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18100	https://www.paramvisions.com/2022/03/how-to-make-temperature-correction-for.html	How to make temperature correction for a steel tape in surveying? / How to calculate true length for temperature correction? ~ PARAM VISIONS Home About us Contact us Disclaimer Privacy policy PARAM VISIONS All about civil construction knowledge- PARAM VISIONS Home Categories-1 » Building materials Causes and reasons Comparison and difference Construction machinery Different types Home and garden Procedure and checklist Terms and definition Catogeries-2 » BBS and cutting length Estimation and calculation Geotechnical and mechanics Instant calculator Quantity estimate Rate analysis and cost Structural design Surveying » Field and lab. test Infrastructure Useful » Useful tips Useful links My youtube channel » Param visions Income brains Home » Surveying » How to make temperature correction for a steel tape in surveying? / How to calculate true length for temperature correction? How to make temperature correction for a steel tape in surveying? / How to calculate true length for temperature correction? param visions29.3.221 comment Eg: A 30m. surveyor's steel tape is correct at 21°C. The distance measured by using this tape on a day when the temperature is 32°C is 230m. Calculate the true distance between the points by applying temperature correction. Coefficient of thermal expansion ∝ = 1.4 х 10⁻⁵ per ° C. To find: The true distance between the points. Calculation: The true length L = [ Measured length + temp. correction.] L = [ L1 + Ct ] Now, The temperature correction is calculated by the formula Ct = [ ∝ х ( Tm - To ) х L1 ] Where, ∝ = Coefficient of thermal expansion = 1.4 х 10⁻⁵ Tm = Field temp. = 32°C To = Standard temp. = 21°C L1 = Measured distance = 230m. Ct = [1.4 х 10⁻⁵ х ( 32°C - 21°C ) х 230m. ] = 0.0354m. The true length between the points L = [ L1 + Ct ] = [230m. + 0.0354m.] L =230.0354m. To understand A to Z of surveying,click here. Thank you for going through these calculation steps❤.Have a good day😄. Share: Email ThisBlogThis!Share to XShare to FacebookShare to Pinterest Related Posts: What is drone surveying?/What are the benefits of drone surveying? How to determine the angle by using measuring tape?/Finding the angle between two sides by measuring tape. How to calculate the horizontal distance using auto level machine? What is contour interval?/ Uses of contour maps in surveying. How to do a layout of the building plan with a 3 - 4 - 5 rule?/ Using 3-4-5 method in construction work. ←Newer PostOlder Post→Home 1 comment: AnonymousFebruary 7, 2024 at 2:26 PM Great ReplyDelete Replies Reply Add comment Load more... Labels BBS and cutting length(32) Building materials(33) Causes and reasons(20) Comparison and difference(26) Construction machinery(14) Different types(46) Estimation and calculation(90) Field and lab. test(9) Geotechnical and mechanics(24) Home and garden(19) Infrastructure(9) Instant calculator(24) Procedure and checklist(31) Quantity estimate(39) Rate analysis and cost(35) Structural design(19) Surveying(33) Terms and definition(60) Useful tips(22) Translate Powered by Translate Popular Tags Blog Archives Discover more tile Building material granite granites Tile Building materials Granite tiles Blog Archive popular posts What is brass in civil construction? Its conversion and weight calculation. Brass is a non-standard unit that is still prevalent in India with a history of 200 years or so. The word brass is used as a measurement un... Minimum space required for car parking in a residential building. When you build a house, you will keep aside some space for the parking of the car. The dimensions of individual cars are different & in... Basic rules for lapping in column reinforcement. / 7 basic rules for providing Lap splice in column. Now, let us go through 7 basic rules while lapping the rebars in columns. 1. Lap length diameter: When we extend the top bar of the column... How to calculate the area of irregular shaped land or plot?/ Irregular shape site area calculation.( part-1 ) If you are looking for a readymade calculator, then click here. For the calculation procedure, go through the following steps. 👇 Now, let... Recent Posts Points to check before purchasing granites./How to buy best quality granites?/Tips for buying granites. Standard height of switch boards for different rooms of residential buildings./Standard height & location of electric switch boards. When do we need DPC in construction?/Plinth beam or DPC, which one is necessary for building?/Plinth beam Vs. DPC in construction. Google search × Custom Search Please visit My youtube channel Copyright © 2025 PARAM VISIONS \| Powered by Blogger Design by FlexiThemes \| BTheme by NewBloggerThemes.com \| Security Guard Course Burlington \| Security Guard Course Thunder Bay Original text Rate this translation Your feedback will be used to help improve Google Translate
18101	https://www.desmos.com/calculator/lx8hyqblop	Graphing Polar Equations \| Desmos Loading... Graphing Polar Equations Save Copy Log In Sign Up r = 6cos(theta) 1 r = 2 + 4cos(theta) 3 Expression 5: "u" equals 0 u=0 0 0 360 3 6 0 5 Example 4a 6 Example 4b 9 Example 4c 12 Example 4d 15 18 powered by powered by "x"x "y"y "a" squared a 2 "a" Superscript, "b" , Baseline a b 7 7 8 8 9 9 divided by÷ functions (( )) less than< greater than> 4 4 5 5 6 6 times× \| "a" \|\|a\| ,, less than or equal to≤ greater than or equal to≥ 1 1 2 2 3 3 negative− A B C StartRoot, , EndRoot pi π 0 0 .. equals= positive+
18102	https://courses.lumenlearning.com/calculus2/chapter/first-order-differential-equations/	First-Order Differential Equations Learning Outcomes Write a first-order linear differential equation in standard form Find an integrating factor and use it to solve a first-order linear differential equation See the example on the introduction page for a first-order linear differential equation. Definition A first-order differential equation is linear if it can be written in the form a(x)y′+b(x)y=c(x), where a(x),b(x), and c(x) are arbitrary functions of x. Remember that the unknown function y depends on the variable x; that is, x is the independent variable and y is the dependent variable. Some examples of first-order linear differential equations are (3x2−4)y′+(x−3)y=sinx(sinx)y′−(cosx)y=cotx4xy′+(3lnx)y=x3−4x. Examples of first-order nonlinear differential equations include (y′)4−(y′)3=(3x−2)(y+4)4y′+3y3=4x−5(y′)2=siny+cosx. These equations are nonlinear because of terms like (y′)4,y3, etc. Due to these terms, it is impossible to put these equations into the same form as the definition. Standard Form Consider the differential equation (3x2−4)y′+(x−3)y=sinx. Our main goal in this section is to derive a solution method for equations of this form. It is useful to have the coefficient of y′ be equal to 1. To make this happen, we divide both sides by 3x2−4. y′+(x−33x2−4)y=sinx3x2−4 This is called the standard form of the differential equation. We will use it later when finding the solution to a general first-order linear differential equation. Returning to the definition, we can divide both sides of the equation by a(x). This leads to the equation y′+b(x)a(x)y=c(x)a(x). Now define p(x)=b(x)a(x) and q(x)=c(x)a(x). Then the definition becomes y′+p(x)y=q(x). We can write any first-order linear differential equation in this form, and this is referred to as the standard form for a first-order linear differential equation. Example: Writing First-Order Linear Equations in Standard Form Put each of the following first-order linear differential equations into standard form. Identify p(x) and q(x) for each equation. y′=3x−4y 3xy′4y−3=2 (here x>0) y=3y′−4x2+5 Show Solution Add 4y to both sides: y′+4y=3x. In this equation, p(x)=4 and q(x)=3x. 2. Multiply both sides by 4y−3, then subtract 8y from each side: 3xy′4y−3=23xy′=2(4y−3)3xy′=8y−63xy′−8y=−6. Finally, divide both sides by 3x to make the coefficient of y′ equal to 1: y′−83xy=−23x. This is allowable because in the original statement of this problem we assumed that x>0. (If x=0 then the original equation becomes 0=2, which is clearly a false statement.) In this equation, p(x)=−83x and q(x)=−23x. 3. Subtract y from each side and add 4x2−5: 3y′−y=4x2−5. Next divide both sides by 3: y′−13y=43x2−53. In this equation, p(x)=−13 and q(x)=43x2−53. Watch the following video to see the worked solution to Example: Writing First-Order Linear Equations in Standard Form. You can view the transcript for “4.5.1” here (opens in new window). try it Put the equation (x+3)y′2x−3y−4=5 into standard form and identify p(x) and q(x). Hint Multiply both sides by the common denominator, then collect all terms involving y on one side. Show Solution y′+15x+3y=10x−20x+3;p(x)=15x+3 and q(x)=10x−20x+3 Try It Integrating Factors We now develop a solution technique for any first-order linear differential equation. We start with the standard form of a first-order linear differential equation: y′+p(x)y=q(x). The first term on the left-hand side of y′+b(x)a(x)y=c(x)a(x) is the derivative of the unknown function, and the second term is the product of a known function with the unknown function. This is somewhat reminiscent of the power rule from the Differentiation Rules section. If we multiply y′+p(x)y=q(x) by a yet-to-be-determined function μ(x), then the equation becomes μ(x)y′+μ(x)p(x)y=μ(x)q(x). The left-hand side of y′+p(x)y=q(x) can be matched perfectly to the product rule: ddx[f(x)g(x)]=f′(x)g(x)+f(x)g′(x). Matching term by term gives y=f(x),g(x)=μ(x), and g′(x)=μ(x)p(x). Taking the derivative of g(x)=μ(x) and setting it equal to the right-hand side of g′(x)=μ(x)p(x) leads to μ′(x)=μ(x)p(x). This is a first-order, separable differential equation for μ(x). We know p(x) because it appears in the differential equation we are solving. Separating variables and integrating yields μ′(x)μ(x)=p(x)∫μ′(x)μ(x)dx=∫p(x)dxln\|μ(x)\|=∫p(x)dx+Celn\|μ(x)\|=e∫p(x)dx+C\|μ(x)\|=C1e∫p(x)dxμ(x)=C2e∫p(x)dx Here C2 can be an arbitrary (positive or negative) constant. This leads to a general method for solving a first-order linear differential equation. We first multiply both sides of y′+p(x)y=q(x) by the integrating factor μ(x). This gives μ(x)y′+μ(x)p(x)y=μ(x)q(x). The left-hand side of the above equation can be rewritten as ddx(μ(x)y). ddx(μ(x)y)=μ(x)q(x). Next integrate both sides with respect to x. ∫ddx(μ(x)y)dx=∫μ(x)q(x)dxμ(x)y=∫μ(x)q(x)dx Divide both sides by μ(x): y=1μ(x)[∫μ(x)q(x)dx+C]. Since μ(x) was previously calculated, we are now finished. An important note about the integrating constant C: It may seem that we are inconsistent in the usage of the integrating constant. However, the integral involving p(x) is necessary in order to find an integrating factor for y′+b(x)a(x)y=c(x)a(x). Only one integrating factor is needed in order to solve the equation; therefore, it is safe to assign a value for C for this integral. We chose C=0. When calculating the integral inside the brackets in ddx(μ(x)y)=μ(x)q(x), it is necessary to keep our options open for the value of the integrating constant, because our goal is to find a general family of solutions to y′+b(x)a(x)y=c(x)a(x). This integrating factor guarantees just that. Problem-Solving Strategy: Solving a First-order Linear Differential Equation Put the equation into standard form and identify p(x) and q(x). Calculate the integrating factor μ(x)=e∫p(x)dx. Multiply both sides of the differential equation by μ(x). Integrate both sides of the equation obtained in step 3, and divide both sides by μ(x). If there is an initial condition, determine the value of C. Example: Solving a First-order Linear Equation Find a general solution for the differential equation xy′+3y=4x2−3x. Assume x>0. Show Solution To put this differential equation into standard form, divide both sides by x: y′+3xy=4x−3. Therefore p(x)=3x and q(x)=4x−3. 2. The integrating factor is μ(x)=e∫(3x)dx=e3lnx=x3. 3. Multiplying both sides of the differential equation by μ(x) gives us x3y′+x3(3x)y=x3(4x−3)x3y′+3x2y=4x4−3x3ddx(x3y)=4x4−3x3. 4. Integrate both sides of the equation. ∫ddx(x3y)dx=∫4x4−3x3dxx3y=4x55−3x44+Cy=4x25−3x4+Cx−3. 5. There is no initial value, so the problem is complete. Analysis You may have noticed the condition that was imposed on the differential equation; namely, x>0. For any nonzero value of C, the general solution is not defined at x=0. Furthermore, when x<0, the integrating factor changes. The integrating factor is given by μ(x)y′+μ(x)p(x)y=μ(x)q(x) as f(x)=e∫p(x)dx. For this p(x) we get e∫p(x)dx=e∫(3x)dx=e3ln\|x\|=\|x\|3, since x<0. The behavior of the general solution changes at x=0 largely due to the fact that p(x) is not defined there. Watch the following video to see the worked solution to Example: Solving a First-Order Linear Equation. For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end. You can view the transcript for this segmented clip of “4.5.2” here (opens in new window). try it Find the general solution to the differential equation (x−2)y′+y=3x2+2x. Assume x>2. Hint Use the method outlined in the problem-solving strategy for first-order linear differential equations. Show Solution y=x3+x2+Cx−2 Now we use the same strategy to find the solution to an initial-value problem. Example: A First-order Linear Initial-Value Problem Solve the initial-value problem y′+3y=2x−1,y(0)=3. Show Solution This differential equation is already in standard form with p(x)=3 and q(x)=2x−1. The integrating factor is μ(x)=e∫3dx=e3x. Multiplying both sides of the differential equation by μ(x) gives e3xy′+3e3xy=(2x−1)e3xddx[ye3x]=(2x−1)e3x. Integrate both sides of the equation: ∫ddx[ye3x]dx=∫(2x−1)e3xdxye3x=e3x3(2x−1)−∫23e3xdxye3x=e3x(2x−1)3−2e3x9+Cy=2x−13−29+Ce−3xy=2x3−59+Ce−3x. 4. Now substitute x=0 and y=3 into the general solution and solve for C: y=23x−59+Ce−3x3=23(0)−59+Ce−3(0)3=−59+CC=329. Therefore the solution to the initial-value problem is y=23x−59+329e−3x. Watch the following video to see the worked solution to Example: A First-Order Linear Initial-Value Problem. For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end. You can view the transcript for this segmented clip of “4.5.2” here (opens in new window). try it Solve the initial-value problem y′−2y=4x+3,y(0)=−2. Show Solution y=−2x−4+2e2x Try It Candela Citations CC licensed content, Original 4.5.1. Authored by: Ryan Melton. License: CC BY: Attribution 4.5.2. Authored by: Ryan Melton. License: CC BY: Attribution CC licensed content, Shared previously Calculus Volume 2. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at Licenses and Attributions CC licensed content, Original 4.5.1. Authored by: Ryan Melton. License: CC BY: Attribution 4.5.2. Authored by: Ryan Melton. License: CC BY: Attribution CC licensed content, Shared previously Calculus Volume 2. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at
18103	https://ocw.mit.edu/courses/3-091sc-introduction-to-solid-state-chemistry-fall-2010/pages/electronic-materials/13-band-theory-of-solids/	Browse Course Material Course Info Instructor Prof. Donald Sadoway Departments Materials Science and Engineering As Taught In Fall 2010 Level Undergraduate Topics Engineering Chemical Engineering Materials Science and Engineering Science Chemistry Learning Resource Types grading Exams with Solutions notes Lecture Notes theaters Lecture Videos assignment_turned_in Problem Sets with Solutions theaters Problem-solving Videos grading Exams assignment Problem Sets Supplemental Exam Materials Download Course search GIVE NOW about ocw help & faqs contact us 3.091SC \| Fall 2010 \| Undergraduate Introduction to Solid State Chemistry Electronic Materials 13. Band Theory of Solids « Previous \| Next » Session Overview \| \| \| --- \| \| Modules \| Electronic Materials \| \| Concepts \| properties of metals and insulators, band theory of solids (Drude; Bloch; Heitler and London), band gaps in metals, semiconductors, and insulators \| \| Keywords \| metallic bonding, free electron gas, band gap, electrical conductivity, Bloch wave, photoexcitation, charge carrier, metal, insulator, semiconductor, thermal conductivity, valence band, conduction band, antibonding orbital, bonding orbital, carrier mobility, absorption edge, thermal excitation, electron, hole, current, Paul Drude, Felix Bloch, Walter Heitler, Fritz London \| \| Chemical Substances \| copper (Cu), beryllium (Be), diamond (C), silicon (Si), germanium (Ge), tin (Sn), lead (Pb) \| \| Applications \| photovoltaics, photosensors, light-emitting diodes (LEDs), temperature sensors \| Prerequisites Before starting this session, you should be familiar with prior topics from Structure of the Atom (Session 1 through Session 7) and Bonding & Molecules (Session 8 through Session 12), specifically: Electron orbital filling order, energy levels, and the Schrödinger equation Linear combination and hybridization of orbitals Electromagnetic radiation, particularly the visible spectrum, and how to convert between wavelength, frequency, and energy Learning Objectives After completing this session, you should be able to: Describe the “free electron gas” model and its shortcomings in explaining the physical properties of metals. Derive the band structure of a solid, starting from the orbital diagrams of individual atoms. Calculate the absorption edge, carrier density, and electrical conductivity of a material, and predict how incident photons of given energies or wavelengths will interact with a material. Explain how electronic structure and bonding affects the thermal conductivity, electrical conductivity, optical behavior, and other bulk properties of solids. Classify materials as metals, insulators, or semiconductors, and sketch a schematic band diagram for each one, with key features labeled. Describe what happens during photoexcitation and thermal excitation. Reading Archived Lecture Notes #3 (PDF) \| Book Chapters \| Topics \| --- \| \| [Saylor] 12.5, “Correlation Between Bonding and the Properties of Solids.” \| Ionic solids; molecular solids; covalent solids; metallic solids \| \| [Saylor] 12.6, “Bonding in Metals and Semiconductors.” \| Band theory; requirements for metallic behavior; insulators; semiconductors \| \| [JS] 2.4, “The Metallic Bond.” \| Metallic bonding; delocalized electrons; electronegativity \| \| [JS] 15.1, “Charge Carriers and Conduction.” \| Holes and electrons; Ohm’s Law; resistivity/conductivity; carrier mobility and drift velocity \| \| [JS] 15.2, “Energy Levels and Energy Bands.” \| Pauli exclusion, Hund’s rule, and orbital filling; energy bands and gaps; the Fermi function; thermal promotion; metals, insulators, and semiconductors \| Lecture Video View video page Download video Download transcript Resources Lecture Slides (PDF) Lecture Summary Prof. Ron Ballinger (homepage) gives today’s lecture, explaining how the behavior of electrons in aggregate solids determines their electrical and thermal conductivities, optical absorption, and other physical properties. He derives the valence and conduction band structures for electrons in metals (e.g. Cu, Be) using LCAO-MO, and then extends this approach to insulators (e.g. C) and semiconductors (e.g. Si, Ge), which exhibit band gaps. Electrons are promoted across the band gap by photoexcitation or thermal excitation, leaving holes behind. Controlling the population and flow of charge carriers is the fundamental principle underlying modern semiconductor engineering. Homework Problems (PDF) Solutions (PDF) For Further Study Supplemental Readings Bloch, Felix. “Über die Quantenmechanik der Elektronen in Kristallgittern.” Zeitschrift für Physik A: Hadrons and Nuclei 52 (1928): 555-600. (Note: this article is in German.) Heitler, Walter, and Fritz London. “Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik.” Zeitschrift für Physik A: Hadrons and Nuclei 44 (1927): 455-472. (Note: this article is in German.) People Felix Bloch – 1952 Nobel Prize in Physics Paul Drude Walter Heitler Fritz London Other OCW and OER Content \| Content \| Provider \| Level \| Notes \| --- --- \| \| Introduction to Semiconductors \| DoITPoMS \| Undergraduate \| See “Introduction to Energy Bands.” \| \| 5.112 Principles of Chemical Science \| MIT OpenCourseWare \| Undergraduate (first-year) \| Start - 8:15 in Lecture 34: Bonding in Metals and Semiconductors \| « Previous \| Next » Course Info Instructor Prof. Donald Sadoway Departments Materials Science and Engineering As Taught In Fall 2010 Level Undergraduate Topics Engineering Chemical Engineering Materials Science and Engineering Science Chemistry Learning Resource Types grading Exams with Solutions notes Lecture Notes theaters Lecture Videos assignment_turned_in Problem Sets with Solutions theaters Problem-solving Videos grading Exams assignment Problem Sets Supplemental Exam Materials Download Course Over 2,500 courses & materials Freely sharing knowledge with learners and educators around the world. 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18104	https://fiveable.me/key-terms/ap-enviro/parts-per-million-ppm	Parts per Million (PPM) - (AP Environmental Science) - Vocab, Definition, Explanations \| Fiveable \| Fiveable ap study content toolsprintables upgrade All Key Terms AP Environmental Science Parts per Million (PPM) ♻️ap environmental science review key term - Parts per Million (PPM) Citation: MLA Definition PPM is a unit of measurement used to express the concentration of one substance in a solution or mixture. It represents the number of parts of a particular substance per million parts of the whole. Related terms Parts per Billion (PPB):Similar to PPM, PPB is another unit of measurement used for even smaller concentrations. It represents the number of parts of a substance per billion parts. Concentration:Concentration refers to the amount or proportion of a substance present in a given volume or space. Dilution:Dilution is the process of reducing the concentration of a solute in a solution by adding more solvent. 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18105	https://www.allaboutcircuits.com/textbook/alternating-current/chpt-8/low-pass-filters/	Network Sites: Latest News Technical Articles Latest News Technical Articles Market Insights Education Latest Projects Education Log In Join Textbook Low-pass Filters Join our Engineering Community! Sign-in with: Home Textbook Alternating Current (AC) Filters Low-pass Filters Filters What is a Filter? Low-pass Filters High-pass Filters Band-pass Filters Band-stop Filters Resonant Filters Summary of Filters Vol. Alternating Current (AC) Chapter 8 Filters Low-pass Filters PDF Version By definition, a low-pass filter is a circuit offering easy passage to low-frequency signals and difficult passage to high-frequency signals. There are two basic kinds of circuits capable of accomplishing this objective, and many variations of each one: The inductive low-pass filter in (Figure below) and the capacitive low-pass filter in (Figure also below). Inductive Low-Pass Filter Inductive low-pass filter The inductor’s impedance increases with increasing frequency. This high impedance in series tends to block high-frequency signals from getting to the load. This can be demonstrated with a SPICE analysis: (Figure below) ``` inductive lowpass filter v1 1 0 ac 1 sin l1 1 2 3 rload 2 0 1k .ac lin 20 1 200 .plot ac v(2) .end ``` The response of an inductive low-pass filter falls off with increasing frequency. Capacitive Low-Pass Filter Capacitive low-pass filter The capacitor’s impedance decreases with increasing frequency. This low impedance in parallel with the load resistance tends to short out high-frequency signals, dropping most of the voltage across series resistor R1. (Figure below) ``` capacitive lowpass filter v1 1 0 ac 1 sin r1 1 2 500 c1 2 0 7u rload 2 0 1k .ac lin 20 30 150 .plot ac v(2) .end ``` The response of a capacitive low-pass filter falls off with increasing frequency. The inductive low-pass filter is the pinnacle of simplicity, with only one component comprising the filter. The capacitive version of this filter is not that much more complex, with only a resistor and capacitor needed for operation. However, despite their increased complexity, capacitive filter designs are generally preferred over inductive because capacitors tend to be “purer” reactive components than inductors and therefore are more predictable in their behavior. By “pure” I mean that capacitors exhibit little resistive effects than inductors, making them almost 100% reactive. Inductors, on the other hand, typically exhibit significant dissipative (resistor-like) effects, both in the long lengths of wire used to make them, and in the magnetic losses of the core material. Capacitors also tend to participate less in “coupling” effects with other components (generate and/or receive interference from other components via mutual electric or magnetic fields) than inductors, and are less expensive. However, the inductive low-pass filter is often preferred in AC-DC power supplies to filter out the AC “ripple” waveform created when AC is converted (rectified) into DC, passing only the pure DC component. The primary reason for this is the requirement of low filter resistance for the output of such a power supply. A capacitive low-pass filter requires an extra resistance in series with the source, whereas the inductive low-pass filter does not. In the design of a high-current circuit like a DC power supply where additional series resistance is undesirable, the inductive low-pass filter is the better design choice. On the other hand, if low weight and compact size are higher priorities than low internal supply resistance in a power supply design, the capacitive low-pass filter might make more sense. Cutoff Frequency All low-pass filters are rated at a certain cutoff frequency. That is, the frequency above which the output voltage falls below 70.7% of the input voltage. This cutoff percentage of 70.7 is not really arbitrary, all though it may seem so at first glance. In a simple capacitive/resistive low-pass filter, it is the frequency at which capacitive reactance in ohms equals resistance in ohms. In a simple capacitive low-pass filter (one resistor, one capacitor), the cutoff frequency is given as: Scroll to continue with content Inserting the values of R and C from the last SPICE simulation into this formula, we arrive at a cutoff frequency of 45.473 Hz. However, when we look at the plot generated by the SPICE simulation, we see the load voltage well below 70.7% of the source voltage (1 volt) even at a frequency as low as 30 Hz, below the calculated cutoff point. What’s wrong? The problem here is that the load resistance of 1 kΩ affects the frequency response of the filter, skewing it down from what the formula told us it would be. Without that load resistance in place, SPICE produces a Bode plot whose numbers make more sense: (Figure below) ``` capacitive lowpass filter v1 1 0 ac 1 sin r1 1 2 500 c1 2 0 7u note: no load resistor! .ac lin 20 40 50 .plot ac v(2) .end ``` For the capacitive low-pass filter with R = 500 Ω and C = 7 µF, the Output should be 70.7% at 45.473 Hz. fcutoff = 1/(2πRC) = 1/(2π(500 Ω)(7 µF)) = 45.473 Hz When dealing with filter circuits, it is always important to note that the response of the filter depends on the filter’s component values and the impedance of the load. If a cutoff frequency equation fails to give consideration to load impedance, it assumes no load and will fail to give accurate results for a real-life filter conducting power to a load. Application of Low-Pass Filter One frequent application of the capacitive low-pass filter principle is in the design of circuits having components or sections sensitive to electrical “noise.” As mentioned at the beginning of the last chapter, sometimes AC signals can “couple” from one circuit to another via capacitance (Cstray) and/or mutual inductance (Mstray) between the two sets of conductors. A prime example of this is unwanted AC signals (“noise”) becoming impressed on DC power lines supplying sensitive circuits: (Figure below) Noise is coupled by stray capacitance and mutual inductance into “clean” DC power. The oscilloscope-meter on the left shows the “clean” power from the DC voltage source. After coupling with the AC noise source via stray mutual inductance and stray capacitance, though, the voltage as measured at the load terminals is now a mix of AC and DC, the AC being unwanted. Normally, one would expect Eload to be precisely identical to Esource, because the uninterrupted conductors connecting them should make the two sets of points electrically common. However, power conductor impedance allows the two voltages to differ, which means the noise magnitude can vary at different points in the DC system. If we wish to prevent such “noise” from reaching the DC load, all we need to do is connect a low-pass filter near the load to block any coupled signals. In its simplest form, this is nothing more than a capacitor connected directly across the power terminals of the load, the capacitor behaving as a very low impedance to any AC noise, and shorting it out. Such a capacitor is called a decoupling capacitor: (Figure below) Decoupling capacitor, applied to load, filters noise from DC power supply. A cursory glance at a crowded printed-circuit board (PCB) will typically reveal decoupling capacitors scattered throughout, usually located as close as possible to the sensitive DC loads. Capacitor size is usually 0.1 µF or more, a minimum amount of capacitance needed to produce a low enough impedance to short out any noise. Greater capacitance will do a better job at filtering noise, but size and economics limit decoupling capacitors to meager values. REVIEW: A low-pass filter allows for easy passage of low-frequency signals from source to load, and difficult passage of high-frequency signals. Inductive low-pass filters insert an inductor in series with the load; capacitive low-pass filters insert a resistor in series and a capacitor in parallel with the load. The former filter design tries to “block” the unwanted frequency signal while the latter tries to short it out. The cutoff frequency for a low-pass filter is that frequency at which the output (load) voltage equals 70.7% of the input (source) voltage. Above the cutoff frequency, the output voltage is lower than 70.7% of the input, and vice versa. RELATED WORKSHEETS: Active Filters Worksheet Passive Filter Circuits Worksheet What is a Filter? Textbook Index High-pass Filters Lessons in Electric Circuits Volumes » Direct Current (DC) Alternating Current (AC) ##### Chapters » 1Basic AC Theory 2Complex Numbers 3Reactance and Impedance—Inductive 4Reactance and Impedance—Capacitive 5Reactance and Impedance—R, L, And C 6Resonance 7Mixed-Frequency AC Signals 8Filters Pages » What is a Filter? Low-pass Filters High-pass Filters Band-pass Filters Band-stop Filters Resonant Filters Summary of Filters 9Transformers 10Polyphase AC Circuits 11Power Factor 12AC Metering Circuits 13AC Motors 14Transmission Lines 15Contributors List Semiconductors Digital Circuits EE Reference DIY Electronics Projects Advanced Textbooks Practical Guide to Radio-Frequency Analysis and Design Designing Analog Chips Related Content What Is a Low Pass Filter? A Tutorial on the Basics of Passive RC Filters Analyzing the Moog Filter AVX Thin Film Harmonic Low Pass Filter \| Tech Specs From Filter Specs to Window Parameters in FIR Filter Design How to Select the Cutoff Frequency of Your Low-Pass Filter Phase Line Filter Design Published under the terms and conditions of the Design Science License You May Also Like #### Innovate, Anticipate and Automate with Future Electronics In Partnership with Future Electronics #### Programmable Power Supplies Keep Racing Team on Track by AMETEK Programmable Power #### OpenGMSL Association Launches to Enable In-Vehicle GMSL Connectivity by Jake Hertz #### Zuken Brings AI Updates to Flagship PCB CAD Systems by Duane Benson #### Modified Agile for Electronics Development: A Smarter Path to High-Value Solutions by Altium
18106	https://www.youtube.com/watch?v=aJCP6Y3gW0I	constructing parallel lines (rhombus method) - geometry vinteachesmath 44900 subscribers 122 likes Description 15025 views Posted: 16 Oct 2021 In this video I show how to construct parallel lines with the rhombus method. The specific question covered involves constructing a line parallel to given line through a given point. This technique is a quick, efficient way to construct parallel lines. I prefer this technique over the other method which involves the copying angles construction technique. Outside of this computer software, this is my favorite type of compass to use: If this video was helpful, please LIKE and SUBSCRIBE. If you have any requests, leave the topics you want me to cover in the COMMENT section. Thanks for watching! 17 comments Transcript: what's up i'm vin and today i want to show how to construct parallel lines using the rhombus method but first i want to talk about why this process works so our goal is going to be to construct a quadrilateral with four equal sides and that's going to be enough to establish that we have a rhombus and once we know we have a rhombus a rhombus is a parallelogram and opposite sides of a parallelogram are parallel so what i want to do is i'm going to extend the compass in such a way that i know that it's going to hit this line l at two locations so you could imagine here if i swing it like this it's going to hit line l in two locations and for this entire construction i'm not going to change the length of the compass at all because noah rhombus has four sides with equal measure so notice here we get a few intersection points that this arc is going to hit line l at two locations but i'm going to label this location here as a so i started this construction by placing the compass on point b but now i'm going to move it from b over to point a and i'm going to swing another arc remember i'm not changing the length of the compass at all but i swing the arc and it hits this line l at another location which i'm going to call point c and this represents the third point of our rhombus and now i'm going to move the compass once more over to point c and i'm going to swing an arc for one last time until it hits that original arc and that last point of intersection here we're going to call point d so now a b c d if i connect all these four points would make a rhombus but the only points i really need to connect here are points b and d because that's going to form the straight line that's going to be parallel to line l so if we look here and let's just say i aim this really carefully when i connect points b and d this line that i've just made is going to be parallel to line l so i'm going to draw my arrows in and the conclusion that we could draw here let's say i call this line m i could say that line l is parallel to line m okay well this is going to conclude this video on constructing parallel lines with the rhombus method if this video was helpful please like and subscribe it really helps me grow the channel and if you got any requests just leave the topics you want me to cover in the comment section below and thanks for watching
18107	https://pubmed.ncbi.nlm.nih.gov/25244284/	Structure-activity relationship study around guanabenz identifies two derivatives retaining antiprion activity but having lost α2-adrenergic receptor agonistic activity - PubMed Clipboard, Search History, and several other advanced features are temporarily unavailable. Skip to main page content An official website of the United States government Here's how you know The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site. The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely. 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Structure-activity relationship study around guanabenz identifies two derivatives retaining antiprion activity but having lost α2-adrenergic receptor agonistic activity Phu Hai Nguyen1,Hassan Hammoud,Sophie Halliez,Yanhong Pang,Justine Evrard,Martine Schmitt,Nassima Oumata,Jean-Jacques Bourguignon,Suparna Sanyal,Vincent Beringue,Marc Blondel,Frédéric Bihel,Cécile Voisset Affiliations Expand Affiliation 1 Inserm UMR 1078, Université de Bretagne Occidentale , Faculté de Médecine et des Sciences de la Santé; Etablissement Français du Sang (EFS) Bretagne; CHRU Brest, Hôpital Morvan, Laboratoire de Génétique Moléculaire, 29200 Brest, France. PMID: 25244284 DOI: 10.1021/cn5001588 Item in Clipboard Structure-activity relationship study around guanabenz identifies two derivatives retaining antiprion activity but having lost α2-adrenergic receptor agonistic activity Phu Hai Nguyen et al. ACS Chem Neurosci.2014. Show details Display options Display options Format ACS Chem Neurosci Actions Search in PubMed Search in NLM Catalog Add to Search . 2014 Oct 15;5(10):1075-82. doi: 10.1021/cn5001588. Epub 2014 Sep 22. Authors Phu Hai Nguyen1,Hassan Hammoud,Sophie Halliez,Yanhong Pang,Justine Evrard,Martine Schmitt,Nassima Oumata,Jean-Jacques Bourguignon,Suparna Sanyal,Vincent Beringue,Marc Blondel,Frédéric Bihel,Cécile Voisset Affiliation 1 Inserm UMR 1078, Université de Bretagne Occidentale , Faculté de Médecine et des Sciences de la Santé; Etablissement Français du Sang (EFS) Bretagne; CHRU Brest, Hôpital Morvan, Laboratoire de Génétique Moléculaire, 29200 Brest, France. PMID: 25244284 DOI: 10.1021/cn5001588 Item in Clipboard Cite Display options Display options Format Abstract Guanabenz (GA) is an orally active α2-adrenergic agonist that has been used for many years for the treatment of hypertension. We recently described that GA is also active against both yeast and mammalian prions in an α2-adrenergic receptor-independent manner. These data suggest that this side-activity of GA could be explored for the treatment of prion-based diseases and other amyloid-based disorders. In this perspective, the potent antihypertensive activity of GA happens to be an annoying side-effect that could limit its use. In order to get rid of GA agonist activity at α2-adrenergic receptors, we performed a structure-activity relationship study around GA based on changes of the chlorine positions on the benzene moiety and then on the modifications of the guanidine group. Hence, we identified the two derivatives 6 and 7 that still possess a potent antiprion activity but were totally devoid of any agonist activity at α2-adrenergic receptors. Similarly to GA, 6 and 7 were also able to inhibit the protein folding activity of the ribosome (PFAR) which has been suggested to be involved in prion appearance/maintenance. Therefore, these two GA derivatives are worth being considered as drug candidates. Keywords: Antiprion compounds; PrPSc prion protein; guanabenz; structure−activity relationship study; yeast model for prion diseases; α2-adrenergic agonist. PubMed Disclaimer Similar articles Antihypertensive drug guanabenz is active in vivo against both yeast and mammalian prions.Tribouillard-Tanvier D, Béringue V, Desban N, Gug F, Bach S, Voisset C, Galons H, Laude H, Vilette D, Blondel M.Tribouillard-Tanvier D, et al.PLoS One. 2008 Apr 23;3(4):e1981. doi: 10.1371/journal.pone.0001981.PLoS One. 2008.PMID: 18431471 Free PMC article. Protein folding activity of the ribosome (PFAR) -- a target for antiprion compounds.Banerjee D, Sanyal S.Banerjee D, et al.Viruses. 2014 Oct 23;6(10):3907-24. doi: 10.3390/v6103907.Viruses. 2014.PMID: 25341659 Free PMC article.Review. The various facets of the protein-folding activity of the ribosome.Voisset C, Saupe SJ, Blondel M.Voisset C, et al.Biotechnol J. 2011 Jun;6(6):668-73. doi: 10.1002/biot.201100021. Epub 2011 May 12.Biotechnol J. 2011.PMID: 21567961 Review. Protein folding activity of ribosomal RNA is a selective target of two unrelated antiprion drugs.Tribouillard-Tanvier D, Dos Reis S, Gug F, Voisset C, Béringue V, Sabate R, Kikovska E, Talarek N, Bach S, Huang C, Desban N, Saupe SJ, Supattapone S, Thuret JY, Chédin S, Vilette D, Galons H, Sanyal S, Blondel M.Tribouillard-Tanvier D, et al.PLoS One. 2008 May 14;3(5):e2174. doi: 10.1371/journal.pone.0002174.PLoS One. 2008.PMID: 18478094 Free PMC article. The toll-like receptor agonist imiquimod is active against prions.Oumata N, Nguyen PH, Beringue V, Soubigou F, Pang Y, Desban N, Massacrier C, Morel Y, Paturel C, Contesse MA, Bouaziz S, Sanyal S, Galons H, Blondel M, Voisset C.Oumata N, et al.PLoS One. 2013 Aug 16;8(8):e72112. doi: 10.1371/journal.pone.0072112. eCollection 2013.PLoS One. 2013.PMID: 23977222 Free PMC article. See all similar articles Cited by Protein misfolding, amyotrophic lateral sclerosis and guanabenz: protocol for a phase II RCT with futility design (ProMISe trial).Bella ED, Tramacere I, Antonini G, Borghero G, Capasso M, Caponnetto C, Chiò A, Corbo M, Eleopra R, Filosto M, Giannini F, Granieri E, Bella V, Lunetta C, Mandrioli J, Mazzini L, Messina S, Monsurrò MR, Mora G, Riva N, Rizzi R, Siciliano G, Silani V, Simone I, Sorarù G, Volanti P, Lauria G.Bella ED, et al.BMJ Open. 2017 Aug 11;7(8):e015434. doi: 10.1136/bmjopen-2016-015434.BMJ Open. 2017.PMID: 28801400 Free PMC article.Clinical Trial. The antibiotic robenidine exhibits guanabenz-like cytoprotective properties by a mechanism independent of protein phosphatase PP1:PPP1R15A.Claes Z, Jonkhout M, Crespillo-Casado A, Bollen M.Claes Z, et al.J Biol Chem. 2019 Sep 6;294(36):13478-13486. doi: 10.1074/jbc.RA119.008857. Epub 2019 Jul 23.J Biol Chem. 2019.PMID: 31337709 Free PMC article. Ribosomal RNA Modulates Aggregation of the Podospora Prion Protein HET-s.Pang Y, Kovachev P, Sanyal S.Pang Y, et al.Int J Mol Sci. 2020 Sep 1;21(17):6340. doi: 10.3390/ijms21176340.Int J Mol Sci. 2020.PMID: 32882892 Free PMC article. Therapeutic strategies for identifying small molecules against prion diseases.Uliassi E, Nikolic L, Bolognesi ML, Legname G.Uliassi E, et al.Cell Tissue Res. 2023 Apr;392(1):337-347. doi: 10.1007/s00441-021-03573-x. Epub 2022 Jan 6.Cell Tissue Res. 2023.PMID: 34989851 Review. Prion therapeutics: Lessons from the past.Shim KH, Sharma N, An SSA.Shim KH, et al.Prion. 2022 Dec;16(1):265-294. doi: 10.1080/19336896.2022.2153551.Prion. 2022.PMID: 36515657 Free PMC article.Review. 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18108	https://teachchemistry.org/classroom-resources/topics/nuclear-chemistry?q%5Bgrade_level_ratings_grade_level_id_eq%5D=3&q%5Bresource_topics_topic_id_in%5D%5B%5D=88	Classroom Resources \| Nuclear Chemistry \| AACT 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. 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Classroom Resources: Nuclear Chemistry Filter by: Subtopics Half Lives Types Grade Level Advanced Chemistry Sort by: Filter Clear All Filters 1 – 19 of 19 Classroom Resources Radioactive Isotopes, Half Lives, Isotopes \| High School Activity: Isotope Sisters PuzzleMark as Favorite (5 Favorites) In this activity, students are given clues about various sister isotopes in order to complete a crossword puzzle. Students will become familiar with isotope names, symbols, and mass numbers as they consider descriptions of the isotopes and information about their uses to solve the puzzle. Radioactive Isotopes, Half Lives \| High School Activity: What Can We Learn From Glaciers?Mark as Favorite (8 Favorites) In this activity, students will learn how scientists study Earth’s atmosphere and climate history through the analysis of ice cores obtained from glaciers. They will then analyze radioisotope data from ice core samples to determine the age of the samples based on the decay of radioactive Kr-81 isotopes. Students will also research other isotopes that can be used to determine ages of different types of samples. Half Lives, Radioactive Isotopes, Radiation \| High School Activity: Simulation Activity: Half-Life InvestigationMark as Favorite (33 Favorites) In this simulation, students will have the opportunity to investigate the decay of two samples of unstable atoms. Students will interact with the simulation in order to decay the unstable samples resulting in a visual and graphical interpretation of half-life. Alpha/Beta/Gamma Decay, Half Lives, Radiation, Radioactive Isotopes, Isotopes, Atomic Mass, Subatomic Particles \| High School Activity: Radiological Applications of IsotopesMark as Favorite (56 Favorites) In this lesson, students will apply their knowledge of nuclear notation using trading cards to investigate and discuss the applications of isotopes in the medical field. The conclusion of the activity includes a summative assessment where students must advertise the radiological services using their knowledge of isotopes and their medical applications Half Lives \| High School Activity: Nuclear Medicine Half-LivesMark as Favorite (28 Favorites) In this activity, students will model two half-life scenarios related to nuclear medicine. Through this activity they will learn how to describe half-lives through explanations, calculations, particulate diagrams, and graphs as well as analyze the benefits of long and short half-lives through the context of nuclear medicine. Half Lives, Alpha/Beta/Gamma Decay, Radioactive Isotopes \| High School Activity: Graphical Analysis of Nuclear DecayMark as Favorite (37 Favorites) In this activity, students analyze a series of graphs and data points to discover a pattern, and realize the meaning of a half-life. During this investigation, students will make connections between the concepts of nuclear decay, radiation and the Law of Conservation of Mass. Radioactive Isotopes, Half Lives, Alpha/Beta/Gamma Decay \| High School Activity: Nuclide Stability InvestigationMark as Favorite (10 Favorites) In this activity, students will examine the relationship between the stability of an isotope, its half-life, and the make-up of its nucleus. Radioactive Isotopes, Half Lives \| High School Activity: Using Dice to Explore Radioactive DecayMark as Favorite (34 Favorites) In this activity, students will use dice to simulate the radioactive “decay” of samples of two different elements with two different half-lives. At the end of the simulation, all the groups will pool their data (by round) and then the class results will be graphed. The graphs will be analyzed to illustrate the process of radioactive decay and to determine the half-life of each element in the fictitious time units of “rounds”. Radioactive Isotopes, Half Lives \| High School Activity: Radioactive Decay and Seafloor DataMark as Favorite (11 Favorites) In this activity, students will apply their understanding of radioactive decay to analyze and interpret the meaning of Atlantic seafloor isotope data. Students will then use their results to suggest past changes that have occurred with the seafloor Half Lives, Radioactive Isotopes, Radiation, Phase Changes \| High School Activity: Radioactive Dating: The Demise of FrostyMark as Favorite (36 Favorites) In this activity students will investigate the idea that carbon dating is based on gathering evidence in the present and extrapolating it to the past. Students will use a simple graph to extrapolate data to its starting point and then pool the data to make a graph that simulates half-life. Students will be introduced to solving mathematical problems that involve half-life. Radioactive Isotopes, Half Lives \| High School Activity: Radioactive Decay and Peat BogsMark as Favorite (2 Favorites) In this activity, students will apply their understanding of radioactive decay to establish that radiometric dating (specifically C-14) can be used to reliably determine the age of Earth’s materials. Half Lives, Radioactive Isotopes \| High School Activity: Investigating Exponential DecayMark as Favorite (4 Favorites) In this activity, students will learn about radioactivity, exponential decay, and half-life through two hands-on experiences. Radioactive Isotopes, Radiation, Half Lives, Atomic Structure, Subatomic Particles, Model of the Atom, History \| Middle School, High School Activity: Marie Curie Video QuestionsMark as Favorite (25 Favorites) In this activity, students will watch a short video and learn about Marie Curie, her Nobel Prizes, radiation experiments, and discovery of new elements. Half Lives, Radioactive Isotopes, Radiation \| High School Simulation: Half-Life InvestigationMark as Favorite (15 Favorites) In the March 2017 simulation, students will have the opportunity to investigate the decay of two samples of unstable atoms. Students will interact with the simulation in order to decay the unstable samples resulting in a visual and graphical interpretation of half-life. Radioactive Isotopes, Half Lives, History \| High School Lesson Plan: Radiocarbon Dating and Willard LibbyMark as Favorite (8 Favorites) In this lesson, students will learn about the development and application of radiocarbon dating through an article reading. There are a series of activities to help promote literacy in the science classroom related to the reading. This lesson could be easily used as plans for a substitute teacher, as most of the activities are self-guided. Radioactive Isotopes, Half Lives, Subatomic Particles, Periodic Table \| High School Activity: Why are Some Isotopes Radioactive?Mark as Favorite (66 Favorites) In this activity, students use periodic trends and data to make predictions about what makes an isotope radioactive. They will then verify or refine their predictions using a PhET simulation. Radioactive Isotopes, Radiation, Half Lives, History \| Elementary School, Middle School, High School Video: Marie Curie VideoMark as Favorite (28 Favorites) This video tells the story about Marie Curie, including her Nobel Prizes, radiation experiments, and discovery of new elements. Irene Curie is also mentioned. Half Lives \| High School, Middle School Lab: Half-LifeMark as Favorite (36 Favorites) In this lab, students visualize the random nature of atomic decay (or first order chemical reactions) and also helps them realize the important difference between macroscale and microscale phenomena. Half Lives, Graphing \| High School Lab: Twizzler Half-LifeMark as Favorite (47 Favorites) In this lab, students will better understand the concept of half-lives. 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18109	https://people.math.harvard.edu/~knill/teaching/math22a2018/handouts/lecture34.pdf	LINEAR ALGEBRA AND VECTOR ANALYSIS MATH 22A Unit 34: Stokes Applications Topology 34.1. A region E in Rn is called simply connected if it is connected and for every closed loop C in E there is a continuous deformation Cs of C within G such that C0 = C and C1(t) = P is a point. For example, C(t) = [cos(t), sin(t), 0] can be deformed in E = R3 to a point with Cs(t) = [(1 −s) cos(t), (1 −s) sin(t), 0] as C1(t) = P = [0, 0, 0] for all t. Each Euclidean space Rn is simply connected. The region G = {x2+y2 > 0} ⊂R3 is not simply connected as the circle C : r(t) = [cos(t), sin(t), 0] winding around the z-axis can not be pulled together to a point within G. The region G = {x2 + y2 + z2 > 0} ⊂R3 is simply connected, but G = {x2 + y2 > 0} in R2 is not. Remember that F was called irrotational if curl(F) = 0 everywhere. Theorem: If F is irrotational on a simply connected E then F = ∇f in E. 34.2. Proof: since E is simply connected and curl(F) = 0, every closed loop C can be ﬁlled in by a surface S = S 0≤s≤1 Cs which has the boundary C. Stokes theorem gives R S F · dr = RR S curl(F) · dS = 0. The closed loop property implies path independence. A potential f can be obtained by ﬁxing a base point p in E, then deﬁne for any other point x a path Cpx going from p to x. The potential function f is then deﬁned as f(x) = R Cpx F · dr. QED 34.3. The ﬁeld F(x, y, z) = [−y/(x2 + y2), x/(x2 + y2), 0] is deﬁned everywhere except on the z-axis. The domain E, where F is deﬁned is not simply connected. There is no global function f which is a potential for F. 34.4. The notion of “simply connectedness” is important in topology. The ﬁrst solved Millenium problem, the Poincar´ e conjecture, is now a theorem. It tells that a 3-dimensional manifold which is simply connected is topologically equivalent to the 3-sphere {x2 + y2 + z2 + w2 = 1} ⊂R4. In two dimensions, the result was known for a long time already, because the structure of 2-dimensional connected manifolds is known. Electromagnetism 34.5. The Maxwell-Faraday equation in electromagnetism relates the electric ﬁeld E and the magnetic ﬁeld B with the partial diﬀerential equation curl(E) = −d dtB. Given a surface S, the ﬂux integral RR S B · dS is called the magnetic ﬂux Linear Algebra and Vector Analysis of B through the surface. If we integrate the Maxwell-Faraday equation, we see that RR S curl(E)·dS is equal to minus the rate of change of the magnetic ﬂux −d dt RR S B ·dS. Stokes theorem now assures that RR S curl(E) · dS = R C E · dr is the line integral of the electric ﬁeld along the boundary. But this is electric potential or voltage. We see: We can generate an electric potential by changing the magnetic ﬂux. 34.6. Changing the magnetic ﬂux can happen in various ways. We can generate a changing magnetic ﬁeld by using alternating current. This is how transformers work. An other way to change the ﬂux is to rotate a wire in a ﬁxed magnetic ﬁeld. This is the principle of the dynamo: Figure 1. The dynamo, implemented using the ray tracer Povray. Electric current is generated by moving a wire in a ﬁxed magnetic ﬁeld. 34.7. The vector ﬁeld A(x, y, z) = [−y,x,0] (x2+y2+z2)3/2 is called the vector potential of a magnetic ﬁeld B = curl(A). The picture shows some ﬂow lines of this magnetic dipole ﬁeld B. Problem: Find the ﬂux of B through the lower half sphere x2 + y2 + z2 = 1, z ≤0 oriented downwards. Solution: Since we have an integral of the curl of the vector ﬁeld A, we use Stokes theorem and integrate A(r(t)) along the boundary curve r(t) = [cos(t), −sin(t), 0]. First of all, we have A(r(t)) = [sin(t), cos(t), 0]. The velocity is r ′(t) = [−sin(t), cos(t), 0]. The integral is R 2π 0 −1 dt = −2π. Figure 2. The ﬂux of the magnetic ﬁeld B through a surface can be computed with Stokes by computing a line integral of the vector potential A. 34.8. Here are all the four magical Maxwell equations for the electric ﬁeld E and magnetic ﬁeld B related to the charge density σ and the electric current j. The constant c is the speed of light. (By using suitable coordinates, one can assume c = 1.) div(E) = 4πσ, div(B) = 0, c · curl(E) = −Bt, c · curl(B) = Et + 4πj . Fluid dynamics 34.9. If F is the ﬂuid velocity ﬁeld and C is a closed curve, then R C F · dr is called the circulation of F along C. The curl of F is called the vorticity of F. A vortex line is a ﬂow line of curl(F). Given a curve C, we can let any point in C ﬂow along the vorticity ﬁeld. This produces a vortex tube S. The ﬂux of the vorticity though a surface S is the vortex strength of F through S. Stokes theorem implies the Helmholtz theorem. Theorem: If Cs ﬂows along F, then R Cs F · dr stays constant. 34.10. Proof: Let C be a closed curve and Cs(t) be the curve after letting it ﬂow using a deformation parameter s. The deformation produces a tube surface S = St s=0 Cs which has the boundary C and Ct. Since the curl of F is always tangent to the surface S, the ﬂux of the curl of F through S is zero. Stokes theorem implies that R C F ·dr − R Cs F ·dr = 0. The negative sign is because the orientation of Cs is diﬀerent from the orientation of C if the surface has to be to the left. Figure 3. Helmholtz theorem assures that the circulation along a ﬂux tube is constant. This is a direct application of Stokes theorem: because the curl of F is tangent to the tube, there is no ﬂux through the tube. Complex analysis 34.11. An application of Green’s theorem is obtained, when integrating in the complex plane C. Given a function f(z) = u(z) + iv(z) from C →C and a closed path C parametrized by r(t) = x(t) + iy(t) in C, deﬁne the complex integral R b a (u(x(t) + iy(t)) + iv(x(t) + iy(t)))(x′(t) + iy′(t)) dt. This is R b a u(r(t))x′(t) −v(r(t))y′(t) dt + i R b a v(r(t))x′(t) + u(r(t))′(t) dt. These are two line integrals. The real part is F = [u, −v], the imaginary part is F = [v, u]. Assume C bounds a region G, then Green’s theorem tells that the ﬁrst integral is RR G −vx −uydxdy and the second integral is RR G ux −vy dxdy. It turns out now that for nice functions f like polynomials, the Cauchy-Riemann diﬀerential equations ux = vy, vx = −ux hold so that these line integrals are zero. We have therefore Theorem: If f is a polynomial and C a closed loop, R C f(z) dz = 0 Linear Algebra and Vector Analysis Homework: thanksgiving quickies Problem 34.1: We can measure how many magnetic monopoles there are in the interior of a closed surface S by computing RR S B · dS. We see that B = curl(A) for a magnetic potential A, which is a vector ﬁeld. What is RR S B · dS? (We will see in the next lecture why this tells about the amount of magnetic monopoles inside S.) Problem 34.2: a) Deﬁne div([P, Q, R]) = Px + Qy + Rz. Check that div(curl(F)) = 0. b) Is div(grad(f)) = 0 for all functions? c) Is curl(curl(F)) = [0, 0, 0] for all ﬁelds? d) Which of the regions in Figure 4 are simply connected? e) Which of the capital letters A −Z are not simply connected? Figure 4. Complement B \T of the solid torus T in a ball B, the solid {1 < x2 + y2 + z2 < 4} or the complement of two small balls in a larger ball. Problem 34.3: Let S be the torus r(u, v) = [(3 + cos(u)) cos(v), (3 + cos(u)) sin(v), sin(u)] and F the vector ﬁeld F(x, y, z) = [−y, x, 0]. What is the ﬂux of F through S? (No computation and no Stokes theorem is needed). Problem 34.4: If F is a vector ﬁeld, which is everywhere perpendicular to a surface S pointing in the normal direction of S, and \|F(x, y, z)\| = 1. What is RR S F · dS? Problem 34.5: a) Can you ﬁnd a vector ﬁeld F with curl(F) = [0, x2, 0]? b) Can you ﬁnd a vector ﬁeld F with curl(F) = [0, 0, x2]? c) Can you ﬁnd a vector ﬁeld F = [P, Q, R] such that div(F) = x2? d) Can you ﬁnd a gradient ﬁeld F = ∇(f) such that div(F) = x2? e) Given a function g(x, y, z), ﬁnd F such that div(F) = g. Oliver Knill, knill@math.harvard.edu, Math 22a, Harvard College, Fall 2018
18110	https://mathoverflow.net/questions/261115/number-of-perfect-matchings-in-bipartite-graph-with-given-minimum-degree	Skip to main content Number of perfect matchings in bipartite graph with given minimum degree Ask Question Asked Modified 8 years, 6 months ago Viewed 3k times This question shows research effort; it is useful and clear 3 Save this question. Show activity on this post. Let G be a spanning subgraph of Kn,n with minimum degree δ(G)≥n/2. It's easy to show using Hall's theorem that G has a perfect matching, and the example of two disjoint copies of K⌊n/2⌋+1,⌈n/2⌉−1 side by side shows that n/2 is sharp. This extremal example "almost" has lots of perfect matchings—it has lots of matchings covering almost all of the vertices. If δ(G)≥n/2, how many perfect matchings must G contain? For k-regular bipartite graphs the answer to the corresponding question is due to Schrijver. A closely related question did not focus on the range of degrees where perfect matchings are guaranteed· co.combinatorics graph-theory matching-theory perfect-matchings Share CC BY-SA 3.0 Improve this question Follow this question to receive notifications edited Apr 13, 2017 at 12:58 CommunityBot 122 silver badges33 bronze badges asked Feb 1, 2017 at 16:49 Ben BarberBen Barber 4,66922 gold badges2727 silver badges3838 bronze badges Add a comment \| 1 Answer 1 Reset to default This answer is useful 7 Save this answer. Show activity on this post. It's a theorem of Marshall Hall that in a bipartite graph of minimum degree r, where there exists a perfect matching, there will be at least r! perfect matchings. In your case the conditional disappears and you get a lower bound of ⌈n2⌉! perfect matchings. This lower bound is tight, and it's achieved by the graph on vertices u1,v1,…,un,vn, where there is an edge betweenn ui and vj for all i,j≤⌈n2⌉, and an edge for all i≤j. A perfect matching in this graph must contain edges uivi for all i>⌈n2⌉, therefore it contains exactly ⌈n2⌉! perfect matchings. Share CC BY-SA 3.0 Improve this answer Follow this answer to receive notifications edited Feb 2, 2017 at 7:12 bof 14.8k22 gold badges4848 silver badges7171 bronze badges answered Feb 1, 2017 at 18:10 Gjergji ZaimiGjergji Zaimi 86.3k44 gold badges243243 silver badges407407 bronze badges 3 Sorry if I misread, but doesn't the vertex un have degree 1? – monkeymaths Commented Feb 2, 2017 at 9:36 That's beautifully simple. The problem with choosing greedily in the original setting is that after your first choice both n and the minimum degree might have dropped by 1. But if you first replace the minimum degree condition by (Philip) Hall's condition, then restrict to a minimal set of vertices where Hall's condition is tight (which must have size at least r), it now plays nicely with induction. – Ben Barber Commented Feb 2, 2017 at 10:23 @monkeymaths Yes, which is related to (Marshall) Hall's result only needing the existence of a perfect matching and a one-sided minimum degree. It's not a sharp answer to the question as I happened to phrase it, but it still says that there are lots of perfect matchings. – Ben Barber Commented Feb 2, 2017 at 10:26 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 co.combinatorics graph-theory matching-theory perfect-matchings See similar questions with these tags. Linked 2 Counting matchings in middle levels of the Boolean lattice 1 On Schrijver's lower bound for the number of perfect matchings Related 8 Condition on a bipartite graph to have an m-factor 7 Perfect matching in a vertex-transitive hypergraph 2 Bipartite dimension of an almost crown graph 7 Graph to Bipartite conversion preserving number of perfect matchings 2 Number of distinct perfect matchings/near perfect matchings in an induced subgraph 7 Disjoint perfect matchings in complete bipartite graph Question feed By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
18111	https://www.ijntr.org/download_data/IJNTR08090019.pdf	International Journal of New Technology and Research (IJNTR) ISSN: 2454-4116, Volume-8, Issue-8, August 2022 Pages 06-08 6 www.ijntr.org  Abstract— In mathematics, mathematicians and statistics have always been fascinated by the nature of numbers. Since the birth of mathematics, a lot has been there that can be done by the use of numbers. There is still more to do with numbers. This involves finding patterns that can be used to relate numbers. This makes the numbers more interesting to play with and discover more fascinating formulas and ideas. In this article, a lot will be discussed on number patterns, the relationship between several number progressions, and the formulas associated with them. More specifically, the sum of harmonic progression and its applications. To be specific, harmonic progression is a number sequence generated by obtaining the reciprocals of the real numbers in an arithmetic progression. Index Terms— Harmonic Progression, statistics. I. INTRODUCTION A number pattern is a trend followed by a list of numbers in a given sequence. Patterns are what define the relationship between two numbers. Solving a sequence requires an understanding of the rule applied in getting the next term or the sum of given terms. The understanding is required in solving all the types of number series in the field of mathematics. According to (Andrews, 1990), a number pattern is a generalization of the rule followed by a series of numbers. Before we understand harmonic progression, a scholar will always want to know what progression is. A progression is a pattern of numbers. For instance, gives a progression since there is an observed pattern that the numbers follow. In this progression, the next number is gotten by adding 2 to the previous number. However, a pattern depends on the type of progression. As indicated, a progression is a real number list exhibiting a particular trend. Every term in a progression is given by a generalized rule called the nth term denoted by . For our example of a progression, the nth term can be given by the generalized formula . When , then the first term is , when , then the second term becomes 4, and so on, this is a perfect generalized rue for obtaining that next or nth term in this progression. With or focus on harmonic progression, we have several other types of progressions that includes; arithmetic progression, and geometric progression (Panagiotou, 2011). Because the harmonic sequence is the reciprocal of the arithmetic progression, we need to understand the arithmetic progression to define the harmonic progression perfectly. A given sequence is said to be a harmonic progression if and Siddhant Gvalani, KC College only if its terms are the multiplicative inverse of an arithmetic progression that does not include a zero. Defining Harmonic progression and the sum of the terms harmonic progression The Harmonic Sequence is the term given to the reciprocal form of the Arithmetic Sequence, which uses numbers that are never going to equal 0. Furthermore, the total of such a sequence is referred to as the Harmonic Series. II. GREY AREAS IN THE STUDY OF HARMONIC PROGRESSION. The relationship between the arithmetic progression and the harmonic progression. Interestingly, we can form a harmonic progression from arithmetic progression by taking the reciprocals of the terms in the arithmetic regression. While we have a common ratio between two adjacent numbers in a geometric progression, in a harmonic progression, we have a common difference between two adjacent terns. A common raion in a geometric progression is denoted by r while a common difference in arithmetic series is denoted by d. in every sequence, the first term is denoted by a. this implies that we can get the next terms can be calculated given the common difference and or common ratio and one of the terms in the series. This can be done by a series of computations. The nthterm in the arithmetic series An arithmetic progression takes the general form; The nth term is given by the formula , where a is the first term, d is the common difference and n is the nth term requested. This is an indication that the later progression, is an arithmetic regression with 2 as both the common difference and the first term. A general rule suggests that; with two given terms of a series, if G.P, A.P, and H.P are specified as geometry, arithmetic, and harmonic progression, then, . Similarly The nth term in a harmonic series The harmonic series is the reciprocal of the arithmetic series. People will always confuse harmonic series and arithmetic series. Harmonic series take the form; By inspecting the trending and relating it to the arithmetic progression, one will realize that the nth term is given by the formula . Likewise, a and d are the first and nth requested respectively.According to (Yadav, 2008), The nth term of the Harmonic series = 1/ (nth term of the equivalent Arithmetic series). In this case, the first term of a harmonic progression obtained from the arithmetic progression, will be, The first term is, Deriving the Sum of a Harmonic Progression Siddhant Gvalani Deriving the Sum of a Harmonic Progression 7 www.ijntr.org . However, taking the reciprocal does away with the common difference. This is one of the grey areas in the study of harmonic series. It is worth noting that a harmonic progression does not have a common difference. Example1: Given the harmonic series, , find the fourth term Solution We use the formula; . Remember that d is the common difference for the corresponding arithmetic series In any computation, the first term a and the common difference d, one must always use those in the corresponding arithmetic progression. In the above series, the graph of the harmonic series will be as in the table below. A sample graph for the sum of the nth term in harmonic series The fourth term is and the first term is The sum of n terms harmonic series. In this case, we consider a general form of a harmonic progression, . The sum will be given by the formula; . Here, a is the first term of arithmetic progression, d is the common difference of arithmetic progression, and “ln” is the natural log. This formula is very different from that of the arithmetic series. The common difference d in an ammonic series remains the common difference of the corresponding arithmetic progression. For instance, the common difference in the harmonic series is 2 which is the common difference for the corresponding arithmetic progression, . Example. Compute the sum of the first four terms in the harmonic series Solution We compute this using the formula; A sampe graph for the sum of n terms harmonic series Using a graph, we get the same answer. Understanding the difference between harmonic progression and arithmetic progression Arithmetic and harmonic progressions are two different but almost similar and confusing. We have to find a way of defining the two series. Understanding arithmetic progression is very important in addressing and grasping key concepts on harmonic progression. Below is a table showing the key and noticeable differences. Table 1: Key differences between arithmetic and harmonic progressions. International Journal of New Technology and Research (IJNTR) ISSN: 2454-4116, Volume-8, Issue-8, August 2022 Pages 06-08 8 www.ijntr.org Criteria Arithmetic progression Harmonic progression Calculation of we nth term We add fixed common differences d to the previous tern of the series. ; We add fixed common differences d to the denominator of the previous tern of the series; General form It has the general form Has the general form nth tern formula The nth term is; The nth term is ; The sum of nth tern formula The formula for the sum of n terms is; The formula for the sum of n terms is; Example 2, 4, 6, 8… H.P is the reciprocal of the arithmetic series. Example; Zero rule Might contain zero Does not contain zero III. IMPACT OF THE HARMONIC PROGRESSION ON LIFE SITUATIONS. Every mathematical concept learned is mos probably applicable in real-life situations. These are some of the things that many people do not know. Like in the case of arithmetic mathematics. Harmonic progression has been applied in the music industry in establishing theories on sound and the study of sound. To study the growth of the animals in a park, A simple harmonic progression is applied. For an instance, given that thee were 6 million deers in a park, increasing at a rate of 2000 per month. This information can be used to predict the number of deers in the park in the next 10 years. It may come as a surprise to learn that the investigation of the harmonic sequence dates somewhere back to the sixth century when the Greek mathematician Pythagoras investigated the characteristics of the universe. The study of music was his first application for it. A harmonic series is an example of an infinite series, which does not have any limits and is characterized by the fact that the sum of progressively aspects tends to infinity. REFERENCES (2022). Retrieved 25 July 2022, from Andrews, P. (1990). Generalising number patterns. Mathematics in School, 19(4), 9-13. Harmonic Progressions: Concept & Tricks. Hitbullseye.com. (2022). Retrieved 25 July 2022, from Harmonic Sequence \| Harmonic Series \| Harmonic Sequence Formula, Graph and Poperties. Cuemath. (2022). Retrieved 25 July 2022, from quence/#Sum-of-Harmonic-Sequence. harmonic sequence \| mathematics. Encyclopedia Britannica. (2022). Retrieved 25 July 2022, from . Môn Hughes, G. (2022). Harmonic progression. Music Teacher, 101(4), 38-39. Panagiotou, E. N. (2011). Using history to teach mathematics: The case of logarithms. Science & Education, 20(1), 1-35. Progression - Definition, Meaning \| Formulas of AP, GP, HP. Cuemath. (2022). Retrieved 25 July 2022, from Roza, N., Arnawa, I., &Yerizon, Y. (2018). Practicality of mathematics learning tools based on discovery learning for topic sequence and series. International Journal of Scientific dan technology Research, 7(5), 236-241. Yadav, D. K. GENERAL STUDY ON TWO-DIMENSIONAL GENERALIZED ARITHMETIC PROGRESSION. Zelator, K. (2009). Exploring Progressions: A Collection of Problems. arXiv preprint arXiv:0904.3855.
18112	https://www.youtube.com/playlist?list=PLlKrT5nQa5mw4Q2ie7YoBcwoC93koHx4X	Voter Education - YouTube Back Skip navigation Search Search with your voice Sign in Home HomeShorts ShortsSubscriptions SubscriptionsYou YouHistory History Play all Voter Education by Electoral Commission of South Africa (IEC) • Playlist•30 videos•466 views Play all PLAY ALL Voter Education 30 videos 466 views Last updated on Feb 21, 2025 Save playlist Shuffle play Share Show more Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) Subscribe Play all Voter Education by Electoral Commission of South Africa (IEC) Playlist•30 videos•466 views Play all 1 4:49:15 4:49:15 Now playing Youth Civic Education and Activation at Msunduzi Athletics Stadium, uMgungundlovu Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) • 153 views • Streamed 7 months ago • 2 0:39 0:39 Now playing Youth Civic Education and Activation at Msunduzi Athletics Stadium, uMgungundlovu Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) • 5 views • Streamed 7 months ago • 3 1:01 1:01 Now playing IEC Topic 6 I Seat Allocation Provincial Legislature Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) • 44 views • 9 months ago • 4 1:02 1:02 Now playing IEC Topic 5 I Seat Allocation National Proportional Ballot Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) • 25 views • 9 months ago • 5 1:00 1:00 Now playing IEC Topic 4 I Regional Votes Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) • 29 views • 9 months ago • 6 1:01 1:01 Now playing IEC Topic 3 I Three Ballots Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) • 45 views • 9 months ago • 7 1:01 1:01 Now playing IEC Topic 2 I Candidates Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) • 22 views • 9 months ago • 8 1:01 1:01 Now playing IEC Topic 1 I What has changed elections 2024 Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) • 55 views • 9 months ago • 9 8:10 8:10 Now playing Election Matters \| Schools Democracy Programme: Moagisi Sibanda weighs in SABC News SABC News • 601 views • 1 year ago • 10 5:43 5:43 Now playing IEC starts School Democracy Programme encouraging young people to vote Newzroom Afrika Newzroom Afrika • 229 views • 1 year ago • 11 7:26 7:26 Now playing IEC launches Schools Democracy Programme in North West Newzroom Afrika Newzroom Afrika • 171 views • 1 year ago • 12 2:02 2:02 Now playing 3 Ballot Features - Balloting Education Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) • 643K views • 1 year ago • 13 0:20 0:20 Now playing Section 24A (S24A) Votes \| Voting Outside Your District Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) • 931 views • 1 year ago • 14 1:01 1:01 Now playing #KnowYourBallots Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) • 117K views • 1 year ago • 15 2:32 2:32 Now playing 2024 Elections \| Low youth voter turnout leaves IEC concerned SABC News SABC News • 1.1K views • 1 year ago • 16 2:07 2:07 Now playing 2024 Elections \| The importance of the youth vote eNCA eNCA • 919 views • 1 year ago • 17 2:05 2:05 Now playing 2024 Elections \| IEC steps up campaign to woo youth voters eNCA eNCA • 389 views • 1 year ago • 18 3:28 3:28 Now playing IEC tells youth: your vote matters Newzroom Afrika Newzroom Afrika • 386 views • 1 year ago • 19 0:57 0:57 Now playing Online Voter Registration Electoral Commission of South Africa (IEC) Electoral Commission of South Africa (IEC) • 1.8M views • 1 year ago • 20 1:01 1:01 Now playing What has changed for Elections 2024? 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18113	https://www.reddit.com/r/learnmath/comments/qhxu67/high_school_maths_how_to_compare_exponents_with/	[High School Maths] How to compare exponents with different bases : r/learnmath Skip to main content[High School Maths] How to compare exponents with different bases : r/learnmath Open menu Open navigationGo to Reddit Home r/learnmath A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to learnmath r/learnmath r/learnmath Post all of your math-learning resources here. Questions, no matter how basic, will be answered (to the best ability of the online subscribers). 397K Members Online •4 yr. ago MashedSwede [High School Maths] How to compare exponents with different bases I got this question on my study sheet: Compare these expressions without a calculator, and beyond all doubt, which one is bigger. 15•2¹¹•8³⁷ or 10•4⁴•3⁷⁷. My first attempt was to get the same base on each exponent. I took 8³⁷ and turned it into 2³°³⁷, which equals 2¹¹¹. Then I added it with 2¹¹ to get 2¹²². Then I moved to the other expression and turned 4⁴ into 2²°⁴, which equals 2⁸. Afterwards, I solved 2⁸ to 256 and multiplied it by 10. Now I am stuck on comparing 15•2¹²² and 2560•3⁷⁷. I also tried using logarithms, however I feel like that wouldn't work for a 'no calculator' question. The method I used was 122log(2)≈36,73 and 77log(3)≈36,74. Then I multiplied them by 15 and 2560 respectively. My question is, how do I solve this question without a calculator, and is there a method to solve these types of questions in the future? Read more Share New to Reddit? Create your account and connect with a world of communities. Continue with Google Continue with Google. Opens in new tab Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy. Public Anyone can view, post, and comment to this community Top Posts Reddit reReddit: Top posts of October 28, 2021 Reddit reReddit: Top posts of October 2021 Reddit reReddit: Top posts of 2021 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation
18114	https://www.scmp.com/news/china/politics/article/3270075/east-chinas-suzhou-latest-join-tighten-belts-call-beijing-leads-austerity-drive	East China’s Suzhou latest to join ‘tighten belts’ call as Beijing leads austerity drive Spending cuts, judicious use of government assets and focus on green energy among new rules rolled out by Suzhou government The cutbacks include not replacing official cars until they are 10 years old or have done 100,000km (over 62,000 miles), and no government or rental cars for business trips to destinations along the high-speed rail system. The set of rules, published on the city’s official website last week, also call for government assets to be used in flexible ways, including renting out or auctioning idle land and housing, and sharing the assets between agencies and regions. All conference rooms and public services facilities should be shared between offices by the year-end, and some car parks and outdoor bathrooms should even be open to the public, according to the directives. Suggested energy-saving measures include choosing electric vehicles where possible for new or replacement official cars, launching a photovoltaic project, and cutting waste – including in cafeterias.
18115	https://simple.wikipedia.org/wiki/Mathematical_induction	Mathematical induction - Simple English Wikipedia, the free encyclopedia Jump to content [x] Main menu Main menu move to sidebar hide Getting around Main page Simple start Simple talk New changes Show any page Help Contact us About Wikipedia 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. Give to Wikipedia Create account Log in [x] Personal tools Give to Wikipedia Create account Log in [x] Toggle the table of contents Contents move to sidebar hide Beginning 1 Examples of proof by inductionToggle Examples of proof by induction subsection 1.1 Sum of the first n natural numbers 1.2 The sum of the interior angles of a polygon 2 Inductive definition 3 Related pages 4 References Mathematical induction [x] 67 languages العربية Asturianu Azərbaycanca বাংলা 閩南語 / Bân-lâm-gí Башҡортса Беларуская Български Català Чӑвашла Čeština Cymraeg Dansk Deutsch Ελληνικά English Español Esperanto Euskara فارسی Français Galego 한국어 Հայերեն हिन्दी Hrvatski Bahasa Indonesia Interlingua Íslenska Italiano עברית ಕನ್ನಡ Қазақша Latina Latviešu Lietuvių Magyar Македонски മലയാളം Nederlands 日本語 Norsk bokmål Norsk nynorsk Oʻzbekcha / ўзбекча Polski Português Română Русский Shqip සිංහල Slovenčina Slovenščina کوردی Српски / srpski Srpskohrvatski / српскохрватски Suomi Svenska தமிழ் Татарча / tatarça ไทย Türkçe Українська Tiếng Việt 吴语粵語中文 Руски Change links Page Talk [x] English Read Change Change source View history [x] Tools Tools move to sidebar hide Actions Read Change Change source View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Sandbox Edit interlanguage links Print/export Make a book Download as PDF Page for printing In other projects Wikimedia Commons Wikidata item From Simple English Wikipedia, the free encyclopedia Mathematical induction is a special way of proving a mathematical truth. It can be used to prove that something is true for all the natural numbers (or all positive numbers from a point onwards). The idea is that if: Something is true for the first case (base case); Whenever that same thing is true for a case, it will be true for the next case (inductive case), then That same thing is true for every case by induction. In the careful language of mathematics, a proof by induction often proceeds as follows: State that the proof will be by induction over n{\displaystyle n}. (n{\displaystyle n} is the induction variable.) Show that the statement is true when n{\displaystyle n} is 1. Assume that the statement is true for any natural number n{\displaystyle n}. (This is called the induction step.) Show then that the statement is true for the next number, n+1{\displaystyle n+1}. Because it is true for 1, then it is true for 1+1 (=2, by the induction step), then it is true for 2+1 (=3), then it is true for 3+1 (=4), and so on. Examples of proof by induction [change \| change source] Sum of the first n natural numbers [change \| change source] Prove that for all natural numbersn: 1+2+3+....+(n−1)+n=1 2 n(n+1){\displaystyle 1+2+3+....+(n-1)+n={\tfrac {1}{2}}n(n+1)} Proof: First, the statement can be written as: 2∑k=1 n k=n(n+1){\displaystyle 2\sum {k=1}^{n}k=n(n+1)} (for all natural numbers _n) By induction on n, First, for n=1: 2∑k=1 1 k=2(1)=1(1+1){\displaystyle 2\sum _{k=1}^{1}k=2(1)=1(1+1)}, so this is true. Next, assume that for some n=n 0 the statement is true. That is,: 2∑k=1 n 0 k=n 0(n 0+1){\displaystyle 2\sum {k=1}^{n{0}}k=n_{0}(n_{0}+1)} Then for n=n 0+1: 2∑k=1 n 0+1 k{\displaystyle 2\sum {k=1}^{{n{0}}+1}k} can be rewritten as 2(∑k=1 n 0 k+(n 0+1)){\displaystyle 2\left(\sum {k=1}^{n{0}}k+(n_{0}+1)\right)} Since 2∑k=1 n 0 k=n 0(n 0+1),{\displaystyle 2\sum {k=1}^{n{0}}k=n_{0}(n_{0}+1),} 2∑k=1 n 0+1 k=n 0(n 0+1)+2(n 0+1)=(n 0+1)(n 0+2){\displaystyle 2\sum {k=1}^{n{0}+1}k=n_{0}(n_{0}+1)+2(n_{0}+1)=(n_{0}+1)(n_{0}+2)} Hence the proof is complete by induction. The sum of the interior angles of a polygon [change \| change source] Mathematical induction is often stated with the starting value 0 (rather than 1). In fact, it will work just as well with a variety of starting values. Here is an example when the starting value is 3: "The sum of the interior angles of a n{\displaystyle n}-sided polygon is (n−2)180{\displaystyle (n-2)180} degrees." The initial starting value is 3, and the interior angles of a triangle is (3−2)180{\displaystyle (3-2)180} degrees. Assume that the interior angles of a n{\displaystyle n}-sided polygon is (n−2)180{\displaystyle (n-2)180} degrees. Add on a triangle which makes the figure a n+1{\displaystyle n+1}-sided polygon, and that increases the count of the angles by 180 degrees (n−2)180+180=(n+1−2)180{\displaystyle (n-2)180+180=(n+1-2)180} degrees. Since both the base case and the inductive case are handled, the proof is now complete. There are a great many mathematical objects for which proofs by mathematical induction works. The technical term is a well-ordered set. Inductive definition [change \| change source] The same idea can work to define a set of objects, as well as to prove statements about that set of objects. For example, we can define n{\displaystyle n}th degree cousin as follows: A 1{\displaystyle 1}st degree cousin is the child of a parent's sibling. A n+1{\displaystyle n+1}st degree cousin is the child of a parent's n{\displaystyle n}th degree cousin. There is a set of axioms for the arithmetic of the natural numbers which is based on mathematical induction. This is called "Peano's Axioms". The undefined symbols are \| and =.The axioms are \| is a natural number. If n{\displaystyle n} is a natural number, then n\|{\displaystyle n\|} is a natural number. If n\|=m\|{\displaystyle n\|=m\|} then n=m{\displaystyle n=m}. One can then define the operations of addition and multiplication and so on by mathematical induction. For example: m+\|=m\|{\displaystyle m+\|=m\|} m+n\|=(m+n)\|{\displaystyle m+n\|=(m+n)\|} Related pages [change \| change source] Mathematical proof Proof by contradiction References [change \| change source] ↑"The Definitive Glossary of Higher Mathematical Jargon". Math Vault. 2019-08-01. Retrieved 2020-09-23. ↑"3.4: Mathematical Induction - An Introduction". Mathematics LibreTexts. 2018-04-25. Retrieved 2020-09-23. ↑"Induction". discrete.openmathbooks.org. Retrieved 2020-09-23. Retrieved from " Category: Mathematics This page was last changed on 14 January 2024, at 21:36. Text is available under the Creative Commons Attribution-ShareAlike License and the GFDL; additional terms may apply. See Terms of Use for details. Privacy policy About Wikipedia Disclaimers Code of Conduct Developers Statistics Cookie statement Mobile view Edit preview settings Search Search [x] Toggle the table of contents Mathematical induction 67 languagesAdd topic
18116	https://www.youtube.com/watch?v=a3V5QZwXBlE	Art of Problem Solving: Triangle Inequality Introduction Art of Problem Solving 103000 subscribers 61 likes Description 16609 views Posted: 23 Dec 2011 Art of Problem Solving's Richard Rusczyk introduces the Triangle Inequality. This video is part of our AoPS Prealgebra curriculum. Take your math skills to the next level with our Prealgebra materials: 📚 AoPS Prealgebra Textbook: 🖥️ AoPS Prealgebra 1 Course (Textbook Chapters 1-7): 🖥️ AoPS Prealgebra 2 Course (Textbook Chapters 8-15): 🔔 Subscribe to our channel for more engaging math videos and updates Transcript:
18117	https://www.slideshare.net/slideshow/water-purification-processes-in-natural-systems-lecture-2/65248513	Change Language Water purification processes in natural systems lecture 2 Streams have the ability to purify themselves through natural processes like dilution, dispersion, sedimentation, oxidation, and reactions driven by temperature, sunlight, and microorganisms. When wastewater is discharged into a stream, there are zones of degradation, active decomposition, and recovery before the stream reaches a clear water zone. Dissolved oxygen levels in a stream typically follow a deoxygenation curve as organic matter is broken down, followed by a reoxygenation curve as oxygen is replenished, resulting in an overall DO sag curve. Recommended More Related Content What's hot Viewers also liked Similar to Water purification processes in natural systems lecture 2 More from Munira Shahbuddin Recently uploaded Water purification processes in natural systems lecture 2
18118	https://pmc.ncbi.nlm.nih.gov/articles/PMC2050825/	Function, structure and therapeutic potential of complement C5a receptors - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide New Try this search in PMC Beta Search PERMALINK Copy As a library, NLM provides access to scientific literature. 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Learn more: PMC Disclaimer \| PMC Copyright Notice Br J Pharmacol . 2007 Jul 2;152(4):429–448. doi: 10.1038/sj.bjp.0707332 Search in PMC Search in PubMed View in NLM Catalog Add to search Function, structure and therapeutic potential of complement C5a receptors P N Monk P N Monk 1 Academic Neurology Unit, School of Medicine and Biomedical Science, University of Sheffield Sheffield, UK Find articles by P N Monk 1,, A-M Scola A-M Scola 1 Academic Neurology Unit, School of Medicine and Biomedical Science, University of Sheffield Sheffield, UK Find articles by A-M Scola 1, P Madala P Madala 2 Centre for Drug Design and Development, Institute for Molecular Bioscience, University of Queensland Brisbane, Queensland, Australia Find articles by P Madala 2, D P Fairlie D P Fairlie 2 Centre for Drug Design and Development, Institute for Molecular Bioscience, University of Queensland Brisbane, Queensland, Australia Find articles by D P Fairlie 2 Author information Article notes Copyright and License information 1 Academic Neurology Unit, School of Medicine and Biomedical Science, University of Sheffield Sheffield, UK 2 Centre for Drug Design and Development, Institute for Molecular Bioscience, University of Queensland Brisbane, Queensland, Australia Author for correspondence: p.monk@shef.ac.uk Received 2006 Dec 15; Revised 2007 Apr 27; Accepted 2007 Apr 27; Issue date 2007 Oct; Collection date 2007 Oct 15. Copyright 2007, Nature Publishing Group PMC Copyright notice PMCID: PMC2050825 PMID: 17603557 Abstract Complement fragment (C)5a is a 74 residue pro-inflammatory polypeptide produced during activation of the complement cascade of serum proteins in response to foreign surfaces such as microorganisms and tissue damaged by physical or chemical injury. C5a binds to at least two seven-transmembrane domain receptors, C5aR (C5R1, CD88) and C5L2 (gpr77), expressed ubiquitously on a wide variety of cells but particularly on the surface of immune cells like macrophages, neutrophils and T cells. C5aR is a classical G protein-coupled receptor that signals through G α i and G α 16, whereas C5L2 does not appear to couple to G proteins and has no known signalling activity. Although C5a was first described as an anaphylatoxin and later as a leukocyte chemoattractant, the widespread expression of C5aR suggested more general functionality. Our understanding of the physiology of C5a has improved significantly in recent years through exploitation of receptor knockout and knockin mice, C5 and C5a antibodies, soluble recombinant C5a and C5a analogues and newly developed receptor antagonists. C5a is now also implicated in non-immunological functions associated with developmental biology, CNS development and neurodegeneration, tissue regeneration, and haematopoiesis. Combined receptor mutagenesis, molecular modelling, structure-activity relationship studies and species dependence for ligand potency on C5aR have been helpful for identifying ligand binding sites on the receptor and for defining mechanisms of receptor activation and inactivation. This review will highlight major developments in C5a receptor research that support C5aR as an important therapeutic target. The intriguing possibilities raised by the existence of a non-signalling C5a receptor are also discussed. Keywords: complement, C5a, G protein, receptor, inflammation, immunity, antagonist Formation of C5a Human complement is a complex network of soluble and membrane-associated serum proteins that form a highly regulated, exquisitely directed and normally measured humoral and cellular immune response to infectious organisms (bacteria/viruses/parasites), to tissue damaged by chemical, physical, radiation or neoplasia insults and to other foreign surfaces not recognized as ‘self'. The complement system is an ancient part of the innate immune system that has existed and adapted for >350 million years, components having been identified in organisms as primitive as the Horseshoe Crab (Carcinoscorpius rotundicauda) (Zhu et al., 2005). After encountering pathogen-associated molecular patterns, activation of the complement system proceeds through a stepwise hierarchy of proteolytic activation events, each proteolytic enzyme catalytically cleaving downstream members of the cascade. Complement activation occurs through three different pathways (Figure 1). The classical activation pathway, initially described as a ‘complement' to specific antibody lysis of bacteria (Bordet, 1895), is a response to the formation of immune complexes (IC) of complement fixing IgG1 and IgM antibodies. More recently, low affinity IgM antibodies involved in defence against infection and cancer and increasingly recognized as an important part of the innate immune system have also been shown to activate complement through the classical pathway (Vollmers and Brandlein, 2006). A second lectin activation pathway is initiated by lectins, which recognize the sugar structures that decorate the surfaces of infectious organisms. The third activation mechanism is the alternative pathway, which relies on the continuous degradation of component C3 that occurs on pathogen and host cell surfaces. Further complement activation is usually inhibited by control factors such as decay-accelerating factor and CD59 but the lack of these control factors on ‘non-self' surfaces leads to a rapidly amplified complement cascade activation (Thurman and Holers, 2006). In primitive organisms, the complement cascade is primarily opsonic, leading to the phagocytosis of targets. In higher organisms such as mammals, there are more than 30 serum components in the cascade, reflecting the complex effector pathways that lead not just to opsonization but also to the formation of a lytic membrane attack complex that perforates membranes of microorganisms causing cell death. Among other products are small protein anaphylatoxin fragments C3a, C4a and C5a. Figure 1. Open in a new tab Three pathways for complement activation. Each of the three pathways produces C5a and C5b, the latter assembling with C6, C7, C8 and C9 serum proteins to form the membrane attack complex. The cascade is highly regulated to avoid stepwise amplification but uncontrolled or aberrant regulation, resulting in protracted complement activation, can cause disease. Serum is a reservoir of the precursors of the complement fragments and so, even in the early stages of the innate immune response, high concentrations of these fragments may be produced and sustained for prolonged periods. Unlike C3a, for which even resting concentrations are high (>100 n M) because of the continual degradation of C3, there is almost no detectable C5a in the resting state (<1 n M) of healthy individuals. After activating human serum with cobra venom factor, concentrations of C5a can reach ∼285 n M. Interestingly, complement fragments can also be directly generated by proteases unrelated to the complement cascade; C5 degradation by thrombin, a participant in the coagulation cascade, causes C5a production even in animals with a genetic deficiency of the upstream complement protein C3 (Huber-Lang et al., 2002). Similarly, proteases found in allergenic house dust mite (Dermatophagoides farinae) faeces have been shown to generate anaphylatoxins from purified human C3 and C5, suggesting a possible route to C5a (and C3a) production in asthma (Maruo et al., 1997). Pro-inflammatory amorphous silica (Governa et al., 2005) and asbestos fibres (Governa et al., 2000) have also been shown to activate C5 directly. This review focuses on the function, structure and therapeutic potential of the cell surface receptors for one of these human complement fragments, namely C5a. Structure and function of C5a Human C5a (12–14.5 kDa) is composed of 74-amino acids, including Asn64, which has an N-linked carbohydrate moiety that is not essential for biological activity but very likely regulates C5a activity in vivo. It is missing from the highly homologous (69%) but equipotent porcine C5a. The solution structure (Zhang et al., 1997; Zuiderweg and Fesik, 1989; Zuiderweg et al., 1989) of human C5a (Figure 2) shows an antiparallel 4-helix bundle (residues 1–63), the four different helical segments (4–12, 18–26, 32–39, 46–63) being stabilized by three disulphide bonds (Cys 21-Cys 47, Cys 22-Cys 54, Cys 34-Cys 55) and connected by loop segments 13–17, 27–33 and 40–45. The 63-residue helix bundle fragment is highly cationic and confers high affinity for the cell surface. The C-terminal residues 69–74 also form a bulky helical turn connected to the 4-helix bundle by a short loop. Reducing disulphide bonds or selectively removing residues before the N-terminal disulphide from C5a1 to 74 substantially decreases function. The fragment C5a1–69 missing the C-terminal pentapeptide binds to cells but has no agonist activity, consistent with the N-terminal helix bundle conferring affinity, while the C-terminus alone is the receptor-activating domain. Loop 1 (residues C5a12–20, including four Lys residues 12, 14, 19, 20), loop 3 (C5a39–46) and the C-terminal 6–8 residues (especially Arg74) are important for binding to C5a receptor (C5aR) and agonist potency. Neutralizing antibodies to C5a have implicated the region Lys20-Arg37 as important for receptor binding. Figure 2. Open in a new tab Solution structure of human C5a determined from 1H NMR spectroscopy in H 2 0/D 2 O (Zhang et al., 1997), showing (left) : four helices MLQK 4 KIEEIAAK 12 YKH (blue), SVV 18 KKCCYDGA 26 CVNN (orange), DE 32 TCEQRAA 39 RISLGP (black), R 46 CIKAFTQCCVVA 63 SQ (violet) joined by loops (green) and a C terminal D 69 GLGR 74 (red), which adopts a 1.5 turn helix joined to the four helix bundle by a short loop (green); (right): electrostatic map showing residues with charged acidic (red) or basic (blue) side chains. C5a is readily metabolized by serum and cell-surface carboxypeptidases (Bokisch and Muller-Eberhard, 1970) that remove the C-terminal arginine to form ‘C5a des Arg', reducing potency to only 3–10% for promoting neutrophil chemotactic activity and to <1% in inducing a spasmogenic response from ileal tissue. Further removal of the C-terminal pentapeptide by carboxypeptidase Y inactivates the molecule (<1%) for both chemotactic and spasmogenic activity. The enzyme-linked immunoadsorbent assays used to measure serum C5a detect both forms equally well. The high activity levels of carboxypeptidases mean that most, if not all, of the C5a detected is actually C5a des Arg. C5a was first described as a classical anaphylatoxin, capable of stimulating the secretion of histamine from mast cells (Friedberger, 1910), and later identified as a potent neutrophil (Snyderman et al., 1970; Becker, 1972) and macrophage (Snyderman et al., 1975) chemoattractant. Now C5a is recognized as a pleiotropic molecule that can modulate the activity of many cell types, with a broad range of biological functions both inside and outside of the immune system. All cells of the myeloid lineage, including eosinophils (Kay et al., 1973), basophils (Hook et al., 1975) and neutrophils, sub-populations of monocytes (Falk and Leonard, 1980) and most, if not all, tissue macrophage types (including alveolar macrophages (McCarthy and Henson, 1979), liver Kuppfer cells (Laskin and Pilaro, 1986) and microglia in the central nervous system (Yao et al., 1990)) respond to C5a. Although some types of lymphoid cells have been shown to respond to C5a, this is not universally accepted. Both B and T lymphocytes were initially reported to migrate towards C5a (El-Naggar et al., 1980); helper T cells were shown to have an increased antigen-induced proliferative response in the presence of C5a (Morgan et al., 1983) and germinal centre (Kupp et al., 1991), and naïve tonsillar (Ottonello et al., 1999) B cells migrate in response to C5a. In contrast, a study using fluorescently labelled C5a failed to show significant binding to more than 6% of lymphocytes (van Epps and Chenoweth, 1984), and anti-C5aR antibodies did not bind to murine lymphoid cells (Soruri et al., 2003b). However, low levels of the C5aR have been detected by flow cytometry on CD3+ human T cells, particularly after lectin stimulation (0.6% rising to 14.4% positive for C5aR) and on the Jurkat T-cell line (Nataf et al., 1999). The same study showed that CD3+ T cells would migrate towards C5a. Thus, it appears that subsets of lymphocytes may be responsive to C5a and this percentage increases following activation. Mast cells also show highly variable responsiveness to C5a; for instance, skin mast cells respond to C5a, whereas lung and intestinal mast cells do not (Lawrence et al., 1987). More recently, expression of a receptor for C5a has been shown to discriminate between mast cell subsets, which also show distinct differences in protease expression, suggesting that C5a responsiveness is programmed into mast cell development (Oskeritzian et al., 2005). Although C5a has long been known to induce smooth muscle contraction, this has been thought to be secondary to the release of histamine and arachidonic acid-derived mediators (Regal et al., 1983). However, evidence has accumulated to show that C5a may also have direct effects because smooth muscle cells (SMC) have been shown to express low levels of anaphylatoxin receptors (Haviland et al., 1995; Gasque et al., 1998; Zwirner et al., 1999). However, there are no data on the function of C5a in SMC. In liver, hepatic stellate cells have been shown to undergo a small fibrotic response to C5a (Schlaf et al., 2004), and C5a can act as a growth factor in regenerating rat hepatocytes (Daveau et al., 2004). In fact, the absence of C5 or the blockade of C5aR both lead to impairment of liver regeneration, and the reconstitution of C5-deficient mice with C5a can restore this function (Mastellos et al., 2001). Endocrine and folliculostellate cells of the anterior pituitary gland express both forms of C5aR, and C5a stimulates mitogen-activated protein kinase (MAPK) activation in a mouse pituitary cell line (Francis et al., 2005), suggesting a possible role for C5a in the regulation of the hypothalamic-pituitary-adrenal (HPA) axis. Elevated levels of C5a have been found in the serum of patients with inflammatory disorders. Overexpression or underregulation of C5a is implicated in human and/or experimental models of inflammatory conditions, such as rheumatoid arthritis (RA) (Grant et al., 2002; Woodruff et al., 2002) and osteoarthritis, adult respiratory distress syndrome (Hammerschmidt et al., 1980b), inflammatory bowel diseases (Woodruff et al., 2003), lupus, ischaemia/reperfusion injury (Arumugam et al., 2003; Martin et al., 1988; Proctor et al., 2004; Woodruff et al., 2004), chronic obstructive pulmonary disease (Marc et al., 2004), sepsis (Huber-Lang et al., 2002), IC disorders (Strachan et al., 2000, 2001) and peritonitis (Godau et al., 2004), asthma and allergy (Abe et al., 2001; Baelder et al., 2005; Gerard and Gerard, 2002; Lambrecht, 2006), psoriasis (Kapp and Schopf, 1985), gingivitis (Okada and Silverman, 1979), atherosclerosis (Hammerschmidt et al., 1981), tissue rejection (Gaca et al., 2006), extracorporeal bypass (Tofukuji et al., 1998), glomerulonephritis (Kondo et al., 2001), meningitis, pancreatitis (Bhatia, 2002), fibrotic conditions (Hillebrandt et al., 2005; Jones et al., 1998), lung injury (Mulligan et al., 1996), neurodegeneration and macular degeneration (Kijlstra et al., 2005; van Beek et al., 2003), cystic fibrosis (Fick et al., 1986), fetal rejection (Girardi et al., 2006), systemic lupus erythematosus (Hammerschmidt et al., 1980a), anaphylactic and haemorrhagic (Harkin et al., 2004) shock, and following major trauma (Sewell et al., 2004), burns (Piccolo et al., 1999) and infection. Excessive complement activation may thus affect many hundreds of millions of people. Bioavailable C5aR antagonists could conceivably have potent anti-inflammatory properties in many diseases, while agonists could be valuable immunostimulants, enhancing humoral and cellular immunity. Receptors for C5a The first human C5aR was cloned in 1991 (Boulay et al., 1991; Gerard and Gerard, 1991). The second human C5aR, C5L2, was identified in 2000 (Ohno et al., 2000). Both genes are localized to the same region of chromosome 19, q13.33 and encoded in a two exon structure, with the 5′ untranslated region and initiating codon in the first exon, and the remainder of the coding sequence and the 3′ untranslated region in the second (Gerard et al., 1993). This is typical of the members of the chemoattractant receptor family. The sequences of C5aR and C5L2 are shown in Figure 3. C5aR is categorized in the peptide receptor subfamily of class A rhodopsin-like receptors. In a recent analysis based on the sequences of the transmembrane (TM) helices (Surgand et al., 2006), C5aR and C5L2 clustered with other chemoattractant receptors, such as type-II angiotensin-II receptor, bradykinin receptors, the formyl peptide receptor family, ChemR23 and several orphan G-protein-coupled receptors (GPCRs). Similarly, Joost and Methner (2002) placed C5aR and C5L2 in the GPCR family A8, with formyl peptide receptors, ChemR23 and the orphan receptors GPR1, 15 and 44 based on the sequences from the TM regions. A small number of single-nucleotide polymorphisms (SNP) have been found in both receptor genes. In the promoter region of C5aR, an SNP at position −245 (T/C) has been discovered (Barnes et al., 2004) and the coding region C5aR has two non-synonymous SNP at 4G/A (Asp/Asn at amino-acid position 2) and 859G/T (Asn/Lys at position 278) and two synonymous SNP: 72T/C, 727G/A (Birney et al., 2006). C5L2 has two synonymous SNP at 614G/A and 860C/T (Birney et al., 2006). There are no known associations between these SNP and human disease. Mice in which either C5aR (Hopken et al., 1996) or C5L2 (Gerard et al., 2005) has been genetically deleted are fully viable, but show alterations in many of the disease processes that involve C5a. The deletion of C5aR demonstrated a non-redundant role for this receptor in mucosal defence (Hopken et al., 1996) and in one model of RA (Ji et al., 2002) and a role in the reverse passive Arthus response (Hopken et al., 1997), contact sensitivity (Tsuji et al., 2000), glomerulonephritis (Welch et al., 2002), pulmonary hypersensitivity (Shushakova et al., 2002), granuloma formation in response to Mycobacterium infection (Borders et al., 2005) and in mast cell-mediated neutrophil accumulation in peritonitis (Mullaly and Kubes, 2007). In contrast, the course of experimental autoimmune encephalomyelitis was unaltered (Reiman et al., 2002) and Th2 cytokines, high serum IgE levels and substantial recruitment of inflammatory cells were actually increased after pulmonary allergen challenge in C5aR-deficient mice (Kohl et al., 2006). C5L2 deficiency has been reported to enhance responses to C5a in vivo (Gerard et al., 2005) but to diminish the responsiveness to C5a of neutrophils in vitro (Chen et al., 2007), suggesting multiple roles for this receptor. Figure 3. Open in a new tab Sequences of the C5a receptors, C5aR and C5L2, with potential glycosylation sites asterisked. Both receptors (42–45 kDa) also have characteristic arrays of Asp and Tyr residues at the N-termini; overall sequence identity is 35%. The degree of conservation for individual residues is shown by the depth of shading on the C5L2. Residues identified in site-directed or random saturation mutagenesis studies as having an important role in ligand binding, and/or signalling by C5aR are shown in blue. Sequences critical for G-protein coupling in C5aR, which are changed in C5L2, are shown in red. Control of receptor expression Although initially thought to be restricted to mast cells and cells of the myeloid lineage, C5aR expression is now known to be widespread (Table 1). Northern blot analysis has shown the range of expression of C5L2 to be broadly similar to that of C5aR. C5L2 is expressed in various cells and tissues, such as astrocytes, neutrophils/macrophages, mast cells, immature dendritic cells, as well as in the brain, lung, heart, kidney, liver, ovary or testis (Gavrilyuk et al., 2005; Lee et al., 2001; Ohno et al., 2000; Okinaga et al., 2003; Otto et al., 2004). The control of receptor expression at the gene level has not been thoroughly explored, but distinct transcriptional control mechanisms have been shown to occur in murine microglial cells and astrocytes (Martin et al., 2006). At the cellular level, the CCAAT box nuclear factor Y binding site (−96) is involved in lipopolysaccharide (LPS)-induced transcriptional upregulation of C5aR in murine macrophages and endothelial cells (Hunt et al., 2005) with minor contributions from a GATA site (−298) and a CP2 site (−155). Prostaglandin E2 upregulates C5aR in monocyte-derived dendritic cells (Weinmann et al., 2003), whereas interleukin-4 (IL-4) downregulates C5aR expression in monocytes (Soruri et al., 2003a). C5aR is upregulated by IL-6 in the lung, liver, kidney and heart of septic rats (Riedemann et al., 2003) and in the brain by tumour necrosis factor-α (TNF α) in closed head injury and Listeria infection in mice (Stahel et al., 1997, 2000). Monocytic differentiation of HL-60 cells, in the presence of vitamin D3, is associated with increased expression of C5aR (Zahn et al., 1997); C5aR is also upregulated in myeloblastic cell lines by dibutyryl-cAMP, phorbol ester and interferon (IFN)γ (Burg et al., 1996; Rubin et al., 1991). Less is known about the control of C5L2 expression, but total cellular C5L2 expression has been shown to decrease on neutrophils after exposure to C5a (Huber-Lang et al., 2005). A single study has reported regulation of C5L2 expression on cell lines: dibutyryl-cAMP and IFN γ induced upregulation of this receptor on U937 and HL-60 cells, but TNF α had no effect. In the epithelial HeLa cell line, constitutive expression of a low level of C5L2 but not C5aR was detected, and treatment with IFN γ and TNF α drastically reduced C5L2expression (Johswich et al., 2006). C5aR is known to rapidly internalize after treatment with C5a (Huey and Hugli, 1985) and is then recycled to the cell surface (Van Epps et al., 1990). Recycling has been proposed to be important for directed cellular migration in a gradient of C5a (Naik et al., 1997) but is not apparently related to the clearance of C5a from plasma (Oppermann and Gotze, 1994). Table 1. C5aR expression on non-myeloid cell lines \| Loci \| Cell Type \| Reference \| :--- \| Circulatory system \| Mouse microvascular endothelial cells \| (Laudes et al., 2002) \| \| CNS \| Microglia, reactive astrocytes \| (Gasque et al., 1997) \| \| \| Neural stem cells \| (Rahpeymai et al., 2006) \| \| \| Neurons \| (O'Barr et al., 2001) \| \| \| Oligodendrocytes \| (Nataf et al., 2001) \| \| Connective tissue \| Mast cell subtypes \| (Oskeritzian et al., 2005) \| \| \| Synoviocytes \| (Yuan et al., 2003) \| \| \| Human synovial mast cells \| (Kiener et al., 1998) \| \| \| Human articular chondrocytes \| (Onuma et al., 2002) \| \| Eye \| Retinal pigment epithelial cell line \| (Fukuoka and Medof, 2001) \| \| Circulatory system \| Septic cardiomyocytes \| (Niederbichler et al., 2006) \| \| Immune system \| CD3+ murine T cells \| (Connelly et al., 2006) \| \| \| Human tonsillar B cells \| (Ottonello et al., 1999) \| \| \| Rat thymocytes \| (Riedemann et al., 2002a) \| \| \| Plasmacytoid dendritic cells \| (Gutzmer et al., 2006) \| \| Kidney \| Cultured human renal glomerular mesangial cells \| (Braun and Davis, 1998) \| \| \| Human renal proximal tubular cells \| (Fayyazi et al., 2000) \| \| Liver \| HepG2 cells \| (Buchner et al., 1995; Haviland et al., 1995; McCoy et al., 1995) \| \| \| Hepatic stellate Kupfer cells \| (Schlaf et al., 2003) \| \| \| Stimulated hepatocytes \| (Koleva et al., 2002; Schlaf et al., 2003; Schieferdecker et al., 2000) \| \| Lung \| Human and mouse bronchial epithelial and smooth muscle \| (Drouin et al., 2001; Floreani et al., 1998) \| \| \| Rat alveolar epithelia cells \| (Riedemann et al., 2002b) \| \| Skin \| Inflamed keratinocytes \| (Fayyazi et al., 1999; Zwirner et al., 1999) \| Open in a new tab C5a binding to C5aR The human C5aR binds C5a with a K d of 1 n M but has an affinity for C5a desArg that is 10 to 100-fold lower. Ribosomal protein S19 and bacterial chaperone Skp have both been reported to bind to C5aR, although only one laboratory has reported these findings to date (Shrestha et al., 2004; Nishiura et al., 1996), and the receptor-binding mechanism remains obscure. The potential roles of these non-complement-derived C5aR ligands have recently been reviewed (Yamamoto, 2007). The ligand-binding sites on C5aR have been mapped by a number of methods. Antibodies directed against the N-terminal domain have been shown to inhibit the binding of C5a (Morgan et al., 1993; Oppermann et al., 1993), and deletion of the N-terminus also prevents C5a binding (Mery and Boulay, 1993; DeMartino et al., 1994). A chimeric form of C5aR, with the N-terminus of the receptor for the closely related anaphylatoxin C3a, C3aR, also loses the ability to bind C5a (Crass et al., 1999a). However, in all of these cases, peptide analogues of the C-terminus of C5a have still been able to activate the receptor, indicating the presence of an additional binding site. To identify this site, C5aR chimerae containing the analogous domains of the formyl peptide receptor showed that the second and third extracellular loops (ECLs) of C5aR were essential for ligand binding (Pease et al., 1994). A series of powerful genetic studies using a yeast selection system has provided a great deal of evidence for the roles of the ECLs and the TM helices in the formation of the ligand-binding site. These experiments coupled the human C5aR to endogenous G proteins that normally mediate responses to mating pheromones, driving the expression of HIS3 and allowing growth on media lacking histidine (Baranski et al., 1999) when C5aR is activated by co-expressed human C5a. Using this system, functional receptors have been selected from large libraries of C5aR molecules that have undergone random saturation mutagenesis. Helix by helix and loop by loop, those residues critical for ligand binding, receptor oligomerization, activation and G-protein coupling have been described (Baranski et al., 1999; Geva et al., 2000; Gerber et al., 2001b; Floyd et al., 2003; Klco et al., 2003, 2005, 2006; Hagemann et al., 2006; Matsumoto et al., 2007b). It is not clear how sensitive this system is, and residues that make a small contribution to signal transduction by the ligand may be missed. In addition, the need for C5a to be produced by yeast, rather than being added exogenously because of the impermeability of the cell membrane, may lead to very high concentrations of C5a in the proximity of C5aR, which could further reduce sensitivity. However, despite these minor caveats, this powerful and elegant system has produced many exciting results, some of which are discussed below. Taken together, the data obtained from all of these experimental approaches have led to the two-site model of receptor activation in which there is a primary high affinity contact between basic residues in the core of C5a (Figure 2) and acidic residues in the N-terminus of C5aR (Figure 3) plus a secondary interaction between the C-terminus of C5a and a binding pocket formed by hydrophobic residues in the TM domains and charged residues at the base of the ECLs. The contributions of these different regions are discussed below. Binding sites on the C5aR N-terminus Mapping of the interaction site at the N-terminus of C5aR has been performed in a number of ways, with antibodies targeted to this region and N-terminal deletions having similar inhibitory effects on ligand binding to hC5aR. Identification of the actual residues involved has been problematic, however. A multiple mutant of hC5aR (Asp15,16,18,21Asn) showed a 40-fold decrease in hC5a affinity, and hC5aR(Asp10,15,16,18,21Asn) showed a 133-fold reduction (DeMartino et al., 1994). In contrast, the single mutations of Asp10Asn and Asp27Asn or a double mutation (Asp21,27Asn) had no effect on hC5a binding, whereas the multiple substitutions hC5aR(Asp10, 15, 16Asn) or hC5aR(Asp15,16,21,27Asn) showed no detectable hC5a binding (Mery and Boulay, 1994). Similarly, a nuclear magnetic resonance (NMR) study on the hC5aR N-terminus highlighted the importance of residues 21–30 in hC5a binding (Chen et al., 1998). O-sulphation of tyrosine residues has been shown to be important for the formation of the ligand-binding site in several GPCR, such as CXCR4 and CCR5 (Hsu et al., 2005). In hC5aR, residues Tyr11 and Tyr14 have been shown to be sulphated; the mutation Tyr11Phe showed almost complete loss of C5a binding and Tyr14Phe showed ∼50% loss of binding affinity, whereas mutation of Tyr8 had no effect on either sulphation or ligand binding, suggesting that sulphation is essential for the formation of the ligand-binding site on hC5aR (Farzan et al., 2001). Providing some support for these findings, a yeast random saturation mutagenesis (RSM) screening study on the N-terminus also found that residues 24–30 were likely to be important for C5a binding (Hagemann et al., 2006) but that no single Asp residue was critical. However, this study also found that Tyr11 and Tyr14 could be substituted by a range of other amino acids and so were unlikely to be involved in ligand binding in apparent contradiction of the mutagenesis data. This is probably because yeast lack protein tyrosine O-sulphation machinery (Moore, 2003), and the maintenance of ligand binding in the yeast system may suggest that the high periplasmic concentrations of C5a that occur could be compensating for a low affinity of binding by non-sulphated C5aR. Binding sites on the C5aR ECLs Point mutagenesis studies have identified several critical residues in the juxtamembrane regions of these loops, including Arg175, Glu199, Arg206, Asp282. It has been proposed that the receptor interaction site for the C-terminal carboxylate of C5aR agonists is at Arg206, a residue at the extracellular face of helix 5 (Gerber et al., 2001a). Mutation of Arg206 to Ala has only a small effect on receptor activation by C5a (Cain et al., 2001). Taken together with the observation that C5a des-Arg74 binds to, but does not activate, Arg206Ala-C5aR (Cain et al., 2001), it is possible that mutation of this receptor residue perturbs the global structure of the receptor rather than disrupting specific ligand interactions. This view is further supported by the finding that a ligand-independent constitutively active C5aR mutant (Ile124Asn/Leu127Gln) can be completely deactivated by substitution of Arg206 by His (Gerber et al., 2001a). Another potential receptor site for interaction with the C-terminal carboxylate is Arg175, located either on the extracellular face of helix 4 or in the adjacent loop. The analogous residue (Arg161) in the closely related C3a receptor has been proposed to interact with the C-terminal carboxylate of C3a (Sun et al., 1999). We have previously shown that although C5aR mutated at Arg175 is only weakly activated by C5a, it can be strongly activated by a mutant form of C5a des-Arg74 isolated from a randomly mutated C5a des-Arg74 library (Cain et al., 2003), suggesting that a specific and important interaction between C5aR and C5a is lost when Arg 175 is mutated to either Ala or Asp. A possible explanation of the data is that the peptide carboxylate makes interactions with both Arg206 and Arg175 at different points in the receptor binding and activation process. Asp282, at the extracellular face of helix 7, has been shown to interact with the side chain of Arg74 of C5a, and with the C-terminal Arg in peptide analogues (Cain et al., 2001, 2003). The mutation Glu199Lys has a complete lack of responsiveness to agonists lacking a C-terminal Arg, such as C5a des-Arg74 and C5a[Ala74], suggesting that in addition to a previously demonstrated interaction between Lys68 of C5a and receptor Glu199 (Monk et al., 1995; Crass et al., 1999b), the side chain of the C-terminal Arg74 residue interacts with Glu199. However, the loss of this interaction following mutation of Glu 199 has no effect on the responsiveness to C5a, possibly suggesting only a transient interaction between Arg74 and Glu199, with a more important interaction occurring between Arg74 and Asp282. This is clearly shown by the mutation Asp282Arg, which has a very low responsiveness to C5a, but a relatively normal response to C5a des-Arg74 and similar ligands (Cain et al., 2001, 2003). Highly conserved Cys residues in loop 1 (Cys109) and loop 2 (Cys188) have been shown to be critical for receptor expression, probably owing to the formation of a stabilizing disulphide linkage (Kolakowski et al., 1995). The yeast RSM screening system described above has confirmed the identification of Arg206 as a key residue, since the only allowed mutation is to Lys. Similarly, the importance of Asp282, where no substitutions were detected, is also confirmed (Baranski et al., 1999; Klco et al., 2006). Yeast screening of the second ECL, in particular, has provided a plethora of information on this key structure: Arg175 is relatively highly conserved, although only Cys188 was regarded as critical, most probably because of the disulphide linkage that this residue makes with Cys109. However, even more interesting was the finding that several of the mutated receptor sequences were constitutively active, suggesting that EC2 is a negative regulator of receptor activation (Klco et al., 2005). Genetic mapping of the first ECL revealed the importance of receptor activation of the Trp-Phe-X-Gly motif that is highly conserved in the GPCR superfamily (Klco et al., 2006), although these residues do not contribute to the formation of the ligand-binding site. Binding sites on the TM helices Several mutagenesis studies have investigated the role of residues in the TM helices. Asp82 in TM-II has been shown to be critical for signalling but not ligand binding by C5aR (Bubeck et al., 1994; Monk et al., 1994b; Kolakowski et al., 1995). A systematic analysis of Pro and Cys residues in the helices determined several that were critical for ligand binding (Pro170, Cys221) and/or signalling (Pro36, Pro170, Pro214, Cys86, Cys157, Cys285). The yeast RSM screening system enabled the identification of a TM residue, Ile116 in TM-III, as being involved in receptor antagonism. The key role of a binding site in the vicinity of Ile116 was recently confirmed using site-specific disulphide capture, a technique in which potentially interacting amino acids in both ligand and receptor are substituted by Cys residues. The formation of a disulphide linkage indicates that these residues are in close proximity during the binding process. In this way, Leu117, Pro113 and Gly262, residues predicted to be near Ile116 in C5aR models, have been identified as interacting with ligands (Buck et al., 2005). The site-specific disulphide capture methodology has also been used to screen a library of thiol-containing small molecules for C5a mimics (Buck and Wells, 2005). Other TM residues identified by the high degree of conservation in yeast RSM screens include Tyr222 and Leu112 (Baranski et al., 1999), which are also suggested to be important in receptor function due to conservation in other GPCR. In fact, by assuming that residues with side chains located at helix/helix interfaces are likely to be most highly conserved because of the complementary shapes required to pack helices together, the likely relative orientations of the helices can be mapped (Geva et al., 2000). Patches of preserved residues on helices I and II have also suggested a potential interaction site for other membrane proteins or specialized lipids; alternatively, this region could be involved in homodimer formation (Geva et al., 2000). Ligand binding by C5L2 Human C5L2 is a high affinity receptor for C5a that also binds C5a des Arg with a much higher affinity than C5aR (EC 50 values for C5a and C5a des Arg are 7 and 36 n M, respectively), whereas mouse C5L2 binds mouse C5a des Arg with a 4000-fold higher affinity than mouse C5a (Scola et al., 2007). Although C5L2 binding of C5a and C5a des Arg has been confirmed by several groups (Cain and Monk, 2002; Okinaga et al., 2003; Johswich et al., 2006), the reported ability of C5L2 to bind other anaphylatoxins such as C3a des Arg (Kalant et al., 2003) remains a controversial issue. However, a recent paper (Johswich et al., 2006) has suggested that the binding of C3a des Arg may have been an artefact of the binding protocol rather than specific binding to C5L2. C5L2 has a similar pattern of tyrosine and acidic N-terminal residues to the C5aR, which have been shown to be a major feature of extracellular binding of C5a (Figure 1b). C5L2 also shares similarities with the C5aR in the number of charged and hydrophobic residues in the loops and TM regions, which are involved in the interaction with the C-terminus of C5a. Despite these common features, ligand binding by the two receptors is clearly different. Antibodies directed against the N-terminal domain or mutation of tyrosine and acidic residues in the C5L2 N-terminus significantly inhibit C5a des Arg binding but have little effect on the interaction with C5a (Scola et al., 2007). Peptide and peptidomimetic ligands for C5aR Peptide and peptidomimetic compounds have been developed as small molecule regulators of C5aR. The full agonist activity of C5a is located in the C-terminal 8 residues (Kawai et al., 1991) and Abbott researchers derived synthetic peptide analogues as agonists at C5aR that inhibit C5a binding with Ki values of ∼300 μ M. A decapeptide analogue, Tyr-Ser-Phe-Lys-Pro-Met-Pro-Leu-D Ala-Arg, is a full agonist against C5aR at low μ M concentrations (Finch et al., 1997) that also binds to the C3a receptor, C3aR (Proctor et al., 2004). L156,602 (Figure 5) is a cyclic peptide produced by Streptomyces with a weak ability to inhibit C5a binding (IC 50=2 μ M), but toxicity has prevented further development as a C5a antagonist (Tsuji et al., 1995). Poly-L-Arg and protamine were expected to inhibit C5a binding owing to the presence of the basic residues from C5a but lack of selectivity has prevented these compounds from being used further (Olsen et al., 1988). Structure/activity studies by Abbott researchers on the C5a sequence resulted in a very active hexapeptide agonist, N-MethylPhe-Lys-Pro-D Cha-Cha-D Arg-CO 2 H, which has an IC 50 of ∼25 n M against C5aR of isolated polymorphonuclear leukocyte (PMN) membranes (Kawai et al., 1992). Substitutions at position 5 by Merck researchers resulted in a partial agonist when cyclohexylalanine (Cha) was replaced with Phe (Drapeau et al., 1993) but a full antagonist when replaced by Trp (Konteatis et al., 1994), namely N-MethylPhe-Lys-Pro-D Cha-Trp-D Arg-CO 2 H (Trp5). Figure 5. Open in a new tab Modelled interaction between 3D53 (green) and human C5aR (orange) showing ligand-binding pocket (left: side view, right: top view) with Arg, Trp, dCha and AcPhe components of 3D53 fitting between helices of C5aR with key receptor residues labelled according to Ballesteros and Palczewski (2001). Figure updated since Higginbottom et al. (2005). The latter compound was the first pure antagonist with no agonist activity even at 100 μ M. It was a potent antagonist (inhibiting binding IC 50∼200 n M), receptor activation by 100 n M C5a (IC 50∼85 n M), on human PMN (Wong et al., 1998; Finch et al., 1999). In 1995, in the Centre for Drug Design and Development (3D Centre), it was suspected (Fairlie et al., 1995) that the Pro-D Cha pairing in Trp5 might favour a tight reverse turn motif around Pro as in other macrocycles (Fairlie et al., 1995; Chalmers and Marshall, 1995) and that it might be possible to stabilize this through a Lys side chain to C-terminus cyclization. Although Trp5 had no discernible structure in water, 1H NMR spectra subsequently showed in early 1996 that Trp5 had a well-defined gamma turn structure in the dipolar aprotic solvent DMSO (Wong et al., 1998), which we had previously used with some success to predict structure of short peptides in membrane environments. Based on our notion that this may be the active conformation, we deployed the unusual side chain–main chain cyclization constraint, resulting in mid-1996 in much more potent and chemically stable antagonists (Wong et al., 1998, 1999a; Finch et al., 1999; March et al., 2004). Among these cyclic antagonists was 3D53, AcPhe[Orn-Pro-DCha-Trp-Arg], cyclized through the side chain of Orn and the terminal carboxylate (Wong et al., 1999b), with an IC 50 of 60 n M for the inhibition of C5a binding to whole PMNs and 30 n M for the inhibition of PMN degranulation (Finch et al., 1999). This peptide has been the most intensively evaluated C5a antagonist (Taylor and Fairlie, 2005) with high affinity for dog, cat and rat PMN C5aR (IC 50=40 n M) but lower affinity for mouse PMN C5aR (IC 50>10 μ M) (Woodruff et al., 2001) and antagonist activity at cells transfected with human, gerbil (Meriones unguiculatus) but not mouse C5aR (Waters et al., 2005). It was stable to rat serum, gastric fluid and gastric enzymes (Taylor and Fairlie, 2005). Of great interest was the demonstration that 3D53 and analogues were potent inhibitors of C5a-induced neutrophil chemotaxis and cytokine production from macrophages in vitro (Haynes et al., 2000), and these properties were also consistently evident in vivo. Interestingly, 3D53 showed little ability to block C5a or C5a des Arg binding to C5L2 (Otto et al., 2004), reinforcing suggestions that the two C5aRs have different ligand-binding mechanisms. Although only 1% orally bioavailable or slightly more for analogues without the amide bond connecting the cycle to N-terminal appendages (March et al., 2004), a little indiscriminant in binding to GPCRs resulting in off-target side effects (Schnatbaum et al., 2006), and expensive to manufacture, 3D53 did show significant efficacy following i.v., p.o., s.c. and t.d. administration in a variety of rat models of inflammatory disease (Table 2) (Kohl, 2006). It was licensed as PMX53 for clinical development by Promics Ltd (subsequently taken over by Peptech Pty Ltd). Other cyclic analogues with more favourable pharmacokinetics, for example JPE-1375 (Schnatbaum et al., 2006) and JSM-7717 ( have been developed but results in clinical trials are not yet available. Table 2. C5aR cyclic peptide antagonist 3D53 and analogues in disease models \| Disease \| Animal model \| Dose/delivery route \| References \| :--- :--- \| \| Arthritis \| Rat Monoarticular \| 1–3 mg/kg/day p.o. \| (Woodruff et al., 2002) \| \| \| Antigen-Induced \| \| \| \| \| Rat Adjuvant-Induced \| 1 mg/kg/day p.o. \| Unpublished \| \| \| Rat Collagen-Induced \| 1 mg/kg/day p.o. \| Unpublished \| \| \| Rat Paw Oedema \| 1 mg/kg/day p.o. \| Unpublished \| \| Fetal Miscarriage \| Mouse Antiphospholipid Abs \| 50 μ g/mouse i.p \| (Girardi et al., 2003) \| \| Cardiac Fibrosis \| Rat Hypotensive (DOCA) \| 1 mg/day p.o. \| (Mirkovic et al., 2002) \| \| Glomerulonephritis \| Rat Antibody-induced \| 1–10 mg/kg i.v./p.o. \| Unpublished \| \| Haemorrhagic Shock \| Rat Aorta Aneurysm \| 1 mg/kg i.v. \| (Harkin et al., 2004) \| \| Huntington's Disease \| Rat Neuronal Damage \| 10 mg/kg p.o. \| (Woodruff et al., 2006) \| \| Immune Complex Disorder \| Rat Arthus \| 1 mg/kg i.v. \| (Short et al., 1999) \| \| \| Rat Peritoneal Arthus \| 1–10 mg/kg p.o. \| \| \| \| Rat Dermal Arthus \| 0.4–1 mg t.d., 1–10 mg/kg p.o. \| (Strachan et al., 2000, 2001) \| \| Inflammatory BowelDisease \| Rat (TNBS-induced) \| 10 mg/kg p.o., 0.3 mg/kg s.c. \| (Woodruff et al., 2003) \| \| Influenza \| Mouse \| 1 mg/kgi.p. \| (Kim et al., 2004) \| \| Liver Injury \| Mouse \| 1 mg/kg i.p. \| (Strey et al., 2003) \| \| Lung Injury \| Mouse \| 1 mg/kg intratracheally. \| (Huber-Lang et al., 2002) \| \| Lupus Nephritis \| Mouse SLE \| 1 mg/kg/day s.c. \| (Bao et al., 2005) \| \| Reperfusion Injury \| Rat Intestinal \| 1 mg/kg i.v.; 10 mg/kg p.o. \| (Arumugam et al., 2002) \| \| \| Mouse Intestinal \| 25 μ g/mouse i.v. \| Fleming et al., 2003 \| \| \| Rat Kidney \| 1 mg/kgi.v., 10 mg/kg p.o. \| (Arumugam et al., 2003) \| \| \| Rat Liver \| 1 mg/kg i.v. 10 mg/kg p.o. \| (Arumugam et al., 2004) \| \| \| Rat Limb \| 1 mg/kg i.v., 10 mg/kg p.o. \| (Woodruff et al., 2004) \| \| Sepsis \| Rat Neutropaenia (C5a, LPS, cobra venom factor) \| 0.3–3 mg/kg i.v. \| (Saatvedt et al., 1996; Short et al., 1999; Taylor and Fairlie, 2005) \| \| \| \| 10 mg/kg p.o. \| (Strachan et al., 2001) \| \| \| Mouse Caecal Ligation \| 50 mg/kg topical \| Unpublished \| \| \| \| 1–3 mg/kgi.v., 10 mg/kg p.o. \| (Huber-Lang et al., 2002) \| Open in a new tab Abbreviations: DOCA: deoxycorticosterone acetate; TNBS: trinitrobenzene sulphonic acid; SLE: systemic lupus erythematosus; LPS: lipopolysaccharide. Non-peptidic ligands for C5aR To overcome problems associated with peptides, some development of cheaper, orally more bioavailable and more target-selective non-peptidic compounds as either agonists or antagonists has taken place, with at least five groups known to have non-peptidic compounds in development. Early non-peptidic ligands (Figure 4) were of only low-moderate affinity antagonists for human C5aR, such as Merck's aminoquinolines (Lanza et al., 1992) and Rhone-Poulenc's phenylguanidines such as RPR121154 (IC 50=0.8 μ M), which completely inhibited the respiratory burst response of human neutrophils to 100 n M C5a (Astles et al., 1997). The basic nature of RPR121154 suggests that it may mimic a positively charged receptor-binding site in the core domain of C5a, although there is no evidence for this mechanism. Merck reported several other structural types of antagonists with submicromolar potencies (De Laszlo et al., 1997), but they were not developed further due to partial agonist responses. Interestingly, the hydantoin shown in Figure 4 was a potent full agonist, EC 50=20 n M (De Laszlo et al., 1997). Figure 4. Open in a new tab Structures of small molecule ligands for C5aR. The optimization of a series of substituted phenylguanidines led to Mitsubishi Pharma's tetrahydronaphthalene-based compound W54011 (Figure 4), which is a competitive non-peptidic C5aR antagonist ([125I]hC5a IC 50=2.2 n M) that inhibits intracellular Ca 2+ mobilization, chemotaxis and production of reactive oxygen species with IC 50=3.1, 2.7 and 1.6 n M, respectively (Sumichika et al., 2002). The combination of potency and oral availability appeared promising but its substantial hydrophobicity and problems with species specificity (active for human, cynomolgus monkey and gerbil but not mouse, rat, guinea pig, rabbit or dog neutrophils) complicated pre-clinical studies. NDT9520492 (Figure 4) is a member of a large series of compounds developed by Neurogen Corp with C5aR antagonist activity (Waters et al., 2005) at human and gerbil but not mouse C5aR. A similar compound, NGD 2000–1 had no therapeutic effect in an asthma study and, while a Phase II trial in RA showed some promise, the compound inhibited cytochrome P450 3A4 and development was halted ( Among other non-peptidic C5a antagonists is Jerini JSM7431, which appears to have been discontinued. Binding of small molecule ligands to C5aR Unlike the binding of proteins to C5aR, small molecule ligands bind primarily in the TM region of the receptor. Recently, the putative binding site on C5aR has been reported for the linear antagonist Trp5 (Gerber et al., 2001b; Higginbottom et al., 2005) and the cyclic antagonist 3D53 (Higginbottom et al., 2005), based on a combination of studies that included molecular modelling of the receptor, molecular docking of NMR structures of the ligands into the homology model of the receptor, site directed mutagenesis of the receptor and structure-activity studies for various ligands binding to wild type versus mutant C5aR on PMNs. Mapping of ligand binding using these methods suggested that Trp5 and 3D53 bind at the same (or slightly overlapping) location in the TM region of C5aR near the extracellular interface (Figure 5). Key receptor residues were thought to be on TM-III: Ile116 (3.32), Tyr121 (3.37), TM-V: Glu199, (5.35), Arg206 (5.42), Phe211 (5.47), Leu215 (5.51)), TM-VI: Phe251 (6.44), Trp255 (6.48) and TM-VII: Asp282 (7.35), Val286 (7.39), Tyr290 (7.43); the numbers in brackets show residue positions according to Ballesteros and Palczewski (2001) The model reflects the importance within 3D53 of Arg, D-Cha and Ac-Phe components as binding residues and the Trp as an antagonist-determining residue. It also places the Ac-Phe appendage on the cycle in the vicinity of extracellular loop two (not shown), which is thought to act as a lid on the ligand-binding active site. Flexible peptide agonists reversibly enter the hydrophobic ‘pit' in the TM region of the receptor, but Trp5 and especially 3D53 occupy the cavity may hold the ECL2 lid down. This may be the reason why it is difficult to dissociate 3D53 from the receptor (slow off rate) and why it has an insurmountable mechanism of antagonism. Non-peptidic ligands such as W-54011 (Sumichika et al., 2002) and NDT9520492 (Waters et al., 2005) appear to bind in very similar locations within the TM region of C5aR as the Trp and D-Cha side chains of 3D53, based at this time on scant evidence from effects of species dependence in neutrophil C5aR or site-directed mutation of C5aR residues (for example Trp213 (5.49)). Thus, W-54011 potently inhibits C5a-induced intracellular Ca 2+ mobilization in neutrophils of cynomolgus monkeys and gerbils but not mice, rats, guinea pigs, rabbits and dogs. It is important to point out that the IC 50 values reported for competitive reversible antagonists W54011 and NDT952492 are biased by the low n M concentrations of C5a that they were measured against. Neither non-peptidic compound is as effective as the insurmountable cyclic peptide antagonist 3D53 at higher C5a concentrations, a distinction that we attribute to the unique ability of 3D53 to fill the hydrophobic C5aR cleft and close the ECL2 loop lid on the cleft. The smaller non-peptidic compounds reported to date simply do not occupy enough space to interact strongly with the lid of the cavity while being anchored in their binding sites and are readily displaced. Intracellular signalling via C5aR C5aR primarily couples to G α i2 (Sheth et al., 1991; Skokowa et al., 2005), a pertussis toxin (PT)-sensitive G protein. However, ectopically expressed C5aR, and also C5aR in some haemopoietic cell types such as monocytes, can also couple to G α 16 (Monk and Partridge, 1993; Buhl et al., 1994; Kalant et al., 2003), a PT-insensitive G protein. The loss of G β 1 and G β 2 (effectively no G β expression at all) in J774 mouse macrophages eliminates all responsiveness to C5a (Hwang et al., 2005), whereas the loss of G β 1 alone does not affect chemotaxis (Hwang et al., 2004). Recently, C5aR has been shown to be able to couple to a wide range of G proteins when key intracellular residues are mutated, suggesting that the regulation of the G-protein coupling range occurs by a mechanism of repression rather than by positive promotion of interactions (Matsumoto et al., 2007b). C5aR also couples directly or indirectly to a small range of other intracellular proteins (Figure 6). The Wiskot–Aldrich syndrome protein (WASP) was detected as a binding partner of the C-terminus of C5aR using a yeast 2 hybrid assay (Tardif et al., 2003). WASP binding was strongly potentiated in the presence of active cdc42, a small guanine-5′-triphosphate (GTP)-binding protein, suggesting that the association occurs after C5aR activation. WASP is a multifunctional protein with a role in the regulation of actin dynamics (Ochs and Notarangelo, 2005), and so could be involved in the chemotactic response to C5a. Figure 6. Open in a new tab The ‘interactome' of C5aR. C5aR interacts directly or indirectly with kinases (purple), GTP binding/regulatory proteins (red), transcription factors (pink), other signalling enzymes (blue) or structural proteins (grey). Internalization of C5aR is mediated by clathrin, which associates with receptor-bound β-arrestin (Ar) and the actin cytoskeleton. Proteins, such as hsp27, phosphorylated by MAP kinase-activated protein kinase 2 (MAPKAP-K2), may regulate the actin cytoskeleton. MAPKAP-K2 is itself activated by the mitogen-activated kinase (MAPK/ERK/JNK) cascade, in turn activated by kinase Akt (also known as PK-B) or by p21-associated protein kinase (PAK) complexed with Rac/Cdc42 guanine nucleotide exchange factor PIX α, cdc42 and G-protein-coupled receptor kinase-interactor 2 (GIT2). G-protein α-subunits are deactivated by regulator of G-protein signalling 1 (RGS1) that stimulates GTP conversion to GDP; in the GDP-bound state, α-subunits can bind to and modulate the activity of the NADPH-oxidase component p67 phox. βγ-subunits directly activate PAK and indirectly activate PK-C β by increasing diacylglycerol and intracellular Ca 2+ ([Ca 2+]i) through phospholipase C β (PLC β). βγ may be sequestered by G-protein-coupled receptor kinase (GRK), which also phosphorylates C5aR along with PK-C β. Transcription factors signal transducer and activator of transcription 3 (STAT3), cAMP responsive element binding protein (CREB) and nuclear factor (NF)-κ B are activated at the convergence of the kinase pathways, and apoptosis inhibited by phosphorylation of Bcl-associated death promoter (BAD) and upregulation of caspase degradation. JNK, c-Jun N-terminal kinase; NADPH, nicotinamide adenine dinucleotide phosphate. Activated C5aR has also been shown to associate with two of the four mammalian β-arrestins (β-arrestin 1, 2), which have different dependencies on the phosphorylation status of the receptor (Braun et al., 2003). The arrestins have multiple roles, being involved in receptor trafficking and the regulation of signalling (Gurevich and Gurevich, 2006). G-protein receptor kinases (GRK) are thought to control the phosphorylation levels of C5aR and most likely GRK2 and GRK3, which are found co-expressed with C5aR in cell lines such as HMC-1 (Langkabel et al., 1999). However, overexpression of GRK2 (and 6) failed to alter phosphorylation patterns of C5aR (Milcent et al., 1999) and so it is possible that only GRK3 is involved in C5aR phosphorylation in vivo. Apart from their role in phosphorylating GPCR, GRK also interact with a range of other signalling molecules, including Akt, MAPK/ERK kinase (MEK) and phosphatidylinositol 3-kinase (PI3K), suggesting a wider role in connecting GPCR with diverse signalling pathways (Ribas et al., 2007). RGS1, an effective GTPase-activating protein (GAP) for G subunits of the Gi and the Gq family has been shown to be involved in C5aR desensitization (Denecke et al., 1999). C5aR has been reported to activate phospholipase C (PLC)β 2 but not PLC β 3 in a PT-sensitive manner (Jiang et al., 1996) in Cos7 cells, although the ability of transfected human C5aR to stimulate PLC activity in rat basophilic leukaemia cells, which express only PLC β 3 (Ali et al., 1997), suggests that C5aR may also couple to this isoform. In neutrophils, C5a leads to causes downstream activation of p21-activated kinases (PAK), which are downstream effectors of cdc42 and rac GTPases (Huang et al., 1998) as well as G protein γ subunits; PAK family members are involved in altering cell morphology/chemotaxis, the activation or potentiation of several distinct MAPK cascades and the activation of nuclear factor-κ B (NF-κ B) in macrophages (Bokoch, 2003). Interestingly, the PAK-associated guanine nucleotide exchange factor (PIX α) also binds to PAK1 and, in association with G βγ subunits, forms a complex that can activate cdc42 (Li et al., 2003) in a positive feedback loop. GIT2, a GAP that regulates Arf activity, also associates with PAK and is indispensable for direction sensing in C5a-stimulated neutrophils (Mazaki et al., 2006); GIT2 is further involved in controlling the production of superoxide anions during chemotaxis and in orienting superoxide production towards the source of chemoattractant (Mazaki et al., 2006). C5a can activate the transcription factor, cAMP response element-binding protein (CREB), by phosphorylation at the convergence of two pathways, PI3K/Akt and extracellular signal-regulated kinase (ERK) signalling (Perianayagam et al., 2006); CREB activation has been proposed to be a part of the mechanism by which C5a can delay neutrophil apoptosis (Perianayagam et al., 2002, 2004) and prolong an inflammatory response. p38a MAPK is activated by PAK1/PAK2 and, in turn, activates MAPK-activated protein kinase 2 (MAPKAP-K2); thus, in primary macrophages from MAPKAP-K2 deficient mice, chemotaxis to C5a is impaired (Rousseau et al., 2006); heat-shock protein HSP-27 is a likely substrate of MAPKAP-K2 in these cells. The p38 MAPK inhibitor, SB203580, can inhibit C5a-induced migration in a mouse acute lung injury model (Nash and Heuertz, 2005). RhoG in murine neutrophils may be involved in Rac1 and Rac2 activation, leading to nicotinamide adenine dinucleotide phosphate oxidase activation (Condliffe et al., 2006). C5a activates the PI3K/Akt signalling pathway and induces the phosphorylation of the p38a MAPK, ERK and c-Jun N-terminal kinase, leading to suppression of IL-12 production in human monocytes (la Sala et al., 2005) and mouse macrophages (Hawlisch et al., 2005). In human erythroleukaemia cells, signal transducers and activators of transcription (STAT3) phosphorylation can be stimulated by C5a in a PTX-insensitive manner, most likely through G α 16 and the Ras/Raf/MEK/ERK and c-Src/JAK pathways (Lo et al., 2003); in contrast, STAT3 phosphorylation occurs only through an ERK pathway in C5a-stimulated neutrophils (Kuroki and O'Flaherty, 1999). In endothelial cells but not leukocytes, C5a-induced motility can be blocked by inhibitors of the epidermal growth factor (EGF) receptor (EGFR) and by neutralizing antibodies against the EGFR and heparin-binding EGF-like factor (Schraufstatter et al., 2002); transactivation of EGFR by several GPCRs has been reported and is thought to lead to the amplification of responses. C5aR can form homodimers (Geva et al., 2000; Floyd et al., 2003; Klco et al., 2003), probably by associations between helices I and II from the partner receptors, and can also complex with other GPCRs in heterooligomers, for instance with CCR5 (Huttenrauch et al., 2005). The consequences of these interactions for C5aR are, as yet, unclear; oligomers form early in the biosynthetic pathway and are known to be important during transport to the plasma membrane (Milligan et al., 2003) and studies on oxytocin and vasopressin receptors reveal a complex pharmacology after oligomerization (Albizu et al., 2006). Ligand binding by C5aR in homo- or hetero-oligomers has not yet been investigated but CCR5 co-expressed with C5aR was found to be phosphorylated after C5a addition, suggesting a role in the control of other chemoattractants (Huttenrauch et al., 2005). Intracellular protein binding sites on C5aR The sites on C5aR that control localization and signalling have been investigated in a small number of studies using point mutagenesis of intracellular regions of C5aR. The mutation of Arg68 in intracellular loop 1 or Trp230, Thr235, Thr238 in loop 3 diminishes the ability of C5aR to signal in Cos7 cells that co-express G α 16 (Kolakowski et al., 1995). Deletion of the C-terminal 23 amino acids of C5aR had little effect on G α i protein-dependent signalling in transfected mammalian cells (Monk et al., 1994a), although internalization was dependent on residues 335–350, as shown by a series of C-terminal truncations (Bock et al., 1997). This inhibitory effect on internalization is probably due to a loss of phosphorylation sites in the C-terminal domain, as the simultaneous mutation of Ser332, 334 and 338 to Ala also caused an 80% reduction of phosphate incorporation into C5aR (Giannini et al., 1995). This reduction leads to a significant retardation of ligand-induced internalization (Naik et al., 1997), most likely by loss of an association with other components of the internalization machinery, β-arrestin, clathrin and dynamin (Braun et al., 2003). Phosphorylation of Ser334 was found to be critical for the sequential phosphorylation of the other Ser residues in this triplet; mutation of Ser334 to Asp allowed phosphorylation to occur as normal at Ser332, 338 (Christophe et al., 2000), followed by the remaining C-terminal Ser residues, 314, 317 and 327. Interestingly, the same authors found that loss of phosphorylation at Ser332 and 334 prevented receptor desensitization even when internalization was still normal, resulting in prolonged responses to C5a. The regions of C5aR involved in coupling to G proteins have been broadly defined using peptide analogues of the intracellular loops and the C-terminus fused to a cell permeant sequence derived from Kaposi fibroblast growth factor (Auger et al., 2004). This work showed the proximal region of the C-terminus to be a major G-protein-binding site, with loop 3 having a role in G-protein activation. The intracellular regions of C5aR have also been analysed using the yeast screening method. These studies have given the most profound insight into the coupling between C5aR and G proteins, providing information on how G-protein specificity occurs and on the mechanism of G-protein activation. The C-terminus of C5aR was shown to be dispensable for G-protein coupling, and there were a minority of preserved residues in the first (2/16 residues studied were completely resistant to substitution) and second (6/17 resistant residues) intracellular loops. In contrast, the third intracellular loop was quite highly conserved (11/21 residues) (Matsumoto et al., 2007a). Interestingly, an analysis of mutants selected on the basis of efficient coupling to G α i in yeast showed that many mutations, particularly in the C-terminus and second intracellular loop broaden G-protein specificity and mutants that couple well to G α q and G α s-like G proteins were characterized. It was concluded that the normal sequence of C5aR contains negative regulators of specificity that can be disrupted by mutation (Matsumoto et al., 2007b). Signalling via C5L2 Although C5L2 has the conventional structure of a GPCR, studies have found that C5L2 does not couple to G proteins. This is thought to be owing to the lack of a highly conserved Asp-Arg-Tyr motif, found in the third TM domain, which in C5L2 is replaced with delocalized lipophilic cation. In the presence of C5a, cells transfected with C5L2 show no increase in cytosolic calcium levels or activation of the MAP kinase pathway, and C5L2 transfected RBL cells failed to degranulate upon stimulation with C5a or C5a des Arg (Cain and Monk, 2002; Okinaga et al., 2003; Johswich et al., 2006). When the Asp-Leu-Cys motif of C5L2 is mutated to Asp-Arg-Cys, the binding of C5a can induce a small increase in intracellular calcium levels, suggesting an incomplete restoration of G-protein coupling (Okinaga et al., 2003). Other intracellular and TM sequences that are not conserved between C5L2 and C5aR, for instance the Asn-Pro-X-X-Tyr motif in TM-VII and a deleted polar tripeptide in intracellular loop 3 (Figure 3), may also contribute to the inability of C5L2 to couple to signalling pathways. The ability of C5L2 to bind anaphylatoxins without signalling has led to the suggestion that C5L2 may have a role as an anaphylatoxin decoy receptor, thereby regulating the availability of C5a and C5a des Arg. Rat neutrophils stimulated with C5a and LPS, in the presence of a C5L2 blocking antibody, produce dramatically increased levels of IL-6 compared to controls (Gao et al., 2005). C5L2-deficient mice produce neutrophils with an increased response to both C5a and C5a des Arg and show a 2- to 3-fold increased influx of neutrophils into the lung of −/− C5L2 animals and higher levels of TNF-α and IL-6 when compared to wt-mice in a model of pulmonary IC injury (Gerard et al., 2005). A comprehensive study of sepsis patients found higher C5L2 content in PMN obtained from patients who survived the observation period compared to patients who failed to survive; low C5L2 expression seemed to correlate with sepsis induced multi-organ failure, suggesting an important role for C5L2 in sepsis (Huber-Lang et al., 2005). In contrast to these data, a recent report has suggested that C5L2 is a positive modulator for signalling through C5aR (Chen et al., 2007). Neutrophils from C5L2-deficient mice responded less strongly to C5a compared to cells from wild-type animals, and reduced numbers of peritoneal macrophages were elicited by thioglycollate. Airway hyper-responsiveness and inflammatory cell infiltration was reduced in C5L2-deficient mice, although these mice were more susceptible to the lethal effects of LPS. Several studies found that cell lines transfected with C5L2 do not show net loss of receptor from the membrane after ligand binding, suggesting that C5L2 does not undergo ligand-induced internalization (Cain and Monk, 2002; Okinaga et al., 2003). The difference in results may be owing to the short time of exposure of C5L2 to ligand used in these in vitro studies (5–15 min) compared to patients with sepsis where C5L2 expressing cells would be exposed to anaphylatoxins on a much longer (hours/days) time scale. Therefore, the regulation of C5L2 expression seems to be variable, depending on both cell type and level of exposure to anaphylatoxins, and the mechanisms involved in regulating C5L2 expression have yet to be elucidated. Although C5L2 does not appear to signal using the traditional mechanisms employed by GPCRs, several studies suggest that C5L2 has the ability to induce cellular effects. A recent study (Gavrilyuk et al., 2005) found that noradrenaline could upregulate C5L2 message and protein in rat astrocytes, and this correlated with an anti-inflammatory response induced by noradrenaline; transfection of astrocytes by C5L2 down regulated NF-κ B activity, whereas antisense oligonucleotides against C5L2 caused the reverse effect. This observation suggests that the presence of C5L2 may exert some inhibitory effects within the cell, although the mechanisms behind such responses are currently unknown. The suggestions that C5L2 can both prevent ligand from interacting with C5aR and downregulate pro-inflammatory signalling raise the possibility that treatments that increase C5L2 expression could be used as part of an anti-inflammatory strategy. Conclusions There is now strong evidence of a pathogenic role for C5a from studies in numerous disease models using antibodies to C5a or C5aR, soluble receptor sCR1 and C5aR-knockout and knockin transgenic mice (Weisman et al., 1990; Bozic et al., 1996; Goodfellow et al., 1997; Hopken et al., 1997; Mohr et al., 1998; Lee et al., 2006), and especially from studies of small molecule antagonists (for example Table 2). Alexion now has Phase III data for a C5 antibody (eculizumab) to treat haemolytic anaemia; Avant has Phase IIb data for sCR1 in cardiac bypass surgery (Ratner, 2006); Promics has Phase IIa data for 3D53 in rheumatoid arthritis. However, as therapeutics for chronic inflammatory diseases, such biologics are compromised by high cost, low bioavailability, metabolic instability and the need for repeated injections. The development of effective small molecule antagonists for C5aR is an attractive alternative, and compounds generated to date have accelerated our understanding of the central involvement of C5a in many inflammatory disease states, albeit so far mainly through the use of rodent models of disease. Those studies have demonstrated profound immunoregulatory effects for C5aR antagonists in vivo and encouraging benefits in animal models of human inflammatory diseases. One caveat concerning the use of C5aR antagonists should be made, however. The recent demonstration of a protective role for C5aR in the sensitization phase of asthma (reviewed by Kohl and Wills-Karp, 2007) suggests that, although the use of powerful C5aR antagonists may be beneficial for existing inflammatory conditions, patients may become more easily sensitized to new pulmonary allergens. The discovery of C5L2 as an inhibitory C5a/C5a des Arg receptor has also raised the intriguing possibility of the use of this receptor as a novel anti-inflammatory strategy, although further work is required to determine the full functionality of this protein. Acknowledgments PNM acknowledges the support of the Wellcome Trust (project grants 007521 and 072231). DPF acknowledges support from the National Health and Medical Research Council and the Australian Research Council. 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18119	https://eduseed.in/math/understanding-place-value/	Skip to content Larger Numbers: Indian vs. International System As we move to larger numbers, the way numbers are grouped and named differs between the Indian number system and the International number system. While both systems rely on place value, the periods (groups of digits) and their names differ slightly. Indian Number System In the Indian number system, commas are placed differently to help read large numbers, breaking them into groups (called periods) of hundreds, thousands, lakhs and crores. The comma is placed after every two digits starting from the right, after the hundreds place. The Indian place value chart looks like this. For example, 1,23,45,678 is read as one crore, twenty-three lakh, forty-five thousand, six hundred seventy-eight. 52,34,567 is read as fifty-two lakh, thirty-four thousand, five hundred sixty-seven. 70,89,12,345 is read as seventy crore, eighty-nine lakh, twelve thousand, three hundred forty-five. Now, place commas and try reading the below number in the Indian System. 384672915 International Number System In the International system, commas are placed after every three digits, making it easier to work with millions and billions, as used in many countries worldwide. This system groups digits into ones, thousands, and millions. The International place value chart looks like this. For example, 12,345,678 is read as twelve million, three hundred forty-five thousand, six hundred seventy-eight. 3,456,789 is read as three million, four hundred fifty-six thousand, seven hundred eighty-nine. 987,654,321 is read as nine hundred eighty-seven million, six hundred fifty-four thousand, three hundred twenty-one. Now, place commas and try reading the below number in the International System. 7835491206 Why is Place Value Important? Reading and Writing Numbers: Understanding place value allows children to correctly read and write large numbers in both systems. Without this knowledge, even small numbers can become confusing. Performing Arithmetic: Place value plays a key role in addition, subtraction, multiplication, and division. When solving problems, numbers need to be lined up according to their place values (ones under ones, tens under tens, etc.). Comparing Numbers: Place value helps students compare large numbers. For example, 9,876,543 is greater than 8,765,432 because the digit in the millions place is larger. Understanding Patterns: Place value also helps children recognize patterns when multiplying or dividing numbers by 10, 100, or 1,000. Each multiplication by 10 shifts the digits one place to the left, increasing their value. Fun Ways to Learn Place Value Learning place value can be enjoyable and engaging for young minds when taught with activities. Here are a few activities that help students master this concept: Place Value Blocks: Children can use blocks or counters to represent ones, tens, hundreds, and more. Grouping blocks into larger units helps them visualize place values. Place Value Charts: A place value chart helps students break down large numbers into their components. It’s a great tool for practicing reading and writing numbers. Expanded Form Games: Breaking numbers into their expanded form (e.g., 5,678 as 5,000 + 600 + 70 + 8) helps children see the value of each digit. These can be turned into fun games to keep kids engaged. Place value is the foundation of understanding numbers and performing arithmetic operations. Mastering this concept early allows children to build strong mathematical skills and makes them feel confident working with numbers of all sizes in both the Indian and International number systems. Practice Quiz on Place Value FAQs on Place Value What is the difference between place value and face value? Place value: The value of a digit depending on its position in the number (e.g., the place value of 4 in 4,567 is 4,000). Face value: The value of the digit itself, regardless of its position (e.g., the face value of 4 in 4,567 is simply 4). How do place values work in decimals? In decimal numbers, place values are extended to the right of the decimal point. The first place after the decimal is tenths, the second is hundredths, and so on. For example, in 0.56, 5 is in the tenths place (0.5), and 6 is in the hundredths place (0.06). What is the place value of digits in a number like 12,345.678? The place value of 1 is 10,000 (ten thousands place). The place value of 2 is 2,000 (thousands place). The place value of 3 is 300 (hundreds place). The place value of 4 is 40 (tens place). The place value of 5 is 5 (ones place). The place value of 6 is 0.6 (tenths place). The place value of 7 is 0.07 (hundredths place). The place value of 8 is 0.008 (thousandths place). What is the place value of zero? Zero has no value, but it plays an important role in place value. It acts as a placeholder to indicate the absence of a digit in a particular place. For example, in 204, the zero in the tens place indicates there are no tens. How can place value help in rounding numbers? Understanding place value is crucial for rounding numbers. To round a number, you look at the digit in a specific place value and round based on the value of the next lower place value (e.g., rounding 473 to the nearest ten gives 470 because 3 is less than 5). How does place value relate to expanded form? Expanded form expresses a number by showing the value of each digit according to its place value. For example, the expanded form of 345 is 300 + 40 + 5. How does place value help in comparing numbers? Place value allows you to compare numbers starting from the largest place value. For example, when comparing 789 and 695, start by comparing the digits in the hundreds place: 7 is greater than 6, so 789 is larger than 695. Why is the place value of 8 in the number 1287 not 8, but 80? The place value of a digit in a number depends on its position within the number. In the number 1287, the digit 8 is in the tens place. Here’s how the place value system works: The digit 7 is in the ones place. The digit 8 is in the tens place. The digit 2 is in the hundreds place. The digit 1 is in the thousands place. Since 8 is in the tens place, its place value is 8 tens, or 80. If the 8 were in the ones place, its place value would be just 8, but because it is in the tens position, it is multiplied by 10, resulting in 80. So, the place value of 8 in 1287 is 80. × We're here to help!
18120	https://en.namu.wiki/w/%EC%97%90%EB%AA%AC%20%ED%94%84%EB%A0%88%EC%9D%B4	Emmon Frey - NamuWiki RecentChanges Emmon Frey Last Modified: 2025-01-28 14:52:43 Category A Song of Ice and Fire/Characters 1. outline2. track 1.outline Emmon Frey A character in A Song of Ice and Fire and a member of House Frey . He is the second son of Lord Walder Frey and is said to be bald, short and nervous. He is the husband of Jenna Lannister and has a son , Cleos Frey . 2.track He was afraid of Tywin Lannister , who was younger than himself , and gave the impression that he was intimidated by the Lannister family, such as being held tightly to his wife, Jenna. During the War of the Five Kings, he sided with his wife, the Lannisters, rather than his parents, the House of Frey, and sent his children to the Lannisters. After the blood wedding, he received the Riveran of the Touli family as a king. Because of this, he has been Hee -hee, who has been a great place for Liberland for a while, but he is dissatisfied when Peter Bailes is appointed as a lord of Haren Hall and the British Lord of Riverland. And when Jamie put a crush on the wall when he attacked Riveran with Jamie Lannister,Tommen Baratheon gave her a letter signed as king, and Jamie was very annoying. Currently, I somehow managed to get River Run and capture Edmoure Tully , but first of all, there is a lack of troops, and above all , important figures such as Brynden Tully and Jane Westerling have escaped, and the current situation in the Seven Kingdoms is returning to the opening scene once again. The status of the current trend, Emon, is also quite unstable. This document is available under CC BY-NC-SA 2.0 KR. (except for some documents and illustrations where licenses are specified) The copyright of the contributed document belongs to each contributor, and each contributor owns the copyright of the part they contribute. RecentChanges namu.wiki Contáctenos Términos de uso Operado por umanle S.R.L. Hecho con ❤️ en Asunción, República del Paraguay Su zona horaria es GMT Impulsado por the seed engine 한국어 This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply. This site is protected by hCaptcha and its Privacy Policy and Terms of Service apply.
