects the line $CA\;$ at $B_{1}\;$ and $B_{2},\;$ and the circle passing through $H\;$ with centre the midpoint of $AB\;$ intersects the line $AB\;$ at $C_{1}\;$ and $C_{2}.\;$ Show that $A_{1},\;$ $A_{2},\;$ $B_{1},\;$ $B_{2},\;$ $C_{1},\;$ $C_{2}\;$ lie on a circle. Solution 1 One solution stems from the observation that the orthocenter serves as the radical center of the three circles whilst the altitudes are the radical axes of the circles taken in pairs. The situation is reminscent of the circles constructed on the sides of a triangle as diameters. Consider the circles centered at the midpoints $M_{a}\;$ and $M_{b}\;$ of sides $BC\;$ and $AC,\;$ respectively. Since $CH\perp AB\;$ we also have $M_{a}M_{b}\perp CH.\;$ But the line joining the centers of two intersecting circles is perpendicular to their common chord. In particular, this is what $CH\;$ is - the line containing the common chord of the two circles. Which exactly means that $CH\;$ is their radical axis. The same holds for the other two pairs of circles. Let $r_{a},\;$ $r_{b},\;$ $r_{c}\;$ be the radii of the circles centered at $M_{a},\;$ $M_{b},\;$ $M_{c},\;$ which we denote $C(M_{a}, r_{a}),\;$ $C(M_{b}, r_{b}),\;$ $C(M_{c}, r_{c}),\;$ respectively. Let also set $a = BC,$ $b = AC,$ $c = AB.$ Since $C\;$ lies on the radical axis of $C(M_{a}, r_{a})\;$ and $C(M_{b}, r_{b}),\;$ we may apply the Pythagorean theorem to the triangles formed by tangents of common length $t\;$ from $C\;$ to the two circles, half-sides of $BC\;$ and $AC\;$ and the radii of the circles: (1) $(a/2)^{2} - r_{a}^{2} = t^{2} = (b/2)^{2} - r_{b}^{2}.\;$ On the other hand, if the six points $A_{1},\;$ $A_{2},\;$ $B_{1},\;$ $B_{2},\;$ $C_{1},\;$ $C_{2}\;$ lie on a circle, the latter is bound to be centered at the circumcenter O of $\Delta ABC.\;$ So lets show that $O\;$ is equidistant from the six points. To this end, denote $s_{a} = OM_{a},$ $s_{b} = OM_{b},$ $s_{c} = OM_{c}.$ Since, $O\;$ is the circumcenter of $\Delta ABC,\;$ (2) $(a/2)^{2} + s_{a}^{2} = R^{2} = (b/2)^{2} + s_{b}^{2},\;$ $R\;$ being the circumradius of $\Delta ABC.\;$ Subtracting (1) from (2) we see that (3) $r_{a}^{2} + s_{a}^{2} = r_{b}^{2} + s_{b}^{2},\;$ implying that $O\;$ is equidistant from $A_{1},\;$ $A_{2},\;$ $B_{1},\;$ $B_{2}.\;$ The four points $A_{1},\;$ $A_{2},\;$ $B_{1},\;$ $B_{2}\;$ are thus concyclic. Similarly, $A_{1},\;$ $A_{2},\;$ $C_{1},\;$ $C_{2}\;$ are equidistant from $O\;$ which puts $C_{1}\;$ and $C_{2}\;$ on the same circle. Observe now that the requirement for $\Delta ABC\;$ to be acute is spurious. The six points $A_{1},\;$ $A_{2},\;$ $B_{1},\;$ $B_{2},\;$ $C_{1},\;$ $C_{2}\;$ are concyclic for any $\Delta ABC.\;$ A further generalization replaces the altitudes with arbitrary concurrent cevians. Which shows that both requirements are nothig but red herrings. Solution 2 This solution is by Jack D'Aurizio. We may put a reference system with its centre at the circumcentre of $\Delta ABC,\;$ so that, by Euler's theorem, we simply have $H=A+B+C.\;$ We have to show $CA_1\cdot CA_2 = CB_1\cdot CB_2\;$ and so on, but: $CA_1\cdot CA_2 = \frac{1}{4}\|B-C\|^2 -\frac{1}{4}\|B+C+2A\|^2,\;$ hence the problem boils down to proving: $ \|B+C\|^2 - \|B+C+2A\|^2 = \|A-C\|^2 - \|A+C+2B\|^2. $ That is straightforward, since both the LHS and the RHS equal: $\begin{eqnarray} && -4R^2\left(1+\cos(2A)+\cos(2B)+\cos(2C)\right)\&=&-4R^2\left(4-2\sin^2 A-2\sin^2 B-2\sin^2 C\right)\&=&2(a^2+b^2+c^2)-16R^2=2(R^2-OH^2).\end{eqnarray}$ What Is Red Herring On the Difference of Areas Area of the Union of Two Squares Circle through the Incenter Circle through the Incenter And Antiparallels Circle through the Circumcenter Inequality with Logarithms Breaking Chocolate Bars Circles through the Orthocenter 100 Grasshoppers on a Triangular Board Simultaneous Diameters in Concurrent Circles An Inequality from the 2015 Romanian TST Schur's Inequality Further Properties of Peculiar Circles Inequality with Csc And Sin Area Inequality in Trapezoid Triangles on HO From Angle Bisector to 120 degrees Angle A Case of Divergence An Inequality for the Cevians through Spieker Point via Brocard Angle An Inequality In Triangle and Without Problem 3 from the EGMO2017 Mickey Might Be a Red Herring in the Mickey Mouse Theorem A Cyclic Inequality from the 6th IMO, 1964 Three Complex Numbers Satisfy Fermat's Identity For Prime Powers Probability of Random Lines Crossing Planting Trees in a Row Two Colors - Three Points Radical Axis and Radical Center How to Construct a Radical Axis A Property of the Line IO: A Proof From The Book Cherchez le quadrilatere cyclique II Circles On Cevians Circles And Parallels Circles through the Orthocenter Coaxal Circles Theorem Isosceles on the Sides of a Triangle Properties of the Circle of Similitude Six Concyclic Points Radical Axis and Center, an Application Radical axis of two circles Radical Axis of Circles Inscribed in a Circular Segment Radical Center Radical center of three circles Steiner's porism Stereographic Projection and Inversion Stereographic Projection and Radical Axes Tangent as a Radical Axis Two Circles on a Side of a Triangle Pinning Butterfly on Radical Axes Two Lines - Two Circles Two Triples of Concurrent Circles Circle Centers on Radical Axes Collinearity with the Orthocenter Six Circles with Concurrent Pairwise Radical Axes Six Concyclic Points on Sides of a Triangle Line Through a Center of Similarity |Activities| |Contact| |Front page| |Contents| |Geometry| Copyright © 1996-2018 Alexander Bogomolny
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https://genomics.princeton.edu/storeylab/papers/fdringenomics.pdf
Statistical Significance for Genome-Wide Studies John D. Storey∗† and Robert Tibshirani‡ January 2003; Revised May 2003 Abstract: With the increase in genome-wide experiments and the sequencing of mul-tiple genomes, the analysis of large data sets has become commonplace in biology. It is often the case that thousands of features in a genome-wide data set are tested against some null hypothesis, where a number of features are expected to be sig-nificant. Here we propose an approach to measuring statistical significance in these genome-wide studies based on the concept of the false discovery rate. This approach offers a sensible balance between the number of true positives and false positives that is automatically calibrated and easily interpreted. In doing so, a measure of statistical significance called the q-value is associated with each tested feature. The q-value is similar to the well known p-value, except it is a measure of significance in terms of the false discovery rate rather than the false positive rate. Our approach avoids a flood of false positive results, while offering a more liberal criterion than what has been used in genome scans for linkage. Keywords: false discovery rates, genomics, multiple hypothesis testing, p-values, q-values Abbreviations: FDR, false discovery rate; pFDR, positive false discovery rate. ∗To whom correspondence should be sent. Email: storey@stat.berkeley.edu. †Department of Statistics, University of California, Berkeley CA 94720. ‡Department of Health Research & Policy and Department of Statistics, Stanford University, Stanford CA 94305. 1 Introduction Some of the earliest genome-wide studies involved testing for linkage at loci spanning a large portion of the genome. Since a separate statistical test is performed at each locus, traditional p-value cut-offs of 0.01 or 0.05 had to be made stricter to avoid an abundance of false positive results. The threshold for significance in linkage analysis is usually chosen so that the probability of any single false positive among all loci tested is less than or equal to 0.05. This strict criterion is used mainly because one or very few loci are expected to show linkage in any given study [1, 2]. Due to the recent surge in high-throughput technologies and genome projects, many more types of genome-wide studies are now underway. The analyses of these data also involve performing statistical tests on thousands of features in a genome. As opposed to the linkage case, it is expected that many more than one or two of the tested features are statistically significant. Guarding against any single false positive occurring is often going to be much too strict and will lead to many missed findings. The goal is therefore to identify as many significant features in the genome as possible, while incurring a relatively low proportion of false positives. We are specifically concerned with situations in which a well-defined statistical hypothesis test is performed on each of thousands of features represented in a genome. These “features” can be genes, all nucleotide words of a certain length, SNP markers, etc. Several motivating examples are given below. For each feature, a null hypothesis is tested against an alternative hypothesis. In this work, we say that a feature is truly null if the null hypothesis is true, and a feature is truly alternative if the alternative hypothesis is true. If a feature is called significant, then the null hypothesis is rejected in favor of the alternative hypothesis. The goal is to propose and estimate a measure of significance for each feature that meets the practical goals of the genome-wide study, and that is easily interpreted in terms of the simultaneous testing of thousands of features. We propose that the recently introduced q-value [3,4] is a well suited measure of significance for this growing class of genome-wide tests of significance. The q-value is an extension of a quantity called the “false discovery rate” , which has received much recent attention in the statistics literature [6, 7, 8, 9, 10, 11]. A false discovery rate method has been used in detecting differential gene expression in DNA microarray experiments , which can be shown to be equivalent to under certain assumptions. Also, ideas similar to false discovery rates have appeared in the genetics literature [1,13]. Similarly to the p-value, the q-value gives each feature its own individual measure of significance. Whereas the p-value is a measure of significance in terms of the false positive rate, the q-value is a measure in terms of the false discovery rate. The false positive rate and false discovery rate are often mistakenly equivocated, but their difference is actually very important. Given a rule 2 for calling features significant, the false positive rate is the rate that truly null features are called significant. The false discovery rate is the rate that significant features are truly null. For example, a false positive rate of 5% means that on average 5% of the truly null features in the study will be called significant. A false discovery rate of 5% means that among all features called significant, 5% of these are truly null on average. The q-value provides a measure of each feature’s significance, automatically taking into account the fact that thousands are simultaneously being tested. Suppose that features with q-values less than or equal to 5% are called significant in some genome-wide test of significance. This results in a false discovery rate of 5% among the significant features. A p-value threshold of 5% yields a false positive rate of 5% among all null features in the data set. In light of the definition of the false positive rate, a p-value cut-offsays little about the content of the features actually called significant. The q-values directly provide a meaningful measure among the features called significant. Since significant features will likely undergo some subsequent biological verification, a q-value threshold can be phrased in practical terms as the proportion of significant features that turn out to be false leads. Here we show that the false discovery rate is a sensible measure of the balance between the number of true positives and false positives in many genome-wide studies. We motivate our pro-posed approach in the context of several recent and prominent papers in which awkwardly chosen p-value cut-offs were used in an attempt to at least qualitatively achieve what the q-value directly achieves. We also introduce a fully automated method for estimating q-values, with an initial treatment of dependence issues between the features and guidelines as to when the estimates are accurate. The proposed methodology is applied to some gene expression data taken from cancer tumors , supporting previously shown results as well as providing some new information. Motivating Examples Consider the following four recent articles in which thousands of features from a genome-wide data set were tested against a null hypothesis. In each case, p-values thresholds were employed to decide which features to call significant, although the ultimate goal was to identify many truly alternative features without including too many false positives. Example 1: Detecting differentially expressed genes. A common goal in DNA microarray ex-periments is to detect genes that show differential expression across two or more biological condi-tions . This is an important question to answer since it allows one to discover genes involved in differentiating complex biological states. In this scenario, the “features” are the genes, and they are tested against the null hypothesis that there is no differential gene expression. One of the goals 3 of Hedenfalk et al. is to find genes that are differentially expressed between BRCA1-mutation-positive tumors and BRCA2-mutation-positive tumors by obtaining several microarrays from each cell type. In their analysis they compute a modified F-statistic and use it to assign a p-value to each gene. A p-value cut-offof 0.001 was selected to find 51 genes out of 3226 that show differential gene expression. A rough calculation shows that about 3 false positives are expected with this cut-off. They later use a threshold of 0.0001 and conclude that 9 to 11 genes are differentially expressed. Example 2: Identifying exonic splicing enhancers. Exonic splice enhancers (ESEs) are short oligonucleotide sequences that enhance pre-mRNA splicing when present in exons . Fairbrother et al. analyzed human genomic DNA in order to predict ESEs based on the statistical analysis of exon-intron and splice site composition. They assessed the statistical significance of all 4096 possible hexamers, the null hypothesis being a mathematical formulation of a hexamer not being an ESE. A statistic is formed based on the location of the hexamers in 4817 human genes where the exon-intron structure has been well characterized. The end product is a p-value associated with each of the 4096 hexamers. A p-value cut-offof 10−4 was used based on the rationale that at most 4096 × 10−4 < 1 false positive is expected under this criterion. This cut-offyields 238 significant hexamers, a number of which were subsequently biologically verified. Example 3: Genetic dissection of transcriptional regulation. Global monitoring of gene expres-sion and large scale genotyping were recently used to study transcriptional regulation in yeast. Brem et al. crossed two strains of yeast, where many genes appeared to be differentially ex-pressed between these two strains. For 40 of the resulting haploid progeny, the expression levels of 6215 genes were measured using microarrays. Linkage was tested between 3312 markers spanning the genome and each of these 6215 “quantitative traits.” A statistically significant linkage between a gene’s expression level and a marker indicates that a regulator for that gene is located in the re-gion of the marker. In analyzing these data, one can perform a statistical test for each gene-marker combination, resulting in millions of p-values, or one can test each gene for showing linkage to at least one locus, resulting in 6215 p-values. Taking the latter approach and using a p-value cut-off of 8.5 × 10−3, the authors report that 507 genes show linkage to at least one locus, where 53 are expected by chance. A cut-offof 1.6 × 10−4 yields 205 genes showing linkage to at least one locus, where 1 is expected by chance. The p-values are calculated according to a permutation scheme in order to capture the dependence between adjacent markers . The above cut-offs correspond to respective thresholds of 5 × 10−5 and 2 × 10−6 when testing every gene-marker combination. Several other p-value cut-offs with similar pieces of information are given throughout the article. Example 4: Finding binding sites of transcriptional regulators. Transcriptional regulatory pro-teins bind to specific promoter sequences to participate in the regulation of gene expression. The availability of complete genome sequences and the development of a method for genome-wide bind-4 ing analysis has allowed the characterization of genomic sites bound by specific transcriptional reg-ulators. Lee et al. used genome-wide location analysis to investigate how yeast transcriptional regulators bind to promoter sequences across the genome. Specifically, binding of 106 transcrip-tional factors was measured across the genome. At each genomic location, a p-value was calculated under the null hypothesis that no binding occurs, resulting in the consideration of thousands of p-values. Lee et al. “generally describe results obtained at a p-value threshold of 0.001 because [their] analysis indicates that this threshold maximizes inclusion of legitimate regulator-DNA in-teractions and minimizes false positives.” They estimate that among the 3985 interactions found to be significant at this threshold, about 6% to 10% are false positives. Reasonable p-value thresholds were sought in each of the four examples. Three of them used four or more cut-offs in an attempt to circumvent the inherent difficulty in interpreting a p-value threshold in a genome-wide study. The significance of the results is consequently obfuscated by the multiple cut-offs that are applied to the p-values. Two pieces of information make such analyses more straightforward and universally interpretable. The first is an estimate of the overall proportion of features that are truly alternative, even if these cannot be precisely identified. For example, what proportion of the 3226 genes in Example 1 are differentially expressed? The second is a measure of significance that can be associated with each feature so that thresholding these numbers at a particular value has an easy interpretation. We provide both of these in our proposed approach. Note that in Example 1, one could just as well work with the modified F-statistic and threshold it directly. This is equivalent to thresholding the p-values described above. The proposed methodology described in terms of the original statistics can be intuitively pleasing for certain cases, proving that p-values are not a necessary intermediate step. However, in other cases, such as Example 2 and Example 3, the test statistics and null distributions are much more complicated, and p-values provide a convenient numerical measure of the strength of evidence against the null for each feature. For this reason, we describe our proposal in terms in p-values rather than test statistics. It is also preferable to present the q-value estimates in terms of p-values to make the method widely applicable. However, working with the original test statistics and null distributions will lead to the same q-value estimates . Proposed Method and Results The dilemma of how to consider, say, m p-values is seen more clearly by considering the various outcomes that occur when a significance threshold is applied to them. Table 1 lists these outcomes: specifically, F is the number of false positives, T is the number of true positives, and S is the total number of features called significant. Also, m0 is the number of truly null features in the 5 study, and m1 = m −m0 is the number of truly alternative features. These quantities can be used to form an overall error measure for any given p-value cut-off. Regardless of whether the p-value threshold is fixed or data-dependent, the quantities F, T, and S are random variables. Therefore, it is common statistical practice to write the overall error measure in terms of an expected value, which we denote by E[·]. Table 1 about here. If the false positive rate is the error measure used, then a simple p-value threshold is employed. A p-value threshold of 0.05, for example, guarantees only that the expected number of false positives is E[F] ≤0.05 · m. This number is much too large for all the examples we have considered, and the false positive rate is too liberal. The error measure that is typically controlled in genome scans for linkage is the family-wise error rate, which can be written as Pr(F ≥1). (Note that we can guarantee that Pr(F ≥1) ≤α by calling all features significant with p-values less that or equal to α/m, which is the well known Bonferroni correction.) Controlling Pr(F ≥1) is practical when very few features are expected to be truly alternative (e.g., in the linkage case) since any false positive can lead to a large waste of time. However, the family-wise error rate is much too conservative for many of the genome-wide studies currently being performed, including the four examples we considered where many features are expected to be truly alternative. It is therefore useful to find an error measure in between these, specifically, one that provides a sensible balance between the number of false positive features F and the number of true positive features T. This balance can efficiently be achieved by considering the ratio #false positive features #significant features = F F + T = F S , which can be stated in words as the proportion of false positive features among all of those called significant. We are particulary interested in the false discovery rate, which is defined to be the expected value of this quantity: FDR = E  F F + T  = E F S  . To be completely rigorous, there is the possibility that S = 0 in which case F/S is undefined, so some adjustment has to be made to this definition (see Remark A in the Appendix). The FDR can also be written in terms of the well known specificity, (m0 −F)/m0, and sensitivity, T/m1: FDR = E  m0 · [1 −specificity] m0 · [1 −specificity] + m1 · sensitivity  . 6 It is clear that the false discovery rate is a useful measure of the overall accuracy of a set of significant features for the examples we described and for many other genome-wide studies. But one would also like a measure of significance that can be attached to each individual feature. The q-value is a measure designed to reflect this. Suppose that we list the features in order of their evidence against the null hypothesis. It is practical to arrange the features in this way since calling one feature significant means that any other feature with more evidence against the null should also be called significant. Hence we list the features from smallest to largest p-value. If a threshold value is chosen, we call all features significant up through that threshold. The q-value for a particular feature is the expected proportion of false positives incurred when calling that feature significant. Therefore, calculating the q-values for each feature and thresholding them at q-value level α pro-duces a set of significant features so that a proportion of α are expected to be false positives. Typically the p-value is described as the probability of a null feature being as or more extreme than the observed one. “As or more extreme” in the above set-up means that it would appear higher on the list. The q-value of a particular feature can be described as the expected proportion of false positives among all features as or more extreme than the observed one. The q-value has a special probabilistic relationship to the p-value (yielding the origin of its name) that is briefly explained in Remark A of the Appendix. As a concrete example, we considered the data from to identify genes that are differentially expressed between BRCA1-mutation-positive tumors and BRCA2-mutation-positive tumors. Using a two-sample t-statistic, we calculated a p-value for each of 3170 genes under the null hypothesis of no differential gene expression. See Remark C in the Appendix for specific details. Figure 1 shows a density histogram of the 3170 p-values. The dashed line is the density we would expect if all genes were null (not differentially expressed), so it can be seen that many genes are differentially expressed. Figure 1 about here. Given the definition of the q-value, it makes sense to begin by estimating the FDR when calling all features significant whose p-value is less than or equal to some threshold t, where 0 < t ≤1. Denote the m p-values by p1, p2, . . . , pm, and let F(t) = #{null pi ≤t; i = 1, . . . , m} and S(t) = #{pi ≤t; i = 1, . . . , m}. 7 We then want to estimate FDR(t) = E F(t) S(t)  . Since we are considering many features (i.e., m is very large), it can be shown that FDR(t) = E F(t) S(t)  ≈E[F(t)] E[S(t)] . (1) A simple estimate of E[S(t)] is the observed S(t); that is, the number of observed p-values less than or equal to t. In estimating E[F(t)], recall that p-values corresponding to truly null hypotheses should be uniformly distributed∗. Therefore, the probability a null p-value is less than or equal to t is simply t, and it follows from Table 1 that E[F(t)] = m0 · t. Since the total number of truly null features m0 is unknown it has to be estimated. Equivalently one can estimate the (more interpretable) proportion of features that are truly null, which we denote by π0 ≡m0/m. It is difficult to estimate π0 without specifying the distribution of the truly alternative p-values. However, exploiting the fact that null p-values are uniformly distributed, a reasonable estimate can be formed. From Figure 1 we can see that the histogram density of p-values beyond 0.5 looks fairly flat, which indicates that there are mostly null p-values in this region. The height of this flat portion actually gives a conservative estimate of the overall proportion of null p-values. This can be quantified with b π0(λ) = #{pi > λ; i = 1, . . . , m} m(1 −λ) , which involves the tuning parameter λ. Setting λ = 0.5, we estimate that 67% of the genes in the data are not differentially expressed. Note that through significance tests, prediction models, and various other techniques, it has been qualitatively argued that BRCA1-mutation-positive tumors and BRCA2-mutation-positive tumors can be distinguished by their genetic profiles . Our estimate of 67% provides a direct measurement of this: we estimate that at least 33% of the examined genes are differentially expressed between these two tumor types. Using traditional p-value cut-offs, Hedenfalk et al. were only comfortable with concluding that 9 to 11 genes are differentially expressed out of over 3000. The rationale behind the estimate of π0 is that p-values of truly alternative features will tend to be close to zero, whereas p-values of null features will be uniformly distributed among [0, 1]. “Most” of the p-values we observe near 1 will be null then. If we were able to count only null p-values, then #{null pi>λ} m(1−λ) would be an unbiased estimate of π0. The inclusion of a few alternative p-values only makes this estimate conservative. If we take λ = 0, then b π0(λ) = 1, which is usually going ∗If the null p-values are not uniformly distributed, then one wants to err in the direction of overestimating p-values (i.e, underestimating significance). Correctly calculated p-values is important assumption underlying our methodology. See also Remark D in the Appendix. 8 to be much too conservative in genome-wide data sets, where a sizable proportion of features are expected to be truly alternative. However, as we set λ closer to 1 the variance of b π0(λ) increases, making the estimated q-values more unreliable. By examining the data in Figure 1, a common sense choice for λ was λ = 0.5. In general, it is useful to automate this choice. We introduce a novel and fully automated method in Remark B of the Appendix for estimating π0 that borrows strength across a range of b π0(λ). This automated method also happens to result in b π0 = 0.67. By plugging these quantities into the right hand side of 1, FDR(t) is estimated by [ FDR(t) = b π0m · t S(t) = b π0m · t #{pi ≤t}. The more mathematical definition of the q-value is the minimum FDR that can be attained when calling that feature significant (see Remark A in the Appendix). Thus, the q-value of feature i is mint≥pi FDR(t), where we have simply considered all thresholds t ≥pi. We can estimate the q-value of feature i by simply plugging [ FDR(t) into the above definition: b q(pi) = min t≥pi [ FDR(t). Note that this guarantees that the estimated q-values are increasing in the same order as the p-values. This method is presented in an easily implemented and fully automated algorithm in Remark B of the Appendix. We mention two mathematical results concerning the accuracy of the estimated q-values that hold for large m under what we call “weak dependence” of the p-values (or features). Weak dependence can loosely be described as any form of dependence whose effect becomes negligible as the number of features increases to infinity (see Remark D in the Appendix and ref. ). The first result is the following: if we call all features significant with q-values less than or equal to α, then for large m the false discovery rate will be less than or equal to α. The second result is that the estimated q-values are simultaneously conservative for the true q-values. This means that the estimated q-value of each feature is greater than or equal to its true q-value, across all features at once. Under this result, one can consider each feature’s significance simultaneously without worrying about inducing bias. In a sense, the second result implies that one can consider all α cut-offs simultaneously, which is a much stronger generalization of the first result. These conservative properties are desirable because one does not want to underestimate the true q-values or the true proportion of false positives. We hypothesize that the most likely form of dependence between features in a genome-wide data set will meet the weak dependence requirement, although this has to be considered for each application. Specifically for DNA microarray data, we argue that since genes behave dependently in small groups (i.e., pathways), with each group essentially being independent of the others, then the dependence can be modeled in blocks in such a way to satisfy 9 the mathematical conditions. More specific details of these mathematical results can be found in Remark D of the Appendix. Given this potentially valuable theoretical justification for considering all q-values simultane-ously, even in the presence of weak dependence, it is possible to use several plots to calibrate the q-value cut-offone would want to apply in a study. (On the other hand, a single cut-offis not always necessary; each feature’s estimated q-value could simply be reported.) Figure 2a shows a plot of the q-values versus their t-statistics from the data. Figure 2b is a plot of the q-values versus their p-values. One can see the expected proportion of false positives for different p-value cut-offs from this plot. Figure 2c shows the number of significant genes for each q-value. Notice that for estimated q-values slightly greater than 0.02, there is a sharp increase in the number of significant genes over a small increase in q-value. This allows one to easily see that a slightly larger q-value cut-offresults in many more significant genes. Finally, Figure 2d shows the expected number of false positives as a function of the number of genes called significant. In general, these last three plots can be used concurrently to give the researcher a comprehensive view of what features to examine further. Figure 2 about here. In our analysis, thresholding genes with q-values less than 0.05 yields 160 genes significant for differential expression. This means that about 8 of the 160 genes called significant are expected to be false positives. It has been previously been noticed that a large block of genes are over-expressed in BRCA1-mutation-positive tumors, particularly genes involved in DNA repair and apoptosis . We find that 117 of the 160 called significant at q-value level 0.05 are over-expressed in BRCA1-mutation-positive tumors, quantitatively supporting their claim. The 0.05 q-value cut-off is arbitrary, and we do not recommend that this value necessarily be used. Considering particular genes allows us to examine their individual q-values. For example, the MSH2 gene (clone 32790) is the eighth most significant gene for differential expression with a q-value of 0.013 and a p-value of 5.05 × 10−5. This gene is over-expressed in the BRCA1-mutation-positive tumors indicating increased levels of DNA repair . MSH2’s p-value of 5.05×10−5 says that the probability a null (non-differentially expressed) gene would be as or more extreme than MSH2 is 5.05×10−5. But MSH2’s statistic could also be unlikely for a differentially expressed gene. The q-value allows a quantification of this: the estimated q-value for MSH2 is 0.013, meaning that about 0.013 of the genes as or more extreme than MSH2 are false positives. The PDCD5 gene (clone 502369) is the 47th most significant gene with a q-value of 0.022 and p-value of 4.79 × 10−4. This gene, associated with inducing apoptosis , is also over-10 expressed in BRCA1-mutation-positive tumors. The CTGF gene (clone 38393) is the 159th most significant gene for differential expression (q-value = 0.049, p-value=0.0036), and is over-expressed in BRCA2-mutation-positive. Activity of this gene is associated with suppressing apoptosis , further supporting earlier claims . Therefore our results support the previous observation that many genes are over-expressed in BRCA1-mutation-positive tumors, particularly genes involved in DNA repair and apoptosis. A full list of genes with their q-values, p-values and fold-change is available at It is a common mistake to state that the p-value is the probability a feature is a false positive. We stress that the q-value is also not the probability that the feature is a false positive. In the above example MSH2 has q-value equal to 0.013 – this does not imply that MSH2 is a false positive with probability 0.013. Rather, 0.013 is the expected proportion of false positives incurred if we call MSH2 significant. Since the q-value measure includes genes that are possibly much more significant than MSH2, the probability that MSH2 is itself a false positive may be substantially higher. In terms of the false discovery rate approach, this probability can also be thought of as a “local false discovery rate” [8, 3, 24]. Statistical significance involves making a decision between null and alternative hypotheses. When assigning multiple measures of statistical significance, it is necessary to account for the fact that decisions are made for m features simultaneously. The q-value accomplishes this by conditioning on the fact that every feature “as or more extreme” will also be called significant, while a local false discovery rate does not. However, the latter quantity clearly provides very useful information, and ideally one would have both estimates available for the analysis of a genome-wide study. Discussion We have proposed the q-value as a useful false discovery rate based measure of significance for genome-wide studies. The methodology we have proposed is the only methodology theoretically shown to be conservative (over all q-values) in situations plausibly encountered in genomics (see Remark D of the Appendix and ). The proposed methodology is easy to implement and inter-pret, and it is fully automated. The original FDR methodology is too conservative for genomics applications since it assumes π0 = 1. For example, controlling the FDR at 0.03, 0.05, or 0.07 in the expression data finds 80, 160, or 231 significant genes, respectively, using our proposed method. The methodology in only finds 21, 88, or 153, respectively, indicating this earlier method’s esti-mates are too conservative and result in a substantial loss of power. The approach in also forces one to choose a single acceptable FDR level before any data are seen, which is often going to be impractical and too restrictive. 11 The q-value of a particular feature in a genome-wide data set is the expected proportion of false positives incurred when calling that feature significant. One may use the q-values as an exploratory guide for which features to investigate further. One may also take all features with q-values less than or equal to some threshold α to attain a false discovery rate less than or equal to α. Most importantly, a systematic use of q-values in genome-wide tests of significance will yield a clear balance of false positives to true positive results and give a standard measure of significance that can be universally interpreted. The methodology we presented also provides an estimate b π0 of the proportion of features following the null hypothesis. The quantity b π1 = 1 −b π0 estimates a lower bound on the proportion of truly alternative features. For example, among the 3170 genes we examined from , we found that at least 33% are differentially expressed between BRCA1-mutation-positive tumors and BRCA2-mutation-positive tumors. Similar estimates from the other examples we considered would be interesting to compute. The software qvalue can be downloaded at This program takes a list of p-values and computes their q-values and b π0. A version of Figure 2 is also generated. Appendix Remark A. FDR, pFDR, and the Q-value In this article, we have used “false discovery rate” and FDR = E[F/S] somewhat loosely. It will almost always be the case that S = 0 with positive probability, which implies that E[F/S] is undefined. The quantity E[F/S|S > 0] · Pr(S > 0) was proposed as a solution to this problem , which is the result of setting F/S = 0 whenever S = 0 in the original E[F/S]. This quantity is technically called the false discovery rate (FDR) in the statistics literature. In our case we want to place a measure of significance on each feature, which is done under the assumption that the feature is called significant. Thus, the inclusion of Pr(S > 0) is somewhat awkward. An alternative quantity, called the positive false discovery rate (pFDR), was recently proposed , which is simply defined as pFDR = E[F/S|S > 0]. The q-value is most technically defined as the minimum pFDR at which the feature can be called significant . Since m is large in genome-wide studies, we have that Pr(S > 0) ≈1 and FDR ≈pFDR ≈E[F]/E[S], so the distinction is not crucial here. Also, the estimate we use is easily motivated for either quantity [4,10]. Suppose that each feature’s statistic probabilistically follows a random mixture of a null dis-tribution and an alternative distribution. Then under a fixed significance rule, the pFDR can be written as Pr(feature i is truly null | feature i is significant), for any i = 1, . . . , m . Similarly, the false positive rate can be written as Pr(feature i is significant | feature i is truly null), for any i = 1, 2, . . . , m. Notice that the similarity between the pFDR and false positive rate – the argu-12 ments have simply been swapped in the conditional probabilities. This connection is the motivation for calling our proposed quantity “q-value.” Indeed, the p-value of a feature is technically defined to be the minimum possible false positive rate when calling that feature significant . Likewise, the q-value is based on the minimum possible pFDR. Remark B. General Algorithm for Estimating Q-values There is a trade-offbetween bias and variance in choosing the λ to use in b π0(λ). For well formed p-values, it should be the case that the bias of b π0(λ) decreases with increasing λ, the bias being the smallest when λ →1 . Therefore, the method we use here is to estimate limλ→1 b π0(λ) = b π0(1). In doing so, we will borrow strength across the b π0(λ) over a range of λ, giving an implicit balance between bias and variance. Consider Figure 3, where we have plotted b π0(λ) versus λ for λ = 0, 0.01, 0.02, . . . , 0.95. By fitting a natural cubic spline to these data (solid line), we have estimated the overall trend of b π0(λ) as λ increases. We purposely set the degrees of freedom of the natural cubic spline to 3; this means we limit its curvature to be like a quadratic function, which is suitable for our purposes. It can be seen from Figure 3 that the natural cubic spline fits the points quite well. The natural cubic spline evaluated at λ = 1 is our final estimate of π0. For a variety of simulations and forms of dependence (data not shown), this method performed well, often eliminating all bias in b π0. Figure 3 about here. The following is the general algorithm for estimating q-values from a list of p-values. 1. Let p(1) ≤p(2) ≤. . . ≤p(m) be the ordered p-values. This also denotes the ordering of the features in terms of their evidence against the null hypothesis. 2. For a range of λ, say R = {0, 0.01, 0.02, . . . , 0.95}, calculate b π0(λ) = #{pj > λ} m(1 −λ) . 3. Let ˆ f be the natural cubic spline with 3 degrees of freedom of b π0(λ) on λ. 4. Set the estimate of π0 to be b π0 = ˆ f(1). 5. Calculate b q(p(m)) = min t≥p(m) b π0m · t #{pj ≤t} = b π0 · p(m). 13 6. For i = m −1, m −2, . . . , 1, calculate b q(p(i)) = min t≥p(i) b π0m · t #{pj ≤t} = min b π0m · p(i) i , b q(p(i+1))  . 7. The estimated q-value for the ith most significant feature is b q(p(i)). Remark C. Analysis of the Hedenfalk et al. Data The data from can be obtained at Supplement/. The data consist of 3226 genes on n1 = 7 BRCA1 arrays and n2 = 8 BRCA2 arrays, along with some arrays from sporadic breast cancer which we did not use. If any gene had one or more measurement exceeding 20, then this gene was eliminated. A value of 20 is several IQR’s (interquartile range) away from the IQR of all the data, and did not seem trustworthy for this example. This left m = 3170 genes. We tested each gene for differential expression between these two tumor types by using a two-sample t-statistic. Let the log2 expression value from the jth array and the ith gene be denoted by xij. Then xi2 = 1 n2 P j∈BRCA2 xij and s2 i2 = 1 n2−1 P j∈BRCA2 (xij −xi2)2 are the sample mean and variance for gene i among the arrays taken from BRCA2 tumors. We can similarly define xi1 and s2 i1 to be the sample mean and variance for the ith gene among the BRCA1 tumor arrays. The two sample t-statistic for the ith gene, allowing for the possibility that the tumors have different variances, is then ti = xi2 −xi1 q s2 i1 n1 + s2 i2 n2 for i = 1, 2, . . . , 3170. We next calculated null versions of t1, t2, . . . , t3170 when there is no differential gene expression. Since it is not clearly valid to assume that the ti follow a t distribution, we calculate these by a permutation method. Consider all possible ways to assign n = 15 arrays to n1 = 7 arrays from BRCA1 and n2 = 8 arrays from BRCA2. Under the assumption that there is no differential gene expression, the t-statistic should have the same distribution regardless of how we make these assignments. Specifically, the labels on the arrays are randomly scrambled, and the t-statistics are recomputed. Therefore, for B = 100 permutations of the array labels we get a set of null statistics t0b 1 , . . . , t0b 3170, b = 1, . . . B. The p-value for gene i, i = 1, 2, . . . , 3170 was calculated by pi = B X b=1 #{j : |t0b j | ≥|ti|, j = 1, . . . , 3170} 3170 · B . We estimated the q-values for differential gene expression between the BRCA1 and BRCA2 tumors using the above algorithm. All results, including the computer code used to analyze the data can be found at 14 Remark D. Theoretical Properties Several mathematical results hold under “weak dependence” of the p-values (or features in the genome). These mathematical results indicate that our method yields conservative q-value esti-mates. The conservative property is desirable because one does not want to underestimate the true q-values (for the same reason one would not want to underestimate a p-value). Suppose that with probability 1, we have S(t)/m →G(t) and F(t)/m0 →G0(t) for each t ∈[0, 1] as m →∞, where G and G0 are continuous functions. In words, this says that the empirical distribution functions of the observed p-values and null p-values converge point-wise to some continuous functions. Weak dependence is defined as dependence that allows this pointwise convergence. (As a rule of thumb, the more local the dependence is, the more likely it is to meet the weak the dependence criterion.) Also suppose that G0(t) ≤t (i.e., uniform distribution or more conservative), and that m0/m converges. If we constrain b π0 ≥minλ∈R b π0(λ) (which should usually be the case), then it can be shown that for any δ > 0, lim m→∞min pi≥δ [b q(pi) −q-value(pi)] ≥0. This means that the estimated q-values are simultaneously conservative for the true q-values, even when taking the worst case scenario over [δ, 1] for arbitrarily small δ. Also, we can conclude that lim m→∞ #{false positive b q(pi) ≤α} #{b q(pi) ≤α} ≤α which means that if we call all genes with q-values less than or equal to α, then in the long run the false discovery rate will be less than or equal to α. The proofs of these claims follow from minor modifications to some of the main results in . References Morton, N. E. (1955) Am. J. Hum. Gen. 7, 277–318. Lander, E. S. & Kruglyak, L. (1995) Nat. Genetics 11, 241–247. Storey, J. D. (2001) The positive false discovery rate: A Bayesian interpretation and the q-value. Ann. Stat., in press. Storey, J. D. (2002) J. Roy. Stat. Soc., Ser. B 64, 479–498. Benjamini, Y. & Hochberg, Y. (1995) J. Roy. Stat. Soc., Ser. B 85, 289–300. Yekutieli, D. & Benjamini, Y. (1999) J. Stat. Plan. Inf. 82, 171–196. 15 Benjamini, Y. & Hochberg, Y. (2000) J. Ed. Behav. Stat. 25, 60–83. Efron, B., Tibshirani, R., Storey, J. D., & Tusher, V. (2001) J. Am. Stat. Assoc. 96, 1151–1160. Genovese, C. & Wasserman, L. (2002) J. Roy. Stat. Soc., Ser. B 64, 499–517. Storey, J. D., Taylor, J. E., & Siegmund, D. (2002) Strong control, conservative point estima-tion, and simultaneous conservative consistency of false discovery rates: A unified approach. J. Roy. Stat. Soc., Ser. B, in press. Tzeng, J. Y., Byerley, W., Devlin, B., Roeder, K., & Wasserman, L. (2003) J. Am. Stat. Assoc. 98, 236–246. Tusher, V., Tibshirani, R., & Chu, C. (2001) Proc. Natl. Acad. Sci. USA 98, 5116–5121. Tatusov, R. L., Altschul, S. F., & Koonin, E. V. (1994) Proc. Natl. Acad. Sci. USA 91, 12091–12095. Hedenfalk, I., Duggan, D., Chen, Y. D., Radmacher, M., Bittner, M., Simon, R., Meltzer, P., Gusterson, B., Esteller, M., Kallioniemi, O. P., Wilfond, B., Borg, A., & Trent, J. (2001) N. Engl. J. Med. 344, 539–548. Slonim, D. K. (2002) Nat. Genetics 32, 502–508 (supp). Blencowe, B. J. (2000) Trends Bioch. Sci. 25, 106–110. Fairbrother, W. G., Yeh, R. F., Sharp, P. A., & Burge, C. B. (2002) Science 297, 1007–1013. Brem, R. B., Yvert, G., Clinton, R., & Kruglyak, L. (2002) Science 296, 752–755. Churchill, G. A. & Doerge, R. W. (1994) Genetics 138, 963–971. Lee, T. I., Rinaldi, N. J., Robert, F., Odom, D. T., Bar-Joseph, Z., Gerber, G. K., Hannett, N. M., Harbison, C. T., Thompson, C. M., Simon, I., Zeitlinger, J., Jennings, E. G., Murray, H. L., Gordon, D. B., Ren, B., Wyrick, J. J., Tagne, J. B., Volkert, T. L., Fraenkel, E., Gifford, D. K., & Young, R. A. (2002) Science 298, 799–804. Kolodner, R. (1996) Genes Dev. 10, 1433–1442. Liu, H. T., Wang, Y. G., Zhang, Y. M., Song, Q. S., Di, C. H., Chen, G. H., Tang, J., & Ma, D. L. (1999) Bioch. Biophys. Res. Comm. 254, 203–210. Hishikawa, K., Oemar, B. S., Tanner, F. C., Nakaki, T., Luscher, T. F., & Fujii, T. (1999) J. Biol. Chem. 274, 37461–37466. 16 Efron, B. & Tibshirani, R. (2002) Gen. Epi. 23, 70–86. Lehmann, E. L. (1986) Testing Statistical Hypotheses. (Springer-Verlag, New York), Second edition. 17 Table and Figure Captions Table 1: Possible outcomes from thresholding m features for significance. Figure 1: A density histogram of the 3170 p-values from the Hedenfalk et al. data. The dashed line is the density histogram we would expect if all genes were null (not differentially expressed). The dotted line is at the height of our estimate of the proportion of null p-values. Figure 2: Results from the Hedenfalk et al. data. (a) The q-values of the genes versus their respective t-statistics. (b) The q-values versus their respective p-values. (c) The number of genes occurring on the list up through each q-value versus the respective q-value. (d) The expected number of false positive genes versus the total number of significant genes given by the q-values. Figure 3: The b π0(λ) versus λ for the Hedenfalk et al. data. The solid line is a natural cubic spline fit to these points to estimate b π0(λ = 1). 18 Table 1: Called Called Significant Not Significant Total Null True F m0 −F m0 Alternative True T m1 −T m1 Total S m −S m 19 Density of observed p−values 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 Figure 1: 20 −8 −6 −4 −2 0 2 4 6 0.0 0.2 0.4 0.6 a t−statistics q−values 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 b p−values q−values 0.00 0.02 0.04 0.06 0.08 0.10 0 50 150 250 350 c q−value number of significant genes 0 100 200 300 400 500 0 20 40 60 d number of significant genes number of expected false positives Figure 2: 21 0.0 0.2 0.4 0.6 0.8 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 PSfrag replacements λ π0  λ Figure 3: 22
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https://www.cs.virginia.edu/~robins/cs302/Sipser_2006_Second_Edition_Problems.pdf
INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION MICHAEL SIPSER Massachusetts Institute of Technology THOMSON COURSE TECHNOLOGY Australia Canada Mexico Singapore Spain United Kingdom United States THOIVISON COURSE TECHNOLOGY Introduction to the Theory of Computation, Second Edition by Michael Sipser Senior Product Manager: Alyssa Pratt Executive Editor: Mac Mendelsohn Associate Production Manager: Aimee Poirier Senior Marketing Manager: Karen Seitz COPYRIGHT © 2006 Thomson Course Technology, a division of Thomson Learning, Inc. Thomson LearningTM is a trademark used herein under license. Printed in the United States of America 1 2 3 45 67 89 QWT 0908070605 For more information, contact Thomson Course Technology 25 Thomson Place Boston, Massachusetts, 02210. Or find us on the World Wide Web at: www.course.com ALL RIGHTS RESERVED. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means graphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, or information storage and retrieval systems without the written permission of the publisher. Associate Product Manager: Mirella Misiaszek Editorial Assistant: Jennifer Smith Senior Manufacturing Coordinator: Trevor Kallop Cover Designer: Steve Deschesne For permission to use material from this text or product, submit a request online at Any additional questions about permissions can be submitted by e-mail to thomsonrights~thomson.com Disclaimer Thomson Course Technology reserves the right to revise this publication and make changes from time to time in its content without notice. ISBN 0-534-95097-3 CONTENTS Preface to the First Edition xi To the student .... ...................................... xi To the educator ..................................... xii The first edition ..... .................................... xiii Feedback to the author ............................... xiii Acknowledgments ...... .................................. xiv Preface to the Second Edition xvii 0 Introduction 1 0.1 Automata, Computability, and Complexity . . . . . . . . . . . . . 1 Complexity theory ................................... 2 Computability theory ................................. 2 Automata theory ..................................... 3 0.2 Mathematical Notions and Terminology ..................... 3 Sets ... .............................................. 3 Sequences and tuples ..... .................................. 6 Functions and relations ..... ................................ 7 G raphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Strings and languages . . . . . . . . . . . . . . . . . . . . . . . 13 Boolean logic .......... ..................... 14 Summary of mathematical terms . . . . . . . . . . . . . . . . . 16 0.3 Definitions, Theorems, and Proofs .. ........................ 17 Finding proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 0.4 Types of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Proof by construction .......................... 221 Proof by contradiction . . . . . . . . . . . . . . . . . . . . . . . 21 Proof by induction .............................. 22 Exercises, Problems, and Solutions ....................... .. 25 v Vi CONTENTS Part One: Automata and Languages 1 Regular Languages 1.1 Finite Automata.................... Formal definition of a finite automaton ...... Examples of finite automata .............. Formal definition of computation .......... Designing finite automata ............... The regular operations .................. 1.2 Nondeterminism ....................... Formal definition of a nondeterministic finite aut Equivalence of NFAs and DFAs ........... Closure under the regular operations ........ 1.3 Regular Expressions ..................... Formal definition of a regular expression .... Equivalence with finite automata .......... 1.4 Nonregular Languages ................. The pumping lemma for regular languages . . . Exercises, Problems, and Solutions ............. 2 Context-Free Languages 2.1 Context-free Grammars ................... Formal definition of a context-free grammar . . Examples of context-free grammars ......... Designing context-free grammars .......... Ambiguity . . . . . . . . . . . . . . . . . . . . . Chomsky normal form .................. 2.2 Pushdown Automata ..................... Formal definition of a pushdown automaton . . . Examples of pushdown automata .......... Equivalence with context-free grammars ...... 2.3 Non-context-free Languages ............... The pumping lemma for context-free languages. Exercises, Problems, and Solutions ............. Part Two: Computability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . tomaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 31 31 35 37 40 41 44 47 53 54 58 63 64 66 77 77 82 99 100 102 103 104 105 106 109 111 112 115 123 123 128 135 3 The Church-Turing Thesis 137 3.1 Turing Machines ........................... 137 Formal definition of a Turing machine ................. 139 Examples of Turing machines ........................ .142 3.2 Variants of Turing Machines . . . . . . . . . . . . . . . . . . . . . 148 Multitape Turing machines ........................ .148 Nondeterministic Turing machines ...................... . 150 Enumerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 CONTENTS Vii Equivalence with other models ..... ......................... 153 3.3 The Definition of Algorithm ..... ............................ 154 Hilbert's problems .................................. 154 Terminology for describing Turing machines . . . . . . . . . . 156 Exercises, Problems, and Solutions .......................... 159 4 Decidability 165 4.1 Decidable Languages ................................. 166 Decidable problems concerning regular languages ........... 166 Decidable problems concerning context-free languages . . . . . 170 4.2 The Halting Problem .... ................................. 173 The diagonalization method . . . . . . . . . . . . . . . . . . . 174 The halting problem is undecidable . . . . . . . . . . . . . . . 179 A Turing-unrecognizable language . . . . . . . . . . . . . . . . 181 Exercises, Problems, and Solutions .......................... 182 5 Reducibility 187 5.1 Undecidable Problems from Language Theory .............. 188 Reductions via computation histories .................... 192 5.2 A Simple Undecidable Problem .......................... 199 5.3 Mapping Reducibility . . . . . . . . . . . . . . . . . . . . . . . . 206 Computable functions . . . . . . . . . . . . . . . . . . . . . . . 206 Formal definition of mapping reducibility . ................ 207 Exercises, Problems, and Solutions .......................... 211 6 Advanced Topics in Computability Theory 217 6.1 The Recursion Theorem . . . . . . . . . . . . . . . . . . . . . . . 217 Self-reference ................................... .218 Terminology for the recursion theorem . . . . . . . . . . . . . 221 Applications ....................................... . 222 6.2 Decidability of logical theories ..... .......................... 224 A decidable theory ................................. 227 An undecidable theory ............................... 229 6.3 Turing Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . 232 6.4 A Definition of Information . . . . . . . . . . . . . . . . . . . . . 233 Minimal length descriptions ..... .......................... 234 Optimality of the definition . . . . . . . . . . . . . . . . . . . . 238 Incompressible strings and randomness ... .................. 239 Exercises, Problems, and Solutions .......................... 242 Part Three: Complexity Theory 245 7 Time Complexity 247 7.1 Measuring Complexity .................................... . 247 Big-O and small-o notation ........................... 248 Viii CONTENTS Analyzing algorithms ............................ .. 251 Complexity relationships among models ... .................. 254 7.2 The Class P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Polynomial time . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Examples of problems in P ........................... .258 7.3 The Class NP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Examples of problems in NP ........................... 267 The P versus NP question ........................... .269 7.4 NP-completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Polynomial time reducibility ........................... 272 Definition of NP-completeness ......................... .276 The Cook-Levin Theorem . . . . . . . . . . . . . . . . . . . . 276 7.5 Additional NP-complete Problems ....................... 283 The vertex cover problem . . . . . . . . . . . . . . . . . . . . . 284 The Hamiltonian path problem ... ........................ 286 The subset sum problem ............................. 291 Exercises, Problems, and Solutions .......................... 294 8 Space Complexity 303 8.1 Savitch's Theorem ................................. .305 8.2 The Class PSPACE ................................. .308 8.3 PSPACE-completeness . . . . . . . . . . . . . . . . . . . . . . . 309 The TQBF problem . . . . . . . . . . . . . . . . . . . . . . . . 310 Winning strategies for games . . . . . . . . . . . . . . . . . . . 313 Generalized geography .. .............................. 315 8.4 The Classes L and NL ................................. 320 8.5 NL-completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Searching in graphs ................................. 325 8.6 NL equals coNl ................................. .. 326 Exercises, Problems, and Solutions .......................... 328 9 Intractability 335 9.1 Hierarchy Theorems .................................. 336 Exponential space completeness ..... ........................ 343 9.2 Relativization ..................................... 348 Limits of the diagonalization method ... .................... 349 9.3 Circuit Complexity ................................. 351 Exercises, Problems, and Solutions .......................... 360 10 Advanced topics in complexity theory 365 10.1 ApproximationAlgorithms ............................. 365 10.2 Probabilistic Algorithms ............................... 368 The class BPP ................................. .. 368 Prim ality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Read-once branching programs ............................ . 376 10.3 Alternation ...................................... 380 Alternating time and space . . . The Polynomial time hierarchy . 10.4 Interactive Proof Systems ...... Graph nonisomorphism ...... Definition of the model ...... IP = PSPACE ............. 10.5 Parallel Computation....... Uniform Boolean circuits .... The class NC ............. P-completeness . . . . . . . . . 10.6 Cryptographyh.y............ Secret keys ............... Public-key cryptosystems . ... One-way functions ......... Trapdoor functions ......... Exercises, Problems, and Solutions . . Selected Bibliography Index CONTENTS IX 381 386 387 387 388 390 399 400 402 404 405 405 407 407 409 411 415 421 PREFACE TO THE FIRST EDITION TO THE STUDENT Welcome! You are about to embark on the study of a fascinating and important subject: the theory of computation. It comprises the fundamental mathematical proper-ties of computer hardware, software, and certain applications thereof. In study-ing this subject we seek to determine what can and cannot be computed, how quickly, with how much memory, and on which type of computational model. The subject has obvious connections with engineering practice, and, as in many sciences, it also has purely philosophical aspects. I know that many of you are looking forward to studying this material but some may not be here out of choice. You may want to obtain a degree in com-puter science or engineering, and a course in theory is required-God knows why. After all, isn't theory arcane, boring, and worst of all, irrelevant? To see that theory is neither arcane nor boring, but instead quite understand-able and even interesting, read on. Theoretical computer science does have many fascinating big ideas, but it also has many small and sometimes dull details that can be tiresome. Learning any new subject is hard work, but it becomes easier and more enjoyable if the subject is properly presented. My primary ob-jective in writing this book is to expose you to the genuinely exciting aspects of computer theory, without getting bogged down in the drudgery. Of course, the only way to determine whether theory interests you is to try learning it. xi Xii PREFACE TO THE FIRST EDITION Theory is relevant to practice. It provides conceptual tools that practition-ers use in computer engineering. Designing a new programming language for a specialized application? What you learned about grammars in this course comes in handy. Dealing with string searching and pattern matching? Rememberfinite automata and regular expressions. Confronted with a problem that seems to re-quire more computer time than you can afford? Think back to what you learned about NP-completeness. Various application areas, such as modern cryptographic protocols, rely on theoretical principles that you will learn here. Theory also is relevant to you because it shows you a new, simpler, and more elegant side of computers, which we normally consider to be complicated ma-chines. The best computer designs and applications are conceived with elegance in mind. A theoretical course can heighten your aesthetic sense and help you build more beautiful systems. Finally, theory is good for you because studying it expands your mind. Com-puter technology changes quickly. Specific technical knowledge, though useful today, becomes outdated in just a few years. Consider instead the abilities to think, to express yourself clearly and precisely, to solve problems, and to know when you haven't solved a problem. These abilities have lasting value. Studying theory trains you in these areas. Practical considerations aside, nearly everyone working with computers is cu-rious about these amazing creations, their capabilities, and their limitations. A whole new branch of mathematics has grown up in the past 30 years to answer certain basic questions. Here's a big one that remains unsolved: If I give you a large number, say, with 500 digits, can you find its factors (the numbers that di-vide it evenly), in a reasonable amount of time? Even using a supercomputer, no one presently knows how to do that in all cases within the lifetime of the universe! The factoring problem is connected to certain secret codes in modern cryptosys-tems. Find a fast way to factor and fame is yours! TO THE EDUCATOR This book is intended as an upper-level undergraduate or introductory gradu-ate text in computer science theory. It contains a mathematical treatment of the subject, designed around theorems and proofs. I have made some effort to accommodate students with little prior experience in proving theorems, though more experienced students will have an easier time. My primary goal in presenting the material has been to make it clear and interesting. In so doing, I have emphasized intuition and "the big picture" in the subject over some lower level details. For example, even though I present the method of proof by induction in Chapter 0 along with other mathematical preliminaries, it doesn't play an im-portant role subsequently. Generally I do not present the usual induction proofs of the correctness of various constructions concerning automata. If presented clearly, these constructions convince and do not need further argument. An in-duction may confuse rather than enlighten because induction itself is a rather sophisticated technique that many find mysterious. Belaboring the obvious with PREFACE TO THE FIRST EDITION Xiii an induction risks teaching students that mathematical proof is a formal manip-ulation instead of teaching them what is and what is not a cogent argument. A second example occurs in Parts Two and Three, where I describe algorithms in prose instead of pseudocode. I don't spend much time programming Turing machines (or any other formal model). Students today come with a program-ming background and find the Church-Turing thesis to be self-evident. Hence I don't present lengthy simulations of one model by another to establish their equivalence. Besides giving extra intuition and suppressing some details, I give what might be called a classical presentation of the subject material. Most theorists will find the choice of material, terminology, and order of presentation consistent with that of other widely used textbooks. I have introduced original terminology in only a few places, when I found the standard terminology particularly obscure or confusing. For example I introduce the term mapping reducibility instead of many-one reducibility. Practice through solving problems is essential to learning any mathemati-cal subject. In this book, the problems are organized into two main categories called Exercises and Problems. The Exercises review definitions and concepts. The Problems require some ingenuity. Problems marked with a star are more difficult. I have tried to make both the Exercises and Problems interesting chal-lenges. THE FIRST EDITION Introduction to the Theory of Computation first appeared as a Preliminary Edition in paperback. The first edition differs from the Preliminary Edition in several substantial ways. The final three chapters are new: Chapter 8 on space complex-ity; Chapter 9 on provable intractability; and Chapter 10 on advanced topics in complexity theory. Chapter 6 was expanded to include several advanced topics in computability theory. Other chapters were improved through the inclusion of additional examples and exercises. Comments from instructors and students who used the Preliminary Edition were helpful in polishing Chapters 0-7. Of course, the errors they reported have been corrected in this edition. Chapters 6 and 10 give a survey of several more advanced topics in com-putability and complexity theories. They are not intended to comprise a cohesive unit in the way that the remaining chapters are. These chapters are included to allow the instructor to select optional topics that may be of interest to the serious student. The topics themselves range widely. Some, such as Turing reducibility and alternation, are direct extensions of other concepts in the book. Others, such as decidable logical theories and cryptography, are brief introductions to large fields. FEEDBACK TO THE AUTHOR The internet provides new opportunities for interaction between authors and readers. I have received much e-mail offering suggestions, praise, and criticism, XIV PREFACE TO THE FIRST EDITION and reporting errors for the Preliminary Edition. Please continue to correspond! I try to respond to each message personally, as time permits. The e-mail address for correspondence related to this book is sipserbookfmath.mit.edu. A web site that contains a list of errata is maintained. Other material may be added to that site to assist instructors and students. Let me know what you would like to see there. The location for that site is ACKNOWLEDGM ENTS I could not have written this book without the help of many friends, colleagues, and my family. I wish to thank the teachers who helped shape my scientific viewpoint and educational style. Five of them stand out. My thesis advisor, Manuel Blum, is due a special note for his unique way of inspiring students through clarity of thought, enthusiasm, and caring. He is a model for me and for many others. I am grateful to Richard Karp for introducing me to complexity theory, to John Addison for teaching me logic and assigning those wonderful homework sets, to Juris Hartmanis for introducing me to the theory of computation, and to my father for introducing me to mathematics, computers, and the art of teaching. This book grew out of notes from a course that I have taught at MIT for the past 15 years. Students in my classes took these notes from my lectures. I hope they will forgive me for not listing them all. My teaching assistants over the years, Avrim Blum, Thang Bui, Andrew Chou, Benny Chor, Stavros Cos-madakis, Aditi Dhagat, Wayne Goddard, Parry Husbands, Dina Kravets, Jakov Kucan, Brian O'Neill, loana Popescu, and Alex Russell, helped me to edit and expand these notes and provided some of the homework problems. Nearly three years ago, Tom Leighton persuaded me to write a textbook on the theory of computation. I had been thinking of doing so for some time, but it took Tom's persuasion to turn theory into practice. I appreciate his generous advice on book writing and on many other things. I wish to thank Eric Bach, Peter Beebee, Cris Calude, Marek Chrobak, Anna Chefter, Guang-len Cheng, Elias Dahlhaus, Michael Fischer, Steve Fisk, Lance Fortnow, Henry J. Friedman, Jack Fu, Seymour Ginsburg, Oded Goldreich, Brian Grossman, David Harel, Micha Hofri, Dung T. Huynh, Neil Jones, H. Chad Lane, Kevin Lin, Michael Loui, Silvio Micali, Tadao Murata, Chris-tos Papadimitriou, Vaughan Pratt, Daniel Rosenband, Brian Scassellati, Ashish Sharma, Nir Shavit, Alexander Shen, Ilya Shlyakhter, Matt Stallmann, Perry Susskind, Y. C. Tay, Joseph Traub, Osamu Watanabe, Peter Widmayer, David Williamson, Derick Wood, and Charles Yang for comments, suggestions, and assistance as the writing progressed. The following people provided additional comments that have improved this book: Isam M. Abdelhameed, Eric Allender, Shay Artzi, Michelle Ather-PREFACE TO THE FIRST EDITION XV ton, Rolfe Blodgett, Al Briggs, Brian E. Brooks, Jonathan Buss, Jin Yi Cai, Steve Chapel, David Chow, Michael Ehrlich, Yaakov Eisenberg, Farzan Fallah, Shaun Flisakowski, Hjalmtyr Hafsteinsson, C. R. Hale, Maurice Herlihy, Vegard Holmedahl, Sandy Irani, Kevin Jiang, Rhys Price Jones, James M. Jowdy, David M. Martin Jr., Manrique Mata-Montero, Ryota Matsuura, Thomas Minka, Farooq Mohammed, Tadao Murata, Jason Murray, Hideo Nagahashi, Kazuo Ohta, Constantine Papageorgiou, Joseph Raj, Rick Regan, Rhonda A. Reumann, Michael Rintzler, Arnold L. Rosenberg, Larry Roske, Max Rozenoer, Walter L. Ruzzo, Sanatan Sahgal, Leonard Schulman, Steve Seiden, Joel Seiferas, Ambuj Singh, David J. Stucki, Jayram S. Thathachar, H. Venkateswaran, Tom Whaley, Christopher Van Wyk, Kyle Young, and Kyoung Hwan Yun. Robert Sloan used an early version of the manuscript for this book in a class that he taught and provided me with invaluable commentary and ideas from his experience with it. Mark Herschberg, Kazuo Ohta, and Latanya Sweeney read over parts of the manuscript and suggested extensive improvements. Shafi Goldwasser helped me with material in Chapter 10. I received expert technical support from William Baxter at Superscript, who wrote the LATEX macro package implementing the interior design, and from Larry Nolan at the MIT mathematics department, who keeps everything run-ning. It has been a pleasure to work with the folks at PWS Publishing in creat-ing the final product. I mention Michael Sugarman, David Dietz, Elise Kaiser, Monique Calello, Susan Garland and Tanja Brull because I have had the most contact with them, but I know that many others have been involved, too. Thanks to Jerry Moore for the copy editing, to Diane Levy for the cover design, and to Catherine Hawkes for the interior design. I am grateful to the National Science Foundation for support provided under grant CCR-9503322. My father, Kenneth Sipser, and sister, Laura Sipser, converted the book di-agrams into electronic form. My other sister, Karen Fisch, saved us in various computer emergencies, and my mother, Justine Sipser, helped out with motherly advice. I thank them for contributing under difficult circumstances, including insane deadlines and recalcitrant software. Finally, my love goes to my wife, Ina, and my daughter, Rachel. Thanks for putting up with all of this. Cambridge, Massachusetts Michael Sipser October, 1996 PREFACE TO THE SECOND EDITION Judging from the email communications that I've received from so many of you, the biggest deficiency of the first edition is that it provides no sample solutions to any of the problems. So here they are. Every chapter now contains a new Selected Solutions section that gives answers to a representative cross-section of that chapter's exercises and problems. To make up for the loss of the solved problems as interesting homework challenges, I've also added a variety of new problems. Instructors may request an Instructor's Manual that contains addi-tional solutions by contacting the sales representative for their region designated at www. course. comn. A number of readers would have liked more coverage of certain "standard" topics, particularly the Myhill-Nerode Theorem and Rice's Theorem. I've par-tially accommodated these readers by developing these topics in the solved prob-lems. I did not include the Myhill-Nerode Theorem in the main body of the text because I believe that this course should provide only an introduction to finite automata and not a deep investigation. In my view, the role of finite automata here is for students to explore a simple formal model of computation as a prelude to more powerful models, and to provide convenient examples for subsequent topics. Of course, some people would prefer a more thorough treatment, while others feel that I ought to omit all reference to (or at least dependence on) finite automata. I did not include Rice's Theorem in the main body of the text be-cause, though it can be a useful "tool" for proving undecidability, some students might use it mechanically without really understanding what is going on. Using xvii XViii PREFACE TO THE SECOND EDITION reductions instead, for proving undecidability, gives more valuable preparation for the reductions that appear in complexity theory. I am indebted to my teaching assistants, Ilya Baran, Sergi Elizalde, Rui Fan, Jonathan Feldman, Venkatesan Guruswami, Prahladh Harsha, Christos Kapout-sis, Julia Khodor, Adam Klivans, Kevin Matulef, loana Popescu, April Rasala, Sofya Raskhodnikova, and Iuliu Vasilescu who helped me to craft some of the new problems and solutions. Ching Law, Edmond Kayi Lee, and Zulfikar Ramzan also contributed to the solutions. I thank Victor Shoup for coming up with a simple way to repair the gap in the analysis of the probabilistic primality algorithm that appears in the first edition. I appreciate the efforts of the people at Course Technology in pushing me and the other parts of this project along, especially Alyssa Pratt and Aimee Poirier. Many thanks to Gerald Eisman, Weizhen Mao, Rupak Majumdar, Chris Umans, and Christopher Wilson for their reviews. I'm indebted to Jerry Moore for his superb job copy editing and to Laura Segel of ByteGraphics (lauras~bytegraphics. com) for her beautifully precise rendition of the fig-ures. The volume of email I've received has been more than I expected. Hearing from so many of you from so many places has been absolutely delightful, and I've tried to respond to all eventually-my apologies for those I missed. I've listed here the people who made suggestions that specifically affected this edition, but I thank everyone for their correspondence. Luca Aceto, Arash Afkanpour, Rostom Aghanian, Eric Allender, Karun Bak-shi, Brad Ballinger, Ray Bartkus, Louis Barton, Arnold Beckmann, Mihir Bel-lare, Kevin Trent Bergeson, Matthew Berman, Rajesh Bhatt, Somenath Biswas, Lenore Blum, Mauro A. Bonatti, Paul Bondin, Nicholas Bone, Ian Bratt, Gene Browder, Doug Burke, Sam Buss, Vladimir Bychkovsky, Bruce Carneal, Soma Chaudhuri, Rong-Jaye Chen, Samir Chopra, Benny Chor, John Clausen, Alli-son Coates, Anne Condon, Jeffrey Considine,JohnJ. Crashell, Claude Crepeau, Shaun Cutts, Susheel M. Daswani, Geoff Davis, Scott Dexter, Peter Drake, Jeff Edmonds, Yaakov Eisenberg, Kurtcebe Eroglu, Georg Essl, Alexander T. Fader, Farzan Fallah, Faith Fich, Joseph E. Fitzgerald, Perry Fizzano, David Ford, Jeannie Fromer, Kevin Fu, Atsushi Fujioka, Michel Galley, K. Gane-san, Simson Garfinkel, Travis Gebhardt, Peymann Gohari, Ganesh Gopalakr-ishnan, Steven Greenberg, Larry Griffith, Jerry Grossman, Rudolf de Haan, Michael Halper, Nick Harvey, Mack Hendricks, Laurie Hiyakumoto, Steve Hockema, Michael Hoehle, Shahadat Hossain, Dave Isecke, Ghaith Issa, Raj D. Iyer, Christian Jacobi, Thomas Janzen, Mike D. Jones, Max Kanovitch, Aaron Kaufman, Roger Khazan, Sarfraz Khurshid, Kevin Killourhy, Seungjoo Kim, Victor Kuncak, Kanata Kuroda, Suk Y. Lee, Edward D. Legenski, Li-Wei Lehman, Kong Lei, Zsolt Lengvarszky, Jeffrey Levetin, Baekjun Lim, Karen Livescu, Thomas Lasko, Stephen Louie, TzerHung Low, Wolfgang Maass, Arash Madani, Michael Manapat, Wojciech Marchewka, David M. Martin Jr., Anders Martinson, Lyle McGeoch, Alberto Medina, Kurt Mehlhorn, Nihar Mehta, Albert R. Meyer, Thomas Minka, Mariya Minkova, Daichi Mizuguchi, G. Allen Morris III, Damon Mosk-Aoyama, Xiaolong Mou, Paul Muir, Ger-PREFACE TO THE SECOND EDITION XIX man Muller, Donald Nelson, Gabriel Nivasch, Mary Obelnicki, Kazuo Ohta, Thomas M. Oleson, Jr., Curtis Oliver, Owen Ozier, Rene Peralta, Alexander Perlis, Holger Petersen, Detlef Plump, Robert Prince, David Pritchard, Bina Reed, Nicholas Riley, Ronald Rivest, Robert Robinson, Christi Rockwell, Phil Rogaway, Max Rozenoer, John Rupf, Teodor Rus, Larry Ruzzo, Brian Sanders, Cem Say, Kim Schioett, Joel Seiferas, Joao Carlos Setubal, Geoff Lee Seyon, Mark Skandera, Bob Sloan, Geoff Smith, Marc L. Smith, Stephen Smith, Alex C. Snoeren, Guy St-Denis, Larry Stockmeyer, Radu Stoleru, David Stucki, Hlisham M. Sueyllam, Kenneth Tam, Elizabeth Thompson, Michel Toulouse, Fric Tria, Chittaranjan Tripathy, Dan Trubow, Hiroki Ueda, Giora Unger, Kurt L. Van Etten, Jesir Vargas, Bienvenido Velez-Rivera, Kobus Vos, Alex Vrenios, Sven Waibel, Marc Waldman, Tom Whaley, Anthony Widjaja, Sean Williams, Joseph N. Wilson, Chris Van Wyk, Guangming Xing, Vee Voon Yee, Cheng Yongxi, Neal Young, Timothy Yuen, Kyle Yung, Jinghua Zhang, Lilla Zollei. Most of all I thank my family-lna, Rachel, and Aaron-for their patience, understanding, and love as I sat for endless hours here in front of my computer screen. Cambridge, 1lassachusetts Michael Sipser December, 2004 EXERCISES 25 Pk. we know that Pk+1 = PkM- Y Therefore, using the induction hypothesis to calculate Pk, Multi g Pkl =[PM ( M I )] Y. Multiplying through by At and rewriting Y yields Pk+1 = PM'~" - Y (M IM (M-= pMk+l _ y (mk+l -I) Thus the formula is correct for t = k + 1, which proves the theorem. ........................................................................................................................................................................ Problem 0.14 asks you to use the preceding formula to calculate actual mort-gage payments. EXERCISES 0.1 Examine the following formal descriptions of sets so that you understand which members they contain. Write a short informal English description of each set. a. {1,3,5,7, ... } b. {..., -4, -2,0,2,4,...} c. {nj n = 2m for some m in AN} d. {nj n = 2m for some m in XA, and n = 3k for some k in AV} e. {wl w is a string of Os and is and w equals the reverse of w} f. {nj n is an integer and n = n + 1} 0.2 Write formal descriptions of the following sets a. The set containing the numbers 1, 10, and 100 b. The set containing all integers that are greater than 5 c. The set containing all natural numbers that are less than 5 d. The set containing the string aba e. The set containing the empty string f. The set containing nothing at all 26 CHAPTER 0/ INTRODUCTION 0.3 Let A be the set {x, y, z} and B be the set {x, y}. a. Is A a subset of B? b. Is B a subset of A? c. What is A U B? d. WhatisA nB? e. What is A x B? f What is the power set of B? 0.4 If A has a elements and B has b elements, how many elements are in A x B? Explain your answer. 0.5 If C is a set with c elements, how many elements are in the power set of C? Explain your answer. 0.6 Let X be the set {1, 2, 3, 4, 5} and Y be the set {6, 7, 8, 9, 10}. The unary function f: X- Y and the binary function g: X x Y- Y are described in the following tables. n f(n) g 6 7 8 9 10 1 6 1 10 10 10 10 10 2 7 2 7 8 9 10 6 3 6 3 7 7 8 8 9 4 7 4 9 8 7 6 10 5 6 5 6 6 6 6 6 a. What is the value of f (2)? b. What are the range and domain off? c. What is the value of g(2, 10)? d. What are the range and domain of g? e. What is the value ofg(4, f(4))? 0.7 For each part, give a relation that satisfies the condition. a. Reflexive and symmetric but not transitive b. Reflexive and transitive but not symmetric c. Symmetric and transitive but not reflexive 0.8 Consider the undirected graph G= (V, E) where V, the set of nodes, is {1, 2,3, 4} and E, the set of edges, is {{1,2}, {2,3}, {1,3}, {2,4}, {1,4}}. Draw the graph G. What is the degree of node 1? of node 3? Indicate a path from node 3 to node 4 on your drawing of G. PROBLEMS 27 0.9 Write a formal description of the following graph. PROBLEMS 0.10 Find the error in the following proof that 2 = 1. Consider the equation a = b. Multiply both sides by a to obtain a 2 ab. Subtract b2 from both sides to get a2 - b2 = ab -b 2 . Now factor each side, (a + b) (a -b) = b(a -b), and divide each side by (a -b), to get a + b = b. Finally, let a and b equal 1, which shows that 2 = 1. 0.11 Find the error in the following proof that all horses are the same color. CLAIM: In any set of h horses, all horses are the same color. PROOF: By induction on h. Basis: For h = 1. In any set containing just one horse, all horses clearly are the same color. Induction step: For k > I assume that the claim is true for h = k and prove that it is true for h = k + 1. Take any set H of k + 1 horses. We show that all the horses in this set are the same color. Remove one horse from this set to obtain the set H 1 with just k horses. By the induction hypothesis, all the horses in H, are the same color. Now replace the removed horse and remove a different one to obtain the set H 2. By the same argument, all the horses in H2 are the same color. Therefore all the horses in H must be the same color, and the proof is complete. 0.12 Show that every graph with 2 or more nodes contains two nodes that have equal degrees. A0. 13 Ramsey's theorem. Let G be a graph. A clique in G is a subgraph in which every two nodes are connected by an edge. An anti-clique, also called an independent set, is a subgraph in which every two nodes are not connected by an edge. Show that every graph with n nodes contains either a clique or an anti-clique with at least 2' log2 n nodes. 28 CHAPTER 0 / INTRODUCTION AO. 1 4 Use Theorem 0.25 to derive a formula for calculating the size of the monthly pay-ment for a mortgage in terms of the principal P, interest rate I, and the number of payments t. Assume that, after t payments have been made, the loan amount is reduced to 0. Use the formula to calculate the dollar amount of each monthly pay-ment for a 3 0-year mortgage with 3 60 monthly payments on an initial loan amount of $100,000 with a 5% annual interest rate. SELECTED SOLUTIONS 0.13 Make space for two piles of nodes, A and B. Then, starting with the entire graph, repeatedly add each remaining node x to A if its degree is greater than one half the number of remaining nodes and to B otherwise, and discard all nodes to which x isn't (is) connected if it was added to A (B). Continue until no nodes are left. At most half of the nodes are discarded at each of these steps, so at least log2 n steps will occur before the process terminates. Each step adds a node to one of the piles, so one of the piles ends up with at least 2 log2 n nodes. The A pile contains the nodes of a clique and the B pile contains the nodes of an anti-clique. 0.14 We let Pt = 0 and solve for Y to get the formula: Y = PMt(MJ - 1)/(M' -1). For P = $100, 000, I = 0.05, and t = 360 we have M = 1 + (0.05)/12. We use a calculator to find that Y $536.82 is the monthly payment. EXERCISES 83 EXERCISES A1 .1 The following are the state diagrams of two DFAs, M1 and M2 . Answer the follow-ing questions about each of these machines. bb b b a Ml M2 a. What is the start state? b. What is the set of accept states? c. What sequence of states does the machine go through on input aabb? d. Does the machine accept the string aabb? e. Does the machine accept the string e? Al .2 Give the formal description of the machines M1 and M2 pictured in Exercise 1.1. 