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15100	https://www.thestudentroom.co.uk/showthread.php?t=1692916	Applying to uni this year? Find your applicant group here It's two against one, our Year 12 chat thread is here>> Start a Grow Your Grades blog and join our supportive community! More info here >> The doctor will see you now - Which Medical School Should You Apply To >> Applying to uni this year? Find your applicant group here It's two against one, our Year 12 chat thread is here>> Get The Student Room app Easier than ever to use, with dark mode included. Find out more This discussion is now closed. Check out other Related discussions Maths question differentiation What is logarithms (chemistry help) Definite integration A level maths integration question Edexcel integration questions a level Maclaurin series need help understanding my mistakes - parametric equations challenge STEP Maths I, II, III 1989 solutions Why does solving differential equations not give the modulus of x? OCR A-level Further Mathematics B (MEI) Core pure - 22nd May 2025 [Exam Chat] STEP Maths I,II,III 1987 Solutions OCR A-level Mathematics A Paper 2 - 12th June 2025 [Exam Chat] Another math problem. integration AQA A-level Mathematics Paper 1 - 4th June 2025 [Exam Chat] AQA A-level Further Mathematics Paper 1 - 22nd May 2025 [Exam Chat] Numbers of Arrangements STEP maths I, II, III 1991 solutions Isaac Physics A Vertical Throw Separable differential equation logs edexcel show that ln(x) =< x-1 ?? A anzerftum how do i show that ln(x) is less or equal to x-1? we can assume that e^x is larger or equal to x+1. it doesn't say we can assume anything else (ie, inverse of e is ln, algebra of logs etc..) that's why i'm stuggling to prove this without extra assumptions Reply 1 A anzerftum OP sorry, bad notation. we can assume that e^y is larger or equal to y+1. Reply 2 A mmmpie 12 It doesn't mean you can't use . It's not an assumption. Reply 3 A qgujxj39 17 We can definitely assume that exp and ln are inverses, this is basically the definition of ln. Reply 4 A Iron&Wine Original post by anzerftum how do i show that ln(x) is less or equal to x-1? we can assume that e^x is larger or equal to x+1. it doesn't say we can assume anything else (ie, inverse of e is ln, algebra of logs etc..) that's why i'm stuggling to prove this without extra assumptions Reply 5 A anzerftum OP Original post by mmmpie It doesn't mean you can't use . It's not an assumption. so just plug in x=e^y? simple as? Reply 6 A anzerftum OP Original post by mmmpie It doesn't mean you can't use . It's not an assumption. to get e^x>x+1 which we're assuming true? Reply 7 A qgujxj39 17 Original post by anzerftum so just plug in x=e^y? simple as? Pretty much, although we need to be careful, you're probably going to have to use the fact that it's an increasing function, so the inequality is preserved. Reply 8 A Zuzuzu 12 Given this question on it's own I'd consider ln(x) - x + 1 either side of a certain value. Given the hint, though, try making a substitution. Reply 9 A anzerftum OP Original post by tommm Pretty much, although we need to be careful, you're probably going to have to use the fact that it's an increasing function, so the inequality is preserved. so okay, we're using the substitution to show the assumption e^y>y+1 which we're assuming true? shouldn't we use the assumption to show that ln(x) =< x-1 ? the above seems kind of "inside-out" Reply 10 A mmmpie 12 Original post by anzerftum Original post by anzerftum so just plug in x=e^y? simple as? Where did you get that from? You're trying to prove Unparseable LaTeX formula: \ln x \leq x-1 , so I'd begin with Unparseable LaTeX formula: e^{\ln x} \leq e^{x-1} EDIT: Oh, right, I see the substitution. Senior moment there. Reply 11 A anzerftum OP Original post by mmmpie Where did you get that from? You're trying to prove Unparseable LaTeX formula: \ln x \leq x-1 , so I'd begin with Unparseable LaTeX formula: e^{\ln x} \leq e^{x-1} EDIT: Oh, right, I see the substitution. Senior moment there. thank you Related discussions Maths question differentiation What is logarithms (chemistry help) Definite integration A level maths integration question Edexcel integration questions a level Maclaurin series need help understanding my mistakes - parametric equations challenge STEP Maths I, II, III 1989 solutions Why does solving differential equations not give the modulus of x? 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15101	https://chem.libretexts.org/Courses/Chabot_College/Chem_12A%3A_Organic_Chemistry_Fall_2022/10%3A_Properties_and_Reactions_of_Alkenes/10.04%3A_Degrees_of_Unsaturation	10.4: Degrees of Unsaturation - Chemistry LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode 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Home 2. Campus Bookshelves 3. Chabot College 4. Chem 12A: Organic Chemistry Fall 2022 5. 10: Properties and Reactions of Alkenes 6. 10.4: Degrees of Unsaturation Expand/collapse global location Chem 12A: Organic Chemistry Fall 2022 Front Matter 1: Introduction to Organic Chemistry- review of atoms and molecules 2: Polarity, Intermolecular Forces, and Physical Properties of Molecules 3: Representations of Organic Molecules 4: Acids and Bases 5: Alkanes and Conformations 6: Stereoisomerism 7: Introduction to Organic Reactions 8: Nucleophilic Substitution Reactions 9: Elimination Reactions 10: Properties and Reactions of Alkenes 11: Properties and Reactions of Alkynes 12: Free Radical Reactions 13: Properties and Reactions of Alcohols 14: Infrared Spectroscopy and Mass Spectrometry 15: Nuclear Magnetic Resonance Spectroscopy Back Matter 10.4: Degrees of Unsaturation Last updated Jul 11, 2022 Save as PDF 10.3: The Alkene Double Bond and Stereoisomerism 10.5: The E/Z System (when cis/trans does not work) Page ID 391372 ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. Saturated and Unsaturated Molecules 2. Calculating The Degree of Unsaturation (DU) 1. References Contributors and Attributions Learning Objectives calculate the Degrees of Unsaturation (DU) and apply it to alkene structure Saturated and Unsaturated Molecules In the lab, saturation may be thought of as the point when a solution cannot dissolve anymore of a substance added to it. In terms of degrees of unsaturation, a molecule only containing single bonds with no rings is considered saturated. CH 3 CH 2 CH 31-methyoxypentane Unlike saturated molecules, unsaturated molecules contain double bond(s), triple bond(s) and/or ring(s). CH 3 CH=CHCH 33-chloro-5-octyne There are many ways one can go about determining the structure of an unknown organic molecule. Although, nuclear magnetic resonance (NMR) and infrared radiation (IR) are the primary ways of determining molecular structures, these techniques require expensive instrumentation and are not always readily available. Fortunately, calculating the degrees of unsaturation provides useful information about the structure. The degree of unsaturation indicates the total number of pi bonds and rings within a molecule which makes it easier for one to figure out the molecular structure. DU = Degrees of Unsaturation = (number of pi bonds) + (number of rings) Alkenes (R 2 C=CR 2) and alkynes (R–C≡C–R) are called unsaturated hydrocarbons because they have fewer hydrogen atoms than does an alkane with the same number of carbon atoms, as is indicated in the following general formulas: Calculating The Degree of Unsaturation (DU) If the molecular formula is given, plug in the numbers into this formula: (7.2.1)D⁢o⁢U=2⁢C+2+N−X−H 2 C is the number of carbons N is the number of nitrogens X is the number of halogens (F, Cl, Br, I) H is the number of hydrogens The molecular formula of a hydrocarbon provides information about the possible structural types it may represent. A saturated molecule contains only single bonds and no rings. Another way of interpreting this is that a saturated molecule has the maximum number of hydrogen atoms possible to be an acyclic alkane. Thus, the number of hydrogens can be represented by 2C+2, which is the general molecular representation of an alkane. As an example, for the molecular formula C3H4 the number of actual hydrogens needed for the compound to be saturated is 8 [2C+2=(2x3)+2=8.] The compound needs 4 more hydrogens in order to be fully saturated (expected number of hydrogens-observed number of hydrogens=8-4=4). Degrees of unsaturation is equal to 2, or half the number of hydrogens the molecule needs to be classified as saturated. Hence, the DoB formula divides by 2. The formula subtracts the number of X's because a halogen (X) replaces a hydrogen in a compound. For instance, in chloroethane, C2H5Cl, there is one less hydrogen compared to ethane, C2H6. For example, consider compounds having the formula C5H8. The formula of the five-carbon alkane pentane is C5H12 so the difference in hydrogen content is 4. This difference suggests such compounds may have a triple bond, two double bonds, a ring plus a double bond, or two rings. Some examples are shown here, and there are at least fourteen others! For a compound to be saturated, there is one more hydrogen in a molecule when nitrogen is present. Therefore, we add the number of nitrogens (N). This can be seen with C 3 H 9 N compared to C 3 H 8. Oxygen and sulfur are not included in the formula because saturation is unaffected by these elements. As seen in alcohols, the same number of hydrogens in ethanol, C 2 H 5 OH, matches the number of hydrogens in ethane, C 2 H 6. The following chart illustrates the possible combinations of the number of double bond(s), triple bond(s), and/or ring(s) for a given degree of unsaturation. Each row corresponds to a different combination. One degree of unsaturation is equivalent to 1 ring or 1 double bond (1 π bond). Two degrees of unsaturation is equivalent to 2 double bonds, 1 ring and 1 double bond, 2 rings, or 1 triple bond (2 π bonds). When the DU is 4 or greater, the presence of benzene rings is very likely. \| DU \| Possible combinations of rings/ bonds \| --- \| \| \| # of rings \| # of double bonds \| # of triple bonds \| \| 1 \| 1 \| 0 \| 0 \| \| \| 0 \| 1 \| 0 \| \| 2 \| 2 \| 0 \| 0 \| \| \| 0 \| 2 \| 0 \| \| \| 0 \| 0 \| 1 \| \| \| 1 \| 1 \| 0 \| Remember, the degrees of unsaturation only gives the sum of double bonds, triple bonds and/or rings. For instance, a degree of unsaturation of 3 can contain 3 rings, 2 rings+1 double bond, 1 ring+2 double bonds, 1 ring+1 triple bond, 1 double bond+1 triple bond, or 3 double bonds. Example: Benzene What is the Degree of Unsaturation for Benzene? Solution The molecular formula for benzene is C 6 H 6. Thus, DU= 4, where C=6, N=0,X=0, and H=6. 1 DoB can equal 1 ring or 1 double bond. This corresponds to benzene containing 1 ring and 3 double bonds. However, when given the molecular formula C 6 H 6, benzene is only one of many possible structures (isomers). The following structures all have DU of 4 and have the same molecular formula as benzene. However, these compounds are very rare, unlike benzene. We will learn more about the reasons for benzen's high stability when we studey aromaticity in later chapters. Exercises Are the following molecules saturated or unsaturated: (b.) (c.) (d.) C 10 H 6 N 4 Using the molecules from (1) above, give the degrees of unsaturation for each. Calculate the degrees of unsaturation, classify the compound as saturated or unsaturated, and list all the ring/pi bond combination possible for the following molecular formulas: (a.) C 9 H 20(b.) C 7 H 8(c.) C 5 H 7 Cl (d.) C 9 H 9 NO 4 Calculate degrees of unsaturation (DoU) for the following, and propose a structure for each. a) C 5 H 8 b) C 4 H 4 Calculate the degree of unsaturation (DoU) for the following molecules a) C 5 H 5 N b) C 5 H 5 NO 2 c) C 5 H 5 Br The following molecule is caffeine (C 8 H 10 N 4 O 2), determine the degrees of unsaturation (DoU). Answer 1. (a.) unsaturated (E ven though rings only contain single bonds, rings are considered unsaturated.) (b.) unsaturated (c.) saturated (d.) unsaturated If the molecular structure is given, the easiest way to solve is to count the number of double bonds, triple bonds and/or rings. However, you can also determine the molecular formula and solve for the degrees of unsaturation by using the formula. (a.)2 (b.) 2(one double bond and the double bond from the carbonyl) (c.) 0 (d.) 10 3. (a.) DU =0; saturated(Remember-a saturated molecule only contains single bonds) (b.) DU = 4; unsaturatedThe molecule can contain any of these combinations of rings and pi bonds that add up to 4, such as(i) 4 double bonds (ii) 4 rings (iii) 2 double bonds+2 rings (iv) 1 double bond+3 rings (v) 3 double bonds+1 ring (vi) 1 triple bond+2 rings (vii) 2 triple bonds (viii) 1 triple bond+1 double bond+1 ring (ix) 1 triple bond+2 double bonds (c.) DU = 2; unsaturated (i) 1 triple bond (ii) 1 ring+1 double bond (iii) 2 rings (iv) 2 double bonds (d.) DU = 6; (i) 3 triple bonds (ii) 2 triple bonds+2 double bonds (iii) 2 triple bonds+1 double bond+1 ring (iv)...(As you can see, the degrees of unsaturation only gives the sum of double bonds, triple bonds and/or ring. Thus, the formula may give numerous possible structures for a given molecular formula.) 4. 5.a)4 b) 4 c) 3 6.DU =6 References Vollhardt, K. P.C. & Shore, N. (2007). Organic Chemistry(5 th Ed.). New York: W. H. Freeman. (473-474) Shore, N. (2007). Study Guide and Solutions Manual for Organic Chemistry(5 th Ed.). New York: W.H. Freeman. (201) Contributors and Attributions Dr. Dietmar Kennepohl FCIC (Professor of Chemistry, Athabasca University) Prof. Steven Farmer (Sonoma State University) Kim Quach (UCD) 10.4: Degrees of Unsaturation is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts. Back to top 10.3: The Alkene Double Bond and Stereoisomerism 10.5: The E/Z System (when cis/trans does not work) Was this article helpful? Yes No Recommended articles 10: Properties and Reactions of Alkenes Article typeSection or PageShow Page TOCno on pageTranscludedyes Tags source-chem-45189 © Copyright 2025 Chemistry LibreTexts Powered by CXone Expert ® ? The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. 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15102	https://www.reddit.com/r/statistics/comments/lrg6cr/q_evaluating_predictive_model_on_test_set_with/	[Q] Evaluating predictive model on test set with adjusted R squared. : r/statistics Skip to main content[Q] Evaluating predictive model on test set with adjusted R squared. : r/statistics Open menu Open navigationGo to Reddit Home r/statistics A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to statistics r/statistics r/statistics /r/Statistics is going dark from June 12-14th as an act of protest against Reddit's treatment of 3rd party app developers. This community will not grant access requests during the protest. Please do not message asking to be added to the subreddit. 605K Members Online •5 yr. ago vasili111 [Q] Evaluating predictive model on test set with adjusted R squared. Question I am trying to evaluate predictive model on test set with adjusted R squared using first formula from here: My question is: Should I use n and p of the test set or the train set? Read more Share Related Answers Section Related Answers Best software for statistical analysis Applications of Bayesian statistics Differences between R and Python for stats Common pitfalls in data interpretation How to visualize complex datasets effectively 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 February 24, 2021 Reddit reReddit: Top posts of February 2021 Reddit reReddit: Top posts of 2021 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation
15103	https://books.google.com/books/about/Heat_Transfer.html?id=GajCQgAACAAJ	Heat Transfer - Jack Philip Holman - Google Books Sign in Hidden fields Try the new Google Books Books Add to my library Try the new Google Books Check out the new look and enjoy easier access to your favorite features Try it now No thanks Try the new Google Books My library Help Advanced Book Search Get print book No eBook available Amazon.com Barnes&Noble.com Books-A-Million IndieBound Find in a library All sellers» ### Get Textbooks on Google Play Rent and save from the world's largest eBookstore. Read, highlight, and take notes, across web, tablet, and phone. Go to Google Play Now » My library My History Heat Transfer Jack Philip Holman McGraw-Hill, 2002 - Science - 665 pages As one of the most popular heat transfer texts, Jack Holman's HEAT TRANSFER is noted for its clarity, accessible approach, and inclusion of many examples and problem sets. The new Ninth Edition retains the straight-forward, to-the-point writing style while covering both analytical and empirical approaches to the subject. Throughout the book, emphasis is placed on physical understanding while, at the same time, relying on meaningful experimental data in those situations that do not permit a simple analytical solution. New examples and templates provide students with updated resources for computer-numerical solutions. More » From inside the book Contents CHAPTER 1 3 CHAPTER 5 Dimensions 73 Copyright 39 other sections not shown Other editions - View all Heat Transfer Jack Philip Holman Snippet view - 2002 Heat Transfer Jack Philip Holman No preview available - 2002 Common terms and phrases aluminumanalysisAssumeboundary conditionsboundary layerBtu/hC₁Calculate the heatCalculate the temperaturecondensationconstant heat fluxconvection boundaryconvection coefficientconvection environmentconvection heat transferconvection heat-transfer coefficientcoolingcoppercylinderdiffusiondiskemissivityenergy balanceFigure Exampleflat plateflow ratefluidfree convectionft²heat conductedheat exchangerheat flowheat lossheat-transfer ratehorizontalincrementsinsideinsulationisothermalkg/m³kg/slaminarliquidlostm²/smaintainednodal equationsNusseltobtainone-dimensionaloverall heat-transfer coefficientpipeproblempropertiesradiosityReynolds numbersemi-infiniteshape factorshown in Figuresolidsolutionspherestainless steelsteady-statesuddenly exposedsurface temperatureT₁Tabletemperaturetemperature differencetemperature distributionthermal conductivitythermal radiationthermal resistancethicknesstotal heatTP+1tube wallturbulentunit lengthvelocityverticalW/m²wall temperatureΔτдудх Bibliographic information Title Heat Transfer McGraw-Hill series in mechanical engineering AuthorJack Philip Holman Edition 9, illustrated Publisher McGraw-Hill, 2002 Original from the University of Michigan Digitized Aug 26, 2011 ISBN 0071122303, 9780071122306 Length 665 pages SubjectsScience › Mechanics › Thermodynamics Science / Mechanics / Thermodynamics Technology & Engineering / Mechanical Technology & Engineering / Power Resources / General Export CitationBiBTeXEndNoteRefMan About Google Books - Privacy Policy - Terms of Service - Information for Publishers - Report an issue - Help - Google Home
15104	https://arxiv.org/pdf/2009.03412	Analysis of landscape hierarchy during coarsening and aging in Ising spin glasses Stefan Boettcher and Mahajabin Rahman Department of Physics, Emory University, Atlanta, GA 30322, USA We use record dynamics (RD), a coarse-grained description of the ubiquitous relaxation phe-nomenology known as "aging", as a diagnostic tool to find universal features that distinguish between the energy landscapes of Ising spin models and the ferromagnet. According to RD, a non-equilibrium system after a quench relies on fluctuations that randomly generate a sequence of irreversible record-sized events (quakes or avalanches) that allow the system to escape ever-higher barriers of meta-stable states within a complex, hierarchical energy landscape. Once these record events allow the system to overcome such barriers, the system relaxes by tumbling into the following meta-stable state that is marginally more stable. Within this framework, a clear distinction can be drawn between the coarsening dynamics of an Ising ferromagnet and the aging of the spin glass, which are often put in the same category. To that end, we interpolate between the spin glass and ferromagnet by varying the admixture p of ferromagnetic over anti-ferromagnetic bonds from the glassy state (at 50% each) to wherever clear ferromagnetic behavior emerges. The accumulation of record events grows logarithmic with time in the glassy regime, with a sharp transition at a specific admixture into the ferromagnetic regime where such activations saturate quickly. We show this ef-fect both for the Edwards-Anderson model on a cubic lattice as well as the Sherrington-Kirkpatrick (mean-field) spin glass. While this transition coincides with a previously observed zero-temperature equilibrium transition in the former, that transition has not yet been described for the latter. I. INTRODUCTION The morphology of complex energy landscapes [1, 2], and the parameters that control it, are of continuing in-terest in a large variety of scientific endeavors, from pro-tein folding and evolutionary landscapes in biology [3–7], the design of amorphous materials [8–18], to the hard-ness of combinatorial optimization problems [5, 19–23]. The challenges encountered in describing the geometry of the extremely high-dimensional space of attainable con-figurations are enormous [1, 2, 12, 23–28]. The structure of such energy landscapes hugely impacts the dynamics of statistical systems evolving through them. While re-laxation in simple, smooth landscapes is rapid, like the exponential cooling of a cup of coffee , relaxation in complex energy landscapes can possess a myriad of metastable states to temporarily or permanently trap any dynamic process. In turn, simple relaxation processes can serve as diagnostic tools to explore features of land-scapes [19, 24, 25, 27, 30–32]. It is particularly enticing when it is possible to discover universal aspects of such landscapes that allow to categorize those features and, ultimately, predict and control dynamic behavior. Variations in temperature can be used to take a full measure of landscapes. At high temperature correspond-ingly higher echelons in energy get explored, while an-nealing or quenching is used to trace out a descent through the landscape towards configurations of lower en-ergy. A conceptually simple protocol consists of prepar-ing a system at a high temperature, where it equilibrates easily, and then instantaneously quenching it down to a fixed, low temperature, to explore how it relaxes towards equilibrium thereafter. Such an “aging” protocol , when applied to systems in a complex energy landscape, elicits quite subtle relaxation behaviors which, unlike for the coffee mentioned above, keeps the system far from a new equilibrium for very long times. Anomalously slow relaxation and full aging in a complex landscape ensues when downward paths are obstructed by barriers, ener-getic or entropic, that trap the system in neighborhoods with many local minima. The aging phenomenology is associated with memory effects by which the current activity is imprinted by a dependence on the waiting time tw since the quench. For a wide class of systems, generally considered to be glassy, it is found that correlations, instead of being time trans-lational invariant, G(t, t w) ∼ f (t − tw), roughly depend on a ratio, G ∼ f (t/t w). Although memory effects in out-of-equilibrium systems are generally of interest, that fact alone is not sufficient to categorize its energy landscape as complex or glassy. To emphasize this fact, and to pro-vide a deeper insight into the relation between landscape morphology and aging dynamics, we investigate here the aging in families of models that interpolate between a well-known spin glass [34, 35] and the corresponding fer-romagnet . Albeit glass and ferromagnet exhibit sim-ilar scaling with age t, it stands to reason that the aging dynamics of a homogeneous ferromagnet differs signifi-cantly from that of a glass. In contrast to the hierarchi-cal, multimodal energy landscape of a glass [37–41], that of a ferromagnet is smooth. Yet, much of the literature leads to the impression that relaxation via coarsening in a ferromagnet and glassy aging are synonymous [42–44]. Technically, one could argue that the fact that in either extreme a growing length-scale emerges is indicative of coarsening domains. We posit that the process by which those length-scales grow with age, logarithmically in the glassy case and with a power-law for a ferromagnet, is fundamentally different. As discussed in Ref. , energy barriers that scale with the size of a domain to be flipped imply that fur-ther growth in those domains is curtailed to be merely arXiv:2009.03412v2 [cond-mat.dis-nn] 7 Jan 2021 2logarithmic in time. Such a feedback does not emerge in the coarsening of a ferromagnetic Ising system, where energy barriers remain insensitive to the size of the do-main to be flipped. Accordingly, the landscape of a glassy system has a hierarchical structure in that, the lower an energy it has reached, the higher the barriers get, and thus, the harder it becomes to escape local minima . In a homogeneously coarsening system, energy barriers remain largely independent of the depth reached within the landscape, providing some roughness and metastabil-ity but of bounded scale beyond which the structure is relatively smooth. Within the aging process, this differ-ence manifests itself dramatically in the manner that the relaxing system responds to fluctuations, as illustrated in Fig. 2. In a ferromagnet, average fluctuations in energy, beyond some low, fixed threshold, suffice to cross typ-ical barriers, often followed by disproportionately large expulsions of heat. In contrast, to advance glassy sys-tems with diverging energy barriers, mere average fluc-tuations become ineffective. To be able to relax, those fluctuations have to produce ever new records in their size to overcome ever steeper barriers, which is the basis of what is called Record Dynamics (RD) . Such record-production, decorrelated by a wide separation in time, is known to unfold only on a logarithmic scale [48, 49]. Although irrelevant for the cases studied in Ref. (and here), it should be noted that entropic effects can become dominant and may entail diverging free-energy barriers with domain size, even in an otherwise homoge-nous system. One example is a 3-spin Ising ferromag-net . Systems driven by entropic barriers, such as the free volume in a hard-core colloidal system, are referred to as “structural” glasses. In those systems, a hierarchical free-energy landscape emerges dynamically . In the following, we define a simple coarse-graining pro-cedure, counting the number of “valleys” traversed in the energy landscape, that effectively probes the impact of fluctuations on the aging dynamics. Our focus on land-scape morphology reveals the nature of the irreversible, intermittent events that allow the expulsion of excess en-ergy from the system. It shows a dynamical transition between a glassy and a ferromagnetic relaxation regime based on this measure. In that, we reproduce similar findings using scaling exponents of two-time correlation functions in the thermodynamic limit , which indi-cated that this dynamical transition is closely related with a zero-temperature equilibrium transition between a glass and a ferromagnet . That this transition tran-scends into the non-equilibrium realm highlights the fact that aging in a glassy system is a distinct process from what is found in homogeneous systems, characteristic of a distinct, hierarchical landscape. Our paper is organized as follows: In the next Sec-tion II, we introduce the families of Ising spin models we employ in our study. In Sec. III, we will discuss RD and the measures we will apply to detect record fluctuations. In Sec. IV, we present the results of our investigation, and we conclude in Sec. V. II. MODELS Ising spin systems, consisting of spin variables σi = ±1, have been widely used, first of all as ferromagnets, to model spontaneous symmetry breaking and contin-uous phase transitions . With the random admix-ture of anti-ferromagnetic bonds, they have also served as models for disordered materials and glasses gener-ally [34, 35, 53, 54]. The relevance of such spin models reaches far beyond physics, into biological and sociologi-cal applications, for example . Here, we are employ-ing families of such spin models that interpolate between the randomly disordered spin glass on a cubic lattice, called the Edwards-Anderson model (EA) , as well as its mean-field version, the Sherrington-Kirkpatrick model (SK) , on one side and the respective homogeneous ferromagnetic systems on the other. Each system consists of a random mixture of ferromagnetic and anti-ferromagnetic bonds J between neighboring spins σi and σj that are drawn from a distribution P (J) we have cho-sen to be bi-modal, i.e., Jij = ±J0, with energy units such that J0 = 1 in 3D and J0 = 1 /√N in the mean field case. A fraction p of ferromagnetic bonds is balanced out with a fraction 1−p of anti-ferromagnetic bonds such that P (J) = pδ (J − J0) + (1 − p)δ (J + J0) . (1) For each, the Hamiltonian (without external field) reads H = − ∑ 〈ij 〉 Jij σiσj , (2) where 〈ij 〉 refers to all extant bonds between neighboring spins σi and σj , either on a cubic lattice for EA or all mutual pairs of spins for SK. In the family EA of models we study on the cubic lat-tice [36, 51], we change the admixture of bonds by varying p between 12 ≤ p ≤ 1, from the pure glass with an equal mix of bonds ( p = 12 ) to a pure ferromagnet when all bonds are ferromagnetic ( p = 1 ). The situation is more complicated for SK, where already a sub-extensive excess of ferromagnetic bonds, away from the pure glass, results in ferromagnetic behavior. Specifically, since all N spins are mutually connected, there are 12 N (N − 1) bonds, and it only takes an imbalance between either type of bond, merely of order v √N , to achieve ferromagnetic ordering. Thus, we define a family of mean-field models parametrized by α with p = 12 + α√N , varying between 0 ≤ α ≤ 2 to explore the full range of behaviors . III. AGING AND RECORD DYNAMICS A. Simulation of Quenches in Spin Glasses The distinction between slow relaxation in glassy ver-sus homogeneous systems is succinctly analyzed in the simplest conceivable protocol of a hard quench from an 3E1 E3 E5 B3 B5 E4 B4 E2 B2B1 Figure 1. Illustration of the definition of valleys. The trace through an energy landscape produces a time sequence of en-ergy records ( Ei) and of barrier records ( Bj), relative to the most recent “ Ei” [30, 31]. Only the highest and lowest records of the “ Ei" and “ Bj”are kept to give a strictly alternat-ing sequence “ . . . Ei0Bj1Ei1Bj2Ei2. . . ”. Then, any sequence “Bj1Ei1Bj2” demarcates a valley (vertical lines). easily equilibrated high-temperature state into an or-dered phase, whether glassy or ferromagnetic, crossing a phase transition in the process. Such a pure ag-ing protocol has been studied extensively in the last 40 years [33, 44, 48, 57, 58]. In this process, the system is thrown far out of equilibrium, left with an enormous amount of excess heat to be released to the bath to be able to descent deeper into its energy landscape to reach states with the appropriate (equilibrium) energy. To facilitate such a quench for the family of Ising spin models considered in our study, for each instance at time t = 0 , we initiate with randomly assigned spins, either σi = ±1, which corresponds to T = ∞, and run the sim-ulation for t > 0 at a low, finite temperature. For our family of models on the cubic lattice, the critical temper-ature for a transition into an ordered state varies from Tc ≈ 1.1J0 in EA , to about Tc ≈ 4.5J0 for the ferro-magnet . In our Monte Carlo simulations, we quench to Tq = 0 .7J0 for all p, similar to Ref. , and monitor the aging process for about 10 5 sweeps. For the fam-ily of mean field models, we only vary the admixture of ferromagnetic bonds minutely, so that the transition tem-perature does not deviate much from that of SK, which is known to be Tc = J0 . Here, we also quench to Tq = 0 .7J0 throughout. For each value of p in our study, we have averaged results over at least 10 4 realizations. B. Valleys in an Energy Landscape Key to our analysis of coarsening versus glassy relax-ation is the definition of a measure that can serve to distinguish the effect of fluctuations on the irreversible events by which a system relaxes. One such measure has been provided by Dall and Sibani in Ref. . There, the internal energy of an entire system of finite size is mon-itored to observe its time-trace for the ensuing quench. Since the system is expelling energy into the bath to re-lax, on average, the energy gradually decreases, albeit via localized, intermittent events , in line with experi-mental observations of glassy systems [43, 62–64]. In par-ticular, record-sized fluctuations are needed for a glassy system to relax, according to RD [46, 47]. As illustrated in Fig. 1, Dall and Sibani defined the “valley” production as an observable. in the follow-ing way: Let E be the up-to-now lowest energy value encountered up to time t and let E(t) be the instan-taneous energy. In turn, let the “barrier” B be the up-to-now highest energy attained, relative to the most recent E, i.e., B = E(t) − E. An energy trace then maps into a random sequence of symbols, like in Fig. 1, . . . E1B1B2B3E2E3B4B5E4E5 . . . . Note that the trace can generate a sub-sequence of records in the lowest en-ergy, i.e., multiple E’s in a row, before it encounters its next higher (record) barrier B, and also a sub-sequence of such B’s before it meets the next E, and so on. Clearly, it is the latest E or B in either type of sub-sequence that is significant: Each prior one is merely transitory, while the last one supersedes each prior one as record that reaches its ultimate significance only after a new record fluctuation in the opposite direction is attained. Thus, we squash the entire sequence into a strict alternation between E and B, as the stricken letters in Fig. 1 imply, which then yield: . . . E1B3E3B5E5 . . . . Then, a “valley” is defined as the part of the trace between two consecutive record barrier-crossings, as indicated by vertical dashed lines there. If the ground state were to be reached, i.e., no further energy minimum could be found, the sequence would terminate, of course. To focus truly on locally correlated record barrier crossings, it would be useful to refine this definition of valley . However, unless a system gets too large, with too many simultaneous but spatially distant quakes, by considering a small enough system these events become sufficiently rare to dominate the fluctuations in the entire system trace, instead of being “washed out” by overlap-ping ones. This point illustrates also that, to understand a thermodynamic system out-of-equilibrium , it is often not helpful to take the thermodynamic limit. Examples of a valley sequence from our simulations is shown for single energy traces in Fig. 2 for the EA spin glass (top) and the corresponding ferromagnetic system (bottom) on a cubic lattice. These plots exemplify the stark difference in the effect of fluctuations on either type of system that we discuss in the following. 4-1.78 Energy Density Spin Glass (EA) Energy Density Random Ferromagnet Figure 2. Typical trajectories of an aging process through the energy landscape of the spin glass model on a 3d -lattice with L3= 16 3spins and a fraction pof ferromagnetic bonds and 1−panti-ferromagnetic bonds, here with p= 0 .5(top) and p= 0 .85 (bottom). Energy ( H) and barrier ( N) records, as defined in Fig. 1, are marked along each trajectory, where the vertical dashed lines indicate the transition between consecu-tive valleys. While the energy decreases, on average, gradually as a logarithm in time with an ongoing but random produc-tion of further records in the glassy case ( p= 0 .5), the more ferromagnetic system ( p= 0 .85 ) expels energy in a few large events which appear to be triggered by typical fluctuations, record-sized fluctuations are seemingly irrelevant. C. Dynamics Driven by Record-Sized Fluctuations As alluded to in the introduction, glassy and other-wise homogeneous systems such as a ferromagnet distin-guish themselves in the manner fluctuations affect their relaxation dynamics. In the latter, barriers are compa-rably low and remain invariant independent of the depth within the landscape and, thus, of the age of the process. As Fig. 2 exemplifies, large releases of energy are pre-ceded by typical fluctuations at any stage of the process. Fewer events, like the evaporation of a domain in coars-ening, happen not because individual events become so much harder but rather because so many fewer events can happen when only few domains is left. Larger domains may take a little more time to evaporate, as meandering interfaces need to find each other and collide, but such an entropy barrier does not dominate the otherwise domain-size independent energetic barriers . Yet, ordinary fluctuations suffice to bring those interfaces together. In the glassy system, however, it is the barrier height growing with domain size that decelerates the event-rate. Although many domains remain available even after a long aging time, few muster the chance fluctuation re-quired to break up. In a landscape with those barriers, ordinary fluctuations become ineffective to drive the re-laxation process. They merely “rattle” the system during increasingly longer quasi-equilibrium interludes. Only rare, extraordinary large, in fact, record-size fluctuations manage to scale such barriers to expel excess heat, ad-vance the relaxation, and grow domain size, minutely. These features, widely shared across many disordered materials, have inspired the phenomenological descrip-tion at the basis of RD . The definition of valleys in the preceding section provides an especially adapt mea-sure for this phenomenology. In RD, the relaxation pro-cess of a non-equilibrium system after a hard quench is determined by large (i.e., record-sized), irreversible fluc-tuations which move the system from one meta-stable state to the next (usually only marginally more sta-ble than the last one) within its complex energy land-scape [46, 66]. This can be thought of as the system overcoming energy barriers in a hierarchical energy land-scape [37–41]. The rate λ(t) of such record events, also termed “quakes”, decelerates with time as 1/t . Hence, the expected number of events in a time interval [ t, tw], is 〈n(t, t w)〉 t ˆ tw λ (t′) dt ′ ln ( ttw ) (3) implying that the dynamics of the system is self-similar in the logarithm of time. That time-homogeneity is a common feature of many aging systems [44, 57, 67]. In our studies here, we are more concerned with the rate of events λ(t) and the logarithmic growth of observables in time. The dependence on waiting time tw has been the focus elsewhere [46, 61, 66]. IV. NUMERICAL RESULTS A. Edwards-Anderson Model Applying the measure of a valley number defined in Sec. III B to the cubic Ising spin model introduced in Sec. II provides a notable distinction between glassy and homogeneous coarsening behavior, as Fig. 3 shows. For all p < p c ≈ 0.77 , the critical threshold found in Ref. , we find that the valley count progresses logarithmically in time (in fact, like the root of that logarithm ), consis-tent with Eq. (3). For larger values of p, the valley count slows ever more significantly to eventually plateau at a finite value, apparently. All the results shown here were obtained for systems with N = 16 3 = 8096 spins, using 510 1 10 2 10 3 10 4 Sweeps 1 2 3 4 5 6 7 8 9 10 11 12 Valleys p = 0.5 p = 0.55 p = 0.6 p = 0.65 p = 0.7 p = 0.75 p = 0.8 p = 0.85 p = 0.9 p = 0.95 p = 1.0 Figure 3. Average number of valleys in EA, as defined in Fig. 1, that are traversed with time after a quench to T = 0 .7J0 in a Ld = 16 3 spin glass with a fraction p of ferromagnetic bonds and 1 − p anti-ferromagnetic bonds. For p ≤ 0.75 , the generation of valleys evolves essentially inde-pendent of p, while for a larger admixture of ferromagnetic bonds valley generation progresses to cease ever more rapidly and the number of valleys reached plateaus. 10 1 10 2 10 3 10 4 Sweeps 0 0.2 0.4 0.6 Magnetization p = 0.5 p = 0.55 p = 0.6 p = 0.65 p = 0.7 p = 0.75 p = 0.8 p = 0.85 p = 0.9 p = 0.95 p = 1.0 Figure 4. Average magnetization per spin in EA, 〈m〉, ob-served with time after a quench during the ensuing aging process, as described in Fig. 3. Like there, systems with p ≤ 0.75 behave glassy in a p-independent manner with little discernible magnetic ordering, while the more ferromagnetic systems become increasingly more ordered. periodic boundaries, since we found very little variation with system size for larger N .The fact that the underlying ordered state is either glassy or ferromagnetic affords us to also measure the increase in magnetization with time, as demonstrated in Fig. 4. This measure actually exhibits a more dramatic transition between the glassy and the ferromagnetic case, as consecutive snapshots of both, the valley count as well as the magnetization, are shown in Fig. 5 for a progres-sion of times that increases by a factor of 8. In these plots, we have also marked the zero-temperature transi-tion at pc ≈ 0.77 , which proves consistent asymptotically 0.5 0.6 0.7 0.8 0.9 1 density p 0 0.2 0.4 0.6 2 4 6 8 10 12 Valleys t = 2 14 t = 2 11 t = 2 8 Figure 5. Finite-time snapshots for EA of the numbers of valleys generated (top) and the corresponding magnetization per spin, 〈m〉 (bottom), as a function of ferromagnetic bond fraction p for three different times, taken from the data at T =0.7J0 shown in Fig. 3 and Fig. 4, respectively. The vertical line at pc = 0 .77 indicates the zero-temperature transition found in Ref. between a glassy and a ferromagnetic phase. 10 1 10 2 10 3 10 4 Sweeps 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 Event Rate p = 0.5 p = 0.55 p = 0.6 p = 0.65 p = 0.7 p = 0.75 p = 0.8 p = 0.85 p = 0.9 p = 0.95 p = 1.0 Figure 6. Instantaneous rate of record barrier crossing events in EA, as defined in Fig. 1, with time after the quench, as described in Fig. 3. Asymptotically, for larger times, that rate varies as a power-law with a seemingly hyperbolic decline, ∼ 1/t (dotted line), for all p < 0.75 to an almost quadratic decline, ∼ 1/t 2 (dash-dotted line), for larger p. with the transition out of the glassy relaxation behavior. Finally, we can also look at the instantaneous rate of barrier crossing events, effectively the derivative of the valley production, i.e, inverting the integral in Eq. (3). Indeed, throughout the glassy regime, the rate deceler-ates roughly hyperbolically, in accordance with the RD predictions. [Note that this could miss a minor logarith-mic correction, such as λ(t) ∼ 1/(t√ln t), for instance, needed to get √ln t for the valley production in Fig. 3.] For p > p c, in the ferromagnetic coarsening regime, we notice that the rate falls off increasingly sharper, ulti-mately about as ∼ 1/t 2. Consequently, its integral stalls 6 Figure 7. Number of valleys traversed during relaxation en-suing after a quench of SK for different bond fractions αfrom a high temperature T=∞to T= 0 .7J0, averaged over an ensemble of trajectories for N= 2048 spins. In the range 0.0≤α≤0.6,the number of valleys traversed grows loga-rithmically and largely independent of α, indicating that the regime is glassy. out into the plateaus seen in Fig. 3. Apparently, do-main mergers occur more rapidly, on a power-law scale, in coarsening ferromagnets. Despite the rapid drop in the event rate, the average domain size manages to in-crease as a power-law , because later mergers expel larger amounts of excess heat, see Fig. 2. In case of the glass, each event expels on average a fixed amount of heat, roughly. Therefore, both valley production and domain growth proceed similarly (logarithmically), as an integral of the event rate, since each activation has the same impact. B. Sherrington-Kirkpatrick Model Using the valley counts defined in Sec. III B as an or-der parameter, we find a clear transition from a glassy regime to a ferromagnetic one in the mean field as well. However, unlike for EA on a cubic lattice, extending the neighborhood of each spin to all others in the case of SK changes the dynamics, and we have to explore the critical threshold at which the spin glass to ferromag-netic transition takes place on a different scale. Mutual connections between all spins require the number of fer-romagnetic bonds to only slightly exceed the number of antiferromagnetic bonds, in order to tip the system into becoming ordered. The transition to the ferromagnetic regime occurs almost immediate beyond a bond density of p = 0 .5, with a strong system size dependence, forcing us to adapt a different scale to observe it. To properly describe the behavior of SK, we therefore reparametrize the bond density in terms of α via p = 12 + α√N . Then, within the range of 0 ≤ α ≤ 2.0, we can localize a tran-sition that varies only slowly with size. Figure 8. Average magnetization for SK in the same simula-tions shown in Fig. 7. According to this measurement, the system begins to order at αc≈0.6, since a non-zero magne-tization in the long-time limit indicates that majority of the spins have ferromagnetically ordered. The transition in mag-netization shown here is far more dramatic than in the valley counts, but nevertheless affirms the same critical threshold. Figure 9. Instantaneous average valley counts and magneti-zation in SK as function of αat different sweep-times t= 16 ,256 and 4096 from left to right, each for three different system sizes indicated on the legend. The first row shows the aver-age number of valleys, and the second row shows the average magnetization. According to this data, the valley production is time dependent as the sharpness of the transition becomes more pronounced in the later sweeps. In contrast, the magne-tization appears to be saturated already early on, predicting the critical threshold within 16 sweeps. Additionally, we see no system size effects when using αas the parameter. Similar to Fig. 3 for EA, in Fig. 7 we show the numbers of valleys found in a SK system with N = 2048 spins. There appears to be a critical threshold at αc ≈ 0.6. For α ≤ 0.6, the valley production increases about as log (t), essentially uniform with bond density, given the nearly perfect overlap in the data. This is no longer case when α > 0.6, where the production of valleys decreases gradually before plateauing completely. While domains in the sense of geometric regions of a certain length do not exist in a mean field system with long-range interactions, individual spins develop clusters of increasingly ordered 7 Figure 10. Instantaneous rates for the number of record bar-rier crossings as a function of time, for every α-value in SK. The instantaneous rate decreases as a power-law for all but the highest admixture values. In the glassy regime, the de-celerations is essentially hyperbolic (dotted line), while the rate drops more sharply for α > 0.6, up to roughly t−1.5 at α= 1 .6(dash-dotted line), beyond which further record events become immeasurably rare. local fields with some of their neighbors that entrench the system into deeper valleys. It becomes increasingly more difficult for the system to overcome the energy barrier of flipping the entire cluster, causing the relaxation process to evolve logarithmically . That said, evidence of a critical threshold suggests that beyond αc, the system changes its landscape dramati-cally. It exhibits an inclination to order rapidly, facil-itated by the fact that local fields of individual spins immediately affect all others, as the evolution of mag-netization in Fig. 8 suggests. Flat interfaces between such clusters, as they may exist between domains in low-dimensional lattices like EA, are absent here and any imbalance in size quickly erodes inferior clusters. Therefore, despite the quantitative differences pertaining to local structure between the Edwards-Anderson and Sherrington-Kirkpatrick spin glass, our results suggest that the glassy behavior in both can be attributed to the hierarchical nature of the energy landscape, and the lack of it beyond the transition to ferromagnetic order, seen both in Fig. 4 and Fig. 8. We have also checked the evolution of valley counts across different system sizes and found only a minimal dependence of the transition on larger size, as shown in Fig. 9. While the relationship (or lack thereof) between the number of valleys encountered and the bond admix-ture exhibits time dependence, the critical threshold with regard to ordering already emerges after about two hun-dred sweeps. There is clearly an agreement between val-ley statistics and the ferromagnetic order parameter in suggesting αc ≈ 0.6 as the critical threshold. Lastly, we look at the deceleration of the rate of record barrier crossing events in Fig. 10. As shown in Fig. 6for the Edwards-Anderson model, the rate decays with a power of time t. While there is a steeper decelera-tion in the barrier crossing events for larger α-values, the difference between the exponents is quite subtle on this time scale within our simulations. In the glassy regime, α < α c ≈ 0.6, the rate clearly decays hyperbolically, whereas it falls off steeper above αc. However, for values α > 1.6, the fall-off becomes so significant that new val-leys are not encountered beyond the first ∼ 100 sweeps. V. CONCLUSION Our study explores the distinction between glassy re-laxation and ordinary coarsening, which is often ignored in the description and analysis of aging systems. Focus-ing on families of models that interpolate between either extreme, we not only apply measures [30, 31] that clearly indicate the difference but also show a rather sharp tran-sition in the non-equilibrium behavior between those ex-tremes that, for the Edwards-Anderson model on a cu-bic lattice, appears to coincide with the (equilibrium) zero-temperature transition between spin glass and fer-romagnet . The corresponding transition we find at a sub-extensive scale in SK seems to have been unnoticed. While the distinction we are making between a coars-ening (ferromagnetic) and an aging (glassy) regime can be seen as semantic, considering that both, algebraic as well as logarithmic growing domains, are commonly portrayed as coarsening , the difference in dynamic behavior after a quench is profound. The picture that emerges is one of a largely convex landscape on one side with invariant energetic barriers in the case of coarsening, a system that despite its often complex network of frac-tal interfaces locally homogenizes rather quickly. On the other side, we find a hierarchical landscape [37–41] with energetic (and potentially entropic) barriers that grow with deeper entrenchment within the landscape, render-ing all but record fluctuations ineffective for relaxation. H. Frauenfelder, ed., Landscape Paradigms in Physics and Biology (Elsevier, Amsterdam, 1997). D. J. Wales, Energy landscapes (Cambridge University Press, Cambridge, 2003). S. A. Kauffman and E. D. Weinberger, Journal of Theo-retical Biology 141 , 211 (1989). H. Frauenfelder, S. Sligar, and P. Wolynes, Science 254 ,1598 (1991). P. F. Stadler, Europhysics Letters 20 , 479 (1992). J. D. Bryngelson, J. N. Onuchic, N. D. Socci, and P. G. Wolynes, Proteins: Structure, Function, and Genetics 21 , 167 (1995). 8 V. Mustonen and M. Lässig, Trends in genetics 25 , 111 (2009). K. D. Ball, R. S. Berry, R. E. Kunz, F. Y. Li, A. Proykova, and D. J. Wales, Science 271 , 259 (1996). M. A. C. Wevers, J. C. Schön, and M. Jansen, J. Phys.:Condens. Matter 11 , 6487 (1999). J. C. Schön and P. Sibani, Europhys. Lett. 49 , 196 (2000). J. Schön, J. Phys. Chem. A 106 , 10886 (2002). S. Schubert and K. H. Hoffmann, Computer Physics Communications 174 , 191 (2006). T. Aspelmeier, R. A. Blythe, A. J. Bray, and M. A. Moore, Phys. Rev. B 74 , 184411 (2006). G. J. Rylance, R. L. Johnston, Y. Matsunaga, C. B. Li, A. Baba, and T. Komatsuzaki, PNAS 103 , 18551 (2006). A. Fischer, K. H. Hoffmann, and P. Sibani, Phys. Rev. E 77 , 041120 (2008). P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, and F. Zamponi, Nature Communications 5, 3725 (2013). Q. Liao and L. Berthier, Phys. Rev. X 9, 011049 (2019). J.-C. O. dit Biot, P. Soulard, S. Barkley, E. R. Weeks, T. Salez, E. Raphael, and K. Dalnoki-Veress, Physical Review Research 2, 023070 (2020). S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Science 220 , 671 (1983). S. Mertens, Phys. Rev. Lett. 84 , 1347 (2000). M. Mézard and A. Montanari, Constraint Satisfaction Networks in Physics and Computation (Oxford Univer-sity Press, Oxford, 2006). A. Percus, G. Istrate, and C. Moore, Computational Complexity and Statistical Physics (Oxford University Press, New York, 2006). P. S. Peter Salamon and R. Frost, Facts, conjectures and improvements for simulated annealing (SIAM, Philadel-phia, 2002). J. C. Schön, H. Putz, and M. Jansen, J. Phys.: Condens. Matter 8, 143 (1996). P. Sibani, R. van der Pas, and J. C. Schön, Computer Physics Communications 116 , 17 (1999). A. K. Hartmann, Euro. Phys. J. B 13 , 539 (2000). P. Sibani and J. C. Schön, in Applied Parallel Comput-ing, Lecture Series in Computer Science, Springer Ver-lag , edited by J. Fagerholm (2002). A. Heuer, J. Phys.: Condens. Matter 20 , 373101 (2008). H. Gould, An introduction to computer simulation meth-ods : applications to physical systems (Addison-Wesley, Reading, Mass, 1996). J. Dall and P. Sibani, Eur. Phys. J. B 36 , 233 (2003). S. Boettcher and P. Sibani, Eur. Phys. J. B 44 , 317 (2005). J.-P. Bouchaud, V. Dupuis, J. Hammann, and E. Vin-cent, Phys. Rev. B 65 , 024439 (2001). L. Struik, Physical aging in amorphous polymers and other materials (Elsevier Science Ltd, New York, 1978). S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975). D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35 ,1792 (1975). M. Manssen and A. K. Hartmann, Phys. Rev. B 91 ,174433 (2015). R. G. Palmer, D. L. Stein, E. Abraham, and P. W. Anderson, Phys. Rev. Lett. 53 , 958 (1984). S. Teitel and E. Domany, Phys. Rev. Lett. 55 , 2176 (1985). P. Sibani, Phys. Rev. B 34 , 3555 (1986). K. H. Hoffmann and P. Sibani, Phys. Rev. A 38 , 4261 (1988). P. Sibani and K. H. Hoffmann, Physical Review Letters 63 , 2853 (1989). T. Komori, H. Yoshino and H. Takayama, J. Phys. Soc. Japan 68 , 3387 (1999). P. Mayer, H. Bissig, L. Berthier, L. Cipelletti, J. P. Gar-rahan, P.Sollich and V. Trappe, Phys. Rev. Lett. 93 ,115701 (2004). G. Biroli, Journal of Statistical Mechanics: Theory and Experiment 5, P05014 (2005). J. D. Shore, M. Holzer, and J. P. Sethna, Physical Re-view B 46 , 11376 (1992). D. M. Robe, S. Boettcher, P. Sibani, and P. Yunker, EuroPhys. Lett. 116 , 38003 (2016). P. Sibani and J. Dall, Europhys. Lett. 64 , 8 (2003). P. Sibani and H. J. Jensen, Stochastic Dynamics of Com-plex Systems (Imperial College Press, 2013). P. Sibani, S. Boettcher, and H. J. Jensen, (arXiv:2008.12684). M. Mezard, Physica A 306 , 25 (2002). A. K. Hartmann, Physical Review B 59 , 3617 (1999). M. Plischke and B. Bergersen, Equilibrium Statistical Physics, 2nd edition (World Scientifc, Singapore, 1994). M. Mézard, G. Parisi, and M. A. Virasoro, Spin glass theory and beyond (World Scientific, Singapore, 1987). K. H. Fischer and J. A. Hertz, Spin Glasses (Cambridge University Press, Cambridge, 1991). D. L. Stein and C. M. Newman, Spin Glasses and Com-plexity (Princeton University Press, Princeton, 2013). Note that in the bond matrix Jij of SK there are ∼ N 2 2 bonds, thus, α skews an equal mix of ferromagnetic and anti-ferromagnetic bonds with an imbalance of O(N 32 ),a vanishing fraction of all bonds but significantly larger than random fluctuations between bond types of O(N ). J. M. Hutchinson, Progress in Polymer Science 20 , 703 (1995). Eric Vincent, Jacques Hammann, Miguel Ocio, Jean-Philippe Bouchaud, and Leticia F. Cugliandolo, SPEC-SACLAY-96/048 (1996). M. Baity-Jesi, R. A. Baños, A. Cruz, L. A. Fernan-dez, J. M. Gil-Narvion, A. Gordillo-Guerrero, D. Iñiguez, A. Maiorano, F. Mantovani, E. Marinari, V. Martin-Mayor, J. Monforte-Garcia, A. M. Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, M. Pivanti, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Y. and, Physical Review B 88 , 224416 (2013). P. Butera and M. Comi, Phys. Rev. B 62 , 14837 (2000). P. Sibani and H. J. Jensen, Europhys. Lett. 69 , 563 (2005). H. Bissig, S. Romer, L. Cipelletti, V. Trappe, and P. Schurtenberger, Phys. Chem. Comm. 6, 21 (2003). L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, and S. Mazoyer, J. Phys.:Condens. Matter 15 , S257 (2003). P. Yunker, Z. Zhang, K. B. Aptowicz, and A. G. Yodh, Phys. Rev. Lett. 103 , 115701 (2009). P. Sibani and S. Boettcher, Phys. Rev. B 98 , 054202 (2018). N. Becker, P. Sibani, S. Boettcher, and S. Vivek, J. Phys.: Condens. Matter 26 , 505102 (2014). C. Chamon, M. P. Kennett, H. E. Castillo, and L. F. Cugliandolo, Phys. Rev. Lett. 89 , 217201 (2002).
15105	https://sentence.yourdictionary.com/succumbed	Make Our Dictionary Yours Sign up for our weekly newsletters and get: By signing in, you agree to our Terms and Conditions and Privacy Policy. Success! We'll see you in your inbox soon. Succumbed Sentence Examples He looked to Sarah, who had also succumbed to the craving. The readiness with which the American Indian succumbed to disease is well known. He seems to have touched at the island of Tortugas, so named on account of the large number of turtles found there, and to have landed at several places, but many of his men succumbed to disease and he himself was wounded in an Indian attack, dying soon afterward in Cuba. Dresden was the last great victory of the First Empire, By noon on the 27th August the Austrians and Russians were completely beaten and in full retreat, the French pressing hard behind them, but meanwhile Napoleon himself again succumbed G Beereri B eip \ ii g? The Ionians in turn succumbed to the Dorians of Argos, who, according to the legend, were led by Deiphontes; and from that time the city continued to preserve its Dorian character. Or maybe he'd succumbed to the weird draw around the woman. In 1849 Garibaldi's wife Anita, who had accompanied him on his retreat from Rome, succumbed to fatigue in the marshes near Ravenna. They succumbed to the Persian dynasty of the Sassanids, who ruled successfully for about four centuries, established the Zoroastrian faith as their state religion, and maintained a creditable conflict with the East Roman empire. In the crypt is the grave of a traveller, who succumbed to excessive drinking of the local wine known as Est, est, est. When Caesar invaded Britain 54 B.C. they joined him against their domestic rivals and it is possible (though not certain) that half a century after Caesar's departure they succumbed to them. In Prussia at least the medieval system of local self-government had succumbed completely to the centralizing policy of the monarchy, and when it was revived it was at the will and for the purposes of the central authorities, as subsidiary to the bureaucratic system. At least, she'd thought this until Dusty succumbed to it. But they succumbed before the advance of the Medo-Persian power in 606 B.C., whereas it was not till 555 that Cyrus took Babylon. In 1644 the Ming succumbed to the attacks of the Manchus, a northern tribe who captured Peking and founded the present imperial house. Later, it allied itself with the Mongols and fought against the Mamelukes, to whom, however, it finally succumbed in 1375. During the later part of their history they were in continual contact with Assyria, and, as a Syrian power, and perhaps also as a Cappadocian one, they finally succumbed to Assyrian pressure. In the middle ages Arezzo was generally on the Ghibelline side; it succumbed to Florence in 1289 at the battle of Campaldino, but at the end of the century recovered its strength under the Tarlati family. The rationalist spirit is, of course, coeval with human evolution; religion itself began with a rational attempt to maintain amicable relations with unknown powers, and each one of the dead religions succumbed before the development of rationalist inquiry into its premises. The Lombard princes, who had frequently defended their city against the Saracens, succumbed before Robert Guiscard, who took the castle after an eight months' siege and made Salerno the capital of his new territory. Under such disheartening conditions it is not surprising that this body was totally unable to cope with Sickingens insurrection, and that a few weeks after its meeting at Nuremberg in 1524 it succumbed to a series of attacks and disappeared from the history of Germany. The despotate of Epirus succumbed in 1449, the duchy of Athens in 1456; in 1453 Constantinople was taken and the decrepit Byzantine empire perished; the greater part of Bosnia submitted in 1463; the heroic resistance of the Albanians under Scanderbeg collapsed with the fall of Croia (1466), and Venetian supremacy in Upper Albania ended with the capture of Scutari (1478). It was, perhaps inevitable; when Patrick Stewart's Shakespearean-trained voice intoned the opening voiceover, most of us quickly succumbed. That's how long my little prize remained with me until my darling succumbed to the trials and tribulations of life on the road, with me. Monty, as he became known during his brief public life, succumbed to his injuries. There are still many magneto exchanges in existence, but when new exchanges are erected only the very smallest are equipped for magneto working, that system having succumbed to the common battery one in the case of all equipments of moderate and large dimensions. During the milder interglacial period some southern types, such as Rhododendron ponticum, still held their own, but ultimately succumbed. During his brief reign he set on foot some domestic reforms, and sought to revive the authority of the senate, but, after a victory over the Goths in Cilicia, he succumbed to hardship and fatigue (or was slain by his own soldiers) at Tyana in Cappadocia. And so the old limitations of Israel's popular religion, - the same limitations that encumbered also the religions of all the neighbouring races that succumbed in turn to Assyria's invincible progress, - now began to disappear. Two assaults were repulsed after hours of hand-to-hand fighting; and when, after a fresh bombardment, the garrison saw that their case was hopeless, they killed their women and children, and only succumbed at last to a third assault because every man of them was either killed or mortally wounded. The anxiety, fatigue and cold to which he was thus exposed, affecting a constitution naturally weak, laid the foundation of the disease to which he afterwards succumbed. This spirit gave way to the physicians, who regarded " chemistry as the art of preparing medicines," a denotation which in turn succumbed to the arguments of Boyle, who regarded it as the " science of the composition of substances," a definition which adequately fits the science to-day. During the wars of the French Revolution, it was taken by Dumouriez in 1793, evacuated soon after and retaken by Pichegru in 1795, after the whole of Holland had already succumbed to the French. These were no longer numerous, many having succumbed to the hardships and sufferings of all kinds to which they had been exposed. Walton illustrates Herbert's kindness to the poor by many touching anecdotes, but he had not been three years in Bemerton when he succumbed to consumption. The compatibility of Christian and later Neo-Platonic ideas is evidenced by the writings of Synesius, bishop of Ptolemais, and though Neo-Platonism eventually succumbed to Christianity, it had the effect, through the writings of Clement and Origen, of modifying the tyrannical fanaticism and ultradogmatism of the early Christian writers. They only succumbed when the weight of the archduke Maximilian was thrown into the scale against them (1484). But little by little he succumbed to his milieu, the atmosphere of false confidence and passivity created around him by Alexeiev. In 1899 her grandson, the hereditary prince of Coburg, had succumbed to phthisis, and in 1900 his father, the duke of Coburg, the queen's second son, previously known as the duke of Edinburgh, also died (July 30). Sadik Beg soon repented of having asked for a Khoja, and eventually marched against Kashgar, which by this time had succumbed to Buzurg Khan and Yakub Beg, but was defeated and driven back to Khokand. Why Neoplatonism succumbed in the conflict with Christianity is a question which the historians have never satisfactorily answered. In 1 453 the king succumbed, Alvaro was arrested, tried and condemned by a process which was a mere parody of justice, and executed at Valladolid on the 2nd of June 1453. The Hohenstaufen succumbed to it, and the papacy itself received a terrible shock, which shook its vast empire to the foundations. The Arab tribes in Mesopotamia were Christian, and Heraclius at Edessa hoped for their support; but Karkisiya and Hit succumbed (636), and then Tekrit; and Heraclius retired to Samosata. He succumbed to pressure from the boss. Meanwhile the other independent principalities of Gondwana had in turn succumbed. Lavoisier adequately recognized and acknowledged how much he owed to the researches of others; to himself is due the co-ordination of these researches, and the welding of his results into a doctrine to which the phlogistic theory ultimately succumbed. But can a historian separate the opinions which rose to authority in the church from the other opinions which succumbed? Three more of her children, as well as her husband, quickly caught the disease, and the youngest, "May," succumbed on the 16th. The sultan's policy had been consistently directed to crushing the overgrown power of his vassals; in the spring of 1831 two rebellious pashas, Hussein of Bosnia and Mustafa of Scutari, had succumbed to his arms; and, since he was surrounded and counselled by the personal enemies of the pasha of Egypt, it was likely that, so soon as he should feel himself strong enough, he would deal in like manner with Mehemet Ali. On the return journey Dr. Wulff and Olsen succumbed to the privation of scanty food and bad weather, and the survivors had difficulty in reaching Etah. It succumbed to the Indo-Scythian Empire of the Kushana, who had obtained the sovereignty of Bactria as early as about A.D. Julian pressed forward to Ctesiphon but succumbed to a wound; and his successor Jovian soon found himself in such straits, that he could only extricate himself and his army by a disgraceful peace at the close of 363, which ceded the possessions on the Tigris and the great fortress of Nisibis, and pledged Rome to abandon Armenia and her Arsacid protg, Arsaces III., to the Persian. Eventually he succumbed to a conspiracy of his magnates, at whose head stood the general Bahram Cobin, who had defeated the Turks, but afterwards was beaten by the Romans. The historical bent thus given to the drama was continued by the versatile Mendes Leal, by Gomes da Amorim and by Pinheiro Chagas, who all however succumbed more or less to the atmosphere and machinery of ultra-Romanticism, while the plays of Antonio Ennes deal with questions of the day in a spirit of combative liberalism. At the very outset of a promising career he suddenly succumbed to an attack of smallpox on the 6th of November 1650, his son William III. During the Northern War between Sweden and Russia, it was courageously defended (1700), but after the battle of Poltava it succumbed, and was taken in July 1710 by the Russians. Such was the situation when the president, early in July 1850, was stricken by the disease to which he succumbed on the 9th. The kingdom probably succumbed to the Huns established in the neighbourhood. His election to the papacy, on the 29th of October 1591, was brought about by Philip II., who profited little by it, however, inasmuch as Innocent soon succumbed to age and feebleness, dying on the 30th of December 1591. This disaster, though partly retrieved in the campaign of the following year, had a serious effect upon his vitality; henceforth he declined in health and in 1180 succumbed to a slow fever. About 250 B.C. Diodotus (Theodotus), governor of Bactria under the Seleucidae, declared his independence, and commenced the history of the Greco-Bactrian dynasties, which succumbed to Parthian and nomadic movements about 126 B.C. After this came a Buddhist era which has left its traces in the gigantic sculptures at Bamian and the rock-cut topes of Haibak. He strongly upheld in the House of Commons the measures taken, first by Mr. Macpherson and then by Sir Hamar Greenwood, to restore law and order in that country; and definitely refused to interfere in the case of the Lord Mayor of Cork who, sentenced to imprisonment for conducting a rebel organization, went on hunger-strike and eventually succumbed in gaol. When the Carrara family succumbed in 1405, Este voluntarily surrendered to Venice and was allowed its independence, under a podesta; and thenceforth it followed the fortunes of Venetia. But by the middle of October the Chinese army was decisively defeated; Peking was occupied; those British and French prisoners who had not succumbed to the hardships of their confinement were liberated. A year later the Emperor was stricken down by illness, and succumbed to it on July 30 1912. By 1208, however, the Kadambas had been overthrown by the Rattas, who in their turn succumbed to the Yadavas of Devagiri in 1250. But two weeks after his arrival he succumbed to dysentery, and was buried at the age of eighty-three in the church of the Annunziata. On the 15th of July 1895 he was attacked and barbarously mutilated by a band of Macedonian assassins in the streets of Sofia, and succumbed to his injuries three days later. For a time he thought of responding to the appeal of some of the Polish revolutionaries, but Warsaw succumbed (September 1831) before he could set out. A man of action and not of cunning shifts, he succumbed on the 10th of July to the blows of his own government, which had passed from his hands into those of Robespierre, his ambitious and crafty rival. Politically moribund, it succumbed to the attacks of its virile southern neighbours, who, having emerged from foreign tutelage, developed according to the natural laws of their own genius and environment. He had just become connected with the Revue de Paris, when his delicate constitution succumbed to a slight attack of illness on the 19th of October 1894. Its northern and southern extremities have been named Cape Costigan and Cape Molyneux, in memory of two explorers who were among the first in modern times to navigate the sea and succumbed to the consequent fever and exhaustion. The station succumbed to disrepair many years before is closed. Mice in the study that received treatment remained healthy for almost a year after untreated mice succumbed to the disease. The baby succumbed to illness suddenly just weeks after birth. His cheerful demeanor succumbed to the stress of his bad marriage. He succumbed to defeat. She succumbed to heat stroke and awaits a new air conditioner to feel better. He finally succumbed to his illness this past December. I tried to resist but eventually succumbed and agreed to edit the paper. Those without hats nearly succumbed to frostbite. To educate a child regarding the dangers of alcohol was deemed more beneficial than waiting until they had succumbed. In 1996, Monsanto's pest-resistant Bt-cotton succumbed to a heat wave in the southern US and was destroyed by bollworms and other pests. Now sadly it has succumbed to the competition of the chain bookstores. Two bulbs have succumbed to a rot that luckily did not spread all through the whole potful. Like many before him he has succumbed to the wallet draining world of high power rocketry and is UKRA Level 1 certified. He died in 45 1; some years earlier Nestorius, the ex-patriarch, had succumbed perhaps to his persecution and to old age, in the neighbourhood of Akhmim. Once Howie succumbed to slumber, his sleep was anything but peaceful. Alice was a geranium Cynthia had lovingly rescued from certain death by frost last September when the rest of the couple's first-year garden succumbed to the advancing seasons. Alice was a geranium Cynthia had lovingly rescued from certain death by frost last September when the rest of their first year garden succumbed to the advancing seasons. In 1544 the Indians, so far as they had not succumbed to the labour of the mines and fields to which they were put by the Spaniards, were proclaimed emancipated. So far the Hevea plantations in Ceylon and the East have not been seriously troubled by insect or fungoid pests, and those which have occurred have succumbed to proper treatment. Soon after marriage his wife was attacked by a lingering illness, to which she succumbed, Lagrange devoting all his time, and a considerable store of medical knowledge, to her care. At last he succumbed to the repeated requests of Girolamo or Geronimo Cardano, who swore that he would regard them as an inviolable secret. He succumbed to leprosy on the 15th of April 1889. He succumbed to fear and choked up. He succumbed under the weight he had to carry. The outcome depended on Bill Rice who unfortunately succumbed to defeat. After the exodus, which perhaps took place about 1300 B.C., they moved northwards again and founded a state of modest dimensions, which attained a short-lived unity under Solomon, but succumbed to internal dissensions and to the attacks of Assyria and Babylon. Shortly after his accession he was threatened with invasion by Cambyses, the Persian conqueror of Egypt, but (according to his own account) destroyed the fleet sent by the invader up the Nile, while (as we learn from Herodotus) the land-force succumbed to famine (see Cambyses). It succumbed to the ceaseless alternation of tolerance and persecution which characterized the Arab rule in Egypt, and the mass of the Coptic people became unfaithful to the Church. In spite of all their bravery, they succumbed to the Greek phalanx, when once the generalship of a Miltiades or a Pausanias had brought matters to a hand to hand conflict; and it was with justice that the GrecksAeschylus, for instance viewed their battles against the Persian as a contest between spear and bow. On the fields of Marathon and Plataea, the Persian archers succumbed to the Greek phalarn of hoplites; but the actual decision was effected by Themistocles who had meanwhile created the Athenian fleet which at Salamis proved its superiority over the Perso-Phoenician armada, anc thus precluded beforehand the success of the land-forces. Here the Graeco-Bactrian and Graeco-Indian kingdoms held their own, till, in 139 B.C., they succumbed before the invading Mongolian and Scythian tribes (see BACTRIA and works quoted there). But this state of affairs was too insecure even for these rovers, and they would speedily have succumbed had not a refuge been found for them by the fortunate conquest of Jamaica in 1655 by the navy of the English Commonwealth. The substation of SR origin formerly resided on the right, but had succumbed many years before the station 's closure. When Christian ideas succumbed in the 18th century to rationalist ideas, feudal society fought its death battle with the then revolutionary bourgeoisie. Mice in the 17 month study treated with mAbs remain clinically healthy almost a year after the untreated mice succumbed to the disease. Sir Digby Jones, director general of the CBI, said the government has succumbed to pressure from the unions. By the end, the hero of this story seems to have finally succumbed to defeat. It succumbed to heat stroke on Monday and awaits ministration, and probably a new fan, in a dark corner of a room. The chances are that, no appliances being at hand to assist him, he succumbed under the weight he had to carry. He finally succumbed to his illness on 12 December. I tried and tried and eventually succumbed to editing the kernel. The outcome depended on Bill Rice who unfortunately succumbed to a straight set defeat. Those with bob hats went home freezing those without nearly succumbed to frost bite of the ear lobes. A few oft given reasons are crazy schedules or the relationship succumbed to the harsh light of the paparazzi's constant spotlight. John Belushi - Another celebrity who succumbed to a drug overdose, this Saturday Night Live alum died in 1982 from a fatal injection of cocaine and heroin. However, all of that changed once she succumbed to a well-publicized nose job that stalled her career instead of helping it. Married in 1984, the pair only had five short years together before Radner succumbed to cancer. McClanahan was a breast cancer survivor, but succumbed to a stroke on June 3, 2010. After their father's defrocking and the divorce of their parents, the brothers abandoned the road in favor of Nashville where they succumbed to the forbidden allure of music and began collaborating with songwriter Angelo Petraglia. Many people do not understand when they should get help for someone that has succumbed to alcohol poisoning. Henry, who had stayed behind with his injured girlfriend, eventually succumbed to the heat, as did she. While other trailers succumbed to the economic woes of the World War II era, the Airstream endured although its luxury label meant that few of its motor homes were sold. Some are open to the public, while others are either closed to the public - sometimes they still exist as private residences - or have succumbed to the demands of time and progress. The painter, knowing that making such an agreement with such an evil entity was the wrong thing to do, succumbed to his desperate greed and accepted the offer. Unfortunately, flash has succumbed to a lot of bootlegging in recent years. An example of this was Pedro Zamora, who was openly homosexual during filming of the San Francisco house, and eventually succumbed to AIDS in 1994, after a long battle with the disease and plenty of activism credited to his name. Of greater historical interest are the Chams, who are to be found for the most part in southern Annam and in Cambodia, and who, judging from the numerous remains found there, appear to have been the masters of the coast region of Cochin-China and Annam till they succumbed before the pressure of the Khmers of Cambodia and the Annamese. The last Greek prince, Hermaeus, seems to have succumbed about 30 B.C. It was just at this time that the Graeco-Roman world of the West was consolidated as the Roman Empire, and, though Greek rule in India had disappeared, active commercial intercourse went on between India and the Hellenistic lands. Their sufferings on the route were dreadful; many succumbed and were abandoned. As a fortress, Metz has always been of the highest importance, and throughout history down to 1870 it had never succumbed to an enemy, thus earning for itself the name of La pucelle. After proclaiming his intention of conferring on his subjects the blessings of peace, he joined in 1798 an Anglo-Austrian coalition against France; but when Austria paid more attention to her own interests than to the interests of monarchical institutions in general, and when England did not respect the independence of Malta, which he had taken under his protection, he succumbed to the artful blandishments of Napoleon and formed with him a plan for ruining the British empire by the conquest of India. In 1107 B.C., however, he sustained a temporary defeat at the hands of Merodach-nadin-akhi (Marduknadin-akhe) of Babylonia, where the Kassite dynasty had finally succumbed to Elamite attacks and a new line of kings was on the throne. Here on the 28th of December 1825 he succumbed to the combined effects of climate and of opium. She covered her eyes with one forearm and succumbed to tears. Fierce opposition ensued, and the pari passu compromise was adopted to which reference is made in the section on Education above; Mr Savona was an able organizer, and began the real emancipation of the Maltese masses from educational ignorance; but he succumbed to agitation before accomplishing substantial results. Browse other sentences examples The word usage examples above have been gathered from various sources to reflect current and historical usage. They do not represent the opinions of YourDictionary.com. Related Articles You may know William Makepeace Thackeray as the author of the epic tome Vanity Fair (not to be confused with the magazine) and The Luck of Barry Lyndon. In a time period dominated by the likes of Charles Dickens and his gritty look at English society, Thackeray took a more colorful approach and created some of English literature’s most iconic antiheroes in the process. Aside from his revolutionary characterization, Thackeray’s writing style was at once witty, insightful and melodic, which makes for some delightful quotes that stand on their own. Your boss who always corrects you; online commenters who lead with “Actually”; that guy at the supermarket who reminded you that you shouldn’t end your sentence with a preposition — chances are, you know a lot of pompous blowhards. They’re pretentious and annoying (and in the case of the grocery store guy, wrong), but you have to deal with them anyway. So why not insult them by using sciolist, a word they likely don’t know, to add insult to injury? Also Mentioned In Words near succumbed in the Dictionary
15106	https://encyclopediaofmath.org/wiki/Laguerre_polynomials	Log in www.springer.com The European Mathematical Society Navigation Main page Pages A-Z StatProb Collection Recent changes Current events Random page Help Project talk Request account Tools What links here Related changes Special pages Printable version Permanent link Page information Namespaces Page Discussion Variants Views View View source History Actions Laguerre polynomials From Encyclopedia of Mathematics Jump to: navigation, search Chebyshev–Laguerre polynomials Polynomials that are orthogonal on the interval $ ( 0 , \infty ) $ with weight function $ \phi ( x) = x ^ \alpha e ^ {-x}$, where $ \alpha > - 1 $. The standardized Laguerre polynomials are defined by the formula $$ L _ {n} ^ \alpha ( x) = \ \frac{x ^ {- \alpha } e ^ {x} }{n!} \frac{d ^ {n} }{dx ^ {n} } ( x ^ {\alpha + n } e ^ {-x} ) ,\ \ n = 0 , 1 , . . . . $$ Their representation by means of the gamma-function is $$ L _ {n} ^ \alpha ( x) = \ \sum _ { k= 0}^{ n } \frac{\Gamma ( \alpha + n + 1 ) }{\Gamma ( \alpha + k + 1 ) } \frac{( - x ) ^ {k} }{k ! ( n - k ) ! } . $$ In applications the most important formulas are: $$ ( n + 1 ) L _ {n+1} ^ \alpha ( x) = \ ( \alpha + 2n + 1 - x ) L _ {n} ^ \alpha ( x) - ( \alpha + n ) L _ {n-1} ^ \alpha ( x) , $$ $$ x L _ {n-1} ^ {\alpha + 1 } ( x) = ( n + \alpha ) L _ {n-1} ^ \alpha ( x) - n L _ {n} ^ \alpha ( x) , $$ $$ ( L _ {n} ^ \alpha ( x) ) ^ \prime = - L _ {n-1} ^ {\alpha + 1 } ( x) . $$ The polynomial $ L _ {n} ^ \alpha ( x) $ satisfies the differential equation (Laguerre equation) $$ x y ^ {\prime\prime} + ( \alpha - x + 1 ) y ^ \prime + n y = 0 ,\ n = 1 , 2 , . . . . $$ The generating function of the Laguerre polynomials has the form $$ \frac{e ^ {- x t / ( 1 - t ) } }{( 1 - t ) ^ {\alpha + 1 } } = \ \sum _ { n=0}^\infty L _ {n} ^ \alpha ( x) t ^ {n} . $$ The orthonormal Laguerre polynomials can be expressed in terms of the standardized polynomials as follows: $$ \widehat{L} {} _ {n} ^ \alpha ( x) = (- 1) ^ {n} L _ {n} ^ \alpha ( x) \sqrt { \frac{\Gamma ( n + 1 ) }{\Gamma ( \alpha + n + 1 ) } } . $$ The set of all Laguerre polynomials is dense in the space of functions whose square is integrable with weight $ \phi ( x) $ on the interval $ ( 0 , \infty ) $. Laguerre polynomials are most frequently used under the condition $ \alpha = 0 $; these were investigated by E. Laguerre , and are denoted in this case by $ L _ {n} ( x) $( in contrast to them, the $ L _ {n} ^ \alpha ( x) $ are sometimes known as generalized Laguerre polynomials). The first few Laguerre polynomials $ L _ {n} ( x) $ have the form $$ L _ {0} ( x) = 1 ,\ L _ {1} ( x) = 1 - x , $$ $$ L _ {2} ( x) = 1 - 2 x + \frac{x ^ {2} }{2} , $$ $$ L _ {3} ( x) = 1 - 3 x + \frac{3 x ^ {2} }{2} - \frac{x ^ {3} }{6} , $$ $$ L _ {4} ( x) = 1 - 4 x + 3 x ^ {2} - \frac{2 x ^ {3} }{3} + \frac{x ^ {4} }{24} . $$ The Laguerre polynomial $ L _ {n} ^ \alpha ( x) $ is sometimes denoted by $ L _ {n} ( x ; \alpha ) $. References \| \| \| --- \| \| \| E. Laguerre, "Sur le transformations des fonctions elliptiques" Bull. Soc. Math. France , 6 (1878) pp. 72–78 \| \| \| V.A. Steklov, Izv. Imp. Akad. Nauk. , 10 (1916) pp. 633–642 \| \| \| G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975) \| \| \| P.K. Suetin, "Classical orthogonal polynomials" , Moscow (1979) (In Russian) \| Comments Laguerre polynomials can be written as confluent hypergeometric functions (cf. Confluent hypergeometric function) and belong to the classical orthogonal polynomials. They have a close connection with the Heisenberg representation: as matrix elements of irreducible representations and as spherical functions on certain Gel'fand pairs (cf. Gel'fand representation) associated with the Heisenberg group. See the references given in [a1], Chapt. 1, §9. References \| \| \| --- \| \| [a1] \| G.B. Folland, "Harmonic analysis in phase space" , Princeton Univ. Press (1989) \| How to Cite This Entry: Laguerre polynomials. Encyclopedia of Mathematics. URL: This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article Retrieved from " Categories: TeX auto TeX done This page was last edited on 31 August 2020, at 18:26. Privacy policy About Encyclopedia of Mathematics Disclaimers Copyrights Impressum-Legal Manage Cookies
15107	https://chem.libretexts.org/Bookshelves/Organic_Chemistry/Organic_Chemistry_(Morsch_et_al.)/10%3A_Organohalides/10.02%3A_Preparing_Alkyl_Halides_from__Alkanes_-_Radical_Halogenation	Skip to main content 10.2: Preparing Alkyl Halides from Alkanes - Radical Halogenation Last updated : Mar 18, 2024 Save as PDF 10.1: Names and Properties of Alkyl Halides 10.3: Preparing Alkyl Halides from Alkenes - Allylic Bromination Page ID : 31495 Jim Clark, Steven Farmer, Dietmar Kennepohl, Layne Morsch, William Reusch, Kristen Kelley, & Britt Farquharson LibreTexts ( \newcommand{\kernel}{\mathrm{null}\,}) Objectives After completing this section, you should be able to explain why the radical halogenation of alkanes is not usually a particularly good method of preparing pure samples of alkyl halides. 2. use bond energies to account for the fact that in radical chlorinations, the reactivity of hydrogen atoms decreases in the order 3. predict the approximate ratio of the expected products from the monochlorination of a given alkane. Study Notes The following terms are synonymous: methyl hydrogens, primary hydrogens, and 1° hydrogens. methylene hydrogens, secondary hydrogens, and 2° hydrogens. methine hydrogens, tertiary hydrogens, and 3° hydrogens. Note that in radical chlorination reactions, the reactivity of methine, methylene and methyl hydrogens decreases in the ratio of approximately 5 : 3.5 : 1. This will aid in the prediction of expected products from the monochlorination of a given alkane. Radical Halogenation Alkanes (the simplest of all organic compounds) undergo very few reactions. One of these reactions is halogenation , or the substitution of a single hydrogen on the alkane for a single halogen (Cl2 or Br2) to form a haloalkane. This reaction is very important in organic chemistry because it functionalizes alkanes which opens a gateway to further chemical reactions. General Reaction Radical Chain Mechanism The reaction proceeds through the radical chain mechanism which is characterized by three steps: initiation, propagation, and termination. Initiation requires an input of energy but after that the reaction is self-sustaining. Step 1: Initiation During the initiation step free radicals are created when ultraviolet light or heat causes the X-X halogen bond to undergo homolytic to create two halogen free radicals. It is important to note that this step is not energetically favorable and cannot occur without some external energy input. After this step, the reaction can occur continuously (as long as reactants provide) without input of more energy. Step 2: Propagation The next two steps in the mechanism are called propagation steps. In the first propagation step , a chlorine radical abstracts hydrogen atom from methane. This gives hydrochloric acid (HCl, the inorganic product of this reaction) and the methyl radical . In the second propagation step , the methyl radical reacts with more of the chlorine starting material (Cl2). One of the chlorine atoms becomes a radical and the other combines with the methyl radical to form the alkyl halide product. Step 3: Termination In the three termination steps of this mechanism , radicals produced in the mechanism an undergo radical coupling to form a sigma bond. These are called termination steps because a free radical is not produced as a product, which prevents the reaction from continuing. Combining the two types of radicals produced can be combined to from three possible products. Two chlorine radicals and couple to form more halogen reactant (Cl2). A chlorine radical and a methyl radical can couple to form more product (CH3Cl). An finally, two methyl radicals can couple to form a side product of ethane (CH3CH3). This reaction is a poor synthetic method due to the formation of polyhalogenated side products. The desired product occurs when one of the hydrogen atoms in the methane has been replaced by a chlorine atom. However, the reaction doesn't stop there, and all the hydrogens in the methane can in turn be replaced by chlorine atoms to produce a mixture of chloromethane, dichloromethane, trichloromethane and tetrachloromethane. Energetics Why do these reactions occur? Is the reaction favorable? A way to answer these questions is to look at the change in enthalpy ΔH that occurs when the reaction takes place. ΔH = (Energy put into reaction) – (Energy given off from reaction) If more energy is put into a reaction than is given off, the ΔH is positive, the reaction is endothermic and not energetically favorable. If more energy is given off in the reaction than was put in, the ΔH is negative, the reaction is said to be exothermic and is considered favorable. The figure below illustrates the difference between endothermic and exothermic reactions. ΔH can also be calculated using bond dissociation energies (ΔH°): Let’s look at our specific example of the chlorination of methane to determine if it is endothermic or exothermic : Since, the ΔH for the chlorination of methane is negative, the reaction is exothermic . Energetically this reaction is favorable. In order to better understand this reaction we need to look at the mechanism ( a detailed step by step look at the reaction showing how it occurs) by which the reaction occurs. Chlorination of Other Alkanes When alkanes larger than ethane are halogenated, isomeric products are formed. Thus chlorination of propane gives both 1-chloropropane and 2-chloropropane as mono-chlorinated products. The halogenation of propane discloses an interesting feature of these reactions. All the hydrogens in a complex alkane do not exhibit equal reactivity. For example, propane has eight hydrogens, six of them being structurally equivalent primary, and the other two being secondary. If all these hydrogen atoms were equally reactive, halogenation should give a 3:1 ratio of 1-halopropane to 2-halopropane mono-halogenated products, reflecting the primary/secondary numbers. This is not what we observe. Light-induced gas phase chlorination at 25 ºC gives 45% 1-chloropropane and 55% 2-chloropropane. CH3-CH2-CH3 + Cl2 → 45% CH3-CH2-CH2Cl + 55% CH3-CHCl-CH3 These results suggest strongly that 2º-hydrogens are inherently more reactive than 1º-hydrogens, by a factor of about 3.5:1. Further experiments showed that 3º-hydrogens are about 5 times more toward halogen atoms 1º-hydrogens. Thus, light-induced chlorination of 2-methylpropane gave predominantly (65%) 2-chloro-2-methylpropane, the substitution product of the sole 3º-hydrogen, despite the presence of nine 1º-hydrogens in the molecule . (CH3)3CH + Cl2 → 65% (CH3)3CCl + 35% (CH3)2CHCH2Cl The Relative Reactivity of Hydrogens to Radical Chlorination This difference in reactivity can only be attributed to differences in C-H bond dissociation energies. In our previous discussion of bond energy we assumed average values for all bonds of a given kind, but now we see that this is not strictly true. In the case of carbon-hydrogen bonds, there are significant differences, and the specific dissociation energies (energy required to break a bond homolytically) for various kinds of C-H bonds have been measured. These values are given in the following table. \| R (in R –H) \| methyl \| ethyl \| i-propyl \| t-butyl \| --- --- \| Bond Dissociation Energy (kcal/mole) \| 103 \| 98 \| 95 \| 93 \| This data shows that a tertiary C-H bond (93 kcal/mole) is easier to break than a secondary (95 kcal/mole) and primary (98 kcal/mole) C-H bond. These bond dissociation energies can be used to estimate the relative stability of the radicals formed after homolytic cleavage. Because a tertiary C-H bond requires less energy to undergo homolytic cleavage than a secondary or primary C-H bond, it can be inferred that a tertiary radical is more stable than secondary or primary. Relative Stability of Free Radicals Exercise Write out the complete mechanism for the chlorination of methane. Answer : The answer to this problem is actually above in the initiation, propagation and termination descriptions. Exercise Explain, in your own words, how the first propagation step can occur without input of energy if it is energetically unfavorable. Answer : Since the second step in propagation is energetically favorable and fast, it drives the equilibrium toward products, even though the first step is not favorable. Exercise Which step of the radical chain mechanism requires outside energy? What can be used as this energy? Answer : Initiation step requires energy which can be in the form of light or het. Exercise Having learned how to calculate the change in enthalpy for the chlorination of methane apply your knowledge and using the table provided below calculate the change in enthalpy for the bromination of ethane. \| \| \| --- \| \| Compound \| Bond Dissociation Energy (kcal/mol) \| \| CH3CH2-H \| 101 \| \| CH3CH2-Br \| 70 \| \| H-Br \| 87 \| \| Br2 \| 46 \| Answer : To calculate the enthalpy of reaction, you subtract the BDE of the bonds formed from the BDE of the bonds broken. Bonds broken are C-H and Br-Br. Bonds formed are H-Br adn C-Br. Bonds broken - bonds formed = change in enthalpy (101 kcal/mol + 46 kcal/mol) - (87 kcal/mol + 70 kcal/mol) = change in enthalpy -10 kcal/mol = change in enthalpy for bromination of ethane. Exercise 1) Predict the mono-substituted halogenated product(s) of chlorine gas reacting with 2-methylbutane. 2) Predict the relative amount of each mono-brominated product when 3-methylpentane is reacted with Br2. Consider 1°, 2°, 3° hydrogen. 3) For the following compounds, give all possible monochlorinated derivatives. Answer 10.1: Names and Properties of Alkyl Halides 10.3: Preparing Alkyl Halides from Alkenes - Allylic Bromination