18121	https://web.njit.edu/~tyson/P111_chapter5.pdf	Copyright © 2012 Pearson Education Inc. PowerPoint® Lectures for University Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Chapter 5 Applying Newton’s Laws Copyright © 2012 Pearson Education Inc. Goals for Chapter 5 • To use Newton’s first law for bodies in equilibrium • To use Newton’s second law for accelerating bodies • To study the types of friction and fluid resistance • To solve problems involving circular motion Copyright © 2012 Pearson Education Inc. Introduction • We’ll start with equilibrium, in which a body is at rest or moving with constant velocity. • Next, we’ll study objects that are not in equilibrium and deal with the relationship between forces and motion. • We’ll analyze the friction force that acts when a body slides over a surface. • We’ll analyze the forces on a body in circular motion at constant speed. Copyright © 2012 Pearson Education Inc. Particles in Equilibrium If the acceleration of an object that can be modeled as a particle is zero, the object is said to be in equilibrium • The model is the particle in equilibrium model Mathematically, the net force acting on the object is zero 0 0 and 0       x y F F F Copyright © 2012 Pearson Education Inc. Two-dimensional equilibrium : Example • A car engine hangs from several chains. •Find the expression for the tension in each of three chains in terms of w Copyright © 2012 Pearson Education Inc. Two dimensional equilibrium: Example Solution: T1=w T3 cos600 =T2 T3 sin600=T1 Copyright © 2012 Pearson Education Inc. Equilibrium, Example 2 Find tension in each cable if the weight of street lamp is 122 N Copyright © 2012 Pearson Education Inc. Copyright © 2012 Pearson Education Inc. Copyright © 2012 Pearson Education Inc. Inclined Planes Forces acting on the object: • The normal force acts perpendicular to the plane • The gravitational force acts straight down Choose the coordinate system with x along the incline and y perpendicular to the incline Replace the force of gravity with its components Copyright © 2012 Pearson Education Inc. A car on an inclined plane An car of weight w rests on a slanted ramp attached to a trailer. Only a cable running from the trailer prevents the car form rolling off the ramp. Find the tension in the cable and the force that the ramp exerts on the car’s tires. W = mg T- mgsinα = 0 T = mgsinα n-mgcosα = 0 n = mgcosα Copyright © 2012 Pearson Education Inc. Incline plane. Object not in equilibrium: ΣF =ma • What is the acceleration of a toboggan loaded with students of total mass of 500 kg sliding down a friction-free slope? Assume α = 150 . mgsinα = ma a = gsinα = 9.8m/s2 sin150 =2.53m/s2 Copyright © 2012 Pearson Education Inc. Multiple Objects. Bodies connected by a cable and pulley Copyright © 2012 Pearson Education Inc. Multiple Objects, Example Copyright © 2012 Pearson Education Inc. Frictional forces • When a body rests or slides on a surface, the friction force is parallel to the surface. • Friction between two surfaces arises from interactions between molecules on the surfaces. Copyright © 2012 Pearson Education Inc. Kinetic and static friction • Kinetic friction acts when a body slides over a surface. • The kinetic friction force is fk = µkn. • Static friction acts when there is no relative motion between bodies. • The static friction force can vary between zero and its maximum value: fs ≤ µsn. Copyright © 2012 Pearson Education Inc. Static friction followed by kinetic friction • Before the box slides, static friction acts. But once it starts to slide, kinetic friction acts. Copyright © 2012 Pearson Education Inc. Some approximate coefficients of friction Copyright © 2012 Pearson Education Inc. Friction in horizontal motion To start a 500-N crate moving across a level floor you have to pull with a 230-N horizontal force. Once the crate starts to move, you can keep it moving at constant speed with only 200N. What are the coefficients of static and kinetic friction? Before the crate moves, static friction acts on it. After it starts to move, kinetic friction acts. Copyright © 2012 Pearson Education Inc. Static friction can be less than the maximum Copyright © 2012 Pearson Education Inc. Friction - Example Find the coefficient of static friction force if the block starts to slide down at an angle of 12.50 Start sliding when fs = fs,max fs,max =µsn=µsmgcosθ mgsinθ = µsmgcosθ µs = tan q  tan12.50 =0.22 Copyright © 2012 Pearson Education Inc. Motion on a slope having friction What is the acceleration of a toboggan loaded with students of total mass of 500 kg sliding down a slope? Assume α=150 and coefficient of kinetic friction between the toboggan and the slope µk=0.08. mgsinα –fk= ma fk =µkn n=mgcos α fk =µkmgcosα mgsinα - µkmgcosα = ma a=gsinα - µkgcosα = (sin150-0.08cos150)9.8m/s2 = 1.78m/s2 Copyright © 2012 Pearson Education Inc. Friction, Example A hockey puck on a frozen pond is given an initial speed of 12 m/s. If the puck slides 60 m before coming to rest, determine the coefficient of kinetic friction between the puck and ice. Copyright © 2012 Pearson Education Inc. Friction, Example Copyright © 2012 Pearson Education Inc. Pulling a crate at an angle •The angle of the pull affects the normal force, which in turn affects the friction force. •How hard must you pull the crate to keep it moving with constant velocity? Assume µk = 0.4 Copyright © 2012 Pearson Education Inc. Fluid resistance and terminal speed • The fluid resistance on a body depends on the speed of the body. • A falling body reaches its terminal speed when the resisting force equals the weight of the body. • The figures at the right illustrate the effects of air drag. Copyright © 2012 Pearson Education Inc. Dynamics of circular motion • If a particle is in uniform circular motion, both its acceleration and the net force on it are directed toward the center of the circle. • The net force on the particle is Fnet = mv2/R. Copyright © 2012 Pearson Education Inc. What if the string breaks? • If the string breaks, no net force acts on the ball, so it obeys Newton’s first law and moves in a straight line. Copyright © 2012 Pearson Education Inc. Force in uniform circular motion • A sled on frictionless ice is kept in uniform circular motion by a rope. • F = Copyright © 2012 Pearson Education Inc. A conical pendulum A bob at the end of a wire moves in a horizontal circle with constant speed. Copyright © 2012 Pearson Education Inc. A car rounds a flat curve Copyright © 2012 Pearson Education Inc. A car rounds a banked curve • At what angle should a curve be banked so a car can make the turn even with no friction? Copyright © 2012 Pearson Education Inc. Uniform motion in a vertical circle • A person on a Ferris wheel moves in a vertical circle. Top: n-w = -mac n = w-mac Bottom: n-w = mac n = w + mac
18122	https://www.allmathtricks.com/unit-digit-number/	Published Time: 2018-07-28T18:28:27+00:00 How to Find Unit Digit of a Power Number \| Unit Digit Problems with Solutions - All Math Tricks Close Menu FacebookX (Twitter)Instagram FacebookX (Twitter)LinkedInPinterestRSS Home Blog About Us Contact Us Math Tricks Pure Mathematics number system Algebra Polynomials Progressions Geometry Coordinate geometry Quantitative Aptitude Interest Calculations percentage Ratio proportion and variation Calculus Derivatives limits and integrals Math Reasoning You are at:Home»Pure Math»number system»How to Find Unit Digit of a Power Number \| Unit Digit Problems with Solutions number system How to Find Unit Digit of a Power Number \| Unit Digit Problems with Solutions BysivaalluriJuly 28, 2018 Updated:February 23, 2025No Comments6 Mins Read Table of Contents Toggle Finding the Last Digit of any Number With Power \| Unit place of a Number Find the last digit of a number with power Last digit of a number of questions Find the last digit of a large exponent Unit Digit problems with solutions Finding the Last Digit of any Number With Power \| Unit place of a Number In Quantitative aptitude, questions asked to find the last digit and last two digits of a power or large expression. This article explained different types of tools to serve as shortcuts to finding the last digits of an expanded power. Find the last digit of a number with power First, identify the pattern last digit (unit place) for the power of numbers “N” Digit N 1N 2N 3N 4N 5N 6N 7N 8N 9 11 1 1 1 1 1 1 1 24 8 6 2 4 8 6 2 39 7 1 3 9 7 1 3 46 4 6 4 6 4 6 4 55 5 5 5 5 5 5 5 66 6 6 6 6 6 6 6 79 3 1 7 9 3 1 7 84 2 6 8 4 2 6 8 91 9 1 9 1 9 1 9 From the above table, we can observe as following The last digit of power of1, 5 & 6 is always comes same number as a unit place. The last digit of power of2 repeats in a cycle of numbers –4, 8, 6 & 2 The last digit of power of 3 repeats in a cycle of numbers – 9, 7, 1 & 3 The last digit of power of4 repeats in a cycle of numbers – 6 & 4 The last digit of the power of 7 repeats in a cycle of numbers – 9, 3, 1, & 7 The last digit of the power of 8 repeats in a cycle of numbers –4, 2, 6 & 8 The last digit of the power of 9 repeats in a cycle of numbers – 1 & 9 Explanation: If the Last digit ( Unit place ) of numbers having 1 , 5 & 6 ( – – – – 1)n = ( – – – – 1) (- – – – -5) n = ( – – – – 5) (- – – – -6) n = (- – – – -6) If the unit place ( Last digit ) of any number “ A n ”having2, 3, 7 or 8, then the unit place of that number depends upon the value of power“ n”and follows Expressed power “ n”Unit Place of ( – – -2)nUnit Place of ( – – -3)nUnit Place of ( – – -7)nUnit Place of ( – – -8)n 4x6 1 1 6 4x + 12 3 7 8 4x + 24 9 9 4 4x + 38 7 3 2 If the unit place ( Last digit ) of any number “ A n ”having4 & 9then the unit place of that number depends upon the value of power “ n”and follows Expressed power “ n”Unit Place of ( – – -4)nUnit Place of ( – – -9)n 2x (Even number)6 1 2x + 1 (Odd number)4 9 Last digit of a number of questions Examples – 1 : Find the last digit of the number 3 2015 Solution:The power 2015 can be written as [ (503 x 4) + 3 ] So from the above table unit digit of a given number is – 7 Examples – 2: Find the last digit of the number 4444 2015 Solution: Here power value is an odd number So the last digit of the given number is 4 Hint: The last digit of any number having “4” then power has an even number then unit place becomes 6 and power has an odd number then unit place becomes 4 Example 3 : What is the last digit of the number 4 2012 Solution: Here power value is an even number. So unit digit of the given number is 6 Examples – 4 :Find the last digit of the number 11 123+5 Solution:Here The unit place has ” 1″ so the final number also comes ” 1″ as a unit place Examples – 5 :Find the digit at the unit place of the number 19 25 Hint: The last digit of any number having “9” then power has an even number then unit place becomes 1 and power has an odd number then unit place becomes 9 Solution: Here power has an odd number so the final number unit place comes ” 9″ Examples – 6:Find the digit at the unit place of the number Solution: First find the unit place of 3 99-3 Hint: Here the pattern of the last digits are 1 , 3, 9, 7, 1, 3 , 9 , 7 . . . . . . . for the powers 4x , 4x+1 , 4x+2 , 4x+3 . . . . . respectively. = 3 96 here 96 multiple of 4 so the last digit comes as 1 = ( – – – – 1 )50 = ( – – – – – – – – 1) i.e unit digit having 1 so the final number unit place also comes 1 Find the last digit of a large exponent It is aremainder theorem application – The last digit of an expression equals to the remainder of that expression divided by 10. Unit Digit problems with solutions Examples – 7: Find the unit digit of the expression123 x 587 x 987 x 78 Solution: Here given expression 123 x 587 x 987 x 78 divided by 10 and find the remainder So unit digit of the given expression is 6 Examples – 8:Find the unit digit of the expression 578497 x 87548 x 25417 Solution: Here give the expression 578497 x 87548 x 25417 divided by 10 and find the remainder = 578497 x 87548 x 25417 / 10 = 7 x 8 x 7 / 10 So unit digit of the given expression is 2 Related Topics : Number Categories Topics in Quantitative aptitude math for all types of exams Shortcut Math Tricks for helpful to improve speed in all calculations Hi friends Thanks for reading. I hope you like it. 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18123	https://mathworld.wolfram.com/RankMatrix.html	Rank Matrix -- from Wolfram MathWorld TOPICS AlgebraApplied MathematicsCalculus and AnalysisDiscrete MathematicsFoundations of MathematicsGeometryHistory and TerminologyNumber TheoryProbability and StatisticsRecreational MathematicsTopologyAlphabetical IndexNew in MathWorld Discrete Mathematics Graph Theory Graph Properties Graph Matrices Rank Matrix If the rank polynomial of a graph is given by , then is the number of subgraphs of with rank and co-rank , and the matrix is called the rank matrix of . For example, the rank matrix of the complete bipartite graph, which has rank polynomial (1) is given by (2) (Biggs 1993, p.73), and the rank matrix of the Petersen graph is (3) (Godsil and Royle 2001, p.356). See also Rank Polynomial Explore with Wolfram\|Alpha More things to try: rank matrix arcsin 2 Frobenius number {4, 7, 12} References Biggs, N.L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, p.73, 1993.Godsil, C. and Royle, G. Algebraic Graph Theory. New York: Springer-Verlag, 2001. Referenced on Wolfram\|Alpha Rank Matrix Cite this as: Weisstein, Eric W. "Rank Matrix." From MathWorld--A Wolfram Resource. Subject classifications Discrete Mathematics Graph Theory Graph Properties Graph Matrices 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
18124	https://arxiv.org/pdf/2008.00318	A characterization of the strong law of large numbers for Bernoulli sequences LUÍSA BORSATO 1, EDUARDO HORTA 2, and RAFAEL RIGÃO SOUZA 2,† 1Institute of Mathematics and Statistics, Universidade de São Paulo, São Paulo, Brazil. E-mail: luisabborsato@gmail.com 2Institute of Mathematics and Statistics, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil. E-mail: eduardo.horta@ufrgs.br ; †rafars@mat.ufrgs.br The law of large numbers is one of the most fundamental results in Probability Theory. In the case of independent sequences, there are some known characterizations; for instance, in the independent and identically distributed setting it is known that the law of large numbers is equivalent to integrability. In the case of dependent sequences, there are no known general characterizations — to the best of our knowledge. We provide such a characterization for identically distributed Bernoulli sequences in terms of a product disintegration. Keywords: law of large numbers, random measure, disintegration, conditional independence. 1. Introduction It is somewhat intuitive to most people 1 that if a coin is thrown independently a large number of times, then the observed proportion of heads should not be far from the parameter of unbalancedness θ ∈ [0 , 1] (this quantity being understood as representing the probability, or ‘chance’, of observing heads in any one individual throw). In the Theory of Probability, the law of large numbers supports, generalizes and also provides a precise mathematical meaning to this intuition — an intuition which can be traced back at least to Cardano’s 16th-century Liber de ludo aleae . In his 1713 treatise Ars Conjectandi , Jacob Bernoulli gave the first proof of the fact that (in modern notation) if X is a Binomial random variable with parameters n ∈ N and 0 ≤ p ≤ 1, then one has the inequality P(\|n−1X − p\| > ε ) ≤ (1 + c)−1, provided n is large enough, where ε and c are arbitrarily prescribed positive constants . This is a typical weak law statement — although it was not until the time of Poisson that the name “loi des grands nombres” was coined [11, p.7]. See [13, 14] for a compelling historical perspective on the law of large numbers, a history which culminated in ‘the’ strong law for independent and identically dis-tributed sequences, according to which the almost sure convergence of the sequence of sample means to the (common) expected value is equivalent to integrability. Also, still in the context of independent sequences, we highlight the importance of Kolmogorov’s strong law for independent sequences whose partial sums have variances satisfying a summability condition. 1 One may argue that most people interpret probability — at least when it comes to coin-throwing — in a Popperian sense, i.e. seeing probability statements as utterances which quantify the physical propensity of a given outcome in a given experiment, in lieu of an epistemic view where such statements only measure the degree to which we are uncertain about said outcome . To us the propensity interpretation seems adequate in the framework of coin-throwing, as it is meaningful to establish a connection between the coin’s physical center of mass and the propensity of it landing ‘heads’ in any one given throw: recalling that a coin throw is governed by classical (deterministic) mechanics, we could for instance let Ω denote the set of all possible initial conditions (angle, speed, spin, etc) and then make the requirement that the subset comprised of all initial conditions whose corresponding outcome is ‘heads’ be a measurable set, with measure p ∈ [0 , 1] . Clearly such p is a function of the coin’s center of mass. 1 arXiv:2008.00318v1 [math.PR] 1 Aug 2020 2Outside the realm of independence, things get trickier. As famously put by Michel Loève [9, p.6], “martingales, Markov dependence and stationarity are the only three dependence concepts so far iso-lated which are sufficiently general and sufficiently amenable to investigation, yet with a great number of deep properties”. The contemporary probabilist would likely add uncorrelatedness, m-dependence, exchangeability and mixing properties to that list. In any case, the ways through which independence may fail to hold are manifold, and thus one might infer that dependence is too wide a concept, which means we should not expect to easily obtain a characterization of the law of large numbers for depen-dent sequences. Indeed, there are many scenarios where one can give sufficient conditions under which a law of large numbers holds for such sequences — to cite just a few examples: the weak law for pair-wise uncorrelated sequences of random variables; the strong law for mixing sequences [7, 8]; the strong law for exchangeable sequences ; some very interesting results concerning decay of correlations (see, for example, ) — but, to the best of our knowledge, no characterization has been provided so far 2. In this paper, we provide one such characterization for sequences of identically distributed Bernoulli random variables, in terms of the concept of a product disintegration . Our main result shows that, to a certain degree, independence is an inextricable aspect of the law of large numbers. Our conceptualization derives from — and generalizes — the notion of an exchangeable sequence of random variables, to which we shall recall the precise definition shortly. First, let us get back to heuristics. The intuition underlying the coin-throwing situation depicted above remains essentially the same if we assume that, before fabricating the coin, the parameter of unbalancedness will be chosen at random in the interval [0 , 1] . In this case, conditionally on the value of the randomly chosen ϑ (let us say that the realized value is θ), the long run proportion of heads definitely ought to approach θ. The natural follow-up is to consider the not so evident scenario in which we choose at random (possibly distinct) parameters of unbalancedness ϑ0, . . . , ϑ n, . . . and then, given a realization of these random variables (say, θ0, . . . , θ n, . . . ), we fabricate distinct coins accordingly, that is, each corresponding to one of the sampled parameters of unbalancedness, and then sequentially throw them, independently from one another. Our main result implies that, if the sequence (ϑn) is stationary and satisfies a law of large numbers, then the long run proportion of heads in the latter scenario will approach Eϑ0. Moreover, we show that the converse is also true: if a stationary sequence of coin throws has the property that the proportion of heads in the first n throws approaches, with certainty, the parameter of unbalancedness, then the coin throws are conditionally independent, where the conditioning is on a sequence of random parameters of unbalancedness satisfying themselves a law of large numbers. As a byproduct stemming from our effort to provide a rigorous proof to Theorem 2.1, we developed the framework of product disintegrations , which provides a model for sequences of random variables that are conditionally independent — but not necessarily identically distributed — thus being a gen-eralization of exchangeability. In this context, we highlight the importance of Theorem 3.7, which constitutes the fundamental step in proving Theorem 2.1 and also yields several examples that illus-trate applications of both mathematical and statistical interest. The paper is organized as follows. In the next section we state our main result, Theorem 2.1, and provide some heuristics connecting our conceptualization to the theory of exchangeable sequences of random variables and to de Finetti’s Theorem. In section 3, we develop the theory in a slightly more general framework, introducing the concept of a product disintegration as a generalization of exchangeability. We then state and prove our auxiliary results, of which Theorem 2.1 is an immediate corollary. Section 4 provides a few examples. 2It is well known that the problem can be translated — although not ipsis litteris — to the language of Ergodic Theory, and there are many characterizations of ergodicity of a dynamical system. The law of large numbers for stationary sequences is indeed implied by the Ergodic Theorem, but the converse implication does not hold in general. A characterization of the strong LLN for Bernoulli sequences 3 2. Main result and its relation to exchangeability We now state our main result. The proof is postponed to section 3. Theorem 2.1. Let X := ( X0, X 1, . . . ) be a sequence of Bernoulli (p) random variables, where 0 ≤ p ≤ 1. Then one has lim n→∞ 1 n n−1 ∑ i=0 Xi = p, almost surely (1) if and only if there exists a sequence ϑ = ( ϑ0, ϑ 1, . . . ) of random variables taking values in the unit interval such that: 1. almost surely, for all n ≥ 0 and all x0, x 1, . . . , x n ∈ { 0, 1} one has P(X0 = x0, . . . , X n = xn \| ϑ) = n ∏ i=0 ϑxi i (1 − ϑi)1−xi , (2) and 2. almost surely, it holds that lim n→∞ 1 n n−1 ∑ i=0 ϑi = p. (3) Remark 2.2 . The above theorem says that a sequence of coin throws has the property that the pro-portion of heads in the first n throws approaches, with certainty, the “parameter of unbalancedness” p ∈ [0 , 1] if and only if the coin throws are conditionally independent, where the conditioning is on a sequence of random parameters of unbalancedness whose corresponding sequence of sample means converges to p. Thus, for sequences of identically distributed Bernoulli (p) random variables, the strong law of large numbers holds precisely when the experiment can be described as the outcome of a two-step mechanism, in which the first step encapsulates dependence and convergence of the sample means, whereas in the second step the random variables are realized in an independent manner. The conditional independence expressed in equation (2) is closely related to the notion of exchange-ability . Recall that a sequence X := ( X0, X 1, . . . ) of random variables is said to be exchangeable iff for every n ≥ 1 and every permutation σ of {0, . . . , n } it holds that the random vectors (X0, . . . , X n) and (Xσ(0) , . . . , X σ(n)) are equal in distribution. An important characterization of exchangeability, de Finetti’s Theorem states that a necessary and sufficient condition for a sequence of random variables to be exchangeable is that it is conditionally independent and identically distributed . To be precise, in the context of a sequence X := ( X0, X 1, . . . ) of Bernoulli (p) random variables, exchangeability is equivalent to existence of a random variable ϑ taking values in the unit interval such that, almost surely, for all n ≥ 0 and all x0, . . . , x n ∈ { 0, 1} one has P (X0 = x0, . . . , X n = xn \| ϑ) = n ∏ i=0 ϑxi (1 − ϑ)1−xi . (4) Moreover, ϑ is almost surely unique and given by ϑ = lim n→∞ n−1 ∑n−1 i=0 Xi. In fact, the above equivalence holds with greater generality — see [6, Theorem 11.10]. 4In view of de Finetti’s Theorem, one is tempted to ask what happens when the random product mea-sure (4) characterizing exchangeable sequences — whose factors are all the same random probability measure — is substituted by an arbitrary random product measure (whose factors are not necessarily the same). This led us to introduce the concept of a product disintegration , which we develop below, and which ultimately provided us with the framework yielding Theorem 2.1. 3. General theory and proof of Theorem 2.1 We now proceed to developing a slightly more general theory — one that will lead us to Theorem 3.7, of which Theorem 2.1 is a corollary. Let us begin by establishing some terminology and notation. In all that follows, S is a compact, metrizable space . We let M1(S) denote the set of Borel probability mea-sures on S. The former is itself a compact metrizable space when endowed with the topology of weak convergence — according to which a sequence (μn) of probability measures converges to a given μ ∈ M1(S) if and only if ∫ f (x) μn(d x) → ∫ f (x) μ(d x), for each continuous function f : S → R.In particular M1(S) admits a Borel σ-field — see Theorem A.3. If (Ω , F , P) is a probability space and ξ : Ω → M1(S) is a Borel measurable mapping, we call ξ a random probability measure on S,whose value (which is a fixed probability measure) at a point ω ∈ Ω we shall denote by ξω and ξ(ω, ·) interchangeably. Meas b(S) denotes the space of measurable, bounded maps from S to R, and C (S) denotes the subspace of Meas b(S) comprised of continuous maps from S to R. Given f ∈ Meas b(S) and μ ∈ M1 (S) we shall write ∫ f (x) μ(d x), μ (f ) and ˆf (μ) interchangeably. If ξ is a random prob-ability measure on S, the baricenter of ξ is defined as the unique element Eξ ∈ M1 (S) such that the equality ∫ Ω ∫ S f (x)ξω(d x)P(d ω) = ∫ S f (x)Eξ(d x) holds for all f ∈ C (S). The baricenter Eξ is also known as the Pettis integral of ξ with respect to P, or as the P-expectation of ξ, and its existence is guaranteed by the Riesz-Markov Theorem A.21. As usual, we write PY for the distribution of a ran-dom variable Y with values in a measurable space M , that is, PY (B) = P (Y ∈ B), for any measurable subset B ⊆ M . In what follows N denotes the set of nonnegative integers. Definition 3.1 (Product Disintegration) . Let X := ( X0, X 1, . . . ) be a sequence of random variables taking values in a compact metric space S. We say that a sequence ξ := ( ξ0, ξ 1, . . . ) of random proba-bility measures on S is a product disintegration of X iff, with probability one, the equality P [X0 ∈ A0, . . . , X n ∈ An \| ξ] = ξ0 (A0) · · · ξn (An) (5) holds for each n ∈ N and each family A0, . . . , A n of measurable subsets of S. If ξ is a stationary sequence, then we say that ξ is a stationary product disintegration .The definition above says that, conditionally on ξ, the sequence X := ( X0, X 1, . . . ) is independent — or, to be more precise, that for almost all elementary outcome ω in the sample space, it holds that the conditional probability P(X ∈ · \| ξ)ω is a product measure on SN. See the standard construction below for more details, where a justification for the terminology disintegration is provided. Also, notice that if ξ is stationary, then clearly X is stationary as well. The following result is an important characterization of product disintegrations. It allows us to work with the seemingly weaker requirement that the identity (5) hold only on a set Ω[ n; A0, . . . , A n] having P-measure 1, for each n ∈ N and each family A0, . . . , A n of measurable subsets of S. Lemma 3.2. Let X := ( X0, X 1, . . . ) be a sequence of random variables taking values in a compact metric space S, and let ξ = ( ξ0, ξ 1, . . . ) be a sequence of random probability measures on S. Then ξ is a product disintegration of X if and only if for each n and each (n + 1) -tuple A0, . . . , A n of measurable subsets of S, the equality (5) holds almost surely. A characterization of the strong LLN for Bernoulli sequences 5 Proof. The ‘only if’ part of the statement is trivial. For the ‘if’ part, let S N denote the product σ-field on SN. By Lemma A.14, S N coincides with the Borel σ-field corresponding to the product topology on SN, and therefore SN is a Borel space. By Theorem A.12, there exists an event Ω∗ ⊆ Ω with P(Ω ∗) = 1 such that A 7 → P(X ∈ A \| ξ)ω is a probability measure on S N for each ω ∈ Ω∗.Now let C := {Ak : k ∈ N} be a countable collection of sets of the form Ak = Bk 0 × · · · × Bkn(k) × S × · · · which generates S N (see Corollary A.15). By assumption, for each k there is an event Ωk ⊆ Ω with P(Ω k) = 1 such that P(X ∈ Ak \| ξ)ω = ξω 0 (Bk 0 ) · · · ξωn(k)(Bkn(k)) holds for ω ∈ Ωk. Thus, for ω ∈ Ω′ := ( ⋂∞ k=0 Ak) ∩ Ω∗, with P(Ω ′) = 1 , the probability measures P(X ∈ · \| ξ)ω and ∏∞ n=0 ξωn agree on a π-system which generates S N, and therefore they agree on S N. This establishes the stated result. Now we prove that product disintegrations always exist: Lemma 3.3. Any sequence X := ( X0, X 1, . . . ) of S-valued random variables admits a product dis-integration. Proof. For n ∈ N and ω ∈ Ω, let ξωn = δXn(ω), where δx is the Dirac measure at x ∈ S. Now fix n ∈ N and let A0, . . . , A n be measurable subsets of S. We first prove that the map ω 7 → ξω 0 (A0) · · · ξωn (An) ≡ I[X0∈A0,...,X n∈An] (ω) (6) is σ (ξ)-measurable and integrable: by Theorem A.1, the maps fAi : M1(S) → R defined by fAi (μ) := μ(Ai), are measurable and thus, by the Doob-Dynkin Lemma A.20, the map ω 7 → ξωi (Ai) = fAi ◦ ξi(ω) is measurable with respect to σ(ξi) ⊆ σ(ξ). Thus (6) defines a σ(ξ)-measurable map, as stated. Moreover, for B ∈ σ (ξ) we have E {ξ0 (A0) · · · ξn (An) IB } = E { I[X0∈A0,...,X n∈An,B ] } = P {X0 ∈ A0, . . . , X n ∈ An, B } , and therefore ξ0 (A0) · · · ξn (An) is a version of P [X0 ∈ A0, . . . , X n ∈ An \| ξ]. Now it is only a matter of applying Lemma 3.2. We shall call the sequence δ = (δX0 , δ X1 , . . . ) appearing in the above lemma the canonical prod-uct disintegration of X. Notice, in particular, that product disintegrations are not unique (see Exam-ple 4.1). Also, it is clear that stationarity of X entails stationarity of δ. We now argue that , without loss of generality, one can take the underlying probability space Ω to be the compact metric space SN ⊗ M1(S)N, endowed with its Borel σ-field F , and equipped with the probability measure defined, for Borel subsets A ⊆ SN and B ⊆ M1(S)N, by P(A × B) = ∫ B ρ(λ, A) Q(d λ) (7) where Q is a probability measure defined on M1(S)N (that is, Q ∈ M1(M1(S)N)) and ρ(λ, A) := (∏ i∈N λi ) (A), λ ∈ M1(S)N, A ⊆ SN measurable . In this construction, the random variables X and ξ can be defined as projections by putting, for ω =(x, λ) ∈ Ω, X(ω) := x and ξ(ω) := λ, where x = ( x0, x 1, . . . ) and λ = ( λ0, λ 1, . . . ). The next lemma ensures that, in the probability space (Ω , F , P), indeed ξ is a product disintegration of X, with Pξ = Q.For convenience, we shall call this the standard construction .6 Lemma 3.4. ρ is a probability kernel from M1(S)N to SN. Proof. It is sufficient to prove that the map λ 7 → ρ(λ, ·) ≡ ∏ i∈N λi from M1(S)N to M1(SN) is measurable. Let (λn) be a sequence in M1(S)N, i.e., for each n, λn = ( λn 0 , λ n 1 , . . . ) with λni ∈ M1(S),for each i, such that lim n→∞ λn = λ = ( λ0, λ 1, . . . ) ∈ M1(S)N; that is, lim n→+∞ λni = λi, for all i. Also, let A = A0 × A1 × · · · × AL × S × S × . . . be an open set in SN. Since lim n→+∞ λni = λi, we know, by the Portmanteau Theorem, that lim inf n→+∞ λni (Ai) ≥ λi(Ai). Now, ρ (λn, A) = (∏ j∈N λnj ) (A) = ∏Lj=0 λnj (Aj ). This implies lim inf n→+∞ ρ (λn, A) = lim inf n→+∞ L ∏ j=0 λnj (Aj ) = L ∏ j=0 lim inf n→+∞ λnj (Aj ) ≥ L ∏ j=0 λj (Aj ) = ρ (λ, A) , which proves that λ 7 → ρ(λ, ·) is continuous and, a fortiori , measurable. Interestingly, the standard construction evinces the fact that the joint law of a sequence of random variables with values in S can always be written as the baricenter of a random product measure on SN.Indeed, as product disintegrations always exist (Lemma 3.3), if we let X = ( X0, X 1, . . . ) be such a sequence (and seeing X as a SN-valued random variable) with product disintegration ξ = ( ξ0, ξ 1, . . . ),then, writing ρ(λ) ≡ ρ(λ, ·), we have PX = E (∏ ∞ n=0 ξn ) = ∫ ρ ◦ ξ(ω) P(d ω) = ∫ ρ(λ) Pξ(d λ) and, of course, Pξ{λ : ρ(λ) is a product measure } = 1 . Moreover, the standard construction justifies the adoption of the terminology product disintegration ; indeed, in this setting the family of probability measures (ηω : ω ∈ Ω) defined on (Ω , F ) via ηω(A × B) := ρ(ξ(ω), A) Iξ∈B ≡ P(A × B \| ξ)ω, for measurable sets A ⊆ SN and B ⊆ M1(S)N, provides a disintegration of P with respect to σ(ξ).See the definition 10.6.1 in and also the proof of Theorem 3.7 for more details. Theorem 2.1 is a direct consequence of Theorem 3.7 below. The ‘if’ part of this proposition is inspired by a similar result that has appeared — albeit in a different framework — in [5, Theorem 1]. 3 Its proof relies on the following disintegration theorem . Theorem 3.5. Let Ω and Λ be compact metric spaces, let P be a Borel probability measure on Ω,and let ξ : Ω → Λ be a Borel mapping. Then there exists a collection (ηλ : λ ∈ Λ) of Borel probability measures on Ω such that 1. the functions λ 7 → ηλ(E) are Borel measurable, for each measurable subset E ⊆ Ω.2. one has ηλ{ω : ξ (ω) 6 = λ} = 0 , for every λ ∈ range( ξ).3. for all measurable subsets E ⊆ Ω and L ⊆ Λ one has P(E ∩ ξ−1(L)) = ∫ L ηλ(E) Pξ(d λ). Proof. This is a direct consequence of Proposition 10.4.12 in . 3The reasoning used by the authors in their proof is essentially the same as the one we apply here, although their statement corresponds to a weak law whereas ours is a strong law. We also made an effort to provide the measure theoretic details in the argument. A characterization of the strong LLN for Bernoulli sequences 7 Remark 3.6 . In the context of the above theorem, it is commonplace to write ηλ(E) =: P(E \| ξ = λ),in which case the above theorem yields the substitution principle , P{ω : g(ω, ξ(ω)) = g(ω, λ) \| ξ = λ} = 1 for all λ ∈ range( ξ) and all measurable functions g defined on Ω × Λ. The probability kernel appearing in the above theorem is essentially unique: indeed, if (ηλ 1 : λ ∈ Λ) is another such kernel, then it is easy to see that ηλ 1 = ηλ for λ on a set of total Pξ-measure. Theorem 3.7. Let X = ( X0, X 1, . . . ) be a sequence of S-valued random variables. Assume ξ =(ξ0, ξ 1, . . . ) is a product disintegration of X, and let f ∈ C(S). Then it holds that lim n→∞ 1 n n−1 ∑ i=0 (f ◦ Xi − ξi(f )) = 0 (8) almost surely. In particular, the limit X∞(f ) := lim n→∞ n−1 n−1 ∑ i=0 f ◦ Xi exists almost surely if and only if the limit ξ∞(f ) := lim n→∞ n−1 n−1 ∑ i=0 ξi(f ) exists almost surely, in which case one has X∞(f ) = ξ∞(f ) almost surely. Remark 3.8 . Notice that, in the theorem above, no additional assumptions are imposed on the product disintegration ξ. In particular, Theorem 3.7 holds when ξ is the canonical product disintegration of X.This is crucial for the ‘only if’ part of Theorem 2.1. Remark 3.9 . For simplicity, we just ask f ∈ C(S) in the statement of Theorem 3.7, and in fact this is all we need in the following results and also in the examples of section 4, but we remark that the result also holds if f is only assumed to be measurable and bounded. The corollary below is an immediate consequence of Theorem 3.7, by taking S as a compact subset of the real line and f as the identity map (in which case ξi(f ) = ∫ S x ξ i(d x) = E(Xi \| ξ)), and shows how the product disintegration can be used to assure the validity of the strong law of large numbers for a sequence of uniformly bounded random variables. Corollary 3.10. Suppose S is a compact subset of the real line, and assume ξ := ( ξ0, ξ 1, . . . ) is a product disintegration of X := ( X0, X 1, . . . ), where the Xi are random variables with values in S. Then the limit X∞ := lim n→∞ n−1 ∑n−1 i=0 Xi exists almost surely if and only if the limit ξ∞ := lim n→∞ n−1 ∑n−1 i=0 E(Xi \| ξ) exists almost surely, in which case X∞ = ξ∞ a.s. If moreover ξω ∞ does not depend on ω (almost surely), then the strong law of large numbers holds for X. Proof of Theorem 3.7. Write Zi := f ◦ Xi − ξi(f ). We have P ( lim n→∞ ∣∣∣n−1 ∑n−1 i=0 Zi ∣∣∣ = 0 ) = E { P ( lim n→∞ ∣∣∣n−1 ∑n−1 i=0 Zi ∣∣∣ = 0 ∣∣∣∣ ξ )} . (9) 8The idea now is that (Zn \| ξ : n ∈ N) is an independent sequence, with E [Zn \| ξ] = 0 and sup n Var ( Zn \| ξ) ≤ 4‖f ‖2 ∞ < ∞, and therefore Kolmogorov’s strong law (Theorem A.22) ensures that, with probability one, the conditional probability inside the expectation in (9) is equal to 1. To make this argument precise, take Ω, P, X and ξ as in the standard construction discussed above, and let (ηλ : λ ∈ M1(S)N) be given as in Theorem 3.5, with Λ = M1(S)N. In this setting it is easy to see that, for E ⊆ Ω of the form E = A × B, with A ⊆ SN and B ⊆ M1(S)N, we have ηλ(E) = ρ(λ, A)IB (λ). Indeed, here we have (A × B) ∩ ξ−1(L) = A × (B ∩ L) and then, by (7), P((A × B) ∩ ξ−1(L)) = ∫ B∩L ρ(λ, A) Pξ(d λ) = ∫ L ρ(λ, A) IB (λ) Pξ(d λ). In particular, ηλ(A × M1(S)) = ρ(λ, A). (10) Now let E = {ω : lim ∣∣∣n−1 ∑n−1 i=0 Zi(ω) ∣∣∣ = 0 }. By Theorem 3.5, we have P(E) = ∫ ηλ(E) Pξ(d λ) and ηλ(E) = ηλ{ω : lim \|n−1 ∑n−1 i=0 f ◦ Xi(ω) − λi(f )\| = 0 }, (11) Thus, writing Aλ = {x ∈ SN : lim \|n−1 ∑n−1 i=0 f (xi) − λi(f )\| = 0 }, we see that the following equal-ity of events holds Aλ × M1(S)N = {ω : lim \|n−1 ∑n−1 i=0 f ◦ Xi(ω) − λi(f )\| = 0 }. Therefore, by (10) and (11), we obtain ηλ(E) = ρ(λ, Aλ) = 1 , where the rightmost equality follows from Kolmogorov’s strong law, as ρ(λ, ·) is the law of a sequence of independent, zero mean random variables with uniformly bounded variances. This establishes (8). The second part of the statement now follows trivially. Proof of Theorem 2.1. Recall that S = {0, 1}. The idea is that in this setting M1(S) is isomorphic to the unit interval. First, notice that given any two probability measures λ, μ ∈ M1(S), we have that λ 6 = μ iff λ{1} 6 = μ{1}. Thus, the mapping λ 7 → f1(λ) := λ{1} is one-to-one from M1(S) onto [0 , 1] .As Theorem A.1 tells us that this mapping is measurable, we can apply Kuratowski’s range and inverse Theorem A.23 to conclude that its inverse is also measurable. For the ‘if’ part of the theorem, let ξωn be the unique probability measure in M1(S) for which ξωn {1} = ϑn(ω), n ∈ N, ω ∈ Ω. The reasoning in the preceding paragraph then tells us that σ(ϑn) = σ(ξn) for all n and consequently σ(ϑ) = σ(ξ). Therefore, we have that P(X ∈ · \| ξ) = P(X ∈ · \| ϑ), which tells us that ξ is a product disintegration of X since the righthandside in this equality is a product measure on SN (with probability 1), by assumption. As we have, again by assumption, that lim n→∞ n−1 ∑n−1 i=0 ξi(f ) = p almost surely, with f = I{1} (which is a continuous function on S), Theorem 3.7 then tells us that (1) holds. For the ‘only if’ part, let now ξ = ( ξ0, ξ 1, . . . ) denote the canonical product disintegration of X, and write ϑn := ξn{1} for all n. It is clear (again using the fact that σ(ξ) = σ(ϑ)) that (2) holds. Also, we have by assumption that lim n→∞ n−1 ∑n−1 i=0 f ◦ Xi = p, with f = I{1} (as Xi = I[Xi=1] ), and since ξi(f ) ≡ ξi{1} = ϑi, Theorem 3.7 tells us that the limit in (3) holds. This completes the proof. A characterization of the strong LLN for Bernoulli sequences 9 4. Examples 4.1. Product disintegrations per se Example 4.1 (Product disintegrations are not (necessarily) unique) . Let ϑ = ( ϑn : n ∈ N) be a sequence of independent and identically distributed random variables, uniformly distributed in the unit interval [0 , 1] , and let, for n ∈ N, ξn be the random probability measure on S := {0, 1} de-fined via ξωn (1) := ϑn(ω), where for simplicity we write ξn(x), x ∈ S, instead of ξn({x}). Assume further that ξ is a product disintegration of a given sequence X of Bernoulli random variables — if necessary, proceed with the standard construction. As argued in the proof of Theorem 2.1, we have σ(ξ) = σ(ϑ) and, in particular, it holds that conditionally on ξ each Xn is a Bernoulli ran-dom variable with parameter ϑn. That is, for each n ∈ N we have P(Xn = 1 \| ξ) = ϑn. Now de-fine ˆξn : Ω → M1(S) by ˆξωn (1) := δXn(ω)(1) = I[Xn=1] (ω), so that ˆξ := ( ˆξ0, ˆξ1, . . . ) is the canonical product disintegration of X. Clearly ξ and ˆξ are different since ˆξωn is equal either to δ{0} or δ{1}, whereas this is not true of ξn. Indeed, for θ ∈ [0 , 1) , we have P(ξn(1) ≤ θ) = P(ϑn ≤ θ) = θ, whereas P( ˆξn(1) ≤ θ) = P(I[Xn=1] ≤ θ) = P(I[Xn=1] = 0) = P(Xn = 0) . Example 4.2 (Random Walk as a two-stage experiment with random jump probabilities) . In the same setting as Example 4.1, let Zn := 2 Xn − 1, n ∈ N. Clearly Z = ( Z0, Z 1, . . . ) is an independent and identically distributed sequence of standard Rademacher random variables, i.e., for each n ∈ N it holds that P(Zn = +1) = P(Zn = −1) = 1/2. Indeed, for any x0, x 1, . . . , x n ∈ { 0, 1}, we have P(Z0 =2x0 − 1, . . . , Z n = 2 xn − 1) = P (X0 = x0, . . . , X n = xn) = E ∏nj=0 ξj (xj ) = ∏nj=0 Eξj (xj ), where the last equality follows from the assumption that the ϑn’s are independent. Moreover, Eξj (xj ) = 1 /2 since the left-hand side in this equality is either Eϑj or 1 − Eϑj . Now let S0 := 0 and Sn = Z0 + · · · + Zn−1 for n ≥ 1. By the above derivation, (Sn : n ≥ 0) is the symmetric random walk on Z.Therefore, although — unconditionally — at each step the process (Sn) jumps up or down with equal probabilities, we have that conditionally on ξ it evolves according to the following rule: at step n,sample a Uniform [0 , 1] random variable ϑn independent of anything that has happened before (and of anything that will happen in the future), and go up with probability ϑn, or down with probability 1 − ϑn. Example 4.3 . Let X = ( X0, X 1, . . . ) be an exchangeable sequence of Bernoulli (p) random variables. In particular, X satisfies equation (4) for some random variable ϑ taking values in the unit interval. Then, defining the random measures ξn via ξn({1}) := ϑ for all n, it is clear that (ξ0, ξ 1, . . . ) =: ξ is a stationary product disintegration of X — again using the fact that σ(ξ) = σ(ϑ). In particular, in this scenario, an unconditional strong law of large numbers does not hold for X, unless when ϑ is a constant. See also Theorem 2.2 in , which provides a characterization of the strong law for the class of integrable, exchangeable sequences. This example illustrates that the existence of a product disintegration is not sufficient for the law of large numbers to hold (indeed, by Proposition 3.3, any sequence of random variables admits a product disintegration). Example 4.4 (Concentration inequalities) . One important consequence of the notion of a product dis-integration is that it allows us to easily translate certain concentration inequalities (such as the Chernoff bound, Hoeffding’s inequality, Bernstein’s inequality, etc) from the independent case to a more general setting. Recall that the classical Hoeffding inequality says that, if X = ( X0, X 1, . . . ) is a sequence of independent random variables with values in [0 , 1] , then one has the bound P (Sn ≥ t) ≤ exp (−2t2/n ) for all t > ESn, where Sn := ∑n−1 i=0 Xi.10 Theorem 4.5 (Hoeffding-type inequality) . Let X = ( X0, X 1, . . . ) be a sequence of random variables with values in the unit interval S := [0 , 1] , and let ξ = ( ξ0, ξ 1, . . . ) be a product disintegration of X.Then, for any t > 0, it holds that P (Sn ≥ t \| E(Sn \| ξ) < t ) ≤ exp (−2t2/n ) , where Sn := ∑n−1 i=0 Xi. Proof. From the classical Hoeffding inequality applied to the probability measures P(· \| ξ)ω, we have P (Sn ≥ t \| ξ) I{E(Sn \| ξ)<t } ≤ exp (−2t2/n ) I{E(Sn \| ξ)<t }. Taking the expectation on both sides of the above inequality, and dividing by P(E(Sn \| ξ) < t ), yields the stated result. Notice that if ξ is the canonical product disintegration of X, then the above theorem is not very useful: indeed in this case we have E(Sn \| ξ) = Sn, so the left-hand side in the inequality is zero. The above theorem also tells us that, for t > 0, P(Sn ≥ t) = P(Sn ≥ t ∣∣ E(Sn \| ξ) < t ) × P(E(Sn \| ξ) < t ) P(Sn ≥ t ∣∣ E(Sn \| ξ) ≥ t) × P(E(Sn \| ξ) ≥ t) ≤ exp ( − 2t2 n ) P(E(Sn \| ξ) ≥ t) so the rate at which P(Sn ≥ t) → 0 as t → ∞ is governed by the rate at which P (E(Sn ∣∣ ξ) ≥ t) → ∞ as t → ∞ . To illustrate, let us consider two extreme scenarios, one in which ξn = ξ0 for all n (so that X is exchangeable) and one in which the ξn’s are all mutually independent: in the first case, we have that E(Sn \| ξ) = n ∫ 10 x ξ 0(d x), and thus the rate at which P(Sn ≥ t) → 0 as t → ∞ depends only on the distribution of the random variable ∫ 10 x ξ 0(d x). On the other hand, if the ξn’s are independent, then we have that E(Sn \| ξ) = ∑n−1 i=0 ∫ 10 x ξ n(d x), and in this case the summands are independent random variables with values in the unit interval. Therefore, we can apply the classical Hoeffding inequality to these random variables to obtain the upper bound P(Sn ≥ t) ≤ 2 exp( −2t2/n ) for t > ESn (in fact, we already know that the upper bound exp( −2t2/n ) holds, since independence of the ξn’s entails independence of the Xn’s). Example 4.6 . Let S := [ a, b ]d where d is a positive integer and a < b ∈ R. Given a sequence X = ( X1, X 2, . . . ) of S-valued random variables, we shall write Xn = ( X1 n , . . . , X dn). Suppose ξ = ( ξ0, ξ 1, . . . ) is a product disintegration of X. Equation (5) then yields, for all measurable sets Aji ⊆ [a, b ], with i ∈ { 0, . . . , n } and j ∈ { 1, . . . , d }, the equality P(X10 ∈ A10, . . . X d 0 ∈ Ad 0 , . . . , X 1 n ∈ A1 n , . . . , X dn ∈ Adn \| ξ) = ξ0(A10 ×· · ·× Ad 0 ) · · · ξn(A1 n ×· · ·× Adn). An identity as above appears naturally in statistical applications, for instance when one observes sam-ples of size d, (X1 n , . . . , X dn), n = 0 , 1, . . . , from distinct “populations” ξ0, ξ 1, . . . — we refer the reader to and references therein for details. 4.2. Convergence Example 4.7 (Regime switching models) . Let S = {− 1, 1} and put M ′ := {μ, λ } ⊆ M1(S) with μ(1) > λ (1) . The measures μ and λ are to be interpreted as 2 distinct “regimes” (for example, expan-sion and contraction, in which case one would likely assume μ(1) > 1/2 > λ (1) ). Let (Qij : i, j ∈ M ′) be a row stochastic matrix with stationary distribution π = ( πμ, π λ). Let ξ := ( ξ0, ξ 1, . . . ) be a A characterization of the strong LLN for Bernoulli sequences 11 Markov chain with state space M ′, initial distribution π and transition probabilities (Qij ). Notice that Eξn = μπ μ + λπ λ for all n.Assume X := ( X0, X 1, . . . ) is a sequence of S-valued random variables and that ξ is a product disintegration of X. Then we have, for x ∈ {− 1, 1}, that P(Xn = x) = Eξn(x) = μ(x)πμ + λ(x)πλ.We also have, for x0, x 1 ∈ {− 1, 1}, P(X0 = x0, X 1 = x1) = Eξ0(x0)ξ1(x1)= μ(x0)μ(x1)πμQμμ + μ(x0)λ(x1)πμQμλ λ(x0)μ(x1)πλQλμ + λ(x0)λ(x1)πλQλλ (12) This shows that in general it may be difficult to compute the finite-dimensional distributions of the process (X0, X 1, . . . ) — although this process inherits stationarity from ξ. Also, an easy check tells us that generally speaking X is not a Markov chain. Nevertheless, assuming Q is irreducible and positive recurrent (i.e., πμ /∈ { 0, 1}), we have by the ergodic theorem for Markov chains, that lim n→∞ 1 n n−1 ∑ k=0 h ◦ ξk = πμh(μ) + πλh(λ) = Eh ◦ ξ0, a.s, (13) for any bounded h : M ′ → R. Now let f : S → R and consider the particular case where h ◦ ξ := ξ(f ).Equation 13 becomes lim n→∞ 1 n n−1 ∑ k=0 ξk(f ) = πμμ(f ) + πλλ(f ) = Eξ0(f ), a.s. (14) Therefore, using Theorem 3.7 and then (14), we have that lim n→∞ 1 n n−1 ∑ k=0 f ◦ Xk = Eξ0(f )= μ(f )P[ξ0 = μ] + λ(f )P[ξ0 = λ]= ( f (1) μ(1) + f (−1) μ(−1)) πμ + ( f (1) λ(1) + f (−1) λ(−1)) πλ = f (1) (μ(1) πμ + λ(1) πλ ) + f (−1) (μ(−1) πμ + λ(−1) πλ ) holds almost surely. In particular this is true with f = I{1}; thus, even though the ‘ups and downs’ of X are governed by a law which can be rather complicated (as one suspects by inspecting equation (12)), we can still estimate the overall (unconditional) probability of, say, the expansion regime by computing the proportion of ups in a sample (X0, . . . , X n): lim n→∞ 1 n n−1 ∑ k=0 I{1}(Xk) = μ(1) πμ + λ(1) πλ. Example 4.8 . Suppose ϑ = ( ϑ0, ϑ 1, . . . ) is a submartingale, with range( ϑn) ⊆ [0 , 1] for all n. By the Martingale Convergence Theorem, there exists a random variable ϑ∞ such that lim ϑn = ϑ∞ almost surely (thus, we can assume without loss of generality that 0 ≤ ϑ∞ ≤ 1). Furthermore, let S := {0, 1}12 0 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 n ϑn, Xn and ¯Xn Figure 1 . A sample path of the submartingale (ϑn). and, for n ∈ N, let ξn denote the random probability measure on S defined via ξn({1}) = ϑn, and ξn({0}) = 1 − ϑn. We have ξn(I{1}) = ϑn → ϑ∞ a.s. Assume further that ξ = ( ξ0, ξ 1, . . . ) is a prod-uct disintegration of a sequence X = ( X0, X 1, . . . ) of random variables with values in S. Using The-orem 3.7 we have lim n→∞ 1 n n−1 ∑ i=0 I{1}(Xi) = lim n→∞ 1 n n−1 ∑ i=0 ξi(I{1}) = lim n→∞ 1 n n−1 ∑ i=0 ϑi = ϑ∞ a.s. which means, the proportion of 1’s in (X0, . . . , X n) approaches ϑ∞ with probability one. To illustrate, let (Un : n ≥ 1) be a sequence of independent and identically distributed Uniform [0 , 1] random variables. Let ϑ0 = U0/2 and, for n ≥ 1, define ϑn := ϑn−1 + 2 −(n+1) Un. Figure 1 displays, in blue, a simulated sample path of the submartingale (ϑn : n ∈ N) up to n = 20 . The ◦’s represent the successive outcomes of the coin throws (where the probability of ‘heads’ in the nth throw is ϑn). In purple are displayed the sample path of the means (n−1Sn : n ∈ N), where Sn is the partial sum ∑n−1 i=0 Xn. In this model, even if we only observe the outcomes of the coin throws, we can still estimate the value of ϑ∞: all we need to do is to compute the proportion of heads in X0, . . . , X n, with n large. Example 4.9 . We now show that a certain class of stochastic volatility models can be accommodated into our framework of product disintegrations. Stochastic volatility models are widely used in the financial econometrics literature, as they provide a parsimonious approach for describing the volatility dynamics of a financial asset’s return — see and and references therein for an overview. A basic specification of the model 4 is as follows: let Z = ( Zt : t ∈ Z) and W = ( Wt : t ∈ Z) be centered iid 4Which can be relaxed by putting g◦Htin place of eHt/2, and allowing Hto evolve according to more flexible dynamics. A characterization of the strong LLN for Bernoulli sequences 13 sequences, independent from one another, and define X and H via the stochastic difference equations Xt = e Ht/2Zt, t ∈ N, and Ht = α + βH t−1 + Wt, t ≥ 1, where α and β are real constants and where H0 follows some prescribed distribution. The random vari-able Xt is interpreted as the return (log-price variation) on a given financial asset at date t, and the Ht’s are latent (i.e, unobservable) random variables that conduct the volatility of the process X. Usually this process is modelled with Gaussian innovations, that is, with Wt and Zt normally distributed for all t.In this case the random variables Xt are supported on the whole real line, so we need to consider other distributions for Z and W if we want to ensure that the Xt’s are compactly supported. Our objective is to show how Theorem 3.7 can be used to estimate certain functionals of the latent volatility process H in terms of the observed return process X. To begin with, notice that if \|β\| < 1 and if H0 is defined via the series H0 := (1 − β)−1α + ∑∞ k=0 βkW−k, then H (and X) is strictly stationary and ergodic, in which case we have that lim n→∞ 1 n n−1 ∑ t=0 g ◦ Ht = Eg ◦ H0 (15) almost surely, for any PH0 -integrable g : SH → R, where we write SH := supp H0. Also, no-tice that, by construction, we have for all n, all measurable A0, . . . , A n ⊆ S := supp X0 and all h = ( h0, h 1, . . . ) ∈ SN H , P(X0 ∈ A0, . . . , X n ∈ An \| H = h) = P(eH0/2Z0 ∈ A0, . . . , eHn/2Zn ∈ An \| H = h) (∗) = P(eh0/2Z0 ∈ A0, . . . , ehn/2Zn ∈ An \| H = h) (∗∗ ) = P(eh0/2Z0 ∈ A0, . . . , ehn/2Zn ∈ An ) = n ∏ t=0 P(eht/2Zt ∈ At ) (∗∗∗ ) = n ∏ t=0 P(Xt ∈ At \| H = h). (16) Where (∗) is yielded by the substitution principle, (∗∗ ) follows from the fact that Z and H are in-dependent (as H only depends on W ), and (∗ ∗ ∗ ) is just a matter of repeating the previous steps. A reasoning similar to the one used in the proof of Lemma 3.2 then tells us that P(X ∈ · \| H)ω is a prod-uct measure on SN for almost all ω. Also, notice that in particular we have that P(Xt ∈ A \| H = h) = P(e ht/2Zt ∈ A) for all t. In fact, let ϕ : SH → M1(S) be defined via ϕ(h, A ) := P(e h/ 2Z0 ∈ A), for h ∈ SH and measurable A ⊆ S, where we write ϕ(h, A ) in place of ϕ(h)( A) for convenience. Since the Zt’s are identically distributed, we have in particular that ϕ(h, A ) = P(e h/ 2Zt ∈ A) for all t. The preceding derivations now allow us to conclude that ϕ(Ht(ω), A ) = P(Xt ∈ A \| Ht)ω = P(Xt ∈ A \| H)ω. (17) We are now in place to introduce a product disintegration of X, by defining ξωt (A) := P(Xt ∈ A \| H)ω for measurable A ⊆ S, t ∈ Z and ω ∈ Ω. To see that ξ = ( ξ0, ξ 1, . . . ) is indeed a product disintegration of X, first notice that ξ0(A0) · · · ξn(An) is σ(ξ)-measurable for every n and every 14 (n + 1) -tuple A0, . . . , A n of measurable subsetes of S. Moreover, defining ψ : SN H → M1(S)N via ψ(h0, h 1, . . . ) = ( ϕ(h0), ϕ (h1), . . . ), we obtain, by equations (16) and (17), E(ξ0(A0) · · · ξn(An)I[ξ∈B] ) = E(ϕ(H0, A 0) · · · ϕ(Hn, A n)I[H∈ψ−1(B)] ) = P(X0 ∈ A0, . . . , X n ∈ An, H ∈ ψ−1(B)) = P(X0 ∈ A0, . . . , X n ∈ An, ξ ∈ B), whence P(X0 ∈ A0, . . . , X n ∈ An \| ξ) = ξ0(A0) · · · ξn(An), and then Lemma 3.2 tells us that ξ is — voilà — a product disintegration of X.Now, since ϕ is continuous and one-to-one, we have that ϕ is a homeomorphism from SH onto its range whenever SH is compact (in particular, range( ϕ) is compact, hence measurable, in M1(S)). Also, as ξt = ϕ ◦ Ht for all t, we have that Ht = ϕ−1 ◦ ξt is well defined. Suppose now that f : S → R is a given continuous function. We have ξωt (f ) = ∫ S f (x) ξωt (d x) = ∫ S f (x) ϕ(Ht(ω), dx) =: g(Ht(ω)) and, as H is ergodic, it holds that lim n→∞ n−1 ∑n−1 t=0 ξt(f ) = Eg ◦ H0, where we know that the expectation is well defined, as E\|g ◦ H0\| ≤ E (∫ S \|f (x)\| ξ0(d x)) < ∞, with the expected value given by Eg ◦ H0 = ∫ S f (x) PX0 (d x). We can now apply Theorem 3.7 to see that Eg ◦ H0 = lim n→∞ n−1 n−1 ∑ t=0 f ◦ Xt. The conclusion is that, for suitable g of the form g(h) = ∫ S f (x) ϕ(h, dx), we can estimate Eg ◦ H0 by the data (X0, X 1, . . . , X n) as long as n is large enough, even if we cannot observe H. Of course, this follows from ergodicity of X, but it is interesting anyway to arrive at this result from an alternate perspective; moreover, one can use Hoeffding type inequalities as in Example 4.4 to easily derive a rate of convergence for sample means of X based on the rate of convergence of sample means of H. 5. Concluding remarks In this paper we prove that a sequence of Bernoulli (p) random variables satisfies the strong law of large number if and only if the sequence is conditionally independent, where the conditioning is on a sequence of [0 , 1] -valued random variables, whose corresponding sequence of sample means converges almost surely. As a byproduct, we introduce the concept of a product disintegration , which generalizes exchangeability. Some applications of the concept are illustrated in Section 4. Further applications of product disintegrations and of Theorem 3.7 appear as a possible path to be pursued in future work. A road not taken. At some point, during the development of the present paper, we delved into the pos-sibility of translating our approach to the language of Ergodic Theory. This proved more difficult than we first thought, but we did come up with a conjecture: consider the left–shift operator T acting on SN,given by (T x)i = xi+1 for x = ( x0, x 1, . . . ) ∈ SN, and define ˜T : M1 (S)N → M1 (S)N analogously. Recall that ρ(λ) := ∏ i∈N λi for λ ∈ M1(S)N. Conjecture 1. Let S be a compact metric space. A T -invariant measure q ∈ M1(SN) is T –ergodic if and only if there exists a ˜T –ergodic measure Q ∈ M1(M1 (S)N) such that q = ∫ ρ (λ) Q (d λ).A characterization of the strong LLN for Bernoulli sequences 15 Auxiliary results Given a topological space S, we will write B ≤ S to mean that B belongs to the σ-field generated by the topology of S (i.e., that B is a Borel subset S). A.1. Spaces of measures For a compact metric space S endowed with its Borel σ-field, let Meas b(S) denote the set of measur-able, bounded maps f : S → R, let C(S) ⊆ Meas b(S) denote the set of continuous maps from S to R, and let M (S) denote the set of finite Borel measures on S. As in the main text, M1(S) ⊆ M (S) denotes the set of Borel probability measures on S. For f ∈ Meas b(S), we define the evaluation map ˆf : M (S) → R by ˆf (μ) := ∫ f (x) μ(d x), for μ ∈ M (S).There are a few manners through which one can introduce a σ-field on M (S) (and, a fortiori, on M1(S)). The most commonly adopted approach is to consider in M (S) the weak topology relative to C(S) (in conventional probabilistic jargon, this is simply called the weak topology), that is, the coarsest topology on M (S) for which, for every f ∈ C(S), its evaluation map ˆf is continuous. The following theorem presents some very useful results. Item 2 is related to Prokhorov’s compactness criterion, but is not restricted to probability measures. The last three items show that, if the aim is to obtain a σ-field in M (S), there is no need for topological considerations (on M (S)). Theorem A.1. Let S be a compact metric space. Then 1. M (S) is Polish (i.e. is separable and admits a complete metrization) in the weak topology. 2. A set K ⊆ M (S) is weakly relatively compact if and only if sup μ∈K ˆf (μ) < ∞ for all nonneg-ative f ∈ C(S).3. The Borel σ-field relative to the weak topology on M (S) coincides with the σ-field σ(C ), where C can be taken as any one of the following classes: i. C = { ˆf : f ∈ C(S), f ≥ 0}. ii. C = { ˆf : f = IB , B ≤ S}. iii. C = { ˆf : f ∈ Meas b(S)}. Remark A.2 . In summary, item (3) above says the following: if we write τ ( ˆf : f ∈ C(S)) for the topology on M1(S) generated by the mappings ( ˆf : f ∈ C(S)) (that is, the weak topology), then σ ( τ ( ˆf : f ∈ C(S))) = σ ( ˆf : f ∈ C(S) ) , etc. Proof. For the first two items, and sub-items i. and ii. of the last item, see Theorem A2.3 in . The proof will be complete once we establish the identity σ{ ˆf : f ∈ Meas b(S)} = σ{ ˆf : f = IB , B ≤ S}. Clearly the inclusion σ{ ˆf : f ∈ Meas b(S)} ⊇ σ{ ˆf : f = IB , B ≤ S} holds. For the converse inclusion, it is enough to show that, for every g ∈ Meas b(S), one has ˆg ∈ σ{ ˆf : f = IB , B ≤ S} =: ˜B. If g = IB for some B ≤ S, then clearly ˆg ∈ ˜B. If g is simple, with standard represen-tation g(x) = ∑nj=1 aj IAj (x), then ˆg(λ) = ∑nj=1 aj ˆgj (λ), where gj = IAj . Thus, ˆg ∈ ˜B as it is a lin-ear combination of elements of ˜B. For the general g ∈ Meas b(S), let (gn) be a sequence of simple func-tions with \|gn\| ≤ \| g\| and gn → g. Then the Dominated Convergence Theorem gives ˆg(λ) = lim ˆ gn(λ) and hence ˆg ∈ ˜B, which concludes the result. 16 Since M1(S) = {μ ∈ M (S) : ˆf (μ) = 1 } = ˆf −1({1}), with f = IS ∈ C(S), clearly M1(S) is a weakly closed (hence measurable) subset of M (S). By item 2 in Theorem A.1, M1(S) is weakly compact. Indeed, more can be said: M1(S) is a compact metrizable space. Usually, this fact is stated in terms of the so called Lévy-Prokhorov metric, which works for quite general S but suffers from a “lack of interpretability”. Conveniently, when S is compact there is an equivalent metric generating the weak topology, given by d(μ, ν ) = ∑ k≥1 2−k ∣∣∣ ˆfk(μ) − ˆfk(ν) ∣∣∣ , where {fk}k≥1 is a dense and countable subset of the unit ball in C(S). The following result is an immediate corollary to Theorem 8.3.2 in : Theorem A.3. The weak topology on M1(S) is metrizable. A.2. Random probability measures Definition A.4 . A random probability measure on S is defined to be a Borel measurable map ξ : Ω → M1(S). We shall denote the value of a random probability measure ξ at the point ω by ξω and, for a Borel subset B ⊆ S, we will use the notation ξω(B) and ξ(ω, B ) undistinguishedly. The latter notation is justified in Theorem A.10 below. Lemma A.5 (measurability of ξ). A map ξ : Ω → M1(S) is measurable if and only if the map ω 7 → ∫ f dξω = ˆf ◦ ξ(ω) is a random variable for every f ∈ C , where C can be taken as any one of the sets C(S), Meas b(S) or {IB : B ≤ S}. Proof. By Theorem A.1, the Borel σ-field on M1(S) is given by σ{ ˆf : f ∈ C(S)} = σ{ ˆf : f ∈ Meas b(S)} = σ{ ˆf : f = IB , B ≤ S}. The ‘only if’ part follows immediately. For the ‘if’ part, notice that σ{ ˆf : f ∈ C } is the smallest σ-field containing the sets ˆf −1(E), with f ∈ C and E ≤ R. Now ˆf ◦ ξ is measurable for every f ∈ C iff ( ˆf ◦ ξ)−1(E) ∈ F for every f ∈ C and every E ≤ R iff ξ−1(G) ∈ F for every G of the form ˆf −1(E) with f ∈ C and E ≤ R. Since the class of such G generates the Borel σ-field on M1(S), the result follows. Theorem A.6 (existence of Baricenter) . Let (Ω , F , P) be a probability space, S a compact metric space, and let ξ be a random probability measure on S. Then there exists a unique element ¯μ ∈ M1(S) such that the equality ∫ S f (x) ¯ μ(d x) = ∫ Ω ∫ S f (x) ξω(d x) P(d ω) holds for all f ∈ C(S). Proof. Let ϕ : C(S) → R be defined by ϕ(f ) := ∫ Ω ∫ S f (x) ξω(d x) P(d ω). Clearly ϕ(f ) ≥ 0 if f ≥ 0, ϕ(αf + g) = αϕ (f ) + ϕ(g), and ϕ(1) = 1 . Thus, by the Riesz-Markov Theorem, there is an element ¯μ ∈ M1(S) such that the stated equality holds. Definition A.7 . The unique element ¯μ yielded by Theorem A.6 is called the baricenter of ξ (analo-gously: the P-expectation of ξ; analogously: the baricenter of Pξ). Notation: ¯μ =: ∫ ξ dP =: EPξ. We also write simply Eξ in place of EPξ when P is understood from context. A characterization of the strong LLN for Bernoulli sequences 17 Lemma A.8. Let ξ be a random probability measure on S and let Eξ be its baricenter. Then 1. (commutativity) For each measurable subset B ⊆ S, the equality Eξ (B) = E (ξ (B)) holds; 2. (maximal support) there exists a set Ω0 with P (Ω 0) = 0 such that, for ω / ∈ Ω0, the relation supp ξω ⊆ supp Eξ holds. Proof. For the first item, let λ(B) := E(ξ(B)) , B ⊆ S measurable. Clearly we have λ(B) ≥ 0 and λ(Ω) = 1 . Moreover, if (Bj ) is a sequence of measurable subsets of S which are pairwise disjoint such that B = ⋃∞ j=1 Bj , then for each ω we have ξω(B) = lim n→∞ ∑nj=1 ξω(Bj ) ≤ 1. Thus, by the Dominated Convergence Theorem (DCT), we have λ(B) = ∑∞ j=1 λ(Bj ). Therefore λ is a probability measure on S. Now let K ⊆ S be closed and let (fn) be a sequence of continuous functions on S such that 1 ≥ fn(x) → IK (x), x ∈ S. On the one hand we have Eξ(K) = lim n→∞ ∫ fn(x) Eξ(d x),by DCT. On the other hand, for each ω it holds that 0 ≤ ξω(K) = lim n→∞ ∫ fn(x) ξω(d x) ≤ ‖ fn‖∞ ≤ 1, again by DCT. Applying the DCT once more yields λ(K) = lim n→∞ ∫ ∫ fn(x) ξω(d x) P(d ω) = lim n→∞ ∫ fn(x) Eξ(d x) = Eξ(K), where the second equality follows from the definition of the baricenter. Thus, Eξ and λ are measures on S whose values on closed sets coincide, and this implies Eξ = λ, as asserted. For the second item, let U := S \ supp ( Eξ). Then ξ (U ) ≥ 0 and E (ξ (U )) = Eξ (U ) = 0 , by item 1. Hence ξ (U ) = 0 almost surely. A.3. Probability Kernels Definition A.9 (see , page 20) . Given two measurable spaces (Ω , F ) and (S, S ), a map ξ : Ω × S → R is said to be a probability kernel from (Ω , F ) to (S, S ) iff D1 For each ω ∈ Ω, the map B → ξ(ω, B ) is a probability measure on S.D2 For each B ≤ S, the map ω 7 → ξ(ω, B ) is F -measurable. Kernels play an important role in probability theory, appearing in many forms, for example random measures, conditional distributions, Markov transition functions, and potentials . Indeed, in many circumstances, one feels more comfortable working with probability kernels instead of random prob-ability measures as defined above, given the prevalence of the former concept in the literature. The following result connects the two concepts, showing that they are indeed equivalent: Theorem A.10. Fix two measurable spaces (Ω , F ) and (S, S ), and assume S = σ(C ) for some π-system C . Let ξ : Ω × S → R be such that ξ(ω, ·) is a probability measure on S, for every ω ∈ Ω.Then the following conditions are equivalent: 1. ξ is a probability kernel from (Ω , F ) to (S, S ).2. ω 7 → ξ(ω, ·) is an F -measurable mapping from Ω to M1(S).3. ω 7 → ξ(ω, E ) is an F -measurable mapping from Ω to [0 , 1] for every E ∈ C .In particular, the above equivalences hold with C = F .18 Proof. This is just a restatement of Lemma 1.40 in , by noticing that the Borel σ-field on M1(S) coincides with σ( ˆf : f = IB , B ≤ S) as ensured by Theorem A.1. Definition A.11 . A kernel ξ from (Ω , F0) to (S, S ) is said to be a regular conditional distribution of a random variable X : Ω → S given a σ-field F0 ⊆ F iff the equality ∫ F ξ(ω, B ) P(d ω) = P([ X ∈ B] ∩ F ) holds for all B ≤ S and F ∈ F0. In particular, for each B ≤ S the random variable ξ(·, B ) is a version of P(X ∈ B \| F0). Theorem A.12 (Regular conditional distribution — , Theorem 6.3) . Let (S, S ) and (T, T ) be measurable spaces, and let X and ξ be random variables taking values in S and T respectively. Assume further that S is Borel. Then there exists a probability kernel η from T to S such that P(X ∈ B \| ξ)ω = η(ξ(ω), B ) for all B ∈ S and all ω in a set Ω∗ ⊆ Ω with P(Ω ∗) = 1 . Moreover, η is unique almost everywhere-Pξ. Remark A.13 . In the conditions of the above Theorem, one can introduce a probability kernel η′ from (Ω , σ (ξ)) to (S, S ) by putting η′(ω, B ) := η(ξ(ω), B ). Also, if ξ is the identity map and (T, T ) = (Ω , F0), where F0 ≤ F , then automatically η is a kernel from Ω to S which is a regular version of P(X ∈ · \| F0). A.4. Product spaces Lemma A.14 (Product and Borel σ-fields — , Lemma 1.2) . Let S have topology τ and let S := σ (τ ) be the Borel σ-field on S. Let τ N be the product topology on SN and let S N be the product (cylindrical) σ–field on SN. If S is metrizable and separable, then σ(τ N) = S N, that is, S N is the Borel σ-field on SN. Corollary A.15. Let S be a separable metric space with topology τ , and let B be a countable basis for τ which is stable under finite intersections. For each n ∈ N and each B0, . . . , B n ∈ B, define C (n; B0, . . . , B n) := B0 × · · · × Bn × S × · · · ⊆ SN. (18) Let C denote the collection of all sets of the form (18) . Then C is a countable π-system which generates the Borel σ-field on SN. Remark A.16 . The set B above can be obtained as follows: let D be a countable, dense subset of S, and let D be the collection of all balls with centers in D and rational radii. Now let Bn, n ≥ 1, be the collection formed by all intersections of n elements of D, that is, B ∈ Bn iff there exist x1, . . . , x n ∈ D and r1, . . . , r n ∈ Q such that B = ⋂ni=1 ball( xi; ri). Clearly, each Bn is countable. Now let B := ⋃ n≥1 Bn. Proof of Corollary A.15. We begin by proving that C is indeed a π-system. Clearly, C is non-empty. Now, let A0, . . . , A m, B 0, . . . , B n ∈ B and consider C (m; A0, . . . , A m), C (n; B0, . . . , B n). Without loss of generality, suppose n ≥ m. Then C (m; A0, . . . , A m) ∩ C (n; B0, . . . , B n) = A0 ∩ B0 × · · · × Am ∩ Bm × · · · × Bn × S × . . . = C (n; A0 ∩ B0, . . . , A m ∩ Bm, . . . , B n).A characterization of the strong LLN for Bernoulli sequences 19 Since B is stable under finite intersections, Ai ∩ Bi ∈ B, for each i ∈ { 0, . . . , m }, and the result follows. It remains to show that C generates the Borel σ-field on SN. Clearly any A ∈ C is a Borel set in SN. For the reverse inclusion, by Lemma A.14 and the facts that σ(τ ) = σ(B) and σ(τ N) = σ(BN), it suffices to prove that given {Ai}+∞ i=0 a sequence of elements in B, it holds that A = A0 × A1 × · · · × An × An+1 × · · · ∈ σ(C). For each m ∈ N, define A(m) = A0 × · · · × Am × S × . . . , i.e., A(m) = C (m; A0, . . . , A m). Surely, for each m ∈ N, A(m) ∈ σ(C). Furthermore, note that A = ∩+∞ m=0 A(m),so that A ∈ σ(C). A.5. Additional auxiliary results Definition A.17 . Two measurable spaces (M, M ) and (N, N ) are said to be Borel isomophic if there exists a bijection h : M → N such that both h and h−1 are measurable. A measurable space (M, M ) is said to be a Borel space if it is Borel isomorphic to a Borel subset of the interval [0 , 1] . Definition A.18 . A topological space M is said to be a Polish space iff it is separable and admits a complete metrization. Theorem A.19 (, Theorem A1.2) . Let M be a Polish space. Then every Borel subset of M is a Borel space. Lemma A.20 (Doob-Dynkin Lemma — , Lemma 1.13) . Let (M, M ) and (N, N ) be measurable spaces, and let f : Ω → M and g : Ω → N be any two given functions. If M is Borel, then f is σ(g)-measurable if and only if there exists a measurable mapping h : N → M such that f = h ◦ g, Theorem A.21 (Riesz-Markov) . Let S be a locally compact Hausdorff space and ϕ a positive linear functional on Cc(S). Then there is a unique Radon measure μ on the Borel σ-field of S for which ϕ(f ) = ∫ S f (x) μ(d x) for all f ∈ Cc(S). In particular, if S is compact and ϕ(1) = 1 , then μ is a probability measure. Theorem A.22 (Kolmogorov’s strong law of large numbers) . Let X := ( X0, X 1, . . . ) be an indepen-dent sequence of random variables such that sup n Var ( Xn) < ∞. Then it holds that lim n→∞ n−1 n−1 ∑ i=0 (Xi − EXi) = 0 almost surely. Theorem A.23 (range and inverse, Kuratowski — , Theorem A1.3) . Let f be a measurable bijec-tion between two Borel spaces S and T . Then the inverse f −1 is again measurable. Acknowledgements The author Luísa Borsato is supported by grant 2018/21067-0, São Paulo Research Foundation (FAPESP). The author Eduardo Horta wishes to thank MCTIC/CNPq (process number 438642/2018-0) for fi-nancial support. 20 References Bernoulli, J. (2005). On the law of large numbers . [Translation by O.B. Sheynin into English of the Pars Quarta of Ars Conjectandi.] Available at www.sheynin.de/download/bernoulli. pdf Bogachev, V. (2007) Measure Theory , Springer-Verlag, Berlin. doi:10.1007/ 978-3-540-34514-5 . Cardano, G. (2015) The book on games of Chance: the 16th-century treatise on probability . Courier Dover Publications. Davis, R. and Mikosch, T. (2009) Probabilistic Properties of Stochastic Volatility Models. In T. Mikosch, J.P. Kreiß, R. A. Davis and T. G. Andersen (eds.), Handbook of Financial Time Series (pp. 255–267). Berlin/Heidelberg: Springer. doi:10.1007/978-3-540-71297-8_11 . Horta, E. and Ziegelmann, F. (2018) Conjugate processes: Theory and application to risk fore-casting, Stochastic Processes and their Applications 128 (3) 727–755. doi:10.1016/j.spa. 2017.06.002 . Kallenberg, O. (2002) Foundations of Modern Probability , Probability and its Applications, Springer-Verlag, New York. doi:10.1007/b98838 . Kuczmaszewska, A. (2011) On the strong law of large numbers for ϕ-mixing and ρ-mixing random variables, Acta Mathematica Hungarica 138 174–189. doi:10.1007/ s10474-011-0089-z . Kontorovich, A. and Brockwell, A. (2014) A Strong Law of Large Numbers for Strongly Mixing Processes, Communications in Statistics - Theory and Methods 43 (18) 3777–3796. doi:10. 1080/03610926.2012.701696 . Loève, M. (1973). Paul Lévy, 1886-1971. The Annals of Probability , 1 (1) 1–8. doi:10.1214/ aop/1176997021 . Petersen, A. and Müller, H-G. (2016). Functional data analysis for density functions by transfor-mation to a Hilbert space. Annals of Statistics , 44 (1) 183–218. doi:10.1214/15-AOS1363 . Poisson, S.D. (1837). Recherches sur la probabilité des jugemens en matière criminelle at en matière civile, précédés des règles générales du calcul des probabilités . Paris: Bachelier. Popper, K. (1959) The propensity interpretation of Probability. The British Journal for the Phi-losophy of Science 10 (37) 25–42. Seneta, E. (1992) On the history of the Strong Law of Large Numbers and Boole’s inequality. Historia Mathematica 19 (1) 24–39. doi:10.1016/0315-0860(92)90053-E . Seneta, E. (2013) A Tricentenary history of the law of large numbers. Bernoulli 19 (4) 1088–1121. doi:10.3150/12-BEJSP12 . Shephard, N. and Andersen, T. G. (2009) Stochastic Volatility: Origins and Overview. In T. Mikosch, J.P. Kreiß, R. A. Davis and T. G. Andersen (eds.), Handbook of Financial Time Series (pp. 233–254). Berlin/Heidelberg: Springer. doi:10.1007/978-3-540-71297-8_10 . Taylor, R. L. and Hu, T-C. (2018) On laws of large numbers for exchangeable random variables, Stochastic Analysis and Applications 5 (3) 323–334. doi:10.1080/07362998708809120 . Hu, T.C., Rosalsky, A. and Volodin, A. (2008) On convergence properties of sums of dependent random variables under second moment and covariance restrictions, Statistics and Probability Let-ters 78 1999–2006. doi:10.1016/j.spl.2008.01.073 .
18125	https://byjus.com/maths/how-to-solve-linear-differential-equation/	A linear equation or polynomial, with one or more terms, consisting of the derivatives of the dependent variable with respect to one or more independent variables is known as a linear differential equation. A general first-order differential equation is given by the expression: dy/dx + Py = Q where y is a function and dy/dx is a derivative. The solution of the linear differential equation produces the value of variable y. dy/dx + 2y = sin x dy/dx + y = ex \| \| \| Table of contents: Definition Solution Solving First Order Differential Equation Examples \| Linear Differential Equations Definition A linear differential equation is defined by the linear polynomial equation, which consists of derivatives of several variables. It is also stated as Linear Partial Differential Equation when the function is dependent on variables and derivatives are partial. A differential equation having the above form is known as the first-order linear differential equation where P and Q are either constants or functions of the independent variable (in this case x) only. Also, the differential equation of the form, dy/dx + Py = Q, is a first-order linear differential equation where P and Q are either constants or functions of y (independent variable) only. To find linear differential equations solution, we have to derive the general form or representation of the solution. Non-Linear Differential Equation When an equation is not linear in unknown function and its derivatives, then it is said to be a nonlinear differential equation. It gives diverse solutions which can be seen for chaos. Solving Linear Differential Equations For finding the solution of such linear differential equations, we determine a function of the independent variable let us say M(x), which is known as the Integrating factor (I.F). Multiplying both sides of equation (1) with the integrating factor M(x) we get; M(x)dy/dx + M(x)Py = QM(x) …..(2) Now we chose M(x) in such a way that the L.H.S of equation (2) becomes the derivative of y.M(x) i.e. d(yM(x))/dx = (M(x))dy/dx + y (d(M(x)))dx … (Using d(uv)/dx = v(du/dx) + u(dv/dx) ⇒ M(x) /(dy/dx) + M(x)Py = M (x) dy/dx + y d(M(x))/dx ⇒M(x)Py = y dM(x)/dx ⇒1/M'(x) = P.dx Integrating both sides with respect to x, we get; (\begin{array}{l} log M (x) = \int Pdx (As \int \frac {f'(x)}{f(x)} ) = log f(x) \end{array} ) (\begin{array}{l} M(x) = e^{\int Pdx}= I.F\end{array} ) Now, using this value of the integrating factor, we can find out the solution of our first order linear differential equation. Multiplying both the sides of equation (1) by the I.F. we get (\begin{array}{l} e^{\int Pdx}\frac{dy}{dx} + yPe^{\int Pdx} = Qe^{\int Pdx} \end{array} ) This could be easily rewritten as: (\begin{array}{l} \frac {d(y.e^{\int Pdx})}{dx} = Qe^{\int Pdx} (Using \frac{d(uv)}{dx} = v \frac{du}{dx} + u\frac{dv}{dx} ) \end{array} ) Now integrating both the sides with respect to x, we get: (\begin{array}{l} \int d(y.e^{\int Pdx }) = \int Qe^{\int Pdx}dx + c \end{array} ) (\begin{array}{l} y = \frac {1}{e^{\int Pdx}} (\int Qe^{\int Pdx}dx + c )\end{array} ) where C is some arbitrary constant. How to Solve First Order Linear Differential Equation Learn to solve the first-order differential equation with the help of steps given below. Rearrange the terms of the given equation in the form dy/dx + Py = Q where P and Q are constants or functions of the independent variable x only. To obtain the integrating factor, integrate P (obtained in step 1) with respect to x and put this integral as a power to e. (\begin{array}{l} e^{\int Pdx} = I.F\end{array} ) Multiply both the sides of the linear first-order differential equation with the I.F. (\begin{array}{l} e^{\int Pdx} \frac{dy}{dx} + yPe^{\int Pdx} = Qe^{\int Pdx} \end{array} ) The L.H.S of the equation is always a derivative of y × M (x) i.e. L.H.S = d(y × I.F)/dx d(y × I.F)dx = Q × I.F In the last step, we simply integrate both the sides with respect to x and get a constant term C to get the solution. (\begin{array}{l}\therefore y \times I.F = \int Q \times I.F dx + C,\end{array} ) where C is some arbitrary constant Similarly, we can also solve the other form of linear first-order differential equation dx/dy +Px = Q using the same steps. In this form P and Q are the functions of y. The integrating factor (I.F) comes out to be and using this we find out the solution which will be (\begin{array}{l}(x) \times (I.F) = \int Q \times I.F dy + c \end{array} ) Now, to get a better insight into the linear differential equation, let us try solving some questions. where C is some arbitrary constant. \| \| \| --- \| \| Related Links \| \| \| Solve Separable Differential Equations \| Differential Equations Applications \| \| Differential Equations for Class 12 \| Differential Calculus \| \| Formation Differential Equations Whose General Solution Given \| Ordinary Differential Equations \| Solved Examples Example 1: Solve the LDE = dy/dx = [1/(1+x3)] – [3x2/(1 + x2)]y Solution: The above mentioned equation can be rewritten as dy/dx + [3x2/(1 + x2)] y = 1/(1+x3) Comparing it with dy/dx + Py = O, we get P = 3x2/1+x3 Q= 1/1 + x3 Let’s figure out the integrating factor(I.F.) which is, (\begin{array}{l} e^{\int Pdx} \end{array} ) (\begin{array}{l}I.F = e^{\int \frac {3x^2}{1 + x^3}} dx = e^{ln (1 + x^3)} \end{array} ) ⇒I.F. = 1 + x3 Now, we can also rewrite the L.H.S as: d(y × I.F)/dx, ⇒ d(y × (1 + x3)) dx = [1/(1 +x3)] × (1 + x3) Integrating both the sides w. r. t. x, we get, ⇒ y × ( 1 + x3) = x ⇒ y = x/(1 + x3) ⇒ y = [x/(1 + x3) + C Example 2: Solve the following differential equation: dy/dx + (sec x)y = 7 Solution: Comparing the given equation with dy/dx + Py = Q We see, P = sec x, Q = 7 Now lets find out the integrating factor using the formula (\begin{array}{l} e^{\int Pdx}= I.F \end{array} ) (\begin{array}{l} e^{\int secdx}= I.F. \end{array} ) (\begin{array}{l} I.F. = e^{ln \|sec x + tan x \|} = sec x + tan x \end{array} ) Now we can also rewrite the L.H.S as d(y × I.F)/dx}, i.e . d(y × (sec x + tan x )) ⇒d(y × (sec x + tan x ))/dx = 7(sec x + tan x) Integrating both the sides w. r. t. x, we get, (\begin{array}{l} \int d ( y × (sec x + tan x )) = \int 7(sec x + tan x) dx \end{array} ) (\begin{array}{l} \Rightarrow y × (sec x + tan x) = 7 (ln\|sec x + tan x\| + log \|sec x\| ) \end{array} ) (\begin{array}{l} y =\frac {7(ln\|sec x + tan x\| + log\|sec x\| }{(sec x + tan x)} + c \end{array} ) Example 3: A curve is passing through the origin and the slope of the tangent at a point R(x,y) where -1<x<1 is given as (x4 + 2xy + 1)/(1 – x2). What will be the equation of the curve? Solution: We know that the slope of the tangent at (x,y) is, tanƟ= dy/dx = (x4 + 2xy + 1)/1 – x2 Reframing the equation in the form dy/dx + Py = Q , we get dy/dx = 2xy/(1 – x2) + (x4 + 1)/(1 – x2) ⇒dy/dx – 2xy/(1 – x2) = (x4 + 1)/(1 – x2) Comparing we get P = -2x/(1 – x2) Q = (x4 + 1)/(1 – x2) Now, let’s find out the integrating factor using the formula. (\begin{array}{l} e^{\int Pdx}= I.F \end{array} ) (\begin{array}{l} e^{\int \frac{-2x}{1-x^2}}dx = e^{ln (1 – x^2)} = 1 – x^2 =I.F \end{array} ) Now we can also rewrite the L.H.S as (\begin{array}{l} \frac {d(y × I.F)}{dx}, \end{array} ) (\begin{array}{l} i.e.,\frac{d(y × (1 – x^2))}{dx} = \frac{x^4 + 1}{1 – x^2} × 1 – x^2 \end{array} ) Integrating both sides w. r. t. x, we get, (\begin{array}{l} \int d(y × (1 – x^2)) = \int \frac{x^4 + 1}{1 – x^2} × (1 – x^2 )dx \end{array} ) (\begin{array}{l} \Rightarrow y × (1 – x^2) = \int x^4 + 1 dx …(1) \end{array} ) x (1 – x2) = x5/5 + x + C ⇒ y = x5/5 + x/(1 – x2) + C It is the required equation of the curve. Also as the curve passes through origin; substitute the values as x = 0, y = 0 in the above equation. Thus, C = 0. Hence, equation of the curve is: ⇒ y = x5/5 + x/(1 – x2) Frequently Asked Questions – FAQs Q1 What is a linear differential equation? A linear equation or polynomial, with one or more terms, consisting of the derivatives of the dependent variable with respect to one or more independent variables is known as a linear differential equation. Q2 What is the example of a linear differential equation? Examples of linear differential equations are: xdy/dx+2y = x2 dx/dy – x/y = 2y dy/dx + ycot x = 2x2 Q3 How to solve the first order differential equation? First write the equation in the form of dy/dx+Py = Q, where P and Q are constants of x only Find integrating factor, IF = e∫Pdx Now write the solution in the form of y (I.F) = ∫Q × I.F C Q4 What is the difference between linear and nonlinear equations? A linear equation will always exist for all values of x and y but nonlinear equations may or may not have solutions for all values of x and y. Q5 What is the difference between linear and nonlinear differential equations? A linear differential equation is defined by a linear equation in unknown variables and their derivatives. A nonlinear differential equation is not linear in unknown variables and their derivatives. Comments Leave a Comment Cancel reply
18126	https://www.xconvert.com/unit-converter/horsepower-(metric)-to-foot-pounds-per-second	Horsepower (metric) (PS) to Foot-pounds per second (ft-lb/s) conversion Horsepower (metric) to Foot-pounds per second conversion table \| Horsepower (metric) (PS) \| Foot-pounds per second (ft-lb/s) \| --- \| \| 0 \| 0 \| \| 1 \| 542.47603863681 \| \| 2 \| 1084.9520772736 \| \| 3 \| 1627.4281159104 \| \| 4 \| 2169.9041545473 \| \| 5 \| 2712.3801931841 \| \| 6 \| 3254.8562318209 \| \| 7 \| 3797.3322704577 \| \| 8 \| 4339.8083090945 \| \| 9 \| 4882.2843477313 \| \| 10 \| 5424.7603863681 \| \| 20 \| 10849.520772736 \| \| 30 \| 16274.281159104 \| \| 40 \| 21699.041545473 \| \| 50 \| 27123.801931841 \| \| 60 \| 32548.562318209 \| \| 70 \| 37973.322704577 \| \| 80 \| 43398.083090945 \| \| 90 \| 48822.843477313 \| \| 100 \| 54247.603863681 \| \| 1000 \| 542476.03863681 \| How to convert horsepower (metric) to foot-pounds per second? Let's explore the conversion between metric horsepower and foot-pounds per second. This conversion deals with power, which is the rate at which work is done or energy is transferred. Understanding the Conversion Horsepower (metric), often denoted as PS (from the German "Pferdestärke"), and foot-pounds per second are both units of power. Converting between them involves understanding the relationship between the metric and imperial systems of measurement. Conversion Formulas The key to converting between metric horsepower and foot-pounds per second lies in knowing the conversion factor. 1 Horsepower (metric) to Foot-pounds per second: 1 Foot-pounds per second to Horsepower (metric): Step-by-Step Conversions Let’s break down the conversion process. Converting 1 Horsepower (metric) to Foot-pounds per second: Start with 1 PS: You're given 1 metric horsepower. Apply the conversion factor: Therefore, 1 metric horsepower is approximately equal to 542.48 foot-pounds per second. Converting 1 Foot-pounds per second to Horsepower (metric): Start with 1 ft⋅lb/s: You're given 1 foot-pound per second. Apply the conversion factor: Therefore, 1 foot-pound per second is approximately equal to 0.00184 metric horsepower. Historical Context and Facts The concept of horsepower was originally developed by James Watt, a Scottish inventor and mechanical engineer, in the late 18th century. Watt sought a way to market the power of his steam engines, so he compared them to the power of horses, which were a common source of power at the time. Watt defined one horsepower as the power required to lift 33,000 pounds by one foot in one minute. TechTarget - horsepower (hp) Metric horsepower, while similar, is defined differently. One metric horsepower is the power required to lift 75 kilograms against the Earth's gravitational force over a distance of one meter in one second. Real-World Examples Here are some real-world scenarios where you might convert between metric horsepower and foot-pounds per second. Car Engines: Automotive engineers often work with both metric and imperial units. When comparing the power output of engines designed in different countries, converting between PS and ft⋅lb/s helps in making direct comparisons. Electric Motors: In industrial applications, electric motors are used to power various machines. Their power output can be specified in either metric horsepower or foot-pounds per second, depending on the region or industry standards. Hydraulic Systems: Hydraulic systems, used in construction equipment and other heavy machinery, rely on power to perform work. Converting between metric horsepower and foot-pounds per second is necessary when analyzing and designing these systems. See below section for step by step unit conversion with formulas and explanations. Please refer to the table below for a list of all the Foot-pounds per second to other unit conversions. What is Horsepower (metric)? This section will provide a comprehensive overview of metric horsepower, including its definition, origins, calculation, and real-world applications. Definition and Origin Metric horsepower (PS, PferdeStärke in German, or cheval-vapeur in French) is a unit of power defined as the power required to raise a mass of 75 kilograms against Earth's gravitational force over a distance of one meter in one second. It is slightly less than the imperial horsepower. Calculation The value of one metric horsepower is: Therefore, approximately: (Imperial Horsepower) Historical Context The term "horsepower" was originally coined by James Watt to compare the output of steam engines to the power of draft horses. While Watt's original definition is related to the imperial horsepower, the metric horsepower evolved separately on the European continent, primarily for similar comparisons involving machinery and animal power. Real-World Examples Automobiles: Engine power is commonly specified in metric horsepower in many parts of the world. For example, a typical family car might have an engine rated at 150 PS. Motorcycles: Motorcycle engine power is also frequently stated in metric horsepower. Agricultural Machinery: Tractors and other farming equipment often have their power output measured in PS. Industrial Pumps and Motors: The power of pumps, fans, and electric motors used in industrial applications can be rated in metric horsepower. For example a pump may be rated at 5 PS. Fun fact While not commonly named after any specific person or law, it's interesting to know how metric horsepower is used across Europe. It gives the user a good understanding of the "power" of a machine. What is foot-pounds per second? Foot-pounds per second is a unit of power, commonly used in mechanical engineering and physics, especially in the United States. It represents the amount of work done (in foot-pounds) per unit of time (in seconds). Let's break it down. Definition of Foot-Pounds per Second Foot-pounds per second (ft⋅lb/s) is a unit of power that expresses the rate at which work is performed. One foot-pound is the amount of energy required to raise a one-pound object a distance of one foot against gravity. Therefore, foot-pounds per second tell you how quickly that work is being done. Understanding the Components Foot-pound (ft⋅lb): This is a unit of energy or work. It's calculated as the force in pounds multiplied by the distance in feet. Second (s): This is the unit of time. Combining these gives you: Conversion to Other Units Foot-pounds per second can be converted to other common units of power: Watts (W): The standard SI unit of power. Horsepower (hp): A common unit of power, especially for engines. Therefore: Historical Context While there isn't a specific "law" tied directly to foot-pounds per second, the concept of power and its measurement is closely related to the work of James Watt. He improved the steam engine, and horsepower was originally defined to compare the power of steam engines to that of horses. While horsepower is more commonly associated with Watt, foot-pounds per second provides a more granular and fundamental way to express power. Real-World Examples Lifting Objects: A motor lifting a 100-pound object 5 feet in 2 seconds is doing work at a rate of: Pumping Water: A pump lifting water 20 feet at a rate of 10 pounds per second is performing work at a rate of: Small Electric Motors: Many small electric motors are rated in terms of horsepower or watts, but you can convert those ratings to foot-pounds per second to understand the rate at which they can perform work. For example, a motor rated at 1/4 horsepower is approximately 137.5 ft⋅lb/s. Importance Foot-pounds per second are valuable for calculating the rate at which machines perform work, enabling engineers to design and analyze mechanical systems. Understanding this unit provides a fundamental grasp of power and its relationship to work and time. Complete Horsepower (metric) conversion table \| Convert 1 PS to other units \| Result \| --- \| \| Horsepower (metric) to Watts (PS to W) \| 735.49875 \| \| Horsepower (metric) to Milliwatts (PS to mW) \| 735498.75 \| \| Horsepower (metric) to Kilowatts (PS to kW) \| 0.73549875 \| \| Horsepower (metric) to Megawatts (PS to MW) \| 0.00073549875 \| \| Horsepower (metric) to Gigawatts (PS to GW) \| 7.3549875e-7 \| \| Horsepower (metric) to British thermal units per second (PS to Btu/s) \| 0.6971182104441 \| \| Horsepower (metric) to Foot-pounds per second (PS to ft-lb/s) \| 542.47603863681 \| \| Horsepower (metric) to Horsepower (British) (PS to hp) \| 0.9863200702488 \| Power conversions Horsepower (metric) to Watts (PS to W) Horsepower (metric) to Milliwatts (PS to mW) Horsepower (metric) to Kilowatts (PS to kW) Horsepower (metric) to Megawatts (PS to MW) Horsepower (metric) to Gigawatts (PS to GW) Horsepower (metric) to British thermal units per second (PS to Btu/s) Horsepower (metric) to Foot-pounds per second (PS to ft-lb/s) Horsepower (metric) to Horsepower (British) (PS to hp)
18127	https://www.fullerlittleminds.com/home/8-low-prep-addition-fluency-activities	Categories Sep 1 8 Low Prep Addition Fluency Activities We all know that a student’s ability to add fluently will help them succeed in so many other areas of math throughout their schooling. That’s why it is so important to give students many opportunities to practice their addition facts and build fluency at an early age. I’m going to show you 8ways students can practice their addition facts and build fluency without tons of extra prep for you! What is Addition Fluency? The first thing we need to understand is that fluency DOES NOT mean memorization. Students can be fluent in addition without having all of their addition facts memorized. Fluency means that the student can use their understandings of place value and addition properties (commutative property, associative property, etc.) to efficiently solve an addition problem. For example, let’s say a student is asked to solve 9 + 4. The student does not have 9+4 memorized, but within a few seconds comes up with the solution. The student says, “The answer is 13!” When asked how the student knew the answer, they responded, “I took 1 away from the 4 to give to the 9. That meant I had 10 + 3. I know that 10+3 = 13.” This student used their understanding of place value and adding on to decade numbers to solve this problem efficiently, even though they did not have the fact 9+4 memorized. So How Do We Practice Addition Fluency? In order for students to become fluent in addition, they need a LOT of practice. In my first grade classroom, I had an addition/subtraction fluency station in my math rotations for the WHOLE YEAR. Every day, students practiced addition and subtraction facts, no matter what we were learning in math that day. There are two things to keep in mind when planning this much fact practice for your students. Keep it fresh! Change up the fluency activities so students are engaged and excited to practice their addition facts. Differentiate! Some students will become fluent in their math facts faster than others. Make sure you have built-in opportunities for differentiation to challenge all students. With those thoughts in mind, let’s get to the low-prep fluency activities! Activity #1: Whiteboard Addition with Dice or Spinner This one is an oldie, but a goodie! Sometimes simple is best. Materials Needed: whiteboard whiteboard marker & eraser dice or spinner How: Students will roll two dice (or spin the spinner twice) to find the addends for their addition problem. Students write the addends on their whiteboard and find the sum. Repeat! Differentiation Ideas: Use 10 or 12 sided dice instead of 6 sided dice. Use a spinner with teen numbers. Use three dice instead of two for addition with three addends. Keep this Activity Fresh: Use big dice or inflatable dice. Use colored dry erase markers. Activity #2: Dice and Spinner Addition Fluency Game This activity is just like activity #1, but provides a little more structure if your students need some guidance! Click the image to grab this game! Materials Needed: addition fluency game directions addition fluency game board whiteboard marker or pencil dice or spinner How: Choose a game board for your students. Choose between a dice and spinner game, using a dry erase marker or pencil, and rolling/spinning two addends or one addend. If you choose a game board with one addend already filled in, students will roll one dice (or spin one time) to find the second addend and then find the sum. If you choose the game board with no addends, students will roll or spin to find both addends and then solve to find the sum. Differentiation Ideas: Use 10 or 12 sided dice instead of 6 sided dice. Use a spinner with teen numbers. Keep this Activity Fresh: Use big dice or inflatable dice. Use colored dry erase markers. Another way to keep this activity fresh is to change the game! Here are some other no-prep addition games to practice addition fluency. Click the images below! Activity #3: Playing Card Addition Partner Game This activity adds the excitement of working with a partner! Students can compete or work together to practice addition facts. Materials Needed: playing cards How: Each student will have their own deck of playing cards. Each partner will turn over one playing card. Students can either compete to find the sum or take turns finding the sum of the two cards that were turned over. Differentiation Ideas: Turn over 3 cards for 3 addends. Use the Jack, Queen, and King as teen numbers. Keep this Activity Fresh: Use a timer and see how fast they can go through the entire deck. See if they can beat their own time! Activity #4: Interactive Digital Activities It’s no secret that students love digital activities. If you are 1:1 or have enough devices for a digital math station, I highly recommend using digital activities to practice addition fluency. Click the image to get this interactive addition activity for Google Slides. Materials Needed: tablet or computer Google Slides How: For this interactive digital activity, students will open the activity in Google Slides. Students will move the interactive pieces to represent the addends and then find the sum. Differentiation Ideas: Use my Make Ten digital activity for students who are not ready to practice addition within 20. Use my Missing Addend digital activity or Three Addends digital activity for students in need of a challenge. Keep this Activity Fresh: Keep students interested in this activity by offering a different version such as my Halloween interactive addition activity for Google Slides. Activity #5: Handfuls of Counters or Cubes This activity is great because it can be used with almost any math manipulative and it can be an independent or partner game! Materials Needed: two-sided counters or unifix cubes whiteboard or paper whiteboard marker or pencil How: If playing with a partner, each student will grab a handful of counters or cubes. Students will count the objects in their handful. If one student grabbed 5 counters and another student grabbed 7 counters, the students will write 5 + 7 on their whiteboard or paper. Then, students will find the sum. If playing independently, the student will grab two handfuls and follow the same procedure. Differentiation Ideas: To make this game easier you can limit the number of manipulatives the students have access to. For example, if a student is only adding within 10, only give them 10 counters. To make this game more challenging try using three handfuls for three addends. Keep this Activity Fresh: Use different kinds of manipulatives! (Yes, this is the perfect time to break out those Target Dollar Spot erasers I know you have in your cupboard!) Something as simple as exchanging counters for mini pizza erasers really does keep students excited about the activity! Activity #6: Color by Addition Worksheets The past few years I have had classes that LOVE to draw and color. Color by addition worksheets are one of my favorite low-prep activities to practice addition fluency. Materials Needed: color by addition worksheet crayons, colored pencils, or markers How: Print a color by addition worksheet of your choice! I suggest mine. ;) Students will find sums for addition facts and color the picture according to the code. Differentiation Ideas: My color by addition worksheets come with three levels: add within 10, add within 15, and add within 20. You can print the worksheets that fit the needs of each student. Provide manipulatives for students who need that support. Keep this Activity Fresh: Use different themes throughout the year to keep students excited about color by addition worksheets! Click the images below to try some themed worksheets. Activity #7: Whiteboard Reveal Partner Game This is another simple but effective activity. Students love practicing with a partner and as a teacher, I love not prepping ANYTHING for this activity! Materials Needed: whiteboards whiteboard markers & erasers How: Each student will have a whiteboard and hide their whiteboard from their partner. Each student will write a number on their whiteboard. Then, the partners will count down from 3 and reveal their numbers. Students will take the two revealed numbers and write an addition fact. For example, if student #1 writes 3 and student #2 writes 8, the students will be solving 3 + 8. Students will solve their addition fact on their whiteboards and then compare answers. Repeat! Differentiation Ideas: Give students an upper or lower limit for their number. For example, let a pair of students know they have to write a number larger than 5. Keep this Activity Fresh: Use colored whiteboard markers. Let students add a noun after their number. Partners can take turns picking a noun. For example, student #1 chooses “popsicles” as the noun, student #1 might write 4 popsicles and student #2 might write 7 popsicles. Then, the sum is 11 popsicles. This gives students a chance to be silly with their noun choice to make it more fun, but it’s also a good reminder to label their answer when solving word problems! Activity #8: Task Cards My favorite thing about task cards is that they are so versatile! My favorite way to use task cards is to hang them around the room and allow students to use a clipboard and “solve the room.” Materials Needed: task cards recording sheet pencil How: Students will solve the addition problem on each task card and write the sum on their recording sheet. Differentiation Ideas: Use missing addend or three addend task cards for students who are ready for a challenge. Allow students to use notecards to create and solve their own task cards. Keep this Activity Fresh: Use task cards in different ways to keep students engaged. Maybe this week you will hang task cards around the room but next week you will use them to play Scoot. Change it up! Try other addition fluency task cards. Click the images below to try some other addition fluency task cards! …and more! Of course there are so many more ways to practice addition facts and build addition fluency, but I hope I’ve sparked some ideas and you’re ready to help your students practice, practice, practice! Remember, fluency is not about memorization, but repetition is important! Make sure your students have plenty of opportunities to practice addition facts and keep it fun! Want more from Fuller Little Minds? Join the FLM E-Mail List (and Get Exclusive Free Resources!) Follow on Instagram Check out More Blog Posts Follow on Teachers Pay Teachers Hello, I’m Becca! I’m a teacher, military wife, dog mom, and resource creator. I love to create simple, low-prep resources that are engaging and save teachers time! Latest Article Free Resources Sign-up to get access to my Free Resource Vault! You will be able to access all of my free products and even get some exclusive freebies! No spam. Unsubscribe at anytime. Sep 15 Resource Overview: Halloween Color by Addition and Subtraction Sep 15 Resource Overview: Halloween Color by Addition and Subtraction Mar 11 How I do (Low-Prep) Math Rotations Mar 11 How I do (Low-Prep) Math Rotations Related Posts Mar 11 How I do (Low-Prep) Math Rotations Sep 15 Resource Overview: Halloween Color by Addition and Subtraction Dec 3 5 Reasons to Use Never-Ending Math Games in Your Math Centers @fullerlittleminds Vocabulary parade…or an excuse to wear sweatpants to work?! Definitely both. 😂 Vocabulary parade…or an excuse to wear sweatpants to work?! Definitely both. 😂 Student: “Mrs.Fuller I need to calm down, can I get some sand?” Me: “Great idea! I also need to calm down, can you get some for me too?” I’m not going to lie, it helped. 😂 Anyone else having one of those weeks?! Student: “Mrs.Fuller I need to calm down, can I get some sand?” Me: “Great idea! I also need to calm down, can you get some for me too?” I’m not going to lie, it helped. 😂 Anyone else having one of those weeks?! Weekend mood 😴 Teaching has been DRAINING lately. I hope you are able to take the weekend to step away from work and recharge those batteries. ✨ Today’s battery recharging activities included: ☕️coffee with the hubs and pups 💻creating a ne Weekend mood 😴 Teaching has been DRAINING lately. I hope you are able to take the weekend to step away from work and recharge those batteries. ✨ Today’s battery recharging activities included: ☕️coffee with the hubs and pups 💻creating a ne Sometimes I type up a nice label and sometimes I scribble on a sticky note, slap it on, and call it good. 🤷🏻‍♀️ Here’s your reminder that you can do both. You can want your classroom to look nice and also not want to spend a lot of time Sometimes I type up a nice label and sometimes I scribble on a sticky note, slap it on, and call it good. 🤷🏻‍♀️ Here’s your reminder that you can do both. You can want your classroom to look nice and also not want to spend a lot of time
18128	https://www.reddit.com/r/learnmath/comments/esnoy8/finding_where_fx_is_increasing_or_decreasing/	Finding where f(x) is increasing or decreasing using f'(x) : r/learnmath Skip to main contentFinding where f(x) is increasing or decreasing using f'(x) : r/learnmath Open menu Open navigationGo to Reddit Home r/learnmath A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to learnmath r/learnmath r/learnmath Post all of your math-learning resources here. Questions, no matter how basic, will be answered (to the best ability of the online subscribers). 403K Members Online •6 yr. ago RytheGuy97 Finding where f(x) is increasing or decreasing using f'(x) The question I'm dealing with right now gives me the equation f'(x)=3x^2+2x+5 and is asking me to find where f(x) is increasing or decreasing. My idea is just figure out what f(x) is through reversing the power rule, getting an equation of f(x)=x^3+x^2+5x, then go through the process of finding the zeros of f(x), putting them on a sign chart, and plugging them into f'(x) to see if it's positive or negative. This seems to make sense to me but I can't shake the feeling that this isn't what they're trying to get us to do, or that there's some mistake that I'm overlooking. Read more Share Related Answers Section Related Answers Meaning of f(x) in function relationships How to find f(x) values from a graph How to identify when a function is increasing How to interpret the graph of a function f(x) Effective strategies for mastering algebra New to Reddit? Create your account and connect with a world of communities. Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy. Public Anyone can view, post, and comment to this community 0 0 Top Posts Reddit reReddit: Top posts of January 23, 2020 Reddit reReddit: Top posts of January 2020 Reddit reReddit: Top posts of 2020 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation
18129	https://chrisyeh96.github.io/2019/07/18/bias-variance-decomposition.html	Bias-Variance Decomposition of Mean Squared Error \| Chris Yeh Chris Yeh About Blog CV Projects Publications Bias-Variance Decomposition of Mean Squared Error Posted: Jul 18, 2019. Last updated: Dec 5, 2022. Tags:ML I derive the bias-variance decomposition of mean squared error for both estimators and predictors, and I show how they are related for linear models. Contents Bias and Variance of an Estimator Maximum Likelihood Estimation Example: Variance Estimator of Gaussian Data Example: Linear Regression Main Results Comparing linear regression and ridge regression estimators Bias and Variance of a Predictor (or Model) Setup Decomposition Discussion Example: Linear Regression and Ridge Regression Main Results Decomposing Bias for Linear Models Further areas of investigation Appendix Mean squared error (MSE) is defined in two different contexts. The MSE of an estimator quantifies the error of a sample statistic relative to the true population statistic. The MSE of a regression predictor (or model) quantifies the generalization error of that model trained on a sample of the true data distribution. This post discusses the bias-variance decomposition for MSE in both of these contexts. To start, we prove a generic identity. Theorem 1: For any random vector X∈R p and any constant vector c∈R p, E[‖X−c‖2 2]=tr Cov[X]+‖E[X]−c‖2 2. Proof All of the expectations and the variance are taken with respect to P(X). Let μ:=E[X]. tr Cov(X)+‖μ−c‖2 2=∑i=1 p Var(X i)+‖μ−c‖2 2=∑i=1 p E[(X i−μ i)2]+‖μ−c‖2 2=E[∑i=1 p(X i−μ i)2]+‖μ−c‖2 2=E[(X−μ)T(X−μ)]+(μ−c)T(μ−c)=E[X T X]−2 E[X]T μ+μ T μ+μ T μ⏟=0−2 μ T c+c T c=E[X T X−2 X T c+c T c]=E[‖X−c‖2 2] We write the special case where X and c are scalars instead of vectors as a corollary. Corollary 1: For any random variable X∈R and any constant c∈R, E[(X−c)2]=Var[X]+(E[X]−c)2. Bias and Variance of an Estimator Consider a probability distribution P T(X) parameterized by T, and let D={x(i)}i=1 N be a set of samples drawn i.i.d. from P T(X). Let T^=T^(D) be an estimator of T that has variability due to the randomness of the data from which it is computed. For example, T could be the mean of P T(X). The sample mean T^=1 N∑i=1 N x(i) is an estimator of T. For the rest of this section, we will use the abbreviation E T≡E D∼P T(X) and similarly for variance. The mean squared error of T^ decomposes nicely into the squared-bias and variance of T^ by a straightforward application of Theorem 1 where T is constant. M S E(T^)=E T[‖T^(D)−T‖2 2]=‖E T[T^]−T‖2 2+tr Cov T[T^]=‖Bias[T^]‖2 2+tr Cov[T^] Terminology for an estimator T^: The standard error of a scalar estimator T^ is S E(T^)=Var[T^]. If T^ is a vector, then the standard error of its i-th entry is S E(T^i)=Var[T^i]. T^ is unbiased if Bias[T^]=0. T^ is efficient if Cov[T^] equals the Cramer-Rao lower bound I(T)−1/N where I(T) is the Fisher Information matrix of T. T^ is asymptotically efficient if it achieves this bound asymptotically as the number of samples N→∞. T^ is consistent if T^→T in probability as N→∞. Sources Fan, Zhou. Lecture Notes from STATS 200 course, Stanford University. Autumn 2016. link. “What is the difference between a consistent estimator and an unbiased estimator?” StackExchange. link. Maximum Likelihood Estimation It can be shown that given data D sampled i.i.d. from P T(X), the maximum likelihood estimator T^M L E=arg max T^P T^(D)=arg max T^∏i=1 N P T^(x(i)) is consistent and asymptotically efficient. See Rice 8.5.2 and 8.7. Sources Rice, John A. Mathematical statistics and data analysis. 3rd ed., Cengage Learning, 2006. General statistics reference. Good discussion on maximum likelihood estimation. Example: Variance Estimator of Gaussian Data Consider data D sampled i.i.d. from a univariate Gaussian distribution N(μ,σ 2). Letting x¯=1 N∑i=1 N x(i) be the sample mean, the variance of the sampled data is S N 2=1 N∑i=1 N(x(i)−x¯)2. The estimator S N 2 is both the method of moments estimator (Fan, Lecture 12) and maximum likelihood estimator (Tobago) of the population variance σ 2. Nonetheless, even though it is consistent and asymptotically efficient, it is biased (proof on Wikipedia). E[S N 2]=N−1 N σ 2≠σ 2 Correcting for the bias yields the usual unbiased sample variance estimator. S N−1 2=N N−1 S N 2=1 N−1∑i=1 N(x(i)−x¯)2 Interestingly, although S N−1 2 is an unbiased estimator of the population variance σ 2, its square-root S N−1 is a biased estimator of the population standard deviation σ. This is because the square root is a strictly concave function, so by Jensen’s inequality, E[S N−1]=E[S N−1 2]<E[S N−1 2]=σ 2=σ. The variances of the two estimators S N 2 and S N−1 2 are also different. The distribution of N−1 σ 2 S N−1 2 is χ N−1 2 (chi-square) with N−1 degrees of freedom (Rice 6.3), so Var[S N−1 2]=Var[σ 2 N−1 χ N−1 2]=σ 4(N−1)2 Var[χ N−1 2]=2 σ 4 N−1. Instead of directly calculating the variance of S N 2, let’s calculate the bias and variance of the family of estimators parameterized by k. S k 2=1 k∑i=1 N(x(i)−x¯)2=N−1 k S N−1 2 E[S k 2]=N−1 k E[S N−1 2]=N−1 k σ 2 Bias[S k 2]=E[S k 2]−σ 2=N−1−k k σ 2 Var[S k 2]=(N−1 k)2 Var[S N−1 2]=2 σ 4(N−1)k 2 M S E[S k 2]=Bias[S k 2]2+Var[S k 2]=(N−1−k)2+2(N−1)k 2 σ 4 Although S N 2 is biased whereas S N−1 2 is not, S N 2 actually has lower mean squared error for any sample size N>2, as shown by the ratio of their MSEs. M S E(S N 2)M S E(S N−1 2)=σ 4(2 N−1)/N 2 2 σ 4/(N−1)=(2 N−1)(N−1)2 N 2 In fact, within the family of estimators of the form S k 2, the estimator with the lowest mean squared error is actually k=N+1. In most real-world scenarios, though, any of S N−1, S N, and S N+1 works well enough for large N. Sources Giles, David. “Variance Estimators That Minimize MSE.” Econometrics Beat: Dave Giles’ Blog, 21 May 2013. link. Taboga, Marco. “Normal Distribution - Maximum Likelihood Estimation.” StatLect. link. “Variance.” Wikipedia, 18 July 2019. link. Example: Linear Regression In the setting of parameter estimation for linear regression, the task is to estimate the coefficients w∈R p that relate a scalar output Y to a vector of regressors X∈R p. It is typically assumed that Y and X are random variables related by Y=w T X+ϵ for some noise ϵ∈R. However, we will take the unusual step of not necessarily assuming that the relationship between X and Y is truly linear, but instead that their relationship is given by Y=f(X)+ϵ for some arbitrary function f:R p→R. Suppose that the noise ϵ∼E is independent of X and that E is some arbitrary distribution with mean 0 and constant variance σ 2. One example of such a noise distribution is ϵ∼N(0,σ 2), although our following analysis does not require a Gaussian distribution. Thus, for a given x, E y∼P(Y\|X=x)[y]=f(x)Var y∼P(Y\|X=x)[y]=Var ϵ∼E[ϵ]=σ 2 Note that if σ 2=0, then Y is deterministically related to X, i.e. Y=f(X). We aim to estimate a linear regression function f^ that approximates the true f over some given training set of N labeled examples D={(x(i),y(i))}i=1 N sampled from an underlying joint distribution P(X,Y). In matrix notation, we can write D=(X,y) where X∈R N×p and y∈R N have the training examples arranged in rows. We can factor P(X,Y)=P(Y∣X)P(X). We have assumed that P(Y∣X) has mean f(X) and variance σ 2. However, we do not assume anything about the marginal distribution P(X) of the inputs, which is arbitrary depending on the dataset. For the rest of this post, we use the following abbreviations for the subscripts of expectations and variances: ϵ≡ϵ∼E y∣x≡y∼p(Y∣X=x)y∣X≡y∼p(Y∣X=X)x≡x∼P(X)D≡D∼P(X,Y) and the following shorthand notations: Z X=(X T X)−1 X T∈R p×N Z X,α=(X T X+α I d)−1 X T∈R p×N w^X,f=Z X f(X)=(X T X)−1 X T f(X)∈R p w^X,f,α=Z X,α f(X)=(X T X+α I d)−1 X T f(X)∈R p h X(x)=Z X T x=X(X T X)−1 x∈R N h X,α(x)=Z X,α T x=X(X T X+α I d)−1 x∈R N. Main Results The ordinary least-squares (OLS) and ridge regression estimators for w are w^=arg min w‖f(X)−X w‖2=(X T X)−1 X T y w^Ridge=arg min w‖f(X)−X w‖2+α‖w‖2 2=(X T X+α I d)−1 X T y. Their bias and variance properties are summarized in the table below. Note that in the case of arbitrary f, the bias of an estimator is technically undefined, since there is no “true” value to compare it to. (See highlighted cells in the table.) Instead, we compare our estimators to the parameters w⋆ of the best-fitting linear approximation to the true f. When f is truly linear, i.e. f(x)=w T x, then w⋆=w. The derivation for w⋆ can be found here. w⋆=arg min w E x[(f(x)−w T x)2]=E x[x x T]−1 E x[x f(x)]. We separately consider 2 cases: The training inputs X are fixed, and the training targets are sampled from the conditional distribution y∼P(Y∣X=X). Both the training inputs and targets are sampled jointly (X,y)∼P(X,Y). We also show that the variance of the ridge regression estimator is strictly less than the variance of the linear regression estimator when X are considered fixed. Furthermore, there always exists some choice of α such that the mean squared error of w^Ridge is less than the mean squared error of w^. \| \| arbitrary f(X) \| linear f(X) \| --- \| \| fixed X \| E D \| fixed X \| E D \| \| OLS \| Bias \| 0 \| E X[w^X,f]−w⋆ \| 0 \| 0 \| \| Variance \| σ 2(X T X)−1 \| Cov X[w^X,f]+σ 2 E X[(X T X)−1] \| σ 2(X T X)−1 \| σ 2 E X[(X T X)−1] \| \| Ridge Regression \| Bias \| Z X,α X w⋆−w⋆ \| E X[Z X,α X]w⋆−w⋆ \| w^X,f,α−w \| E X[w^X,f,α]−w \| \| Variance \| σ 2 Z X,α Z X,α T \| Cov X[w^X,f,α]+σ 2 E X,ϵ[Z X,α Z X,α T] \| σ 2 Z X,α Z X,α T \| Cov X[w^X,f,α]+σ 2 E X,ϵ[Z X,α Z X,α T] \| Linear Regression Estimator for arbitrary f Details First, we consider the case where the training inputs X are fixed. In this case, the estimator w^ is unbiased relative to w⋆. E y\|X[w^]=E y\|X[(X T X)−1 X T y]=(X T X)−1 X T E y\|X[y]=(X T X)−1 X T f(X)=w^X,f=n(X T X)−1⋅1 n X T f(X)=(1 n∑i=1 N x(i)x(i)T)−1 1 n∑i=1 N x(i)f(x(i))=E x[x x T]−1 E x[x f(x)]=w⋆Cov[w^∣X]=Cov y\|X[w^]=Cov y\|X[(X T X)−1 X T y]=Z Cov y\|X[y]Z T=Z(σ 2 I n)Z T=σ 2 Z Z T=σ 2(X T X)−1 However, if the training inputs X are sampled randomly, then the estimator is no longer unbiased, and the variance term also becomes dependent on f. E D[w^]=E X[E y\|X[w^]]=E X[w^X,f]Bias[w^]=E X[w^X,f]−w⋆ We prove by counterexample that E X[w^X,f]≠w⋆. Suppose X∼Uniform[0,1] is a scalar random variable, let f(x)=x 2, and consider a training set of size 2: D={a,b}∼P(X). We evaluate the integral by WolframAlpha. Otherwise, we can also compute the integral manually by splitting the fraction and doing a u-substitution with u=a 2. w⋆=E x[x x T]−1 E x[x f(x)]=E x[x 2]−1 E x[x 3]=(1/3)−1(1/4)=3/4 E D[w^]=E X[(X T X)−1 X T f(X)]=E a,b∼P(X)[(a 2+b 2)−1(a 3+b 3)]=∫0 1∫0 1 a 3+b 3 a 2+b 2 d a d b=1 6(2+π−ln⁡4)≈0.63 The derivation for the variance of w^ relies heavily on the linearity of expectation for matrices (see Appendix). Cov[w^]=Cov D[w^]=Cov X,ϵ[(X T X)−1 X T(f(X)+ϵ)]=Cov X,ϵ[w^X,f+Z X ϵ]=E X,ϵ[(w^X,f+Z X ϵ)(w^X,f+Z X ϵ)T]−E X,ϵ[w^X,f+Z X ϵ]E X,ϵ[w^X,f+Z X ϵ]T=E X,ϵ[w^X,f w^X,f T+w^X,f(Z X ϵ)T+Z X ϵ w^X,f T+Z X ϵ(Z X ϵ)T]−E X[w^X,f]E X[w^X,f]T=E X[w^X,f w^X,f T]+0+0+E X,ϵ[Z X ϵ ϵ T Z X T]−E X[w^X,f]E X[w^X,f]T=Cov X[w^X,f]+E X[Z X E ϵ[ϵ ϵ T]⏟=σ 2 I N Z X T]=Cov X[w^X,f]+σ 2 E X[(X T X)−1] Linear Regression Estimator for linear f In this setting, we assume that f(x)=w T x for some true w. As a special case, if the noise is Gaussian distributed ϵ∼N(0,σ 2), then w^ is the maximum likelihood estimator (MLE) for w, so it is consistent and asymptotically efficient. Details If X is fixed, then the least-squares estimate is unbiased. Bias[w^∣X]=E y∣X[w^]−w=w^X,f−w=(X T X)−1 X T X w−w=0. Therefore, the expectation of the bias over the distribution of X is also 0. If X is fixed, then variance of the least-squares estimate is the same for linear or nonlinear f, since it does not depend on f. However, when the training inputs X are sampled randomly, the variance does depend on f. Subsitituting f(X)=X w into the variance expression derived for arbitrary f yields Cov X[w^X,f]=Cov X[(X T X)−1 X T f(X)]=Cov X[(X T X)−1 X T(X w)]=Cov X[w]=0 so Cov[w^]=σ 2 E X[(X T X)−1]. Ridge Regression Estimator The ridge regression estimator w^Ridge is a linear function of the least-squares estimator w^. w^Ridge=(X T X+α I d)−1 X T y=(X T X+α I d)−1(X T X)(X T X)−1⏟=I d X T y=(X T X+α I d)−1(X T X)w^=Z X,α X w^ Details If f is arbitrary and X is fixed, then the expectation of the ridge regression estimator is not equal to w⋆, so it is biased. The inequality on the first line comes from the fact that Z X,α X=(X T X+α I d)−1(X T X)≠I d. E y\|X[w^Ridge]=E y\|X[Z X,α X w^]=Z X,α X E y\|X[w^]=Z X,α X w⋆≠w⋆Bias[w^Ridge∣X]=E y∣X[w^Ridge]−w⋆=Z X,α X w⋆−w⋆Cov y\|X[w^Ridge∣X]=Cov y\|X[Z X,α X w^]=Z X,α X Cov y\|X[w^]X T Z X,α T=σ 2 Z X,α X(X T X)−1 X T Z X,α T=σ 2 Z X,α Z X,α T If f was truly linear so w⋆=w and f(X)=X w, then we can simplify the bias. However, the variance expression does not depend on f, so it is the same regardless of whether f is linear or not. Bias[w^Ridge∣X]=Z X,α X w−w=Z X,α f(X)−w=w^X,f,α−w. If the training inputs X are sampled randomly with arbitrary f, then the bias and variance are as follows. The variance derivation follows a similar proof to the ordinary linear regression. E D[w^Ridge]=E X[E y∣X[w^Ridge]]=E X[Z X,α X]w⋆Bias[w^Ridge]=E X[Z X,α X]w⋆−w⋆Cov[w^Ridge]=Cov X,ϵ[w^X,f,α+Z X,α ϵ]=E X[w^X,f,α w^X,f,α T]+E X,ϵ[Z X,α ϵ ϵ T Z X,α T]−E X[w^X,f,α]E X[w^X,f,α]T=Cov X[w^X,f,α]+σ 2 E X,ϵ[Z X,α Z X,α T]. If f is truly linear, then Bias[w^Ridge]=E X[w^X,f,α]−w. Comparing linear regression and ridge regression estimators For any α>0 and assuming the training inputs X are fixed and full-rank, the ridge regression estimator has lower variance than the standard linear regression estimator without regularization. This result holds regardless of whether f is linear or not. Because the estimators w^ and w^Ridge are vectors, their variances are really covariance matrices. Thus, when we compare their variances, we actually compare the definiteness of their covariance matrices. One way to see this is that the MSE formula only depends on the trace of the covariance matrix. For any two vectors a and b, Cov[a]−Cov[b]≻0⟹tr(Cov[a]−Cov[b])>0⟺tr(Cov[a])>tr(Cov[b]). The first implication relies on the fact that if a matrix is positive definite, its trace is positive. Thus, showing that Cov[w^∣X]≻Cov[w^Ridge∣X] establishes that the w^ has a larger variance term in its MSE decomposition. For linear models, comparing the definiteness of the covariance matrices is also directly related to the variance of the predicted outputs. This makes more sense when we discuss the variance of the ridge regression predictor later in this post. Theorem: If we take the training inputs X∈R n×d with n≥d to be fixed and full-rank while the training labels y∈R N have variance σ 2, then the variance of any ridge regression estimator with α>0 has lower variance than the standard linear regression estimator without regularization. In other words, ∀α>0.Cov[w^Ridge∣X]≺Cov[w^∣X]. Proof Let S=X T X and W=(X T X+α I)−1. Both S and W are symmetric and invertible matrices. Note that S≻0 because z T S z=‖X z‖2 2>0 for all non-zero z (since X has linearly independent columns). Then, W−1=(S+α I)≻0 because I≻0 and α>0. Since the inverse of any positive definite matrix is also positive definite, S−1≻0 and W≻0 as well. Cov[w^Ridge∣X]=σ 2 Z X,α Z X,α T=σ 2 W X T X T W=σ 2 W S W Cov[w^∣X]=σ 2(X T X)−1=σ 2 S−1 Cov[w^∣X]−Cov[w^Ridge∣X]=σ 2(S−1−W S W) We will show that S−1−W S W≻0 (positive definite), which implies that Cov[w^Ridge∣X]≺Cov[w^∣X]. We first show W−1 S−1 W−1−S=(S+α I)S−1(S+α I)−S=(I+α S−1)(S+α I)−S=2 α I+α 2 S−1 which is clearly positive definite since I≻0, S−1≻0, and α>0. We can then expand S−1−W S W=W W−1 S−1 W−1 W−W S W=W(W−1 S−1 W−1−S)W=α W(2 I+α S−1)W which is positive definite. This is because z T W(2 I+α S−1)W z>0 for all W z≠0 (since the matrix inside the parentheses is positive definite), and W is invertible so W z≠0⟺z≠0. Having shown that the ridge regression estimator is biased but has lower variance than the unbiased least-squares estimator, the obvious next question is whether the decrease in variance is greater than the bias. Indeed, the following theorem shows that the ridge regression estimator is always able to achieve lower mean squared error. Theorem: Assume that the training inputs X are fixed and that f(x)=w T x is truly linear. Then M S E[w^Ridge]<M S E[w^] if and only if 0<α<2 σ 2‖w‖2 2. As the proof for this is quite involved, we refer readers to Theorem 1.2 of Wieringen, 2015 or Theorem 4.3 of Hoerl and Kennard, 1970 for different proofs of this theorem. Sources Hoerl, Arthur E., and Robert W. Kennard. “Ridge regression: Biased estimation for nonorthogonal problems.” Technometrics 12.1 (1970): 55-67. link. Proves that the MSE of ridge regression estimator is less than the MSE of the least-squares estimator for certain values of α. “Prove that the variance of the ridge regression estimator is less than the variance of the OLS estimator.” StackExchange. link. Taboga, Marco. “Ridge Regression.” StatLect. link. Provides alternative proof for why the ridge regression estimator has lower variance than the ordinary linear regression estimator. van Wieringen, Wessel N. “Lecture notes on ridge regression.” arXiv preprint arXiv:1509.09169 (2018). link. Reference for bias and variance of linear and ridge regression estimators. Only discusses the case where training inputs are fixed. Bias and Variance of a Predictor (or Model) Setup We consider the same setup discussed in the Linear Regression section above. To recap, we have 3 random variables X∈R p, Y∈R, and ϵ∈R, related by Y=f(X)+ϵ for some function f:R p→R. The noise ϵ∼E is independent of X and is distributed with mean 0 and variance σ 2. Decomposition The bias-variance decomposition of mean squared error is a method for analyzing a deterministic model’s behavior when trained on different training sets drawn from the same underlying distribution. To do this, we fix some test point x and then iterate the following procedure many times: Sample y∼p(Y∣X=x). Equivalently, sample ϵ∼E, then set y=f(x)+ϵ. Sample a training set D from P(X,Y). Fit f^D to the training set. Predict f^D(x) as our estimate of y. The mean squared error of our model f^D on a particular test point x is then M S E(x)=E y\|x[E D[(y−f^D(x))2]]=(Bias[f^(x)])2+Var[f^(x)]+Noise where Bias[f^(x)]=E D[f^D(x)]−f(x)Var[f^(x)]=Var D[f^D(x)]Noise=σ 2. Proof M S E(x)=E y\|x[E D[(y−f^D(x))2]]=E y\|x[Var D[f^D(x)]+(E D[f^D(x)]−y)2]=Var D[f^D(x)]+E y\|x[(E D[f^D(x)]−y)2]=Var D[f^D(x)]+Var p(y\|x)[y]+(E D[f^D(x)]−E y\|x[y])2=Var D[f^D(x)]+σ 2+(E D[f^D(x)]−f(x))2=Var[f^(x)]+Noise+(Bias[f^(x)])2 The 2nd equality comes from applying Corollary 1 where y is constant w.r.t. D. The 4th equality comes again from applying Corollary 1, but this time E D[f^D(x)] is constant w.r.t. y. Thus we have decomposed the mean squared error into 3 terms: bias, variance, and noise. Notice that if there is no noise (σ 2=0), then the mean squared error decomposes strictly into bias and variance. The mean squared error at x is also known as expected prediction error at x, commonly written as E P E(x). Discussion The noise term σ 2, also known as irreducible error or aleatoric uncertainty, is the variance of the target Y around its true mean f(x). It is inherent in the problem and it does not depend on the model or training data. If the data generation process is known, then we may know σ 2. Otherwise, we may estimate σ 2 with the sample variance of y at duplicated (or nearby) inputs x. However, the bias and variance components do depend on the model. A misspecified model, i.e. a model that does not match the true distribution of the data, will generally have bias. Thus, a model with high bias may underfit the data. On the other hand, more complex models have lower bias but higher variance. Such models have a tendency to overfit the data. In many circumstances it is possible to achieve large reductions in the variance term Var D[f^D(x)] with only a small increase in bias, thus reducing overfitting. We show this explicitly in the setting of linear models by comparing linear regression with ridge regression. In general, we are unable to exactly calculate the bias and variance of a learned model without knowing the true f. However, we can estimate the bias, variance, and MSE at a test point x by taking bootstrap samples of the dataset to approximate drawing different datasets D. Example: Linear Regression and Ridge Regression Consider a linear model f^(x)=w^T x over p-dimensional inputs x∈R p, where the intercept is included in w^. The relationship between the bias/variance of an estimator and the bias/variance of the model is straightforward for a linear model. Thus, we can readily use the results derived for the bias and variance of linear and ridge regression estimators. Bias[f^(x)]=E D[f^D(x)]−f(x)=E D[w^D T x]−w T x=(E D[w^D]−w)T x=Bias[w^]T x Var[f^(x)]=Var D[f^D(x)]=Var D[x T w^D]=x T Cov D[w^D]x However, when f is arbitrary, we cannot use the estimator bias results directly because they were derived relative to w⋆. Here, we are interested in the bias of w^T x vs. f(x), as opposed to w⋆T x. As before, we separately consider the cases where the true f is an arbitrary function and when f is perfectly linear in x. We also consider whether or not the training inputs X are fixed. The training targets y are always sampled from P(Y∣X). Main Results \| \| arbitrary f(X) \| linear f(X) \| --- \| \| fixed X \| E D \| fixed X \| E D \| \| OLS \| Bias \| x T w^X,f−f(x) \| x T E X[w^X,f]−f(x) \| 0 \| 0 \| \| Variance \| σ 2‖h X(x)‖2 2 \| x T Cov X[w^X,f]x+σ 2 E X[‖h X(x)‖2 2] \| σ 2‖h X(x)‖2 2 \| σ 2 E X[‖h X(x)‖2 2] \| \| Ridge Regression \| Bias \| Z X,α X w⋆T x−w⋆T x \| E X[Z X,α X]w⋆T x−w⋆T x \| w^X,f,α T x−w T x \| E X[w^X,f,α]T x−w T x \| \| Variance \| σ 2‖h X,α(x)‖2 2 \| x T Cov X[w^X,α,f]x+σ 2 E X[‖h X,α(x)‖2 2] \| σ 2‖h X,α(x)‖2 2 \| x T Cov X[w^X,α,f]x+σ 2 E X[‖h X,α(x)‖2 2] \| Decomposing Bias for Linear Models Before discussing the bias and variance of the linear and ridge regression models, we take a brief digression to show a further decomposition of bias for linear models. While there may exist similar decompositions for other model families, the following decomposition explicitly assumes that our model f^(x) is linear. Let w⋆=arg min w E x∼P(X)[(f(x)−w T x)2] be the parameters of the best-fitting linear approximation to the true f, which may or may not be linear. Then, the expected squared bias term decomposes into model bias and estimation bias. E x[(Bias[f^(x)])2]=E x[(E D[f^D(x)]−f(x))2]=E x[(w⋆T x−f(x))2]+E x[(E D[w^D T x]−w⋆T x)2]=Average[(Model Bias)2]+Average[(Estimation Bias)2] The model bias is the error between the best-fitting linear approximation w⋆T x and the true function f(x). Note that w⋆ is exactly defined as the parameters of a linear model that minimizes the average squared model bias. If f is not perfectly linear, then the squared model bias is clearly positive. The estimation bias is the error between the average estimate E D[w^D T x] and the best-fitting linear approximation w⋆T x. For example, if the true function was quadratic, then there would be a large model bias. However, if f is linear, then the model bias is 0; in fact, both the model bias and the estimation bias are 0 at all test points x, as shown in the next section. On the other hand, ridge regression has positive estimation bias, but reduced variance. Proof For any arbitrary w^, (f(x)−w^T x)2=(f(x)−w⋆T x+w⋆T x−w^T x)2=(f(x)−w⋆T x)2+(w⋆T x−w^T x)2+2(f(x)−w⋆T x)(w⋆T x−w^T x). The expected value of the 3rd term (with respect to x) is 0. E x[(f(x)−w⋆T x)(w⋆T x−w^T x)]=E x[(f(x)−w⋆T x)x T(w⋆−w^)]=(E x[f(x)x T]−E x[w⋆T x x T])(w⋆−w^)=(E x[f(x)x T]−E x[f(x)x T]E x[x x T]−1 E x[x x T])(w⋆−w^)=0 Since this result holds for arbitrary w^, we can choose in particular w^=E D[w^D] and get our desired result E x[(Bias[f^(x)])2]=E x[(f(x)−E D[f^D(x)])2]=E x[(f(x)−E D[w^D]T x)2]=E x[(f(x)−w⋆T x)2]+E x[(w⋆T x−E D[w^D]T x)]. Sources “Decomposition of average squared bias.” StackExchange. link. Hastie, Trevor, et al. The Elements of Statistical Learning. 2nd ed., Springer, 2009. link. Discussion leading up to equation (2.27), and Sections 7.1-7.3. Linear Regression for Arbitrary f Beyond deriving the values in the chart, we also prove that if the training data X are fixed, then the average in-sample variance is 1 N∑i=1 N Var[f^(x(i))]=p N σ 2. Details The model prediction at a test point x can be expressed as a linear combination of the input targets y. f^(x)=w^T x=y T X(X T X)−1 x=y T h X(x) First, we consider the case where the training inputs X to be fixed while the training labels y have variance σ 2. In other words, we treat X as the marginal distribution P(X). In this setting, although there is model bias if f is not linear, the average estimation bias is 0 because w⋆=w^X,f, as shown previously. Bias[f^(x)∣X]=f(x)−E y\|X[f^(x)]=f(x)−x T E y\|X[w^]=f(x)−x T w^X,f Var[f^(x)∣X]=x T Cov[w^∣X]x=σ 2 x T(X T X)−1 x=σ 2 h X(x)T h X(x)=σ 2‖h X(x)‖2 2 Taking the training data X as an approximation of the true distribution P(X) over inputs, we can compute the average in-sample variance 1 N∑i=1 N Var[f^(x(i))∣X]=1 N∑i=1 N σ 2 x(i)T(X T X)−1 x(i)=1 N σ 2 tr(X(X T X)−1 X T)=1 N σ 2 tr(X T X(X T X)−1)=1 N σ 2 tr(I p)=p N σ 2 Thus, variance of a linear regression model increases linearly with the input dimension p and decreases as 1/N in the training set size. However, if the training inputs X are not fixed but also sampled randomly, then the bias and variance are as follows. Notably, the estimation bias is not necessarily 0, because E X[w^X,f]≠w⋆, as shown previously. In other words, linear regression has estimation bias under model misspecification. Bias[f^(x)]=f(x)−E D[f^D(x)]=f(x)−x T E D[w^D]=f(x)−x T E X[w^X,f]Var[f^(x)]=Var D[f^D(x)]=x T Cov D[w^D]x=x T(Cov X[w^X]+σ 2 E X[(X T X)−1])x=x T Cov X[w^X]x+σ 2 E X[x T Z X Z X T x]=x T Cov X[w^X]x+σ 2 E X[‖h X(x)‖2 2] Linear Regression for Linear f For linear f, in addition to proving the bias and variance results in the table above, we show that for large N and assuming E[X]=0, the expected variance is E x[Var[f^(x)]]=p N σ 2. Then, the expected MSE is E x[M S E(x)]=σ 2+E x[Bias D[f^D(x)]2]+E x[Var D[f^D(x)]]=σ 2+0+p N σ 2=σ 2(1+p N). Details Since the linear regression estimators are unbiased when f is linear, the model also has no bias. Note that this means the model has zero model bias and zero estimation bias. Bias[f^(x)∣X]=Bias[w^∣X]T x=0 Bias[f^(x)]=Bias[w^]T x=0 For the variance of the model, if the training inputs X are fixed, the variance of a linear regression model for arbitrary f is σ 2‖h X(x)‖2 2 which does not depend on f. Thus, it is the same regardless of whether f is actually linear or not. However, when the training inputs are sampled randomly, the variance does depend on f. When deriving the variance of the linear regression estimator for linear f, we saw that Cov X[w^X]=0. Therefore, the variance of the model with randomly sampled inputs is Var[f^(x)]=σ 2 E X[‖h X(x)‖2 2]=σ 2 x T E X[(X T X)−1]x. If N is large and assuming E[X]=0, then 1 N X T X→Cov(X) (see Appendix), so E X[(X T X)−1]≈1 N Cov(X)−1. E x[Var[f^(x)]]=E x[σ 2 x T E X[(X T X)−1]x]≈1 N σ 2 E x[x T Cov(X)−1 x]=1 N σ 2 E x[tr(x T Cov(X)−1 x)]=1 N σ 2 E x[tr(x x T Cov(X)−1)]=1 N σ 2 tr(E x[x x T]Cov(X)−1)=1 N σ 2 tr(Cov(X)Cov(X)−1)=1 N σ 2 tr(I p)=p N σ 2 Sources Weatherwax, John L., and David Epstein. A Solution Manual and Notes for: The Elements of Statistical Learning. 2019. link. Chapter 2, variance of linear regression model when actual data relationship is linear. Ridge Regression The bias and variance expressions for ridge regression come as a straightforward application of the equations (copied again below) that use the existing results for the bias and variance of the ridge regression estimators. Bias[f^(x)]=Bias[w^]T x Var[f^(x)]=x T Cov D[w^D]x These equations also make it clear why we compare the definiteness of the covariance matrices between different estimators. Since we know that Cov[w^Ridge]≺Cov[w^], then by definition ∀x.x T(Cov[w^]−Cov[w^Ridge])x>0⟺∀x.x T Cov[w^]x−x T Cov[w^Ridge]x>0⟺∀x.Var[f^(x)]>Var[f^Ridge(x)]. Further areas of investigation When writing this post, I was unable to determine whether the variance of w^Ridge is lower than the variance of w^ when the training inputs X are sampled randomly. I was only able to find a proof assuming X are fixed. If you happen to have any ideas, please let me know through a GitHub issue! Appendix Linearity of Expectation for Matrices Suppose A∈R n×k is a random matrix and X∈R k×k is a random matrix, where A and X are independent. Then, E A,X[A X A T]=∑A∑X P(A,X)A X A T=∑A∑X P(A)P(X)A X A T=∑A P(A)A(∑X P(X)X)A T=∑A P(A)A E X[X]A T=E A[A E X[X]A T] Covariance Let X be a random vector, and let X be a matrix containing N i.i.d. samples of X: X=[−x(1)−⋮−x(N)−] Then, the covariance of X is defined as Cov(X)=E[X X T]−E[X]E[X]T. If E[X]=0, then the covariance can be approximated by Cov(X)≈1 N∑i=1 N x(i)x(i)T=1 N X T X for large N. Contents Bias and Variance of an Estimator Maximum Likelihood Estimation Example: Variance Estimator of Gaussian Data Example: Linear Regression Main Results Comparing linear regression and ridge regression estimators Bias and Variance of a Predictor (or Model) Setup Decomposition Discussion Example: Linear Regression and Ridge Regression Main Results Decomposing Bias for Linear Models Further areas of investigation Appendix This page was generated by GitHub Pages. © Christopher Yeh, 2025
18130	https://proofwiki.org/wiki/Limit_of_(Cosine_(X)_-_1)_over_X_at_Zero	Limit of (Cosine (X) - 1) over X at Zero From ProofWiki Jump to navigation Jump to search Contents 1 Theorem 2 Proof 1 3 Proof 2 4 Proof 3 5 Proof 4 Theorem : $\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x = 0$ Proof 1 This proof works directly from the definition of the cosine function: \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- \| \| \| \| \| \| (\ds \cos x) \| (=) \| \| \| \| (\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}) \| \| \| Definition of Real Cosine Function \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds \paren {-1}^0 \cdot \frac {x^{2 \cdot 0} } {\paren {2 \cdot 0}!} + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}) \| \| \| \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds 1 + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}) \| \| \| Definition of Zero Factorial and Definition of Zeroth Power \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- \| \| \| \| \| \| (\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x) \| (=) \| \| \| \| (\ds \lim_{x \mathop \to 0} \frac {1 + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} - 1} x) \| \| \| \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds \lim_{x \mathop \to 0} \frac {\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} } x) \| \| \| \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds \lim_{x \mathop \to 0} \frac {\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n - 1} } {\paren {2 n - 1}!} } 1) \| \| \| Power Series is Differentiable on Interval of Convergence and L'Hôpital's Rule \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds \lim_{x \mathop \to 0} \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {x^{2 n - 1} } {\paren {2 n - 1}!}) \| \| \| \| \| Now let: : $\map {f_n} x = \paren {-1}^n \dfrac {x^{2 n - 1} } {\paren {2 n - 1}!}$ Then for every $n \in \N_{> 0}$, and for all $x \in \closedint {\dfrac 1 2} {\dfrac 1 2}$: \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- \| \| \| \| \| \| (\ds \map {f_n} x) \| (\le) \| \| \| \| (\ds \size {\paren {-1}^n \frac {x^{2 n - 1} } {\paren {2 n - 1}!} }) \| (\ds \; = \frac { {\size x}^{2 n - 1} } {\paren {2 n - 1}!}) \| \| Absolute Value Function is Completely Multiplicative \| \| \| \| \| \| \| \| (\ds ) \| (\le) \| \| \| \| (\ds \frac 1 {2^{2 n - 1} \paren {2 n - 1}!}) \| \| \| Power Function is Strictly Increasing over Positive Reals \| \| \| \| \| \| \| \| (\ds ) \| (\le) \| \| \| \| (\ds \frac 1 {2^{2 n - 1} }) \| \| \| because the factorial is strictly increasing \| \| \| \| \| \| \| \| (\ds ) \| (\le) \| \| \| \| (\ds \frac 1 {2^n}) \| \| \| because $n \ge 1 \iff 2 n - 1 \ge n$ \| \| But from Sum of Infinite Geometric Sequence: : $\ds \sum_{n \mathop = 1}^\infty \frac 1 {2^n} = 2 < \infty$ By the Weierstrass M-Test, $\ds \sum_{n \mathop = 1}^\infty \map {f_n} x$ converges uniformly to some function $f$ on $\closedint {\dfrac 1 2} {\dfrac 1 2}$. But from Real Polynomial Function is Continuous, and the Uniform Limit Theorem $f$ is continuous on $\closedint {\dfrac 1 2} {\dfrac 1 2}$. So: : $\ds \lim_{x \mathop \to 0} \map f x = \map f 0 = \sum_{n \mathop = 1}^\infty \paren {-1} \frac {0^{2 n - 1} } {\paren {2 n - 1}!} = 0$ $\blacksquare$ Proof 2 This proof assumes the truth of the Derivative of Cosine Function: From Cosine of Zero is One: : $\cos 0 = 1$ From Derivative of Cosine Function: : $\map {D_x} {\cos x} = -\sin x$ and by Derivative of Constant: : $\map {D_x} {-1} = 0$ So by Sum Rule for Derivatives: : $\map {D_x} {\cos x - 1} = -\sin x$ By Sine of Zero is Zero: : $\sin 0 = 0$ From Derivative of Identity Function: : $\map {D_x} x = 1$ Thus L'Hôpital's Rule applies and so: : $\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x = \lim_{x \mathop \to 0} \frac {-\sin x} 1 = \frac {-0} 1 = 0$ $\blacksquare$ Proof 3 \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- \| \| \| \| \| \| (\ds \lim_{x \mathop \to 0} \frac {\cos x - 1} x) \| (=) \| \| \| \| (\ds \lim_{x \mathop \to 0} \frac {\paren {\cos x - 1} \paren {\cos x + 1} } {x \paren {\cos x + 1} }) \| \| \| \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds \lim_{x \mathop \to 0} \frac {\cos^2 x - 1} {x \paren {\cos x + 1} }) \| \| \| \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds \lim_{x \mathop \to 0} \frac {-\sin^2 x} {x \paren {\cos x + 1} }) \| \| \| Sum of Squares of Sine and Cosine \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds \paren {\lim_{x \mathop \to 0} \frac {\sin x} x} \paren {\lim_{x \mathop \to 0} \frac {-\sin x} {\cos x + 1} }) \| \| \| Product Rule for Limits of Real Functions \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds 1 \times \paren {\lim_{x \mathop \to 0} \frac{-\sin x} {\cos x + 1} }) \| \| \| Limit of $\dfrac {\sin x} x$ at Zero \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds \frac {\ds \lim_{x \mathop \to 0} \paren {-\sin x} } {\ds \lim_{x \mathop \to 0} \paren {\cos x + 1} }) \| \| \| Quotient Rule for Limits of Real Functions \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds \frac 0 2) \| \| \| \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds 0) \| \| \| \| \| $\blacksquare$ Proof 4 \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- --- --- \| \| \| \| \| \| (\ds \frac {\cos x - 1} x) \| (=) \| \| \| \| (\ds \frac {\cos x - \cos 0} x) \| \| \| Cosine of Zero is One \| \| \| \| \| \| \| \| (\ds ) \| (\to) \| \| \| \| (\ds \valueat {\dfrac \d {\d x} \cos x} {x \mathop = 0}) \| \| \| as $x \to 0$, from Definition of Derivative of Real Function at Point \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds \bigvalueat {\sin x} {x \mathop = 0}) \| \| \| Derivative of Cosine Function \| \| \| \| \| \| \| \| (\ds ) \| (=) \| \| \| \| (\ds 0) \| \| \| Sine of Zero is Zero \| \| $\blacksquare$ Retrieved from " Categories: Proven Results Differential Calculus Cosine Function Examples of Limits of Real Functions Limit of (Cosine (X) - 1) over X at Zero Navigation menu Search
18131	https://peopleadmin.com/blog/what-is-the-difference-between-a-tenured-professor-and-an-adjunct-professor/	What is the difference between a tenured professor and an adjunct professor? What is the difference between a tenured professor and an adjunct professor? Being a tenured or adjunct professor is a job that is unique to higher education, and the different types of faculty roles leave some asking: what is a tenured professor? What is an adjunct professor? How does academic tenure work? There is a lot going on in the world of HigherEd faculty, so in this article, we’ll break down the difference between tenured and adjunct professors, the role of faculty at an institution, and how the tenure process works. What is a tenured professor? A tenured professor is a faculty member at a college or university who has a full-time teaching position and strong job security that protects academic freedom. Their jobs are protected so that they can conduct potentially controversial or unpopular research and teach freely without fear of being fired—though they can still be fired for causes like misconduct, hate speech, or incompetence, usually accompanied by a peer review. Professors with tenure often have indefinite contracts and receive higher salaries than adjunct professors. They teach, conduct research in their fields, serve on college committees, and mentor students. These professors usually have the highest degree in their field, which is frequently a Ph.D, and have conducted significant research, demonstrated scholarly achievement, taught and mentored successfully, and made an impact on their academic field. The percentage of tenured professors has fallen consistently at institutions across the United States, from nearly 80% in 1970 to 20% in 2018. What is an adjunct professor? An adjunct or contingent professor is a part-time or contingent college or university professor who works on a short-term contract. 70% of adjunct faculty work on per-semester contracts, and 25% hold jobs outside academia to supplement their work and incomes since they generally have a lower salary than tenured professors. Similar to tenured professors, adjunct professors generally hold a doctorate or a graduate degree. Today, they make up the majority of professors on any college campus. Adjunct professors teach courses and mentor students, but are not usually expected to conduct research, publish papers, or serve on committees. What is the tenure process? The tenure process is long and difficult. The first step is securing a tenure-track role, meaning a role where a professor is teaching while working towards the requirements for tenure (distinct from an adjunct or part-time role). That is generally an assistant professor role, which is considered a probationary period. Assistant professors then must demonstrate excellence in teaching, research, and service during the next 5-10 years in order to be considered for tenure. After that probationary period, tenure review begins. This is usually a year-long review by administrators and by peer faculty members to determine if a professor’s work qualifies them for tenure. Tenure review is a stressful and complex process that requires professors to collect and share years worth of research, publications, teaching and work history, and more. If, at the end of the year, tenure is awarded, then that professor becomes an associate professor with tenure. Eventually, associate professors may be promoted to full professors later in their career, and sometimes take on administrative roles. If an assistant professor does not get tenure after their review, they might stay at that institution another year or so but need to look for another role. Being denied tenure at one higher education institution doesn’t necessarily mean a professor won’t get tenure at another institution, but it is still a difficult path. PeopleAdmin’s faculty solutions PeopleAdmin is constantly innovating to solve the challenges faced by HigherEd, and that includes streamlining and simplifying the tenure review process. Faculty Information System (FIS) collects and organizes faculty’s academic output in a user-friendly, centralized system. When tenure review comes around, faculty can easily collect and share their work history, academic history, research, papers, course history, and more without digging through files and systems. FIS also eases the burden on review committees with centralized, digitized materials, custom workflows, and confidential recommendations, removing the need for paper binders and back-and-forth emails. Learn more about how PeopleAdmin can support faculty and academic processes at your institution! 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18132	https://papers.nips.cc/paper_files/paper/2023/file/5491280797f3192b895bce84eb83df8d-Supplemental-Conference.pdf	Likelihood Ratio Confidence Sets for Sequential Decision Making Nicolas Emmenegger∗ ETH Zürich Mojmír Mutný∗ ETH Zürich Andreas Krause ETH Zürich Abstract Certifiable, adaptive uncertainty estimates for unknown quantities are an essential ingredient of sequential decision-making algorithms. Standard approaches rely on problem-dependent concentration results and are limited to a specific combi-nation of parameterization, noise family, and estimator. In this paper, we revisit the likelihood-based inference principle and propose to use likelihood ratios to con-struct any-time valid confidence sequences without requiring specialized treatment in each application scenario. Our method is especially suitable for problems with well-specified likelihoods, and the resulting sets always maintain the prescribed coverage in a model-agnostic manner. The size of the sets depends on a choice of estimator sequence in the likelihood ratio. We discuss how to provably choose the best sequence of estimators and shed light on connections to online convex opti-mization with algorithms such as Follow-the-Regularized-Leader. To counteract the initially large bias of the estimators, we propose a reweighting scheme that also opens up deployment in non-parametric settings such as RKHS function classes. We provide a non-asymptotic analysis of the likelihood ratio confidence sets size for generalized linear models, using insights from convex duality and online learning. We showcase the practical strength of our method on generalized linear bandit problems, survival analysis, and bandits with various additive noise distributions. 1 Introduction One of the main issues addressed by machine learning and statistics is the estimation of an unknown model from noisy observations. For example, in supervised learning, this might concern learning the dependence between an input (covariate) x and a random variable (observation) y. In many cases, we are not only interested in an estimate ˆ θ of the true model parameter θ⋆, but instead in a set of plausible values that θ⋆could take. Such confidence sets are of tremendous importance in sequential decision-making tasks, where uncertainty is used to drive exploration or risk-aversion needs to be implemented, and covariates are iteratively chosen based on previous observations. This setting includes problems such as bandit optimization, reinforcement learning, or active learning. In the former two, the confidence sets are often used to solve the exploration-exploitation dilemma and more generally influence the selection rule (Mukherjee et al., 2022), termination rule (Katz-Samuels and Jamieson, 2020), exploration (Auer, 2002) and/or risk-aversion (Makarova et al., 2021). When we interact with the environment by gathering data sequentially based on previous confidence sets, we introduce correlations between past noisy observations and future covariates. Data collected in this manner is referred to as adaptively gathered (Wasserman et al., 2020). Constructing estimators, confidence sets, and hypothesis tests for such non-i.i.d. data comes with added difficulty. Accordingly, and also for its importance in light of the reproducibility crisis (Baker, 2016), the task has attracted significant attention in the statistics community in recent years (Ramdas et al., 2022). ∗Equal contribution. 37th Conference on Neural Information Processing Systems (NeurIPS 2023). (a) Gaussian L (b) Laplace L (c) Gaussian L in RKHS Figure 1: (a) and (b) show examples of confidence sets defined via level sets of the log-likelihood function in 2D at two dataset sizes, for Gaussian (a) and Laplace (b) likelihoods respectively. The sets inherit the geometry of the likelihood, and are not always ellipsoidal. (c) shows confidence bands on an RKHS function in a bandit game searching for the optimum. We compare prior work on confidence sets (Abbasi-Yadkori et al., 2011), our LR sets, and a common heuristic (orange). Our sets are nearly as small as the commonly used heuristic, but have provable coverage and can vastly improve sequential decision-making tasks such as bandits by quickly eliminating hypotheses. Instead of deriving explicit concentration inequalities around an online estimator, we construct confidence sets implicitly defined by an inclusion criterion that is easy to evaluate in a computationally efficient manner and requires little statistical knowledge to implement. Roughly speaking, given a model pθ(y \| x) that describes the conditional dependence of the observation y given the covariate x under parameter θ, we will build sets based on a weighted modification of the sequential likelihood ratio statistic (Robbins et al., 1972; Wasserman et al., 2020) Rt(θ) := Lt({ˆ θs}t s=1) Lt(θ) := Qt s=1 pws ˆ θs (ys \| xs) Qt s=1 pws θ (ys \| xs) , (1) where {ˆ θs}s is a running estimator sequence that we are free to choose, but which may only depend on previously collected data. Parameters θ for which this statistic is small, i.e., for which Rt(θ) ≤1/α will be included in the set (and considered plausible). Examples of sets in a parametric and non-parametric setting are shown in Figure 1. The weighting terms ws ∈(0, 1] are crucial for dealing with inherent irregularities of many conditional observation models but can be flexibly chosen. Classically, these are set to ws = 1. The full exposition of our method with choice of estimators and weights is given in Section 2. Apart from being easy to use and implement, our approach also comes with performance guarantees. These sets maintain a provable 1 −α coverage – a fact we establish using Ville’s inequality for supermartingales (Ville, 1939), which is known to be essentially tight for martingales (see Howard et al., 2018, for a discussion). Therefore, in stark contrast to alternate methods, our confidence sequence is fully data-dependent, making it empirically tighter than competing approaches. Despite the rich history of sequential testing and related confidence sets going back to Wald (1945) and Robbins et al. (1972), these sets have found little use in the interactive machine learning community, which is a gap we fill in the present paper. Contributions In this work, we revisit the idea of using likelihood ratios to generate anytime-valid confidence sets. The main insight is that whenever the likelihood of the noise process is known, the likelihood ratio confidence sets are fully specified. They inherit their geometry from the likelihood function, and their size depends on the quality of our estimator sequence. We critically evaluate the likelihood ratio confidence sets and, in particular, we shed light on the following aspects: Firstly, for generalized linear models, we theoretically analyze the geometry of the LR confidence sets under mild assumptions. We show their geometry is dictated by Bregman divergences of exponential families (Chowdhury et al., 2022). Secondly, we show that the size of the confidence set is dictated by an online prediction game. The size of these sets depends on a sequence of estimators {ˆ θs}t s=1 that one uses to estimate the unknown parameter θ⋆. We discuss how to pick the estimator sequence in order to yield a provably small radius of the sets, by using the Follow-the-Regularized-Leader algorithm, which implements a regularized maximum-likelihood estimator. We prove that the radius of the con-fidence sets is nearly-worst-case optimal, and accordingly, they yield nearly-worst-case regret bounds when used in generalized linear bandit applications. However, due to their data-dependent nature, they can be much tighter than this theory suggests. Thirdly, we analyze the limitations of classical (un-weighted) LR sets when the underlying conditional observation model is not identifiable. In this case, the resulting (inevitable) estimation bias unnecessarily increases the size of the confidence sets. To mitigate this, we propose an adaptive reweighting scheme that decreases the effect of uninformed early 2 bias of the estimator sequence on the size of the sets downstream. The reweighting does not affect the coverage guarantees of our sets and utilizes an elegant connection to (robust) powered likelihoods (Wasserman et al., 2020). Finally, thanks to the adaptive reweighting scheme, our sets are very practi-cal as we showcase experimentally. We demonstrate that our method works well with exponential and non-exponential family likelihoods, and in parametric as well as in kernelized settings. We attribute their practical benefits to the fact that they do not depend on (possibly loose) worst-case parameters. 2 The Likelihood Method The sequential likelihood ratio process (LRP) in (1) is a statistic that compares the likelihood of a given model parameter, with the performance of an adaptively chosen estimator sequence. As noted above, we generalize the traditional definition, which would have ws ≡1, and define a corresponding confidence set as Ct = {θ \| Rt(θ) ≤1/α} . (2) The rationale is that the better a parameter θ is at explaining the data {(xs, ys)}t s from the true model θ⋆, the smaller this statistic will be, thereby increasing its chances to be included in Ct. When we construct Rt, the sequence of xs, ws and ˆ θs cannot depend on the noisy observation ys. Formally, consider the filtration (Fs)∞ s=0 with sub-σ-algebras Fs = σ(x1, . . . , y1, . . . xs, ys, xs+1). We require that ˆ θs and ws are Fs−1-measurable. Under these very mild assumptions and with arbitrary weights ws ∈(0, 1], we can show coverage, i.e., our (weighted) confidence sets uniformly track the true parameter with probability 1 −α. Theorem 1. The stochastic process Rt(θ⋆) in (1) is a non-negative supermartingale with respect to the filtration (Ft) and satisfies R0(θ⋆) ≡1. In addition, the sequence Ct from (2) satisfies Pθ⋆(∃t : θ⋆̸∈Ct) ≤α. The last statement follows by applying Ville’s inequality for super-martingales on Rt(θ⋆). The proof closely follows Wasserman et al. (2020). While coverage is always guaranteed irrespective of the estimator sequence {ˆ θs}, we would like to make the sets as small as possible at fixed coverage, which we do by picking a well-predicting estimator sequence. 2.1 The Estimator Sequence Game The specification of the LR process (LRP) allows us to choose an arbitrary estimator sequence {ˆ θs}s. To understand the importance of the sequence, let us introduce θ⋆to the definition of Rt in (1), and divide by Lt({ˆ θs}t s=1). This gives the equivalent formulation Ct := ( θ Lt(θ⋆) Lt(θ) ≤1 α Lt(θ⋆) Lt({ˆ θs}t s=1) ←confidence parameter ) . We see that the predictor sequence does not influence the geometry of the confidence set, which is fully specified by the likelihood function. We also observe that the ratio on the right-hand side serves as a confidence parameter controlling the size (radius) of the confidence sets measured under the likelihood ratio distance to θ⋆. If the confidence parameter goes to zero, only θ⋆is in the set. The better the estimator sequence is at predicting the data, the smaller the inclusion threshold, and hence the smaller the sets will ultimately be. Specifically, taking the log, we would like to minimize Rt := log Lt(θ⋆) Lt({ˆ θs}t s=1) = t X s=1 −log(pws ˆ θs (ys \| xs)) − t X s=1 −log(pws θ⋆(ys \| xs)). (3) The quantity Rt corresponds to a regret in an online prediction game, as will become apparent below. Online Prediction Game Online optimization is a mature field in interactive learning (Cesa-Bianchi and Lugosi, 2006; Orabona, 2019). The general goal is to minimize a sequence of loss functions as in Eq. (3) and compete against a baseline, which typically is the best-in-hindsight prediction, or – in our case – given by the performance of the fixed parameter θ⋆. Specifically, at every timestep s, iteratively, the agent chooses an action ˆ θs based on Fs−1, and a loss function fs(θ) is revealed. In most of the online optimization literature, fs can be chosen adversarially. In our prediction game, we know the whole form of loss function fs(θ) = −log(pws θ (ys \| xs)), as can be seen in (3), and not just fs(ˆ θs). Opposed to traditional assumptions in online prediction, in our case, fs are non-adversarial, but have a stochastic component due to ys. Also, contrary to most instances of online prediction, we do not com-pare against the best-in-hindsight predictor, but θ⋆instead, as this is more meaningful in our setting. 3 Online Optimization Algorithms Generally, we seek an algorithm that incurs low regret. Here, we focus on Follow-the-Regularized Leader (FTRL), which corresponds exactly to using regularized max-imum likelihood estimation, making it a natural and computationally practical choice. The update rule is defined in Alg. 1 (Line 3). While other algorithms could be considered, FTRL enjoys the optimal regret rate for generalized linear regression as we show later, and is easily implemented. In order to run the algorithm, one requires a sequence of strongly convex regularizers. For now, let us think of it as ψs(θ) = λ\|\|θ\|\|2 2, which we use in practice. However, one can derive a tighter analysis for a slightly modified, time-dependent regularization strategy for generalized linear models as we show in Sec. 3.3. 2.2 Adaptive Reweighting: Choosing the Right Loss There is yet more freedom in the construction of the LR, via the selection of the loss function. Not only do we select the predictor sequence, but also the weights of the losses via wt. This idea allows controlling the influence of a particular data point (xt, yt) on the cumulative loss based on the value of xt. For example, if we know a priori that for a given xt our prediction will be most likely bad, we can opt out of using the pair (xt, yt) by setting wt = 0. Below we will propose a weighting scheme that depends on a notion of bias, which captures how much of the error in predicting yt is due to our uncertainty about ˆ θt (compared to the uncertainty we still would have knowing θ⋆). Sometimes this bias is referred to as epistemic uncertainty in the literature, while the residual part of the error is referred to as aleatoric. Putting large weight on a data point heavily affected by this bias might unnecessarily increase the regret of our learner (and hence blow up the size of the confidence set). Note that, conveniently, even if we put low weight (zero) on a data point, nothing stops us from using this sample point to improve the estimator sequence in the next prediction round. As we will show below, our reweighting scheme is crucial in defining a practical algorithm for Reproducing Kernel Hilbert Space (RKHS) models and in high signal-to-noise ratio scenarios. Since we do not know θ⋆, our strategy is to compute an estimate of the bias of the estimator ˆ θt and its effect on the value of the likelihood function for a specific x that we played. We use the value of the bias to rescale the loss via wt such that its effect is of the same magnitude as the statistical error (see Algorithm 1; we call this step BIAS-WEIGHTING). Intuition To give a bit more intuition, suppose we have a Gaussian likelihood. Then the negative log-likelihood of (xt, yt) with weighting is proportional to wt σ2 (yt −x⊤ t ˆ θt)2. Now, if xt does not lie in the span of the data points {xs}t−1 s=1 used to compute ˆ θt, it is in general unavoidable to incur large error, inversely proportional to σ2. To see this, let us decompose the projection onto xt as x⊤ t (ˆ θt −θ⋆) = x⊤ t (ˆ θt −E[ˆ θt]) \| {z } statistical error + x⊤ t (E[ˆ θt] −θ⋆) \| {z } biasxt(ˆ θt) , (4) where the first term represents the statistical error up to time t, while the second, bias, is deterministic, and independent of the actual realization y, depending only θ⋆. Estimators with non-zero bias are biased. Plugging this into the likelihood function, we see that in expectation 1 σ2 E[(yt −x⊤ t ˆ θt)2\|Ft−1] ≲ 1 σ2 bias2 xt(ˆ θt) + ϵ2 + C t , where ϵ2 is the unavoidable predictive error in expectation (due to a noisy objective) and is a constant independent of σ2. C t is the statistical error, and C is independent of σ2. Note that the bias term scales inversely with the variance, and leads to unnecessarily big confidence parameters for small σ2. In fact, the problem is that we use the likelihood to measure the distance between two parameters, but this is only a “good“ distance once the deterministic source of the error (bias) vanishes. For this reason, without weighting, the incurred regret blows up severely in low-noise settings. To counter this, we balance the deterministic estimation bias and noise variance via proper selection of wt. In this case, it turns out that wt = σ2 σ2+bias2 xt(ˆ θt) ensures that the overall the scaling is independent of σ2. While the choice of weights {ws}t s influences the geometry of the confidence sets, with a good data collection and estimation strategy the bias asymptotically decreases to zero, and hence the weights converge to 1. Bias estimation In order to generalize this rule beyond Gaussian likelihoods, we need a proper generalization of the bias. Our generalization is motivated by our analysis of generalized linear models, but the method can be applied more broadly. The role of the squared statistical error (variance) is played by the inverse of the smoothness constant of the negative log-likelihood functions fs, denoted by L. This is the usual smoothness, commonly seen in the convex optimization literature. 4 We consider penalized likelihood estimators with strongly convex regularizers (Alg. 1, line 3). For this estimator class, we define the bias via a hypothetical stochastic-error-free estimate ˆ θ× t , had we access to the expected values of the gradient loss functions (a.k.a. score). We use the first-order optimality conditions and the indicator function of the set Θ, iΘ, to define the error-free-estimate ˆ θ× t , and the bias of the estimator ˆ θt as bias2 xt(ˆ θt) = (x⊤ t (θ⋆−ˆ θ× t ))2 with E "t−1 X s=1 ∇log pˆ θ× t (ys\|xs) # −∇ψt(ˆ θ× t ) + iΘ(ˆ θ× t ) = 0, (5) where the expectation denotes a sequence of expectations conditioned on the prior filtration. This notion of bias coincides with the definition of bias in Eq. (4) for the Gaussian likelihood. This quantity cannot be evaluated in general, however, we prove a computable upper bound. Theorem 2 (Bias estimate). Let the negative log-likelihood have the form, −log pθ(ys\|xs) = g(x⊤ s θ), where g : R →R is µ strongly-convex and let the regularizer be ψt(θ) = λ\|\|θ\|\|2 2 making the overall objective strongly convex. Then, defining Vµ;λ t = Pt s=1 µxsx⊤ s + λI, we can bound bias2 x(ˆ θt) ≤2λ\|\|θ⋆\|\|2 2x⊤(Vµ;λ t )−1x. (6) The proof is deferred to App. A.3, and requires elementary convex analysis. This leads us to propose the weighting scheme wt = 1/L bias2 xt(ˆ θt)+1/L. We justify that this is a sensible choice by analyzing the confidence set on the GLM class in Section 3, which satisfies the smoothness and strong-convexity conditions. We show that this rule properly balances the stochastic and bias components of the error in the regret as in (3). However, this rule is more broadly applicable beyond the canonical representation of GLM or the GLM family altogether. Algorithm 1 Constructing the LR Confidence Sequence 1: Input: convex set Θ ⊂Rd, confidence level α > 0, likelihood pθ(y\|x), regularizers {ψt}t 2: for t ∈N0 do 3: ˆ θt = arg minθ∈Θ Pt−1 s=1 −log pθ(ys \| xs) + ψt(θ) ▷FTRL 4: wt = ( 1/L 1/L+bias2 xt(ˆ θt) THIS WORK 1 CLASSICAL ▷BIAS-WEIGHTING biasxt(ˆ θt) in Eq. (5) or Eq.(6) 5: Ct = θ ∈Θ Qt s=1 pws ˆ θs (ys \| xs) pws θ (ys \| xs) ≤1 α . ▷Confidence set 6: end for 3 Theory: Linear Models While the coverage (i.e., “correctness”) of the likelihood ratio confidence sets is always guaranteed, their worst-case size (affecting the “performance”) cannot be easily bounded in general. We analyze the size and the geometry of the LR confidence sequence in the special but versatile case of generalized linear models. 3.1 Generalized Linear Models We assume knowledge of the conditional probability model pθ(y\|x), where the covariates x ∈X ⊂Rd, and the true underlying model parameter lies in a set Θ ⊂Rd. If t is indexing (discrete) time, then xt is acquired sequentially, and the – subsequently observed – yt is sampled from an exponential family distribution parametrized as pθ(y \| xt) = h(y) exp T(y) · x⊤ t θ −A(x⊤ t θ) . (7) Here, A is referred to as the log-partition function of the conditional distribution, and T(y) is the suf-ficient statistic. The function h is the base measure, and has little effect on our further developments, as it cancels out in the LR. Examples of commonly used exponential families (Gaussian, Binomial, Poisson or Weibull) with their link functions can be found in Table 1 in App. A.1. In order to facilitate theoretical analysis for online algorithms, we make the following assumptions about the covariates x ∈X and the set of plausible parameters Θ. 5 Assumption 1. The covariates are bounded, i.e., supx∈X \|\|x\|\|2 ≤1, and the set Θ is contained in an ℓ2-ball of radius B. We will also assume that the log-partition function is strongly convex, that is, that there exists µ := infz∈[−B,B] A′′(z), and that A is L-smooth, i.e. L := supz∈[−B,B] A′′(z). These assumptions are common in other works addressing the confidence sets of GLMs (Filippi et al., 2010; Faury et al., 2020), who remark that the dependence on µ is undesirable. However, in contrast to these works, our confidence sets do not use these assumptions in the construction of the sets. We only require these for our theoretical analysis. As these are worst-case parameters, the practical performance can be much better for our sets. 3.2 Geometry and Concentration Before stating our results, we need to define a distance notion that the convex negative log-likelihoods induce. For a continuously differentiable convex function f, we denote the Bregman divergence as Df(a, b) := f(a) −f(b) −∇f(b)⊤(a −b). The ν-regularized sum of log-partition functions is defined as Zν t (θ) := t X s=1 wsA(x⊤ s θ) + ν 2\|\|θ\|\|2 2. (8) This function will capture the geometry of the LR confidence sets. The confidence set size depends mainly on two terms. One refers to a notion of complexity of the space referred to as Bregman information gain: Γν t (˜ θt) = log R Rd exp(−ν 2 \|\|θ\|\|2 2)dθ R Rd exp(−DZν t (θ,˜ θt))dθ , first defined by Chowdhury et al. (2022) as a generalization of the information gain of Srinivas et al. (2009), γν t = log det(P i=1 µ ν xix⊤ i + I) for Gaussian likelihoods. We will drop the superscript whenever the regularization is clear from context and simply refer to γt. This term appears because one can relate the decay of the likelihood as a function of the Bregman Divergence from θ⋆with the performance of a (regularized) maximum likelihood estimator via convex (Fenchel) duality. In particular, if ˜ θt is a regularized MLE, Γν t := Γν t (˜ θt) will asymptotically scale as O(d log t) (cf. Chowdhury et al., 2022, for further discussion). For Gaussian likelihoods and ws ≡1, it coincides with the classical information gain independent of ˜ θt. The second term that affects the size is the regret Rt of the online prediction game over t rounds we introduced previously in (3). These two parts together yield the following result: Theorem 3. Let ν > 0 and α, δ ∈(0, 1). For the level 1 −α confidence set Ct defined in (2) under the GLM in (7), with probability 1 −δ, for all t ≥1, any θ ∈Ct satisfies DZν t (θ, θ⋆) ≤4L µ ξt + 2 log 1 δ + 2Rt, (9) where ξt = log 1 α + νB2 + Γν t and L, µ are defined as above and finally Rt is the regret of the game in Eq. (3). The set defined via the above divergence does not coincide with the LR confidence set. It is slightly larger due to a term involving ν (as in Eq. (8)). This is a technical consequence of our proof technique, where the gradient of Zν t needs to be invertible, and regularization is added to this end. We note that this ν > 0 can be chosen freely. Note that the theorem involves two confidence levels, α and δ: α is a bound on the Type I error – coverage of the confidence sets – while δ upper bounds the probability of a large radius – and is therefore related to the power and Type II error of a corresponding hypothesis test. The proof of the theorem is deferred to App. B.2. To give more intuition on these quantities, let us instantiate them for the Gaussian likelihood case with ws ≡1. In this scenario, Zν t (θ) = Pt s=1 1 2σ2 \|\|θ\|\|2 xsx⊤ s + ν 2\|\|θ\|\|2 2, and the (in this case symmetric) Bregman divergence is equal to DZν t (θ⋆, θ) = 1 2\|\|θ −θ⋆\|\|2 Vσ−2;ν t , where Vµ;ν t = Pt s=1 µxsx⊤ s + νI, which means that our confidence sets are upper bounded by a ball in the same norm as those in the seminal work on linear bandits (Abbasi-Yadkori et al., 2011). 3.3 Online Optimization in GLMs: Follow the Regularized Leader The size of the confidence sets in Theorem 3 depends on the regret of the online prediction game involving the estimator sequence. We now bound this regret when using the Follow-the-Regularized-Leader (FTRL) algorithm in this setting. This high probability bound is novel to the best of our 6 knowledge and may be of independent interest. We state in a weight-agnostic manner first, and then with our particular choice. The latter variant uses a specifically chosen regularizer. In this case, we can track the contribution of each time-step towards the regret separately. Theorem 4. Let ψt(θ) = λ\|\|θ\|\|2 2. Assume Assumption 1, and additionally that A is L-smooth everywhere in Rd, and let wt ∈[0, 1] be arbitrary. Then, with probability 1 −δ the regret of FTRL (Alg. 1) satisfies for all t ≥1 Rt ≤λB2 + L µ (γλ t + 2 log(1/δ)) + 2L2B2 µ γλ t . (10) The regret bounds are optimal in the orders of γλ t , matching lower bounds of Ouhamma et al. (2021), as for linear models γt = O(d log t). Combining results of Thm. 4 with Thm. 3, we get a confidence parameter that scales with O(√γt), for confidence sets of the form \|\|θ −θ⋆\|\|Vt, which coincides with the best-known confidence sets in this setting in the worst-case (Abbasi-Yadkori et al., 2012). The requirement of global L−smoothness can be relaxed to L−smoothness over Θ. With a more elaborate (but less insightful) analysis, we can show that we achieve a ˜ O(γt) bound even in this case. The proofs of these results are deferred to App. C.4, App. C.5 and App. C.6 respectively. Regret, Weighting and Estimation Bias Interestingly, the term in Thm. 4 involving the (crude) proxy to the bias – the bound B – is not scaled by the same L/µ factors as the other terms in the regret bound (10) and in Theorem 3. Namely, the prefactor is L2/µ instead of L/µ. This extra dependence manifests itself in the unnecessary penalization through the estimation bias we introduced in Sec. 2.2, particularly in low-noise settings. We addressed this issue by picking the weights {wt}. While the above theorem holds for any valid weighting, it does not exhibit the possible improvement from using specific weights. We argued earlier that the error in prediction should not be measured by the likelihood function if there is deterministic error, since initially, we are fully uncertain about the value of θ⋆ ⊤(·) outside the span of previous observations. Of course, if our goal would be to purely pick weights to minimize Rt, then ws = 0 would lead to zero regret and hence be optimal. However, the likelihood ratio would then be constant, and uninformative. In other words, the associated log-partition Bregman divergence in Theorem 3 would be trivial and not filter out any hypotheses. Clearly, some balance has to be met. With this motivation in mind, we proposed a nonzero weighting that decreases the regret contribution of the bias, namely wt = 1/L 1/L+bias2 xt(ˆ θt). The advantage of this choice becomes more apparent when we use the regularizer ψt(θ) = λ\|\|θ\|\|2 + A(x⊤ t θ) to obtain the following result. Theorem 5. Let ψs(θ) = λ\|\|θ\|\|2 + A(x⊤ s θ). Assume Assumption 1, and additionally that A is L-smooth everywhere in Rd, and choose ws = 1/L 1/L+biasxs(ˆ θs)2 . Additionally, let the sequence of xs be such that, P s(1 −ws)(fs(θ⋆) −fs(¯ θs+1)) ≤L/µγλ t , where ¯ θs is the FTRL optimizer with the regularizer λ\|\|θ\|\|2 2 from Theorem 4 2. Then, with probability 1 −δ the regret of FTRL (Alg. 1) satisfies for all t ≥1 Rt ≤λB2 + 2L µ γλ t + log 1 δ + L µ t X s=1 B2 1/L + bias2 xs(ˆ θs) ∆γλ s , where ∆γλ s = γλ s+1 −γλ s . One can see that for points where the information gain ∆γs is large (corresponding to more unexplored regions of the space, where the deterministic source of error is then large), the weighting scheme will make sure that the multiplicative contribution of B2 is mitigated, along with having the correct prefactor L/µ. The reader may wonder how this result is useful when we replace bias2 xs(ˆ θs) with the upper bound from Thm. 2. While instructive, our bound still only makes the bias proxy B2 appear in front of the information gain ∆γt, instead of the more desireable bias itself. In the latter case, we could also directly make use of the upper bound and get an explicit result only using an upper bound on the bias. We leave this for future work. We point out that this choice of ψs(θ) in Theorem 5 corresponds to the Vovk-Azoury-Warmuth predictor (Vovk, 2001; Azoury and Warmuth, 1999) in the online learning literature. This choice is helpful in order to track the bias contribution more precisely in our proof. 