1.3 The formal description of a DFA M is ({ql,q2,q3,q4,q5},{u,d},6,q3,{q3}), where 3 is given by the following table. Give the state diagram of this machine. u d q1 qi q2 q2 qi q3 q3 q2 q4 q4 q3 q5 q5 q4 q5 1.4 Each of the following languages is the intersection of two simpler languages. In each part, construct DFAs for the simpler languages, then combine them using the construction discussed in footnote 3 (page 46) to give the state diagram of a DFA for the language given. In all parts Z = {a, b}. a. {wj en has at least three a's and at least two b's} Ab. {wI w has at exactly two a's and at least two b's} c. {fwl w has an even number of a's and one or two b's} Ad. {wl w has an even number of a's and each a is followed by at least one b} e. {w I w has an even number of a's and one or two b's} f. {wI in has an odd number of a's and ends with a b} g. {w I w has even length and an odd number of a's} CHAPTER I / REGULAR LANGUAGES 1.5 Each of the following languages is the complement of a simpler language. In each part, construct a DFA for the simpler language, then use it to give the state diagram of a DFA for the language given. In all parts E = {a, b}. Aa. {wl w does not contain the substring ab} Ab. {wl w does not contain the substring baba} c. {wl w contains neither the substrings ab nor ba} d. {w| w is any string not in ab } e. {w| w is any string not in (ab+)1 f. {w| w is any string not in a U b} g. { w w is any string that doesn't contain exactly two a's} h. {wI w is any string except a and b} 1.6 Give state diagrams of DFAs recognizing the following languages. In all parts the alphabet is {0,1 } a. {wI w begins with a 1 and ends with a } b. {wl w contains at least three 1s} c. {wl w contains the substring 0101, i.e., w = xOiOiy for some xr and y} d. {w I w has length at least 3 and its third symbol is a 0} e. {wl wi starts with 0 and has odd length, or starts with 1 and has even length} f {wI wi doesn't contain the substring 1101 g. {uw the length of w is at most 5} h. {wl w is any string except 11 and 1111 i. {wj every odd position of w is a ll j. {w I w contains at least two Os and at most one 11 k. {s, 0} 1. {w I w contains an even number of Os, or contains exactly two ls} m. The empty set n. All strings except the empty string 1.7 Give state diagrams of NFAs with the specified number of states recognizing each of the following languages. In all parts the alphabet is {0,1}. Aa. The language {w I w ends with 001 with three states b. The language of Exercise 1.6c with five states c. The language of Exercise 1.61 with six states d. The language {O} with two states e. The language 01 O+ with three states Af The language 1(001+) with three states g. The language {e} with one state h. The language O with one state 1.8 Use the construction given in the proof of Theorem 1.45 to give the state diagrams of NFAs recognizing the union of the languages described in a. Exercises 1.6a and 1.6b. b. Exercises 1.6c and 1.6f. 84 EXERCISES 85 1.9 Use the construction given in the proof of Theorem 1.47 to give the state diagrams of NFAs recognizing the concatenation of the languages described in a. Exercises 1.6g and 1.6i. b. Exercises 1.6b and 1.6m. 1.10 Use the construction given in the proof of Theorem 1.49 to give the state diagrams of NFAs recognizing the star of the language described in a. Exercise 1.6b. b. Exercise 1.6j. c. Exercise 1.6m. A 1.11 Prove that every N FA can be converted to an equivalent one that has a single accept state. 1.12 Let D {wI w contains an even number of a's and an odd number of b's and does not contain the substring ab}. Give a DFA with five states that recognizes D and a regular expression that generates D. (Suggestion: Describe D more simply.) 1.13 Let F be the language of all strings over {0,1} that do not contain a pair of is that are separated by an odd number of symbols. Give the state diagram of a DFA with 5 states that recognizes F. (You may find it helpful first to find a 4-state NFA for the complement of F.) 1.14 a. Show that, if M is a DFA that recognizes language B, swapping the accept and nonaccept states in M yields a new DFA that recognizes the complement of B. Conclude that the class of regular languages is closed under comple-ment. b. Show by giving an example that, if M is an NFA that recognizes language C, swapping the accept and nonaccept states in M doesn't necessarily yield a new NFA that recognizes the complement of C. Is the class of languages recognized by NFAs closed under complement? Explain your answer. 1.15 Give a counterexample to show that the following construction fails to prove The-orem 1.49, the closure of the class of regular languages under the star operation.8 Let N1 = (Qj, E, 61, qj, F 1) recognize Al. Construct N = (Qj, E, 6, qj, F) as follows. N is supposed to recognize A. a. The states of N are the states of N1. b. The start state of N is the same as the start state of N1. c. F ={q}UF,. The accept states F are the old accept states plus its start state. d. Define 6 so that for any q C Q and any a C E, f61(q,a) q Fora & (q a) d(q, a) U {qj} q (E F, and a = e. (Suggestion: Show this construction graphically, as in Figure 1.50.) 81n other words, you must present a finite automaton, N1, for which the constructed automaton N does not recognize the star of N1 's language. 86 CHAPTER 1 / REGULAR LANGUAGES 1.16 Use the construction given in Theorem 1.39 to convert the following two nonde-terministic finite automata to equivalent deterministic finite automata. (a) (b) 1.17 a. Give an NFA recognizing the language (01 U O00 U 010). b. Convert this NFA to an equivalent DFA. Give only the portion of the DFA that is reachable from the start state. 1.18 Give regular expressions generating the languages of Exercise 1.6. 1.19 Use the procedure described in Lemma 1.55 to convert the following regular ex-pressions to nondeterministic finite automata. a. (0 U 1)000(0 U 1) b. (((00)(11)) U 01) C. 0 1.20 For each of the following languages, give two strings that are members and two strings that are not members-a total of four strings for each part. Assume the alphabet E {a,b} in all parts. a. asb e. EaE'baE b. a(ba)b f. aba U bab c. a U b g. (e U a)b d. (aaa) h. (aUbaUbb)E 1.21 Use the procedure described in Lemma 1.60 to convert the following finite au-tomata to regular expressions. a b (a) (b) EXERCISES 87 1.22 In certain programming languages, comments appear between delimiters such as /# and #/. Let C be the language of all valid delimited comment strings. A member of C must begin with /# and end with #/ but have no intervening #V. For simplic-ity, we'll say that the comments themselves are written with only the symbols a and b; hence the alphabet of C is E = {a, b, /I #}. a. Give a DFA that recognizes C. b. Give a regular expression that generates C. A1. 2 3 Let B be any language over the alphabet S. Prove that B = B iff BB C B. 1.24 Afinite state transducer (FST) is a type of deterministic finite automaton whose output is a string and not just accept or reject. The following are state diagrams of finite state transducers T1 and T2. °l/O2/1a/O ai 0/0 i/i i/ 2/1 /0 /0 2 1q b/i q 0/0 a/i T, T2 Each transition of an FST is labeled with two symbols, one designating the input symbol for that transition and the other designating the output symbol. The two symbols are written with a slash, /, separating them. In T1, the transition from q1 to q2 has input symbol 2 and output symbol 1. Some transitions may have multiple input-output pairs, such as the transition in T1 from q1 to itself. When an FST computes on an input string w, it takes the input symbols WV. w, one by one and, starting at the start state, follows the transitions by matching the input labels with the sequence of symbols wi ... w, = w. Every time it goes along a transition, it outputs the corresponding output symbol. For example, on input 2212011, machine T1 enters the sequence of states q1, q2, q2, q2, q2, q1, q1, q1 and produces output 1111000. On input abbb, T2 outputs 1011. Give the sequence of states entered and the output produced in each of the following parts. a. T, on input 0il e. T2 on input b b. Ti on input 211 f. T2 on input bbab c. T, on input 121 g. T2 on input bbbbbb d. T1 on input 0202 h. T2 on input e 1.25 Read the informal definition of the finite state transducer given in Exercise 1.24. Give a formal definition of this model, following the pattern in Definition 1.5 (page 3 5). Assume that an FST has an input alphabet E and an output alphabet r but not a set of accept states. Include a formal definition of the computation of an FST. (Hint: An FST is a 5-tuple. Its transition function is of the form 6: Q x e ~Q x F.) 88 CHAPTER I / REGULAR LANGUAGES 1.26 Using the solution you gave to Exercise 1.25, give a formal description of the ma-chines T1 and T2 depicted in Exercise 1.24. 1.27 Read the informal definition of the finite state transducer given in Exercise 1.24. Give the state diagram of an FST with the following behavior. Its input and output alphabets are {0,1}. Its output string is identical to the input string on the even positions but inverted on the odd positions. For example, on input 0000111 it should output 1010010. 1.28 Convert the following regular expressions to NFAs using the procedure given in Theorem 1.54. In all parts E = {a, b}. a. a(abb) U b b. a' U (ab)+ c. (a U b t )a'b' 1.29 Use the pumping lemma to show that the following languages are not regular. Aa Al {0=10n2 | n > 0} b. A2 = {wwwl w E {a,b}"} Ac. A3 = {a2 6 1 n > 0} (Here, a2" means a string of 2n a's.) 1.30 Describe the error in the following "proof" that 0 1 is not a regular language. (An error must exist because 0 1 is regular.) The proof is by contradiction. Assume that 0 1 is regular. Let p be the pumping length for 0 1 given by the pumping lemma. Choose s to be the string OP P. You know that s is a member of 0 1, but Example 1.73 shows that s cannot be pumped. Thus you have a contradiction. So 0' 1' is not regular. PROBLEMS 1.31 For any string aw iWiW2 .W,,, the reverse of w, written wR, is the string w in reverse order, Wn ... W2Wi 1 . For any language A, let AR = {w'h w E A}. Show that if A is regular, so is ARt. 1.32 Let E3 contains all size 3 columns of Os and is. A string of symbols in E3 gives three rows of Os and is. Consider each row to be a binary number and let B = {w X 31I the bottom row of w is the sum of the top two rows}. For example, [O] [oB] [] e B, but [°] [B] X B. Show that B is regular. (Hint: Working with BR is easier. You may assume the result claimed in Problem 1.31.) PROBLEMS 89 1.33 Let 2 = 0[o , [o]:, Here, E2 contains all columns of Os and is of height two. A string of symbols in E2 gives two rows of Os and is. Consider each row to be a binary number and let C = f 2 I the bottom row of w is three times the top row}. For example, [ [] [ [" ] [c] 6 C, but [ 1 ] [] [ 1 ] C. Show that C is regular. (You may assume the result claimed in Problem 1.31.) 1.34 Let Z2 be the same as in Problem 1.33. Consider each row to be a binary number and let D = {w E E2 I the top row of w is a larger number than is the bottom row}. For example, y [ [O]J [] [ ] E D, but [L] [ ] [°] 0 D. ShowthatDis regular. 1.35 Let E2 be the same as in Problem 1.33. Consider the top and bottom rows to be strings of Os and is and let E = fw e E2 I the bottom row of w is the reverse of the top row of wl. Show that E is not regular. 1.36 Let B, = {ak I where k is a multiple of n}. Show that for each n > 1, the language B, is regular. 1.37 Let Ct = {jx r is a binary number that is a multiple of n}. Show that for each n > 1, the language C. is regular. 1.38 An all-NFA Al is a 5-tuple (Q, Z, 6, qo, F) that accepts x G E if every possible state that M could be in after reading input x is a state from F. Note, in contrast, that an ordinary NFA accepts a string if some state among these possible states is an accept state. Prove that all-NFAs recognize the class of regular languages. 1.39 The construction in Theorem 1.54 shows that every GNFA is equivalent to a GNFA with only two states. We can show that an opposite phenomenon occurs for DFAs. Prove that for every k > I a language Ak C {o,i} exists that is recognized by a DFA with k states but not by one with only k -I states. 1.40 Say that string x is a prefix of string y if a string z exists where xz = y and that x is a proper prefix of y if in addition x =$ y. In each of the following parts we define an operation on a language A. Show that the class of regular languages is closed under that operation. Aa. NOPREFIX(A)= {w E AlnoproperprefixofwisamemberofA}. b. NOEXTEND(A)= {w Al w is not the proper prefix of any string in A}. 1.41 For languages A and B, let the perfect sbuffle of A and B be the language {wl w = aib. akbk, where a, ... ak E A and bi ... bk E B, each aj, bi e El Show that the class of regular languages is closed under perfect shuffle. 1.42 For languages A and B, let the shuffle of A and B be the language {wl w = albi ... akbk, where ai ... ak C A and bi ... bk E B, each aj, bi e CY} Show that the class of regular languages is closed under shuffle. 90 CHAPTER I / REGULAR LANGUAGES 1.43 Let A be any language. Define DROP-OUT(A) to be the language containing all strings that can be obtained by removing one symbol from a string in A. Thus, DROP-OUT(A) ={rzl xzyz C A where x, z E E, y E E}. Show that the class of regular languages is closed under the DROP-OUT operation. Give both a proof by picture and a more formal proof by construction as in Theorem 1.47. A1 .44 Let B and C be languages over Y= {0. I}. Define B C = {w E BI for some y e C, strings w and y contain equal numbers of is}. Show that the class of regular languages is closed under the 2- operation. 1.45 Let A/B = {wl wx E A for some x e B}. Show that if A is regular and B is any language then A/B is regular. 1.46 Prove that the following languages are not regular. You may use the pumping lemma and the closure of the class of regular languages under union, intersection, and complement. a. {OnmOnI m n > 0} Ab. {001 |m:An} c. {wI w E {0,1} is not a palindrome}9 d. {wtwi wot £ {0,1}+} 1.47 Let E = {I, #} and let Y = {wl W = #2#... #Xk for k > 0, each xi e l, and xi :A xj for i 7 j. Prove that Y is not regular. 1.48 Let E = {0,} and let D = {wl w contains an equal number of occurrences of the substrings 01 and 10}. Thus 101 C D because 101 contains a single 01 and a single 10, but 1010 , D because 1010 contains two l0s and one 01. Show that D is a regular language. 1.49 a. Let B = { 5 y y c {0, I} and y contains at least k is, for k > 1}. Show that B is a regular language. b. Let C = { 1 ky y E {0, 1} and y contains at most k is, for k > 1}. Show that C isn't a regular language. A1 .50 Read the informal definition of the finite state transducer given in Exercise 1.24. Prove that no FST can output wuv for every input w if the input and output alpha-bets are {0,1j. 1.51 Let x and y be strings and let L be any language. We say that x and y are distin-guishable by L if some string z exists whereby exactly one of the strings xz and yz is a member of L; otherwise, for every string z, we have xz e L whenever yz e L and we say that x and y are indistinguishable by L. If x and y are indistinguishable by L we write x - L Y. Show that -L is an equivalence relation. 9Apalindrome is a string that reads the same forward and backward. PROBLEMS 91 A1. 52 Myhill-Nerode theorem. Refer to Problem 1.51. Let L be a language and let X be a set of strings. Say that X is pairwise distinguishable by L if every two distinct strings in X are distinguishable by L. Define the index of L to be the maximum number of elements in any set that is pairwise distinguishable by L. The index of L may be finite or infinite. a. Show that, if L is recognized by a DFA with k states, L has index at most k. b. Show that, if the index of L is a finite number k, it is recognized by a DFA with k states. c. Conclude that L is regular iff it has finite index. Moreover, its index is the size of the smallest DFA recognizing it. 1.53 Let E = {0, 1, +, =} and ADD = yx=y+zl x, y, z are binary integers, and x is the sum of y and z}. Show that ADD is not regular. 1.54 Consider the language F = {a'bi c I i, j, k > D and if i = I then j = kJ. a. Show that F is not regular. b. Show that F acts like a regular language in the pumping lemma. In other words, give a pumping length p and demonstrate that F satisfies the three conditions of the pumping lemma for this value of p. c. Explain why parts (a) and (b) do not contradict the pumping lemma. 1.55 The pumping lemma says that every regular language has a pumping length p, such that every string in the language can be pumped if it has length p or more. If p is a pumping length for language A, so is any length p' > p. The minimum pumping length for A is the smallest p that is a pumping length for A. For example, if A = 01, the minimum pumping length is 2. The reason is that the string s = 0 is in A and has length 1 yet s cannot be pumped, but any string in A of length 2 or more contains a 1 and hence can be pumped by dividing it so that x = O. y = 1, and z is the rest. For each of the following languages, give the minimum pumping length and justify your answer. Aa. 0001 f e Ab. 01 g. 10101 c. 001 UO1 h. 10(110)0 Ad. 01+0+1 u 101 i. 1011 e. (01) j. E 1.56 If A is a set of natural numbers and k is a natural number greater than 1, let Bk (A) = {w I w is the representation in base k of some number in Al. Here, we do not allow leading Os in the representation of a number. For example, B2({3, 5}) = {11, 101} and B3({3, 5}) = {10, 12}. Give an example of a set A for which B2 (A) is regular but B3 (A) is not regular. Prove that your example works. 92 CHAPTER I / REGULAR LANGUAGES 1.57 If A is any language, let A, be the set of all first halves of strings in A so that A 1 -{x for some y, xJ = yJ and xy e A}. Show that, if A is regular, then so is A 1 1.58 If A is any language, let A i -be the set of all strings in A with their middle thirds removed so that A l -{ = {xzz for some y, IZx = -yI = IzI and xyz e A}. Show that, if A is regular, then A 1 -is not necessarily regular. 1.59 Let M =(Q,,6,qo,F) be a DFA and let h be a state of Al called its "home". A synchronizing sequence for M and h is a string s C E where oi(q, s) = h for every q e Q. (Here we have extended d to strings, so that 6(q, S) equals the state where M ends up when M starts at state q and reads input s.) Say that M is syncbronizable if it has a synchronizing sequence for some state h. Prove that, if M is a k-state synchronizable DFA, then it has a synchronizing sequence of length at most k3. Can you improve upon this bound? 1.60 Let = {a, b}. For each k > 1, let Ck be the language consisting of all strings that contain an a exactly k places from the right-hand end. Thus Ck = Ea~k-1. Describe an NFA with k + 1 states that recognizes Ck, both in terms of a state diagram and a formal description. 1.61 Consider the languages Ck defined in Problem 1.60. Prove that for each k, no DFA can recognize Ck with fewer than 2k states. 1.62 Let E = {a, b}. For each k > 1, let Dk be the language consisting of all strings that have at least one a among the last k symbols. Thus Dk = Ea(E U E) 5 Describe an DFA with at most k + 1 states that recognizes Dk, both in terms of a state diagram and a formal description. 1.63 a. Let A be an infinite regular language. Prove that A can be split into two infinite disjoint regular subsets. b. Let B and D be two languages. Write B c- D if B C D and D contains infinitely many strings that are not in B. Show that, if B and D are two regular languages where B ca D, then we can find a regular language C where B c C c D. 1.64 Let N be an NFA with k states that recognizes some language A. a. Show that, if A is nonempty, A contains some string of length at most k. b. Show that, by giving an example, that part (a) is not necessarily true if you replace both A's by A. c. Show that, if A is nonempty, A contains some string of length at most 2k d. Show that the bound given in part (b) is nearly tight; that is, for each k, demonstrate an NFA recognizing a language Ak where Ak is nonempty and where Wk's shortest member strings are of length exponential in k. Come as close to the bound in (b) as you can. SELECTED SOLUTIONS 93 '1.65 Prove that, for each n > 0, a language B, exists where a. B, is recognizable by a NFA that has n states, and b. if B, = A1 U U. uAk, for regular languages Ai, then at least one of the Ai requires a DFA with exponentially many states. SELECTED SOLUTIONS 1.1 For M1: (a) qj; (b) {q2}; (c) qi, q2, q3, qj, qj; (d) No; (e) No For M2: (a) qj; (b) {qj, q4}; (c)ql,ql,ql,q2, q4; (d)Yes; (e)Yes 1.2 M 2 = ({qlq2,q3},{a,b}, 5i,ql,{q2}). M 3 = ({ql,q2, q3,q4}, {a,b}, 2,ql,{qi, q4}). The transition functions are 1 a b q1 q2 ql q2 q3 q3 qD q2 qj 32 a b q1 qi q2 q2 q3 q4 q3 q2 qj q4 q3 q4 1.4 (b) The following are DFAs for the two languages { w w has exactly three a's} and { w| w has at least two b's}: b b b ab a a a,b _ b b Combining them using the intersection construction gives the DFA: -i a 4^ a~b b b b Though the problem doesn't request you to simplify the DFA, certain states can be combined to give CHAPTER 1 / REGULAR LANGUAGES b b b (d) These are DFAs for the two languages {wl w has an even number of a's} and { wj each a is followed by at least one b}: b a b a b a ab b Combining them using the intersection construction gives the DFA: Though the problem doesn't request you to simplify the DFA, certain states can be combined to give ab 94 SELECTED SOLUTIONS 95 1.5 (a) The left-hand DFA recognizes {wl~ U contains ab}. The right-hand DFA recog-nizes its complement, {wl w doesn't contain ab}. b a ab b a aab a bQ Qa _ b (b) This DFA recognizes twl w contains baba}. a b b ab a This DFA recognizes {wl w does not contain baba}. a b b a b a 1. 7 (a) 0,1 (i 1.11 Let N (Q, Z, 6, qo, F) be any NFA. Construct an NFA N' with a single accept state that accepts the same language as N. Informally, N' is exactly like N except it has E-transitions from the states corresponding to the accept states of N, to a new accept state, qccept. State qaccepr has no emerging transitions. More formally, N' = (Q U {qaccpt} En, 6', qo, {qamept}), where for each q e Q and a E E '(q, a) { (q, a) if a e or q , F Ij(q, a) U {qaccpt} if a E and q C F and 6'(qaccept, a) = 0 for each a e El. 1.23 We prove both directions of the "iff." (a) Assume that B = B+ and show that BB C B. For every language BB C B' holds, so if B = B', then BB C B. (-) Assume that BB C B and show that B = B'. For every language B C B', so we need to show only B' C B. If w E B', then w = X1X2 Xk where each xri G B and k > 1. Because X1, X2 E B and BB C B, we have X1X2 c B. Similarly, because xr12 is in B and X3 is in B, we have X1X2X3 C B. Continuing in this way, x 1 ... Xrk e B. Hence w C B, and so we may conclude that B' C B. 96 CHAPTER 1 / REGULAR LANGUAGES The latter argument may be written formally as the following proof by induction. Assume that BB C B. Claim: For each k > 1, if 1 , . k E B, then xi Xk e B. Basis: Prove for k = 1. This statement is obviously true. Induction step: For each k > 1, assume that the claim is true for k and prove it to be true for k + 1. If Xi, . . ., Xk, Xk+1 E B, then by the induction assumption, xi ... Xk E B. There-fore XI Xkrk+1 e BB, but BB C B, so XI -Zk+1 G B. That proves the induction step and the claim. The claim implies that, if BB C B, then B+ C B. 1.29 (a) Assume that Al = {O'i12' n > 0} is regular. Let p be the pumping length given by the pumping lemma. Choose s to be the string OPiP22P. Because s is a member of Ai and s is longer than p, the pumping lemma guarantees that s can be split into three pieces, s = xyz, where for any i > 0 the string xy'z is in A1 . Consider two possibilities: 1. The string y consists only of Os, only of is, or only of 2s. In these cases the string xyyz will not have equal numbers of Os, is, and 2s. Hence xyyz is not a member of Al, a contradiction. 2. The string y consists of more than one kind of symbol. In this case xyyz will have the Os, is, or 2s out of order. Hence xyyz is not a member of Al, a contradiction. Either way we arrive at a contradiction. Therefore, A, is not regular. (c) Assume that A3 {a2' n > 01 is regular. Let p be the pumping length given by the pumping lemma. Choose s to be the string a2P. Because s is a member of A1 and s is longer than p, the pumping lemma guarantees that s can be split into three pieces, s = xyz, satisfying the three conditions of the pumping lemma. The third condition tells us that Ixyl < p. Furthermore, p < 2P and so uyl < 2P. Therefore IxyyzI = IxzlI + wyI < 2P + 2P = 2P+'. The second condition requires Iy I > 1 so 2P < Ixyyz I < 2P+1. The length of xyyz cannot be a power of 2. Hence xyyz is not a member of A3, a contradiction. Therefore, A3 is not regular. 1.40 Let M = (Q, 2, 6, qo, F) be an NFA recognizing A, where A is some regular language. Construct M' (Q', 2, 6', go', F') recognizing NOPREFIX(A) as follows: . Q'= Q. 2. For r E Q' and a £E2 define 6'(r, a) = p(, ) if r F 3. go' = go. 4. F' F. SELECTED SOLUTIONS 97 1.44 Let MB = (QB, a ,B, qB, FB) and MC = (Qc, a Yc, qc, Fc) be DFAs recog-nizing B and C respectively. Construct NFA M (Q, a, 6, qo, F) that recognizes B # C as follows. To decide whether its input w is in B 2- C, the machine M checks that w E B, and in parallel, nondeterministically guesses a string y that contains the same number of is as contained in w and checks that y G C. 1.Q =QBXQC. 2. For (q, r) E Q and a E Z define ({(6B(q,0),r)} if a = 0 6((q,r),a) = {(6B(q,l), 6c(r,l))} if a= 1 f{(q, 6c(r, 0))} if a = E. 3. qo = (qB, qc)-4. F = FB X FC. 1.46 (b) Let B = {0fo l' m # n}. Observe that Bn o1= {QklkI k > 0}. If B were regular, then B would be regular and so would 13 n o 1 . But we already know that {okk1 I k > 01 isn't regular, so B cannot be regular. Alternatively, we can prove B to be nonregular by using the pumping lemma di-rectly, though doing so is trickier. Assume that B ={omlnI m 7# n} is regular. Let p be the pumping length given by the pumping lemma. Observe that p! is di-visible by all integers from 1 to p, where p! = p(p - I)(p - 2)... 1. The string s = 0P1P+P1 E B, and IsI > p. Thus the pumping lemma implies that s can be di-vided as xyz with x = 0A, y = 0b, and z = 0c1P+P!, where b > 1 and a + b + c = p. Let s' be the string xy'i+z, where i = p!/b. Then yi = OP! so Yi+1 = ob+p!, and so Xyz = oa+bc+p 1+Pp That gives xyz = OP+P! 1 p+P V B, a contradiction. 1.50 Assume to the contrary that some FST T outputs wiz on input u;. Consider the input strings 00 and 01. On input 00, T must output 00, and on input 01, T must output 10. In both cases the first input bit is a 0 but the first output bits differ. Operating in this way is impossible for an FST because it produces its first output bit before it reads its second input. Hence no such FST can exist. 1.52 (a) We prove this assertion by contradiction. Let M be a k-state DFA that recog-nizes L. Suppose for a contradiction that L has index greater than k. That means some set X with more than k elements is pairwise distinguishable by L. Because M has k states, the pigeonhole principle implies that X contains two distinct strings x and y, where 6(qo, x) = 6(qo, y). Here 6(qo, x) is the state that M is in after start-ing in the start state qo and reading input string r. Then, for any string z C E, 6(qo, xz) = 6(qo, yz). Therefore either both xz and yz are in L or neither are in L. But then x and y aren't distinguishable by L, contradicting our assumption that X is pairwise distinguishable by L. (b) Let X = {s1, . ., Sk} be pairwise distinguishable by L. We construct DFA M = (Q, E, 6, qo, F) with k states recognizing L. Let Q = {ql, . .. , qk and define 6(qi, a) to be qj, where sj -L sia (the relation -L is defined in Prob-lem 1.51). Note that sj -L sia for some sj E X; otherwise, X U sia would have k + 1 elements and would be pairwise distinguishable by L, which would contra-dict the assumption that L has index k. Let F = {qi Isi E L}. Let the start state qo be the qj such that si -L S. M is constructed so that, for any state qj, {sl 6(qo, s) = qi} = {sI S -L si}. Hence M recognizes L. 98 CHAPTER I / REGULAR LANGUAGES (c) Suppose that L is regular and let k be the number of states in a DFA recognizing L. Then from part (a) L has index at most k. Conversely, if L has index k, then by part (b) it is recognized by a DFA with k states and thus is regular. To show that the index of L is the size of the smallest DFA accepting it, suppose that L's index is exactly k. Then, by part (b), there is a k-state DFA accepting L. That is the smallest such DFA because if it were any smaller, then we could show by part (a) that the index of L is less than k. 1.55 (a) The minimum pumping length is 4. The string 000 is in the language but cannot be pumped, so 3 is not a pumping length for this language. If s has length 4 or more, it contains is. By dividing s onto xyz, where x is 000 and y is the first 1 and z is everything afterward, we satisfy the pumping lemma's three conditions. (b) The minimum pumping length is 1. The pumping length cannot be 0 because the string a is in the language and it cannot be pumped. Every nonempty string in the language can be divided into xyz, where x = a and y is the first character and z is the remainder. This division satisfies the three conditions. (d) The minimum pumping length is 3. The pumping length cannot be 2 because the string 11 is in the language and it cannot be pumped. Let s be a string in the language of length at least 3. If s is generated by 01+0+1, we can write it as ryz, where x is the empty string, y is the first symbol of s, and z is the remainder of s. Breaking s up in this way shows that it can be pumped. If s is generated by 10 1, we can write it as xyz, where 2 = 1 and y = 0 and z is the remainder of S. This division gives a way to pump s. 128 CHAPTER 2 / CONTEXT-FREE LANGUAGES EXERCISES 2.1 Recall the CFG G 4 that we gave in Example 2.4. For convenience, let's rename its variables with single letters as follows. E -E+TIT T -T x F F F -(E) a Give parse trees and derivations for each string. a. a b. a+a c. a+a+a d. ((a)) 2.2 a. UsethelanguagesA ={a m b'c" m,n>O} andB ={a'b c tm rn,n>O} together with Example 2.36 to show that the class of context-free languages is not closed under intersection. b. Use part (a) and DeMorgan's law (Theorem 0.20) to show that the class of context-free languages is not closed under complementation. A2 .3 Answer each part for the following context-free grammar G. R -XRX I S S -aTb I bTa T XTX J X e X a aIb What are the variables of G? What are the terminals of G? Which is the start variable of G? Give three strings in L(G). Give three strings not in L(G). True or False: T => aba. True or False: T => aba. True or False: T =~. T. 1. J. k. 1. m. n. 0. True or False: T = T. True or False: XXX =4 aba. True or False: X 4> aba. True or False: T = XX. True or False: T 4 XXX. True or False: S => E. Give a description in English of L(G). 2.4 Give context-free grammars that generate the following languages. In all parts the alphabet E is {0,1}. Aa. {wl w contains at least three ls} b. {w I w starts and ends with the same symbol} c. {wI the length of w is odd} Ad. {wI the length of w is odd and its middle symbol is a 0} e. {w) w = w R, that is, w is a palindrome} f. The empty set a. b. C. d. e. f. g. h. EXERCISES 129 2.5 Give informal descriptions and state diagrams of pushdown automata for the lan-guages in Exercise 2.4. 2.6 Give context-free grammars generating the following languages. Aa. The set of strings over the alphabet {a,b} with more a's than b's b. The complement of the language {anbn I n > 0} AC. {W#.r WI is asubstringofx for w,x E {0,i}'} d. {X1#X2# .. #XkI k > 1, each xi G {a, b}l, and for some i and j, xi = xr'} A 2 .7 Give informal English descriptions of PDAs for the languages in Exercise 2.6. A 2 .8 Show that the string the girl touches the boy with the flower has two different leftmost derivations in grammar G2 on page 101. Describe in English the two different meanings of this sentence. 2.9 Give a context-free grammar that generates the language k A = {a'bj c Ii = j or j = k where i,j, k > 0}. Is your grammar ambiguous? Why or why not? 2.10 Give an informal description of a pushdown automaton that recognizes the lan-guage A in Exercise 2.9. 2.11 Convert the CFG G 4 given in Exercise 2.1 to an equivalent PDA, using the proce-dure given in Theorem 2.20. 2.12 Convert the CFG G given in Exercise 2.3 to an equivalent PDA, using the procedure given in Theorem 2.20. 2.13 Let G = (V, Z, R, S) be the following grammar. V = {S, T, U}; Y {0, #}; and R is the set of rules: S -TT U T -OT To j U -4 OUoO a. Describe L(G) in English. b. Prove that L(G) is not regular. 2.14 Convert the following CFG into an equivalent CFG in Chomsky normal form, using the procedure given in Theorem 2.9. A -BAB I B B -00 2.15 Give a counterexample to show that the following construction fails to prove that the class of context-free languages is closed under star. Let A be a CFL that is generated by the CFG G = (V, , R. S). Add the new rule S -SS and call the resulting grammar G'. This grammar is supposed to generate A. 2.16 Show that the class of context-free languages is closed under the regular operations, union, concatenation, and star. 2.17 Use the results of Problem 2.16 to give another proof that every regular language is context free, by showing how to convert a regular expression directly to an equiv-alent context-free grammar. 130 CHAPTER 2/ CONTEXT-FREE LANGUAGES PROBLEMS A2.18 a. Let C be a context-free language and R be a regular language. Prove that the language C n R is context free. b. Use part (a) to show that the language A = {wI w e {a, b, c} and contains equal numbers of a's, b's, and c's} is not a CFL. 2.19 Let CFG G be S aSb I bY I Ya Y bY b aY I E Give a simple description of L(G) in English. Use that description to give a CFG for L(G), the complement of L(G). 2.20 Let A/B = {wI wx E A for some x e B}. Show that, if A is context free and B is regular, then A/B is context free. 2.21 Let E = {a,b}. Give a CFG generating the language of strings with twice as many a's as b's. Prove that your grammar is correct. 2.22 Let C = {zx#y x, y G {0,l} and x 4 y}. Show that C is a context-free language. 2.23 Let D = {xylx, y G {0,1} and xj = Myj butx 7# y}. Show that D is a context-free language. 2.24 Let E = {aib I i 0 j and 2i 54 j}. Show that E is a context-free language. 2.25 For any language A, let SUFFIX(A) = {vI uV E A for some string u}. Show that the class of context-free languages is closed under the SUFFIX operation. 2.26 Show that, if G is a CFG in Chomsky normal form, then for any string w e L(G) of length n > 1, exactly 2n - 1 steps are required for any derivation of w. 2.27 Let G = (V, A, R, (STMT)) be the following grammar. (STMT) (ASSIGN) I (IF-THEN) I (IF-THEN-ELSE) (IF-THEN) -a if condition then (STMT) (IF-THEN-ELSE) - if condition then (STMT) else (STMT) (ASSIGN) a:=1 = {if, condition, then, else, a: =1}. V {(STMT), (IF-THEN), KIF-THEN-ELSE), (ASSIGN)} G is a natural-looking grammar for a fragment of a programming language, but G is ambiguous. a. Show that G is ambiguous. b. Give a new unambiguous grammar for the same language. 2.28 Give unambiguous CFGs for the following languages. a. {wl in every prefix of w the number of a's is at least the number of b's} b. {wl the number of a's and b's in w are equal} c. {wl the number of a's is at least the number of b's} 2.29 Show that the language A in Exercise 2.9 is inherently ambiguous. PROBLEMS 131 2.30 Use the pumping lemma to show that the following languages are not context free. a. {Ott0"V I n n tl| > 0} Ab {fo#0 2,?# I l n > 0} Ac. {w#tj w is a substring oft, where w,t e {a,b}} d. {t7 #t2# #tkJ k > 2, each t, E {a,b}, and t, = tj for some i : j} 2.31 Let B be the language of all palindromes over {0,1} containing an equal number of Os and is. Show that B is not context free. 2.32 Let E = {1, 2, 3, 4} and C = {v c E I in vw, the number of is equals the number of 2s, and the number of 3s equals the number of 4s}. Show that C is not context free. 2.33 Show that F {ab2 I i 7 kj for every positive integer k} is not context free. 2.34 Consider the language B = L(G), where G is the grammar given in Exercise 2.13. The pumping lemma for context-free languages, Theorem 2.34, states the exis-tence of a pumping length p for B. What is the minimum value of p that works in the pumping lemma? Justify your answer. 2.35 Let G be a CFG in Chomsky normal form that contains b variables. Show that, if G generates some string with a derivation having at least 2h steps, L(G) is infinite. 2.36 Give an example of a language that is not context free but that acts like a CFL in the pumping lemma. Prove that your example works. (See the analogous example for regular languages in Problem 1.54.) '2.37 Prove the following stronger form of the pumping lemma, wherein both pieces v and y must be nonempty when the string s is broken up. If A is a context-free language, then there is a number k where, if s is any string in A of length at least k, then s may be divided into five pieces, s - uvxyz, satisfying the conditions: a. for each i > 0, uvixyiz C A, b. v e and yp7 E, and c. |vxyl < k. A238 Refer to Problem 1.41 for the definition of the perfect shuffle operation. Show that the class of context-free languages is not closed under perfect shuffle. 2.39 Refer to Problem 1.42 for the definition of the shuffle operation. Show that the class of context-free languages is not closed under shuffle. 2.40 Say that a language is prefix-closed if the prefix of any string in the language is also in the language. Let C be an infinite, prefix-closed, context-free language. Show that C contains an infinite regular subset. 2.41 Read the definitions of NOPREFIX(A) and NOEXTEND(A) in Problem 1.40. a. Show that the class of CFLs is not closed under NOPREFIX operation. b. Show that the class of CFLs is not closed under NOEXTEND operation. 2.42 Let E = {i.#} and Y = {W| v = t1 #t2# #tk for k > 0, each ti E 1, and ti $ tj whenever i - j}. Prove that Y is not context free. 132 CHAPTER 2 / CONTEXT-FREE LANGUAGES 2.43 For strings w and t, write w t if the symbols of w are a permutation of the symbols of t. In other words, wt if t and w have the same symbols in the same quantities, but possibly in a different order. For any string w, define SCRAMBLE(w) {tI t w}. For any language A, let SCRAMBLE(A) = {tI t E SCRAMBLE(w) for some w G A}. a. Show that, if E= {0, 1}, then the SCRAMBLE of a regular language is context free. b. What happens in part (a) if E contains 3 or more symbols? Prove your answer. 2.44 If A and B are languages, define A o B = {xyl x E A and y E B and IJ = IyI}. Show that if A and B are regular languages, then A o B is a CFL. 2.45 Let A = w wtw ,l W. t E {0, 1} and JwI t }. Prove that A is not a context-free language. SELECTED SOLUTIONS 2.3 (a) R, X, S, T; (b) a, b; (c) R; (d) Three strings in G are ab, ba, and aab; (e) Three strings not in G are a, b, and e; (f) False; (g) True; (h) False; (i) True; (j) True; (k) False; (I) True; (m) True; (n) False; (o) L(G) consists of all strings over a and b that are not palindromes. 2.4 (a) S R1R1RR (d) S - l S0 1 OS S I S O 1 I1Sl R OR I 1RIe 2.6 (a) S TaT (c) S -TX T TT|aTb{bTa|a| T -OTO|1T1I#X T generates all strings with at least as X -OX I ix I s many a's as b's, and S forces an extra a. 2.7 (a) The PDA uses its stack to count the number of a's minus the number of b's. It enters an accepting state whenever this count is 0. In more detail, it operates as follows. The PDA scans across the input. If it sees a b and its top stack symbol is a a, it pops the stack. Similarly, if it scans a a and its top stack symbol is a b, it pops the stack. In all other cases, it pushes the input symbol onto the stack. After the PDA scans the input, if b is on top of the stack, it accepts. Otherwise it rejects. (c) The PDA scans across the input string and pushes every symbol it reads until it reads a #. If # is never encountered, it rejects. Then, the PDA skips over part of the input, nondeterministically deciding when to stop skipping. At that point, it compares the next input symbols with the symbols it pops off the stack. At any disagreement, or if the input finishes while the stack is nonempty, this branch of the computation rejects. If the stack becomes empty, the machine reads the rest of the input and accepts. SELECTED SOLUTIONS 133 2.8 Here is one derivation: (SENTENCE) => (NOUN-PHRASE)(VERB-PHRASE) = (CMPLX-NOUN)(VERB-PHRASE) => (CMPLX-NOUN)(CMPLX-VERB)(PREP-PHRASE) > (ARTICLE) (NOUN) (CMPLX-VERB)(PREP-PHRASE) => The boy (VERB) (NOUN-PHRASE)(PREP-PHRASE) => The boy (VERB)(NOUN-PHRASE)(PREP)(CMPLX-NOUN)=z The boy touches (NOUN-PHRASE) (PREP)(CMPLX-NOUN) = The boy touches (CMPLX-NOUN)(PREP)(CMPLX-NOUN)=n The boy touches (ARTICLE)(NOUN)(PREP)(CMPLX-NOUN)=' The boy touches the girl with (CMPLX-NOUN) = The boy touches the girl with (ARTICLE)(NOUN)=s The boy touches the girl with the flower Here is another derivation: (SENTENCE) X (NOUN-PHRASE)(VERB-PHRASE) X (CMPLX-NOUN)(VERB-PHRASE) =t (ARTICLE)(NOUN)(VERB-PHRASE) X The boy (VERB-PHRASE) => The boy (CMPLX-VERB)=> The boy (VERB)(NOUN-PHRASE) =:. The boy touches (NOUN-PHRASE) => The boy touches (CMPLX-NOUN)(PREP-PHRASE)=. The boy touches (ARTICLE)(NOUN)(PREP-PHRASE)=1 The boy touches the girl (PREP-PHRASE) =:i The boy touches the girl (PREP)(CMPLX-NOUN)=> The boy touches the girl with (CMPLX-NOUN) =# The boy touches the girl with (ARTICLE) (NOUN) =z The boy touches the girl with the flower Each of these derivations corresponds to a different English meaning. In the first derivation, the sentence means that the boy used the flower to touch the girl. In the second derivation, the girl is holding the flower when the boy touches her. 2.18 (a) Let C be a context-free language and R be a regular language. Let P be the PDA that recognizes C, and D be the DFA that recognizes R. If Q is the set of states of P and Q' is the set of states of D, we construct a PDA P' that recognizes C n R with the set of states Q x Q'. P' will do what P does and also keep track of the states of D. It accepts a string w if and only if it stops at a state q E Fp x FD, where FP is the set of accept states of P and FD is the set of accept states of D. Since C n R is recognized by P', it is context free. (b) Let R be the regular language ab'c. If A were a CFL then A n R would be a CFL by part (a). However, A n R = {arb'c' | n > 0}, and Example 2.36 proves that A n R is not context free. Thus A is not a CFL. 2.30 (b) Let B = {On#02n#03, n > 0J. Let p be the pumping length given by the pumping lemma. Let s = op#o 2p #o3p. We show that s = uvxyz cannot be pumped. Neither v nor y can contain #, otherwise rv2 wy 2 z contains more than two #s. Therefore, if we divide s into three segments by #'s: o0, 2p and 0 3p, at least one of the segments is not contained within either v or Y. Hence xv 2 Wy 2 z is not in B because the 1 : 2 : 3 length ratio of the segments is not maintained. 134 CHAPTER 2 / CONTEXT-FREE LANGUAGES (c) Let C = {w#tI w is a substring of t, where w, t C {a, b} }. Let p be the pumping length given by the pumping lemma. Let s = aPbP#aPbl. We show that the string s = uvxyz cannot be pumped. Neither v nor y can contain #, otherwise uvozyoz does not contain # and therefore is not in C. If both v and y are nonempty and occur on the left-hand side of the #, the string UV2 Xy 2 z cannot be in C because it is longer on the left-hand side of the #. Similarly, if both strings occur on the right-hand side of the #, the string uvOxy 0z cannot be in C because it is again longer on the left-hand side of the #. If only one of v and y is nonempty (both cannot be nonempty), treat them as if both occurred on the same side of the # as above. The only remaining case is where both v and y are nonempty and straddle the #. But then v consists of b's and y consists of a's because of the third pumping lemma condition lvxyl < p. Hence, uv2 xy2 z contains more b's on the left-hand side of the #, so it cannot be a member of C. 2.38 Let A be the language {Okik I k > 0} and let B be the language {akb3k Ik > 01. The perfect shuffle of A and B is the language C = {(Oa)k(Ob)k(1b)2 kI k > 0}. Languages A and B are easily seen to be CFLs, but C is not a CFL, as follows. If C were a CFL, let p be the pumping length given by the pumping lemma, and let s be the string (0a)P(Ob)P(1b) 2 P. Because s is longer than p and s C C, we can divide a = uvxyz satisfying the pumping lemma's three conditions. Strings in C contain twice as many is as a's. In order for uv2Xy 2z to have that property, the string vry must contain both is and a's. But that is impossible, because they are separated by 2p symbols yet the third condition says that cVZyI < p. Hence C is not context free. EXERCISES 159 Now M scans the list of edges. For each edge, M tests whether the two underlined nodes n1 and n2 are the ones appearing in that edge. If they are, .A dots n1, removes the underlines, and goes on from the beginning of stage 2. If they aren't, M checks the next edge on the list. If there are no more edges, {n,, n2} is not an edge of G. Then M moves the underline on n2 to the next dotted node and now calls this node n2. It repeats the steps in this paragraph to check, as before, whether the new pair {n,, n2} is an edge. If there are no more dotted nodes, n1 is not attached to any dotted nodes. Then M sets the underlines so that n1 is the next undotted node and n2 is the first dotted node and repeats the steps in this paragraph. If there are no more undotted nodes, Al has not been able to find any new nodes to dot, so it moves on to stage 4. For stage 4, Al scans the list of nodes to determine whether all are dotted. If they are, it enters the accept state; otherwise it enters the reject state. This completes the description of TM M. EXERCISES 3.1 This exercise concerns TM M2 whose description and state diagram appear in Ex-ample 3.7. In each of the parts, give the sequence of configurations that M2 enters when started on the indicated input string. a. 0. Ab. 00. C. 000. d. oooooo. 3.2 This exercise concerns TM M 1 whose description and state diagram appear in Ex-ample 3.9. In each of the parts, give the sequence of configurations that Al 1 enters when started on the indicated input string. Aa. 1. b. 1#1. C. 1##1. d. 10#11. e. 10#10. A 3.3 Modify the proof of Theorem 3.16 to obtain Corollary 3.19, showing that a lan-guage is decidable iff some nondeterministic Turing machine decides it. (You may assume the following theorem about trees. If every node in a tree has finitely many children and every branch of the tree has finitely many nodes, the tree itself has finitely many nodes.) 3.4 Give a formal definition of an enumerator. Consider it to be a type of two-tape Turing machine that uses its second tape as the printer. Include a definition of the enumerated language. 160 CHAPTER 3/ THE CHURCH-TURING THESIS A 3.5 Examine the formal definition of a Turing machine to answer the following ques-tions, and explain your reasoning. a. Can a Turing machine ever write the blank symbol L on its tape? b. Can the tape alphabet F be the same as the input alphabet E? c. Can a Turing machine's head ever be in the same location in two successive steps? d. Can a Turing machine contain just a single state? 3.6 In Theorem 3.21 we showed that a language is Turing-recognizable iff some enu-merator enumerates it. Why didn't we use the following simpler algorithm for the forward direction of the proof? As before, s1, S2, . . . is a list of all strings in E. E = "Ignore the input. 1. Repeat the following for i = 1, 2, 3, .... 2. RunMonsi. 3. If it accepts, print out si,." 3.7 Explain why the following is not a description of a legitimate Turing machine. MSad = "The input is a polynomial p over variables xi, . . ., Xk. 1. Try all possible settings of x1 , ... , Xk to integer values. 2. Evaluate p on all of these settings. 3. If any of these settings evaluates to 0, accept; otherwise, reject." 3.8 Give implementation-level descriptions of Turing machines that decide the follow-ing languages over the alphabet {0,1}. Aa. {fl w contains an equal number of Os and is} b. {wf w contains twice as many Os as ls} c. {wJ w does not contain twice as many Os as is} PROBLEMS 3.9 Let a k-PDA be a pushdown automaton that has k stacks. Thus a O-PDA is an NFA and a 1-PDA is a conventional PDA. You already know that 1-PDAs are more powerful (recognize a larger class of languages) than O-PDAs. a. Show that 2-PDAs are more powerful than 1-PDAs. b. Show that 3-PDAs are not more powerful than 2-PDAs. (Hint: Simulate a Turing machine tape with two stacks.) A3 .10 Say that a write-once Turing machine is a single-tape TM that can alter each tape square at most once (including the input portion of the tape). Show that this variant Turing machine model is equivalent to the ordinary Turing machine model. (Hint: As a first step consider the case whereby the Turing machine may alter each tape square at most twice. Use lots of tape.) PROBLEMS 161 3.11 A Turing machine with doubly infinite tape is similar to an ordinary Turing ma-chine, but its tape is infinite to the left as well as to the right. The tape is initially filled with blanks except for the portion that contains the input. Computation is defined as usual except that the head never encounters an end to the tape as it moves leftward. Show that this type of Turing machine recognizes the class of Turing-recognizable languages. 3.12 A Turing machine with left reset is similar to an ordinary Turing machine, but the transition function has the form 6: Q x row x F x {RRESET}. If 6(q, a) = (r, b, RESET), when the machine is in state q reading an a, the ma-chine's head jumps to the left-hand end of the tape after it writes b on the tape and enters state r. Note that these machines do not have the usual ability to move the head one symbol left. Show that Turing machines with left reset recognize the class of Turing-recognizable languages. 3.13 A Turing machine with stay put instead of left is similar to an ordinary Turing machine, but the transition function has the form 6: Q x F- Q x rF x {R,S}. At each point the machine can move its head right or let it stay in the same position. Show that this Turing machine variant is not equivalent to the usual version. What class of languages do these machines recognize? 3.14 A queue automaton is like a push-down automaton except that the stack is replaced by a queue. A queue is a tape allowing symbols to be written only on the left-hand end and read only at the right-hand end. Each write operation (we'll call it a push) adds a symbol to the left-hand end of the queue and each read operation (we'll call it a pull) reads and removes a symbol at the right-hand end. As with a PDA, the input is placed on a separate read-only input tape, and the head on the input tape can move only from left to right. The input tape contains a cell with a blank symbol following the input, so that the end of the input can be detected. A queue automaton accepts its input by entering a special accept state at any time. Show that a language can be recognized by a deterministic queue automaton iff the language is Turing-recognizable. 3.15 Show that the collection of decidable languages is closed under the operation of Aa. union. d. complementation. b. concatenation. e. intersection. c. star. 3.16 Show that the collection of Turing-recognizable languages is closed under the op-eration of Aa. union. c. star. b. concatenation. d. intersection. 3.17 Let B = {(Mi), (M 2), . } be a Turing-recognizable language consisting of TM descriptions. Show that there is a decidable language C consisting of TM descrip-tions such that every machine described in B has an equivalent machine in C and vice versa. 162 CHAPTER 3/ THE CHURCH-TURING THESIS 3.18 Show that a language is decidable iff some enumerator enumerates the language in lexicographic order. 3.19 Show that every infinite Turing-recognizable language has an infinite decidable subset. 3.20 Show that single-tape TMs that cannot write on the portion of the tape containing the input string recognize only regular languages. 3.21 Let cix9 + C2X-I + --+ cnx + cn±1 be a polynomial with a root at x = x0. Let cmax be the largest absolute value of a ci. Show that xooI < (n+1) ecn A3 .2 2 Let A be the language containing only the single string s, where 0 if life never will be found on Mars. 1 if life will be found on Mars someday. Is A decidable? Why or why not? For the purposes of this problem, assume that the question of whether life will be found on Mars has an unambiguous YES or No answer. SELECTED SOLUTIONS 3.1 (b) q100, uq2O, uxq3u, uq 5 xu, qiuxu, uq 2 xu, uxq2u, uxuqaccept 3.2 (a) qi 11, xq31, xlq3u, xluqrejct. 3.3 We prove both directions of the "iff." First, if a language L is decidable, it can be decided by a deterministic Turing machine, and that is automatically a nondeter-ministic Turing machine. Second, if a language L is decided by a nondeterministic TM N, we construct a deterministic TM D2 that decides L. Machine D2 runs the same algorithm that appears in the TM D described in the proof of Theorem 3.16, with an additional Stage 5: Reject if all branches of the nondeterminism of N are exhausted. We argue that D2 is a decider for L. If N accepts its input, D2 will eventually find an accepting branch and accept, too. If N rejects its input, all of its branches halt and reject because it is a decider. Hence each of the branches has finitely many nodes, where each node represents one step of N's computation along that branch. Therefore N's entire computation tree on this input is finite, by virtue of the theorem about trees given in the statement of the exercise. Consequently D will halt and reject when this entire tree has been explored. 3.5 (a) Yes. The tape alphabet F contains u. A Turing machine can write any characters in F on its tape. (b) No. E never contains u, but r always contains u. So they cannot be equal. (c) Yes. If the Turing machine attempts to move its head off the left-hand end of the tape, it remains on the same tape cell. (d) No. Any Turing machine must contain two distinct states qaccept and qrejmct. So, a Turing machine contains at least two states. SELECTED SOLUTIONS 163 3.8 (a) "On input string Iv: 1. Scan the tape and mark the first 0 which has not been marked. If no unmarked 0 is found, go to stage 4. Otherwise, move the head back to the front of the tape. 2. Scan the tape and mark the first 1 which has not been marked. If no unmarked 1 is found, reject. 3. Move the head back to the front of the tape and go to stage 1. 4. Move the head back to the front of the tape. Scan the tape to see if any unmarked is remain. If none are found, accept; otherwise, reject." 3.10 We first simulate an ordinary Turing machine by a write-twice Turing machine. The write-twice machine simulates a single step of the original machine by copying the entire tape over to a fresh portion of the tape to the right-hand side of the currently used portion. The copying procedure operates character by character, marking a character as it is copied. This procedure alters each tape square twice, once to write the character for the first time and again to mark that it has been copied. The position of the original Turing machine's tape head is marked on the tape. When copying the cells at, or adjacent to, the marked position, the tape contents is updated according to the rules of the original Turing machine. To carry out the simulation with a write-once machine, operate as before, except that each cell of the previous tape is now represented by two cells. The first of these contains the original machine's tape symbol and the second is for the mark used in the copying procedure. The input is not presented to the machine in the format with two cells per symbol, so the very first time the tape is copied, the copying marks are put directly over the input symbols. 3.15 (a) For any two decidable languages LI and L2, let M1 and Al 2 be the TMs that decide them. We construct a TM M' that decides the union of L1 and L2: "On input w: 1. Run M1 on w. If it accepts, accept. 2. Run M2 on w. If it accepts, accept. Otherwise, reject." M' accepts w if either Ml or M2 accepts it. If both reject, M' rejects. 3.16 (a) For any two Turing-recognizable languages L1 and L2, let M1 and M2 be the TMs that recognize them. We construct a TM M' that recognizes the union of Li and L2: "On input w: 1. Run M1 and M2 alternatively on w step by step. If either accept, accept. If both halt and reject, reject." If either Ml and Al2 accept w, M' accepts te because the accepting TM arrives to its accepting state after a finite number of steps. Note that if both Al' and M2 reject and either of them does so by looping, then M' will loop. 3.22 The language A is one of the two languages, {0} or { 1 }. In either case the language is finite, and hence decidable. If you aren't able to determine which of these two languages is A, you won't be able to describe the decider for A, but you can give two Turing machines, one of which is A's decider. 182 CHAPTER 4 / DECIDABILITY machine M is a decider for A. M = "On input W: 1. Run both M1 and M 2 on input w in parallel. 2. If M1 accepts, accept; if M 2 accepts, reject." Running the two machines in parallel means that M has two tapes, one for simu-lating M1 and the other for simulating M2. In this case M takes turns simulating one step of each machine, which continues until one of them accepts. Now we show that M decides A. Every string w is either in A or A. Therefore either M1 or M2 must accept w. Because M halts whenever M1 or M 2 accepts, M always halts and so it is a decider. Furthermore, it accepts all strings in A and rejects all strings not in A. So M is a decider for A, and thus A is decidable. COROLLARY 4.23 ................................................ ATM is not Turing-recognizable. PROOF We know that ATM is Turing-recognizable. If ATM also were Turing-recognizable, ATM would be decidable. Theorem 4.11 tells us that ATM is not decidable, so ATM must not be Turing-recognizable. EXERCISES A4.1 Answer all parts for the following DFA M and give reasons for your answers. I a. Is (M, 0100) e ADFA? b. Is (M, 011) e ADFA? c. Is (M) e ADFA? d. Is (M, 0100) E AREX? e. Is (M) £ EDFA? f. Is (M, M) E EQDFA? PROBLEMS 183 4.2 Consider the problem of determining whether a DFA and a regular expression are equivalent. Express this problem as a language and show that it is decidable. 4.3 Let ALLDFA ={ (A)I A is a DFA and L(A) = } . Show that ALLDFA is decidable. 4.4 Let AECFG = {(G) I G is a CFG that generates E}. Show that AECFG is decidable. 4.5 Let X be the set {1, 2, 3, 4, 5} and Y be the set {6, 7, 8, 9, 10}. We describe the functions f: X -Y and g: X Y in the following tables. Answer each part and give a reason for each negative answer. n f (n) n g(n) 1 6 1 10 2 7 2 9 3 6 3 8 4 7 4 7 5 6 5 6 Aa. Is f one-to-one? Ad. Is g one-to-one? b. Is f onto? e. Is g onto? c. Is f a correspondence? f. Is g a correspondence? 4.6 Let B be the set of all infinite sequences over {0,1}. Show that 13 is uncountable, using a proof by diagonalization. 4.7 Let T = { (i, j, k) I i, j, k E A}. Show that T is countable. 4.8 Review the way that we define sets to be the same size in Definition 4.12 (page 175). Show that "is the same size" is an equivalence relation. PROBLEMS A4.9 Let INFINITEDFA ={(A)l A is a DFA and L(A) is an infinite language}. Show that INFINITEDFA is decidable. 4.10 Let INFINITEPDA = { (M) I M is a PDA and L(M) is an infinite language}. Show that INFINITEPDA is decidable. A4.11 Let A = {(M)J M is a DFA which doesn't accept any string containing an odd number of Is}. Show that A is decidable. 4.12 Let A = {(R, S) I R and S are regular expressions and L(R) C L(S)}. Show that A is decidable. A 4.13 Let Z = {0,1}. Show that the problem of determining whether a CFG generates some string in 1 is decidable. In other words, show that {(G)I G is a CFG over {0,I} and l n L(G) - O} is a decidable language. '4.14 Show that the problem of determining whether a CFG generates all strings in l is decidable. Inotherwords,showthat{(G)l GisaCFGover{0,} and l C L(G)} is a decidable language. 184 CHAPTER 4 / DECIDABILITY 4.15 Let A = {(R)I R is a regular expression describing a language containing at least one string w that has 111 as a substring (i.e., w = x 1 ly for some x and y)}. Show that A is decidable. 4.16 Prove that EQDFA is decidable by testing the two DFAs on all strings up to a certain size. Calculate a size that works. 4.17 Let C be a language. Prove that C is Turing-recognizable iff a decidable language D exists such that C = xIl y (Kx, y) G D)}. 4.18 Let A and B be two disjoint languages. Say that language C separates A and B if A C C and B C C7. Show that any two disjoint co-Turing-recognizable languages are separable by some decidable language. 4.19 Let S = {(M) I M is a DFA that accepts w R whenever it accepts w}. Show that S is decidable. 4.20 A language is prefix-free if no member is a proper prefix of another member. Let PREFIX-FREEREx = {RI R is a regular expression where L(R) is prefix-free}. Show that PREFIX-FREEREX is decidable. Why does a similar approach fail to show that PREFIX-FREEcFG is decidable? A4. 2 1 Say that an NFA is ambiguous if it accepts some string along two different com-putation branches. Let AMBIGNFA = {KN)I N is an ambiguous NFA}. Show that AMBIGNFA is decidable. (Suggestion: One elegant way to solve this problem is to construct a suitable DFA and then run EDFA on it.) 4.22 A useless state in a pushdown automaton is never entered on any input string. Con-sider the problem of determining whether a pushdown automaton has any useless states. Formulate this problem as a language and show that it is decidable. A4.2 3 Let BALDFA = {(M)I M is a DFA that accepts some string containing an equal number of Os and 1s4. Show that BALDFA is decidable. (Hint: Theorems about CFLs are helpful here.) 4.24 Let PALDFA = {(M)I M is a DFA that accepts some palindrome}. Show that PALDFA is decidable. (Hint: Theorems about CFLs are helpful here.) 4.25 Let E = {(M) I M is a DFA that accepts some string with more Is than 0s}. Show that E is decidable. (Hint: Theorems about CFLs are helpful here.) 4.26 Let C = { (G, x) J G is a CFG that generates some string w, where x is a substring of w }. Show that C is decidable. (Suggestion: An elegant solution to this problem uses the decider for ECFG.) 4.27 Let CCFG = {I G, k)I L(G) contains exactly k strings where k > 0 or k = o}. Show that CCFG is decidable. 4.28 Let A be a Turing-recognizable language consisting of descriptions of Turing ma-chines, {(Mi), KM2), . .}. , where everyMi is a decider. Prove that some decidable language D is not decided by any decider Mi whose description appears in A. (Hint: You may find it helpful to consider an enumerator for A.) SELECTED SOLUTIONS 185 SELECTED SOLUTIONS 4.1 (a) Yes. The DFA M accepts 0100. (b) No. M doesn't accept 011. (c) No. This input has only a single component and thus is not of the correct form. (d) No. The first component is not a regular expression and so the input is not of the correct form. (e) No. M's language isn't empty. (0 Yes. M accepts the same language as itself. 4.5 (a) No, f is not one-to-one because f (1) = f (3). (d) Yes, g is one-to-one. 4.9 The following TM I decides INFINITEDFA. I = "On input (A) where A is a DFA: 1. Let k be the number of states of A. 2. Construct a DFA D that accepts all strings of length k or more. 3. Construct a DFA M such that L(M) = L(A) n L(D). 4. Test L(M) = 0, using the EDFA decider T from Theorem 4.4. 5. If T accepts, reject; if T rejects, accept." This algorithm works because a DFA which accepts infinitely many strings must accept arbitrarily long strings. Therefore this algorithm accepts such DFAs. Con-versely, if the algorithm accepts a DFA, the DFA accepts some string of length k or more, where k is the number of states of the DFA. This string may be pumped in the manner of the pumping lemma for regular languages to obtain infinitely many accepted strings. 4.11 The following TM decides A. "On input (M): 1. Construct a DFA 0 that accepts every string containing an odd number of is. 2. Construct DFA B such that L(B) = L(M) n L(O). 3. Test whether L(B) = 0, using the EDFA decider T from Theo-rem 4.4. 4. If T accepts, accept; if T rejects, reject." 4.13 You showed in Problem 2.18 that, if C is a context-free language and R is a regular language, then C n R is context free. Therefore 1 n L(G) is context free. The following TM decides A. "On input (G): 1. Construct CFG H such that L(H) = 1 n L(G). 2. Test whether L(H) = 0, using the ECFG decider R from Theo-rem 4.8. 3. If R accepts, reject; if R rejects, accept." 186 CHAPTER 4 / DECIDABILITY 4.21 The following procedure decides AMBIGNFA. Given an NFA N, we design a DFA D that simulates N and accepts a string iff it is accepted by N along two different computational branches. Then we use a decider for EDFA to determine whether D accepts any strings. Our strategy for constructing D is similar to the NFA to DFA conversion in the proof of Theorem 1.39. We simulate N by keeping a pebble on each active state. We begin by putting a red pebble on the start state and on each state reachable from the start along E transitions. We move, add, and remove pebbles in accordance with N's transitions, preserving the color of the pebbles. Whenever two or more pebbles are moved to the same state, we replace its pebbles with a blue pebble. After reading the input, we accept if a blue pebble is on an accept states of N. The DFA D has a state corresponding to each possible position of pebbles. For each state of N, three possibilities occur: it can contain a red pebble, a blue pebble, or no pebble. Thus, if N has n states, D will have 3' states. Its start state, accept states, and transition function are defined to carry out the simulation. 4.23 The language of all strings with an equal number of Os and is is a context-free language, generated by the grammar S -iSOS I oSiS I E. Let P be the PDA that recognizes this language. Build a TM M for BALDFA, which operates as follows. On input KB), where B is a DFA, use B and P to construct a new PDA R that recognizes the intersection of the languages of B and P. Then test whether R's language is empty. If its language is empty, reject; otherwise, accept. EXERCISES 211 EXERCISES 5.1 Show that EQCFG is undecidable. 5.2 Show that EQCFG is co-Turing-recognizable. 5.3 Find a match in the following instance of the Post Correspondence Problem. ab ] [b] [aba] [aal} l abab | a] b La I 5.4 If A <m B and B is a regular language, does that imply that A is a regular lan-guage? Why or why not? A 5 .5 Show that ATM is not mapping reducible to ETM. In other words, show that no computable function reduces ATM to ETM. (Hint: Use a proof by contradiction, and facts you already know about ATM and ETM.) A 5.6 Show that <,11 is a transitive relation. A 5 .7 Show that if A is Turing-recognizable and A <rn A, then A is decidable. A 5 .8 In the proof of Theorem 5.15 we modified the Turing machine M so that it never tries to move its head off the left-hand end of the tape. Suppose that we did not make this modification to M. Modify the PCP construction to handle this case. PROBLEMS 5.9 Let T = { (M) I A/l is a TM that accepts w R whenever it accepts wu}. Show that T is undecidable. A 5 .10 Consider the problem of determining whether a two-tape Turing machine ever writes a nonblank symbol on its second tape when it is run on input w. Formulate this problem as a language, and show that it is undecidable. A 5 .11 Consider the problem of determining whether a two-tape Turing machine ever writes a nonblank symbol on its second tape during the course of its computation on any input string. Formulate this problem as a language, and show that it is undecidable. 5.12 Consider the problem of determining whether a single-tape Turing machine ever writes a blank symbol over a nonblank symbol during the course of its computation on any input string. Formulate this problem as a language, and show that it is undecidable. 5.13 A useless state in a Turing machine is one that is never entered on any input string. Consider the problem of determining whether a Turing machine has any useless states. Formulate this problem as a language and show that it is undecidable. 212 CHAPTER 5/ REDUCIBILITY 5.14 Consider the problem of determining whether a Turing machine M on an input w ever attempts to move its head left when its head is on the left-most tape cell. Formulate this problem as a language and show that it is undecidable. 5.15 Consider the problem of determining whether a Turing machine M on an input w ever attempts to move its head left at any point during its computation on w. Formulate this problem as a language and show that it is decidable. 5.16 Let r = {o, 1, u4 be the tape alphabet for all TMs in this problem. Define the busy beaverfunction BB: A(-M[ as follows. For each value of k, consider all k-state TMs that halt when started with a blank tape. Let BB(k) be the maximum number of is that remain on the tape among all of these machines. Show that BB is not a computable function. 5.17 Show that the Post Correspondence Problem is decidable over the unary alphabet E = {}. 5.18 Show that the Post Correspondence Problem is undecidable over the binary alpha-bet Z = {o,1}. 5.19 In the silly Post Correspondence Problem, SPCP, in each pair the top string has the same length as the bottom string. Show that the SPCP is decidable. 5.20 Prove that there exists an undecidable subset of {I}. 5.21 LetAMBIGCFG = {(G)I G is an ambiguous CFG}. Show thatAMBIGcFG is unde-cidable. (Hint: Use a reduction from PCP. Given an instance P [ ] 2 [k]. of the Post Correspondence Problem, construct a CFG G with the rules S T I B T tiTa, -tkTak |tial ,- tkak B biBai bkBak |biaas| bk ak, where al, ... , ak are new terminal symbols. Prove that this reduction works.) 5.22 Show that A is Turing-recognizable iff A <mr ATM. 5.23 Show that A is decidable iff A <m O 1'. 5.24 Let J = {wl either w = ox for some x i ATM, or w = ly for some y E ATM }. Show that neither J nor J is Turing-recognizable. 5.25 Give an example of an undecidable language B, where B <.., B. 5.26 Define a two-headedfinite automaton (2DFA) to be a deterministic finite automa-ton that has two read-only, bidirectional heads that start at the left-hand end of the input tape and can be independently controlled to move in either direction. The tape of a 2DFA is finite and is just large enough to contain the input plus two ad-ditional blank tape cells, one on the left-hand end and one on the right-hand end, that serve as delimiters. A 2DFA accepts its input by entering a special accept state. For example, a 2DFA can recognize the language {aflbncrl n > 0}. a. Let A2DFA = {M, x) I M is a 2DFA and M accepts x4. Show that A2DFA is decidable. b. Let E2DFA = {M)I M is a 2DFA and L(M) = 0}. Show that E2DFA is not decidable. PROBLEMS 213 5.27 A two-dimensionalfinite automaton (2DIM-DFA) is defined as follows. The input is an m x n rectangle, for any m, n > 2. The squares along the boundary of the rectangle contain the symbol # and the internal squares contain symbols over the input alphabet E. The transition function is a mapping Q x E -Q x {L, R, U, D} to indicate the next state and the new head position (Left, Right, Up, Down). The machine accepts when it enters one of the designated accept states. It rejects if it tries to move off the input rectangle or if it never halts. Two such machines are equivalent if they accept the same rectangles. Consider the problem of determin-ing whether two of these machines are equivalent. Formulate this problem as a language, and show that it is undecidable. A '5.28 Rice's theorem. Let P be any nontrivial property of the language of a Turing machine. Prove that the problem of determining whether a given Turing machine's language has property P is undecidable. In more formal terms, let P be a language consisting of Turing machine descrip-tions where P fulfills two conditions. First, P is nontrivial-it contains some, but not all, TM descriptions. Second, P is a property of the TM's language-whenever L(M1 ) = L(M2 ), we have (M1) E P iff (M2) E P. Here, M1 and M2 are any TMs. Prove that P is an undecidable language. 5.29 Show that both conditions in Problem 5.28 are necessary for proving that P is undecidable. 5.30 Use Rice's theorem, which appears in Problem 5.28, to prove the undecidability of each of the following languages. Aa. INFINITETM= {(M)I M is a TM and L(M) is an infinite language}. b. {(M)I MisaTM and 1011 EL(M)}. c. ALLTM= {(M)J MisaTMandL(M) =E. 5.31 Let E J 3x + t for odd x f x/2 for even x for any natural number x. If you start with an integer x and iterate f, you obtain a sequence, x, f(x), f(f(x)),)... Stop if you ever hit 1. For example, if x = 17, you get the sequence 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Extensive computer tests have shown that every starting point between 1 and a large positive integer gives a sequence that ends in 1. But, the question of whether all positive starting points end up at 1 is unsolved; it is called the 3x + 1 problem. Suppose that ATM were decidable by a TM H. Use H to describe a TM that is guaranteed to state the answer to the 3x + 1 problem. 5.32 Prove that the following two languages are undecidable. a. OVERLAPcFrc {(G, H)I G and H are CFGs where L(G) n L(H) 0}. (Hint: Adapt the hint in Problem 5.21.) b. PREFLV-FkEECF= {GI G is a CFG where L(G) is prefix-free}. 5.33 Let S ={(M) M is a TM and L(M)= {(M)} }. Show that neither S nor S is Turing-recognizable. 5.34 Consider the problem of determining whether a PDA accepts some string of the form {wwj w c {0,1}>} . Use the computation history method to show that this problem is undecidable. 214 CHAPTER 5/ REDUCIBILITY 5.35 Let X = {KM, w) M is a single-tape TM that never modifies the portion of the tape that contains the input w}. Is X decidable? Prove your answer. SELECTED SOLUTIONS 5.5 Suppose for a contradiction that ATM <m ETM via reduction f. It follows from the definition of mapping reducibility that ATM <m ETM via the same reduction func-tion f. However ETM is Turing-recognizable and ATM is not Turing-recognizable, contradicting Theorem 5.28. 5.6 Suppose A <m B and B f(x) C B and y G B g(y) e C. Consider the composition function h(x) = g(f(x)). We can build a TM that computes h as follows: First, simulate a TM for f (such a TM exists because we assumed that f is computable) on input x and call the output y. Then simulate a TM for g on y. The output is h(x) = g(f(x)). Therefore h is a computable function. Moreover, x G A 4=z h(x) e C. Hence A <m C via the reduction function h. 5.7 Suppose that A <m A. Then A <,,, A via the same mapping reduction. Because A is Turing-recognizable, Theorem 5.28 implies that A is Turing-recognizable, and then Theorem 4.22 implies that A is decidable. 5.8 You need to handle the case where the head is at the leftmost tape cell and attempts to move left. To do so add dominos [#qal [#rb for every q, r e Q and a, b e F, where 6(q, a) = (r, b, L). 5.10 Let B = {(M, w) l M is a two-tape TM which writes a nonbank symbol on its second tape when it is run on w}. Show that ATM reduces to B. Assume for the sake of contradiction that TM R decides B. Then construct TM S that uses R to decide ATM. S = "On input (M, w): 1. Use M to construct the following two-tape TM T. T = "On input x: 1. Simulate M on x using the first tape. 2. If the simulation shows that M accepts, write a non-blank symbol on the second tape." 2. Run R on (T, w) to determine whether T on input w writes a nonblank symbol on its second tape. 3. If R accepts, M accepts w, therefore accept. Otherwise reject." SELECTED SOLUTIONS 215 5.11 Let C = { (M) I Mis a two-tape TM which writes a nonblank symbol on its second tape when it is run on some input}. Show that ATM reduces to C. Assume for the sake of contradiction that TM R decides C. Construct TM S that uses R to decide ATM. S = "On input (M, w): 1. Use M and w to construct the following two-tape TM T_ TW = "On any input: 1. Simulate M on w using the first tape. 2. If the simulation shows that M accepts, write a non-blank symbol on the second tape." 2. Run Ron (Tw) to determine whether T, ever writes a nonbank symbol on its second tape. 3. If R accepts, M accepts w, therefore accept. Otherwise reject." 5.28 Assume for the sake of contradiction that P is a decidable language satisfying the properties and let Rp be a TM that decides P. We show how to decide ATM using Rp by constructing TM S. First let T0 be a TM that always rejects, so L(Ta) = 0. You may assume that (TO) 0 P without loss of generality, because you could pro-ceed with P instead of P if (T0) E P. Because P is not trivial, there exists a TM T with (T) E P. Design S to decide ATM using Rp's ability to distinguish between To and T. S = "On input (M, W): 1. Use M and w to construct the following TM M,. MW, = "On input x: 1. Simulate M on w. If it halts and rejects, reject. If it accepts, proceed to stage 2. 2. Simulate T on x. If it accepts, accept." 2. Use TM RP to determine whether (Mar) E P. If YES, accept. If NO, reject." TM M. simulates T if M accepts w. Hence L(Mm) equals L(T) if M accepts w and 0 otherwise. Therefore (M, w) E P iff M accepts w. 5.30 (a) INFINITETM is a language of TM descriptions. It satisfies the two conditions of Rice's theorem. First, it is nontrivial because some TMs have infinite languages and others do not. Second, it depends only on the language. If two TMs recognize the same language, either both have descriptions in INFINITETM or neither do. Consequently, Rice's theorem implies that INFINITETM is undecidable. 242 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY EXERCISES 6.1 Give an example in the spirit of the recursion theorem of a program in a real pro-gramming language (or a reasonable approximation thereof) that prints itself out. 6.2 Show that any infinite subset of MINTM is not Turing-recognizable. A 6.3 Show that if A <T B and B <T C then A <T C. 6.4 Let ATM' = {(M, w)I M is an oracle TM and MATM accepts w}. Show that ATM' is undecidable relative to ATM. A 6.5 Is the statement IxVy [x+Y= y a member of Th(.Af,+)? Why or why not? What about the statement 3x Vy [+x =x ? PROBLEMS 6.6 Describe two different Turing machines, M and N, that, when started on any input, M outputs (N) and N outputs (M). 6.7 In the fixed-point version of the recursion theorem (Theorem 6.8) let the trans-formation t be a function that interchanges the states qaccept and qreject in Turing machine descriptions. Give an example of a fixed point for t. 6.8 Show that EQTM Am EQTM. A6 .9 Use the recursion theorem to give an alternative proof of Rice's theorem in Prob-lem 5.28. A 6 . 1 0 Give a model of the sentence 0eq = VX [ Rl (Z. x)] AqVx, [Rl(,y) x R(yx)] A Vxy,z [ (Ri (x, y) A Ri (y, z)) -R (x, z)] 6.11 Let ,eq be defined as in Problem 6.10. Give a model of the sentence Olt = Oeq AVx,y [Ri(-,y) -'R2(X,) A Vx,y [-R (x, y) -(R2 (X,Y) EDR2 (Y, ))] AVx,y,z [ (R2(X, y) A R2 (y, z)) R 2 (X, z)1 A Vxy [R2(X,Y)I. A 6 .12 Let (N, <) be the model with universe V and the "less than" relation. Show that Th(AK, <) is decidable. SELECTED SOLUTIONS 243 6.13 For each m >1 let Z,= {0, 1,2, ... ,m -1} and let Y = (Zm,+, x) be the model whose universe is Zm and that has relations corresponding to the + and x relations computed module m. Show that for each m the theory Th(.Fm) is decidable. 6.14 Show that for any two languages A and B a language J exists, where A <-X J and B < X J. 6.15 Show that for any language A, a language B exists, where A K(x) + K(y) + c. SELECTED SOLUTIONS 6.3 Say that MB decides A and M2C decides B. Use an oracle TM M3 , where M 3C decides A. Machine 113 simulates Mi. Every time M1 queries its oracle about some string ., machine M3 tests whether x C B and provides the answer to M1. Because machine M 3 doesn't have an oracle for B and cannot perform that test directly, it simulates M2 on input x to obtain that information. Machine M3 can obtain the answer to M2 's queries directly because these two machines use the same oracle, C. 6.5 The statement Hz Vy [ X+y=y ] is a member of Th(K, +) because that statement is true for the standard interpretation of + over the universe AN'. Recall that we use ' = {0, 1, 2, ... } in this chapter and so we may use r = 0. The statement =Ir Vy [ r+y=x ] is not a member of Th(A, +) because that statement isn't true in this model. For any value of x, setting y = 1 causes x+y=x to fail. 244 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY 6.9 Assume for the sake of contradiction that some TM X decides a property P, and P satisfies the conditions of Rice's theorem. One of these conditions says that TMs A and B exist where (A) C P and (B) 0 P. Use A and B to construct TM R: R = "On input w: 1. Obtain own description KR) using the recursion theorem. 2. Run X on (R). 3. If X accepts (R), simulate B on w. If X rejects (R), simulate A on w." If (R) e P, then X accepts (R) and L(R) = L(B). But (B) 0 P, contradicting (R) e P, because P agrees on TMs that have the same language. We arrive at a similar contradiction if (R) M P. Therefore our original assumption is false. Every property satisfying the conditions of Rice's theorem is undecidable. 6.10 The statement 0,, gives the three conditions of an equivalence relation. A model (A, R1), where A is any universe and R1 is any equivalence relation over A, is a model of ,eq. For example, let A be the integers Z and let R1 = {(i, i)I i G Z}. 6.12 Reduce Th(AJ, <) to Th(fV, +), which we've already shown to be decidable. To do so, show how to convert a sentence 0, over the language of Th(A(, <), to a sentence 02 over the language of Th(AF, +) while preserving truth or falsity in the respective models. Replace every occurrence of i < j in X1 by the formula ]k [ (i+k j) A (k+kok) ] in 02, where k is a different new variable each time. Sentence P2 is equivalent to X1 because "i is less than j" means that we can add a nonzero value to i and obtain j. Putting 02 into prenex-normal form, as re-quired by the algorithm for deciding Th(JV, +), requires a bit of additional work. The new existential quantifiers are brought to the front of the sentence. To do so, these quantifiers must pass through Boolean operations that appear in the sen-tence. Quantifiers can be brought through the operations of A and V without change. Passing through - changes 3 to V and vice-versa. Thus -3k 0 becomes the equivalent expression Vk -'y, and -Vk V) becomes 3k -'f . 294 CHAPTER 7 / TIME COMPLEXITY yi or zi for each i, but not both. Now we make the satisfying assignment. If the subset contains yi, we assign xi TRUE; otherwise, we assign it FALSE. This assignment must satisfy X because in each of the final k columns the sum is always 3. In column cj, at most 2 can come from gj and hj, so at least 1 in this column must come from some yi or zi in the subset. If it is yi, then xi appears in cj and is assigned TRUE, so cj is satisfied. If it is zi, then x, appears in cj and xi is assigned FALSE, so cj is satisfied. Therefore X is satisfied. Finally, we must be sure that the reduction can be carried out in polynomial time. The table has a size of roughly (k + 1)2, and each entry can be easily calculated for any X. So the total time is 0(n 2 ) easy stages. ........................................................................................................................................................................ EXERCISES 7.1 Answer each part TRUE or FALSE. a. 2n = 0(n). Ad. nlogn 0(n 2 ). b. n2 = 0(n). e. 37^ = 20(n). Ac. n2 = 0(7t log 2 n). f. 22 - 0(22") 7.2 Answer each part TRUE or FALSE. a. n = o(2n). Ad. 1 = o(n). b. 2n = o(n2). e. n = o(logn). Ac. 2' = o(3). f. 1 = o(1/n). 7.3 Which of the following pairs of numbers are relatively prime? Show the calcula-tions that led to your conclusions. a. 1274 and 10505 b. 7289 and 8029 7.4 Fill out the table described in the polynomial time algorithm for context-free lan-guage recognition from Theorem 7.16 for string w = baba and CFG G: S -RT R -TRI a T -TRIb 7.5 Is the following formula satisfiable? (xVy) A (xVy) A (iVy) A (Y Vy) 7.6 Show that P is closed under union, concatenation, and complement. PROBLEMS 295 7.7 Show that NP is closed under union and concatenation. 7.8 Let CONNECTED = K(G) G is a connected undirected graph}. Analyze the algorithm given on page 157 to show that this language is in P. 7.9 A triangle in an undirected graph is a 3-clique. Show that TRIANGLE E P, where TRIANGLE = {f(G) I G contains a triangle}. 7.10 Show that ALLDFA is in P. 7.11 Call graphs G and H isomorphic if the nodes of G may be reordered so that it is identical to H. Let ISO {(G, H) I G and H are isomorphic graphs}. Show that ISO E NP. PROBLEMS 7.12 Let MODEXP { (a, b, c, p) a, b, c, and p are binary integers such that ab - c (mod p)}. Show that MODEXP E P. (Note that the most obvious algorithm doesn't run in polynomial time. Hint: Try it first where b is a power of 2.) 7.13 A permutation on the set {1, ... , k} is a one-to-one, onto function on this set. When p is a permutation, p' means the composition of p with itself t times. Let PERM-POWER -{ (p, q, t) I p = qt where p and q are permutations on {1, . .. , k} and t is a binary integer}. Show that PERM-POWER E P. (Note that the most obvious algorithm doesn't run within polynomial time. Hint: First try it where t is a power of 2). 7.14 Show that P is closed under the star operation. (Hint: Use dynamic programming. On input y =yi ... y, for yi G A, build a table indicating for each i < j whether the substring yi ... yj e A for any A e P.) A7 .15 Show that NP is closed under the star operation. 7.16 Let UNARY-SSUM be the subset sum problem in which all numbers are repre-sented in unary. Why does the NP-completeness proof for SUBSET-SUM fail to show UNARY-SSUM is NP-complete? Show that UNARY-SSUM e P. 7.17 Show that, if P = NP, then every language A E P, except A = 0 and A = E, is NP-complete. '7.18 Show thatPRIMES -{mImisaprimenumberinbinary} e NP. (Hint: Forp > 1 the multiplicative group Z; = {xrI x is relatively prime to p and 1 < x < p} is both cyclic and of order p -1 iff p is prime. You may use this fact without justifying it. The stronger statement PRIMES G P is now known to be true, but it is more difficult to prove.) 7.19 We generally believe that PATH is not NP-complete. Explain the reason behind this belief. Show that proving PATH is not NP-complete would prove P 5 NP. 296 CHAPTER 7 / TIME COMPLEXITY 7.20 Let G represent an undirected graph. Also let SPATH { (G, a, b, k) G contains a simple path of length at most k from a to bl, and LPATHI- { (G, a, b, k) G contains a simple path of length at least k from a to b}. a. Show that SPATH C P. b. Show that LPATH is NP-complete. You may assume the NP-completeness of UHAMPATH, the Hamiltonian path problem for undirected graphs. 7.21 Let DOUBLE-SAT = { (X) I ¢ has at least two satisfying assignments}. Show that DOUBLE-SAT is NP-complete. A7 .22 Let HALF-CLIQUE = {(G)I G is an undirected graph having a complete sub-graph with at least m/2 nodes, where m is the number of nodes in G}. Show that HALF-CLIQUE is NP-complete. 7.23 Let CNFk = {(X) X is a satisfiable cnf-formula where each variable appears in at most k places}. a. Show that CNF2 G P. b. Show that CNF3 is NP-complete. 