15108	https://www.imaios.com/en/e-anatomy/anatomical-structures/axilla-1536887744	MY ACCOUNT Articles talking about IMAIOS and its products What our users say about us Our commitments Get help with your subscription, account and more Human anatomy 2 Human anatomy 1 Human neuroanatomy Axilla Axilla Definition Muhammad A. Javaid The axilla, also known as the arm pit, is a pyramid-shaped space situated below the shoulder joint. Its apex, called the axillary inlet, faces upwards, while the base, known as the floor, points downward. Inside the axilla there are numerous vessels, nerves, lymph nodes, and adipose tissue. It serves as a pathway for nerves and vessels to travel from the neck to the arm. The axillary inlet has a triangular shape and is bounded by the first rib on the medial side, the clavicle anteriorly, and the superior border of the scapula and coracoid process posteriorly. The trunks of brachial plexus, subclavian artery, and vein cross over the first rib as they pass through the axillary inlet into the axillary space. Inside the axilla, the trunks branch into divisions and cords, while the subclavian vessels transform into the axillary artery and vein. The axillary space is surrounded by anterior, posterior, medial, and lateral walls, which are formed by muscles or bone. For example: The pectoralis major and minor muscles form the anterior wall, and the pectoralis major also creates the anterior axillary fold. The subscapularis, teres minor, teres major, latissimus dorsi, and long head of triceps muscles form the posterior wall. The serratus anterior muscle and upper ribs form the medial wall. The intertubercular sulcus of the humerus forms the lateral wall. Please note that the short head of the biceps and coracobrachialis, both inserting into the coracoid process of the scapula, also pass through and ascend in the axilla. The main artery in the axilla is the axillary artery, which is a continuation of the subclavian artery. It exits the axilla (distal to the teres major muscle) to enter the arm where it becomes the brachial artery. The primary nerve structure within the axilla is the brachial plexus, formed by the C5-T1 roots (or anterior rami) of spinal nerves. Its branches provide innervation to the axillary, shoulder, and upper limb regions. The innervation of axillary muscles is as follows: The pectoralis major and minor muscles are innervated by the medial and lateral pectoral nerves (branches of the medial and lateral cords of the brachial plexus). The serratus anterior muscle is innervated by the long thoracic nerve (derived from the C5, C6, C7 roots of the brachial plexus). The coracobrachialis and short head of the biceps femoris muscles are innervated by the musculocutaneous nerve (a branch of the medial cord of the brachial plexus). The subscapularis muscle is Innervated by the upper and lower subscapular nerves (branches of the posterior cord of the brachial plexus). The teres major muscle is Innervated by the lower subscapular nerve (a branch of the posterior cord). The latissimus dorsi muscle is Innervated by the thoracodorsal nerve (a branch of the posterior cord). References Gordon, A. and Alsayouri, K. Anatomy, Shoulder and Upper Limb, Axilla. [Updated 2022 Jul 25]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2022 Jan-. Available from: Drake, R.L., Vogl, A.W. and Mitchell, A.W.M. (2009). ‘Chapter 7: Upper Limb’ in Gray’s anatomy for Students. (2nd ed.) Philadelphia PA 19103-2899: Elsevier, pp. 684-710. Gordon, A. and Alsayouri, K. Anatomy, Shoulder and Upper Limb, Axilla. [Updated 2022 Jul 25]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2022 Jan-. Available from: Drake, R.L., Vogl, A.W. and Mitchell, A.W.M. (2009). ‘Chapter 7: Upper Limb’ in Gray’s anatomy for Students. (2nd ed.) Philadelphia PA 19103-2899: Elsevier, pp. 684-710. Gallery Human anatomy 2 Underlying structures: Human anatomy 1 Underlying structures: There are no anatomical children for this anatomical part Spotted a mistake? Don't hesitate to suggest a correction, translation or content improvement. Report a problem Your feedback helps us to improve the content. Do not hesitate to suggest a correction, we will examine it carefully. Please could you describe the error IMAIOS is a company which aims to assist and train human and animal practitioners. Serving healthcare professionals through interactive anatomy atlases, medical imaging, collaborative database of clinical cases, online courses... Cookie preferences IMAIOS and selected third parties, use cookies or similar technologies, in particular for audience measurement. 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15109	https://www.m-hikari.com/ams/ams-2014/ams-169-172-2014/sustekAMS169-172-2014.pdf	Applied Mathematical Sciences, Vol. 8, 2014, no. 172, 8601 - 8609 HIKARI Ltd, www.m-hikari.com On Hessians of Composite Functions ˇ Cestm´ ır B´ arta, Martin Kol´ aˇ r and Jan ˇ Sustek Department of Mathematics, Faculty of Science The University of Ostrava, 701 03 Ostrava Czech Republic Copyright c ⃝2014 ˇ Cestm´ ır B´ arta, Martin Kol´ aˇ r and Jan ˇ Sustek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we will study the Hessian matrices of composite func-tions F which are in the form F(x) = f(g(x)) with f : R →R and g: Rn →R. We present an explicit formulas for the Hessian of F and for the inverse of the Hessian matrix. Mathematics Subject Classiﬁcation: 26B05 Keywords: Hessian, composite function 1 Introduction For a function f : Rn →R the Hessian matrix Hf(x) is deﬁned as the n × n matrix of the second-order partial derivatives, Hf(x) =     ∂2f ∂x2 1 . . . ∂2f ∂x1∂xn . . . ... . . . ∂2f ∂xn∂x1 . . . ∂2f ∂x2 n    . The determinant of the above matrix is called Hessian. The Hessian matrix plays an important role in many areas of mathematics. It is important in neural computing, for instance several non-linear optimization algorithms used for training neural networks are based on considerations of the second-order properties of the error surface, which are controlled by the Hessian matrix . 8602 ˇ Cestm´ ır B´ arta, Martin Kol´ aˇ r and Jan ˇ Sustek The inverse of the Hessian matrix has been used to identify the least signiﬁcant weights in a network as a part of network ‘pruning’ algorithms, or it can also be used to assign error bars to the predictions made by a trained network . Hessian matrices are also applied in Morse theory . Throughout this paper we will study the Hessian matrices of composite functions F which are in the form F(x) = f(g(x)) with f : R →R and g: Rn → R. We present an explicit formulas for the Hessian of F and for the inverse of the Hessian matrix. In previous years, several papers were published with such formulas, but only for special functions F, see e.g. or . These results follow easily from our general results. We will use the following notation for simple work with vectors and matrices col a(i) =    a(1) . . . a(n)   , row a(i) = a(1) . . . a(n) , mat a(i, j) =    a(1, 1) . . . a(1, n) . . . ... . . . a(n, 1) . . . a(n, n)   , diag a(i) =    a(1) 0 ... 0 a(n)   . Using this notation we have ∇g(x) = col ∂g ∂xi and I = diag 1 = mat δij , where δij is the Kronecker symbol, i.e. δii = 1 and δij = 0 for i ̸= j. 2 Results Our ﬁrst result concerns the determinant of the Hessian matrix. Theorem 1 Let f : R →R and g: Rn →R be real functions and let F(x) = f(g(x)) be their composition. Then the Hessian of the function F equals det HF(x) = 1 + f ′′(g(x)) f ′(g(x)) ∇g(x)T · Hg(x)−1 · ∇g(x) f ′(g(x))n det Hg(x) . The following result concerns the inverse of the Hessian matrix. Theorem 2 Let f : R →R and g: Rn →R be real functions and let F(x) = f(g(x)) be their composition. Then the inverse of the Hessian matrix of the function F equals HF(x)−1 = I − Hg(x)−1 · ∇g(x) · ∇g(x)T f′(g(x)) f′′(g(x)) + ∇g(x)T · Hg(x)−1 · ∇g(x) · Hg(x)−1 f ′(g(x)) . On Hessians of composite functions 8603 Theorem 3 is a consequence of previous theorems for a particular class of functions g. Theorem 3 Let αi ∈R for i = 1, . . . , n and let f : R →R be a real function. Put g(x) = n Q k=1 xαk k and F(x) = f(g(x)). Then the Hessian of F equals det HF(x) = 1+g(x)f ′′(g(x)) f ′(g(x)) \|α\| \|α\| −1 f ′(g(x))n(−1)n+1(\|α\|−1) n Y k=1 αkxnαk−2 k , where \|α\| = n P k=1 αk. The inverse of the Hessian matrix is HF(x)−1 = 1 g(x)f ′(g(x)) mat 1 \|α\| −1− 1 f′(g(x)) g(x)f′′(g(x))(\|α\| −1)2 + \|α\|(\|α\| −1) −δij αi xixj . Example 1 Chen considered function of the form F(x) = f n P k=1 hk(xk) and computed inductively its Hessian. We obtain the same result directly using Theorem 1. We have g(x) = n P k=1 hk(xk). Hence ∇g(x) = col h′ i(xi) , Hg(x) = diag h′′ i (xi) , det Hg(x) = n Y k=1 h′′ k(xk) , Hg(x)−1 = diag 1 h′′ i (xi) . Then Theorem 1 implies that det HF(x) = 1 + f ′′(g(x)) f ′(g(x)) row h′ i(xi) · diag 1 h′′ i (xi) · col h′ i(xi) f ′(g(x))n n Y k=1 h′′ k(xk) = 1 + f ′′(g(x)) f ′(g(x)) n X k=1 h′ k(xk)2 h′′ k(xk) f ′(g(x))n n Y k=1 h′′ k(xk) . Moreover, using Theorem 2, we obtain the inverse of the Hessian matrix. HF(x)−1 = I − diag 1 h′′ i (xi) · col h′ i(xi) · row h′ i(xi) f′(g(x)) f′′(g(x)) + row h′ i(xi) · diag 1 h′′ i (xi) · col h′ i(xi) diag 1 h′′ i (xi) f ′(g(x)) = mat δij − h′ i(xi)h′ j(xj) h′′ i (xi) f′(g(x)) f′′(g(x)) + n P k=1 h′ k(x2 k) h′′ k(xk) ! · diag 1 f ′(g(x))h′′ i (xi) = mat δij − h′ i(xi)h′ j(xj) h′′ i (xi) f′(g(x)) f′′(g(x)) + n P k=1 h′ k(x2 k) h′′ k(xk) ! 1 f ′(g(x))h′′ j(xj) 8604 ˇ Cestm´ ır B´ arta, Martin Kol´ aˇ r and Jan ˇ Sustek Example 2 Trojovsk´ y and Hlad´ ıkov´ a considered function F(x) = exp n Q k=1 xk and computed its Hessian as the sum of 2n simpler determinants. We obtain the same result directly using Theorem 3. We have f(t) = et and αi = 1 for every i. Hence f ′(t) = f ′′(t) = et. Then Theorem 3 implies that det HF(x) = 1 + g(x) n n −1 eng(x)(−1)n+1(n −1)g(x)n−2 = (−1)n+1 n −1 + n n Y k=1 xk e n n Q k=1 xk n Y k=1 xn−2 k . Moreover we obtain the inverse of the Hessian matrix. HF(x)−1 = 1 g(x)eg(x) mat 1 n −1 − 1 (n−1)2 g(x) + n(n −1) −δij xixj = mat 1 e n Q k=1 xk n Q k=1 xk 1 n −1 − n Q k=1 xk (n −1)2 + n(n −1) n Q k=1 xk −δij xixj Example 3 Consider the function F(x) = n s n Q k=1 xk, i.e. the geometric mean. Using the notation of Theorem 3, we have f(t) = n √ t and αi = 1 for every i. Hence f ′(t) = 1 nt 1 n −1 and f ′′(t) = 1 n 1 n −1 t 1 n −2. Then Theorem 3 implies that det HF(x) = 1+g(x) 1 n 1 n −1 g(x) 1 n −2 1 ng(x) 1 n −1 n n −1 1 nng(x)1−n(−1)n+1(n−1)g(x)n−2 = 0 . 3 Proofs In our proofs we will need the following two lemmas. We present them also with their short proofs. Lemma 1 (Matrix determinant lemma) (, Lemma 1.1) Let A be an invert-ible n × n matrix and let u, v ∈Rn be column vectors. Then det(A + uvT) = (1 + vTA−1u) det A . Proof. By a direct computation we easily obtain the identity A 0 vT 1 I + A−1uvT A−1u 0T 1 I 0 −vT 1 = A u 0T 1 + vTA−1u . On Hessians of composite functions 8605 Taking determinants of both sides we immediately obtain det(A + uvT) = det A · det(I + A−1uvT) = det A · (1 + vTA−1u) . □ Lemma 2 (Sherman-Morrison formula) (, Section 2.7.1) Let A be an in-vertible n × n matrix and let u, v ∈Rn be column vectors. Suppose that the matrix A + uvT is invertible. Then (A + uvT)−1 = I − A−1uvT 1 + vTA−1u A−1 . Proof. From the assumption and Lemma 1 it follows that 1 + vTA−1u ̸= 0. Then I = I + A−1uvT −A−1uvT + (vTA−1u)A−1uvT 1 + vTA−1u = I + A−1uvT −A−1uvT + A−1u(vTA−1u)vT 1 + vTA−1u = A−1(A + uvT) −A−1uvTA−1 1 + vTA−1u (A + uvT) = A−1 −A−1uvTA−1 1 + vTA−1u (A + uvT) We obtain the result by multiplying by (A + uvT)−1. □ Using these lemmas we now prove our theorems. Proof of Theorem 1. The second-order partial derivatives of F are ∂2F ∂xi∂xj = f ′′(g(x)) ∂g ∂xi (x) ∂g ∂xj (x) + f ′(g(x)) ∂2g ∂xi∂xj (x) . (1) Hence the Hessian matrix can be written as HF(x) = A + uvT, where A = f ′(g(x))Hg(x) , u = f ′′(g(x))∇g(x) , v = ∇g(x) . (2) From Lemma 1 we obtain det HF(x) = 1+∇g(x)T· f ′(g(x))Hg(x) −1·f ′′(g(x))∇g(x) det f ′(g(x))Hg(x) = 1 + f ′′(g(x)) f ′(g(x)) ∇g(x)T · Hg(x)−1 · ∇g(x) f ′(g(x))n det Hg(x) . □ 8606 ˇ Cestm´ ır B´ arta, Martin Kol´ aˇ r and Jan ˇ Sustek Proof of Theorem 2. As in the proof of Theorem 1 we obtain (1) and (2). Then Lemma 2 implies HF(x)−1 = I − f ′(g(x))Hg(x) −1 · f ′′(g(x))∇g(x) · ∇g(x) T 1 + ∇g(x) T · f ′(g(x))Hg(x) −1 · f ′′(g(x))∇g(x) · f ′(g(x))Hg(x) −1 = I − Hg(x)−1 · ∇g(x) · ∇g(x)T f ′(g(x)) f ′′(g(x)) + ∇g(x)T · Hg(x)−1 · ∇g(x) · Hg(x)−1 f ′(g(x)) . □ Proof of Theorem 3. The derivatives of the function g are ∂g ∂xi = αi xi g(x) , ∂2g ∂x2 i = αi(αi −1) x2 i g(x) , ∂2g ∂xi∂xj = αiαj xixj g(x) . Therefore its gradient is equal to ∇g(x) = g(x) col αi xi and its Hessian matrix is Hg(x) = mat αiαj xixj −diag αi x2 i g(x) = −g(x)(A + uvT) , where A = diag αi x2 i , u = −col αi xi , v = col αi xi . Then Lemma 1 implies that det Hg(x) = (−g(x))n 1 −row αi xi · diag x2 i αi · col αi xi n Y k=1 αk x2 k = (−g(x))n(1 −\|α\|) n Y k=1 αk x2 k = (−1)n(1 −\|α\|) n Y k=1 αkxnαk−2 k . (3) Lemma 2 implies that Hg(x)−1 = −1 g(x) diag x2 i αi + diag xi αi · col αi xi · row αi xi · diag x2 i αi 1 −row αi xi · diag x2 i αi · col αi xi ! = −1 g(x) diag x2 i αi + mat xixj 1 −\|α\| . (4) On Hessians of composite functions 8607 We will use the following identity row αi xi · diag x2 i αi · col αi xi + row αi xi · mat xixj 1 −\|α\| · col αi xi = \|α\| + \|α\|2 1 −\|α\| = \|α\| 1 −\|α\| . (5) Then (3), (4), (5) and Theorem 1 imply that det HF(x) = 1 + f ′′(g(x)) f ′(g(x)) g(x) row αi xi · −1 g(x) diag x2 i αi + mat xixj 1 −\|α\| · g(x) col αi xi × f ′(g(x))n(−1)n(1 −\|α\|) n Y k=1 αkxnαk−2 k = 1 −g(x)f ′′(g(x)) f ′(g(x)) row αi xi · diag x2 i αi · col αi xi + row αi xi · mat xixj 1 −\|α\| · col αi xi × f ′(g(x))n(−1)n+1(\|α\| −1) n Y k=1 αkxnαk−2 k = 1 + g(x)f ′′(g(x)) f ′(g(x)) \|α\| \|α\| −1 f ′(g(x))n(−1)n+1(\|α\| −1) n Y k=1 αkxnαk−2 k . By a direct computation, using (5), we obtain ∇g(x)T · Hg(x)−1 · ∇g(x) = g(x) row αi xi · −1 g(x) diag x2 i αi + mat xixj 1 −\|α\| · g(x) col αi xi = −g(x) row αi xi · diag x2 i αi · col αi xi + row αi xi · mat xixj 1 −\|α\| · col αi xi = g(x) \|α\| \|α\| −1 (6) and Hg(x)−1 · ∇g(x) · ∇g(x)T = −1 g(x) diag x2 i αi + mat xixj 1 −\|α\| · g(x) col αi xi · g(x) row αi xi = −g(x) diag x2 i αi · col αi xi · row αi xi + 1 1 −\|α\| mat xixj · col αi xi · row αi xi = g(x) \|α\| −1 mat αjxi xj . (7) 8608 ˇ Cestm´ ır B´ arta, Martin Kol´ aˇ r and Jan ˇ Sustek Then (6), (7) and Theorem 2 imply HF(x)−1 = I − Hg(x)−1 · ∇g(x) · ∇g(x)T f′(g(x)) f′′(g(x)) + ∇g(x)T · Hg(x)−1 · ∇g(x) ! · Hg(x)−1 f ′(g(x)) = I − g(x) \|α\|−1 mat αjxi xj f′(g(x)) f′′(g(x)) + g(x) \|α\| \|α\|−1 ! · −1 g(x)f ′(g(x)) diag x2 i αi + mat xixj 1 −\|α\| = 1 g(x)f ′(g(x)) β mat αjxi xj −I · diag x2 i αi + mat xixj 1 −\|α\| , (8) where β = g(x) \|α\|−1 f′(g(x)) f′′(g(x)) + g(x) \|α\| \|α\|−1 . Now we have β mat αjxi xj · diag x2 i αi = β mat xixj , (9) β mat αjxi xj · mat xixj 1 −\|α\| = β\|α\| 1 −\|α\| mat xixj , (10) −I · diag x2 i αi = −mat δij αi xixj , (11) −I · mat xixj 1 −\|α\| = 1 \|α\| −1 mat xixj . (12) From the identity β + β\|α\| 1−\|α\| = β 1−\|α\| we obtain that (9) + (10) = −1 f′(g(x)) g(x)f′′(g(x))(\|α\| −1)2 + \|α\|(\|α\| −1) mat xixj . (13) Finally, (8), (11), (12) and (13) imply that HF(x)−1 = 1 g(x)f ′(g(x)) mat 1 \|α\| −1− 1 f′(g(x)) g(x)f′′(g(x))(\|α\| −1)2 + \|α\|(\|α\| −1) −δij αi xixj . □ Acknowledgement The authors were supported by grant GAˇ CR P201/12/2351 of the Czech Sci-ence Foundation and by grant SGS08/Pˇ rF/2014 of the University of Ostrava. On Hessians of composite functions 8609 References Ch.M. Bishop. Neural networks for pattern recognition. Oxford University Press, 1995. B.-Y. Chen. An Explicit Formula of Hessian Determinants of Composite Functions and Its Applications. Kragujevac Journal of Mathematics, 36 (1), 2012, 27–39. J. Ding, A. Zhou. Applied Mathematics Letters, 20 (12), 2007, 1223–1226. J. Milnor. Morse theory. Princeton University Press, 1969. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery. Numerical Recipes: The Art of Scientiﬁc Computing (3rd ed.), Cambridge University Press, New York, 2007. P. Trojovsk´ y, E. Hlad´ ıkov´ a. On the Hessian of the Exponential Function with n Variables. International Journal of Pure and Applied Mathematics, 66 (3), 2011, 287–296. Received: October 19, 2014; Published: December 1, 2014
15110	https://www.sigmaaldrich.com/US/en/product/sigald/161527?srsltid=AfmBOop9zRC1VfAVz1YK6yFH1WfF6sAsqZTEvj5SJmkjPiUF7Xw6DjE4	Sodium hydrosulfide 207683-19-0 Skip to Content Products Cart 0 IL EN Products Products Applications Services Documents Support Analytical ChemistryCell Culture & AnalysisChemistry & BiochemicalsClinical & DiagnosticsFiltration Greener Alternative Products Industrial MicrobiologyLab AutomationLabwareMaterials ScienceMolecular Biology & Functional GenomicsmAbs Development & ManufacturingmRNA Development & ManufacturingPharma & Biopharma ManufacturingProtein BiologyWater Purification Analytical ChemistryCell Culture & AnalysisChemistry & SynthesisClinical & DiagnosticsEnvironmental & Cannabis TestingFood & Beverage Testing & ManufacturingGenomicsMaterials Science & EngineeringMicrobiological TestingmAbs Development & ManufacturingmRNA Development & ManufacturingPharma & Biopharma ManufacturingProtein BiologyResearch & Disease AreasWater Purification Contract ManufacturingContract TestingCustom ProductsDigital Solutions for Life ScienceIVD Development & ManufacturingProduct ServicesSupport Safety Data Sheets (SDS) Certificates of Analysis (COA) Certificates of Origin (COO) Certificates of Quality (COQ) Customer Support Contact Us Get Site Smart FAQ Quality & Regulatory Calculators & Apps Webinars Login Order Lookup Quick Order Cart 0 Products Lab Chemicals 161527 +1 Key Documents SDS COA COO Specification Sheet View All Documentation Skip To Compare Similar Items Safety Information Documentation Peer Reviewed Papers Questions & Answers Reviews 161527 Share Sodium hydrosulfide hydrate Sign Into View Organizational & Contract Pricing Select a Size Change View About This Item Linear Formula: NaSH·xH 2 O CAS Number: 207683-19-0 Molecular Weight: 56.06 (anhydrous basis) MDL number: MFCD00149796 UNSPSC Code: 12352302 PubChem Substance ID: 329751132 NACRES: NA.55 Form: chips flakes Technical Service Need help? Our team of experienced scientists is here for you. Let Us Assist Recommended Products Slide 1 of 10 1 of 3 Z2000010 Zinc acexamate impurity A Quick View Sigma-Aldrich 225541 Diisopropyl azodicarboxylate Quick View Sigma-Aldrich 207942 Hydrazine monohydrate Quick View Sigma-Aldrich A8625 N-Acetyl-D-glucosamine Quick View Sigma-Aldrich W332208 Thiamine hydrochloride Quick View Sigma-Aldrich B17905 Benzyl bromide Quick View Sigma-Aldrich 62528 Lithium hydroxide monohydrate Quick View Sigma-Aldrich 450197 Lithium hydroxide monohydrate Quick View Sigma-Aldrich L4533 Lithium hydroxide monohydrate Quick View Sigma-Aldrich 205249 Di-tert-butyl dicarbonate Quick View Properties form chips flakes Quality Level 100 concentration ≥60% (by Na 2 S 2 O 3, titration) mp 52-54°C (lit.) SMILES string [Na]S.[H]O[H] InChI 1S/Na.H2O.H2S/h;21H2/q+1;;/p-1 InChI key ZNKXTIAQRUWLRL-UHFFFAOYSA-M Looking for similar products? Visit Product Comparison Guide Related Categories Lab Chemicals Salts Compare Similar Items View Full Comparison Show Differences [x] 1 of 1 \| ###### This Item \| 157953 \| 27029 \| 464422 \| --- --- \| \| Sigma-Aldrich 161527 Sodium hydrosulfide hydrate Quick View \| Sigma-Aldrich 157953 Sodium hydrosulfite Quick View \| Sigma-Aldrich 27029 Sodium cholate hydrate Quick View \| Sigma-Aldrich 464422 2-Ketobutyric acid sodium salt hydrate Quick View \| \| form chips, flakes \| form powder \| form powder \| form powder \| \| Quality Level 100 \| Quality Level 200 \| Quality Level 200 \| Quality Level 100 \| \| mp 52-54°C (lit.) \| mp - \| mp - \| mp 210°C (dec.) (lit.) \| \| concentration ≥60% (by Na 2 S 2 O 3, titration) \| concentration - \| concentration - \| concentration - \| Description General description Sodium hydrosulfide hydrate is a hydrated inorganic salt of sodium. It participates in the synthesis of (E)-2-cyano-2-(thiazolidin-2-ylidene)ethanethioamide. Application It may be used as a sulfur nucleophile to induce the C-S bond formation in α,β-dichloro vinyl ketones to form 5- to 8-membered cyclic thioethers. Sodium hydrosulfide hydrate may be used in the synthesis of following: benzothiazole 4-methoxybenzothioamide 2-(4-methoxyphenyl)imidazoline 7-chloro-4′-methoxythioflavone Safety Information Pictograms GHS06,GHS05,GHS09 Signal Word Danger Hazard Statements H301,H314,H400 Precautionary Statements P260 - P273 - P280 - P301 + P310 + P330 - P303 + P361 + P353 - P305 + P351 + P338 Hazard Classifications Acute Tox. 3 Oral - Aquatic Acute 1 - Skin Corr. 1B Storage Class Code 6.1D - Non-combustible acute toxic Cat.3 / toxic hazardous materials or hazardous materials causing chronic effects WGK WGK 3 Flash Point(F) 194.0 °F - closed cup Flash Point(C) 90 °C - closed cup Documentation SDS Specification Sheet Certificate of Analysis Certificate of Origin Choose from one of the most recent versions: Certificates of Analysis (COA) Lot/Batch Number SHBR6131 SHBR3997 SHBQ7513 SHBR0193 SHBP0761 Don't see the Right Version? If you require a particular version, you can look up a specific certificate by the Lot or Batch number. Search for a COA Already Own This Product? Find documentation for the products that you have recently purchased in the Document Library. Visit the Document Library Customers Also Viewed Slide 1 of 3 1 of 1 Sigma-Aldrich SML0100 GYY4137 Dichloromethane complex Quick View Sigma-Aldrich 307823 Sodium hydrogen sulfate Quick View Sigma-Aldrich T62405 Silver trifluoroacetate Quick View Peer Reviewed Papers Practical and Versatile Synthesis of Thioflavones from 2-Bromobenzoyl Chlorides. Lee JI and Choi JS. J. Korean Chem. Soc., 59(3) (2015) Design, synthesis, and in vitro antitumor evaluation of novel diaryl ureas derivatives. Min Sun et al. European journal of medicinal chemistry, 45(6), 2299-2306 (2010-02-26) Two series of novel diaryl ureas have been designed and synthesized, with their in vitro antitumor effect screened on human non-small cell lung cancer (NSCLC) cell line A549 and human breast cancer cell line MDA-MB-231. Some target compounds demonstrated significant Synthesis of Benzothiazoles through Copper-Catalyzed One-Pot Three-Component Reactions with Use of Sodium Hydrosulfide as a Sulfur Surrogate. Park N, et al. European Journal of Organic Chemistry, 10, 1984-1993 (2012) Inhalation of sodium hydrosulfide (NaHS) alleviates NO2-induced pulmonary function and hematological impairment in rats. Zili Zhang et al. Life sciences, 232, 116650-116650 (2019-07-16) Inhalation of NO2 leads to a progressive airflow limitation and the development of emphysema-like lesions. We report on the efficacy of hydrogen sulfide (NaHS) for alleviating NO2-induced pulmonary impairment. Sprague Dawley rats were exposed to 20 ppm NO2 for 6 h over Exogenous hydrogen sulfide mitigates NLRP3 inflammasome-mediated inflammation through promoting autophagy via the AMPK-mTOR pathway. Honggang Wang et al. Biology open, 8(7) (2019-07-19) The aim of this study was to investigate whether exogenous hydrogen sulfide (H2S) could mitigate NLRP3 inflammasome-mediated inflammation through promoting autophagy via the AMPK-mTOR pathway in L02 cells. L02 cells were stimulated with different concentrations of oleic acid (OA), then View All Related Papers Technical Service Our team of scientists has experience in all areas of research including Life Science, Material Science, Chemical Synthesis, Chromatography, Analytical and many others. 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All Posts Psychiatrist ADHD Transcranial Magnetic Stimulation Medical Marijuana Evaluation RI Mental Health Therapy LGBTQ+ BIPOC Anxiety Communication Parenting college student Navigating Loneliness Building connections depression Suicide Prevention Bipolar Disorder Holidays Veterans PTSD Setting goals Mental wellness New Year's resolutions Seasonal Affective Disorder Teen Mental Health Depression Panic Attacks Black Mental Health mental health Genesight Testing Search Does Weather Affect Your Mood--The Science Behind Weather and Mood Revive Therapeutic Services Jan 21 3 min read Updated: Feb 4 As spring approaches and daylight stretches longer into the evening, many of us begin to notice a shift—not just in the seasons, but in our moods. The warm breeze, blossoming flowers, and extended sunlight bring a renewed sense of energy and positivity. But why does this happen? How exactly does weather—and specifically daylight—affect our mood? In this blog, we’ll explore the fascinating connection between weather, daylight, and our mood, and answer the question: does weather affect mood? The Science Behind Weather Affecting Mood For centuries, people have believed in the profound impact weather has on emotions. Modern science confirms this, with numerous studies revealing the intricate link between environmental factors like sunlight, temperature, and even humidity, and our mental health. One of the most significant influences is daylight, which plays a crucial role in regulating our circadian rhythms—the natural 24-hour cycles that govern sleep, energy levels, and mood. When we experience more daylight, our brains produce less melatonin, a hormone that induces sleepiness, and more serotonin, often referred to as the “happiness hormone.” This biochemical shift explains why longer days in spring and summer often lead to improved mood, increased energy, and a greater sense of well-being. Seasonal Affective Disorder (SAD) On the flip side, the lack of daylight during winter months can lead to Seasonal Affective Disorder (SAD), a form of depression that affects millions of people worldwide. Symptoms of SAD include low energy, difficulty concentrating, and feelings of sadness or hopelessness. As spring approaches and daylight hours increase, many individuals with SAD notice a significant improvement in their symptoms, underscoring the powerful relationship between sunlight and mental health. The Role of Weather in Affecting Our Mood While daylight is a key factor, other weather elements can also influence mood. Warm, sunny days are often associated with feelings of happiness and relaxation, while gloomy, rainy days might evoke sadness or introspection. Interestingly, moderate temperatures have been linked to higher productivity and creativity, suggesting that weather’s impact extends beyond just mood. However, it’s important to note that individual responses to weather can vary. While some people thrive in the heat of summer, others feel their best during cooler autumn days. Personal preferences, lifestyle, and even cultural factors can shape how weather affects each person’s mood. Daylight and Its Impact on Mental Health One of the most striking effects of increased daylight is its ability to combat symptoms of depression and anxiety. Exposure to natural light has been shown to: Boost Vitamin D Levels:Sunlight helps our bodies produce Vitamin D, which is essential for bone health and has been linked to improved mood. Improve Sleep Quality:By regulating melatonin production, daylight helps align our sleep-wake cycles, leading to more restful sleep. Enhance Cognitive Function:Studies suggest that exposure to natural light can improve focus, memory, and overall cognitive performance. For those who spend most of their time indoors, incorporating more natural light into daily routines can make a noticeable difference. Whether it’s taking a walk during lunch or working near a window, these small changes can help harness the mood-enhancing benefits of daylight. Tips for Embracing Spring and Its Mood-Boosting Benefits As we welcome the arrival of spring, there are several ways to make the most of the season’s longer days and better weather: Spend Time Outdoors:Engage in activities like hiking, gardening, or simply enjoying a picnic in the park to soak up natural light. Exercise Regularly:Physical activity releases endorphins, which complement the mood-enhancing effects of daylight. Practice Mindfulness:Take a moment to appreciate the beauty of spring, whether it’s listening to birdsong, admiring blooming flowers, or feeling the warmth of the sun. Maintain a Balanced Routine:Align your daily schedule with natural light patterns by waking up earlier and winding down as the sun sets. Does Weather Affect Mood? Absolutely! In conclusion, the arrival of spring and the lengthening daylight hours offer a natural remedy for winter blues. Weather, and particularly daylight, undeniably influences mood, affecting everything from energy levels to mental health. By understanding and embracing this connection, we can take proactive steps to harness the positive effects of longer days and warmer weather. So as the season changes, step outside, feel the sun on your skin, and let the vibrant energy of spring elevate your mood. And remember, whether it’s a sunny afternoon or a cool, breezy morning, each moment spent in nature can contribute to a happier, healthier you. 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15112	http://hyperphysics.phy-astr.gsu.edu/hbase/Particles/neutrino.html	\| \| \| \| \| \| \| \| \| \| \| --- --- --- --- --- \| \| Electron Neutrinos and Antineutrinos The history of a particle that appeared to have no charge and no mass is an interesting one. The electron neutrino (a lepton) was first postulated in 1930 by Wolfgang Pauli to explain why the electrons in beta decay were not emitted with the full reaction energy of the nuclear transition. The apparent violation of conservation of energy and momentum was most easily avoided by postulating another particle. Enrico Fermi called the particle a neutrino and developed a theory of beta decay based on it, but it was not experimentally observed until 1956. This elusive particle, with no charge and almost no mass, could penetrate vast thicknesses of material without interaction. The mean free path of a neutrino in water would be on the order of 10x the distance from the Earth to the Sun. In the standard Big Bang model, the neutrinos left over from the creation of the universe are the most abundant particles in the universe. This remnant neutrino density is put at 100 per cubic centimeter at an effective temperature of 2K (Simpson). The background temperature for neutrinos is lower than that for the microwave background (2.7K) because the neutrino transparency point came earlier. The sun emits vast numbers of neutrinos which can pass through the earth with little or no interaction. This leads to the statement "Solar neutrinos shine down on us during the day, and shine up on us during the night!". Bahcall's modeling of the solar neutrino flux led to the prediction of about 5 x 106 neutrinos/cm2s. A remarkable opportunity for observing neutrinos came with Supernova 1987A when the Japanese observing team detected neutrinos almost coincident with the discovery of the light from the supernova. Neutrinos interact only by the weak interaction. Their interactions are usually represented in terms of Feynman diagrams. \| \| \| \| --- \| Neutrinos as leptons \| Role in supernova \| Other neutrino types \| \| \| \| --- \| \| Detection of neutrinos \| Does the neutrino have any mass? \| \| \| \| Why do we say that neutrinos are left-handed? \| \| \| \| Neutrino cross-section for interaction \| \| \| \| Neutrinos in the early universe \| \| Index References Kearns, et al. Simpson Bahcall \| \| \| \| \| --- \| \| HyperPhysics Quantum Physics \| R Nave \| \| Go Back \| \| \| \| \| \| \| --- --- \| Detection of Neutrinos The first experimental observation of the neutrino interacting with matter was made by Frederick Reines, Clyde Cowan, Jr, and collaborators in 1956 at the Savannah River Plant in South Carolina. Their neutrino source was a nuclear reactor (it actually produced antineutrinos from beta decay). \| \| \| --- \| \| \| Modern neutrino detectors at IMB in Ohio and Kamiokande in Japan detected neutrinos from Supernova 1987A. A new neutrino detector at Sudbury, Ontario began collecting data in October of 1999. Another Japanese neutrino detector called Super Kamiokande became operational in April 1996. \| An early set of experiments with a facility called the solar neutrino telescope, measured the rate of neutrino emission from the sun at only one third of the expected flux. Often referred to as the Solar Neutrino Problem, this deficiency of neutrinos has been difficult to explain. Recent results from the Sudbury Neutrino Observatory suggest that a fraction of the electron neutrinos produced by the sun are transformed into muon neutrinos on the way to the earth. The observations at Sudbury are consistent with the solar models of neutrino flux assuming that this "neutrino oscillation" is responsible for observation of neutrinos other than electron neutrinos. \| \| \| Cherenkov Radiation \| \| Index Reference McDonald, Klein & Wark \| \| \| \| \| --- \| \| HyperPhysics Quantum Physics \| R Nave \| \| Go Back \| \| \| \| \| \| \| \| --- --- --- \| \| Sudbury Neutrino Observatory The new Sudbury Neutrino Observatory (SNO) consists of a 1000 metric ton bottle of heavy water suspended in a larger tank of light water. The apparatus is located in Sudbury, Ontario, Canada at a depth of about 2 km down in a nickel mine. A 18 m diameter geodesic array of 9,500 photomultiplier tubes surrounds the heavy water to detect Cerenkov radiation from the neutrino interaction which dissociates deuterium: \| \| \| \| --- \| \| \| \| \| Show other detection reactions for SNO \| \| The distinctive characteristic of the heavy water observatory is that it can measure both the electron neutrino flux and the total neutrino flux (electron, muon and tau neutrinos). It should allow them to determine whether neutrinos change flavors. If so, it could explain the solar neutrino problem and would show that the neutrinos have mass. SNO began operating in production mode in October, 1999, and as of Summer 2000 had collected a sizable number of neutrino events both from the sun (the main focus of the experiment) and from atmospheric events with pions and muons. The Cerenkov cones of the solar neutrinos center about the direction opposite the sun, showing about the same flux at night as during the day. This was an expected result, since the mean free path of a neutrino in matter is about 22 lightyears in lead and having the earth in the path makes little difference. A sizable number of the atmospheric neutrino events come from below, having traveled all the way through the earth and forming the Cerenkov cone in the photomultiplier tubes at the top of the spherical heavy-water ball. These Cerenkov cones are scattered all around the sphere, while the solar ones of course show a precise anti-solar direction. The depth of the detector protects it from the intense bombardment of cosmic ray muons which reaches the earth's surface. The detector measures only about 70 muon events per day, and they are easily distinguished from neutrino events since the muon interacts by the electromagnetic interaction and produces a much larger signal in the detector array. In order to detect the ring of light which is the signature of Cerenkov radiation, the responses of all the photomultiplier tubes (PMTs) are monitored with a very short time scale. In order to be counted as an "event" in the detector, at least 20 PMTs must be triggered within an interval of 100 nanoseconds. \| \| \| How SNO detects neutrinos \| \| Index Reference Feder Simpson McDonald, Klein & Wark \| \| \| \| \| --- \| \| HyperPhysics Quantum Physics \| R Nave \| \| Go Back \| \| \| \| \| \| --- --- \| \| The Solar Neutrino Telescope Raymond Davis of Brookhaven National Laboratory constructed a neutrino detector 1.6 km underground in the Homestake Gold Mine in Lead, South Dakota. The detector consists of a 378,000 liter tank of perchloroethylene, which is further isolated by being submerged in water. Theoretical expections were about one neutrino-chlorine interaction per day, but the measured solar neutrino events were about a third of that, raising serious questions about the abundance of solar neutrinos (the Solar Neutrino Problem). The detection of neutrinos by this instrument was based on the interaction of neutrinos with chlorine nuclei to produce argon. The argon can be removed from the tank and measured so that the number of neutrinos captured in a given time interval can be determined. The argon decays back to the chlorine isotope from which it was created by the process of electron capture. The detection of this transition is aided by the definite energy of the x-ray emitted during the electron capture process. This mine experiment was able to detect about 15 argon atoms a month, according to Simpson. \| \| \| --- \| \| \| Perchloroethylene is ordinary dry-cleaning fluid, but 400,000 gallons is a lot of cleaning fluid. Davis denies the story that he was besieged by wire coat-hanger salesmen after the large purchase. \| \| Index Reference Simpson \| \| \| \| \| --- \| \| HyperPhysics Quantum Physics \| R Nave \| \| Go Back \| \| \| \| \| \| --- --- \| \| Detection of Supernova Neutrinos Since the neutrino can pass through the entire Earth without interaction, it takes specialized techniques to detect one. After being postulated by Pauli in 1930 to explain anomalies in beta decay, they were not actually detected until 1956 by Reines and Cowan. Detection of neutrinos is now well developed and a classic opportunity for neutrino detection occurred with Supernova 1987A. A burst of ten neutrinos was detected within a time interval of about 15 seconds at a neutrino detector deep in a mine in Japan. They had to penetrate the Earth to get to the detector. \| \| \| --- \| \| \| More detail Energies in eV \| \| Index \| \| \| \| \| --- \| \| HyperPhysics Quantum Physics \| R Nave \| \| Go Back \| \| \| \| --- \| \| Neutrino Mass? No definite mass has been measured for the neutrino, and the standard comment about most experiments is "the results are consistent with zero mass for the neutrino". But this raises certain theoretical problems and there have been many attempts to set a range for the mass of the neutrino. Since its mass is evidently very small, if non-zero, the mass is usually stated in terms of its energy equivalent in electron volts. Most experiments conclude that the mass equivalent of the neutrino is less than 50 eV. One of the recent pieces of information about neutrino mass came from the neutrinos observed from Supernova 1987A. Ten neutrinos arrived within 15 seconds of each other after traveling 180,000 light years, and they differed by a up to factor of three in energy. This limits the neutrino rest mass energy to less than about 30 eV (Rohlf). New experimental evidence from the Super-Kamiokande neutrino detector in Japan represents the strongest evidence to date that the mass of the neutrino is non-zero. Models of atmospheric cosmic ray interactions suggest twice as many muon neutrinos as electron neutrinos, but the measured ratio was only 1.3:1. The interpretation of the data suggested a mass difference between electron and muon neutrinos of 0.03 to 0.1 eV. Presuming that the muon neutrino would be much more massive than the electron neutrino, then this implies a muon neutrino mass upper bound of about 0.1 eV. The recent neutrino measurements at the Sudbury Neutrino Observatory are consistent with the modeled total neutrino flux and add evidence for neutrino oscillation, a process which can only occur if the neutrinos have mass. \| Index References Rohlf Kearns, et al. \| \| \| \| \| --- \| \| HyperPhysics Quantum Physics \| R Nave \| \| Go Back \| \| \| \| --- \| \| Some Neutrino History The electron neutrino (a lepton) was first postulated in 1930 by Wolfgang Pauli to explain why the electrons in beta decay were not emitted with the full reaction energy of the nuclear transition. The apparent violation of conservation of energy and momentum was most easily avoided by postulating another particle. Enrico Fermi called the particle a neutrino and developed a theory of beta decay based on it, but it was not experimentally observed until 1956. Wolfgang Pauli introduced the neutrino to the world of physics in 1930 with a famous letter to "Liebe Radioacktive Damen und Herren" (Dear radioactive ladies and gentlemen) at the Tubingen meeting of radioactivity researchers. Pauli's first public discussion of the neutrino was at the 7th Solvay Conference in Brussels in 1933. References: Wolfgang Pauli and Modern Physics Wiki on Wolfgang Pauli \| Index References Rohlf Kearns, et al. \| \| \| \| \| --- \| \| HyperPhysics Quantum Physics \| R Nave \| \| Go Back \|
15113	https://emorysurgicalfocus.com/2020/10/23/afferent-loop-syndrome/	Afferent loop syndrome \| Surgical Focus Skip to primary content Surgical Focus Emory University Department of Surgery Search Main menu Home About Essential Articles EUH Surgery A EUH Surgery B/EUHM/ESJH Genral Surgery Hepatobiliary Surgery Colorectal Surgery ACCS Vascular Surgery Trauma Surgery Burn Surgery Critical Care Surgical Oncology Pediatric Surgery Thoracic Surgery Breast, Melanoma, and Endocrine Surgery Transplant Surgery Post navigation ← PreviousNext → Afferent loop syndrome Posted on October 23, 2020 by E. Lawson Termsinsuk P, Chantarojanasiri T, Pausawasdi N. Diagnosis and treatment of the afferent loop syndrome.Clin J Gastroenterol. 2020 Oct;13(5):660-668. “ALS is a rare condition with the incidence ranging from 0.2 to 1.0% depending on the type of operation and anastomotic limb reconstruction. ALS has been reported in 0.3–1.0% of patients after total gastrectomy with Billroth II or Roux-en-Y reconstruction, 1% after laparoscopic distal gastrectomy with Billroth II reconstruction, and 0.2% after distal gastrectomy with Roux-en-Y reconstruction [4–6]. Other operations of which ALS can occur include total gastrectomy with loop esophagojejunostomy with simple or pouch Roux-en-Y reconstruction and pancreaticoduodenectomy with conventional loop and Roux-en-Y reconstruction; nonetheless, the data on incidence were limited .” Dumon K and Dempsey DT. (2019). Postgastrectomy Syndromes. In Charles J Yeo (Ed.) Shackelford’s Surgery of the Alimentary Tract, 8th ed.: 719-734. Elsevier, Philadelphia. Full-text for Emory users. “Afferent loop obstruction, also called afferent loop syndrome, is a mechanical complication that infrequently occurs following construction of a GJ. The creation of a GJ leaves a segment of proximal small bowel (duodenum and proximal jejunum) upstream from the anastomosis. With Billroth II or loop GJ the afferent limb conducts bile, pancreatic juices, and other proximal intestinal secretions toward the GJ 51 ; with Roux-en-Y the afferent limb conducts the succus toward the jejunojejunostomy and is also called the biliopancreatic limb. The operations most commonly associated with afferent loop obstruction are Billroth II and Roux-en-Y GJ (distal gastrectomy or gastric bypass), and Roux-en-Y esophagojejunostomy (total gastrectomy). 52 The incidence of significant afferent loop obstruction after these procedures is low (0.3% to 1.0%) and is similar after open and laparoscopic surgery.” FIGURE 62.7.Causes of afferent loop syndrome include (A) kinking and angulation of the afferent limb, (B) internal herniation of the afferent limb behind the efferent limb, (C) stenosis of the gastrojejunal anastomosis, (D) redundancy of the afferent limb leading to volvulus, or (E) adhesions involving the afferent limb. (Modified from Miller TA, Mercer DW. Derangements in gastric function secondary to previous surgery. In: Miller TA, ed.Modern Surgical Care: Physiologic Foundations and Clinical Applications.2nd ed. St. Louis: Quality Medical; 1998:402.) Share this: Click to share on X (Opens in new window)X Click to share on Facebook (Opens in new window)Facebook Click to share on LinkedIn (Opens in new window)LinkedIn Click to email a link to a friend (Opens in new window)Email Like Loading... This entry was posted in Gastroenterology and tagged Afferent Loop Syndrome by E. Lawson. Bookmark the permalink. 1 thought on “Afferent loop syndrome” Pingback: Digest for July 4-10, 2022 \| Surgical Focus Leave a comment Cancel reply Δ Comment Reblog SubscribeSubscribed Surgical Focus Join 86 other subscribers Sign me up Already have a WordPress.com account? Log in now. Surgical Focus SubscribeSubscribed Sign up Log in Copy shortlink Report this content View post in Reader Manage subscriptions Collapse this bar Loading Comments... Write a Comment... Email (Required) Name (Required) Website %d
15114	https://pmc.ncbi.nlm.nih.gov/articles/PMC5944396/	Gender Dysphoria in Adults: An Overview and Primer for Psychiatrists - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer \| PMC Copyright Notice Transgend Health . 2018 May 18;3(1):57–A3. doi: 10.1089/trgh.2017.0053 Search in PMC Search in PubMed View in NLM Catalog Add to search Gender Dysphoria in Adults: An Overview and Primer for Psychiatrists William Byne William Byne 1 Mental Illness Research Education and Clinical Center, James J Peters VA Medical Center, Bronx, New York. 2 Department of Psychiatry, Icahn School of Medicine at Mount Sinai and Center for Transgender Medicine and Surgery at Mount Sinai, New York, New York. Find articles by William Byne 1,,2,,, Dan H Karasic Dan H Karasic 3 Department of Psychiatry, University of California, San Francisco, San Francisco, California. Find articles by Dan H Karasic 3, Eli Coleman Eli Coleman 4 Program in Human Sexuality, Department of Family Medicine and Community Health, University of Minnesota Medical School, Minneapolis, Minnesota. Find articles by Eli Coleman 4, A Evan Eyler A Evan Eyler 5 Departments of Psychiatry and Family Medicine, University of Vermont College of Medicine, Burlington, Vermont. Find articles by A Evan Eyler 5, Jeremy D Kidd Jeremy D Kidd 6 Department of Psychiatry, Division on Substance Use Disorders, College of Physicians and Surgeons of Columbia University, New York, New York. Find articles by Jeremy D Kidd 6, Heino FL Meyer-Bahlburg Heino FL Meyer-Bahlburg 7 Division of Gender, Sexuality, and Health, New York State Psychiatric Institute/Department of Psychiatry, College of Physicians and Surgeons of Columbia University, New York, New York. Find articles by Heino FL Meyer-Bahlburg 7, Richard R Pleak Richard R Pleak 8 Department of Psychiatry, Division of Child and Adolescent Psychiatry, Hofstra North Shore-LIJ School of Medicine, Albert Einstein College of Medicine, Zucker Hillside Hospital, Ambulatory Care Pavilion, Glen Oaks, New York. Find articles by Richard R Pleak 8, Jack Pula Jack Pula 9 Department of Psychiatry, Division of Gender, Sexuality and Health, College of Physicians and Surgeons of Columbia University, New York, New York. Find articles by Jack Pula 9 Author information Article notes Copyright and License information 1 Mental Illness Research Education and Clinical Center, James J Peters VA Medical Center, Bronx, New York. 2 Department of Psychiatry, Icahn School of Medicine at Mount Sinai and Center for Transgender Medicine and Surgery at Mount Sinai, New York, New York. 3 Department of Psychiatry, University of California, San Francisco, San Francisco, California. 4 Program in Human Sexuality, Department of Family Medicine and Community Health, University of Minnesota Medical School, Minneapolis, Minnesota. 5 Departments of Psychiatry and Family Medicine, University of Vermont College of Medicine, Burlington, Vermont. 6 Department of Psychiatry, Division on Substance Use Disorders, College of Physicians and Surgeons of Columbia University, New York, New York. 7 Division of Gender, Sexuality, and Health, New York State Psychiatric Institute/Department of Psychiatry, College of Physicians and Surgeons of Columbia University, New York, New York. 8 Department of Psychiatry, Division of Child and Adolescent Psychiatry, Hofstra North Shore-LIJ School of Medicine, Albert Einstein College of Medicine, Zucker Hillside Hospital, Ambulatory Care Pavilion, Glen Oaks, New York. 9 Department of Psychiatry, Division of Gender, Sexuality and Health, College of Physicians and Surgeons of Columbia University, New York, New York. Address correspondence to: William Byne, MD, PhD, James J. Peters VA Medical Center, Research Bldg, Rm 5F-04B, Bronx, NY 10468 william.byne@mssm.edu This report was prepared by the American Psychiatric Association (APA) Workgroup on Gender Dysphoria with oversight by the APA Council on Quality Care. This article has been extracted and revised from a larger document, “Gender Dysphoria and Gender-variant Patients: A Primer for Psychiatrists,” which was approved by the APA Board of Trustees as an APA Resource Document. A course based on this content was also presented as APA Course 315, Transgender and Intersex for the Practicing Psychiatrist, given at the 168th Annual Meeting of the APA, May 16, 2015, Toronto, Canada, and Course 4196 by the same name given at the 169th Annual Meeting of the APA, May 24, 2016, Atlanta, GA. Collection date 2018. © William Byne et al. 2018; Published by Mary Ann Liebert, Inc. This Open Access article is distributed under the terms of the Creative Commons License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. PMC Copyright notice PMCID: PMC5944396 PMID: 29756044 Abstract Regardless of their area of specialization, adult psychiatrists are likely to encounter gender-variant patients; however, medical school curricula and psychiatric residency training programs devote little attention to their care. This article aims to assist adult psychiatrists who are not gender specialists in the delivery of respectful, clinically competent, and culturally attuned care to gender-variant patients, including those who identify as transgender or transsexual or meet criteria for the diagnosis of Gender Dysphoria (GD) as defined by The Diagnostic and Statistical Manual of Mental Disorders (5th edition). The article will also be helpful for other mental health professionals. The following areas are addressed: evolution of diagnostic nosology, epidemiology, gender development, and mental health assessment, differential diagnosis, treatment, and referral for gender-affirming somatic treatments of adults with GD. Keywords: assessment, gender dysphoria, gender transition, mental health, psychiatry, intersex, transgender Introduction Individuals who would likely be considered transgender today are evident throughout the historical record.1 The historical and sociocultural conceptualizations of gender variance, and their evolution within mental health professions over the past century and a half are reviewed elsewhere.2 Nineteenth and 20th century theories of gender variance and views of appropriate treatment were pathologizing and highly stigmatizing to transgender people.2 While mainstream psychiatry is now more affirming of gender variance, transgender individuals often are aware of the history in this area and many are likely to have encountered providers who adhere to outdated stigmatizing theories and approaches to treatment.3 Today's mental health professionals should, therefore, be familiar with the history in this area as it is not unusual for gender-variant patients to have apprehensions about seeking mental healthcare or to raise questions about their providers' views and approach to treatment considering that history. Between 1963 and 1979, over 20 university-based gender identity clinics opened in the United States.2,4 These clinics provided interdisciplinary care that included psychiatrists and other mental health professionals and played an important role in the provision of medical services to transgender people and in promoting research to improve their care.2,4 The majority of these clinics closed following a 1981 decision of the U.S. Department of Health and Human Services (HHS) that labeled sex reassignment surgery as experimental,5 a decision what was overturned by HHS in 2014 in a determination that concluded that the 1981 decision was “unreasonable and contrary to contemporary science and medical standards of care.”6 With the closure of the academic gender clinics, transgender people in the United States came to rely on a loose network of medical and mental health providers, often affiliated with the Harry Benjamin International Gender Dysphoria Association (HBIGDA), which was subsequently renamed the World Professional Association for Transgender Health (WPATH). HBIGDA/WPATH developed and successively revised standards of care (SOC) for gender transition, which are currently in their seventh revision as the WPATH SOC7.7 In the WPATH SOC7, mental health professionals are tasked with determining whether those interested in gender-affirming treatments meet eligibility criteria, have capacity for informed consent, and have adequately anticipated the psychosocial impacts of their transition. The WPATH SOC also provide clinical guidance for health professionals to assist transgender people in their search for psychological well-being in their gendered selves. In the absence of other comprehensive English language guidelines, U.S. providers and their professional associations came to rely heavily on the HBIDGA/WPATH SOC.8–10 Similarly, insurance carriers and tax courts employ WPATH SOC criteria in evaluating the medical necessity of transition treatments for determination of reimbursable and tax-deductible medical expenses.11–14 With transition services offered outside of university-based clinics, U.S. medical schools and residency training programs offered little exposure to the provision of transition services, leaving psychiatrists and other physicians poorly prepared for the growth in demand for these services seen in recent years.15 This article aims to assist adult psychiatrists and other mental health professionals who are not gender specialists in the care of these individuals. Detailed information on the assessment and treatment of gender dysphoria in children and adolescents can be found elsewhere.16–19 A glossary of transgender-related terms is found in Table 1. Providers should be respectful of their patients' identity labels; however, due to the rapid evolution of gender terminology, they may need to clarify how both their patients and colleagues employ particular terms. Table 1. Glossary Assigned gender: the initial gender attributed to an individual after birth; for most individuals, this corresponds to the sex on their original birth certificate, aka assigned gender, birth sex. Cisgender: a term for individuals whose experienced and expressed gender are congruent with their gender assigned at birth, that is, those who are not transgender. Experienced gender: one's sense of belonging or not belonging to a particular gender, aka gender identity. Expressed gender: how one expresses one's experienced gender. Gender: a person's social status as male (boy/man) or female (girl/woman), or alternative category. Gender-affirming surgery: surgical procedures intended to alter a person's body to affirm their experienced gender identity, aka sex reassignment surgery, gender reassignment surgery, and gender-confirming surgery. Gender assignment: assignment of a gender to an individual. In typically developed newborns, the initial gender assignment (aka “birth-assigned gender”) is usually made on the basis of the appearance of the external genitalia. Gender binary: a gender-categorization system limited to the two options, male and female. Individuals who identify outside the gender binary may use a variety of gender identity labels, including genderqueer or nonbinary. Gender dysphoria (not capitalized): distress caused by the discrepancy between one's experienced/expressed gender and one's assigned gender and/or primary or secondary sex characteristics. Gender Dysphoria (GD) (capitalized): a diagnostic category in DSM-5, with specific diagnoses defined by age group-specific sets of criteria. This article addresses only GD in adults. Gender identity: one's identity as belonging or not belonging to a particular gender, whether male, female, or a nonbinary alternative, aka experienced gender. Gender Identity Disorder (GID) a diagnostic category in DSM-III and DSM-IV that was replaced in DSM-5 by GD. Gender incongruence (not capitalized): incongruence between experienced/expressed gender and assigned gender, and/or psychical gender characteristics. Gender Incongruence (capitalized): a diagnostic category (analogous to GD in DSM-5) proposed for ICD-11. Gender role: cultural/societal definition of the roles of males and females (or of alternative genders). Gender transition: the process through which individuals alter their gender expression and/or sex characteristics to align with their sense of gender identity. Gender variance: any variation of experienced or expressed gender from socially ascribed norms within the gender binary. Gendered behavior: behavior in which males and females differ on average. Genderqueer: an identity label used by some individuals whose experienced and/or expressed gender does/do not conform to the male/female binary or who reject the gender binary. Intersex conditions: a subset of the somatic conditions known as “disorders of sex development” or “differences of sex development “in which chromosomal sex is inconsistent with genital sex, or in which the genital or gonadal sex is not classifiable as either male or female. Some individuals who report their identity as “intersex” do not have a verifiable intersex condition. Sex: a person's categorization as biologically male or female, usually on the basis of the genitals and reproductive tract. Sex assigned at birth: the sex or gender first assigned to an individual after birth. Also known as “natal gender,” “birth-assigned sex,” and “gender assigned at birth.” Often queried as “What sex was listed on your original birth certificate?” Sexual orientation: a person's pattern of sexual attraction and physiological arousal to others of the same, other, both, or neither sex. Sexual orientation cannot be inferred from one's gender identity. As a show of respect, we recommend that the sexual orientation of transgender individuals be expressed in relation to their gender identity rather than their gender assigned at birth; however, all gender scholars do not follow that convention. Ambiguity in charting can be avoided by using terms such as sexually attracted to men, women, both, or neither. Transgender: an umbrella term usually referring to persons whose experienced or expressed gender does not conform to normative social expectations based on the gender they were assigned at birth. Transsexual: a term often reserved for the subset of transgender individuals who desire to modify, or have modified, their bodies through hormones or surgery to be more congruent with their experienced gender. Open in a new tab On official documents such as birth certificates, driver's licenses, and passports, the traditional category “sex” is equivalent to “gender” in current psychological terminology. “Trans” (also “Trans”) More recent umbrella terms being increasingly used to avoid distinguishing between transgender and transsexual individuals. DSM, Diagnostic and Statistical Manual of Mental Disorders; GD, Gender Dysphoria; GID, Gender Identity Disorder; ICD, International Classification of Diseases. Diagnostic and Statistical Manual of Mental Disorders and Transgender-Related Nosology The first two editions of the Diagnostic and Statistical Manual of Mental Disorders (DSM) published in 1952 and 1968, respectively, did not include any gender diagnosis.20 The diagnosis, “Transsexualism” (sic), first appeared in 1975 in the ninth revision of the International Classification of Diseases (ICD)-921 and subsequently, in the DSM-III in 1980 under the parent category, Sexual Deviations.22 The defining characteristics of this diagnosis were as follows: (1) discomfort about one's assigned sex; (2) “cross-dressing,” in reality or fantasy, as the other sex, but not for the purpose of sexual excitement; and (3) the desire to get rid of one's primary and secondary sex characteristics and to acquire those of the other sex. DSM-III also included “Gender Identity Disorder of Childhood” (GIDC). Both transsexualism and GIDC were carried over into DSM-IIIR, but were no longer categorized as sexual deviations. Instead, they were placed within the parent category, Disorders Usually First Evident in Infancy, Childhood, or Adolescence.23 This category also included disruptive behavior disorder, eating disorders, and tic disorders. Under this parent category, DSM-IIIR added a new diagnosis, Gender Identity Disorder of Adolescence and Adulthood Nontranssexual Type (GIDAANT). These changes recognized that gender identity disorder (GID) often begins in childhood, may or may not persist into adolescence and adulthood, and when it does persist, it may not entail a desire for the primary or secondary sexual characteristics of the other sex. With DSM-IV, the diagnoses of Transsexualism and GIDAANT were discontinued, but GIDC and GIDAA were retained and placed under a new parent category, Sexual and Gender Identity Disorders, a category that also included the unrelated sexual dysfunctions and paraphilias.24 Individuals with somatic intersex conditions, who experienced dysphoria attributable to dissatisfaction with their gender assigned at birth, could be diagnosed with Gender Identity Disorder Not Otherwise Specified. Retention of the diagnosis by the DSM and its new name, including the word “disorder,” was perceived by many as stigmatizing and contributing to societal discrimination against transgender individuals.25 By analogy to homosexuality, much of the distress and functional impairment associated with being transgender, and required for the diagnosis of GID, could derive from social stigmatization rather than from being transgender, per se. On the other hand, removal of a coded diagnosis for medical classification and billing purposes would limit access to transition care, deny the full impact of gender dysphoria, and prove harmful to transgender individuals.2,26 Ultimately, the diagnosis was retained by DSM-5,27 but its name was changed to Gender Dysphoria (GD), simultaneously removing the stigmatizing “disorder” from its name and shifting the focus to dysphoria as the target symptom for intervention and treatment, rather than gender identity itself.27,28 GD was also moved out of the parent category that included sexual dysfunctions and paraphilias, with which it has nothing in common, and into a separate parent category, also named Gender Dysphoria. Use of the diagnostic label, GD, requires that a person meets the full criteria specified in DSM-5. This is distinctly different from the historical generic use of the term, gender dysphoria, which refers to the distress caused by a discrepancy between one's experienced gender and assigned gender, whether or not full DSM criteria for GD are met. For clarity here, references to the diagnosis will be capitalized or abbreviated (i.e., Gender Dysphoria or GD) while references to the symptom will not be capitalized or abbreviated (i.e., gender dysphoria). The DSM is a manual on mental disorders and, therefore, despite the name change, GD retains its classification as a mental disorder. In contrast, the ICD is not limited to only mental disorders. In its forthcoming eleventh iteration, ICD-11, the diagnosis of Gender Incongruence (GI) (corresponding to GD in DSM-5 terminology) will most likely be moved out of the section on mental disorders. Instead, it has been proposed to place it in a separate section tentatively named Conditions Related to Sexual Health or Sexual and Gender Health.29 Placing GI in this section will declassify it as a mental disorder, while maintaining a diagnosis that will facilitate access to care through third party reimbursement, and could eventually lead to American Psychiatric Association (APA) removing GD from the DSM. Importantly, the GD diagnosis does not apply automatically to people who identify as transgender but is given only to those who either exhibit clinically significant distress or impairment associated with a perceived incongruence between their experienced/expressed gender and their assigned gender or who, after transition, no longer meet full criteria, but require ongoing care (e.g., hormonal replacement therapy). In DSM-5, this latter group is given a “post-transition” specifier. Unlike previous versions of the DSM, in DSM-5, gender-dysphoric individuals with somatic intersex conditions, who were previously excluded from the diagnosis, can now receive the diagnosis with a specifier to indicate the presence of the intersex condition. DSM-5 is also the first DSM to recognize the legitimacy of gender identities outside the gender binary such that individuals with GD are no longer described as identifying simply as “the other gender,” but as “the other gender (or some alternative gender different from one's assigned gender).” Examples of alternative genders include eunuch, genderqueer, and nonbinary. Epidemiology Epidemiological research has employed different measures of transgender populations, resulting in varying estimates of prevalence.30,31 Some studies assessed the fraction of a population, which had received the DSM-IV diagnosis of GID or the ICD 10 diagnosis of transsexualism, both of which were limited to clinical populations who sought binary transition (male-to-female or female-to-male). For example, the prevalences reported in DSM-5 (0.005–0.014% for birth-assigned males; 0.002–0.003% for birth-assigned females) are based on people who received a diagnosis of GID or transsexualism, and were seeking hormone treatment and surgery from gender specialty clinics,25 and, therefore, do not reflect the prevalence of all individuals with gender dysphoria or who identify as transgender. The prevalence of transgender people receiving gender specialty care in the Netherlands has been estimated at 0.008% for transgender women and 0.003% for transgender men.32 More recent data for those obtaining surgery in Belgium were similar.33 In Sweden, point prevalence in 2010 was estimated to be 0.013% for transgender women and 0.008% for transgender men.34 A higher percentage, 0.023%, received a diagnosis of GID recorded in the health records of the U.S. Veteran's Administration.35 Other studies, rather than measuring the proportion of a population that received a clinical diagnosis, have reported on those who self-identified as transgender or gender incongruent, and found that measuring self-identity yields much higher numbers. In 2016, data from the Center for Disease Control's Behavioral Risk Factor Surveillance System suggested that 0.6% of U.S. adults identify as transgender, double the estimate utilizing data from the previous decade.36 In a large Massachusetts population-based phone survey, 0.5% of the population (age 18–64 years) identified as transgender.37 In another large population-based survey in the Netherlands, 1.1% of those assigned male at birth (age 15–70 years) reported an incongruent gender identity (stronger identification with a gender other than the one assigned at birth), as did 0.8% of those assigned female at birth.38 Recent surveys of youth showed even higher numbers. In New Zealand, 1.2% of high school students surveyed identified as transgender.39 In a survey of San Francisco middle school students (grades 6–8), 1.3% identified as transgender.40 More study is needed, but these larger numbers indicate that many transgender people have not been counted in clinical studies, including those with nonbinary identities, those not seeking transition care, those receiving hormones outside of clinics specializing in transgender care or by self-administration, and others who identify as transgender when surveyed, but do not report gender dysphoria to clinicians. Gender Development Biological considerations Animal research has established that sex differences in the phenotype of both body and brain as well as behaviors are the result of multiple, sex-biasing factors. These include hormonal, sex-chromosomal,41 genetic, and epigenetic contributions.42 The sensitivity of brain tissues to organizational effects of sex hormones appears to be particularly high at prenatal/perinatal stages of development and gradually declines toward young adulthood.43 The timing of hormonal secretions in the course of development, however, gives the impression of three discrete sensitive periods: (1) pre/perinatal; (2) pubertal44; and (3) for females, the first pregnancy.45 In humans, statistical sex differences in brain structure are well documented,46 and findings of sensitive periods for sexual differentiation of the brain appear to parallel those seen in other mammals.47,48 The evidence for brain/behavior effects of prenatal androgenization is particularly strong,49–51 much of which derives from studies of individuals with somatic intersex conditions and varying degrees of functional androgen exposure.51–53 Androgenization of the brain depends not only upon the level of androgen to which a fetus is exposed but also upon numerous other factors, including the presence of enzymes to convert androgens to the specific metabolites required by particular brain cells, their steroid receptors, and their postreceptor mechanisms that are involved in the full response to androgens. Receptor structure, which can influence sensitivity, is genetically determined, while the activity of genes for receptors and postreceptor mediators is subject to epigenetic modulation.54 As the period of genital differentiation largely precedes the sexual differentiation of the brain,55 it is conceivable that GD in individuals without somatic intersex conditions could reflect a brain-limited intersex condition (i.e., a lack of concordance between the sexually differentiated state of the brain and body). That hypothesis has been tested in a variety of ways, including searching for features of the brain in individuals with GD that more closely match their experienced gender than their birth-assigned gender.56 Investigations in this regard have included postmortem morphometric and stereological studies,57 as well as in vivo morphometric,58 functional magnetic resonance imaging,59 and diffusion tensor imaging studies of the brain,60–62 and examination of otoacoustic emissions.63 As reviewed elsewhere,53,56,64 while some positive findings in the predicted direction have been reported,56,64 inferences are currently limited. This is because few findings have been replicated and few studies have adequately controlled for potentially confounding variables such as age, sexual orientation, transition status (including history of gender-affirming hormonal treatment, if any), and hormonal status at the time of study (or of death in the case of postmortem studies).53 Much of what is known about the role of early hormonal exposure on the development of gender identity in humans derives from studies of gender outcomes in individuals with somatic intersex conditions. Early guidelines for initial gender assignment for such infants relied heavily on the surgical potential to achieve concordance between the gender assigned and the appearance and functional potential of the external genitalia, in particular, the capacity of penile-vaginal intercourse.9 Current guidelines, however, emphasize what is known about the long-term gender outcomes of individuals with intersex conditions on a syndrome by syndrome basis.52 Overall, these data suggest that regardless of genetic constitution, or gonadal or genital development at birth, individuals prenatally exposed to a full complement of masculinizing hormonal influences (i.e., androgen exposure and the cellular mechanisms for responding fully to androgens as described above) have an increased likelihood of GD when assigned female.51,52 Conversely, most reported 46,XY individuals with complete androgen insensitivity syndrome (and hence no functional androgenization of the brain) have developed a female gender identity, despite having a Y chromosome as well as normally developed and functioning testes.51,52 To date, however, no brain marker of sexual differentiation has been validated to guide the initial gender assignment of infants with intersex conditions. Psychosocial factors influencing gender expression In mammals, and particularly in humans, psychological and social factors have a major additional influence on behavioral outcome.65 In humans, these psychosocial processes include verbal labeling (e.g., “boy” and “girl”) and nonverbal gender-cuing (e.g., gender-specific clothing and haircuts) of children by parents and others in their social environment, as well as the shaping of children's gendered behavior by positive and negative reinforcement and later by explicit statements of gender-role expectations. Related processes in developing children include gender-selective observational learning/imitation, the formation of gender stereotypes and of related self-concepts, and self-socialization. The effects on gender development have been documented in a vast body of research in developmental psychology.65 The impact of such psychosocial factors, however, is not determinative. This is evidenced by individuals in whom gender identity is discordant with the initial gender assignment and gender of rearing, for example, transgender individuals and a higher than expected proportion of individuals with particular intersex conditions (i.e., 46,XY individuals with high degrees of somatic hypomasculinization and 46,XX individuals with high degrees of somatic hypermasculinization66,67). Factors in gender-identity development Systematic data on gender identity development are much more limited than those on gendered behavior. Yet, the data available, especially for those with intersex conditions, lead to the conclusion that, while early androgenization plays a role, a definitive biological predetermination of gender identity seems unlikely. Not a single biological factor, but multiple factors (i.e., biological, psychological, and social) appear to influence the development of gender identity.50 The need to transition gender is even less understood in individuals without, compared to those with, intersex conditions.68 Along with the dramatically increased referrals of gender-variant individuals to specialized clinics in Western Europe and North America over the last two decades,69,70 there has been a diversification of presentations beyond the original “transsexual” who sought (or was perceived by providers to seek) change to the “other” gender through treatment with gender-affirming hormones and genital surgeries. Currently, many transgender people seek chest, but not genital surgery, or only gender-affirming hormones, or only a social transition without any medical changes. Others may simply desire flexibility in gender expression without transition to “the other gender,” identifying, for example, as nonbinary or genderqueer.71,72 Prospective follow-up studies of children, who before puberty had met criteria for the DSM-IV diagnosis of GID, showed that the majority of those diagnosed with GID in early or early middle childhood “desisted,” meaning that they subsequently identified as their birth-assigned gender and did not meet criteria for GID. As adults, many identified as lesbian, gay, or bisexual.73–75 Some “desisters,” however, subsequently transitioned later in life.73 The data available do not allow a clear prediction before puberty of which child will persist and transition permanently, and which child will not.75 With the introduction of stricter criteria for the diagnostic category of Gender Dysphoria in DSM-5, the persistence rate likely will be higher,73 but this needs to be tested by future long-term follow-up studies. For example, the degree of gender nonconformity and whether a child believes they are, as opposed to wishes to be, “the other” gender have been proposed as predictors of persistence.76,77 Those in whom GD persists from childhood into adolescence are likely to experience an exacerbation of dysphoria with the emergence of (or with the anticipation of) undesired secondary sexual characteristics at puberty, in which case pubertal suspension should be considered.10 Regardless of their initial sexual orientation, during and after transitioning to express their experienced gender, some individuals retain their pretransition sexual attraction patterns, while others change.7 In some transgender women, the desire to transition gender is preceded by fantasizing themselves as women, sometimes with sexual arousal.78 This phenomenon has been controversially interpreted by some as fetishism.79 Importantly, neither a history of fetishistic arousal nor one's sexual orientation precludes one from meeting the criteria for the diagnosis of GD27 or eligibility for gender transition services.7,80 Mental Health Assessment and Treatment This section addresses the assessment and treatment of adults with gender identity or expression concerns in the absence of an intersex condition. GD in individuals with intersex conditions is addressed in the Appendix. Treatment of GD in prepubescent children, where there is currently less consensus,81 is addressed elsewhere as is treatment of adolescents, including selection of candidates for pubertal suspension.81,82 The primary roles of the mental health professional in assessing and treating patients with GD are based on expert consensus,7,8,10,20 summarized in Table 2 and described more fully below in the broader context of gender variance. Table 2. Roles of the Psychiatrist Assess and diagnose gender concerns according to current DSM criteria and see that they are addressed. Assess and diagnose any coexisting psychopathology and see that it is addressed. Assess eligibility for hormonal and/or surgical treatments, or refer to professionals capable of making such assessments. Assess capacity to give informed consent for hormonal and surgical treatments. Ensure that eligible individuals are aware of the full range of treatment options and their physical, psychological, and social implications, including risks, benefits, and impact on sexual functioning and reproductive potential. Ensure adequate psychological and social preparation for transition treatments. Refer patients for hormonal or surgical treatments, collaborating with providers as needed. Ensure continuity of mental healthcare as indicated throughout transition and beyond. Open in a new tab Expert consensus regarding the treatment of adults has been arrived at after many years of clinical experience. Attempts to engage individuals in psychotherapy to change their gender identity or expression are currently not considered fruitful by the mental health professionals with the most experience working in this area7,9,83 and legal bans of therapies aimed at changing sexual orientation have recently been extended to therapies aimed at changing gender identity or expression in a number of U.S. states and Canadian provinces.84,85 Currently, psychotherapeutic involvement with adults with GD is primarily used to assist in clarifying their desire for, and commitment to, changes in gender expression and/or somatic treatments to minimize discordance with their experienced gender, and to ensure that they are aware of and have considered alternatives.7 Gender questioning, gender-variant, and transgender adults present to mental health services for a variety of reasons. Some presentations may relate explicitly to gender. For example, patients may wish to explore their gender identity, consider transition options and concerns (e.g., coming out to family or coworkers), or request evaluation for hormonal or surgical treatments. The latter may include requests for referrals for such treatments, including requests for mental health referral letters as specified by the WPATH SOC7 or required by their providers of transition treatments and/or insurance carriers.7,11–13 According to WPATH SOC7, as an alternative to an evaluation by a mental health professional, primary care providers who are competent in the assessment of GD may evaluate patients for hormone therapy, particularly in the absence of significant coexisting mental health concerns and when working in the context of a multidisciplinary specialty team.7 Patients may also seek couples or family therapy before, during, or after transition to address the impact of the transition on interpersonal or family dynamics. Alternatively, many transgender patients seek or are referred to psychiatric services for reasons that are either unrelated to gender identity or expression (e.g., management of primary psychiatric illnesses), or only partially related (e.g., sequela of childhood trauma as a result of minority stress due to gender nonconformity). A careful evaluation for a history and psychological sequela of gender-related stigma and abuse, from childhood on, is crucial given the high rates of violence and bullying experienced by gender-variant individuals, as well as the high rates of discrimination, unemployment, homelessness, sex work, and HIV infection.3,86 High rates of depressive, anxiety, and substance use disorders, as well as suicidal ideation and completed suicide have been linked to such gender minority stress.87–89 In addition to these mental health disparities, the transgender population also exhibits marked general health disparities.90 Few of these disparities are linked to sexually transmitted infections or hormonal or surgical transition treatments,7,10,90 but are instead linked to financial barriers to care as well as avoidance of healthcare due to experienced and/or anticipated stigma and discrimination in healthcare settings, and the widespread belief among transgender individuals that medical professionals are poorly trained to meet their needs,3 a belief that appears to be well-founded.15 Extensive guidance on overcoming these barriers to care, including creating a welcoming clinical environment, can be found elsewhere.91 Assessment of gender concerns Treatment should be patient centered and tailored to the needs and individuality of each patient. Patients should be asked what names and pronouns they use and should be addressed by those names and pronouns regardless of their stage of transition. Those who transitioned many years ago and are seeking treatment for another problem typically need much less focus on gender history than those who are questioning their gender identity, just beginning gender transition, or exploring options for gender expression. When gender is not the primary concern, devoting the appropriate amount of attention to gender-related issues is important, balancing against an overemphasis on gender that can feel inadvertently stigmatizing to the patient or distract from adequate focus on the chief complaint. While it is important to avoid the assumption that coexisting psychiatric symptoms are due to gender variance, the impact of past and present gender-related stigma should be considered in the biopsychosocial evaluation. This is particularly important in light of the stress-diathesis model of psychiatric illness and its exacerbations.8,92 Suicidality should always be assessed, as should protective factors such as social and family supports.93 Suicidal ideation3,94 and completed suicide90 are dramatically increased in this population and GD may be a risk factor for suicidality, independent of other psychiatric conditions.94,95 Up to 47% of transgender adults have considered or attempted suicide.93 Assessment of suicide risk is especially important during periods of heightened vulnerability, such as when transgender identity is disclosed to family and more broadly.9,83 The gender assessment should include the age and circumstances when the patient first became aware of a sense of difference from peers of the same sex assigned at birth as well as experiences of negative affect or self-perception related to that sense of difference.8,20 Any history of peripubertal and/or pubertal distress due to the anticipation and/or emergence of unwanted secondary sex characteristics should also be explored, as should past experiences of gender-related stigmatization, discrimination, harassment, and violence.8,20 The patient's history of coping mechanisms and support systems should also be examined.8,20 Gender expression (e.g., pronoun use, name changes, manner of dress, and bodily modifications) over time should be explored as well as what has and has not been helpful in improving the sense of well-being. It is important to clarify each patient's goals and plans for social and/or medical transition, degree of commitment, and expectations.7,96 For those who do not wish to transition, assessing current psychosocial challenges and formulating with the patient how to best address them (e.g., psychotherapy, group therapy, and social support) should not be neglected. Recommendations regarding psychiatric assessment of individuals with GD have focused largely on assessment of eligibility for and decision-making capacity related to medical and surgical gender transition services.7,8,10 Eligibility for both gender-affirming hormone therapy and surgeries requires persistent gender dysphoria, a documented diagnosis of GD based on DSM-5 criteria, and the capacity to give informed consent.7 In addition, any significant medical or psychiatric concerns must be sufficiently controlled so that they do not interfere with the patient's ability to safely adhere to the treatment regimen. The current standard of care in major clinics, the WPATH SOC7, and insurance requirements for reimbursement of services follow a flexible progression of transition steps, which may begin with completely reversible steps (e.g., change of pronouns, name, and manner of dress), followed by partially reversible changes (e.g., gender-affirming hormones), and then irreversible gender-affirming surgeries.7,10–14,97 There is flexibility in this process given that some people do not pursue all of these interventions or may prefer to do so in a different sequence. For example, transgender men may wish to undergo mastectomy or male breast construction before initiating masculinizing hormones.7 Before gonadectomy, 12 months of continuous hormone therapy consistent with the patient's gender goals are recommended, unless hormones are clinically contraindicated for the individual. The aim of hormone therapy before gonadectomy is primarily to allow the individual to experience a period of gender-affirming hormones, before irreversible surgical intervention.7 Before masculinizing or feminizing genital reconstructive surgeries, the WPATH SOC7 also recommend 12 continuous months of living in a gender role that is congruent with the patient's gender identity.7 Diagnosis of gender dysphoria The DSM-5 diagnostic criteria for GD in adolescents and adults are shown in Table 3. Diagnosing GD in adults by these criteria is usually straightforward, especially for those with overt manifestations in childhood, exacerbation of distress with pubertal changes, and persistence into adulthood in the absence of significant coexisting mental health concerns.8,9 Table 3. Diagnostic Criteria for Gender Dysphoria in Adolescents and Adults A marked incongruence between an individual's experienced/expressed gender and assigned sex as evidenced by two of the below, which have been present after the onset of puberty for at least 6 months: A marked incongruence between one's experienced/expressed gender and primary and/or secondary sex characteristics (or the anticipated secondary sex characteristics in young adolescents). A strong desire to be rid of one's primary and/or secondary sex characteristics because of a marked incongruence with one's experienced/expressed gender (or a desire to prevent the development of the anticipated secondary sex characteristics in young adolescents). A strong desire for the primary and/or secondary sex characteristics of another gender. A strong desire to be of a gender different from one's assigned gender. A strong desire to be treated as a gender different from one's assigned gender. A strong conviction that one has the typical feelings and reactions of a gender different from one's assigned gender. The condition is associated with distress or impairment in social, occupational, or other important areas of functioning that are clinically significant. Open in a new tab Adapted from DSM-5.27 Assessment of patients who are seeking transition services, but do not clearly meet criteria for GD, may require more time and exploratory therapy9 (e.g., a patient desiring hormonal or surgical treatment to transition to another gender, who does not clearly experience incongruence between their experienced gender and their gender assigned at birth). The same is true for those with the onset of gender dysphoria in the context of a psychiatric disturbance (e.g., psychosis, dissociative disorder, and autism spectrum disorder) or recent trauma9,98,99; those who are ambivalent about their gender identity or desired sex characteristics; and those who exhibit marked exacerbations and remissions of dysphoria over time. The psychiatrist must assess whether some factor other than GD accounts for the expressed desire to transition. If not, coexisting mental illness is not a contraindication to supporting transition if it is sufficiently controlled to not interfere with the patient's capacity for decision-making or ability to safely adhere to the demands of the desired treatment.7,9,98 Differential diagnosis Few conditions can be mistaken for GD. Simple nonconformity to gender roles can be differentiated from GD based on the degree of associated distress and whether or not the individual identifies as the sex assigned to them at birth. GD can be differentiated from body dysmorphic disorder (BDD), in which an individual may wish a body part to be removed or altered because it is viewed as deformed.27 In contrast, in GD alterations are sought for anatomical characteristics that are incongruent with one's gender identity. BDD and GD can, however, coexist and the presence of BDD is not an absolute contraindication for gender-confirming surgery.27 Transvestic disorder is characterized by significant distress or impairment due to sexual arousal in the context of cross-dressing fantasies, urges, or behavior. It may exist independently or co-occur with GD,27 and is not a contraindication to supporting transition in those who