2Note that this assumption was missing in an earlier version. ¯ θs+1 corresponds to a regularized MLE that did observe the data pair (xs, ys). 7 4 Application: Linear and Kernelized Bandits Our main motivation to construct confidence sets is bandit optimization. A prototypical bandit algorithm – the Upper Confidence Bound (UCB) (Auer, 2002) – sequentially chooses covariates xs in order to maximize the reward Pt s=1 rθ⋆(xs), where rθ⋆is the unknown pay-off function parametrized by θ⋆. UCB chooses the action xs which maximizes the optimistic estimate of the reward in each round, namely xs = arg max x∈X max θ∈Cs−1 rθ(x), (11) where Cs−1 is some confidence set for θ⋆, and can be constructed with Algorithm 1 from the first s −1 data points. An important special case is when rθ⋆is linear (Abe and Long, 1999) or modelled by a generalized linear model (Filippi et al., 2010). In that case, the inner optimization problem is convex as long as Cs−1 is convex. The outer optimization is tractable for finite X. In the applications we consider, our confidence sets are convex, and we easily solve the UCB oracle using convex optimization toolboxes. Extension to RKHS We introduced the framework of LR confidence sets only for finite-dimensional Euclidean spaces. However, it can be easily extended to Reproducing Kernel Hilbert Spaces (RKHS) (Cucker and Smale, 2002). The definition of the LR process in (1) is still well-posed, but now the sets are subsets of the RKHS, containing functions f ∈Hk. An outstanding issue is how to use these sets in downstream applications, and represent them tractably as in Figure 1. Conveniently, even with infinite-dimensional RKHSs, the inner-optimization in (11) admits a Lagrangian formulation, and the generalized representer theorem applies (Schölkopf et al., 2001; Mutný and Krause, 2021). In other words, we can still derive a pointwise upper confidence band as ucb(x) = maxf∈Hk,\|\|f\|\|k≤B,f∈Cs ⟨f, k(x, ·)⟩in terms of {xj}s j=1 ∪{x}, leading to a s + 1-dimensional, tractable optimization problem. We also point out that the weighting is even more paramount in the RKHS setting, as the bias never vanishes for many infinite dimensional Hilbert spaces (Mutný and Krause, 2022). For this purpose, our weighting is of paramount practical importance, as we can see in Figure 2a), where the gray arrow represents the significant improvement from reweighting. 4.1 Instantiation of the Theory for Linear Bandits Before going to the experiments, we instantiate our theoretical results from Sec. 3 to the important and well-studied special case of linear payoffs. In that case, rθ(x) = ⟨x, θ⟩and the agent observes ys = ⟨xs, θ⋆⟩+ ηs upon playing action xs, where ηs ∼N(0, σ2). We are interested in minimizing the so-called cumulative pseudo-regret, namely, Rt = Pt s=1[⟨x⋆, θ⋆⟩−⟨xs, θ⋆⟩], where x⋆refers to the optimal action. Using the set from (2) along with Theorem 3 and the FTRL result of Theorem 4 we can get a regret bound for the choice ws ≡1. Theorem 6. Let ws ≡1. For any λ ≥ 1 σ2 , with probability at least 1 −3δ, for all t ∈N we have Rt ≤6 q tγλ t σ q log(1/δ) + γλ t + σλ1/2B + B q γλ t . Our results are optimal in both d and t up to constant and logarithmic factors. The proof is deferred to App. D, but is an instantiation of the aforementioned theorems, along with a standard analysis. There, we also compare to the seminal result of Abbasi-Yadkori et al. (2011), which does not suffer from the dependence on B√γt. We attribute this to the incurred bias in the absence of the reweighting scheme. For the weighted likelihood ratio, we can obtain a result similar to the above, but multiplied by an upper bound on sups≥1 w−1 s . This is undesirable, as our experiments will show that the reweighting scheme vastly improves performance. While this could be somewhat mitigated by using the Theorem 5 instead of Theorem 4 to bound the FTRL regret, a better result should be achievable using our weighting scheme that improves upon Theorem 6 and possibly even matches Abbasi-Yadkori et al. (2011) exactly in the worst-case. We leave this for future work. 4.2 Experimental Evaluation In this subsection, we demonstrate that the practical applicability goes well beyond the Gaussian theoretical result from the previous subsection. In the examples below, we always use the UCB 8 Figure 2: Bandit experiments: On the y-axis we report cumulative regret, while the x-axis shows the number of iterations. In a) and b) we report the results for linear models with different parametric additive noise. In c) we report the results on a survival analysis with a log-Weibull distribution (p = 2) and in d) we showcase Poisson bandits. See App. E for more details. Heuristic methods are dashed, while provable are solid. Our sets perform the best among all provable methods. Notice in a) the difference in gray and black represents the improvement due to adaptive weighting over ws = 1 for all s ∈[t]. For each experiment we did 10 reruns, median values are plotted. algorithm but employ different confidence sets. In particular, we compare our LR confidence sets for different likelihood families with alternatives from the literature, notably classical sub-family confidence sets (Abbasi-Yadkori et al., 2011; Mutný and Krause, 2021), and the robust confidence set of Neiswanger and Ramdas (2021). In practice, however, the radius of these confidence sets is often tuned heuristically. We include such sets as a baseline without provable coverage as well. The main take-home message from the experiments is that among all the estimators and confidence sets that enjoy provable coverage, our confidence sets perform the best, on par with successful heuristics. For all our numerical experiments in Figure 2, the true payoff function is assumed to be an infinite dimensional RKHS element. For further details and experiments, please refer to App. E. Additive Noise Models Suppose that rθ⋆is linear and we observe ys = x⊤ s θ⋆+ ηs, where ηs is additive noise, and θ⋆is an element of a Hilbert space. We consider classical Gaussian noise as well as Laplace noise in Fig. 2[a), b)]. Notice that in both cases our confidence sets yield lower regret than any other provably valid method. In both cases they are performing as good as heuristic confidence sets with confidence parameter βt ≡2 log(1/δ). The sub-Gaussian confidence sets of Abbasi-Yadkori et al. (2011) (AY 2011) are invalid for the Laplace distribution as it is not sub-Gaussian but only sub-Exponential. For this reason, we compare also with sub-exponential confidence sets derived similarly to those of (Faury et al., 2020). The confidence sets of (Neiswanger and Ramdas, 2021) (NR 2021) perform similarly on Gaussian likelihood, but are only applicable to this setting, as their generalization to other likelihood families involves intractable posterior inference. We note also the difference be-tween the unweighted LR and the weighted one. The examples in Fig. 2 use the true payoff functions r(x) = −(1.4 −3x) sin(18x), which we model as an element of a RKHS with squared exponential kernel lengthscale γ = 6×10−2 on [0, 1.2], which is the baseline function no. 4 in the global optimiza-tion benchmark database infinity77 (Gavana, 2021). Additional experiments can be found in App. E. Poisson Bandits A prominent example of generalized linear bandits (GLB) are Poisson bandits, where the linear payoff is scaled by an exponential function. We instantiate our results on a common benchmark problem, and report the results in Fig. 2d). We improve the regret of UCB for GLBs compared to two alternative confidence sets: one that uses a Laplace approximation with a heuristic confidence parameter, and one inspired by considerations in Mutný and Krause (2021) (MK 2021), also with a heuristic confidence parameter. Note that we cannot compare directly to their provable results in their original form as they do not state them in the canonical form of the exponential family. 9 Survival Analysis Survival analysis is a branch of statistics with a rich history that models the lifespan of a service or product (Breslow, 1975; Cox, 1997; Kleinbaum and Klein, 2010). The classical approach postulates a well-informed likelihood model. Here, we use a specific hazard model, where the survival time T is distributed with a Weibull distribution, parametrized by λ and p. The rate λθ(x) = exp(x⊤θ) differs for each configuration x, and p – which defines the shape of the survival distribution – is fixed and known. We assume that the unknown part is due to the parameter θ which is the quantity we build a confidence set around to use within the UCB Algorithm. In particular, the probability density of the Weibull distribution is P(T = t\|x) = λθ(x)ptp−1 exp(−tpλθ(x)). In fact, with p = 2, the confidence sets are convex and the UCB rule can be implemented efficiently. Interestingly, this model admits an alternate linear regression formulation. Namely upon using the transformation Y = log T, the transformed variables Y \|x follow a Gumbel-type distribution, with the following likelihood that can be obtained by the change of variables P(Y = y\|x) = λθ(x)p exp(y)p exp(−exp(y)pλθ(x)). The expectation over Y allows us to express it as a linear regression problem since E[Y \|x] = −(θ⊤x + γ)/p, where γ is the Euler-Mascheroni constant. More importantly, Y \|x is sub-exponential. Hence, this allows us to use confidence sets for sub-exponential variables constructed with the pseudo-maximization technique inspired by Faury et al. (2020). More details on how these sets are derived can be found in App. E. However, these approaches necessarily employ crude worst-case bounds and as can be seen in Figure 2c) the use of our LR-based confidence sequences substantially reduces the regret of the bandit learner. 5 Related Work and Conclusion Related Work The adaptive confidence sequences stem from the seminal work of Robbins et al. (1972), who note that these sets have α-bounded Type I error. The likelihood ratio framework has been recently popularized by Wasserman et al. (2020) for likelihood families without known test statistics under the name universal inference. This approach, although long established, is surprisingly uncommon in sequential decision-making tasks like bandits. This might be due to the absence of an analysis deriving the size of the confidence sets (Mutný and Krause, 2021), a necessary ingredient to obtain regret bounds. We address this gap for generalized linear models. Another reason might be that practitioners might be interested in non-parametric sub-families – a scenario our method does not cover. That being said, many fields such as survival analysis (Cox, 1997) do have well-informed likelihoods. However, most importantly, if used naively, this method tends to fail when one departs from assumptions that our probabilistic model is identifiable (i.e., pθ(· \| x) = p˜ θ(· \| x) even if θ ̸= ˜ θ). We mitigate this problem by introducing the scaling parameters wt in Eq. (1) to deal with it. Prevalent constructions of anytime-valid confidence intervals rely on carefully derived concentration results and for a specific estimator such as the least-squares estimator and noise sub-families such as sub-Gaussian, sub-Bernoulli and sub-Poisson Abbasi-Yadkori et al. (2011); Faury et al. (2020); Mutný and Krause (2021). Their constructions involve bounding the suprema of collections of self-normalized stochastic processes (Faury et al., 2020; Mutný and Krause, 2021; Chowdhury et al., 2022). To facilitate closed-form expressions, worst-case parameters are introduced that prohibitively affect the size of the sets – making them much larger than they need to be. Chowdhury et al. (2022) use the exact form of the likelihood to build confidence sets for parameters of exponential families. However, their approach is restricted to exponential family distributions. They use self-normalization and mixing techniques to explicitly determine the size of the confidence set and do not use an online learning subroutine as we do here. Neiswanger and Ramdas (2021) use likelihood ratios for bandit optimization with possibly misspecified Gaussian processes but is not tractable beyond Gaussian likelihoods. The relation between online convex optimization and confidence sets has been noted in so-called online-to-confidence conversions (Abbasi-Yadkori et al., 2012; Jun et al., 2017; Orabona and Jun, 2021; Zhao et al., 2022), where the existence of a low-regret learner implies a small confidence set. However, these sets still use potentially loose regret bounds to define confidence sets. Our definition is implicit. We do not necessarily need a regret bound to run our method, as the radius will depend on the actual, instance-dependent performance of the learner. Conclusion In this work, we generalized and analyzed sequential likelihood ratio confidence sets for adaptive inference. We showed that with well-specified likelihoods, this procedure gives small, any-time valid confidence sets with model-agnostic and precise coverage. For generalized linear models, we quantitatively analyzed their size and shape. We invite practitioners to explore and use this very versatile and practical methodology for sequential decision-making tasks. 10 Acknowledgments and Disclosure of Funding We thank Wouter Koolen and Aaditya Ramdas for helpful discussions as well as for organizing the SAVI workshop where these discussions took place. NE acknowledges support from the Swiss Study Foundation and the Zeno Karl Schindler Foundation. MM has received funding from the Swiss National Science Foundation through NFP75. This publication was created as part of NCCR Catalysis (grant number 180544), a National Centre of Competence in Research funded by the Swiss National Science Foundation. References Abbasi-Yadkori, Y., Pál, D., and Szepesvari, C. (2011). Improved algorithms for linear stochastic bandits. In Advances in Neural Information Processing Systems, pages 2312–2320. Abbasi-Yadkori, Y., Pal, D., and Szepesvari, C. (2012). Online-to-confidence-set conversions and application to sparse stochastic bandits. 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Starting with E [Rt(θ⋆) \| Ft−1] = E " Rt−1(θ⋆) pwt ˆ θt (yt \| xt) pwt θ⋆(yt \| xt) Ft−1 # = Rt−1(θ⋆) Z pwt ˆ θt (y \| xt) pwt θ⋆(y \| xt)pθ⋆(y \| xt)dy = Rt−1(θ⋆)e((wt−1)Dr wt(pθ⋆(xt),pˆ θt(xt)) ≤Rt−1(θ⋆). The second equality is due to the fact that Rt−1(θ⋆) only depends on x1, y1 through xt−1, yt−1. Since ˆ θt is Ft−1 measurable by assumption, pˆ θt is a density, and if wt < 1, the integral is equal to an exponential of the Rényi-divergence Dr wt(·, ·). The negativity of the exponent follows from wt < 1 and the non-negativity of the divergence. Note that in the "degenerate" case of wt = 1, we can easily see that the integral is over a density (cancellation), and hence also bounded by 1. The last part of the statement follows easily by using Ville’s inequality for supermartingales. All the elements of the above proof appear in Wasserman et al. (2020) albeit separately, and not with time-varying powered robust likelihoods. A.3 Proof of Theorem 2 (Bias) We will need a gradient characterization of strong-convexity, which we prove in the following lemma. Lemma 1 (Convexity: Gradient). Defining Ft(θ) = − t X s=1 E θ⋆[∇logθ p(ys \| xs) \| Fs−1], under the assumption pθ(ys\|xs) = −g(x⊤ s θ) and g is µ-strongly convex, we have for any θ ∈Θ: (Ft(θ) −Ft(θ⋆))⊤(θ −θ⋆) ≥\|\|θ −θ⋆\|\|2 Vµ;0 t . Proof. We assume that g is µ-strongly convex. Therefore, for any s ≤t, we get the two inequalities g(x⊤ s θ) −g(x⊤ s θ⋆) ≥g′(x⊤ s θ⋆)(x⊤ s θ −x⊤ s θ⋆) + µ 2 \|\|x⊤ s θ⋆−x⊤ s θ\|\|2 2 g(x⊤ s θ⋆) −g(x⊤ s θ) ≥g′(x⊤ s θ)(x⊤ s θ⋆−x⊤ s θ) + µ 2 \|\|x⊤ s θ⋆−x⊤ s θ\|\|2 2. Adding these two together, we obtain 0 ≥(g′(x⊤ s θ⋆) −g′(x⊤ s θ))(x⊤ s (θ −θ⋆)) + µ\|\|θ −θ⋆\|\|2 xsx⊤ s . Observing that −∇θpθ(ys \| xs) = −g′(x⊤ s θ)xs, we can equivalently write 0 ≥(∇log pθ(ys \| xs) −∇log pθ⋆(ys \| xs))⊤(θ −θ⋆)) + µ\|\|θ −θ⋆\|\|2 xsx⊤ s . 14 This holds for any realization of ys, and hence taking expectations yields (E[∇log pθ⋆(ys \| xs) \| Fs−1] −E[∇log pθ(ys \| xs) \| Fs−1])⊤(θ −θ⋆) ≥µ\|\|θ −θ⋆\|\|2 xsx⊤ s . Summing up over s ≤t and using the definition of Ft we get (Ft(θ) −Ft(θ⋆))⊤(θ −θ⋆) ≥\|\|θ −θ⋆\|\|2 Vµ t ;0. Notice that the estimator in Alg. 1 has to fulfil the KKT conditions. We will denote the condition for belonging to the set as h(θ) ≤B2, where h is a squared and twice-differentiable norm (there are many choices beyond \|\| · \|\|2 2). The KKT conditions are t X s=1 −∇θ log pθ(ys\|xs) + ∇ψt(θ) + l∇h(θ) = 0 (12) l(h(θ) −B2) = 0 l ≥0, where the second and third conditions represent a complementary slackness requirement. Notice that the system of these equations has a unique solution due to the strong-convexity of the objective, and has to attain a unique minimum on a compact convex subset of Rd. Adding the same quantity on both sides of (12) yields t X s=1 −Eθ⋆[∇θ log pθ(ys\|xs)\|Fs−1] + ∇ψt(θ) + l∇h(θ) = t X s=1 [∇θ log pθ(ys\|xs) −Eθ⋆[∇θ log pθ(ys\|xs)\|Fs−1]] \| {z } :=Et . (13) This line motivates the definition of the error-free estimator in θ× t (6), where Et is set to zero. We will also make use of a fundamental property of the score (gradient of log-likelihood), namely Eyt∼pθ⋆(· \| x)[∇log pθ⋆(yt\|xt)\|Ft−1] = 0. (14) A classical textbook reference for this is e.g. McCullagh (2018) but any other classical statistics textbook should contain it. Using these observations, we can already prove Theorem 2. Proof of Theorem 2. Using the optimality conditions of θ× t , h(θ) = \|\|θ\|\|2 2 and ψt(θ) = \|\|θ\|\|2 2, we obtain the following statements: t X s=1 −Eθ⋆[∇log pθ× t (ys\|xs)\|Fs−1] + λθ× t + 2lθ× t = 0 = ⇒ t X s=1 −Eθ⋆[∇log pθ× t (ys\|xs)\|Fs−1] + Eθ⋆[∇log pθ⋆(ys\|xs)\|Fs−1] + λθ× t + 2lθ× t = 0, where in the last line we used the property (14). Now, notice that since we know θ⋆is generating the data, the best possible explanation without enforcing the constraint and the regularization would be to set θ× t = θ⋆as the cross-entropy is minimized at this point, and the above is just the optimality condition for optimizing the cross-entropy between these two distribution. Of course, this is only in the absence of regularization or constraints i.e. λ = 0. Now, with the regularization constraint, as the true θ⋆lies inside the constraint h(θ) ≤B2, and both the regularization and constraints induce star-shaped sets, their effect is to make θ× t smaller in norm than θ∗. This holds generally for any h which is a norm. As a consequence of this consideration, \|\|θ× t \|\|2 < B, and then the complementary slackness dictates that l = 0. We can therefore proceed with this simplification. Let us use the shorthand Ft(θ) = Pt s=1 −Eθ⋆[∇log pθ(ys\|xs)\|Fs−1] and compute Ft(θ× t ) −F(θ⋆) + λ(θ× t −θ⋆) = −λθ⋆ 15 = ⇒(θ× t −θ⋆)⊤(Ft(θ× t ) −F(θ⋆) + λ(θ× t −θ⋆)) = −λ(θ× t −θ⋆)⊤θ⋆ Lemma 1 = ⇒\|\|θ× t −θ⋆\|\|2 Vµ,λ t ≤−λ(θ× t −θ⋆)⊤θ⋆. (15) It suffices to apply the Cauchy-Schwarz Inequality and invoke (15): biasxs(ˆ θs)2 = (x⊤ s (ˆ θ× t −θ⋆))2 ≤ \|\|xs\|\|2 (Vµ,λ t )−1\|\|θ× t −θ⋆\|\|2 Vµ,λ t ≤ λ\|\|xs\|\|2 (Vµ,λ t )−1λ(θ× t −θ⋆)⊤(−θ⋆) ≤ λ\|\|xs\|\|2 (Vµ,λ t )−1\|\|(θ× t −θ⋆)\|\|2\|\|θ⋆\|\|2 ≤ 2λ\|\|xs\|\|2 (Vµ,λ t )−1\|\|θ⋆\|\|2 2, where in the last inequality we used \|\|θ× t \|\|2 ≤\|\|θ⋆\|\|2, due to the regularizer, as explained above. GLM models Let us define the processes St = t X s=1 xsT(ys) and Wt = t X s=1 xsA′(x⊤ s θ⋆). In this scenario, an equivalent of (13) then involves the gradient of the regularized (unweighted) log-partition function Zλ t we defined in (8) and is equal to t X s=1 A′(x⊤ s θ)xs + ∇ψt(θ) + l∇h(θ) = ˜ Et, (16) where ˜ Et = St for ˆ θt and ˜ Et = Wt for θ× t . B Proof of Theorem 3 (Bregman Ball Confidence Set) Proof sketch We give a quick sketch of the proof. To bound the size of the sets, we will draw inspiration from the i.i.d. parameter estimation analysis of Wasserman et al. (2020) and separate out the likelihood ratio in a part that relates the true parameter with the estimator sequence (i.e. regret), and a part that is independent of the estimator and characterized by a supremum of a stochastic process. We want to show that any point which is far away from the true parameter will eventually not be included in the confidence set anymore. Defining L(t)({ˆ θs}t s=1) as Qt i=1 pwi ˆ θi (yi \| xi), we wish to show that for any θ far from θ⋆, we have log 1 Rt(θ) = log L(t)(θ) L(t)(θ⋆) + log L(t)(θ⋆) L(t)({ˆ θs}t s=1) ≤log(α), which is equivalent to saying that θ ̸∈Ct. The second term corresponds to our notion of regret exactly (Rt, as discussed above). The first term is what we will focus on. We will bound the supremum of log L(t)(θ) L(t)(θ⋆) for all θ sufficiently far away from θ⋆. "Far away" will be measured in the Bregman divergence outlined above. Note that this quantity can be expected to be negative, in general, (especially for "implausible" parameters), since with enough data, θ⋆should appear much more likely. Writing this ratio out, we will observe that it is equal to −DZ0 t (θ, θ⋆) + ⟨θ −θ⋆, t X s=1 wsxs(T(ys) −E θ⋆[T(ys)])⟩ \| {z } ≈˜ St . At this point, it will be sufficient to bound the cross term (second term) over the whole of Θ. We view this supremum as part of the Legendre Fenchel transform of the function Bt(λ) = DZν t (θ⋆+ λ, θ⋆): sup λ∈Rd λ⊤˜ St −Bt(λ) = (Bt)⋆( ˜ St) and harness duality properties of the Bregman divergence, along with known concentration arguments (Chowdhury et al., 2022, Theorem A.1). 16 B.1 Technical Lemmas We need to introduce the concept of Legendre functions: Definition 1. Let f : Rd →R be a convex function and C = int(dom(f)). Then, a function is called Legendre if it satisfies 1. C is non-empty. 2. f is differentiable and strictly convex on C. 3. limn→∞\|\|∇f(xn)\|\| = ∞for any sequence (xn)n with xn ∈C for all n and limn→∞xn = x for some x ∈∂C. This means that the gradient has to blow up near the edge of the domain. Note as well that the boundary condition is vacuous if there is no boundary. Legendre functions have some nice properties, most importantly regarding the bijectivity of their gradients (see e.g. Lattimore and Szepesvári (2020)): Lemma 2. For a Legendre function f : Rd →R 1. ∇f is a bijection between int(dom(f)) and int(dom(f ∗)) with the inverse (∇f)−1 = ∇f ∗. 2. Df(x, y) = Df ∗(∇f(y), ∇f(x)) for all x, y ∈int(dom(f)). 3. The Fenchel conjugate f ∗is also Legendre. With this, we can prove a slightly extended result, that appears as Lemma 2.1. in Chowdhury et al. (2022). Lemma 3. For a Legendre function f we have the identity Df(x, y) = (Df,x)∗(∇f(y) −∇f(x)) where we define Df,x(λ) = Df(x + λ, x). Notational Shorthands Remember the model (7), with log-partition function A. We define As(θ) = wsA(x⊤ s θ) and Ts(y) := wsxsT(y) to denote the log-partition function and the response function of the same exponential family distribution, but parametrized by θ instead of x⊤ s θ. That this is a valid parametrization can easily be seen from the likelihood definition. Indeed, denote by pEF β the exponential family reward distribution with parameter β. Then our model (7) can be seen to satisfy pθ(y \| xs) = px⊤ s θ(y) = h(y) exp(T(y)x⊤ s θ −A(x⊤ s θ)) = h(y) exp(Ts(y)⊤θ −As(θ)). Exponentiating the likelihood with a weighting ws gives rise to another exponential family distribution. We can see that pws θ (y \| xs) = hws(y) exp(wsT(y)x⊤ s θ −wsA(x⊤ s θ)) = hws(y) exp(Ts(y)⊤θ −As(θ)). Note that this does not necessarily integrate to one, but it is easy to see that there is a normalization function ˜ h that makes it integrate to one. Therefore, the following is a valid parametrization of an exponential family distribution: ˜ h(y) exp(Ts(y)⊤θ −As(θ)). Additionally, let A0(θ) = ν 2\|\|θ\|\|2 2 be defined on Rd (i.e. a Legendre Function). We will also define the estimator ˜ θt = (∇Zν t )−1 t X s=1 Ts(ys) ! . This is a well-defined quantity because the gradient will be invertible, by Lemma 2 above. Conveniently, Chowdhury et al. (2022) prove the following Theorem 7 using an elegant application of the method of mixtures. 17 Proposition 1 (Theorem 7 in Chowdhury et al. (2022)). With probability 1 −δ, for all t ∈N DZν t (θ⋆, ˜ θt) ≤log(1/δ) + A0(θ⋆) + Γν t , where Γν t = log R Rd exp(−1 2\|\|θ\|\|2 2)dθ R Rd exp(−DZν t (θ, ˜ θt))dθ ! . Lastly, we will need the (one-argument) function Bt(λ) = DZν t (θ + λ, θ), i.e. a shortcut for the Bregman divergence of Zν t at θ. We use this one-argument function as we will be interested in its dual. We will also need a lemma on the sub-homogeneity properties of this object. Lemma 4. Under Assumption 1, for θ ∈Θ and λ such that θ + λ ∈Θ, we have for any γ ≤ µ 2L Bt(γλ) ≤1 2γBt(λ), i.e. function g(γ) = Bt(γλ) is sub-homogeneous with contraction parameter 1 2 on [0, µ 2L]. See Appendix B.3 for a proof. B.2 Proof of Theorem 3 As mentioned in the main paper, our proof will show that all θ sufficiently far from θ⋆will be excluded from Ct eventually. Equation (B) in the main text specifies the exclusion criterion, i.e. θ ̸∈Ct if and only if 1 Rt(θ) = log L(t)(θ) L(t)(θ⋆) + log L(t)(θ⋆) L(t)({ˆ θs}t s=1) ≤log(α). (17) The second term is bounded by the regret of the online learner. And therefore, a sufficient condition for θ ̸∈Ct is log L(t)(θ) L(t)(θ⋆) ≤log(α) −Rt. Henceforth, we will be interested in having an explicit set ˜ Ct such that we can upper bound sup θ / ∈˜ Ct log L(t)(θ) L(t)(θ⋆) . (18) This will imply that that ˜ Cc t ⊂Cc t , or in other words, Ct ⊂˜ Ct. Without further ado, let us derive a more convenient form of the ratio in question log L(t)(θ) L(t)(θ⋆) = log Qt s=1 h(ys) exp wsx⊤ s θT(ys) −wsA(x⊤ s θ) Qt s=1 h(ys) exp (wsx⊤ s θT(ys) −wsA(x⊤ s θ⋆)) ! = t X s=1 wsx⊤ s θT(ys) −wsA(x⊤ s θ) −wsx⊤ s θ⋆T(ys) + wsA(x⊤ s θ⋆) = t X s=1 ⟨θ −θ⋆, wsT(ys)xs⟩+ wsA(x⊤ s θ⋆) −wsA(x⊤ s θ) = t X s=1 ⟨θ −θ⋆, wsT(ys)xs⟩− wsA(x⊤ s θ) −wsA(x⊤ s θ⋆) −x⊤ s (θ −θ⋆)wsA′(x⊤ s θ⋆) + x⊤ s (θ −θ⋆)wsA′(x⊤ s θ⋆) = t X s=1 ⟨θ −θ⋆, wsT(ys)xs⟩−wsDA(x⊤ s θ, x⊤ s θ⋆) −x⊤ s (θ −θ⋆)wsA′(x⊤ s θ⋆) 18 = − t X s=1 wsDA(x⊤ s θ, x⊤ s θ⋆) + t X s=1 ⟨θ −θ⋆, wsT(ys)xs −xswsA′(x⊤ s θ⋆)⟩. We can switch parametrizations as described above: log L(t)(θ) L(t)(θ⋆) = −DZ0 t (θ, θ⋆) + t X s=1 ⟨θ −θ⋆, Ts(ys) −∇As(θ⋆)⟩ = −DZ0 t (θ, θ⋆) + t X s=1 ⟨θ −θ⋆, Ts(ys) −E θ⋆[Ts(ys)]⟩ = −DZ0 t (θ, θ⋆) + ⟨θ −θ⋆, St⟩, (19) where we define St := Pt s=1 (Ts(ys) −Eθ⋆[Ts(ys)]). Zν t is strictly convex whenever ν ̸= 0, and convex otherwise (it might also be strictly convex otherwise, corresponding to some cases where the xs span the full d-dimensional Euclidean space and ws > 0, which will be satisfied uniformly. We note that since dom(Zν t ) = Rd, Zν t is, therefore, Legendre, and its gradient is invertible. We will relate our problem to this estimator via the duality properties developed above. First, note that by the well-known fact Eθ⋆[Ts(ys)] = ∇As(θ⋆) and by the definition of ˜ θt, we have St = ∇Zν t (∇Zν t )−1 t X s=1 Ts(ys) !! −∇Z0 t (θ⋆) = ∇Zν t ˜ θt −∇Zν t (θ⋆) \| {z } =: ˜ St + ∇A0(θ⋆) \| {z } =νθ⋆ . (20) Now, we leverage the duality properties: We can write sup λ∈Rd λ⊤˜ St −Bt(λ) (i) = (Bt)⋆( ˜ St) (20) = (Bt)⋆(∇Zν t (˜ θt) −∇Zν t (θ⋆)) Lemma 3 = DZν t (θ⋆, ˜ θt), (21) where (i) is simply the definition of the Legendre-Fenchel transform. Why did we do all this work? Well, we are interested in the supremum in Equation (18). It is sufficient to bound the supremum over all θ ∈Θ of terms of the form (see Equation (19)) ⟨θ −θ⋆, St⟩. While we could do a covering type argument (carefully relaxing the i.i.d. data assumptions typical in empirical process theory), it is much easier to relate this supremum to the estimator via duality. With probability at least 1 −δ, Proposition 1 gives us a high-probability time-uniform bound on DZν t (θ⋆, ˜ θt) ≤log(1/δ) + A0(θ⋆) + Γν t , and therefore, by plugging into Equation (21) and making the reparametrization λ = γ(θ −θ⋆) for some positive γ, it gives us ∀t ≥0 ∀θ ∈Rd ∀γ ∈R+ : γ ˜ S⊤ t (θ −θ⋆) −Bt(γ(θ −θ⋆)) ≤log(1/δ) + A0(θ⋆) + Γν t . Therefore, for all t ≥0 and all θ ∈Rd, the following holds: S⊤ t (θ −θ⋆) = ˜ S⊤ t (θ −θ⋆) + ∇A0(θ⋆)⊤(θ −θ⋆) ≤1 γ log(1/δ) + 1 γ A0(θ⋆) + 1 γ Γν t + 1 γ Bt(γ(θ −θ⋆)) + ∇A0(θ⋆)⊤(θ −θ⋆). Since A0(θ) = ν 2\|\|θ\|\|2 2, restricting our uniform bound over θ ∈Θ gives us ∀t ≥0 ∀θ ∈Θ: S⊤ t (θ −θ⋆) ≤1 γ log(1/δ) + ν 2γ B2 + 1 γ Γν t + 1 γ Bt(γ(θ −θ⋆)) + νB2. 19 Now, we note that under Assumption 1, Lemma 4 kicks in and we have for any t ≥0, θ ∈Θ and γ = µ 2L S⊤ t (θ −θ⋆) ≤1 γ log(1/δ) + ν 2γ B2 + 1 γ Γν t + 1 2Bt(θ −θ⋆) + νB2. (22) Finally, we can use this in (19) to obtain log L(t)(θ) L(t)(θ⋆) ≤−Bt(θ −θ⋆) + ⟨θ −θ⋆, St⟩ (22) ≤−1 2Bt(θ −θ⋆) + 1 γ log(1/δ) + ν 2γ B2 + 1 γ Γν t + νB2 ≤−1 2Bt(θ −θ⋆) + 2L µ log(1/δ) + νB2 2 + Γν t + νB2. (23) It remains to investigate the full likelihood ratio in (17): 1 Rt(θ) −log(α) = log L(t)(θ) L(t)(θ⋆) + log L(t)(θ⋆) L(t)({ˆ θs}t s=1) + log(1/α) (23) & (3) ≤ −1 2Bt(θ −θ⋆) + 2L µ log(1/δ) + νB2 2 + Γν t + νB2 + log(1/α) + Rt. (24) Note that crucially for θ ∈Θ, we have θ ̸∈Ct ⇐ ⇒ 1 Rt(θ) −log(α) ≤0. This is implied by BZν t (θ, θ⋆) ≥4L µ log(1/δ) + νB2 2 + Γν t + 2νB2 + 2 log(1/α) + 2Rt, or, since L ≥µ, more compactly by BZν t (θ, θ⋆) ≥4L µ log(1/δ) + νB2 + Γν t + 2 log(1/α) + 2Rt. B.3 Proof of Technical Lemmas First, we will prove Lemma 3. The proof exactly follows Chowdhury et al. (2022), we include it here for convenience because it is very short. Proof. By definition (Df,x)∗(∇f(y) −∇f(x)) = sup a∈Rd (⟨a, ∇f(y) −∇f(x)⟩−Df,x(a)) = sup a∈Rd (⟨a, ∇f(y) −∇f(x)⟩−Df(x + a, x)) = sup a∈Rd (⟨a, ∇f(y) −∇f(x)⟩−f(x + a) + f(x) + ⟨∇f(x), a⟩) = sup a∈Rd (⟨a, ∇f(y)⟩−f(x + a) + f(x)) . Since f is strictly convex and differentiable, first-order optimality conditions imply that the optimal a satisfies ∇f(y) −∇f(x + a) = 0 (a is unconstrained). Since the gradient is invertible, we must have a = y −x. If we plug this into the above, we have (Df,x)∗(∇f(y) −∇f(x)) = ⟨y −x, ∇f(y)⟩−f(y) + f(x) = f(x) −f(y) −⟨∇f(y), x −y⟩ = Df(x, y). 20 Now we prove Lemma 4. To this end, we will do a reduction to the one-dimensional case, and prove the one-dimensional result below. Lemma 5. Under Assumption 1, for any a ∈[B, B], any γ ∈(0, µ 2L] and any ∆with a+∆∈[B, B] A(a + γ∆) −A(a) −A′(a)γ∆≤1 2γ [A(a + ∆) −A(a) −A′(a)∆] . We prove that this implies the desired sublinearity of the full Bregman difference. Proof. (of Lemma 4). Let θ and λ be such that θ, θ + λ ∈Θ. We will first show that for any s ∈{0, . . . , t}, BAs(θ,θ+·) is sublinear, and then the result follows by the linearity of the Bregman divergence. Define as = x⊤ s θ and ∆s = x⊤ s λ. Then we have \|∆s\| = \|x⊤ s (λ)\| ≤\|\|xs\|\|\|\|λ\|\| ≤ \|\|xs\|\|(\|\|θ\|\| + \|\|θ + λ\|\| ≤(B + B) ≤2B. Similarly we have \|as\| ≤B. Hence we satisfy the premise of Lemma 5 and we deduce that DAs(θ + γλ, θ) = wsA(x⊤ s θ + γx⊤ s λ) −wsA(x⊤ s θ) + ⟨xswsA′(x⊤ s θ), γλ⟩ = ws(A(as + γ∆s) −A(as) + A′(as)γ∆s) ≤ws 2 γ [A(as + γ∆s) −A(as) + A′(as)γ∆s] = 1 2γ wsA(x⊤ s θ + x⊤ s λ) −wsA(x⊤ s θ) + wsA′(x⊤ s θ)x⊤ s λ = 1 2γDAs(θ + λ, θ). We also note that for γ ≤ µ 2L ≤1 2, DA0(θ + γλ, θ) = ν 2\|\|γλ\|\|2 = γ2ν 2 \|\|λ\|\|2 ≤γν 4 \|\|λ\|\|2 = 1 2γDA0(θ + λ, θ). (25) Therefore, by summing up the terms, we obtain Bt(γλ) ≤1 2γBt(λ). Then it remains to prove that Assumption 1 implies Lemma 5. Proof. (Lemma 5) L-Lipschitzness of A′ implies smoothness of A. Additionally, µ’s existence implies strong convexity of A. With this, we can write for any a and ∆with a + ∆∈[−B, B] A(a + ∆) ≥A(a) + A′(a)∆+ µ 2 ∆2 = ⇒A(a + ∆) −A(a) −A′(a)∆≥µ 2 ∆2. Similarly, A(a + γ∆) −A(a) −A′(a)γ∆≤L 2 γ2∆2. Putting this together, we have A(a + γ∆) −A(a) −A′(a)γ∆≤L 2 γ2∆2 = Lγ2 µ µ 2 ∆2 ≤Lγ µ γ[A(a + ∆) −A(a) −A′(a)∆]. The question is therefore: when is Lγ µ ≤1 2? Clearly, choosing γ0 = µ 2L makes Lγ µ ≤1 2 for all γ ≤γ0. 21 C FTRL Results: Proofs C.1 Technical Lemmas I: Exponential Families Lemma 6 (MGF for Exponential family). E[exp(T(y)u)\|x] = exp(A(θ⋆ ⊤x + u) −A(θ⋆ ⊤x)). Proof. E[exp(T(y)u)\|x] = Z y exp(T(y)u)h(y) exp(T(y)θ⋆ ⊤x −A(θ⋆ ⊤x))dy = Z exp(T(y)(θ⋆ ⊤x + u))h(y) exp(−A(θ⋆ ⊤x)) × exp(−A(θ⋆ ⊤x + u)) exp(A(θ⋆ ⊤x + u))dy = exp(A(θ⋆ ⊤x + u) −A(θ⋆ ⊤x)), where the last step follows because the density of a new exponential family distribution with parameter θ⋆ ⊤x + u also integrates to 1. C.2 Technical Lemmas II: Elliptical Potential Lemma We will repeatedly use instantiations of the following key lemma, known as the elliptical potential lemma. We will use the version from Hazan et al. (2006). Other variants are stated in Abbasi-Yadkori et al. (2011) or Carpentier et al. (2020). Lemma 7 (Lemma 11 in Hazan et al. (2006)). Let us ∈Rd be a sequence of vectors such that \|\|us\|\| ≤r. Define ¯ Vt = Pt s=1 usu⊤ s + λI. Then t X s=1 \|\|us\|\|2 ¯ V−1 s ≤log det ¯ Vt det λI ≤d log r2t λ + 1 . We will also need a result where the time indices of the matrix are shifted. For this, note that if λ ≥r2, then usu⊤ s ⪯r2I ⪯λI, and so we get ¯ Vs ≤¯ Vs−1 + usu⊤ s ⪯¯ Vs−1 + λI ⪯2 ¯ Vs−1. Under our conditions, it follows that t X s=1 \|\|us\|\|2 ¯ V−1 s−1 ≤2 t X s=1 \|\|us\|\|2 ¯ V−1 s Corollary 1. We have the following bounds: γλ t = log det(P s=1 µxsx⊤ s + λI) det(λI) ≤d log µt λ + 1 , and t X s=1 \|\|xs\|\|2 (Vµ;λ s−1)−1 ≤2 µγλ t . Proof. The first bound is trivial by instantiating us = √µxs. The second bound is by noting t X s=1 \|\|xs\|\|2 (Vµ;λ s−1)−1 = 1 µ t X s=1 \|\|us\|\|2 (Vµ;λ s−1)−1 ≤2 µ t X s=1 \|\|us\|\|2 (Vµ;λ s )−1 ≤2 µγλ t . 22 C.3 Technical Lemmas III: Supermartingales Lemma 8 (Martingale Increment). Define the parametrized random processes Mj(r) = exp(∇fj(θ⋆)⊤r −A′(x⊤ j θ⋆)x⊤ j r −A(x⊤ j θ⋆−x⊤ j r) + A(x⊤ j θ⋆)) and Nj(r) = exp(∇fj(θ⋆)⊤r −L 2 r⊤xjx⊤ j r). Then, under Assumption 1 we have for any r ∈Rd that E[Mj(r) \| Fj−1] = 1 and E[Nj(r) \| Fj−1] ≤ 1. Proof. First, using the form of the exponential family and and recalling that ∇θfj(θ) = −∇θ log pθ(yj \| xj) = ∇θ[A(x⊤ j θ) −T(yj)x⊤ j θ] we obtain E[exp(∇fj(θ⋆)⊤r) \| Fj−1] = Z y exp(∇fj(θ⋆)⊤r) × h(y) exp(T(y)x⊤ j θ⋆−A(x⊤ j θ⋆))dy = Z y exp(−T(y)x⊤ j r + A′(x⊤ j θ⋆)x⊤ j r) × h(y) exp(T(y)x⊤ j θ⋆−A(x⊤ j θ⋆))dy = exp(A′(x⊤ j θ⋆)x⊤ j r) Z y h(y) exp(T(y)(x⊤ j θ⋆−x⊤ j r)) exp(−A(x⊤ j θ⋆−x⊤ j r))dy \| {z } =1 × exp(A(x⊤ j θ⋆−x⊤ j r)) exp(−A(x⊤ j θ⋆)) = exp(A′(x⊤ j θ⋆)x⊤ j r) exp(A(x⊤ j θ⋆−x⊤ j r)) exp(−A(x⊤ j θ⋆)), which finishes the proof. The second statement follows by using L-smoothness on the last equation and therefore noting that Nj(r) ≤Mj(r). Lemma 9. (Sequential Mixing) Define the martingale process, Mt(r1, . . . rt) = t Y s=1 Ns(rs), and recursively define the mixture martingale, ¯ Ms = ¯ Ms−1 × Z r Ns(r)ps(r)dr, where ps is a probability distribution equal N(0, H−1 s ), Hs = Ps−1 j=1 Lxjx⊤ j + Iλ L µ , and ¯ M0 = 1. Then the following statements hold • { ¯ Ms}s is an adapted super-martingale with respect to the usual filtration. • ¯ Mt = exp( µ L Pt s=1 ∇fs(θ⋆)⊤(Vµ;λ s )−1∇fs(θ⋆)) q det(Iλ) det(Vµ;λ s ). where Vµ;λ s = s X j=1 µxjx⊤ j + λI. Proof. The first point follows from the fact that ps(r) is deterministic conditioned on the sub-σ-algebra Fs−1 (since ps makes use of xs but not xs+1). Therefore, under mild regularity conditions E[ ¯ Ms \| Fs−1] = E ¯ Ms−1 Z ps(r)Ns(r)dr \| Fs−1 = ¯ Ms−1 Z r ps(r) E[Ns(r) \| Fs−1]dr ≤¯ Ms−1. 23 In other words, mixing does not affect the supermartingale properties. For the second point, we derive an explicit form of the mixture martingale. Note that we can write out Z r Ns(r)ps(r)dr = 1 q (2π)d det(H−1 s ) Z r exp ∇fs(θ⋆)⊤r −L 2 \|\|r\|\|2 xsx⊤ s −1 2r⊤Hsr dr. (26) We can complete the square to obtain ∇fs(θ⋆)⊤r −L 2 \|\|r\|\|2 xsx⊤ s −1 2r⊤Hsr = 1 2\|\|∇fs(θ⋆)\|\|2 (Hs+Lxsx⊤ s )−1 −1 2\|\|r −(Hs + Lxsx⊤ s )−1∇fs(θ⋆)\|\|2 Hs+Lxsx⊤ s . The second term is the exponent of a exponent of a Gaussian integral with covariance H−1 s+1, and therefore results in Z r exp −1 2\|\|r −(Hs + Lxsx⊤ s )−1∇fs(θ⋆)\|\|2 Hs+Lxsx⊤ s dr = q (2π)d det(H−1 s+1). Plugging this into (26) we get Z r Ns(r)ps(r)dr = s det Hs det Hs+1 exp 1 2\|\|∇fs(θ⋆)\|\|2 H−1 s+1 . By multiplying the individual steps, we can see that the determinant terms cancel in a telescoping product. This leads to the formulation ¯ Mt = exp t X s=1 ∇fs(θ⋆)⊤H−1 s+1∇fs(θ⋆) ! s det λL µ I det Ht+1 . To conclude the proof, note that Hs+1 = L µ Vµ;λ s . Lemma 10. Under assumption of Lemma 9, P t X s=1 \|\|∇fs(θ⋆)\|\|2 (Vµ;λ s )−1 ≤L µ log det(Vµ;λ s ) det(Iλ) + L µ log 1 δ ! ≤δ. (27) with probability 1 −δ. Proof. The statement, follows by applying Ville’s inequality for supermartingales, applying the logarithm, and rearranging. Namely, P( ¯ Mt ≥δ) = P(log( ¯ Mt) ≥log(δ)) ≤δ. The following results allow us to upper bound the weighted regret by the unweighted regret: Lemma 11 (Weighting Reduction). Let {θs}t s=1 be a sequence of vectors adapted to the filtration {Fs−1}s. Define ∆t({θs}) = t X s=1 ws(fs(θs) −fs(θ⋆)) −fs(θs) + fs(θ⋆) = t X s=1 (1 −ws)(fs(θ⋆) −fs(θs)). Then, Pt = exp(∆t({θs}s)) is a non-negative super-martingale for any choice of adapted {ws}, and hence, t X s=1 ws(fs(θs) −fs(θ⋆)) ≤ t X s=1 (fs(θs) −fs(θ⋆)) + log 1 δ with probability 1 −δ for all t ≥0. 24 Proof. E[Pt \| Ft−1] = Eθ⋆ " exp t X s=1 −(1 −ws)fs(θs) + (1 −ws)fs(θ⋆) ! Ft−1 # = Pt−1 E yt∼P⋆exp(−(1 −wt)ft(θt) + (1 −wt)ft(θ⋆)) = Pt−1 Z yt exp(−(1 −wt)ft(θt) + (1 −wt)ft(θ⋆)) exp(−ft(θ⋆))dyt = Pt−1 Z yt exp(−(1 −wt)ft(θt) −wtft(θ⋆))dyt = Pt−1 Z yt pθt(yt \| xt)1−wtpθ⋆(yt \| xt)wtdyt = Pt−1 exp(−(1 −wt)Dwt(θ⋆, θt)) ≤Pt−1. We have used here the definition of the Renyi-divergence and the fact that it is always non-negative, namely Dw(θ1, θ2) = 1 w −1 log Z y pθ1(y \| x)1−wpθ1(y \| x)wdy ≥0, for 0 < w ̸= 1.3 The rest follows by the application of Ville’s inequality. C.4 FTRL Proof: the Unweighted Case Proof of Theorem 4 (first part). We define the function that FTRL minimizes in each step (to pick ˆ θt) as, gt(θ) = Pt−1 s=1 −log pθ(ys \| xs) + λ\|\|θ\|\|2 2. We can rewrite this objective as ˆ θt = arg min θ∈Θ gt(θ) = arg min θ∈Θ t−1 X s=1 fs(θ) + λ\|\|θ\|\|2 2 = arg min θ∈Θ t−1 X s=1 ms(θ) + ϕt(θ), where we recall that4 fs(θ) = A(x⊤ s θ) −T(ys)x⊤ s θ −log h(ys), and we have introduced the shorthands ms(θ) = −T(ys)x⊤ s θ and ϕt(θ) = t−1 X s=1 A(θ⊤xs) + λ\|\|θ\|\|2 2. In essence, we have shifted some of the objective into what is commonly looked at as the regularizer. By a standard telescoping sum argument, we obtain for any u t X s=1 (ms(ˆ θs) −ms(u)) = ϕt+1(u) −min θ ϕ1(θ) + t X s=1 [gs(ˆ θs) −gs+1(ˆ θs+1) + ms(ˆ θs)] + gt+1(ˆ θt+1) −gt+1(u) \| {z } ≤0 ≤ ϕt+1(u) + t X s=1 [gs(ˆ θs) −gs+1(ˆ θs+1) + ms(ˆ θs)] = ϕt+1(u) + t X s=1 [gs(ˆ θs) −gs+1(ˆ θs+1) + gs+1(ˆ θs) −ϕs+1(ˆ θs) −gs(ˆ θs) + ϕs(ˆ θs)] 3The case wt = 1 is trivial for us. 4The log h(ys) term does not play any role in the regret nor the FTRL objective. 25 = ϕt+1(u) + t X s=1 [gs+1(ˆ θs) −gs+1(ˆ θs+1) −ϕs+1(ˆ θs) + ϕs(ˆ θs)]. Now we use the strong-convexity of gs+1 under the norm \|\|·\|\|Vµ;λ s where Vµ;λ s = Ps j=1 µxsx⊤ s +λI, t X s=1 (ms(ˆ θs) −ms(u)) ≤ ϕt+1(u) + t X s=1 [(ˆ θs −ˆ θs+1)⊤∇gs+1(ˆ θs) −1 2(ˆ θs −ˆ θs+1)⊤Vµ;λ s (ˆ θs −ˆ θs+1) −ϕs+1(ˆ θs) + ϕs(ˆ θs)] ≤ ϕt+1(u) + t X s=1 [(ˆ θs −ˆ θs+1)⊤∇fs(ˆ θs) −1 2(ˆ θs −ˆ θs+1)⊤Vµ;λ s (ˆ θs −ˆ θs+1) −ϕs+1(ˆ θs) + ϕs(ˆ θs)] ≤ ϕt+1(u) + t X s=1 1 2\|\|∇fs(ˆ θs)\|\|2 (Vµ;λ s )−1 −ϕs+1(ˆ θs) + ϕs(ˆ θs) , where in the second inequality we used that ∇gs(ˆ θs)⊤(x −ˆ θs) ≥0 due to the first-order optimality conditions for convex constrained minimization. Lastly, we optimized the resulting quadratic function over ˆ θs+1 (over Rd) to get a worst case bound involving the dual-norm. Note that for the shorthands we defined above: t X s=1 [−ϕs+1(ˆ θs) + ϕs(ˆ θs)] = t X s=1 −A(ˆ θ⊤ s xs). Using our previous observations and the definition of ϕt+1(θ⋆), we get for the overall regret: Rt = t X s=1 fs(ˆ θs) −fs(θ⋆) = t X s=1 ms(ˆ θs) −ms(θ⋆) + t X s=1 A(x⊤ s ˆ θs) −A(x⊤ s θ⋆) = t X s=1 A(x⊤ s θ⋆) −A(x⊤ s ˆ θs) + t X s=1 A(x⊤ s ˆ θs) −A(x⊤ s θ⋆) + 1 2 t X s=1 \|\|∇fs(ˆ θs)\|\|2 (Vµ;λ s )−1 + λ\|\|θ⋆\|\|2 ≤1 2 t X s=1 \|\|∇fs(ˆ θs)\|\|2 (Vµ;λ s )−1 + λ\|\|θ⋆\|\|2 ≤1 2 t X s=1 \|\|T(ys)xs −A′(x⊤ s ˆ θs)xs\|\|2 (Vµ;λ s )−1 + λ\|\|θ⋆\|\|2 ≤ t X s=1 h \|\|T(ys)xs −A′(x⊤ s θ⋆)xs\|\|2 (V µ;λ s )−1 + \|\|(A′(x⊤ s ˆ θs) −A′(x⊤ s θ⋆))xs\|\|2 (Vµ;λ s )−1 i + λ\|\|θ⋆\|\|2 ≤ t X s=1 h \|\|T(ys)xs −A′(x⊤ s θ⋆)xs\|\|2 (V µ;λ s )−1 + 2L2B2\|\|xs\|\|2 (Vµ;λ s )−1 i + λ\|\|θ⋆\|\|2 ≤ t X s=1 h \|\|∇fs(θ⋆)\|\|2 (Vµ;λ s )−1 + 2L2B2\|\|xs\|\|2 (Vµ;λ s )−1 i + λB2 26 ≤λB2 + L µ γλ t + log 1 δ + t X s=1 2L2B2\|\|xs\|\|2 (Vµ;λ s )−1 ≤λB2 + L µ γλ t + log 1 δ + 2L2B2 µ γλ t . The last line follows because of Lemma 7, and the second to last one follows because of Lemma 10. Notice that if we wish to deal with arbitrary weights {wt}, we can simply resort to Lemma 11 and bound the weighted case with the unweighted case. In that case, we incur an additional additive log(1δ) term. C.5 FTRL Analysis: the Weighted Case (Vovk-Azoury-Warmuth Forecaster) Proof. We define the function that FTRL minimizes in each step (to pick ˆ θt) as ˜ gt(θ) = Pt−1 s=1[A(x⊤ s θ) −T(ys)x⊤ s θ] + ψt(θ) with ψt(θ) = A(x⊤ t θ) + λ\|\|θ\|\|2 2. We can rewrite this ob-jective as ˆ θt = arg min θ∈Θ ˜ gt(θ) = arg min θ∈Θ t−1 X s=1 ms(θ) + ϕt(θ), by introducing the shorthands ms(θ) = −T(ys)x⊤ s θ and (notice the difference in time index of the second sum when compared to the proof in the previous subsection): ϕt(θ) = t X s=1 A(θ⊤xs) + λ\|\|θ\|\|2 2. In addition consider the objective gt from the classical FTRL analysis in Section C.4. It is not used to run the online algorithm, but is helpful in our analysis. With our new components, it is equal to gt(θ) = t−1 X s=1 ms(θ) + t−1 X s=1 A(θ⊤xs) + λ\|\|θ\|\|2 2 = t−1 X s=1 ms(θ) + ϕt−1(θ), and its minimizer is ¯ θt = arg minθ∈Θ gt(θ). Also, consider a weighted version of the regularizer ¯ ϕt(θ) = t X s=1 wsA(x⊤ s θ) + λ\|\|θ\|\|2 2, which will be useful. We use a variant of a similar telescoping sum argument as in the previous proof of Section C.4. We specifically use θ⋆as the comparator to compete against. Notice that we insert a telescoping sum involving the objective gs, which is not the objective that our estimator is minimizing: t X s=1 ws(ms(ˆ θs) −ms(θ⋆)) (∗) = ¯ ϕt(θ⋆) −ϕ0(¯ θ1) + t X s=1 [gs(¯ θs) −gs+1(¯ θs+1) + wsms(ˆ θs)] + gt+1(¯ θt+1) −gt+1(θ⋆) \| {z } ≤0 + t X s=1 (1 −ws)fs(θ⋆) ≤ ¯ ϕt(θ⋆) + t X s=1 [ws(gs(¯ θs) −gs+1(¯ θs+1)) + wsms(ˆ θs)] + t X s=1 (1 −ws)(gs(¯ θs) −gs+1(¯ θs+1) + fs(θ⋆)) 27 (∗∗) = ¯ ϕt(θ⋆) + t X s=1 ws[gs(¯ θs) −gs+1(¯ θs+1) + gs+1(ˆ θs) −ϕs(ˆ θs) −gs(ˆ θs) + ϕs−1(ˆ θs)] t X s=1 (1 −ws)[gs(¯ θs) −gs+1(¯ θs+1) + fs(θ⋆)] (∗∗∗) ≤ ¯ ϕt(θ⋆) + t X s=1 ws[−gs+1(¯ θs+1) + gs+1(ˆ θs) −ϕs(ˆ θs) + ϕs−1(ˆ θs)] + ˜ ∆t. In (∗), we used the shorthands and definitions introduced above. In (∗∗), we used the identity gs+1(ˆ θs)−ϕs(ˆ θs)−gs(ˆ θs)+ϕs−1(ˆ θs) = ms(ˆ θs). Finally, for (∗∗∗), recall that ¯ θs is the minimizer of gs, and hence, gs(¯ θs)−gs(ˆ θs) ≤0. Next, define ˜ ∆t = Pt s=1(1−ws)[gs(¯ θs)−gs+1(¯ θs+1)+fs(θ⋆)]. We will bound this term later. Now, we use the strong-convexity of gs+1(θ) under the norm \|\|·\|\|Vµ;λ s where Vµ;λ s = Ps j=1 µxjx⊤ j + λI, namely gs+1(¯ θs+1) ≥gs+1(ˆ θs+1) + ∇gs+1(ˆ θs)⊤(¯ θs+1 −ˆ θs) + 1 2\|\|¯ θs+1 −ˆ θs\|\|2 Vµ;λ s . We can then proceed as follows: t X s=1 ws(ms(ˆ θs) −ms(u)) ≤ ˜ ∆t + ¯ ϕt(θ⋆) + t X s=1 ws[∇gs+1(ˆ θs)⊤(ˆ θs −¯ θs+1) −1 2\|\|¯ θs+1 −ˆ θs\|\|2 Vµ;λ s −ϕs(ˆ θs) + ϕs−1(ˆ θs)] ≤ ˜ ∆t + ¯ ϕt(θ⋆) + t X s=1 ws[(∇˜ gs(ˆ θs) + ∇ms(ˆ θs))⊤(ˆ θs −¯ θs+1) −1 2\|\|¯ θs+1 −ˆ θs\|\|2 Vµ;λ s −ϕs(ˆ θs) + ϕs−1(ˆ θs)] ≤ ˜ ∆t + ¯ ϕt(θ⋆) + 1 2 t X s=1 ws\|\|∇ms(ˆ θs)\|\|2 (Vµ s )−1 −ws(ϕs(ˆ θs) + ϕs−1(ˆ θs)), (28) where in the second to last line we used that ∇˜ gs(ˆ θs)⊤(x −ˆ θs) ≥0 for any x, due to the optimality of ˆ θs for the FTRL objective. In the last line, we optimized over ¯ θs+1 to get a worst-case bound on the quadratic function involving it. Also, note that for the shorthands we defined above: t X s=1 ws(−ϕs(ˆ θs) + ϕs−1(ˆ θs)) = t X s=1 ws(−A(ˆ θ⊤ s xs)). Our goal here is to upper bound the overall regret: Rt = t X s=1 ws(fs(ˆ θs) −fs(θ⋆)) = t X s=1 ws(ms(ˆ θs) −ms(θ⋆)) + t X s=1 ws(A(x⊤ s ˆ θs) −A(x⊤ s θ⋆)) ≤ ¯ ϕt(θ⋆) − t X s=1 wsA(x⊤ s ˆ θs) + 1 2 t X s=1 ws\|\|∇ms(ˆ θs)\|\|2 (Vµ s )−1 + ˜ ∆t + t X s=1 ws(A(x⊤ s ˆ θs) −A(x⊤ s θ⋆)) 28 = t X s=1 ws(A(x⊤ s θ⋆)) − t X s=1 wsA(x⊤ s ˆ θs) + 1 2 t X s=1 ws\|\|∇ms(ˆ θs)\|\|2 (Vµ s )−1 + λ\|\|θ⋆\|\|2 + ˜ ∆t + t X s=1 ws(A(x⊤ s ˆ θs) −A(x⊤ s θ⋆)) = ˜ ∆t + 1 2 t X s=1 ws\|\|∇ms(ˆ θs)\|\|2 (Vµ s )−1 + λ\|\|θ⋆\|\|2. (29) The first inequality follows by plugging in (28). We return back to the term ˜ ∆t, ˜ ∆t = t X s=1 (1 −ws)[gs(¯ θs) −gs+1(¯ θs+1) + fs(θ⋆)] = t X s=1 (1 −ws)[gs(¯ θs) −gs(¯ θs+1) −fs(¯ θs+1) + fs(θ⋆)] ≤ t X s=1 (1 −ws)(fs(θ⋆) −fs(¯ θs+1)) ≤L µ (γλ t + log(1/δ)), (30) where the second to last line is by the optimality of ¯ θs and the last one by the assumption in the theorem. Carrying on with the analysis, i.e. with (29), we insert the definition of ms and obtain Rt ≤∆t + 1 2 t X s=1 ws\|\|−T(ys)xs\|\|2 (V µ;λ s )−1 + λ\|\|θ⋆\|\|2 ≤∆t + t X s=1 h ws\|\|T(ys)xs −A′(x⊤ s θ⋆)xs\|\|2 (V µ;λ s )−1 + ws\|\|(A′(x⊤ s θ⋆))xs\|\|2 (V µ;λ s )−1 i + λ\|\|θ⋆\|\|2 (∗) ≤∆t + t X s=1 \|\|T(ys)xs −A′(x⊤ s θ⋆)xs\|\|2 (V µ;λ s )−1 + L2B2 t X s=1 ws\|\|xs\|\|2 (V µ;λ s )−1 + λ\|\|θ⋆\|\|2 (∗∗) = ∆t + t X s=1 \|\|∇fs(θ⋆)\|\|2 (Vµ s )−1 + L t X s=1 B2 1/L + bias2 xs(ˆ θs) \|\|xs\|\|2 (V µ;λ s )−1 + λB2 (∗∗∗) ≤ λB2 + 2L µ γλ t + log 1 δ + L t X s=1 B2 1/L + bias2 xs(ˆ θs) \|\|xs\|\|2 (V µ;λ s )−1. In (∗), we use ws ≤1 and the Lipschitzness of A′. In (∗∗), we use the definition of the weights. Finally, in (∗∗∗), we used Lemma 10 and (30). By substituting ∆γs = µ\|\|xs\|\|2 (V µ;λ s )−1, we finish the proof. The event in Lemma 10 holds with probability 1 −δ, completing the proof. C.6 FTRL Analysis: Beyond Global Smoothness In this subsection, we give alternative analysis which avoids the necessity to impose a global smoothness condition our likelihood; instead strong convexity within a bounded domain suffices, and we will only assume that ϵs := E x⊤ s θ⋆ [T(ys)] −T(ys) are sub-Exponential random variables, setting us apart from Zhao et al. (2022) which assume sub-Gaussianity. In particular, we can show the following theorem 29 Theorem 7. With probability 1 −δ, uniformly over time t ∈N, we have Rt ≤cd log2(t/δ)) log(t), where the universal constant c hides all constants independent of t, d and δ. C.6.1 Lemmas We state the following result on sub-Exponential random variables. Proposition 2 (Theorem 2.13 in Wainwright (2019)). If X is a centered sub-Exponential variable with some finite variance proxy, then there exist constants c1, c2 > 0 such that for any t > 0 P(\|X\| ≥a) ≤c1e−c2a. By some careful union bounds (akin to a stitching argument), we can also provide upper bounds on anytime-valid upper bounds on the process St = maxs≤t ϵs. Lemma 12. For any sequence (ϵs)∞ s=1 of sub-Exponential-variables, there exists a constant ˜ c independent of t such that P(∃t : max s≤t \|ϵs\| ≥˜ c log(s/δ)) ≤δ. Proof. (of Lemma 12) By Proposition 2, there exists c1, c2 > 0 such that we have P(\|ϵs\| ≥a) ≤ c1e−c2a for any fixed s. Note that c1e−c2a ≤δ is satisfied for a ≥ 1 c2 log(c1/δ) =: c3 log(c1/δ). Let us denote by Ei the event all j ∈[2i, 2i+1) ∩N satisfy the inequality \|ϵj\| < c3 log(c1(22i+1)/δ). For a single j, this happens with probability at least 1 −δ/22i+1. Therefore, by a union bound, as \|[2i, 2i+1) ∩N\| = 2i, we can bound the probability of the complement, namely P(Ec i ) ≤2i δ 22i+1 = δ 2i+1 . Now, by another union bound, we can conclude that P(∪∞ i=0Ec i ) ≤ ∞ X i=0 δ 2i+1 = δ 2 1 1 −1 2 = δ. Now we also have for any j in this range that 22i+1 ≤2j2, and therefore, if Ei holds, we have for any j ∈[2i, 2i+1) ∩N: ϵj ≤c3 log(c1(2j2)/δ) ≤2c3 log(2c1j/δ) ≤˜ c log(j/δ). We can immediately see that this implies P(∃t : max s≤t \|ϵs\| ≥˜ c log(s/δ)) ≤δ. as desired. C.6.2 Proof of Theorem 7 Our proof initially follows the FTRL regret bound proofs in the adversarial setting Hazan (2016); Orabona (2019). It also has overlap with the proof in Zhao et al. (2022). We define the function that FTRL minimizes in each step as (to pick ˆ θt) gt(θ) = t−1 X s=1 −log pθ(ys \| xs) + ϕ(θ) for convenience. We initially use the same steps as in Theorem 4 to see that for any u ∈Θ t X s=1 (fs(ˆ θs) −fs(u)) ≤ϕ(u) −min θ ϕ(θ) + t X s=1 [gs(ˆ θs) −gs+1(ˆ θs+1) + fs(ˆ θs)] + gt+1(ˆ θt+1) −gt+1(u) 30 ≤λB2 + t X s=1 [gs(ˆ θs) −gs+1(ˆ θs+1) + fs(ˆ θs)]. Similarly to the proof of Theorem 4 in Appendix C.4 we bound these increments by the dual norm of the gradient of the objective. gs(ˆ θs) −gs+1(ˆ θs+1) + fs(ˆ θs) ≤ \|\|∇fs(ˆ θs)\|\|2 (Vµ;λ s )−1 2 . (31) Now we note that ∇fs(θ) = A′(x⊤ s θ)xs −T(ys)xs. Using properties of the exponential family, we deduce that ∇fs(θ) = E x⊤ s θ[T(ys)] −T(ys) xs = E x⊤ s θ[T(ys)] − E x⊤ s θ⋆ [T(ys)] + E x⊤ s θ⋆ [T(ys)] −T(ys) xs. From here on out, we proceed more crudely than in our previous analyses, since we are only concerned with asymptotic behavior when d and t are large. Let us define U := sup θ∈Θ \| E x⊤ s θ[T(ys)] − E x⊤ s θ⋆ [T(ys)]\|, which is a model-dependent, deterministic quantity. Let us define the noise variables ϵs := E x⊤ s θ⋆ [T(ys)] −T(ys). We bound \|\|∇fs(θ)\|\|2 (Vµ;λ s )−1 ≤2(U 2 + ϵ2 s)\|\|xs\|\|2 (Vµ;λ s )−1. Note that the ϵs are centered, independent sub-Exponential variables, and as such are guaranteed to satisfy P(∃t : max s≤t \|ϵs\| ≥c4 log(s/δ)) ≤δ, by Lemma 12. This tells us that conditional on this event, we can upper bound for any t Rt ≤λB2 + 1 µ(U 2 + ˜ c2 log2(t/δ)) t X s=1 \|\|xs\|\|2 (Vµ;λ s )−1. for some constant ˜ c independent of t. By Lemma 7, there is thus a constant c′ independent of t and d such that Rt ≤c′d log2(t/δ)) log(t) = O(d log3(t)), with probability 1 −δ uniformly over t ∈N. D Regret Consequences for Stochastic Linear Bandits As a corollary of our analysis, we provide the regret for stochastic linear bandits that use our confidence sets within the LinUCB algorithm. Proof. We proceed in two parts: first, we instantiate Theorem 3 and then we follow the classical regret analysis for stochastic linear bandits. 31 Specializing the Bregman divergence results By Theorem 3, we know that for any ν > 0, we have that with probability 1 −δ, DZν t (θ, θ⋆) ≤4L µ ξt + 2 log 1 δ + 2Rt, (32) for all t, where ξt = log 1 α + νB2 + Γν t and Rt is the online convex optimization regret. We also recall that Zν t (θ) = t X s=1 wsA(x⊤ s θ) + ν 2\|\|θ\|\|2 2. In the Gaussian case, where A(z) = z2/(2σ2). This implies that ∇Zν t (θ) = t X s=1 ws σ2 xsx⊤ s θ + νθ = Wσ−2;ν t θ, where we have defined a weighted version of Vσ−2;ν t as Wσ−2;ν t = Pt s=1 wsxsx⊤ s σ2 +νI and therefore the Bregman divergence is given by DZν t (θ, θ⋆) = 1 2\|\|θ −θ⋆\|\|2 Wσ−2;ν t . We can also see that the Bregman information gain is given by Γν t = log R Rd exp(−1 2\|\|θ\|\|2 2)dθ R Rd exp(−DZν t (θ, ˜ θt))dθ ! = log   R Rd exp(−1 2\|\|θ\|\|2)2dθ R Rd exp(−1 2\|\|θ −θ⋆\|\|2 Wσ−2;ν t )dθ  . These Gaussian integrals are straightforward to evaluate. We know that Z Rd exp −1 2\|\|θ\|\|2 2 dθ = (2π)d/2p det((νId)−1). Similarly, Z Rd exp −1 2\|\|θ −θ⋆\|\|2 Wσ−2;ν t dθ = (2π)d/2 q det((Wσ−2;ν t )−1). Then, we can compute Γν t = log det(Wσ−2;ν t ) det(νI) ! = log det t X s=1 wsxsx⊤ s σ2ν + I !! . In the unweighted case, with which we proceed, we have Γν t = γν t , that is we recover the classical upper bound on the information gain (Srinivas et al., 2009). To summarize, we have specialized the bound (32) to say that for any θ ∈Ct, we have (since L = µ = 1/σ2) \|\|θ −θ⋆\|\|2 Vσ−2;ν t ≤8 log(1/α) + νB2 + γν t + 4 log(3/δ)) + 4Rt. Now, we instantiate the regret of the online learner using Theorem 4. With probability 1−δ, uniformly over t, we have Rt ≤λB2 + L µ γλ t + log 1 δ + 2L2B2 µ γλ t . (33) We get by chosing α = δ, and setting ν = λ that \|\|θ −θ⋆\|\|2 Vσ−2;λ t ≤8 log(1/δ) + νB2 + γν t + 4 log(1/δ)) + 4λB2 + 4(γλ t + log(1/δ)) + 8B2 σ2 γλ t ≤16 log(1/δ) + 12λB2 + 8 B2 σ2 + 1 γλ t =: βt. 32 Linear bandit regret analysis We are ready to proceed with the bandit analysis for the UCB Algorithm. We follow Lattimore and Szepesvári (2020) and bound the pseudo-regret, letting x⋆be the optimal action. We bound the instantaneous regret at step 0 ≤s ≤t as rs = ⟨θ⋆, x⋆−xs⟩ = ⟨θ⋆, x⋆⟩−⟨θ⋆, xs⟩ ≤max θ∈Cs−1⟨θ, x⋆⟩−⟨θ⋆, xs⟩ ≤max x∈X max θ∈Cs−1⟨θ, x⟩−⟨θ⋆, xs⟩ (∗) = max θ∈Cs−1⟨θ, xs⟩−⟨θ⋆, xs⟩ (∗∗) ≤\|\|˜ θs −θ⋆\|\|Vσ−2;λ s−1 \|\|xs\|\|(Vσ−2;λ s−1 )−1 ≤ p βs−1\|\|xs\|\|(Vσ−2;λ s−1 )−1 ≤ p βt\|\|xs\|\|(Vσ−2;λ s−1 )−1. The first inequality replaces θ⋆by the upper confidence bound for action x⋆, which is valid with probability 1 −α = 1 −δ uniformly over time. Then, (∗) uses the fact that xt is chosen to maximize the upper confidence bound. Finally (∗∗) defines the UCB parameter ˜ θt. By Corollary 1, we have t X s=1 \|\|xs\|\|2 (Vσ−2;λ s−1 )−1 ≤2σ2γλ t . Plugging all this together and using an ℓ1/ℓ2-norm inequality, we get Rt = t X s=1 rs ≤ p βt v u u tt t X s=1 \|\|xs\|\|2 (Vσ−2;λ s−1 )−1 ≤ q 2tβtσ2γλ t ≤ q 2σ2t(16 log(1/δ) + 12λB2 + 8γλ t + 8B2/σ2γλ t )γλ t ≤6√tγt σ q log(1/δ) + γλ t + σλ1/2B + B q γλ t . To summarize and to justify why this bound holds with probability 1 −3δ uniformly over time, note that we have bounded the probability of the FTRL bound (33) not holding for some t by δ. Then, the probability of (32) not holding for some t is at most δ. Finally, the anytime Type I error of our sets is also bounded by δ. A union bound therefore concludes the proof. D.1 Comparison to Abbasi-Yadkori et. al. (2011) We compare our result to the one from Abbasi-Yadkori et al. (2011). Under the assumption that λ ≥1, they show that the regret satisfies Rt ≤4 p td log(λ + t/d) √ λB + σ p 2 log(1/δ) + d log(1 + t/(λd) . Observe that there is a reparametrization for the regularizer to get even more similar bounds. If we take λ = ˜ λ/σ2 for some ˜ λ ≥1, our bound reads as 6 q tγ ˜ λ/σ2 t σ q log(1/δ) + γ ˜ λ/σ2 t + ˜ λ1/2B + B q γ ˜ λ/σ2 t . 33 Given that by Corollary 1, we have γ ˜ λ/σ2 t ≤d log t ˜ λ + 1 , we get almost matching bounds, up to an additional B q γ ˜ λ/σ2 t term blowing up the regret, which we attribute to the accumulation of bias without the reweighting scheme. The remaining differences are down to using slightly different versions of the elliptical potential lemma, trading off generality and tightness (Abbasi-Yadkori et al., 2011; Hazan et al., 2006; Lattimore and Szepesvári, 2020). E Experimental Details E.1 Calibration Plots In Figure 3 we report the calibration of heuristics as well as other theoretically motivated works. The other theoretically motivated works are very pessimistic and are not appropriately calibrated. Note that one caveat of reporting calibration is that it is very much influenced by the data collection scheme in the sequential regime. In our case we use a bandit algorithm to collect the data. Arguably, in this setting, regret might be a better measure rather than looking at the calibration of the confidence sets. Additionally, the calibration depends on the true value θ⋆. We report the results for zero parameter and a random parameter from a unit ball. We also report results for i.i.d. data. E.2 Baselines and Details In the following section, we describe the baseline we used in the comparison. The details of parameters used in the experiments can be found in Table 2. As there is no explicit statement for sub-exponential variables, we give a formal derivation in Appendix F. E.2.1 Sub-Exponential Random Variables: Confidence Sets For this baseline, we will assume a linear model with additive sub-exponential noise, namely that there is θ∗∈Θ such that yt = ⟨θ∗, xt⟩+ ηt, where ηt is (ν, γ)-conditionally sub-exponential (Wainwright, 2019). We let as usual Vν−2;λ t = t X s=1 xsx⊤ s ν2 λI. With this, one can prove the following time-uniform concentration result: Proposition 3. For any k ∈(0, 1), the following holds: P ∃t : \|\|ˆ θt −θ∗\|\|Vν−2;λ t ≥ p βSE ≤δ, where p βSE = √ λ\|\|θ⋆\|\|2 + √ λkB + d √ λkB log 1 1 −k + 1 √ λkB log (det(Vν−2;λ t ))1/2 δ det( √ λI) ! . The proof is very similar to prior work Faury et al. (2020); Mutný and Krause (2021), and can be found in Appendix F. This can readily be applied in the survival analysis (after a suitable transformation explained in the main paper) and Laplace noise experiments. E.2.2 Poisson Heuristics We implement two heuristics. One is a Bayesian formulation due to the Laplace method, and one is due to considerations in Mutný and Krause (2021) of how to construct a valid confidence set using the Fisher information. The Laplace method creates an ellipsoidal confidence set using the second-order information evaluated at the penalized maximum likelihood estimator. Namely, the second derivative of the likelihood evaluated at the maximum penalized likelihood ˆ θt is Vlaplace = t X s=1 exp(ˆ θ⊤ t xs)xsx⊤ s . 34 0.2 0.4 0.6 0.8 Confidence level 0.0 0.2 0.4 0.6 0.8 1.0 Empirical Coverage Gaussian Likelihood AY (2011) LR (Our Alg.1) LR (no weights) Heuristic = 2log(1/ ) ideal (a) ADAPTIVE Bandit sequence, random \|\|θ⋆\|\|2 = 1, σ = 0.1 0.2 0.4 0.6 0.8 Confidence level 0.0 0.2 0.4 0.6 0.8 1.0 Empirical Coverage Gaussian Likelihood AY (2011) LR (Our Alg.1) LR (no weights) Heuristic = 2log(1/ ) ideal (b) ADAPTIVE Bandit sequence, random \|\|θ⋆\|\|2 = 1, σ = 0.01 0.2 0.4 0.6 0.8 Confidence level 0.0 0.2 0.4 0.6 0.8 1.0 Empirical Coverage Gaussian Likelihood AY (2011) LR (Our Alg.1) LR (no weights) Heuristic = 2log(1/ ) ideal (c) ADAPTIVE Bandit sequence, θ⋆= 0, σ = 0.1 0.2 0.4 0.6 0.8 Confidence level 0.0 0.2 0.4 0.6 0.8 1.0 Empirical Coverage Gaussian Likelihood AY (2011) LR (Our Alg.1) LR (no weights) Heuristic = 2log(1/ ) ideal (d) ADAPTIVE Bandit sequence θ⋆= 0, σ = 0.01 0.2 0.4 0.6 0.8 Confidence level 0.0 0.2 0.4 0.6 0.8 1.0 Empirical Coverage Gaussian Likelihood AY (2011) LR (Our Alg.1) LR (no weights) Heuristic = 2log(1/ ) ideal (e) IID sequence, random \|\|θ⋆\|\|2 = 1, σ = 0.1 0.2 0.4 0.6 0.8 Confidence level 0.0 0.2 0.4 0.6 0.8 1.0 Empirical Coverage Gaussian Likelihood AY (2011) LR (Our Alg.1) LR (no weights) Heuristic = 2log(1/ ) ideal (f) IID sequence, random \|\|θ⋆\|\|2 = 1, σ = 0.01 0.2 0.4 0.6 0.8 Confidence level 0.0 0.2 0.4 0.6 0.8 1.0 Empirical Coverage Gaussian Likelihood AY (2011) LR (Our Alg.1) LR (no weights) Heuristic = 2log(1/ ) ideal (g) IID sequence, θ⋆= 0, σ = 0.01 0.2 0.4 0.6 0.8 Confidence level 0.0 0.2 0.4 0.6 0.8 1.0 Empirical Coverage Gaussian Likelihood AY (2011) LR (Our Alg.1) LR (no weights) Heuristic = 2log(1/ ) ideal (h) IID sequence θ⋆= 0, σ = 0.01 Figure 3: We plot the calibration diagram for data collected from a bandit game trying to optimize a ground truth function using the same model as in Fig. 2a). Instead of the test function, we use an explicit member of the confidence set to avoid a potential mismatch between models. We check after T = 15 whether θ⋆∈Ct and average over 200 runs. We see that the (LR) are more conservative than the ideal calibration, however, they are provably valid and substantially better than any theoretically motivated confidence sets. We also see that the heuristic is not calibrated and fails many times. We see that for i.i.d. data, our sets are somewhat conservative since the data is not adapted, and our approach is not necessary. We note that the results depend on θ⋆. We use this to define ellipsoidal confidence sets as \|\|ˆ θt −θ\|\|2 Vlaplace ≤2 log(1/δ). The other heuristic suggests using the worst-case parameter instead, namely Vmutny = t X s=1 exp(B)xsx⊤ s . This method would have provable coverage with a proper confidence parameter. Its derivation is beyond the scope of this paper. 35 Benchmark function dim \|X\| γ B λ Gaussian/Laplace σ/b 1D 1 26 0.06 4 1 0.15 Camelback 2 102 0.2 2 1 0.10 Table 2: Summary of experimental parameters E.2.3 NR (2021) This method follows from Neiswanger and Ramdas (2021). Per se, this method was not developed to be versatile in terms of likelihood but instead provides a confidence set on f, even if it originates from a misspecified prior. Nevertheless, it provides a likelihood-aware confidence sequence that is anytime-valid and doesn’t employ worst-case parameters, and hence is a good benchmark for our analysis. The confidence sets are of the form {θ \| log L(θ) ≤log(1/δ) + log(p(D))}, where log(p(D)) is the current log-evidence of the data given the Gaussian prior. For more informa-tion, see Neiswanger and Ramdas (2021). E.3 Additive Models We implemented two likelihoods, namely Gaussian and Laplace. We implemented the discretization of the domain \|X\|, and in the implementation we used Nystrom features defined on \|X\| providing the exact representation of the RKHS on the X, The points were chosen to be on a uniform grid. Notice that for the regularized estimator, we chose the rule of thumb λ = 1/B as is motivated in (Mutný and Krause, 2022). The laplace parameter was picked as b = 0.15 likewise. Note that Laplace distribution is sub-exponential with parameters (b, 2b2). We use 1/σ2 or 1/b respectively for the value L. Strictly speaking, the Laplace likelihood is not smooth, but a smoothed objective would most likely inherit a value depending on b. As we maintain coverage with any choice of weighting, we do not risk invalidity by using a heuristic choice for L. E.4 Survival Analysis We implemented the method exactly as specified, using the Weibull likelihood with parameter p = 2. Upon log-transformation, the Gumbel distribution is sub-exponential. To determine the parameter, consider the moment-generating function of the Gumbel distribution (β = 1/p in the canonical parameterization): E[eXt] = Γ(1 −t/2) exp(t) ≤exp(t2/2) for t < 1/2, hence, the sub-exponentiality parameter is 1, and we can use the above sub-exponential confidence sets with value b = 1. For the likelihood ratio code, we used L = exp(B), as this is the leading term of the Hessian of log-likelihood. The function is not smooth everywhere, but on a bounded domain, this turns out to be an appropriate scaling. E.5 Poisson Bandits In this case, we implemented a bandit game, where we used the parametrization rθ(x) ∼ Poisson(exp(−θ⊤Φ(x))), where Φ(x) is the RKHS evaluation operator, and θ is the unknown value. We used L = exp(B), as this is the leading term of the Hessian of log-likelihood in this parametriza-tion. The function is not smooth everywhere but on a bounded domain this is an appropriate scaling. E.6 Additional Benchmark Functions We focus on an additional baseline function: Camelback in 2D, a standard BO benchmark function. The results can be see in Figure 4. 36 Figure 4: Camelback function. 10 repeats with median and standard quantiles plotted. Note that our method is the best method with provable coverage. F Proof of Proposition 3 Define St and the shorthand Vt St = t X s=1 ηs xs ν2 and Vt := Vν−2;λ t = t X s=1 xsx⊤ s ν2 + λI. and the parametrized process Mt(x) = exp(⟨x, St⟩−1 2\|\|x\|\|2 Vt). Lemma 13. If ηt is (ν, γ)-conditionally sub-Exponential, then Mt(x) is a super-martingale on the ball {x ∈Rd \| \|\|x\|\|2 ≤ν2 γ } with M0(x) ≤1. Note that for γ →0 (sub-Gaussian case), this recovers Lemma 20.2. in Lattimore and Szepesvári (2020). Proof. It is easy to observe that for any x, we have exp S⊤ 0 x −1 2\|\|x\|\|2 V0 = exp −1 2\|\|x\|\|2 λI ≤1. For the first part, we can write E[Mt(x)\|Ft−1] = E[exp(⟨x, St⟩−1 2\|\|x\|\|2 Vt) \| Ft−1] = E[exp(⟨x, St−1⟩−1 2\|\|x\|\|2 Vt−1) exp( 1 ν2 ⟨x, ηtxt⟩− 1 2ν2 \|\|x\|\|2 xtx⊤ t ) \| Ft−1] = Mt−1(x) E[exp( 1 ν2 ηt⟨x, xt⟩) \| Ft−1] exp(−1 2ν2 \|\|x\|\|2 xtx⊤ t ), where in the last step we use that xt is Ft−1-measurable. Now, as long as 1 ν2 \|⟨x, xs⟩\| ≤1 γ , we can apply our definition of conditional sub-Exponential noise to bound E[exp(ηt 1 ν2 ⟨x, xt⟩)] ≤exp (⟨x, xt⟩)2ν2 2ν4 ≤exp \|\|x\|\|2 xtx⊤ t 2ν2 ! . 37 From this we directly conclude E[Mt(x)\|Ft−1] ≤Mt−1(x). By Cauchy-Schwarz, a sufficient condition is \|\|x\|\|2 ≤ν2 γ as this implies (with our assumptions on the actions) \|⟨x, xt⟩\| ≤\|\|x\|\|2\|\|xt\|\|2 ≤\|\|x\|\|2 ≤ν2 γ . In the following, will use this result to prove any-time confidence estimates for the parameter θ using the technique of pseudo-maximization, closely following Mutný and Krause (2021). Recall that Mt(x) is defined on the ball of radius ν2 γL. This allows us some freedom in choosing the radius of the ball on which we integrate. In particular, let K be this radius. While we need K ≤ν2 γL, we can make K larger by choosing larger ν2 (increasing ν2 only makes the set of noise distributions larger). Ultimately, we wish to bound (following Lattimore and Szepesvári (2020)) \|\|ˆ θt −θ∗\|\|Vt = \|\|V−1 t X1:t(X1:tθ⋆+ η1:t) −θ⋆\|\|Vt = \|\|V−1 t X1:tX1:tθ⋆−θ⋆+ V−1 t St\|\|Vt ≤\|\|St\|\|V−1 t + √ λ\|\|θ⋆\|\|2. We can not control the second term, so we focus on the first: the self-normalized residuals. Via fenchel duality, one can motivate that the right object to study is the supremum of the martingale Mt(x) over all x ∈R.5. Define ˜ Mt to be the martingale Mt from above but with λ = 0, i.e. no regularisation term. Similarly, let ˜ Vt = Vt −λI be the design matrix without the regularisation term. Slightly counterintuitively, we will study ¯ Mt = Z \|\|x\|\|2≤K ˜ Mt(x)dh(x), where h is the probability density function of a truncated normal distribution with inverse variance λ, that is with covariance matrix 1 λI. By Lemma 20.3 in Lattimore and Szepesvári (2020), ¯ Mt is also a super-martingale with ¯ M0 ≤1. Then we have ¯ Mt = 1 N(h) Z \|\|x\|\|2≤K exp x⊤St −1 2\|\|x\|\|2 ˜ Vt exp −1 2x⊤λIx dx = 1 N(h) Z \|\|x\|\|2≤K exp x⊤St −1 2\|\|x\|\|2 Vt dx. We will define the shorthand ft(x) = x⊤St −1 2x⊤Vtx = ft(x∗) + ∇ft(x∗)⊤(x −x∗) −1 2(x − x∗)⊤Vt(x −x∗) (by Taylor’s theorem), where x∗= arg max\|\|x\|\|≤kK ft(x), k ∈(0, 1) will be chosen later. We can lower bound ¯ Mt by ¯ Mt = 1 N(h) Z \|\|x\|\|2≤K exp x⊤St −1 2\|\|x\|\|2 Vt dx = exp(ft(x∗)) N(h) Z \|\|x\|\|2≤K exp ∇ft(x∗)⊤(x −x∗) −1 2(x −x∗)⊤Vt(x −x∗) dx = exp(ft(x∗)) N(h) Z \|\|y+x∗\|\|2≤K exp ∇ft(x∗)⊤y −1 2y⊤Vty dy (34) ≥exp(ft(x∗)) N(h) Z \|\|y\|\|2≤(1−k)K exp ∇ft(x∗)⊤y −1 2y⊤Vty dy (35) 5But that is in our case ill-defined 38 = exp(ft(x∗)) N(h) Z \|\|y\|\|2≤(1−k)K exp ∇ft(x∗)⊤y exp −1 2y⊤Vty dy = exp(ft(x∗))N(g) N(h) E y∼g exp ∇ft(x∗)⊤y ≥exp(ft(x∗))N(g) N(h) exp E y∼g[∇ft(x∗)⊤y] (36) = exp(ft(x∗))N(g) N(h) . where in step (34) we used the change of variables x = y + x∗. In (35) we use that if \|\|y\|\|2 ≤(1 −k)K, then \|\|x∗+ y\|\|2 ≤\|\|x∗\|\|2 + \|\|y\|\|2 ≤(1 −k)K + kK = K. Finally, in (36), we used Jensen’s inequality. The last inequality follows from symmetry. Note that we implicitly defined g to be a truncated normal distribution with covariance matrix V−1 t on the ball of radius (1 −k)K. This puts us in a position to put Ville’s inequality to good use: δ ≥P ∃t : log( ¯ Mt) ≥log(1/δ) ≥P ∃t : ft(x∗) + log N(g) N(h) ≥log(1/δ) ≥P ∃t : ft(x∗) ≥log N(h) N(g)δ . We now wish to recover \|\|St\|\|Vt. Recall the definition of ft(x∗) as the maximum over all x in a ball of radius kK. Consequently, we can choose x = V−1 t St \|\|St\|\|V−1 t √ λkK, which has norm bounded by kK. We have ft(x∗) ≥ft V−1 t St \|\|St\|\|V−1 t √ λkK ! = \|\|St\|\|V−1 t √ λkK −λk2K2, which immediately yields P \|\|St\|\|V−1 t ≥ √ λkK + 1 √ λkK log N(h) N(g)δ ≤δ. The only thing that remains is bounding log N(h) N(g) . We give the following Lemma that is a slightly generalized version of Mutný and Krause (2021) and originally inspired by Faury et al. (2020). Lemma 14. The normalizing constants satisfy log N(h) N(g) ≤d log 1 1 −k + log (det(Vt))1/2 det( √ λI) . We can use the bound from Lemma 14 to conclude that P \|\|St\|\|V−1 t ≥ √ λkK + d √ λkK log 1 1 −k + 1 √ λkK log (det(Vt))1/2 δ det( √ λI) ≤δ. We stated earlier that \|\|ˆ θt −θ∗\|\|Vt ≤\|\|St\|\|V−1 t + √ λ\|\|θ⋆\|\|2. Combining this with our analysis, we get the Proposition 3. We may now choose the parameters k, K and λ. Note that to get sub-Gaussian rates as in Abbasi-Yadkori, one needs to pick a regularization parameter of the order of λ = d log(T). 39 Proof of Lemma 14 We give a proof of the Lemma for completeness, and because the additional generality makes for a slightly different proof, even though the bound stays the same. Proof. We have N(h) = Z \|\|x\|\|2≤K exp(−λ\|\|x\|\|2 2)dx = 1 \|det( √ 2λI)\| Z \|\|x\|\|2≤K exp −1 2\|\| √ 2λx\|\|2 2 \|det( √ 2λI)\|dx = 1 \|det( √ 2λI)\| Z \|\|x\|\|2≤ √ 2λK exp −1 2\|\|x\|\|2 2 dx. Further we have N(g) = Z \|\|x\|\|2≤(1−k)K exp(−1 2x⊤Vtx)dx = 1 \|det(V1/2 t )\| Z \|\|x\|\|2≤(1−k)K exp(−1 2\|\|V1/2 t x\|\|2 2)\|det(V1/2 t )\|dx = 1 \|det(V1/2 t )\| Z S exp(−1 2\|\|x\|\|2 2)dx, where S = {V1/2 t x \| \|\|x\|\| ≤(1 −k)K} = {x \| \|\|V−1/2 t x\|\| ≤(1 −k)K} = {x \| x⊤V−1 t x ≤ (1 −k)K}. Note that Vt ⪰λI and so V−1 t ⪯1 λI. Therefore if \|\|x\|\|2 2 ≤(1 −k)K √ λ, we have q x⊤V−1 t x ≤ 1 √ λ \|\|x\|\|2 ≤(1 −k)K = ⇒x ∈S. Thus {x \| \|\|x\|\|2 ≤(1 −k) √ λK} ⊆S and N(g) ≥ 1 \|det(V1/2 t )\| Z \|\|x\|\|2≤(1−k) √ λK exp(−1 2\|\|x\|\|2 2)dx. We may therefore bound N(g) N(h) ≤(det Vt)1/2 (det √ 2λI) R \|\|x\|\|2≤ √ 2λK exp −1 2\|\|x\|\|2 2 dx R \|\|x\|\|2≤(1−k) √ λK exp(−1 2\|\|x\|\|2 2)dx. By a rather crude bound (as 1 −k ≤ √ 2 in any case) we get R \|\|x\|\|2≤ √ 2λK exp −1 2\|\|x\|\|2 2 dx R \|\|x\|\|2(1−k) √ λK exp(−1 2\|\|x\|\|2 2)dx ≤ R \|\|x\|\|2≤(1−k) √ λK exp −1 2\|\|x\|\|2 2 dx + R (1−k) √ λK≤\|\|x\|\|2≤ √ 2λK exp −1 2\|\|x\|\|2 2 dx R \|\|x\|\|2≤(1−k) √ λK exp(−1 2\|\|x\|\|2 2)dx = 1 + R (1−k) √ λK≤\|\|x\|\|2≤ √ 2λK exp −1 2\|\|x\|\|2 2 dx R \|\|x\|\|2≤(1−k) √ λK exp(−1 2\|\|x\|\|2 2)dx ≤1 + exp −1 2(1 −k)2λK2 exp −1 2(1 −k)2λK2 R (1−k) √ λK≤\|\|x\|\|2≤ √ 2λK dx R \|\|x\|\|2≤(1−k) √ λK dx = 1 + vold( √ 2λK) −vold((1 −k) √ λK) vold((1 −k) √ λK) = vold( √ 2λK) vold((1 −k) √ λK) 40 = (1 −k)−d√ 2 d. We can put this together to obtain N(h) N(g) ≤(1 −k)−d√ 2 d (det(Vt))1/2 det( √ 2λI) = (1 −k)−d (det(Vt))1/2 det( √ λI) . 41
18133	https://www.studiobinder.com/blog/what-is-a-hyperbole-definition-examples/	Skip to content Have you ever felt at a loss for words to describe what you are feeling or the point you are making? Odds are you’ve probably turned to using hyperbole. Hyperbole is used throughout common conversations, speech, rhetoric, film, and literature. What is a hyperbole? Why is it so commonly used and what is it effective at communicating? Let’s take a look at the function of this very specific and useful tool. Watch: What is Hyperbole? Subscribe for more filmmaking videos like this. Subscribe on YouTube hyperbole definition and examples First, let’s define hyperbole Hyperbole, along with many literary devices, are techniques to better communicate our ideas. What is the definition of hyperbole and how do we distinguish this from other literary devices? Before we present some classic and effective examples, let’s take a look at the hyperbole definition. Hyperbole Definition What is a hyperbole? Hyperbole is a literary device used to draw emphasis through extreme exaggeration. Hyperbole is not meant to be taken literally, but rather understood as a means of communicating something specific. Those who hear or read the hyperbole should understand that it is an exaggeration. You’ve probably heard common hyperboles in everyday conversations such as “I’m so hungry, I could eat a horse,” “I’ve seen this movie a hundred times,” or “It cost an arm and a leg.” What is hyperbole used for? Describe a feeling Emphasize a point Comedic delivery Examples of Hyperbole in a Sentence Hyperbole examples Before we break down the many uses of hyperbole, let's quickly review some classic hyperbole examples. As you run through these, you'll be able to see just how common and effective this figure of speech really is. I'm so hungry, I could eat a horse My feet are killing me That plane ride took forever This is the best book ever written I love you to the moon and back The pen is mightier than the sword I've told you this 20,000 times Cry me a river As you can see, we either use or encounter hyperbolic language on a daily basis in our everyday speech. So, what does that the hyperbole mean for writers? Whether you're crafting the next great American novel, writing a comedic screenplay, or advertising a product in a commercial, you'll need to have this technique in your toolkit. If you're writing dialogue, for example, it usually helps to use language that actually sounds how people talk. This is just one way to make that happen. Now, let's look at some specific applications and examples of hyperbole in a sentence. “My parents would have about two hemorrhages apiece if I told anything pretty personal about them.” J.D. Salinger, The Catcher in the Rye Here, Holden Caulfield is emphasizing just how much his parents would freak out if he divulged any information about them. The hyperbole (“two hemorrhages apiece”) not only heightens his point, but also accentuates Caulfield’s distinctive voice. “The brightness of her cheek would shame those stars.” William Shakespeare, Romeo and Juliet Romeo uses hyperbole several times when describing Juliet and his love for her. It’s all very fitting, since their love is so intense. There was a firestorm out there. Dresden was one big flame. The one flame ate everything organic, everything that would burn.” — Kurt Vonnegut, Slaughterhouse-Five Vonnegut employs hyperbole to get across just how all-consuming the firebombing of Dresden was, and how horrifying it felt to be in the middle of. Related Posts The Ultimate Guide to Literary Devices → Analyzing Symbolism in Literature and Film → What is a Metaphor? Definition and Examples → What is a hyperbole used for? How to use hyperbole in writing We either use or encounter hyperbolic language on a daily basis in our everyday speech. So, what does that mean for writers? Whether you’re crafting the next great American novel, writing a comedic screenplay, or advertising a product in a commercial, you’ll need to have this technique in your toolkit. If you’re writing dialogue, for example, it usually helps to use language that actually sounds how people talk. Let’s take a look at the use of hyperbole to that effect in this scene from National Lampoon’s Vacation, which we imported into StudioBinder’s screenwriting software: In this scene, Ellen is fed up with Clark. His tireless pursuit for a great vacation has made him ignore the actual dynamics and needs of the family around him. To cut through to him, Ellen uses hyperbole in this sentence to throw his pursuit in his face: “Tomorrow you’ll kill the desk clerk, hold up a McDonalds…” Of course, Ellen doesn’t actually think Clark will do this. But she’s making a point: his grand plans are constantly backfire, and are the opposite of fun. So the next time you’re writing, consider using some hyperbole to spice up your work. If you do, it will be the greatest thing ever written, invariably causing readers to faint with excitement. Related Posts Defining and Analyzing the Oxymoron → What is an Allegory? Definition and Examples → How Are Flashbacks Used in Film and Literature? → UP NEXT What is a Simile? Along with hyperbole, the simile is a common literary device used to emphasize a point. Rather than exaggeration, similes use comparison. Learn more about the simile in our next article where we analyze the simile definition as well as examples from film and literature. Up Next: What is a Simile? → Write and produce your scripts all in one place Write and collaborate on your scripts FREE. 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18134	https://www.mathsisfun.com/y_intercept.html	Y Intercept of a Straight Line Equation of a Straight Line Gradient (Slope) of a Straight Line Show Ads Hide Ads \| About Ads We may use Cookies OK Home Algebra Data Geometry Physics Dictionary Games Puzzles [x] Algebra Calculus Data Geometry Money Numbers Physics Activities Dictionary Games Puzzles Worksheets Hide Ads Show Ads About Ads Donate Login Close Y-Intercept of a Straight Line Where a line crosses the y-axis of a graph Just find the value of y when x equals 0: Example: In the above diagram the line crosses the y axis at y = 1 Example: Here the line crosses the y axis at y = −2 Point The y-intercept is an (x,y) point with x=0, so we show it like this (try dragging the points): geometry/images/geom-line-equn.js?mode=intery Mathopolis:Q1)Q2)Q3)Q4)Q5)Q6)Q7)Q8)Q9)Q10) Equation of a Straight LineGradient (Slope) of a Straight LineTest YourselfStraight Line Graph CalculatorGraph Index Donate ○ Search ○ Index ○ About ○ Contact ○ Cite This Page ○ Privacy Copyright © 2025 Rod Pierce
18135	https://www.wikihow.com/Find-the-Inverse-of-a-Quadratic-Function	Log in The Inverse of Quadratic Functions: Visual Steps & Solutions Last Updated: July 17, 2025 This article was reviewed by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been viewed 373,830 times. Inverse functions can be very useful in solving numerous mathematical problems. Being able to take a function and find its inverse function is a powerful tool. With quadratic equations, however, this can be quite a complicated process. First, you must define the equation carefully, be setting an appropriate domain and range. You then have a choice of three methods to calculate the inverse function. The choice of method is mostly up to your personal preference. Finding an Inverse Quadratic Function Starting with a function in the form y = ax2 + c, combine like terms to simplify the equation. Find the domain and range and switch x and y to invert the equation. Rewrite the inverted equation in terms of y, then find the range and domain of the inverse function. Test your function to make sure it works. Steps Finding the Inverse of a Simple Function Completing the Square to Determine the Inverse Function Using the Quadratic Formula Community Q&A Tips You Might Also Like References About This Article To find the inverse of a quadratic function, start by simplifying the function by combining like terms. Then, determine the domain and range of the simplified function. Once you have the domain and range, switch the roles of the x and y terms in the function and rewrite the inverted equation in terms of y. Finally, determine the domain and range of the inverse function. To learn how to find the inverse of a quadratic function by completing the square, scroll down! Did this summary help you?YesNo Reader Success Stories Dylan Truitt Dec 8, 2020 Did this article help you? Dylan Truitt Dec 8, 2020 G. M. May 27, 2017 Quizzes Do I Have a Dirty Mind Quiz Personality Analyzer: How Deep Am I? Am I a Good Kisser Quiz Rizz Game: Test Your Rizz "Hear Me Out" Character Analyzer What's Your Red Flag Quiz You Might Also Like How to Complete the Square to Solve a Quadratic Equation How to Find the Maximum or Minimum Value of a Quadratic Function Easily How to Find the Domain and Range of a Function How to Solve a Quadratic Equation: A Step-by-Step Guide Featured Articles How to Stop Instagram from Suggesting Adult Content Will I Get Together With My Crush Quiz The Easiest Way to Clean Your Room from Top to Bottom Wondering if Someone Likes You Online? 11 Important Signs to Watch Out For How to Increase Your Self Confidence with Positive Daily Practices How to Fix Painful Shoes Trending Articles How Rare Is Your Name? 5 Different Types of Butts: Find Your Shape How Will I Die Quiz How Many People Have a Crush On Me Quiz Labubu Blind Box Generator: Unbox & Collect Your Own Online Lababus How to Know if a Person Is Interested in You Featured Articles How to Use Castor Oil to Promote Healthy Hair How to Apply for an Internship Tips for Trimming Long Hair Evenly (Your Own or Someone Else’s) How to Whiten Teeth With Baking Soda 130+ Sexy, Sweet, & Seductive Messages for Him Something Bit Me! Common Insect Bites (with Pictures) Featured Articles Do I Have Common Sense Quiz 50 Cute & Flirty Knock-Knock Jokes to Make Them Smile How to Calm an Aggressive Cat How to Get a Flat Stomach in 7 Days: Exercises, Diet Tips, & More How Do You Know if She Likes You? 15+ Signs She’s Into You How to Flirt Watch Articles How to Remove Hard Water Spots from Glass: DIY Tips & Tricks How to Sew a Button How to Cook Egg-Fried Rice at Home How to Purposely Shrink Any Type of Clothing in the Washer and Dryer How to Become a Vegetarian How to Make Your Own Deep Conditioner Trending Articles What Is My Mental Age Quiz What Is Your Soul Animal Quiz What Animal Do I Look Like Quiz Never Have I Ever Generator Halloween Costume Idea Generator: Find the Perfect Trendy or Classic Costume The Ultimate Guide to Women’s Underwear & Panty Types Quizzes Am I Smart Quiz How Insecure Am I Quiz What Disney Princess Am I Quiz Do I Have a Phobia Quiz Guess My Age Quiz Am I a Genius Quiz Follow Us Get all the best how-tos! Sign up for wikiHow's weekly email newsletter
18136	https://www.mathmammoth.com/practice/area-model-multiplication	Multiplying with the Area Model (grades 4-5) MM Practice MM Practice × Multiplication Multiplication TablesMultiplication Matching GameWith ZerosSquare Roots and Squaring (Cube roots too)ExponentsMultiples of numbersMulti-digit MultiplicationMultiplying with the Area ModelMathy's Berry Picking AdventureBingo!Integers BingoMake Multiplications Card GameFruity MathInteractive Multiplication ChartMake Number Sentences Division Division FactsDivision Facts Matching GameWith RemaindersDivide Whole Numbers Ending in ZerosLong DivisionThe Sieve of EratosthenesFind FactorsGCF & LCMBingo!Integers BingoMake Number SentencesFruity Divisibility Add & Subtract Connection between Addition and SubtractionSingle-DigitTwo-Digit (Mental Math)Fact FamiliesAddition Hidden Picture GameSubtraction Hidden Picture GameIntegers Hidden Picture Game"7 Up" Addition FactsMathy's Berry Picking AdventureBingo!Integers BingoFruity MathMake Number SentencesNumber BondsVertical Addition Colors GameVertical Subtraction Colors Game Place Value RoundingPlace ValueEven or Odd100-Chart GameSkip-count in a 100-chartOrdering NumbersOrdering DecimalsPlot numbers and integers on the number line Fractions & Decimals Fraction MatcherFractions on the Number LineEquivalent Fractions with Visual ModelsEquivalent Fractions Matching GameMixed NumbersAdd FractionsMultiply Fractions and Mixed NumbersDivide Fractions and Mixed NumbersOrder Fractions on a Number LineComparing FractionsFractions & Decimals Matching GameDecimal Addition & Subtraction Hidden Picture GameMultiply/Divide Decimals by Powers of 10Fruity Math — DecimalsDecimals BingoMake Decimal Number SentencesMultiply Rational NumbersAdd and Subtract Integers Measurement AnglesElapsed TimeTelling TimeTelling Time - Past & TillArea BuilderMeasurement Units Matching GameCalendar Activities Statistics & Probability Virtual Dice RollerVirtual Coin TosserMean and Median Integers Plot numbers and integers on the number lineAddition & Subtraction of integers on a number lineIntegers BingoIntegers Hidden Picture GameMake Number SentencesDistance between integers Pre-Algebra Unit RatesExpression ExchangeRational NumbersPercentOrder of OperationsSlope-Intercept EquationEquality ExplorerComplex FractionsFunction Builder Money Counting MoneyShopping GameMaking Change Kindergarten Sorting GameBeach Comparisons Geometry Geometric Transformations Multiplying with the Area Model Online activity and game for grades 4-5 In this activity, you will see multiplication and partial products illustrated with an area model, and you can practice using this model to multiply numbers (4th and 5th grade math). The sides of the rectangle represent the two numbers to be multiplied, and the area of the rectangle is the product (the answer). The activity has three parts. The first part, Explore, lets you drag the sides of the rectangle, and see the corresponding area on the grid. You can choose to reveal the partial products, and the step-by-step calculation of the total area. The second part, "Generic", allows you to input any numbers to be the side lengths of the rectangle, in two parts. Again, you can reveal the partial products, and the step-by-step calculation of the total area. The last part is the Game with six levels (of increasing difficulty) where you can really put your skills to practice! You are asked to find the total area, one of the missing side lengths, or one of the partial areas. Start Credit: This activity is created by PhET Screenshots from the game and activity: Leave some feedback for the developer Rate this activity: Name: Email Address (optional unless you want a response): Feedback about the activity for the developer (this does not go to your teacher): [x] Allow my comment to be posted on this site Submit Copyright © 2025 MathMammoth.com This website uses cookies to ensure you get the best experience on our website. Learn more in our privacy policy. Got it
18137	https://math.stackexchange.com/questions/4004559/truncated-taylor-series-of-the-exponential	Skip to main content Truncated Taylor series of the exponential Ask Question Asked Modified 4 years, 6 months ago Viewed 1k times This question shows research effort; it is useful and clear 5 Save this question. Show activity on this post. Let N∈N∗, δ>0, t>0 and consider f(t,δ,N)=e−t/δ∑k=0N−1(t/δ)kk!.(1) Let now N=⌈δ−γ⌉, with 0<γ<1. Can we conclude that limδ→0f(t,δ,⌈δ−γ⌉)=0, for all t>0? The reasoning is that if N→∞ (for δ fixed), then the sum tends to et/δ and therefore f(t,δ,N)→1. If on the other hand δ→0, for N fixed, we have that f(t,δ,N)→0 since the exponential goes faster to zero than the polynomial to infinity. In this case, we let δ→0 and simultaneously N→∞, but with a "slower pace". Can we conclude that the limit is zero? If yes, how? Edit: After some reasoning I came up with f(t,δ,N)=Γ(N,t/δ)(N−1)!, with Γ(⋅,⋅) the upper incomplete Gamma function. I implemented this in Matlab, and the interesting behaviour is that if N=δ−1, then the limit (1) is 1/2 if 01. If N=δ−γ, with γ∈(0,1), the limit is 0, and if N=δ−γ, with γ>1, then the limit is 1, independently of t. calculus limits taylor-expansion exponential-function Share CC BY-SA 4.0 Follow this question to receive notifications edited Jan 29, 2021 at 15:28 G. Gare asked Jan 29, 2021 at 14:14 G. GareG. Gare 1,5551010 silver badges2828 bronze badges 9 my bad, I misread – NHL Commented Jan 29, 2021 at 15:11 1 Speaking intuitively, the question comes down to identifying the dominant terms in the series for the exponential. If you think about this, the dominant terms are necessarily around tδ−1: until that point, the terms are rapidly growing, and after that they are rapidly decaying. – Ian Commented Jan 29, 2021 at 15:30 Sorry Ian but I did not understand your comment. In my intuition (and some computations seem to confirm this) if γ is different from 1 then t does not play a role in the result – G. Gare Commented Jan 29, 2021 at 15:36 1 If γ is different from 1 and t is different from 0 then asymptotically you are not cutting off the sum anywhere near tδ−1 and so you either have "all the important terms" (γ>1) or "none of the important terms" (0<γ<1). – Ian Commented Jan 29, 2021 at 15:37 Ok gotcha. How would you rigorously prove this? – G. Gare Commented Jan 29, 2021 at 15:39 \| Show 4 more comments 1 Answer 1 Reset to default This answer is useful 1 Save this answer. Show activity on this post. For simplicity let x=t/δ for the moment. The main point is that xkk! is within a polynomial factor of (xek)k, by Stirling. (It's within a bounded factor of (xe)kkk+1/2 but this is overkill for this application.) So for 0<c<1, the sum of all the terms with k>x(1+c) behaves basically like (e1+c)x(1+c)c+1c. This is the dominant term times 11−r where r=11+c is the common ratio of the bounding geometric series. Similarly for the first x(1−c) terms you have a bound like (e1−c)x(1−c)1c. Returning to your setting, if γ>1 then you exclude only the terms past x(1+c) where c→∞ as δ→0, and so the error goes to zero. If γ<1 then you exclude all the terms past x(1−c) for c→1 as δ→0 and so the terms you are including become negligible relative to ex, i.e. the error goes to 1. If instead you look at N∼Cδ−1 then the above bounds properly come into play. Share CC BY-SA 4.0 Follow this answer to receive notifications edited Jan 30, 2021 at 2:14 answered Jan 29, 2021 at 18:51 IanIan 105k55 gold badges100100 silver badges171171 bronze badges 2 Thanks. I may be slow, but I'm not grasping it. Could you be a bit more precise? "The error" is et/δ−∑∞k=N⋯? – G. Gare Commented Jan 30, 2021 at 17:01 @G.Gare The error I'm referring to is 1−e−t/δ∑…. – Ian Commented Jan 30, 2021 at 17:04 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 calculus limits taylor-expansion exponential-function See similar questions with these tags. 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18138	https://stats.oarc.ucla.edu/r/seminars/intro-to-linear-regression-r/	Introduction to Linear Regression in R MENU HOME SOFTWARE ► R Stata SAS SPSS Mplus Other Packages ► GPower SUDAAN Sample Power RESOURCES ► Annotated Output Data Analysis Examples Frequently Asked Questions Seminars Textbook Examples Which Statistical Test? SERVICES ► Remote Consulting Services and Policies ► Walk-In Consulting Email Consulting Fee for Service FAQ Software Purchasing and Updating Consultants for Hire Other Consulting Centers ► Department of Statistics Consulting Center Department of Biomathematics Consulting Clinic ABOUT US Skip to primary navigation Skip to main content Skip to primary sidebar stats.oarc.ucla.edu Statistical Methods and Data Analytics Search this website HOME SOFTWARE R Stata SAS SPSS Mplus Other Packages GPower SUDAAN Sample Power RESOURCES Annotated Output Data Analysis Examples Frequently Asked Questions Seminars Textbook Examples Which Statistical Test? SERVICES Remote Consulting Services and Policies Walk-In Consulting Email Consulting Fee for Service FAQ Software Purchasing and Updating Consultants for Hire Other Consulting Centers Department of Statistics Consulting Center Department of Biomathematics Consulting Clinic ABOUT US Introduction to Linear Regression in R Welcome! In this seminar, we will cover several fundamental topics related to linear regression models, including: The linear regression model equation. The ordinary least squares method. Simple linear regression with both continuous and categorical predictors. Multiple linear regression. Modeling interactions between two variables. Regression diagnostics. Throughout the seminar, we will utilize R to fit linear regression models, interpret model estimates, and conduct regression diagnostics. To participate fully, please ensure you have the latest versions of R and RStudio installed. You can download both R and RStudio for free from the RStudio website, where links to both downloads are available. In addition, please install the following packages before the workshop: install.packages(c("knitr", "tidyverse", "skimr", "kableExtra")) You can download the seminar slides here and the .rmd file with the code for the workshop here. If you are new to R, watch this seminar. Primary Sidebar Click here to report an error on this page or leave a comment Your Name (required) Your Email (must be a valid email for us to receive the report!) Comment/Error Report (required) Δ How to cite this page UCLA OARC © 2024 UC REGENTS HOME CONTACT
18139	https://flexbooks.ck12.org/cbook/ck-12-middle-school-physical-science-flexbook-2.0/section/7.11/primary/lesson/acid-base-neutralization-ms-ps/	Neutralization Reaction \| CK-12 Foundation AI Teacher Tools – Save Hours on Planning & Prep. Try it out! Skip to content What are you looking for? Search Math Grade 6 Grade 7 Grade 8 Algebra 1 Geometry Algebra 2 PreCalculus Science Earth Science Life Science Physical Science Biology Chemistry Physics Social Studies Economics Geography Government Philosophy Sociology Subject Math Elementary Math Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Interactive Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Conventional Math 6 Math 7 Math 8 Algebra I Geometry Algebra II Probability & Statistics Trigonometry Math Analysis Precalculus Calculus What's the difference? 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Learn. Interact. eXplore. CCSS Math Concepts and FlexBooks aligned to Common Core NGSS Concepts aligned to Next Generation Science Standards Certified Educator Stand out as an educator. Become CK-12 Certified. Webinars Live and archived sessions to learn about CK-12 Other Resources CK-12 Resources Concept Map Testimonials CK-12 Mission Meet the Team CK-12 Helpdesk FlexLets Know the essentials. Pick a Subject Donate Sign InSign Up Start Practice 7.11 Neutralization Reaction 1. FlexBooks 2.0> 2. CK-12 Physical Science for Middle School> 3. Neutralization Reaction Written by:Jean Brainard, Ph.D. Fact-checked by:The CK-12 Editorial Team Last Modified: Aug 01, 2025 Lesson Review Asked on Flexi Related Content Lesson [Figure 1] What is one of the most important characteristics of a referee? A referee must be neutral. They can’t favor one team over the other. In chemistry, being neutral means not being an acid or a base. Pure water is an example of a neutral substance. In some chemical reactions, an acid and a base combine to form neutral products, including water. You’ll see how this happens when you read this article. Acids, Bases, and Ions An acid is a compound that produces positive hydrogen ions (H+) and negative nonmetal ions when it dissolves in water. (Ions are atoms that have become charged by losing or gaining electrons.) Hydrochloric acid (HCl) is an example of an acid. When it dissolves in water, it produces positive hydrogen ions and negative chloride ions (Cl-). This can be represented by the chemical equation: HCl →H 2 O H+ + Cl- A base is a compound that produces negative hydroxide ions (OH-) and positive metal ions when it dissolves in water. For example, when the base sodium hydroxide (NaOH) dissolves in water, it produces negative hydroxide ions and positive sodium ions (Na+). This can be represented by the chemical equation: NaOH →H 2 O OH- + Na+ Q: If you were to combine acid and base solutions, what products do you think would be produced? A: Combining acid and base solutions produces water and a neutral ionic compound. Reactions of Acids and Bases When an acid and a base react, the reaction is called a neutralization reaction. That’s because the reaction produces neutral products. Water is always one product, and a salt is also produced. A salt is a neutral ionic compound. Let’s see how a neutralization reaction produces both water and a salt, using as an example the reaction between solutions of hydrochloric acid and sodium hydroxide. The overall equation for this reaction is: NaOH + HCl → H 2 O and NaCl Now let’s break this reaction down into two parts to see how each product forms. Positive hydrogen ions from HCl and negative hydroxide ions from NaOH combine to form water. This part of the reaction can be represented by the equation: H+ + OH- → H 2 O Positive sodium ions from NaOH and negative chloride ions from HCL combine to form the salt sodium chloride (NaCl), commonly called table salt. This part of the reaction can be represented by the equation: Na+ + Cl- → NaCl Another example of a neutralization reaction can be seen in the Figurebelow. [Figure 2] These antacid tablets contain the base calcium carbonate (CaCO 3). The base reacts with hydrochloric acid (HCl) in the stomach. The reaction neutralizes the acid to relieve acid indigestion. Q: What products are produced when antacid tablets react with hydrochloric acid in the stomach? A: The products are water and the salt calcium chloride (CaCl 2). Carbon dioxide (CO 2) is also produced. The reaction is represented by the chemical equation: CaCO 3 + 2HCl → H 2 O + CaCl 2 + CO 2 Summary When acid and base solutions react, they produce water and a neutral ionic compound called a salt. The reaction is called a neutralization reaction. Review Describe a neutralization reaction. What is a salt? Give an example. Fill in the missing products in the chemical equation below. It represents a neutralization reaction between solutions of nitric acid (HNO 3) and potassium hydroxide (KOH): HNO 3 + KOH → ____ + _____ Resources Add Note CancelSave Discuss with Flexi NOTES / HIGHLIGHTS Please Sign In to create your own Highlights / Notes Add Note Edit Note Remove Highlight Image Attributions Asked by Students Here are the top questions that students are asking Flexi for this concept: What is the quickest way to neutralize stomach acid? The quickest way to neutralize stomach acid is by taking an antacid. Antacids are over-the-counter medications that help reduce the amount of acid in your stomach, alleviating symptoms of indigestion, acid reflux, or heartburn. They contain basic compounds (like calcium carbonate, aluminum hydroxide, and magnesium hydroxide), which immediately neutralize the stomach acid upon contact, providing rapid relief. How do you neutralize muriatic acid? Muriatic acid, also known as hydrochloric acid (HCl), can be neutralized using a base such as sodium bicarbonate (baking soda, NaHCO 3) or sodium hydroxide (NaOH). The neutralization reaction is as follows: HCl + NaHCO 3 → NaCl + H 2 O + CO 2 or HCl + NaOH → NaCl + H 2 O Always remember to add acid to water, not the other way around, to prevent a violent reaction. Also, use appropriate personal protective equipment (gloves, goggles, lab coat) when handling acids and bases. Does vinegar neutralize acid? Vinegar, which is a weak acid itself (acetic acid), does not neutralize other acids. Instead, a base or alkali is needed to neutralize acids. How can citric acid in the stomach be neutralized using baking soda? Citric acid in the stomach can be neutralized using baking soda (sodium bicarbonate) through a chemical reaction. The reaction is as follows: C 6 H 8 O 7 + 3NaHCO 3 → 3H 2 O + 3CO 2 + Na 3 C 6 H 5 O 7 In this reaction, citric acid (C 6 H 8 O 7) reacts with sodium bicarbonate (NaHCO 3) to produce water (H 2 O), carbon dioxide (CO 2), and sodium citrate (Na 3 C 6 H 5 O 7). The carbon dioxide is released as a gas, which can cause burping. The sodium citrate remains in the solution and is less acidic than the original citric acid, thus neutralizing the acidity. Is KCl acidic basic or neutral? KCl is considered neutral when dissolved in water. This is because it is formed from a strong acid (HCl) and a strong base (KOH), which completely dissociate in water, producing neutral ions (K+ and Cl-). Overview An acid produces positive hydrogen ions and negative nonmetal ions when dissolved in water. A base produces negative hydroxide ions and positive metal ions when dissolved in water. The reaction between an acid and a base is called neutralization, producing water and a salt. In a neutralization reaction, hydrogen ions from the acid combine with hydroxide ions from the base to form water, and metal ions from the base combine with nonmetal ions from the acid to form a salt. Antacid tablets containing calcium carbonate react with hydrochloric acid in the stomach, producing water, calcium chloride, and carbon dioxide. Vocabulary chemistry chemical reaction ion nonmetal chemical equation metal ionic compound neutralization reaction salt Test Your Knowledge Question 1 The reaction of an acid with a base usually results in the production of _____. a O H− b H 2 O c N o n e o f t h e a b o v e d H 3 O+ Check It When an acid and a base react, the reaction is called a neutralization reaction. That’s because the reaction produces neutral products. Water is always one product, and a salt is also produced. A salt is a neutral ionic compound. FlexCard™ Question 2 The neutralization of an acid with a base invariably results in the production of _____. a H 3 O+ b H 2 O c N o n e o f t h e a b o v e d O H− Check It When acid and base solutions react, they produce water and a neutral ionic compound called a salt. The reaction is called a neutralization reaction. FlexCard™ Asked by Students Ask your question Here are the top questions that students are asking Flexi for this concept: What is the quickest way to neutralize stomach acid? The quickest way to neutralize stomach acid is by taking an antacid. Antacids are over-the-counter medications that help reduce the amount of acid in your stomach, alleviating symptoms of indigestion, acid reflux, or heartburn. They contain basic compounds (like calcium carbonate, aluminum hydroxide, and magnesium hydroxide), which immediately neutralize the stomach acid upon contact, providing rapid relief. How do you neutralize muriatic acid? Muriatic acid, also known as hydrochloric acid (HCl), can be neutralized using a base such as sodium bicarbonate (baking soda, NaHCO 3) or sodium hydroxide (NaOH). The neutralization reaction is as follows: HCl + NaHCO 3 → NaCl + H 2 O + CO 2 or HCl + NaOH → NaCl + H 2 O Always remember to add acid to water, not the other way around, to prevent a violent reaction. Also, use appropriate personal protective equipment (gloves, goggles, lab coat) when handling acids and bases. Does vinegar neutralize acid? Vinegar, which is a weak acid itself (acetic acid), does not neutralize other acids. Instead, a base or alkali is needed to neutralize acids. How can citric acid in the stomach be neutralized using baking soda? Citric acid in the stomach can be neutralized using baking soda (sodium bicarbonate) through a chemical reaction. The reaction is as follows: C 6 H 8 O 7 + 3NaHCO 3 → 3H 2 O + 3CO 2 + Na 3 C 6 H 5 O 7 In this reaction, citric acid (C 6 H 8 O 7) reacts with sodium bicarbonate (NaHCO 3) to produce water (H 2 O), carbon dioxide (CO 2), and sodium citrate (Na 3 C 6 H 5 O 7). The carbon dioxide is released as a gas, which can cause burping. The sodium citrate remains in the solution and is less acidic than the original citric acid, thus neutralizing the acidity. Is KCl acidic basic or neutral? KCl is considered neutral when dissolved in water. This is because it is formed from a strong acid (HCl) and a strong base (KOH), which completely dissociate in water, producing neutral ions (K+ and Cl-). Related Content Wet or Stinky Back to Neutralization Reaction \| Image \| Reference \| Attributions \| --- \| \| [Figure 1] \| Credit:Flickr: Harris Walker Source: License:CC BY 2.0 \| \| \| [Figure 2] \| Credit:User:Midnightcomm/Wikimedia Commons Source: License:CC BY 2.5 \| Ask me anything! 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18140	https://math.stackexchange.com/questions/2083457/closed-interval-method	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Closed Interval Method Ask Question Asked Modified 1 year, 11 months ago Viewed 5k times 1 $\begingroup$ From the book "Calculus" by James Stewart, The Closed Interval Method is used to find the absolute(global) maximum and minimum values of a continuous function on a close interval $[a,b]$. 1) Find the values of $f$ at the critical numbers of $f$ in $(a,b)$. 2) Find the values of $f$ at the endpoints of the interval. 3) The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value. I have problem understanding step 1. Fermat's Theorem states that if function $f$ has local maximum/minimum at point $x=a$, then $a$ is a critical number. Also the converse is not true as the function $y=x^3$ has a critical point at $x=0$ but $x=0$ is not a local minimum/maximum. Also, from my understanding, absolute minimum/maximum on an closed interval can either be a local min/max on the closed interval or at the end points. Hence, we should find the local minimum/maximum and compare the values with the value of function at the endpoints. But from the example of $y=x^3$, doing step 1) as suggested in the book can derive point that is a critical number but not a local minimum/maximum. Perhaps, it is proven such points can never be the absolute maximum/minimum? Edit: knowing such point $a$ is not a local minimum/maximum tells us that there exist $x$ such that $f(x) > a$ and $f(x) < a$ and the absolute maximum/minimum would either be another critical number or at the end points. real-analysis calculus maxima-minima Share edited Jun 30, 2019 at 16:13 user9464 asked Jan 4, 2017 at 14:53 Little RookieLittle Rookie 1,23611 gold badge1818 silver badges4040 bronze badges $\endgroup$ Add a comment \| 2 Answers 2 Reset to default 1 $\begingroup$ Step 1 is just part of the whole story. And you are right that Step 1 would give points that are not absolute min/max. That's exactly why you need Step 2 and Step 3. In the case of $f(x)=x^3$ (say in $[-1,1]$), it is true that you will get $x=0$ as a critical number. However, Step 2 and Step 3 will rule it out. [Added to answer the question in the comment:] Because global min/max must also be local min/max. If a critical point is not a local min/max, then it cannot be global min/max. Share edited Jan 4, 2017 at 16:09 answered Jan 4, 2017 at 15:00 user9464user9464 $\endgroup$ 7 $\begingroup$ why is it that Step 2 and Step 3 will definitely rule out such points? Can you explain in more details? $\endgroup$ Little Rookie – Little Rookie 2017-01-04 15:21:33 +00:00 Commented Jan 4, 2017 at 15:21 $\begingroup$ In step 1, you get $f(0)=0$. But in Step 2, you get $f(-1)=-1$ and $f(1)=1$. Now by step 3, what you get? $\endgroup$ user9464 – user9464 2017-01-04 15:24:47 +00:00 Commented Jan 4, 2017 at 15:24 $\begingroup$ How is it that step 2 and step 3 will rule out such points in general? Not only in the case of $y=x^3$ $\endgroup$ Little Rookie – Little Rookie 2017-01-04 15:28:55 +00:00 Commented Jan 4, 2017 at 15:28 $\begingroup$ It is because by definition so. In Step 3, one compares all the points one gets in Step 1 and Step 2 so that one can identify the absolute max and absolute min. The bad points you described would not "win" in the comparison because they cannot be the absolute max/min. $\endgroup$ user9464 – user9464 2017-01-04 15:31:44 +00:00 Commented Jan 4, 2017 at 15:31 $\begingroup$ Intuitively, they cannot be the absolute maximum/minimum. But is there a formal proof that i can refer to? $\endgroup$ Little Rookie – Little Rookie 2017-01-04 15:53:21 +00:00 Commented Jan 4, 2017 at 15:53 \| Show 2 more comments 1 $\begingroup$ Just a thought, since the end points of a closed interval function say [a,b] can be a local extremum and at the same time, Fermat's theorem says that all local extremums occur at critical points. Does it mean to say that x=a and x=b is a critical point? For Eg, f(x)=x^3 where x is defined in the interval [-3,4]. x=-3 and x=4 are both local extremums but why are they considered critical points? f'(a) and f'(b) is not zero and is defined. The definition of a critical point is such that if x=c is a critical point, f'(c) is either 0 or undefined. Share answered Oct 26, 2023 at 2:49 JusJus 30511 silver badge88 bronze badges $\endgroup$ 1 $\begingroup$ It is a convention. Usually by definition endpoints are not considered critical, however you can choose to consider them critical depending how you define critical numbers. $\endgroup$ Henry – Henry 2025-02-26 15:58:01 +00:00 Commented Feb 26 at 15:58 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 real-analysis calculus maxima-minima 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 0 Relationship between local max/min and absolute max/min Related Max-min of a function on closed, bounded interval using EVT 0 find maximum and minimum values for f(x) function on a closed interval 1 Example of a calculus optimization problem where the answer occurs at an endpoint 1 Find the absolute extrema of the function $f(x)=x^2-2x-2$ on $[0,1]$ 1 Does Stewart's calculus consider endpoints of the domain of a (nice) function to be critical numbers? 1 Confusion regarding Fermat's Theorem and Closed Interval Method 1 How to prove that a local extrema is an Absolute extrema on an open interval? 0 Using closed interval method to find global maximum and minimum values of $f(x)= \sqrt{4-x^2}$ Hot Network Questions The geologic realities of a massive well out at Sea What meal can come next? 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18141	https://www.youtube.com/watch?v=9B2L55qDfEw	The Cofunction Identities The Math Sorcerer 1220000 subscribers 24 likes Description 1073 views Posted: 17 May 2018 Please Subscribe here, thank you!!! The Cofunction Identities Transcript: in this video we're briefly going to talk about the cofunction identities so cofunction identities so we'll do this in degrees you can also do it in radians so we're going to let theta be an acute angle so let theta be acute that means that it's measure is strictly between 0 and 90 degrees and the first cofunction identity is for the sine function so the sine of theta is equal to the cosine of 90 degrees minus theta if you are doing this in radians you would replace 90 degrees with PI over 2 and then the cosine of theta it's pretty easy to memorize it's just sine instead of 90 degrees minus theta so sine and cosine are Co functions hence the name cosine you can do the same thing with secant and cosecant so the secant of theta is equal to the cosecant of 90 degrees minus theta likewise the cosecant of theta is equal to the secant of 90 degrees minus theta and the very last ones are tangent and cotangent so the tangent of theta is equal to the cotangent really easy they're all really easy to memorize 90 minus theta and then the cotangent of theta as you probably guessed is the tangent of 90 degrees minus theta so again if theta is in radians you would replace the 90 degrees with PI over 2 let's go ahead and do a couple of simple examples so you see how this works it's actually very very simple once you know how to do it it's easy if you don't know how to do it it's impossible all right so the question will say to express in terms of its cofunction so Express in terms of its cofunction co-function alright so part a let's try secant of 39 degrees so the secant of 39 degrees well how would you do this well the comb function is cosecant so this would be cosecant and then this is your theta here right this is your theta so be 90 minus theta would be 90 degrees minus 39 degrees okay and 90 minus 39 is 51 to be cosecant of 51 degrees and that's it that's the final answer so the secant of 39 degrees is equal to the cosecant of 51 degrees and we leave the answer like this because it says expressed in terms of its cofunction so that's exactly what we have done let's try our harder one tangent of PI over 3 so the cofunction for tangent is cotangent so this would be cotangent but now it's not degrees it's radians so this is our theta so here we have to put not 90 but PI over 2 minus and then our theta which is PI over 3 that's our theta right so it's kind of like 90 minus theta except it's PI over 2 minus theta ok so this is the cotangent now you have to subtract these so we have to have the same denominator so the common denominator is 6 so we're going to multiply this one by 3 over 3 and this one by 2 over 2 it's going to give us 3 PI over 6 minus 2 PI over 6 so that's equal to the cotangent and then so 3 PI over 6 minus 2 PI over 6 is simply PI over 6 so that would be the final answer for that case one more let's do another one with sequined so say secant of one so whenever you write one and there's no degree symbol you have to assume it's in radians so this would simply be the cosecant so we're in radians so we use PI over two instead of 90 degrees minus theta and our theta here is simply one so it's just PI over two minus one and that's it you leave it just like that I hope this video made sense
18142	https://resources.finalsite.net/images/v1697572831/audubonschoolsorg/vqftmpbgm2rwiapykjhg/apchemistry.pdf	a Course: AP Chemistry Unit Name: Thermodynamics Grade Lev el: 1 1-1 2 Content Statements Entropy Entropy of formation Laws of Thermodynamics Gibbs Free Energy Enthalpy Heat Specific Heat Capacity Joules Hess’ Law Formation Reactions Enthalpy of Formation Enthalpy of fusion / vaporization Standard Molar Enthalpy of formation Summation method NJSLS: 5.1.12.A-D: All NJSLS RST.11-12.1-10 ∙ Overarching Essential Questions How much energy and entropy is released / absorbed during a chemical or physical reaction? Overarching Enduring Understandings In all chemical and physical changes energy is released or absorbed in the form of enthalpy and entropy. This energy is quantifiable and an essential component when deciding if a reaction will occur. Unit Essential Questions What is temperature a measure of? In what direction is energy transferred between 2 bodies? Is the total energy between multiple systems fixed? What are the main processes that chemical reactions use to change their energy? How can Hess’s law be used to determine the enthalpy change of a reaction? What is calorimetry and what does it measure? How is the net energy change of a chemical reaction related to the bond energy of the reactants and products? Using thermodynamic data, how can it be determined if a chemical reaction is spontaneous over a specific temperature range? Unit Enduring Understandings Two systems with different temperatures that are in thermal contact will exchange energy. The quantity of thermal energy transferred from one system to another is called heat. Energy is neither created nor destroyed, but only transformed from one form to another. Breaking bonds requires energy, and making bonds releases energy. Chemical or physical processes are driven by a decrease in enthalpy or an increase in entropy, or both. Unit Rationale To measure the energy changes of chemical and physical reactions and to use it to predict the spontaneity of a reaction. Unit Overview The laws of thermodynamics describe the essential role of energy and explain and predict the direction of changes in matter. Resources Chemistry and Chemical Reactivity, Kotz and Treichel, Saunders College Publishing Chemistry and Chemical Reactivity Student Solutions Manual, Saunders College Publishing Chemistry and Chemical Reactivity, Study Guide, Saunders College Publishing Chemistry and Chemical Reactivity, Pocket Guide, Saunders College Publishing www.wolframalpha.com Suggested Student Activities Lab Activity – Verifying Hess’ Law Course: AP Chemistry Unit Name: Chemical Kinetics Grade Lev el: 1 1-1 2 Content Statements Order Rate law Rate law constant Integrated rate law Molecularity Catalyst Unimolecular Bimolecular NJSLS 5.1.12.A-D: All NJSLS RST.11-12.1-10 Termolecular Rate equation Activation Energy Energy Diagram Spectrophotometer Overarching Essential Questions How do we mathematically describe the concentration – time – rate relationship between chemical species in a chemical reaction? Overarching Enduring Understandings Rates of chemical reactions are determined by details of the molecular collisions. Unit Essential Questions What factors influence the rate of a chemical reaction? How is a rate law determined from experimental data? What are the units of the rate law constant? How is the rate law of an elementary step in a reaction determined? What is activation energy and how is it overcome? What is the rate determining step in a multistep reaction? What is an intermediate and how is it identified in a multi step reaction? Unit Enduring Understandings The rate of a reaction is influenced by the concentration or pressure of reactants, the phase of the reactants and products, and environmental factors such as temperature and solvent. The rate law shows how the rate depends on reactant concentrations. The magnitude and temperature dependence of the rate of reaction is contained quantitatively in the rate constant. Elementary reactions can be unimolecular or involve collisions between two or more molecules. Not all collisions are successful. To get over the activation energy barrier, the colliding species need sufficient energy. Also, the orientations of the reactant molecules during the collision must What effect does a catalyst have on activation energy? What are some specific examples of catalysts? allow for the rearrangement of reactant bonds to form product bonds. A successful collision can be viewed as following a reaction path with an associated energy profile. The mechanism of a multistep reaction consists of a series of elementary reactions that add up to the overall reaction. In many reactions, the rate is set by the slowest elementary reaction, or rate-limiting step. Reaction intermediates, which are formed during the reaction but not present in the overall reaction, play an important role in multistep reactions. Catalysts function by lowering the activation energy of an elementary step in a reaction mechanism, and by providing a new and faster reaction mechanism. Important classes in catalysis include acid-base catalysis, surface catalysis, and enzyme catalysis. Unit Rationale Chemical reactions can be mathematically modeled and predicted in such a way that concentration and time are related. Unit Overview Mathematical models can be developed through the collection of experimental data to accurately predict the behavior of systems. Resources Chemistry and Chemical Reactivity, Kotz and Treichel, Saunders College Publishing Chemistry and Chemical Reactivity Student Solutions Manual, Saunders College Publishing Chemistry and Chemical Reactivity, Study Guide, Saunders College Publishing Chemistry and Chemical Reactivity, Pocket Guide, Saunders College Publishing www.wolframalpha.com Suggested Student Activities Lab - Spectrophotometric analysis of the dichromate ion. Catalytic decomposition of hydrogen peroxide using the iodide ion Manganese dioxide as a catalyst in a chemical reaction Course: AP Chemistry Unit Name: Chemical Reactions Grade Lev el: 1 1-1 2 Content Statements Oxidation Reduction Synthesis Metathesis Decomposition Acid Base Reaction Bronsted-Lowry Lewis Theory NJSLS : 5.1.12.A-D: All NJSLS RST.11-12.1-10 Galvanic Cell Gas Forming Reactions Electrochemistry Electron Transfer Proton Transfer Titration Buffer Solution Henderson-Hasslebach Conjugate Acid-Base Pair ICE Chart Equilibrium constants Equilibrium expressions Solubility product constants Acid and Base constants Activity series Reduction table Qualitative analysis Overarching Essential Questions What are the main classifications of chemical reactions? What are the main differences between types of chemical reactions? How can the amounts of products and reactants within a chemical reaction be stoichiometrically related? Overarching Enduring Understandings Changes in matter involve the rearrangement and/or reorganization of atoms and/or the transfer or electrons Unit Essential Questions How are chemical changes represented? Unit Enduring Understandings Chemical changes are represented by a balanced chemical equation that identifies the ratios with which reactants reach and products form. How can quantitative information be derived from stoichiometric calculations that utilize the mole ratios from the balanced chemical equations? What is the role of stoichiometry in real-world applications? What are synthesis, decomposition, neutralization and oxidation – reduction reactions? How are they classified? How do they differ from one another? In oxidation-reduction reactions, how does the transfer of electrons help identified what is oxidized and what is reduced? What are examples of evidence for the occurrence of a chemical reaction? How can net changes in energy for a chemical reaction be classified? What kinds of reactions involve the conversion between chemical and electrical energy? Chemical reactions can be classified by considering what the reactants are, what the products are, or how they change from one into the other. Classes of chemical reactions include synthesis, decomposition, acid-base, and oxidation reduction reactions. Chemical and physical transformations may be observed in several ways and typically involve a change in energy. Unit Rationale Chemical reactions are the backbone of industry, medicine, and many other fields. The ability to fully describe and quantify these reactions is of the utmost importance to society. Unit Overview Through the process of experimentation and data collection involving the various types of chemical reactions, the quantities of reactants, products, and energies can be quantified and predicted using stoichiometry. Resources Chemistry and Chemical Reactivity, Kotz and Treichel, Saunders College Publishing Chemistry and Chemical Reactivity Student Solutions Manual, Saunders College Publishing Chemistry and Chemical Reactivity, Study Guide, Saunders College Publishing Chemistry and Chemical Reactivity, Pocket Guide, Saunders College Publishing www.wolframalpha.com Suggested Student Activities Lab – Qualitative analysis of anions Lab – Net Ionic equations Lab – Electrochemical Series Lab – Redox Titration Lab – Electrolysis, the faraday, and Avogadro’s number Lab – Synthesis of aspirin and oil of wintergreen Lab – Synthesis of a coordination compound Lab – Analysis of a coordination compound Lab – Synthesis of esters Course: AP Chemistry Unit Name: Equilibrium Grade Lev el: 1 1-1 2 Content Statements Equilibrium ICE Chart Equilibrium constants LeChatelier’s Principle NJSLS : 5.1.12.A-D: All Equilibrium expressions Product and Reactant favored Solubility product constant Acid Base equilibrium Titration pH pOH Ka, Kb, Kw, Ksp Buffer NJSLS RST.11-12.1-10 Overarching Essential Questions How is the process of equilibrium related different categories of reactions? How equilibrium reactions are mathematically modeled and quantified? Overarching Enduring Understandings Any bond or intermolecular attraction that can be formed can be broken. These two processes are in a dynamic competition, sensitive to the initial conditions and external perturbations. Unit Essential Questions In what classes of reactions are both forward and reverse processes considered? How does the reaction quotient Q help to determine if a system is at equilibrium? How do variables such as molarity, pressure, and temperature effect a system at equilibrium? How can the equilibrium constant, K, relate to the amount of product and reactant present in the system? Unit Enduring Understandings Chemical equilibrium is a dynamic, reversible state in which rates of opposing processes are equal Systems at equilibrium are responsive to external perturbations, with the response leading to a change in the composition of the system. Chemical equilibrium plays an important role in acid-base chemistry and in solubility The equilibrium constant is related to temperature and the difference in Gibbs free energy between reactants and products. What is LeChatelier’s principle? In which direction will a system shift if Q and K are not equal? How can equilibrium concepts be used to describe the proton transfer of acid-base reactions? How is pH and pOH calculated? How can pH and pOH be related to pKa and pKb? How is the solubility of a substance understood in terms of chemical equilibrium? How is the Gibbs free energy related to the equilibrium constant? Unit Rationale Chemical equilibrium can be observed in many types of reactions including acid-base and precipitations. The process of equilibrium is far reaching and must be investigated in these various types of reactions. Unit Overview The process of equilibrium for chemical reactions will be mathematically modeled so predictions for the amounts of reactants and products can be accurately calculated. Resources Chemistry and Chemical Reactivity, Kotz and Treichel, Saunders College Publishing Chemistry and Chemical Reactivity Student Solutions Manual, Saunders College Publishing Chemistry and Chemical Reactivity, Study Guide, Saunders College Publishing Chemistry and Chemical Reactivity, Pocket Guide, Saunders College Publishing www.wolframalpha.com Suggested Student Activities Lab – Spectrophotometric determination of an equilibrium constant Lab – LeChatelier’s Principle Lab – Standardization of NaOH using KHP Lab – Determination of concentration by acid-base titration Lab – Preparation of buffer solutions Course: AP Chemistry Unit Name: Atomic theory and the mole concept Grade Lev el: 1 1-1 2 Content Statements Atom Mole Molecule Electron Proton Neutron Spectrophotometry Mass Spectrometer Atomic Theory NJSLS : 5.1.12.A-D: All NJSLS RST.11-12.1-10 Periodicity Subatomic particles Quantum Theory Isotopes Conservation of atoms Overarching Essential Questions How is matter understood in terms of chemical elements, the fundamental building block of all matter? Overarching Enduring Understandings The chemical elements are fundamental building materials of all matter, and all matter can be understood in terms of arrangements of atoms. These atoms retain their identity in chemical reactions. Unit Essential Questions What are molecules composed of, and how do elements combine? How is chemical analysis used to determine the atoms and composition of a substance? What is the mole and how is it used to count atoms? What are the particles that compose the atom and what are their properties? How can the electronic structure of the atom be described? What are the main periodic trends of elements? Unit Enduring Understandings All matter is made of atoms. There are a limited number of types of atoms; these are elements. The atoms of each element have unique structures arising from interactions between electrons and nuclei. Elements display periodicity in their properties when the elements are organized according to increasing atomic number. This periodicity can be explained by the regular variations that occur in the electronic structures of atoms. Periodicity is a useful principle for understanding properties and predicting trends in properties. Its modern day uses range from examining the composition of materials to generating ideas for designing new materials. Atoms are so small that they are difficult to study directly; atomic models are constructed to explain experimental data on collections of atoms. What is the best currently accepted atomic model? Explain how the theoretical model of the atom is not an exact description, but rather a work in progress open for refinement. How has mass spectrometry refined the past atomic models? How can spectrophotometers be used to probe the structure of atoms and molecules? How are chemical reactions represented using symbols? How can the conservation of atoms be used to compute the masses of substances involved in reactions? Atoms are conserved in physical and chemical processes. Unit Rationale The current accepted atomic model helps us understand the structure of the atom and how molecules and atoms behave during chemical and physical reactions. Unit Overview Matter will be investigated from the subatomic level through the macroscopic scale, with mathematical relationships linking both. Resources Chemistry and Chemical Reactivity, Kotz and Treichel, Saunders College Publishing Chemistry and Chemical Reactivity Student Solutions Manual, Saunders College Publishing Chemistry and Chemical Reactivity, Study Guide, Saunders College Publishing Chemistry and Chemical Reactivity, Pocket Guide, Saunders College Publishing www.wolframalpha.com Suggested Student Activities Lab – Electrolysis, the Faraday and Avogadro’s number Lab – Molecular models and Lewis structures Lab – Periodic trends in elements Course: AP Chemistry Unit Name: Intermolecular f orces and the properties of materials Grade Lev el: 1 1-1 2 Content Statements Chemical Properties Physical Properties Particle Spacing Ions Dipole Induced dipole Van der Waals forces Intermolecular forces Coulomb’s Law London Dispersion forces Hydrogen bonding Metallic bonding NJSLS : 5.1.12.A-D: All NJSLS RST.11-12.1-10 VSEPR model Ionic Solids Covalent network solids Molecular solids Overarching Essential Questions How can the physical and chemical properties of matter be described and predicted from the arrangement of atoms, ions or molecules and the forces between them. Overarching Enduring Understandings Chemical and Physical properties of materials can be explained by the structure and the arrangement of atoms, ions or molecules and the forces between them Unit Essential Questions How can the different properties of solids and liquids be explained at the atomic and macroscopic levels? What mathematical relationships can describe the gaseous state of matter? What are London dispersion forces and how can their relative strengths be predicted? What are dipole forces and how do they vary from hydrogen bonding forces? How can intermolecular forces be used to predict the properties of substances? What is electronegativity, and how is it used to describe covalent bonding? Unit Enduring Understandings Matter can be described by its physical properties. The physical properties of a substance generally depend on the spacing between the particles that make up the substance and the forces of attraction among them. Forces of attraction between particles are important in determining many macroscopic properties of a substance, including how the observable physical state changes with temperature. The strong electrostatic forces of attraction holding atoms together in a unit are called chemical bonds. The type of bonding in the solid state can be deduced from the properties of the solid state. What is ionic bonding, and how can it be used to describe a crystal lattice? What is metallic bonding and how does it describe the unique properties of metals? What is the VSEPR model and how is it used to predict the Lewis diagrams of molecules? What are ionic solids and their general properties? What are metallic solids and their general properties? What are covalent network solids and their general properties? What are molecular solids and their general properties? Unit Rationale Intermolecular forces can be predicted using the current atomic model and molecular geometries. These IM forces can then be used to make predictions on properties and behaviors of matter. Unit Overview Intermolecular forces will be described using molecular and atomic theories, then those predictions will be applied to various materials. Resources Chemistry and Chemical Reactivity, Kotz and Treichel, Saunders College Publishing Chemistry and Chemical Reactivity Student Solutions Manual, Saunders College Publishing Chemistry and Chemical Reactivity, Study Guide, Saunders College Publishing Chemistry and Chemical Reactivity, Pocket Guide, Saunders College Publishing www.wolframalpha.com Suggested Student Activities Lab – Chromatography Lab – Enthalpy of vaporization and fusion. Reapproved June 2017 Appendix Differentiation Enrichment ● Utilize collaborative media tools ● Provide differentiated feedback ● Opportunities for reflection ● Encourage student voice and input ● Model close reading ● Distinguish long term and short term goals Intervention & Modification ● Utilize “skeleton notes” where some required information is already filled in for the student ● Provide access to a variety of tools for responses ● Provide opportunities to build familiarity and to practice with multiple media tools ● Leveled text and activities that adapt as students build skills ● Provide multiple means of action and expression ● Consider learning styles and interests ● Provide differentiated mentors ● Graphic organizers ELLs ● Pre-teach new vocabulary and meaning of symbols ● Embed glossaries or definitions ● Provide translations ● Connect new vocabulary to background knowledge ● Provide flash cards ● Incorporate as many learning senses as possible ● Portray structure, relationships, and associations through concept webs ● Graphic organizers 21st Century Skills ● Creativity ● Innovation ● Critical Thinking ● Problem Solving ● Communication ● Collaboration Integrating Technology ● Chromebooks ● Internet research ● Online programs ● Virtual collaboration and projects ● Presentations using presentation hardware and software
18143	https://artofproblemsolving.com/wiki/index.php/Vieta%27s_Formulas?srsltid=AfmBOorbDZX-fcai-wa8E0OoWHFSa9Ulj2qa9VKwoqZYKgtGiykMYmQo	Art of Problem Solving Vieta's Formulas - 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 Vieta's Formulas Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Vieta's Formulas In algebra, Vieta's formulas are a set of results that relate the coefficients of a polynomial to its roots. In particular, it states that the elementary symmetric polynomials of its roots can be easily expressed as a ratio between two of the polynomial's coefficients. It is among the most ubiquitous results to circumvent finding a polynomial's roots in competition math and sees widespread usage in many math contests/tournaments. Contents 1 Statement 2 Proof 3 Problems 3.1 Introductory 3.2 Intermediate 4 Advanced 5 See also Statement Let be any polynomial with complex coefficients with roots , and let be the elementary symmetric polynomial of the roots. Vieta’s formulas then state that This can be compactly summarized as for some such that . Proof Let all terms be defined as above. By the factor theorem, . We will then prove Vieta’s formulas by expanding this polynomial and comparing the resulting coefficients with the original polynomial’s coefficients. When expanding the factorization of , each term is generated by a series of choices of whether to include or the negative root from every factor . Consider all the expanded terms of the polynomial with degree ; they are formed by multiplying a choice of negative roots, making the remaining choices in the product , and finally multiplying by the constant . Note that adding together every multiplied choice of negative roots yields . Thus, when we expand , the coefficient of is equal to . However, we defined the coefficient of to be . Thus, , or , which completes the proof. Problems Here are some problems with solutions that utilize Vieta's quadratic formulas: Introductory 2005 AMC 12B Problem 12 2007 AMC 12A Problem 21 2010 AMC 10A Problem 21 2003 AMC 10A Problem 18 2021 AMC 12A Problem 12 Intermediate 2017 AMC 12A Problem 23 2003 AIME II Problem 9 2008 AIME II Problem 7 2021 Fall AMC 12A Problem 23 2019 AIME I Problem 10 Advanced 2020 AIME I Problem 14 See also Polynomial Retrieved from " Categories: Algebra Polynomials Theorems 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.
18144	https://www.youtube.com/watch?v=eRbF7VZJ3PU	Vector Equation of a Line in 2-space (full lesson) \| MCV4U JensenMath 250000 subscribers 121 likes Description 15324 views Posted: 12 May 2020 Go to for a copy of the lesson and practice questions. In this lesson you will learn how to write the vector equation of a line by finding a resultant vector that can point to any point on the line. The resultant is the sum of a position vector and a scalar multiple of a direction vector. You will also learn what the parametric equation of a line is as well. 7 comments Transcript: Intro we're now onto the last unit of vectors this is lesson 1 vector equation of a line in two space now in grade 9 you've learned about how to write equations of lines and you learned how to write them in a couple different formats in slope y-intercept form right that's the form y equals MX plus B and also standard form where you rearranged to have all the variables on one side and had it set equal to 0 so you should be familiar with those two formats from earlier in your high school career what we're going to do in this course is learn about two new ways we could write the equation of a line now that we know what vectors are we can actually write the equation of a line as a sum of vectors and that's called the vector equation of a line so I'll teach you what all of this means in this lesson and once we know how to create the vector equation of a line we can create parametric equations of a line by separating the components so once you're ready to learn about the vector equation and parametric equation of a line head on over to Jensen math dot CA so you can get a copy of the lesson you'll be able to find a copy of the lesson in the calculus course like I said where in the last vectors unit now so unit 3 of vectors called lines and planes and we're in the first lesson of this unit once you're here you've get a blank copy of the lesson here and you can get practice questions by clicking on worksheet and after you're done the worksheet you can check your solutions by clicking on worksheet solutions so when you're ready let's get started [Music] okay let's start the lesson by learning about the vector equation of a line so in your lesson you should see this diagram beside this paragraph that explains the diagram before I read the paragraph let me just orient you by describing what these vectors you see in this diagram are now our goal is to try and come up with an equation for that line and what we end up doing with vectors is we define that line by coming up with an equation for a resultant vector that could point to any point on that line we wanted it to and how can we come up with that resultant vector we write that resultant vector as of sum of two vectors a position vector which points to a point on the line plus some scalar multiple of a direction vector now the direction vector down here is a vector that's parallel to the line so if we add a scalar multiple of the direction vector with the position vector what we get is a resultant vector that could point to any point on the line depending on what we set the scalar multiple to be so let's read this Vector Equation of a Line paragraph here which is going to explain what I just tried to show you in the diagram in slope y-intercept form and standard flow and we have equations that define all points on a line a vector of equation of line is an equation that describes resultant vectors that start at the origin and end at a point on the line in order to create these resultant vectors we need the position vector that gives us a point on the line we call that position vector vector R naught and we also need a direction vector parallel to the line we call that vector M by adding the position vector and the direction vector together we get a resultant vector that has its tip on a point on the line by multiplying the direction vector by some scalar T we get a resultant vector that can have its tip at any point on the line and that's what makes it the vector equation of a line so in this diagram the blue one is the position vector it points to a point on the line the green one is a direction vector it's parallel to the line when we add the position vector and some scalar multiple of the direction vector we get a resultant vector that has tip at a point on the line and by varying that value of the scalar multiple T we can get this resultant to point to any point on that line we want that's why that resultant vector defines the line let me show you in geogebra how that works so here's a line here we can define that line as a sum of two vectors we can define it as a sum of a position vector so in order to do that we would need to know some point on the line and the position vector is the vector that starts at the origin and has its tip at that point we can add that vector with a scalar multiple of a vector that is parallel to the line so we need to know a direction vector a vector that is parallel to the line so if I add this direction vector with the position vector the resultant of those two would have its tip at a point on the line so let me reposition the direction vector so it's tip-to-tail with the position vector so we can see what it would look like when we add them together and the resultant of these two would look like this notice the tip is at a point on the line now if I vary the scalar multiple of the direction vector notice the direction vector is still going to be parallel to the line it's just going to change its magnitude so that's why the resultant vector of the position vector and any scalar multiple of the direction vector always has its tip at a point on the line notice the red vector is always pointing to a point on the line that's why that resultant vector defines that line so how does a vector equation of a line look it looks like this a resultant vector is equal to the sum two vectors the position vector plus a scalar multiple of the direction vector so down here we have definitions of what all those variables stand for so our resultant vector is a position vector to any unknown point on the line the resultant vector always has its tip at a point on the line vector R naught is a position vector to any one known point on the line and vector M is the direction vector it's a vector that's parallel to the line and what we do is we do a scalar multiple of the direction vector using the variable T where T can be any real number often you'll see this equation written in this form when working in 2-space right now so when working in two space a vector has two components in X and y components so the resultant vector is going to have an ax and a y component and so is the position vector it's going to have an X and Y component and the direction vector is also going to have an X and a Y component so you'll see a vector equation of a line written either in that format or this format but they mean the same thing now that we know how to write the vector equation of a line let's try example 1 for a line that goes Example 1 Geogebra through these two points come up with the vector equation of the line so let's take a look at what we have in geogebra and then it'll help you understand how we're going to do it algebraically so it gave us two points on the line and there's only one line that can go through those two points I have it drawn what's new about this question is we need to be able to write the equation of this line as a vector equation of a line so we need to write it as a resultant of two vectors a position vector and a scalar multiple of a direction vector the position vector can point to any point on the line so it gave us two points point a and B we'll just pick one of them I'll choose my position factor to point to point B so it has its tail at the origin and tip at point B that's my position vector and now to that I need to add a scalar multiple of a direction vector and a direction vector is a vector that's parallel to this line so to get a vector parallel to this line I could use the position vector a B so let's find the position vector that would connect point A to point B and that would look like this notice how that vector is parallel to the line so we call that our direction vector if I add that direction vector to the position vector I would get a resultant that has its tip at a point on the line so let me translate this direction vector so it's so that it's tip-to-tail with this position vector and notice how if I were to add those and get a resultant vector its tip is at a point on the line and if I multiply that Direction vector by any scalar I still have a resultant that has its tip at a point on the line so our vector equation is going to be the position vector plus a scalar multiple of the direction vector so over here let's come up with our position vector we call that vector or not so the position vector I decided to point to point B but we could have done to point a but I'll do point B so the position vector is vector 3 1 and my direction vector I decided is going to be the position vector a B and I can get the X and y components of that by doing tip - tail so the tip is at point B the tail is at Point a so to get the X component do X component tip - X component tail I get - and to get the Y component of the direction vector do the y coordinate of the tip minus the y coordinate of the tail 1 minus 4 is negative 3 so 2 negative 3 is position vector a B so the vector equation of this line I'll write it in this format X y equals the position vector 3 1 plus a scalar multiple of the direction vector and there we have it there's a vector equation of a line no matter what real number I plug in for T the sum of these two vectors would give me a position vector that points to a point on the line alright Part B says determine three more position vectors two points on the line and then graph the line so basically it wants us to come up with three resultant vectors that have their tips at points on the line so we'll use this vector equation of the line to come up with those I know the resultant vector will always have its tip at a point on the line as long as I choose a real value for the scalar T so let's just pick some values for T and then calculate some resultant vectors so I'll choose some easy values for T just how about one two and three and now all we have to do is sub each of those values into this vector equation of the line and we'll get three different resultant vectors which will each have their tip at a point on the line and then I'll show that to you on the graph down here so let's start with t equals 1 and now we can use our skills from adding algebraic vectors and multiplying by scalars to come up with the resultant vector by adding these two vectors together so when adding the vectors I have to add the X component of the position vector plus the scalar multiple of the X component from the direction vector so if 3 plus 1 times 2 is 5 and then to get the Y component of the resultant I do the Y component of the position vector plus the scalar multiple of the Y component of the direction vector so 1 plus negative 3 is negative 2 so the position vector 5 negative 2 should point to a point on our line let me draw that position vector and blue for us let's come up with two more position vectors let's let T equal two so now when I'm finding the resultant vector I just have to add the X component of the position plus two times the X component of the direction so three plus two times two that's seven and then to get the Y component of the resultant I do the y component of the position plus two times the Y component of the direction vector so 1 plus 2 times negative 3 that's negative 5 so let me graph that position vector as well now I'm just realizing that a scalar multiple of three is going to give us a position vector which won't be able to fit on our graph so let me choose a different value for my third value for T let's choose negative one so when I calculate this resultant I'll plug in negative 1 for T and now I will find the resultant of these two so 3 minus 1 times 2 that's 1 and 1 minus 1 times negative 3 that's 4 so I get the resultant vector 1 4 so if we've done this properly all three of these position vectors should point to a point on the line and what did the original question tell us about the line it gave us two points they gave us the point 1 4 & 3 1 1 4 oh that's that point right there and 3 1 that's this point right here if I connect those points with a line all three of the position vectors should have their tips at a point on the line so let's try there we go so notice that the tip of that vector is on the line the tip of that vector is on the line and the tip of that vector is on the line so hopefully you can see now how these vector equations of lines work Part C says determine if the point Example 2 Vector Equation Part C 2 3 is on the line so let me start by rewriting the vector equation of the line here if point 2 3 is on the line there is some value of T that will get me a resultant vector it has an X component of 2 and a Y component of 3 so let me set the X component equal to 2 and the y component equal to 3 and see if I can solve for a value of T that makes that true for both the X and the y components now it'd be useful to separate our work into X component and Y component to get the X component of the resultant vector 2 that would have to come from the X component of the position plus a scalar multiple of the X component of the direction vector so let me make an equation from that information to would be equal to 3 plus 2t and now to get the Y component of the resultant 3 that would come from the Y component of the position vector plus a scalar multiple of the Y component of the direction vector so let me make an equation from that information 3 equals 1 minus 3t now if point 2 3 is on the line there should be a single value for T that makes both of these equations true so let's find out if I were to solve this equation I would get negative 1 equals 2t T equals negative 1/2 if I were to solve this equation I would get 2 equals negative 3t T equals negative 2/3 since there are different values negative 2/3 and negative 1/2 are clearly not the same value so that tells us that the 0.23 is not on the line because there's not a single value of T that gets us a resultant vector 2 3 therefore 2 3 is not on the line ok so you now know how to write the vector equation of a line no problem Example 1 Parametric Equations it's the sum of two vectors the sum of a position vector and a scalar multiple of a direction vector but what we're going to move on to now is another way we can write an equation of a line it's called parametric equation of a line and we can come up with that easily if we have the vector equation we just separate the vector equation into its X&Y components so the X component of the resultant equals the X component of the position vector plus the scalar multiple of the X component of the direction vector that's what this says right here x equals x naught plus T times m1 and same with the Y component the Y component of the resultant is equal to the Y component of the position vector plus a scalar multiple of the Y component of the direction vector that's where this equation comes from so these down here are called parametric equations of the line and I should also mention in this lesson we're only working in two space will be working in three space in the future but right now the parametric equations there's only two of them for a line we only need an equation for the X and for the y components so if we think back to example 1 this was the vector equation we could write the parametric equations by breaking into the two equations one for the X and 1 for the Y so for the X component of the resultant it was calculated by doing 3 plus a scalar multiple of 2 so x equals 3 plus 2t and the Y component of the resultant was calculated by doing 1 plus a scalar multiple of negative 3 so y equals 1 minus 3t so these are the parametric equations of the line from example 1 let's try example 2 where you're given the parametric equations of the line let's start by seeing if we can find the Example 2 Parametric Equations coordinates of two points on the line so in order to find two points on the line well the points both have x and y coordinates we can get them by just picking values for T so let's pick some easy values for T let's choose T equals zero and T equals 1 when T equals 0 X would equal 3 plus 2 times 0 so X would equal 3 and Y would equal negative 5 plus 4 times 0 so Y would equal negative 5 so what we would get is a resultant factor 3 negative 5 and we know the resultant vector has its tip at a point on the line so we know the point 3 negative 5 is on the line let's get another point let's let T equal 1 the X component would be 3 plus 2 times 1 so x equals 5 and the Y component of the resultant would be negative 5 plus 4 times 1 so y equals negative 1 so the resultant vector would be the vector 5 negative 1 and I know that resultant vector points to a point on the line so that tells me that the point 5 negative 1 is on the line so I've come up with 2 points on this line 3 negative 5 and 5 negative 1 let's check if that's right so here in geogebra I have the line set up it's here in blue and I have that line written as a sum of two vectors a position vector and a scalar multiple of Direction vector and that gives me this resultant vector which points to a point on the line so we let T equal 0 and we set it pointed to the point 3 negative 5 let's try that out let's set T to 0 and notice yes the resultant does point to 3 negative 5 so 3 negative 5 yes is on the line and we let T equal 1 when T is equal to 1 we said that the vector pointed to the point 5 negative 1 and it does therefore the point 5 negative 1 yes is on the line Part B says write a vector equation of Example 3 Vector Equation the line so from the parametric equation let me just copy and paste that over we need to know that these numbers come from the X&Y components of the position vector and these numbers come from the X&Y components of the direction vector so our position vector R naught is equal to the vector 3 negative 5 that's our position vector and our direction is equal to the vector two for now I haven't talked about this yet but we always want to reduce our direction vector as much as possible remember a direction vector just has to be parallel to the line so I can create a parallel vector to this which actually has reduced values any scalar multiple of this is going to be parallel to it so if I multiply this vector by a half I would get a vector that is parallel to it but the numbers are reduced a half times two four would give me the vector one two you can think of it like reducing fractions so vector 1/2 would be my reduced version of the direction vector so I would use that so the vector equation would be X y equals my position vector three negative five and make sure you're always using square brackets here to indicate that we are working with vectors this is not a point this is a vector so vector three negative five plus a scalar multiple of the direction vector one two so there's my final answer for the vector equation of the line just make sure you use a reduced version of the direction vector if possible Part C says write the scalar equation of the line so that's actually Part C another way of saying write the standard form equation of the line now standard form equations for lines are fine into space but we won't be able to do that in three space because that's how we write the equation of planes but I'm jumping ahead a little bit we'll get there in a few lessons in two space we can write the standard form equation of a line and it's called the scalar equation and to do that what we do is in the parametric equations we isolate T and both of the equations and then we set the equations equal and rearrange in the standard form by setting equal to zero so here are the parametric equations of that line so what we have to do is each of those parametric equations we have to isolate for T so I'll separate into the X equation and the Y equation and now I'll isolate both these equations for T so subtract the 3 and divide the two and over here add the 5 and then divide the 4 now I have two equations with T isolated so I know T equals this and T also equals this so I know I can set those two things equal to each other and now I just have to rearrange this into standard form so let me start by getting rid of these fractions by multiplying both sides of this equation by 4 and I would get 2 times X minus 3 on the left and just y plus 5 on the right expand the left 2x minus 6 equals y plus 5 and let's move everything to the left 2x minus y minus 11 equals 0 so we call this the scalar equation of the line it's the equation of the line in standard form and the notice the general format it follows is ax plus B y plus some constant equals 0 that's the format of the scalar equation of a line Part D Part D says determine if line 1 is parallel to line 2 where line 1 is that line we've been working with up till now so I've just copy and pasted that over so it wants to see are these two lines parallel to each other well these lines would be parallel to each other if the direction vectors were scalar multiples of each other so what was the direction vector for line 1 Direction vector for line 1 was 2 before I get that from these numbers right here and we could use the reduced version 1/2 but I'll just leave it like that for now and Direction vector 2 I'll get from line 2 is the vector 312 so these lines are parallel to each other if these vectors are scalar multiples of each other because that would mean that these vectors are collinear they fall on the same line so let's check if they're scalar multiples let's check if the X components and the Y components are both the same scalar multiples of each other so 2 is equal to a scalar multiple of 3 and 4 is equal to a scalar multiple of 12 are they the same scalar multiple so for the X and the y components is their scalar multiple okay that makes both of these true so K equals 2/3 from this equation and K equals 1/3 from this equation so since they're not scalar multiples since I got different values for K I would say that the lines are not parallel therefore l1 and l2 are not parallel example three gives us the scalar equation of a line and it wants us to start by graphing the line so to graph a line in standard form we there's a couple ways we could do it we could rearrange into slope y-intercept form or I could just find two points on this line plot them and connect them the easiest way to find two points I think the easiest two points to find would be the X and y intercepts so let's find the x intercept by setting y to 0 and solving so 4 X plus 5 times 0 plus 20 equals 0 if I solve this I get X is negative 5 so the x intercept is here at negative 5 and now I'll also find the y intercept by setting X to 0 and solving so 4 times 0 plus 5y plus 20 equals 0 I would get y equals negative 4 if I self-thought so to graph the line I just have to connect those two points there's only one line that would go through those two points ok so there's the line we have a craft Part B says determine a position vector that is perpendicular to Part B Example the line so there's a couple ways we could do this how I'm going to do this is I'm going to find a direction vector that is parallel to the line so I'm going to find a vector that connects these two points so I'll call this direction vector M and the vector that connects those two points well at this point is the point negative 5 0 and this point is the point 0 negative 4 I'll make this detail and this the tip so I'll get a position vector that connects those two points the X component by doing tip- tail so 0 minus negative 5 is 5 and the y component do tip- tail negative 4 minus 0 is negative 4 so the position vector 5 negative 4 will be parallel to that line there's the position vector 5 negative 4 notice it's parallel to the line now if I want a vector perpendicular to the line I know that it would be perpendicular to this direction vector which means it would have a dot product of 0 with this direction vector remember that vectors that a dot product of zero are perpendicular to each other so I know there is some normal vector normal meaning perpendicular that has a dot product of zero with the direction vector so I'm going to let m dot n equal zero m-meaning Direction vector n meaning normal vector the perpendicular vector so five negative four dot some vector XY is equal to zero so when doing dot product we multiply the X components so 5x plus we multiply the Y components negative 4y so I could just write 5x minus 4y equals to 0 now there's an infinite number of solutions to this so we can pick any value we want for either X or Y and then solve for the other that makes it true to make the numbers work out nicely and so that you can see the shortcut after I'm going to choose Y to be 5 so 5x minus 4 times 5 equals 0 so that mean x equals 4 so the x value is 4 the Y value is 5 so the normal vector is the vector 4/5 remember the shortcut for getting a perpendicular vector we didn't have to go through all of this work all we had to do was swap these two values and change one of their signs so put the negative 4 first the 5 second and then change the negative 4 to positive 4 and that works that gives you a perpendicular vector so all of this work is unnecessary but this is just kind of showing you why that works so there's the line and the normal vector we just found was the vector 4/5 so it's that vector notice that that is perpendicular to the direction vector we had which was dr. 5 negative 4 so we figured out that those two are perpendicular to each other that means that this normal vector if translated to be on this line would also be perpendicular with the line Part C is the interesting part how does the position vector from Part B compared to Part C Example the scalar equation so how does this normal vector the normal vector remember was the vector four five how does this factor specifically the x and y components compared to the standard form equation it gave us in part a but look right here for five so we could say the component of the normal vector correspond to the coefficient of x and y in the scalar form equation Part D says write a vector equation of the line well we have all the necessary components for that we have the direction vector we got in Part B it was vector five negative four and we got multiple points on the line in Part A we had the x-intercept and the y-intercept the x-intercept was the point negative 5 0 so a position vector would just point to that point so it would be the vector negative 5 0 and the vector equation would just be a sum of the position vector plus a scalar multiple of the direction vector so my vector equation would be x y equals the position vector negative 5 0 plus a scalar multiple of the direction vector 5 negative 4 so if we look back at the graph here here's the position vector I pointed to that point and then we're adding a scalar multiple of this direction vector that we found earlier and that would give us a resultant vector which points to a point on the line and if we did a different scalar multiple of the direction vector it would point to it would just point to a different point on the line so we did position vector plus a scalar multiple of the direction vector and that gives us our resultant vector example 4 says find a scalar equation for this line and this line is written as the vector equation of a line so we want to translate this into the scalar equation of a line to do that it needs to be in the form ax plus B y plus C equals 0 where remember a and B correspond to the X and y components of a normal vector to the line so a vector perpendicular to the line so how do we find a vector perpendicular to the line well the direction vector tells us a vector parallel to the line so the direction vector is the vector 2 negative 1 and the shortcut for finding a perpendicular vector which we call a normal vector is just switch the X and y components and then change one of their signs either make the to negative or make the one positive so I'll do that version of it so this vector is perpendicular to the direction vector which then tells me that this vector is perpendicular to this line let me just prove that to you in geogebra so here we have the line the vector equation was the position vector plus a scalar multiple the direction vector and that gives us our resultant that points to a point on the line no matter what the scalar multiple of the direction vector is what we did was we took the direction vector and we switched the X and y components and changed one of their signs and we got the normal vector of one - known as that vector it forms a 90 degree angle with the line so when we write the scalar equation of the line we just take the X component of the normal plug it in for a the Y component of the normal plug it in for B and then any point we know on the line we know this point is on the line because a position vector points to a point on the line plug that in for X and Y and then we can solve for C so I'll do this in a couple parts let me first replace the coefficients a and B with the x and y components of the normal vector so 1 X plus 2y plus C equals 0 now I have to substitute in for X&Y any point that I know is on the line well I know the position vector points to a point on the line so I know the point zero three is all on the line so I can take its x-coordinate and its y-coordinate and sub it in for x and y so one times zero plus two times three plus C equals zero and now I can solve for that constant C and I would get C equals negative six so the scalar equation of the line have a B and C substitute it in so 1 X plus 2y minus 6 equals 0 that's the scalar equation of the line
18145	https://www.youtube.com/watch?v=34GkWcAc9jM	Using Zeros to Graph I LOVE YOU Crystal Clear Maths 12 subscribers 57 likes Description 2984 views Posted: 31 Mar 2013 It is possible to use the concept of zeros (when the product of a number of factors equals zero) to produce graphs composed of many component parts. In this video I show how to create just ONE EQUATION that produces the words "I LOVE YOU" on graph paper. This might be a very "nerdy" way to express your love for someone :-) I decided to post the entire video as it was recorded with hardly any editing. One reason is that I am explaining various steps all the way through it. Another is that, if you wish, you can simply 'jump' to the end of the video if you only want to see the result. You don't HAVE to watch it all ... just watch enough to understand. You might now ask another question: what kind of graph is created if the product of factors is NOT equal to zero? I will be producing a few videos illustrating this concept as soon as I have finished the rest of the videos and worksheets for this set. If you would like to download a copy of the full equation from the video, you will find it here ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you wish to be informed of each video as I produce it, please subscribe to my channel. It would be appreciated if you would like and/or comment on this video as well -- especially if these suggestions have helped you. For more information about mathematics or how to study visit my website, Crystal Clear Mathematics, at If you wish to be kept up to date with what I am producing on the website (ad free, spam free, cost free mathematics and study materials), please add your name to the mailing list there. Download my FREE 32 page PDF "How to Study" booklet at Best wishes with your study and your mathematics! Thank you. 18 comments Transcript: Welcome to Crystal Clear Mathematics where it is easier than you think. I'm your host, Graeme Henderson. This particular video is a rather special and unique one. In the last few videos I've been showing you how to draw the graphs of polynomial equations (when they're factorised) by studying their roots. And I've also shown you how to construct polynomial equations given their graph ... by studying their roots and their y-intercept. All of this skill has been based on our understanding of zeros. That is, that when we have a product of a number of factors making zero, then each of those factors may be zero. We're going to use that information, plus an understanding of another simple curve, and another principle (a third principle), to construct what I hope will be a rather unique thing for you. And that is, I'm going to attempt in this one video and perhaps even without editing (as an uninterrupted video) to construct the words "I LOVE YOU" on graph paper with just one equation. So, follow me carefully if you will. The first principle is this. If I wish to construct an ellipse (a long, thin one), that goes through a and -a, and b and -b, it has something like the equation for a circle, x-squared plus y-squared equals radius-squared (I'm going to use 1). And, the way we get x to go from a to -a, is to do this. You can see that, when y is worth zero, when x is worth a we get a-squared over a-squared which is one. And, when x is -a, we get -a all squared which is plus a-squared over a-squared, which is one also. This is a rather clever way of distorting the x-axis. And similarly, to distort the y-axis, we do this. So, I think you can see that, when y is equal to b (and x is zero because we're on the y-axis), when y = b, we get b-squared over b-squared is one ... and the same occurs at -b. So, the question is, "How can I construct a line that appears to be a single, vertical straight line?" Well, obviously, I need to bring these in, and I need a to be a very, very tiny number. So, if a is a very tiny number, like one thousandth for example, then you can see that, when I'm dividing by a fraction, this denominator appears up the top so I would have something like 1000-squared, x-squared, plus y-squared over ... (I haven't decided what I want to do with b yet ... let's make it go from 2 to -2) ... then this would be 2-squared equals 1. I can probably write this a little more compactly by changing a thousand squared, which is a million, into 10-to-the-power-6, because I'm going to need every bit of room I can get. And, I think you'll agree that, if I bring the 1 over to this side, that that also is a true form for the equation of this particular ellipse. So, this ellipse now goes from -2 to +2 and from plus-one-thousandth to minus-one-thousandth, so it's basically a line interval ... at least, to the naked eye ... and that's what I'm trying to do [achieve]. So, that's step number one ... to create a line interval, like the letter I or the descender on the letter L. That's what we need. The second thing is, "How do we move graphs around the coordinate plane (the Cartesian plane)?" The fact is that we have to distort the x-axis again, simply by adding and subtracting values. To move it down to here so the line would appear down here (for example) through -5, our adjustment would be, not to do anything on the y-axis, but simply to do this ... and this particular ellipse has now moved here. So, we now have a stratagem, or a strategy, whereby we can create what appears to be a line interval and we can move it. So far, so good! Well, I did have to stop and clean the board, but let's continue. I'm going to plan out, on a very long x-axis, where my letters will go. I'm going to need quite a few gaps ... this is unplanned unrehearsed, and I've never done it before ... it's just an idea I had a few days ago to share with you ... but let's try it. Minus 2,3,4,5,6,7,8,9,10,11,12,13,14 ... we'll do that ... -14,-12,-10,-8,-6,-4,-2 ... just marking every second one. 1,2,3,4,5,6,7,8,9,10,11,12,13 ... Ok ... Let's try [to] put ... we want "I LOVE YOU" ... we can probably get "LOVE" here ... let's try to put the "E" there, for example. Now I told you the "U" at this stage will have ... the "V" will have to look a bit like a "U." There's our "O." There's our "L." Leave a gap of perhaps three ... "I LOVE." Leave another gap of three between the words and we have Y ... O ... (it'll go up to 14) ... U. And let's see if we can generate one equation that produces all of this material. Now you can see that I'm going to have to write fairly small. This vertical line ... y = (you're going to enjoy this) ... this vertical line is that same ellipse that we were taking about before. So, it's going to be 10^6 ... and to move the x-value down to -11, it'll be (x+11)^2 ... and it's going to be y^2 on 2^2 minus 1. If this expression is zero ... sorry, I don't need that (you can see this is very unrehearsed) ... but if this expression is zero, then it will construct that graph. The next one, I want this line, and it's going to look the same ... 10^6 ... going through -8, so it's (x+8)^2 plus y^2 on 2^2 minus 1. If that expression there ... if that factor is equal to zero, it will create that line. Let's do the next vertical line. This one goes through -5. You can see this is going to get a little bit boring, so I might actually speed this up on the camera [I chose not to]. The next vertical line is through -3 ... (x+3)^2 plus y^2 on 2^2 minus 1. Where are we ... we've done that one, we've got the one through -2 ... the one through zero (that's just going to be our ordinary vertical ellipse) ... one through 1 ... I'll just do the long ones first ... so, one through 8 ... if I can squeeze it in here we'll get the one through 9 ... one through 11, 12 and 14. That's got all those vertical ones done. Now, this one's a bit more tricky. We want it to start down at the origin going from ... instead of +2 to -2 we just want a +1 to -1 ... and we're going to shift it so its centre moves from the origin to there. And it's going to look like ... same width (so the x-value [coefficient] won't change) ... it's going to move to x=6 ... the y-value is going to move up to y=1 ... and (I'll write it in but it's going to go from plus to minus one, so that's a bit redundant). That has now got that descender there so, we've got all of our descenders in place. Let's now deal with the horizontals. If we want a horizontal, then we want the ellipse to have a very small vertical dimension and to be quite long horizontally. So, we're going to swap these two coefficients around and I want it only to be two units long so x^2 will be over 1^1 (I'll write it in). And this time we're going to have 10^6 y^2 minus one ... but, I want it moved to here, for example, where the centre is x=-7 and y=-2. So, I want (x+7)^2 and (y+2)^2. I hope that made sense to you. I haven't really taught you how to do this yet. I'll be creating some videos describing it. And now we want to create these other horizontal lines. So, we've got one centred at (-4, +2), so it's going to be (x+4)^2 ... I won't write the 1^2 ... plus 10^6 (y-2)^2 -1. I should be using parentheses and brackets, but I won't start now. We'll just embed parentheses. This one here (let's speed up) is going to be (x+4)^2 plus 10^6 (y+2)^2 -1, which makes it go through (-4, -2). There it is. This one here is through (-1, -2) so we're going to have (x+1)^2 + 10^6 (y+2)^2 -1. We've done that one. We've still got a few to go. This one here's at (2, 2) ... (x-2)^2 and 10^6 (y-2)^2 -1. This one here at (2, -2) ... (x-2)^2 + 10^6 (y+2)^2 -1. I'll come back to this one. This one here has only moved up to x=7, y=0, so it'll be (x-7)^2 + 10^6 y^2 -1. This one here at (7, -2) ... (x-7)^2 + 10^6 (y+2)^2 -1. Three and a half to go! This one here's at (10, 2) ... so, we've got ... (x-10)^2 + 10^6 (y-2)^2 -1. (10, -2) ... y ... x ... oh, I'm obviously getting tired (this is a long equation), ah ... (x-10)^2 ... that's correct ... plus 10^6 (y+2)^2 -1. This one here is at (13, -2) so it'll be ... (x-13)^2 + 10^6 (y+2)^2 -1. And, we've just got this one to do. Now this one, the centre is going to be at (1.5, 0) and it's going to be extended horizontally half a unit in each direction. So (I think this'll just show on the video), we want ... (x-1.5)^2 ... that will locate it correctly. To get the right dimensions we want to divide by half-squared, so I'm going to put a 4 out the front ... + 10^6 y^2 -1. And I'll set it all equal to zero. That is one equation! Now, why does it work? Well, the principle is this: if I choose any particular x or y value, every single one of these terms, every single one of these factors has a value! That is, the entire expression is fully defined for every x and y value. That's important. But, if you take any particular factor ... because here we have a multitude of factors multiplying to make zero ... so, any one of them could be zero ... if you made that one zero, then it creates (or it defines) a graph where x is 12 and y is zero ... 12 and zero ... and it creates that vertical line. If we chose this one here, it's when x is -4 and y is -2 ... -4, -2 ... and it creates this horizontal line. So, every part of this equation has a job to do, and creates a separate little part of the graph. And I hope you found that an interesting exercise. I don't recommend it ... well, you can do what you like with it! But it does show you the principle that, when you have an equation fully factorised, each factor when it equals zero can create an interesting part of the curve for you. Now, I know that's been a bit of a fumble. I decided to do it so you could see me 'ad lib.' I hope you've enjoyed the experience and learned from it. There'll be an interesting worksheet created which will encourage you to draw graphs like circles with crosses through [them] and all that sort of business. If you're interested in doing that and interested in learning the skills, then please look at the description below the video and download the worksheet. It's a PDF file ... there's no charge. Just have fun and enjoy your mathematics. Thank you for watching.