7.24 Let 0 be a 3cnf-formula. An 7&-assignment to the variables of X is one where each clause contains two literals with unequal truth values. In other words, an 7&-assignment satisfies X without assigning three true literals in any clause. a. Show that the negation of any +-assignment to 0 is also an +-assignment. b. Let 5$SAT be the collection of 3cnf-formulas that have an 7-assignment. Show that we obtain a polynomial time reduction from 3SAT to :ASAT by replacing each clause ci (Yl V Y2 V Y3) with the two clauses (Y1 v y 2 V zi) and (zT v y 3 V b), where zi is a new variable for each clause ci and b is a single additional new variable. c. Conclude that 7&SAT is NP-complete. 7.25 A cut in an undirected graph is a separation of the vertices V into two disjoint subsets S and T. The size of a cut is the number of edges that have one endpoint in S and the other in T. Let MAX-CUT = { (G, k) I G has a cut of size k or more}. Show that MAX-CUT is NP-complete. You may assume the result of Prob-lem 7.24. (Hint: Show that 7&SAT 0, such that elements of S can be colored red or blue so that no Ci has all its elements colored with the same color.} Show that SET-SPLITTING is NP-complete. 7.29 Consider the following scheduling problem. You are given a list of final exams F1 , .. . , Fk to be scheduled, and a list of students Si, . . . , SI. Each student is taking some specified subset of these exams. You must schedule these exams into slots so that no student is required to take two exams in the same slot. The problem is to determine if such a schedule exists that uses only h slots. Formulate this problem as a language and show that this language is NP-complete. 298 CHAPTER 7 / TIME COMPLEXITY 7.30 This problem is inspired by the single-player game Minesweeper, generalized to an arbitrary graph. Let G be an undirected graph, where each node either contains a single, hidden mine or is empty. The player chooses nodes, one by one. If the player chooses a node containing a mine, the player loses. If the player chooses an empty node, the player learns the number of neighboring nodes containing mines. (A neighboring node is one connected to the chosen node by an edge.). The player wins if and when all empty nodes have been so chosen. In the mine consistency problem you are given a graph G, along with numbers labeling some of G's nodes. You must determine whether a placement of mines on the remaining nodes is possible, so that any node v that is labeled m has exactly rn neighboring nodes containing mines. Formulate this problem as a language and show that it is NP-complete. A7 .31 In the following solitaire game, you are given an mn x mn board. On each of its n2 positions lies either a blue stone, a red stone, or nothing at all. You play by removing stones from the board so that each column contains only stones of a sin-gle color and each row contains at least one stone. You win if you achieve this objective. Winning may or may not be possible, depending upon the initial con-figuration. Let SOLITAIRE = { (G) I G is a winnable game configuration}. Prove that SOLITAIRE is NP-complete. 7.32 Let U = {(M, , #t) I TM M accepts input x within t steps on at least one branch}. Show that U is NP-complete. 7.33 Recall, in our discussion of the Church-Turing thesis, that we introduced the lan-guage D = { (p) I p is a polynomial in several variables having an integral root}. We stated, but didn't prove, that D is undecidable. In this problem you are to prove a different property of D-namely, that D is NP-hard. A problem is NP-hard if all problems in NP are polynomial time reducible to it, even though it may not be in NP itself. So, you must show that all problems in NP are polynomial time reducible to D. 7.34 A subset of the nodes of a graph G is a dominating set if every other node of G is adjacent to some node in the subset. Let DOMINATING-SET = {f(,G k) C G has a dominating set with k nodes}. Show that it is NP-complete by giving a reduction from VERTEX-COVER. 7.35 Show that the following problem is NP-complete. You are given a set of states Q {qo, q, . . ., q, } and a collection of pairs {(s, ri), .. ., (Sk, rk)} where the si are distinct strings over E {0, 1}, and the ri are (not necessarily distinct) members of Q. Determine whether a DFA M = (Q, Z, 6, qo, F) exists where h(qo, si) = ri for each i. Here, 6(q, s) is the state that M enters after reading a, starting at state q. (Note that F is irrelevant here). 7.36 Show that if P = NP, a polynomial time algorithm exists that produces a satisfying assignment when given a satisfiable Boolean formula. (Note: The algorithm you are asked to provide computes a function, but NP contains languages, not func-tions. The P = NP assumption implies that SAT is in P, so testing satisfiability is solvable in polynomial time. But the assumption doesn't say how this test is done, and the test may not reveal satisfying assignments. You must show that you can find them anyway. Hint: Use the satisfiability tester repeatedly to find the assignment bit-by-bit.) PROBLEMS 299 7.37 Show that if P = NP, you can factor integers in polynomial time. (See the note in Problem 7.36.) A7.3 8 Show that if P = NP, a polynomial time algorithm exists that takes an undirected graph as input and finds a largest clique contained in that graph. (See the note in Problem 7.36.) 7.39 In the proof of the Cook-Levin theorem, a window is a 2 x 3 rectangle of cells. Show why the proof would have failed if we had used 2 x 2 windows instead. 7.40 Consider the algorithm MINIMIZE, which takes a DFA M as input and outputs DFA M'. MINIMIZE = "On input (M), where M (Q. A, 6, qo, A) is a DFA: 1. Remove all states of M that are unreachable from the start state. 2. Construct the following undirected graph G whose nodes are the states of M. 3. Place an edge in G connecting every accept state with every nonaccept state. Add additional edges as follows. 4. Repeat until no new edges are added to G: 5. For every pair of distinct states q and r of M and every a E: 6. Add the edge (q, r) to G if (6(q, a), 5(r, a)) is an edge of G. 7. For each state q, let [q] be the collection of states [q] = {r c QI no edge joins q and r in G}. 8. Form a new DFA M' = (Q', A, 6', qo', A') where Q' = {[q]I q E Q}, (if [q] = [r], only one of them is in Q'), 61([q], a) = [6(q, a)], for every q e Q and a G A, go' = [go], and A' = {[q]J q E A}. 9. Output (M')." a. Show that M and M' are equivalent. b. Show that M' is minimal-that is, no DFA with fewer states recognizes the same language. You may use the result of Problem 1.52 without proof. c. Show that MINIMIZE operates in polynomial time. 7.41 For a cnf-formula X with m variables and c clauses, show that you can construct in polynomial time an NFA with O(cm) states that accepts all nonsatisfying assign-ments, represented as Boolean strings of length m. Conclude that NFAs cannot be minimized in polynomial time unless P = NP. 7.42 A 2cnf-formula is an AND of clauses, where each clause is an OR of at most two literals. Let 2SAT = { (¢) I 0 is a satisfiable 2cnf-formula}. Show that 2SAT G P. 7.43 Modify the algorithm for context-free language recognition in the proof of The-orem 7.16 to give a polynomial time algorithm that produces a parse tree for a string, given the string and a CFG, if that grammar generates the string. 7.44 Say that two Boolean formulas are equivalent if they have the same set of variables and are true on the same set of assignments to those variables (i.e., they describe the same Boolean function). A Boolean formula is minimal if no shorter Boolean formula is equivalent to it. Let MIN-FORMULA be the collection of minimal Boolean formulas. Show that, if P = NP, then MIN-FORMULA E P. 300 CHAPTER 7 / TIME COMPLEXITY 7.45 The difference hierarchy DiP is defined recursively as a. DP =NP and b. DiP={AIA=B\CforBinNPandCinDi-1P}. (Here B \ C = B n c.) For example, a language in D2P is the difference of two NP languages. Sometimes D2 P is called DP (and may be written Dv). Let Z = { (G1, k1 , G2 , k2)1 GI has a ki-clique and G2 doesn't have a k2-cliquel. Show that Z is complete for DP. In other words, show that every language in DP is polynomial time reducible to Z. 7.46 Let MALX-CLIQUE = { (G, k) I the largest clique in G is of size exactly k}. Use the result of Problem 7.45 to show that MAX-CLIQUE is DP-complete. 7.47 Let f: 'V Af be any function wheref(n) = o(n log n). Show that TIME(f(n)) contains only the regular languages. 7.48 Call a regular expression star-free if it does not contain any star operations. Let EQSF REX = { (R, S) I R and S are equivalent star-free regular expressions}. Show that EQSF-REX is in coNP. Why does your argument fail for general regular ex-pressions? 7.49 This problem investigates resolution, a method for proving the unsatisfiability of cnf-formulas. Let q5 = C 1 A C2 A ... A Cm be a formula in cnf, where the Ci are its clauses. Let C = {Ci I Ci is a clause of 0}. In a resolution step we take two clauses C. and Cb in C which both have some variable x, occurring positively in one of the clauses and negatively in the other. Thus C, = (x V Y1 V Y2 V ... V Yk) and Cb = (x V zi V Z2 V ... V zt), where the y, and zi are literal. We form the new clause (yi V Y2 V Y. Vk z1 v Z2 V ... V Z1) and remove repeated literals. Add this new clause to C. Repeat the resolution steps until no additional clauses can be obtained. If the empty clause ( ) is in C then declare 0 unsatisfiable. Say that resolution is sound if it never declares satisfiable formulas to be unsatisfi-able. Say that resolution is complete if all unsatisfiable formulas are declared to be unsatisfiable. a. Show that resolution is sound and complete. b. Use part (a) to show that 2SAT e P. SELECTED SOLUTIONS 7.1 (c)FALSE; (d)TRUE. 7.2 (c) TRUE; (d) TRUE. SELECTED SOLUTIONS 301 7.15 Let A G NP. Construct NTM M to decide A in nondeterministic polynomial time. M = "On input w: 1. Nondeterministically divide w into pieces w =XI 2 ... Xk. 2. For each xi, nondeterministically guess the certificates that show xi e A. 3. Verify all certificates if possible, then accept. Otherwise if verification fails, reject." 7.22 We give a polynomial time mapping reduction from CLIQUE to HALF-CLIQUE. The input to the reduction is a pair (G, k) and the reduction produces the graph (H) as output where H is as follows. If G has m nodes and k = m/2 then H = G. If k < m/2, then H is the graph obtained from G by adding j nodes, each connected to every one of the original nodes and to each other, where j = m -2k. Thus H has m + j 2m - 2k nodes. Observe that G has a k-clique iff H has a clique of size k +j m -k and so (G, k) E CLIQUE iff (H) c HALF-CLIQUE. If k > 2m, then H is the graph obtained by adding j nodes to G without any additional edges, where j = 2k -m. Thus H has m + j = 2k nodes, and so G has a k-clique iff H has a clique of size k. Therefore (G, kk) C CLIQUE iff (H) e HALF-CLIQUE. We also need to show HALF-CLIQUE E NP. The certificate is simply the clique. 7.31 First, SOLITAIRE E NP because we can verify that a solution works, in polynomial time. Second, we show that 3SAT n, the space complexity class SPACE(f (n)) is the same whether you define the class by using the single-tape TM model or the two tape read-only input TM model. 8.2 Consider the following position in the standard tic-tac-toe game. Let's say that it is the X -player's turn to move next. Describe the winning strategy for this player. (Recall that a winning strategy isn't merely the best move to make in the current position. It also includes all the responses that this player must make in order to win, however the opponent moves.) 8.3 Consider the following generalized geography game wherein the start node is the one with the arrow pointing in from nowhere. Does Player I have a winning strat-egy? Does Player II? Give reasons for your answers. 8.4 Show that PSPACE is closed under the operations union, complementation, and star. A8 .5 Show that NL is closed under the operations union, intersection, and star. 8.6 Show that any PSPACE-hard language is also NP-hard. A 8.7 Show that ADFA E L. PROBLEMS 8.8 Let EQREX = { (R, S) I R and S are equivalent regular expressions}. Show that EQREX e PSPACE. 329 330 CHAPTER 8 / SPACE COMPLEXITY 8.9 A ladder is a sequence of strings si, 82, . .. , Sk, wherein every string differs from the preceding one in exactly one character. For example the following is a ladder of English words, starting with "head" and ending with "free": head, hear, near, fear, bear, beer, deer, deed, feed, feet, fret, free. Let LADDERDFA = { (M. a, t) I M is a DFA and L(M) contains a ladder of strings, starting with s and ending with t}. Show that LADDERDFA is in PSPACE. 8.10 The Japanese game go-moku is played by two players, "X" and "O," on a 19 x 19 grid. Players take turns placing markers, and the first player to achieve 5 of his markers consecutively in a row, column, or diagonal, is the winner. Consider this game generalized to an n x n board. Let GM { (B) I B is a position in generalized go-moku, where player "X" has a winning strategy}. By a position we mean a board with markers placed on it, such as may occur in the middle of a play of the game, together with an indication of which player moves next. Show that GM C PSPACE. 8.11 Show that, if every NP-hard language is also PSPACE-hard, then PSPACE = NP. 8.12 Show that TQBF restricted to formulas where the part following the quantifiers is in conjunctive normal form is still PSPACE-complete. 8.13 Define ALBA = {(M, w) I M is an LBA that accepts input w}. Show that ALBA is PSPACE-complete. 8.14 Consider the following two-person version of the language PUZZLE that was de-scribed in Problem 7.26. Each player starts with an ordered stack of puzzle cards. The players take turns placing the cards in order in the box and may choose which side faces up. Player I wins if, in the final stack, all hole positions are blocked, and Player II wins if some hole position remains unblocked. Show that the problem of determining which player has a winning strategy for a given starting configuration of the cards is PSPACE-complete. 8.15 The cat-and-mouse game is played by two players, "Cat" and "Mouse," on an arbi-trary undirected graph. At a given point each player occupies a node of the graph. The players take turns moving to a node adjacent to the one that they currently occupy. A special node of the graph is called "Hole." Cat wins if the two players ever occupy the same node. Mouse wins if it reaches the Hole before the preceding happens. The game is a draw if a situation repeats (i.e., the two players simultane-ously occupy positions that they simultaneously occupied previously and it is the same player's turn to move). HAPPY-CAT {(G,c, m,h)I G,c,om,h, are respectivelya graph, and positions of the Cat, Mouse, and Hole, such that Cat has a winning strategy if Cat moves first}. Show that HAPPY-CAT is in P. (Hint: The solution is not complicated and doesn't depend on subtle details in the way the game is defined. Consider the entire game tree. It is exponentially big, but you can search it in polynomial time.) PROBLEMS 331 8.16 Read the definition of MIN-FORMULA in Problem 7.44. a. Show that MIN-FORMULA E PSPACE. b. Explain why this argument fails to show that MIN-FORMULA C coNP: If 0 f MIN-FORMULA, then 0 has a smaller equivalent formula. An NTM can verify that X E MIN-FORMULA by guessing that formula. 8.17 Let A be the language of properly nested parentheses. For example, (0) and ( (0()) () are in A, but ) ( is not. Show that A is in L. 8.18 Let B be the language of properly nested parentheses and brackets. For example, ( [() 0(] () [] ) is in B but ( [)] is not. Show that B is in L. 8.19 The game of Nim is played with a collection of piles of sticks. In one move a player may remove any nonzero number of sticks from a single pile. The players alternately take turns making moves. The player who removes the very last stick loses. Say that we have a game position in Nim with k piles containing s1 , -. ., Sk sticks. Call the position balanced if, when each of the numbers si is written in binary and the binary numbers are written as rows of a matrix aligned at the low order bits, each column of bits contains an even number of is. Prove the following two facts. a. Starting in an unbalanced position, a single move exists that changes the position into a balanced one. b. Starting in a balanced position, every single move changes the position into an unbalanced one. Let NIM ={ (81, . . ., Sk) I each si is a binary number and Player I has a winning strategy in the Nim game starting at this position}. Use the preceding facts about balanced positions to show that NIM C L. 8.20 Let MULT = {a#b#cl where a, b, c are binary natural numbers and a x b = c}. Show that MULT e L. 8.21 For any positive integer x, let xz be the integer whose binary representation is the reverse of the binary representation of x. (Assume no leading Os in the binary representation of x.) Define the function 7e+: .V-AB where R+ (x) = + xl. a. Let A 2 = {(x, y) I 7Z+ (x) = y}. Show A2 E L. b. Let A 3 = {(,y)|I IZ+ (+(x)) = y}. Show A 3 G L. 8.22 a. Let ADD = { (x, y, z) I x, y, z > 0 are binary integers and x + y z}. Show that ADD e L. b. Let PAL-ADD {(x, y)I x, y > 0 are binary integers where x + y is an integer whose binary representation is a palindrome}. (Note that the binary representation of the sum is assumed not to have leading zeros. A palin-drome is a string that equals its reverse). Show that PAL-ADD C L. 8.23 Define UCYCLE = {(G)I G is an undirected graph that contains a simple cycle}. Show that UCYCLE e L. (Note: G may be a graph that is not connected.) 8.24 For each n, exhibit two regular expressions, R and S, of length poly(n), where L(R) # L(S), but where the first string on which they differ is exponentially long. In other words, L(R) and L(S) must be different, yet agree on all strings of length 2" for some constant e > 0. 332 CHAPTER 8 / SPACE COMPLEXITY 8.25 An undirected graph is bipartite if its nodes may be divided into two sets so that all edges go from a node in one set to a node in the other set. Show that a graph is bipartite if and only if it doesn't contain a cycle that has an odd number of nodes. Let BIPARTITE = { (G) I G is a bipartite graph} . Show that BIPARTITE C NL. 8.26 Define UPATH to be the counterpart of PATH for undirected graphs. Show that BIPARTITE <L UPATH. (Note: As this edition was going to press, 0. Rein-gold announced that UPATH G L. Consequently, BIPARTITE G L, but the algorithm is somewhat complicated.) 8.27 Recall that a directed graph is strongly connected if every two nodes are connected by a directed path in each direction. Let STRONGLY-CONNECTED ={ (G) I G is a strongly connected graph}. Show that STRONGLY-CONNECTED is NL-complete. 8.28 Let BOTHNFA = {(Ml,M 2 )1 M1 andM2 are NFAs where L(Ml) n L(M2 ) 78 0}. Show that BOTHNFA is NL-complete. 8.29 Show that ANFA is NL-complete. 8.30 Show that EDFA is NL-complete. 8.31 Show that 2SAT is NL-complete. 8.32 Give an example of an NL-complete context-free language. A 8. 33 Define CYCLE= {(G)I G is a directed graph that contains a directed cycle}. Show that CYCLE is NL-complete. SELECTED SOLUTIONS 8.5 Let Ai and A 2 be languages that are decided by NL-machines N1 and N2. Con-struct three Turing machines: N, deciding Al U A 2; No deciding Al o A 2; and N. deciding AT. Each of these machines receives input w. Machine Nu nondeterministically branches to simulate N1 or to simulate N2 . In either case, Nu accepts if the simulated machine accepts. Machine N. nondeterministically selects a position on the input to divide it into two substrings. Only a pointer to that position is stored on the work tape-insufficient space is available to store the substrings themselves. Then N. simulates N1 on the first substring, branching nondeterministically to simulate N1 's nonde-terminism. On any branch that reaches N1 's accept state, N. simulates N2 on the second substring. On any branch that reaches N2 's accept state, N. accepts. Machine N. has a more complex algorithm, so we describe its stages. N, = "On input w: 1. Initialize two input position pointers pi and P2 to 0, the position immediately preceding the first input symbol. 2. Accept if no input symbols occur after p2. 3. Move p2 forward to a nondeterministically selected input posi-tion. SELECTED SOLUTIONS 333 4. Simulate N1 on the substring of w from the position following pi to the position at P2, branching nondeterministically to sim-ulate N1 's nondeterminism. 5. If this branch of the simulation reaches N1 's accept state, copy P2 to pi and go to stage 2." 8.7 Construct a TM M to decide ADFA. When M receives input (A, w), a DFA and a string, M simulates A on w by keeping track of A's current state and its current head location, and updating them appropriately. The space required to carry out this simulation is O(log n) because M can record each of these values by storing a pointer into its input. 8.33 Reduce PATH to CYCLE. The idea behind the reduction is to modify the PATH problem instance (G, s, t) by adding an edge from t to s in G. If a path exists from s to t in G, a directed cycle will exist in the modified G. However, other cycles may exist in the modified G because they may already be present in G. To handle that problem, first change G so that it contains no cycles. A leveled directed graph is one where the nodes are divided into groups, A1 , A2 ... . Ak, called levels, and only edges from one level to the next higher level are permitted. Observe that a leveled graph is acyclic. The PATH problem for leveled graphs is still NL-complete, as the following reduction from the unrestricted PATH problem shows. Given a graph G with two nodes s and t, and m nodes in total, produce the leveled graph G' whose levels are m copies of G's nodes. Draw an edge from node i at each level to node j in the next level if G contains an edge from i to j. Additionally, draw an edge from node i in each level to node i in the next level. Let s' be the node s in the first level and let t' be the node t in the last level. Graph G contains a path from s to t iff G' contains a path from s' to t'. If you modify G' by adding an edge from t' to s', you obtain a reduction from PATH to CYCLE. The reduction is computationally sim-ple, and its implementation in logspace is routine. Furthermore, a straightforward procedure shows that CYCLE E NL. Hence CYCLE is NL-complete. EXERCISES 361 EXERCISES A 9 .1 Prove that TIME(2') TIME(2"'). A 9.2 Prove that TIME(2') c TIME(22 ,). A 9.3 Prove that NTIME(n) C PSPACE. 9.4 Show how the circuit depicted in Figure 9.26 computes on input 0110 by showing the values computed by all of the gates, as we did in Figure 9.24. 9.5 Give a circuit that computes the parity function on three input variables and show how it computes on input 011. 9.6 Prove that if A E P then pA = p. 9.7 Give regular expressions with exponentiation that generate the following languages over the alphabet {0,1}. Aa. All strings of length 500 Ab. All strings of length 500 or less AC. All strings of length 500 or more Ad. All strings of length different than 500 e. All strings that contain exactly 500 is f. All strings that contain at least 500 is g. All strings that contain at most 500 is h. All strings of length 500 or more that contain a 0 in the 500th position i. All strings that contain two Os that have at least 500 symbols between them 9.8 If R is a regular expression, let R{mn} represent the expression Rm U Rm+lU ... U R. Show how to implement the RIm"I} operator, using the ordinary exponentiation operator, but without " . 9.9 Show that if NP = pSAT, then NP = coNP. 9.10 Problem 8.13 showed that ALBA is PSPACE-complete. a. Do we know whether ALBA E NL? Explain your answer. b. Do we know whether ALBA E P? Explain your answer. 9.11 Show that the language MAX-CLIQUE from Problem 7.46 is in PST. PROBLEMS 9.12 Describe the error in the following fallacious "proof" that P+NP. Assume that P=NP and obtain a contradiction. If P=NP, then SAT C P and so for some k, SATE TIME(nk). Because every language in NP is polynomial time reducible to SAT, you have NP C TIME(nk). Therefore P C TIME(nk). But, by the time hierarchy theorem, TIME(nk+1) contains a language that isn't in TIME(nk), which contradicts P C TIME(nk). Therefore P 5# NP. 362 CHAPTER 9 / INTRACTABILITY 9.13 Consider the function pad: E x JV-E# that is defined as follows. Let pad (s, 1) = s#j, where j = max(O, 1 - m) and m is the length of s. Thus pad(s, 1) simply adds enough copies of the new symbol # to the end of s so that the length of the result is at least 1. For any language A and function f: .V-\ N define the language pad(A, f(m)) as pad(A, f(m)) = {pad(s, f(m))I where s e A and m is the length of s}. Prove that, if A G TIME(n 6), then pad(A, n2 ) C TIME(n3). 9.14 Prove that, if NEXPTIME :A EXPTIME, then P # NP. You may find the function pad, defined in Problem 9.13, to be helpful. A9 .1 5 Define pad as in Problem 9.13. a. Prove that, for every A and natural number k, A C P iff pad(A, nk) P. b. Prove that P 4 SPACE(n). 9.16 Prove that TQBF , SPACE(n' /3) '9.17 Read the definition of a 2DFA (two-headed finite automaton) given in Prob-lem 5.26. Prove that P contains a language that is not recognizable by a 2DFA. 9.18 Let EREO-T {(R) R is a regular expression with exponentiation and L(R) = 0}. Show that EREGC C P. 9.19 Define the unique-sat problem to be USAT = { (0) I ¢ is a Boolean formula that has a single satisfying assignment}. Show that USAT C pSAT. 9.20 Prove that an oracle C exists for which NPC 7 coNPC. 9.21 A k-query oracle Turing machine is an oracle Turing machine that is permitted to make at most k queries on each input. A k-oracle TM M with an oracle for A is written MA k and pAtk is the collection of languages that are decidable by polynomial time k-oracle A TMs. a. Show that NP U coNP C pSATjl b. Assume that NP 54 coNP. Show that P U coNP c pSATl, 9.22 Suppose that A and B are two oracles. One of them is an oracle for TQBF, but you don't know which. Give an algorithm that has access to both A and B and that is guaranteed to solve TQBF in polynomial time. 9.23 Define the function parity, as in Example 9.25. Show that parity, can be com-puted with 0(n) size circuits. 9.24 Recall that you may consider circuits that output strings over {0,1} by designating several output gates. Let add : {0,1} 2,-_ {O,10, 1 take the sum of two n bit binary integers and produce the n + 1 bit result. Show that you can compute the add, function with 0(n) size circuits. SELECTED SOLUTIONS 363 9.25 Define the function majority.: {O,1}"-{O,1} as majority,(-i, . X.7)= { xi <n/2; Thus the majority function returns the majority vote of the inputs. Show that majority can be computed with a. 0(n 2 ) size circuits. b. 0(n log n) size circuits. (Hint: Recursively divide the number of inputs in half and use the result of Problem 9.24.) 9.26 Define the problem majority, as in Problem 9.25. Show that it may be computed with 0(n) size circuits. SELECTED SOLUTIONS 9.1 The time complexity classes are defined in terms of the big-0 notation, so constant factors have no effect. The function 2 n1 is 0(2n) and thus A c TIME(2n) iff A c TIME(2n+'). 9.2 The containment TIME(2") C TIME(2 2") holds because 2' < 22,. The con-tainment is proper by virtue of the time hierarchy theorem. The function 22, is time constructible because a TM can write the number 1 followed by 2n Os in 0(22,) time. Hence the theorem guarantees that a language A exists that can be decided in 0(22,) time but not in 0(22n/ log 22n) = 0(22n/2n) time. Therefore A E TIME(22n) but A X TIME(2n). 9.3 NTIME(n) C NSPACE(n) because any Turing machine that operates in time t(n) on every computation branch can use at most t(n) tape cells on every branch. Furthermore NSPACE(n) C SPACE(n2 ) due to Savitch's theorem. However, SPACE(n 2 ) C SPACE(n3 ) because of the space hierarchy theorem. The result follows because SPACE(n3 ) C PSPACE. 9.7 (a) E500; (b) (E U E) 500; (c) E500; (d) (E Ue )4 9 9 u E501E 9.15 (a) Let A be any language and k c AK. If A G P, then pad(A, nk) e P because you can determine whether w G pad(A, nk) by writing w as s#1 where s doesn't contain the # symbol, then testing whether IwI = Slk, and finally testing whether s E A. Implementing the first test in polynomial time is straightforward. The second test runs in time poly(jwl), and because Jwl is poly(lsl), the test runs in time poly(IsD) and hence is in polynomial time. If pad(A, nk) E P, then A e P because you can determine whether w ( A by padding w with # symbols until it has length WIk and then testing whether the result is in pad(A, nk). Both of these tests require only polynomial time. 364 CHAPTER 9 / INTRACTABILITY (b) Assume that P = SPACE(n). Let A be a language in SPACE(n2 ) but not in SPACE(n) as shown to exist in the space hierarchy theorem. The language pad(A, n2 ) C SPACE(n) because you have enough space to run the 0(n 2 ) space algorithm for A, using space that is linear in the padded language. Because of the assumption, pad(A, n2) E P, hence A G P by part (a), and hence A E SPACE(n), due to the assumption once again. But that is a contradiction. EXERCISES 411 We can use a trapdoor function such as the RSA trapdoor function, to con-struct a public-key cryptosystem as follows. The public key is the index i gener-ated by the probabilistic machine G. The secret key is the corresponding value t. The encryption algorithm breaks the message m into blocks of size at most log N. For each block w the sender computes fi. The resulting sequence of strings is the encrypted message. The receiver uses the function h to obtain the original message from its encryption. EXERCISES 10.1 Show that a circuit family with depth 0(log n) is also a polynomial size circuit family. 10.2 Show that 12 is not pseudoprime because it fails some Fermat test. 10.3 Prove that, if A <L B and B is in NC, then A is in NC. 10.4 Show that the parity function with n inputs can be computed by a branching pro-gram that has 0(n) nodes. 10.5 Show that the majority function with n inputs can be computed by a branching program that has 0(n2 ) nodes. 10.6 Show that any function with n inputs can be computed by a branching program that has 0(2n) nodes. A10.7 Show that BPP C PSPACE. PROBLEMS 10.8 Let A be a regular language over {0,1}. Show that A has size-depth complexity (O(n), 0(log n)). '10.9 A Boolean formula is a Boolean circuit wherein every gate has only one output wire. The same input variable may appear in multiple places of a Boolean formula. Prove that a language has a polynomial size family of formulas iff it is in NC'. Ignore uniformity considerations. 10.10 A k-head pushdown automaton (k-PDA) is a deterministic pushdown automaton with k read-only, two-way input heads and a read/write stack. Define the class PDAk = {AI A is recognized by a k-PDA}. Show that P = Uk PDAk. (Hint: Recall that P equals alternating log space.) 412 CHAPTER 10/ ADVANCED TOPICS IN COMPLEXITY THEORY 10.11 Let M be a probabilistic polynomial time Turing machine and let C be a language where, for some fixed 0 < cl < 62 < 1, a. w 9' C implies Pr[M accepts w] < ci, and b. w e C implies Pr[M accepts w] > 62. Show that C E BPP. (Hint: Use the result of Lemma 10.5.) 10.12 Show that, if P = NP, then P = PH. 10.13 Show that, if PH = PSPACE, then the polynomial time hierarchy has only finitely many distinct levels. 10.14 Recall that NPSAT is the class of languages that are decided by nondeterminis-tic polynomial time Turing machines with an oracle for the satisfiability problem. Show that NPSAT = S 2P. 10.15 Prove Fermat's little theorem, which is given in Theorem 10.6. (Hint: Consider the sequence a', a2 .... What must happen, and how?) A 10.16 Prove that, for any integer p > 1, if p isn't pseudoprime, then p fails the Fermat test for at least half of all numbers in Z,. 10.17 Prove that, if A is a language in L, a family of branching programs (B1 , B 2,... exists wherein each Bw accepts exactly the strings in A of length n and is bounded in size by a polynomial in n. 10.18 Prove that, if A is a regular language, a family of branching programs (Bi, B 2 ,... exists wherein each B,, accepts exactly the strings in A of length n and is bounded in size by a constant times n. 10.19 Show that, if NP C BPP then NP = RP. 10.20 Define a ZPP-machine to be a probabilistic Turing machine which is permitted three types of output on each of its branches: accept, reject, and ?. A ZPP-machine M decides a language A if M outputs the correct answer on every input string w (accept if w E A and reject if w X A) with probability at least 2 and M never outputs the wrong answer. On every input, M may output ? with probability at most j. Furthermore, the average running time over all branches of M on ul must be bounded by a polynomial in the length of to. Show that RP n coRP = ZPP. 10.21 Let EQBP {(B1 , B 2)1 B1 and B2 are equivalent branching programs}. Show that EQBP is coNP-complete 10.22 Let BPL be the collection of languages that are decided by probabilistic log space Turing machines with error probability 3. Prove that BPL C P. SELECTED SOLUTIONS 10.7 If M is a probabilistic TM that runs in polynomial time, we can modify M so that it makes exactly no coin tosses on each branch of its computation, for some con-stant r. Thus the problem of determining the probability that M accepts its input string reduces to counting how many branches are accepting and comparing this number with 12(nr). This count can be performed by using polynomial space. SELECTED SOLUTIONS 413 10.16 Call a a witness if it fails the Fermat test for p, that is, if aP 0 1 (mod p). Let Z; be all numbers in { 1, . . l, p- } that are relatively prime to p. If p isn't pseudoprime, it has a witness a in Z;. Use a to get many more witnesses. Find a unique witness in Zp for each nonwit-ness. If d G Z is a nonwitness, you have dP-1 1 (mod p). Hence da mod p $ 1 (mod p) and so da mod p is a witness. If d1 and d2 are distinct nonwitnesses in zP then dia mod p :$ d2 a mod p. Otherwise (d -d2)a -0 (mod p), and thus (d1 - d2 )a = Cp for some integer c. But di and d2 are in Zp, and thus (di - d2 ) < P. so a = cp/(di -d 2) and p have a factor greater than 1 in common, which is im-possible because a and p are relatively prime. Thus the number of witnesses in Z, must be as large as the number of nonwitnesses in Z and consequently at least half of the members of Z are witnesses. Next show that every member b of Zp that is not relatively prime to p is a witness. If b and p share a factor, then be and p share that factor for any e > 0. Hence bP-1 $ 1 (mod p). Therefore you can conclude that at least half of the member of Zp are witnesses. SELECTED BIBLIOGRAPHY I. ADLEMAN, L. Two theorems on random polynomial time. In Proceedings of the Nineteenth IEEE Symposium on Foundations of Computer Science (1978), pp.7 5 -83. 2. ADLEMAN, L. M., AND HUANG, M. A. Recognizing primes in random polynomial time. In Proceedings of the Nineteenth Annual ACM Symposium on the Theory of Computing (1987), pp. 462-469. 3. ADLEMAN, L. M., POMERANCE, C., AND RUMELY, R. S. On distinguish-ing prime numbers from composite numbers. Annals of Mathematics 117 (1983),173-206. 4. AGRAWAL, M., KAYAL, N., AND SAXENA, N. PRIMES is in P. (2002), 5. AHo, A. V, HOPCROFT, J. E., AND ULLMAN, J. D. Data Structures and Algorithms. Addison-Wesley, 1982. 6. AHO, A. V., SETHI, R., AND ULLMAN, J. D. Compilers: Principles, Tech-niques, Tools. Addison-Wesley, 1986. 7. AKL, S. G. The Design and Analysis of Parallel Algorithms. Prentice-Hall International, 1989. 8. ALON, N., ERDOS, P., AND SPENCER, J. H. The Probabilistic Method. John Wiley & Sons, 1992. 9. ANGLUIN, D., AND VALIANT, L. G. Fast probabilistic algorithms for Hamiltonian circuits and matchings. Journal of Computer and System Sciences 18 (1979),155-193. 10. ARORA, S., LUND, C., MOTWANI, R., SUDAN, M., AND SZEGEDY, M. Proof verification and hardness of approximation problems. In Proceedings of the Thirty-third IEEE Symposium on Foundations of Computer Science (1992), pp.14-23. 11. BAASE, S. Computer Algorithms: Introduction to Design and Analysis. Addison-Wesley, 1978. 12. BABAI, L. E-mail and the unexpected power of interaction. In Proceedings of the Fifth Annual Conference on Structure in Complexity Theory (1990), pp. 30-44. 13. BACH, E., AND SHALLIT, J. Algorithmic Number Theory, Vol. 1. MIT Press, 1996. 415 416 14. BALCAZAR, J. L., DIAZ, J., AND GABARR6, J. Structural Complexity I, II. EATCS Monographs on Theoretical Computer Science. Springer Verlag, 1988 (I) and 1990 (II). 15. BEA-ME, P. W., COOK, S. A., AND HOOVER, H. J. Log depth circuits for division and related problems. SIAM Journal on Computing 15, 4 (1986), 994-1003. 16. BLUM, M., CHANDRA, A., AND WEGMAN, M. Equivalence of free boolean graphs can be decided probabilistically in polynomial time. Information Pro-cessing Letters 10 (1980), 80-82. 17. BRASSARD, G., AND BRATLEY, P. Algorithmics: Theory and Practice. Pren-tice-Hall, 1988. 18. CARMICHAEL, R. D. On composite numbers p which satisfy the Fermat congruence aPe 1- p. American Mathematical Monthly 19 (1912), 22-2 7. 19. CHOMSKY, N. Three models for the description of language. IRE Trans. on Information Theory 2 (1956), 113-124. 20. COBHAm, A. The intrinsic computational difficulty of functions. In Pro-ceedings of the International Congress for Logic, Methodology, and Philosophy of Science, Y. Bar-Hillel, Ed. North-Holland, 1964, pp. 24-30. 21. COOK, S. A. The complexity of theorem-proving procedures. In Proceed-ings of the Third AnnualACM Symposium on the Theory of Computing (1971), pp. 151-158. 22. CORMEN, T., LEISERSON, C., AND RIVEST, R. Introduction to Algorithms. MIT Press, 1989. 23. EDMONDs, J. Paths, trees, and flowers. Canadian Journal of Mathematics 17 (1965), 449-467. 24. ENDERTON, H. B. A Mathematical Introduction to Logic. Academic Press, 1972. 25. EVEN, S. Graph Algorithms. Pitman, 1979. 26. FELLER, W. An Introduction to Probability Theory and Its Applications, Vol. 1. John Wiley & Sons, 1970. 27. FEYNMAN, R. P., HEY, A. J. G., AND ALLEN, R. W Feynman lectures on computation. Addison-Wesley, 1996. 28. GAREY, M. R., AND JOHNSON, D. S. Computers and Intractability-A Guide to the Theory of NP-completeness. W. H. Freeman, 1979. 29. GILL, J. T. Computational complexity of probabilistic Turing machines. SIAMJournal on Computing 6, 4 (1977), 675-695. 30. GODEL, K. On formally undecidable propositions in Principia Mathematica and related systems I. In The Undecidable, M. Davis, Ed. Raven Press, 1965, pp. 4-38. 31. GOEMANS, M. X., AND WILLIAMSON, D. P. .878-approximation algo-rithms for MAX CUT and MAX 2SAT. In Proceedings of the Twenty-sixth Annual ACM Symposium on the Theory of Computing (1994), pp. 422-431. 32. GOLDWASSER, S., AND MICALI, S. Probabilistic encryption. Journal of Computerand System Sciences (1984), 270-229. 417 33. GOLDWASSER, S., MICALI, S., AND RACKOFF, C. The knowledge com-plexity of interactive proof-systems. SIAM journal on Computing (1989), 186-208. 34. GREENLAW, R., HOOVER, H. J., AND Ruzzo, W. L. Limits to Parallel Computation: P-completeness Theory. Oxford University Press, 1995. 35. HARARY, F. Graph Theory, 2d ed. Addison-Wesley, 1971. 36. HARTMANIS, J., AND STEARNS, R. E. On the computational complexity of algorithms. Transactions of the American Mathematical Society 117 (1965), 285-306. 37. HILBERT, D. Mathematical problems. Lecture delivered before the In-ternational Congress of Mathematicians at Paris in 1900. In Mathematical Developments Arisingfrom Hilbert Problems, vol. 28. American Mathematical Society, 1976, pp. 1-34. 38. HOFSTADTER, D. R. Goedel, Escher, Bach: An Eternal Golden Braid. Basic Books, 1979. 39. HOPCROFT, J. E., AND ULLMAN, J. D. Introduction to Automata Theory, Languages and Computation. Addison-Wesley, 1979. 40. JOHNSON, D. S. The NP-completeness column: Interactive proof systems for fun and profit. Journal ofAlgorithms 9, 3 (1988), 426-444. 41. KARP, R. M. Reducibility among combinatorial problems. In Complexity of Computer Computations (1972), R. E. Miller and J. W Thatcher, Eds., Plenum Press, pp. 85-103. 42. KARP, R. M., AND LIPTON, R. J. Turing machines that take advice. EN-SEIGN: L'Enseignement Mathematique Revue Internationale 28 (1982). 43. LAWLER, E. L. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, 1991. 44. LAWLER, E. L., LENSTRA, J. K., RINooY KAN, A. H. G., AND SHMOYS, D. B. The Traveling Salesman Problem. John Wiley & Sons, 1985. 45. LEIGHTON, F. T. Introduction to Parallel Algorithms and Architectures: Array, Trees, Hypercubes. Morgan Kaufinann, 1991. 46. LEVIN, L. Universal search problems (in Russian). Problemy Peredachi Infor-matsii 9, 3 (1973), 115-116. 47. LEWIS, H., AND PAPADIMITRIOU, C. Elements ofthe Theory of Computation. Prentice-Hall, 1981. 48. Li, M., AND VITANYI, P. Introduction to Kolmogorov Complexity and its Appli-cations. Springer-Verlag, 1993. 418 49. LICHTENSTEIN, D., AND SIPSER, M. GO is PSPACE hard. Journal of the ACM (1980), 393-401. 50. LUBY, M. Pseudorandomness and Cryptographic Applications. Princeton Uni-versity Press, 1996. 51. LUND, C., FORTNOw, L., KARLOFF, H., AND NISAN, N. Algebraic meth-ods for interactive proof systems. Journal of the ACM 39, 4 (1992), 859-868. 52. MILLER, G. L. Riemann's hypothesis and tests for primality. Journal of Computer and System Sciences 13 (1976), 300-317. 53. NIVEN, I., AND ZUCKERMAN, H. S. An Introduction to the Theory of Num-bers, 4th ed. John Wiley & Sons, 1980. 54. PAPADIMITRIOU, C. H. Computational Complexity. Addison-Wesley, 1994. 55. PAPADIMITRIOU, C. H., AND STEIGLITZ, K. Combinatorial Optimization (Algorithms and Complexity). Prentice-Hall, 1982. 56. PAPADIMITRIOU, C. H., AND YANNAKAKIS, M. Optimization, approxi-mation, and complexity classes. Journal of Computer and System Sciences 43, 3 (1991), 425-440. 57. POMERANCE, C. On the distribution of pseudoprimes. Mathematics of Com-putation 37, 156 (1981), 587-593. 58. PRATT, V. R. Every prime has a succinct certificate. SIAMJournal on Com-puting4, 3 (1975), 214-220. 59. RABIN, M. 0. Probabilistic algorithms. In Algorithms and Complexity: New Directions and Recent Results, J. E Traub, Ed. Academic Press, 1976, pp. 21-39. 60. REINGOLD, 0. Undirected st-connectivity in log-space. (2004), 61. RIVEST, R. L., SHAMIR, A., AND ADLEMAN, L. A method for obtaining digital signatures and public key cryptosytems. Communications of the ACM 21, 2 (1978), 120-126. 62. ROCHE, E., AND SCHABES, Y. Finite-State Language Processing. MIT Press, 1997. 63. SCHAEFER, T. J. On the complexity of some two-person perfect-infor-mation games. Journal of Computer and System Sciences 16, 2 (1978), 185-225. 64. SEDGEWICK, R. Algorithms, 2d ed. Addison-Wesley, 1989. 65. SHAMIR, A. IP = PSPACE. Journal of theACM 39, 4 (1992), 869-877. 66. SHEN, A. IP = PSPACE: Simplified proof. Journal of the ACM 39, 4 (1992), 878-880. 67. SHOR, P. W Polynomial-time algorithms for prime factorization and dis-crete logarithms on a quantum computer. SJAMJournal on Computing 26, (1997), 1484-1509. 68. SIPSER, M. Lower bounds on the size of sweeping automata. Journal of Computer and System Sciences 21, 2 (1980), 195-202. 419 69. SIPSER, M. The history and status of the P versus NP question. In Proceed-ings of the Twenty-fourth Annual ACM Symposium on the Theory of Computing (1992), pp. 603-618. 70. STINSON, D. R. Cryptography: Theory and Practice. CRC Press, 1995. 71. TARJAN, R. E. Data structures and network algorithms, vol. 44 of CBMS-NSF Reg. Conf. Ser. Appl. Math. SIAM, 1983. 72. TURING, A. M. On computable numbers, with an application to the Entscheidungsproblem. In Proceedings, London Mathematical Society, (1936), pp. 230-265. 73. ULLMAN, J. D., AHO, A. V., AND HOPCROFT, J. E. The Design and Analysis of Computer Algorithms. Addison-Wesley, 1974. 74. VAN LEEUWEN, J., Ed. Handbook of Theoretical Computer Science A: Algo-rithms and Complexity. Elsevier, 1990. INDEX Symbols (natural numbers), 4, 227 (real numbers), 157, 176 (nonnegative real numbers), 249 (empty set), 4 (element), 3 (not element), 3 (subset), 3 (proper subset), 4 (proper subset), 328 (union operation), 4, 44 (intersection operation), 4 (Cartesian or cross product), 6 (empty string), 13 (reverse of w), 14 (negation operation), 14 (conjunction operation), 14 (disjunction operation), 14 (exclusive OR operation), 15 (implication operation), 15 (equality operation), 15 (reverse implication), 18 = (implication), 18 (logical equivalence), 18 o (concatenation operation), 44 (star operation), 44 + (plus operation), 65 P(Q) (power set), 53 E (alphabet), 53 ( U {e}), 53 (.) (encoding), 157, 259 u (blank), 140 <m (mapping reduction), 207 <T (Turing reduction), 233 <L (log space reduction), 324 <p (polynomial time reduction), 272 d(z) (minimal description), 236 Th(M) (theory of model), 226 K(x) (descriptive complexity), 236 V (universal quantifier), 310 3 (existential quantifier), 310 { (exponentiation), 343 O(f (n)) (big-O notation), 249-250 o(f (n)) (small-o notation), 250 421 1z -R+ 0 c c C U n x e U7? A V (T 422 INDEX Accept state, 34, 35 Acceptance problem for CFG, 170 for DFA, 166 for LBA, 194 for NFA, 167 for TM, 174 Accepting computation history, 193 Accepting configuration, 141 Accepts a language, meaning of, 36 ACFG, 170 Acyclic graph, 376 ADFA, 166 Adjacency matrix, 259 Adleman, Leonard M., 415, 418 Agrawal, Manindra, 415 Aho, Alfred V, 415, 419 Akl, Selim G., 415 ALBA, 194 Algorithm complexity analysis, 248-253 decidability and undecidability, 165-182 defined, 154-156 describing, 156-159 Euclidean, 261 polynomial time, 256-263 running time, 248 ALLCFG, 197 Allen, Robin W, 416 Alon, Noga, 415 Alphabet, defined, 13 Alternating Turing machine, 381 Alternation, 380-386 Ambiguity, 105-106 Ambiguous NFA, 184 grammar, 105, 212 Amplification lemma, 369 AND operation, 14 ANFA, 167 Angluin, Dana, 415 Anti-clique, 27 Approximation algorithm, 365-367 AREX, 168 Argument, 8 Arithmetization, 394 Arity, 8, 225 Arora, Sanjeev, 415 ASPACE(f(n)), 382 Asymptotic analysis, 248 Asymptotic notation big-O notation, 249-250 small-o notation, 250 Asymptotic upper bound, 249 ATIME(t(n)), 382 ATM, 174 Atomic formula, 225 Automata theory, 3, see also Context-free language; Regular language. Average-case analysis, 248 Baase, Sara, 415 Babai, Laszlo, 415 Bach, Eric, 415 Balcazar,Jos6 Luis, 416 Basis of induction, 23 Beame, Paul W, 416 Big-O notation, 248-250 Binary function, 8 Binary operation, 44 Binary relation, 8, 9 Bipartite graph, 332 Blank symbol u, 140 Blum, Manuel, 416 Boolean circuit, 351-359 depth, 400 gate, 352 size, 400 uniform family, 400 wire, 352 Boolean formula, 271, 310 minimal, 349 quantified, 311 Boolean logic, 14-15 Boolean matrix multiplication, 401 Boolean operation, 14, 225, 271 Boolean variable, 271 Bound variable, 310 Branching program, 376 read-once, 377 Brassard, Gilles, 416 Bratley, Paul, 416 Breadth-first search, 256 Brute-force search, 257, 260, 264, 270 Cantor, Georg, 174 Carmichael number, 372 Carmichael, R. D., 416 INDEX 423 Cartesian product, 6, 46 CD-ROM, 321 (Certificate, 265 CFG, see Context-free grammar CFL, see Context-free language Chaitin, GregoryJ., 236 Chandra, Ashok, 416 (Characterisdc sequence, 178 Checkers, game of, 320 Chernoff bound, 370 Chess, game of, 320 Chinese remainder theorem, 373 Chomsky normal form, 106-109, 130, 170, 263 Chomsky, Noam, 416 (Church, Alonzo, 2, 155, 227 Church-Turing thesis, 155-156, 253 CIRCUIT-SAT, 358 Circuit-satisfiability problem, 358 CIRCUIT-VALUE, 404 Circular definition, 65 Clause, 274 Clique, 27, 268 CLIQUE, 268 Closed under, 45 Closure under complementation context-free languages, non-, 128 P, 294 regular languages, 85 Closure under concatenation context-free languages, 129 NP, 295 P, 294 regular languages, 47, 60 Closure under intersection context-free languages, non-, 128 regular languages, 46 Closure under star context-free languages, 129 NP, 295 P, 295 regular languages, 62 Closure under union context-free languages, 129 NP, 295 P, 294 regular languages, 45, 59 CNF-formula, 274 Co-Turing-recognizable language, 181 Cobham, Alan, 416 Coefficient, 155 Coin-flip step, 368 Complement operation, 4 Complexity class ASPACE(f(n)), 382 ATIME(t(n)), 382 BPP, 369 coNL, 326 coNP, 269 EXPSPACE, 340 EXPTIME, 308 IP, 389 L, 321 NC, 402 NL, 321 NP, 264-270 NPSPACE, 308 NSPACE(f(n)), 304 NTIME(f(n)), 267 P, 256-263, 269-270 PH, 386 PSPACE, 308 RP, 375 SPACE(f(n)), 304 TIME(f(n)), 251 ZPP, 412 Complexity theory, 2 Church-Turing thesis, 155-156, 253 Composite number, 265, 371 Compositeness witness, 373 COMPOSITES, 265 Compressible string, 239 Computability theory, 2 decidability and undecidability, 165-182 recursion theorem, 217-224 reducibility, 187-211 Turing machines, 137-1 54 Computable function, 206 Computation history context-free languages, 197-198 defined, 192 linear bounded automata, 193-197 Post correspondence problem, 199-205 reducibility, 192-205 Computational model, 31 Computer virus, 222 424 INDEX Concatenation of strings, 14 Concatenation operation, 44, 47, 60-61 Configuration, 140, 141, 322 Conjunction operation, 14 Conjunctive normal form, 274 coNL, 326 Connected graph, 11, 157 coNP, 269 Context-free grammar ambiguous, 105, 212 defined, 102 Context-free language decidability, 170-172 defined, 101 efficient decidability, 262-263 inherently ambiguous, 106 pumping lemma, 123-128 Cook, Stephen A., 271, 359, 402, 416 Cook-Levin theorem, 271-360 Cormen, Thomas, 416 Corollary, 17 Correspondence, 175 Countable set, 175 Counterexample, 18 Counting problem, 392 Cross product, 6 Cryptography, 405-411 Cut edge, 367 Cut, in a graph, 296, 367 Cycle, 11 Davis, Martin, 155 Decidability, see also Undecidability. context-free language, 170-172 of ACFG, 170 of ADFA, 166 of AREX, 168 of ECFG, 171 of EQDFA, 169 regular language, 166-170 Decidable language, 142 Decider deterministic, 142 nondeterministic, 152 Decision problem, 366 Definition, 17 Degree of a node, 10 DeMorgan's laws, example of proof, 20 Depth complexity, 400 Derivation, 100 leftmost, 106 Derives, 102 Descriptive complexity, 236 Deterministic computation, 47 Deterministic finite automaton acceptance problem, 166 defined, 35 emptiness testing, 168 minimization, 299 DFA, see Deterministic finite automaton Diagonalization method, 174-181 Diaz, Josep, 416 Difference hierarchy, 300 Digital signatures, 407 Directed graph, 12 Directed path, 12 Disjunction operation, 14 Distributive law, 15 Domain of a function, 7 Dynamic programming, 262 ECFG, 171 EDFA, 168 Edge of a graph, 10 Edmonds, Jack, 416 ELBA, 195 Element distinctness problem, 147 Element of a set, 3 Emptiness testing for CFG, 171 for DFA, 168 for LBA, 195 for TM, 189 Empty set, 4 Empty string, 13 Encoding, 157, 259 Enderton, Herbert B., 416 Enumerator, 152-153 EQCFG, 172 EQDFA, 169 EQREx, 344 EQTM Turing-unrecognizability, 210 undecidability, 192 Equality operation, 15 Equivalence relation, 9 Equivalent machines, 54 Erd6s, Paul, 415 Error probability, 369 ETM, undecidability, 189 INDEX 425 Euclidean algorithm, 261 Even, Shimon, 416 EXCLUSIVE OR operation, 15 Existential state, 381 Exponential bound, 250 Exponential, versus polynomial, 257 EXPSPACE, 340 EXPSPACE-completeness, 343-349 EXPTIME, 308 Factor of a number, 371 Feller, William, 416 Fermat test, 372 Fermat's little theorem, 371 Feynman, Richard P., 416 Final state, 35 Finite automaton automatic door example, 32 computation of, 40 decidability, 166-170 defined, 35 designing, 41-44 transition function, 35 two-dimensional, 213 two-headed, 212 Finite state machine, see Finite automaton. Finite state transducer, 87 Fixed point theorem, 223 Formal proof, 230 Formula, 225, 271 FORMULA-GAME, 314 Fortnow, Lance, 418 Free variable, 225 FST, see Finite state transducer Function, 7-9 argument, 8 binary, 8 computable, 206 domain, 7 one-to-one, 175 one-way, 408 onto, 7, 175 polynomial time computable, 272 range, 7 space constructible, 336 time constructible, 340 transition, 35 unary, 8 Gabarr6, Joaquim, 416 Gadget in a completeness proof, 283 Game, 313 Garey, Michael R., 416 Gate in a Boolean circuit, 352 Generalized geography, 316 Generalized nondeterministic finite automaton, 70-76 converting to a regular expression, 71 defined, 70, 73 Geography game, 315 GG (generalized geography), 317 Gill, John T., 416 GNFA, see Generalized nondeterministic finite automaton GO, game of, 320 Go-moku, game of, 330 Gbdel, Kurt, 2, 227, 230, 416 Goemans, Michel X., 416 Goldwasser, Shafi, 416, 417 Graph acyclic, 376 coloring, 297 cycle in, 11 degree, 10 directed, 12 edge, 10 isomorphism problem, 295, 387 k-regular, 21 labeled, 10 node, 10 strongly connected, 12 sub-, 11 undirected, 10 vertex, 10 Greenlaw, Raymond, 417 Halting configuration, 141 Halting problem, 173-181 unsolvability of, 174 HALTTM, 188 Hamiltonian path problem, 264 exponential time algorithm, 264 NP-completeness of, 286-291 polynomial time verifier, 265 HAMPATH, 264, 286 Harary, Frank, 417 Hartmanis, Juris, 417 426 INDEX Hey, AnthonyJ. G., 416 Hierarchy theorem, 336-347 space, 337 time, 341 High-level description of a Turing machine, 157 Hilbert, David, 154, 417 Hofstadter, Douglas R., 417 Hoover, H. James, 416, 417 Hopcroft, John E., 415, 417, 419 Huang, Ming-Deh A., 415 iff, 18 Implementation description of a Turing machine, 157 Implication operation, 15 Incompleteness theorem, 230 Incompressible string, 239 Indegree of a node, 12 Independent set, 27 Induction basis, 23 proof by, 22-25 step, 23 Induction hypothesis, 23 Inductive definition, 65 Infinite set, 4 Infix notation, 8 Inherent ambiguity, 106 Inherently ambiguous context-free language, 106 Integers, 4 Interactive proof system, 387-399 Interpretation, 226 Intersection operation, 4 IS0,387 Isomorphic graphs, 295 Johnson, David S., 416, 417 k-ary function, 8 k-ary relation, 8 k-clique, 267 k-optimal approximation algorithm, 367 k-tuple, 6 Karloff, Howard, 418 Karp, Richard M., 417 Kayal, Neeraj, 415 Kolmogorov, Andrei N., 236 L, 321 Labeled graph, 10 Ladder, 330 Language co-Turing-recognizable, 181 context-free, 101 decidable, 142 defined, 14 of a grammar, 101 recursively enumerable, 142 regular, 40 Turing-decidable, 142 Turing-recognizable, 142 Turing-unrecognizable, 181 Lawler, Eugene L., 417 LBA, see Linear bounded automaton Leaf in a tree, 11 Leeuwen, Jan van, 419 Leftmost derivation, 106 Leighton, F. Thomson, 417 Leiserson, Charles E., 416 Lemma, 17 Lenstra, Jan Karel, 417 Leveled graph, 333 Levin, Leonid A., 271, 359, 417 Lewis, Harry, 417 Lexical analyzer, 66 Lexicographic ordering, 14 Li, Ming, 417 Lichtenstein, David, 418 Linear bounded automaton, 193-197 Linear time, 253 Lipton, Richard J., 417 LISP, 154 Literal, 274 Log space computable function, 324 Log space reduction, 324, 404 Log space transducer, 324 Luby, Michael, 418 Lund, Carsten, 415, 418 Majority function, 363 Many-one reducibility, 206 Mapping, 7 Mapping reducibility, 206-211 polynomial time, 272 Markov chain, 33 Match, 199 Matijasevi6, Yuri, 155 MAX-CLIQUE, 300, 361 INDEX 427 MAX-CUT, 296 Maximization problem, 367 Member of a set, 3 Micali, Silvio, 416, 417 Miller, Gary L., 418 MIN-FORMULA, 383 Minesweeper, 298 Minimal Boolean formula, 349 Minimal description, 236 Minimal formula, 383 Minimization of a DFA, 299 Minimization problem, 366 Minimum pumping length, 91 Model, 226 MODEXP, 295 Modulo operation, 8 Motwani, Rajeev, 415 Multiset, 4, 269 Multitape Turing machine, 148-150 Myhill-Nerode theorem, 91 NL, 321 NL-complete problem PATH, 322 NL-completeness defined, 324 Natural numbers, 4 NC, 402 Negation operation, 14 NFA, see Nondeterministic finite automaton Nim, game of, 331 Nisan, Noam, 418 Niven, Ivan, 418 Node of a graph, 10 degree, 10 indegree, 12 outdegree, 12 Nondeterministic computation, 47 Nondeterministic finite automaton, 47-58 computation by, 48 defined, 53 equivalence with deterministic finite automaton, 55 equivalence with regular expression, 66 Nondeterministic polynomial time, 266 Nondeterministic Turing machine, 150-152 space complexity of, 304 time complexity of, 255 NONISO, 387 NOT operation, 14 NP, 264-270 NP-complete problem 3SAT, 274, 359 CIRCUIT-SAT, 358 HAMPATH, 286 SUBSET-SUM, 292 3COLOR, 297 UHAMPATH, 291 VERTEX-COVER, 284 NP-completeness, 271-294 defined, 276 NP-hard, 298 NP-problem, 266 NpA, 348 NPSPACE, 308 NSPACE(f(n)), 304 NTIME(f(n)), 267 NTM, see Nondeterministic Turing machine o(f (n)) (small-o notation), 250 One-sided error, 375 One-time pad, 406 One-to-one function, 175 One-way function, 408 One-way permutation, 408 Onto function, 7, 175 Optimal solution, 366 Optimization problem, 365 OR operation, 14 Oracle, 232, 348 Oracle tape, 348 Outdegree of a node, 12 P,256-263,269-270 P-complete problem CIRCUIT-VALUE, 404 P-completeness, 404 pA 348 Pair, tuple, 6 Palindrome, 90, 128 Papadimitriou, Christos H., 417, 418 Parallel computation, 399-404 Parallel random access machine, 400 Parity function, 353 Parse tree, 100 428 INDEX Parser, 99 Pascal, 154 Path Hamiltonian, 264 in a graph, 11 simple, 11 PATH, 259, 322 PCP, see Post correspondence problem. PDA, see Pushdown automaton Perfect shuffle operation, 89, 131 PH, 386 Pigeonhole principle, 78, 79, 124 Pippenger, Nick, 402 Polynomial, 154 Polynomial bound, 250 Polynomial time algorithm, 256-263 computable function, 272 hierarchy, 386 verifier, 265 Polynomial verifiability, 265 Polynomial, versus exponential, 257 Polynomially equivalent models, 257 Pomerance, Carl, 415, 418 Popping a symbol, 110 Post correspondence problem (PCP), 199-205 modified, 200 Power set, 6, 53 PRAM, 400 Pratt, Vaughan R., 418 Prefix notation, 8 Prefix of a string, 89 Prefix-free language, 184 Prenex normal form, 225, 310 Prime number, 265, 295, 371 Private-key cryptosystem, 407 Probabilistic algorithm, 368-380 Probabilistic function, 408 Probabilistic Turing machine, 368 Processor complexity, 400 Production, 100 Proof, 17 by construction, 21 by contradiction, 21-22 by induction, 22-25 finding, 17-20 necessity for, 77 Proper subset, 4, 328 Prover, 389 Pseudoprime, 372 PSPACE, 308 PSPACE-complete problem FORMULA-GAME, 314 GG, 317 TQBF, 311 PSPACE-completeness, 309-320 defined, 309 Public-key cryptosystem, 407 Pumping lemma for context-free languages, 123-128 for regular languages, 77-82 Pumping length, 77, 91, 123 Pushdown automaton, 109-122 context-free grammars, 115-122 defined, 111 examples, 112-114 schematic of, 110 Pushing a symbol, 110 Putnam, Hilary, 155 PUZZLE, 297, 330 Quantified Boolean formula, 311 Quantifier, 310 in a logical sentence, 225 Query node in a branching program, 376 Rabin, Michael O., 418 Rackoff, Charles, 417 Ramsey's theorem, 27 Range of a function, 7 Read-once branching program, 377 Real number, 176 Recognizes a language, meaning of, 36, 40 Recursion theorem, 217-224 fixed-point version, 223 terminology for, 221 Recursive language, see Decidable language. Recursively enumerable, see Turing-recognizable. Recursively enumerable language, 142 Reducibility, 187-211 mapping, 206-211 polynomial time, 272 via computation histories, 192-205 INDEX 429 Reduction, 187, 207 mapping, 207 Reflexive relation, 9 Regular expression, 63-76 defined, 64 equivalence to finite automaton, 66-76 examples Of, 65 Regular language, 31-82 closure under concatenation, 47, 60 closure under intersection, 46 closure under star, 62 closure under union, 45, 59 decidability, 166-170 defined, 40 Regular operation, 44 REGULARTM, 191 Reingold, Omer, 418 Rejecting computation history, 193 Rejecting configuration, 141 Relation, 8, 225 binary, 8 Relatively prime, 260 Relativization, 348-351 RELPRIME, 261 Reverse of a string, 14 Rice's theorem, 191, 213, 215, 242, 244 Rinooy Kan, A. H. G., 417 Rivest, Ronald L., 416, 418 Robinson, Julia, 155 Roche, Emmanuel, 418 Root in a tree, 11 of a polynomial, 155 Rule in a context-free grammar, 100, 102 Rumely, Robert S., 415 Ruzzo, Walter L., 417 SAT, 276, 308 #SAT, 392 Satisfiability problem, 271 Satisfiable formula, 271 Savitch's theorem, 305-307 Saxena, Nitin, 415 Schabes, Yves, 418 Schaefer, Thomas J., 418 Scope, 310 Scope, of a quantifier, 225 Secret key, 405 Sedgewick, Robert, 418 Self-reference, 218 Sentence, 311 Sequence, 6 Sequential computer, 399 Set, 3 countable, 175 uncountable, 176 Sethi, Ravi, 415 Shallit, Jeffrey, 415 Shamir, Adi, 418 Shen, Alexander, 418 Shmoys, David B., 417 Shor, Peter W, 418 Shuffle operation, 89, 131 Simple path, 11 Sipser, Michael, 418, 419 Size complexity, 400 Small-o notation, 250 SPACE(f(n)), 304 Space complexity, 303-333 Space complexity class, 304 Space complexity of nondeterministic Turing machine, 304 Space constructible function, 336 Space hierarchy theorem, 337 Spencer, Joel H., 415 Stack, 109 Star operation, 44, 62-63, 295 Start configuration, 141 Start state, 34 Start variable, in a context-free grammar, 100, 102 State diagram finite automaton, 34 pushdown automaton, 112 Turing machine, 144 Stearns, Richard E., 417 Steiglitz, Kenneth, 418 Stinson, Douglas R., 419 String, 13 Strongly connected graph, 12, 332 Structure, 226 Subgraph, 11 Subset of a set, 3 SUBSET-SUM, 268, 292 Substitution rule, 100 Substring, 14 430 INDEX Sudan, Madhu, 415 Symmetric difference, 169 Symmetric relation, 9 Synchronizing sequence, 92 Szegedy, Mario, 415 Tableau, 355 Tarjan, Robert E., 419 Tautology, 382 Term, in a polynomial, 154 Terminal, 100 Terminal in a context-free grammar, 102 Th(M), 226 Theorem, 17 Theory, of a model, 226 3COLOR, 297 3SAT, 274, 359 Tic-tac-toe, game of, 329 TIME(f(n)), 251 Time complexity, 247-294 analysis of, 248-253 of nondeterministic Turing machine, 255 Time complexity class, 267 Time constructible function, 340 Time hierarchy theorem, 341 TM, see Turing machine TQBF, 311 Transducer finite state, 87 log space, 324 Transition, 34 Transition function, 35 Transitive closure, 401 Transitive relation, 9 Trapdoor function, 410 Tree, 11 leaf, 11 parse, 100 root, 11 Triangle in a graph, 295 Tuple, 6 Turing machine, 137-154 alternating, 381 comparison with finite automaton, 138 defined, 140 describing, 156-159 examples of, 142-147 marking tape symbols, 146 multitape, 148-150 nondeterministic, 150-152 oracle, 232, 348 schematic of, 138 universal, 174 Turing reducibility, 232-233 Turing, Alan M., 2, 137, 155, 419 Turing-decidable language, 142 Turing-recognizable language, 142 Turing-unrecognizable language, 181-182 EQTM, 210 Two-dimensional finite automaton, 213 Two-headed finite automaton, 212 2DFA, see Two-headed finite automaton 2DIM-DFA, see Two-dimensional finite automaton 2SAT, 299 Ullman, Jeffrey D., 415, 417, 419 Unary alphabet, 52, 82, 212 function, 8 notation, 259, 295 operation, 44 Uncountable set, 176 Undecidability diagonalization method, 174-181 of ATM, 174 Of ELBA, 195 of EQTM, 192 of ETM, 189 of HALTTM, 188 of REGULARTM, 191 of EQcFG, 172 of Post correspondence problem, 200 via computation histories, 192-205 Undirected graph, 10 Union operation, 4, 44, 45, 59-60 Unit rule, 107 Universal quantifier, 310 Universal state, 381 Universal Turing machine, 174 Universe, 225, 310 Useless state in PDA, 184 in TM, 211 INDEX 431 Valiant, Leslie G., 415 Variable Boolean, 271 bound, 310 in a context-free grammar, 100, 102 start, 100, 102 Venn diagram, 4 Verifier, 265, 388 Vertex of a graph, 10 VETRTEK-COVER, 284 Virus, 222 Vitanyi, Paul, 417 Wegman, Mark, 416 Well-formed formula, 225 Williamson, David P., 416 Window, in a tableau, 279 Winning strategy, 314 Wire in a Boolean circuit, 352 Worst-case analysis, 248 XOR operation, 15, 354 Yannakakis, Mihalis, 418 Yields for configurations, 141 for context-free grammars, 102 Zuckerman, Herbert S., 418 SS ; 0I M -ta -UP 0 S 000 ] az li lE mS S Sl -. -. 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https://en.wikipedia.org/wiki/Brow_ridge
Jump to content Brow ridge العربية বাংলা Brezhoneg Català Čeština Deutsch Español Français Bahasa Indonesia Italiano Nederlands 日本語 Norsk bokmål Polski Português Русский Suomi ไทย 中文 Edit links From Wikipedia, the free encyclopedia Bony ridge located above the eye sockets of all primates | Brow ridge | | --- | | Frontal bone. Outer surface. Brow ridge labelled as "superciliary arch" at center right). | | | Details | | | Identifiers | | | Latin | arcus superciliaris | | TA98 | A02.1.03.005 | | TA2 | 524 | | FMA | 52850 | | Anatomical terms of bone [edit on Wikidata] | | The brow ridge, or supraorbital ridge known as superciliary arch in medicine, is a bony ridge located above the eye sockets of all primates and some other animals. In humans, the eyebrows are located on their lower margin. Structure [edit] The brow ridge is a nodule or crest of bone situated on the frontal bone of the skull. It forms the separation between the forehead portion itself (the squama frontalis) and the roof of the eye sockets (the pars orbitalis). Normally, in humans, the ridges arch over each eye, offering mechanical protection. In other primates, the ridge is usually continuous and often straight rather than arched. The ridges are separated from the frontal eminences by a shallow groove. The ridges are most prominent medially, and are joined to one another by a smooth elevation named the glabella. Typically, the arches are more prominent in men than in women, and vary between different human populations. Behind the ridges, deeper in the bone, are the frontal sinuses. Terminology [edit] The brow ridges, being a prominent part of the face in some human populations and a trait linked to sexual dimorphism, have a number of names in different disciplines. In vernacular English, the terms eyebrow bone or eyebrow ridge are common. The more technical terms frontal or supraorbital arch, ridge or torus (or tori to refer to the plural, as the ridge is usually seen as a pair) are often found in anthropological or archaeological studies. In medicine, the term arcus superciliaris (Latin) or the English translation superciliary arch. This feature is different from the supraorbital margin and the margin of the orbit. Some paleoanthropologists distinguish between frontal torus and supraorbital ridge. In anatomy, a torus is a projecting shelf of bone that unlike a ridge is rectilinear, unbroken and goes through glabella. Some fossil hominins, in this use of the word, have the frontal torus, but almost all modern humans only have the ridge. Development [edit] Spatial model [edit] The Spatial model proposes that supraorbital torus development can be best explained in terms of the disparity between the anterior position of the orbital component relative to the neurocranium. Much of the groundwork for the spatial model was laid down by Schultz (1940). He was the first to document that at later stages of development (after age 4) the growth of the orbit would outpace that of the eye. Consequently, he proposed that facial size is the most influential factor in orbital development, with orbital growth being only secondarily affected by size and ocular position. Weindenreich (1941) and Biegert (1957, 1963) argued that the supraorbital region can best be understood as a product of the orientation of its two components, the face and the neurocranium. The most composed articulation of the spatial model was presented by Moss and Young (1960), who stated that "the presence… of supraorbital ridges is only the reflection of the spatial relationship between two functionally unrelated cephalic components, the orbit and the brain" (Moss and Young, 1960, p282). They proposed (as first articulated by Biegert in 1957) that during infancy the neurocranium extensively overlaps the orbit, a condition that prohibits brow ridge development. As the splanchnocranium grows, however, the orbits begin to advance, thus causing the anterior displacement of the face relative to the brain. Brow ridges then form as a result of this separation. Bio-mechanical model [edit] The bio-mechanical model predicts that morphological variation in torus size is the direct product of differential tension caused by mastication, as indicated by an increase in load/lever ratio and broad craniofacial angle. Research done on this model has largely been based on earlier work of Endo. By applying pressure similar to the type associated with chewing, he carried out an analysis of the structural function of the supraorbital region on dry human and gorilla skulls. His findings indicated that the face acts as a pillar that carries and disperses tension caused by the forces produced during mastication. Russell and Oyen et al. elaborated on this idea, suggesting that amplified facial projection necessitates the application of enhanced force to the anterior dentition in order to generate the same bite power that individuals with a dorsal deflection of the facial skull exert. In more prognathic individuals, this increased pressure triggers bone deposition to reinforce the brow ridges, until equilibrium is reached. Oyen et al. conducted a cross-section study of Papio anubis in order to ascertain the relationship between palate length, incisor load and Masseter lever efficiency, relative to torus enlargement. Indications found of osteoblastic deposition in the glabella were used as evidence for supraorbital enlargement. Oyen et al.’s data suggested that more prognathic individuals experienced a decrease in load/lever efficiency. This transmits tension via the frontal process of the maxilla to the supraorbital region, resulting in a contemporary reinforcement of this structure. This was also correlated to periods of tooth eruption. In a later series of papers, Russell developed aspects of this mode further. Employing an adult Australian sample, she tested the association between brow ridge formation and anterior dental loading, via the craniofacial angle (prosthion-nasion-metopion), maxilla breadth, and discontinuities in food preparation such as those observed between different age groups. Finding strong support for the first two criteria, she concluded that the supraorbital complex is formed as a result of increased tension due to the widening of the maxilla, thought to be positively correlated with the size of the masseter muscle, as well as with the improper orientation of bone in the superior orbital region. Function [edit] See also: Human skeletal changes due to bipedalism Some researchers have suggested that brow ridges function to protect the eyes and orbital bones during hand-to-hand combat, given that they are a dimorphic trait. Paleolithic humans [edit] Pronounced brow ridges were a common feature among paleolithic humans. Early modern people such as those from the finds from Jebel Irhoud and Skhul and Qafzeh had thick, large brow ridges, but they differ from those of archaic humans like Neanderthals by having a supraorbital foramen or notch, forming a groove through the ridge above each eye, although there were exceptions, such as Skhul 2 in which the ridge was unbroken, unlike other members of her tribe. This splits the ridge into central parts and distal parts. In current humans, almost always only the central sections of the ridge are preserved (if preserved at all). This contrasts with many archaic and early modern humans, where the brow ridge is pronounced and unbroken. Other animals [edit] The size of these ridges varies also between different species of primates, either living or fossil. The closest living relatives of humans, the great apes and especially gorillas or chimpanzees, have a very pronounced supraorbital ridge, which has also been called a frontal torus, while in modern humans and orangutans, it is relatively reduced. The fossil record indicates that the supraorbital ridge in early hominins was reduced as the cranial vault grew; the frontal portion of the brain became positioned above rather than behind the eyes, giving a more vertical forehead. Supraorbital ridges are also present in some other animals, such as wild rabbits, eagles and certain species of sharks. The presence of a supraorbital ridge in the Korean field mouse has been used to distinguish it among related species. See also [edit] Anatomy portal Occipitofrontalis muscle Supraciliary scale Frontal sinus References [edit] This article incorporates text in the public domain from page 135 of the 20th edition of Gray's Anatomy (1918) ^ Patton, Kevin T.; Thibodeau, Gary A. (2018). Anthony's Textbook of Anatomy & Physiology - E-Book. Elsevier Health Sciences. p. 276. ISBN 9780323709309. ^ American Heritage Dictionary. supraorbital Archived 2007-12-14 at the Wayback Machine ^ torus Archived 2007-12-15 at the Wayback Machine ^ Jump up to: a b Sollas, W.J. (1925). "The Taungs Skull". Nature. 115 (2902): 908–9. Bibcode:1925Natur.115..908S. doi:10.1038/115908a0. S2CID 4125405. ^ Aloisi, Massimiliano (2000). The Origin of Humankind: Conference Proceedings of the International Symposium, Venice, 14-15 May 1998. IOS Press. ISBN 9781586030308. ^ Oyen and Russell, 1984, p. 368-369 ^ Endo, B (1965). "Distribution of Stress and Strain Produced in the Human Facial Skeleton by the Masticatory Force". The Journal of Anthropological Society of Nippon. 73 (4): 123–136. doi:10.1537/ase1911.73.123. ^ Endo, B (1970). "Analysis of Stresses around the Orbit Due to Masseter and Temporalis Muscles Respectively". The Journal of Anthropological Society of Nippon. 78 (4): 251–266. doi:10.1537/ase1911.78.251. ^ Endo, B (1973). "Stress analysis of the gorilla face". Primates. 14: 37–45. doi:10.1007/bf01730514. S2CID 23751360. ^ Endo B (July 1966). "A biomechanical study of the human facial skeleton by means of strain-sensitive lacquer". Okajimas Folia Anatomica Japonica. 42 (4): 205–17. doi:10.2535/ofaj1936.42.4_205. PMID 6013426. S2CID 16122254. ^ Jump up to: a b Russell, MD (1985). "The supraorbital torus: "A most remarkable peculiarity."". Current Anthropology. 26: 337. doi:10.1086/203279. S2CID 146857927. ^ Russell MD (May 1982). "Tooth eruption and browridge formation" (PDF). Am J Phys Anthropol. 58 (1): 59–65. doi:10.1002/ajpa.1330580107. hdl:2027.42/37614. PMID 7124915. ^ Oyen, OJ, Rice, RW, and Cannon, MS (1979). "Brow ridge structure and function in extant primates and Neanderthals". American Journal of Physical Anthropology. 51: 88–96. doi:10.1002/ajpa.1330510111.{{cite journal}}: CS1 maint: multiple names: authors list (link) ^ Oyen OJ, Walker AC, Rice RW (September 1979). "Craniofacial growth in olive baboons (Papio cynocephalus anubis): browridge formation". Growth. 43 (3): 174–87. PMID 116911. ^ Shea, Brian T.; Russell, Mary D. (July 1986). "On skull form and the supraorbital torus in primates". Current Anthropology. 27 (3): 257–260. doi:10.1086/203427. JSTOR 2742880. S2CID 145273372. ^ Carrier, David; Morgan, Michael H. (2015). "Protective buttressing of the hominin face" (PDF). Biological Reviews. 90 (1): 330–346. doi:10.1111/brv.12112. PMID 24909544. S2CID 14777701. ^ "Homo sapiens - H. sapiens (Anatomically Modern Humans - AMH) are the species we belong to". Archived from the original on 2011-09-08. Retrieved 2019-01-29. ^ Bhupendra, P. "Forehead Anatomy". Medscape references. Medscape. Retrieved 11 December 2013. ^ "How to ID a modern human?". News, 2012. Natural History Museum, London. Retrieved 11 December 2013. ^ H. H. Kolb (June 1992). "The supraorbital ridge as an indicator of age in wild rabbits (Oryctolagus cuniculus)". Journal of Zoology. 227 (2): 334–338. doi:10.1111/j.1469-7998.1992.tb04830.x. Retrieved 16 February 2023. ^ "Learn About Eagles from A to Z". National Eagle Center. Retrieved 16 February 2023. ^ J. Douglas Ogilby (1893). "Description of a new shark from the Tasmanian coast". Records of the Australian Museum. 2 (5): 62–63. doi:10.3853/j.0067-1975.2.1893.1194. Retrieved 16 February 2023. ^ "First Report of the Herb Field Mouse, Apodemus uralensis (Pallas, 1811) from Mongolia". Mongolian Journal of Biological Sciences: 36. ISSN 2225-4994. Retrieved 16 February 2023. Further reading [edit] Endo, B (1965). "Distribution of Stress and Strain Produced in the Human Facial Skeleton by the Masticatory Force". The Journal of Anthropological Society of Nippon. 73 (4): 123–36. doi:10.1537/ase1911.73.123. Endo, B (1970). "Analysis of Stresses around the Orbit Due to Masseter and Temporalis Muscles Respectively". The Journal of Anthropological Society of Nippon. 78 (4): 251–66. doi:10.1537/ase1911.78.251. Endo, B (1973). "Stress analysis of the gorilla face". Primates. 14: 37–45. doi:10.1007/bf01730514. S2CID 23751360. Russell, Mary Doria (June 1985). "The Supraorbital Torus: 'A Most Remarkable Peculiarity'". Current Anthropology. 26 (3): 337–360. doi:10.1086/203279. S2CID 146857927. Oyen, Ordean J.; Rice, Robert W.; Cannon, M. Samuel (July 1970). "Browridge structure and function in extant primates and Neanderthals". American Journal of Physical Anthropology. 51 (1): 83–95. doi:10.1002/ajpa.1330510111. External links [edit] Wikimedia Commons has media related to Superciliary arch. The Frontal Bone, California State University at Chico site. | v t e Neurocranium of the skull | | --- | | Occipital | | | | --- | | Squamous part | external + Inion/External occipital protuberance + Occipital bun + External occipital crest + Nuchal lines + Suprainiac fossa planes + Occipital + Nuchal internal + Cruciform eminence + Internal occipital protuberance + Internal occipital crest + Groove for transverse sinus | | Lateral parts | Condyle + Condyloid fossa + Condylar canal Hypoglossal canal jugular + Jugular process + Jugular tubercle | | Basilar part | Pharyngeal tubercle Clivus | | Other | Foramen magnum + Basion + Opisthion | | | Parietal | Parietal eminence Temporal line Parietal foramen Sagittal sulcus Sagittal keel Sagittal crest | | Frontal | | | | --- | | Squamous part | Frontal suture Frontal eminence external + Superciliary arches + Glabella foramina + Supraorbital foramen + Brow ridge + Foramen cecum Zygomatic process internal + Sagittal sulcus + Frontal crest | | Orbital part | Ethmoidal notch Fossa for lacrimal gland Trochlear fovea Frontal sinus Frontonasal duct | | | Temporal | | | | --- | | Squamous part | Articular tubercle Suprameatal triangle Mandibular fossa Petrotympanic fissure Zygomatic process | | Mastoid part | Mastoid foramen Mastoid process (Mastoid cells) Mastoid notch Occipital groove Sigmoid sulcus Mastoid antrum (Aditus) | | Petrous part | Carotid canal Facial canal + Hiatus Internal auditory meatus Cochlear aqueduct Stylomastoid foramen fossae + Subarcuate fossa + Jugular fossa canaliculi + Inferior tympanic + Mastoid Styloid process Petrosquamous suture (note: ossicles in petrous part, but not part of temporal bone) | | Tympanic part | Suprameatal spine | | | Sphenoid | | | | --- | | Surfaces | Superior surface: Sella turcica + Dorsum sellae + Tuberculum sellae + Hypophysial fossa + Posterior clinoid processes Ethmoidal spine Chiasmatic groove Middle clinoid process Petrosal process Clivus Lateral surface: Carotid groove Sphenoidal lingula Anterior surface: Sphenoidal sinuses | | Great wings | foramina + Rotundum + Ovale + Vesalii + Spinosum Spine Infratemporal crest Sulcus of auditory tube | | Small wings | Superior orbital fissure Anterior clinoid process Optic canal | | Pterygoid processes | fossae + Pterygoid + Scaphoid pterygoid plates + Lateral + Medial Pterygoid canal Hamulus | | Other | Body Sphenoidal conchae | | | Ethmoid | | | | --- | | Plates | Cribriform plate + Crista galli + Olfactory foramina Perpendicular plate | | Surfaces | Lateral surface Orbital lamina Uncinate process Medial surface Supreme nasal concha Superior nasal concha Superior meatus Middle nasal concha Middle meatus | | Labyrinth | Ethmoid sinus ethmoidal foramina + Posterior + Anterior | | | | | --- | | Authority control databases | Terminologia Anatomica | Retrieved from " Categories: Wikipedia articles incorporating text from the 20th edition of Gray's Anatomy (1918) Primate anatomy Bones of the head and neck Facial features Hidden categories: Webarchive template wayback links CS1 maint: multiple names: authors list Articles with short description Short description matches Wikidata Commons category link is on Wikidata
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https://emedicine.medscape.com/article/1123213-treatment