meet criteria for GD.7 Gender-themed delusions have been reported to occur in up to 20% of those with psychotic disorders.100 Such delusions can usually be easily differentiated from GD by their content (i.e., if they do not entail the belief that one's gender differs from that assigned at birth), as well as by their presence only during psychotic phases of illness, and the absence of other DSM criteria required for the diagnosis of GD.98 Importantly, GD and psychotic disorders may coexist and patients with both diagnoses can benefit from gender-affirmative treatment and appropriate hormonal and/or surgical gender interventions.98 Timely diagnosis of GD may be impeded when it is first overtly expressed in adolescence or early adulthood coincident with, or shortly following, the first psychotic episode.98 Mental health treatment Statements in this section are based on the cited studies supplemented by the authors' cumulative clinical experience treating patients with GD. Psychotherapy can be useful for patients with GD; however, many successfully transition or decide against transition with little or no psychotherapy. Psychotherapy may be helpful at different times and for different reasons across the lifespan.7 Many transgender people seek mental health treatment on an intermittent basis, while contemplating gender transition, at key points in the transition process, or post-transition if symptoms recur or worsen. Participation in transgender support groups, including peer-led groups, and other interactions with transgender individuals or the transgender community are often useful in clarifying the goals of those who experience ambivalence about transition. With patients who are otherwise eligible for transition treatments, but express ambivalence about transition, the therapist should maintain a stance of neutrality, creating a safe therapeutic space in which the patient can weigh all options and arrive at a decision in their own time. Many transgender adults need some combination of hormonal treatment and/or surgical procedures for relief of GD, but some experience relief with a change in gender expression without any medical treatment.7 Strengthening resilience factors identified in the transgender population93 should be a focus, particularly, in patients with suicidal ideation. Although treatment with exogenous estrogen or testosterone carries a risk for medical side effects,10 both have been associated with improvement with respect to anxiety, mood, and mood stability, as well as overall satisfaction and quality of life for both transgender women and transgender men.101–104 Similarly, review of the available literature9 demonstrates the benefits of surgery in alleviating GD and the rarity of postsurgical regret. Emotional changes may occur with use of either androgen or estrogen supplementation, although these are usually subtle.9 An increase in libido usually occurs with androgen use with female to male transition.10 Although decreased libido due to antiandrogen and/or estrogen treatment in individuals transitioning male to female is common,10 some may experience a stronger interest in sex, perhaps due to the affirming aspects of attaining desired bodily changes. Safer sex information and instruction in self-protective negotiation in sexual settings should be provided and tailored to the anatomy, needs, and experiences of transgender persons.9 Masculinizing hormones have been associated with a possible destabilization of psychotic and bipolar disorders, especially with supraphysiological blood levels of testosterone7 in both cisgender and transgender men.105–106 The likelihood of such episodes can, therefore, be minimized by careful dosing and monitoring. Detailed information on specific gender-affirming surgical procedures can be found elsewhere.7,107 Psychiatrists should collaborate with other providers (e.g., endocrinologists, surgeons, psychotherapists, primary care providers, social workers, and other mental health professionals) to ensure that patients have the knowledge required to adequately evaluate the benefits, risks, and limitations of desired treatments and their alternatives. This is necessary not only for informed consent but also to ensure adequate preparation for surgery and postsurgical needs (e.g., convalescent period, period of sexual abstinence, and vaginal dilatation in the case of vaginoplasty). Helping the patient anticipate and prepare for psychosocial impacts of treatment (e.g., impact on social relationships and employment) is also essential. Importantly, transition treatments target GD, not coexisting psychiatric diagnoses, and coexisting diagnoses are likely to require ongoing attention after transition, although symptom severity may be ameliorated.98,100,102 Referrals for hormones and surgery Whether the initial evaluation for hormones is done by the hormone prescriber or by a mental health professional, criteria for starting hormones are the same: the presence of persistent GD, the ability to give informed consent, and relative mental health stability.7 Insurance carriers and surgeons require mental health evaluation before transition-related surgeries to assess and document eligibility, readiness, and medical necessity of the requested procedure.7,10–14 The specific requested content of referral letters varies among surgical providers and insurance plans. To avoid unnecessary delays in treatment, letter writers should be aware of such differences and ensure that their letters meet the requirements of all relevant parties. The content requested by most providers and insurance carriers is similar to that outlined in the WPATH SOC7. Genital and gonadal surgeries usually require documentation from two licensed mental health professionals, while chest surgeries generally require just one evaluation and referral.7,108 Although not requirements of WPATH SOC7, some insurers require one letter from a psychiatrist or other doctoral level mental health provider, or may specify a minimal duration of mental healthcare.13 Such requirements vary by health system, insurance carrier, and state, and raise challenges for those without access to reimbursement for mental healthcare. Current Social Issues: Stigmatization and Access to Care Transgender health advocates have worked to address societal discrimination against transgender people, including stigmatization of identity, discrimination in schools, workplaces, and healthcare, and to improve access to care. Increasingly, this advocacy has been embraced by major institutional and governmental agencies. One large online survey, the National Transgender Discrimination Survey88 found that rejection, discrimination, victimization, and violence against transgender people occur in a multitude of settings and negatively affect transgender people across the life span. Transgender youth are often harassed and assaulted in schools, which is associated with dropping out and subsequent impoverishment. Many transgender people are harassed at work or lose jobs due to their gender identity and expression. Discrimination extends to healthcare settings, where patients may be refused care or treated disrespectfully, or do not have access to care.88 U.S. public policy has contributed to the lack of access to care. A report by the National Center for Health Care Technology of the HHS Public Health Service issued in 1981, titled “Evaluation of Transsexual Surgery,” deemed these procedures “experimental,” and recommended that Medicare not cover transition-related care. This was formalized in a 1989 Health Care Financing Administration National Coverage Determination.5 Exclusion of transgender healthcare in private insurance as well as Medicaid and Medicare was near universal in the decades to come. A lack of funding for clinical care and research led to the closing of transgender care programs at academic institutions in the years following the 1981 report. Many transgender health insurance exclusions have been removed recently. This trend started with increasing numbers of employers in the last 15 years adding transition care to health coverage. Starting in 2013, some states have ruled that transgender healthcare exclusions are discriminatory and have banned them from state-regulated health insurance plans. In 2014, the 1981 Medicare policy was reversed, removing categorical exclusions for transgender care.6 In 2015, the HHS moved to end categorical exclusions for transgender care from all insurance and care providers who accept federal funding or reimbursement;109 and since 2016, insurers in the Federal Employees Health Benefits Program must include transition-related coverage for transgender federal employees.110 During this same period, executive orders and other guidance from the Obama administration conferred increased protection against discrimination to transgender individuals in workplace and educational settings,111 the ban on open military service of transgender individuals was lifted,112 and changes at the HHS and the National Institutes of Health (NIH) facilitated research to better define and address the health needs of transgender individuals.111 Much work remains, however, to fully actualize these policy changes. In addition, progress has been slowed on the federal level by the change in presidential administrations and legal actions.113 WPATH SOC7 7 has attempted to improve access to care by including the informed consent model for hormone administration. In multidisciplinary clinics providing transgender care, primary care providers can assess for and diagnose longstanding GD that might benefit from treatment with hormones and administer hormones without referral from a mental health professional. However, patients with cooccurring mental health conditions should be referred to mental health providers when appropriate. WPATH has advocated for the depathologization of transgender identity, the medical necessity of transgender care, and improved access to legal gender change.7 The APA has also attempted to reduce stigma and improve access to care. As discussed previously, the DSM-IV diagnosis of GID, regarded as stigmatizing by many transgender health and advocacy groups, was replaced with GD in DSM-5.114 In addition, the APA approved position articles on discrimination and access to care. Its statement on discrimination against transgender and gender-variant individuals115 opposes all private and public discrimination against transgender individuals, and its statement on access to care for transgender and gender-variant individuals116 urged the removal of all categorical healthcare exclusions for transgender people and advocated for the expansion of access to care. Increased access to care must be accompanied by culturally competent research in transgender health, recommended by the Institute of Medicine86 and outlined in the NIH's Strategic Plan to Advance Research on the Health and Well-being of Sexual and Gender Minorities.117 Expanded and improved education of healthcare providers is necessary, and the American Association of Medical Colleges has produced guidelines for curricular and climate change to improve transgender health.118 Principles of culturally competent care for transgender and nonbinary patients should be included in residency training as well, including psychiatric residency programs. Conclusions Transgender, nonbinary, and gender questioning people are sufficiently common that even psychiatrists whose practice does not focus on gender are likely to encounter patients who have transitioned gender, are planning or considering transition, or are questioning their gender identity. Gender concerns are only one of the reasons these individuals may seek psychiatric care and, regardless of their area of specialization, psychiatrists should be adept at conducting respectful, culturally sensitive, and affirming gender assessments without placing an undue emphasis on gender when it is not the patient's presenting concern. Mental health professionals must fully appreciate that the focus of treatment for GD is on the dysphoria, not the gender identity. At the same time, they must appreciate the role of minority stress in gender minority mental health disparities, screen for related manifestations, including anxiety disorders, depression, and suicidality, and consider resilience factors in treatment planning. Psychiatrists should also be competent in the provision of routine psychiatric care that is gender affirming to gender variant patients with serious mental illnesses without assuming that the gender variance is a manifestation of the illness. They should not expect coexisting serious mental illness, especially in the context of strong genetic loading, to fully resolve with successful treatment of GD and should assist the patient in formulating realistic expectations. If not included in their residency or fellowship training, or supervised clinical experience, psychiatrists should familiarize themselves with the standards of care for gender transition as described in the WPATH SOC7 and outlined in this article, as well as the roles and minimal competencies of mental health professionals working with adults with GD.7 In addition to the minimal competencies, WAPTH SOC7 recommends that health professionals take steps to sustain or augment their cultural competency to work with transgender and other gender minority patients by participating in continuing education and becoming knowledgeable about community, advocacy, and public policy issues that affect transgender individuals and their families.7 All providers should work within their sphere of competency and refer patients when necessary. Board-certified psychiatrists should be competent in the diagnosis of GD by the criteria of the most current DSM and in assuring that any coexisting psychiatric disorder is appropriately diagnosed and adequately controlled.118 In the absence of additional training, they should refer to other providers or seek supervision in fulfilling the other tasks of mental health professionals in addressing the gender concerns of transgender and other gender diverse patients. Providers from all disciplines should work within their professional organizations to ensure that training in gender-affirmative care is integrated throughout all levels of the training curriculum.119 Abbreviations Used APA American Psychiatric Association BDD Body Dysmorphic Disorder DSM Diagnostic and Statistical Manual of Mental Disorders GD Gender Dysphoria GID Gender Identity Disorder GIDAANT Gender Identity Disorder of Adolescence and Adulthood Nontranssexual Type GIDC Gender Identity Disorder of Childhood HBIGDA Harry Benjamin International Gender Dysphoria Association HHS U.S. Department of Health and Human Services ICD International Classification of Diseases NIH National Institutes of Health SOC Standards of Care WPATH World Professional Association for Transgender Health Appendix Gender Dysphoria in Patients with Intersex Conditions As reviewed elsewhere,A1 Gender Dysphoria (GD) and patient-initiated gender transition occur with increased frequency in individuals with intersex conditions. Because Diagnostic and Statistical Manual of Mental Disorders-5 now allows gender-dysphoric individuals with somatic intersex conditions to receive the diagnosis of GD, psychiatrists need to be aware of assessment- and treatment-relevant characteristics of such individuals that differ from gender-dysphoric individuals without somatic intersexuality.A2 Intersex conditions are a subset of conditions relatively recently designated as “disorders of sex development”A3 or “differences of sex development” (DSD).A4 We use the term “intersex” in this document as our focus is on that subset of individuals with DSDs who were born with atypical external genitalia or lack of concordance among various sex characteristics such as sex chromosomes, gonads, or external genitalia so that questions often arise as to which gender should be assigned at birth. GD may develop from late preschool age through late adulthood with a range from 0% to ∼70% depending on the specific intersex syndrome, its severity (degree of androgen insensitivity, degree of 21-hydroxylase deficiency, degree of genital atypicality, etc.), the gender originally assigned, and the postnatal history of exposure to both endogenous and exogenous sex hormones.A5 Persons with the combination of GD and intersex condition encounter fewer barriers to legal gender reassignment, and the barriers to hormonal and surgical treatments are much lower.A1 This is because, depending on the particular condition, individuals with an intersex condition may have been gonadectomized (often due to concern about risk of malignancy) before puberty so that administration of exogenous hormones is required as part of routine care to induce puberty. In addition, infertility is quite common whether due to the condition itself or to gonadectomy, and genital surgery has often been done in infancy or childhood with the intent of affirming, both to the patient and the parents, the gender to which the individual was assigned. Furthermore, such early procedures may have been followed by additional surgical modifications in adolescence or young adulthood. Decisions regarding hormonal and surgical procedures are complicated by the highly variable somatic presentations of the various intersex conditions. Thus, to be fully effective, the mental health provider needs to be informed about the medical and surgical history of the individual,A6,A7 the available data on long-term gender development (e.g., contentment vs. dysphoria in the assigned gender), and other psychological outcomes of patients on a syndrome by syndrome basis.A5,A8 Moreover, intersex conditions are frequently associated with stigma, even in medical settings, which may result in shame and maladaptive coping mechanisms on the part of the patients as well as their parents.A9–A12 Providers need to be aware of the many ways in which some individuals with intersex conditions report having been stigmatized by their treatment by clinicians and parents (e.g., failure of age-appropriate disclosure of their condition, attempts to modify their gender expression, and repeated genital examinationsA9,A13). Efforts are under way to develop decision-making tools and clinical checklists to ensure that parents and affected children are adequately assessed and informed as active participants in decision-making processes and that the intersex condition and its ramifications are disclosed to the affected individual in an age-appropriate manner.A14 Gender evaluation The questionnaires and interview schedules developed for the assessment of gender development in transgender individuals who do not have an intersex conditionA15,A16 apply to those with intersex conditions as well, but need to be complemented by detailed medical, surgical, and related psychosocial histories, including the histories of disclosure to the patient of her/his medical condition, efforts made to reinforce the initial gender assignment, and responses by parents and providers to behaviors perceived as atypical with respect to the gender assignment. Mental health providers should also assess the patient's knowledge of their surgical history, their understanding of the implications with respect to fertility and gender-affirming hormonal and surgical procedures, and any history of shaming or other stigma due to their condition, or perceived gender atypicality with respect to their gender assigned at birth. Decisions regarding gender transition For individuals with intersex conditions, GD usually raises the question of transition to a different gender, and all issues of relevance to transgender persons without these conditions should also be considered here. Yet, the situation is often more complex than in GD in the absence of an intersex condition. Factors contributing to the desire to transition may include the awareness of the discrepancy between assigned gender and genetic factors such as the karyotype, anatomic factors such as the type of gonads, and secondary sex characteristics like breast development in men or hirsutism and masculine habitus in women. Related psychosocial influences may derive from being misidentified as the “other” gender or from frank stigmatization due to gender-atypical physical features. Different cultures and even subcultures within a given country may differ in the roles (including rights) associated with one's gender, and in the salience and weight of criteria used in decision-making on gender reassignment.A17,A18 When discussing gender options, clinicians need to consider the legal regulations of the country in which they work as well as the religious and other ideologies that can influence the gender perspectives of patients (and of caregivers for minors). These considerations are also very important when doing clinical work with visitors or immigrants from foreign countries. Thus, the viewpoints of patients (and caregivers) within their cultural contexts should be explored in detail and taken into consideration when these individuals are provided with psychoeducation about gender and other issues related to their intersex conditions.A19 As with other transgender patients, when working with patients with an intersex condition and GD, clinicians should engage the patient in a detailed discussion of their expectations from the gender transition: the social effects of public gender change as well as the medical and social effects of the attendant change in hormone treatment and, if desired, of genital or chest surgery. Some of their expectations may be unrealistic, and after detailed discussion, some patients may modify the hormonal and/or surgical treatments they desire or decide against medical treatments or legal gender change, and pursue other ways of finding authenticity in their gender expression. Patients may be happy with their gender-atypical bodies and/or adapt a nonbinary gender identity such as “intersex.” Mental health providers should not assume that patients would benefit from conforming to fit within a gender binary, physically or with respect to gender identity. Empathic listening is especially important in working with intersex individuals, perhaps particularly with those who have inadvertently discovered their intersex status in adolescence or adulthood, and may have been stigmatized for gender nonconformity or homosexuality, or subjected to irreversible hormonal or surgical treatments consistent with their assigned rather than their experienced gender. Upon discovery of their biological status, such patients may feel betrayed by their parents and physicians, feeling they colluded to keep them in ignorance of their medical condition, damaged their bodies, or punished or stigmatized them for their gendered behaviors. Such patients need empathic validation of their feelings. Assurance that parents and providers had their best intentions at heart, while usually true, is likely to be experienced as an empathic failure and negatively impact the formation of a therapeutic alliance. As is often seen in many individuals with uncommon medical conditions, many people with intersex conditions experience varying degrees of isolation and loneliness.A1 Therefore, linking them to existing intersex support groups by internet or face-to-face meetings can be very beneficial. Despite the emotional relief that support groups can provide, such contacts may sometimes cause additional concerns. For instance, the composition of the group (e.g., the syndromes represented within the group, the personalities of some group members, or the goals of the group) may not meet the individual's expectations, and the information provided may not always be accurate. Thus, some monitoring of the patient's experience with the chosen group is recommended. Hormonal and surgical treatments As reviewed elsewhere,A1 many individuals with both an intersex condition and GD will be agonadal in later adolescence or adulthood, either because they were born that way (e.g., in syndromes involving gonadal dysgenesis) or due to surgery, for instance, for the prevention of gonadal malignancy. In those with intact gonads (especially 46,XX congenital adrenal hyperplasia raised female), loss of fertility may be another issue of concern. Persons who are agonadal are usually on hormone replacement therapy by the time of late adolescence. Cessation of that treatment, change to treatment with hormones congruent with their gender identity, patient education for informed consent, and the monitoring of treatment effects are tasks of the endocrinologist. Also, the technical aspects of genital surgery are more complex than in patients receiving gender-confirming genital surgeries, who do not have intersex conditions. Both the external genitalia and the internal reproductive tract in intersex conditions typically differ from what most surgeons are familiar with in transgender patients without these conditions. In addition, many patients with intersex conditions have already undergone one or more genital surgeries by late adolescence. The resulting postsurgical anatomy constitutes an additional challenge for the surgeon performing gender-confirming surgery, and a good sex-functional outcome may be more difficult to achieve. Mental health providers should also be aware that not all individuals who identify their gender or gender identity as intersex have a somatic intersex condition, and should ensure that those who do have an intersex condition are receiving adequate medical care, including hormones (to prevent osteoporosis) and cancer screenings, as appropriate to their particular condition.A3 Without challenging a patient's identity label, this distinction can usually be made by inquiring about the name of the patient's condition, when and how they learned of it, and any history of related surgeries, hormonal replacement, or ongoing follow-up evaluations. If there is any doubt, appropriate referrals should be made to ensure that the patient is receiving adequate follow-up and treatment. Appendix References A1. Byne W, Bradley SJ, Coleman E, et al. Report of the American Psychiatric Association task force on treatment of gender identity disorder. Arch Sex Behav. 2012;41:759–796 [DOI] [PubMed] [Google Scholar] A2. Meyer-Bahlburg HFL. Variants of gender differentiation in somatic disorders of sex development: recommendations for Version 7 of the World Professional Association for Transgender Health's Standards of Care. Int J Transgend. 2009;11:226–237 [Google Scholar] A3. Hughes IA, Houk C, Ahmed SF, et al. Consensus statement on management of intersex disorders. J Pediatr Urol. 2006;2:148–162 [DOI] [PubMed] [Google Scholar] A4. Johnson EK, Rosoklija I, Finlayson C, et al. Attitudes towards “disorders of sex development” nomenclature among affected individuals. J Pediatr Urol. 2017;13:402–413 [DOI] [PubMed] [Google Scholar] A5. Meyer-Bahlburg HFL. Introduction: gender dysphoria and gender change in persons with intersexuality. Arch Sex Behav. 2005;34:371–373 [DOI] [PubMed] [Google Scholar] A6. Rey RA, Josso N. Diagnosis and treatment of disorders of sexual development. In: Endocrinology: Adult and Pediatric, 7th ed. Vol II. (Jameson JL, De Groot LJ; eds). Philadelphia, PA: Elsevier/Saunders, 2016, pp. 2086–2118 [Google Scholar] A7. New MI, Lekarev O, Parsa A, et al. (eds). Genetic Steroid Disorders. London, UK: Academic Press/Elsevier, 2014 [Google Scholar] A8. Meyer-Bahlburg HFL. Psychoendocrinology of congenital adrenal hyperplasia. In: Genetic Steroid Disorders (New MI, Lekarev O, Parsa A, et al.; eds). London, UK: Academic Press/Elsevier, 2014, pp. 285–300 [Google Scholar] A9. Consortium on the Management of Disorders of Sex Development. Handbook for Parents. Rohnert Park, CA: Intersex Society of North America, 2006 [Google Scholar] A10. Meyer-Bahlburg HFL, Khuri J, Reyes-Portillo J, et al. Stigma associated with classical congenital adrenal hyperplasia in women's sexual lives. Arch Sex Behav. 2018;47:943–951 [DOI] [PubMed] [Google Scholar] A11. Meyer-Bahlburg HF, Reyes-Portillo JA, Khuri J, et al. Syndrome-related stigma in the general social environment as reported by women with classical congenital adrenal hyperplasia. Arch Sex Behav. 2017;46:341–351 [DOI] [PubMed] [Google Scholar] A12. Meyer-Bahlburg HFL, Khuri J, Reyes-Portillo J, New MI. Stigma in medical settings as reported retrospectively by women with congenital adrenal hyperplasia (CAH) for their childhood and adolescence. J Pediatr Psychol. 2017;42:496–503 [DOI] [PMC free article] [PubMed] [Google Scholar] A13. Consortium on the Management of Disorders of Sex Development. Clinical Guidelines for the Management of Disorders of Sex Development. Rohnert Park, CA: Intersex Society of North America, 2006 [Google Scholar] A14. Siminoff LA, Sandberg DE. Promoting shared decision making in disorders of sex development (DSD): decision aids and support tools. Horm Metabol Res. 2015;47:335–339 [DOI] [PubMed] [Google Scholar] A15. Zucker KJ. Measurement of psychosexual differentiation. Arch Sex Behav. 2005;34:375–388 [DOI] [PubMed] [Google Scholar] A16. Meyer-Bahlburg HFL. Gender monitoring and gender reassignment of children and adolescents with a somatic disorder of sex development. Child Adolesc Psychiatr Clin N Am 2011;20:639–649 [DOI] [PubMed] [Google Scholar] A17. Lang C, Kuhnle U. Intersexuality and alternative gender categories in non-Western cultures. Horm Res. 2008;69:240–250 [DOI] [PubMed] [Google Scholar] A18. Meyer-Bahlburg HF. Introduction to the special section on culture and variants of sex/gender: gias and stigma. Arch Sex Behav. 2017;46:337–339 [DOI] [PubMed] [Google Scholar] A19. Meyer-Bahlburg HF, Baratz Dalke K, Berenbaum SA, et al. Gender assignment, reassignment and outcome in disorders of sex development: update of the 2005 Consensus Conference. Horm Res Paediatr. 2016;85:112–118 [DOI] [PubMed] [Google Scholar] Author Disclosure Statement No author has any conflict of interest to report. Cite this article as: Byne W, Karasic DH, Coleman E, Eyler AE, Kidd JD, Meyer-Bahlburg HFL, Pleak RR, Pula J (2018) Gender dysphoria in adults: an overview and primer for psychiatrists, Transgender Health 3:1, 57–A3, DOI: 10.1089/trgh.2017.0053. References Weismantel M. Towards a transgender archeology: a queer rampage through prehistory. In: The Transgender Studies Reader 2 (Stryker S, Whittle S; eds). New York: Routledge, 2013, pp. 319–334 [Google Scholar] Drescher J. Queer diagnoses: parallels and contrasts in the history of homosexuality, gender variance, and the diagnostic and statistical manual. Arch Sex Behav. 2010;39:427–460 [DOI] [PubMed] [Google Scholar] Grant JM, Mottet LA, Tanis J, et al. Injustice at Every Turn: A Report of the National Transgender Discrimination Survey. Washington, DC: National Center for Transgender Equality, 2010. Available at www.thetaskforce.org/static_html/downloads/reports/reports/ntds_full.pdf Accessed February11, 2018 [Google Scholar] Meyerowitz J. How Sex Changed: A History of Transsexuality in the United States. Boston, MA: Harvard University Press, 2004 [Google Scholar] Center for Medicare and Medicaid Services. NCD 140.3, Transsexual Surgery, 1989 Department of Health and Human Services Departmental Appeals Board Appellate Division. NCD 140.3, Transsexual Surgery. Docket No. A-13-87. Decision No. 25762014 Coleman E, Bockting W, Botzer M, et al. Standards of care for the health of transsexual, transgender, and gender-nonconforming people, version 7. Int J Transgend. 2011;13:165–232 [Google Scholar] American Psychological Association. Guidelines for psychological practice with transgender and gender nonconforming people. Am Psychol. 2015;70:832–864 [DOI] [PubMed] [Google Scholar] Byne W, Bradley SJ, Coleman E, et al. Report of the American Psychiatric Association Task Force on Treatment of Gender Identity Disorder. Arch Sex Behav. 2012;41:759–796 [DOI] [PubMed] [Google Scholar] Hembree WC, Cohen-Kettenis PT, Gooren L, et al. Endocrine treatment of gender-dysphoric/gender-incongruent persons: an endocrine society clinical practice guideline. J Clin Endocrinol Metab. 2017;102:3869–3903 [DOI] [PubMed] [Google Scholar] Lusztig TB. Deducting the cost of sex reassignment surgery: how O'Donnabhain v. Commissioner can help us make sense of medical expense deduction. Columbia J Tax Law. 2012;3:86–112 [Google Scholar] Aetna. Gender reassignment surgery. Available at www.aetna.com/cpb/medical/data/600_699/0615.html Accessed January17, 2018 Medicaid New York State. Transgender related care and services. Available at Accessed January17, 2018 Massachusetts Blue Cross Blue Shield. Medical policy transgender services. Available at www.bluecrossma.com/common/en_US/medical_policies/189%20Transgender%20Services%20prn.pdf Accessed January17, 2017 Obedin-Maliver J, Goldsmith ES, Stewart L, et al. Lesbian, gay, bisexual, and transgender-related content in undergraduate medical education. JAMA. 2011;306:971–977 [DOI] [PubMed] [Google Scholar] Menvielle E. A comprehensive program for children with gender variant behaviors and gender identity disorders. J Homosex. 2012;59:357–368 [DOI] [PubMed] [Google Scholar] Zucker KJ, Wood H, Singh D, Bradley SJ. A developmental, biopsychosocial model for the treatment of children with gender identity disorder. J Homosex. 2012;59:369–397 [DOI] [PubMed] [Google Scholar] de Vries AL, Cohen-Kettenis PT. Clinical management of gender dysphoria in children and adolescents: the Dutch approach. J Homosex. 2012;59:301–320 [DOI] [PubMed] [Google Scholar] Edwards-Leeper L, Spack NP. Psychological evaluation and medical treatment of transgender youth in an interdisciplinary “Gender Management Service” (GeMS) in a major pediatric center. J Homosex. 2012;59:321–336 [DOI] [PubMed] [Google Scholar] Drescher J, Byne W. Gender variance and transsexuality. In: Comprehenisve Textbook of Psychiatry, IX ed. (Sadock BJ, Sadockl VA, Ruiz PR; eds). Baltimore, MD: Williams and Wilkins, 2017, pp. 2023–2039 [Google Scholar] World Health Organization. International Classification of Diseases, Ninth Revision. Geneva, Switzerland: World Health Organization, 1975 [Google Scholar] American Psychiatric Association. Diagnostic and Statistical Manual of Mental Disorders III. Washington, DC: American Psychiatric Association, 1980 [Google Scholar] American Psychiatric Association. Diagnostic and Statistical Manual of Mental Disorders, 3rd ed., rev. Washington, DC: American Psychiatric Association, 1987 [Google Scholar] American Psychiatric Association. Diagnostic and Statistical Manual of Mental Disorders IV. Washington, DC: American Psychiatric Association, 1994 [Google Scholar] Karasic D, Drescher J. Introduction. In: Sexual and Gender Diagnoses of the Diagnostic and Statistical Manual (DSM) (Karasic D, Drescher J; eds). New York: Routledge, 2005, pp. 1–5 [Google Scholar] Drescher J. Controversies in gender diagnoses. LGBT Health. 2014;1:10–14 [DOI] [PubMed] [Google Scholar] American Psychiatric Association. Diagnostic and Statistical Manual of Mental Disorders, 5th ed. Arlington, VA: American Psychiatric Publishing, 2013 [Google Scholar] Zucker KJ, Cohen-Kettenis PT, Drescher J, et al. Memo outlining evidence for change for gender identity disorder in the DSM-5. Arch Sex Behav. 2013;42:901–914 [DOI] [PubMed] [Google Scholar] Reed GM, Drescher J, Krueger RB, et al. Disorders related to sexuality and gender identity in the ICD-11: revising the ICD-10 classification based on current scientific evidence, best clinical practices, and human rights considerations. World Psychiatry. 2016;15:205–221 [DOI] [PMC free article] [PubMed] [Google Scholar] Deutsch MB. Making it count: improving estimates of the size of transgender and gender nonconforming populations. LGBT Health. 2016;3:181–185 [DOI] [PubMed] [Google Scholar] Winter S, Diamond M, Green J, et al. Transgender people: health at the margins of society. Lancet. 2016;388:390–400 [DOI] [PubMed] [Google Scholar] Bakker A, van Kesteren PJ, Gooren LJ, Bezemer PD. The prevalence of transsexualism in the Netherlands. Acta Psychiatr Scand. 1993;87:237–238 [DOI] [PubMed] [Google Scholar] De Cuypere G, Van Hemelrijck M, Michel A, et al. Prevalence and demography of transsexualism in Belgium. Eur Psychiatry. 2007;22:137–141 [DOI] [PubMed] [Google Scholar] Dhejne C, Öberg K, Arver S, Landén M. An analysis of all applications for sex reassignment surgery in Sweden, 1960–2010: prevalence, incidence, and regrets. Arch Sex Behav. 2014;43:1535–1545 [DOI] [PubMed] [Google Scholar] Blosnich JR, Brown GR, Shipherd JC, et al. Prevalence of gender identity disorder and suicide risk among transgender veterans utilizing veterans health administration care. Am J Public Health. 2013;104:S532–S534 [DOI] [PMC free article] [PubMed] [Google Scholar] Flores AR, Herman JL, Gages GJ, Brown TNT. How many adults identify as transgender in the United States. 2016. Available at Accessed June10, 2017 Conron KJ, Scott G, Stowell GS, Landers SJ. Transgender health in Massachusetts: results from a household probability sample of adults. Am J Public Health. 2012;102:118–122 [DOI] [PMC free article] [PubMed] [Google Scholar] Kuyper L, Wijsen C. Gender identities and gender dysphoria in the Netherlands. Arch Sex Behav. 2014;43:377–385 [DOI] [PubMed] [Google Scholar] Clark TC, Lucassen MF, Bullen P, et al. The health and well-being of transgender high school students: results from the New Zealand adolescent health survey (Youth’2012). J Adolesc Health. 2014;55:93–99 [DOI] [PubMed] [Google Scholar] Shields JP, Cohen R, Glassman JR, et al. Estimating population size and demographic characteristics of lesbian, gay, bisexual, and transgender youth in middle school. J Adolesc Health. 2013;52:248–250 [DOI] [PubMed] [Google Scholar] Arnold AP. A general theory of sexual differentiation. J Neurosci Res. 2017;95:291–300 [DOI] [PMC free article] [PubMed] [Google Scholar] Nugent BM, Wright CL, Shetty AC, et al. Brain feminization requires active repression of masculinization via DNA methylation. Nat Neurosci. 2015;18:690–697 [DOI] [PMC free article] [PubMed] [Google Scholar] Schulz KM, Molenda-Figueira HA, Sisk CL. Back to the future: the organizational-activational hypothesis adapted to puberty and adolescence. Horm Behav. 2009;55:597–604 [DOI] [PMC free article] [PubMed] [Google Scholar] Sisk CL, Zehr JL. Pubertal hormones organize the adolescent brain and behavior. Front Neuroendocrinol. 2005;26:163–174 [DOI] [PubMed] [Google Scholar] Brunton PJ, Russell JA. The expectant brain: adapting for motherhood. Nat Rev Neurosci. 2008;9:11–25 [DOI] [PubMed] [Google Scholar] Ritchie SJ, Cox SR, Shen X, et al. Sex differences in the adult human brain: evidence from 5,216 UK Biobank participants. bioRxiv 2017. Available at Accessed February11, 2018 [DOI] [PMC free article] [PubMed] Hoekzema E, Barba-Muller E, Pozzobon C, et al. Pregnancy leads to long-lasting changes in human brain structure. Nat Neurosci. 2017;20:287–296 [DOI] [PubMed] [Google Scholar] Schulz KM, Sisk CL. The organizing actions of adolescent gonadal steroid hormones on brain and behavioral development. Neurosci Biobehav Rev. 2016;70:148–158 [DOI] [PMC free article] [PubMed] [Google Scholar] Zucker KJ. Intersexuality and gender identity differentiation. Annu Rev Sex Res. 1999;10:1–69 [PubMed] [Google Scholar] Meyer-Bahlburg HF, Baratz Dalke K, Berenbaum SA, et al. Gender assignment, reassignment and outcome in disorders of sex development: update of the 2005 Consensus Conference. Horm Res Paediatr. 2016;85:112–118 [DOI] [PubMed] [Google Scholar] Byne W. In: the sexed and gendered brain. In: Gender Specific Medicine (Legato MJ; ed). New York, NY: Academic Press, 2010, pp. 101–112 [Google Scholar] Houk CP, Lee PA. Intersexed states: diagnosis and management. Endocrinol Metab Clin North Am. 2005;34:791–810 [DOI] [PubMed] [Google Scholar] Smith ES, Junger J, Derntl B, Habel U. The transsexual brain—a review of findings on the neural basis of transsexualism. Neurosci Biobehav Rev. 2015;59:251–266 [DOI] [PubMed] [Google Scholar] Bramble MS, Roach L, Lipson A, et al. Sex-specific effects of testosterone on the sexually dimorphic transcriptome and epigenome of embryonic neural stem/progenitor cells. Sci Rep. 2016;6:3691–6. [DOI] [PMC free article] [PubMed] [Google Scholar] Goy RW, McEwen BS. Sexual Differentiation of the Brain. Cambridge, MA: MIT Press, 1980 [Google Scholar] Guillamon A, Junque C, Gomez-Gil E. A review of the status of brain structure research in transsexualism. Arch Sex Behav. 2016;45:1615–1648 [DOI] [PMC free article] [PubMed] [Google Scholar] Swaab DF, Garcia-Falgueras A. Sexual differentiation of the human brain in relation to gender identity and sexual orientation. Funct Neurol. 2009;24:17–28 [PubMed] [Google Scholar] Luders E, Toga AW. Sex differences in brain anatomy. Prog Brain Res. 2010;186:3–12 [DOI] [PubMed] [Google Scholar] Staphorsius AS, Kreukels BP, Cohen-Kettenis PT, et al. Puberty suppression and executive functioning: an fMRI-study in adolescents with gender dysphoria. Psychoneuroendocrinol. 2015;56:190–199 [DOI] [PubMed] [Google Scholar] Rametti G, Carrillo B, Gomez-Gil E, et al. White matter microstructure in female to male transsexuals before cross-sex hormonal treatment. A diffusion tensor imaging study. J Psychiatr Res. 2011;45:199–204 [DOI] [PubMed] [Google Scholar] Rametti G, Carrillo B, Gomez-Gil E, et al. The microstructure of white matter in male to female transsexuals before cross-sex hormonal treatment. A DTI study. J Psychiatr Res. 2011;45:949–954 [DOI] [PubMed] [Google Scholar] Hahn A, Kranz GS, Kublbock M, et al. Structural connectivity networks of transgender people. Cereb Cortex. 2015;25:3527–3534 [DOI] [PMC free article] [PubMed] [Google Scholar] Burke SM, Menks WM, Cohen-Kettenis PT, et al. Click-evoked otoacoustic emissions in children and adolescents with gender identity disorder. Arch Sex Behav. 2014;43:1515–1523 [DOI] [PubMed] [Google Scholar] Mueller SC, De Cuypere G, T'Sjoen G. Transgender research in the 21st century: a selective critical review from a neurocognitive perspective. Am J Psychiatry. 2017;174:1155–1162 [DOI] [PubMed] [Google Scholar] Blakemore JE, Berenbaum S, Liben LS. Gender Development. New York, NY: Psychology Press, 2008 [Google Scholar] de Vries AL, Doreleijers TA, Cohen-Kettenis PT. Disorders of sex development and gender identity outcome in adolescence and adulthood: understanding gender identity development and its clinical implications. Pediatr Endocrinol Rev. 2007;4:343–351 [PubMed] [Google Scholar] Meyer-Bahlburg H (ed). Hormonal and Genetic Basis of Sexual Differentiation Disorders and Hot Topics in Endocrinology: Proceedings of the 2nd World Conference (Advances in Experimental Medicine and Biology, Vol 707) New York, NY: Springer, 2011 [Google Scholar] Steensma TD, Kreukels BP, de Vries AL, Cohen-Kettenis PT. Gender identity development in adolescence. Horm Behav. 2013;64:288–297 [DOI] [PubMed] [Google Scholar] Wood H, Sasaki S, Bradley SJ, et al. Patterns of referral to a gender identity service for children and adolescents (1976–2011): age, sex ratio, and sexual orientation. J Sex Marital Ther. 2013;39:1–6 [DOI] [PubMed] [Google Scholar] Aitken M, Steensma TD, Blanchard R, et al. Evidence for an altered sex ratio in clinic-referred adolescents with gender dysphoria. J Sex Med. 2015;12:756–763 [DOI] [PubMed] [Google Scholar] Bockting W. Psychotherapy and the real-life experience: from gender dichotomy to gender diversity. Sexologies. 2008;17:211–224 [Google Scholar] Kuper LE, Nussbaum R, Mustanski B. Exploring the diversity of gender and sexual orientation identities in an online sample of transgender individuals. J Sex Res. 2012;49:244–254 [DOI] [PubMed] [Google Scholar] Steensma TD, Cohen-Kettenis PT. More than two developmental pathways in children with gender dysphoria. J Am Acad Child Adolesc Psychiatry. 2015;54:147–148 [DOI] [PubMed] [Google Scholar] Drummond KD, Bradley SJ, Peterson-Badali M, et al. Behavior problems and psychiatric diagnoses in girls with gender identity disorder: a follow-up study. J Sex Marital Ther. 2018;44:172–187 [DOI] [PubMed] [Google Scholar] Wallien MS, Cohen-Kettenis PT. Psychosexual outcome of gender-dysphoric children. J Am Acad Child Adolesc Psychiatry. 2008;47:1413–1423 [DOI] [PubMed] [Google Scholar] Zucker KJ, Bradley SJ, Sullivan CB, et al. A gender identity interview for children. J Pers Assess. 1993;61:443–456 [DOI] [PubMed] [Google Scholar] Steensma TD, McGuire JK, Kreukels BP, et al. Factors associated with desistence and persistence of childhood gender dysphoria: a quantitative follow-up study. J Am Acad Child Adolesc Psychiatry. 2013;52:582–590 [DOI] [PubMed] [Google Scholar] Doorduin T, van Berlo W. Trans people's experience of sexuality in the Netherlands: a pilot study. J Homosex. 2014;6:654–672 [DOI] [PubMed] [Google Scholar] Veale JF, Clarke DE, Lomax TC. Sexuality of male-to-female transsexuals. Arch Sex Behav. 2008;37:586–597 [DOI] [PubMed] [Google Scholar] Nieder TO, Elaut E, Richards C, Dekkar A. Sexual orientation of trans adults is not linked to outcome of transition-related health care, but worth asking. Int Rev Psychiatry. 2016;28:103–111 [DOI] [PubMed] [Google Scholar] Drescher J, Byne W. (eds). Treating Transgender Children and Adolescents. New York, NY: Routledge, 2013 [Google Scholar] American Academy of Child and Adolescent Psychiatry Committee on Quality Issues. Practice parameter on gay, lesbian, or bisexual sexual orientation, gender nonconformity, and gender discordance in children and adolescents. J Am Acad Child Adolesc Psychiatry. 2012;51:957–974 [DOI] [PubMed] [Google Scholar] Bockting WO, Knudson G, Goldberg JM. Counseling and mental health care for transgender adults and loved ones. Int J Transgend. 2006;9:185–208 [Google Scholar] Drescher J, Schwartz A, Casoy F, et al. The growing regulation of conversion therapy. J Med Regul. 2016;102:7–12 [PMC free article] [PubMed] [Google Scholar] Byne W. Regulations restrict practice of conversion therapy. LGBT Health. 2016;3:97–99 [DOI] [PubMed] [Google Scholar] Institute of Medicine. The Health of Lesbian, Gay, Bisexual, and Transgender People: Building a Foundation for Better Understanding. Washington, DC: The National Academies Press, 2011 [PubMed] [Google Scholar] Meyer IH. Prejudice, social stress, and mental health in lesbian, gay, and bisexual populations: conceptual issues and research evidence. Psychol Bull. 2003;129:674–697 [DOI] [PMC free article] [PubMed] [Google Scholar] Cohen JM, Blasey C, Barr Taylor C, et al. Anxiety and related disorders and concealment in sexual minority young adults. Behav Ther. 2016;47:91–101 [DOI] [PMC free article] [PubMed] [Google Scholar] Reisner SL, Greytak EA, Parsons JT, Ybarra ML. Gender minority social stress in adolescence: disparities in adolescent bullying and substance use by gender identity. J Sex Res. 2015;52:243–256 [DOI] [PMC free article] [PubMed] [Google Scholar] Brown GR, Jones KT. Mental health and medical health disparities in 5135 transgender veterans receiving healthcare in the Veterans Health Administration: a case-control study. LGBT Health. 2016;3:122–131 [DOI] [PubMed] [Google Scholar] The Joint Commission. Advancing Effective Communication, Cultural Competence, Patient and Family Centered Care for the Lesbian, Gay, Bisexual and Transgender (LGBT) Community: A Field Guide. Oak Brook, IL: The Joint Commission, 2011. Available at Accessed February11, 2018 [Google Scholar] Mandelli L, Nearchou FA, Vaiopoulos C, et al. Neuroticism, social network, stressful life events: association with mood disorders, depressive symptoms and suicidal ideation in a community sample of women. Psychiatry Res. 2015;226:38–44 [DOI] [PubMed] [Google Scholar] Moody C, Smith NG. Suicide protective factors among trans adults. Arch Sex Behav. 2013;42:739–752 [DOI] [PMC free article] [PubMed] [Google Scholar] Terada S, Matsumoto Y, Sato T, et al. Suicidal ideation among patients with gender identity disorder. Psychiatry Res. 2011;190:159–162 [DOI] [PubMed] [Google Scholar] Heylens GEE, Kreukels BP, Paap MC, et al. Psychiatric characteristics in transsexual individuals: multicentre study in four European countries. Br J Psychiatry. 2014;204:151–156 [DOI] [PubMed] [Google Scholar] Karasic DH. Transgender and gender nonconforming patients. In: Clinical Manual of Cultural Psychiatry, 2nd ed. (Lim RF; ed). Washington, DC: American Psychiatric Publishing, 2015, pp. 397–410 [Google Scholar] Hembree WC, Cohen-Kettenis P, Delemarre-van de Waal HA, et al. Endocrine treatment of transsexual persons: an endocrine society clinical practice guideline. J Clin Endocrinol Metab. 2009;94:3132–3154 [DOI] [PubMed] [Google Scholar] Meijer JH, Eeckhout GM, van Vlerken RH, de Vries AL. Gender dysphoria and co-existing psychosis: review and four case examples of successful gender affirmative treatment. LGBT Health. 2017;4:106–114 [DOI] [PubMed] [Google Scholar] Jacobs LA, Rachlin K, Erickson-Schroth L, Janssen A. Gender dysphoria and co-occurring autism spectrum disorders: review, case examples, and treatment considerations. LGBT Health. 2014;1:277–282 [DOI] [PubMed] [Google Scholar] à Campo J, Nijman H, Merckelbach H, Evers C. Psychiatric comorbidity of gender identity disorders: a survey among Dutch psychiatrists. Am J Psychiatry. 2003;160:1332–1336 [DOI] [PubMed] [Google Scholar] Gorin-Lazard A, Baumstarck K, Boyer L, et al. Hormonal therapy is associated with better self-esteem, mood, and quality of life in transsexuals. J Nerv Ment Dis. 2013;201:996–1000 [DOI] [PubMed] [Google Scholar] Davis S, Meier CS. Effects of testosterone treatment and chest reconstruction surgery on mental health and sexuality in female-to-male transgender people. Int J Sex Health. 2014;26:113–128 [Google Scholar] Gomez-Gil E, Zubiaurre-Elorza L, Esteva I, et al. Hormone-treated transsexuals report less social distress, anxiety and depression. Psychoneuroendocrinol. 2012;37:662–670 [DOI] [PubMed] [Google Scholar] White HJM, Reisner SL. A systematic review of the effects of hormone therapy on psychological functioning and quality of life in transgender individuals. Transgend Health. 2016;1:21–31 [DOI] [PMC free article] [PubMed] [Google Scholar] Onakomaiya MM, Henderson LP. Mad men, women and steroid cocktails: a review of the impact of sex and other factors on anabolic androgenic steroids effects on affective behaviors. Psychopharmacol. 2016;233:549–569 [DOI] [PMC free article] [PubMed] [Google Scholar] Moore E, Wisniewski A, Dobs A. Endocrine treatment of transsexual people: a review of treatment regimens, outcomes, and adverse effects. J Endocrinol Metab. 2003;88:3467–3473 [DOI] [PubMed] [Google Scholar] Ettner R, Monstrey S, Coleman E. (eds). Principles of Transgender Medicine and Surgery. New York, NY: Routledge, 2016 [Google Scholar] Karasic DH. Mental Health Care and Assessment of Transgender Adults. 2015. Available at www.lgbthealtheducation.org/topic/transgender-health Accessed February1, 2018 U.S. Department of Health and Human Services. Non-discrimination in Health Programs and Activities Proposed Rule. Available at www.hhs.gov/civil-rights/for-individuals/section-1557/nondiscrimination-health-programs-and-activities-proposed-rule/index.html Accessed March28, 2018 FEHB Program Carrier Letter 2015-12. U.S. Office of Personnel Management. Available at www.opm.gov/healthcare-insurance/healthcare/carriers/2015/2015-12.pdf Accessed March28, 2018 Byne W. Sustaining progress toward LGBT health equity: a time for vigilance, advocacy, and scientific inquiry. LGBT Health. 2017;4:1–3 [DOI] [PubMed] [Google Scholar] Rosenberg M. Transgender people will be allowed to serve openly in military. New York Times, 2016. Available at www.nytimes.com/2016/07/01/us/transgender-military.html Accessed February11, 2018 [Google Scholar] Byne W. A year into the Trump administration: LGBT health equity fighting to stand ground. LGBT Health. 2018;5:1–529324176 [Google Scholar] Cohen-Kettenis PT, Pfäfflin F. The DSM diagnostic criteria for gender identity disorder in adolescents and adults. Arch Sex Behav. 2010;39:499–513 [DOI] [PubMed] [Google Scholar] American Psychiatric Association. Position Statement on Discrimination Against Transgender and Gender Variant Individuals: American Psychiatric Association. 2012. Available at www.wapsychiatry.org/assets/documents/2017/position-2012-transgender-gender-variant-discrimination.pdf Accessed February11, 2018 Drescher J, Haller E. (eds). Position Statement on Access to Care for Transgender and Gender Variant Indivicuals. Washington, DC: American Psychiatric Association, 2012 [Google Scholar] National Institutes of Health. NIH FY 2016–2020 Strategic Plan to Advance Research on the Health and Well-being of Sexual and Gender Minorities. Bethesda, MD: National Institutes of Health, 2015 [Google Scholar] The Accreditation Council for Graduate Medical Education and The American Board of Psychiatry and Neurology. The Psychiatry Milestone Project, 2015. Available at www.acgme.org/Portals/0/PDFs/Milestones/PsychiatryMilestones.pdf Accessed February11, 2018 Association of American Medical Colleges. Implementing Curricular and Institutional Climate changes to Improve Health Care for Individuals Who Are LGBT, Gender Nonconforming, or Born with DSD. Washington, DC: Association of American Medical Colleges, 2014 [Google Scholar] Articles from Transgender Health are provided here courtesy of Mary Ann Liebert, Inc. ACTIONS View on publisher site PDF (1.2 MB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Abstract Introduction Diagnostic and Statistical Manual of Mental Disorders and Transgender-Related Nosology Epidemiology Gender Development Mental Health Assessment and Treatment Current Social Issues: Stigmatization and Access to Care Conclusions Abbreviations Used Appendix Author Disclosure Statement References Cite Copy Download .nbib.nbib Format: Add to Collections Create a new collection Add to an existing collection Name your collection Choose a collection Unable to load your collection due to an error Please try again Add Cancel Follow NCBI NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top
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15116	http://www.irisa.fr/dionysos/pages_perso/sericola-old/PAPIERS/IJOC18.pdf	INFORMS JOURNAL ON COMPUTING Vol. 00, No. 0, Xxxxx 0000, pp. 000–000 issn 0899-1499\|eissn 1526-5528\|00\|0000\|0001 INFORMS doi 10.1287/xxxx.0000.0000 c ⃝0000 INFORMS Authors are encouraged to submit new papers to INFORMS journals by means of a style ﬁle template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named jour-nal. INFORMS journal templates are for the exclusive purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print or online or to submit the papers to another publication. Probabilistic Analysis of Rumor Spreading Time Yves Mocquard Universit´ e de Rennes 1 - IRISA, France. yves.mocquard@irisa.fr Bruno Sericola Inria Rennes - Bretagne Atlantique, France. bruno.sericola@inria.fr Emmanuelle Anceaume CNRS - IRISA, Rennes, France. emmanuelle.anceaume@irisa.fr The context of this work is the well studied dissemination of information in large scale distributed networks through pairwise interactions. This problem, originally called rumor mongering, and then rumor spreading has mainly been investigated in the synchronous model. This model relies on the assumption that all the nodes of the network act in synchrony, that is, at each round of the protocol, each node is allowed to contact a random neighbor. In this paper, we drop this assumption under the argument that it is not realistic in large scale systems. We thus consider the asynchronous variant, where at random times, nodes successively interact by pairs exchanging their information on the rumor. In a previous paper, we performed a study of the total number of interactions needed for all the nodes of the network to discover the rumor. While most of the existing results involve huge constants that do not allow us to compare diﬀerent protocols, we provided a thorough analysis of the distribution of this total number of interactions together with its asymptotic behavior. In this paper we extend this discrete time analysis by solving a conjecture proposed previously and we consider the continuous time case, where a Poisson process is associated to each node to determine the instants at which interactions occur. The rumor spreading time is thus more realistic since it is the real time needed for all the nodes of the network to discover the rumor. Once again, as most of the existing results involve huge constants, we provide tight bound and equivalent of the complementary distribution of the rumor spreading time. We also give the exact asymptotic behavior of the complementary distribution of the rumor spreading time around its expected value when the number of nodes tends to inﬁnity. Key words : rumor spreading time, pairwise interactions, Poisson process, Markov chain, analytic performance evaluation 1 Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time 2 INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS 1. Introduction Randomized rumor spreading is an important mechanism that allows the dissemination of information in large and complex networks through pairwise interactions. This mechanism initially proposed by Demers et al. (1987) for the update of a database replicated at diﬀerent sites, has then been adopted in many applications ranging from resource discovery as in Harchol-Balter et al. (1999), data-aggregation as in Kempe et al. (2003), complex distributed applications as in Censor-Hillel et al. (2012), or virus propagation in computer networks as in Berger et al. (2005), to mention just a few. A lot of attention has been devoted to the design and study of randomized rumor spread-ing algorithms. Initially, some rumor is placed on one of the nodes of a given network, and this rumor is propagated to all the nodes of the network through pairwise interactions between nodes. One of the important questions raised by these protocols is the spreading time, that is time it needs for the rumor to be known by all the nodes of the network. Several models have been considered to answer this question. The most studied one is the synchronous push-pull model, also called the synchronous random phone call model. This model assumes that all the nodes of the network act in synchrony, which allows the algorithms designed in this model to divide time in synchronized rounds. During each synchronized round, each node i of the network selects at random one of its neighbor j and either sends to j the rumor if i knows it (push operation) or gets the rumor from j if j knows the rumor (pull operation). In the synchronous model, the spreading time of a rumor is deﬁned as the number of synchronous rounds necessary for all the nodes to know the rumor. In one of the ﬁrst papers dealing with the push operation only, Frieze and Grimmet (1985) proved that when the underlying graph is complete, the ratio of the number of rounds over log2(n) converges in probability to 1 + ln(2) when the number n of nodes in the graph tends to inﬁnity. Further results have been established (see for example Pittel (1987), Karp et al. (2000) and the references therein), the most recent ones resulting from the observation that the rumor spreading time is closely related to the conductance of the graph of the network, see Giakkoupis (2011). Investigations have also been done in diﬀerent topologies of the network as in Chierichetti et al. (2011), Daum et al. (2016), Fountoulakis and Panagiotou (2013), Panagiotou et al. (2015), in the presence of link or nodes failures as in Feige et al. Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS 3 (1990), in dynamic graphs as in Clementi et al. (2015) and spreading with node expansion as in Giakkoupis (2014). In distributed networks, and in particular in large scale distributed systems, assuming that all nodes act synchronously is unrealistic. Several authors have recently dropped this assumption by considering an asynchronous model. In the discrete time case, Acan et al. (2015) study the rumor spreading time for any graph topology. They show that both the average and guaranteed spreading time are Ω(nln(n)), where n is the number of nodes in the network. Angluin et al. (2008) analyze the spreading time of a rumor by only considering the push operation (which they call the one-way epidemic operation), and show that with high probability, a rumor injected at some node requires O(nln(n)) interactions to be spread to all the nodes of the network. This result is interesting, nevertheless the constants arising in the complexity are not determined. In the continuous time case, Ganesh (2015) considers the propagation of a rumor when there are n independent unit rate Poisson processes, one associated with each node. At a time when there is a jump of the Poisson process associated with node i, this node becomes active, and chooses another node j uniformly at random with which to communicate. Ganesh (2015) analyzes the mean and the variance of the spreading time of the rumor on general graphs and Panagiotou and Speidel (2017) proposes a thorough study for spreading a rumor on particular Erd¨ os-R´ enyi random graphs. In Daley and Kendall (1965) the authors propose a diﬀerent model in which, in addition to spreaders and ignorants, is introduced the notion of stiﬂers. A stiﬂer learns the rumor but does not propagate it. A stiﬂer results from the interaction between two spreaders, or between a spreader and a stiﬂer. These authors have conjectured that the number of stiﬂers is asymptotically normal with mean and variance linear in n, where n is the size of the system. This conjecture has been proved in Pittel (1990). This model has been generalized by Lebensztayn et al. (2011) where the authors assume moreover that each spreader ceases to propagate the rumour right after being involved in a random number of stiﬂing experiences. Under a general initial conﬁguration they establish the asymptotic behaviour of the ultimate proportion of ignorants as the population size grows to inﬁnity. In Comets et al. (2014), the authors propose a model in which spreaders have a random emission capital that decreases at each emission. They study the proportion of ignorants that receive the information before the emission capital of all the spreaders Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time 4 INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS is exhausted, as well as the exhaustion time. This work is extended Erd¨ os-R´ enyi random graphs in Comets et al. (2016). In the present paper we consider the rumor spreading time in the asynchronous push-pull model for both the discrete and continuous time cases. This model provides minimal assumptions on the computational power of the nodes. In the discrete time case, nodes interact by pairs at random and if at least one node possesses the rumor, the other one also gets informed of it. In this case, the spreading time is deﬁned by the number of interactions needed for all the nodes of the network to learn the rumor. In the continuous time case, as suggested by Ganesh (2015), a Poisson process is associated with each node and at a jump occurrence of Poisson process of a node, this node contacts randomly a neighbor to interact with it as in the discrete time case, i.e. to get informed of the rumor if one of these two nodes possesses the rumor. The n Poisson processes are supposed to be independent with the same rate. In Mocquard et al. (2016) we analyzed the rumor spreading time in the discrete time asynchronous push-pull model. In the present paper we extend the results obtained in Mocquard et al. (2016) in two ways. First, we prove the conjecture formulated therein and second, we deal with the continuous time asynchronous push-pull model. The remainder of this paper is organized as follows. Section 2 presents the main results obtained in Mocquard et al. (2016) in the discrete time model needed to solve the con-tinuous time model. We also prove in this section the conjecture formulated in Mocquard et al. (2016). More precisely, if Tn denotes the total number of interactions needed for all the n nodes to get the rumor then, limn− →∞P{Tn > E(Tn)} ≈0.448429663727, where E(Tn) = (n −1)Hn−1 and Hk is the harmonic series truncated at step k. In Section 3, we consider the continuous time model. A Poisson process is associated with each node and each jump of these independent Poisson processes correspond to an interaction between two diﬀerent nodes. In this model, the time needed for all the n nodes to get the rumor is denoted by Θn. We ﬁrst give simple expressions of the expected value and variance of Θn. Then we give an explicit expression of its distribution and we obtain a simple bound of its complementary distribution which is proved to also be an equivalent of its tail. It is also shown that this bound is much more tight than already known bounds. Finally, we give the limiting distribution of the ratio Θn/E(Θn) when the number n of nodes tends to inﬁnity. Finally, Section 4 concludes the paper. Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS 5 2. The discrete time case We recall in this section the main results obtained in Mocquard et al. (2016) needed to deal with the continuous time case. We also prove the conjecture formulated in Mocquard et al. (2016). In the discrete time case, the total number of interactions needed so that all the n nodes get the rumor is denoted by Tn. We suppose without any loss of generality that among the n nodes, a single one initially knows the rumor. The case where the number of initial nodes possessing the rumor is greater than one has been considered in Mocquard et al. (2016). A value 0 or 1 is associated with each node. A node with value 1 means that this node knows the rumor and a node with value 0 means that it is not aware of the rumor. For every t ≥0, we denote by C(i) t the value (0 or 1) of node i at time t. At time 0, all the C(i) 0 are equal to 0 except one which is equal to 1 and which corresponds to the node initially knowing the rumor. At each discrete instant t, two distinct indexes i and j are successively chosen among the set of nodes {1,...,n} randomly. We denote by Xt the random variable representing this choice and we suppose that this choice is uniform, i.e we suppose that P{Xt = (i,j)} = 1 n(n −1)1{i̸=j}. Once the couple (i,j) is chosen at time t ≥1, we have C(i) t = C(j) t = max { C(i) t−1,C(j) t−1 } and C(m) t = C(m) t−1 for m ̸= i,j. The random variable Tn, deﬁned by Tn = inf { t ≥0 \| C(i) t = 1, for every i ∈{1,...,n} } , represents the number of interactions needed for all the nodes in the network to know the rumor. We introduce the discrete time stochastic process Y = {Yt, t ≥0} with state space {1,...,n} deﬁned, for all t ≥0, by Yt = { i ∈{1,...,n} \| C(i) t = 1 } . Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time 6 INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS The random variable Yt represents the number of nodes knowing the rumor at time t. The stochastic process Y is then a homogeneous Markov chain with n states, states 1,...,n−1 being transient and state n absorbing. The random variable Tn can then be written as Tn = inf{t ≥0 \| Yt = n}. It is well-known, see for instance Sericola (2013), that the distribution of Tn is given, for every k ≥0, by P{Tn > k} = αQk1, (1) where α is the row vector containing the initial probabilities of states 1,...,n −1, that is αi = P{Y0 = i} = 1{i=1}, Q is the matrix obtained containing the transition probabilities between transient states, that is, as shown in Mocquard et al. (2016), Qi,i = 1 −2i(n −i) n(n −1) for i ∈{1,··· ,n −1} and Qi,i+1 = 2i(n −i) n(n −1), for i ∈{1,··· ,n −2} (2) and 1 is the column vector of dimension n −1 with all its entries equal to 1. For i ∈{0,...,n}, we introduce the notation pi = 2i(n −i) n(n −1) and we denote by Hk the harmonic series deﬁned by H0 = 0 and Hk = ∑k ℓ=1 1/ℓ, for k ≥1. If we denote by Si, for i ∈{1,...,n −1}, the total time spent by the Markov chain Y in state i, then Si has a geometric distribution with parameter pi and we have Tn = n−1 ∑ i=1 Si. 2.1. Analysis of the spreading time The mean time E(Tn) needed so that all the nodes get the rumor is then given by E(Tn) = α(I −Q)−11, (3) where I is the identity matrix. Its explicit value has been obtained in Mocquard et al. (2016). It is given, for every n ≥1, by E(Tn) = (n −1)Hn−1 ∼ n− →∞nln(n). (4) Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS 7 In the same way, the explicit value of the variance Var(Tn) can be found in Mocquard et al. (2016). It is given by Var(Tn) = (n −1)2 2 n−1 ∑ ℓ=1 1 ℓ2 −n −1 n Hn−1 ∼ n− →∞ π2n2 12 . An explicit expression of the distribution of Tn, for n ≥2, has been obtained in the following theorem wich will used to deal with the continuous time case. Theorem 1. For every n ≥1, k ≥0, we have P{Tn > k} = ⌊n/2⌋ ∑ j=1 (cn−1,j(1 −pj) + kdn−1,j)(1 −pj)k−1, where the coeﬃcients cn−1,j and dn−1,j, which do not depend on k, are given, for j ∈ {1,...,n −1}, recursively by c1,j = 1{j=1} and d1,j = 0 and for i ∈{2,...,n −1} by                                        ci,j = pici−1,j pi −pj −pidi−1,j (pi −pj)2 for i ̸= j,n −j, di,j = pidi−1,j pi −pj for i ̸= j,n −j, ci,i = 1 − ⌊n/2⌋ ∑ j=1,j̸=i ci,j for i ≤⌊n/2⌋, ci,n−i = 1 − ⌊n/2⌋ ∑ j=1,j̸=n−i ci,j for i > ⌊n/2⌋, di,i = pici−1,i for i ≤⌊n/2⌋, di,n−i = pici−1,n−i for i > ⌊n/2⌋. (5) Proof. See Mocquard et al. (2016). 2.2. Bounds and asymptotic analysis of the distribution of Tn The following bound and equivalent of the complementary distribution of Tn will be used in the continuous time case to obtain similar bound and equivalent. Theorem 2. For all n ≥2 and k ≥1 we have P{Tn > k} ≤ ( 1 + 2k(n −2)2 n )( 1 −2 n )k−1 , P{Tn > k} ∼ k− →∞ ( 1 + 2k(n −2)2 n )( 1 −2 n )k−1 . Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time 8 INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS Proof. See Mocquard et al. (2016). Recall that E(Tn) = (n −1)Hn−1, where Hk is the harmonic series. We proved in Moc-quard et al. (2016) that for all real c ≥0, we have lim n→∞P{Tn > cE(Tn)} =    0 if c > 1 1 if c < 1. (6) For c = 1, this result was formulated in Mocquard et al. (2016) as a conjecture. We are now able to give a proof of it. Theorem 3. lim n− →∞P{Tn > E(Tn)} = 1 −2e−γK1 ( 2e−γ) ≈0.448429663727. where γ is the Euler’s constant given by γ = limn− →∞(Hn −ln(n)) ≈0.5772156649 and K1 is the modiﬁed Bessel function of the second kind of order 1 given, for z > 0, by K1(z) = z 4 ∫+∞ 0 t−2e−t−z2/4tdt. Proof. See Online Supplement in Mocquard et al. (2018). Relation (6) shows that for large values of n (n − →∞) and for all ε > 0, we have Tn ≤ (1+ε)E(Tn) with probability 1, Tn > (1−ε)E(Tn) with probability 1. Moreover Theorem 3 shows that for large values of n (n − →∞), we have Tn > E(Tn) with probability 0.44843 and thus Tn ≤E(Tn) with probability 0.55157. 3. The continuous time case As in the discrete time case, we suppose without any loss of generality that among the n nodes, a single one initially knows the rumor and a value 0 or 1 is associated with each node. A node with value 1 means that this node knows the rumor and a node with value 0 means that it is not aware of the rumor. For every t ≥0, we denote by C(i) t the value (0 or 1) of node i at time t. At time 0, all the C(i) 0 are equal to 0 except one which is equal to 1 and which corresponds to the node initially knowing the rumor. In the continuous time case, a Poisson process is associated with each node. These n Poisson processes are independent and have the same rate λ > 0. When the Poisson process associated with node i has a jump, this node chooses another node j randomly, with a Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS 9 given distribution to interact with node i. This is equivalent to consider a single Poisson process with rate nλ at the jumps of which two distinct nodes are randomly chosen to interact with a given distribution. Then as in the discrete time case, the two nodes change their value with the maximum value of each node. Again, we want to evaluate the time needed to spread the rumor that is the time needed so that all the nodes get value 1. We denote by (τℓ)ℓ≥0 the successive jumps of the Poisson process with rate nλ, with τ0 = 0. Then once the couple (i,j) is chosen at time τℓ, we have C(i) t = C(j) t = max { C(i) τℓ−1,C(j) τℓ−1 } and C(m) t = C(m) τℓ−1 for m ̸= i,j and t ∈[τℓ,τℓ+1). For every ℓ≥1, we denote by Xℓthe random variable representing this choice at time τℓand we suppose that this choice is uniform, i.e. we suppose that, for all ℓ≥1, we have P{Xℓ= (i,j)} = 1 n(n −1)1{i̸=j}. We consider the random variable Θn deﬁned by Θn = inf { t ≥0 \| C(i) t = 1, for every i ∈{1,...,n} } , which represents the time needed for all the nodes in the network to know the rumor. We introduce the continuous time stochastic process Z = {Zt, t ∈R+} with state space {1,...,n} deﬁned, for all t ≥0, by Zt = { i ∈{1,...,n} \| C(i) t = 1 } . The random variable Zt represents the number of nodes knowing the rumor at time t. The stochastic process Z is then a homogeneous Markov chain with transition rate matrix B. The non zero entries of matrix B are given, for i ∈{1,...,n}, by    Bi,i = −nλpi, Bi,i+1 = nλpi, for i ̸= n. Indeed, when Zt = i, the next node is activated with rate nλ. In order for process Z to reach state i + 1 from state i, this activated node, say node ℓ, either possesses the rumor (probability i/n) and the node contacted by ℓ, say m, does not possess the rumor (probability (n −i)/(n −1)) or node ℓdoes not possess the rumor (probability (n −i)/n) Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time 10 INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS and it contacts node m which possesses the rumor (probability i/(n−1)). This means that, for i ∈{1,...,n −1}, the rate Bi,i+1 is given by Bi,i+1 = nλ2i(n −i) n(n −1) = nλpi. The states 1,...,n −1 of Z are transient and state n is absorbing. The random variable Θn can then be written as Θn = inf{t ≥0 \| Zt = n}. It is well-known, see for instance Sericola (2013), that the distribution of Θn is given, for every t ≥0, by P{Θn > t} = αeRt1, (7) where α is the row vector containing the initial probabilities of states 1,...,n −1, that is αi = P{Z0 = i} = 1{i=1}, R is the sub-matrix obtained from B by deleting the row and the column corresponding to absorbing state n and 1 is the column vector of dimension n −1 with all its entries equal to 1. For every i ∈{1,...,n −1} we denote by Ui the sojourn time of process Z in state i, that is the time during which the system counts exactly i nodes knowing the rumor. The random variables Ui are independent and exponentially distributed with rate µi = nλpi and we have Θn = n−1 ∑ i=1 Ui. 3.1. Expectation and variance of Θn The expected value and the variance of Θn were obtained by Molchanov and Whitmeyer (2010) in the push model case. We extend these results to the push-pull model in the following two lemmas. Lemma 1. For all n ≥2, we have E(Θn) = (n −1)Hn−1 nλ and E(Θn) ∼ n− →∞ ln(n) λ . Proof. We have E(Θn) = n−1 ∑ i=1 E(Ui) = 1 nλ n−1 ∑ i=1 1 pi = 1 nλE(Tn) = (n −1)Hn−1 nλ . The rest of the proof is evident since Hn−1 ∼ n− →∞ln(n). Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS 11 Lemma 2. For all n ≥2, we have Var(Θn) = (n −1)2 2n2λ2 (n−1 ∑ i=1 1 i2 + 2Hn−1 n ) ≤1 λ2 (π2 12 + Hn−1 n ) and lim n− →∞Var(Θn) = π2 12λ2. Proof. The random variables Uℓbeing independent, we have Var(Θn) = n−1 ∑ i=1 Var(Ui) = 1 n2λ2 n−1 ∑ i=1 1 p2 i = (n −1)2 4λ2 n−1 ∑ i=1 1 i2(n −i)2 = (n −1)2 4n2λ2 n−1 ∑ i=1 (1 i + 1 n −i )2 = (n −1)2 4n2λ2 (n−1 ∑ i=1 1 i2 + n−1 ∑ i=1 1 (n −i)2 + 2 n−1 ∑ i=1 1 i(n −i) ) = (n −1)2 4n2λ2 ( 2 n−1 ∑ i=1 1 i2 + 2 n n−1 ∑ i=1 (1 i + 1 n −i )) = (n −1)2 4n2λ2 ( 2 n−1 ∑ i=1 1 i2 + 4Hn−1 n ) ≤1 λ2 (π2 12 + Hn−1 n ) . The rest of the proof is evident since Hn−1 ∼ n− →∞ln(n). Note that the diﬀerence betwen the push model and the push-pull model is due to simply a factor of 2 in the transition probabilities, giving corresponding factors of 2 in the mean and 4 in the variance. 3.2. Explicit expression of the distribution of Θn The distribution of Θn, for n ≥2, which is given by Relation (7) can be easily computed as follows. We make use of the uniformization technique, see for instance Sericola (2013). We introduce the uniformized Markov chain associated with the Markov chain Z which is characterized by its uniformization rate ν and by its transition probability matrix G. The uniformization rate ν must satisfy ν ≥maxi∈{1,...,n}(−Bi,i) and matrix G is related to the inﬁnitesimal generator R by G = In + B/ν, where In denotes the identity matrix of order n. We denote by Nt the number of transitions occurring during the interval [0,t]. The process Nt is a Poisson process with rate ν and since B = −ν(In −G), we have R = −ν(In−1 −P), where P is the sub-matrix obtained from G by deleting the row and the column corresponding to absorbing state n. Relation (7) can then be written as P{Θn > t} = αeRt1 = ∞ ∑ k=0 e−νt(νt)k k! αP k1. Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time 12 INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS It is easily checked that max i∈{1,...,n}(−Ri,i) = max i∈{1,...,n}(nλpi) ≤nλ. By taking ν = nλ, we get, from Relation (2), P = Q and thus, using (1), this leads to P{Θn > t} = ∞ ∑ k=0 e−nλt(nλt)k k! P{Tn > k} = ∞ ∑ k=0 e−nλt(nλt)k k! αQk1. (8) Using this expression we obtain the following explicit expression of the distribution of Θn. Theorem 4. For every n ≥1, t ≥0, we have P{Θn > t} = ⌊n/2⌋ ∑ j=1 (cn−1,j + nλtdn−1,j)e−nλpjt, where the coeﬃcients cn−1,j and dn−1,j are given by Relations (5). Proof. From Theorem 1, we have for every n ≥1 and k ≥0, P{Tn > k} = ⌊n/2⌋ ∑ j=1 (cn−1,j(1 −pj) + kdn−1,j)(1 −pj)k−1, where the coeﬃcients cn−1,j and dn−1,j are given by Relations (5). Using now Relation (8), we obtain P{Θn > t} = ∞ ∑ k=0 e−nλt(nλt)k k!   ⌊n/2⌋ ∑ j=1 cn−1,j(1 −pj)k + ⌊n/2⌋ ∑ j=1 kdn−1,j(1 −pj)k−1   = ⌊n/2⌋ ∑ j=1 cn−1,je−nλpjt + nλt ⌊n/2⌋ ∑ j=1 dn−1,je−nλpjt, which completes the proof. 3.3. Bounds and tail behavior of the distribution of Θn We obtain in this section a very simple bound of the complementary distribution of Θn and we show that this bound is also an equivalent of its tail. This bound and equivalent of the quantity P{Θn > t} are derived from Theorem 2. Theorem 5. For all n ≥3 and t ≥0 we have P{Θn > t} ≤ [ 2(n −2)2λt + n n −2 ] e−2λt, P{Θn > t} ∼ t− →∞ [ 2(n −2)2λt + n n −2 ] e−2λt. Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS 13 Note that for n = 2, we have Θ2 = U1 which is exponentially distributed with rate µ1 = 2λ and thus P{Θ2 > t} = e−2λt. Proof. From Theorem 2, we have for n ≥2 and k ≥1, P{Tn > k} ≤ ( 1 + 2k(n −2)2 n )( 1 −2 n )k−1 . Since P{Tn > 0} = 1, this leads to P{Θn > t} = ∞ ∑ k=0 e−nλt(nλt)k k! P{Tn > k} ≤e−nλt + ∞ ∑ k=1 e−nλt(nλt)k k! ( 1 + 2k(n −2)2 n )( 1 −2 n )k−1 = e−nλt + ∞ ∑ k=1 e−nλt(nλt)k k! ( 1 −2 n )k−1 + 2(n −2)2λt ∞ ∑ k=1 e−nλt((n −2)λt)k−1 (k −1)! = e−nλt + ne−nλt ( e(n−2)λt −1 ) n −2 + 2(n −2)2λte−nλte(n−2)λt = [ 2(n −2)2λt + n n −2 ] e−2λt − 2 n −2e−nλt ≤ [ 2(n −2)2λt + n n −2 ] e−2λt. which completes the ﬁrst part of the proof. On the one hand since p1 < pj for j ∈{2,...,⌊n/2⌋}, we have, from Theorem 1, P{Tn > k} ∼ k− →∞dn−1,1k ( 1 −2 n )k−1 . On the other hand, from Theorem 2, we have P{Tn > k} ∼ k− →∞ ( 1 + 2k(n −2)2 n )( 1 −2 n )k−1 . These two results imply that dn−1,1 = 2(n −2)2 n . In the same way, from Theorem 4, we get P{Θn > t} ∼ t− →∞dn−1,1nλte−nλp1t = 2(n −2)2λte−2λt ∼ t− →∞ [ 2(n −2)2λt + n n −2 ] e−2λt, which completes the proof. Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time 14 INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS We give in the following two diﬀerent bounds for the quantity P{Θn > cE(Θn)}, with c ≥1. These bounds will be compared and used to obtain the limiting behaviour of this quantity when the number n of nodes goes to inﬁnity. Recalling that E(Θn) = (n −1)Hn−1/(nλ), a ﬁrst bound is obtained by an immediate application of Theorem 5.1 of Janson (2014), which leads, for all n ≥3 and for all real number c ≥1, to P{Θn > cE(Θn)} ≤1 c exp ( −2(n −1)Hn−1(c −1 −ln(c)) n ) . (9) Note that the right-hand side term is equal to 1 when c = 1. Applying Theorem 5 at point cE(Θn), we obtain the following second bound. P{Θn > cE(Θn)} ≤ [ 2(n −2)2λcE(Θn) + n n −2 ] e−2λcE(Θn) = [2c(n −2)2(n −1)Hn−1 n + n n −2 ] exp ( −2c(n −1)Hn−1 n ) . From now on we denote this bound by φ(c,n) and in the same way, we denote by ψ(c,n) the bound of P{Θn > cE(Θn)} obtained in (9). We then have, for n ≥3 and c ≥1, φ(c,n) = [2c(n −2)2(n −1)Hn−1 n + n n −2 ] exp ( −2c(n −1)Hn−1 n ) , ψ(c,n) = 1 c exp ( −2(n −1)Hn−1(c −1 −ln(c)) n ) . These two bounds are compared in the next theorem. Theorem 6. For every n ≥5, there exists a unique c∗≥1 such that φ(c∗,n) = ψ(c∗,n) and we have    φ(c,n) > ψ(c,n) for all 1 ≤c < c∗ φ(c,n) < ψ(c,n) for all c > c∗. (10) Furthermore, lim c− →∞ φ(c,n) ψ(c,n) = 0. Proof. Let us introduce the quantities An = (n −1)Hn−1 n ,Bn = 2(n −2)2An and Cn = n n −2. We then have φ(c,n) ψ(c,n) = ( Bnc2 + Cnc ) e−2An(1+ln(c)) = ( Bnc2−2An + Cnc1−2An) e−2An. Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS 15 It is easily checked that the sequence An is strictly increasing and that A3 = 1. It follows that for n ≥5, we have An > 1 and so 1 −2An < 2 −2An < 0. This implies that for every n ≥5, the function φ(c,n)/ψ(c,n) is strictly decreasing with c on [1,+∞) and that lim c− →∞ φ(c,n) ψ(c,n) = 0. Consider now the sequences xn and yn deﬁned for n ≥5, by xn = φ(1,n) ψ(1,n) = (Bn + Cn)e−2An and yn = 2e−2(n −2)2An (n −1)2 . The sequence An being increasing, it is easily checked that sequence yn is increasing too. Moreover, we have xn ≥Bne−2(1+ln(n−1)) = e−2Bn (n −1)2 = 2e−2(n −2)2An (n −1)2 = yn. A simple computation shows that we have y34 > 1. The sequence yn being increasing, we obtain yn > 1 for every n ≥34. It follows that we also have xn > 1 for all n ≥34. A numerical computation gives xn > 1 for n ∈{5,...,33} which means that for all n ≥5, we have xn = φ(1,n)/ψ(1,n) > 1. The function φ(c,n)/ψ(c,n) being strictly decreasing with c on [1,+∞), we deduce that there exists a unique solution, called c∗, to the equation φ(c,n)/ψ(c,n) = 1 and (10) follows. This theorem shows that our bound φ(c,n) is much more tight than the one obtained using the result of Janson (2014), which has been denoted by ψ(c,n), for c > c∗, not only because the ratio φ(c,n)/ψ(c,n) decreases with c and tends to 0 when c tends to inﬁnity, but also because for every value of n, the value of c∗is very close to 1 as shown in Table 1. Moreover, from Theorem 5, our bound is optimal in the sense that P{Θn > cE(Θn)} ∼ c− →∞φ(c,n). Table 2 and Figure 1 illustrate, for a network composed of n = 1000 nodes, the behavior of the bounds φ(c,1000) and ψ(c,1000), as a function of c, compared to the exact value Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time 16 INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS Table 1 Values of c∗for diﬀerent network sizes n. n 10 102 103 104 105 106 107 108 109 c∗ 1.253 1.163 1.128 1.109 1.095 1.085 1.078 1.071 1.066 Table 2 Values of P{Θ1000 > cE(Θ1000)}, φ(c,1000) and ψ(c,1000) for diﬀerent values of c. c 1 1.2 1.4 1.6 1.8 2 P{Θ1000 > cE(Θ1000)} 0.446 0.063 0.005 3.9 × 10−4 2.6 × 10−5 1.6 × 10−6 φ(c,1000) ≥1 0.288 0.017 9.7 × 10−4 5.5 × 10−5 3 × 10−6 ψ(c,1000) 1.0 0.634 0.276 0.089 0.023 0.005 of complementary distribution function of Θ1000 at point cE(Θ1000), computed using The-orem 4. Table 2 illustrates clearly the result of Theorem 5. Indeed the values of our bound φ(c,1000) are very close to the real value of the complementary distribution function, while the values of ψ(c,1000) tend to move away from this real value even for small values of c. Note that when c = 1 both bounds are useless and the real value P{Θ1000 > E(Θ1000)} is very close to the limit obtained in Theorem 9 of next section. Figure 1 shows the large gap between the bounds φ(c,1000) and ψ(c,1000) when c is greater than c∗whose value 0 0.2 0.4 0.6 0.8 1 c 1 1.2 1.4 1.6 1.8 2 c ψ(c,1000) φ(c,1000) P{Θ1000 > cE(Θ1000)} Figure 1 Bounds ψ(c,1000), φ(c,1000) and real value of P{Θ1000 > cE(Θ1000} as a function of c. The point at which the bounds are equal is c∗= 1.12819634. Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS 17 is c∗= 1.12819634. Moreover this large gap increases when n increases since the value of c∗decreases to 1 when n increases, as shown in Table 1. 3.4. Asymptotic analysis of the distribution of Θn We analyze in this section the behavior of the complementary distribution of Θn at point cE(Θn) when the number n of nodes in the network tends to inﬁnity, in function of the value of c. We prove in the following theorem that the bounds φ(c,n) and ψ(c,n), obtained from Theorem 5 and Relation (9) respectively with t = cE(Tn), both tend to 0 when n goes to inﬁnity. Theorem 7. For all real number c > 1, we have lim n− →∞φ(c,n) = 0 and lim n− →∞ψ(c,n) = 0. Proof. It is easily checked that φ(c,n) ∼ n− →∞ 2cn2 ln(n) n2c which tends to 0 when n tends to inﬁnity. Concerning ψ(c,n) we have ψ(c,n) ∼ n− →∞ 1 ce−ln(n)(c−1−ln(c)). For c > 1 we have c −1 −ln(c) > 0 which implies that ψ(c,n) tends to 0 when n tends to inﬁnity. Theorem 8. For all real c ≥0, we have lim n→∞P{Θn > cE(Θn)} =    0 if c > 1 1 if c < 1. Proof. From Theorem 7, both bounds φ(c,n) and ψ(c,n) of P{Θn > cE(Θn)} tend to 0 when n tends to inﬁnity, for c > 1. So using either φ(c,n) or ψ(c,n) we deduce that lim n− →∞P{Θn > cE(Θn)} = 0 for all c > 1. In the case where c < 1, Theorem 5.1 of Janson (2014) leads to P{Θn > cE(Θn)} ≥1 −exp (−2(n −1)Hn−1(c −1 −ln(c)) n ) . Since c −1 −ln(c) > 0 for all c ∈[0,1), the right-hand side term of this inequality tends to 1 when n − →∞. Thus, limn− →∞P{Θn > cE(Θn)} = 1 when c < 1. Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time 18 INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS The following theorem considers the case c = 1. Note that the result is identical to the one of Theorem 3 in the discrete time case. Theorem 9. lim n− →∞P{Θn > E(Θn)} = 1 −2e−γK1 ( 2e−γ) ≈0.448429663727. where γ is the Euler’s constant given by γ = limn− →∞(Hn −ln(n)) ≈0.5772156649 and K1 is the modiﬁed Bessel function of the second kind of order 1 given, for z > 0, by K1(z) = z 4 ∫+∞ 0 t−2e−t−z2/4tdt. Proof. See Online Supplement in Mocquard et al. (2018). Remark. Some possible extensions of this work are the following. 1. We have supposed that the initial number of nodes knowing the rumor is equal to 1. The case where this number is equal to ℓ, with ℓ≥2, has been dealt with in Mocquard et al. (2016) in the discrete time case. This extension to the continuous time case is almost straightforward since it suﬃces to redeﬁne the random variable Θn as Θn = Uℓ+ ··· + Un. 2. Instead of considering the total time needed for all the nodes to obtain the rumor, one could be interested in the total time needed for a ﬁxed percentage, say ρ, of the nodes to obtain the rumor. In that case the random variable Θn to consider should be redeﬁned as Θn = U1 +···+U⌈ρn⌉. Of course this extension could also be combined with the ﬁrst one above. 3. The instants at which the interactions between nodes occur have been modeled by a Poisson process. This could be generalized by considering, instead of a Poisson process, a Phase-type renewal process which preserves the Markov property and can approximate every point process. Acknowledgement. We would like to thank Professor Philippe Carmona for his expert advice concerning the proof of Theorem 3. 4. Conclusion In this paper we have provided a thorough analysis of the rumor spreading time in the asynchronous push-pull model in the continuous time case by completing and extending the results already obtained in the discrete time case. Such a precise analysis is a step towards the design of more complex problems such as, for instance, the leader election Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS 19 in large distributed systems. Our analysis concerning the tail distribution of the rumor spreading time and its limiting behavior when the number of nodes goes to inﬁnity has never been done in such detail before. It shows that the evaluation of the ﬁrst moment of the rumor spreading time is far from suﬃcient to provide a global control of the system. References Acan H, Collevecchio A, Mehrabian A, Nick W (2015) On the push&pull protocol for rumour spreading. Proceedings of the ACM Symposium on Principles of Distributed Systems (PODC). Angluin D, Aspnes J, Eisenstat D (2008) Fast computation by population protocols with a leader. Distributed Computing 21(2):183–199. Berger N, Borgs C, Chayes JT, Saberi A (2005) On the spread of viruses on the internet. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). Censor-Hillel K, Haeupler B, Kelner J, Maymounkov P (2012) Global computation in a poorly connected world: Fast rumor spreading with no dependence on conductance. Proceedings of the Annual ACM Symposium on Theory of Computing (STOC). Chierichetti F, Lattanzi S, Panconesi A (2011) Rumor spreading in social networks. Theoretical Computer Science 412(24):2602–2610. Clementi A, Crescenzi P, Doerr C, Fraigniaud P, Pasquale F, Silvestri R (2015) Rumor spreading in random evolving graphs. Random structures and Algorithms 48(2):290–312. Comets F, Delarue F, Schott R (2014) Information transmission under random emission constraints. Envi-ronmental Modelling & Software 23(6):973–1009. Comets F, Gallesco C, Popov S, Vachkovskaia M (2016) Constrained information transmission on Erd¨ os-R´ enyi graphs. Markov Processes and Related Fields 22:111–138. Daley D, Kendall DG (1965) Stochastic rumours. IMA Journal of Applied Mathematics 1(1):42–55. Daum S, Kuhn F, Maus Y (2016) Rumor spreading with bounded indegree. Proceedings of the International Colloquium on Structural Information and Communication Complexity (SIROCCO). Demers A, Gealy M, Greene D, Hauser C, Irish W, Larson J, Shenker S, Sturgis H, Swinehart D, Terry D (1987) Epidemic algorithms for replicated datbase maintenance. Proceedings of the ACM Syposium on Principles of Distributed Systems (PODC). Feige U, Peleg D, Raghavan P, Upfal E (1990) Randomized broadcast in networks. Random Structures and Algorithms 1(4):447–460. Fountoulakis N, Panagiotou K (2013) Rumor spreading on random regular graphs and expanders. Random Structures and Algorithms 43(2):201–220. Frieze A, Grimmet G (1985) The shortest-path problem for graphs with random arc-lengths. Discrete Applied Mathematics 10(1):57–77. Y. Mocquard, B. Sericola and E. Anceaume: Probabilistic Analysis of Rumor Spreading Time 20 INFORMS Journal on Computing 00(0), pp. 000–000, c ⃝0000 INFORMS Ganesh AJ (2015) Rumour spreading on graphs. Technical report, URL uk/~maajg/teaching/complexnets/rumours.pdf. Giakkoupis G (2011) Tight bounds for rumor spreading in graphs of a given conductance. Proceedings of the International Symposium on Theoretical Aspects of Computer Science (STACS). Giakkoupis G (2014) Tight bounds for rumor spreading with vertex expansion. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). Harchol-Balter M, Leighton T, Lewin D (1999) Resource discovery in distributed networks. Proceedings of the ACM Syposium on Principles of Distributed Systems (PODC). Janson S (2014) Tail bounds for sums of geometric and exponential variables. Technical report, URL http: //www2.math.uu.se/~svante/papers/sjN14.pdf? Karp R, Schindelhauer C, Shenker S, Vocking B (2000) Randomized rumor spreading. Proceedings of the Annual Symposium on Foundations of Computer Science (FOCS). Kempe D, Dobra A, Gehrke J (2003) Gossip-based computation of aggregate information. Proceedings of the Annual IEEE Symposium on Foundations of Computer Science (FOCS). Lebensztayn E, Machado AF, Rodrguez PM (2011) On the behaviour of a rumour process with random stiﬂing. Environmental Modelling & Software 26:517–522. Mocquard Y, Robert S, Sericola B, Anceaume E (2016) Analysis of the propagation time of a rumour in large-scale distributed systems. Proceedings of the 15th IEEE International Symposium on Network Computing and Applications (NCA). Mocquard Y, Sericola B, Anceaume E (2018) Online supplement for Probabilistic analysis of rumor spreading time. Molchanov S, Whitmeyer JM (2010) Two Markov Models of the Spread of Rumors. Journal of Mathematical Sociology 34:157–166. Panagiotou K, Perez-Gimenez X, Sauerwald T, Sun H (2015) Randomized rumor spreading: the eﬀect of the network topology. Combinatorics, Probability and Computing 24(2):457–479. Panagiotou K, Speidel L (2017) Asynchronous rumor spreading on random graphs. Algorithmica 78(3):968– 989. Pittel B (1987) On spreading a rumor. SIAM Journal on Applied Mathematics 47(1):213–223. Pittel B (1990) On a daley-kendall model of random rumours. Journal of Applied Probability 27(1):14–27. Sericola B (2013) Markov Chains. Theory, Algorithms and Applications. Applied stochastic methods series (Wiley).