18146	https://web.ma.utexas.edu/users/m408s/m408d/2016/LM10-5-3.html	Ellipses can be elegantly described in four ways. \| \| \| 1. Via Cartesian (rectangular) coordinates. 2. In terms of distances to two foci (plural of focus). 3. In terms of distances to a focus and a directrix. 4. In polar coordinates. \| We will do the first two on this page, and the third and fourth later on. The simplest description of an ellipse is as a squashed or stretched circle. Start with the unit circle $x^2 + y^2 =1$, and stretch it by a factor of $a$ in the $x$ direction and $b$ in the $y$ direction to get: \| \| \| The standard formula for an ellipse in rectangular coordinates is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$ \| The points $(\pm a,0)$ (and sometimes the points $(0,\pm b)$) are called vertices. If $a > b > 0$, then the major axis is the line segment from $(-a,0)$ to $(a,0)$ and the semi-major axis is the line segment from the origin to $(a,0)$. Likewise, the minor axis runs from $(0,-b)$ to $(0,b)$ and the semi-minor axis runs from the origin to $(0,b)$. If $b > a > 0$, then the major and semi-major axes are vertical and the minor and semi-minor axes are horizontal. For now we'll stick with the case that $a > b$, so that the ellipse is short and fat. The origin is the center of the ellipse. \| \| \| --- \| \| Now let $c = \sqrt{a^2-b^2}$. The points $(\pm c, 0)$ are called foci. These points are extremely important in astronomy, since planets follow elliptical orbits with the sun at a focus, not with the sun at the center. Let $F_1(-c,0)$ and $F_2(c,0)$ be the two foci, let $P(x,y)$ be an arbitrary point on the ellipse. Let $L_1$ be the distance from $F_1$ to $P$, and let $L_2$ be the distance from $F_2$ to $P$, as in the figure on the right. \| \| \| \| \| Amazing fact: The ellipse is the set of all points where $L_1 + L_2 = 2a$. \| This fact gives elliptical rooms amazing acoustic properties. If you whisper at one focus of such a room, the sound waves from your voice will bounce off the walls and converge at the other focus -- that's why it is called a focus. The same goes for light reflecting off elliptical mirrors. To understand the amazing fact, let's convert the equation $L_1 + L_2 = 2a$ to rectangular coordinates: \begin{eqnarray} L_1 + L_2 & = & 2a \cr\cr L_1 & = & 2a-L_2 \cr \cr \sqrt{(x+c)^2+y^2} & = & 2a -\sqrt{(x-c)^2 + y^2} \cr\cr (x+c)^2 + y^2 & = & 4a^2 + (x-c)^2 + y^2 - 4a \sqrt{(x-c)^2 + y^2}\cr\cr 4a\sqrt{(x-c)^2 + y^2}&=& 4a^2-4cx \cr \cr a \sqrt{(x-c)^2 + y^2} &=& a^2-cx \cr \cr a^2(x-c)^2+ a^2 y^2 &=& a^4+c^2x^2 -2a^2cx \cr \cr a^2x^2 + a^2c^2 -2a^2cx + y^2 &=& a^4 + c^2x^2 -2a^2cx \cr \cr (a^2-c^2)x^2 + a^2 y^2 &=& a^2(a^2-c^2) \cr \cr b^2 x^2 + a^2 y^2 &=& a^2b^2 \cr \cr \frac{x^2}{a^2} + \frac{y^2}{b^2} &=& 1,\end{eqnarray} where we have used the fact that $b^2=a^2-c^2$. That's a long and messy calculation for a simple and elegant result. You should be able to construct the equation of an ellipse given any two of $a$, $b$ and $c$, since you can get the third from $c^2=a^2-b^2.$ \| \| \| --- \| \| Example 1: Find the location of the foci of the ellipse $\displaystyle{\frac{x^2}{25} + \frac{y^2}{9}=1}$. \| Solution: We have $a=5$ and $b=3$, so $c = \sqrt{a^2-b^2} = 4$. The foci are at $(\pm 4,0)$. \| \| \| \| --- \| \| Example 2: Find the equation of an ellipse with foci at $(\pm 3,0)$ if $b=4$. \| Solution: Since $c=3$ and $b=4$, $a^2=3^2+4^2=25$, so $a=5$. This makes the equation $$\frac{x^2}{25} + \frac{y^2}{16} = 1.$$ \| \| \| \| --- \| \| The ratio $c/a$ is called the eccentricity of the ellipse, and is usually denoted $e$. Note that $e < 1$. A circle can be viewed as an ellipse with eccentricity zero, and with both foci at the origin. \| \| \| \| \| It is easy to plot an ellipse as a parametrized curve. Just take $$x = a \cos(t); \qquad y = b\sin(t),$$ with the parameter $t$ running from $0$ to $2\pi$. \|
18147	https://allen.in/jee/maths/graphical-method-linear-programming	Published Time: 2025-02-24T11:14:50.707Z Graphical Method in Linear Programming: Overview & Steps Courses NEET Class 11th Class 12th Class 12th Plus JEE Class 11th Class 12th Class 12th Plus Class 6-10 Class 6th Class 7th Class 8th Class 9th Class 10th View All Options Online Courses Distance Learning Hindi Medium Courses International Olympiad Test Series NEET Class 11th Class 12th Class 12th Plus JEE (Main+Advanced) Class 11th Class 12th Class 12th Plus JEE Main Class 11th Class 12th Class 12th Plus Classroom Results NEW NEET 2025 2024 2023 2022 JEE 2025 2024 2023 2022 Class 6-10 Scholarships TALLENTEX AOSAT ALLEN E-Store More ALLEN for Schools About ALLEN Blogs News Careers Request a call back Book home demo Login Algebra SetsRelations and FunctionsComplex NumbersProbabilityBinomial TheoremDeterminants and MatricesSequence & SeriesQuadratic Equations & ExpressionLogarithm Calculus Differential Calculus Limits & Continuity Continuity and Differentiability Integral Calculus Definite Integral Indefinite Integral Area under Curves Differential Equations Coordinate Geometry Conic SectionPoint and Straight Lines3D Geometry Vectors Coplanar Vectors Trigonometry Inverse Trigonometric Functions Exams JEE Main ExamJEE Advanced Exam JEE PhysicsJEE Chemistry HomeJEE MathsGraphical Method Linear Programming Graphical Method in Linear Programming Linear programming is a powerful technique used in optimization, where the goal is to maximize or minimize a linear objective function, subject to a set of linear constraints. One of the simplest methods of solving linear programming problems is the Graphical Method. This method is particularly effective when dealing with two variable problems, as it allows us to visualize the solution and better understand the relationship between constraints. 1.0 What is Linear Programming? Linear programming (LP) is a mathematical technique for optimization. It involves finding the best possible outcome (such as maximum profit or minimum cost) in a mathematical model whose requirements are represented by linear relationships. These relationships are typically expressed through an objective function (which we aim to optimize) and constraints (which define the limits or restrictions). For example, a company may want to maximize its profit by deciding how much of two products to produce, subject to constraints such as limited resources, time, or production capacity. 2.0 The Graphical Method: An Overview The Graphical Method is used to solve linear programming problems with two variables. It involves plotting the constraints on a graph, identifying the feasible region, and then determining the optimal point that maximizes or minimizes the objective function. Key Elements of a Graphical Method: Decision Variables: These are the variables we need to solve for, often denoted as x 1 and x 2 in two-variable problems. Objective Function: A linear equation that needs to be maximized or minimized, typically in the form Z=c 1x 1+c 2x 2. Constraints: These are linear inequalities that limit the values of the decision variables (e.g., x 1+2 x 2≤6). 3.0 Steps in the Graphical Method Step 1: Formulate the Linear Programming Problem Start by defining the objective function and the constraints for the problem. For example, consider the following problem: Objective: Maximize Z=3 x 1+4 x 2 Constraints: x 1+x 2≤4 2 x 1+x 2≤5 x 1≥0,x 2≥0 (non-negativity constraints) Step 2: Graph the Constraints Each constraint is a linear equation that can be plotted on a graph. To graph the constraint, first convert the inequality into an equation by replacing the inequality sign with an equal sign. For example, for the constraint x 1+x 2≤4, graph the line x 1+x 2=4. This line divides the plane into two regions. The feasible region will be one side of the line, depending on whether the inequality is ≤ or ≥. Repeat this process for each constraint, plotting them on the same graph. Step 3: Identify the Feasible Region The feasible region is the area that satisfies all the constraints. It is typically a polygon (or sometimes unbounded) where all the constraint lines intersect. This region represents all possible solutions that meet the problem's requirements. Step 4: Locate the Corner Points The optimal solution to a linear programming problem in the graphical method will always occur at one of the corner points (also called vertices) of the feasible region. These points can be found by identifying where the constraint lines intersect. Step 5: Evaluate the Objective Function Now that we have the corner points, evaluate the objective function at each of these points. Substitute the values of x 1 and x 2 into the objective function Z=3 x 1+4 x 2and calculate the corresponding values of Z. Step 6: Choose the Optimal Solution The solution that gives the highest (for maximization problems) or lowest (for minimization problems) value of the objective function is the optimal solution. If there are multiple corner points with the same value, there may be multiple optimal solutions. 4.0 Solved Example: A Simple Linear Programming Problem Problem: Maximize Z=3 x 1+2 x 2 Subject to: x 1+x 2≤6 2 x 1+x 2≤8 x 1≥0,x 2≥0 Plot the constraints: x 1+x 2=6(L in e 1) 2 x 1+x 2=8(L in e 2) Find the feasible region: The feasible region will be bounded by these lines and the non-negative axes. Locate the corner points: Identify the intersections of the lines, which are the possible solutions. Evaluate the objective function at each corner point: At (0, 6), Z = 3(0) + 2(6) = 12 At (4, 0), Z = 3(4) + 2(0) = 12 At (2, 4), Z = 3(2) + 2(4) = 14 Optimal solution: The maximum value of Z occurs at (2, 4), where Z = 14. Example 2:An aeroplane of an airline can carry a maximum of 200 passengers. A profit of Rs 400 is made on each first-class ticket and a profit of Rs 300 is made on each economy-class ticket. The airline reserves at least 20 seats for first class. However, at least 4 times as many passengers prefer to travel by economy class than by first class. Determine how many of each type of tickets must be sold in order to maximize the profit for the airline. What is the maximum profit? Solution: Let x tickets of first class and y tickets of economy class be sold to maximize the profit. Then, x≥20, y≥4 x, x≥80 and x+y≤200. The profit function is given by Z = 400x + 300y. PointsCorner pointsCorresponding Value Z A(20, 80)Rs 32000 B(40, 160)Rs 64000 → Maximise C(20, 180)Rs 62000 So, for a maximum profit first class ticket should be 40 and economy class should be 160. Example 3:A firm manufactures two types of products, A and B, and sells them at a profit of Rs 5 per unit of type A and Rs 3 per unit of type B. Each product is processed on two machines, M 1 and M 2. One unit of type A requires one minute of processing time on M 1 and two minutes of processing time on M 2; whereas one unit of type B requires one minute of processing time on M 1 and one minute on M 2. Machines M 1 and M 2 are respectively available for at most 5 hours and 6 hours in a day. Find out how many units of each type of product should the firm produce a day in order to maximize the profit. Solve the problem graphically. Solution: Let x units of A and y units of B be produced in order to have a maximum profit. Then, clearly x ≥ 0 and y ≥ 0. x units of A and y units of B will take (x + y) minutes on M 1. ∴ x + y ≤ 300. x units of A and y units of B will take (2x + y) minutes on M 2. ∴ 2x + y ≤ 360. Let Z be the profit function. Then, Z = 5x + 3y. We have to maximize Z= 5x + 3y , Subject to the constraints x + y ≤ 300 2x + y ≤ 360 x ≥ 0 y ≥ 0. PointsCorner pointsCorresponding Value Z O(0, 0)0 A(180, 0)900 B(60, 240)1020 → Maximise C(0, 300)900 So, for maximum profit x = 60 and y = 240. 5.0 Advantages of the Graphical Method Visual Insight: It gives a clear visual representation of the feasible region, helping you understand the relationship between constraints and objective. Simple for Two Variables: The method is straightforward and easy to apply for problems with only two variables. Intuitive: It is an excellent tool for teaching and understanding the basics of linear programming. 6.0 Limitations of the Graphical Method Limited to Two Variables: The graphical method is only feasible for problems with two decision variables. For larger problems (three or more variables), other methods such as the Simplex Method are needed. Less Practical for Large Problems: As the number of constraints or decision variables increases, the graphical method becomes cumbersome and impractical. Table of Contents 1.0 What is Linear Programming? 2.0 The Graphical Method: An Overview 3.0 Steps in the Graphical Method 4.0 Solved Example: A Simple Linear Programming Problem 5.0 Advantages of the Graphical Method 6.0 Limitations of the Graphical Method CONTENTS Frequently Asked Questions What is the Graphical Method in Linear Programming? The Graphical Method is a technique used to solve linear programming problems with two decision variables. It involves graphing the constraints, identifying the feasible region, and then finding the optimal solution by evaluating the objective function at the corner points of the feasible region. What type of problems can be solved using the Graphical Method? The Graphical Method is used to solve linear programming problems with two decision variables. It works best for small-scale problems, where visualizing the constraints and feasible region is possible. How do you graph constraints in the Graphical Method? To graph constraints, convert each inequality into an equation by replacing the inequality sign with an equal sign. Then, plot the resulting lines on the coordinate plane. The feasible region is the area where all the constraints overlap. What is the feasible region in linear programming? The feasible region is the area on the graph that satisfies all the constraints of the linear programming problem. It is the region where the solution to the problem exists. The optimal solution will always lie at one of the corner points (vertices) of the feasible region. What is the role of corner points in the Graphical Method? The corner points (or vertices) of the feasible region are the key points in the graphical method. The optimal solution will always be at one of these points. To find the optimal solution, evaluate the objective function at each corner point and choose the one that maximizes or minimizes the function, depending on the problem. How do you evaluate the objective function using the Graphical Method? After identifying the corner points of the feasible region, substitute the values of the decision variables at each corner point into the objective function to find the corresponding values. The point that gives the highest (for maximization) or lowest (for minimization) value is the optimal solution. Join ALLEN! (Session 2025 - 26) Name Mobile Number Class Choose class Goal Choose your goal Preferred Programs Preferred Mode State Choose State [x] I agree to Terms & conditions [x] I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address. 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18148	https://apps.dtic.mil/sti/tr/pdf/ADA327984.pdf	Factoring Polynomials Modulo Composites Adam Klivans May 8, 1997 Abstract This paper (characterizes all the factorizations of a polynomial with coeffi-cients in the ring Z, where n is a composite number. We give algorithmn to compute such factorizations along with algebraic classifications. 19970806 082 Contents 1 Introduction 3 1.1 Circuit complexity theory ............................ 3 2 Some Important Tools in Z•4xI 4 2.1 The Z,-[x] phenomena .............................. 4 2.2 The Chinese Remainder Theorem ....................... 5 2.3 Irreducibility criteria in Z [.] ..................... 7 2.4 Hensel's Lemma ................................. 9 2.5 A naive approach to factoring ......................... 11 3 The Case of Small Discriminants 12 3.1 The p-adic numbers ....... ......................... 12 3.2 Resultants ...................................... 15 3.3 The correspondence to factoring over the p-adics ........... 18 3.4 An improved factorization method ..................... 20 4 Factoring when the Discriminant is Zero 21 4.1 Lifting conditions ................................. 22 4.2 Some examples ................................... 23 5 Further Algebraic Considerations 25 5.1 Local rings ....... .............................. 25 5.2 Hensel's Lemma generalized .......................... 26 5.3 Ideal decomposition ....... ......................... 27 5.4 The umique factorization theorem ....... .................. 29 6 Conclusions and Questions 29 6.1 Some conclusions .................................. 29 6.2 Questions ..................................... 30 7 Acknowledgements 31 8 Appendix 31 2 1 Introduction This paper attempts to understand the computational and algebraic differences between polynomials over a field and polynomials over a ring. Polynomials over a field are well understood. Many important polynomial time algorithms in computational algebra have been developed by taking advantage of the un-derlying field structure. In the case where the polynomials are over a ring, however, very little seems to be known. In this paper we try to umderstand the algebraic and computational complexity of polynomials over the ring of the integers modulo n where n is a composite. In particular, we will attempt to compute and characterize all factorizations of a univariate polynomial into irreducibles. Perhaps understanding this polynomial ring will lead to a deeper understanding of the computational limits of circuits as well as faster algorithms in computational algebra. In the course of our discussion we will point out the many differences between working over fields versms over rings to illustrate the severity of the existing gap. 1.1 Circuit complexity theory A motivation for this study comes from circuit complexity theory which is the study of determining the hardness (or relative ease) of a given problem by analyzing the circuits that represent it. A circuit can be thought of as a directed acyclic graph where the nodes are called gates. The edges leading into a node can be thought of as inputs, and the edges leaving a node carry the output of that gate on its given inputs. For the purposes of this paper, we are concerned only with boolean circuits, namely each input can take on a value equal to either 0 or 1. The nodes with zero in-degree should be thought of as inputs. When the inputs are set to some initial vector, the values will trickle through the circuit producing 1 or more output values. Now it is clear how a circuit could be used to decide membership in an arbitrary set. We say that a circuit decides membership in a set S if for every candidate encoded in zeros and ones, our circuit outputs a '1' on that input if and only if the candidate is in S. In this context, we wish to think of a family of circuits, one for each different input length. We can also measure the depth of a circuit in the obvioms way. A family of circuits has constant depth if each circuit in the family has depth at most k regardless of the length of the input. This model of computation has lead to many interesting results such as the fact that the parity function cannot be computed by polynomial size constant depth circuits [FSS84]. We can make this model even more interesting by allowing gates other than simply AND, OR, and NOT. In fact, it is known that constant depth circuits which have MOD, gates cannot compute the MODq function for any q that is not 3 a power of p. What is the computational signifigance of having MOD, gates where n is a composite? We further restrict our model of computation to polynomials which repre-sent boolean functions. We say that a polynomial f in n variables represents the OR function if, when restricted to inputs of O's and l's, 1. f (. 1 ... xn) 6 0 when xi = l for some i < n. 2. f(X1... Xn) = 0 when xi = 0 for all i < n. We measure the complexity ofpolynomial f by its degree. Recall the degree of a multivariate polynomial is the maximum over all monomials of the sum of the powers of the indeterminates in that monomial. It is known that for polynomials over a field (namely over the integers mod p) the lowest degree polynomial representing the OR function on N variables has degree fN/Ip- 1)] [Smo87]. However, the bounds on the degree of a polynomial over a ring (the integers mod n) are not as precise. The best known lower bound on the degree of a polynomial representing the OR function mod n is Q(log N) [TB94], and fairly recently a surprising upper bound of O(N1/r) where r is the number of distinct primes dividing n was discovered [BBR94]. In [BBR94] we learn that a low a degree polynomial for OR would imply the existence of small, low-depth mod n circuits for the AND function. 2 Some Important Tools in Z,,[x] 2.1 The Zn[x] phenomena Definition 2.1 Let Z denote the ring of integers and Zn Z/nZ the ring of integers modulo n. Definition 2.2 Let Zn[x] denote the ring of polynomials with coefficients from Zn. We first examine a few instances of weirdness in the ring Z,[x] with a few examples. The presence of zero divisors in the following rings allows for very strange constructions. Amazingly, for example, the polynomial x is not neces-sarily irreducible in Zn [x]! In particular we can write the following factorization: x - (4x + 3)(3x + 4) mod 6 Here a congruence f - g mod n between polynomials means that f - g has all 4 coefficients congruent to 0 mod n. We show later how to prove that this is a factorization into irreducibles. Also note that X2 + 9 7-- (x+ 1)(x+7) =- (x+3)(x+5) mod 8 All four factors above are in fact irreducible, and so there is no unique factor-ization in the composite case. We turn next to the first important tool needed here: the Chinese Remainder Theorem. 2.2 The Chinese Remainder Theorem Theorem 2.3 Let R be a commutative ring with identity. Let A1, A2 ... Ak be ideals in R. Then the map R -4 R/A xf R/A 2 x ... x R/Ak defined by r ý- (r+Ai,r+A 2,... , r+Ak) is a ring homomorphism with kernel.A1.... .Ak. If the ideals are pairwise comaximal (i.e., for each i,j E {1, 2,... k} we have Ai + Aj = R), then the map is surjective, and we may assert R/(A 1 A 2 ... A.) ftR/A 1 x R/A 2 x ... x R/Ak. (A proof can be found in any abstract algebra book, for example [DFP0].) In particular we may take R to be Z,4z] and its corresponding comaximal ideals to = ki k2 k be the ideals Zk [.x] for each prime factor pi dividing n where n = Pt p'2 P•"' This gives us a nicer representation for polynomials in Zn[x]. For a given f E Zn, [x], we can write f as the following tuple: f =-(fl,f2,... ,fs) where f equals f mod p• Operations on these tuples are pointwise, since the mapping is an isomorphism. FRom this, we see that an irreducible factor g of f corresponds to the following tuple: (1 1,1 .... 1,g .... ,1 ,1) where gi is irreducible mod pi. Clearly no two tuples can multiply together to result in g since gi is irreducible. Thus, every factorization in R must produdce products of tuples of the above form. We ignore umits for the time being since they only trivially modify the above factorizations. This discussion gives us an immediate corollary: Corollary 2.4 [vzGH96a] Let f E Z•[x] and n = rll<i<,pki. The number of irreducible factors of f E Zz[x] is the sum of the number of irreduicble factors of fi E Z t, [T]. Pi 5 We show given a factorization into these tuples how we can reconstruct a fac-torization in Zn[x]. Proposition 2.5 Given Z,[x] ' Z i., [x] x ... x Zgk [x] and f I .. fn a factor-ization of f into irreducibles where each fi has the tuple form above, we can reconstruct a factorization in Zn[x] in polynomial time. Proof: Let fi = (gy, g2,.... , g,). Let Coeff(h,i) denote the coefficient of the ith power in the polynomial h.Let M be the degree of the largest polynomial in fi. More precisely, we look at all the polynomial entries in the s-tuple for fi and let M be the degree of the highest degree polynomial entry. Let fi(k) denote the kth entry in the s-tuple corresponding to fi. The corresponding coefficient of each power of x in fi's representation in Z,[x] can be reconstructed by looking at its coefficient in each entry of fi's s-tuple in the following manner. M f, = CRT(fij).c' j=O where CRT(fi, j) denotes the solution to the following set of equations: y = Coeff(fi( 1),j) mod pkl Y= Coeff(f,( 2),yj) mod pý~' y = Coeff(fW'),j) mod pk This can be calculated using the Chinese Remainder Theorem. The notation is complicated but the idea is simple. Given an s-tuple representing a polyno-mial, we can reconstruct its representation in Z,4[x by applying the Chinese Remainder Theorem coordinatewise. 0 With this result in hand, we can show that. factoring polynomials in Z,[x] is quite difficult. Theorem 2.6 [Sha93] There is a polynomial time reduction from factoring integers to factoring polynomials in Zn[x] Proof: Given some n E Z we attempt to factor it by examining the polynomial f = x over Zn[x]. Let n = (-1)kpki ... p.l. Recall that from the Chinese Remainder Theorem, f has an equivalent form as the s-tuple (X, X,... , x, x). Assume that we can factor f into irreducibles so that (up to a unit) we have 6 f = ff2f...h where each fi is irreducible thus having the form fi = (1, 1'..... 1, X,l,... ,1) where x is in the ith position. So over Zn, each fl is of the form aix + bi mod n with ai,bi E Z and so 0 mod pji lmod4p' for i j Thus the gc.d(n, bi) = pki for 1 < i < s. So our factorization of z immediately gives ms a factorization m 1 m2 ... ran. We need only figure out the prime and exponent for each wi. Assume that rni = p4i. Then ki is bounded by [log mni. We can take jth roots where j varies from 2 to [log mil. If none of the jth roots are in Z, we know m is prime. If one of the jth roots is in Z we can repeat the procedure on the result until a prime is reached and then easily reconstruct the exponent. Since the size of the exponent is logarithmic in rn this a polynomial time procedure. Thms, the existence of a polynomial time algorithm to factor in Z[x] is unlikely. (Compare this with the many randomized polynomial time algorithms (See [Ber70J) to factor in F[.T] where F is a field to see the contrast between rings and fields.) N 2.3 Irreducibility criteria in Zpk[X] The Chinese Remainder Theorem reduces the problem to working over rings of the form Z 1 k[x]. Let r = pk from now on. We would like to determine what factors of a polynomial are actually irreducible. We establish some criterion to determine if a polynomial in Z., [x] is irreducible. A nice observation is the following: Proposition 2.7 Given an f E ZpA,[x] not equal to 0 mod p we can write f as f' + pg where p does not divide /' Proof: Let 7 f E~l+ a-)z i=O where each ai < p and j(i) < k. FRom this we see n nz i=0 i=O P9 f Now we can deduc~e the following remark: Proposition 2.8 Let f E Z ,,[1 not equal to 0 mod p. If f is irreducible in Zp[x] then f is irreducible in Zp,,[x] for all k > 1. Proof. Assume f is reducible in Zpx., for some k. Then f =- gh mod pk and usning the above observation, f =(g1 +P92) (hl +ph2) mod p, Hence, f =- g.,h, rood p contradicting irreducibility mod p. This formula also gives a characterization off all the units in Zn[x]. Proposition 2.9 Let f E Zi..,[x] such that f 0£ 0 mod p. Then f is a. unit, in Zp• [,ri] if and only if f is of the form ,a + pg where ,a E Zp.-Proof. Every polynomial f can be written in the form ft +p f2 where we have ge'd(fi,p) = 1. Assume f is a unit and a~ssumne fj has degree > 1. Since f is a unit, there exists h = (hi + ph2) such that fh = 1. This implies fjhl + p((f2hi + flh2) + pf2h2) = 1 + p.-O But, fj has degree strictly bigger than 0, and both gcd(fi,, p) = I and gcd(hlt, p)= 1. Thlm the monomial of highest degree in f, cannot have cancelled out. So fjhj could not possibly be the constant polynomial 1. Hence for f to be a unit, it mlmt be of the above form. If f = a + pg where a E Zp then f = (I - a- 1(-p)g). The following familiar identity is helpful: I =I+ h+ h 2+ ... + 1 7- h and thus f 1 + a- (-p)y + (a+'(-p)g) + (a-l(-P)g)k-Notice that after the k - 1 term, all of the terms have a factor of pk in them which zero out. Our inverse is thus a well defined element of Zp. [x,]. 0 This also tells us that if f mod p is a unit, then f mod pk is a unit for all k > 1. 2.4 Hensel's Lemma In order to further our analysis of irreducibility as well as develop a method of factorization, we introduce the most important mathematical tool of the paper: Theorem 2.10 [Hensel's Lemma] Let p be a prime, k > 1,and let f, g,h E Z[x] such that f - gh j 0 modp and gcd(g mod p,h modp) = 1 in Z,[x]. Then there exist polynomials § and h such that f = §h mod pk with g g mod p, h =- h mod p. Proof: [BS96] We give an algorithm to construct g' and h' and prove its cor-rectness. Step 1. Find A and 11 E Zp[x] such that Ag + ph = 1. (We know sudh Ak and IL exist since g and h are relatively prime. We can find them easily by using the Extended Euclidean Algorithm for polynomials.) Step 2. Iteratively construct polynomials g' and h' according to the following for loop: for i = 2 to k do q := (f - gh)/(p'-') mod p u:= qp mod g v:= qA mod h g g + pi-1u h h + pl-1v end Return(g' g, h' = h) The proof of correctness is by induction on i. Assume that f = gh mod pi-1 (g and h are also monic). Notice that the (:onstruction of q makes sense since f-gh - 0 mod p'-1. We need only check that (g+p'-1u)(h+p'-'v) f mod p'. Hence, we have 9 (g +jp-L) (h + ptlv) gh + p- 1 (uh + vy) + p~ 2 'uVmod pt -gh + p-' (oh +)g) rmod pt but notice uh+vg uh modg q/uh mod g q(1 - Ag) mod g q mod g Similarly we (:an see that uh+vg =- q mod h. Since h and g are coprime, by the Chinese Remainder Theorem we see that uh + ng = q. Hence in our original equation we have (g± ÷jp-t )(h +pi-v) gh +pi-lq mod pi f modp t Thus 4- = (g + pi'-u) and h- = (h + pi-) are as required. From Proposition 2.8, if g and h are irreducible then § and h are irreducible. Now we can show why we only care about monic polynomials. Corollary 2.11 Let f E Zpi-[] with k > 1. Finding the irreducible factors of f reduces to the case where f is monic. Proof: [vzGH96a] We can write f as p"g where gcd(p,g) = 1. Then g 2 eomo modp where e0 is a unit mod p and thls mod pk. Since gcd(eO,m)= 1, we can use Hensel's Lemma to find a lifting such that g =- em mod pk-v where e =- e0 mod p and m = rmn mod p where r, is monic. But since we have factored out p" from f, every factorization of f corresponds to a factorization of g mod pk-•,. Thus we need only look at the irreducible factors of p' (which are trivial) and the irreducible factors of m iup to units, but m is monic. Hence, we need only consider monic polynomials from now on. M Now it is somewhat clearer as to how to go about finding one factorization of a polynomial mod n. We first look at. the irreducible factors of f mod 10 p and use Hensel's Lemma for each factor and for each prime divisor of n. Then we reconstruct the factorization mod n using the Chinese Remainder Theorem. This leaves us with two important questions. First, what happens if f = gk mod p for some irreducible g (i.e., how do we lift in this case)? Secondly, how do we compute all the different factorizatious of f? 2.5 A naive approach to factoring At some point, all known methods for computing all of the factorizations of a polynomial require solving a system of linear equations. We will illustrate this by constructing an extremely poor factoring algorithm. Assume we want to compute all the factorizations of a polynomial f E Z[x] mod pk. Let ms also assume that we are not interested in factorizations where any given factor has degree greater than or equal to the given polynomial. One way to do this is to solve a complicated system of equations (via the method of undetermined coefficients) with the knowledge that every factorization mod pk corresponds to a unique factorization mod p. For example: Example 2.12 Let f E Z[z] where f = gh mod p. We wish to compute all the factorizations off mod p2 . Assume that f factors mod p into linear polynomials so that g = go + gjx and h = ho -+ hx. Now notice that all factorizations mod p2 must satisfy the following system of equations: f - (g +pG)(h +pH) modp 2 where G and H are some unknown linear polynomials E Zp[x]. Then let G -Go + Gjx and H = Ho + Hlx. Expanding the above equation gives 1us f gh +p(Hogo + Hlgox + Hog x + Hjg x2 + Goh0 + Glhoz + Gohlx + Glhlx 2) + p2 (... ) mod p2 Since we are working mod p2 the last term drops oat. We only need the coefficient of the p term to be zero for our factorization to work out properly. Hence we need GlhC +Hlgl -Omodp Gtho + Hlgo + H og 0 +Go h l -Omodp Hogo +G oho -Omodp Notice that h0, h,, go, gy are fixed values since we compute the factorization of f mod p. Hence, we have a system of linear equations which can be solved 11 rather easily. This approach begins to break down as we need to factor moduo larger powers of p as well as if we need to compute factors with larger degrees. The next section will give us a better approach to this process. 3 The Case of Small Discriminants The problem of computing all factorizations of a polynomials can be divided into two radically different cases. The case when the discriminant is small requires important properties of the p-adic numbers. Abstractly, every factor-ization mod pk of a polynomial whose discriminant is 'small' corresponds to a imique factorization over the p-adics. Thus, with a factorization from the p-adics our problem is greatly simplified as we shall see. We follow development partially outlined in [vzGH96a]. 3.1 The p-adic numbers Kurt Hensel invented the p-adic numbers in the early twentieth century in order to solve number theoretic problems. Since then they have been an important tool in both analysis and algebra for many different problems. We give some brief introductory material for concreteness concerning the p-adics (see [BS66] for a complete treatment of this material). Definition 3.1 Fix some prime p. A p-adic number, denoted {fx}, is a se-quence of integers satisying Xn -- xn-1 mod pn. Two sequences {f xm} and {x',} determine the same p-adic integer if and only if xn '- X'n mod pn+l. It is easy to see that each p-adic integer has the following canonical form: {f.} = {lao,a +a p,a 0 + alp + (22P.... } where each ai E (0... p - 1}. Let Z(,) denote the ring of p-adic- integers where the addition and multiplication operations are performed coordinate-wise. It easy to see that for x, y E Z(,), xy and x + y are p-adic integers and so our ring is well defined. We will introdce the more conventional notation for a p-adic integer, namely an infinite sum of the form a = Zi>O p'ai where ai <. p for all i later in this section. We now aim to show a fairly simple property, namely that 12 Z(p)[z] are a lmique factorization domain. Compare this with earlier examples that show Zpk,[i] is not a UFD. The following theorem can be found in any book on abstract algebra: Theorem 3.2 If a ring R is a UFD then R[x] is a UFD Lemma 3.3 If a p-adic integer {zn,} is a unit then zo 0 0 mod p. Proof: If {fx, is a unit then there exists a {yj such that {xnyn} = 1 Vn. In particular zoy0 = 1 mod p Hence To must be relatively prime to p. 0 Theorem 3.4 Every p-adic integer, distinct from zero, has a unique represen-tation in the form a = pbE where E is a unit. Proof: [BS66] Let a E Z(,). Then if a is a unit, take k = 0. If a is not a unit then let k be the smallest index for which Xk U 0 mod pk From the definition of p-adic numbers, xk+.= Xk-1 = 0 mod pk. Let y. =--for all s > 0. Notice that, P ky.s - P y.- = Xk+., -Xk+.s-1 = 0 mod pk+. and thLs yV V ys-i1 mod p5 Hence, {fys} determines p-adic unit. Clearly {fX} = pk~ y. Theorem 3.5 Z(p) is a UFD. Proof: Consider some a E Z(,). Then from Theorem 3.4, a = pkE for some imit E. Hence, a = p... p E. But p is trivially irreducible, so this could be the k times only factorization up to umits. Hence, Z( 1 ) is a UFD. u FRom Theorem 3.2, Z(p) [x] is a UFD. Now that we have established that Z(p) x] is a UFD, we need to determine the relationship between factorizations in Z(,)[x] and factorizations in Zz[x]. (This is done in Section 3.3) To do this we introduce a non-archimidean metric as well as an alternate way of viewing p-adic numbers. 13 Definition 3.6 A metric 6 is called non-archimidean if and only if 6(x + y) < max(6(x), 6(y)) Definition 3.7 We define the function vp by the following equation v ( if a 5 0 and pU is the largest power of p dividing a oo ifa=O The fimction v, is usually called a valuation. It is easy to see that the function p-l)P(a) defines a non-archimidean metric on the p-adic integers. Let 6P repre-sent this p-adic metric. With this metric in hand, we can form a more convenient representation of a p-adic integer. For any a = (01, a 2 ,... ) E Z(,) we can write a as the following slim: at -- Z fipZ i>O where fli = ai -ai- 1 and Po = a• 0 . Normally, this series would diverge, but with our p-adic metric, larger powers of p result in smaller values from the p-adic metric. Let Sn denote the sum of the first n terms of a. Then 6ip(Sn) = -Hence our sum converges and our representation for a is well defined. 14 3.2 Resultants Recall that in order to use Hensel's Lemma to lift a factorization of some polynomial f, we require f to have a factorization into a product of at least two relatively prime polynomials mod p. We would like to be able to detect the 'difficult' cases where f is a power of a single irreducible polynomial mod p. The following material is outlined in [CL092]. For an excellent description of how resultants, discriminants, and polynomial greatest common divisors are computed, see [Akr89I. Lemma 3.8 Let f , g E Z[xI be polynomials of degrees 1 > 0 and m > 0 respec-tively. Then f and g have a common factor if and only if there are polynomials A, B E Z[xl such that 1. A and B are not both zero 2. A has degree at most m - 1 and B has degree at most I - 1 3. Af +Bg = 0. Proof: [CLO921 Assume f and g have a common factor h E Z[xJ. Then f = hfl and g = hg9 where fl,gi E Z[x]. We see gif + (-fl)g = g9 1hfl - fh hg- = 0. A = gy and B = -fl are as required. Now assume that. polynomials A and B have the three above properties. By (1), B $ 0. Proceed by contradiction and assume that f and g have no common factor. Then they are relatively prime and we can find polynomials A' and B' such that A'f + B'g = 1. Multiplying by B and keeping in mind the fact that, Bg = -Af we see that B = (A'f +B'g)B = ABf +B'Bg -A'Bf - B'Af -(A'B - B'A)f But B is nonzero and from the last equation muLst have degree at least that of f, namely 1. This contradicts (2). Hence, f and g muLst have a common factor of positive degree. 0 Now given f and g we would like to see if we can compute such an A and B to determine if they do indeed have a common factor. This problem reduces to solving the following system of linear equations. Let 15 A = co -1 r + + Crn1 B = doxr-1 + +dt-1 where the coefficients of the polynomials should be thought, of as unknowns. We want to find a solution such that the equation Af + Bg = 0 holds. To do this we can also write out f and g f = aox t + + at, ao 0 0 g =bo.T' + ... + b,,, bo 0 0 substituting appropriately we achieve the following rather large system of linear equations: aoxo + bodo = 0 coefficient of xl+m-1 alto + aocl + bdo + bod 1 = 0 coefficient of xl+m-2 atom-1 + bmd-1 0 coefficient of xA. This is an appropriate time to introduce the Sylvester Matrix. Definition 3.9 Given polynomials f,g as above, the Sylvester matrix off and g is the coefficient matrix of the above system of equations. We denote this Sylvester matrix as S(f, g) by the following (1 + m) x (1 + m) matrix a, bm a11 bin-i bm S(f,g) = ao at : b- C R(l+m)x(l+m) aq-1 bo ao b 0 the empty spaces are filled by zeros. The Sylvester matrix is the coefficient matrix of the above system of equations. The resultant of f and g with respect to x denoted Res(f,g) is the determinant of the Sylvester matrix. Hence, Res(f,g) = det(S(f,g)) 16 An immediate result of the preceding discussion is the following proposition: Proposition 3.10 Given f,g E Z[x] of positive degree, the resultant Res(f, g) E Z is an integer polynomial in the coefficients of f and g. Ebrthermore, f and g have a common factor E Z[x] if and only if Res(f,g) = 0. Proof: [CL092] The resultant is zero -the coefficient matrix of equations has zero determinant -the system of equations has a nonzero sohltion. a Another important consequence of resultants is the following proposition: Proposition 3.11 Given fg E Z[x] of positive degree, there are, polynomials A,B E Z[z] such that Af + Bg = Res(f,g). Proof: We have previously analyzed a case where we were searching for a solution to the equation Af + Bg = 0. Now we analyze the case where we want a solution to the equation A'f + B'g = 1. We form the following similar system of equations: ao.xo + bodo = 0 coefficient of ,I+m-I aico + aoci + bid o + bodi = 0 coefficient of xI+m-2 alcm.1 + bmdl_1 - 1 coefficient of x0. Cramer's rule can be used to solve this system of equations resulting in the following solution (the details are worked out in CLO). A'- A Res(fg) B' 1 B Res(f,g) Multiplying through by Res(f~g) we see that Af + By = Res(f,g) To summarize, given f,g E Z[x] the Res(f,g) $ 0 if and only if f and g are coprime. We also know that we can find polynomials A and B si(Ii that Af + Bg = Res(f,g). Porthermore, for any polynomial h E Z[x] with deg(h)< 1 + m there exist uniquely determined polynomials A and B such that res(f, g)h = Af + Bg. The uniqueness of A and B comes from the fact that h has degree less than 1 + m. The degree of h insures that the system of equations we have to solve is similar to the two others introduced in this section. 17 3.3 The correspondence to factoring over the p-adics Definition 3.12 Let f = aox' + ... + a1 E Z[x]. The discriminant of f is defined as follows: disc(f) - (-l)zc--)/2 Res (f, fi) ao where f' is the derivative of f. It is well known that f is square-free if and only if its discriminant is non-zero. Notation 3.13 Let g,h E Z[zI. Then r(g,h) = v,,(res(g,h)) and d(g) = vp(disc(g)), where disc(g) is the discriminant of g. Now we can prove the major technical theorem of this section: Theorem 3.14 [Hensel's Lemma II] Let p E Z prime k E N and f,u,w E Z[x] be polynomials of degrees n+m, n, and m respectively with the following properties 1. f =uw mrod pk and the leading coefficients (lc) of f and uIw are equal 2. the resultant res(u, w) is nonzero 3. k > 2r(u,w) Then there are polynomials g, h E Z(,) [H such that f = gh E Z(,)[x], g - u mod pk-r(w), h = w mod pk-r(u,7,) Proof: [vzGH96a] Set p = r(u, w). We will inductively construct polynomials pi and ¢i E Z[x] such that if f ab mod pk+i-I with a, b E Z[z] such that a u mod pk-P and b = u mod pk-p then f = (a + pk-P+i-lV,) (b + pk-p+i-lWo) mod pk+i Note that if we can do this then we will have proved the claim. If we have for every i > 0 such a polynomial, then we can sum over all positive i, and we will have a polynomial with p-adic: coefficients that satisfies the above claims. It is important to realize that the infinite sum does not result in an element of the ring of formal power series. This is because Wi and Oi have bounded degrees, and only the coefficients in our resulting sum can be thought of as an infinite 18 slim. Assume that f -= um mod pki > 1, and a, b E Z[x] are already constructed such that f = ab modpk+i- 1. Then f = ab + pk+i-ll where I E Z[x] and deg(l) < n + m since lc(ab) = hc(f). Notice that a =- u mod pk-P and b -w mod pk-P. Also, k - p > p so a is equivalent to u and b is equivalent to w modulo a higher power than the largest power of p dividing the resultant of u and w. Thus r(a, b) can be no larger than r(u, in) (If it were larger, then we could calculate r(a, b) and mod out by pk-p to find a larger r(u, in)). Since they are equivalent modulo p'-p r(a, b) Ž r(u, un). Hence r(a, b) = r(u, iv). Now we can use Proposition 3.11 to find Wi and V¢i E Z[.] of degrees less than m, n such that pPI = api + bi and thms pPl =- api + tb4j mod pP'1 Then we see f -(a +pk-p+i-l¢i) (b + pk-p+i-l1i) f- ab _ pk-p+i-l(api + bV/i) _ p2k-2P+ 2i-2Wjip, pk+i-1 - pk-p+i-lpp - P2k-2p+2i-2(Oii 0 mod pk+i becaulse i > 1 and k > 2p. We do this for all i > 0 in order to construct the following polynomials: g = It +-E Pk-p+i-l ) i> 1 h = un + Zpk-p+i-11i i>1 Expanding out the above sums reveals that g and h have coefficients which are inifinite sums that correspond to a p-adic integer. Almost magically, f = gh over Z(p) [x] since f -gh mod pk for all k. By our above construction, g -u mod pk-P and h =- in mod pk-p. M 19 Theorem 3.15 Condition (c) is true if k > disc(f). Proof: The proof, found in both [vzGH96a] and [BS66], goes as follows: Let f = gh with g, h E Z(,)[.x]. Then disc(f) = disc(gh) = disc(g)disc(f)res(g, h)2 Thus, d(f) d(g) + d(h) + 2r(g, h) Ž 2r(g, h). Since the discriminant and the resultant are polynomials in the coefficients of f, g, h, the same is true for factorizations over Z5 k. N Hence, for any polynomial whose discriminant is smaller compared to the power of the prime, we know the following: Any factorization of f = gh mod pk corresponds to a unique factorization over the p-adics. This factorization f = §h E Z(p)[x] is equivalent to gh mod pk-p(g,h). In essence, given any two factorizations f - gh mod pk and f = g'h' mod pk, gh ý- g'h' mod pk-p(g,h) We note von zur Gathen formalizes this in the following way: Proposition 3.16 Let f = Y11<i<1gi over Z(p) with disc(f) # 0, 1 > 1 and gi E Z(p)[z] monic and irreducible for 1 < i < 1. Let f = gh mod pk with g, h E R[x] monic and k > d(f). Then there exists a partition {1,... , 1} = SUS' such that g IES 9gi mod pk-P and h -I-j, sgj with p = r(ES gi, HIjEys,ygj). If g is irreducible over Zpk[x] then there exists 1 < i < I such that g y gi mod pk-r(gi,fIji6i g9i) Proof: The proof follows immediately from Theorem 3.14. Given some fac-torization f = gh mod p , we can lift this to a factorization f = jh,. But factorization over Z(p)[xl is unique, hence the irreducible factors of f are par-titioned among f and j and hence their respective projections mod pk-r(g,h). U 3.4 An improved factorization method Now we can give a much better algorithm for computing all of the factorizations of some f mod pk. First we need to calculate one factorization into irreducibles of f mod pk. Sometimes this can be done by a complicated set of lifting procedures (See Appendix A) or by Chistov's algorithm [Chi94] for computing the factorization of a polynomial over a local ring (namely the p-adics in this case). Chistov's algorithm gives us a factorization in Z(,,) [x], but we can simply mod all of the factors by p k to retrieve a factorization into irreducibles mod pk. 20 In order to determine all factorizations we need to solve some systems of linear equations. They are considerably simpler, however, becauise of Theorem 3.14. Given f E Z[x] and a factorization f l<I<_j gi mod pk we know for each irreducible factor u of f over Zx•,[x], u gi mod pk-r(oih) where h = IjH 11 9j. Hence any factorization of f must correspond to a sobition of the equation found in [vzGH96a]. S (g + pk-r(9,h) (h + pk-r(g,h)p) modpk _ pk-r(gh)(ý,h + Ogb) -P2k-2r(gsh)o¢- 0 mod pk -vh + bgi( 1 0mod pr(s•,h) S(gi, h) o 0 mod Pr(o h) 0.-1 - 1mdp~lh ¢0 where O<i<m ]<_i<:n Any solution to the above equation corresponds to a factorization mod pk. After finding all solutioms, we can set gi = g9+1 and h = h/gi+l, and solve another system of equations until we have found all possible irreducible factors. If at each step there are at most N different solutions found then we could conceivably have N' distinct factorizations into irreducibles. Since choosing any set of I factors (1 from a possible N at every step) will result in a factorization of f mod pk. Fortunately, there are polynomial time algorithms to put the above Sylvester matrix in Smith normal form, giving us a relatively easy method for solving the system of equations and preserving solutions mod pk. 4 Factoring when the Discriminant is Zero When k is not bigger than 2p we cannot use the above machinery to help us in finding factorizations. As long as disc(f) is non-zero (as long as our polynomials 21 have at least two coprime factors mod p) we have some way of computing at least one factorization. If the discriminant of our polynomial f is zero, i.e., f _ ge mod p for some irreducible polynomial g, it is not clear how to even lift this factorization to one mod pk. This section will look at these rather umfortunate cases outlined in [vzGH96b]. 4.1 Lifting conditions Theorem 4.1 [vzGH96b] Let f w- w modpk = ge modp, g irreducible over Z5 [x] and e > 2, k > 1 with u,w E Z[x] monic and u -g mod p,w e-1 mod p for some I < f. Then the following are equivalent: 1. E Z[x] over Z, divisible by g'. 2. For every cp E Z[x] with deg(W) < deg(u) there exists a polynomial V) E Z[x. with deg(b) < deg(w) such that f - (1 + pkW)(w +pkO) mod pk+l. 3. There exist polynomials W, 0 E Z[x] with deg(W) < deg(u), and deg(O) < deg(w) such that f =- (u + pkp)(11 + pko) mod pk+l. 4. There exist polynomials ,o, 4 E z[x] with f =- (IL + pkw) (Iw + pko) mod pk+1 Proof: (i) = (ii). Let La""= g•a modp with a E Z[x], and (p,V E Z[x] with deg(W) < deg(u), and 4 - a - ge-21p mod p. Notice f - (u + pký)(11, + Pk4) = f _ Pk•(V + Oq,) = f - fU) _ pk (•ge-I + (a -" _ V2l)g,) = f - uwl -pk g1a 0- (m od pk+1 (ii) >(iii) =ý- (iv). We are left with (iv) • (i). Let W,4 E Z[x] with f (u + pkp) (w + pko) mod pk+l. Then f -w 1111 p. _ og•-1 + V)g1 _ g 1 (-ge-il + 4') mod p. 22 With this in hand we (can prove show a certain class of polynomials to be irreducible mod pk for all k > 2. Definition 4.2 We call f = -o<i<. aixi E Z[x] an Eisenstein polynomial if if a,,=1 and ai = O mod p for O < i < n and ao $ O mod p2. Corollary 4.3 Let p E Z be prime and f E Z[x] an Eisenstein polynomial. Then f is irreducible modpk for all k > 2. Proof: [vzGH96b] Since f is Eisenstein, f =- m' mod p. In this case, g = x. Let 1<l<B. Then f-XIT-I f -. n_ ap O<i 3 we could simply mod the factors by p2 and find a factorization mod p2 . 4.2 Some examples We can use the lifting criterion to create an (admittedly slow) algorithm for computing all the factorizations of a polynomial f that equals g' mod p for some irreducible g mod p. Say we want to find all factorizations mod pk. We choose 1 < I < f starting at 1 = 1 and apply see if the factorization mod p can be lifted to p2 by computing Lv" and applying Theorem 4.1. At the lifting step for p2 we make an arbitrary choice, namely we choose Wp such that. so has degree less than u. This could be an unfortunate choice, however, becamse our choice of Vo may preclude the possibility of lifting mod p'. In [vzGH96b], we see some interesting examples: Example 4.4 Let f = x 2 + 27in + 162. Then f = X 2 mod 3 and f-"2 0 mod 3. Assume that we have chosen some (o as above to lift this to a factorization mod 9. Then we have f - (i + 3 wo)(i + 3(-w)) (i + 3wo)(i + 6w) mod 9 where 0 < o < 2. Then 23 f - (x + 3w)(x + 6w) • (p+q2 od3 fU+9(x6 ) w2(W +V 2) mod 3. 9 So we can only lift this factorization to one mod 33 if 2((9 + V2) = 0 mod 3. This happens only when W = 0 or 2. Thus, had we chosen (P = 1, we would not be able to lift this factorization. Unfortunately, this procedure can get rather complicated as the next example will illustrate: Example 4.5 Let p = 3, f = 510, and I = 10. After two lifting steps, we obtain the following factorization: f =- mW mod 81, where U = X4 + 393.3 + 392-2 + 9cpiX + 99O and 1) = x6 + 7893.,5 + (78902 + 992)X4 + (72W1 + 189293 + 5493)x3 + (729o + 9W2 + 549pjW93)X2 + (549o093 + 54plO9 2 )X + 54Woy2 + 54W3 and 0 <9i <27 for i E {2,3}, and 0 < 9o < 9 for i E {0, 1}. Then f-MD 293x9 + 292X + (2V3 + 2yi)27 + (29P + 29o + ±4)X6 81 S+ 929tX + (2v' + 2 3 23 +(2W9cpj + 9i92)x3 + (p2 + y4)X 2 mod 3 From the above lemma, we can only lift this factorization if g' divides f -nw/pk mod p. Hence we need the following to be true: 2~109 + w -9 0 rmod 3 IN2+W2 0 mod3. These equations turn out to be satisfied if and only if 92 = 0 mod 3 and 90 =- 0 mod 3. As the degree of f gets larger, the difficulty of solving these equations to find all factorizatioms grows quickly. In fact, the biggest obstacle to computing these factorizations is to determine which parameters will allow for liftings to higher powers of p. It is not clear how to simultaneously satisfy the all of the parameters at each step. Hence, the best algorithm known runs in exponential time, simply trying out all possible values for each parameter. 24 5 Further Algebraic Considerations In this section we attempt to give some further purely algebraic. considerations of factorizations in Zi#, [x] partially outlined in [McD74]. We will prove reslilts for a more general ring than Zpx, [x] and show that all results apply to our case. All rings in this section are commutative and have identity. (a) denotes the principal ideal generated by a. 5.1 Local rings Definition 5.1 A local ring is a ring with a unique maximal ideal. Example 5.2 Z(,), Zp. and Zpi, [x] are all local rings whose unique maximal ideal in all cases is (p). Recall that R/m where m is a maximal ideal of R is actually a field. The field that results from taking R/m where m is our unique maximal ideal is called a local field. Let k = R/m. Define the natural projection from R[x] to k[x] by It. In Zp. [x], p takes a polynomial in ZP. [x] and reduces all of its coefficients moduilo p. We need the following long string of definitions to continue this development. Some of the definitions are repeated from previous sections for clarity. Definition 5.3 Let f and g E R[x] Then " f is nilpotent if there is an integer n such that fn _ 0. " f is a unit if there is a polynomial h with fh = 1. " f is regular is f is not a zero divisor. " f is prime if (f) is a proper prime ideal. " f is irreducible if f is not a unit and whenever f = gh then g or h is a Imit. " f is primary if (f) is a primary ideal. " f and g are associated if (f) = (g). "• f and g are coprime if R[x] = (f) + (g) The following proposition gives ls some simple characterizations for the above definitions: Proposition 5.4 [McD74] Let f = ao + alx +-... + anxn E R[x]. 1. The following are equivalent 25 (a) f is a unit (b) psf is a unit. (c) ao is a unit and al ... an are nilpotent 2. The following are equivalent (a) f is nilpotent (b) Iuf = 0 (c) ao,... ,an are nilpotent (d) f is a zero divisor (e) there is a non-zero a E R with af = 0. 3. The following are equivalent (a) f is regular (b) (aO, al, ... ,an) = R (c) ai is a unit for some 0 < i < n (d) pf 0 0 Proof: The proof of parts (a) and (b) follow immediately from Proposition 2.9 in the first section. Part (c) is quite easy as well. If f is regular then it is not a zero divisor. Hence, we cannot 'factor' out p from one of the coefficients. This implies that some ai ý (p). Since (p) is our unique maximal ideal, ai must be a unit. Since some ai is a unit, (ao, al,... , an) = R. Furthermore, since not all the coefficients are in (p), our projection onto m[ix] must be non-zero. Hence, lif $40. 0 5.2 Hensel's Lemma generalized Now we can restate Hensel's Lemma in a more general setting: Theorem 5.5 [Generalized Hensel's Lemma] Let f E R[x] and lif = 9ýi... g-n where ffi,... , gn are pair-wise coprime. Then there exist gl,... , gn E R[x] such that 1. g-,... ,ggn are pair-wise coprime 2. t~gi = f for 1 < i < n. Proof: The proof is identical to that of the first Hensel's Lemma. All of the details are in [McD74]. 0 26 5.3 Ideal decomposition We build towards a nice characterization of all factorizations of a polynomial in such a ring. In order to do this, we need to apply some theorems from Primary Ideal Decomposition found in [Hun74]. Before this, we make a few observations. Lemma 5.6 Let I, J be comaximal ideals of a ring R. I + J = I n J. Proof: Recall that IJ is the set of all finite sums of the form ij such that i E I andj E J. IJ C I f J since for a E IJ, a = rij for some r E R, and thus a = (ri)j and a = (rj)i. Now let a E I fl J. Since I, J are comaximal, there exist rl,r 2 E R such that r 1i+r 2j = 1. Hence arli+ar2j = a. But a = ci and a = bj for some c, b E R. Thum bjr i + cir 2j = c • ij(br 1 + cr 2) = a. Hence a E IJ. M Corollary 5.7 Let 1,, 12,... , In be pair-wise comaximal ideals. Then it follows that Iu ... In = I, nI2 n .f.. n In. Definition 5.8 Let I be an ideal of R. The radical of I, denoted Rad I, is the intersection over all prime ideals P that contain I. If the set of prime ideals containing I is empty then Rad I is defined to be R. Definition 5.9 If Q is a primary ideal in a commutative ring R, then the radical P of Q is called the associated prime ideal of Q. We say that Q is P primary. Lemma 5.10 Let R be a local ring. Then if (jig) is a primary ideal then (g) is a primary ideal. Proof: Let ab E (g). We assume that. b 0 (g). We need to show that an E (g) for some n.We know (jig) is a primary ideal by assumption. ILg = g + M where M is our unique maximal ideal. Since ab E (g), ji(ab) E (11g) •. ab + M = (a + M)(b + M) E (g + M). But (g + M) is primary. Hence (a + M)k = (ak + M) e (g + M) (ak + M) = ug + M -ak -ug E M. This implies that ak = ug + m for some m E M and u E R. Now let d be the nilpotency of m and we see: 27 IE I akd' = (ug~m)d i=O -gY+md -gy where d is the nilpotency of m and Y is what remains after factoring out a g from the above sum. Hence akd E (g) so (g) is primary. u Corollary 5.11 Let R = ZI[x], e E N. Let g be an irreducible non-zero polynomial mod p and h and arbitrary element of Zpk[x]. Then (g + ph) is a primary ideal. In particular, (ge + ph) is (g) primary. Proof: Notice that ge + ph mod p y_ ge mod p which is trivially a primary polynomial. By the above lemma, g' + ph must be primary. N We introduce the next two definitions and theorem for the proof of the main theorem of this section: Definition 5.12 We say an ideal C of R has a primary decomposition if C A1 n A2 n ... n An with each Ai a Pi primary ideal of R for some prime ideal Pi of R. If no Ai contains A 1 n A2 n ... .A and if the ideals P 1,... , Pn are distinct then the primary decomposition is said to be reduced. Definition 5.13 Let C, Aj, and Pi as above. If Pi 9 Pj for all j 0 i then Pi is said to be an isolated prime ideal of C. Theorem 5.14 Let C be an ideal of R with two reduced primary decomposi-tions A, AA 2 n...fnAk = C = A AA2 nl... nfA' where Ai is Pi primary and Aý is P! primary. Then k=s and (after reordering) Pi = Pj' for i = 1, 2,... , k. Futhermore if Ai and Aý both are Pi primary and Pi is an isolated prime then Ai = AX. The original statement. of the theorem and proof can be found in [Hun74]. It is stated originally for R-modiiles, but we view a ring R as an R-module over itself and so everything applies naturally. 28 5.4 The unique factorization theorem Now we can prove the much anticipated major theorem of this section. Theorem 5.15 [McD74] Let f be a regular polynomial in R[z]. Then 1. f = 6gi ..'. g, where J is a unit and g1 "" gn are regular primary coprime polynomials. 2. Iff = 6g- ... gn = flhl .. hm where J and /3 are units and {gi} and {hi} are regular primary coprime polynomials then n = m. and, after reordering (hi) = (gi), 1 < i < n. Proof: First we prove (1). Let f be reguflar in R[x]. Then jif is non-zero. Hence jif = 3j1"" ... where the gi's are irreducible coprime polynomials in k[.]. In other words, we have projected our polynomial mod p to find its fac-torization into powers of irreducible c(oprime polynomials. Now, using Hensel's Lemma, we (:an find a factorization f = 6gl ... gn where /i6 = 3 and ligi = gij'. Notice that. each gi = g' +ph for some irreducible polynomial g and some poly-nomial h E R[x]. Thus by Lemma 5.10 (gj)'s and similary (hi)'s are primary. Now we prove (2). Since we have f = gi ... gn = hi ... hn we have the following series of equations: (f) = (g(""gn)=(hi...hn) But since the (gij)s are pairwise comaximal we have that (91)(92) ... (9g) -(g9) n (g2) n ... nl (gn) and similarly for the (hi)'s. The umderlying prime ideal for each (gi) = (ge + ph) is simply (g). Trivially, for g, h distinct irreducible polynomials mod p, (g) 0 (h). Hence, every underlying prime ideal in our product is isolated. Thus, we have found two reduced primary decompositions for f where every Pi is isolated for every Pi primary ideal in the product. By the Theorem 5.14 after renumbering, the individual ideals must be equal. Thus our factorizations are unique up to ideals 6 Conclusions and Questions 6.1 Some conclusions The discriminant of a polynomial determines whether or not it is hard to calcu-late all of its factorizations mod pk. In all cases we can use a umique factorization 29 modulo p to help find all the factorizations. This information alone is not very helpful. If the prime power we are factoring over is much larger than the dis-criminant, we can use the correspondence with the p-adic( integers to form a relatively simple method to solve a system of equations in polynomial time. If the discriminant is zero, we have difficulty characterizing the factoriza-tions of our polynomials, because we cannot easily lift the factorization. This case results in a complicated systems of diophantine equations. The Primary Decomposition Theory provides us with a nice characteriza-tion of the factorizations of a polynomial. Although the factorization of a polynomial is not unique in Zp, [z], it is unique up to the ideals generated by the coprime factors. We would like to take advantage of this algebraic situation and come up with an algorithm that exploits it. Unfortumately, all of the known ideal membership problems rely upon a Griibner Basis algorithm which runs in exponential time. These results could be applied the multivariate (case were it not for our current inability to lift multivariate factorizations. Applying this in the multi-variate case could result in new bounds for polynomials representing boolean functions modlilo n. 6.2 Questions We would like to use the results to get bounds on the degree of a polynomial representing a boolean function. This c(oluld be done by examining its factor-ization over the p-adics. Unfortunately, these polynomials are all multi-variate, and our results do not directly apply. The problem is that when two mul-tivariate polynomials f, g are relatively prime, there do not necessarily exist polynomials f ', g' such that fg'+ gf ' = 1. Thus, Hensel's Lemma breaks down. An interesting problem is determining whether or not a multivariate factoriza-tion can be lifted and if so, how? This would provide us with a way to use all of the machinery developed for the univariate case. It is also unclear as to how Primary Decomposition Theory can be used, outside of Gribner Basis algorithms, to provide some insight on factorizations. Exploiting this natural algebraic structure seems quite possible. Is there a feasible way of implementing/verifying Chistov's algorithm for factoring polynomials over Z(,)[xJ in polynomial time? Currently, it seems far beyond what we can implement. 30 7 Acknowledgements I greatly appreciate my advisor Professor Dana Scott's support throughout the entire process of writing this thesis. From the outset, Professor Scott allowed me to pursue mathematical topics of my own independent interest; his guidance enabled my success. Many thanks go to Andrej Bauer who suffered through the many technical details of this paper and selflessly devoted hours to assist me. His mathematical knowledge and Mathematica prowess were invaluable. I also appreciate Professor Steven Rudich's constant enthusiasm and inspiration which motivated the writing of this paper. I had useful conversations with Professor Ravindran Kannan and Glenn Durfee. 8 Appendix This Appendix contains code for the Mathematica Symbolic Computation Pack-age. It includes a function, CompFactor, which takes as input a polynomial in Z[x] and will produce a factorization mod n for a specified composite. If the polynomial is of the form g' mod p for some prime p dividing n, then the algorithm will not compute a factorization. This ca.se corresponds to the case where the discriminant of f is zero and thuLs cannot be lifted without a tedious exponential time algorithm. Otherwise, the polynomial is factored into coprime factors using Hensel Lifting and the Chinese Remainder Theorem. Get ["Numbermheory'NumberTheoryFunctions"']; Get ["Algebra'PolynomialPowerMod' "]; Get ["Algebra'PolynomialExtendedGCD'")]; ExtraCoeff [aList, i_]: = If [a == (then) (else) Prepend[ExtraCoeff [Rest [a] i], Coefficient [First [First[a]] ],xW] I ExtraConCoeff [a-List]:= If [a =- f, (then) 31 (else), PrependEExtraConCoeff [Rest [al]],PolynomialMod [First [First [a]] ,x]] I Extral~oduli [a..List]: If [a ==fl (then) (else) PrependEExtraModuli [Rest [al]],First [Rest [First [al]]] I (Given the list {{faci,mj},{fac2,m-2}} we can reconstruct the polynomial with this decomposition ) ChinesePolyRem[aList ,nj]: Module [{ModuliList ,pp,ResPoly}, ModuliList =Extraj~oduli [a]; For [pp=O ,pp< (n+1),p+ If [pp==O, ( then ) ResPoly=ChineseRemainderTheorem[ExtraConCoeff [a] ,ModuliList], (else ) ResPoly = (ResPoly + (Chine seRemaindermheorem [ ExtraCoeff [a~pp] ,ModuliList ) x-pp)]]; {Re sPa ly}] (This takes a polynomial f, its two factors mod p (g and h) as well as p and the degree to lift to and produces a lifted factorization Based on Eric Bach's Algorithmic Number Theory book-- see Bibliography) HenselLift[f_,g_,h,p.,kJ :=Module[{t,a,b,q,u,v,gg,hh}, t=PolynomialExtendedGCD [g ,h,Modulus->p]; a--t[E[2,t]11] b~t] /This creates a tuple of n I's with the irred polynomial in the kth position, i.e. {i,i,l,irred,1,i,1} It corresponds to an irreducible factor in the product ring / CreateIrreducible [irrerL n_,k_,mmlistj :=Module [{final}, final= {}; For [oo=i, oop]]; (This takes a polynomial f, a list of its irreducible factors mod p 33 {{pl,exp},p2,exp2},{pS,exp3} .. I and lifts it to a complete factorization mod p-k. n corresponds to the number of irred factors LiftFactors~f-,a-List,n.,p-,k-j: Module [{productsof ar,TempPolyList ,LiftedList)-, TempPolyList a; If[(aE[2,2]J I= && Lemgthua] = ( else ) If [Length [a] ==2, Print ["Failure"], ( else ) LiftedList={}; Tempf = f productsofar=PolyProd [a]; AppemdTo [LiftedList ,{1 ,11]; TempPolyList = Rest [TempPolyList]; Firstfac =TempPolyList; Secomdfac =PolynomialQuotient [productsofar,Firstfac ,x,Modulus->p]; For [jj=O,jj~n+1,jj++, (Primt[jj]; If [Length [TempPolyList]1==, ( them ) Return [{LiftedList ,p-k}], ( else ) With[{FLiftFac -First [HenselLift [Tempf ,Firstfac ,Secondfac,p,k]], SLiftFac =HenselLift [Tempf ,Firstfac,Secondfac ,p,k] 1, If [Length[TempPolyList] -2, ( only 2 factors to lift ( them ) (AppendTo [LiftedList ,{FLiftFac, 1)-; AppendTo [LiftedList, {SLiftFac ,il TempPolyList =Rest [TempPolyList], ( else ) (TempPolyList =Rest [TempPolyList]; AppendTo [LiftedList ,{FLiftFac, 11]; Tempf = SLit tFac; Firstfac =TempPolyList; Secondfac= PolynomialQuotient [Secondfac,Firstfac,z,Modulus->p] ;)]]])] ;];] (CreateMasterList takes a polynomial f, and a list of factors (fac) of some modulus. It reduces f by each element of fac and factors it using 34 previous procedures. Returned is a list of the following type: This corresponds to f's factorization nod 5 and nod 7 CreateMasterList[f., facList]: Module [{TenpFL,GoalList ,n,currentp ,currentexp,FacList}, TenpFL = fac; n = Length [f ac; GoalList ={}; For [ii=O, iicn, ii++, currentp = First[First[TenpFL]]; currentexp = First [Rest [First [TenpFL]]]; If [((FactorList[f,Modulus->currentp]) > 1 && Length [FactorListEf ,Modulus->currentp]] == 2) ,Abort [] ,Print[ ["Liftable"] FacList =MyFactorList[f,currentp]; t = Length[FacList]; If[t==2, (then ) AppendTo [GoalList ,{{{i i} ,{FacList ); If [Rest [TenpFL] == {},Return[GoalList) ,TenpFL=Rest [TenpFL]hl;] ( Final List takes the list created by CreateMasterList and expands everything by converting it into irreducibles of the formCii1,f,,) and sending it to the poly chinese renainder theorem. It then reconstructs the correct factors and spits out our factorization It gets the length of this tuple fron deg ) FinalList [Master.List,degj]: Module [{TMaster,MModuliList ,Finalautput ,Outerloop,Innerloop,Interoutput, Innerlist}, T~aster = Master; MModuliList = Map[Last,Master]; FinalOutput={}; Outerloop = Length[Master); For [iii=i,iii<Outerloop+1, iii++, Inneroutput{}l; Innerloop = Length[First[First[TMaster]] -1]; Innerlist = First [First [TMaster]); 35 ForEj j=2,jj<Innerloop-Ii ,jj++, (AppendTo[lnneroutput, First [ChinesePolyRem [Createlrreduc ibis [Innerlist ,Outerloop,jj-i,MModuliList] ,deg]] ;)]; Mhaster = Rest[TMaster]); AppendTo EFinalOutput ,Inneroutput];]; Flatten [FinalOutput]] ( This gives the actual factorization. The naster function MasterFactor [t.,deg.,nj]: FinalList [Ore ateMasterListEf ,Factorlnteger En]],deg] ConpFactor [poly. ,noddj := MasterFactor [poly, 2Exponent [poly ,x] ,nodd] 36 References [Akr89] Alkiviadis G. Akritas. Elements of Computer Algebra with Appli-cations. John Wiley and Sons, New York, 1989. [BBR94] David A. Mix Barrington, Richard Beigel, and Steven Rudi('. Rep-resenting boolean functions as polynomials modulo composite num-bers. Computational Complexity, 4:367-382, 1994. [Ber70] E.R. Berlekamp. Factoring polynomials over large finite fields. Math. Comp., 24, 1970. [BS66] Z. I. Borevich and I.R. Shafarevich. Number Theory. Academic Press, New York, 1966. [BS96] Eric Bach and Jeffrey Shallit. Algorithmic Number Theory, vol-ume 1. The MIT Press, Camrbidge, Massachusetts, 1996. [Chi94] A.L. Chistov. Efficient factorization of polynomials over local fields. J. Math. Sciences, 70, 1994. [CLO92] David Cox, John Little, and Donal O'Shea. Ideals, Varieties, and Algorithms. Springer-Verlag, New York, 1992. [DF90] Arthur Dummit and Thomas Foote. Abstract Algebra. Math Books, New York, 1990. [FSS84] M. Furst, J.B. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17:13-27, April 1984. [Hun74] Thomas W. Hungerford. Algebra. Springer-Verlag, New York, 1974. [Mc:D74] Bernard R. McDonald. Finite Rings with Identity. Marcel Dekker, Inc., New York, 1974. [Sha93] A. Shamir. On the generation of polynomials which are hard to fac-tor. In 25th Annual ACM Symposium on the Theory of Computing, 1993. [Smo87] R. Smolensky. On interpretation by analytic functions with special properties and some weak lower bounds on the size of circuits with symmetric gates. Proceedings of the 19th Annual A CM Symposium on Theory of Computing, pages 77-82, 1987. [TB94] G. Tardos and D. A. M. Barrington. A lower bound on the mod 6 degree of the OR function, April 1994. [vzGH96a] Joachim von zur Gathen and Silke Hartlieb. Factoring modular polynomials. Proc. ISSA C, 1996. [vzGH96b] Joachim von zur Gathen and Silke Hartlieb. Factorization of Poly-nomials Modulo Small Prime Powers. Technical report, University of Paderborn, Germany, 1996. 37
18149	https://www.hurricanescience.org/science/basic/oceancirulation/index.html	Hurricanes: Science and Society: Surface Ocean Circulation Home Science Hurricane Science Hurricane Observations Hurricane Forecasting and Modeling Basic Science Hurricanes & Society Hurricane Hazards and Impacts Mitigation and Preparation to Response and Recovery History Interactive History Timeline Featured Storms Hurricane Case Studies Resources Teachers Downloads Hurricane Resource Links Science and Education Symposium 2010 2013 Hurricane Webinar Series for Educators NHC: 4-8th Grade Webinars Galleries Glossary About Hurricane Science Hurricane Observations Hurricane Forecasting and Modeling Basic Science Atmosphere Hydrologic Cycle Pressure Gradient Convection Coriolis General Atmospheric Circulation Ocean Water Properties Sea Surface Temperature Surface Ocean Circulation Tides Home>Science>Basic Science>Surface Ocean Circulation Surface Ocean Circulation Ocean surface currents are driven by the wind. When the wind blows over the ocean, energy is transferred to the ocean surface through friction between the air and the water. This energy results in movement of the water in a direction to the right of the wind direction because of the Coriolis force (in the Northern Hemisphere). The ocean surface circulation is characterized by gyres in each ocean basin. The subtropical gyres are the strongest. The currents in each gyre are not all the same. Map showing the major ocean surface currents and subtropical gyres. American Meteorological Society. In the North Atlantic, the western boundary current is the Gulf Stream. The Gulf Stream begins by Florida and travels up the East Coast of the US. The Gulf Stream is the boundary between warmer surface waters to the southeast and colder surface waters along the US coast. The Gulf Stream often meanders and produces many eddies in the ocean, small offshoots from the current with a rotating circulation. Eddies can bring warmer or colder water across the Gulf Stream boundary, affecting hurricanes that might travel over them. Another current in North Atlantic that can affect hurricanes is the Loop Current. The Loop Current is located in the Gulf of Mexico on the boundary between Caribbean waters and Gulf of Mexico waters. The loop current transports warm water and has a variable path, sometimes extending quite far into the Gulf of Mexico. The path of the Loop Current, and eddies from the Loop Current, can bring warmer than typical water into the Gulf of Mexico. This movie shows sea-surface height in the Gulf of Mexico beginning in January 1993 through 2004. The red regions are "high" relative to the surroundings; below the figure is a height scale, in cm; the flow is clockwise around a high, counter-clockwise around a low. In the open ocean, satellite-derived sea-surface height can be used to identify areas where the warm layer in the upper ocean is thick (high sea-surface height) and areas where this warm layer is thin (low sea-surface height). A thicker warm layer is more favorable for developing and supporting intense hurricanes than a thinner warm layer. Animation credit: Robert Leben, CU/CCAR. Copyright © 2020, University of Rhode Island. Disclaimer Please address comments and questions to hurricane@etal.uri.edu Lorem ipsum dolor Lorem ipsum dolor sit amet, consectetur adipiscing elit. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Praesent eget mauris vitae purus aliquet pretium. Nullam nibh neque, consectetur vel, iaculis vitae, volutpat et, mi. Aliquam vel justo id purus facilisis ultricies. Nulla facilisi.
18150	https://math.libretexts.org/Under_Construction/Purgatory/MAT_1320_Finite_Mathematics/05%3A_Sets_and_Counting/5.03%3A_Permutations	Skip to main content 5.3: Permutations Last updated : Jan 3, 2021 Save as PDF Fundamental Trigonometric Identities Default Text Page ID : 40145 ( \newcommand{\kernel}{\mathrm{null}\,}) Learning Objectives In this section you will learn to: Perform calculations using factorials. Count the number of possible permutations (ordered arrangement) of n items taken r at a time. Count the number of possible permutations when there are conditions imposed on the arrangements. Prerequisite Skills Before you get started, take this prerequisite quiz. How many three-letter word sequences can be formed using the letters { A, B, C } if no letter is to be repeated? Click here to check your answer : sequences can be formed. If you missed this problem, review Section 5.2. (Note that this will open in a new window.) A California license plate consists of a number from 1 to 5, then three letters followed by any three digits. Repetition is possible. How many such plates are possible? Click here to check your answer : different plates are possible. If you missed this problem, review Section 5.2. (Note that this will open in a new window.) How many different 4-letter radio station call letters can be made if the first letter must be K or W and no letters can be repeated? Click here to check your answer : station names can be made. If you missed this problem, review Section 5.2. (Note that this will open in a new window.) Factorials When working with the multiplication axiom, we will often need to multiply sequential, descending numbers as we did in Example 5.2.5. We have a special notation for that calculation, which we will use a great deal in this as well as in the next chapter. Definition: Factorial is read as "n factorial." where is a natural number. Example Calculate 5! Solution Using the definition of a factorial, . Multiplying these numbers gives a value of . Permutations In Example 5.2.6, we were asked to find the word sequences formed by using the letters { A, B, C } if no letter is to be repeated. The tree diagram gave us the following six arrangements. ABC, ACB, BAC, BCA, CAB, and CBA. Arrangements like these, where order is important and no element is repeated, are called permutations. Definition: Permutations A permutation of a set of elements is an ordered arrangement where each element is used once. Example How many three-letter word sequences can be formed using the letters { A, B, C, D }? Solution There are four choices for the first letter of our word, three choices for the second letter, and two choices for the third. \| \| \| \| --- \| 4 \| 3 \| 2 \| Applying the multiplication axiom, we get different arrangements. Example How many permutations of the letters of the word ARTICLE have consonants in the first and last positions? Solution In the word ARTICLE, there are 4 consonants. Since the first letter must be a consonant, we have four choices for the first position, and once we use up a consonant, there are only three consonants left for the last spot. We show as follows: \| \| \| \| \| \| \| \| --- --- --- \| 4 \| \| \| \| \| \| 3 \| Since there are no more restrictions, we can go ahead and make the choices for the rest of the positions. So far we have used up 2 letters, therefore, five remain. So for the next position there are five choices, for the position after that there are four choices, and so on. We get \| \| \| \| \| \| \| \| --- --- --- \| 4 \| 5 \| 4 \| 3 \| 2 \| 1 \| 3 \| So the total permutations are . Example Given five letters { A, B, C, D, E }. Find the following: The number of four-letter word sequences. The number of three-letter word sequences. The number of two-letter word sequences. Solution The problem is easily solved by the multiplication axiom, and answers are as follows: The number of four-letter word sequences is . The number of three-letter word sequences is . The number of two-letter word sequences is . We often encounter situations where we have a set of n objects and we are selecting r objects to form permutations. We refer to this as permutations of n objects taken r at a time, and we usually write it as nPr. Definition: nPr The Number of Permutations of n Objects Taken r at a Time or where and are natural numbers. Note that different texts may use different notation for permutations. , or all represent permutations. Example Given five letters { A, B, C, D, E }. Use the permutation formula to find the following: The number of four-letter word sequences. The number of three-letter word sequences. The number of two-letter word sequences. Solution Since we are selecting from 5 different letters in the set, . The value for is the number of letters we use in each sequence. The number of four-letter word sequences is 5P4 = The number of three-letter word sequences is 5P3 = The number of two-letter word sequences is 5P2 = Example An auto service station has 6 employees and 3 bays. How many different ways can the employees be placed at the three bays if each bay only gets one employee? Solution We will identify and in each case and solve using the formulas provided. Since we have 6 employees to select from, . Since we are placing only 3 of them at a time, . Therefore we are trying to calculate 6P3. Next we consider some more permutation problems to get further insight into these concepts. Example In how many different ways can 4 people be seated in a straight line if two of them insist on sitting next to each other? Solution Let us suppose we have four people A, B, C, and D. Further suppose that A and B want to sit together. For the sake of argument, we tie A and B together and treat them as one person. The four people are CD. Since is treated as one person, we have the following possible arrangements. Note that there are six more such permutations because A and B could also be tied in the order BA. And they are So altogether there are 12 different permutations. Let us now do the problem using the multiplication axiom. After we tie two of the people together and treat them as one person, we can say we have only three people. The multiplication axiom tells us that three people can be seated in 3! ways. Since two people can be tied together 2! ways, there are 3! 2! = 12 different arrangements Example You have 4 math books and 5 history books to put on a shelf that has 5 slots. In how many ways can the books be shelved if the first three slots are filled with math books and the next two slots are filled with history books? Solution We first do the problem using the multiplication axiom. Since the math books go in the first three slots, there are 4 choices for the first slot, 3 choices for the second and 2 choices for the third. The fourth slot requires a history book, and has five choices. Once that choice is made, there are 4 history books left, and therefore, 4 choices for the last slot. The choices are shown below. \| \| \| \| \| \| --- --- \| 4 \| 3 \| 2 \| 5 \| 4 \| Therefore, the number of permutations are . Alternately, we can see that is really same as 4P3, and is 5P2. So the answer can be written as (4P3) (5P2) = 480. Clearly, this makes sense. For every permutation of three math books placed in the first three slots, there are 5P2 permutations of history books that can be placed in the last two slots. Hence the multiplication axiom applies, and we have the answer (4P3) (5P2). We summarize the concepts of this section: 1. Factorial Where is a natural number. 2. Permutations A permutation of a set of elements is an ordered arrangement where each element is used once. 3. Permutations of n Objects Taken r at a Time or where and are natural numbers. Fundamental Trigonometric Identities
18151	https://www.cs.huji.ac.il/~noam/econcs/2002/dkl.pdf	Substitutes, complemen ts and equilibrium in t w o-sided mark et mo dels Danilo v V.I. , K oshev o y , G. A. y and C. Lang zx 5th April 2001 Abstract There are t w o sets of agen ts: buy ers B and sellers S . Eac h t yp e of agen t is allo w ed to trade with as man y agen ts on the opp osite side it wishes. Agen ts' decision pro cess is determined b y a mark et price system p , where p = (p(s; b); (s; b) 2 S B ) . Namely , a seller s solv es the task max B B [p(s; B ) c(s; B )] , where c(s; B ) is the cost incurred b y seller s when he con tracts with a set B of buy ers. A buy er b , similarly , will solv e for max S S [u(S; b) p(S; b)] , where u(S; b) is the utilit y of buy er b after con tracting with S sellers. W e examine the existence of comp etitiv e equilibrium in this mark et. W e sho w that equilibria exist in those mark ets in whic h all the go o ds on sale are pure substitutes, or in whic h all go o ds are pure complemen ts or nally in whic h there are appropriate com bi-nations of substitutes and complemen ts. W e also establish results ab out the structure of the sets of equilibrium prices and allo cations. W e sho w that the substitution and comple-men tarit y requiremen ts are in timately related with the discrete con v exit y (or conca vit y) requiremen ts imp osed on the corresp onding cost and utilit y functions of the mark et agen ts. Cen tral Institute of Economics and Mathematics, Russian A cadem y of Sciences, Mosco w, Russia. The nancial supp ort of RFBR gran t # 00-15-98873 is gratefully ac kno wledged. y Cen tral Institute of Economics and Mathematics, Russian A cadem y of Sciences, Mosco w, Russia. z Visiting CEMI & Genev a Univ ersit y , Switzerland. The nancial supp ort # 8210-053488 of the Swiss National Science F oundation is gratefully ac kno wledged. x The authors wish to thank t w o anon ymous referees for their remarks. 1 1 . In tro duction W e consider here a t w o-sided mark et, in whic h agen ts are divided in t w o complemen tary groups, sa y , sellers S and buy ers B . W e allo w here b oth that a seller trade with a few buy ers and that a buy er trade with a few sellers. T rades b et w een agen ts b elonging to a same group are forbidden. A trade b et w een a seller s and a buy er b consists in the transfer (from s to b ) of some giv en item (s; b) (and, p ossibly , of a transfer of money from b to s ). This mark et is a mark et with "indivisible" go o ds since a trade b et w een a pair of agen ts in v olv es at most a single item 1 (either agen ts conclude a deal or they do not). W e shall assume here that utilit y is transferable, and that money transfers b et w een agen ts are allo w ed. In this pap er, w e study the issue of existence of comp etitiv e equilibrium and in v estigate the structures of the sets of equilibrium prices and allo cations. T o talk ab out comp etitiv e outcomes in this mark et, w e shall ha v e to assume that for ev ery item p oten tially transferred from s to b a price p(s; b) emerged. This price system p = (p(s; b)) determines buy ers' demand sc hedules, on the one hand, and sellers' supply sc hedules, on the other. W e dene a comp etitiv e equilibrium in the standard w a y . In mark ets with indivisible go o ds, equilibria need not to obtain in the man y-to-man y case in con trast to the one-to-one setup. W e giv e conditions to w arran t existence of comp etitiv e equilibrium. Roughly sp eaking, w e require some kind of "discrete con v exit y" of utilities. Namely , w e sho w that there exist comp etitiv e equilibria in mark ets in whic h all go o ds are pure substitutes, or in whic h all go o ds are pure complemen ts, or nally in whic h there are sp ecic com binations of substitutes and complemen ts. Let us no w briey recall some salien t asp ects of t w o-sided mark et mo dels and discuss within this setup the substitutes and complemen ts cases. The early studies of t w o-sided mark ets mo dels fo cused on one-to-one setups: -the as-signmen t problem (K o opmans and Bec kmann (1957), Shapley and Sh ubik (1972)) and -the marriage matc hing problem (Gale and Shapley (1962)). In b oth these setups, no require-men ts w ere needed for the existence of solutions, moreo v er sets of solutions exhibited lattice structures. As w e shall p oin t out later on, the gross substitution prop ert y w as automatically satised in this setup. In a one-to-man y (ev en a man y-to-man y) framew ork, Cra wford and Kno er (1981) pro v ed existence of equilibrium imp osing the separabilit y of utilit y or pro duction functions under capacit y constrain ts. Kelso and Cra wford (1982) in tro duced the gross substitution prop ert y to establish existence in the one-to-man y sub case. Gul and Stacc hetti (1999) prop osed t w o conditions equiv alen t to the gross substitution prop ert y . They claim that gross substitution is necessary and sucien t to ensure existence of W alrasian equilibria. This is a sligh t o v erstate-men t. As it turns out, Danilo v, K oshev o y , and Murota (1998) sho w that discrete conca vit y is the appropriate condition. Moreo v er, gross substitution is one particular instance of discrete conca vit y . W e clarify this relationship in the presen t w ork. W e sho w rst that GS -functions (in a b o olean con text 2 ) are nothing else than p olymatroidal (PM ) conca v e functions. If w e 1 A more general setup migh t b e one in whic h trades b et w een opp osed agen ts w ould in v olv e a few commo di-ties, p ossibly divisible. 2 W e call b o olean the con text in whic h agen ts are allo w ed to consume or pro duce no more than one item of 2 allo w agen ts to consume or pro duce ev en tually more than one item of a giv en t yp e, then the GS condition turns out to b e w eak er than the PM condition and y et to o w eak for existence. The case of pure complemen ts also has a resp ectable tradition in the economic literature and it turns out it is also an instance of discrete conca vit y . W e shall not dw ell to o deeply up on this issue no w. But w e can recall, for instance, that Sam uelson (1947) asso ciated the idea of complemen tarit y with that of sup ermo dularit y (see T opkis (1978)). W e follo w this practice. Some time ago Danilo v, K oshev o y , and Sotsk o v (1994, 1997) relied up on sub/sup ermo dularit y conditions to pro v e the existence of equilibria in an econom y with in tellectual (or informa-tional) go o ds. In the presen t w ork, w e shall also establish existence in a mixed case in whic h the mark et go o ds are partitioned in t w o groups. The go o ds of the rst group are m utual substitutes, and the go o ds of the second are m utual complemen ts. It will b e imp ortan t here that buy ers (consumers) and sellers (pro ducers) ha v e consensual views ab out this partition in the sense that they all agree ab out whic h go o ds are substitutes and whic h are complemen ts. This requiremen t will b e called the Compatibilit y Principle. In the job mark et mo del of Kelso and Cra wford (1982), the Compatibilit y Principle will sp ell as follo ws. The w ork ers p erceiv e the rms as substitutes, th us b y compatibilit y , rms should p erceiv e the w ork ers as substitutes as w ell. 2 . A t w o-sided mark et mo del Let there b e t w o (nite) sets of agen ts S and B . W e shall call them sel lers and buyers 3 . Eac h agen t is allo w ed to form a partnership (or a deal) with agen ts from the opp osite side. Moreo v er, an agen t can ha v e as man y partners as he w an ts. In other w ords, w e are in a man y-to-man y setup. W e in tro duce the follo wing denition whic h inciden tally accoun ts for this m ultiplicit y of p ossible deals. Denition A matching is an arbitrary subset S B . Denote b y (b) = fs; (s; b) 2 g and (s) = fb; (s; b) 2 g . An elemen tary deal consists in a transfer of at most one item (s; b) from the seller s to the buy er b 4 . When buy er b gets hold of a set S S of items, his utilit y increases b y the amoun t u(S; b) . Similarly , when seller s assigns his items to a set B B of buy ers, he exp eriences a loss of utilit y . In our opinion, it is con v enien t to view sellers as pro ducers. The pro duction cost of a set B of items for s is denoted b y c(s; B ) . a giv en t yp e. 3 Of course, w e can think of them as "rms and w ork ers" (lik e in Kelso and Cra wford (1982) or Roth (1984)), or as "colleges and studen ts" (lik e in Gale and Shapley (1962)), or, if w e allo w ourselv es to pursue the allegory of marriage, as "men and w omen" (though, in promiscuous marriages), nally , as "service-pro viding facilities and customers". 4 One can assume that the go o ds oered b y a giv en seller are somewhat similar: one seller supplies houses, another -cars, the third planes, and so on.... F urthermore, w e assume that eac h buy er needs no more than one item of an y giv en t yp e. 3 It seems rather natural to p ose b oth that c(s; ;) = 0 and u(;; b) = 0 . W e dene no w the total gain of matc hing to b e equal to the sum of buy ers' gains min us the sum of sellers' pro duction costs, v () = X b u((b); b) X s c(s; (s)): Denition A c or e outc ome (or stable outc ome ) is a pair (; x) , where is some matc hing and x : N = S [ B ! R is a v ector of agen ts utilities, whic h satisfy the follo wing conditions: 1. v () = x(N ) , 2. F or an y coalition K = (S [ B ) N and for an y matc hing 0 S B , w e ha v e x(K ) v ( 0 ) . W e dene x(K ) = P k 2K x(k ) . In particular, x(i) 0 for eac h agen t i 2 N . Example 1 Empty c or e with heter o gene ous buyers. W e consider a mark et with t w o sellers s; s 0 and t w o buy ers b; b 0 . Seller s has t w o b ottles of gin g ; g 0 on sale. Seller s 0 has t w o b ottles of tonic w ater t; t 0 on sale. The cost functions of s (resp. s 0 ) are: c(s; g ) = c(s; g 0 ) = 0; c(s; fg ; g 0 g) = +1 (or 100); c(s 0 ; t) = c(s 0 ; t 0 ) = 0; c(s 0 ; ft; t 0 g) = +1 (or 100): In this example, the items pro duced b y an y one of the sellers are p erfect substitutes with resp ect to pro duction. Moreo v er eac h seller has a capacit y constrain t: he can sell (and pro duce) at most one b ottle. The utilit y functions of the buy ers are: u(b; g ) = u(b; t) = 0; u(s; fg ; tg) = 1; u(b 0 ; g 0 ) = u(b 0 ; t 0 ) = 1; u(b 0 ; fg 0 ; t 0 g) = 1: W e notice that b lik es to consume gin and tonic together and th us for him these go o ds are complemen ts. The second buy er b 0 will b e happ y to drink just an ything. He is indieren t with resp ect to gin, tonic, and gin & tonic. Easy computations sho w that: v (fs; s 0 ; bg) = 1 [a] , v (fs; b 0 g) = 1 [b] , v (fs 0 ; b 0 g) = 1 [c] and v (fs; s 0 ; b; b 0 g) = 1 [d] . Supp ose that x is a pa y o v ector in the core. Then from [a] and [d], w e ha v e x(b 0 ) = 0 . F rom [b] and [d], w e ha v e x(s 0 ) = x(b) = 0 . F rom [c] and [d], w e ha v e x(s) = x(b) = 0 . Th us x 0 . This con tradicts the P areto optimalit y of x . Example 2 The sep ar able c ase. In the separable (or additiv e) case, seller s 's cost function c(s; B ) is giv en as the sum c(s; b) o v er b 2 B (w e dene the utilit y of a buy er b in a similar 4 fashion). An y trade is then decomp osed in a series of separate and elemen tary trades (s; b) . No w, clearly , a trade (s; b) obtains as so on as u(s; b) c(s; b) ; the buy er b then transfers the amoun t of money p(s; b) to the seller s , and u(s; b) p(s; b) c(s; b) . Eac h deal (s; b) is concluded at a price p(s; b) . W e can alw a ys assume that the price p(s; b) of an item (s; b) w ould lie in b et w een u(s; b) and c(s; b) ev en if the deal (s; b) do es not nally materialize. Th us p = (p(s; b)) is the mark et price system at equilibrium. The stable outcomes coincide with the comp etitiv e allo cations at equilibrium in this separable case. It is no w time to dene what is a comp etitiv e equilibrium in this setup. Let there b e a price system (p(s; b); s 2 S ; b 2 B ) , in our mark et. The seller s solv es the follo wing problem max [p(s; B ) c(s; B )]; for B B ; (1) while the buy er b solv es for max [u(S; b) p(S; b)]; for S S : (2) W e ha v e an equilibrium when the solutions to (1) and (2) are consisten t. More precisely , Denition An e quilibrium is a pair (; p) , where is a matc hing and p = (p(s; b)) is a price system suc h that (s) solv es (1), for eac h s 2 S , and (b) solv es (2), for eac h b 2 B . The comp etitiv e allo cations b elong to the core. Precisely , giv en a pair (; p) , w e dene the utilit y x(s) of seller s to b e x(s) = p(s; (s)) c(s; (s)) (and similarly for buy er b ). Prop osition 1 . Given an e quilibrium (; p) , the p air (; x) is a stable outc ome. The pro of is standard, th us omitted. The con v erse of this Prop osition is not true as sho wn in the follo wing example. Example 3 Cor e without e quilibria. Again there are t w o sellers and t w o buy ers. The buy ers are as in Example 1: the rst buy er desires gin and tonic together, the second will b e happ y with either. Ho w ev er the cost functions are dieren t. No w the pro duction cost of the rst b ottle is equal to 1 , while the second b ottle is pro duced at no cost. This con text obtains in the case of informational go o ds in whic h t ypically the cost of pro ducing the rst sp ecimen is signican tly larger than that of duplicates (Danilo v, K oshev o y , and Sotsk o v (1994)). It is easy to see that v (K ) = 0 for an y coalition K . Th us the core consists of the unique p oin t (0; 0; 0; 0) . Let us sho w that there is no equilibrium. Supp ose that the equilibrium prices of seller s 's items are equal to p; p 0 and that those of seller s 0 are equal to q ; q 0 . Since an equilibrium allo cation is stable, it yields a net surplus of 0 to ev ery agen t on the mark et. W e ha v e therefore the follo wing system of inequalities: 5 1: p 1; p 0 1; p + p 1 2: q 1; q 0 1; q + q 1 3: 0 p; 0 q ; 1 p + q 4: 1 p 0 ; 1 q 0 ; 1 p 0 + q 0 : The inequalities 0 p , 1 p 0 and p + p 0 1 giv e p = 0 and p 0 = 1 . Similarly , 0 q , 1 q 0 and q + q 0 1 giv e q = 0 and q 0 = 1 . Ho w ev er this is nev er compatible with 1 p + q . Examples 1 and 3 sho w that, in con trast to the one-to-one setup, equilibria need not obtain in the man y-to-man y setup. In the sequel, w e pro vide the conditions whic h w arran t existence of equilibrium (and, consequen tly , of stable outcomes 5 ). W e consider t w o main conditions. The rst is the gross substitution condition as in tro duced b y Kelso and Cra wford (1982) 6 . The second is the p olar (in some sense) condition of complemen tarit y . W e shall consider the case of substitutes and that of complemen ts separately and then briey discuss a mixed case. W e start with a few generalities ab out the existence and the structure of the set of equilibria in the transferable setup. 3 . Generalities ab out equilibria In the transferable case, the existence issue is related to an optimization problem. T o this end, w e aggregate all buy ers in to a single consumer and all sellers in to a single pro ducer. Namely , let = S B . Consider the follo wing t w o function U and C on the set 2 : U () = X b u((b); b); C () = X s c(s; (s)); where is a matc hing. U () is the aggregate utilit y deriv ed from , and C () is the aggregate pro duction cost of . W e are in terested in the aggregate surplus U C . The maxim um of this function is equal to v (N ) . The set of optimal matc hings is denoted b y M . Prop osition 1 states that an y equilibrium matc hing b elongs to M . An optimal matc hing will b e a comp etitiv e allo cation, whenev er it can b e supp orted b y some price system p . A price p supp orts a matc hing with resp ect to consumption if U () U ( ) p() p( ) . That is: p is a sup ergradien t to U at the p oin t . Similarly , a price p 5 The conditions under whic h stable outcomes can b e decen tralized remain y et to b e found. Some results w ere obtained b y Kanek o (1982), Quinzii (1984) and Kelso and Cra wford (1982). 6 Of course, the notion of gross substitutabilit y w as form ulated and in v estigated long ago. (One can trace it bac k to Hic ks. Morishima, Negishi, Arro w and Hahn, and man y others made imp ortan t con tributions.) The classical denition has b een form ulated in the case of single-v alued demands. In economies with indivis-ible commo dities, demands at certain prices are una v oidably m ulti-v alued. Th us one has to pro vide for an appropriate form ulation of this condition. Note that P oltero vic h and Spiv ak (1982) prop osed an alternativ e form ulation to that of Kelso and Cra wford (1982). 6 supp orts a matc hing with resp ect to pro duction if C () C ( ) p() p( ) . That is: p is a subgradien t to U at the p oin t . Of course, p supp orts with resp ect to consumption and pro duction if and only if p separates the functions U U ( ) and C C ( ) , that is, U U ( ) p p( ) C C ( ): Prop osition 2 A p air ( ; p) is an e quilibrium if and only if p sep ar ates the functions U U ( ) and C C ( ) . Pr o of Supp ose that ( ; p) is an equilibrium. Then, for ev ery buy er b , the follo wing in-equalit y holds, u(; b) p(; b) u( (b); b) p( (b); b): A dding these inequalities, w e obtain U () p() U ( ) p( ) . Similarly for C . Con v ersely , supp ose that p supp orts with resp ect to consumption, th us U () U ( ) p() p( ) . This inequalit y is equiv alen t to the follo wing series of inequalities u(; b) u( (b); b) p(; b) p( (b); b); b 2 B : Eac h inequalit y implies that (b) is optimal for buy er b at the price p(; b) . Similarly for the sellers. Corollary The set of e quilibria has the form M P , wher e P is the set of pric es sep ar ating the functions U and C + v (N ) . The set P can b e empt y , as in Example 1 and 3. In order to exhibit a linear functional p separating the "osculating" functions U and C + max (U C ) (that is in order to exhibit a "sandwic h") some kind of "conca vit y" requiremen t is needed for the functions U and C . Danilo v and K oshev o y (2000) (and, see also Danilo v, K oshev o y , and Murota (1998)) dev elop this p oin t at length. W e giv e here some a v or of their w ork, restraining ourselv es to functions giv en on the b o olean lattice 2 . First, let us iden tify the set 2 with the set f0; 1g in the v ector space R . T o this end, w e asso ciate the c haracteristic v ector 1 A : ! R , 1 A (! ) = 1; ! 2 A 0; ! = 2 A: to ev ery subset A . The functions U and C are no w dened on the set of v ertices of the unit cub e Q := [0; 1] = co (f0; 1g . Denote b y co(U ) the conca vication of U , i.e., the minimal conca v e 7 function on the cub e Q whic h is sup erior or equal to U . Ob viously , co(U )(X ) = U (X ) at ev ery v ertex X of Q . A price system p is view ed, th us, as the linear functional on the space R , whic h is equal to p(! ) at the p oin t 1 ! . No w to an y linear functional p 2 (R ) , w e asso ciate the con v ex set, D (U; p) := Argmaxfco (U ) pg in the cub e Q . It is an in teger p olytop e. (A p olytop e is in teger if its v ertices are in teger.) This p olytop e will b e called a c el l (or an anity ar e a ) of the function U 7 . The cell D (U; p) is the set of p oin ts of Q , for whic h the function co(U ) coincides with the ane function p + max(U p) . Lemma If D (U; p) is a c el l of U , and D 0 is a fac e of D (U; p) , then D 0 is also a c el l of U . Pr o of The face D 0 is the set of p oin ts in D (U; p) , where a certain linear functional q on R attains its maxim um. Then D 0 = D (U; p "q ) , where " is a small and p ositiv e n um b er. The follo wing prop osition sho ws that the crucial reason for the existence of a linear func-tional separating U and C ("a sandwic h") is that the in tersections of cells of the functions U and C are in teger p olytop es. Prop osition 3 (Sandwic h) L et U and C b e functions on 2 , and let C U . Supp ose that, for every p , the interse ction of the p olytop es D (U; p) and D (C ; p) is an inte ger p olytop e. Then U and C ar e sep ar ate d by some line ar functional. Pr o of Since C U , w e ha v e U C 0 . W e pro v e the follo wing stronger claim. Claim coU + co (C ) 0: Let x b e an arbitrary p oin t in the cub e Q . Let D b e a cell of U con taining x , and D 0 b e a cell of C con taining x as w ell. The in tersection D \ D 0 is an in teger p olytop e b y assumption. Let X 1 ; ; X n b e the v ertices of this p olytop e. Then x is obtained as a con v ex com bination of these v ertices, x = P i i X i , where i 0 , and P i i = 1 . W e assert that co(U )(x) = P i i U (X i ) and co(C )(x) = P i i C (X i ) . W e pro v e the rst equalit y (the second is obtained similarly). Since co (U ) is ane on D and X 1 ; : : : ; X n 2 D , then co(U )(x) = P i i co(U )(X i ) . Ho w ev er co (U ) coincides with U at the v ertices of the cub e Q , therefore co (U )(X i ) = U (X i ) and P i i U (X i ) = co (U )(x) . No w, co(U )(x) + co(C )(x) = X i i U (X i ) X i i C (X i ) = X i i [U (X i ) C (X i )] 0: 7 A ctually it is a cell of co(U ) . F or brevit y , w e write "cell of U ". 8 This terminates the pro of of our claim. Let us return to the pro of of Prop osition 3. W e ha v e the follo wing inequalit y co (U ) + co(C ) 0 , that is co(C ) co (U ) . The function on the left side of this inequalit y is con v ex, while that on the righ t side is conca v e. By a classical separation of con v ex sets argumen t, there exists b oth a linear function q and a real suc h that co(C ) q + coU: Since co (U ) U , q (X ) + U (X ) for an y in teger p oin t X of Q . Similarly , C (X ) q (X ) + , for an y in teger p oin t X in Q . Remark 1. If, in Prop osition 3, the function U is monotone, then there exists a monotone separating functional p . T o start with, note that co (U ) is a monotone function on the cub e Q . No w instead of considering co(U ) , w e consider its monotone extension F on the whole p ositiv e orthan t R + . It is dened as follo ws: F (x) = co(U )(min (x; 1 )) . F is conca v e and is ev erywhere (in the cub e Q ) inferior or equal to co(C ) . In this case, an y separating functional p will b e non-negativ e. Remark 2. The theory of Discrete Conca vit y (Danilo v and K oshev o y (2000)) c haracterizes the classes of discrete functions for whic h Prop osition 3 holds true. In the next section, w e sho w that GS -functions form a class of discrete conca vit y on the b o olean cub e. 4 . The Gross Substitution Prop ert y Let b e an arbitrary nite set of items. W e iden tify here a bund le to a subset A . A utilit y function is a mapping u : 2 ! R for whic h u(;) = 0 . A price functional is the simplest example of suc h a map. Let p : ! R represen t a price sc hedule. Then the v alue of a bundle A for the price sc hedule p is giv en b y p(A) = P a2A p(a) . Giv en a utilit y function u and a price p , the net utility deriv ed from A is dened b y u(A) p(A) . A consumer with utilit y u selects the bundles yielding the highest p ossible net utilit y at prices p . Denote b y D (u; p) the set of optimal bundles, that is: D (u; p) = fA ; u(A) p(A) u(A 0 ) p(A 0 ) for an y A 0 g: D (u; p) is the demand set at the price p . (Henceforth, w e shall drop the parameter u when no confusion is p ossible.) Denition (Kelso and Cra wford) The utilit y function u is said to gener ate gr oss substi-tutable demands or (in short) is a GS-function if, for an y pair of prices p; q , suc h that q p , and for an y A 2 D (p) , there exists B 2 D (q ) suc h that f! 2 A; p(! ) = q (! )g B . 9 In other w ords: supp ose that in the pro cess of going from the price system p to the price system q , some prices increase, while others remain unc hanged. If some item ! w as demanded at prices p , and if ! 's price sta y ed put in q , then it migh t b e demanded at prices q . One can exp ect that this prop ert y hold true when the items in are substitutable. In tu-itiv ely , this excludes the complemen tarit y-t yp e relationship b et w een go o ds, that is evidenced in suc h commo dit y bundles as the bundle "k ey and k ey-lo c k" or the bundle "left and righ t glo v e". Inciden tally Gul and Stacc hetti (1999) formalize the absence of complemen tarit y b e-t w een go o ds in their Single Impro v emen t prop ert y . Denition (Gul and Stacc hetti) Supp ose A = 2 D (p) , hence A do es not maximize u p . Then there exists a b etter bundle B (with u(B ) p(B ) > u(A) p(A) ), suc h that jA B j 1 and jB Aj 1 . A utilit y function u whic h satises this condition for all p is said to ha v e the SI -pr op erty. In other w ords, if the bundle A is not optimal at prices p , then a b etter bundle B can b e found, whic h is deriv ed from A b y p erforming an y of the follo wing three op erations: remo ving an item from A , or adding an item to A , or nally doing b oth. Th us w e ma y impro v e up on a bundle b y p erforming these elemen tary op erations. Gul and Stacc hetti (1999) in their Theorem 1 pro v e that the GS and SI prop erties are equiv alen t. W e, in turn, prop ose an alternativ e c haracterization of the GS -prop ert y . T o this end, w e in tro duce the follo wing notions. Denition (i) A r o ot in the space R is an y of the v ectors 1 a ; 1 a 1 b , where a; b 2 . (ii) A con v ex p olytop e in the space R is a g-p olymatr oid if an y of its edges is parallel to some ro ot. (iii) A function u , dened on the set 2 , is called PM-c onc ave if its cells are g -p olymatroids. Theorem 1 A function on the set 2 is a GS-function if and only if it is PM-c onc ave. Pr o of Supp ose that u is a GS -function. Let us c hec k that eac h edge of a cell of u is parallel to some ro ot, and this for eac h cell. Consider a giv en edge. A ccording to Lemma 1, w e can assume that this edge is some cell D (co(u); p) of the function co(u) . Let this edge connect the t w o v ertices 1 A and 1 B of the cub e Q . Then the bundles A and B are the only optimal bundles at the price p . As B 6= A , w e can assume that there exists b 0 2 B n A . Let us sligh tly increase the price of item b 0 . Then at this resulting price system p 0 , bundle A remains the sole optimal bundle. B whic h isn't optimal an y more, is nev ertheless preferred to an y other non-optimal bundle. By the SI -prop ert y , bundle A can b e obtained from bundle B pro vided one p erforms one of the follo wing elemen tary op erations: remo ving some item b , or remo ving some b and adding some a . This means that the v ectors 1 A and 1 B dier b y a ro ot. Con v ersely , supp ose u is a PM -function. Let us c hec k that the SI prop ert y holds. Let A b e a sub optimal bundle at prices p . Giv en a ro ot r , denote b y d(r ) the deriv ativ e of function co(u) p at the p oin t A in the direction r . By PM -conca vit y of co (u) , d(r ) > 0 , for some ro ot r . This is b ecause the edges of the cells of co(u) are parallel to ro ots. 10 Inequalit y d(r ) > 0 implies that the (op en) segmen t (A; A + r ) in tersects some cell D of u . Since, b oth the (closed) segmen t [A; A + r ] and D are in tegral g -p olymatroids, b y the Edmonds-F rank theorem (b elo w), their in tersection is an in tegral p olytop e namely [A; A + r ] . No w this implies that the function co(u) p is ane on this segmen t. In particular, the v alue of this function at the p oin t B := A + r is larger than its v alue at the p oin t A . Remark 1. The functions considered ab o v e, w ere giv en on the v ertices of the unit cub e. But it w ould b e more appropriate to dev elop the theory of GS and PM functions for functions dened on the whole of the lattice Z of in teger p oin ts of R . W e exp ose this theory in Danilo v, K oshev o y , and Lang (2001). Remark 2. Up to no w, follo wing in this Gul and Stacc hetti (1999), w e considered only those functions on 2 , whose v alues w ere nite. But there are go o d reasons to consider functions whic h ma y tak e innite v alues. In particular, when dealing with "utilit y" functions, it is con v enien t to consider the v alue 1 . Similarly when w e are dealing with "cost functions", w e'd lik e to consider the v alue +1 . Of course, the cells of u , in this case, will co v er the con v ex h ull of its eectiv e domain co (dom (u)) , and not the whole cub e Q . Theorem 1 still holds in this case. Theorem 2 (Sandwic h theorem) L et U and V b e PM-c onc ave functions on f0; 1g . Supp ose that V U . Then ther e exists a line ar functional p and a r e al numb er such that V p + U . This result (pro v en rst b y Murota (1996)) follo ws from b oth Prop osition 3 and the p oly-matroid in tersection theorem giv en b elo w (see, for instance, F rank and T ardös (1988)). Theorem (Edmonds-F rank) The interse ction of two inte ger g -p olymatr oids 8 is an inte-ger p olytop e. 5 . Mark ets with pure substitutes Let us return to our t w o-sided mark et whose sellers s 2 S ha v e cost functions c(s; ) on 2 B , and whose buy ers b 2 B ha v e utilit y functions u(; b) on 2 S . W e imp ose the follo wing assumption. Assumption (Gross Substitution) u(; b) is a GS-function for every buyer b 2 B and c(s; ) is a GS-function for every sel ler s 2 S . Theorem 3 A two-side d market in which the Gr oss Substitution Assumption is satise d has e quilibria. Pr o of W e pro v e existence b y an aggregation argumen t. Let U and C b e the aggregate utilit y and cost functions on 2 , where = S B . W e assert that U and C are b oth GS -functions. Indeed, 8 Note that Edmonds and F rank dene a g -p olymatroid as a p olytop e giv en b y sp ecic linear inequalities (see F rank and T ardös (1988)). Their denition and the one giv en ab o v e are equiv alen t (see Danilo v and K oshev o y (2000)). 11 D (U; p) = Y b D (u(; b); p(; b)); and similarly for C . It is straigh tforw ard to c hec k that D (U; p) satises the gross substi-tution prop ert y . It is done either b y referring to the denition or b y c hec king that the edges of the asso ciated p olytop e co(D (U; p)) = Q b co(D (u(; b); (p(b))) ha v e the required form. No w all follo ws from Prop osition 2, Theorem 1 and Theorem 2. W e are no w ready to state t w o assertions ab out the structures of the set of equilibrium prices and of the set of equilibrium allo cations. F rom Corollary 1, w e kno w that the set of equilibria has the form M P . If 2 M and p 2 P then M = Argmax (U p ) \ Argmax(p C ); P = @ co U ( ) \ @ co(C )( ): Here @ denotes the sup erdieren tial of a conca v e function. Recall that the sup erdieren tial of a function at a p oin t is the set of all sup ergradien ts to this function at this p oin t. Moreo v er the sets M and P are endo w ed with sp ecic structures. Denition W e sa y that a subset M 2 is in-b etwe en-c onvex set if, giv en an y t w o elemen ts and suc h that , it con tains all in termediate as w ell (that is it con tains all suc h that ) 9 . Theorem 4 In a two-side d market satisfying the Gr oss Substitution Assumption, a) M is an in-b etwe en-c onvex subset of 2 and b) P is a sublattic e of R . Pr o of a) Since the in tersection of in-b et w een-con v ex subsets is in-b et w een-con v ex, it suces to pro v e that Argmax(U p) is in-b et w een-con v ex. The function U p is a GS -function, th us is submo dular (Gul and Stacc hetti (1999), Lemma 6). Then to get assertion a) it suces to pro v e that the set of maxima of a submo dular function f is in-b et w een-con v ex. Indeed, let and b e t w o maxima of f and let . Denote b y 0 = ( ) [ . Clearly , \ 0 = and [ 0 = . Then, on the one hand, f () f ( ) = f ( ) and f ( 0 ) f ( ) = f ( ): On the other hand, b y submo dularit y , f () + f ( 0 ) f ( \ 0 ) + f ( [ 0 ) = f ( ) + f ( ) f () + f ( 0 ): 9 Analogously , this notion can b e form ulated for an y ordered set. 12 The inequalities hold with equalit y . In particular, also b elongs to Argmax (f ) . b) The in tersection of sublattices is a sublattice, th us w e need only pro v e that @ f (x) is a sublattice for ev ery PM -conca v e function f . Without loss of generalit y , w e ma y assume that x = 0 and that the function f is homogeneous (of degree one). One-dimensional cells of this function, i.e. ra ys, are generated b y ro ots. Hence the sup erdieren tial is giv en b y a system of linear inequalities of the form c(a) p(a) c 0 (a) or of the form c(a; b) p(a) p(b) c 0 (a; b) , where c(a); c 0 (a); : : : are constan ts. Ev ery inequalit y of this kind denes a sublattice, as w ell as the whole system. T o conclude this section, w e discuss a few examples of GS -functions (or alternativ ely of PM -conca v e functions). Example 4 Consider a seller whic h can not pro duce more than one item, i.e., his cost c(B ) equals +1 as so on as B con tains more than one elemen t. Then c is PM -conca v e b ecause its cells are g -p olymatroids. Indeed, the cells of c are faces of the unit simplex, whose v ertices are constituted b y 0 and the basis v ectors 1 b , for b 2 B . If all sellers are of this t yp e (as are the w ork ers in Kelso and Cra wford (1982)) the gross substitution prop ert y is fullled automatically on the sellers' side. The mark et econom y has comp etitiv e equilibria if moreo v er w e require gross substitution on the buy ers' side. Example 5 Consider no w a buy er who is not eager to consume more than one item. His utilit y then tak es the form, u(S ) = max s2S u(s): The function u is PM -conca v e. This holds b ecause the monotone exten tion of a PM -conca v e function is PM -conca v e. Therefore the gross substitution prop ert y is automatically fullled on the buy ers' side if all buy ers are of this t yp e. If, on top, w e require gross substitution on the sellers' side, w e ha v e existence. Note that Kanek o (1982) considered the particular sub case in whic h cost functions w ere additiv e. Observ e, nally , that if sellers cannot pro duce more than one item and buy ers ha v e no need for more than one item, then the gross substitution prop ert y is fullled automatically on b oth sides of the mark et and clearly comp etitiv e equilibria alw a ys obtain in this kind of t w o-sided mark et. A ctually this is the mark et in v estigated b y Shapley and Sh ubik (1972). Example 6 An y separable (or additiv e, or linear) function on 2 is a GS -function. This explains wh y the mark et in Example 2 has comp etitiv e equilibria. W e can devise a more in teresting instance in whic h separable functions are used, lik e in Cra wford and Kno er (1981). They consider separable cost functions with capacit y constrain ts. Let l b e a linear function and k capacit y constrain t. A cost function with capacit y constrain t is a function c whic h coincides with l as long as jB j k , and is equal to +1 elsewhere. 13 (Example 4 can b e view ed as a sp ecial case of capacit y constrain t, where the capacit y is k = 1 .) Cra wford and Kno er (1981) pro v ed existence of equilibria in this con text. Of course, their result is claried when one realizes that suc h functions are PM -conca v e. (More generally , an y PM -conca v e function to whic h w e imp ose a capacit y constrain t remains a PM -conca v e function.) Example 7 Bevia, Quinzii, and Silv a (1999) considered another in teresting case. They sho w ed that a function of the form P b u b (x b ) + ( P b x b ) , where is a conca v e function of one v ariable, is a GS -function. When (t) is equal to 0 for t k and is equal to 1 for t > k , w e are bac k to the discussion in Example 6. The previous examples w e considered, are particular cases of a more general construction (see Danilo v, K oshev o y , and Murota (1998) or Danilo v and Lang (2000)) whic h w e describ e no w. Supp ose T is a family of subsets of . This family is called laminar if, for ev ery A; B 2 T , one of the three conditions hold: A B , B A or A \ B = ; . Consider no w the collection of conca v e functions of one v ariable A , indexed b y A 2 T . W e construct the new function U on the b o olean set 2 dening, for X , that U (X ) = X A2T A (jX \ Aj): Then U is a PM -conca v e function. One can also use the fact that the (inmal) con v olution of PM -conca v e functions is a PM -function to deriv e new PM -conca v e functions. Other, in order to c hec k the GS prop ert y , one can use the follo wing c haracterization: u is GS i its F enc hel conjugate u is sup ermo dular (Danilo v and Lang (2000)). 6 . Mark ets with pure complemen ts W e consider no w the p olar case to a mark et with substitutes, that of a mark et with pure complemen tary go o ds. Recall that if u is a GS -function, then it is submo dular and th us has "decreasing marginal utilit y". Con v ersely , a sup ermo dular function u has "increasing marginal utilit y". This means that the dierence u(A) u(A n a) is a monotone function of A , whic h means that the incremen t of utilit y deriv ed from adding an item a to a bundle A is greater the larger the bundle A . This prop ert y of the utilit y function p oin ts out to the existence of some complemen tarit y among the go o ds added-to-the-bundle and those en tering-the-bundle. W e require submo dularit y of cost functions to mo del complemen tarit y in the pro duction pro cesses. W e imp ose here the follo wing assumption. 14 Assumption (Complemen tarit y) u(; b) is a sup ermo dular function for every buyers b 2 B and c(s; ) is a submo dular function for every sel ler s 2 S . Theorem 5 Consider a two-side d market in which the Complementarity assumption is sat-ise d. Then a) this market has e quilibria, b) the set M of e quilibrium al lo c ations is a sublattic e of 2 , c) the set P of e quilibrium pric es is an in-b etwe en-c onvex subset of R . Pr o of The pro of is v ery similar to that whic h w as dev elop ed in the pure substitutes case. The aggregate utilit y function U is clearly sup ermo dular, while the aggregate cost function C is submo dular. The appropriate v ersion of Prop osition 3, whic h states the existence of a separation b et w een sup ermo dular and submo dular functions, used here, is due to F rank (1982) (or else see Danilo v, K oshev o y , and Sotsk o v (1994)). The existence of a separation again rests on the shap e of the cells of the relev an t functions. W e only discuss the matter for the sup ermo dular function U . Its conca vication co (U ) is linear o v er simplexes of the standard triangulation of the unit cub e Q = [0; 1] . Namely , let b e a w eak order on the set . The corresp onding simplex ( ) consists of all monotone maps ( ; ) ! [0; 1] . These simplexes ( ( )) constitute the standard simplicial decomp osition of the cub e Q when runs through the set of all w eak orders on . (The same holds true for the function C .) The in tersection of an y of these simplexes is also a simplex of this standard triangulation, th us it is an in teger p olytop e. This completes the pro of of F rank's Sandwic h Theorem and th us pro v es p oin t a) . W e obtain assertion b) b y remarking that the set of maxima of a sup ermo dular function is a sublattice. F or what concerns p oin t c) , then P is the in tersection of t w o sup erdieren tials (of the functions U and C ) at an y optimal matc hing . The in tersection of in-b et w een-con v ex sets is in-b et w een-con v ex. Th us w e only need to sho w the follo wing lemma. Lemma The sup er dier ential of a sup ermo dular function is in-b etwe en-c onvex. Pr o of Let f b e a sup ermo dular function and p 0 p 00 b e t w o sup ergradien ts to f at a p oin t x . W e ha v e to c hec k that, for an y p suc h that p 0 p p 00 , there holds f (y ) f (x) p(y ) p(x): (3) Since p 0 and p 00 are subgradien ts, there holds f (y _ x) f (x) p 0 (y _ x) p 0 (x) and f (y ^ x) f (x) p 00 (y ^ x) p 00 (x): No w, y _ x x and p p 0 imply p 0 (y _ x) p 0 (x) p(y _ x) p(x) , and y ^ x x and p p 00 imply p 00 (y ^ x) p 00 (x) p(y ^ x) p(x) . Th us, w e ha v e f (y _ x) f (x) + f (y ^ x) f (x) p(y _ x) p(x) + p(y ^ x) p(x): (4) 15 Since p is a mo dular function p(y _ x) + p(y ^ x) = p(y ) + p(x) holds, th us (4) can b e rewritten as follo ws f (y _ x) + f (y ^ x) 2f (x) p(y ) p(x): (5) F rom (5) and from sup ermo dularit y of f , i.e. f (y ) + f (x) f (y _ x) + f (y ^ x) , w e obtain f (y ) f (x) = f (y ) + f (x) 2f (x) f (y _ x) + f (y ^ x) 2f (x) p(y ) p(x): Therefore, (3) is v eried. This completes the pro of of the lemma and hence of p oin t c) . In con trast to PM -conca v e functions, sup ermo dular functions ha v e already a resp ectable tradition in economics. They app ear in the con text of con v ex co op erativ e games (a concept due to Shapley (1971)) as w ell as in a n um b er of in v en tory problems (see T opkis (1978)). W e giv e here a few examples of sub/sup ermo dular functions and suggest ho w they can b e constructed. The task is sligh tly simpler than in the case of PM -conca v e functions, for the set of sup ermo dular functions forms a con v ex cone. It is not dicult to devise sup ermo dular func-tions dep ending on t w o or three v ariables; summing suc h functions yields new sup ermo dular functions. W e can examine another application of this summation principle. Let b e a non-negativ e function on 2 . Dene the function u on 2 , where, for A , u(A) = X B A (B ): u is a sup ermo dular function. Finally , it is simple to construct anon ymous sup ermo dular functions. Let b e a mono-tone con v ex function of a single v ariable. Then the function U on 2 , U (A) = (jAj) , is sup ermo dular. More: w e can substitute the n um b er of elemen ts j j b y an arbitrary p ositiv e measure on (see Shapley (1971)). The functions just dened ab o v e are particular instances of the follo wing observ ation: the comp osition Æ U of a monotone sup ermo dular function U with a monotone con v ex function is sup ermo dular (T opkis (1978), see also Lo vász (1983)). 7 . Mark ets with b oth substitutes and complemen ts W e consider no w the follo wing mixed case in whic h part of the go o ds on sale are substitutes, whereas the remainder are complemen ts. T o this end, w e assume that the sellers are divided in to t w o groups S s and S c . The go o ds supplied b y sellers of the rst group are m utual substitutes, whereas those supplied b y the sellers of the second group are m utual complemen ts. W e imp ose the follo wing three conditions on b oth utilit y and cost functions: Assumption (The Compatibilit y Principle) 16 (S s ) the function c(s; ) is PM-c onc ave for any s 2 S s ; (S c ) the function c(s; ) is sup ermo dular for any s 2 S c ; (B ) the utility function u(; b) of any buyer b is a sum of a PM-c onc ave function u s (; b) of variables fr om the set S s and a sup ermo dular function u c (; b) of variables fr om the set S c . Theorem 6 A two-side d market satisfying the Comp atibility Principle has e quilibria. Pr o of The rationale of the pro of is again the same as in the preceding cases. The aggregate utilit y function U is the sum of t w o functions U s and U c , where the rst is a PM -conca v e function of the v ariables in the set S s B , while the second is a sup ermo dular function of the v ariables in the set S c B . The cells of U are Cartesian pro ducts of t w o t yp es of p olytop es : in teger p olymatroids in the space R S s B and simplexes of the standard triangulation of the unit cub e in the space R S c B . The aggregate cost function C is similarly the sum of a PM -con v ex function of the v ari-ables in the set S s B and of a submo dular function of the v ariables in the set S c B . There, as w ell, the cells of C are Cartesian pro ducts of t w o t yp es of p olytop es : in teger p olymatroids in the space R S s B and simplexes of the standard triangulation of the unit cub e in the space R S c B . The in tersection of t w o suc h cells is an in teger p olytop e. Therefore, the Prop osition 3 obtains for the functions U and C , whence existence of equilibria. One readily sees that the set M of equilibrium matc hings is the cartesian pro duct of an in-b et w een-con v ex subset of 2 S s B and of a sublattice in 2 S c B . The set P of equilibrium prices is the cartesian pro duct of a sublattice of R S s B and of an in-b et w een-con v ex subset of R S c B . The same results hold true in the more general con text, in whic h the set of all go o ds is partionned in to t w o groups -a group of substitutes and a group of complemen ts. In this case, w e also need to mak e use of a mo died Compatibilit y Principle, whic h imp oses that if t w o go o ds are substitutes with resp ect to consumption purp oses, then they should b e so with resp ect to pro duction purp oses, and iden tically for complemen ts. Nev ertheless, w e ha v e no satisfactory explanation ab out wh y the set of go o ds could turn out to b e partionned in suc h a w a y . Last, this principle of compatibilit y justies the recourse to a gross substitution argumen t in Kelso and Cra wford's one-to-man y job matc hing mo del. Indeed. Since w ork ers are allo w ed to w ork for a single rm only , their cost functions are of the GS -t yp e (see Example 4). The principle of compatibilit y requires that utilit y functions of rms (actually their gross pro duct functions) b e of the GS -t yp e as w ell. Similarly , when buy ers need at most one item, then their utilit y functions are of the GS -t yp e (see Example 5). In con trast, observ e that the Compatibilit y Principle is in fact (and not surprisingly so) violated in Examples 1 and 3, where w e fail to exhibit a comp etitiv e equilibrium. 17 References Bevia, C., M. Quinzii, and J. Silv a (1999). Buying sev eral indivisible go o ds. Mathematic al So cial Scienc es 37, 123. Cra wford, V. P . and E. Kno er (1981). Job matc hing with heterogeneous rms and w ork ers. Ec onometric a 49, 437450. Danilo v, V. and G. K oshev o y (2000). Discrete con v exit y and unimo dularit y i. T ec hnical rep ort, Cen tral Institute of Economics and Mathematics. Danilo v, V., G. A. K oshev o y , and A. Sotsk o v (1994). Equilibrium in a mark et of in tellectual go o ds. Mathematic al So cial Scienc es 27, 133144. Danilo v, V. I., G. K oshev o y , and K. Murota (1998). Equilibria in economies with indi-visible go o ds and money . Mimeo, Researc h Institute for Mathematical Sciences, Ky oto Univ ersit y , Ky oto Japan. Danilo v, V. I., G. A. K oshev o y , and C. Lang (2001). Gross substitution, discrete con v exit y and submo dularit y . Jan uary 2001. submitted to the Journal of Discrete Applied Math-ematics, CEMI, Russian A cadem y of Sciences, Mosco w. Danilo v, V. I. and C. Lang (2000). The gross substitution prop ert y in piece-wise ane economies (sv o jstv o v alo v o j zamenimosti dlja kuso c hno-linejn yx ek onomik). Decem-b er 2000. submitted to Mathematic heskie i Ek onomic heskie Meto di, CEMI, Russian A cadem y of Sciences, Mosco w. F rank, A. (1982). An algorithm for submo dular functions on graphs. A nnals of Discr ete Mathematics 16, 97120. F rank, A. and E. T ardös (1988). Generalized p olymatroids and submo dular o ws. Mathe-matic al Pr o gr amming 42, 489563. Gale, D. and L. Shapley (1962). College admissions and the stabilit y of marriage. A meric an Mathematic al Monthly 69, 915. Gul, F. and E. Stacc hetti (1999). W alrasian equilibrium with gross substitutes. Journal of Ec onomic The ory 87, 95124. Kanek o, M. (1982). The cen tral assignmen t game and the assignmen t mark ets. Journal of Mathematic al Ec onomics 10, 14831504. Kelso, A. and V. Cra wford (1982). Job matc hing, coalition formation and gross substitutes. Ec onometric a 50, 14831504. K o opmans, T. and M. J. Bec kmann (1957). Assignmen t problems and the lo cation of eco-nomic activities. Ec onometric a 25, 5376. Lo vász, L. (1983). Submo dular functions and con v exit y . In M. G. A.Bac hem and B. K orte (Eds.), Mathematic al Pr o gr amming: The State of A rt, pp. 235257. Springer-V erlag. Murota, K. (1996). Con v exit y and Steinitz's exc hange prop ert y . A dvanc es in Mathemat-ics 124, 272311. P oltero vic h, V. M. and V. Spiv ak (1982). Mappings with gross substitution in the theory of economic equilibrium (Otobrazhenja s v alo v o j zamenimost'ju v teorii ek onomic hesk ogo ra vno v esija). In Curr ent Pr oblems in Mathematics, pp. 111154. V seso juznij Institut Nauc hno j i T ekhnic hesk o j Informatsii. 18 Quinzii, M. (1984). Core and comp etitiv e equilibria with indivisibilities. International Jour-nal of Game The ory 13, 4160. Roth, A. (1984). Stabilit y and p olarization of in terests in job matc hing. Ec onometric a 52, 4757. Sam uelson, P . (1947). F oundations of Ec onomic A nalysis. N.Y.: A theneum. Shapley , L. S. (1971). Cores of con v ex games. International Journal of Game The ory 1, 1126. Shapley , L. S. and M. Sh ubik (1972). The assignmen t game I: the core. International Journal of Game The ory 1, 111130. T opkis, D. M. (1978). Minimizing a submo dular function on a lattice. Op er ations R e-se ar ch 26, 305321. 19
18152	https://www.rasch.org/rmt/rmt101g.htm	Disattenuating Correlation Coefficients When two sets of measures, {x} and {y}, are correlated, measurement error lowers the correlation coefficient below the level it would have reached had the measures been precise. The reliability (Cronbach Alpha, KR-20, Rasch, etc.) of a set of measures is the proportion of observed variance not due to measurement error, rxx for set {x} and ryy for set {y}. Measurement error can be removed from a correlation coefficient, rxy, to estimate the correlation coefficient disattenuated of measurement error, Rxy, by the formula (Spearman 1904, 1910): Rxy = rxy / sqrt (rxx ryy) For two sets of person scores or measures, use the person "test" reliabilities.For two sets of item p-values or measures, use the item (not the "test") reliabilities. If you have the standard error of each score or measure, then the reliability of the set of scores or measures is:rxx = [(observed variance of the measures) - sum(SE² of each measure)/(count of measures)] / [(observed variance of the measures)] Disattenuated values greater than 1.00 indicate that measurement error is not randomly distributed. Report the disattenuated correlation as 1.0. Muchinsky (1996) summarizes features of the disattenuated correlation coefficient: 1.Disattenuation does not change the quality of the measures or their predictive power. 2.Disattenuated correlations are not directly comparable with uncorrected correlations. 3.Disattenuated correlations are not suited to statistical hypothesis testing. 4.Disattenuation is not a substitute for precise measurement. 5.But, disattenuation tells us whether the correlation between two sets of measures is low because of measurement error or because the two sets are really uncorrelated. Randall E. Schumacker Muchinsky P.M. (1996) The correction for attenuation. Educational & Psychological Measurement 56:1, 63-75. Spearman C. (1904) The proof and measurement of association between two things. American Journal of Psychology, 15, 72-101. Spearman C. (1910) Correlation calculated from faulty data. British Journal of Psychology, 3, 271-295 Zimmerman, D. W., & Williams, R. H. (1997). Properties of the Spearman correction for attenuation for normal and realistic non-normal distributions. Applied Psychological Measurement, 21, 253-270. The reliabilities, rxx and ryy can be computed from tables of measures with standard errors:rxx = ( S.D.(measures for set(x))2 - RMSE(set(x))2 ) / S.D.(measures for set(x))2ryy = ( S.D.(measures for set(y))2 - RMSE(set(y))2 ) / S.D.(measures for set(y))2 \| Table of Disattenuated of Correlation Coefficients \| \| Reliability (Test 1)multiplied byReliability (Test 2) \| Reported Test 1 x Test 2 Correlation Coefficient \| \| .05 \| .10 \| .15 \| .20 \| .25 \| .30 \| .35 \| .40 \| .45 \| .50 \| .55 \| .60 \| .65 \| .70 \| .75 \| .80 \| .85 \| .90 \| .95 \| \| .05 \| .22 \| .45 \| .67 \| .89 - - - - - - - \| .10 \| .16 \| .32 \| .47 \| .63 \| .79 \| .95 - - - - - - \| .15 \| .13 \| .26 \| .39 \| .52 \| .65 \| .77 \| .90 - - - - - - \| \| .20 \| .11 \| .22 \| .34 \| .45 \| .56 \| .67 \| .78 \| .89 - - - - - \| .25 \| .10 \| .20 \| .30 \| .40 \| .50 \| .60 \| .70 \| .80 \| .90 - - - - - \| \| .30 \| .09 \| .18 \| .27 \| .37 \| .46 \| .55 \| .64 \| .73 \| .82 \| .91 - - - - \| .35 \| .08 \| .17 \| .25 \| .34 \| .42 \| .51 \| .59 \| .68 \| .76 \| .85 \| .93 - - - - \| \| .40 \| .08 \| .16 \| .24 \| .32 \| .40 \| .47 \| .55 \| .63 \| .71 \| .79 \| .87 \| .95 - - - \| .45 \| .07 \| .15 \| .22 \| .30 \| .37 \| .45 \| .52 \| .60 \| .67 \| .75 \| .82 \| .89 \| .97 - - - \| \| .50 \| .07 \| .14 \| .21 \| .28 \| .35 \| .42 \| .49 \| .57 \| .64 \| .71 \| .78 \| .85 \| .92 \| .99 - - \| .55 \| .07 \| .13 \| .20 \| .27 \| .34 \| .40 \| .47 \| .54 \| .61 \| .67 \| .74 \| .81 \| .88 \| .94 - - \| .60 \| .06 \| .13 \| .19 \| .26 \| .32 \| .39 \| .45 \| .52 \| .58 \| .65 \| .71 \| .77 \| .84 \| .90 \| .97 - - \| \| .65 \| .06 \| .12 \| .19 \| .25 \| .31 \| .37 \| .43 \| .50 \| .56 \| .62 \| .68 \| .74 \| .81 \| .87 \| .93 \| .99 - \| .70 \| .06 \| .12 \| .18 \| .24 \| .30 \| .36 \| .42 \| .48 \| .54 \| .60 \| .66 \| .72 \| .78 \| .84 \| .90 \| .96 - \| .75 \| .06 \| .12 \| .17 \| .23 \| .29 \| .35 \| .40 \| .46 \| .52 \| .58 \| .64 \| .69 \| .75 \| .81 \| .87 \| .92 \| .98 - \| \| .80 \| .06 \| .11 \| .17 \| .22 \| .28 \| .34 \| .39 \| .45 \| .50 \| .56 \| .61 \| .67 \| .73 \| .78 \| .84 \| .89 \| .95 - \| \| .85 \| .05 \| .11 \| .16 \| .22 \| .27 \| .33 \| .38 \| .43 \| .49 \| .54 \| .60 \| .65 \| .71 \| .76 \| .81 \| .87 \| .92 \| .98 \| .90 \| .05 \| .11 \| .16 \| .21 \| .26 \| .32 \| .37 \| .42 \| .47 \| .53 \| .58 \| .63 \| .69 \| .74 \| .79 \| .84 \| .90 \| .95 \| .95 \| .05 \| .10 \| .15 \| .21 \| .26 \| .31 \| .36 \| .41 \| .46 \| .51 \| .56 \| .62 \| .67 \| .72 \| .77 \| .82 \| .87 \| .92 \| .97 \| "The correlation coefficient corrected for attenuation between two tests x and y is the correlation between their true scores [or true measures]. If, on the basis of a sample of examinees, the corrected coefficient is near unity, the experimenter concludes that the two tests are measuring the same trait." (p. 117) in Joreskog, K.G.(1971) Statistical analysis of sets of congeneric tests, Psychometrica 36, 109-133 Disattenuating correlation coefficients. Schumacker RE, Muchinsky PM. … Rasch Measurement Transactions, 1996, 10:1 p.479 \| \| \| \| \| Rasch Books and Publications \| \| Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang \| Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene \| Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver \| Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. 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18153	https://www.bruegel.org/system/files/wp_attachments/Working-Paper-2021_09.pdf	WORKING PAPER \| ISSUE 09/2021 \| 14 JUNE 2021 Keywords: Collusion, Vertical dierentiation, Nash bargaining JEL Classications: D43, L13, L40, K21 Thanos Athanasopoulos (athanasios.athanasopoulos@dmu.ac.uk) is a lecturer at De Montfort University Burak Dindaroglu (burakdindaroglu@iyte.edu.tr) is an Assistant Professor of Economics at Izmir Institute of Technology Georgios Petropoulos (georgios.petropoulos@bruegel.org) is a Research Fellow at Bruegel and an Associate at MIT Recommended citation: Athanasopoulos, T., B. Dindaroglu and G. Petropoulos (2021) ‘Stability of collusion and quality differentiation: a Nash bargaining approach’ , Working Paper 09/2021, Bruegel THANOS ATHANASOPOULOS, BURAK DINDAROGLU AND GEORGIOS PETROPOULOS How do incentives to collude depend on how asymmetric firms are? In many markets product quality is an important parameter that determines firms’ market strategies. We study collusion in a quality-differentiated duopoly and we adopt a Nash bargaining approach to compute the collusive equilibrium and assess its stability. We derive collusive and deviation strategies as continuous functions of quality asymmetry. We obtain novel and surprising results. Stability of collusion is associated with quality differentiation in a non-monotonic way. For low levels of differentiation, an increase in quality difference makes collusion less stable. The opposite holds for high levels of differentiation. Also, while low quality firms are more likely to leave the cartel for small quality differences, high quality firms determine cartel stability when the quality difference is suffciently high. Our results have implications for empirical research, and antitrust enforcement. STABILITY OF COLLUSION AND QUALITY DIFFERENTIATION: A NASH BARGAINING APPROACH 1 Introduction The relationship between ﬁrms’ asymmetries and collusive behavior has been at the center of attention for antitrust practitioners as well as strategy theorists. In this paper, we investigate cartel stability in quality diﬀerentiated industries. Many detected cartels refer to industries that exhibit market-share asymmetries1, which are often due to vertical diﬀerentiation in product performance, brand image, or reputation. Quality is an important parameter which plays an important role in market decisions.2 In digital and technology markets where production costs are falling3 and big data analytics are extensively used, cost asymmetry becomes less relevant, and quality diﬀerentiation emerges as one of the important parameters for deﬁning market strategies. Two important questions relate to how the degree of quality diﬀerentiation aﬀects the stability of cartels, and which ﬁrm - the innovative leader or a technological laggard - is more likely to abandon the collusive agreement. The scarce literature on the topic (H¨ ackner, 1994; Symeonidis, 1999; Bos and Marini, 2019, and; Ecchia and Lambertini, 1997) has investigated these questions by adopting a static joint proﬁt maximization approach and imposing further ad hoc assumptions so that the solution is implementable (i.e., full market coverage, ﬁxed market shares under collusion and competition). It concludes that quality asymmetry and stability of collusion have a monotonic relationship. Static joint proﬁt maximization is not an appropriate method for computing collusive agree-ments in many settings, especially under asymmetries and in the absence of inter-ﬁrm payments (Bain, 1948; Harrington, 1991). This is because it allocates production towards the most eﬃcient ﬁrm without considering the dynamic stability of collusion, which depends on how this allocation aﬀects the incentives of less eﬃcient ﬁrms to participate. If the resulting allocation does not provide suﬃcient incentives for the latter to collude, it is not implementable. 1Ganslandt et al. (2012) report that in 43 cartel cases investigated by the European Commission between 2002 and 2007, the size of the second-largest ﬁrm was on average 70% of the size of the largest ﬁrm. Davies and Lyons (1996) ﬁnd similar results for earlier years in the EU. 2In the Coty case (see Press Release No. 132/17 Luxembourg, 6 December 2017, Judgment in Case C-230/16 Coty Germany GmbH v. Parfumerie Akzente GmbH), the European Court of Justice concluded that market competition is multidimensional in online commerce and apart from the price component there are other relevant dimensions such as product quality and brand image. 3OECD (2015) observes declining costs along the data value chain which shift the attention from cost considerations to data-induced quality aspects in (online) commerce. The rise of cloud computing also contributed to the fall in production costs in a symmetric way, even if this fall is still diﬃcult to be accurately captured by the oﬃcial statistics (Byrne, Corrado and Sichel, 2018). 1 Given the trade-oﬀbetween static joint proﬁt maximization and dynamic stability when ﬁrms are asymmetric, a more appropriate method to study collusion is the Nash bargaining approach (Nash, 1950). Nash bargaining allows us to focus on the set of implementable subgame perfect Nash equilibrium collusive strategies. When the set of subgame perfect equilibria is large, as it is often the case in repeated game settings, it is natural to consider the ﬁrms to engage in bargaining over the set of potential outcomes (Harrington, 1991). We adopt an inﬁnite time horizon and we consider two ﬁrms with diﬀerent quality levels (which we call the leader with high quality and the follower with low quality) that either compete in prices or collude, without any inter-ﬁrm payments. We analytically derive the Nash bargaining solution that jointly determines the level and the division of collusive proﬁts.4 Nash bargaining allows for the endogenous derivation of implementable collusive prices weighting both static proﬁts and dynamic incentives for collusion, without having to rely on additional assumptions that may be diﬃcult to justify.5 Assessing the stability of the derived collusive agreement requires to specify the optimal pun-ishment mechanism for potential deviators. We show that in our case grim trigger strategies (Friedman, 1971) constitute the optimal punishment: The deviator enjoys a period of a deviation proﬁt followed by (Nash equilibrium) price competition for all the remaining periods. By constructing a speciﬁc parameter that denotes the degree of quality asymmetry we can, in turn, express competitive, collusive, and deviation strategies as functions of quality asymmetry. Our approach allows us to measure stability as a continuous function of quality diﬀerence. We show that this approach brings new insights that unravel how ﬁrms strategically respond to changes in the degree of quality asymmetry. We ﬁnd that the stability of collusion is related to the degree of quality diﬀerentiation in a non-monotonic way. For low levels of quality diﬀerentiation, an increase in quality asymmetry leads to less stable collusive agreements. But, the opposite holds for high degrees of quality diﬀerentiation. 4Alternatively, it is possible to consider the two stage approach (static joint proﬁt maximization, followed by Nash bargaining) used by Schmalensee (1987) to compute the collusive equilibrium under cost asymmetry. The ﬁrst stage is also illustrated in the exercise 6.1 of Tirole (1988). With the help of software packages we can derive the analytical solution of this two-stage problem under quality asymmetry. However, relying on Nash bargaining alone leads to identical results with this two stage approach. Since inter-ﬁrm payments are not allowed, it is the Nash bargaining problem that fully characterizes the collusive equilibrium. Nash bargaining, by deﬁnition, gives a solution at the Pareto frontier. 5Both Symeonidis (1999) and H¨ ackner (1994) state that they resort to static proﬁt maximization as an ad hoc way to compute the collusive equilibrium, as they were unable to derive the Nash Bargaining solution due to the involved computational diﬃculties. 2 We also show that it is the follower (leader) who has higher incentives to deviate from the collusive agreement if the quality diﬀerence between the two ﬁrms’ goods is relatively low (high). As the quality diﬀerence increases, the collusive price of the leader and the follower diverges. As more consumers will prefer the high-quality product, the collusive agreement adjusts the two prices so that the follower will keep its consumer base and will be able to derive suﬃcient proﬁts so that it still wants to participate in the collusive agreement. Despite this price adjustment, collusive proﬁts also diverge with vertical diﬀerentiation. They are reallocated from the follower to the leader. Moreover, both ﬁrms have proﬁtable deviations that attract the rival’s consumer base in the deviation period. These one-period deviations are more proﬁtable (in comparison to the equilibrium collusive proﬁt) if diﬀerentiation is low because consumers are more tempted to switch to the deviator. This eﬀect is stronger for the follower since deviation gives this ﬁrm access to higher-valuation consumers that are served by the leader under collusion. In contrast, for large quality diﬀerences, the follower ﬁnds it more diﬃcult to capture additional consumers in the deviation period since more consumers prefer to consume the high quality good rather than switching, even though the follower’s price is lower. At the same time, the one-period beneﬁts from deviation relative to collusion decrease for the leader, as it serves all high valuation consumers through the collusive mechanism. Deviation to capture the low valuation consumers served by the follower is a less attractive option. In fact, for each ﬁrm, both one-period deviation proﬁts and competitive proﬁts (which determine the strength of the punishment after deviation) get closer to collusive proﬁts with quality asym-metry. Hence, as vertical diﬀerentiation increases we observe two countervailing eﬀects: one-shot deviations become less attractive, but, at the same time, the punishment following the deviation is less severe. For low degrees of diﬀerentiation, it is the latter eﬀect that dominates. Therefore, collusion becomes less stable with quality asymmetry. For large degrees of diﬀerentiation, it is the former eﬀect that dominates, and collusion becomes more stable as the quality diﬀerence rises. To our knowledge, we are the ﬁrst to report the non-monotonic relationship between the stability of collusion and the degree of asymmetry between participating ﬁrms in vertically diﬀerentiated industries. We are also able to identify the ﬁrm that has the greatest incentives to deviate from the collusive agreement as a function of the exact degree of quality diﬀerentiation. Our results show that the eﬃcient ﬁrm does not necessarily have stronger incentives to deviate from a collusive 3 agreement than a less eﬃcient ﬁrm, as it has been found by the literature. The Nash bargaining solution has already been implemented in the literature that deals with cost asymmetry (Schmalensee, 1987, Harrington, 1991, Miklos-Thal, 2011). The main conclusion in these papers is that cost asymmetry hinders collusion and that it is the least eﬃcient ﬁrm that has more incentives to deviate from the collusive equilibrium. But, quality diﬀerentiation, unlike cost asymmetry, directly aﬀects consumer preferences, which we model explicitly. Under cost asymmetry, consumers have to choose among identical products and their product choices are driven only by the price level of each ﬁrm. In our model, quality is an additional parameter that aﬀects consumer choice, hence the collusive equilibrium and deviation strategies.6 We undertake a numerical exercise in which the two ﬁrms do not only diﬀer in product quality but also in the marginal cost of production. This exercise suggests that our results on the non-monotonicity of cartel stability are retained with the addition of diﬀerent marginal costs, while all our results remain qualitatively the same for small cost diﬀerences. We also consider the case where direct monetary transfers are feasible as an extension.7 For suﬃciently high degrees of diﬀerentiation, we ﬁnd that collusion is more stable when inter-ﬁrm payments are not feasible. This contradicts the conventional wisdom that collusion with side-payments leads to more stable collusive structures. While side-payments require explicit coordination, our results when inter-ﬁrm payments are absent can be the outcome of either tacit or explicit coordination.8 We use the terms cartel and collusion interchangeably throughout the paper. The rest of the paper is organized as follows. We introduce our model in Section 2, and study the competitive equilibrium in Section 3. Section 4 characterizes the collusive equilibrium and states our main results. We study a model incorporating diﬀerent marginal costs for diﬀerent quality levels in Section 5. Section 6 studies the implications of the availability of side payments, 6Interestingly, we show that when ﬁrms compete in prices, quality asymmetry in our framework generates diﬀerent optimal punishment mechanism than the one we expect under price competition with cost asymmetry (Mikl os-Thal, 2011). 7There is evidence that some cartels have used side payments. For example, as Pesendorfer (2000) reports, a bid-rigging scheme in Florida used side payments in the provision of school milk and dairies to compensate cartel members for refraining from bidding. Asker (2010) reports another cartel formed in stamp auctions in New York, where side payments were used for a similar reason. Probably, a more prominent example is the case of vitamin cartels (Igami and Sugaya, 2018): The heads of the vitamin divisions of big pharmaceutical companies agreed to freeze market shares at pre-determined levels and split sales according to these quotas. Side payments were arranged in the form of compensating sales from under-quota members to those who exceeded their quotas. 8Melkonyan et al. (2018) illustrate how ﬁrms can solve a (virtual) bargaining problem and collude tacitly. 4 and Section 7 concludes. 2 The Model There are two ﬁrms, the leader (L) and the follower (F), interacting repeatedly in the same market over an inﬁnite, discrete-time horizon. The stage game models a vertically diﬀerentiated industry setting in the tradition of Shaked and Sutton (1982). Each ﬁrm supplies a single product whose quality is given by qi, i = L, F, with qL ≥qF . Our primary interest is in cases with qL > qF , while we brieﬂy study the symmetric case with qL = qF ≡q0. We assume quality levels to be exogenously given. Firms simultaneously choose prices pL and pF to maximize the discounted sum of period proﬁts ΠL and ΠF . The marginal costs of production for all products are normalized to zero9, and ﬁrms have a common discount factor δ ∈(0, 1). We denote the degree of diﬀerentiation by k ≡qL/qF . There is a continuum of heterogeneous consumers who diﬀer in their valuations θ for product quality, where θ is uniformly distributed in the interval [0,1]. Each consumer has unit demand and obtains net utility U(θ) = ⎧ ⎨ ⎩ θqi −pi when buying from ﬁrm i 0 when not buying. (1) Customers observe qualities and prices before making their purchasing decisions. 3 Competition We begin by analyzing competitive prices and proﬁts. If ﬁrms are symmetric (qL = qF ≡q0), Bertrand competition leads to marginal cost pricing, so that pL = pF ≡p∗ 0 = 0 and ΠL = ΠF ≡ Π∗ 0 = 0. It is straightforward to characterize the equilibrium of the stage game under asymmetry. Given qualities qL > qF and prices pL > pF , there is an indiﬀerent consumer situated at ˆ θ, given by 9We consider positive and distinct marginal costs in Section 5. 5 ˆ θqL −pL = ˆ θqF −pF , so that ˆ θ = pL −pF qL −qF . (2) Consumers with θ > ˆ θ buy from L as long as θ > pL/qL ≡θL, and those with θ < ˆ θ buy from F as long as θ > pF /qF ≡θF . Hence, the demand for L is given by max{0, 1 −max{θL, ˆ θ}}, and the demand for F is max{min{ˆ θ, 1}−θF , 0}. Assuming that prices satisfy 1 > ˆ θ ≥θL ≥θF > 0, proﬁts can be written as ΠL = pL(1 −ˆ θ) and ΠF = pF (ˆ θ −θF ). Firms determine prices by maximizing proﬁts, which leads to best response functions pbr L = pF 2 + qF (k −1) 2 , pbr F = pL 2k . (3) Solving the best response functions jointly, we get Nash equilibrium prices p∗ L = 2qL(qL −qF ) 4qL −qF , p∗ F = qF (qL −qF ) 4qL −qF , (4) and proﬁts Π∗ L = 4q2 L(qL −qF ) (4qL −qF )2 , Π∗ F = qLqF (qL −qF ) (4qL −qF )2 . (5) Note that the indiﬀerent consumer in equilibrium is given by θ ˆ∗ = (2qL − qF )/(4qL − qF ) and that the assumption 1 > θ ˆ∗ > θ∗ L > θ∗ F > 0 is satisﬁed in equilibrium. 4 Collusion We consider collusion in prices when side payments between ﬁrms are prohibited or infeasible. By applying the Folk theorem, any collusive outcome is sustainable if there is an inﬁnite time horizon, the discount factor is suﬃciently high, and there is an eﬃcient punishment mechanism for deviators. The collusive agreement typically cannot be enforced through legal instruments. Hence, we need to rely on the concept of subgame perfect Nash equilibrium in an inﬁnitely repeated game setting where there is an underlying one period game with one or more Nash equilibria. Firms collude until one of them deviates, after which a grim trigger punishment phase occurs: In each period of 6 the punishment phase, ﬁrms earn their competitive proﬁts. The sustainability of collusion requires Πc i ≥(1 −δ)Πd i + δΠ∗ i , (6) for each i = L, F, where the superscript () denotes the punishment phase, (c) denotes collusion, and (d) denotes deviation. Condition (6) implies that ﬁrm i does not deviate as long as δ ≥ˆ δi = Πd i −Πc i Πd i −Π∗ i , (7) where ˆ δi is a ﬁrm-speciﬁc threshold discount factor measuring the incentives of ﬁrm i = L, F to deviate from the collusive agreement. Hence, the stability of collusion is determined by the discount factor ˆ δ ≡max{ˆ δF , ˆ δL}. Proposition 1 shows that in our framework the grim trigger punishment (Friedman, 1971) is the optimal punishment mechanism in the sense of Abreu (1986, 1988), and therefore dominates any form of stick-and-carrot punishment. Proposition 1. The optimal punishment mechanism is the grim trigger punishment. Following the deviation, ﬁrms revert to the static Nash equilibrium for all the subsequent periods. Proof. We deﬁne as the optimal mechanism, the one that i) minimizes the expected payoﬀof the deviator; 2) it is credible such that the payoﬀof the non-deviator in the punishment phase is suﬃciently high to implement that punishment. It suﬃces to show that there cannot be a more severe punishment for the deviator which is at the same time credible for the non-deviator. Let us assume that the leader deviates. The payoﬀof the leader and the follower under the grim trigger strategies, in the punishment phase, will be: Π∗ L 1−δ and Π∗ F 1−δ, respectively. Following Abreu (1986) a natural candidate mechanism will be the one which punishes harshly the deviator for the ﬁrst τ periods of the punishment phase (the stick). Given expression (3), the most harsh punishment for the leader will be the follower to set pF = 0 for the ﬁrst τ periods. Then, for each period t ≤τ, the leader gets payoﬀqF (k−1) 4 which is smaller than Π∗ L. For t > τ, let the follower charge price po F > 0. This mechanism can be optimal only if the following two 8  7 conditions are satisﬁed: qF (k −1) 4 (1 −δτ) + δτΠL(pbr L , po F ) < Π∗ L, (8) and δτΠF (pbr L , po F ) ≥Π∗ F . (9) It is easy to see that there is no price po F that satisﬁes both conditions simultaneously for any value of τ. From (3) we know that ΠL(pL br, po F ) is monotonically increasing in po F . Hence, the condition (8) takes the minimum value for the lowest possible po F for which condition (9) is satisﬁed with equality. At that minimum value, (8) is still violated, since the leader has higher payoﬀ than in the grim trigger strategy. The proof follows analogous steps in case we consider less harsh punishments in which pF ∈ (0, p∗ F ) in the ﬁrst τ periods. Same logic and results apply in the case the follower is the deviator. 4.1 Symmetric benchmark Collusion under symmetry (qL = qF ≡ q0) is straightforward to characterize. Setting prices to p0 leads to demand 1 − p0/q0 and proﬁts Πi = p0(1 − p0/q0). Proﬁt maximization gives pc 0 = q0/2 and total proﬁts Πc 0 = q0/4, which are shared equally to give individual proﬁts q0/8 to each ﬁrm. The optimal deviation strategy is to slightly undercut the opponent and obtain monopoly proﬁts Π0 d = q0/4 for one period. Then, the joint proﬁt maximization gives the threshold discount factor δ ˆ0 = 1/2. 4.2 Collusive equilibrium In asymmetric environments, collusion requires specifying an agreement as to how collusive proﬁts will be allocated among players. We consider an agreement whereby the joint cartel decision emerges from bilateral bargaining, where the disagreement point is the competitive proﬁt allocation. This 8 Figure 1: Collusive prices of leader and follower and monopoly price of leader as functions of the quality diﬀerentiation (k). leads to the following Nash bargaining sharing rule: max pc L,pc F {(Πc L −Π∗ L)(Πc F −Π∗ F )} (10) s.t. Πc L > Π∗ L, Πc F > Π∗ F , where Πc L = pc L 1 −pc L −pc F qL −qF , Πc F = pc F pc L −pc F qL −qF −pc F qF , (11) and pc L and pc F denote equilibrium collusive prices. The bargaining problem in (10) leads to analytical solutions for collusive prices pc L and pc F .10 The marginal consumers ˆ θc, θc L and θc F that determine demand functions are calculated using collusive prices, and satisfy ˆ θc > θc L > θc F . Figure 1 depicts pc L qF and pc F qF as well as pm L qF , where pm L is the monopoly price of the leader.11 The leader’s collusive price is increasing in quality diﬀerentiation k while the respective price for the follower is decreasing. Thus, as the quality advantage increases, so does the equilibrium price diﬀerential. A higher price diﬀerential allows the follower to keep its base of consumers despite the increased quality asymmetry. As a result, the follower retains its collusive proﬁt at a level that makes the collusive equilibrium sustainable for a range of the common discount factor δ. An 10Expressions are too lengthy to be reported here. The supplementary Mathematica ﬁle includes all calculations. 11Price pm L is computed from the ﬁrst order condition of the maximization problem maxpL{pL(1 − pL qF k)}. 9 interesting feature of the collusive equilibrium is that the leader charges a price that exceeds its monopoly price. The leader is willing to forgo a part of the monopoly proﬁt by charging a higher price so that the follower has suﬃcient incentives to participate in the collusive equilibrium without deviating. The diﬀerence between the leader’s collusive and monopoly price is increasing in k. 4.3 Deviation strategies The optimal deviation strategy for each ﬁrm i = L, F is to select the price that maximizes its proﬁts given the rival ﬁrm’s collusive price pc j, where j = L, F and j ̸= i. The deviator’s best response to the other ﬁrm playing its collusive equilibrium strategy could potentially be an interior price choice - coming from the ﬁrst-order conditions of its proﬁt maximization problem - or a price that could force the competitor to have zero demand. This leads to: pd L = ⎧ ⎪ ⎨ ⎪ ⎩ pc F 2 + qF (k−1) 2 if pc F ≤qF (k−1) 2k−1 , kpc F if qF (k−1) 2k−1 < pc F , (12) for the leader, and pd F = ⎧ ⎪ ⎨ ⎪ ⎩ pc L 2k if pc L ≤qF 2k(k−1) 2k−1 , pc L −qF (k −1) if qF 2k(k−1) 2k−1 < pc L, (13) for the follower.12 When the deviation does not violate the constraint θF ≤ˆ θ ≤1, it is best for a ﬁrm to deviate according to the best response functions in (3), by maximizing own proﬁts holding rival’s price at its collusive level. These are stated by the ﬁrst interval in the deviation functions above. However, price levels may be such that the deviating ﬁrm can push its rival to have zero demand in the deviation period. For the follower, this occurs if the best response function leads ˆ θd, the consumer that is indiﬀerent between the two products, to be equal to θc F , essentially leaving the leader with zero demand. In this price range, the follower undertakes a form of limit pricing to keep the leader’s demand at zero and serve all consumers with θ ∈[θF , 1] in the deviation period. A similar, but slightly diﬀerent strategy exists for the leader, whose limit pricing deviation leads to the binding 12Note that equilibrium collusive prices satisfy pc F ≤ qF 2 and pc L ≤qF 2k−1 2k−1, ∀k > 1. We present the deviation strategies that may arise given the collusive equilibrium strategies. If we also include deviations oﬀequilibrium paths in the analysis, there is a third deviation strategy for the leader (when follower’s collusive price is greater than qF 2 ) and the follower (when leader’s collusive price greater than 2k−1 2k−1) for which the deviator charges its monopoly price. 10 (a) (b) Figure 2: Collusive, competitive and deviation proﬁts for (a) the follower and (b) the leader as a function of k. constraint ˆ θ = θF , which eﬀectively keeps F out of the market in the deviation period. 4.4 Stability of collusion The collusive, competitive, and deviation proﬁts for the leader and the follower are depicted in Figure 2. Collusive and one-shot deviation proﬁts are monotonically increasing (decreasing) for the leader (follower) as quality diﬀerentiation (k) increases, while competitive proﬁts of both ﬁrms are increasing functions of k. Note that when the degree of diﬀerentiation is low, both ﬁrms have proﬁtable one-shot deviations that allow them to serve the other’s customers during the deviation period. However, the resulting incentives for one-shot deviations diminish as diﬀerentiation is larger since stealing the rival’s consumers becomes more diﬃcult and costly as it requires a larger price cut. In contrast, competitive proﬁts become more attractive for the colluding ﬁrms as k increases. The threshold discount factors, which we call ˆ δc L, ˆ δc F , are calculated using (7). These are depicted in Figure 3 as functions of k. The stability of collusion is determined by ˆ δc = max{ˆ δc L, ˆ δc F }. The following two propositions summarize our main results: Proposition 2. For each of the two ﬁrms, the relationship between the threshold discount factor and quality diﬀerentiation (k) is an inverted-U. The follower has stronger incentives to deviate from the collusive agreement if 1 < k < 1.869, while the leader has stronger incentives to deviate if k > 1.869. The incentives of both the leader and the follower to deviate from the collusive equilibrium follow 11 Figure 3: The critical discount factors of the leader (blue) and the follower (orange) as functions of quality diﬀerentiation (k). an inverted-U pattern with k. The peak occurs at a lower value of k for the follower. For each ﬁrm, as k increases, while the one-period deviation becomes a relatively less attractive option, the punishment (competitive) payoﬀ becomes relatively more attractive. For low-quality diﬀerences, it is the latter eﬀect that dominates and collusion becomes less stable with k. For high-quality diﬀerences, it is the former eﬀect that dominates and hence collusion becomes more stable. Indeed, the diﬀerence between the one-period deviation proﬁt and static collusive proﬁt declines at a lower (higher) rate than the diﬀerence between static proﬁts under collusion and competition for both ﬁrms when quality diﬀerentiation is low (high). Furthermore, the ﬁrm that determines cartel stability depends on the degree of quality diﬀer-entiation. More precisely, for lower values of the quality diﬀerence, the follower is more tempted to deviate from the collusive agreement compared to its counterpart. This is because, for low degrees of diﬀerentiation, deviation allows the low-quality ﬁrm to steal high-valuation consumers that are served by the leader under collusion. As the degree of diﬀerentiation rises, however, the follower ﬁnds it less attractive to deviate, as the leader’s high-valuation customers optimally purchase the high-quality product even if the follower tries to lure them away with a lower price. This dimin-ishing tendency of the follower to leave the cartel, as the degree of diﬀerentiation rises, makes the leader more prone to deviate since its punishment payoﬀ is higher. Looking at the overall picture, the non-monotonic relationship between ﬁrm’s incentives to deviate from the cartel and the degree of vertical diﬀerentiation is naturally passed on to cartel 12 stability. So, the relationship between cartel stability and k is non-monotonic as well: Proposition 3. There exist cutoﬀs k = 1.426, ˜ k = 1.829, ˆ k = 2.65, such that the cartel becomes (a) more stable with increased quality diﬀerentiation when k < k < ˜ k or k > ˆ k, (b) less stable with vertical diﬀerentiation when k < k or ˜ k < k < ˆ k. These results deviate from the literature in vertically diﬀerentiated industries, according to which i) there is a monotonic relationship between the quality asymmetry and collusion (i.e., H¨ ackner, 1994; Symeonidis, 1999; Ecchia and Lambertini, 1997), and ii) a single ﬁrm has uni-formly higher incentives to abandon the cartel: either the high-quality ﬁrm (H¨ ackner, 1994) or the technological laggard (Symeonidis, 1999; Bos and Marini, 2019). Adopting a Nash bargaining approach that allows to determine endogenously the collusive equilibrium instead of computing this equilibrium in an ad hoc way provides new insights on ﬁrms’ equilibrium strategies that have direct implications for incentives to collude. 5 Diﬀerent marginal costs In this section, we incorporate non-zero marginal costs of production to our baseline model presented in the previous sections. We characterize the collusive equilibrium and its stability when the two ﬁrms have diﬀerent marginal costs, denoted cL (leader) and cF (follower). It is natural to consider marginal costs of production to increase with product quality, hence to assume cF ≤cL. 13 In the presence of non-zero marginal costs, the following two constraints need to be satisﬁed for competitive proﬁts to be non-negative: cL ≤qF (k −1) 2k 2k −1 + cF k 2k −1, cF ≤qF k −1 2k −1 + cL 2k −1. (14) We study the eﬀects of various cost conﬁgurations for each k under the constraints stated above. To illustrate the main results from this analysis, we present results on the stability of collusion with the following simpliﬁcations: we normalize cF to zero, qF to one, and use the cost speciﬁcation cL = c(k −1) for the leader where c is a given constant. This allows the marginal cost of the leader 13The collusive equilibrium under diﬀerent marginal costs is obtained numerically. Details of the model are not presented for space considerations. A detailed description of the model as well as the Octave/Matlab ﬁles that are used to generate solutions are available as supplementary ﬁles. 13 (a) (b) Figure 4: Threshold discount factors of (a) the follower and (b) the leader as functions of k and c. to increase with product quality and allows us to vary parameter c while respecting (14). The eﬀect of increasing marginal cost diﬀerences is shown in Figure 4, which depicts threshold deviations for both ﬁrms as functions of k and c. The results reveal that the inverted-U relationship between quality diﬀerentiation and cartel stability is retained for each ﬁrm as the cost diﬀerence increases. For small cost diﬀerences, the relative incentives to deviate for the leader and the follower are similar to our ﬁnding stated in Proposition 2: the follower has higher incentives to deviate for small k, while the leader has higher incentives to deviate for large k values. This ranking of incentives is overturned for larger cost diﬀerences. As in our baseline model, in-dividual incentives and the overall stability of collusion is determined by the interplay between two mechanisms; one relating to the desirability of deviation, and the other the threat of punishment. As cL increases, so does the leader’s collusive price, pc L. As a result, θ ˆc increases and approaches one, hence the leader serves an increasingly smaller fraction of the highest-valuation consumers. Accordingly, deviation gives this ﬁrm access to a larger fraction of additional high-valuation con-sumers below its demand threshold, which renders deviation more attractive. At the same time, the relative value of deviation for the follower is diminished, since θ ˆc is higher and the leader’s con-sumers are more diﬃcult to divert from consuming its product. As a result, the incentive structure that led to the threshold functions in Figure 3 is reversed. This deviation eﬀect is stronger for small k values. On the other hand, an increase in cL reduces (increases) the competitive proﬁts of 14 the leader (follower), hence makes the deterrence eﬀect stronger (weaker) for this ﬁrm. Hence, the leader has weaker, while the follower has stronger incentives to deviate compared to our baseline model. This eﬀect is more dominant for larger k values. Figure 4 exhibits the outcome of the combined eﬀect: the reversal in incentives, as well as the eﬀect of increasing cost diﬀerences for a given value of the quality diﬀerential. The non-monotonic relationship between cartel stability and diﬀerentiation stated in Proposi-tion 3 is qualitatively retained for each value of the marginal cost diﬀerence. 6 Collusion with side payments We now consider the case in which intra-ﬁrm payments are feasible. Under such conditions, the colluding ﬁrms can attempt to maximize joint period proﬁts and implement this solution using side payments. Unlike our baseline model, closed-form solutions for the collusive equilibrium and threshold discount factors are easy to obtain. The joint proﬁt maximization problem of the cartel can be written as Πsp = max pF ,pL Πsp L + Πsp F , where Πsp L = psp L 1 −psp L −psp F qL−qF and Πsp F = psp F psp L −psp F qL−qF −psp F qF . The ﬁrst order conditions for pL and pF give psp F = qF 2 , psp L = psp F + qL −qF 2 = qL 2 , (15) which also imply ˆ θsp = θsp L = θsp F = 1/2, and give total cartel proﬁts equal to Πsp = qL/4. All sales are made by the leader. As a consequence, this outcome can only be implemented using side payments. We again consider a Nash bargaining rule for the sharing of total collusive proﬁts: max Πsp L ,Πsp F (Πsp L −Π∗ L)(Πsp F −Π∗ F ) s.t. Πsp L + Πsp F = qL 4 , (16) 15 which leads to equilibrium collusive proﬁts Πsp L = 8q2 L −5qLqF 8(4qL −qF ) , Πsp F = 3qLqF 8(4qL −qF ). (17) Note that the collusive participation constraints are satisﬁed for all qL and qF with qL > qF > 0. To implement the strategy, the leader makes all sales and pays an amount equal to Πsp F in (17) to the follower in each period. It is easy to see that the leader’s optimal deviation from the collusive agreement is to refuse to make the side payment to the follower, which gives the deviation proﬁt Πd,sp L = Πsp = qL/4. The optimal deviation strategy of the follower is derived in an analogous way to the previous section. This leads to the two-part deviation proﬁts Πd,sp F = ⎧ ⎪ ⎨ ⎪ ⎩ qL(2qF −qL) 4qF if 1 < k < 3 2, qLqF 16(qL−qF ) if 3 2 ≤k. (18) Note that Πd,sp L > Πsp L for all qL > qF > 0. However, Πd,sp F > Πsp F only if k < 5 2. For higher diﬀerentiation with k ≥5 2, the follower never deviates from the collusive agreement. The critical discount factors for both ﬁrms can then be computed using (7) as ˆ δsp L = 3 4 1 − 3 1 + 8k (19) and ˆ δsp F = ⎧ ⎪ ⎨ ⎪ ⎩ 1 − 4k+5 12−42k+80k2−32k3 if 1 < k < 3 2, (4k−1)(5−2k) 3(8k−5) if 3 2 ≤k, (20) respectively. Note that ˆ δsp L is strictly increasing in k; the leader ﬁnds deviation to be more attractive as diﬀerentiation between the two ﬁrms increases. However, ˆ δsp F exhibits an inverted-U relationship with k. There is a critical value, kcr F (numerical value 1.234) that maximizes the follower’s incentives to deviate. Let ˆ δsp = max{ˆ δsp F , ˆ δsp L }. The follower has stronger incentives to deviate from the collusive agreement if 1 < k < 5 4 (ˆ δsp = ˆ δsp F ), while the leader is more likely to deviate from the cartel arrangement if 5 4 < k (ˆ δsp = ˆ δsp L ). 16 Figure 5: Critical discount factors for the stability of collusion with and without side payments as a function of quality diﬀerentiation (k). Comparing the stability of collusion, δ ˆc of our baseline model above with the side payments case, δ ˆsp (Figure 5) we see that: Proposition 4. There is a cutoﬀ value for quality asymmetry, k∗ = 1.708, above which collusion is more stable in the absence of side payments. This indicates that side payments can lead to the destabilization of the collusive agreement for high levels of quality diﬀerentiation. This is because the leader has stronger incentives to deviate and capture monopoly proﬁts by not providing the side payment to the follower. Higher degrees of diﬀerentiation also guarantee larger competitive proﬁts in the punishment phase that will follow. In contrast, when side payments are not feasible, the leader’s deviation can never be as proﬁtable as in the side payments case. This is particularly true for high degrees of quality diﬀerentiation. 7 Conclusion In this paper, we investigate cartel stability in a quality diﬀerentiated duopoly. We deviate from ad hoc assumptions that have been commonly used in the literature by relying instead on the Nash bargaining approach. We ﬁnd that the relationship between cartel stability and quality diﬀerentiation is non-monotonic. In addition, a low quality (high quality) ﬁrm has higher incentives to deviate from the collusive 17 agreement for low (high) degrees of diﬀerentiation between competitors. We also ﬁnd that side payments can render collusion more stable only if product qualities in the industry are suﬃciently close to one another. Understanding the incentives to collude is important for organizing deterrence mechanisms that promote competition. In this respect, our model predictions shed light on the incentives of market leaders and followers to collude, in cases the quality of products and services is an important strategic variable (as in digital ecosystems and technology markets). In many instances, deterrence of collusive agreements relies on identifying potential whistle-blowers within the ﬁrms that only have weak incentives to collude. Our approach and results have important implications for future research. The literature on the relationship between collusion and innovation largely deals with cost-reducing innovation. Our analysis paves the way for investigating the relationship between collusion and innovation when innovation improves a product in technological performance or in use-value. Extending our model to study the relationship between R&D competition and collusion on a learning curve (e.g., by adding a quality investment step in each ﬁrm’s decision problem per period) is part of our current research eﬀorts. The computational diﬃculties introduced by the general market setting restricted our eﬀorts to the case of a duopoly. The generalization of our model to an oligopoly with an arbitrary number of ﬁrms is also part of our ongoing research. 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18154	https://en.wikipedia.org/wiki/Complement_component_5a	Jump to content Search Contents (Top) 1 Structure 2 Functions 3 Binding process 4 Diseases 5 References 6 External links Complement component 5a العربية Galego Српски / srpski Srpskohrvatski / српскохрватски Українська Edit links Article Talk Read Edit View history Tools Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Download as PDF Printable version In other projects Wikidata item Appearance From Wikipedia, the free encyclopedia Protein fragment \| complement component 5 \| \| Schematic representation of three-dimensional structure of complement 5a \| \| Identifiers \| \| Symbol \| C5 \| \| NCBI gene \| 727 \| \| HGNC \| 1331 \| \| OMIM \| 120900 \| \| RefSeq \| NM_001735 \| \| UniProt \| P01031 \| \| Other data \| \| Locus \| Chr. 9 q34.1 \| \| \| Search for \| \| Structures \| Swiss-model \| \| Domains \| InterPro \| \| C5a is a protein fragment released from cleavage of complement component C5 by protease C5-convertase into C5a and C5b fragments. C5b is important in late events of the complement cascade, an orderly series of reactions which coordinates several basic defense mechanisms, including formation of the membrane attack complex (MAC), one of the most basic weapons of the innate immune system, formed as an automatic response to intrusions from foreign particles and microbial invaders. It essentially pokes microscopic pinholes in these foreign objects, causing loss of water and sometimes death. C5a, the other cleavage product of C5, acts as a highly inflammatory peptide, encouraging complement activation, formation of the MAC, attraction of innate immune cells, and histamine release involved in allergic responses. The origin of C5 is in the hepatocyte, but its synthesis can also be found in macrophages, where it may cause local increase of C5a. C5a is a chemotactic agent and an anaphylatoxin; it is essential in the innate immunity but it is also linked with the adaptive immunity. The increased production of C5a is connected with a number of inflammatory diseases. Structure [edit] Human polypeptide C5a contains 74 amino acids and has 11kDa. NMR spectroscopy proved that the molecule is composed of four helices and connected by peptide loops with three disulphide bonds between helix IV and II, III. There is a short 1.5 turn helix on N-terminus but all agonist activity take place in the C-terminus. C5a is rapidly metabolised by a serum enzyme carboxypeptidase B to a 72 amino acid form C5a des-Arg without C terminal arginine. Functions [edit] C5a is an anaphylatoxin, causing increased expression of adhesion molecules on endothelium, contraction of smooth muscle, and increased vascular permeability. C5a des-Arg is a much less potent anaphylatoxin. Both C5a and C5a des-Arg can trigger mast cell degranulation, releasing proinflammatory molecules histamine and TNF-α. C5a is also an effective chemoattractant, initiating accumulation of complement and phagocytic cells at sites of infection or recruitment of antigen-presenting cells to lymph nodes. C5a plays a key role in increasing migration and adherence of neutrophils and monocytes to vessel walls. White blood cells are activated by upregulation of integrin avidity, the lipoxygenase pathway and arachidonic acid metabolism. C5a also modulates the balance between activating versus inhibitory IgG Fc receptors on leukocytes, thereby enhancing the autoimmune response. Binding process [edit] C5a interact with receptor protein C5a Receptor 1 (C5aR1) on the surface of target cells such as macrophages, neutrophils and endothelial cells. C5aR1 is a member of the G-protein-coupled receptor superfamily of proteins, predicted to have seven transmembrane helical domains of largely hydrophobic amino acid residues, forming three intra- and three extra-cellular loops, with an extracellular N-terminus and an intracellular C-terminus. C5a binding to the receptor is a two-stage process: an interaction between basic residues in the helical core of C5a and acidic residues in the extracellular N-terminal domain allows the C-terminus of C5a to bind to residues in the receptor transmembrane domains. The latter interaction leads to receptor activation, and the transduction of the ligand binding signal across the cell plasma membrane to the cytoplasmic G protein Gi type GNAI2. Sensitivity of C5aR1 to C5a stimulation is enhanced by lipopolysaccharides exposure. C5a, acting via C5aR1, is shown to differentially modulate lipopolysaccharides-induced inflammatory responses in primary human monocytes versus macrophages, yet this is not due to C5aR1 upregulation. C5L2 is another C5a receptor that is thought to regulate the C5a-C5aR1 effects. There is apparently contradictory evidence showing decoy receptor activity conferring anti-inflammatory properties and also signalling activity conferring pro-inflammatory properties. Diseases [edit] C5a is a powerful inflammatory mediator, and seems to be a key factor in the development of pathology of many inflammatory diseases involving the complement system such as sepsis, rheumatoid arthritis, inflammatory bowel disease, systemic lupus erythemotosis, psoriasis. The inhibitor of C5a that can block its effects would be helpful in medical applications. Another candidate is PMX53 or PMX205 that is highly specific for CD88 and effectively reduces inflammatory response. C5a has been identified as a key mediator of neutrophil dysfunction in sepsis, with antibody blockade of C5a improving outcomes in experimental models. This has also been shown in humans, with C5a-mediated neutrophil dysfunction predicting subsequent nosocomial infection and death from sepsis. Recent data demonstrates that C5a not only impairs phagocytosis by neutrophils but also impairs phagosomal maturation, inducing a marked alteration in the neutrophil phosphoproteomic response to bacterial targets. C5a binding to C5aR1 and C5aR2 (C5L2) mediates the formation of neutrophil extracellular traps and release of cytotoxic histones to the extracellular space, which is believed to act as a pathogenetic process of acute respiratory distress syndrome (ARDS) and promote tumor growth and metastasis. References [edit] ^ a b c Manthey HD, Woodruff TM, Taylor SM, Monk PN (November 2009). "Complement component 5a (C5a)". The International Journal of Biochemistry & Cell Biology. 41 (11): 2114–2117. doi:10.1016/j.biocel.2009.04.005. PMID 19464229. ^ Klos A, Wende E, Wareham KJ, Monk PN (January 2013). "International Union of Basic and Clinical Pharmacology. [corrected]. LXXXVII. Complement peptide C5a, C4a, and C3a receptors". Pharmacological Reviews. 65 (1): 500–543. doi:10.1124/pr.111.005223. PMID 23383423. ^ Ward PA (February 2004). "The dark side of C5a in sepsis". Nature Reviews. Immunology. 4 (2): 133–142. doi:10.1038/nri1269. PMID 15040586. S2CID 22630287. ^ Seow V, Lim J, Cotterell AJ, Yau MK, Xu W, Lohman RJ, et al. (April 2016). "Receptor residence time trumps drug-likeness and oral bioavailability in determining efficacy of complement C5a antagonists". Scientific Reports. 6 (1) 24575. Bibcode:2016NatSR...624575S. doi:10.1038/srep24575. PMC 4837355. PMID 27094554. ^ Gerard NP, Gerard C (February 1991). "The chemotactic receptor for human C5a anaphylatoxin". Nature. 349 (6310): 614–617. Bibcode:1991Natur.349..614G. doi:10.1038/349614a0. PMID 1847994. S2CID 4338594. ^ Fujita T (14 October 1999). Boulay F (ed.). "PROW and IWHLDA present the GUIDE on: CD88". Protein Reviews on the Web. Archived from the original on 2008-07-24. ^ Seow V, Lim J, Iyer A, Suen JY, Ariffin JK, Hohenhaus DM, et al. (October 2013). "Inflammatory responses induced by lipopolysaccharide are amplified in primary human monocytes but suppressed in macrophages by complement protein C5a". Journal of Immunology. 191 (8): 4308–4316. doi:10.4049/jimmunol.1301355. PMID 24043889. S2CID 207429042. ^ Raby AC, Holst B, Davies J, Colmont C, Laumonnier Y, Coles B, et al. (September 2011). "TLR activation enhances C5a-induced pro-inflammatory responses by negatively modulating the second C5a receptor, C5L2". European Journal of Immunology. 41 (9): 2741–2752. doi:10.1002/eji.201041350. PMC 3638321. PMID 21630250. ^ Klos A, Wende E, Wareham KJ, Monk PN (January 2013). "International Union of Basic and Clinical Pharmacology. [corrected]. LXXXVII. Complement peptide C5a, C4a, and C3a receptors". Pharmacological Reviews. 65 (1): 500–543. doi:10.1124/pr.111.005223. PMID 23383423. ^ Woodruff TM, Crane JW, Proctor LM, Buller KM, Shek AB, de Vos K, et al. (July 2006). "Therapeutic activity of C5a receptor antagonists in a rat model of neurodegeneration". FASEB Journal. 20 (9): 1407–1417. doi:10.1096/fj.05-5814com. PMID 16816116. S2CID 9206660. ^ Jain U, Woodruff TM, Stadnyk AW (January 2013). "The C5a receptor antagonist PMX205 ameliorates experimentally induced colitis associated with increased IL-4 and IL-10". British Journal of Pharmacology. 168 (2): 488–501. doi:10.1111/j.1476-5381.2012.02183.x. PMC 3572573. PMID 22924972. ^ Huber-Lang MS, Younkin EM, Sarma JV, McGuire SR, Lu KT, Guo RF, et al. (September 2002). "Complement-induced impairment of innate immunity during sepsis". Journal of Immunology. 169 (6): 3223–3231. doi:10.4049/jimmunol.169.6.3223. PMID 12218141. ^ Conway Morris A, Kefala K, Wilkinson TS, Dhaliwal K, Farrell L, Walsh T, et al. (July 2009). "C5a mediates peripheral blood neutrophil dysfunction in critically ill patients". American Journal of Respiratory and Critical Care Medicine. 180 (1): 19–28. doi:10.1164/rccm.200812-1928OC. PMC 2948533. PMID 19324972. ^ Morris AC, Brittan M, Wilkinson TS, McAuley DF, Antonelli J, McCulloch C, et al. (May 2011). "C5a-mediated neutrophil dysfunction is RhoA-dependent and predicts infection in critically ill patients". Blood. 117 (19): 5178–5188. doi:10.1182/blood-2010-08-304667. PMID 21292772. ^ Conway Morris A, Datta D, Shankar-Hari M, Stephen J, Weir CJ, Rennie J, et al. (May 2018). "Cell-surface signatures of immune dysfunction risk-stratify critically ill patients: INFECT study". Intensive Care Medicine. 44 (5): 627–635. doi:10.1007/s00134-018-5247-0. PMC 6006236. PMID 29915941. ^ Conway Morris A, Anderson N, Brittan M, Wilkinson TS, McAuley DF, Antonelli J, et al. (November 2013). "Combined dysfunctions of immune cells predict nosocomial infection in critically ill patients". British Journal of Anaesthesia. 111 (5): 778–787. doi:10.1093/bja/aet205. PMID 23756248. ^ Unnewehr H, Rittirsch D, Sarma JV, Zetoune F, Flierl MA, Perl M, et al. (April 2013). "Changes and regulation of the C5a receptor on neutrophils during septic shock in humans". Journal of Immunology. 190 (8): 4215–4225. doi:10.4049/jimmunol.1200534. PMID 23479227. ^ Wood AJ, Vassallo AM, Ruchaud-Sparagano MH, Scott J, Zinnato C, Gonzalez-Tejedo C, et al. (August 2020). "C5a impairs phagosomal maturation in the neutrophil through phosphoproteomic remodeling". JCI Insight. 5 (15). doi:10.1172/jci.insight.137029. PMC 7455072. PMID 32634128. ^ Bosmann M, Grailer JJ, Ruemmler R, Russkamp NF, Zetoune FS, Sarma JV, et al. (December 2013). "Extracellular histones are essential effectors of C5aR- and C5L2-mediated tissue damage and inflammation in acute lung injury". FASEB Journal. 27 (12): 5010–5021. doi:10.1096/fj.13-236380. PMC 3834784. PMID 23982144. ^ Ortiz-Espinosa S, Morales X, Senent Y, Alignani D, Tavira B, Macaya I, et al. (March 2022). "Complement C5a induces the formation of neutrophil extracellular traps by myeloid-derived suppressor cells to promote metastasis". Cancer Letters. 529: 70–84. doi:10.1016/j.canlet.2021.12.027. PMID 34971753. S2CID 245556050. External links [edit] Complement+C5a at the U.S. National Library of Medicine Medical Subject Headings (MeSH) \| v t e Complement system \| \| Pathways \| C L A \| \| Activators/enzymes \| \| \| \| --- \| \| Early \| C: C1 + C1q + C1r + C1s C4 + C4a + C4b C2 L: MASP1/MASP2 MBL A: Factor B Factor D Factor P/Properdin \| \| Middle \| C3 + C3a + C3b/iC3b C5 + C5a + C5b C3-convertase C5-convertase \| \| Late \| MAC + C5b + C6 + C7 + C8 + C9 \| \| \| Inhibitors \| CLA: C1-inhibitor Decay-accelerating factor/CD59 Factor I CL: C4BP A: Factor H \| \| Complement receptors \| CR1 CR2 CR3 CR4 CD11b/CD11c/CD18 Anaphylatoxin + C3a + C5a \| \| Function \| Cytotoxicity(by MAC) immune adherence Inducing inflammation Opsonization \| Retrieved from " Categories: Genes on human chromosome 9 Complement system Molecular biology Hidden categories: Articles with short description Short description matches Wikidata Complement component 5a Add topic
18155	https://ashpublications.org/blood/article/144/Supplement%201/5348/526789/Hemoglobin-S-Chocowinity-A-De-Novo-Compound-Beta	Published Time: 2024-11-05 Hemoglobin S/Chocowinity: A De Novo Compound Beta Globin Heterozygosity Resulting in Severe Sickle Cell Anemia \| Blood \| American Society of Hematology Skip to Main Content Advertisement intended for health care professionals Open Menu Close ASH Open Menu ASH Home Open External Link ASH Store Open External Link Advocacy Open External Link Education Open External Link Meetings Open External Link Publications Open External Link Research Open External Link ASH Clinical News ASH Image Bank Open External Link ASH News Daily ASH-SAP Blood Journals Open Menu Blood Blood Advances Blood Global Hematology Blood Immunology & Cellular Therapy Blood Neoplasia Blood Red Cells & Iron Blood Vessels, Thrombosis & Hemostasis Hematology The Hematologist International Open Menu Blood Chinese Edition Open External Link Blood Italian Edition Open External Link Blood Latin America Edition Open External Link Blood Spanish Edition Open External Link Cart User Tools Dropdown Cart Sign In Open Menu Search Dropdown Menu header search search input Search input auto suggest filter your search Search Toggle Menu Menu Issues Open Menu Current Issue All Issues First edition Abstracts Open Menu 2024 Annual Meeting 2024 Late Breaking 2023 Annual Meeting 2023 Late Breaking 2022 Annual Meeting 2022 Late Breaking 2021 Annual Meeting 2020 Annual Meeting 2020 Late Breaking All Meeting Abstracts Collections Open Menu Blood Cover Contest Blood Podcast Collections Special Collections Multimedia Author Center Open Menu Alerts Author Guide Style Guide Submit Open External Link Why Submit to Blood? About Open Menu About Blood Alerts Blood Classifieds Copyright Editorial Board Publications Staff Public Access Subscriptions Skip Nav Destination Article Navigation 114.Sickle Cell Disease, Sickle Cell Trait, and Other Hemoglobinopathies, Excluding Thalassemias: Clinical and Epidemiological\|November 5, 2024 Hemoglobin S/Chocowinity: A De Novo Compound Beta Globin Heterozygosity Resulting in Severe Sickle Cell Anemia Free Beng R. Fuh, Beng R. Fuh 1 Sickle Cell Center, East Carolina University, Greenville, NC Search for other works by this author on: This Site PubMed Google Scholar Kelli A Clemons, Kelli A Clemons 2 East Carolina University Medical Center, Greenville, NC Search for other works by this author on: This Site PubMed Google Scholar Ashish A Khanchandhani, Ashish A Khanchandhani 2 East Carolina University Medical Center, Greenville, NC Search for other works by this author on: This Site PubMed Google Scholar Chelsea Rivenbark Chelsea Rivenbark 3 ECU Sickle Cell Center, East Carolina University, Greenville, NC Search for other works by this author on: This Site PubMed Google Scholar Blood (2024) 144 (Supplement 1): 5348. Split-Screen Share Icon Share Facebook X LinkedIn Email Bluesky Tools Icon Tools Open Menu Request Permissions Cite Icon Cite Search Site Open the PDF for in another window Citation Beng R. Fuh, Kelli A Clemons, Ashish A Khanchandhani, Chelsea Rivenbark; Hemoglobin S/Chocowinity: A De Novo Compound Beta Globin Heterozygosity Resulting in Severe Sickle Cell Anemia. _Blood_ 2024; 144 (Supplement 1): 5348. doi: Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search Search Sickle cell disease usually results from homozygosity of the HbS beta globin gene, HbS/beta thalassemia or from compound heterozygosity of the HbS gene and another mutation such as HbC. These mutations are usually inherited from parents who either have these conditions or are carriers of the traits. Here we report a subtype of sickle cell disease that results from compound heterozygosity of the HbS trait and a de novo beta globin gene mutation, hemoglobin chocowinity, that results in severe sickle cell anemia. Infant female was born to parents with known HbS trait in the mother and no known hemoglobinopathy in the father. Newborn screening showed Hemoglobin F and Hemoglobin S indicating sickle cell anemia. Paternity testing performed for other reasons confirmed paternity. Quantitative hemoglobin electrophoresis at 2 months of age showed HbF of 61.9%, HbS of 35.8%, and HbA2 of 2.3%. Blood counts in the following months showed worsening microcytosis so sickle cell beta thalassemia zero was suspected. Beta globin gene sequencing of the Beta 120 through the first two positions of 125 showed a mutation of the beta globin gene with deleted, CAGTGCCAA inserted HGVS: c.361_377delinsCAGTGCCAA, p.K121Qfs17 Genomic: g.5246895_5246911delinsTTGGCACTG that has not been previously reported. Hemoglobin electrophoresis via HPLC did not identify any unknown variant of hemoglobin indicating that this novel beta globin variant did not produce a stable hemoglobin. Bone marrow biopsy showed a grossly normocellular marrow with erythroid hyperplasia. Hemoglobin electrophoresis on the bone marrow did not identify any abnormal hemoglobin variant indicating that the gene product from this gene variant was not present even in early red blood cell progenitors. Testing for common alpha globin gene mutation was negative (normal). Sickle cell trait was confirmed in the mother. Paternal testing showed normal blood counts with a normal hemoglobin, mean corpuscular volume, and other red blood cell indices. Paternal hemoglobin electrophoresis was normal with no findings to suggest beta thalassemia. Beta globin gene sequencing on the father showed normal wild type beta globin and specifically, the mutation identified in the patient was not identified in the father. This indicates that this is a de novo mutation in the patient. The patient is now 3 years old and phenotypically, has high baseline hemolysis with severe baseline anemia. At baseline, hemoglobin is ~6g/dL, absolute reticulocyte counts of ~400k/uL, LDH ~1100U/L, Total bilirubin of ~2mg/dL. Severe sickle cell complications to date have included recurrent severe splenic sequestration for which splenectomy is planned and aplastic anemia. The patient was started on hydroxyurea with modest improvement in anemia and hemolysis indices. In conclusion, we have identified a de novo mutation of the beta globin gene mutation, hemoglobin Chocowinity, which results in either a nonsense mutation or produces an extremely labile hemoglobin. Individuals who are compound heterozygous for this mutation and sickle cell trait, have a phenotype like hemoglobin beta thalassemia zero and have severe sickle cell anemia. Given the de novo nature of this mutation, close monitoring of the clinical course of this patient will be needed to further understand the clinical implications of this novel hemoglobinopathy. Disclosures Fuh:Pfizer: Honoraria. This content is only available as a PDF. Open the PDF for in another window 2024 Sign in via your Institution Add comment Close comment form modal Submit a comment Name Please enter your name. Affiliations Please enter your affiliations Comment title Please supply a title for your comment. Comment This field is required [x] I agree to the terms and conditions.You must accept the terms and conditions. Read the terms and conditions You have entered an invalid code. Submit Cancel Thank you for submitting a comment on this article. Your comment will be reviewed and published at the journal's discretion. Please check for further notifications by email. Close Comment not saved. Please try again. This feature is available to Subscribers Only Sign In or Create an AccountClose Modal Close login modal My Account Sign In Volume 144, Issue Supplement 1 November 5 2024 Previous Article Next Article Advertisement intended for health care professionals Sign in via your Institution Potential Articles of Interest Hemoglobin A2 Values in Sickle Cell Disease Patients Reveal Alarming HeterozygosityDinesh Pendharkar, Garima Nirmal, Blood, 2024 An Electronic Database of Human Hemoglobin Variants on the World Wide WebChui, Blood, 1998 Variation in the Amount of Hemoglobin S in a Patient with Sickle Cell Trait and Megaloblastic AnemiaBlood, 1963 Mild sickle-cell anaemia in Iran associated with high levels of fetal haemoglobin.M Haghshenass, J Med Genet, 1977 De novo heterozygous Hb G-Waimanalo (α64(E13)Asp>Asn, CTG>CCG; HBA1:c.193G>A) variant in a sickle cell disease patient of an Indian tribeRavindra Kumar, J Clin Pathol, 2021 Cascade testing effectively identifies undiagnosed sickle cell disease in The Gambia: a quality improvement projectEtienne Deans-Louis, Angela Allen, Stephen Allen, Archives of Disease in Childhood, 2024 Powered by Privacy policy Google Analytics settings View Metrics ×Close Modal Cited By Google Scholar Email alerts Article Activity Alert First Edition Alert Latest Issue Alert Close Modal Advertisement intended for health care professionals Advertisement intended for health care professionals About Blood Abstracts Advertising in Blood Alerts All Issues Author Guide Blood Classifieds Collections Contact Us Current Issue First edition Newsroom Permissions Subscriptions Submit to Blood American Society of Hematology 2021 L Street NW, Suite 900 Washington, DC 20036 TEL +1 202-776-0544 FAX +1 202-776-0545 ASH Publications Blood Blood Advances Blood Global Hematology Blood Immunology & Cellular Therapy Blood Neoplasia Blood Red Cells & Iron Blood Vessels, Thrombosis & Hemostasis Hematology, ASH Education Program ASH Clinical News ASH-SAP The Hematologist American Society of Hematology ASH Home ASH Image Bank ASH Store Advocacy Education Meetings Research Copyright 2025 by American Society of Hematology Cookie Settings Privacy Policy Cookie Policy Terms of Use Contact Us Close Modal Close Modal This Feature Is Available To Subscribers Only Sign In or Create an Account Close Modal Close Modal
18156	https://tutors.com/lesson/how-to-find-the-perimeter-of-a-pentagon	Tutoring jobs 20+ tutors near you & online ready to help. Find a tutor TABLE OF CONTENTS How to Find The Perimeter of a Pentagon Written by Malcolm McKinsey Fact-checked by Paul Mazzola Perimeter of a pentagon The perimeter, P, of a pentagon is the distance around its five straight sides. To find the perimeter of a pentagon, you must add the length of all 5 sides together. For regular pentagons the formula is P = 5 x s, where s equals side length. How you find the perimeter of a pentagon depends on what type of pentagon you have and what is known about it. Get free estimates from geometry tutors near you. Search Regular and irregular pentagons Regular pentagons have five congruent sides, five congruent interior angles, and five congruent exterior angles. Like all regular polygons, all the sides must be the same length, and all the angles must be the same measurement. Irregular pentagons have five sides and five angles, but neither the sides nor the angles are congruent. An irregular polygon can have sides of five different lengths and angles of five different measures. It is easier to find the perimeter of a regular pentagon since we have a formula. To find the perimeter of an irregular pentagon, you must measure and add up the five sides. Perimeter of a pentagon formula Using the perimeter of a pentagon formula, you can find the perimeter of a regular pentagon with relative ease. To find the perimeter of a regular pentagon with sides of length, s, you use this formula: P=5×s In our formula, 5 is the number of sides, and s is the length of the side that we know. Just like with the perimeter of a square, or the perimeter of a polygon in general, you find the perimeter of a pentagon by adding all the sides together. If you are finding the perimeter of a regular pentagon, then you know that all five sides are equal lengths, so you can simplify the formula using multiplication instead of addition. If you prefer to use the addition method your perimeter for a regular pentagon would look like this: P=s+s+s+s+s You have an s for each side of the pentagon. Perimeter of a regular pentagon example Let's pretend that we have a regular pentagon and that we know one side length is 3 cm3 cm. What is the perimeter of the pentagon? If we know one side of a regular pentagon, then we know the length of each side because a regular pentagon has equal sides. We can simply plug our known side into our formula: P=5×s P=5×3 P=15 Knowing that the length of a side is 3 cm, we used the perimeter formula of a pentagon, we found that the perimeter of this regular pentagon is 15 cm. Another important part of a pentagon is the apothem and the area. Learn how to find the area of a pentagon using the area formula. You can find the area of a regular pentagon or an irregular pentagon. Related articles Find geometry tutors in your area Geometry Tutors New York Geometry Tutors Los Angeles Geometry Tutors Chicago Geometry Tutors Houston Geometry Tutors Phoenix Geometry Tutors Philadelphia Geometry Tutors San Antonio Geometry Tutors Dallas Geometry Tutors San Diego Geometry Tutors San Jose Geometry Tutors Detroit Geometry Tutors San Francisco Geometry Tutors Jacksonville Geometry Tutors Indianapolis Geometry Tutors Austin Geometry Tutors Columbus Geometry Tutors Fort Worth Geometry Tutors Charlotte Geometry Tutors Memphis Geometry Tutors Baltimore Find tutors nearby Geometry Tutors near me Math Tutors near me Algebra Tutors near me Algebra 2 Tutors near me Calculus Tutors near me Online Math Tutors near me College Algebra Tutors near me Precalculus Tutors near me College Math Tutors near me Home Tutors near me ACT Math Tutors near me AP Statistics Tutors near me AP Calculus Tutors near me Elementary Math Tutors near me Pre-Algebra Tutors near me Discrete Math Tutors near me Arithmetic Tutors near me Linear Algebra Tutors near me Middle School Math Tutors near me Statistics Tutors near me Trigonometry Tutors near me
18157	https://www.britannica.com/animal/house-mouse	SUBSCRIBE Ask the Chatbot Games & Quizzes History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture ProCon Money Videos house mouse rodent Print Also known as: Mus musculus Written and fact-checked 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. They write new content and verify and edit content received from contributors. The Editors of Encyclopaedia Britannica Article History Related Topics: : mouse : knockout mouse : crest-tailed marsupial mouse See all related content house mouse, (Mus musculus), rodent native to Eurasia but introduced worldwide through association with humans. Highly adaptive, the house mouse has both behavioral and physiological traits—such as the ability to survive in buildings and aboard ships, a tendency to move into agricultural fields and leave when the habitat changes, and a rapid rate of reproduction—that allow it to thrive wherever humans do. The house mouse has thin whiskers, narrow hind feet, and short, sharp claws; its long, slender, scantily haired tail and prominent, thinly furred ears appear naked, but on the rest of the body the fur is short and soft. Domesticated laboratory strains may be white (true albinos), black, patterned with black and white, or blond, whereas native populations have tawny-brown upperparts and white bellies with shorter, bicoloured tails. Introduced feral populations, on the other hand, have dark, grayish brown upperparts paling to gray on the sides; underparts are similar to the sides and sometimes tinged with buff, and the tail is uniformly dark gray. The animal has a distinctive strong, musky odour. Generally weighing 12 to 30 grams (0.4 to 1.1 ounces), the house mouse has a small, slender body 6 to 11 cm (2.4 to 4.3 inches) long, and its tail length equals its body length. All these dimensions, however, can vary among different populations around the world. House mice are primarily nocturnal and terrestrial. Nervously active, they are agile climbers and jumpers and are also good swimmers. Outdoors, they excavate burrows in which to build nests of dry grass, but they will also den among rocks and crevices. House mice living outdoors eat insects and seeds, including grains, which makes them pests in some areas. Indoor house mice are also considered pests; essentially omnivorous, they construct nests in any protected place and can contaminate food and damage property. Indoor house mice breed throughout the year, but outdoor populations at temperate latitudes breed only from early spring until late fall. Gestation lasts 19 to 21 days, and each female of these prolific rodents can produce up to 14 litters per year (5 to 10 is usual); 5 or 6 young per litter is normal, although litters of up to 12 are sometimes produced. Life span can be as long as three years in laboratory mice but is considerably shorter among free-living mice. Britannica Quiz Wild Words from the Animal Kingdom Vocabulary Quiz Eurasia is the modern natural range of house mice, but researchers speculate that this is the result of migration from a likely habitat of origin in the grasslands of the northern Indian subcontinent. In tropical Asia, where their natural habitats are occupied by other, closely related species of Mus, house mice live only in buildings. Populations at temperate latitudes, however, can inhabit buildings (either seasonally or throughout the year) or live outside in grasslands, fallow fields, croplands, grassy coastal dunes, or shrubby deserts. When fields are plowed or crops harvested, these mice move into other fields or houses but not into forests. Western Europe is the primary source of house mice introduced into the United States, but a small population in southern California came from Asia. Humans eventually learned to domesticate and breed laboratory mice, which are an inbred genetic mosaic of European, Japanese, and Chinese stocks used in biomedical and genetic research. House mice are one of 38 species in the genus Mus, a member of the subfamily Murinae in the mouse family Muridae within the order Rodentia. The Editors of Encyclopaedia Britannica mouse Table of Contents Introduction General features Natural history Geographic distribution and habitat Classification and evolutionary history References & Edit History Quick Facts & Related Topics Images For Students mouse summary Quizzes Animal Group Names Deadliest Animals Quiz Wild Words from the Animal Kingdom Vocabulary Quiz Match the Baby Animal to Its Mama Quiz Animal Factoids mouse rodent genus Also known as: Mus Written by Guy Musser Archbold Curator Emeritus Vertebrate Zoology and Mammalogy, American Museum of Natural History, New York City, U.S. Guy Musser Fact-checked 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. They write new content and verify and edit content received from contributors. The Editors of Encyclopaedia Britannica Last Updated: • Article History Key People: : Ryuzo Yanagimachi : Charles Elton Related Topics: : Java shrew-mouse : Thomas’s pygmy mouse : gray-bellied pygmy mouse : flat-haired mouse : fawn-coloured mouse On the Web: : A-Z Animals - Mouse (Aug. 07, 2025) See all related content mouse, (genus Mus), the common name generally but imprecisely applied to rodents found throughout the world with bodies less than about 12 cm (5 inches) long. In a scientific context, mouse refers to any of the 38 species in the genus Mus, which is the Latin word for mouse. The house mouse (Mus musculus), native to Central Asia, has established itself with human populations in many other parts of the world. All rodents with a mouselike or ratlike body, regardless of body size or diagnostic traits, were described as species of Mus between 1758 and the late 1800s. Subsequent study shifted most of those species into many different groups, leaving Mus as a smaller, clearly defined genus with a particular combination of traits. Within the genus there are four distinctive groups: spiny mice (subgenus Pyromys), shrew-mice (subgenus Coelomys), rice field mice and the house mouse (subgenus Mus), and African mice (subgenus Nannomys). General features Mice have a slender body, blunt or tapered muzzle, scantily haired, prominent ears, narrow hind feet with bald soles, and sharp, small claws. The thinly furred tail appears hairless; it may be about as long as the head and body, or it can be much shorter. One of the largest is the flat-haired mouse (M. platythrix) of peninsular India, weighing about 18 grams (0.6 ounce), with a body 10 to 12 cm (4 to 4.7 inches) long and a shorter tail (7 to 8 cm [2.8 to 3.1 inches]). The smallest is probably the pygmy mouse (M. minutoides) of sub-Saharan Africa, weighing 3 to 12 grams (0.11 to 0.42 ounce), with a body 6 to 8 cm (2.3 to 3.1 inches) long and a short tail of 3 to 6 cm (1.2 to 2.3 inches). There is considerable variation in fur texture and colour among the species of Mus. At one extreme are the spiny-furred species in the subgenus Pyromys, whose upperparts and undersides are covered with flat, channeled spines nestled in soft underfur (juveniles are not spiny). At the other extreme are the shrew-mice from Sumatra (M. crociduroides) and Java (M. vulcani), whose soft, short, and dense coat appears woolly or velvety. All the other species have a soft or slightly coarse, moderately thick coat with short or long hairs. A colour combination common to many mice is gray to brown upperparts, white underparts, white feet, and a tail that is dark above and white below. Variations of this pattern include upperparts of buff, bluish gray, blackish gray, reddish brown, or chocolate brown, with underparts ranging from white to various shades of gray, sometimes tinged with silver or buff. The feet may be white or the same colour as the upperparts, and the tail may be bicoloured or uniformly dark gray to dark brown. Natural history Mice in their natural habitats are primarily nocturnal, although some will occasionally forage during the day. They are ground dwellers, although some species are also agile climbers and leapers as well as capable swimmers. A few are specialized burrowers rarely seen above ground. Most species, especially those living in savannas and grasslands, excavate burrows and chambers in which they build globular nests of dry vegetation. In an intact ecosystem, species of Mus, along with other small-bodied rodents, are preyed upon, sometimes to an appreciable degree, by reptiles, mammals, and birds (especially owls). Britannica Quiz Match the Baby Animal to Its Mama Quiz The simple but effective excavation technique of mice is exemplified by the Ryukyu mouse (M. caroli). This mouse loosens soil with its incisor teeth, carrying a load of debris in its mouth and piling it outside the burrow entrance or sometimes stacking loose soil inside the burrow and then pushing the pile out with its hind feet. In the diked rice fields of Thailand, small piles of soil below holes in the dike signal the presence of Ryukyu mice. Each hole is the opening to a tunnel extending upward to a nest chamber above water level, then to another opening on the other side of the dike. Forest species may also burrow, but most of them construct nests in rock crevices or beneath rotting tree trunks and brush piles on the forest floor. The gray-bellied pygmy mouse (M. triton) of sub-Saharan Africa, for example, apparently does not burrow but uses pathways made by larger rodents. Diet varies among species. Outdoors the house mouse consumes seeds and insects; indoors it eats nearly anything digestible. Most other species eat a combination of plant parts (especially seeds), insects, and other invertebrates. Stomachs of gray-bellied pygmy mice caught in East Africa, for example, contained plant parts, pieces of bark, insects (mostly adult beetles), and worms. Access for the whole family! Bundle Britannica Premium and Kids for the ultimate resource destination. Subscribe Depending upon the species and geographic region, mice may breed throughout the year or only during the wet seasons in southern latitudes and from spring to fall in northern latitudes. Except for the house mouse, which can produce up to 14 litters per year (1 to 12 offspring per litter), there is little information about the reproductive biology of most species. In the deserts of India, the little Indian field mouse (M. booduga) bears from 1 to 13 young per litter and breeds throughout the year. In Southeast Asia, the fawn-coloured mouse (M. cervicolor) has been reported to produce litters of two to six young in July and December. In East Africa, the pygmy mouse breeds during the wet seasons from April to June and September to December and bear litters of two to eight young. Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. 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 The Editors of Encyclopaedia Britannica. "house mouse". Encyclopedia Britannica, 2 May. 2025, Accessed 12 August 2025. Share Share to social media Facebook X External Websites Internet Center for Wildlife Damage Management - House Mice Animal Diversity Web - House mouse Nature - Heredity - Evolutionary and dispersal history of Eurasian house mice Mus musculus clarified by more extensive geographic sampling of mitochondrial DNA PestWorld.org - House Mice University of Kentucky - Entomology - Control of Mice Purdue University Extension - The House Mouse UC IPM - House Mouse National Center for Biotechnology Information - PubMed Central - Insights into mammalian biology from the wild house mouse Mus musculus WebMD - Mice: Health Risks, Habits, and Extermination BMC - Frontiers in Zoology - Lifetime development of behavioural phenotype in the house mouse (Mus musculus) Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. 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 Musser, Guy. "mouse". Encyclopedia Britannica, 7 Aug. 2025, Accessed 12 August 2025. Share Share to social media Facebook X URL External Websites National Center for Biotechnology Information - PubMed Central - The Mighty Mouse: The Impact of Rodents on Advances in Biomedical Research American Physiological Society - Physiology - Probing Pedomorphy and Prolonged Lifespan in Naked Mole-Rats and Dwarf Mice A-Z Animals - Mouse Animal Corner - Mice LiveScience - Mouse Facts: Habits, Habitat & Types of Mice Sheffield Hallam University - The best laid schemes oï¿½ mice and men : the evolution of the computer mouse Britannica Websites Articles from Britannica Encyclopedias for elementary and high school students. mouse - Children's Encyclopedia (Ages 8-11) mouse - Student Encyclopedia (Ages 11 and up)
18158	https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0205687	Prevalence of hypothyroidism in patients with hyponatremia: A retrospective cross-sectional study \| PLOS One Skip to main content Advertisement plos.org Create account Sign in Publish About Browse Searchadvanced search Browse Topics Browse Subject Areas ? Click through the PLOS taxonomy to find articles in your field. For more information about PLOS Subject Areas, click here. 24 Save Total Mendeley and Citeulike bookmarks. 13 Citation Paper's citation count computed by Dimensions. 5,419 View PLOS views and downloads. 1 Share Sum of Facebook, Twitter, Reddit and Wikipedia activity. Open Access Peer-reviewed Research Article Prevalence of hypothyroidism in patients with hyponatremia: A retrospective cross-sectional study Takanobu Nagata ,Contributed equally to this work with: Takanobu Nagata, Shoko Nakajima, Atsushi Fujiya, Hiroshi Sobajima, Makoto Yamaguchi Roles Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing E-mail:nagata-t@hotmail.co.jp Affiliation Department of Nephrology, Yokkaichi Municipal Hospital, Yokkaichi, Japan ⨯ Shoko Nakajima ,Contributed equally to this work with: Takanobu Nagata, Shoko Nakajima, Atsushi Fujiya, Hiroshi Sobajima, Makoto Yamaguchi Roles Formal analysis, Investigation, Methodology, Supervision Affiliation Department of Diabetology, Yokkaichi Municipal Hospital, Yokkaichi, Japan ⨯ Atsushi Fujiya ,Contributed equally to this work with: Takanobu Nagata, Shoko Nakajima, Atsushi Fujiya, Hiroshi Sobajima, Makoto Yamaguchi Roles Formal analysis, Investigation, Methodology, Supervision Affiliation Department of Diabetology and Nephrology, Ogaki Municipal Hospital, Ogaki, Japan ⨯ Hiroshi Sobajima ,Contributed equally to this work with: Takanobu Nagata, Shoko Nakajima, Atsushi Fujiya, Hiroshi Sobajima, Makoto Yamaguchi Roles Formal analysis, Methodology, Supervision Affiliation Department of Diabetology and Nephrology, Ogaki Municipal Hospital, Ogaki, Japan ⨯ Makoto YamaguchiContributed equally to this work with: Takanobu Nagata, Shoko Nakajima, Atsushi Fujiya, Hiroshi Sobajima, Makoto Yamaguchi Roles Formal analysis, Investigation, Methodology, Supervision Affiliation Department of Nephrology, Yokkaichi Municipal Hospital, Yokkaichi, Japan ⨯ Prevalence of hypothyroidism in patients with hyponatremia: A retrospective cross-sectional study Takanobu Nagata, Shoko Nakajima, Atsushi Fujiya, Hiroshi Sobajima, Makoto Yamaguchi x Published: October 11, 2018 Article Authors Metrics Comments Media Coverage Abstract Introduction Materials and methods Results Discussion Conclusion Supporting information Acknowledgments References Reader Comments Figures Abstract Objective Hypothyroidism has been suggested to be an uncommon cause of hyponatremia. However, little is known about the prevalence of hypothyroidism in patients with different levels of hyponatremia. The objective of this study was to investigate the prevalence of hypothyroidism among patients with hyponatremia of varying severity while taking into consideration potential confounders associated with thyroid function. Methods All data on thyrotropin (TSH), free thyroxine (T4), and serum sodium (Na) levels were retrospectively collected from medical records at two Japanese tertiary hospitals. The main outcome measure was overt hypothyroidism, defined as TSH > 10.0 μIU/mL and free T4 < 1.01 ng/dL. Results Of 71,817 patients, 964 patients (1.3%) had overt hypothyroidism. The prevalence of overt hypothyroidism in each category of hyponatremia (Na ≥136, 130–135, and ≤129 mEq/L) was 1.2% (787/65,051), 2.4% (124/5,254) and 3.5% (53/1,512), respectively. A significant increase in prevalence was observed as the severity of hyponatremia increased (P < 0.001 for trend). Multivariate logistic regression with adjustment for age, sex, kidney function, and serum albumin level showed that the odds ratios for overt hypothyroidism increased with increasing severity of hyponatremia when compared with Na ≥ 136 mEq/L (130–135 mEq/L: 1.43, 95% confidence interval [CI], 1.15 to 1.78, P = 0.001; ≤129 mEq/L: 1.87, 95% CI, 1.32 to 2.63, P < 0.001; P< 0.001 for trend). Conclusion The prevalence of overt hypothyroidism was significantly higher as the severity of hyponatremia progressed, even after adjusting for potential confounders. Hypothyroidism should be differentiated in patients with hyponatremia. Figures Citation:Nagata T, Nakajima S, Fujiya A, Sobajima H, Yamaguchi M (2018) Prevalence of hypothyroidism in patients with hyponatremia: A retrospective cross-sectional study. PLoS ONE 13(10): e0205687. Editor:Sun Young Lee, Boston University School of Medicine, UNITED STATES Received:May 21, 2018; Accepted:September 28, 2018; Published: October 11, 2018 Copyright: © 2018 Nagata et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability:All relevant data are within the paper and its Supporting Information files. Funding:The authors received no specific funding for this work. Competing interests: The authors have declared that no competing interests exist. Introduction Hypothyroidism is often referred to as a cause of hyponatremia, but several reports have shown that the association between thyroid function and serum sodium levels is very weak and of marginal clinical relevance[1–6]. Retrospective cross-sectional analyses have shown that the prevalence of hyponatremia and distribution of serum sodium levels were similar among euthyroid and hypothyroid patients. A recent review and clinical practice guideline for hyponatremia have mentioned that even though hypothyroidism is one possible cause of hyponatremia, it should only be attributed to severe hypothyroidism, as in myxedema coma[4,6,7]. However, past reports had simply compared the prevalence of hyponatremia between patients with and without hypothyroidism[1,5,8]. In addition, some confounders such as sex, age, kidney function, and serum albumin level, which are reported to be associated with thyroid function[9–12], were not considered. Furthermore, no information was available about the prevalence of hypothyroidism with varying severity of hyponatremia. The objective of the present retrospective cross-sectional study was to investigate the prevalence of hypothyroidism by severity of hyponatremia, and to clarify whether the association between the severity of hyponatremia and hypothyroidism is affected by confounding factors associated with thyroid function. Materials and methods The study protocol was approved by the ethics committee of Yokkaichi Municipal Hospital and Ogaki Municipal Hospital. The study was conducted in accordance with the Declaration of Helsinki. The ethics committee approved waiver of informed consent for this study. The approval number of Yokkaichi Municipal Hospital was 2017–8 and that of Ogaki Municipal Hospital was 20161222–4. All data on thyrotropin (TSH) levels between January 2008 and December 2017 were retrospectively collected from medical records at Yokkaichi Municipal Hospital in Yokkaichi, Japan, and Ogaki Municipal Hospital, in Ogaki, Japan. Both are tertiary hospitals in their respective medical districts. For each patient, initial TSH data during the study period and free thyroxine (T4) data from the same day were obtained. Patient age, sex, and levels of serum creatinine, blood urea nitrogen, potassium, chloride, total protein, albumin, and free triiodothyronine (T3) on the same day as TSH testing were also obtained. Next, we extracted the minimum serum sodium (Na) level within 3 days of TSH testing because TSH testing was unavailable at night and during holidays. Consequently, hyponatremia could have already been treated by the time of TSH testing. We excluded patients aged 17 years or younger. The reference ranges for TSH and free T4 were 0.27 to 4.20 μIU/mL and 1.01 to 1.79 ng/dL, respectively. Estimated glomerular filtration rate (eGFR) was calculated using the equation from the Japanese Society of Nephrology: eGFR (mL/min/1.73 m 2) = 194 × serum creatinine-1.094 × age-0.287 × 0.739 (for female patients). We defined hypothyroidism as TSH > 4.20 μIU/mL, and overt hypothyroidism as TSH > 10.0 μIU/mL plus free T4 < 1.01 ng/dL. Hyponatremia was defined as Na ≤ 135 mEq/L. Hyponatremia was classified as mild (130–135 mEq/L) or moderate to profound (≤129 mEq/L) according to guideline. Statistical analysis Continuous variables were expressed as medians and interquartile ranges and compared using the Kruskal-Wallis test. Categorical variables were expressed as percentages and compared using Fisher’s exact test. Linear trends in proportions were assessed using the Cochran-Armitage test. Pearson’s correlation coefficients were used for correlation analysis. Univariate and multivariate logistic regression models were used to compute odds ratios and 95% confidence intervals (CIs) for hypothyroidism or overt hypothyroidism in each category of hyponatremia, with Na ≥ 136 mEq/L as the referent group. Multivariate models were adjusted for sex, age, eGFR, and serum albumin, which are factors that have previously been reported to be associated with thyroid function[9–12,15]. Univariate and multivariate linear trend tests were performed using each Na category as an ordinal variable. All P values were two-tailed, and P< 0.05 was considered statistically significant. All statistical analyses were performed using EZR (Saitama Medical Center, Jichi Medical University, Saitama, Japan), which is a graphical user interface for R (The R Foundation for Statistical Computing, Vienna, Austria). Results Overall, data from 71,817 patients were included in this study. Patient characteristics are shown in Table 1. In total, 4,710 patients (6.6%) had hypothyroidism and 964 patients (1.3%) had overt hypothyroidism. The proportion of patients with hypothyroidism among those with a Na level of ≥136 mEq/L, 130–135 mEq/L, and ≤129 mEq/L was 6.1% (3,975 patients), 10.3% (539 patients), and 13.0% (196 patients), respectively. The proportion of overt hypothyroidism in the same categories was 1.2% (787 patients), 2.4% (124 patients), and 3.5% (53 patients), respectively. Download: PPT PowerPoint slide PNG larger image TIFF original image Table 1. Patient characteristics by severity of hyponatremia. The Cochran-Armitage trend test showed a significant increase in the proportion of patients with hypothyroidism and overt hypothyroidism as the severity of hyponatremia increased (both P< 0.001). The Pearson’s correlation coefficient for TSH and Na for the entire study population was -0.021 (95% CI, -0.028 to -0.013, P< 0.001) (Fig 1). Among patients with hypothyroidism and overt hypothyroidism, the values were -0.007 (95% CI, -0.035 to 0.021, P = 0.624) and 0.029 (95% CI, -0.034 to 0.092, P = 0.365) (Fig 1). A statistically significant but weak correlation was observed between TSH and Na in the entire study population. No statistically significant relationship was observed between TSH and Na among patients with hypothyroidism and overt hypothyroidism, respectively. Download: PPT PowerPoint slide PNG larger image TIFF original image Fig 1. Correlation between thyrotropin and serum sodium levels in all patients (a), patients with hypothyroidism (b), and overt hypothyroidism (c). Pearson’s correlation coefficient: a, -0.021 (95% confidence interval [CI], -0.028 to -0.013, P< 0.001); b, -0.007 (95% CI, -0.035 to 0.021, P = 0.624); c, 0.029 (95% CI, -0.034 to 0.092, P = 0.365). Hypothyroidism was defined as TSH > 4.20 μIU/mL. Overt hypothyroidism was defined as TSH > 10.0 μIU/mL plus free T4 < 1.01 ng/dL. The unadjusted and adjusted odds ratios for hypothyroidism and overt hypothyroidism in each hyponatremia category are shown in Table 2. The adjusted odds ratios for hypothyroidism among patients with Na of 130–135 mEq/L and ≤129 mEq/L compared with Na ≥136 mEq/L were 1.11 (95% CI, 0.99 to 1.24, P = 0.054) and 1.32 (95% CI, 1.11 to 1.58, P = 0.003), respectively. The odds ratios for overt hypothyroidism were 1.43 (95% CI, 1.15 to 1.78, P = 0.001) and 1.87 (95% CI, 1.32 to 2.63, P< 0.001). A significant trend of increase in adjusted odds ratios was observed among patients with both hypothyroidism and overt hypothyroidism (P< 0.001 for trend). Download: PPT PowerPoint slide PNG larger image TIFF original image Table 2. Associations between hyponatremia category and hypothyroidism. Discussion This retrospective cross-sectional study investigated the prevalence of hypothyroidism in patients with hyponatremia. The correlation between TSH and Na levels was very weak, but the prevalence of hypothyroidism and overt hypothyroidism were significantly higher among patients with hyponatremia compared with patients with Na ≥ 136 mEq/L. Furthermore, a significant trend of increase in the prevalence of hypothyroidism and overt hypothyroidism was observed as the severity of hyponatremia increased, even after adjusting for potential confounders associated with thyroid function. Past studies have reported a weak association between TSH and Na levels. Wolf et al. reported a very mild positive correlation between TSH and Na levels (-0.022; P = 0.046) in patients with newly diagnosed hypothyroidism in a single-center retrospective analysis. They concluded that this weak association was not clinically significant. Schwarz et al. reported that TSH and Na levels were not significantly correlated (-0.02; P> 0.05) in patients whose TSH and Na levels were examined in the emergency department in a single center retrospective analysis. The correlation coefficients in the present study were similar. The reason for this weak association could be that most patients with hypothyroidism were normonatremic. Therefore, the correlation could be weakened because the majority of patients had normonatremia when TSH and Na levels were analyzed as continuous variables. However, we thought that it was necessary to investigate the prevalence of hypothyroidism in patients with hyponatremia to understand the direction of the causal relationship. It has been reported that there was no significant difference in distribution of Na levels between hypothyroid patients and controls, but every 10 mU/L rise in TSH is associated with a 0.14 mmol/L fall in Na. Consequently, a recent review and clinical practice guideline for hyponatremia suggested that hypothyroidism rarely causes hyponatremia and hyponatremia should not be attributed to hypothyroidism except in cases of severe hypothyroidism, such as myxedema coma[4,6,7]. However, the present study showed that the prevalence of hypothyroidism and overt hypothyroidism in patients with hyponatremia tended to increase as the severity of hyponatremia increased. This result indicated that some patients with hyponatremia could have concomitant hypothyroidism. This result seemed plausible given the pathophysiology of hyponatremia in patients with hypothyroidism. The putative mechanism by which hypothyroidism affects Na level is that hypothyroidism induces decreased cardiac output, which results in compensatory elevation of antidiuretic hormone levels and decreased GFR[16,17]. That may induce free water retention and decrease excretion by decreasing water delivery to the diluting segment of the nephron[6,18]. It is therefore reasonable that hypothyroidism could be a cause of hyponatremia, if not alone, even without severe hypothyroidism did not exist. One strength of the present study was that confounders that could affect the prevalence of hypothyroidism were considered. An epidemiologic study revealed that hypothyroidism was more common in females and elderly individuals, and several reports have shown a higher prevalence of hypothyroidism in patients with chronic kidney disease[11,19]. In addition, it has been reported that serum albumin levels are significantly correlated with thyroid hormone levels. Since the association between hyponatremia and hypothyroidism was significant even after adjusting for these factors, the results of present study can be considered more convincing. This study has several limitations. First, this was a retrospective study. Since all TSH and free T4 testing in the present study was performed in patients who were suspected to have thyroid disorder by their attending physicians, the actual prevalence of hypothyroidism in patients with hyponatremia remains unknown. Second, patients had different comorbidities of various etiologies that could have caused hyponatremia, but patients’ medical records were not reviewed in this study. Third, information about the use of medications that could cause hyponatremia, such as diuretics, was not available because TSH testing occurred during the first visit for most patients. Thus, information about medications prescribed by other clinics could not be obtained. Forth, we could not investigate the causality between hypothyroidism and hyponatremia because of the cross-sectional study design. A prospective and longitudinal study is needed to clarify these issues. Conclusion Although the correlation between TSH and Na was very weak, the prevalence of overt hypothyroidism increased significantly as the severity of hyponatremia progressed. Furthermore, the association remained even after adjusting for potential confounders associated with thyroid function. Thus, hypothyroidism should be differentiated in patients with hyponatremia. Supporting information Anonymous data set of 71817 patients. Skip to figshare navigation \| \| A \| B \| C \| D \| E \| F \| G \| H \| I \| J \| K \| L \| M \| N \| --- --- --- --- --- --- --- \| 1 \| id \| age \| m0f1 \| TSH \| fT4 \| fT3 \| Na \| K \| Cl \| UA \| BUN \| Cre \| TP \| Alb \| \| 2 \| 1 \| 41 \| 0 \| 0.295 \| 1.28 \| NA \| 99 \| 4.3 \| 92 \| 3.2 \| 7 \| 0.65 \| 7.1 \| 4.7 \| \| 3 \| 2 \| 68 \| 1 \| 0.562 \| 1.93 \| 2.37 \| 99 \| 2.7 \| 68 \| 0.8 \| 8.1 \| 0.43 \| 7.6 \| 4.6 \| \| 4 \| 3 \| 55 \| 0 \| 0.94 \| 1.14 \| 1.2 \| 100 \| 3.4 \| 70 \| 6.2 \| 23.2 \| 1.5 \| 5 \| 2.42 \| \| 5 \| 4 \| 78 \| 1 \| 1.272 \| 1.39 \| 2.3 \| 100 \| 4.1 \| 70 \| 0.8 \| 7.2 \| 0.35 \| 6.1 \| 3.2 \| \| 6 \| 5 \| 90 \| 1 \| 1.066 \| 0.99 \| 1.17 \| 101 \| 2.9 \| 75 \| 1.4 \| 11.2 \| 0.28 \| 4.5 \| 2.6 \| \| 7 \| 6 \| 48 \| 0 \| 0.262 \| 1.07 \| 2.94 \| 101 \| 3.8 \| 87 \| 2.9 \| 6.7 \| 0.82 \| 7.2 \| 4.5 \| \| 8 \| 7 \| 72 \| 1 \| 1.04 \| 1.62 \| 2.14 \| 101 \| 3.7 \| 70 \| 2.7 \| 20 \| 0.57 \| 7.6 \| 4.59 \| \| 9 \| 8 \| 71 \| 0 \| 1.527 \| 1.69 \| NA \| 101 \| 5 \| 77 \| NA \| 14.7 \| 0.53 \| 7.4 \| 4.8 \| \| 10 \| 9 \| 73 \| 0 \| 2.325 \| 0.76 \| 1.05 \| 102 \| 4.4 \| 79 \| 2.7 \| 10.7 \| 0.38 \| NA \| NA \| \| 11 \| 10 \| 79 \| 1 \| 4.34 \| 0.87 \| 1.6 \| 102 \| 3.8 \| 84 \| NA \| 10.9 \| 0.89 \| NA \| 3.23 \| \| 12 \| 11 \| 43 \| 0 \| 0.446 \| 1.5 \| 3.05 \| 102 \| 3.6 \| 85 \| 1.5 \| 2.3 \| 0.33 \| 7.1 \| 4.8 \| \| 13 \| 12 \| 75 \| 0 \| 0.669 \| 0.8 \| 1.97 \| 103 \| 6.1 \| 79 \| 10.5 \| 85.9 \| 9.34 \| 5.9 \| 3.5 \| \| 14 \| 13 \| 74 \| 0 \| 0.285 \| 1.89 \| 2.67 \| 103 \| 3.8 \| 86 \| 2.2 \| 12.3 \| 0.71 \| 5.7 \| 3.5 \| \| 15 \| 14 \| 74 \| 0 \| 0.906 \| 1.15 \| 2.41 \| 104 \| 3.9 \| 79 \| NA \| NA \| NA \| NA \| NA \| \| 16 \| 15 \| 51 \| 1 \| 0.488 \| 1.28 \| NA \| 104 \| 3.3 \| 77 \| 2.5 \| 3.8 \| 0.36 \| NA \| NA \| \| 17 \| 16 \| 84 \| 1 \| 0.56 \| 1.69 \| 1.92 \| 104 \| 3.6 \| 73 \| 3.2 \| 22.3 \| 0.53 \| 6.5 \| 4 \| \| 18 \| 17 \| 63 \| 1 \| 1.451 \| 0.81 \| 1.77 \| 105 \| 3.8 \| 83 \| 4.9 \| 37 \| 1.41 \| NA \| NA \| \| 19 \| 18 \| 54 \| 1 \| 1.466 \| 0.9 \| 2.43 \| 105 \| 4.5 \| 81 \| 4.3 \| 6.3 \| 0.39 \| 8.6 \| 4.8 \| \| 20 \| 19 \| 87 \| 1 \| 2.403 \| 1.02 \| 2.16 \| 105 \| 4.3 \| 77 \| 1.7 \| 5 \| 0.35 \| NA \| NA \| \| 21 \| 20 \| 69 \| 0 \| 0.46 \| 1.33 \| 1.64 \| 105 \| 4.7 \| 75 \| NA \| 9.8 \| 0.42 \| 6.7 \| 3.7 \| \| 22 \| 21 \| 56 \| 0 \| 0.524 \| 1.35 \| 2.35 \| 105 \| 4.3 \| 82 \| 2.1 \| 7.1 \| 0.53 \| 6 \| 4 \| \| 23 \| 22 \| 75 \| 1 \| 1.287 \| 1.4 \| 2.93 \| 105 \| 4.6 \| 72 \| 4.4 \| 8.9 \| 0.66 \| 7.8 \| 4.8 \| \| 24 \| 23 \| 62 \| 1 \| 0.918 \| 1.6 \| 1.94 \| 105 \| 3.1 \| 82 \| 0.9 \| 14.7 \| NA \| 6.5 \| NA \| \| 25 \| 24 \| 65 \| 0 \| 3.042 \| 1.34 \| 2.61 \| 106 \| 4.8 \| 83 \| 2 \| 10.7 \| 0.75 \| 6.5 \| 2.9 \| \| 26 \| 25 \| 60 \| 1 \| 0.23 \| 1.52 \| 1.54 \| 106 \| 3 \| 77 \| 0.9 \| 8.7 \| 0.37 \| 6.6 \| 3.59 \| \| 27 \| 26 \| 35 \| 0 \| 0.687 \| 0.81 \| 1.93 \| 107 \| 3.9 \| 88 \| 3.2 \| 7.5 \| 0.66 \| 7.2 \| 4.2 \| \| 28 \| 27 \| 57 \| 0 \| 0.664 \| 0.84 \| 2.34 \| 107 \| 4 \| 91 \| 2.3 \| 6.2 \| 0.49 \| 5.6 \| 3.3 \| \| 29 \| 28 \| 72 \| 0 \| 2.443 \| 1.27 \| 2.45 \| 107 \| 4.7 \| 80 \| 2.3 \| 16.5 \| 0.47 \| 7.8 \| 4.2 \| \| 30 \| 29 \| 85 \| 0 \| 0.904 \| 1.31 \| 2.61 \| 107 \| 5.4 \| 77 \| 2.5 \| 22.2 \| 0.84 \| 6.8 \| 3.9 \| \| 31 \| 30 \| 90 \| 1 \| 3.358 \| 1.35 \| 2.2 \| 107 \| 2.9 \| 99 \| 4.9 \| 21.4 \| 0.69 \| 5.4 \| 2.9 \| \| 32 \| 31 \| 74 \| 0 \| 0.836 \| 1.42 \| 2.62 \| 107 \| 2.6 \| 93 \| 5.1 \| 10.6 \| 0.45 \| NA \| NA \| \| 33 \| 32 \| 65 \| 0 \| 0.911 \| 1.66 \| 2.2 \| 107 \| 33.9 \| 105 \| 1.7 \| 24.5 \| 0.83 \| 6.7 \| 3.3 \| \| 34 \| 33 \| 72 \| 0 \| 0.003 \| 2.53 \| 2.5 \| 107 \| 4.3 \| 80 \| 1 \| 13.6 \| 0.52 \| 6.9 \| 4.6 \| \| 35 \| 34 \| 68 \| 1 \| 7.954 \| 0.45 \| 2.78 \| 108 \| 3.7 \| 72 \| 1.8 \| 5.4 \| 0.4 \| 7.7 \| 4.6 \| \| 36 \| 35 \| 91 \| 1 \| 0.614 \| 1.45 \| NA \| 108 \| 3.5 \| 94 \| 5.6 \| 15.7 \| 0.84 \| NA \| NA \| \| 37 \| 36 \| 75 \| 0 \| 2.115 \| 1.52 \| NA \| 108 \| 4.4 \| 77 \| 1.1 \| 4.6 \| 0.54 \| NA \| NA \| \| 38 \| 37 \| 81 \| 1 \| 0.99 \| 1.66 \| 1.49 \| 108 \| 4 \| 77 \| 0.9 \| 4.9 \| 0.3 \| 5.6 \| 3.37 \| \| 39 \| 38 \| 36 \| 1 \| 0.853 \| 1.76 \| 2.89 \| 108 \| 3.8 \| 78 \| NA \| 10.5 \| 0.41 \| 7 \| 4.5 \| \| 40 \| 39 \| 84 \| 0 \| 1.353 \| 2.04 \| 3.55 \| 108 \| 4.8 \| 79 \| 4 \| 17.3 \| 0.67 \| 6.9 \| 4.2 \| \| 41 \| 40 \| 75 \| 1 \| 14.206 \| 0.52 \| 2.8 \| 109 \| 3.2 \| 85 \| 1.7 \| 13 \| 0.6 \| NA \| NA \| \| 42 \| 41 \| 73 \| 1 \| 4.55 \| 0.83 \| 1 \| 109 \| 3.1 \| 87 \| 2.4 \| 11.2 \| 0.26 \| 4.5 \| 2.31 \| \| 43 \| 42 \| 70 \| 0 \| 1.056 \| 0.93 \| 2.05 \| 109 \| 3.8 \| 88 \| 3.8 \| 7.9 \| 0.81 \| 6 \| 3.6 \| \| 44 \| 43 \| 91 \| 1 \| 2.51 \| 0.96 \| 1 \| 109 \| 6.5 \| 78 \| 2.4 \| 21.1 \| 0.6 \| 4.2 \| 2.11 \| \| 45 \| 44 \| 71 \| 0 \| 4.76 \| 0.97 \| 2.04 \| 109 \| 3.9 \| 80 \| 3.2 \| 11.6 \| 0.6 \| 7.2 \| 4.23 \| \| 46 \| 45 \| 77 \| 0 \| 0.447 \| 1.1 \| 2.67 \| 109 \| 3.6 \| 87 \| 3.4 \| 3.7 \| 0.47 \| 6.5 \| 4.1 \| \| 47 \| 46 \| 80 \| 1 \| 0.53 \| 1.17 \| 1.92 \| 109 \| 2.8 \| 87 \| 1.7 \| 10.2 \| 0.5 \| 5.6 \| 3.44 \| \| 48 \| 47 \| 69 \| 1 \| 0.5 \| 1.36 \| 2 \| 109 \| 4.6 \| 84 \| 5.4 \| 15.9 \| 0.6 \| 6.2 \| 4.05 \| \| 49 \| 48 \| 75 \| 1 \| 1.05 \| 1.36 \| 1.63 \| 109 \| 3 \| 84 \| 0.8 \| 6.3 \| 0.3 \| 5.9 \| 3.23 \| \| 50 \| 49 \| 74 \| 0 \| 0.37 \| 1.41 \| 2.67 \| 109 \| 0 \| 0 \| 2.8 \| 13.8 \| 0 \| 7 \| 4.29 \| 13253288.csv ShareDownload figshare Anonymous data set of 71817 patients. (CSV) S1 Table.Anonymous data set of 71817 patients. (CSV) Acknowledgments We are grateful to K. Murakami, A. Hattori, T. Amano, M. Hori, M. Shibata for their collaboration during the early stages of this study. References Croal BL, Blake AM, Johnston J, Glen AC, O’Reilly DS. Absence of relation between hyponatraemia and hypothyroidism. Lancet (London, England). 1997;350: 1402. View Article Google Scholar Warner MH, Holding S, Kilpatrick ES. The effect of newly diagnosed hypothyroidism on serum sodium concentrations: A retrospective study . Clin Endocrinol (Oxf). 2006;64: 598–599. pmid:16649984 View Article PubMed/NCBI Google Scholar Pantalone K, Hatipoglu B. Hyponatremia and the Thyroid: Causality or Association? J Clin Med. 2014;4: 32–36. pmid:26237016 View Article PubMed/NCBI Google Scholar Aylwin S, Burst V, Peri A, Runkle I, Thatcher N. ‘Dos and don’ts’ in the management of hyponatremia. Curr Med Res Opin. 2015;31: 1755–1761. pmid:26173050 View Article PubMed/NCBI Google Scholar Wolf P, Beiglböck H, Smaijs S, Wrba T, Rasoul-Rockenschaub S, Marculescu R, et al. Hypothyroidism and Hyponatremia: Rather Coincidence Than Causality. Thyroid. 2017;27: 611–615. pmid:28351291 View Article PubMed/NCBI Google Scholar Liamis G, Filippatos TD, Liontos A, Elisaf MS. Hypothyroidism-associated hyponatremia: Mechanisms, implications and treatment. Eur J Endocrinol. 2017;176: R15–R20. pmid:27484454 View Article PubMed/NCBI Google Scholar Spasovski G, Vanholder R, Allolio B, Annane D, Ball S, Bichet D, et al. Clinical practice guideline on diagnosis and treatment of hyponatraemia. Nephrol Dial Transplant. 2014;29: 1–39. View Article Google Scholar Schwarz C, Leichtle AB, Arampatzis S, Fiedler GM, Zimmermann H, Exadaktlyos AK, et al. Thyroid function and serum electrolytes: Does an association really exist? Swiss Med Wkly. 2012;142. pmid:22987514 View Article PubMed/NCBI Google Scholar Feinstein EI, Kaptein EM, Nicoloff JT, Massry SG. Thyroid function in patients with nephrotic syndrome and normal renal function. Am J Nephrol. 1982;2: 70–6. Available: pmid:7180903 View Article PubMed/NCBI Google Scholar Vanderpump MPJ, Tunbridge WMG, French JM, Appleton D, Bates D, Clark F, et al. The incidence of thyroid disorders in the community: A twenty-year follow-up of the Whickham Survey. Clin Endocrinol (Oxf). 1995;43: 55–68. View Article Google Scholar Chonchol M, Lippi G, Salvagno G, Zoppini G, Muggeo M, Targher G. Prevalence of subclinical hypothyroidism in patients with chronic kidney disease. Clin J Am Soc Nephrol. 2008;3: 1296–1300. pmid:18550654 View Article PubMed/NCBI Google Scholar Åsvold BO, Bjøro T, Vatten LJ. Association of thyroid function with estimated glomerular filtration rate in a population-based study: The HUNT study. Eur J Endocrinol. 2011;164: 101–105. pmid:20930062 View Article PubMed/NCBI Google Scholar Matsuo S, Imai E, Horio M, Yasuda Y, Tomita K, Nitta K, et al. Revised Equations for Estimated GFR From Serum Creatinine in Japan. Am J Kidney Dis. 2009;53: 982–992. pmid:19339088 View Article PubMed/NCBI Google Scholar Kanda Y. Investigation of the freely available easy-to-use software “EZR” for medical statistics. Bone Marrow Transplant. 2013;48: 452–458. pmid:23208313 View Article PubMed/NCBI Google Scholar Chaker L, Bianco AC, Jonklaas J, Peeters RP. Hypothyroidism. Lancet. 2017;390: 1550–1562. pmid:28336049 View Article PubMed/NCBI Google Scholar Skowsky WR, Kikuchi TA. The role of vasopressin in the impaired water excretion of myxedema. Am J Med. 1978;64: 613–621. pmid:645727 View Article PubMed/NCBI Google Scholar Hanna FWF, Scanlon MF. Hyponatraemia, hypothyroidism, and role of arginine-vasopressin. Lancet. 1997;350: 755–756. pmid:9297992 View Article PubMed/NCBI Google Scholar Derubertis FR, Michelis MF, Bloom ME, Mintz DH, Field JB, Davis BB. Impaired water excretion in myxedema. Am J Med. 1971;51: 41–53. pmid:5570319 View Article PubMed/NCBI Google Scholar Rhee CM, Kalantar-Zadeh K, Streja E, Carrero J-J, Ma JZ, Lu JL, et al. The relationship between thyroid function and estimated glomerular filtration rate in patients with chronic kidney disease. Nephrol Dial Transplant. 2015;30: 282–287. pmid:25246335 View Article PubMed/NCBI Google Scholar Download PDF Citation XML Print Share Reddit Facebook LinkedIn Mendeley Bluesky Email Advertisement Subject Areas ? For more information about PLOS Subject Areas, click here. We want your feedback. Do these Subject Areas make sense for this article? Click the target next to the incorrect Subject Area and let us know. Thanks for your help! HypothyroidismIs the Subject Area "Hypothyroidism" applicable to this article? 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18159	https://www.goodreads.com/author/quotes/5217.George_Bernard_Shaw	, Sign In Join Goodreads helps you follow your favorite authors. Be the first to learn about new releases! Start by following George Bernard Shaw. Follow Author George Bernard Shaw > Quotes (?) Quotes are added by the Goodreads community and are not verified by Goodreads. (Learn more) Showing 1-30 of 885 “Life isn't about finding yourself. Life is about creating yourself.” ― George Bernard Shaw tags: inspirational, life, yourself “A life spent making mistakes is not only more honorable, but more useful than a life spent doing nothing.” ― George Bernard Shaw tags: art, life 8191 likes Like “Make it a rule never to give a child a book you would not read yourself.” ― George Bernard Shaw tags: books, children 7964 likes Like “The reasonable man adapts himself to the world: the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man.” ― George Bernard Shaw, Man and Superman tags: life, philosophy, philosophy-of-life, progress 3999 likes Like “You see things; you say, 'Why?' But I dream things that never were; and I say 'Why not?” ― George Bernard Shaw, Back to Methuselah tags: inspirational 3954 likes Like “If you want to tell people the truth, make them laugh, otherwise they'll kill you.” ― George Bernard Shaw tags: kill, laughter, misattributed-oscar-wilde, truth 3929 likes Like “Animals are my friends...and I don't eat my friends.” ― George Bernard Shaw tags: animals, attributed-no-source, friends, humor, vegetarianism 3096 likes Like “Those who cannot change their minds cannot change anything.” ― George Bernard Shaw tags: adaptation, change, open-mindedness, self-improvement 2777 likes Like “There are two tragedies in life. One is to lose your heart's desire. The other is to gain it.” ― George Bernard Shaw, Man and Superman tags: desire, life, lost, paradox, tragedy 1719 likes Like “Youth is wasted on the young.” ― George Bernard Shaw tags: attributed-no-source 1685 likes Like “If you cannot get rid of the family skeleton, you may as well make it dance.” ― George Bernard Shaw, Immaturity tags: family, humor 1394 likes Like “There is no love sincerer than the love of food.” ― George Bernard Shaw, BBC Radio presents Man and superman tags: food, gluttony, love 1290 likes Like “Never wrestle with pigs. You both get dirty and the pig likes it.” ― George Bernard Shaw tags: bernard-shaw, irish, pigs, pointlessness 1145 likes Like “People are always blaming their circumstances for what they are. I don't believe in circumstances. The people who get on in this world are the people who get up and look for the circumstances they want, and if they can't find them, make them.” ― George Bernard Shaw, Mrs. Warren's Profession tags: motivational, overcoming-obstacles 1126 likes Like “The single biggest problem in communication is the illusion that it has taken place.” ― George Bernard Shaw tags: communication, misattributed, misunderstanding 1089 likes Like “When two people are under the influence of the most violent, most insane, most delusive, and most transient of passions, they are required to swear that they will remain in that excited, abnormal, and exhausting condition continuously until death do them part.” ― George Bernard Shaw, Getting Married 1028 likes Like “The liar's punishment is, not in the least that he is not believed, but that he cannot believe anyone else.” ― George Bernard Shaw, The Quintessence of Ibsenism tags: believe, liar, lie, punishment 1024 likes Like “Why should we take advice on sex from the pope? If he knows anything about it, he shouldn't!” ― George Bernard Shaw 873 likes Like “Success does not consist in never making mistakes but in never making the same one a second time.” ― George Bernard Shaw tags: mistakes, success 818 likes Like “A Native American elder once described his own inner struggles in this manner: Inside of me there are two dogs. One of the dogs is mean and evil. The other dog is good. The mean dog fights the good dog all the time. When asked which dog wins, he reflected for a moment and replied, The one I feed the most.” ― George Bernard Shaw tags: conscience, good-and-evil, humanity, native-americans, self-determination 816 likes Like “I am enclosing two tickets to the first night of my new play; bring a friend ... if you have one." — George Bernard Shaw, playwright (to Winston Churchill) "Cannot possibly attend first night; will attend second, if there is one." — Churchill's response” ― George Bernard Shaw 805 likes Like “Patriotism is, fundamentally, a conviction that a particular country is the best in the world because you were born in it....” ― George Bernard Shaw tags: anti-patriotism, identity, nation, nationalism, politics 777 likes Like “He knows nothing; and he thinks he knows everything. That points clearly to a political career.” ― George Bernard Shaw, Major Barbara tags: arrogance, ignorance, politicians, politics 753 likes Like “You use a glass mirror to see your face; you use works of art to see your soul.” ― George Bernard Shaw, Back to Methuselah tags: art, reminding 744 likes Like “This is the true joy in life, being used for a purpose recognized by yourself as a mighty one. Being a force of nature instead of a feverish, selfish little clod of ailments and grievances, complaining that the world will not devote itself to making you happy. I am of the opinion that my life belongs to the whole community and as long as I live, it is my privilege to do for it what I can. I want to be thoroughly used up when I die, for the harder I work, the more I live. I rejoice in life for its own sake. Life is no brief candle to me. It is a sort of splendid torch which I have got hold of for the moment and I want to make it burn as brightly as possible before handing it on to future generations.” ― George Bernard Shaw 730 likes Like “A pessimist is a man who thinks everybody is as nasty as himself, and hates them for it.” ― George Bernard Shaw tags: funny, humor, optimism, pessimism 707 likes Like “Dancing is a perpendicular expression of a horizontal desire.” ― George Bernard Shaw tags: dance, dancing, definition, innuendo, sex, widely-misattributed, wordplay 703 likes Like “My way of joking is to tell the truth. It's the funniest joke in the world.” ― George Bernard Shaw, John Bull's Other Island tags: humor 676 likes Like “If you have an apple and I have an apple and we exchange these apples then you and I will still each have one apple. But if you have an idea and I have an idea and we exchange these ideas, then each of us will have two ideas.” ― George Bernard Shaw tags: ideas Like “After all, the wrong road always leads somewhere.” ― George Bernard Shaw 533 likes Like « previous 1 2 3 4 5 6 7 8 9 … 29 30 next » All Quotes \| Add A Quote Books by George Bernard Shaw Pygmalion and Three Other Plays 18,205 ratings Pygmalion / My Fair Lady 15,681 ratings Arms and the Man 11,178 ratings Saint Joan 9,537 ratings Open Preview See a Problem? We’d love your help. Let us know what’s wrong with this preview of Saint Joan by George Bernard Shaw. Thanks for telling us about the problem. Not the book you’re looking for? Preview — Saint Joan by George Bernard Shaw Welcome back. Just a moment while we sign you in to your Goodreads account.