For You News & Perspective Tools & Reference Edition English Medscape Editions About You Professional Information Newsletters & Alerts Your Watch List Formulary Plan Manager Log Out Register Log In EN Medscape Editions English Deutsch Español Français Português UK X Univadis from Medscape About You Professional Information Newsletters & Alerts Your Watch List Formulary Plan Manager Log Out Register Log In close Please confirm that you would like to log out of Medscape. If you log out, you will be required to enter your username and password the next time you visit. Log out Cancel processing.... Tools & Reference>Dermatology Lichen Planus Treatment & Management Updated: Jan 28, 2025 Author: Lauren Abigail Hoffpauir; Chief Editor: William D James, MD more...;) Share Print Feedback Close Facebook Twitter LinkedIn WhatsApp Email Sections Lichen Planus Sections Lichen Planus Overview Practice Essentials Background Pathophysiology Etiology Epidemiology Prognosis Patient Education Show All Presentation History Physical Examination Complications Show All DDx Workup Laboratory Studies Imaging Studies Histologic Findings Show All Treatment Medication Corticosteroids, Topical Corticosteroids Retinoid-like Agents Acne Agents, Topical Calcineurin Inhibitors Nitroimidazoles DMARDs, PDE4 Inhibitors 5-Aminosalicylic Acid Derivatives Show All Media Gallery;) References;) Treatment Medical Care Lichen planus (LP) is a self-limited disease that usually resolves over several months but can sometimes take years to do so. Mild cases can be treated with fluorinated topical steroids. More severe cases, especially those with scalp, nail, and mucous membrane involvement, may need more intensive therapy. The first-line treatments of cutaneous LP are topical steroids, particularly class I or II ointments. A second choice would be systemic steroids for symptom control and possibly more rapid resolution. Classic LP frequently is very responsive to modest doses of corticosteroids. Oral metronidazole has been shown to be an effective therapy for some patients. Oral acitretin has been shown to be effective in published studies. Many other treatments, including mycophenolate mofetil at 1-1.5 g twice daily, are of uncertain efficacy, owing to the paucity of experience. In a randomized double-blind study, administration of sulfasalazine at up to 2.5 g/day for 6 weeks led to improvement in lesions (>80%) and pruritus (>90%) in patients with generalized LP. For LP of the oral mucosa, topical steroids are usually tried first. Topical and systemic cyclosporine regimens have been tried with some success ; however, a randomized double-blind study indicated that topical cyclosporine was a less effective but much more costly regimen than clobetasol. Newer topical calcineurin inhibitors (eg, tacrolimus and pimecrolimus) have replaced topical cyclosporine for the treatment of LP. Other options include oral or topical retinoids. Even with these effective treatments, relapses are common. Close monitoring of lipid levels is suggested for patients with LP who are treated with oral retinoid agents because a case control study found that the risk of dyslipidemia in these patients is increased two- to threefold. In fact, according to a meta-analysis of 5242 patients, even those who did not receive retinoids might still have dyslipidemia. Patients with widespread LP may respond to narrowband or broadband ultraviolet (UV)-B therapy. Psoralen with UV-A (PUVA) therapy for 8 weeks has been reported to be effective. PUVA therapy typically consists of oral psoralen 0.6 mg/kg administered 1.5-2 hours before UV-A treatment, which usually starts at 0.5-1 J/cm2 and is increased by 0.5 J/cm2 per visit. Use of topical ointment at the time of UV-A treatment may decrease the effectiveness of PUVA. The risks and benefits of PUVA should be considered. PUVA is carcinogenic. Long-term risks include dose-related actinic degeneration, squamous cell carcinoma, and cataracts. A phototoxic reaction with erythema, pruritus, phytophotodermatitis, and friction blisters could occur. Precautions should be taken for persons with a history of skin cancers or hepatic insufficiency. Apremilast may be an effective treatment for LP, but double-blind controlled trials are lacking. [5, 30] Medication References Atzmony L, Reiter O, Hodak E, Gdalevich M, Mimouni D. Treatments for Cutaneous Lichen Planus: A Systematic Review and Meta-Analysis. Am J Clin Dermatol. 2016 Feb. 17 (1):11-22. [QxMD MEDLINE Link]. Fazel N. Cutaneous lichen planus: A systematic review of treatments. J Dermatolog Treat. 2015 Jun. 26 (3):280-3. [QxMD MEDLINE Link]. Rasi A, Behzadi AH, Davoudi S, Rafizadeh P, Honarbakhsh Y, Mehran M, et al. Efficacy of oral metronidazole in treatment of cutaneous and mucosal lichen planus. J Drugs Dermatol. 2010 Oct. 9 (10):1186-90. [QxMD MEDLINE Link]. Omidian M, Ayoobi A, Mapar MA, Feily A, Cheraghian B. Efficacy of sulfasalazine in the treatment of generalized lichen planus: randomized double-blinded clinical trial on 52 patients. J Eur Acad Dermatol Venereol. 2010 Sep. 24 (9):1051-4. [QxMD MEDLINE Link]. Viswanath V, Joshi P, Dhakne M, Dhoot D, Mahadkar N, Barkate H. Evaluation of the Efficacy and Safety of Apremilast in the Management of Lichen Planus. Clin Cosmet Investig Dermatol. 2022. 15:2593-2600. [QxMD MEDLINE Link].[Full Text]. Arias-Santiago S, Buendía-Eisman A, Aneiros-Fernández J, Girón-Prieto MS, Gutiérrez-Salmerón MT, Mellado VG, et al. Cardiovascular risk factors in patients with lichen planus. Am J Med. 2011 Jun. 124 (6):543-8. [QxMD MEDLINE Link]. Pavlotsky F, Nathansohn N, Kriger G, Shpiro D, Trau H. Ultraviolet-B treatment for cutaneous lichen planus: our experience with 50 patients. Photodermatol Photoimmunol Photomed. 2008 Apr. 24 (2):83-6. [QxMD MEDLINE Link]. Alaizari NA, Al-Maweri SA, Al-Shamiri HM, Tarakji B, Shugaa-Addin B. Hepatitis C virus infections in oral lichen planus: a systematic review and meta-analysis. Aust Dent J. 2016 Sep. 61 (3):282-7. [QxMD MEDLINE Link]. Chuang TY, Stitle L, Brashear R, Lewis C. Hepatitis C virus and lichen planus: A case-control study of 340 patients. J Am Acad Dermatol. 1999 Nov. 41 (5 Pt 1):787-9. [QxMD MEDLINE Link]. Shengyuan L, Songpo Y, Wen W, Wenjing T, Haitao Z, Binyou W. Hepatitis C virus and lichen planus: a reciprocal association determined by a meta-analysis. Arch Dermatol. 2009 Sep. 145 (9):1040-7. [QxMD MEDLINE Link]. Bigby M. The relationship between lichen planus and hepatitis C clarified. Arch Dermatol. 2009 Sep. 145 (9):1048-50. [QxMD MEDLINE Link]. Raslan HM, Ezzat WM, Abd El Hamid MF, Emam H, Amre KS. Skin manifestations of chronic hepatitis C virus infection in Cairo, Egypt. East Mediterr Health J. 2009 May-Jun. 15 (3):692-700. [QxMD MEDLINE Link]. Georgescu SR, Tampa M, Mitran MI, Mitran CI, Sarbu MI, Nicolae I, et al. Potential pathogenic mechanisms involved in the association between lichen planus and hepatitis C virus infection. Exp Ther Med. 2019 Feb. 17 (2):1045-1051. [QxMD MEDLINE Link]. Korkij W, Chuang TY, Soltani K. Liver abnormalities in patients with lichen planus. A retrospective case-control study. J Am Acad Dermatol. 1984 Oct. 11 (4 Pt 1):609-15. [QxMD MEDLINE Link]. Manolache L, Seceleanu-Petrescu D, Benea V. Lichen planus patients and stressful events. J Eur Acad Dermatol Venereol. 2008 Apr. 22 (4):437-41. [QxMD MEDLINE Link]. Copeman PW, Tan RS, Timlin D, Samman PD. Familial lichen planus. Another disease or a distinct people?. Br J Dermatol. 1978 May. 98 (5):573-7. [QxMD MEDLINE Link]. Balasubramaniam P, Ogboli M, Moss C. Lichen planus in children: review of 26 cases. Clin Exp Dermatol. 2008 Jul. 33 (4):457-9. [QxMD MEDLINE Link]. Murphy R, Edwards L. Desquamative inflammatory vaginitis: what is it?. J Reprod Med. 2008 Feb. 53 (2):124-8. [QxMD MEDLINE Link]. Ingafou M, Leao JC, Porter SR, Scully C. Oral lichen planus: a retrospective study of 690 British patients. Oral Dis. 2006 Sep. 12 (5):463-8. [QxMD MEDLINE Link]. Belfiore P, Di Fede O, Cabibi D, Campisi G, Amarù GS, De Cantis S, et al. Prevalence of vulval lichen planus in a cohort of women with oral lichen planus: an interdisciplinary study. Br J Dermatol. 2006 Nov. 155 (5):994-8. [QxMD MEDLINE Link]. Di Fede O, Belfiore P, Cabibi D, De Cantis S, Maresi E, Kerr AR, et al. Unexpectedly high frequency of genital involvement in women with clinical and histological features of oral lichen planus. Acta Derm Venereol. 2006. 86 (5):433-8. [QxMD MEDLINE Link]. Lucchese A, Dolci A, Minervini G, Salerno C, DI Stasio D, Minervini G, et al. Vulvovaginal gingival lichen planus: report of two cases and review of literature. Oral Implantol (Rome). 2016 Apr-Jun. 9 (2):54-60. [QxMD MEDLINE Link]. Robles-Méndez JC, Rizo-Frías P, Herz-Ruelas ME, Pandya AG, Ocampo Candiani J. Lichen planus pigmentosus and its variants: review and update. Int J Dermatol. 2018 May. 57 (5):505-514. [QxMD MEDLINE Link]. Pai VV, Kikkeri NN, Sori T, Dinesh U. Graham-little piccardi lassueur syndrome: an unusual variant of follicular lichen planus. Int J Trichology. 2011 Jan. 3 (1):28-30. [QxMD MEDLINE Link]. Gonzalez-Moles MA, Scully C, Gil-Montoya JA. Oral lichen planus: controversies surrounding malignant transformation. Oral Dis. 2008 Apr. 14 (3):229-43. [QxMD MEDLINE Link]. Knackstedt TJ, Collins LK, Li Z, Yan S, Samie FH. Squamous Cell Carcinoma Arising in Hypertrophic Lichen Planus: A Review and Analysis of 38 Cases. Dermatol Surg. 2015 Dec. 41 (12):1411-8. [QxMD MEDLINE Link]. Cribier B, Frances C, Chosidow O. Treatment of lichen planus. An evidence-based medicine analysis of efficacy. Arch Dermatol. 1998 Dec. 134 (12):1521-30. [QxMD MEDLINE Link]. Lim KK, Su WP, Schroeter AL, Sabers CJ, Abraham RT, Pittelkow MR. Cyclosporine in the treatment of dermatologic disease: an update. Mayo Clin Proc. 1996 Dec. 71 (12):1182-91. [QxMD MEDLINE Link]. Conrotto D, Carbone M, Carrozzo M, Arduino P, Broccoletti R, Pentenero M, et al. Ciclosporin vs. clobetasol in the topical management of atrophic and erosive oral lichen planus: a double-blind, randomized controlled trial. Br J Dermatol. 2006 Jan. 154 (1):139-45. [QxMD MEDLINE Link]. Paul J, Foss CE, Hirano SA, Cunningham TD, Pariser DM. An open-label pilot study of apremilast for the treatment of moderate to severe lichen planus: a case series. J Am Acad Dermatol. 2013 Feb. 68 (2):255-61. [QxMD MEDLINE Link]. Lai YC, Yew YW, Schwartz RA. Lichen planus and dyslipidemia: a systematic review and meta-analysis of observational studies. Int J Dermatol. 2016 May. 55 (5):e295-304. [QxMD MEDLINE Link]. Media Gallery Lichen planus on flexor part of wrist. Close-up view of lichen planus. Lichen planus shows Wickham striae (white, fine, reticular lines). Lichen planus on oral mucosa with ulceration in center of lesion appears with whitish papules and plaques in periphery. Lichen planus lesion. Image from Syed Wali Peeran, with no alterations, Wikimedia Commons ( Intraoral lichen planus lesion. Image from Syed Wali Peeran, with no alterations, Wikimedia Commons ( Bandlike infiltrate of lymphocytes obscures dermoepidermal junction. Eosinophilic apoptotic keratinocytes (colloid bodies) are visible. Image from Clay Cockerell, MD, Cockerell Dermatopathology, Dallas, TX. Higher-power view. Wedge-shaped hypergranulosis and individual degenerated keratinocytes are visible in addition to hyperkeratosis. Vacuolar degeneration is visible. Image from Clay Cockerell, MD, Cockerell Dermatopathology, Dallas, TX. of 8 Tables Back to List Contributor Information and Disclosures Author Lauren Abigail Hoffpauir Doctor of Osteopathy Candidate, Texas College of Osteopathic MedicineLauren Abigail Hoffpauir is a member of the following medical societies: American College of Osteopathic Surgeons, American College of Surgeons, American Society of Plastic SurgeonsDisclosure: Nothing to disclose. Coauthor(s) Edward J Zabawski, Jr, DO, MBA Adjunct Assistant Professor, Clinical SciencesEdward J Zabawski, Jr, DO, MBA is a member of the following medical societies: American Osteopathic Association, New England Dermatological Society, American Academy of Dermatology, Noah Worcester Dermatological Society, European Academy of Dermatology and VenereologyDisclosure: Nothing to disclose. Specialty Editor Board Francisco Talavera, PharmD, PhD Adjunct Assistant Professor, University of Nebraska Medical Center College of Pharmacy; Editor-in-Chief, Medscape Drug ReferenceDisclosure: Received salary from Medscape for employment. Warren R Heymann, MD Head, Division of Dermatology, Professor, Department of Internal Medicine, Rutgers New Jersey Medical SchoolWarren R Heymann, MD is a member of the following medical societies: American Academy of Dermatology, American Society of Dermatopathology, Society for Investigative DermatologyDisclosure: Nothing to disclose. Chief Editor William D James, MD Emeritus Professor, Department of Dermatology, University of Pennsylvania School of MedicineWilliam D James, MD is a member of the following medical societies: American Academy of Dermatology, American Contact Dermatitis Society, Association of Military Dermatologists, Association of Professors of Dermatology, American Dermatological Association, Women's Dermatologic Society, Medical Dermatology Society, Dermatology Foundation, Society for Investigative Dermatology, Pennsylvania Academy of DermatologyDisclosure: Received income in an amount equal to or greater than $250 from: Elsevier Served as a speaker for various universities, dermatology societies, and dermatology departments. Additional Contributors Laura Stitle, MD Staff Physician, Department of Dermatology, Indiana University Medical CenterLaura Stitle, MD is a member of the following medical societies: Alpha Omega AlphaDisclosure: Nothing to disclose. Tsu-Yi Chuang, MD, MPH, FAAD Clinical Professor, Department of Dermatology, Keck School of Medicine of the University of Southern California; Dermatologist, HealthCare PartnersTsu-Yi Chuang, MD, MPH, FAAD is a member of the following medical societies: American Academy of Dermatology, American Society for Dermatologic Surgery, International Society of DermatologyDisclosure: Nothing to disclose. Joshua A Zeichner, MD Assistant Professor, Director of Cosmetic and Clinical Research, Mount Sinai School of Medicine; Chief of Dermatology, Institute for Family Health at North GeneralJoshua A Zeichner, MD is a member of the following medical societies: American Academy of Dermatology, National Psoriasis FoundationDisclosure: Received consulting fee from Valeant for consulting; Received grant/research funds from Medicis for other; Received consulting fee from Galderma for consulting; Received consulting fee from Promius for consulting; Received consulting fee from Pharmaderm for consulting; Received consulting fee from Onset for consulting. Close;) What would you like to print? What would you like to print? Print this section Print the entire contents of Print the entire contents of article Sections Lichen Planus Overview Practice Essentials Background Pathophysiology Etiology Epidemiology Prognosis Patient Education Show All Presentation History Physical Examination Complications Show All DDx Workup Laboratory Studies Imaging Studies Histologic Findings Show All Treatment Medication Corticosteroids, Topical Corticosteroids Retinoid-like Agents Acne Agents, Topical Calcineurin Inhibitors Nitroimidazoles DMARDs, PDE4 Inhibitors 5-Aminosalicylic Acid Derivatives Show All Media Gallery;) References;) encoded search term (Lichen Planus) and Lichen Planus What to Read Next on Medscape Related Conditions and Diseases Lichen Planus Oral Lichen Planus Graham-Little-Piccardi-Lasseur Syndrome Erythema Dyschromicum Perstans Lichen Nitidus Lichen Striatus Dysphagia and Odynophagia in a Woman With Hypothyroidism News & Perspective Ruxolitinib Cream Shows Promise in Treating Lichen Planus Survey Identifies Gaps in Lichen Planus Management Lichen Planus: Study Maps Prevalence and Treatment Approaches These Adverse Events Linked to Improved Cancer Prognosis How to Manage Esophageal Conditions: Expert Pearls Treatment of Frontal Fibrosing Alopecia in Black Patients Drug Interaction Checker Pill Identifier Calculators Formulary Clues in the Oral Cavity: Are You Missing the Diagnosis? 2002 1123213-overview Diseases & Conditions Diseases & Conditions Lichen Planus 2002 1078327-overview Diseases & Conditions Diseases & Conditions Oral Lichen Planus 2001 /viewarticle/ruxolitinib-cream-shows-promise-treating-lichen-planus-2024a1000j36 news news Ruxolitinib Cream Shows Promise in Treating Lichen Planus
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2.2.1对数的运算性质 搜索 我的图书馆 查看信箱 系统消息 官方通知 设置 开始对话 有 11 人和你对话，查看 忽略 历史对话记录 通知设置 发文章 发文工具 撰写网文摘手文档视频思维导图随笔相册原创同步助手 其他工具 图片转文字文件清理AI助手 留言交流 搜索 分享 QQ空间QQ好友新浪微博微信 生成长图转Word打印朗读全屏修改转藏+1 × 微信扫一扫关注 查看更多精彩文章 × 微信扫一扫 将文章发送给好友 2.2.1对数的运算性质 家有学子 2011-10-26 |1211阅读|12 转藏 大 中 小 转藏全屏朗读打印转Word生成长图分享 QQ空间QQ好友新浪微博微信 展开全文 课题：§2.2.1对数的运算性质 教学目的：（1）理解对数的运算性质； （2）知道用换底公式能将一般对数转化成自然对数或常用对数； （3）通过阅读材料，了解对数的发现历史以及对简化运算的作用． 教学重点：对数的运算性质，用换底公式将一般对数转化成自然对数或常用对数 教学难点：对数的运算性质和换底公式的熟练运用． 教学过程： 一、引入课题 1．对数的定义：； 2．对数恒等式：； 二、新课教学 1．对数的运算性质 提出问题： 根据对数的定义及对数与指数的关系解答下列问题： 1 设，，求； 2 设，，试利用、表示·． （学生独立思考完成解答，教师组织学生讨论评析，进行归纳总结概括得出对数的运算性质１，并引导学生仿此推导其余运算性质） 运算性质： 如果，且，，，那么： 1·＋； 2－； 3． （引导学生用自然语言叙述上面的三个运算性质） 学生活动： 1 阅读教材Ｐ 75 例 3、4，； 设计意图：在应用过程中进一步理解和掌握对数的运算性质． 2 完成教材Ｐ 79 练习 1~3 设计意图：在练习中反馈学生对对数运算性质掌握的情况，巩固所学知识． 2．利用科学计算器求常用对数和自然对数的值 设计意图：学会利用计算器、计算机求常用对数值和自然对数值的方法． 思考：对于本小节开始的问题中，可否利用计算器求解的值？从而引入换底公式． 3．换底公式 （，且；，且；）． 学生活动 1 根据对数的定义推导对数的换底公式． 设计意图：了解换底公式的推导过程与思想方法，深刻理解指数与对数的关系． 2 思考完成教材 P 76 问题（即本小节开始提出的问题）； 3 利用换底公式推导下面的结论 （1）； （2）． 设计意图：进一步体会并熟练掌握换底公式的应用． 说明：利用换底公式解题时常常换成常用对数，但有时还要根据具体题目确定底数． 4．课堂练习 1 教材Ｐ 79 练习 4 2 已知 3 试求：的值。（对换5与2，再试一试） 4 5 设,，试用、表示 三、归纳小结，强化思想 本节主要学习了对数的运算性质和换底公式的推导与应用，在教学中应用多给学生创造尝试、思考、交流、讨论、表达的机会，更应注重渗透转化的思想方法． 四、作业布置 1．基础题：教材 P 86 习题 2．2（A 组）第 3~5、11 题； 2．提高题： 1 设,，试用、表示； 2 设,，试用、表示； 3 设、、为正数，且，求证：． 3．课外思考题： 设正整数、、（≤≤）和实数、、、满足： ，， 求、、的值． 对数与对数运算学案 一、复习引入 问题１：ｘ ２＝４，ｘ＝ _ ？ 问题 2：2 ｘ＝４，ｘ＝ _ ？ 问题 3：若我国目前人口总数为 13 亿，按 1%的平均增长率，x 年后的人口总数 y= _______ ？经过？年后，人口总数可达到 18 亿？ 二、对数的定义 1．加法的逆运算是 __ ，乘法的逆运算是 ___ ，平方运算的逆运算是 __ ，指数运算的逆运算是 _ 。 2．一般地，如果，那么： 数ｘ叫做以ａ为底 N 的对数，其中ａ叫 _ ，N 叫 _ 。 3．N 是一个怎样的数？为什么？ 4．例 1：将下列指数式化为对数式，对数式化为指数式： 注：①“”简写为“”叫常用对数 ②“”简写为“”叫自然对数。 5.强化训练 把下列指数式化为对数式，对数式化为指数式： 三、能力提升 1.请完成下列填空： （1） __ ；（2） __ ； （3） __ ；（4） __ ， （5）=___。 2.例 2，求下列各式中的 x 的值： （1）；（2）；（3）；（4）。 3.强化训练 求下列各式的值： 四、小结：今天学了什么？ 答： 五、作业布置 完成 P74 页 A1，A2。 高考资源网 1．log 12 3＋log 12 4 等于() A．7 B．12 C．1 D．log 12 7 【解析】 log 12 3＋log 12 4＝log 12(3×4)＝1.故选 C. 【答案】 C 2．log 5 2·log 2 5 的值为() A.高考资源网 2(1)B．1 C.2(3)D．2 【解析】 log 5 2·log 2 5＝log 5 2·log52(log55)＝1.故选 B. 【答案】 B 3．已知 lg2＝a，lg7＝b，那么 log 8 98＝__. 【解析】 log 8 98＝lg8(lg98)＝lg23(72×2) ＝3lg2(lg72＋lg2)＝3lg2(2lg7＋lg2) ＝3a(2b＋a). 【答案】 3a(2b＋a) 4．设 3 x＝4 y＝36，求 x(2)＋y(1)的值． 【解析】 (1)∵3 x＝36,4 y＝36， ∴x＝log 3 36，y＝log 4 36， ∴x(1)＝log336(1)＝log363(log3636)＝log 36 3， y(1)＝log436(1)＝log364(log3636)＝log 36 4， ∴x(2)＋y(1)＝2log 36 3＋log 36 4 ＝log 36(9×4)＝1. 一、选择题(每小题 5 分，共 20 分) 1．(2009 年湖南卷)log 2 的值为() A．－B. C．－2(1)D.2(1) 【解析】 log 2＝2(1)log 2 2＝2(1).故选 D. 【答案】 D 2．若 lg 2＝a，lg 3＝b，则 lg 12(lg 15)等于() A.2a＋b(1＋a＋b)B.a＋2b(1＋a＋b) C.2a＋b(1－a＋b)D.a＋2b(1－a＋b) 【答案】 C 3．已知 a＝log 3 2，用 a 表示 log 3 8－2log 3 6 是() A．a－2 B．5a－2 C．3a－(1＋a)2 D．3a－a 2－1 【解析】 由 log 3 8－2log 3 6＝3log 3 2－2(log 3 2＋log 3 3)＝3a－2(a＋1)＝a－2. 【答案】 A 4．(log 4 3＋log 8 3)(log 3 2＋log 9 8)等于() A.6(5)B.12(25) C.4(9)D．以上都不对 【解析】 原式＝log38(log33)·log39(log38) ＝3log32(1)·2(3log32) ＝6log32(5)×2(5)log 3 2＝12(25).故选 B. 【答案】 B 二、填空题(每小题 5 分，共 10 分) 5．log 27＝__. 【解析】 log 27＝log()6＝6. 【答案】 6 6．已知 2 x＝5 y＝10，则 x(1)＋y(1)＝__. 【解析】 由 2 x＝5 y＝10 得 x＝log 2 10，y＝log 5 10， x(1)＋y(1)＝log210(1)＋log510(1) ＝lg2＋lg5＝1. 【答案】 1 三、解答题(每小题 10 分，共 20 分) 7．求下列各式的值： (1)(lg 5)2＋lg 50·lg 2； (2)lg 14－2lg 3(7)＋lg 7－lg 18； (3)log 3(1)27－log 3(1)9； (4)log 8 9×log 3 32. 【解析】 (1)原式＝(lg 5)2＋lg(10×5)lg 5(10) ＝(lg 5)2＋(1＋lg 5)(1－lg 5) ＝(lg 5)2＋1－(lg 5)2＝1. (2)方法一：原式＝lg(2×7)－2lg 3(7)＋lg 7－lg(3 2×2) ＝lg 2＋lg 7－2(lg 7－lg 3)＋lg 7－(2lg 3＋lg 2)＝0 方法二：原式＝lg 14＋lg 3(7)2＋lg 7－lg 18 ＝lg 2×18(7)＝lg 1＝0. (3)原式＝log 3(1)9(27)＝log 3(1)3＝－1. (4)原式＝lg8(lg9)×lg3(lg32)＝3lg2(2lg3)×lg3(5lg2)＝3(10). 8．已知 m 2＝a，m 3＝b，m>0 且 m≠1，求 2log m a＋log m b. 【解析】 由 m 2＝a，m 3＝b，m>0 且 m≠1，得 log m a＝2，log m b＝3； ∴2log m a＋log m b＝2×2＋3＝7. 9．(10 分)已知 ln a＋ln b＝2ln(a－2b)，求 log 2 b(a)的值． 【解析】 因为 ln a＋ln b＝2ln(a－2b)，解得 ab＝(a－2b)2. a 2－5ab＋4b 2＝0，解得 a＝b 或 a＝4b， 又 a－2b>0(b>0，)所以 a>2b>0，故 a＝4b，log 2 b(a)＝log 2 4＝2， 即 log 2 b(a)的值是 2. 【答案】 2 w.w.w.k.s.5.u.c.o.m www.ks5u.com 本站是提供个人知识管理的网络存储空间，所有内容均由用户发布，不代表本站观点。请注意甄别内容中的联系方式、诱导购买等信息，谨防诈骗。如发现有害或侵权内容，请点击 一键举报。 转藏分享 QQ空间QQ好友新浪微博微信 献花（1） +1 来自： 家有学子>《高一》 举报/认领 上一篇： 1.3.1函数的单调性 下一篇： 2.2.3对数函数的性质（性质的应用） 猜你喜欢 0 条评论 发表 请遵守用户评论公约 查看更多评论 类似文章更多 高一预科----2.21对数及其运算 ②根据对数定义求loga1和logaa?a>0，且a≠1?的值.。常用对数：我们通常将以10为底的对数叫做常用对数．为了简便，N的常用对数log10N简记作lgN.例如：log105简记作lg5；log103.5简记作lg3.5.1求log... 神奇的对数恒等式、万能的换底公式！最全对数性质硬科普！ log(a,a)=1，log(a,1)=0，log(a,a^2)=2.log(a,1/a)=-1，log(a,√a)=1/2.①log(a,M×N)=log(a,M)+log(a,N)②log(a,M/N)=log(a,M)-log... 导学案：2 (4)(lg2)3＋3lg2·lg5＋(lg5)3.(4)原式＝(lg2＋lg5)[(lg2)2－lg2·lg5＋(lg5)2]＋3lg2·lg5＝(lg2)2＋2lg2·lg5＋(lg5)2＝(lg2＋lg5)2＝1.(1)4lg2＋3lg5－lg；3．lg2＋lg5＝1，lg2＝1... 第4_3对数优质教育课件PPT 选填 《第2章 基本初等函数、导数及其应用》课件2-6 [方法引航]?1?对于多个对数值大小比较，首先利用对数性质分开正、负数?与0比较?再分开?0，1?与?1，＋∞??与1比较??2?解决简单的对数不等式，应先利用对数的运算性质化为同底数的对数值，再利用对数函数... 对数 log(a^n)(b^m) = [m×ln(b)]÷[n×ln(a)] = (m÷n）×{[ln(b)]÷[ln(a)]}纳皮尔的骨头，是苏格兰数学家约翰纳皮尔发明，是一种用来计算乘法与除法，类似算盘的工具。到如今几... 4.3.2 对数的运算（第2课时）课件-2022-2023学年高一上学期数学人教A版（2019）必修第一册 数 学 人 教 版 必 修 第 一 册第四章 指数函数与对数函数4.3.2 对数的运算 换底公式 授课人：魏永祥2022/11/18 15:35:56对数运算性质：知识回顾对数的运算性质把乘积转化为加法，把商转化为减法，把乘... 换底公式 换底公式。换底公式就是log(a)(b)=log(c)(b)/log(c)(a)(a,c均大于零且不等于1）。log(a)(b)=log(c)(b)/log(c)(a)(a,c均大于零且不等于1） 推导过程。如：log（10）（5）=log（5）（5）/log（5）（10）l... 2022年高考复习 2.6 对数与对数函数 (lg2)2＋lg2·lg50＋lg25＝__.解析：因为对数函数y＝logax的图象恒过点(1,0)，所以函数y＝loga(x－1)的图象恒过点(2,0)，所以函数y＝loga(x－1)＋2的图象恒过点(2,2)．.[解析]由题意得，当0... 个图VIP年卡，限时优惠价168元>>x 家有学子 关注对话 TA的最新馆藏 同谷杜甫草堂的兴衰与流变 杜甫在同谷写的诗 [转]柳公权小楷《玄秘塔碑》，法度森严 苏轼 楷书《宸奎阁碑》 王羲之《丧乱帖》珂罗版 柳公权 楷书《玄秘塔碑》 喜欢该文的人也喜欢更多 鬼谷子：真正善于驭人术的人，从不靠权势压人，而是用人心作钩，...阅66 古人给后人20条“处世忠告”阅77 我又发现了13个免费神器，每一个都超级让人心动！阅79 社会到底有多现实阅138 《菜根谭》4句名言，饱含处世智慧，受用终生！阅178 热门阅读换一换 高阶思维包含什么，如何培养？阅27695 项目管理方案和实施方案概述阅26568 物业小区消防应急预案阅27928 公司员工晋升管理制度阅67428 事故反思总结10篇阅76553 复制 打印文章 发送到手机微信扫码，在手机上查看选中内容 全屏阅读 朗读全文 分享文章QQ空间QQ好友新浪微博微信 AI解释 复制 打印文章 adsbygoogle.js 发送到手机微信扫码，在手机上查看选中内容 全屏阅读 朗读全文 AI助手 阅读时有疑惑？点击向AI助手提问吧 联系客服 在线客服： 360doc小助手2 客服QQ： 1732698931 联系电话：4000-999-276 客服工作时间9:00-18:00，晚上非工作时间，请在QQ留言，第二天客服上班后会立即联系您。
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Prime factors of 15 Home Blog eBooks ACCUPLACER Mathematics ACT Mathematics AFOQT Mathematics ALEKS Tests ASVAB Mathematics ATI TEAS Math Tests CBEST Math Test CHSPE Math CLEP College Algebra CLEP College Mathematics Common Core Math DAT Math Tests FSA Tests FTCE General Knowledge Math GED Mathematics GMAS GRE Quantitative Reasoning HiSET Math Exam HSPT Math ISEE Mathematics ParaPro Math Test PARCC Tests Praxis Core Math Test PSAT Math Tests SAT Math Tests SBAC Tests SHSAT Math SSAT Math Tests STAAR Tests TABE Tests TASC Math TSI Mathematics Courses ACCUPLACER ACT ALEKS ASVAB ATI TEAS 6 CLEP College Algebra CLEP College Mathematics DAT Quantitative Reasoning GED HiSET ISEE Lower Level ISEE Middle Level ISEE Upper Level PSAT SAT SHSAT SSAT Lower Level SSAT Middle Level SSAT Upper Level TASC TSI Worksheets Accuplacer Math Worksheets ACT Math Worksheets AFOQT Math Worksheets ALEKS MATH WORKSHEETS ASVAB Math Worksheets CBEST Math Worksheets CHSPE Math Worksheets CLEP College Mathematics Worksheets GED Math Worksheets HiSET Math Worksheets PERT Math Worksheets Pre-Algebra Worksheets PSAT Math Worksheets SAT Math Worksheets SSAT Middle-Level Math Worksheets SIFT Math Worksheets TASC Math Worksheets TABE Math Worksheets THEA Math Worksheets TSI Math Worksheets About Us user Login 0 Shopping Cart $0.00View Cart What is the prime factorization of the number 15? Prime factors of 15 The prime factors are: 3, 5 Prime Factor Decomposition or Prime Factorization is the process of finding which prime numbers can be multiplied together to make the original number. 15 divided by 3, 5 gives no remainder. This means Prime number are numbers that can divide without remainder. How to Calculate Prime Factorization of 15? The main method of prime factorization is start dividing the number by the any divisible prime numbers until the only numbers left are prime numbers (2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,...)(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,...). 15÷3=5 15÷3=5 5÷5=1 5÷5=1 Factorize(15)=(15)=3×5 3×5 Factor Tree of 15 15 3 5 If you want to learn more about prime factorization, take a look at the Wikipedia page. Prime factors of: New releases Buy NowQuick View AFOQT Math in 7 Days ~~$$20.99~~$$14.99 Buy NowQuick View 5 Full Length GED Math Practice Tests ~~$$18.99~~$$14.99 Buy NowQuick View Accuplacer Math Exercise Book ~~$$18.99~~$$14.99 Buy NowQuick View The Most Comprehensive ATI TEAS 6 Math Preparation Bundle ~~$$76.99~~$$36.99 Related Prime Factors Prime factors of 1 Prime factors of 2 Prime factors of 3 Prime factors of 4 Prime factors of 5 Prime factors of 6 Prime factors of 7 Prime factors of 8 Prime factors of 9 Prime factors of 10 Prime factors of 11 Prime factors of 12 Prime factors of 13 Prime factors of 14 Prime factors of 15 Prime factors of 16 Prime factors of 17 Prime factors of 18 Prime factors of 19 Prime factors of 20 FSA Mathematics Workbook for Grade 3 ~~$$18.99~~$$14.99 Buy Now ParaPro Math Practice Workbook ~~$$25.99~~$$14.99 Buy Now Ace the ACT Math in 30 Days ~~$$20.99~~$$14.99 Buy Now algebra 1 Workbook ~~$$18.99~~$$14.99 Buy Now ALEKS Math Exercise Book ~~$$18.99~~$$14.99 Buy Now STAAR Grade 7 Math Comprehensive Prep Bundle ~~$$104.99~~$$36.99 Buy Now FSA Mathematics Workbook for Grade 3 ~~$$18.99~~$$14.99 Buy Now ParaPro Math Practice Workbook ~~$$25.99~~$$14.99 Buy Now Ace the ACT Math in 30 Days ~~$$20.99~~$$14.99 Buy Now algebra 1 Workbook ~~$$18.99~~$$14.99 Buy Now ALEKS Math Exercise Book ~~$$18.99~~$$14.99 Buy Now STAAR Grade 7 Math Comprehensive Prep Bundle ~~$$104.99~~$$36.99 Buy Now Subscribe to get amazing offers, special vouchers, the latest updates, fantastic coupons, and more exclusive promotions Worksheets Accuplacer Math Worksheets ACT Math Worksheets ALEKS Math Worksheets ASVAB Math Worksheets GED Math Worksheets HiSET Math Worksheets Pre Algebra Math Worksheets SAT Math Worksheets TSI Math Worksheets TASC Math Worksheets Top Courses ACCUPLACER Math Complete Course ACT Math Complete Course ALEKS Math Complete Course ASVAB Math Complete Course GED Math Complete Course HiSET Math Complete Course PSAT Math Complete Course SAT Math Complete Course TASC Math Complete Course TSI Math Complete Course Copyrights © 2022 All Rights Reserved by Testinar Inc. info@testinar.com ·
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Verificación de mapeo conforme. - Variable compleja y Análisis de Fourier - Rincón Matemático Typesetting math: 100% Ingresar Registrarse 29 Septiembre, 2025, 08:28 am Menu Ingresar Registrarse Inicio Ingresar Registrarse Rincón Matemático » Matemática » Análisis Matemático » Variable compleja y Análisis de Fourier Tema: Verificación de mapeo conforme. « anteriorpróximo » Imprimir Páginas: Ir Abajo Autor Tema: Verificación de mapeo conforme. 0 Usuarios y 1 Visitante están viendo este tema. 28 Junio, 2020, 06:22 pm Leído 1142 veces Hauss π π π Mensajes: 148 País: Karma: +0/-0 Sexo: Verificación de mapeo conforme. Hola, qué tal, podrían ayudarme con el siguiente problema por favor: Encuentre el mapeo conforme del disco |z−4 i|<2 al semiplano v<u, (w=u+i v) tal que el centro del disco se mapee al punto 4 y el punto 2 i se mapee al origen. He intentado lo siguiente para buscar el mapeo, pero hay una parte en la que tengo duda: Con w 1=z−4 i llevamos el disco al origen, con w 2=w 1 2 lo convertimos al disco unitario, lo llevamos al semiplano superior con w 3=−i w 2+1 w 2+i y con w 4=i log(w 3)α con α=5 π 4 llevamos al semiplano superior al semiplano que buscamos. Por tanto la transformación que hace lo que buscamos es w(z)=w 4(w 3(w 2(w 1(z)))) Aun me falta ver que cumpla con las condiciones requeridas, pero mi duda principalmente es lo que he señalado con rojo, intuyo que cumple con lo que busco, pero no sé si estoy en lo correcto. Agradecería si me dijeran que tengo mal o si ustedes tienen alguna otra forma de resolverlo. Gracias. En línea 29 Junio, 2020, 11:30 am Respuesta #1 Luis Fuentes el_manco Administrador Mensajes: 58,146 País: Karma: +0/-0 Re: Verificación de mapeo conforme. Hola Cita de: ASamuel en 28 Junio, 2020, 06:22 pm Encuentre el mapeo conforme del disco |z−4 i|<2al semiplano v<u, (w=u+i v)tal que el centro del disco se mapee al punto 4y el punto 2 ise mapee al origen. He intentado lo siguiente para buscar el mapeo, pero hay una parte en la que tengo duda: Con w 1=z−4 illevamos el disco al origen, con w 2=w 1 2lo convertimos al disco unitario, lo llevamos al semiplano superior con w 3=−i w 2+1 w 2+iy con w 4=i log(w 3)α con α=5 π 4 llevamos al semiplano superior al semiplano que buscamos. Pero si no me equivoco la transformación que lo lleva en el plano superior está mal. Debería de ser: w 3=−i w 2+1 w 2−1 La siguiente pretende ser un giro de ángulo α=5 π 4. Debería de ser: w 4=e i α w 3. Además tienes que ajustar los puntos. El centro del disco original va al centro del disco unitario. Su imagen en el semiplano superior es: i. El punto 2 i del disco original va al punto −i del disco unitario. Su imagen en el semiplano superior es: 1. A ti te interesaría que esas imágenes fueran respectivamente P=4 e−α i y Q=0. Entonces antes de aplicar el giro intercala una transformación dentro del semiplano superior que lleve −i en P y −1 en 0. Toda transformación del semiplano superior es de la forma: a z+b ç c z+d con a,b,c,d∈R y a d−b c>0. Saludos. CORREGIDO. En línea 29 Junio, 2020, 05:12 pm Respuesta #2 Hauss π π π Mensajes: 148 País: Karma: +0/-0 Sexo: Re: Verificación de mapeo conforme. Hola, al principio he tratado de proceder de manera analoga a como mencionas, pero tuve unos problemas, creo que porque soy nuevo en esto. Aqui mencionas: Citar A ti te interesaría que esas imágenes fueran respectivamente P=4 e−α iy Q=0. ¿A que se debe el signo negativo de P=4 e−α i? Citar Entonces antes de aplicar el giro intercala una transformación dentro del semiplano superior que lleve −ien y −1en 0. Toda transformación del semiplano superior es de la forma: a z+b c z+dcon a,b,c,d∈Ry a d−b c>0. En esta parte tengo dos dudas, la primera, mencionas que buscamos una transformación dentro del plano nos lleve −i en creo que falto una parte y no sé a donde te refieras a llevarlo, y la segunda, como mencione, al principio trate de hacerlo con ese tipo de transformaciones del semiplano, pero he visto ejemplos y necesitaria más puntos para que pueda aplicarlo, ¿no?, o ¿tengo que tomarlos de una manera especifica para poder aplicar estas transformaciones?. Muchas gracias por tus respuestas siempre son de mucha ayuda En línea 30 Junio, 2020, 08:32 pm Respuesta #3 Luis Fuentes el_manco Administrador Mensajes: 58,146 País: Karma: +0/-0 Re: Verificación de mapeo conforme. Hola Vaya por delante que tenía varias erratas. Cita de: ASamuel en 29 Junio, 2020, 05:12 pm ¿A que se debe el signo negativo de P=4 e−α i? Recapitulemos. El enunciado pide llevar el centro del disco en el 4. La dos primeras transformaciones llevan el centro del disco dado en el centro del disco unidad. La siguiente (que transforma el disco unidad en el semiplano superior) lleva el centro en el punto i. Sabemos que terminaremos con una último giro de α grados. Entonces antes de hacerlo quiero modificar ese punto i para que al ser girado α grados nos de 4. Por tanto quiero que, antes del giro, el punto i se transforme en el punto 4 girado −α grados, es decir, en P=4 e−α. Citar En esta parte tengo dos dudas, la primera, mencionas que buscamos una transformación dentro del plano nos lleve −i encreo que falto una parte y no sé a donde te refieras a llevarlo, Eso ya lo arregle: en P. Además recuerda que tenía otra errata. Es el punto i, no el −i. Citar y la segunda, como mencione, al principio trate de hacerlo con ese tipo de transformaciones del semiplano, pero he visto ejemplos y necesitaria más puntos para que pueda aplicarlo, ¿no?, o ¿tengo que tomarlos de una manera especifica para poder aplicar estas transformaciones?. Tienes que buscar a,b,c,d tal que: a i+b c i+d=4 e−α i a+b c+d=0 De la segunda condición, a=−b. Y sin pérdida de generalidad puedes suponer a=−b=1. Entonces te queda: d+c i=1 4 e α i(i−1)=1 4 2–√(−1−i)(i−1)=1 2 2–√ de donde d=1 2 2–√ y c=0. Saludos. P.D. Revisa las cuentas. En línea Imprimir Páginas: Ir Arriba « anteriorpróximo » Rincón Matemático » Matemática » Análisis Matemático » Variable compleja y Análisis de Fourier Tema: Verificación de mapeo conforme. Ir a: SMF 2.0.19 | SMF © 2020, Simple Machines Rincón Matemático © 2025 Theme by SMF Tricks Página generada en 0.153 segundos con 26 consultas. Ingresar × Bienvenido(a), Visitante. Por favor, ingresa o regístrate. Ingresar con nombre de usuario, contraseña y duración de la sesión Ingresar
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Faraday's Law, ε = − N Δϕ/Δt | EMI | Physics | Khan Academy Khan Academy India - English 548000 subscribers 137 likes Description 8301 views Posted: 1 Jun 2022 We know that changing magnetic flux induces emf. In this video, let's figure out the magnitude of that induced emf. Khan Academy is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We offer quizzes, questions, instructional videos, and articles on a range of academic subjects, including math, biology, chemistry, physics, history, economics, finance, grammar, preschool learning, and more. We provide teachers with tools and data so they can help their students develop the skills, habits, and mindsets for success in school and beyond. Khan Academy has been translated into dozens of languages, and 15 million people around the globe learn on Khan Academy every month. As a 501(c)(3) nonprofit organization, we would love your help! Donate here: Created by Vibhor Pandey 15 comments Transcript: Intro we know that changing magnetic flux induces some emf in a coil the goal of this video is to figure out the magnitude or the amount of that amf generated Faradays Law let's start off by bringing back faraday's experiment setup now faraday used an ammeter in place of a lamp here ammeter is an instrument used for measuring current for our purposes let's use a lamp and whenever there is induced emf in the coil we should see the lamp glow now it was observed that when this magnet is moved towards or away from the coil the lamp started glowing which meant that there was some induced emf in the coil let's see the magnetic field lines now when we move the magnet towards and away see how the lamp closed and this led faraday to conclude that whenever magnetic flux through the coil changed there was some emf induced in the coil it was also observed that the bulb glowed brighter when the magnet was moved faster towards or away from the coil so for example if we take four seconds to move this magnet from this point right here to this point the bulb glows and there is some induced emf and it could look like this now if we move the magnet between the same points in two seconds if we reduce the time by half notice how the brightness of the lamp changes i'm sure you must have seen that the brightness of the lamp increases and it turns out that the amf induced in the coil in two seconds was double the amount of emf that was induced when we took four seconds to move the magnet between the same points now if we reduce the time by one fourth if we take one second to move the magnet between the same points again notice how the brightness of the lamp changes and there you go again we saw that the brightness increased and it turns out that the amf induced in this case is four times the amf induced when we took four seconds to move the magnet between the same points this led faraday to conclude that the magnitude or the amount of emf that is induced is equal to the rate at which the magnetic flux through the coil changed and this can be expressed mathematically just like this this right here is called faraday's law of electromagnetic induction which says that the amf induced the amount of emf induced the magnitude of emf induced is equal to the rate of change of magnetic flux through the coil this negative sign over here is called lenses law and this law helps us predict the direction of current in the coil we will not be focusing on lenses law in this video we will talk about lenses law in depth in one of our other videos for this video let's only focus on the magnitude of the amf induced in the coil let me remove the field line so that we can see this clearly there you go now if we have only one loop and the magnetic flux through this loop changes then the emf induce is directly equal to the rate of change of magnetic flux through this loop if we have three loops then it turns out that emf is induced in each and every loop and the total emf induced is equal to the sum of the emf induced in each and every loop so we add the rate of change of flux three times which gives us the total amf induced if we have six loops then again emf is induced in each and every loop and the total emf is equal to the rate of change of flux in each and every loop and if we continue increasing the number of loops that means you have to add the amf induced that many times so if we have n number of loops then you multiply the rate of change of flux n times to get the total emf induced now this equation right here is used for figuring out the average induced emf average because we are figuring out the amf induced over a period of time delta t over some interval of time this could be two seconds three seconds four seconds there is also instantaneous emf which is given by d phi by dt this is calculated when we are interested in figuring out the emf induced at any one particular instant in time in this video we will only be focusing on average induced emf so let's do that with a couple of examples now for our first case we have Examples a region of uniform magnetic field and the strength of this magnetic field is 2 tesla and let's say we have a square loop which is which has a side of 6 meters and it is moving to the right with a speed of 3 meters per second because it is moving to the right after some time this square loop will completely enter the region of uniform magnetic field and will look just like this the question is to figure out the average the magnitude of the average emf induced when the loop has fully entered the region of uniform magnetic field why don't you pause the video and try this one on your own first all right so we know that the magnitude of average emf induced is given by this relation right here is equal to the rate of change of magnetic flux now over here n is just one because this is just one square wire there's no coil over here so n is just one we need to think about the change in magnetic flux and also the time duration for which the magnetic flux is changing so let's begin by thinking about delta t first delta t is the time duration for which the magnetic flux changes and the magnetic flux through the coil starts changing when the loop starts entering the region of uniform magnetic field so kind of like this when the loop is moving to the right and at this very instant right when it starts entering the region that is when we should start recording a time that is when we should start the timer for delta t so we know the speed that is three meters per second that means that in one second the loop travels three meters to the right so that would look like this it has traveled 3 meters to the right and the distance over here is this distance right here it has traveled 3 meters to the right now if we take one more second then in one more second it will take further three meters to the right and the loop will move like this further three meters to the right so now over here three meters is this distance right here that is the point when we should stop the timer because now the loop has fully entered the region of uniform magnetic field and now when it continues to move forward there will be no change in magnetic flux because the number of magnetic field lines through this loop aren't really changing so the time duration for which the magnetic flux changes that comes out to be as two seconds now let's think about the change in magnetic flux that is delta phi so delta phi is the final magnetic flux minus the initial one we can think about the initial to begin with initially the loop is completely outside the region of uniform magnetic field and when that is the case there are no magnetic field lines passing through the loop so the magnetic flux is just zero so that leaves us with the final flux now flux is given by ba cos theta we already know the strength of the magnetic field and the we can figure out the area of this square loop which would be 6 into 6 36 meter square but what about this cos theta over here now theta is the angle between the magnetic field strength and the area vector so in order to figure out what would theta be let's try and look at this setup from a certain angle so let's say this is these are our coordinate axes and here is a loop so we are looking at this setup from a certain angle now if we try and draw the magnetic field vector that will always be pointing down because these magnetic field lines are going away from you they're away from the plane of the screen so the magnetic field vector could look like this this right here is your magnetic field vector and the area vector could either be along the magnetic field vector or directly opposite to it so if it is along the magnetic field vector it would look like this now over here the angle between the area vector and the magnetic field vector is 0 degrees so that would be zero and cos zero is just one it is also possible that the area vector instead of vertically down it's pointing in a direction completely opposite to the magnetic field direction just like this now in this case the angle between the magnetic field and the area vector is 180 degrees so that is -1 now it turns out that the direction of the induced current determines the direction of the area vector and we can predict the direction of the current using lenses law again that is something that we will not go into this video we are only interested in the magnitude of cos theta and in either case the magnitude of cos theta comes out to be as one so now let's plug in the values and if we do that here is your magnetic field strength here is your area and the cos theta is just one so this will give us the final flux and rather the change in flux to be as 72 vapors now when we put all of this the change in flux and the time duration in the first equation that will give us 72 divided by two seconds and finally the emf induced comes out to be as 36 volts Example now in our next example we have a coil between the two poles of a magnet and the magnetic field lines look like this they go from north to south pole and this coil can be rotated in a clockwise manner just like this the strength of these magnetic field lines is two tesla the area of this coil is six meter square and we can say that this coil is made up of ten loops of fire and so ten loops of wire they make up to form a coil so n capital n becomes 10 in this one now this coil rotates from an angle of 60 degrees to an angle of 180 degrees in three seconds this theta is the angle between the magnetic field vector and the area vector of the coil the question is to figure out what is the average induced emf when the coil rotates from 60 degrees to 180 degrees in 3 seconds why don't you pause the video and try this one on your own first all right so the average induced emf is given by the same relation which is n into change in magnetic flux delta phi by delta t and we are interested in the magnitude so you see the mod signs over here now we already know the time interval delta t which is three seconds in this one so we only need to figure out the change in magnetic flux change in flux is final minus initial and flux is b a cos theta so we can write change in flux as b a cos theta f which is the final angle minus b a cos theta i which is the initial angle let's try and understand how do these angles really look like so for that let's place ourselves on top of this coil let's say we are on the top this is our eye and we're looking down on the coil so the magnetic field lines will look just like this they go from the north end to the south end just in this manner and to begin with the angle between the area vector of the coil and the magnetic field lines is 60 degrees so the coil would be in an orientation like this if we show the area vector that would be right over here now theta that is 60 degrees is this angle right here that is the angle between the area vector and the magnetic field once it has rotated by some degrees and finally the angle between the area vector and the magnetic field is 180 degrees so in that case the coil would look just like this along with the area vector you can see how the angle between the magnetic field vector and the area vector is 180 degrees so theta 60 degrees to begin with so that makes cos theta as 1 by 2 and theta 180 degrees after 3 seconds so that makes cos 180 degrees to be as minus 1. now if you plug in these values right over here we should get change in flux to be as 2 multiplied by 6 which is the area multiplied by minus 1 minus 1 because we are starting with the final flux the so it has the final angle theta f so that is 180 degrees therefore this is minus 1 minus the initial flux and that would be 2 into 6 into 1 by 2 cos 60 degrees now when we work this out this comes out to be as minus 12 that is the final flux minus c initial flux which is six this comes out to be as minus 18. so minus 18 if you take the mod it becomes 18 webbers now if you place these values into the first equation that will become 10 into 18 divided by 3 then is the number of turns time interval is 3 seconds and the change in flux came out to be as 18 webbers so the emf induced and this one is 60 volts
6090
https://byjus.com/physics/value-of-electron/
What is the Value of Electrons? An electron is a subatomic particle commonly represented as e– or e. It possesses negative polarity. An electron inherits the properties like – Charge, Mass, Spin, etc. The values of electrons imply the value of the charge of an electron, the mass of the electron, and a quantum mechanical property- spin of the electron along with corresponding units. Charge of Electron The electric charge by an Electron is the unit elementary negative charge. The value is: e– = 1.60217662 × 10-19 coulombs The charge of an electron is responsible for the electric charge. Mass of Electron The mass of an electron is 1/1836 of proton mass. The value of electron mass is: me = 9.10938356 × 10-31 kilograms The values of charge and mass of electrons are commonly used in solving problems in physics. Electron Spin Spin or angular momentum is an intrinsic quantum mechanical property of subatomic particles. The spin of an electron is half-integral values: s = ½ You may also want to check out these topics given below! Mu Naught Value Epsilon Naught Value Relation Between Ev And Joule Value Of hc Summary: The table given below comprises the Mass, Charge and Spin value of electrons along with units. | | | | --- | | Value of electron | Unit | | Electron charge | 1.60217662 × 10-19 | Coulombs | | Electron charge in eV | 1.60217662 × 10-19 | Joule | | Electron mass | 9.10938356 × 10-31 | Kg | | Electron mass in amu | 0.00054858 | Atomic Mass Unit(amu) | | Electron Spin | ½ | | Watch the video and learn about the discovery of electrons. 3,26,269 Hope you have learnt the value of electrons, like- The mass of electrons in Kilograms as well as in amu. The charge of the electron in Coulombs and eV. The value of electron spin. Test Your Knowledge On Value Of Electron! Q5 Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin! Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz Congrats! Visit BYJU’S for all Physics related queries and study materials Your result is as below 0 out of 0 arewrong 0 out of 0 are correct 0 out of 0 are Unattempted Login To View Results Did not receive OTP? Request OTP on Login To View Results Comments Leave a Comment Cancel reply Ahmed Hatem September 17, 2020 at 12:29 am Thank You Reply Register with BYJU'S & Download Free PDFs
6091
https://www.youtube.com/watch?v=FM2iO1nk07Q
Arrange the digits to make a true subtraction problem! Math Puzzles 3000 subscribers 53 likes Description 3709 views Posted: 17 Dec 2020 This numerical challenge is a fun and stimulating way to push your arithmetic skills to their limits. It will force you to exercise your brain by thinking about subtraction in reverse. Try it on your own, but stay tuned for Huzefa's explanation. Check back every Friday for more awesome math puzzles! To learn about math tutoring and standardized test prep, please visit To check out our other math channel, go to 2 comments Transcript: what's up everybody and welcome back to another math puzzle in this one the objective is to arrange the digits correctly to complete the subtraction problem but notice in that top row in the green purple and blue spaces you're only gonna have three two and nine as your options for example in the bottom row you've only got eight two and zero to work with again we have to arrange the digits so that once we complete the subtraction we're left with 724. go ahead and hit that pause button and see if you can figure it out when you're ready hit play and i'll give you the explanation so normally in a problem like this i would encourage you strategically to start from right to left so we're starting with the ones place and then we're moving to the tens and the hundreds but this is a unique problem because i see that the final answer is gonna have a seven in the hundreds column that's really important because there's truly only one way i'm going to get a seven not with the three not with the two they're too small i've gotta have a nine go in this spot and if a nine's gonna go here i know that it has to be a two here it's not gonna be a zero even if even if i had a zero and i borrowed one from the nine the best i would get is an eight i couldn't get a seven so now we're already part of the way there next i'm gonna look at the first column and i need to get a four but that's kind of weird because both of these numbers are less than four so the natural conclusion is we're gonna have to borrow right we're gonna have to borrow to get that subtraction correct so the only thing that's gonna go here can't be a zero because if it's zero i don't need to borrow right so i know that eight has got to go here all right and that's great but what now has to go up here a three or a two well if i put a three up here and i borrowed a one it would be thirteen minus eight which is a five instead it's gotta be the two up top because if i borrowed a one i'd have twelve minus eight which is four this of course now leaves the 3 and the 0 so naturally 3 has to go here and 0 has to go here and the final answer is 932 minus 208 you can check that out yourself that indeed equals 724 i hope you enjoyed this math puzzle and if you did please click that like button if you want to see more math puzzles on the regular make sure to click subscribe thank you guys so much for joining and i will see you in the next video take it easy [Music]
6092
https://www.khanacademy.org/math/cc-fourth-grade-math/imp-addition-and-subtraction-2/imp-rounding-whole-numbers/v/rounding-whole-numbers-2
Use of cookies Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy 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 and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer. More information Manage Consent Preferences Strictly Necessary Cookies Always Active Certain cookies and other technologies are essential in order to enable our Service to provide the features you have requested, such as making it possible for you to access our product and information related to your account. For example, each time you log into our Service, a Strictly Necessary Cookie authenticates that it is you logging in and allows you to use the Service without having to re-enter your password when you visit a new page or new unit during your browsing session. Functional Cookies These cookies provide you with a more tailored experience and allow you to make certain selections on our Service. For example, these cookies store information such as your preferred language and website preferences. Targeting Cookies These cookies are used on a limited basis, only on pages directed to adults (teachers, donors, or parents). We use these cookies to inform our own digital marketing and help us connect with people who are interested in our Service and our mission. We do not use cookies to serve third party ads on our Service. Performance Cookies These cookies and other technologies allow us to understand how you interact with our Service (e.g., how often you use our Service, where you are accessing the Service from and the content that you’re interacting with). Analytic cookies enable us to support and improve how our Service operates. For example, we use Google Analytics cookies to help us measure traffic and usage trends for the Service, and to understand more about the demographics of our users. We also may use web beacons to gauge the effectiveness of certain communications and the effectiveness of our marketing campaigns via HTML emails.
6093
https://www.lmnoeng.com/literature.htm
Open Channel Flow and Pipe Flow Literature LMNO Engineering, Research, and Software, Ltd. Open Channel Flow and Pipe Flow Literature Discussion of Equations for Pressurized Flow and Open Channel Hydraulics LMNO Engineering Home(all calculators)Register to Enable CalculatorsConsultingPast Newsletters Privacy policy Other than to investigate fraud or cooperate with law enforcement, your information is not distributed to any other entity. LMNO Engineering reserves the right to change this policy. Consulting Free newsletters: SubmitLMNO@LMNOeng.com Phone (USA): (740)707‑2614 Open Channel Flow The Manning Equation is the most commonly used equation to analyze open channel flows. The Manning Equation is utilized in our open channel design calculations - Design of Circular Culverts, Design of Rectangular Channels, and Design of Trapezoidal Channels. It is a semi-empirical equation for simulating water flows in channels and culverts where the water is open to the atmosphere, i.e. not flowing under pressure, and was first presented in 1889 by Robert Manning. The channel can be any shape - circular, rectangular, triangular, etc. The units in the Manning equation appear to be inconsistent; however, the value k has hidden units in it to make the equation consistent. The Manning Equation was developed for uniform steady state flow. Uniform means that the channel is prismatic. Prismatic means the channel has constant dimensions (including depth) along its length. Steady state means the flowrate, velocity, and everything else are constant with time. In reality no flow can be uniform and steady. However, for individual channel reaches (e.g. one mile of a 200 mile river) the assumptions may be fairly well achieved. The Manning Equation is also used successfully to simulate "gradually varied flow" where the channel is not prismatic. In this case, S is the slope of the energy grade line. For prismatic flows, S is the slope of the hydraulic grade line which is the slope of the water surface and is the same as the slope of the channel bottom. Flows Under Pressure (Closed Conduits, Pipes) This website has two comprehensive calculations for simulating steady flows under pressure - Design of Circular Water Pipes and Design of Circular Liquid or Gas Pipes. We also have a water hammer calculation that predicts pressures during transient conditions due to closing or opening a valve. In addition, we have calculations for flow measurement using orifices, nozzles, and venturi meters. The website also has several smaller calculations which solve individual portions of the energy equation; these smaller calculations are linked in the following paragraph. The energy equation represents elevation, pressure, and velocity forms of energy. The energy equation for a fluid moving in a closed conduit is written between two locations at a distance (length) L apart. Energy losses for flow through ducts and pipes consist of major losses and minor losses. Major losses are due to friction between the moving fluid and the inside walls of the duct. Minor losses are due to fittings such as valves and elbows. Major losses are computed using either the Darcy-Weisbach friction loss equation (which utilizes the Moody friction factor) or the Hazen-Williams friction loss equation. The Darcy-Weisbach method is generally considered more accurate than the Hazen-Williams method. Additionally, the Darcy-Weisbach method is valid for any liquid or gas; Hazen-Williams is only valid for water at ordinary temperatures (40 to 75 o F). The Hazen-Williams method is very popular, especially among civil engineers, since its friction coefficient (C) is not a function of velocity or duct (pipe) diameter. Hazen-Williams is simpler than Darcy-Weisbach for calculations where you are solving for flowrate, velocity, or diameter. References For open channel flow: Chaudhry, M. H. 1993. Open-Channel Flow. Prentice-Hall, Inc. Chow, V. T. 1959. Open-Channel Hydraulics. McGraw-Hill, Inc. (the classic text) French, R. H. 1985. Open-Channel Hydraulics. McGraw-Hill Book Co. Mays, L. W. editor. 1999. Hydraulic design handbook. McGraw-Hill Book Co. Munson, B.R., D. F. Young, and T. H. Okiishi. 1998. Fundamentals of Fluid Mechanics. John Wiley and Sons, Inc. 3ed. Streeter, V. L., E. B. Wylie, and K. W. Bedford. 1998. WCB/McGraw-Hill. 9ed. For closed conduit flow: Munson, B.R., D. F. Young, and T. H. Okiishi. 1998. Fundamentals of Fluid Mechanics. John Wiley and Sons, Inc. 3ed. Gerhart, P. M, R. J. Gross, and J. I. Hochstein. 1992. Fundamentals of Fluid Mechanics. Addison-Wesly Pubishing Co. 2ed. Hwang, N. H. C. and R. J. Houghtalen. 1996. Fundamentals of Hydraulic Engineering Systems. Prentice Hall. 3ed. Mays, L. W. editor. 1999. Hydraulic design handbook. McGraw-Hill Book Co. Potter, M. C. and D. C. Wiggert. 1991. Mechanics of Fluids. Prentice-Hall, Inc. Roberson, J. A., J. J. Cassidy, and M. H. Chaudhry. 1998. Hydraulic Engineering. John Wiley and Sons, Inc. 2ed. Roberson, J. A. and C. T. Crowe. 1990. Engineering Fluid Mechanics. Houghton Mifflin Co. Streeter, V. L., E. B. Wylie, and K. W. Bedford. 1998. WCB/McGraw-Hill. 8ed. White, F. M. 1979. Fluid Mechanics. McGraw-Hill, Inc. © 1998-2025 LMNO Engineering, Research, and Software, Ltd. All rights reserved. LMNO Engineering, Research, and Software, Ltd. 7860 Angel Ridge Rd. Athens, Ohio 45701 USA Phone: (740)707‑2614 LMNO@LMNOeng.com
6094
https://www.neurology.org/doi/10.1212/WNL.57.8.1416
Randomized trial of vigabatrin in patients with infantile spasms | Neurology Skip to main content Skip to main contentAAN.com AAN Publications Author Center About the Journals Press Releases 0 Cart Search Sign inSUBSCRIBE Quick Search anywhere Enter search term Quick search in Citations Journal Year Volume Issue Page Searching: Anywhere AnywhereCitation Advanced SearchSearch Trending Terms: Cognitive Disorders/Dementia Cerebrovascular Disease/Stroke Multiple Sclerosis Health Disparities navigate the sidebar menu Sign inSUBSCRIBE Quick Search anywhere Enter search term Journals Neurology® Articles Latest Research Latest Non-Research Current Issue Past Issues Editorial Board Masthead Pictorial Directory All Neurology Journals Editorial Boards Topic Collections Submit a manuscript Author Center E-alerts Neurology® Clinical Practice Neurology® Education Neurology® Genetics Neurology® Neuroimmunology & Neuroinflammation Neurology® Open Access Research Articles All Research Articles Null Hypothesis Non-Research Articles All Non-Research Articles Research Methods in Neurology Open Review Resident & Fellow Neurology® Podcast Letters to the Editor Practice Current Continuing Medical Education (CME) Topic Collections Multimedia Blogs Infographics Neurology® Video Journal Club Video Summaries About About the Journals Advertise Contact Us Editorial Boards Ethics Policies For Authors and Reviewers Author Center Information for Reviewers Submit Manuscript Sign Up for E-Alerts Press Releases Terms of Service Privacy Policy The most widely read and highly cited peer-reviewed neurology journal Articles Latest Research Latest Non-Research Current Issue Past Issues Editorial Board Masthead Pictorial Directory All Neurology Journals Editorial Boards Topic Collections Submit a manuscript Author Center E-alerts Reference #1 Articles October 23, 2001 Letter to the Editor Randomized trial of vigabatrin in patients with infantile spasms R. D.Elterman, MD, W. D.Shields, MD, K. A.Mansfield, and J.Nakagawathe US Infantile Spasms Vigabatrin Study GroupAuthors Info & Affiliations October 23, 2001 issue 57 (8) 1416-1421 Letters to the Editor (1) Track CitationsAdd to favorites Get Access Contents Abstract Publication History References Information & Authors Metrics & Citations Get Access References Figures Tables Media Share Abstract _Background:_ Infantile spasms are a rare but devastating pediatric epilepsy that, outside the United States, is often treated with vigabatrin. The authors evaluated the efficacy and safety of vigabatrin in children with recent-onset infantile spasms. _Methods:_ This 2-week, randomized, single-masked, multicenter study with a 3- year, open-label, dose-ranging follow-up study included patients who were younger than 2 years of age, had a diagnosed duration of infantile spasms of no more than 3 months, and had not previously been treated with adrenocorticotropic hormone, prednisone, or valproic acid. Patients were randomly assigned to receive low-dose (18–36 mg/kg/day) or high-dose (100–148 mg/kg/day) vigabatrin. Treatment responders were those who were free of infantile spasm for 7 consecutive days beginning within the first 14 days of vigabatrin therapy. Time to response to therapy was evaluated during the first 3 months, and safety was evaluated for the entire study period. _Results:_ Overall, 32 of 142 patients who were able to be evaluated for efficacy were treatment responders (8/75 receiving low-dose vigabatrin vs 24/67 receiving high doses, p< 0.001). Response increased dramatically after approximately 2 weeks of vigabatrin therapy and continued to increase over the 3-month follow-up period. Time to response was shorter in those receiving high-dose versus low-dose vigabatrin (p = 0.04) and in those with tuberous sclerosis versus other etiologies (p< 0.001). Vigabatrin was well tolerated and safe; only nine patients discontinued therapy because of adverse events. _Conclusions:_ These results confirm previous reports of the efficacy and safety of vigabatrin in patients with infantile spasms, particularly among those with spasms secondary to tuberous sclerosis. Get full access to this article View all available purchase options and get full access to this article. Get Access Already a Subscriber? Sign in as an individual or via your institution References 1. Trevathan E, Murphy CC, Yeargin-Allsopp M. The descriptive epidemiology of infantile spasms among Atlanta children. Epilepsia. 1999; 40: 748–751. Crossref PubMed Google Scholar 2. Wong M, Trevathan E. Infantile spasms. Pediatr Neurol. 2001; 24: 89–98. Crossref PubMed Google Scholar 3. Gidal BE, Privatera MD, Sheth RD, Gilman JT. Vigabatrin: a novel therapy for seizure disorders. Ann Pharmacol. 1999; 33: 1277–1286. Crossref Google Scholar 4. Grove J, Schechter PJ, Tell G, et al. Increased gamma-aminobutyric acid (GABA), homocarnosine and beta-alanine in cerebrospinal fluid of patients treated with gamma-vinyl GABA (4-amino-hex-5-enoic acid). Life Sci. 1981; 28: 2431–2439. Crossref PubMed Google Scholar 5. Schechter PJ, Hanke NF, Grove J, Huebert N, Sjoerdsma A. Biochemical and clinical effects of gamma-vinyl GABA in patients with epilepsy. Neurology. 1984; 34: 182–186. Crossref PubMed Google Scholar 6. Appleton RE, Peters ACB, Mumford JP, Shaw DE. Randomised, placebo-controlled study of vigabatrin as first-line treatment of infantile spasms. Epilepsia. 1999; 40: 1627–1633. Crossref PubMed Google Scholar 7. Chiron C, Dulac O, Beaumont D, Palacois L, Pajot N, Mumford J. Therapeutic trial of vigabatrin in refractory infantile spasms. J Child Neurol. 1991; 6 (suppl 2): 2. Crossref Google Scholar 8. Chiron C, Dumas C, Jambaque I, Mumford J, Dulac O. Randomized trial comparing vigabatrin and hydrocortisone in infantile spasms due to tuberous sclerosis. Epilepsy Res. 1997; 26: 389–395. Crossref PubMed Google Scholar 9. Aicardi J, Sabril IS, Investigator and Peer Review Groups, Mumford JP, Dumas C, Wood S. Vigabatrin as initial therapy for infantile spasms: a European retrospective survey. Epilepsia. 1996; 37: 638–642. Crossref PubMed Google Scholar 10. Dawson-Saunders B, Trapp RG. Basic and clinical biostatistics. Norwalk, CT: Appleton & Lange, 1990. Google Scholar 11. Fejerman N, Cersósimo R, Caraballo R, et al. Vigabatrin as a first-choice drug in the treatment of West syndrome. J Child Neurol. 2000; 15: 161–165. Crossref PubMed Google Scholar 12. Vigevano F, Cilio MR. Vigabatrin versus ACTH as first-line treatment for infantile spasms: a randomized, prospective study. Epilepsia. 1997; 38: 1270–1274. Crossref PubMed Google Scholar 13. Eke T, Talbot JF, Lawden MC. Severe persistent visual field construction associated with vigabatrin. BMJ. 1997; 314: 180–181. Crossref PubMed Google Scholar 14. Gross-Tsur V, Banin E, Shahar E, Shalev RS, Lahat E. Visual impairment in children with epilepsy treated with vigabatrin. Ann Neurol. 2000; 48: 60–64. Crossref PubMed Google Scholar 15. Wild JM, Martinez C, Reinshagen G, Harding GFA. Characteristics of a unique visual field defect attributed to vigabatrin. Epilepsia. 1999; 40: 1784–1794. Crossref PubMed Google Scholar 16. Snead OC III, Benton JW Jr, Hosey LC, et al. Treatment of infantile spasms with high-dose ACTH: efficacy and plasma levels of ACTH and cortisol. Neurology. 1989; 39: 1027–1031. Crossref PubMed Google Scholar 17. Baram TZ, Mitchell WG, Tournay A, Snead OC, Hanson RA, Horton EJ. High-dose corticotropin (ACTH) versus prednisone for infantile spasms: a prospective, randomized, blinded study. Pediatrics. 1996; 97: 375–379. Crossref PubMed Google Scholar 18. Siemes H, Spohr HL, Michael T, Nau H. Therapy of infantile spasms with valproate: results of a prospective study. Epilepsia. 1988; 29: 553–560. Crossref PubMed Google Scholar 19. Bachman DS. Use of valproic acid in treatment of infantile spasms. Arch Neurol. 1982; 39: 49–52. Crossref PubMed Google Scholar 20. Zimmerman HJ, Ishak KG. Valproate-induced hepatic injury: analyses of 23 fatal cases. Hepatology. 1982; 2: 591–597. PubMed Google Scholar 21. Wyllie E, Wyllie R, Cruse RP, Erenberg G, Rothner AD. Pancreatitis associated with valproic acid therapy. AJDC. 1984; 138: 912–914. PubMed Google Scholar 22. Farrell K. Benzodiazepines in the treatment of children with epilepsy. Epilepsia. 1986; 27 (suppl 1): S45-S52. Crossref PubMed Google Scholar 23. Rintahaka PJ, Nakagawa JA, Shewmon DA, Kyyronen P, Shields WD. Incidence of death in patients with intractable epilepsy during nitrazepam treatment. Epilepsia. 1999; 40: 492–496. Crossref PubMed Google Scholar 24. Veggiotti P, Cieuta C, Rey E, Dulac O. Lamotrigine in infantile spasms. Lancet. 1994; 344: 1375–1376. Google Scholar 25. Glauser TA, Clark PO, Stawsburg R. A pilot study of topiramate in the treatment of infantile spasms. Epilepsia. 1998; 39: 1324–1328. Crossref PubMed Google Scholar 26. Glauser TA, Clark PO, McGee K. Long-term response to topiramate in patients with West syndrome. Epilepsia. 2000; 41 (suppl 1): S91–S94. Crossref Google Scholar 27. Yanai S, Hanai T, Narazaki O. Treatment of infantile spasms with zonisamide. Brain Dev. 1999; 21: 157–161. Crossref PubMed Google Scholar 28. Appleton RE. Guideline may help in prescribing vigabatrin. BMJ. 1998; 317: 1322. Crossref PubMed Google Scholar 29. Jambaque I, Chiron C, Dumas C, Mumford J, Dulac O. Mental and behavioral outcome of infantile epilepsy treated by vigabatrin in tuberous sclerosis patients. Epilepsy Res. 2000; 38: 151–160. Crossref PubMed Google Scholar Show all references Letters to the Editor (1) 4 June 2002 Randomized trial of vigabatrin in patients with infantile spasms John P Osborne Andrew L. Lux, Stuart W. Edwards, Eleanor Hancock, Anthony L. Johnson, Colin R. Kennedy, Finbar J K O'Callaghan, Richard W. Newton and Christopher M. Verity The study of Elterman et al. is a significant advance in our understanding of the treatment of infantile spasms, particularly in the clear definition of response and relapse in terms of both the clinical and neurophysiological response, in its size and in the use of actuarial curves (survival curves) for responders . There are, however, some weaknesses in the published data, and in their interpretation and analysis. The results were reported on almost 80% (142/179) of children randomized to receive treatment rather than an intention-to-treat analysis of the 179 children allocated a treatment - as recommended in the CONSORT statement . The data show that 16 results are missing from those exposed to high dose and nine from the low-dose group; two cases were not exposed, and for 10 cases the report form is not available. These numbers (37 of those randomized compared to only 32 responders) are large enough to distort the findings, if the responses in these infants differed from the responses in those reported. The reported primary response rate in children on "high-dose" vigabatrin (normal dose in the UK) was low at 24/67 (36%) and for low-dose vigabatrin it was 8/75 (11%). Of those with tuberous sclerosis (TS), 13/25 (52%) responded. Only 19/117 (16%) with other underlying aetiologies responded. This confirms again that vigabatrin is most efficacious when treating tuberous sclerosis. The proportions on low and high dose are not given by aetiology however and the paper should have stated these and the odds ratios (the Mantel-Haenszel test statistic) of a response for the two different groups: those with and those without tuberous sclerosis . Turning to the actuarial curves, the x-axis should be calibrated as "time from randomly allocating treatment" rather than "days of vigabatrin therapy" since the duration of therapy depends not only on efficacy but also on tolerability. Figure 5 suggests, for example, that all children with TS were responders by 71 days, where in fact two had been withdrawn from treatment and one was a non-responder. Children who stopped treatment within the 3-month period because of lack of treatment effect or presence of adverse effect should, rather than being right-censored, be classified as non-responders. Figure 4 seems to show a persistently poorer outcome in children who were initially treated with low-dose, and the difference persists despite the study protocol allowing a change to higher doses of vigabatrin and the addition of other treatments. This could be due either to a long-term adverse effect of an ineffective initial dose of vigabatrin on the response rate, or to right censoring, or to selection bias or to chance. The placebo controlled study of Appleton et al showed no significant difference in primary outcome but the group initially treated with placebo for 5 days appeared to have a significantly better outcome at 6 months than the group initially treated with vigabatrin . This is the opposite effect to that seen by Elterman et al. We think these differences are most likely due to chance, but the discrepant findings argue potently for further studies. This study ought not to claim that it confirms both the safety and efficacy of vigabatrin. The adverse effect that is potentially most serious and irreversible is vigabatrin-attributed visual field loss. There is no validated method for excluding such losses in children of this age. The best estimate of risk for such losses must be at least one-third - the same as that risk in adults or older children. Most importantly, the reported increase in response rate over the 3- month follow up period could be due to treatments other than vigabatrin since these were allowed after 21 days. Regrettably, no information on other treatments is given. References: Elterman RD, Shields WD, Mansfield KA, Nakagawa J; and the US Infantile Spasms Vigabatrin Study Group. Randomized trial of vigabatrin in patients with infantile spasms. Neurology 2001;57:1416-1421. Moher D, Schulz KF, Altman DG. The CONSORT statement: revised recommendations for improving the quality of reports of parallel-group randomized trials. JAMA 2001;134:657-662. Lang TA, Secic M. How to Report Statistics in Medicine: Annotated Guidelines for Authors, Editors, and Reviewers. Philadelphia, PA: American College of Physicians; 1997:249. Appleton RE, Peters ACB, Mumford JP, Shaw DE. Randomised, placebo -controlled trial of vigabatrin as first-line treatment of infantile spasms. Epilepsia 1999;40:1627-1633. Wild JM, Martinez C, Reinshagen G, Harding GFA. Characteristics of a unique visual field defect attributed to vigabatrin. Epilepsia 1999;40:1784-1794. view more Information & Authors Information Authors Information Published In Neurology® Volume 57 • Number 8 • October 23, 2001 Pages: 1416-1421 PubMed: 11673582 Copyright © 2001. Publication History Received: January 22, 2001 Accepted: June 10, 2001 Published online: October 23, 2001 Published in issue: October 23, 2001 Permissions Request permissions for this article. Request permissions Authors Affiliations & Disclosures Expand All R. D.Elterman, MD View all articles by this author W. D.Shields, MD View all articles by this author K. A.Mansfield View all articles by this author J.Nakagawa View all articles by this author the US Infantile Spasms Vigabatrin Study Group Members of the US Infantile Spasms Vigabatrin Study Group are listed in the Appendix on page 1421. View all articles by this author From the Medical City Dallas Hospital (Drs. Elterman and Mansfield), TX; and Mattel Children’s Hospital (Drs. Shields and Nakagawa), University of California, Los Angeles. Notes Address correspondence and reprint requests to Dr. Roy D. Elterman, Dallas Pediatric Neurology Associates, The Center for Epilepsy Treatment, Medical City Dallas Hospital, 7777 Forest Lane, Suite A307, Dallas, TX 75230; e-mail: RoyDElterman@aol.com Metrics & Citations Metrics Citations 185 Metrics Article Metrics Accesses Citations No data available. 106 185 Total 6 Months 12 Months Total number of accesses and citations for the most recent 6 whole calendar months. Citation information is sourced from Crossref Cited-by service. See more details Referenced in 1 patents Referenced in 1 clinical guideline sources 61 readers on Mendeley Citations Download Citations If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Select your manager software from the list below and click Download. Format [x] Direct Import Cited By Kette D. Valente, Leticia Brito Sampaio, Silvia Vincentiis, Anna Lecticia R. Pinto, Maria Augusta Montenegro, Tuberous Sclerosis Complex: An updated in the treatment of epilepsy for early careers, Epilepsy & Behavior, 167, (110396), (2025). Crossref Rachata Boonkrongsak, Kantapon Trongkamolchai, Sirorat Suwannachote, Somjit Sri‐Udomkajorn, Raviwan Wittawassamrankul, Ravindra Arya, Kullasate Sakpichaisakul, Combination Therapy With Vigabatrin and Prednisolone Versus Vigabatrin Alone for Infantile Spasms, Annals of Clinical and Translational Neurology, 12, 5, (1012-1021), (2025). Crossref Alejandra Vasquez, Anthony L. Fine, Management of Developmental and Epileptic Encephalopathies, Seminars in Neurology, 45, 02, (206-220), (2025). Crossref Debopam Samanta, Evolving treatment strategies for early-life seizures in Tuberous Sclerosis Complex: A review and treatment algorithm, Epilepsy & Behavior, 161, (110123), (2024). Crossref Rujuta Sathe, Gyaneshwar Shrestha, Aria Terango, David Tabibzadeh, Rajsekar R. Rajaraman, Hiroki Nariai, Shaun A. Hussain, Symptomatic vigabatrin‐associated MRI toxicity is associated with simultaneous hormonal therapy among patients with infantile spasms , Epilepsia Open, 10, 1, (314-320), (2024). Crossref Alessandro Iodice, Giuliana Marchiò, Francesca Asta, Katia Rocchetti, Anna Rosati, Vigabatrin-associated hyperkinetic movements in two children with epileptic spasms: Case reports and video phenomenology description, Seizure: European Journal of Epilepsy, 120, (12-14), (2024). Crossref Rajsekar R. Rajaraman, Rachel J. Smith, Shingo Oana, Atsuro Daida, Daniel W. Shrey, Hiroki Nariai, Beth A. Lopour, Shaun A. Hussain, Computational EEG attributes predict response to therapy for epileptic spasms, Clinical Neurophysiology, 163, (39-46), (2024). Crossref Usman Abdulfatai, Stephen Ejeh, Abduljelil Ajala, Samuel Ndaghiya Adawara, Olasupo Sabitu Babatunde, Zakari Ya'u Ibrahim, QSAR, molecular docking, and molecular designs of some anti-epilepsy compounds, Intelligent Pharmacy, 2, 3, (427-434), (2024). Crossref Emma Macdonald-Laurs, Winston Dzau, Aaron E.L. Warren, Matthew Coleman, Cristina Mignone, Sarah E. M. Stephenson, Katherine B. Howell, Identification and treatment of surgically-remediable causes of infantile epileptic spasms syndrome, Expert Review of Neurotherapeutics, 24, 7, (661-680), (2024). Crossref Robert J. Richardson, Steven Petrou, Alexander Bryson, Established and emerging GABAA receptor pharmacotherapy for epilepsy, Frontiers in Pharmacology, 15, (2024). Crossref See more Loading... Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. Share FacebookX (formerly Twitter)LinkedInemail Get Access Get Access Login options Check if you have access through your login credentials or your institution to get full access on this article. Personal loginInstitutional Login Email Password Forgot your password? Reset it here [x] Keep me logged in Login Don’t have an account? Registerhere Save for laterItem saved, go to cart Purchase Options Purchase this article to access the full text. Purchase Access, $39 for 24hr of access USD $39.00 Add to Cart Checkout References References 1. Trevathan E, Murphy CC, Yeargin-Allsopp M. The descriptive epidemiology of infantile spasms among Atlanta children. Epilepsia. 1999; 40: 748–751. Crossref PubMed Google Scholar 2. Wong M, Trevathan E. Infantile spasms. Pediatr Neurol. 2001; 24: 89–98. Crossref PubMed Google Scholar 3. Gidal BE, Privatera MD, Sheth RD, Gilman JT. Vigabatrin: a novel therapy for seizure disorders. Ann Pharmacol. 1999; 33: 1277–1286. Crossref Google Scholar 4. Grove J, Schechter PJ, Tell G, et al. Increased gamma-aminobutyric acid (GABA), homocarnosine and beta-alanine in cerebrospinal fluid of patients treated with gamma-vinyl GABA (4-amino-hex-5-enoic acid). Life Sci. 1981; 28: 2431–2439. Crossref PubMed Google Scholar 5. Schechter PJ, Hanke NF, Grove J, Huebert N, Sjoerdsma A. Biochemical and clinical effects of gamma-vinyl GABA in patients with epilepsy. Neurology. 1984; 34: 182–186. Crossref PubMed Google Scholar 6. Appleton RE, Peters ACB, Mumford JP, Shaw DE. Randomised, placebo-controlled study of vigabatrin as first-line treatment of infantile spasms. Epilepsia. 1999; 40: 1627–1633. Crossref PubMed Google Scholar 7. Chiron C, Dulac O, Beaumont D, Palacois L, Pajot N, Mumford J. Therapeutic trial of vigabatrin in refractory infantile spasms. J Child Neurol. 1991; 6 (suppl 2): 2. Crossref Google Scholar 8. Chiron C, Dumas C, Jambaque I, Mumford J, Dulac O. Randomized trial comparing vigabatrin and hydrocortisone in infantile spasms due to tuberous sclerosis. Epilepsy Res. 1997; 26: 389–395. Crossref PubMed Google Scholar 9. Aicardi J, Sabril IS, Investigator and Peer Review Groups, Mumford JP, Dumas C, Wood S. Vigabatrin as initial therapy for infantile spasms: a European retrospective survey. Epilepsia. 1996; 37: 638–642. Crossref PubMed Google Scholar 10. Dawson-Saunders B, Trapp RG. Basic and clinical biostatistics. Norwalk, CT: Appleton & Lange, 1990. Google Scholar 11. Fejerman N, Cersósimo R, Caraballo R, et al. Vigabatrin as a first-choice drug in the treatment of West syndrome. J Child Neurol. 2000; 15: 161–165. Crossref PubMed Google Scholar 12. Vigevano F, Cilio MR. Vigabatrin versus ACTH as first-line treatment for infantile spasms: a randomized, prospective study. Epilepsia. 1997; 38: 1270–1274. Crossref PubMed Google Scholar 13. Eke T, Talbot JF, Lawden MC. Severe persistent visual field construction associated with vigabatrin. BMJ. 1997; 314: 180–181. Crossref PubMed Google Scholar 14. Gross-Tsur V, Banin E, Shahar E, Shalev RS, Lahat E. Visual impairment in children with epilepsy treated with vigabatrin. Ann Neurol. 2000; 48: 60–64. Crossref PubMed Google Scholar 15. Wild JM, Martinez C, Reinshagen G, Harding GFA. Characteristics of a unique visual field defect attributed to vigabatrin. Epilepsia. 1999; 40: 1784–1794. Crossref PubMed Google Scholar 16. Snead OC III, Benton JW Jr, Hosey LC, et al. Treatment of infantile spasms with high-dose ACTH: efficacy and plasma levels of ACTH and cortisol. Neurology. 1989; 39: 1027–1031. Crossref PubMed Google Scholar 17. Baram TZ, Mitchell WG, Tournay A, Snead OC, Hanson RA, Horton EJ. High-dose corticotropin (ACTH) versus prednisone for infantile spasms: a prospective, randomized, blinded study. Pediatrics. 1996; 97: 375–379. Crossref PubMed Google Scholar 18. Siemes H, Spohr HL, Michael T, Nau H. Therapy of infantile spasms with valproate: results of a prospective study. Epilepsia. 1988; 29: 553–560. Crossref PubMed Google Scholar 19. Bachman DS. Use of valproic acid in treatment of infantile spasms. Arch Neurol. 1982; 39: 49–52. Crossref PubMed Google Scholar 20. Zimmerman HJ, Ishak KG. Valproate-induced hepatic injury: analyses of 23 fatal cases. Hepatology. 1982; 2: 591–597. PubMed Google Scholar 21. Wyllie E, Wyllie R, Cruse RP, Erenberg G, Rothner AD. Pancreatitis associated with valproic acid therapy. AJDC. 1984; 138: 912–914. PubMed Google Scholar 22. Farrell K. Benzodiazepines in the treatment of children with epilepsy. Epilepsia. 1986; 27 (suppl 1): S45-S52. Crossref PubMed Google Scholar 23. Rintahaka PJ, Nakagawa JA, Shewmon DA, Kyyronen P, Shields WD. Incidence of death in patients with intractable epilepsy during nitrazepam treatment. Epilepsia. 1999; 40: 492–496. Crossref PubMed Google Scholar 24. Veggiotti P, Cieuta C, Rey E, Dulac O. Lamotrigine in infantile spasms. Lancet. 1994; 344: 1375–1376. Google Scholar 25. Glauser TA, Clark PO, Stawsburg R. A pilot study of topiramate in the treatment of infantile spasms. Epilepsia. 1998; 39: 1324–1328. Crossref PubMed Google Scholar 26. Glauser TA, Clark PO, McGee K. Long-term response to topiramate in patients with West syndrome. Epilepsia. 2000; 41 (suppl 1): S91–S94. Crossref Google Scholar 27. Yanai S, Hanai T, Narazaki O. Treatment of infantile spasms with zonisamide. Brain Dev. 1999; 21: 157–161. Crossref PubMed Google Scholar 28. Appleton RE. Guideline may help in prescribing vigabatrin. BMJ. 1998; 317: 1322. Crossref PubMed Google Scholar 29. Jambaque I, Chiron C, Dumas C, Mumford J, Dulac O. Mental and behavioral outcome of infantile epilepsy treated by vigabatrin in tuberous sclerosis patients. Epilepsy Res. 2000; 38: 151–160. Crossref PubMed Google Scholar Volume 57 | Number 8 October 23, 2001 advertisement Topic Collections Discover the latest research, podcasts, infographics, and much more on topics that interest you most. All Topic Collections Open Review Pilot Project Explore peer reviews, author responses, and editor comments on selected articles. Learn More Letters to the Editor The Letters section represents an opportunity for ongoing author debate and post-publication peer review. View our submission guidelines for Letters to the Editor before submitting your comment. View the Letters for this article (1) Submit a Letter for this article Recommended Neurology® Research Article 29 Aug 2022 continuing medical education Association of Time to Clinical Remission With Sustained Resolution in Children With New-Onset Infantile Spasms Christopher J. Yuskaitis, John R. Mytinger, Fiona M. Baumer, Bo Zhang, Shanshan Liu, Debopam Samanta, Shaun A. Hussain, Elissa G. Yozawitz, Cynthia G. Keator, Charuta Joshi, Rani K. Singh, Sonal Bhatia, Sonam Bhalla, Renée Shellhaas, Chellamani Harini, and [...] on behalf of the Pediatric Epilepsy Research Consortium +12 authors Neurology® Research Article 15 Jul 2021 Comparative Effectiveness of Initial Treatment for Infantile Spasms in a Contemporary US Cohort Zachary M. Grinspan, Kelly G. Knupp, Anup D. Patel, Elissa G. Yozawitz, Courtney J. Wusthoff, Elaine C. Wirrell, Ignacio Valencia, Nilika S. Singhal, Douglas R. Nordli, John R. Mytinger, Wendy G. Mitchell, Cynthia G. Keator, Tobias Loddenkemper, Shaun A. Hussain, Chellamani Harini, William D. Gaillard, Ivan S. Fernandez, Jason Coryell, Catherine J. Chu, Anne T. Berg, and [...] Renee A. Shellhaas +17 authors Neurology® Articles 1 Aug 1989 Treatment of infantile spasms with high‐dose ACTH: Efficacy and plasma levels of ACTH and cortisol O. C. Snead, J. W. Benton, L. C. Hosey, J. W. Swann, D. Spink, D. Martin, and [...] R. Rej +3 authors Neurology® Special Article 11 Jun 2012 Free Access Evidence-based guideline update: Medical treatment of infantile spasms [RETIRED]: Report of the Guideline Development Subcommittee of the American Academy of Neurology and the Practice Committee of the Child Neurology Society Neurology® Research Article 29 Aug 2022 Prioritizing Hormone Therapy Over Vigabatrin as the First Treatment for Infantile Spasms: A Quality Improvement Initiative John R. Mytinger, William Parker, Steven W. Rust, Stephanie M. Ahrens, Dara V.F. Albert, Christopher W. Beatty, Julie Chrisman, Daniel J. Clark, Andrea Debs, Danielle Denney, Mary Karn, James Herbst, Adam P. Ostendorf, Mary C. Taylor, Jaime D.E. Twanow, Jorge Vidaurre, and [...] Anup D. 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6095
https://www.youtube.com/watch?v=ZO6HVaoDthw
How To Find The Third Angle in a Triangle Given the Other Two Angles The Math Sorcerer 1220000 subscribers 39 likes Description 20903 views Posted: 8 Dec 2020 How To Find The Third Angle in a Triangle Given the Other Two Angles If you enjoyed this video please consider liking, sharing, and subscribing. Udemy Courses Via My Website: My FaceBook Page: There are several ways that you can help support my channel:) Consider becoming a member of the channel: My GoFundMe Page: My Patreon Page: Donate via PayPal: Udemy Courses(Please Use These Links If You Sign Up!) Abstract Algebra Course Advanced Calculus Course Calculus 1 Course Calculus 2 Course Calculus 3 Course Calculus Integration Insanity Differential Equations Course College Algebra Course How to Write Proofs with Sets Course How to Write Proofs with Functions Course Statistics with StatCrunch Course Math Graduate Programs, Applying, Advice, Motivation Daily Devotionals for Motivation with The Math Sorcerer Thank you:) 3 comments Transcript: in this problem we're given a triangle we have 90 degrees here 45 degrees here and then x is here and we have to find the measure of x so the thing with triangles is if you add up all of the angles inside a triangle you're always going to get 180 every single time this is super useful in in math so basically all you have to do in this case is just add up all of these angles and set it equal to 180 so x plus 45 degrees plus 90 degrees and that's equal to 180 degrees so 45 plus 90 is 135 so we have x plus 135 and that's equal to 180 degrees then we can just subtract 135 from both sides and that's going to give us looks like x equals 45 degrees and that would be the final answer
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https://sites.pitt.edu/~kaveh/Matching.pdf
11/15/13 9:29 AM Graph Theory Lecture Notes 11a Page 3 of 6 The algorithm proceeds by growing a tree, starting at s, in the underlying graph, using only usable arcs as frontier edges. If the tree eventually includes t we have found a flow augmenting path, otherwise the tree vertices form one set of a minimum cut. Algorithm: Initialize the tree S = {s}, a counter i = 1, and label s with 0. While S does not contain t do a. Locate a usable arc as a frontier edge with the smallest labeled endpoint. b. If w is the unlabeled endpoint, and v the labeled endpoint Set back(w) = v. Label w with i. Let i := i + 1; Put w in T. If there is no usable arc, return the cut S, V-S. If t is in S, the flow augmenting path is obtained from reading the vector back() starting at t. For a given (s,t)-network, we can find the maximum flow by starting with all arcs having 0 flow, then finding a flow augmenting path and adjusting the flows accordingly. Repeat this until there are no further flow augmenting paths. Matchings, Transversals and Vertex Covers A set M of edges in a graph G is called a matching of G if no two edges in set M have an endpoint in common. A maximum matching of graph G is a matching of G with the greatest number of edges while a maximal matching is a matching which is not contained in any larger matching. While any maximum matching is certainly maximal, the reverse is not generally true. In the above graph, {a}, {b,d}, and {c,e} are all maximal matchings, but only the last two are maximum matchings. Job Assignment Problem revisited Given a bipartite graph (X, Y) we can form an (s,t)-network as follows: Add a vertex s, and join it with arcs going from s to each vertex in X. Orient all the edges in the bipartite graph from X to Y. Finally, add a vertex t, and join it with arcs from each vertex in Y to t. Give all of the arcs leaving s or entering t a capacity of 1, but the arcs between X and Y will get infinite capacity. Call this network N. Notice that any feasible flow of N where all the flow values are integers (necessarily either 0 or 1), corresponds to a matching of the original graph (the matching consists of the edges which correspond to arcs 11/15/13 9:29 AM Graph Theory Lecture Notes 11a Page 4 of 6 between X and Y which have positive flow), and vice versa, any matching of the graph gives rise to a feasible flow. The value of the flow is the number of edges in the matching. Vertex Covers Let G be a graph and C a subset of the vertices of G. The set C is a vertex cover of graph G if every edge of G is incident with at least one vertex in C. A minimum vertex cover is a vertex cover with the least number of vertices. Proposition 12.4.5: Let M be a matching in a graph G, and let C be a vertex cover of G. Then |M| |C|. Corollary 12.4.6: Let M be a matching and let C be a vertex cover of a graph G such that |M| = |C|. Then M is a maximum matching and C is a minimum vertex cover. The converse of this corollary is not necessarily true, but it is true for bipartite graphs as we shall see below. Since we have a relationship between matchings in a bipartite graph and flows in the corresponding network, we should consider what a vertex cover looks like in the network. Let A be a subset of X and B a subset of Y. We will consider the set K = A B. Let S = {s} (X-A) B and T = {t} (Y-B) A, which is clearly a cut in the network. 11/15/13 9:29 AM Graph Theory Lecture Notes 11a Page 5 of 6 Theorem: K is a vertex covering of the graph if and only if (S,T) is a cut of the network of finite capacity. Pf: Suppose (S,T) is a cut of finite capacity. Then no arcs from X to Y are in this cut, in particular there are none from X-A to Y-B in the cut. So, all arcs from A go to Y-B, and all arcs from X-A go to B. Therefore, every edge of the graph has one vertex in A or one vertex in B, and so, K is a covering. Note that the arcs in the cut that start at s all go to A, and those that end at t all start in B, each having capacity 1, so the capacity of the cut is |A B|. Now, suppose that K is a vertex covering of the graph. Let A be the intersection of K with X and B the intersection of K with Y. Form S and T as before. Consider an arc from a vertex in S to one in T. If this arc starts at s, it must go to A (and so, has capacity 1). If it starts in X-A, it must go to B since K is a vertex covering, so such an arc does not go to T. If it starts in B, then it must go to t, and so, has capacity 1. Thus, all arcs from S to T have finite capacity, so the cut (S, T) has finite capacity. Theorem 12.4.7: (König, 1931) Let G be a bipartite graph. Then the size of a maximum matching equals the size of a minimum vertex cover. Pf: Consider the associated network of the bipartite graph. Change the infinite capacities to 2's, this will not change any flows. All capacities are integers, so the maximum flow exists and will have integer values. The value of this flow is the number of edges in a maximum matching, call it n. By the Ford-Fulkerson theorem, n is the minimum capacity of any (S, T) cut. The trivial cut, S = {s}, has finite capacity, so the minimum will have to be finite. By the previous theorem, any cut with the minimum capacity corresponds to a covering with |A B| = n. But now, there exists a cover with n vertices and a matching with n edges, so the cover must be a minimum cover. An interesting application of this result is: 11/15/13 9:29 AM Graph Theory Lecture Notes 11a Page 6 of 6 Theorem 12.4.8: (König-Egerváry, 1931) Let A be a 0-1 matrix. The the maximum number of 1's in the matrix, no two of which lie in the same row or column, is equal to the minimum number of rows and columns that together contain all the 1's in A.