15117	https://en.wikipedia.org/wiki/Computer-supported_cooperative_work	Jump to content Computer-supported cooperative work العربية Dansk Deutsch Ελληνικά Español Français 한국어 Bahasa Indonesia Italiano Latviešu Norsk bokmål Norsk nynorsk Português Suomi ไทย Українська 中文 Edit links From Wikipedia, the free encyclopedia Sorry to interrupt, but our fundraiser won't last long. This Wednesday, we ask you to join the 2% of readers who give. If everyone reading this right now gave just $2.75, we'd hit our goal quickly. $2.75 is all we ask. September 3: Knowledge is human. We're sorry we've asked you a few times recently, but it's Wednesday, September 3—please don't wait until tomorrow to help. We're happy you consult Wikipedia often. If just 2% of our most loyal readers gave $2.75 today, we'd reach our goal quickly. Most readers donate because Wikipedia is useful, others because they realize knowledge needs humans. If you agree, please give. Any contribution helps, whether it's $2.75 one time or monthly. 25 years ago Wikipedia was a dream. A dream built piece by piece by people, not machines. Now, with 65 million articles and 260,000 volunteers across the world, Wikipedia is proof that knowledge is human—a place of free, collaborative, and accessible knowledge. Your donation isn't just supporting a website; it's investing in the world's largest collaborative project of human intelligence—crafted by humans, for humans. Please join the 2% of readers who give what they can to help keep Wikipedia strong and growing. Thank you. Field studying how people work in groups with the support of computing systems Computer-supported cooperative work (CSCW) or computer-supported collaboration is the study of how people utilize technology collaboratively, often towards a shared goal. CSCW addresses how computer systems can support collaborative activity and coordination. More specifically, the field of CSCW seeks to analyze and draw connections between currently understood human psychological and social behaviors and available collaborative tools, or groupware. Often the goal of CSCW is to help promote and utilize technology in a collaborative way, and help create new tools to succeed in that goal. These parallels allow CSCW research to inform future design patterns or assist in the development of entirely new tools. Computer supported cooperative work includes "all contexts in which technology is used to mediate human activities such as communication, coordination, cooperation, competition, entertainment, games, art, and music" (from CSCW 2023). History [edit] The development of this field reaches back to the late 1960s and the visionary assertions of Ted Nelson, Douglas Engelbart, Alan Kay, Glenn Gould, Nicholas Negroponte and others who saw a potential for digital media to ultimately redefine how people work. A very early thinker, Vannevar Bush, even suggested in 1945 As We May Think. The inventor of the computer "mouse", Douglas Engelbart, studied collaborative software (especially revision control in computer-aided software engineering and the way a graphic user interface could enable interpersonal communication) in the 1960s. Alan Kay worked on Smalltalk, which embodied these principles, in the 1970s, and by the 1980s it was well regarded and considered to represent the future of user interfaces. However, at this time, collaboration capabilities were limited. As few computers had even local area networks, and processors were slow and expensive, the idea of using them simply to accelerate and "augment" human communication was eccentric in many situations. Computers processed numbers, not text, and the collaboration was in general devoted only to better and more accurate handling of numbers. This began to change in the 1980s with the rise of personal computers, modems and more general use of the Internet for non-academic purposes. People were clearly collaborating online with all sorts of motives, but using a small suite of tools (LISTSERV, netnews, IRC, MUD) to support all of those motives. Research at this time focused on textual communication, as there was little or no exchange of audio and video representations. Some researchers, such as Brenda Laurel, emphasized how similar online dialogue was to a play, and applied Aristotle's model of drama to their analysis of computers for collaboration. Another major focus was hypertext—in its pre-HTML, pre-WWW form, focused more on links and semantic web applications than on graphics. Such systems as Superbook, NoteCards, KMS and the much simpler HyperTies and HyperCard were early examples of collaborative software used for e-learning. The origins of CSCW as a field are intertwined with the rise and subsequent fall of office automation as response to some of the criticisms, particularly the failure to address the impact human psychological and social behaviors can have. Greif and Cashman created the term CSCW to help employees seeking to further their work with technology. A few years later, in 1987, Charles Findley presented the concept of collaborative learning-work. Computer-supported cooperative work is an interdisciplinary research area of growing interest which relates workstations to digitally advanced networking systems. The first technologies were economically feasible, but their interoperability was lacking which makes understanding a well-tailored supporting system difficult. Due to global markets, more organizations are being pushed to decentralize their corporate systems. When faced with the complexities of today's business issues, a significant effort must be made to improve manufacturing systems' efficiency, improve product quality, and reduce time to market. Audio [edit] In the 1990s, the rise of broadband networks and the dotcom boom presented the internet as mass media to a whole generation. By the late 1990s, VoIP and net phones and chat had emerged. For the first time, people used computers primarily as communications, not "computing" devices. This, however, had long been anticipated, predicted, and studied by experts in the field. Pioneers [edit] Other pioneers in the field included Ted Nelson, Austin Henderson, Kjeld Schmidt, Lucy Suchman, Sara Bly, Randy Farmer, and many "economists, social psychologists, anthropologists, organizational theorists, educators, and anyone else who can shed light on group activity." - Grudin. Politics and business [edit] In this century, the focus has shifted to sociology, political science, management science and other business disciplines. This reflects the use of the net in politics and business and even other high-stakes collaboration situations, such as war. War [edit] Though it is not studied at the ACM conferences, military use of collaborative software has been a very major impetus of work on maps and data fusion, used in military intelligence. A number of conferences and journals are concerned primarily with the military use of digital media and the security implications thereof. COVID-19 [edit] The idea of CSCW or computer-supported cooperative work has become useful over the years since its inception and most especially during the COVID-19 pandemic.[citation needed] The measures to mitigate the virus' spread led to firm closures and increased the rates of remote working and learning. People now share a common workspace, hold virtual meetings, see and hear each other's movements and voices in a common virtual workspace with a group-centered design. Central concerns and concepts [edit] CSCW is a design-oriented academic field that is interdisciplinary in nature and brings together librarians, economists, organizational theorists, educators, social psychologists, sociologists, anthropologists and computer scientists, among others. The expertise of researchers in various and combined disciplines help researchers identify venues for possible development. Despite the variety of disciplines, CSCW is an identifiable research field focused on understanding characteristics of interdependent group work with the objective of designing adequate computer-based technology to support such cooperative work. Essentially, CSCW goes beyond building technology itself and looks at how people work within groups and organizations, as well as the impacts of technology on those processes. CSCW has ushered in a great extent of melding between social scientists and computer scientists. These scientists work together to overcome both technical and non-technical problems within the same user spaces. For example, many R&D professionals working with CSCW are computer scientists who have realized that social factors play an important role in the development of collaborative systems. On the flip side, many social scientists who understand the increasing role of technology in our social world become "technologists" who work in R&D labs developing cooperative systems. Over the years, CSCW researchers have identified a number of core dimensions of cooperative work. A non-exhaustive list includes: Awareness: individuals working together need to be able to gain some level of shared knowledge about each other's activities. Articulation work: cooperating individuals must be able to partition work into units, divide it amongst themselves and, after the work is performed, reintegrate it. Appropriation (or tailorability): how an individual or group adapts a technology to their own particular situation; the technology may be appropriated in a manner completely unintended by the designers. These concepts have largely been derived through the analysis of systems designed by researchers in the CSCW community, or through studies of existing systems (for example, Wikipedia). CSCW researchers that design and build systems try to address core concepts in novel ways. However, the complexity of a domain can make it difficult to produce conclusive results. Articulation work [edit] Articulation work is essentially the work that makes other work exist and possible. It is an effort made to make other work easier, more manageable, and can either be planned or unplanned. Therefore articulation work is an integral part of software process since software processes can sometimes fail or break down. Articulation work is also commonly known as "invisible work" since it is not always noticed. The concept was introduced by Anselm Strauss. He described it as a way to observe the "nature of mutually dependent actors in their division of labour". It was introduced in CSCW by Schmidt and Bannon in 1992, where it would be applied to more realistic work scenarios in society. Articulation work is inherent in collaboration. The idea of articulation work was initially used in relation to computer-supported cooperative work, but it was travelled through other domains of work, such as healthcare. Initially, articulation work was known for scheduling and allocation of resources, but now, extends beyond that. Articulation work can also be seen as the response developers make to adapt to changes due to error or misjudgments in the real world. There are various models of articulation work that help identify applicable solutions to recover or reorganize planned activities. It is also important to note that it can vary depending on the scenario. Oftentimes there is an increase in the need for articulation work as the situation becomes more complex. Because articulation work is so abstract, it can be split into two categories from the highest level: individual activity and collective activity. With individual activity, articulation work is almost always applicable. It is obvious that the subject is required to articulate his / her own work. But when a subject is faced with a new task, there are many questions that must be answered in order to move forward and be successful. This questioning is considered the articulation work to the actual project; invisible, but necessary. There is also articulation of action within an activity. For example, creating to-do lists and blueprints may be imperative to progressing a project. There is also articulation of operation within an action. In terms of software, the user must have adequate knowledge and skill in using computer systems and knowledge about software in order to perform tasks. In a teamwork setting, articulation is imperative for collective activity. To maximize the efficiency of all the people working, the articulation work must be very solid. Without a solid foundation, the team is unable to collaborate effectively. Furthermore, as the size of the team increases, the articulation work becomes more complex. What goes in between the user and the system is often overlooked. But software process modeling techniques as well as the model of articulation work is imperative in creating a solid foundation that allows for improvement and enhancement. In a way, all work needs to be articulated; there needs to be a who, what, where, when and how. With technology, there are many tools that utilize articulation work. Tasks such as planning and scheduling can be considered articulation work. There are also times when the articulation work is bridging the gap between the technology and the user. Ultimately, articulation work is the means that allows for cooperative work to be cooperative, a main objective of CSCW. Matrix [edit] One of the most common ways of conceptualizing CSCW systems is to consider the context of a system's use. One such conceptualization is the CSCW Matrix, first introduced in 1988 by Johansen; it also appears in Baecker (1995). The matrix considers work contexts along two dimensions: whether collaboration is co-located or geographically distributed, and whether individuals collaborate synchronously (same time) or asynchronously (not depending on others to be around at the same time). Same time/same place – face to face interaction [edit] Roomware Shared tables, wall displays Digital whiteboards Electronic meeting systems Single display groupware Group decision support system Same time/different place – remote interaction [edit] Electronic meeting systems Videoconferencing Real-time groupware Messaging (instant messaging, email, chat) Telephoning Different time/same place – continuous task (ongoing task) [edit] Team rooms Large displays Post-it War-rooms Different time/different place – communication and coordination [edit] Electronic meeting systems Blogs Workflow Version control This matrix is an outline of CSCW in different contexts, but it does have its limitations for users who are beginners at understanding CSCW. For example, there is a collaborative mode called multi-synchronous that can not fit the matrix. As the field evolves whether by new social standards or technological development, the simple matrix cannot describe all of CSCW and fields of research within. Model of Coordinated Action (MoCA) [edit] The Model of Coordinated Action, as a framework for analyzing group collaboration, identifies several dimensions of common features of cooperative work that extend beyond the CSCW matrix and allow for more complexity in describing how teams work given certain conditions. The seven total dimensions that constitute the model (MoCA) are used to describe essential "fields of action" seen in existing CSCW research. Rather than existing as a rigid matrix with distinct quadrants, this model is to be interpreted as multidimensional – each dimension existing as its own continuum. These ends of these dimensions' continuums are defined in the following subsections. Synchronicity [edit] This is pertaining to the time at which the collaborative work occurs. This could range from live meetings conducted at exact times to viewing recordings or responding to messages that do not require one or all participants to be active at the time the recording, message, or other deliverable was created. Physical distribution [edit] This covers the distance in which team members could be geographically separated while still being able to collaborate. The least physically distributed cooperative work is a meeting in which all team members are physically present in the same space and communicating verbally, face-to-face. Conversely, technology now allows for more distanced communication that could extend as far as meeting from multiple countries. Scale [edit] The scale of a collaborative project refers to how many individuals comprise the project team. As the number of people involved increases, the division of tasks must become more intricate and complex to ensure that each participant is contributing in some way. Number of communities of practice [edit] A community of practice refers to a group of individuals with shared, common knowledge of a specific subject. This group may be composed of both newcomers and experts. New members will gain knowledge through exposure and immersion and become experts as newer members join, thus expanding the community of practice over time. These groups can be as specific or as broad as their members feel is necessary, as no two people have the same set of knowledge and diversification of perspectives is common. Nascence [edit] Some collaborative projects are designed to be more long-lasting than others, often meaning that their standard practices and actions are more established than newer, less developed projects. Synonymous with "newness", nascence refers to how established a cooperative effort is at a given point in time. While most work is always developing in some way, newer projects will have to spend more time establishing common ground among its team members and will thus have a higher level of nascence. Planned permanence [edit] This dimension encourages teams to establish common practices, terminology, etc. within the group to ensure cohesion and understanding among the work. It is difficult to gauge how long a project will last, therefore establishing these foundations in early stages helps to prevent confusion between group members at later stages when there may be higher stakes or deeper investigation. The notion of planned permanence is essential to the model as it allows for productive communication between individuals who may have different expertise or are members of different communities of practice. Turnover [edit] This dimension is used to describe the rate at which individuals leave a collaborative group. Such events may occur at various rates depending on the impact one's departure may have on the individual and the group. In a well-established collaborative action or a group with a small scale, a team member leaving may have detrimental effects, whereas temporary projects with open membership may have high turnover rates covered by the project's high scale. Crowdsourcing, such as the means by which Wikipedia creates its articles, are an example of an entity with high turnover rates (e.g. a Wikipedian contributes only to one article at one time) that does not face impactful consequences due to the high scale of the collaborative work. Considerations for interaction design [edit] Self-presentation [edit] Self-presentation has been studied in traditional face-to-face environments, but as society has embraced content culture, social platforms have generated new affordances for presenting oneself online. Due to technological growth, social platforms, and their increased affordances, society has reconfigured the way users self-present online due to audience input and context collapse. In an online setting, audiences are physically invisible which complicates the users ability to distinguish their intended audience. Audience input, on social platforms, can range from commenting, sharing, liking, tagging, etc. For example, LinkedIn is a platform who encourages commentary where positive feedback outweighs negative feedback on topics including career announcements. Conversely, audience input can be unwarranted which can lead to real-life implications, especially for marginalized groups who are prone to both warranted and unwarranted commentary on public posts. Context collapse is when separate audiences join together and make curated content for an audience which is visible to unintended audiences. The likelihood of context collapse is especially challenging with the surge of proprietary software which introduces a conflict of interest for the users who have an ideal audience, but the platforms algorithm has a differing one. Collapsed context influences self-presentation when previously separate audiences are merged into one. Affordance [edit] Main article: Affordance As media platforms proliferate, so do the affordances offered that directly influence how users manage their self-presentation. According to researchers, the three most influential affordances on how users present themselves in an online domain include anonymity, persistence, and visibility. Anonymity in the context of social media refers to the separation of an individual's online and offline identity by making the origin of their messages unspecified. Platforms that support anonymity have users that are more likely to depict their offline self accurately online (i.e Reddit). Comparatively, platforms with less constraints on anonymity are subject to users that portray their online and offline selves differently, thus creating a "persona". Facebook, for example, requires its users to abide by its "real-name" policy, further connecting their offline and online identities. Furthermore, being able to unequivocally associate an online persona to a real-life human contributes to how users present themselves online honestly. Platforms which have "content persistence" store content so it may be accessed at a later point in time. Platforms including Instagram and Facebook are highly persistent with their ability to make content available until deleted. Whereas, Snapchat has lower persistence because content is ephemeral causing users to post content that represents their offline self more accurately. This affordance strongly affects users' self-presentation management because they recognize content can be openly accessed on platforms that are highly persistent. On social platforms, visibility is created when information is acquired with search of a word or phrase or even topic name, an example being a hashtag. When content is visible, users become aware of their self-presentation and will adjust accordingly. However, some platforms give their users leverage in specifying how visible their content is, thus affording for visibility control. For example Snapchat and Instagram both allow users to build a "close friends list" and block specific people from viewing content. Nonetheless, intended audiences are never guaranteed. Facebook is an example of a platform that shares content to both primary (e.g. direct friends) and secondary viewers (e.g. friends of friends). The concern of visibility with Facebook's algorithm is notably challenging for marginalized groups because of such blurred visibility mechanisms. In addition, users face privacy concerns relative to visibility given the current era of screenshotting. Boundary object [edit] Main article: Boundary object A boundary object is an informational item which is used differently by various communities or fields of study and may be a concrete, physical item or an abstract concept. Examples of boundary objects include: Most research libraries, as different research groups may use different resources from the same libraries. An interdisciplinary research project, as different business sectors and research groups may have different goals for the project. The outline of a U.S. state's boundaries, which may be drawn on a roadmap for travelers or on an ecological map for biologists. In computer-supported cooperative work, boundary objects are typically used to study how information and tools are transmitted between different cultures or communities. Some examples of boundary objects in CSCW research are: Electronic health records, which pass health information between groups with different priorities (such as doctors, nurses, and medical secretaries). The concept of a "digital work environment", as used in Swedish political debate. Standardization vs. flexibility in CSCW [edit] Standardization is defined as "agile processes that are enforced as a standard protocol across an organization to share knowledge and best practice." Flexibility, on the other hand, is the "ability to customize and evolve processes to suit the aims of an agile team". As CSCW tools, standardization and flexibility are almost mutually exclusive from each other. In CSCW, flexibility comes in two forms, flexibility for future change, and flexibility for interpretation. Everything that is done on the internet has a level of standardization due to the internet standards. In fact, Email has its own set of standards, of which the first draft was created in 1977. No CSCW tool is perfectly flexible, and all lose flexibility in the same three levels. Either flexibility is lost when the programmer makes the toolkit, when the programmer makes the application, and/or when the user uses the application. Standardization in information infrastructure [edit] Information infrastructure requires extensive standardization to make collaboration work. Since data is transferred from company to company and occasionally nation to nation, international standards have been put in place to make communication of data much simpler. Often one company's data will be included in a much larger system, and this would become almost impossible without standardization. With information infrastructure, there is very little flexibility in potential future changes. Due to the fact that the standards have been around for decades and there are hundreds of them, it is nearly impossible to change one standard without greatly affecting the others. Flexibility in toolkits [edit] Creating CSCW toolkits requires flexibility of interpretation; it is important that these tools are generic and can be used in many different ways. Another important part of a toolkit's flexibility is the extensibility, the extent to which new components or tools can be created using the tools provided. An example of a toolkit that is flexible in how generic the tools are is Oval. Oval consists of four components: objects, views, agents, and links. This toolkit was used to recreate four previously existing communication systems: The Coordinator, gIBIS, Lotus Notes, and Information Lens. It proved that, due to its flexibility, Oval was able to create many forms of peer-to-peer communication applications. Applications [edit] Applications in education [edit] There have been three main generations to distance education, starting with the first being through postal service, the second through mass media such as the radio, television, and films, and the third being the current state of e-learning. Technology-enhanced learning, or "e-learning", has been an increasingly relevant topic in education, especially with the development of the COVID-19 pandemic that has caused many schools to switch to remote learning. E-learning is defined as "the use of technology to support and enhance learning practice". It includes the utilization of many different types of information and communication technologies (ICTs) and is limited to the use of intranet and internet in the teaching and learning process. The development of content is mainly through using learning objectives to create activities through Virtual Learning Environments, Content Management Systems, and Learning Management Systems. These technologies have created massive change in their use as CSCW tools, allowing students and teachers to work on the same platforms and have a shared online space in which to communicate in. The delivery of content can be either asynchronous, such as email and discussion forums, or synchronous, like through chat or video conferencing. Synchronous education allows for much more equal interaction between students and instructors and better communication between students for the facilitation of group projects and assignments. Community of inquiry framework [edit] E-learning has been explained by the community of inquiry (COI) framework introduced by Garrison et al. In this framework, there are three major elements: cognitive presence, social presence, and teaching presence. Cognitive presence in this framework is the measure of how well meaning is able to be constructed from the content being taught. It assumes that students have access to a large network from which to gain information from. This includes peers, instructors, alumni, and practicing professionals. E-learning has allowed this network to be easily accessible through the internet, and these connections can be made synchronously through video, audio, or texts. Social presence in the community of inquiry framework is how well participants can connect with one another on a social level and present themselves as "real people". Video conferencing has been shown to increase social presence within students. One study found that "social presence in VC [Virtual Conferencing] can have a positive effect on group efficacy and performance by amplifying group cohesion". This information is greatly useful in designing future systems because it explains the importance of technology like video conferencing in synchronous e-learning. Groups that are able to see each other face to face have a stronger bond and are able to complete tasks faster than those without it. Increasing the social presence in online education environments helps facilitate in the understanding of the content and the ability for the group to solve problems. Teaching presence in the COI framework contains two main functions: creation of content and the facilitation of this content. The creation of content is usually done by the instructor, but students and instructors can share the role of facilitator, especially in higher education settings. The goal of teaching presence is "to support and enhance social and cognitive presence for the purpose of realizing educational outcome". Virtual educational software and tools are becoming more readily used globally. Remote educational platforms and tools must be accessible for various generations, including children as well as guardians or teachers, yet these frameworks are not adapted to be child-friendly. The lack of interface and design consideration for younger users causes difficulty in potential communication between children and older generations utilizing the software. This in turn leads to a decrease in virtual learning participation as well as potential diminished collaboration with peers. In addition, it may be difficult for older teachers to utilize such technology, and communicate with their students. Similar to orienting older workers with CSCW tools, it is difficult to train younger students or older teachers in utilizing virtual technology, and may not be possible for widely spread virtual classrooms and learning environments. Applications in gaming [edit] Collaborative mixed reality games modify the shared social experience, during which players can interact in real-time with physical and virtual gaming environments and with other multiplayer video gamers. This may be done through any means of communication, self-representation, and collaboration. Communication systems [edit] The group members experience effective communication practices following the availability of a common platform for expressing opinions and coordinating tasks. The technology is applicable not only in professional contexts but also in the gaming world. CSCW usually offers synchronous and asynchronous games to allow multiple individuals to compete in a certain activity across social networks. Thus, the tool has made gaming more interesting by facilitating group activities in real-time and widespread social interactions beyond geographical boundaries. Self-presentation [edit] HCI, CSCW, and game studies in MMORPGs highlight the importance of avatar-mediated self-presentation in player experience. These studies have put together known two components of self-presentation in games. First, through personal choice and personalization of avatars, various social values (such as gender roles and social norms) are integrated and reflected in the player's self-image. Second, self-presentation in games conjointly options experimentation of fully new identities or reaffirmation of existing identities. This includes cross-gender play and queerness gameplay. Computer-mediated communication in gaming settings takes place across different channels, which can consist of structured message systems, bulletin boards, meeting rooms, and shared diaries. As such, the players can hold conversations while proceeding with the game to create a lively experience. Thus, the features of video games offer a platform for users to openly express themselves. Collaborations and game design in multi-user video games [edit] The most collaborative and socially interactive aspect of a video game is the online communities. Popular video games often have various social groups for their diverse community of players. For example, in the quest-based multiplayer game World of Warcraft, the most collaborative and socially interactive aspect of the game are the "Guilds", which are alliances of individuals with whom players must join forces. By incorporating Guilds, World of Warcraft creates opportunities for players to work together with their team members who can be from anywhere in the world. WOW players who are associated with a Guild are more likely to play and do quests with the same Guild mates each time which develops a strong bond between players and a sense of community. These bonds and friendships formed from playing with Guild mates, not only improves collaboration within the game, it also creates a sense of belonging and community which is one of the most important attribute of online gaming communities. When it comes to designing a multi-user collaborative game, it involves positive interdependence, personal accountability, and social skills. Positive Interdependence is the dependence of collaboration from members of a group in order to accomplish a task. In video games, this is the idea of players on a team or in a group understanding that working together is beneficial, and that the success and failure of the group is shared equally if all members participate. An example of including a positive interdependence aspect to a video game is creating a common shared goal for the team to increase collaboration. The next guideline is personal accountability, which is the idea that each individual in a group must put forth their best effort for the team's overall success. Personal accountability might be incorporated into video games by including an incentive system where individual players are rewarded with additional points for completing an objective or an action that improves the team's chances of success. The final guideline, social skills, is the most important to consider when designing a collaborative game. An example of developing player social skills through a video game can be creating in-game situations where players have to assign roles, plan, and execute to solve the problem. By following these guidelines, game makers can create gaming-environments which encourage collaboration and social interaction between players. Applications of mobile devices [edit] Mobile devices are generally more accessible than their non-mobile counterparts, with about 41% of the world's population as per a survey from 2019 owning a mobile device. This coupled with their relative ease of transport makes mobile devices usable in a large variety of settings which other computing devices would not function as well. Because of this, mobile devices make videoconferencing and texting possible in a variety of settings which would not be accessible without such a device. The Chinese social media platform WeChat has is utilized to facilitate communication between patients and doctors. WeChat is able to enhance healthcare interactions between patient and doctor by allowing direct communication of the patient's symptoms. Applications in social media [edit] Social media tools and platforms have expanded virtual communication amongst various generations. However, with older individuals being less comfortable with CSCW tools, it is difficult to design social platforms that account for both older and younger generational social needs. Often, these social systems focus key functionality and feature creation for younger demographics, causing issues in adaptability for older generations. In addition, with the lack of scalability for these features, the tools are not able to adapt to fit evolutional needs of generations as they age. With the difficulty for older demographics to adopt these intergenerational virtual platforms, the risk of social isolation is increased in them. While systems have been created specifically for older generations to communicate amongst one another, system design frameworks are not complex enough to lend to intergenerational communication. Applications to ubiquitous computing [edit] Main article: Ubiquitous computing Along the lines of a more collaborative modality is something called ubiquitous computing. Ubiquitous computing was first coined by Mark Weiser of Xerox PARC. This was to describe the phenomenon of computing technologies becoming prevalent everywhere. A new language was created to observe both the dynamics of computers becoming available at mass scale and its effects on users in collaborative systems. Between the use of social commerce apps, the rise of social media, and the widespread availability of smart devices and the Internet, there is a growing area of research within CSCW that how come out of these three trends. These topics include ethnomethodology and conversation analysis (EMCA) within social media, ubiquitous computing, and instant message based social commerce. Ethnomethodology and synchronicity [edit] In You Recommend I Buy: How and Why People Engage in Instant Messaging Based Social Commerce, researchers on this project analyzed twelve users of Chinese Instant Messenger (IM) social commerce platforms to study how social recommendation engines on IM commerce platforms result in a different user experience. The study was entirely on Chinese platforms, mainly WeChat. The research was conducted by a team composed of members from Stanford, Beijing, Boston, and Kyoto. The interviewing process took place in the winter of 2020 and was an entirely qualitative analysis, using just interviews. The goal of the interviews were to probe about how participants got involved in IM based social commerce, their experience on IM based social commerce, the reasons for and against IM based social commerce, and changes introduced by IM based social commerce to their lives. An IM-based service integrates directly with more intimate social experiences. Essentially, IM is real-time texting over a network. This can be both a synchronous or asynchronous activity. IM based social commerce makes the user shopping experience more accessible. In terms of CSCW, this is an example of ubiquitous computing. This creates a "jump out of the box" experience as described in the research because the IM based platform facilitates a change in user behavior and the overall experience on social commerce. The benefit of this concept is that the app is leveraging personal relationships and real-life networks that can actually lead to a more meaningful customer experience, which is founded upon trust. Embeddedness [edit] A second CSCW paper, Embeddedness and Sequentiality in Social Media, explores a new methodology for analyzing social media—another expression of ubiquitous computing in CSCW. This paper used ethnomethodology and conversation analysis (EMCA) as a framework to research Facebook users. In brief, ethnomethodology studies the everyday interactions of people and relates how this pertains to forming their outlook of the world. Conversational analysis delves into the structures of conversations so as to extract information about how people construct their experiences. The team behind this research, hailing from University of Nottingham and Stockholm University, recognized that "moment-by-moment, unfolding, real-time human action" was somewhat missing from the CSCW literature on social media. The significance of this is they felt that by exploring EMCA, it could provide different insights on collaborative social network systems, as opposed to relying solely on recall. Here is a formal definition for EMCA: For EMCA, the activities of everyday life are structured in time—some things routinely happen before others. Fundamentally there is a 'sequentiality' to activity, something that has been vital for developing understanding of the orderly nature of talk and bodily interaction. In other words, EMCA pays attention to the sequence of events, so as to reveal some sort of underlying order about our behavior in our day-to-day interactions. In the bigger picture, this work reveals that time, as one of the dimensions to consider within collaborative systems design, matters. Another major factor would be distance. Does Distance Still Matter? Revisiting the CSCW Fundamentals on Distributed Collaboration is another research article that, as the title suggests, explores under what circumstances distance matters. Most notably, it mentions the "mutual knowledge problem." This problem arises when a group in a distributed collaborative system experiences a breakdown in communication due to the fact that its members lack a shared understanding for the given context they are working in. According to the article, it matters that everyone is in alignment over the nature of what they are doing. Co-located, parallel and sequential activities [edit] The solutions of unresolved issues in ubiquitous computing systems can be explored now that the observations of user experiences in social media, which are normally based on recollection, are no longer needed. Some of the unresolved questions include: "How does social media start being used, stop being used? When is it being used, and how is that usage ordered and integrated into other, parallel activities at the time?" Parallel activities refer to occurrences in co-located groupware and ubiquitous computing technologies like social media. Examining these sequential and parallel activities in user groups on social media networks enables the ability to "[manage] the experience of that everyday life." An important takeaway from this paper on EMCA and sequentiality is that it reveals how the choices made by designers of social media apps ultimately mediates our end-user experience, for better or for worse. It reveals: "when content is posted and sequentially what is associated with it." Ubiquitous computing infrastructures [edit] On the topic of computing infrastructures, Democratizing Ubiquitous Computing – a Right for Locality presents a study from researchers at Lancaster University on ubiquitous computing ("ubicomp") to identify where there exists positive or negative effects on users and society at large. The research specifically focuses on cities or urban areas as they are places where one can expect a lot of technological and social activities to take place. An apparent guiding principle to the research is that the goal of advancing any ubicomp technologies should be to maximize the amount of good to as many people in a society as possible. A key observation is made about the way in which these infrastructures come into being: A ubiquitous computing infrastructure can play an important role in enabling and enhancing beneficial social processes as, unlike electricity, digital infrastructure enhances a society's cognitive power by its ability to connect people and information . While infrastructure projects in the past had the idealistic notion to connect the urban realm and its communities of different ethnicity, wealth, and beliefs, Graham et al. note the increasing fragmentation of the management and ownership of infrastructures. This is because ubicomp has the potential to further disadvantage marginalized communities online. The current disadvantage of ubiquitous computing infrastructures is that they do not best support urban development. Proposals to resolve these social issues include increased transparency about personal data collection as well as individual and community accountability about the data collection process in ubicomp infrastructure. Data at work: supporting sharing in science and engineering is one such paper that goes into greater depth about how to build better infrastructures that enable open data-sharing and thus, empower its users. What this article outlines is that in building better collaborative systems that advance science and society, we are, by effect, "promoting sharing behaviors" that will encourage greater cooperation and more effective outcomes. Essentially, ubiquitous computing will reflect society and the choices it makes will influence those computing systems that are put in place. Ubiquitous computing is huge to the field of CSCW because as the barriers between physical boundaries that separate us break down with the adoption of technology, our relationships to those locations is actually strengthened. However, there remains few potential challenges when it comes to social collaboration and the workplace. Computer-supported collaboration on Art [edit] The romanticized notion of a lone, genius artist has existed since the time of Giorgio Vasari’s Lives of the Artists, published in 1568. Vasari promulgated the idea that artistic skill was endowed upon chosen individuals by gods, which created an enduring and largely false popular misunderstanding of many artistic processes. Artists have used collaboration to complete large scale works for centuries, but the myth of the lone artist was not widely questioned until the 1960s and 1970s. With the appearance of computers, and especially with the invention of the internet, collaboration on art became easier than before. This crowd-sourced creativity online is putting a "new twist" on traditional ideas of artistic ownership, online communication and art production. In some cases, people don't even know they are making contributions to online art. Artists in the computer era are considered more "socially aware" in a way that supports social collaboration on social matters. Art duos, such as the Italian Hackatao duo, collaborate both physically and online while creating their art in order to "create a meeting place between the NFT and traditional art worlds." Crowdsourcing aids with innovation processes, successful implementation and maintenance of ideas generation, thereby providing support for the development of promising innovative ideas. Crowdsourcing has been used in various ways from rousing musical numbers, to choreography, set design, costumes and marketing materials and in some cases was crowdsourced using social media platforms. Challenges [edit] Social – technical gap [edit] The success of CSCW systems is often so contingent on the social context that it is difficult to generalize. Consequently, CSCW systems that are based on the design of successful ones may fail to be appropriated in other seemingly similar contexts for a variety of reasons that are nearly impossible to identify a priori. CSCW researcher Mark Ackerman calls this "divide between what we know we must support socially and what we can support technically" the social-technical gap and describes CSCW's main research agenda to be "exploring, understanding, and hopefully ameliorating" this gap. It is important to analyze 'what we know we must support socially' for a few reasons. The way interaction takes place within an in-person setting is something that cannot be easily changed unlike the way technology is able to be manipulated to fit specific needs today. There are certain norms and standards lived up to within peoples' day to day lives, a certain part of those norms and attitudes carry over into the online world. The problem is mimicking daily communication styles and behavior into an online setting. Schmidt examines this concept within "Mind the Gap", he states "Cooperative work is a tricky phenomenon. We are all engaged in cooperative activities of various sorts in our everyday lives and routinely observe others working together around us. We are all experts from our everyday experience. And yet this quotidian insight can be utterly misleading when applied to the design of systems to support cooperative work". Though in-person communication on a day-to-day basis is natural for most, it does not easily translate over into cooperative work. This highlights the need for adaptability within CSCW systems, Schmidt expands on the "crucial requirement of flexibility that arises from the changing needs of the cooperative work setting". These all tie together to highlight the gaps within CSCW. Leadership [edit] Generally, teams working in a CSCW environment need the same types of leadership as non-CSCW teams. However, research has shown that distributed CSCW teams may need more direction at the time the group is formed than traditional working groups, largely to promote cohesion and liking among people who may not have as many opportunities to interact in person, both before and after the formation of the working group. Adoption of groupware [edit] Groupware goes hand in hand with CSCW. The term refers to software that is designed to support activities of a group or organization over a network and includes email, conferencing tools, group calendars, workflow management tools, etc. While groupware enables geographically dispersed teams to achieve organizational goals and engage in cooperative work, there are also many challenges that accompany use of such systems. For instance, groupware often requires users to learn a new system, which users may perceive as creating more work for them without much benefit. If team members are not willing to learn and adopt groupware, it is highly difficult for the organization to develop the requisite critical mass for the groupware to be useful. Further, research has found that groupware requires careful implementation into a group setting, and product developers have not as yet been able to find the most optimal way to introduce such systems into organizational environments. On the technical side, networking issues with groupware often create challenges in using groupware for CSCW. While access to the Internet is becoming increasingly ubiquitous, geographically dispersed users still face challenges of differing network conditions. For instance, web conferencing can be quite challenging if some members have a very slow connection and others are able to utilize high speed connections. Intergenerational groups [edit] Adapting CSCW tools for intergenerational groups is a prevalent issue within all forms of CSCW. Different generations have different feelings towards technology as well as different ways to utilize technology. However, as technology has become integral to everyday tasks, it must be accessible to all generations of people. With cooperative work becoming increasingly important and diversified, virtual interaction between different generations is also expanding. Given this, many fields that utilize CSCW tools require carefully designed frameworks to account for different generations. Workplace teams [edit] One of the recurring challenges in CSCW environments is development of an infrastructure that can bridge cross-generational gaps in virtual teams. Many companies rely on communication and collaboration between intergenerational employees to be successful, and often this collaboration is performed using various software and technologies. These team-driven groupware platforms range from email and daily calendars to version control platforms, task management software, and more. These tools must be accessible to workplace teams virtually, with remote work becoming more commonplace. Ideally, system designs will accommodate all team members, but orienting older workers to new CSCW tools can often be difficult. This can cause problems in virtual teams due to the necessity of incorporating the wealth of knowledge and expertise that older workers bring to the table with the technological challenges of new virtual environments. Orienting and retraining older workers to effectively utilize new technology can often be difficult, as they generally have less experience than younger workers with learning such new technologies. As older workers delay their retirement and re-enter the workforce, teams are becoming increasingly intergenerational, meaning that the creation of effective intergenerational CSCW frameworks for virtual environments is essential. Tools in CSCW [edit] Collaboration amongst peers has always been an integral aspect to getting something done. Working together not only eases the difficulty of the task at hand, but leads to more effective work that is accomplished. As computers and technology become increasingly important in everyday lives, communication skills change as technology allows individuals to stay connected across many previous barriers. Barriers to communication might have been the end of the work day, being across the country or even slow applications that are more of a hindrance than an aid. With new collaborative tools that have been tried and tested, these previous barriers to communication have been shattered and replaced with new tools that help progress collaboration. Tools that have been integral in shaping computer supported cooperative work can be split into two major categories: communication and organization. Communication: The ability to communicate with others while working is a luxury that has increased the speed and accuracy at which tasks are accomplished. Individuals can also send pictures of code and issues through platforms like Microsoft Teams without anyone needing to change screen monitors. This particular change increased office productivity and communication by almost half. The ability to send more specific information faster gave the employees the ability to get more done with also much less effort for themselves. Tools like Microsoft Teams and Slack also allow people to collaborate with ease even if they are in different time zones or different geographical areas. This means that work is no longer tied to specific offices at a 9 to 5 job, but can be done anywhere because you have the ability to communicate with one or groups of people on a large scale. Organization: Apps such as iCal and Reminders on the iPhone provide time-oriented structure and remind users of the tasks they must complete. Organization and communication go hand in hand with one another, as they help individuals better plan their day because apps warn them when two events overlap, a due date approaches, or whether there is time available for an event. There is reduced hassle to daily scheduling and group coordination. Such apps usually tie into different electronic devices such as computers and tablets, therefore people receive reminders across multiple platforms. If the platform permits, individuals in teams can set reminders for other people. Departmental conflicts [edit] Cross-boundary breakdowns [edit] Cross-boundary breakdowns are when different departments of the same organization unintentionally harm other. They may be caused by failures to coordinate activities across multiple departments, a form of articulation work. Hospitals may experience cross-boundary breakdowns during patient transfers. When a patient is sent from the emergency department to the operation room, the inpatient access department (IPA) must normally be notified, allowing them to track the number and location of available ICU beds. However, when the emergency department fails to notify the IPA, the IPA staff are later unable to find suitable beds for patients. Re-coordinating activities [edit] To restore useful communication between departments after a cross-boundary breakdown, organizations may perform re-coordinating activities. Hospitals may respond to cross-boundary breakdowns by explicitly ranking key resources or assigning "integrator" roles to multiple staff members across different departments. Challenges in research [edit] Differing meanings [edit] In the CSCW field, researchers rely on a variety of sources that include journals and research schools of thought. These different sources may lead to disagreement and confusion, as there are terms in the field that can be used in different contexts ("user", "implementation", etc.) User requirements change over time and are often not clear to participants due to their evolving nature and the fact that requirements are always in flux. Identifying user needs [edit] CSCW researchers often have difficulty deciding which set(s) of tools will benefit a particular group because of the nuances within organizations. This is exacerbated by the fact that it is challenging to accurately identify user/group/organization needs and requirements, since such needs and requirements inevitably change through the introduction of the system itself. When researchers study requirements multiple times, the requirements themselves often change and evolve once the researchers have completed a particular iteration. Evaluation and measurement [edit] The range of disciplinary approaches leveraged in implementing CSCW systems makes CSCW difficult to evaluate, measure, and generalize to multiple populations. Because researchers evaluating CSCW systems often bypass quantitative data in favor of naturalistic inquiry, results can be largely subjective due to the complexity and nuances of organizations themselves. Possibly as a result of the debate between qualitative and quantitative researchers, three evaluation approaches have emerged in the literature examining CSCW systems. However, each approach faces its own unique challenges and weaknesses: \| Evaluation approach \| Usage \| Weakness \| --- \| Methodology-oriented frameworks \| Explain the methods of inquiry available to CSCW researchers \| Not providing guidance for selecting the best method for a particular research question or population \| \| Conceptual frameworks \| Provide guidelines for determining factors that a researcher should consider and evaluate through CSCW research \| Fail to link conceptual constructs with methodological approaches. Thus, while researchers may know what factors are important to their inquiry, they may have difficulty understanding which methodologies will result in the most informative findings \| \| Concept-oriented frameworks \| Provide specific advice for studying isolated aspects of CSCW \| Lack guidance as to how specific areas of study can be combined to form more comprehensive insight \| Diversity, equity, and inclusion [edit] Gender [edit] In computer-supported cooperative work, there are small psychological differences between how men and women approach CSCW programs. This can lead to unintentionally biased systems, due to the majority of software being designed and tested by men. As well, in systems where societal gender differences are not accounted for and countered, men tend to overrepresent women in these online spaces. This can lead to women feeling potentially alienated and unfairly targeted by CSCW programs. In recent years, more studies have been conducted on how men and women interact with each other using CSCW systems. Findings do not indicate that men and women have performance difference when performing CSCW tasks, but rather that each gender approaches and interacts with software and performs CSCW tasks differently. In most findings, men were more likely to explore potential choices and willing to take risks compared to women. In group tasks, women in general were more conservative in voicing their opinions and suggestions on tasks when paired with a male, but inversely were very communicative when paired with another woman. As well, men are found to be more likely to take control of group activities and teamwork, even from a young age, leading to further ostracizing of women speaking up in CSCW group work. Additionally, in CSCW message boards, men on average posted more messages and engaged more frequently than their female counterparts. Increasing female participation [edit] The dynamic of women in the workforce not participating as much is less of a CSCW problem and is prevalent in all workspaces, but software can still be designed to increase female participation in CSCW. In software design, women are more likely to be involved if software is designed to center communication and cooperation. This is one possible method to increasing female participation, and it does not address why CSCW has lower female participation in the first place. In a study, women generally rated themselves as being poor at understanding technology, having difficulty at using mobile programs, and disliked using CSCW software. However, when asked these same questions about specific software in general, they rated themselves just as strongly as the men in the study did. This lack of confidence in software as a whole impacts women's ability to efficiently and effectively use online programs compared to men, and accounts for some of the difficulties women face in using CSCW software. Despite being an active area of research since the 1990s, many developers often do not take gender differences into account when designing their CSCW systems. These issues compound on top of the cultural problems mentioned previously, and lead to further difficulties for women in CSCW. By enabling developers to be more aware of the differences and difficulties facing women in CSCW design, women can be more effective users of CSCW systems through sharing and voicing opinions. Conferences [edit] Since 2010, the Association for Computing Machinery (ACM) has hosted a yearly conference on CSCW, "The ACM Conference on Computer Supported Cooperative Work". The conference is sponsored by the SIGCHI special interest group. The CSCW conference was held bianually from 1986 to 2010 and annually thereafter. By 2010, CSCW researchers were observing that the name "Computer Supported Cooperative Work" no longer reflected the work done in the field. As a result of those debates, the conference would expand its name to "CSCW & Social Computing", incorporating the reality of social computing research within CSCW as a field. The conference is currently held in October or November and features research in the design and use of technologies that affect organizational and group work. With the development of new devices that allow collaboration from different locations and contexts, CSCW includes researchers from both academia and industry to discuss virtual collaboration from both social and technical perspectives. Internationally, the Institute of Electrical and Electronics Engineers (IEEE) sponsors the International Conference on Computer Supported Work in Design, which takes place yearly. In addition, the European Society for Socially Embedded Technologies sponsors the European Conference on Computer Supported Cooperative Work, which has been held every two years since 1989. CSCW panels are a regular component of conferences of the adjacent field of science and technology studies. Related fields [edit] Related fields are collaborative product development, CAD/CAM, computer-aided software engineering (CASE), concurrent engineering, workflow management, distance learning, telemedicine, medical CSCW and the real-time network conferences called MUDs (after "multi-user dungeons," although they are now used for more than game-playing). See also [edit] Citizen science Collaborative development environment Collaborative information seeking Collaborative innovation network Collaborative software Collaborative work systems Collaborative working environment Collaborative working system Commons-based peer production Computer-supported collaboration Computer-supported collaborative learning E-professional Human–computer interaction Integrated collaboration environment List of collaborative software List of project management software Mass collaboration Participatory design Pervasive informatics Remote work Social computing Social peer-to-peer processes Toolkits for User Innovation Ubiquitous computing Virtual research environment Wicked problem References [edit] ^ Jump up to: a b Grudin, Jonathan; Poltrock, Steven (1 January 2024). "Computer Supported Cooperative Work". The Encyclopedia of Human-Computer Interaction, 2nd Ed. Interaction Design Foundation. ^ Carstensen, P.H.; Schmidt, K. (1999). "Computer supported cooperative work: new challenges to systems design". Archived from the original on 2006-09-25. Retrieved 2007-08-03. ^ Wilson, P. (1991). Computer Supported Cooperative Work: An Introduction. Springer Science & Business Media. ISBN 9780792314462. ^ "The 26th ACM Conference On Computer-Supported Cooperative Work And Social Computing". Retrieved 2023-04-24. ^ Jump up to: a b Grudin, J. (1988). "Why CSCW applications fail: problems in the design and evaluation of organization of organizational interfaces". Proceedings of the 1988 ACM conference on Computer-supported cooperative work. ACM Press New York, NY, US. pp. 85–93. ^ Jump up to: a b c Joseph A. Carpini, Sharon K. Parker and Mark A. Griffin, 20 March 2017. "A Look Back and a Leap Forward: A Review and Synthesis of the Individual Work Performance Literature" Academy of Management Annals, Vol. 11, No. 2 ^ Convertino, G.; Farooq, U.; Rosson, M. B.; Carroll, J. M.; Meyer, B. J. F. (July 2007). "Supporting intergenerational groups in computer-supported cooperative work (CSCW)". Behaviour & Information Technology. 26 (4): 275–285. doi:10.1080/01449290601173473. ISSN 0144-929X. S2CID 18381170. ^ Dourish, P.; Bellotti, V. (1992). "Awareness and coordination in shared workspaces". Proceedings of the 1992 ACM conference on Computer-supported cooperative workcc. ACM Press New York, NY, USA. pp. 107–114. ^ Schmidt, K.; Bannon, L. (1992). "Taking CSCW seriously". Computer Supported Cooperative Work. 1 (1–2): 7–40. doi:10.1007/BF00752449. S2CID 13920599. ^ Strauss, A. (1985). "Work and the Division of Labor". The Sociological Quarterly. 26 (1): 1–19. doi:10.1111/j.1533-8525.1985.tb00212.x. S2CID 145533067. ^ MacKay, W.E. (1991). "Patterns of sharing customizable software". Proceedings of the 1990 ACM conference on Computer-supported cooperative work. ACM Press New York, NY, USA. pp. 209–221. ^ Dourish, P. (2003). "The Appropriation of Interactive Technologies: Some Lessons from Placeless Documents". Computer Supported Cooperative Work. 12 (4): 465–490. doi:10.1023/A:1026149119426. S2CID 12993958. ^ Schmidt, K. (1991). "Computer Support for Cooperative Work in Advanced Manufacturing". International Journal of Human Factors in Manufacturing. 1 (4): 303–320. CiteSeerX 10.1.1.142.652. doi:10.1002/hfm.4530010402. ^ Stuart Geiger (2011). Lovink & Tkacz (ed.). "The Lives of Bots" in Critical Point of View: A Wikipedia Reader (PDF). Archived (PDF) from the original on 2013-12-10. Retrieved 2015-10-01. ^ Jump up to: a b Pallesen, Trine; Jacobsen, Peter H. (2018). "Articulation work from the middle—a study of how technicians mediate users and technology". New Technology, Work and Employment. 33 (2): 171–186. doi:10.1111/ntwe.12113. hdl:10398/83b3e3e2-b6ca-425c-814d-6005f7f067a0. ISSN 1468-005X. ^ Jump up to: a b c Peiwei Mi and W. Scacchi, "Modeling Articulation Work in Software Engineering Processes," in Proceedings. First International Conference on the Software Process,, Redondo Beach, Calif., 1991 pp. 188–201. doi: 10.1109/ICSP.1991.664349 ^ Jump up to: a b Fjuk, Annita & Nurminen, Markku & Smørdal, Ole & Centre, Turku. (2002). Taking Articulation Work Seriously – an Activity Theoretical Approach. ^ Ferreira, Jennifer; Sharp, Helen; Robinson, Hugh (2011). "User experience design and agile development: managing cooperation through articulation work". Software: Practice and Experience. 41 (9): 963–974. doi:10.1002/spe.1012. ISSN 1097-024X. S2CID 27990427. Archived from the original on 2022-04-08. Retrieved 2021-04-28. ^ Hampson, Ian; Junor, Anne (2005). "Invisible work, invisible skills: interactive customer service as articulation work". New Technology, Work and Employment. 20 (2): 166–181. doi:10.1111/j.1468-005X.2005.00151.x. ISSN 1468-005X. S2CID 109587446. Archived from the original on 2022-04-08. Retrieved 2021-04-28. ^ Jump up to: a b Johansen, Robert (2020-11-25), "User Approaches to Computer-Supported Teams", Technological Support for Work Group Collaboration, CRC Press, pp. 1–31, doi:10.1201/9781003063940-1, hdl:1721.1/49359, ISBN 978-1-003-06394-0, S2CID 60980034, retrieved 2022-01-20 ^ Baecker, R.M.; others (1995). Readings in Human-Computer Interaction: Toward the Year 2000. Morgan Kaufmann Publishers. ^ Molli, P.; Skaf-Molli, H.; Oster, G.; Jourdain, S. (2002). "SAMS: Synchronous, asynchronous, multi-synchronous environments". The 7th International Conference on Computer Supported Cooperative Work in Design. COPPE/UFRJ. pp. 80–84. doi:10.1109/cscwd.2002.1047653. ISBN 85-285-0050-0. S2CID 1260775. ^ Jump up to: a b c d e f g h Lee, Charlotte P.; Paine, Drew (2015-02-28). "From the Matrix to a Model of Coordinated Action (MoCA)". Proceedings of the 18th ACM Conference on Computer Supported Cooperative Work & Social Computing. CSCW '15. New York, NY, USA: Association for Computing Machinery. pp. 179–194. doi:10.1145/2675133.2675161. ISBN 978-1-4503-2922-4. S2CID 1406466. ^ Jump up to: a b c d e f g h i j Hollenbaugh, E. E. (2020). Self-Presentation in Social Media: Review and Research Opportunities. Review of Communication Research, 9, 80–98. Received January 2020; revised June 2020; accepted July 2020. ^ Jump up to: a b c d e f g h DeVito, M. A., Birnholtz, J., & Hancock, J. T. (2017, February). Platforms, people, and perception: Using affordances to understand self-presentation on social media. In Proceedings of the 2017 ACM conference on computer supported cooperative work and social computing (pp. 740–754). ^ Jump up to: a b c d Haimson, O. L. (2017). Digital and physical barriers to changing identities. XRDS: Crossroads, The ACM Magazine for Students, 24(2), 26–29. ^ Jump up to: a b Huvila, Isto; Anderson, Theresa Dirndorfer; Jansen, Eva Hourihan; McKenzie, Pam; Westbrook, Lynn; Worrall, Adam (2014). "Boundary objects in information science research: An approach for explicating connections between collections, cultures and communities". Proceedings of the American Society for Information Science and Technology. 51 (1): 1–4. doi:10.1002/meet.2014.14505101003. ISSN 1550-8390. ^ Jump up to: a b Star, Susan Leigh (1989), "The Structure of Ill-Structured Solutions: Boundary Objects and Heterogeneous Distributed Problem Solving", Distributed Artificial Intelligence, Elsevier, pp. 37–54, doi:10.1016/b978-1-55860-092-8.50006-x, ISBN 978-1-55860-092-8, archived from the original on 2022-01-20, retrieved 2022-01-19 ^ Light, Ann; Anderson, Theresa Dirndorfer (2009), Wagner, Ina; Tellioğlu, Hilda; Balka, Ellen; Simone, Carla (eds.), "Research Project as Boundary Object: Negotiating the conceptual design of a tool for International Development", ECSCW 2009, London: Springer London, pp. 21–41, doi:10.1007/978-1-84882-854-4_2, ISBN 978-1-84882-853-7, retrieved 2022-01-19 ^ Bossen, Claus; Jensen, Lotte Groth; Udsen, Flemming Witt (February 2014). "Boundary-Object Trimming: On the Invisibility of Medical Secretaries' Care of Records in Healthcare Infrastructures". Computer Supported Cooperative Work. 23 (1): 75–110. doi:10.1007/s10606-013-9195-5. ISSN 0925-9724. S2CID 13916543. ^ Nauwerck, Gerolf; Cowen Forssell, Rebecka (2018). "The Digital Work Environment–a Challenge and an Opportunity for CSCW". Reports of the European Society for Socially Embedded Technologies. doi:10.18420/ecscw2018_4. ISSN 2510-2591. Archived from the original on 2022-04-08. Retrieved 2022-01-20. ^ Jump up to: a b Tendedez, Helena; Ferrario, Maria Angela M.A.F.; Whittle, Jon (2018-11-01). "Software Development and CSCW: Standardization and Flexibility in Large-Scale Agile Development". Proceedings of the ACM on Human-Computer Interaction. 2 (CSCW): 171:1–171:23. doi:10.1145/3274440. ^ Jump up to: a b Hanseth, O.; Monteiro, E.; Hatling, Morten (1996). "Developing Information Infrastructure: The Tension Between Standardization and Flexibility". Science, Technology, & Human Values. 21 (4): 407–426. doi:10.1177/016224399602100402. S2CID 76944. ^ Jump up to: a b Dourish, James Paul (1996). "Open implementation and flexibility in CSCW toolkits". S2CID 37874295. {{cite journal}}: Cite journal requires \|journal= (help) ^ Jump up to: a b Anderson, Terry; Dron, Jon (2011-03-25). "Three generations of distance education pedagogy". The International Review of Research in Open and Distributed Learning. 12 (3): 80–97. doi:10.19173/irrodl.v12i3.890. ISSN 1492-3831. Archived from the original on 2021-04-15. Retrieved 2021-04-28. ^ Ferri, Fernando; Grifoni, Patrizia; Guzzo, Tiziana (2020-11-13). "Online Learning and Emergency Remote Teaching: Opportunities and Challenges in Emergency Situations". Societies. 10 (4): 86. doi:10.3390/soc10040086. ISSN 2075-4698. ^ Jump up to: a b c Vijayalakshmi, V.; Venkatachalapathy, K.; Ohmprakash, V. (2017-12-31). "Analysis of E-Learning Concept". International Journal on Future Revolution in Computer Science & Communication Engineering. 3 (12): 392–396. ISSN 2454-4248. Archived from the original on 2021-04-28. Retrieved 2021-04-28. ^ Bates, A. W.; Bates, Tony (2005). Technology, E-learning and Distance Education. Psychology Press. ISBN 978-0-415-28437-0. ^ Jump up to: a b c d Garrison, D.Randy; Anderson, Terry; Archer, Walter (1999-03-01). "Critical Inquiry in a Text-Based Environment: Computer Conferencing in Higher Education". The Internet and Higher Education. 2 (2–3): 87–105. doi:10.1016/S1096-7516(00)00016-6. hdl:2149/739. ISSN 1096-7516. S2CID 16697934. Archived from the original on 2021-04-20. Retrieved 2021-04-28. ^ Fatani, Tarah H. (2020-10-31). "Student satisfaction with videoconferencing teaching quality during the COVID-19 pandemic". BMC Medical Education. 20 (1): 396. doi:10.1186/s12909-020-02310-2. ISSN 1472-6920. PMC 7602774. PMID 33129295. ^ Jump up to: a b Yoon, Pilhyoun; Leem, Junghoon (January 2021). "The Influence of Social Presence in Online Classes Using Virtual Conferencing: Relationships between Group Cohesion, Group Efficacy, and Academic Performance". Sustainability. 13 (4): 1988. Bibcode:2021Sust...13.1988Y. doi:10.3390/su13041988. ^ Walsh, Greg; Foss, Elizabeth (2015-06-21). "A case for intergenerational distributed co-design". Proceedings of the 14th International Conference on Interaction Design and Children. Boston MA: ACM. pp. 99–108. doi:10.1145/2771839.2771850. ISBN 978-1-4503-3590-4. ^ Alharthi, Sultan A.; Spiel, Katta; Hamilton, William A.; Bonsignore, Elizabeth; Toups, Zachary O. (2018-10-30). "Collaborative Mixed Reality Games". Companion of the 2018 ACM Conference on Computer Supported Cooperative Work and Social Computing. CSCW '18. Jersey City, NJ: Association for Computing Machinery. pp. 447–454. doi:10.1145/3272973.3273013. ISBN 978-1-4503-6018-0. ^ Jump up to: a b Zhang, Xun; Gui, Xinning; Kou, Yubo; Li, Yukun (2020-10-14). "Mobile Collocated Gaming: Collaborative Play and Meaning-Making on a University Campus". Proceedings of the ACM on Human-Computer Interaction. 4 (CSCW2): 142:1–142:24. doi:10.1145/3415213. S2CID 224805104. Archived from the original on 2021-04-29. ^ Wohn, Donghee Yvette; Freeman, Guo (2020-01-01). "Live Streaming, Playing, and Money Spending Behaviors in eSports". Games and Culture. 15 (1): 73–88. doi:10.1177/1555412019859184. ISSN 1555-4120. S2CID 197463195. ^ Li, Lingyuan; Freeman, Guo; Wohn, Donghee Yvette (2020-11-02). "Power in Skin: The Interplay of Self-Presentation, Tactical Play, and Spending in Fortnite". Proceedings of the Annual Symposium on Computer-Human Interaction in Play. CHI PLAY '20. Virtual Event, Canada: Association for Computing Machinery. pp. 71–80. doi:10.1145/3410404.3414262. ISBN 978-1-4503-8074-4. S2CID 226238720. ^ Jump up to: a b Ducheneaut, Nicolas; Yee, Nicholas; Nickell, Eric; Moore, Robert J. (2006-04-22). ""Alone together?"". Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. CHI '06. Montréal, Québec, Canada: Association for Computing Machinery. pp. 407–416. doi:10.1145/1124772.1124834. ISBN 978-1-59593-372-0. S2CID 287487. ^ Jump up to: a b c d e Zea, Natalia Padilla; Sánchez, José Luís González; Gutiérrez, Francisco L.; Cabrera, Marcelino J.; Paderewski, P. (2009-12-01). "Design of educational multiplayer videogames: A vision from collaborative learning". Advances in Engineering Software. 40 (12): 1251–1260. doi:10.1016/j.advengsoft.2009.01.023. ISSN 0965-9978. Archived from the original on 2021-04-28. Retrieved 2021-04-28. ^ Jump up to: a b c Young, Jacob Brian (2020). Removing spatial boundaries in immersive mobile communications (Thesis thesis). University of Otago. Archived from the original on 2022-01-20. Retrieved 2022-01-20. ^ Jump up to: a b Ding, Xianghua; Chen, Yunan; Ding, Zhaofei; Xu, Yiwen (2019-11-07). "Boundary Negotiation for Patient-Provider Communication via WeChat in China". Proceedings of the ACM on Human-Computer Interaction. 3 (CSCW): 157:1–157:24. doi:10.1145/3359259. S2CID 207959454. ^ Gutierrez, Francisco J.; Ochoa, Sergio F.; Vassileva, Julita (2017). "Mediating Intergenerational Family Communication with Computer-Supported Domestic Technology". In Gutwin, Carl; Ochoa, Sergio F.; Vassileva, Julita; Inoue, Tomoo (eds.). Collaboration and Technology. Lecture Notes in Computer Science. Vol. 10391. Springer International Publishing. pp. 132–147. doi:10.1007/978-3-319-63874-4_11. ISBN 978-3-319-63874-4. Archived from the original on 2021-04-28. Retrieved 2021-04-28. ^ Sayago, Sergio (2019), "Editorial Introduction—Perspectives on HCI Research with Older People", Perspectives on Human-Computer Interaction Research with Older People, Human–Computer Interaction Series, Cham: Springer International Publishing, pp. 3–17, doi:10.1007/978-3-030-06076-3_1, ISBN 978-3-030-06075-6, S2CID 151081393, retrieved 2021-04-28 ^ "OECD Glossary of Statistical Terms – Ubiquitous computing Definition". stats.oecd.org. Archived from the original on 2021-03-09. Retrieved 2021-04-19. ^ Cao, Hancheng; Chen, Zhilong; Cheng, Mengjie; Zhao, Shuling; Wang, Tao; Li, Yong (2021-01-23). "You Recommend, I Buy: How and Why People Engage in Instant Messaging Based Social Commerce". arXiv:2011.00191 [cs.CY]. ^ Jump up to: a b c d e f Reeves, Stuart; Brown, Barry (2016-02-27). "Embeddedness and sequentiality in social media". Proceedings of the 19th ACM Conference on Computer-Supported Cooperative Work & Social Computing (PDF). CSCW '16. San Francisco, California, USA: Association for Computing Machinery. pp. 1052–1064. doi:10.1145/2818048.2820008. ISBN 978-1-4503-3592-8. S2CID 19036589. ^ Bjørn, Pernille; Esbensen, Morten; Jensen, Rasmus Eskild; Matthiesen, Stina (2014-11-21). "Does Distance Still Matter? Revisiting the CSCW Fundamentals on Distributed Collaboration". ACM Transactions on Computer-Human Interaction. 21 (5): 27:1–27:26. doi:10.1145/2670534. ISSN 1073-0516. S2CID 9562515. ^ Jump up to: a b c Weise, Sebastian; Hardy, John; Agarwal, Pragya; Coulton, Paul; Friday, Adrian; Chiasson, Mike (2012-09-05). "Democratizing ubiquitous computing". Proceedings of the 2012 ACM Conference on Ubiquitous Computing. UbiComp '12. Pittsburgh, Pennsylvania: Association for Computing Machinery. pp. 521–530. doi:10.1145/2370216.2370293. ISBN 978-1-4503-1224-0. S2CID 13640738. ^ Birnholtz, Jeremy P.; Bietz, Matthew J. (2003-11-09). "Data at work". Proceedings of the 2003 international ACM SIGGROUP conference on Supporting group work. GROUP '03. Sanibel Island, Florida: Association for Computing Machinery. pp. 339–348. doi:10.1145/958160.958215. ISBN 978-1-58113-693-7. S2CID 8748434. ^ Jump up to: a b "Strangers gather on Web to make collective art - CNN.com". CNN. Retrieved 2021-06-08. ^ "NFTs may be the future of art — but are they threatening the future of the planet?". www.cbsnews.com. Retrieved 2021-06-08. ^ Chang, Brittany. "A piece of NFT art just auctioned for a record-breaking $69 million. NFT artists making millions say the craze could permanently change the art world". Business Insider. Retrieved 2021-06-08. ^ "La casa d'aste Christie's debutta nel mercato della crypto art". Forbes Italia (in Italian). 2021-02-24. Retrieved 2021-06-08. ^ Rapkin, Mickey (2021-02-17). "'Beeple Mania': How Mike Winkelmann Makes Millions Selling Pixels". Esquire. Retrieved 2021-06-08. ^ Leimeister, J. M.; Huber, M.; Bretschneider, U.; Krcmar, H. (2009). "Leveraging Crowdsourcing: Activation-Supporting Components for IT-Based Ideas Competition". Journal of Management Information Systems. 26: 197–224. doi:10.2753/MIS0742-1222260108. S2CID 17485373. Archived from the original on 2010-01-23. ^ Stephy Chung. "The cultural moments that defined 2020". CNN. Retrieved 2021-06-08. ^ Ackerman, M. (2000). "The Intellectual Challenge of CSCW: The gap between social requirements and technical feasibility". Human-Computer Interaction. 15 (2): 179–203. CiteSeerX 10.1.1.4.9910. doi:10.1207/S15327051HCI1523_5. S2CID 2676709. ^ Jump up to: a b Schmidt, K.; Simone, C. (2000). "Mind the Gap! Towards a Unified View of CSCW". COOP. S2CID 16452016. ^ Thompson, L.F; Coovert, M.D (2006). Bowers, C.; Salas, E.; Jentsch, F. (eds.). "Understanding and developing virtual computer-supported cooperative work teams". Creating Hi-tech Teams: 213–241. ^ Jump up to: a b c Olson, J.M.; Olson, J.S. (2008). Sears, A.; Jacko, J. A. (eds.). "The human computer interaction handbook: Fundamentals, evolving technologies, and emerging applications". Group Cooperative Work: 545–558. ^ Convertino, G.; Farooq, U.; Rosson, M. B.; Carroll, J. M.; Meyer, B. J. F. (2007-07-01). "Supporting intergenerational groups in computer-supported cooperative work (CSCW)". Behaviour & Information Technology. 26 (4): 275–285. doi:10.1080/01449290601173473. ISSN 0144-929X. S2CID 18381170. ^ Convertino, J.; Farooq, U.; Rosson, M.; Carroll, J.; Meyer, B. (2007). "Supporting intergenerational groups in computer-supported cooperative work (CSCW)". Behaviour & Information Technology. 4. 26 (4): 275–285. doi:10.1080/01449290601173473. S2CID 18381170. ^ Brynjolfsson, Erik; Horton, John J.; Ozimek, Adam; Rock, Daniel; Sharma, Garima; TuYe, Hong-Yi (2020-06-15). "COVID-19 and Remote Work: An Early Look at US Data". Working Paper Series. doi:10.3386/w27344. S2CID 225713945. Archived from the original on 2021-04-23. Retrieved 2021-04-28. {{cite journal}}: Cite journal requires \|journal= (help) ^ Walsh, Greg; Foss, Elizabeth; Yip, Jason; Druin, Allison (2013-04-27). "Facit Pd". Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. CHI '13. Paris: Association for Computing Machinery. pp. 2893–2902. doi:10.1145/2470654.2481400. ISBN 978-1-4503-1899-0. S2CID 335426. ^ Sumita Raghuram, N. Sharon Hill, Jennifer L. Gibbs and Likoebe M. Maruping, 16 January 2019. "Virtual Work: Bridging Research Clusters", Academy of Management Annals, Vol. 13, No. 1 ^ Jump up to: a b Jennifer A. Chatman and Sandra E. Spataro, 1 April 2005 "Using Self-Categorization Theory to Understand Relational Demography–Based Variations in People's Responsiveness to Organizational Culture", Academy of Management Journal, Vol. 48, No. 2 ^ Jump up to: a b Earley, Brigitt (2021-03-08). "These Organization Apps Will Simplify Your Life". Oprah Daily. Archived from the original on 2022-01-20. Retrieved 2022-01-20. ^ Jump up to: a b c d e f g Abraham, Joanna; Reddy, Madhu C. (2013). "Re-coordinating activities". Proceedings of the 2013 conference on Computer supported cooperative work. CSCW '13. San Antonio, Texas: ACM Press. pp. 67–78. doi:10.1145/2441776.2441787. ISBN 978-1-4503-1331-5. S2CID 517363. ^ Grudin, J. (1994). "Computer-Supported Cooperative Work: History and Focus". Computer. 27 (5): 19–26. doi:10.1109/2.291294. S2CID 807626. ^ Koch, M; Gross, T (2006). "Computer-Supported Cooperative Work – Concepts and Trends". Proc Conf of the Association Information and Management AIM. 75: 165–172. ^ Neale, D; Carrol, J; Rosson, M (2004). "Evaluating computer-supported cooperative work". Proceedings of the 2004 ACM conference on Computer supported cooperative work. pp. 112–121. doi:10.1145/1031607.1031626. ISBN 978-1581138108. S2CID 16838267. ^ Jump up to: a b Prinsen, F.R.; Volman, M.L.L.; Terwel, J. (2007-02-13). "Gender-related differences in computer-mediated communication and computer-supported collaborative learning". Journal of Computer Assisted Learning. 23 (5): 393–409. doi:10.1111/j.1365-2729.2007.00224.x. ISSN 0266-4909. ^ Jump up to: a b Koulouri, Theodora; Lauria, Stanislao; Macredie, Robert D. (2017). "The influence of visual feedback and gender dynamics on performance, perception and communication strategies in CSCW". International Journal of Human-Computer Studies. 97: 162–181. doi:10.1016/j.ijhcs.2016.09.003. ISSN 1071-5819. ^ Prinsen, Fleur-Ruth; Volman, Monique; Terwel, Jan (2007). "Gender-related differences in computer-mediated communication and computer-supported collaborative learning". Journal of Computer Assisted Learning. 23 (5): 393–409. doi:10.1111/j.1365-2729.2007.00224.x. ^ Jump up to: a b Bao, Yukun; Xiong, Tao; Hu, Zhongyi; Kibelloh, Mboni (2013). "Exploring Gender Differences on General and Specific Computer Self-Efficacy in Mobile Learning Adoption". Journal of Educational Computing Research. 49 (1): 111–132. arXiv:1402.4211. doi:10.2190/ec.49.1.e. ISSN 0735-6331. S2CID 492259. ^ Jump up to: a b Bernard, M; Mills, M; Friend, C (2000). "Male and female attitudes towards Computer–Mediated group interactions". Usability News. 2: 34–35. ^ Jump up to: a b Grudin, Jonathan (2010). "CSCW". ACM Interactions. No. XVII.4 July + August 2010. p. 38. Retrieved 2025-02-27. ^ "CSCW: Computer Supported Cooperative Work". ACM Digital Library. Archived from the original on 2022-03-05. Retrieved 2022-02-08. ^ Koch, Michael; Schwabe, Gerhard (2015-06-01). "Interview with Jonathan Grudin on "Computer-Supported Cooperative Work and Social Computing"". Business & Information Systems Engineering. 57 (3): 213–215. doi:10.1007/s12599-015-0377-1. ISSN 1867-0202. ^ "International Working Group on CSCW in Design". Archived from the original on 2022-01-27. Retrieved 2022-02-08. ^ "ECSCW – European Conference on Computer-Supported Cooperative Work". Retrieved 23 June 2019. Further reading [edit] Most cited papers This list, the CSCW Handbook Papers, is the result of a citation graph analysis of the CSCW Conference. It was established in 2006 and reviewed by the CSCW community. This list only contains papers published in one conference; papers published at other venues have also had significant impact on the CSCW community. The "CSCW handbook" papers were chosen as the overall most cited within the CSCW conference ... It led to a list of 47 papers, corresponding to about 11% of all papers. Dourish, P.; Bellotti, V. (1992). "Awareness and coordination in shared workspaces". Proceedings of the 1992 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 107–114. Grudin, J. (1988). "Why CSCW applications fail: problems in the design and evaluation of organization of organizational interfaces". Proceedings of the 1988 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 85–93. Root, R.W. (1988). "Design of a multi-media vehicle for social browsing". Proceedings of the 1988 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 25–38. Patterson, J.F.; Hill, R.D.; Rohall, S.L.; Meeks, S.W. (1990). "Rendezvous: an architecture for synchronous multi-user applications". Proceedings of the 1990 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 317–328. Greenberg, S.; Marwood, D. (1994). "Real time groupware as a distributed system: concurrency control and its effect on the interface". Proceedings of the 1994 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 207–217. Nardi, B.A.; Whittaker, S.; Bradner, E. (2000). "Interaction and outeraction: instant messaging in action". Proceedings of the 2000 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 79–88. Hughes, J.A.; Randall, D.; Shapiro, D. (1992). "Faltering from ethnography to design". Proceedings of the 1992 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 115–122. Tang, J.C.; Isaacs, E.A.; Rua, M. (1994). "Supporting distributed groups with a Montage of lightweight interactions". Proceedings of the 1994 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 23–34. Neuwirth, C.M.; Kaufer, D.S.; Chandhok, R.; Morris, J.H. (1990). "Issues in the design of computer support for co-authoring and commenting". Proceedings of the 1990 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 183–195. Crowley, T.; Milazzo, P.; Baker, E.; Forsdick, H.; Tomlinson, R. (1990). "MMConf: an infrastructure for building shared multimedia applications". Proceedings of the 1990 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 329–342. Roseman, M.; Greenberg, S. (1992). "GROUPKIT: a groupware toolkit for building real-time conferencing applications". Proceedings of the 1992 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 43–50. Shen, H.H.; Dewan, P. (1992). "Access control for collaborative environments". Proceedings of the 1992 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 51–58. Gaver, W.W. (1992). "The affordances of media spaces for collaboration". Proceedings of the 1992 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 17–24. Orlikowski, W.J. (1992). "Learning from Notes: organizational issues in groupware implementation". Proceedings of the 1992 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 362–369. Sun, C.; Ellis, C. (1998). "Operational transformation in real-time group editors: issues, algorithms, and achievements". Proceedings of the 1998 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 59–68. Bly, S.A. (1988). "A use of drawing surfaces in different collaborative settings". Proceedings of the 1988 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 250–256. Leland, M.D.P.; Fish, R.S.; Kraut, R.E. (1988). "Collaborative document production using quilt". Proceedings of the 1988 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 206–215. Conklin, J.; Begeman, M.L. (1988). "gIBIS: a hypertext tool for exploratory policy discussion". ACM Transactions on Information Systems. 6 (4): 303–331. doi:10.1145/58566.59297. S2CID 2609461. Bentley, R.; Hughes, J.A.; Randall, D.; Rodden, T.; Sawyer, P.; Shapiro, D.; Sommerville, I. (1992). "Ethnographically-informed systems design for air traffic control". Proceedings of the 1992 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 123–129. Mantei, M. (1988). "Capturing the capture concepts: a case study in the design of computer-supported meeting environments". Proceedings of the 1988 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 257–270. Lantz, K.A. (1986). "An experiment in integrated multimedia conferencing". Proceedings of the 1986 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 267–275. Harrison, S.; Dourish, P. (1996). "Re-place-ing space: the roles of place and space in collaborative systems". Proceedings of the 1996 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 67–76. Roseman, M.; Greenberg, S. (1996). "TeamRooms: network places for collaboration". Proceedings of the 1996 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 325–333. Ishii, H. (1990). "TeamWorkStation: towards a seamless shared workspace". Proceedings of the 1990 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 13–26. Ressel, M.; Nitsche-ruhland, D.; Gunzenhäuser, R. (1996). "An integrating, transformation-oriented approach to concurrency control and undo in group editors". Proceedings of the 1996 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 288–297. Edwards, W.K. (1996). "Policies and roles in collaborative applications". Proceedings of the 1996 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 11–20. Bellotti, V.; Bly, S. (1996). "Walking away from the desktop computer: distributed collaboration and mobility in a product design team". Proceedings of the 1996 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 209–218. Ackerman, M.S. (1998). "Augmenting Organizational Memory: A Field Study of Answer Garden". ACM Transactions on Information Systems. 16 (3): 203–224. CiteSeerX 10.1.1.12.589. doi:10.1145/290159.290160. S2CID 15780647. Abbott, K.R.; Sarin, S.K. (1994). "Experiences with workflow management: issues for the next generation". Proceedings of the 1994 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 113–120. Resnick, P.; Iacovou, N.; Suchak, M.; Bergstrom, P.; Riedl, J. (1994). "GroupLens: an open architecture for collaborative filtering of netnews". Proceedings of the 1994 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 175–186. Prakash, A.; Shim, H.S. (1994). "DistView: support for building efficient collaborative applications using replicated objects". Proceedings of the 1994 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 153–164. Streitz, N.A.; Geißler, J.; Haake, J.M.; Hol, J. (1994). "DOLPHIN: integrated meeting support across local and remote desktop environments and LiveBoards". Proceedings of the 1994 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 345–358. Foster, G.; Stefik, M. (1986). "Cognoter: theory and practice of a colab-orative tool". Proceedings of the 1986 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 7–15. Shen, C.; Lesh, N.B.; Vernier, F.; Forlines, C.; Frost, J. (2002). "Sharing and building digital group histories". Proceedings of the 2002 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 324–333. Sohlenkamp, M.; Chwelos, G. (1994). "Integrating communication, cooperation, and awareness: the DIVA virtual office environment". Proceedings of the 1994 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 331–343. Olson, J.S.; Teasley, S. (1996). "Groupware in the wild: lessons learned from a year of virtual collocation". Proceedings of the 1996 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 419–427. Reder, S.; Schwab, R.G. (1990). "The temporal structure of cooperative activity". Proceedings of the 1990 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 303–316. Fish, R.S.; Kraut, R.E.; Chalfonte, B.L. (1990). "The VideoWindow system in informal communication". Proceedings of the 1990 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 1–11. Haake, J.M.; Wilson, B. (1992). "Supporting collaborative writing of hyperdocuments in SEPIA". Proceedings of the 1992 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 138–146. Hudson, S.E.; Smith, I. (1996). "Techniques for addressing fundamental privacy and disruption tradeoffs in awareness support systems". Proceedings of the 1996 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 248–257. MacKay, W.E. (1990). "Patterns of sharing customizable software". Proceedings of the 1990 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 209–221. Trigg, R.H.; Suchman, L.A.; Halasz, F.G. (1986). "Supporting collaboration in notecards". Proceedings of the 1986 ACM conference on Computer-supported cooperative work. New York: ACM Press. pp. 153–162. Patterson, J.F.; Day, M.; Kucan, J. (1996). "Notification servers for synchronous groupware". Proceedings of the 1996 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 122–129. Myers, B.A.; Stiel, H.; Gargiulo, R. (1998). "Collaboration using multiple PDAs connected to a PC". Proceedings of the 1998 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 285–294. Ackerman, M.S.; Halverson, C. (1998). "Considering an organization's memory". Proceedings of the 1998 ACM conference on Computer supported cooperative work. New York, NY: ACM Press. pp. 39–48. Teasley, S.; Covi, L.; Krishnan, M.S.; Olson, J.S. (2000). "How does radical collocation help a team succeed?". Proceedings of the 2000 ACM conference on Computer supported cooperative work. New York, NY: ACM Press. pp. 339–346. Kuzuoka, H.; Kosuge, T.; Tanaka, M. (1994). "GestureCam: a video communication system for sympathetic remote collaboration". Proceedings of the 1994 ACM conference on Computer supported cooperative work. New York: ACM Press. pp. 35–43. Others Grudin, Jonathan (1994). "Computer-Supported Cooperative Work: History and Focus". Computer. 27 (5). IEEE: 19–26. doi:10.1109/2.291294. ISSN 0018-9162. S2CID 807626. Archived from the original on June 29, 2006. Retrieved 2006-12-11. External links [edit] CSCW Conference, ACM CSCW Conference Series European CSCW Conference Foundation, European CSCW Conference Series GROUP Conference COOP Conference External links [edit] SPARC - Space Physics and Aeronomy Research Collaboratory. Science Of Collaboratories - Science of Collaboratories Project Home, with links to over 100 specific collaboratories Paul Resnick - Professor Paul Resnick's home page ( papers on SocioTechnical Capital, reputation systems, ride share coordination services, recommender systems, collaborative filtering, social filtering). Reticula - Weblogs, Wikis, and Public Health Today. News, professional activities, and academic research. US National Health Information Network News about and links into the US NHIN and efforts to build a nationwide virtual electronic health record to support and facilitate electronic collaboration between clinicians, hospitals, patients, social work, and public health. Political Blogosphere - The Political Blogosphere and the 2004 U.S. Election: Divided They Blog, Adamic L. and Glance N., HP Labs, 2005. ("In this paper, we study the linking patterns and discussion topics of political bloggers. Our aim is to measure the degree of interaction between liberal and conservative blogs, and to uncover any differences in the structure of the two communities.") - CSCW and Groupware Literature Guide: Randy's Reviews, Recommendations, and (Optional) Referrals. \| Authority control databases \| \| --- \| \| National \| Germany United States France BnF data Israel \| \| Other \| IdRef Yale LUX \| ^ CSCW Handbook Papers ^ Jump up to: a b Jacovi, M.; Soroka, V.; Gilboa-freedman, G.; Ur, S.; Shahar, E.; Marmasse, N. (2006). "The chasms of CSCW: a citation graph analysis of the CSCW conference". Proceedings of the 2006 20th anniversary conference on Computer supported cooperative work. ACM Press New York, NY. pp. 289–298. Retrieved from " Categories: Computer-related introductions in 1984 Collaboration Groupware Multimodal interaction Human–computer interaction Instructional design Hidden categories: CS1 errors: missing periodical CS1 Italian-language sources (it) Articles with short description Short description matches Wikidata All articles with unsourced statements Articles with unsourced statements from February 2025