18160	https://www.htmlgoodies.com/javascript/building-a-better-tostring-method/	Building a Better toString() Method \| HTML Goodies ___ Getting Started HTML HTML5 CSS Javascript Guides Mobile Design Webmaster XML ASP Java Database .Net Perl PHP Security CMS SEO Video News Search Monday, September 29, 2025 Facebook Linkedin Twitter Getting Started HTML HTML5 CSS Javascript Guides More Mobile Design Webmaster XML ASP Java Database .Net Perl PHP Security CMS SEO Video News Search HomeJavascript Building a Better toString() Method ByRob Gravelle August 13, 2015 FacebookTwitterLinkedinEmailPrint The humble toString() method is widely implemented across numerous programming languages. It provides a simple and convenient mechanism for representing an object’s state in readable form. Not surprisingly, it’s used extensively in debugging during development. Its other many uses include logging, passing informative error messages to exceptions, and displaying information to the client without any need for specialized formatting and conversion methods. Unfortunately, this powerful tool in debugging and troubleshooting is all-too-often left to its default implementation by developers. And that’s a shame, because it’s really easy to write your own. In today’s article, I’ll show you how to write a dynamic toString() in Java. Java’s Default toString() Implementation The makers of Java have made to String() accessible to all objects by placing it in the root Object from which all other objects inherit from. All that’s required in order to use it is to either invoke it explicitly or pass the object to a method that expects a String argument, like sysout. That causes an object’s toString() to be invoked automagically. Unfortunately, the default Object implementation of toString() is not the most informative. It returns a string consisting of the name of the class, the “at” sign character ‘@’, followed by the hexadecimal representation of the hash code of the object. In other words, this method returns a string equal to the value of: getClass().getName() + '@' + Integer.toHexString(hashCode()) Here’s what was displayed for an instance of the Bank class that we’ll be using as our sample object today: ca.gc.cbsa.banking.models.Bank@fc5408 At least we know that it’s a Bank class! Generally, developers will want to override toString() to display more pertinent details about an object state – i.e. its attributes. That usually leads to something like this: @Override public String toString(){ return "Account Number: " + accountNumber + "n" + "Account Owner: " + accountOwner + "n" + "Overdraft Protection?: " + hasOverdraftProtection + "n" + "Account Balance: " + balance + "n" + "Service Fee: " + serviceFee; } Sure, it produces far superior output to the default implementation, but who want’s to repeat that exercise for every class they create? A Simple toString() Implementation using Reflection Java’s powerful Reflection API allows us to inspect an application’s classes, interfaces, fields and methods at runtime, without knowing their names at compile time. Because Reflection can be utilized to circumvent Object-Oriented programming’s built-in security features – namely encapsulation – reliance on Reflection should be mainly limited to development, testing, and debugging. Of course, there are exceptions, such as mapping objects to tables in a database at runtime, like Butterfly Persistence does. But before we do that, let’s take a look at our test class. The Bank class has three Fields: the branch name, an ArrayList of accounts, and a HashMap that links accounts to their owners. Without a scope modifier, these Fields would have the default visibility of Package. Note that methods have mostly been removed for brevity. package ca.gc.cbsa.banking.models; import java.util.ArrayList; import java.util.HashMap; import com.robgravelle.utils.ObjectUtils; public class Bank { String branchName; ArrayList accounts = new ArrayList(); HashMap> clientMap = new HashMap>(); public Bank(String aBranchName, ArrayList<Account> someAccounts){ branchName = aBranchName; accounts = someAccounts; initializeClientMap(); } public ArrayList<Account> getAccounts() { return accounts; } public void setAccounts(ArrayList<Account> accounts) { this.accounts = accounts; } public String getBranchName() { return branchName; } public void setBranchName(String branchName) { this.branchName = branchName; } } This simple toString() method iterates over a Class’s declared fields and displays them in a nicely formatted way. Reflection methods are exposed via the Class object. It’s a special type of object that provides information about the class. Obtaining a reference to the Class is as easy as calling a class instance’s getClass() method. From there, the getDeclaredFields() method returns an array of declared fields. The Field object in turn has a method to get the field name and value: getName() and get() respectively. @Override public String toString() { StringBuilder result = new StringBuilder(); String newLine = System.getProperty("line.separator"); result.append(this.getClass().getName()); result.append(" Object {"); result.append(newLine); //determine fields declared in this class only (no fields of superclass) Field[] fields = this.getClass().getDeclaredFields(); //print field names paired with their values for (Field field : fields) { result.append(" "); field.setAccessible(true); try { result.append(field.getName()); result.append(": "); //requires access to private field: result.append(field.get(this)); } catch (IllegalAccessException ex) { System.out.println(ex); } result.append(newLine + newLine); } result.append("}"); return result.toString(); } By calling Field.setAcessible(true) you turn off the access checks for the Field instance so that you can access it even if it is private, protected or package scope. Setting it to true doesn’t hurt if the Field has public scope, so there’s really no harm in applying it to every field. Testing Our toString() Method Now we are ready to try out our toString() method. The Bank constructor requires a branch name and an ArrayList of BankAccounts (not shown). Each account in turn contains an account number, the holder’s name, and opening balance. public static void main(String[] args) { ArrayList accounts = new ArrayList() {{ this.add(new BankAccount(1234, "Rob Gravelle", 100.00)); this.add(new BankAccount(2345, "Al Bundy", 0.00)); this.add(new BankAccount(3456, "Sue Bastianich", 1000.00)); }}; Bank bank = new Bank("Main branch", accounts ); System.out.println(bank); } Running the above main() method produces the following informative output: ca.gc.cbsa.banking.models.Bank Object { branchName: Main branch accounts: [Account Number: 1234 Account Holder: Rob Gravelle Account Balance: 100.0 Service Fee: 5.0, Account Number: 2345 Account Holder: Al Bundy Account Balance: 0.0 Service Fee: 5.0, Account Number: 3456 Account Holder: Sue Bastianich Account Balance: 1000.0 Service Fee: 0.0] clientMap: {Sue Bastianich=[Account Number: 3456 Account Holder: Sue Bastianich Account Balance: 1000.0 Service Fee: 0.0], Rob Gravelle=[Account Number: 1234 Account Holder: Rob Gravelle Account Balance: 100.0 Service Fee: 5.0], Al Bundy=[Account Number: 2345 Account Holder: Al Bundy Account Balance: 0.0 Service Fee: 5.0]} } Promoting Reusability For better reusability, you can place the real code in a utility class: package com.robgravelle.utils; import java.lang.reflect.Field; import java.util.ArrayList; import ca.gc.cbsa.banking.interfaces.Account; import ca.gc.cbsa.banking.models.Bank; import ca.gc.cbsa.banking.models.BankAccount; public class ObjectUtils { public static String objToString(Object obj) { StringBuilder result = new StringBuilder(); String newLine = System.getProperty("line.separator"); result.append(obj.getClass().getName()); result.append(" Object {"); result.append(newLine); //determine fields declared in this class only (no fields of superclass) Field[] fields = obj.getClass().getDeclaredFields(); //print field names paired with their values for (Field field : fields) { field.setAccessible(true); result.append(" "); try { result.append(field.getName()); result.append(": "); //requires access to private field: result.append(field.get(obj)); } catch (IllegalAccessException ex) { System.out.println(ex); } result.append(newLine + newLine); } result.append("}"); return result.toString(); } } Now all that the toString() does is call the static delegate method and pass the class instance to it: @Override public String toString() { return ObjectUtils.objToString(this); } Conclusion Even though it’s not hard to do, I wouldn’t recommend writing your own generic toString() method from scratch unless you have very particular requirements. In up-coming articles, we’ll learn how to piggy-back on the shoulders of toString() trail blazers! FacebookTwitterLinkedinEmailPrint Previous article Adobe Flash Player 18 Combats Recent Boycott in Mozilla with .209 Release Next article Provide a JSON Feed from your WordPress Site using the JSON API Plugin Rob Gravelle Rob Gravelle resides in Ottawa, Canada, and has been an IT guru for over 20 years. In that time, Rob has built systems for intelligence-related organizations such as Canada Border Services and various commercial businesses. In his spare time, Rob has become an accomplished music artist with several CDs and digital releases to his credit. Get the Free Newsletter! Subscribe to Developer Insider for top news, trends & analysis Email Address [x] By subscribing, you agree to our Terms of Use and Privacy Policy. 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18161	https://mathoverflow.net/questions/468056/packing-an-upwards-equilateral-triangle-efficiently-by-downwards-equilateral-tri	mg.metric geometry - Packing an upwards equilateral triangle efficiently by downwards equilateral triangles - MathOverflow Join MathOverflow 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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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 Packing an upwards equilateral triangle efficiently by downwards equilateral triangles Ask Question Asked 1 year, 6 months ago Modified1 year, 5 months ago Viewed 2k times This question shows research effort; it is useful and clear 30 Save this question. Show activity on this post. Consider the problem of packing an upwards-pointing unit equilateral triangle "efficiently" by downwards-pointing equilateral triangles, where "efficiently" means that there is little wasted area relative to the perimeter of the triangles used in the packing. The n t h n t h generation of the Sierpinski triangle packs all but (3/4)n(3/4)n of the area of the large upwards triangle by downwards triangles, at the cost of a net perimeter of ≍(3/2)n≍(3/2)n. Thus, if we let ε ε denote the area not packed, then the perimeter of the triangles used in this construction is ≫ε−α≫ε−α for α=log(3/2)log(4/3)=1.409…α=log⁡(3/2)log⁡(4/3)=1.409…. My question is whether this phenomenon is general: given any finite collection of downward equilateral triangles in the upward unit equilateral triangle that is a packing (i.e., interiors are disjoint) and leaves an area of ε ε not covered, is it true that the total perimeter of the triangles used is of the form ≫ε−c≫ε−c for some absolute constant c>0 c>0? For my application I do not need an optimal exponent c c. [EDIT: as pointed out in answers, to make the answer positive for large ε ε, the outer triangle should also count towards the total perimeter.] I think I can establish a bound of the form ≫log 1 ε≫log⁡1 ε (roughly speaking, by arguing that every dyadic scale of triangles between ε ε and 1 1 has to contribute a constant amount of perimeter, otherwise there will be too much waste), but for my application I really need a polynomial lower bound (or maybe exp((log log 1 ε)C)exp⁡((log⁡log⁡1 ε)C) for a large absolute constant C C might suffice). It's intuitively plausible to me that the Sierpinski packing is the "best" packing for this purpose, and that the smaller triangles really have to contribute more than a constant amount of perimeter, but I am finding it surprisingly tricky to locate a rigorous argument. Perhaps this sort of question has already been studied in the literature? One can interpret this question as an exotic form of an isoperimetric inequality, where the region of interest is required to be a disjoint union of downwards pointing equilateral triangles in a fixed upward pointing triangle, but this question seems rather orthogonal to the usual theory of isoperimetric inequalities, so I don't believe that this interpretation is particularly fruitful. mg.metric-geometry packing-and-covering triangles 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 Apr 4, 2024 at 21:59 Terry TaoTerry Tao asked Mar 30, 2024 at 21:22 Terry TaoTerry Tao 118k 34 34 gold badges 479 479 silver badges 564 564 bronze badges 6 1 The trivial lower bound is e c log 1/ε√e c log⁡1/ε. Will that be sufficient?fedja –fedja 2024-03-31 03:23:47 +00:00 Commented Mar 31, 2024 at 3:23 That should suffice I think.Terry Tao –Terry Tao 2024-03-31 03:53:33 +00:00 Commented Mar 31, 2024 at 3:53 When it is not clear whether I am right or wrong, I would never begin a question with "am I being an idiot".Alex M. –Alex M. 2024-04-02 07:56:22 +00:00 Commented Apr 2, 2024 at 7:56 @AlexM. And yet I and many of my British contemporaries might well do exactly that. What is your point?Yemon Choi –Yemon Choi 2024-04-02 10:38:30 +00:00 Commented Apr 2, 2024 at 10:38 1 @AtypicalAnorexic this is the case for the specific packing induced by the Sierpinski triangle, but in general a packing of upwards triangles does not canonically induce a complementary packing of downwards triangles .Terry Tao –Terry Tao 2024-04-02 14:11:53 +00:00 Commented Apr 2, 2024 at 14:11 \|Show 1 more comment 2 Answers 2 Sorted by: Reset to default This answer is useful 11 Save this answer. Show activity on this post. Ok I think this is an argument that the perimeter at least ε−c ε−c for some sufficiently small c c. To avoid special cases where ε ε is large, we use the convention that we count the sides of the upwards-pointing triangle as part of the total perimeter. Claim 1: Consider any packing as above. Let p p be the total perimeter of triangles, and let s s be the side-length of the smallest triangle. Then s=O(ε/p).s=O(ε/p). The proof idea is the following observation. For any downwards-pointing triangle T T consider the set of points at distance at most, say, s/100 s/100 from T T. The majority of these points are not covered, and have T T their closest triangle. Summing this over all triangles gives ε=Ω(p s)ε=Ω(p s), or, equivalently, s=O(ε/p)s=O(ε/p). Claim 2:p≥ε−c p≥ε−c for any sufficiently small c>0 c>0. We prove this by induction on the number of triangles. With the convention that the borders of the upwards-pointing triangle are counted as part of the perimeter, the statement is clearly true for small c>0 c>0 for any packing of exactly one triangle, so the base case is clear. Now suppose the inequality holds for all packings with up to n n triangles. Let T 1,T 2,…,T n+1 T 1,T 2,…,T n+1 denote a packing of n+1 n+1 triangles ordered in decreasing size. For any 1≤i≤n+1 1≤i≤n+1, let ε i ε i denote area not covered by the first i i triangles, p i p i the total perimeter of the first i i triangles, and s i s i the side length of the i i:th triangle respectively. By the induction hypothesis, we know that p n≥ε−c n p n≥ε n−c. Hence p n+1≥p n+3 s n+1≥ε−c n+3 s n+1,p n+1≥p n+3 s n+1≥ε n−c+3 s n+1, and the claim follows if we can show that ε−c n+3 s n+1≥ε−c n+1.ε n−c+3 s n+1≥ε n+1−c. To evaluate this, observe that ε−c n=(ε n+1+(3–√/4)s 2 n+1)−c≥ε−c n+1(1−(3–√/4)c s 2 n+1/ε n+1),ε n−c=(ε n+1+(3/4)s n+1 2)−c≥ε n+1−c(1−(3/4)c s n+1 2/ε n+1), where the last step follows from the convexity of (1+x)−c(1+x)−c. Plugging this into the inequality above, it remains to show 3 s n+1≥(3–√/4)c s 2 n+1/ε 1+c n+1 3 s n+1≥(3/4)c s n+1 2/ε n+1 1+c, or 4 3–√≥c s n+1/ε 1+c n+1 4 3≥c s n+1/ε n+1 1+c. Observe that we can assume that ε n+1/ε n ε n+1/ε n and p n+1/p n p n+1/p n are both Θ(1)Θ(1). The former is because, by Claim 1, ε n+1=Ω(s n+1 p n+1)≥Ω(s 2 n+1).ε n+1=Ω(s n+1 p n+1)≥Ω(s n+1 2). The latter is because n≥1 n≥1 and triangles are ordered decreasingly. In particular, this lets us also assume that p n+1=Θ(ε−c n+1)p n+1=Θ(ε n+1−c). So c s n+1/ε 1+c n+1 c s n+1/ε n+1 1+c is up to a constant factor equal to c s n+1 p n+1/ε n+1 c s n+1 p n+1/ε n+1, which by Claim 1 is O(c)O(c). In particular, choosing c>0 c>0 sufficiently small, this expression is less than 4 3–√4 3, as desired. 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 Apr 3, 2024 at 17:04 Anders MartinssonAnders Martinsson 231 1 1 silver badge 4 4 bronze badges 5 1 "the statement is clearly true for small c>0 c>0 for any packing of exactly one triangle, so the base case is clear." Really? I would say exactly the opposite: for every c>0 c>0 we can start packing with a single triangle of sufficiently small size to make the base false (ε ε is constant and p p is small...)fedja –fedja 2024-04-03 23:19:04 +00:00 Commented Apr 3, 2024 at 23:19 1 The base can be fixed though by waiting until ε=0.000001 ε=0.000001 in which case we know that the perimeter is at least 10 10 or so. The induction step seems correct. Congratulations!fedja –fedja 2024-04-03 23:53:42 +00:00 Commented Apr 3, 2024 at 23:53 Thanks a lot! Concerning the base case, the solution I had for this is to count the border as part of the total perimeter. So p p is always at least 3 3 (crucually, this convention also works in the proof of Claim 1). I think then the base case is clear? More generally, I guess this means there is nothing 'magical' about it being triangles. Packing any shape that satisfies Claim 1 into an area for which the base case is true will also give a ε−c ε−c perimeter.Anders Martinsson –Anders Martinsson 2024-04-04 06:34:01 +00:00 Commented Apr 4, 2024 at 6:34 4 Nice argument! I like how it identifies that Claim 1 is the key geometric feature of the shapes used.Terry Tao –Terry Tao 2024-04-04 22:00:42 +00:00 Commented Apr 4, 2024 at 22:00 I wonder if one can make the constant in Claim 1 sharp in the Sierpinski gasket case - i.e. is the number of points with T T as their closest triangle always at least the side length of T T times the side length of the smallest triangle times the area of an equilateral triangle with side length 1 1? Just visualizing the Voronoi cells this seems right but I don't quite have an argument. And this wouldn't quite make the overall argument sharp in the Sierpinski gasket case.Will Sawin –Will Sawin 2024-05-16 18:00:33 +00:00 Commented May 16, 2024 at 18:00 Add a comment\| This answer is useful 17 Save this answer. Show activity on this post. OK, posting then. I prefer to think of triangles pointing to the right in the triangle pointing to the left. Let δ=e−log 1/ε√δ=e−log⁡1/ε. For each small triangle T T, let I I be the interval of length δ δ times the length of the projection of T T to the horizontal axis with the same endpoint. If the sum of lengths of I I's is noticeable, we are done. Otherwise, for each x x not in the union of I I's, the intersections of the triangles with the corresponding vertical line form a system of intervals J J satisfying dist(J′,J′′)≥δ min(\|J′\|,\|J′′\|)dist⁡(J′,J″)≥δ min(\|J′\|,\|J″\|) and for many x x those intervals cover the cross-section up to ε ε or so. Now assume that we have K=K(x)K=K(x) intervals in this system. All gaps are at most ε ε. Take every fifth gap and remove the adjacent interval of length ≤ε/δ≤ε/δ. We'll remove a fixed portion of the intervals and the gaps will become ≤ε/δ≤ε/δ (well, +2 ε+2 ε, of course, but who cares?). Repeat for 1 2 log 1/ε−−−−−−√1 2 log⁡1/ε steps. Either we will be still left with something, in which case we have K≥e c log 1/ε√K≥e c log⁡1/ε (at each step we remove a fixed portion of all remaining intervals), or we'll eliminate everything using only intervals of length ε√ε or less, which results in K≥ε−1/2 K≥ε−1/2, which is even better (our intervals have to almost cover the cross-section). But then the total perimeter is at least ∫K(x)d x∫K(x)d x over the good set and we are done again. There may be some room for improvement here, but that would require some more thinking :-) 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 Mar 31, 2024 at 4:16 fedjafedja 63.8k 11 11 gold badges 164 164 silver badges 308 308 bronze badges 3 1 Thanks! For my application, it is actually the lower bound on sup x K(x)sup x K(x) which is important, rather than the perimeter, and your argument actually gives a polynomial bound in that case, so I have all that I need for my application, thanks! The general case (where one seeks a polynomial lower bound on the perimeter) still seems intellectually interesting, though.Terry Tao –Terry Tao 2024-03-31 14:57:24 +00:00 Commented Mar 31, 2024 at 14:57 2 For sup x K(x)sup x K(x), couldn't you look at the set of points in the right-most (100 ϵ)(100 ϵ)-fraction of our big triangle? At most 10 percent (say) of these points are uncovered. So by averaging, there is some x x in this very-right region with at least 90 percent of points in the vertical slice covered. However, every right-pointing triangle contributes intervals of length at most O(ϵ)O(ϵ) inside this region. So we need Ω(1/ϵ)Ω(1/ϵ) intervals.Zach Hunter –Zach Hunter 2024-03-31 16:52:48 +00:00 Commented Mar 31, 2024 at 16:52 2 One can also make this little trick work 'away from the boundary'. The idea is that if we do not have polynomially large perimeter, we must have some large triangle T∗T∗ with side-length >ϵ c>ϵ c (for some small choice of c>0 c>0). But then, if we flip T∗T∗ triangle along its vertical base, we can look at the (100 ϵ 1−c)(100 ϵ 1−c)-fraction of right-most points in this flipped area. And the trick works here too, which allows you to find values of x x away from the right-most boundary of the host triangle (since T∗T∗ should appear in lots of places).Zach Hunter –Zach Hunter 2024-03-31 17:55:54 +00:00 Commented Mar 31, 2024 at 17:55 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 mg.metric-geometry packing-and-covering triangles See similar questions with these tags. 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18162	https://www.cirrelt.ca/documentstravail/cirrelt-2013-11.pdf	____ Worst-Case Analysis for New Online Bin Packing Problems Mauro Maria Baldi Teodor Gabriel Crainic Guido Perboli Roberto Tadei February 2013 CIRRELT-2013-11 G1V 0A6 Bureaux de Montréal : Bureaux de Québec : Université de Montréal Université Laval C.P. 6128, succ. Centre-ville 2325, de la Terrasse, bureau 2642 Montréal (Québec) Québec (Québec) Canada H3C 3J7 Canada G1V 0A6 Téléphone : 514 343-7575 Téléphone : 418 656-2073 Télécopie : 514 343-7121 Télécopie : 418 656-2624 www.cirrelt.ca Worst-Case Analysis for New Online Bin Packing Problems Mauro Maria Baldi1, Teodor Gabriel Crainic2,3,, Guido Perboli1,2, Roberto Tadei1 1 Department of Control and Computer Engineering, Politecnicò di Torino, Corso Duca degli Abruzzi, 24 - I-10129 Torino, Italy 2 Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT) 3 Department of Management and Technology, Université du Québec à Montréal, P.O. Box 8888, Station Centre-Ville, Montréal, Canada H3C 3P8 Abstract. We consider two new online bin packing problems, the online Variable Cost and Size Bin Packing Problem (o-VCSBPP) and the online Generalized Bin Packing Problem (o-GBPP). We take two well-known bin packing algorithms to address them, the First Fit (FF) and the Best Fit (BF). We show that both algorithms have an asymptotic worst-case ratio bound equal to 2 for the o-VCSBPP and this bound is tight. When there are enough bins of a particular type to load all items, FF and BF also have an absolute worst-case ratio bound equal to 2 for the o-VCSBPP, and this bound is also tight. In addition, we prove that no worst-case ratio bound of FF and BF can be computed for the o-GBPP. Therefore, we consider a natural evolution of these algorithms, the First Fit with Rejection (FFR) and the Best Fit with Rejection (BFR), able to reject inconvenient bins at the end of the process. Similarly, we prove that no worst-case ratio of these algorithms can be computed for the o-GBPP. Finally, we give sufficient conditions under which algorithms do not admit any performance ratio, and conclude that the worst-case results obtained for the o-VCSBPP and the o-GBPP also hold for the offline variant of these two problems. Keywords. Online bin packing problems asymptotic worst-case ratio absolute worst-case ratio. Acknowledgements. This project has been partially funded by the Natural Sciences and Engineering Council of Canada (NSERC), through its Discovery Grants program, and the Italian Ministry of Education, University, and Research, under the PRIN 2009 project “Methods and Algorithms for the Logistics Optimization”. Results and views expressed in this publication are the sole responsibility of the authors and do not necessarily reflect those of CIRRELT. Les résultats et opinions contenus dans cette publication ne reflètent pas nécessairement la position du CIRRELT et n'engagent pas sa responsabilité. ________ Corresponding author: TeodorGabriel.Crainic@cirrelt.ca Dépôt légal – Bibliothèque et Archives nationales du Québec Bibliothèque et Archives Canada, 2013 © Copyright Baldi, Crainic, Perboli, Tadei and CIRRELT, 2013 1 Introduction The Bin Packing Problem (BPP), both online and oﬄine [12, 7, 4], is a widely studied problem. It consists of ﬁnding the minimum number of bins, with the same capacity, in order to accommodate a set of items satisfying capacity constraints. Johnson proposed the Next Fit (NF) algorithm for this problem and proved that its performance ratio is 2. Johnson et al. showed that the First Fit (FF) and the Best Fit (BF) algorithms have both performance ratios of 17/10. Crainic et al. conducted an asymptotic worst-case analysis on lower bounds for the BPP. The BPP was exploited in many ﬁelds such as computer science and engineering, transportation, logistics, and telecommunications. Due to its theoretical and practical relevance, several variants and richer settings were proposed. Li and Chen studied the variant where all bins have the same capacity but are characterized by a nondecreasing concave cost function of the bin utilization. The authors proved that for this problem the FF and BF algorithms have asymptotic and absolute worst-case ratios equal to 2. Friesen and Langston proposed the Variable Sized Bin Packing Problem (VSBPP), where bins with diﬀerent sizes are available and the goal is to minimize the wasted space. The authors provided one online and two oﬄine algorithms and proved that their worst-case ratios are 2, 3/2, and 4/3, respectively. Kang and Park studied this problem by also considering the bin costs, but assuming that the bin unit cost does not increase as the bin size increases. They provided two oﬄine algorithms and showed that their asymptotic worst-case ratio is equal to 3/2. The most advanced variants of the BPP are the Variable Cost and Size Bin Packing Problem (VCSBPP) and the Generalized Bin Packing Problem (GBPP) [2, 1]. In the VCSBPP, in addition to having diﬀerent sizes, bins also have diﬀerent ﬁxed selection costs, which are not necessarily correlated to the bin sizes. Crainic et al. proposed accurate bounds. No perfomance ratio is available for the VCSBPP, either online or oﬄine. Finally, the GBPP generalizes several packing problems characterized by multiple item and bin attributes, and the presence of both compulsory and non-compulsory items. Exact and approximated methods for this problem were proposed in . No performance ratio exists for this problem either. Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 Although the VCSBPP and the GBPP were studied in their oﬄine variant, to the best of our knowledge, nobody has ever addressed their online variant. For this reason, we de-cided to consider the online VCSBPP (o-VCSBPP) and the online GBPP (o-GBPP), and perform a worst-case analysis of two well-known algorithms to address these problems, the FF and the BF. In this paper, we show that, although the o-VCSBPP is a more general problem than the one studied by Li and Chen , we can still guarantee the same asymptotic worst-case ratio bound equal to 2 and this bound is tight. Furthermore, we prove that, when there are enough bins of a particular type to load all items, we can guarantee for the o-VCSBPP an absolute worst-case ratio bound equal to 2 of both algorithms and this bound is also tight. For the o-GBPP, we prove that no worst-case ratio bound of FF and BF can be computed. We also show that one drawback of the FF and the BF algorithm when applied to the o-GBPP is that, at the end of the process, there might be some bins whose cost is greater than the total proﬁt of the items loaded into them. For this reason, a natural evolution of these algorithms is considered: the First Fit with Rejection (FFR) and the Best Fit with Rejection (BFR). Both are able to reject inconvenient bins at the end of the process. We also prove that no worst-case ratio of these algorithms can be computed. Finally, we give suﬃcient conditions under which algorithms do not admit any per-formance ratio, and conclude that the worst-case results obtained for the o-VCSBPP and the o-GBPP also hold for the oﬄine variant of these two problems. This paper is organized as follows. In Section 2, we introduce the o-VCSBPP and the o-GBPP and, in Section 3, we present the worst-case analysis of FF and BF for the o-VCSBPP and the o-GBPP. 2 The new online bin packing problems 2.1 The o-VCSBPP In the VCSBPP , a set of bins J , with \|J \| = m, and a set of items I, with \|I\| = n, are given. The bins are classiﬁed into types belonging to the set T . Each item i ∈I is characterized by volume wi and each bin of type t ∈T is characterized by cost Ct and capacity Wt. Without loss of generality, when \|T \| > 1 we assume that Ct Wt ≤Ct+1 Wt+1 , ∀t ∈T {\|T \|}. (1) 2 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 The goal is to accommodate all items into proper bins in order to minimize the overall cost, given by the sum of the costs of the selected bins. In the o-VCSBPP, items arrive online to a decision maker. When the decision maker receives an item, the item information is revealed. This information consists of the item volume and whether the incoming item is the last one. 2.2 The o-GBPP In the GBPP , a set of bins J , with \|J \| = m, and a set of items I, with \|I\| = n, are given. The bins are classiﬁed into types belonging to the set T . The set of items I is composed by two subsets, IC and INC. IC is the set of compulsory items, i.e., those items that are mandatory to load, whilst INC is the set of non-compulsory items, i.e., those items that might not be loaded. Each item i ∈I is characterized by volume wi and proﬁt pi, and each bin of type t ∈T is characterized by cost Ct and capacity Wt. The goal is to minimize the net overall cost, i.e., the diﬀerence between the total cost of the selected bins and the total proﬁt of the selected non-compulsory items. Note that, the total proﬁt of the compulsory items is not taken into account because, since compulsory items must be loaded, their total proﬁt would act as a constant within the objective function. In the o-GBPP, items also arrive online to a decision maker. When the decision maker receives an item, the item information is revealed. This information consists of the item volume, the item proﬁt, whether the item is compulsory or non-compulsory, and whether the incoming item is the last one. If the given m bins are not enough to load all items for the o-VCSBPP and all compul-sory items for the o-GBPP, then these two problems are clearly infeasible. Infeasibility can be handled as in by introducing a dummy bin type. In particular, we add a special bin of type v with volume Wv = P i ∈I wi for the o-VCSBPP and Wv = P i ∈IC wi for the o-GBPP, and set its cost Cv to a value much higher than the costs of the other bins in order to discourage its use, e.g., Cv ≫P t ∈T Ct. 3 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 3 Worst-case analysis of FF and BF for the o-VCSBPP and the o-GBPP 3.1 The asymptotic and the absolute worst-case ratios Several denominations and deﬁnitions can be found in the literature for the asymptotic and the absolute worst-case ratio. Sometimes they are named performance ratios or com-petitive ratios. They measure the gap between the solution value found by an algorithm and the optimum in the worst-case. Formally, given a minimization problem Π, an instance I ∈Π of the problem, and an algorithm A, the value of the solution yielded by the algorithm is A(I) and the optimum is OPT(I). The asymptotic worst-case ratio is the smallest positive R such that the following relation holds for any instance of the problem A(I) ≤R · OPT(I) + O(1), ∀I ∈Π (2) The absolute worst-case ratio is the smallest positive ρ such that the following relation holds for any instance of the problem A(I) ≤ρ · OPT(I), ∀I ∈Π (3) 3.2 The FF and the BF algorithms For the o-VCSBPP and the o-GBPP, FF works as follows. Each time an item arrives online, the decision maker tries to load it into the ﬁrst open bin which can contain it. If none among the open bins has enough residual space to accommodate the new item, then a new bin is opened. The type of this new bin is the one with the smallest cost over volume ratio among the available bin types (for the sake of simplicity, we assume this type is unique). According to (1), this also means opening all type 1 bins ﬁrst. If there are not enough bins to accommodate all items, then type 2 bins will be opened, and so on. Similarly, BF works as FF for both problems with the only diﬀerence that, each time an item arrives online, the decision maker tries to load it into the best open bin, i.e., the one with the smallest residual space after depositing the item. If no open bins have enough residual space to accommodate the new item, then a new bin is opened as done 4 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 in FF. Given any instance I of problem Π, with Π ∈{o −V CSBPP, o −GBPP}, and any algorithm A applied to instance I ∈Π, we name • T A ⊆T the set of bin types selected by algorithm A • J A ⊆J the set of bins selected by algorithm A, with p = \|J A\| • J A t ⊆J A the set of bins of type t ∈T A selected by algorithm A, with pt = \|J A t \|. These sets are clearly a partition of J A, i.e., ∪t∈T AJ A t = J A, ∩t∈T AJ A t = ∅, and P t∈T A pt = p • J ∗⊆J the set of bins in an optimal solution of instance I ∈Π, with q = \|J ∗\| • σ : J →T an indicator function such that, given bin j ∈J , σ(j) is its type t ∈T • β(j) the level of bin j ∈J A ∪J ∗, i.e., the total volume of the items loaded into bin j • open bin any used bin. 3.3 Worst-case analysis of FF and BF for the o-VCSBPP Theorem 1 The asymptotic worst-case ratio of the FF and the BF algorithms for the o-VCSBPP has a bound equal to 2 and this bound is tight. Proof. Let A ∈{FF, BF} be either the FF or the BF algorithms. In general, there might not be enough type 1 bins to accommodate all items and more than one bin type is necessary. For each set J A t , there is at most one bin i ∈J A t such that β(i) ≤Wt 2 . If, by contradiction, another bin j ∈J A t existed such that β(j) ≤Wt 2 , then the items in bins i and j could be merged together into a unique bin. Therefore, two cases hold 1. all bins in J A t have a level greater than half of their capacity 2. all bins but one in J A t have a level greater than half of their capacity. Case 1 5 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 We have that β(j) > Wt 2 ∀j ∈J A t , ∀t ∈T A (4) By (4) we have X j∈J A t β(j) = pt X j=1 β(j) > pt X j=1 Wt 2 = Wt 2 pt (5) Case 2 There exists a bin i ∈J A t with β(i) ≤Wt 2 and β(j) > Wt 2 , ∀j ∈J A t {i}, ∀t ∈T A. Moreover, for each bin j ∈J A t {i} it must be β(j) + β(i) > Wt ∀j ∈J A t {i}, ∀t ∈T A (6) otherwise the items of bin i could be merged with the items of another open bin in J A t . We have X j∈J A t β(j) = X j∈J A t {i} β(j) + β(i) > > X j∈J A t {i} (Wt −β(i)) + β(i) = = (pt −1) (Wt −β(i)) + β(i) = = (pt −2) (Wt −β(i)) + (Wt −β(i)) + β(i) = = Wt + (pt −2) (Wt −β(i)) ≥ ≥ Wt + (pt −2) Wt −Wt 2 = Wt 2 pt (7) which is like (5) in Case 1. In both cases, we have X j∈J A β(j) = X t∈T A X j∈J A t β(j) > X t∈T A Wtpt 2 (8) 6 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 Therefore, for a general instance I A(I) = X t∈T A ptCt = 2 X t∈T A Wtpt 2 Ct Wt (9) Note that, by (1), Ct Wt ≤max t∈T A Ct Wt = C\|T A\| W\|T A\| . Consequently, (9) becomes A(I) = 2 X t∈T A Wtpt 2 Ct Wt < 2 X t∈T A Wtpt 2 C\|T A\| W\|T A\| (10) Moreover, there must be a proper ∆≥0 such that C\|T A\| W\|T A\| = C1 W1 + ∆ (11) We have A(I) < 2 X t∈T A Wtpt 2 C\|T A\| W\|T A\| = = 2 X t∈T A Wtpt 2 C1 W1 + ∆ = = 2 C1 W1 + ∆ X t∈T A Wtpt 2 (12) Applying (8) to (12), we get A(I) < 2 C1 W1 + ∆ X t∈T A Wtpt 2 < 2 C1 W1 + ∆ X j∈J A β(j) (13) Since all items are loaded, the sum of the levels of the bins in the solution yielded by algorithm A is equal to the sum of the levels of the bins in an optimal solution and so to the total item volume, i.e. P j∈J A β(j) = P j∈J ∗β(j) = P i∈I wi. Therefore, (13) becomes 7 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 A(I) < 2 C1 W1 + ∆ X j∈J A β(j) = = 2 C1 W1 + ∆ X j∈J ∗ β(j) = = 2 C1 W1 X j∈J ∗ β(j) + 2∆ X j∈J ∗ β(j) = = 2 X j∈J ∗ C1 W1 β(j) + 2∆ X i∈I wi (14) Due to (1), we have C1 W1 ≤Cσ(j) Wσ(j) , ∀j ∈J ∗. Then A(I) < 2 X j∈J ∗ C1 W1 β(j) + 2∆ X i∈I wi ≤2 X j∈J ∗ Cσ(j) Wσ(j) β(j) + 2∆ X i∈I wi (15) As β(j) ≤Wσ(j), ∀j ∈J ∗, we get A(I) < 2 X j∈J ∗ Cσ(j) Wσ(j) β(j)+2∆ X i∈I wi ≤2 X j∈J ∗ Cσ(j) Wσ(j) Wσ(j)+2∆ X i∈I wi = 2 OPT(I)+2∆ X i∈I wi (16) which means that algorithm A has an asymptotic worst-case ratio bound equal to 2. To prove that this bound is tight, let us consider instance I composed by n items with volume w = 1 2 + ϵ, with ϵ > 0, k ≤n type 1 bins with C1 = 1, W1 = 1 + ϵ, and n type 2 bins with C2 = 1 2 + ϵ, W2 = 1 2 + ϵ. Note that C1 W1 = 1 1 + ϵ < 1 = C2 W2 (17) Therefore, A will select k bins of type 1 ﬁrst and then n−k type 2 bins. Note that each type 1 bin is not big enough to accommodate two items because 2w = 1+2ϵ > 1+ϵ = W1. Thus we have A(I) = k C1 + (n −k)C2 = k + (n −k) 1 2 + ϵ (18) 8 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 On the contrary, the optimal solution consists of accommodating each of the n items into a type 2 bin because, for small values of ϵ, C2 < C1. Then OPT(I) = n C2 = n 1 2 + ϵ (19) For the approximation ratio, we get A(I) OPT(I) = k + (n −k) 1 2 + ϵ n 1 2 + ϵ = 2k + (n −k)(1 + 2ϵ) n(1 + 2ϵ) (20) Computing the limit for ϵ →0, we obtain lim ϵ→0 A(I) OPT(I) = lim ϵ→0 2k + (n −k)(1 + 2ϵ) n(1 + 2ϵ) = n + k n (21) When k < n, there are not enough type 1 bins to accommodate all items and A yields n −k bins in common with those in the optimal solution. Therefore, the smaller k is, the more the approximation ratio approaches a value of 1. The worst-case ratio is met when there are enough type 1 bins to accommodate all items, i.e., k = n. In this case, the approximation ratio is equal to 2, which proves the bound tightness. □ Corollary 1 The absolute worst-case ratio of the FF and the BF algorithms for the o-VCSBPP, when there are enough type 1 bins to accommodate all items, has a bound equal to 2 and this bound is tight. Proof. As in Theorem 1, let A ∈{FF, BF} be either the FF or the BF algorithm. When there are enough type 1 bins to accommodate all items, A will just select these bins. Therefore T A = 1, C1 W1 = C\|T A\| W\|T A\| , and, because ∆= 0, (16) becomes A(I) < 2 OPT(I). (22) The proof of the bound tightness is the same of that of Theorem 1. □ 9 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 3.4 Worst-case analysis of FF and BF for the o-GBPP Whilst for the o-VCSBPP we can guarantee performance ratios of the FF and the BF algorithms, this is not true for the o-GBPP. Theorem 2 It is impossible to compute the asymptotic and absolute worst-case ratios of the FF and the BF algorithms for the o-GBPP. Proof. Let us consider instance I(νA, νB, νC), composed of one bin type (\|T \| = 1 and t = 1) with W1 = W, C1 = C, and the set of items I is split into three subsets, A, B, and C, with \|A\| = νA, \|B\| = νB, and \|C\| = νC. An item which belongs to subset X ∈{A, B, C} is called a type X item. Let type A items be compulsory with wA = W, type B items be non-compulsory with wB = W, pB = C + ϵ, and type C items be non-compulsory with wC = W, and pC = C −ϵ, with ϵ > 0 (note that the proﬁt pA is not deﬁned because type A items are compulsory). As before, let A ∈{FF, BF} be either the FF or the BF algorithm. Since all items have a volume equal to W, each bin can accommodate only one item. Consequently, we must open νA bins to accommodate all νA type A compulsory items and νB bins to accommodate all νB type B non-compulsory items, which are taken because pB > C. Since no type C item is proﬁtable and cannot be loaded with any other item, none of them will be loaded. We have OPT(I(νA, νB, νC)) = νAC + νB(C −pB) = νAC −νBϵ (23) Let us call iX an item i which belongs to the subset X and consider the following online item sequence νA times z }\| { iA . . . iA νB times z }\| { iB . . . iB νC times z }\| { iC . . . iC Applying A to the above sequence, we have A(I(νA, νB, νC)) = νAC + νB(C −pB) + νC(C −pC) = νAC −νBϵ + νCϵ (24) According to (2), in order to compute the asymptotic worst-case ratio, we have to ﬁnd a proper constant O(1) and the smallest positive R such that νAC −νBϵ + νCϵ ≤R(νAC −νBϵ) + O(1) (25) 10 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 If we consider instance I(0, 0, νC), (25) becomes νCϵ ≤O(1) (26) which is impossible because a constant cannot be greater than a linear (O(ν)) term. Therefore, the asymptotic worst-case ratio cannot be computed. According to (3), in order to compute the absolute worst-case ratio, we have to ﬁnd the smallest positive ρ such that νAC −νBϵ + νCϵ ≤ρ(νAC −νBϵ) (27) If we consider instance I(1, 0, νC), (27) becomes C + νCϵ ≤ρC (28) which implies ρ →+∞, since νC can be arbitrarily large. On the contrary, if we consider instance I(0, 1, νC), (27) becomes −ϵ + νCϵ ≤−ρϵ (29) which implies ρ →−∞, since νC can be arbitrarily large, and contradicts the previous assumption on ρ. Therefore, the absolute worst-case ratio cannot be computed. □ One drawback of the FF and the BF algorithm when applied to the o-GBPP is that they accept every incoming item, independently of its proﬁt. Therefore, at the end of the process, there might be some bins containing non-compulsory items only, whose cost is greater than the total proﬁt of the items loaded into them. Using these bins is clearly wrong. A natural improvement of FF and BF consists of rejecting, at the end of the process, bins of this kind. These algorithms are named FFR and BFR, where R stands for Rejection. We also prove that for the FFR and the BFR algorithms the asymptotic and absolute worst-case ratios cannot be computed. Theorem 3 It is impossible to compute the asymptotic and absolute worst-case ratios of the FFR and BFR algorithms for the o-GBPP. 11 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 Proof. Let us consider instance I(µ, ν) composed of one bin type (\|T \| = 1 and t = 1), with W1 = W, C1 = C, and where the set of items I is split into three subsets, A, B, and C, with \|A\| = µ, and \|B\| = \|C\| = 2ν. Let type A items be compulsory with wA = W, let type B items be non-compulsory with wB = W 2 and pB = C 2 + ϵ, and let type C items be non-compulsory with wC = W 2 and pC = C 2 −2ϵ, for small positive values of ϵ. Finally, let A ∈{FFR, BFR} be either the FFR or the BFR algorithm. It can be easily veriﬁed that an optimal solution consists of µ bins each containing one type A compulsory item, and ν bins each containing two type B non-compulsory items. Type C non-compulsory items are not selected because they are not proﬁtable. Thus OPT(I(µ, ν)) = µC + ν(C −2pB) = µC −2νϵ (30) Let us consider the following online item sequence µ times z }\| { iA . . . iA 2ν times z }\| { iB iC . . . iB iC which, for both FFR and BFR, yields µ bins each containing one type A compulsory item and ν bins each containing one type B and one type C non-compulsory item. However, the ν bins containing non-compulsory items will be discarded because pB+pC = C−ϵ < C. Therefore A(I(µ, ν)) = µC (31) We show that, according to deﬁnition (2), it is impossible to ﬁnd R and O(1) such that A(I(µ, ν)) ≤R · OPT(I(µ, ν)) + O(1) (32) If we substitute (30) and (31) into (32), we have µC ≤R(µC −2νϵ) + O(1) (33) If we consider instance I(0, ν), (33) becomes 0 ≤−2Rνϵ + O(1) (34) Since ν can be arbitrarily large, then it implies that R ≤0, independently of the constant O(1). 12 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 If we consider instance I(µ, 0), (33) becomes µC ≤RµC + O(1) (35) which implies that R ≥1, independently of constant O(1), in contrast with the previous requirement that R ≤0. Therefore, it is impossible to compute the asymptotic worst-case ratio of A. This result holds independently of constant O(1) and, according to (2) and (3), the absolute worst-case ratio is the particular case when O(1) = 0. As a result, the absolute worst-case ratio of A cannot be computed. □ The results of Theorem 3 can easily be generalized as follows Corollary 2 Given a minimization problem Π and an algorithm A, let I(µ, ν) ∈Π be an instance with µ, ν ∈N, such that A(I(µ, ν)) = αµ, and OPT(I(µ, ν)) = βµ −γν, with α, β, γ > 0. Then, it is impossible to compute the asymptotic and absolute worst-case ratios of algorithm A. Proof. Trivial, similar to that of Theorem 3. □ We observe that FF and BF for the o-VCSBPP and the o-GBPP, and FFR and BFR for the o-GBPP, act in the same way when they address the oﬄine variant of these two problems. We can conclude therefore, that the worst-case results obtained in this paper for the online Variable Cost and Size Bin Packing Problem and the online Generalized Bin Packing Problem also hold for the oﬄine variant of these two problems. 4 Conclusions We introduced two new online bin packing problems, the online Variable Cost and Size Bin Packing Problem and the online Generalized Bin Packing Problem, of great interest to the ﬁelds of computer science and engineering as well as transportation, logistics, and telecommunications. 13 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 The contribution of the paper to the literature is of two orders. First, it yields an asymptotic and absolute worst-case ratio bound of First Fit and Best Fit equal to 2 for the o-VCSBPP and this bound is tight. Second, it proves that no worst-case ratio bound of these algorithms can be computed for the o-GBPP, even considering their natural evolution, i.e., First Fit with Rejection and Best Fit with Rejection. Moreover, the paper gives an interesting generalization of these results which consists of suﬃcient conditions under which algorithms do not admit any performance ratio. Finally, we showed that the worst-case results obtained for the online Variable Cost and Size Bin Packing Problem and the online Generalized Bin Packing Problem also hold for the oﬄine variant of these two problems. Acknowledgments This project has been partially funded by the Natural Sciences and Engineering Council of Canada (NSERC), through its Discovery Grants program, and the Italian Ministry of Education, University, and Research, under the PRIN 2009 project “Methods and Algorithms for the Logistics Optimization”. References M. M. Baldi, T. G. Crainic, G. Perboli, R. Tadei, Branch-and-price and beam search algorithms for the variable cost and size bin packing problem with optional items, Annals of Operations Research, DOI 10.1007/s10479-012-1283-2. M. M. Baldi, T. G. Crainic, G. Perboli, R. Tadei, The generalized bin packing problem, Transportation Research Part E 48 (6) (2012) 1205–1220. T. G. Crainic, G. Perboli, M. Pezzuto, R. Tadei, Computing the asymptotic worst-case of bin packing lower bounds, European Journal of Operational Research 183 (2007) 1295–1303. T. G. Crainic, G. Perboli, M. Pezzuto, R. Tadei, New bin packing fast lower bounds, Computers & Operations Research 34 (2007) 3439–3457. T. G. Crainic, G. Perboli, W. Rei, R. Tadei, Eﬃcient lower bounds and heuristics for the variable cost and size bin packing problem, Computers & Operations Research 38 (2011) 1474–1482. D. K. Friesen, M. A. Langston, Variable sized bin packing, SIAM Journal on Com-puting 15 (1986) 222–230. 14 Worst-Case Analysis for New Online Bin Packing Problems CIRRELT-2013-11 D. S. Johnson, Near-Optimal bin packing algorithms, PhD thesis, Dept. of Mathe-matics, M.I.T., Cambridge, MA, 1973. D. S. Johnson, A. Demeters, J. D. Hullman, M. R. Garey, and R. L. Graham, Worst-case performance bounds for simple one-dimensional packing algorithms, SIAM Journal on Computing, 3 (1974) 299–325. J. Kang, S. Park, Algorithms for the variable sized bin packing problem, European Journal of Operational Research 147 (2003) 365–372. C. L. Li, Z. L. Chen, Bin-packing problem with concave costs of bin utilization, Naval Research Logistics 53 (4) (2006) 298–308. S. Seiden, On the online bin packing problem, Journal of the ACM 49 (5) (2002) 640–671. J. D. Ullman, The performance of a memory allocation algorithm, Tech. 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18163	https://www.rcboe.org/cms/lib/GA01903614/Centricity/Domain/2950/Sec%2012.4.pdf	1 2 3 © Houghton Mifflin Harcourt Publishing Company Name Class Date Explore Identifying Similarity in Right Triangles A Make two copies of the right triangle on a piece of paper and cut them out. B Choose one of the triangles. Fold the paper to find the altitude to the hypotenuse. C Cut the second triangle along the altitude. Label the triangles as shown. Resource Locker Resource Locker Module 12 663 Lesson 4 12.4 Similarity in Right Triangles Essential Question: How does the altitude to the hypotenuse of a right triangle help you use similar right triangles to solve problems? © Houghton Mifflin Harcourt Publishing Company D Place triangle 2 on top of triangle 1. What do you notice about the angles? E What is true of triangles 1 and 2? How do you know? F Repeat Steps 1 and 2 for triangles 1 and 3. Does the same relationship hold true for triangles 1 and 3? Reflect 1. How are the hypotenuses of the triangles 2 and 3 related to triangle 1? 2. What is the relationship between triangles 2 and 3? Explain. 3. When you draw the altitude to the hypotenuse of a right triangle, what kinds of figures are produced? 4. Suppose you draw △ABC such that ∠B is a right angle and the altitude to the hypotenuse intersects hypotenuse _ AC at point P. Match each triangle to a similar triangle. Explain your reasoning. A. △ABC △PAB B. △PBC △CAB C. △BAP △BPC Module 12 664 Lesson 4 © Houghton Mifflin Harcourt Publishing Company Explain 1 Finding Geometric Means of Pairs of Numbers Consider the proportion a __ x = x __ b where two of the numbers in the proportion are the same. The number x is the geometric mean of a and b. The geometric mean of two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such that x = √ ― ab or x 2 = ab. Example 1 Find the geometric mean x of the numbers. A 4 and 25 Write proportion. 4 _ x = x _ 25 Multiply both sides by the product of the denominators. 25x · 4 _ x = 25x · x _ 25 Multiply. 100x _ x = 25 x 2 _ 25 Simplify. 100 = x 2 Take the square root of both sides. √ 100 = √ x 2 Simplify. 10 = x B 9 and 20 Write proportion. _ x = x _ 20 Multiply both sides by the product of the denominators. 20x · _ x = 20x · x _ 20 Multiply. x _ x = 20 x 2 _ 20 Simplify. = x 2 Take the square root of both sides. √ _ = √ x 2 Simplify. = x Reflect 5. How can you show that if positive numbers a and b are such that a __ x = x __ b , then x = √ ― ab ? Your Turn Find the geometric mean of the numbers. If necessary, give the answer in simplest radical form. 6. 6 and 24 7. 5 and 12 Module 12 665 Lesson 4 A D B C x c b h a y © Houghton Mifflin Harcourt Publishing Company Explain 2 Proving the Geometric Means Theorems In the Explore activity, you discovered a theorem about right triangles and similarity. The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. That theorem leads to two additional theorems about right triangles. Both of the theorems involve geometric means. Geometric Means Theorems Theorem Example Diagram The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse. h 2 = xy or h = √ ― xy The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. a 2 = xc or a = √ ― xc b 2 = yc or b = √ ― yc Example 2 Prove the first Geometric Means Theorem. Given: Right triangle ABC with altitude _ BD Prove: CD _ BD = BD _ AD Statements Reasons 1. △ABC with altitude _ BD 1. Given 2. △CBD ∼ △BAD 2. 3. CD _ BD = BD _ AD 3. Reflect 8. Discussion How can you prove the second Geometric Means Theorem? Module 12 666 Lesson 4 5 7 z x y z x y 10 2 © Houghton Mifflin Harcourt Publishing Company Explain 3 Using the Geometric Means Theorems You can use the Geometric Means Theorems to find unknown segment lengths in a right triangle. Example 3 Find the indicated value. A x Write proportion. 2 _ x = x _ 10 Multiply both sides by the product of the denominators. 10x · 2 _ x = 10x · x _ 10 Multiply. 20x _ x = 10 x 2 _ 10 Simplify. 20 = x 2 Take the square root of both sides. √ ― 20 = √ ― x 2 Simplify. 2 √ ― 5 = x B y Write proportion. 10 _ y = y _ 12 Multiply both sides by the product of the denominators. 10 _ y = y _ 12 Multiply. _ y = _ 12 Simplify. = Take the square root of both sides. √ ――― = √ ―― Simplify. = y Reflect 9. Discussion How can you check your answers? Your Turn 10. Find x. Module 12 667 Lesson 4 b a c A B C X b a e c A B C X d b a c A B C © Houghton Mifflin Harcourt Publishing Company Explain 4 Proving the Pythagorean Theorem using Similarity You have used the Pythagorean Theorem in earlier courses as well as in this one. There are many, many proofs of the Pythagorean Theorem. You will prove it now using similar right triangles. The Pythagorean Theorem In a right triangle, the square of the sum of the lengths of the legs is equal to the square of the length of the hypotenuse. Example 4 Complete the proof of the Pythagorean Theorem. Given: Right △ABC Prove: a 2 + b 2 = c 2 Part 1 Draw the altitude to the hypotenuse. Label the point of intersection X. ∠BXC ≅ ∠BCA because . ∠B ≅ ∠B by . So, △BXC ∼ △BCA by . ∠AXC ≅∠ACB because . ∠A ≅∠A by . So, △AXC ∼ △ACB by . Part 2 Let the lengths of the segments of the hypotenuse be d and e, as shown in the figure. Use the fact that corresponding sides of similar triangles are proportional to write two proportions. Proportion 1: △BXC ∼ △BCA, so a _ c = _ a . Proportion 2: △AXC ∼△ACB, so b _ c = _____ b . Module 12 668 Lesson 4 © Houghton Mifflin Harcourt Publishing Company Part 3 Now perform some algebra to complete the proof as follows. Multiply both sides of Proportion1 by ac. Write the resulting equation. Multiply both sides of Proportion12 by bc. Write the resulting equation. Adding the two resulting equations give this: Factor the right side of the equation: Finally, use the fact that e + d = by the Segment Addition Postulate to rewrite the equation as . Reflect 11. Error Analysis A student used the figure in Part 2 of the example, and wrote the following incorrect proof of the Pythagorean Theorem. Critique the student’s proof. △BXC ∼ △BCA and △BCA ∼ △CXA, so △BXC ∼ △CXA by transitivity of similarity. Let CX = f. Since corresponding sides of similar triangles are proportional, e _ f  = f _ d  and f 2 = ed. Because △BXC ∼ △CXA and they are right triangles, a 2 = e 2 + f 2 and b 2 = f 2 + d 2 . Add the equations. a 2 + b 2 = e 2 + 2 f 2 + d 2 Substitute. = e 2 + 2ed + d 2 Factor. = (e + d) 2 Segment Addition Postulate = c 2        Module 12 669 Lesson 4 © Houghton Mifflin Harcourt Publishing Company Elaborate 12. How would you explain to a friend how to find the geometric mean of two numbers? 13. △XYZ is an isosceles right triangle and the right angle is ∠Y. Suppose the altitude to hypotenuse _ XZ intersects _ XZ at point P. Describe the relationships among triangles △XYZ, △YPZ and △XPY. 14. Can two different pairs of numbers have the same geometric mean? If so, give an example. If not, explain why not. 15. Essential Question Check-In How is the altitude to the hypotenuse of a right triangle related to the segments of the hypotenuse it creates? Module 12 670 Lesson 4 Q P S R D E B C Y Z W X 65 25 y x z y z x 30 40 © Houghton Mifflin Harcourt Publishing Company • Online Homework • Hints and Help • Extra Practice Evaluate: Homework and Practice Write a similarity statement comparing the three triangles to each diagram. 1. 2. 3. Find the geometric mean x of each pair of numbers. If necessary, give the answer in simplest radical form. 4. 5 and 20 5. 3 and 12 6. 8 and 13 7. 3.5 and 20 8. 1.5 and 84 9. 2 _ 3 and 27 _ 40 Find x, y, and z. 10. 11. Module 12 671 Lesson 4 y z 9.6 12.8 x a e d b c A C B D © Houghton Mifflin Harcourt Publishing Company 12. Use the diagram to complete each equation. 13. c _ e  = _ d 14. c _ a  = a _ 15. c + d _ b = b _ 16. d _ = e _ c  17. c (c + d) = 2 18. 2 = cd Find the length of the altitude to the hypotenuse under the given conditions. 19. BC = 5 AC = 4 20. BC = 17 AC = 15 21. BC = 13 AC = 12 Module 12 672 Lesson 4 E F G 8 4 D © Houghton Mifflin Harcourt Publishing Company 22. Communicate Mathematical Ideas The area of a rectangle with a length of ℓ and a width of w has the same area as a square. Show that the side length of the square is the geometric mean of the length and width of the rectangle. H.O.T. Focus on Higher Order Thinking 23. Algebra An 8-inch-long altitude of a right triangle divides the hypotenuse into two segments. One segment is 4 times as long as the other. What are the lengths of the segments of the hypotenuse? 24. Error Analysis Cecile and Amelia both found a value for EF in △DEF. Both students work are shown. Which student’s solution is correct? What mistake did the other student make? Cecile: 12 ___ EF = EF ___ 8  So E F 2 = 12 (8) = 96. Then EF = √ 96 = 4 √ 6 . Amelia: 8 ___ EF = EF ___ 4  So E F 2 = 8 (4) = 32. Then EF = √ 32 = 4 √_ 2 . Module 12 673 Lesson 4 © Houghton Mifflin Harcourt Publishing Company Lesson Performance Task In the example at the beginning of the lesson, a $100 investment grew for one year at the rate of 50%, to $150, then fell for one year at the rate of 50%, to $75. The arithmetic mean of +50% and -50%, which is 0%, was not a good predictor of the change, for it predicted the investment would still be worth $100 after two years, not $75. 1. Find the geometric mean of 1 + 50% and 1 - 50%. (Each 1 represents the fact that at the beginning of each year, an investment is worth 100% of itself.) Round to the nearest thousandth. 2. It is the geometric mean, not the arithmetic mean, that tells you what the interest rate would have had to have been over an entire investment period to achieve the end result. You can use your answer to Exercise 1 to check this claim. Find the value of a $100 investment after it increased or decreased at the rate you found in Exercise 1 for two years. Show your work. 3. Copy the right triangle shown here. Write the terms “Year 1 Rate” , “Year 2 Rate” , and “Average Rate” to show geometrically how the three investment rates relate to each other. 4. The geometric mean of n numbers is the nth root of the product of the numbers. Find what the interest rate would have had to have been over 4 years to achieve the result of a $100 investment that grew 20% in Year 1 and 30% in Year 2, then lost 20% in Year 3 and 30% in Year 4. Show your work. Round your answer to the nearest tenth of a percent.       Module 12 674 Lesson 4
18164	https://artofproblemsolving.com/wiki/index.php/Geometric_sequence?srsltid=AfmBOopp3yPCioVRALGBqOIoK2xG0Ryg2Vxt_sRr7yNWQlr3z1-co_Np	Page Toolbox Search Geometric sequence In algebra, a geometric sequence, sometimes called a geometric progression, is a sequence of numbers such that the ratio between any two consecutive terms is constant. This constant is called the common ratio of the sequence. For example, is a geometric sequence with common ratio and is a geometric sequence with common ratio ; however, and are not geometric sequences, as the ratio between consecutive terms varies. More formally, the sequence is a geometric progression if and only if . A similar definition holds for infinite geometric sequences. It appears most frequently in its three-term form: namely, that constants , , and are in geometric progression if and only if . Contents Properties Because each term is a common multiple of the one before it, every term of a geometric sequence can be expressed as the sum of the first term and a multiple of the common ratio. Let be the first term, be the th term, and be the common ratio of any geometric sequence; then, . A common lemma is that a sequence is in geometric progression if and only if is the geometric mean of and for any consecutive terms . In symbols, . This is mostly used to perform substitutions, though it occasionally serves as a definition of geometric sequences. Sum A geometric series is the sum of all the terms of a geometric sequence. They come in two varieties, both of which have their own formulas: finitely or infinitely many terms. Finite A finite geometric series with first term , common ratio not equal to one, and total terms has a value equal to . Proof: Let the geometric series have value . Then Factoring out , mulltiplying both sides by , and using the difference of powers factorization yields Dividing both sides by yields , as desired. Infinite An infinite geometric series converges if and only if ; if this condition is satisfied, the series converges to . Proof: The proof that the series convergence if and only if is an easy application of the ratio test from calculus; thus, such a proof is beyond the scope of this article. If one assumes convergence, there is an elementary proof of the formula that uses telescoping. Using the terms defined above, Multiplying both sides by and adding , we find that Thus, , and so . Problems Here are some problems with solutions that utilize geometric sequences and series. Introductory Intermediate See also Something appears to not have loaded correctly. Click to refresh.
18165	https://people.duke.edu/~rnau/Decision411_2007/411log.htm	The logarithm transformation Linearization property Positivity requirement and choice of base First difference of LOG = percentage change The poor man's deflator Trend in logged units = percentage growth Errors in logged units = percentage errors Linearization property: The LOG function has the defining property that LOG (XY) = LOG(X) + LOG(Y)--i.e., the logarithm of a product equals the sum of the logarithms. Therefore, logging tends to convert multiplicative relationships to additive relationships, and it tends to convert exponential(compound growth) trends to linear trends. By taking logarithms of variables which are multiplicatively related and/or growing exponentially over time, we can often explain their behavior with linear models. For example, here is a graph of LOG(AUTOSALE). Notice that the log transformation converts the exponential growth pattern to a linear growth pattern, and it simultaneously converts the multiplicative (proportional-variance) seasonal pattern to an additive (constant-variance) seasonal pattern. (Compare this with the original graph of AUTOSALE.) (Return to top of page.) Positivity requirement and choice of base: The logarithm transformation can be applied only to data which are strictly positive--you can't take the log of zero or a negative number! Also, there are two kinds of logarithms in standard use: "natural" logarithms and base-10 logarithms. The only difference between the two is a scaling constant, which is not really important for modeling purposes. In Statgraphics, the LOG function is the natural log, and its inverse is the EXP function. (EXP(Y) is the natural logarithm base, 2.718..., raised to the Yth power.) The base-10 logarithm and its inverse are LOG10 and EXP10 in Statgraphics. However, in Excel and many hand-held calculators, the natural logarithm function is written as LN instead, and LOG stands for the base-10 logarithm. (Return to top of page.) First difference of LOG = percentage change: When used in conjunction with differencing, logging converts absolute differences into relative (i.e., percentage) differences. Thus, the series DIFF(LOG(Y)) represents the percentage change in Y from period to period. Strictly speaking, the percentage change in Y at period t is defined as (Y(t)-Y(t-1))/Y(t-1), which is only approximately equal to LOG(Y(t)) - LOG(Y(t-1)), but the approximation is almost exactif the percentage change is small. In Statgraphics terms, this means that DIFF(Y)/LAG(Y,1) is virtually identical to DIFF(LOG(Y)). If you don't believe me, here's a plot of the percent change in auto sales versus the first difference of its logarithm, zooming in on the last 5 years. The blue and red lines are virtually indistinguishable except at the highest and lowest points. (Return to top of page.) The poor man's deflator: Logging a series often has an effect very similar to deflating: it dampens exponential growth patterns and reduces heteroscedasticity (i.e., stabilizes variance). Logging is therefore a "poor man's deflator" which does not require any external data (or any head-scratching about which price index to use). Logging is not exactly the same as deflating--it does not eliminate an upward trend in the data--but it can straighten the trend out so that it can be better fitted by a linear model. (Compare the logged auto sales graph with the deflated auto sales graph.) If you're going to log the data and then fit a model that implicitly or explicitly uses differencing (e.g., a random walk, exponential smoothing, or ARIMA model), then it is usually redundant to deflate by a price index, as long as the rate of inflation changes only slowly: the percentage change measured in nominal dollars will be nearly the same as the percentange change in constant dollars. Mathematically speaking, DIFF(LOG(Y/CPI)) is nearly identical DIFF(LOG(Y)): the only difference between the two is a very faint amount of noise due to fluctuations in the inflation rate. To demonstrate this point, here's a graph of the first difference of logged auto sales, with and without deflation: By logging rather than deflating, you avoid the need to incorporate an explicit forecast of future inflation into the model: you merely lump inflation together with any other sources of steady compound growth in the original data. Logging the data before fitting a random walk model yields a so-called geometric random walk--i.e., a random walk with geometric rather than linear growth. A geometric random walk is the default forecasting model that is commonly used for stock price data. (Return to top of page.) Trend in logged units = percentage growth: Because changes in the natural logarithm are (almost) equal to percentage changes in the original series, it follows that the slope of a trend line fitted to logged data is equal to the average percentage growth in the original series. For example, in the graph of LOG(AUTOSALE) shown above, if you "eyeball" a trend line you will see that the magnitude of logged auto sales increases by about 2.5 (from 1.5 to 4.0) over 25 years, which is an average increase of about 0.1 per year, i.e., 10% per year. It is much easier to estimate this trend from the logged graph than from the original unlogged one! The 10% figure obtained here is nominal growth, including inflation. If we had instead eyeballed a trend line on a plot of logged deflated sales, i.e., LOG(AUTOSALE/CPI), its slope would be the average real percentage growth. Usually the trend is estimated more precisely by fitting a statistical model that explicitly includes a local or global trend parameter, such as a linear trend or random-walk-with-drift or linear exponential smoothing model. When a model of this kind is fitted in conjunction with a log transformation, its trend parameter can be interpreted as a percentage growth rate. Errors in logged units = percentage errors: Another interesting property of the logarithm is that errors in predicting the logged series can be interpreted as percentage errors in predicting the original series, albeit the percentages are relative to the forecast values, not the actual values. (Normally one interprets the "percentage error" to be the error expressed as a percentage of the actual value, not the forecast value, athough the statistical properties of percentage errors are usually very similar regardless of whether the percentages are calculated relative to actual values or forecasts.) Thus, if you use least-squares estimation to fit a linear forecasting model to logged data, you are implicitly minimizing mean squared percentage error, rather than mean squared error in the original units--which is probably a good thing if the log transformation was appropriate in the first place. And if you look at the error statistics in logged units, you can interpret them as percentages. For example, the standard deviation of the errors in predicting a logged series is essentially the standard deviation of the percentage errors in predicting the original series, and the mean absolute error (MAE) in predicting a logged series is essentially the mean absolute percentage error (MAPE) in predicting the original series. Statgraphics tip: In the Forecasting procedure in Statgraphics, the error statistics shown on the Model Comparison report are all in untransformed (i.e., original) units to facilitate a comparison among models, regardless of whether they have used different transformations. (This is a very useful feature of the Forecasting procedure--in most stat software it is hard to get a head-to-head comparison of models with and without a log transformation.) However, whenever a regression model or an ARIMA model is fitted in conjunction with a log transformation, the standard-error-of-the-estimate or white-noise-standard-deviation statistics on the Analysis Summary report refer to the transformed (logged) errors, in which case they are essentially the RMS percentage errors. (Return to top of page.)
18166	https://www.engineeringtoolbox.com/wind-load-d_1775.html	Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications! Wind Load vs. Wind Speed Wind load on surface - Wind load calculator. When moving air - wind - is stopped by a surface - the dynamic energy in the wind is transformed to pressure. The pressure acting the surface transforms to a force Fw = pd A = 1/2 ρ v2 A (1) Fw = wind force (N) A = surface area (m2) pd = dynamic pressure (Pa) ρ = density of air (kg/m3) v = wind speed (m/s) Note - in practice wind force acting on a object creates more complex forces due to drag and other effects. Wind Load Calculator Wind Pressure vs. Wind Speed \| Wind Speed(m/s) \| Wind Load1)(Pa) \| \| 1 \| 0.6 \| \| 2 \| 2.4 \| \| 3 \| 5.4 \| \| 4 \| 9.6 \| \| 5 \| 15 \| \| 6 \| 22 \| \| 7 \| 29 \| \| 8 \| 38 \| \| 9 \| 49 \| \| 10 \| 60 \| \| 11 \| 73 \| \| 12 \| 86 \| \| 13 \| 101 \| \| 14 \| 118 \| \| 15 \| 135 \| \| 16 \| 154 \| \| 17 \| 173 \| \| 18 \| 194 \| \| 19 \| 217 \| \| 20 \| 240 \| \| 21 \| 265 \| \| 22 \| 290 \| \| 23 \| 317 \| \| 24 \| 346 \| \| 25 \| 375 \| \| 26 \| 406 \| \| 27 \| 437 \| \| 28 \| 470 \| \| 29 \| 505 \| \| 30 \| 540 \| \| 31 \| 577 \| \| 32 \| 614 \| \| 33 \| 653 \| \| 34 \| 694 \| \| 35 \| 735 \| \| 36 \| 778 \| \| 37 \| 821 \| \| 38 \| 866 \| \| 39 \| 913 \| \| 40 \| 960 \| \| 41 \| 1009 \| \| 42 \| 1058 \| \| 43 \| 1109 \| \| 44 \| 1162 \| \| 45 \| 1215 \| \| 46 \| 1270 \| \| 47 \| 1325 \| \| 48 \| 1382 \| \| 49 \| 1441 \| \| 50 \| 1500 \| 1) density of air 1.2 kg/m3 1 m/s = 3.6 km/h = 196.85 ft/min = 2.237 mph 1 Pa = 1 N/m2 = 1.4504×10-4 psi (lb/in2) Example - Hurricane Wind Load acting on a Wall Surface A hurricane with wind speed 35 m/s is acting on a 10 m2 wall. The dynamic force can be calculated as Fw = 1/2 ρ v2 A = 1/2 (1.2 kg/m3) (35 m/s)2 (10 m2) = 7350 N = 7.35 kN Or - from the table above the wind load per square metre is 735 N/m2. The total load on the wall can be calculated as (735 N/m2) (10 m2) = 7350 N A hurricane acting on a 10 m2 wall creates a force equal to the weight of aprox. 750 kg. Unit Converter . Make Shortcut to Home Screen? Cookie Settings
18167	https://www.maths4everyone.com/resources/downloads/cumulative-frequency-gcse-9-1-practice-questions-30310.pdf	CUMULATIVE FREQUENCY [ESTIMATED TIME: 35 minutes] GCSE (+ IGCSE) EXAM QUESTION PRACTICE 12 The cumulative frequency graph gives information about the lengths, in minutes, of 80 telephone calls. 5 O 20 40 60 80 10 15 Length of call (minutes) Cumulative frequency 20 25 30 (a) Find an estimate for the number of calls which were longer than 15 minutes. .............................................................. (2) (b) Find an estimate for the interquartile range of the lengths of the 80 calls. ............................................................... minutes (2) (Total for Question 12 is 4 marks) Do NOT write in this space. 1. [4 marks] Questions compiled by: @Maths4Everyone Contains questions which have been reproduced with the kind permission of Pearson Education Limited UK 12 P43074A01224 10 The cumulative frequency graph gives information about the monthly rainfall, in millimetres, in the United Kingdom during 120 months in the years 2001 to 2010. (a) Use the graph to estimate the number of months for which rainfall was less than 50 mm. ......................................... (1) (b) Use the graph to find an estimate for the median monthly rainfall. ......................................... mm (1) (c) Use the graph to find an estimate for the interquartile range of the monthly rainfall. ......................................... mm (2) (Total for Question 10 is 4 marks) Monthly rainfall (mm) Cumulative frequency 140 120 100 80 60 40 20 00 20 40 60 80 100 120 140 160 180 200 220 2. [4 marks] 10 P42070A01020 12 The cumulative frequency table shows information about the diameters of 60 oranges. Diameter (d mm) Cumulative frequency !"#$#d ! 60 12 !"#$#d ! 70 42 !"#$#d ! 80 !% !"#$#d ! 90 !& !"#$#d !"100 !' !"#$#d ! 110 60 (a) On the grid, draw a cumulative frequency graph for the table. (2) (b) Use your graph to find an estimate for the median diameter of the 60 oranges. ........................................... mm (2) (Total for Question 12 is 4 marks) Cumulative frequency Diameter (d mm) 60 40 20 0 !" 60 70 80 90 100 110 3. [4 marks] 1'#S'+ B.#&Y GI !"#$$$%&%-#'(! <?;('L8; OOF 0A'+@,$,.#5%S'+?('Z,'&@/+8(#=A+8%S'+%&?-($#5%-&+#B-,5+5A'+.'&85A+-?+QF+5(''+B(#&@A'; O 10 20 30 40 50 60 40 30 20 10 Cumulative frequency Length (cm) + 6#7+ ^%&2+#&+'5%$#5'+?-(+5A'+$'2%#&+.'&85A; + ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;+@$ IJK + 6B7+ ^%&2+#&+'5%$#5'+?-(+5A'+%&5'(Z,#(5%.'+(#&8'+-?+5A'+.'&85A; + ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;+@$ IJK + 6@7+ ^%&2+#&+'5%$#5'+?-(+5A'+&,$B'(+-?+B(#&@A'+C%5A+.'&85A+-?+$-('+5A#&+QQ+@$; + ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; IOK ROO I<'0-=S.-;N1K 4. [5 marks] Leave blank 13 12. The cumulative frequency graph gives information about the ages of people in India. The cumulative frequency is given as a percentage of all the people in India. (a) Use the cumulative frequency graph to find an estimate for the percentage of people in India who are (i) aged less than 20, ...........................% (ii) aged 54 or over. ...........................% (2) (b) Find an estimate for the interquartile range of the ages of people in India. ..................... years (2) Turn over Q12 (Total 4 marks) N23068A01324 5. [4 marks] 14 P46228A01424 DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA 12 The table gives some information about the incomes, £I, of 100 people in the UK. Income (£I) Frequency 0 < I - 10000 12 10000 < I - 20000 41 20000 < I - 30000 25 30000 < I - 40000 12 40000 < I - 50000 6 50000 < I - 60000 4 (a) Complete the cumulative frequency table. Income (£I) Cumulative frequency 0 < I - 10000 12 0 < I - 20000 0 < I - 30000 0 < I - 40000 0 < I - 50000 0 < I - 60000 (1) 6. [6 marks] 15 P46228A01524 Turn over (b) On the grid, draw a cumulative frequency graph for your table. Cumulative frequency Income (£) 20 40 60 80 100 0 10000 20000 30000 40000 50000 60000 (2) (c) Use your graph to find an estimate for (i) the median, £...................................................... (ii) the interquartile range. £...................................................... (3) (Total for Question 12 is 6 marks) 0 12 P40613A01224 14 The grouped frequency table gives information about the ages of 200 elephants. Age (t years) Frequency t - 10 55 t - 20 t - 30 t - 22 t - 50 13 t - 10 (a) Complete the cumulative frequency table. Age (t years) Cumulative frequency t - 10 t - 20 t - 30 t - t - 50 t - (1) 7. [5 marks] 13 P40613A01324 Turn over (b) On the grid, draw a cumulative frequency graph for your table. (2) (c) Use the graph to find an estimate for the number of elephants with ages of more than \HDUV ........................................... (2) (Total for Question 14 is 5 marks) Do NOT write in this space. x O 50 10 20 30 40 50 60 100 150 Cumulative frequency Age (t years) 200 12 P44619A01220 16 The table shows information about the lengths of time that 120 people spent in a supermarket. Time (t minutes) Frequency 0 < t - 10 8 10 < t - 20 17 20 < t - 30 25 30 < t - 40 40 40 < t - 50 22 50 < t - 60 8 (a) Complete the cumulative frequency table. Time (t minutes) Cumulative frequency 0 < t - 10 0 < t - 20 0 < t - 30 0 < t - 40 0 < t - 50 0 < t - 60 (1) 8. [5 marks] 13 P44619A01320 Turn over (b) On the grid, draw a cumulative frequency graph for your table. (2) (c) Use your graph to find an estimate for the median length of time spent in the supermarket by these people. ......................................... minutes (2) (Total for Question 16 is 5 marks) Time (t minutes) Cumulative frequency 120 100 80 60 40 20 O 10 20 30 40 50 60
18168	http://www.nature.com/scitable/topicpage/discovery-of-dna-as-the-hereditary-material-340	This page has been archived and is no longer updated Discovery of DNA as the Hereditary Material using Streptococcus pneumoniae By: Clare O'Connor, Ph.D. (Biology Department, Boston College) © 2008 Nature Education Citation: O'Connor, C. (2008) Discovery of DNA as the hereditary material using Streptococcus pneumoniae. Nature Education 1(1):104 It was a pleasant surprise to the scientific community when clinical research with the bacterium Streptococcus pneumoniae led to the discovery of DNA as the hereditary material. Aa) Aa) Aa) No one could have predicted that experiments designed to understand bacterial pneumonia would lead to the discovery of DNA as the hereditary material. In the early part of the twentieth century, before the advent of antibiotics, pneumococcal infections claimed many more lives than they do today. Researchers on both sides of the Atlantic were thus actively engaged in studying Streptococcus pneumoniae, the bacterium responsible for clinical infections. Early on, it became apparent that multiple strains of S. pneumoniae were responsible for causing bacterial pnuemonia. Researchers also noted that patients developed antibodies to the particular strain, or serotype, with which they were infected, but these antisera were not universally reactive against pneumococcal strains. However, the bacterial isolates and serum samples from these clinical studies provided the critical reagents for the experiments that ultimately led to the identification of DNA as the hereditary material. Pneumococcal Research Provides Critical Tools in DNA Research Although numerous scientists engaged in pneumococcal research during the first half of the twentieth century, two of these researchers played an especially important role in the course of events that led to the discovery of DNA as the hereditary material. One of these individuals was Oswald Avery. Avery joined the Rockefeller Institute for Medical Research, now the Rockefeller University, in 1913 as part of a team seeking to develop a therapeutic serum for treating lobular pneumonia. Avery believed that knowledge of the chemical composition of the pneumococcus bacterium was essential for understanding and treating the disease. He perfected his biochemical technique by focusing on the chemical composition of the capsule that surrounded virulent S strains of pneumococci. In his early work, Avery helped establish that polysaccharides were a major component of the pneumococcal capsule and that capsules from different serotypes of pneumococci had distinctive polysaccharide compositions. Avery also concerned himself with the role of capsules in pathogenicity, as capsules were notably absent from the surface of nonvirulent R forms of streptococci. Defying the conventional wisdom of the time, Avery hypothesized that the polysaccharides in the capsules were the actual antigens stimulating the production of antibodies in infected patients (Avery & Goebel, 1933). Meanwhile, in England, a scientist named Frederick Griffith was also investigating the different serotypes of pneumococci that appeared in patients. He and others noted that two different forms of the bacteria, named R (rough) and S (smooth) for the appearance of their surfaces, could spontaneously convert to the other form, presumably by mutations that changed the cells' ability to synthesize capsules. The different serotypes of pneumococci, however, were assumed to be unalterable, or stable. Griffith's experiments soon proved that this assumption was incorrect. Specifically, in a 1928 paper, Griffith reported that mice unexpectedly died after he injected them with a mixture of live Type I R and heat-killed Type II S pneumococci, neither of which killed the mice when injected alone (Figure 1). Griffith was also able to recover infectious Type II S pneumococci from the hearts of the dead mice, indicating that the changes were heritable. The reverse experiment gave consistent results as well; in other words, mice injected with a mixture of heat-killed Type I S and living Type II R pneumococci died and produced living Type I S cells. From these results, Griffith hypothesized that a substance capable of withstanding a limited heat treatment could be transferred from nonliving S cells of one serotype to living R cells of another serotype, transforming the recipient R cells into S cells of the donor serotype. The results galvanized the pneumococcal research community and provided Avery with another critical tool for isolating the transforming principle. Purification of the Active Transforming Principle When Avery first became aware of Griffith's results, he treated them with skepticism. Other researchers and laboratories, however, were quick to reproduce and build upon Griffith's data. Within a few years, Sia and Dawson (1931) showed that transformation could be carried out in liquid cultures of pneumococci as well as in mice, allowing more precise control of environmental variables in transformation experiments. In 1932, Alloway further demonstrated that the active transforming principle was present in sterile, cell-free extracts prepared from heat-treated pneumococci by filtration. These additional findings convinced Avery that the transforming principle could be identified, and he applied his considerable biochemical expertise to its purification from pneumococcal extracts (Avery et al., 1944). A critical aspect of any biochemical purification is the development of an assay, or a way to measure the activity of interest. For their experiments, Avery and his colleagues developed conditions under which R cells could be reliably transformed into S cells using extracts of heat-killed Type III S cells. These same conditions could then be used to measure transforming activity in fractions obtained at different steps in the purification process. To quantify the actual amount of transforming principle in a fraction, each sample was tested at a series of increasing dilutions. These data represent four identical experiments in which Avery and his colleagues tested the ability of the purified factor, designated preparation 44, to transform Type II R cells into Type III S cells. The transforming activity was very concentrated in the extract, since it could be diluted ten-thousand-fold without losing its transforming ability. Fractions that maintained transforming activity at the highest dilutions were deemed to possess the highest concentration of transforming activity. Specifically, when at least 0.01μg of the extract was added to cells, transformation was observed. When any less than 0.01μg was added, the transformation was inconsistent (comparing samples 1 and 3 with 2 and 4) (Figure 2). Figure 2: Data from Oswald Avery's work. Creative Commons Avery, O. T., MacLeod, C. M., & McCarty, M. Studies on the chemical nature of the substance inducing transformation of pneumococcal types: Induction of transformation desoxyribonucleic acid fraction isolated from pneumococcus type III. Journal of Experimental Medicine 79, 137–157 (1944). Large quantities of Type III S culture, e.g., 75 liters, were required for the initial extracts, which were then separated into different fractions using a variety of organic solvents and detergents. At each step, extracts were tested for their activity in the transformation assay. Throughout the purification, the transforming principle showed the physical properties expected for a nucleic acid, rather than those expected for a protein, lipid, or other kind of polysaccharide. The investigators described the purified transforming principle as a viscous and slightly cloudy solution that formed fibrous strands when mixed with ethanol, all characteristics associated with DNA. Physical Characterization of the Transforming Principle Avery and his colleagues submitted the purified transforming principle to rigorous physical characterization in order to demonstrate that it possessed the properties expected of DNA (Avery et al., 1944). The elemental composition of the purified transforming compound was close to the theoretical values for DNA (last row, sodium desoxyribonucleate) (Figure 3). Significantly, the purified principle had a high phosphorous content, which is characteristic of DNA, but not of proteins. Figure 3: Isolating DNA. Data from Oswald Avery's experiments. Creative Commons Avery, O. T., MacLeod, C. M., & McCarty, M. Studies on the chemical nature of the substance inducing transformation of pneumococcal types: Induction of transformation desoxyribonucleic acid fraction isolated from pneumococcus type III. Journal of Experimental Medicine 79, 137–157 (1944). Figure Detail Consistent with these results, the factor gave positive reactions in chemical tests for DNA, but negative or weakly positive reactions in tests for proteins and RNA. Other tests indicated that the transforming principle was a very large molecule that absorbed the same spectrum of ultraviolet light as DNA. However, the most definitive proof that the transforming principle was DNA was its sensitivity to specific enzymes, called DNAses, that specifically degrade different kinds of DNA. Avery and his colleagues were able to show that transforming activity was not destroyed by enzymes that degrade proteins or RNA. At the time, Avery could not obtain samples of pure DNAse. Instead, Avery and his colleagues used crude preparations from animal tissues that were known to contain DNAse activity. They then measured the ability of these various crude preparations to destroy the transforming principle in parallel with measurements of phosphatase, esterase, and DNAse activities in the same extracts. In all cases, the ability of the crude extracts to destroy the transforming principle was proportional to their DNAse activity, measured with pure calf thymus DNA as substrate (Figure 4). Figure 4: Oswald Avery's work helped establish DNA as hereditary material. Data from Avery's experiments. Creative Commons Avery, O. T., MacLeod, C. M., & McCarty, M. Studies on the chemical nature of the substance inducing transformation of pneumococcal types: Induction of transformation desoxyribonucleic acid fraction isolated from pneumococcus type III. Journal of Experimental Medicine 79, 137–157 (1944). Figure Detail DNA Has the Properties Expected of Genes In retrospect, the experiments reported in Avery and his colleagues' landmark paper of 1944 provided convincing proof that DNA was the hereditary material. It is not surprising, however, that it took some time for the community to adopt the new "dogma" of DNA as the genetic material. Before the experiments of Avery and Griffith, the dogma of the time was that protein was the genetic material, as it was present in the nucleus in nearly equal amounts as DNA, and was structurally more diverse. It was easier to imagine a genetic "language" of 20 symbols than of merely four repeating symbols. The details of the information transfer from DNA to protein were still undiscovered, and many scientists were reluctant to dismiss proteins, which are more structurally diverse than DNA, as the genetic material. Avery and his colleagues (1944) clearly appreciated the importance of their findings, however. They noted that transformation produced changes that are "predictable, type-specific, and heritable" and that "[n]ucleic acids of this type must be regarded not merely as structurally important but as functionally active in determining the biochemical activities and specific characteristics of pneumococcal cells." References and Recommended Reading Alloway, J. L. The transformation in vitro of R pneumococci into S forms of different specific types by the use of filtered pneumococcus extracts. Journal of Experimental Medicine 55, 91–99 (1932) Avery, O. T., & Goebel, W. F. Chemoimmunological studies on the soluble specific substance of pneumococcus I: The isolation and properties of the acetyl polysaccharide of pneumococcus type I. Journal of Experimental Medicine 58, 731–755 (1933) Avery, O. T., MacLeod, C. M., & McCarty, M. Studies on the chemical nature of the substance inducing transformation of pneumococcal types: Induction of transformation desoxyribonucleic acid fraction isolated from pneumococcus type III. Journal of Experimental Medicine 79, 137–157 (1944) Griffith, F. The significance of pneumococcal types. Journal of Hygiene 27, 113–159 (1928) McCarty, M. Discovering genes are made of DNA. Nature 421, 406 (2003) doi:10.1038/nature01398 (link to article) National Library of Medicine. "Profiles in Science: Oswald T. Avery Collection." (accessed on September 30, 2008) Sia, R. H. P., & Dawson, M. H. In vitro transformation of pneumococcal types II: The nature of the factor responsible for the transformation of pneumococcal types. Journal of Experimental Medicine 54, 701–710 (1931) Steinman, R. M., & Moberg, C. L. A triple tribute to the experiment that transformed biology. 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18169	https://journalofethics.ama-assn.org/article/fallopian-tube-ligation-or-salpingectomy-means-reducing-risk-ovarian-cancer/2015-09	AMA Journal of Ethics® Illuminating the Art of Medicine ## State of the Art and Science Sep 2015 Fallopian Tube Ligation or Salpingectomy as Means for Reducing Risk of Ovarian Cancer J. Brian Szender, MD, MS and Shashikant B. Lele, MD AMA J Ethics. 2015;17(9):843-848. doi: 10.1001/journalofethics.2015.17.9.stas1-1509. The Problem of Ovarian Cancer Ovarian cancer remains the most lethal gynecologic malignancy in the United States, both in rate of fatality (64 percent of patients ultimately die of their disease ) and in overall deaths (14,270 in 2014 ). Although 50-75 percent of patients treated with chemotherapy initially respond to the medications, most will have recurrences of the disease . The driving force behind the poor survival rates is the stage at diagnosis. Approximately 65 percent of patients present with widespread (stages III or IV) disease, at which point cure is uncommon . For patients with stage I disease, on the other hand, five-year survival rates exceed 90 percent . One reason that most patients are diagnosed at late stages is that the clinical symptoms of ovarian cancer usually do not become apparent until the disease has disseminated throughout the peritoneal cavity. Although multiple attempts have been made to develop screening programs aimed at detecting early-stage disease, current screening methods are fraught with low sensitivity and specificity, high false-positive rates, and an unfavorable balance between the risks of early intervention and the benefits of cancer risk reduction [2-4]. Attempts at Ovarian Cancer Screening Because the clinical symptoms of ovarian cancer are vague and often appear late in the course of disease, numerous attempts have been made to initiate screening programs to identify preclinical disease in asymptomatic women . Some methods for screening include pelvic examination, ultrasound, and blood testing. The Prostate, Lung, Colorectal, and Ovarian (PLCO) Cancer Screening Randomized Controlled Trial found that screening did more harm than good with respect to ovarian cancer . Specifically, study subjects underwent unnecessary surgeries that did not diagnose ovarian cancer and were associated with intraoperative and postoperative complications. The United Kingdom Collaborative Trial of Ovarian Cancer Screening, published in 2015, found that serial testing of the cancer antigen (CA) 125 protein, interpreted according to the Risk of Ovarian Cancer Algorithm (ROCA), and ultrasound were better at detecting ovarian cancer than a single threshold CA 125 test . Ultimately, screening for ovarian cancer is not ready for application outside of clinical trials because the results have not been validated in independent cohorts. Clinicians must maintain a high index of suspicion, i.e., consider ovarian cancer a likely possibility, to clinically diagnose it. Due to the absence of an effective screening algorithm for assessing risk or clinical symptoms that develop with early-stage disease, primary prevention strategies are crucial for reducing ovarian cancer-related deaths. Experience from Hereditary Breast and Ovarian Cancer Syndromes Identifying patients at increased risk for ovarian cancer is key to prevention, early detection, and, ultimately, improving survival. Those with BRCA1 mutations have a 39-46 percent lifetime risk of ovarian cancer, those with BRCA2 mutations have a 10-27 percent risk, and up to 24 percent of those with Lynch syndrome will develop ovarian cancer . At this time, the best tools that clinicians have for ovarian cancer prevention are a thorough family history and testing appropriate patients for genetic susceptibility . The Society of Gynecologic Oncologists (SGO) policy statement on genetic counseling says unaffected individuals with increased risk—i.e., relatives with ovarian cancer; a family history suggestive of Lynch syndrome based on Amsterdam Criteria or Bethesda Guidelines; known mutations in the family or a family member diagnosed with breast cancer before age 45; multiple breast cancers, male breast cancer, pancreatic cancer, or aggressive prostate cancer (with a Gleason score of 7 or above)—should be referred for genetic counseling and, potentially, testing for germline mutations in BRCA . If BRCA mutations or Lynch syndrome are identified, the National Comprehensive Cancer Network (NCCN) recommends removal of both fallopian tubes and ovaries between the ages of 35 and 40, based on the particular mutation carried. CA 125 tests and pelvic ultrasound have been considered, but there is not sufficient evidence that these tests are sensitive or specific enough to obviate the need for surgery . Fallopian Origin and Prevention of Ovarian Cancer A proposed model for ovarian carcinogenesis arising in the fallopian tube has emerged over the last decade [9, 10]. This tubal-origin hypothesis has gained traction with identification of pre-invasive lesions in the fallopian tubes of high-risk patients undergoing risk-reducing surgery . Thus, bilateral salpingectomy with ovarian conservation was proposed as a “middle-ground” method of primary prevention, with the benefit of removing potential tissue of origin and without the risks of surgical menopause. This method has been proposed for clinical trials in high-risk patients, but results are not currently available . The SGO in 2013 published a clinical practice statement recommending that a bilateral salpingectomy should be considered “at the time of abdominal or pelvic surgery, hysterectomy, or in lieu of tubal ligation” . The American College of Obstetricians and Gynecologists (ACOG) had a more tempered statement, saying that salpingectomy should be considered for population-risk patients, i.e., those without increased risk based on personal or family history, but they were clear that the approach to pelvic surgery, hysterectomy, or sterilization should not change simply to increase the chances of completing bilateral salpingectomy . Both of these statements were more conservative than the proposed plan of the British Columbia Ovarian Cancer Research Group program, instituted in 2010, which involved performing opportunistic salpingectomy with benign hysterectomy or in lieu of bilateral tubal ligation for permanent contraception. These authors suggested that this approach would yield a 20-40 percent population risk reduction for ovarian cancer over the next 20 years . The estimated risk reduction for any individual person undergoing opportunistic salpingectomy is up to 50 percent . Although this is an appreciable benefit, it must be tempered with a reminder that women at population risk of ovarian cancer have only a 1:70 or 1.4 percent lifetime risk . The significant benefits of opportunistic salpingectomy, besides the risk reduction, are the ease and speed of the procedure, the rarity of complications, the convenience of removing the specimen, and the fact that surgical removal is theoretically the only way to permanently reduce the risk of ovarian cancer (although bilateral tubal ligation without salpingectomy has also been associated with decreased risk ). Whether salpingectomy is more beneficial than tubal ligation has not been established. Unresolved Questions Despite the popularity of salpingo-oophorectomy as a method of reducing risk of ovarian cancer, data from the Nurses’ Health Study suggest that oophorectomy before age 47.5 years may be associated with increased risk of death from other causes, such as cardiovascular disease , and that the actual permanent risk reduction with salpingectomy, as opposed to the theoretical 50 percent reduction , is not entirely clear. Numerous questions remain regarding the optimal timing of salpingectomy, as the timespan during which the ovaries are susceptible to induction of cancer from the fallopian tubes is certainly not infinitely large. A bilateral salpingectomy at age 30 is logically more effective at risk reduction than the same surgery at age 60. Unfortunately, the relationship between time and risk reduction has not been not characterized, and prospective studies of the effect of age at salpingectomy on risk reduction would require prohibitively large cohort sizes and long follow-up periods. Similarly, there are other commonly accepted interventions associated with risk reduction, including oral contraceptive pill use and breastfeeding [2, 15, 16]. It is not known how salpingectomy and oral contraceptive pill use interact with one another, although presumably women with a history of bilateral salpingectomy will use birth control pills less frequently, given that the prevention of unintended pregnancy is no longer a concern. Another unresolved question is whether salpingectomy should be used instead of tubal ligation for a “two birds with one stone” approach to sterilization and risk reduction. Caution should be exercised when choosing salpingectomy over tubal ligation for sterilization, not because of the inability to reverse salpingectomy—tubal ligation also should not be performed on women who may desire future childbearing, and in vitro fertilization is a viable method of achieving pregnancy after salpingectomy or tubal ligation —but because “low-risk” surgery does not equal “no risk.” We should be cautioned by prior experience with opportunistic appendectomy at the time of cesarean section or hysterectomy : with opportunistic appendectomy, stump leaks, bleeding, and infection were all possible. Furthermore, salpingectomy increases the length of the operation, and length of surgery has consistently been identified as an independent risk factor for postoperative morbidity [19-23], so even an opportunistic salpingectomy can increase some risks. Another issue is that payers may be reluctant to authorize the charges for risk-reducing procedures, given the number needed to prevent a single case of ovarian cancer. The theoretical number needed reported by Kwon and colleagues in 2015 was 273 for salpingectomy at the time of hysterectomy and 366 for salpingectomy in lieu of other tubal occlusion methods for sterilization . Although these numbers are on the same order of magnitude as the number needed to vaccinate with the human papilloma virus vaccine in the United States , the costs associated with vaccination are less than the costs of salpingectomy. Conclusions Ultimately, we think ACOG’s recommendation of a discussion about risks and benefits of removing both fallopian tubes at the time of hysterectomy is reasonable. However, we cannot place enough importance on the statement, “the approach to hysterectomy or sterilization should not be influenced by the theoretical benefit of salpingectomy” . In the absence of results from prospective studies, which will not be available for decades, fallopian tubes should be removed when a convenient opportunity arises, but extensive surgery should not be attempted just for that purpose. Read More Diagnosis/Errors Evidence-based practice/Effectiveness Genetics/Genetic counseling References Sopik V, Igbal J, Rosen B, Narod SA. Why have ovarian cancer mortality rates declined? Part II. Case-fatality [published online ahead of print June 14, 2015]. Gynecol Oncol. doi: 10.1016/j.ygyno.2015.06.016. View Article PubMed Google Scholar 2. Sopik V, Igbal J, Rosen B, Narod SA. Why have ovarian cancer mortality rates declined? Part I. Incidence [published online ahead of print June 14, 2015]. Gynecol Oncol. doi: 10.1016/j.ygyno.2015.06.017. Google Scholar 3. Buys SS, Partridge E, Black A, et al. Effect of screening on ovarian cancer mortality: the Prostate, Lung, Colorectal and Ovarian (PLCO) Cancer Screening Randomized Controlled Trial. JAMA. 2011;305(22):2295-2303. View Article PubMed Google Scholar 4. Parker WH, Feskanich D, Broder MS, et al. Long-term mortality associated with oophorectomy compared with ovarian conservation in the nurses’ health study. Obstet Gynecol. 2013;121(4):709-716. View Article PubMed Google Scholar 5. Menon U, Ryan A, Kalsi J, et al. Risk algorithm using serial biomarker measurements doubles the number of screen-detected cancers compared with a single-threshold rule in the United Kingdom Collaborative Trial of Ovarian Cancer Screening. J Clin Oncol. 2015;33(18):2062-2071. View Article PubMed Google Scholar 6. Lancaster JM, Powell CB, Chen LM, Richardson DL; SGO Clinical Practice Committee. Society of Gynecologic Oncology statement on risk assessment for inherited gynecologic cancer predispositions. Gynecol Oncol. 2015;136(1):3-7. View Article PubMed Google Scholar 7. American College of Obstetricians and Gynecologists Committee on Gynecologic Practice. Committee Opinion No. 477: the role of the obstetrician-gynecologist in the early detection of epithelial ovarian cancer. Obstet Gynecol. 2011;117(3):742-746. View Article PubMed 8. National Comprehensive Cancer Network. NCCN clinical practice guidelines in oncology: genetic/familial high-risk assessment: breast and ovarian version 1.2015. 9. Kurman RJ, Shih IeM. The origin and pathogenesis of epithelial ovarian cancer: a proposed unifying theory. Am J Surg Pathol. 2010;34(3):433-443. View Article PubMed Google Scholar 10. Kindelberger DW, Lee Y, Miron A, et al. Intraepithelial carcinoma of the fimbria and pelvic serous carcinoma: evidence for a causal relationship. Am J Surg Pathol. 2007;31(2):161-169. View Article PubMed Google Scholar Greene MH, Mai PL, Schwartz PE. Does bilateral salpingectomy with ovarian retention warrant consideration as a temporary bridge to risk-reducing bilateral oophorectomy in BRCA1/2 mutation carriers? Am J Obstet Gynecol. 2011;204(1):19.e1-6. View Article PubMed Google Scholar 12. Society of Gynecologic Oncology (SGO). SGO clinical practice statement: salpingectomy for ovarian cancer prevention. November 2013. Accessed July 27, 2015. 13. American Congress of Obstetricians and Gynecologists Committee on Gynecologic Practice. Committee opinion no. 620: salpingectomy for ovarian cancer prevention. Obstet Gynecol. 2015;125(1):279-281. View Article PubMed 14. Kwon JS, McAlpine JN, Hanley GE, et al. Costs and benefits of opportunistic salpingectomy as an ovarian cancer prevention strategy. Obstet Gynecol. 2015;125(2):338-345. View Article PubMed Google Scholar 15. Cibula D, Widschwendter M, Májek O, Dusek L. Tubal ligation and the risk of ovarian cancer: review and meta-analysis. Hum Reprod Update. 2011;17(1):55-67. View Article PubMed Google Scholar 16. Cibula D, Widschwendter M, Zikan M, Dusek L. Underlying mechanisms of ovarian cancer risk reduction after tubal ligation. Acta Obstet Gynecol Scand. 2011;90(6):559-563. View Article PubMed Google Scholar 17. Lin YJ, Ou YC, Huang FJ, Lin PY, Kung FT, Lan KC. Ovarian response to gonadotropins in patients with tubal factor infertility: salpingectomy versus nonsalpingectomy. J Minim Invasive Gynecol. 2013;20(5):637-641. View Article PubMed Google Scholar 18. American Congress of Obstetricians and Gynecologists Committee on Gynecologic Practice. ACOG Committee Opinion #323: elective coincidental appendectomy. Obstet Gynecol. 2005;106(5, pt 1):1141-1142. 19. Matulewicz RS, Sharma V, McGuire BB, Oberlin DT, Perry KT, Nadler RB. The effect of surgical duration of transurethral resection of bladder tumors on postoperative complications: an analysis of ACS NSQIP data. Urol Oncol. 2015;33(8):338.e19-338.e24. View Article PubMed Google Scholar 20. Catanzarite T, Saha S, Pilecki MA, Kim JY, Milad MP. Longer operative time during benign laparoscopic and robotic hysterectomy is associated with increased 30-day perioperative complications [published online ahead of print June 9, 2015]. J Minim Invasive Gynecol. doi: 10.1016/j.jmig.2015.05.022. Google Scholar 21. Qin C, de Oliveira G, Hackett N, Kim JY. Surgical duration and risk of urinary tract infection: an analysis of 1,452,369 patients using the National Surgical Quality Improvement Program (NSQIP). Int J Surg. 2015;20:107-112. View Article PubMed 22. Tan TW, Kalish JA, Hamburg NM, et al. Shorter duration of femoral-popliteal bypass is associated with decreased surgical site infection and shorter hospital length of stay. J Am Coll Surg. 2012;215(4):512-518. View Article PubMed Google Scholar 23. Reames BN, Bacal D, Krell RW, Birkmeyer JD, Birkmeyer NJ, Finks JF. Influence of median surgeon operative duration on adverse outcomes in bariatric surgery. Surg Obes Relat Dis. 2015;11(1):207-213. View Article PubMed Google Scholar Also in this Issue Sep 2015 In the Literature After Equipoise: Continuing Research to Gain FDA Approval Allison Kerianne Crockett, MD Medicine and Society The HPV Vaccine: Overcoming Barriers to Acceptance of a Medical Triumph Jennifer Emberger, MD, MPH Case and Commentary Oncofertility for Adolescents: When Parents and Physicians Disagree about Egg Cryopreservation for a Mature Minor Annekathryn Goodman, MD Medical Education Designing an Ethics Curriculum in Obstetrics and Gynecology Matthew Schlumbrecht, MD, MPH View Full Issue
18170	https://www.frontiersin.org/journals/veterinary-science/articles/10.3389/fvets.2022.951300/full	Frontiers \| Plasma atropine concentrations associated with decreased intestinal motility in horses Frontiers in Veterinary Science About us About us Who we are Mission and values History Leadership Awards Impact and progress Frontiers' impact Our annual reports Publishing model How we publish Open access Peer review Research integrity Research Topics FAIR² Data Management Fee policy Services Societies National consortia Institutional partnerships Collaborators More from Frontiers Frontiers Forum Frontiers Planet Prize Press office Sustainability Career opportunities Contact us All journalsAll articlesSubmit your researchSearchLogin Frontiers in Veterinary Science Sections Sections Anesthesiology and Animal Pain Management Animal Behavior and Welfare Animal Nutrition and Metabolism Animal Reproduction - Theriogenology Comparative and Clinical Medicine Livestock Genomics Oncology in Veterinary Medicine One Health Parasitology Veterinary Clinical, Anatomical, and Comparative Pathology Veterinary Dentistry and Oromaxillofacial Surgery Veterinary Emergency and Critical Care Medicine Veterinary Epidemiology and Economics Veterinary Humanities and Social Sciences Veterinary Imaging Veterinary Infectious Diseases Veterinary Neurology and Neurosurgery Veterinary Pharmacology and Toxicology Veterinary Regenerative Medicine Veterinary Surgery Zoological Medicine ArticlesResearch TopicsEditorial board About journal About journal Scope Field chief editors Mission & scope Facts Journal sections Open access statement Copyright statement Quality For authors Why submit? 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Article types Author guidelines Editor guidelines Publishing fees Submission checklist Contact editorial office Submit your researchSearchLogin Your new experience awaits. Try the new design now and help us make it even better Switch to the new experience ORIGINAL RESEARCH article Front. Vet. Sci., 02 September 2022 Sec. Veterinary Pharmacology and Toxicology Volume 9 - 2022 \| This article is part of the Research Topic Rising Stars in Veterinary Pharmacology and Toxicology, 2021: Pharmacokinetics and PKPD ModelingView all 7 articles Plasma atropine concentrations associated with decreased intestinal motility in horses Carl Ekstrand1Peter Michanek1Ronette Gehring1,2Anna Sundell 1Annika Källse 1Mikael Hedeland3Lena Ström4 1 Department of Biomedicine and Veterinary Public Health, Division of Pharmacology and Toxicology, Swedish University of Agricultural Sciences, Uppsala, Sweden 2 Department of Population Health Sciences, Division of Veterinary and Comparative Pharmacology, Utrecht University, Utrecht, Netherlands 3 Department of Medicinal Chemistry, Division of Analytical Pharmaceutical Chemistry, Uppsala University, Uppsala, Sweden 4 Department of Clinical Sciences, Division of Large Animal Surgery, Swedish University of Agricultural Sciences, Uppsala, Sweden Introduction: Atropine is an essential part of the treatment protocol for equine uveitis. Topical atropine administration has been associated with decreased intestinal motility and abdominal pain in horses. Experimental studies have indicated that frequent dosing is associated with a higher risk than dosing every 6 h. Unfortunately, no quantitative pharmacodynamic data for inhibition of the equine gut are published. Materials and methods: Eight standardbred horses were assigned to receive either atropine or saline (control) to be infused over 30 min in a two-treatment cross-over design. Atropine concentrations in plasma were measured using ultra-high-performance liquid chromatography–tandem mass spectrometry. Intestinal motility was measured using borborygmi frequency and electrointestinography (EIG). Experimental data were analyzed using a non-linear mixed effects model. The model was then used to simulate different dosing regimens. Results: Atropine significantly decreased borborygmi response and EIG response. Six horses developed clinical signs of abdominal pain. The pharmacokinetic typical values were 0.31, 1.38, 0.69, and 1.95 L/kg·h for the volumes of the central, the highly perfused, the scarcely perfused compartments, and the total body clearance, respectively. The pharmacodynamic typical values were 0.31 μg/L and 0.6 and 207 nV 2 7 cpm for the plasma concentration at 50% of the maximum response and the maximum response and the baseline of cecal EIG response, respectively. Six different dosing regimens of topical atropine sulfate to the eye (0.4 and 1 mg every hour, every 3 h, and every 6 h) were simulated. Conclusion: The IV PK/PD data coupled with simulations predict that administration of 1 mg of topical atropine sulfate administered to the eye every hour or every 3 h will lead to atropine accumulation in plasma and decreased intestinal myoelectric activity. Administration every 6 h predicted a safe dosing regimen in full-sized horses. Clinical studies would be valuable to confirm the conclusions. For smaller equids and horses put at risk for colic due to othercauses, droplet bottles that deliver 40 μl of 1% atropine sulfate per drop or less may be used to lower the risk further. Introduction Atropine is an alkaloid anti-cholinergic drug acting as a non-selective antagonist at muscarinic receptors (1). It increases heart rate, relaxes smooth muscle cells, and decreases salivation and mucus secretion. Relaxation of smooth muscle cells in the gastrointestinal tract might decrease gastrointestinal motility and impair transport through the intestines. In sensitive species such as horses, clinical signs of abdominal pain is a well-described side effect of atropine administration. In equine ophthalmology, the main use of atropine is as a topical mydriatic and cycloplegic in treatment protocols for uveitis. Uveitis causes ciliary muscle spasms and pupillary contraction (miosis). The spasm is painful and chronic complications may occur, including synechia between tissues in the eye that can cause persistent pupil constriction, glaucoma, and decreased vision (2). Topical atropine (eye drops) reverse ciliary muscle spasm and the pupil dilates, which relieves pain and decreases the risk for synechia and permanently decreased vision. Different dosing regimens have been reported from experimental studies, with or without side effects. Hourly topical administration of 1 mg atropine sulfate to the eye has been associated with clinical signs of abdominal pain in horses (3). The most likely explanation was systemic absorption of atropine that inhibited intestinal motility. If the drug was administered every 6 h instead, clinical signs of abdominal pain were absent, suggesting a difference in systemic atropine exposure between the two dosing regimens (4). Recently, pharmacokinetic modeling and simulation indicated a short atropine plasma terminal half-life and suggested a complete washout from the circulation between administrations in the 6-h protocol (5). In contrast, simulations predicted accumulation of atropine for the 1 mg per h dosing regimen. This could explain the difference in abdominal pain between study results, but the concentration–response relationship between atropine and intestinal motility remains unclear. This study aimed to quantitatively determine the pharmacodynamics of atropine with regard to its effects on intestinal motility to better estimate the risk for abdominal pain and colic in horses following atropine exposure. Materials and methods Animals Eight standardbreds (three geldings and five mares) without known systemic or ophthalmic diseases were included in the study. The horses were 8–18 years and weighed 480–675 kg. During the study, horses were kept in single boxes (their home environment). During washout periods, horses were on pasture during the day time and in single boxes during the nights. Water and hay were available ad libitum during experiments. The study was approved by the Animal Ethics Committee, Uppsala, Sweden. Experimental design The study was a blinded, randomized cross-over design including two intravenous (IV) constant rate infusions, one active treatment and one control treatment, administered over 30 min using an infusion pump (Volumat Agilia, Fresenus Kabi AG, Hamburg, Germany). For active treatment, atropine sulfate (Atropin Mylan 0.5 mg/ml, Mylan AB, Stockholm, Sweden) corresponding to the atropine doses 7.5 μg/kg (horses #1–4) and 10 μg/kg (horses #5–8) was diluted in saline (9 mg/ml, Fresenius Kabi, Uppsala, Sweden). Saline was used for the control treatment. A minimum of 3 weeks washout period was applied between treatments. Before the start of the infusion, horses were exercised by walk at a constant pace during 20–30 min using a horse walker (Pro-walker 18-8, Innovation Sandviken, Sandviken, Sweden) familiar to the horses. One IV catheter (MILA international inc. Florence, KY, United States) was placed in each jugular vein (one for infusion and one for sampling) after desensitization of the skin using a prilocaine + lidocaine cream (EMLA 25 + 25 mg/g, Aspen Nordic, Ballerup, Denmark). To prepare for electrointestinography (EIG), the hair over the right flank and abdomen was clipped and the skin was washed with antiseptic soap. Via transabdominal ultrasonography, the cecum and the right dorsal colon were identified. After cleaning the area with alcohol, foam conductive adhesive gel electrodes (MAXENSOR, disposable ECG Electrodes, MediMaxTech UK Ltd., Surrey, UK) were applied. Active and reference electrodes were placed over the cecum and right dorsal colon, respectively. A ground electrode was placed on the ventrolateral abdomen. Impedance was kept below 5 kΩ in all recordings. Responses were amplified, filtered, and stored using a Powerlab system [Powerlab 8/35 and BioAmp FE 235, ADInstruments (Europe) Ltd, Chalgrove, UK]. The EIG frequency was measured within a range of 1.8–12 cycles per min (cpm). Baseline responses were recorded before the start of the infusion. Thereafter, responses were recorded for 5 min during pre-established regular intervals for 10 h after the start of the infusion (see protocol below). EIG responses were analyzed by running spectrum method with fast Fourier transform (FFT), and the total EIG power (nV 2 · cpm) was evaluated. Borborygmi frequency was monitored through auscultation for 1 min per quadrant and scored as followed: absent (0), intermittent (1), and continuous (2) over the observation period. The sum of the scores for all four quadrants (total scores) was used in statistical analyses. Data collection and blood sampling protocols Blood was collected using EDTA-coated tubes at time 0 (pre-dose), at 5, 10, 20, 30, 32, 35, 40, 45, 50 min and 1, 1.25, 1.5, 1.75, 2, 2.33, 2.67, 3, 3.5, 4, 5, 6, 8, and 10 h after the start of infusion. The samples were centrifuged at 1,500 g for 10 min before plasma was transferred to new tubes and immediately frozen to −20°C. At the end of the day, plasma samples were transferred to −70°C pending analyses. Borborygmi frequency was collected at time 0 (pre-dose), at 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, 3, 3.25, 3.5, 3.75, 4, 4.5, 5, 6, 7, 8, 9, and 10 h. The EIG data were collected at 0, 0.08, and 0.33 h and then followed the same protocol as above from 0.5 h. Horses were constantly monitored throughout the 10-h observation period for behaviors associated with acute abdominal pain, namely, depression, flank watching, weight shifting, restlessness, kicking abdomen, pawing, stretching, sternal recumbency, lateral recumbency, attempt to lie down, and collapse (6). Analytical method Atropine concentration in plasma was quantified at the National Veterinary Institute (SVA) in Uppsala, Sweden, using ultra–high-performance liquid chromatography–tandem mass spectrometry (UHPLC-MS/MS). The system was composed of an Acquity UHPLC coupled to a TQS Micro tandem quadrupole mass spectrometer with an electrospray interface operating in the positive mode (Waters Corporation, Milford, MA, United States). The calibration range was 0.05–60 μg/L plasma. The precision (relative standard deviation) was in the range of 2.1–8.3% and the recovery was 95.5–98%. The analytical method is thoroughly described in Ström et al.'s work (5). Data analyses A non-linear mixed effects (NLME) model was used for Pharmacokinetic (PK) and pharmacodynamic (PD) analyses using Monolix 2020R1 (Lixoft, Antony, France). Model evaluation was performed by graphical inspection of diagnostic plots (individual fits, observed data vs. predicted data, weighted residuals vs. time, weighted residuals vs. concentration and the visual predictive check, VPC), parameter precision, and objective function values (OFVs), that is, −2 × log likelihood (−2LL) and Bayesian Information Criteria (BIC). A three-compartment model with intravenous administration and first-order elimination was fitted to the atropine concentration–time data. Atropine acts as an antagonist on muscarinic receptors. Hence, a direct response (sigmoidal I max) model was fitted to both cecum and colon EIG data. The PK model was parameterized using Clearance (Cl), the volume of the central compartment (V 1), the highly perfused compartment (V 2), the poorly perfused compartment (V 3), inter-compartmental clearance from compartment V 1 to compartment V 2 (Q 1), and inter-compartmental clearance from compartment V 1 to compartment V 3 (Q 2). The PD model was parameterized by means of four parameters: The baseline of response (R 0), the atropine plasma concentration at 50% of the response (IC 50), maximum inhibition (I max), and a sigmoidal parameter (n), also called Hills coefficient. The n-parameter was fixed to 1. All parameters were assumed to be log-normally distributed except for the I max parameter, which was assumed as logit normally distributed. A multiplicative (proportional) residual error model was used. Observations below a lower limit of quantification (LLOQ) were censored, that is, any concentration between 0 and LLOQ could be predicted by the model. The statistical model for between-subject variability (BSV) was described by: θ i=θ t v∙exp(η i)(1)θ i=θ t v•exp(η i)(1) where θ i is the value of the pharmacokinetic parameters in the i th horse, θ tv is the typical population value of the parameter, and η i is the deviation from the corresponding population value associated with the i th horse. The standard deviation of the random effects (ω) reported by Monolix was then transformed to a coefficient of variation (CV%) using Equation (2): C V%=√exp(ω 2)−1∙100(2)C V%=exp(ω 2)−1•100(2) Shrinkage of the random effects (eta) toward the means was described as: s h r i n k a g e=1−v a r(η r)ω 2(3)s h r i n k a g e=1-v a r(η r)ω 2(3) where var( η r ) is the variance of the random effects. When shrinkage for eta was >30%, the random component was not considered to be robustly estimated. Simulation of intestinal response after topical atropine administration as eye drops The plasma concentration–time course and cecum EIG response–time courses after topical atropine administration were simulated in a population of 500 horses based on the PK/PD parameters from this study using Simulx2020R1 (Lixoft, Antony, France). The fitted PK model was adapted for extravascular administration by adding an absorption compartment. The parameter values for the absorption rate constant (k a, 5.95 h−1) and bioavailability (F, 0.69) were collected from Ström et al. (5). The simulated atropine doses were 1.67 and 0.67 μg/kg representing 0.1 and 0.04 ml of 1% atropine sulfate solution to a 500 kg horse, respectively. Both dose levels were used to simulate three different dosing protocols over 24 h: every hour, every 3 h, and every 6 h. Statistical analyses Independent of the PK-/PD-modeling approach, the EIG- and auscultation response data were subjected to conventional statistical hypothesis testing by means of a linear mixed-effects model. Categorical fixed effects were time and dose. The horse was used as a random effect. Data were compared between doses for every timepoint using Tukey's test for pair-wise comparisons. An ad hoc analysis was performed to compare data after atropine and control administration with the pre-administration data. The repeated measures structure of the data was accounted for with respect to both time and individual. Statistical significance was considered when p< 0.05. The analyses were performed using the statistical software JMP pro 16.0.0 (SAS institute inc. Cary, NC, United States). Results Atropine concentration–time course The concentration–time courses were grouped after the two different atropine doses (Figure 1). After dose normalization, data from the two dosing regimens were superimposed. Immediately after the infusion, there was a rapid fall in plasma concentration followed by an intermediate phase and a terminal phase of decreasing concentrations. At 10 h, atropine plasma concentration was quantifiable in only one horse (horse #8). Atropine plasma concentration was below LLOQ (0.05 μg/L) at 8 h in this horse. FIGURE 1 Figure 1. Semi-logarithmic spaghetti plot of observed atropine plasma concentrations over time during and after 7.5 μg/kg (blue lines) and 10 μg/kg (red lines) atropine, administered as a 30 min constant rate infusion to four horses per dosing regimen. The pharmacokinetic three-compartment model fits well into the observed data. The OFVs were −247 and −220 for the three-compartment model compared with −127 and −105 for the two-compartment model for −2LL and BIC, respectively. The pharmacokinetic parameters were estimated with good precision [the relative standard errors (RSE) were below 20%]. The observations vs. predictions were randomly scattered around the line of unity, the vast majority of the weighted residuals were scattered between −2 and 2, and the VPC suggests that the model prediction intervals superimpose the observed data (Figure 2). Shrinkages were below 15% for all PK parameters. The PK model parameters, their RSE, and BSV are given in Table 1. FIGURE 2 Figure 2. Diagnostic plots of the pharmacokinetic model: (A) observations vs. prediction plot, (B) individual weighted residuals vs. time and vs. observed concentration, and (C) visual predictive check (VPC) (C). Filled black circles represent observed data and filled red circles represent model predicted concentrations below the quantification limit. The solid line in (A) represents the line of unity (observation = prediction). The solid lines in (C) represent the 10th, 50th, and 90th empirical percentile, respectively. The gray shaded areas in (C) are the 10th and the 90th prediction intervals and the red shaded area is the median prediction interval. TABLE 1 Table 1. Pharmacokinetic model parameter estimates and secondary parameter estimates after 30 min constant rate infusion of atropine in eight horses. Intestinal motility During control treatment, the borborygmi response (total score summarized for all four quadrants) was constantly >5 in all horses (Figure 3). Intermittent or constant borborygmi were present in all quadrants at all observations. There was a significant effect of time (p< 0.0001), dose (p< 0.0001), and the interaction atropine dose and time (p< 0.001). Compared with control treatment, borborygmi response was significantly lower after atropine administration between 0.5 and 1.25 h (p< 0.0001) and at 1.75 h (p = 0.03). Compared with pre-administration data, atropine decreased borborygmi response at 0.5, 0.75, 1, 1,25 h (p< 0.0001), 1.5 h (p = 0.004), and 1.75 h (p = 0.02). Control treatment did not decrease borborygmi response at any timepoint compared with pre-administration data. FIGURE 3 Figure 3. Median (symbols) and range (error bars) borborygmi response (A), cecum EIG response (B), and colon EIG response (C) during and after atropine (blue) and control (red) administered as a 30 min constant rate infusion to eight horses. Black stars indicate a significant lower response after atropine than after control treatment. A solid horizontal blue line indicates the time when the response was significantly lower after atropine treatment than at 0 h. There was a significant effect for the interaction of atropine treatment and time for both cecum EIG response (p< 0.0001) and colon EIG response (p< 0.0001). Compared with control treatment, cecum EIG response was significantly lower after atropine administration at 0.75 h (p = 0.027) (Figure 3). Compared with pre-administration, atropine decreased cecum EIG response at 0.33, 0.5, 0.75 (p< 0.0001), 1 h (p = 0.004), 1.25 h (p = 0.002), 1.5 h (p = 0.03), and 1.75 h (p = 0.02). Compared with pre-administration, atropine decreased colon EIG response at 0.33 h (p = 0.002), 0.5 h (p = 0.02), and 0.75 h (p< 0.001) (Figure 3). Control treatment did not decrease cecum or colon EIG response compared with pre-administration observations. The PD model was fitted to both cecum and colon EIG response data without major bias (Figures 4, 5). The PD parameters for cecum EIG response were estimated with good precision (RSE below 30%). For colon EIG response, the potency value (IC 50-value) was imprecise (RSE 125%). Other parameters were estimated with acceptable precision. Shrinkages were below 10% for all PD parameters. Pharmacodynamic parameters, their RSE, and BSV are given in Table 2. FIGURE 4 Figure 4. Diagnostic plots of the cecum EIG response pharmacodynamic model: (A) observations vs. predictions plot, (B) weighted residuals vs. time and vs. observed concentration, and (C) visual predictive check (VPC). Filled black circles represent observed data. The solid line in (A) represents the line of unity (observation = prediction). The solid lines in (C) represent the 10th, 50th, and 90th empirical percentile, respectively. The gray shaded areas in (C) are the 10th and the 90th prediction intervals and the red shaded area is the median prediction interval. FIGURE 5 Figure 5. Diagnostic plots of the colon EIG response pharmacodynamic model: (A) observations vs. predictions plot, (B) weighted residuals vs. time and vs. observed concentration, and (C) visual predictive check (VPC). Filled black circles represent observed data. The solid line in (A) represents the line of unity (observation = prediction). The solid lines in (C) represent the 10th, 50th, and 90th empirical percentile, respectively. The gray shaded areas in (C) are the 10th and the 90th prediction intervals and the red shaded area is the median prediction interval. TABLE 2 Table 2. Pharmacodynamic model parameters for inhibition of intestinal motility induced by atropine exposure in horses. Clinical signs of acute abdominal pain Atropine infusion induced behaviors associated with abdominal pain in six horses (75%): two horses treated with 7.5 μg/kg atropine and four horses treated with 10 μg/kg atropine. Behavior onsets were between 15 and 52 min into the experiment and behavior durations were between 0.29 and 75 min. The behaviors observed were depression, flank watching, and weight shifting. Simulation of atropine concentration–time courses and EIG response–time courses Simulations showed that atropine accumulated in plasma after dosing every hour and every 3 h but not after dosing every 6 h (Figures 6, 7). The EIG responses were inhibited in a concentration-related fashion. The dosing regimen 1.67 μg/kg hourly induced the greatest suppression of intestinal myoelectrical activity, both compared with less-frequent dosing and with the lower dose (0.67 μg/kg). FIGURE 6 Figure 6. One simulated example of atropine concentration–time courses (A–C) and cecum EIG response–time courses (D–F) in horses following topical administration of 1.67 μg/kg atropine sulfate as eye drops every hour (left column), every 3 h (middle column), and every 6 h (right column). The dose 1.67 μg/kg represents 100 μl 1% atropine sulfate (corresponding to 835 μg atropine) for a 500 kg horse. The solid black horizontal line in concentration–time plots represent the population value (IC 50-value) for the concentration at 50% of maximal response (0.31 μg/L). FIGURE 7 Figure 7. One simulated example of atropine concentration–time courses (A–C) and cecum EIG response–time courses (D–F) in horses following topical administration of 0.67 μg/kg atropine sulfate as eye drops every hour (left column), every 3 h (middle column), and every 6 h (right column). The dose 0.67 μg/kg represents 40 μl (the average droplet volume delivered by a droplet bottle) 1% atropine sulfate (corresponding to 334 μg atropine) for a 500 kg horse. The solid black horizontal line in concentration–time plots represents the population value (IC 50-value) for the concentration at 50% of maximal response (0.31 μg/L). Discussion This study is the first quantitative PK/PD study investigating the relationship between dose, atropine concentrations, and intestinal response. The resultant PK/PD model was applied to predict the risk for adverse gastrointestinal side effects given different dosing regimens by means of simulations. This provides a valuable tool for clinicians and veterinary pharmacologists to improve the safety and efficacy of atropine in horses. The plasma concentrations increased in direct proportion to the dose. This indicates linear PK within the studied concentration range. The use of a three-compartment PK model described the experimental atropine data well. The goodness-of-fit plots presented in Figure 2 show neither bias in the structural model nor the error model. The observed and model predicted concentrations were randomly scattered around the line of unity, the residuals randomly scattered around zero with the majority of residuals between −2 and 2, and the prediction intervals overlapped the empirical percentiles. The typical values for clearance and volume at a steady state (i.e., the sum of the volumes for the respective compartments) were 1.95 L/kg·h and 2.38 L/kg, respectively. This is similar to 1.9 L/kg·h and 1.7 L/kg previously reported using a two-compartment model (5). This was not surprising since both studies were performed using standardbred horses, atropine concentrations were determined using UHPLC/MS-MS, and data were analyzed using NLME. Atropine administration significantly decreased both borborygmi response and EIG response. Moreover, 75% of the horses developed clinical signs of abdominal pain. Atropine, in vitro and in vivo, has shown to decrease intestinal motility, increase gastrointestinal transit time, and induce clinical signs of abdominal pain in horses (3, 7–11). Borborygmi frequency has commonly been used to evaluate the equine abdomen both clinically and in experimental pharmacological studies (12–17). Borborygmi response decreased after atropine administration compared with both control treatment and 0 h, similar to what has been described in several previous studies (3, 5, 8, 10). The measurement of borborygmi response is subjective. Previous studies have shown that repeated measurements by the same observer tend to be consistent, but that inter-observer variability is higher (18). Therefore, this experiment was blinded and the same researcher performed all auscultations. However, atropine administration induced pupil dilation. This could have compromised the blinding of the observer. Some subjectivity in the results can therefore not be excluded. Electrointestinography data, a more objective measurement, were also recorded and used in PK/PD modeling. Percutaneous recording of intestinal myoelectric activity has been suggested to be clinically applicable and a useful tool to evaluate intestinal motility experimentally (19–21). Baseline EIG data showed variability between individuals with a BSV of 30% for the cecum and 60% for the colon (Figure 3). Intestinal motility and emergence of abdominal pain (colic) also vary depending on management, for example, feeding and housing (22–24). Horses in this study were walked before the start of each experimental leg and fed hay during the experiment, both of which increase intestinal motility. The variability together with the conservative statistical model to avoid type I errors are the most probable reasons that only cecum EIG response at 0.75 h was significantly lower than the control treatment. However, the EIG response was also significantly lower after atropine administration compared with pre-administration data. The pharmacodynamic model was able to quantify the inhibition of intestinal electrical activity induced by atropine exposure following IV administration. The goodness-of-fit plots presented in Figures 4, 5 also suggest a model fit with no bias of either the structural model or the error model. The typical values for the potency (IC 50 values) were 0.31 and 0.45 μg/L for the cecum and the colon, respectively. Cecum EIG response was also significantly lower for a longer period after atropine administration than colon EIG response. This suggests that the cecum is more sensitive to atropine exposure than the colon in fed horses. This is similar to what has previously been shown in fasted horses and after sedation with xylazine (25). No atropine potency values or efficacy values have previously been published in horses. In humans, the PK/PD relationship for atropine was characterized using heart rate and saliva flow as markers for the response (26, 27). The potency values for heart rate and saliva flow were then estimated to be 6.2 and 3.7 μg/L. This is approximately 10- to 20-fold higher than the IC 50 values presented in the present study, that is, the concentration to achieve half maximum EIG response is lower than that needed for cardiovascular or secretory effects in man. This was unexpected. Larger doses are generally required for inhibition of intestinal motility than for decreasing salivary secretion or vagal tone in other species (28). In previous studies which investigated the association between ophthalmic atropine treatment and systemic effects, 100 μl of 1% atropine sulfate (corresponding to 835 μg atropine) was administered topically (3–5). Labeled ophthalmic solutions are generally available in dropper bottles that deliver lower volumes, with an average droplet volume of 40 μl (range 25–70 μl) (29). Hence, two doses were simulated in this study; a full dose, 1.67 μg/kg (835 μg/500 kg) and a 40% dose (0.67 μg/kg). When topical dosing every hour was simulated using the PK/PD data derived after IV dosing in this study combined with literature data from Ström et al. (5), atropine was predicted to accumulate in plasma at concentrations above the IC 50-value for cecum EIG response (0.31 μg/L). Consequently, the intestinal myoelectric activity was predicted to decrease, which would explain why horses developed colic with this dosing regimen (3). Also with simulated topical dosing, every 3 h atropine concentrations were predicted to peak above 0.31 μg/L, and there was an accumulation of atropine in plasma. However, the short half-life resulted in trough concentrations below the IC 50-value, and inhibition of intestinal myoelectric activity was less than compared with hourly dosing. Ström et al. (5) reported similar results; atropine accumulated in plasma and borborygmi response was lowered after repeated topical administration of 1 mg atropine sulfate. These horses did not show any clinical signs of abdominal pain. If the 6-h protocol was simulated, no clinically important drug accumulation was predicted. Atropine was predicted to peak above 0.31 μg/L, but the decrease in intestinal myoelectrical activity was of short duration, and the intestinal function over the dosing interval is unlikely to be affected. Consistent with these results, borborygmi response remained unchanged in other experimental studies using the 6 h dosing regimen (4, 5). This dosing regimen is consistent with administration every 4–24 h, which is one of the current dose recommendations for atropine in the treatment of equine uveitis (4, 30). If a droplet bottle delivering 40 μl per drop is used, the risk for colic decreases further. The lower dose is unlikely to cause colic using the 3- or 6-h dosing interval, based on the simulations performed in this study using IV and literature data. With more frequent dosing, however, the accumulation of atropine in plasma will decrease intestinal motility and might induce clinical signs of abdominal pain during chronic administration. Caution is also advised when horses and ponies smaller than 500 kg are treated since the dose per kg, and consequently, the plasma concentrations, will increase with decreasing weight. Horses with uveitis treated with atropine might be at risk for colic despite the dosing regimen. In a retrospective study on 337 equids, a univariate analyses suggested that topical use of atropine was associated with a higher risk for colic (31). However, when age and hospitalization time were added to the analysis, they became significant predictors of colic risk, and atropine lost its significance. Other factors that are associated with risk for colic are pain, activity level, change in feed, and stabling conditions (13, 22, 23). Accordingly, horses exposed to any of those factors may be more sensitive to plasma atropine exposure, that is, the potency value for the decrease in intestinal myoelectrical activity might be lower than in healthy horses. Atropine dilates the pupil and, therefore, exposure to sunlight should be avoided. Also, uveitis many times induces photophobia and pain. Therefore, horses on atropine treatment due to uveitis are often stabled and their activity levels are decreased. Hospitalized horses might also have a change in their diet. These factors are more likely to cause colic than a low daily dose of topical atropine (e.g., 1 mg atropine sulfate every 6 h or less) is used on horses. Moreover, atropine reverses the painful ciliary muscle spasm that might decrease the risk of colic. In conclusion, the IV PK/PD data coupled with simulations presented here predict that topical administration of 1 mg atropine sulfate every hour or every 3 h leads to drug accumulation and decreased intestinal myoelectric activity. However, topical administration every 6 h was predicted to be a safe option in full-sized horses. Clinical studies would be valuable to confirm these conclusions. For small horses, ponies, and horses put at risk for colic due to other causes (e.g., hospitalization, decreased exercise, environmental stress, pain), droplet bottles that deliver 40 μl 1% atropine sulfate per drop or less may be used to lower the risk for colic further. Data availability statement The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation. Ethics statement The animal study was reviewed and approved by the Regional Animal Ethics Committee, Uppsala, Sweden. Author contributions CE and LS planned the experiment and performed the experiment together with AK, AS, and PM. The data were then analyzed by CE, PM, RG, and MH. CE drafted the manuscript. All authors revised the manuscript and approved the final version of the manuscript. Funding This study was funded by the Petra Lundberg Foundation and the Sveland Foundation for Animal Welfare and Health. Acknowledgments Authors would like to express their sincere gratitude to Mari Wallbring and staff for all their assistance when the experiment was performed. Conflict of interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The handling editor JM declared a past collaboration with the author RG. 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Reduction in drop size of ophthalmic topical drop preparations and the impact of treatment. J Adv Pharm Technol Res. (2011) 2:192–4. doi: 10.4103/2231-4040.85540 PubMed Abstract \| CrossRef Full Text \| Google Scholar 30. McMullen RJ, Fischer BM. Medical and surgical management of equine recurrent uveitis. Vet Clin N Am Equine Pract. (2017) 33:465–81. doi: 10.1016/j.cveq.2017.07.003 PubMed Abstract \| CrossRef Full Text \| Google Scholar 31. Patipa LA, Sherlock CE, Witte SH, Pirie GD, Berghaus RD, Peroni JF. Risk factors for colic in equids hospitalized for ocular disease. J Am Vet Med Assoc. (2012) 240:1488–93. doi: 10.2460/javma.240.12.1488 PubMed Abstract \| CrossRef Full Text \| Google Scholar Keywords: atropine sulfate, colic, equine, pharmacokinetics, pharmacodynamics Citation: Ekstrand C, Michanek P, Gehring R, Sundell A, Källse A, Hedeland M and Ström L (2022) Plasma atropine concentrations associated with decreased intestinal motility in horses. Front. Vet. Sci. 9:951300. doi: 10.3389/fvets.2022.951300 Received: 23 May 2022; Accepted: 25 July 2022; Published: 02 September 2022. Edited by: Jonathan Paul Mochel, Iowa State University, United States Reviewed by: Anthony Blikslager, North Carolina State University, United States Kristen Messenger, North Carolina State University, United States Copyright © 2022 Ekstrand, Michanek, Gehring, Sundell, Källse, Hedeland and Ström. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. Correspondence: Ronette Gehring, r.gehring@uu.nl Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher. 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18171	https://conchovalleyendo.com/blog/our-office/2025/10855/navigating-dental-and-endodontic-emergencies/	Navigating Dental and Endodontic Emergencies: A Comprehensive Guide \| Concho Valley Endodontics (325) 947-3040 2014 W. Beauregard Avenue, San Angelo, TX 76901 Recent Posts Saving Your Natural Smile: How Root Canal Therapy Restores Damaged Teeth Don’t Let a Cracked Tooth Crash Your Summer Plans Cracked Teeth: Unveiling Hidden Dangers and Solutions Mastering Dental Pain Management During Endodontic Procedures Your Summer Oral Care Routine: Staying Healthy in the Heat Blog Archive 2025 2024 2023 2022 2021 2020 2019 Our Office 2014 W. Beauregard Avenue San Angelo, TX 76901 (325) 947-3040 About ### About This is a description that should not be displaying since I have use descriptions set to false. 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Beauregard Avenue, San Angelo, TX 76901 Phone: (325) 947-3040 Social: Patient Login Navigating Dental and Endodontic Emergencies: A Comprehensive Guide July 22, 2025 Imagine finding yourself in a situation where a sudden toothache or unexpected swelling makes you question whether what you’re experiencing is just an inconvenience or something more serious. Dental health can be unpredictable—sometimes the issues we face seem manageable, but other times they demand immediate attention to prevent long-term problems. This discussion delves into the world of endodontic care by exploring when and why certain dental conditions become emergencies. It explains how common symptoms, such as severe tooth pain or visible signs like abscesses, can indicate deeper issues with the tooth pulp. It outlines practical steps that anyone can take to manage discomfort while waiting for professional help. By providing clear guidance on both recognizing these critical situations and taking appropriate first aid measures, this article aims to empower you with knowledge. Understanding what happens behind the scenes in an endodontic emergency can make a significant difference in how effectively you respond during such moments. Decoding Dental Emergencies: What You Need To Know Dental emergencies can strike without warning, leaving patients uncertain about whether to seek immediate care or wait it out. The unpredictability of these situations adds to the stress, making it difficult for individuals to assess their condition accurately. While not all dental issues require urgent attention, certain symptoms should prompt an immediate response. For example, persistent pain that doesn’t subside with over-the-counter medication, swelling in the gums or face, and visible abscesses are clear indicators of a potential emergency. The key to managing any dental emergency effectively is recognizing these critical signs early on. By doing so, you can prevent further complications and ensure timely intervention. This section aims to educate readers about what constitutes an actual emergency versus minor discomfort that might not necessitate urgent care. Endodontic Emergencies Unveiled: Symptoms and Causes An endodontic emergency is a dental issue involving the innermost part of the tooth, known as the pulp. This soft tissue contains nerves and blood vessels, which can become infected or inflamed due to various factors such as untreated cavities, trauma, or previous dental work. Common symptoms of an endodontic emergency include severe tooth pain, swelling around the affected area, sensitivity to hot or cold temperatures, and visible abscesses. These signs often indicate that the infection has spread beyond the pulp into surrounding tissues, potentially leading to more serious health issues if left untreated. Understanding these specific symptoms helps individuals identify when they need professional help right away. By recognizing an endodontic emergency early on, patients can take proactive steps towards preserving their oral health and preventing further damage. Immediate Steps to Take: Managing Pain Until You Reach Us Not everyone has immediate access to a dentist during an emergency situation. In such cases, knowing how to manage symptoms before professional care becomes available is crucial. Some practical first aid measures include rinsing the mouth with warm salt water to reduce bacteria and relieve discomfort. Applying a cold compress to the outside of the cheek can help alleviate swelling. Over-the-counter pain relievers like ibuprofen or acetaminophen can temporarily ease toothache until you reach a professional. While these steps provide temporary relief, they do not replace the need for professional dental treatment. It’s essential to follow up with an endodontist as soon as possible after taking initial measures at home. This ensures that any underlying issues are addressed promptly and effectively. Why Timely Intervention Matters: Preventing Further Damage Seeking prompt dental care for endodontic emergencies is not just about alleviating immediate pain; it’s also about preventing long-term complications. Delays in treatment can lead to more severe health issues, including tooth loss or systemic infections that spread beyond the oral cavity. Endodontic procedures such as root canals are designed to remove infected pulp and seal off the tooth to prevent further infection. By addressing these problems early, patients can avoid more invasive surgeries down the line. This proactive approach helps maintain overall health and well-being by keeping dental issues from escalating into more significant concerns. In addition to preventing physical discomfort and potential tooth loss, timely intervention ensures that any underlying infections are treated before they cause further damage. Early detection and treatment can save both time and resources in the long run, making it a wise investment for maintaining good oral health. Your Next Steps: Contacting Us For Emergency Care The importance of having access to reliable emergency dental care cannot be overstated. When faced with severe tooth pain or swelling, knowing where to turn can make all the difference in how quickly and effectively you receive treatment. At Concho Valley Endodontics, we understand the urgency of these situations and are committed to providing timely intervention when it matters most. Our team of highly trained professionals is equipped to handle even the most complex endodontic emergencies with precision and care. To contact us for emergency care, simply call (325) 947-3040 or visit our website at conchovalleyendo.com. Our dedicated staff will guide you through the process and ensure that you receive the prompt attention you need during critical moments. Frequently Asked Questions What constitutes an endodontic emergency? An endodontic emergency involves any condition that requires immediate attention from our experienced team at Concho Valley Endodontics to save a tooth and prevent further complications. Common signs of an endodontic emergency include severe, persistent pain or pressure in your mouth, jaw, or ear; swelling around the face or neck; a pimple on your gum that won't go away; or a cracked or broken tooth with significant damage. If you're experiencing any of these symptoms, don't hesitate to contact us immediately for prompt care. Our team is available 24/7 to assist you in managing your pain and addressing the underlying issue. How should I manage my symptoms until I can see a dentist? If you're experiencing endodontic pain, there are several steps you can take at home to help manage your symptoms until you can reach our office: Rinse your mouth with warm water to clean the area and remove any food particles. Use an ice pack on the outside of your cheek near the affected area for 15-minute intervals to help reduce swelling. Take over-the-counter pain relievers, such as ibuprofen or acetaminophen, to manage discomfort. Follow the package instructions and do not exceed the recommended dosage. Avoid hot foods or liquids if you have a fever, as they can make your condition worse. While these home remedies may provide temporary relief, it's crucial to seek professional help from Concho Valley Endodontics as soon as possible. We're committed to providing fast and efficient care for our patients in need. What technology does Concho Valley Endodontics use to handle emergencies effectively? At Concho Valley Endodontics, we pride ourselves on using state-of-the-art technology to ensure precise, accurate, and efficient care for our patients. When it comes to handling endodontic emergencies, some of the advanced tools and techniques we employ include: Cone Beam Computed Tomography (CBCT): Our on-site CBCT machine allows us to capture high-resolution 3-D images at low radiation doses, helping us make accurate diagnoses and create detailed treatment plans. Clinical Microscopes: With the help of clinical microscopes, we can see aspects of the tooth that are invisible to the naked eye, enabling us to perform intricate procedures with greater precision and success rates. EdgePRO® Procedure: This innovative technique uses a precision light to enhance visualization during root canal treatments, allowing for a gentler and more comfortable experience for our patients. Is it normal to have some discomfort after an endodontic procedure? It's not uncommon to experience some discomfort or sensitivity immediately following an endodontic procedure, such as a root canal. This is often due to the inflammation and irritation caused by the infection that required treatment. In most cases, any discomfort can be managed with over-the-counter pain relievers and should subside within a few days. However, if you're experiencing severe or persistent pain, swelling, or other unusual symptoms, please don't hesitate to contact our office for further guidance. Our team at Concho Valley Endodontics will provide you with detailed post-operative care instructions to help ensure your comfort and speed up the healing process. If you have any questions or concerns, we're always here to help. What if I have a dental emergency outside of regular office hours? At Concho Valley Endodontics, we understand that dental emergencies don't always occur during our regular business hours. That's why we provide after-hour support for our patients in need. If you're experiencing a dental emergency outside of our regular office hours, please call our main line at (325) 947-3040 and follow the prompts to reach our on-call registered dental assistant. This dedicated team member is available 24/7 to help manage your pain, provide immediate guidance, and arrange prompt care if necessary. We're committed to ensuring that you always have access to the high-quality endodontic care you need, when you need it most. How can I prevent endodontic issues in the future? Maintaining good oral hygiene and attending regular dental check-ups are essential for preventing endodontic problems. Here are some tips to help keep your teeth healthy and strong: Brush and floss daily: Remove plaque and bacteria by brushing at least twice a day and flossing once a day. Use fluoride toothpaste and mouthwash: Fluoride helps strengthen tooth enamel and prevent decay. Avoid hard or sticky foods: Steer clear of foods that can damage teeth, such as ice, hard candies, or popcorn kernels. Wear a mouthguard during sports: Protect your teeth from injury with a custom-fitted mouthguard while playing contact sports. Stay hydrated and eat a balanced diet: A healthy lifestyle supports overall oral health and can help prevent issues like dry mouth, which increases the risk of decay. By following these guidelines, you can significantly reduce your risk of developing endodontic problems in the future. If you have any concerns about your dental health, don't hesitate to contact our team at Concho Valley Endodontics for personalized advice and care. What should I expect during my first visit to Concho Valley Endodontics? Welcome to Concho Valley Endodontics! We're excited to help you achieve a healthy, comfortable smile. Here's what you can expect during your first visit with us: Consultation: Our team will greet you warmly and guide you through our office, introducing you to our state-of-the-art facilities. We'll then discuss your dental history, symptoms, and any concerns you may have. Examination: Dr. McIntosh will perform a thorough examination of your teeth, gums, and jaw, using advanced technology such as digital X-rays or CBCT scans to gain a comprehensive understanding of your oral health. Treatment Planning: Based on the findings from our examination, we'll develop a personalized treatment plan tailored to your specific needs. We'll explain the proposed procedure, expected outcomes, and any associated costs in detail, ensuring you're fully informed throughout the process. Comfort Measures: Our top priority is your comfort. We offer various sedation options to help ease any anxiety you may have during your visit. Our team will work diligently to ensure that you feel at ease and well-cared for throughout your time with us. How do I care for my tooth after a root canal? After a root canal procedure, it's essential to follow our post-operative care instructions to ensure the successful healing of your treated tooth. Here are some tips for caring for your tooth: Avoid chewing on the treated side until the local anesthesia has worn off completely to prevent biting your cheek or tongue and to allow the filling material to set properly. Take any prescribed medication as directed to help manage any discomfort or swelling. Over-the-counter pain relievers can also be used if necessary. Avoid eating hard, chewy, or very hot foods for the first 24 hours after your procedure to minimize the risk of damaging the tooth or disrupting the filling material. Maintain excellent oral hygiene by brushing and flossing gently around the treated area to keep it clean and promote healing. Be sure to use a soft-bristled toothbrush and avoid aggressive scrubbing. Attend your follow-up appointment as scheduled to ensure that the temporary filling is holding securely and that your tooth is healing properly. Once your tooth has healed, we'll recommend placing a crown over it for added protection and strength. Can I still have endodontic treatment if I'm pregnant? Yes, it's generally safe to undergo endodontic treatment while you're pregnant. In fact, it's crucial to address any dental issues promptly to prevent the spread of infection and potential complications for both mother and child. The American Dental Association (ADA) and the American Academy of Obstetricians and Gynecologists (ACOG) recommend that dental care should be provided during pregnancy if necessary, as the benefits outweigh the risks. Our team at Concho Valley Endodontics will take extra precautions to ensure your safety and comfort throughout the procedure: We'll use protective measures, such as a lead apron with a thyroid collar, to minimize radiation exposure. We'll avoid administering local anesthetics that contain epinephrine unless absolutely necessary. In most cases, we can use alternative anesthetic agents that are safe for pregnant patients. We'll keep the procedure as short and comfortable as possible to reduce stress and anxiety. Recent Posts Saving Your Natural Smile: How Root Canal Therapy Restores Damaged Teeth Don’t Let a Cracked Tooth Crash Your Summer Plans Cracked Teeth: Unveiling Hidden Dangers and Solutions Mastering Dental Pain Management During Endodontic Procedures Your Summer Oral Care Routine: Staying Healthy in the Heat Blog Archive 2025 2024 2023 2022 2021 2020 2019 Our Office 2014 W. Beauregard Avenue San Angelo, TX 76901 (325) 947-3040 Recent Posts Saving Your Natural Smile: How Root Canal Therapy Restores Damaged Teeth Don’t Let a Cracked Tooth Crash Your Summer Plans Cracked Teeth: Unveiling Hidden Dangers and Solutions Mastering Dental Pain Management During Endodontic Procedures Your Summer Oral Care Routine: Staying Healthy in the Heat Blog Archive 2025 2024 2023 2022 2021 2020 2019 Our Office 2014 W. Beauregard Avenue San Angelo, TX 76901 (325) 947-3040 Explore More First VisitPatient FormsOur OfficeOur DoctorOur ServicesReviewsRequest AppointmentEmergency CarePatient Login Get in Touch Phone: (325) 947-3040 Email: office@conchovalleyendo.com Address: 2014 W. Beauregard Avenue, San Angelo, TX 76901 Hours of Operation Monday 8:00 am - 5:00 pm Tuesday 8:00 am - 5:00 pm Wednesday 8:00 am - 5:00 pm Thursday 8:00 am - 5:00 pm Friday 8:00 am - 12:00 pm Saturday Closed Sunday Closed Recent Posts Saving Your Natural Smile: How Root Canal Therapy Restores Damaged Teeth Don’t Let a Cracked Tooth Crash Your Summer Plans Cracked Teeth: Unveiling Hidden Dangers and Solutions Mastering Dental Pain Management During Endodontic Procedures Your Summer Oral Care Routine: Staying Healthy in the Heat Blog Archive 2025 2024 2023 2022 2021 2020 2019 Our Office 2014 W. Beauregard Avenue San Angelo, TX 76901 (325) 947-3040 © 2025 Concho Valley Endodontics. All Rights Reserved. \| Accessibility Policy Endodontic Marketing Endodontist in West San Angelo \| West San Angelo Endodontist \| Root Canal Specialist West San Angelo
18172	https://www.dam.brown.edu/people/alcyew/handouts/numdiff.pdf	APMA 0160 (A. Yew) Spring 2011 Numerical diﬀerentiation: ﬁnite diﬀerences The derivative of a function f at the point x is deﬁned as the limit of a diﬀerence quotient: f′(x) = lim h→0 f(x + h) −f(x) h In other words, the diﬀerence quotient f(x + h) −f(x) h is an approximation of the derivative f′(x), and this approximation gets better as h gets smaller. How does the error of the approximation depend on h? Taylor’s theorem with remainder gives the Taylor series expansion f(x + h) = f(x) + hf′(x) + h2 f′′(ξ) 2! where ξ is some number between x and x + h. Rearranging gives f(x + h) −f(x) h −f′(x) = h f′′(ξ) 2 , which tells us that the error is proportional to h to the power 1, so f(x + h) −f(x) h is said to be a “ﬁrst-order” approximation. If h > 0, say h = ∆x where ∆x is a ﬁnite (as opposed to inﬁnitesimal) positive number, then f(x + ∆x) −f(x) ∆x is called the ﬁrst-order or O(∆x) forward diﬀerence approximation of f′(x). If h < 0, say h = −∆x where ∆x > 0, then f(x + h) −f(x) h = f(x) −f(x −∆x) ∆x is called the ﬁrst-order or O(∆x) backward diﬀerence approximation of f′(x). By combining diﬀerent Taylor series expansions, we can obtain approximations of f′(x) of various orders. For instance, subtracting the two expansions f(x + ∆x) = f(x) + ∆x f′(x) + ∆x2 f′′(x) 2! + ∆x3 f′′′(ξ1) 3! , ξ1 ∈(x, x + ∆x) f(x −∆x) = f(x) −∆x f′(x) + ∆x2 f′′(x) 2! −∆x3 f′′′(ξ2) 3! , ξ2 ∈(x −∆x, x) gives f(x + ∆x) −f(x −∆x) = 2∆x f′(x) + ∆x3 f′′′(ξ1) + f′′′(ξ2) 6 , so that f(x + ∆x) −f(x −∆x) 2∆x −f′(x) = ∆x2 f′′′(ξ1) + f′′′(ξ2) 12 Hence f(x + ∆x) −f(x −∆x) 2∆x is an approximation of f′(x) whose error is proportional to ∆x2. It is called the second-order or O(∆x2) centered diﬀerence approximation of f′(x). If we use expansions with more terms, higher-order approximations can be derived, e.g. consider f(x + ∆x) = f(x) + ∆x f′(x) + ∆x2 f′′(x) 2! + ∆x3 f′′′(x) 3! + ∆x4 f(4)(x) 4! + ∆x5 f(5)(ξ1) 5! f(x −∆x) = f(x) −∆x f′(x) + ∆x2 f′′(x) 2! −∆x3 f′′′(x) 3! + ∆x4 f(4)(x) 4! −∆x5 f(5)(ξ2) 5! f(x + 2∆x) = f(x) + 2∆x f′(x) + 4∆x2 f′′(x) 2! + 8∆x3 f′′′(x) 3! + 16∆x4 f(4)(x) 4! + 32∆x5 f(5)(ξ3) 5! f(x −2∆x) = f(x) −2∆x f′(x) + 4∆x2 f′′(x) 2! −8∆x3 f′′′(x) 3! + 16∆x4 f(4)(x) 4! −32∆x5 f(5)(ξ4) 5! Taking 8×(ﬁrst expansion −second expansion)−(third expansion −fourth expansion) cancels out the ∆x2 and ∆x3 terms; rearranging then yields a fourth-order centered diﬀerence approximation of f′(x). Approximations of higher derivatives f′′(x), f′′′(x), f(4)(x) etc. can be obtained in a similar manner. For example, adding f(x + ∆x) = f(x) + ∆x f′(x) + ∆x2 f′′(x) 2! + ∆x3 f′′′(x) 3! + ∆x4 f(4)(ξ1) 4! · · · f(x −∆x) = f(x) −∆x f′(x) + ∆x2 f′′(x) 2! −∆x3 f′′′(x) 3! + ∆x4 f(4)(ξ2) 4! · · · gives f(x + ∆x) + f(x −∆x) = 2f(x) + ∆x2f′′(x) + ∆x4 f(4)(ξ1) + f(4)(ξ2) 24 , so that f(x + ∆x) −2f(x) + f(x −∆x) ∆x2 −f′′(x) = ∆x2 f(4)(ξ1) + f(4)(ξ2) 24 Hence f(x + ∆x) −2f(x) + f(x −∆x) ∆x2 is a second-order centered diﬀerence approximation of the sec-ond derivative f′′(x). Here are some commonly used second- and fourth-order “ﬁnite diﬀerence” formulas for approximating ﬁrst and second derivatives: O(∆x2) centered diﬀerence approximations: f′(x) : f(x + ∆x) −f(x −∆x) /(2∆x) f′′(x) : f(x + ∆x) −2f(x) + f(x −∆x) /∆x2 O(∆x2) forward diﬀerence approximations: f′(x) : −3f(x) + 4f(x + ∆x) −f(x + 2∆x) /(2∆x) f′′(x) : 2f(x) −5f(x + ∆x) + 4f(x + 2∆x) −f(x + 3∆x) /∆x3 O(∆x2) backward diﬀerence approximations: f′(x) : 3f(x) −4f(x −∆x) + f(x −2∆x) /(2∆x) f′′(x) : 2f(x) −5f(x −∆x) + 4f(x −2∆x) −f(x −3∆x) /∆x3 O(∆x4) centered diﬀerence approximations: f′(x) : −f(x + 2∆x) + 8f(x + ∆x) −8f(x −∆x) + f(x −2∆x) /(12∆x) f′′(x) : −f(x + 2∆x) + 16f(x + ∆x) −30f(t) + 16f(x −∆x) −f(x −2∆x) /(12∆x2) In science and engineering applications it is often the case that an exact formula for f(x) is not known. We may only have a set of data points (x1, y1), (x2, y2), . . . , (xn, yn) available to describe the functional dependence y = f(x). If we need to estimate the rate of change of y with respect to x in such a situation, we can use ﬁnite diﬀerence formulas to compute approximations of f′(x). It is appropriate to use a forward diﬀerence at the left endpoint x = x1, a backward diﬀerence at the right endpoint x = xn, and centered diﬀerence formulas for the interior points.