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https://github.com/fpvandoorn/Dagstuhl-tables
GitHub - fpvandoorn/Dagstuhl-tables: Dagstuhl's Happy Diner Problem Skip to content Navigation Menu Toggle navigation Sign in Appearance settings Platform GitHub Copilot Write better code with AI GitHub Spark New Build and deploy intelligent apps GitHub Models New Manage and compare prompts GitHub Advanced Security Find and fix vulnerabilities Actions Automate any workflow Codespaces Instant dev environments Issues Plan and track work Code Review Manage code changes Discussions Collaborate outside of code Code Search Find more, search less Explore Why GitHub Documentation GitHub Skills Blog Integrations GitHub Marketplace MCP Registry View all features Solutions By company size Enterprises Small and medium teams Startups Nonprofits By use case App Modernization DevSecOps DevOps CI/CD View all use cases By industry Healthcare Financial services Manufacturing Government View all industries View all solutions Resources Topics AI DevOps Security Software Development View all Explore Learning Pathways Events & Webinars Ebooks & Whitepapers Customer Stories Partners Executive Insights Open Source GitHub Sponsors Fund open source developers The ReadME Project GitHub community articles Repositories Topics Trending Collections Enterprise Enterprise platform AI-powered developer platform Available add-ons GitHub Advanced Security Enterprise-grade security features Copilot for business Enterprise-grade AI features Premium Support Enterprise-grade 24/7 support Pricing Search or jump to... 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Reload to refresh your session.Dismiss alert {{ message }} fpvandoorn/Dagstuhl-tablesPublic NotificationsYou must be signed in to change notification settings Fork 1 Star 6 Dagstuhl's Happy Diner Problem License Unlicense license 6 stars1 forkBranchesTagsActivity Star NotificationsYou must be signed in to change notification settings Code Issues 0 Pull requests 0 Actions Security### Uh oh! There was an error while loading.Please reload this page. Insights Additional navigation options Code Issues Pull requests Actions Security Insights fpvandoorn/Dagstuhl-tables master BranchesTags Go to file Code Open more actions menu Folders and files | Name | Name | Last commit message | Last commit date | --- --- | | Latest commit ------------- History ------- 156 Commits | | .gitignore | .gitignore | | | | LICENSE | LICENSE | | | | README.md | README.md | | | | SAT.nb | SAT.nb | | | | SAT.txt | SAT.txt | | | | abstract.tex | abstract.tex | | | | dagstuhl.py | dagstuhl.py | | | | literature.md | literature.md | | | | lowerbound.txt | lowerbound.txt | | | | sat.md | sat.md | | | | tables.py | tables.py | | | | View all files | Repository files navigation README Unlicense license Dagstuhl's Happy Diner Problem The Table Assignment Assignment The Problem Statement: What is the minimum number of meals so that each of the n conference participants can share at least one meal with every other participant when eating at tables of at most k persons? We call this number T(n,k). In particular, we have an unlimited number of tables, and we do not require that any two participants have a meal together exactly once, or that every table is fully occupied. Dagstuhl's Table Table | n \ k | 2 | 3 | 4 | 5 | 6 | 7 | 8 | --- --- --- --- | | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | | 3 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | | 4 | 3 | 3 c | 1 | 1 | 1 | 1 | 1 | | 5 | 5 | 3 ↖ | 3 di | 1 | 1 | 1 | 1 | | 6 | 5 | 4 fg | 3 | 3 di | 1 | 1 | 1 | | 7 | 7 | 4 | 3 | 3 | 3 d | 1 | 1 | | 8 | 7 | 4 | 3 B | 3 | 3 | 3 d | 1 | | 9 | 9 | 4 AH | 4 ↘ | 3 ↖ | 3 | 3 | 3 d | | 10 | 9 | 6 c | 4 ↖ | 4 fg | 3 | 3 | 3 | | 11 | 11 | 6 | 5 g | 4 | 3 | 3 | 3 | | 12 | 11 | 6 E | 5 | 4 | 3 B | 3 | 3 | | 13 | 13 | 7 a | 5 | 4 E | 4 ↘ | 3 ↖ | 3 | | 14 | 13 | 7 | 5 | 5 g | 4 | 4 fg | 3 | | 15 | 15 | 7 F | 5 | 5 | 4 | 4 | 3 | | 16 | 15 | 9 c | 5 H | 5 | 4 | 4 | 3 B | | 17 | 17 | 9 | 7 i | 5 ↖ | 4 | 4 | 4 ↘ | | 18 | 17 | 9 G | 7 | 6 j | 4 B | 4 | 4 | | 19 | 19 | 10 a | 7 | 6 | 5 gi↘ | 4 ↖ | 4 | | 20 | 19 | 10 | 7 X | 6 | 5 | 5 g | 4 | | 21 | 21 | 10 F | 8 c | 6 | 5 | 5 | 4 | | 22 | 21 | 12 c | 8 | 6 | 5 L | 5 | 4 E | | 23 | 23 | 12 | 8 | 6 | 6 g | 5 | 5 g | | 24 | 23 | 12 J | 8 J | 6 | 6 | 5 | 5 | | 25 | 25 | 13 a | 9 a | 6 AH | 6 | 5 E | 5 | | 26 | 25 | 13 | 9 | 7-8 a E | 6 ↖ | 6 j | 5 | | 27 | 27 | 13 AH | 9 | 8-9 i | 6-7 | 6 | 5 | | 28 | 27 | 15 c | 9 F | 8-9 | 7 g | 6 | 5 | | 29 | 29 | 15 | 11 i | 8-9 | 7 | 6 | 5 | | 30 | 29 | 15 J | 11 G | 8-9 X | 7 B | 6 E | 5 B | Legend: We use lower-case letters a-z (or ↘) to justify lower bounds and upper-case letters A-Z (or ↖) to justify upper bounds. These bounds are explained under Lower Bounds and Upper Bounds. No explanation is given when n ≤ k or k = 2 or the value can be derived from the inequality T(n,k) ≤ T(n+1,k). We have the relation T(n+1,k+1) ≤ T(n,k) (see Relations). If we use this as an upper bound we write ↖ (the value in this cell is at most the value to the top-left of this cell) and as a lower bound we write ↘ (this value is at least the value to the bottom-right). The bolded values indicate perfect solutions (see Terminology). The upper bound J has not been verified by the authors of this table. Dual table | T \ k | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | --- --- --- --- --- --- | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | | 2 | 2 | 3 c | 4 di | 5 di | 6 d | 7 d | 8 d | 9 d | 10 d | 11 d | 12 d | | 3 | 4 | 5 ← fg | 8 B c→ | 9 ← fg | 12 B → | 13 ← fg | 16 B→ | 17 ← fg | 20 B→ | 21 ← fg | 24 B→ | | 4 | 4 | 9 AH | 10 ← g | 13 E g | 18 B gi→ | 19 ← g | 22 E g | 27 B i | 28 ←g | 29-31 ← g | | | 5 | 6 | 9 c | 16 H | 17 ← j | 22 L g | 25 E j | 32 B gi | 33 ← j | 38 L g | 39-41 ← j | | | 6 | 6 | 12 E a | 16 i | 25 AH a | 26-27 ← g | 31-32 E g | 32-37 g | 41-42 E g | 50 B g | 51-52 ← g | | | 7 | 8 | 15 F | 20 X c | 25-26 i | 30-34 B h | 31-38 ←j | 32-44 j | 45-50 B j | 50-56 j | 51-63 g | | | 8 | 8 | 15 c | 24 J a | 26-30 E i | 36-38 E i | 49 AH a | 50-52 ← g | 54-59 K g | 55-66 ←g | 56-73 ← g | | | 9 | 10 | 18 G a | 28 F | 30-35 X c | 36-42 i | 49-51 i | 64 H a | 65-68 ← g | 66-75 ←j | 67-84 ← g | | | 10 | 10 | 21 F | 28 i | 35-40 A a | 42-48 AB i | 49-57 i | 64-67 i | 81 H a | 82-86 ←g | 83-94 ← g | | | 11 | 12 | 21 c | 32 G a | 35-45 a | 42-54 c | 49-63 i | 64-73 i | 81-85 i | | | | | 12 | 12 | 24 J a | 36 J a | 37-46 ← i | 48-60 B a | 49-70 i | 64-80 i | 81-92 i | | 121 AH a | | | 13 | 14 | 27 AH | 40 F | 41-50 ← i | | | 72-88 B i | 81-100 i | | | | | 14 | 14 | 27 c | 40 i | | 66-68 A i | 77-84 A a | 88-96 A i | 99-108 A i | | | | | 15 | 16 | 30 J a | | 55-60 A a | 66-72 i | | | 99-117 i | | | | | 16 | 16 | 33 J | 48 J a | 65 F a | 66-78 i | 91-93 A i | 104-112 A a | 117-126 A i | | | | | 17 | 18 | 33 c | 52 F | 65-66 i | 78-84 A c | 91-99 i | | | | | | | 18 | 18 | 36 J a | 52 i | 65-70 i | | 91-105 i | 104-123 i | | | | | | 19 | 20 | 39 J | | | | 91-112 i | 104-129 i | | | | | | 20 | 20 | 39 c | 60 J a | | 78-98 i | | 104-136 i | 153-157 A i | | | | | 21 | 22 | 42 J a | 64 H | 85 J a | 102 A i | 119-126 A a | 136-144 A i | 153-164 i | | | | | 22 | 22 | 45 A | 64 i | 85-86 i | 102-108 i | | 136-152 i | 153-172 i | | | | | 23 | 24 | 45 c | | 85-90 i | | 133-135 A i | 152-160 A c | 171-180 A i | | | | | 24 | 24 | 48 J a | 72 J a | | | 133-141 i | | 171-189 i | | | | | 25 | 26 | 51 J | 76 J | 95-100 A a | 114-126 A a | 133-147 i | | 171-198 i | | | | | 26 | 26 | 51 c | 76 i | 105 J a | 114-128 i | 133-154 i | 152-179 i | | | | | | 27 | 28 | 54 J a | | 105-106 i | 114-132 i | | 152-185 i | 207-216 A a | | | | | 28 | 28 | 57 J | | 105-110 i | 114-138 i | | 184-192 A i | | | | | | 29 | 30 | 57 c | 88 J | 115 A c | 138-144 A c | 161-175 A a | 184-200 i | 207-229 i | | | | | 30 | 30 | 60 J a | 88 i | | | 161-177 i | 184-208 i | 207-236 i | | | | | 31 | 32 | 63 A | | 125 AH a | | 161-183 i | | 207-244 i | | | | Dual problem statement: What is the maximum number of conference participants so that each participant can share at least one of the T meals with any other participant when eating at tables of at most k participants? This is the maximal n = n(T,k) such that T(n,k) ≤ T. This table has the same information as the previous one, organized differently. This table is harder to read, but much more informationally dense. If you want to read a value of T(n,k) from this table: Look at the k-th column. Find the smallest T that possibly satisfies n(T,k) ≥ n. Then T(n,k) ≥ T. Find the smallest T' that definitely satisfies n(T',k) ≥ n. Then T(n,k) ≤ T'. For example, to find T(14,3) we see that T = 7 is the first value where n(T,3) ≥ 14 (since n(7,3) = 15), so T(14,3) = 7. For example, to find T(24,6), as of August 2019 we see that n(5,6) is in the range 18-24 and n(6,6) is in the range 26-30. We see that n(5,6) ≥ 24 is possible but not guaranteed and that n(6,6) ≥ 24 is guaranteed. So T(24,6) is either 5 or 6. Using similar logic we can conclude that T(25,6) = T(26,6) = 6. Even though the exact values in the table are not known, we do know that T = 6 is the smallest value where n(T,6) is at least n (when n is 25 or 26). With only this information it is possible that T(27,6) > 6. In this table the upper-case (or ←) letters show why the entry is not smaller (why it is possible to have a solution with this number of participants) and the lower-case (or →) letters show why the entry is not larger (why it is not possible to have a solution with one more partipant). The use of T(n+1,k+1) ≤ T(n,k) is indicated with ← and →. ← means that this cell strictly greater than the cell to the left, while → means that this cell is strictly smaller than the cell to the right. Sometimes two cells point at each other (e.g. (T,k) = (4,4) and (T,k) = (4,5)). This looks circular, but it is not. It means that the upper bound of the left cell follows from the upper bound of the right cell and the lower bound of the right cell follows from the lower bound of the left cell. For k > 3 and T > 12 we only put values where either the lower bound or the upper bound is nontrivial (not obtained by ← → a c B) The upper bound J has not been verified by the authors of this table. Terminology Given a table assignment for 1 or more meals. We say that this is a valid solution if every participant meet every other participant at least once. A valid solution with n participants and table size of at most k is called a (n,k)-solution. Given a valid solution. We say that it is an optimal solution if there is no valid solution (with the same n and k) with fewer days. Given a valid solution. We say that it is a perfect solution if every participant meets every other participant exactly once and all tables have k participants every meal. A distribution is a seating assignment for a single meal. A configuration is an assignment for the number of participants to each table for a single meal (but not stating which participant goes where). We say that a configuration C is dominated if it has two tables with a and b participants and a + b ≤ k. In this case, we can merge these two tables and still have a valid solution, so we may assume we have a solution without dominated configurations. A connection is a pair of people sitting at the same table. A new connection is a connection that was not yet formed on a previous day. The number of connections a table of size l makes is s(l) = l(l-1)/2, not all of which might be new connections. The number of new connections s(c) of a configuration c is the sum. The optimal configurationopt is the configuration with the most connections. It has ⌊ n / k ⌋ tables of size k and one table of size n mod k, and it is unique. If k < n ≤ k^2 - 1 then it is not possible to create 2 s(opt) new connections after two days, since at least one pair of participants has to sit at the same table on both days, or a configuration other than the optimal configuration has to be used. Let s(c1,c2) the maximum number of new connections c2 can make on day 2 is c1 was used on day 1. Properties Relations T(n+1,k) ≥ T(n,k) ≥ T(n+1,k+1) ≥ T(n,k+1). If a value in the table can be derived from the first inequality, no other explanation is given. The second inequality follows by treating two of the n+1 people as a single person (always seating them together), and then applying a (n,k)-solution. T(n,k) ≤ T(n,m) T(m,k). If we have a seating arrangement for n participants at table size m, then we can give a seating arrangement for table size k by simulating tables of size m over T(m,k) meals. This subsumes the relation T(n+1,k) ≥ T(n,k) ≥ T(n,k+1) above since T(k,k+1)=1. If there is a perfect (n,m)-solution and a perfect (m,k)-solution then there is a perfect (n,k)-solution. Perfect Solutions Necessarily, every perfect solution is optimal. Necessary requirements for a perfect (n,k)-solution to exist are k - 1 | n - 1 and k | n (or n = 1), i.e. n ≡ k (mod k(k-1)). A perfect (n,k)-solution exists iff T(n,k) = (n-1)/(k-1). The perfect solutions are in boldface in the tables above. Perfect solutions are also called Kirkman Systems in the literature. Perfect solutions are also characterized by the Social Golfer Problem or resolvable 2-designs. We give various explicit perfect solutions in literature.md The upper bounds F, H and (some of) J all give families of perfect solutions. Special Cases T(n,k) = 1 if n ≤ k trivially. T(n,2) = n if n is odd, T(n,2)=n-1 if n is even. The lower bound is given by a. The upper bound can be obtained as follows. Suppose n = 4m: split it up in 2 groups of size 2m, first do those groups independently in 2m-1 days, then in the next 2m days on day i let person j in group 1 sit with person i+j (mod 2m) in group 2. This also follows from upper bound A. Suppose n = 2m with m odd: For the first m days: on day i let person i sit with i+m and person i+j with i-j. For the next m-1 days, on day 2k-1 and 2k let person i sit with the persons i+2k-1 and i-(2k-1). If i is even it will sit with i+2k-1 first, and if i is odd it will sit with i-(2k-1) first. Suppose n is odd, then we can use the solution for n+1 participants. If k is even and k < n ≤ 2k then T(n,k) = 3. Also, if k is odd and k < n < 2k then T(n,k) = 3. These are all the values for (n,k) where T(n,k) = 3. If n > k then T(n,k) ≥ 3 is explained by the lower bound d. If k is even and n ≤ 2k then T(n,k) ≤ 3 is explained by the upper bound B. If k is odd and n < 2k then T(n,k) ≤ 3 is explained by T(n,k) ≤ T(n-1,k-1) ≤ 3, using the previous bullet point. For smaller n we have T(n,k) = 1 trivially and for larger n we have T(n,k) ≥ 4 by lower bound f. Lower Bounds ↘/→: The relation T(n,k) ≥ T(n+1,k+1) can sometimes be used as a lower bound. See Relations. T(n,k) ≥ (n-1)/(k-1) (special case of a). Every participant can see only k-1 participants per meal, and needs to see n-1 participants. a: Suppose n = mk+l with 0 ≤ l < k. There are n(n-1)/2 pairs. At most mk(k-1)/2+l(l-1)/2 pairs can be formed per meal. So T(n,k) is at least equal to the quotient of these 2. b: obsolete. c: If n = mk+1 then the bound from a is n(n-1)/(mk(k-1)) = n/(k-1). Suppose k - 1 | n (i.e. n ≡ -(k-1) mod k(k-1)) and k > 2. Then T(n,k) ≥ n/(k-1)+1, because if it's possible after n/(k-1) days, we need to form mk(k-1)/2 new connections every meal. This means that Every table needs to be size k, except for 1 table of size 1 every meal. Nobody can meet the same person twice. This means that after every meal, the number of participants participant A has met is divisible by k-1, so it can never equal n-1. d: If n > k then T(n,k) ≥ 3. It cannot be done in 2 days, because on day 1 there are at least 2 tables. All participants on table 1 need to be sit with all participants not on table 1 on day 2, but that means that everyone needs to sit together on day 2. Contradiction. e: proven for this special case, see below. f: If k is odd then T(2k,k) ≥ 4. This is likely a special case of g. This also implies that for even k we have T(2k+1,k) ≥ T(2(k+1),k+1) ≥ 4 Note that all non-dominated distributions use 2 or 3 tables. Suppose this can be done in 3 days. Suppose the assignment for day 1 is {a, b, c} with k ≥ |a| ≥ |b| ≥ |c| ≥ 0. We may assume that |a| is the largest among all table sizes on all days. Let x ∈ a and let x ∈ a₂ ⊆ a, b₂ ⊆ b and c₂ ⊆ c be the participants of the table containing x on day 2. We may assume that b₂ ≠ ∅ (by interchanging day 2 and 3), and from this we can conclude that a₂ ≠ a (otherwise |a| was not largest). Now on day 3, all of the following must be true Everyone in a₂ has to meet everyone in b \ b₂ and c \ c₂, so they must all be at the same table. Everyone in a \ a₂ has to meet everyone in b₂ and c₂. Everyone in b₂ has to additionally meet everyone in c \ c₂. Since not everyone can be at the same table, this means that c₂ = c. The table assignment on day 3 must be { a₂ ∪ (b \ b₂), (a \ a₂) ∪ b₂ ∪ c }. b₂ ≠ b, since otherwise |a₂ ∪ b₂ ∪ c₂| > k. Everyone in b \ b₂ has never seen anyone in c, which means that c = ∅. This means that |a| = |b| = k Everyone in a \ a₂ and everyone in b \ b₂ must have seen each other on day 2, so the table assignment on day 2 was { a₂ ∪ b₂ (a \ a₂) ∪ (b \ b₂) }. Since k is odd, we have max(|a₂|, |a \ a₂|) > k / 2 and max(|b₂|, |b \ a₂|) > k / 2, which means that on either day 2 or day 3 there was a table with more that k participants. Contradiction! g: Improvement of a when k < n ≤ k^2 - 2 The idea for this lower bound is that every day after the first day many connections are already present on day 1. This means that the number of connections you can make on most (all but one) days is much less than calculated at a. On day 1 at most s(opt) connections can be added On subsequent days fewer connections can be added. We can give an upper bound, and run through all (sensible) pairs of non-dominated configurations to find the maximal number s of connections that can be added on day 2 (where we can choose the configuration of day 1 to be as optimal as possible). This means that after T+1 days at most s(opt) + T s connections can be made, which gives a lower bound for the number of days. The Python program that is part of dagstuhl.py tries to compute a better lower bound (an older, worse Mathematica function computing this is given in lowerbound.txt.) h: If there is no perfect (n,k)-solution but k | n and k - 1 | n - 1, then T(n-1, k) > (n - 1) / (k - 1). To see this, first notice that T(n-1, k) ≥ (n - 1) / (k - 1) is given by the lower bound a. The only way equality can hold is if all but 1 table has size k every meal, and seating every participant exactly once with every other participant. This means that every participant sits at a table of size k - 1 exactly once. We can add a participant at the empty seat every meal, and then we get a perfect (n, k)-solution, which contradicts the assumption. For (n,k) = (36,6) there is no perfect solution, see Latin Squares. i: Let n = mk + l, and let d be the lower bound given by a. Suppose that (d-1)(k-1) + (l-1) < n-1, then there is no solution in d days with an optimal configuration. The reason is as follows: Let A be any participant sitting at the smallest table in the optimal configuration: it can meet at most l-1 participants that meal, and at most k-1 participants every other meal, which is not enough. This means that for any solution in d days, the smallest table size is (n-1)-(d-1)(k-1). We can now check whether it is possible to create at least n(n-1) / 2d connections in a single day, with this extra condition on the smallest table size. If not, then T(n,k) > d. We don't use i if a or c applies. This bound is always at least as strong as a or c. When n < k^2 most of the time g is stronger. j: If k < n ≤ k^2 - 2 and the lower bound by g gives a number of days where s(opt) + T s = n(n-1)/2, then all connections starting at day 2 have to be optimal. This is is usually impossible, so that 1 more day is needed. If every day uses the same configuration (which is usually/always the case under the assumption above), then this means that participants that sat together on both days 1 and 2 will sit together every day. The reason is that every day requires the least number of duplicated connections, and the duplicated connections between day 2 and n must be a subset of the duplicated connections between day 1 and n (otherwise day n will form fewer new connections than required, because there is distinct overlap with day 1 and day 2). This means that those groups sit together every day. When we quotient by the groups (i.e. consider the groups as single participants) we get an (n', k')-solution for n' < n and k' < k. We get a contradiction if such a solution doesn't exist. We can also compute how often groups of a particular size will meet, or how often a group that sat at a particular table on day 1 will meet other groups of a certain size. This often leads to contradictions, which are described below. We can also replace all groups of size l by groups of size l', which leads to a contradiction for e.g. T(42, 11). For T(18,5) we work out this argument in detail: There are 153 connections to be made, and the only configurations with at least 30 connections are s(5553)=33 and s(5544)=32. If 5553 is never used, then since s(5544,5544) = 30, the maximum number of connections is 32+430<153, so this is impossible So assume day 1 is 5553 and note that s(5553,5553)=30>s(5553, 5544). This means that every day must be 5553, and every table with 5 participants must create 9 new connections after day 1. Given a table of size 5 on day 5, then for every previous day, 2 people already sat together. Since 9 new connections must be formed, this means that 2 people sat together for all 5 meals. Hence there must be 3 pairs of people that sat together every day. There must be a table where 2 such pairs meet. If this is not on day 1, such a table creates at most 8 new connections, which is a contradiction. Hence there is a table on day 1 without any of the pairs. This table must contain 2 people that also sit together on day 5, contradiction. SAT-solvers can do this with quite some effort (~2 minutes assuming the configuration 5553 exists?). For T(26,7), the only way we can form the required 325 connection in 5 days is to use the optimal configuration 7775 every day, and to form 73 connections on day 1 and 63 connections on days 2-5 (there must be at least 10 duplicate connections). This means that 10 pairs must always sit together, and 3 tables must contain always 3 pairs, and 1 table always 1 pair. This means that 5 pairs only sit on tables with 2 other pairs and 1 single participant, so they will meet 10 pairs and 5 single participants, but they need to meet 9 pairs and 7 single participants. So it's impossible in 5 days. T(34,9) > 5, because otherwise the configuration 9997 must always be used and there must be 3 triples that always sit together on a table of size 9, but the different triples can never meet T(39,7) > 7 because otherwise we need configuration 777774 every day, with 6 pairs always sitting together spread over the 5 tables of size 7. These pairs can only have 7 meetings, while 15 are required. T(45,8) > 7, because otherwise we need configuration 888885 every day, with 10 pairs always sitting together. These pairs only have 35 meetings, while 45 are required. T(51,9) > 7, because otherwise we need configuration 999996 every day, with 15 pairs always sitting together. If we collapse these 15 pairs to single participants, we would get a solution to T(36,6)=7 which doesn't exist. This argument also works for T(57,10) > 7. T(76,10) > 9, because otherwise we need configuration 10,10,10,10,10,10,10,6 every day, with 15 pairs always sitting together. These pairs only have 81 meetings, while 105 are required. T(42,11) > 5, because otherwise we need configuration 11,11,11,9 every day, and the groups consist of 10 triples and 6 pairs. Reducing the size of each group by 1, this would lead to a (26, 7)-solution, which is impossible. On the other hand, T(38,10) = 5 and T(22,6) = 5 are possible (see L). k (currently unused): If k < n ≤ k^2 - 2 and the lower bound by g gives a number of days where d := d(n, k) := s(opt) + T s - n(n-1)/2 is only slightly larger than 0, we can hopefully still sometimes derive a contradiction. Promising candidates are d(27, 6) = 2, d(32, 7) = 4, d(37, 8) = 6, d(42, 9) = 8, d(59, 9) = 6, d(86, 10) = 5. Priority of labels: nothing (it follows from the cell above), then ↘/→, then a/c, then d/f/h/i, then g/j, then e If a label with earlier (higher) priority applies, we don't write this label. We do write multiple labels with the same priority. Upper Bounds ↖/←: The relation T(n+1,k+1) ≤ T(n,k) can be used as an upper bound. See Relations. A: T(km,k) ≤ T(m,k) + m if m is coprime with (k-1)!. Divide the participants into k groups of m people. On the first T(m,k) days, everyone meets every participant of their group. Number the participants in each group using the remainder classes modulo m. On the m days after that, on day i (0≤i<m) make a table with participant j from the first group, j+i from the second group, j+2i from the third group, and so on. If m has no divisor smaller than k, then every participant will meet every participant from another group this way. In particular, this shows that if there is a perfect (m,k)-solution and m is coprime with (k-1)! then there is a perfect (km,k)-solution. In particular, if p is prime there is a perfect (p^k,p)-solution. B: T(nl,kl) ≤ T(n,k). This can be seen by making n groups of l people each and always seating all people in a single group together. C: obsolete. D: obsolete. E: found solution for this special case, see below. (We don't use E if another letter applies.) F, G, X: a solution has been elsewhere, usually in articles. An explicit solution is given in literature.md. See Relation to the Social Golfer Problem. See Relation to Block Designs. X: Some solutions were given on Stack Exchange. H: If k is a prime power and n is a power of k, then there is a perfect (n,k)-solution. Consider the field F of order k, and a vector field V with cardinality n over F. For every 1-dimensional subspace L of V the sets of 1-dimensional affine spaces parallel to L forms a partition of V. This defines a table assignment for a single meal. The set of all table assignments determined by all 1-dimensional subspaces in this way forms a perfect (n,k)-solution. The reason that it is perfect follows from the fact that 1-dimensional affine spaces stand in bijective correspondence to pairs of points in V. This idea is due to Neil Strickland. If k is prime (not just a prime power) and n is a power of k, then the existence of a perfect (n,k)-solution also follows from upper bound A. J: see Kirkman Systems and Relation to Block Designs. L: Given an (n,k)-solution in T days and a subset A ⊆ n of participants. We say that A is i-good if no i+1 participants from A sit at the same table during the same meal. Given an i-good set A with m elements, and b ≥ 1, c ≥ 0, then there is a (bn+cm,bk+ci)-solution in T days, by replacing everyone not in A by b people and everyone in A with b + c people, that always sit together. For c = 0 this gives back upper bound B. There is a (16,4)-solution with a 6-element 2 good set (take the solution to T(22,6) given explicitly below and let A be the set of 6 pairs that always sit together). Therefore (taking c = 1) we have T(16b+6,4b+2) ≤ 5. K: From a perfect solution, we can get new solutions with the table size two larger. If we start with a perfect (n,k)-solution (with k > 2) and a 2-good set A (see upper bound L) such that |A|+(k-2)|A|(|A|-1)/2 < n, then we can find a 2-good set with one more participant. The reason is that every pair of people in A sit at the same table exactly once, with k-2 other people. Therefore, at most |A|+(k-2)|A|(|A|-1)/2 other people cannot be added to A while keeping A 2-good, which means that there is someone we can add to A so that the new set is 2-good. Priority of labels: nothing (it follows from the cell below), then ↖/←, then A/B/H/K, then L, then F/G, then E/X, then J. If a label with earlier (higher) priority applies, we don't write this label. We do write multiple labels with the same priority. Literature Relation to the Social Golfer Problem The Social Golfer Problem is a problem similar to Dagstuhl's Happy Diner Problem: given a group of mk golfers playing in m groups of k golfers each day. No two golfers play together more than once. What is the maximum possible of days they can play? Call this number G(m,k). From a good solution G(m,k) of the Social Golfer Problem we can retrieve a (mk,k)-solution to the Happy Diner Problem. F: There is a perfect (mk,k)-solution iff G(m,k) = (mk - 1) / (k - 1) iff T(mk,k) = (mk - 1) / (k - 1). G: If G(m,k)(k-1) = mk - 2 then T(mk,k) = G(m,k) + 1. This is a lower bound by a and a upper bound using the solution to G(m,k): take the solution to G(m,k) for the first G(m,k) meals. Then everyone has seen all other participants, but 1. For the last meal, have one table for each of the pair of participants which still need to see each other. G: If G(m,k)(k-1) = mk - 3 (and k ≥ 3) then T(mk,k) ≤ G(m,k) + 2. After the solution to G(m,k) every participant still needs to meet 2 other participants, which can be easily achieved in two days, by splitting everyone up in group of 2 or 3 people. We only use the letters F and G if we could find an explicit, valid solution. A trivial upper bound is G(m,k) ≤ (mk-1)/(k-1). We call a solution to the Social Golfer Problem good if it is close to this trivial upper bound. See External Links for more sources on the Social Golfer Problem. In particular many solutions of the Social Golfer Problem can be found in this Mathematica Demonstration. The external links contain the following good solutions (only solutions that give new results for the Dagstuhl Happy Dinner Problem are included here). Explicit solutions are given in literature.md. G(5,3) = 7 gives T(15,3) = 7 (MD). G(6,3) = 8 gives T(18,3) = 9 (MD). G(7,3) = 10 gives T(21,3) = 10 (MD). G(11,3) = 16 gives T(33,3) = 16 (Survey). G(7,4) = 9 gives T(28,4) = 9 (MD). G(8,4) = 10 gives T(32,4) = 11 (MD). G(10,4) = 13 gives T(40,4) = 13 (Survey). G(13,4) = 17 gives T(52,4) = 17 (CRCB). G(13,5) = 16 gives T(65,5) = 16 (Survey). Many of these solutions are also found in the literature, see J. Kirkman Systems J: A perfect (n,3)-solution for n ≥ 3 is called a Kirkman Triple System and is possible iff n ≡ 3 mod 6. This is proven in Solution of Kirkman's schoolgirl problem, Ray-Chaudhuri and Wilson (1971). This shows that G(2k+1,3) = 3k+1. Together with lower bound a, this gives that T(6k+1,3) = T(6k+2,3) = T(6k+3,3) = 3k+1. J: The optimal (6m,3)-solution for m > 1 has 3m days. This follows from the study of Nearly Kirkman Triple Systems. A Nearly Kirkman Triple System is a (6m,3)-solution in 3m days where on the first 3m-1 days, all tables have 3 participants, and on the last day every table has 2 participants. In such a solution, all participants meet all other participants exactly once. A Nearly Kirkman Triple Systems exists for 6m ≥ 18. ANKS states this and gives references. For 6m = 12 there is a optimal (12,3)-solution in 6 days which is not a Nearly Kirkman Triple System. (see E) J: A Kirkman System is an alternative name for an optimal solution. According to On resolvable designs they exist for all (n,4) with n ≡ 4 mod 12. ANKS cites papers that show that for every k there is a Kirkman System for all but finitely many n ≡ k mod k(k-1). J: A Nearly Kirkman System is an (n,k)-solution where all participants meet each other exactly once, on all but 1 days all tables contain k participants and on the last day all tables contain k-1 participants. A necessary condition for a Nearly Kirkman System is that n ≡ 0 mod k(k-1). If a Nearly Kirkman System exists for (n,k) then T(n,k) = n / (k-1) ANKS proves that for every k there is a Nearly Kirkman System for all but finitely many n ≡ 0 mod k(k-1). It also states that for k = 4 and n ≡ 0 mod 12 there is a Nearly Kirkman System for all n, except possibly for n in {12, 84, 132, 264, 372, 456, 552, 660, 804, 852} Latin Squares The Social Golfer Problem G(n,n) is related to finding mutually orthogonal Latin squares. Let L(n) is the maximal number of mutually orthogonal Latin squares of order n, that is L(n) = A001438(n). The relation is G(n,n) = L(n) + 2. The reason is that the first two days specify a bijection between the participants and the squares in a n × n grid (the cell (i,j) corresponds to the participant p(i,j) that was in group i on day 1 and in group j on day 2.) Every day d ≥ 3 produces a Latin square by putting k in cell (i,j) if participant p(i,j) was in group k. The fact that this is a Latin square follows from the fact that no pair of participants was in the same group on day d, 1 and 2. Days d₁, d₂ ≥ 3 have no pair of participants in the same group iff the corresponding Latin Squares are orthogonal. There is no pair of mutually orthogonal squares of order 6, so L(6) = 1 and G(6,6) = 3 (OEIS). In particular there is no perfect (36,6)-solution, so T(36,6) > 7. L(10) is the smallest unknown value. According to this Math Stack Exchange answerL(10) < 9. This implies that there is no perfect (100,10)-solution, so T(100,10) > 11. Relation to Block Designs A resolvable 2-(n,k,1) design is an equivalent characterization of a perfect (n,k)-solution. This is also called a (n,k,1)-RBIBD (resolvable balanced incomplete block design). It is a special case of a BIBD (or 2-design) with (v,b,r,k,λ) = (n,b,r,k,1) (where b = n(n-1)/(k(k-1)) and r = (n-1)/(k-1)). This 2-design is equivalently characterized as the Steiner SystemS(2,k,n). J: In RBIBD5 it is shown that a (n,5,1)-RBIBD exists for all n ≡ 5 mod 20, except possibly for n in {45, 185, 225, 345, 465, 645}. A (K,λ)-RGDD (resolvable group divisible design) is a generalization, where the participants are partitioned in groups, and every pair of participants from two different groups meet exactly λ times and every pair of distinct participants from the same group never meet. The table sizes must all be in the set K. Dagstuhl's Happy diner problem asks to find a ({1,...,k},1)-GDD but where participants can meet each other multiple times. Many solutions to (K,1)-RGDDs will give solutions to Dagstuhl's Happy Diner problem. If all groups have size 1, then the solution is immediate. If all groups are size ≤ k then we get an solution by adding 1 day where all participants meet everyone in their group. Solutions for Individual Cases The solution T(6,3) ≥ 4 is very detailed. Other solutions with the same techniques will have much less explanation. So if you don't understand the reasoning, read T(6,3) ≥ 4 first (See Examples). See here for instructions/information about the SAT-solvers used in this project. The indicated time is the fastest time we found (always produced by the most efficient script). T(12,3) ≤ 6 Solution found by SAT-solvers: 159 278 3AC 46B 12C 345 8AB 679 12B 348 79A 56C 168 25A 39B 47C 14A 236 57B 89C 137 249 BC 58 6A kissat took 0.1s on a laptop It is impossible to do his with only 4 tables for each meal. This was verified by a SAT-solver (after aggressive symmetry breaking can be done in 134 seconds on a laptop). T(17,4) ≤ 7 Solution found by SAT-solvers: 1234 5678 9ABC DEFG H 1469 28EH 7ACG B 35DF 7 48CD 26BF 1AEG 359H 1BDH 8A 6CF 379E 245G 1 45BE 67GH 89DF 23AC 29 CEH 56AD 38BG 147F 36E 158C 9G 27BD 4AFH kissat took 1.4s on a laptop T(13,5) ≤ 4 Solution found with kissat (<0.1s): 12345 6789A BCD 19AB 2367C 458D 18C 239AD 4567B 167D 238B 459AC T(26,5) ≤ 8 Solution found by Adam Zsolt Wagner using AlphaEvolve: 5 9 11 16 20 | 2 10 14 22 25 | 4 12 15 18 21 | 0 1 17 19 23 | 3 6 7 8 13 | 24 8 13 14 15 19 | 0 2 5 6 18 | 11 21 22 23 24 | 3 4 16 17 25 | 1 7 9 10 12 | 20 1 4 14 17 24 | 5 7 8 21 25 | 6 10 15 16 23 | 2 3 11 12 20 | 9 13 18 19 22 | 0 2 5 15 17 21 | 0 4 10 11 13 | 7 14 18 20 23 | 1 6 9 19 25 | 8 12 16 22 24 | 3 1 8 10 11 20 | 0 3 9 14 21 | 2 12 13 16 19 | 4 5 7 17 22 | 6 15 18 24 25 | 23 3 6 18 22 23 | 0 8 10 12 21 | 7 11 15 19 25 | 9 13 17 20 24 | 1 2 5 14 16 | 4 6 11 12 14 17 | 0 15 20 22 25 | 2 4 8 9 23 | 1 13 16 18 21 | 3 5 10 19 24 | 7 8 10 11 17 18 | 1 3 9 15 22 | 4 6 19 20 21 | 0 2 7 16 24 | 5 12 13 23 25 | 14 T(22,6) ≤ 5 Solution found by SAT-solvers: 123456 789ABC DEFGHI JKLM 458FIM 26ABDK 379GHL 1CEJ 3ABFIJ 45CGHK 18DL 2679EM 45ABEL 3CDM 268GHJ 179FIK 38EK 4579DJ 1ABGHM 26CFIL kissat took 7.4s on a laptop This is a special case of L T(36,6) ≤ 8 Solution found by an annealing algorithm (to be committed): 22 4 7 28 9 1 | 8 21 6 29 15 2 | 27 32 19 17 30 16 | 0 11 13 10 5 14 | 31 12 20 18 25 34 | 23 33 26 24 35 3 | 12 30 29 31 0 33 | 26 22 11 24 17 6 | 35 20 14 7 21 2 | 23 19 15 1 28 18 | 32 9 34 10 16 8 | 27 3 13 4 5 25 | 11 20 15 3 32 34 | 16 6 27 33 7 12 | 4 8 30 1 14 24 | 9 2 5 0 18 26 | 19 13 25 22 29 21 | 17 35 10 31 28 23 | 25 26 15 30 7 10 | 6 14 31 19 9 3 | 1 5 34 17 21 33 | 32 13 24 12 28 2 | 23 22 8 20 0 27 | 11 18 4 16 29 35 | 33 32 22 18 14 15 | 30 6 13 23 4 34 | 19 8 35 26 5 12 | 28 0 10 16 21 3 | 20 9 17 24 25 29 | 31 1 27 7 11 2 | 33 8 25 11 28 19 | 31 1 26 13 20 16 | 23 29 9 5 32 7 | 21 6 24 10 18 27 | 0 12 17 4 14 15 | 22 30 34 35 2 3 | 34 19 0 11 7 24 | 22 1 3 29 10 12 | 18 5 28 6 20 30 | 31 26 32 4 8 21 | 2 17 25 16 14 23 | 9 35 15 27 33 13 | 19 33 20 10 2 4 | 34 29 28 14 27 26 | 9 11 21 23 12 30 | 3 7 17 8 18 13 | 24 16 31 5 22 15 | 1 35 25 6 0 32 | T(25,7) ≤ 5 Solution found by Mathematica SAT-solver: 235KLP 78BHJM 16ADEG 49CFINO 25EGIJO 1348BFL 6CMNP 79ADHK 47EFGHP 25689B 3ACDJLN 1IKMO 46FJK 8ABDIOP 1257CHN 39EGLM 367HILO 19JP 245ADFM 8BCEGKN kissat took 1.7s on a laptop T(31,7) ≤ 6 Solution found by Adam Zsolt Wagner using AlphaEvolve: 3 4 5 6 10 14 15 | 9 11 13 23 26 | 0 17 21 22 24 27 | 1 2 7 16 20 29 | 8 12 18 19 25 28 30 7 8 13 14 19 22 29 | 0 2 10 15 24 25 26 | 6 11 12 | 4 5 16 17 23 27 28 | 1 3 9 18 20 21 30 2 11 14 17 18 27 30 | 0 6 7 9 24 28 29 | 1 4 5 8 19 20 26 | 10 12 13 15 16 21 | 3 22 23 25 2 6 8 19 21 23 | 9 14 16 25 | 1 10 11 15 20 22 28 | 3 7 12 17 26 27 29 | 0 4 5 13 18 24 30 0 3 8 11 16 19 24 | 14 21 26 28 | 2 4 5 9 12 22 | 1 6 13 17 20 25 27 | 7 10 15 18 23 29 30 8 9 10 15 17 19 27 | 0 1 12 14 20 23 24 | 2 3 13 28 | 6 16 18 22 26 30 | 4 5 7 11 21 25 29 T(22,8) ≤ 4 Solution found by Mathematica SAT-solver: 12345678 9ABCDEFG HIJKLM 47ACIK 1369BDHM 258EFGJL 258ACHM 136EFGIK 479BDJL 47EFGHM 136ACJL 2589BDIK kissat took 0.8s on a laptop T(41,9) ≤ 6 Solution found with the help of Adam Zsolt Wagner using AlphaEvolve: 1 3 8 24 27 36 | 10 13 18 28 29 30 31 33 | 0 6 7 11 20 26 34 39 40 | 2 4 5 9 12 17 22 23 38 | 14 15 16 19 21 25 32 35 37 3 5 18 19 21 27 33 34 40 | 20 22 26 31 36 37 38 | 6 11 12 13 23 24 25 28 32 | 2 4 7 8 16 29 30 35 | 0 1 9 10 14 15 17 39 0 19 21 22 24 29 30 38 39 | 1 2 4 13 28 34 37 40 | 5 6 8 11 14 15 31 | 7 9 17 18 25 32 33 36 | 3 10 12 16 20 23 26 27 35 12 14 15 23 29 30 34 36 40 | 8 9 13 17 19 20 21 26 28 | 5 7 10 24 37 | 1 6 11 16 18 22 33 35 38 | 0 2 3 4 25 27 31 32 39 9 16 17 24 31 34 35 40 | 2 4 6 10 11 19 21 36 | 1 5 20 25 26 29 30 32 | 3 7 13 14 15 22 27 28 38 | 0 8 12 18 23 33 37 39 0 5 13 16 28 35 36 39 | 2 4 14 15 18 20 24 26 33 | 1 7 12 19 21 23 31 | 8 10 22 25 32 34 38 40 | 3 6 9 11 17 27 29 30 37 Examples Examples of specific upper and lower bounds. These all follow from other values or a general principle. These were found before we had these general principles, and we did them individually. Examples that do not follow from other principles are listed under Solutions for Individual Cases. T(6,3) ≥ 4 This is a special case of f. Suppose there is a solution in 3 days. At least 2 days need configuration (3,3). The reason is that we need to establish 15 connections between participants over 3 days. We can establish at most 6 connections during a single day by configuration (3,3). Configuration (2,2,2) is not dominated, but establishes only 3 connections Any other configuration is dominated by either (3,3) or (2,2,2). If there is at most 1 day with configuration (3,3), then the maximum number of established connections is 6 + 3 + 3 = 12 < 15, which is not enough. Without loss of generality we can assume that the first day has configuration (3,3), distributed as 123 456 (i.e. 1, 2 and 3 sit together and 4, 5 and 6 sit together). Now on the other days, we can establish at most 4 connections. The reason is that if we use configuration (3,3), then 2 participants on the first table already sat together on table 1, and the same for the second table. Therefore, configuration (3,3) gives at most 4 new connections. We already saw that the only other non-dominated configuration gives at most 3 new connections. Therefore, the maximal number of connections we can establish is 6 + 4 + 4 = 14 < 15, which is not enough. So there is no valid (6,3)-solution with 3 days. T(10,4) ≤ 4 This also follows from T(10,4) ≤ T(9,3) = 4. (Relations and either A or H) Solution found by hand: 1234 5678 90 1259 3670 48 1280 4679 35 045 1267 389 T(11,4) ≥ 5 This also follows from g. Suppose there is a solution in 4 days. The only configurations which are not dominated are (4,4,3) and (3,3,3,2). The first adds at most 15 connections, the second at most 10. Therefore, on at least 3 days we need a (4,4,3) configuration, WLOG on day 1. For the other days any table of size 4 has 1 pair in common with day 1, so adds at most 5 new connections. Therefore, at most 13 new connections can be added during each day. This means we cannot get 55 connections, therefore we get a contradiction. T(10,5) ≥ 4 This is a special case of f. Suppose there is a valid solution in 3 days. The only configurations which are not dominated are (5,5) (<= 20 conns), (4,4,2) (<= 13 conns) and (4,3,3) (<= 12 conns). Therefore, we need (5,5) at least once. WLOG day 1 is distributed 01234 56789. From now on (5,5) has at most 12 new conns, (4,4,2) has at most 9 new conns and (4,3,3) has at most 8 new conns. This means we cannot get 45 connections, therefore we have no valid solution in 3 days. T(14,5) ≥ 5 This also follows from g. UNSAT found with kissat (<0.1s). T(13,6) ≥ 4 This is a special case of f. Suppose there is a valid solution in 3 days. The configurations with at least 26 connections which are not dominated are (6,6,1) and (6,5,2), one of which has to occur at least once. Suppose day 1 is (6,6,1). Then no other day can have more than 20 connections. Day 2 has (6,6,1) at most 11+9 = 20 connections (6,5,2) at most 11+8+1 = 20 connections (actually, less) (6,4,3) and (5,5,3) and (5,4,4) also have less than 20 connections, all other configurations are dominated. Suppose no day is (6,6,1). Then every day needs 26 connections exactly, which is impossible. Alternatively, this can be derived from T(13,6) ≥ T(14,7) ≥ 4 T(19,6) ≥ 5 This also follows from g. Suppose there is a valid solution in 4 days. 181 connections have to be made, and at most 45 connections can be made per day, with configuration (6,6,6,1). Every other configuration has at most 41 connections. This means that (6,6,6,1) has to appear, WLOG on day 1. Now on every other day, the configuration (6,6,6,1) can give at most 13+12+12=37 connections, which means that there is no way to get 181 connections in 4 days. T(21,6) ≤ 5 This was found by manually modifying T(16,4) = 5, using the same method as K. K only shows that T(20,6) ≤ 5. If you replace 1H, 2I, 5J, 6K and 9L by single participants, you get a perfect (16,4)-solution 1234HI 5678JK 9ABCL DEFG 179DHL 26AFIK 38CE 45BGJ 18BFH 25CDIJ 37AG 469EKL 15AEHJ 289GIL 36BDK 47CF 16CGHK 27BEI 359FJL 48AD T(14,7) ≥ 4 This is a special case of f. Suppose there is a valid solution in 3 days. The configuration (7,7) has to occur, since there is no way to make at least 61 connections in 2 days otherwise. After (7,7) at most 24 connections can be made per day. So there are at most 42 + 24 + 24 = 90 < 91 connections, which is not enough. T(21,7) ≥ 5 This also follows from T(21,7) ≥ T(20,7) ≥ 5 where the second inequality follows from g. Suppose there is a valid solution in 4 days. 210 connections have to be made, and at most 63 connections can be made per day, with configuration (7,7,7). It is impossible to have a non-dominated solution where 5 tables are used every day. If (7,7,7) appears, WLOG on day 1, then at most 48 connections can be made every other day, but 63+48+48+48=207<210 is not enough. If (7,7,7) doesn't appear, then every day has 4 tables. Then at most 57 connections can be made on day 1, and at most 49 connections on future days (since all 7s lose at least 3 and all 6s lose at least 2 connections). This is also not enough, since 57+49+49+49=204<210. Questions The condition n ≡ k mod k(k-1) is necessary for a perfect (n,k)-solution to exist. To what extent is it sufficient? It is true when k is a prime power and n is a power of k, see H. In the smallest case where k is not a prime power, the first non-trivial example is immediately a counterexample. There is no perfect (36,6)-solution (see Latin Squares). It is true for k = 2 and k = 3. For k = 4 it's true according to On resolvable designs. According to ANKS for every k this is true for all but finitely many n (Theorem 1, citing The existence of resolvable block designs. Survey of combinatorial theory, D Ray-Chaudhuri, R Wilson - 1973). This shows that for a fixed k the asymptotic behavior of T(n,k) is T(n,k) ≤ (n-1)/(k-1) + O(1), i.e. T(n,k) - (n-1)/(k-1) is bounded by a constant (possibly depending on k). For every n and k is there an optimal (n,k)-solution in which, during every meal, at most one table is not completely occupied? This is false. All optimal (8,5)-solutions have at least one day with two tables of four participants. This was found by brute force, but is quite easy to see by hand (to do). It is also false that there always is an optimal (n,k)-solution where there are ⌈ n/k ⌉ tables each day (where ⌈ x ⌉ is the smallest integer which is at least x). The Mathematica SAT-solver easily found a solution that T(12,3) ≤ 6. With the help of other SAT-solvers Bernardo Subercaseaux showed that the problem was unsatisfiable under the additional condition that only 4 tables could be used per day. In all cases we know the following holds: if there is a perfect (n,k) solution, then T(n+1,k) = T(n,k) + 2. Does this always hold? Or maybe T(n+1,k) ≥ T(n,k) + 2? External Links / References Dagstuhl's Happy Diner Problem: We submitted two sequences to the OEIS: A318240 and A318241. The problem is stated on Sarah's Oberwolfach Problem Page. Finding the perfect (n,3)-solutions is a special case of the Oberwolfach Problem. Someone asked the value of T(18,4) - 1 on Math Stack Exchange Social Golfer Problem: Wolfram Mathworld Warwick's result page (2002) Markus Triska's master thesis (2008) Edd Pegg Jr.'s Math Game page (2007) A107431. Mathematica Demonstration Math Stack Exchange This page contains a solution that G(8,5) ≥ 8. Kirkman System: Wolfram Mathworld, Dutch dissertation by Pieter Mulder (1917). Solution of Kirkman's schoolgirl problem, Ray-Chaudhuri and Wilson, 1971. In Proc. of Symp. in Pure Math, Vol 19. Kirkman triple systems and their generalizations: A survey, Rees and Wallis, 2002. Springer Asymptotic Existence of Nearly Kirkman Systems, Hao Chen, Wen-Song Chu Block Designs: A Survey of Resolvable Solutions of Balanced Incomplete Block Designs, Sanpei Kageyama, Rev. Inst. Internat. Statist., 40, 269–273. JSTOR On Cyclically Resolvable Cyclic Steiner 2-Designs, Clement Lam and Ying Miao, Journal of Combinatorial Theory, Series A, Volume 85, Issue 2, February 1999, Pages 194-207. ScienceDirect On resolvable designs, Haim Hanani, D.K. Ray-Chaudhuri, Richard M. Wilson Discrete Mathematics, 1972, 3:343-357. Oberwolfach Problem: Sarah's Oberwolfach Problem Page. Contributing Contributions are welcome! Feel free to add any information. Please provide links or justifications of claims you make. Errata 2025.06.13: argument j was incorrect. 2025.04.10: T(13,5) was incorrectly listed to be 5. About Dagstuhl's Happy Diner Problem Resources Readme License Unlicense license Uh oh! There was an error while loading. Please reload this page. Activity Stars 6 stars Watchers 5 watching Forks 1 fork Report repository Releases No releases published Packages 0 No packages published Contributors 2 fpvandoornFloris van Doorn abooijAuke Booij Languages TeX 37.5% Mathematica 36.1% Python 26.4% Footer © 2025 GitHub,Inc. Footer navigation Terms Privacy Security Status Community Docs Contact Manage cookies Do not share my personal information You can’t perform that action at this time.
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Lesson Plan in Math 5 Using 5E | PDF | Subtraction | Behavior Modification 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 100%(6)100% found this document useful (6 votes) 7K views 3 pages Lesson Plan in Math 5 Using 5E This lesson plan is for a Grade 5 mathematics class on adding and subtracting decimal numbers with or without regrouping. The lesson will take 50 minutes and uses the 5E model of instruction… Full description Uploaded by Mark John A. Valencia AI-enhanced title and description Go to previous items Go to next items Download Save Save Lesson Plan in Math 5 using 5E For Later Share 100%100% found this document useful, undefined 0%, undefined Print Embed Ask AI Report Download Save Lesson Plan in Math 5 using 5E For Later You are on page 1/ 3 Search Fullscreen Republic of the Philippines Department of Education Region X Division of Cagayan de Oro City Southwest 1 District BALUARTE ELEMENTARY SCHOOL S.Y. 2022-2023 LESSON PLAN USING 5E MODEL November 15, 2022 MATHEMATICS – Grade 5 Topic Adding and Subtracting Decimal Numbers withou t or with Regrouping Grade Level Grade 5 Time Allotment 50 minutes Learning Competency Adds and subtracts d ecimal numbers through th ousandths without and with regrouping. (M5NS-IIb-106.1) Objectives At the end of the session, students should be able to:1.Add and su btrac t decima l number s throug h thous andth s withou t and with regrouping.2.Write the sum and diffe rence of d ecima ls.3.Valu e impo rtanc e of linin g up of de cimal p oints a ccu rately. ELICIT Let the learners to round off the following decimals.1)3.4 5 9 3 6 2)3.4 5 9 3 6 3)3.4 5 9 3 6 4)3.4 5 9 3 6 ENGAGE Let the learner play the Traveling Game .Materials: flash cards Mechanics:  Call two learners to stand at the back of the classroom.  The teacher will flash the cards on addition and subtraction.  The learner who gives the correct answer first will advance forward with one pace and is given a point.2. Show a video of how to add and subtract decimals. Then, ask the following questions:  How do you add and subtract decimals?  Why do we need to line up the decimal points when adding and subtracting? EXPLORE Divide the class into fi ve (5) groups. Let them go outside to play Amazing Race. Discuss the standards before out of the classroom. adDownload to read ad-free  Avoid unneces sary noise in getting in and out of the classroom  Be with the group all the time  Not to go out empty-handed. Bring notebook and pen  Work on the assigned task  Be mindful of the time allotted for the activity 2. After the game, tell the learners to go back to the classroom.3. Ask the following questions:  How did you add and subtract decimals? EXPLAIN Present the steps in adding and subtracting decimals. In adding decimals: 1.Arr an ge th e nu mbe rs in col um n,2.Al ig n th e de ci ma l po in t,3.Ad d li ke w ho le n um be r,4.Put the de cimal po int direc tly belo w the deci mal poi nt in the add ends In subtracting decimals: 1.Arr an ge t he n umb er i n co lum n;2.Al ig n th e de ci ma l po in t;3.Su btr act lik e wh ole num ber;4.Put the de cimal po int direc tly belo w the deci mal poi nt in the minu end and subtrahend;5.When th e minue nd has les s deci mal plac e than the s ubtra hend, add the number of zero (0) equal to the decimal places of the subtrahend;6.Su btr act as who le num ber. ELABORATE Ask the learners to solv e the following items to answe r the riddle. 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https://www.govinfo.gov/content/pkg/GOVPUB-C13-5dca61206b094d8b3a54099ebcff1baa/pdf/GOVPUB-C13-5dca61206b094d8b3a54099ebcff1baa.pdf
THERMAL CONDUCTIVITY OF ALUMINUM, COPPER, IRON, AND TUNGSTEN FOR TEMPERATURES FROM 1 K TO THE MELTING POINT National Bureau of Standards U.S. Department of Commerce Boulder, Colorado 80303 June 1984 QC 100 ,1156 84-3007 1984 NBSIR 84-3007 THERMAL CONDUCTIVITY OF ALUMINUM, COPPER, IRON, AND TUNGSTEN FOR TEMPERATURES FROM 1 K TO THE MELTING POINT J. G. Hust A. B. Lankford Chemical Engineering Science Division National Engineering Laboratory National Bureau of Standards U.S. Department of Commerce Boulder, Colorado 80303 June 1984 U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director COMTENTS Page Abstract 1 1. Introduction . , , 1 1.1 Thermal Conductivity 4 1.2 Electrical Resistivity ........ 7 1.3 Wiedemann-Franz-Lorenz (WFL) Law 8 1.4 Literature Review 10 1.5 Analysis , , . 11 1.6 Thermal Expansion Corrections , . ~ . 13 1.7 Acknowledgments 13 1.8 References , ...... 14 2. Copper 15 2.1 General . , 15 2.2 Deviations From Recoirmended Equation . 16 2.2.1 Physical Defect Effects 17 2.2.2 Size Effects 20 2.2.3 Magnetic Field Effects ....... 21 2.3 Electrical Resistivity and Lorenz Ratio 21 2.4 Summary for Copper .... 2.5 Format for Annotated Bibliography for Copper 62 3. Aluminum 3.1 General 3.2 Deviations From Recommended Equation 3.2.1 Physical Defect Effects 3.2.2 Size Effects 3.2.3 Magnetic Field Effects iii Page 3.3 Electrical Resistivity and Lorenz Ratio .......... 91 3.4 Summary for Aluminum , , . , 93 3.5 Format for Annotated Bibliography for Aluminum 122 4. Iron ...... . , , 141 4.1 General 141 4.2 Deviations From Recommended Equation 142 4.2.1 Physical Defect Effects , , . . 143 4.2.2 Size Effects 144 4.2.3 Magnetic Field Effects ........ 144 4.3 Electrical Resistivity and Lorenz Ratio . 145 4.4 Summary for Iron 146 4.5 Format for Annotated Bibliography for Iron 176 5. Tungsten 199 5.1 General . . 199 5.2 Deviations From Recommended Equation 200 5.2.1 Physical Defect Effects 201 5.2.2 Size Effects 202 5.2.3 Magnetic Field Effects ...... 203 5.3 Electrical Resistivity and Lorenz Ratio 204 5.4 Summary for Tungsten .................... 205 5.5 Format for Annotated Bibliography for Tungsten 235 IV THERMAL CONDUCTIVITY OF ALUMINUM, COPPER, IRON, AMO TUNGSTEN FOR TEMPERATURES FROM 1 K TO THE MELTING POINT J. G. Hust and A. B. Lankford Chemical Engineering Science Division National Bureau of Standards Boulder, Colorado 80303 Literature data on the thermal conductivity of commercially Dure alu-minum, copper, iron, and tunqsten specimens have been collected, coded, critically analyzed, and correlated with analytical techniques based on the-oretical and empirical equations. The resulting functions are presented and used to qenerate tables and graphs of thermal conductivity as a function of temperature and residual resistivity ratio (RRR). An annotated biblioq-raphy of references is included. Discussions are included on the variations in thermal conductivity caused by chemical impurities, physical defects, size effects, and maqnetic fields. Smoothed values are presented for tem-peratures from 1 K to near the melting point and for a large ranqe of RRR val ues. I Keywords: aluminum; cooper; electrical resistivity; iron; Lorenz ratio; residual resistivity ratio; thermal conductivity; tungsten 1. Introduction The growth of modern technology has increased the need for and use of ther-mophysical properties reference data on materials, often by engineers not totally familiar with the physical phenomena that influence the properties or the condi-tions under which the data are valid. Such a lack of understanding can lead to serious design errors, especially for low-temperature transport properties, such as thermal conductivity, that are strongly dependent on detailed material charac-teristics as well as environmental conditions. The data explosion appearing in the literature may be a drawback rather than an asset to the non-specialist, who needs ready access to critically evaluated, correlated, and smoothed data with a clear indication of their range of application. Center for Information and Nu-merical Data Analysis and Synthesis (CINDAS) (at Purdue University), the editors of Landol t-Bornstein, the Office of Standard Reference Data of the National Rureau of Standards, and others have made efforts to per form this service. These 1 efforts generally encompass such a wide range of materials and properties that the results are lacking in detail desirable for re ference data. For example, ex-ample, when recommended values are presented, it is not always clear how well they agree with the original data. In cases where recommended values are not qiven, the user must exercise considerable effort to obtain such values. This paper presents both the literature thermal conductivity data as well as recom-mended values for a few metals (aluminum, copper, iron, and tungsten 1 alonq with a clear indication of the agreement of the recommended values and the experi-mental data. The objective of this work is to present reference data for selected metals including the most pure research materials and the more commonly produced commer-cially pure materials. Generally soeakinq this includes impurities up to about the 1 % level. The reference data are based on a critical analysis and correla-tion of the best experimental data. Only two variables (temperature and residual electrical resistivity) are used to correlate the selected experimental data. There are numerous factors that complicate the establishment of the uncertainty of this correlation. First, there is the direct thermal conductivity measurement uncertainty. It has been found that 5-10% uncertainties are common and occasion-ally data are found in error by as much as 50%. Second, literature data are fre-quently found for inadequately characterized specimens, i.e., the investigator did not report such factors as residual resistivity, purity, anneal condition, and specimen size. In addition, these factors are freauently not determined for the exact specimen measured. Third, there are uncertainties introduced by ex-tracting literature data by reading small qraphs and sometimes from extrapolated equations. Finally, the uncertainty of the reference data is dependent on the validity of the correlation equation used. Therefore, the reference data pre-sented here, for a wide range of purities, are probably not as accurate as those that can be obtained from the accurate measurement of a single specimen; hut they 2 do represent the best available data tor the entire family of specimens consid-ered. The reference data are most accurate for all specimens measured at hiqh tem-peratures where impurity effects are small and for the most pure specimens mea-sured at low temperatures where the impurity effects are dramatic. A secondary objective was to include as wide a ranqe of impurities as possible with a single equation. Because of this the uncertainty tends to increase at the hiqher im-purity levels. However, specimen size effects and increased experimental uncer-tainties tend to increase at low temperatures and low impurity levels, therefore the total uncertainty at low temperatures is considerably larqer than at hiqh temperatures. Each major section of this report discusses details of the devia-tions. This work was performed under the auspices of the Committee on DATA for Science and Technology of the International Council of Scientific Unions (CODATA) which seeks to ".... improve the quality, reliability, and accessibility of data of importance to science and technology ....". Therefore, its Task Group on Thermophysical Properties has embarked on preparation of critically evaluated and recommended values of a limited set of properties on materials often used for calibration or reference. The properties are thermal conductivity, electrical resistivity, heat capacity, thermal diffusivity, absolute thermopower, and thermal expansion. Materials include aluminum, alumina, copper, iron, silicon, and tungsten. The above properties are similar in that at temperatures , T, well below the Debye characteristic temperature, q, they vary rapidly with T in a complex manner according to the degree of excitation of the thermal vibrations (phonons) and the electrons in some cases. The properties are dissimilar as far as their dependence on purity is concerned. The transport properties (cher.:l condi'p-tivity, thermal diffusivity, and electrical resistivity) are dominated bv nos' lattice imperfections as temperature approaches absolute zero, whereas specific heat and thermal expansion depend primarily on the host lattice characteri sties and are practically insensitive to lattice imperfections. Thermal conductivity and electrical resistivity of pure metals are closely associated with the same conduction source and limiting (scattering) mechanisms. This correlation is expressed by the Wiedemann-Franz-Lorenz Ratio. Because of this correlation between thermal and electrical resistivity and because the im-purity effect is primarily additive in resistivity, the residual resistivity (or the residual resistivity ratio) is a useful parameter to characterize these prop-erties for a given specimen. A brief discussion of these properties is given below. 1.1. Thermal Conductivity The thermal conductivity, x, of a metal or alloy usually is considered to be the sum of the electronic, A e , and lattice, A g , components: X-Xg + Xg (l.l.l) There are other mechanisms of heat transport; however, they generally are not ap-plicable for metals. The electronic term designates the thermal energy carried by the electrons, while the lattice term designates the energy carried by the quantized lattice vibrations (phonons). In pure metals the lattice term is small (frequently less than 5 and almost always less than 20 percent). Although theory provides a guide for the dependencies of the lattice conductivity and its order of magnitude, it is generally not sufficient for reference data purposes. Theory does provide an adequate formulation for the 1 ow-temperature electronic term. For these reasons, the formulation provided here is based on the theoretical form of the electronic term. Modifications of this form to account for the lattice contribution and higher temperature dependencies are based on the experimental data. 4 Theory shows that at low temperatures the electronic thermal conductivity component is limited mainly by two mechanisms, a) the interaction of the elec-trons with phonons, and b) the interaction of the electrons with physical and chemical imperfections. The interaction of electrons with phonons leads to a resistive term approximately proportional to T^. The proportionality constant, a, is characteristic of the generic type of metal, i.e., it is an intrinsic prop-erty of the base metal. The interaction of electrons with lattice defects leads to a thermal resistivity which is inversely proportional to temperature. The proportionality constant, a, is determined by the type and concentration of the lattice imperfections. This approximation of low-temperature electronic thermal conductivity is written as X = (W + W.)" 1 = (0/T + cjV 1 where n = 2 (1.1.2) and where W0 represents the electron-defect interaction and W-j is the electron-phonon interaction. Cezairliyan and Touloukian presented a revised form of this theoretical equation to account for observed deviations. This re-vised form, reviewed in reference , is based on experimental data that show a) n is usually larger than 2, and b) a is weakly dependent on lattice imperfec-tions. The modified equation is valid only for temperatures up to about 1.5 times the temperature at which the maximum in thermal conductivity occurs. For the metals of interest here, this limit occurs at about 40 K. At higher tempera-tures, thermal conductivity decreases more slowly with increasing temperature than predicted by this equation. Theory predicts that at high temperatures , thermal conductivity of metals should approach a constant. To account for this high temperature behavior, the form presented by Cezairliyan and Touloukian has been further modified-. Finally, evidence has been presented that indicates The references shown in [ ] are listed in Section 1.8. 5 the presence of an interaction term between W0 and W-j . It is denoted by W-j 0 . The base equation selected to represent the predominant thermal conduc-tivity behavior of these metals is therefore. A, = (W + w. + w. r 1 b v o i io' (1.1.3) where W = g/T o (1.1.4) P (p +p ) p W i = PjT 2/(l + P 1 P 3 T 2 4 exp(-(P 5/T) 6 )) + Wc (1.1.5) W io = P 7 W i Wo/(W i + V (1-1-6) where the P-j's are parameters determined by least squares fit of the experi-mental data. The quantity Wc is a temperature dependent term (defined later for each metal) that accounts for mathematical residual deviations in W-j. This equation is intended to describe the thermal conductivity of annealed bulk speci-mens of these metals. Thus it describes the intrinsic thermal conductivity and the limiting effect of the presence of chemical impurities in each metal. The reader should be aware, however, of the existence of other limiting mechanisms, such as physical defects, size effects, and magnetic field effects. Brief dis-cussions of these effects are presented for each metal. Both thermal conductivity and electrical resistivity of a pure metal are strongly dependent on the concentration of lattice imperfections. Indeed, they are both influenced to the same degree by the imperfections at very low tempera-tures. The resulting correlation is referred to as the Wiedemann-Franz-Lorenz Law, discussed in Section 1.3. Since electrical resistivity is much easier to measure accurately than thermal conductivity, it is often used to determine im-perfection concentrations, and therefore thermal conductivity. The following discussions on electrical resistivity and the Wiedemann-Franz-Lorenz Law are in-cluded for that reason. 6 1.2. Electrical Resistivity For discussion purposes the electrical resistivity is described adequately by Matthiessen's rule (MR). This rule, (1.2.1) states that the electrical resis tivity, p, of a metal is the sum of two parts: the intrinsic resistivity, p-j, which is characteri zed by electrons interacting with phonons only, and the re-sidual resistivity, p 0 , which is characterized by electrons interacting with the chemical and physical imperfections of the metal. p(T) = P0 + Pi (T) (1.2.1) The residual resistivity is temperature independent, while the intrinsic resis-tivity increases rapidly with temperature. The specific temperature dependence varies widely both with temperature and base material. Nevertheless, p-j is not dependent significantly upon composition changes for a given base metal. Thus, if one knows p-j(T) for a given base metal, p(T) for any composition of that metal can be approximated from (1.2.1) after measuring only p Q . The value of p 0 is obtained by measuring p at low temperatures , where p-j is negligible (liquid helium temperatures generally are adequate). For several reasons, the ratio of the resistivity at the ice point to the residual resistivity is fre-quently used to characterize the purity of a metal. This ratio is called the re sidual resistivity ratio and is denoted by RRR. Within the limitations of MR, RRR can be written as, RRR = -1— + 1 (1.2.2) Po Although, deviations from MR cause (1.2.2) to be slightly in error, we hav adopted it for conversion between p0 and RRR in this report. This conversio-is necessary because some authors use p 0 for characterization while others use 7 RRR. In either case, the authors frequently do not report the value of p™ It was convenient in this study of thermal conductivity to develop a simple ana-lytical approximation for electrical resistivity. The base function chosen and fitted to selected resistivity data for each metal is where p = p + p . + p . H Mo H io pQ = residual electrical resistivity (1.2.3) (1.2.4) P (P +P ) P P n -= PjT 2/(l + PjPjT 2 4 exp(-(P 5/T) 6 )) + pc (1.2.5) p io = P 7 p i po/(p i + po> (1.2.6) Note the similarity to Eqs. 1.1.3 through 1.1.6. The pj 0 term accounts for observed deviations from MR. The quantity p c is designed to account for the mathematical residual deviations in p^ . It should be noted that Eq. 1.2.3 does not assume the validity of MR as does Eq. 1.2.2. Section 1.5 describes how this apparent contradiction was circumvented. 1.3 Wiedemann-Franz-Lorenz (WFL) Law In 1853 Wiedemann and Franz formulated an empirical law relating the thermal and electrical conductivities of metals, namely, that the ratio of the electrical and thermal conductivities (WF ratio) at a given temperature is the same for all metals. In 1872 Lorenz discovered that the WF ratio is proportional to tempera-ture. The result was the Wiedemann-Franz-Lorenz (WFL) law: £ = Xp = LT (1.3.1) where a = electrical conductivity, L = Lorenz number, and T = absolute tempera-ture. 8 data evaluation process. Graphical illustrations of the dependencies of A, p, and L are given in the latter section of this report. 1.4. Literature Review The existing principal compilations and reviews [2,3] were used as a start-ing point for this compilation. The resulting list of references was updated by searching current literature, abstracting services, and computerized data banks, as well as the reference lists of the most recent publications. The initial em-phasis was directed toward temperatures below 300 K. Later the scope was ex-tended to include temperatures up to near the melting point of each metal. The high temperature compilation was directed toward obtaining the most significant publications rather than a complete listing. Since the principal interest is the dependence of thermal conductivity on temperature and electrical resistivity for relatively pure metals, not all of the literature data for a given base metal is referenced here. For example, numerous publications on the measurement of thermal conductivity at a single temperature have been excluded. Also, all measurements on specimens with more than 1% total impurity concentration were excluded. Each of the selected sources was coded for content, and the data were ex-tracted for computer analysis. When the literature data were presented in graphical form, the graphs were enlarged and read as accurately as possible. The resulting data were then smoothed to reproduce the original curves. Each set of data for a given measured specimen was characterized with values of residual re-sistivity, RRR, chemical impurity concentrations, and thermal /mechanical history. Other details regarding the experimental procedure, purpose of the work, and analysis of data are also coded in the annotated bibliography for the convenient.-of the reader. 10 Drude gave a theoretical derivation of the WFL law in 1900 and obtained a value of 2.228 x 10-8 v2/k 2 for L. Sommerfeld calculated the first order approximation of L from the free electron theory of metals. His value, 2.443 x 1 q-8\/2/k 2 commonly is designated L 0 . It should be pointed out that the theoretically calculated value of L is based on the electron component of thermal conductivity, A e . Electron Lorenz numbers, p\e/T, other than the Sommerfeld value are designated by Le to distinguish them from total Lorenz numbers, L = pA/T. For many pure metals the experimentally determined values of L fall between 2.2 and 3.0 x 10'^V^/K^ at room temperature and only slightly higher at 100 °C. At very low temperatures (liquid helium) the experimental values are near the Sommerfeld value. Thus 3 as defined in Eq. 1.1.4 is equal to p0/Lq . For intermediate temperatures the agreement of L and L0 vanishes. At inter-mediate temperatures, Le and L are strong functions of purity and temperature, with the decrease from L0 greatest for the most pure metals and at about 15 K to 40 K. Deviations of L from L0 at high temperatures are often attributed to the presence of lattice conductivity; however, as pointed out by Touloukian, Powell, Ho, and Klemens , the actual electronic Lorenz number may also deviate from L 0 , even in the absence of lattice conductivity. For the interested reader, detailed theoretical treatments can be found in numerous sources. A few of these are listed in the references [2-13]. The principal reason for consideration of the WFL law in this report is be-cause of its utility in examining experimental data for unusual behavior. The functions used to represent A and p were used to calculate the Lorenz ratio as a function of temperature and RRR. The resulting plots were examined to aid the 9 establish values of p 0 and RRR for each reported thermal conductivity measure-ment in the literature: 1 . The value of p 0 that produces the best agreement of Eq. 1.1.3 and the reported thermal conductivity data at low temperatures was selected. The corresponding residual resistivity ratio, RRR, was calculated from Eq. 1 . 2 . 2 . In addition, at this stage of the analysis, the Sommerfeld value of the Lorenz ratio, L0 = 2.443 x 10" 8 , was assumed to be valid to ob-tain the corresponding value of p = p0 /L0 . The values of p ^-/2 used i n Eq. 1.2.2 for the four metals investigated are given below. Metal Pi 273 (bflm) Copper 15.4 A1 uminum 24.8 Iron 87.0 Tungsten 48.4 The values listed in this table are our best estimates based on literature values and a variety of published and unpublished measurements performed at this labora-tory over a period of 25 years. 2. Upon completion of the data fits and comparisons, plots were made of RRR (selected) versus RRR (reported) for each metal. These plots were used to determine the Lorenz ratio adjustment necessary for each metal and the range of RRR for which Eq. 1.1.3 is valid. Recommended values as a function of T and RRR are given in the form of equations, tables, and graphs for each metal . 