15118	https://fiveable.me/complex-analysis/unit-3/harmonic-functions/study-guide/dZMC3Jrs6fGk3ojz	Harmonic functions \| Complex Analysis Class Notes \| Fiveable \| Fiveable new!Printable guides for educators Printable guides for educators. Bring Fiveable to your classroom ap study content toolsprintablespricing my subjectsupgrade 📐Complex Analysis Unit 3 Review 3.4 Harmonic functions All Study Guides Complex Analysis Unit 3 – Analytic Functions Topic: 3.4 📐Complex Analysis Unit 3 Review 3.4 Harmonic functions Written by the Fiveable Content Team • Last updated September 2025 Written by the Fiveable Content Team • Last updated September 2025 print study guide copy citation APA 📐Complex Analysis Unit & Topic Study Guides Introduction to Complex Numbers Complex Functions and Mappings Analytic Functions 3.1 Limits and continuity 3.2 Differentiability and analyticity 3.3 Cauchy-Riemann equations 3.4 Harmonic functions Elementary Functions Complex Integration Analytic Function Series Representations Residue Theory and Applications Conformal Mappings Harmonic Functions Entire and Meromorphic Functions Riemann Surfaces Special Topics and Applications print guide report error Harmonic functions are the real-valued cousins of analytic functions. They satisfy the Laplace equation and share many cool properties with analytic functions, like the mean value property and maximum principle. They're key players in complex analysis. The real and imaginary parts of analytic functions are harmonic, and we can use harmonic functions to build analytic ones. This connection helps us solve problems in physics, like figuring out electric fields or heat flow. It's a powerful tool in our complex analysis toolkit. Harmonic Functions and Analytic Functions Properties of Harmonic Functions Harmonic functions are real-valued functions that satisfy the Laplace equation in a given domain If a function $f(z) = u(x, y) + iv(x, y)$ is analytic in a domain $D$, then both $u(x, y)$ and $v(x, y)$ are harmonic functions in $D$ The real and imaginary parts of an analytic function are harmonic functions and are related by the Cauchy-Riemann equations Harmonic functions have many properties similar to analytic functions Mean value property: the value of a harmonic function at any point is equal to the average of its values on any circle centered at that point and lying within the domain Maximum principle: a non-constant harmonic function cannot attain its maximum or minimum value within its domain Relationship between Harmonic and Analytic Functions The real and imaginary parts of an analytic function are harmonic functions If $f(z) = u(x, y) + iv(x, y)$ is analytic, then $u(x, y)$ and $v(x, y)$ are harmonic Harmonic functions can be used to construct analytic functions If $u(x, y)$ is harmonic, there exists a harmonic conjugate $v(x, y)$ such that $f(z) = u(x, y) + iv(x, y)$ is analytic The Cauchy-Riemann equations connect the partial derivatives of the real and imaginary parts of an analytic function $\partial u/\partial x = \partial v/\partial y$ and $\partial u/\partial y = -\partial v/\partial x$ Harmonic Functions and the Laplace Equation The Laplace Equation The Laplace equation for a function $u(x, y)$ in two dimensions is $\partial²u/\partial x² + \partial²u/\partial y² = 0$ A real-valued function is harmonic if and only if it satisfies the Laplace equation in its domain To determine if a function is harmonic, compute its second partial derivatives and check if their sum equals zero Examples of harmonic functions include: Real and imaginary parts of the complex exponential: $e^z = e^x \cos y + i e^x \sin y$ Real and imaginary parts of the complex logarithm: $\log z = \ln\|z\| + i \arg z$ Real and imaginary parts of complex polynomials: $z^n = (x + iy)^n$ Solving the Laplace Equation The Laplace equation is a second-order partial differential equation Solutions to the Laplace equation are called harmonic functions Techniques for solving the Laplace equation include: Separation of variables: assume the solution is a product of functions of each variable separately Green's functions: express the solution as an integral involving a special function (the Green's function) and boundary data Conformal mappings: transform the problem to a simpler domain, solve, and map back to the original domain Finding Harmonic Conjugates Definition and Properties of Harmonic Conjugates If $u(x, y)$ is a harmonic function, then there exists a unique harmonic function $v(x, y)$, called the harmonic conjugate of $u$, such that $f(z) = u(x, y) + iv(x, y)$ is analytic The harmonic conjugate $v(x, y)$ is unique up to an additive constant If $u(x, y)$ and $v(x, y)$ are harmonic conjugates, then the curves $u(x, y) = c₁$ and $v(x, y) = c₂$ are orthogonal for any constants $c₁$ and $c₂$ Finding Harmonic Conjugates using the Cauchy-Riemann Equations The harmonic conjugate $v(x, y)$ can be found by integrating the Cauchy-Riemann equations: $\partial v/\partial x = \partial u/\partial y$ $\partial v/\partial y = -\partial u/\partial x$ To find $v(x, y)$, integrate one of the Cauchy-Riemann equations with respect to the appropriate variable Integrate $\partial v/\partial x = \partial u/\partial y$ with respect to $x$ to find $v(x, y)$ up to a function of $y$ Integrate $\partial v/\partial y = -\partial u/\partial x$ with respect to $y$ to find $v(x, y)$ up to a function of $x$ The resulting function $v(x, y)$ will be the harmonic conjugate of $u(x, y)$ Applications of Harmonic Functions Boundary Value Problems Boundary value problems involve finding a harmonic function that satisfies given conditions on the boundary of a domain The Dirichlet problem seeks a harmonic function with prescribed values on the boundary The Poisson integral formula provides a solution to the Dirichlet problem for a harmonic function in a disk, given its boundary values The Neumann problem involves prescribed normal derivatives on the boundary Green's functions can be used to solve boundary value problems by expressing the solution as an integral involving the boundary data and the Green's function Conformal mappings can be employed to transform a boundary value problem in a complicated domain to a simpler domain, solve the problem there, and then map the solution back to the original domain Physical Applications Harmonic functions appear in various physical contexts, such as: Electrostatics: the electric potential in a charge-free region is a harmonic function Fluid dynamics: the velocity potential of an irrotational, incompressible flow is a harmonic function Heat conduction: the steady-state temperature distribution in a region without heat sources or sinks is a harmonic function The properties of harmonic functions, such as the mean value property and the maximum principle, have physical interpretations in these contexts In electrostatics, the mean value property implies that the electric potential at a point is the average of the potential on any sphere centered at that point In heat conduction, the maximum principle states that the maximum and minimum temperatures occur on the boundary of the region 3.3 BackNext Study Content & Tools Study GuidesPractice QuestionsGlossaryScore Calculators Company Get $$ for referralsPricingTestimonialsFAQsEmail us Resources AP ClassesAP Classroom every AP exam is fiveable history 🌎 ap world history🇺🇸 ap us history🇪🇺 ap european history social science ✊🏿 ap african american studies🗳️ ap comparative government🚜 ap human geography💶 ap macroeconomics🤑 ap microeconomics🧠 ap psychology👩🏾‍⚖️ ap us government english & capstone ✍🏽 ap english language📚 ap english literature🔍 ap research💬 ap seminar arts 🎨 ap art & design🖼️ 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15119	https://math.stackexchange.com/questions/4191443/cubic-equation-problem-fracx33-x-k	functions - Cubic equation problem $\frac{x^3}{3}-x=k$ - 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 Cubic equation problem x 3 3−x=k x 3 3−x=k Ask Question Asked 4 years, 2 months ago Modified4 years, 2 months ago Viewed 196 times This question shows research effort; it is useful and clear 4 Save this question. Show activity on this post. The cubic function x 3 3−x=k x 3 3−x=k has three different roots α,β,γ α,β,γ about the real number k. Let's call the minimum value of \|α\|+\|β\|+\|γ\|\|α\|+\|β\|+\|γ\| as m m. FInd the value of m 2 m 2. My approach is as follow x 3 3−x=k⇒f(x)=x 3 3−x−k x 3 3−x=k⇒f(x)=x 3 3−x−k f′(x)=x 2−1=0 f′(x)=x 2−1=0 Hence x=±1 x=±1 f(1)=1 3−1−k=−(k+2 3)&f(−1)=−1 3+1−k=−(k−2 3)f(1)=1 3−1−k=−(k+2 3)&f(−1)=−1 3+1−k=−(k−2 3) For real roots f(1)f(−1)<0 f(1)f(−1)<0 Therefore −(k+2 3)×(−(k−2 3))<0−(k+2 3)×(−(k−2 3))<0 k∈(−2 3,2 3)k∈(−2 3,2 3) We know that α+β+γ=0 α+β+γ=0 But how we will find the minimum values of the sum of the modulus of the roots. functions cubics Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications asked Jul 6, 2021 at 5:53 Samar Imam ZaidiSamar Imam Zaidi 9,272 4 4 gold badges 31 31 silver badges 78 78 bronze badges 8 3 Can the root be complex?Asher2211 –Asher2211 2021-07-06 06:03:24 +00:00 Commented Jul 6, 2021 at 6:03 No that is why I have used k∈(−2 3,2 3)k∈(−2 3,2 3)Samar Imam Zaidi –Samar Imam Zaidi 2021-07-06 06:07:06 +00:00 Commented Jul 6, 2021 at 6:07 The condition was not mentioned in the question.Asher2211 –Asher2211 2021-07-06 06:07:51 +00:00 Commented Jul 6, 2021 at 6:07 3 @SamarImamZaidi I think what Asher is saying is that the problem never mentions that the three roots of the polynomial must be real, only that k k must be real. You've shown that, if the roots are real, then −2/3<k<2/3−2/3<k<2/3. However, if the roots are allowed to be complex, we can no longer assume this restriction on k k, and the problem becomes more difficult.Carl Schildkraut –Carl Schildkraut 2021-07-06 06:12:36 +00:00 Commented Jul 6, 2021 at 6:12 3 @SamarImamZaidi For a complex number z=a+b i z=a+b i it is a standard definition that \|z\|=a 2+b 2−−−−−−√\|z\|=a 2+b 2. Distance from the origin and all that. That is why others are asking about the possibility of complex roots.Jyrki Lahtonen –Jyrki Lahtonen 2021-07-06 06:27:34 +00:00 Commented Jul 6, 2021 at 6:27 \|Show 3 more comments 3 Answers 3 Sorted by: Reset to default This answer is useful 6 Save this answer. Show activity on this post. Hint −− using α β+β γ+γ α=−3 α β+β γ+γ α=−3: m 2=(\|α\|+\|β\|+\|γ\|)2=α 2+β 2+γ 2+2(\|α β\|+\|β γ\|+\|γ α\|)=(α+β+γ)2−2(α β+β γ+γ α)+2(\|α β\|+\|β γ\|+\|γ α\|)=6+2(\|α β\|+\|β γ\|+\|γ α\|)≥6+2\|α β+β γ+γ α\|=12 m 2=(\|α\|+\|β\|+\|γ\|)2=α 2+β 2+γ 2+2(\|α β\|+\|β γ\|+\|γ α\|)=(α+β+γ)2−2(α β+β γ+γ α)+2(\|α β\|+\|β γ\|+\|γ α\|)=6+2(\|α β\|+\|β γ\|+\|γ α\|)≥6+2\|α β+β γ+γ α\|=12 This gives a lower bound on m 2 m 2. To prove it's an actual minimum, it is enough to find a k k such that \|α\|+\|β\|+\|γ\|=2 3–√\|α\|+\|β\|+\|γ\|=2 3, which turns out not to be too hard. Note: the above assumes the roots are real (per OP's comments), in order for \|α\|2=α 2\|α\|2=α 2 to hold. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Jul 6, 2021 at 6:53 answered Jul 6, 2021 at 6:18 dxivdxiv 78k 6 6 gold badges 69 69 silver badges 127 127 bronze badges 2 1 @Asher2211 What that part proves is (only) the lower bound m 2≥12 m 2≥12.dxiv –dxiv 2021-07-06 06:37:02 +00:00 Commented Jul 6, 2021 at 6:37 1 Thanks for pointing out. For k=0 k=0 the conditions are satisfied.Asher2211 –Asher2211 2021-07-06 07:00:26 +00:00 Commented Jul 6, 2021 at 7:00 Add a comment\| This answer is useful 2 Save this answer. Show activity on this post. Note: This is assuming that the question intends to ask that each of {α,β,γ}{α,β,γ} must be real in addition to k k. If they are allowed to be non-real, the situation is trickier. The sum \|α\|+\|β\|+\|γ\|\|α\|+\|β\|+\|γ\| is preserved if k k is replaced by −k−k, so we can without loss of generality assume that k<0 k<0, so that there are two positive roots and one negative root. Let the positive roots be α>β α>β, and let α(k)α(k) and β(k)β(k) represent the functions mapping k k to the largest root and to the second largest root of x 3/3−x−k x 3/3−x−k, respectively. We seek to minimize 2 α(k)+2 β(k)2 α(k)+2 β(k). See if you can do this via calculus, writing the derivative of 2 α(k)+2 β(k)2 α(k)+2 β(k) in terms of α(k)α(k), β(k)β(k), and k k. Then you only need to check the endpoints (k=0 k=0 and k=−2/3 k=−2/3), and any places where the derivative may be 0 0. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Jul 6, 2021 at 6:14 Carl SchildkrautCarl Schildkraut 37.9k 2 2 gold badges 51 51 silver badges 94 94 bronze badges 1 You may want to check what I did. Tracking the sum of two roots is unnecessarily complicated. If you modify accordingly I will delete my answer and upvote yours.Oscar Lanzi –Oscar Lanzi 2021-07-06 09:08:53 +00:00 Commented Jul 6, 2021 at 9:08 Add a comment\| This answer is useful 2 Save this answer. Show activity on this post. Note: this is similar to another answer, except that by recognizing only one root needs to be tracked instead of the other two the calculation is simplified. First observe that if we reverse the sign of k k then all roots are also reversed in sign with no effect on the set of absolute values, thus no impact on the sum \|α\|+\|β\|+\|γ\|\|α\|+\|β\|+\|γ\|. So we can cover all values of this sum by considering just the case where k k is nonpositive. Then only one root is negative by Descartes' Rule of Signs. Calling that root α α we then have \|α\|+\|β\|+\|γ\|=−α+β+γ\|α\|+\|β\|+\|γ\|=−α+β+γ =−2 α∵α+β+γ=0=−2 α∵α+β+γ=0 Thus find the nonpositive value of k k that minimizes the absolute value of the negative root α α. Given that k=0 k=0 gives α=−3–√α=−3 and k<0 k<0 must give α<−3–√α<−3 (why?), you then get your answer using the blue equation above. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Jul 6, 2021 at 20:04 answered Jul 6, 2021 at 8:58 Oscar LanziOscar Lanzi 50.2k 2 2 gold badges 55 55 silver badges 135 135 bronze badges 1 1 Thanks @514, it worked perfectly!Oscar Lanzi –Oscar Lanzi 2021-07-06 18:24:01 +00:00 Commented Jul 6, 2021 at 18:24 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 functions cubics See similar questions with these tags. 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15120	https://fiveable.me/ap-calc/unit-5/sketching-graphs-functions-their-derivatives/study-guide/aT1iYD0w3cZ4vq9YNoLG	Sketching Graphs of Functions and Their Derivatives - AP Calc Study Guide \| Fiveable \| Fiveable ap study content toolsprintables upgrade ♾️AP Calculus AB/BC Unit 5 Review 5.8 Sketching Graphs of Functions and Their Derivatives All Study Guides AP Calculus AB/BC Unit 5 – Analytical Applications of Differentiation Topic: 5.8 ♾️AP Calculus AB/BC Unit 5 Review 5.8 Sketching Graphs of Functions and Their Derivatives Written by the Fiveable Content Team • Last updated September 2025 Verified for the 2026 exam Verified for the 2026 exam•Written by the Fiveable Content Team • Last updated September 2025 print study guide copy citation APA ♾️AP Calculus AB/BC Unit & Topic Study Guides AP Calculus AB/BC Exams Unit 1 – Limits and Continuity Unit 2 – Fundamentals of Differentiation Unit 3 – Composite, Implicit, and Inverse Functions Unit 4 – Contextual Applications of Differentiation Unit 5 – Analytical Applications of Differentiation Unit 5 Overview: Analytical Applications of Differentiation 5.1 Using the Mean Value Theorem 5.2 Extreme Value Theorem, Global vs Local Extrema, and Critical Points 5.3 Determining Intervals on Which a Function is Increasing or Decreasing 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema 5.6 Determining Concavity 5.7 Using the Second Derivative Test to Determine Extrema 5.8 Sketching Graphs of Functions and Their Derivatives 5.9 Connecting a Function, Its First Derivative, and its Second Derivative 5.10 Introduction to Optimization Problems 5.11 Solving Optimization Problems 5.12 Exploring Behaviors of Implicit Relations Unit 6 – Integration and Accumulation of Change Unit 7 – Differential Equations Unit 8 – Applications of Integration Unit 9 – Parametric Equations, Polar Coordinates, and Vector–Valued Functions (BC Only) Unit 10 – Infinite Sequences and Series (BC Only) Frequently Asked Questions Previous Exam Prep Study Tools Exam Skills AP Cram Sessions 2021 Live Cram Sessions 2020 practice questions print guide report error 5.8 My Combo Previously you learned all about utilizing a function’s derivatives to find its increasing and decreasing intervals, relative and global extrema, concave up and concave down intervals, and more! Can we combine all this to form a complete image of a function’s graph? Yes, we can! Today, you’ll learn how to utilize all these tools in your toolbox to form an overall picture of a function. more resources to help you study practice questionscheatsheetscore calculator 📈 Sketching Graphs Drawing the graphs of functions and their derivatives is very useful in determining key features of the function. From a graph, you can identify discontinuities, find critical points, and extrema, and discern many other important elements of a function. There are seven steps to sketching a graph. This may seem like a lot, but once you break it down, you’ll get a hang of it! 🔎 〰️ Find the domain of the function and determine if there are any discontinuities. 🎯 Identify key features of the function, such as intercepts and symmetry. 💯 Find critical points. 📶 Determine where the function increases and decreases. 🤔 Find the extrema of the function. 🥈 An alternative to steps 4) and 5) is to use the Second Derivative Test to determine the extrema of the function. ✏️ Determine points of inflection and intervals of concavity. And there we have it! An overview of what the entire function looks like! You can now use all these interesting points to sketch a graph of the function. 😄 🎨 Sketching Graphs Walkthrough Now, let’s delve deeper into what each of these steps entails and sketch the following function! f(x)=(x+2)2(x−1)f(x)=(x+2)^2(x-1)f(x)=(x+2)2(x−1) 〰️ Step 1) Find the domain and look for discontinuities. A useful fact to memorize is that all polynomial functions have a domain consisting of all real numbers. Since we’re graphing a polynomial here, we know that the domain consists of all real numbers. When looking for discontinuities, note if the function is rational or has any points where f(x)f(x)f(x) is not defined. Since there seem to be no points where f(x)f(x)f(x) is not defined, we can conclude that this function is continuous everywhere in its domain. 🎯 Step 2) Identify key features of the function, such as intercepts and symmetry. First, let’s look for x-intercepts, where f(x)=0 f(x)=0 f(x)=0. 0=(x+2)2(x−1)0=(x+2)^2(x-1)0=(x+2)2(x−1) We can quickly do this by setting each factor equal to zero and solving for x. (x+2)2=0(x+2)^2=0(x+2)2=0 x+2=0 x+2=0 x+2=0 x=−2 x=-2 x=−2 And now, to set x−1 x-1 x−1 equal to 0. (x−1)=0(x-1)=0(x−1)=0 x=1 x=1 x=1 This gives us two x-intercepts: x=−2 x=-2 x=−2 and x=1 x=1 x=1. The coordinates of these x-intercepts are (−2,0)(-2,0)(−2,0) and (1,0)(1,0)(1,0). Great! We got our first points we can plot. Let’s now look for y-intercepts, where x=0 x=0 x=0. f(0)=(0+2)2(0−1)f(0)=(0+2)^2(0-1)f(0)=(0+2)2(0−1)f(0)=(2)2(−1)=−4 f(0)=(2)^2(-1)=\boxed{-4}f(0)=(2)2(−1)=−4 We have another point we can jot down to plot later! The coordinates of the y-intercept are (0,−4)(0,-4)(0,−4). Lastly, let’s look for symmetry. A function is symmetric with respect to a line if, when reflected across that line, the function looks the same as the original function. There are two types of symmetry a function may have: Even - the function is symmetrical across the y-axis. f(−x)=f(x)f(-x)=f(x)f(−x)=f(x) for all x x x in the domain of the function. Odd - the function is symmetric about the origin. f(−x)=−f(x)f(-x)=-f(x)f(−x)=−f(x) for all x x x in the domain of the function. If neither of these statements is true, the function has no symmetry. With our function, this is the case, so our function has no symmetry. An image displaying even, odd, or no symmetry. Image Courtesy of The Organic Chemistry Tutor ⚡ So far, we know the following: The function is continuous. We can plot (−2,0)(-2,0)(−2,0), (1,0)(1,0)(1,0), and (0,−4)(0,-4)(0,−4). 💯 Step 3) Find critical points. If you recall from earlier sections of the unit, the first step of using a function’s derivative to determine its behavior involved finding critical points! These points occur when the function’s first derivative is equal to zero or is undefined. The first derivative of our function is the following, using the product rule. f′(x)=2(x+2)(x−1)+(x+2)2 f'(x)=2(x+2)(x-1)+(x+2)^2 f′(x)=2(x+2)(x−1)+(x+2)2 Setting this equal to zero and solving for x, we get x=−2 x=-2 x=−2 and x=0 x=0 x=0 as the function’s critical points. We can use this information in the next step! 📶 Step 4) Determine where the function increases and decreases. Using these critical points, we can determine the intervals where the function increases and decreases. ➕=📈 If the derivative of the function on the interval between two adjacent/nearest critical points is negative, then the function is decreasing on that interval. ➖=📉 If the derivative of the function is positive, then the function is increasing on that interval. Please revisit our AP Calculus - 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing guide for a more in-depth explanation! Let’s evaluate the first derivative around each of our critical points: \| Interval \| $$$x$ \| f′(x)f'(x)f′(x) \| Verdict \| --- --- \| \| (−∞,−2)(-\infin,-2)(−∞,−2) \| x=−3 x=-3 x=−3 \| f′(−3)=9 f'(-3)=9 f′(−3)=9 \| f f f is increasing \| \| (−2,0)(-2,0)(−2,0) \| x=−1 x=-1 x=−1 \| f′(−1)=−3 f'(-1)=-3 f′(−1)=−3 \| f f f is decreasing \| \| (0,∞)(0,\infin)(0,∞) \| x=1 x=1 x=1 \| f′(1)=9 f'(1)=9 f′(1)=9 \| f f f is increasing \| Whew! Now we know that f f f is increasing between (−∞,−2)(-\infin,-2)(−∞,−2) and $(0,\infin)$ and decreasing between (−2,0)(-2,0)(−2,0). 🤔 Step 5) Find the extrema of the function. Based on the information we discovered in the last step, we can easily apply the First Derivative Test to find the extrema of the function! ⬇️ The critical points where the function is decreasing on its left and increasing on its right are minimums. ⬆️ The critical points where the function is increasing on its left and decreasing on its right are maximums. Visit our AP Calculus - 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema guide if you need a refresher! We know that to the left of x=−2 x=-2 x=−2, the function is increasing, and to the right of x=−2 x=-2 x=−2, the function is decreasing. Therefore, x=−2 x=-2 x=−2 is a local maximum. Vice versa is true for our critical point at x=0 x=0 x=0. Since the function is decreasing to its left and increasing to its right, x=0 x=0 x=0 is a local minimum. 🥈 Alternative to Steps 4 & 5 An alternative to steps 4) and 5) is to use the Second Derivative Test to determine the extrema of the function. Recall that this test states that… If the second derivative of a function at a critical point is negative, then the function has a relative maximum at that point (and it’s concave down). If the second derivative of a function at a critical point is positive, then the function has a relative minimum at that point (and it’s concave up). Please see our AP Calculus - 5.7 Using the Second Derivative Test to Determine Extrema guide. ✏️ Step 6) Determine points of inflection and concavity. A function has a possible point of inflection if its second derivative is equal to zero at the point. To verify it is indeed a point of inflection, we must check to see if the function changes from concave up to concave down or vice versa (aka the second derivative changes from negative to positive or vice versa) on the two sides of the point. For more information, revisit our AP Calculus - 5.6 Determining Concavity of Functions over Their Domains guide! Here’s the second derivative of our function, f(x)f(x)f(x). f′′(x)=6 x+6 f''(x)=6x+6 f′′(x)=6 x+6 Let’s find the possible point of inflection by setting this equal to 0. 0=6 x+6 0=6x+6 0=6 x+6−6=6 x-6=6x−6=6 x x=−1 x=-1 x=−1 Great! Now, we have to analyze the sign of the second derivative to see if this is a true point of inflection. \| x x x \| f′′(x)f''(x)f′′(x) \| Concavity \| --- \| −2-2−2 \| −6-6−6 \| Concave down \| \| 0 0 0 \| 6 6 6 \| Concave up \| Since the second derivative’s sign switches and the function’s concavity changes, x=−1 x=-1 x=−1 is a point of inflection. 🎨 Putting it All Together! Compiling all of the information we found, we know that: The function is continuous. We can plot (−2,0)(-2,0)(−2,0), (1,0)(1,0)(1,0), and (0,−4)(0,-4)(0,−4). f f f is increasing between (−∞,−2)(-\infin,-2)(−∞,−2) and (0,∞)(0,\infin)(0,∞). f f f is decreasing between (−2,0)(-2,0)(−2,0). x=−2 x=-2 x=−2 is a local maximum. x=0 x=0 x=0 is a local minimum. Concavity changes at x=−1 x=-1 x=−1 from concave down, to concave up. And here we have it! Image of $f(x)=(x+2)^2(x-1)$, created with Desmos. You may not have needed all of the information we obtained, but having it all helps you double-check that you graphed correctly. Great work! 👏 📝 Sketching Graphs Practice Try to sketch the following function on your own! What does the graph of f(x)f(x)f(x) look like? f(x)=x 3+3 x 2+3 f(x)=x^{3}+3x^{2}+3 f(x)=x 3+3 x 2+3 Be sure to go through each step on your own before you scroll down further and see the answer! ✅ Sketching Graphs Question Solution We’ll briefly go through each step and then show you the graph. Instead of going through steps 4 and 5 this time, we’ll try the alternative. 〰️ Find the domain of the function and determine if there are any discontinuities. Since this is a polynomial, the domain is all real numbers and there are no discontinuities. 🎯 Identify key features of the function, such as intercepts and symmetry. The x-intercept is (−3.279,0)(-3.279,0)(−3.279,0) and the y-intercept is (0,3)(0,3)(0,3). This function has neither even nor odd symmetry. 💯 Find critical points. By setting f′(x)=3 x 2+6 x f'(x)=3x^2+6x f′(x)=3 x 2+6 x equal to 0 and factoring, we know that the function has critical points at x=−2 x=-2 x=−2 and x=0 x=0 x=0. 🥈 Alternative method: use the Second Derivative Test to determine the extrema of the function. f′′(x)=6 x+6 f''(x)=6x+6 f′′(x)=6 x+6 f′′(−2)=−6 f''(-2)=-6 f′′(−2)=−6. Since −6<0-6<0−6<0, this means the function is concave down at this point, and thus by the Second Derivative Test is a maximum. f′′(0)=6 f''(0)=6 f′′(0)=6. Since 6>0 6>0 6>0, this means the function is concave up at this point, and thus by the Second Derivative Test is a minimum. ✏️ Determine points of inflection and intervals of concavity. x=−1 x=-1 x=−1 is the only possible point of inflection. When we check concavity of the function to its left and right, we find that the function is concave down to the left of the point and concave up to the right of the point. Since the concavity of the function changes at x=−1 x=-1 x=−1, it is a point of inflection. 🎨 Putting it All Together! The y-intercept is (0,3)(0,3)(0,3). x=−2 x=-2 x=−2 is a maximum. x=0 x=0 x=0 is a minimum. x=−1 x=-1 x=−1 is a point of inflection (concave down to the left, concave up to the right) We can plug these points into f(x)f(x)f(x) to get their exact y-values. With this information, we see that f(x)=x 3+3 x 2+3 f(x)=x^{3}+3x^{2}+3 f(x)=x 3+3 x 2+3 looks like the following: Graph of f(x)=x 3+3 x 2+3 f(x) = x^3 + 3x^2 + 3 f(x)=x 3+3 x 2+3. Image Created with Desmos. ⭐ Closing You can follow these same steps for drawing the graph of the derivative of a function, and the second derivative. While you may not have to draw a graph from scratch on the AP exam, you will likely have to know how to determine the key information about a function from its graph, and drawing the graph can give you an in-depth understanding of these key features. ✍️ Frequently Asked Questions How do I know if a function is increasing or decreasing from its derivative graph? Look at the graph of f′—its sign tells you whether f is increasing or decreasing. - If f′(x) > 0 on an interval, f is increasing there; if f′(x) < 0, f is decreasing. - Points where f′ = 0 or f′ is undefined are critical points. Make a sign chart for f′ around each critical point to see if the sign changes. - If f′ changes + → − at a point, f has a local maximum. - If f′ changes − → +, f has a local minimum. - If f′ doesn’t change sign, no local extremum (horizontal tangent). - Use f′′ (or the slope of f′) to check concavity: f′ increasing ⇒ f is concave up; f′ decreasing ⇒ concave down. Inflection points occur where f′ changes from increasing to decreasing (or vice versa). This is exactly what the AP CED asks you to do for FUN-4.A (use f′ and f′′ to predict f). For extra practice and examples, see the Topic 5.8 study guide ( and thousands of practice problems ( What's the difference between f'(x) = 0 and f''(x) = 0 on a graph? f'(x) = 0 means the slope of f is zero at x—a horizontal tangent and a critical point. That point could be a local max, local min, or just a flat point (no extremum). To tell which you use the first-derivative sign chart (change from + to − → max, − to + → min) or the second-derivative test: if f'(c)=0 and f''(c)>0 ⇒ local min; if f''(c)<0 ⇒ local max; if f''(c)=0 the second-derivative test is inconclusive. f''(x) = 0 means the concavity of f might be changing at x (possible inflection point). But f''(c)=0 alone does NOT guarantee an inflection—concavity must actually switch (f'' changes sign across c). So: - f'(c)=0 → horizontal tangent/critical point (check for max/min). - f''(c)=0 → possible inflection (verify sign change of f''). These distinctions are exactly what Topic 5.8 tests (critical points, inflection, concavity). For more examples and practice, see the Topic 5.8 study guide ( and lots of practice problems ( When do I use the first derivative vs the second derivative to sketch graphs? Use f' when you want to know where f is increasing/decreasing, locate critical points, and classify local extrema; use f'' when you want concavity and inflection points. Concretely: - First derivative (f'): find where f'=0 or undefined → potential local max/min or horizontal tangents. Use a sign chart or the First Derivative Test to tell whether f changes from + to − (local max) or − to + (local min). This gives increasing/decreasing intervals (CED FUN-4.A.10, FUN-4.A.9). - Second derivative (f''): find where f''=0 or undefined → possible inflection points. Use the Second Derivative Test at critical points: if f'(c)=0 and f''(c)>0 → local min; if f''(c)<0 → local max. More generally, f''>0 means concave up, f''<0 means concave down (CED keywords: concavity, inflection). On the AP exam you’ll often combine both: use f' to get extrema and sign charts, then f'' to confirm concavity/inflection (Topic 5.8). For extra guided practice see the Topic 5.8 study guide ( Unit 5 overview ( and plenty of practice problems ( How do I find critical points and inflection points step by step? Step-by-step: Critical points 1. Compute f′(x). Find where f′(x)=0 or f′ is undefined—those x are candidates (critical points). Include endpoints of your domain for absolute extrema. 2. Make a sign chart for f′: pick test points in each interval between candidates to see where f′>0 (f increasing) or f′<0 (f decreasing). 3. Use the First Derivative Test: if f′ changes +→− at a candidate, f has a local max; −→+ gives a local min; no sign change → no local extremum. You can also use the Second Derivative Test when f′(c)=0: if f″(c)>0 ⇒ local min; if f″(c)<0 ⇒ local max; if f″(c)=0 the test is inconclusive. 4. Don’t forget points where f is not differentiable (cusps/vertical tangents)—they can be critical and must be checked directly. Step-by-step: Inflection points / concavity 1. Compute f″(x). Inflection candidates are where f″(x)=0 or f″ is undefined (and where f is continuous). 2. Make a sign chart for f″ to see where concave up (f″>0) or concave down (f″<0). 3. An inflection point occurs where concavity changes sign (concave up ↔ concave down). Verify f is continuous there. On the AP exam you’ll need clear justification (show f′/f″ work and sign charts). For worked examples and practice, see the Topic 5.8 study guide ( and try problems at ( I'm confused about concave up vs concave down - how do I tell from f''(x)? Concavity is just what f'' tells you about the shape of f: - If f''(x) > 0 on an interval, f is concave up there (graph opens like a cup). Example: f(x)=x^2 has f''(x)=2>0. - If f''(x) < 0, f is concave down there (graph bends downward). Example: f(x)=-x^2 has f''(x)=-2<0. Interpretation: concave up means f' is increasing; concave down means f' is decreasing. So checking f'' is the same as checking whether the slope is getting larger or smaller. Inflection points: x = c is a possible inflection point if f''(c)=0 or undefined AND f'' changes sign at c. Just f''(c)=0 alone doesn’t guarantee an inflection. Second derivative test for extrema (useful on the AP): if f'(c)=0 and f''(c)>0 → local min; if f''(c)<0 → local max; if f''(c)=0 → test is inconclusive, use first-derivative test. For AP Topic 5.8 you’ll often combine sign charts for f' and f'' to justify behavior (see the Topic 5.8 study guide ( For extra practice, try problems at ( Can someone explain how to go from f'(x) to the original function f(x) graph? Start by reading f′ like a sign/shape map of f. - Sign of f′ tells increasing/decreasing: where f′>0, f rises; where f′<0, f falls (first-derivative test gives local max/min where f′ changes sign). - Slope of f′ (i.e. f″) tells concavity: if f′ is increasing (f″>0) then f is concave up; if f′ is decreasing (f″<0) then f is concave down. Inflection points occur where f′ changes from increasing to decreasing (or vice versa). - Horizontal tangents on f correspond to zeros of f′. Zeros where f′ crosses the axis are candidates for local extrema; zeros where f′ just touches the axis might be saddle points. Watch for cusps/vertical tangents—f′ might be undefined there. - To get actual y-values: integrate f′. ∫ f′ = f + C, so the graph of f is the antiderivative family of f′ (vertical shift depends on initial value). Also, definite integrals of f′ over intervals give net change in f (use when you know one point). This is exactly what the CED expects: use f′ and f″ to justify behavior (increasing/decreasing, concavity, extrema, inflection). For a focused review, see the Topic 5.8 study guide ( and more practice problems at ( What does it mean when f'(x) is positive but f''(x) is negative? If f′(x) > 0 but f″(x) < 0 at x (or on an interval), it means f is increasing there but concave down. In plain terms: the function’s y-values are rising (slopes of tangent lines are positive), but the slope is getting smaller as x increases—the graph bends downward like the top of a hill but still goes up. Key AP-CED language: f′ > 0 → increasing interval; f″ < 0 → concave down. This combination does not give a local max or min by itself (you need a sign change in f′ for that), but it does tell you the slope is decreasing—tangent lines have positive y-values but become less steep. Use this info when sketching f from f′ and f″ or when applying the first/second derivative tests in Topic 5.8 (Sketching Graphs of Functions and Their Derivatives). For a quick visual and practice, check the Topic 5.8 study guide ( and try related problems on Fiveable ( How do I sketch f(x) if I'm only given the graph of f'(x)? Start by reading f′ as a sign-and-slope map of f. - Increasing/decreasing: where f′>0, f rises; where f′<0, f falls. Mark intervals from the graph of f′. - Critical points / local extrema: zeros of f′ (and places f′ undefined). Use the first-derivative test: if f′ changes +→− at a zero, f has a local max; −→+ gives a local min; no sign change → no local extremum. - Concavity & inflection: concave up where f′ is increasing (f′′>0), concave down where f′ is decreasing (f′′<0). Inflection points occur where f′ has a local extremum (slope of f′ changes sign). - Horizontal tangents on f correspond to points where f′=0. Vertical tangents/cusps on f would show as vertical asymptotes or blow-ups in f′. Practical sketching steps: mark key x-values where f′=0 or undefined; record sign of f′ on intervals; note whether f′ is rising/falling to get concavity; draw a smooth f consistent with those features (choose any vertical shift—AP never requires the absolute y unless given). This matches CED keywords (critical points, first/second derivative tests, concavity). For a guided walk-through and examples, see the Topic 5.8 study guide ( For more unit review and 1000+ practice problems, check the Unit 5 page ( and practice list ( Why does my graph look wrong when I try to connect the critical points? If your sketch looks wrong when you just “connect the critical points,” it’s because critical points (where f′ = 0 or undefined) are just landmarks—they don’t determine the shape between them. You still must use f′ and f″ to fill in the behavior: - Use a sign chart for f′ to know where f is increasing/decreasing, not just the x-values of critical points. (First Derivative Test → local max/min.) - Use f″ (or the slope of f′) to get concavity between critical points. Inflection points occur where concavity changes, so the curve can bend even if f′ stays positive. - Make the curve smooth: horizontal tangents at f′=0, sharp turns if f′ is undefined (cusps), or vertical tangents if slopes blow up. - Don’t join points with straight lines—draw continuous curves with the correct slope and concavity at each point. On the AP exam you should justify behavior with derivative sign/concavity reasoning (FUN-4.A). For a quick refresher, check the Topic 5.8 study guide ( and practice problems ( to get more examples. What's the relationship between where f'(x) = 0 and the max/min points on f(x)? When f′(x) = 0 you’ve found a critical point (a horizontal tangent). That’s a candidate for a local max, local min, or neither—f′(x)=0 alone doesn’t guarantee an extremum. Use sign tests: - If f′ changes from + to − at the point → local maximum. - If f′ changes from − to + → local minimum. - If f′ doesn’t change sign (e.g., + to + or − to −) → not an extremum (inflection/horizontal tangent). You can also use the second-derivative test at a critical point where f″ exists: - f′(c)=0 and f″(c)>0 ⇒ local minimum (concave up). - f′(c)=0 and f″(c)<0 ⇒ local maximum (concave down). - f″(c)=0 ⇒ inconclusive—go back to the first-derivative sign chart. Remember critical points also include where f′ is undefined (cusps, vertical tangents)—those can be extrema, so check endpoints and where f′ fails to exist. For practice and AP-style examples, see the Topic 5.8 study guide ( and try sign-chart problems from the unit practice ( I don't understand how to use the second derivative test - when does it fail? The second-derivative test: if f′(c)=0 and f″(c)>0, f has a local min at c; if f″(c)<0, f has a local max. It fails (is inconclusive) when f″(c)=0 or f″(c) does not exist. In those cases you can’t conclude anything from f″ alone—use the first-derivative test (check sign of f′ on each side of c) or look at higher derivatives/Taylor expansion. Quick examples: - f(x)=x^3 at x=0: f′(0)=0 and f″(0)=0, but 0 is an inflection point (not a max or min) → second-derivative test fails. - If f′(c) is undefined (cusp, vertical tangent), the second-derivative test doesn’t apply at all. On the AP exam you should justify your conclusion (FUN-4.A): if f″(c)=0 say it’s inconclusive and show a first-derivative sign chart or evaluate higher derivatives. For a short review, see Topic 5.8 study guide ( For more practice, use the unit page ( and practice problems ( How do I know if a critical point is a local max, local min, or neither? Look at f′ and f″ around the critical point. - First Derivative Test (best on AP): find where f′ = 0 or undefined, then check sign of f′ just left and right. - f′ changes + → − : local maximum. - f′ changes − → + : local minimum. - f′ keeps same sign: neither (inflection or plateau). - Second Derivative Test (quick when f′(c)=0): compute f″(c). - f″(c) > 0 ⇒ concave up ⇒ local minimum. - f″(c) < 0 ⇒ concave down ⇒ local maximum. - f″(c) = 0 ⇒ inconclusive → use first-derivative sign chart or higher derivatives. Also watch for endpoints (use Candidates Test for absolute extrema) and non-differentiable critical points (cusps/corners): those can be maxima/minima, so check values or one-sided behavior. On the AP exam, justify your conclusion (show sign chart or f″ calculation) to earn full points. For more worked examples and practice, see the Topic 5.8 study guide ( and try problems at ( What are the steps to analyze a function completely using derivatives? Step-by-step to analyze a function completely with derivatives (CED-aligned): 1. Domain & continuity: find domain, endpoints, vertical asymptotes, and places f is not differentiable. 2. Compute f′ and f″ (simplify). Note where they exist and zeros/undefined points → critical points and possible inflection candidates. 3. Sign chart for f′: determine intervals where f′>0 (increasing) and f′<0 (decreasing). Use first-derivative test at critical points to classify local maxima/minima. 4. Sign chart for f″ (or derivative of f′): determine concave up (f″>0) and concave down (f″<0). Confirm inflection points where concavity changes. Use second-derivative test when convenient to classify stationary critical points (if f″≠0). 5. Check endpoints & asymptotic/behavior at infinity for global extrema (Candidates Test). 6. Identify other features: horizontal tangents, vertical tangents, cusps, and sketch using intercepts, slope behavior, inflection, and asymptotes. 7. Justify each conclusion with derivative sign reasoning (FUN-4.A & FUN-4.A.10). For AP-style practice and review, see the Topic 5.8 study guide ( the Unit 5 overview ( and extra practice problems ( How can I tell where a function has inflection points just from looking at f'(x)? Look for where f''(x) changes sign—but using only f'(x) that means you want x-values where the graph of f' changes from increasing to decreasing or vice versa. Practically: - Inflection points of f occur where f' has a local extremum (a local max or min). At those x, f'' = (slope of f') = 0 or undefined, and f'' typically changes sign. - So scan f' for peaks/troughs (horizontal tangents) or places where f' goes from rising to falling (f'' changes from + to −) or falling to rising (f'' changes from − to +). - Important: also check that f is defined there (and that f'' actually changes sign). A zero of f' is not automatically an inflection point—only if f' switches monotonicity across it. This is exactly the FUN-4.A idea: use f' (and implicitly f'') to justify concavity/inflection. For more worked examples and AP-style practice, see the Topic 5.8 study guide ( and try problems at ( When f'(x) changes from positive to negative, what happens to the original function? If f′(x) changes from positive to negative at x = c (and f is continuous/differentiable there), then f goes from increasing to decreasing at c—so f has a local maximum at x = c by the First Derivative Test. In CED terms, f′ > 0 on the left of c (increasing), f′ < 0 on the right (decreasing), so c is a critical point that’s a local max. You can also check f″(c): if f″(c) < 0 the Second Derivative Test confirms concave down and a local max, but the first-derivative sign change is enough. 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Vertebrate Animals in Research IACUC: Engage in Contracted / Subcontracted Vertebrate Animal Research IACUC: Modify an Existing Protocol IACUC: PHS AWA / USDA Registration IACUC: Protocol Review Process IACUC: Resources IACUC: Training and Qualifications Export Control Export Control Compliance Manual Export Control Glossary Export Control Training Materials Export Control: Frequently Asked Questions Export Controlled Chemicals Export Controlled Items High Performance Computing and Export Control at Lehigh Sanctioned Countries Research Security Financial Conflicts of Interest in Research and Sponsored Programs Determining if an SFI is Related LIRA Disclosure Process and Resources Public Accessibility of a COI Related to Public Health Service-Funded Research fCOI: Additional Resources fCOI: Frequently Asked Questions fCOI: Glossary Breadcrumb Home Policies, Guidance & Forms Guidance: Justifying The Number of Animals In Research Guidance: Justifying the Number of Animals in Research \| Responsible Office \| Office of Research Integrity \| --- \| \| Originally Issued \| 24-Apr-2015 \| \| Last Revised \| n/a \| \| Author \| N. Coll \| The intent of this Institutional Animal Care and Use Committee (IACUC) guidance is to assist researchers in preparing a response to 7g on the IACUC Protocol Form: “Provide the number of animals requested and the rationale for how the number of animals was determined to be appropriate. Whenever possible, the number of animals requested must be justified statistically”. The guidance is designed to assist researchers in preparing application for review by the IACUC and to help ensure compliance with the institution’s animal welfare Assurance and informed by regulations outlined in the Guide. Introduction and Background Investigators are required to use the minimum number of animals necessary to obtain valid results. Question 7g of the IACUC Protocol Form addresses this requirement by providing the rationale for the number of animals requested. Generally, answering this question requires a power analysis. Power refers to the probability of avoiding a Type II error, or, the ability of a statistical test to detect true differences when they exist. The power of a test generally depends on four variables: sample size, effect size to be detected (medium), the Type I error rate (alpha, usually .05), and the variability of the sample. Power is usually specified at 0.80, that is, 80% likely to be right. Alternatively, if animal number to be used are based on previous work or publications, detailed citations are acceptable. Statistical tests provide a way to estimate whether the differences measured between groups of animals treated differently in one experiment are “real” (i.e. will be reproducible nearly all the time), or have just occurred by chance. Most often, this is stated as the p-value. P < 0.05 means the “result” could occur by chance less than 5% of the time. From simple t-tests to the most complicated analyses, the mathematical assumptions underlying statistical tests require that the methods of analysis, the p-value, and the minimal difference between groups you want to find, all be decided upon beforehand. Procedures for Providing Statistical Justification The IACUC requires the following information about how you a) determine sample sizes, and b) analyze your data: A brief description of the experimental design, including the control and experimental groups and their sample sizes, if applicable. “For example, there will be 4 groups, including one control group that receives vehicle, and three groups, each of which receives 1, 2, and 4 mg of the drug (n = 10 per group, n = 40 total).” A description of the statistical methods for determining the sample size, e.g., a Power Analysis, if applicable. Please specify the variable used in the power analysis, and the results of the power analysis. If these have been determined previously, cite the publication. For example, "Using Cohen's d, we determine that a sample size of 10 per group is necessary to detect statistically significant effects of treatment." A statement of the probability value used to detect significant differences (i.e., the P-value, the alpha significance levels). For example “Differences between groups will be considered statistically significant if P < 0.05.” A statement of the effect size that will be considered substantive. For example, “Statistically significant differences will be considered small if the Coefficient of Determination (r 2 ) is greater than 0.1 but less than 0.3, medium if r 2 is great than .3 but less than 0.5, and large or substantive if 2 is greater than 0.5.” The IACUC does not insist on any given alpha or beta levels, and it will evaluate arguments for deviation of either of these from traditionally used values. Additional Details When the animal is not the experimental unit, determine the appropriate sample size for your unit of study (e.g., the number of neurons needed) and extrapolate to determine the maximum number of animals required. Include information regarding the number of experimental units expected to be derived from each individual animal. When statistical justification is not possible, briefly explain why. Provide a rationale for the proposed number of animals, such as complete citations of previous research or experience. For pilot studies or teaching protocols, the researcher may indicate that statistical justification does not apply because a hypothesis is not being tested. A firm number (i.e. not a range) must be provided for a three-year period. A maximum number may be listed. It is recommended that a 5-10% overage is considered in order to cover cases where animals must be removed from studies for nonexperimental reasons. For renewal submissions, provide an updated justification of the number of animals required for the next three years; including all breeding, control, and experimental animals. Total animals requested for a renewal may need to be modified from the previously approved protocol version. More than one justification may be applicable to the proposed study, depending on the mix of observations and experiments being conducted. Combine non-statistical and statistical justifications as appropriate. Resources for Completing Power Analyses There are free web sources available for completing a power analysis for many simple and common experimental designs: References Guide for the Care and Use of Laboratory Animals, 8 th Ed., National Research Council; National Academy Press: Washington, DC, 2011. Animal Welfare Act and Animal Welfare Regulations. Part 2 – Regulations; Subpart C – Research Facilities; § 2.31, e, 1-2. 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15122	https://english.stackexchange.com/questions/91935/synonym-for-do-you-mean-without-negative-connotations	errors - Synonym for "do you mean" without negative connotations - English Language & Usage Stack Exchange Join English Language & Usage 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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You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Synonym for "do you mean" without negative connotations [closed] Ask Question Asked 12 years, 10 months ago Modified12 years, 10 months ago Viewed 12k times This question shows research effort; it is useful and clear 0 Save this question. Show activity on this post. It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center. Closed 12 years ago. Whenever I use the phrase "do you mean to say", I notice that the word "mean" has a variety of negative connotations (cruelty, harshness, etc.) Is there any alternative for this phrase that doesn't have such unpleasant connotations? ("Do you mean to say this" sounds very similar to "what you said was mean", despite having a completely different meaning - that's why I'm concerned about the connotation.) connotation errors Share Share a link to this question Copy linkCC BY-SA 3.0 Improve this question Follow Follow this question to receive notifications edited Nov 21, 2012 at 23:55 Anderson GreenAnderson Green asked Nov 21, 2012 at 22:38 Anderson GreenAnderson Green 427 2 2 gold badges 5 5 silver badges 12 12 bronze badges 9 2 The verb "mean" has no such unpleasant connotation of cruelty, and would not by any reader or hearer be taken to have anything to do with the adjective "mean" or the adverb "meanly".StoneyB on hiatus –StoneyB on hiatus 2012-11-21 23:02:13 +00:00 Commented Nov 21, 2012 at 23:02 I think this is Too Localised. I mean to say - how many people really think I'm being mean, just because I try to say what I mean?FumbleFingers –FumbleFingers 2012-11-21 23:11:52 +00:00 Commented Nov 21, 2012 at 23:11 2 ...further to which, I would just add that for me at least, saying to someone "Are you trying to say [whatever]?" definitely does have negative overtones, in that it implicitly accuses the other person of being inarticulate.FumbleFingers –FumbleFingers 2012-11-21 23:13:54 +00:00 Commented Nov 21, 2012 at 23:13 1 Rhyme has no predictable connotation, though a writer/speaker may exploit rhyme to impose a connotation. The reason mean,adj. and mean,vb. don't cross-connote is that their contexts don't overlap, precisely because they play different syntactic roles.StoneyB on hiatus –StoneyB on hiatus 2012-11-22 00:01:29 +00:00 Commented Nov 22, 2012 at 0:01 3 @Anderson Green: You'd have problems with happy and unhappy, then. I'm sure that homographs and homophones do intrude into our perception and have some effect on our 'feel' for a word. The word for that relative of the cormorant, the shag, for instance, still causes me unwarranted trouble. We have to draw the line somewhere, though, and not be too precious, mollycoddling ourselves or others. Oh, and mean (= signify), mean (= stingy or cruel) and mean (= a type of average) are three different words.Edwin Ashworth –Edwin Ashworth 2012-11-22 00:04:14 +00:00 Commented Nov 22, 2012 at 0:04 \|Show 4 more comments 4 Answers 4 Sorted by: Reset to default This answer is useful 4 Save this answer. Show activity on this post. Though I can't imagine someone (unless it's a non-native audience) taking "mean" in the wrong sense, here are a couple other ways to ask the same question: Is your intention to say (x)? What I'm hearing you say is (x) I'm unclear on your meaning (hopefully meaning is not misconstrued as "mean") Can you clarify that please? Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications answered Nov 21, 2012 at 23:08 Kristina LopezKristina Lopez 26.7k 6 6 gold badges 59 59 silver badges 114 114 bronze badges 5 1 Whenever I hear someone say "Did you mean to say that?", it reminds me of the sentence "Why are you so mean?" or "What you said was really mean." I know that the meanings are completely different, but they still sound similar.Anderson Green –Anderson Green 2012-11-21 23:34:05 +00:00 Commented Nov 21, 2012 at 23:34 @AndersonGreen, that's the beauty of language, there is usually another way to say something and you can just choose not to use a certain word or expression without apologizing to anyone for your reasons! :-)Kristina Lopez –Kristina Lopez 2012-11-21 23:38:18 +00:00 Commented Nov 21, 2012 at 23:38 Still, this question got downvoted for some reason - does that mean that this question isn't even worthy of discussion?Anderson Green –Anderson Green 2012-11-21 23:52:04 +00:00 Commented Nov 21, 2012 at 23:52 1 @AndersonGreen: I can imagine all sorts of things that the name Winnie the Pooh reminds you of :D Armen Ծիրունյան –Armen Ծիրունյան 2012-11-22 00:24:00 +00:00 Commented Nov 22, 2012 at 0:24 @AndersonGreen, Actually, this is meant to be a question and answer site, not a discussion site - but as you can see by the comments - discussions do occur. As for the downvote, ideally, downvoters should acknowledge the reason for the downvote.Kristina Lopez –Kristina Lopez 2012-11-22 00:29:12 +00:00 Commented Nov 22, 2012 at 0:29 Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. Sorry, but "Do you mean to say this?" doesn't sound at all similar to "Do you say this meanly?". Although a few of the words are the same, the ideas expressed are as distant as Durban and Detroit. "Do you mean to say this?" is always a negative statement because it implies one of two possibilities: (1) What you said wasn't clear enough for me to understand. Did you really want to say "ABC" instead of "XYZ"? or (2) I'm sorry, but I'm not very good at understanding what other people say unless it's said at my level. Did you really want to say "ABC" instead of "XYZ"? Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications answered Nov 21, 2012 at 23:06 user21497 user21497 4 1 + 0.98 Detroit isn't nearly so distant as Durban where I'm sitting.StoneyB on hiatus –StoneyB on hiatus 2012-11-21 23:09:05 +00:00 Commented Nov 21, 2012 at 23:09 1 @StoneyB: "Approximate distance as the crow flies in miles from Detroit United States to Durban South Africa is 8747 miles or 14073.92 Kilometers" & "The circumference of the earth at the equator is 24,901.55 miles (40,075.16 kilometers)." "Approximate distance as the crow flies in miles from Detroit United States to Taipei Taiwan is 7532 miles or 12118.99 Kilometers" & "Approximate distance as the crow flies in miles from Durban to Taipei Taiwan is 7064.3 miles or 11376.3 Kilometers."user21497 –user21497 2012-11-21 23:25:29 +00:00 Commented Nov 21, 2012 at 23:25 1 I meant that this sounds very odd to me. Perhaps I'm dating myself, or showing myself provincial, or simply ignorant, but "distant" without "from" to me means "distant from here"; I'd write "as far apart as D & D" or "as distant from each other as D from D".StoneyB on hiatus –StoneyB on hiatus 2012-11-21 23:50:15 +00:00 Commented Nov 21, 2012 at 23:50 @StoneyB: O, IC. I'll invoke Chomsky: My S is naturally ambiguous because it's an elision of one of these deep structures: "as distant from me as are Durban and Detroit" & "as distant from each other as are Durban and Detroit". That's why he invented trace theory. I'm 5 years your senior, so you're not dating yourself. You're hardly provincial or ignorant & prove yourself to be anything but in all of your answers & comments. I have to take the blame for your misunderstanding. That's what I do when journal reviewers complain about my English: I wasn't clear enough. Sorry about that. :-)user21497 –user21497 2012-11-22 02:27:44 +00:00 Commented Nov 22, 2012 at 2:27 Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. What is the context of the situation? For me, it would mean( here: the word used as a verb ) either that 1) you want to make sure that someone you are/were talking to meant something you thought about or 2) you want to correct someone's incorrect use of phrase/word etc. 1) A: I think she is grumpy. B: Do you mean that she complains a lot today? 2) A: Could I have expresso, please? B: Do you mean ESpresso? The second situation may be read as you are being rude, as people simply do not like to be corrected by others. Otherwise it shouldn't have negative connotations. And of course, a second answer to your question may be that you confuse a verb : to mean - to express or represent something such as an idea, thought, or fact ( Cambridge Dictionary ) and a rather colloquial use of an adjective mean example: You are mean to me! which can be interpreted as someone is being not nice/rude/cruel to the other person. Then, the connotation with the ADJECTIVE - mean - would be negative. I hope I did help. Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications edited Nov 22, 2012 at 7:56 answered Nov 21, 2012 at 23:17 CanelaCanela 31 4 4 bronze badges 0 Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. Yesterday, I had just answered a question about dysphemistic euphemism - the use of gentle phrases pejoratively due to the deteriorated effects of the euphemism. In this case, we are looking at a similar effect - aggressive amelioration. Normally, we would use ameliorative phrases and words to be polite. e.g. the use of the word please. However, due to the authoritarian projection of such ameliorative expressions, they have taken on an aggressive impression: Can you please sit down?! With all due respect, you have no authority here. Do you mean to say our corporation would survive without positive revenue? I would prefer that you sat down. Would you mind taking your shoes off? One common step to sustain the ameliorative effect of a phrase is the use of subjunctive (which is effective mostly only on native speakers, who normally understand the etiquette of using the subjunctive). Otherwise, using the past tense rather than the present tense: Could you take your shoes off at the mud-room before entering the living room? The subjunctive denotes possibility rather than affirmative action - meaning that the host of the home gives you a choice to remove your shoes, hoping that you would reciprocate his/her politeness. Did you mean to say that our company could survive without positive revenue? Meaning, I might have heard it wrongly a moment ago, but could you confirm that it is you opinion (giving the other party a wider leeway to affirm the opposite). The word please has taken on a rather aggressive ameliorative effect lately and frequently should not even be included if a statement is meant to be sincerely ameliorative. Could you please take your shoes off?! vs Could you take your shoes off? The placing of the word please affects the mood Please, could you take your shoes off? / imo, polite authoritative / Could you please take your shoes off? / imo, aggressive authoritative / Could you take your shoes off, please? / imo, pleading / Do you mean to say taking a left would lead us to the highway? vs Did you say that taking a left would lead us to the highway? Would you say that taking a left would lead us to the highway? Let us presume that we took a left. Would you confirm again that would lead us to the highway? I agree that taking a left leads us to the highway. Please let me know if I am wrong. I don't quite agree that taking a left would lead us to the highway. I am still not convinced that taking a left would lead us to the highway. Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications answered Nov 22, 2012 at 0:37 Blessed GeekBlessed Geek 9,701 21 21 silver badges 35 35 bronze badges Add a comment\| Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions connotation errors See similar questions with these tags. 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15123	https://www.nature.com/articles/eye2011321	Visual and anatomical outcomes following vitrectomy for complications of diabetic retinopathy: The DRIVE UK Study \| Eye Your privacy, your choice We use essential cookies to make sure the site can function. We also use optional cookies for advertising, personalisation of content, usage analysis, and social media. By accepting optional cookies, you consent to the processing of your personal data - including transfers to third parties. Some third parties are outside of the European Economic Area, with varying standards of data protection. See our privacy policy for more information on the use of your personal data. Manage preferences for further information and to change your choices. Accept all cookies Skip to main content Thank you for visiting nature.com. You are using a browser version with limited support for CSS. 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Advertisement View all journals Search Search Search articles by subject, keyword or author Show results from Search Advanced search Quick links Explore articles by subject Find a job Guide to authors Editorial policies Log in Explore content Explore content Research articles Reviews & Analysis News & Comment Current issue Collections Follow us on Twitter Subscribe Sign up for alerts RSS feed About the journal About the journal Journal Information Open Access Fees and Funding About the Editors Journal News Special Issues About the Partner EYE Covers Contact For Advertisers Subscribe Publish with us Publish with us For Authors & Referees Language editing services Open access funding Submit manuscript Subscribe Sign up for alerts RSS feed nature eye clinical study article Clinical Study Published: 06 January 2012 Visual and anatomical outcomes following vitrectomy for complications of diabetic retinopathy: The DRIVE UK Study B Gupta1, S Sivaprasad1, R Wong2, A Laidlaw2, T L Jackson1, D McHugh1& … T H Williamson2 Show authors Eyevolume 26,pages 510–516 (2012)Cite this article 3361 Accesses 82 Citations 3 Altmetric Metrics details Abstract Introduction End-stage diabetic eye disease is an important cause of severe visual impairment in the working-age group. With the increasing availability of refined surgical techniques as well as the early diagnosis of disease because of screening, one would predict that the prevalence of this condition is decreasing and the visual outcome is improving. Aim To study the prevalence and visual outcome following vitrectomy for complications of diabetic retinopathy. Materials and methods This study identified the patients who underwent vitrectomy from January 2007 to December 2009 because of diabetes-related complications in South East London. Data collected included baseline demographics, best-corrected visual acuity, indication for the vitrectomy, complication, outcome, and duration of follow-up. Results The prevalence of people requiring vitrectomy who are registered in the diabetes register of this region was 2 per 1000 people with diabetes. Vitrectomy was required in 185 eyes of 158 patients during this period. These included 83 Caucasians, 51 Afro-Caribbeans, 17 South Asians, and 7 from other ethnic groups. There were 58 patients with type I diabetes and 100 with type II, with a mean duration of diabetes of 23 and 16.5 years, respectively. The reason for vitrectomy included tractional retinal detachment (TRD) in 109 eyes, non-clearing vitreous haemorrhage (NCVH) in 68 eyes, and other causes in 8 eyes. In all, 50% of the eyes with TRD and NCVH, and 87% of the eyes with NCVH improved by at least three ETDRS lines at 12 months. Poor predictors of visual success included longer duration of diabetes (OR: 0.69), use of insulin (OR: 0.04), presence of ischaemic heart disease (OR: 0.04), delay in surgery (OR: 0.59), and the failure to attend clinic appointments (OR: 0.58). Preoperative use of intravitreal bevacizumab in eyes with TRD undergoing vitrectomy showed a marginal beneficial effect on co-existent maculopathy (P=0.08) and required less laser intervention post procedure, but did not affect the number of episodes of late-onset vitreous haemorrhage post vitrectomy (P=0.81). Conclusion Visual outcome has improved significantly in eyes with complications due to diabetic retinopathy compared with the previously reported Diabetic Vitrectomy Study. Similar content being viewed by others Safety and effectiveness of pre-emptive diabetic vitrectomy in patients with severe, non-fibrotic retinal neovascularisation despite panretinal photocoagulation Article 21 July 2022 Referrals for proliferative diabetic retinopathy from two UK diabetic retinopathy screening services: a 10-year analysis of visual outcomes, requirement for vitrectomy, and mortality Article Open access 23 April 2024 Diabetic retinal disease Article 28 August 2025 Login or create a free account to read this content Gain free access to this article, as well as selected content from this journal and more on nature.com Access through your institution or Sign in or create an account Continue with Google Continue with ORCiD References Resnikoff S, Pascolini D, Etya’ale D, Kocur I, Pararajasegaram R, Pokharel GP et al. Global data on visual impairment in the year 2002. Bull World Health Organ 2004; 82 (11): 844–851. PubMedPubMed CentralGoogle Scholar Zimmet P, Alberti KG, Shaw J . Global and societal implications of the diabetes epidemic. Nature 2001; 414 (6865): 782–787. ArticleCASPubMedGoogle Scholar Fine SL, Patz A . Ten years after the Diabetic Retinopathy Study. Ophthalmology 1987; 94 (7): 739–740. ArticleCASPubMedGoogle Scholar The Diabetic Retinopathy Vitrectomy Study Research Group. Early vitrectomy for severe vitreous hemorrhage in diabetic retinopathy. Two-year results of a randomized trial. Diabetic Retinopathy Vitrectomy Study report 2. Arch Ophthalmol 1985; 103 (11): 1644–1652. ArticleGoogle Scholar Williams DF, Williams GA, Hartz A, Mieler WF, Abrams GW, Aaberg TM . Results of vitrectomy for diabetic traction retinal detachments using the en bloc excision technique. Ophthalmology 1989; 96 (6): 752–758. ArticleCASPubMedGoogle Scholar Arevalo JF, Wu L, Sanchez JG, Maia M, Saravia MJ, Fernandez CF et al. Intravitreal bevacizumab (Avastin) for proliferative diabetic retinopathy: 6-months follow-up. Eye (Lond) 2009; 23 (1): 117–123. ArticleCASGoogle Scholar Gregori NZ, Feuer W, Rosenfeld PJ . Novel method for analyzing snellen visual acuity measurements. Retina 2010; 30 (7): 1046–1050. ArticlePubMedGoogle Scholar Yorston D, Wickham L, Benson S, Bunce C, Sheard R, Charteris D . Predictive clinical features and outcomes of vitrectomy for proliferative diabetic retinopathy. Br J Ophthalmol 2008; 92: 365–368. ArticleCASPubMedGoogle Scholar Mason III JO, Colagross CT, Haleman T, Fuller JJ, White MF, Feist RM et al. Visual outcome and risk factors for light perception and no light perception vision after vitrectomy for diabetic retinopathy. Am J Ophthalmol 2005; 140 (2): 231–235. ArticlePubMedGoogle Scholar Smiddy WE, Feuer W, Irvine WD, Flynn Jr HW, Blankenship GW . Vitrectomy for complications of proliferative diabetic retinopathy. Functional outcomes. Ophthalmology 1995; 102 (11): 1688–1695. ArticleCASPubMedGoogle Scholar Yorston D, Wickham L, Benson S, Bunce C, Sheard R, Charteris D . Predictive clinical features and outcomes of vitrectomy for proliferative diabetic retinopathy. Br J Ophthalmol 2008; 92 (3): 365–368. ArticleCASPubMedGoogle Scholar Thompson JT, de BS, Michels RG, Rice TA, Glaser BM . Results of vitrectomy for proliferative diabetic retinopathy. Ophthalmology 1986; 93 (12): 1571–1574. ArticleCASPubMedGoogle Scholar DRVS. Early vitrectomy for severe proliferative diabetic retinopathy in eyes with useful vision. Results of a randomized trial—Diabetic Retinopathy Vitrectomy Study Report 3. The Diabetic Retinopathy Vitrectomy Study Research Group. Ophthalmology 1988; 95 (10): 1307–1320. ArticleGoogle Scholar Rizzo S, Genovesi-Ebert F, di BE, Vento A, Miniaci S, Williams G . Injection of intravitreal bevacizumab (Avastin) as a preoperative adjunct before vitrectomy surgery in the treatment of severe proliferative diabetic retinopathy (PDR). Graefes Arch Clin Exp Ophthalmol 2008; 246 (6): 837–842. ArticleCASPubMedGoogle Scholar da RL, Ribeiro JA, Costa RA, Barbosa JC, Scott IU, de Figueiredo-Pontes LL et al. Intraoperative bleeding during vitrectomy for diabetic tractional retinal detachment with versus without preoperative intravitreal bevacizumab (IBeTra study). Br J Ophthalmol 2009; 93 (5): 688–691. ArticleGoogle Scholar Steel DH, Connor A, Habib MS, Owen R . Entry site treatment to prevent late recurrent postoperative vitreous cavity haemorrhage after vitrectomy for proliferative diabetic retinopathy. Br J Ophthalmol 2010; 94 (9): 1219–1225. ArticleCASPubMedGoogle Scholar Bandello F, Battaglia PM, Lanzetta P, Loewenstein A, Massin P, Menchini F et al. Diabetic macular edema. Dev Ophthalmol 2010; 47: 73–110. ArticlePubMedGoogle Scholar Kook D, Wolf A, Kreutzer T, Neubauer A, Strauss R, Ulbig M et al. Long-term effect of intravitreal bevacizumab (avastin) in patients with chronic diffuse diabetic macular edema. Retina 2008; 28 (8): 1053–1060. ArticlePubMedGoogle Scholar Soheilian M, Ramezani A, Obudi A, Bijanzadeh B, Salehipour M, Yaseri M et al. Randomized trial of intravitreal bevacizumab alone or combined with triamcinolone versus macular photocoagulation in diabetic macular edema. Ophthalmology 2009; 116 (6): 1142–1150. ArticlePubMedGoogle Scholar Stefansson E . The therapeutic effects of retinal laser treatment and vitrectomy. A theory based on oxygen and vascular physiology. Acta Ophthalmol Scand 2001; 79 (5): 435–402. ArticleCASPubMedGoogle Scholar Download references Author information Authors and Affiliations Department of Opthalmology, Laser and Retinal Research Unit, King's College Hospital, London, UK B Gupta,S Sivaprasad,T L Jackson&D McHugh St. Thomas’ Hospital, London, UK R Wong,A Laidlaw&T H Williamson Authors 1. B GuptaView author publications Search author on:PubMedGoogle Scholar 2. S SivaprasadView author publications Search author on:PubMedGoogle Scholar 3. R WongView author publications Search author on:PubMedGoogle Scholar 4. A LaidlawView author publications Search author on:PubMedGoogle Scholar 5. T L JacksonView author publications Search author on:PubMedGoogle Scholar 6. D McHughView author publications Search author on:PubMedGoogle Scholar 7. T H WilliamsonView author publications Search author on:PubMedGoogle Scholar Corresponding author Correspondence to B Gupta. Ethics declarations Competing interests The authors declare no conflict of interest. Additional information Meeting Presentation: Poster Presentation: SN- ARVO: Diabetic update, Chennai, India, September 2010. Rights and permissions Reprints and permissions About this article Cite this article Gupta, B., Sivaprasad, S., Wong, R. et al. Visual and anatomical outcomes following vitrectomy for complications of diabetic retinopathy: The DRIVE UK Study. Eye26, 510–516 (2012). Download citation Received: 03 May 2011 Accepted: 09 November 2011 Published: 06 January 2012 Issue Date: April 2012 DOI: Share this article Anyone you share the following link with will be able to read this content: Get shareable link Sorry, a shareable link is not currently available for this article. Copy shareable link to clipboard Provided by the Springer Nature SharedIt content-sharing initiative Keywords pars plana vitrectomy proliferative diabetic retinopathy tractional retinal detachment non-clearing vitreous haemorrhage Subjects Diabetes complications Surgery This article is cited by The BElfast Retinal Tear and detachment Score (BERT Score) in vitreous haemorrhage Mohamad Baba Richard Best Eye (2024) Patient-reported outcome measures in vitreoretinal surgery: a systematic review Anusha Yoganathan Teresa Sandinha David Steel Eye (2023) Safety and effectiveness of pre-emptive diabetic vitrectomy in patients with severe, non-fibrotic retinal neovascularisation despite panretinal photocoagulation Shi Zhuan Tan David H. Steel David Alistair H. Laidlaw Eye (2023) Visual outcomes and complications following one-way air-fluid exchange technique for vitreous hemorrhage post vitrectomy in proliferative diabetic retinopathy patients Qun Wang Jie Zhao Yifei Huang BMC Ophthalmology (2021) Persistent subretinal fluid following diabetic tractional retinal detachment repair: risk factors, natural history, and management outcomes Ahmed Algethami Mohammed Talea Sulaiman M. Alsulaiman International Ophthalmology (2021) Sections References Abstract References Author information Ethics declarations Additional information Rights and permissions About this article This article is cited by Advertisement Resnikoff S, Pascolini D, Etya’ale D, Kocur I, Pararajasegaram R, Pokharel GP et al. Global data on visual impairment in the year 2002. Bull World Health Organ 2004; 82 (11): 844–851. PubMedPubMed CentralGoogle Scholar Zimmet P, Alberti KG, Shaw J . Global and societal implications of the diabetes epidemic. Nature 2001; 414 (6865): 782–787. ArticleCASPubMedGoogle Scholar Fine SL, Patz A . Ten years after the Diabetic Retinopathy Study. Ophthalmology 1987; 94 (7): 739–740. ArticleCASPubMedGoogle Scholar The Diabetic Retinopathy Vitrectomy Study Research Group. Early vitrectomy for severe vitreous hemorrhage in diabetic retinopathy. Two-year results of a randomized trial. Diabetic Retinopathy Vitrectomy Study report 2. Arch Ophthalmol 1985; 103 (11): 1644–1652. ArticleGoogle Scholar Williams DF, Williams GA, Hartz A, Mieler WF, Abrams GW, Aaberg TM . Results of vitrectomy for diabetic traction retinal detachments using the en bloc excision technique. Ophthalmology 1989; 96 (6): 752–758. ArticleCASPubMedGoogle Scholar Arevalo JF, Wu L, Sanchez JG, Maia M, Saravia MJ, Fernandez CF et al. Intravitreal bevacizumab (Avastin) for proliferative diabetic retinopathy: 6-months follow-up. Eye (Lond) 2009; 23 (1): 117–123. ArticleCASGoogle Scholar Gregori NZ, Feuer W, Rosenfeld PJ . Novel method for analyzing snellen visual acuity measurements. Retina 2010; 30 (7): 1046–1050. ArticlePubMedGoogle Scholar Yorston D, Wickham L, Benson S, Bunce C, Sheard R, Charteris D . Predictive clinical features and outcomes of vitrectomy for proliferative diabetic retinopathy. Br J Ophthalmol 2008; 92: 365–368. ArticleCASPubMedGoogle Scholar Mason III JO, Colagross CT, Haleman T, Fuller JJ, White MF, Feist RM et al. Visual outcome and risk factors for light perception and no light perception vision after vitrectomy for diabetic retinopathy. Am J Ophthalmol 2005; 140 (2): 231–235. ArticlePubMedGoogle Scholar Smiddy WE, Feuer W, Irvine WD, Flynn Jr HW, Blankenship GW . Vitrectomy for complications of proliferative diabetic retinopathy. Functional outcomes. Ophthalmology 1995; 102 (11): 1688–1695. ArticleCASPubMedGoogle Scholar Yorston D, Wickham L, Benson S, Bunce C, Sheard R, Charteris D . Predictive clinical features and outcomes of vitrectomy for proliferative diabetic retinopathy. Br J Ophthalmol 2008; 92 (3): 365–368. ArticleCASPubMedGoogle Scholar Thompson JT, de BS, Michels RG, Rice TA, Glaser BM . Results of vitrectomy for proliferative diabetic retinopathy. Ophthalmology 1986; 93 (12): 1571–1574. ArticleCASPubMedGoogle Scholar DRVS. Early vitrectomy for severe proliferative diabetic retinopathy in eyes with useful vision. Results of a randomized trial—Diabetic Retinopathy Vitrectomy Study Report 3. The Diabetic Retinopathy Vitrectomy Study Research Group. Ophthalmology 1988; 95 (10): 1307–1320. ArticleGoogle Scholar Rizzo S, Genovesi-Ebert F, di BE, Vento A, Miniaci S, Williams G . Injection of intravitreal bevacizumab (Avastin) as a preoperative adjunct before vitrectomy surgery in the treatment of severe proliferative diabetic retinopathy (PDR). Graefes Arch Clin Exp Ophthalmol 2008; 246 (6): 837–842. ArticleCASPubMedGoogle Scholar da RL, Ribeiro JA, Costa RA, Barbosa JC, Scott IU, de Figueiredo-Pontes LL et al. Intraoperative bleeding during vitrectomy for diabetic tractional retinal detachment with versus without preoperative intravitreal bevacizumab (IBeTra study). Br J Ophthalmol 2009; 93 (5): 688–691. ArticleGoogle Scholar Steel DH, Connor A, Habib MS, Owen R . Entry site treatment to prevent late recurrent postoperative vitreous cavity haemorrhage after vitrectomy for proliferative diabetic retinopathy. Br J Ophthalmol 2010; 94 (9): 1219–1225. ArticleCASPubMedGoogle Scholar Bandello F, Battaglia PM, Lanzetta P, Loewenstein A, Massin P, Menchini F et al. Diabetic macular edema. Dev Ophthalmol 2010; 47: 73–110. ArticlePubMedGoogle Scholar Kook D, Wolf A, Kreutzer T, Neubauer A, Strauss R, Ulbig M et al. Long-term effect of intravitreal bevacizumab (avastin) in patients with chronic diffuse diabetic macular edema. Retina 2008; 28 (8): 1053–1060. ArticlePubMedGoogle Scholar Soheilian M, Ramezani A, Obudi A, Bijanzadeh B, Salehipour M, Yaseri M et al. Randomized trial of intravitreal bevacizumab alone or combined with triamcinolone versus macular photocoagulation in diabetic macular edema. Ophthalmology 2009; 116 (6): 1142–1150. ArticlePubMedGoogle Scholar Stefansson E . The therapeutic effects of retinal laser treatment and vitrectomy. A theory based on oxygen and vascular physiology. Acta Ophthalmol Scand 2001; 79 (5): 435–402. 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15124	https://pmc.ncbi.nlm.nih.gov/articles/PMC11353533/	Molecular-Scale Liquid Density Fluctuations and Cavity Thermodynamics - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. 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Learn more: PMC Disclaimer \| PMC Copyright Notice Entropy (Basel) . 2024 Jul 24;26(8):620. doi: 10.3390/e26080620 Search in PMC Search in PubMed View in NLM Catalog Add to search Molecular-Scale Liquid Density Fluctuations and Cavity Thermodynamics Attila Tortorella Attila Tortorella 1 Scuola Superiore Meridionale, Via Mezzocannone, 4, 80138 Naples, Italy; attila.tortorella@unina.it 2 Department of Chemical Sciences, University of Naples Federico II, Via Cintia, 4, 80126 Naples, Italy Investigation, Data curation, Writing – original draft Find articles by Attila Tortorella 1,2, Giuseppe Graziano Giuseppe Graziano 3 Department of Science and Technology, University of Sannio, Via Francesco de Sanctis, snc, 82100 Benevento, Italy Conceptualization, Writing – original draft, Writing – review & editing, Project administration Find articles by Giuseppe Graziano 3, Editor: Tomaž Urbić Author information Article notes Copyright and License information 1 Scuola Superiore Meridionale, Via Mezzocannone, 4, 80138 Naples, Italy; attila.tortorella@unina.it 2 Department of Chemical Sciences, University of Naples Federico II, Via Cintia, 4, 80126 Naples, Italy 3 Department of Science and Technology, University of Sannio, Via Francesco de Sanctis, snc, 82100 Benevento, Italy Correspondence: graziano@unisannio.it; Tel.: +39-0824-305133 Roles Attila Tortorella: Investigation, Data curation, Writing – original draft Giuseppe Graziano: Conceptualization, Writing – original draft, Writing – review & editing, Project administration Tomaž Urbić: Academic Editor Received 2024 Jun 19; Revised 2024 Jul 17; Accepted 2024 Jul 23; Collection date 2024 Aug. © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( PMC Copyright notice PMCID: PMC11353533 PMID: 39202090 Abstract Equilibrium density fluctuations at the molecular level produce cavities in a liquid and can be analyzed to shed light on the statistics of the number of molecules occupying observation volumes of increasing radius. An information theory approach led to the conclusion that these probabilities should follow a Gaussian distribution. Computer simulations confirmed this prediction across various liquid models if the size of the observation volume is not large. The reversible work required to create a cavity and the chance of finding no molecules in a fixed observation volume are directly correlated. The Gaussian formula for the latter probability is scrutinized to derive the changes in enthalpy and entropy, which arise from the cavity creation. The reversible work of cavity creation has a purely entropic origin as a consequence of the solvent-excluded volume effect produced by the inaccessibility of a region of the configurational space. The consequent structural reorganization leads to a perfect compensation of enthalpy and entropy changes. Such results are coherent with those obtained from Lee in his direct statistical mechanical study. Keywords: density fluctuations, maximum entropy principle, Gaussian distribution, cavity distribution, solvent-excluded volume effect 1. Introduction A theoretical analysis of solvation (i.e., the transfer of a solute molecule from a fixed position in the gas phase to a fixed position in the liquid phase, according to the so-called Ben–Naim standard ) indicates the need to take into account the cavity creation process in the solution [2,3,4,5,6,7]. The need descends from the recognition that, since a liquid is a condensed state of the matter, a suitable space (i.e., a cavity) must be created, at a fixed position, to host the solute molecule. This need is well understood by theoreticians but not so well by experimentalists. The latter claim that any liquid possesses a lot of void space, around 50% of the total volume, and so cavity creation should be unnecessary. However, the void volume in a liquid is divided into many tiny pieces whose dimensions are not able to host real solute molecules [8,9,10,11,12] (for instance, the average dimensions of such small pieces depend on the diameter of liquid molecules, which can be understood by thinking of the voids left in a box filled by tennis balls or by ping-pong balls). The process of cavity creation can be examined through theoretical approaches or computer simulations using appropriate models for different liquids. The reversible work associated with cavity creation is a positive and large quantity for every liquid system. [2,3,4,5,6,7]. Moreover, it is accepted that the cavity creation reversible work possesses its largest value in water compared to other liquid systems [13,14,15], which is also the reason for the low solubility of nonpolar molecules in water [2,3,4,5,6,7]. Cavity creation was analyzed by means of statistical mechanics 40 years ago by Lee , and there were two findings. (a) The reversible work of cavity creation has an entropic origin, since cavity creation reduces the number of configurations available to liquid molecules (i.e., only the liquid configurations in which the described cavity is present can be chosen). Thus, the statistical ensemble size is reduced, leading to an entropy decrease in all liquids, which can be described as a solvent-excluded volume effect. (b) There is a cavity enthalpy change coming from a structural reorganization of liquid molecules (cavity creation is a perturbation that pushes the liquid molecules to assume positions which render possible the cavity existence). This structural reorganization differs from the solvent-excluded volume effect and produces also an entropy change that exactly compensates the cavity enthalpy change [16,17]. The creation of cavities is only driven by entropy, which is a fundamental characteristic that applies to all cavities formed by molecular-scale equilibrium density fluctuations. A fundamental theorem in statistical mechanics relates the reversible work of creating a cavity to the logarithm of the probability, P(0; v), of finding no liquid molecules within the volume of the desired cavity . In order to exploit this connection, P(0; v) must be known. Pratt and colleagues [19,20,21] devised an elegant information theory approach to arrive at P(0; v) by determining the probabilities of finding the centers of n molecules P(n; v) inside a randomly positioned volume v, which corresponds to the solvent-excluded volume of the specified cavity (i.e., a spherical cavity has a van der Waals radius r c and a solvent-excluded radius R c = r c + r s, where r s is the radius which describes the solvent as spheres; in other words, r c is the radius of the spherical volume where no part of the solvent molecules can be found, while R c is the radius of a sphere where no center of the solvent molecules can be found ). Using a flat default model in an approach based on the maximum entropy principle resulted in a discrete Gaussian distribution for the P(n; v) probabilities . This theoretical result is in line with the fluctuation theory in statistical mechanics where a Gaussian approximation holds for the fluctuation in the number of particles in the grand-canonical ensemble . Moreover, it has been supported by computer simulations of liquid models (i.e., hard sphere fluids , Lennard–Jones liquids , n-hexane , dimethyl sulfoxide , and several models of water [19,27,28,29]). The P(n; v) probabilities are well described by Gaussian distributions for not so large observation volumes (i.e., when the ratio of observation volume radius to liquid molecule radius is smaller than two, the Gaussian distribution holds regardless of the nature of the energetic interactions among liquid molecules). For instance, it has been shown that the so-called monoatomic water model is characterized by equilibrium density fluctuations that follow a Gaussian distribution up to a cavity radius R c ≈ 4 Å . It should be clear that there is no compelling reason to expect that the P(n; v) probabilities should obey a Gaussian distribution. Indeed, by increasing the radius of the observation volume in water models, there are large deviations from Gaussian values in the low-number tail of the distribution [27,28,29,31,32,33]. Moreover, a different distribution, called a binomial cell model, has been proposed to analyze molecular-scale density fluctuations in water models . Notwithstanding the studies published on this matter, the basic relationships between the probability distribution of number density fluctuations and cavity thermodynamics have not yet spelled out in detail except for the analysis by one of us . In this study, we want to demonstrate that the entropic basis of the reversible work of cavity creation directly emerges by the P(0; v) formula provided by the Gaussian distribution. 2. Theoretical Foundation A liquid possesses a huge ensemble of molecular configurations, and a statistical mechanical description is unavoidable. Assuming that X is a multidimensional vector accounting for the coordinates of every spherical molecule which is present in the liquid, the probability density function associated with a specific liquid configuration, in the NPT ensemble, is ρ(X) = exp[−H(X)/k T]/∫ exp[−H(X)/k T]dX(1) where H(X) = E(X) + P·V(X) is the enthalpy of the configuration X, E(X) represents the total interaction energy of liquid molecules in the configuration X, V(X) is the volume of the configuration X, P is the pressure of the liquid, k is the Boltzmann constant, and the denominator is the isobaric–isothermal configurational partition function. The probability of cavity occurrence in the liquid is the chance of finding no molecular centers in the solvent-excluded volume, v, of the cavity, or the chance that the centers of all the N spherical molecules of the liquid are located outside the solvent-excluded volume, v, of the cavity. This probability is obtained by integrating ρ(X) over all the configurations having the centers of all the N spherical molecules in the volume − v [2,35]. In this respect, it is important to note that (a) the position of the cavity must be fixed but can be arbitrarily located within the liquid volume since the liquid density is uniform at equilibrium; (b) although the total volume is not strictly constant in the NPT ensemble, it is reasonable to assume that the total volume of a macroscopic system will always be close to the ensemble average value ⟨V⟩. Thus, one has P(0; v) = P(N; − v) = ∫ρ(X) dX − v(2) where the integration domain has a clarified meaning. The calculation of P(0; v) corresponds to picking out only a very small fraction of the total liquid configurations, i.e., the ones having the cavity of the requested solvent-excluded volume. This selection strongly reduces the amount of molecular configurations available to the liquid and leads to a liquid entropy loss (i.e., entropy is an extensive thermodynamic function, and its magnitude depends on the size of the statistical ensemble [16,18,23]). This entropy loss holds for any liquid with no regard for the interactions between the liquid molecules. Cavity creation in a determined position within a liquid, keeping fixed NPT, leads to an increment of the average volume of the system by an extent equal to the van der Waals volume of the cavity, v vdW. Nevertheless, the presence of a void sphere of solvent-excluded volume v implies that the spherical shell given by the difference (v − v vdW) becomes inaccessible to the centers of liquid molecules. This solvent-excluded volume effect is a constraint for every molecule of the liquid whose centers, during their continuous translations, cannot enter the cavity solvent-excluded volume if the cavity must exist (see Figure 1). The inaccessible shell may be approximated by the solvent accessible surface area of the cavity in all the solvents. It is interesting to note that a geometric entropy, linearly proportional to the superficial area of the solvent-excluded volume, emerged in a theoretical approach based on a density field theory . Figure 1. Open in a new tab The cavity creation (i.e., the inner circle), at constant NPT, increases the volume of the system by an amount which corresponds to the van der Waals cavity volume. As a consequence, a spherical shell corresponding to the difference between the solvent-excluded volume of the cavity and its van der Waals volume (i.e., the space between the outer circle and the inner one) becomes inaccessible to the center of liquid molecules (the filled blue circle represents one liquid molecule) if the cavity is to exist. This geometric effect comes from the solvent-excluded volume associated with cavity creation. As underscored by Tolman , there is an exact statistical mechanical relationship between the occurrence probability of a constrained configuration of a thermodynamic system and the reversible work to produce that constrained configuration: P(0; v) = exp−W(0; v)/k T where W(0; v) is the reversible work (i.e., the Gibbs free energy change) to create a cavity of solvent-excluded volume equal to v, W(0; v) = ΔG c(v; R c), where R c is the solvent-excluded cavity radius. It is important to underscore that at 300 K and 1 atm, Equation (3) implies that ΔG c = 40.0 kJ mol−1 corresponds to P(0; v) = 1.1 10−7, and ΔG c = 60.0 kJ mol−1 corresponds to P(0; v) = 2.0 10−9, regardless of the liquid. These numbers highlight how large the decrease in available liquid configurations caused by cavity creation is. Now, the assumption that equilibrium density fluctuations at a molecular level follow a Gaussian distribution can be scrutinized to shed further light on the entropy loss associated with cavity creation. 3. Gaussian Fluctuations In compliance with the results of computer simulations of various liquids [19,20,21,24,25,26,27,28,29,31], the probability of finding the centers of exactly n molecules within an observation volume v, when the liquid number density is ρ ≡ N Av/v m (i.e., N Av is the Avogadro’s number and v m is the molar volume of the liquid), and pressure and temperature are held constant, is well described by a Gaussian distribution if the observation volume is not large (see above). Therefore, one has P(n; v) = (2π·σ n 2)−1/2 · exp(−δ n 2/2σ n 2)(4) where δ n = n − <n>, <n> is the average number of molecular centers in the volume v, <n> = ρ·v, and σ n 2 = <δ n 2> = <n 2> − <n>2 is the variance of the Gaussian distribution, i.e., the mean square fluctuation in the number of molecular centers inside the volume v. It is important to recognize that the first two moments of the Gaussian distribution are related to the number density and the radial distribution function of the liquid, respectively, which are quantities that are experimentally measurable [19,20,21]. Since we are looking for the probability of finding a cavity in the liquid, we need the probability P(0; v) of finding no molecular centers in the volume v: P(0; v) = (2π·σ n 2)−1/2 · exp(−<n>2/2σ n 2) = (2π·σ n 2)−1/2 · exp(−ρ 2 v 2/2σ n 2)(5) The probability of cavity occurrence is related to equilibrium density fluctuations on a molecular level, underscoring that the creation of a cavity is a special process, which only depends on the properties of the pure liquid. Introducing Equation (5) into Equation (3), one obtains the following: ΔG c(v; R c) = (k T/2)·ln(2π·σ n 2) + (k T· ρ 2 v 2/2σ n 2)(6) The σ n 2 value depends on the v size and can solely be determined by means of computer simulations on a molecularly detailed model of the liquid of interest [19,20,21]. According to the values reported by Sulimov and co-workers in the case of the TIP4P water model , for a cavity whose solvent-excluded volume is suitable to host methane, R c = 3.3 Å, <n> = 5.11, σ n 2 = 1.39 , and using Equation (6), ΔG c = 26.1 kJ mol−1 at 300 K; for a cavity whose solvent-excluded volume is suitable to host neopentane, R c = 4.4 Å, <n> = 11.77, σ n 2 = 2.62 , and using Equation (6), ΔG c = 69.4 kJ mol−1 at 300 K . These ΔG c values are in line with those calculated by direct computer simulations, as it can readily be controlled upon looking at Table 3 in . Moreover, it is easy to verify that on increasing the cavity solvent-excluded volume, the logarithmic term in Equation (6) becomes smaller and smaller in comparison to the other one. So it is possible to state that ΔG c is inversely proportional to the variance of the Gaussian distribution, ΔG c ∝ 1/σ n 2. We do not want to perform calculations with Equation (6) but rather to deepen its statistical and thermodynamic features and consequences. The joint probability of having both a cavity of solvent-excluded volume v 1 and a cavity of solvent-excluded volume v 2 can be given by a Gaussian distribution of solvent-excluded volume v 1 + v 2. The latter Gaussian distribution, however, is not the product of the two Gaussian functions describing the probability of zero occupancy in v 1 and zero occupancy in v 2: P(0; v 1 + v 2) ≠ P(0; v 1)·P(0; v 2)(7) The two events do not appear to be independent of each other because the zero occupancy of v 1 depends on the occupancy number of v 2, since the number of liquid molecules is fixed in the system. This feature of the Gaussian functions highlights an interesting physical point . When the volume of interest corresponds to the molar volume of the liquid, the variance of the Gaussian distribution is proportional to the isothermal compressibility, σ n 2 ∝ β T , and this implies that ΔG c ∝ 1/β T. The reversible work of cavity creation, measuring the entropy loss due to the solvent-excluded volume effect, is inversely proportional to the isothermal compressibility of the liquid. This relationship was originally obtained by Pratt and colleagues [19,20,21]. Water has the smallest β T value among all common liquids (i.e., at 25 °C, β T(in atm−1·10 5) = 4.58 for water, 9.80 for benzene, 11.55 for c-hexane, 16.27 for n-hexane, 10.81 for carbon tetrachloride, 14.79 for methanol, and 10.26 for ethanol ), and, in fact, it has the largest ΔG c value for a given van der Waals cavity radius among all common liquids, as demonstrated by computer simulation results [13,14,15]. The isothermal compressibility is a macroscopic thermodynamic quantity, and a closer scrutiny is necessary to single out the microscopic difference between water and the other liquids. In compliance with statistical mechanics, β T measures the fluctuations in the liquid number density in the grand canonical ensemble, β T = v m·σ n 2/<n>2·k T . Water has the lowest β T value among all common liquids, since the molar volume of water is the smallest among those of all common liquids: at 25 °C and 1 atm, v m(in cm 3 mol−1) = 18.07 for water , 89.41 for benzene, 108.75 for c-hexane, 131.62 for n-hexane, 97.09 for carbon tetrachloride, 40.73 for methanol, and 58.68 for ethanol . This fact stems from the effective size of liquid molecules. The effective size of water molecules is the smallest among those of all common liquids; at 25 °C, the effective hard sphere diameter is 2.80 Å for water , 5.26 Å for benzene, 5.63 Å for c-hexane, 5.92 Å for n-hexane, 5.37 Å for carbon tetrachloride , 3.83 Å for methanol, and 4.44 for ethanol Å . The effective hard sphere diameter of water molecules is even smaller than their van der Waals diameter as a consequence of the bunching up effect due to the strength of H bonds . Using fundamental relationships of equilibrium thermodynamics to Equation (6) allows the derivation of the changes of both enthalpy and entropy which arise from cavity creation: ΔH c = −T 2[∂(ΔG c/T)/∂ T]P = −(k T 2/2)·{∂[ln(2π·σ n 2) + (ρ 2 v 2/σ n 2)]/∂ T}P = = −(k T 2/2σ n 2)·{[1 − ((ρ 2 v 2/σ n 2)]·(∂ σ n 2/∂ T)P + 2ρv 2·(∂ ρ/∂ T)P}(8) and ΔS c = −(∂ ΔG c/∂ T)P = −(k/2)·ln(2π·σ n 2) − (k ρ 2 v 2/2σ n 2) + −(k T/2σ n 2)·{[1 − ((ρ 2 v 2/σ n 2)]·(∂ σ n 2/∂ T)P + 2ρv 2·(∂ ρ/∂ T)P}(9) In performing the derivatives, the σ n 2 quantity has been considered a single variable function of temperature, and the v quantity has been considered temperature independent . It is worth noting that these “approximations” affect in the same manner both ΔH c and ΔS c and do not alter the analysis below. Equations (8) and (9) illustrate the following points: (a) the enthalpy change associated with cavity formation is completely offset by a corresponding entropy change, resulting in no net enthalpic contribution to the reversible work of cavity creation; (b) the cavity entropy change includes an additional term, expressed as −ΔG c/T, which quantifies the solvent-excluded volume effect related to cavity formation in a liquid. This term represents the entropy reduction due to the decreased size of the liquid’s statistical ensemble when selecting configurations that include the desired cavity. The above sentences may appear a circular argument unless Equation (8) is identified as the enthalpy change due to cavity creation in an independent manner. In water, the σ n 2 quantity depends little on temperature because the isothermal compressibility of water is nearly constant in the temperature range 0–100 °C . Thus, the quantity (∂ σ n 2/∂ T)P should be negligible, and Equation (8) can be rearranged to ΔH c ≅ −(k T 2 ρv 2/σ n 2)·(∂ ρ/∂ T)P = k T 2 ρ 2 v 2 α P/σ n 2(10) where α P = −(1/ρ)·(∂ ρ/∂ T)P is the isobaric thermal expansion coefficient of the liquid. According to Equation (10), ΔH c ∝ α P, agreeing with the equation originally derived by Pierotti within the framework of scaled particle theory . In the other liquids, the quantity σ n 2 depends on temperature, but the factor [1 − (ρ 2 v 2/σ n 2)] occurring in Equation (8) should be small; thus, Equation (10) is going to be a not-bad approximation for every liquid. By assuming that v = v m and using the statistical mechanical definition of the isothermal compressibility in Equation (10), the latter becomes ΔH c = α P·T·v m/β T(11) Both α P and β T are thermodynamic response functions , and it is reliable to associate them with a process such as cavity creation that implies a structural reorganization of the pure liquid. In the configurations possessing the desired cavity, liquid molecules must have special spatial distributions that produce changes in both enthalpy and entropy. This structural reorganization can be described by a proper function of α P and β T of the pure liquid, because there is no solute molecule inserted in the liquid when the cavity is created. On the basis of Equation (11), it is correct to state that the structural reorganization (which is distinct from the solvent-excluded volume effect) associated with cavity creation is characterized by a complete enthalpy–entropy compensation. A qualitative picture of the various thermodynamic quantities associated with cavity creation in water, at room temperature and 1 atm, is shown in Figure 2. Figure 2. Open in a new tab A qualitative bar plot of the various thermodynamic quantities associated with cavity creation in water at room temperature and 1 atm. The cavity entropy change has been divided in the solvent-excluded volume contribution, labeled x, and the liquid structural reorganization contribution, labeled r. The latter exactly compensates the enthalpy term. 4. Conclusions Equilibrium density fluctuations at a molecular scale follow a Gaussian distribution in several liquids when the observation volumes are not large. This makes it possible to arrive at an analytical relationship for the probability of finding no molecules in a solvent-excluded volume corresponding to the desired cavity [19,20,21]. A careful analysis of this relationship leads to formulas for the changes in Gibbs free energy, enthalpy and entropy which occur upon cavity creation. These formulas demonstrate that (a) the Gibbs free energy required for the creation of a cavity is purely entropic due to the reduction in the dimensions of the statistical ensemble caused by the solvent-excluded volume effect; (b) there is a complete compensation between the enthalpy and entropy changes stemming from the rearrangement of solvent molecules around the cavity. This thermodynamic scenario matches the one determined by Lee with a general statistical mechanical approach. Author Contributions Conceptualization, G.G.; formal analysis and investigation, A.T. and G.G.; writing—original draft preparation, A.T. and G.G.; writing—review and editing, A.T. and G.G. All authors have read and agreed to the published version of the manuscript. Data Availability Statement The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. Conflicts of Interest The authors declare no conflicts of interest. Funding Statement This research was supported by internal funds of the University of Sannio; it has not received external funding. Footnotes Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. References 1.Ben-Naim A. Solvation Thermodynamics. Springer; Boston, MA, USA: 1987. [Google Scholar] 2.Ben-Naim A. Water and Aqueous Solutions. Springer; Boston, MA, USA: 1974. [Google Scholar] 3.Lee B. The physical origin of the low solubility of nonpolar solutes in water. Biopolymers. 1985;24:813. doi: 10.1002/bip.360240507. [DOI] [PubMed] [Google Scholar] 4.Tomasi J., Persico M. Molecular Interactions in Solution: An Overview of Methods Based on Continuous Distributions of the Solvent. Chem. Rev. 1994;94:2027–2094. doi: 10.1021/cr00031a013. [DOI] [Google Scholar] 5.Gallicchio E., Kubo M.M., Levy R.M. Enthalpy−Entropy and Cavity Decomposition of Alkane Hydration Free Energies: Numerical Results and Implications for Theories of Hydrophobic Solvation. J. Phys. Chem. B. 2000;104:6271–6285. doi: 10.1021/jp0006274. [DOI] [Google Scholar] 6.Graziano G. Shedding Light on the Hydrophobicity Puzzle. Pure Appl. Chem. 2016;88:177–188. doi: 10.1515/pac-2015-1003. [DOI] [Google Scholar] 7.Silverstein T.P. The Hydrophobic Effect: Is Water Afraid, or Just Not That Interested? ChemTexts. 2020;6:26. doi: 10.1007/s40828-020-00117-8. [DOI] [Google Scholar] 8.Pohorille A., Pratt L.R. Cavities in Molecular Liquids and the Theory of Hydrophobic Solubilities. J. Am. Chem. Soc. 1990;112:5066–5074. doi: 10.1021/ja00169a011. [DOI] [PubMed] [Google Scholar] 9.Pratt L.R., Pohorille A. Theory of Hydrophobicity: Transient Cavities in Molecular Liquids. Proc. Natl. Acad. Sci. USA. 1992;89:2995–2999. doi: 10.1073/pnas.89.7.2995. [DOI] [PMC free article] [PubMed] [Google Scholar] 10.Tang K.E.S., Bloomfield V.A. Excluded Volume in Solvation: Sensitivity of Scaled-Particle Theory to Solvent Size and Density. Biophys. J. 2000;79:2222–2234. doi: 10.1016/S0006-3495(00)76470-8. [DOI] [PMC free article] [PubMed] [Google Scholar] 11.Stone M.T., In’t Veld P.J., Lu Y., Sanchez I.C. Hydrophobic/Hydrophilic Solvation: Inferences from Monte Carlo Simulations and Experiments. Mol. Phys. 2002;100:2773–2792. doi: 10.1080/00268970210139912. [DOI] [Google Scholar] 12.Graziano G. Water: Cavity Size Distribution and Hydrogen Bonds. Chem. Phys. Lett. 2004;396:226–231. doi: 10.1016/j.cplett.2004.07.126. [DOI] [Google Scholar] 13.Tomás-Oliveira I., Wodak S.J. Thermodynamics of Cavity Formation in Water and N-Hexane Using the Widom Particle Insertion Method. J. Chem. Phys. 1999;111:8576–8587. doi: 10.1063/1.480199. [DOI] [Google Scholar] 14.Graziano G. Comment on “Reevaluation in Interpretation of Hydrophobicity by Scaled Particle Theory”. J. Phys. Chem. B. 2002;106:7713–7716. doi: 10.1021/jp014558k. [DOI] [Google Scholar] 15.Sedov I., Magsumov T. The Gibbs Free Energy of Cavity Formation in a Diverse Set of Solvents. J. Chem. Phys. 2020;153:134501. doi: 10.1063/5.0021959. [DOI] [PubMed] [Google Scholar] 16.Lee B. A Procedure for Calculating Thermodynamic Functions of Cavity Formation from the Pure Solvent Simulation Data. J. Chem. Phys. 1985;83:2421–2425. doi: 10.1063/1.449287. [DOI] [Google Scholar] 17.Graziano G. Scaled Particle Theory Study of the Length Scale Dependence of Cavity Thermodynamics in Different Liquids. J. Phys. Chem. B. 2006;110:11421–11426. doi: 10.1021/jp0571269. [DOI] [PubMed] [Google Scholar] 18.Tolman R.C. The Principles of Statistical Mechanics. Oxford University Press; London, UK: 1938. [Google Scholar] 19.Hummer G., Garde S., García A.E., Pohorille A., Pratt L.R. An Information Theory Model of Hydrophobic Interactions. Proc. Natl. Acad. Sci. USA. 1996;93:8951–8955. doi: 10.1073/pnas.93.17.8951. [DOI] [PMC free article] [PubMed] [Google Scholar] 20.Hummer G., Garde S., García A.E., Paulaitis M.E., Pratt L.R. Hydrophobic Effects on a Molecular Scale. J. Phys. Chem. B. 1998;102:10469–10482. doi: 10.1021/jp982873+. [DOI] [Google Scholar] 21.Pratt L.R. Molecular Theory of Hydrophobic Effects: “She Is Too Mean to Have Her Name Repeated”. Annu. Rev. Phys. Chem. 2002;53:409–436. doi: 10.1146/annurev.physchem.53.090401.093500. [DOI] [PubMed] [Google Scholar] 22.Jaynes E.T. Information Theory and Statistical Mechanics. Phys. Rev. 1957;106:620–630. doi: 10.1103/PhysRev.106.620. [DOI] [Google Scholar] 23.McQuarrie D. Statistical Mechanics. Harper & Row; New York, NY, USA: 1976. [Google Scholar] 24.Crooks G.E., Chandler D. Gaussian Statistics of the Hard-Sphere Fluid. Phys. Rev. E. 1997;56:4217–4221. doi: 10.1103/PhysRevE.56.4217. [DOI] [Google Scholar] 25.Huang D.M., Chandler D. Cavity Formation and the Drying Transition in the Lennard-Jones Fluid. Phys. Rev. E. 2000;61:1501–1506. doi: 10.1103/PhysRevE.61.1501. [DOI] [PubMed] [Google Scholar] 26.Peter C., van der Vegt N.F.A. Solvent Reorganization Contributions in Solute Transfer Thermodynamics: Inferences from the Solvent Equation of State. J. Phys. Chem. B. 2007;111:7836–7842. doi: 10.1021/jp0712708. [DOI] [PubMed] [Google Scholar] 27.Gomez M.A., Pratt L.R., Hummer G., Garde S. Molecular Realism in Default Models for Information Theories of Hydrophobic Effects. J. Phys. Chem. B. 1999;103:3520–3523. doi: 10.1021/jp990337r. [DOI] [Google Scholar] 28.Head-Gordon T., Lynden-Bell R.M., Dowdle J.R., Rossky P.J. Predicting Cavity Formation Free Energy: How Far Is the Gaussian Approximation Valid? Phys. Chem. Chem. Phys. 2012;14:6996–7004. doi: 10.1039/c2cp00046f. [DOI] [PubMed] [Google Scholar] 29.Cerdeiriña C.A., González-Salgado D. Temperature, Pressure, and Length-Scale Dependence of Solvation in Water-like Solvents. I. Small Solvophobic Solutes. J. Phys. Chem. B. 2021;125:297–306. doi: 10.1021/acs.jpcb.0c09569. [DOI] [PubMed] [Google Scholar] 30.Molinero V., Moore E.B. Water Modeled As an Intermediate Element between Carbon and Silicon. J. Phys. Chem. B. 2009;113:4008–4016. doi: 10.1021/jp805227c. [DOI] [PubMed] [Google Scholar] 31.Ashbaugh H.S. Gaussian and Non-Gaussian Solvent Density Fluctuations within Solute Cavities in a Water-like Solvent. J. Chem. Theory Comput. 2024;20:1505–1518. doi: 10.1021/acs.jctc.3c00387. [DOI] [PMC free article] [PubMed] [Google Scholar] 32.Patel A.J., Varilly P., Chandler D., Garde S. Quantifying Density Fluctuations in Volumes of All Shapes and Sizes Using Indirect Umbrella Sampling. J. Stat. Phys. 2011;145:265–275. doi: 10.1007/s10955-011-0269-9. [DOI] [PMC free article] [PubMed] [Google Scholar] 33.Rego N.B., Patel A.J. Understanding Hydrophobic Effects: Insights from Water Density Fluctuations. Annu. Rev. Condens. Matter Phys. 2022;13:303–324. doi: 10.1146/annurev-conmatphys-040220-045516. [DOI] [Google Scholar] 34.Alexandrovsky V.V., Basilevsky M.V., Leontyev I.V., Mazo M.A., Sulimov V.B. The Binomial Cell Model of Hydrophobic Solvation. J. Phys. Chem. B. 2004;108:15830–15840. doi: 10.1021/jp031189e. [DOI] [Google Scholar] 35.Graziano G. Cavity Thermodynamics in the Gaussian Model of Particle Density Fluctuations. Chem. Phys. Lett. 2007;446:313–316. doi: 10.1016/j.cplett.2007.08.063. [DOI] [Google Scholar] 36.Callaway D.J.E. Surface Tension, Hydrophobicity, and Black Holes: The Entropic Connection. Phys. Rev. E. 1996;53:3738–3744. doi: 10.1103/PhysRevE.53.3738. [DOI] [PubMed] [Google Scholar] 37.Kell G.S. Density, Thermal Expansivity, and Compressibility of Liquid Water from 0.Deg. to 150.Deg. Correlations and Tables for Atmospheric Pressure and Saturation Reviewed and Expressed on 1968 Temperature Scale. J. Chem. Eng. Data. 1975;20:97–105. doi: 10.1021/je60064a005. [DOI] [Google Scholar] 38.Wilhelm E., Battino R. Estimation of Lennard-Jones (6,12) Pair Potential Parameters from Gas Solubility Data. J. Chem. Phys. 1971;55:4012–4017. doi: 10.1063/1.1676694. [DOI] [Google Scholar] 39.Madan B., Lee B. Role of Hydrogen Bonds in Hydrophobicity: The Free Energy of Cavity Formation in Water Models with and without the Hydrogen Bonds. Biophys. Chem. 1994;51:279–289. doi: 10.1016/0301-4622(94)00049-2. [DOI] [PubMed] [Google Scholar] 40.Lee B. Enthalpy-Entropy Compensation in the Thermodynamics of Hydrophobicity. Biophys. Chem. 1994;51:271–278. doi: 10.1016/0301-4622(94)00048-4. [DOI] [PubMed] [Google Scholar] 41.Pierotti R.A. A Scaled Particle Theory of Aqueous and Nonaqueous Solutions. Chem. Rev. 1976;76:717–726. doi: 10.1021/cr60304a002. [DOI] [Google Scholar] 42.Reiss H. Advances in Chemical Physics. John Wiley & Sons, Ltd.; Hoboken, NJ, USA: 1965. Scaled Particle Methods in the Statistical Thermodynamics of Fluids; pp. 1–84. [Google Scholar] Associated Data This section collects any data citations, data availability statements, or supplementary materials included in this article. Data Availability Statement The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. Articles from Entropy are provided here courtesy of Multidisciplinary Digital Publishing Institute (MDPI) ACTIONS View on publisher site PDF (647.8 KB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Abstract 1. Introduction 2. Theoretical Foundation 3. Gaussian Fluctuations 4. 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15125	https://www3.dbu.edu/naugle/pdf/2302_handouts/parts_of_syllogism.pdf	1 -Philosophy 2302 Intro to Logic Dr. Naugle Distribution of Terms Parts of a Syllogismi Introduction: Every syllogism is made up of propositions and every proposition is made up of two terms: subject and predicate. These terms are related to each other by is/is not and are/are not. Thus there are four possible types of propositions. Now we want to talk about the two basic terms of categori-cal propositions under the headings of distributed and undistributed. I. Distribution of Terms A. Definition of the distribution of terms The distribution of terms is concerned with two basic points: (1) the classes designated by the subject and predicate terms (roses, redness); and (2) the extent to which these classes are occupied or distributed (all or only part). 1. Classes: reference is made in all four types of categorical propositions to various classes designated by the two primary terms, the subject and the predicate. What we need to know is whether the reference is to the whole of the class or only to part of the class. Distributed: If the reference is to the whole of the class, then the class is said to be distributed. A term is distributed when it refers to all the members of the class (fully occupied). Distribution can be designated by a stated or implied all. Undistributed: If the reference is only to part of the class, then the class is said to be undistributed. A term is undistributed when it refers to less than all the members of its class (not fully occupied). B. The relation of distributed subject and predicate terms to the quantity of propositions (universal and/or particular): Terms have distribution; propositions have quantity which itself depends on the distribution of the subject. First we look for the distribution of the subject-class and then seek the distribution of the - 2 -predicate-class. The distribution of terms follows a set, consistent pattern for the four types of categorical propositions (A, E, I, O). II. Distribution of Terms in the Four Types of Categorical Propositions A. Type A propositions: All S is P (universal affirmative) {S= Distributed, P = undistributed} The A proposition asserts that every member of the subject class is a member of (but not the whole of) the predicate class. Since ref-erence is made to every member of the subject class (All S…), the subject is said to be distributed. But is reference being made to every member of the predicate class? NO. For example, if you say: "All artists are eccentric." you are not saying that only artists are eccentric, nor are you saying that artists make up the whole class of eccentric people. You are only saying that if a person is an artist, he is a member of the class of eccentric people (which includes, but goes beyond artists; philosophers are eccentric too!). So, the predicate term of an A proposition is undistributed. In other words, the sum total of all artists (distributed!) is only a part of the class of eccentric people (undistributed). To demonstrate the undistributed nature of the predicate, this proposition cannot be converted to say: "All eccentric people are artists" since this would be jumping from a knowledge of some things (all artists who are eccentric) to a presumed knowledge of all things (all eccentrics are artists). Try another example: "All horses are four-legged animals." B. Type E propositions: No S is P (universal negative) {S = distributed, P = distributed} The E proposition's quantifier (No S…) makes reference in a neg-ative way to every member of the subject class. Thus it is universal. E propositions also state that not a single member of the S class is a member of the P class, and thus the reference is to the whole of the predicate class. This could be only if the whole of the P class were surveyed and no S were found. Therefore, the predicate of E propositions is distributed. For example, if you say: "No cats are dogs." - 3 -you would have to be aware of every member of the predicate class dogs to make sure there were no cats in that class. So not only is the subject in this case distributed, but so also is the predicate. Because both terms are distributed, this E proposition converts simply: "No dogs are cats." Like in math, the function of addition is transitive: "4 + 2 = 6 and 2 + 4 = 6." Or try another example: No republicans are pacifists. C. Type I propositions: Some S are P (particular affirmative) {S = undistributed, P = undistributed} The quantifier makes it clear that only some members of the sub-ject class are being referred to, so the subject is undistributed (Some S …). Therefore, the proposition as a whole is particular. But is the predicate class similarly undistributed? YES, because reference is being made to only some of the members of that class not the whole of it. For example, if you say: "Some men are wealthy." you are identifying only some members of the wealthy class who are members of the subject class (i.e., men). You are not con-cerned with the rest of the P class (the wealthy) who are of another kind (women who are wealthy). Hence, in I propositions, both the subject class and the predicate class are undistributed, and con-sequently such a proposition can be converted simply: "Some of the wealthy are men." D. Type O propositions: Some S is not P (particular negative) {S = undistributed, P = distributed} The quantifier "some" in type O propositions indicates that refer-ence is being made to only some of the subject class (Some S …). The subject term of the O propositions is therefore undistributed and the proposition as a whole is particular. Is the predicate class also undistributed? NO, it is distributed, because to say that Some S is not P, you have to know the sum total of the P class to make this assertion. For example, if you say: "Some registered voters are not property owners." - 4 -you have to know the sum total of property owners to assert that some registered voters do not belong or are not found anywhere in the class of property owners. If you deny that something is inside a certain circle (property owners), you have to deny that it can be found anywhere in that circle (you have to know the contents of the whole circle!). You have to refer to the whole circle, not just part of it. Hence, in type O propositions, the subject is always undistributed and the predicate is always distributed and for this reason, type O propositions cannot be converted. Try this one: "Some people are not happy." Another source explains it like this: "The particular negative (O) propositions asserts that at least one member of S is not a member of P. Since the other members of S may or may not be outside of P, it is clear that the statement "Some S are not P does not make a claim about every member of S, so S is undistributed. But, as may be seen from the diagram, the statement does assert that the entire P class is separated from this one member of S that is outside; that is, it does make a claim about every member of P. Thus, in the particular negative (O) propositions, P is distributed and S is undistributed. E. Summary 1. Universal subjects and negative predicates are distributed. Particular subjects and affirmative predicates are undistributed. Proposition Form Subject Term Predicate Term A (+) D U E (- ) D D I (+) U U O (-) U D i NB: This material is taken from several logic texts authored by N. Geisler, H. Kahane, and others. I make not claim to originality in this material.