18173	https://arxiv.org/pdf/1505.00001	1 Rule based lexicographical permutation sequences Asbjørn Brændeland Abstract In a permutation sequence built by means of sub permutations, the transitions between successive permutations are subject to a set of n(n–1)/2 rules that naturally group into n–1 matrices with a high degree of regularity. By means of these rules, the sequence can be produced in O(3 n!) time and O( n3) space. To generate all permutations of a given set one can go for (among other things) a minimum of changes from one permutation to the next , , , or lexicographic order , at the cost of having to perform more changes . Here, we go for the latter. A lexicographic order can be established by recursively producing successively larger sub-permutations, starting from the right, as in the following example on six elements. This makes the permuted elements move back and forth according to a set of rules , from one permutation to the next. 1 ( a b c d e f ) 705 ( f e b c a d ) 2 ( a b c d f e ) 706 ( f e b c d a ) 3 ( a b c e d f ) 707 ( f e b d a c ) 4 ( a b c e f d ) 708 ( f e b d c a ) 5 ( a b c f d e ) 709 ( f e c a b d ) 6 ( a b c f e d ) 710 ( f e c a d b ) 7 ( a b d c e f ) 711 ( f e c b a d ) 8 ( a b d c f e ) 712 ( f e c b d a ) 9 ( a b d e c f ) … 713 ( f e c d a b ) 10 ( a b d e f c ) 714 ( f e c d b a ) 11 ( a b d f c e ) 715 ( f e d a b c ) 12 ( a b d f e c ) 716 ( f e d a c b ) 13 ( a b e c d f ) 717 ( f e d b a c ) 14 ( a b e c f d ) 718 ( f e d b c a ) 15 ( a b e d c f ) 719 ( f e d c a b ) 16 ( a b e d f c ) 720 ( f e d c b a ) Here is one way of producing the sequence: Make a 6  6 matrix M with identical rows = [ a b c d e f ], take the downwards diagonal D and the 5  6 matrix N = M \ D, make the permutations of each row in N recursively and prepend D[i] to each permutation of N[i]. The principle is illustrated in Figure 1 , with four instead of six elements. Figure 1. The method, generalized to n-element sets, is easily implemented but not optimal. It requires more than 5( n – 1) n! concatenations, and, since the entire set of sub-permutations must be in place before the final concatenations can be made, the space requirement is nn!. However, the number of permu-tation rules is only (𝑛 2) so if we can find these rules we can get an implementation that runs in O(3 n!) time and requires O( n3) space. 2 Let Pi be the permutation sequence. The transition from for example P12 to P13 above is given by the rule [0, 0, 2, 2, –2, –2] which assigns the following moves to the elements e1, …, e6 in P12 : Leave e1 and e2 in place, move e3 and e4 two places to the right and move e5 and e6 two places to the left. For a 6 elements permutation there are 15 rules. One set of rules brings a new element to the front. Since one of the 6 elements starts at the front there must be 5 such rules that apply to the permutations with indices 5!, 2 5!, 3 5!, 4 5! and 5 5!, and, except for these rules, for each successive segment of length 5! the rules must be the same. Within the first segment, a set of 4 rules brings a new element to position 2. These apply to indices 4!, 2 4!, 3 4! and 4 4! as well as to the corresponding indices in the subsequent segments of length 5!. Then there is a set of 3 rules that apply to indices 3!, 2 3!, 3 3!, and to the corresponding indices in the other length 4! segments, a set of 2 rules that apply to indices 2!, and 2 2!, in the length 3 segments and finally one rule that apply to index 1 and every other odd index. We let the index of a rule R be the index of the first permutation to which R applies. It follows from the above reasoning that the permutations to which R applies are evenly spaced, such that for Ri the set of relevant indices is i + j(i + 1)!, for j  0; and ( i + 1)! is then the step length of Ri. All the rules in a set have the same step length. The step length s for one set of rules Sk equals the in-dex of the first rule in the next set, Sk+1 , and s is also the distance between the rule indices of Sk+1 . For R1 the step length is 2, for R2 and R4 the step length is 6, for R6, R12 and R18 the step length is 24, etc.. Now, we could reason about the contents of each rule, or we can simply observe them —and a striking pattern emerges: Each set of rules, leading zeros excluded, induces a matrix: a 1  2 matrix for the odd indices, a 2  3 matrix for the indices divisible by 2! but not by 3!, a 3  4 matrix for the indices divisible by 3! but not by 4!, etc. Let RM m be an m  m + 1 matrix, let C be the first column in RM m , let M be the m  m matrix RM m \ C, and let D be the upwards diagonal in M. Then C[i] = i, D[ i] = i – m – 1, and otherwise M[i, j] = m + 1 – 2j, when i and j are the row and column indices, respectively. Figure 2. Rules, step lengths and the indices of the permutations to which the rules apply. 3 Complexity The processing of one permutation involves searching for the pertinent rule and performing the moves. Since the search keys are constantly changed, as explained below, the average search time = (𝑛 2)/2. For each number of moves m there are m – 1 rules. Each 2-moves-rule applies to n!/2! permutations, each 3-moves-rule applies to n!/3! permutations, etc, and each n-moves-rule apply to n!/ n! = 1 permu-tation. This gives the following number of moves: 12n!/2! + 2 3n!/3! + 3 4n!/4! + … + ( n – 1) nn!/ n! = n!(1 + 1 + 1/ 2 + 1/2! + 1/3! + … + 1/( n – 2)!)  3n!, and the total time is about n!(3 + (𝑛 2)/2). Implementation For an actual generation of a permutation sequence we can represent each rule by an object containing the rule R, a permutation index p and a step length s and, to avoid an inordinate amount of rule tracking computation, we place the objects in some searchable structure. Together p and s give all the permu-tation indices to which R applies. Given permutation Pp, we look up the object O such that O.p = p and produce Pp+1 by means of O.R , and before we go to the next permutation, we set O.p to O.p + O.s , so that O is searchable when the next permutation index to which O.R applies, comes around. The implementation is written in Racket , a dialect of Scheme . Permutations by rules Since lists are immutable in Racket, and we need to update the rule index regularly, we use a vector rather than a list to represent the rule object. (define (make-rule-object index step rule) (vector index step rule)) (define (get-index rule-object) (vector-ref rule-object 0)) (define (get-step rule-object) (vector-ref rule-object 1)) (define (get-rule rule-object) (vector-ref rule-object 2)) (define (update-index! rule-object) (vector-set! rule-object 0; The permutation index is the zero’th vector element. (+ (get-index rule-object) ; Update the current index (get-step rule-object)))) ; by adding the step length. (define (move-elements n elements rule) (let ((perm-vector (make-vector n))) ; Use a temporary vector for the new permutation. (for-each (lambda (elem move pos) (vector-set! perm-vector (+ pos move) elem)) elements rule (enumerate 0(- n 1))) ; Current positions. † (vector->list perm-vector))) ; Return a list with the new permutation. (define (make-perm-rule qin) (define moves (cons i; The row indices fill the first column in the m  q matrix. (map (lambda (j) ; m  m matrix column index. (if (= j(-q i)) (- i q) ; [i, j] is in the upwards diagonal in the m  m matrix. (- q ( 2 j)))) (enumerate 1 (-q 1)))) ; m  m matrix column indices. † (front-pad-list 0n moves)) ; Fill inn leading zeros ‡4 (define (make-perm-matrix mn) ; Permutation matrix index and permutee size. (define p(factorial m)) ; Index of first rule in matrix. (define q(+ m1)) ; Matrix width. (define s( pq)) ; Step length. (map (lambda (p r) (make-rule-object p s r)) (enumerate p( pm) p) ; Permutation indices. † (map (lambda (i) (make-perm-rule q i n)) ; Rule. (enumerate 1m)))) ; Matrix row indices † Here, we use a simple list for the permutation rules, which gives an average search time = (𝑛 2)/2. Since the key values, the indices, are constantly changed, there are no obvious alternatives. (define (set-up-perm-rules n) (flatten (map (lambda (m) (make-perm-matrix m n)) (enumerate 1 (-n 1))))) ; † (define (find-perm-rule perm-index rules) (let ((rule-object (first rules))) (if (= perm-index (get-index rule-object)) ; Found the right object, so (begin (update-index! rule-object) ; prepare it for next search, and (get-rule rule-object)) ; return the rule. (find-rule perm-index (rest rules))))) ; Keep searching (define (permute-by-rules permutee) (define n(length permutee)) (define max-index (factorial n)) (define rules (set-up-permutation-rules n)) (define (iterate perm perm-index) (displayln perm) ; Show the current permutation (if (>= perm-index max-index) 'permutations-by-rules-completed (iterate (move-elements nperm (find-perm-rule perm-index rules)) (+ perm-index 1)))) (iterate permutee 1)) ———————————————————————————————————————————————————————————————————————————————————————— ; † enumerate takes two range arguments a and b and an optional step argument s and returns a list with the number sequence a, a + s, a + 2 s, … , a + ks , where a + ks  b < a + ( k + 1) s. The default step value = 1. ; ‡ front-pad-list takes 3 arguments: a pad , a full length , and a paddee , and returns a list with the padde following a sufficient number of leading pads to fill the given length. References Heap, B. R. (1963). "Permutations by Interchanges". The Computer Journal 6 (3): 293 –4. Johnson, Selmer M. (1963), "Generation of permutations by adjacent transposition", Mathematics of Computation 17: 282 –285. Sedgewick, R (1977). "Permutation generation methods" (PDF). Computing Surveys 9: 137 –164. Trotter, H. F. (August 1962), "Algorithm 115: Perm", Communications of the ACM 5 (8): 434 –435, doi:10.1145/368637.368660.
18174	https://www.quora.com/How-do-I-write-an-equation-that-describes-the-axis-of-symmetry-from-a-graph	Something went wrong. Wait a moment and try again. Axis of Symmetry Plot Construction Parabolas (geometry) Linear Equations Functions (mathematics) Basic Algebra 5 How do I write an equation that describes the axis of symmetry from a graph? Sioban Snybee MS from Mathematics · Author has 200 answers and 177.5K answer views · 4y · Graphically look for where an imaginary mirror could be placed so that one side of the “mirror” line is a true reflection of the other side of the “mirror” line. Once you graphically determine where that mirror line is placed, you have to determine the equation for that line. It can be a vertical line like x=2 : Or a horizontal line like y= -2 : or a line at some other angle like y=x : In order to find the equation of an axis of symmetry (AOS) in any of these cases, you have to find a point that the AOS goes through and then ask your self what else you know about the line. If it’s a vertical line, and you know it goes through point (c, d), then the AOS formula would be x=c. If it’s a horizontal line, and you know it goes through point (c, d), then the AOS formula would be y=d. If it’s neither vertical nor horizontal you have to determine the slope of the AOS. This process varies depending on the graph. Some AOSs intersect the graph in two points, and given two points you can use the slope formula to find the slope and the point-slope form of a line to get the AOS’s equation, for example: The AOS goes through (-2,2) and (2,-2), so the slope is and the equation for the AOS is (y-2) = -(x-(-2)) or y = -x It’s also important to note that graphs can have more than one AOS: And some graphs have apparent symmetry, but pinning down an exact formula for an axis of symmetry is hard. Calculus can help sometimes, but I’m not sure if that is a discussion you are ready for. 11.5K views · View upvotes · Sponsored by Bigin by Zoho CRM Hard to follow-up with prospects across multiple inboxes and DMs? Track emails, calls, WhatsApp messages, and social DMs in one place and respond instantly. Related questions How do you write the axis of symmetry as an equation from a graph? What is the equation of the axis of symmetry of the graph of? What is the formula to find the axis of symmetry? How do I write an equation that describes the parabola's axis of symmetry using the two points (3,-2) and (7,-2)? How do you find the axis of symmetry of a third degree equation? Robert Colburn Former Student Assistant IV at Cabrillo College (1998–2001) · Author has 3K answers and 2.4M answer views · 4y Related How do I find the equation of axis of symmetry if I was given a graph? I’m given a parabola with x intercepts of (-6,0) and(2,0) and a vertex of (-2,-4) and I don’t know how to put it into an equation of axis of symmetry. How do I find the equation of axis of symmetry if I was given a graph? I’m given a parabola with x intercepts of (-6,0) and(2,0) and a vertex of (-2,-4) and I don’t know how to put it into an equation of axis of symmetry. The axis of symmetry is an imaginary line that goes through the vertex and “cuts the parabola in half”. It’s a great tool for graphing a parabola because you can calculate points on one side of this line and then reflect those points to the other side, thereby getting twice as many points with less calculations. The axis of symmetry for your parabola is just a vertical line goi How do I find the equation of axis of symmetry if I was given a graph? I’m given a parabola with x intercepts of (-6,0) and(2,0) and a vertex of (-2,-4) and I don’t know how to put it into an equation of axis of symmetry. The axis of symmetry is an imaginary line that goes through the vertex and “cuts the parabola in half”. It’s a great tool for graphing a parabola because you can calculate points on one side of this line and then reflect those points to the other side, thereby getting twice as many points with less calculations. The axis of symmetry for your parabola is just a vertical line going through your vertex at (- 2, - 4). The equation of any vertical line is x = c where c is the x-coordinate of any of the points on that line. The equation of the axis of symmetry for your parabola is then x = - 2 Gary Ward MaEd in Education & Mathematics, Austin Peay State University (Graduated 1997) · Author has 4.9K answers and 7.6M answer views · 2y Related How do I sketch the graph of y=-1-x^2? From the graph, what is the equation of the line of symmetry? How do I sketch the graph of y=-1-x^2? From the graph, what is the equation of the line of symmetry? Rewrite it as y = (-1)x² + (0)x (-1) in the form ax² = bx + c The axis of symmetry is x = -b/(2a) = -(0)/2(-1) = 0 Since c is where the graph crosses the y-axis when x = 0, the vertex is at (0, -1) a = -1 so the parabola opens down. Starting at the vertex, go right 1, down 1, right 1, down 3, right 1, down 5, right 1, down 7, … Starting at the vertex, go left 1, down 1, left 1, down 3, left1, down 5, left1, down 7, … Draw a smooth line connecting the points with an arrowhead at each end pointing down. How do I sketch the graph of y=-1-x^2? From the graph, what is the equation of the line of symmetry? Rewrite it as y = (-1)x² + (0)x (-1) in the form ax² = bx + c The axis of symmetry is x = -b/(2a) = -(0)/2(-1) = 0 Since c is where the graph crosses the y-axis when x = 0, the vertex is at (0, -1) a = -1 so the parabola opens down. Starting at the vertex, go right 1, down 1, right 1, down 3, right 1, down 5, right 1, down 7, … Starting at the vertex, go left 1, down 1, left 1, down 3, left1, down 5, left1, down 7, … Draw a smooth line connecting the points with an arrowhead at each end pointing down. Assistant Bot · Sep 7 An axis of symmetry from a graph is a vertical or horizontal line (most commonly vertical) that splits a figure so each side is a mirror image. For typical graphs you’ll write it as a simple linear equation: x = c (vertical axis) or y = c (horizontal axis). Determine the constant c from the graph using these steps. Identify the type of symmetry Vertical symmetry (common for parabolas and many functions): axis is a vertical line x = c. Horizontal symmetry (less common): axis is a horizontal line y = c. Rotational or other symmetries are not expressed as a single vertical/horizontal line. For a parabola y = ax^2 + bx + c (graph opening up or down) Axis of symmetry: x = -b/(2a). If you only have the graph, find the x-coordinate of the vertex; the axis is x = (that x-coordinate). For functions given graphically without formula Locate the vertex (or central line of mirror-image points). Read its x-coordinate (for vertical axis) or y-coordinate (for horizontal axis). Write the equation x = value or y = value, using exact values when possible (fractions) or a rounded decimal if necessary. For symmetric sets of points Pair symmetric points across the suspected axis. If point (x1,y) pairs with (x2,y), axis is vertical line x = (x1 + x2)/2. If (x,y1) pairs with (x,y2), axis is horizontal line y = (y1 + y2)/2. Examples Parabola with equation y = 2x^2 - 8x + 5: axis x = -(-8)/(2·2) = 8/4 = 2, so axis is x = 2. Graph shows vertex at (3, -1): axis is x = 3. Two symmetric points (1,4) and (5,4): axis is x = (1+5)/2 = 3 → x = 3. Symmetric points (2,1) and (2,-3) indicate horizontal axis y = (1 + (-3))/2 = -1 → y = -1. Tips for accuracy Use exact coordinates from grid intersections when possible. For noisy or approximate graphs, average multiple symmetric pairs for a better estimate. For conic sections other than vertical parabolas, determine orientation first; rotated axes require advanced methods (not a simple x = c or y = c). Concise summary: identify the symmetry type, read or compute the central coordinate, and write the corresponding linear equation x = c (vertical) or y = c (horizontal). Ask follow-up questions on Poe, a free app from Quora By continuing you agree to create a Poe account, and to its Terms of Service & Privacy Policy Tell me more Can you explain how to find the vertex of a parabola from its graph? What if the graph has multiple axes of symmetry? How does symmetry apply to other types of functions besides parabolas? 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. 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Related questions What is the axis of symmetry for the equation, y=−4x2+20x+15? Is there always symmetry in an equation? How can I find the vertex and axis of symmetry? How do I write an equation for the given graph? How do we obtain an equation from a graph? Alex Thomas Knows English · Author has 56 answers and 20.9K answer views · 3y Originally Answered: How do you write the axis of symmetry as an equation from a graph? · If you have a graph of a function, the axis of symmetry is the line that divides the graph into two mirror images. This line will go through the vertex of the graph. To find the equation of the axis of symmetry, you can use the formula x = -b/2a. Gary Ward MaEd in Education & Mathematics, Austin Peay State University (Graduated 1997) · Author has 4.9K answers and 7.6M answer views · 2y Related How do you graph the parabola and find the y-intercept, axis of symmetry, and vertex x^2+6x-40? How do you graph the parabola and find the y-intercept, axis of symmetry, and vertex of y = x^2+6x-40? y = ax² + bx + c so a = 1; b = 6; and c = -40 C is always the y-intercept for a vertical parabola where the axis of symmetry is parallel to the y-axis, so the y-intercept is -40 or the point (0, -40). The axis of symmetry can be found using [math]\, x = \frac{-b}{2a} \,[/math] so that [math]\, x = \frac{-(6)}{2(1)} = -3 \,[/math] is the axis of symmetry. Put the x-value for the axis of symmetry into the equation to find the y-value. y = (-3)² +6(-3) - 40 = -49, so the vertex is (-3, -49). For every unit left or right of th How do you graph the parabola and find the y-intercept, axis of symmetry, and vertex of y = x^2+6x-40? y = ax² + bx + c so a = 1; b = 6; and c = -40 C is always the y-intercept for a vertical parabola where the axis of symmetry is parallel to the y-axis, so the y-intercept is -40 or the point (0, -40). The axis of symmetry can be found using [math]\, x = \frac{-b}{2a} \,[/math] so that [math]\, x = \frac{-(6)}{2(1)} = -3 \,[/math] is the axis of symmetry. Put the x-value for the axis of symmetry into the equation to find the y-value. y = (-3)² +6(-3) - 40 = -49, so the vertex is (-3, -49). For every unit left or right of the axis of symmetry, y will increase or decrease by x² coefficient times the series of odd numbers starting with 1 (a[1,3,5,7, …]). The roots can be found by factoring, y = (x + 10)(y - 4) so that … x + 10 = 0 → x = -10 and y - 4 = 0 → y = 4 giving roots (-10, 0) and (4, 0). Graphing this equation looks better if the axes are not equal. In many cases you will be allowed to plot a parabola using just the roots, vertex and y-intercept. Some teachers will ask you to make a table of values and plot the points. Table below the graph. Note we started at the vertex and went left and right adding a · odd numbers that appear in the third column. Sponsored by Grammarly Is your writing working as hard as your ideas? Grammarly’s AI brings research, clarity, and structure—so your writing gets sharper with every step. Pavel Juranek Technician, Programmer, Analyst, Consultant · Author has 3.7K answers and 1.3M answer views · 3y Related How do you write the axis of a symmetry equation? The axis of symmetry has two properties a) it is a line, say L b) all midpoints M defined by: P(M) = (P(A) + P(sym(A))) / 2 lies in L where sym(A) is a point symmetric (in the considered sense) to some point A P(X) is positional vector of X Use formula of the symmetry in b) and gets L. Philip Lloyd Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist. · Author has 6.8K answers and 52.8M answer views · 6y Related How would I determine the axes of symmetry of a hyperbola? ASYMPTOTES TO HYPERBOLAE. Start with simple cases and emphasise the logic. The previous 4 examples are all centred at (0, 0). A Hyperbola can be moved (or translated) up or down is the same way that the parabola y = x^2 can be moved 4 units to the right if we change the equation to y = (x – 4)^2 Consider the Hyperbola: Note: I never like to use the standard formulae for these asymptotes. The above method emphasises logical thinking. as below… We need to move the asymptotes so that they now go through (6, – 1) If y = ⅔x + c and y = – ⅔x + d We subs x = 6, y = – 1 so c = – 5 and d = +3 The equations of t ASYMPTOTES TO HYPERBOLAE. Start with simple cases and emphasise the logic. The previous 4 examples are all centred at (0, 0). A Hyperbola can be moved (or translated) up or down is the same way that the parabola y = x^2 can be moved 4 units to the right if we change the equation to y = (x – 4)^2 Consider the Hyperbola: Note: I never like to use the standard formulae for these asymptotes. 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Please draw two points A(3;-2) ; and ; B(7;-2) ; now draw the perpendicular bisector of the line AB ; huraaaaa this is axis of symmetry and its equation is:x=(xA+xB)/2 ; Mention: as a matter of fact if two points by coordinates: A(a;k) ; B(b;k) have been located on a parabloa the line x=(a+b)/2 or two other points M(k;a) andN(k;b)two points of a parabloa the line : y=(a+b)/2 are their axix of symmetry. Have a nice time and good luck. Related questions How do you write the axis of symmetry as an equation from a graph? What is the equation of the axis of symmetry of the graph of? What is the formula to find the axis of symmetry? How do I write an equation that describes the parabola's axis of symmetry using the two points (3,-2) and (7,-2)? How do you find the axis of symmetry of a third degree equation? What is the axis of symmetry for the equation, y=−4x2+20x+15? Is there always symmetry in an equation? How can I find the vertex and axis of symmetry? How do I write an equation for the given graph? How do we obtain an equation from a graph? Consider the equation y^2 = 8x + 48. What is the axis of symmetry? Algebra: Why do we graph an equation? How do you find the equation of the line of symmetry of a quadratic graph? How do you write the axis of a symmetry equation? The equation of the axis of symmetry of the graph of a quadratic function is x=-1. The graph passes through the points (0,3) and (-3, 9). What is the equation of its function? About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
18175	http://cavanaughmath.pbworks.com/w/file/fetch/63967678/Precalc%20Lesson%207.5.pdf	Copyright © Cengage Learning. All rights reserved. Analytic Trigonometry Copyright © Cengage Learning. All rights reserved. 7.5 More Trigonometric Equations 3 Objectives ► Solving Trigonometric Equations by Using Identities ► Equations with Trigonometric Functions of Multiples of Angles 4 More Trigonometric Equations In this section we solve trigonometric equations by first using identities to simplify the equation. We also solve trigonometric equations in which the terms contain multiples of angles. 5 Solving Trigonometric Equations by Using Identities 6 Solving Trigonometric Equations by Using Identities In the next example we use trigonometric identities to express a trigonometric equation in a form in which it can be factored. 7 Example 1 – Using a Trigonometric Identity Solve the equation 1 + sinθ = 2 cos2θ. Solution: We first need to rewrite this equation so that it contains only one trigonometric function. To do this, we use a trigonometric identity: 1 + sinθ = 2 cos2θ 1 + sinθ = 2(1 – sin2θ ) 2 sin2θ + sinθ – 1 = 0 (2 sinθ – 1) (sinθ + 1) = 0 Given equation Pythagorean identity Put all terms on one side Factor 8 Example 1 – Solution 2 sinθ – 1 = 0 or sinθ + 1 = 0 sinθ = or sinθ = –1 θ = or θ = Because sine has period 2π, we get all the solutions of the equation by adding integer multiples of 2π to these solutions. cont’d Set each factor equal to 0 Solve for sinθ Solve for sinθ in interval [0, 2π) 9 Example 1 – Solution Thus the solutions are θ = + 2kπ θ = + 2kπ θ = + 2kπ where k is any integer. cont’d 10 Example 3 – Squaring and Using an Identity Solve the equation cosθ + 1 = sinθ in the interval [0, 2π). Solution: To get an equation that involves either sine only or cosine only, we square both sides and use a Pythagorean identity: cosθ + 1 = sinθ cos2θ + 2 cosθ + 1 = sin2θ cos2θ + 2 cosθ + 1 = 1 – cos2θ 2 cos2θ + 2 cosθ = 0 Given equation Pythagorean identity Square both sides Simplify 11 Example 3 – Solution 2 cosθ = 0 or cosθ + 1 = 0 cosθ = 0 or cosθ = –1 θ = or θ = π Because we squared both sides, we need to check for extraneous solutions. From Check Your Answers we see that the solutions of the given equation are π /2 and π. cont’d Set each factor equal to 0 Solve for cosθ Solve for θ in [0, 2π) 12 Example 3 – Solution Check Your Answer: θ = θ = θ = π cos + 1 = sin cos + 1 = sin cos π + 1 = sin π 0 + 1 = 1 0 + 1 ≟ –1 –1 + 1 = 0 cont’d 13 Example 4 – Finding Intersection Points Find the values of x for which the graphs of f (x) = sin x and g (x) = cos x intersect. Solution 1: Graphical The graphs intersect where f (x) = g (x). In Figure 1 we graph y1 = sin x and y2 = cos x on the same screen, for x between 0 and 2π. (a) (b) Figure 1 14 Example 4 – Solution 1 Using or the intersect command on the graphing calculator, we see that the two points of intersection in this interval occur where x ≈ 0.785 and x ≈ 3.927. Since sine and cosine are periodic with period 2π, the intersection points occur where x ≈ 0.785 + 2kπ and x ≈ 3.927 + 2kπ where k is any integer. cont’d 15 Example 4 – Solution 2 Algebraic To find the exact solution, we set f (x) = g (x) and solve the resulting equation algebraically: sin x = cos x Since the numbers x for which cos x = 0 are not solutions of the equation, we can divide both sides by cos x: = 1 tan x = 1 cont’d Equate functions Divide by cos x Reciprocal identity 16 Example 4 – Solution 2 The only solution of this equation in the interval (–π /2, π /2) is x = π /4. Since tangent has period π , we get all solutions of the equation by adding integer multiples of π : x = + kπ where k is any integer. The graphs intersect for these values of x. You should use your calculator to check that, rounded to three decimals, these are the same values that we obtained in Solution 1. cont’d 17 Equations with Trigonometric Functions of Multiples of Angles 18 Equations with Trigonometric Functions of Multiples of Angles When solving trigonometric equations that involve functions of multiples of angles, we first solve for the multiple of the angle, then divide to solve for the angle. 19 Example 5 – A Trigonometric Equation Involving Multiple of an Angle Consider the equation 2 sin 3θ – 1 = 0. (a) Find all solutions of the equation. (b) Find the solutions in the interval [0, 2π). Solution: (a) We first isolate sin 3θ and then solve for the angle 3θ. 2 sin 3θ – 1 = 0 2 sin 3θ = 1 sin 3θ = Given equation Add 1 Divide by 2 20 Example 5 – Solution 3θ = cont’d Solve for 3θ in the interval [0, 2π) (see Figure 2) Figure 2 21 Example 5 – Solution To get all solutions, we add integer multiples of 2π to these solutions. So the solutions are of the form 3θ = + 2kπ 3θ = + 2kπ To solve for θ, we divide by 3 to get the solutions where k is any integer. cont’d 22 Example 5 – Solution (b) The solutions from part (a) that are in the interval [0, 2π) correspond to k = 0, 1, and 2. For all other values of k the corresponding values of θ lie outside this interval. So the solutions in the interval [0, 2π) are cont’d 23 Example 6 – A Trigonometric Equation Involving a Half Angle Consider the equation . (a) Find all solutions of the equation. (b) Find the solutions in the interval [0, 4π). Solution: (a) We start by isolating tan : Given equation Add 1 24 Example 6 – Solution Since tangent has period π, to get all solutions we add integer multiples π of to this solution. So the solutions are of the form Divide by Solve for in the interval cont’d 25 Example 6 – Solution Multiplying by 2, we get the solutions θ = + 2kπ where k is any integer. (b) The solutions from part (a) that are in the interval [0, 4π) correspond to k = 0 and k = 1. For all other values of k the corresponding values of x lie outside this interval. Thus the solutions in the interval [0, 4π) are cont’d
18176	https://journals.lww.com/anatomicpathology/fulltext/2025/05000/modern_approach_to_nodal_t_cell_lymphomas.4.aspx	Advances in Anatomic Pathology Log in or Register Subscribe to journal Subscribe Get new issue alerts Get alerts;;) Subscribe to eTOC;;) ### Secondary Logo Enter your Email address: Privacy Policy ### Journal Logo Articles Advanced Search Toggle navigation SubscribeRegisterLogin Browsing History Articles & Issues Current Issue Previous Issues Published Ahead-of-Print For Authors Submit a Manuscript Information for Authors Language Editing Services Author Permissions Journal Info About the Journal Editorial Board Advertising Open Access Subscription Services Reprints Rights and Permissions Articles Advanced Search May 2025 - Volume 32 - Issue 3 Previous Article Next Article Outline AT WHAT POINT DO WE BELIEVE THAT A BIOPSY IS T-CELL LYMPHOMA RATHER THAN A REACTIVE PROLIFERATION? HOW DOES FLOW CYTOMETRY ASSIST IN A DIAGNOSIS OF T-CELL LYMPHOMA? WHAT IS THE BEST STEPWISE APPROACH FOR DIAGNOSING NODAL T-CELL LYMPHOMA? HOW DOES EPSTEIN-BARR VIRUS TESTING IMPACT THE DIAGNOSIS OF T-CELL LYMPHOMA? IN WHICH DIAGNOSTIC SITUATIONS IS NEXT GENERATION SEQUENCING MOST CONTRIBUTORY AND ARE CLONALITY STUDIES RECOMMENDED FOR ALL CASES? WHEN IS A DIAGNOSIS OF FOLLICULAR HELPER T-CELL LYMPHOMA APPROPRIATE? WHAT CHARACTERISTICS ARE MOST USEFUL TO DIFFERENTIATE ANAPLASTIC LARGE CELL LYMPHOMA FROM PERIPHERAL T-CELL LYMPHOMA, NOT OTHERWISE SPECIFIED? CAN NODAL PERIPHERAL T-CELL LYMPHOMA BE ACCURATELY DIAGNOSED ON NEEDLE BIOPSIES? CONCLUSION REFERENCES Images Slideshow Gallery Export PowerPoint file Download PDF EPUB Cite Copy Export to RIS Export to EndNote Share Email Facebook X LinkedIn Favorites Permissions More Cite Permissions Image Gallery Article as EPUB Export All Images to PowerPoint FileAdd to My Favorites Email to Colleague Colleague's E-mail is Invalid Your Name: Colleague's Email: Separate multiple e-mails with a (;). Message: Your message has been successfully sent to your colleague. Some error has occurred while processing your request. Please try after some time. Export to End Note Procite Reference Manager [x] Save my selection Review Articles Modern Approach to Nodal T-Cell Lymphomas Ondrejka, Sarah L. DO; de Leval, Laurence MD, PhD† Author Information Pathology and Laboratory Medicine Institute, Cleveland Clinic, Cleveland, OH †Department of Laboratory Medicine and Pathology, Institute of Pathology, Lausanne University Hospital and Lausanne University, Lausanne, Switzerland The authors have no funding or conflicts of interest to disclose. Reprints: Sarah L. Ondrejka, DO, Department of Pathology and Laboratory Medicine, Diagnostics Institute, Cleveland Clinic, 9500 Euclid Avenue, L-30, Cleveland 44195, OH (e-mail: ondrejs@ccf.org); Laurence de Leval, MD, PhD, Department of Laboratory Medicine and Pathology, Institute of Pathology, Lausanne University Hospital, 25 rue du Bugnon, Lausanne 1011, Switzerland (e-mail: laurence.deleval@chuv.ch). All figures can be viewed online in color at www.anatomicpathology.com. This is an open access article distributed under the Creative Commons Attribution License 4.0 (CCBY), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Advances In Anatomic Pathology 32(3):p 220-238, May 2025. \| DOI: 10.1097/PAP.0000000000000492 Open Abstract In recent decades, there have been many meaningful contributions to the pathology literature with respect to T-cell lymphoma pathogenesis and biology and improved diagnostics. We know more about disease classification, clinical characteristics, immunophenotype, and genetics than ever before, and yet diagnosis of nodal T-cell lymphomas continues to be a challenging exercise. Complicating interpretation are the many non-neoplastic mimickers of peripheral T-cell lymphoma including drug effects, viruses, autoimmune, and idiopathic conditions, that must be considered when faced with an abnormal lymph node biopsy. The number of immunohistochemical stains required to make a diagnosis of T-cell lymphoma is not standardized and may be exhaustive, requiring judicious use of tissue sections. Clonality studies may contribute to the diagnosis, though questions remain about test modality, when to exercise interpretive caution, and what to do if a clone cannot be demonstrated. Use of next generation sequencing in the diagnosis of nodal T-cell lymphomas is increasing, but how the data can be practically applied to diagnosis is still under examination. The goal of this paper is to consider nodal T-cell lymphoma diagnosis and classification in a modern context, using a question-and-answer format to capture the interest of the reader and address common pathology consultation queries. The recently published International Consensus Classification1,2 provides a framework for lymphoma diagnosis and highlights some relevant changes for peripheral T-cell lymphomas (PTCLs) including the unification of follicular helper T-cell lymphoma (TFH lymphoma) as a single entity that encompasses 3 subtypes: angioimmunoblastic-type (angioimmunoblastic T-cell lymphoma), follicular-type, and not otherwise specified (NOS). Primary nodal Epstein-Barr virus+ T-cell/NK-cell lymphoma, which was introduced in the 2017 WHO classification as a variant of peripheral T-cell lymphoma, NOS, is now considered a provisional entity. Within anaplastic large cell lymphoma, ALK-negative (ALK-negative ALCL), DUSP22-rearranged (DUSP22-R) ALK-negative ALCL is defined as a genetic subtype of systemic ALK-negative ALCL. Other published updates in T-cell lymphoma diagnosis apply to extranodal entities of T-cell lymphoma including those arising primarily in the gastrointestinal tract or skin and soft tissue, and are beyond the scope of this review.1 Despite subtle differences in terminology the 5th edition of the WHO classification (WHO-HAEM5) applies a similar rubric, converging on a fundamental understanding of TFH lymphoma and acknowledging the molecular heterogeneity within ALK-negative ALCL.3–5 Once the diagnosis of a nodal T-cell lymphoma is established, the boundary between some entities is sometimes blurred. For example, differentiating anaplastic large cell lymphoma, ALK negative, or adult T-cell leukemia/lymphoma from peripheral T-cell lymphoma, not otherwise specified, requires many considerations given the morphologic and immunophenotypic overlap. Establishing a follicular helper T-cell immunophenotype may help in diagnosis of a follicular helper T-cell (TFH) lymphoma, but appropriate diagnosis of this entity requires more than expression of TFH antigens. Each disease entity may encompass a broad spectrum of histopathologic features and slight variations to the immunophenotype, so a wide recognition of the subtypes of peripheral T-cell lymphoma is necessary. Finally, small-volume biopsies are increasing in popularity but create an obstacle to proper diagnosis and classification due to insufficient tissue for histologic review and ancillary studies, and the associated risk of diagnostic error. It is important to bear in mind that lymph nodes may be secondarily involved by mature T-cell neoplasms that are more typically found in the peripheral blood or bone marrow, such as T-cell prolymphocytic leukemia or adult T-cell leukemia/lymphoma (ATLL), or by disseminated T-cell lymphomas that are primary to extranodal locations such as skin (mycosis fungoides) or the intestine. These topics will be periodically discussed as they pertain to the differential diagnosis of nodal T-cell lymphoma. The focus of this paper is to provide a practical approach to lymph node-based PTCL and to concentrate on the entities that are considered primary to lymph nodes, mainly, ALCL, TFH lymphoma, PTCL, not otherwise specified (NOS), and primary nodal EBV-positive T-cell/NK-cell lymphoma (Table 1). We begin with when to consider a T-cell lymphoma in a lymph node biopsy rather than a reactive proliferation, discuss immunophenotypic, flow cytometric, and molecular adjuncts to diagnosis, and consider difficulties in the differential diagnosis among T-cell lymphoma entities. TABLE 1 - Summary of Disease Characteristics of the Primary Nodal T-Cell Lymphoma Entities Scroll left or right to view entire table. \| Entity \| Epidemiology \| Morphology \| Immunophenotype \| Genetics \| :---: :---: \| ALK+ ALCL \| 16% of PTCL M>F Median age 30-35 y \| Hallmark cells Variable patterns: Common Lymphohistiocytic Small cell Hodgkin-like Spindle cell \| CD30+, ALK+, EMA+, often CD2+, CD4+, CD43+; usually loss of several T-cell antigens (CD3, CD5, CD7), EBV-, cytotoxic markers+phospho-STAT3+ \| NPM1::ALK1 fusion is most common, ALK1 has many alternative partners JAK-STAT signaling activation \| \| ALK- ALCL \| 8%-9% of PTCL M>F Median age 54 y \| Hallmark cells Common pattern \| CD30+ (uniform, strong), loss of some T-cell antigens, EMA +/-, EBV-, cytotoxic markers +/-, phospho-STAT3 + or - \| DUSP22-R (20-30%) TP63-R (2-8%) JAK-STAT signaling activation± mutations Rarely other tyrosine kinase domain gene fusions \| \| TFH lymphoma AI-type Follicular-type NOS \| 27% of PTCL M>F Median age 60-65 y \| AITL-type: patterns I, II, III, often diffuse with polymorphic inflammatory infiltrate and FDC expansion Follicular-type: Follicular lymphoma-like or PTGC-like NOS: Diffuse and no FDC expansion or T-zone pattern \| CD3+, CD4+, TFH positive (PD1, CXCL13, ICOS, CD10, BCL6), loss of other T-cell markers (CD7 or CD5), scattered EBV positivity CD21 helps identify FDC meshwork pattern Reactive CD8+ cells can be relatively abundant \| Mutations in epigenetic regulators (TET2, DNMT3A) Hotspot RHOA G17V mutation and IDH2 R172 mutations Fusions involving CD28 and ICOS or CTLA4, ITK::SYK Gain-of-function mutations (PLCG1, CD28, CD28, FYN, PIK3, CARD11) \| \| PTCL, NOS \| 26%-27% of PTCL M=F Median age 60 y \| Variable cytology, small irregular or medium-large lymphocytes Diffuse growth pattern Variable microenvironment Lennert pattern <10% \| Positive for pan-T-cell antigens (CD3, CD2, CD5, CD7) with reduced/absent expression of some antigens CD4+ CD8+, CD4+CD8+ or CD4-CD8-, CD30+/-, TCRαβ>γδ, LEF1+, cytotoxic +/- \| CDKN2A and PTEN deletions, TP53 alterations Low prevalence of IRF4, TP63, VAV1, FYN, CD28 fusions Molecular subgroups defined by gene expression profiling: PTCL-TBX21 PTCL-GATA3 \| \| Primary nodal EBV+ T/NK-cell lymphoma \| Rare, 1%-10% of PTCL with geographic variation and a higher incidence in Asian countries Elderly patients \| Monomorphic large cell morphology Lack of angiocentricity or necrosis \| CD3+, CD5+/-, activated cytotoxic phenotype, EBV diffusely+with type II latency pattern \| Recurrent mutations in TET2, DNMT3A, STAT3, PIK3CD, DDX3X 14q11.2 loss \| AI indicates angioimmunoblastic; ALCL, anaplastic large cell lymphoma; ALK, anaplastic lymphoma kinase; EBV, Epstein-Barr virus; NOS, not otherwise specified; PTCL, peripheral T-cell lymphoma; PTGC, progressive transformation of germinal centers; R, rearranged. AT WHAT POINT DO WE BELIEVE THAT A BIOPSY IS T-CELL LYMPHOMA RATHER THAN A REACTIVE PROLIFERATION? The most important consideration when making a diagnosis of a nodal T-cell lymphoma is to be certain that a reactive lymphoproliferation is not mistaken as a neoplasm. It is recommended that the presence of several features associated with malignancy including the clinical presentation, lymph node histology, immunophenotype, and genetics are present and in alignment with the diagnosis (Table 2). Paracortical hyperplasia is one of the most common and nonspecific reactive patterns of lymphadenopathy and is composed of interfollicular expansion by histiocytes, plasma cells, eosinophils, immunoblasts, and vascular proliferation. As T-cell lymphomas can exhibit polymorphic inflammation with a T-zone or interfollicular pattern and eosinophilia, paracortical hyperplasia including with atypical features (partial effacement, occasionally a small clonal T-cell gene rearrangement) can be mistaken for lymphoma (Fig. 1). Paracortical hyperplasia can be seen in the setting of recent viral infection, vaccination, anticonvulsant therapy or other drug-induced lymphadenopathies (phenytoin, carbamazepine, amlodipine, fluoxetine, allopurinol, sulfonamides, methimazole, lamotrigine, gabapentin, nevirapine, monoclonal antibody therapy, among others).6 Reactive lymphoid proliferations usually lack complete architectural effacement of the lymph node and the polymorphous infiltrate is composed of recognizable immune cell types without significant atypia. The lymph node sinuses should remain patent and B-cell follicles are not widely separated and are composed of primary and secondary follicles in different stages of evolution. In a reactive lymph node, follicular dendritic cell meshworks are intact without attenuation or expansion. In contrast, complete effacement of the architecture with compressed or obliterated lymph node sinuses and the presence of a cytologically abnormal lymphoid population is more suggestive of lymphoma.7 Diffuse paracortical hyperplasia with subtotal effacement of the lymph node architecture and immunoblastic proliferation is the most common morphologic pattern in viral lymphadenitis. Infectious mononucleosis represents a common mimicker of T-cell lymphoma; it contains a robust interfollicular infiltrate of immunoblasts and isolated Reed-Sternberg-like cells and occasionally areas of necrosis, in a background of a high proportion cytotoxic CD8+ T cells.7,8 Caution is advised in that a small clonal T-cell receptor gene rearrangement may be detected in the setting of benign, self-limited infectious mononucleosis.9,10 TABLE 2 - Features of Malignancy in Nodal T-Cell Lymphoma Scroll left or right to view entire table. \| Feature \| Favors malignant \| Favors benign \| :---: \| Clinical features \| B symptoms, systemic lymphadenopathy, syndromic features of AITL (polyclonal hypergammaglobulinemia, hemolytic anemia, skin rash, rheumatologic symptoms) Hemophagocytic lymphohistiocytosis (although this has many underlying etiologies) Elevated LDH \| Localized lymphadenopathy Lack of B symptoms Medication record: anticonvulsants, amlodipine, fluoxetine, monoclonal antibody therapy Recent vaccination \| \| Lymph node histology/architecture \| Effaced Proliferated and disrupted follicular dendritic cell meshworks Open subcapsular sinus with perinodal infiltrate Sinusoidal involvement by large neoplastic cells Hallmark cells, atypical lymphoid cells (ie, irregular nuclei, clear cells) \| Paracortical expansion Preserved follicle density Open sinuses Polymorphous interfollicular infiltrate (small lymphocytes, histiocytes, plasma cells, eosinophils) Immunoblasts near high endothelial venules \| \| Flow cytometry \| Loss of surface CD3 Loss of CD7 or CD5 CD3+/CD4+/CD10+ T cells Monotypic TRBC1/TRBC2 \| Elevated CD4:CD8 ratio without loss of other T-cell antigens Identification of a very small TFH subset \| \| Immunophenotype \| Strong/diffuse CD30 positivity Relative increase in PD1+/CD10+/ICOS+/CXCL13+/BCL6+ T cells Abnormal CD21+ FDC meshwork pattern Loss of T-cell markers Cross-antigen expression (eg. CD20+ T cells, CD4+/TIA+ T cells) \| CD30+ immunoblastic proliferation Relative increase in T cells expressing less than 2 TFH markers or positive for TFH markers with a dim level of expression Preserved CD21+ FDC meshwork pattern Retention of T-cell markers \| \| Genetics \| Clonal TR rearrangement (beta or gamma or both) Monoclonal IGH or IGK rearrangements may be detected in T-cell lymphoma with a B-cell component (TFH lymphomas) Hotspot pathogenic mutations (ie, RHOA G17V, IDH2, etc.) or pathogenic mutations in known drivers of malignancies Rearrangements of DUSP22, TP63, ITK::SYK, CD28::ICOS, CD28::CTLA4 \| Polyclonal TR rearrangement or isolated tube C in TR-beta testing Caution is needed if only TET2 or DNMT3a variants are identified, as these may indicate nonspecific clonal hematopoiesis \| Caution is advised if a specimen has only a few characteristics favoring malignancy. FIGURE 1: Reactive lymph node with paracortical expansion in a 74-year-old man with chronic thoracic and cervical lymphadenopathy. A, A 2 cm cervical lymph node was excised and showed overall preserved architecture with capsular fibrosis. B, Hyperplastic follicles were separated by enlarged paracortical areas containing histiocytic aggregates. C, The paracortical areas contained a mixed lymphoid population with scattered eosinophils. D, CD21 showed tight follicular dendritic cell meshworks restricted to the follicles. E, CD20 stained the follicles and a minority of the cells outside. CD3 stained the paracortical areas (F) where the lymphoid cells were predominantly CD4+ (G). Many cells in the paracortex were positive for PD1 (H) or ICOS (I), with a staining intensity lower than in the follicular helper T cells in the germinal centers. This biopsy was initially mistakenly interpreted as follicular helper T-cell lymphoma, but clonality and sequencing analyses failed to demonstrate a monoclonal TR gene rearrangement or somatic mutations. Drug-induced lymphadenopathy has been described especially in the setting of anticonvulsant therapy and causes generally less prominent architectural distortion as it also encompasses germinal center hyperplasia, eosinophilia, and immunoblastic proliferation. Depending on the duration of drug therapy from hypersensitivity reactions to longer duration effects, the lymph node may have features from focal obliteration with eosinophilia and necrosis, to a burned-out phase with follicular depletion.11 Knowledge of the medical chart is helpful since certain immune-mediated drug reactions such as drug rash with eosinophilia and systemic symptoms (DRESS) is characteristically accompanied by fever, rash, elevated liver enzymes, and C-reactive protein, occurring 1 to 8 weeks after drug introduction.12 IgG4-related lymphadenopathy occurs in the spectrum of IgG4-related disease, affecting adult patients with mass-forming lesions or organ symptoms of fibrosclerosis and elevated serum IgG4. Involvement of lymph nodes can mimic a TFH lymphoma particularly in cases characterized by diffuse interfollicular expansion with plasma cells, immunoblasts and plasmablasts, and eosinophils, and occasionally follicular regression (Fig. 2).13 This can appear similar to a histologic pattern II angioimmunoblastic T-cell lymphoma, with regressive germinal centers and inflammatory interfollicular proliferation of immune cells.14 FIGURE 2: IgG4 lymphadenopathy. This lymph node is characterized by marked enlargement of the paracortex (A) which shows prominent vessels and a polymorphous infiltrate including large lymphoid cells, eosinophils, and plasma cells (B). Many lymphoid cells are CD4+ (C). Many plasma cells are IgG4+ (D). Mimics of PTCL include inborn errors of immunity with lymph node manifestation such as autoimmune lymphoproliferative syndrome (ALPS), due to mutations in the FAS/FAS-L and defective apoptosis. These patients may demonstrate lymph node enlargement with paracortical expansion by mature, double negative (CD4-/CD8-) cytotoxic αβ T cells and increased interfollicular vascularity. Distinguishing features of ALPS lymph nodes from PTCL include a spectrum of reactive germinal center changes, an increase in CD5-coexpressing B cells, and florid plasmacytosis and considering the clinical context. Significant increases in double negative (CD4-/CD8-) T cells with an accompanying polyclonal B-cell lymphocytosis by peripheral blood flow cytometry may be identified in patients with ALPS.15 Histiocytic necrotizing lymphadenitis (Kikuchi-Fujimoto lymphadenitis) causes sometimes near-total lymph node effacement by paracortical hyperplasia of histiocytes and activated T cells that can be large and immunoblast-like with a high Ki-67 proliferative fraction, with confluent areas of necrosis and karyorrhexis, overall worrisome for T-cell lymphoma (Fig. 3). An important clue is the clinical context of Kikuchi-Fujimoto disease often occurring in young women with cervical lymphadenopathy. Recognition of features of Kikuchi-Fujimoto lymphadenitis such as crescentic, myeloperoxidase-positive histiocytes, increased plasmacytoid dendritic cells, and few/absent plasma cells and absent neutrophils can be helpful, as well as understanding the progression of histologic stages that occur. These include an early proliferative lesion with abundant atypical mononuclear cells, a necrotizing phagocytic lesion, and a late xanthomatous form.16 Since PTCL can be composed of large neoplastic lymphoid cells in some cases with intermixed histiocytes and areas of necrosis, it is important to keep Kikuchi-Fujimoto lymphadenitis in mind especially in smaller biopsies in which the full morphologic and geographic spectrum of lymphadenitis may not be revealed. FIGURE 3: Kikuchi-Fujimoto. A, The lymph node shows a completely effaced architecture. B, It contains a lymphohistiocytic infiltrate with large atypical lymphoid cells. CD8 stains many of the lymphoid cells including the large ones (C) and perforin demonstrates a cytotoxic phenotype (D). E, Many histiocytes express myeloperoxidase. Conversely, there are some types and growth patterns of nodal T-cell lymphoma that may not be so obviously recognized and could be considered false-negatives. TFH lymphoma, AI-type pattern 1 features large hyperplastic germinal centers, often with absent or attenuated mantle zones and perifollicular abnormal clear cells merging into a polymorphous infiltrate with increased vascularity in the paracortex, and pattern 2 is characterized by regressed follicles mimicking Castleman disease (Fig. 4). Follicular dendritic cell (FDC) meshworks are not expanded and are relatively intact, unlike the FDC pattern seen in other types of TFHL AITL-type.17 Examples of TFH lymphoma AI-type have been described including several presented at the 2022 EAH4P workshop with low tumor burden and indolent, paucisymptomatic clinical behavior despite disseminated low-level disease. The lymph nodes from these patients demonstrated mostly preserved architecture with expanded mantle zones and follicular involution, and subtle infiltrates of abnormal TFH cells around the outer edge of follicles or in the paracortex.18 Regarding PTCL, NOS, some cytologic variants are composed of predominantly small cells. Clues to diagnosis include an effaced growth pattern, abnormal phenotype, and molecular data. FIGURE 4: Follicular helper T-cell lymphoma, angioimmunoblastic-type, patterns 1 (A–E) and 2 (F). In pattern 1, the low-power view of the lymph node shows large reactive germinal centers (A) with attenuated mantle zones and atypical clear cells at the periphery (B); IgD confirms paucity of residual mantle cells (C), ICOS stains the atypical cells forming a rim surrounding germinal center (D) and CD21 shows a tight follicular dendritic cell meshwork restricted to the germinal center (E). In pattern 2, atypical clear cells are seen in association with a regressed follicle (F). HOW DOES FLOW CYTOMETRY ASSIST IN A DIAGNOSIS OF T-CELL LYMPHOMA? Flow cytometric immunophenotyping is a sensitive diagnostic tool that contributes to a diagnosis of T-cell lymphoma by analysis of a fresh single-cell suspension of lymph node for deviations in antigen expression from normal T cells and, in some cases, demonstration of clonality. Careful attention to tissue processing is necessary to ensure cellular disaggregation while minimizing cell loss and nonspecific antibody binding, and optimizing antigenicity and viability.19 The T-cell immune response is complex and requires understanding of the major physiological T-cell subsets in addition to subtle alterations in antigen expression that can expand in various disease states. For example, prior work focused on analyzing T-cell subsets in spleen demonstrated small reproducible subsets of normal CD4+CD7- T cells (late memory T cells), subsets of T cells with altered CD5 expression or intensity (naive CD8+ cells), and minor CD4+ CD8+ populations.20 Successful performance of T-cell analysis by flow cytometry requires skilled technical staff, sensitive well-maintained instrumentation, tight processing protocols, and careful choice of antibodies and tagged fluorochromes in panel design.21 Lymphoma cells may exhibit increased autofluorescence as compared with background/normal T cells, and thresholds for positivity or negativity must be carefully considered. Nonetheless, features suggestive of T-cell neoplasia including a significantly abnormal CD4-to-CD8 ratio beyond what is considered physiological, a lack of expected T-cell markers (eg, CD2, CD5, CD7), CD10 expression, or abnormal staining intensity for any T-cell marker can be adjunctively helpful to making a diagnosis. In expert hands, sensitivity reaching as high as 92% for immunophenotypic aberrancy in PTCL samples can be achieved by cluster analysis and comparison of populations to laboratory-established reference ranges for known non-neoplastic T-cell subsets.22 In a large retrospective analysis of solid tissues comparing flow cytometry with histopathologic diagnosis that included 73 cases of nodal T-cell lymphoma, there was agreement between flow cytometry interpretation and final histopathology in 66.7% to 88.2% of cases depending on the T-cell lymphoma entity, and results were comparable even in cases with <20% neoplastic cells. Cases with nonagreement were attributed to the inability to detect an aberrant population by flow cytometry, suggesting that an abnormal population is helpful when present, but a normal flow cytometry result does not exclude the diagnosis of T-cell lymphoma.23 Prior studies of AITL by flow cytometry have provided predictable aberrant patterns to look for in assessment of TFH lymphoma. The neoplastic T cells vary in proportion to the background non-neoplastic T cells, ranging from 23% to 29% of lymphocytes. Absent or dim positivity for surface CD3 is a significant immunophenotypic aberrancy for mature T cells, which in combination with antigens such as CD4 and/or CD5 would still permit recognition as T cells rather than NK cells (which would also lack surface CD3 and express CD2 and CD7).21 The majority of cases express CD2, CD4,CD5, CD10+/-, and CD45, and negative or dim surface CD3. Absence of CD7 expression on a significantly expanded subset of CD4+ T cells is another common abnormal feature.24,25 Identification of CD3-/dim CD4+ aberrant T cells by flow cytometry can be a helpful feature in excluding morphologic mimics of TFH lymphoma such as reactive hyperplasia or Hodgkin lymphoma (Fig. 5).26,27 It is important to mention that CD10 is present on a small subset of normal TFH cells which make up a minor component of reactive germinal centers, so very small populations of CD4+CD10+ T cells without other phenotypic aberrancies should be interpreted with caution and with histologic correlation. Flow cytometry is not always able to detect an immunophenotypically abnormal T-cell population in AITL/TFH lymphoma; in one large study of 155 cases, an aberrant T-cell population was detected in 97 cases representing 0.5% to 90% of lymphocytes.24 Another caveat is that rarely cases of TFH lymphoma, AI-type or PTCL have occurred in association with EBV+ diffuse large B-cell proliferations interpreted as DLBCL, or with a monoclonal B-cell expansion that can take a number of forms and may be detected by flow cytometry. In such cases, the abnormal T-cell subset may be overlooked, especially if comprising a low percentage of overall cellularity, or if it cannot be detected at all and a monoclonal B-cell population is detected, the working diagnosis may stray farther from a PTCL.28,29 FIGURE 5: Flow cytometry dot plots of an example case of follicular helper T-cell lymphoma. Compared with normal T cells (red), the abnormal T cells (green) lack surface CD3, and are positive for CD4, CD2, CD5, and CD7. Although TRBC1 appears uniformly negative, it cannot be interpreted in the setting of surface CD3 loss. Analysis of the T-cell receptor beta chain constant region (TRBC) by flow cytometry can be a useful method for evaluation of T-cell clonality, since its 2 mutually exclusive isoforms (TRBC1 and TRBC2) can be interpreted in a manner analogous to kappa and lambda light chains. Several studies have demonstrated the utility of adding TRBC1 and/or TRBC2 to an 8-color or 10-color flow cytometry panel with other T-cell antigens to evaluate clonality in αβ T cells.30–32 A limitation to this approach is that γδ T cells lack TRBC1 and TRBC2, so gating analysis is limited to surface CD3 expressing, αβ T cells in order to avoid overinterpretation as a TRBC1-negative clonal T-cell population.33 In some specimens, single staining for TRBC1 can produce monotypic TRBC1-dim subsets that can be difficult to interpret within laboratory-defined thresholds (eg, <15% or >85% TRBC1-positive events).33 A small subset of dim monotypic TRBC1 expressing T cells can be incidentally detected in patients with reactive or comorbid medical conditions and should be interpreted with caution.34 Larger T-cell populations with dim monotypic TRBC1 have been shown by additional studies to represent TRBC2-positive T cells.31 The addition of TRBC2 to the panel can help to resolve an abnormal subset in a substantial proportion of cases. Some mature αβT-cell neoplasms that might appear TRBC1-negative on single staining (and therefore possibly TRBC2 restricted), may prove to be negative for both TRBC1 and TRBC2, suggesting that dual staining may provide more reliable information for determination of TRBC isoform expression.31 WHAT IS THE BEST STEPWISE APPROACH FOR DIAGNOSING NODAL T-CELL LYMPHOMA? To accommodate an initial differential diagnosis that is comprehensive of T-cell lymphoma, we refer the reader to an algorithm for the workup and classification of nodal PTCLs (Fig. 6, adapted from Campo et al1) which provides a general framework for routine use. It begins with consideration of HTLV1 status which would contribute strongly to classification as ATLL despite a wide diversity of morphologic and immunophenotypic features that nearly match any other subtype of PTCL without HTLV1 association (Fig. 7).35 FIGURE 6: Algorithm for the workup and classification of nodal PTCLs. Once a T-cell lymphoma is established, this hierarchical orientation to subclassification of specific entities is helpful to the pathologist. Modified with permission from Campo et al.1 Adaptations are themselves works protected by copyright. So in order to publish this adaptation, authorization must be obtained both from the owner of the copyright in the original work and from the owner of copyright in the translation or adaptation. FIGURE 7: Adult T-cell leukemia/lymphoma, lymphomatous form. A, The lymph node is diffusely involved by a CD3+ population. B, The infiltrate consists of large pleomorphic atypical cells with numerous mitoses. The cells are positive for FOXP3 (C) and CD25 (D). The approach to immunohistochemistry may consist of an initial limited panel of stains to assist in excluding B-cell lymphomas, Hodgkin lymphoma, and reactive processes from suspected T-cell lymphoma. This might include a few T-cell stains initially (CD3, CD5, perhaps CD4, and CD8 upfront if suspicion is high), CD10, CD20, CD21, CD30, CD45, PAX5 followed by additional T-cell antigens (CD2, CD4, CD7, CD8, TFH markers, cytotoxic markers, TCRγ, TCRβF1), specialized stains (ALK1, TCL1, TdT, CD56), additional FDC meshwork stains, and EBER in situ hybridization.36 CD3 is quite robust by immunohistochemistry and serves as an initial starting point with additional markers of T-cell subsets including CD4, CD8, and pan-T-cell markers CD2, CD5, and CD7 comprising a backbone for initial characterization. CD30 is a key marker for the diagnosis of T-cell lymphomas and can be used as an initial branch point for further subclassification.37,38 If the pattern of CD30 staining is strong and uniform and suggestive of ALCL, ALK protein expression is the next step to establish a diagnosis of ALK+ versus ALK- ALCL. ALK+ ALCL may be positive for CD4, CD43, CD8 (smaller subset of cases), and cytotoxic molecules, and is commonly negative for CD3, CD5, and TCR proteins (Fig. 8).38,39 If ALK is negative, the differential diagnosis remains broad but includes ALK- ALCL, along with ATLL, mycosis fungoides with large cell transformation involving a lymph node, other extranodal T-cell lymphomas involving a lymph node, and a subset of PTCL NOS with CD30 expression in >80% of lymphoid cells. Cases of ALK- ALCL, should show strong and uniform expression of CD30 and resemble the common pattern of ALK-positive ALCL. Variant patterns of ALK- ALCL are not recognized.2 FISH testing for DUSP22 rearrangement is recommended by the ICC and enables the identification of this subgroup, which is important biologically and for prognosis.1,40 ALK- ALCL with DUSP22 rearrangement (Fig. 9) are usually negative for cytotoxic markers and EMA, and have unique cytologic features including less pleomorphism with uniform hallmark cells and doughnut cells.40,41 Strong and uniform LEF1 expression had a high positive and negative predictive value for DUSP22 rearrangement in ALK-negative ALCL.42 The characteristic morphologic and immunophenotypic features that help identify possible DUSP22-rearranged cases further support this entity as a distinct clinicopathologic subset of ALCL. Like ALK+ ALCL, ALK-, ALCL typically has aberrant loss of pan–T-cell antigens (CD3, CD5, and CD7), while CD43 and CD2 are more likely to be positive.42 ALCL is characterized by sinusoidal growth initially until it may take over geographic regions of the lymph node (Fig. 8).44 Infiltrates of ALCL may be subtle and missed if high magnification is not used or misinterpreted as metastatic carcinoma if confirmatory immunohistochemistry is not applied. FIGURE 8: ALK-positive anaplastic large cell lymphoma. A, Low-power view of a lymph node showing diffuse involvement and intrasinusoidal dissemination. B, High-power view showing large anaplastic cells. C, The cells show strong homogeneous paranuclear dot-like and membranous CD30 expression. D, CD3 stains small reactive T cells and is negative in the large neoplastic cells. E, Nuclear and cytoplasmic positivity for ALK reflects a t(2:5) NPM1::ALK fusion. The neoplastic cells are positive for EMA (F) and perforin (G). FIGURE 9: DUSP22-rearranged ALK-negative anaplastic large cell lymphoma. A, The large neoplastic cells form cohesive sheets and comprise many hallmark cells. The lymphoma cells are strongly positive for CD30 (B), positive for MUM1 (C), negative for granzyme B (D), positive for CD8 (dim) (E), positive for CD2 (F), and show downregulated BCL2 expression (G). If ALCL is not an appropriate diagnosis, CD4 and CD8 immunohistochemistry provide another divergence to evaluate for a TFH lymphoma, since TFH lymphomas are derived from a functional subset of CD4+ T-helper cells with similarities in genetic landscape.45–49 The 5 markers recommended and most widely used to define this phenotype are PD1, CXCL13, ICOS, BCL6, and CD10,50 though other TFH markers include CXCR5, CD57, and SAP.51 A TFH immunophenotype is established when the lymphoma cells are positive for ideally 3 TFH markers, but a minimum of 2 TFH markers with convincing expression is acceptable in the right context (Fig. 10).1 In addition to assessing the immunophenotype, the distribution of TFH cells and intensity of staining should be considered, since TFH cells are normally present in lymph nodes in germinal centers and paracortex, and PD1 intensity may vary depending on the anatomic compartment. PD1 positive T cells may be increased in reactive conditions including viral lymphadenitis (Fig. 1).52 CD10 and CXCL13 may be positive in only a small subset of the neoplastic cells (Fig. 10E), and are generally less sensitive but more specific for a diagnosis of TFH lymphoma. PD1 and ICOS are much more sensitive, but less specific with overlap in reactive lymphadenopathies, and PD1 expression should be strong if it is used as a marker to rule in a TFH phenotype.29,53 Follicular dendritic cell meshwork stains are important to evaluate the lymph node architecture and lymphoma growth pattern; since AI-type and other types of TFH lymphoma have characteristic meshwork morphologies.54 It is important to recognize that TFH cells can be abundant in small B-cell neoplasms, specifically marginal zone lymphoma and follicular lymphoma, and represent a pitfall in the diagnosis of TFH lymphoma.55,56 FIGURE 10: Follicular helper T-cell lymphoma. In this follicular helper T-cell lymphoma of the angioimmunoblastic type (A), the neoplastic cells are positive for CD4 (B), PD1 (C), ICOS (D), CD10 (E), and CXCL13 (F). The strong expression of follicular helper T-cell markers goes along with clear cell morphology of the neoplastic cells (A) in an IDH2-mutated case. If the T-cell lymphoma is CD8+, CD4-/CD8-, or CD4+ without a TFH immunophenotype, cytotoxic markers TIA1, granzyme B, and/or perforin should be performed with a positivity threshold of >50% tumor cells. In normal cells, the TIA1 protein is constitutively expressed in cytotoxic cells whereas perforin and granzymes are inducible upon activation,57 as such, cohorts of cytotoxic PTCL demonstrate TIA expression at a slightly higher rate.58,59 If cytotoxic molecules are present, nodal involvement by an extranodal PTCL or NK/T-cell lymphoma should be excluded. EBV testing can differentiate between a cytotoxic PTCL, NOS and primary nodal EBV+ T/NK-cell lymphoma. If cytotoxic molecules are negative, the diagnosis defaults to PTCL, NOS (Fig. 11).1 If a diagnosis of PTCL, NOS is established, an immunohistochemical algorithm could be performed to approximate the cell-of-origin from Th1 or Th2 functional subsets.60 The algorithm was proposed after gene expression profiling studies of PTCL, NOS identified 2 molecular subgroups with distinct transcriptomes and different prognosis: PTCL-TBX21 (Th1) and PTCL-GATA3 (Th2). Compared with PTCL-GATA3, PTCL-TBX21 has a better prognosis, except for a subset of the PTCL-TBX21 subgroup with a cytotoxic profile. PTCL-TBX21 has less genomic complexity, a higher frequency of epigenetic modifying genes, and enrichment in interferon and NF-ĸB pathways, while PTCL-GATA3 generally has a poorer prognosis, greater genomic complexity, and upregulation of the MYC-induced and PI3K-induced pathways.61,62 The proposed immunohistochemical algorithm begins with the determination of TBX21 expression and is followed by assessment of CXCR3, GATA3, and CXCR4, with thresholds for positivity for TBX21 and CXCR3 at 20%, and for GATA3 and CCR4 at 50%.60 A follow-up study showed that the algorithm could correctly classify 87% of cases compared with the digital GEP classification, and identified differences in microenvironment, tumor cell size, and immunophenotype between PTCL-TBX21 and PTCL-GATA3.63 FIGURE 11: Peripheral T-cell lymphoma, NOS. The lymph node is diffusely involved by a proliferation of relatively monotonous medium-sized atypical lymphoid cells (A–B), which are positive for CD4 (C), CD2 (D), negative for CD7 (E), and aberrantly coexpress CD79a (F). G, Only few PAX5-positive cells are present. Stains for cytotoxic markers were negative (not shown). If the diagnosis of PTCL NOS is under consideration and the lymphoma cells appear to be monotonous, medium-sized, blast-like, or expressing dual positivity or negativity for CD4 and CD8, the possibility of a T-lymphoblastic lymphoma needs to be excluded. TdT, CD34, and CD1a stains can be helpful. The differential diagnosis with T-lymphoblastic lymphoma (T-LBL) in a lymph node includes indolent T-lymphoblastic proliferation, usually forming aggregates of monotonous lymphoid cells with an immature thymocyte phenotype that are polyclonal and must be differentiated from T-LBL as clinical behavior is effectively benign.64 HOW DOES EPSTEIN-BARR VIRUS TESTING IMPACT THE DIAGNOSIS OF T-CELL LYMPHOMA? Examination for EBER RNAs by in situ hybridization or EBV by immunohistochemistry in either the neoplastic cells or the surrounding microenvironment impacts the diagnosis of nodal T-cell lymphoma and is a necessary step in diagnosis. Primary EBV+ nodal T/ NK-cell lymphoma (Fig. 12) is an entity distinct from extranodal NK/T-cell lymphoma nasal type (ENNKTCL) that is listed as a provisional entity in the ICC classification. It is a primary nodal disease and must be distinguished from other T/NK EBV+ lymphoproliferative disorders that may infiltrate lymph nodes. Clinical criteria for diagnosis require exclusion of nasal involvement, while dissemination to extranodal sites such as bone marrow, liver, or other viscera is permitted. EBV testing is mandatory to demonstrate EBV positivity (usually latency pattern II) in virtually all the tumor cells. It can be distinguished from ENNKTCL by pathologic features including monomorphic large cell morphology, lack of angiocentric growth and necrosis, negativity for CD56 and positivity for CD8, and lineage corresponding to T cells more closely than NK cells. The tumor occurs in older adults from East Asia and has a dismal prognosis even when compared with ENNKTCL or PTCL, NOS.65–68 The differential diagnosis with ENKTCL comes up because cervical lymph node involvement by ENKTCL may be detected before identification of a nasal mass.69 Some cases of primary EBV-positive nodal T-cell lymphoma can have moderate to high expression of CD30, so EBER positivity in the majority of the tumor cells is a helpful diagnostic feature to exclude CD30+ PTCL, NOS, or ALK- ALCL.38 FIGURE 12: Primary nodal EBV+ T/NK-cell lymphoma. A, A lymph node biopsy in a 36-year-old man with chronic active EBV disease who developed lymphadenopathy shows an effaced architecture with infiltration of the hilar fat. The tissue infiltrate consists of large atypical lymphoid cells (B) which are positive for CD3 (C) and perforin (D). E, Most cells are positive for EBV by situ hybridization with EBER probes. A common feature of TFHL is the presence of EBV+ clonal B-cell and plasma cell proliferations including large B cells expressing CD30 and resembling Hodgkin/Reed-Sternberg (HRS) cells.70,71 The B-cell proliferation is polyclonal or clonal and heterogeneous, and TFHL especially AITL is described with a variable number of EBV+ B cells, often as isolated or small clusters of B blasts to focally confluent large B cells that can mask an underlying of TFH lymphoma (AITL) (Fig. 13).72 In some circumstances, TFH lymphoma of AI-type may have EBV-negative HRS-like cells or EBV-negative clonal plasma cell proliferations, so the absence of EBV does not argue against the phenomenon but is helpful when present.71,73 Some patients with TFH of AI-type can develop aggressive B-cell proliferations with latency patterns II or III, similar to those seen in immunosuppressed individuals.72 While previous publications have reported EBV-positive proliferations in PTCL, NOS, it is likely a lower occurrence than previously reported due to earlier definitions of PTCL, NOS that may have included cases of TFH lmphoma.68 FIGURE 13: EBV-positive large B-cell proliferation in association with follicular helper T-cell lymphoma, angioimmunoblastic type. An enlarged inguinal lymph node in a 74-year-old woman with generalized lymphadenopathy has an effaced architecture (A) and contains a pleomorphic proliferation of medium sized and large lymphoid cells with abundant vessels (B). C, CD21 stains irregularly enlarged follicular dendritic cell meshworks. ICOS highlights atypical medium-sized cells (D) and CD79a stains many large blastic cells (E). F, Many cell are positive for EBV by in situ hybridization with EBER probes. Double stains for EBER and CD20 (brown and red, G) and EBER and CD3 (dark blue and brown, H) show that the EBV-positive cells belong to the B-cell lineage. Polyclonal IG gene rearrangements were found in this case. IN WHICH DIAGNOSTIC SITUATIONS IS NEXT GENERATION SEQUENCING MOST CONTRIBUTORY AND ARE CLONALITY STUDIES RECOMMENDED FOR ALL CASES? T-cell receptor (TR) gene rearrangement studies have been useful as markers of lineage and clonality in T-cell neoplasms for many years. The BIOMED-2 primer PCR tubes were reported to detect clonal TR gene rearrangements in >90% of T-cell malignancies and had a reasonable rate of polyclonal testing in reactive lesions.74 Though this is not absolute, the complete IG/TR gene rearrangement pattern of a lymphoid malignancy might support lineage assignment. For example, TFH lymphoma AI-type might harbor both TR- and IG-rearrangements. Along those lines, detection of a solitary IG-rearrangement without a detectable T-cell clone does not completely exclude the possibility of a T-cell lymphoma. It should be noted that TR gene studies in reactive tissues may show clonal T-cell populations but these are usually small clones and are not synonymous with malignancy.75,76 Next generation sequencing-based technology for analysis of the TR gene has demonstrated high efficiency and reliability in detecting clonal rearrangements in T-cell lymphomas, but this is not (yet) in widespread use.77,78 Nonetheless, identification of a clonal TR gene rearrangement in the appropriate clinical and pathologic context can be very helpful in establishing a diagnosis of T-cell lymphoma, while a polyclonal pattern can support a reactive lymphoproliferation. An algorithmic study of T-cell lymphoma diagnosis in North America determined that after a tiered comprehensive immunohistochemistry approach, the addition of T-cell receptor PCR studies provided additional positive value to achieving a WHO 4th ed. diagnosis, yet gene rearrangement contributed to a change in only 8% of reviews.36 Targeted gene panels for high-throughput sequencing (HTS) have been adapted to formalin-fixed paraffin-embedded tissue, though classification of T-cell lymphoma is still primarily accomplished by morphologic assessment and immunophenotyping. Nonetheless, there are some diagnostic scenarios where clonality testing and HTS may be effective. For T-cell lymphoproliferations with HRS-like cells in a T-cell rich background, clonality analysis is useful because a monoclonal TR rearrangement supports a diagnosis of T-cell lymphoma and argues against CHL or B-cell lymphomas. Monoclonal IG-rearrangements may be variably demonstrated in CHL, nodular lymphocyte-predominant B-cell lymphoma, and T-cell/histiocyte rich large B-cell lymphoma as well as in TFHL with an associated B-cell component. Demonstration of mutations in genes commonly mutated in T-cell lymphomas (CARD11, CD28, DNMT3A, IDH2, PLCG1, RHOA, STAT3, and TET2) supports that diagnosis.79 Up to 80% of patients with TFHL have underlying clonal hematopoiesis, with shared TET2 and/or DNMT3A mutations in the early progenitor cells as well as the lymphoma, so identification of these types of mutations without more specific variants (eg, RHOA and IDH2) may not be as meaningful to a diagnosis.80–82 Another scenario in which HTS can be helpful is in the evaluation of T-cell expansions with a TFH immunophenotype. In this setting, demonstration of a monoclonal TR gene rearrangement or somatic mutation in relevant genes is useful in the distinction between reactive versus neoplastic expansions of TFH cells. During evaluation of a TFH expansion, the identification of mutations in genes related to B-cell lymphomas favoring marginal zone or follicular lymphoma would potentially be a cause for re-examination of the specimen and help in supporting a reactive TFH proliferation.79 RNA-based expression profiling has emerged as a new tool for determining PTCL molecular subtypes and determining biological signatures and microenvironment but routine use of this technology has not been adopted widely for diagnostic practice.45,61 WHEN IS A DIAGNOSIS OF FOLLICULAR HELPER T-CELL LYMPHOMA APPROPRIATE? It is helpful when knowledge of the clinical presentation including coexisting disorders and chronic infections and laboratory findings are available to the pathologist. A characteristic clinical syndrome of autoimmune manifestations, disseminated lymphadenopathy, skin rashes, effusions, and B symptoms may help to guide the diagnosis.83 Exclusion of reactive causes of TFH proliferation including drug reactions, vaccinations, viral infections, and other immune conditions is necessary, and exclusion of nodal involvement by cutaneous T-cell lymphoma is required by the ICC and WHO5 classification schemes.1,4 Serology testing for human lymphotropic virus 1 (HTLV1) should be performed in principle, because up to 30% of ATLLs express TFH markers to some extent.84–86 TFHL is comprised of 3 morphologic subtypes: AI-type, follicular-type, and not otherwise specified. Recognition of the histologic pattern of these types will assist in diagnosis. AITL-type is composed of a polymorphous infiltrate including variable proportions of neoplastic cells, with intermixed small lymphocytes, histiocytes, immunoblasts, eosinophils, and plasma cells. The abnormal lymphoid cells are usually small to medium in size with nuclear atypia and clear cytoplasm. There is usually a stromal component with arborizing high endothelial venules and typically disrupted and proliferated follicular dendritic cell (FDC) meshworks (Figs. 13C, 14). In some cases, histiocytes, large B-cell immunoblasts or HRS-like cells might be abundant.29,68,87 TFHL follicular type is least common and is comprised of pale aggregates of medium-sized abnormal T cells within expanded IgD+ mantle zones (progressive transformation of germinal center-like pattern) or resembles a follicular lymphoma-like pattern with nodular aggregates of neoplastic cells with an abnormal TFH T-cell phenotype within a meshwork of follicular dendritic cells. These cases lack the extrafollicular proliferation of FDC and increased vascular density of AITL-type.88,89 TFH lymphoma, NOS, fits the definition of TFHL by immunophenotype and may have some characteristics of AITL-type, but may present as what was formerly called the “T-zone variant” of PTCL, NOS with a TFH immunophenotype, or with perifollicular involvement, or with an entirely diffuse growth pattern without the FDC meshworks characteristic of AITL-type.68,90 FIGURE 14: Core needle biopsy with follicular helper T-cell lymphoma, angioimmunoblastic type. This core needle biopsy of relatively good size (A) shows prominent vessels and a polymorphous infiltrate including lymphoid cells with clear cytoplasm (B). C, CD23 highlights a diffuse and dense meshwork of follicular dendritic cells which is key in establishing the diagnosis. The atypical cells are positive for CD5 (D), with expression of CD10 (E) and BCL6 (F). WHAT CHARACTERISTICS ARE MOST USEFUL TO DIFFERENTIATE ANAPLASTIC LARGE CELL LYMPHOMA FROM PERIPHERAL T-CELL LYMPHOMA, NOT OTHERWISE SPECIFIED? It is necessary to carefully consider how to differentiate ALCL and PTCL, NOS. Patients with CD30+ PTCL, NOS appear to have an inferior prognosis than patients with ALCL, ALK-.91 Some cases of PTCL NOS can express CD30 strongly in a substantial proportion of cells. In one study of 141 PTCL NOS, more than 20% of cases showed CD30 positivity in >50% of the tumor cells.92 The small cell variant of ALCL, ALK+ (5% to 10% of cases of ALK+ ALCL) follows a different CD30 expression pattern than other variants of ALCL and can be confused with PTCL, NOS if immunohistochemistry for ALK is not performed. A minority of the neoplastic cells are large with hallmark morphology, and these tend to cluster in a perivascular location (Fig. 15). Some of the lymphoma cells are quite small and weak to negative for CD30, which could lead to a mistaken diagnosis of PTCL NOS. For this reason, ALK immunohistochemistry must be applied in any case of T-cell lymphoma with any degree of CD30 expression.38 FIGURE 15: Small-cell pattern of ALK+ anaplastic large cell lymphoma. A, The neoplastic infiltrate consists of small to medium-sized cells with clear cytoplasm and scattered larger cells. The atypical cells are positive for CD3 (B), CD30 stains strongly the perivascular cells and is weak to negative in the other cells (C); similarly, ALK is mostly expressed in the cells surrounding the vessels (D). A subgroup of CD30+ PTCLs with anaplastic morphology and Hodgkin-like features including a polymorphous inflammatory background and eosinophilia originally described in 2003 may cause confusion with classic Hodgkin lymphoma or other CD30 positive T-cell lymphomas.93 Similar cases were later found to harbor recurrent JAK2 rearrangements.94 In a different study, a cohort of CD30+CD15+ PTCL, NOS were compared with ALK- ALCL by RNA sequencing and showed no clear segregation between the 2 groups, and a subset of PTCL, NOS cases demonstrated DUSP22 rearrangements by FISH, suggesting that these cases may fit better within the spectrum of ALK- ALCL than PTCL, NOS.83,95 While TP63 rearrangements have been described with prognostic association in a subset of ALCL, ALK-,40TP63 rearrangements have been reported in both ALCL, ALK- and PTCL, NOS, so this finding does not necessarily favor ALK- ALCL if other features are equivalent.94 Some cases of PTCL, NOS with a cytotoxic phenotype have IRF4 rearrangements [resulting from t(6;14)(p25;q11.2)] that may be detected using a FISH probe for DUSP22/IRF4 locus. These are CD30-negative and should not be confused with DUSP22-R ALK- ALCL.97 CAN NODAL PERIPHERAL T-CELL LYMPHOMA BE ACCURATELY DIAGNOSED ON NEEDLE BIOPSIES? Core needle biopsy lymph node samples (CNB) are a viable option and are increasingly performed for reasons of cost, waiting time for the procedure, and less morbidity. Nonetheless, obtaining material for lymphoma diagnosis through a surgical excisional biopsy is considered ideal because collection of more tissue allows the assessment of architecture, performance of immunohistochemistry and molecular tests, and residual archival tissue that can be used later for research or clinical trial enrollment. A study performed by the French Lymphopath Network involving a large inventory of lymph node CNB and surgical excision samples from patients with suspected lymphoma, from 2010 to 2018, saw an increase in CNB from 25% to 40% during the study period. Although CNB provided a definitive diagnosis in most of the patients with suspected lymphoma, it showed an increased risk of nondefinitive diagnosis with CNB samples and suggested the need for expert review in cases of lymphoma suspicion in CNB specimens. The diagnostic performance of expert pathology review was higher in excisional biopsies compared with CNB.98,99 Another study performed by colleagues in Brazil identified a high rate of definitive diagnosis with CNB but an increased percentage of nondiagnostic biopsies in T-cell lymphomas (30%), followed by classic Hodgkin lymphoma (10.6%).100 The polymorphic background and sparse large HRS cells in HL, and the microenvironmental features in TFHL (which can occasionally contain HRS-like cells) can be difficult to evaluate on CNB without careful assessment of the lymph node architecture. The growth pattern of TFHL AI-type often requires an expanded meshwork of follicular dendritic cells (Fig. 14) and evaluation of the distribution of TFH cells within the context of paracortex and follicles.98 CNB may not always provide a faster diagnosis due to the risk of one or several procedural attempts before achieving adequate material for diagnosis. Adequate CNB sampling including a wide gauge needle and multiple cores may be sufficient for diagnostic testing in the majority of cases and might even allow for molecular tests including high-throughput sequencing. However, some patients with a final diagnosis of TFHL made on excision may have one or several needle core biopsies preceding the contributive excisional biopsy. TFHL on final excisional biopsy may be preceded by an erroneous diagnosis (HL, EBV+ B-cell lymphoproliferative disorder) or a nonconclusive diagnosis as seen in a small cohort of workshop cases.18 In CNB samples of suspected lymphoma that demonstrate diffuse growth with a monomorphic large cell morphology and abnormal T-cell immunophenotype, a diagnosis of a specific T-cell lymphoma entity may be possible. Other CNB samples with a heterogeneous background, limited visual assessment of lymph node architecture, poor conceptualization of follicular dendritic cell meshwork patterns, or samples containing large HRS-like cells are problematic and a diagnosis of atypical lymphoproliferation (more or less suspicious for malignancy) may be advisable with the recommendation to perform an excisional biopsy. CONCLUSION When encountering a patient with suspected nodal T-cell lymphoma, the stepwise approach derived from recent classification schemes provide a useful framework for diagnosis. 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18177	https://bmcr.brynmawr.edu/2024/2024.01.24/	Aristotle’s syllogism and the creation of modern logic: between tradition and innovation, 1820s-1930s – Bryn Mawr Classical Review Skip to content Bryn Mawr Classical Review ==========================BMCR ==== Bryn Mawr Classical ReviewAdvanced Search Archive For Reviewers Books Available for Review Guidelines for Reviewers Statement on Publication Ethics For Publishers About BMCR Editorial Board Statement on Publication Ethics Bryn Mawr Commentaries History of BMCR Privacy Policy BMCR 30th Anniversary Celebration For Editors FAQ Subscribe Search (BMCR ID BMCR 2024.01.24 Aristotle’s syllogism and the creation of modern logic: between tradition and innovation, 1820s-1930s Lukas M. Verburgt, Matteo Cosci, Aristotle's syllogism and the creation of modern logic: between tradition and innovation, 1820s-1930s. Bloomsbury studies in the Aristotelian tradition. London: Bloomsbury, 2022. Pp. 256. ISBN 9781350228849. Review by Davide Falessi, SNSF; University of Lucerne; École Pratique des Hautes Études-PSL. davide.falessi@unilu.ch; davide.falessi@etu.ephe.psl.eu [Authors and titles are listed at the end of the review] Aristotle’s Syllogism and the Creation of Modern Logic is dedicated to the memory of John Concoran and aims to do “for modern logic what Concoran did for the work of George Boole, namely to make sense of and do justice to the idea that Aristotelian syllogistic logic contributed to its creation” (1). Thus, tracing the history of syllogistic from Richard Whately’s syllogistic through the work of Mill, Venn, Bolzano, Brentano, Hilbert, Frege and many other important logicians, up to the 1930s, the authors want to show that modern logic “has reformed and expanded rather than abandoned the Aristotelian heritage” (4). In particular, the aims are: to single out the role of Aristotelian syllogistic logic in the creation of modern logic; to explain the “different attitudes toward syllogistic” at the basis of the different development of modern logic and syllogistic itself (1); to outline the debate on the “proper nature, scope and method of logic” so as to delineate the “cross-pollination” of different traditions (3). The structure is as follows: after lists of figures and tables used in the book and a list of contributors and the acknowledgements, there is a brief and enlightening introduction, followed by fourteen chapters and, at the end, by a useful index of names and concepts quoted in the several contributions. In the introduction (“History of Modern Logic in a New Key”) Verburgt and Cosci tackle the so-called “standard narrative on the history of modern logic” as the main target of their enquiry (1). Indeed, it is usually said that the development of modern logic, particularly in the period chosen by the authors, is due entirely to a refutation of Aristotelian logic. According to Verburgt and Cosci, this is clearly expressed by the “Quine-Putnam” explanation of the origin of modern logic. Quine claims that the Aristotelian logic is to modern logic what the ‘arithmetic of primitives tribes’ is to modern mathematics. In other words, ancient logic would be a “prescientific fragment” of the whole logical theory developed in the nineteenth century. Quine also maintains that there is a discrete change between ancient and modern logic, started by Frege’s Begriffschrift, and similarly Putnam thinks that there is a fixed and precise starting point of modern logic, which is not Frege but Boole’s Mathematical Analysis of Logic and Laws of Thought. However, as Verbugt and Consci point out, the so-called “mathematical turn in logic was not a development beginning in the second half of the nineteenth century, let alone an accomplishment of Frege and Boole alone” (3). The main issue is thus the description of the origin of modern logic, which implies in turn the description of the decline of syllogistic. The question is: given that there is a decline, is it a sudden disappearance or a slow reformulation? It is of course true that in the nineteenth century logic and mathematics had an amazing development, and there was also a revolution based on the rejection of the past. During the 1800s, the domain of the Aristotelian syllogistic was eclipsed. Nevertheless, it must also be said that “the collapse of the Aristotelian empire did not happen overnight” (2). As Rome was not made in a day, it is also true that Rome was not undone in a day. The same holds for the millenarian empire of Aristotelian logic and hence it is not surprising that between the 1820s and the 1930s “there was an explosion of attempts to rethink logic” (2), as in Hilbert’s case – something shown in William Ewald’s chapter, “Hilbert’s Use of the Syllogism”, with the attempt to provide an “internalization” or “nostrification” of the syllogism. So, the “standard narrative” offers discontinuous account of the development of modern logic. In other words, Quine and Putman’s “search for a discrete event (or even an exact year) obscured the fact that the mathematical turn itself was part of a broader process of the lingering demise of the syllogistic, that is, of the gradual downfall of what for over two millennia had been logic’s paradigm” (3). The authors of this book reject this discontinuous account, and trace a continuous line from Aristotle to Gentzen, showing: how modern theories are bound to Aristotelian logic or influenced by syllogistic reasoning. For example, Calvin Jongsma in his chapter “Richard Whately’s Revitalization of the Syllogistic Logic” states that “Whately’s logic became the leading traditional text of its time” precisely because it provides a “vigorous defence of syllogistic reasoning” (28). Or, to take another example, Sun-Joo Shin, in Chapter Five, entitled “Logic of Relations by De Morgan and Peirce: A Case Study for the Refinement of Syllogism”, states that a “disconnection” (108) between Aristotelian syllogism and De Morgan and Peirce’s logic cannot be truly sustained without compromising a correct understanding of the logic of relations. how syllogistic is influenced by modern theories. For example, the Vorlesungen über die Algebra der Logik by Ernst Schröder, presented by Peckhaus in Chapter Six, “Ernst Schröder’s Algebra of Logic and the ‘Logic of the Ancient’”, provided a fundamental tool for translating the rules of Aristotelian logic into algebraic language. This translation guarantees a reconsideration of syllogistic by which, for example, Schröder can select and rearrange fifteen valid forms of traditional syllogism. In this way, a “parallel development of modern logicians reshaping the syllogism and reflections on the syllogism shaping modern logic” (1) are perfectly described. As a collection of different essays by different authors, the work could have become disorganised, as each author has his or her own focus and approaches the chapter differently from the others. However, care was taken to ensure unity and continuity, and to highlight the precedents and legacy of each author. For example, Verbugt’s chapter “Mill and the British Tradition of Inductive Logic: The Role of Syllogism” has a paragraph entirely devoted to the relationship between Mill and Whately, and thus ties in with Calvin Jongsma’s chapter. Matteo Cosci also devotes several paragraphs of his chapter “Brentano and Hillebrand on Syllogism: Development and Reception of the ‘Idiogenetic’ Theory” to the reception of Brentano’s logic and to the influence of Leibniz on Brentano (Cosci shows that, although Brentano does not mention Leibniz in his work, he refers to him in his lectures). Moreover, several parallels between different authors help to give immediate clues to their “different attitudes towards syllogistic” while also giving the impression of a continuous history, instead of an abrupt change from ancient to modern logic. To name a few: the parallelism proposed by Verbugt between Mill and Venn on the petitio principii of syllogisms proper to the empiricist (and the sceptical) rejection of syllogistic logic; the parallelism between Hilbert and Quine on the analysis of syllogism, provided by William Ewald in his “Hilbert’s Use of the Syllogism”; the relationship between Frege’s and Aristotle’s logic proposed by Erich H. Reck in his “Frege’s Relation to Aristotle and the Emergence of Modern Logic”. Finally, the historical background set out in the general overview proves useful to help guide the reader smoothly through the gradual evolution of syllogistic. Therefore, this work has the unquestionable merit of presenting not only the value of Aristotelian logic in modern logic, but also of emphasising the multitude of interpretations of syllogistic provided by many, sometimes overlooked, authors – such Hugh McColl, presented by Jean-Marie C. Chevalier in “Hugh MacColl: Never Twist the Syllogism Again” – and even the role of syllogistic in the structure of modern logic studies, identifying how syllogistic influenced new theories (as in the case of José Ferreirós’s chapter “The Role of Syllogistic Logic in Early Set Theory” dedicated to an analysis of the cursus studiorum of logic in the syllabus of the German Gymnasium). Moreover, the book reveals the pivotal role of syllogism not only at the formal level, as in the case of the Bolzano’s “objective” logic of “sentences in themselves”, studied by Mark Siebel in “The Aristotelian Roots of Bolzano’s Logic”, or in the case of Boole’s syllogistic as presented by David E. Dunning in Chapter Four (“George Boole and the ‘Pure Analysis’ of the Syllogism”), but also at the epistemological level, as in case of Mill’s discussion of the epistemological status acquired by syllogistic reasoning, presented in Verburgt’s contribution. Finally, in-depth reflection is also successfully conducted on the important topic of how deductive reasoning was progressively distinguished from syllogism and on the role of syllogistic in the development of the notion of deducibility, for example in Paola Cantù’s chapter “Syllogism and Beyond in the Peano School”, and in the last chapter of the book, “The Fate of the Syllogism in the Göttingen School”, by Curtis Franks. This work fills a gap that has so far existed in the detailed history of modern syllogistic. It also offers suggestions for further research topics, such as the work on philosophical and formal logic by logicians such as E. E. Constance Jones, Sophie Bryant and Augusta Klein. As a final remark, it can be said that the main thesis of the book seems entirely agreeable and indisputable. One cannot be so naive as to assume that the course of history is discontinuous: even change presupposes that what is negated is considered in the very act of negating it for, trivially, to know whether a negation is true or false one must already know the truth value of the proposition that is negated. To use the words of our authors: “in order to establish a break with tradition, the pioneers of modern logic had to engage with that very tradition” (3). Surprisingly enough, the authors found themselves forced to repeat what is rather clear for everyone who has done a basic course in history. The fact that historical changes are not abrupt, as thunder at night, is an unquestionable commonplace and, needless to say, this holds not only for historical change in general but also for those particular forms of historical change that are part of larger historical change, e.g. the “mathematical turn” in logic. By this I do not mean to say that the authors have based their book on a triviality so that it appears to be useless. On the contrary, the fact that they felt they had to remind scholars of this elementary principle of all historical research speaks volumes about the state of studies in the history of logic, especially where (and perhaps this is another “standard narrative”) it is believed that the complete understanding of an author’s logical theory can be fully achieved even while ignoring the context and history surrounding it. Authors and Titles Introduction, Lukas M. Verburgt (NIAS/Leiden University, Netherlands) & Matteo Cosci Richard Whately’s Revitalization of Syllogistic Logic, Calvin Jongsma Mill and the British Tradition of Inductive Logic: The Role of Syllogism, Lukas M. Verburgt The Aristotelian Roots of Bolzano’s Logic, Mark Siebel George Boole and the “Pure Analysis” of the Syllogism, David E. Dunning Logic of Relations by De Morgan Peirce: A Case Study for the Refinement of Syllogism, Sun-Joo Shin Ernst Schröder’s Algebra of Logic and the “Logic of the Ancient”, Volker Peckhaus Brentano and Hillebrand on Syllogism: Development and Reception of the “Idiogenetic” Theory, Matteo Cosci Hugh MacColl: Never Twist the Syllogism Again, J.M.C. Chevalier Frege’s Relation to Aristotle and the Emergence of Modern Logic, Erich H. Reck Christine Ladd-Franklin’s Antilogism, Francine F. Abeles Syllogism and Beyond in the Peano School, Paola Cantù Hilbert’s Use of the Syllogism, William Ewald The Role of Syllogistic Logic in Early Set Theory, José Ferreirós The Fate of the Syllogism in the Göttingen School, 1910-1940, Curtis Franks Get BMCR sent to your inbox Subscribe to BMCR Contact Us bmcr@bmcreview.org Login
18178	https://fiar.me/node/279	¿Qué es un cártel? \| FIAR Inicio Elementos Motivos Madre Maestra Fútbol + IA Qué es el 'cártel editorial' Protección de datos Instrucción, PIB y deuda Social Petición de alta en FIAR Suscripción al boletín Slack Agenda Contacto Sin encriptar Encriptado ¿Qué es un cártel? ¿Qué es un cártel?¿Cómo funciona el mercado de los libros de texto?¿Qué ha sancionado y desarticulado la CNMC?¿Qué ha presentado FIAR en la Fiscalía Provincial de Valencia?Noticias sobre el cártel editorialFAQ ¿Qué es y qué implica un cártel? La CNMC define un cártel como “la actividad que consista en coordinar el comportamiento de una empresa en el mercado o influir en los parámetros de competencia a través de conductas tales como la fijación, directa o indirecta, de precios, de otras condiciones comerciales o de servicio, de cuotas de producción o de ventas, los intercambios de información sobre precios a aplicar o cantidades proyectadas; el reparto de mercados, incluidas las pujas fraudulentas, la restricción de las importaciones o las exportaciones o los boicots colectivos, todas ellas comprendidas en el concepto de cártel“. Por tanto las acciones que supondrían una actividad de cártel por parte de las empresas serían, entre otras: Fijación de precios: cualquier acuerdo relacionado con los precios. Acuerdos directos sobre el precio final de venta al consumidor, porcentajes de incrementos de precios, comisiones, suplementos, descuentos, plazos de pagos, acuerdo de respeto de listas de precios… Fijación de condiciones de mercado:coordinación del comportamiento de los competidores con relación a las otras empresas con el fin de maximizar beneficios. Fijación de cuotas de producción:se fija la cantidad de producto o de ofertas de servicios en relación a las cuotas de mercado o limitando la capacidad de producción mediante acuerdos de producción conjunta. Fijación de ventas:fijar la cantidad de producción y limitar las ventas favorece a las empresas para poner un precio más elevado. Intercambios de información sobre precios a aplicar e intercambios de información sobre cantidades proyectadas:cualquier tipo de intercambio de información con la competencia con el objetivo de eliminar la competencia del sector a través de la cooperación entre empresas. Reparto de mercados:cada operador del mercado se reparte una zona geográfica en la que no pueden intervenir los demás, convirtiéndose en un monopolio de ese determinado territorio. Pujas fraudulentas:los competidores se ponen de acuerdo para designar quién se llevará cada puja. Restricción de las importaciones o de las exportaciones:acuerdo o serie de acuerdos entre empresas dirigidos a la restricción de la libre entrada de mercancías en un país mediante imposición de aranceles o cuotas; y acuerdos o serie de acuerdos dirigidos al cobro de un precio determinado por las exportaciones y/o al reparto de los mercados de exportación o acuerdos de exportación en exclusiva. Boicots colectivos:acción por parte de varias empresas de un sector de negarse a comprar, vender o a practicar algún tipo de relación comercial con un tercero con el objetivo de conseguir ventajas relacionadas al parón de su actividad empresarial. Tal como expone laweb de la CNMCen el apartado deconductas prohibidas, existen algunos acuerdos que, si bien cumplen los requisitos del artículo 1 de la Ley 15/2007, no son sancionables por considerarse que conllevan efectos favorables para los consumidores, mejoras en la producción, la distribución o la comercialización, fomento del progreso técnico que contrarrestan sus efectos perjudiciales desde el punto de vista de la competencia. Un ejemplo de estos acuerdos son los intercambios de información entre competidores en el marco de la coordinación de algunos registros de morosos, que facilitan un mejor funcionamiento de las relaciones empresa-clientes en determinado sector. Nuevo concepto de cártel en España (modificación de la LDC por Real Decreto-ley 9/2017, de 26 de mayo) Uno de los conceptos básicos del Derecho de la Competencia es sin duda el de cártel, propio de las conductas más graves contra la competencia, merecedoras del mayor reproche antitrust. Mientras en España carecíamos de un delimitación legal, la doctrina ya se refería al mismo como "todo acuerdo o medida horizontal de origen concertado y limitativa de la competencia" (Costas Comesaña, J.,Los cárteles de crisis, Madrid, 1997, pg. 28). Posteriormente, la disposición adicional cuarta de la Ley española de Defensa de la Competencia incorpora un concepto legal en 2007: "todo acuerdo secreto entre dos o más competidores cuyo objeto sea la fijación de precios, de cuotas de producción o de venta, el reparto de mercados, incluidas las pujas fraudulentas, o la restricción de las importaciones o las exportaciones". Ciertamente esta delimitación presentaba importantes deficiencias, puestas de manifiesto por la doctrina científica. En el ámbito internacional el cártel ya era definido en 1998, año en el que la OCDE se refiere al mismo como aquel “acuerdo restrictivo de la competencia, práctica concertada o concierto realizado por competidores para fijar precios, realizar ofertas colusorias, establecer restricciones o cuotas de producción, o dividir los mercados mediante asignación de clientes, proveedores, territorios o líneas comerciales”. De forma análoga se pronunció ocho años más tarde la Comunicación de la Comisión sobre clemencia y en 2015 el considerando 2 del Reglamento (UE) 2015/1348 de la Comisión, de 3 de agosto de 2015, que modifica el Reglamento (CE) no 773/2004 relativo al desarrollo de los procedimientos de la Comisión con arreglo a los artículos 81 y 82 del Tratado CE a los efectos de la inclusión de aspectos generales relacionados con el programa de clemencia. Como puede observarse, las diferencias del concepto de cártel entre el Derecho español y el derivado de la Unión Europea son evidentes. La importancia de la unificación entre ambos es grande, sobre todo por los efectos derivados del mismo (recuérdese, por ejemplo, que la clemencia es sólo aplicable en supuestos de "cártel"). Ante esta realidad, el 27 de mayo de 2017 el BOE publicó el Real Decreto-ley 9/2017 de 26 de mayo, por el que se transponen directivas de la Unión Europea en los ámbitos financiero, mercantil y sanitario, y sobre el desplazamiento de trabajadores (aquí). Junto a la importante transposición de la llamada "Directiva de Daños", se modifican diversos apartados de la disposición adicional cuarta de la Ley 15/2007, de 3 de julio, de Defensa de la Competencia, incorporando al Derecho español un nuevo concepto de cártel: "Todo acuerdo o práctica concertada entre dos o más competidores cuyo objetivo consista en coordinar su comportamiento competitivo en el mercado o influir en los parámetros de la competencia mediante prácticas tales como, entre otras, la fijación o la coordinación de precios de compra o de venta u otras condiciones comerciales, incluso en relación con los derechos de la propiedad intelectual e industrial; la asignación de cuotas de producción o de venta; el reparto de mercados y clientes, incluidas las colusiones en licitaciones, las restricciones de las importaciones o exportaciones o las medidas contra otros competidores contrarias a la competencia". Entre las primeras reflexiones que pueden realizarse sobre este nuevo concepto de cártel podrían destacarse las siguientes: El nuevo concepto de cártel amplía su ámbito objetivo, al incluir nuevas conductas y considerar la enumeración como ejemplificativa. Antes de la reforma era evidente que el ámbito objetivo del concepto de cártel en España no coincidía con el europeo. En primer lugar, junto a las conductas colusorias claramente constitutivas del cártel (v. gr. fijación de precio o reparto de mercados), se observaban ciertas prácticas que solo eran consideradas como tales por algunas normas europeas o comunicación internas. Concretamente, en el ámbito europeo se han incluido las "medidas anticompetitivas contra los competidores" y la "fijación de condiciones comerciales distintas del precio". En el contexto interno la Comunicación CNC (2013) incluía estas últimas y los "boicots colectivos", además de los "intercambios de información sobre precios a aplicar o cantidades proyectadas". La nueva delimitación del cártel acerca al mismo a su delimitación europea al incorporarlas "prácticas concertadas". E n este sentido, mientras en Derecho interno definía antes el cártel simplemente como “acuerdo”, en el ámbito comunitario se extiende, además, a las “prácticas concertadas”. A pesar de ello, la Comunicación de 19 de junio de 2013, de la CNC, sobre el programa de clemencia, ya se alineaba con el ámbito europeo, al confirmar que el cártel “puede constituir un acuerdo en sentido estricto, una práctica concertada o, lo que es más habitual en la práctica tanto nacional como comunitaria, una forma de colusión compleja y compuesta”. Se elimina el carácter secreto del cártel. Creemos que esta supresión es acertada, al existir cárteles no secretos. No obstante, dicho carácter secreto seguirá siendo importante cuando se pretenda aplicar el programa de clemencia. En este sentido, debe recordarse que la justificación de la exigencia del carácter secreto en dicho ámbito reside, como derivaba también de la Comunicación europea sobre clemencia, de “la dificultad inherente en detectar y poner fin a conductas anticompetitivas muy graves que, por llevarse a cabo con ocultación y disimulo, resultan mucho más difíciles de perseguir e investigar en todo su alcance y magnitud sin la colaboración de empresas o personas físicas involucradas en dicha infracción y, por tanto, conocedoras de los pormenores del cártel y capaces de aportar pruebas de éste” (punto 10 in fine de la Comunicación CNC). ¿Cómo funciona el mercado de los libros de texto? ¿Qué ha sancionado y desarticulado la CNMC? ¿Qué ha presentado FIAR en la Fiscalía Provincial de Valencia? ¿Cómo funciona el mercado de los libros de texto? › Más información UE OCDE McKinsey DigitalES AMETIC ADIGITAL AEPIA Informes sobre IA Dónde estamos x SUSCRIBIRSE PARA RECIBIR EL BOLETÍN DE NOTICIAS Email validated! 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18179	https://www.researchgate.net/publication/285652554_Goodman_Gilman's_The_Pharmacological_Basis_of_Therapeutics	Published Time: 2005-01-01 Goodman & Gilman's The Pharmacological Basis of Therapeutics \| Request PDF Article Goodman & Gilman's The Pharmacological Basis of Therapeutics January 2005 Authors: A. Burke A. Burke This person is not on ResearchGate, or hasn't claimed this research yet. E. M. Smyth E. M. Smyth This person is not on ResearchGate, or hasn't claimed this research yet. Gerald Fitzgerald University College Cork Request full-text PDF To read the full-text of this research, you can request a copy directly from the authors. Request full-text Download citation Copy link Link copied Request full-textDownload citation Copy link Link copied To read the full-text of this research, you can request a copy directly from the authors. Citations (176) Discover the world's research 25+ million members 160+ million publication pages 2.3+ billion citations Join for free No full-text available To read the full-text of this research, you can request a copy directly from the authors. Request full-text PDF Citations (176) References (0) ... 18 T h e s a f e t y a n d t o l e r a b i l i t y o f Nimesulide has well been established in double-blind, multicenter studies. 17, A post-marketing study of Nimesulide o n a s a m p l e o f 4 0 1 p a t i e n t s h a d reported nausea (9.0%), vomiting (8.5%), heartburn (7.5%) as a major captured adverse event. 18 In our study, only 1% of patients reported nausea and dyspepsia complaints post-Nimesulide and 88% with no noted events. ... ... 23,24 The onset of analgesic and antipyretic action of oral Nimesulide is 15 to 60 minutes 25 reaching peak plasma concentration in 1.2 to 2.7 hours 20 while that for Paracetamol is within 30 to 60 min. 21 Nimesulide has longer duration of antipyretic action (10-12 h) 25 as compared to paracetamol (3-6 h). 21 Nimesulide, with its preferential activity on COX-2 along with a short half-life, contributes to a rapid onset of antipyretic action, a potent antiinflammatory, and analgesic activities. ... ... 21 Nimesulide has longer duration of antipyretic action (10-12 h) 25 as compared to paracetamol (3-6 h). 21 Nimesulide, with its preferential activity on COX-2 along with a short half-life, contributes to a rapid onset of antipyretic action, a potent antiinflammatory, and analgesic activities. It also reduces the levels of matrix metalloproteases and other biomarkers of joint destruction, which decreases the disease progression. ... Effectiveness of Nimesulide in Acute Fever Management in Adults: Retrospective Electronic Medical Records Database Study Outcome in Outpatient Department Article Full-text available Jul 2021 J Assoc Phys India S Arulrhaj Mangesh Tiwaskar Mudit Sabharwal K K Aggarwal Background: Various clinical trials have established anti-inflammatory and antipyretic properties of Nimesulide in a controlled setting, however, the fever management in real-world settings is quite different. Objective: To assess the effectiveness of Nimesulide in acute fever management in real-world clinical practice. Methodology: A retrospective, multicenter study was conducted on electronic medical records (EMR) of 302 patients visiting out-patient departments at three centers between Jan 2016 and Jan 2020 and were prescribed Nimesulide for acute fever. The effectiveness of Nimesulide was analyzed as a change in fever from baseline to follow-up visit within 14 days and tolerability as the number of side effects captured post-Nimesulide ingestion. Results: The provisional diagnosis at the baseline visit reported major complaints like fever, fever with abdominal pain, body-ache, cough and myalgia. The mean baseline body temperature was 103.2±1.5°F with a mean duration of 4.4±2.8 days significantly (p 0.0001) decreased to 99.7±1.8°F on the administration of Nimesulide. The liver and the renal profiles were found to be normal on records, and the side effects such as nausea and dyspepsia were reported only in 2% of patients. Conclusion: Nimesulide was found to be well-tolerated and effective as an antipyretic for acute fever management in adults during short-term use in real-world clinical practice. View Show abstract ... However, our results show that aspirin, at low concentration (10 -7 M), increases cell viability, diminishing necrosis and apoptosis. Burke et al. (2006) reported that human daily ingestion of aspirin, 80 to 350 mg, produces plasma salicylate levels of 2.4 to 9.7 µg/ml, corresponding to 1.3 × 10 -6 to 5.4 x 10 -5 M of aspirin in the culture medium . According to these results, the concentrations used in our experiments correspond to doses lower than 70 mg/day, considered as low doses of aspirin . ... ... However, our results show that aspirin, at low concentration (10 -7 M), increases cell viability, diminishing necrosis and apoptosis. Burke et al. (2006) reported that human daily ingestion of aspirin, 80 to 350 mg, produces plasma salicylate levels of 2.4 to 9.7 µg/ml, corresponding to 1.3 × 10 -6 to 5.4 x 10 -5 M of aspirin in the culture medium . According to these results, the concentrations used in our experiments correspond to doses lower than 70 mg/day, considered as low doses of aspirin . ... ... However, our results show that aspirin, at low concentration (10 -7 M), increases cell viability, diminishing necrosis and apoptosis. Burke et al. (2006) reported that human daily ingestion of aspirin, 80 to 350 mg, produces plasma salicylate levels of 2.4 to 9.7 µg/ml, corresponding to 1.3 × 10 -6 to 5.4 x 10 -5 M of aspirin in the culture medium . According to these results, the concentrations used in our experiments correspond to doses lower than 70 mg/day, considered as low doses of aspirin . ... Effects of aspirin on inflammation and oxidative stress induced by Aβ1-42 in astrocytes in primary culture Article Full-text available Mar 2021 FREE RADICAL BIO MED Maria Dolores Mauricio Constanza Aldasoro Adrian Jorda Soraya L. Valles Aspirin has been used as anti-inflammatory and anti-aggregate for decades but the precise mechanism(s) of action after the presence of the toxic peptide Aβ1-42 in cultured astrocytes remains poorly resolved. Here we use low-doses of aspirin (10-7 M) in astrocytes in primary culture in presence or absence of Aβ1-42 toxic peptide. We noted an increase of cell viability and proliferation with or without Aβ1-42 peptide presence in aspirin treated cells. In addition, a decrease in apoptosis, determined by Caspase 3 activity and the expression of Cyt c and Smac/Diablo, were detected. Also, aspirin diminished necrosis process (LDH levels), pro-inflammatory mediators (IL-β and TNF-α) and NF-ᴋB protein expression, increasing anti-inflammatory PPAR-γ protein expression, preventing Aβ1-42 toxic effects. Aspirin inhibited COX-2 and iNOS without changes in COX-1 expression, increasing anti-oxidant protein (Cu/Zn-SOD and Mn-SOD) expression in presence or absence of Aβ1-42. Taken together, our results show that aspirin, at low doses increases cell viability by decreasing inflammation and oxidative stress, preventing the deleterious effects of the Aβ1-42 peptide on astrocytes in primary culture. The use of low doses of aspirin may be more suitable for Alzheimer's disease. View Show abstract ... However, our results show that aspirin, at low concentration (10 -7 M), increases cell viability, diminishing necrosis and apoptosis. Burke et al. (2006) reported that human daily ingestion of aspirin, 80 to 350 mg, produces plasma salicylate levels of 2.4 to 9.7 µg/ml, corresponding to 1.3 × 10 -6 to 5.4 x 10 -5 M of aspirin in the culture medium . According to these results, the concentrations used in our experiments correspond to doses lower than 70 mg/day, considered as low doses of aspirin . ... ... However, our results show that aspirin, at low concentration (10 -7 M), increases cell viability, diminishing necrosis and apoptosis. Burke et al. (2006) reported that human daily ingestion of aspirin, 80 to 350 mg, produces plasma salicylate levels of 2.4 to 9.7 µg/ml, corresponding to 1.3 × 10 -6 to 5.4 x 10 -5 M of aspirin in the culture medium . According to these results, the concentrations used in our experiments correspond to doses lower than 70 mg/day, considered as low doses of aspirin . ... ... However, our results show that aspirin, at low concentration (10 -7 M), increases cell viability, diminishing necrosis and apoptosis. Burke et al. (2006) reported that human daily ingestion of aspirin, 80 to 350 mg, produces plasma salicylate levels of 2.4 to 9.7 µg/ml, corresponding to 1.3 × 10 -6 to 5.4 x 10 -5 M of aspirin in the culture medium . According to these results, the concentrations used in our experiments correspond to doses lower than 70 mg/day, considered as low doses of aspirin . ... Action of low doses of Aspirin in Inflammation and Oxidative Stress induced by aβ 1-42 on Astrocytes in primary culture Article Full-text available Mar 2020 Int J Med Sci Adrian Jorda Martin Aldasoro Constanza Aldasoro Soraya L. Valles Aspirin has been used as anti-inflammatory and anti-aggregate for decades but the precise mechanism(s) of action after the presence of the toxic peptide Aβ1-42 in cultured astrocytes remains poorly resolved. Here we use low-doses of aspirin (10-7 M) in astrocytes in primary culture in presence or absence of Aβ1-42 toxic peptide. We noted an increase of cell viability and proliferation with or without Aβ1-42 peptide presence in aspirin treated cells. In addition, a decrease in apoptosis, determined by Caspase 3 activity and the expression of Cyt c and Smac/Diablo, were detected. Also, aspirin diminished necrosis process (LDH levels), pro-inflammatory mediators (IL-β and TNF-α) and NF-ᴋB protein expression, increasing anti-inflammatory PPAR-γ protein expression, preventing Aβ1-42 toxic effects. Aspirin inhibited COX-2 and iNOS without changes in COX-1 expression, increasing anti-oxidant protein (Cu/Zn-SOD and Mn-SOD) expression in presence or absence of Aβ1-42. Taken together, our results show that aspirin, at low doses increases cell viability by decreasing inflammation and oxidative stress, preventing the deleterious effects of the Aβ1-42 peptide on astrocytes in primary culture. The use of low doses of aspirin may be more suitable for Alzheimer's disease. View Show abstract ... The reverse transcriptase-polymerase chain reactions (RT-PCR) from swabs for SARS-CoV-2 were noted and analyzed. Respiratory samples were collected every 1 to 2 days until 2 sequential negative results were obtained. The viral duration defined as the interval from the first day of positive nucleic acid tests to the first day of continuous negative tests. ... ... Aspirin has an ability to directly inhibit the prostaglandin synthesis by irreversible inactivation of cyclooxy-genase 1 and 2 (COX-1/-2), whereas other NSAIDs are competitive inhibitors. Several studies suggest that aspirin may reduce the risk of deadly infections. But latest studies argued that the NSAIDs have powerful effects on the immune system and may have association with the poor prognosis after infection with COVID-19. ... Effect of low-dose aspirin on mortality and viral duration of the hospitalized adults with COVID-19 Article Full-text available Feb 2021 MEDICINE Qiang Liu Na Huang Anni Li Xiaolin Zhou To clarify the effect of aspirin on mortality and viral duration in adults infected with respiratory syndrome coronavirus 2 (SARS-Cov-2). After propensity score-matched (PSM) case-control analyses 24 pairs of patients were enrolled and followed up for 2 months. Both 30-day and 60-day mortality in the aspirin group were significantly lower than that in the non-aspirin group (P = .021 and P = .030, respectively). The viral duration time between the 2 groups was not significantly different (P = .942). Among adults (with hypertension, cardiovascular diseases) infected with SARS-Cov-2, low-dose aspirin medication (100 mg/day) was associated with lower risk of mortality compared with non-aspirin users. View Show abstract ... Inflammation is a response by living tissue to any kind of injury and is characterized by pain, redness, heat and swelling. 6 Vasoactive chemicals increase the permeability of the arterioles which then allows blood cells, chemical substances, blood proteins and fluids to accumulate in that region, and that fluid accumulation causes swelling. 6 The study investigated the HRBC membrane hemolysis and membrane stabilization, antifungal and antibacterial potentials of various extracts of the leaves of Justicia gendarussa. ... ... 6 Vasoactive chemicals increase the permeability of the arterioles which then allows blood cells, chemical substances, blood proteins and fluids to accumulate in that region, and that fluid accumulation causes swelling. 6 The study investigated the HRBC membrane hemolysis and membrane stabilization, antifungal and antibacterial potentials of various extracts of the leaves of Justicia gendarussa. ... Tropical Journal of Natural Product Research Original Research Article Antimicrobial, Antifungal and HRBC Membrane Hemolysis and Membrane Stabilization Properties of Various Extracts of Justicia gendarussa Article Full-text available Oct 2020 M. Amin Mir Muhammad Waqar Ashraf Himani Himani Bilal Ahmad Mir Diseases of microbial origin are of important and utmost public health concern. This heightened concern is due to the increasing use of antimicrobial drugs and increasing resistance to antimicrobial agents. Presently, scientists have shown increased interest in the use of plant products against microbial diseases. Inflammation due to injury or age-related natural inflammation is due to denaturation of proteins. The plant Justicia gendarussa is used as antimicrobial and anti-inflammatory medicine. Consequently, this study evaluated Justicia gendarussa leave extracts as antibacterial, antifungal and anti-inflammatory agents. The study evaluated the growth inhibition of many bacterial and fungal strains as well as the membrane stabilizing effect by various extracts of this plant. The extracts inhibited the growth of Escherichia coli, Micrococcus luteus, Bacillus pumilus, B. cereus, B. lecheniformis, Salmonella typhi, Streptococcus mutans bacterial strains, with the ethanol extract showing the highest inhibition as compared to water extract, whereas the petroleum ether extract inhibited the growth of only Micrococcus luteus, Bacillus pumilus, B. lecheniformis. and J. gendarussa. Petroleum ether, ethanol and water extracts also inhibited the growth of many fungi species, viz, Aspergillus niger, Fusarium, Nigrospora oryza, and Aspergillus flavus. Among the extracts, the ethanol extracts showed the greatest inhibition against the growth of Aspergillus niger and Nigrospora oryza. All the extracts of Justicia gendarussa showed marked membrane stabilizing effect, among which the ethanol extract showed more membrane stabilization followed by water and petroleum ether extracts. View Show abstract ... Paracetamol (PAR) is chemically N-(4-hydroxyphenyl)acetamide . It is used as an analgesic and antipyretic agent . Orphenadrine citrate (Or.cit) is chemically (RS)-N,N-Dimethyl-2-[(2methylphenyl)phenylmethoxy]ethanamine dihydrogen 2-hydroxypropane-1,2,3-tricarboxylate . ... ... The identical parts of spectra provide an excellent range for PAR determination in its high concentrations in a lot of wavelengths neglecting the nonlinear part from 200-215 and 228-265 nm. Zero-order absorption spectra of 20 µg/mL PAR (1), 15 µg/mL Or.cit(2), and 5 µg/mL CAF (3). ... Simultaneous Determination of Paracetamol, Orphenadrine Citrate, and Caffeine Ternary Mixture by Different Spectrophotometric Methods Article Sep 2019 Shereen Ahmed Aya Soudi Eman Elzanfaly Hala E Zaazaa The resolving power of spectrophotometric assisted mathematical techniques was demonstrated for the simultaneous determination of the challenging ternary mixture of paracetamol (PAR), orphenadrine citrate (Or.cit), and caffeine (CAF), where PAR, which has the highest absorptivity, is formulated in a very high concentration compared with Or.cit and CAF. Simultaneous area ratio subtraction (SARS), derivative spectrophotometry (D), constant multiplication (CM), and dual wavelength (DW) resolution techniques were able to resolve the three drugs. In SARS, PAR concentration was determined directly at 304 nm in the division spectra using PAR normalized spectrum as a divisor. After subtraction of the constant and multiplying by PAR normalized spectrum; CAF and Or.cit can be determined by first and fourth derivative methods, respectively, from the division spectra. Also, Or.cit and CAF can be obtained by the DW method and CM method, respectively, upon the division of the obtained ratio spectra by CAF spectrum. The proposed methods were validated with good accuracy and precision over the concentration ranges of 20–90, 3–40, and 1–20 μg/mL for PAR, Or.cit, and CAF, respectively. Specificity of the proposed methods was assessed by analyzing laboratory prepared mixtures containing different ratios of the cited drugs. The proposed methods were applied successfully for the determination of PAR, Or.cit, and CAF in their dosage form without any interference. View Show abstract ... Although it is a defence mechanism taking part in the inflammatory reaction, it can induce, maintain or aggravate many diseases. 1 Cassia fistula linn belonging to family Caesalpiniaceae has been used for years, traditionally, by tribals and locals in India for the treatment of various inflammatory conditions like the diseases of the heart, leprosy, inflammation, as antipyretic, in rheumatism, in kapha, skin diseases, liver complaints, diseases of the eye, throat trouble and chest complaints. 2 Preliminary phytochemical screening of the methanolic extract of the leaves as carried out andthe presence of flavonoids, glycosides, tannins and phenolics were detected. ... ... Flavonoids are found to be a group of polyphenolic compounds responsible for the various biological effects as anti-inflammatory, antihepatotoxic and antiulcer. 1,2 Lipoxygenases forms a family consisting of non-heme iron-containing enzymes. These enzymes catalyzes the polyenic fatty acids like the arachidonic acid to its corresponding lipid hydroperoxideproducts. ... Evaluation of anti-lipoxygenase activity of Cassia fistula linn leaves using in vitro methods Article Full-text available Aug 2018 Anju Gopi Manju Jisho Background: Numerous plants are claimed to possess anti-inflammatory phytoconstituents in folk medicine, however, one among them is Cassia fistula linn leaves. The tree is 6-9 m high with straight trunk and smooth bark. It is pale green when young and gets rough and dark when old. The leaves are 23-30cm long and have got 4-8 pairs of oblong leaflets. Due to lack of specific scientific reports regarding its use for its anti-lipoxygenase property, this particular plant was selected for this particular study with the aim to bring scientific evidence for its therapeutic use.Methods: The anti-lipoxygenase study as carried by using 5-lipoxygenase(5-LOX) assay and 12-lipoxygenase(12-LOX) assay. In both the methods, absorbance of various concentrations of the tests and the control solutions were measured at 234nm.Results: Preliminary phytochemical study showed the presence of flavonoids, glycosides, tannins and phenolics. It was found that both the 5-LOX and 12-LOX were inhibited by the extract with a 50% inhibitory concentration (IC50) of 6.23mg/ml obtained for the 5-LOX assay and an IC50 of 3.22mg/ml attained for the 12-LOX assay.Conclusions: The methanolic extract of the plant’s leaves showed anti-lipoxygenase activity similar to Indomethacin, thus ensuring that it could be used as an effective anti-inflammatory medicine. View Show abstract ... Since NSAIDs, or steroidal anti-inflammatory drugs, have been around for a while, prolonged use of them has resulted in negative side effects and damage to humans. [3,4] The initiation of inflammation is a protective reaction to harmful external substances, like pathogens, viruses, dust particles, irritants, and damaged cells. This process involves several stages, beginning with an initiation phase, followed by a peak of inflammation, and concluding with a resolution phase. ... Plant-Based Anti-Inflammatory Agents: A Scientific Review of Bioactive Compounds and Mechanisms Article Apr 2025 Kavitha Samiappan Jyothsna Chalakoth View ... Diclofenac is the most well-known NSAID in the world and a member of the Non-Steroidal Anti-Inflammatory Class of Drugs (NSAIDS) . Diclofenac is frequently used to treat mild pain and inflammation, which are typical signs of many illnesses. ... Comparative Biochemical and Pharmacological Assessment of Expired and Unexpired Diclofenac on Egg Albumin-Induced Inflammation in Wistar Rats Article Full-text available Jan 2025 Adebayo Br¹ Oladosu Ma¹ Moses Adondua Abah Ezeja Ch A non-steroidal anti-inflammatory medicine (NSAID) called diclofenac is recommended for treatment in rheumatoid arthritis and several non-rheumatoid illnesses that cause pain and inflammation. The relative effectiveness of drugs after their stated expiration dates is up for debate. Furthermore, studies reveal that drugs lose their effectiveness after expiration because the active ingredients may become less potent or more unstable. The purpose of this study was to examine how the medicine diclofenac, both expired and unexpired, affected the inflammation caused by egg albumin in Wistar rats. For the purpose of the efficacy trial, twenty-four wistar rats weighing between 120 and 150 grams were split up into four groups of six rats each. Group C was given outdated diclofenac medication, Group D was given expired diclofenac medication, Group B was given inflammation without treatment, and Group A (control group) was given 0.5 mL/kg distilled water. Rats' paw edema caused by fresh egg albumin was used to measure anti-inflammatory responses. A vernier caliper was positioned at the edge of the phalanges and metatarsals to measure and record the animals' paw thickness. The test medications and controls were administered orally to the rats. Before being administered, the uncoated tablets were broken up and dissolved. A vernier caliper was used to measure and record the anti-inflammatory activity one hour after treatment began. This process was repeated one, two, three, four, and five hours later. Ocular punctures were used to obtain blood, which was then centrifuged to extract serum. Assessments of haematological parameters, biochemical parameters, oxidative stress biomarkers, and histology were conducted. According 2 Current Trends in Pharmacology and Clinical Trials to the findings, the control and unexpired diclofenac groups had considerably higher levels of oxidative stress indicators like SOD, CAT, and GSH than the other groups. Compared to the control and unexpired diclofenac groups, the induced untreated and expired groups had significantly higher MDA levels. HDL concentration was significantly higher in the control and unexpired drug group than in the induced untreated group, but liver function parameters (ALT, AST, ALP) and lipid profile parameters (TAG, CHOL, LDL) were significantly higher in the induced untreated group than in the control, unexpired, and expired groups. All of the groups had different levels of urea, uric acid, and total bilirubin. The control and unexpired groups had considerably higher hematological results (WBC, RBC, neutrophils, hemoglobin, and monocytes) than the diclofenac-expired and induced untreated groups. When compared to the control and group that received unexpired diclofenac, the histopathology results also examined liver damage in both the induced untreated group and the groups that received expired medication. According to the study's findings, the medication diclofenac that has not expired has the strength to perform anti-inflammatory, analgesic, and antipyretic effects. Additionally, there are still some active ingredients in outdated diclofenac that have the same function. View Show abstract ... Diclofenac is the most well-known NSAID in the world and a member of the Non-Steroidal Anti-Inflammatory Class of Drugs (NSAIDS) . Diclofenac is frequently used to treat mild pain and inflammation, which are typical signs of many illnesses. ... Medicinal Herbs and Phytoconstituents Proved for Anticancer Activity-A Comprehensive Review Research Jan 2025 Anthony Olalekan Akande A non-steroidal anti-inflammatory medicine (NSAID) called diclofenac is recommended for treatment in rheumatoid arthritis and several non-rheumatoid illnesses that cause pain and inflammation. The relative effectiveness of drugs after their stated expiration dates is up for debate. Furthermore, studies reveal that drugs lose their effectiveness after expiration because the active ingredients may become less potent or more unstable. The purpose of this study was to examine how the medicine diclofenac, both expired and unexpired, affected the inflammation caused by egg albumin in Wistar rats. For the purpose of the efficacy trial, twenty-four wistar rats weighing between 120 and 150 grams were split up into four groups of six rats each. Group C was given outdated diclofenac medication, Group D was given expired diclofenac medication, Group B was given inflammation without treatment, and Group A (control group) was given 0.5 mL/kg distilled water. Rats' paw edema caused by fresh egg albumin was used to measure anti-inflammatory responses. A vernier caliper was positioned at the edge of the phalanges and metatarsals to measure and record the animals' paw thickness. The test medications and controls were administered orally to the rats. Before being administered, the uncoated tablets were broken up and dissolved. A vernier caliper was used to measure and record the anti-inflammatory activity one hour after treatment began. This process was repeated one, two, three, four, and five hours later. Ocular punctures were used to obtain blood, which was then centrifuged to extract serum. Assessments of haematological parameters, biochemical parameters, oxidative stress biomarkers, and histology were conducted. According 2 Current Trends in Pharmacology and Clinical Trials to the findings, the control and unexpired diclofenac groups had considerably higher levels of oxidative stress indicators like SOD, CAT, and GSH than the other groups. Compared to the control and unexpired diclofenac groups, the induced untreated and expired groups had significantly higher MDA levels. HDL concentration was significantly higher in the control and unexpired drug group than in the induced untreated group, but liver function parameters (ALT, AST, ALP) and lipid profile parameters (TAG, CHOL, LDL) were significantly higher in the induced untreated group than in the control, unexpired, and expired groups. All of the groups had different levels of urea, uric acid, and total bilirubin. The control and unexpired groups had considerably higher hematological results (WBC, RBC, neutrophils, hemoglobin, and monocytes) than the diclofenac-expired and induced untreated groups. When compared to the control and group that received unexpired diclofenac, the histopathology results also examined liver damage in both the induced untreated group and the groups that received expired medication. According to the study's findings, the medication diclofenac that has not expired has the strength to perform anti-inflammatory, analgesic, and antipyretic effects. Additionally, there are still some active ingredients in outdated diclofenac that have the same function. View Show abstract ... This gives redness due to the raised blood circulation toward the area. Inflammation can be either acute or chronic inflammation. Acute inflammation may be an initial response of the body to harmful stimuli. ... Anti-inflammatory activity of zinc oxide nanoparticles prepared using amla fruits Article Full-text available May 2019 Rajeshkumar Shanmugam Rangeela Manoharan Anitha Roy Lakshmi Thangavelu Aim: Anti-inflammatory activity of zinc oxide (ZnO) nanoparticles prepared using amla fruit. Objective: Preparation of ZnO nanoparticles using amla fruit extract and its anti inflammatory activity. Materials and Methods: Collection and preparation of amla fruit extract, synthesis of ZnO nanoparticles using amla fruit extract, collection of NPs using centrifugation, antiinflammatory activity of ZnO nanoparticle using ultraviolet-visible spectroscopy, and inhibition of albumin denaturation assay were used. Results: Aspirin, diclofenac, and ibuprofen are the most commonly used drugs for the inflammation which belongs to nonsteroidal anti-inflammatory group of drug. These drugs result in adverse side effects and damage the human biological system such as liver and gastrointestinal tract and may also cardiovascular system. According to the research, ZnO particles produced from amla fruit showed the anti-inflammatory activity and thus can be considered as a potential candidate as an anti-inflammatory agent, thus reducing the major health problems. Conclusion: The ZnO nanoparticles were synthesized from the amla fruit. All these data with results show that ZnO nanoparticle produced from amla fruit potent antiinflammatory property. The present study shows that ZnO nanoparticle can be used for various medicinal purposes. Further investigations are required for the development of new classes of analgesics and anti-inflammatory drugs from amla fruit. View Show abstract ... inhibited paw edema as compared to negative control and 100mg/kg extract. The observed edema inhibition was pronounced in the later stages of inflammation, and it was comparable to the effects of non-steroidal anti-inflammatory medications like indomethacin . The finding of this study were in line with that of the methanol extract of C. petasites that had shown significant dose-dependent edema reduction in the carrageenan induced animal model . ... Evaluation of wound healing and anti-inflammatory activity of hydro-alcoholic extract and solvent fractions of the leaves of Clerodendrum myricoides (Lamiaceae) in mice Article Full-text available Jul 2024 PLOS ONE Alemante Tafese Beyna Assefa Kebad Mengesha Ermias Teklehaimanot Yefter Wubayehu Kahaliw Background Wounds significantly affect people’s quality of life and the clinical and financial burden of healthcare systems around the world. Many of the current drugs used to treat wounds have problems such as; allergies and drug resistance. Hence, the exploration of new therapeutic agents from natural origin may avert this problem. Clerodendrum myricoides have long been used to treat wounds in Ethiopia. Despite this, nothing has so far been reported about the wound healing and anti-inflammatory activity of C. myricoides. This study aimed to evaluate the wound healing and anti-inflammatory activity of 80% methanol extract and solvent fractions of C. myricoides leaves in mice. Methods Leaves of C. myricoides were extracted using the maceration technique. The extract was formulated as 5% and 10% w/w ointments. The wound healing activity of the extract was evaluated using excision, incision, and burn wound models whereas the healing activities of solvent fractions were evaluated using the excision wound model. A carrageenan-induced paw edema model was used for the anti-inflammatory test. Results In the dermal toxicity test, 2000 mg/kg of 10% extract was found to be safe. In excision and burn wound models, treatment with 10% and 5% extract showed a significant (p<0.001) wound contraction. Solvent fractions of the extract significantly reduced wound contraction. A significant reduction in periods of epithelialization and favorable histopathology changes were shown by extract ointments. In incision wounds, 10% (p<0.001) and 5% (p<0.01) extracts significantly increase skin-breaking strength. After one hour of treatment, 400 mg/kg (p<0.001) and 200 mg/kg (p<0.05) showed significant reduction in paw edema. Conclusion Results of this study indicate that 80% methanol extract and the solvent fraction of the leaves of C. myricoides possess wound-healing and anti-inflammatory activity and support traditional claims. View Show abstract ... This finding is in agreement with the research study of Yuen & Lai, who concluded that protection against liver injury was as a result of the administered extracts of Ganoderma lucidum in the wistar rats. Furthermore, Burke et al, submitted that this noble mushroom successfully reversed the elevated bilirubin in acetaminophen-induced liver injury in wistar rats. Unconjugated bilirubin which is a waste product of haemoglobin breakdown after 120 days is taken up by the kupffer cells of the liver, where it is converted by the enzyme uridine diphosphosphoglucuronate glucuronosyltransferase (UDT) into soluble conjugated bilirubin. ... Effects of Garnoderma lucidum on Acetaminophen-Induced Liver Injury in Wistar Rats Article Full-text available Mar 2024 Osahon S. Usiobeigbe Airhomwanbor Kingsley Lucky Eromosele Omolumen Daniella Damilola Ogunsina Introduction: Ganoderma lucidum is considered to be a medicinal mushroom, widely used to prevent or treat different types of diseases including cancer, cardiovascular disease and hepatic dysfunction. This study aimed to evaluate the effect of Ganoderma lucidum on acetaminophen-induced liver injury in wistar rats. Methods: Forty (40) male wistar rats were used for this study. Hepatoxicity was induced by oral administration of acetaminophenn (3000 mg/kg of body weight) for the last 21 consecutive days of the dietary of regimen Ganoderma lucidum. These rats were divided into eight cages each containing Five rats. Control Group 1 fed on feed and water only throughout the study, Group 2 received acetaminophen only, Group 3 received Acetaminophen + Standard drug (silymarin), Group 4 received Acetaminophen +100 mg/kg body weight of Ganoderma lucium extract, Group 5 received Acetaminophen + 200 mg/kg body weight of Ganoderma lucidum extract, Group 6 received Acetaminophen + 300 mg/kg body weight of Ganoderma lucidum extract , Group 7 received 100 mg/kg of Ganoderma lucidum extract, Group 8 received Acetaminophen+Standard Drug (silymarin) + 300 mg/kg body weight of Ganderma lucidum extract. Blood samples was collected via cardiac puncture within 24 hours of Sacrifice. The extent of the liver injury was determined by assessing the plasma levels of Tumor Necrosis Factor Alpha (TNF-α), Alpha Feto protein, Alanine aminotransferase (ALT), Aspartate aminotransferase (AST), Alkaline phosphatase (ALP), Total bilirubin (TB), Conjugated bilirubin (CB), Unconjugated Bilirubin (UB), Albumin, Gamma Glutamyl Transaminase (GGT) and total protein (TP) using spectrophotometric method and ELISA as appropriate. Results: Oral administration of Acetaminophen significantly increased the plasma levels of the parameters accessed, suggesting severe liver damage in the rats. However, the treatment of Ganoderma lucidum decreased these hepatotoxic indices at a significant level of P <0.01 for TNF-α, AFP, ALT, AST, UB, TB, GGT and ALP, while Albumin and Conjugated Bilirubin were significantly decreased at a level of (P<0.05) in the Ganoderma lucidum + Acetaminophen-administered group compared to those of the control group. Conclusion: Thus, the results of the present investigation demonstrates that the Ganoderma lucidum provides significant hepatoprotective activity against acetaminophen-induced liver injury in wistar rats. View Show abstract ... Prostaglandins, tumor necrosis factor-alpha (TNF-α), and interleukin-1 beta (IL-1β) were assessed as markers of gastric mucosal activity. Aspirin and other non-steroidal anti-inflammatory drugs inhibit the biosynthesis of prostaglandins, which are critical protective factors against gastric acid irritation (Burke et al., 2006). Prostaglandin inhibition leads to early damage to the mucosal cells, reduced mucosal blood flow, decreased mucus and bicarbonate secretion, and increased acid secretion, ultimately resulting in ulcer formation (Rajkapoor et al., 2002). ... Phytochemical Evaluation And Protective Effect Of Tithoniadiversifolia (Hemsl.) Leaf Extract Against Aspirin-Induced Ulcer In Wistar Rats Article Full-text available Jan 2023 Okereke Stanley C. Rutherford Ikenna Esiaba Caleb Joel Nwaogwugwu Ijeoma Chukwuemeka Objectives: The objective of this study was to evaluate the phytochemical composition and gastroprotective properties of Tithonia diversifolia extract. Methods: Gas Chromatography-Mass Spectrophotometry (GC-MS) was used to quantify the chemical constituents present in the extract. Fourier Transform Infrared (FT-IR) spectroscopy was employed to identify functional groups in the extract. An acute toxicity study was conducted on rats to assess the safety of the extract. For the gastroprotective study, Wistar rats were divided into different groups and pretreated with different doses of T. diversifolia extract before ulcer induction with aspirin. Results: GC-MS analysis revealed the presence of twenty-eight chemical constituents in the extract,with benzyl alcohol, p-hydroxy-alpha-[(methylamino) methyl] being the most prominent. FT-IR analysis identified several functional groups in the extract. The acute toxicity study showed no signs of toxicity or mortality. Both the T. diversifolia extract and omeprazole (standard drug) significantly reduced ulcer parameters, antioxidant activity, phase Journal of Namibian Studies, 34 S2(2023): 2538-2573 ISSN: 2197-5523 (online) 2539 II enzymes, matrix metalloproteinase activity, prostaglandin E2 levels, and inflammatory biomarkers compared to the negative control group. The percentage inhibition of ulcers followed a dose-dependent trend. Conclusions: The findings of this study suggest that the decoction of Tithonia diversifolia has potential gastroprotective properties against aspirin-induced ulcers. The extract exhibited significant reduction in ulcer parameters and modulation of various biochemical markers associated with ulcers. These results support the traditional use of T. diversifolia in ethnomedicine for the treatment of ulcers. Further studies are warranted to explore the specific mechanisms of action and potential clinical applications of T. diversifolia extract in gastroprotection. View Show abstract ... Indomethacin, a non-steroidal antiinflammatory drug (NSAID), has been observed to mediate its analgesic effects by inhibiting cyclooxygenase enzymes I and II, a process that converts arachidonic acid to its unstable intermediates (PGH2 and PGG2). The inhibition of this process by indomethacin administration thus results in depleted thromboxane A2 and a variety of prostaglandins production as well as downregulation of pain receptors sensitization (Burke et al., 2006;Roberts & Morrow, 2002), which are characterized by improved mobility as experienced in the present study. Also, the ability of the ogiri samples (AAC70, AAC65, AAC22, AAH22, AAD10, AAD51, and AAD30) to act as analgesic may be attributed to their Dribose yield and concentrations via mechanisms to attenuate and prevent inflammatory responses (Pan et al., 2010) and as well to act as potent cyclooxygenase enzymes I inhibitors and anti-phlogistic agents (Hossain et al., 2012). ... Anti-inflammatory and analgesic potentials of fermented Citrullus vulgaris with mutant and non-mutant strains of Bacillus subtilis to produce condiment (ogiri) Article Dec 2023 Catherine Yemisi Babatuyi Olusegun V. Oyetayo F.A. Akinyosoye Idowu Sunday Oyeleye Background: Oxidative stress leading to degenerative diseases has become the leading cause of health issues. The use of non-steroidal anti-inflammatory drugs have be associated with complications. The development of nu-traceuticals/functional foods that can modulate oxidative stress becomes very necessary. Mutated Bacillus subtilis (B. subtilis) have been reported to produce D-ribose, having ameliorative effects on pro-inflammatory cytokine and increased ATP synthesis. Aims: The present study investigated analgesic and anti-inflammatory effects of ogiri samples fermented with mutant (modified) and non-mutant strains of B. subtilis in α-carrageenan-induced rats. Methods: Inflammation and pain were induced via α-carrageenan, while indomethacin and ogiri samples were used for treatment. Mobility and inflammatory volumes, pro-inflammatory cytokines, E-NTPDase activities, antioxidant activities and levels, hepatic and renal markers were investigated. Results: It was shown that some of the ogiri fermented with mutant strains of Bacillus subtilis samples ameliorated α-carrageenan-induced inflammatory oedema and painful movements. The percentage inhibition ranged from 18.69 ± 1.98-66.35 ± 2.97, with high percentage inhibition of oedema observed with samples IDC22, AAC70, AAC65, AAH22, AAD10, AAD51, and AAD30 (67.02 ± 1.98, 66.35 ± 2.97, 61.21 ± 2.31, 63.55 ± 1.32, 61.22 ± 2.31, 59.81 ± 0.66 and 59.35 ± 2.32) respectively. The reduction rate and mobility ranged from 5.30 to 10.40 mm and 169.00-288.50 min. The samples mentioned above also showed better amelioration with reduction rates (10.4, 9.85, 9.65, 10.1, 9.85. 9.85 and 9.70), and mobility values (257.00, 273.50, 261.00, 268.00, 271.50, 258.50, and 257.50) than other samples. The E-NTPDase activity ranged from 10.191 to 13.369 and showed energy restoration on some ogiri samples AAC70, AAC65, AAC22, AAH22, AAD10, AAD51, and AAD30 (12.558, 11.882, 11.288, 11.242, 12.324, 11.807 and 11.616) compared to the standard drug (9.798). Pro-inflammatory cytokines, inflammatory markers, antioxidants, hepatic and renal enzymes, and concentrations altered were restored in the same samples mentioned above. Conclusion: The modified ogiri samples, especially AAC70, can be applied as analgesic and anti-inflammatory agents. View Show abstract ... The most important four indicators of inflammation were pain, redness, heat or warmness, and swelling. Injury to any part of the body the arterioles present around the tissue enlarges due to this circulation of blood raises at that injured part causing redness , and it leads to loss of function the increased blood circulation increases the formation of intracellular spaces due to these leukocytes, protein, and fluids move to inflamed regions of the injury [38,39]. Inflammation is differentiated into two main types they were acute and chronic. ... In-vitro Evaluation of Anti-oxidant and Anti-inflammatory Potential of Bio-synthesized Silver Nanoparticles from Endemic Medicinal Plant Species Terminalia pallida Article Full-text available Apr 2022 P. Rama Mohan N. Savithramma View ... headache, toothache, neuralgias, migraine, menstrual pains, pain due to cold and flu infections, muscle and joint pain, pain due to surgical operations or injuries. It ranks high in analgesic poisoning due to its low price and availability as it can be sold without a prescription . ... Evaluation of the effects of curcumin, erdosteine, vitamin E and vitamin C on paracetamol toxicity Article Full-text available Jan 2022 Nurşah Başol Paracetamol toxicity is one of the most common causes of drug induced toxicity in the world. This study aims to investigate the efficacy of curcumin, erdosteine, vitamin E and vitamin C administration in paracetamol-induced liver damage in comparison with N-acetyl cysteine (NAC) in the treatment and prevention of liver toxicity due to paracetamol poisoning. 49 Wistar-Albino rats were used for this study. The rats were randomly divided into 7 groups. Group 1, which was the control group, received no drug administration. All the other groups received the minimum toxic dosage of paracetamol (1 gr/kg). Group 2 was not administered any other drug. Curcumin (100 mg/kg) was administered to Group 3. Group 4 received Vitamin E (170 mg/kg) and Group 5 received Vitamin C (300 mg/kg). Erdosteine (150 mg/kg) was administered to Group 6 and the last group (group 7) was received NAC for 2 days. After 72 hours, the experiment was completed. Liver and kidney tissues and blood samples were collected. AST and ALT values were higher in PCT group compared with the control group. Additionally, in PCT group, SOD and GSH-PX levels were lower while MDA levels were found to be higher in comparison to the control group. In the treatment groups, curcumin proved to be the most efficient agent, with NAC and erdosteine following. Histopathological images supported that curcumin played a key role in preventing liver damage. On the other hand, the results indicated no significant contribution of vitamin E and C in reducing paracetamol induced liver damage. In the light of the results, it is indicated that oxidative stress and lipid peroxidation are principal mechanisms of paracetamol induced liver damage, whereas curcumin and erdosteine are efficient agents in preventing said damage. Furthermore, the findings of the study suggest that curcumin in particular can be used as an alternative drug to NAC after further research on humans. View Show abstract ... Aspirin blockade of cyclooxygenase-1(C ox-1) and (Cox-II) results in reduction of prostaglandin synthesis. The interruption of prostaglandin synthesis results in impairment of mucosal damage repair, thus facilitating mucosal injury (Burke et al., 2006). Aspirin and related non-steroidal antiinflammatory drugs and alcohol can aggravate or interfere with the healing of peptic ulcers. ... Effects of Methanolic Extract of the Rind of Lanatus (Watermelon) in Aspirin Induced Gastric Ulceration in Male Wistar Rats Article Full-text available Jul 2016 Beatrice Olatundun Oluwatayo Catherine Adigwe Wali Kolawole Tolu Datonye Victor Dapper The u Organization (WHO) has long recognized and drawn the attention of countries to the ever increasing interest of the public in the use of medicinal plants and their products in the treatment of various ailments. is a common global health problem with increasing incidence and prevalence. approach to control. extract of induced gastric ulceration in rats. 180g five groups using 0.2gm/ follows: Group 1 served as the control (re Groups 2, 3 and 4 received 100mg/kg, methanolic extract of the rind of 5 receive day 22, percentage ulcer that, compared to control r the rind effect by significantly reducing the gastric juice volume, gastric acidity, ulcer index (p<0.05) in a dose dependent manner wh (p<0.05) and the percentage ulcer inhibition were significantly increased (p<0.05) in rats treated with the extract when compared with the control and the effect is simila oxide dismutase (while extract that the ulceration in male wistar rats. extract of the rind of ameliorative aspirin ulceration. View Show abstract ... This study chose the outcome of temperature reduction at 1 and 3 hours after acetaminophen administration for the reason that acetaminophen plasma concentrations usually peaked at 30 to 60 minutes after oral administration and 3 hours after rectal administration. 15,16 The previous meta-analysis, which included the data analysis of both adults and children, revealed that the decrease in temperature at 1 hour after administration had no significant difference between rectal and oral administration (WMD, À0.14 C; 95% CI, À0.36 C to 0.08 C; P = .49). There was also no difference of the decline in temperature at 3 hours after administration of acetaminophen (WMD, À0.10 C; 95% CI, À0.41 C to 0.21 C; P = .84). 9 In addition, this previous meta-analysis study had done the comparison of the maximum decline in temperature between rectal and oral administration, which also indicated no difference (WMD, À0.10 C; 95% CI, À0.24 C to 0.04 C; P > .99), ... Antipyretic Effectiveness of Oral Acetaminophen Versus Rectal Acetaminophen in Pediatric Patients With Fever Article May 2022 Nessa Tantivit Sittinun Thangjui Angkawipa Trongtorsak BACKGROUND AND OBJECTIVE Acetaminophen, one of the routine medicines used for temperature reduction in febrile children, is available in multiple routes of administration, including oral and rectal routes. Our objective is to compare the antipyretic effectiveness of oral acetaminophen versus rectal acetaminophen in pediatric patients with fever in terms of temperature reduction. METHODS Medline and Embase databases were searched from inception to August 2021. Cohort studies, case-control studies, experimental studies, and randomized controlled trial studies comparing oral and rectal administered acetaminophen in pediatric patients were included. Two reviewers independently extracted data. RESULTS A total of 5 randomized studies (n = 362) were included in the meta-analysis. No significant difference was found between oral and rectal acetaminophen in temperature reduction at 1 hour (weighted mean difference [WMD], 0.04oC; 95% confidence interval [CI], −0.10oC to 0.19oC; P = .501) or 3 hours (WMD, −0.14oC; 95% CI, −0.37oC to 0.10oC; P = .212) after administration (WMD, −0.14oC; 95% CI, −0.37oC to 0.10oC; P = .212). CONCLUSION Oral and rectal acetaminophen have no significant difference in antipyretic effectiveness at 1 and 3 hours after administration. If both options are available, oral acetaminophen would be preferred because of a more predictable drug level after administration. However, for febrile children with specific circumstances for whom oral acetaminophen could not be administered, rectal acetaminophen may be an alternative option for a short period of time (<48 hours). View Show abstract ... When a section of a b c d e the body is injured, the arterioles in the surrounding tissue widen. As a result of the increased blood flow to the region, redness develops . Acute and chronic inflammations are two types of inflammation. ... Tragia involucrata Leaf-Mediated ZnO NPs: Biomedical Applications, Ointment Formulation and Electrochemical Studies Article Full-text available Mar 2022 APPL BIOCHEM BIOTECH Udhayan S Ramanathan Prof.Udayakumar Gurusamy K Gopalasatheeskumar Kasiramar Zinc oxide (ZnO) NPs, owing to their broad biomedical applications, have recently attracted the scientific community with incredible interest as therapeutic agents. So, the present study aims at preparation of ZnO NPs, using Tragia involucrata leaf extract and exploring their capability as antioxidant, anticancer and anti-inflammatory agents. Besides, the ointment formulation and electrochemical studies were also carried out in this work. The antioxidant activity of the synthesized ZnO NPs was evaluated using DPPH assay method and the results clearly showed higher inhibition of about 70% and lower inhibition of about 14% for 100 µg/ml and 25 µg/ml concentrations, respectively. The cytotoxic effects of ZnO NPs were evaluated against human cancer cell lines such as A549 (lungs), HeLa (cervical), HeP-2 (laryngeal) and MCF-7 (breast). The outcome of this investigation confirmed the effectiveness of the synthesized NPs against HeP-2 even at the lowest concentration. The anti-inflammatory activity was measured by the inhibition of protein denaturation assay. A higher inhibition of about 54% was noticed at the concentration of 100 µg/ml. In the case of the ointment formulation study, the pastes prepared using the biosynthesized ZnO NPs and commercially available ZnO powder were compared and evaluated using the parameters such as pH, spreadability, moisture content, extrudability, foamability and physical examinations. As it has been noticed that all the observed parameters were matching well with those of the commercially available ZnO powder, ZnO NPs, synthesized using Tragia involucrata, may be suggested for the clinical trials. Cyclic voltammetry was used to measure the specific capacitance of the synthesized ZnO NPs for different scan rates. The results of this study showed the gradual decrease in specific capacitance value for the corresponding increase in scan rates. Therefore, the results of present study indicated that ZnO NPs prepared using Tragia involucrata leaves were found to be effective for all the above chosen applications and hence, have multifunctional capacity. View Show abstract ... One specific genotype causing slow sulfoxidation increases the risk of toxic side effects by ninefold (Ayesh et al., 1987). The arrival of new drugs like monoclonal antibodies and biologics has practically eliminated gold as an antirheumatic substance in the Western world (Burke et al., 2006;N.I.H., 2012). Another form of Au, auranofin, was used as an equally potent but oral alternative to Au-TM (Madeira et al., 2012). ... Immunotoxicology of metals Chapter Jan 2022 Per Hultman Kenneth Michael Pollard Our understanding of the effect of metals (including ions and their compounds) on the immune system continues to evolve. Observed effects include immunosuppression, immune stimulation, hypersensitivity, and autoimmunity. Many metals show a paradoxical dose-response pattern comprising stimulation of immune function at low doses and suppression at higher doses, but global immune function is often preserved due to the redundancy and the reserve capacity of the immune system, and clinically relevant effects are uncommon. Clinically relevant hypersensitivity reactions due to metals are dominated by T cell-mediated allergic contact dermatitis, particularly in response to exposure to beryllium, cobalt, chromium, gold, mercury, and nickel. Immediate (type I) hypersensitivity reactions dominated by airways symptoms occur infrequently, and then most often with platinum, but rarely with nickel or chromium. The induction of metal-induced autoimmunity, including the formation of immune-complex deposits, is well documented in humans, but the number of recognized cases is few. Studies in rodents using mercury and gold have increased our knowledge of the mechanisms of metal-induced autoimmunity. Of special importance is the unraveling of genetic factors that regulate susceptibility to mercury-induced autoimmunity, including the uptake and retention of mercury, as well as the threshold metal concentrations for eliciting autoimmunity. Recently mercury, lead, and cadmium have been shown to accelerate and/or exacerbate autoimmunity in autoimmune-prone animal models. The importance of metal exposure for inducing and/or accelerating autoimmunity in humans remains to be determined. View Show abstract ... Aspirin blockade of cyclooxygenase-1(C ox-1) and (Cox-II) results in reduction of prostaglandin synthesis. The interruption of prostaglandin synthesis results in impairment of mucosal damage repair, thus facilitating mucosal injury (Burke et al., 2006). Aspirin and related non-steroidal antiinflammatory drugs and alcohol can aggravate or interfere with the healing of peptic ulcers. ... Effects of Methanolic Extract of the Rind of Lanatus (Watermelon) Ulceration Article Full-text available Jun 2016 Datonye Victor Dapper Kolawole Tolu Beatrice Olatundun Oluwatayo View ... This gives a raised blood circulation towards the area (redness). Inflammation is either acute or chronic inflammation. Acute inflammation may be an initial response of the body to harmful stimuli. ... Medicinal plants with anti-inflammatory activity Article Aug 2016 Sunita Verma Inflammation is part of the body's immune response. There can be four primary indicators of inflammation: pain, redness, heat or warmness and swelling. Plants have the ability to synthesize a wide verity of phytochemical compounds as secondary metabolites which shows anti-inflammatory activity. In the present review an attempt has been made to investigate the anti-inflammatory activity of some medicinal plants. View Show abstract ... The arterioles in the surrounding tissue dilate when a part of the human body is injured. This results in increased blood flow to the affected area (redness) . Inflammation is a ubiquitous process that happens in a disturbed state of homeostasis such as damage, exposure to contaminating substances and infection, as well as is triggered by innate immune system receptors for the removal of pathogens when they are identified . ... Medicinal plants with non-steroidal anti-inflammatory-like activity Article Full-text available Jan 2021 Mediterr J Pharm Pharm Sci Abdul Sami Muhammad Akram View ... Unlike opioids tolerance, withdrawl and respiratory depression do not occur with ketorolac. It is one of the few NSAIDS approved for parenteral administration . ... Comparison of Analgesic Effect of Intravenous Magnesium Sulphate & Ketorolac in Management of Acute Pain in Upper Limb Orthopaedic Trauma Article Full-text available Jan 2015 Rajmala Jaiswal Teena Bansal Navdeep Kaur Geeta Ahlawat View ... Bayer found that heroin can be rapidly metabolized into morphine in the body 13,14 . The concept of prodrug was identified during the structural modification of chloramphenicol to reduce the unpleasant taste and improve solubility in aqua. Two types of chloramphenicol prodrugs (Fig. 2) were synthesized. ... Highly Efficient Prodrugs: Design and Therapeutic Applications Article Full-text available Dec 2020 Ashutosh Pal Bimal Krishna Banik Prodrug is a very powerful way for the improvement of biopharmaceutical, physicochemical, or pharmacokinetic possessions of pharmacologically dynamic mediators. Prodrug is a pharmacologically not an active compound, which can be converted into an active drug by biotransformation which is metabolic and such process the efficiency of drugs gets improved with specific target delivery. The conversion of a prodrug to drug may happen before concentration, after concentration, or at a precise part of the physique. This approach has many advantages over drug administration which is in our convention. In this review, different types of carriers, which can be used for prodrug synthesis are summarized. Examples of both marketed and investigational prodrugs from several promoieties are discussed not only for their advantages and uses but also their prospects. The purpose of this review is to introduce in detail the foundation behind the use of the prodrug methodology from past to present, and at the same time, to consider the possible consequences, which may evolve from insufficient initiation of prodrugs. Furthermore, the concept of prodrug and the classifications of prodrugs will be discussed in this article and it is expected that this review will be helpful for medicinal chemists for their research in the upcoming days. View Show abstract ... In a study conducted by Abessi et al wherein they compared the efficacy of Difluprednate 0.05% and Loteprednol Gel 0.5% After Cataract Surgery, they found that the anti inflammatory action of both loteprednol and difluprednate was comparable in uncomplicated cases. 16 Similarly in a meta analytical survey conducted by Duan P et al., they concluded that topical NSAID's were effective in controlling postoperative anterior chamber inflammation. 17 In a study on an Indian population conducted by Bannale et al., to compare Efficacy and the Safety of Topical Loteprednol Etabonate and Topical Flurbiprofen Sodium in Patients with Post-Operative Inflammation after Cataract Extraction, the results were comparable. 2 No study has been conducted in the Indian population exploring the role of steroids in intraoperative mydriasis, owing to their inhibitory action on prostaglandins. ... A comparative study of the efficacy of preoperative loteprednol versus flurbiprofen in cataract surgery Article Full-text available Dec 2020 Tek Chand Arvind Chauhan To evaluate the efficacy and cost-effectiveness of preoperative loteprednol versus flurbiprofen, when combined with moxifloxacin in maintaining intraoperative mydriasis and reducing postoperative complications. 200 cataract patients were divided into 2 groups. Group 1 receiving preoperative Flurbiprofen(0.03%) with moxifloxacin(0.5%) and Group 2 receiving Loteprednol(0.5%) with moxifloxacin(0.5%), three times a day 24 hrs prior to cataract surgery,followed by postoperative loteprednol(0.5%) and moxifloxacin(0.5%) in both groups. Intraoperative mydriasis and post operative complications were analysed. Patients were followed up on day 1, 1 week, 3 weeks and 7 weeks. Parameters evaluated included inflammation, pain, corneal edema, anterior chamber reaction (including cells and flare), intraocular pressure, CME. Intraoperative mydriasis obtained in both the groups was comparable with no significant difference. Postoperative visual outcome was also comparable but postoperative inflammation was slightly more in Group 1 (p <0.05). Pain was more in group 1 on first postoperative day which was not statistically significant but this difference did not persist during further follow ups. 3 patients in group 1 and 2 patients in Group 2 developed CME on the last follow up which was not statistically significant. The rest of the parameters were comparable in the 2 groups with a much better compliance in group 2. This hospital based study in western Rajasthan revealed Loteprednol to be equally potent as flurbiprofen in maintaining intraoperative mydriasis along with significant reduction of postoperative complications while maintaining cost effectiveness and compliance as a single eye drop was used before and after surgery, with no steroid related complications. View Show abstract ... Aspirin blockade of cyclooxygenase-1(C ox-1) and (Cox-II) results in reduction of prostaglandin synthesis. The interruption of prostaglandin synthesis results in impairment of mucosal damage repair, thus facilitating mucosal injury (Burke et al., 2006). Aspirin and related non-steroidal antiinflammatory drugs and alcohol can aggravate or interfere with the healing of peptic ulcers. ... Tissue Protein Carbonylation in Aging: A Strategic Analysis of Age-Related Protein Modification Article Full-text available May 2019 Kolawole Tolu Beatrice Olatundun Oluwatayo Arthur Nwafor Ogadinma Ilochi Free radicals generated in a variety of biological systems have been implicated in mechanisms of aging and age-related pathologies. This study strategically revealed the varying levels of carbonylated proteins in 3 different tissues of 40 wistar rats of varying ages. Their ages include 25-30, 45-50 and 65-70 days. The brain, heart and kidney tissue homogenates were prepared and biochemically analyzed for products of protein oxidation using 2,4-dinitrophenylhydrazones and autoantibodies against carbonylated proteins. This study revealed a direct proportional relationship between age and protein carbonylation in brain, heart and kidney tissue homogenates. The level of carbonylated proteins were significantly (P≤0.05) increased in the assayed tissues as all test groups advanced in age. Oxidative modification of proteins in brain and kidney tissues showed similar trend. This age-related biochemical manifestation may be as a result of increased generation of free radicals at mitochondrial level or decreased anti-oxidant defenses as living organisms advance in age. View Show abstract ... 19 Ibuprofen is a known inhibitor of Cyclo-oxygenase (COX-2), and by inhibiting COX, it reduces the production of prostaglandins (PGs), thereby causing vasodilation. 20,21 On the other hand, macrolides inhibit the endothelial injury caused by oxygenderived free radicals, and thus prevent an increase in vascular permeability. Thus, though macrolides does not have established analgesic property, both the drug groups affect hemodynamic component of inflammation, albeit through different routes. ... Aich TK et al. Macrolide Antibiotic in Opioid Article Jan 2019 Aich Tk Introduction: While receiving opioid detoxification treatment with standard Detoxification protocol, one patient was additionally prescribed Roxithromycin for his associated skin infection. Upon withdrawal of roxithromycin following improvement in skin infection, very next day same patient complained of heightened opioid withdrawal symptoms. These symptoms relieved again following reintroduction of Roxithromycin! Material And Method: It was a naturalistic study of 72 Opioid Dependent patients who received inpatient detoxification treatment at the Universal College of Medical Sciences, Bhairahawa, Nepal. A total of 16 patients received standard opioid detoxification protocol before the incidence of above mentioned patient. Subsequently, a total of 56 patients received Roxithromycin, in addition to the standard protocol, during their inpatient stay with a maximum for a period of 10 days. Necessary ethical clearance from the institute's ethical clearance committee was taken before carrying out this trial. Duration of inpatient stay and follow-up pattern after discharge was taken as the objective assessment of the efficacy of Roxithromycin in reducing severity of withdrawal symptoms. All other treatment parameters were similar in two groups. Results: Two groups did not differ in most demographic and clinical variables compared. Opioid dependence patients who received Roxithromycin had significantly longer (t=2.5; p=0.01) voluntary hospitalization stay (10.6 days, SD=6.2) vis-à-vis patients who did not receive it (6.4 days, SD=3.9). They also reported significantly more number of follow-ups after discharge (Fisher's Exact Test= 0.02). Conclusion: Roxithromycin, besides being an antibiotic, also possibly act as an anti-inflammatory and immune-modulatory agent by regulating leukocyte function View Show abstract ... 19 Ibuprofen is a known inhibitor of Cyclo-oxygenase (COX-2), and by inhibiting COX, it reduces the production of prostaglandins (PGs), thereby causing vasodilation. 20,21 On the other hand, macrolides inhibit the endothelial injury caused by oxygenderived free radicals, and thus prevent an increase in vascular permeability. Thus, though macrolides does not have established analgesic property, both the drug groups affect hemodynamic component of inflammation, albeit through different routes. ... Macrolide Antibiotic in Opioid Detoxification: A Serendipitous discovery? Article Full-text available Nov 2019 Tapas Kumar Aich J Haider Manoj Dhungana Introduction: While receiving opioid detoxification treatment with standard Detoxification protocol, one patient was additionally prescribed Roxithromycin for his associated skin infection. Upon withdrawal of roxithromycin following improvement in skin infection, very next day same patient complained of heightened opioid withdrawal symptoms. These symptoms relieved again following reintroduction of Roxithromycin! Material And Method: It was a naturalistic study of 72 Opioid Dependent patients who received inpatient detoxification treatment at the Universal College of Medical Sciences, Bhairahawa, Nepal. A total of 16 patients received standard opioid detoxification protocol before the incidence of above mentioned patient. Subsequently, a total of 56 patients received Roxithromycin, in addition to the standard protocol, during their inpatient stay with a maximum for a period of 10 days. Necessary ethical clearance from the institute’s ethical clearance committee was taken before carrying out this trial. Duration of inpatient stay and follow-up pattern after discharge was taken as the objective assessment of the efficacy of Roxithromycin in reducing severity of withdrawal symptoms. All other treatment parameters were similar in two groups. Results: Two groups did not differ in most demographic and clinical variables compared. Opioid dependence patients who received Roxithromycin had significantly longer (t=2.5; p=0.01) voluntary hospitalization stay (10.6 days, SD=6.2) vis-à-vis patients who did not receive it (6.4 days, SD=3.9). They also reported significantly more number of follow-ups after discharge (Fisher’s Exact Test= 0.02). Conclusion: Roxithromycin, besides being an antibiotic, also possibly act as an anti-inflammatory and immune-modulatory agent by regulating leukocyte function. View Show abstract ... When there is an injury to any part of the human body, the arterioles in the encircling tissue dilate. This provides a raised blood circulation towards the area (redness) (1). Inflammation is either acute or chronic inflammation. ... Assessment of In vitro anti-inflammatory activity of ethanol extract of Petiveria alliacea L. (Phytolaccaceae) Article Full-text available Jan 2019 A. RAJESH Doss A. Dr. Tresina Soris P Veerabahu Ramasamy Mohan The aim of the present study, is to study the phytochemical and In-vitro anti-inflammatory activity of ethanol extract of Petiveria alliacea L. whole plant. Phytochemical screening of whole plant ethanol extract revealed the presence of alkaloid, flavonoid, saponin, steroid, phenol, tannin, glycoside and terpenoid. In-vitro anti-inflammatory activity was screened against inhibition of albumin denaturation, proteinase inhibitory activity, heat induced haemolysis, hypotonicity induced haemolysis and antilipoxygenase activity. Aspirin was used as standard drug. The results showed that Petiveria alliacea whole plant at a concentration of 500 µg/ml significantly (P<0.01) protect the heat induced protein denaturation, proteinase inhibitory action and heat induced haemolysis of erythrocyte. Hypotonicity induced haemolysis and lipoxygenase activity were significantly (P<0.01) inhibited at the concentration of 500 µg/ml. the results obtained in the present study indicate that ethanol extract of Petiveria alliacea whole plant can be a potential source of anti-inflammatory agents. View Show abstract ... 1 Nonsteroidal anti-inflammatory drugs (NSAIDs) are the main stay of treatment of pain. 2 It is known fact that the risk of gastrointestinal bleeding and other side effects are associated with acute and chronic use of non-steroidal anti-inflammatory drugs (NSAIDs). 3 Keeping in view the gravity of adverse effects of NSAIDs it is necessary to search for new drugs with less adverse effects. ... Evaluation of dose dependent analgesic response by extracts of Myristica fragrans on albino wistar rats: an experimental study Article Full-text available Aug 2019 Priyamvada Sharma Imran Zaheer Syed Ziaur Rahman Rahat Ali Khan Background: The objective of the study was to evaluate analgesic activity of ethanolic extract, methanol and benzene fraction of Myristica fragrans on wistar albino rats.Methods: The present study was carried out in the department of pharmacology JNMC AMU and F.H. Medical College, Agra. The analgesic activity was evaluated by employing the Eddy’s hot plate method and tail flick response method. In both the tests, Rats of either sex weighing 150-200 g were used. The total number of animals n=36 were allocated to six groups. Each group consist of six animals each. The response noted in animals that were tested by hot plate method was reaction time for licking/biting of both the paws before and after administration of control & test drugs. However in Tail flick test, the pain threshold response was recorded before and after administration of control & test drugs. The statistical analysis was done by using one-way ANOVA. The data is expressed as Mean±SEM. P<0.05 was considered to be statistically significant.Results: Ethanolic extracts and methanol fraction of M. fragrans showed statistically significant (p<0.001) increase in reaction time for licking/biting in hot plate method. On the contrary a significant increase in pain threshold was also recorded in tail flick response test. It is interesting to note that no significant degree of analgesia related to any dose of benzene fraction was observed.Conclusions: The present study reveals the dose dependent significant analgesic activity of the extracts of M. fragrans i.e. ethanolic extracts and methanol fraction in both the test. However, the degree of analgesia was recorded significantly higher in groups received higher doses of extracts of M. fragrans. View Show abstract ... Therefore, this drug is frequently used as a model in studies on in vivocytoprotective activity of new substances or compounds 20,21 . Administration of aspirin or other non-steroidal anti-inflammatory drugs inhibits the biosynthesis of prostaglandins, particularly; prostaglandin E2 (PGE2) and prostaglandin I2 (PGI2) 22,23 , which are protective factors (mucosal resistance factors) against irritation of the stomach by gastric acid 24,25 . Inhibition of prostaglandins results in early damage to the mucosal, parietal, and endothelial cells 26 , decrease mucosal blood flow, decrease secretion of mucus and bicarbonate, and simultaneous increase in acid secretion 27 , thus leading to the formation of an ulcer. ... A11-A.10.V.8.I.1.1124 Article Full-text available Dec 2018 Chidi Ijeoma Nosiri Arunsi Uche Okuu The gastroprotective effect of Annona muricata (Linn.) seed extract on aspirin induced ulcer model in wistar rats was evaluated using standard analytical methods. A total of twenty-five (25) Wistar rats used in the study was divided into 5 groups of 5 rats each: 1, Normal control (received 0.2ml distilled water); 2,Negative control (received 400mg/kg Aspirin); 3, Positive controla (pretreated with omeprazole 20mg/kg);4, positive controlb(pretreated with A. muricataseed extract 200mg/kg); and 5,positive controlc (pretreated withA. muricataseed extract400mg/kg extract).The result divulged that the administration of Aspirin led to significant increase (p<0.05) in ulcer score, ulcer index, gastric volume, total acidity and pepsin activity with means and standard deviations of 14.92±0.09; 10.10±0.17, 7.12±0.23mL; 12.14±0.21mEq/L; and 98.82±0.17mEq/L respectively. These ulcerogenic indices were lowered significantly (p<0.05) in groups pretreated with omeprazole and ethanolic seed extract of A. muricata. Furthermore, Omeprazole offered higher percentage gastroprotection than A. muricata seed extract. The degree of gastroprotection is in the order: A. muricata (200mg/kg)<A. muricata (400mg/kg)<Omeprazole (20mg/kg). The findings of the present study demonstrated that ethanolic seed extract of A.muricata manifested comparable gastroprotection effect against aspirin-induced ulceration vis-à-vis the reference drug. Omeprazole and the different doses of the extracts in this study may have suppressed gastric acid secretion via inhibition of H+/K+ ATPase or enhanced prostaglandin synthesis. View Show abstract ... The coexistence of analgesic and antiinflammatory activities is well defined for various non-steroidal anti-inflammatory drugs (NSAIDs), particularly salicylates and their congeners . The principal therapeutic effects of NSAIDs derive from their ability to inhibit prostaglandin G/H synthase (cyclooxygenase or COX) which convert arachidonic acid to the unstable intermediates PGG2 and PGH2 and leads to the production of thromboxane A2 and a variety of prostaglandins . Prostaglandins are also known to cause pain and NSAIDs are particularly effective when inflammation has caused sensitization of pain receptors to normally painless mechanical or chemical stimuli . ... Evaluation of Ant-inflamatory Activity of Fumaria Officinalis Linn. Herb Extract on Experimental Animal Research Full-text available Jun 2010 uday Raj Sharma Venkata Surendra Sajal Kumar Jha Surendra Vada Fumaria officinalis Linn. is a local medicinal plant used in ethnomedicine for the treatment of constipation, bronchitis and asthma. The aqueous decoction and the ethanolic extracts were subjected to anti-inflammatory activity using experimental animal model, in the presence of the positive control drugs. The inflammation was induced by carrageenan. From the results obtained the ethanolic extract showed significant activity (P < 0.001) comparable to the reference drug used. At the different dose range used (100, 200 and 500 mg/kg), there was significant differences in their anti-inflammatory activity hence they were dose-dependent. The results of the study showed the justification of the use of the plant in the treatment of inflammatory disease. View Show abstract ... The probable mechanism of action of carrageenan-induced inflammation is biphasic; the first phase is attributed to the release of histamine, serotonin and kinins in the first hour; while the second phase is attributed to the release of prostaglandins and lysosome enzymes in the second to third hour [11,12]. The principal therapeutic effects of NSAIDs derive from their ability to inhibit prostaglandin G/H synthase (cyclooxygenase or COX) which convert arachidonic acid to the unstable intermediates PGG2 and PGH2 and leads to the production of thromboxane A2 and a variety of prostaglandins . Prostaglandins are also known to cause pain and NSAIDs are particularly effective when inflammation has caused sensitization of pain receptors to normally painless mechanical or chemical stimuli . ... Evaluation of Ant-inflammatory activity of Rhododendron Arboreum herb extract on experimental animal Research Full-text available Jul 2009 uday Raj Sharma Venkata Surendra Sajal Kumar Jha Surendra Vada Rhododendron Arboreum is a local medicinal plant used in ethnomedicine for the treatment of constipation, bronchitis and asthma. The aqueous decoction and the ethanolic extracts were subjected to anti-inflammatory activity using experimental animal model, in the presence of the positive control drugs. The inflammation was induced by carrageenan. From the results obtained the aqueous extract showed significant activity (P < 0.001) comparable to the reference drug used. At the different dose range used (40, 60 and 100 mg/kg), there was no significant differences in their anti-inflammatory activity hence they were not dose-dependent. However, the ethanolic extract did not show any appreciable activity and were also not dose-dependent. The results of the study showed the justification of the use of the plant in the treatment of inflammatory disease conditions, and the active chemical constituents when isolated will be added to the present anti-inflammatory agents. View Show abstract ... However, side-effects, and also occasionally very severe possible adverse effects, can occur. There are a number of anti-inflammatory herbs that could help to achieve similar results without the harmful effect . Inflammation is either acute or chronic inflammation. ... Evaluation of Anti-inflammatory Activity of Ethanolic Extract of the Root of Chamaecyparis pisifera (Siebold and Zucc.) Endl. Article Jan 2019 Arnt Win Aye Mon Thida Nyo Thida Cho View ... Synthetically prepared drugs are utilized worldwide for the treatment of various diseases. However, these drugs have several side effects such as gastric ulcer, kidney damage as well as cardiac abnormities . Plants have been used as an important source of medicine for centuries. ... ANTIOXIDANT ACTIVITY, PHYTOCHEMICAL ANALYSIS AND TOTAL POLYPHENOLICS CONTENT OF ESSENTIAL OIL, METHANOL EXTRACT AND METHANOL FRACTIONS FROM COMMELINA NUDIFLORA Article Full-text available Aug 2018 Muhammad dawood Shah Mohammad Iqbal Objective: In the present study, the essential oil, methanol extract, and methanol fractions (n-hexane, chloroform, ethyl acetate, and n-butanol) obtained from Commelina nudiflora were investigated for the free radical scavenging effects and phytochemical analysis.Methods: The antioxidative effect of the essential oil, methanol extracts and methanol fractions were evaluated using 2, 2 diphenyl-1-picrylhydrazyl (DPPH) radical scavenging assay. Total phenolic and ﬂavonoid contents were determined using Folin-Ciocalteau and aluminium chloride reagents respectively. The phytochemical analyses of the essential oil, methanol extracts and methanol fractions were performed by gas chromatography and mass spectrometry (GCMS). Results: The antioxidant, total phenolic and total flavonoid contents of butanol, ethyl acetate and chloroform fractions were higher followed by methanol extract, hexane fraction and essential oil. Phytochemical analysis indicated the presence of alkaloid, saponin, steroid, phytosterols, triterpenoids and tannins etc. The identified bioactive constituents of essential oil, methanol extract and methanol fractions of C. nudiflora were indole, 2-methoxy-4-vinylphenol, 2-pentadecanone, 6,10,14-trimethyl, phenol, benzyl alcohol, eugenol, phenol, 2, 4-bis (1,1-dimethylethyl), hexadecanoic acid, ethyl ester (palmitic acid ester), n-hexadecanoic acid (palmitic acid), 9, 12-octadecadienoic acid, (linoleic acid) and phytol. All identified bioactive compounds and their derivatives were generally reported with antimicrobial, antioxidant, anti-inflammatory and antitumor properties.Conclusion: The obtained data suggest that the essential oil, methanol extract and methanol fractions of C. nudiflora possess remarkable antioxidant activities and vital phytochemicals. Thus the plant can be a utilized as a potential source of nutraceutical with antioxidant activity. View Show abstract ... Anti-arthritic activity and evaluation 10 In the experiment, one group of animals were kept as control and given only vehicle while another group or received a standard drug for comparison of activity and authenthicity of the experiment. Paw volume was measuredat different time intervals using a volume differential meter model7101 .mean ... Anti-Arthritic Activity of Plant AcalyphaIndica Extract Article Jan 2016 Mathew George Lincy Joseph Kamal Singh Vyas Saira Susan Varghese View ... Its inhibition in gastric mucosa by NSAIDs, such as aspirin, caused elevation in gastric acid secretion and reduced mucosal blood flow, mucus, and bicarbonate secretion. 47,48 It is therefore seemed that MEVA probably prevented the inhibitory action of aspirin on prostaglandin synthesis in the gastric mucosa cells. ... Protective Effects of Methanol Extract of Vernonia amygdalina ( del .) Leaf on Aspirin-Induced Gastric Ulceration and Oxidative Mucosal Damage in a Rat Model of Gastric Injury Article Full-text available Jul 2018 Modinat Adebukola Adefisayo Rufus Ojo Akomolafe Akinsomisoye Stephen Kehinde P. Olamilosoye This study investigated the quantitative polyphenolic constituents and gastroprotective effects of methanol extract of Vernonia amygdalina leaf (MEVA) against aspirin-induced gastric ulcer in rats. Ulceration was induced by 3 days’ oral administration of aspirin (150 mg/kg body weight). Wistar rats were pretreated with cimetidine (reference drug) at a dose of 100 mg/kg body weight and MEVA at 200, 300, and 400 mg/kg body weight once daily for 28 days prior to ulcer induction. At the end of the experiment, gastric secretions, antioxidant status, and histopathological alteration were evaluated. We observed that the significantly increased ulcer index, gastric volume, free and total acidity, malondialdehyde level, and pepsin activity were effectively reduced following treatment with 200 and 300 mg/kg MEVA. The extract also markedly attenuated the reduced activity of superoxide dismutase and reduced glutathione level as well as pH and mucin content in the ulcerated rats. Administration of the extract also significantly attenuates necrosis of the stomach tissue of the ulcerated rats. The results suggested that the MEVA leaf, preferably at 200 and 300 mg/kg body weight, ameliorated aspirin-induced gastric ulceration via antioxidative and H2 receptor antagonist. View Show abstract ... Paracetamol is a drug that has been used safely in baby, children and adults for many years as an analgesic and antipyretic . Intoxication is frequently encountered because paracetamol is easily accessible. ... Sıçanlarda CYP2E1 İnihibisyonu İle Parasetamol İntoksikasyonunda Punica granatum Çekirdek Yağı Ektraktının Hepatoprotektif Etkisi Article Full-text available Sep 2018 KAFKAS UNIV VET FAK Damla Cetin Pinar Aksu Kiliçle Gulname FINDIK GUVENDI Muhammed Yayla Punica granatum seed oil enriched by punikalagins has been shown to have a therapeutic effect against antidiabetic, anticancer, antiinflammatory and some organ toxicities. We aimed to reveal both protective and therapeutic effects of punica granatum seed oil, which has strong antioxidant and anti-inflammatory activity on paracetamol-induced hepatic toxicity via biochemically, molecularly and pathologically in our study. Our study, 64 albino wistar rats were fasted for 24 h and then divided into 8 equal groups. Group 1: Healthy, Group 2: 2 g/kg of paracetamol (2a: 24 h, 2b: 48 h) (orally), Group 3: 140 mg/kg of n-acetylcysteine (orally) + paracetamol, Group 4: 0.32 mg/mL Punica granatum (i.p) + paracetamol, Group 5: 0.64 mg/mL Punica granatum + paracetamol, Group 6: Paracetamol + 0.32 mg/mL Punica granatum, Group 7: Paracetamol + 0.64 mg/mL Punica granatum, Group 8: Paracetamol + 140 mg/kg n-acetylcysteine. The study was terminated at 24 and 48 h after paracetamol administration. Serum ALT and AST levels were significantly increased at 24th and 48th h of paracetamol administration according to toxicity. While malondialdehyde, CYP2E1 and TNF-α levels also increased in the liver, superoxide dismutase and glutathione peroxidase levels decreased significantly. Increased ALT, AST levels with malondialdehyde and TNF-α levels significantly decreased by punica granatum seed oil (low doses) application and antioxidant levels were also significantly improved. Punica granatum seed oil may be used as a potential therapeutic agent in the future by strengthening the antioxidant system and preventing inflammation, especially liver toxicity due to overdose of paracetamol in suicide- battered individuals. View Show abstract ... After substantial investigation, the source of toxicity was eventually linked to residues of diclofenac present in the vulture food chain as a result of livestock being on palliative diclofenac treatment at the time of their death (Oaks et al., 2004;Shultz et al., 2004;Green et al., 2004). Diclofenac (2-[2-(2,6-dichlorophenyl amino) phenyl]acetic acid) is a non-steroidal anti-inflammatory drug (NSAID), developed in the late 1970s for the treatment of pain, fever and inflammatory conditions such as osteoarthritis, rheumatoid arthritis, and ankylosing spondylosis in humans (Scully et al., 1993;Burke et al., 2006). The drug also has merit in veterinary medicine for the management of livestock disease due to its analgesic and anti-inflammatory properties. ... The use of liver slices from the Cape vulture ( Gyps coprotheres ) to better understand the role of liver toxicity of Non-steroidal anti-inflammatory drugs (NSAIDs) in vultures Article Jul 2018 ENVIRON TOXICOL PHAR Emmanuel Oluwasegun Adawaren Lillian Mukandiwa Emmanuel Mfotie Njoya Vinny Naidoo Diclofenac, a non-steroidal anti-inflammatory drug (NSAID) was responsible for the death of millions of vultures on the Asian subcontinent, following the consumption of diclofenac contaminated carcasses. The aim of this research was to establish if liver slices could serve as an alternate means of predicting the toxicity of NSAIDs in Gyps vultures. The Cape vulture liver slices was prepared and incubated with four NSAIDs for 6 hours. A percent clearance of 1.0 ± 0.253, 0.58 ± 0.153, 0.961 ± 0.312 and 1.242 ± 0.406 (%/hg) was attained for diclofenac, carprofen, ketoprofen and meloxicam respectively. Both meloxicam and diclofenac exerted toxic effects on the hepatic cells. Protein content indicated that the vulture tissue had lower enzyme levels than expected for an animal of its size. The poor distinction between the ex vivo hepatic percent clearance of meloxicam and diclofenac indicates that liver slices is not an ideal model to investigate NSAIDs toxicity in Cape vulture. View Show abstract ... The prostamides and their structural analogs are structurally, pharmacologically, and functionally distinct from prostaglandins and prostaglandin analogs. The free acid of bimatoprost is identical to that of latanoprost, the only exception being a double, instead of single, bond at the carbon 13-14 positions. Bimatoprost exerts its effects by stimulating the prostamide receptor, which is pharmacologically distinct from F prostanoid (FP) receptors. ... Bimatoprost in Dermatology Article Full-text available May 2018 Abhijeet Kumar Jha Bimatoprost is a prostamide analogue used for treatment of glaucoma in ophthalmology. Surprisingly, the side effects such as increased pigmentation of eyelids and hypertrichosis in patients being treated with prostaglandin analogues for glaucoma have opened new areas of application in various dermatological disorders such as alopecia mainly affecting eyelashes, eyebrows, and vitiligo. View Show abstract ... Pain is induced by injection of irritants into the peritoneal cavity of mice (Koster et al., 1959;Singh and Majumdar, 1995). The animals react with a characteristic stretching behavior which is called writhing (Burke and Fitzgerald, 2006). Writhing is defined as a stretch, tension to one side, extension of hind legs, or contraction of the abdomen so that the abdomen of the mice touches the floor, or turning of the trunk (twist). ... Potent antibacterial flavonoids from Pseudathria hookeri Wight & Arn (Fabaceae) Conference Paper Apr 2018 Joseph Tchamgoue Simeon F. Kouam Jean Paul Dzoyem Udo Bakowsky Despite the extensive use of antibiotics and vaccination programs, microbial infections continue to be a leading cause of morbidity and mortality worldwide. This is exacerbated by widespread antibiotic resistance, the emergence of new pathogens in addition to the resurgence of old ones, and the lack of effective new therapeutics. Plants represent a rich source of new molecules with pharmacological properties, which may be used as lead compounds for the development of new drugs to fight against such resistance. In this light, we carried out the chemical investigation of Pseudarthria hookeri, a plant used in many African countries for the treatment of several ailments including microbial infections viz pneumonia, cough, abdominal pains, and diarrhea. Our research led us to the isolation and characterization of three new flavonoids and several known compounds which were assessed for their antibacterial activity against pathogenic bacterial strains involved in diarrhea and respiratory infections. The antibacterial activity was assessed by determining the minimum inhibitory concentration (MIC) and minimal bactericidal concentration (MBC) using the broth microdilution method. Five flavonoids showed signiﬁcant antibacterial activity against all the bacterial strains tested except Enterococcus faecalis. The flavonoids pseudarflavone A and 6-prenylpinocembrin showed the highest antibacterial effect with minimum inhibitory concentration (MIC) values ranging from 16 to 32 and 8 to 64 µg/mL, against Gram-positive and Gram negative bacteria, respectively. The MBC/MIC ratio of pseudarflavone A against Klebsiella pneumoniae and the MBCs of 6-prenylpinocembrin against Escherichia coli and Staphylococcus aureus were similar to those obtained with the reference antibiotic ciproﬂoxacin. These compounds emerged as promising drug candidates, highlighting their potential as lead compounds. Moreover, the tested compounds showed no toxic effect on MIN-6 and 3T3 cell lines up-to 400 µM, suggesting their safety profile. To the best of our knowledge, this is the first chemical investigation of this plant. View Show abstract ... There are a number of anti-inflammatory herbs that could help to achieve similar results without the harmful effect. Inflammation is a severe response by living tissue to any kind of injury. It is a complex process which is frequently associated with pain, involves occurrences such as increased vascular permeability, increase of protein denaturation and membrane alterations. ... INVITRO ANTI-INFLAMMATORY ACTIVITY OF HYDROMETHANOLIC SEED, FRUIT AND LEAVE EXTRACTS OF CAPSICUM CHINENSE (RED PEPPER) Article Full-text available Jan 2015 Angela Nnenna Ukwuani-Kwaja I B Hassan The present investigation was carried out to evaluate the in vitro anti-inflammatory potential of Capsicum chinense. Hydromethanolic extracts of leave, seed, and fruits of Capsicum chinense (Red pepper) was assessed for its in vitro anti-inflammatory activity using heat and hypotonic solution induced membrane stabilization activity and protein denaturation assay at different concentration. Diclofenac sodium was used as standard drugs. The results showed a dose dependent (p<0.05) RBC membrane stabilization and inhibition of protein denaturation activity. Hydromethanolic leaves extract (400µg/ml) exhibited the highest percent protection of 70.76%, 58.14% and 59.38% against heat and hypotonic solution induced membrane stabilization activity and protein denaturation respectively. From the above study, it was concluded that hydromethanolic seed, fruit and leaves extract of C. chinense has significant membrane stabilization property which was comparable to the standard drug Diclofenac. This study provides invitro anti-inflammatory evidence for the use of this plant in the treatment of inflammatory conditions. However, these effects need to be confirmed using in vivo models. View Show abstract ... Arylpropionic acid derivatives such as ibuprofen, ketoprofen and flurbiprofen represent an important class of non-steroidal anti-inflammatory drugs. 1 They are widely used in medicines as the racemic mixture, however, only the S-enantiomer is biologically active. 2 Therefore, researchers have taken great interest in resolving the enantiomers of these compounds via enzyme catalysis using lipases, specifically the industrially established Novozyme 435 (supported lipase B from Candida Antarctica), which has been the focus of many publications. Chiral resolution coupled with direct esterification with an alcohol is considered more desirable than transesterification or hydrolysis as the number of processing steps can be reduced. ... Towards sustainable kinetic resolution, a combination of bio-catalysis, flow chemistry and more sustainable bio-based solvents Article Dec 2017 Andree Iemhoff James Sherwood Con Robert McElroy Andrew Hunt The esterification of 2-phenylpropionic acid was investigated as a model system for enzyme catalysed (CALB, Novozyme 435) reactions in bio-based solvents. A multi-parameter correlation taking into account solvent parameters was... View Show abstract Design, Synthesis, and Biological Evaluation of New Ureido (Thioureido) Anthranilic Acid Isosteres: Molecular Docking, In Silico ADMET Predictions, and In Vivo Anti‐Inflammatory Activity Article Full-text available Apr 2025 CHEM BIODIVERS Kamel Harrouche Boutaoui Nassima Katia Mohand Saidi Smail Khelili A novel series of anthranilic acid isosteres were designed and synthesized as antiinflammatory agents. The in silico absorption, distribution, metabolism, excretion, and toxicity (ADMET) study predicted a favorable pharmacokinetic profile and respect for Lipinski's rule of five. Density functional theory (DFT) calculations revealed an improvement in some target compounds’ electronic parameters compared to diclofenac (DCF) and aspirin (ASA), predicting an improvement in their biological activity. Docking investigations demonstrated a strong affinity toward the cyclooxygenase (COX)‐1 and COX‐2 enzymes, with a relative preference for COX‐2, predicting antiinflammatory activity. The MolDock scores were between −140.59 and −91.81 kcal/mol for COX‐1 and between −148.10 and −108.9 kcal/mol for COX‐2. The experimental pharmacological investigation confirmed these theoretical findings. Indeed, target compounds demonstrated a significant inhibition of the carrageenan‐induced paw edema in rats and probable inhibition of COX. Particularly, compounds 4e and 4h devoid of COOH group, which provoke serious gastrointestinal irritation, exhibited antiinflammatory activity comparable to that of salicylic acid (ASA) and surpassed the effectiveness of DCF. Cpmpounds 4e and 4h showed 91.72% inhibition after 3h, against 91.03% and 83.44% for ASA and DCF, respectively, with a greater onset effect, and also surpassing the reference compounds after 1 and 2 h. The results also indicate good pharmacokinetic profile of the target compounds similar to ASA and DCF. View Show abstract Anti-Inflammatory and Anti-Arthritic Activity of some Indigenous plants: A Review Article Aug 2023 P. Manimekalai S. Ajina A. Meena Jesiliya Inflammatory and arthritic conditions are among those which are treated using traditional remedies, with considerable success. The aim of the present review is to collect all the current reliable data on experiments reporting the anti-inflammatory and anti-arthritic effects of medicinal plants and natural products. Investigation was carried out by analyzing recognized books and peer-reviewed papers. This article mainly focused on experimental research conducted on medicinal plants, particularly with anti-inflammatory and anti-arthritic activity with their bioactive compounds. A total of 10 plant species have been identified as active or promising sources of phytochemicals with anti-inflammatory and anti-arthritic properties. View Show abstract Nonsteroidal Anti-inflammatory Drugs (NSAIDs) Chapter Jun 2016 Laura L. Wayman View KETOPROFEN;: AN EFFECTIVE AGENT AGAINST RHEUMATOID ARTHRITIS Article Jan 2018 Farya Zafar Huma Ali Ghazala Raza Naqvi Huma Shareef Ketoprofen is effectively useful in managing arthritis, rheumatoid arthritis,osteoarthritis and ankylosing spondylitis. This article covers the pharmacological uses,toxicology, contraindications, food – drug, drug-drug interactions and associated side effects ofKetoprofen that have been reported in literature in earlier years. View Show abstract Evaluation of Analgesic Activity of Extracts of Delphinium denudatum in Animal Models: A Dose Dependent Pre-Clinical Trial Article Full-text available Nov 2018 Imran Zaheer Syed Ziaur Rahman Rahat Ali Khan Pain is one common health problem with substantial socioeconomic impact because of its high incidence. It is associated with a number of diseases and is estimated that 80-100% of the population experience back pain at least once in the life-time . Non-Steroidal Anti-Inflammatory Drugs (NSAIDs) are the mainstay of treatment of pain . It was reported that the risk of gastrointestinal bleeding was significantly associated with the acute use of NSAIDs like regular-dose aspirin, diclofenac, ketorolac and Piroxicam etc., increased the risk of bleeding in both acute and chronic therapy . Therefore, it is necessary to search for new drugs with less adverse effects. Medicinal plants have been used in the development of new drugs and continue to play an invaluable role in the progress of drug discovery . Recently, many natural medicines derived from medicinal plants such as Capsicum annuum, Cannabis sativa and Papaver somniferum were considered as effective and safer for treatment of various diseases including pain . Delphinium denudatum family Ranunculaceae is a medicinal herb commonly known as jadwar and used in Unani Medicine. The roots are reported to be useful in a variety of ailments such as paralysis, epilepsy, facial palsy, insanity, mania, hysteria, atony, migraine, numbness, tremors, infantile convulsions, aconite poisoning, snake bite, scorpion sting, arthritis, cardiac weakness, palpitation, rheumatism, toothache. Delphinium plants have been medicinally used for centuries [6,7]. Use of its root as an analgesic is found in Unani medicine . Earlier studies showed that ethanolic extract of Delphinium denudatum has analgesic activity [9,10]. The present study was performed to validate the earlier study and to screen additionally the effect of its methanol fraction. View Show abstract Show more ResearchGate has not been able to resolve any references for this publication. Recommended publications Discover more Article Pharmacological activities of Hypericum scabrum L. January 2011 · European Review for Medical and Pharmacological Sciences Nabavi S.M. et al Read more Article Amylin: pharmacology and physiology January 2005 A. Young Read more Article A survey of studies on the chemical constituents pharmacological activities of Dictamnus L January 2003 X.W. Fan X. Zhang S.M. Wang Read more Chapter Bisphosphonate: Eine neue Stoffklasse zur Therapie von Knochenkrankheiten. Pharmakologische Grundlag... January 1992 H Fleisch Im folgenden ist der derzeitige Wissenstand zur Pharmakologie der Bisphosphonate und ihrer Anwendung bei Knochenkrankheiten kurz zusammengefaßt. Die Literaturhinweise sind absichtlich auf ein Minimum beschränkt. Sollten weitergehende Informationen über die Bisphosphonate im allgemeinen erwünscht sein, so sei der Leser auf eine neuere Übersicht verwiesen (Fleisch 1988). Read more Looking for the full-text? You can request the full-text of this article directly from the authors on ResearchGate. Request full-text Already a member? 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18180	https://www.mathreference.com/set,pigeon.html	The Pigeonhole Principle Sets, The Pigeonhole Principle SearchSite mapContact us Main Page Sets Use the arrows below to step through Sets. Custom Links: The Pigeonhole Principle If set S has more members than set T (both finite), there is no 1-1 function F mapping S into T. Take the smallest counterexample and delete some member of S and its image in T. This gives a 1-1 function on smaller sets. At the base of the inductive argument, T has one element and S has more, whence the 1-1 function cannot exist. The name "pigeonhole Principle" was coined when this counting argument was aplied to pigeons, as they occupied a set of protected havens (pigeonholes). When there is an excess of birds, a 1-1 function mapping pigeons to pigeonholes is not possible. Some must remain outside, or share a hole. Depending on your mood, this concept may seem too obvious to bother proving, yet one should not underestimate its power. Consider an old favorite. Select 9 different lattice points in R3. Prove that one of the 36 segments determined by these 9 points contains another lattice point. Remember that a lattice point is a point with integer coordinates. A parity argument combined with the pigeonhole principle proves the result. A lattice point in R3 possesses 1 of 8 parity combinations; each coordinate is even or odd. Since 9 points were selected, two have the same parity combination by the pigeonhole principle. Their average, or midpoint,is another lattice point. Custom Links: © 2002-2023 Karl Dahlke