12 1.5. Analysis Selected data sets (primary data) were used to establish best values over the entire range of temperature and p0 or RRR. The primary data were chosen on the basis of proven laboratory techniques, as applied to pure and annealed speci-mens. These data are believed to be accurate to within 5 to 10%. The primary data were then used to optimize the parameters in the selected equation for each metal. The p 0 or RRR value assigned to each data set was not necessarily that reported by the author. The values were selected to minimize the thermal conduc-tivity deviations in the low temperature range, i.e., the range below the peak in the curve. If data did not exist below the peak in the \ vs. T curve, the author's value of RRR (or p0 ) was used. If a value of RRR (or p0 ) was not reported for a high temperature data set, it was estimated by considering purity and anneal conditions. The relationship of the selected values of RRR (or p0 ) and reported values are shown in each of the following major sections. The fitted equation was then compared to other data sets, including those for the less pure and unannealed specimens. The comparisons were examined for deviations varying systemati cal ly with RRR and temperature. The results of this analysis are described in the section for each metal (Sections 2 through 5). As briefly discussed in Section 1.2, the relationship between RRR and p0 is not uniquely defined for a given specimen in the absence of other measure-ments. For very pure metals RRR is frequently reported because the determination of RRR can be done without a knowledge of the form factor, s,/ A, i.e., the length to cross-sectional area ratio of the specimen. Such specimens are frequently very thin; and consequently, the accurate determination of i/k and p0 is diffi-cult. In the absence of MR deviations, Eq. 1.2.2 may be used to define the rela-tionship between p 0 and RRR. However, MR deviations are known to exist and, thus Eq. 1.2.2 is also inexact. We have chosen the following procedure to 11 1.6 Thermal Expansion Corrections It would be desirable to specify that the recommended values in this report are based on the actual dimensions of the specimens at any given temperature, i.e., they are corrected for thermal expansion. However, only a small fraction of the authors indicated whether their reported data are corrected or uncorrected for thermal expansion. Therefore, the reported data were not modified for thermal expansion effects. This, although an undesirable situation, is not serious in view of the relatively large measurement uncertainties. It is also noted that, although a linear correction is usually appropriate, the proper cor-rection depends on the nature of the measurement method employed. For the convenience of the reader, estimates of the magnitude of the thermal expansion correction are given here. Room temperature is used as the basis for this correction since specimen dimensions are normally determined at room tem-perature. The four metals discussed have positive thermal expansion coeffi-cients, thus the thermal conductivity values corrected for thermal expansion are smaller than the uncorrected values at temperatures above room temperature, and larger at temperatures below room temperature. 1.7 Acknowledgments This work began several years ago and has progressed at a low level of ef-fort. It was funded in part by the Air Force Office of Scientific Research. During these several years numerous NBS staff members have contributed to the tasks involved. In particular, we wish to acknowledge the assistance of Dr. Nancy Simon, Mr. Bruce Howry, and Mr. Jeff Wood. We also express apprecia-tion to Dr. M. L. Minges (USAF-AFML), Dr. G. K. White (CSIRO-Austral ia) and other members of the CODATA Task Group on Thermophysical Properties for their helpful suggestions and reviews of this manuscript. 13 1.8 References 1. Cezairliyan, A. and Touloukian, Y. S. , Correlation and prediction of thermal conductivity of metals through the application of the principle of corres-ponding states, in Advances in Thermophysical Properties at Extreme Tempera-tures and Pressures, Third Symposium on Thermoohysical Properties, ASME (1965), 301-313. 2. Touloukian, Y. S. , Powell, R. W. , Ho, C. Y. , and Klemens, P. G. , Thermo-physical properties of matter, Vol . 1, Thermal Conductivity (Metallic ele-ments and alloys), Plenum Press, N.Y. (1970), 1469 pp. 3. Childs, G. E., Ericks, L. J., and Powell, R. L., Thermal conductivity of solids at room temperature and below (A review and compilation of the lit-erature), National Bureau of Standards, Monograph No. 131 (1973), 608 pp. 4. Parrott, J. E. and Stukes, A. D. , Thermal conductivity of solids, J. W. Arrowsmith Ltd., Bristol, England (1975), 157 pp. 5. Berman, R. , Thermal conduction in solids, Clarendon Press, Oxford, England (1976), 193 pp. 6. Klemens, P. D. , Theory of the thermal conductivity of solids, in Thermal Conductivity, Vol. 1, R. P. Tye, Editor, Academic Press, N.Y. (1969), 1-68. 7. Ziman, J. M. , Electrons and Phonons (The theory of transport phenomena in solids). Clarendon Press, Oxford, England (1960), 554 pp. 8. Wilson, A. H. , The theory of metals, Cambridge Press, England (1958), 346 pp. 9. Klemens, P. G. , Thermal conductivity of solids at low temperatures , in Hand-buch der Physik, S. Flugge, Editor, Vol. 14, Spri nger-Verl ag, Berlin ( 1956), 198-281. 10. Klemens, P. G. , Thermal conductivity and lattice vibrational modes, in Solid State Physics, Vol. 7, Academic Press, N.Y. (1958), 1-99. 11. Mendelssohn, K. and Rosenberg, H. M. , The thermal conductivity of metals at low temperatures, in Solid State Physics, Vol. 12, Academic Press, N.Y. (1961), 223-274. 12. Mott, N. F. and Jones, H. , Theory of the properties of metals and alloys. Clarendon Press, Oxford, England (1936), 326 pp. 13. Peierls, R. E. , Quantum theory of solids. Clarendon Press, Oxford, England (1955), 229 pp. 14 2. Copper 2.1 General Copper is discussed first because it is the most extensively measured metal. A total of 44 references were selected for inclusion in the annotated bibliog-raphy. The following references from the annotated bibliography (Section 2.5) represent the primary data sets: 4, 7, 8, 14, 16, 18, 19, 21, 23, 24, 26, 27, 28, 29, 30, 33, 34, 35, 38, 41, 42, and 43. The primary data cover a range of temperatures from 0.2 to 1250 K, and a range of RRR from 20 to 1800. These data are illustrated in Figs. 2.1.1 through 2.1.6. These data are illustrated seven sets to a figure to avoid figures that are overly crowded and to aid in identifying the source of each data set. For additional convenience a composite of all data is also given. Fig. 2.1.6, but without source identification. For comparison, the electrically purest copper ever produced has an RRR of about 50,000. These high RRR values are not indica-tive of the actual impurity content. They were produced by oxygen anneal of high purity copper. The impurity content under these conditions remains unchanged, but the impurities (especially iron) chemically combine with oxygen to form less effective scattering centers. The RRR of typical commercially pure copper wire is in the range of 50 to 500. Very high purity copper, produced routinely, may have an RRR as high as 2000. Equation 1.1.3 was fitted to represent the primary copper data over the en-tire temperature range. The values of the parameters , P-j ,i = 1, 2, ..., 7, obtained by nonlinear least squares fit are 15 p 1.754 x 10 -8 1 P 2 2.763 P 6 1.765 P 3 = 1102 P 7 = 0.838/B^ 1661 P 4 = -0.165 where 3 r = e/0.0003. All units are SI. The data at high RRR were examined for systematic residuals as a function of temperature. These residuals were represented by the Wc term in Eq. 1.1.5. The result is where Wc and T are in SI units. 2.2 Deviations From Recommended Equation The deviations of the primary data from Eq. 1.1.3 are illustrated in Figs. 2.2.1 through 2.2.6. Figure 2.2.6 is a composite of all deviations without data source identification. Five data sets exhibit differences of greater than +10%. In most of these cases, the deviations are significantly higher than the stated or implied uncertainty of the source document. It is not clear if these deviations are the result of underestimated uncertainties or the result of real differences between specimens. Although temperature dependent deviations do exist for individual data sets, the overall pattern is random in nature. No sys-tematic trends with RRR were identified. The primary data were selected from the literature data on relatively large, well annealed specimens. Therefore, the deviations exhibited in Figs. 2.2.1 W c = -0.00012 £n(T/420) exp(-(£n(T/470)/0.7) 2 ) -0.00016 £n(T/73) exp(-Un(T/87/0.45) 2 ) -0.00002 £n(T/18) exp(-Un(T/21)/0.5) 2 ) 16 through 2.2.6 are indicative of the combined effect of (a) experimental measure-ment errors and (b) the inability of Eq. 1.1.3 to account for the effects of chemical impurity variations. The effects of physical defect variations, small specimen size variations, and magnetic fields are exhibited, in part, by the de-viations of the secondary data. The thermal conductivity variations, caused by other than chemical impurity variations, are not expected to be represented as well by Eq. 1.1.3. However, the RRR (or p0 ) correlating parameter does account for an appreciable part of these variations. Some users may find this to be an adequate representation and, therefore, discussions of these comparisons are in-cluded for completeness. The deviations of the secondary data sets (other than the primary sets) are illustrated in Figs. 2.2.7 through 2.2.16. These plots are divided into two groups, according to the magnitudes of the deviations. Figures 2.2.7 through 2.2.12 and Figs. 2.2.13 through 2.2.16. Figures 2.2.12 and 2.2.16 are composite graphs for each group, respecti vely. Finally, it was of interest to compare the values calculated from Eq. 1.1.3 to the most widely used references (13,39) of recommended thermal conductivity values. Figure 2.2.17 shows this comparison. Within the combined uncertainties of this work and references 13,39 the differences are not significant. It should be noted however that references 13,39 give values only for a single value of RRR (approximately 1800), while Eq. 1.1.3 covers a wide range of RRR. 2.2.1 Physical Defect Effects Investigations of physical defects in copper have produced several note-worthy references 5,10,12,15,22,26,33,41,44. We shall discuss each of them bel ow. Reference 10 shows the effects of mechanical deformation. The peak value of the thermal conductivity of the unstrained specimen was 2750 Wm-1 K 1 . After 17 a 30.8% elongation, the value at the peak was 1050 W-irT^K" 1 . The devia-tions from Eq. 1.1.3 for the unstrained condition were within +4%. Reference 22 reports the effects of mechanical deformation on two specimens of different purities. For the 99.9% pure Cu specimen, the unstrained peak value of the thermal conductivity was 2700 Wm_1 K _1 , and after a 30.8% elongation, the peak value was 1100 W'nT^K" 1 . For the 99.99% Cu speci-men, the unstrained peak value was 5800 W-m -1 K -1 , and after a 28.4% elongation, the peak value was 1300 Wm_1 K _1 . The deviations from Eq. 1.1.3 for the 99.9% copper specimen are within +12% (unstrained condition). Similarly, the deviations for the 99.99% copper specimen are within +3% (unstrained condition). The strained conditions were not compared. Another study of the effect of mechanical deformation is given in refer-ence 15 in which a specimen was torsi onally deformed (nd/£ = 1.29, where n is the number of turns on the specimen). The specimen was 99.55% copper and had strained thermal conductivity values of 24 W'lrT-^K"-'-at 20 K, and 50 W , m"lK --'-at 40 K. After these measurements were taken, the specimen was annealed in helium at 450 °C for three hours and measurements were taken again. After the anneal, the values were 36 Wm“^K _1 at 20 K and 60 Wm"-'-K'--at 40 K. The thermal conductivities of a control specimen were also measured. The deviations from Eq. 1.1.3 for the control specimen were within +16%. The deviations for the deformed specimen were within +26%, while for the annealed specimen, the deviations were within +_20%. A possible explanation for these large deviations may be that the specimen's RRR is close to the lower validity limit of Eq. 1.1.3. References 26 and 33 demonstrate the effect on thermal conductivity of annealing a specimen, then drawing it. The specimen was annealed in vacuum and the thermal conductivity was determined. A similar specimen was cold -drawn 26% 18 and its thermal conductivity was measured. The maximum thermal conductivity for the annealed specimen was 14000 WtfT-'-’K” 1 , while for the drawn specimen it was 2500 W'nT^IC 1 . The deviations from Eq. 1.1.3 for the annealed specimen were within +_9%, while for the cold-drawn specimen, the deviations were within +6%. Reference 41 reports the differences in thermal conductivity of a specimen in the "as-drawn" state and in the annealed state. The peak value of the thermal conductivity for the drawn condition was 1540 WttT-’K" 1 , and that for the annealed condition was 5300 WnT^!C"l. The deviations from Eq. 1.1.3 for the as-drawn state were within +7%, while for the annealed state, they were within + 10%. Reference 12 investigated the effects on thermal conductivity of different annealing conditions and sizes. For an oxygen anneal, the peak conductivity was found to be 56500 WtitI'K"! for a specimen of 12.5 ym thickness, and 55300 W'm'^K' 1 for a specimen of 125 ym. After a high vacuum anneal, these peaks are 15500 WnT^K _ -and 5000 , respec-tively. The deviations from Eq. 1.1.3 for the smaller specimen were within j+30% for the oxygen anneal, +35% for the high vacuum anneal. Those for the larger specimen were within +30% for the oxygen anneal, +120% for the high vacuun an-neal. A possible explanation for these very large deviations may be that size corrections applied to the data were too large. Reference 44 showed the effect on thermal conductivity of a specimen in an unannealed (as fabricated) state and in an annealed state. The peak conduct:.':, in the unannealed state was 390 WttT-^K"! , while for the annealed state, it was 570 Wm“l.«-l t The deviations from Eq. 1.1.3 for both the unar,-nealed and annealed specimens were within +5%. 19 Reference 5 reported the effect of neutron irradiation on thermal conduc-tivity. Neutron irradiation produces physical defects within the lattice. The neutrons had energies up to 10^ eV while the specimens had a maximum exposure of 6.5 x 1()23 neutrons per square meter. The damage, as measured by thermal resistivity, was linear with neutron exposure. The maximum thermal conductivity for the unirradiated specimen was 16700 Wm_1 K _ ^, while for the maximum exposure specimen, it was 1700 Wm_1 K _1 . The deviations from Eq. 1.1.3 for the unirradiated specimen were within +35%, while those for the maximum ex-posure were within +20%. The nature of the deviations for these specimens are not understood. Although the temperature dependence of physical defect scattering mechanism is different from that due to impurity scattering, Eq. 1.1.3 generally represents the unannealed specimen data quite well (within +20%). This indicates that the residual electrical resistivity characterizes both types of scattering for the range of RRR included here. 2.2.2 Size Effects Nath showed that the thermal conductivity was a strong function of specimen thickness below about 0.1 pm while it is nearly independent of thickness for specimens larger than 0.4 pm. The variation of the thermal conductivity with thickness was studied for temperatures above 80 K and was found to be significant even at 500 K. A 50% reduction in thermal conductivity below that of bulk copper was observed for a film of 0.01 pm thickness. Reference 12 contains thermal conductivity data that has al ready been cor-rected for size effects on two copper crystals of 12.5, 125 pm thickness. ^Nath, P. and Chopra, K, L. , Thermal Conductivity of Copper Films, Thin Solid Films, 20, 53-62 (1974). 20 2.2.3 Magnetic Field Effects Although magnetic field effects on thermal conductivity were not studied ex-plicitly, reference 24 showed that a 6 T field increases the thermal resistivity by 33% at 3 K, decreasing to 18% at 31 K for the same field. The increase in thermal resistivity changes linearly with the field while the slope decreases with increasing temperature. Sparks^ measured the effect of magnetic fields up to 8 T on two specimens of copper of differing purities. The magnetothermal resistivity introduced into the high conductivity specimen exhibited a strong temperature dependence below a field of 2 T. Above this value, the temperature dependence was much less. The value of the magnetothermal resistivity at high fields was about 0.0004 mKW -1 near 4 K. Up to 8 T, the lower conductivity specimen ex-hibited a more uniform behavior, and the magnetothermal resistivity introduced near 4 K was appreciably higher, 0.002 nrK'W 1 . The relative change in thermal resistivity, however, is greater for the high purity specimen at a given field. Fevrier and Morize^ reported measurement on two specimens of pure copper at fields up to 5 T at 4.2 K. The maximum field induced a factor of four increase in resistivity. Nevertheless, the ratio of the thermal and electrical conductivities was reported to be a constant for both longitudinal and transverse magnetic fields. 2.3 Electrical Resistivity and Lorenz Ratio During this investigation it was frequently helpful to examine the Lorenz ratio as a function of temperature and RRR. However, to do this we needed values ^Sparks, L. L. , Magnetothermal Conductivity of Selected Pure Metals and Alloys, Semi-Annual Technical Report on Materials Research in Support of Superconducting Machinery- IV , NBSIR 75-828 (1975). ^Fevrier, A. and Morize, D. , The effect of Magnetic Field on the Thermal Con-ductivity and Electrical Resistivity of Different Materials, Cryogenics, n, 603-6 (1973. 21 of electrical resistivity. Therefore, we selected those data sources from the primary data set that contained electrical resistivity data. The electrical resistivity data used is shown in Figs. 2.3.1 through 2.3.3. Figure 2.3.3 is a composite of all data. The parameters for Eq. 1.2.3 are: P x = 0.1171 x 10" 16 P 2 = 4.49 P 3 = 3.841 x 10 10 P4 = 1.14 P 5 = 50 P 6 = 6.428 P 7 = 0.4531 All units are SI. The deviations of the experimental data from this equation are illustrated in Figs. 2.3.4 through 2.3.6. It is clear that in the midrange of the £nT plots (10 to 100 K) there is a fairly large uncertainty in the representation (+10 to 15%). Smooth curves of p vs. T at selected RRR values are plotted in Fig. 2.3.7. From the p(T,RRR) and x(T,RRR) equations values of L(T,RRR) were calculated and plotted in Fig. 2.3.8. Irregul arities are greatly magnified on this plot (+10% equals about 1 cm on the L scale). It is noted that the bumps in the curves in the vicinity of 30 to 80 K correspond to the region of greatest uncertainty in both the p(T,RRR) and x(T,RRR) representati ons. However, it is also noted that the bumps correspond to less than +10% irregularity in L from what is normally expected and this corresponds to the combined irregularity of the p and X repre-sentations. In Section 1.5 we discuss the procedure for selecting values of p 0 and calculating RRR for each thermal conductivity data set. These values of p 0 22 along with the Sommerfeld value of Lorenz ratio were used to best fit each low temperature data set. The resulting values of RRR obtained by this procedure are compared to the values reported in the references in Fig. 2.3.9 and are listed in Table 2.3.1. Figure 2.3.9 shows values of RRR (calc), those values from the above procedure, versus RRR (obs), those values reported in the references listed in the annotated bibliography. Also shown in this figure is the line that repre-sents RRR (calc) = RRR (obs). Systematic deviations from this line indicate ranges in which the derived Eq. 1.1.3 is invalid. Figure 2.3.9 indicates that the fitted equation becomes progressi vely less valid below RRR values of 20. Four points in the region 200 to 1600 lie appreciably (20%) from the line. Dis-cussions with these authors suggest that large uncertainties were present in the p 0 measurements for these specimens. In addition, the data listed in Table 2.3.1 confirm the validity of using the Sommerfeld value of the Lorenz ratio for copper. 2.4 Summary for Copper Equation 1.1.3 represents the primary copper data to within +15% of the ex-perimental values. Deviations for unannealed specimens (i.e., those containing physical defects) are within 20%. Based on the deviations illustrated in Figs. 2.2.1 through 2.2.16 and the large changes that occur in thermal conductivity due to the introduction of chemical defects, physical defects, size effect, and magnetic fields, it is clear that a large proportion of these effects is reflected by the residual electrical resistivity. The incorporati on of RRR (or p 0 ) in Eq. 1.1.3 produces an equa-tion that represents the data for a wide range of copper specimens and environ-ments . Equation 1.1.3 with the parameters listed here was used to generate the^a, conductivity values for selected temperatures and values of RRR. These vain' are listed in Table 2.4.1 and plotted in Fig. 2.4.1. 23 List of Tables and Figures for Copper Tables Page Table 2.3.1 Comparison of Calculated and Observed RRR Values for Copper 27 Table 2.4.1 Thermal Conductivity Values for Copper Calculated From Eq. 1.1.3 at Selected Temperatures and RRR Values 28 Figures Page Figure 2.1.1 Experimental thermal conductivity data selected from the following primary references in the copper annotated bibliography: (4,7,8,14,16,18) 29 Figure 2.1.2 Experimental thermal conductivity data selected from the following primary references in the copper annotated bibliography: (19,20,21,23,24,27) 30 Figure 2.1.3 Experimental thermal conductivity data selected from the following primary references in the copper annotated bibliography: (26,27,28,29,30,33) 31 Figure 2.1.4 Experimental thermal conductivity data selected from the following primary references in the copper annotated bibliography: (35,38,41,43) 32 Figure 2.1.5 Experimental thermal conductivity data selected from the following primary reference in the copper annotated bibliography: (42) 33 Figure 2.1.6 Composite of the data in Figs. 2.1.1 through 2.1.5 .... 34 Figure 2.2.1 Thermal conductivity deviations of the copper data from the following primary references compared to Eq. 1.1.3: (4,7,8,14,16,18) 35 Figure 2.2.2 Thermal conductivity deviations of the copper data from the following primary references compared to Eq. 1.1.3: (19,20,21,23,24,27) 36 Figure 2.2.3 Thermal conductivity deviations of the copper data from the following primary references compared to Eq. 1.1.3: (26,27,28,29,30,33) 37 Figure 2.2.4 Thermal conductivity deviations of the copper data from the following primary references compared to Eq. 1.1.3: (35,38,41,43) 38 24 Page Figure 2.2.5 Thermal conductivity deviations of the copper data from the following primary reference compared to Eq. 1.1.3: (42) 39 Figure 2.2.6 -Composite of the deviations in Figs. 2.2.1 through 2.2.5 40 Figure 2.2.7 Thermal conductivity deviations of the copper data from the following secondary references compared to Eq. 1.1.3: (1,2, 3, 6, 9) 41 Figure 2.2.8 Thermal conductivity deviations of the copper data from the following secondary references compared to Eq. 1.1.3: (10,15,17,22,25) 42 Figure 2.2.9 Thermal conductivity deviations of the copper data from the following secondary references compared to Eq. 1.1.3: (26,31,32,33,37) 43 Figure 2.2.10 Thermal conductivity deviations of the copper data from the following secondary reference compared to Eq. 1.1.3: (37) 44 Figure 2.2.11 Thermal conductivity deviations of the copper data from the following secondary references compared to Eq. 1.1.3: (40,43,44) 45 Figure 2.2.12 Composite of the deviations in Figs. 2.2.7 through 2.2.11 46 Figure 2.2.13 Thermal conductivity deviations of the copper data from the following secondary references compared to Eq. 1.1.3: (5,7,12) 47 Figure 2.2.14 Thermal conductivity deviations of the copper data from the following secondary references compared to Eq. 1.1.3: (11,12,15,27,36) 48 Figure 2.2.15 Thermal conductivity deviations of the copper data from the following secondary references compared to Eq. 1.1.3: (36,44) 49 Fiqure 2.2.16 Composite of the deviations in Figs. 2.2.13 through 2.2.15 50 Figure 2.2.17 Comparison of Eq. 1.1.3 to the values recommended for copper in the following references: (13,39) 51 Figure 2.3.1 Experimental electrical resistivity data for copper selected from the following references in the copper annotated bibliography: (4,14,21,23,24,26,29) 52 25 Page Figure 2.3.2 Experimental electrical resistivity data for copper selected from the following references in the copper annotated bibliography: (33,38,43) .... 53 Figure 2.3.3 Composite of the electrical resistivity data in Figs. 2.3.1 and 2.3.2 54 Figure 2.3.4 Electrical resistivity deviations of the copper data from the following references compared to Eq. 1.2.3: (4,14,21,23,24,26,29) .... 55 Figure 2.3.5 Electrical resistivity deviations of the copper data from the following references compared to Eq. 1.2.3: (33,38,43) 56 Figure 2.3.6 Composite of the electrical resistivity deviations shown in Figs. 2.3.4 and 2.3.5 57 Figure 2.3.7 Electrical resistivity for copper as a function of temperature calculated from Eq. 1.2.3 at selected values of RRR 58 Figure 2.3.8 Lorenz ratio for copper as a function of temperature calculated from Eq. 1.2.3 and Eq. 1.1.3 at selected values of RRR 59 Figure 2.3.9 RRR values calculated as per Section 1.5, RRR(CALC) versus reported RRR values, RRR(OBS), for copper 60 Figure 2.4.1 Thermal conductivity for copper as a function of temperature calculated from Eq. 1.1.3 at selected values of RRR 61 26 14 16 18 23 24 26 33 33 42 43 43 1 9 10 15 15 15 26 33 36 36 36 37 37 37 37 37 37 37 37 37 37 37 Comparison of Calculated and Observed RRR Values for Copper. RRR (obs.) RRR (calc.) Primary Data 274.0 45.5 214.0 900.0 270.0 900.0 49.2 1530.0 1450.0 400.0 1780.0 19.5 38.8 271.0 44.3 216.0 900.0 190.0 900.0 51.5 1070.0 1060.0 306.0 1800.0 20.5 42.8 Secondary Data 29.0 2.5 140.0 1.6 1.6 1.6 115.0 350.0 870.0 1390.0 2680.0 1.9 2.1 2.5 2.5 3.2 3.1 3.9 3.6 3.8 4.2 4.5 38.0 3.4 120.0 2.0 1.9 1.6 102.0 102.0 880.0 1350.0 2630.0 2.6 3.0 3.1 3.3 4.0 4.4 4.5 4.6 5.0 5.5 5.9 27 Table 2.4.1. Thermal Conductivity Values for Copper Calculated from Eq. 1.1.3 at Selected Temperatures and RRR Values. X( Wm -1 •K -'' ) (K) RRR = 30 RRR = 100 RRR = 300 RRR = 1000 RRR = 3000 1 46 156 471 1574 4726 2 91 312 942 3147 9434 3 137 468 1413 4710 14044 4 183 624 1880 6243 18380 5 228 779 2343 7715 22170 6 274 933 2796 9075 25084 7 319 1085 3232 10260 26834 8 365 1235 3642 11197 27328 9 409 1380 4015 11836 26756 10 454 1520 4343 12172 25496 12 541 1778 4844 12127 22264 14 624 2002 5144 11544 19150 16 703 2186 5267 10725 16398 18 777 2324 5231 9771 13924 20 843 2408 5054 8727 11683 25 960 2381 4215 6135 7271 30 999 2119 3245 4151 4573 35 970 1784 2436 2859 3028 40 900 1467 1841 2047 2122 45 814 1205 1423 1531 1568 50 731 1002 1135 1196 1216 60 597 740 799 824 832 70 513 601 634 647 651 80 465 526 549 557 560 90 437 485 502 508 510 100 421 461 475 480 482 150 396 419 426 429 430 200 391 407 413 414 415 250 388 401 405 407 407 300 386 397 400 401 402 400 383 391 393 394 394 500 379 385 387 388 388 600 374 379 381 381 381 700 368 372 374 374 374 800 362 365 367 367 367 900 356 359 360 360 360 1000 350 352 353 353 354 1100 344 347 347 348 348 1200 339 341 342 342 342 1300 335 337 337 338 338 28 THERMRL CONDUCT I VITY,W/m. Figure 2.1.1 Experimental thermal conductivity data selected f rom the following primary ref erences In the copper annotated bibliography : (4,7,8,14,16,181 O-(43, A-(7), -17 ), V-(83, O-(143, +-(16), X-(18) 29 THERMRL CONDUCTIVITY, H/m. 0.3 1 3 10 30 100 300 1000 TEMPERATURE, K Figure 2.1.2 Experimental thermal conductivity data selected from the following primary references In the copper annotated bibliography : (19,20,21,23,24,27) O-(19), A-(20), -(21), V-(23), O-(24), +-(27), X-(27) 30 THERMRL CONDUCTIVITY, W/m. Figure 2.1.3 Experimental thermal conductivity data selected f rom the following primary references In the copper annotated bibliography : (26,27,28,23,30,33) O-(27), A-(28), -(26), V-(29), O-(30), +-(33), X-(33) 31 THERMRL CONDUCTIVITY, W/m. Figure 2.1.4 Experimental thermal conductivity data selected from the following primary references In the copper annotated bibliography: (35,38,41,431 O-(351, A-(38), -(41), V-(41), O-(41), +-(43), X-(43) 32 THERMRL CONDUCTIVITY,W/m. 0.3 1 3 10 30 100 300 1000 TEMPERATURE, K Figure 2.1.5 Experimental thermal conductivity data selected from the following primary reference In the copper annotated bibliography: (42) THERMAL CONDUCTIVITY,W/m. 0.3 1 3 10 30 100 300 1000 TEMPERRTURE,K Figure 2.1.6 Composite of the data In figs. 2.1.1 through 2.1.5 Figure 2.2.1 Thermal conductivity deviations of the copper data ' the following primary references compared to eq. (4,7,8,14, 16,18) O-(4), A-(7), -(7), V-(8), O-(14), +-(16), X-(18) 35 0.3 1 3 10 30 100 300 1000 TEMPERRTURE,K Figure 2.2.2 Thermal conductivity deviations of the copper data from the following primary references compared to eq. (1.1.3): (19,20,21,23,24,27) O -(19), A-(20), -(21), V” (23), O-(24), + -(27), X-(27) 36 Figure 2.2.3 Thermal conductivity deviations of the copper data from the following primary references compared to eq. (1.1.3): (26,27,28,29,30,33) O-(27), A-(28), -(26), V-(29), O-(30), +-(33), X-(33) 37 Figure 2.2.4 Thermal conductivity deviations of the copper data from the following primary ref erences compared to eq. (1.1.3): (35,38,41 ,43) O-(35), A-(38), “ (41), V-(41), O-(41), +-(43), X-(43) 38 0.3 1 3 10 30 100 300 1000 TEMPERATURE, K Figure 2.2.5 Thermal conductivity deviations of the copper data from the following primary reference compared to eq. (1.1.3): (42) O-(42) 39 0.3 3 10 30 100 300 1000 TEMPERATURE, K Figure 2.2.6 Composite of the deviations In figs. 2.2.1 through 2.2.5 40 Figure 2.2.7 Thermal conductivity deviations of the copper data from the following secondaru references compared to eq. (1.1.3):(1,2,3,6,9) O-[1), A -(2), -(3), V-(63, O-(63 , +-(63 , X-(9) 41 0.3 1 3 10 30 100 300 TEMPERATURE, K figure 2.2.8 Thermal conductivity deviations of the copper data from the following secondary references compared to eg. (1. 1. 3) : (10,15,17,22,25) O -(101, A-(15), -(15), V-(17), O-(22), +-(22), X-(25) 42 Figure 2.2.9 Thermal conductivity deviations of the copper data from the following secondcry references compared to eq. ( 1 . 1 .3) : (26,31 ,32,33,37) O-(26), A-(31), -(32), V-(33), O-(37), +-(37), X-(37) 43 Figure 2.2.10 Thermal conductivity deviations of the copper data from the following secondary reference compared to eq. (1.1.3) s C37) O- (37), A-(37), -(37), V-(37), O- (37), +-(37), X-(37) 44 Figure 2.2.11 Thermal conductivity deviations of the copper data from the following secondary references compared to eq. (1.1.3) : (40,43,44) O-(40), A-(40), -(43), V-(44), O-(44) 45 Figure 2.2.12 Composite of the deviations In figs. 2. 2. 7 through 2.2.11 46 100 f ! r i i ' i i i i | i i i i m 1 i I r 50 -O J CC 0 1 CO CD O 0 0 0 00 -50 --K>ocP “7&-QJ O E-h O Q -100 --150 --200 0.2 0.4 2 4 10 20 TEMPERRTURE,K 40 Figure 2.2.13 Thermal conductivity deviations of the copper data from the following secondary references compared to eq. (1.1.3) : (5,7,12) O-(5), A-(5), -(5), V-(5), O-(7), +-(12), X-(12) 47 PCT. DEV. (OBS.-CRLC. 0.2 0.4 1 2 4 10 20 40 TEMPERRTURE,K Figure 2.2.14 Thermal conductivity deviations of the copper data from the following secondary references compared to eq. (1.1.3) : (11,12,15,27,36) O-(12), A-(12), - (ID, V-(11), O-(15), +-(27), X-(36) 48 Figure 2.2.15 Thermal conductivity deviations of the copper data from the following secondary references compared to eq. (1.1.3): (36,441 O-(36), A-(36), -(44) 49 PCI . DEV. (OBS.-CRLC. Figure 2.2. 16 Composite of the deviations In figs. 2.2.13 through 2.2.15 50 PCT. DEV. (OBS.-CALC. Figure 2.2.17 Comparison of eq. (1.1.31 to the values recommended for copper In the following references :( 13,39 ) O- (39), A-(13) 51 ELECTRICAL RESISTIVITY, nil . TEMPERATURE, K Figure 2.3.1 Experimental electrical resistivity data for copper selected from the following references In the copper annotated bibliography: (4,14,21,23,24,26,291 O-(41, A-(141, O-(241, +-(261, -(21 1 , V-(231, X-(291 52 ELECTRICAL RESISTIVITY, nfi. I ' I 1 "T 1 'I I T T rrrf -30 10 V o V V V V VVV v V W o o oo ooooooo^ 0.3 -0.1 -A O A A 0.03 -0.01 ® 1 |Q| 1 1 1 1 1 1 i i i . i 10 20 40 100 TEMPERATURE, K 200 Figure 2.3.2 Experimental electrical resistivity data for copper selected from the following references In the copper annotated bibliography: (33,38,43) O-(33), A-(33), -(38), V-(43), O-(43) xUJl.1 I I 400 53 ELECTRICAL RESISTI VITY,nC . Figure 2.3.3 Composite of the electrical resistivity data In figs. 2.3.1 and 2.3.2 54 1 2 4 10 20 40 100 200 400 TEMPERRTURE,K Figure 2.3.4 Electrical resistivity deviations of the copper data from the following references compared to eq . (1.2.3) :(4, 14, 21, 23, 24, 26 O-14), A-(14), -(21) O-(24), + -(26), X-(29) 29) V-(23) / 55 1 2 4 10 20 40 100 200 400 temperature:, k figure 2.3.5 Electrical resistivity deviations of the copper data from the following references compared to eq. (1.2.3) : (33,38,43) O-(33), A-(33), -(38), V-(43), O-(43) 56 1 2 4 10 20 40 100 200 400 TEMPERRTURC^K Figure 2.3.6 Composite of the electrical resistivity deviations shown In figs. 2.3.4 and 2.3.5 57 ELECTRICRL RESISTIVITY, nfi. TEMPERRTURE,K Figure 2.3.7 Electrical resistivity for copper as a function of temperature calculated from eq. (1.2.3) at selected values of RRR. 58 LORENZ RATIO, 10 V /K TEMPERATURE, K Figure 2.3.8 Lorenz ratio for copper as a function of temperature calculated from eq. (1.2.3) and eq. (1.1.3) at selected values of RRR. 59 RRR(CflLC) Figure 2.3.9 RRR values calculated as per Section 1.5, RRR(CflLC), versus reported RRR values, RRR(OBS), for copper. O “ Primary, A" Secondary, ~ Secondary 60 THERMAL CONDUCTIVITY, W/m. TEMPERATURE, K Figure 2.4.1 Thermal conductivity for copper as a function of temperature calculated from eq. (1.1.3) at selected values of RRR. 61 V 2.5 FORMAT FOR ANNOTATED BIBLIOGRAPHY OF COPPER REFERENCE AUTHOR, TITLE, CITATION ANNOTATION PURPOSE SPECIMEN a) Dimensions/Shape; b) Crystal Status; c) Thermal ,/Mech. History; d) Purity Specification; e) RRR; f) p 0 ; g) Other Characteri zation Data APPARATUS a) Type; b) Thermometry/Cal ibration/Anchoring; c) Thermal Isolation; d) Other (Q meas. ) DATA a) Temperature Range/Difference; b) Content of Tables, Figures and Equations/Data Extraction; c) Uncertainty/ Impreci sion ; d) Disputable Corrections to Measurements by Authors; e) Errata (by Author or Reviewer) ANALYSIS a) Comparisons; b) Conclusions 1 ty 62 Allen, J. F. and Mendoza, E. , Thermal Conductivity of Copper and German Silver at Liquid Helium Temperatures, Proc. Cambridge Philos. Soc., 4j4, 280-3 (1948) PURPOSE To describe a longitudinal apparatus and perform x measurements on copper and German silver. SPECIMEN a) /rod; c) annealed in air/machined before and after annealing; d) 0.003% Ag, 0.003% Ni , 0.003% Pb, 02 free; f) po = 5.5 x 10“ 8 qcrn; g) source: Johnson, Matthey and Co., No. 1562. APPARATUS a) longitudinal; b) phosphor-bronze thermometers/cal ibrated in terms of vapor pressure of He bath using 1932 scale/thermometer leads thermally grounded to can; c) radiation shield, vacuum insulation. DATA a) 1.8 to 4.1 K/0.01 K maximum; b) figure 2 -X; c) uncertainty: ±2%. ANALYSIS b) in agreement with Makinson's theory. Andrews, F. A., Webber, R. T., and Spohr, D. A., Thermal Conductivity of Copper, Aluminum and Tin at Helium Temperatures , ASTI A AD No. 147716; Contract No. AF 33(616) -56-8 PURPOSE To measure X for Cu, A1 , Sn and for Sn in low magnetic field, and determine an emperical relationship. SPECIMEN a) /rod; b) polycrystal 1 i ne ; d) 99.998% Cu; g) source: Johnson, Matthey and Co. APPARATUS a ) longitudinal ; b) gas thermometers; c) radiation shield, vacuum insul ation. DATA a) 2.5 to 4.6 K; b) figure 2 -X; c) uncertainty: ±4% at T < 4.5 K, > 4% at T > 4.5 K; d) corrections for second virial coefficient and oil volume change for gas thermometry. ANALYSIS b) results in agreement with Allen, J. F. and Mendoza, E. , Proc. Cambridge Philos. Soc. , 44, 280 (1948). 63 Balcerek, K. , Lipinski, L. , Mucha, J., Rafalowicz, J., Wlosewicz, D., Grosse, G. and Hegenbarth, E., Thermal Conductivity of Copper in Temperature Range 15 to 60 K, Acta Phys. Pol. A, 4-9, 417-20 (1976) PURPOSE To verify the CINDAS X equation for pure polycrystal 1 ine Cu. SPECIMEN a) 3.05 mm dia./rod; b) polycrystal 1 ine; c) /mechanically worked; d) 99.999% Al; g) 6 = 1.33 cm-K ? -W-l. APPARATUS a) longitudinal; b) Cu-Au + 0.03 at. % Fe low temperature thermocouples of alloy and Cu wi res//thermocoupl es glued to specimen surface. DATA a) 14.5 to 58 K; b) figure 1 - x/data points taken from private communication with Rafalowicz, J.; c) uncertainty: larger than ±10%. ANALYSIS b) Ke = (a'Tn + b/T)" 1 , n = 2.21, m = 2.63, B = 0.0237 cm.K 2 .W"l, a" = 0.0423 x 10~ 4 cm-K( n_m-mn+ ^ )/( n+ l ) . W“1 ; CINDAS equation verified in the range of maximum X. Berman, R. , MacDonald, D. K. C. , The Thermal and Electrical Conductivity of Copper at Low Temperatures, Proc. Roy. Soc. London, Ser. A, 211, 122-8 (1952) PURPOSE To measure x and p at low temperatures for Cu. SPECIMEN a) 1.12 mm, 0.45 mm dia./rods; c) drawn, annealed 6 h in He at 450 °C; d) 0.0005% Ag, < 0.0003% Ni , < 0.0004% Pb; f) P4.2 = 5.5 nfi-cm; g) source: Johnson, Matthey and Co., No. 4234. APPARATUS a) longitudinal; b) gas thermometers ; c) radiation shield. DATA a) 3 to 90 K; b) figure 1 -X, figure 2 -p; c) uncertainty: ±2%; d) data corrected for conduction through glass thermometers. ANALYSIS b) W(cm K W-l) = 0.212/T + 2.55 x 10- 5 T2 for 12 < T < 30 K, 1/p (10“ 6 n-cm) = 1/189.6 + 2.64 x 10“1°T 5 for 30 > T > 12; the Weidemann Franz Law is not strictly obeyed at temperatures < 4 K, minimum of X given in Makinson's theory not found. 64 Bowman, H. F. , Ziebold, T. D. , Smith, J. L. , Jr., Low Temperature Thermal Conductivity of Copper Irradiated with Reactor Neutrons, Proceedings of the Eighth Thermal Conductivity Conf . , 175-84 (Oct 7-10, 1968) PURPOSE To measure x for radiated and irradiated Cu samples. SPECIMEN a) 2.00 in. (5.08 cm) long, 0.120 in. (0.305 cm) square/bar; d) 99.999 + % Cu. APPARATUS a ) longitudinal ; b) precision Ge thermometers/instrument leads thermally anchored to reference heat sink, Ge sensors anchored to pick-up shoes. DATA a) 5 to 100 K/0.4 K at 5 K, 4 K at 100 K; b) figure 4 -X; c) uncertainty: ±3%. ANALYSIS b) the influence of each neutron collision is roughly 10 to 30 times greater than the influence of each impurity atom. Davey, G. and Mendelssohn, K., Heat Conductivity of Pure Metals Below 1 K, Phys. Lett., 1(3), 183-4 (1963) PURPOSE To measure X for Cu, Au, Ti , Fe below 1 K. SPECIMEN a) /wire; b) polycrystalline; c) Cu 3 annealed for 3 h at 625 °C; d) Cu 1: commercial wire, Cu 2, Cu 3: 99.999% Cu. APPARATUS a) not given. DATA a) 0.4 to 1.0 K; b) figure 1 -X. ANALYSIS a) Phillips, N. , Phys. Rev., 100, 1719 (1955), Jericho, M. H. , Thesis, Cambridge (1963); b) anomalous deviations from T 1 for x. 65 T7] Dupre, A., Van Itterbeek, A., and Michiels, L. , Heat Conductivity of Copper Below 1 K, Phys. Lett., 8(2), 99-100 (1964) PURPOSE To measure X of Cu samples from 0.2 K to 0.7 K. SPECIMEN a) 3 mm dia./rods; d) Cu 1: <10 ppm; Cu 3: 1 ppm Co, 2 ppm Fe, 0.5 ppm Mn, 1 ppm Ni , 2 ppm Si; Cu 5: 97 ppm Co, 110 ppm Fe, 130 ppm Mn , 140 ppm Ni , 160 ppm Si; g) sources: Cu 1: Metallurgical Society of Hoboken (Belgium); Cu 3: Johnson, Matthey and Co., CB 8; Cu 5: Johnson, Matthey and Co. CB 2; Cu 1 : 3 = 0.075 cm-K^W" 1 ; Cu 3 : 3 = 1.73 cm.K 2W"l; Cu 5: 3 = 17.3 cm K 2W~1 . APPARATUS a) not given. DATA a) 0.2 to 0.7 K; b) figure 1 -x(Cu 1), figure 2 - (Cu 3, Cu 5). ANALYSIS a) Davey, G. and Mendelssohn, K., Phys. Lett., 7> 183 (1963); b) results verify x = T/3; no anomalies found as in Davey, G. and Mendelssohn, K. Fletcher, R. , The Nernst-Etti nghausen Coefficient and the Kondo Effect in Copper and Gold, Philos. Mag., 25(1), 87-95 (1972) PURPOSE To measure the isothermal Nernst-Etti nghausen coefficient over a wide temperature range. SPECIMEN a) cut from 7 x 5 x 0.025 cm/plate; c) annealed for 6 hr. at 950 °C; d) spectroscopical ly pure; e) RRR = 45.5; g) source: Johnson, Matthey and Co. APPARATUS a ) 1 ongitudi nal ; d) Q 1 measured. DATA a) 5 to 42 K; b) figure 2 -X (zero field). 66 Garber, M. , Scott, B. W. , and Blatt, F. J., Thermal Conductivity of Dilute Copper Alloys, Phys. Rev., 130 (6) , 2188-92 (1963) PURPOSE To measure x of Cu and find its lattice component. SPECIMEN a) 0.25 cm swaged; d) for di a. , 5-10 cm specimen 14 = specimen supplied by long/rod; c) annealed at 0.23% Cd, 0.76% In; f) American Smelting and 600 °C for 3 h, then = 1.06 uft’crn; g) Cu Co. Re?ini ng APPARATUS a ) 1 ongi tudi nal ; b) 2 Au 2.1% Co vs. Ag 37% Au thermocoupl es/cal ibrated against Pt resistance thermometer/sol dered to small Cu fittings placed along specimen ci rcumference. DATA a) 12.5 to 93 K; b) figure 2 -X (total).. ANALYSIS b) below 20 K, the lattice component is reduced relative to the residual electrical resistivity, in agreement with Pippard's theory. Gladun, C. H. R. and Holzhauser, W. , Work on Thermal Conductivity at Low Temperatures , Monatsber. Dtsch. Akad. Wiss. Berl in , 6(1 ) , 311-3 (1961) PURPOSE To measure X and p of metal specimens from 17 to 100 K. SPECIMEN a) rod; d) electrolytic Cu; f) pn = 1.1 x 10" 8 f2 cm; g) mechanical deformation ranges from 0 to 30.8%. APPARATUS a) longitudi nal ; b) Pb resistance thermometers ; c) radiation shield, vacuum i nsul ation. DATA a) 19 to 83 K; b) figure 1 -X (no deformation) . ANALYSIS b) measurements show a strong lowering of the maximum x and a correspond-ing increase in residual resistance for increasing deformation. 67 Groger, V. and Stangler, F. , Der Warmewiderstand infolge Elektron -Phonon-streuung in Ideal -Rei nem Kupfer, Acta Phys. Austriaca, 40, 145-9 (1974) PURPOSE To determine W-j of high purity Cu due to electron-phonon interactions. SPECIMEN a) specimen 1: 12.5 ym thick, specimen 2: 125. ym thick/foil; c) annealed for 4 h at 950 °C; d) specimen 1=3 ppm Fe, specimen 2 = 99.999% Cu. APPARATUS a) longitudinal; b) carbon resistance thermometer. DATA a) 8 to 37 K; b) figure 1 -X; c) uncertainty: less than ±5%. ANALYSIS b) Wi = aT 2 - 98 . Groger, V. and Stangler, F. , Verbesserung der Warmelei tfahigkeit von Reinstkupfer durch G1 uhbehandl ung unter geringem sauerstoffparti aldruck, Z. Metal Ikd. , 65(5), 333-6 (1974) PURPOSE To investigate the effect of different annealing conditions on x of high purity Cu. SPECIMEN a) specimen 1: 12.5 ym thick, specimen 2: 0.125 ym thick/foil; c) three states of anneal for each specimen: O2 anneal , high vacuum anneal, unan-nealed; d) ultrahigh purity (see Table 3). APPARATUS a) longitudinal ; b) carbon resistance thermometers. DATA a) 5 to 70 K; b) figure 1 -X (specimen 1), figure 2 -X (specimen 2); c) uncertainty: less than ±5%; d) size effect, lattice conductivity correc-tions. ANALYSIS a) Anderson, H. H. and Nielsen, M. , Riso, Rep. No. 77; b) Lorenz number 15% higher after high vacuum anneal than Lo for small Fe content; for large Fe content, L is 22% higher than Lo. 68 Ho, C. Y., Powell, R. W. , Li ley, P. E. , Thermal Conductivity of the Ele-ments: A Comprehensive Review, J. Phys. Chem. Ref. Data, 3 , Supplement No. 1, 342-257 (1974) PURPOSE To provide a comprehensive listing of data on x of the elements. SPECIMEN d) high purity; f) p 0 = 5.79 x 10 “--^ ^’cm for T below 100 K. APPARATUS a) not given. DATA a) 0 to 8500 K; b) Table 48 -X, recommended values; c) uncertainty: ±2% near room temperatue, ±4% at low and high temperatures, and ±15% for the molten phase up to 2000 K. The values above 2000 K are provisional. Hust, J. G. and Giarratano, P. J., Semi-Annual Report on Materials Research in Support of Superconducting Machinery: Thermal Conductivity, NB SIR 74-393, 1-35 (1974) PURPOSE To measure x for materials for use in superconducting machinery. SPECIMEN a) 23 cm long, 3.26 mm dia./rod; c) swaged, vacuum annealed at 650 °C for 1 h; e) RRR = 14; g) 0FHC. APPARATUS a) longitudinal ; b) thermocouples/cal ibrated by Pt + Ge resistance thermometers; c) sample surrounded by insulation and radiation shield. DATA a) 8 to 300 K; b) Table 2 -X, p, L; c) uncertainty in X: from 4 to 20 K: ±2%; from 20 to 200 K: ±1%; from 200 to 300: ±2%; uncertainty in p: ±0 . 2%. ANALYSIS a) data within 10% of predicted values found in previous report ( NBSIR 74-359). 59 Kemp, W. R. G. , Klemens, P. G. , and Tainsh, R. J., The Thermal Conductivity of Copper Alloys: Effect of Plastic Deformation and Annealing, Philos. Mag., 4(43), 845-57 (1959) PURPOSE To study the lattice component of a as a function of torsional deformation. SPECIMEN a) 8 cm long, 3.25 mm dia./rods; c) drawn, annealed at 450 °C; No. 0: control specimen, No. 1: torsionally deformed to 1.29 with no anneal. No. 4: No. 1 annealed at 450 °C in He; d) 99.55% Cu, 0.35% As, 0.05% P; f) No. 0 (p 0 = 2.71 yQ • cm ) , No. 1 (p 0 = 2.79 yftcm), No. 4 (p 0 = 2.77 yftcm). APPARATUS a) longitudinal; b) He thermometers ; c) radiation shield, insulation. DATA a) 7 to 92 K; b) figure 1 -X, Table 1 -p0 . ANALYSIS b) the observed change in Ag due to removal of dislocations during recrystal 1 i zation. Laubitz, M. J., Transport Properties of Pure Metals at High Temperatures, Can. J. Phys., 45, 3677-96 (1967) PURPOSE To report measurements of high-temperature properties of monovalent metal s. SPECIMEN a) 1.9 cm dia., 20 cm long/rod; c) annealed for 2 h at 700 K/machined from 1 in. dia. x 15 in. bar; d) 99.999 + % Cu; e) RRR = 900. APPARATUS a) modified Forbes Bar method; b) Pt/Pt-10% Rh thermocouples/calibrated at NRC/peened into specimen; c) several radiation shields, argon atmosphere of 10-25 cm Hg. DATA a ) 300 to 1200 K; b) equation 4 -A; c) uncertainty: ±0.9% at 300 K to ±1.5% at 1200 K. ANALYSIS a) Schofield, F. H. , Proc. Roy. Soc. London, Ser. A, 107 , 206 (1925), Midryukov, V. E., Vestn. Mosk. Univ., Ser. Mat., Mekh., Astron., Fiz., Khim., 12, 57 (1957); b) A = 4.182(7) -0.631(6) x 10“ 3 T; results agree with Mikryukov to within combined experimental error. 70 17] Lees, C. H. , the Effects of Temperature and Pressure on the Thermal Conduc-tivities of Solids -Part II: The Effects of Low Temperatures on the Thermal and vlectrical Conductivities of Certain Approximately Pure Metals and Alloys, Philos. Trans. Roy. Soc. London, Ser. A, 208, 381-443 (1908) PURPOSE To measure x and p of several materials from 107 to 299 K. SPECIMEN a) 7 cm long, 0.585 cm dia./rod; c) /soft-drawn, turned. APPARATUS a) longitudinal ; b) Pt resistance thermometer. DATA a ) 107 to 299 K; b) Table (pp. 406-7) -X, Table (p. 432) -p; c) uncer-tainty: less than ±5%; d) corrections for heat leaks. ANALYSIS b) x increases with increasing temperature. Lindenfeld, P., Lynton, E. A., and Soulen, R., Metallic Heat Conductivity Below 1 K, Phys. Lett., 19(4), 265 (1965) PURPOSE To measure X of Cu below 1 K. SPECIMEN a) 0.05 cm thick/foil; c) annealed for 3 h at 530 °C; e) RRR = 270; g) elec-trolytic tough pitch from Chase Brass and Copper Co. APPARATUS a) longitudi nal ; b) carbon resi stors/cal ibrated against the vapor pressure (above 1.2 K), susceptibility of ruby probe, germanium resistance thermom-eters (below 1.2 K). DATA a ) 0.4 to 1.5 K; b) figure 1 -X/T vs. T; c) uncertainty: less than ±14. ANALYSIS b) no anomaly found as in Davey, G. and Mendelssohn, K. , Phys. Lett., 7 _ (1963). 71 Lucks, C. F. and Deem, H. W. , Thermal Properties of Thirteen Metals, Am. Soc. Test. Mater., Spec. Tech. Publ . , No. 227, 1-29 (1958) PURPOSE To report high temperature data on several metals. SPECIMEN a) 3/4 in. dia. x 6 in. /rod; c) cold drawn/electrolytic tough pitch; g) fed-eral specification QQ-C-502; source: Revere Copper and Brass, Inc. APPARATUS a) longitudinal; b) specimen soldered to heater block, Fe standard; c) guard tubes, insulation. DATA a) 366 to 1255 K; b) Table VI -X. ANALYSIS b) Smith, C. S. and Palmer, E. W. , Am. Inst. Min. Metal 1. Pet. Eng. , 124 (1937); Schofield, F. H., Proc. Roy. Soc. London, Ser. A, 107 (1925). Mendelssohn, K. and Rosenberg, H. M. , The Thermal Conductivity of Metals at Low Temperatures. I. The Elements of Groups 1, 2, and 3, Proc. Phys. Soc., London, Sect. A, 6_5, 385 (1952); see Rosenberg, H. M. , The Thermal Conductivity of Metals at Low Temperatures , Philos. Trans. Roy. Soc. London, 247 , 441-497 (Mar 1955) Mikryukov, V. E. , Vestn. Mosk. Univ., Ser. Mat., Mekh., Astron., Fiz., Khim. , jl(2) , 53-70 (1956) PURPOSE To measure X and p of Cu. SPECIMEN b) polycrystal 1 i ne ; d) 99.99% Cu. APPARATUS a) not given. DATA a) 320 to 773 K; b) Table 1 -X, p; c) uncertainty: ±3%. 72 Misiorek, H. , Zakrzewski , T. , and Rafalowicz, J., Influence of Plastic Deformation on the Thermal Conductivity Maximum of Copper and Aluminum in the Temperature Range 4.2 to 70 K, Phys. Status Sol i d i A, _47, K137 (1978) PURPOSE To determine the effect of increasing deformation on the thermal conductivity maximum. SPECIMEN a) /rod; c) deformed, undeformed specimens; d) 99.9%, 99.99% Cu. APPARATUS a) longitudinal. DATA a) 15 to 65 K (undeformed); b) figures 3,4; e) captions for all figures switched: figure 1 should be labeled as 4, figure 2 as 3, figure 3 as 1, figure 4 as 2. ANALYSIS b) with increasing deformation, the maximum decreases in value and is shifted toward higher temperatures. Moore, J. P. , McElroy, D. L. , and Graves, R. S. , Thermal Conductivity and Electrical Resistivity of High-Purity Copper from 78 to 400 K, Can. J. Phys., 45, 3849-65 (1967) PURPOSE To compare measurements of a guarded longitudinal technique with other data. SPECIMEN a) /rods; b) polycrystal 1 ine ; d) 99.999% Cu; e) RRR = 900; g) average grain size: 574 ym, hardness of 40 (DPH) with 0.1 kg load. APPARATUS a) longitudinal ; b) Chromel-P, constantan thermocouples/cal ibrated spool s/electrical ly insulating epoxy; c) Au plated radiation shield, spun A1 2 O3 placed between polished specimen and shield. DATA a) 85 to 400 K; b) Table 2 -X, p (both smoothed); c) uncertainty: ± 1 . 86 %; d) x, p uncorrected for thermal expansion. ANALYSIS a) Laubitz, M. J., Can. J. Phys., 45^, 3677 (1967); b) x(W cm-ik" 1 ) = 4.1631 -5.904 x 10 ' 4 T + 7.0872 x 10 5 T" 3 300 < T < 1200. 73 Nelson, W. E. and Hoffman, A. R. , Measurements of Temperatures and Mag-netic Field Dependence of Electrical Resistivity and Thermal Conductivity in QFHC Copper, Proceedings of the Fourteenth Thermal Conductivity Conf., 73-80 (Jun 2-4, 1975) PURPOSE To measure X and p on Cu in a longitudinal magnetic field. SPECIMEN a) 2.5 cm long; c) soldered at T = 160 °C/machined from 0.5 in. rod; d) commercial grade OFHC, 3 ppm P, 1 ppm Zn , 10 ppm each Pb, 0, Te, 10 ppm total As, Bi , Mn, Sb, Sn; f) (p 0 = 3.15 x 10~8 ft. cm); g) source: Admiral Brass and Copper Co., Alloy 101. APPARATUS a) longitudinal; b) carbon resistor thermocouples/cal ibrated against com-mercial Ge resistance thermometer and fit to a modified resistance vs. temperature expression (Clemment, J. R. and Quinnell, E. H. , Rev. Sci . Instr. , 24 (1952) 213). DATA a) 4.05 to 34.9 K; b) Table 1 -X, p, L; c) uncertainty in X: ±5%, uncer-tainty in p: ±1.5%. ANALYSIS b) X- 1 (0 field) = aT 2 + eT” 1 , p (0 field) = a + bT 4 - 42 . Nicol, J. and Tseng, T. P., Thermal Conductivity of Copper between 0.25 and 4.2 K, Phys. Rev., 92(4), 1062-3 (Nov 15, 1953) PURPOSE To measure low temperature x for Cu. SPECIMEN a) 27.2 cm long, 0.025 cm dia./wi re; b) polycrystal 1 ine; d) commercial grade high-purity magnet wire; g) source: General Electric Co. APPARATUS a) longitudinal; b) T < 1 K: Cr-K-al urn paramagnetic thermometry, ac bridge. T > 1 K: carbon thermometers//T < 1 K: contact between salts and mounting by high pressure salt molding. DATA a) 0.25 to 4.2 K; b) figure 1 -X. ANALYSIS b) x = aT: a(W-cm”l K“ 2 ) = 1.76; the observed linear dependence of X on temperature indicates X is electronic in character. 74 Powell, R. L. , Roder, H. M. , and Hall, W. J., Low-Temperature Transport Properties of Copper and Its Dilute Alloys: Pure Copper, Annealed and Cold-Drawn, Phys. Rev., 115 (2) , 314-23 (Jul 15, 1959) PURPOSE To measure x, p and therinoel ectric power of two high purity Cu specimens. SPECIMEN a) specimen 1: 0.07 in. (0.18 cm) dia., specimen 2: 0.0816 in. (0.207 cm) dia./rod; c) specimens 1,2 swaged and drawn from 3/8 in. (0.95 cm) rod/specimens 1,2: annealed in vacuum at 400 °C, 2 h, after swaging and before drawing. Specimen 1 annealed at 400 °C, 2 h, after drawing; d) specimens 1,2: 99.999% Cu; e) specimen 1 (RRR = 1530), specimen 2 (RRR = 115); f) specimen 1 (p 0 = 1.01 x 10" 9 ftcm), specimen 2 (p 0 = 1.30 x 10"° ftcm); g) source: CRL of American Smelting and Refining Co. APPARATUS a) longitudinal ; b) Au-Co vs. Cu thermocouples, Pt resistance thermometer. DATA a) 4 to 105 K; b) Table 1 -X, Table 2 -p n and RRR, figure 4 -p, figure 8 -L. ANALYSIS a) experimental results agree qualitatively with theory and previous work; b) observed p deviation from Matthiessen 's rule; L curve flattens out considerably below Sommerfeld valve when extrapolated to 0 K. Powell, R. L., Roder, H. M., and Rogers, W. M., Low Temperature Thermal Conductivity of Some Commercial Copper, J. Appl . Phys., 28(11), 1282-8 (Nov 1957) — PURPOSE To measure X for some commercial coppers. SPECIMEN a) all specimens 0.144 in. (0.366 cm) dia./rod: c) all specimens ground from 0.25 in. (0.64 cm) to 0.144 in. (0.366 cm); d) coalesced: 99.98% Cu; Electrolytic Tough Pitch: 99.98% Cu; Tellurium Cu: 99.4% Cu; Phosphorus Deoxidized Cu: 99.62% Cu; g) more detailed characteri zation given in Table 1. APPARATUS a) longitudinal ; b) 8 Au-Co vs. Cu thermocouples/cal ibrated from 4 to 300 K in separate apparatus; c) temperature controlled radiation shield and vacuum insulation. DATA a ) 5 to 37 K, 4 to 33 K, 6 to 84 K, 5 to 80 K for coalesced, electrolytic , free cutting, phosphorized respecti vely/1 to 60 K; b) figure 2 -X. ANALYSIS b') K/ r = (1/6) + (T/D), B(cm.K2. w-l) = 54(±2), 190(±10), D(cm.|<3.w"l) = 7(±1) x 10°, 2.2(±0.5) x 10^ for phosphorized and free cutting respectively; X of work hardened Cu-alloys is predominantly electronic thermal conduction as limited by the imperfection scattering. 75 Powell, R. L. , Rogers, W. M. , and Coffin, D. 0., An Apparatus for Measure-ment of Thermal Conductivity of Solids at Low Temperatures , J. Res. Nat. Bur. Stand., 59(5), 349-55 (Nov 1957) PURPOSE To describe an apparatus for X measurements of solids between 4 and 300 K. SPECIMEN a) 23.2 cm long, 0.367 cm dia./rod; c) annealed in He for 4 h, 400 °C, cooled to 200 °C for 8 h; d) 13 ppm 0o> 8 ppm Pb, 7 ppm Ni , Fe < 5 ppm; g) density = 8.90 g/cnP, hardness on Vickers diamond point with 10 kg weight: 54.1 -longitudinal, 48.8 - transverse, source: Phelps Dodge Copper Products Corp. APPARATUS a) longitudinal; b) Au-Co vs. Cu thermocouples, Pt resistance reference thermometer/Pt resistance thermometer calibrated down to 12 K by Tempera-ture Measurements Section of NBS; c) vacuum insulation. DATA a) 9 to 142 K; b) figure 8 -X; c) uncertainty: ±5% (maximum). Powell, R. W. and Tye, R. P., New Measurements on Thermal Conductivity Reference Materials, Internat. J. Heat and Mass Transfer , H)(5) , 581-95 (1967) PURPOSE To provide information on x and p of possible reference materials. SPECIMEN a) 15 cm long, 7 mm dia./rod; c) heat treated to 900 °C; d) 0FHC; g) source: Johnson, Matthey and Co. No. 4351. APPARATUS a) longitudinal . DATA a) 293 to 1173 K; b) Table 1 -X; c) uncertainty: not given. ANALYSIS a) see Table 2 for data comparison at 50 °C; b) present data agree well with the two highest values of Kannuluik, W. G. and Laby, T. H., Proc. Roy. Soc., A121 , 640 (1928); Mikryukov, V. E. , Mosk. Gos. Univ., Ser. Mat., Mekh. Astron. Fiz. i Khim. _U(2), 53 (1956). 76 Powers, R. W. , Schwartz, D. , and Johnston, H. L. , The Thermal Conductivity of Metals and Alloys at Low Temperatures. I. Apparatus for Measurements Between 2 5 and 300 K, USAF TR-264-5, Cryogenic Lab., Dept, of Chem. , O.S.U., Columbus, Ohio (1950) PURPOSE To describe an apparatus for measuring x for metals from 25 to 300 K in small temperature increments. SPECIMEN a) 20 in. (51 cm) long, 0.5 in. (1.3 cm) dia./rod; d) 0FHC; g) source: American Brass Co. APPARATUS a ) longitudi nal ; b) Cu-constantan thermocoules/cal ibrated against a He gas thermometer; c) 3 monel radiation shields. DATA a) 23 to 245 K; b) Table 4 -X; c) uncertainty: ±0.7% from 100 to 250 K, ±1.9% at 30 K, Quick, R. W. , Child, C. D. , and Lanphear, B. S. , Thermal Conductivity of Copper. II. Conductivity at Low Temperatures, Phys. Rev., 3(1), 1-20 (1895) PURPOSE To extend observations of X from 219 to 260 K. SPECIMEN d) electrolytic Cu. APPARATUS a) Forbes bar method. DATA aT~219 to 260 K; b) Table 10 -X. ANALYSIS b) An increase in T corresponds to an increase in x over the temperature range measured. 77 Rhodes, B. L. , Moeller, C. E. , and Saver, H. J., An Apparatus for Determining Thermal Conductivity of Solids From 20 to 600 K, Cryogenics, 5(1), 17-20 (1965) PURPOSE To discuss a X apparatus with which a large temperature gradient is not required over the specimen. SPECIMEN a) 10 cm long, 6 mm dia./rod; d) 99.999% Cu. APPARATUS a) longitudinal ; b] Cu-constantan thermocouples; c) radiation shield, 2 x 10 -5 mm (3 x 10“^ Pa) insulation. DATA a) 98 to 573 K; b) Table 2 - average X; c) uncertainty: ±3% (maximum) for T > 48, ±12% at 20 K. ANALYSIS a) Roder, Powell, and Hall, Phys. Rev., 115 , 314 (1959); White, Aust. J. Phys., 6 , 397 (1953); Berman and MacDonald, Proc. Roy. Soc. London, Ser. A, 211 , 122 (1952); b) data agree to within ±6% of reference data. Roder, H. M. , Powell, R. L., and Hall, W. .J., Thermal and Electrical Conductivity of Pure Copper, Low Temperature Physics and Chemistry, 364-6 (Aug 1958) PURPOSE To measure X and p for pure Cu. SPECIMEN a) /rod ; c) specimens 1,2: annealed, specimen 3: not annealed/ specimen 3: drawn to 25% less cross sectional area; d) specimens 1,3: 99.999% Cu, specimen 2: 343 ppm Ag; e) specimen 1 ( RRR = 1450), specimen 2 (RRR = 400), specimen 3 (RRR > 350); g) source: American Smelting and Refining Co. APPARATUS a) not given, but probably longitudinal. DATA a) specimen 1: 4 to 86 K, specimen 2: 4 to 64 K, specimen 3: 5 to 30 K; b) figure 1 -X, figure 2 -p, figure 3 -L. ANALYSIS b) W = ATn + B/T. 78 Rosenberg, H. M. , The Thermal Conductivity of Metals at Low Temperatures , Philos. Trans. Roy. Soc. London, 247 , 441-97 (Mar 1955); see also Mendel ssohn , K., and Rosenberg, H. M. , Proc. Phys. Soc., London, Sect. A, 5J5, 385 (1952) PURPOSE To measure x for many metals at low temperatures. SPECIMEN a) 2.99 cm long, 2.99 mm dia./rod; b) polycrystal 1 i ne ; c) vacuum annealed, 800 °C for several h/turned down from 3.02 mm dia.; d) 99.999% Cu; g) P293 /P 20 = 85.3, source: Johnson, Matthey and Co., No. 4234. APPARATUS a) longitudinal ; b) gas thermometers/cal ibrated at liquid He and H 2 tem-peratures/thermometer capillaries anchored to liquefier top and vacuum jacket; c) Cu radiation shield, vacuum insulation. DATA a) 2.5 to 41 K; b) figure 4 -X; c) uncertainty: ±3% (maximum). ANALYSIS a) Berman and MacDonald (1952) curve of Cu from same batch agrees for tem-peratures above the maximum and a = 2.5 x 10" 5 also agrees; b) Matthiessen 1 s rule is not strictly valid in the region of x maximum; 1/x = aT2 -3/T. Schofield, F. H. , The Thermal and Electrical Conductivities of Some Pure Metals, Proc. Roy. Soc. (London), 107 , 206-27 (1924) PURPOSE To measure x and p of several metals. SPECIMEN a) 3/4 in. (1.905 cm) dia./rod; c) hot rolled to 1 in. dia., drawn to 7/8 in. dia., machined and polished to 3/4 in. dia., then annealed; d) 99.9%; g) source: T. Bolton and Sons, Ltd., Oakmoor. APPARATUS a) Forbes bar method; b) Pt/10%Ir-Pt thermocouples; c) insulation between specimen and guard ring. DATA a) 369 to 898 K; b) Appendix II -X, p; c) uncertainty: not given. ANALYSIS a) Lees (1908); Jaeger and Diesselhorst (1900). 79 Schriempf, J. T. , Deviations from Matthiessen 1 s Rule in the Low Temperature Thermal and Electrical Resistivities of Very Pure Copper, Proceedings of the Seventh Thermal Conductivity Conf., 249-52 (Nov 13-16, 1967) PURPOSE To use x and p data to show the deviation of Matthiessen 1 s rule at low temperatures. SPECIMEN a) Cu 1: 0.119 in. (0.302 cm) dia., Cu 2 and Cu 2-0: 0.076 in. (0.19 cm) dia./rods; c) Cu 1 : annealed 12 h at 1000 °C in air, 10" 3 mm (0.1 Pa), Cu 2: annealed 3 h at 530 °C, 10~6 mm (10 -4 Pa), Cu 2-0: Cu 2 annealed 22 h at 1000 °C in air, 5 x 10“4 mm (7 x 10 -3 Pa)/Cu 1: swaged from 3/8 (0.95 cm) to 0.125 in. (0.318 cm) dia., annealed, etched, Cu 2: swaged to 0.080 in. (0.20 cm) dia., etched, annealed; d) Cu 1: 100 ppm Mn, Cu 2 and Cu 2-0; Mn < 10 ppm; f) Cu 1 (p 0 = 1,73 x 10 -11 f2m), Cu 2 (p 0 = 0.579 x 10“ 1J-£2m), Cu 2-0 (p 0 = 1.12 x 10 -11 fMm); g) source: American Smelting and Refining Co. APPARATUS a) longitudinal; b) thermocouple difference thermometer/referenced by Ge thermometer. DATA a) 2 to 20 K; b) figure 1 -X; c) uncertainty: ±3%. ANALYSIS b) results agree with White, G. K. and Tainsh, R. J., Phys. Rev., 119 , 1869 (1960), but disagree with Powell, R. L., Roder, H. M., and Hall, W. J., Phys. Rev., 115, 314 (1959). 80 Scott, B. W. , Transport Properties of Dilute Copper Alloys, Ph.D. Thesis, Michigan State Univ. (1962), University Microfilms Inc., Ann Arbor, Michigan PURPOSE To measure x, p and thermoel ectri c power of dilute Cu alloys and compare with existing theory. SPECIMEN a) all specimens 5 to 10 cm long, 0.25 cm dia./rods; c) all specimens cast into 3/16 x 2 in. (0.48 x 5 cm) slugs from 99.999% Cu, homogenized at 600 °C for six days and annealed in sealed vycor capsules, each step in 2/3 atm. (6.8 x 10^ Pa) He (specimen 8 high vacuum). Specimen 3 annealed at 600 °C for 5 h, 750 °C for 1 h, specimens 14, 15, 17 annealed at 600 °C for 3 h, remaining annealed at 750 °C for 2 h/all swaged between homogeniza tion and annealing, and given acid etch before casting and annealing; d) specimen 8: 0.61% Sn, specimen 14: 0.225% Cd, 0.762% In, specimen 15: 0.104% Cd, 0.389% In, specimen 17: 0.477% Cd, 0.294% In, specimen 101: 0.47% Zn, 0.294% In, specimen 102: 0.49% Zn, 0.77% In, specimen 103: 0.416% In, 0.995% Zn, specimen 104: 0.331% Zn, 0.35% Ga, specimen 105: 0.34% Zn, 0.805% Ga , specimen 106: 0.856% Zn, 0.37% Ga; f) specimen 8 (p 0 = 1.75 x 10" 6 ftcm), specimen 14 (p~ = 1.061 x 10"® Qcm), specimen 15 (p 0 = 0.549 x 10"® ftcm], specimen 101 (p 0 = 0.481 x 10"® ft cm) , specimen 102 (pp = 1.03 x 10"® ftcm), specimen 103 (p 0 = 0.73 x 10"® ftcm), specimen 104 (p 0 = 0.585 x 10"® ftcm), specimen 105 (pp = 1.418 x 10"® ftcm), specimen 106 (p 0 = 0.718 x 10"® ftcm); g) source: American Smelting and Refining Co. APPARATUS a) longitudinal; b) Au -2.1 at. % Cu vs. Ag -37 at. % Au thermocouples/ calibrated against Pt resistance thermometer certified by NBS; c) vacuum in sul ation. DATA a) 7 to 100 K/l to 5 K; b) Tables on pp. 142-149, 162-164, 166-170 -X and p, all specimens; c) uncertainty: ±3% at 20 K, ±7% at 50 K. ANALYSIS b) low temperature lattice conductivity of dilute alloys is explained by Pippard's theory. At higher temperatures the anisobaric effect is ap-parently masked by scattering. 81 Siu, M. C. I., Carroll, W. L. , and Watson, T. W. , Thermal Conductivity and Electrical Resistivity of Six Copper-Base Alloys, NBSIR 76-1003, 1-18 (1976) PURPOSE To measure A and p at high temperatures. SPECIMEN a) 37 cm long, 0.64 cm dia./rod; c) extruded at 1233 K and aged at 693 K in cracked natural gas atmosphere; d) 0.002% Fe, 0.662% Ni , 0.20% Zr. APPARATUS a) longitudinal; b) Pt/Pt -10% Rh; c) heaterguard, vacuum, alumina powder surrounds specimen and guard. DATA a) 373 to 923 K; b) Table 4 -X, Table 5 -p; c) uncertainty: ±2%. ANALYSIS b) the measured values of A, p conform to the Smith-Palmer equation to within 10%. Touloukian, Y. S. , Powell, R. W. , Ho, C. Y., and Klemens, P. G. , Thermo-physical Properties of Matter, Volume 1: Thermal Conductivity, Metallic Ele-ments and Alloys, 68-81 (1970) PURPOSE To provide an extensive list of data for x of the metallic elements and al 1 oys. SPECIMEN d) 99.999%; f) p0 = 8.51 x lO' 10 si- cm. APPARATUS a) not given. DATA a) 0 to 1356 K; b) Figure and Table 12R -X, recommended values; c) uncer-tainty: ±3% near room temperature, ±3 to 5% at other temperatures ; e) the values below 1.5 Tm are calculated to fit the experimental data by using n = 2.40, a = 0.19, m = 2.59, a" = 4.16 x 10~ 6 , and 3 = 0.0348. 82 Tseng, Tse-Pei , Properties of Matter at Very Low Temperatures , Ph.D. Thesis, Ohio State Univ. (1954) PURPOSE To provide x data on commercial grade Cu. SPECIMEN a) 0.025 cm dia./wire; d) commercial grade; g) source: General Electric Co. APPARATUS a) longitudinal; b) C resistance thermometers/resi stance extrapolated to zero power of measuring current for calibration; c) 10 " 6 mm (10"4 Pa) insul ation. DATA a) 1.4 to 4.2 K; b) Table 2 -X; d) corrections for thermometer and heater lead conduction. ANALYSIS b) results are comparable to Mendelssohn, K. and Rosenberg, H. M. , Proc. Phys. Soc., London, Sect. A, 65_, 385 (1952). White, G. K., The Thermal and Electrical Conductivity of Copper at Low Temperatures, Aust. J. Phys., 6^(4), 397-404 (1953) PURPOSE To measure X and p for pure Cu. SPECIMEN a) Cu 1, 2: 2 mm dia., Cu 3: 1 mm di a . , al 1 5 cm long/ rods; c) Cu 2 is Cu 1 after vacuum annealing at 550 °C for 3 h/Cu 1 , 3 in "as drawn" condition; d) all specimens, 0.0005% Ag, < 0.0003% Ni , 0.0004% Pb, trace of Ga, Fe; e) Cu 1 (p 0 = 0.051 yncm); Cu 2 (p 0 = 0.0576 yfl»cm); Cu 3 (p0 = 0.00458 pQcm); g) source: Johnson, Matthey and Co., No. JM4272. APPARATUS a) longitudinal ; b) gas thermometers ; c) vacuum insulation. DATA aT~Cu 1: 2.5 to 116 K, Cu 2: 5 to 58 K, Cu 3: 2.03 to 160 K; b) figure 2 -X; c) uncertainty: ±4% from 5 to 15 K, ±1% for all others; d) correction for radiative heat loss at higher temperatures. ANALYSIS b) results in agreement with Berman, R. and MacDonald, D. K. C. , sample of pure annealed Cu; measurements support theory of non-additivity of impurity and ideal resistances. S3 T42] Whits, G. K. and Tainsh, R. J., Lorenz Number for High Purity Copper, Phys. Rev., 119(6), 1869-71 (Sep 1960) PURPOSE To investigate L for very pure copper. SPECIMEN a) 0.03 in. (0.08 cm) dia., 8 cm long/wire; c) annealed at 530 °C in vacuo/ rolled and drawn from 0.75 in. (1.9 cm) dia.; d) < 0.0001% Fe, Sb, Se each, < 0.0002% Te, As each; f) pg = 0.87 x 10"^ ft.cm; g) source: American Smelting and Refining Co. APPARATUS a) longitudinal; b) He gas thermometers. DATA a) 2 to 55 K; b) figure 1 -X; c) uncertainty: ±4% for 2 < T < 15; ±1% T > 15. ANALYSIS b) X = A/T + BTn : A = 0.035 cm«K W" 1 compared to 0.059 for Powell, R. L. , Roder, H. M., and Hall, W. J., Phys. Rev., 115, 314 (1959). White, G. K. and Woods, S. B. , Thermal and Electrical Conductivities of Solids at Low Temperatures, Can. J. Phys., 33, 58-73 (1955); The Lattice Thermal Conductivity of Dilute Copper Alloys, Philos. Mag., 45, 1343-5 (1954) PURPOSE To describe an apparatus for measuring X and p of solids between 2 and 300 K. SPECIMEN a) 1 to 2 mm dia., 6 cm long/rods; c) Cu 1, 2, 3 annealed at 700 °C for 2 h/drawn; d) Cu 1: 0.02% Ge, Cu 2: 0.056% Fe. Cu 3: 0.0043% Fe; f) Cu 1 (p 0 =0.084 x 10 - 6 ft. cm), Cu 2 (p 0 = 0.53 x 10"° ft.cm), Cu 3 (p 0 = 0.041 x I0“ 6 ft. cm). APPARATUS a) longitudinal; b) gas thermometers//al 1 leads to specimen are anchored to Cu pillars; c) temperature controlled radiation shield. DATA a) 1 .5 to 142 K; b) figure 2 -p, figure 3 -X, Table 1 -p0 ; c) uncer-tainty: ±5% for high X specimens from 4 to 15 K, all others: ±1%; d) cor-rection for radiation. ANALYSIS b) X = AT" 1 + B T2 - 4 . 84 Zavaritskii, N. V. and Zel'dovich, A. G. , Thermal Conductivity of Technical Materials at Low Temperatures, Sov. Phys. Tech. Phys. , 1, 1970-4 (1956) PURPOSE To describe an apparatus for measuring x for solid materials. SPECIMEN a) 20 and 6 mm long, 10 and 6 mm dia./rods; c) M-3-anneal ed: annealed at 800 °C/M-3-anneal ed : cut from tube; d) M-3-annealed and unannealed: Bi < 0.003, Sb < 0.05, As < 0.05, Fe 0.05, Ni < 0.20, Pb < 0.05, Sn < 0.05, S < 0.01, 0 _< 0.1%, Cupalloy: 0.61% Cr, 0.18% Ag. APPARATUS a) longitudinal ; b) carbon resistance thermometers/cal ibrated from 2 to 4.2 K and from 14 to 20 K from vapor pressures of He and H 2 respectively and from 20 to 110 K from standard Pt resistance thermometer; c) radiation shield and vacuum insulation. DATA a) 2 to 100 K; b) figure 2 -X; c) uncertainty: ±5%. ANALYSIS b) results agree with White, G. K. and Woods, S. B. , Can. J. Phys., 33., 58 (1955). 85 3. Aluminum 3.1 General Aluminum has also been measured extensively but not as completely as copper A total of 35 publications are included in the annotated bibliography (Sec-tion 3.5). The following of these data sets were selected as primary data: 1, 9, 11, 12, 17, 25, 26, 29, and 33. The primary data contain only annealed specimens, and covers a range of tern perature from 2 to 873 K, and a range of RRR from 13 to 16800. Thermal conduc-tivity values from these sources are shown in Figs. 3.1.1 through 3.1.3, where the last graph is a composite of the data. Although the RRR of the most elec-trically pure aluminum ever produced is comparable to that for copper (50,000), aluminum with RRR values in the range of 10,000 to 20,000 is more readily ob-tai ned. Equation 1.1.3 was fitted to the primary data set over the entire tempera-ture range. The values of the parameters, P-j , i = 1,2 ... 7 obtained by non-linear least squares fit are: = 4.716 x 10" 8 P 5 = 130.9 P 2 = 2.446 P 6 = 2.5 P 3 = 623.6 P ? = 0.8168 P4 = -0.16 with all units in SI. The data at high RRR were then examined for systematic residual deviations as a function of temperature. These residuals were represented by the Wc term in Eq. 1.1.5. The resulting equation for Wc is: 87 W = -0.0005 £n( T/330) exp(-(£n(T/380)/0.6) 2 ) -0.0013 £n(T/110) exp(-Un(T/94)/0.5) 2 ), where Wc and T are in SI units. 3.2 Deviations from Recommended Equation Equation 1.1.3 represented the overall primary aluminum data to yield random deviations. Again some of the individual data sets exhibit systematic trends. The deviations of these data from Eq. 1.1.3 are shown in Figs. 3.2.1, 3.2.2, and 3.2.3. Although temperature dependent deviations exist for individual data sets, the overall pattern is random in nature. No systematic trends with RRR were noted. The primary data were selected from the literature data on relatively large, well annealed specimens. Therefore, the deviations exhibited in Figs. 3.2.1 through 3.2.3 are indicative of the combined effect of a) experimental measure-ment errors and b) the inability of Eq. 1.1.3 to account for the effects of chemical impurity variations. The effects of physical defect variations, small specimen size variations, and magnetic fields are exhibited, in part, by the de-viations of the secondary data. The thermal conductivity variations caused by other than chemical impurity variations are not expected to be represented as well by Eq. 1.1.3. However, the RRR (or p 0 ) correlating parameter does account for an appreciable part of these variations. Some users may find this to be an adequate representation and, therefore, discussions of these comparisons are in-cluded for completeness. The deviations of the secondary data sets are illustrated in Figs. 3.2.4 through 3.2.10. Figures 3.2.7 and 3.2.10 are composite deviation plots. Eq. 1.1.3 is compared to the most commonly used sources of reference data by Ho and Touloukian, references 14A,32A, respecti vely, in Fig. 3.2.11. The agreement 88 is within the combined uncertainties except in the region around 60 K. In this region data by Cook, reference 5, were published after both data sets and is the reason for the large differences. 3.2.1 Physical Defect Effects Investigations of physical defects in aluminum have produced only a few ref-erences 21,25. Each of these references will be discussed below. Changes in thermal conductivity due to elongation are studied in refer-ence 21. The conclusion is that increasing deformation lowers the thermal con-ductivity maximum and shifts it to a higher temperature. Unfortunatel y , it ap-pears that the figure captions do not correspond to the proper figures; caption 3 belongs to Fig. 2 and caption 4 to Fig. 1. Note that in Fig. 2, the nondeformed specimen has the wrong temperature dependence above the temperature of the maxi-mum thermal conductivity. The deviations from Eq. 1.1.3 for this specimen were within +_60%. The deviations from Eq. 1.1.3 for the specimens in Fig. 1 were within +10% for the undeformed specimen. The deformed specimens were not com-pared. Reference 25 showed the effects on the thermal conductivity of an "as fabri-cated" specimen and an annealed specimen of the same stock. Also included was a high purity specimen. The peak value of the thermal conductivity for this speci-men was about 1580 Wm - ^ -IC 1 , for the "as fabricated" specimen, 389 Wm“lK"l, and for the annealed specimen, the peak value was 332 The authors comment on the effect of the anneal , in-dicating that the decrease in peak value of the annealed specimen it is due to the large amount of impurities present in the original stock. The deviations from Eq. 1.1.3 for the high purity specimen were within +7% as were those for the annealed specimen. The deviations for the "as fabricated" specimen were within +8%. 89 Although the temperature dependence of physical defect scattering mechanisms is different from that due to impurity scattering. Eg. 1.1.3 represents the unan-nealed specimen data to within +10% . This indicates that the residual electrical resistivity characteri zes both types of scattering for the range of RRR included here. 3.2.2 Size Effects Amundsen and Olsen^ studied the size effects of several aluminum speci-mens. As the specimen size decreases, the peak value of the resistivity in-creases. Notice that magnetic fields were used to find the thermal resistivity of each specimen. The ratio of the thermal and electrical resistivities was shown to be constant, except at the highest fields, where the thermal resistivity increased faster than the electrical resistivity. Although not specifically studied, we can make some general comments regard-ing the thermal conductivity of several specimens at low temperatures as a func-tion of specimen size. The peak thermal conductivity is shifted toward lower temperatures for the larger specimens. The justification of this effect resides in its observation in other metals. 