15126	https://pubmed.ncbi.nlm.nih.gov/38038886/	LAMB2 gene: broad clinical spectrum in Pierson syndrome - 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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LAMB2 gene: broad clinical spectrum in Pierson syndrome Emre Leventoğlu1,Emine Dönmez2,Bahriye Uzun Kenan3,Burcu Yazıcıoğlu3,Bahar Büyükkaragöz3,Kibriya Fidan3,Sevcan A Bakkaloğlu3,Oğuz Söylemezoğlu3 Affiliations Expand Affiliations 1 Department of Pediatric Nephrology, Faculty of Medicine, Gazi University, Ankara, Turkey. dremrelevent@gmail.com. 2 Department of Pediatrics, Faculty of Medicine, Gazi University, Ankara, Turkey. 3 Department of Pediatric Nephrology, Faculty of Medicine, Gazi University, Ankara, Turkey. PMID: 38038886 PMCID: PMC11294298 DOI: 10.1007/s13730-023-00838-y Item in Clipboard Case Reports LAMB2 gene: broad clinical spectrum in Pierson syndrome Emre Leventoğlu et al. CEN Case Rep.2024 Aug. Show details Display options Display options Format CEN Case Rep Actions Search in PubMed Search in NLM Catalog Add to Search . 2024 Aug;13(4):258-263. doi: 10.1007/s13730-023-00838-y. Epub 2023 Dec 1. Authors Emre Leventoğlu1,Emine Dönmez2,Bahriye Uzun Kenan3,Burcu Yazıcıoğlu3,Bahar Büyükkaragöz3,Kibriya Fidan3,Sevcan A Bakkaloğlu3,Oğuz Söylemezoğlu3 Affiliations 1 Department of Pediatric Nephrology, Faculty of Medicine, Gazi University, Ankara, Turkey. dremrelevent@gmail.com. 2 Department of Pediatrics, Faculty of Medicine, Gazi University, Ankara, Turkey. 3 Department of Pediatric Nephrology, Faculty of Medicine, Gazi University, Ankara, Turkey. PMID: 38038886 PMCID: PMC11294298 DOI: 10.1007/s13730-023-00838-y Item in Clipboard Cite Display options Display options Format Abstract Pierson syndrome (PS) is a rare autosomal recessive disease, characterized by congenital nephrotic syndrome (CNS), and ocular and neurologic abnormalities. In affected cases, there is abnormal b-2 laminin which is compound of the several basement membranes caused by inherited mutations in the LAMB2 gene. Although patients have mutations in the same gene, the phenotype is highly variable. In this case series, the relationship between genotype and phenotype is emphasized, and information about the clinical follow-up of the patients is presented. Hereby, we report four pediatric cases with PS as a result of mutation in the LAMB2 gene. Clinical spectrum of LAMB2-associated disorders varies from mild-to-severe ocular, kidney, and neurologic involvement. Since genotype-phenotype correlation in PS has not been clearly demonstrated, we recommend that all patients with ophthalmic anomalies and glomerular proteinuria should be tested for LAMB2 mutations. Keywords: Congenital nephrotic syndrome; Ocular anomaly; Pediatrics. © 2023. The Author(s), under exclusive licence to Japanese Society of Nephrology. PubMed Disclaimer Conflict of interest statement The authors have no conflicts of interest to disclose. Similar articles A novel LAMB2 gene mutation associated with a severe phenotype in a neonate with Pierson syndrome.Zemrani B, Cachat F, Bonny O, Giannoni E, Durig J, Fellmann F, Chehade H.Zemrani B, et al.Eur J Med Res. 2016 Apr 30;21:19. doi: 10.1186/s40001-016-0215-z.Eur J Med Res. 2016.PMID: 27130041 Free PMC article. LAMB2 mutation with different phenotypes in China .Zhang H, Cui J, Wang F, Xiao H, Ding J, Yao Y.Zhang H, et al.Clin Nephrol. 2017 Jan;87 (2017)(1):33-38. doi: 10.5414/CN108979.Clin Nephrol. 2017.PMID: 27925579 A new mutation associated with Pierson syndrome.Kulali F, Calkavur S, Basaran C, Serdaroglu E, Kose M, Saka Guvenc M.Kulali F, et al.Arch Argent Pediatr. 2020 Jun;118(3):e288-e291. doi: 10.5546/aap.2020.eng.e288.Arch Argent Pediatr. 2020.PMID: 32470267 English, Spanish. Alport syndrome and Pierson syndrome: Diseases of the glomerular basement membrane.Funk SD, Lin MH, Miner JH.Funk SD, et al.Matrix Biol. 2018 Oct;71-72:250-261. doi: 10.1016/j.matbio.2018.04.008. Epub 2018 Apr 16.Matrix Biol. 2018.PMID: 29673759 Free PMC article.Review. Mutations in the human laminin beta2 (LAMB2) gene and the associated phenotypic spectrum.Matejas V, Hinkes B, Alkandari F, Al-Gazali L, Annexstad E, Aytac MB, Barrow M, Bláhová K, Bockenhauer D, Cheong HI, Maruniak-Chudek I, Cochat P, Dötsch J, Gajjar P, Hennekam RC, Janssen F, Kagan M, Kariminejad A, Kemper MJ, Koenig J, Kogan J, Kroes HY, Kuwertz-Bröking E, Lewanda AF, Medeira A, Muscheites J, Niaudet P, Pierson M, Saggar A, Seaver L, Suri M, Tsygin A, Wühl E, Zurowska A, Uebe S, Hildebrandt F, Antignac C, Zenker M.Matejas V, et al.Hum Mutat. 2010 Sep;31(9):992-1002. doi: 10.1002/humu.21304.Hum Mutat. 2010.PMID: 20556798 Free PMC article.Review. See all similar articles Cited by A systematic review of associations between the environment, DNA methylation, and cognition.Glover S, Illyuk J, Hill C, McGuinness B, McKnight AJ, Hunter RF.Glover S, et al.Environ Epigenet. 2024 Dec 16;11(1):dvae027. doi: 10.1093/eep/dvae027. eCollection 2025.Environ Epigenet. 2024.PMID: 39882510 Free PMC article.Review. References Matejas V, Hinkes B, Alkandari F, et al. Mutations in the human laminin β2 (LAMB2) gene and the associated phenotypic spectrum. Hum Mutat. 2010;31:992–1002. 10.1002/humu.21304 - DOI - PMC - PubMed Büyükkaragöz B, Bakkaloğlu SA, Özmen C, Ezgü FS. Pierson syndrome characterized by mild renal variant: a case report. Gazi Med J. 2021;32:461–3. Durbeej M. Laminins. Cell Tissue Res. 2010;339:259–68. 10.1007/s00441-009-0838-2 - DOI - PubMed Aydin B, Ipek MS, Ozaltin F, et al. A novel mutation of laminin β-2 gene in Pierson syndrome manifested with nephrotic syndrome in the early neonatal period. Genet Couns. 2013;24(1):41–7. - PubMed Zenker M, Aigner T, Wendler O, et al. Human laminin b-2 deficiency causes congenital nephrosis with mesangial sclerosis and distinct eye abnormalities. Hum Mol Genet. 2004;13:2625–32. 10.1093/hmg/ddh284 - DOI - PubMed Show all 18 references Publication types Case Reports Actions Search in PubMed Search in MeSH Add to Search MeSH terms Abnormalities, Multiple / diagnosis Actions Search in PubMed Search in MeSH Add to Search Abnormalities, Multiple / genetics Actions Search in PubMed Search in MeSH Add to Search Child Actions Search in PubMed Search in MeSH Add to Search Child, Preschool Actions Search in PubMed Search in MeSH Add to Search Eye Abnormalities / diagnosis Actions Search in PubMed Search in MeSH Add to Search Eye Abnormalities / genetics Actions Search in PubMed Search in MeSH Add to Search Eyelids / abnormalities Actions Search in PubMed Search in MeSH Add to Search Eyelids / pathology Actions Search in PubMed Search in MeSH Add to Search Female Actions Search in PubMed Search in MeSH Add to Search Genetic Association Studies / methods Actions Search in PubMed Search in MeSH Add to Search Humans Actions Search in PubMed Search in MeSH Add to Search Infant Actions Search in PubMed Search in MeSH Add to Search Infant, Newborn Actions Search in PubMed Search in MeSH Add to Search Laminin / genetics Actions Search in PubMed Search in MeSH Add to Search Male Actions Search in PubMed Search in MeSH Add to Search Mutation Actions Search in PubMed Search in MeSH Add to Search Myasthenic Syndromes, Congenital / diagnosis Actions Search in PubMed Search in MeSH Add to Search Myasthenic Syndromes, Congenital / genetics Actions Search in PubMed Search in MeSH Add to Search Nephrotic Syndrome / diagnosis Actions Search in PubMed Search in MeSH Add to Search Nephrotic Syndrome / genetics Actions Search in PubMed Search in MeSH Add to Search Phenotype Actions Search in PubMed Search in MeSH Add to Search Pupil Disorders / diagnosis Actions Search in PubMed Search in MeSH Add to Search Pupil Disorders / genetics Actions Search in PubMed Search in MeSH Add to Search Substances Laminin Actions Search in PubMed Search in MeSH Add to Search laminin beta2 Actions Search in PubMed Search in MeSH Add to Search Supplementary concepts Pierson syndrome Actions Search in PubMed Search in MeSH Add to Search Related information MedGen PubChem Compound (MeSH Keyword) [x] Cite Copy Download .nbib.nbib Format: Send To Clipboard Email Save My Bibliography Collections Citation Manager [x] NCBI Literature Resources MeSHPMCBookshelfDisclaimer The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). 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15127	https://www.learner.org/series/insights-into-algebra-1-teaching-for-learning-2/properties/lesson-plan-1-the-x-factor-trinomials-and-algebra-tiles/	Social Studies: Click here for tools on how to talk to teens about social media → Insights Into Algebra 1: Teaching for Learning Professional Development > Insights Into Algebra 1: Teaching for Learning > 5. Properties > 5.1 Lesson Plan 1: The X Factor – Trinomials and Algebra Tiles 6-8, 9-12 Properties Lesson Plan 1: The X Factor – Trinomials and Algebra Tiles Overview: This lesson will teach students to factor trinomial expressions of the form x2 + bx + c. Students will use algebra tiles to identify the binomial factors and the graphing calculator to verify the result. In addition, students will identify the x-intercepts and y-intercepts of each trinomial function and explore relationships between the trinomial x2 + bx + c and its factored form (x + m)(x + n). Time Allotment: One 50-minute class period Subject Matter: Factoring Learning Objectives: Students will be able to: Factor trinomials of the form x2 + bx + c into two binomial factors. Identify the relationships that exist between b, c, m, and n when x2 + bx + c is factored as (x + m)(x + n). Standards: Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000. NCTM Algebra Standard for Grades 6-8 NCTM Algebra Standard for Grades 9-12 Teacher Supplies Supplies: Teachers will need the following: Factoring Gift handout Students will need the following: A set of algebra tiles – click here for a printable page Graphing calculator Teachers Activities and Assignment Steps Introductory Activity: 1. Ask a student to define the word “factor.” Elicit that “factor,” when used as a verb, means to “to rewrite a number or expression as a product of two or more numbers or expressions.” 2. Have students, in pairs or small groups, talk about ways to factor 30. Students should come up with four ways: 1 x 30, 2 x 15, 3 x 10, and 5 x 6. Note that students might also include methods that break 30 into more than 2 factors, such as 5 x 3 x 2 or 10 x 3 x 1. 3. Explain that the factors of 30 are “numerical factors” and that during today’s lesson, students will look at algebraic factors. Learning Activities: 1. Give some examples of algebraic expressions that can be written in factored form: 3x2 – 6x = 3x (x – 2) a2 + 7a + 12 = (a + 4)(a + 3) 2.Use the distributive property to multiply the factors and obtain a product, and then have students verify that both of these equations are true. 3.Reinforce the definition of factoring by asking, “Which side is in factored form?” Students should conclude that the right side is in factored form, because factoring means to rewrite an expression as a product or as a multiplication problem. 4.In groups, have students use algebra tiles to write the first trinomial on Factoring Gift in factored form. That is, explain that you would like them to find two binomials that, when multiplied, give this trinomial: x2 + 6x + 8. Reinforce that the algebra tiles must be arranged to form a rectangle with no gaps. It’s important to note that students may arrange the tiles in any formation that results in a rectangle (for example: below, left). However, students must be able to identify the side length of the rectangle, which may be easier if the rectangle is arranged in a systematic manner (below, right). Students should understand that they can find the area of the rectangle by using the area formula, namely A = lw = (x + 4)(x + 2), or by adding the individual pieces inside. These pieces are x2 + x + x + x + x + x + x + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = x2+ 6x + 8. Because the two expressions measure the same area, they must be equivalent. 5.Once students have used the tiles to find the factored form (x + 2)(x + 4),have them verify the product using the distributive property. 6.Have students graph the trinomial x2 + 6x + 8 and its factored form, (x + 2)(x + 4), on the graphing calculator. Students should see that the graphs are identical. 7.Ask students to identify the values of x at which the graph crosses the x-intercept. Using the TRACE feature, students should identify (-2, 0) and (-4, 0)as the x-intercepts. 8.Give students 30 seconds to determine where the y-intercept occurs. Using the TRACE feature, students should identify (0, 8) as the y-intercept. 9.Have students use the same process for the second and third trinomials on the Factoring Gift handout: x2 + 7x + 6 and x2 + 8x + 12. Factor the trinomial using algebra tiles, and write it in factored form. Verify the product with the distributive property. Identify the x-intercepts. Identify the y-intercept. On the chalkboard or overhead projector, create a table that lists all of the information that has so far been obtained: 10.Give students one minute to discuss, in pairs or small groups, any patterns or relationships that exist in the table. Encourage them to look for several different patterns, and explain that there are many to find. Students should identify the following relationships: The y coordinate of the y-intercept is equal to the constant term in the trinomial; that is, if the polynomial is x2 + bx + c, the y-intercept occurs at (0, c). The x coordinates of the x-intercepts are equal to the opposite of the constant terms when the trinomial is written in factored form; that is, if the polynomial can be expressed as (x + m)(x + n), the x-interceptsoccur at (-m, 0) and (-n, 0) because x + m = 0 or x + n = 0. If a trinomial x2 + bx + c can be written as (x + m)(x + n), then b = m + n and c = m x n. Note: The final pattern listed above, that b = m + n and c = m x n, is one of the keystones of this lesson. Students must realize that this relationship always holds, and that it is the key to factoring trinomials. If students have not identified this relationship by this point of the lesson, have them continue to factor trinomials from the Factoring Gift handout until they identify this pattern. 11.Using the patterns identified in the table, students should factor the fourth trinomial from the Factoring Gift handout, x2 + 7x + 10, without algebra tiles or the graphing calculator. At this point, students should have discovered that they must find two numbers for which the product is 10 and the sum is 7, resulting in (x + 2)(x + 5). 12.Ask students to factor x2 + 4x + 6. To factor this trinomial, students must identify two numbers that have a product of 6 and a sum of 4. Because no real numbers exist for which this is true, students should conclude that this trinomial cannot be factored. Define such a polynomial as a “prime trinomial.” 13.Have students graph x2 + 4x + 6 on graphing calculator. To make sure students understand this visual representation, which shows why this expression can’t be factored, point out or elicit that the graph does not cross the x-axis, so it has no x-intercepts and consequently no real factors. Culminating Activity/Assessment: In their math journals, have students identify all positive values of C such that x2 + 6x + C is a factorable trinomial. Students should identify at least three values for C, namely 5, 8, and 9. Students may also notice that C = 0 is a solution, but it produces a slightly different form than this lesson addresses. 9 x2 + 6x + 9 = (x + 3)(x + 3) 8 x2 + 6x + 8 = (x + 2) (x + 4) 5 x2 + 6x + 5 = (x + 1)(x + 5) 0 x2 + 6x = x (x + 6) Related Standardized Test Questions The questions below dealing with factoring polynomials have been selected from various state and national assessments. Although the lesson above may not fully equip students to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other state assessment questions. Taken from the New York Regents High School Examination, Mathematics (August 2002): A rectangular park is three blocks longer than it is wide. The area of the park is 40 square blocks. If w represents the width, write an equation in terms of w for the area of the park. Find the length and the width of the park. Solution: w(w + 3) = 40, w2 + 3w – 40 = 0, (w + 8)(w – 5) = 0, w + 8 = 0 or w – 5 = 0, w = -8 or w = 5. The width must be positive, so w = 5 and l = 8. The width of the park is five blocks and the length of the park is 8 blocks. Taken from the New York Regents High School Examination (January 2003): What are the factors of x2 – 10x – 24? A. (x – 4)(x + 6)B. (x – 4)(x – 6)C. (x – 12)(x + 2) (correct answer) D. (x + 12)(x – 2) Taken from the Virginia Standards of Learning Assessment (Spring 2002): Which property justifies the following statement? If 3a + 3b = 12, then 3(a + b) = 12 A. Commutative property of multiplicationB. Distributive property for multiplication over addition (correct answer) C. Multiplicative identity property D. Associative property of addition Taken from the Virginia Standards of Learning Assessment, Algebra I (Spring 2002): Which is the complete factorization of 2x2 + 5x + 3? A. (2x + 1)(x + 2)B. (2x + 1)(x + 3)C. (2x + 2)(x + 1)D. (2x + 3)(x + 1) (correct answer) Taken from the Virginia Standards of Learning Assessment, Algebra I (Spring 2002): Taken from the Virginia Standards of Learning Assessment, Algebra II (Spring 2001): Which is a zero of the function f(x) = x2 + 6x + 8? A. -8B. -4 (correct answer) C. 2 D. 4 Student Work: Factoring Assignment Teacher Commentary: First of all, I am impressed with the student’s neatness and organization. The notes are easy to follow. There is evidence that the student understands what it means to factor a number and to factor an expression. He/she appears to understand how to apply the distributive property. The student might not fully understand what is meant by an expression being prime. It is unclear that this student understood the last problem where he/she had to decide on the values for C to make the trinomial factorable. I would have preferred some written explanation such as, “we are looking for two numbers whose product is 8 and whose sum is 6.” Perhaps next time I will ask the students to summarize their findings in words and write them down on paper. I would make one change to this handout (remember I call it a “gift”): on #16, I would stipulate that C can be greater than or equal to zero. Series Directory Insights Into Algebra 1: Teaching for Learning 1 Variables and Patterns of Change 1 Lesson Plan 1: Miles of Tiles – The Pool Border Problem 2 Lesson Plan 2: Cups and Chips 3 Teaching Strategies: Cooperative Learning 4 Teaching Strategies: Manipulatives 2 Linear Functions and Inequalities 1 Lesson Plan 1: The Phone Bill Problem – Linear Functions 2 Lesson Plan 2: Hot Dog Sales – Solving Linear Equations and Inequalities 3 Teaching Strategies: Worthwhile Mathematical Tasks 4 Teaching Strategies: Appropriate Use of Technology 3 Systems of Equations and Inequalities 1 Lesson Plan 1: Left Hand, Right Hand – Solving Systems of Equations 2 Lesson Plan 2: Hassan’s Pictures – Linear Programming and Profit Lines 3 Teaching Strategies: Strategies for Teaching English Language Learners 4 Teaching Strategies: Building Understanding 4 Quadratic Functions 1 Lesson Plan 1: Up, Down, Right, Left – Function Families 2 Lesson Plan 2: Bouncing Ball – Function Families 3 Teaching Strategies: Developing a Community of Learners 4 Teaching Strategies: Alternative Assessment 5 Properties 1 Lesson Plan 1: The X Factor – Trinomials and Algebra Tiles 2 Lesson Plan 2: Curses and Re-Curses! It’s Happening Again. 3 Teaching Strategies: Rule of Four 4 Teaching Strategies: Patterns 6 Exponential Functions 1 Lesson Plan 1: Overrun by Skeeters – Exponential Growth 2 Lesson Plan 2: Bigger and Smaller – Exponent Rules 3 Teaching Strategies: Affective Domain 4 Teaching Strategies: Instructional Decision Making 7 Direct and Inverse Variation 1 Lesson Plan 1: Be Direct – Oil Spills on Land 2 Lesson Plan 2: Very Varied – Inverse Variation 3 Teaching Strategies: Questioning Techniques 4 Teaching Strategies: Concepts First, Skills Later 8 Mathematical Modeling 1 Lesson Plan 1: Mathematical Modeling, Circular Movement and Transmission Ratios 2 Lesson Plan 2: Skeeter Populations and Exponential Growth 3 Teaching Strategies: Listening to Students 4 Teaching Strategies: Lesson Study Credits Produced by Thirteen/WNET. 2004. Closed Captioning ISBN: 1-57680-740-1 Sections ### 5.1 Lesson Plan 1: The X Factor – Trinomials and Algebra Tiles ### 5.2 Lesson Plan 2: Curses and Re-Curses! It’s Happening Again. ### 5.3 Teaching Strategies: Rule of Four ### 5.4 Teaching Strategies: Patterns ### PDF: Workshop 5 Guide Printout ### PDF: Factoring Gift Printout ### PDF: A Set of Algebra Tiles Worksheet Workshops ### Workshop 1 Variables and Patterns of Change In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations. ### Workshop 2 Linear Functions and Inequalities In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra. ### Workshop 3 Systems of Equations and Inequalities In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities. ### Workshop 4 Quadratic Functions ### Workshop 5 Properties In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations. ### Workshop 6 Exponential Functions In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases. ### Workshop 7 Direct and Inverse Variation In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes. ### Workshop 8 Mathematical Modeling This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.
15128	https://math.stackexchange.com/questions/259001/a-continuous-surjection-that-is-not-a-quotient-map	general topology - A continuous surjection that is not a quotient map - 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 continuous surjection that is not a quotient map Ask Question Asked 12 years, 9 months ago Modified12 years, 1 month ago Viewed 2k times This question shows research effort; it is useful and clear 5 Save this question. Show activity on this post. I was going through an old topology prelim, and encountered a question which I'm really not sure how I should work out. Here it is: Suppose we let X=R×{3,4,…}⊂R 2 X=R×{3,4,…}⊂R 2. Now let L θ⊂R 2 L θ⊂R 2 be the line through the origin with slope tan θ tan⁡θ, i.e. the directed angle from the positive x x-axis to L θ L θ is θ θ. Further, we let Y=⋃i≥3 L π/i.Y=⋃i≥3 L π/i. Also, we define g:X→Y g:X→Y by g(x,i)=(x,x tan(π/i))g(x,i)=(x,x tan⁡(π/i)). We have to show that g g is a continuous surjection, but not a quotient map. Any ideas how I should approach this? general-topology Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Dec 15, 2012 at 21:55 LibertronLibertron asked Dec 15, 2012 at 0:32 LibertronLibertron 4,503 2 2 gold badges 38 38 silver badges 83 83 bronze badges 0 Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 5 Save this answer. Show activity on this post. Proving that g g is a continuous surjection is straightforward. Note also that g g is almost a bijection: the origin is the only point of Y Y that has more than one pre-image under g g. Thus, you can expect that this point will be significant in showing that g g is not a quotient map. To show this, let A={⟨x,x tan π k⟩∈Y:\|x\|<1 k},A={⟨x,x tan⁡π k⟩∈Y:\|x\|<1 k}, and show that A A is not open in Y Y, but g−1[A]g−1[A] is open in X X. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Aug 14, 2013 at 23:38 answered Dec 15, 2012 at 0:46 Brian M. ScottBrian M. Scott 633k 57 57 gold badges 824 824 silver badges 1.4k 1.4k bronze badges 8 Just for my understanding, is there a theorem that characterizes quotient maps in the Euclidean setting by being open? Because for general topological spaces being open is not necessary for a continuous surjection to be a quotient map.erlking –erlking 2012-12-15 00:50:40 +00:00 Commented Dec 15, 2012 at 0:50 2 @erlking: No, that’s not the point at all. The point is that by definition a set is open in the quotient topology iff its preimage is open in the original space.Brian M. Scott –Brian M. Scott 2012-12-15 00:52:59 +00:00 Commented Dec 15, 2012 at 0:52 I don't understand the inequality \|k\|<1 k\|k\|<1 k. I believe k k is an integer greater than or equal to 3 3, and that is why I am somewhat confused. Can you please clarify that?Libertron –Libertron 2013-08-14 23:32:49 +00:00 Commented Aug 14, 2013 at 23:32 1 @Libertron: No, it does not mean showing that g g is continuous. It means only what it says: show that the set g−1[A]g−1[A] is open in X X. Do this by seeing exactly what the set g−1[A]g−1[A] is, and verifying that it is indeed open in the space X X.Brian M. Scott –Brian M. Scott 2013-08-15 16:48:10 +00:00 Commented Aug 15, 2013 at 16:48 1 @SubhajitPaul: Yes, ⟨x,y⟩⟨x,y⟩ is my preferred notation for the ordered pair; it is quite common in set theory, and I prefer it elsewhere because parentheses have more than enough other meanings in mathematics. And yes, k k ranges over the integers greater than 2 2.Brian M. Scott –Brian M. Scott 2021-06-06 19:21:09 +00:00 Commented Jun 6, 2021 at 19:21 \|Show 3 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 general-topology See similar questions with these tags. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Linked 5Is projection of locally connected compact subset locally connected? 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15129	https://en.wikipedia.org/wiki/Sierra_Nevada	Jump to content Sierra Nevada Afrikaans العربية Azərbaycanca বাংলা Башҡортса Беларуская Български Brezhoneg Català Cebuano Čeština Cymraeg Dansk Deutsch Eesti Ελληνικά Español Esperanto Euskara فارسی Français Frysk Gaeilge Gàidhlig Galego 한국어 Հայերեն हिन्दी Hrvatski Bahasa Indonesia Íslenska Italiano עברית ქართული Kiswahili Latina Latviešu Lietuvių Magyar Македонски മലയാളം मराठी مصرى Bahasa Melayu Nederlands 日本語 Nordfriisk Norsk bokmål Norsk nynorsk Occitan Oʻzbekcha / ўзбекча پنجابی Polski Português Romnă Русский Scots Shqip Simple English Slovenčina Српски / srpski Srpskohrvatski / српскохрватски Suomi Svenska தமிழ் ไทย Türkçe Українська Tiếng Việt Winaray 吴语粵語中文 Edit links Coordinates: 38°00′N 119°30′W / 38.000°N 119.500°W / 38.000; -119.500 From Wikipedia, the free encyclopedia Mountain range in the United States For other uses, see Sierra Nevada (disambiguation). "Range of Light" redirects here. For the S. Carey album, see Range of Light (album). \| Sierra Nevada \| \| --- \| \| The Sierra's Mills Creek cirque (center) is on the west side of the Sierra Crest, south of Mono Lake (top, blue). \| \| \| Highest point \| \| \| Peak \| Mount Whitney \| \| Elevation \| 14,505 ft (4,421 m) \| \| Coordinates \| 36°34′42.9″N 118°17′31.2″W / 36.578583°N 118.292000°W / 36.578583; -118.292000 \| \| Dimensions \| \| \| Length \| 400 mi (640 km) north-south from Fredonyer Pass to Tehachapi Pass \| \| Width \| 80 mi (130 km) \| \| Area \| 24,370 sq mi (63,100 km2) \| \| Naming \| \| \| Etymology \| 1777: Spanish for "snowy mountain range" \| \| Nicknames \| the Sierra the High Sierra Range of Light (1894, John Muir) \| \| Geography \| \| \| Position of Sierra Nevada inside California \| \| \| Country \| United States \| \| States \| California Nevada \| \| Range coordinates \| 38°00′N 119°30′W / 38.000°N 119.500°W / 38.000; -119.500 \| \| Geology \| \| \| Rock age \| Mesozoic \| \| Rock types \| batholith igneous \| The Sierra Nevada (/siˌɛrə nɪˈvædə, -ˈvɑːd-/ see-ERR-ə nih-VA(H)D-ə)[a] is a mountain range in the Western United States, between the Central Valley of California and the Great Basin. The vast majority of the range lies in the state of California, although the Carson Range spur lies primarily in Nevada. The Sierra Nevada is part of the American Cordillera, an almost continuous chain of mountain ranges that forms the western "backbone" of the Americas. The Sierra runs 400 mi (640 km) north-south, and its width ranges from 50 mi (80 km) to 80 mi (130 km) across east–west. Notable features include the General Sherman Tree, the largest tree in the world by volume; Lake Tahoe, the largest alpine lake in North America; Mount Whitney at 14,505 ft (4,421 m), the highest point in the contiguous United States; and Yosemite Valley sculpted by glaciers from one-hundred-million-year-old granite, containing high waterfalls. The Sierra is home to three national parks, twenty-six wilderness areas, ten national forests, and two national monuments. These areas include Yosemite, Sequoia, and Kings Canyon National Parks, as well as Devils Postpile National Monument. More than one hundred million years ago during the Nevadan orogeny, granite formed deep underground. The range started to uplift less than five million years ago, and erosion by glaciers exposed the granite and formed the light-colored mountains and cliffs that make up the range. The uplift caused a wide range of elevations and climates in the Sierra Nevada, which are reflected by the presence of five life zones (areas with similar plant and animal communities). Uplift continues due to faulting caused by tectonic forces, creating spectacular fault block escarpments along the eastern edge of the southern Sierra. The Sierra Nevada has played an important role in the history of California and the United States. The California gold rush occurred in the western foothills from 1848 through 1855. Due to its inaccessibility, the range was not fully explored until 1912.: 81 Name and etymology [edit] Used in 1542 by Juan Rodríguez Cabrillo to describe a Pacific Coast Range (Santa Cruz Mountains), the term "Sierra Nevada" was a general identification of less familiar ranges toward the interior. In 1776, Pedro Font's map applied the name to the range currently known as the Sierra Nevada. The literal translation is "snowy mountains", from sierra "a range of mountains", 1610s, from Spanish sierra "jagged mountain range", lit. "saw", from Latin serra "a saw"; and from the Spanish adjective nevado "snowy". While many mountain ranges are unanimously referred to in the plural (Smokies, Rockies, Cascades, etc.), some locals who live in "the Sierra" are not hesitant to admonish those who refer to the area as "the Sierras". However, there are historical and literary references that use the plural, such as the 1871 collection of Joaquin Miller poems, Songs of the Sierras. Ansel Adams, in response to a publication of his photographs under the title Parmelian Prints of the High Sierras, commented, "To add an s is a linguistic, Californian, and mountaineering sin." Geography [edit] The Sierra Nevada lies primarily in Central and Eastern California, with the Carson Range, a small but historically important spur, extending into Nevada. West-to-east, the Sierra Nevada's elevation increases gradually from 500 feet (150 m) in the Central Valley to more than 14,000 feet (4,300 m) atop the highest peaks of its crest 50 to 75 miles (80 to 121 km) to the east. The east slope forms the steep Sierra Escarpment. Unlike its surroundings, the range receives a substantial amount of snowfall and precipitation due to orographic lift. Setting [edit] The Sierra Nevada's irregular northern boundary stretches from the Susan River and Fredonyer Pass to the North Fork Feather River. It represents where the granitic bedrock of the Sierra Nevada dives below the southern extent of Cenozoic igneous surface rock from the Cascade Range. The range is bounded on the west by California's Central Valley, on the east by the Basin and Range Province, and on the southeast by the Mojave Desert. The southern boundary is at Tehachapi Pass. Physiographically, the Sierra is a section of the Cascade–Sierra Mountains province, which in turn is part of the larger Pacific Mountain System physiographic division. The California Geological Survey states that "the northern Sierra boundary is marked where bedrock disappears under the Cenozoic volcanic cover of the Cascade Range." Watersheds [edit] The range is drained on its western slope by the Central Valley watershed, which discharges into the Pacific Ocean at San Francisco. The northern third of the western Sierra is part of the Sacramento River watershed (including the Feather, Yuba, and American River tributaries), and the middle third is drained by the San Joaquin River (including the Mokelumne, Stanislaus, Tuolumne, and Merced River tributaries). The southern third of the range is drained by the Kings, Kaweah, Tule, and Kern rivers, which flow into the endorheic basin of Tulare Lake, which rarely overflows into the San Joaquin during wet years. The eastern slope watershed of the Sierra is much narrower; its rivers flow out into the endorheic Great Basin of eastern California and western Nevada. From north to south, the Susan River flows into intermittent Honey Lake, the Truckee River flows from Lake Tahoe into Pyramid Lake, the Carson River runs into Carson Sink, the Walker River into Walker Lake; Rush, Lee Vining and Mill Creeks flow into Mono Lake; and the Owens River into dry Owens Lake. Although none of the eastern rivers reach the sea, many of the streams from Mono Lake southwards are diverted into the Los Angeles Aqueduct which provides water to Southern California. Elevation [edit] The height of the mountains in the Sierra Nevada increases gradually from north to south. Between Fredonyer Pass and Lake Tahoe, the peaks range from 5,000 feet (1,500 m) to more than 9,000 feet (2,700 m). The crest near Lake Tahoe is roughly 9,000 feet (2,700 m) high, with several peaks approaching the height of Freel Peak (10,881 ft or 3,317 m). Farther south, the highest peak in Yosemite National Park is Mount Lyell (13,120 ft or 3,999 m). The Sierra rises to almost 14,000 feet (4,300 m) with Mount Humphreys near Bishop, California. Finally, near Lone Pine, Mount Whitney is at 14,505 feet (4,421 m), the highest point in the contiguous United States. South of Mount Whitney, the elevation of the range quickly dwindles. The crest elevation is almost 10,000 feet (3,000 m) near Lake Isabella, but south of the lake, the peaks reach only a modest 8,000 feet (2,400 m). Notable features [edit] There are several notable geographical features in the Sierra Nevada: Lake Tahoe is a large, clear freshwater lake in the northern Sierra Nevada, with an elevation of 6,225 ft (1,897 m) and an area of 191 sq mi (490 km2). Lake Tahoe lies between the main Sierra and the Carson Range, a spur of the Sierra. Hetch Hetchy Valley, Yosemite Valley, Kings Canyon, and Kern Canyon are examples of many glacially-scoured canyons on the west side of the Sierra. Yosemite National Park is filled with notable features such as waterfalls, granite domes, high mountains, lakes, and meadows. Groves of giant sequoias Sequoiadendron giganteum occur along a narrow band of altitude on the western side of the Sierra Nevada. Giant sequoias are the largest trees in the world. Two of the largest rivers in California, which form the Central Valley and drain into San Francisco Bay, derive most of their flow from the western slopes of the Sierra Nevada. The northern of the two is the Sacramento River (which also drains the adjacent Cascade Range and Klamath Range); the southern one is the San Joaquin River. Communities [edit] Communities in the Sierra Nevada include Paradise, South Lake Tahoe, Truckee, Grass Valley, Lee Vining, Mammoth Lakes, Sonora, Nevada City, Placerville, Pollock Pines, Portola, Auburn, Colfax, Kennedy Meadows and Shaver Lake. Protected areas [edit] Main article: List of protected areas of the Sierra Nevada Much of the Sierra Nevada consists of federal lands and is either protected from development or strictly managed. The mountain range is home to three National Parks – Yosemite, Kings Canyon, and Sequoia – and two national monuments – Devils Postpile and Giant Sequoia. Ten national forests span much of the mountain range's remaining area. Within these national parks, monuments, and forests lie 26 wilderness areas, which together protect 15.4% of the Sierra's 63,118 km2 (24,370 sq mi) from logging, development, and wheeled vehicle use. The United States Forest Service and the Bureau of Land Management currently control 52% of the land in the Sierra Nevada. Logging and grazing are generally allowed on land controlled by these agencies, under federal regulations that balance recreation and development on the land. The California Bighorn Sheep Zoological Area near Mount Williamson in the southern Sierra was established to protect the endangered Sierra Nevada bighorn sheep. Starting in 1981, hikers were unable to enter the Area from May 15 through December 15, in order to protect the sheep. As of 2010, the restriction has been lifted and access to the Area is open for the whole year. Geologic history [edit] For central Sierra Nevada geology, see Geology of the Yosemite area. The earliest rocks in the Sierra Nevada are metamorphic roof pendants of Paleozoic age, the oldest being metasedimentary rocks from the Cambrian in the Mount Morrison region. These dark-colored hornfels, slates, marbles, and schists are found in the western foothills (notably around Coarsegold, west of the Tehachapi Pass) and east of the Sierra Crest. The earliest granite of the Sierra started to form in the Triassic period. This granite is mostly found east of the crest and north of 37.2°N. In the Triassic and into the Jurassic, an island arc collided with the west coast of North America and raised a chain of volcanoes, in an event called the Nevadan orogeny. Nearly all subaerial Sierran Arc volcanoes have since disappeared; their remains were redeposited during the Great Valley Sequence and the subsequent Cenozoic filling of the Great Valley, which is the source of much of the sedimentary rock in California. In the Cretaceous, a subduction zone formed at the edge of the continent. This means that an oceanic plate started to dive beneath the North American Plate. Magma, formed through the subduction of the ancient Farallon Plate, rose in plumes (plutons) deep underground, their combined mass forming what is called the Sierra Nevada batholith. These plutons formed at various times, from 115 Ma to 87 Ma. The earlier plutons formed in the western half of the Sierra, while the later plutons formed in the eastern half of the Sierra. At this time, the Sierra Nevada formed the western ramp of a high plateau to the east, the Nevadaplano. During this period, rivers cut deep canyons into the range, generating topographic relief similar to the modern Sierra Nevada. This period of incision was halted approximately 30 million years ago by vast outpourings of pyroclastic flows from Nevada which filled the northern Sierran valleys with volcanic deposits. These pyroclastic flows, which continued for about 10 million years, were followed by andesitic lahars which nearly completely buried the northern Sierran landscape such that only the tallest peaks emerged above a volcanic plain. This second period of volcanism appears to have been triggered by crustal extension associated with extension of the Basin and Range Province. As this andesitic volcanism began waning about five million years ago, the rivers were able to begin eroding away the 100s of meters of volcanic deposits and resume the incision that had been halted by the first period of volcanism. Some studies have argued that this recent incision is a sign of recent tectonic uplift. Other geologists claim that the elevations of many of the modern rivers flowing down the range are only 100–300 meters (300–1,000 ft) lower than their ancient counterparts from 30–40 million years ago and the overall elevation and bedrock topography of the northern Sierra Nevada has changed little since at least 30–40 million years ago. About 2.5 Ma, the Earth's climate cooled, and ice ages started. Glaciers carved out characteristic U-shaped canyons throughout the Sierra. The combination of river and glacier erosion exposed the uppermost portions of the plutons emplaced millions of years before, leaving only a remnant of metamorphic rock on top of some Sierra peaks. Extension of the Basin and Range continues today, leading to downdropping of crustal blocks just east of the Sierra Nevada during large earthquakes, such as the Lone Pine earthquake of 1872. Sierra Escarpment viewed from the east. In the foreground is Tinemaha Reservoir in the Owens Valley. Climate and meteorology [edit] The climate of the Sierra Nevada is influenced by the Mediterranean climate of California. During the fall, winter and spring, precipitation in the Sierra ranges from 20 to 80 in (510 to 2,030 mm) where it occurs mostly as snow above 6,000 ft (1,800 m). Precipitation is highest on the central and northern portions of the western slope between 5,000 and 8,000 feet (1,500 and 2,400 m) elevation, due to orographic lift.: 69 Above 8,000 feet (2,400 m), precipitation diminishes on the western slope up to the crest, since most of the precipitation has been wrung out at lower elevations. Most parts of the range east of the crest are in a rain shadow, and receive less than 25 inches of precipitation per year. While most summer days are dry, afternoon thunderstorms are common, particularly during the North American Monsoon in mid and late summer. Some of these summer thunderstorms drop over an inch of rain in a short period, and the lightning can start fires. Summer high temperatures average 42–90 °F (6–32 °C). Winters are comparatively mild, and the temperature is usually only just low enough to sustain a heavy snowpack. For example, Tuolumne Meadows, at 8,600 feet (2,600 m) elevation, has winter daily highs about 40 °F (4 °C) with daily lows about 10 °F (−12 °C). The growing season lasts 20 to 230 days, strongly dependent on elevation. The highest elevations of the Sierra have an alpine climate. The Sierra Nevada snowpack is the major source of water and a significant source of electric power generation in California. Many reservoirs were constructed in the canyons of the Sierra throughout the 20th century, Several major aqueducts serving both agriculture and urban areas distribute Sierra water throughout the state. However, the Sierra casts a rain shadow, which greatly affects the climate and ecology of the central Great Basin. This rain shadow is largely responsible for Nevada being the driest state in the United States. Precipitation varies substantially from year to year. It is not uncommon for some years to receive precipitation totals far above or below normal. The height of the range and the steepness of the Sierra Escarpment, particularly at the southern end of the range, produces a wind phenomenon known as the "Sierra Rotor". This is a horizontal rotation of the atmosphere just east of the crest of the Sierra, set in motion as an effect of strong westerly winds. The Sierra Nevada is home to the Mono winds, strong, dry downslope winds that primarily affect the western slopes, especially in the central region, and are most common from late fall to spring. With gusts reaching over 80 miles per hour, these winds can cause widespread disruption, uprooting trees, damaging infrastructure, and making mountain passes hazardous for drivers. Because of the large number of airplanes that have crashed in the Sierra Nevada, primarily due to the complex weather and atmospheric conditions such as downdrafts and microbursts caused by geography there, a portion of the area, a triangle whose vertices are Reno, Nevada; Fresno, California; and Las Vegas, Nevada, has been dubbed the "Nevada Triangle", in reference to the Bermuda Triangle. Some counts put the number of crashes in the triangle at 2,000, including millionaire and record-breaking flyer Steve Fossett. Hypotheses that the crashes are related in some way to the United States Air Force's Area 51, or to the activities of extra-terrestrial aliens, have no evidence to support them. Ecology [edit] Main article: Ecology of the Sierra Nevada The Sierra Nevada is divided into a number of biotic zones, each of which is defined by its climate and supports a number of interdependent species. Life in the higher elevation zones adapted to colder weather, and to most of the precipitation falling as snow. The rain shadow of the Sierra causes the eastern slope to be warmer and drier: each life zone is higher in the east. A list of biotic zones, and corresponding elevations, is presented below: The western foothill zone, 1,000–2,500 ft (300–760 m),: 92 with grassland, oak-grass savanna and chaparral-oak woodland. Gray pine (also known as Foothill pine) is intermixed with the oak woodland.: 95 The Pinyon pine-Juniper woodland, 5,000–7,000 ft (1,500–2,100 m) east side only.: 92 The Sierra Nevada lower montane forest (indicator species: Ponderosa pine, Jeffrey pine), 2,500–7,000 ft (760–2,130 m) west side, 7,000–9,000 ft (2,100–2,700 m) east side.: 92 This biotic zone is notable for containing giant sequoia. The Sierra Nevada upper montane forest (indicator species: Lodgepole pine, Red fir) 7,000–9,000 ft (2,100–2,700 m) west side, 9,000–10,500 ft (2,700–3,200 m) east side.: 92 The Sierra Nevada subalpine zone (indicator species: Whitebark pine) 9,000–10,500 ft (2,700–3,200 m) west side, 10,500–11,500 ft (3,200–3,500 m) east side: 92 The alpine region at greater than 10,500 ft (3,200 m), and greater than 11,500 ft (3,500 m) east side.: 92 History [edit] Native Americans [edit] Main article: Great Basin tribes Archaeological excavations placed Martis people of Paleo-Indians in northcentral Sierra Nevada during the period of 3,000 BCE to 500 CE. The earliest identified sustaining indigenous people in the Sierra Nevada were the Northern Paiute tribes on the east side, with the Mono tribe and Sierra Miwok tribe on the western side, and the Kawaiisu and Tübatulabal tribes in the southern Sierra. Today, some historic intertribal trade route trails over mountain passes are known artifact locations, such as Duck Pass with its obsidian arrowheads. The California and Sierra Native American tribes were predominantly peaceful, with occasional territorial disputes between the Paiute and Sierra Miwok tribes in the mountains. Washo and Maidu were also in this area prior to the era of European exploration and displacement. Initial European-American exploration [edit] See also: History of the Yosemite area and California Trail American exploration of the mountain range started in 1827. Although prior to the 1820s there were Spanish missions, pueblos (towns), presidios (forts), and ranchos along the coast of California, no Spanish explorers visited the Sierra Nevada. The first Americans to visit the mountains were amongst a group led by fur trapper Jedediah Smith, crossing north of the Yosemite area in May 1827, at Ebbetts Pass. In 1833, a subgroup of the Bonneville Expedition led by Joseph Reddeford Walker was sent westward to find an overland route to California. Eventually the party discovered a route along the Humboldt River across present-day Nevada, ascending the Sierra Nevada, starting near present-day Bridgeport and descending between the Tuolumne and Merced River drainage. The group may have been the first non-indigenous people to see Yosemite Valley. The Walker Party probably visited either the Tuolumne or Merced Groves of giant sequoia, becoming the first non-indigenous people to see the giant trees, but journals relating to the Walker party were destroyed in 1839, in a print shop fire in Philadelphia. Starting in 1841, emigrants from the United States started to move to California via Sonora and Walker Passes. In the winter of 1844, Lt. John C. Frémont, accompanied by Kit Carson, was the first European American to see Lake Tahoe. The Frémont party camped at 8,050 ft (2,450 m). Gold rush [edit] Main article: California Gold Rush The California Gold Rush began at Sutter's Mill, near Coloma, in the western foothills of the Sierra. On January 24, 1848, James W. Marshall, a foreman working for Sacramento pioneer John Sutter, found shiny metal in the tailrace of a lumber mill Marshall was building for Sutter on the American River. Rumors soon started to spread and were confirmed in March 1848 by San Francisco newspaper publisher and merchant Samuel Brannan. Brannan strode through the streets of San Francisco, holding aloft a vial of gold, shouting "Gold! Gold! Gold from the American River!" On August 19, 1848, the New York Herald was the first major newspaper on the East Coast to report the discovery of gold. On December 5, 1848, President James Polk confirmed the discovery of gold in an address to Congress.: 80 Soon, waves of immigrants from around the world, later called the "forty-niners", invaded the Gold Country of California or "Mother Lode". Miners lived in tents, wood shanties, or deck cabins removed from abandoned ships. Wherever gold was discovered, hundreds of miners would collaborate to put up a camp and stake their claims. Because the gold in the California gravel beds was so richly concentrated, the early forty-niners simply panned for gold in California's rivers and streams.: 198–200 However, panning cannot take place on a large scale, and miners and groups of miners graduated to more complex placer mining. Groups of prospectors would divert the water from an entire river into a sluice alongside the river, and then dig for gold in the newly exposed river bottom.: 90 By 1853, most of the easily accessible gold had been collected, and attention turned to extracting gold from more difficult locations. Hydraulic mining was used on ancient gold-bearing gravel beds on hillsides and bluffs in the gold fields.: 89 In hydraulic mining, a high-pressure hose directed a powerful stream or jet of water at gold-bearing gravel beds. It is estimated that by the mid-1880s, 11 million troy ounces (340 metric tons) of gold (worth approximately US$16 billion in 2020 prices) had been recovered by "hydraulicking". A consequence of these extraction methods was that large amounts of gravel, silt, heavy metals, and other pollutants were washed into streams and rivers.: 32–36 As of 1999[update], many areas still bear the scars of hydraulic mining, since the resulting exposed earth and downstream gravel deposits do not support plant life.: 116–121 It is estimated that by 1855, at least 300,000 gold-seekers, merchants, and other immigrants had arrived in California from around the world.: 25 The huge numbers of newcomers brought by the Gold Rush drove Native Americans out of their traditional hunting, fishing and food-gathering areas. To protect their homes and livelihood, some Native Americans responded by attacking the miners, provoking counter-attacks on native villages. The Native Americans, out-gunned, were often slaughtered. Thorough exploration [edit] The Gold Rush populated the western foothills of the Sierra Nevada, but even by 1860, most of the Sierra was unexplored. The state legislature authorized the California Geological Survey to officially explore the Sierra (and survey the rest of the state). Josiah Whitney was appointed to head the survey. Men of the survey, including William H. Brewer, Charles F. Hoffmann and Clarence King, explored the backcountry of what would become Yosemite National Park in 1863. In 1864, they explored the area around Kings Canyon. In 1869, John Muir started his wanderings in the Sierra Nevada range, and in 1871, King was the first to climb Mount Langley, mistakenly believing he had summited Mount Whitney, the highest peak in the range. In 1873, Mount Whitney was climbed for the first time by 3 men from Lone Pine, California, on a fishing trip. From 1892 to 1897 Theodore Solomons made the first attempt to map a route along the crest of the Sierra. Other people finished exploring and mapping the Sierra. Bolton Coit Brown explored the Kings River watershed in 1895–1899. Joseph N. LeConte mapped the area around Yosemite National Park and what would become Kings Canyon National Park. James S. Hutchinson, a noted mountaineer, climbed the Palisades (1904) and Mount Humphreys (1905). By 1912, the USGS published a set of maps of the Sierra Nevada, and the era of exploration was over.: 81 Logging [edit] See also: Logging in the Sierra Nevada Logging in the Sierra Nevada has significantly impacted the landscape. The logging industry in the Sierra Nevada started in the early 1800s, when settlers relied on hand tools and ox-teams.: 103, 127 Before the California Gold Rush, the industry was relatively small, and most of the lumber used in the state was imported. However, as the demand for lumber to support the mining industry increased, logging became a major industry in the region. Initially, most of the lumber produced in California was used in mining. The Comstock Lode was a major center for logging, with operations supplying lumber for the construction of mine structures, such as tunnels, shafts, and buildings, as well as fuel for the mines. Dan DeQuille observed in 1876, "the Comstock Lode may truthfully be said to be the tomb of the forests of the Sierra. Millions upon millions of feet of lumber are annually buried in the mines, nevermore to be resurrected." In the late 1800s, the logging industry moved westward due to the depletion of white pine forests in the upper Midwest.: 9–14 This shift was encouraged by the positive portrayal of the Sierra Nevada as a promising timber region. In 1859, Horace Greely marveled, "I never saw anything so much like good timber in the course of any seventy-five miles' travel as I saw in crossing the Sierra Nevada." The logging industry experienced significant growth in the late 1800s due to several factors. The Timber and Stone Act of 1878 allowed individuals to claim ownership of old-growth timber tracts, which were later consolidated under joint-stock companies, such as those founded by Midwestern lumber magnates.: 142–144 These companies had the financial resources to transport timber from remote locations and build sawmills near the tracks of the Southern Pacific railroad which connected the San Joaquin Valley to the rest of the state in the 1870s. This facilitated the nationwide distribution of lumber. In addition, technological advancements, such as the shay locomotive and the v-shaped log flume, made it easier to transport lumber across mountainous terrain. Conservation [edit] See also: Protected areas of the Sierra Nevada The tourism potential of the Sierra Nevada was recognized early in the European history of the range. Yosemite Valley was first protected by the federal government in 1864. The Valley and Mariposa Grove were ceded to California in 1866 and turned into a state park. John Muir perceived overgrazing by sheep and logging of giant sequoia to be a problem in the Sierra. Muir successfully lobbied for the protection of the rest of Yosemite National Park: Congress created an Act to protect the park in 1890. The Valley and Mariposa Grove were added to the Park in 1906. In the same year, Sequoia National Park was formed to protect the Giant Sequoia: all logging of the Sequoia ceased at that time. In 1903, the city of San Francisco proposed building a hydroelectric dam to flood Hetch Hetchy Valley. The city and the Sierra Club argued over the dam for 10 years, until the U.S. Congress passed the Raker Act in 1913 and allowed dam building to proceed. O'Shaughnessy Dam was completed in 1923. Between 1912 and 1918, Congress debated three times to protect Lake Tahoe in a national park. None of these efforts succeeded, and after World War II, towns such as South Lake Tahoe grew around the shores of the lake. By 1980, the permanent population of the Lake Tahoe area grew to 50,000, while the summer population grew to 90,000. The development around Lake Tahoe affected the clarity of the lake water. In order to preserve the lake's clarity, construction in the Tahoe basin is currently regulated by the Tahoe Regional Planning Agency. As the 20th century progressed, more of the Sierra became available for recreation; other forms of economic activity decreased. The John Muir Trail, a trail that followed the Sierra crest from Yosemite Valley to Mount Whitney, was funded in 1915 and finished in 1938. Kings Canyon National Park was formed in 1940 to protect the deep canyon of the Kings River. In the 1920s, automobile clubs and nearby towns started to lobby for trans-Sierra highways over Piute Pass (which would have closed the gap in SR 168) and other locations. However, by end of the 1920s, the Forest Service and the Sierra Club decided that roadless wilderness in the Sierra was valuable, and fought the proposal. The Piute Pass proposal faded out by the early 1930s, with the Forest Service proposing a route over Minaret Summit in 1933. The Minaret Summit route was lobbied against by California's Governor Ronald Reagan in 1972. The expansion of the John Muir and Ansel Adams Wildernesses in the 1980s sealed off the Minaret Summit route. A trans-Sierra route between Porterville and Lone Pine was proposed by local businessmen in 1923. Eventually, a circuitous route across the Sierra was built across Sherman Pass by 1976. By 1964, the Wilderness Act protected portions of the Sierra as primitive areas where humans are simply temporary visitors. Gradually, 20 wilderness areas were established to protect scenic backcountry of the Sierra. These wilderness areas include the John Muir Wilderness (protecting the eastern slope of the Sierra and the area between Yosemite and Kings Canyon Parks), and wilderness within each of the National Parks. The Sierra Nevada still faces a number of issues that threaten its conservation. Logging occurs on both private and public lands, including controversial clearcut methods and thinning logging on private and public lands. Grazing occurs on private lands as well as on National Forest lands, which include Wilderness areas. Overgrazing can alter hydrologic processes and vegetation composition, remove vegetation that serves as food and habitat for native species, and contribute to sedimentation and pollution in waterways. A recent increase in large wildfires, like the Rim Fire in Yosemite National Park and the Stanislaus National Forest and the King Fire on the Eldorado National Forest, has prompted concerns. A 2015 study indicated that the increase in fire risk in California may be attributable to human-induced climate change. A study looking back over 8,000 years found that warmer climate periods experienced severe droughts and more stand-replacing fires and concluded that as climate is such a powerful influence on wildfires, trying to recreate presettlement forest structure may be difficult in a warmer future. See also [edit] Geography portal Mountains portal California portal Nevada portal Bibliography of the Sierra Nevada List of Sierra Nevada road passes List of Sierra Nevada topics Sierra Nevada (Spain) Explanatory notes [edit] ^ Spanish pronunciation: [ˈsjera neˈβaða]; lit. 'snowy range'. ^ The ship was named after Mount Kearsarge in New Hampshire, see "Kearsarge (BB-5)". Dictionary of American Naval Fighting Ships. Naval History & Heritage Command (NHHC). February 23, 2005. Archived from the original on September 21, 2015. Retrieved December 15, 2012. References [edit] ^ Jump up to: a b "Mount Whitney". NGS Data Sheet. National Geodetic Survey, National Oceanic and Atmospheric Administration, United States Department of Commerce. ^ Jump up to: a b "Sierra Nevada". Ecological Subregions of California. United States Forest Service. Archived from the original on December 5, 2010. ^ Jump up to: a b "Sierra Nevada". SummitPost.org. Archived from the original on September 17, 2020. Retrieved May 29, 2010. ^ Jump up to: a b c "The Sierra Nevada Region". USCB Biogeography lab. Archived from the original on July 20, 2011. ^ Muir, John (1894). "Chapter 1: The Sierra Nevada". The Mountains of California. Archived from the original on April 10, 2014. Retrieved May 29, 2010. ^ Wells, John C. (2008). Longman Pronunciation Dictionary (3rd ed.). Longman. ISBN 978-1-4058-8118-0. ^ Carlson, Helen S. (1976). Nevada Place Names: A Geographical Dictionary. University of Nevada Press. p. 215. ISBN 978-0-87417-094-8. ^ "Cascade-Sierra Mountains Province (U.S. National Park Service)". www.nps.gov. Archived from the original on February 12, 2022. Retrieved February 12, 2022. ^ Jump up to: a b c d e f Roper, Steve (1997). Sierra High Route: Traversing Timberline Country. The Mountaineers Press. ISBN 978-0-89886-506-6. ^ Farquhar, Francis P. (1926). "K". Place Names of the Sierra Nevada. San Francisco: Sierra Club. Archived from the original on March 13, 2006.{{cite book}}: CS1 maint: bot: original URL status unknown (link) ^ Farquhar, Francis P. (March 1925). "Exploration of the Sierra Nevada". California Historical Society Quarterly. 4 (1): 3–58. doi:10.2307/25177743. hdl:2027/mdp.39015049981668. JSTOR 25177743. Archived from the original on April 30, 2011. ^ Farquhar, Francis P. (1926). "S". Place Names of the Sierra Nevada. San Francisco: Sierra Club. Archived from the original on May 25, 2024.{{cite book}}: CS1 maint: bot: original URL status unknown (link) ^ "Sierra". Etymology Online. Archived from the original on August 6, 2011. Retrieved February 27, 2011. ^ "Nevada". Etymology Online. Archived from the original on August 29, 2011. Retrieved February 27, 2011. ^ Jump up to: a b Moon, Freda (July 19, 2021). "Is it 'The Sierra' or 'The Sierras'? Californians can't agree". SFGATE. Archived from the original on January 29, 2023. Retrieved January 28, 2023. ^ "POET OF THE SIERRAS, JOAQUIN MILLER, DIES; His Body to be Burned on Pyre at Mountain Home and Ashes Borne by Winds". The New York Times. February 18, 1913. ISSN 0362-4331. Archived from the original on July 30, 2021. Retrieved July 30, 2021. ^ Adams, Ansel; Mary Street Alinder (1996). Ansel Adams: An Autobiography. NY: Little, Brown & Co. pp. 65–66. ISBN 0-8212-2241-4. ^ Jump up to: a b c "Chapter 33-Ecological subregions of the United States, Sierran Steppe - Mixed Forest - Coniferous Forest". United States Forest Service. Archived from the original on January 2, 2014. Retrieved August 30, 2013. ^ "Subsection M261Eb: Fredonyer Butte – Grizzly Peak". Archived from the original on December 5, 2010. Retrieved August 2, 2010. ^ "Sierra Nevada". Peakbagger.com. Archived from the original on May 15, 2011. Retrieved August 7, 2010. ^ "California Geomorphic Provinces" (PDF). California Geological Survey. 2002. Archived from the original (PDF) on July 21, 2004. ^ "California Geologic Provinces" (PDF). California Geological Survey. p. 2. Note 36. Archived from the original (PDF) on December 22, 2016. ^ "Google terrain map". Archived from the original on September 25, 2023. Retrieved May 29, 2010. ^ Jump up to: a b "Facts about Lake Tahoe". USGS. Archived from the original on July 21, 2011. Retrieved May 12, 2007. ^ "The General Sherman Tree". U.S. National Park Service. Archived from the original on March 15, 2010. ^ Sierra National Forests Indicator Species Amendment Final Environmental Impact Statement (PDF) (Report). p. 5. Archived from the original (PDF) on August 16, 2021. Retrieved May 10, 2020. ^ "Forest Service Proposes to Change Designation of Bighorn Sheep Zoological Areas". United States Forest Service. September 25, 2010. Archived from the original on October 5, 2011. Retrieved January 23, 2011. ^ Jump up to: a b Stevens, CH; Greene, DC (2000). "Geology of Paleozoic rocks in eastern Sierra Nevada roof pendants, California". Geological Society of America. Field Guide 2. ^ "Geology and Mineral Deposits of the Mount Morrison Quadrangle, Sierra Nevada, California" (PDF). United States Geological Survey. Archived (PDF) from the original on September 20, 2015. Retrieved December 12, 2014. ^ Jump up to: a b Unger, Tanya S. "Mesozoic Plutonism in the Sierra Nevada Batholith". Archived from the original on September 23, 2015. Retrieved June 1, 2010. ^ Shaffer, Jeffrey. "Evolution of the Yosemite Landscape – The Nevadan Orogeny". One Hundred Hikes in Yosemite. Archived from the original on April 24, 2011. ^ Blakely, Ron. "Geologic History of Western US". Archived from the original on June 22, 2010. Retrieved June 1, 2010. ^ Jump up to: a b c d e f g h i j k Schoenherr, Allan A. (1995). A Natural History of California. UC Press. ISBN 978-0-520-06922-0. ^ Ernst, W. G. (July 1, 2009). "Rise and fall of the Nevadaplano". International Geology Review. 51 (7–8): 583–588. Bibcode:2009IGRv...51..583E. doi:10.1080/00206810903063315. ISSN 0020-6814. S2CID 129541879. Archived from the original on September 20, 2021. Retrieved September 20, 2021. ^ Henry, C. D.; Hinz, N. H.; Faulds, J. E.; Colgan, J. P.; John, D. A.; Brooks, E. R.; Cassel, E. J.; Garside, L. J.; Davis, D. A.; Castor, S. B. (February 1, 2012). "Eocene-Early Miocene paleotopography of the Sierra Nevada-Great Basin-Nevadaplano based on widespread ash-flow tuffs and paleovalleys". Geosphere. 8 (1): 1–27. Bibcode:2012Geosp...8....1H. doi:10.1130/GES00727.1. ISSN 1553-040X. ^ Bateman, P.C.; Wahrhaftig, C. (1966). Geology of the Sierra Nevada in Geology of Northern California. California Division of Mines and Geology. pp. 107–172. ^ Joel Michaelsen. "Basin and Range (Transierra) Region Physical Geography". Archived from the original on July 27, 2011. Retrieved May 7, 2010. ^ Wakabayashi, J. (2013). "Paleochannels, stream incision, erosion, topographic evolution, and alternative explanations of paleoaltimetry, Sierra Nevada, California". Geosphere. 9 (2): 191–215. Bibcode:2013Geosp...9..191W. doi:10.1130/GES00814.1. ^ Beeson, Helen W; McCoy, Scott W (2022). "Disequilibrium river networks dissecting the western slope of the Sierra Nevada, California, USA, record significant late Cenozoic tilting and associated surface uplift". GSA Bulletin. 134 (11–12): 2809–2853. Bibcode:2022GSAB..134.2809B. doi:10.1130/B35463.1. ^ Mulch, Andreas; Graham, Stephan A; Chamberlain, C. Page (2006). "Hydrogen isotopes in Eocene river gravels and paleoelevation of the Sierra Nevada". Science. 313 (5783): 87–89. Bibcode:2006Sci...313...87M. doi:10.1126/science.1125986. PMID 16825568. ^ Gabet, E.J.; Miggins, D. (2020). "Minimal net incision of the northern Sierra Nevada (California, USA) since the Eocene-early Oligocene". Geology. 48 (10): 1023–1027. Bibcode:2020Geo....48.1023G. doi:10.1130/G47902.1. ^ Cassel, E.J.; Graham, S.A.; Chamberlain, C.P. (2009). "Cenozoic tectonic and topographic evolution of the northern Sierra Nevada, California, through stable isotope paleoaltimetry in volcanic glass". Geology. 37 (6): 547–550. Bibcode:2009Geo....37..547C. doi:10.1130/G25572A.1. ^ "1872 Lone Pine Earthquake". Sierra Nevada Virtual Museum. Archived from the original on May 22, 2011. Retrieved May 31, 2010. Few people ever see a mountain range grow, but on March 26, 1872, the 300 residents of Lone Pine, California, did. ^ "Average Annual Precipitation". Sierra Nevada Photos. Archived from the original on February 22, 2008. Retrieved January 2, 2014. ^ "Weather". Yosemite. National Park Service. Archived from the original on October 9, 2016. Retrieved October 8, 2016. ^ "Water—Most of California's Water Comes from the Sierra Nevada" (PDF). Sierra Nevada Conservancy. Archived from the original (PDF) on June 18, 2010. Retrieved June 9, 2010. ^ "Climatology by state based on climate division data: 1971–2000". NOAA Earth Systems Research Laboratory. Archived from the original on April 21, 2013. Retrieved July 11, 2010. ^ Grubišic, Vanda; Billings, Brian J. (2006). Sierra Rotors: A Comparative Study of Three Mountain Wave and Rotor Events (PDF). 12th Conference on Mountain Meteorology. American Meteorological Society. Archived (PDF) from the original on July 5, 2011. Retrieved May 8, 2010. ^ Ruscha, Charles P. Jr. (February 1976). "Forecasting the Mono Wind" (PDF). National Oceanic and Atmospheric Administration. NWS WR-105. Retrieved November 22, 2021. ^ Schoenmann, Joe. "The Nevada Triangle: A Graveyard For Planes". knpr.org. Archived from the original on March 2, 2019. Retrieved March 18, 2019. ^ Winter, Stuart (January 3, 2010). "Mystery of the Nevada Triangle". Sunday Express. Archived from the original on August 20, 2015. Retrieved September 15, 2015. ^ Pupp, Martin (director) (December 1, 2014). The Missing Evidence: Nevada Triangle (TV series episode). Archived from the original on September 2, 2015. Retrieved September 15, 2015. ^ Fites-Kauffman, J.; P. W. Rundel; N. Stephenson; D. A. Weixelman (2007). "Montane and subalpine vegetation of the Sierra Nevada and Cascade Ranges". In Barbour, M.G.; Keeler-Wolf, T.; Schoenherr, A.A. (eds.). Terrestrial vegetation of California (3rd ed.). Berkeley, CA, USA: University of California Press. pp. 460–501. ^ Hoffmann, Charles F. (1868). "Notes on Hetch-Hetchy Valley". Proceedings of the California Academy of Sciences. 1 (3:5): 368–370. Archived from the original on May 9, 2011. Retrieved September 27, 2006. ^ Drake, Bill (2000). "Ancient petroglyph makers of the Northern Sierra". sierrarockart.org. Archived from the original on May 16, 2008. ^ "Prehistoric Context" (PDF). Idaho-Maryland Mine Project, Master Environmental Assessment. cityofgrassvalley.com. June 2006. p. 2. Archived from the original (PDF) on July 5, 2010. Retrieved August 15, 2008. ^ Jump up to: a b c Wuerthner, George (1994). Yosemite: A Visitors Companion. Stackpole Books. pp. 13–14. ISBN 978-0-8117-2598-9. ^ Jump up to: a b c Schaffer, Jeffrey P. (1999). Yosemite National Park: A Natural History Guide to Yosemite and Its Trails. Berkeley: Wilderness Press. ISBN 978-0-89997-244-2. ^ Kiver, Eugene P.; Harris, David V. (1999). Geology of U.S. Parklands (5th ed.). New York: John Wiley & Sons. ISBN 978-0-471-33218-3. ^ Farquhar, Francis P. (March 1925). "Exploration of the Sierra Nevada". California Historical Society Quarterly. 4 (1): 3–58. doi:10.2307/25177743. hdl:2027/mdp.39015049981668. JSTOR 25177743. Archived from the original on October 19, 2022. Retrieved December 27, 2022. ^ Frémont's "Long Camp". 2007 . Archived from the original on August 19, 2017. Retrieved May 29, 2010. ^ "California Historic Gold Mines" (PDF). State of California. Archived from the original (PDF) on December 14, 2006. ^ Jump up to: a b Bancroft, Hubert Howe (1889). History of California, Volume 23: 1843–1850. San Francisco: The History Company. pp. 32–34. ^ Jump up to: a b c Starr, Kevin (2005). California: a history. New York: The Modern Library. ^ Holliday, J. S. (1999). Rush for riches; gold fever and the making of California. Oakland, California, Berkeley and Los Angeles: Oakland Museum of California and University of California Press. p. 60. ^ Brands, H. W. (2003). The age of gold: the California Gold Rush and the new American dream. New York: Anchor (reprint ed.). ^ Jump up to: a b c d Rawls, James J.; Orsi, Richard J., eds. (1999). A golden state: mining and economic development in Gold Rush California (California History Sesquicentennial Series, 2). Berkeley and Los Angeles: University of California Press. ^ "Mining History and Geology of the Mother Lode". Archived from the original on December 3, 2006. ^ Moore, James G. (2000). Exploring the Highest Sierra. Stanford University Press. ISBN 978-0-8047-3703-6. ^ Muir, John (1911). My First Summer in the Sierra. Houghton Mifflin. ISBN 978-1-883011-24-6. {{cite book}}: ISBN / Date incompatibility (help) ^ Leonard, Brendan (n.d.). "Famous U.S. Summits: Mount Whitney, California". REI Co-op Journal. www.rei.com/blog: REI Co-op. Archived from the original on July 16, 2018. Retrieved July 16, 2018. ^ Jump up to: a b Johnston, Hank (1997). The Whistles Blow No More. Stauffer Publishing. ISBN 0-87046-067-6. ^ Straka, Tom; Wynn, Bob (January 17, 2018). "Square-Set Timbering and the V-Flume Kept the Comstock Lode Running Strong". History.net. HistoryNet LLC. Archived from the original on December 27, 2022. Retrieved December 27, 2022. The Comstock Lode may truthfully be said to be the tomb of the Sierras. Millions upon millions of feet of lumber are annually buried in the mines, nevermore to be resurrected. When once it is planted in the lower levels, it never again sees the light of day. …For a distance of 50 or 60 miles, all the hills of the eastern slope of the Sierras have been to a great extent denuded of trees of every kind; those suitable only for wood as well those fit for the manufacture of lumber for use in the mines. ^ Johnston, Hank (2011). Rails to the Minarets: The Story of the Sugar Pine Lumber Company (Fourth Edition (Revised) ed.). Fish Camp, California: Stauffer Publishing. ISBN 978-0-9846848-0-9. ^ Horace, Greely (1859). Overland Journey: New York to San Francisco the Summer of 1859. New York: C.M. Saxton, Barker & Company. p. 280. Archived from the original on March 7, 2023. Retrieved December 31, 2022. ^ Zimmerman, Robert (Fall 1998). "Log Flume". American Heritage's Invention and Technology. American Heritage. Archived from the original on November 20, 2022. Retrieved December 23, 2022. ^ McDougall Weiner, Jackie (2009). Timely Exposures: The Life and Images of C.C. Curtis, Pioneer California Photographer. Tulare, California: Tulare County Historical Society. ^ Simpson, John W. (2005). Dam!: Water, Power, Politics, and Preservation in Hetch Hetchy and Yosemite National Park. Pantheon Books. ISBN 978-0-375-42231-7. ^ Righter, Robert W. (2005). The Battle over Hetch Hetchy: America's Most Controversial Dam and the Birth of Modern Environmentalism. Oxford University Press. ISBN 978-0-19-531309-3. ^ "Stream and Ground-Water Monitoring Program, Lake Tahoe Basin, Nevada and California". USGS. Archived from the original on May 30, 2010. Retrieved May 31, 2010. ^ "Construction Monitoring". Tahoe Regional Planning Agency. Archived from the original on July 16, 2011. ^ Starr, Walter A. (November 1947). "Trails". Sierra Club Bulletin. 32 (10). ^ Jump up to: a b c Marsh, Steve (2015). "The High Sierra Piute Highway" (PDF). US Forest Service. Archived from the original (PDF) on August 15, 2021. Retrieved December 31, 2020. ^ "Trail Over Mountains Supported". Los Angeles Times. June 15, 1923. p. II10. ^ "See It All in the Sierra". The Fresno Bee. October 24, 1976. ^ Jump up to: a b "Forest Issues - CSERC". CSERC. December 16, 2014. Archived from the original on January 21, 2016. Retrieved January 28, 2016. ^ "2014 Grazing Report Released by CSERC - CSERC". CSERC. Archived from the original on February 2, 2016. Retrieved January 28, 2016. ^ Yoon, Jin-Ho; Wang, S.-Y. Simon; Gillies, Robert R.; Hipps, Lawrence; Kravitz, Ben; Rasch, Philip J. (2015). "Extreme Fire Season in California: A Glimpse Into the Future?". Bulletin of the American Meteorological Society. 96 (11): S5 – S9. Bibcode:2015BAMS...96S...5Y. doi:10.1175/BAMS-D-15-00114.1. ISSN 1520-0477. OSTI 1240234. Archived from the original on February 1, 2016. Retrieved September 26, 2016. ^ Pierce, Jennifer L.; Meyer, Grant A.; Timothy Jull, A. J. (November 4, 2004). "Fire-induced erosion and millennial-scale climate change in northern ponderosa pine forests". Nature. 432 (7013): 87–90. Bibcode:2004Natur.432...87P. doi:10.1038/nature03058. ISSN 0028-0836. PMID 15525985. S2CID 1452537. 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15131	https://math.stackexchange.com/questions/551994/prove-a-statement-complements-of-unions	elementary set theory - prove a statement (complements of unions) - 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 prove a statement (complements of unions) Ask Question Asked 11 years, 11 months ago Modified11 years, 11 months ago Viewed 6k times This question shows research effort; it is useful and clear 2 Save this question. Show activity on this post. I want to prove this statement: (A 1∪A 2)c=A 1 c∪A 2 c(A 1∪A 2)c=A 1 c∪A 2 c where the c c means the complement. Any help would be greatly appreciated. elementary-set-theory Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Nov 5, 2013 at 18:27 Lord_Farin 17.9k 9 9 gold badges 52 52 silver badges 132 132 bronze badges asked Nov 4, 2013 at 19:56 user1960836user1960836 211 3 3 silver badges 7 7 bronze badges 7 3 For future reference: meta.math.stackexchange.com/questions/5020/…shade4159 –shade4159 2013-11-04 19:57:49 +00:00 Commented Nov 4, 2013 at 19:57 3 You can not prove it! Its not right. Either the Left Hand Side or the Right Hand Side needs to have 'intersection' in place of 'union' but not both simultaneously!This is true only in the case that both A 1 A 1 and A 2 A 2 are empty sets!wannadeleteacct –wannadeleteacct 2013-11-04 20:01:20 +00:00 Commented Nov 4, 2013 at 20:01 @Manasi is right. Look up DeMorgan's Laws. Under complementation, unions become intersections and vice versa.Cameron L. Williams –Cameron L. Williams 2013-11-04 20:02:54 +00:00 Commented Nov 4, 2013 at 20:02 Sorry, my bad. It was supposed to be an intersection at the right hand side. I corrected it user1960836 –user1960836 2013-11-04 20:05:36 +00:00 Commented Nov 4, 2013 at 20:05 1 Don't correct it: you got two good answers. Others will have the same question as you and find the answers here. Leave this question as is, and ask a new question if you have one. Mho.msh210 –msh210 2013-11-04 20:06:45 +00:00 Commented Nov 4, 2013 at 20:06 \|Show 2 more comments 3 Answers 3 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. You will have trouble proving that (A 1∪A 2)c=A c 1∪A c 2(A 1∪A 2)c=A 1 c∪A 2 c, since it is not true, in general. (In fact, it holds precisely when A 1=A 2 A 1=A 2.) However, the following are true in general: (A 1∪A 2)c=A c 1∩A c 2(A 1∪A 2)c=A 1 c∩A 2 c (A 1∩A 2)c=A c 1∪A c 2(A 1∩A 2)c=A 1 c∪A 2 c Edit: In response to your (temporary) correction, let me say that you are on the right track. Since x∉A 1,x∉A 1, then by definition, x∈???x∈??? Since x∉A 2,x∉A 2, then x∈???x∈??? Consequently, what can we say? For the other inclusion, you'll basically be doing the same thing, but in reverse. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications edited Nov 5, 2013 at 20:16 answered Nov 4, 2013 at 20:01 Cameron BuieCameron Buie 105k 10 10 gold badges 106 106 silver badges 245 245 bronze badges Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. This is one of De Morgan's Laws. We want to prove that (A∪B)c=A c∩B c(A∪B)c=A c∩B c. Let x∈(A∪B)c x∈(A∪B)c. Then x∉A∪B x∉A∪B. So x∉A x∉A and x∉B x∉B. Therefore, x∈A c x∈A c and x∈B c x∈B c. It follows that x∈A c∩B c x∈A c∩B c. Thus (A∪B)c⊆A c∩B c(A∪B)c⊆A c∩B c. Now, let x∈A c∩B c x∈A c∩B c. Then x∈A c x∈A c and x∈B c x∈B c. So x∉A x∉A and x∉B x∉B. Therefore x∉A∪B x∉A∪B. It follows that x∈(A∪B)c x∈(A∪B)c. Thus A c∩B c⊆(A∩B)c A c∩B c⊆(A∩B)c. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Nov 5, 2013 at 19:03 1233dfv1233dfv 5,773 1 1 gold badge 28 28 silver badges 43 43 bronze badges Add a comment\| This answer is useful 0 Save this answer. Show activity on this post. This is false in the general case. For example, if A 1 A 1 is the set of integers ({…,−2,−1,0,1,2,…}{…,−2,−1,0,1,2,…}) and A 2 A 2 is the set of positive real numbers, and the universe is the set of all real numbers, then (A 1∪A 2)c(A 1∪A 2)c doesn't contain −1−1 or 1 2 1 2, but A c 1∪A c 2 A 1 c∪A 2 c contains them. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Nov 4, 2013 at 20:02 msh210msh210 3,958 2 2 gold badges 25 25 silver badges 38 38 bronze badges 4 1 Do you want me to start a new thread with almost the same question? I just made a typo. I'm not sure people are gonna benefit from that typo? It will rather be a waste of space to start a new thread I think user1960836 –user1960836 2013-11-04 20:12:30 +00:00 Commented Nov 4, 2013 at 20:12 I am still stuck on how to prove this statement, so help or hints is appreciated user1960836 –user1960836 2013-11-05 07:03:35 +00:00 Commented Nov 5, 2013 at 7:03 Are you not planning to revert your 'fix' to your question? Let me know, please, so that I can delete this now-rendered-wrong answer. Re "I am still stuck on how to prove this statement, so help or hints is appreciated", see the other answer. See also how to get answers to your questions; one suggestion that should perhaps be on that page is "don't drastically change the question after it has good answers".msh210 –msh210 2013-11-05 07:43:40 +00:00 Commented Nov 5, 2013 at 7:43 I see what you mean. I will revert it, so the answers provided match the old question that had a crucial typo, and create a new one with the right question user1960836 –user1960836 2013-11-05 17:48:11 +00:00 Commented Nov 5, 2013 at 17:48 Add a comment\| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions elementary-set-theory See similar questions with these tags. 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15132	https://dictionary.cambridge.org/us/dictionary/learner-english/pertinent	PERTINENT \| definition in the Cambridge Learner’s Dictionary Dictionary Translate Grammar Thesaurus +Plus Cambridge Dictionary +Plus Games Shop Cambridge Dictionary +Plus My profile +Plus help Log out {{userName}} Cambridge Dictionary +Plus My profile +Plus help Log out Log in / Sign up English (US) Learner’s Dictionary {{word}} {{#beta}} Beta{{/beta}} English Grammar English–Spanish Spanish–English Definition of pertinent – Learner’s Dictionary pertinent adjective formal uk Your browser doesn't support HTML5 audio /ˈpɜːtɪnənt/us Your browser doesn't support HTML5 audio Add to word listAdd to word list relatingdirectly to a subject: a pertinent question (Definition of pertinent from the Cambridge Learner's Dictionary © Cambridge University Press) Translations of pertinent in Chinese (Traditional) 有關的，直接相關的… See more in Chinese (Simplified) 有关的，直接相关的… See more in Spanish pertinente… See more in Portuguese pertinente… See more in more languages in Polish in Turkish in Russian stosowny, na temat… See more doğrudan bir konuyla bağlantılı/alakalı/ilgili, ilişkili, ilişkin… See more относящийся к делу, по существу… See more Need a translator? Get a quick, free translation! Translator tool Browse persuade persuasion persuasive pertain to sth pertinent perturbed Peru peruse pervade Word of the Day Victoria sponge UK Your browser doesn't support HTML5 audio /vɪkˌtɔː.ri.ə ˈspʌndʒ/ US Your browser doesn't support HTML5 audio /vɪkˌtɔːr.i.ə ˈspʌndʒ/ a soft cake made with eggs, sugar, flour, and a type of fat such as butter. 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15133	https://taxfoundation.org/data/all/state/sales-tax-rates/	2025 Sales Tax Rates \| Sales Taxes by State Skip to content U.S. Europe EN Subscribe Donate Research Research Areas Federal Tax State Tax Global Tax EU Tax All Research Meet Our Experts Featured Projects Tariff Tracker FAQ: One Big Beautiful Bill Act Tax Calculator Growth and Opportunity Tax Plan Federal Tax Modeling State Reforms for Mobility and Modernization State Tax Competitiveness Index International Tax Competitiveness Index European Tax Policy Scorecard Featured Issues Tariffs & Trade One Big Beautiful Bill Act Property Taxes Global Tax Deal Debt & Deficits Cost Recovery Taxes on Savers & Investors State Tax Trends Latest Research ### Trump Tariffs: Tracking the Economic Impact of the Trump Trade War 42 min read Experts Data Data Areas Federal Tax State Tax Global Tax EU Tax All Data Meet Our Experts Featured Projects Facts & Figures: How Does Your State Compare? 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Home•Data•State Taxes•State and Local Sales Tax Rates, Midyear 2025 See previous versions of this post State and Local Sales Tax Rates, Midyear 2025 July 8, 2025 August 6, 2025 17 min read By: Jared Walczak Download 2025 Data Print Retail sales taxes are an essential part of most states’ revenue toolkits, responsible for 32 percent of state taxA tax is a mandatory payment or charge collected by local, state, and national governments from individuals or businesses to cover the costs of general government services, goods, and activities. collections and 13 percent of local tax collections (24 percent of combined collections). They also benefit from being more pro-growth than the other major state tax, the individual income taxAn individual income tax (or personal income tax) is levied on the wages, salaries, investments, or other forms of income an individual or household earns. The U.S. imposes a progressive income tax where rates increase with income. The Federal Income Tax was established in 1913 with the ratification of the 16th Amendment. Though barely 100 years old, individual income taxes are the largest source, because they introduce fewer economic distortions. Forty-five states collect statewide sales taxes, while consumers also face local sales taxes in 38 states, including Alaska, which does not impose a statewide tax. These local rates can be substantial, and in some cases can rival or even exceed state rates, which means some states with moderate statewide sales taxA sales tax is levied on retail sales of goods and services and, ideally, should apply to all final consumption with few exemptions. Many governments exempt goods like groceries; base broadening, such as including groceries, could keep rates lower. A sales tax should exempt business-to-business transactions which, when taxed, cause tax pyramiding. rates actually impose quite high combined state and local rates compared to other states. The five states with the highest average combined state and local sales tax rates are Louisiana (10.11 percent), Tennessee (9.61 percent), Arkansas (9.48 percent), Washington (9.47 percent), and Alabama (9.44 percent). The five states with the lowest average combined rates are Alaska (1.82 percent), Hawaii (4.50 percent), Maine (5.50 percent), Wyoming (5.56 percent), and Wisconsin (5.72 percent). Nationwide, the population-weighted average sales tax rate is 7.52 percent, up from 7.49 percent in January. Excluding the five states without statewide sales taxes, the weighted average rate has riven from 7.68 to 7.72 percent. Sales tax rate differentials can induce consumers to shop across borders. Sales tax bases also impact how much revenue is collected from a tax and how the tax affects the economy. Sales taxes are just one part of an overall tax structure and should be considered in context. For example, Tennessee has high sales taxes but no income tax, whereas Oregon has no sales tax but high income taxes. While many factors influence business location and investment decisions, sales taxes are something within policymakers’ control that can have immediate impacts. The data below provides a population-weighted average of local sales taxes as of July 1, 2025, to give a sense of the average local rate for each state. The table provides a full state-by-state listing of state and local sales tax rates. 