18181	https://www.xconvert.com/unit-converter/nanometers-to-centimeters	Nanometers (nm) to Centimeters (cm) conversion Nanometers to Centimeters conversion table \| Nanometers (nm) \| Centimeters (cm) \| --- \| \| 0 \| 0 \| \| 1 \| 1e-7 \| \| 2 \| 2e-7 \| \| 3 \| 3e-7 \| \| 4 \| 4e-7 \| \| 5 \| 5e-7 \| \| 6 \| 6e-7 \| \| 7 \| 7e-7 \| \| 8 \| 8e-7 \| \| 9 \| 9e-7 \| \| 10 \| 0.000001 \| \| 20 \| 0.000002 \| \| 30 \| 0.000003 \| \| 40 \| 0.000004 \| \| 50 \| 0.000005 \| \| 60 \| 0.000006 \| \| 70 \| 0.000007 \| \| 80 \| 0.000008 \| \| 90 \| 0.000009 \| \| 100 \| 0.00001 \| \| 1000 \| 0.0001 \| How to convert nanometers to centimeters? Converting between nanometers and centimeters involves understanding the relationship between these units of length. Here's a breakdown of how to perform these conversions: Understanding the Conversion Both nanometers (nm) and centimeters (cm) are units of length in the metric system. The key is knowing their relationship to the base unit, the meter (m): 1 meter (m) = nanometers (nm) 1 meter (m) = centimeters (cm) From these relationships, we can derive the direct conversion factor between nanometers and centimeters. Converting Nanometers to Centimeters To convert nanometers to centimeters, you need to know that 1 cm is equal to 10,000,000 nm (i.e., nm). Therefore, to convert from nm to cm, you divide by . Formula: Step-by-Step Conversion (1 nm to cm): Start with the value in nanometers: 1 nm Divide by : cm Result: Converting Centimeters to Nanometers To convert centimeters to nanometers, you multiply the number of centimeters by . Formula: Step-by-Step Conversion (1 cm to nm): Start with the value in centimeters: 1 cm Multiply by : nm Result: Examples Thickness of a Coating: A very thin coating on a material might be 50 nm thick. Converting to centimeters: . Wavelength of Light: Blue light has a wavelength of approximately 450 nm. Converting to centimeters: . Interesting Facts The nanometer scale is crucial in nanotechnology and materials science. Many modern technologies, such as semiconductor manufacturing, rely on precise control at the nanometer level. The ability to create structures and devices at this scale has led to advances in electronics, medicine, and various other fields. Richard Feynman, a Nobel laureate in Physics, gave a famous lecture in 1959 titled "There's Plenty of Room at the Bottom," which is considered one of the inspirations for the field of nanotechnology. He discussed the possibility of manipulating individual atoms and molecules to create new materials and devices. See below section for step by step unit conversion with formulas and explanations. Please refer to the table below for a list of all the Centimeters to other unit conversions. What is Nanometers? A nanometer is a unit of length in the metric system, crucial for measuring extremely small distances. It's widely used in nanotechnology, materials science, and other fields dealing with nanoscale phenomena. Definition and Formation A nanometer (nm) is equal to one billionth of a meter. The prefix "nano-" comes from the Greek word "νᾶνος" (nanos), meaning dwarf. It indicates a factor of . So, when we say something is a nanometer in size, we mean it's incredibly tiny. Connection to Light and Wavelengths Light's wavelength is frequently measured in nanometers. The range of visible light, for instance, falls between 400 nm (violet) and 700 nm (red). The color of light we perceive is determined by its wavelength in this range. Applications and Examples Nanotechnology: A primary field using nanometers, designing and manipulating materials and devices at the atomic and molecular level. For example, transistors in modern CPUs are measured in nanometers (e.g., 5nm, 3nm process). Materials Science: Characterizing the size of nanoparticles and thin films. For example, the thickness of graphene, a single layer of carbon atoms, is about 0.34 nm. Biology: Measuring the size of viruses, DNA, and other biological structures. For instance, the diameter of a DNA molecule is roughly 2 nm. Manufacturing: Fabricating microchips and other nanoscale devices. For example, Extreme Ultraviolet (EUV) lithography uses light with a wavelength of 13.5 nm to create intricate patterns on microchips. Key Figures and Laws While there isn't a single law named after nanometers, the field is deeply intertwined with quantum mechanics and materials science. Scientists like Richard Feynman, with his famous 1959 lecture "There's Plenty of Room at the Bottom," helped inspire the field of nanotechnology. His ideas on manipulating individual atoms and molecules laid the groundwork for much of the nanoscale research happening today. Interesting Facts A human hair is about 80,000-100,000 nm wide. Nanomaterials can exhibit unique properties compared to their bulk counterparts due to quantum mechanical effects and increased surface area. Nanoparticles are being explored for various applications, including drug delivery, solar cells, and catalysts. What is centimeters? Here's information about centimeters, suitable for inclusion on your website. What is Centimeters? Centimeters (cm) are a unit of length in the metric system. They are commonly used for everyday measurements and technical applications alike. Understanding their relationship to other units and their practical applications is key. Centimeter Definition and Formation A centimeter is defined as one-hundredth of a meter. The prefix "centi-" indicates a factor of . Therefore: The metric system, including centimeters, originated in France during the French Revolution in the late 18th century, aiming for a standardized and rational system of measurement. Relationship to Other Units Here's how centimeters relate to some other common units of length: Millimeter (mm): 1 cm = 10 mm Meter (m): 1 m = 100 cm Inch (in): 1 in = 2.54 cm (exactly) Foot (ft): 1 ft = 30.48 cm (exactly) Common Uses and Examples Centimeters are used in a variety of contexts: Clothing: Measuring body dimensions (e.g., waist, inseam) for clothing sizes. Construction: Measuring lengths of building materials, room dimensions. Electronics: Specifying the size of electronic components or device dimensions. Maps: Indicating scale on maps, representing distances on the ground. For example, a map might have a scale where 1 cm represents 1 kilometer. Everyday objects: The width of a standard pen is approximately 1 cm. A credit card is roughly 8.5 cm long and 5.4 cm wide. Medical field: Wound measurement and monitoring of growth. Notable Associations While no specific law is named after the centimeter, its importance stems from its place within the widely adopted metric system. The metric system's adoption has been a key factor in scientific progress, enabling standardized communication and calculations. The International System of Units (SI), which defines the meter and therefore the centimeter, is maintained by the International Bureau of Weights and Measures (BIPM). Complete Nanometers conversion table \| Convert 1 nm to other units \| Result \| --- \| \| Nanometers to Micrometers (nm to μm) \| 0.001 \| \| Nanometers to Millimeters (nm to mm) \| 0.000001 \| \| Nanometers to Centimeters (nm to cm) \| 1e-7 \| \| Nanometers to Decimeters (nm to dm) \| 1e-8 \| \| Nanometers to Meters (nm to m) \| 1e-9 \| \| Nanometers to Kilometers (nm to km) \| 1e-12 \| \| Nanometers to Mils (nm to mil) \| 0.00003937008 \| \| Nanometers to Inches (nm to in) \| 3.937008e-8 \| \| Nanometers to Yards (nm to yd) \| 1.0936133333333e-9 \| \| Nanometers to US Survey Feet (nm to ft-us) \| 3.2808334383331e-9 \| \| Nanometers to Feet (nm to ft) \| 3.28084e-9 \| \| Nanometers to Fathoms (nm to fathom) \| 5.4680666666667e-10 \| \| Nanometers to Miles (nm to mi) \| 6.2137121212121e-13 \| \| Nanometers to Nautical Miles (nm to nMi) \| 5.3995641955722e-13 \| Length conversions Nanometers to Micrometers (nm to μm) Nanometers to Millimeters (nm to mm) Nanometers to Centimeters (nm to cm) Nanometers to Decimeters (nm to dm) Nanometers to Meters (nm to m) Nanometers to Kilometers (nm to km) Nanometers to Mils (nm to mil) Nanometers to Inches (nm to in) Nanometers to Yards (nm to yd) Nanometers to US Survey Feet (nm to ft-us) Nanometers to Feet (nm to ft) Nanometers to Fathoms (nm to fathom) Nanometers to Miles (nm to mi) Nanometers to Nautical Miles (nm to nMi)
18182	https://www.youtube.com/watch?v=sGb_c00wCa8	Given f(x)=2x+1 and g(x)=x^(2)+2x-1, find (f-g) (x). Then evaluate the difference when x = 2. \| ... Doubtnut 3940000 subscribers 2 likes Description 171 views Posted: 4 Jan 2023 Given f(x)=2x+1 and g(x)=x^(2)+2x-1, find (f-g) (x). Then evaluate the difference when x = 2. Class: 14 Subject: MATHS Chapter: FUNCTIONS Board:GOVT You can ask any doubt from class 6-12, JEE, NEET, Teaching, SSC, Defense and Banking exam on Doubtnut App or You can Whatsapp us at - 8400400400 Link - Join our courses to improve your performance and Clear your concepts from basic for Class 6-12 School and Competitive exams (JEE/NEET) - Contact Us: 👉 Have Any Query? Ask Us. 🤙 Call: 01247158250 💬 WhatsApp: 8400400400 📧 Email: info@doubtnut.com 🌐 Website: Welcome to Doubtnut. Doubtnut is World’s Biggest Platform for Video Solutions of Physics, Chemistry, Maths and Biology Doubts with over 5 million+ Video Solutions. Doubtnut is a Q&A App for Maths, Physics, Chemistry and Biology (up to JEE Advanced and NEET Level), Where You Can Ask Unlimited Questions by Clicking a Picture of Doubt on the Doubtnut App and Get Instant Video Solution. Subscribe Our YouTube Channels: ✿ Doubtnut: ✿ Class 11-12, JEE & NEET (Hindi): ✿ Class 11-12, JEE & NEET (English):: ✿ Class 6-10 (Hindi): ✿ Class 6-10 (English): ✿ Doubtnut Govt. Exams: Follow Us: 🔔 Facebook: 🔔 Instagram: 🔔 Telegram: 🔔 Twitter: 🔔 LinkedIn: Our Telegram Pages: 🔔 Doubtnut Official: 🔔 Doubtnut IIT JEE: 🔔 Doubtnut NEET: 🔔 Doubtnut CBSE Boards: 🔔 Doubtnut UP Boards: 🔔 Doubtnut Bihar Boards: 🔔 Doubtnut Government Exams: class 14 class 14 physics class 14 chemistry class 14 english class 14 maths class 14 biology cbse class 14 result class 14 economics class 14 accountancy class 14 syllabus physics cbse class 14 class 14 english grammar class 14 syllabus cbse class 14 history class 14 geography class 14 ncert class 14 syllabus cbse class 14 maths cbse class 14 english cbse class 14 physics cbse class 14 chemistry class 14 grammar cbse class 14 biology cbse class 14 commerce cbse class 14 accountancy class 14 syllabus chemistry class 14 grammar syllabus class 14 syllabus maths class 14 latest syllabus class 14 syllabus english class 14 syllabus biology class 14 syllabus of physics class 14 syllabus of chemistry class 14 syllabus science class 14 syllabus ncert class 14 syllabus of english class 14 syllabus commerce Transcript: foreign then evaluate the difference when X is equal to 2 so you have the value of x we need to find f of x minus G of x where X is equal to 2 okay let's begin f of x is equal to 2x plus 1 so F of 2 will be 2 multiplied by 2 plus 1 which is equal to 4 plus 1 5 F of 2 okay similarly G of X is equal to x square plus 2X minus 1. so G of 2 will be 2 whole Square plus 2 multiplied by 2 minus 1. whole Square sorry so this is 4 plus 4 minus 1 8 minus 1 is 7 okay now we need to find F of 2 minus G of 2 okay which is equal to 5 minus 7 minus 2 This Is The Answer hence the value of f minus G of X will be minus 2 thank you need IIT Jee Mains or Advanced levels is other students
18183	https://www.youtube.com/watch?v=eth3BtfKo3s	Recurrence Relations for Recursive Sequences in Discrete Math Intermation 38800 subscribers 61 likes Description 2527 views Posted: 13 Jul 2021 Calculating the elements of a sequence using an expression based on the element's position in the sequence is a straightforward process. Sometimes, however, elements of a sequence are best computed using one or more previous terms. In this lesson, we discuss the benefits and drawbacks of these recursive sequences and how to prove the recurrence relation is true. We also get a glimpse at the Method of Differences. Timestamps 00:00 \| Intro 00:14 \| Spreadsheet Example 06:57 \| Converting Spreadsheet Example to a Recursive Sequence 09:36 \| Fibonacci Sequence Recurrence Relation 11:54 \| Factorial Sequence as a Recursive Sequence 13:20 \| Potential Computational Savings with Recursive Sequence 16:47 \| Method of Differences Recursive Sequence 18:53 \| Proving the Recurrence Relation Hashtags recurrence #recursive #sequence Transcript: Intro i think that frank bunker gilbreth had me in mind when he came up with the saying give the hardest problems to the laziest people and they'll figure out the easiest way to do them [Music] Spreadsheet Example one of the face-to-face courses that i teach comes on tuesdays and thursdays and at the beginning of the semester when i'm coming up with the syllabus i usually put it into a spreadsheet and come up with the days and what the reading is and the topic and the assignments and so forth so i can have a schedule for the students so i bring up the spreadsheet and i start doing things like okay on the 20 i don't know maybe maybe this semester starts on the 25th of august i don't know so we have 25 august that'd be a tuesday so 27 august would be a thursday which would mean we'll crime any does august have 31 or 30 30 days half sept oh it may i don't know maybe it's 31 days so right so that would be september first would be i have to add five to get to the next tuesday and then you add two to get to the thursday and then you add five to get to the next tuesday and you get the idea right uh i think i've got that right i don't know first three weeks heck the semester's a little bit longer than that so what i want to do is i want to take advantage of the spreadsheet i want to take advantage of my laziness and i want to come up with a formula now when we were talking about explicit sequences we talked about a sequence where we had a sub n where a sub n was defined in terms of n so you had an expression some sort of an expression that said okay for a sub 5 it's equal to and there was an expression that had only n as a parameter so let's see if we can't figure out an explicit express an explicit sequence that defines the days for my syllabus and so we've got this starting day this and we're going to call that starting day a constant so every semester we'll come up with a different starting day you know 25th of august is probably not the starting day for this particular semester but on 27th of august that's equal to 25 august plus two right so we have a tuesday and a thursday but this one september this is equal to 25 august plus seven and then we've got 25 august plus 9 and then 25 august plus 14 and then 25 august plus 16. and hopefully that will give us the right expressions you know this is not really going to work very well uh and well it's going to be difficult to come up with an expression that lends itself to just terms of n but let's go ahead and try so what we've got here is we've got the tuesday thursday tuesday thursday tuesday thursday the tuesdays are pretty easy it looks like every tuesday is a multiple of seven added to the 25th of august so what we've got is our n right and so this is like this is an n we'll just put n here so this guy is n of zero uh one two three four five so there's our n's now how can we come up with something that represents the tuesdays that that will be based on this n well if you divide n by two you're going to get the week number so this is week zero this is week one this is week two and so and n divided by two will give you what week it's in well for tuesday we want to have one week added for the n equals two for the tuesday for n equals four we wanna have two weeks added so it's going to be for example two divided by two that's one times seven will tell us how much we need to add to twenty-five when when n is equal to four four divided by two that's two so we have two weeks or two times seven so what i've got is whenever i'm on a tuesday i want to take n divided by 2 and multiply it by 7 to get this number here so let's just go ahead and this will give us a way to calculate the tuesdays based on and in fact you could even do it up on this guy so this is 2 5 august plus 0. and so let's see we can actually take this and simply put plus n divided by two times seven n divided by two times seven n divided by two times seven that gives us the tuesdays but what about the thursdays you know we're not going to be able to use this same expression because it's going to have to add another 2 to it plus n divided by 2 is going to give you like for example in the case of n is equal to 1 you're going to get 0.5 so 0.5 times 7 not quite right so for thursdays what we're going to have is n minus 1 divided by 2 times seven plus two this is really getting ugly isn't it so whenever n is equal to one this becomes zero divided by two zero times seven is zero plus two there's your plus two right and then whenever n is equal to three we've got three minus one that's two two divided by two is one so it's one times seven plus two which is nine so it looks like this expression works and so for the thursdays we've got n minus 1 divided by 2 times 7 plus 2. so we've got n minus 1 divided by 2 times 7 plus 2 you know this explicit sequence doesn't really look like a lot of fun so let's come up with something a little bit easier to represent or an easier way to calculate out what each one of these values is Converting Spreadsheet Example to a Recursive Sequence now we're not necessarily going to worry about the ends in an explicit way what we're going to do is use them as a reference to go back and calculate a previous calculate our current value from previous value for example this date here 1 september is actually 7 added to this one 3 september is 7 added to 27 august 8th september is 7 added to 1 september 10 september is 3 september plus 7. so all i really need to do is simply say okay i have determined that these are the starting dates so we so we go ahead and just determine initial values and it may be just one of them it may be two of them maybe three of them but there's just a couple of them at the beginning that we're going to determine so once we've determined our initial values this guy right here is a sub 0. this guy is a sub 1 this is a sub 2 a sub 3 a sub 4 a sub 5. and so all we need to do is say okay a sub 2 that's equal to a sub 0 plus 7 it's equal to this week august 25th plus 7. a sub 3 adds 7 to a sub 1. so we have a sub 1 plus 7. a sub 4 is equal to a sub 2. it's one week after this state right here so this tuesday the following tuesday is just 7 added to a sub 2. a sub 5 is equal to a sub 3 plus 7 and we have a very consistent very easy way in order to determine what our next a sub whatever is the problem is as you determine initial values and in order to get to a sub 5 you will have to have determined a sub 3. this is called a recurrence relation for a sequence and basically what it means is that a sub n is determined based on the values of any combination of a sub 0 all the way up to a sub n minus 1. it requires initial values however let's take a look at some more recursive sequences Fibonacci Sequence Recurrence Relation now if you spent any time in computing you have probably seen this sequence called the fibonacci sequence it's probably one of the most popular or well-known recurrence relations for a sequence that there is out there now it starts out with two initial values a sub zero is set up to zero and a sub one is set up to 1. those are your initial values now the recurrence relation is that a sub n is equal to a sub n minus 1 plus a sub n minus 2. now this sequence may seem simple to you but its currents in nature is just incredible you'll see it everywhere in terms of the way petals on plants come out or flowers in terms of like the sizes of the segments of a of a of some sort of shell seashells and so forth so it's really cool that we have this sequence so that we can kind of examine nature in a mathematical way but the sequence goes like this so you have 0 1 and then the next one one plus zero is one one plus one is two two plus one is three two three plus two is five then you have three plus five is eight and then eight plus five is thirteen and then eight plus uh is 21 and so on now there is a problem if i want to figure out a sub let's say a sub 42 a sub 42 in order to calculate that in the fibonacci sequence and there are ways to create an explicit formula an explicit formula based only on n for the fibonacci sequence but basically what you're looking at is a sub 41 plus a sub 40 right well in order to determine these two what do you have to do you have to determine the previous you know the the the 39 and the 38 and the 37 all the way down until you get to the predefined values that may or may not take more time than trying to determine based on the explicit sequence but let's try another sequence Factorial Sequence as a Recursive Sequence earlier in this series of lessons we talked about something called a factorial so n factorial is equal to n times n minus 1 times n minus 2 times all the way down until you get to times two right actually it's times one also but one times anything is itself right so if i've got say four factorial that's equal to four times three times two times 1 and once again the 1 is not necessary necessarily necessary but so we have 4 times 3 is 12 times 2 is 24. all right now if i were to have the factorial the the the explicit sequence that's based on the factorial i could have something like a sub n is equal to n factorial so we would get this sequence so 1 factorial is 1 2 factorial is 2 3 factorial is 6 4 factorial is 24 and so on now it turns out that we can have so this was the explicit right turns out we can have a recursive relation too and so a sub n is equal to we'll just say n times a sub n minus 1. that means that if we know a sub n minus 1 we can calculate a sub n so this is the recursive all right now there is a drawback as we Potential Computational Savings with Recursive Sequence said before for the recursive i in order to calculate a sub n i have to calculate all a sub n minus 1 elements there is a benefit however let's take a look at this mathematically if i want to calculate assume we assume we need the first 10 elements of this sequence now i realize that that what i'm about to do is very inefficient but let's just talk about this from the point of view of the computer if i were to calculate this out just using the explicit sequence just using the explicit sequence then a sub and and what i'm looking at now is the number of computations or specifically the number of multiplications so we'll actually change that and we'll just say the number of multiplications well if i were to do this explicitly then factorial of one is no multiplications factorial of two no multiplications factorial of three we have one multiplication factorial of four we have two multiplications factorial well well yeah it's four times three times two that that's um two multiplications we don't really have to worry about the one but then we have five six seven eight nine 10. so the total number of multiplications if we figured out each one of the elements of the sequence where the sequence is equal to the sequence of factorials if we used an explicit operation right we used explicitly figured out how to do that computation you have 1 plus 2 is 3 plus 3 is 6 10 15 21 28 36 this equals 36 multiplications all right now what if we did it so this guy right here is the explicit what if we did it with the recursive well with recursive what you've got is i need no multiplications for one factorial no multiplications for two factorial one multiplication for three factorial but then when i go to 4 factorial i use the result from 3 factorial multiply it once by 4 i go to the 5 factorial multiply it once 4 5 then go to 6 i need to only multiply so i'm only doing one multiplication for each one of these elements of the sequence past two so i have three four five six seven eight nine ten that is eight multiplications all right significantly fewer that's that's that's more than 75 percent fewer multiplications whenever i do it recursively alright so we've actually got a savings in terms of the number of multiplications that are required in order to determine a sequence for the factorial Method of Differences Recursive Sequence one more quick example here there is something that many years ago in mathematics that was developed called the method of differences charles babbage used this for his first machine in the 1800s use something called the method of differences in order to determine powers of numbers and so i've got a sub n is equal to n squared all right that sequence is pretty easy to figure out so a sub n is equal to n squared let's just start at one let's start at n equals one so n equals one let's just have n and then n squared we have one 2 3 4 5 6 7 and then n squared is 1 4 9 16 25 36 49 right all right now it turns out the method of differences allows us to do something that only requires addition and subtraction in order to determine the sequence and in that case the explicit c excuse me the recursive sequence becomes a sub n is equal to 2 times a sub n minus 1. so 2 times the previous element minus a sub n minus 2 plus 2. and if you don't believe me well let's take a look let's see we're trying to figure out what a sub 5 is a sub 5 would be 5 squared so we're looking for this 25 can we work it out with this recursive expression here so 2 times a sub 4 2 times 16 32 minus a sub and so that would be a sub 3 so minus 9 that would give us what 23 plus 2 that would give us 25. so it turns out that it looks like at least for a sub 5 that this expression works but does it work for everything well what i'd like to do is Proving the Recurrence Relation simply use this expression right here and substitute it in for the a sub n minus 1 and the a sub n minus 2 to see if we can get what a sub n is equal to so we've got um let's just start out this expression 2 times a sub n minus 1 minus a sub n minus 2 plus 2. so a sub n minus 1 well that's equal to n minus 1 squared right minus uh a sub n minus 2 that's n minus 2 squared plus 2. let's go ahead and square these two terms here so we've got n squared minus two n plus one minus and we'll square that guy we've got n squared minus four n plus four and then we have the plus two hanging out there at the end multiply the 2 through we get 2 n squared minus 4 n plus 2 minus n squared and then we we distribute the minus plus 4 n minus 4 and then plus 2. all right now what we can see is that right up front we've got minus n squared combined with 2n squared gives us just n squared so that takes care of those two and then we've got minus 4m plus 4n these two cancel each other out and then we've got plus 2 minus 4 that's negative 2 plus 2 that's equal to 0. looks like it all reduced down to n squared so a sub n is equal to n squared that was our original that was our original explicit definition for the sequence so there you go we've done a couple of things here first of all i've shown you kind of what a recursive a recurrence relation for a sequence is how you develop a recurrence relation for a sequence how it differs from an explicit what its requirements are what its drawbacks are one of the benefits is that we can actually determine if we're going to have to determine the whole sequence anyway we can use it in order to determine uh things with fewer fewer operations in many cases and i've also shown how if you have both the explicit and the recursive how you can prove they're true if you stick around you may learn some other ways that we are using nowadays in order to improve the performance of our computing solutions
18184	https://en.wikipedia.org/wiki/Tangent%E2%80%93secant_theorem	Tangent–secant theorem - 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 References 2 External links Tangent–secant theorem [x] 8 languages Deutsch Ελληνικά Español Français Italiano Magyar Română தமிழ் 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 Wikidata item From Wikipedia, the free encyclopedia Geometry theorem relating line segments created by a secant and tangent line Beginning with the alternate segment theorem, ⟹∠P G 2 T=∠P T G 1⟹△P T G 2∼△P G 1 T⟹\|P T\|\|P G 2\|=\|P G 1\|\|P T\|⟹\|P T\|2=\|P G 1\|⋅\|P G 2\|{\displaystyle {\begin{array}{cl}\implies &\angle PG_{2}T=\angle PTG_{1}\[4pt]\implies &\triangle PTG_{2}\sim \triangle PG_{1}T\[4pt]\implies &{\frac {\|PT\|}{\|PG_{2}\|}}={\frac {\|PG_{1}\|}{\|PT\|}}\[2pt]\implies &\|PT\|^{2}=\|PG_{1}\|\cdot \|PG_{2}\|\end{array}}} In Euclidean geometry, the tangent-secant theorem describes the relation of line segments created by a secant and a tangent line with the associated circle. This result is found as Proposition 36 in Book 3 of Euclid's Elements. Given a secant g intersecting the circle at points G 1 and G 2 and a tangent t intersecting the circle at point T and given that g and t intersect at point P, the following equation holds: \|P T\|2=\|P G 1\|⋅\|P G 2\|{\displaystyle \|PT\|^{2}=\|PG_{1}\|\cdot \|PG_{2}\|} The tangent-secant theorem can be proven using similar triangles (see graphic). Like the intersecting chords theorem and the intersecting secants theorem, the tangent-secant theorem represents one of the three basic cases of a more general theorem about two intersecting lines and a circle, namely, the power of point theorem. References [edit] S. Gottwald: The VNR Concise Encyclopedia of Mathematics. Springer, 2012, ISBN9789401169820, pp. 175-176 Michael L. O'Leary: Revolutions in Geometry. Wiley, 2010, ISBN9780470591796, p. 161 Schülerduden - Mathematik I. Bibliographisches Institut & F.A. Brockhaus, 8. Auflage, Mannheim 2008, ISBN978-3-411-04208-1, pp.415-417 (German) External links [edit] Tangent Secant Theorem at proofwiki.org Power of a Point Theorem auf cut-the-knot.org Weisstein, Eric W."Chord". 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18185	https://math.stackexchange.com/questions/3972995/a-sequence-where-the-nth-term-is-the-cumulative-sum-over-n-1	A sequence where the nth term is the cumulative sum over n-1 - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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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 A sequence where the nth term is the cumulative sum over n-1 Ask Question Asked 4 years, 8 months ago Modified4 years, 8 months ago Viewed 154 times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. Consider the following sequence: u n=−u 1 n−β γ n(u 1+...+u n−1)u n=−u 1 n−β γ n(u 1+...+u n−1), where u 1 u 1, β β and γ γ are some none zero constant. In this sequence, can I obtain the expression for u n u n as a function of u 1 u 1? If I can, what would it be? At first, it seems pretty straightforward but I couldn't get a nice closed form. sequences-and-series Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications edited Jan 4, 2021 at 21:39 SSPSSP asked Jan 4, 2021 at 21:07 SSPSSP 13 3 3 bronze badges 2 Hint: (n+1)u n+1−n u n=?(n+1)u n+1−n u n=?achille hui –achille hui 2021-01-04 21:47:41 +00:00 Commented Jan 4, 2021 at 21:47 2 Why give us two constants β,γ β,γ when one will do?TonyK –TonyK 2021-01-04 23:14:06 +00:00 Commented Jan 4, 2021 at 23:14 Add a comment\| 2 Answers 2 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. Clearly your recurrence relation holds for only for n>1.n>1. Otherwise at n=1,n=1, we get u 1=−u 1 u 1=−u 1 and consequently u n=0 u n=0 for all n∈N.n∈N. In particular at n=2,n=2, we have u 2=−u 1 2(1+β γ).u 2=−u 1 2(1+β γ). Thanks for Achille hui to pointing this in comments. There are few different ways that you can look at this question. One way is note that the quantity n u n+β γ(u 1+⋯+u n−1)=−u 1 n u n+β γ(u 1+⋯+u n−1)=−u 1 is a constant. By rewriting the same equation for n+1,n+1, we get (n+1)u n+1+β γ(u 1+⋯+u n−1+u n)=n u n+β γ(u 1+⋯+u n−1),(n+1)u n+1+β γ(u 1+⋯+u n−1+u n)=n u n+β γ(u 1+⋯+u n−1), which simplifies to u n+1=(n−β γ n+1)u n∀n≥2.u n+1=(n−β γ n+1)u n∀n≥2. Clearly if u 1=0 u 1=0 then the sequence is trivial. if β γ β γ is an integer other than 1,1, then all u β γ+1=u β γ+2=⋯=0.u β γ+1=u β γ+2=⋯=0. Otherwise u n+1=u 2 u 3 u 2⋯u n u n−1 u n+1 u n=u 2∏k=2 n(k−β γ k+1)=−u 1(1+β γ)(2−β γ)⋯(n−β γ)(n+1)!u n+1=u 2 u 3 u 2⋯u n u n−1 u n+1 u n=u 2∏k=2 n(k−β γ k+1)=−u 1(1+β γ)(2−β γ)⋯(n−β γ)(n+1)! for all n≥2.n≥2. In fact, this last formula contain above two remarks as special cases. Another way to solve this recurrence is compute first few terms of the sequence by hand and identify a pattern among them. Then we can use mathematical induction to justify our pattern. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Jan 5, 2021 at 0:30 answered Jan 4, 2021 at 23:00 BumblebeeBumblebee 19k 5 5 gold badges 53 53 silver badges 94 94 bronze badges 0 Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. Here is an approach via generating functions (though it mostly proves that this method is not always the best one!). We define U(x):=∑∞n=1 u n x n U(x):=∑n=1∞u n x n, and for convenience I'll introduce r:=β γ r:=β γ as well. To obtain a relation on this GF, we first rearrange the recurrence as u n+u 1+r(u 1+⋯+u n−1)=0.u n+u 1+r(u 1+⋯+u n−1)=0. We now multiply both sides of the recurrence by x n−1 x n−1 and sum from n=2 n=2 to infinity to obtain ∑n=2∞n u n x n−1+u 1∑n=2∞x n−1+∑n=2∞r(u 1+⋯+u n−1)x n−1=0.∑n=2∞n u n x n−1+u 1∑n=2∞x n−1+∑n=2∞r(u 1+⋯+u n−1)x n−1=0. We deal with these sums one-by-one. For the first, we use the power rule for derivatives to write ∑n=2∞n u n x n−1=d d x∑n=2∞n u n x n=d d x[U(x)−u 1 x]=U′(x)−u 1.∑n=2∞n u n x n−1=d d x∑n=2∞n u n x n=d d x[U(x)−u 1 x]=U′(x)−u 1. For the second, we have the geometric sum u 1∑n=2∞x n−1=u 1 x 1−x.u 1∑n=2∞x n−1=u 1 x 1−x. For the third, we may rewrite the series as ∑n=2∞r(u 1+⋯+u n−1)x n−1=∑n=1∞r(u 1+⋯+u n)x n=∑n=1∞∑k=1 n r u k x n=∑1≤k≤n≤∞r u k x n=∑k=1∞∑n=k∞r u k x n=∑k=1∞r u k x k 1−x=r 1−x U(x)∑n=2∞r(u 1+⋯+u n−1)x n−1=∑n=1∞r(u 1+⋯+u n)x n=∑n=1∞∑k=1 n r u k x n=∑1≤k≤n≤∞r u k x n=∑k=1∞∑n=k∞r u k x n=∑k=1∞r u k x k 1−x=r 1−x U(x) Therefore we have the differential equation U′(x)−u 1+u 1 x 1−x+r 1−x U(x)=0.U′(x)−u 1+u 1 x 1−x+r 1−x U(x)=0. This may be solved by seeking an appropriate integrating factor. Indeed, we have (1−x)−r d d x[(1−x)r U(x)]=U′(x)+r 1−x U(x)=u 1−u 1 x 1−x=u 1 1−2 x 1−x(1−x)−r d d x[(1−x)r U(x)]=U′(x)+r 1−x U(x)=u 1−u 1 x 1−x=u 1 1−2 x 1−x We move (1−x)−r(1−x)−r over, integrate both sides with base point x=0 x=0, and move (1−x)r(1−x)r over. (Note that U(0)=0 U(0)=0.) We obtain U(x)=(1−x)−r u 1∫0(1−2 x)(1−x)r−1 d x=u 1(1+r)−2 r x−(1+r)(1−x)r r(r−1).U(x)=(1−x)−r u 1∫0(1−2 x)(1−x)r−1 d x=u 1(1+r)−2 r x−(1+r)(1−x)r r(r−1). If we fearlessly expand this using the binomial expansion, we obtain U(x)=u 1 r(r−1)[(1+r)−2 r x−(1+r)∑k=0∞(r k)(−x)k]=u 1[x+∑k=2∞(r k)(−x)k]U(x)=u 1 r(r−1)[(1+r)−2 r x−(1+r)∑k=0∞(r k)(−x)k]=u 1[x+∑k=2∞(r k)(−x)k] Hence we finally conclude that, for n≥2 n≥2, we have u n=u 1(−1)n(r n)=u 1(−1)n r(r−1)⋯(r−n+1)n!=−u 1 r(1−r)⋯(n−1−r)n!u n=u 1(−1)n(r n)=u 1(−1)n r(r−1)⋯(r−n+1)n!=−u 1 r(1−r)⋯(n−1−r)n! which agrees with the (much shorter) deduction by Bumblebee. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Jan 5, 2021 at 7:26 SemiclassicalSemiclassical 18.7k 4 4 gold badges 39 39 silver badges 99 99 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 sequences-and-series See similar questions with these tags. 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18186	https://www.pnas.org/doi/10.1073/pnas.2412493121	Origin and evolution of auxin-mediated acid growth \| PNAS ARTICLESCurrent IssueLatest ArticlesSpecial FeaturesList of IssuesPNAS Nexus Front Matter AUTHORSInformation for AuthorsEditorial and Journal PoliciesSubmission ProceduresPublication Charges Topics Physical Sciences Social Sciences Biological Sciences Featured Topics Physics Chemistry Sustainability Science Sustainable Development Goals Articles By Topic Applied Mathematics Applied Physical Sciences Astronomy Biophysics and Computational Biology Computer Sciences Earth, Atmospheric, and Planetary Sciences Engineering Environmental Sciences Mathematics Statistics Featured Topics Anthropology Sustainability Science Sustainable Development Goals Articles By Topic Demography Economic Sciences Environmental Sciences Political Sciences Psychological and Cognitive Sciences Social Sciences Featured Topic Anthropology Sustainability Science Sustainable Development Goals Articles By Topic Agricultural Sciences Applied Biological Sciences Biochemistry Biophysics and Computational Biology Cell Biology Developmental Biology Ecology Environmental Sciences Evolution Genetics Immunology and Inflammation Medical Sciences Microbiology Neuroscience Pharmacology Physiology Plant Biology Population Biology Psychological and Cognitive Sciences Systems Biology 0 Cart Sign inRegister Individual LoginInstitutional Login Submit Quick Search anywhere Enter journal, DOI, article type, keywords, authors, etc. Quick search in Citations Journal Year Volume Issue Page/eLocation ID Required field Quick Search in Journals Enter journal, DOI, article type, keywords, authors, etc. Quick Search in Journals Enter journal, DOI, article type, keywords, authors, etc. Searching: Anywhere AnywhereCitationPNAS NexusProceedings of the National Academy of Sciences Advanced SearchSearch SUGGESTED SEARCHES: Covid-19Artificial IntelligenceMpoxClimate ChangeGun Violence Sign inRegister Individual LoginInstitutional Login Quick Search anywhere Enter journal, DOI, article type, keywords, authors, etc. ARTICLES Current Issue Latest Articles Special Features List of Issues PNAS Nexus Front Matter AUTHORS Information for Authors Editorial and Journal Policies Submission Procedures Publication Charges Topics ContactSite MapTerms & Privacy PolicyAccessibility backPhysical Sciences Featured Topics Physics Chemistry Sustainability Science Sustainable Development Goals Articles By Topic Applied Mathematics Applied Physical Sciences Astronomy Biophysics and Computational Biology Computer Sciences Earth, Atmospheric, and Planetary Sciences Engineering Environmental Sciences Mathematics Statistics backSocial Sciences Featured Topics Anthropology Sustainability Science Sustainable Development Goals Articles By Topic Demography Economic Sciences Environmental Sciences Political Sciences Psychological and Cognitive Sciences Social Sciences backBiological Sciences Featured Topic Anthropology Sustainability Science Sustainable Development Goals Articles By Topic Agricultural Sciences Applied Biological Sciences Biochemistry Biophysics and Computational Biology Cell Biology Developmental Biology Ecology Environmental Sciences Evolution Genetics Immunology and Inflammation Medical Sciences Microbiology Neuroscience Pharmacology Physiology Plant Biology Population Biology Psychological and Cognitive Sciences Systems Biology backTopics Physical Sciences Social Sciences Biological Sciences Reference #1 Research Article Plant Biology Free access Share on Origin and evolution of auxin-mediated acid growth Hai Yue Zeng, Shiyu Deng, Congcong Jin, +10, Zhiyun Shang, Le Chang, Jiajun Wang, Xue Han Ao Wang, Dan Jin, Yubo Wang, Hang He, Lanxin Li Xing Wang Deng and Ning Weiweining@swu.edu.cn-10Authors Info & Affiliations Contributed by Xing Wang Deng; received June 21, 2024; accepted October 28, 2024; reviewed by Daniel J. Cosgrove, Richard A. Dixon, and Loren Rieseberg December 10, 2024 121 (51) e2412493121 2,738 4 Metrics Total Views 2,738 Last 12 Months 2,738 Total Citations 4 Last 12 Months 4 View all metrics Track CitationsAdd to Reading List PDFEPUB Contents Vol. 121 \| No. 51 Significance Abstract Results Discussion Materials and Methods Data, Materials, and Software Availability Acknowledgments Supporting Information References Information & Authors Metrics & Citations View Options References Figures Tables Media Share Significance Although the molecular mechanism underlying acid growth has been basically elucidated in Arabidopsis, its origin and scope in plants remain unclear. Here, we revealed that core genes of acid growth mainly originated in Charophyta and functionally evolved in land plants. However, we found that the PM H+-ATPase in Charophyceae algal Chara braunii is regulated by light instead of auxin, indicating that light control of PM H+-ATPase activity occurred before the regulation by auxin. In addition, we found that auxin elicits transcriptional response and cell elongation independently of canonical auxin receptor TRANSPORT INHIBITOR RESPONSE 1/AUXIN SIGNALING F-BOX (TIR1/AFB) and acid growth in C. braunii. Our study sheds light on how a major growth regulatory pathway such as acid growth might be acquired and developed during plant terrestrialization. Abstract The classical acid growth theory suggests that auxin stimulates cell expansion by triggering apoplast acidification via plasma membrane (PM)-localized H+-ATPase. Here, we reconstructed the origin and evolutionary history of auxin-mediated acid growth. Comparative phylogenomic analysis showed that most core components of acid growth originated in Charophyta and then underwent subclass expansion and functional innovation during plant terrestrialization. In Charophyceae algae Chara braunii, we found that PM H+-ATPase has formed a core regulatory module with TMK and PP2C.D, which can be activated by photosynthesis-dependent phosphorylation through light rather than auxin. Despite the lack of canonical auxin receptor TRANSPORT INHIBITOR RESPONSE 1/AUXIN SIGNALING F-BOX (TIR1/AFB), auxin elicits significant internodal elongation and transcriptional reprogramming in C. braunii, implying the existence of an ancient auxin-mediated growth mechanism. We propose that the evolution of acid growth represents a neofunctional adaptation to terrestrial environments, in which PM H+-ATPase in carbon concentrating for photosynthesis was utilized to acidify apoplast for cell expansion, and the core components responsible for acid growth eventually established a regulatory network in land plants by connecting with the TIR1/AFB pathway. Sign up for PNAS alerts. Get alerts for new articles, or get an alert when an article is cited. Manage alerts Auxin plays integral roles in plant growth, development, and morphogenesis, in part by regulating cell expansion. First proposed in the 1970s, the acid growth theory suggests that auxin activates the plasma membrane (PM)-localized P-type H+-ATPase, resulting in H+ efflux across the PM (1, 2). Subsequent apoplast acidification activates a series of pH-dependent cell wall remodeling proteins such as expansin (EXP), leading to cell wall loosening (3). Furthermore, increases in H+ efflux cause PM hyperpolarization, which drives solute absorption including K+ influx along with subsequent water uptake, resulting in high turgor pressure necessary for cell expansion (4). In recent decades, the molecular mechanism by which auxin regulates PM H+-ATPase has gradually been elucidated in the model plant Arabidopsis thaliana. Arabidopsis responds to auxin mainly through two pathways. The canonical auxin receptor TRANSPORT INHIBITOR RESPONSE 1/AUXIN SIGNALING F-BOX (TIR1/AFB) induces genome-wide transcriptional reprogramming by releasing the repression of AUXIN/INDOLE-3-ACETIC ACID (Aux/IAA) to transcriptional factors AUXIN RESPONSE FACTOR (ARF) (5). On the other hand, the cupin protein AUXIN-BINDING PROTEIN 1 (ABP1) or ABP1-LIKE (ABL) recognizes auxin in conjunction with the extracellular domain of TRANSMEMBRANE KINASE (TMK) at the PM, triggering rapid phosphoproteomic changes (6, 7). The above two respective signals direct D-CLADE TYPE 2C PROTEIN PHOSPHATASE (PP2C.D) and TMK to antagonistically regulate the phosphorylation status of the penultimate threonine (pT) on the C terminus of PM H+-ATPases (8, 9). When pT is phosphorylated, the 14-3-3 protein binds to this site and leads to the stimulation of PM H+-ATPase catalytic activity (10, 11). Additionally, in response to auxin, members of the SMALL AUXIN UP RNA (SAUR) family, induced by ARF-associated transcriptional factor complexes, interact with PP2C.D and alter phosphatase activity to indirectly regulate PM H+-ATPase, allowing the intensity control of apoplast acidification (9, 12) (SI Appendix, Fig. S1). Although much comprehension of auxin-mediated cell expansion has been derived from studies in Arabidopsis, it remains unclear when and how the overall mechanism originated and evolved. It is assumed that plant ancestors first arose in marine environments, later migrated to freshwater, and finally colonized dry land approximately 600 million years ago (13). During this evolutionary process, plants developed plastic and diversified morphology in part via spatial-temporal control of cell division and cell expansion through sophisticated phytohormone regulatory systems to adapt to environmental changes (14–16). Studies show that the intracellular auxin receptor TIR1/AFB is exclusively present in land plants (17–19). Although ARF originated from early-diverging charophytes, the auxin-response A/B-ARF classes only occur in land plants (20, 21). Nevertheless, the auxin efflux carrier PIN-FORMED (PIN) and the polar auxin transportation are widely detected in Charophyta (22–25), as well as extensive transcriptional reprogramming and phosphoproteomic responses induced by exogenous IAA were observed (26–28). Combined with the fact that auxin triggers many cellular responses in algae, including cell elongation, PM polarity regulation, and morphogenesis (28–33), it indicates that an ancient auxin system already existed before TIR1/AFBs. Elucidating the origin and evolution of auxin-mediated acid growth can enhance comprehension of plant terrestrialization at the molecular level. Presently, cell elongation through a conserved ethylene-mediated transcriptional regulation in charophytes has been considered an adaptive response in aquatic and semiaquatic environments that could facilitate plant terrestrialization (34, 35). Similarly, the application of auxin (IAA) affects cell elongation in Klebsormidium nitens (Klebsormidiophyceae) and some Chara species (Charophyceae) (26, 32, 36). However, how auxin regulates cell elongation in algae remains unclear. Apart from hormonal signal input, PM H+-ATPases as the key regulator of acid growth could be different between aquatic algae and land plants. PM H+-ATPases coexist with Na+ pump to maintain PM electric potential in algae, while those of land plants evolved with more complex C-terminal regulatory sites to respond to multiple signals and gradually lost Na+ pumps (37). Here, we reconstructed the evolutionary history of auxin-mediated acid growth via multispecies phylogenies across plant lineage, examining the sequence-based evolution and relationships of core components. The results showed that all the components of acid growth can be found in C. braunii of Charophyta, but they have not yet been constructed into an auxin-mediated regulatory pathway. Instead of auxin, the PM H+-ATPases are activated by light-mediated phosphorylation, which works for photosynthesis and possibly in light-stimulated cell expansion. Additionally, the auxin-induced internodal cell elongation and transcriptional reprogramming in C. braunii support the existence of an ancient auxin regulatory system. This study provides insights into the evolutionary history and the incremental steps to the formation of the auxin-mediated acid growth pathways that prevail in land plants. Results The Complete Set of Acid Growth Components Originated in Charophyta and Expanded in Land Plants. In this study, we focused on the acid growth induced by TIR1/AFB-Aux/IAA-ARF and ABP1/ABL-TMK pathways (SI Appendix, Fig. S1). We selected P-type (PM) H+-ATPase, PP2C.D, TMK, SAUR, EXP, and ABP1/ABL as core components of the acid growth response, then identified their orthologs based on sequence similarity and domain scan in 30 representative plant genomes with high-quality annotations, ranging from Rhodophyta, Chlorophyta, Charophyta, Bryophyta, Pteridophyta, Gymnospermae, and Angiospermae (Datasets S1 and S2). The genome-based phylogenetic tree depicts evolutionary relationships among the included species (Fig. 1 A). Fig. 1. Open in Viewer Origin and evolution of core components responsible for auxin-mediated acid growth. (A) Identification of the orthologs of core components responsible for acid growth in 30 representative plant species. Circle size represents gene copy number. The ancient whole-genome duplication/triplication events were labeled on the different branches of the phylogenetic tree based on previous reports, and the named duplication events are shown alongside their Greek letter (38, 39). (B) Gradual coevolution model of auxin-mediated acid growth. The emergence of prominent features across various evolutionary stages is illustrated. Among the selected core components of acid growth, PM H+-ATPase is the most conserved component present in all algae and land plants. Before TIR1/AFB emerged in the common ancestor of land plants (17), TMK, PP2C.D, and SAUR had originated in middle-diverging charophytes (Klebsormidiophyceae and Charophyceae). EXP and ABP1 can be traced to earlier Chlorophyta stages but are absent from early-diverging Charophyta (Mesostigmatophyceae and Chlorokybophyceae). The copy numbers of EXP expand at the Charophyta stage, while those of PM H+-ATPase, SAUR, and PP2C.D tend to expand in land plants, including in Marchantia polymorpha that has not undergone whole-genome duplication/triplication (WGD/WGT) events (38, 39). This suggests that these genes may be replicated through WGD/WGT and small-scale duplication events (40) and contribute to physiological complexity and morphological diversity in land plants after selection and retention. In contrast, the copy numbers of ABP1/ABL and TMK, which form the auxin coreceptor complex, are relatively stable (1 to 4), implying that they probably follow a dose balance to maintain functional stability (41). Land plants have all core components of auxin-mediated acid growth, including the ubiquitous presence of the TIR1/AFB-Aux/IAA-ARF auxin nuclear transcription pathway in land plants (17). The only exception is the absence of an orthologous ABP1/ABL in M. polymorpha (Fig. 1 A and Dataset S2). We identified the Charophyceae agal Chara braunii as the most primitive species where the acid growth pathway could originate because it possesses all core components of acid growth except TIR1/AFB (Fig. 1 B), suggesting that Charophyceae represents a critical stage for the origin of acid growth. The clonal strain NIES-1587 of C. braunii was subsequently utilized for further investigation. PM H+-ATPase of C. braunii Is Activated through Phosphorylation by Light but not Auxin. PM H+-ATPase serves as the cornerstone of acid growth. Among land plants, the pT site at the C-terminal autoinhibitory regulatory domain (R domain) is the key molecular switch for auxin-mediated acid growth (10). The phylogenetic tree of PM H+-ATPase was constructed based on the maximum likelihood method. It exhibited three high support subgroups (bootstrap > 80%) which reflect different pT distributions (Fig. 2 A). Subgroup-I consists of algae lacking pT, while subgroup-II encompasses both algae and bryophytes including K. nitens where pT-containing H+-ATPases first appeared. Subsequently, the pT-containing H+-ATPases expanded in later-diverging Zygnematophyceae within subgroup II, which ultimately became the predominant form among land plants in subgroup III. Further sequence alignment revealed that the occurrence of pT is accompanied by the 14-3-3 protein recognition motif YpTV (11), which initially emerged in K. nitens and was positively selected and retained in all Zygnematophyceae and land plants (SI Appendix, Fig. S2). These results suggest that the evolution of the pT site in PM H+-ATPase correlates with the transition from aquatic to terrestrial environments. Fig. 2. Open in Viewer Functional evolution of PM H+-ATPases from aquatic to terrestrial. (A) Maximum likelihood tree of P-type H+-ATPases isolated and reconstructed from a monophyletic group of similar-sequence proteins. AtAHAs and CbHAs are highlighted. Additional representative P-type H+-ATPases from fungi were regarded as outgroups. Branches with bootstrap values greater than 50 are displayed. (B) The phosphorylation level of PM H+-ATPase of C. braunii in response to IAA and white light. The microsomal fractions were extracted from 12-hour-dark-adapted C. braunii thalli, which were pretreated in the liquid mSWC-2 medium containing 0.1% dimethyl sulfoxide (DMSO)/1 μM IAA/10 μM DCMU for 1 h, then were illuminated with white light at 1000 lx or kept in the dark for 6 h. The immunoblot assay was analyzed on an Mn 2+-based phos-assay or a standard SDS-PAGE gel and probed with an anti-PM H+-ATPase antibody. The graph represents the phosphorylation level of PM H+-ATPase, quantified according to the signal intensity ratio of the phosphorylated H+-ATPase band to the sum of phosphorylated and unphosphorylated H+-ATPase bands. Values represent the mean and SEMs of three independent experiments. Paired t tests were used to calculate significant differences (P< 0.05). (C) PM H+-ATPase hydrolysis activities in C. braunii thalli under different treatments. Vanadate-sensitive ATP hydrolysis was measured by determining nicotinamide adenine dinucleotide (NADH) consumption. Values represent the means of three independent biological replications with SEMs. One-way ANOVA with Tukey’s test was used to calculate significant differences. Although the first pT-containing H+-ATPase was found in K. nitens, all PM H+-ATPase isoforms of C. braunii (CbHA1-4) belong to subgroup I and lack a pT site (Fig. 2 A and SI Appendix, Fig. S2). PM H+-ATPase of Chara were also insensitive to fusicoccin (42, 43), a fungal wilting toxin which activates the PM H+-ATPase by promoting 14-3-3 protein binding to the phosphorylated pT-containing YpTV motif. By RT-qPCR, we found that CbHA2 is most abundantly expressed among four CbHA isoforms (SI Appendix, Fig. S3 A), which implies this isoform may dominate the physiological effects in C. braunii. Unlike the structures of 10 transmembrane helices in most plant PM H+-ATPase (44), the transmembrane topology shows that CbHA2 lacks the first N-terminal transmembrane helix, which causes its N terminus to be localized in the extracellular space (SI Appendix, Fig. S3 B). This marked difference together with its dissimilar C-terminal intracellular region compared to land plants (45) possibly indicates a distinct regulatory mechanism. In Charophyceae algae, light induces the formation of alternating extracellular acid bands and base bands along the length of internodal cells by promoting the regional formation of their unique organelles at PM, known as charasomes. Charasomes are closely attached with a high density of PM H+-ATPases, which secrete H+ to acidify the apoplast. This process converts bicarbonate (HCO 3-) in the water into readily permeable carbon dioxide (CO 2) for photosynthesis (46–49). To better understand the function of PM H+-ATPase in Chara, we treated C. braunii under dark and white light conditions and extracted their microsome fractions to detect the phosphorylation level and ATP hydrolysis activity. The result showed that light increased the phosphorylation level and enhanced the activity of PM H+-ATPase of C. braunii, which can be inhibited by the photosynthesis inhibitor 3-(3,4-dichlorophenyl)-1,1-dimethylurea (DCMU) (Fig. 2 B and C), implying that level of photosynthesis may act as positive feedback to further promote carbon enrichment. In contrast, the application of IAA did not result in any detectable changes in phosphorylation or transcription level of CbHAs (Fig. 2 B and SI Appendix, Fig. S3 A). These results indicate that PM H+-ATPase was regulated by light and photosynthetic status, consistent with a role in carbon concentration of photosynthesis in Charophyceae, but was not regulated by auxin. TMK and PP2C.D Underwent Domain Recombination and Formed a PM H+-ATPase-TMK/PP2C.D Interacting Module in Charophyta. In Arabidopsis, TMK and PP2C.D constitute a pair of antagonistic kinases and phosphatases that activate and inhibit PM H+-ATPase via phosphorylation and dephosphorylation at pT, respectively. Together, PM H+-ATPase-TMK/PP2C.D forms a core module that functions as a main switch of H+ efflux into apoplast during acid growth. TMK belongs to the LEUCINE-RICH REPEAT RECEPTOR-LIKE KINASE (LRR-RLK) family, containing one signal peptide, multiple LRR repeats, one transmembrane region, and one kinase domain (50). We isolated monophyletic TMKs from candidate LRR-RLK proteins and detected a homologous protein in C. braunii (CbTMK-like) near the root of this monophyletic group (Fig. 3 A). Its N terminus is characterized by a long sequence, in which one region is annotated as the domain of unknown function 641 (DUF641). Except for lacking a signal peptide, other domains of CbTMK-like are consistent with the domain structure of complete TMKs (Fig. 3 A). Considering that members of the LRR-RLK family have frequently lost or gained domains during evolution (51), we suggest canonical TMKs could be evolved from such TMK-like protein through N-terminal disruption and the acquisition of a signal peptide. Fig. 3. Open in Viewer TMK and PP2C.D originated and formed a PM H+-ATPase-PP2C.D/TMK interacting module in Charophyta. (A) Maximum likelihood tree of TMKs and their domain architecture. Bootstrap values greater than 50 are indicated. (B) Maximum likelihood tree of the PP2C family and domain recombination events. Colors indicated by dots represent the ancestral species categories included in PP2C clades. PP2C.Ds are compared with PP2C.Cs. The table shows the distribution of PP2C.D-specific AMPC isoforms and their proportion in PP2C.Ds of each species. (C) Membrane-based yeast two-hybrid assays showing the interactions of CbTMKLC and CbPP2C.D with CbHA2. (D) Co-IP assays showing the association of the CbHA2 with CbTMKLC and CbPP2C.D in rice protoplasts transiently expressing indicated epitope-tagged fusion proteins. PP2C.D is the D-clade of the PP2C family with 12 main clades (A–L) (52). Phylogenetic analysis of PP2C indicated that PP2C.D and PP2C.C are the closest and most recent clades derived from Charophyta, whereas other subfamilies arose from Rhodophyta or Chlorophyta in earlier periods (Fig. 3 B). Notably, PP2C.C has conserved tandem PP2C domains in both charophytes and land plants, while many PP2C.Ds in charophytes and bryophytes are specifically combined with an ammonium transporter (AMT) domain in the N terminus, a construction that we have named AMPC (AMT-PP2C) (Fig. 3 B and SI Appendix, Fig. S4). Intriguingly, we noticed that some bacterial species possess similar protein structure comprising AMT with a PP2C-like domain, SpoIIE (a bacterial developmental phosphatase) (53) (Fig. 3 B and Dataset S3). Since AMT activity is regulated by C-terminal phosphorylation for both plants and bacteria (54, 55) and PP2C can dephosphorylate AMT (56), it seems possible that the AMPC/AMT-SpoIIE proteins may enable autoregulation of AMT activity by their phosphatase domain. It was found that plant-specific AMPC and bacterial-specific AMT-SpoIIE proteins are phylogenetically distinct in both full-length and individual domain sequences (SI Appendix, Fig. S5), suggesting that they emerged independently during plant and bacterial evolution, respectively. Importantly, PP2C.Ds in vascular plants completely lost the AMT domain, or the AMPC subtype lineage was lost, as PP2C.D with a single PP2C domain became the sole form (Fig. 3 B). Perhaps this is an important feature for PP2C.Ds, but not other PP2Cs, to become unique targets of regulation by SAUR proteins in vascular plants such as Arabidopsis. To test the formation of PM H+-ATPase-PP2C.D/TMK module in C. braunii, we examined the protein–protein interactions of CbPP2C.D (the single PP2C domain isoform of PP2C.Ds in C. braunii) and the CbTMK-like with CbHA2. Membrane-based yeast two-hybrid and co-immunoprecipitation (CO-IP) assays showed that both the C-terminal intracellular region of CbTMK-like (CbTMKLC) and CbPP2C.D physically interacted with CbHA2 (Fig. 3 C and D). These protein interaction results suggest that C. braunii not only contains primitive forms of PM H+-ATPase, TMK, and PP2C.D, these proteins interact in a way similar to a functional PM H+-ATPase-PP2C.D/TMK module of land plants. However, the phosphorylation site(s) in CbHA2 regulated by CbPP2C.D/TMK has not been identified. SAUR Family Originated in Charophyta, Expanded in Bryophyta, and Developed Functionally in Vascular Plants. The plant-specific SAUR family comprises crucial auxin-responsive factors that induce acid growth by interacting with and inhibiting PP2C.D (9). We found that the earliest orthologous SAUR was present in C. braunii (CbSAUR). Compared with AtSAURs, CbSAUR exhibits the highest homology (37% amino acid identity) to AtSAUR78, which has two highly homologous isoforms AtSAUR76 and AtSAUR77 (Fig. 4 A and SI Appendix, Fig. S6). CbSAUR and another charophyte SAUR (SmSAUR in S. muscicola) both have an extended N-terminal domain, but the sequences and secondary structures of their SAUR domains are consistent with those of AtSAUR78 subfamily. Based on phylogenetic analysis across the plant lineage, the SAUR family can be divided into three subgroups (A–C) consisting of different species distributions (Fig. 4 B and SI Appendix, Fig. S7). The unclustered C-SAUR occupies the basal position and is conserved in Charophyta and all land plants, while A-SAUR and B-SAUR are two monophyletic clades (bootstrap > 60%) and only present in vascular plants. Notably, in Arabidopsis, many members of the A-SAUR and B-SAUR subgroups (e.g., AtSAUR40 and AtSAUR19) physically interact with and inhibit PP2C.Ds (9). However, AtSAUR76-78 and CbSAUR, which belong to the C-SAUR subgroup, showed no interactions with PP2C.Ds in the yeast two-hybrid assays (Fig. 4 B). The specific functions and mechanisms of AtSAUR76-78 are still unclear. It is possible that the ancient C-SAURs including CbSAUR and AtSAUR76-78 probably do not directly regulate PP2C.D, and the regulatory relationship of the SAUR-PP2C.D complex was established later in evolution with the expansion of the SAUR family. Fig. 4. Open in Viewer Gene duplication–mediated neofunctionalization facilitates the establishment of acid growth in land plants. (A) Maximum likelihood tree of SAUR family. Bootstrap values are shown at key nodes. Colors indicated by dots represent the species categories included in SAUR family subgroups. (B) Nuclear-based yeast two-hybrid assays testing interactions between SAURs and PP2C.Ds. Plates were incubated for 24 h to visualize color differences. (C and D) Relative expression levels of CbSAUR and AtSAUR76-78 under treatment with 1 μM IAA or 5 ppm ethylene, as determined by RT-qPCR. RNA samples were extracted from 7-d-old Arabidopsis seedlings and vegetative stages of C. braunii thalli. Values are shown as means ± SEM; n = 3. Two-way ANOVA was used to calculate significant differences (P< 0.05) within each gene. (E) Maximum likelihood tree of the EXP superfamily in plants, bacteria, and fungi. The bootstrap values are shown for key nodes. Colors indicated by dots/branches represent the species categories included in each family. Species of Chlorophyta/Charophyta EXPs in the three clades are listed on the right. Notably, the transcriptional levels of AtSAUR76-78 and AtPP2C.D1 are regulated by ethylene (57–60), a phytohormone that formed a complete signaling pathway in C. braunii (22). We identified multiple binding sites for ethylene signaling transcriptional factors ETHYLENE-INSENSITIVE 3 (EIN3)/EIN3-LIKE (EIL) and downstream ETHYLENE RESPONSE FACTORS (ERFs) in the promoter regions of CbSAUR and CbPP2C.D (Dataset S4). The expression levels of CbSAUR and CbPP2C.D were unaffected by treatment with 1 μM IAA, but significantly down-regulated by treatment with 5 ppm ethylene (Fig. 4 C). Importantly, in wild-type Arabidopsis (Col-0), treatment with either 1 μM IAA or 5 ppm ethylene significantly up-regulated AtSAUR76 and AtSAUR77, but it down-regulated AtSAUR78. In the ein3 eil1 mutant, the ethylene responses of AtSAUR76 and AtSAUR77 were completely abolished, and the ethylene response of AtSAUR78 exhibited further downregulation. The IAA-induced and the basal expression levels of AtSAUR76, 77 were also diminished in ein3 eil1, while the IAA response in AtSAUR78 was revered (Fig. 4 D). These findings suggest that both CbSAUR and AtSAUR76-78 primarily respond to ethylene and that auxin regulates the expression of the AtSAUR78 subfamily in part through ethylene transcription factors EIN3/EIL1. Taken together, our results suggest that the conserved C-SAUR is an originally ancient subgroup ranging from Charophyta to land plants. This subgroup was first regulated by ancient ethylene signaling in Charophyta, then regulated also by auxin through cross talk with ethylene in land plants. We speculate that, through multiple duplication events, certain C-SAURs in vascular plants underwent functional diversification to generate A/B-SAUR, which gained the capacity to interact with PP2C.D thereby becoming a crucial part of acid growth. It should be noticed that the expansion of the SAUR family occurred in Bryophyta, which correlates with the emergence of the TIR1/AFB pathway (17), reinforcing the idea that the SAURs serve as downstream output effectors of the TIR1/AFB nuclear transcription pathway to mediate auxin-induced acid growth. Charophyta Possess the Unique Transitional Type of EXP Superfamily. EXPs are the earliest identified and strongest acid growth effector proteins (61, 62). The EXP superfamily in land plants includes four families: EXPA/B and EXP-LIKE A/B (EXLA/B). It has been proven that the EXPA family and some members of the EXPB family can induce cell wall loosening by acidic pH (63). We identified canonical two-domain EXPs in whole plant lineage, also in bacteria and fungi where they are presented, and constructed a phylogeny (Fig. 4 E and SI Appendix, Fig. S8). It was shown that the EXP superfamily can be divided into three monophyletic clades containing bacterial/fungal EXP (known as EXLX), the EXPB family (accompanied by EXLA and EXLB families), and the EXPA family, respectively. Notably, the basal groups of these three clades are composed mainly of Charophyta EXPs, suggesting that they are early forms of EXPs. Chlorophyta EXPs may be the ancestors of the EXP superfamily, which are located in the basal group of the EXLX clade together with Charophyta EXPs. The EXLX clade includes EXPs of Chlorophyta, Charophyta, bacteria, and fungi, indicating the likely horizontal gene transfer (HGT) events between bacterial/fungal EXLXs and algae EXPs. We used Alienness (64) to identify the best donor from C. braunii and best recipient from Stigmatella erecta in the EXLX clade, computationally obtained strong support of the HGT event (HGT index = 108.50, alien index = 94.15) (Fig. 4 E). In the EXPB clade, Charophyta EXPs evolved into the land-plant-specific EXPB family and the seed-plant-specific EXLA/B family. The basal group of the EXPA clade consists exclusively of later-diverging Zygnematophyceae EXPs, which are the progenitors of land-plant-specific EXPAs (Fig. 4 E and SI Appendix, Fig. S8). It has been previously observed that a Zygnematophyceae EXP located outside of EXPA can alter cell wall structure despite the considerable diverge in gene architecture (14). This suggests that EXP experienced extensive gene duplication and neofunctionalization at the Charophyta stage and may play a role in the functional transition to an acid-response regulation during plant terrestrialization. Sequence evolution is the basis for neofunctionalization. To further understand the pattern of sequence evolution in the EXP superfamily, we selected a representative set of 60 EXPs from all species for in-depth analysis, whose reconstructed phylogeny showed consistent architecture compared with all EXP sequences (SI Appendix, Fig. S9 A). Sequence alignment showed that the regions of two domains were relatively conserved, but the differences were more variable between species at the N and C terminus (SI Appendix, Fig. S10). Further motif analysis showed that the EXP superfamily has multiple conserved motifs, but different families also have their unique motifs and arrangements. For example, Chlorophyta-originated motif 9 exists almost exclusively in the EXPA clade while motif 12 is mostly present in EXLX; motif 10 presents only in the seed plant-specific EXLA/B family; motif 11 exists only in the EXPB and EXLA/B families (SI Appendix, Fig. S9). These differences might contribute to the diverse functions of EXP families, such as EXPAs acting on cellulose–cellulose junctions while EXPB can act on xylans (62). ABP1 or Its Core Elements are Partially Absent in some Plant Species. ABP1 and ABLs belong to the cupin superfamily, they bind auxin and form a coreceptor complex with TMK which phosphorylate pT of PM H+-ATPase to activate acid growth (6, 7). Despite similar tertiary structures, ABP1 and ABL exhibit minimal sequence similarity except for the auxin-binding motif (7) in the Cupin_2 domain (PF07883). We reconstructed the phylogeny of Cupin_2 domain-containing proteins of plants along with rare bacterial and archaeal species which were assumed to possess ABP1 (65), then isolated and reconstructed the monophyletic groups containing AtABP1 and AtABL1/2 (Fig. 5 A). ABP1 presents in algae, land plants, and bacteria, the phylogenetic relationships and calculated index support the good HGT event from Charophyta to bacteria (HGT index = 82.00, alien index = 71.51). However, the previously reported archaea ABP1 (65) is more like a paralogous cupin protein. In contrast, the ABL is only detected in seed plants (Fig. 1 A and Fig. 5 A). Fig. 5. Open in Viewer The presence and absence of critical elements for auxin-binding and subcellular localization of ABP1/ABL. (A) Maximum likelihood tree of ABP1/ABL isolated and reconstructed from a monophyletic group of Cupin_2 domain-containing proteins. Red stars indicate an archaeal protein previously regarded as ABP1 (Sulfolobus acidocaldarius, YP_255873.1). Branches with bootstrap values greater than 45 are displayed. Solid circles on the right indicate the presence of a signal peptide, auxin-binding motif, and KDEL motif; hollow circles represent the absence of these elements. (B) Amino acid sequence alignment of auxin-binding motifs in representative ABP1/ABL sequences. Red asterisks represent conserved metal-core sites, and blue asterisks indicate conserved hydrophobic sites. Notably, although the auxin-binding motif of ABP1 is relatively conserved, some species (e.g., M. polymorpha) do not contain ABP1, and the signal peptides responsible for extracellular secretion are frequently missing (e.g., C. braunii), which might be caused by the variant N terminus. Moreover, the KDEL motif needed for ER retention (66) is only present in Angiosperms (Fig. 5 A and B and SI Appendix, Fig. S11). Based on the yeast signal peptide assay (67), we confirmed that the signal peptides of KnABP1 (K. nitens) and AtABP1 (Arabidopsis) are essential for extracellular secretion (SI Appendix, Fig. S12). Due to ABP1 has minimal affinity for auxin in the high-pH cytoplasmic environment (6, 68), signal peptide-mediated extracellular localization may be necessary for the auxin-binding capacity of ABP1. Since ABP1 and ABL exhibit a distant phylogenetic relationship (Fig. 5 A), and ABL was identified by reverse genetic screening (7), certain unknown cupin proteins may also potentially perform auxin-binding function in some species that lack ABP1/ABL or their signal peptides. Auxin Promotes Cell Elongation in C. braunii Independently of PM H+-ATPase. Previous study has shown that auxin failed to induce H+ efflux and apoplast acidification in Chara (42). On the other hand, auxin responses have been observed. It was shown that 1 μM IAA promoted rhizoid elongation in Chara globularis (36) and internodal elongation in Chara vulgaris (32). Moreover, Chara exhibited land plant-like polar auxin transportation (23), probably mediated by putative PIN orthologs (22). We determined the elongation of the apical three internodal cells in response to 1 μM IAA in C. braunii. All three internodes were consistently elongated after 6 h of IAA treatment compared with the mock condition (Fig. 6 A). The effect in the 1 st short node was strongest and observed only after 1 h of treatment. The application of N-1-naphthylphthalamic acid (NPA), which specifically blocks PIN-mediated auxin efflux (69), inhibited growth in the 2 nd and 3 rd internodes, but the 1 st internode was unaffected, suggesting that polar transport of apical-synthesized auxin is involved in lower internodal cell elongation in C. braunii. Combined with the result that auxin does not affect CbHAs phosphorylation (Fig. 2 B), our findings showed auxin regulates cell elongation in C. braunii, although it seems not connected to PM H+-ATPase. Intriguingly, a similar phenomenon in which auxin promotes cell elongation without regulating PM H+-ATPase has been observed in the semiaquatic fern Regnellidium diphyllum (70, 71), implying that acid-growth-independent auxin-induced cell expansion also exists in land plants. Fig. 6. Open in Viewer Auxin-induced internodal cell elongation and transcriptional reprogramming in C. braunii. (A) Elongation curves of the apical three internodal cells of C. braunii were examined under treatment with 0.1% DMSO (mock), 1 μM IAA, and 50 μM NPA. The algae were grown under a 10 h light: 14 h dark cycle. Each treatment was started after 1 h light period. Values are shown as means ± SEMs; n = 14. The positions of the three internodal cells are indicated on the Left. (Scale bars, 1 cm.) (B) Heatmap illustrating the expression levels of representative DEGs at 6 h. Normalized expression values (TPM) are shown in the cells. The cell colors correspond to the column value normalized as a relative value (Z score) indicated by the scale. (C) RT-qPCR verification of IAA-mediated CbRGLG2a gene upregulation in C. braunii thalli exposed or not exposed to 1 μM IAA and 10 mg/mL CHX for 6 h. Values are shown as means ± SEMs; n = 3. Unpaired t tests were used to calculate significant differences within CHX− and CHX+ groups. Auxin Triggers Genome-Wide Transcriptional Reprogramming in C. braunii. We carried out a transcriptome analysis of auxin response in C. braunii thalli. RNA-Seq showed that IAA treatment elicited a significant change in gene expression, with 146 differentially expressed genes (DEGs; 122 up-regulated and 24 down-regulated) in 1 h and 830 DEGs (245 up-regulated and 585 down-regulated) in 6 h (Datasets S5 and S6). Among DEGs up-regulated at 6 h, Gene Ontology (GO) enrichment analysis identified significant terms related to phytohormones, growth/development, membrane components, and transcriptional regulation. The DEGs include genes involved in hormonal metabolic processes and transport, as well as genes whose products localize to the PM and cell wall (SI Appendix, Fig. S13). In contrast to land plants, we did not detect classical auxin-response genes, such as SAUR, Aux/IAA, and GRETCHEN HAGEN 3 [GH3] in C. braunii, consistent with the lack of a canonical auxin TIR1/AFB pathway in Chara. Nor were IAA-up-regulated EXPs detected (Fig. 6 C), in contrast to the ethylene-mediated cell elongation in Zygnematophyceae, which up-regulates the expression of many EXPs (16). In addition, many genes associated with abiotic stress responses, particularly members of the HEAT SHOCK PROTEIN (HSP) family, were generally down-regulated at 6 h (Fig. 6 C and Dataset S6), indicating that IAA promotes coordination between growth and stress responses in C. braunii. We hypothesized that specific auxin-response genes promote cell elongation of C. braunii (Fig. 6 C). Calcium pectate cycles have been implicated in regulating the cell growth rate of Chara (72), and our results indicate that the PECTATE LYASE-LIKE (PPL) genes of C. braunii encoding pectate lyase were up-regulated both 1 h and 6 h after IAA treatment (Fig. 6 C and Datasets S5 and S6), which may facilitate cell elongation by modifying the structure of pectin and cell wall (73). It is noteworthy that the expression of PPLs in land plants can also be induced by auxin and participate in cell wall remodeling and cell expansion (74–76). For genes involved in transmembrane transport, it shows that sugar transporter SUGARS WILL EVENTUALLY BE EXPORTED TRANSPORTER (SWEET) up-regulated by auxin, which may regulate cell elongation by mediating sugar transport, energy signaling, osmotic regulation, and even PIN-mediated auxin polarity transport (77, 78). CALMODULIN-BINDING TRANSCRIPTION ACTIVATOR 3 (CAMTA3) is thought to be required for extracellular ATP (eATP) signaling (79), and eATP signaling recently was found to regulate auxin transcriptional response (80). In particular, we focused on a cluster of overall up-regulated genes in the RING DOMAIN LIGASE 2 (RGLG2) subfamily (Fig. 6 C and Dataset S6). RGLG2 acts as a ubiquitin E3 ligase and has a near-copy (RGLG1) in Arabidopsis (81). The rglg1 rglg2 double mutant Arabidopsis exhibits defects in auxin synthesis and PIN-mediated polar transport, as well as the loss of apical dominance and alterations in phyllotaxy (82). Likewise, CbRGLG2 may participate in auxin-related physiological processes in C. braunii. Multiple 5’-CCTG-3’ motifs are present in the promoter region of the highly expressed isoform CbRGLG2a (CBR_g31557). In K. nitens, this motif is bound by the transcription factor RELATED TO ABI3/VP1 (RAV) which activates certain auxin transcriptional responses (27). RT-qPCR experiments on CbRGLG2a revealed IAA-mediated upregulation regardless of exposure to the protein translation inhibitor cycloheximide (CHX) (Fig. 6 D), suggesting this is a primary transcriptional response to auxin, and the expression level of CbRGLG2a is inhibited by certain unknown protein. Our transcriptomic data are consistent with the hypothesis that auxin promotes a cell expansion mechanism that may involve pectin dynamics before its connection to the acid-growth module. Discussion The acid growth theory was proposed in the 1970s to explain auxin-mediated cell expansion, and its molecular mechanisms have been gradually elucidated in land plants like Arabidopsis over the last decade. In Arabidopsis, auxin-mediated acid growth machinery is composed of the intracellular TIR1/AFB-Aux/IAA-ARF-SAUR-PP2C.D-PM H+-ATPase pathway and the extracellular ABP1/ABL-TMK-PM H+-ATPase pathway (6, 8, 9). Based on the comparative phylogenomic analysis, we found that PM H+-ATPases are present in all algae and land plants, probably because they perform essential functions on cell membranes (83). ABP1 and EXP originated early in Chlorophyta, whereas the core components of TMK, PP2C.D, and SAUR, which respond to auxin and regulate PM H+-ATPase, appeared in Charophyta. However, despite the important effects of auxin on cell expansion and morphogenesis in algae (29–31, 33) and that acid-growth components with the core PM H+-ATPase-TMK/PP2C.D interacting module have appeared in C. braunii, our data showed that auxin does not regulate these components nor stimulate PM H+-ATPases in this algae. Evidence for this includes that CbHAs lack pT site, IAA cannot stimulate phosphorylation of CbHAs, CbSAUR is insensitive to auxin and does not interact with CbPP2C.D. We speculate that the neofunctional subclasses of SAUR and EXP specifically associated with auxin-mediated acid growth were generated by gene duplication after plant landing. Together with the emergence of TIR1/AFB (17), these SAUR and EXP genes became output effectors of auxin in cell expansion in a complex regulatory network of auxin-mediated acid growth in land plants. Our study also revealed that PM H+-ATPases of C. braunii are insensitive to auxin but activated by light through photosynthesis-dependent phosphorylation, which is consistent with the light-induced charasome and acid band formation in Chara (46, 47). Related to the finding that photosynthesis inhibitor DCMU inhibited charasome formation (84), we found that DCMU also inhibited light-mediated PM H+-ATPase phosphorylation. We propose that the state of photosynthesis, probably through sugar signaling as in land plants (85), positively feedback-regulates PM H+-ATPase and extracellular acidification for photosynthesis. Light regulation of charasome and CbHAs is somewhat reminiscent of light-mediated phosphorylation and activation of PM H+-ATPase in stomatal guard cells of Arabidopsis, especially for red light which activates the PM H+-ATPase in a photosynthesis-dependent manner, facilitating the opening of the stomata (86, 87). Noticeably, PP2C.Ds are involved in the dephosphorylation of light/photosynthesis-regulated PM H+-ATPase in stomata guard cells of Arabidopsis (87–89). Since PP2C.D originates from Charophyta and interacts with PM H+-ATPase, this may imply that the PM H+-ATPase-PP2C.D regulatory module initially developed in response to the light/photosynthesis state before being tailored for the auxin-mediated acid growth pathway. It remains to be examined whether TMK is involved in photosynthesis. Although the PM H+-ATPase activity of Charophyta is insensitive to auxin, both auxin and acid seem to promote its cell elongation independently. On the one hand, we found that auxin elicits a pronounced transcriptional response and cell elongation in C. braunii, which is independent of PM H+-ATPase activation and auxin receptor TIR1/AFB. Similar to the auxin-mediated transcriptional reprogramming in K. nitens and the TIR1/AFB pathway in land plants (26, 27, 90), we observed the primary transcriptional regulation in C. braunii which does not require protein synthesis, indicating an ancestral auxin system controls cell expansion in algae. On the other hand, acid-induced cell elongation has been observed in the Charophyceae algae Nitella. It was shown that the growth rate of PM H+-ATPase-rich acid bands was higher than that of base bands in the living Nitella internodal cell, which could be weakened by alkaline buffers and vice versa (91). Notably, the acid-enhanced extensibility was also observed in boiled isolated cell walls of Nitella (92, 93), indicating EXPs and other cell wall proteins in Charophyta may not be required in this physical acid growth. The different mechanisms between Charophyceae algae and land plants in cell elongation may be related to their structure and composition of the cell walls. The cell walls of land plants have complex extracellular hydrophobic polymers, while algae do not have a true cuticle, making them more susceptible to external pH changes (15, 94). Previous experiments also confirmed that the cell wall of Nitella has a considerable ability to bind H+ (91). In addition, C. braunii is deficient in several cell wall proteins compared to land plants, including canonical arabinogalactan-proteins which are key functional cell wall proteins common to land plants (95), as well as xyloglucan transglycosylase/hydrolase and endo-glucanase (15) which play crucial roles in remodeling cell walls and regulating cell elongation in Arabidopsis (96, 97). Consequently, the physicochemical properties and regulation of cell walls in Charophyceae algae may differ significantly from those in land plants. As the functional innovation of cell walls, the formation of TIR1/AFB-mediated auxin signaling, and the establishment of acid growth network, the regulatory relationship between auxin and acid growth was eventually linked in land plants. Materials and Methods Plant materials, growth conditions, and phytohormone/chemical treatment are described in SI Appendix, Materials and Methods. The detailed procedures of bioinformatic analysis, immunoblot assay, ATP hydrolysis assay, yeast two-hybrid assays, co-immunoprecipitation assay, RT-qPCR, yeast secretion assay, cell elongation measurement, and RNA-Seq are provided in SI Appendix, Materials and Methods. The primers used in this study are listed in Dataset S7. Data, Materials, and Software Availability RNA-Seq data have been deposited in NCBI Sequence Read Archive (PRJNA1022317) (98). Raw files of phylogenetic analysis data have been deposited in figshare ( (99). All study data are included in the article and/or supporting information. Acknowledgments We express our gratitude to Takayuki Kohchi (Kyoto University) and Ryuichi Nishihama (Tokyo University of Science) for the valuable discussions. This study was supported by the National Natural Science Foundation of China (NSFC) Key Program (32230006) to X.W.D. and (32250710144) to N.W., as well as the Key R&D Program of Shandong Province, China (ZR202211070163 to X.W.D.). Author contributions H.Y.Z., H.H., X.W.D., and N.W. designed research; H.Y.Z., S.D., C.J., Z.S., L.C., J.W., X.H., A.W., D.J., Y.W., and L.L. performed research; H.Y.Z. and N.W. analyzed data; and H.Y.Z., L.L., X.W.D., and N.W. wrote the paper. Competing interests The authors declare no competing interest. Supporting Information Appendix 01 (PDF) Download 16.64 MB Dataset S01 (XLSX) Download 17.41 KB Dataset S02 (XLSX) Download 43.64 KB Dataset S03 (XLSX) Download 14.47 KB Dataset S04 (XLSX) Download 53.94 KB Dataset S05 (XLSX) Download 21.06 KB Dataset S06 (XLSX) Download 76.24 KB Dataset S07 (XLSX) Download 10.96 KB References 1 D. L. Rayle, R. Cleland, Enhancement of wall loosening and elongation by Acid solutions. Plant. Physiol.46, 250–253 (1970). Go to reference Crossref PubMed Google Scholar 2 A. Hager, H. Menzel, A. Krauss, Versuche und Hypothese zur Primärwirkung des Auxins beim Streckungswachstum. Planta100, 47–75 (1971). Go to reference Crossref PubMed Google Scholar 3 S. McQueen-Mason, D. M. Durachko, D. J. Cosgrove, Two endogenous proteins that induce cell wall extension in plants. Plant. Cell4, 1425–1433 (1992). Go to reference PubMed Google Scholar 4 M. Claussen, H. Lüthe, M. Blatt, M. Böttger, Auxin-induced growth and its linkage to potassium channels. 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Data, Materials, and Software Availability RNA-Seq data have been deposited in NCBI Sequence Read Archive (PRJNA1022317) (98). Raw files of phylogenetic analysis data have been deposited in figshare ( (99). All study data are included in the article and/or supporting information. Submission history Received: June 21, 2024 Accepted: October 28, 2024 Published online: December 10, 2024 Published in issue: December 17, 2024 Keywords acid growth cell expansion auxin evolution plant terrestrialization Acknowledgments We express our gratitude to Takayuki Kohchi (Kyoto University) and Ryuichi Nishihama (Tokyo University of Science) for the valuable discussions. This study was supported by the National Natural Science Foundation of China (NSFC) Key Program (32230006) to X.W.D. and (32250710144) to N.W., as well as the Key R&D Program of Shandong Province, China (ZR202211070163 to X.W.D.). Author contributions H.Y.Z., H.H., X.W.D., and N.W. designed research; H.Y.Z., S.D., C.J., Z.S., L.C., J.W., X.H., A.W., D.J., Y.W., and L.L. performed research; H.Y.Z. and N.W. analyzed data; and H.Y.Z., L.L., X.W.D., and N.W. wrote the paper. Competing interests The authors declare no competing interest. Notes Reviewers: D.J.C., The Pennsylvania State University; R.A.D., University of North Texas; and L.R., The University of British Columbia. Authors Affiliations Expand All Hai Yue Zeng School of Life Sciences, Southwest University, Chongqing 400715, China State Key Laboratory of Wheat Improvement, Peking University Institute of Advanced Agricultural Sciences, Shandong Laboratory of Advanced Agricultural Sciences, Weifang 261000, China State Key Laboratory of Wheat Improvement, School of Advanced Agricultural Sciences and School of Life Sciences, and Academy for Advanced Interdisciplinary Studies, Peking-Tsinghua Center for Life Sciences, Peking University, Beijing 100871, China View all articles by this author Shiyu Deng School of Life Sciences, Southwest University, Chongqing 400715, China View all articles by this author Congcong Jin State Key Laboratory of Wheat Improvement, Peking University Institute of Advanced Agricultural Sciences, Shandong Laboratory of Advanced Agricultural Sciences, Weifang 261000, China View all articles by this author Zhiyun Shang State Key Laboratory of Wheat Improvement, Peking University Institute of Advanced Agricultural Sciences, Shandong Laboratory of Advanced Agricultural Sciences, Weifang 261000, China State Key Laboratory of Wheat Improvement, School of Advanced Agricultural Sciences and School of Life Sciences, and Academy for Advanced Interdisciplinary Studies, Peking-Tsinghua Center for Life Sciences, Peking University, Beijing 100871, China View all articles by this author Le Chang State Key Laboratory of Wheat Improvement, School of Advanced Agricultural Sciences and School of Life Sciences, and Academy for Advanced Interdisciplinary Studies, Peking-Tsinghua Center for Life Sciences, Peking University, Beijing 100871, China View all articles by this author Jiajun Wang School of Life Sciences, Southwest University, Chongqing 400715, China State Key Laboratory of Wheat Improvement, School of Advanced Agricultural Sciences and School of Life Sciences, and Academy for Advanced Interdisciplinary Studies, Peking-Tsinghua Center for Life Sciences, Peking University, Beijing 100871, China View all articles by this author Xue Han State Key Laboratory of Wheat Improvement, Peking University Institute of Advanced Agricultural Sciences, Shandong Laboratory of Advanced Agricultural Sciences, Weifang 261000, China View all articles by this author Ao Wang Beijing Key Laboratory of Development and Quality Control of Ornamental Crops, Department of Ornamental Horticulture, College of Horticulture, China Agricultural University, Beijing 100193, China View all articles by this author Dan Jin College of Agronomy and Biotechnology, Southwest University, Chongqing 400715, China View all articles by this author Yubo Wang Beijing KeJianYiYan Technology Co., Ltd, Beijing 100000, China View all articles by this author Hang He State Key Laboratory of Wheat Improvement, Peking University Institute of Advanced Agricultural Sciences, Shandong Laboratory of Advanced Agricultural Sciences, Weifang 261000, China State Key Laboratory of Wheat Improvement, School of Advanced Agricultural Sciences and School of Life Sciences, and Academy for Advanced Interdisciplinary Studies, Peking-Tsinghua Center for Life Sciences, Peking University, Beijing 100871, China View all articles by this author Lanxin Li Beijing Key Laboratory of Development and Quality Control of Ornamental Crops, Department of Ornamental Horticulture, College of Horticulture, China Agricultural University, Beijing 100193, China View all articles by this author Xing Wang Deng1 State Key Laboratory of Wheat Improvement, Peking University Institute of Advanced Agricultural Sciences, Shandong Laboratory of Advanced Agricultural Sciences, Weifang 261000, China State Key Laboratory of Wheat Improvement, School of Advanced Agricultural Sciences and School of Life Sciences, and Academy for Advanced Interdisciplinary Studies, Peking-Tsinghua Center for Life Sciences, Peking University, Beijing 100871, China View all articles by this author Ning Wei1weining@swu.edu.cn School of Life Sciences, Southwest University, Chongqing 400715, China View all articles by this author Notes 1 To whom correspondence may be addressed. Email: deng@pku.edu.cn or weining@swu.edu.cn. Metrics & Citations Metrics Citations Metrics Article usage Views Citations No data available. 2,738 4 Total First 90 Days 6 Months Total number of views and citations Note: The article usage is presented with a three- to four-day delay and will update daily once available. Due to ths delay, usage data will not appear immediately following publication. Citation information is sourced from Crossref Cited-by service. Altmetrics See more details Picked up by 1 news outlets Posted by 19 X users Referenced by 2 Bluesky users 9 readers on Mendeley Citations Cite this article H.Y. Zeng, S. Deng, C. Jin, Z. Shang, L. Chang, J. Wang, X. Han, A. Wang, D. Jin, Y. Wang, H. He, L. Li, X.W. Deng, [...] &N. Wei, +0 authors Origin and evolution of auxin-mediated acid growth, Proc. Natl. Acad. Sci. U.S.A. 121 (51) e2412493121, Copy Copied! Copying failed. Export the article citation data by selecting a format from the list below and clicking Export. Format [x] Direct import Cited by Loading... View Options View options PDF Download this article as a PDF file. PDF eReader View this article with eReader. eReader Figures Open all in viewer Fig. 1. Origin and evolution of core components responsible for auxin-mediated acid growth. (A) Identification of the orthologs of core components responsible for acid growth in 30 representative plant species. Circle size represents gene copy number. The ancient whole-genome duplication/triplication events were labeled on the different branches of the phylogenetic tree based on previous reports, and the named duplication events are shown alongside their Greek letter (38, 39). (B) Gradual coevolution model of auxin-mediated acid growth. The emergence of prominent features across various evolutionary stages is illustrated. Go to FigureOpen in Viewer Fig. 2. Functional evolution of PM H+-ATPases from aquatic to terrestrial. (A) Maximum likelihood tree of P-type H+-ATPases isolated and reconstructed from a monophyletic group of similar-sequence proteins. AtAHAs and CbHAs are highlighted. Additional representative P-type H+-ATPases from fungi were regarded as outgroups. Branches with bootstrap values greater than 50 are displayed. (B) The phosphorylation level of PM H+-ATPase of C. braunii in response to IAA and white light. The microsomal fractions were extracted from 12-hour-dark-adapted C. braunii thalli, which were pretreated in the liquid mSWC-2 medium containing 0.1% dimethyl sulfoxide (DMSO)/1 μM IAA/10 μM DCMU for 1 h, then were illuminated with white light at 1000 lx or kept in the dark for 6 h. The immunoblot assay was analyzed on an Mn 2+-based phos-assay or a standard SDS-PAGE gel and probed with an anti-PM H+-ATPase antibody. The graph represents the phosphorylation level of PM H+-ATPase, quantified according to the signal intensity ratio of the phosphorylated H+-ATPase band to the sum of phosphorylated and unphosphorylated H+-ATPase bands. Values represent the mean and SEMs of three independent experiments. Paired t tests were used to calculate significant differences (P< 0.05). (C) PM H+-ATPase hydrolysis activities in C. braunii thalli under different treatments. Vanadate-sensitive ATP hydrolysis was measured by determining nicotinamide adenine dinucleotide (NADH) consumption. Values represent the means of three independent biological replications with SEMs. One-way ANOVA with Tukey’s test was used to calculate significant differences. Go to FigureOpen in Viewer Fig. 3. TMK and PP2C.D originated and formed a PM H+-ATPase-PP2C.D/TMK interacting module in Charophyta. (A) Maximum likelihood tree of TMKs and their domain architecture. Bootstrap values greater than 50 are indicated. (B) Maximum likelihood tree of the PP2C family and domain recombination events. Colors indicated by dots represent the ancestral species categories included in PP2C clades. PP2C.Ds are compared with PP2C.Cs. The table shows the distribution of PP2C.D-specific AMPC isoforms and their proportion in PP2C.Ds of each species. (C) Membrane-based yeast two-hybrid assays showing the interactions of CbTMKLC and CbPP2C.D with CbHA2. (D) Co-IP assays showing the association of the CbHA2 with CbTMKLC and CbPP2C.D in rice protoplasts transiently expressing indicated epitope-tagged fusion proteins. Go to FigureOpen in Viewer Fig. 4. Gene duplication–mediated neofunctionalization facilitates the establishment of acid growth in land plants. (A) Maximum likelihood tree of SAUR family. Bootstrap values are shown at key nodes. Colors indicated by dots represent the species categories included in SAUR family subgroups. (B) Nuclear-based yeast two-hybrid assays testing interactions between SAURs and PP2C.Ds. Plates were incubated for 24 h to visualize color differences. (C and D) Relative expression levels of CbSAUR and AtSAUR76-78 under treatment with 1 μM IAA or 5 ppm ethylene, as determined by RT-qPCR. RNA samples were extracted from 7-d-old Arabidopsis seedlings and vegetative stages of C. braunii thalli. Values are shown as means ± SEM; n = 3. Two-way ANOVA was used to calculate significant differences (P< 0.05) within each gene. (E) Maximum likelihood tree of the EXP superfamily in plants, bacteria, and fungi. The bootstrap values are shown for key nodes. Colors indicated by dots/branches represent the species categories included in each family. Species of Chlorophyta/Charophyta EXPs in the three clades are listed on the right. Go to FigureOpen in Viewer Fig. 5. The presence and absence of critical elements for auxin-binding and subcellular localization of ABP1/ABL. (A) Maximum likelihood tree of ABP1/ABL isolated and reconstructed from a monophyletic group of Cupin_2 domain-containing proteins. Red stars indicate an archaeal protein previously regarded as ABP1 (Sulfolobus acidocaldarius, YP_255873.1). Branches with bootstrap values greater than 45 are displayed. Solid circles on the right indicate the presence of a signal peptide, auxin-binding motif, and KDEL motif; hollow circles represent the absence of these elements. (B) Amino acid sequence alignment of auxin-binding motifs in representative ABP1/ABL sequences. Red asterisks represent conserved metal-core sites, and blue asterisks indicate conserved hydrophobic sites. Go to FigureOpen in Viewer Fig. 6. Auxin-induced internodal cell elongation and transcriptional reprogramming in C. braunii. (A) Elongation curves of the apical three internodal cells of C. braunii were examined under treatment with 0.1% DMSO (mock), 1 μM IAA, and 50 μM NPA. The algae were grown under a 10 h light: 14 h dark cycle. Each treatment was started after 1 h light period. Values are shown as means ± SEMs; n = 14. The positions of the three internodal cells are indicated on the Left. (Scale bars, 1 cm.) (B) Heatmap illustrating the expression levels of representative DEGs at 6 h. Normalized expression values (TPM) are shown in the cells. The cell colors correspond to the column value normalized as a relative value (Z score) indicated by the scale. (C) RT-qPCR verification of IAA-mediated CbRGLG2a gene upregulation in C. braunii thalli exposed or not exposed to 1 μM IAA and 10 mg/mL CHX for 6 h. Values are shown as means ± SEMs; n = 3. Unpaired t tests were used to calculate significant differences within CHX− and CHX+ groups. Go to FigureOpen in Viewer Tables Media Share Share Share article link COPY LINK Copied! Copying failed. Share on social media FacebookX (formerly Twitter)LinkedInGmailemail References References 1 D. L. Rayle, R. 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18187	https://www.science.org/doi/10.1126/sciadv.abj0790	Skip to main content Main content starts here The prevalence and specificity of local protein synthesis during neuronal synaptic plasticity Chao Sun Andreas Nold, [...] , Claudia M. Fusco Vidhya Rangaraju [...] , Tatjana Tchumatchenko Mike Heilemann and Erin M. Schuman authors +2 authors fewerAuthors Info & Affiliations Science Advances 17 Sep 2021 Vol 7, Issue 38 DOI: 10.1126/sciadv.abj0790 NotificationsBookmark Abstract To supply proteins to their vast volume, neurons localize mRNAs and ribosomes in dendrites and axons. While local protein synthesis is required for synaptic plasticity, the abundance and distribution of ribosomes and nascent proteins near synapses remain elusive. Here, we quantified the occurrence of local translation and visualized the range of synapses supplied by nascent proteins during basal and plastic conditions. We detected dendritic ribosomes and nascent proteins at single-molecule resolution using DNA-PAINT and metabolic labeling. Both ribosomes and nascent proteins positively correlated with synapse density. Ribosomes were detected at ~85% of synapses with ~2 translational sites per synapse; ~50% of the nascent protein was detected near synapses. The amount of locally synthesized protein detected at a synapse correlated with its spontaneous Ca2+ activity. A multifold increase in synaptic nascent protein was evident following both local and global plasticity at respective scales, albeit with substantial heterogeneity between neighboring synapses. INTRODUCTION Biological compartments can function and adapt with autonomy by localizing cell biological organelles and machinery. Perhaps the best example of this is the neuronal synapse, where local control of protein production and distribution can allow, in principle, for the compartmentalization of synaptic function and plasticity (1–4). However, the prevalence, utility, and specificity of locally synthesized proteins are still not well understood. For example, is local protein synthesis constitutive during basal synaptic activity? And how are nascent proteins distributed among synapses that vary both in density and in activity level (5, 6)? In addition, the relationship between the machines (ribosomes), locally synthesized proteins, and synaptic features are not well understood. All of these questions are particularly important during plasticity, where local or global activity manipulations can result in a local or global change in synaptic strengths (3, 7). During plasticity, the underlying proteome remodeling concerns not only a few receptor complexes but also thousands of protein species (8). If plasticity is induced globally but proteins are supplied locally, what are the rules to allot nascent proteins to the synapse population? When plasticity is induced locally, how local is the change in protein distribution? Does it occur at the level of individual synapses, short segments of dendritic branches containing synaptic clusters, whole dendritic branches, etc.? To address these questions, we first visualized and quantified the dendritic protein synthesis sites by labeling and measuring both assembled ribosomes and nascent proteins in mature, cultured rat hippocampal neurons using quantitative, multiplexed single-molecule localization imaging (DNA-PAINT) (9, 10). We monitored a cohort of newly synthesized proteins and correlated their spatial distribution with that of assembled ribosomes over time, following locally and globally induced plasticity. We observed widespread protein synthesis near synapses under both basal and stimulated conditions. During normal synaptic transmission, the amount of locally synthesized proteins detected at a synapse was correlated with its level of ongoing spontaneous activity. Plasticity induced by single-spine stimulations or by a global activity manipulation both resulted in a significant increase in local protein synthesis at respective spatial scales. However, the elevated nascent protein is not specific to stimulated spines but rather spread to neighboring synapses. RESULTS The ribosomal large and small subunits assemble during mRNA translation (11). To localize the sites of protein synthesis in synapses and dendrites of cultured hippocampal neurons, we colocalized ribosomal large and small subunits using quantitative, multiplexed, single-molecule localization microscopy (DNA-PAINT; Fig. 1A; see also fig. S1A and Materials and Methods) (12–14). To tag each subunit for single-molecule localization, we immunolabeled the endogenous ribosomal proteins RPS11 (epitope for 40S small subunit) and RPL36a (epitope for 60S large subunit) using primary antibodies and secondary antibodies, each labeled with a distinct single-strand DNA “docking” oligo for sequential multiplexed imaging [see fig. S1 (B to D) and Materials and Methods]. Tagged ribosomal subunits were detected by repeated, brief fluorescence detection events representing the transient hybridization between the docking and imager (complementary and fluorescent) oligos to form localization clusters (Fig. 1A), as previously described (12). Coincident detection of localization clusters from both subunits further identified assembled ribosomes (Fig. 1A, bottom right; see Materials and Methods). Using sucrose gradients that enrich ribosomes of different compositions (fig. S2A; see also Materials and Methods) (15), our method reliably distinguished between the enriched small-subunit fraction [isolated 40S clusters; Fig. 1B, left (green)], the enriched monosome [colocalizations that appeared isolated and uniform in size; Fig. 1B, middle; see also fig. S2 (B and C)], and the enriched polyribosome fraction (colocalizations that appeared in clusters, with an average magenta/green localization ratio of 0.95; Fig. 1B, right; see Materials and Methods). A significant difference in cluster size was also detected between the 40S fraction, 80S fraction, and polyribosome fraction (Fig. 1C and fig. S2, D and E). To validate the in situ sensitivity of ribosome detection, we induced the disassembly of polysomes in neurons with two different protein synthesis inhibitors (Fig. 1D and fig. S3A) (15, 16) and observed the predicted loss of the larger ribosome clusters (Fig. 1D). Furthermore, the degree of large and small subunit colocalization that we detected was significantly different from the colocalization obtained by chance (fig. S3, B to D). Together, these data indicate that our method can be used to reliably visualize translation sites in dendrites. We quantified the extent to which the detected ribosomes populated the dendrites and localized near synapses by coimmunolabeling with a synaptic marker (anti-Bassoon; Fig. 1E). The dendritic arbor shown in Fig. 1E was populated with synapses at an average density of ~1 per micrometer of dendrite (Fig. 1E; see also fig. S3E), as previously reported (17). Ribosomes populated the entire dendritic arbor; we detected ~2 ribosome clusters per micrometer of dendrite, corresponding to ~8 translating ribosomes/μm (the average ribosome cluster containing ~4 ribosomes; see Materials and Methods). This density is greater than that previously reported for polyribosomes using electron microscopy (~1 per μm with the requirement of ≥3 ribosomes detected in a cluster) (5). The detection and inclusion of monosomes (15) here [excluded in electron microscopy (EM) studies] likely contributed to the higher ribosome occupancy observed. We found that ~85% of synapses had one or more ribosome clusters associated with it (within a 1-μm radius from the local maxima of a Bassoon puncta; Fig. 1F; see Materials and Methods). Using heat and contour maps to depict the synapse and ribosome density distributions, respectively, we found that synapses and ribosomes tended to covary in their abundance (Fig. 1G), suggesting that local hotspots of translation form around synapse clusters (Fig. 1E, inset). We found a positive correlation (r = 0.67; Fig. 1H) between ribosome density and synapse density within individual dendritic segments that constitute a dendritic branch [see Fig. 3E (inset) for examples; see fig. S3F and Materials and Methods for dendrite segmentation]. Hence, local sites of protein synthesis congregate near synapses along dendrites, giving rise to ~2 translational sites per synapse. The proximity of ribosomes to synapses suggests that locally translated proteins may support the maintenance of the existing synaptic protein population. To address this question, proteins made from these local sites of translation must be labeled and visualized. We thus combined nascent protein metabolic labeling with DNA-PAINT in addition to ribosome localization. To tag nascent proteins for quantitative single-molecule localization, we used a brief (15 min) pulse of the noncanonical amino acid azidohomoalanine (AHA) [BioOrthogonal Non-Canonical Amino Acid Tagging (BONCAT); Fig. 2A; see Materials and Methods] (10, 18). At the cost of labeling all nascent proteins, a brief labeling period was used since prolonged labeling causes signal crowding that confounds individual clustering of nascent protein localizations (fig. S4A). After AHA labeling, neurons were fixed and AHA-tagged nascent proteins were labeled with a single-stranded DNA docking oligo using copper-free click chemistry (see Materials and Methods) (19). The samples were then immunolabeled for ribosome subunits (Fig. 1A) and dendritic and synaptic reference markers [using anti-microtubule-associated protein (MAP2) and anti-Bassoon antibodies]. Tagged nascent proteins and ribosomes in dendrites and synapses were detected sequentially using respective imager oligos (complementary and fluorescent) (Fig. 2A). As previously reported (18), without AHA treatment, we observed only a low level of background signal in the dendrites [Fig. 2 (B and C) and fig. S4 (B and C) indicate the corresponding dendrites]. In neurons treated with AHA, nascent proteins appeared as clusters of fluorescent localizations in dendrites (Fig. 2D). Our analyses indicated that a single copy of nascent protein was equivalent to ~7 ± 4 localizations (fig. S5; see also Materials and Methods) (20). We observed that nearly 30% of the nascent proteins were within 1 μm of a ribosome(s) and over 90% of the nascent proteins were detected within 5-μm radius of a nearby ribosome(s) (Fig. 2E and fig. S6A). In addition, local ribosome hotspots appeared to frequently overlap with high-density nascent protein domains (Fig. 2, E and F; see also Fig. 2I), while dendritic segments with low-ribosome occupancy often showed low levels of nascent proteins (Fig. 2E). We observed a significant, positive correlation (r = 0.73) between the position and level of nascent proteins and ribosomes among the dendritic segments (Fig. 2G). Overall, local differences in ribosome density appeared to account for local differences in nascent protein levels, suggesting that most of the proteins detected could arise from a local ribosome pool. If the above-described nascent proteins were synthesized by nearby ribosomes, then imposing a chase (30 min of no label) following the brief metabolic label should diminish the spatial correlation because nascent proteins will have diffused. To test this, we compared the nearest-ribosome distances of individual nascent proteins in dendrites with and without a 30-min chase immediately following metabolic labeling. A significant increase in the nearest-ribosome distances was observed following the 30-min chase (Fig. 2H). We also noted that the local maxima of nascent protein contours were displaced from nearby ribosome hotspots [Fig. 2I, 30-min chase; see also fig. S6 (B to D)] following the chase. These data support the idea that on brief time scales, there is a tight spatial relationship between dendritic ribosomes and their nascent protein products. To address how the above locally translated proteins distribute to synapses, we quantified the extent to which individual locally translated proteins occupied excitatory synapses during normal, ongoing network activity. Using synapse templates created by an excitatory synaptic marker [anti–postsynaptic density 95 (PSD95); Fig. 3, A and B; see also fig. S7], we extracted nascent synaptic proteins over a 6400-μm2 field of view and detected and quantified the nascent protein at each synapse on a dendrite (Fig. 3C). Although our 15-min labeling protocol inevitably resulted in an underdetection of nascent protein copies (see Materials and Methods and Discussion), we still detected ~8 copies of tagged nascent proteins per micrometer of dendrite (on average, 55 ± 11 localizations/μm with ~7 localization per copy; Fig. 3A), with 51 ± 6% detected within a 500-nm radius of a synapse (the PSD95 maxima) and 10 ± 5% overlapping with the PSD95. The number of nascent proteins present at individual synapses ranged from 0 to 10 (0 to 70 localizations) for the 1289 synapses measured from six cell replicates under control conditions. The number of nascent proteins detected within dendritic branches did not exhibit a gradient reflecting proximity to the cell soma (fig. S5C; see also Fig. 2B), consistent with the idea that they are synthesized locally. To examine the local protein allocation to synapses during basal activity, we recorded basal, spontaneous synaptic Ca2+ transients (500 frames at 1 Hz) before metabolic labeling and DNA-PAINT (Fig. 3D; see also Materials and Methods). Neurons were transfected with GCaMP6s (Ca2+-transient indicator; Fig. 3E, green) and PSD95-mCherry (synaptic reference marker; Fig. 3E, magenta) before Ca2+ imaging. We detected 4562 Ca2+ events distributed across 110 synapses in ~8 min of imaging. The subsequent detection of nascent protein (Fig. 3E) indicated that synapses that exhibited higher activity levels (Fig. 3, F and G) also had higher levels of locally synthesized protein (Fig. 3H). Overall, we found a significant correlation between the nascent synaptic proteins and basal synaptic Ca2+ events (Fig. 3I). Synaptic activity is thus a useful predictor of the locally synthesized protein levels observed under basal conditions. The above data indicate that local protein synthesis is prevalent during the ongoing maintenance of synapses, even in the absence of overt stimulations. The above data suggest that nascent protein levels at synapses might sense activity levels. We next queried whether the local protein supply is altered by activity-dependent plasticity. We first examined a form of plasticity that is induced and expressed across the entire neuron, synaptic upscaling (7). Cultured hippocampal neurons were treated with tetrodotoxin (TTX; 2 μM) for 1 or 24 hours, representing an early “induction” time point (1 hour) and a time point when synaptic scaling had been established (24 hours) (2). At the end of the (1- or 24-hour) upscaling induction, neurons were metabolically labeled (4 mM AHA) for the last 15 min before fixation and processing to identify dendrites and synapses, as described above (Fig. 3, A and B). As observed previously (21), synapse density increased slightly after 1 and 24 hours of activity blockade (fig. S8) with a small but significant increase in mean synapse size (fig. S7) (22). We quantified the amount of nascent protein present and detected a significant, multifold increase in both dendrites (Fig. 4A) and synapses (Fig. 4B; see also fig. S9) after both 1 and 24 hours of upscaling. This indicates that local protein synthesis is globally elevated during synaptic scaling. While, at the population level, there was a scaling-induced increase in protein level at synapses and along dendrites, we observed significant diversity in synaptic protein levels within the dendrite (Fig. 4, C to E, and fig. S9, D to F). High-protein synapses did not appear clustered (Fig. 4, C to E; examples indicated by purple arrows) but were frequently neighbored by synapses with little to no detectable nascent protein [also schematized in Fig. 4F (right inset)]. Consequently, the average protein per synapse within a dendritic segment was very similar across all the segments (Fig. 4F) despite the high variability between individual synapses within a segment. Dendritic segments with highly clustered synapses [purple box in Fig. 4F (right inset)] became the local translational “hotspots” (fig. S10, lighter-colored segments). Hence, local translational hotspots are likely driven by the local clustering of synapses in number rather than the clustering of high-protein synapses [schematized in Fig. 4F (right inset)]. To better visualize the contribution of local protein supply at the synapse population level, for each synapse, we plotted its nascent protein level (x), the local synapse density (y), and the average nascent synaptic protein level present at its neighboring synapses (z) in a three-dimensional (3D) parameter space (Fig. 4, G to I, and fig. S11; see Materials and Methods). Overall, the greater than 1000 synapses that we measured did not segregate into discreet clusters in the 3D space. After 1 and 24 hours of upscaling (Fig. 4, H and I, respectively), the synapse population dilated significantly along all three axes in comparison to control, consistent with Fig. 4B and fig. S8 (synapse density increase), with the largest increase seen in the z axis, representing the increase in the neighboring synaptic protein level. Driven by a small number of extremely high-protein synapses, the population after 1 hour of upscaling appeared slightly bifurcated (indicated by two gray arrows in Fig. 4H): Extremely high-protein synapses occupied the high-protein-per-synapse region of the parameter space (bottom right corner), while their neighboring synapses extended into the high-local-protein-density region (top front corner). This distribution showed that protein was far from evenly distributed among neighboring synapses (as shown in Fig. 4C), reminiscent of competition between nearby synapses (see fig. S12 for illustrations of proposed modes of protein distribution in 3D) (23, 24). Compared to 1 hour of upscaling (Fig. 4E), the population contracted in the parameter space after 24 hours of upscaling, driven by a loss of extremely high-protein synapses (Fig. 4I). These observations were robust across dendrites and neurons within the same treatment group (fig. S12, A to C). Overall, these data indicate that synaptic upscaling elicited a global increase in local protein production, with significant heterogeneity in nascent protein levels at individual synapses. How are locally synthesized proteins allocated during locally induced forms of plasticity? To explore this, we elicited morphological plasticity at single spines using two-photon glutamate uncaging (Fig. 5A) combined with stimulation of the protein kinase A pathway, as previously described (23, 25). As before, neurons were transfected to express GCaMP6s and PSD95-mCherry before local spine stimulation, metabolic labeling, and DNA-PAINT (Fig. 5A; see Materials and Methods). As expected, local stimulation of an individual spine resulted in a Ca2+ increase in the stimulated spine, but not adjacent spines (fig. S13). As observed previously (25), this plasticity protocol resulted in an increase in spine size (Fig. 5, C and D). While spine size increased less than twofold (Fig. 5D, x axis), local protein supply to synapses increased by several-fold (Fig. 5D, y axis). We nevertheless detected a significant correlation between the spine and spine-base nascent protein level (indicated by dashed boxes in Fig. 5C) and the induced spine size increase (Fig. 5D). To investigate whether the increase in nascent protein at the stimulated synapses was specific, all the identified synapses within 15-μm distance from the stimulated spine were analyzed (Fig. 5, B and C). Figure 5C shows the distribution of nascent proteins (magenta) within this dendritic segment after local stimulation (the same set of synapses encircled as in Fig. 5B). To reveal the protein distribution among these neighboring synapses, their nascent protein allocation (Fig. 5E; see Materials and Methods) was plotted relative to their distances from the corresponding stimulated spines (each color series represents a set of neighbors near a stimulated spine replicate; seven stimulated spine replicates with at least five neighbors within 5 μm are shown; see Materials and Methods). Neighboring synapses nearer to the stimulated spine (e.g., <3 μm) showed heterogeneous but significantly higher protein allocation on average compared to those farther away (Fig. 5E, inset). Consistently, at a larger scale, dendritic segments containing stimulated spines exhibited significantly higher mean protein (per synapse) compared to control untreated segments (Fig. 5F; see also fig. S14) across the dendrite. At the same time, neighboring segments did not significantly rise above or sink below the remaining untreated segments (fig. S14), implying locally elevated protein synthesis rather than a depletion of proteins from surrounding segments. Together, these data indicate that although the spine size changes were specific to the stimulated spine, the nascent proteins stimulated by plasticity induction were not specific. DISCUSSION Decades of research have detected thousands of mRNAs, the essential elements of the protein synthesis machinery, and evidence for protein synthesis in neuronal processes (1, 5, 15, 26, 27). Locally synthesized nascent proteins have been visualized (28, 29). However, a persistent open question has been the prevalence of local protein synthesis and whether locally synthesized proteins contribute to the proteome to offset ongoing protein turnover. In addition, the specificity of locally synthesized proteins during plasticity has not been addressed. Here, we used metabolic labeling and DNA-PAINT to spatially dissect and quantify the contribution of local protein synthesis at neuronal synapses during basal activity and following plasticity. Although our method required subsampling (low concentration of AHA for just 15 min), we detected nascent proteins throughout the dendrites in the absence of any exogenous stimulation and ribosomes at 85% of all synapses. A multifold increase in nascent proteins was observed at synapses following both global synaptic upscaling and local single-spine plasticity. Rather than being synapse specific, the local protein supply appeared to be produced and redistributed among neighboring synapses. Local protein synthesis thus occurs during maintenance while harboring a multifold capacity to supply proteins for synapses during plasticity. How abundant is local translation in neurons? Here, we detected ~8 ribosome copies per micrometer of dendrite and ~8 tagged copies of nascent proteins per micrometer of dendrite within 15 min of metabolic labeling. Note that since ribosomes cluster around synapses, these numbers could increase by nearly 10-fold at synapses. Taking the necessary subsampling into consideration, these estimates give a lower bound of the local translation that occurs in dendrites. Furthermore, each neuron establishes to 10,000 synapses with other neurons, giving rise to the active local translation of 103 to 104 transcripts within 15 min. Last, we tagged ~1 copy of nascent protein per ribosome copy within a 15-min labeling period. Considering that a translating ribosome makes, on average, one copy of protein per minute (~5 amino acids per second; ~300 amino acids for an average protein) (30), we estimated a sample rate of ~10% for locally synthesized proteins (also see Materials and Methods). Taking this into account, we further estimate that roughly 80 protein copies are locally synthesized per micrometer of dendrite within just 15 min. Where do locally synthesized proteins go after being made by local ribosomes? While biophysical measurements and simulations have provided important information on how proteins move in the cytosol (31, 32), endogenous proteins made from local ribosomes may exhibit different movements because of their spatial confinement and molecular crowding (e.g., in spines versus in dendritic shafts) (33), as well as local protein-protein interaction mechanisms engaged to capture them (34). Thus far, it has been difficult to visualize and resolve the distribution (at a proteome-wide scale) of individual, locally synthesized, endogenous proteins. To directly probe this question, we localized ribosomes together with the nascent proteins tagged in distal dendrites using brief metabolic labeling. Hence, the locally synthesized protein cohort was spatially and temporally “dissected” from the somatic influx of nascent proteins. While some fast-diffusing proteins from the soma may have reached our fields of view, we did not observe a gradient of nascent proteins as a function of distance from the cell body; rather, we detected a tight spatial relationship between dendritic ribosomes and nascent proteins. These suggest that, although individual protein species may differ markedly in their addressing from local sources, the locally synthesized proteome, as a whole, tends to distribute locally. How “local” is local protein synthesis? While early thinking posited the idea of a dedicated toolkit of mRNAs and ribosomes for an individual synapse (35), emerging data rather suggest a local sharing of resources: a local “neighborhood” of synapses populated by translating ribosomes, mRNAs (36, 37), and potentially other essential organelles such as mitochondria (25) and endoplasmic reticulum (4). In our experiments, despite the observation that most nascent proteins were initially detected close to ribosomes, chase experiments revealed that a significant fraction of the locally synthesized proteome appeared somewhat mobile and spread over time. Previous studies have found that even scaffold proteins such as PSD95 are constantly exchanged between neighboring synapses (38). The neighborhood specificity of local protein synthesis is further evident during plasticity: Local clusters of synapses obtained similar level of proteins on average during global upscaling, while single-spine stimulation elicited an increase of local protein supply that “spilled over” to neighboring spines. This local sharing of protein supply may prime adjacent synapses for subsequent plasticity (8). Many studies have suggested that mRNAs are not present in sufficient copy numbers to populate all synapses (e.g., one calcium/calmodulin-dependent protein kinase II mRNA per 5 μm, rather than per synapse) (36). The local sharing of nascent protein is therefore implied by both the densities of mRNAs and ribosomes and the nascent proteins detected in our single-spine plasticity experiments. These provide a potential molecular basis for clustered and heterosynaptic plasticity observed to underlie various forms of plasticity and memory (6, 39, 40). MATERIALS AND METHODS Cell culture Dissociated rat hippocampal neuron cultures were prepared and maintained as previously described (28). Briefly, rat hippocampi were dissected from postnatal day 0 to 1 pups of either sex (Sprague-Dawley strain; Charles River Laboratories), dissociated with papain (Sigma-Aldrich), and then plated at a density of 30 × 103 cells/cm2 on poly-d-lysine–coated glass-bottom petri dishes (MatTek). Neurons were maintained (until day of experiment) in a humidified atmosphere at 37°C and 5% CO2 in growth medium (Neurobasal-A supplemented with B27 and GlutaMAX-I; Life Technologies) for 13 to 21 days in vitro (DIV) to ensure synapse maturation. For polysome profiling, because of the larger quantity of cells needed, dissociated rat cortical neuron cultures were prepared and maintained similar to above. All experiments complied with National Animal Care Guidelines and the guidelines issued by the Max Planck Society and were approved by local authorities. Ribosome profiling Harringtonine (LKT Laboratories) was prepared at a final concentration of 2 μM (2 mM stock, 100% ethanol) and incubated with cells for 30 min at 37°C. Puromycin (Thermo Fisher Scientific) was prepared at a final concentration of 500 μM and incubated with cells for 10 min at 37°C. After treatments, cells were immediately placed on ice, washed with ice-cold phosphate-buffered saline (PBS) containing cycloheximide (100 μg/ml), lysed, and scraped in polysome lysis buffer [20 mM tris (pH 7.4), 150 mM NaCl, 5 mM MgCl2, TURBO DNase (24 U/ml), cycloheximide (100 μg/ml), 1% Triton X-100, 1 mM dithiothreitol (DTT), RNasin Plus RNase Inhibitor (200 U/ml), and 8% glycerol]. After scraping, the lysates were pipetted up and down until homogenization was clear with a 0.4 × 20 mm syringe needle (HSW FINE-JECT) on ice. The lysates were then centrifuged at 10,000g for 10 min at 4°C. The supernatant was used for ribosome fractionation. For sucrose gradients, all solutions were prepared in gradient buffer [20 mM tris (pH 7.5), 8% glycerol, 150 mM NaCl, 5 mM MgCl2, cycloheximide (100 μg/ml), and 1 mM DTT]. Gradients were prepared by sequentially adding different sucrose concentrations (in order from first added to last: 8 ml of 55%, 0.5 ml of 50%, 0.5 ml of 40%, 0.5 ml of 30%, 0.5 ml of 20%, and 0.5 ml of 10%) into the same thin-wall polypropylene tube (Beckman, catalog no. 331372). After the addition of each sucrose solution, tubes were placed at −80°C to freeze the content before the next layer. The gradients were stored at −80°C and left to equilibrate at 4°C overnight. Then, 0.5 to 1.5 optical density (measured with NanoDrop at 260 nm) of the lysates were loaded on top of the gradients and spun at 36,000 rpm at 4°C for 2 hours with a SW 41 Ti rotor (Beckman). Fractions from each sample were collected every 7 s using a density gradient fractionation system (Teledyne ISCO; intensity used, 1), chased by 60% sucrose and 10% glycerol in water at 850 μl/min, and continuous monitoring at 254 nm using a UA-6 detector. Fractions of 125 μl corresponding to the 40S, 80S, or the polysome peaks were collected and pooled as the enriched 40S, 80S, and polysome fractions, respectively. For plating ribosome fractions, diluted fractions were incubated in (1:100 dilution in the above lysis buffer without TURBO DNase, glycerol, and protease inhibitor cocktails) poly-d-lysine–coated MatTek dishes for 2 hours before the liquid was removed, and 4% paraformaldehyde in PBS containing cycloheximide (100 μg/ml) was added to fix the samples. After fixation, the dishes were washed three times using PBS buffer (pH 7.4). Metabolic labeling with AHA and synaptic scaling Neurons (18 to 21 DIV) on MatTek dishes were incubated in the growth medium described above (for upscaling, 2 μM TTX was used for 1 or 24 hours) at 37°C and 5% CO2. Neurons were incubated in methionine-free Neurobasal-A (custom-made by Life Technologies) supplemented with 4 mM AHA for 15 min (10). For upscaling, this step started 15 min before each treatment ended, and 2 μM TTX was added. In methionine control experiments, AHA was replaced by 4 mM methionine (Sigma-Aldrich). Subsequently, cells were washed twice with Neurobasal-A, fixed in paraformaldehyde-sucrose (4% paraformaldehyde; Alfa Aesar) and 4% sucrose in PBS supplemented with Mg2+ and Ca2+ at room temperature for 20 min, washed again, permeabilized with 0.5% Triton X-100 in 1 × PBS (pH 7.4) for 15 min, and blocked with blocking buffer (4% goat serum in 1× PBS) for 1 hour. To optimize conditions for the subsequent click reaction, neurons were equilibrated by washes with 1× PBS (pH 7.8). The BONCAT part of the assay was performed as described previously (28) with the following modification: We used a previously reported single-stranded DNA oligo sequence (P1 docking oligo) (12) modified to carry a more reactive alkyne, dibenzocyclooctyne (DBCO; GeneLink), as a tag in the copper-free azide-alkyne cycloaddition click reaction (19). For the Cu-free click reaction, 0.4 μM P1-DBCO tag was prepared in 1× PBS (pH 7.8) before application to the cells and the click chemistry was performed overnight at room temperature. After the click reaction, cells were washed two times with PBS (pH 7.8) and 0.5% Triton in PBS and three times with PBS (pH 7.4) before immunofluorescence labeling with various markers (see below). We estimated the undersampling rate of our metabolic labeling on the basis of the measured synaptic nascent protein localizations (see below the “Data analyses” section). Immunofluorescence labeling Neurons were washed three times in PBS before blocking in PBS containing 4% goat serum (Gibco) for 1 hour. Neurons were incubated overnight with guinea pig antibodies, anti-MAP2 (1:2000; 188004, Synaptic Systems), or chicken antibodies, anti-MAP2 (1:2000; ab5392, Abcam), and mouse antibodies, anti-PSD95 (1:1000; MA1-046, Thermo Fisher Scientific), or guinea pig antibodies, anti-Bassoon (1:1000; 141004, Synaptic Systems), in PBS containing 4% goat serum (Gibco) at 4°C to stain the dendrites for morphology and synapses, respectively. To stain ribosomal subunits, mouse antibodies, RPL36a (1:500 overnight, or 1:1000, 3 hours for subsampling; sc-100831, Santa Cruz Biotechnology), and rabbit antibodies, RPS11 (1:500 overnight, or 1:1000, 3 hours for subsampling; A303-936A, Bethyl Laboratories), were used. The samples were then washed three times in PBS (5 min each) before incubation for 1 hour with anti–guinea pig antibody conjugated with Alexa Fluor 488 (1:1000; Nanoprobes) or anti-chicken antibody conjugated with Alex Fluor 405 (1:1000; Nanoprobes) and anti-mouse antibody with Alexa Fluor 546 (1:1000; Nanoprobes) or anti–guinea pig antibody with Alexa Fluor 488 (1:1000; Nanoprobes). For secondary antibody staining of ribosome for DNA-PAINT, anti-mouse antibodies conjugated with P1 imager oligo and anti-rabbit antibodies conjugated with P5 imager oligo were used (1:1000; custom-made, as previously reported) (9). For ribosome-nascent protein colocalization, anti-mouse antibodies conjugated with P3 imager oligo and anti-rabbit antibodies conjugated with P5 imager oligo were used for ribosomal subunits. Neurons were washed three times in PBS (5 min each). All steps were performed at room temperature. Neurons were then stored in PBS at 4°C for up to 3 weeks until DNA-PAINT imaging. Transfection, Ca2+ imaging, and local spine stimulation Experiments were conducted similar to previously described (25). Briefly, transfections were carried out 12 DIV with CombiMag (OZ Biosciences) and Lipofectamine 2000 (Invitrogen) following the manufacturer’s instructions. Live cell imaging was carried out using 13 DIV hippocampal neurons. All live cell imaging was performed at 32°C, in E4 imaging buffer [120 mM NaCl, 3 mM KCl, 10 mM Hepes (pH 7.4), 3 mM CaCl2, 1 mM MgCl2, and 10 mM glucose]. Glutamate uncaging experiments used a modified E4 buffer lacking MgCl2 and containing 4 mM CaCl2. All live cell imaging used an inverted spinning disk confocal microscope (3i imaging systems; model CSU-X1) using the SlideBook 5.5 software. Images were acquired with a Plan-Apochromat ×63/1.4 oil differential interference contrast objective at laser powers 1.1 mW (488 nm) and 0.8 mW (561 nm) for basal Ca2+ imaging and glutamate uncaging experiments with an Evolve 512 camera (Photometrics). We used 488-nm excitation and 525/30-nm emission filters for imaging GCaMP6s, and 561-nm excitation and 617/73-nm emission filters were used for imaging PSD95-mCherry. Images were analyzed using ImageJ (see the “Data analyses” section for details). OriginPro 2019 was used for data analysis, statistical testing, and plotting graphs. Transfected neurons were identified by GCaMP6s fluorescence (i.e., calcium transients). For basal Ca2+ activity imaging, 500 frames were acquired at 1 Hz (five planes each spanning 5 μm in Z) in E4 buffer containing 1 μM TTX followed by 15-min metabolic labeling with AHA (in E4 buffer; 4 mM) before fixation. Subsequent steps are as described in the “Metabolic labeling with AHA and synaptic scaling” section. For glutamate uncaging experiments, spines were identified using PSD95-mCherry fluorescence. Immediately before glutamate uncaging, neurons were treated with 1 μM TTX (citrate salt, 2 mM stock made in water), 50 μM forskolin (100 mM stock made in dimethyl sulfoxide; Tocris Bioscience), 4 mM AHA, and 2 mM 4-methoxy-7-nitroindolinyl–caged l-glutamate (100 mM stock made in E4 buffer; Tocris Bioscience) in modified E4 buffer lacking Mg2+ (see above). Glutamate uncaging was carried out with a 720-nm multiphoton laser (Chameleon, Coherent) and a Pockels cell (Conoptics) for controlling the uncaging pulses. Spines at least 50 μm away from the cell body were chosen for uncaging experiments. To test a spine’s response to an uncaging pulse, an uncaging spot (∼2 μm2) close to a spine head was selected, and two to three uncaging pulses at 10-ms pulse duration per pixel and 2.5-mW power were delivered to confirm spine-specific calcium transients. During uncaging, 60 uncaging pulses at 0.5 Hz with 10-ms pulse duration per pixel at 2.5-mW power were used. After uncaging, neurons were left in the same AHA-containing buffer for ~13 min, adding up to a total of 15 min of AHA incubation before washing with modified E4 buffer and fixation. Following fixation, neurons were processed as described in the “Metabolic labeling with AHA and synaptic scaling” section. Superresolution microscopy For DNA-PAINT imaging, the imaging buffer contains 500 nM P1, P3, or P5 conjugated with Atto655 (Eurofins Genomics) in 500 mM NaCl in PBS (pH 7.4) (20). For ribosome-nascent protein colocalization, 2 nM P1 was used for nascent protein PAINT to shorten the duration of multiplexed imaging. Immunolabeled cultured neurons containing 90-nm gold fiducial markers (A1190, Nanoparz) were imaged on an N-STORM system (Nikon, Japan): an Eclipse Ti-E inverted microscope, equipped with a Perfect Focus System and a motorized x-y stage. Total internal reflection fluorescence (TIRF) and highly inclined and laminated optical sheet (HILO) (20) configurations were adjusted using a motorized TIRF illuminator in combination with a ×100 oil-immersion objective [CFL Apo TIRF; 1.49 numerical aperture (NA)] with a final pixel size of 158 nm. For imaging, 647-nm excitation wavelength was used and housed in a MLC400B (Agilent) laser combiner. An optical fiber guided the laser beam to the microscope body and via a dichroic mirror (T660LPXR, Chroma) to the sample plane. Fluorescence emission was separated from excitation light via a band-pass filter (ET705/72m, Chroma) and detected by an iXon Ultra electron multiplying charge-coupled device (EMCCD) camera (DU-897U-CS0-23 #BV, Andor). The software NIS-Elements Ar/C and μManager were used to control the setup and the camera. Alternatively, immunolabeled cultured neurons prepared similarly were imaged on a Leica DMi8 S system with Infinity TIRF HP, Infinity Scanner, and an iXon Ultra 888 EMCCD camera (Andor). Oil-immersion objective (×100; HC PL APO CORR TIRF; 1.47 NA) was used in combination with a motorized TIRF illuminator. For imaging, 638-nm excitation wavelength was used (150 mW), and LAS X software package was used for image acquisition. Wide-field micrographs of the MAP2 and PSD95 (or Bassoon) reference markers (see the “Immunofluorescence labeling” section) were obtained for nonproximal dendrites (at least one branching point away from the soma) before DNA-PAINT as summed projections of 2-μm-thick Z-stacks to capture the entire dendritic volume. The chosen fields of view typically contained clearly separated dendrites that originated from the same neuron. HILO illumination was used for superresolution acquisition with a power of 30 to 40 mW, which was determined directly after the objective and under wide-field configuration. For the N-STORM system, the intensity density (45% intensity of 647-nm laser) was 0.9 kW/cm2. Time-lapse datasets with 50,000 frames and 16-bit depth were acquired for nascent protein localization at 5-Hz frame rate and 5-MHz camera readout bandwidth (preamplification, 3; electron multiplying gain, 4). For the DMi8 S system, time-lapse datasets with 50,000 frames and 16-bit depths were acquired for ribosomal subunit localization (25,000 frames for nascent protein PAINT in ribosome-nascent protein colocalization) at 5-Hz frame rate and 10-MHz camera readout bandwidth (gain list, 2; electron multiplying gain, 100). Data analyses DNA-PAINT acquisitions were reconstructed with Picasso:Localize, a module of the Picasso software (12), by applying a minimal net gradient of 2500 (for N-STORM) or 15,000 (for DMi8 S). With Picasso:Render, drift corrections were applied in two subsequent fashions: First, a drift correction based on the redundant cross-correlation, with a segmentation of 200, was applied. Second, fiducial markers (gold beads) were manually selected, localized, and used for drift correction. Drift-corrected data were filtered using Picasso:Filter. Raw localizations within the same location were further filtered temporally on the basis of their average frame numbers and SD of frame numbers to eliminate background signal due to the unspecific binding of the imager oligo to a random target. These background signals are often clustered temporally rather than distributed through the imaging course, resulting in lower or higher average frame numbers and lower frame number variance compared to real signal. Afterward, raw localizations within a maximal distance of 6× measured localization precision and showing a maximum number of transient dark frames of 20 were linked together, resulting in a single, linked localization event (referred to as “localization”). Localization precision (Nearest neighbor based analysis values) was determined to be 13.1 nm, as previously described using nearest neighbor–based analyses (41). Coincidence detection thresholds the interdistances between small- and large-subunit clusters at 100 nm based on the estimates of oligo docking site interdistance (maximum, ~50 nm), localization precision (see above paragraph), and corrected image drift (based on fiducial marker separation after drift correction). Without primary or secondary antibodies, only a low level of background signal was observed in the dendrites (fig. S1G). Prolonged labeling created too much signal, making it impossible to isolate and analyze individual clusters of ribosomal localizations (fig. S1E). Subsampling was thus optimized in situ (fig. S1F). Using purified monosomes plated on a coverslip (fig. S2, B and C), subsampling is estimated to detect 31% RPS11, 33% RPL36a, 10% of monosomes (i.e., individual 80S ribosomes), and, by calculation, most polysomes (e.g., a calculated ~70% for a polysome with four 80S ribosomes, an average ribosome cluster). Synaptic regions were determined using a custom-written algorithm and the diffraction-limited PSD95 or Bassoon immunolabeling signal. To identify the positions of excitatory synapses, local PSD95 (or Bassoon) puncta maxima and minima were identified and normalized to the same intensity range (0 for minima and 255 for maxima). Pixels of PSD95 puncta with over 12% intensity (30 in normalized intensity) of its associated, normalized local maximum (255 in normalized intensity) were selected to define the synaptic compartments. These local intensity thresholds resulted in puncta size estimates that were less affected by local intensity and background differences (e.g., the phenomena where brighter puncta appear larger by eye while less bright puncta appear smaller; fig. S3, right table) because of heterogeneity in staining or focusing. Puncta were eliminated if they are on the soma or more than 2 μm away from a dendritic shaft marked by MAP2 signal; Puncta with sizes smaller than 0.15 μm2 were excluded from size-based analyses because of significant inaccuracy of size measurements at smaller spatial scales. Dendrites with a high level of background signal outside MAP2-labeled regions, dendrites containing overlapping signals with AHA labeling from glia, out-of-focus dendritic branches, and branches shorter than 5 μm were excluded from analyses. All in all, this created a synapse mask with a measured, average synapse density and synapse size distribution consistent with published values (17). This mask effectively enriched protein signals allocated into the synaptic area by excluding adjacent regions such as nearby proteins in the shaft (e.g., spine base). To evaluate adjacent regions such as the spine neck and base, a mask was generated to include an area of 1 μm in diameter centered on the PSD intensity maxima; analyses showed qualitatively consistent results. All localizations <2.5 μm from a skeletonized line that traversed the center of the MAP2-labeled dendrites were included as total dendritic localizations. Synaptic nascent protein localizations were further identified by a custom-written, noise-tolerant cluster identification algorithm (Density-based spatial clustering of applications with Noise based) (42), the spatial-distance parameter (r) of which was determined using the measured localization precisions for each dendrite. In particular, any cluster with <3 localizations within r was excluded from clustering, which is more selective than DBSCAN. It is therefore more robust against the inclusion of noise localizations. The PAINT signal of a single docking oligo (further approximated as a single protein copy due to the low incorporation efficiency of AHA) was determined as follows: We plotted the number of linked localization distributions (see fig. S2) for all the identified synaptic clusters and attributed the smallest population of clusters as the signal of a single protein copy; we further determined the average dark time for a single-protein copy from this population (containing ≤11 linked localizations) (fig. S2) and found it comparable with previously reported benchmark test of a single docking oligo based on DNA origami (with identical setup) (20); hence, we calculated that a single protein copy corresponds to roughly 7 ± 4 linked localizations. Because of these approximations and the known difficulties in obtaining accurate molecule counting (43), in the main text, we directly use the number of linked localizations as a proxy measure for protein copy numbers. In parallel, the widely used DBSCAN cluster identification was also implemented to double-check the robustness of our subsequent analyses. Afterward, Picasso:Render and ImageJ were used for dendritic-branch level analyses. A 1D dendritic tree was constructed on the basis of the MAP2-immunolabeling signal. Synapses and localization positions were then mapped onto the closest dendritic branch. The local synapse and synaptic nascent protein densities per unit length were obtained by applying smoothing with a Gaussian kernel with parameter σ = 2 μm. Note that for protein density, this contribution is proportional to its synaptic localization counts. To define dendritic segments, borders were drawn at the turning points of the smoothed synapse density, such that segments with a range of low and high synapse densities were identified (fig. S3F). The majority of dendritic segments were <10 μm in length, containing <10 synapses. We constrain segment size to be at least 0.5 μm and the slope of the synapse density at the turning points to be >0.1 synapse/μm2 for partitioning into a new segment. The synapse densities and synaptic nascent protein densities were measured using the corresponding total numbers divided by the dendritic lengths. To estimate the undersampling rates due to metabolic labeling and click chemistry, we note that in untreated groups, synaptic nascent protein localization was ~1 to 2 per synapse. A synapse contains approximately 104 to 105 protein copies with an average half-life of 5 DIV (44). This gives an estimate of 1 to 10 proteins being renewed every 15 min, which amounts to ~7 to 70 linked localizations per synapse (fig. S5; assuming ~1 localization per synapse in untreated). This yields an undersampling estimate of 86 to 98.6%, meaning that we likely acquire data from 1.4 to 14% of the synaptic nascent protein pool. Undersampling is crucial for avoiding high-density, overlapping localizations, which may introduce inaccuracies in quantification (see fig. S4). To count the synaptic Ca2+ events during GCaMP6s imaging, a maximum intensity projection image was generated from the GCaMP6s time-lapse images. A region of interest (ROI) was chosen for each PSD95-mCherry–positive spine, and a temporal fluorescence profile over 500 frames of imaging was generated for each ROI. Using automatic peak identification in OriginLab 2019, the number of Ca2+ events was counted for each ROI. To measure changes in spine size, a line crossing the center of the spine head was drawn, as described previously (25). The pixel intensity profile along the line was fit to a Gaussian to obtain the full width at half maximum, which is defined as the spine head width and used as a proxy for spine head size changes. For glutamate uncaging experiments, the shape of the uncaged spine was traced using the GCaMP6s signal, including 500 nm into the dendritic shaft, ±1 μm upstream and downstream. δF was calculated by subtracting the average projection GCaMP6s signal before uncaging (F) from the maximum projection GCaMP6s signal during uncaging. δF/F was normalized by the stimulated spine area and used as a measure for Ca2+ transients due to uncaging. The number of nascent protein localizations (20,000 frames; 5 nM P1 imager) within the defined spine and base area was counted. As control, unstimulated PSD95-mCherry–positive spines from adjacent dendritic branches were measured similarly. Constructs PSD95-mCherry construct was obtained in-house (45); GCaMP6s was purchased from Addgene (plasmids 40753). Reagents Unless noted otherwise, all substances were molecular biology or cell culture grade and purchased from Sigma-Aldrich or Roth. TTX citrate was used from stock solutions (Tocris Bioscience; 2 mM in H2O and 2 μM final). Statistical analysis Two-sample t tests were used for comparisons between two groups. Comparisons between more than two groups were conducted using analysis of variance (ANOVA) with Bonferroni post hoc analysis. One-tailed tests were used for Pearson’s correlations. For all statistical tests, a P value of less than 0.05 was considered significant. For experiments that involved comparisons between drug treatments, at least six cell replicates from four preparations were measured for each treatment group. Otherwise, at least three cell replicates from three preparations were measured for each experiment. Acknowledgments We thank G. Tushev, F. Kretschmer, and L. Anneser for assistance in coding; A. R. Dörrbaum and S. T. Dieck for instructions during metabolic labeling and synaptic upscaling; C. Böger and A. S. Hafner for advice on image acquisition and analysis of DNA-PAINT; I. Bartnik, N. Fuerst, and D. Vogel for the preparation of cultured neurons; A. Schwartz for EM imaging; the Max Planck Institute for Biophysics for sharing EM facility; A. Koegel and E. Conti for the gift of purified ribosomes; and G. Laurent for inputs during manuscript preparation. Funding: C.S. is supported by an EMBO long-term postdoctoral fellowship (EMBO ALTF 860-2018) and HFSP Cross-Disciplinary Fellowship (LT000737/2019-C). C.S. and A.N. are both supported by Joachim Herz Stiftung Add-on Fellowship for Interdisciplinary Life Science (project number 850027). T.T. is funded by the Max Planck Society, the DFG CRC1080: Molecular and Cellular Mechanisms of Neural Homeostasis, and the fellowship of the Behrens-Weise-Foundation. M.H. is funded by the German Science Foundation (grants SFB 807, SFB902, HE 6166/11-1, and EXC115) and the Bundesministerium für Bildung und Forschung (BMBF:eBio). E.M.S. is funded by the Max Planck Society, an Advanced Investigator award from the European Research Council (grant 743216), DFG CRC 1080: Molecular and Cellular Mechanisms of Neural Homeostasis, and DFG CRC 902: Molecular Principles of RNA-based Regulation. Author contributions: Conceptualization: C.S. and E.M.S. Experimentation and image analyses: C.S., V.R., and C.M.F. Data analyses: C.S. and A.N. Coding: A.N. Manuscript writing: C.S. and E.M.S. Manuscript editing: all authors. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Supplementary Materials This PDF file includes: Figs. S1 to S14 Download 1.91 MB View/request a protocol for this paper from Bio-protocol. REFERENCES AND NOTES 1 A.-S. Hafner, P. G. Donlin-Asp, B. Leitch, E. Herzog, E. M. 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This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited. Submission history Received: 19 April 2021 Accepted: 27 July 2021 Permissions Request permissions for this article. Request permissions Acknowledgments We thank G. Tushev, F. Kretschmer, and L. Anneser for assistance in coding; A. R. Dörrbaum and S. T. Dieck for instructions during metabolic labeling and synaptic upscaling; C. Böger and A. S. Hafner for advice on image acquisition and analysis of DNA-PAINT; I. Bartnik, N. Fuerst, and D. Vogel for the preparation of cultured neurons; A. Schwartz for EM imaging; the Max Planck Institute for Biophysics for sharing EM facility; A. Koegel and E. Conti for the gift of purified ribosomes; and G. Laurent for inputs during manuscript preparation. Funding: C.S. is supported by an EMBO long-term postdoctoral fellowship (EMBO ALTF 860-2018) and HFSP Cross-Disciplinary Fellowship (LT000737/2019-C). C.S. and A.N. are both supported by Joachim Herz Stiftung Add-on Fellowship for Interdisciplinary Life Science (project number 850027). T.T. is funded by the Max Planck Society, the DFG CRC1080: Molecular and Cellular Mechanisms of Neural Homeostasis, and the fellowship of the Behrens-Weise-Foundation. M.H. is funded by the German Science Foundation (grants SFB 807, SFB902, HE 6166/11-1, and EXC115) and the Bundesministerium für Bildung und Forschung (BMBF:eBio). E.M.S. is funded by the Max Planck Society, an Advanced Investigator award from the European Research Council (grant 743216), DFG CRC 1080: Molecular and Cellular Mechanisms of Neural Homeostasis, and DFG CRC 902: Molecular Principles of RNA-based Regulation. Author contributions: Conceptualization: C.S. and E.M.S. Experimentation and image analyses: C.S., V.R., and C.M.F. Data analyses: C.S. and A.N. Coding: A.N. Manuscript writing: C.S. and E.M.S. Manuscript editing: all authors. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Authors Affiliations Chao Sun Max Planck Institute for Brain Research, Frankfurt, Germany. Roles: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Validation, Visualization, Writing - original draft, and Writing - review & editing. View all articles by this author Andreas Nold Max Planck Institute for Brain Research, Frankfurt, Germany. Institute of Experimental Epileptology and Cognition Research, Life and Brain Center, Universitätsklinikum Bonn, Venusberg-Campus 1, 53127 Bonn, Germany. Roles: Formal analysis and Software. View all articles by this author Claudia M. Fusco Max Planck Institute for Brain Research, Frankfurt, Germany. Roles: Conceptualization, Investigation, and Methodology. View all articles by this author Vidhya Rangaraju Max Planck Institute for Brain Research, Frankfurt, Germany. Roles: Conceptualization, Investigation, Methodology, Validation, and Writing - review & editing. Present address: Max Planck Florida Institute for Neuroscience, Jupiter, FL, USA. View all articles by this author Tatjana Tchumatchenko Max Planck Institute for Brain Research, Frankfurt, Germany. Institute of Experimental Epileptology and Cognition Research, Life and Brain Center, Universitätsklinikum Bonn, Venusberg-Campus 1, 53127 Bonn, Germany. Roles: Formal analysis, Funding acquisition, Methodology, Supervision, Validation, Visualization, and Writing - review & editing. View all articles by this author Mike Heilemann Institute of Physical and Theoretical Chemistry, Goethe University, Frankfurt, Germany. Roles: Formal analysis, Funding acquisition, Methodology, and Resources. View all articles by this author Erin M. Schuman erin.schuman@brain.mpg.de Max Planck Institute for Brain Research, Frankfurt, Germany. Roles: Conceptualization, Funding acquisition, Methodology, Project administration, Resources, Supervision, Visualization, Writing - original draft, and Writing - review & editing. View all articles by this author Funding Information European Molecular Biology Organization: EMBO ALTF 860 H2020 European Research Council: 743216 H2020 European Research Council: 743216 H2020 European Research Council: EMBO ALTF 860-2018 H2020 European Research Council: 850027 H2020 European Research Council: DFG CRC1080 Human Frontier Science Program: LT000737/2019-C German Science Foundation: HE 6166/11-1 Joachim Herz Stiftung Add on Fellowship for Interdisciplinary Life Science Project: 850027 H2020 European Research Council: SFB 807 H2020 European Research Council: SFB 902 H2020 European Research Council: SFB902 Deutsche Forschungsgemeinschaft: SFB 902 Deutsche Forschungsgemeinschaft: CRC 1080 Notes Corresponding author. Email: erin.schuman@brain.mpg.de Metrics & Citations Metrics Article Usage Note: The article usage is presented with a three- to four-day delay and will update daily once available. Due to this delay, usage data will not appear immediately following publication. Citation information is sourced from Crossref Cited-by service. 54 citation in Crossref 51 citation in Web of Science Altmetrics See more details Picked up by 2 news outlets Posted by 39 X users Highlighted by 1 platforms 126 readers on Mendeley Citations Cite as Chao Sun et al. , The prevalence and specificity of local protein synthesis during neuronal synaptic plasticity.Sci. Adv.7,eabj0790(2021).DOI:10.1126/sciadv.abj0790 Export citation Select the format you want to export the citation of this publication. Cited by Takeshi Chujo, Kazuhito Tomizawa,Neurological Diseases Caused by Loss of Transfer RNA Modifications: Commonalities in Their Molecular Pathogenesis, Journal of Molecular Biology, 437, 16, (169047), (2025). Crossref 2. Claudia M. Fusco, Anja Staab, Ashley M. Bourke, Georgi Tushev, Kristina Desch, Erico Moreto Lins, Elena Ciirdaeva, Susanne tom Dieck, Nina Kaltenschnee, Alexander Heckel, Julian D. Langer, Erin M. Schuman,Neuronal processes contain the essential components for the late steps of ribosome biogenesis, Proceedings of the National Academy of Sciences, 122, 31, (2025). Crossref 3. Raul Portugal, Beatriz Rodrigues, Ricardo A. Leitão, Mariline Silva, Paulo S. Pinheiro, Ana Luisa Carvalho,Shaping the synapse through neuronal activity-regulated miRNAs, Trends in Neurosciences, (2025). Crossref 4. Chiara Fiorenzani, Adele Mossa, Silvia De Rubeis,DEAD/DEAH-box RNA helicases shape the risk of neurodevelopmental disorders, Trends in Genetics, 41, 5, (437-449), (2025). Crossref 5. Constanze Depp, Jordan L. Doman, Maximilian Hingerl, Judy Xia, Beth Stevens,Microglia transcriptional states and their functional significance: Context drives diversity, Immunity, 58, 5, (1052-1067), (2025). Crossref 6. 