3.2.3 Magnetic Field Effects Although magnetic field effects on thermal conductivity were not studied ex-plicitly, reference 14 showed that the peak conductivity value in a 0.5 T mag-netic field was 5400 WnTK"l, while in a zero field, the value was 10500 Wm 1 K_1 . It is interesting to notice that this peak disappears completely for a field somewhere between 0.5 and 1 T. 1 Amundsen, T. and Olsen, T. , Size Dependent Thermal conductivity in Aluminum Films, Phil. Mag., _U, 564-74 (1965). 90 Reference 31 reported that the peak conductivity value in a 1 T field was 3800 while in a zero field the value was 5800 W’iW'K - 1 . This study supports the statement that the peak in thermal conductivity disappears between 0.5 and 1 T. Reference 32 reported low temoerature thermal conductivity data for fields uo t.o 5 T. At zero field and a temperature of 20 K, the specimen had a conduc-tivity value of 6500 W'm -1 ‘K" 1 . At a field of 5 T, the conductivity of the same specimen dropped to about 1800 Wm _ ^K"--. This report indicates that the peak in thermal conductivity disappears for a field between 1 and 2 T. Amundsen and Souik^ reported measurements on two single crystal specimens of pure aluminum in fields up to 1.3 T and temperatures below 4 K. The increase in thermal resistivity was linear in both specimens above 0.3 T, falling sharply off below this field. A maximum increase in resistivity of 1.5 times was induced by the magnetic field, but the ratio of thermal resistivity to electrical resistivity was essentially independent of the field (less than 10% variation). Sparks^ reported measurements on two specimens with similar thermal his-tories from 5 to 20 K. There were large differences in RRR and in thermal con-ductivity values due to different impurity concentrations. At 6 K, and zero field, a factor of two difference is found in the thermal conductivities. For a field of 8 T, he found a decrease of 29% at 5 K, and a 50% decrease at 20 K. 3.3 Electrical Resistivity and Lorenz Ratio It was desirable to examine the Lorenz ratio of aluminum during the course of this investigation. Therefore, an approximation of electrical resistivity was ^Amundsen, T and Souik. R. P. , Measurements of the Thermal Magnetoresi stance of Aluminum, J. Low Temp. Phys., 2jl), 121-9 (1970). ^Sparks, L. L. , Magnetic Field Effect on Thermal Conductivity of Selected Metals, Advances in Cryogenic Engineering, 24, 224-31, Plenum Press (1977). 91 needed. To do this, we selected those sources from this thermal conductivity compilation that also contained electrical resistivity data and fitted Eq. 1.2.3 to the data. The electrical resistivity data used is shown in Figs. 3.3.1 and 3.3.2 with a composite of the data in Fig. 3.3.3. The parameters for Eq. 1.2.3 are P l = 0.09052 x 10" 16 P 5 = 40. P 2 = 4.551 P 6 = 13.64 P 3 = 5.173 x 10 10 P 7 = 0.7416 P4 = 1.26 All units are SI. The deviations of the experimental data from this equation are illustrated in Figs. 3.3.4 through 3.3.6. Again, as for copper, there is considerable spread in the deviations in the range 10-80 K. It is recognized that this is partly due to the large range of p values represented (several orders of magnitude) as well as the inadequacy of the simple equation used for representation. Nevertheless, it is felt that for this purpose the approximation is useful. Smooth values of p and T at selected RRR values are plotted in Fig. 3.3.7. From the p(T,RRR) and x(T,RRR) equations smooth values of L(T,RRR) are plotted in Fig. 3.3.8. As for copper, the irregul arities in the shape of these curves from that expected are within the combined uncertainties of the two equations used. In Section 1.5, we discuss the procedure for selecting values of p 0 and calculating RRR for each thermal conductivity data set. These values of p 0 along with the Sommerfeld value of Lorenz ratio were used to best fit each low temperature data set. The resulting values of RRR obtained by this procedure are compared to the values reported in the references in Fig. 3.3.9 and are listed in Table 3.3.1. Figure 3.3.9 shows values of RRR (calc), those values from the 92 above procedure, versus RRR (obs), those values reported in the references listed in the annotated bi bl iography. Also shown in this figure is the line that repre-sents RRR (calc) = RRR (obs). Systematic deviations from this line indicate ranges in which the derived Eq. 1.1.3 is invalid. With the exception of four points from the secondary data sets, all of the points lie near the line. Equa-tion 1.1.3 is thus valid for the low temperatures from RRR of 10 to 10,000. In-spection of Table 3.3.1 indicates that the calculated values of RRR are generally smaller than the observed values. This suggests that the Lorenz ratio of aluminum at low temperatures may be slightly smaller than the Sommerfeld value. 3.4 Summary for Aluminum Equation 1.1.3 represents the thermal conductivity of an annealed specimen of aluminum over the whole temperature range. The deviations from the primary set are shown in Figs. 3. 2. 1-3. 2. 3. Deviations for specimens in which physical defects are important (unannealed specimens) are within +20%. Based on the deviations illustrated in Figs. 3.2.1 through 3.2.10 and the large range that occurs in low temperature thermal conductivity of aluminum (nearly three orders of magnitude) due to the introduction of chemical impurities and physical defects, it is clear that a large proportion of these effects is re-flected by the residual electrical resistivity. The incorporat ion of RRR (or p 0 ) in Eq. 1.1.3 produces an equation that represents the data for a wide range of aluminum specimens. Equation 1.1.3 with the parameters listed here was used to generate thermal conductivity values for selected temperatures and values of RRR. These values are listed in Table 3.4.1 and plotted in Fig. 3.4.1. 93 List of Tables and Figures for Aluminum Tabl es Page Table 3.3.1 Comparison of Calculated and Observed RRR Values for Aluminum 96 Table 3.4.1 Thermal Conductivity Values for Aluminum Calculated from Eq. 1.1.3 at Selected Temperatures and RRR Values 97 Figures Page Figure 3.1.1 Experimental thermal conductivity data selected from the following primary references in the aluminum annotated bibliography: (1,5,9,11) 98 Figure 3.1.2 Experimental thermal conductivity data selected from the following primary references in the aluminum annotated bibl iography : (12,17,25,26,29,33) 99 Figure 3.1.3 Composite of the data in figures 3.1.1 and 3.1.2 100 Figure 3.2.1 Thermal conductivity deviations of the aluminum data from the following primary references compared to Eq. 1.1.3: (1,5,9,11) 101 Figure 3.2.2 Thermal conductivity deviations of the aluminum data from the following primary references compared to Eq. 1.1.3: (12,17,25,26,29,33) 102 Figure 3.2.3 Composite of the deviations in figures 3.2.1 and 3.2.2 103 Figure 3.2.4 Thermal conductivity deviations of the aluminum data from the following secondary references compared to Eq. 1.1.3: (2,3,6,7,10,14) 104 Figure 3.2.5 Thermal conductivity deviations of the aluminum data from the following secondary references compared to Eq. 1.1.3: (15,16,19,20,21,23) 105 Figure 3.2.6 Thermal conductivity deviations of the aluminum data from the following secondary references compared to Eq. 1.1.3: (23,25,26,27,28) 106 Figure 3.2.7 Composite of the deviations in figures 3.2.4 through 3.2.6 107 94 Page Figure 3.2.8 Thermal conductivity deviations of the aluminum data from the following secondary references compared to Eq. 1.1.3: (3,13,18,21,22,24,30) 108 Figure 3.2.9 Thermal conductivity deviations of the aluminum data from the following secondary references compared to Eq. 1.1.3: (30,31,32) 109 Figure 3.2.10 Composite of the deviations in figures 3.2.8 and 3.2.9 110 Figure 3.2.11 Comparison of Eq. 1.1.3 to the values recommended for aluminum in the following references : (14A,32A) Ill Figure 3.3.1 Experimental electrical resistivity data for aluminum selected from the following references in the aluminum annotated bibliography: (4,9,11,25,26) 112 Figure 3.3.2 Experimental electrical resistivity data for aluminum selected from the following reference in the aluminum annotated bibl iography : (33) 113 Figure 3.3.3 Composite of the electrical resistivity data in figures 3.3.1 and 3.3.2 114 Figure 3.3.4 Electrical resistivity deviations of the aluminum data from the following references compared to Eq. 1.2.3: (4,9,11,25,26) 115 Figure 3.3.5 Electrical resistivity deviations of the aluminum data from the following reference compared to Eq. 1.1.3: (33) . . . 116 Figure 3.3.6 Composite of the electrical resistivity deviations shown in figures 3.3.4 and 3.3.5 . . 117 Figure 3.3.7 Electrical resistivity for aluminum as a function of temperature calculated from Eq. 1.2.3 at selected values of RRR 118 Figure 3.3.8 Lorenz ratio for aluminum as a function of temperature calculated from Eq. 1.2.3 and Eq. 1.1.3 at selected values of RRR 119 Figure 3.3.9 RRR values calculated as per Section 1.5, RRR(CALC), versus reported RRR values, RRR(OBS), for aluminum .... 120 Figure 3.4.1 Thermal conductivity for aluminum as a function of temperature calculated from Eq. 1.1.3 at selected values of RRR 121 95 Table 3.3.1. Comparison of Calculated and Observed RRR Values for Aluminum. Reference RRR (obs.) RRR (calc.) Primary Data 1 467.0 420.0 1 676.0 605.0 1 840.0 720.0 5 11600.0 11600.0 11 2750.0 2700.0 11 4370.0 4100.0 25 12.8 12.8 25 97.0 91 Secondary Data 14 2800.0 1550.0 14 4500.0 1720.0 22 520.0 520.0 23 147.0 147.0 23 86.0 86.0 23 36.0 36.0 23 17.0 17.0 30 15000.0 11900.0 30 24000.0 18000.0 96 le 3.4.1. Thermal Conducti vity Val ues for Aluminum Cal cul ated from Eq. 1.1 .3 at Selected Temperatures and RRR Values. A(W-m" 1 "K” 1 ) T (K) RRR = 30 RRR = 100 RRR = 300 RRR = 1000 RRR = 3000 RRR = 10000 1 29 98 295 984 2954 9842 2 57 195 589 1956 5892 19521 3 86 292 833 2941 8765 28499 4 114 390 1175 3897 11475 35887 5 143 487 1463 4817 13885 40840 6 171 583 1746 5577 15853 43072 7 200 678 2020 6452 17272 42980 8 228 772 2232 7116 18109 41300 9 256 864 2528 7651 18406 38717 10 284 953 2755 8044 18260 35708 12 338 1122 3133 3420 17095 29474 14 391 1272 3398 8340 15360 23801 16 442 1400 3544 7960 13478 19074 18 489 1500 3582 7418 11658 15308 20 532 1572 3534 6801 9997 12366 25 617 1628 3178 5227 6706 7527 30 662 1542 2666 3843 4492 4791 35 664 1373 2130 2756 3036 3151 40 631 1172 1652 1972 2096 2143 45 581 980 1274 1440 1497 1519 50 526 817 997 1087 1117 1128 60 430 588 664 696 706 710 70 361 454 492 507 512 513 80 312 372 394 403 405 406 90 278 320 334 340 341 342 100 255 286 297 300 301 302 150 223 239 244 245 246 246 200 222 234 237 238 239 239 250 224 233 235 236 237 237 300 226 234 236 237 237 237 400 231 237 239 239 239 239 500 230 235 237 237 237 237 600 226 230 231 231 231 231 700 220 229 224 224 224 224 800 214 217 217 218 218 218 900 209 212 212 212 212 212 97 THERMAL CONDUCTIVITY, W/m. TEMPERATURE, K Figure 3.1.1 Experimental thermal conductivity data selected from the following primary references In the aluminum annotated bibliography: (1,5,9,11) O-(1), A-(1), -(1), V-(5), O-(9), +-(11), X-(11) 98 THERMAL CONDUCTIVITY, W/m . Figure 3.1.2 Experimental thermal conductivity data selected from the following primary references In the aluminum annotated bibliography : (12,17,25,26,29,33) O-(12), A-(17), -(25), V-(25), O-(26), +-(29), X-(33) 99 THERMRL CONDUCTIVITY,W/m. 2 4 10 20 40 100 200 400 TEMPERRTURE,K Figure 3.1.3 Composite of the data In figs. 3.1.1 and 3.1.2 100 Figure 3.2.1 Thermal conductivity deviations of the aluminum data from the following primary references compared to eq. (1.1.3): (1,5,9,11) O-(1), A-(1), -(1), V-(5), O-(9), + -(11), X-(11) 101 20 -1 1 1 .1 . i » j_ i i_i_i . -i . i, , i i 2 4 10 20 40 100 200 400 TEMPERRTURE,K Figure 3.2.2 Thermal conductivity deviations of the aluminum data from the following primary ref erences compared to eq. (1.1.3): (12,17,25,26,29,33) O-(12), A-(17), -(25), V-(25), O-(26), +-(29), X-(33) in? 2 4 10 20 40 100 200 400 TEMPERATURE ,K Figure 3.2.3 Composite of the deviations In figs. 3.2.1 and 3.2.2 103 2 4 10 20 40 100 200 400 TEMPERRTURE,K Figure 3.2.4 Thermal conductivity deviations of the aluminum data from the following secondary references compared to eq. (1.1.3): (2,3,6,7,10,14) O-(2), A-(3), -(6), V-(7), O-(10), +-(14), X-(14) 104 2 4 10 20 40 100 200 400 TEMPERATURE, K Figure 3.2.5 Thermal conductivity deviations of the aluminum data from the following secondary references compared to eq. (1.1.3): (15,16,19,20,21,23) O-(15), A-(16), -(19), V-(20), O-(21), +-(23), X-(23) 105 2 4 10 20 40 100 200 400 TEMPERRTURE,K Figure 3.2.6 Thermal conductivity deviations of the aluminum data from the following secondary references compared to eq. ( 1 . 1 .3) : (23,25,26,27,28) O-(23), A-(23), -(25), V-(26), O-(26), +-(27), X-(28) 106 Figure 3.2.7 Composite of the deviations In figs. 3. 2. 4 through 3.2.6 107 Figure 3.2.8 Thermal conductivity deviations of the aluminum data from the following secondary references compared to eg. (1.1.3) : (3,13,18,21,22,24,30) O-(3), A-(13), -(18), V-(21), O-(22), +-(24), X-(30) 103 Figure 3.2.9 Thermal conductivity deviations of the aluminum data from the following secondary references compared to eq. (1.1.3) : (30,31,32) O-(30), A-(31), -(32) 1C3 Figure 3.2.10 Composite of the deviations In figs. 3.2.8 and 3.2.9 110 Figure 3.2.11 Comparison of eq. (1.1.3) to the values recommended for aluminum In the following references ! ( 14fl,32F!) O- (HR), A-(32fi) 111 ELECTRICAL RESISTIVITY, nfi. 2 4 10 20 40 100 200 400 TEMPERATURE, K Figure 3.3.1 Experimental electrical resistivity data for aluminum selected from the following references In the aluminum annotated bibliography: (4,9,11,25,26) O-(4), A-(9), - (ID, O-(25), + -(25), X-(26) V-(11), 112 ELECTRICAL RESISTIVITY, nQ. Figure 3.3.2 Experimental electrical resistivity data for aluminum selected from the following reference In the aluminum annotated bibliography: (33) O-(33) 113 ELECTRICRL RESISTIVITY, nfl. 2 4 10 20 40 100 200 400 TEMPERRTURE,K Figure 3.3.3 Composite of the electrical resistivity data In figs. 3.3.1 and 3.3.2 2 4 10 20 40 100 200 400 TEMPERATURE, K Figure 3.3.4 Electrical resistivity deviations of the aluminum data from the following references compared to eq. (1.2. 3)1(4,9,11, 25, 26) O-( 4 ) , A-(9), -(11), V-(11), O-(25), +-(25), X-(26) 115 Figure 3.3.5 Electrical resistivity deviations of the aluminum data from the following reference compared to eq. ( 1 .2.3) s (33) O-(33) 116 Figure 3.3.6 Composite of the electrical resistivity deviations shown In figs. 3.3.4 and 3.3.5 117 TEMPERATURE, K Figure 3.3.7 Electrical resistivity for aluminum as a function of temperature calculated from eq. (1.2.3) at selected values of RRR. 118 LORENZ RRTIO, 10 °VVK TEMPERRTURE,K Figure 3.3.8 Lorenz ratio for aluminum as a calculated from eq. (1.2.3) and values of RRR. f unction of temperature eq. (1.1.3) at selected 119 RRR(CflLC) Figure 3.3.9 RRR values calculated as per Section 1.5, RRR(CFILC), versus reported RRR values, RRR(OBS), for aluminum. O - Primary, A - Secondary, - Secondary 120 THERMAL CONDUCTIVITY, W/m. TEMPERATURE, K Figure 3.4.1 Thermal conductivity for aluminum as a function of temperature calculated from eq. (1.1.3) at selected values of RRR. 121 3.5 FORMAT FOR ANNOTATED BIBLIOGRAPHY OF ALUMINUM REFERENCE AUTHOR, TITLE, CITATION ANNOTATION PURPOSE SPECIMEN a) Dimensions/Shape; b) Crystal Status; c) Thermal /Mech. History; d) Purity Specification; e) RRR; f) p 0 ; g) Other Characteri zation Data APPARATUS a) Type; b) Thermometry/Cal i brat i on/Anchoring; c) Thermal Isolation; d) Other (Q meas.) DATA a) Temperature Range/Difference; b) Content of Tables, Figures and Equa-tions/Data Extraction; c) Uncertainty/Imprecision; d) Disputable Corrections to Measurements by Authors; e) Errata (by Author or Reviewer) ANALYSIS a) Comparisons; b) Conclusions 122 Andrews, F. A., Webber, R. T. and Spohr, D. A., Thermal Conductivities of Pure Metals at Low Temperatures. I. Aluminum, Phys. Rev., 84(5), 994-6, (1951) PURPOSE lo measure X for three high purity A1 specimens. SPECIMEN a) 4 in. (10 cm) long, 0.15 in. (0.38 cm) dia./rod; b) A1 1,2: single crystals, A1 3: polycrystal 1 i ne; c) A1 3: annealed; d) A1 1,2: 0.001% Mg, 0.001% Si, 0.0006% Fe, 0.0004% Cu, 0.0004% Na. A1 3: 0.002% Mg, < 0.001% Si, < 0.0005% Fe. < 0.0005% Cu, faint trace of Na; e) A1 1 (RRR = 840), A1 2 ( RRR = 676), A1 3 (RRR = 467); f) A1 1 (p 0 = 3.04 nft.cm), A1 2 (3.85 n«cm), A1 3 (p 0 = 5.51 nq.cm). APPARATUS aj longitudinal ; b) gas thermometers ; c) 0.05 ym of Hg (6.7 x 10" J Pa) vacuum. DATA aTT to 27 K/0.03 K at 2 K, 0.15 K at 20 K; b) figure 1 -X/data points are listed in TPRC data series; c) uncertainty: from 2 to 4.2 and 14.5 to 20.5 K: ±4%; from 4.2 to 14.5 K: ±10%; d) corrections due to departures of the thermometer system from ideal behavior at liquid helium temperatures were taken from Hulm, J. K. , Proc. Roy. Soc. London, Ser. A, 204 , 98 (1950). ANALYSIS b) X" 1 = (AT)’ 1 + BT2 . Bailey, L. C. , The Thermal Conductivities of Certain Approximately Pure Metals and Alloys at High Temperatures, Proc. Roy. Soc. London, Ser. A, 134, 57-76 (1932) PURPOSE lo continue the work of C. H. Lees on the effect of temperatures between -160 °C and 15 °C, on x of nine metals and six alloys. SPECIMEN a ) / to 8 cm long, 0.585 cm dia./rods; c) turned down from larger rods; dj 99% A1 ; g) specimens same as Lees, C. H., Philos. Trans. Roy. Soc. London, Ser. A, 208 , 381-443 (1908)/density = 2.7 g/crrr at 293 K. APPARATUS If) longitudinal ; b) thermocouples (chromel -al umel )/hot junction calibrated at: melting ice, steam, aniline vapor, benzophenone vapor, sulphur vapor. Cold junction: 293 K. Main current balanced against standard cell; c) powdered magnesia between guard tube and specimen, asbestos wool between guard tube and vessel . DATA aj^5 to 554 °C; b) Table 1 -X. ANALYSIS a) Schofield, F. H. , Proc. Roy. Soc. London, Ser. A, 107 , 206 (1925); Griffiths, E., Proc. Roy. Soc. London, Ser. A, 1_15, 2TJ5 ( 1 927 ) ; x reaches a maximum at 225 °C. 123 Bid we 11, C. C. and Hogan, C. L. , Thermal Conductivity of Aluminum; Solid and Liquid States, J. Appl . Phys., 18, 775 , (Aug 1947) PURPOSE To investigate A for A1 up to the melting point and beyond, with a modified Forbes bar method. SPECIMEN a) 25 cm long, 2.5 cm dia./rod; d) specimen 1 = 99.2% Al, 0.10% Si, 0.67% Fe, 0.01% Cu; Mn, Mg < 0.01%, specimen 2 = 99.95% Al ; g) source: Aluminum Company of America. APPARATUS a) modified Forbes bar method; b) thermocouples; c) finely screened Sil-o-cel insulation. DATA a) specimen 1, 25 to 590 °C, specimen 2, 25 to 900 °C; b) data points taken from text p. 779; d) correction for temperature drift. ANALYSIS a) data on specimen 1 is in agreement with Konno, S. , Philos. Mag., 40, 542 (1920); b) the data are consistent with k/pC = K/t + K'. Cook, J . G. , Moore, J. P., Matsumura, T. , and Van der Meer, M. P., The Thermal and Electrical Conductivity of Aluminum, Proceedings of the Fourteenth Thermal Conductivity Conf. (Storrs, Conn., June 2-4, 1975) P. G. Klemens and T. K. Chu, eds., Plenum Press, New York (1976) pp. 65-71 PURPOSE To measure A, p, and the absolute Seebeck coefficient of pure Al from 80 to 400 K. SPECIMEN e) specimen A (RRR = 8500), specimen B (RRR = 11000), specimen C (RRR = 950), specimen D (RRR = 17); f) specimen A ( pq = 2.8 x 10"^ uPcm), specimen B (p Q = 2.1 x 10 -4 pn»cm), specimen C (p 0 = 2.5 x 10“4 jjficm), specimen D (p Q = 0.14 uftcm). DATA a) 20 to 400 K; b) Table 1 -specimen characteri zation , Table 2 -A; c) un-certainty: ±1.2%; d) correction for impurities and thermal expansion. ANALYSIS a) data does not agree with Seeberg. P. and Olsen, T. , Phys. Norv., 2, 197 (1967); b) A = A + BT‘ 4 + CT 2 + DT" 1 ; Ag 1 = T/17 + 5000 T' 2 . 124 Cook, J. 6. , Moore, J. P. , Matsumura, T. , and Van der Meer, M. P. , The Thermal and Electrical Conductivity of Aluminum, 0RNL-5079, (Sep 1975) PURPOSE To measure X, p, and the absolute Seebeck coefficient of pure A1 from 80 to 400 K. SPECIMEN e) specimen A ( RRR = 8500), specimen B ( RRR = 11000), specimen C (RRR = 950), specimen D (RRR = 17); f) specimen A (pg = 2.8 x 10~ 4 ufi-crn), specimen B (p 0 = 2.1 x 10~4 yfi»cm), specimen C (p 0 = 8.5 x 10“ 4 uficm), specimen D (p 0 = 0.14 |jf2 cm). DATA a) 20 to 400 K; b) Table 1 -specimen characteri zation , Table 2 -X; c) uncertainty: ±1.2%; d) correction for impurities and thermal expansion. ANALYSIS a) data does not agree with Seeberg. P., Olsen, T. , Phys. Norv., 2, 197 (1967); b) X = A + BT' 4 + CT2 + DT“1; Xg 1 = T/G + HT“ 2 . COMMENT The data set contained in Reference 22 was not used because of comments made in this paper. De Nobel, J., Heat Conductivity of Steels and a Few Other Metals at Low Temperatures, Physica (Utrecht) , 17(5) , 551-52 (May 1951) PURPOSE To measure X for A1 , Fe, Monel metal, Ni-Cr steel and Mn-Cr steel by two methods and compare the results. SPECIMEN a) /rod; c) /rolled; g) Brinell hardness 17. APPARATUS a) longitudinal ; b) I gas thermometers, II resistance thermometers ; c) vacuum insulation. DATA a) 16 to 87 K; b) Table II -X. ANALYSIS b) for pure metals (> 99.93%), X is proportional to T at very low temperatures and a maximum occurs between 3 and 20 K. 125 Der Nigohossian, G. , Optimization of Electrical Leads for Cryogenic Apparatus, Commissariat a L'Energie Atomique, Saclay, France, Centre d'Etudes Nucleaires de Saclay, Rep. No. CEA-R-3167 (Feb 1967) 120 pp PURPOSE To optimize the geometry of electrical leads to cryogenic containers so that heat leakage is minimized. SPECIMEN a) 30 mm; b) single crystal . APPARATUS a) longitudinal; b) Au-Co vs. Cu thermocouples for 70 to 300 K, carbon resistance thermometers for lower temperatures; c) vacuum insulation, gas cooled shield. DATA a) 4 to 295 K; b) figure p. 75 -p, figure p. 76 -X; c) uncertainty: ±10% (measurements with carbon resistance thermometers) . ANALYSIS a) x data agrees with Roder, Powell, and Hall, Conference of Low Temperature Physics and Chemistry, Madison, Wise., 364-367 (1958). Donth, E. and Gladun, C. , Measurement of Thermal Conductivity at Low Temperatures by a Non-Stationary Method, Cryogenics, 2 , 223-5 (Jun 1962) PURPOSE To present a method of non-stationary x measurement at low temperatures. SPECIMEN a) /rod; d) 99.5% A1 . APPARATUS a) longitudinal; b) lead resistance thermometers ; c) vacuum insulation, radiation shield. DATA a) 22 to 90 K; b) figure 3 -X. ANALYSIS b) the method allows the determination, continuously and quickly (15-120 minutes), of x over a wide temperature range. 126 Duggi'n, M. J., The Thermal Conductivities of Aluminum and Platinum, J. Phys. D: Appl . Phys., 3^> L21-23 (1970) PURPOSE To measure x of A1 , Pt. SPECIMEN a) /slab; d) 99.99%. APPARATUS a) guarded axial heat flow. DATA a) 380 to 600 K; b) Table 2 -X, p, L; c) uncertainty: +2.5%. ANALYSIS a) results agree with recommended values from TPRC (Powell, et al . , 1966). Erdman, C. A., A Dynamic Technique for Measuring Thermal Conductivity in Cylindrical Geometry, Designed for Use in Radiation Damage Studies, Ph.D. Thesis in nuclear engineering, Univ. of Illinois, Urbana-Champaign (1971) PURPOSE To develop a valid technique for measuring x which involves a small amount of equipment at the sample location. SPECIMEN a) 6 in. long (15 cm), 0.625 in. (1.59 cm) dia./rod; b) single crystal ; c) /machined from 1.25 in. (3.18 cm) dia./rod; d) 99.995% Al . APPARATUS a ) radial ; b) thermocouples anchored to brass holder; c) vacuum insula-tion and shield. DATA a) 100 to 300 K; b) Table 9 -X; c) uncertainty: ±6.6% at 296 K, ±8.9% at 105 K; d) corrections for radiation and conduction. ANALYSIS a) data agree with recommended values for high purity polycrystalline Al from the Seventh Thermal Conductivity Conference (Gai thersburg, MD, 1967). 127 Fenton, E. W. , Rogers, J. S. , and Woods, S. 8., Lorenz Numbers of Pure Aluminum, Silver and Gold at Low Temperatures , Can. J. Phys., 41, 2026-33 (Jul 1963) PURPOSE To attain accurate values of L for A1 , Au, Ag and compare with the Sommerfeld value. SPECIMEN a) length/area (cm-1 ): specimen 1 -3500, specimen 2 -1500, specimens 1,2 6 cm long; c) specimens 1,2 acid etched and annealed at 550 °C in air for 10 mi n. /specimens cut from 0.010 in. (0.025 cm) sheet, rod specimens rolled square and drawn once; d) 99.9999% A1 ; f) specimen 1 (p 0 = 9.03 x 10“ 10 flcm), specimen 2 (p 0 = 5.68 x 10" 10 ncm); g) source: Consolidated Mining and Smelting Co. of Canada. APPARATUS a ) longitudinal ; b) gas thermometers//al 1 leads to specimen are anchored to Cu pillars; c) temperature controlled radiation shield. DATA a) 2 to 50 K; b) figure 4 -L, Table 2 -pQ , p-j/X data points listed in TPRC data series; c) uncertainty: ±1.5%. ANALYSIS b) results agree with theoretical Sommerfeld value to about 1 1/2%. Flynn, D. R., private communication (1965) DATA a ) 120 to 720 K; b) /data points listed in TPRC data series. Gladun, C. and Holzhauser, W., Studies of Heat Conductivity at Low Temperatures , Monatsber. Dtsch. Akad. Wiss. Berl i n , 6_(4) , 310-3 (1964) PURPOSE To report development of an improved non-equilibrium apparatus and data for Cu, A1 , and Ti . SPECIMEN d) 99.99% A1 . APPARATUS a) longitudinal. DATA a) 4 to 55 K; b) figure 4 -X. ANALYSIS b) X -v T2 , p x T5 for T < 30 K. 128 Gostishcher, V. I. and Drozd, A. A., Heat Conductivity of Aluminum in Strong Transverse Magnetic Fields, Phys. Met. Metal logr. (USSR) (Engl. Transl.), 39(6), 168-70 (1975), Transl. of Fiz. Met. Metalloved., 39(6), 1307 (1975) PURPOSE To investigate the effect of strong transverse magnetic fields on x for high-purity aluminum, in view of its use as a sheath for superconductors . SPECIMEN a) 3.5 cm long, 0.15 cm dia., 0.3 cm dia. at thermocouple, thermometer and heater locations/rod; b) polycryst«ll i ne; c) /extruded through a round die; e) run 1 (RRR = 2800), run 2 (RRR = 4500); f) run 1, (p 0 = 8.55 x 10-1° ncm), run 2 (p 0 = 5.32 x 10”^ ftcm); g) run 1, 15 days after extrusion, run 2, 3.5 months after extrusion. APPARATUS a ) longitudi nal ; b) carbon resistance thermometers and Cu-Ag thermocouples c) radiation shield with maintained thermal gradient similar to specimen. DATA a) 6 to 48 K; b) figure 1 -X. ANALYSIS b) x in strong fields is independent of density of imperfections but determined by topology of the Fermi surface. [14A] Ho, C. Y. , Powell, R. W. , Liley, P. E. , Thermal Conductivity of the Ele-ments: A Comprehensive Review, J. Phys. Chem. Ref. Data, ]3, Supplement No. 1, 242-257 (1974) PURPOSE To provide a comprehensive listing of data on x of the elements. SPECIMEN d) high purity; f) p 0 = 5.94 x 10“ 10 frcm for T below 150 K. APPARATUS a) not given. DATA a) 0 to 8500 K; b) Table 2 -X, recommended values; c) uncertainty: ±5% below room temperature, ±3% above room temperature, and ±8% for the molten phase up to 1273 K. The values above 1273 K are provisional. 129 Hogan, C. L. , The Thermal Conductivity of Metals at High Temperature, Lehigh Univ., Bethlehem, PA., Ph.D. Thesis, (1950) 42 pp. PURPOSE To measure X for metals and alloys in the temperature range 0 to 1000 °C. SPECIMEN a) 12 in. (30.5 cm) long, 0.25 in. (0.64 cm) bore/tube; d) 99.996% A1 ; g) source: Norton RA 98 material. APPARATUS a) Forbes bar method; b) Chromel -A1 umel thermocouples glued with Alundum; c) Sil-o-cel insulation. DATA a) 0 to 790 °C ; b) /data points listed in TPRC data series; c) uncertainty: ±5%. ANALYSIS b) the basic theory of Wilson and Makinson is found to be valid. Lees, C. H., The Effects of Temperature and Pressure on the Thermal Conduc-tivities of Solids. Part II, Philos. Trans. Roy. Soc. London, Ser. A, 208, 381 443 (1908) PURPOSE To measure x and p for certain metals and alloys, and compare the results with electronic theories. SPECIMEN a) 7 to 8 cm long, 0.585 cm dia./rod; c) /specimen turned down from larger rod; d) 99% A1 ; g) density at 20 °C = 2.70 g/cm^, source: Johnson, Matthey and Co. APPARATUS a) longitudinal; b) Pt resistance thermometer -oil contact/cal ibrated at boiling point of 0 2 , icepoint and boiling point of H2 0. DATA a) -166 to 24 °C; b) Table on p. 413 -X, Graph on p. 432 -p; d) radiative heat loss, resistance of thermometer leads, offset of Pt resistor temperature from bar temperature. ANALYSIS a) Jager and Diesselhorst measurements , (Abh. Phys. Tech. Reichsanstal t , 3, p. 269 (1900), compared at 18 °C; b) found little variation in x in temperature range investigated. 130 Mendelssohn, K. and Rosenberg, H. M. , The Thermal Conductivity of Metals at Low Temperatures. I. The Elements of Groups 1, 2, 3, Proc. Phys. Soc., London, 65^(6), 385-8, (Jun 1952) PURPOSE To measure x for metals at low temperatures . SPECIMEN a) 15 cm long; 1 to 2 mm dia./rod; b) polycrystal 1 ine; c) annealed; d) 99.994% A1 ; g) source: Johnson, Matthey and Co. APPARATUS a ) longitudi nal ; b) gas thermometers. DATA a) 4 to 50 K; b) figure 4 - x/data points listed in TPRC data series; c) uncertainty: ±3% (maximum). ANALYSIS a) results disagree with those of Webber, R. T.. Andrews, F. A., and Spohr, D. A., Phys. Rev., 84, 994 (1951); b) X- = aT? + 3/ T. Merisov, B. A., Khotkevich, V. I., Zlobintsev, G. M., and Kozinets, V. V. , Thermal Conductivity of Some Metals and Alloys at 4.2 to 273 K, J. Eng. Phys. (USSR) (Engl. Transl . ) , 12(5) , 364-6 (1967) PURPOSE To report a thermal potentiometer method of measuring x in which the radiation correction is experimental ly determined. SPECIMEN d) 0.05% Cu, 0.03% Fe, 0.35% Si, 0.10% others. APPARATUS a) thermal potentiometer method; b) Ge resistance thermometer from 4.2 to 20 K, Cu-constantan thermocouple from 20 to 273 K. DATA a) 6 to 273 K; b) figure 2 -X/data points taken from Table in Merisov, et al.. Thermophysical Properties of Substances at Low Temperatures, First All-Union Meeting, Feb. 16-19, 1971, 85-88 (1972); c) uncertainty: ±5%; d) experimentally determined correction for radiation heat loss. 131 Mikryukov, V. E. , Thermal and Electrical Properties of Cu, Ag, A1 , and the Alloy System Cu-Be, .J. Moscow Univ. JJ?(6), 57-67 (1957), Vestn. Mosk. 'Jni v . , Ser. Mat., Mekh., Astron., Fiz., Khim., 12(6), 57-67 (1957 ) PURPOSE To present data on \ and p for some metals. SPECIMEN d) 99.99% A1 . APPARATUS a) not given. DATA a) 65 to 523 °C; b) Table 4 -A and p. ANALYSIS b) the Wiedemann-Franz law is not valid in this temperature range. Mikryukov, V. E. and Karagezyan, A. G., Thermal and Electrical Properties of Alloys of the Systems A1 -Mg and A1 -Cu , Inzh. Fiz. Zh., 4(12), 90-3 (1961 ) PURPOSE To investigate the temperature dependence of X and p, using the Wiedemann-Franz law for the systems Al-Cu and Al-Mg from room temperature to melting point. SPECIMEN a) 300 mm long, 3 mm dia.; c) annealed in vacuum at 430 to 520 °C; d) 99.9% A1 . DATA a) 60 to 480 °C; b) ,/data points listed in TPRC data series. A NALYSIS a) Wiedeman-Franz ratio in agreement with theoretical values; b) in the investigated alloys, thermal transport is basically by electrons. 132 Misiorek, H. , Zarrzewski , T. , and Rafalowicz, J., Influence of Plastic Deformation on the Thermal Conductivity Maximum of Copper and Aluminum in the Temperature Range 4.2 to 70 K, Phys. Status Solidi A, 47, K137-40 (1978) PURPOSE To determine the effect of increasing plastic deformation on A maximum of Cu and A1 in the range of 4.2 to 70 K. SPECIMEN c) both specimens annealed at 540 °C for 4 h in He atmosphere/deformed by extension at 77 K; d) 99.999% and 99.99% A1 . APPARATUS a) longitudinal steady state. DATA a) 5 to 50 K; b) figures 3, 4 -X (undeformed); e) captions for all figures switched: figure 1 should be labelled as 4, figure 2 as 3, figure 3 as 1, figure 4 as 2. ANALYSIS b) with increasing deformation, maximum A decreases in value and is shifted toward higher temperatures. Moore, J. P. , McElroy, D. L. , and Barisoni, M. , Thermal Conductivity Mea-surements Between 78 and 340 on Aluminum, Iron, Platinum, and Tungsten, Proceed-ings of the Sixth Thermal Conductivity Conf. (1966) PURPOSE To measure A, p and the Seebeck coefficient for A1 , Fe, Pt, and W between 78 and 340 K. SPECIMEN c) /machined from stock; d) 99.999% A1 ; e) RRR = 520; g) P273 = 2.440 yftcm, source: Reynolds Aluminum Co. APPARATUS a) longitudinal; b) Chromel-P and constantan thermocoupl es/cal ibrated against Pt resistance thermometers/thermocouple wires thermally grounded on guard cylinder; c) guard cylinder. DATA a) 100 to 380 K; b) Table 3 -A. ANALYSIS a) A values reported by Powell, et al . (1965) agreed to within 1/4%; c' the experimental A curve has a minimum which disagrees with the theoretical curve in magnitude and temperature. COMMENT This data set not used because of reported error in technique. See Coo 1 , J. G., Moore, J. P., Matsumura, T, and Van der Meer, M. P. , The Ther d ar : Electrical Conductivity of Aluminum, 0RNL-5079 (1975). 133 Mucha, J. and Rafalowicz, J., Thermal Conductivity Minimum of Aluminum, Phys. Status Sol i di A, 48, 221-4 (1978) PURPOSE To measure X minima for six different pure aluminum specimens, and to test Wilson's equation. SPECIMEN a ) 10.0 to 20.0 cm long, 0.8 cm dia./rod; b) polycrystal 1 ine; c) annealed at 500 °C, 48 h cooldown; d) specimen HI: 99.861%, H2: 99.526%, R0: 99.9931%, R6 : 99.97%; e) specimen HI (RRR = 36), H2 (RRR = 17), R0 ( RRR = 147), R6 (RRR = 86). APPARATUS a ) longitudinal ; b) constantan-mangani n thermocouples; c) 10"^ mm (10-5 p a ) insulation, screen at specimen temperature. DATA a) 80 to 380 K; b) figure 1 -X. ANALYSIS b) deviation of 6 to 7% of experimental data for least squares fit of W = e 0 /L 0 T + AT^ -BT^. The Wilson equation fails to describe the X curve in the minimum range. Mucha, J., Wlosowicz, D. , and Rafalowicz, J., Thermal Conductivity of Con-structional Aluminum at Temperatures Ranging from 77 to 300 K, Chlodnictwo, 4(11), 7-8 (1974) PURPOSE To report X for 99.8% and 98.5% Aluminum (constructional) from 77 to 300 K. SPECIMEN d) 99.8% A1 , 0.07% Fe, 0.057% Si, 0.006% Cu, 0.001% Ti , 0.008% V, 0.004% Mn, 0.001% Zn, 0.007% Cr, 0.006% Mg. APPARATUS a) longitudinal; b) room to N 2 temperatures : Cu-constantan thermo-couples; He temperatures : carbon resi stors/thermocouples scaled separately for room and N£ temperatures with a Hg thermometer, carbon resistors. DATA a) 77 to 300 K; b) Graph p. 7 -X; c) uncertainty: ±5% (maximum). 134 Powell, D . L. , Hall, W. J., and Roder, H. M. , Low Temperature Transport Properties of Comercial vetals and Alloys. II. Aluminums, J. Appl . Phys., 31(3), 496-503 (1960) >'JRP0SE '"o o^esent data ^or x, p, L, thermoel ectn'c force and thermoelectric power from 4 to 120 K. SPECIMEN a' specimen A, 3, C: 3.66 mm dia./rods; b) specimen A: single crystal; specimen B, C: polycrystalline; c) specimen A: ground from 3.68 to 3.55 mm, annealed in vacuum at 400 °C for 2 h; specimen 3: turned a^d d-awn from 0.5 in. '1.25 cm) sheet to 3.65 mm rod; specimen C: turned and d rawn from 0.5 in. '1.25 cm) sheet to 3.65 mr rod, annealed in vacuum at 350 °C for 1 n; d) specimen A: 99.995% A1 ; specimen 3: 99.308%; specimen C: 98.917%; g) other characteristics found in ^able 1. APPARATUS a) longitudinal; b) Au-Co vs. Cu thermocouples. DATA a) 4 to 120 K; b) Table II -X, figure 1 -p, figure 5 -L. ANALYSIS b~j L for high x samples were considerably below the Sommerfeld value while those for the low x samples were somewhat above it. Powell, R. W. , Tye, R. P. , and Woodman, M. J., 'Hie Thermal Conductivity o Pure and -.lloyed Aluminum. I. Solid Aluminum as a Reference Material, -d.ances ^nermophysical Properties at Extreme Temperatures a^d "ressu res , Tnird Sy^pos 1 ^ on ’’hermophysical Properties, ASME, 227-88 ' 1S55 ) PURPOSE To investigate the suitability of high purity and other aluminums as a X reference standard. SPECIMEN Low temperature: a) specimen S.P. 'super pure): 8 cm x 0.44 cm x 0.44 cm/bar; d) specimen S.P.: 99.993% A1 . High temperature: a) specimen S.P.: 28.0 cm x 2.81 cm dia.; specimen C51 : 27.4 c^ x 3.17 an dia. /rods; d) specimen S.P.: 99.993% A1 ; specimen C51 : 99.6% A1 . APPARATUS a) longitudinal; b) Ni-Cr, constantan, 90% Pt, 10% 3 n themocouD'es. DATA Low temperature: a) specimen S. D .: 123 to 323 K; b) Table III -X, p, and ; d) radiat'/e heat loss correction. High temperature: a) specimen S.P., C51: 323 to 873 K; b) Tab<e III -x, p, and d) radiative heat loss correction. ANALYSIS b") X = ATp + 8. 135 Powers, R. W. , Schwartz, D., and Johnston, H. L. , The Thermal Conductivity of Metals and Alloys at Low Temperatures, Ohio State Univ., Columbus, Rep. No. TR 264-5 (Apr 1951) Contract No. W33-038-AC-14794 (16243) 23 pp. PURPOSE To describe an apparatus for x measurements and discuss the results for pure A1 , Cu, and Ni from 25 to 300 K. SPECIMEN a ) 20 in. (51 cm) long, 0.5 in. (1.3 cm) di a . ; c) cold-drawn, 55% reduc-tion; d) 99.99+% purity; g) source: Aluminum Company of America. APPARATUS a) longitudinal; b) Cu-constantan thermocoupl es/thermocoupl es calibrated with a gas thermometer c) 3 gold plated temperature controlled shields and 10 _ 5 mm of Hg (10~3 Pa) insulation. DATA a) 25 to 238 K; b) Table II and Table III -X; c) uncertainty: ±0.6% at 250 K, ±0.7% at 100 K, ±1.8% at 30 K; d) corrections for radiation and con duction. Roberts, R. B. and Crisp, R. S. , Thermoel ectric Power and Thermal Conduc-tivity an Integral Method - Aluminum, Philos. Mag., 36(1), 81-9 (1977) PURPOSE To develop and evaluate an apparatus for the simultaneous measurement of x and thermoel ectric power of metal wire using an integral method from 2 to 300 K. SPECIMEN a) 18 cm long, 0.25 and 0.50 mm dia., for T < 70 K and T > 70 K respectively/wire; c) annealed at 600 °C at 10-6 mm of Hg (10"4 Pa) 24 h, slow cooled/1 mm wire drawn to 0.5 run and 0.25 mm dia. for two specimens; d) 99.9999% A1 . APPARATUS a) integral method; b) carbon resistance thermometer, thermocouple/thermo-couple calibrated against Pt thermometer calibrated by CSIR0 Sydney to the IPTS 68, carbon resistor was calibrated against He x and boiling point and against Pt thermometer at 15 K; c) thin-walled stainless steel tube heat shield and A1 Mylar surrounded complete assembly. DATA a) 2 to 300 K; b) /data points supplied by author; c) uncertainty: less than ±1.0%. ANALYSIS a) data agrees with TPRC recommended values for 100 to 300 K; b) the inte-gral method was shown to be capable of yielding accurate measurements of X 136 Rosenberg, H. M. , The Thermal Conductivity of Metals at Low Temperatures , Philos. Trans. Roy. Soc. London, Ser. A, 247 (933) , 441-97 (1955) PURPOSE To investigate and report on x for 32 metallic elements in the 2 to 40 or 90 K range, to give a general picture of X at low temperatures. Also to measure p so the Wiedemann-Franz relation could be studied. SPECIMEN a) 2.97 cm long, 0.394 cm dia./rod; b) polycrystall i ne; c) annealed in vacuum at 600 °C, several h; d) 99.994% A1 ; e) p 293/ p 20 = 279; g) source: Johnson, Matthey and Co. APPARATUS a) longitudinal; b) gas thermoineters//thermal ly anchored to liquefier and vacuum jacket; c) 10~ 5 mm of Mg (10“ p Pa) insulation. DATA a) 2 to 42 K; b) figure 12 - X/data points listed in TPRC data series; c) uncertainty: ±3%; d) corrections for external volume of gas thermom-eter. ANALYSIS a) compared to Andrews, Webber, and Spohr (1951); b) X“1 = aT^ + bT~1 ; WQ seems to vary as T^2 over large temperature range. Seeberg, P. and Olsen, T. , The Thermal Conductivity of Pure Aluminum at Low Temperatures, Phys. Norv.,_2(3), 197-201 (1967) PURPOSE To measure x for super pure aluminum and determine how the phonon scatter-ing term (W-j ) varies with purity. SPECIMEN a) specimens 1,2 -102, 65 cm long, 2.98, 2.03 mm dia. respect i vel y/ rods ; c) air annealed at 480 °C, 24 h/cold rolled, drawn, wound into helix; d) zone refined high purity A1 ; e) specimen 1 (RRR = 24,000), specimen 2 (RRR = 15,000); f) specimen 1 (p 0 = 1.3 x 10"^ ftcm), specimen 2 (p^ = 1.9 x 10“10 ft. cm); g) electron mean free path, specimen 1 (0.75 mm], specimen 2 (0.4 mm) . APPARATUS a) longitudinal; b) < 4 K: carbon resistors, > 4 K: Ge resistance ther-mometers/< 4 K: calibrated against He vapor pressure. DATA a) 2.5 to 33.5 K; b) figure 1 -X; c) uncertainty: ±2% (maximum), T < 4; ±5% (maximum), 5 < T < 30. ANALYSIS b) x_1 = aT^ + gT-1; Matthi essen ' s rule fails to provide a satisfactory description of the electronic x in metals, and the deviation of A1 follows a pattern similar to tin and indium. 137 Sirota, N. N. , Drozd, A. A., and Gostishcher, V. I., Measurement of Elec-trical and Thermal Conductivity of Metals in Strong Magnetic Fields, Thermo-physical Properties of Substances at Low Temperatures (Proceedings of the First All-Union Meeting, Feb. 16-19, 1971) M. P. Orlova, ed., All-Union Scientific Re-search Institute of Physico-Technical and Radiotechnical Studies (Moscow, 1972) pp. 149-58 PURPOSE To measure x for aluminum in strong, perpendicular magnetic fields. SPECIMEN d) 99.999% A1 . APPARATUS a) longitudinal-stationary heat flow; b) carbon resistance thermometers/ calibrated in strong magnetic fields (both perpendicular and parallel); c) temperature controlled shield. DATA a) 7 to 73 K; b) figure 6 -X, figures 5 and 7 p and L. Sirota, N. N., Gostishcher, V. I., and Drozd, A. A., Thermal Conductivity of Aluminum in Strong Magnetic Fields at Low Temperatures , JETP Lett., 16(4) , 170-2 (Aug 1972) PURPOSE To study x for metals in strong magnetic fields at low temperatures to separate electronic and lattice components. SPECIMEN a) 60 x 3 x 3 mm/bar; b) single crystal; c) /cut from ingot; e) RRR = 6000; f) P 0 = 1.2 x 10-10 s . cm . APPARATUS a ) longitudinal ; b) resistance thermometers ; c) radiation shield with tem-perature gradient similar to specimen. DATA a) 6 to 57 K; b) figure 1 - x(H=0); c) uncertainty: ±2%; e) RRR = 6000 is inconsistent with data, closer to 600. ANALYSIS b) temperature dependence of the lattice component of x in region of maxi-mum: Xg = AT^e-S'. A transverse magnetic field exerts a strong influence on x for high purity A1 . 138 [32A] Touloukian, Y. S. , Powell, R. W. , Ho, C. Y. , and Klemens, P. G. , Thermo-physical Properties of Matter, Volume 1: Thermal Conductivity, Metallic Clements and Alloys, 68-81 (1970) PURPOSE To provide an extensive list of data for A of the metallic elements and al loys. SPECIMEN d) 99.9999%; f) p0 = 5.93 x lO " 10 a cm. APPARATUS a) not given. DATA a) 0 to 8000 K; b) higure and Table 1R -A, recommended values; c) uncer-tainty: ± 3 % near room temperature, ±3 to 5% at other temperatures ; e) the values below 1.5 Tm are calculated to fit the experimental data by using n = 2.00, a = 0.61, m = 2.61, a !1 = 4.87 x 10‘ 5 , and 6 = 0.0245. Wilkes, K. E., Thermal Conductivity Measurements Between 77 K and 373 K on Iron, Cobalt, Aluminum, and Zinc, M.S. Thesis, Purdue University, (1968) 93 pp. PURPOSE To resolve discrepancies and fill gaps in the literature for A for Co, Al , and Zn. SPECIMEN a) 10.16 cm long, 1.225 cm dia./rod; b) polycrystalline; c) unannealed; d) 0.5 ppm Cu, 0.5 ppm Si, and 0.1 ppm Mg; f) P 273 = 0.02425 pfl.m, p 77 = 0.0021 pPm; g) density = 2698 Kg/m “ 3 at 23 °C. APPARATUS a) longitudinal; b) Chrome! -constantan thermocoupl es/T < 273 brated with a resistance thermometer and tables from Powell, Sparks, L. L. , MBS Report 9249, T > 273 K: rhodium thermocoupl e/al 1 thermocouple leads K: R. cal i-L., and rods; c) 5 x 105 mm of Hg (7 x 10 '^ pa) insulation. calibrated against a platinur-from specimen tied to support DATA a) 88 to 283 K; b) Table 4 -A, figure 13 -p/p data points generated from equation of line on figure 13; d) corrections for radiation and conduc-tion. ANALYSIS a) p curve is very close to Flynn; Powell, Tye, Woodman; and Moore, McElroy, Barisoni but A curves differ; b) L is well below theoretical but appears to approach theoretical value well above ''oom ter Denature . 139 4. Iron 4.1 General A total of 41 references on iron are included in the annotated bibliography of Section 4.5. The following references represent the primary data sets: 3A, 8, 9, 11, 13, 15, 16, 18, 23, 25, 28, 29, 31, 33, and 34. The primary data cover a range of temperatures from 1.5 to 1000 K, and a range of RRR from 4 to 200. These data are illustrated in Figs. 4.1.1 through 4.1.3 with a composite of the data given in Fig. 4.1.4. Iron produced in bulk form is generally of much lower electrical purity than either copper or aluminum. RRR values above 550 are not reported. Equation 1.1.3 was used to represent the iron data over the entire tempera-ture range. The values for the parameters, P-j , i = 1, 2 ... 7, obtained by nonlinear least squares fit are: P 1 166.9 x 10" 8 on li 238.6 It CM CL 1.868 P 6 1.392 P 3 1.503 x 10 5 P? = 0.0 P4 -1.22 with all units in SI. The data at high RRR 'were examined for systematic residual deviations as a function of temperature. The residuals were represented by the Wc term in Eq. 1.1.5 with the following equation: Wc = -0.004 Jin(T/440) exp( - ( An ( T/650/0.8) 2 ) -0.002 £n(T/90) exp(-Un(t/90)/0.45) 2 ) where Wc and T are in SI units. 141 4.2 Deviations From Recommended Equation The deviations of the primary data from Eq. 1.1.3 are illustrated in Figs. 4.2.1 through 4.2.3 with a composite shown in Fig. 4.2.4. Only three data sets exhibit differences of greater than +10%. Although deviations, systematic with temperature, exist for individual data sets, the overall pattern is random in nature. No systematic trends varying with RRR were identified. The primary data were selected from the literature data on relatively large, well annealed specimens. Therefore, the deviations exhibited in Figs. 4.2.1 through 4.2.4 are indicative of the combined effect of a) experimental measure-ment errors and b) the inability of Eq. 1.1.3 to account for the effects of chemical impurity variations. The effects of physical defect variations, small specimen size variations, and magnetic fields are exhibited, in part, by the deviations of the secondary data. The thermal conductivity variations caused by other than chemical impurity variations are not expected to be represented as well by Eq. 1.1.3. However, the RRR (or p 0 ) correlating parameter does account for an appreciable part of these variations. Some users may find this to be an adequate representation and, therefore, discussions of these comparisons are in-cluded for completeness. The deviations of the secondary data sets are divided into two groups ac-cording to the magnitude of the deviations. They are illustrated in Figs. 4.2.5 through 4.2.10. The equation developed here is also compared to the reference data in the following references 9A, 12, 20, and 32A in Fig. 4.2.11. The differences are within the combined uncertainties of the sources. It is noted that we did not include data within our primary set for RRR above 200. Therefore the comparison to the data in references 9A, 32A are an extrapolation of our equation. 142 4.2.1 Physical Defect Effects Investigations of physical defects in iron have produced only a few refer-ences 20, 29, 33. Each of these references will be discussed below. Reference 33 reports the effects of annealing on thermal conductivity. The peak value of the thermal conductivity of the unannealed specimen was 99 WnfK", while for the annealed specimen of the same Armco iron stock, it was 112 Wm“^K"^. The deviations from Eq. 1.1.3 for the unannealed specimen were within +9%, while those for the annealed specimen were within +8%. Although not directly related to physical defects, references 20 and 29 re-port on the effects of impurities. Reference 20 states that as much as 7% varia tion in thermal conductivity is possible at 298 K for a specimen of Armco iron. This difference is caused by variations of impurity concentrations in the Armco stock. Reference 29 reports that the peak conductivity of pure (99.99%) iron is 287 Wm_1 K_i at 32 K, while for SAE 1020 steel 99.48% Fe, peak value is 65 WttT-^K' 1 at 160 K. The deviations from Eq. 1.1.3 the pure iron were within +4%, while those for the steel were within +5%. Reference 20 also reports on the effect of ice-water quenching on thermal conductivity. A 1% increase was reported for specimens of thickness greater tha 8 mm. For a specimen of less than this thickness, ice-water quenching should produce a greater effect on the thermal conductivity. No thermal strain effect was produced. Although the temperature dependence of the physical defect scattering mechan ism is different from that due to impurity scattering, Eq. 1.1.3 represents the unannealed specimen data quite well (within +10%). This indicates that the re-sidual electrical resistivity characteri zes both types of scattering for the range of RRR included here. 143 4.2.2 Size Effects No size effect studies were found in our literature on iron. 4.2.3 Magnetic Field Effects Although magnetic field effects on thermal conductivity were not studied ex-plicitly, some interesting changes occur for iron (and ferromagnetic metals in general). We will discuss these effects below. Reference 3 shows that the thermal resistivity of a single crystal , oriented in the direction, decreases as the magnetic field increases from 0 to 12 KOe. Fickett 1 reports that ferromagnetic metals can show a decrease in re-sistivity with increasing fields. This decrease in a pure ferromagnetic metal is rather large. Note that the opposite behavior is seen in nonferromagnetic metal s. Takaki amd Igaki 2 report that the electrical resistivity decreases rapidly when a magnetic field of less than 0.8 kA/m is applied. The resistivity reaches a minimum at 32 and 40 kA/m in single crystal specimens whose orientations are and [Oil], respecti vely. This effect also occurs for polycrystal 1 i ne specimens in longitudinal fields of 56 to 64 kA/m. One of the conclusions drawn in this paper indicates that a modified residual resistivity ratio, RRR^, should be used to characterize the purity of iron (RRR h = p ( 298 ) /p^ where is the minimum value of p for a specimen in a longitudinal magnetic field of 56-64 kA/m). iFic.kett, F. R. , Electrical Properties of Materials and Their Measurement at Low Temperatures , NBS Technical Note 1053, National Bureau of Standards, Boulder, Colorado, p. 41 (1982). 2 Takaki , S. and Igaki , K, Electrical Resistivity of High Purity Iron at 4.2 K, Trans. Jpn. Inst. Met. JJ7, 353-9 (1976). 144 Equation 1.1.3 is expected to represent the thermal conductivity of specimens in a magnetic field reasonably ,vel 1 for temperatures above 30 X. 3ul ow this tem-perature, the deviations are expected to increase dramatical ly. 4.3 Electrical Resistivity and Lorenz Ratio It was desirable to examine the Lorenz ratio of iron during the course of this investigation. Therefore an approximation of electrical resistivity as a function of temperature md RRR was required. To obtain an approximate equation we selected those sources from the primary set that also reported electrical ''o-sistivity data and fitted Eqs. 1.2.3 to those data. The electrical resistivity data used here are shown in Figs. 4.3.1 to 4.3.3. The parameters obtained for this equation are: ? 1 = 41.47 x 10" 16 P 5 = 180.3 P 0 = 3.241 P,-= 1.947 i o P 3 = 7.638 x 10U ?! = 0.1867 P4 = 1.95 The systematic residuals, p c , obtained from this fit wera than represented by -3 x 10" 9 in (T/ 105 ) exp(-(in(T/120)/0.45) 2 ) All units are SI. The deviations of the data fron this fit are shown in Figs. 4.3.4 to 4.3.6. Again the spread in the mid-range (40 to 100 K) is relatively large with a cor-respondingly large uncertainty, however, for the purpose at hand this is con-sidered adequate. Smooth values of p vs. T at selected RRR values are plotted in Fig. 4.3.7. From Eqs. 1.1.3 and 1.2.3 smooth values of L(T,RRR) are plotted in Fig. 4.3.8. No unexpected irrejul unities appear in Fig. 4.3.8. In Section 1.5 we discuss the procedure for selecting values of p 0 and cal-culating RRR for each thermal conductivity data set. These values of p 0 along with the Sominerfel d value of Lorenz ratio were used to best fit each low tempera-ture data set. The resulting values of RRR obtained by this procedure are com-pared to the values reported in the references in Fig. 4.3.9 and are listed in Table 4.3.1. Figure 4.3.9 shows values of RRR (calc), those values from the above procedure, versus RRR (obs), those values reported in the references listed in the annotated bibl iography. Also shown in this figure is the line that repre-sents RRR (calc) = RRR (obs). Systematic deviations from this line indicate ranges in which the derived Eq. 1.1.3 is invalid. For iron, the calculated values of RRR are within 10% of the observed values for the range 13 to 200. Also, Table 4.3.1 indicates that the Sommerfeld value of the Lorenz ratio is valid for iron. 4.4 Summary for Iron Equation 1.1.3 represents the primary iron data to within +10'% of the experi-mental value at a given temperature. Deviations for unannealed specimens (i.e., those containing physical defects) are within +10%. Based on the deviations illustrated in Figs. 4.2.1 through 4.2.3 it is clear that the use of RRR in Eq. 1.1.3 accounts for a large proportion of the impurity effect in. iron. Equation 1.1.3 with the parameters listed here was used to gen-erate smoothed values of thermal conductivity as a function of temperature and selected values of RRR. These are listed in Table 4.4.1 and Fig. 4.4.1. 146 List of Tables and Figures for Iron Tab! es Page Table 4.3.1 Comparison of Calculated and Observed RRR Values for Iron 149 Table 4.4.1 Thermal Conductivity Values for Iron Calculated From Eq. 1.1.3 at Selected Temperatures and RRR Values 150 Figures Page Figure 4.1.1 Experimental thermal conductivity data selected from the following primary references in the iron annotated bibliography: (8,9,11,13,15,16) 151 Figure 4.1.2 Experimental thermal conductivity data selected from the following primary references in the iron annotated bibliography: (3A, 18, 23, 25, 28) 152 Figure 4.1.3 Experimental thermal conductivity data selected from the following primary references in the iron annotated bibliography: (29,31,33,34) 153 Figure 4.1.4 Composite of the data in figures 4.1.1 through 4.1.3 . . . 154 Figure 4.2.1 Thermal conductivity deviations of the iron data from the following primary references compared to Eq. 1.1.3: (8,9,11,13,15,16) 155 Figure 4.2.2 Thermal conductivity deviations of the iron data from the following primary references compared to Eq. 1.1.3: (3A, 18, 23, 25, 28) 156 Figure 4.2.3 Thermal conductivity deviations of the iron data from the following primary references compared to Eq. 1.1.3: (29,31,33,34) 157 Figure 4.2.4 Composite of the deviations in figures 4.2.1 through 4.2.3 158 Figure 4.2.5 Thermal conductivity deviations of the iron data from the following secondary references compared to Eq. 1.1.3: (1,3,4,5,6,10) 159 Figure 4.2.6 Thermal conductivity deviations of the iron data from the following secondary references compared to Eq. 1.1.3: (11,12,14,17,19,22) 160 147 Page Figure 4.2.7 Thermal conductivity deviations of the iron data from the following secondary references compared to Eq. 1.1.3: ( 21 ,22A,24 ,26 ,27 ,30 ,32 ) 161 Figure 4.2.8 Thermal conductivity deviations of the iron data from the following secondary refeences compared to Eq. 1.1.3: (33,34,35,36) 162 Figure 4.2.9 Composite of the deviations in figures 4.2.6 through 4.2.8 163 Figure 4.2.10 Thermal conductivity deviations of the iron data from the following secondary references compared to Eq. 1.1.3: (2,7,22B) 164 Figure 4.2.11 Comparison of Eq. 1.1.3 to the values recommended for iron in the following references: (9A,12,20,32A) .... 165 Figure 4.3.1 Experimental electrical resistivity data for iron selected from the following references in the iron annotated bibliography: (3A, 8, 11 ,15,16,25) 166 Figure 4.3.2 Experimental electrical resistivity data for iron selected from the following references in the iron annotated bibl iography : (31,34) 167 Figure 4.3.3 Composite of the electrical resistivity data in figures 4.3.1 and 4.3.2 168 Figure 4.3.4 Electrical resistivity deviations of the iron data from the following references compared to Eq. 1.2.3: (3A, 8, 11, 15, 16, 25) 169 Figure 4.3.5 Electrical resistivity deviations of the iron data from the following references compared to Eq. 1.2.3: (31,34) 170 Figure 4.3.6 Composite of the electrical resistivity deviations shown in figures 4.3.4 and 4.3.5 171 Figure 4.3.7 Electrical resistivity for iron as a function of temperature calculated from Eq. 1.2.3 at selected values of RRR 172 Figure 4.3.8 Lorenz ratio for iron as a function of temperature calculated from Eq. 1.2.3 and Eq. 1.1.3 at selected values of RRR 173 Figure 4.3.9 RRR values calculated as per Section 1.5, RRR(CALC), versus reported RRR values, RRR(OBS), for iron 174 Figure 4.4.1 Thermal conductivity for iron as a function of temperature calculated from Eq. 1.1.3 at selected values of RRR 175 148 le en 3A 3A 3A 8 11 13 15 15 16 10 11 11 12 14 36 Comparison of Calculated and Observed RRR Values for Iron. RRR (obs.) RRR (calc Primary Data 20.89 20.89 21.24 21.24 21.55 21.55 23.0 23.0 13.0 12.7 23.33 24.1 98.0 90.0 95.0 103.0 40.3 40.3 Secondary Data 189.0 189.0 13.0 13.9 13.0 12.8 23.0 23.4 29.4 29.4 36.4 36.4 149 Table 4.4.1. Thermal Conductivity Values for Iron Calculated from Eq. 1.1.3 at Selected Temperatures and RRR Values. X( W'liT 1 •K~ 1 ) T (K) RRR = 10 RRR = 30 RRR = 100 RRR = 300 1 2.5 8.1 28 84 2 5.1 16.3 56 168 3 7.6 24 83 251 4 10.1 32 111 333 5 12.6 41 138 414 6 15.2 49 166 492 7 17.7 57 192 567 8 20 65 218 637 9 23 73 244 702 10 25 81 269 761 12 30 96 315 858 14 35 111 357 925 16 40 125 393 961 18 45 139 422 970 20 49 152 445 957 25 61 179 471 863 30 71 198 462 735 35 79 208 429 609 40 86 210 384 500 45 91 204 336 410 50 94 194 292 340 60 96 170 225 247 70 94 150 183 195 80 92 133 156 164 90 89 122 138 144 100 87 114 126 130 150 81 94 100 102 200 78 88 91 92 250 76 82 85 85 300 72 77 79 79 400 64 67 68 69 500 58 60 60 61 600 52 53 54 54 700 46 47 47 47 800 41 41 42 42 900 36 37 37 37 1000 32 32 33 33 150 THERMAL CONDUCTIVITY ,W/m . 1 2 4 10 20 40 100 200 400 TEMPERATURE, K FLgure 4.1.1 Experimental thermal conductivity data selected from the following primary ref erences In the Iron annotated bibliography: (8,9,11,13,15,16) 0= (8), A-(9), -(11), V -(13), O-(15), +-(15), X-(16) 151 THERMAL CONDUCT I V I TY , W/m . Figure 4.1.2 Experimental thermal conductivity data selected from the following primary ref erences In the Iron annotated bibliography: ( 3R , 1 8 , 23 , 25 , 28 ) O-(18), A-(3fl) , -(23), V-(25), O-(3R) , +-(3fl), X» (28) 152 THERMRL CONDUCTIVITY, W/m. Figure 4.1.3 Experimental thermal conductivity data selected f rom the following primary ref erences In the Iron annotated bibliography: (29,31,33,34) O-(29), A-(29), -(31), V-(31), O-(33), + -(34) 153 THERMAL CONDUCTIVITY, W/m. 1 2 4 10 20 40 100 200 400 TEMPERATURE, K Figure 4.1.4 Composite of the data In figs. 4.1.1 through 4.1.3 154 1 2 4 10 20 40 100 200 400 TEMPERATURE, K Figure 4.2.1 Thermal conductivity deviations of the Iron data from the following primary references compared to eq. (1.1.3): (8,9,11,13,15,16) O-(8), A-(9), -(11), V-(13), O-(15), +-(15), X-(16) 155 Figure 4.2.2 Thermal conductivity deviations of the Iron data from the following primary references compared to eq. (1.1.3): (3FI, 18,23,25,28) O-(18), A-( 3R ) , -(23), V-(25), O -(3R) , +-(3fl), X-(28) 156 1 2 4 10 20 40 100 200 400 TEMPERRTURE,K Figure 4.2.3 Thermal conductivity deviations of the Iron data f rom the following primary ref erences compared to eq. (1.1.3): (29,31 ,33,34) O-(29), A-(29), -(31), V-(31), O -(33), +-(34) 157 o -) cr 0 1 CO CD O bJ CD E- CD Cl. 15 10 -5 -V x" 7 + +^ 7 \ +M -o O ° -srOBaa^^^y . £- r -<^--5 J--10 -x -15 --20 XC33 ^ ^ A S aa A aO O + 10 20 40 100 TEMPERATURE, K 200 400 Figure 4.2.4 Composite of the deviations In figs. 4.2.1 through 4.2.3 158 Figure 4.2.5 Thermal conductivity deviations of the Iron data from the following secondary references compared to eq. ( 1 . 1 .3) :( 1,3,4,5,6,10) O-(1), A- (3), -(4), V-(5), O-(6), +-(6), X-(10) 159 Figure 4.2.6 Thermal conductivity deviations of the Iron data from the following secondary ref erences compared to eg. U. 1. 3) : (11,12,14,17,19,22) O-(11), A- dll, -(12), V -(14), O-(17), + -(19), X-(22) 150 2 4 10 20 40 100 200 400 1000 TEMPERATURE, K Figure 4.2.7 Thermal conductivity deviations of the Iron data from the following secondary ref erences compared to eq. (1.1.3) : (21 ,22R,24,26,27,30,32 ) O-(21), A-(22R) , -(24), V-(26), O " (27), +-(30), X-(32) 161 Figure 4.2.8 Thermal conductivity deviations of the Iron data f rom the following secondary ref erences compared to eg. (1.1.3) : (33,34,35,36) O-(33), A-(33), -(34), V-(35), <0-(36) 152 20 — i 1 i • i i I i i | CD _J az 0 1 • CD CD O QJ O f- CD Q 15 10 5 0 -5 -10 15 h a a aa + Z—• A _ A A O &0 A tr =f V + S -k-—Q-A° < q _ . o l -20 10 20 40 100 200 TEMPERATURE, K 400 1000 Figure 4.2.9 Composite of the deviations In figs. 4. 2. 6 through 4.2.8 163 PCT. DEV. (OBS.-CPILC. Figure 4.2.10 Thermal conductivity deviations of the Iron data from the following secondary references compared to eq. (1 . 1 .3) : (2,7,22B) O-(2) , A -(2), -(2), V-(7 ) , O-(22B) 164 Figure 4.2. 11 Comparison of eq. (1.1.3) to the values recommended for Iron In the following references; (9fl, 12,20, 32FI) O- (9m , A-(9fl) , -(12), V-(20), O-(32R) 165 ELECTRICAL RESISTIVITY, nfl. 2 4 10 20 40 100 200 400 TEMPERATURE, K Figure 4.3.1 Experimental electrical resistivity data for Iron selected from the following references In the Iron annotated bibliography: (3fl,8, 1 1 , 15, 16,25) O-(8), O -(25), A-(11), + -(3R) , -(15), X-(30) V-(16), 166 ELECTRICRL RESISTIVITY, nfi. Figure 4.3.2 Experimental electrical resistivity data for Iron selected from the following references In the Iron annotated bibliography: (31,34) O-(31), A-(34) 167 ELECTRICRl RESISTIVITY, nfl. 2 4 10 20 40 100 200 400 TEHPERRTURE,K Figure 4.3.3 Composite of the electrical resistivity data In figs. 4.3.1 and 4.3.2 2 4 10 20 40 100 200 400 TEMPERATURE, K Figure 4.3.4 Electrical resistivity deviations of the Iron data f rom the following ref erences compared to eq . (1.2.3) : (38,8,11,15,16,25) o-(8) , A-(11), -(15), V-(16), O-(25) , + -(3fl) , X-(3R) 169 Figure 4.3.5 Electrical resistivity deviations of the Iron data from the following references compared to eg. (1.2.3) : (31,34) O-(31), A-(34) 170 Figure 4.3.6 Composite of the electrical resistivity deviations shown In figs. 4.3.4 and 4.3.5 171 ELECTRICAL RESISTIVITY, nfh TEMPERATURES Figure 4.3.7 Electrical resistivity for Iron as a function of temperature calculated from eq. (1.2.3) at selected values of RRR. 172 JLill I ,> 1, >l,i hi AiJLl— l.i t ,U LORENZ RATIO, 10 °VVK TEMPERATURE, K Figure 4.3.8 Lorenz ratio for Iron as a function of temperature calculated from eq. (1.2.3) and eq. (1.1.3) at selected values of RRR. 173 RRR(CRLC) Figure 4.3.9 RRR values calculated as per Section 1.5, RRR(CRLC), versus reported RRR values, RRR(0BS),for Iron. O” Primary, A = Secondary, ” Secondary 174 THERMRL. CONDUCTIVITY,W/m. TEMPERflTURE,K Figure 4.4.1 Thermal conductivity for Iron as a function of temperature calculated from eq. (1.1.3) at selected values of RRR. 175 4.5 FORMAT FOR ANNOTATED BIBLIOGRAPHY OF IRON AUTHOR, TITLE, CITATION REFERENCE ANNOTATION PURPOSE SPECIMEN a) Dimensions/Shape; b) Crystal Status; c) Thermal /Mech . History; d) Purity Specification; e) RRR; f) p0 ; g) Other Characteri zation Data APPARATUS a) Type; b) Thermometry/Cal ibration/Anchoring; c) Thermal Isolation; d) Other (Q meas. ) DATA a) Temperature Range/Difference; b) Content of Tables, Figures and Equa-tions/Data Extraction; c) Uncertainty/Imprecision; d) Disputable Corrections to Measurements by Authors; e) Errata (by Author or Reviewer) ANALYSIS a) Comparisons; b) Conclusions 176 [ 1 ] Androulakis, J. G. and Kosson, R. L. , Experimental Determination of the Thermal Conductivity of Solids Between 90 and 200 K, 7th Conference on Thermal Conductivity, Nat. Bur. Stand. (U.S.), Spec. Publ . No. 302, 337-348 (1968) PURPOSE To describe details of an apparatus for measuring A which can be used for good and poor conductors by varying sample geometry. SPECIMEN c) /machined from bar; g) SAE 1020 steel. APPARATUS a) longitudinal; b) Cu-constantan thermocouples; c) feedback controlled guard, radiation shields; 105 -106 pa vacuum. DATA a) 78.8 to 154 K; b) figure 7 -A; c) uncertainty: ± 8% at 89 K, ±5% at 200 K. ANALYSIS a) results agree with Powell, R. L. and Blanpied, W. A., NBS, Circular 556 , and Scott, R. B., Cryogenic Engineering, D. Van Nostrand Co., New York (1959). Arajs, S. , Oliver, B. F. , and Dunmyre, G. R. , Thermal Conductivity of High-Purity Iron at Low Temperatures , J. Appl . Phys., 36, No. 7, 2210-2 (1965) PURPOSE To study A of a high purity iron as a function of temperature and compare the results with theory. SPECIMEN a) specimen A: 15 cm long, 1.75 cm dia./rod; b) specimen B: polycrystal-line; c) specimen A: distilled and reduced to metal with H2 , electron beam zone refined, annealed in Pd purified Ho, specimen B: annealed/ specimen A: swaged and cleaned with chemical polish before anneal; d) specimen A: 99.997%, specimen B: 99.926%. APPARATUS a) longitudi nal . DATA a) specimen A (run 1 ): 6 to 198 K, specimen A (run 2): 6 to 78.4 K, speci-men B: 5 to 193 K; b) figure 1 -A. ANALYSIS a) A maximum higher than those of Rosenberg, H. M. , Phil. Trans. R. Soc . (London), A247 , 441 (1955) and Kemp, W. R. G. , Klemens, P. G., and Tainsh, R. J., Ann. Phys. (Leipzig), _5, 35 (1959). 177 Beitchman, J. G. , Trussel , C. W. , and Coleman, R. V., Electron Transport and Lorenz Number in Iron, Phys. Rev. Lett., 25(18), 1291-94 (Nov 1970) PURPOSE To measure p and W of Fe single crystals from 2 to 77 K with and without magnetic fields and investigate theoretical predictions for L and electron transport theory. SPECIMEN b) single crystal , <1 1 1 > axial orientation; e) 700 < RRR < 2000. DATA a) 2.2 to 20 K; b) figure 1 - W(H = 0, 95 A/m). ANALYSIS b) intrinsic temperature dependence of p is aT^, and behavior of single domain state (with H / 0) is not too different from non-transition metals. [3A] Berman, R. , Hardy, N. D. , Sahota, M. , Hust, J. G. , and Tainsh, R. J., Standard Reference Materials for Thermal Conductivity Below 100 K, Proceedings of the Seventeenth Conference on Thermal Conductivity, Gai thersburg , MD, 105-16 (1983) PURPOSE To report results of a CODATA round-robin investigation involving standard reference materials of stainless steel, tungsten, and electrolytic iron. SPECIMEN from Leeds University a) SRM 734; b) approximately 1/4 in. (dia.) x 2 in. (6 x 50 mm); c) RRR = 21.24. SPECIMEN from National Measurement Laboratory a) SRM 734; b) approximately 1/4 in. (dia.) x 2 in. (6 x 50 mm); c) RRR = 21.55. APPARATUS a) not described. DATA from Leeds University a) 3 to 100 K; b) Table -A/data made available by private communication. DATA from CSIR0 a) 4 to 90 K; b) Table -A, p, L/data made available by private communica-tion. 178 Bideau, 0., Troadec , J. P. , Meury, J. L. , Rosse, G., and Dang Tran Quan, Si-multaneous Measurements of Thermoel ectri c Power and Thermal Conductivity of Small Samples from 100 to 300 K, Rev. Phys. Appl . , 13(8) , 415-8 (Aug 1978) PURPOSE To simultaneously measure thermoel ectric power and x on small samples from 100 to 300 K, and present evaluation of apparatus. SPECIMEN a) 6 mm long, 6.4 mm diam./rod; c) /faces polished; d) SRM 734 electrolytic Fe. APPARATUS a) longitudinal ; b) Cu-constantan thermocouples//^ tempered specimen insul ation. leads; c) radiation shield, 10~6 mm foil of for thermal contact Hg (10-4 Pa), DATA a) 100 to 300 K; b) figure 6 -X; c) uncertainty: <±3%. ANALYSIS a) results deviate <3% from Hust, J. G. , Giarratano, P. J., Nat. Bur. Stand. (U.S.), Spec. Publ . No. 260 (1975). Cason, J. L. , Jr., Thermal Conductivity Measurements of Eu 0 across the Curie Point, S.D. School of Mines and Technol., Rapid City, Dept, of Physics, Rept. No. TR-22 (Oct 1967), Contract No. NONR-2964 (01), 85 pp., M.S. Thesis PURPOSE To measure X of Fe as a standard for other measurements. SPECIMEN a) 19 mm long, 3 mm dia./rod; g) source: Battelle Memorial Institute; Armco. APPARATUS a) absolute longitudinal; b) Cu-constantan thermocouples on specimen, GRT on heat sink, PRT in bath; c) radiation shield, heater temperature monitor, vacuum insulation. DATA a) 76.6 to 160 K; b) figure 13 -X; c) total uncertainty: ±20% at 68 K, ±14% at 297 K. ANALYSIS b) results agree within 6% of Powell, R. W. , Progress in Internati onal Re -search on Thermodynamic and Transport Properties, Academic Press, New York (1962). 179 deNobel , J., Heat Conductivity of Steels and a Few Other Metals at Low Tem-peratures, Physica (Utrecht), 17, 551-62 (1951) PURPOSE To measure X for some metals and alloys used in cryostat construction. SPECIMEN c) specimen 6936: forged, specimen 1 1 66 A/4 : annealed at 800 °C, furnace cooled; d) specimen 6936: 99.93% Fe, specimen 1 166 A/4 : 0.14% C, 0.08% Si, 0.07% Mn; g) source: Sir R. A. Hadfield, specimen 1 166 A/4 : mild steel, 104 hardness B., specimen 6936: 103 hardness 3. APPARATUS a ) longitudi nal ; b) gas thermometers/cal ibrated against Pt thermometer in bath; c) 10"6 mm of Hg (10 / ^ Pa) vacuum. DATA a) 16 to 93 K; b) Table 2 -X; d) correction for thermometry heat leak. ANALYSIS b) X is proportional to T at low temperatures. Fieldhouse, I. B., Hedge, J. C., and Lang, J. I., Measurements of Thermal Properties, WADC Tech. Rep. 58-274, 20 pp; ASTIA 206892 (1958) PURPOSE To calibrate a x apparatus. SPECIMEN a) 0.625 in. (1.59 cm) I.D. x 3.0 in. (7.6 cm) 0.0. x 1.0 in. (2.5 cm) thick/disks; g) source: Armco. APPARATUS a ) radial ; b) Pt vs. Pt + 10% Rh thermocouples; c) bubble alumina, end guard heaters, water-cooled steel housing. DATA a) 208 to 2410 K; b) Table 1 -X; c) uncertainty: ±5%. ANALYSIS a) results agree with Powell, R. W. , Proc. Phys. Soc., London, 4(5, pp. 659-74 (1934). 180 Fulkerson, W. , Moore, J. P. , and McElroy, D. L. , Comparison of the Thermal Conductivity, Electrical Resistivity and Seebeck Coefficient of a High-Purity Iron and an Armco Iron to 1000 °C, J. Appl . Phys., 37_(7), 2639-53 (1966) PURPOSE To make a comparative investigation of ingot and high purity Fe for x, p, and S up to 1000 °C SPECIMEN a) 3 in. (7.6 cm) dia., 1.13 to 1.45 in. (2.87-3.68 cm) thick, 9 in. (23 cm) tall stack/disks; e) RRR = 9.6, 23.0; g) source: Armco. APPARATUS a) radial; b) Pt -10% Rh vs. Pt annealed reference grade thermocouple; c) guard disks. DATA a) 323 to 1273 K; b) Table III -x, Table IV -p; c) uncertainty in X: ±1.5%; uncertainty in p: ±0.3%. ANALYSIS a) results agree with previous measurements. Godfrey, T. G. , Fulkerson, W., Kollie, T. G., Moore, J. P., and McElroy, D. L. , Thermal Conductivity of Uranium Dioxide and Armco Iron by an Improved Radial Heat Flow Technique, U.S.A.E.C. Rep. 0RNL-3556, 1-67 (1964) PURPOSE To describe an improved apparatus (radial heat flow) for making measure-ments on solids from -57 to 1100 °C and report results and analysis on UO 2 and ingot Fe. SPECIMEN a) 1.375 in. (3.493 cm) outer radius, 5/8 in. (1.59 cm) inner radius, 1 in. (2.54 cm) thick; d) < 99.4%. APPARATUS a ) radi al ; b) Pt vs. Pt-10% Rh reference grade thermocouple; c) He atm, end guard heaters, granular alumina, insulation. DATA a) 385 to 990 K; b) Table El -X/data points listed in TPRC data series; c) uncertainty: ±1.5%. ANALYSIS a) over the temperature range 100 to 1000 °C, results agree to ±3% with Powell, R. W., ASME, N.Y. (1962), Laubitz, M. J., Can. J. Phys. 38(7), 837 (1960), and Cody, G. D. , Abeles, B. , and Beers, D. S. , Trans. Metal 1 . Soc. AIME , 221(2), 25 (1961). 