2025 Data 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 Sales Tax Rates as of July 1, 2025 Expand or Collapse Table State & Local Sales Tax Rates as of July 1, 2025 \| State \| State Tax Rate \| State Tax Rank \| Avg. Local Tax Rate \| Max Local \| Combined Tax Rate \| Combined Rank \| --- --- --- \| Alabama \| 4.00% \| 40 \| 5.44% \| 11.00% \| 9.44% \| 5 \| \| Alaska \| 0.00% \| 46 \| 1.82% \| 7.85% \| 1.82% \| 46 \| \| Arizona \| 5.60% \| 28 \| 2.92% \| 5.30% \| 8.52% \| 11 \| \| Arkansas \| 6.50% \| 9 \| 2.98% \| 6.125% \| 9.48% \| 3 \| \| California (a) \| 7.25% \| 1 \| 1.73% \| 5.25% \| 8.98% \| 7 \| \| Colorado \| 2.90% \| 45 \| 4.96% \| 8.30% \| 7.86% \| 16 \| \| Connecticut \| 6.35% \| 12 \| 0.00% \| 0.00% \| 6.35% \| 33 \| \| Delaware \| 0.00% \| 46 \| 0.00% \| 0.00% \| 0.00% \| 47 \| \| Florida \| 6.00% \| 17 \| 1.02% \| 2.00% \| 7.02% \| 24 \| \| Georgia \| 4.00% \| 40 \| 3.44% \| 5.00% \| 7.44% \| 19 \| \| Hawaii (b) \| 4.00% \| 40 \| 0.50% \| 0.50% \| 4.50% \| 45 \| \| Idaho \| 6.00% \| 17 \| 0.03% \| 3.00% \| 6.03% \| 37 \| \| Illinois \| 6.25% \| 13 \| 2.67% \| 4.75% \| 8.92% \| 8 \| \| Indiana \| 7.00% \| 2 \| 0.00% \| 0.00% \| 7.00% \| 25 \| \| Iowa \| 6.00% \| 17 \| 0.94% \| 2.00% \| 6.94% \| 29 \| \| Kansas \| 6.50% \| 9 \| 2.28% \| 4.25% \| 8.78% \| 9 \| \| Kentucky \| 6.00% \| 17 \| 0.00% \| 0.00% \| 6.00% \| 38 \| \| Louisiana \| 5.00% \| 32 \| 5.11% \| 7.00% \| 10.11% \| 1 \| \| Maine \| 5.50% \| 29 \| 0.00% \| 0.00% \| 5.50% \| 44 \| \| Maryland \| 6.00% \| 17 \| 0.00% \| 0.00% \| 6.00% \| 38 \| \| Massachusetts \| 6.25% \| 13 \| 0.00% \| 0.00% \| 6.25% \| 35 \| \| Michigan \| 6.00% \| 17 \| 0.00% \| 0.00% \| 6.00% \| 38 \| \| Minnesota \| 6.875% \| 6 \| 1.26% \| 3.00% \| 8.13% \| 15 \| \| Mississippi \| 7.00% \| 2 \| 0.06% \| 1.00% \| 7.06% \| 23 \| \| Missouri \| 4.225% \| 38 \| 4.19% \| 5.875% \| 8.41% \| 12 \| \| Montana (c) \| 0.00% \| 46 \| 0.00% \| 0.00% \| 0.00% \| 47 \| \| Nebraska \| 5.50% \| 29 \| 1.48% \| 2.00% \| 6.98% \| 28 \| \| Nevada \| 6.85% \| 7 \| 1.39% \| 1.525% \| 8.24% \| 13 \| \| New Hampshire \| 0.00% \| 46 \| 0.00% \| 0.00% \| 0.00% \| 47 \| \| New Jersey (d) \| 6.625% \| 8 \| -0.02% \| 3.3125% \| 6.60% \| 30 \| \| New Mexico (b) \| 4.875% \| 35 \| 2.79% \| 4.5625% \| 7.67% \| 17 \| \| New York \| 4.00% \| 40 \| 4.54% \| 4.875% \| 8.54% \| 10 \| \| North Carolina \| 4.75% \| 36 \| 2.25% \| 2.75% \| 7.00% \| 27 \| \| North Dakota \| 5.00% \| 32 \| 2.08% \| 3.50% \| 7.08% \| 22 \| \| Ohio \| 5.75% \| 27 \| 1.55% \| 2.25% \| 7.30% \| 21 \| \| Oklahoma \| 4.50% \| 37 \| 4.55% \| 7.00% \| 9.05% \| 6 \| \| Oregon \| 0.00% \| 46 \| 0.00% \| 0.00% \| 0.00% \| 47 \| \| Pennsylvania \| 6.00% \| 17 \| 0.34% \| 2.00% \| 6.34% \| 34 \| \| Rhode Island \| 7.00% \| 2 \| 0.00% \| 0.00% \| 7.00% \| 25 \| \| South Carolina \| 6.00% \| 17 \| 1.49% \| 3.00% \| 7.49% \| 18 \| \| South Dakota (b) \| 4.20% \| 39 \| 1.91% \| 4.50% \| 6.11% \| 36 \| \| Tennessee \| 7.00% \| 2 \| 2.61% \| 2.75% \| 9.61% \| 2 \| \| Texas \| 6.25% \| 13 \| 1.95% \| 2.00% \| 8.20% \| 14 \| \| Utah (a) \| 6.10% \| 16 \| 1.32% \| 4.70% \| 7.42% \| 20 \| \| Vermont \| 6.00% \| 17 \| 0.39% \| 1.00% \| 6.39% \| 32 \| \| Virginia (a) \| 5.30% \| 31 \| 0.47% \| 2.70% \| 5.77% \| 41 \| \| Washington \| 6.50% \| 9 \| 2.97% \| 4.10% \| 9.47% \| 4 \| \| West Virginia \| 6.00% \| 17 \| 0.58% \| 1.00% \| 6.58% \| 31 \| \| Wisconsin \| 5.00% \| 32 \| 0.72% \| 2.90% \| 5.72% \| 42 \| \| Wyoming \| 4.00% \| 40 \| 1.56% \| 3.00% \| 5.56% \| 43 \| \| District of Columbia \| 6.00% \| \| 0.00% \| 0.000% \| 6.00% \| \| Note: City, county and municipal rates vary. Local rates are weighted by population to compute an average local tax rate. (a) Three states levy mandatory, statewide, local add-on sales taxes at the state level: California (1.25%), Utah (1.25%), and Virginia (1%). We include these in their state sales tax. (b) The sales taxes in Hawaii, New Mexico, and South Dakota have broad bases that include many business-to-business services. (c) Special taxes in local resort areas are not counted here. (d) Salem County, N.J., is not subject to the statewide sales tax rate and collects a local rate of 3.3125%. New Jersey’s local score is represented as a negative. Sources: Sales Tax Clearinghouse; Tax Foundation calculations; State Revenue Department websites. Download Data Data compiled by Jared Walczak Sales Tax Rates as of January 1, 2025 Expand or Collapse Table 2025 Sales Tax Rates by State Combined State & Average Local Sales Tax Rates, January 2025 \| State \| State Tax Rate \| State Tax Rank \| Avg. Local Tax Rate \| Max Local \| Combined Tax Rate \| Combined Rank \| --- --- --- \| Alabama \| 4.00% \| 40 \| 5.427% \| 8.00% \| 9.427% \| 5 \| \| Alaska \| 0.00% \| 46 \| 1.821% \| 7.85% \| 1.821% \| 46 \| \| Arizona \| 5.60% \| 28 \| 2.814% \| 5.30% \| 8.414% \| 11 \| \| Arkansas \| 6.50% \| 9 \| 2.960% \| 6.125% \| 9.460% \| 3 \| \| California (a) \| 7.25% \| 1 \| 1.552% \| 4.75% \| 8.802% \| 8 \| \| Colorado \| 2.90% \| 45 \| 4.957% \| 8.30% \| 7.857% \| 16 \| \| Connecticut \| 6.35% \| 12 \| 0.000% \| 0.00% \| 6.350% \| 33 \| \| Delaware \| 0.00% \| 46 \| 0.000% \| 0.00% \| 0.000% \| 47 \| \| Florida \| 6.00% \| 17 \| 0.948% \| 2.00% \| 6.948% \| 28 \| \| Georgia \| 4.00% \| 40 \| 3.418% \| 5.00% \| 7.418% \| 19 \| \| Hawaii (b) \| 4.00% \| 40 \| 0.500% \| 0.50% \| 4.500% \| 45 \| \| Idaho \| 6.00% \| 17 \| 0.027% \| 3.00% \| 6.027% \| 37 \| \| Illinois \| 6.25% \| 13 \| 2.640% \| 4.75% \| 8.890% \| 7 \| \| Indiana \| 7.00% \| 2 \| 0.000% \| 0.00% \| 7.000% \| 24 \| \| Iowa \| 6.00% \| 17 \| 0.942% \| 2.00% \| 6.942% \| 29 \| \| Kansas \| 6.50% \| 9 \| 2.273% \| 4.25% \| 8.773% \| 9 \| \| Kentucky \| 6.00% \| 17 \| 0.000% \| 0.00% \| 6.000% \| 38 \| \| Louisiana \| 5.00% \| 32 \| 5.116% \| 7.00% \| 10.116% \| 1 \| \| Maine \| 5.50% \| 29 \| 0.000% \| 0.00% \| 5.500% \| 43 \| \| Maryland \| 6.00% \| 17 \| 0.000% \| 0.00% \| 6.000% \| 38 \| \| Massachusetts \| 6.25% \| 13 \| 0.000% \| 0.00% \| 6.250% \| 35 \| \| Michigan \| 6.00% \| 17 \| 0.000% \| 0.00% \| 6.000% \| 38 \| \| Minnesota \| 6.875% \| 6 \| 1.250% \| 3.00% \| 8.125% \| 15 \| \| Mississippi \| 7.00% \| 2 \| 0.062% \| 1.00% \| 7.062% \| 22 \| \| Missouri \| 4.225% \| 38 \| 4.185% \| 5.875% \| 8.410% \| 12 \| \| Montana (c) \| 0.00% \| 46 \| 0.000% \| 0.00% \| 0.000% \| 47 \| \| Nebraska \| 5.50% \| 29 \| 1.472% \| 2.00% \| 6.972% \| 27 \| \| Nevada \| 6.85% \| 7 \| 1.386% \| 1.53% \| 8.236% \| 13 \| \| New Hampshire \| 0.00% \| 46 \| 0.000% \| 0.00% \| 0.000% \| 47 \| \| New Jersey (d) \| 6.625% \| 8 \| -0.024% \| 3.313% \| 6.601% \| 30 \| \| New Mexico (b) \| 4.875% \| 35 \| 2.752% \| 4.563% \| 7.627% \| 17 \| \| New York \| 4.00% \| 40 \| 4.532% \| 4.875% \| 8.532% \| 10 \| \| North Carolina \| 4.75% \| 36 \| 2.246% \| 2.75% \| 6.996% \| 26 \| \| North Dakota \| 5.00% \| 32 \| 2.050% \| 3.50% \| 7.050% \| 23 \| \| Ohio \| 5.75% \| 27 \| 1.483% \| 2.25% \| 7.233% \| 21 \| \| Oklahoma \| 4.50% \| 37 \| 4.505% \| 7.00% \| 9.005% \| 6 \| \| Oregon \| 0.00% \| 46 \| 0.000% \| 0.00% \| 0.000% \| 47 \| \| Pennsylvania \| 6.00% \| 17 \| 0.341% \| 2.00% \| 6.341% \| 34 \| \| Rhode Island \| 7.00% \| 2 \| 0.000% \| 0.00% \| 7.000% \| 24 \| \| South Carolina \| 6.00% \| 17 \| 1.499% \| 3.00% \| 7.499% \| 18 \| \| South Dakota (b) \| 4.20% \| 39 \| 1.914% \| 4.50% \| 6.114% \| 36 \| \| Tennessee \| 7.00% \| 2 \| 2.556% \| 2.75% \| 9.556% \| 2 \| \| Texas \| 6.25% \| 13 \| 1.951% \| 2.00% \| 8.201% \| 14 \| \| Utah (a) \| 6.10% \| 16 \| 1.219% \| 4.20% \| 7.319% \| 20 \| \| Vermont \| 6.00% \| 17 \| 0.366% \| 1.00% \| 6.366% \| 32 \| \| Virginia (a) \| 5.30% \| 31 \| 0.471% \| 2.70% \| 5.771% \| 41 \| \| Washington \| 6.50% \| 9 \| 2.929% \| 4.10% \| 9.429% \| 4 \| \| West Virginia \| 6.00% \| 17 \| 0.569% \| 1.00% \| 6.569% \| 31 \| \| Wisconsin \| 5.00% \| 32 \| 0.702% \| 2.90% \| 5.702% \| 42 \| \| Wyoming \| 4.00% \| 40 \| 1.441% \| 2.00% \| 5.441% \| 44 \| \| District of Columbia \| 6.00% \| \| 0.000% \| 0.000% \| 6.000% \| \| Note: City, county, and municipal rates vary. These rates are weighted by population to compute an average local tax rate. The sales taxes in Hawaii, New Mexico, and South Dakota have broad bases that include many business-to-business services. D.C.'s rank does not affect states' ranks, but the figure in parentheses indicates where it would rank if included. (a) Three states levy mandatory, statewide, local add-on sales taxes at the state level: California (1.25%), Utah (1.25%), and Virginia (1%). We include these in their state sales tax. (b) The sales taxes in Hawaii, New Mexico, and South Dakota have broad bases that include many business-to-business services. (c) Special taxes in local resort areas are not counted here. (d) Salem County, N.J., is not subject to the statewide sales tax rate and collects a local rate of 3.3125%. New Jersey’s local score is represented as a negative. Sources: Sales Tax Clearinghouse; Tax Foundation calculations; State Revenue Department websites. Download Data Data compiled by Jared Walczak Sales Tax Rates as of July 1, 2024 Data compiled byJared Walczak See Prior Analysis Sales Tax Rates as of January 1, 2024 Data compiled byJared Walczak See Prior Analysis Sales Tax Rates as of July 1, 2023 See Prior Analysis Sales Tax Rates as of January 1, 2023 Data compiled by Janelle Fritts See Prior Analysis Sales Tax Rates as of July 1, 2022 Data compiled by Janelle Fritts See Prior Analysis Sales Tax Rates as of January 1, 2022 Data compiled by Janelle Fritts See Prior Analysis Sales Tax Rates as of July 1, 2021 Data compiled by Janelle Fritts See Prior Analysis Sales Tax Rates as of January 1, 2021 Data compiled by Janelle Fritts See Prior Analysis Sales Tax Rates as of July 1, 2020 Data compiled by Janelle Fritts See Prior Analysis Sales Tax Rates as of January 1, 2020 Data compiled by Janelle Fritts See Prior Analysis Sales Tax Rates as of July 1, 2019 Data compiled by Janelle Fritts See Prior Analysis Sales Tax Rates as of January 1, 2019 Data compiled by Janelle Fritts See Prior Analysis Sales Tax Rates as of July 1, 2018 Data compiled by Jared Walczak, Scott Drenkard See Prior Analysis Sales Tax Rates as of January 1, 2018 Data compiled by Jared Walczak, Scott Drenkard See Prior Analysis Sales Tax Rates as of July 1, 2017 Data compiled by Jared Walczak, Scott Drenkard See Prior Analysis Sales Tax Rates as of January 1, 2017 Data compiled by Jared Walczak, Scott Drenkard See Prior Analysis Sales Tax Rates as of July 1, 2016 Data compiled by Jared Walczak, Scott Drenkard See Prior Analysis Sales Tax Rates as of January 1, 2016 Data compiled by Nicole Kaeding, Scott Drenkard See Prior Analysis Sales Tax Rates as of July 1, 2015 Data compiled by Jared Walczak, Scott Drenkard See Prior Analysis Sales Tax Rates as of January 1, 2015 Data compiled by Jared Walczak, Scott Drenkard See Prior Analysis Sales Tax Rates as of July 1, 2014 Data compiled by Scott Drenkard, Liz Emanuel, Jordan Yahiro See Prior Analysis Sales Tax Rates as of January 1, 2014 Data compiled by Scott Drenkard See Prior Analysis Sales Tax Rates as of July 1, 2013 Data compiled by Scott Drenkard See Prior Analysis Sales Tax Rates as of January 1, 2013 Data compiled by Scott Drenkard See Prior Analysis State Sales Tax Rates Five states forego statewide sales taxes: Alaska,Delaware,Montana,New Hampshire, and Oregon. Of these, only Alaska allows localities to impose local sales taxes. California has the highest state-level sales tax rate, at 7.25 percent.Four states tie for the second-highest statewide rate, at 7 percent:Indiana,Mississippi,Rhode Island, and Tennessee. The lowest non-zero state-level sales tax is in Colorado, which has a rate of 2.9 percent. Five states follow with 4 percent rates: Alabama,Georgia, Hawaii,New York, and Wyoming. Louisiana is the most recent state to raise its sales tax rate. The state rate increased from 4.45 to 5.0 percent in January, reversing a prior reduction implemented in July 2018. This rate increase was part of a broader tax reform package that yielded a 3 percent flat individual income tax, a 5.5 percent corporate income taxA corporate income tax (CIT) is levied by federal and state governments on business profits. Many companies are not subject to the CIT because they are taxed as pass-through businesses, with income reportable under the individual income tax., full expensingFull expensing allows businesses to immediately deduct the full cost of certain investments in new or improved technology, equipment, or buildings. It alleviates a bias in the tax code and incentivizes companies to invest more, which, in the long run, raises worker productivity, boosts wages, and creates more jobs., and franchise tax repeal. Prior to that,South Dakotacut its state sales tax rate in 2023, a reduction set to expire after 2026, and New Mexico lowered the rate of its state-level sales tax—a hybrid tax the state refers to as itsgross receipts taxGross receipts taxes are applied to a company’s gross sales, without deductions for a firm’s business expenses, like compensation, costs of goods sold, and overhead costs. Unlike a sales tax, a gross receipts tax is assessed on businesses and applies to transactions at every stage of the production process, leading to tax pyramiding.—from 5.125 percent to 5 percent in July 2022. Notably, if the revenue from the gross receipts tax in any single fiscal year from 2026 to 2029 is less than 95 percent of the previous year’s revenue, then the state’s rate will return to 5.125 percent on the following July 1. Before that, the most recent statewide rate reduction was Louisiana’s now-reversed cut in July 2018. Sales tax rate reductions have been relatively rare in recent years, as state lawmakers have instead prioritized income tax cuts, which yield more economic benefit, reducing individual or corporate income tax rates (or both) in 28 states since 2021. The continued erosion of sales tax bases has also been a factor. Local Sales Tax Rates The five states with the highest average local sales tax rates are Alabama (5.44 percent), Louisiana (5.11 percent), Colorado (4.96 percent), Oklahoma (4.55 percent), and New York (4.54 percent). The first half of 2025 represents an unusually busy period for local sales tax rate changes, with many localities across the country raising rates and only a few lowering them or allowing higher rates to sunset. These increases were largest on net in California, Wyoming, Arizona, Utah, Florida, Ohio, and Tennessee. In California, a countywide 0.25 percent increase in Los Angeles’s sales tax, intended to address homelessness and fund affordable housing initiatives, helped drive the average local rate upward, with Butte (1 percent), Humboldt (0.5 percent), and Sonoma (0.25 percent) also raising county rates, while Mariposa County’s rate declined by 0.5 percent. Many cities also raised rates, with Fontana, Moreno Valley, Escondido, Clovis, San Marcos, San Ramon, Buena Park, Napa, Apple Valley, Davis, and others raising rates by 1 percentage point. Most changes took effect April 1, though a few increases were effective July 1. Illinois similarly had rate increases across a wide range of jurisdictions, although generally much smaller ones. Two counties and 48 other municipal governments increased rates since January 1, with McLean County (1 percent increase) the largest population jurisdiction to do so. Other notable increases include 1 percent in the city of Rock Island, 0.75 percent in the village of Bartlett, and 0.5 percent in Whiteside County. Washington likewise saw a raft of local tax increases. Among the most significant were a 0.1 percent increase in Spokane’s sales tax for public safety and a 0.2 percent increase in Vancouver for cultural access programs. Sales tax rates increased countywide in Lewis and Kittitas (both 0.2 percent) as well, and in many other jurisdictions, often for transportation benefit districts. An unusual situation in Florida has been wound down. Previously, two Hillsborough County discretionary sales surtaxes totaling 1 percent had been temporarily suspended to offset collections under a Hillsborough transportation surtaxA surtax is an additional tax levied on top of an already existing business or individual tax and can have a flat or progressive rate structure. Surtaxes are typically enacted to fund a specific program or initiative, whereas revenue from broader-based taxes, like the individual income tax, typically cover a multitude of programs and services. that was collected for three years before being struck down as unconstitutional in 2021. Revenues from the unconstitutional surtax were intended to be used as temporary replacement funding sources for the suspended surtaxes. That process came to a close on May 31, after which the county’s 1 percent higher rate was restored. Other states saw fewer rate changes, though some were significant. In Arizona, Phoenix’s sales tax rate increased by 0.5 percent on July 1. Tennessee’s Davidson County, home of Nashville, also saw a 0.5 percent tax increase, bringing Nashville’s rate to 9.75 percent as of February 1. North Dakota saw sales tax increases in Bismarck (1.5 to 2 percent), Fargo (2 to 2.25 percent), and Hillsboro (2.5 to 3 percent). Seven small jurisdictions in New Mexico raised rates, the largest of which were Las Cruces (a 0.325 percentage point increase) and Roswell (0.375 percent). The Central Ohio Transit Authority imposed a 0.5 percent sales tax rate increase for Franklin County, Ohio, along with the portions of four other counties that fall within the transit district. In Oklahoma, Cotton and Tillman counties raised rates from 2 to 3 percent, while Muskogee County’s rate rose from 0.65 to 1.499 percent. Utah’s Cache and Servier counties exercised the authority to impose an additional 0.3 percent sales tax to fund transportation, while Hatch County implemented a 1.1 percent resort community tax, and Huntsville, Mapleton, and Saratoga Springs all adopted their own sales tax increases. And in Wyoming, reduced rates in Crook and Fremont counties were offset by 1 percent increases in Teton County and in the city of Casper, which adopted its increase on a temporary basis. Some cities in New Jersey are in “Urban Enterprise Zones,” where qualifying sellers may collect and remit at half the 6.625 percent statewide sales tax rate (3.3125 percent), a policy designed to help local retailers compete with neighboring Delaware, which forgoes a sales tax. We represent this anomaly as a negative 0.03 percent statewide average local rate (adjusting for population as described in the methodology section below), and the combined rate reflects this subtraction. Despite the slightly favorable impact on the overall rate, this lower rate represents an implicit acknowledgment by New Jersey officials that their 6.625 percent statewide rate is uncompetitive with neighboring Delaware’s lack of a sales tax. The Role of Competition in Setting Sales Tax Rates Avoidance of sales tax is most likely to occur in areas where there is a significant difference between jurisdictions’ rates.Researchindicatesthatconsumers can and do leave high-tax areas to make major purchases in low-tax areas, such as from cities to suburbs.For example,evidencesuggests that Chicago-area consumers make major purchases in surrounding suburbs or online to avoid Chicago’s 10.25 percent sales tax rate. At the statewide level, businesses sometimes locate just outside the borders of high sales-tax areas to avoid being subjected to their rates. A stark example of this occurs in New England, where even though I-91 runs up theVermontside of theConnecticutRiver, many more retail establishments choose to locate on the New Hampshire side to avoid sales taxes. Onestudyshows that per capita sales in border counties in sales tax-free New Hampshire have tripled since the late 1950s, while per capita sales in border counties in Vermont have remained stagnant. At one time, Delaware actuallyused its highway welcome signto remind motorists that Delaware is the “Home of Tax-Free Shopping.” State and local governments should be cautious about raising rates too high relative to their neighbors because doing so will yield less revenue than expected or, in extreme cases, revenue losses despite the higher tax rate. Sales Tax Bases: The Other Half of the Equation Sales tax rates differ by state, but sales tax bases also impact how much revenue is collected from a tax and how the tax affects the economy. This report ranks states based on tax rates and does not account for differences in tax bases (e.g., the structure of sales taxes, defining what is taxable and nontaxable). Statescan vary greatlyin this regard. For instance, most states exempt groceries from the sales tax, others tax groceries at a limited rate, and still others tax groceries at the same rate as all other products. Some states exempt clothing or tax it at a reduced rate. Tax expertsgenerally recommendthat sales taxes apply to all final retail sales of goods and services but not intermediate business-to-business transactions in the production chain. These recommendations would result in a tax system that is not only broad-based but also “right-sized,” applying once and only once to each product the market produces.Despite agreement in theory, the application of most state sales taxes isfar from this ideal, and occasionally gets worse. Ideally, states wouldmodernize their sales tax regimesto better align with personal consumption in a changing economy. Hawaii has thebroadestsales tax in theUnited States, but it taxes many products multiple times and, by one estimate, ultimately taxes 119 percent of the state’s personal income.This base is far wider than thenational median, where the sales tax applies to 36 percent of personal income. Methodology Sales Tax Clearinghouse publishes quarterly sales tax data at the state, county, and city levels by ZIP code. We weight these numbers according to the most recent Census population figures to give a sense of the prevalence of sales tax rates in a particular state. This is a change from previous editions, where we used figures available every decade. While changes due to the new weighting were mostly trivial, we show changes in rank based on January 1 figures recalculated under the new population weighting. Due to the updated population weighting, this report is not strictly comparable to previously published editions, though differences amount to minor rounding errors. It should also be noted that while the Census Bureau reports population data using a five-digit identifier that looks much like a ZIP code, this is actually a ZIP Code Tabulation Area (ZCTA), which attempts to create a geographical area associated with a given ZIP code. This is done because a surprisingly large number of ZIP codes do not actually have any residents. For example, the National Press Building in Washington, DC, has its own ZIP code solely for postal reasons. For our purposes, ZIP codes that do not have a corresponding ZCTA population figure are omitted from calculations. These omissions result in some amount of inexactitude but overall do not have a significant effect on resultant averages because proximate ZIP code areas that do have ZCTA population numbers capture the tax rate of those jurisdictions. Stay informed on the tax policies impacting you. Subscribe to get insights from our trusted experts delivered straight to your inbox. Subscribe This number includes mandatory add-on taxes that are collected by the state but distributed to local governments. Because of this, some sources will describe California's sales tax as 6.0 percent. A similar situation exists in Utah and Virginia. The sales taxes in Hawaii, New Mexico, and South Dakota have bases that include many business services, so they are not strictly comparable to other sales taxes. Share this article Twitter LinkedIn Facebook Email Timeline of Activity Back to Top Previous Versions Data July 9, 2024 February 4, 2025 State and Local Sales Tax Rates, Midyear 2024 8 min read Data February 6, 2024 February 4, 2025 State and Local Sales Tax Rates, 2024 9 min read Data July 17, 2023 February 4, 2025 State and Local Sales Tax Rates, Midyear 2023 8 min read Data February 7, 2023 February 4, 2025 State and Local Sales Tax Rates, 2023 11 min read Data July 19, 2022 February 4, 2025 State and Local Sales Tax Rates, Midyear 2022 12 min read Data February 3, 2022 February 4, 2025 State and Local Sales Tax Rates, 2022 12 min read Data July 8, 2021 February 4, 2025 State and Local Sales Tax Rates, Midyear 2021 12 min read Data January 6, 2021 February 4, 2025 State and Local Sales Tax Rates, 2021 12 min read Data July 8, 2020 February 4, 2025 State and Local Sales Tax Rates, Midyear 2020 12 min read Data January 15, 2020 February 4, 2025 State and Local Sales Tax Rates, 2020 12 min read Data July 10, 2019 February 4, 2025 State and Local Sales Tax Rates, Midyear 2019 13 min read Data January 30, 2019 February 4, 2025 State and Local Sales Tax Rates, 2019 11 min read Data July 16, 2018 February 4, 2025 State and Local Sales Tax Rates, Midyear 2018 12 min read Data February 13, 2018 February 4, 2025 State and Local Sales Tax Rates, 2018 11 min read Data July 5, 2017 February 4, 2025 State and Local Sales Tax Rates, Midyear 2017 11 min read Data January 31, 2017 February 4, 2025 State and Local Sales Tax Rates, 2017 11 min read Data July 5, 2016 February 4, 2025 State and Local Sales Tax Rates, Midyear 2016 11 min read Data March 9, 2016 February 4, 2025 State and Local Sales Tax Rates, 2016 11 min read Data July 9, 2015 February 4, 2025 State and Local Sales Tax Rates, Midyear 2015 13 min read Data April 8, 2015 February 4, 2025 State and Local Sales Tax Rates, 2015 13 min read Data September 16, 2014 February 4, 2025 State and Local Sales Tax Rates, Midyear 2014 13 min read Data March 18, 2014 February 4, 2025 State and Local Sales Tax Rates, 2014 11 min read Data August 28, 2013 February 4, 2025 State and Local Sales Tax Rates, Midyear 2013 11 min read Data February 11, 2013 December 16, 2024 State and Local Sales Tax Rates in 2013 10 min read All Related Articles Data September 2, 2025 August 29, 2025 Gas Taxes by State, 2025 California levies the highest tax on gasoline at 70.9 cents per gallon (cpg), followed by Illinois at 66.4 cpg and Washington at 59.0 cpg. 8 min read Blog February 28, 2025 Proposed Estate Tax Changes in Oregon: A Long-Overdue Reform? While evaluating new estate tax bills this legislative session, Oregon legislators should consider the state’s competitive tax landscape and interstate migration patterns. 4 min read Data November 26, 2024 August 5, 2025 Cigarette Taxes and Cigarette Smuggling by State, 2022 Growing cigarette tax levels and differentials have made cigarette smuggling both a national problem and a lucrative criminal enterprise. 16 min read Topics Data Individual and Consumption Taxes Local Option Sales Taxes Online Sales Taxes Sales Taxes Tags Tags: Local Taxes Sales Tax Exclusions & Exemptions State and Local Tax Collections State Tax Trends Locations Locations: Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota State Tennessee Texas The District of Columbia United States Utah Vermont Virginia Washington West Virginia Wisconsin Wyoming Authors Expert Jared Walczak Vice President of State Projects Stay informed on the tax policies impacting you. 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15134	https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:exp/x2ec2f6f830c9fb89:exp-properties/v/simplifying-exponent-expression-with-division	Use of cookies Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy Privacy Preference Center When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer. More information Manage Consent Preferences Strictly Necessary Cookies Always Active Certain cookies and other technologies are essential in order to enable our Service to provide the features you have requested, such as making it possible for you to access our product and information related to your account. For example, each time you log into our Service, a Strictly Necessary Cookie authenticates that it is you logging in and allows you to use the Service without having to re-enter your password when you visit a new page or new unit during your browsing session. Functional Cookies These cookies provide you with a more tailored experience and allow you to make certain selections on our Service. For example, these cookies store information such as your preferred language and website preferences. Targeting Cookies These cookies are used on a limited basis, only on pages directed to adults (teachers, donors, or parents). We use these cookies to inform our own digital marketing and help us connect with people who are interested in our Service and our mission. We do not use cookies to serve third party ads on our Service. Performance Cookies These cookies and other technologies allow us to understand how you interact with our Service (e.g., how often you use our Service, where you are accessing the Service from and the content that you’re interacting with). Analytic cookies enable us to support and improve how our Service operates. For example, we use Google Analytics cookies to help us measure traffic and usage trends for the Service, and to understand more about the demographics of our users. We also may use web beacons to gauge the effectiveness of certain communications and the effectiveness of our marketing campaigns via HTML emails. Cookie List Consent Leg.Interest label label label
15135	https://math.stackexchange.com/questions/tagged/quadratics	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 Questions tagged [quadratics] Ask Question Questions about quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$. Learn more¦ Top users Synonyms (1) 5,594 questions Newest Active Bountied Unanswered Bountied 0 Unanswered Frequent Score Trending Week Month Unanswered (my tags) -2 votes 1 answer 75 views Explain Alpern's method for solving a quadratic modular equation $a¢x^2 + b¢x + c ¡ 0 \pmod n$ [closed] This question will be about Dario Alpern's calculator for $a¢x^2 + b¢x + c ¡ 0 \pmod n$. (Implemented in the function quadmod.c). I have tried to understand it using ChatGPT. First I have asked to ... quadratics R. S. 110 5 votes 1 answer 516 views Prove that $\frac{2a}{a^2+2b^2+3}+\frac{2b}{b^2+2c^2+3}+\frac{2c}{c^2+2a^2+3} \leq 1$ for real numbers How do we prove that for all $a, b, c \in \mathbb{R}$, $$\frac{2a}{a^2+2b^2+3}+\frac{2b}{b^2+2c^2+3}+\frac{2c}{c^2+2a^2+3} \leq 1.$$ I haven't really made much progress in finding a way to tackle this.... algebra-precalculus inequality quadratics symmetric-polynomials uvw MilesB 920 4 votes 0 answers 98 views Rational pre-images of a quadratic polynomial Suppose I'm given rational numbers ${y_1,\dots,y_m} \subset \mathbb{Q}$. It is easy to construct a quadratic polynomial $f(x) = ax^2+bx+c$ such that all of the values $y_i$ are the $y$-coordinate of ... number-theory polynomials systems-of-equations quadratics diophantine-equations MathManiac5772 860 0 votes 3 answers 198 views An upper and lower bound of $abcd$ Problem. Given $a,\,b,\,c,\,d$ be non-negative real numbers. Denote $$u = \frac{a+b+c+d}{4}, \quad v^2 = \frac{ab+ac+ad+bc+bd+cd}{6}.$$ Prove that $$12u^2v^2-8u^4-3v^4-8u(u^2-v^2)\sqrt{u^2-v^2} ... inequality quadratics symmetric-polynomials discriminant rolles-theorem Nguyenhuyen_AG 6,174 1 vote 1 answer 81 views Find scalar that minimises spectral radius of a matrix I have a matrix $A$ that is a quadratic function of a real scalar $\beta$ and real constant matrices $B,C,D$: $$ A = B + \beta C + \beta^2 D $$ I want to find the value of $\beta$ that minimises the ... matrices optimization quadratics implicit-function-theorem spectral-radius Jake Levi 245 1 vote 2 answers 111 views Prove $\frac{1}{(2a+1)(2b+1)}+\frac{1}{(2b+1)(2c+1)}+\frac{1}{(2c+1)(2a+1)} \geqslant \frac{3}{3+2(ab+bc+ca)}.$ Problem. Let $a,b,c$ are positive real numbers. Prove that $$\frac{1}{(2a+1)(2b+1)}+\frac{1}{(2b+1)(2c+1)}+\frac{1}{(2c+1)(2a+1)} \geqslant \frac{3}{3+2(ab+bc+ca)}.$$ The inequality is equivalent to $$... inequality quadratics symmetric-polynomials discriminant uvw user1656827 2 votes 3 answers 150 views Given a quadratic $f(x) = ax^2 + bx + c$ with $a > 100$, what is the maximum number of integers $x$ satisfying $\|f(x)\| \leq 50$? I'm currently stuck on an interesting problem involving quadratic functions. Here's the setup: Given a quadratic $f(x) = ax^2 + bx + c$ with $a > 100$, what is the maximum number of integers $x$ ... polynomials quadratics SpaceGu 307 -1 votes 2 answers 150 views Do we need to specify "exception for $x=-5$ and $x=3$" when asking about real roots of equations? [closed] I want to write a question but I am not sure whether the first version is enough. Determine the range of $p$ such that $px^2+(2p-1)x -15p+4=0$ has real roots. Or should I rewrite it as follows: ... functions polynomials roots quadratics real-numbers D G 420 -2 votes 4 answers 137 views Determine $m \in \mathbb Z$ for which $x_1, x_2\in \mathbb Z$ [closed] Consider the equation $(m-1)x^2-(3-m)x-m=0$ with m real numbers $m$ different from $1$, having roots $x_1, x_2$. Determine $m \in \mathbb Z$ for which $x_1, x_2\in \mathbb Z$. my ideas So I was able ... quadratics integers parametric Pam Munoz Ryan 328 3 votes 6 answers 206 views If the equation $\ln\left(x^2+5x\right)-\ln(x+a+3)=0$ has exactly one solution for $x$, then possible integral value of $a$ is If the equation $\ln\left(x^2+5x\right)-\ln(x+a+3)=0$ has exactly one solution for $x$, then find interval of values of $a$ My Approach: Domain of logarithmic function $\ln(x^2+5x)$ is $x\in (-\infty, ... algebra-precalculus quadratics mathophile 4,668 -3 votes 1 answer 129 views Fill in missing details in derivation of square roots of any nonzero complex number [duplicate] In "Complex analysis, 3rd edition" by Joseph Bak and Donald J. Newman, there is a derivation of square roots of any nonzero complex number $a+bi$. I cannot see how the last expression was ... algebra-precalculus complex-numbers quadratics radicals ezyman 103 0 votes 2 answers 171 views Minimize $a^2+b^2+c^2+d^2$ subject to three quadratic conditions [duplicate] Let real numbers $a$, $b$, $c$ satisfy $\begin{cases}(a+b)(c+d)=25,\(a+c)(b+d)=20,\(a+d)(b+c)=7\end{cases}$, find the minimum of $a^2+b^2+c^2+d^2$. This is the last problem in the O level exam of ... inequality optimization contest-math quadratics maxima-minima youthdoo 4,587 1 vote 2 answers 80 views Find the set of values of $k(k\in R)$ for which the equation $x^2 - 4\|x\| + 3 - \|k-1\| = 0$ will have exactly 4 real roots. The question: Find the set of values of $k(k\in R)$ for which the equation $x^2 - 4\|x\| + 3 - \|k-1\| = 0$ will have exactly 4 real roots. Now, I reasoned that if we remove the modulus on the $x$ and it ... polynomials quadratics Supernerd411 701 1 vote 1 answer 55 views Find all values of $k\in\Bbb R$ for which $(k-1)x^3 - 4x^2 + (k+2)x$ has two roots That's problem statement: Find all values of $k\in\Bbb R$ for which the polynomial $W(x) = (k-1)x^3 - 4x^2 + (k+2)x$ has an even number of roots. We factorize by $x$, so we get $W(x) = x[(k-1)x^2 - ... solution-verification quadratics parametric 233 1 vote 2 answers 108 views If $\alpha$ and $\beta$ are roots of a trigonometric equation, are $\tan\alpha$ and $\tan\beta$ are the roots of the equation as well? [duplicate] So, the original question is this: If $\alpha$ and $\beta$ satisfy the equation $a\tan\theta + b\sec\theta = c$, find $\tan(\alpha+\beta)$. Now, this equation can be simplified to: $$ (a^2-b^2)\tan^... trigonometry quadratics 21 15 30 50 per page 2 3 4 5 373 Next 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 Hot Network Questions Quantizing EM field by imposing canonical commutation relations The geologic realities of a massive well out at Sea Another way to draw RegionDifference of a cylinder and Cuboid Exchange a file in a zip file quickly Singular support in Bezrukavnikov's equivalence What "real mistakes" exist in the Messier catalog? Two calendar months on the same page Where is the first repetition in the cumulative hierarchy up to elementary equivalence? Does clipping distortion affect the information contained within a frequency-modulated signal? How can I show that this sequence is aperiodic and is not even eventually-periodic. 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15136	https://en.wikipedia.org/wiki/Nearest_neighbor_search	Jump to content Search Contents (Top) 1 Applications 2 Methods 2.1 Exact methods 2.1.1 Linear search 2.1.2 Space partitioning 2.2 Approximation methods 2.2.1 Greedy search in proximity neighborhood graphs 2.2.2 Locality sensitive hashing 2.2.3 Nearest neighbor search in spaces with small intrinsic dimension 2.2.4 Projected radial search 2.2.5 Vector approximation files 2.2.6 Compression/clustering based search 3 Variants 3.1 k-nearest neighbors 3.2 Approximate nearest neighbor 3.3 Nearest neighbor distance ratio 3.4 Fixed-radius near neighbors 3.5 All nearest neighbors 4 See also 5 References 5.1 Citations 5.2 Sources 6 Further reading 7 External links Nearest neighbor search العربية Català فارسی Français 한국어 עברית Magyar 日本語 Русский Српски / srpski Srpskohrvatski / српскохрватски ไทย Українська 中文 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 Wikimedia Commons Wikidata item Appearance From Wikipedia, the free encyclopedia Optimization problem in computer science Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point. Closeness is typically expressed in terms of a dissimilarity function: the less similar the objects, the larger the function values. Formally, the nearest-neighbor (NN) search problem is defined as follows: given a set S of points in a space M and a query point q ∈ M, find the closest point in S to q. Donald Knuth in vol. 3 of The Art of Computer Programming (1973) called it the post-office problem, referring to an application of assigning to a residence the nearest post office. A direct generalization of this problem is a k-NN search, where we need to find the k closest points. Most commonly M is a metric space and dissimilarity is expressed as a distance metric, which is symmetric and satisfies the triangle inequality. Even more common, M is taken to be the d-dimensional vector space where dissimilarity is measured using the Euclidean distance, Manhattan distance or other distance metric. However, the dissimilarity function can be arbitrary. One example is asymmetric Bregman divergence, for which the triangle inequality does not hold. Applications [edit] The nearest neighbor search problem arises in numerous fields of application, including: Pattern recognition – in particular for optical character recognition Statistical classification – see k-nearest neighbor algorithm Computer vision – for point cloud registration Computational geometry – see Closest pair of points problem Cryptanalysis – for lattice problem Databases – e.g. content-based image retrieval Coding theory – see maximum likelihood decoding Semantic search Data compression – see MPEG-2 standard Robotic sensing Recommendation systems, e.g. see Collaborative filtering Internet marketing – see contextual advertising and behavioral targeting DNA sequencing Spell checking – suggesting correct spelling Plagiarism detection Similarity scores for predicting career paths of professional athletes. Cluster analysis – assignment of a set of observations into subsets (called clusters) so that observations in the same cluster are similar in some sense, usually based on Euclidean distance Chemical similarity Sampling-based motion planning Methods [edit] Various solutions to the NNS problem have been proposed. The quality and usefulness of the algorithms are determined by the time complexity of queries as well as the space complexity of any search data structures that must be maintained. The informal observation usually referred to as the curse of dimensionality states that there is no general-purpose exact solution for NNS in high-dimensional Euclidean space using polynomial preprocessing and polylogarithmic search time. Exact methods [edit] Linear search [edit] The simplest solution to the NNS problem is to compute the distance from the query point to every other point in the database, keeping track of the "best so far". This algorithm, sometimes referred to as the naive approach, has a running time of O(dN), where N is the cardinality of S and d is the dimensionality of S. There are no search data structures to maintain, so the linear search has no space complexity beyond the storage of the database. Naive search can, on average, outperform space partitioning approaches on higher dimensional spaces. The absolute distance is not required for distance comparison, only the relative distance. In geometric coordinate systems the distance calculation can be sped up considerably by omitting the square root calculation from the distance calculation between two coordinates. The distance comparison will still yield identical results. Space partitioning [edit] Since the 1970s, the branch and bound methodology has been applied to the problem. In the case of Euclidean space, this approach encompasses spatial index or spatial access methods. Several space-partitioning methods have been developed for solving the NNS problem. Perhaps the simplest is the k-d tree, which iteratively bisects the search space into two regions containing half of the points of the parent region. Queries are performed via traversal of the tree from the root to a leaf by evaluating the query point at each split. Depending on the distance specified in the query, neighboring branches that might contain hits may also need to be evaluated. For constant dimension query time, average complexity is O(log N) in the case of randomly distributed points, worst case complexity is O(kN^(1-1/k)) Alternatively the R-tree data structure was designed to support nearest neighbor search in dynamic context, as it has efficient algorithms for insertions and deletions such as the R tree. R-trees can yield nearest neighbors not only for Euclidean distance, but can also be used with other distances. In the case of general metric space, the branch-and-bound approach is known as the metric tree approach. Particular examples include vp-tree and BK-tree methods. Using a set of points taken from a 3-dimensional space and put into a BSP tree, and given a query point taken from the same space, a possible solution to the problem of finding the nearest point-cloud point to the query point is given in the following description of an algorithm. \| \| \| This article may be confusing or unclear to readers. Please help clarify the article. There might be a discussion about this on the talk page. (November 2021) (Learn how and when to remove this message) \| (Strictly speaking, no such point may exist, because it may not be unique. But in practice, usually we only care about finding any one of the subset of all point-cloud points that exist at the shortest distance to a given query point.) The idea is, for each branching of the tree, guess that the closest point in the cloud resides in the half-space containing the query point. This may not be the case, but it is a good heuristic. After having recursively gone through all the trouble of solving the problem for the guessed half-space, now compare the distance returned by this result with the shortest distance from the query point to the partitioning plane. This latter distance is that between the query point and the closest possible point that could exist in the half-space not searched. If this distance is greater than that returned in the earlier result, then clearly there is no need to search the other half-space. If there is such a need, then you must go through the trouble of solving the problem for the other half space, and then compare its result to the former result, and then return the proper result. The performance of this algorithm is nearer to logarithmic time than linear time when the query point is near the cloud, because as the distance between the query point and the closest point-cloud point nears zero, the algorithm needs only perform a look-up using the query point as a key to get the correct result. Approximation methods [edit] An approximate nearest neighbor search algorithm is allowed to return points whose distance from the query is at most times the distance from the query to its nearest points. The appeal of this approach is that, in many cases, an approximate nearest neighbor is almost as good as the exact one. In particular, if the distance measure accurately captures the notion of user quality, then small differences in the distance should not matter. Greedy search in proximity neighborhood graphs [edit] Proximity graph methods (such as navigable small world graphs and HNSW) are considered the current state-of-the-art for the approximate nearest neighbors search. The methods are based on greedy traversing in proximity neighborhood graphs in which every point is uniquely associated with vertex . The search for the nearest neighbors to a query q in the set S takes the form of searching for the vertex in the graph . The basic algorithm – greedy search – works as follows: search starts from an enter-point vertex by computing the distances from the query q to each vertex of its neighborhood , and then finds a vertex with the minimal distance value. If the distance value between the query and the selected vertex is smaller than the one between the query and the current element, then the algorithm moves to the selected vertex, and it becomes new enter-point. The algorithm stops when it reaches a local minimum: a vertex whose neighborhood does not contain a vertex that is closer to the query than the vertex itself. The idea of proximity neighborhood graphs was exploited in multiple publications, including the seminal paper by Arya and Mount, in the VoroNet system for the plane, in the RayNet system for the , and in the Navigable Small World, Metrized Small World and HNSW algorithms for the general case of spaces with a distance function. These works were preceded by a pioneering paper by Toussaint, in which he introduced the concept of a relative neighborhood graph. Locality sensitive hashing [edit] Locality sensitive hashing (LSH) is a technique for grouping points in space into 'buckets' based on some distance metric operating on the points. Points that are close to each other under the chosen metric are mapped to the same bucket with high probability. Nearest neighbor search in spaces with small intrinsic dimension [edit] See also: Intrinsic dimension The cover tree has a theoretical bound that is based on the dataset's doubling constant. The bound on search time is O(c12 log n) where c is the expansion constant of the dataset. Projected radial search [edit] In the special case where the data is a dense 3D map of geometric points, the projection geometry of the sensing technique can be used to dramatically simplify the search problem. This approach requires that the 3D data is organized by a projection to a two-dimensional grid and assumes that the data is spatially smooth across neighboring grid cells with the exception of object boundaries. These assumptions are valid when dealing with 3D sensor data in applications such as surveying, robotics and stereo vision but may not hold for unorganized data in general. In practice this technique has an average search time of O(1) or O(K) for the k-nearest neighbor problem when applied to real world stereo vision data. Vector approximation files [edit] In high-dimensional spaces, tree indexing structures become useless because an increasing percentage of the nodes need to be examined anyway. To speed up linear search, a compressed version of the feature vectors stored in RAM is used to prefilter the datasets in a first run. The final candidates are determined in a second stage using the uncompressed data from the disk for distance calculation. Compression/clustering based search [edit] The VA-file approach is a special case of a compression based search, where each feature component is compressed uniformly and independently. The optimal compression technique in multidimensional spaces is Vector Quantization (VQ), implemented through clustering. The database is clustered and the most "promising" clusters are retrieved. Huge gains over VA-File, tree-based indexes and sequential scan have been observed. Also note the parallels between clustering and LSH. Variants [edit] There are numerous variants of the NNS problem and the two most well-known are the k-nearest neighbor search and the ε-approximate nearest neighbor search. k-nearest neighbors [edit] k-nearest neighbor search identifies the top k nearest neighbors to the query. This technique is commonly used in predictive analytics to estimate or classify a point based on the consensus of its neighbors. k-nearest neighbor graphs are graphs in which every point is connected to its k nearest neighbors. Approximate nearest neighbor [edit] In some applications it may be acceptable to retrieve a "good guess" of the nearest neighbor. In those cases, we can use an algorithm which doesn't guarantee to return the actual nearest neighbor in every case, in return for improved speed or memory savings. Often such an algorithm will find the nearest neighbor in a majority of cases, but this depends strongly on the dataset being queried. Algorithms that support the approximate nearest neighbor search include locality-sensitive hashing, best bin first and balanced box-decomposition tree based search. Nearest neighbor distance ratio [edit] Nearest neighbor distance ratio does not apply the threshold on the direct distance from the original point to the challenger neighbor but on a ratio of it depending on the distance to the previous neighbor. It is used in CBIR to retrieve pictures through a "query by example" using the similarity between local features. More generally it is involved in several matching problems. Fixed-radius near neighbors [edit] Fixed-radius near neighbors is the problem where one wants to efficiently find all points given in Euclidean space within a given fixed distance from a specified point. The distance is assumed to be fixed, but the query point is arbitrary. All nearest neighbors [edit] For some applications (e.g. entropy estimation), we may have N data-points and wish to know which is the nearest neighbor for every one of those N points. This could, of course, be achieved by running a nearest-neighbor search once for every point, but an improved strategy would be an algorithm that exploits the information redundancy between these N queries to produce a more efficient search. As a simple example: when we find the distance from point X to point Y, that also tells us the distance from point Y to point X, so the same calculation can be reused in two different queries. Given a fixed dimension, a semi-definite positive norm (thereby including every Lp norm), and n points in this space, the nearest neighbour of every point can be found in O(n log n) time and the m nearest neighbours of every point can be found in O(mn log n) time. See also [edit] Ball tree Closest pair of points problem Cluster analysis Content-based image retrieval Curse of dimensionality Digital signal processing Dimension reduction Fixed-radius near neighbors Fourier analysis Instance-based learning k-nearest neighbor algorithm Linear least squares Locality sensitive hashing Maximum inner-product search MinHash Multidimensional analysis Nearest-neighbor interpolation Neighbor joining Principal component analysis Range search Similarity learning Singular value decomposition Sparse distributed memory Statistical distance Time series Voronoi diagram Wavelet References [edit] Citations [edit] ^ Cayton, Lawerence (2008). "Fast nearest neighbor retrieval for bregman divergences". Proceedings of the 25th International Conference on Machine Learning. pp. 112–119. doi:10.1145/1390156.1390171. ISBN 9781605582054. S2CID 12169321. ^ Qiu, Deyuan, Stefan May, and Andreas Nüchter. "GPU-accelerated nearest neighbor search for 3D registration." International conference on computer vision systems. Springer, Berlin, Heidelberg, 2009. ^ Becker, Ducas, Gama, and Laarhoven. "New directions in nearest neighbor searching with applications to lattice sieving." Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms (pp. 10-24). Society for Industrial and Applied Mathematics. ^ a b Bewley, A.; Upcroft, B. (2013). Advantages of Exploiting Projection Structure for Segmenting Dense 3D Point Clouds (PDF). Australian Conference on Robotics and Automation. ^ Weber, Roger; Schek, Hans-J.; Blott, Stephen (1998). "A quantitative analysis and performance study for similarity search methods in high dimensional spaces" (PDF). VLDB '98 Proceedings of the 24rd International Conference on Very Large Data Bases. pp. 194–205. ^ Andrew Moore. "An introductory tutorial on KD trees" (PDF). Archived from the original (PDF) on 2016-03-03. Retrieved 2008-10-03. ^ Lee, D. T.; Wong, C. K. (1977). "Worst-case analysis for region and partial region searches in multidimensional binary search trees and balanced quad trees". Acta Informatica. 9 (1): 23–29. doi:10.1007/BF00263763. S2CID 36580055. ^ Roussopoulos, N.; Kelley, S.; Vincent, F. D. R. (1995). "Nearest neighbor queries". Proceedings of the 1995 ACM SIGMOD international conference on Management of data – SIGMOD '95. p. 71. doi:10.1145/223784.223794. ISBN 0897917316. ^ Andoni, A.; Indyk, P. (2006-10-01). "Near-Optimal Hashing Algorithms for Approximate Nearest Neighbor in High Dimensions". 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06). pp. 459–468. CiteSeerX 10.1.1.142.3471. doi:10.1109/FOCS.2006.49. ISBN 978-0-7695-2720-8. ^ a b Malkov, Yury; Ponomarenko, Alexander; Logvinov, Andrey; Krylov, Vladimir (2012), Navarro, Gonzalo; Pestov, Vladimir (eds.), "Scalable Distributed Algorithm for Approximate Nearest Neighbor Search Problem in High Dimensional General Metric Spaces", Similarity Search and Applications, vol. 7404, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 132–147, doi:10.1007/978-3-642-32153-5_10, ISBN 978-3-642-32152-8, retrieved 2024-01-16 ^ a b Malkov, Yury; Yashunin, Dmitry (2016). "Efficient and robust approximate nearest neighbor search using Hierarchical Navigable Small World graphs". arXiv:1603.09320 [cs.DS]. ^ a b Malkov, Yu A.; Yashunin, D. A. (2020-04-01). "Efficient and Robust Approximate Nearest Neighbor Search Using Hierarchical Navigable Small World Graphs". IEEE Transactions on Pattern Analysis and Machine Intelligence. 42 (4): 824–836. arXiv:1603.09320. Bibcode:2020ITPAM..42..824M. doi:10.1109/TPAMI.2018.2889473. ISSN 0162-8828. PMID 30602420. ^ Arya, Sunil; Mount, David (1993). "Approximate Nearest Neighbor Queries in Fixed Dimensions". Proceedings of the Fourth Annual {ACM/SIGACT-SIAM} Symposium on Discrete Algorithms, 25–27 January 1993, Austin, Texas.: 271–280. ^ Olivier, Beaumont; Kermarrec, Anne-Marie; Marchal, Loris; Rivière, Etienne (2006). "Voro Net: A scalable object network based on Voronoi tessellations" (PDF). 2007 IEEE International Parallel and Distributed Processing Symposium. Vol. RR-5833. pp. 23–29. doi:10.1109/IPDPS.2007.370210. ISBN 1-4244-0909-8. S2CID 8844431. ^ Olivier, Beaumont; Kermarrec, Anne-Marie; Rivière, Etienne (2007). "Peer to Peer Multidimensional Overlays: Approximating Complex Structures". Principles of Distributed Systems. Lecture Notes in Computer Science. Vol. 4878. pp. 315–328. CiteSeerX 10.1.1.626.2980. doi:10.1007/978-3-540-77096-1_23. ISBN 978-3-540-77095-4. ^ Malkov, Yury; Ponomarenko, Alexander; Krylov, Vladimir; Logvinov, Andrey (2014). "Approximate nearest neighbor algorithm based on navigable small world graphs". Information Systems. 45: 61–68. doi:10.1016/j.is.2013.10.006. S2CID 9896397. ^ Toussaint, Godfried (1980). "The relative neighbourhood graph of a finite planar set". Pattern Recognition. 12 (4): 261–268. Bibcode:1980PatRe..12..261T. doi:10.1016/0031-3203(80)90066-7. ^ A. Rajaraman & J. Ullman (2010). "Mining of Massive Datasets, Ch. 3". ^ Weber, Roger; Blott, Stephen. "An Approximation-Based Data Structure for Similarity Search" (PDF). S2CID 14613657. Archived from the original (PDF) on 2017-03-04. {{cite journal}}: Cite journal requires \|journal= (help) ^ Ramaswamy, Sharadh; Rose, Kenneth (2007). "Adaptive cluster-distance bounding for similarity search in image databases". ICIP. ^ Ramaswamy, Sharadh; Rose, Kenneth (2010). "Adaptive cluster-distance bounding for high-dimensional indexing". TKDE. ^ Arya, S.; Mount, D. M.; Netanyahu, N. S.; Silverman, R.; Wu, A. (1998). "An optimal algorithm for approximate nearest neighbor searching" (PDF). Journal of the ACM. 45 (6): 891–923. CiteSeerX 10.1.1.15.3125. doi:10.1145/293347.293348. S2CID 8193729. Archived from the original (PDF) on 2016-03-03. Retrieved 2009-05-29. ^ Clarkson, Kenneth L. (1983), "Fast algorithms for the all nearest neighbors problem", 24th IEEE Symp. Foundations of Computer Science, (FOCS '83), pp. 226–232, doi:10.1109/SFCS.1983.16, ISBN 978-0-8186-0508-6, S2CID 16665268. ^ Vaidya, P. M. (1989). "An O(n log n) Algorithm for the All-Nearest-Neighbors Problem". Discrete and Computational Geometry. 4 (1): 101–115. doi:10.1007/BF02187718. Sources [edit] Andrews, L. (November 2001). "A template for the nearest neighbor problem". C/C++ Users Journal. 19 (11): 40–49. ISSN 1075-2838. Arya, S.; Mount, D.M.; Netanyahu, N. S.; Silverman, R.; Wu, A. Y. (1998). "An Optimal Algorithm for Approximate Nearest Neighbor Searching in Fixed Dimensions". Journal of the ACM. 45 (6): 891–923. CiteSeerX 10.1.1.15.3125. doi:10.1145/293347.293348. S2CID 8193729. Beyer, K.; Goldstein, J.; Ramakrishnan, R.; Shaft, U. (1999). "When is nearest neighbor meaningful?". Proceedings of the 7th ICDT. Chen, Chung-Min; Ling, Yibei (2002). "A Sampling-Based Estimator for Top-k Query". ICDE: 617–627. Samet, H. (2006). Foundations of Multidimensional and Metric Data Structures. Morgan Kaufmann. ISBN 978-0-12-369446-1. Zezula, P.; Amato, G.; Dohnal, V.; Batko, M. (2006). Similarity Search – The Metric Space Approach. Springer. ISBN 978-0-387-29146-8. Further reading [edit] Shasha, Dennis (2004). High Performance Discovery in Time Series. Berlin: Springer. ISBN 978-0-387-00857-8. External links [edit] Wikimedia Commons has media related to Nearest neighbours search. Nearest Neighbors and Similarity Search – a website dedicated to educational materials, software, literature, researchers, open problems and events related to NN searching. Maintained by Yury Lifshits Similarity Search Wiki – a collection of links, people, ideas, keywords, papers, slides, code and data sets on nearest neighbours Retrieved from " Categories: Approximation algorithms Classification algorithms Data mining Discrete geometry Geometric algorithms Mathematical optimization Search algorithms Hidden categories: CS1: long volume value CS1 errors: missing periodical Articles with short description Short description is different from Wikidata Wikipedia articles needing clarification from November 2021 All Wikipedia articles needing clarification Commons category link is on Wikidata Nearest neighbor search Add topic
15137	https://www.youtube.com/watch?v=8wIodo1HD4Y	Introduction to reaction rates \| Kinetics \| AP Chemistry \| Khan Academy Khan Academy Organic Chemistry 236000 subscribers 1866 likes Description 666587 views Posted: 11 Oct 2014 The rate of a chemical reaction is defined as the rate of change in concentration of a reactant or product divided by its coefficient from the balanced equation. A negative sign is used with rates of change of reactants and a positive sign with those of products, ensuring that the reaction rate is always a positive quantity. In most cases, concentration is measured in moles per liter and time in seconds, resulting in units of M/s for the reaction rate. View more lessons or practice this subject at Khan Academy is a nonprofit organization with the mission of providing a free, world-class education for anyone, anywhere. We offer quizzes, questions, instructional videos, and articles on a range of academic subjects, including math, biology, chemistry, physics, history, economics, finance, grammar, preschool learning, and more. We provide teachers with tools and data so they can help their students develop the skills, habits, and mindsets for success in school and beyond. Khan Academy has been translated into dozens of languages, and 15 million people around the globe learn on Khan Academy every month. As a 501(c)(3) nonprofit organization, we would love your help! Donate or volunteer today! Donate here: Volunteer here: 88 comments Transcript: The rate of a chemical reaction is defined as the change in the concentration of a reactant or a product over the change in time, and concentration is in moles per liter, or molar, and time is in seconds. So we express the rate of a chemical reaction in molar per second. Molar per second sounds a lot like meters per second, and that, if you remember your physics is our unit for velocity. So, average velocity is equal to the change in x over the change in time, and so thinking about average velocity helps you understand the definition for rate of reaction in chemistry. If we look at this applied to a very, very simple reaction. So we have one reactant, A, turning into one product, B. Now, let's say at time is equal to 0 we're starting with an initial concentration of A of 1.00 M, and A hasn't turned into B yet. So at time is equal to 0, the concentration of B is 0.0. Let's say we wait two seconds. So, we wait two seconds, and then we measure the concentration of A. Obviously the concentration of A is going to go down because A is turning into B. Let's say the concentration of A turns out to be .98 M. So we lost .02 M for the concentration of A. So that turns into, since A turns into B after two seconds, the concentration of B is .02 M. Right, because A turned into B. So this is our concentration of B after two seconds. If I want to know the average rate of reaction here, we could plug into our definition for rate of reaction. Change in concentration, let's do a change in concentration of our product, over the change in time. So, the Rate is equal to the change in the concentration of our product, that's final concentration minus initial concentration. So the final concentration is 0.02. So, we write in here 0.02, and from that we subtract the initial concentration of our product, which is 0.0. So, 0.02 - 0.0, that's all over the change in time. That's the final time minus the initial time, so that's 2 - 0. So the rate of reaction, the average rate of reaction, would be equal to 0.02 divided by 2, which is 0.01 molar per second. So that's our average rate of reaction from time is equal to 0 to time is equal to 2 seconds. We could do the same thing for A, right, so we could, instead of defining our rate of reaction as the appearance of B, we could define our rate of reaction as the disappearance of A. So the rate would be equal to, right, the change in the concentration of A, that's the final concentration of A, which is 0.98 minus the initial concentration of A, and the initial concentration of A is 1.00. So 0.98 - 1.00, and this is all over the final time minus the initial time, so this is over 2 - 0. Now this would give us -0.02. - 0.02 here, over 2, and that would give us a negative rate of reaction, but in chemistry, the rate of reaction is defined as a positive quantity. So we need a negative sign. We need to put a negative sign in here because a negative sign gives us a positive value for the rate. So, now we get 0.02 divided by 2, which of course is 0.01 molar per second. So we get a positive value for the rate of reaction. All right, so we calculated the average rate of reaction using the disappearance of A and the formation of B, and we could make this a little bit more general. We could say that our rate is equal to, this would be the change in the concentration of A over the change in time, but we need to make sure to put in our negative sign. We put in our negative sign to give us a positive value for the rate. So the rate is equal to the negative change in the concentration of A over the change of time, and that's equal to, right, the change in the concentration of B over the change in time, and we don't need a negative sign because we already saw in the calculation, right, we get a positive value for the rate. So, here's two different ways to express the rate of our reaction. So here, I just wrote it in a little bit more general terms. Let's look at a more complicated reaction. Here, we have the balanced equation for the decomposition of dinitrogen pentoxide into nitrogen dioxide and oxygen. And let's say that oxygen forms at a rate of 9 x 10 to the -6 M/s. So what is the rate of formation of nitrogen dioxide? Well, if you look at the balanced equation, for every one mole of oxygen that forms four moles of nitrogen dioxide form. So we just need to multiply the rate of formation of oxygen by four, and so that gives us, that gives us 3.6 x 10 to the -5 Molar per second. So, NO2 forms at four times the rate of O2. What about dinitrogen pentoxide? So, N2O5. Look at your mole ratios. For every one mole of oxygen that forms we're losing two moles of dinitrogen pentoxide. So if we're starting with the rate of formation of oxygen, because our mole ratio is one to two here, we need to multiply this by 2, and since we're losing dinitrogen pentoxide, we put a negative sign here. So this gives us - 1.8 x 10 to the -5 molar per second. So, dinitrogen pentoxide disappears at twice the rate that oxygen appears. All right, let's think about the rate of our reaction. So the rate of our reaction is equal to, well, we could just say it's equal to the appearance of oxygen, right. We could say it's equal to 9.0 x 10 to the -6 molar per second, so we could write that down here. The rate is equal to the change in the concentration of oxygen over the change in time. All right, what about if we wanted to express this in terms of the formation of nitrogen dioxide. Well, the formation of nitrogen dioxide was 3.6 x 10 to the -5. All right, so that's 3.6 x 10 to the -5. So you need to think to yourself, what do I need to multiply this number by in order to get this number? Since this number is four times the number on the left, I need to multiply by one fourth. Right, so down here, down here if we're talking about the change in the concentration of nitrogen dioxide over the change in time, to get the rate to be the same, we'd have to multiply this by one fourth. All right, finally, let's think about, let's think about dinitrogen pentoxide. So, we said that that was disappearing at -1.8 x 10 to the -5. So once again, what do I need to multiply this number by in order to get 9.0 x 10 to the -6? Well, this number, right, in terms of magnitude was twice this number so I need to multiply it by one half. I need to get rid of the negative sign because rates of reaction are defined as a positive quantity. So I need a negative here. So that would give me, right, that gives me 9.0 x 10 to the -6. So for, I could express my rate, if I want to express my rate in terms of the disappearance of dinitrogen pentoxide, I'd write the change in N2, this would be the change in N2O5 over the change in time, and I need to put a negative one half here as well. All right, so now that we figured out how to express our rate, we can look at our balanced equation. So, over here we had a 2 for dinitrogen pentoxide, and notice where the 2 goes here for expressing our rate. For nitrogen dioxide, right, we had a 4 for our coefficient. So, the 4 goes in here, and for oxygen, for oxygen over here, let's use green, we had a 1. So I could've written 1 over 1, just to show you the pattern of how to express your rate.
15138	https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2420_Calculus_II/06%3A_Power_Series/6.03%3A_Taylor_and_Maclaurin_Series	Search x Text Color Text Size Margin Size Font Type selected template will load here Error This action is not available. 6.3: Taylor and Maclaurin Series ( \newcommand{\vecs}{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } ) ( \newcommand{\vecd}{\overset{-!-!\rightharpoonup}{\vphantom{a}\smash {#1}}} ) ( \newcommand{\id}{\mathrm{id}}) ( \newcommand{\Span}{\mathrm{span}}) ( \newcommand{\kernel}{\mathrm{null}\,}) ( \newcommand{\range}{\mathrm{range}\,}) ( \newcommand{\RealPart}{\mathrm{Re}}) ( \newcommand{\ImaginaryPart}{\mathrm{Im}}) ( \newcommand{\Argument}{\mathrm{Arg}}) ( \newcommand{\norm}{\\| #1 \\|}) ( \newcommand{\inner}{\langle #1, #2 \rangle}) ( \newcommand{\Span}{\mathrm{span}}) ( \newcommand{\id}{\mathrm{id}}) ( \newcommand{\Span}{\mathrm{span}}) ( \newcommand{\kernel}{\mathrm{null}\,}) ( \newcommand{\range}{\mathrm{range}\,}) ( \newcommand{\RealPart}{\mathrm{Re}}) ( \newcommand{\ImaginaryPart}{\mathrm{Im}}) ( \newcommand{\Argument}{\mathrm{Arg}}) ( \newcommand{\norm}{\\| #1 \\|}) ( \newcommand{\inner}{\langle #1, #2 \rangle}) ( \newcommand{\Span}{\mathrm{span}}) ( \newcommand{\AA}{\unicode[.8,0]{x212B}}) ( \newcommand{\vectorA}{\vec{#1}} % arrow) ( \newcommand{\vectorAt}{\vec{\text{#1}}} % arrow) ( \newcommand{\vectorB}{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } ) ( \newcommand{\vectorC}{\textbf{#1}} ) ( \newcommand{\vectorD}{\overrightarrow{#1}} ) ( \newcommand{\vectorDt}{\overrightarrow{\text{#1}}} ) ( \newcommand{\vectE}{\overset{-!-!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} ) ( \newcommand{\vecs}{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } ) ( \newcommand{\vecd}{\overset{-!-!\rightharpoonup}{\vphantom{a}\smash {#1}}} ) Learning Objectives In the previous two sections we discussed how to find power series representations for certain types of functions––specifically, functions related to geometric series. Here we discuss power series representations for other types of functions. In particular, we address the following questions: Which functions can be represented by power series and how do we find such representations? If we can find a power series representation for a particular function (f) and the series converges on some interval, how do we prove that the series actually converges to (f)? Overview of Taylor/Maclaurin Series Consider a function (f) that has a power series representation at (x=a). Then the series has the form [\sum_{n=0}^∞c_n(x−a)^n=c_0+c_1(x−a)+c_2(x−a)^2+ \dots. \label{eq1}] What should the coefficients be? For now, we ignore issues of convergence, but instead focus on what the series should be, if one exists. We return to discuss convergence later in this section. If the series Equation \ref{eq1} is a representation for (f) at (x=a), we certainly want the series to equal (f(a)) at (x=a). Evaluating the series at (x=a), we see that [\sum_{n=0}^∞c_n(x−a)^n=c_0+c_1(a−a)+c_2(a−a)^2+\dots=c_0.\label{eq2}] Thus, the series equals (f(a)) if the coefficient (c_0=f(a)). In addition, we would like the first derivative of the power series to equal (f′(a)) at (x=a). Differentiating Equation \ref{eq2} term-by-term, we see that [\dfrac{d}{dx}\left( \sum_{n=0}^∞c_n(x−a)^n \right)=c_1+2c_2(x−a)+3c_3(x−a)^2+\dots.\label{eq3}] Therefore, at (x=a,) the derivative is [\dfrac{d}{dx}\left( \sum_{n=0}^∞c_n(x−a)^n \right)=c_1+2c_2(a−a)+3c_3(a−a)^2+\dots=c_1.\label{eq4}] Therefore, the derivative of the series equals (f′(a)) if the coefficient (c_1=f′(a).) Continuing in this way, we look for coefficients (c_n) such that all the derivatives of the power series Equation \ref{eq4} will agree with all the corresponding derivatives of (f) at (x=a). The second and third derivatives of Equation \ref{eq3} are given by [\dfrac{d^2}{dx^2} \left(\sum_{n=0}^∞c_n(x−a)^n \right)=2c_2+3⋅2c_3(x−a)+4⋅3c_4(x−a)^2+\dots\label{eq5}] and [\dfrac{d^3}{dx^3} \left( \sum_{n=0}^∞c_n(x−a)^n \right)=3⋅2c_3+4⋅3⋅2c_4(x−a)+5⋅4⋅3c_5(x−a)^2+⋯.\label{eq6}] Therefore, at (x=a), the second and third derivatives [\dfrac{d^2}{dx^2} \left(\sum_{n=0}^∞c_n(x−a)^n\right)=2c_2+3⋅2c_3(a−a)+4⋅3c_4(a−a)^2+\dots=2c_2\label{eq7}] and [\dfrac{d^3}{dx^3} \left(\sum_{n=0}^∞c_n(x−a)^n\right)=3⋅2c_3+4⋅3⋅2c_4(a−a)+5⋅4⋅3c_5(a−a)^2+\dots =3⋅2c_3\label{eq8}] equal (f''(a)) and (f'''(a)), respectively, if (c_2=\dfrac{f''(a)}{2}) and (c_3=\dfrac{f'''(a)}{3⋅2}). More generally, we see that if (f) has a power series representation at (x=a), then the coefficients should be given by (c_n=\dfrac{f^{(n)}(a)}{n!}). That is, the series should be [\sum_{n=0}^∞\dfrac{f^{(n)}(a)}{n!}(x−a)^n=f(a)+f′(a)(x−a)+\dfrac{f''(a)}{2!}(x−a)^2+\dfrac{f'''(a)}{3!}(x−a)^3+⋯] This power series for (f) is known as the Taylor series for (f) at (a.) If (x=0), then this series is known as the Maclaurin series for (f). Definition (\PageIndex{1}): Maclaurin and Taylor series If (f) has derivatives of all orders at (x=a), then theTaylor series for the function (f) at (a) is [\sum_{n=0}^∞\dfrac{f^{(n)}(a)}{n!}(x−a)^n=f(a)+f′(a)(x−a)+\dfrac{f''(a)}{2!}(x−a)^2+⋯+\dfrac{f^{(n)}(a)}{n!}(x−a)^n+⋯] The Taylor series for (f) at 0 is known as the Maclaurin series for (f). Later in this section, we will show examples of finding Taylor series and discuss conditions under which the Taylor series for a function will converge to that function. Here, we state an important result. Recall that power series representations are unique. Therefore, if a function (f) has a power series at (a), then it must be the Taylor series for (f) at (a). Uniqueness of Taylor Series If a function (f) has a power series at a that converges to (f) on some open interval containing (a), then that power series is the Taylor series for (f) at (a). The proof follows directly from that discussed previously. To determine if a Taylor series converges, we need to look at its sequence of partial sums. These partial sums are finite polynomials, known as Taylor polynomials. Taylor Polynomials The nth partial sum of the Taylor series for a function (f) at (a) is known as the nth Taylor polynomial. For example, the 0th, 1st, 2nd, and 3rd partial sums of the Taylor series are given by [\begin{align} p_0(x) =f(a) \[4pt] p_1(x) =f(a)+f′(a)(x−a) \[4pt]p_2(x) =f(a)+f′(a)(x−a)+\dfrac{f''(a)}{2!}(x−a)^2\ \[4pt]p_3(x) =f(a)+f′(a)(x−a)+\dfrac{f''(a)}{2!}(x−a)^2+\dfrac{f'''(a)}{3!}(x−a)^3 \end{align}] respectively. These partial sums are known as the 0th, 1st, 2nd, and 3rd Taylor polynomials of (f) at (a), respectively. If (x=a), then these polynomials are known as Maclaurin polynomials for (f). We now provide a formal definition of Taylor and Maclaurin polynomials for a function (f). Definition (\PageIndex{2}): Maclaurin polynomial If (f) has n derivatives at (x=a), then the nth Taylor polynomial for (f) at (a) is [p_n(x)=f(a)+f′(a)(x−a)+\dfrac{f''(a)}{2!}(x−a)^2+\dfrac{f'''(a)}{3!}(x−a)^3+⋯+\dfrac{f^{(n)}(a)}{n!}(x−a)^n.] The nth Taylor polynomial for (f) at 0 is known as the nth Maclaurin polynomial for (f). We now show how to use this definition to find several Taylor polynomials for (f(x)=\ln x) at (x=1). Example (\PageIndex{1}): Finding Taylor Polynomials Find the Taylor polynomials (p_0,p_1,p_2) and (p_3) for (f(x)=\ln x) at (x=1). Use a graphing utility to compare the graph of (f) with the graphs of (p_0,p_1,p_2) and (p_3). Solution To find these Taylor polynomials, we need to evaluate (f) and its first three derivatives at (x=1). (f(x)=\ln x) (f(1)=0) (f′(x)=\dfrac{1}{x}) (f′(1)=1) (f''(x)=−\dfrac{1}{x^2}) (f''(1)=−1) (f'''(x)=\dfrac{2}{x^3}) (f'''(1)=2) Therefore, [\begin{align} p_0(x) =f(1)=0,\[4pt]p_1(x) =f(1)+f′(1)(x−1) =x−1,\[4pt]p_2(x) =f(1)+f′(1)(x−1)+\dfrac{f''(1)}{2}(x−1)^2 = (x−1)−\dfrac{1}{2}(x−1)^2 \[4pt]p_3(x) =f(1)+f′(1)(x−1)+\dfrac{f''(1)}{2}(x−1)^2+\dfrac{f'''(1)}{3!}(x−1)^3=(x−1)−\dfrac{1}{2}(x−1)^2+\dfrac{1}{3}(x−1)^3 \end{align}] The graphs of (y=f(x)) and the first three Taylor polynomials are shown in Figure (\PageIndex{1}). Exercise (\PageIndex{1}) Find the Taylor polynomials (p_0,p_1,p_2) and (p_3) for (f(x)=\dfrac{1}{x^2}) at (x=1). Find the first three derivatives of (f) and evaluate them at (x=1.) [ p_0(x)=1] [p_1(x)=1−2(x−1)] [p_2(x)=1−2(x−1)+3(x−1)^2] [p_3(x)=1−2(x−1)+3(x−1)^2−4(x−1)^3] We now show how to find Maclaurin polynomials for (e^x, \sin x,) and (\cos x). As stated above, Maclaurin polynomials are Taylor polynomials centered at zero. Example (\PageIndex{2}): Finding Maclaurin Polynomials For each of the following functions, find formulas for the Maclaurin polynomials (p_0,p_1,p_2) and (p_3). Find a formula for the nth Maclaurin polynomial and write it using sigma notation. Use a graphing utility to compare the graphs of (p_0,p_1,p_2) and (p_3) with (f). Solution Since (f(x)=e^x),we know that (f(x)=f′(x)=f''(x)=⋯=f^{(n)}(x)=e^x) for all positive integers n. Therefore, [f(0)=f′(0)=f''(0)=⋯=f^{(n)}(0)=1 \nonumber] for all positive integers n. Therefore, we have (p_0(x)=f(0)=1,) (p_1(x)=f(0)+f′(0)x=1+x,) (p_2(x)=f(0)+f′(0)x+\dfrac{f''(0)}{2!}x^2=1+x+\dfrac{1}{2}x^2), (p_3(x)=f(0)+f′(0)x+\dfrac{f''(0)}{2}x^2+\dfrac{f'''(0)}{3!}x^3=1+x+\dfrac{1}{2}x^2+\dfrac{1}{3!}x^3), (p_n(x)=f(0)+f′(0)x+\dfrac{f''(0)}{2}x^2+\dfrac{f'''(0)}{3!}x^3+⋯+\dfrac{f^{(n)}(0)}{n!}x^n=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+⋯+\dfrac{x^n}{n!}=\sum_{k=0}^n\dfrac{x^k}{k!}). The function and the first three Maclaurin polynomials are shown in Figure 2. b. For (f(x)=\sin x), the values of the function and its first four derivatives at (x=0) are given as follows: (f(x)=\sin x) (f(0)=0) (f′(x)=\cos x) (f′(0)=1) (f''(x)=−\sin x) (f''(0)=0) (f'''(x)=−\cos x) (f'''(0)=−1) (f^{(4)}(x)=\sin x) (f^{(4)}(0)=0). Since the fourth derivative is (\sin x,) the pattern repeats. That is, (f^{(2m)}(0)=0) and (f^{(2m+1)}(0)=(−1)^m) for (m≥0.) Thus, we have (p_0(x)=0,) (p_1(x)=0+x=x,) (p_2(x)=0+x+0=x,) (p_3(x)=0+x+0−\dfrac{1}{3!}x^3=x−\dfrac{x^3}{3!},) (p_4(x)=0+x+0−\dfrac{1}{3!}x^3+0=x−\dfrac{x^3}{3!}), (p_5(x)=0+x+0−\dfrac{1}{3!}x^3+0+\dfrac{1}{5!}x^5=x−\dfrac{x^3}{3!}+\dfrac{x^5}{5!}), and for (m≥0), (p_{2m+1}(x)=p_{2m+2}(x)=x−\dfrac{x^3}{3!}+\dfrac{x^5}{5!}−⋯+(−1)^m\dfrac{x^{2m+1}}{(2m+1)!}=\sum_{k=0}^m(−1)^k\dfrac{x^{2k+1}}{(2k+1)!}). Graphs of the function and its Maclaurin polynomials are shown in Figure 3. c. For (f(x)=\cos x), the values of the function and its first four derivatives at (x=0) are given as follows: (f(x)=\cos x) (f(0)=1) (f′(x)=−\sin x) (f′(0)=0) (f''(x)=−\cos x) (f''(0)=−1) (f'''(x)=\sin x) (f'''(0)=0) (f^{(4)}(x)=\cos x) (f^{(4)}(0)=1.) Since the fourth derivative is (\sin x), the pattern repeats. In other words, (f^{(2m)}(0)=(−1)^m) and (f^{(2m+1)}=0) for (m≥0). Therefore, (p_0(x)=1,) (p_1(x)=1+0=1,) (p_2(x)=1+0−\dfrac{1}{2!}x^2=1−\dfrac{x^2}{2!}), (p_3(x)=1+0−\dfrac{1}{2!}x^2+0=1−\dfrac{x^2}{2!}), (p_4(x)=1+0−\dfrac{1}{2!}x^2+0+\dfrac{1}{4!}x^4=1−\dfrac{x^2}{2!}+\dfrac{x^4}{4!}), (p_5(x)=1+0−\dfrac{1}{2!}x^2+0+\dfrac{1}{4!}x^4+0=1−\dfrac{x^2}{2!}+\dfrac{x^4}{4!}), and for (n≥0), (p_{2m}(x)=p_{2m+1}(x)=1−\dfrac{x^2}{2!}+\dfrac{x^4}{4!}−⋯+(−1)^m\dfrac{x^{2m}}{(2m)!}=\sum_{k=0}^m(−1)^k\dfrac{x^{2k}}{(2k)!}). Graphs of the function and the Maclaurin polynomials appear in Figure 4. Exercise (\PageIndex{2}) Find formulas for the Maclaurin polynomials (p_0,p_1,p_2) and (p_3) for (f(x)=\dfrac{1}{1+x}). Find a formula for the nth Maclaurin polynomial. Write your answer using sigma notation. Evaluate the first four derivatives of (f) and look for a pattern. (p_0(x)=1;p_1(x)=1−x;p_2(x)=1−x+x^2;p_3(x)=1−x+x^2−x^3;p_n(x)=1−x+x^2−x^3+⋯+(−1)^nx^n=_{k=0}^n(−1)^kx^k) Taylor’s Theorem with Remainder Recall that the nth Taylor polynomial for a function (f) at a is the nth partial sum of the Taylor series for (f) at a. Therefore, to determine if the Taylor series converges, we need to determine whether the sequence of Taylor polynomials ({p_n}) converges. However, not only do we want to know if the sequence of Taylor polynomials converges, we want to know if it converges to (f). To answer this question, we define the remainder (R_n(x)) as [R_n(x)=f(x)−p_n(x).] For the sequence of Taylor polynomials to converge to (f), we need the remainder (R_n) to converge to zero. To determine if (R_n) converges to zero, we introduce Taylor’s theorem with remainder. Not only is this theorem useful in proving that a Taylor series converges to its related function, but it will also allow us to quantify how well the nth Taylor polynomial approximates the function. Here we look for a bound on (\|R_n\|.) Consider the simplest case: (n=0). Let (p_0) be the 0th Taylor polynomial at a for a function (f). The remainder (R_0) satisfies (R_0(x)=f(x)−p_0(x)=f(x)−f(a).) If (f) is differentiable on an interval I containing (a) and (x), then by the Mean Value Theorem there exists a real number c between a and x such that (f(x)−f(a)=f′(c)(x−a)). Therefore, [R_0(x)=f′(c)(x−a).] Using the Mean Value Theorem in a similar argument, we can show that if (f) is n times differentiable on an interval I containing a and x, then the nth remainder (R_n) satisfies [R_n(x)=\dfrac{f^{(n+1)}(c)}{(n+1)!}(x−a)^{n+1}] for some real number c between a and x. It is important to note that the value c in the numerator above is not the center a, but rather an unknown value c between a and x. This formula allows us to get a bound on the remainder (R_n). If we happen to know that (∣f^{(n+1)}(x)∣) is bounded by some real number M on this interval I, then [\|R_n(x)\|≤\dfrac{M}{(n+1)!}\|x−a\|^{n+1}] for all x in the interval I. We now state Taylor’s theorem, which provides the formal relationship between a function (f) and its nth degree Taylor polynomial (p_n(x)). This theorem allows us to bound the error when using a Taylor polynomial to approximate a function value, and will be important in proving that a Taylor series for (f) converges to (f). Taylor’s Theorem with Remainder Let (f) be a function that can be differentiated (n+1) times on an interval I containing the real number a. Let (p_n) be the nth Taylor polynomial of (f) at a and let [R_n(x)=f(x)−p_n(x)] be the nth remainder. Then for each x in the interval I, there exists a real number c between a and x such that [R_n(x)=\dfrac{f^{(n+1)}(c)}{(n+1)!}(x−a)^{n+1}]. If there exists a real number (M) such that (∣f^{(n+1)}(x)∣≤M) for all (x∈I), then [\|R_n(x)\|≤\dfrac{M}{(n+1)!}\|x−a\|^{n+1}] for all (x) in (I). Proof Fix a point (x∈I) and introduce the function g such that [g(t)=f(x)−f(t)−f′(t)(x−t)−\dfrac{f''(t)}{2!}(x−t)^2−⋯−\dfrac{f^{(n)}(t)}{n!}(x−t)^n−R_n(x)\dfrac{(x−t)^{n+1}}{(x−a)^{n+1}}.] We claim that (g) satisfies the criteria of Rolle’s theorem. Since (g) is a polynomial function (in t), it is a differentiable function. Also, g is zero at (t=a) and (t=x) because [ \begin{align} g(a) =f(x)−f(a)−f′(a)(x−a)−\dfrac{f''(a)}{2!}(x−a)^2+⋯+\dfrac{f^{(n)}(a)}{n!}(x−a)^n−R_n(x) \[4pt] =f(x)−p_n(x)−R_n(x) \[4pt] =0, \[4pt] g(x) =f(x)−f(x)−0−⋯−0 \[4pt] =0. \end{align}] Therefore, g satisfies Rolle’s theorem, and consequently, there exists c between a and x such that (g′(c)=0.) We now calculate (g′). Using the product rule, we note that [\dfrac{d}{dt}[\dfrac{f^{(n)}(t)}{n!}(x−t)^n]=−\dfrac{f^{(n)}(t)}{(n−1)!}(x−t)^{n−1}+\dfrac{f^{(n+1)}(t)}{n!}(x−t)^n.] Consequently, [g′(t)=−f′(t)+[f′(t)−f''(t)(x−t)]+[f''(t)(x−t)−\dfrac{f'''(t)}{2!}(x−t)^2]+⋯+[\dfrac{f^{(n)}(t)}{(n−1)!}(x−t)^{n−1}−\dfrac{f^{(n+1)}(t)}{n!}(x−t)^n]+(n+1)R_n(x)\dfrac{(x−t)^n}{(x−a)^{n+1}}]. Notice that there is a telescoping effect. Therefore, [g′(t)=−\dfrac{f^{(n+1)}(t)}{n!}(x−t)^n+(n+1)R_n(x)\dfrac{(x−t)^n}{(x−a)^{n+1}}]. By Rolle’s theorem, we conclude that there exists a number c between a and x such that (g′(c)=0.) Since [g′(c)=−\dfrac{f^{(n+1})(c)}{n!}(x−c)^n+(n+1)R_n(x)\dfrac{(x−c)^n}{(x−a)^{n+1}}] we conclude that [−\dfrac{f^{(n+1)}(c)}{n!}(x−c)^n+(n+1)R_n(x)\dfrac{(x−c)^n}{(x−a)^{n+1}}=0.] Adding the first term on the left-hand side to both sides of the equation and dividing both sides of the equation by (n+1,) we conclude that [R_n(x)=\dfrac{f^{(n+1)}(c)}{(n+1)!}(x−a)^{n+1}] as desired. From this fact, it follows that if there exists M such that (∣f^{(n+1)}(x)∣≤M) for all x in I, then [\|R_n(x)\|≤\dfrac{M}{(n+1)!}\|x−a\|^{n+1}]. □ Not only does Taylor’s theorem allow us to prove that a Taylor series converges to a function, but it also allows us to estimate the accuracy of Taylor polynomials in approximating function values. We begin by looking at linear and quadratic approximations of (f(x)=\dfrac{x}) at (x=8) and determine how accurate these approximations are at estimating (\dfrac{11}). Example (\PageIndex{3}): Using Linear and Quadratic Approximations to Estimate Function Values Consider the function (f(x)=\sqrt{x}). Solution: a. For (f(x)=\sqrt{x}), the values of the function and its first two derivatives at (x=8) are as follows: (f(x)=\sqrt{x}) (f(8)=2) (f′(x)=\dfrac{1}{3x^{2/3}}) (f′(8)=\dfrac{1}{12}) (f''(x)=\dfrac{−2}{9x^{5/3}}) (f''(8)=−\dfrac{1}{144.}) Thus, the first and second Taylor polynomials at (x=8) are given by (p_1(x)=f(8)+f′(8)(x−8)) (=2+\dfrac{1}{12}(x−8)) (p_2(x)=f(8)+f′(8)(x−8)+\dfrac{f''(8)}{2!}(x−8)^2) (=2+\dfrac{1}{12}(x−8)−\dfrac{1}{288}(x−8)^2). The function and the Taylor polynomials are shown in Figure 4. b. Using the first Taylor polynomial at (x=8), we can estimate [\dfrac{11}≈p_1(11)=2+\dfrac{1}{12}(11−8)=2.25.] Using the second Taylor polynomial at (x=8), we obtain [\sqrt{11}≈p_2(11)=2+\dfrac{1}{12}(11−8)−\dfrac{1}{288}(11−8)^2=2.21875.] c. By Note, there exists a c in the interval ((8,11)) such that the remainder when approximating (\sqrt{11}) by the first Taylor polynomial satisfies [R_1(11)=\dfrac{f''(c)}{2!}(11−8)^2.] We do not know the exact value of c, so we find an upper bound on (R_1(11)) by determining the maximum value of (f'') on the interval ((8,11)). Since (f''(x)=−\dfrac{2}{9x^{5/3}}), the largest value for (\|f''(x)\|) on that interval occurs at (x=8). Using the fact that (f''(8)=−\dfrac{1}{144}), we obtain (\|R_1(11)\|≤\dfrac{1}{144⋅2!}(11−8)^2=0.03125.) Similarly, to estimate (R_2(11)), we use the fact that (R_2(11)=\dfrac{f'''(c)}{3!}(11−8)^3). Since (f'''(x)=\dfrac{10}{27x^{8/3}}), the maximum value of (f''') on the interval ((8,11)) is (f'''(8)≈0.0014468). Therefore, we have (\|R_2(11)\|≤\dfrac{0.0011468}{3!}(11−8)^3≈0.0065104.) Exercise (\PageIndex{3}): Find the first and second Taylor polynomials for (f(x)=\sqrt{x}) at (x=4). Use these polynomials to estimate (\sqrt{6}). Use Taylor’s theorem to bound the error. Evaluate (f(4),f′(4),) and (f''(4).) (p_1(x)=2+\dfrac{1}{4}(x−4);p_2(x)=2+\dfrac{1}{4}(x−4)−\dfrac{1}{64}(x−4)^2;p_1(6)=2.5;p_2(6)=2.4375;) (\|R_1(6)\|≤0.0625;\|R_2(6)\|≤0.015625) Example (\PageIndex{4}): Approximating (\sin x) Using Maclaurin Polynomials From Example b., the Maclaurin polynomials for (\sin x) are given by [p_{2m+1}(x)=p_{2m+2}(x)=x−\dfrac{x^3}{3!}+\dfrac{x^5}{5!}−\dfrac{x^7}{7!}+⋯+(−1)^m\dfrac{x^{2m+1}}{(2m+1)!} \nonumber] for (m=0,1,2,….) Solution a. The fifth Maclaurin polynomial is [p_5(x)=x−\dfrac{x^3}{3!}+\dfrac{x^5}{5!}]. Using this polynomial, we can estimate as follows: [\sin(\dfrac{π}{18})≈p_5(\dfrac{π}{18})=\dfrac{π}{18}−\dfrac{1}{3!}(\dfrac{π}{18})^3+\dfrac{1}{5!}(\dfrac{π}{18})^5≈0.173648.] To estimate the error, use the fact that the sixth Maclaurin polynomial is (p_6(x)=p_5(x)) and calculate a bound on (R_6(\dfrac{π}{18})). By Note, the remainder is [R_6(\dfrac{π}{18})=\dfrac{f^{(7)}(c)}{7!}(\dfrac{π}{18})^7] for some c between 0 and (\dfrac{π}{18}). Using the fact that (∣f^{(7)}(x)∣≤1) for all (x), we find that the magnitude of the error is at most [\dfrac{1}{7!}⋅(\dfrac{π}{18})^7≤9.8×10^{−10}.] b. We need to find the values of (x) such that [\dfrac{1}{7}!\|x\|^7≤0.0001.] Solving this inequality for (x), we have that the fifth Maclaurin polynomial gives an estimate to within (0.0001) as long as (\|x\|<0.907.) Exercise (\PageIndex{4}) Use the fourth Maclaurin polynomial for (\cos x) to approximate (\cos(\dfrac{π}{12}).) The fourth Maclaurin polynomial is (p_4(x)=1−\dfrac{x^2}{2!}+\dfrac{x^4}{4!}). 0.96593 Now that we are able to bound the remainder (R_n(x)), we can use this bound to prove that a Taylor series for (f) at a converges to (f). Representing Functions with Taylor and Maclaurin Series We now discuss issues of convergence for Taylor series. We begin by showing how to find a Taylor series for a function, and how to find its interval of convergence. Example (\PageIndex{5}): Finding a Taylor Series Find the Taylor series for (f(x)=\dfrac{1}{x}) at (x=1). Determine the interval of convergence. Solution For (f(x)=\dfrac{1}{x},) the values of the function and its first four derivatives at (x=1) are (f(x)=\dfrac{1}{x}) (f(1)=1) (f′(x)=−\dfrac{1}{x^2}) (f′(1)=−1) (f''(x)=\dfrac{2}{x^3}) (f''(1)=2!) (f'''(x)=−\dfrac{3⋅2}{x^4}) (f'''(1)=−3!) (f^{(4)}(x)=\dfrac{4⋅3⋅2}{x^5}) (f^{(4)}(1)=4!). That is, we have (f^{(n)}(1)=(−1)^nn!) for all (n≥0). Therefore, the Taylor series for (f) at (x=1) is given by (\displaystyle \sum_{n=0}^∞\dfrac{f^{(n)}(1)}{n!}(x−1)^n=\sum_{n=0}^∞(−1)^n(x−1)^n). To find the interval of convergence, we use the ratio test. We find that (\dfrac{\|a_{n+1}\|}{\|a_n\|}=\dfrac{∣(−1)^{n+1}(x−1)n^{+1}∣}{\|(−1)^n(x−1)^n\|}=\|x−1\|). Thus, the series converges if (\|x−1\|<1.) That is, the series converges for (0<x<2). Next, we need to check the endpoints. At (x=2), we see that (\displaystyle \sum_{n=0}^∞(−1)^n(2−1)^n=\sum_{n=0}^∞(−1)^n) diverges by the divergence test. Similarly, at (x=0,) (\displaystyle \sum_{n=0}^∞(−1)^n(0−1)^n=\sum_{n=0}^∞(−1)^{2n}=\sum_{n=0}^∞1) diverges. Therefore, the interval of convergence is ((0,2)). Exercise (\PageIndex{5}) Find the Taylor series for (f(x)=\dfrac{1}{2}) at (x=2) and determine its interval of convergence. (f^{(n)}(2)=\dfrac{(−1)^nn!}{2^{n+1}}) (\dfrac{1}{2}\displaystyle \sum_{n=0}^∞(\dfrac{2−x}{2})^n). The interval of convergence is ((0,4)). We know that the Taylor series found in this example converges on the interval ((0,2)), but how do we know it actually converges to (f)? We consider this question in more generality in a moment, but for this example, we can answer this question by writing [ f(x)=\dfrac{1}{x}=\dfrac{1}{1−(1−x)}.] That is, (f) can be represented by the geometric series (\displaystyle \sum_{n=0}^∞(1−x)^n). Since this is a geometric series, it converges to (\dfrac{1}{x}) as long as (\|1−x\|<1.) Therefore, the Taylor series found in Example does converge to (f(x)=\dfrac{1}{x}) on ((0,2).) We now consider the more general question: if a Taylor series for a function (f) converges on some interval, how can we determine if it actually converges to (f)? To answer this question, recall that a series converges to a particular value if and only if its sequence of partial sums converges to that value. Given a Taylor series for (f) at a, the nth partial sum is given by the nth Taylor polynomial pn. Therefore, to determine if the Taylor series converges to (f), we need to determine whether (\displaystyle \lim_{n→∞}p_n(x)=f(x)). Since the remainder (R_n(x)=f(x)−p_n(x)), the Taylor series converges to (f) if and only if (\displaystyle \lim_{n→∞}R_n(x)=0.) We now state this theorem formally. Convergence of Taylor Series Suppose that (f) has derivatives of all orders on an interval (I) containing (a). Then the Taylor series [\sum_{n=0}^∞\dfrac{f^{(n)}(a)}{n!}(x−a)^n] converges to (f(x)) for all (x) in (I) if and only if [\lim_{n→∞}R_n(x)=0] for all (x) in (I). With this theorem, we can prove that a Taylor series for (f) at a converges to (f) if we can prove that the remainder (R_n(x)→0). To prove that (R_n(x)→0), we typically use the bound [\|R_n(x)\|≤\dfrac{M}{(n+1)!}\|x−a\|^{n+1}] from Taylor’s theorem with remainder. In the next example, we find the Maclaurin series for (e^x) and (\sin x) and show that these series converge to the corresponding functions for all real numbers by proving that the remainders (R_n(x)→0) for all real numbers (x). Example (\PageIndex{6}): Finding Maclaurin Series For each of the following functions, find the Maclaurin series and its interval of convergence. Use Note to prove that the Maclaurin series for (f) converges to (f) on that interval. Solution a. Using the nth Maclaurin polynomial for (e^x) found in Example a., we find that the Maclaurin series for (e^x) is given by (\displaystyle \sum_{n=0}^∞\dfrac{x^n}{n!}). To determine the interval of convergence, we use the ratio test. Since (\dfrac{\|a_{n+1}\|}{\|a_n\|}=\dfrac{\|x\|^{n+1}}{(n+1)!}⋅\dfrac{n!}{\|x\|^n}=\dfrac{\|x\|}{n+1}), we have (\displaystyle \lim_{n→∞}\dfrac{\|a_{n+1}\|}{\|a_n\|}=\lim_{n→∞}\dfrac{\|x\|}{n+1}=0) for all (x). Therefore, the series converges absolutely for all (x), and thus, the interval of convergence is ((−∞,∞)). To show that the series converges to (e^x) for all (x), we use the fact that (f^{(n)}(x)=e^x) for all (n≥0) and (e^x) is an increasing function on ((−∞,∞)). Therefore, for any real number (b), the maximum value of (e^x) for all (\|x\|≤b) is (e^b). Thus, (\|R_n(x)\|≤\dfrac{e^b}{(n+1)!}\|x\|^{n+1}). Since we just showed that (\displaystyle \sum_{n=0}^∞\dfrac{\|x\|^n}{n!}) converges for all x, by the divergence test, we know that (\displaystyle \lim_{n→∞}\dfrac{\|x\|^{n+1}}{(n+1)!}=0) for any real number x. By combining this fact with the squeeze theorem, the result is (\lim_{n→∞}R_n(x)=0.) b. Using the nth Maclaurin polynomial for (\sin x) found in Example b., we find that the Maclaurin series for (\sin x) is given by (\displaystyle \sum_{n=0}^∞(−1)^n\dfrac{x^{2n+1}}{(2n+1)!}). In order to apply the ratio test, consider (\dfrac{\|a_{n+1}\|}{\|a_n\|}=\dfrac{\|x\|^{2n+3}}{(2n+3)!}⋅\dfrac{(2n+1)!}{\|x\|^{2n+1}}=\dfrac{\|x\|^2}{(2n+3)(2n+2)}). Since (\displaystyle \lim_{n→∞}\dfrac{\|x\|^2}{(2n+3)(2n+2)}=0) for all (x), we obtain the interval of convergence as ((−∞,∞).) To show that the Maclaurin series converges to (\sin x), look at (R_n(x)). For each (x) there exists a real number (c) between (0) and (x) such that (R_n(x)=\dfrac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}). Since (∣f^{(n+1)}(c)∣≤1) for all integers (n) and all real numbers(c), we have (\|R_n(x)\|≤\dfrac{\|x\|^{n+1}}{(n+1)!}) for all real numbers (x). Using the same idea as in part a., the result is (\displaystyle \lim_{n→∞}R_n(x)=0) for all (x), and therefore, the Maclaurin series for (\sin x) converges to (\sin x) for all real (x). Exercise (\PageIndex{6}) Find the Maclaurin series for (f(x)=\cos x). Use the ratio test to show that the interval of convergence is ((−∞,∞)). Show that the Maclaurin series converges to (\cos x) for all real numbers (x). Use the Maclaurin polynomials for (\cos x.) (\sum_{n=0}^∞\dfrac{(−1)^nx^{2n}}{(2n)!}) By the ratio test, the interval of convergence is ((−∞,∞).) Since (\|R_n(x)\|≤\dfrac{\|x\|^{n+1}}{(n+1)!}), the series converges to (\cos x) for all real (x). Proving that E is Irrational In this project, we use the Maclaurin polynomials for (e^x) to prove that e is irrational. The proof relies on supposing that e is rational and arriving at a contradiction. Therefore, in the following steps, we suppose (e=r/s) for some integers r and s where (s≠0.) Key Concepts Key Equations (\sum_{n=0}^∞\dfrac{f^{(n)}(a)}{n!}(x−a)^n=f(a)+f′(a)(x−a)+\dfrac{f''(a)}{2!}(x−a)^2+⋯+\dfrac{f^{(n)}(a)}{n!}(x−a)^n+⋯) Glossary for a function (f) and the nth Taylor polynomial for (f) at (x=a), the remainder (R_n(x)=f(x)−p_n(x)) satisfies (R_n(x)=\dfrac{f^{(n+1)}(c)}{(n+1)!}(x−a)^{n+1}) for some c between x and a; if there exists an interval I containing a and a real number M such that (∣f^{(n+1)}(x)∣≤M) for all x in I, then (\|R_n(x)\|≤\dfrac{M}{(n+1)!}\|x−a\|^{n+1}) Contributors Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at This page titled 6.3: Taylor and Maclaurin Series is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform. Recommended articles The LibreTexts libraries are Powered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement. For more information contact us atinfo@libretexts.org.