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Schwarz, Naguib Mechawar, Gustavo Turecki, Markus M. Nöthen, Anita C. Hansson, Christian C. Witt, Marcella Rietschel, Philipp Koch, Rainer Spanagel, Lea Zillich, Stephanie H. Witt,A multi-omics and cell type-specific characterization of the ventral striatum in human cocaine use disorder, Cell Reports, 44, 2, (115332), (2025). Crossref See more Citation information is sourced from Crossref Cited-by service. View Options View options PDF format Download this article as a PDF file Download PDF Figures Fig. 1. Quantitative, multiplexed, single-molecule localization of assembled ribosomes in neuronal dendrites and synapses. (A) A eukaryotic ribosome with RPL36a (magenta) and RPS11 (green; Protein Data Bank: 4V88). Bottom right: The coincidence detection of RPL36a and RPS11 by DNA-PAINT. (B) Representative images showing localizations of both subunits detected in a plated small-subunit fraction (left) and colocalizations detected in 80S (middle) and polyribosome fractions (right). Scale bar, 500 nm (C) Violin plots showing the size distributions of 40S localization clusters from the 40S fraction (0.014 ± 0.010 μm2), colocalized clusters from the 80S (0.026 ± 0.015 μm2), and the polyribosome fraction (0.062 ± 0.057 μm2). Analysis of variance (ANOVA) with post hoc Bonferroni test indicated significant differences (in μm2) among the three (P< 0.001). (D) Ribosome size distribution (number of localizations per cluster) following puromycin (dark gray) or homoharringtonine (light gray) treatment. (E) Representative image of ribosome occupancy in dendrites (4300 μm2) with anti-Bassoon (synapse, blue) and ribosomal small (green) and large subunits (magenta). Scale bar, 2 μm. Dashed box indicates a branch at higher magnification (right inset; transecting dashed lines indicate segment boundaries). Scale bar, 1 μm. (F) A cumulative plot showing synapse fractions (y axis) in (E) that had ribosomes within a certain radius (x axis). (G) Representative dendritic segments (and branch point) illustrated as an overlay of synapse density (heatmap) and ribosome localizations (contour map). Scale bar, 2 μm. (H) Scatter plot showing the Pearson’s correlation between ribosome localization density and synapse density among 31 dendritic segments in (E) (r = 0.67; one-tailed test, P < 0.001). GO TO FIGURE Fig. 2. Quantitative, multiplexed, single-molecule localization of locally synthesized nascent proteins and assembled ribosomes in neuronal dendrites and synapses. (A) Nascent proteins were metabolically labeled with AHA and subsequently conjugated with a single-strand DNA barcode for visualization. Right scheme shows that recurring, transient, fluorescent DNA hybridization enables single-molecule localization of nascent proteins. TTX, tetrodotoxin. (B) Representative image showing nascent proteins detected following AHA labeling (top image). Box indicates the region enlarged in (D) (see fig. S4B for MAP2 immunostaining). Scale bar, 6 μm. (C) Representative image showing no nascent proteins detected following control treatment with methionine (see fig. S4C). Scale bar, 6 μm. (D) A dendritic segment (outlined by black dashed lines) containing clusters of nascent protein localizations. Scale bar, 0.5 μm. (E) Representative distributions of nascent protein (contour map) and ribosome localizations (heatmap) in a dendrite (4300 μm2). Scale bar, 2.5 μm. (F) Representative three-color colocalizations of nascent protein (blue) and ribosomal large- (magenta) and small-subunit (green) clusters. Black indicates overlap. Scale bar, 300 nm. (G) Scatter plot showing the Pearson’s correlation between nascent proteins and ribosomes (r = 0.73; 143 dendritic segments and more than six cell replicates; one-tailed test, P < 0.001). (H) Violin plots of nearest-ribosome distances (y) for 999 nascent protein clusters detected with 30-min chase (lavender) after metabolic labeling and 985 clusters detected without (gray) chase . Two-sample t test indicated a significant increase in distance following chase (P < 0.01). (I) Representative dendritic segments showing the coincidence of ribosome hotspots (dark green) and nascent protein local maxima (magenta triangles) without and with chase following metabolic labeling. Scale bar, 2 μm. GO TO FIGURE Fig. 3. During basal activity, levels of nascent synaptic protein correlated with synaptic Ca2+ activity. (A) Wide-field micrograph of a neuronal dendrite after 24 hours of upscaling immunolabeled with dendritic and synaptic reference markers (MAP2 in green and PSD95 in magenta). Scale bar, 5 μm. (B) Example of a synaptic mask generated by a custom-written local thresholding algorithm (see Materials and Methods) for the dendritic arbor shown in (A). (C) Synaptic nascent protein localizations (magenta) and dendritic and synaptic reference fluorescence (gray). Scale bar, 4 μm. (D) Cultured rat hippocampal neurons were transfected with GCaMP6s and PSD95-mCherry (see Materials and Methods) before basal Ca2+ activity imaging (followed by 15-min AHA labeling; top row). Subsequent fixation, labeling, and single-molecule localization were as described in Fig. 2. (E) Wide-field micrograph showing a representative dendrite with the overlay of GCaMP6s (green), PSD95-mCherry (magenta), and nascent proteins (black). Scale bar, 4 μm. (F) Higher-magnification image of a dendritic segment [green; indicated by box in (E)] and associated synapses (magenta). (G) Sum Ca2+ activity heatmap of the dendritic segment shown in (F) [also indicated by box in (E)]. (H) Overlay of PSD95-mCherry (magenta) and nascent proteins (black) for the dendritic segment shown in (F) and (G) [also indicated by box in (E)]. Scale bar, 0.5 μm. (I) Scatter plot showing significant correlations between synaptic protein localizations (Y) and synaptic Ca2+ events during basal Ca2+ activity imaging [see (A) and Materials and Methods] for 110 synapses with corresponding adjusted Pearson’s correlation coefficient r and P value ranges for one-tailed tests (P < 0.001). GO TO FIGURE Fig. 4. A global increase in local protein supply was detected heterogeneously among synapses but homogeneously among dendritic segments of synapses during synaptic upscaling. (A) Scatter plots indicating the tagged protein (locally synthesized, nascent; same below) density (localization per micrometer) for 23 dendritic branches from six untreated cells (gray), 34 branches from six cells with 1 hour of upscaling (lavender), and 29 branches from six cells with 24 hours of upscaling (purple). ANOVA with post hoc Bonferroni analysis indicated significant increases after both 1 and 24 hours of upscaling [P < 0.001; same in (B)]. (B) Scatter plots indicating the mean synaptic protein, with significant increases after upscaling. n.s., not significant. (C to E) Example traces depicting the proteins at each synapse distributed along dendritic branches (y axis: nascent protein level; x axis: the series of synapses compressed and lined along the dendritic branches) from untreated and 1- and 24-hour-upscaled neurons. Purple arrows indicate example synapses with high nascent protein levels themselves but neighbored by synapses with low nascent protein levels. (F) Violin plots showing the wide distributions in nascent protein levels between synapses and the much tighter distributions (mean protein per synapse) between dendritic segments of synapses. White lines indicate mean values. Right inset depicts that local protein supply is similar among dendritic segments (pie chart indicates that segments containing more synapses obtain more proteins) but heterogeneous between neighboring synapses. (G to I) 3D scatter plots of (G) 1289 synapses from six untreated neurons, (H) 2955 synapses from six 1-hour upscaling neurons, and (I) 2625 synapses from six 24-hour upscaling neurons. Gray arrows in (H) indicate an apparent bifurcation of the synapse population. GO TO FIGURE Fig. 5. Single-spine stimulation induced a local increase in nascent protein among nearby synapses. (A) Cultured rat hippocampal neurons [12 days in vitro (DIV)] were transfected with GCaMP6s and PSD95-mCherry (see Materials and Methods) before single-spine stimulation (two-photon glutamate uncaging in AHA-containing buffer). Subsequent fixation, labeling, and single-molecule localization were as described in Fig. 2 (see also Materials and Methods). (B) Spinning disk micrograph with overlay of GCaMP6s (maximum projection in Z) and PSD95-mCherry showing the spine of interest before local stimulation (dashed box) and its neighboring synapses (encircled dark puncta). Scale bar, 2 μm. (C) Overlay of spinning disk GCaMP6s micrograph (maximum projection in time; gray) and single-molecule localization micrograph of locally translated nascent proteins (magenta). Dashed boxes indicate the spine and spine base of the stimulated spine. Dashed circles indicate its neighboring synapses. (D) Scatter plot showing significant correlations between nascent protein (localizations of locally synthesized proteins) in stimulated spine and spine base (Y) and δF/F GCaMP6s fluorescence during the stimulation of 17 spines from six preparations (X; see Materials and Methods) with corresponding adjusted Pearson’s correlation coefficient r and P value ranges for one-tailed tests (P < 0.05). (E) Scatter plots showing the nascent protein (localizations of locally synthesized proteins; Y) in the neighboring synapses of six stimulated spines (one color for each set of neighboring synapses) distributed along the dendrite (X). Inset box plot: two-sample t tests indicated a significant increase of local protein supply among neighboring synapses within 3 μm of the stimulated spine compared to those greater than 3 μm away from the stimulated spine (P < 0.01). (F) Box plot showing the mean synaptic protein (per synapse; normalized per cell) within each segment for 9 segments containing stimulated spines and 119 untreated segments. Two-sample t tests indicated a significant increase of local protein supply for segments containing stimulated spines (P < 0.01). GO TO FIGURE References References 1 A.-S. Hafner, P. G. Donlin-Asp, B. Leitch, E. Herzog, E. M. Schuman, Local protein synthesis is a ubiquitous feature of neuronal pre- and postsynaptic compartments. Science 364, eaau3644 (2019). Crossref PubMed Web of Science Google Scholar a [...] of synaptic function and plasticity b [...] for protein synthesis in neuronal processes 2 M. A. Sutton, H. T. Ito, P. Cressy, C. Kempf, J. C. Woo, E. M. 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18188	https://math.stackexchange.com/questions/4747797/linear-functional-equations	reference request - Linear functional equations - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Linear functional equations Ask Question Asked 2 years, 1 month ago Modified2 years, 1 month ago Viewed 312 times This question shows research effort; it is useful and clear 3 Save this question. Show activity on this post. Linear functional equations like (x+1)P(x)=(x−10)P(x+1)(x+1)P(x)=(x−10)P(x+1) are fairly common in competition math, and there are some general techniques for proving things about the solutions, such as degree of polynomial solutions, highest or lowest degree coefficients, etc. One could imagine a somewhat general theory for functional equations of the form L f=g L f=g, with f f the unknown, g g a given function, and L L a linear operator composed of the basic operations of multiplication by x x, i.e. f(x)↦x f(x)f(x)↦x f(x), and shifts f(x)↦f(x+a)f(x)↦f(x+a). Are there any published works on the theory of such equations? linear-algebra reference-request functional-equations Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications asked Aug 5, 2023 at 0:33 YlyYly 15.9k 4 4 gold badges 36 36 silver badges 76 76 bronze badges 2 They are commonly referred to as "finite difference equations".Ivan Neretin –Ivan Neretin 2023-08-05 10:17:49 +00:00 Commented Aug 5, 2023 at 10:17 The book A = B and the chapter on generating functions in Concrete Mathematics contain techniques that are useful. The particular example that you wrote is an hypergeometric term. The first book contains algorithms to deal with sums and difference equations of those, to determine when the solution can be written as a constant sum of hypergeometric terms. Things like that.NDB –NDB 2023-08-05 12:47:20 +00:00 Commented Aug 5, 2023 at 12:47 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. You're talking about Difference Equations. But you could also interpret it as a Recurrence Equation or as a Finite Difference Equation. You could also interperate it as a 0 0 th-order Difference / Delay Differential Equation, or a delay equation (but that's probably less said). In Applications of the Symbolic Toolbox (Chapter 10.13 10.13) by George Lindfield and John Penny they show a good method to solve such equations with constant coefficients. They use the unilateral Z Z transform with its usefull property of linearity and (where Z xy(x)=Y(z)Z xy(x)=Y(z) and k∈Z k∈Z): Z xy(x+k)=z k⋅Y(z)−∑m=0 k−1[z k−m⋅y(m)]Z xy(x+k)=z−k⋅Y(z)+∑m=1 k[z m−k⋅y(−m)]Z xy(x+k)=z k⋅Y(z)−∑m=0 k−1[z k−m⋅y(m)]Z xy(x+k)=z−k⋅Y(z)+∑m=1 k[z m−k⋅y(−m)] An example that they give is "6⋅y(x)−5⋅y(x−1)+y(x−2)=4−x 6⋅y(x)−5⋅y(x−1)+y(x−2)=4−x" and their solution is "y(x)=5 2⋅(1 2)x−2⋅(1 3)x+1 2⋅(1 4)x y(x)=5 2⋅(1 2)x−2⋅(1 3)x+1 2⋅(1 4)x" wich is correct - see here for prove. This method is also mentioned in TRANSFORM METHODS (Chapter Z Z-transforms) by S. Braun. Another method to search for the homogeneous solution is reduction to the characteristic equation as discussed in Theory and Applications of Numerical Analysis (Chapter 13 13 - ORDINARY DIFFERENTIAL EQUATIONS) by G.M. PHILLIPS and P.J. TAYLOR. You can find this method applied here too. More generally applied, you can find this method in Probability (Chapter 7 7 - DIFFERENCE EQUATIONS). A work focused directly on polynomial solutions of difference equations is Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials by Olha Shkaravska and Marko van Eekelen (this article is even free to read): This article addresses the problem of computing an upper bound of the degree d d of a polynomial solution P(x)P(x) of an algebraic difference equation of the form G(x)(P(x−τ 1),…,P(x−τ s))+G 0(x)=0 G(x)(P(x−τ 1),…,P(x−τ s))+G 0(x)=0 when such P(x)P(x) with the coefficients in a field K K of characteristic zero exists and where G G is a non-linear s s-variable polynomial with coefficients in K[x]K[x] and G 0 G 0 is a polynomial with coefficients in K K. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Aug 5, 2023 at 15:59 user1103878 user1103878 Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. 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18189	https://www.vendavo.com/glossary/revenue-optimization/	Skip to content Revenue Optimization The definition of revenue optimization can be simplified into achieving two key objectives: sell more and sell more profitably. Commercial Excellence Assessment Winning Pricing Strategies What is Revenue Optimization? Revenue optimization is a cohesive process that organizations use to effectively manage pricing structures, inventory levels, customer demand, and distribution channels in order to boost long-term revenue growth. The core of revenue optimization focuses on employing data-driven strategies like price optimization, consumer trend analysis, demand forecasting, and other models to improve the delivery of products and services to the right customers at the ideal price. Fundamentally, the definition of revenue optimization can be simplified into achieving two key objectives: sell more and sell more profitably. 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To achieve this balance, Vendavo provides a suite of growth and profitability solutions for pricing, profit analytics, selling, and rebate management. These solutions help businesses not only drive growth in their top-line revenue (81% of recently surveyed organizations report increased revenue in the last 12 months) but also ensure that they capture the full value of their offerings, thereby enhancing their bottom-line profitability. Key Components of Revenue Optimization Revenue optimization is a multifaceted, dynamic approach to increasing a business’s revenue streams. Each component plays a vital role in ensuring cohesive, lasting results from the strategies employed. Pricing Strategies By definition, revenue optimization relies on balancing the art and science of pricing strategies. More than just setting a monetary value on a product or service, it’s about taking into consideration the nuances of market demand, competition, production costs, and the perceived value by customers. Whether businesses opt for dynamic pricing, where prices adapt based on demand, or market penetration pricing to capture market share, the appropriate pricing strategy can pave the way for maximized revenues. Sales Funnel Optimization Beyond pricing, there’s the customer journey to consider. From the moment a potential customer becomes aware of a product to the instant they decide to purchase, every touchpoint matters. Sales funnel optimization is all about refining this journey. It could be the pull of a well-designed website landing page, the precision of tailored marketing messages, or the ease of a seamless checkout process. Not only does fine-tuning these elements ensure stronger sales conversions, but creates an experience that customers will want to revisit. Customer Retention Strategies Depending on the model, acquiring new customers can cost between five and 25 times more than retaining existing customers. In turn, customer retention strategies are critical in the realm of revenue optimization. Retaining customers is about building bridges with loyalty programs, fostering connections with personalized engagement, and consistently delivering unparalleled value. When businesses get this right, they not only convert more sales, but they create valuable brand ambassadors who amplify their offerings. Cross-selling and Upselling Delving deeper into transactional nuances, there are subtle techniques to enhance the value of each sale. Cross-selling is the art of complimenting a purchase with an additional offering. Imagine buying a laptop and being introduced to a sleek laptop bag or an ergonomic mouse. That’s cross-selling in action. Similarly, upselling nudges customers toward an upgrade on their existing purchase. If you’ve ever considered buying a phone and been presented with a higher storage option or an extended warranty, you’ve experienced upselling. Both strategies, when employed judiciously with cross-sell and upsell best practices, can amplify revenues without extensive marketing efforts. In essence, weaving these components of revenue optimization of a business’s overarching framework can tap into avenues for growth and prosperity. How to Optimize Your Revenue To effectively leverage revenue optimization, businesses must integrate aspects of acquisition, retention, expansion, and pricing into a unified strategy. This comprehensive approach relies on aligning various departments within an organization (marketing, sales, finance, and customer service) and encouraging these teams to collectively pool their data and insights. To elaborate, here are some of the core fundamentals by which businesses can optimize their revenue: Extensive Data Analysis: Modern businesses generate vast amounts of data. By leveraging analytics and employing data-driven decision-making processes, businesses can identify patterns, preferences, and pain points of their customers. These insights can guide pricing decisions, marketing campaigns, and product developments. Adaptive Pricing Strategies: Implement flexible pricing models that can adjust based on factors like demand fluctuations, seasonality, inventory levels, and competitor pricing. Techniques like dynamic pricing optimization, value-based pricing, or tiered pricing can be applied as per business needs. Enhanced Customer Experiences: A positive customer journey can lead to higher conversion rates and increased customer loyalty. This involves optimizing the sales funnel, streamlining the checkout process, ensuring post-sales support, and creating a feedback loop to continuously refine the experience. Focus on Customer Retention and Expansion: Beyond acquiring new customers, retaining existing customers is a critical driver of revenue growth. Implementing loyalty programs, exclusive offers, personalized content, and responsive customer service can increase repeat business. Diversify Distribution Channels: Don’t rely on just one platform or sales channel. Building a portfolio of sales channels can help minimize risks associated with overdependence on one strategy. This approach ensures that the right products are sold to the right customers at the right time and for the right price. Collaborate Across Departments: Encourage departments such as marketing, sales, finance, and customer service to share their data within a central place to improve business and customer outcomes. A collaborative approach can lead to more effective revenue optimization strategies and ensures a unified vision, optimizing strategies from various angles. Monitor and Adjust: Revenue optimization is an ongoing process that adapts to changes in the market, competition, and customer preferences. Regularly analyze your data, forecasts, and customer behavior to identify areas for improvement and make the necessary adjustments to your strategies based on performance metrics, customer feedback, and market research. Manage Inventory Efficiently: Balancing supply with demand is critical. Overstock results in holding costs, while stockouts can mean lost sales. Employ inventory management systems to optimize stock levels, predict demand, and minimize costs. Streamline Operational Costs: Revenue optimization isn’t just about increasing sales. By reducing operational inefficiencies and cutting unnecessary costs, businesses can increase the profitability of their net revenue. By integrating these tactics into their operations, businesses can not only optimize their revenue but also ensure sustainability and growth in the long run. Best Practices for Revenue Optimization While it’s critical to have the right tools and processes, the human element of revenue optimization is the underlying variable to success. Your software and decision-making are only as good as the people behind it. So, to effectively optimize for long-term revenue growth, it’s important to consider these best practices. Consider Profitability alongside Topline Revenue While topline revenue provides a snapshot of sales figures, true business success comes from profitability. Especially in industries with varying costs, it’s imperative to look beyond revenue numbers. By assessing revenue alongside associated costs, businesses can ensure they’re genuinely optimizing their returns. Account for Cost Variability One costly mistake is to overlook cost variations. Whether due to product specifications, differing customer needs, or fluctuating market conditions, costs can differ significantly. By factoring in these variations during revenue optimization processes, businesses can make better-informed pricing decisions, enhancing their profitability. Align with Market Demand Pricing and sales strategies need to resonate with market demand and customer preferences. It’s not just about selling in high quantities. It’s about ensuring each sale contributes positively to the bottom line. In turn, optimizing for profit, rather than just volume, is essential. Initiate Optimization with Available Data Perfect data is ideal, but waiting for it can stagnate growth. Instead of stalling revenue optimization efforts due to data concerns, businesses should work with what they have. Starting with the current data can lead to iterative improvements, offering insights that refine processes and get the ball rolling. Continuously Analyze the Market The market landscape is continually evolving. To stay ahead, businesses should regularly analyze market trends, competitor actions, and customer feedback. This continuous analysis allows for timely adjustments to strategies, ensuring they remain relevant and effective. Invest in Technology and Tools Modern revenue optimization often requires sophisticated tools that can handle vast amounts of data and provide actionable insights. Investing in the right technologies, like pricing software and sales optimization tools, can greatly enhance revenue optimization efforts. Regularly Review and Refine Strategies What worked yesterday might not work tomorrow. Regularly reviewing and refining revenue optimization strategies, based on performance metrics and market changes, ensures businesses remain agile and effective. By adopting these best practices, businesses can ensure that their revenue optimization efforts are not only effective but also sustainable, driving long-term growth and profitability. Future Trends for Revenue Optimization As technology and market dynamics evolve, revenue management and optimization will continue to witness transformative trends. Here’s a look at some of the anticipated future trends: Hyper-Personalization Beyond traditional personalized marketing, hyper-personalization will use real-time data to create an individualized customer experience. By understanding each customer’s journey and preferences, businesses can better tailor their offerings and pricing strategies. Advancements in AI and Machine Learning While AI and ML are already present in today’s revenue optimization platforms, advanced algorithms will continue to predict customer behavior more accurately, allowing businesses to adjust their strategies proactively. These technologies can dynamically price products in real time based on demand, inventory, and competitor prices. Augmented Reality (AR) and Virtual Reality (VR) These technologies will influence the shopping experience and, by extension, the pricing. Virtual storefronts and immersive product experiences can command premium pricing and offer opportunities for upselling. Sustainability-Driven Pricing As consumers become more eco-conscious, businesses will incorporate sustainability into their pricing strategies. Products and services that highlight sustainability might command higher prices due to increasing demand from environmentally-conscious consumers. Greater Emphasis on Data Privacy and Security As data becomes central to revenue optimization, ensuring its privacy and security will be paramount. Compliance with regulations and transparent data handling practices will influence customer trust and, in turn, revenue. Increased Focus on Experience Over Products More businesses will realize that consumers are often buying experiences as much as, if not more than, products. This shift will necessitate a rethinking of pricing strategies, emphasizing the value of the experience FAQs Why is Revenue Optimization Important? Revenue optimization is crucial for businesses because it ensures that a company is maximizing its profits from available resources and market opportunities. By analyzing data, refining pricing strategies, streamlining operations, and improving customer interactions, revenue optimization can help to increase the bottom line for businesses while enhancing operational efficiency and customer satisfaction. In a competitive marketplace, businesses that prioritize revenue optimization are better poised to capitalize on emerging trends, adapt to market fluctuations, and achieve sustainable growth. How Can You Analyze Your Revenue? To analyze your revenue effectively, you can follow these steps: Define your revenue analysis goals: Determine what you want to achieve with your revenue analysis, such as identifying top-performing products, understanding customer segments, or improving profitability. Choose the right revenue analysis type: There are various types of revenue analysis, including sales revenue analysis, product-based analysis, and forecasting. Select the type that aligns with your goals and business needs. Gather and organize your data: Collect relevant data on sales, products, customers, and other factors that impact your revenue. Ensure your data is accurate and up-to-date. Use appropriate analysis techniques: Apply techniques like interactive tables, pivot tables, time series, and bar charts to analyze your revenue data. These techniques can help you pinpoint trends, patterns, and insights. Break down your revenue by category: Analyze your revenue by different categories, such as monthly, quarterly, or yearly revenue, to understand the performance of your business over time. Calculate key financial ratios: Use profitability-centric financial ratios, such as return on revenue, to measure your company’s financial health and performance. Allocate revenue to underlying sales: Determine which products or services generated the underlying sales and allocate the revenue accordingly. This information can guide your future marketing strategies and product promotions. Consider external factors: Take into account external factors like economic downturns, new regulations, or global events that can impact your revenue. Ignoring these factors can lead to skewed interpretations of your analysis. Project future trends: Use your revenue analysis to project present trends into the future. This can help you make informed decisions and plan for potential growth or challenges. Continuously improve your analysis: Regularly review and update your revenue analysis techniques, tools, and processes to ensure accuracy and relevance. Avoid using outdated tools that may hinder your analysis. By following these steps, you can gain valuable insights into your revenue performance, identify areas for improvement, and make informed decisions to maximize your profits and business growth. What are the Levers of Revenue Optimization? There are four primary levers that businesses need to grasp in order to know which ones to pull to effectively optimize their revenue and profit margins. These include: Pricing: The most immediate and apparent lever, pricing involves setting the right price for products or services. This includes considerations like dynamic pricing (adjusting prices based on real-time demand and other factors), discount strategies, and psychological pricing. The goal is to find the price point that maximizes profitability while ensuring customer value. Inventory Management: This lever focuses on ensuring the right products or services are available at the right time. For tangible goods, this might mean maintaining optimal stock levels to meet demand without overstocking. In industries like hospitality, it might mean ensuring room availability during peak times. Effective inventory management ensures that opportunities for revenue aren’t lost due to stockouts and that money isn’t wasted on overstock. Distribution Channel Management: Businesses often sell through multiple channels, such as direct sales, third-party retailers, online platforms, and more. Channel management effectively involves choosing where and how to sell products or services for maximum reach and profitability. It also includes strategies for managing relationships with third-party vendors or distributors. Market Mix: This lever pertains to the mixture of customer segments that a business serves. By understanding different customer segments (e.g., business travelers vs. leisure travelers in the hotel industry) and their respective profitability, businesses can target their marketing and service efforts more effectively. It’s about attracting the right kind of customer, not just any customer. When synchronized effectively, these four levers allow businesses to maximize their revenue potential across different facets of their operations. About The Author Mitch is VP, Product Marketing, and a Profit Evangelist at Vendavo with 25+ years of experience in the technical, operational, marketing, and commercial arenas of the process industry. Prior to Vendavo, Mitch was with BASF and Orica in product marketing and business management, driving operational optimization, pricing excellence, and margin improvement, as well as personal engagement in high value sales negotiations. Mitch also has deep experience with raw materials supplier portfolio management having negotiated large scale and long-term agreements with global suppliers. Related Resources #### Price Smarter with Competitive Intelligence in B2B Get practical strategies to strengthen your pricing with market insight. Discover how B2B leaders are applying competitive intelligence to win… Read More > #### A 2025 Aftermarket Playbook: Disruption, Opportunity, and the Path Forward #### Growth + Profitability Summit 2025 Amsterdam Read More >
18190	https://www.sciencedirect.com/science/article/abs/pii/S0022283604012197	The Optimal Fraction of Hydrophobic Residues Required to Ensure Protein Collapse - ScienceDirect Skip to main contentSkip to article Journals & Books Access throughyour organization Purchase PDF Search ScienceDirect Article preview Abstract Introduction Section snippets References (53) Cited by (15) Journal of Molecular Biology ---------------------------- Volume 344, Issue 3, 26 November 2004, Pages 797-811 The Optimal Fraction of Hydrophobic Residues Required to Ensure Protein Collapse Author links open overlay panel Jiangbo Miao a, Judith Klein-Seetharaman a b, Hagai Meirovitch c Show more Add to Mendeley Share Cite rights and content The hydrophobic interaction is the main driving force for protein folding. Here, we address the question of what is the optimal fraction, f of hydrophobic (H) residues required to ensure protein collapse. For very small f (say f<0.1), the protein chain is expected to behave as a random coil, where the H residues are “wrapped” locally by polar (P) residues. However, for large enough f this local coverage cannot be achieved and the thermodynamic alternative to avoid contact with water is burying the H residues in the interior of a compact chain structure. The interior also contains P residues that are known to be clustered to optimize their electrostatic interactions. This means that the H residues are clustered as well, i.e. they effectively attract each other like the H-monomers in Dill's HP lattice model. Previously, we asked the question: assuming that the H monomers in the HP model are distributed randomly along the chain, what fraction of them is required to ensure a compact ground state? We claimed there that f≈p c, where p c is the site percolation threshold of the lattice (in a percolation experiment, each site of an initially empty lattice is visited and a particle is placed there with a probability p. The interest is in the critical (minimal) value, p c, for which percolation occurs, i.e. a cluster connecting the opposite sides of the lattice is created). Due to the above correspondence between the HP model and real proteins (and assuming that the H residues are distributed at random) we suggest that the experimental f should lead to percolating clusters of H residues over the highly dense protein core, i.e. clusters of the core size. To check this theory, we treat a simplified model consisting of H and P residues represented by their α-carbon atoms only. The structure is defined by the C α–C α virtual bond lengths, angles and dihedral angles, and the X-ray structure is best-fitted onto a face-centered cubic lattice. Percolation experiments are carried out for 103 single-chain proteins using six different hydrophobic sets of residues. Indeed, on average, percolating clusters are generated, which supports our theory; however, some sets lead to a better core coverage than others. We also calculate the largest actual hydrophobic cluster of each protein and show that, on average, these clusters span the core, again in accord with our theory. We discuss the effect of protein size, deviations from the average picture, and implications of this study for defining reliable simplified models of proteins. Introduction The hydrophobic interaction is the main driving force for protein folding.1 Therefore, a great deal of work has been done toward understanding the thermodynamic basis of hydrophobicity,2, 3 as well as the effect of this phenomenon on the details of protein structures.4, 5, 6, 7 The ability of a protein chain to organize itself in a stable compact structure is expected to depend strongly on the fraction, f, of the hydrophobic (H) residues. For small f (say f<0.1) of randomly distributed H residues, an H residue could become “wrapped” locally by several hydrophilic (P) residues to form a “blob”. This would lead to an effectively shorter random coil chain of blobs connected by flexible segments, which gains further stability from its high entropy (see Figure 1). However, when f is large enough, the local coverage of the H residues cannot be achieved any more and the only thermodynamic alternative to avoid contact with water is burying them in the interior of a compact chain structure. Obviously, if f is too large the molecule will precipitate and therefore the optimal value observed in real proteins will be a balance between these effects and others. It has been shown that most of the H residues are located in the interior of protein structures, while the exterior is populated mostly by P residues, which interact favorably with the surrounding water.8, 9, 10, 11, 12, 13, 14, 15, 16 More specifically, in an inner sphere of radius R around the center of mass, where R is the radius of gyration, the concentration of the H residues is larger than their fraction, f, in the entire sequence; this concentration decreases significantly in concentric spherical layers of increasing radii, i.e. in going from the core towards the surface, whereas an opposite trend is observed for the P residues.12, 13, 14 The interior contains P residues that are clustered in groups to optimize their electrostatic interactions. Therefore, the H residues are clustered as well,15 and even though the H residues only seek to avoid the contact with water, effectively they can be viewed as attracting each other. This picture is the basis for the HP model proposed by Dill.17, 18 Moreover, at least one H cluster should span most of the core, because if all the H clusters were localized (i.e. each of them surrounded by P residues), it would mean that a chain of blobs would provide the most stable solution rather than a compact structure. One objective of this work is to examine whether core-size H clusters exist in the interior of proteins. However, our main objective is to explain the experimental fraction f of H residues in terms of percolation theory that, as argued later, provides a relation between f, the clustering of H residues, and hence the compactness of protein structures. In its basic form, this theory is developed for a simple experiment carried out on the sites of a large empty lattice (say, a square lattice) as follows:19, 20 each site is visited and a particle is placed there with probability p or the site remains vacant with probability 1−p (using a random number). After completing this experiment for the entire lattice, one asks whether percolation has occurred, i.e. whether a cluster of occupied sites connecting (bridging) the opposite sides of the lattice has been created. It has been shown that for each lattice a critical probability, p c, exists (called the site percolation threshold), where p c is the minimal probability such that for p≥p c, percolation will always occur.19 For a square lattice p c≈0.59, and p c deceases as the coordination number of the lattice increases; thus, p c≈0.31, 0.25 and 0.18 for simple cubic, body-centered cubic, and face-centered cubic (fcc) lattices, respectively (see Figure 2).19 We seek to establish a connection between f and p c. Previously, we took the first step in this direction,21 by applying percolation theory to the simplified HP model.17, 18 In the HP model, a protein is described by a self-avoiding chain on a lattice consisting of N monomers (i.e. N–1 bonds) of two kinds, H and P. Two non-bonded H monomers that are nearest neighbors on the lattice interact with an attractive energy ε (ε=−\|ε\|), where the interaction of PP and HP contacts is zero. Thus, for a given distribution of the H monomers, the ground state (which might be degenerate) is a chain configuration with the lowest possible energy, E=n max ε, where n max is the maximal number of HH contacts. Assuming that the H-monomers are distributed at random along the chain, we have asked the following question: what should be their minimal fraction, f that would lead to a compact collapsed ground state? To answer this question, we first considered 21 a self-avoiding chain consisting only of P monomers and arranged it in a perfect compact structure (a square shape in Figure 3); then, each of the P monomers was visited and changed to an H monomer with probability f. This process is exactly a percolation experiment that for f≥p c would lead to a percolating cluster of the H monomers over the compact (square) chain structure, where in this case p c is the percolation threshold of both the perfect compact structure and the square lattice. However, the cluster is not symmetric, in the sense that only contacts of H monomers that reside on parallel (or anti-parallel) segments of the chain (horizontal HH contacts in Figure 3) contribute to the energy (hence to the stability of the structure), while HH contacts along the chain (perpendicular in Figure 3) do not contribute to the energy (see further discussion in Methods). Elsewhere, we have argued that such a percolating cluster is of low energy, “holding” the perfect compact structure together;21 thus, we have suggested that f≈p c is approximately the minimal fraction of H monomers required to guarantee a collapsed ground state. Indeed, simulations of the HP model on square and simple cubic lattices at low temperatures have supported this idea. Elsewhere, we have given heuristic arguments that this picture also applies to globular proteins,21 and here the connection between the HP model and real proteins is established further. While for f=p c the perfect compact structure introduced above is of low energy, in most cases this energy would not be the lowest possible for the given sequence, i.e. this structure is not the ground state with the maximal number of HH contacts (see Figure 4). Typically, the ground state is characterized by a higher concentration of H monomers in the interior than in the periphery (surface) and an opposite distribution of the P monomers. Correspondingly, the chain loses its perfect compact shape (square in Figure 3), which is manifested by a ramified periphery, while the maximal density, characteristic of the perfect compact structure, remains only in the interior. We call this region of maximal density, “the core” (circled in Figure 4). It should be pointed out that a percolation experiment can be carried out over compact chain configurations with a ramified surface such as the ground state in Figure 4; in this case, the site percolation threshold (p c) will increase because of the decrease in the effective coordination number of the ramified part. However, the percolation threshold for the core alone remains the same as for the perfect compact structure. The distinction between a percolation experiment over the core and over the entire structure will become important in what follows. We have argued earlier that, due to their clustering, the H residues in a real protein effectively attract each other; that is, they behave basically as in the HP model. Therefore, the main conclusions drawn for the HP model should apply to real proteins. Thus, a folded protein structure (i.e. an X-ray structure from the Protein Data Bank (PDB)) corresponds to the ground state of the HP model (for f=p c). Like the latter, the folded structure has a ramified surface and a core defined approximately as the spherical region of highest density around the center of mass. On the other hand, the perfect compact structure of a protein is unknown. One can assume, however, that the density of such a structure would be approximately the same as that of the core. Thus, to test our hypothesis that the experimental fraction, f of randomly distributed H residues is approximately equal to the percolation threshold for the perfect compact structure, we can carry out percolation experiments based on the experimental f on the protein core. To apply our analysis to real proteins, a simplified model of a protein is used, where an amino acid residue is represented by its α-carbon atoms and the structure is thus defined by the C α–C α virtual bond lengths, virtual bond angles, and virtual dihedral angles.22, 23 To keep the lattice picture alive (we shall argue that this is not mandatory), the PDB structure is best-fitted onto an fcc lattice (as described in Methods),23 the core region is defined, and percolation experiments are performed. The size of the largest percolation cluster is compared with the core size to determine whether percolation has occurred. We identify the largest H cluster to compare its size with respect to the core size. Note the distinction between the largest H cluster, which is based on the actual distribution of the (relatively concentrated) H residues in the core, and the clusters generated by the percolation procedure based on the smaller experimental fraction f of H residues in the entire sequence. Percolation experiments are carried out for 103 single-chain proteins using six different hydrophobic sets of residues. Indeed, on average, percolating clusters are generated, which supports our theory; however, some sets lead to a better core coverage than others. We calculate the largest actual hydrophobic cluster of each protein and show that, on average, these clusters span the core, again in accord with our theory. We discuss the effect of protein size, deviations from the average picture, and implications of this study for defining reliable simplified models of proteins. Our analysis is based on the assumption that the H residues are distributed at random over the sequence. Indeed, an early study of protein sequences by White and Jacobs supports this assumption,24 while in a later study these authors have found a slight bias toward the creation of shorter consecutive blocks of H residues than would be anticipated from a random distribution.25 Similar conclusions were found by Schwartz et al.,26 who studied sequences of proteins that are known to fold in aqueous solutions, and by others.27, 28 Therefore, our assumption of random distribution is an approximation, which is justified within the accuracy of our approach. It should also be pointed out that Stauffer29, 30 and de Gennes 31 have applied percolation theory to describe the cluster creation in sol–gel transition in polymers.32 Access through your organization Check access to the full text by signing in through your organization. Access through your organization Section snippets Fitting a protein structure to a lattice The first step in our calculations requires fitting the PDB X-ray structures to an fcc lattice (see Methods). Structures of 103 single-chain proteins are used in the calculations presented. In Table 1 results are shown for the root-mean-square deviation (RMSD) between the best-fitted structures of eight of the longer proteins and the corresponding X-ray structures using two different methods, systematic search (SYS) and Monte Carlo search (MCS). Table 1 reveals that the fittings obtained by MCS Summary and Discussion In this work we have asked the question: what is the optimal fraction f of hydrophobic (H) residues required to ensure protein collapse? We have argued that an f that is too small is expected to lead to a random coil chain of blobs, while for f that is large enough the only thermodynamic alternative to avoid the contact of the H residues with water is burying them in the interior of a compact chain structure; if f is too large, the molecule will precipitate. Indeed, the fraction of H residues Fitting protein structures onto a lattice To conform to the lattice picture of the HP model, protein structures are fitted to a lattice. As in the work of Covell & Jernigan,23 we use a simplified protein model where the amino acid residues are represented by their backbone α-carbon atoms connected successively by virtual bonds;22 the structure is thus defined by the virtual bond lengths, virtual bond angles, and virtual torsion angles, where the correlations between these parameters as reflected in known protein structures have been Acknowledgements The authors gratefully acknowledge financial support from National Science Foundation Information Technology Research grants NSF 0225656 and NSF 0225636, and NIH grant GM61916. Recommended articles References (53) W. Kauzmann Some factors in the interpretations of protein denaturation Advan. Protein Chem. (1959) B.M. Broome et al. Nature disfavors sequences of alternating polar and non-polar amino acids: implications for amyloidogenesis J. Mol. Biol. (2000) I.M. Klotz Comparison of molecular structures of proteins: helix content; distribution of apolar residues Arch. Biochem. Biophys. (1970) W.R. Krigbaum et al. Local interactions as a structure determinant for protein molecules:II Biochim. Biophys. Acta (1979) Z. Alexandrowicz Critically branched chains and percolation clusters Phys. Letters A (1980) M. Levitt A simplified representation of protein conformations for rapid simulation of protein folding J. Mol. Biol. (1976) S.H. White et al. Statistical distribution of hydrophobic residues along the length of protein chains: implications for protein folding and evolution Biophys. J. (1990) C.J. Chothia The nature of the accessible and buried surfaces in proteins J. Mol. Biol. (1976) J. Kyte et al. A simple method for displaying the hydropathic character of a protein J. Mol. Biol. (1982) Y. Nozaki et al. The solubility of amino acids and two glycine peptides in aqueous ethanol and dioxane solutions. Establishment of a hydrophobicity scale J. Biol. Chem. (1971) D.A. Hinds et al. Exploring conformational space with a simple lattice model for protein structure J. Mol. Biol. (1994) J. Liang et al. Are proteins well-packed Biophys. J. (2001) E.I. Shakhnovich Protein design: a perspective from simple tractable models Fold. Des. (1998) K.A.T. Silverstein et al. A simple model of water and the hydrophobic effect J. Am. Chem. Soc. (1998) K.A.T. Silverstein et al. Molecular model of hydrophobic solvation J. Chem. Phys. (1999) Y. Madel-Gutfreund et al. On the significance of alternating patterns of polar and non-polar residues in beta-strands J. Mol. Biol. (2002) J. Hennetin et al. Non-intertwined binary patterns of hydrophobic/nonhydrophobic amino acids are considerably better markers of regular secondary structures than nonconstrained patterns Proteins: Struct. Funct. Genet. (2003) G. Némethy et al. The structure of water and hydrophobic bonding in proteins:III. The thermodynamic properties of hydrophobic bonds in proteins J. Phys. Chem. (1962) B. Lee et al. The interpretation of protein structures: estimation of static accessibility J. Mol. Biol. (1971) I.D. Kuntz Tertiary structure in carboxypeptidase J. Am. Chem. Soc. (1972) C. Chothia Structural invariants in protein folding Nature (1975) H. Meirovitch et al. Empirical studies of hydrophobicity. 1. Effect of protein size on the hydrophobic behavior of amino acids Macromolecules (1980) H. Meirovitch et al. Empirical studies of hydrophobicity. 2. Distribution of the hydrophobic, hydrophilic, neutral, and ambivalent amino acids in the interior and exterior layers of native proteins Macromolecules (1980) H. Meirovitch et al. Empirical studies of hydrophobicity. 3. Radial distribution of clusters of hydrophobic and hydrophilic amino acids Macromolecules (1981) G.D. Rose et al. Hydrophobic basis of packing in globular proteins Proc. Natl Acad. Sci. USA (1980) K.A. Dill Theory for the folding and stability of globular proteins Biochemistry (1985) View more references Cited by (15) The effect of loops on the structural organization of α-helical membrane proteins 2009, Biophysical Journal Citation Excerpt : The results reported in this article are based on a hydrophobicity scale, where Ile, Leu, Val, Phe, Trp, Met, Pro, Ala, and Tyr are defined as hydrophobic (H) and the remaining amino acids as polar (19). This set of H residues has been chosen as the preferred set in ref (20) by applying six different hydrophobicity scales to a set of 103 soluble proteins. All helices were assigned initially using the STRIDE software (21), which does not distinguish between TM helices and loop helices, and sometimes might introduce helix breaks in the middle due to helix kinks. Show abstract Loops connecting the transmembrane (TM) α-helices in membrane proteins are expected to affect the structural organization of the thereby connected helices and the helical bundles as a whole. This effect, which has been largely ignored previously, is studied here by analyzing the x-ray structures of 41 α-helical membrane proteins. First we define the loop flexibility ratio, R, and find that 53% of the loops are stretched, where a stretched loop constrains the distance between the two connected helices. The significance of this constraining effect is supported by experiments carried out with bacteriorhodopsin and rhodopsin, in which cutting or eliminating their (predominately stretched) loops has led to a decrease in protein stability, and for rhodopsin, in most cases, also to the destruction of the structure. We show that for nonstretched loops in the extramembranous regions, the fraction of hydrophobic residues is comparable to that for soluble proteins; furthermore (as is also the case for soluble proteins), the hydrophobic residues in these regions are preferentially buried. This is expected to lead to the compact structural organization of the loops, which is transferred to the TM helices, causing them to assemble. We argue that a soluble protein complexed with a membrane protein similarly promotes compactness; other properties of such complexes are also studied. We calculate complementary attractive interactions between helices, including hydrogen bonds and van der Waals interactions of sequential motifs, such as GXXXG. The relative and combined effects of all these factors on the association of the TM helices are discussed and protein structures with only a few of these factors are analyzed. Our study emphasizes the need for classifying membrane proteins into groups according to structural organization. This classification should be considered when procedures for structural analysis or prediction are developed and applied. Detailed analysis of each structure is provided at ### The twilight zone between protein order and disorder 2008, Biophysical Journal Citation Excerpt : However, the boundary between regions of order and disorder in the amino acid composition space of the lattice models also shifts to lower hydrophobicities and higher charges as chain length increases, a phenomenon not observed with real proteins. In fact, the hydrophobic fractions of real proteins depend little on chain length (53,55), although a maximum somewhere between 200 and 300 residues was found by Bastolla and Demetrius (56). On the other hand, the native energy per residue also tends to be nearly constant for proteins of various sizes (56,57) despite the fact that the number of contacts per residue increases (56). Show abstract The amino acid composition of intrinsically disordered proteins and protein segments characteristically differs from that of ordered proteins. This observation forms the basis of several disorder prediction methods. These, however, usually perform worse for smaller proteins (or segments) than for larger ones. We show that the regions of amino acid composition space corresponding to ordered and disordered proteins overlap with each other, and the extent of the overlap (the “twilight zone”) is larger for short than for long chains. To explain this finding, we used two-dimensional lattice model proteins containing hydrophobic, polar, and charged monomers and revealed the relation among chain length, amino acid composition, and disorder. Because the number of chain configurations exponentially grows with chain length, a larger fraction of longer chains can reach a low-energy, ordered state than do shorter chains. The amount of information carried by the amino acid composition about whether a protein or segment is (dis)ordered grows with increasing chain length. Smaller proteins rely more on specific interactions for stability, which limits the possible accuracy of disorder prediction methods. For proteins in the “twilight zone”, size can determine order, as illustrated by the example of two-state homodimers. ### Pandoravirus celtis illustrates the microevolution processes at work in the giant Pandoraviridae genomes 2019, Frontiers in Microbiology ### On the inverse temperature transition and development of an entropic elastomeric force of the elastin mimetic peptide [LGGVG] 3,7 2012, Journal of Chemical Physics ### Producing high-accuracy lattice models from protein atomic coordinates including side chains 2012, Advances in Bioinformatics ### Grafting short peptides onto polybutadiene-block-poly(ethylene oxide): A platform for self-assembling hybrid amphiphiles 2006, Angewandte Chemie International Edition View all citing articles on Scopus View full text Copyright © 2004 Elsevier Ltd. All rights reserved. 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18191	https://chem.libretexts.org/Bookshelves/General_Chemistry/ChemPRIME_(Moore_et_al.)/03%3A_Using_Chemical_Equations_in_Calculations/3.04%3A_Percent_Yield	3.4.1 Skip to main content 3.4: Percent Yield Last updated : Jun 18, 2023 Save as PDF 3.3.10: Sports, Physiology, and Health- Sodium Silicide Fueled Bicycles 3.4.1: Environment- Synthesis of Biodiesel Fuel Page ID : 49271 Ed Vitz, John W. Moore, Justin Shorb, Xavier Prat-Resina, Tim Wendorff, & Adam Hahn Chemical Education Digital Library (ChemEd DL) ( \newcommand{\kernel}{\mathrm{null}\,}) Not all chemical reactions are as simple as the ones we have considered, so far. Quite often a mixture of two or more products containing the same element is formed. For example, when octane (or gasoline in general) burns in an excess of air, the reaction is 2C8H18+25O2→16CO2+18H2O 2C8H18+25O2→16CO2+18H2O(3.4.1) If oxygen is the limiting reagent, however, the reaction does not necessarily stop short of consuming all the octane available. Instead, some carbon monoxide (CO) forms: 2C8H18+24O2→14CO2+2CO+18H2O 2C8H18+24O2→14CO2+2CO+18H2O Burning gasoline in an automobile engine, where the supply of oxygen is not always as great as that demanded by the stoichiometric ratio, often produces carbon monoxide, a poisonous substance and a major source of air pollution. In other cases, even though none of the reactants is completely consumed, no further increase in the amounts of the products occurs. We say that such a reaction does not go to completion. When a mixture of products is produced or a reaction does not go to completion, the effectiveness of the reaction is usually evaluated in terms of percent yield of the desired product. A theoretical yield is calculated by assuming that all the limiting reagent is converted to product. The experimentally determined mass of product is then compared to the theoretical yield and expressed as a percentage: Percent yield=actual yieldtheoretical yield×100 percent Percent yield=actual yieldtheoretical yield×100 percent The video below (modeled after the octane example given earlier in the chapter) demonstrates visually what the percent yield is, first showing the theoretical yield, then showing the actual yield (where the reaction doesn't go to completion) and finally comparing the actual yield to the theoretical yield to find the percent yield. Example 3.4.13.4.1 : Percent Yield When 100.0 g N2 gas and 25.0 g H2 gas are mixed at 350°C and a high pressure, they react to form 28.96 g NH3 (ammonia) gas. Calculate the percent yield. Solution: We must calculate the theoretical yield of NH3, and to do this, we must first discover whether N2 or H2 is the limiting reagent. For the balanced equation N2+3H2⟶2NH3 N2+3H2⟶2NH3 the stoichiometric ratio of the reactants is S(H2N2)=3 mol H21 mol N2 S(H2N2)=3 mol H21 mol N2 Now, the initial amounts of the two reagents are and nH2(initial)=25.0 g H2×1 mol H22.016 g H2=12.4 mol H2nN2(initial)=100.0 g N2×1 mol N228.02 g N2=3.569 mol N2nH2(initial)nN2(initial)=25.0 g H2×1 mol H22.016 g H2=12.4 mol H2=100.0 g N2×1 mol N228.02 g N2=3.569 mol N2(3.4.2)(3.4.3)(3.4.4) The ratio of initial amounts is thus nH2(initial)nN2(initial)=12.4 mol H23.569 mol N2=3.47 mol H21 mol N2 nH2(initial)nN2(initial)=12.4 mol H23.569 mol N2=3.47 mol H21 mol N2 Since this ratio is greater than S(H2N2), there is an excess of H2. N2 is the limiting reagent. Accordingly we must use 3.569 mol N2 (rather than 12.4 mol H2) to calculate the theoretical yield of NH3. We then have nNH3(theoretical)=3.569 mol N2×2 mol NH31 mol N2=7.138 mol NH3 so that mNH3(theoretical)=7.138 mol NH3×17.03 g NH31 mol NH3=121.6 g NH3 The percent yield is then Percent yield=actual yieldtheoretical yield×100 percent =28.96 g121.6 g×100 percent=23.81 percent Combination of nitrogen and hydrogen to form ammonia is a classic example of a reaction which does not go to completion. Commercial production of ammonia is accomplished using this reaction in what is called the Haber process. Even at the rather unusual temperatures and pressures used for this industrial synthesis, only about one-quarter of the reactants can be converted to the desired product. This is unfortunate because nearly all nitrogen fertilizers are derived from ammonia and the world has come to rely on them in order to produce enough food for its rapidly increasing population. Ammonia ranks third [after sulfuric acid (H2SO4) and oxygen (O2)] in the list of most-produced chemicals, worldwide. It might rank even higher if the reaction by which it is made went to completion. Certainly ammonia and the food it helps to grow would be less expensive and would require much less energy to produce if this were the case. 3.3.10: Sports, Physiology, and Health- Sodium Silicide Fueled Bicycles 3.4.1: Environment- Synthesis of Biodiesel Fuel
18192	https://nrich.maths.org/problems/birthday-sharing	Birthday Sharing \| NRICH Skip to main content Problem-Solving Schools can now access the Hub! Contact us if you haven't received login details Main navigation Teachersexpand_more Early years Primary Secondary Post-16 Professional development Studentsexpand_more Primary Secondary Post-16 Parentsexpand_more Early years Primary Secondary Post-16 Problem-Solving Schoolsexpand_more What is the Problem-Solving Schools initiative? Becoming a Problem-Solving School Charter Hub Resources and PD Events About NRICHexpand_more About us Impact stories Support us Our funders Contact us search menu search close Search NRICH search Or search by topic Number and algebra Properties of numbers Place value and the number system Calculations and numerical methods Fractions, decimals, percentages, ratio and proportion Patterns, sequences and structure Coordinates, functions and graphs Algebraic expressions, equations and formulae Geometry and measure Measuring and calculating with units Angles, polygons, and geometrical proof 3D geometry, shape and space Transformations and constructions Pythagoras and trigonometry Vectors and matrices Probability and statistics Handling, processing and representing data Probability Working mathematically Thinking mathematically Mathematical mindsets Advanced mathematics Calculus Decision mathematics and combinatorics Advanced probability and statistics Mechanics For younger learners Early years foundation stage Birthday sharing It's Sahila's birthday and she is having a party. How could you answer these questions using a picture, with things, with numbers or symbols? Age 5 to 7 Challenge level Primary Number - Multiplication, Division and Ratio Problem Solving Exploring and noticingWorking systematicallyConjecturing and generalisingVisualising and representingReasoning, convincing and proving Being curiousBeing resourcefulBeing resilientBeing collaborative Problem Getting Started Student Solutions Teachers' Resources Problem It's Sahila's birthday and she is having a party. Show us how you could answer these questions using: words pictures numbers objects other ways... Image Sahila has 18 cupcakes for the party tea and she would like to share them out equally onto two plates for the table. How many cakes will go on each plate? Image Sahila has invited nine children to her party. They are going to play a game in pairs. Each pair will need a balloon. How many balloons will they need? Image Sahila is going to give everyone five juggling balls to take home after the party. Will 55 balls be enough? Getting Started Try using counters or Multilink cubes or something else to show the cakes or balloons or juggling balls and make sense of the problem. You could use Lego people to show the children at the party. Student Solutions Rowan from St Andrews tried this at home and the adult with him sent in pictures of his work. The adult wrote out Rowan's explanations in green: Image Image Image Well done Rowan - you have explained your reasoning very clearly. You may be able to find other ways of getting to an answer. Teachers' Resources Birthday Sharing It's Sahila's birthday and she is having a party. Show us how you could answer these questions using: words pictures numbers objects other ways... Image Sahila has 18 cupcakes for the party tea and she would like to share them out equally onto two plates for the table. How many cakes will go on each plate? Image Sahila has invited nine children to her party. They are going to play a game in pairs. Each pair will need a balloon. How many balloons will they need? Image Sahila is going to give everyone five juggling balls to take home after the party. Will 55 balls be enough? Why do this problem? At first glance this looks like any collection of word problems about division. However each different scenario draws attention to a different way of thinking about the idea of division: sharing, grouping successive subtraction or the inverse of multiplication. An article which explores different ways of thinking about division can be found here. Children will be curious that there are different ways to represent their thinking, and discussing their different ways of working will offer insights and deepend their understanding of division. Possible approach In each case it would be possible to use a range of representations of the situation to help solve the problem and children should be encouraged to explain how they have tackled the problem and arrived at their solution using different resources to help them. Look and listen carefully to hear how they make sense of the question and develop a strategy for solving it. This may be somewhat different from the 'taught' approach. Given the opportunity, children often have their own ways of working. Talk to the children about how different people do things in different ways and explain that this actvity is all about that - it's important that the children don't presume that there is one way and one way only to see the calculation. Working within a pair or small group and tackling one problem at a time can help children to focus more deeply on one task rather than racing through them. You could suggest that they think and talk about what they are going to do before they actually begin. They may decide to enact with objects or make a picture and just record the answer. Or use these as a prompt to transfer the problem to a calculation which they would record horizontally. Whichever, your observations will allow you to reflect on the children's confidence, language and understanding and possibly what misconceptions they hold. Key questions Tell me how you are working this out. What do you think of _____'s way of doing this? How did you know what to do first? Is the way you've done this one different from the way you have done the others? Possible extension Ask the pupils to create stories that involve calculations for their partners to do. Give the pupils a written form of a division question eg 18÷?=3 and challenge them to create a story around it. Ask the children to create their own division problems based on stories about sharing, grouping, 'undoing' multiplication or successive subtraction. Additional links to other division problems can be found in this article. Possible support Some pupils will benefit from working with some small toys/dolls so that they can enact the situation before being able to think about the calculation, for example the children could use Lego figures to show the children at the party and counters for the cakes with circles of paper as plates. The physical act of moving objects in different ways while enacting the story can often help less confident children to work out how to think about the division in an appropriate way that makes sense to them. 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18193	https://cdn.mises.org/3_3_2_0.pdf	William Graham Sumner: Critic of Progressive Liberalism by Jonathan Marshall Department o f History, CorneN University In America today, as throughout the West, most people fundamentally accept the "welfare state." Republican Presidents live happily with huge deficits in government accounts, while conservative politicians no longer challenge Medicare or Social Security. The State has beconte a pervasive force in every individual's life, from cradle to grave; it consumes an ever- growing share of national product and employs a sizeable percentage of the labor force. Yet the "positive" state that so many now take for granted is a remarkably recent phenomenon. In the United States, the transformation to a modern welfare state really began only in the Progressive era-less even in terms of the substantive reforms then enacted than in the growth of a climate of opinion, in political and intellectual circles, favorable to State interven- tion. Until the Progressive era, laissez-faire reigned supreme, in accepted theory at least, as the principle by which social and political life ought to be organized. The profoundly important ideological shift that took place around the turn of the century has already become the focus of much scholarly research; this paper, however, seeks to revive and explore the opinions of one of the last, and certainly the most articulate, spokesman for laissez-faire in an era when more and more people were championing the cause of State interference in spheres of life formerly reserved to the individ- ual or to private corporations. William Graham Sumner personifies the classical liberal viewpoint against which the new "progressive liberals," as I have called them, were reacting. As a challenge to the modern liberal synthesis, Sumner's views are even today of unusual interest. I. The Progressive-Liberal Mind By "progressive liberals" I mean those intellectuals identified with Progres- sive reforms or party politics who articulated the assumptions upon which modern liberalism is based. These liberals shared the optimism, but little of the substantive philosophy, of classical liberals in the early English Liberal 262 THE JOURNAL OF LIBERTARIAN STUDIES Party-Manchester School tradition. Instead they championed "taking into the hands of the state the business of the individual man," as William Gladstone sorrowfully described a tendency among members of his own Liberal party near the turn of the century.' They were not content to rely on the laws of God or the market place for social progress. "We can no longer treat life as something that has trickled down to us," insisted Walter Lipp- mann, one of the most self-consciously "liberal" progressives, in 1914. "We have to deal with it deliberately, devise its social organization, alter its tools, formulate its method, educate and control it."' With State power and the techniques of social science at their disposal, they were impatient to break the shackles of the Jeffersonian tradition of limited government and embark on an ambitious program of social reform. Drawing inspiration from uto- pian socialists, advocates of the social gospel, German-school economists, and others, the progressive liberals forged a new, activist conception of the State, in place of the laissez-faire ideal of the State as a mere poker of men and contracts. The ideology of "progressive liberalism," which can he traced back to the birth of the republic, did not, of course, spring unheralded on America only after the turn of the century. "This battle between State-interference and laissez-faire," one writer commented as early as 1884, "is now upon us; it will be waged through all the near future."'Years before the 19th century closed, reformers and socialists such as Henry George, Edward Bellamy, and Henry Demarest Lloyd gained wide audiences with their advocacy of state-sponsored solutions to social problems; the rise of vast new urban and industrial problems seemed to cry out for solutions on an equally grand scale. Contributing to this attitude was the transformation taking place within American Christianity in its attitudes toward poverty. The common 19th century view had been uncompromisingly expressed by the Reverend Henry Ward Beecher: "No man in this land suffers from poverty unless it he more than his fault-unless it he his sin."4The new advocates of the social gospel took a more sympathetic approach to the poor. A leader of this movement, the Monsignor John A. Ryan, devoted numerous speeches and writings to the theme that suffering, poverty, and indigence stemmed from social causes, not individual moral failings. In his 1912 presidential address to the Minne- sota State Conference of Charities and Correction he offered a solution: The State, and only the State, can prevent a large part, probably the larger part, of the social distress which is due primarily to the environ- ment. . . . [I]t can and ought to provide suitable economicconditions by enforcing reasonable minimum standards of labor and livelihood.' Ryan's own views had been heavily influenced by Richard Ely, leader of a school of young German-trained economists who had been heavily imbued 263 WILLIAM GRAHAM SUMNER with the ideology of Bismark's nationalist-welfare state. Ely in particular infused his scientific work with a strong social gospel spirit. With his like- minded colleagues, Ely founded the American Economic Association in 1885, with the unwritten proviso that it must "not include men of the Sumner type. . . ."6 They agreed on a statement of principles that left no question as to their ideological commitments. "We regard the state as an educational and ethical agency whose positive aid is an indispensible condi- tion of human progress," it read. ". . .[Tlhe doctrine of laissez-faire is unsafe in politics and unsound in morals. . . . " 7 Ely's influence spread wide; he taught Woodrow Wilson, profoundly influenced LaFollette and the develop- ment of the "Wisconsin Idea," and Theodore Roosevelt paid tribute to him as the man who "first introduced me to radicalism in economics and then made me sane in my radicalism."" The new political economists also influenced the most articulate and self- conscious spokesmen for progressive liberalism-Herbert Croly, Walter Weyl, and Walter Lippmann, founders of New Republic maga~ine.~ Croly's The Promise o f American Life (1909) was perhaps the central statement of its time in favor of an enlarged role for the State. For Croly, the fundamen- tal social problem was how, in the face of divergent interests and inequali- ties of wealth and achievement, to keep "such a highly differentiated society fundamentally sound and whole." He sought a solution in a program of nationalized democracy to replace the "chaotic individualism" which ever threatened to rend the social fabric. The State would have to play the central organizing role in this process, taking on responsibility "for the suhordina- lion of the individual to the demand of a dominant and constructive national purpose. . . ." If Croly's reverence for the State clashed with traditional American principles, then "the fault in that case lies with the democratic tradition; and the erroneous and misleading tradition must yield before the march of a constructive national democracy."'" Theodore Roosevelt, the first President to put progressive principles into practice, admired Croly and shared many of his views. Undoubtedly, TR's ideas were influenced by his own ambitions and energies: "1 believe in a strong executive," he said while President; "I believe in power."ll But on a more theoretical plane he accepted Croly's rejection of the Jeffersonian tradition. ". . . [W]e must abandon definitely the laissez-faire theory of political economy," he wrote in Outlook in 191 1, "and fearlessly champion a system of increased Governmental control, paying no heed to the cries of worthy people who denounce this as Socialistic."l His rejection of anti-trust laws in favor of close and sympathetic national regulation of the trusts perfectly reflected this stand. The 1912 election campaign illustrates how far the progressive consensus had developed. Although many historians have observed in that campaign a clash between TR's Hamiltonian theory of government and Wilson's indi- 264 THE JOURNAL OF LIBERTARIAN STUDIES vidualism, these differences should not be exaggerated.') For Wilson himself had come to completely reject the Jeffersonian policy of limited government. "We used to say," he observed, ". . . that the best government was the government that did as little governing as possible. . . . But we are coming now to realize that life is so complicated . . . that the law has to step in and create the conditions which will make it tolerable for us to live."'4 He repeated the theme throughout the campaign. "Without the watchful inter- ference, the resolute interference of the government," the future President insisted, "there can be no fair play between individuals and such powerful institutions as the trusts. Freedom today is something more than being let alone. The program of a government of freedom must in these days be positive, not negative merely."ls 11. Sumner's Critique of the Positive State William Graham Sumner, more than any other man, resisted and challenged these intellectual trends. Born in 1840 the son of a laboring English immi- grant, Sumner owed his reputation as the "archenemy of the advocates of social reform"l6 to his extraordinary ability and energy as a publicist, public speaker, and professor of political and social science at Yale. Until his death in 1910, he lectured the nation on the evils of tariff protection, industrial regulation, and militarism. Even as the object of national controversy, and b2te noire of the conservative Yale alumni, Sumner never faltered in his crusade to roll back the State. He lived according to his conviction that every citizen had a patriotic and civic duty to resist the encroachment of the State." Sumner adopted the core of his opinions at an early age. In his early teens, he devoured Harriet Martineau's Illustration of Political Economy, an economics text in story form. Martineau unwaveringly opposed any interfer- ence with the free market, from strikes to poor relief. Any restriction "on the natural direction of labor and capital," she wrote, "is ultimately injurious to every class in the community." In particular, she championed free trade, the defense of which occupied Sumner throughout much of his adult life: "as the general interest of each nation requires that there should be perfect liberty in the exchange of commodities, any restriction on such liberty, for the sake of benefiting any particular class or classes, is the sacrifice of a larger interest to a smaller-that is, a sin in government." Sumner took her words to heart, admitting later that they, even more than his formal training, were responsi- ble for his conceptions of "capital, labor, money, and trade. . . ."In Certainly Sumner's religious training at Yale, by reinforcing his Christian belief in the responsibility and sanctity of the individual moral agent, must have contrib- uted to the laissez-faire attitudes he adopted towards economic and social problems. 265 WILLIAM GRAHAM SUMNER Sumner also borrowed from the great English social theorist and individu- alist, Herbert Spencer, whose Social Darwinian notions reached powerfully across the Atlantic. Aside from Spencer's commitment to laissez-faire and his conservative faith in the inefficacy of reform, Sumner was particularly influenced by Spencer's Social Darwinian doctrine of the "law of conduct and consequence" which held that to insure the survival of the human species, society must distribute rewards according to merit and "fitness."l9 Sumner's Social Darwinism stemmed naturally from the Malthusian traditions of classical political economy, which gave economics its preoccu- pation with scarcity and its reputation as "the dismal science." In the face of scarcity, life is a persistent struggle to wrest from nature the means of subsistence. Some men, by virtue of character or skill, are particularly successful. "The millionaires are a product of natural selection," Sumner explained, skirting close to tautology. ". . . It is because they are thus selected that wealth-both their own and that entrusted to them-aggregates under their hands."20 In a world of scarcity, unfortunately, not everyone can compete success- fully. Sumner, hardly complacent about this fact, admitted that "it is frightful to know of the poverty which some people endure," but classified poverty along with disease, physical defects, and accidents as an act of nature which interferes with man's enjoyment of life. In speaking of abolish- ing poverty, "we might as well talk of abolishing storms, excessive heat and cold, tornadoes, pestilences, diseases, and other ills. Poverty belongs to the struggle for existence, and we are all born into that struggle.21 Socialists and reformers, by blinding themselves to these laws of nature, "bring forward complaints which are really to be made, if at all, against the author of the universe for the hardships which man has to endure in his struggle with nature." In the long run, their schemes would promote the "deterioration of society" by burdening the fit and successful members of society with the task of propping up "the bad ones. The law of the survival of the fittest was not made by man and cannot be abrogated by man. We can only, by interfering with it, produce the survival of the unfittest."22 Yet Sumner did not write off the poor; with his pre-sociological concep- tion of poverty, he offered them a way out, through the Protestant Ethic, in which he had been deeply engrained by his father. Social reform was a mere phantasm compared with individual self-improvement. "The only two things which really tell on the welfare of men on earth," said the preacher Sumner, "are hard work and selfdenial. . . ."23 He was confident, despite the pessi- mism of his Darwinian attitudes, that poverty could "he abolished in a few generations" if everyone acted industriously and brought up their children to do the same.z4 The State should support this process, not through grandiose reforms, but through protection of property, contracts, and life- what Sumner called civil liberty. "What civil liberty does is to turn the 266 THE JOURNAL OF LIBERTARIAN STUDIES competition of man with man from violence and brute force into an indus- trial competition under which men vie with one another for the acquisition of material goods by industry, energy, skill, frugality, prudence, temperance, and other industrial virtues."2s Despite his rather Victorian way of putting things, Sumner was making an important point-that in an economy of scarcity, poverty can be overcome only through production, and not simply by redistribution. Socialist schemes, Sumner believed, would unjustly penalize men who lifted themselves out of poverty by dint of their hard labor and self- sacrifice-thus in the long run undermining social advancement. Why, he asked in his famous essay on "The Forgotten Man," should the industrious man be taxed and penalized to raise the station of those less virtuous and successful? Sumner emerged as the spokesman for the middle classes, who were "always forgotten by sentimentalists, philanthropists, reformers, enthu- siasts. . .. [They] have kept our attention for a long time on the. .. good-for-nothing people, as if they alone deserved our attention." Sumner's moralistic tone, of course, stemmed from his assumption that poverty was a reflec- tion of individual character. "The whole system of social regulation by boards, commissioners, and inspectors," he complained, "consists in reliev- ing negligent people of the consequences of their negligence and so leaving them to continue negligence without correction." Why should the ''forgotten man" be asked to pay for the negligence of others?" Despite his hard and unsophisticated tone, Sumner was making several important points. First, he questioned the beneficence of self-professed philanthropists who agreed to tax third parties in order to support "the poor," "the weak," or other adopted social pets. Sumner himself frequently gave to charities. But, "[wlhat I choose to do by way of exercising my own sympathies under my own reason and conscience is one thing; what another man forces me to do of a sympathetic character, because his reason and conscience approve it, is quite another thing."27 In criticizing the moral basis of redistribution, Sumner revealed himself as primarily concerned with questions of justice and liberty, even more than with Social Darwinian principles. In rejecting the egalitarian state as a "servant of envy," Sumner advanced a fundamental principle of justice: "I am entitled to make the most I can of myself without hindrance from anybody, but I am not entitled to any guarantee that I should make as much of myself as somebody else makes of himself." The real problem with a policy of "survival of the unfittest" is that it can only be achieved by "destroying liberty."28 Sumner believed, further, that structural reforms of society could not really be achieved even at the expense of liberty, for society is much too complex a mechanism to permit man-made tinkering. In an essay on the "Absurd Effort to Make the World Over," he observed that social forces "will have changed the whole problem before our interferences have time to 267 WILLIAM GRAHAM SUMNER make themselves felt."29 Only too aware of the inadequacy of his own knowledge, Sumner naturally resented the "reformers, philanthropists, hu- manitarians, and would-be managers-in-general of society" who fancied themselves experts in social science. Like quack doctors, he observed, they always begin with the question of remedies, and they go at this without any diagnosis or any knowledge of the anatomy or physiology of society. . . . It generally troubles them not a whit that their remedy implies a complete reconstruction of society, or even a reconstruction of human nature. Against all such social quackery the obvious injunction to the quacks is, to mind their own business.JO Contrary to liberals who thought the role of the State should grow in proportion to the size of social problems, Sumner believed that the very complexity of modern society militated more than ever against the success of reform programs. Unwise legislation, passed without sufficient study, tended to stay on the books, shackling future generations with their changed condi- tions.ll Reform programs are doubly fallible, Sumner argued, because of the nature of the agency called on to enact and execute them. The State, far from being "a tutelary genius over us all," was simply a little group of men chosen in a very haphazard way by the majority of us to perform certain services for all of us. The majority do not go about their selection very rationally, and they are almost always disappointed by the results of their own operations. Hence "the State," instead of offering resources of wisdom, right reason, and pure moral sense beyond what the average of us possess, generally offers much less of all those things.32 The reformers were equally unrealistic in their conception of the State as neutral and even-handed. In a society ridden with competing interests, the State becomes a natural arena for their struggles. As Sumner warned in 1909, those who have been "defeated in the competition of life" will seek to "fight over again, on the political domain, what they have lost on the economic domain."33 Sumner was astounded at the naivetC with which reformers believed the "legislative device" would be an unchallenged tool in their hands. "They never appear to remember that the device, when once set up, will itself become the prize of a struggle," Sumner observed, " . . . so that after all the only serious question is: who will get it?" Could advocates of State interference really he certain that their enemies-the railroads, the liquor sellers, the trusts-would not seize control of the very institutions they had set up for less noble ends?" At least in civil society, the forces of competition exercise a sort of check over the actions of businesses and individuals; even commercial monopolies face the discipline of potential competition and must limit their profits accordingly. But the State faces no competition at all; it is "the greatest 268 THE JOURNAL OF LIBERTARIAN STUDIES monopoly of all; it can brook no rival or colleague in its domain," and thus potentially becomes "the most powerful engine by which some men may exploit others." What the reformers forget is that the very power of the State to d o good can also become an unparalleled power to do harm.35 Far from apologizing for big businessj Sumner was simply following to a logical conclusion the findings of the muckrakers whose studies were proof of the political power of big capital. "Can anyone imagine," Sumner asked, "that the rnasterfulness, the overbearing disposition, the greed of gain, and the ruthlessness of methods, which are the faults of the master of industry at his worst, would cease when he was a functionary of the State, which had relieved him of risk and endowed him with authority? Can anyone imagine that politicians would no longer be corruptly fond of money, intriguing, and crafty when they were charged, not only with patronage and government contracts, hut also with factories, stores, ships, and railroadsT"6 Sumner feared that American democracy was sliding into the hands of a new ruling class. The expansion of State power simply permitted the most powerful element in society-capital-to cement its domination. As he repeated endlessly during the Progressive era, the plutocrats, those who invested their money in politics rather than in industry, in lobbyists and in election rigging, had taken control of the reins of State. In 1907, looking hack on this trend, he summed up the problem in stark, even radical, terms: The history of the nineteenth century. . . plainly showed the power of capital in the modern state. Special legislation, charters, and franchises proved to be easy legislative means of using the powers of the state for the pecuniary benefit of the few. . ..The history is disgraceful, and it is a permanent degradation of popular government that power could not be found, or did not exist, in the system to subjugate this abuse and repress this corruption of state power. 7he protective-tariff system is simply an elaborate system by which certain interests inside of a country get control of legislation in order to tax their fellow-citizens for their own benefit. . . . It is the supreme test of a system of government whether its machinery is adequate lor repressing the selfish undertakings of cliques formed of special interests and saving the public from raids and plunder- ers. The modern democratic states fail under this test. . . . Financial scandal is the curse of all modern parliamentary states with a wide suffrage. They give liberty and security, with open chances for individual enterprise, . . . but the political machinery offers opportunities lor manipulation and corrupt abuse. They educate their citizens to seek advantages in the industrial organization by legislative devices, and to use them to the uttermost. . . .We hear ofplutocracy and tainted money, of the power of wealth, and the wickedness of corporations. The disease is less specific. It is constitutional.37 Sumner applied this critique of the State to a number of specific cases, including the problem of regulatory commissions and the entire system of 269 WILLIAM GRAHAM SUMNER trade protection. In the case of commissions, Sumner foreshadowed a whole school of modern critics in pointing to their tendency either to "sink into nonentity" or to become captured by an interest group. Referring to a study of the Interstate Commerce Commission which cast doubt on the compe- tence of its members, Sumner observed that "if a good man is appointed, the railroads presently invite him to come over to them, and they give him two or three times the salary." When such commissions failed in their purpose, Sumner complained, the public sought only to strengthen them further, never to scrap them. He would have preferred, in any case, to hold corpora- tions more strictly to the law rather than sloughing their responsibility off onto an irresponsible commission.38 Above all, Sumner was a tireless opponent of the protective tariff, his paradigm of the abuses to which government power can be put. As vice- president of the American Free Trade League, he fearlessly incurred the wrath of Republican conservatives with his steady stream of speeches and articles condemning protectionism. "Protection arouses my moral indigna- tion," he explained in one of the last speeches before his death. "It is a subtle, cruel, and unjust invasion of one man's rights by another. . . . The moral indignation which it causes is the motive which draws me away from the scientific pursuits which form my real occupation, and forces me to take part in a popular agitation." Industries, he explained, sought protection to save themselves "the trouble and annoyance of business competition and . . . be assured profits in their undertakings by the State, that is, at the expense of their fellow citizens." Sumner objected to protection not simply because it encouraged the unfit to survive, but because it violated his principles of justice. Recalling the campaigns against tax abuse which made up the history of American civil liberties, Sumner called protectionism the worst such abuse, a government "license to certain interests to go out and encroach on others." Protectionism encouraged the corruption of politics and, per- haps worst of all, undermined the work ethic by teaching "us to believe that a man needs a 'pull' of some kind or other to make any industry a success. . . . That is the doctrine of pure graft."39 After years of fruitless campaigning, Sumner was a disillusioned man. In 1906, surveying the failures even of such "reformers" as Theodore Roosevelt to press for tariff reform, Sumner concluded that "we are being governed at the present time by a combination of these protected interests which have got control of the machinery . . . and . ..the personnel of the government to such an extent that it is impossible, practically, to make any breach in this system at all."'"hese words sound more like the polemics of a muckraker than the apologetics of a conservative, but then Sumner was a radical when it came to the defense of individual liberty. He still saw hope in the "very great revolt in the public mind against graft and political and business 270 THE JOURNAL OF LIBERTARIAN STUDIES corruption" that had emerged since the turn of the century-hut only if that revolt could be channeled into libertarian rather than Statist ends. "The way to minimize the dangers to democracy," he never tired of repeating, " . . . is to reduce to the utmost its functions, the number of its officials, the range of its taxing power, the variety of its modes of impinging on the individual, the amount and range of its expenditures, and, in short, its total weight. . . When all was said and done, Sumner's philosophy rested on a profound appreciation for personal liberty, rather than on the cold arguments of the Social Darwinians. He diagnosed many of the same social problems high- lighted by the progressives, but refused to seek answers in the aggrandize- ment of the State; "[wlhenever we try to get paternalized," he warned, "we only succeed in getting policed."42 Whatever one thinks of his philosophical stance, no one can deny his sophistication and prescience in warning of the uses to which new State agencies would be put, by the very interests they were designed to reform or regulate. The history of the United States since his time, from Teapot Dome to Watergate, has provided the raw material for thousands of muckraking accounts of American politics. Sumner's warnings still today provide a powerful antidote to the optimism of liberals every- where as to the beneficence of state power. 111. Sumner: Opponent of Militarism and Imperialism Sumner's laissez-faire doctrine and the emerging progressive-liberal synthe- sis nowhere clashed more sharply than over the issue of imperialism. The differences which put men like Sumner and Theodore Roosevelt so at odds over the Spanish-American War and over later examples of American imperialism, were not accidental products of personal temperament, hut stemmed directly and crucially from their fundamentally opposed concep- tions of the State. While not all progressives agreed, a dominant wing of the movement favored an activist and expansionist foreign policy. Theodore Roosevelt, leader of the Progressive Party, was the archetypal imperialist-the man who championed war with Spain, led the Rough Riders in Cuba, seized the Canal Zone, extended the Monroe Doctrine, and sent the fleet around the world. Roosevelt couched his advocacy of imperialism in the moralistic terms of an international reformer. The United States, he wrote in December 1899, could not "compromise with unrighteousness." Like other colonial powers, it had a duty to conquer "barbarian" races in the cause of civiliza- tion and peace, for "every expansion of a great civilized power means a victory for law, order, and righteousness." Just as the U. S. had warred against the "savages or half-savages" who peopled the continent before the advent of the white man, "the same will be trueof the Philippines." By imposing a "stable and orderly government" there, "one more fair spot of the WILLIAM GRAHAM SUMNER 27 1 world's surface shall have been snatched from the forces of darkne~s."~~ Progressive-liberal intellectuals joined politicians like Roosevelt and Sen- ator Albert Beveridge in promoting imperialism. Herbert Croly, whose dream of a Hamiltonian-nationalist State Roosevelt shared, believed that a vigorous and imperial foreign policy could bind the nation together with a common purpose and thus "constitute a beneficial and a necessary stimulus to that better realization of the Promise of our domestic life." He defended the Spanish-American War for the "tremendous impulse" it gave "to the work of national reform. It made Americans more sensitive to a national idea and more conscious of their national responsibilities." In practical terms, Cuba, a "center of disorder," had to be "pacified" in the interests of the establishment of the "American international system."M The Progressive-liberal defense of imperialism was no anomaly; on the other side of the ocean, Fabians were defending imperialism as a tool for producing national reform and "international civilization."4s Progressives shared Wilson's dream of making the world safe for democracy-through force and occupation if necessary. "We Progressives preach within our own nation the doctrine of social consciousness," Roosevelt told a group of Progressive Party friends in 1912, as part of a defense of the Monroe Doctrine. "So likewise we preach the doctrine of international social con- sciousness. . . . [W]e intend to do all we can to help all the nations of mankind . . . to rise . . . toward an orderly and self-respecting and law- abiding civilization. . . ."46 And as he wrote in Ourlook magazine that year, I feel that the Progressive Party owes no small part of its strength to the fact that it not only stands for the most far reaching measures of social and industrial reform, hut. . . also for the right and duty of this nation to take a position of self-respecting strength among the nations of the world, to . . . show that it has both the spirit and the strength to repel injustice from abroad.4' The reformist imperialism of the Progressives flowed naturally from their advocacy of statist intervention at home. William Leuchtenburg, in his study of the relationship between progressivism and imperialism, concludes that both were "expressions of the same philosophy of government, . . . a worship of definitive action for action's sake."48 Perhaps the English sociologist and political philosopher L. T. Hobhouse, who himself pioneered the transition of English liberalism away from its original commitment to laissez-faire, best expressed this theoretical connection: The socialist development of Liberalism oaved the wav for imoerialism. So non-interwnt~o" ahroad wcnt hy the board alone with l a ~ r , ~ ~ . - ! ~ ~ r e at homc; national l~bcrt! u,ac rankcd with compctltnc industr~al~m an ; , exploded superstition; a positive theory of the State in domestic affairs was matched by a positive theory of Empire, and the way was made straight for imperialism.4~ 272 THE JOURNAL OF LIBERTARIAN STUDIES William Graham Sumner, in rejecting the erosion of liberty and individu- alism at home, was no less fervent and staunch in his critique of militarism and state intervention abroad. Sumner was one of several vice-presidents of the Anti-Imperialism League, an organization dedicated to reversing the expansionist tide inaugurated by the Spanish-American War. Significantly, the AIL leadership almost without exception shared the laissez-faire eco- nomic doctrines characteristic of such English anti-imperialists as Cobden and Bright before them. The logical relation between their economic and anti-imperialist doctrines was not lost on ardent expansionists such as TR's friend Henry Cabot Lodge, who condemned the "theory of the Manchester school" for holding that "territorial expansion or national expansion must be stopped because they were likely to interfere with complete freedom of trade."50 Sumner firmly established his reputation as an anti-imperialist even before the Spanish-American War; in 1896, sensitive to the rise in imperialist sentiment reflected in attempts to annex Hawaii and other Pacific islands, he warned that the costly attempt to acquire new territories would "lessen liberty and require discipline. It will increase taxation and all the pressure of government. It will divert the national energy from the provision of self- maintenance and comfort for the people, and will necessitate stronger and more elaborate governmental machinery. All this will be disastrous to republican institutions and to democracy."5l But nowhere did Sumner show more force or eloquence than in his famous and controversial 1898 address to the Yale Phi Beta Kappa chapter on "The Conquest of the United States by Spain."52 Sumner's thesis, as his provocative title suggests, was that despite America's military victory over the decadent and backward Spanish empire, "we are submitting to be conquered by her on the field of ideas and policies." In adopting the false doctrines of national glory and mercantilism which brought Spain to ruin, Sumner felt, America was threatened with the same fate.33 Sumner perceived that in the course of conquering the Filipinos, "our institutions, our most sacred traditions, and our best established maxims have been trampled underfoot." Americans have believed from the time.of their independence that life, liberty, and the pursuit of happiness are natural rights, common to all men by virtue of their humanity. But apparently the Filipinos were to be an exception; at the first test of our principles we throw that doctrine away and adopt the Spanish doctrine. We are told by all imperialists that these people are not fit for liberty and self- government; that it is rebellion for them to resist our beneficence, that we must send fleets and armies to kill them if they do it,. . .that we may buy them or sell them as we please, and dispose of their "trade" for our own advantage. What is that but the policy of Spain to her dependen- cies?54 273 WILLIAM GRAHAM SUMNER Sumner echoed a theme common to the anti-imperialists, many of whom were prominent in the struggle for civil rights at home:5 if the United States could not insure rights to its own people, how could it be confident of spreading civilization to the Philippines? Sumner's commitment to liberty regardless of race or nationality comes through powerfully in his observa- tion that When the negro postmaster's house was set on fire in the night in South Carulina, and not only he, hut his wife and children, were murdered as they came out, and when, moreover, this incident passed without legal investigation or punishment, it was a bad omen for the extension of liberty, etc., to Malays and Tagals by simply setting over them the American flag.56 Sumner's complaint that despite "talk of civilizing lower races . . . we have exterminated them" was certainly borne out in the Phillippines by America's systematic use of concentration camps, torture of prisoners, burning of villages, and the indiscriminate killing of civilians.57 Ironically, it was Spain's commission of just such abuses as these that provided the moral impetus for American entry into war with Spain in the first place. The repercussions of imperialism would be great both abroad and at home, Sumner predicted: abroad, because the logic of imperialism required that the U.S. move on to controlever more distant areas in order to "secure" its new possessions. "Of course this means that, on the doctrine, we must take the whole earth in order to be safe on any part of it. . . ." Not only was the doctrine absurd, but it would lead the United States into dangerous competition with other strong colonial powers.58 Just as important for Sumner, however, were the domestic implications of imperialism. The sensationalism and jingoism which accompanied Ameri- ca's entry into the war had stifled intelligent debate and prevented "due formulation of public opinion." True patriotism, he objected, "is being prostituted into a nervous intoxication which is fatal to the apprehension of truth"-and he guessed that this climate had been artificially stimulated to "win the consent of classes who would never consent to either financial or political jobbery."sq In this connection, he warned that "militarism, expansion, and imperial- ism will all favor plutocracy," by diverting taxpayers'money into "the hands of a few schemers" and by distracting public attention from the activities of plutocrats at home. Militarism, he predicted, would sap the energies and savings of the population, preventing them from giving "their attention to the problems of their own welfare and . . .their strength to the education and comfort of their own children."60 National prosperity and security lay not in the direction of military glory and imperialism-the false values which brought down the old European empires-but in "domestic development, peace, industry, free trade with everybody, low taxes, industrial power."6l 274 THE JOURNAL OF LIBERTARIAN STUDIES Sumner wanted men to win the struggle against nature, not to engage in fruitless and costly struggles with each other. Sumner predicted in 1900 that "the political history of the United States for the next fifty years will date from the Spanish war of 1898."62Like many sweeping generalizations, Sumner's is not free from objection, yet it contains an important element of truth. America's record of foreign involvements, culminating in the militarist epoch of the cold war and Vietnam war, have borne out many of Sumner's predictions. The growth of the "national security state," a logical culmination of the process Sumner described, has reduced liberties at home and abroad, interfered with the democratic pro- cess, distracted public attention from serious social problems at home, and continues to soak up vast resources that might otherwise he used to tackle those problems. Sumner's insights stemmed not from any special ability as a clairvoyant, but rather from his theoretical appreciation. of the conse-quences to society of a massive growth in state power. IV. Conclusion Despite the cogency and incisiveness of Sumner's critique of the State, he failed to stem the tide of growing government intervention. The last decade of his life, in particular, saw the emergence of "progressivism" and the gradual replacement of laissez-faire doctrines with the ideology of the welfare state. Both intellectuals and politicians found grand reform pro- grams and imperial glory more to their liking than the unexciting, hands-off program Sumner advocated. In his own time, Sumner became a reviled figure-among the imperialists who thought him weak and cowardly, among the German-school economists who thought him dangerously outmoded and "cantankerous,"63 and, of course, among socialists, including Upton Sinclair who referred to Sumner as the "prime minister in the empire of plutocratic education" who "took a ghoulish delight in the glorifying of commercialism . . . and . . . never wearied of pouring out ridicule upon the man who imagined he could do anything to make society better."64 Thanks to this intellectual attack, Sumner is little read today, with a reputation for conservatism and complacency, worthy of only brief mention in texts largely as a spokesman for the curious 19th century doctrine of Social Darwinism. Yet these stereotypes are far from the truth. Despite his cautious and pessimistic attitude towards social planning, Sumner was no conservative, much less a reactionary; a staunch rationalist and individual- ist, he ridiculed those who yearned for an old order based on status or "sentimental relations," while he defended the free society precisely because it enabled men to change their social and economic circumstances to meet their needs.65 Far from being smug or complacent, his dedication to liberty and reform led him into a lifelong battle against plutocracy, protective 275 WILLIAM GRAHAM SUMNER tariffs, and imperialism. His 1909 attack on the Republican Party for taking on "the character of a conspiracy to hold power and to use it for plutocratic ends"66 and the long agitation of Republican notables and alumni to remove Sumner from his post at Yale suggest that his writings were not merely a defense of the established order. He simply refused to "reform" that order by assenting to yet another increase in state power. What makes Sumner's thought endure, curiously enough, is the old- fashioned emphasis on liberty and rights that shines through the cold, positivist guise of his Social Darwinism. "A thoroughly consistent evolution- ist," Richard Hofstadter observes, ". . .would not have been so disturbed by the decline of laissez-faire . . ."67 In Sumner's case, a commitment to liberty came first. Today, few can sympathize with arguments taking as their premise the "survival of the fittest," but the libertarian component of his argument remains strikingly relevant to modern conditions: If a black man is told that the only status allowed by social institutions to him is that of a slave, no black man can work out into realization the powers which he may possess. If the status of women is fixed by custom and law, no woman can show her power to do anything outside of the limits. The social arrangement which sets free individual energy is liberty; for under this each one may prove what he is by what he does, and the society profits by the expansion and evolution of all the power there is in it.68 Today the welfare state is so deeply engrained that few Americans would find much to support in his position that the State owes nothing "to anybody except peace, order, and the guarantee of rights."69 The truth is that Sumner, even in his own time, was a radical; and his modern libertarian descendants, such as Friedrich von Hayek, John Hospers, and Murray Rothbard, are equally so in challenging the assumption shared by both liberals and conser- vatives that the state has a right to control the individual's destiny. NOTES 1. Quoted in Richard Crackatt, "American Liberalism and the Atlantic World, 1916-1917," Journol ofArnerienn Studies. I1 (April, 1977): p. 126-143. 2. Walter Lippmann, Drfl and Mastery (New York: M. Kennerley, 1914). p. 267. 3. Quoted in Sidney Fine, Loisre; Fnire ond rhe General-Welfare Stole (Ann Arbor: Univer- sity o f Michigan Press, 1956). p. 373. 4. Quoted in C. Resek, ed., The Progressives (Indianapolis: Bobbs-Merrill, 1967). p. mi. 5. Ibid.. pp. 125, 131-132. 6. Joseph Dorfman, The Economic Mind in Arnericon Civili:alion, 111 (New York: Viking Press, 1949). p. 206. Cf. R. Ely, Ground Under Our Feet (New Yark: Macmillan, 1938),pp. 132-159 on the founding o f the AEA. Sumner, in turn, was none too fond of the German-school economists, whose historical method he considered devoid of scientific interest. See Daniel M. Fox, The discover.^ of Abundance (Ithaca, N.Y.: Cornell University Press, 1967). p. 185". 276 THE JOURNAL O F LIBERTARIAN STUDIES 7. Fine, Loisse; Foire, p. 216. 8. Ibid., p. 240. 9. Walter Weyl, in particular, was deeply influenced by his teacher Simon Patten, a German- trained economist and colleague of Ely who believed that in an age of abundance, coopera- tian could replace competition as a mode of social organization. On the influence and importance of the New Republic group in progressive-liberal thought, see Crockatt, "American Liberalism," pp. 123-143, and David Noble, "The New Republic and the Idea of Progress, 1914-20," 38 Mississippi Volley Historical Review (December, 1951): pp. 387-402, and Charles Forcey, The Crossroods ofLlberalism (New York: Oxford University Press, 1961). 10. Herbert Croly, me Promise o/Americnn Lfe (New York: Macmillan, 1909), pp. 139,23, 276. I I . Quoted in 1 . M. Blum, The Republicon Rooievelr (Cambridge, Mass.: Harvard University Press, 1954), p. 107. 12. E. C. Razwenc, ed., Roosevelr, Wilson, ondrhe Trusrs (Boston: Heath, 1950). 13. Their differences were unavoidably highlighted in a campaign where two fundamentally progressive candidates needed to assen the uniqueness of their positions. TR frequently made Wilson out to be an unthinking advocate of outmoded Jeffersonian individualism, but this was a campaign exaggeration. 14. Woodrow Wilson, The New Freedom (Englewood Cliffs, N.J.: Prentice-Hall, 1961), p. 27. 15. Ibid., p. 164. 16. Fine, Lnirrez Foire, p. 79. 17. Sumner expressed this conviction in "State Interference," in Wor ond Other Essoys (New Haven: Yale University Press, 1911), p. 225. 18. Harris Starr, William Groham Sumner (New York: Holt, 1925), p. 22; cf. Fine, Lnisrez Faire. o. 10. 19. ~ine,'bisse: Foire, p. 552. Sumner nearly lost his post at Yale thanks to his insistence on using Spencer's Study ofSociology in his classes. See Richard Hofstadler, Social Dorwin- ism in Americon Thoughr (New Yark: G. Braziller, 1959), p. 20. 20. W. G. Sumner, "The Concentration of Wealth-Its Economic Justification," in The Challenge of Facts and Others Essays (New Haven: Yale University Press, 1914), p. 90. 21. Sumner, "Reply to a Socialist," 1904, in Challenge o f Facts, pp. 56-57. 22. Sumner, Wor and Other ESSIIYS, pp. 176-177. 23. Quoted in Starr, Sumner, p. 492.~ 24. Quoted in Fine, Loisse: Foire, pp. 82-83. 25. Sumner. "Challenae of Facts." in Challenne dFocrs. o. 26 26. ~ u m n e r ; "The or gotten an," in The F&;ten ~ o n ' a n d ~ t h e r Essays (New Haven: Yale University Press, 1919), pp. 493, 482. 27. Sumner, "On the Case of a Certain Man Who is Never Thought of," in War, pp. 247-249; Maurice Davie, Willinm Graham Sumner (New York: Crowell, 1963), p. 41. 28. Quotes from Starr, Sumner, pp. 427-430. 29. Sumner, War. pp. 209-210. 30. Quoted in Davie, Sumner, pp. 37-38. 31. Sumner, "Some Points in the New Social Creed," in Earth Hunger and Other Ersoys (New York: Books far Libraries Press, 1970), pp. 207-208; "The State and Monopoly," in ibid., p. 277; "Federal Legislation on Railroads," in Chnllengr o f Fads, p. 182. 32. Quoted in Starr, Sumner, p. 446. 33. Sumner, "The Mores of the Present and the Future," in War, p. 160. 34. Sumner, "Democracy and Plutocracy," in Eorrh Hunger, p. 287. 35. Sumner, "Separation of State and Market," in Eorrh Hunger, p. 310. 36. Sumner, "Absurd EBort . . ."in War, pp. 206-207. 37. Sumner, Folkwoys(New Yark: Dover, 1959), pp. 170-171, 38. Sumner, "The State and Monopoly," in Eorrh Hunger, pp. 277-278. 39. Sumner, "Protectionism: the Ism which teaches that Waste Makes Wealth," in Forgotren Mm, pp. 10-1 1,79, 110-1 11; "Protectionism Twenty Years After," in ibid., p. 134. 40. Sumner, "Protectionism Twenty Years After," in ibid., pp. 136-137. 277 WILLIAM GRAHAM SUMNER 41. Quoted in Davie, Sumner, p. 33. 42. Ibid., p. 27. 43. Theodore Roasevelt, "Expansion and Peace," in William Harbaugh, ed., Writings of 7heodore Roosevell (Indianapolis: Bobbs-Merrill, 1967), pp. 28-34. In 1901 TR told an audience, along much the same lines, "It is our duty toward the people living in barbarism to see that they are freed from their chains." See H. K. Beale, Theodore Roosevelt and the Rise ofAmerico to World Power (Baltimore: Johns Hopkins Press, 1956), p. 34. 44. Croly, Promise, pp. 289, 169, 297. Among the German-school economists, Simon Patten, Walter Weyl's teacher, was a particularly ardent imperialist. See Fox, Discovery, pp. 115-116. 45. Arthur Ekirch, Progressivism in America (New York: New Viewpoints, 1974). p. 182. 46. Quoted in Harbaugh. Writings, p. 77. 47. TR, "How I Became a Progressive," Outlook (October 12, 1912) reprinted in Resek, Progressives, p. 336. 48. William Leuchtenburg, "Progressivism and Imperialism," 39 Mississippi Valley Historical Review (December, 1952): pp. 483-504. 49. Ekirch, Progresrivism, pp. 185-186. 50. E. Berkeley Tompkins, Anti-lmperiolism in the United Slales (Philadelphia: University of Philadelphia Press, 1970). pp. 148-149. 51. Sumner, "The Fallacy of Territorial Expansion," 1896, in Wor, p. 292. 52. The outraged New York Sun called Sumner's speech "typical of the smart and shallow tcaehine bv which that institution lYale1 has suffered him so lone to oervert the intellieence . - . of its s&dnts." Quoted in Starr. Sumner, pp. 297-299. 53. Sumner, Wor, p. 297. 54. Ibid., pp. 302, 309-310. 55. Many of the anti-imperialists-most notably William Lloyd Garrison, son of the famous abolitionist publicist-had a strong background in abolitionist and past-bellum civil rights activities. See Tampkins, Anti-Imperialism, p. 151. 56. Sumner, War, p. 331. These quotes should be sufficient to refute the view that Sumner, like TR, argued out the issue of imperialism an Social Darwinist grounds and came to an anti- imperialist conclusion only because of his fears that "~bsor&on of inferior peoples would mongrelize the race and dilute the purity upon which its supposed superiority depended." This erroneous interpretation is from Lloyd Gardner, el oL, Cmtion of the Americon Empire (New York: Rand McNally, 1973), p. 222. 57. See Stuart Miller, "Our Mylai of 1900: Americans in Philippine Insurrection," in M. B . Young,ed., American Expansionism (Boston: Little, Brown, 1973). pp. 103-1 16. 58. Sumner, "The Predominant Issue," in Wor, p. 351. 59. Ibid., pp. 298-299, 300-301. M). Ibid., pp. 324-325. 61. Ibid., p. 351. 62. Ibid., p. 337. 63. Dorfman, Economic Mind, p. xxviii, quoting John Bates Clark. 64. Quoted in Starr, Sumner, p. 256. 65. Hofstadter, Social Darwinism, p. 8. Hofstadter, with an eye for paradox, writes, "We may wonder whether, in the entire history of thought, there was ever a conservative so utterly progressive as this." 66. Sumner, "The Mores of the Present and the Future," in War, p. IM). 67. Hofstadter, Sociol Dorwinirm, p. 65. 68. Sumner. "Separation of State and Market," in Earth Hunger, p. 308. 69. Quoted in Davie. Surnner, p. 42.
18194	https://imomath.com/index.cgi?page=quadraticCongruencesPrimeModuli	Quadratic Congruences with Prime Moduli IMOmath Olympiads Book Training Calculator Quadratic congruences (home) 1.Quadratic congruences with prime moduli 2.Quadratic congruences with composite moduli 3.Some sums of Legendre’s symbols 4.Problems Quadratic Congruences with Prime Moduli Definition Let (m,n) and (a) be integers, (m > 1), (n\geq1) and ((a,m)=1). We say that (a) is a residue of (n)-th degree modulo (m) if congruence (x^n\equiv a) (mod (m)) has an integer solution; else (a) is a nonresidue of (n)-th degree. Specifically, for (n=2,3,4) the residues are called quadratic, cubic, biquadratic, respectively. This text is mainly concerned with quadratic residues. Theorem 1 Given a prime (p) and an integer (a), the equation (x^2\equiv a) has zero, one, or two solutions modulo (p). Show proof. ;) Suppose that the considered congruence has a solution (x_1). Then so clearly is (x_2=-x_1). There are no other solutions modulo (p), because (x^2\equiv a\equiv x_1^2) (mod (p)) implies (x\equiv\pm x_1.) As a consequence of the above simple statement we obtain: Theorem 2 For every odd positive integer (p), among the numbers (1,2,\dots,p-1) there are exactly (\frac{p-1}2) quadratic residues (and as many quadratic nonresidues). Definition Given a prime number (p) and an integer (a), Legendre's symbol (\left(\frac ap\right)) is defined as [\left(\frac ap\right)=\left{\begin{array}{cl}1,&\mbox{if } p\nmid a\mbox{ and } a \mbox{ is a quadratic residue (mod }p);\newline -1, &\mbox{if }p\nmid a \mbox{ and }a \mbox{ is a quadratic nonresidue (mod } p);\newline 0,&\mbox{if }p\mid a.\end{array}\right.] Example 1 Obviously, (\left(\frac{x^2}p\right) =1) for each prime (p) and integer (x), (p\nmid x). Example 2 Since 2 is a quadratic residue modulo 7 ((3^2\equiv 2)), and 3 is not, we have (\left(\frac27\right)=1) and (\left(\frac37\right)=-1). From now on, unless noted otherwise, (p) is always an odd prime and (a) an integer. We also denote (p\prime=\frac{p-1}2). Clearly, (a) is a quadratic residue modulo (p) if and only if so is (a+kp) for some integer (k). Thus we may regard Legendre's symbol as a map from the residue classes modulo (p) to the set ({-1,0,1}). Fermat's theorem asserts that (a^{p-1}\equiv1) (mod (p)), which implies ( a^{p\prime}\equiv\pm1) (mod (p)). More precisely, the following statement holds: Theorem 3 (Euler's Criterion) (a^{p\prime}\equiv\left(\frac ap\right)) (mod (p)). Show proof. ;) The statement is trivial for (p\mid a). From now on we assume that (p\nmid a). Let (g) be a primitive root modulo (p). Then the numbers (g^i), (i=0,1, \dots,p-2) form a reduced system of residues modulo (p). We observe that ((g^i)^{p\prime}= g^{ip\prime}\equiv1) if and only if (p-1\mid ip\prime), or equivalently, (2\mid i). On the other hand, (g^i) is a quadratic residue modulo (p) if and only if there exists (j\in{0,1,\dots,p-2}) such that ((g^j)^2\equiv g^i) (mod (p)), which is equivalent to (2j\equiv i) (mod (p-1)) . The last congruence is solvable if and only if (2\mid i), that is, exactly when ((g^i)^{p\prime}\equiv1) (mod (p)). The following important properties of Legendre's symbol follow directly from Euler's criterion. Theorem 4 Legendre's symbol is multiplicative, i.e. (\left(\frac{ab}p\right)=\left(\frac ap\right) \left(\frac bp\right)) for all integers (a,b) and prime number (p > 2). Problem 1 There exists a natural number (a < \sqrt p+1) that is a quadratic nonresidue modulo (p). Show solution. ;) Consider the smallest positive quadratic nonresidue (a) modulo (p) and let ( b=\left[\frac pa\right]+1). Since (0 < ab-p < a), (ab-p) must be a quadratic residue. Therefore [1=\left(\frac{ab-p}p\right)=\left(\frac ap\right)\left(\frac ap \right)=-\left(\frac bp\right).] Thus (b) is a quadratic nonresidue and hence ( a\leq b < \frac pa+1), which implies the statement. Theorem 5For every prime number (p > 2), ( \left(\frac{-1}p\right)=(-1)^{\frac{p-1}2}). In other words, the congruence (x^2\equiv-1) modulo a prime (p) is solvable if and only if (p=2) or (p\equiv1) (mod 4). Problem 2 If (p) is a prime of the form (4k+1), prove that (x=(p\prime)!) is a solution of the congruence (x^2+1\equiv0) (mod (p)). Show solution. ;) Multiplying the congruences (i\equiv-(p-i)) (mod (p)) for (i=1,2,\dots,p\prime) yields ((p\prime)!\equiv(-1)^{p\prime} (p\prime+1)\cdots(p-2)(p-1)). Note that (p\prime) is even by the condition of the problem. We now have [x^2=(p\prime)!^2\equiv(-1)^{p\prime}p\prime\cdot (p\prime+1)\cdots(p-2)(p-1)=(-1)^{p\prime}(p-1)!\equiv(-1)^{p\prime+1}=-1\mbox{ (mod }p)] by Wilson's theorem. One can conclude from Problem 1 that every prime factor of number (x^2+y^2) (where (x,y\in\mathbb{N}) are coprime) is either of the form (4k+1), (k\in\mathbb{N}), or equal to 2. This conclusion can in fact be generalized. Theorem 6 Let (x,y) be coprime integers and (a,b,c) be arbitrary integers. If (p) is an odd prime divisor of number (ax^2+bxy+cy^2) which doesn't divide (abc), then [D=b^2-4ac] is a quadratic residue modulo (p). In particular, if (p\mid x^2-Dy^2) and ((x,y)=1), then (D) is a quadratic residue (mod (p)). Show proof. ;) Denote (N=ax^2+bxy+cy^2). Since (4aN=(2ax+by)^2- Dy^2), we have [(2ax+by)^2\equiv Dy^2\;\;\mbox{(mod }p).] Furthermore, (y) is not divisible by (p); otherwise so would be (2ax+by) and therefore (x) itself, contradicting the assumption. There is an integer (y_1) such that (yy_1\equiv1) (mod (p)). Multiplying the above congruence by (y_1^2) gives us ((2axy_1+byy_1)^2 \equiv D(yy_1)^2\equiv D) (mod (p)), implying the statement. For an integer (a), (p\nmid a) and (k=1,2,\dots,p\prime) there is a unique (r_k\in{-p\prime,\dots,-2,-1,1,2,\dots,p\prime}) such that (ka\equiv r_k) (mod (p)). Moreover, no two of the (r_k) 's can be equal in absolute value; hence (\|r_1\|,\|r_2\|,\dots,\|r_{p\prime}\|) is in fact a permutation of ({1,2,\dots,p\prime}). Then [a^{p\prime}=\frac{a\cdot2a \cdot\dots\cdot p\prime a}{1\cdot2\cdot\dots\cdot p\prime}\equiv \frac{r_1r_2\dots r_{p\prime}}{1\cdot2\cdot\dots\cdot p\prime}.] Now, setting (r_k=\epsilon_k\|r_k\|) for (k=1,\dots,p\prime), where (\epsilon_k=\pm1), and applying Euler's criterion we obtain: Theorem 7 (\left(\frac ap\right)= \epsilon_1\epsilon_2\cdots\epsilon_{p\prime}). Observe that (r_k=-1) if and only if the remainder of (ka) upon division by (p) is greater than (p\prime), i.e. if and only if (\left[ \frac{2ka}p\right]=2\left[\frac{ka}p\right]+1). Therefore, ( r_k= (-1)^{\left[\frac{2ka}p\right]}). Now Theorem 7 implies the following statement. Theorem 8 (Gauss' Lemma) (\left( \frac ap\right)=(-1)^S), where ( S=\sum_{k=1}^{p\prime} \left[\frac{2ka}p\right]). Gauss' lemma enables us to easily compute the value of Legendre's symbol (\left(\frac ap\right)) for small (a) or small (p). If, for instance, (a=2), we have (\left(\frac2p\right)=(-1)^S), where ( S=\sum_{k=1}^{p\prime}\left[\frac{4k}p\right]). Exactly (\left[ \frac12p\prime\right]) summands in this sum are equal to 0, while the remaining ( p\prime-\left[\frac12p\prime\right]) are equal to 1. Therefore ( S=p\prime-\left[\frac12p\prime\right]=\left[\frac{p+1}4\right]), which is even for (p\equiv\pm1) and odd for (p\equiv\pm3) (mod 8). We have proven the following Theorem 9(\left(\frac2p\right)=(-1)^{\left[ \frac{p+1}4\right]}). In other words, 2 is a quadratic residue modulo a prime (p > 2) if and only if (p\equiv\pm1) (mod 8). The following statements can be similarly shown. Theorem 10 (a)-2 is a quadratic residue modulo (p) if and only if (p\equiv1) or (p\equiv 3) (mod 8); (b) -3 is a quadratic residue modulo (p) if and only if (p\equiv1) (mod 6); (c) 3 je quadratic residue modulo (p) if and only if (p\equiv\pm1) (mod 12); (d) 5 is a quadratic residue modulo (p) if and only if (p\equiv\pm1) (mod 10). Problem 3 Show that there exist infinitely many prime numbers of the form (a) (4k+1); (b) (10k+9). Show solution. ;) (a) Suppose the contrary, that (p_1,p_2,\dots, p_n) are all such numbers. Then by Theorem 5, all prime divisors of (N=(2p_1p_2\cdots p_n)^2+1) are of the form (4k+1). However, (N) is not divisible by any of (p_1,p_2,\dots,p_n), which is impossible. Part (b) is similar to (a), with number (N=5(2p_1p_2\cdots p_n)^2-1) being considered instead. Problem 4 Prove that for (n\in\mathbb{N}) every prime divisor (p) of number (n^4-n^2+1) is of the form (12k+1). Show solution. ;) We observe that [n^4-n^2+1=(n^2-1)^2+n^2 \hspace{5mm}\mbox{i}\hspace{5mm}n^4-n^2+1=(n^2+1)^2-3n^2.] In view of theorems 5, 6, and 10, the first equality gives us (p\equiv1) (mod 4), whereas the other one gives us (p\equiv\pm1) (mod 12). These two congruences together yield (p\equiv1) (mod 12). Problem 5 Evaluate [\left[\frac1{2003}\right] +\left[\frac{2}{2003}\right]+\left[\frac{2^2}{2003}\right]+\cdots+ \left[\frac{2^{2001}}{2003}\right].] Show solution. ;) Note that 2003 is prime. It follows from Euler's criterion and Theorem 10 that (2^{1001}\equiv\left(\frac2{2003} \right)=-1) (mod (2003)). Therefore (2003\mid 2^i(2^{1001}+1)=2^{1001+i}+2^i); since (2^i) and (2^{1001+i}) are not multiples of (2003), we conclude that [\left[\frac{2^i}{2003}\right]+\left[\frac{2^{1001+i}}{2003} \right]=\frac{2^i+2^{1001+i}}{2003}-1.] Summing up these equalities for (i=0,1,\dots,1000) we obtain that the desired sum equals [\frac{1+2+2^2+\cdots+2^{2001}}{2003}-1001= \frac{2^{2002}-1}{2003}-1001.] The theory we have presented so far doesn't really facilitate the job if we need to find out whether, say, (814) is a quadratic residue modulo (2003). That will be done by the following theorem, which makes such a verification possible with the amount of work comparable to that of the Euclidean algorithm. Theorem 11 (Gauss' Reciprocity Law) For any different odd primes (p) and (q), [\left(\frac pq\right)\left(\frac qp\right)=(-1)^{p\prime q\prime},] where (p\prime=\frac{p-1}2) and (q\prime=\frac{q-1}2). Show proof. ;) Define ( S(p,q)=\sum_{k=1}^{q\prime}\left[ \frac{kp}q\right]). We start by proving the following auxiliary statement. Lemma. (S(p,q)+S(q,p)=p\prime q\prime). Proof of the Lemma. Given (k\in\mathbb{N}), we note that (\left[ \frac{kp}q\right]) is the number of integer points ((k,l)) in the coordinate plane with (0 < l < kp/q), i.e. such that (0 < ql < kp). It follows that the sum (S(p,q)) equals the number of integer points ((k,l)) with (0 < k < p\prime) and (0 < ql < kp). Thus (S(p,q)) is exactly the number of points with positive integer coordinates in the interior or on the boundary of the rectangle (ABCD) that lie {\it below} the line (AE), where (A(0,0)), (B(p\prime,0)), (C(p\prime,q\prime)), (D(0,q\prime)), (E(p,q)). Analogously, (S(q,p)) is exactly the number of points with positive integer coordinates in the interior or on the boundary of the rectangle (ABCD) that lie {\it above} the line (AE). Since there are (p\prime q\prime) integer points in total in this rectangle, none of which is on the line (AE), it follows that (S(p,q)+S(q,p)=p\prime q\prime). (\triangledown) We now return to the proof of the theorem. We have [S(p+q,q)-S(p,q)=1+ 2+\dots+p\prime=\frac{p^2-1}8.] Since Theorem 9 is equivalent to ( \left(\frac2p\right)=(-1)^{\frac{p^2-1}8}), Gauss' lemma gives us [\left(\frac2q\right)\left(\frac pq\right)=\left(\frac{2p}q\right) =\left(\frac{2(p+q)}q\right)=\left(\frac{\frac{p+q}2}q\right)= (-1)^{S(p+q,q)}=\left(\frac2q\right)(-1)^{S(p,q)},] hence ( \left(\frac pq\right)=(-1)^{S(p,q)}). Analogously, (\left(\frac qp\right)=(-1)^{S(q,p)}). Multiplying the last two inequalities and using the lemma yields the desired equality. Let us now do the example mentioned before the Reciprocity Law. Example 3 (\left(\frac{814}{2003}\right)= \left(\frac{2}{2003}\right)\left(\frac{11}{2003}\right) \left(\frac{37}{2003}\right)=-\left(\frac{11}{2003}\right) \left(\frac{37}{2003}\right)). Furthermore, the Reciprocity Law gives us [\left(\frac{11}{2003}\right)=-\left(\frac{2003}{11}\right)= \left(\frac{1}{11}\right)=1\quad\mbox{and}\quad\left(\frac{37}{2003} \right)=\left(\frac{2003}{37}\right)=\left(\frac{5}{37}\right)= \left(\frac{37}{5}\right)=-1.] Thus (\left(\frac{814}{2003}\right)= 1), i.e. (814) is a quadratic residue modulo (2003). Problem 6 Prove that an integer (a) is a quadratic residue modulo every prime number if and only if (a) is a perfect square. Show solution. ;) Suppose that (a) is not a square. We may assume w.l.o.g. (why?) that (a) is square-free. Suppose that (a > 0). Then (a=p_1p_2\cdots p_k) for some primes (p_1, \dots,p_k). For every prime number (p) it holds that [\left(\frac ap\right)=\prod_{i=1}^k\left(\frac{p_i}p\right) \quad\mbox{and}\quad\left(\frac{p_i}p\right)= (-1)^{p_i\prime p\prime}\left(\frac p{p_i}\right).\quad\quad\quad\quad\quad(1)] If (a=2), it is enough to choose (p=5). Otherwise (a) has an odd prime divisor, say (p_k). We choose a prime number (p) such that (p\equiv1) (mod 8), (p\equiv 1) (mod (p_i)) for (i=1,2,\dots,k-1), and (p\equiv a) (mod (p_k)), where (a) is an arbitrary quadratic nonresidue modulo (p_k). Such prime number (p) exists according to the Dirichlet theorem on primes in an arithmetic progression. Then it follows from (1) that (p_1,\dots,p_{k-1}) are quadratic residues modulo (p), but (p_k) is not. Therefore (a) is a quadratic nonresidue modulo (p). The proof in the case (a < 0) is similar and is left to the reader. GradeYard
18195	https://ecampusontario.pressbooks.pub/businessfinancialmath/chapter/4-2-calculating-future-value/	Skip to content Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices. 4.2 Calculating the Future Value LEARNING OBJECTIVES Calculate future value for compound interest. Your company has an employee assistance plan through which employees can borrow funds at 12% compounded semi-annually, much like a loan from a bank, then repay the money at their convenience. An employee who borrowed $4,000 two years ago has been unable to repay the loan until today. As the human resources manager in charge of the assistance plan, you must tell him the exact amount he needs to pay to return his balance to zero. How do you do this? In the above scenario, you need to tell the employee the future value of their loan. The future value ([latex]FV[/latex]) is the accumulated value or maturity value of a loan or an investment at the end of the term of the loan or investment. The future value includes the principal of the loan or investment plus all of the interest the loan or investment has accumulated over the term. The principal of the loan or investment is called the present value ([latex]PV[/latex]). The present value is the amount of money borrowed for a loan or the amount of money invested for an investment at the start of the term. Calculating the Future Value Continuing with the scenario described above, suppose $4000 was borrowed two years ago at 12% compounded semi-annually. At the end of the two years, the borrower will owe two years of compound interest in addition to the original principal of $4,000. The amount of money borrowed for the loan is the present value, so [latex]PV=\$4,000[/latex]. The compounding frequency is semi-annually, or twice per year, which makes the periodic interest rate [latex]i=\frac{j}{m}=\frac{12\%}{2}=6\%[/latex]. This means that every six months, the borrower accumulates 6% of current principal in interest charges. The total number of compoundings is [latex]n=m \times t =2 \times 2=4[/latex], which means that interest will be calculated and added four times over the term of the loan. At the end of the first six months (one compounding period) of the loan, the current balance or future value of the loan is [latex]\begin{eqnarray} \mbox{Balance after first compounding period} & = & \mbox{Principal}+\mbox{Interest} \ FV & = & PV+ i \times PV \ & = & 4,000+0.06 \times 4,000 \ & = & \$4,240 \end{eqnarray}[/latex] In the above calculation, note that the equation [latex]FV=PV+i \times PV[/latex] can be factored and rewritten as [latex]FV=PV(1+i)[/latex], which can be used to calculate the future value for each compounding period. The above process will continue for each of the compounding periods: 6% of the current balance will be added to the balance at the end of each compounding period. For the second compounding period, the present value is the future value from the previous compounding period and the future value is the balance on the loan at the end of the second compounding period. [latex]\begin{eqnarray} FV_{\mbox{ after two compounding periods}} & = & PV\times (1+ i ) \ & = & 4,240\times (1+0.06) \ & = & \$4,494.40 \end{eqnarray}[/latex] Recall that the [latex]PV=\$4,240[/latex] was the result of the future value calculation of the first compounding period. In fact, [latex]4,240=4000 \times (1+0.06)[/latex], which can be substituted into the previous calculation. [latex]\begin{eqnarray} FV_{\mbox{ after two compounding periods}} & = & PV\times (1+ i ) \times (1+i) \ & = & 4,000\times (1+0.06) \times (1+0.06) \ & = & \$4,494.40 \end{eqnarray}[/latex] The equation [latex]FV=PV \times (1+i) \times (1+i)[/latex] can be simplified to the following [latex]\displaystyle{FV=PV \times (1+i) \times (1+i)=PV \times (1+i)^2}[/latex] Continuing the calculations for each of the four compounding periods results in the following. \| \| \| \| --- \| Compounding Period \| Present Value at the Beginning of the Compounding Period \| Future Value at the End the Compounding Period \| \| 1 \| [latex]\small{\$4,000}[/latex] \| [latex]\small{4,000 \times (1+0.06)=\$4,240}[/latex] \| \| 2 \| [latex]\small{\$4,240}[/latex] \| [latex]\small{\begin{eqnarray} 4,240 \times (1+0.06) & = & 4,000 \times (1+0.06) \times (1+0.06) \ & = & 4,000 \times (1+0.06)^2 \ & = & \$4,494.40 \end{eqnarray}}[/latex] \| \| 3 \| [latex]\small{\$4,494.40}[/latex] \| [latex]\small{\begin{eqnarray} 4,494.40 \times (1+0.06) & = & 4,000 \times (1+0.06) \times (1+0.06) \times (1+0.06) \ & = & 4,000 \times (1+0.06)^3 \ & = & \$4,764.06 \end{eqnarray}}[/latex] \| \| 4 \| [latex]\small{\$4,764.06}[/latex] \| [latex]\small{\begin{eqnarray} 4,764.06 \times (1+0.06) & = & 4,000 \times (1+0.06) \times (1+0.06) \times (1+0.06) \times (1+0.06) \ & = & 4,000 \times (1+0.06)^4 \ & = & \$5,059.91 \end{eqnarray}}[/latex] \| Do you notice a pattern in these calculations? With the first compounding period, the formula has only one latex[/latex], with two compounding periods, the formula has two latex[/latex]‘s in the form of latex^2[/latex], with three compounding periods, the formula has three latex[/latex]‘s in the form latex^3[/latex], and so on. In general, the exponent of latex[/latex] equals the number of the compounding period. The Future Value Formula The future value for compound interest is [latex]\displaystyle{FV=PV \times (1+i)^n}[/latex] where [latex]FV[/latex] is the future value. The future value includes the principal plus all of the interest accumulated over the term. [latex]PV[/latex] is the present value or principal. The present value is the starting amount upon which compound interest is calculated. [latex]i[/latex] is the periodic interest rate. The periodic interest rate is the interest rate per compounding period: [latex]i=\frac{j}{m}[/latex] where [latex]j[/latex] is the nominal interest rate and [latex]m[/latex] is the compounding frequency. [latex]n[/latex] is the total number of compounding periods over the term. [latex]n=m \times t[/latex] where [latex]m[/latex] is the compounding frequency and [latex]t[/latex] is the length of the term in years. The amount of interest ([latex]I[/latex]) accumulated by a loan or investment is [latex]\displaystyle{I=FV-PV}[/latex] EXAMPLE If you invested $5,000 for 10 years at 9% compounded quarterly, how much money would you have? What is the interest earned during the term? Solution: The timeline for the investment is below. Step 1: The given information is [latex]\begin{eqnarray} PV & = & \$5,000 \ j & = & 9\% \ m & = & 4 \ t & = & 10 \mbox{ years} \end{eqnarray}[/latex] Step 2: Calculate the periodic interest rate. [latex]\begin{eqnarray} i & = & \frac{j}{m} \ & = & \frac{9\%}{4} \ & = & 2.25\% \ & = & 0.0225 \end{eqnarray}[/latex] Step 3: Calculate the total number of compoundings. [latex]\begin{eqnarray} n & = & m \times t \ & = & 4 \times 10 \ & = & 40 \end{eqnarray}[/latex] Step 4: Calculate the future value. [latex]\begin{eqnarray} FV & = & PV \times (1+i)^n \ & = & 5,000 \times (1+0.0225)^{40} \ & = & 5,000 \times (1.0225)^{40} \ & = & \$12,175.94 \end{eqnarray}[/latex] Step 5: Calculate the interest earned. [latex]\begin{eqnarray} I & = & FV-PV \ & = & 12,175.94-5,000 \ & = & \$7,175.94 \end{eqnarray}[/latex] After 10 years, the principal grows to $12,175.94, which includes your $5,000 principal and $7,175.94 of interest. Using a Financial Calculator A financial calculator, like the TI BAII Plus, has built-in functions to solve compound interest problems. These functions use the “time value of money” buttons on the calculator. Throughout the remainder of this book, we will focus on using the financial calculator to solve problems involving compound interest, instead of using the formulas as illustrated above. USING THE TI BAII PLUS CALCULATOR TO FIND THE FUTURE VALUE FOR COMPOUND INTEREST The time value of money buttons are located in the TVM row (the third row from the top) of the calculator. The five buttons located on the third row of the calculator are five of the seven variables required for time value of money calculations. This row’s buttons are different in colour from the rest of the buttons on the keypad. The other two variables are in a secondary menu above the I/Y key and are accessed by pressing 2nd I/Y. Altogether, there are seven variables required to complete time value of money calculations. Note that P/Y and C/Y are not main button keys in the TVM row. The P/Y and C/Y variables are located in the secondary function accessed by pressing 2nd I/Y. \| \| \| --- \| \| Variable \| Meaning \| \| N \| Total number of compounding periods. This is the same value as [latex]n[/latex] in the future value formula. [latex]N=\mbox{time in years} \times \mbox{compounding frequency}[/latex] \| \| I/Y \| Interest rate per year (i.e. the nominal interest rate). The interest rate is entered in percent form (without the % sign). For example, 5% is entered as 5. \| \| PV \| Present value or principal. \| \| PMT \| Periodic annuity payment. For compound interest only calculations, PMT=0. (Note: in later chapters you will learn about annuities where PMT will not be 0.) \| \| FV \| Future value or maturity value. \| \| P/Y \| Payments frequency for annuity payment. For compound interest only calculations, P/Y is set to the same value as C/Y. (Note: in later chapters you will learn about annuities where P/Y will be set to the frequency of the payments.) \| \| C/Y \| Compounding frequency. This is the value of [latex]m[/latex]. \| To enter values into the calculator: For the main button keys in the TVM row (i.e. N, I/Y, PV, PMT, FV), enter the number first and then press the corresponding button. For example, to enter N=34, enter 34 on the calculator and then press N. For P/Y and C/Y, press 2nd I/Y. At the P/Y screen, enter the value for P/Y and then press ENTER. Press the down arrow to access the C/Y screen. At the C/Y screen, enter the value for C/Y and then press ENTER. Press 2nd QUIT (the CPT button) to exit the menu. For example, to enter P/Y=4 and C/Y=4, press 2nd I/Y. At the P/Y screen, enter 4 and press ENTER. Press the down arrow. At the C/Y screen, enter 4 and press ENTER. Press 2nd QUIT to exit. After all of the known quantities are loaded into the calculator, press CPT and then FV to solve for the future value. NOTES The calculator automatically sets C/Y equal to whatever is entered for P/Y. That is, if you enter 4 for P/Y, the calculator will set C/Y=4. This is what we need to complete compound interest only calculations. When we learn about annuities in later chapters, we will have instances where P/Y and C/Y will need to be different. Your calculator has permanent memory. Once you enter data into any of the time value buttons it is permanently stored until You override it by entering another piece of data and pressing the button; or You clear the memory of the time value buttons by pressing 2nd CLR TVM before proceeding with another question; or The reset button on the back of the calculator is pressed. Compound Interest (Present and Future Values) by Joshua Emmanuel [6:56] (transcript available). Cash Flow Convention Signs An investment and a loan are very different. An investment earns interest and the principal increases over time. A loan is charged interest but is usually paid off through payments, resulting in the principal decreasing over time. Notice that nowhere on the calculator is there a button to enter this critical piece of information. How does the calculator distinguish between the two? You must apply a principle called the cash flow sign convention to the PV, PMT, and FV buttons. If you have money leaving your possession and going somewhere else (such as being put into an investment or making a payment against a loan), you must enter the number as a NEGATIVE number. If you have money coming into your possession and you are receiving it (such as borrowing money from the bank or having an investment mature and pay out to you), you must enter the number as a POSITIVE number. When doing financial calculations it is important to “be somebody” in the transaction. In any compound interest scenario, two parties are always involved—somebody is investing and somebody is borrowing. From this standpoint, think about whether the money leaves you or comes at you. This will help you place the correct sign in front of the PV, PMT, and FV when using your calculator. Who you are will not change the numbers of the transaction, just the cash flow sign convention. If you borrow money from your friend and then pay it back, the initial loan is received by you and hence entered as a positive PV for you. As you pay back the loan, money leaves you and therefore the FV is negative for you. Taking the other side of the coin and being your friend, the loaning of money is a negative PV for him. When you repay the loan, your friend receives it and therefore results in a positive FV for him. \| \| \| --- \| \| Sign of FV \| Sign of PV \| \| Investments \| When you receive your matured investment at the end of the term this is considered as a cash-inflow for you and the future value should be entered as a positive amount. \| When money is invested (paid-out), this amount is considered as a cash-outflow and this amount has to be entered as a negative number for PV. \| \| Loans \| When the loan is repaid at the end of the term this is considered as a cash-outflow for you and the future value should be entered as a negative amount. \| When money is received (loaned), this amount is considered as a cash-inflow and this amount has to be entered as a positive number for PV. \| NOTES Notice that the PV and FV always have opposite signs. When you compute solutions on the BAII Plus calculator, one of the most common error messages displayed is “Error 5.” This error indicates that the cash flow sign convention has been used in a manner that is financially impossible. Some examples of these financial impossibilities include loans with no repayment or investments that never pay out. In these cases, the PV and FV have been incorrectly set to the same cash flow sign. EXAMPLE If you invested $5,000 for 10 years at 9% compounded quarterly, how much money would you have? Solution: The timeline for the investment is below. \| \| \| --- \| \| N \| [latex]4 \times 10=40[/latex] \| \| PV \| [latex]-5,000[/latex] \| \| FV \| ? \| \| PMT \| [latex]0[/latex] \| \| I/Y \| [latex]9[/latex] \| \| P/Y \| [latex]4[/latex] \| \| C/Y \| [latex]4[/latex] \| [latex]FV=\$12,175.94[/latex] TRY IT Find the future value if $53,000 is invested at 6% compounded monthly for 4 years and 3 months. How much interest did the investment make? Click to see Solution \| \| \| --- \| \| N \| [latex]12 \times \frac{51}{12}=51[/latex] \| \| PV \| [latex]-53,000[/latex] \| \| FV \| ? \| \| PMT \| [latex]0[/latex] \| \| I/Y \| [latex]6[/latex] \| \| P/Y \| [latex]12[/latex] \| \| C/Y \| [latex]12[/latex] \| [latex]\displaystyle{FV=\$68,351.02}[/latex] [latex]\begin{eqnarray} I & = & FV-PV \ & = & 68,351.02-53,000 \ & = & \$15,351.02 \end{eqnarray}[/latex] Future Value Calculations with Variable Changes What happens if a variable such as the nominal interest rate, compounding frequency, or even the principal changes somewhere in the middle of the transaction? When any variable changes, you must break the timeline into separate time fragments at the point of the change. To arrive at the solution, you need to work from left to right one time segment at a time using the future value formula. Read and understand the problem. Identify the present value. Draw a timeline broken into separate time segments at the point of any change. For each time segment, identify any principal changes, the nominal interest rate, the compounding frequency, and the length of the time segment in years. Starting with the present value in the first time segment (starting on the left), solve for the future value. Let the future value calculated in the previous step become the present value for the next step. If the principal changes, adjust the new present value accordingly. Calculate the future value of the next time segment. Repeat the previous steps until you obtain the final future value from the final time segment. NOTE Transforming the future value from one time segment into the present value of the next time segment does not require re-entering the computed value. Instead, apply the following technique. Load the calculator with all known compound interest variables for the first time segment. Compute the future value at the end of the segment. With the answer still on your display, adjust the principal if needed, change the cash flow sign by pressing the [latex]\pm[/latex] key, and then store the unrounded number back into the present value button by pressing PV. Change the N, I/Y, and C/Y as required for the next segment. Return to step 2 for each time segment until you have completed all time segments. EXAMPLE Five years ago Coast Appliances was supposed to upgrade one of its facilities at a quoted cost of $48,000. The upgrade was not completed, so Coast Appliances delayed the purchase until now. The construction company that provided the quote indicates that prices rose 6% compounded quarterly for the first 1.5 years, 7% compounded semi-annually for the following 2.5 years, and 7.5% compounded monthly for the final year. If Coast Appliances wants to perform the upgrade today, what amount of money does it need? Solution: The timeline below shows the original quote from five years ago until today. Step 1: Calculate the future value at the end of the first segment. \| \| \| --- \| \| N \| [latex]4 \times 1.5=6[/latex] \| \| PV \| [latex]-48,000[/latex] \| \| FV \| ? \| \| PMT \| [latex]0[/latex] \| \| I/Y \| [latex]6[/latex] \| \| P/Y \| [latex]4[/latex] \| \| C/Y \| [latex]4[/latex] \| [latex]\displaystyle{FV_1=\$52,485.27667....}[/latex] Step 2: Calculate the future value at the end of the second segment. The future value from the first segment becomes the present value for the second segment: [latex]FV_1=\$52,485.27667....=PV_2[/latex]. \| \| \| --- \| \| N \| [latex]2 \times 2.5=5[/latex] \| \| PV \| [latex]-52,485.27667...[/latex] \| \| FV \| ? \| \| PMT \| [latex]0[/latex] \| \| I/Y \| [latex]7[/latex] \| \| P/Y \| [latex]2[/latex] \| \| C/Y \| [latex]2[/latex] \| [latex]\displaystyle{FV_2=\$62,336.04435....}[/latex] Step 3: Calculate the future value at the end of the third segment. The future value from the second segment becomes the present value for the third segment: [latex]FV_2=\$62,336.04435....=PV_3[/latex]. \| \| \| --- \| \| N \| [latex]12 \times 1=12[/latex] \| \| PV \| [latex]-62,336.04435...[/latex] \| \| FV \| ? \| \| PMT \| [latex]0[/latex] \| \| I/Y \| [latex]7.5[/latex] \| \| P/Y \| [latex]12[/latex] \| \| C/Y \| [latex]12[/latex] \| [latex]\displaystyle{FV_3=\$67,175.35}[/latex] Coast Appliance requires $67,175.35 to perform the upgrade today. EXAMPLE Two years ago Lorelei placed $2,000 into an investment earning 6% compounded monthly. Today she makes a deposit to the investment in the amount of $1,500. What is the maturity value of her investment three years from now? Solution: The timeline for the investment is below. Step 1: Calculate the future value at the end of the first segment. \| \| \| --- \| \| N \| [latex]12 \times 2=24[/latex] \| \| PV \| [latex]-2,000[/latex] \| \| FV \| ? \| \| PMT \| [latex]0[/latex] \| \| I/Y \| [latex]6[/latex] \| \| P/Y \| [latex]12[/latex] \| \| C/Y \| [latex]12[/latex] \| [latex]\displaystyle{FV_1=\$2,254.31955....}[/latex] Step 2: Add the $1,500 deposit to [latex]FV_1[/latex] to get the present value for the second segment. [latex]\displaystyle{PV_2=2,254.31955...+1,500=\$3,754.31955...}[/latex] Step 3: Calculate the future value at the end of the second segment. \| \| \| --- \| \| N \| [latex]12 \times 3=36[/latex] \| \| PV \| [latex]-3,754.31955...[/latex] \| \| FV \| ? \| \| PMT \| [latex]0[/latex] \| \| I/Y \| [latex]6[/latex] \| \| P/Y \| [latex]12[/latex] \| \| C/Y \| [latex]12[/latex] \| [latex]\displaystyle{FV_2=\$4,492.72}[/latex] Three years from now Lorelei will have $4,492.72. This represents $3,500 of principal and $992.72 of compound interest. TRY IT Find the future value if $24,500 is invested at 4.1% compounded annually for 4 years; then 5.15% compounded quarterly for 1 year, 9 months; then 5.35% compounded monthly for 1 year, 3 months. Click to see Solution \| \| \| \| \| --- --- \| \| N \| [latex]4[/latex] \| [latex]7[/latex] \| [latex]15[/latex] \| \| PV \| [latex]-24,500[/latex] \| [latex]-28,771.930...[/latex] \| [latex]-31,467.335...[/latex] \| \| FV \| [latex]\textcolor{blue}{28,771.930...}[/latex] \| [latex]\textcolor{blue}{31,467.335...}[/latex] \| [latex]\textcolor{blue}{33,638.67}[/latex] \| \| PMT \| [latex]0[/latex] \| [latex]0[/latex] \| [latex]0[/latex] \| \| I/Y \| [latex]4.1[/latex] \| [latex]5.15[/latex] \| [latex]5.35[/latex] \| \| P/Y \| [latex]1[/latex] \| [latex]4[/latex] \| [latex]12[/latex] \| \| C/Y \| [latex]1[/latex] \| [latex]4[/latex] \| [latex]12[/latex] \| [latex]\displaystyle{FV=\$33,638.67}[/latex] TRY IT Nirdosh borrowed $9,300 4.25 years ago at 6.35% compounded semi-annually. The interest rate changed to 6.5% compounded quarterly 1.75 years ago. What amount of money today is required to pay off this loan? Click to see Solution \| \| \| \| --- \| N \| [latex]5[/latex] \| [latex]7[/latex] \| \| PV \| [latex]9,300[/latex] \| [latex]10,873.1489...[/latex] \| \| FV \| [latex]\textcolor{blue}{-10,873.1489...}[/latex] \| [latex]\textcolor{blue}{12,171.92}[/latex] \| \| PMT \| [latex]0[/latex] \| [latex]0[/latex] \| \| I/Y \| [latex]6.35[/latex] \| [latex]6.5[/latex] \| \| P/Y \| [latex]2[/latex] \| [latex]4[/latex] \| \| C/Y \| [latex]2[/latex] \| [latex]4[/latex] \| Exercises What is the future value of a $7,500 investment at 8% compounded quarterly for 3 years? Click to see Answer Ruth borrowed $53,000 at 6% compounded quarterly 4 years and 3 months ago. How much does Ruth have to pay today to clear the loan? How much interest did Ruth pay? Click to see Answer FV=$68,351.02, I=$15,351.02 3. You invest $19,000 in a savings account at 5.75% compounded semi-annually. How much is in your account 6.5 years from now? Click to see Answer 4. You invest $3,750 in an investment that earns 4.75% compounded annually for the first three years and then 5.5% compounded semi-annually for the next two years. How much money do you have at the end of the five years? How much interest did your investment earn? Click to see Answer FV=$4,804.20, I=$1,054.20 5. James took out a $11,375 loan today. He will pay interest at 7.5% compounded monthly for the first two years and nine months, and then 8.25% compounded quarterly for the next three years and three months. How much money does James owe six years from now? Click to see Answer 6. You are planning a 16-day African safari to Rwanda to catch a rare glimpse of the 700 remaining mountain gorillas in the world. The estimated cost of this once-in-a-lifetime safari is $15,000 including the tour, permits, lodging, and airfare. Upon your graduation from college, your parents have promised you a $10,000 graduation gift. You intend to save this money for five years in a long-term investment earning 8.3% compounded semi-annually. If the cost of the trip will be about the same, will you have enough money five years from now to pay for your trip? Click to see Answer $15,017.33; yes 7. Your investment of $9,000 that you started six years ago earned 7.3% compounded quarterly for the first 3¼ years, followed by 8.2% compounded monthly after that. How much interest has your investment earned so far? Click to see Answer 8. What is the maturity value of your $7,800 investment after three years if the interest rate was 5% compounded semiannually for the first two years, 6% compounded quarterly for the last year, and 2½ years after the initial investment you made a deposit of $1,200? How much interest is earned? Click to see Answer FV=$10,374.33, I=$1,374.33 9. Cristy borrowed $4,800 from a family friend 2½ years ago at 7% compounded annually for the first year and 8% compounded semi-annually thereafter. She made a payment 1½ years into the loan for $2,500. How much should Cristy pay today to clear her loan? Click to see Answer 10. You invested $5,000 10 years ago and made two further deposits of $5,000 each after four years and eight years. Your investment earned 9.2% compounded quarterly for the first two years, 8.75% compounded monthly for the next six years, and 9.8% compounded semi-annually for the remaining years. As of today, how much interest has your investment earned? Click to see Answer 11. Suppose you placed $10,000 into each of the following investments. Rank the maturity values after five years from highest to lowest. 1. 8% compounded annually for two years followed by 6% compounded semi-annually. 2. 8% compounded semi-annually for two years followed by 6% compounded annually. 3. 8% compounded monthly for two years followed by 6% compounded quarterly. 4. 8% compounded semi-annually for two years followed by 6% compounded monthly. Click to see Answer c is largest with FV=$14,023.26, d is second largest with FV=$13,999.47, b is third largest with FV=$13,933.20, a is smallest with FV=$13,927.43 You made the following three investments: $8,000 into a five-year fixed rate investment earning 5.65% compounded quarterly. $6,500 into a five-year variable rate investment earning 4.83% compounded semi-annually for the first 2½ years and 6.5% compounded monthly for the remainder. $4,000 into a five-year variable rate investment earning 4.75% compounded monthly for the first two years and 5.9% compounded quarterly thereafter, with a $4,000 deposit made 21 months into the investment. What is the total maturity value of all three investments after the five years, and how much interest is earned? Click to see Answer $29,270.56, $6,770.56 Attribution “9.2: Determining the Future (Maturity) Value” from Business Math: A Step-by-Step Handbook Abridged by Sanja Krajisnik; Carol Leppinen; and Jelena Loncar-Vines is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted. “9.2: Determining the Future Value” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted. License Business and Financial Mathematics Copyright © 2022 by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted. Share This Book
18196	https://www.youtube.com/watch?v=Ilx_BbQXD0A	GCSE Maths exam Q17: Ratio Simultaneous Equations CHALLENGE Maths with Sarah T 108 subscribers 6 likes Description 466 views Posted: 21 Jan 2024 CHALLENGE - a really nasty question here, you're given various pieces of information in the form of two ratios. You then have to use those ratios to set up two simultaneous equations. Which then need solving... Question taken from GCSE Maths, Edexcel, November 2018, Paper 2H, Q 17 Transcript: hello welcome to another math video by math with Sarah today we're going to be looking um at another math question from November 2018 it's again a higher paper it's paper two so you can use a calculator although as seems to be fairly standard for some of these more tricky questions I'm not sure that you actually need a calculator um it's question 17 so it's definitely a pretty tricky one um but yeah let's give it a go um so where start with this one as with all of these I encourage you to pause the video here have a read through the question give it a little bit of a go yourself if you can't get very far then fine what I'm hoping is that I can there's I think the first part of this question is the most tricky part hope and what I'm hoping is that I can kind of give you that little hint and kind of show you how to do that first part and then actually you'll realize the question isn't that difficult um but it's just making that jump the first jump I think is one of the hardest ones okay so let's give it a go we have got p and Q which are two different numbers p is bigger than Q When you subtract five from p and subtract five from Q the answers are in the ratio 5 to one when you add 20 to p and you add 20 to Q the ratio the answer in the ratio of 5 to2 and we want to find the ratio of P to q and give our answer in its simplic form so you might not believe me this is actually a simultaneous equation question we've got two unknowns and we've got to figure out what they are the problem is we've got to build the equations first so let's get started Let's do let's deal with one first so we've got this one here the first sentence we've been given we we've been told that when we subtract five from P so we've got P subtract five uh and subtract five from Q so then we've got Q subtract five and we've been told that that is in the ratio of five to one okay how do we get from a Rao to an equation at the moment this is tricky we need what we ideally need is to turn this into um is to turn this into an equation okay so what I'm going to do is this we have got I'm going to use some ratio boxes so we have got this is one compared to five hopefully you should all be familiar with ratio box if you haven't seen them before I have got them in in some other videos as well I explain them in a little bit more detail because of the complexity of this one I'm going I'm going to leave it just f um okay let's look at this now then so now we've got we know that this box here is equal to Q minus 5 that therefore means that every other box in this ratio set up is also equal to Q minus 5 so each one of these boxes is Q - 5 q - 5 and Q - 5 okay what you might now be realizing is that we have five lots of Q minus 5 and that that is equal to P minus 5 this is where I think the biggest jump is understanding that that's what's going on here okay hopefully because I've tried to draw it quite visually you can you can imagine what's going on here we've got five lots of Q minus 5 being equal to one lot of P minus 5 Okay so let's set that up as an equation we've got P - 5 is equal to five lots of Q - five we can then obiously expand those brackets so that we've got slightly easier looking equation so we can end up with 5 q minus 25 okay then we can we can actually imply a little bit further by adding five to both sides so that we end up with p on its own being equal to 5 q adding five to this side as well 5 q is 20 rather than- 25 and it's a little bit easier to to make as simple as possible we can draw some method line so we can add five to both sides okay let's now look at the second equation see if we can set up the second equation okay the second equation again obviously if you feel now that now you've got this kind of first hint of where to go that making that initial jump and you think you can give the second one a go then by all means give that a try yeah and you can always check back on the video because I'll go through the second as well so when we add 20 to P p+ 20 and we add 20 to Q the answers are in the ratio five to 2 dra miles further over something El 5 to2 okay this one obviously is a step further it's a little bit more complicated but that doesn't mean that we can't do it it just means that we've got to work a little bit harder so again another set of ratio boxes we can draw them five to two and then obviously this one will be split in half this one will be split into five not as equal as I might have liked but it's fine okay what next then so Q + 20 is OB split between these two boxes isn't that yeah so we know that two lots of Q + 20 sorry that's not true five lots of Q + 20 is equal to two lots of P p + 20 we say that again two lots of p + 20 is equal to five lots of Q + 20 yeah so if we put Q + 20 in each of these and P plus 20 in each of these we can understand those two are equal that's what we've got here okay I'm going to write that out we have got five Lots of Q + 20 is equal to two lots of p + 20 I'm going to come back to this in a second but I'm just going to use a similar ratio box situation as we've got here to try and demonstrate that a little bit better so here we have got basically if we're splitting this Q Plus 20 across the two boxes we've basically got half a q + 10 in each box this is Q over 2 plus 10 in each of these boxes yep and I would have to write that in each of those but just for time sake I'm write okay so now if we want to go back to this one we can say okay well p + 20 I this part here is equal to five Lots of Q 2 + 10 okay Q over 2 + 10 okay so if you didn't quite understand where I was going with this first slot talking about two lots of p plus Q equaling the same as sorry two lots of p plus 20 equaling the same as 5 q + 20 hopefully these ratio boxes will make it easier we've got five lots of Q over2 + 10 which is what each box is is equal to is equal to one lot of p + 20 okay if we now um kind of untangle this equation we end up with p + 20 is equal to 5 q over 2 + 50 so I'm going to be honest pretty nasty little looking equation but it works for now if we' done this if we'd um used this equation we'd have ended up exactly the same we just wouldn't have ended up with a fraction we' have ended up with 2 p + 40 is equal to 5 q + 100 so it's on on the opposite sides of the equals sign but hopefully you can see that these two equations are actually equals just this one here is half of the one above okay we've got this equation over here now so we've got two different equations we've got this equation here and we've got this equation here now I'm going to just alter this equation slightly so that both of them are in the form P equals something because that's going to help me um in case you hadn't realized at this point we're now just doing simultaneous equations essentially we need to find out a P value and a q value the easiest way of doing that is to eliminate one of them I'm going to set both equations equal to p and then I can eliminate P if we do that we end up with 5 q over 2 plus I'm taking away 20 from both sides just to give me p on its own plus 30 okay as I said we've now got our two equations set up that was by far the hardest part of this if you can get to these equations by yourself massive well done because this that definitely the most tricky part of this so now the last part is the simultaneous equation part we've been given we have got we've worked out 5 q take away 20 so that this part here which we know is equal p is equal to this P here of 5 q over 2 plus 30 so I'm going to start by getting rid of the plus 30 obviously now we're just literally putting method lines in just solving this equation working out what Q is I'm going to subtract 30 from both sides as much as I don't like negative numbers I want to deal with this fraction so we're going to end up with 5 q minus 50 is equal to 5 q over 2 I'm now going to multiply everything by two get rid of this annoying fraction which is going to give me remember you've got to multiply everything by Q so we're going to end up with 10 Q minus 100 is equal to 5 q I'm running out space a little bit here so I apologize this ended a little bit squatched um I am then going to take away 5 q from each side again obviously this is just solving an equation so hopefully you're relatively all right with this um and obviously there are many different ways of doing this we going to take away 5 q so I'm going to end up with 5 q and then at the same time I'm going to add 100 to both sides so that I have something side rather than leave with zero so again I'm getting rid of this so I'm going to end up with five Q is equal to 100 and then lastly divide by five to end up with Q is equal to 20 great it's a good start we found out what Q is equal to now obviously as with as with any of these simultaneous equations we can then choose which equation we use to find out what Q is this has got a fraction in it I'm not going anywhere near this one if I can help it so I'm now going to use the first equation which is this one over here um I'm going to do p is equal to 5 Q which is 20 same as 5 20 which is 100 take away 20 so I'm using this equation over here I'm substituting my qals 20 value into here so I'm doing 5 20 which is 100 take away 20 uh so then my P value is going to be equal to 100 take away 20 which is of course 80 okay okay now I feel like I'm pretty much done but I'm going to go back to the questions to just check that I've answered everything and everything fits okay we've got p and Q are two numbers such that P is greater than Q good p is 80 Q is 20 that fits that first bit when you subtract five from p and subtract five from Q so we going to end up with 75 and 15 the ratios are the answer in the ratio 5 to one good they are when you add 20 to P 100 and add 20 to Q we're ending up with 40 the ratios the answer in the ratio of five to two again they are it fits okay we can simplify this down to five to two and then if we've got 75 to 15 again obviously we can simplify that to 5 to one this is looking good so far we've managed to check our answers stick the criteria we've been given okay let's look at the LA last bit find the ratio of P to Q okay so we need to write this in terms of a ratio so we're going to write it as P to Q is 8 oh sorry 8 to 20 okay we've done that bit give your answer in its simplest form we need to do that start off by dividing each by 10 eight to two and then we can divide them further each of those by two to give us four to one so we've now put it in its simplest form our p Q the ratio is 4:1 and this is our final answer okay definitely a tricky question and I think by far the hardest bit is working out what's going on with these how to turn the ratios into an equation that you can solve once you get to the simultaneous equation bit it's a much lower grade question like this this part here is actually relatively straightforward but I think the really tough part of this is getting from a ratio to how we can turn that into an equation that we can then use to solve okay so absolutely massive well done if you if you for one thing followed it but secondly if you managed to do it yourself absolutely amazing um essentially we're just kind of cross multiplying but if it helps you to use the ratio boxes I think they're brilliant tool for being able to visualize what what's going on okay tough question but actually a relatively simple answer and we can check our answer quite nicely answer okay hope that all made sense um do feel free to ping me any questions um or if there's any requests for specific questions or videos either from this paper or from other papers um I hope you have a great day and um please like And subscribe uh to look at if you want any other videos
18197	https://www.youtube.com/watch?v=wWoivCAZRsE	Period, Frequency, and Angular Frequency Jennifer Cash 12600 subscribers 512 likes Description 48167 views Posted: 30 Jul 2015 This video introduces the variables for period, frequency, and angular frequency for oscillations. The equations and units are shown along with a simulation of mass on springs to show different frequencies. 29 comments Transcript: Oscillations So now we're going to talk about period frequency and angular frequency and this is in the context of oscillations and for oscillations it's any sort of repeating pattern of motion for example going around and around and around in a circle. Now we've been looking at a specific case of oscillations called simple harmonic motion. And we have a specific equation for that that we've been working with in physics. But instead of working with the equations, let's take a look at a simulation. And this is from the PH site. And we've used that throughout the semester for a few things. So here's a simulation, masses and springs that I've got preset up the way I wanted to. And I've just got two different masses hanging out here on springs oscillating up and down. And my repeating pattern of motion is the up and down cycle that it's going through. We'll come back to this a little bit later, but you'll notice these two have different motions between them. Period Now, we get to the period. The period is the time for one oscillation cycle. And if I was thinking about this in terms of circular motion, which is where I first define period, it's the time to go once around the circle. It's got a symbol of capital T. And make sure you're using capital T, not lowerase T. And the units for this is going to be seconds for our standard metric unit, but it could really be any time type units. You could use minutes or hours or whatever you need to use to express it. Flipping back over to my simulation here real quick. Notice that these don't have the same period. The amount of time it takes mass number one to go up and down is a much longer, larger time than the shorter amount of time it takes spring number three and mass number three to go up and down. So this would have a short period. This would have a long period. Angular Frequency Then we get into angular frequency. Angular frequency is also called angular speed if we're talking about circular motion. How fast it goes around that circle. And it's related to the period by the equation omega = 2i t where omega is the angular frequency. T is my period that I talked about. If I think about the units for this, I'm going to have radians per second where the second comes from the period and the radian comes from the 2 pi because once around a cycle is 2 pi radians. Then we've got frequency, just regular frequency, one word. It's also related to the period by this equation f= 1 / t where f is now my frequency. t is still my period. If I wanted to think about the units, this equation would imply that I've got 1 / seconds. And that's a perfectly fine unit to use for frequency. But a more general equation for my frequency might be the cycles per time. So if I go through multiple cycles, how many cycles do I go through and how much time? When I think about the equation that way, I could also represent the units as cycles per second. Now whether you think of it as 1 over seconds or cycles per second there's a new unit called a hertz hz which is used to represent frequency. Relationships Now these quantities are related to each other. So the angular frequency is related to the period and the frequency is related to the period. And combining those we can see that the angular frequency is related to the frequency where my angular frequency is equal to 2 pi times my frequency. Now if I pop back over to my simulation really quick. Notice that a short period has to do with a high frequency. it oscillates up and down more frequently than my slower one over here. So a long period, a large number for t represents a low frequency or a small number for the frequency. So we can see two different periods and two different frequencies between these two oscillators. So that is our period and frequency introduction.
18198	https://jmol.sourceforge.net/	Home \| Wiki \| Demonstration pages \| Download Documentation \| FAQ \| History \| Project pages Jmol: an open-source Java viewer for chemical structures in 3D with features for chemicals, crystals, materials and biomolecules \| \| \| --- \| \| \| JSmol is the HTML5 modality of Jmol, able to be embedded into web pages. All the functionality of Jmol (as a standalone application) is also present in JSmol. JSmol is an interactive web browser object. This is a still image, but you can see several animated displays of Jmol abilities by clicking here or on the image itself. (The JSmol library may take some seconds to load. Please, wait and do not reload the page in the meantime.) Here you can see the JSmol object in action, running a script written in the Jmol scripting language. It is not a movie, slide show, or animated image file ... JSmol is an interactive web browser object. 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18199	https://math.stackexchange.com/questions/548806/a-finite-set-always-has-a-maximum-and-a-minimum	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams A finite set always has a maximum and a minimum. Ask Question Asked Modified 4 years ago Viewed 47k times $\begingroup$ I am pretty confident that this statement is true. However, I am not sure how to prove it. Any hints/ideas/answers would be appreciated. real-analysis elementary-set-theory order-theory Share edited Feb 13, 2015 at 18:31 Martin Sleziak 56.3k2020 gold badges211211 silver badges391391 bronze badges asked Nov 2, 2013 at 8:42 CoffeeIsLifeCoffeeIsLife 2,09744 gold badges2323 silver badges3535 bronze badges $\endgroup$ 5 2 $\begingroup$ See Every finite set... $\endgroup$ J. W. Perry – J. W. Perry 2013-11-02 08:49:08 +00:00 Commented Nov 2, 2013 at 8:49 2 $\begingroup$ Suppose there is no maximal element then what happens? For every element $x \in S$,you can always find $ y \in S$ such that $x $\endgroup$ wannadeleteacct – wannadeleteacct 2013-11-02 08:49:30 +00:00 Commented Nov 2, 2013 at 8:49 3 $\begingroup$ You mean a nonempty finite set? $\endgroup$ bof – bof 2013-11-02 09:11:55 +00:00 Commented Nov 2, 2013 at 9:11 $\begingroup$ isn't this weierstrass theorem? $\endgroup$ Ant – Ant 2013-11-02 13:31:40 +00:00 Commented Nov 2, 2013 at 13:31 $\begingroup$ See also math.stackexchange.com/questions/24996/… $\endgroup$ Martin Sleziak – Martin Sleziak 2015-02-13 18:32:25 +00:00 Commented Feb 13, 2015 at 18:32 Add a comment \| 5 Answers 5 Reset to default 42 $\begingroup$ Let $S = {s_1, \ldots,s_n}$ be a nonempty finite set of size $n > 0$. We will show by induction on $n \in \mathbb N$ that there exist some $m,M \in S$ such that for all $s \in S$, we have that $m \leq s \leq M$. Base Case: For $n=1$, we have $S = {s_1}$, so taking $m = s_1$ and $M=s_1$ trivially satisfies the required condition. Induction Hypothesis: Assume that the claim holds for $n=k$, where $k \geq 1$. It remains to prove that the claim holds true for $n = k+1$. To this end, choose any set $S$ with $k+1$ elements, say $S = {s_1 ,\ldots,s_k,s_{k+1}}$. Now by the induction hypothesis, the subset: $$ S' = S \setminus {s_{k+1}} = {s_1 ,\ldots,s_k} $$ has a minimum element and a maximum element. That is, we know that there exists some $m',M' \in S'$ such that for all $s' \in S'$, we have that $m' \leq s' \leq M'$. Now observe that $s_{k+1}$ must fall under $1$ of $3$ cases: Case 1: Suppose that $s_{k+1} < m'$. Then take $m = s_{k+1}$ and $M=M'$. To see why this works, observe that any element in $S$ is either $s_{k+1}$ or some $s' \in S'$, and: $$ m = s_{k+1} < m' \leq s' \leq M' =M $$ Case 2: Suppose that $m' \leq s_{k+1} \leq M'$. Then take $m = m'$ and $M=M'$. To see why this works, observe that any element in $S$ is either $s_{k+1}$ or some $s' \in S'$, and: $$ m = m' \leq s_{k+1} \leq M' = M $$ $$ m = m' \leq s' \leq M' = M $$ Case 3: Suppose that $s_{k+1} > M'$. Then take $m =m'$ and $M=s_{k+1}$. To see why this works, observe that any element in $S$ is either $s_{k+1}$ or some $s' \in S'$, and: $$ m = m' \leq s' \leq M' < s_{k+1} = M $$ Hence, we have shown that $S$ has a minimum and maximum element, as desired. Share answered Nov 2, 2013 at 9:18 AdrianoAdriano 42k44 gold badges5353 silver badges8787 bronze badges $\endgroup$ 8 $\begingroup$ I see where you are coming from, and your proof is through. Thank you for taking the time for answering this. However, I am not really sure whether I can use proof for induction when such elements are not in $\mathbf{N}$. Some time in the past, I tried to use a similar method for proving a statement about $\mathbf{R}$ and was told that I couldn't use proof by induction. I found this link which says otherwise and is pretty through: math.stackexchange.com/questions/4202/induction-on-real-numbers. Would you mind elaborating on why do you think induction would work in this case? $\endgroup$ CoffeeIsLife – CoffeeIsLife 2013-11-02 21:09:43 +00:00 Commented Nov 2, 2013 at 21:09 5 $\begingroup$ It is true that you cannot use induction on a set where well-ordering does not hold, such as the real numbers. However, notice that my proof uses induction on $n \in \mathbb N$, and induction holds for the natural numbers due to well-ordering. We are NOT inducting on the elements of $S$ (for all we know, each $s_i \in S$ could be real numbers, or matrices, or purple hippos). $\endgroup$ Adriano – Adriano 2013-11-03 01:48:27 +00:00 Commented Nov 3, 2013 at 1:48 $\begingroup$ Are you then implying that a finite set on $\mathbf{R}$ does not always have a maximum and a minimum? $\endgroup$ CoffeeIsLife – CoffeeIsLife 2013-11-03 21:13:45 +00:00 Commented Nov 3, 2013 at 21:13 7 $\begingroup$ No, I'm not. As I have proven, finite sets on $\mathbb R$ always have a maximum and a minimum. I suspect that you are confusing the set that we are induction on (that is, the natural numbers, since finite sets always have a cardinality that belongs to $\mathbb N$) with the set that the elements of $S$ belong to (say, the real numbers). $\endgroup$ Adriano – Adriano 2013-11-04 11:52:12 +00:00 Commented Nov 4, 2013 at 11:52 3 $\begingroup$ I guess the proof here assumes there's an order relation on the elements of S. $\endgroup$ RJ Acuña – RJ Acuña 2018-12-06 20:14:16 +00:00 Commented Dec 6, 2018 at 20:14 \| Show 3 more comments 3 $\begingroup$ Let $F$ be a finite set. if $F$ is ${x} $ then we are done since we vacouly have $x \geq x $ and hence $x = \max { x } $. If $F = { a_1 ,... a_n } $. assume they are different, otherwise we are back again to singleton case. Now, take $a_1$. IF $a_1 $ is greater than any other $a_i$ then set $a_1 = \max F $ and we are done. IF not, take $a_2$, and repeat previous step. Continue in this manner inductively. Eventually, we get the max. Share answered Nov 2, 2013 at 8:50 ILoveMathILoveMath 11.1k99 gold badges5757 silver badges128128 bronze badges $\endgroup$ Add a comment \| 1 $\begingroup$ One needs to be careful about the set where to work, and the definition of the ordering. If the set is well ordered, then the above proof works fine. Otherwise, it can happen that there is a supremum, but not a maximum. We can, for example, consider the partial ordering on the set of $2\times 2$ real-valued matrices, where two matrices $A$ and $B$ are such that $A\le B$ if and only if all the entries of $B-A$ are non-negative. With the above-mentioned order relation and set, we consider the subset $$ E = \left{ \begin{pmatrix} 1 & 0 \ 0& 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \ 0& 1 \end{pmatrix} \right} .$$ We can see that $E$ is bounded from above (by $ \begin{pmatrix} 1 & 0 \ 0& 1 \end{pmatrix}$ for example), and from below( by $\begin{pmatrix} 0 & 0 \ 0& 0 \end{pmatrix}$ for example). However, the two elements of $E$ are not comparable using the defined order relation, so none of them is the maximum. Share answered Sep 3, 2021 at 13:07 Rabearivony Rabearivony 1122 bronze badges $\endgroup$ 2 $\begingroup$ If you really want to argue this silly point, a simpler and clearer example is ${ \text{potato}, \text{bicycle}}$. $\endgroup$ MJD – MJD 2021-09-03 13:14:23 +00:00 Commented Sep 3, 2021 at 13:14 $\begingroup$ That's a good point! One must be careful when dealing with order theory. By the way, did you mean "if the set is totally ordered" ? Instead of "well ordered" $\endgroup$ niobium – niobium 2023-06-01 09:16:09 +00:00 Commented Jun 1, 2023 at 9:16 Add a comment \| 0 $\begingroup$ I would just say based on Suppes, that use of induction to prove this conjecture begs the question for the following reasons: The minimality or maximality of the elements within a set, A, cannot be defined without specifying an ordering relation, R, on A. Different R's will yield different minimal elements, called R-minimal elements. Set A only has a unique R-minimal element iff all its non-empty subsets have unique R-minimal elements, R is connected, and R is asymmetric. (i.e. R is a well ordering) Having a unique R-minimal element does not guarantee a unique R-maximal element unless A is finite. Set A is finite iff every non-empty family of subsets of A has a minimal element [ordered by strict inclusion '$\subset$']- A. Tarski via Suppes Furthermore, if one such family of subsets, F, has a minimal element , then it must also have a maximal element via the following argument: If F has a minimal element, then construct G from the elements of F where z is an element of G iff for some y in F, z is Union F - y. If z is minimal in G, then y is maximal in F, and vice versa. If F does not have a maximal element, then G cannot have a minimal element, but G does have a minimal element, hence contradiction, and the theorem is proved. Share edited Feb 8, 2019 at 6:18 answered Feb 1, 2019 at 9:56 xxxx0xxxxxxxx0xxxx 10544 bronze badges $\endgroup$ 6 $\begingroup$ Sorry, but this proves nothing. Besides, proving the statement in the question by induction is the way to go. $\endgroup$ egreg – egreg 2019-02-01 12:59:59 +00:00 Commented Feb 1, 2019 at 12:59 $\begingroup$ Sorry, but induction is actually proved from this theorem, so using it begs the question. $\endgroup$ xxxx0xxxx – xxxx0xxxx 2019-02-01 20:06:43 +00:00 Commented Feb 1, 2019 at 20:06 $\begingroup$ Suppes uses Tarski's definition of a finite set, which is: A set is finite iff all its non-empty families of subsets have a minimal element. Thus G and F must have minimal elements if the set is finite. But if one doesn't have a maximal element, then the other can't have a minimal element, which contradicts Tarski's definition. $\endgroup$ xxxx0xxxx – xxxx0xxxx 2019-02-01 20:22:00 +00:00 Commented Feb 1, 2019 at 20:22 $\begingroup$ And I quote from Suppes et al. "The proof of the theorem about induction for finite sets is facilitated by having available the already defined notion of maximal element of a family of subsets." $\endgroup$ xxxx0xxxx – xxxx0xxxx 2019-02-01 20:49:39 +00:00 Commented Feb 1, 2019 at 20:49 $\begingroup$ There's no hint in the question that Tarski's definition is used and I find it unlikely. Anyway, the argument you sketch should be better written. $\endgroup$ egreg – egreg 2019-02-01 21:24:02 +00:00 Commented Feb 1, 2019 at 21:24 \| Show 1 more comment -8 $\begingroup$ I suppose you mean a set of numbers, $S={s_1,s_2,\dots,s_n}$ and $n$ is the finite size, just let $$ m=\min(s_1,s_2,\dots,s_n) $$ $$ M=\max(s_1,s_2,\dots,s_n) $$ Q.E.D. Share edited Nov 23, 2015 at 16:44 answered Nov 23, 2015 at 16:35 GeorgyGeorgy 1,4751313 silver badges2323 bronze badges $\endgroup$ 9 1 $\begingroup$ How do you know that $m$ and $M$ even exist? This exactly what the question asks. $\endgroup$ Asaf Karagila – Asaf Karagila ♦ 2015-11-23 20:48:37 +00:00 Commented Nov 23, 2015 at 20:48 1 $\begingroup$ How do you prove they are well-defined? The question is essentially requiring to show that you can do that. The answer by ILoveMath use an approach similar to yours, but with an actual proof that this is well-defined for any finite number of arguments. You can't just reduce something to an equivalent problem and call it a proof, especially if you don't actually solve that equivalent problem. $\endgroup$ Asaf Karagila – Asaf Karagila ♦ 2015-11-30 13:49:57 +00:00 Commented Nov 30, 2015 at 13:49 1 $\begingroup$ Yes, that's fine, but it is an inductive definition, and the question is about a rigorous proof that a finite set has minimum and maximum. So the inductive argument needs to be brought up. You can't avoid it, and your answer sweeps this under the rug. $\endgroup$ Asaf Karagila – Asaf Karagila ♦ 2016-07-30 18:02:25 +00:00 Commented Jul 30, 2016 at 18:02 2 $\begingroup$ Which appears in details in the two answers there were there two years before you posted your answer; and does not appear in your answer at all. $\endgroup$ Asaf Karagila – Asaf Karagila ♦ 2016-07-30 18:06:17 +00:00 Commented Jul 30, 2016 at 18:06 4 $\begingroup$ Sure, and I wouldn't expect anyone to prove to me that this is well-defined when I referee a paper. I would, however, expect someone who is asking why a finite set has a minimum and maximum (and clearly, the real analysis tag means this is a set of real numbers) expect the answer to be more than "well, just apply the minimum and maximum functions to the set" which is circular and terrible. And if someone would have written me this answer in a set theory exam, they would have received exactly 0 points for it. $\endgroup$ Asaf Karagila – Asaf Karagila ♦ 2016-07-30 19:31:59 +00:00 Commented Jul 30, 2016 at 19:31 \| Show 4 more comments 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 real-analysis elementary-set-theory order-theory See similar questions with these tags. 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