181 [9A] Ho, C. Y., Powell, R. W. , and Li ley, P. E. , Thermal Conductivity of the Elements: A Comprehensive Review, J. Phys. Chem. Ref. Data, 3 , Supplement No. 1, 242-257 (1974) PURPOSE To provide a comprehensive listing of data on x of the elements. SPECIMEN d) high purity; f) p 0 = 1.43 x 10" 8 ncm for T below 200 K. APPARATUS a) not given. DATA a) 0 to 6500 K; b) Table 78 -x, recommended values; c) uncertainty: ±5% below 100 K, ±3% from 100 K to room temperature, ±2% from room temperature to about 1000 K, increasing to ±8% at 1600 K, and ±15% at the melting point. Holder, T. K., Thermal Conductivity, Electrical Resistivity and Seebeck Co-efficient of High Purity Iron and Selected Iron Alloys from 90 to 400 K, O.R.N.L., Tenn., Metals and Ceramics Div., Rep. No. ORNL/TM-5539 , Contract No. W-7405-Eng-26. Also M.S. Thesis, 110 pp. (Jun 1977) PURPOSE To report x, p, and S of pure Fe and some Fe alloys from 90 to 400 K, cal-culate xe and Xg and compare to theory. SPECIMEN a) 7.6 cm long, 0.65 cm dia./rod; d) 99.994%; e) RRR = 189; g) zone re-fined. APPARATUS a ) longitudinal ; b) chromel-P: constantan thermocoupl es//al 1 specimen leads tempered; c) temperature controlled guard, vacuum insulation. DATA a) 90 to 300 K; b) Table 2 -X, p. Table 12 -L; c) uncertainty: ±1.2%; d) isothermal corrections due to small thermocouple differences. ANALYSIS b ) all oys deviated from Matthiessen 1 s rule. 182 Hust, J. G. , Thermal Conductivity Standard Reference Materials from 4 to 300 K. I. Armco Iron, National Bureau of Standards, Boulder, Colo., Report No. 9740 (Aug 1969) 97 pp. PURPOSE To measure A, p, L and thermopower of several specimens of Armco Fe from 4 to 300 K to check usefulness as an SRM. SPECIMEN a) 23 cm long, 3.6 mm dia./rod; c) annealed at 870 °C in gas-heated air muffle for 1/2 h then in 10“^ mm for 1.5 h at 875 °C, held at 150 °C for 24 h and repeated/machined between anneals; d) 0.015% C, 0.028% Mn , 0.005% P, 0.025% S, 0.003% Si, 0.04 Cu; e) RRR (mean) = 13; g) source: Battelle Memorial Institute, hardness B: 37.1, grain size: 0.064 mm. APPARATUS a) longitudinal; b) 8 thermocouples; c) glass fiber around specimen, tem-perature controlled shield. DATA a) specimens 2, 2a, 4: 6 to 300 K; b) Tables 20, 21 , 22 -A, p, L; c) un-certainty: ±2.5% at 300 K, decreases as T^ to 200 K, 0.7% at 200 to 50 K increasing as 1/T to 1.5% at 4 K. ANALYSIS a) RRR mean values are 5.5% below values of Lucks, C. F. , rms deviation is 3.6% compared to Luck's 6.5%. Hust, J. G. and Giarratano, P. J., Thermal Conductivity and Electrical Re-sistivity Standard Reference Materials: Electrolytic Iron SRM's 734 and 797 from 4 to 1000 K, Nat. Bur. Stand. (U.S.), Spec. Publ . No. 260-50, 32 pp. (Jun 1975) PURPOSE To review development of SRM's, give selection criteria and compile and compare data on A and p for electrolytic Fe and other similar specimens. SPECIMEN a) 23 cm long, 3.6 mm dia./rod; c) annealed at 1000 °C for 2 h, held at 800 °C for 2 h; d) 99.90%; e) mean RRR = 23; g) density = 7.867 g/cm^, Rockwell hardness and grain size: B24, 0.05 mm respectively. APPARATUS a) longitudinal ; b) thermocouples; c) glass fiber, temperature controlled shield, insulation. DATA a) 6 to 1000 K; b) Table 1 -A, p, L; c) uncertainty: 2.5% below 280 K, 3% above 280 K. 183 Hust, J. G. and Sparks, L. L. , Thermal Conductivity Standard Reference Materials from 4 to 300 K. II. OSRM Iron-1265, National Bureau of Standards, Boulder, Colo., Report No. 9771 (Oct 1970) 35 pp. PURPOSE To measure X, p, and L for OSRM Fe-1265 from 4 to 300 K and study the vari-ability by RRR measurements. SPECIMEN a) 23 cm long, 3.6 mm dia./rod; c) annealed at 1000 °C in vacuum or He for 2 h; d) < 99.87%; e) RRR = 23.33 ±0.24; g) density: 7.867 g/cm^, hard-ness B: 23.5 Rockwell, Grain size: 0.0507 mm, OSRM Fe-1265. APPARATUS a) longitudinal; b) 8 thermocouples; c) glass fiber around specimen, tem-perature controlled shield. DATA a) 6 to 280 K; b) Table 8 -A, p, L; c) uncertainty: ±2.5% at 300 K, de-creases as T^ to ±0.7% at 200 K, ±0.7% from 200 to 50 K, increases in-versely with T to ±1.5% at 4 K. ANALYSIS b) Mathiessen's rule application shows dependence of p-j on p 0 . Karweil, J. and Schafer, K. , The Thermal Conductivity of Some Poorly Con-ducting Alloys between 3 and 20 K, Ann. Phys. (Leipzig), 36.» 567-77 (1939) PURPOSE To determine X, p, and L for some important alloys from 3 to 20 K. SPECIMEN a) 2.54 mm dia., 12 cm long/rod; c) /drawn; d) electrolytic Fe; f) RRR = 29.4. APPARATUS a) longitudinal; b) gas thermometers , thermocouples/thermocouples cali-brated at L 02 /leads to specimen run through bath; c) radiation shield, i nsul ation. DATA a) 4.9 to 82 K; b) graph p. 575 -X. ANALYSIS b) Lorenz ratio is reasonably constant at low temperatures. 184 Kemp, W. R. G. , Klemens, P. G. , and Tainsh, R. J., Thermal and Electrical Conductivities of Rhodium and Iron, Ann. Phys. (Leipzig), _5(7), 35-41 ( 1959) PURPOSE To measure cr and X for pure Fe and Rh specimens from 2 to 90 K. SPECIMEN a) specimen 1: 28 x 2.5 x 2.5 mm, specimen 2: 2.4 x 1.7 x 30 mm/bars; c) specimens 1,2: annealed at 950 °C and reannealed/cut from a precipitated plate, compressed between anneals, specimen 2 face ground; e) specimen 1 (p 0 = 0.09 pfl.cm), specimen 2 (p 0 = 0.092 yftcm); g) similar material as used by Griineisen, E., doubly refined electrolytic Fe. APPARATUS a ) longitudi nal ; b) gas thermometers ; c) vacuum, temperature controlled shield, insulation. DATA a) specimen 1: 6.5 to 90 K, specimen 2: 7.7 to 91 K; b) figure 3 -X, Tables 1,2 -p 0 , p, respecti vely. ANALYSIS a) results of p 0 /p 273 higher than Griineisen, E. and Goens, E. , Z. Phys. 44, 615 (1927). Kemp, W. R. G. , Klemens, P. G. , and White, G. K., Thermal and Electrical Conductivities of Iron, Nickel, Titanium and Zirconium at Low Temperatures, Aust. J. Phys., 9(2), 180-8 (1956) PURPOSE To measure x for several metals down to 2 K and determine Xg, X e and L. SPECIMEN a) 2 mm dia./rod; c) annealed in vacuum at 750 °C for 4 h; d) 99.995%; e) RRR = 40.3; f) p0 = 0.248 yft.cm; g) source: Johnson Matthey. APPARATUS a) longitudinal ; b) gas thermometers ; c) temperature controlled shield, vacuum insulation. DATA a) 1.5 to 128 K; b) figures 1,2,4 -X, L, p, respecti vel y. ANALYSIS a) results agree with Rosenberg, R. M. , Philos. Trans. Roy. Soc. London, Ser. A, 247, 441 (1955). 135 Kohlhaas, R. and Kierspe, W. , The Thermal Conductivity of Pure Iron and Some Ferritic and Austenitic Steels Between the Temperature of Liquid Air and Room Temperature, Arch. Eisenhuettenwes. , 36(4), 301-9 (1965) PURPOSE To measure A and p of pure Fe and steels between liquid air and room temperature, determine Ae and Ag, estimate X from p and examine effect of alloying elements on X. SPECIMEN g) 0.5 cm dia, 15 cm long/rod; d) 99.93%. APPARATUS a) longitudinal; b) Fe-constantan thermocouples; c) vacuum. DATA a) 88 to 300 K; b) Tables 2, 3 -X, p; c) uncertainty: ±3.5%. ANALYSIS b) xe , Xg are temperature dependent. Laubitz, M. J., Thermal and Electrical Properties of Armco Iron at High Temperatures , Can. J. Phys., 38(7), 887-907 (1960) PURPOSE To accurately determine x, a and thermoel ectric power of Armco Fe from 0 to 1000 °C in an effort to solve discrepancies in the literature. SPECIMEN a) 7.156 cm long, 2.324 cm dia. /rod; c) annealed at 850 °C 1/2 h/specimen cut from 12 in. (30.5 cm) long sect.; g) source: Battelle Memorial Institute. APPARATUS a) longitudinal; b) 6 Pt/Pt -10% Rh thermocouples//thermocouple leads anchored to guard; c) radiation shields, alumina guard and powder, vacuum i nsul ation DATA a) 303 to 973 K; b) Table 2 -X, Table 3 -p, L; c) uncertainty: ±2.5%: d) non-uniform power distribution of heater, deviations from thermal equilibrium, average correction: 0.06%. ANA LYSIS a) results agree with Powell, R. W. , Proc. Phys. Soc. 46, 659 (1934), but disagree with Hattori , D., Sci. Rep. Tohoku Univ. 2Q, 190 (1937), and Maurer, E. , Arch. Ei senhuettenwes. , 10, 1945 (1936). 186 Lees, C. H. , The Effects of Temperature and Pressure on the Thermal Conduc-tivities of Solids -Part II. The Effects of Low Temperatures on the Thermal and Electrical Conductivities of Certain Approximately Pure -fetal s and Alloys, Philos. Trans. Roy. Soc. London, Ser. A, 208 , 381-443 (1908) PURPOSE To make measurements of x for metals and alloys below 0 °C and to determine the variation with temperature. SPECIMEN a) 7-8 cm long, 0.585 cm dia./rod; c] /turned from bar; d) 99.99%, ("best scrap Fe M ); g) density = 7.74 gm/cm"^ at 21 °C. APPARATUS a) longitudinal ; b) Pt resist thermometers/cal ibrated at boiling point of O 2 , ice point and boil ding point H 2 O. DATA a) 113 to 286 K; b) Table p. 416 -X, graph p. 425 -p; d) correction for radiation heat loss, conduction and offset of Pt temperature from rod tem-perature. ANALYSIS a) Jager and Diesselhorst compared at 18 °C; b) little variation in x over temperature range investigated; L doesn't agree with theory. Lucks, C. F. , Armco Iron: New Concept and Broad-Data Base Justify Its Use as a Thermal Conductivity Reference Material, J. of Testing and Evaluation, 1 , No. 5, 522-31 (1973) PURPOSE Evaluate published data on ingot iron to determine its continued use as a reference materi al . DATA a) 73 to 1273 K (no new data). ANALYSIS Recomnends continued use of ingot iron as reference material. 187 Lucks, C. F. and Deem, H. W. , Thermal Properties of Thirteen Metals, ASTM Spec. Tech. Pub! . No. 227 (1958) PURPOSE To measure A of ingot Fe as a standard for other measurements. SPECIMEN a) 6 in. (15 cm) long, 3/4 in. (1.9 cm) dia./rod; c) /hot rolled; g) source: U. S. Steel Co. APPARATUS a) comparative longitudinal; b) No. 36 B and S gage: (Cu-constantan) , (chromel -al umel ) , (Pt 6% Rh -Pt 30% Rh) or No. 30 B and S gage Pt-Pt 10% Rh thermocouples/ingot Fe as reference; c) double-walled guard tube, 5 x 10~5 mm (7 x 10"3 pa) vacuum. DATA a) specimen 1: 116 to 293 K; b) Table 12 -A. ANALYSIS a) results agree with Am. Mach. 8£, pp. 869-80 (1938). Lucks, C. F., Thompson, H. 8., Smith, A. R., Curry, F. P., Deem, H. W., and Bing, G. F. , The Experimental Measurement of Thermal Conductivities, Specific Heats and Densities of Metallic, Transparent and Protective Materials. Part I., USAF TR 6145-1, 1-127 (1951) [ATI 117 715] PURPOSE To survey literature and make experimental determinations of A, specific heat and densities of metallic, transparent and protective materials used in missiles and supersonic aircraft. SPECIMEN a) 15 cm long, 2 cm dia./rod; c) /hot rolled; d) 99.44%; g) SAE 1020 steel from U.S. Steel Corp. APPARATUS a) comparative longitudinal; b) Cu-constantan thermocouples/cal ibrated to ingot Fe standard on hot end of specimen; c) double guard tube, vacuum in-sul ation. DATA a) 111 to 265 K; b) Table 12 -A (experimental). Table 19 -A (interpo-1 ated). ANALYSIS a) results agree with Armstrong, L. D. , and Dauphinee, T. M. , Can. J. Res., Sect. A, 25,' (Nov 1947) pp. 357-74, Shelton, S. M. , J. Res. Nat. Bur. Stand., 12 (RP 669), (Apr 1934) pp. 441-50, Hattori , D., Sci. Rep. Tohoku Imp. Univ., 26, (1937) pp. 190-205, and Kannuluik, W. G., Proc. Roy. Soc. London, 141,71933) pp. 159-68. 188 [22A] Maurer, E. , Heat Conductivity of Chrome Steels at High Temperatures , Arch. Ei senhlittenw, _10(4) , 145-54 (1936) PURPOSE SPECIMEN II a) 98.7% APPARATUS a) longitudi nal DATA a ) 300 to 970 K/data points taken from TPRC. [22B] McDonald, W. J., Jr., The thermal and Electrical Conductivities of High Purity Iron at Low Temperatures , M.S. Thesis, Univ. of Texas, Austin, Tex. (1962) PURPOSE To report low temperature X and a for a sample of pol ycrystal 1 i ne Fe. SPECIMEN a) 0.062 x 0.062 x 1/5 in. (0.16 x 0.16 x 3.8 cm)/bar; b) polycrystall i ne; c) zone refined/machined from ingot; g) Rengstorff and Goodwin, J. Met. _7, 647 (1955). APPARATUS a) longitudi nal ; b) Au-Co vs. manganin thermocoupl es/cal i brated in place/ specimen leads anchored to exchange gas chamber; c) radiation shield, vacuum insulation. DATA a) 6 to 107 K; b) figure 6 -X, figure 7 -p, figure 10 -L. ANALYSIS b) W L = W -aT" 1 ; X % T3 for T < 10 K. 189 Mendelssohn, K. and Rosenberg, H. M. , The Thermal Conductivity of Metals at Low Temperatures. II. The Transition Elements, Proc. Phys. Soc., London, Sect. A, 65, 388-94 (1952) PURPOSE To measure A for the transition elements from 2 to 40 K and determine their temperature dependences. SPECIMEN a) 5 cm long, 1-2 mm dia./rod; c) annealed; d) 99.99% Fe; g) source: Johnson Matthey. APPARATUS a) longitudinal ; b) gas thermometers . DATA a) 2.3 to 32 K; b) figure 2 -A; c) uncertainty: ±3% maximum. ANALYSIS b) 1/A = aT2 + 0/T. Moak, D. P. , Thermal Energy Storage Supporting Research, Final Tech. Rep. NASA-CR-80058 N67-12043, 1-300 (1966) PURPOSE To measure A of ingot Fe for comparative measurements of some potential thermal energy storage materials for spacecraft. SPECIMEN a) 0.75 in. (1.9 cm) dia., 5.00 in. (12.7 cm) long/rod; d) 99.9% Fe; g) source: Armco. APPARATUS a ) longitudinal ; b) chromel -al umel thermocouples; c) vacuum, guard shield and bubbled alumina insulation. DATA a ) 107 to 280 K; b) Table G-8 -A; c) uncertainty: ±5%. ANALYSIS a) results agree with previous measurements and with Powell, R. W. , et al . , Armco Iron as a Thermal Conductivity Standard, Review of Published Data, p. 454, ASME and Academic Press, New York (1962). 190 Moore, J. P. , McElroy, D. L. , and Barisoni, M. , Thermal Conductivity Mea-surements Between 78 and 340 K on Aluminum, Iron, Platinum and Tungsten, Sixth Thermal Conductivity Conference, Dayton, Ohio, (Oct 1966), Air Force Materials Lab., Wri ght-Patterson AFB, Ohio, pp. 737-78 PURPOSE To make X, p, and S measurements from 78 to 340 K for A1 , W, Pt , Fe, and compare to theory of Makinson for electronic component. SPECIMEN e) RRR = 201: g) electron beam zone refined (3 passes), density = 7.824 g/cm 3 , source: Materials Research Corp. APPARATUS a ) longitudinal ; b) chromel-p vs. constantan thermocoupl es/from calibrated spools/tempered to guard cylinder; c) temperature controlled guard cylinder, Au plated 5 x 10"? mm of Hg (7 x 10 -3 Pa) vacuum. DATA a) 90 to 300 K; b) Table 5 -X, p; c) uncertainty: ±1.8%. ANALYSIS a) results agree with Richter, U. F. and Kohlhaas, R. , Z. Naturforsch. , Teil A, [19, (1964) p. 1632, and Powell, R. W., Hickman, M. J., Tye, R. P. , and Woodman, M. J., ASME (1962), pp. 466-73. National Physical Laboratory, The Thermal Conductivity of Iron, NPL, England, Report for 1964, Basic Physics Division, 128-30 (1965) PURPOSE To show that the results of Lucks, Deem (1958) are inconsistant with present and previous values at high temperatures. SPECIMEN a) 10.45 cm dia. with 1.27 cm dia. axial hole/disks; d) 99.97% Fe; g) source: N.P.L. Metallurgy Division. APPARATUS a ) radial ; b) thermocouple; c) 2 x 10" 5 mm (3 x 10" 3 Pa) vacuum in-sulation, end guards DATA a) 373 to 973 K; b) Table 2 - x/data points listed in TPRC data series. ANALYSIS a) Powel 1 , R. W. , Proc. Phys. Soc., London,^, 407 (1939), Lucks, C. F. , and Deem, H. W. , Am. Soc. Test. Mater., Spec. Tech. Publ . No. 227 (1958), Powell, R. W. , Hickman, M. J., Tye, R. P. , and Woodman, M. J., Progress in Internati onal Research on Thermodynamic and Transport Properties , Academic Press, 466 (1962), Laubitz, M. J., Can. J. Phys., 41, 1663 (1963). 191 Powell, R. W. , Armco Iron as a Thermal Conductivity Standard, ASME 2nd Sym-posium of Thermophysical Properties, 454-65 (1962) PURPOSE To review published data on ingot iron to determine if it is useful as a thermal conductivity standard. DATA a) -180 to 1320 °C (no new data). ANALYSIS Conclude that ingot iron is a good standard up to 800 °C. Above 800 °C data spread is large. Powell, R. W. , Hickman, M. J., Tye, R. P., and Woodman, M. J., Armco Iron as a Thermal Conductivity Standard, 43 Mew Determinations at N.P.L., Progress in International Research on Thermodynamic and Transport Properties , ASME 2nd Sym-posium on Thermophysical Properties, 466-73 (1962) PURPOSE To report new measurements on X and p for ingot iron, recommended values to 1000 °C for use as a standard SPECIMEN a) 10 cm long, 0.63 cm dia./rod; c) machined to specifications; d) 99.84%; g) source: Battelle Memorial Institute. APPARATUS a) absolute longitudinal; b) 42 SWG Ni-Cr and constantan thermocouples; c) radiation shield, insulating powder, vacuum insulation. DATA a) 73 to 1273 K; b) Table 4 -A, p, L; d) heat dissipation in heater wire not connected to rod, heat transfer to or from connecting leads, radiated heat. ANALYSIS a) agrees with Lucks, C. F. and Deem, H. W. , Am. Soc. Test. Mater., Spec. Tech. Publ . No. 227 (1958); b) recommended for thermal conductivity standard. 192 Powers, R. W. , Ziegler, J. B. , and Johnston, H. L. , The Thermal Conduc-tivity of Metals and Alloys at Low Temperatures : II Data on Iron and Several Steels Between 25 and 300 K. Influence of Alloying Constituents, USAF TR 264-5, 20 pp. (1951) [ATI 105923] PURPOSE To provide x data on pure Fe and commercially important steels from 25 to 300 K. SPECIMEN a) specimens 1, 2: 20 in. (51 cm) long, 0.5 in. (1.3 cm) dia./rods; d) specimen 1: 99.99%, specimen 2: 99.5%; g) specimen 1: pure Iron, source: Johnston-MacKay , Ltd., specimen 2: SAE 1020, source: Carnegie Illinois Steel Corp. APPARATUS a) longitudinal; b) Cu-constantan thermocouples; c) double radiation shield, vacuum insulation DATA a) 26 to 300 K; b) Table 2 -specimen 1 -X, Table 3 - specimen 2 -X; c) uncertainty: ±1.9% at 30 K, ±1.1% at 100 K, ±1.0% at room temperature. ANALYSIS a) agrees with Armstrong, L. D. and Dauphinee, T. M. , Can. J. Res., Sect. A, 2_5, 356 (1947), Powell, R. W. , J. Iron Steel Inst., London, CLIV, No. 2, 105 (1946). Richter, F. and Kohlhaas, R. , Thermal Conductivity of Pure Iron Between -180 and 1000 °C with Special Regard to Phase Transformations, Arch. Eisenhuettenwes. , 30(11), 827-833 (1965) PURPOSE To describe a process for calculating the absolute value of X for T > 300 °C, and report measurements for x and p for pure Fe from -180 to 1000 °C. SPECIMEN a) 10-30 mm thick, 63 mm dia./disk; c) vacuum melted; d) 99.94%. APPARATUS a) longitudinal ; b) Fe-constantan thermocouples. DATA a) 93 to 973 K; b) Equations 2,3, Table 2 -X, p, L; c) uncertainty: ±8.7%. ANALYSIS b) X = a + ST-1 T > 400 K. 193 Rosenberg, H. M. , The Thermal Conductivity of Metals at Low Temperatures , Philos. Trans. Roy. Soc. London, Ser. A, 247 , 441-97 (1955) PURPOSE To investigate and report on X for 32 metals in the 2-40 or 90 K range, and to measure p so that the Wiedemann-Franz Law could be studied. SPECIMEN a) 2.89 cm long, 0.202 cm dia./rod; b) polycrystal 1 i ne; c) annealed in vacuum for several hours; d) 99.78%; e) RRR = 63; g) source: Johnson-Matt hey, Co. APPARATUS a) longitudinal ; b) gas thermometers ; c) vacuum insulation DATA a) run 2: 2.0 to 93 K; b) figure 29/ taken from TPRC data series; c) uncer-tainty: ±3%; d) correction made for external volume of gas thermometer. ANALYSIS a) previous measurements not in agreement with present results; previous measurements discarded; b) 1/x = aT^ + g/T. Silverman, L., Thermal Conductivity Data Presented for Various Metals and Alloys up to 900 °C, J. Met., 5, 631-2 (1953) PURPOSE To present thermal conductivity data between room temperature and 900 °C for various metals and alloys used in the electron tube industry. SPECIMEN a) dimensions not given/rod; c) annealed at 900 °C; d) 99.89%; g) source: Svea Iron. APPARATUS a) comparative longitudinal; b) thermocouples/Pb was the primary standard with advance (55 Cu-45 Ni) as the working standard; c) guard tube. DATA a ) 50 to 900 °C , ; b) Table 2 -X; c) uncertainty: ±2%. ANALYSIS b) X decreases with temperature. 194 [32A] Touloukian, Y. S. , Powell, R. W. , Ho, C. Y., and Klemens, P. G. , Thermo-physical Properties of Matter, Volume 1: Thermal Conductivity, Metallic Elements and Alloys, 68-81 (1970) PURPOSE To provide an extensive list of data for x of the metallic elements and alloys. SPECIMEN d) 99.998%; f) p0 = 3.27 x 10‘ 8 a-crn. APPARATUS a) not given. DATA a) 0 to 6500 K; b) c igure and Table 24R-1 -x, recommended values; c) uncer tainty: ±3% near room temperature, ±3 to 8% at other temperatures ; e) the values below 1.5 Tm are calculated to fit the experimental data by using n = 2.10, a = 0.37, m = 2.47, a" = 2.05 x 10' 5 , and & = 1.34. Vuillermoz, P. L. and Pinard, P. , Conductibil ite Thermique du fer a basse temperature, C. R. .Acad. Sci . , Ser. B Paris, 277 , 493-5 (1973) PURPOSE To test the accuracy of an apparatus designed to measure the thermal conduc tivity of small samples of semiconductors. SPECIMEN a) specimens 1, 2: 8.9 x 3.74 x 4.44 mm, specimen 3: 10.78 x 3.04 x 2.95 mm/bars; b) polycrystalline; c) specimen 1: annealed for 30 min at 870 °C, again for 90 min at 875 °C in vacuum, and 24 h at 150 °C in vacuum, specimens 2, 3: no anneal; d) specimen 3: 99.78%; g) specimens 1, 2: Armco. APPARATUS a) longitudinal . DATA a) 4 to 210 K; b) Fig. 1 -X; c) uncertainty: ±6%. ANALYSIS a) Must, J. G. , Nat. Bur. Stand. Rep., private communication; b' conclude that the apparatus is sufficiently precise. 195 Watson, T. W. , Flynn, D. R. , Robinson, H. E. , Thermal Conductivity and Electrical Resistivity of Armco Iron, J. Res. Nat. Bur. Stand., Sect. C, 71(4) , 285-91 (1967) PURPOSE To present X and p data on samples of Armco Iron. SPECIMEN a) specimen 1: 37 cm long, 2.386 cm dia./rod; c) specimen 1: annealed for 1/2 h at 870 °C/specimen 2: cold worked; d) specimens 1 and 2: 99.67%; g) specimen 1 source: Battel le round-robin program, specimen 2 source: Redstone Arsenal . APPARATUS a) longitudinal; b) chrome! P-al umel /cal ibrated at NBS; c) heat shield, diatomaceous earth insulation. DATA a) specimen 1: -160 to 640 °C, specimen 2: -150 to 200 °C; b) speci-men 1: Table 2 -X, p, L, specimen 2: Table 3 -X, p, L; c) corrections for heat exchange with surroundi ngs. ANALYSIS b) expected specimen 2 to have greater x values, lower p values, but found the opposite to be true. Watson, T. W. , Robinson, H. E. , Thermal Conductivity of Some Commercial Iron-Nickel Alloys, J. Heat Transfer, 83, 403-8 (1961) PURPOSE To present X data for some Fe-Ni alloys from -150 to 540 °C. SPECIMEN a) 37 cm long, 2.54 cm dia./rod; d) 99.15%; g) AISI 1015 steel supplied by International Nickel Co. APPARATUS a) longitudinal; b) chrome! -al umel thermocouples; c) radiation shield, diatomaceous earth insulation DATA iy~^150 to 540 °C ; b) Table 2 -X. ANALYSIS a ) agrees with Shelton, S. M. , J. Res. Nat. Bur. Stand., 12_(4), RP 669 (1934), Powell, R. W., Research, (London) , 7_(12) (1954), Powell, R. L. and Blanpied, W. A., Nat. Bur. Stand. (U.S.), Circ. No. 556 (1954). 196 Wilkes, K. E. , Thermal Conductivity Measurements between 77 K and 373 K on Iron, Cobalt, Aluminum, and Zinc, M.S. Thesis, Purdue Univ. (1968) PURPOSE To measure pure Fe as a check on the accuracy of apparatus. SPECIMEN a) 10.44 cm long, 1.247 cm dia./rod; d) 99.96%; e) RRR = 36.4; g) density: 7.879 g/cm. APPARATUS a) longitudinal ; b) chrome! -constantan thermocoupl es/cal ibrated against Pt resistance thermometer; c) vacuum insulation. DATA a) 82 to 372 K; b) Table 2 -X, p; c) uncertainty: ±1.6%; d) radiation, gas conduction corrections. ANALYSIS a) agrees with Moore, J. P. , McElroy, D. L. , and Barisoni, K., Sixth Conf. on Thermal Conductivity, 737 (1966), Richter, Kohlhaas (1964). 197 198 5. Tungsten 5.1 General The annotated bibliography for tungsten (Section 5.5) includes 39 references. The following data sets were selected as primary data: 4A, 26A, 11, 14, 20, 21, 22, 23, 25, 27, 29, 32, and 33. The primary data ranges in temperature from 2 to 3000 K, and in RRR from 30 to 170. The primary data are shown in Figs. 5.1.1 through 5.1.4. As for the other metals, the data are divided into groups of seven and the last figure shows a composite of all data. Because high purity, single crystal tungsten specimens exhibit unusual be-havior at the lowest temperatures ; these data are not included in the primary data set. As a consequence, the range of RRR included in the fit of the data is more restricted than the total range of data. Equation 1.1.3 was fitted to the primary data over the entire range of tem-peratures. The values of the parameters, P-j , i = 1, 2, ... 7, obtained by non-linear least squares fit are P x = 31.70 x 10‘ 8 P 5 = 69.94 P 2 = 2.29 P 6 = 3.557 P 3 = 541.3 P 7 = 0.0 P4 = -0.22 with all units in SI. The systematic residuals from this equation were then represented by the Wc term in Eq. 1.1.5. The result is 199 W c = -0.00035 £n( T/130) exp(-Un(T/230)/0.7) 2 ) + 0.00015 exp(-(£n(T/35OO)/0.8) 2 ) + 0.0006 2 n( T/90) exp(-Un(T/30)/0.4) 2 ) + 0.0003 Jin(T/24) exp( - (£n( T/33)/0. 5) 2 ) where Wc and T are in SI units. 5.2 Deviations From Recommended Equation The deviations of the primary data from Eq. 1.1.3 with these parameters are illustrated in Figs. 5.2.1 through 5.2.4. No deviations exhibit differences greater than + 1%. Although there are systematic trends with respect to tempera-ture, the overall pattern is random in nature. No systematic trends varying with RRR were identified. The primary data were selected from the literature data on relatively large, well annealed specimens. Therefore, the deviations exhibited in Figs. 5.2.1 through 5.2.4 are indicative of the combined effect of a) experimental measure-ment errors and b) the inability of Eq. 1.1.3 to account for the effects of chemical impurity variations. The effects of physical defect variations, small specimen size variations, and magnetic fields are exhibited, in part, by the de-viations of the secondary data. The thermal conductivity variations caused by other than chemical impurity variations are not expected to be represented as well by Eq. 1.1.3. However, the RRR (or p 0 ) correlating parameter does account for an appreciable part of these variations. Some users may find this to be an adequate representation and, therefore, discussions of these comparisons are in-cluded for completeness. 200 The deviations of the secondary data sets are illustrated in Figs. 5.2.5 through 5.2.10. These data are divided into two subgroups according to the mag-nitude of the deviations. The composite plots for each group are 5.2.7 and 5.2.10, respecti vely. Again it was of interest to compare this equation to existing reference data. The deviations of these data are illustrated in Fig. 5.2.11. The data from ref-erence 15 are for Standard Reference Material (designated as SRM 730) at values of RRR = 50, 75, and 100. The reference data from references 12A,27A are for an RRR of 2850 and therefore represent an extrapolation of the equation as given here. Note that the differences in deviations between references 12A and 27A are about 30% between 10 and 20 K. The differences otherwise are within the combined uncertai nties of the reference data. 5.2.1 Physical Defect Effects Investigations of physical defects in tungsten have produced some interesting references 3, 14, 22. Each of these references will be discussed below. Reference 14 reports on the effects of annealing NBS sintered tungsten. The peak conductivity for the unannealed condition was 399 WnT 1 K _ 1 , while anneal-ing produced a peak value of 635 Wvrf^K" 1 . The deviations from Eq. 1.1.3 for both conditions were within +3%. Reference 3 shows the effects of surface condition on thermal conductivity. A polished specimen was measured to have a conductivity of 23000 WvtT^K' at 2 K and 33000 at 4 K. The specimen was then etched, and the resulting measurements were 25000 (at 2 K) and 35000 Wm -1 ’K -1 (at 4 K). The specimen was repolished and measured to have conductivities of 30000 W’m'^'K - at 2 K, 40000 WvrT-'-'K'-'-at 4 K. The specimen was then sandblasted, and the conductivities were found to be 33000 Wm-1 K -1 at 2 K, 42000 W’nT^K -1 at 4 K. Notice that the repolished specimen gave conductivities that were 201 comparable to the original condition, indicating that surface condition is ap-proximately reversible. Although not directly related to physical defects, reference 22 describes the effects of purity on thermal conductivity. Three specimens were involved: a high purity specimen, a "radial" specimen (99.8% W) and a W + 2% Ta specimen. The maximum decrease of 10% occurs at 90 K between the high purity and "radial" specimens. The W + 2 Ta specimen's conductivity was lowered by a factor of two at 90 K relative to the high purity specimen. The deviations from Eq. 1.1.3 for the high purity specimen were within +6%, while the "radial" specimen deviations were within +2%. The W + 2 Ta specimen was not compared to Eq. 1.1.3 because of our requirement that specimens must have less than 1% total impurities. Although the temperature dependence of the physical defect scattering mech-anism is different from that due to impurity scattering, Eq. 1.1.3 represents the unannealed specimen data quite well (+5%). This indicates that the residual electrical resistivity characterizes both types of scattering for the range of RRR included here. 5.2.2 Size Effects In reference 31, the authors demonstrate that specimen size can cause a noticeable change in thermal conductivity values. For a specimen of 3.0 mm diameter (specimen W-7) the peak conductivity was 73000 while for one of 1.5 mm (W-5) the corresponding value was 36000 W'nT^K". For a 1.0 mm diameter specimen (W-4) the value decreased to 15000 W'm --- •!<"-. There is a definite correlation between thermal conductivity and specimen size. The deviations from Eq. 1.1.3 of the 3.0 mm specimen were within +28%, those for the 1.5 mm diameter specimen were within +4%. The deviations for the 1.0 mm diameter specimen were within +12%. 202 Reference 4 shows a similar effect. For a 3.2 mm diameter specimen, the peak conductivity was 86000 while for a 1.4 mm diameter specimen, the value was 58000 For a 0.8 mm diameter specimen, the peak value was 42000 The deviations from Eq. 1.1.3 of the 3.2 mm specimen were within +50%, while those for the 1.4 mm diameter specimen were within +48%. The deviations for the 0.8 mm diameter specimen were within +36%. The maximum deviations indicated above have some common characteri sties. They are all positive, which implies that the calculated value of Eq. 1.1.3 is much too small. The temperature range in which these deviations occur is between 5 and 25 K. 5.2.3 Magnetic Field Effects Although magnetic field effects on thermal conductivity were not explicitly studied, reference 8 shows that an increase in the field decreases the specimen conductivity. For a zero field at 15 K, the conductivity was 8500 Wm”^K“^, while for a field of 2.58 T, the value at 15 K was 61 W'nf^'K”. The authors note that the field dependence increases towards lower temperatures. Refer-ence 9 extends these measurements to higher magnetic fields. The conductivity of the specimen field at 15 K in a field of 3.64 T was 38 Wm_1, K_1 . Reference 30 also showed that there is a decrease in longitudinal magneto-resistance with increasing magnetic fields. The author states that for a field of 1.3 T, the resistance at 4 K had increased by five orders of magnitude rela-tive to its value in a zero field. Thermal conductivity values were reported for specimens in the zero field. Reference 4 confirms that there are five orders of magnitude difference between a zero and a 2.3 T field. 203 5.3 Electrical Resistivity and Lorenz Ratio During this investigation, it was frequently helpful to examine the Lorenz ratio as a function of RRR and temperature. For this reason it was necessary to obtain an approximation of the electrical resistivity as a function of tempera-ture and RRR. For this approximation, we selected those data sources from the primary data that also contained electrical resistivity data. The data used is illustrated in Figs. 5.3.1 through 5.3.3. Figure 5.3.3 is a composite for all of the electrical resistivity data. These data were represented via a nonlinear least squares fit of Eq. 1.2.3. The resulting parameters are P. = 4.801 x 10" 16 P c = 55.63 1 5 P 9 = 3.839 P 6 = 2.391 P 3 = 1.88 x 10 10 P = 0.0 P4 = 1.22 where all units are SI. The systematic residuals from this equation were subsequently represented by the p c term in Eq. 1.2.4 as follows: P c = 0.7 x 10“ 8 £n( T/560) exp ( - ( an ( T/1000 ) /0 . 6) 2 ) The deviations of the experimental data from this equation are illustrated in Figs. 5.3.4 through 5.3.6. Smooth curves are calculated and plotted in Fig. 5.3.7 at RRR values of 30, 100, and 300. From the (T,RRR) and p(T,RRR) equations, values of L(T,RRR) were calculated at the same RRR values and are plotted in Fig. 5.3.8. No unusual behavior in this plot is observed. In Section 1.5 we discuss the procedure for selecting values of p 0 and cal-culating RRR for each thermal conductivity data set. These values of p 0 along with the Sommerfeld value of Lorenz ratio were used to best fit each low 204 temperature data set. The resulting values of RRR obtained by this procedure are compared to the values reported in the references in Fig. 5.3.9 and are listed in Table 5.3.1. Figure 5.3.9 shows values of RRR (calc), those values from the above procedure, versus RRR (obs), those values reported in the references listed in the annotated bibl iography. Also shown in the figure is the line that repre-sents RRR (calc) = RRR (obs). Systematic deviations from this line indicate ranges in which the derived Eq. 1.1.3 is invalid. The primary data for tungsten extend only from 30 to 400 in RRR. These values agree to within 10%. The secondary data at RRR near 60,000 disagree from the line by as much as 30%. Equation 1.1.3 is considered valid only from 30 to 400. In this range it appears that the Sommerfeld value is valid for the Lorenz ratio of tungsten. 5.4 Summary for Tungsten Equation 1.1.3 represents the primary tungsten data to within +10% of the experimental value at a given temperature. Deviations for unannealed specimens (i.e., those containing physical defects) are also in this range. Based on the observed deviations of the primary data set, the uncertainty of the recommended values is as follows. The uncertainty of the low temperature \ values is estimated to be +10% for RRR values from 30 to 300. At RRR values out-side this range, the uncertainty is larger. At temperatures above 200 K, the un-certainty is smaller (+5%) and is expected to be valid for much larger RRR val ues. Equation 1.1.3, with the parameters listed, was used to calculate thermal conductivity values for selected temperatures and RRR. These values are listed in Table 5.4.1 and plotted in Fig. 5.4.1. 205 List of Tables and Figures for Tungsten Tables Page Table 5.3.1 Comparison of Calculated and Observed RRR Values for Tungsten 208 Table 5.4.1 Thermal Conductivity Values for Tungsten Calculated From Eq. 1.1.3 at Selected Temperatures and RRR Values 209 Figures Page Figure 5.1.1 Experimental thermal conductivity data selected from the following primary references in the tungsten annotated bibliography: (4A, 11 ,14,20,22) 210 Figure 5.1.2 Experimental thermal conductivity data selected from the following primary references in the tungsten annotated bibliography: (4A,21 ,22,23,25,26A) 211 Figure 5.1.3 Experimental thermal conductivity data selected from the following primary references in the tungsten annotated bibliography: (27,29,32,33) 212 Figure 5.1.4 Composite of the data in figures 5.1.1 through 5.1.3 . . . 213 Figure 5.2.1 Thermal conductivity deviations of the tungsten data from the following primary references compared to Eq. 1.1.3: (4A, 11 ,14,20,22) 214 Figure 5.2.2 Thermal conductivity deviations of the tungsten data from the following primary references compared to Eq. 1.1.3: (4A,21 ,22 ,23 ,25 ,26A) 215 Figure 5.2.3 Thermal conductivity deviations of the tungsten data from the following primary references compared to Eq. 1.1.3: (27,29,32,33) 216 Figure 5.2.4 Composite of the deviations in figures 5.2.1 through 5.2.3 217 Figure 5.2.5 Thermal conductivity deviations of the tungsten data from the following secondary references compared to Eq. 1.1.3: (1,2,5,7,8,10) 218 Figure 5.2.6 Thermal conductivity deviations of the tungsten data from the following secondary references compared to Eq. 1.1.3: (10,13,16,19,26) 219 206 Page Figure 5.2.7 Composite of the deviations in figures 5.2.5 and 5.2.6 . . 220 Figure 5.2.8 Thermal conductivity deviations of the tungsten data from the following secondary references compared to Eq. 1.1.3: (3,4, 4B ,11 ,28) 221 Figure 5.2.9 Thermal conductivity deviations of the tungsten data from the following secondary reference compared to Eq. 1.1.3: (31) 222 Figure 5.2.10 Composite of the deviations in figures 5.2.8 and 5.2.9 . . 223 Figure 5.2.11 Comparison of Eq. 1.1.3 to the values recommended for tungsten in the following references: (12A,15,27A) . . . 224 Figure 5.3.1 Experimental resistivity data for tungsten selected from the following references in the tungsten annotated bibliography: (5A, 11 ,14,22) 225 Figure 5.3.2 Experimental electrical resistivity data for tungsten selected from the following references in the tungsten annotated bibliography: (4A,21 ,23,25, 26A, 33) 226 Figure 5.3.3 Composite of the electrical resistivity data in figures 5.3.1 and 5.3.2 227 Figure 5.3.4 Electrical resistivity deviations of the tungsten data from the following references compared to Eq. 1.2.3: (5A, 11, 14, 22) 228 Figure 5.3.5 Electrical resistivity deviations of the tungsten data from the following references compared to Eq. 1.2.3: (4A,21 ,23,25, 26A, 33) 229 Figure 5.3.6 Composite of the electrical resistivity deviations shown in figures 5.3.4 and 5.3.5 230 Figure 5.3.7 Electrical resistivity for tungsten as a function of temperature calculated from Eq. 1.2.3 at selected values of RRR 231 Figure 5.3.8 Lorenz ratio for tungsten as a function of temperature calculated from Eq. 1.2.3 and Eq. 1.1.3 at selected values of RRR 232 Figure 5.3.9 RRR values calculated as per Section 1.5, RRR(CALC), versus reported RRR values, RRR(OBS), for tungsten .... 233 Figure 5.4.1 Thermal conductivity for tungsten as a function of temperature calculated from Eq. 1.1.3 at selected values of RRR 234 207 Table 5.3.1. Comparison of Calculated and Observed RRR Values for Tungsten Reference RRR (obs.) RRR (calc.) Primary Data 4A 70.0 70.0 4A 75.4 75.4 4A 131.0 131.0 11 75.0 75.0 14 39.8 40.8 14 74.6 77.6 22 31.4 30.0 22 400.0 400.0 25 150.0 150.0 25A 75.0 75.0 29 46.6 46.6 33 155.0 155.0 Secondary Data 3 40300.0 30000.0 4 96000.0 86000.0 7 2780.0 4850.0 11 75.0 75.0 28 16300.0 16300.0 31 8460.0 8400.0 31 27200.0 26000.0 31 39300.0 36500.0 31 69600.0 56500.0 31 85500.0 71000.0 Table 5.4.1. Thermal Conducti vity Val ues for Tungsten Calculated from Eq. 1.1.3 at Selected Temperatures and RRR Values. x(Wm'V 1 ) T (K) RRR = 30 RRR = 100 RRR = 300 1 14.6 50 151 2 29 100 302 3 44 150 452 4 59 200 602 5 73 249 749 6 88 299 894 7 102 347 1033 8 117 395 1166 9 131 442 1291 10 145 488 1404 12 173 574 1595 14 201 651 1730 16 227 718 1802 18 251 768 1803 20 273 799 1734 25 311 786 1378 30 325 692 1020 35 321 586 768 40 306 494 600 45 285 418 483 50 262 357 398 60 226 281 302 70 211 250 264 80 204 236 246 90 199 225 234 100 195 217 224 150 184 197 201 200 180 139 191 250 175 182 134 300 169 174 176 400 155 158 159 500 143 145 146 600 135 136 137 700 129 130 130 800 124 125 125 900 121 122 122 1000 118 119 119 1100 115 116 116 1200 113 114 114 1300 111 111 112 1400 109 110 110 1500 107 108 108 1600 106 106 106 1800 103 103 103 2000 100 101 101 2200 98 99 99 2400 95 97 97 2600 95 95 95 2800 93 93 93 3000 92 92 92 209 THERMAL CONDUCTI VITY,W/m . figure 5.1.1 Experimental thermal conductivity data selected from the following primary references In the tungsten annotated bibliography: (4fl, 1 1 ,14,20,22) O-(11), A- (11), - (14), V- (14), O -(4fl) , +-(20), X-(22) 210 THERMRL CONDUCTIVITY, W/m . Figure 5.1.2 Experimental thermal conductivity data selected from the following primary references In the tungsten annotated bibliography: (4R,21 ,22,23,25,26R) O-(22) , A-(21) , -(4fl), V-(4fl), O-(23), +-(25), X-(26R) 211 THERMAL CONDUCTIVITY, W/m . Figure 5.1.3 Experimental thermal conductivity data selected from the following primary references In the tungsten annotated bibliography: (27,29,32,33) O-(27), A-(29), -(32), V-(33) 212 THERMAL CONDUCTIVITY, W/m. 1000 -600 r 400 200 -i i i i'i m ii ri ' lT i'T"'" ,|T^7 , rn , »T >rnTr i v V Vv V v V V V v V V V V 7 V V V V « V .+ & v a v ++ ^ 7 V V ^ v+ V D V 30 100 300 TEMPERATURE, K 3000 Figure 5.1.4 Composite of the data In figs. 5.1.1 through 5.1.3 213 1 3 10 30 100 300 1000 3000 TEMPERRTURE,K Figure 5.2.1 Thermal conductivity deviations of the tungsten data from the following primary references compared to eq. (1.1.3): (4R, 11, 14, 20, 22) O- (11), A-(11), - (14), V-(14), O-(4fl) , +-(20), X-(22) 214 Figure 5.2.2 Thermal conductivity deviations of the tungsten data from the following primary references compared to eq. (1.1.3): (4R,21,22,23,25,26R) O- (22), A- (21), -(4R) , V- (4R), O- (23), +- (25), X-(26R) 215 1 3 10 30 100 300 1000 3000 TEIiPERRTURE,K Figure 5.2.3 Thermal conductivity deviations of the tungsten data from the following prlmarg references compared to eq. (1.1.3) (27,29,32,33) O-(27), A-(29), -(32), V-(33) 216 Figure 5.2.4 Composite of the deviations In figs. 5.2.1 through 5.2.3 217 0.3 1 3 10 30 100 300 1000 TEMPERATURE, K Figure 5.2.5 Thermal conductivity deviations of the tungsten data from the following secondary references compared to eq. (1.1.3): (1,2,5,7,8,10) O-(1), A-(2) , -(5), V-(7), O-(8) , + - (10), X-(10) 218 0.3 1 3 10 30 100 300 1000 TEMPERRTURE,K Figure 5.2.6 Thermal conductivity deviations of the tungsten data from the following secondary references compared to eq. (1.1.3):(10,13,16,19,26) O-(10), A-(13), -(16), V-(16), O-(16), +-(19), X-(26) 219 0.3 1 3 10 30 100 300 1000 TEMPERRTURE,K Figure 5.2.7 Composite of the deviations In figs. 5. 2. 5 and 5.2.6 220 13 10 30 100 300 1000 TEMPERATURE ,K Figure 5.2.8 Thermal conductivity deviations of the tungsten data from the following secondary references compared to eq. (1.1.3):(3,4,4B,11,28) O-(11), A-(3), -(4), V-(4), O-(4), +-(4B) , X-(28) 221 Figure 5.2.9 Thermal conductivity deviations of the tungsten data from the following secondary reference compared to eq. (1.1.3) = (31) O- (31), A- (31), -(31), V- (31), O- (31), +-(31) 1 3 10 30 100 300 1000 TEMPERRTURE,K Figure 5.2.10 Composite of the deviations In figs. 5.2.8 and 5.2.9 223 Figure 5.2.11 Comparison of eq. (1.1.3) to the values recommended for tungsten In the following references! ( 12FI, 15, 27fl) O-(15), A-(15), -(15), V-( 27R ) , O- (12R) 724 ELECTRICAL RESISTIVITY, nfl. 4 10 20 40 100 200 400 1000 2000 TEMPERATURE, K Figure 5.3.1 Experimental electrical resistivity data for tungsten selected from the following references In the tungsten annotated bibliography: (5R, 1 1 , 14,22) O-(11), A-(11), -(5fl), V-(14), <0-(14), + -(22), X-(22) 225 ELECTRICAL RESISTIVITY, nfl. Figure 5.3.2 Experimental electrical resistivity data for tungsten selected from the following references In the tungsten annotated bibliography: (4R,21 ,23,25,26fi,33) O-(21), A-(4R) , -(23), V-(25), O- (26R) , + -(33) 226 ELECTRICAL RESISTIVITY, nil. Figure 5.3.3 Composite of the electrical resistivity data In figs. 5.3.1 and 5.3.2 ?27 Figure 5.3.4 Electrical resistivity deviations of the tungsten data from the following references compared to eq. tl.2.3):(5fl,ll,14,22) O-(11), A-(11), -(5fl), V-(14), O-(14), +-(22), X-(22) 228 Figure 5.3.5 Electrical resistivity deviations of the tungsten data from the following references compared to eq. (1.2.3):(4fl,21,23,25,26fl,33) O-(21), A-(4R), -(23), V-(25), O-( 26R ) +-(33) Figure 5.3.6 Composite of the electrical resistivity deviations shown In figs. 5.3.4 and 5.3.5 ELECTRICAL RESISTIVITY, nH. 1000 E 300 100 30 10 3 1 0.3 0.1 — l. i.,uu i,i,g i.n.l,,. ,. !., t4i a 1 1 l.u 1 3 10 30 100 300 1000 3000 TEMPERATURE, K Figure 5.3.7 Electrical resistivity for tungsten as a function of temperature calculated from eq. (1.2.3) at selected values of RRR. 231 Uil LORENZ RATIO, 10 °YVK TEMPERATURE, K Figure 5.3.8 Lorenz ratio for tungsten as a function of temperature calculated from eq. (1.2.3) and eq. (1.1.3) at selected values of RRR. 23 ? RRR(CRLC) Figure 5.3.9 RRR values calculated as per Section 1.5, RRR(CRLC), versus reported RRR values, RRR ( OBS 3 , for tungsten. O” Primary, A - Secondary, ~ Secondary 233 THERMAL CONDUCTIVITY, W/m. TEMPERATURE, K Figure 5.4.1 Thermal conductivity for tungsten as a function of temperature calculated from eq. (1.1.3) at selected values of RRR. 234 5.5 FORMAT FOR ANNOTATED BIBLIOGRAPHY OF TUNGSTEN REFERENCE AUTHOR, TITLE, CITATION ANNOTATION PURPOSE SPECIMEN a) Dimensions/Shape; b) Crystal Status; c) Thermal /Mech . History; d) Purity Specification; e) RRR; f) p0 ; g) Other Characteri zation Data APPARATUS a) Type; b) Thermometry/Cal i brat i on/Anchoring; c) Thermal Isolation; d) Other (Q meas.) DATA a) Temperature Range/Difference; b) Content of Tables, Figures and Equations/Data Extraction; c) Uncertai nty/Impreci sion; d) Disputable Corrections to Measurements by Authors; e) Errata (by Author or Reviewer) ANALYSIS a) Comparisons; b) Conclusions 235 Backlund, N. G. , Measurement and Analysis of the Thermal Conductivity of Tungsten and Molybdenum at 100 to 400 K, Thermal Conductivity Conference, National Physical Laboratory, Teddington, England, 15-17 July (1964) PURPOSE To measure X for W and Mo from 100 to 400 K. SPECIMEN a) 4 mm dia., 10 cm long/rod; g) source: Johnson, Matthey Co. APPARATUS a) longitudi nal ; b) Fe-constantan thermocouples soldered to specimen/cali-brated with earlier measurements (Backlund, J. Phys. Chem. Solids 20 (1961)); c) German silver shielding, isolating bricks between can and base plate, vacuum insulation. DATA a) 95.9 to 280 K; b) figures 3, 5 -p, X respectively; c) uncertainty: ±2%. ANALYSIS b) Xg is proportional to T. Backlund, N. G. , Measurement and Analysis of the Thermal Conductivity of Tungsten and Molybdenum at 100 to 400 K, J. Phys. Chem. Solids, 28, 2219-23 (1967) PURPOSE To obtain information regarding the scattering processes of phonons and electrons in W and Mo. SPECIMEN a) 10 cm long, 4 mm dia. /rod; f) p0 = 1.24 yft-cm; g) source: Johnson, Matthey and Co. APPARATUS a) longitudinal; b) Fe-constantan thermocouples soldered to specimen; c) German silver shielding, isolating bricks between can and base plate; vacuum insulation. DATA a) 87.3 to 282 K; b) figure 5 - X/data points listed in TPRC data series; c) uncertainty: ±1.5%. ANALYSIS b) scattering by electrons is expected to be constant, while Umklapp scattering is found only at higher temperatures. 236 Baer, D. R. and Wagner, 0. K. , Effect of Surface Condition on the Transport Properties of Tungsten, J. Low Temp. Phys., 13(5/6), 445-69 ( 1973) PURPOSE To determine the effect of surface condition on p, W and L. SPECIMEN W-12G a) 0.74 mm dia./rod; b) single crystal; c) el ectropol i shed ; f) p0 = 1.26 x 10 10 acm; g) crystal oriented to , P 2gg/p 0 = 43000. APPARATUS a) longitudinal; b) C resistance thermometers/cal i brated against a standard Ge resistance thermometer; c) adsorbent in vacuum chamber to adsorb residual He. DATA a) 1.46 to 5.31 K; b) figure 12 -1/X; c) uncertainty: ±1%. ANALYSIS b) surface scattering is specular for the el ectropol i shed surface, while diffuse for the el ectroetched and sandblasted surfaces. Batdalov, A. B. , Tamarchenko, V. I., and Shalyt, S. S. , Manifestation of Hydrodynamic Effect in the Thermal Conductivity of Tungsten, JETP Lett. (USSR), 20(6), 171-3 (Sep 1974) PURPOSE To determine x of a compensated metal. SPECIMEN a) specimens 1, 2, 3: 3.2, 1.4, 0.8 mm dia. respective! y/rods; b) single crystals; c) /electrical etching reduced specimen 1 into 2 and 3; e) specimen 1 ( RRR = 86000). DATA a) specimen 1: 2 to 102 K, specimen 2: 2.15 to 12.4 K, specimen 3: 2.23 to 9.0 K; b) figure 1 -X. ANALYSIS b) the hydrodynamic contribution to the total thermal conductivity behaves 1 ike T"34. 237 [4A] Berman, R. , Hardy, N. D., Sahota, M. Hust, J. G. , and Tainsh, R. J., Stand-ards Reference Materials for Thermal Conductivity Below 100 K, Proceedings of the Seventeenth Conference on Thermal Conductivity, Gaithersburg , MD, 105-16 (1983) PURPOSE To report results of a CODATA round-robin investigation involving standard reference materials of stainless steel, tungsten, and electrolytic iron. SPECIMEN from Leeds University a) SRM 730; b) approximately 1/4 in. (dia) x 2 in. (6 x 50 mm); c) RRR = 70. SPECIMEN from National Measurement Laboratory a) SRM 730; b) approximately 1/4 in. (dia) x 2 in. (6 x 50 mm); c) RRR = 75.4, 131. APPARATUS a) not discussed DATA from Leeds University a) 1 to 95 K; b) Table -X/data made available by private communication. DATA from National Measurement Laboratory, CSIR0 a) 2 to 90 K; b) Table -X/data made available by private communication. [4B] Binkele, L., Zur Frage der Hochtemperatur-Lorenzzahl bei Wolfram-eine Analyse neuer Messwerte der thermischen und elektrischen Leitfahi gkei f im Tem-peratubereich 300 bis 1300 K, submitted for publication in High Temperature-High Pressure (1982) PURPOSE SPECIMEN a) SRM 730 APPARATUS a) Details not given. DATA iy~300 to 1300 K 238 Bremmer, H. and DeHaas, W. J., On the Conduction of Heat by Some Metals at Low Temperatures , Physica, 3 , 672 (1936) PURPOSE To determine \ for Pb, Cu and W. SPECIMEN a) RRR = 2174; g) source: N. V. Philips Lampworks Eindhoven. APPARATUS a) longitudi nal ; b) Pt resistance thermometers. DATA a) 1.55 to 21.8 K; b) Table 4 - W/data points listed in TPRC data series. ANALYSIS a) results disagree with Gruneisen, E. and Goens, E. , Z. Phys., 44, 615 (1927). [5A] Cezairliyan, A. and McClure, J. L. , High Speed (Subsecond) Measurement of Heat Capacity, Electrical Resistivity, and Thermal Radiation Properties of Tungsten in the Range 2000 to 3600 K, J. Research NBS, 75A(4) (1971) PURPOSE To apply a high speed measurement technique on a tungsten specimen in the temperature range 2000 to 3600 K. SPECIMEN a) 4 in. length (101 mm), outside dia. 0.25 in. (6.3 mm), wall thickness 0.02 in. (0.5 mmj/tube; c) specimen produced from rod by el ectroerosion , an nealed up to 3200 K; e) RRR = 41; g) polished outer surface, density at 293 K: 19.23 x 10 3 kg-m“ 3 . APPARATUS a) subsecond transient; b) photoelectric pyrometer/cal ibrated against W fil ament standard lamp; c) 10“^ mm of Hg (10“ 3 Pa) vacuum. DATA a) 2000 to 3600 K; b) Table 2, Fig. 3 -p; c) uncertainty: ±1% between 2000 and 3600 K. ANALYSIS a) Results agree with Osborn (1941), Platunov (1964), Neimark (1968), Jones (1926); b) p = 14.08 + 3.515 x 10"^ T where p is in units of 10“° nm and T in K. 239 Cox M. , Thermal and Electrical Conductivities of Tungsten and Tantalum, Phys. Rev., 64, 241 (Oct 1943) PURPOSE To measure A and of pure W and Ta. SPECIMEN a) specimens 2,8: 0.10 in. (0.025 cm) dia., 40 cm long/rods; c) specimen 2: aged 370 h at 2400, 2600 °C, specimen 8: aged 370 h at 2300 °C; g) source: General Electric Co. APPARATUS a) direct current heating; Hg thermometer and barometer for boiling point of N 2 > 02 , H2 O and ice bath. DATA a) 77.4 to 274 K; b) Table 3 -A, p, L/A data listed in TPRC data series. ANALYSIS a) p results agree with Hoi born, L. (1919), Henning, F. (1921), Geiss, W. (1923), Forsythe, W. E. (1925), Meissner, W. (1930), Barratt, T. (1914), Weber, S. (1917), Griineisen, E. ( 1927), Kannuluik, W. G. (1933), and DeHaas, W. J. (1938). A results disagree with Griineisen, E. (1927), Kannuluik, W. G. (1933), and DeHaas, W. J. (1938). DeHaas, W. J. and Biermasz, T. H., Sur la Conductibi 1 ite Thermique aux Basses Temperatures , Rapports et Communications, No. 24, 7e, Congres Inter-national du Froid, Supplement No. 82b, 204 (1936) PURPOSE To determine A for several metals at low temperatures. APPARATUS a) longitudinal; b) Pt resistance thermometer. DATA a) 15.5 to 21.8 K; b) Table 8 -W, L/L data from Table 8, A data points listed in TPRC data series. 240 DeHaas, W. J. and deNobel, J., The Thermal and Electrical Resistance of a Tungsten Single Crystal at Low Temperatures and in Magnetic Fields, Physica, 5^ 449 (1938) PURPOSE To separate x of W into xe and Xn at low temperatures in strong magnetic fields. SPECIMEN a) 30 mm long/hexagonal rod; b) single crystal; e) RRR = 2780; g) source: Philips Works, axis is in direction. APPARATUS a) longitudinal; b) Pb thermorneters/cal ibrated against a Pt resistance ther mometer in bath; c) < 5 x 10"^ mm of Mg (7 x 10“^ Pa) vacuum. DATA a) 15.3 to 88.4 K; b) Tables 1, 3 -X, p/p data from Table 3, X data points listed in TPRC data series. ANALYSIS b) W-F-L holds for p and xe in magnetic fields. deNobel, J., Thermal and Electrical Resistance of a Tungsten Single Crystal at Low Temperatures and in High Magnetic Fields, Physica, _15(5-6), 532-40 (Jul 1949) PURPOSE To investigate the relationship of the W-F-L law to p and x due to elec-trons. SPECIMEN a) 30 mm long/hexagonal rod; b) single crystal; e) RRR = 2780; g) source: Philips Works, axis is in direction, same specimen as used by DeHaas, W. J. (1938). APPARATUS a) longitudinal; b) Pb thermometers. DATA a) 15 to 20 K; b) Table 1 - x/data points listed in TPRC data series, run 1 4: H = 0.82, 2.1, 2.6, 2,8 kA/m, perpendicul ar. ANALYSIS b) W-F-L is not valid in strong magnetic fields for a and xe , and it is not possible to separate Xg and xe . 241 deNobel , J., Thermal and Electrical Resistivity of Some Tungsten Single Crystals at Low Temperatures and in Strong Magnetic Fields, Physica, 23^ 261 (1957) PURPOSE To gain information about Ae and Ag from the change in anisotropy with increasing field strength. SPECIMEN b) single crystals, A oriented in direction, 1-38 and B oriented in (±5°) direction; g) P2(/ p 273 = 26 x 10 -4 . APPARATUS a) longitudinal ; b) phosphor bronze resistance thermometers. DATA a) specimen A: 15.3 to 20.3 K, specimen B: 3.86 to 20.2 K, specimen 1-38: 3.38 to 75.9 K; b) Table 1 -A/data points listed in TPRC data series. ANALYSIS b) A and B anisotropy ratio: 1.13, 1.23 respectively. 242 Fitzer, E. , Thermophysical Properties of Solid Materials. Project Section II. Cooperative Measurements on Heat Transport Phenomena of Solid Mate-rials at High Temperature, AGARD Report No. 606, 1973 PURPOSE To improve the data base used in the design and fabrication of high tempera-ture equipment and systems under development by the NATO countries. SPECIMEN Participant 5 a) /wire; c) sintered. SPECIMEN Participant 31 a) /wire; c) sintered. SPECIMEN Participant 41 a) /wire; c) arc cast; d) 99.55% W; g) p(T). APPARATUS Participant 5 a) comparative APPARATUS Participant 31 a) direct electrical heating APPARATUS Participant 41 a) direct electrical heating; b) pyrometer; d) argon environment. DATA Participant 5 a) 300 to 900 K; b) Table 33 -X. DATA Partaicipant 31 a) 2100 to 2800 K; b) Table 33 -X. DATA Participant 41 a) 1500 to 2750 K; b) Table 35 -X, Table 51 -p; c) uncertainty: ±2%. ANALYSIS Participant 5 b) TPRC Recommended curves (1970, 1972). ANALYSIS Participant 41 b) Jun, et al . , High Temperature-Hi gh Pressure, 43 (1970); TPRC Recommended Curves (1970, 1972). 243 Griineisen, E. and Adenstedt, H. , Anisotropie der Warmeleitung und Thermo-kraft Regul arer Metal le (Wolfram) im Transversal en Magnetfeld bei 20 K, Ann. Phys. (Leipzig) , 29, 597-604 (1937) PURPOSE To determine X and thermopower of W in a transverse magnetic field. SPECIMEN a) 0.106 cir^ x 7.0 cm; g) specimen axis 8° to the direction. APPARATUS a) longitudinal b) Const^ntan-mangani n thermocouples; c) 10"^ to 10"^ mm of Hg (10"2 to 10"3 pa) vacuum. DATA a) 21.5 to 91.2 K; b) Tables 1,2-X/data points listed in TPRC data series, H = 0. ANALYSIS a) agrees with Griineisen, E. and Goens, E. , Z. Phys., 44, 615 (1927). [12A] Ho, C. Y., Powell, R. W. , Liley, P. E., Thermal Conductivity of the Ele-ments: A Comprehensive Review, «J. Phys. Chem. Ref. Data, 3, Supplement No. 1, 242-257 (1974) PURPOSE To provide a comprehensive listing of data on x of the elements. SPECIMEN d) high purity; f) p 0 = 1.70 x 10“^ crcm for T below 200 K. APPARATUS a) not given. DATA a) 0 to 22273 K; b) Table 171 -X, recommended values; c) uncertainty: ±2% near room temperature, ±3% from 300 to 1500 K, ±5% from 100 to 300 K and 1500 to 3000 K, and ±10% below 100 K and above 3000 K. The values above 3660 K are provisional. 244 Must, J. G. , Thermal Conductivity Reference Materials, Progress Report, Proceedings of the Thirteenth Thermal Conductivity Conf . , 22-4 (Nov 5-7, 1973) PURPOSE To present low temperature X measurements for sintered W. SPECIMEN c) sintered. APPARATUS a) longitudinal ; b) vapor pressure of LHe for T < 20 K, Pt reference ther-mometer for T > 20 with differential thermocouples; b) controlled tempera-ture glass fiber shield, vacuum insulation. DATA a) 10 to 250 K; b) Table 1 -X, p, L. ANALYSIS b) SRM 734 can be extended to 800 °C and SRM 735 to 1000 °C. Hust, J. G., Thermal Conductivity Standard Reference Materials from 6 to 280 K: VI. N.B.S. Sintered Tungsten, NBSIR 73-351 (Jan 1974) PURPOSE To present measurements on the transport properties of NBS sintered W. SPECIMEN a) 23 cm long, 3.1 mm dia./rod; c) one specimen annealed at 2020 °C for 1 h, one unannealed; e) annealed (RRR = 74.6), unannealed (RRR = 39.8); f) annealed: p ? = 0.6493 nfi-m, unannealed: pc = 1.229 nn»m; g) density = (19.23 ± 0.05) g/crn^, DPH hardness 1 kg = 405 and 514 for annealed and unannealed respecti vely. APPARATUS a) longitudinal; b) vapor pressure of LHe for T < 20 K, Pt reference ther-mometer for T > 20 with differential thermocouples; c) controlled tempera-ture glass fiber shield, vacuum insulation. DATA a) annealed: 7 to 90 K, unannealed: 6 to 280 K; b) Table 12 - unannealed: X, p, L, Table 13 - annealed: X, p, L; c) uncertainty: X -2.5% at 300 K, decreasing as T^ to 0.70% at 200 K, 0.70% from 200 to 50 K, increasing inversely with temperature to 1.5% at 4 K, p -0.5%. ANALYSIS b) X = 2^ a.[nT] i+1 . i =1 1 245 Hust, J. G. and Giarratano, P. J., Thermal Conductivity and Electrical Re-sistivity Standard Reference Materials: Tungsten SRM's 730 and 799, from 4 to 3000 K, Nat. Bur. Stand. Spec. Publ . No. 260-52, 37 pp. (Sep 1975) PURPOSE To compile, correlate, and analyze X and p data for arc cast and sintered W. SPECIMEN a) 0.51 to 1.27 cm dia./rods; c) W powder vacuum arc melted into billet, an-nealed at 1700 K for 1/2 h/billet machined into rod, acid etched and final swage before anneal; d) < 99.956%; e) specimens 1-3 (RRR = 50, 75, 100 re-spectively); f) specimens 1-3 (p 0 = 0.97, 0.65. 0.49 nft.m respectively); g) AFML arc cast density: (19.20 ± 0.05) g/cnr. APPARATUS a) longitudinal. DATA a) 4 to 3000 K; b) Tables 4, 5 - recommended values of p , X for AFML arc cast and NBS sintered W, respecti vely ; c) uncertainty: in \ is 2.5% at 300 K, decreasing to 0.7% at 200 K, 0.7% from 200 to 50 K increasing in-versely with temperature to 1.5% at 4 K; in p is 0.5%. ANALYSIS 2 b) X = l/(cT n + sCyT) + AT/p [e' 01 T + Be (6 2 /T) 2 J Jun, C. K. and Hoch, M. , Thermal Conductivity of Tantalum, Tungsten, Rhenium, Ta-lOW, Tin, ^222 » W-25 Re in the Temperature Range 1500 to 2800 K, Proceedings of the Sixth Thermal Conductivity Conference, 933-49 (Oct 19-21, 1966), Air Force Materials Lab., Wri ght-Patterson AFB, Ohio PURPOSE To report measurements on X for Ta, W, Re and Ta-lOW, T^, T222 anc W-25 Re alloys. SPECIMEN a) specimens 1,2,3: 2.5339, 2.4785, 2.0801 cm dia., 0.2999, 0.2714, 0.2700 cm thick, respectively/disks; c) /specimen 2 machined from speci-men 1, specimen 3 machined from specimen 2; d) 99.998%; g) source: Fansteel Metallurgical Corp., density: specimens 1,2,3 = 18.89, 19.03, 19.23 g/cm 3 , respecti vely. APPARATUS c) vacuum insulation. DATA a) specimen 1: 1513 to 1930 K, specimen 2: 1572 to 1905 K, specimen 3: 1836 to 2608 K; b) Table 7 -X. 246 Kannuluik, W. G. , On the Thermal Conductivity of Some Metal Wires, Proc. R. See. London, Ser. A, 131 , 320-35 (1931) PURPOSE To investigate x for metals and alloys by an electrical steady state method. SPECIMEN a) 0.1022 cm dia./wire; c) annealing -specimen 1: 220 °C, specimen 2: 1300 °C ; g) source: General Electric Co. APPARATUS a) direct electric heating; b) ice point, steambaths; c) 10" ^ mm of Hg (10 -2 Pa) vacuum. DATA a) 273 to 286 K; b) Tables 4, 5 -x, L/L data from Table 5, X data points listed in TPRC data series; c) uncertainty: ±2%. ANALYSIS a) results disagree with Weber, S. (1917), Barratt, Winter (1914). Kannuluik, W. G., Eddy, C. E., and Oddie, T. H., The Thermal and Electrical Conductivities of Several Metals Between -18 °C and 100 °C, Proc. Roy. Soc. London, Ser. A, 141, 159-68 (1933) PURPOSE To extend observations of \ for metals from -18 °C to 100 °C. SPECIMEN a) specimen 1 : 7.846 cm x 0.01053 cm 2 , specimen 2: 7.940 cm x 0.01022 cm 2/rectangul ar , hexagonal bars respectively; b) single crys-tals; d) 0.001% Co, Cr, In and 0s, each; g) specimen 1: axis in di-rection, specimen 2: axis in direction. APPARATUS a) direct electrical heating; b) ice point, steambaths; c) 10‘ 4 mm of Hg (10“ 2 Pa) vacuum. DATA a) 90.1 to 273 K; b) Table 2 -X, p/p data points from Table 2, X data points listed in TPRC data series; d) correction for radiative heat loss. ANALYSIS a) results agree with Griineisen, E. and Goens, E. , Z. Phys., 44, 615 (1927). 247 Langmuir, I. and Taylor, J. B. , The Heat Conductivity of Tungsten and the Cooling Effects of Leads Upon Filaments at Low Temperatures , Phys. Rev., 50, 68-87 (Jul 1, 1936) PURPOSE To obtain voltage and resistance measurements across a filament, thereby calculating A and the temperature distribution along the wire. SPECIMEN a) specimens 1, 2, 3: 25.82, 12.86, 5.87 cm long respectively, all speci-mens 0.00499 cm dia. /wires; c) all tubes baked at 450 °C, filaments heated to 2000 K for 2 min., 2400 K for 4 h and 2800 K for 30 s; d) thoriated W fi 1 aments. APPARATUS a) direct electrical heating; b) N? bath; c) vacuum insulation. DATA a) 240 to 300 K; b) Table 8 -A/data points listed in TPRC data series. ANALYSIS a) results agree with Barratt, T. , Proc. Phys. Soc., London, 26, 347 (1914) and Kannuluik, W. G. , Proc. Roy. Soc. London, Ser. A, 131 , 320 (1931) 141 , 159 (1933). Mendelssohn, K. and Rosenberg, H. M. , The Thermal Conductivity of Metals at Low Temperatures. II. The Transition Elements, Proc. Phys. Soc., London, Sect. A, 65, 388-94 (1952) PURPOSE To measure A of transition elements at low temperatures. SPECIMEN a) 15 cm long; 1 to 2 mm dia. /rod; b) polycrystal 1 ine; c) annealed; d) 99.99% W; g) source: Johnson, Matthey and Co. APPARATUS a) longitudinal; b) He gas thermometers. DATA a) 2 to 43 K; b) Fig. 2 -A/data taken from TPRC data series; c) uncer-tainty: ±3%. ANALYSIS a) results agree with Hu 1 in , J. K . , Proc. Roy. Soc. London, Ser. A, 204 , 98 (1950). 248 Moore, J. P., Graves, R. S. , Fulkerson, W. , and McElroy, D. L. , The Thermal Properties of Tungsten, Proceedings of the Fifth Thermal Conductivity Conference, Univ. of Denver, Vol . 2, V-G-l - V-G-35 (1965) PURPOSE To present the temperature dependence of several physical properties of poly-crystal 1 i ne tungsten. SPECIMEN a) 3 in. (7.6 cm) dia., 5/16 in. (0.79 cm) dia. hole/disk; c) /machined from 3.5 in. (8.9 cm) dia. pressed and sintered powder billet which had been hot extruded at 1800 °C for 3 to 1 reduction; d) 99.98%; e) p^ nn /p d = 35; g) density: 19.077 g/cm^. APPARATUS a ) radial ; b) annealed Pt^^ Rh^ -Pt/cal ibrated at 3 standard melting points. DATA a) 323 to 1273 K; b) Table 2 -p, X; c) uncertainty: ±1.5% at 1000 °C. ANALYSIS a) results agree with Langmuir, I. and Taylor, J. B. , Phys. Rev. Z. , 50_, 68-87 (1936) and Tye, R. P., Nb, Ta, Mo and W , A. G. Quarrel 1, Ed., Elsevier Publishing Co., 169-79 (1961) between 300 and 600 K. Moore, J. P., McElroy, D. L. and Barisoni, M. , Thermal Conductivity Measure-ments Between 78 and 340 K on Aluminum, Iron, Platinum, and Tungsten, Proceedings of the Sixth Thermal Conductivity Conference, Wri ght-Patterson AFB, Ohio, 737-78 (1966) PURPOSE To describe an apparatus which is capable of accurately measuring x, o, and S for metals between 78 and 340 K. SPECIMEN a) 5 to 8 cm long/rod; c) high purity: electron beam melted; d) 98%, radial: 99.98% W; e) high purity (RRR > 400), radial (RRR = 31.4), 98% (RRR = 4.11; f) radial (p 0 = 0.1593 yftcm), 98% (p 0 = 1.560 yficm); g) density (g/cm3): high purity: 19.29, radial: 19.077, 98% = 19.19. APPARATUS a) longitudinal; b) chromel-P and constantan thermocoupl es/from calibrated spools/thermocouples attached to specimen by discharge welding or epoxy, leads thermally grounded to guard cylinder; c) guard cylinder, 5 x 10' mm of Hg (7 x 10"5 pa) mm vacuum. DATA a) high purity: 80 to 300 K, radial: 100 to 300 K; b) Table 4 -p, x (smoothed values); c) uncertainty: ±1.8%. 249 Osborn, R. H. , Thermal Conductivities of Tungsten and Molybdenum at Incan-descent Temperatures , J. Opt. Soc. Am., _31 , 428 (1941) PURPuot To determine X of W, Mo in filament form. SPECIMEN a) (2.5 to 5.0) x 10" 3 rm dia./wire; c) annealed at 2700 K for 2 h. APPARATUS a) direct electrical heating; b) disappearing filament optical pyrometer; c) vacuum sealed pyrex tube. DATA a) 1100 to 2000 K; b) Table 1 -X, Fig. 3 -p/data points listed in TPRC data series. ANALYSIS a) results agree with Forsythe, W. E. , J. Opt. Soc. Am., 24, 114 (1934). Powell, R. L., Harden, J. L., and Gibson, E. F. , Low Temperature Transport Properties of Commercial Metals and Alloys. IV. Reactor Grade Be, Mo, and W, J. Appl. Phys., 31(7), 1221-4 (1960) PURPOSE To study the low temperature mechanical and transport properties of several reactor materials. SPECIMEN a) 13 mm long; 3.67 mm dia./rod; d) 97.9%; f) p0 = 0.16 x 10"^ ftm. APPARATUS a) longitudinal; b) Au-Co vs. Cu thermocouples; c) vacuum insulation. DATA a) 4 to 70 K; b) figure 1 -X, figure 2 -p, figure 3 -L; d) correction for thermal contraction. ANALYSIS a) deNobel , J., Physica, 25, 261 ( 1957); 23, 349 (1957) values of Xg for single crystal W are three times larger than these results and his values below 20 K show no T 2 dependence as expected theoretically. 250 Powell, R. W. and Tye, R. P. , New Measurements on Thermal Conductivity Ref-erence Materials, Int. J. Heat Transfer, _10(5), 581-96 (1967) PURPOSE To provide further data on x and p of materials suggested for use as stand-ard reference materials. SPECIMEN a) 10 cm long; 0.4 cm dia./rod; d) 99.99% W; e) RRR = 150; g) source: Johnson Matthey and Co. APPARATUS a) longitudinal, ingot Fe standard used for energy outflow measurements. DATA a) run 1: 300 to 700 K, run 2: 450 to 760 K, run 3: 400 to 1000 K; b) figure 3, Table 6 (smoothed). ANALYSIS a) results agree with Laubitz, M. J., Can. J. Phys., 41 (1963) and Flynn, D. R. and Robinson, H. E., private communication by Laubitz, M. J. Sharma, J. K. N. , Heat Conductivities Below 1 K I, Cryogenics, 7^(3), 141-56 (1967) PURPOSE To determine if the anomalies found in X below 1 K represent a general be-havior of metals. SPECIMEN a) /wire; b) polycrystal 1 ine; g) p^/pj 5 Ltd. = 30, source : Lamp and Metals APPARATUS a) longitudinal; b) carbon resi stors/cal ibrated against heat sink tempera-ture for each run. DATA a) 0.5 to 1.0 K; b) figure 9 -X; c) uncertainty: ±2%; corrected for thermomolecul ar pressure. ANALYSIS a) comparison with Davey, G. and Mendelssohn, K., Phys. Lett., 7» 183 (1963). [26A] Taylor, R. E. , Thermal Properties of Tungsten SRM's 730 and 799. J. Heat Transfer, 100(2), 330-3 (1978) PURPOSE To compare A results on sintered tungsten with previous results on arc-cast tungsten. SPECIMEN a) SRM 730, 99.98%; b) sintered; c) RRR = 75, density: 19.23 + 0.05 g/cm®. APPARATUS a) longitudinal . DATA a) 1300 to 2600 K; b) Table 3 -A, p, L; c) corrected for thermal expansion; f) uncertainty: +5% from 300 to 2000 K, +8% above 2000 K. ANALYSIS b) results lie between NBS, TPRC values; d) A(w/mK) = 0.144644 x 10"® T® + 0.08 from 1200 to 3000 K. Timrot, D. L. and Poletskii, E., Use of Heating by Electron Bombardment to Investigate the Coefficient at Heat Conductivity in High Melting Point Alloys and Compounds, High Temp. (USSR), 1_, 147 (1963) PURPOSE To study the thermophysical properties of solids at high temperatures. SPECIMEN a) /rod; d) 99.9+% W; g) 5 < length/dia. ratio < 6. APPARATUS a) electron bombardment; b) optical pyrometer; c) vacuum insulation. DATA a) 1200 to 3000 K; b) text (smoothed), figure 3 -A/data points listed in TPRC data series; c) uncertainty: ±10%. ANALYSIS a) comparison with Powell, R. W. and Schofield, F. H. , Proc. Phys. Soc., London, 51^, 153 (1939); Worthing, A. G. , Phys. Rev., 4-, 6 (1914); Osborn, R. H. , J. Opt. Soc. Am., _3I> 428 (1941); Gumenyuk, V. S. and Lebedev, V. V., Fiz. Met. Metal 1 oved. , _U, 1 (1961). 252 [27A] Touloukian, Y. S. , Powell, R. W. , Ho, C. Y. , and Klemens, P. G. , Thermo-physical Properties of Matter, Volume 1: Thermal Conductivity, Metallic Elements and Alloys, 68-81 (1970) PURPOSE To provide an extensive list of data for x of the metallic elements and al 1 oys. SPECIMEN d) 99.99+%; f) p0 = 1.70 x 10" 9 ncm. APPARATUS a) not given. DATA a ) 0 to 8500 K; b) Figure and Table 63R -X, recommended values; c) uncer-tainty: ±3% near room temperature, ±3 to 8% at other temperatures ; e) the values below 1.5 Tm are calculated to fit the experimental data by using n = 2.40, a' = 2.06 x 10" 5 , and 3 = 0.0696. Trodahl , H. J., The Thermopower of Pure Tungsten Below 9 K, J. Phys. F, 3 , 1972-6 (1973) PURPOSE To make measurements of S on a pure W crystal in order to identify electron-electron scattering effects. SPECIMEN a) 3 cm long; 3 mm dia./rod; b) single crystal; e) RRR = 16300; g) axis along geometric axis, supplied by Materials Research Corp. APPARATUS a) longitudinal; b) matched Ge thermometers . DATA a) 0 to 9 K; b) figure 1 -X. ANALYSIS a) comparison with Wagner, D. K. , Garland, J. C. and Bowers, R. , Phys. Rev. B, 3, 3141 (1971), and White, G. K. and Woods, S. B., Can. J. Phys., 35, 656 11957). 253 Van Witzenburg, W. and Laubitz, M. J., Magnetoresistances and the Phonon Conductivity of Metals, Can. J. Phys. , 46(17) , 1887-94 (1968) PURPOSE To measure the magnetoresi stances of Cu, Ag, Au, and W. SPECIMEN a) 1.6 mm dia./wire; b) annealed in vacuum at 1620 K for 2 h; d) 99.95% W; e) RRR = 46.6; f) p0 = 0.11 y£2cm; g) source: United Min. Chem. Corp. APPARATUS a) longitudinal ; b) Pt resistance thermometers ; c) radiation shield. DATA a) 80 to 150 K; b) Table 4 -X; c) uncertainty: ±3%; d) correction applied to temperature derived from Pt thermometers. ANALYSIS a) compared with Fulkerson, W. , private communication (1968); b) concludes that lattice conductivity accounts for 40% at x for the temperature range i nvestigated. Wagner, D. K. , Lattice Thermal Conductivity and High-Field Electrical and Thermal Magnetoconductivities of Tungsten, Phys. Rev. B, 5^, 3.36-47 (1972) PURPOSE To investigate the temperature dependence of and x for W in a strong mag-netic field to provide further information about low temperature scattering mechani sms. SPECIMEN c) specimen spark cut from zone-refined crystal; e) original crystal: p 2 gg/p 0 63000; g) rod axis parallel to direction. APPARATUS a) longitudinal , H oriented normal to rod axis along direction; b) matched carbon resi stors/cal ibrated against stardard Ge resistance ther-mometer; c) adsorbent, vacuum insulation. DATA a) 1.8 to 6.5 K; b) figures 1,2 -H 2 /pxx, H^/WxxT respectively/p, H (kG) for runs 7-13 = 2.66, 5.32, 7.98, 10.6, 13.3, 16.0, 18.6. ANALYSIS a) results agree with Long, J. R. , Phys. Rev. B, 3 > 2476 (1971). 254 Wagner, D. K. , Garland, J. C. , and Bowers, R. , Low Temperature Electrical and Thermal Resistivities of Tungsten, Phys. Rev. B, 3 , 3141 (1971) PURPOSE To study the temperature dependence of p and WT in a number of high purity W single crystals. SPECIMEN a) W-3 , W-5 , W-8: 1.5 mm dia., W-4: 1.0 mm dia., W-7: 3.0 mm dia. /rods; b) single crystals; c) electron beam zone melted; f) p0 (W-3) = 1.231 x 10”^ ftcm, p 0 (W-4) = 1.780 x.10"^ flcm, p 0 ( W-5 ) = 5.724 x 10" 10 ftcm, pp (W-7) = 0.566 x 10"10 ft. cm, p 0 (W-8) = 0.695 x 10"10 ftcm; g) all oriented to . APPARATUS a) longitudinal ; b) carbon resistance thermometers/cal i brated against stand-ard Ge resistance thermometer; c) adsorbent, vacuum insulation. DATA a) 0 to 6 K/< 30 mK; b) figure 3 -(WT-p0 /L0 ), figure 2 -(p-p0 ) ; c) uncertainty: ±1% for 1.5 mm dia., ±8% for 3.0 mm dia. ANALYSIS a) L results agree with Bennett, A. J. and Rice, M. J., Phys. Rev., 185 , 968 (1969) and Rice, M. J., Phys. Rev. Lett.,^0, 1439 (1968), but are signifi-cantly below Herring, C. Wheeler, M. J., Thermal Diffusivity at Incandescent Temperatures by a Modu-lated Electron Beam Technique, Br. J. Appl . Phys., 16, 365 (1965) PURPOSE To assess the performance of an apparatus by means of measuring K and A of Pt, Ta, Mo, W. SPECIMEN a) 1.5 mm thick/disk; c) /cut from swaged rod; d) 99.5% W; g) source: Gen-eral Electric Co., Osram Lamp Works. APPARATUS a) periodic heat flow; b) optical pyrometer; c) vacuum insulation. DATA a) 1200 to 3000 K; b) Fig. 10/data points listed in TPRC data series; c) un-certainty: ±5%; d) corrected for losses in viewing window of pyrometer, spectral emissivity of specimen. ANALYSIS a) A results agree with Vines, R. F. (1941), Malter, L. and Langmuir, D. B. , Phys. Rev., .55, 745 (1939), Worthing, A. G., Phys. Rev., 28, 190 (1926), and Worthing, A. G. and Forsythe, W. E. , Astrophys. J., 61, 147 (1925). 255 White, G. K. and Woods, S. B., Low Temperature Resistivity of the Transi-tion Elements: Cobalt, Tungsten, and Rhenium, Can. J. Phys., 315, 656 ( 1957) PURPOSE To report experimental values of p and W for Co, W, Rh. SPECIMEN a) Wlb: 4 mm dia./rod; c) annealed at 1350 °C for several hours in vacuum, then kept at 600 °C for several more hours.; d) 0.01% Mo, traces Fe, SI, Cu ; f) p 0 = 3.15 x 10"° ftcm; g) source: Johnson, Matthey and Co. APPARATUS a) longitudinal; b) He gas thermometers ; c) radiation shield, thermometers Au plated to reduce radiation transfer. DATA a) 2 to 127 K; b) figure 2 - A/data listed in TPRC data series; c) uncer-tainty: ±1%. ANALYSIS a) comparison with Rosenberg, H. M. , Philos. Trans. Roy. Soc. London, 247 , 441 (1955), DeHaas, W. J., and deNobel , J., Physica,J5, 449 (1938), and Kannaluik, W. G. , Proc. Roy. Soc. London, Ser. A, 141 , 159 (1933). 256 U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publ icacion Date BIBLIOGRAPHIC DATA SHEET (See instructions) REPORT NO. NBSIR 84-3007 June 1984 4. TITLE AND SUBTITLE THERMAL CONDUCTIVITY OF ALUMINUM, COPPER, IRON, AND TUNGSTEN FOR TEMPERATURES FROM 1 K TO THE MELTING POINT 5. AUTHOR(S) J. G. Hust and A. B. Lankford 6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s) 7. Contract/Grant No. national bureau of standards DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered WASHINGTON, D.C. 20234 9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP) 10. SUPPLEMENTARY NOTES ^ Document describes a computer program; SF-185, FIPS Software Summary, is attached. 11. ABSTRACT (A 200-word or less factual summary of most significant information. I f document includes a significant bibliography or literature survey, mention it here) Literature data on the thermal conductivity of commercially pure aluminum, copper, iron, and tungsten specimens have been collected, coded, critically analyzed, and correlated with analytical techniques based on theoretical and empirical equations. The resulting functions are presented and used to generate tables and graphs of thermal conductivity as a function of temperature and residual resistivity ratio (RRR). An annotated bibliography of references is included. Discussions are included on the variations in thermal conductivity caused by chemical impurities, physical defects, size effects, and magnetic fields. Smoothed values are presented for temperatures from 1 K to near the melting point and for a large range of RRR values. 12. KEY WORDS ( Six to twelve entries; alphabetical order; capitalize on!y proper names; and separate key orcs aluminum; copper; electrical resistivity; iron; Lorenz ratio; residual resistivity ratio; thermal conductivity; tungsten 13. AVAILABILITY u nlimited 14. NO. OF PRINTED PAGES I For Official Distribution. Do Not Release to NTIS Order From Superintendent of Documents, U.S. Government Printing Office, Wash mgton, D. V 20402. 262 15. Puce X Order From National Technical Information Service (NTIS), Springfield. VA. 2216 $ 22.00