15139	https://www.mi-hms.org/sites/default/files/UTI%20Guideline-6.9.21.pdf	Guidelines for Treatment of Urinary Tract Infections 2 This document details the Michigan Hospital Medicine Safety (HMS) Consortium preferred antibiotic choices for treatment of uncomplicated and complicated lower urinary tract infections, pyelonephritis, and urinary tract infections with bacteremia. The treatment recommendations highlighted in this document are not meant to be a comprehensive guideline. This guideline also addresses the appropriate management of asymptomatic bacteriuria which accounts for a substantial burden of unnecessary antimicrobial use. • The recommendations within this guideline are intended to address the management of positive urine cultures in non-pregnant and non-ICU patients. • This guideline is not intended for patients undergoing urologic procedures during their hospitalization, patients who have undergone urinary diversion surgery, or have urinary stents or percutaneous nephrostomy tubes, or meet criteria of severe sepsis or septic shock. • Hospitals choice of preferred antibiotics among options provided should also be based on antimicrobial stewardship/infectious diseases recommendations, hospital formulary restrictions, and hospital antibiograms (especially urine antibiograms when available). Intended Use Overview 3 National guidelines recommend against testing for asymptomatic bacteriuria, except in select circumstances In the absence of signs or symptoms (see below) attributable to a urinary tract infection, patients with a positive urine culture and/or pyuria should not be treated with antibiotics irrespective of high bacterial colony count, or a multi-drug resistant organism. Altered mental status in the absence of signs or symptoms (see below) should not be treated empirically with antibiotics for 48-72 hours while working up alternative causes (e.g., medication side effects, dehydration, constipation, etc.). See Appendix B for algorithm regarding these patients. Urine Testing Do NOT Send Urinalysis or Urine Culture if none of these symptoms are present or there is an alternative cause for the symptom Signs & Symptoms without alternative cause Fever >38o C or rigors Urgency, frequency, dysuria Suprapubic pain or tenderness Costovertebral pain or tenderness New onset mental status changes with leukocytosis (>10,000 cells/mm3), hypo tension (<90mmHg Systolic), or >/= 2 SIRS criteria1 Acute hematuria Spasticity or autonomic dysreflexia in patients with spinal cord injury Examples of symptoms that are NOT indicative of a UTI include: Cloudy/Dirty Urine, Foul-smelling urine, sediment in urine, etc. Symptom-based screening may not be reliable in the setting of renal transplants or urinary diversion. Additionally, please use your clinical judgement in patients with severe sepsis/septic shock or with baseline cognitive or functional impairment with new functional decline or falls who are hemodynamically unstable without alternative etiology. Urine culture alone is appropriate for febrile neutropenia and ASB screening for pregnancy or prior to urologic procedures. Asymptomatic Bacteriuria 4 • Empiric antibiotic choice should take into consideration recent previous culture results, prior antibiotic use, antibiotic allergies, local antibiotic susceptibilities, and severity of presenting illness. Empiric antibiotic choice cannot take into account scenarios that are outside of the scope of these guidelines. • Final antibiotic choice should be based on antibiotic susceptibilities of the pathogen and take into consideration antibiotic allergies of the patient. • Recommended duration of treatment is for an effective antibiotic based on culture results. • Remember good documentation practices at discharge including: documentating stop/start dates, accounting for inpatient AND outpatient duration when calculating total duration, and educating patients on their antibiotic treatment. Empiric Treatment Recommendations for Lower Urinary Tract Infections, Pyelonephritis, and Urinary Tract Infections with Bacteremia DEFINITIONS Uncomplicated Lower Urinary Tract Infection or Cystitis^ Female patients without catheters and without any of the co-morbid conditions listed under complicated lower urinary tract infections Complicated Lower Urinary Tract Infection or Cystitis^ Patients with catheter associated-urinary tract infections (CA-UTI) and non-CAUTI associated urinary tract infection in the following categories: • Men • Women with the following co-morbid conditions: • nephrolithiasis • urologic surgery • urinary obstruction • urinary retention • spinal cord injury • asplenia • receiving chemotherapy for a malignancy or malignancy not in remission • moderate/severe liver disease • hemiplegia • congestive heart failure • cardiomyopathy • moderate/severe chronic kidney disease or on hemodialysis • structural lung disease (moder ate-severe COPD, bronchiectasis, home oxygen) • sickle cell disease • chronic anti-coagulation • bedridden or using wheel chair • diabetes mellitus with Hgb A1C >8 % • immunodeficiency or immunosuppressive treat ments Uncomplicated Pyelonephritis Female patients with pyelonephritis without catheters or any of the co-morbid conditions listed in the definition for complicated lower UTI Complicated Pyelonephritis Patients with pyelonephritis not meeting the definition for uncomplicated pyelonephritis Excluding patients with pyelonephritis, bacteremia, or severe sepsis ^For Cystitis, avoid fluoroquinolones at discharge when alternative agents are available. 5 Antibiotic Duration Considerations Nitrofurantoin2 5 days Avoid in CrCl < 30ml/min Trimethoprim-sulfamethoxazole 3 days Increasing E. Coli resistance Alternative Fosfomycin3 1 dose Cost ~$60/dose May not be available at some retail phar macies May consider extending duration to 3-5 doses IV beta-lactam4 or Oral beta-lactam5 3-7 days You may also consider extending the duration of therapy for fosfomycin to mirror complicated UTI with 3-5 doses, as a recent study showed inferior rates of symptom improvement compared to nitrofurantoin in this population. Fluoroquinolones should be reserved for uncomplicated cystitis when other oral antibiotic options are not feasible because of their propensity for collateral damage (antibiotic resistance, C.difficile infection, and other adverse effects6). When a fluoroquinolone is used, the duration of treatment is 3 days. Uncomplicated Lower Urinary Tract Infection or Cystitis Complicated Lower Urinary Tract Infections or Cystitis Antibiotic Duration Considerations Nitrofurantoin2 7 days Avoid in CrCl < 30ml/min Fosfomycin3 Q 48 hrs X 3-5 doses •Cost ~$60 / dose •May not be available at some retail pharmacies Trimethoprim-sulfamethoxazole 7 days Increasing E. Coli resistance IV beta-lactam4 , Oral beta-lac tam5, or Aztreonam in setting of severe PCN or Cephalosporin allergy 7 days Total antibiotic duration of 7 days (oral, IV, or combination) is usually appropriate, but delayed response to therapy may warrant 10-14 days of therapy. A single dose of Fosfomycin or a 3-day treatment course for other antibiotics can be considered for women < 65 years who develop a CA-UTI without upper urinary tract symptoms after the indwelling catheter has been removed. Fluoroquinolones should be reserved for complicated lower UTI when other oral antibiotic options are not feasible because of their propensity for collateral damage (antibiotic resistance, C.difficile infection, and other adverse effects6). When a fluoroquinolone is used, the duration of treatment is 5-7 days unless there is a delayed response to therapy. 6 Pyelonephritis and Urinary Tract Infections Associated with Bacteremia Uncomplicated Pyelonephritis Antibiotic Duration Trimethoprim-sulfamethoxazole 7-14 days Fluoroquinolones 5-7 days Beta-lactams IV beta-lactam therapy4: 7 days IV beta-lactam therapy4 followed by oral beta-lactam5 or oral trimethoprim-sulfamethoxazole therapy: 7-14 days Complicated Pyelonephritis and UTI with Bacteremia Complicated Pyelonephritis: 7-14 days UTI with Bacteremia: 7-14 days Shorter courses of therapy (7-days) with a fluoroquinolone or IV beta-lactam can be considered in female patients without co-morbid conditions who are bacteremic secondary to pyelonephritis or cystitis/lower UTI who have rapid clinical response to therapy. Antibiotic choice is based on multiple factors and will defer to individual institutions choice. • Nitrofurantoin and Fosfomycin should not be used for pyelonephritis, upper urinary tract infection, or patients with bacteremia. • Due to potential complications from PICC lines (e.g. DVT, CLABSI), oral fluoroquinolones are preferred over PICC line placement for IV antibiotics when the urinary pathogen is susceptible and there are no contraindications to fluoroquinolones. • Oral beta-lactams are associated with lower efficacy and higher relapse rates compared to trimethoprim-sulfamethoxazole and fluoroquinolones. If a beta-lactam is used then initial therapy should be IV therapy followed by oral beta-lactam (assuming uropathogen is susceptible). • A shorter course of therapy (<14 days) is not appropriate for Staphylococcus Aureus bacteremia and another source of infection (outside of the genitourinary tract) should be considered. 7 Antibiotic Dose Trimethoprim-sulfamethoxazole (160 mg/800 mg) 1 DS tablet po BID Nitrofurantoin 100 mg po BID Fosfomycin 3 g dose (see tables for complicated and uncom plicated lower UTI) Amoxicillin-clavulanate 875mg po BID Uncomplicated Cystitis: 500 mg po BID Cephalexin 500 mg po BID-QID Uncomplicated Cystitis: 500 mg po BID Cefpodoxime 100-200 mg po BID Uncomplicated Cystitis: 100 mg po BID Cefdinir 300 mg po BID Cefazolin 1-2g IV q 8 hr Cefuroxime 500 mg po BID 750 mg-1.5g IV q 8 hr Uncomplicated Cystitis: 250 mg po BID Piperacillin-tazobactam 3.375 g IV q 6 hr or 4.5 g IV q 6-8 hr Ceftriaxone 1-2 g IV once daily Cefepime 1-2 g IV q 8-12 hr Aztreonam 1-2 g IV q 8 hr Ertapenem 1 gm IV QD Meropenem 500 mg IV q6 hr or 1g IV q 8 hr Levofloxacin 250-750 mg QD Uncomplicated Cystitis: 250 mg po QD Uncomplicated Pyelonephritis: 7-day duration: 500 mg po QD 5-day duration: 750 mg po QD Ciprofloxacin 250-750 mg po BID 400 mg IV q12 hr Uncomplicated Cystitis: 250 mg po BID Uncomplicated Pyelonephritis: 500 mg po BID Dose adjustment needed based on renal function Dose depends on disease state (Uncomplicated UTI, Complicated UTI, Pyelonephritis), severity of presentation (e.g. septic shock, severe sepsis), presence of bacteremia, and susceptibilities of the pathogen Appendix A 8 Appendix B A A AS S SS S SE E ES S SS S SI I IN N NG G G F F FOR R R U U UR R RI I IN N NA A AR R RY Y Y T T TR R RA A AC C CT T T I I IN N NF F FE E EC C CT T TI I IO ON N N I I IN N N E E EL L LD D DE E ER R RL L LY Y Y I I IN N NP P PA A AT TI I IE E EN N NT T TS S S W W WI I IT T TH H H A A AC C CU U UT T TE E EL L LY Y Y A A AL L LT TE E ER R RE E ED D D M M ME E EN N NT T TA A AL L L S S ST T TA A AT TU U US S S ( ( (A A AM M MS S S) ) ) M M Mo o od d d e el l l e e ed d d b b b a a a s s se ed d d o o on n n M M Mo o od d d y y y, , , L L L ( ( ( 2 2 2 0 0 0 1 1 1 4 4 4 ) ) ) J JA A AM M MA A A 3 3 3 1 1 1 1 1 1 ( ( ( 8 8 8 ) ) ) : : : 8 8 8 4 4 4 4 4 4 ---8 8 8 5 5 5 4 4 4 . . . d d d o o oi i i : : : 1 1 1 0 0 0 . . .1 1 1 0 0 0 0 0 0 1 1 1 / / /j j j a a am m ma a a . . .2 2 2 0 0 0 1 1 1 4 4 4 . . .3 3 3 0 0 0 3 3 3 DO NOT TREAT with antibiotics (Regardless of Culture result) Continue to evaluate for other etiologies and do not treat with antibiotics Do NOT send urine testing for test of cure. If presenting symptoms persist, consider evaluation of other possible etiologies. Consider antibiotic therapy What was the result of the urine culture? Have symptoms resolved? SEND URINE SAMPLE FOR URINE CULTURE STOP FURTHER EVALUATION FOR UTI SEND URINE SAMPLE FOR URINAL YSIS UTI-Specific Signs and Symptoms (Without Alternative Cause) Does the patient have one or mor e of the following (without an alternative explanation)?: SIRS Criteria: Temp >38C or <36C, RR> 20 or PaCO2 <32 mmHg, abnormal WBC (>12,000 or <4,000) or >10% immature [band] forms Elevated WBC (> 10,000 cells/mm3) Hypotension (< 90 mmHg Systolic) Two or more SIRS Criteria Examples: Periods of altered perception, disorganized speech, lethargy , etc. Symptoms Include: Urgency, frequency , dysuria, costovertebral pain or tenderness, ﬂank pain, suprapubic pain or tenderness, acute hematuria, and/or fever (>38C) or rigors. Yes Yes UA POSITIVE UA NEGATIVE Does the patient still have altered mental status after 24-48 hours OR develop UTI-speciﬁc signs/symptoms? Evaluate for other causes of altered mental status Attempt hydration (oral or IV) Evaluate medications for new medications, potential interactions and polypharmacy , or adverse events Consider changes to current medication regimen Acutely observe for prompt resolution within 24-48 hours No Yes Yes No NEGATIVE POSITIVE Please use your clinical judgement in patients with baseline cognitive or functional impairment with new functional decline or falls who are hemodynamically unstable without alternative etiology. Change in Mental Status (Only Symptom) 9 1. SIRS Criteria: Heart rate greater than 90bpm, respiratory rate greater than 20 breaths per minute, temperature less than 36o C, white blood count less than 4,000 cells/mm3, temperature greater than 38o C, white blood count greater than 12,000 cells/mm3. 2. The Beers Criteria recommends avoiding use in geriatric patients >65 with a CrCl< 30 mL/min. (American Geriatric Society 2015, Updated Beers Criteria for Potentially Inappropriate Medication Use in Older Adults. J Am Geriatr Soc. 2015). 3. Fosfomycin susceptibilities may not be routinely available as part of standard antimicrobial susceptibility testing. Fosfomycin susceptibilities have only been established for E.coli and Enterococcus species by the Clinical and Laboratory Standards Institute, but there is data and clinical experience supporting use of the same susceptibility breakpoints for other members of the Enterobacteriaceae group. 4. Examples of IV beta-lactams include but are not limited to Cefazolin, Ceftriaxone, Cefuroxime, Piperacillin-Tazobactam, Cefepime. 5. Examples of oral beta-lactams include, but are not limited to Amoxicillin-Clavulanate, Cephalexin, Cefdinir, Cefuroxime, and Cefpodoxime. 6. In the United States, there are high rates of fluoroquinolone resistance among outpatient and inpatient urinary E.coli isolates. IDSA guidelines advise against empiric use of fluoroquinolones when E.coli resistance exceeds 20%. Other notable adverse effects of fluoroquinolones include - QT interval prolongation and arrhythmia, peripheral neuropathy, tendinopathy, and tendon rupture. In 2016, the FDA placed a black box warning to limit fluoroquinolone use in uncomplicated UTIs due to potential side effects. Footnotes 10 Key References 1. Hooton TM, et al. Diagnosis, prevention, and treatment of catheter-associated urinary tract infection in adults: 2009 international clinical practice guidelines from the Infectious Diseases Society of America. Clin Infect Dis 2010;60:625-663. 2. Nicolle L, et al. Infectious Diseases Society of America Guidelines for the diagnosis and treatment of asymptomatic bacteriuria in adults. Clin Infect Dis 2005; 40:643-54. 3. Gupta K. International clinical practice guidelines for the treatment of acute uncomplicated cystitis and pyelonephritis in women: A 2010 update by the Infectious Diseases Society of America and the European Society for Microbiology and Infectious Diseases. Clin Infect Dis 2011;52:e103-120. 4. Fox Miriam T, et al. A seven-day course of TMP-SMX may be as effective as a seven-day course of ciprofloxacin for the treatment of pyelonephritis. Am J of Medicine 2017;130: 842-845. 5. Moustafa F, et al. Evaluation of the efficacy and tolerance of a short 7 day third generation cephalosporin treatment in the management of acute pyelonephritis in young women in the emergency department. J Antimicrob Chemother 2016;71:1660-1664. 6. Eliakim-Raz N, et al. Duration of antibiotic treatment for acute pyelonephritis and septic urinary tract infection – 7 days or less versus longer treatment: systematic review and meta-analysis of randomized controlled trials. J Antimicrob Chemother 2013;68:2183-91. 7. Chotiprasitsakul D, et al. Comparing the outcomes of adults with enterobacteriaceae bacteremia receiving short-course versus prolonged-course antibiotic therapy in a multicenter, propensity score-matched cohort. Clin Infect Dis 2018;66:172-177. 8. Johnson JR, et al. Acute Pyelonephritis in Adults. N Engl J Med 2018;378:48-59. 9. Huttner A et al. Effect of 5-Day Nitrofurantoin vs Single-Day Fosfomycin on Clinical Resolution of Uncomplicated Lower Urinary Tract Infection in Women: A Randomized Clinical Trial. JAMA. 2018;319(17):1781-1789. Support for HMS is provided by Blue Cross and Blue Shield of Michigan and Blue Care Network as part of the BCBSM Value Partnerships program. Although Blue Cross Blue Shield of Michigan and HMS work collaboratively, the opinions, beliefs and viewpoints expressed by the author do not necessarily reflect the opinions, beliefs and viewpoints of BCBSM or any of its employees. 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15140	https://forum.digikey.com/t/how-to-convert-gauss-to-tesla/78	How to convert Gauss to Tesla One Tesla is equivalent to ten thousand Gauss. If your device is specified in Gauss and you need it in Tesla, then divide its Gauss rating by 10,000. If your device is in Tesla and you need it in Gauss, multiple the Tesla rating by 10,000 instead. These formulas will help you remember. G = T x 10,000 T= G / 10,000 Related topics \| Topic \| \| Replies \| Views \| Activity \| --- --- \| ガウスをテスラに変換する方法技術的なヒント \| 0 \| 586 \| August 31, 2020 \| \| Digi-Key Magnets and Surface Gauss Sensors Transducers terminology , faqs , automation-and-control , sensors-transducers \| 0 \| 3152 \| December 8, 2017 \| \| סיומת "T" או "R" במק"טי Texas Instruments פורום בעברית \| 0 \| 753 \| November 17, 2019 \| \| Digi-Keyの磁石と表面ガウスよくある質問 \| 0 \| 606 \| May 29, 2021 \| \| Site Data Errors for part AH3372 Site Help, Information and Feedback \| 1 \| 702 \| January 20, 2022 \| Powered by Discourse, best viewed with JavaScript enabled We use cookies to provide our visitors with an optimal site experience. View our privacy notice and cookie notice to learn more about how we use cookies and how to manage your settings. By proceeding on our website you consent to the use of cookies.
15141	https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introduction_to_Organic_and_Biochemistry_(Malik)/01%3A_Bonding_in_organic_compounds/1.04%3A_Representing_organic_compounds	1.4: Representing organic compounds - Chemistry LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter Reader Mode 1: Bonding in organic compounds Introduction to Organic and Biochemistry (Malik) { } { "1.01:_What_is_organic_chemistry" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.02:_What_is_a_chemical_bond" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.03:_Hybridization_of_orbitals_and_3D_structures_of_simple_organic_compounds" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.04:_Representing_organic_compounds" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.05:_Formal_Charge" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.06:_Resonance" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } { "00:_Front_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "01:_Bonding_in_organic_compounds" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_Nomenclature_and_physical_properties_of_organic_compounds" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_Stereochemistry" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Organic_reactions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_Carbohydrates" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "06:_Lipids" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "07:_Proteins" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "08:_Nucleic_acids" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "09:_Food_to_energy_metabolic_pathways" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "zz:_Back_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } Thu, 21 Sep 2023 16:54:15 GMT 1.4: Representing organic compounds 394087 394087 Delmar Larsen { } Anonymous Anonymous User 2 false false [ "article:topic", "molecular formula", "license:publicdomain", "authorname:mmalik", "Lewis formula", "condensed formula", "skeletal formula" ] [ "article:topic", "molecular formula", "license:publicdomain", "authorname:mmalik", "Lewis formula", "condensed formula", "skeletal formula" ] Search site Search Search Go back to previous article Sign in Username Password Sign in Sign in Sign in Forgot password Expand/collapse global hierarchy 1. Home 2. Bookshelves 3. Introductory, Conceptual, and GOB Chemistry 4. Introduction to Organic and Biochemistry (Malik) 5. 1: Bonding in organic compounds 6. 1.4: Representing organic compounds Expand/collapse global location 1.4: Representing organic compounds Last updated Sep 21, 2023 Save as PDF 1.3: Hybridization of orbitals and 3D structures of simple organic compounds 1.5: Formal Charge Page ID 394087 Muhammad Arif Malik Hampton University, Hampton, VA ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. Learning Objectives 2. Molecular formula 3. Lewis formula 4. Condensed formula 5. Skeletal formula 1. 1. Some terms related to the primary classification of organic compounds 2. Skeletal formulas of n-alkanes/01:_Bonding_in_organic_compounds/1.04:_Representing_organic_compounds#Skeletal_formulas_of_n-alkanes) 1. Stem names of organic compounds/01:_Bonding_in_organic_compounds/1.04:_Representing_organic_compounds#Stem_names_of_organic_compounds) 2. Homologous series/01:_Bonding_in_organic_compounds/1.04:_Representing_organic_compounds#Homologous_series) 3. General formula of n-alkanes/01:_Bonding_in_organic_compounds/1.04:_Representing_organic_compounds#General_formula_of_n-alkanes) 4. Variations in the structural formula to represent different configurations/01:_Bonding_in_organic_compounds/1.04:_Representing_organic_compounds#Variations_in_the_structural_formula_to_represent_different_configurations) 3. Condensed and skeletal formulas of branched-alkanes/01:_Bonding_in_organic_compounds/1.04:_Representing_organic_compounds#Condensed_and_skeletal_formulas_of_branched-alkanes) 4. Skeletal formulas of alkenes/01:_Bonding_in_organic_compounds/1.04:_Representing_organic_compounds#Skeletal_formulas_of_alkenes) 5. Skeletal formulas of alkynes/01:_Bonding_in_organic_compounds/1.04:_Representing_organic_compounds#Skeletal_formulas_of_alkynes) 6. Skeletal formulas of organic compounds containing heteroatoms/01:_Bonding_in_organic_compounds/1.04:_Representing_organic_compounds#Skeletal_formulas_of_organic_compounds_containing_heteroatoms) 7. Variations in the formulas representing organic compounds/01:_Bonding_in_organic_compounds/1.04:_Representing_organic_compounds#Variations_in_the_formulas_representing_organic_compounds) Examples of drawing Lewis, condensed, and skeletal formulae Example 1.4.1 Solution Example 1.4.2 Example 1.4.3 Learning Objectives Read and draw molecular, Lewis, condensed, and structural formulae of simple organic compounds. Understand the mixed versions of and slight variations in drawing the formulae of simple organic compounds. The molecular formula, Lewis formula, condensed formula, skeletal formula, or a combination of these can represent an organic compound. These ways of representing organic compound and what they mean is described below. Molecular formula The molecular formula tells the symbols of the elements that compose the compound, and the subscript to the element symbol denotes how many atoms of that element are in the molecule. For example, (CH⁢A 4) is a molecule formula of methane which means there is one carbon and four hydrogen atoms in a methane molecule. C⁢A 2⁢H⁡A 6 is a molecule formula of ethane, which means the ethane molecule has two carbon and six hydrogen atoms. Molecular formulas do not tell about the molecule's bonds and shapes. Lewis formula The Lewis structure or Lewis formula shows all the bonding electron pairs as lines (bonds) and lone pairs (non-bonding electron pairs) as pairs of dots around each atom in a molecule. For example, Lewis formula of ethane (C⁢A 2⁢H⁡A 6) is: H−C H\|\|H−C H\|\|H−H that shows each carbon is bonded with one C and three H′⁡s by single bonds. Similarly, Lewis formula of formaldehyde (CH⁢A 2⁢O) is: H−C\|H=O⋅⋅: that shows C is bonded with two H′⁡s by single bonds and with one O by a double bond and O has two lone pairs on it. The lone pairs are usually omitted from Lewis structures except when needed to emphasize their presence. For example, Lewis formula for methanol (CH⁢A 4⁢O) with lone pairs is: H−C H\|\|H−O⋅⋅⋅⋅−H, but it can also be shown as H−C H\|\|H−O−H where the lone pairs are not shown on O but it is understood that two lone pairs are there. Condensed formula The Lewis formulas become complicated and time-consuming for larger organic compounds. Condensed formula simplifies the Lewis formula by writing each C followed by H⁡A′⁢s attached with it. Subscripts are used to show more than one H⁡A′⁢s. If there is a heteroatom, i.e., any atom other than C or H in the chain, it is condensed like C, except for halogens which are condensed like H. For example, the Lewis formula of ethane H−C H\|\|H−C H\|\|H−H is condensed as CH⁢A 3⁢CH⁢A 3. Lewis formula for methanol H−C H\|\|H−O−H is condensed as CH⁢A 3⁢OH. Lewis formula for ethanamine H−C H\|\|H−C H\|\|H−N\|H−H is condensed as CH⁢A 3⁢CH⁢A 2⁢NH⁢A 2. Lewis formula for 2-chloropropane H−C H\|\|H−C H\|\|C⁢l−C H\|\|H−H, is condensed as CH⁢A 3⁢CHClCH⁢A 3. Skeletal formula The Lewis and condensed formulas do not show the geometry of the organic compounds. Further, the Lewis and condensed formulas become complicated and time-consuming for large organic compounds. Skeletal formulas or line-angle formulas overcome these drawbacks by simplifying the representation of organic compounds by omitting C⁢A′⁢s and H⁡A′⁢s in the formula and showing only the skeleton of the compound by C-to-C bonds as lines in a geometry that is closer to the actual geometry. Any heteroatom and H⁡A′⁢s attached to the heteroatom are shown in the skeletal formula. Some terms related to the primary classification of organic compounds Before learning skeletal formulas, it is essential to understand the following terms related to the primary type of organic compounds. Organic compounds containing only C⁢A′⁢s and H⁡A′⁢s are called hydrocarbons. For example, ethane (CH⁢A 3⁢CH⁢A 3), ethene (CH⁢A 2⁢CH⁢A 2), and ethyne (CHCH) described in previous sections are hydrocarbons. The hydrocarbons contain only σ-bonds are called alkanes. For example, methane (CH⁢A 4) and ethane (CH⁢A 3⁢CH⁢A 3) are alkanes. Alkanes containing a chain of C⁢A′⁢s where C⁢A′⁢s are connected with either one or two C⁢A′⁢s are called straight chain alkanes or normal-chain alkanes (n-alkanes). For example, n-alkane having four carbons is n-butane ((CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3), five carbons is n-pentane (CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3), six carbons is n-hexane (CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3), and so on. Alkanes in which at least on C connected with three or four other C⁢A′⁢s are called branched-chain alkanes. For example, isopropane H−C H\|\|H−C H\|\|H−C\|H−H−C H\|\|H−H is a branched-chain alkane The hydrocarbons containing at least one double bond, i.e., a σ-bond and a π-bond together, are called alkenes. For example, ethene H−C\|H=C\|H−H is an alkene. The hydrocarbons containing at least one triple-bond, i.e., a σ-bond and two π-bonds together, are called alkynes. For example, ethyne H−C≡C−H is an alkyne. The hydrocarbons with a planer cyclic structure having alternating odd numbers of double bonds are a special class of hydrocarbons called aromatic hydrocarbons that will be described later. Organic compounds containing at least one heteroatom, i.e., O, N, S, P, etc. are not hydrocarbons. For example, methanol CH⁢A 3⁢OH is not a hydrocarbon. Several classes of organic compounds are not hydrocarbons, which will be described later. Skeletal formulas of n-alkanes Methane (CH⁢A 4) is the simplest alkane that has a tetrahedral geometry around its C, as illustrated in Figure 1.4.1 a. Gray lines in Figure 1.4.1 a show the outline of the tetrahedron shape, and the black lines show C−H bonds. The plane defined by C, H on the top, and H on the right is in the plane of the paper in this perspective drawing (Figure 1.4.1 a). The H on the hashed wedge is going below, and the H on the solid wedge is coming above the plane of the paper. The plane of the paper cuts through the middle of two H⁡A′⁢s on the left of the drawing. The point of view is slightly above C towards the top-right corner. Figure 1.4.1 b shows the same structure without the tetrahedron layout drawing. All bonds are equal, and all bond angles are 109.5 o. Figure 1.4.1 c shows the model of CH⁢A 4 molecule from approximately the same view. Note that the geometry of CH⁢A 4 molecule is two V's of 109.5 o internal angle, placed perpendicular to each other, and joined at the vertex. Figure 1.4.1 d shows the same structure rotated such that the two H⁡A′⁢s in the plane of the paper are on a straight line at the bottom of the drawing. Figure 1.4.1 e shows the model rotated in the same orientation as the perspective drawing in Figure 1.4.1 d. a)b)c)d)e) Figure 1.4.1: a) Perspective drawing of methane (CH⁢A 4) with the outlines of the tetrahedron drawing with gray lines. The plane of the paper (or the screen) is along the plane of C and two H⁡A′⁢s to the right and cuts through the middle of the two H⁡A′⁢s on the left (angel of view is slightly below the top-right corner) b) perspective drawing of CH⁢A 4 in the same view without drawing of the outlines of the tetrahedron, c) model of CH⁢A 4 from the same view, d) perspective drawing of CH⁢A 4 rotated to bring the two H⁡A′⁢s in the plane of the page at the base making an inverted bigger V-shape, and e) model of CH⁢A 4 in the same view as the perspective drawing in d. (Copyright; Public domain) Replacing any H with another sp 3 hybridized C results in ethane (CH⁢A 3⁢CH⁢A 3) as shown in row two of Table 1. A line drawn to represent C−C bond in ethane is the skeletal formula of ethane, as shown in row two of Table 1. When 2nd H of methane is also replaced with another sp 3 hybridized C, it results in propane (CH⁢A 3⁢CH⁢A 2⁢CH⁢A 3), as shown in row three of Table 1. Two H⁡A′⁢s of methane in the plane of the page have been replaced with C⁢A′⁢s in this case resulting in C−C−C bonds in an inverted V-shape in the plane of the page. The skeletal formula representing a propane molecule is two lines connected in an inverted V-shape, as shown in row three of Table 1. Replacing any H of a terminal C of a propane with another sp 3 hybridized C results in butane (CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3) as shown in 4th row of Table 1. Remember rotation around C−C single bond happens. Therefore, the 4th C of butane can be placed at any angle relative to the plane defined by the other three C⁢A′⁢s. However, placing all four C⁢A′⁢s of butane in the same plane is the most stable arrangement because the terminal C⁢A′⁢s, which are the bulkiest groups attached to the internal C⁢A′⁢s, are farthest apart in this arrangement. Three lines connected by zigzag is the skeletal formula representing butane, as shown in row 4th of Table 1. The zigzag lines representing a chain of 4 C⁢A′⁢s can be extended to represent a chain of 5 C⁢A′⁢s, by adding one line to represent a chain of 6 C⁢A′⁢s by adding two lines, and so on, as shown in Table 1 for the cases of n-alkanes having a chain of 2 to 12 C⁢A′⁢s. In summary, the skeletal formula of n-alkane is a line or lines connected zigzag representing C−C bonds. It is understood that: the terminals (end) and corners (bends) of the lines are C⁢A′⁢s, and each C has four bonds, so the bonds which are not shown by the lines are the bonds to H⁡A′⁢s. With this knowledge, it is clear that a line or lines connected zigzag way are the structural formulas that represent structures of n-alkanes without showing C⁢A′⁢s and H⁡A′⁢s in the formula. Table 1: Names, molecular formulas, condensed formulas, models, and skeletal formulas of n-alkanes containing 2 to 12 C⁢A′⁢s in a chain. (Note: Methane has no structural formula as it has no C−C bond. The C⁢A′⁢s are black and H⁡A′⁢s are white color in the models))\| # of C⁢A′⁢s \| Name \| Molecular formula \| Model, Condensed formula, and Structural formula \| \| 1 \| Methane \| CH⁢A 4 \| CH⁢A 4 \| \| 2 \| Ethane \| C⁢A 2⁢H⁡A 6 \| CH⁢A 3⁢CH⁢A 3 \| \| 3 \| Propane \| C⁢A 3⁢H⁡A 8 \| CH⁢A 3⁢CH⁢A 2⁢CH⁢A 3 \| \| 4 \| Butane \| C⁢A 4⁢H⁡A 10 \| CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3 \| \| 5 \| Pentane \| C⁢A 5⁢H⁡A 12 \| CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3 \| \| 6 \| Hexane \| C⁢A 6⁢H⁡A 14 \| CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3 \| \| 7 \| Heptane \| C⁢A 7⁢H⁡A 16 \| CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3 \| \| 8 \| Octane \| C⁢A 8⁢H⁡A 18 \| CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3 \| \| 9 \| Nonane \| C⁢A 9⁢H⁡A 20 \| CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3 \| \| 10 \| Decane \| C⁢A 10⁢H⁡A 22 \| CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3 \| \| 11 \| Undecane \| C⁢A 11⁢H⁡A 24 \| CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3 \| \| 12 \| Dodeane \| C⁢A 12⁢H⁡A 26 \| CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3 \| Stem names of organic compounds Names of n-alkanes without the last syllable, i.e., without -ane, are the stem names that represent the number of C⁢A′⁢s in the organic compound. For example, meth- from methane represents one C, eth- from ethane represents two C⁢A′⁢s, prop- from propane represents three C⁢A′⁢s, and so on. These stem names will be described later in reference to name organic compounds. Homologous series It is evident from comparing the molecular formals of n-alkanes shown in Table 1 that n-alkanes differ from each other by a CH⁢A 2 or a multiple of CH⁢A 2 units. For example, add a CH⁢A 2 to methane (CH⁢A 4) to convert it to propane (C⁢A 2⁢H⁡A 6), and two CH⁢A 2 units to convert it to butane (C⁢A 4⁢H⁡A 10 ) and so on. Series of organic compounds that differ from each other by a CH⁢A 2 or multiple of CH⁢A 2's are homologous series. All of the n-Alkanes shown in Table 1 are members of a homologous series. General formula of n-alkanes The general formula of n-alkanes is C⁢A n⁢H⁡A 2⁢n+2 where n is a counting number, i.e., 1, 2, 3,... For example, when n = 1, it is methane C⁢A 1⁢H⁡A 2×1+2 = CH⁢A 4 (Recall: that when the subscript to the element symbol in the formal is 1, it is not written.). When n = 2, it is ethane C⁢A 2⁢H⁡A 2×2+2 = C⁢A 2⁢H⁡A 6, and so on Variations in the structural formula to represent different configurations The structural formulas in Table 1 represent the linear conformation of n-alkanes which is the most stable confirmation. Remember: rotation is possible around any single bond. For example, n-hexane shown in Figure 1.4.2 a is rotated around the middle C−C bond to acquire new confirmation shown in Figure 1.4.2 b and rotated further along the 2nd C−C from left to acquire another confirmation shown in Figure 1.4.2 c. All three structural formulas are shown in Figure 1.4.2 a, b, and c represent the same molecule. Remember: different shapes of the same molecule obtained by rotation around a single bond are different configurations of the same molecule. a) b)c) Figure 1.4.2: a) n-hexane in a linear configuration and its model in the same configuration, b) the n-hexane in a bent configuration rotated ~120 o around the middle C−C bond indicated by the arrow, i.e., between C⁢A′⁢s in red circles and the model in the same configuration, and c) the n-hexane in an other bent configuration rotated ~120 o again around the 2nd C−C bond indicated by the arrow, from the left, i.e., between C⁢A′⁢s in red circles and the model in the same configuration. (Copyright; Public domain). Condensed and skeletal formulas of branched-alkanes Replacing one or both H⁡A′⁢s of a non-terminal carbon of straight chain alkane (n-alkane) with a C or a C chain results in a branched-alkane. Replacing hydrogen on the middle carbon of propane ( H−C H\|\|H−C H\|\|H−C H\|\|H−H) with a C results in isopropane ( H−C H\|\|H−C H\|\|H−C\|H−H−C H\|\|H−H) which is a branched chain alkane. There are two ways to show the condensed formula of the branched alkane: show the condensed formula of the branch within small brackets next to the carbon it is attached to, e.g., CH⁡A 3⁢CH⁡(CH⁡A 3)⁢CH⁡A 3 is the condensed formula of isopropane; show the condensed formula of the branch hanging above or below the carbon it is attached, e.g., C⁢H⁡3−C\|C⁢H⁡3⁢H−CH⁢A 3 is the condensed formula of isopropane. The skeletal formula of the branched alkane shows the skeletal formula of the branch with the terminal connected to the carbon in the main branch to which it is attached. For example, is the skeletal formula of isopropane. Figure 1.4.3 shows another example of branched alkane and its condensed and skeletal formulas. a)b) or CH⁡A 3⁢CH⁡A 2⁢CH⁡A 2⁢CH⁡(CH⁡A 2⁢CH⁡A 3)⁢CH⁡A 2⁢CH⁡A 2⁢CH⁡A 2⁢CH⁡A 3 c) Figure 1.4.3: 4-Ethyloctane -a branched alkane: a) Lewis formula, b) two ways of writing condensed formulas, and c) its skeletal formula. (Copyright; Public domain). Skeletal formulas of alkenes Hydrocarbons containing at least one double bond (C=C bond) are called alkene. Since the two sp 2 C⁢A′⁢s at the double bond have trigonal planer geometry with bond angles 120 o, the double bond between the two sp 2 C⁢A′⁢s fit in the zigzag skeletal structure, like alkanes, except that two lines are drawn where there is a double bond in the chain. Table 2 shows the names, Lewis structures, models, condensed formulas, and skeletal formulas of two alkene examples. Their nomenclature will be explained later. Table 2: Names, Lewis structures, models, condensed formulas, and skeletal formulas of some alkene examples. (Note: C⁢A′⁢s are black and H⁡A′⁢s are white color in the models)\| Name \| Lewis structure \| Model, Condensed formula, and Structural formula \| \| Hept-2-ene \| H−C H\|\|H−C\|H=C\|H−C H\|\|H−C H\|\|H−C H\|\|H−C H\|\|H−H \| CH⁢A 3⁢CHCHCH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3 \| \| Propene \| H−C\|H=C\|H−C H\|\|H−H \| CH⁢A 2⁢CHCH⁢A 3 \| Skeletal formulas of alkynes Hydrocarbons containing at least one triple bond (C≡C) bond are called alkyne. Since the two sp C⁢A′⁢s at the triple-bond have linear geometry, three lines show the triple bond and the two single bonds to the other atoms attached to them are drawn in line with the triple bond. The zigzag skeletal structure shows the rest of the structure as usual. Table 3 shows some alkyne examples' names, Lewis structures, models, condensed formulas, and skeletal formulas. Their nomenclature will be explained later. Table 3: Names, Lewis structures, models, condensed formulas, and skeletal formulas of some alkyne examples. (Note: C⁢A′⁢s are black and H⁡A′⁢s are white color in the models)\| Name \| Lewis structure \| Model, Condensed formula, and Structural formula \| \| Propyne \| H−C≡C−C H\|\|H−C H\|\|H−H \| CHCCH⁢A 3 \| \| Pent-2-yne \| H−C H\|\|H−C≡C−C H\|\|H−C H\|\|H−H \| CH⁢A 3⁢CCCH⁢A 2⁢CCH⁢A 3 \| \| Non-3-yne \| H−C H\|\|H−C H\|\|H−C≡C−C H\|\|H−C H\|\|H−C H\|\|H−C H\|\|H−C H\|\|H−H \| CH⁢A 3⁢CH⁢A 2⁢CCCH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3 \| Skeletal formulas of organic compounds containing heteroatoms If any heteroatom, i.e., an atom other than C or H, is present in an organic compound: its symbol in the skeletal formula shows it and any H present on the heteroatom is also shown next to the heteroatom as in the condensed formulas. The skeletal structure shows the rest of the structure as usual. Table 4 shows the names, Lewis structures, models, condensed formulas, and skeletal formulas of some example organic compounds containing heteroatoms. Their nomenclature will be explained later. Table 4: Names, Lewis structures, models, condensed formulas, and skeletal formulas of some organic compounds containing heteroatoms. (Note: C⁢A′⁢s are black, H⁡A′⁢s are white, and heteroatom are in other colors in the models)\| Name \| Lewis structure \| Model, Condensed formula, and Structural formula \| \| 2-Chloropropane \| H−C H\|\|H−C H\|\|C⁢l−C H\|\|H−H \| CH⁢A 3⁢CHClCH⁢A 3 \| \| Ethanamine \| H−C H\|\|H−C H\|\|H−N\|H−H \| CH⁢A 3⁢CH⁢A 2⁢NH⁢A 2 \| \| Diethyl ether \| H−C H\|\|H−C H\|\|H−O−C H\|\|H−C H\|\|H−H \| CH⁢A 3⁢CH⁢A 2⁢OCH⁢A 2⁢CH⁢A 3 \| \| Acetone \| H−C H\|\|H−C\|O\|−C H\|\|H−H \| CH⁢A 3⁢COCH⁢A 3 or C⁢H⁡3⁢C\|O\|⁢CH3 \| Variations in the formulas representing organic compounds The condensed and the skeletal formulas are usually presented in organic chemistry, as described in the previous sections. However, they may be varied slightly according to the need. If needed, mixed Lewis, condensed, and skeletal structures emphasize a particular part of the molecule. For example, the H⁡A′⁢s are usually written to the right of C or to the right of heteroatoms in the condensed formula, but sometimes this order may be reversed. For example, Lewis formula for methanol H−C H\|\|H−O−H is presented in condensed form as CH⁢A 3⁢OH, but it may be condensed as H⁡A 3⁢COH, or as H⁡A 3⁢C−OH to emphasize the C−O-bond. Similarly, Lewis formula of acetone H−C H\|\|H−C\|O\|−C H\|\|H−H is changed to skeletal formula as , but it may also be shown as to emphasize the C−H bond. Sometimes the formula is further condensed by placing the repeating units within a bracket and subscript outside the bract to show the number of repeat units. For example, the condensed formula of isopropane is C⁢H⁡3−C\|C⁢H⁡3⁢H−CH⁢A 3 that has three CH⁢A 3 units attached to the central C. It can be further condensed as (CH⁢A 3)⁢A 3⁢CH. Similarly, the condensed formula of decane CH⁢A 3⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 2⁢CH⁢A 3 contains eight CH⁢A 2 units and can be condensed further as CH⁢A 3⁢(CH⁢A 2)⁢A 8⁢CH⁢A 3. Examples of drawing Lewis, condensed, and skeletal formulae Example 1.4.1 Draw molecular, Lewis, condensed, and skeletal formulas of butane. Solution But- in butane means four C⁢A′⁢s and -ane means it is an alkane. Substituting n = 4 in the general formula of alkanes (C⁢A n⁢H⁡A n×2+2) gives: molecular formula = (C⁢A 4⁢H⁡A 4×2+2=C⁢A 4⁢H⁡A 10. For Lewis, condensed, and skeletal formulas follow the steps shown in the figure below. Example 1.4.2 Draw Lewis, condensed, and skeletal formulas for methylethylamine Solution Meth- means one C, eth-means two C⁢A′⁢s, and aminie indicates a N two which the carbon chains are attached. With this information, follow the steps shown in the figure below. Example 1.4.3 Draw Lewis, condensed, and skeletal formula of 1-chlorobutane Solution But- means four C⁢A′⁢s chain and 1-chloro- means there is chlorine at the terminal C. The nomenclature will be explained later. With this information, follow the steps shown in the figure below. This page titled 1.4: Representing organic compounds is shared under a Public Domain license and was authored, remixed, and/or curated by Muhammad Arif Malik. Back to top 1.3: Hybridization of orbitals and 3D structures of simple organic compounds 1.5: Formal Charge Was this article helpful? Yes No Recommended articles 1: Bonding in organic compounds Article typeSection or PageAuthorMuhammad MalikLicensePublic Domain Tags condensed formula Lewis formula molecular formula skeletal formula © Copyright 2025 Chemistry LibreTexts Powered by CXone Expert ® ? 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15142	http://business.baylor.edu/Tom_Kelly/MANKIW4-5.htm	Chapters 4 & 5: Economic Growth Goal: To understand forces causing differences in income over time and across countries. · Economic growth is based upon the production function of the economy. · Differences in income and growth over time must come from differences or changes in capital, labor, and technology · Changes in the production function results in shifts in the aggregate supply curve (to the right) so that higher output occurs for a given price level. Our Primary Task is to develop the Solow model of economic growth. Our Secodary Task is to examine how economic policy can influence the level and growth in our standard of living. Chapter 4: The Solow Growth Model without Technology Change 1. The Supply of Goods and the Production Function a. The Solow model assumes that the supply of goods and services depends upon a production function with constant returns to scale. zY = F(zK, zL) b. Hence, with substitution if z = 1/L, then Y/L = F(K/L, 1), i.e. output per worker depends upon capital per worker. c. With constant returns, changes in output per worker depends upon changes in capital per worker. d. Let y = Y/L and k = K/L, then y = f(k) e. The marginal productivity of capital MPK = f(k+1) - f(k) will diminish if the amount of labor and technology is fixed. (See Figure 4-1) Clearly changes in the stock of capital will result in an increase in the supply of goods and services, even with the supply of labor and technology fixed. B. Investment, Consumption, and the Demand for Goods. a. The Solow model assumes that the demand for goods depends upon consumption and investment. b. If expressed in terms of consumption and investment per worker, then y = c + i c. Consumption is assumed to depend upon income, ie. c = (1-s)y where s is the marginal propensity to save (change in savings per worker for a given change in income per worker) (1-s) is the marginal propensity to consume if income is either consumed or saved. d. Hence, total desired spending (demand) is given by y = (1-s)y + i e. Solving this equation results in investment that is proportional to income, i = sy The Evolution of Capital and the Steady State a. We have seen that y = f(k) and i = sy; therefore, i = s f(k). The higher the capital stock the higher the level of output and investment. Also, the saving rate determines the allocation of output between consumption and savings for every value of k. b. Net investment adds to the capital stock. This occurs only if gross investment is greater than depreciation. If depreciation is given by d k, then change in the capital stock is s f(k) - d k Approaching the Steady State a. The steady state occurs when investment equals depreciation. Then no net investment occurs. s f(k) = d k where k is the steady-state level of capital. b. Changes in the savings rate will increase growth until a new steady state is reached at a higher level of output. c. Rich countries have higher rates of savings and investment than poor countries. The Golden Rule Level of Capital a. The steady state that leads to the highest level of consumption per capita is called the Golden Rule level of capital accumulation. b. The steady state level of consumption c = f(k) - d k (output minus investment). Since output is not changing investment is equal to depreciation. c. Consumption is maximized when the slope of f(k) = d k (the largest vertical distance between f(k) and d k) d. This occurs when the MPK = d Transition to the Golden Rule Steady State a. If more capital than the golden rule, then consumption can be increased with lower saving rate and rate of investment. Over time output per capita will fall along with investment and savings, but consumption per capita will rise above the initial steady state. b. If less capital than the golden rule, then current consumption must be sacrificed in order to increase savings and investment to benefit future generations. c. The US is in this latter condition so that present consumers would have to sacrifice for future generations in order to improve the long-run rate of consumption. Population Growth a. Increases output but lower income per capita b. The change in capital must now be sufficient to meet depreciation plus the erosion in the productivity of capital due to less capital per worker as the number of workers increase. c. Hence, the steady state level of capital per worker falls with more population as the slope depreciation plus employment curve rotates upwards (see Fig. 4-11) d. The Golden Rule level of capital accumulation also falls until the MPK equals d + n. Less capital accumulation results in lower output and consumption per capita in more populated countries. Chapter 5: Technological Progress in the Solow Model a. Solow explains technological progress as the residual between the rate of growth in output and the amount explained by the rate of growth in capital and the rate of growth in labor inputs. This is called the “Solow Residual.” b. Technological progress is assumed to increase the efficiency of labor and, hence, is labor-augmenting. L x E is the efficiency unit of labor, so that k = K/(LxE) is the amount of capital per efficiency unit. c. An increase in labor augmenting technology has the same influence as an increase in population. Now capital must accommodate depreciation, greater population, and labor augmenting technology. d. The Golden Rule level of capital MPK - d = n + g where n is population growth and g is the growth in technology. e. Previously we noted that capital per efficiency unit is constant in the steady state. Then, y = f(k) is also constant in the steady state. But the number of efficiency units per worker is increasing at a rate g. Hence, total output grows at a rate n + g. Therefore, only technological progress can explain persistently rising living standards. Savings, Growth and Economic Policy a. The MPK - d in the US is presently about 8 percent per year well in excess of the current growth rate of below 3 percent per year. b. Policies to change the rate of savings c. Policies to allocate investment d. Policies to encourage technological progress Beyond the Solow Model: Endogenous Growth Theory a. The basic model: Y = AK Growth in Y = sA – d What determines A? b. The role of knowledge (human capital) with constant or increasing returns. How are ideas tuned into innovations? c. “Standing on shoulders” versus “stepping on toes”
15143	http://www.chem.ualberta.ca/~vederas/2006-2007_Chem_161/outlines/notes/Oct_10_06.pdf	Chem 161 Oct. 10, 2006 Nomenclature Alkenes and Alkynes - continued 2-butyl 1,3-butadiene 2-butyl-1,3-butadiene 1-ethyl 1,3,5,7-cyclooctatetraene 1-ethyl-1,3,5,7-cyclooctatetraene n-butane same molecules cis-2-butene trans-2-butene not same molecules stereoisomers E / Z nomenclature 1. look at 1st end of double bond 2. decide which atoms have largest atomic number 3. if cannot decide, go to the next set of atoms (until you can reach a decision) 4. do the same at other end of the double bond 5. Z (Zusammen – together)  same side E (Entgegen – opposite)  opposite side Eg. Br I Br A B C C A CH2 CH2 A Br H high priority Br C B high priority carbon A carbon B - high priority groups are on opposite sides - so the molecule has E configuration 1 2 3 4 Br I Br 4-bromo 3-methyl 2-bromo 1-iodo 2-butene (E)-2,4-dibromo-1-iodo-3-methyl-2-butene or (E)-1,3-dibromo-4-iodo-2-methyl-2-butene H H cis-cyclohexene or Z-cyclohexene normally called just cyclohexene as double bond always cis (Z) Br H E-1-bromocyclohexene normally called just 1-bromocyclohexene as double bond geometry always fixed Br note: in small ring system without any side groups (n < 8) - always cis (Z) double bond Nomenclature of alkynes Rule: - find the longest chain with maximum number of triple bonds, side groups, etc. - number to give 1st multiple bonded (double bond / triple bond) position the lowest number - drop “ane” and add “yne” - for multiple triple bonds, drop “ne” and add “diyne”,” triyne”, etc. eg. H H - ethyne / acetylene (common name) - sp-hybridized carbon atoms, linear structure - σ bond between C and H - one σ and two π bonds between the two carbons - explosive gas CH3 H propyne methylacetylene (common name) H CH3 1-butyne ethylacetylene H3C CH3 2-butyne dimethylacetylene 1 13 5,7,9-triyne no geometry for double bond E-double bond E-double bond Trideca-1,3[E],11[E]-trien-5,7,9-triyne In nature : - alkenes  very common - alkynes  > 1000 alkynes known (often defense substances in plants) Hydrocarbons  C and H only - alkanes – most non-polar - alkenes – intermediate polarity between alkanes and alkynes - alkynes – more polar - overall, they all are very non-polar - density less than water (1.0 g/cm3) – float on water - immiscible with water - dissolve well in non-polar solvents (like-dissolves-like) - low mp, bp compared to other organic molecules of similar size with more electronegative atoms - London (dispersion) forces control self association Isoprenes or Isoprenoids (also known as – terpenes / terpenoids) isoprenoid unit C5 unit myrcene 2 X C5 units 2 X C5 units !-pinene - myrcene – perfume – a monoterpene (C10 unit) - pinene is also a monoterpene (two isoprene units) isoprenoid unit C5 unit limonene 2 X C5 units citral 2 X C5 units O H aldehyde group Degrees of Unsaturation: - all non-cyclic alkanes  have the general formula of CnH2n+2 2,4-dimethylhexane C8H18 has no (zero) degree of unsaturation 1,2-dimethylcyclohexane C8H16 has one degree of unsaturation C10H18 has 2 degrees of unsaturation 1,4-pentadiene C5H8 has 2 degrees of unsaturation !-pinene C10H16 has 3 degrees of unsaturation no stereoisomers possible - in a molecule, a double bond or a ring system represents one degree of unsaturation Reactions of alkene General reaction  addition reaction C C A B A B reverse reaction is elimination reaction ! ! ! A=B=H  hydrogenation reaction – addition of hydrogen to double bond - syn or cis addition of H2  addition of hydrogen from same side of the double bond H H no catalyst no reaction H H catalyst Pt or Ni or Pd C C H H - catalyst helps to break the H-H bond and interacts with alkene electrons, which lowers the activation energy of the reaction Catalyst: - lowers activation energy but remains unchanged overall Reaction Coordinate SM : alkene and H2 Product : alkane (after hydrogenation) Ni H2 Ni H2 same products trans cis Ni H2 1,2-dimethylcyclohexene cis-1,2-dimethylcyclohexane - hydrogenation was syn (cis) addition, giving the cis-product as above Ea (uncatalyzed) Ea (catalyzed) SM Product
15144	https://mathalino.com/reviewer/fluid-mechanics-and-hydraulics/total-hydrostatic-force-plane-and-curved-surfaces	MATHalinoEngineering Math Review Search form Login • Register Home Recent Glossary About Algebra Derivation of Formulas Engineering Economy General Engineering Trigo Spherical Trigonometry Geometry Solid Geometry Analytic Geometry Calculus Integral Calculus Differential Equations Advance Engineering Mathematics Mechanics Strength of Materials Structural Analysis CE CE Board: Math CE Board: Hydro Geo CE Board: Design Surveying Hydraulics Timber Design Reinforced Concrete Geotechnical Engineering Courses Exams Old MCQ Forums Basic Engineering Math Calculus Mechanics General Discussions Blogs Breadcrumbs Total Hydrostatic Force on Surfaces Contents Total Hydrostatic Force on Plane Surfaces Total Hydrostatic Force on Curved Surfaces Back to top Total Hydrostatic Force on Plane Surfaces For horizontal plane surface submerged in liquid, or plane surface inside a gas chamber, or any plane surface under the action of uniform hydrostatic pressure, the total hydrostatic force is given by $F = pA$ where p is the uniform pressure and A is the area. In general, the total hydrostatic pressure on any plane surface is equal to the product of the area of the surface and the unit pressure at its center of gravity. $F = p_{cg}A$ where pcg is the pressure at the center of gravity. For homogeneous free liquid at rest, the equation can be expressed in terms of unit weight γ of the liquid. $F = \gamma \bar{h} A$ where $\bar{h}$ is the depth of liquid above the centroid of the submerged area. Derivation of Formulas The figure shown below is an inclined plane surface submerged in a liquid. The total area of the plane surface is given by A, cg is the center of gravity, and cp is the center of pressure. The differential force dF acting on the element dA is $dF = p \, dA$ $dF = \gamma h \, dA$ From the figure $h = y \sin \theta$,$dF = \gamma(y \sin \theta) \, dA$ Integrate both sides and note that γ and θ are constants, $\displaystyle F = \gamma \sin \theta \int y \, dA$ Recall from Calculus that $\displaystyle \int y \, dA = A\bar{y}$$F = (\gamma \sin \theta) A\bar{y}$ $F = \gamma (\bar{y} \sin \theta)A$ From the figure, $\bar{y} \sin \theta = \bar{h}$, thus, $F = \gamma \bar{h} A$ The product $\gamma \bar{h}$ is a unit pressure at the centroid at the plane area, thus, the formula can be expressed in a more general term below. $F = p_{cg} A$ Location of Total Hydrostatic Force (Eccentricity) From the figure above, S is the intersection of the prolongation of the submerged area to the free liquid surface. Taking moment about point S. $\displaystyle Fy_p = \int y \, dF$ : Where $dF = \gamma(y \sin \theta) \, dA$ $F = \gamma (\bar{y} \sin \theta)A$ $\displaystyle [ \, \gamma (\bar{y} \sin \theta)A \, ]y_p = \int y \, [ \, \gamma(y \sin \theta) \, dA \, ]$ $\displaystyle (\gamma \sin \theta) A \bar{y} \, y_p = (\gamma \sin \theta) \int y^2 \, dA$ $\displaystyle A \bar{y} \, y_p = \int y^2 \, dA$ Again from Calculus, $\displaystyle \int y^2 \, dA$ is called moment of inertia denoted by I Since our reference point is S, $A \bar{y} \, y_p = I_S$ Thus, $y_p = \dfrac{I_S}{A \bar{y}}$ By transfer formula for moment of inertia $I_S = I_g + A{\bar{y}}^2$, the formula for yp will become $y_p = \dfrac{I_g + A{\bar{y}}^2}{A \bar{y}}$ or $y_p = \bar{y} + \dfrac{I_g}{A \bar{y}}$ From the figure above, $y_p = \bar{y} + e$, thus, the distance between cg and cp is Eccentricity, $e = \dfrac{I_g}{A \bar{y}}$ Back to top Total Hydrostatic Force on Curved Surfaces In the case of curved surface submerged in liquid at rest, it is more convenient to deal with the horizontal and vertical components of the total force acting on the surface. Note: the discussion here is also applicable to plane surfaces. Horizontal Component The horizontal component of the total hydrostatic force on any surface is equal to the pressure on the vertical projection of that surface. $F_H = p_{cg}A$ Vertical Component The vertical component of the total hydrostatic force on any surface is equal to the weight of either real or imaginary liquid above it. $F_V = \gamma V$ Total Hydrostatic Force $F = \sqrt{{F_H}^2 + {F_V}^2}$ Direction of $F$ $\tan \theta_x = \dfrac{F_V}{F_H}$ Case 1: Liquid is above the curve surface The vertical component of the hydrostatic force is downward and equal to the volume of the real liquid above the submerged surface. Case 2: Liquid is below the curve surface The vertical component of the hydrostatic force is going upward and equal to the volume of the imaginary liquid above the surface. 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15145	http://mrsbonn6.weebly.com/uploads/5/5/9/8/55985471/medievalcareerspagesandsquires_(1).pdf	PAGES AND SQUIRES By Sarah Peterson HANDS ON HISTORY MEDIEVAL CAREERS: Medieval Careers - Pages and Squires Page 2 Medieval Careers: Pages and Squires Discussion #1: Pages and Squires Training to be a knight started very young. Technically, any free boy could become a knight, but because it costs so much for training, usually only a boy born into a wealthy family could become a knight. For a boy lucky enough to be born the son of a nobleman, training to be a knight began at a young age. Around the age of six or seven, the young boy was sent to the estate of a wealthy lord or knight to be a page and to learn how to ride, fight, hunt and behave. Training included playing strategic games (such as backgammon); learning manners; and participating in physical challenges such as climbing, archery, and swordplay (usually with wooden swords and shields.) For about seven years a boy was trained as a page. In addition to his training, he also served the lord and lady of the castle, taking care of various duties. At the age of 14 or 15, a page would become a squire. A squire continued to serve a knight, but had more responsibilities. He may be required to carry the knight’s armor or sword, or to help the knight dress in his armor. Frequently, the squire would care for the knight’s horses and would be responsible for delivering messages. The training became more intense and focused on preparing for battle. Training would include learning to fight with a sword, mace and axe and also learning the rules of jousting. At the age of 20 or 21, some, but not all, squires became knights. Medieval Times Job Opening! We are currently seeking a young person who is interested in serving as a page. Applicants must be serious about training and willing to dedicate their childhood to the instruction required to ultimately become a squire, and then a Knight. Applicant must be willing to attend to and serve his lordship. If you are willing to leave your family, home and everything you know, this job may be for you! Medieval Careers - Pages and Squires Page 3 Activity #1: Complete a Job Application Job application is on page 5. Students can share responses if they are in a classroom or co-op setting. Activity #2: Train to be a Page Strategic Games – Students learn the rules to a strategic game such as Checkers, Backgammon, or Page in a Palace. If you do not have enough checkers or backgammon games for all students to play, you may prefer to play Page in a Palace - played like Pig in a Pen- and it only requires pencil and paper. The rules and game are on page 6. Activity #3: Train to be a Squire Part of being a good squire was developing memory skills. Students can work on their memory skills by playing a game of “messenger.” The rules are the same as for “operator” or “telephone” except that the students spread out around the room, playground or yard so that there is a distance between them. Attached is a list of medieval messages for the squires to deliver (see page 7). This activity can be very fun with a lot of movement by the students. 1. Spread the students out around the room or yard (the outer parameter works best so that the squires can move in an orderly fashion). 2. Every student will have a chance to deliver the message every round. 3. The first squire will be the only one that will read the message (all others will receive the message verbally). 4. The first squire delivers the message to another student by whispering it so the other students cannot hear. 5. That student then delivers the message to the next student, and so on, until all students have received the message and delivered the message. 6. The last student announces the message out loud for all the students to hear. 7. Some of the messages are rather long and subject to some misinterpretations and can become rather silly. I always remind the students to be appropriate! 8. Students usually like to hear what the original message was. Medieval Careers - Pages and Squires Page 4 Discussion #2: The Melee and the Joust During a time of peace, knights would occasionally participate in a Melee. The purpose of the Melee was to prepare the knights for battle. A Melee was a type of tournament that was fought on a field with regular weapons and could be as perilous as a real battle. The Melee was usually a team sport but sometimes it was just a free-for-all, with knights charging at one another in a bloody combat. Over time, the rules of a melee changed in an attempt to reduce the number of casualties. Jousting was another way for knights to engage in battle during a time of peace. During the jousting tournament, each knight held a lance – a long, sharp weapon similar to a spear or bayonet. The knights, dressed in armor, would ride toward each other. Sometimes they were on opposites sides of a short wall but not always. The goal was to knock their opponent off of his horse or break their rival’s lance. Occasionally, the joust would result in a death, but even that did not reduce the popularity of the event. In preparing for the joust, the knights would occasionally spear rings that were suspended in the trees. Even though jousting began as a way to prepare for battle, it eventually became a popular form of entertainment in its own right. Squires did not actually participate in melees or jousting, but they did provide support for their knight by having horses and weapons ready. Participating in the tournaments in this capacity was a great learning experience for squires who eventually became knights. Activity #4: Participate in a Melee Instead of knocking the opponents off their horses with blunt swords, the student’s “melee” will be more like “Capture the Flag.” Many students will have played this game but for those who have not, the rules can be found on page 8. Activity #5: Pages and Squires Worksheet Complete worksheet entitled “Pages and Squires” found on page 9. Medieval Careers - Pages and Squires Page 5 Pages and Squires Applicant’s Name: Applicant’s Age: Are you a Free Child? Are you related to any Wealthy Knights? To whom do you owe your loyalty? Your Best Friend Your Teacher Your Parents Your King What will you do if you are trapped in a castle siege? Hide in the wine cellar Fight the enemy for as long as I can Take shelter in the chapel Pretend to be blind What do you like to do in your Spare Time? Eat Cake Hunt Wild Boar Juggle Travel What are some other plans you currently have for your childhood? Medieval Careers - Pages and Squires Page 6 Page in a Palace Students play in teams of two. Each player takes a turn connecting one dot to an adjacent dot by drawing a line. Diagonal lines are not allowed. If a player's line completes a square, they have “captured the palace” and he or she marks the inside of the box with his/her initial. Then he/she takes another turn. The player can capture as many palaces as possible during his or her turn. If no palace is captured then it is the other player’s turn. When the game is over, each player counts his/her captured palaces. The player with the highest score wins. Medieval Careers - Pages and Squires Page 7 Long Messages to Deliver The King has asked that you bring his favorite horse, Lancaster, around to the moat to meet his trainer, Donahue. The peasants in the North Valley are revolting against the taxes imposed by King Edward. Taxes are usually collected by the end of each month, but due to the hail and freezing rain, the roads were turned to mud and the horses couldn’t get through. The Queen’s sister, Mary, will be coming to the castle this summer for an extended stay. Please be sure to have her quarters ready by the first day of June! There will be a jousting tournament in May. We expect that the knights will be trained by the end of April. Training will take place at Chivalry Field. The Knight has asked all of his pages to assemble at the stable. He wants to announce that the enemy is approaching from the east. Medieval Careers - Pages and Squires Page 8 Participate in a Melee The objective of the game is for team members to sneak across the border and capture the enemy flag AND then get back to their side without being caught. You will need two flags – one for each team. The game should be played in an area with plenty of running space. The area should have natural or artificial borders. Divide the playing space into two equal sides, with a clear border separating the two. Divide into two teams of three or more people. Each side places their flag on the far end of their side of the field. The flag must be visible and accessible. Players can either attack (where they try to capture the flag) or defend (where they try to prevent the other team from capturing their flag). Countdown to zero and the players begin. Attacking Players: cross the border to the other team’s side and try to capture their flag. Defending Players: tag opposing players as they cross over to capture your flag. If a player is tagged, they are thrown in the dungeon (a small designated area on the far end of the playing field). A player can only leave the dungeon if one of his free teammates tags him while he is in the dungeon. The former prisoner cannot be re-tagged during their escape to their own side. A team wins by capturing the enemy’s flag and crossing back to their side of the border. Medieval Careers - Pages and Squires Page 9 Worksheet: Pages and Squires 1. How old must a boy be to become a page? 2. Where did a page live while he was being trained? a. In the Royal Palace! b. With the Royal Horses! c. With a Wealthy Lord or Knight! d. In the Dungeon! 3. What were three of the activities a page must do as part of his training? a. b. c. 4. At what age could a page become a squire? 5. What was the purpose of a Melee? a. To Prepare for Battle b. To Train a Horse c. To Harm Their Opponent d. To Test the Coat of Armor True False All squires became knights. True False Squires competed in jousts and melees. True False Jousts were a great source of entertainment. Medieval Careers - Pages and Squires Page 10 Worksheet: Pages and Squires Teacher’s Copy 1. How old must a boy be to become a page? A boy must be six or seven years old. 2. Where did a page live while he was being trained? a. In the Royal Palace! b. With the Royal Horses! c. With a Wealthy Lord or Knight! d. In the Dungeon! 3. What were three of the activities a page must do as part of his training? a. Ride a Horse b. Fight c. Hunt Strategic Games, Learning Good Manners, Climbing, and Archery are also acceptable (students only need three answers) 4. At what age could a page become a squire? A page could become a squire at fourteen or fifteen years old. 5. What was the purpose of a Melee? a. To prepare for battle b. To train a horse c. To harm their opponent d. To test the coat of armor True False All Squires became Knights. True False Squires competed in jousts and Melees. True False Jousts were a source of entertainment. Medieval Careers - Pages and Squires Page 11 Additional Activities: Individual or Classroom These activities work individually OR in a group/classroom setting Coloring Page – Page-in-Training Students complete coloring page found on page 12. Life As a Knight: An Interactive History Adventure by Rachael Hanel Ages: 8+ Yes, this is a book, but it is great to use individually or in a group. Individually, a student can read the chapters independently, choosing the paths to take. In a classroom setting, the instructor reads the text until it is time to make a choice about what direction to take. The class then votes (by show of hands) which path to take. In the event of a tie, the teacher chooses. Allow for extra time for this activity – students frequently want to go back to find out the outcomes of paths not originally taken. Try a Musical Instrument Training for a page frequently included learning to play a musical instrument. In a classroom setting, students can use percussion instruments or even make their own drums. If the students play an instrument, they can perform for the other students. In one class, a student brought her guitar and showed the other students chords, which she was kind enough to let them try! Play Darts Surprisingly, you can find darts and dartboards for a relatively cheap price. Students can be divided into two or more teams for a competition (scoring can be as simple or complex as you like). Many of the students will have a difficult time even hitting the dartboard – but it is very fun! This doesn’t have to be a competition between students, it could also be students improving and trying to get their best score. Writing Prompt Students complete the following writing prompt. This is a work of fiction so the students can be creative! If in a classroom situation, the students can share responses. You are a squire who has served the same knight since you were a seven-year-old page. It is time for your lord to decide whether you should be promoted to knight or stay a squire. How do you convince him? Medieval Careers - Pages and Squires Page 12 Page-in-Training: Learning to Shoot Arrows Medieval Careers - Pages and Squires Page 13 Additional Classroom Activities These activities work best in a group or classroom setting Play Castle Keep This game involves a lot of strategy, which is part of a page’s and a squire’s training. In this clever game of medieval maneuvers, build a castle with walls, towers, and a keep by matching color, shape, or both. Will you use your game tiles to strengthen your fortress or to attack vulnerable opponents? Choose wisely, or you may find yourself in royal ruins! The first player to build a complete castle rules the land. This board game can be found at educational stores or online. 2 to 4 Players. Playing Time: 20 Minutes Review Game of Bingo Type the Bingo words into the Bingo Maker (free bingo cards!) Words are on page 14. To make “Bingo” a review game, instead of saying the bingo word, give students the clue that appears on the table. There are only 24 words – with the middle space being free. Medieval Careers - Pages and Squires Page 14 PAGE and SQUIRE BINGO Word on Bingo Card Clue or Hint to Give to Students Page A Young Boy Training to be a Squire Squire A Teen Boy Who is Training to be a Knight Knight A Soldier of the King Joust A Tournament in Which Individuals Fight Melee A Tournament in Which Teams Fight Armor Protective Covering Worn by Knights 6 or 7 Years Old The Age a Boy Becomes a Page 14 or 15 Years Old The Age a Boy Becomes a Squire Opponent Another Word for Enemy Tournament A Competition Held With an Audience Strategic Games These are Used to Teach Strategy Messenger A Person Who Delivers Messages Manners Behaving Appropriately Physical Challenges Part of the Training Requiring Movement and Effort Lance A Long Spear-Like Weapon Archery Training Involving a Bow and Arrow Climbing Training Involving Scaling a Wall or Tree Swordplay Training to Learn How to Use a Sword A Field The Location of Jousts and Melees Estate of Wealthy Lord The Location of Page and Squire Training Dress the Knight One of the Duties of a Squire involving Armor Learning How to Battle The Most Important Part of a Squire’s Training Hunting Training Involving Killing Animals for Food or Sport Caring for Horses One of the Duties of a Squire Medieval Careers - Pages and Squires Page 15 Book Suggestions Fiction: The Making of a Knight: How Sir James Earned His Armor by Patrick O’Brien Ages 6+ James begins his journey at age seven as a page, working his way up to squire and, after proving himself brave in battle, a knight. Through James the reader learns about chivalry, weapons and suiting up for battle. This book is a great Classroom read! 32 pages The Squire’s Tale by Gerald Morris Ages 10+ Terence is an orphan who becomes a squire for Gawain, one of King Arthur’s most famous Knights. This story follows his adventures with Kings, Wars and Damsels in Distress. 224 Pages The Squire, His Knight, and His Lady by Gerald Morris Ages 10+ This is a sequel to the book “The Squire’s Tale.” 240 Pages Castle Diary: The Journal of Tobias Burgess by Richard Platt Ages 9+ Eleven-year-old Tobias Burgess is an English Page in the 13th Century. His journal gives us a view of castle life as he waits on patrons, learns to joust and fight with a sword, and even fights an illness. 128 pages Non-Fiction: Knight: A Noble Guide for Young Squires by Dugald A Steer Ages 6-9. This book could be used as a classroom activity; the book contains colorful illustrations, diagrams, pop-ups, pull tabs, and foldouts all with information guiding the young squire. 32 Pages
15146	https://www.quora.com/There-are-15-points-in-a-plane-set-of-which-only-6-are-in-a-straight-line-then-how-many-different-straight-lines-can-be-made-and-how-many-triangles-can-be-made	There are 15 points in a plane set of which only 6 are in a straight line, then how many different straight lines can be made, and how many triangles can be made? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Mathematics Lines Area Combinations Ma Planes (geometry) Construction of Triangles Euclidean Geometry Combination Permutations and Combinat... Different Lines 5 There are 15 points in a plane set of which only 6 are in a straight line, then how many different straight lines can be made, and how many triangles can be made? All related (33) Sort Recommended Assistant Bot · 1y To solve this problem, we need to determine how many different straight lines and triangles can be formed with the given points. Calculating the Number of Different Straight Lines The total number of points is 15, and we know that 6 of these points are collinear (i.e., they lie on the same straight line). Step 1: Total Lines from Points The total number of lines that can be formed by selecting any 2 points from 15 is given by the combination formula: Total Lines=(15 2)=15×14 2=105 Total Lines=(15 2)=15×14 2=105 Step 2: Subtracting Collinear Lines Since there are 6 points that are collinear, t Continue Reading To solve this problem, we need to determine how many different straight lines and triangles can be formed with the given points. Calculating the Number of Different Straight Lines The total number of points is 15, and we know that 6 of these points are collinear (i.e., they lie on the same straight line). Step 1: Total Lines from Points The total number of lines that can be formed by selecting any 2 points from 15 is given by the combination formula: Total Lines=(15 2)=15×14 2=105 Total Lines=(15 2)=15×14 2=105 Step 2: Subtracting Collinear Lines Since there are 6 points that are collinear, they can only form 1 line. The number of lines that can be formed by selecting any 2 points from these 6 points is: Collinear Lines=(6 2)=6×5 2=15 Collinear Lines=(6 2)=6×5 2=15 Since all these combinations form the same line, we only count this as 1 line. Step 3: Total Unique Lines Thus, the total number of unique lines formed is: Unique Lines=Total Lines−(Collinear Lines−1)=105−(15−1)=105−14=91 Unique Lines=Total Lines−(Collinear Lines−1)=105−(15−1)=105−14=91 So, the total number of different straight lines that can be formed is 91. Calculating the Number of Triangles To form a triangle, we need to select 3 points. We need to ensure that not all 3 points are collinear. Step 1: Total Triangles from 15 Points The total number of triangles that can be formed by selecting any 3 points from 15 is given by: Total Triangles=(15 3)=15×14×13 3×2×1=455 Total Triangles=(15 3)=15×14×13 3×2×1=455 Step 2: Subtracting Collinear Triangles Next, we need to subtract the triangles formed by the 6 collinear points. The number of ways to select 3 points from these 6 collinear points is: Collinear Triangles=(6 3)=6×5×4 3×2×1=20 Collinear Triangles=(6 3)=6×5×4 3×2×1=20 Step 3: Total Unique Triangles Thus, the total number of triangles that can be formed is: Unique Triangles=Total Triangles−Collinear Triangles=455−20=435 Unique Triangles=Total Triangles−Collinear Triangles=455−20=435 So, the total number of triangles that can be formed is 435. Summary Different Straight Lines: 91 Triangles: 435 Upvote · Doug Dillon Ph.D. Mathematics · Author has 12.4K answers and 11.4M answer views ·Updated 6y There are 15 points in a plane set of which only 6 are in a straight line, then how many different straight lines can be made, and how many triangles can be made? To draw a straight line, choose 2 2 points and connect them. One line is already there. If we don’t include the 6 6 collinear point there are (9 2)=36(9 2)=36 ways to connect points. If the 6 6 are included, there are 9 9 off the line and 6 6 on the Continue Reading There are 15 points in a plane set of which only 6 are in a straight line, then how many different straight lines can be made, and how many triangles can be made? To draw a straight line, choose 2 2 points and connect them. One line is already there. If we don’t include the 6 6 collinear point there are (9 2)=36(9 2)=36 ways to connect points. If the 6 6 are included, there are 9 9 off the line and 6 6 on the line for a total of 54 54 lines. Altogether, that’s 90 90 lines. Also we can add one line by connecting all the points which are in straight line .So the answer would be 91. As for triangles, you choose three points. If none of the 6 6 are used, we have (9 3)=84(9 3)=84 triangles. If one of the six is used, you can ... Upvote · Promoted by The Penny Hoarder Lisa Dawson Finance Writer at The Penny Hoarder ·Updated Sep 16 What's some brutally honest advice that everyone should know? Here’s the thing: I wish I had known these money secrets sooner. They’ve helped so many people save hundreds, secure their family’s future, and grow their bank accounts—myself included. And honestly? Putting them to use was way easier than I expected. I bet you can knock out at least three or four of these right now—yes, even from your phone. Don’t wait like I did. 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Hence no. of required lines ={ C(15, 2) - C(6, 2) + 1} = { (15!/2! 13!) - (6!/2! 4!) + 1} = 91 . And no. of required triangles = { C(15, 3) - C(6, 3)} = 455 - 20 = 435 . Upvote · 9 2 9 2 9 1 Related questions More answers below Out of 15 points in a plane, no 3 are in straight line except 8 points which are collinear. How many triangles can be formed by joining them? How many soldiers walk in a straight line? Out of four lines, how many triangles can be formed? How many straight lines can be formed from 10 points if no three points are collinear? How many points are there on a line? Glenn Clemens Author has 1.7K answers and 966.5K answer views ·4y Related There are 15 points in a plane. 3, 4 and 5 points are colinear with respectively (A,B), (B,C) and (C,A). How many triangles can be made with these 15 points with their values being positive? I interpret your question to mean we have a triangle ABC with three points between A and B, four points between B and acute, and five points between C and A. We want to know how many triangles can be formed with thee points. (IDK what you mean by “their values being positive so I’m ignoring that.) I am goin to key off of the number of vertices of ABC that are used. Also, when I speak of choosing points from one side of ABC, I am referring to interior points, not the endpoints. Suppose none of A, B, or C are used. Our triangle can be made by choosing a. one point from each side of ABC: (3)(4)(5) = Continue Reading I interpret your question to mean we have a triangle ABC with three points between A and B, four points between B and acute, and five points between C and A. We want to know how many triangles can be formed with thee points. (IDK what you mean by “their values being positive so I’m ignoring that.) I am goin to key off of the number of vertices of ABC that are used. Also, when I speak of choosing points from one side of ABC, I am referring to interior points, not the endpoints. Suppose none of A, B, or C are used. Our triangle can be made by choosing a. one point from each side of ABC: (3)(4)(5) = 60 OR b. two points from AB and one from BC: (3C2)(4) = 12 OR c. two points from AB and one from CA: (3C2)(5) = 15 OR d. two points from BC and one from AB: (4C2)(3) = 18 OR e. two points from BC and one from AC: (4C2)(5) = 30 OR f. two points from AC and one from AB (5C2)(3) = 30 OR g. two points from AC and one from BC: (5C2)(4) = 40. Subtotal: 205 Suppose we use A only (not B or C). Then we can choose a. one point from AB and one from BC: (3)(4) = 12 OR b. one point from AB and one from AC: (3)(5) = 15OR c. one point from AC and one from BC: (5)(4) = 20 OR d. two points from BC: 4C2 = 6 Subtotal: 53 We ca treat B and C each in a similar way getting Subtotal for B: 57 Subtotal for C: 50 Now suppose we use A and B but not C. We can choose one point from either AC or BC: 5 + 4 = 9 If we use A and C but not B: 3 + 4 = 7 If we use B and C but not A: 3 + 5 = 8 Subtotal: 24 Finally, we can use A, B, and C: 1 triangle. Total: 390. I’m not super confident of this answer. Possible sources of error include my arithmetic, my typing, and my reasoning. I’m curious to see if anyone will confirm or correct my answer. Upvote · 9 6 Marek Čtrnáct Translator From English · Author has 821 answers and 938.4K answer views ·Mar 29 Related In a plane, there are 15 points, each pinned with a nail, forming a rectangular grid of 3 rows and 5 columns with 1 cm intervals. Now there are many rubber bands. How many triangles with an area of 1 square centimeter can be formed? Well, the question is not entirely clear, so this is my interpretation. There are three things it might ask: how many triangles you can form at once, how many triangles can be formed if different positions and orientations are treated as distinct, or how many triangles can be formed if we are just interested in the shapes of triangles. On the balance of probabilities, I think the middle option is most likely. So, whenever we talk about polygons formed on a square grid, the first thing you should check is Pick’s theorem: Pick's theorem - Wikipedia Formula for area of a grid polygon Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of 1 + ⁠ 96 / 2 ⁠ − 1 = 48 [ 1 ] In geometry , Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899. [ 2 ] It was popularized in English by Hugo Steinhaus in the 1950 edition of his book Mathematical Snapshots . [ 3 ] [ 4 ] It has multiple proofs, and can be generalized to formulas for certain kinds of non-simple polygons. i = 7 , b = 8 , A = i + ⁠ b / 2 ⁠ − 1 = 10 Suppose that a polygon has integer coordinates for all of its vertices. Let i {\displaystyle i} be the number of integer points interior to the polygon, and let b {\displaystyle b} be the number of integer points on its boundary (including both vertices and points along the sides). Then the area A {\displaystyle A} of this polygon is: [ 5 ] [ 6 ] [ 7 ] [ 8 ] A = i + b 2 − 1. {\displaystyle A=i+{\frac {b}{2}}-1.} The example shown has i = 7 {\displaystyle i=7} interior points and b = 8 {\displaystyle b=8} boundary points, so its area is A = 7 + 8 2 − 1 = 10 {\displaystyle A=7+{\tfrac {8}{2}}-1=10} square units. Via Euler's formula [ edit ] One proof of this theorem involves subdividing the polygon into triangles with three integer vertices and no other integer points. One can then prove that each subdivided triangle has area exactly 1 2 {\displaystyle {\tfrac {1}{2}}} . Therefore, the area of the whole polygon equals half the number of triangles in the subdivision. After relating area to the number of triangles in this way, the proof concludes by using Euler's polyhedral formula to relate the number of triangles to the number of grid points in the polygon. [ 5 ] Tiling of the plane by copies of a triangle with three integer vertices and no other integer points, as used in the proof of Pick's theorem The first part of this proof shows that a triangle with three integer vertices and no other integer points has area exactly 1 2 {\displaystyle {\tfrac {1}{2}}} , as Pick's formula states. The proof uses the fact that all triangles tile the plane , with adjacent triangles rotated by 180° from each other around their shared edge. [ 9 ] For tilings by a triangle with three integer vertices and no other integer points, each point of the integer grid is a vertex of six tiles. Because the number of triangles per grid point (six) is twice the number of grid points per triangle (three), the triangles are twice as dense in the plane as the grid points. Any scaled region of the plane contains twice as many triangles (in the limit as the scale factor goes to infinity) as the number of grid points it contains. Therefore, each triangle has area 1 2 {\displaystyle {\tfrac {1}{2}}} , as needed for the proof. [ 5 ] A different proof that these triangles have area 1 2 {\displaystyle {\tfrac {1}{2}}} is This allows you to calculate an area of ar Continue Reading Well, the question is not entirely clear, so this is my interpretation. There are three things it might ask: how many triangles you can form at once, how many triangles can be formed if different positions and orientations are treated as distinct, or how many triangles can be formed if we are just interested in the shapes of triangles. On the balance of probabilities, I think the middle option is most likely. So, whenever we talk about polygons formed on a square grid, the first thing you should check is Pick’s theorem: Pick's theorem - Wikipedia Formula for area of a grid polygon Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of 1 + ⁠ 96 / 2 ⁠ − 1 = 48 [ 1 ] In geometry , Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899. [ 2 ] It was popularized in English by Hugo Steinhaus in the 1950 edition of his book Mathematical Snapshots . [ 3 ] [ 4 ] It has multiple proofs, and can be generalized to formulas for certain kinds of non-simple polygons. i = 7 , b = 8 , A = i + ⁠ b / 2 ⁠ − 1 = 10 Suppose that a polygon has integer coordinates for all of its vertices. Let i {\displaystyle i} be the number of integer points interior to the polygon, and let b {\displaystyle b} be the number of integer points on its boundary (including both vertices and points along the sides). Then the area A {\displaystyle A} of this polygon is: [ 5 ] [ 6 ] [ 7 ] [ 8 ] A = i + b 2 − 1. {\displaystyle A=i+{\frac {b}{2}}-1.} The example shown has i = 7 {\displaystyle i=7} interior points and b = 8 {\displaystyle b=8} boundary points, so its area is A = 7 + 8 2 − 1 = 10 {\displaystyle A=7+{\tfrac {8}{2}}-1=10} square units. Via Euler's formula [ edit ] One proof of this theorem involves subdividing the polygon into triangles with three integer vertices and no other integer points. One can then prove that each subdivided triangle has area exactly 1 2 {\displaystyle {\tfrac {1}{2}}} . Therefore, the area of the whole polygon equals half the number of triangles in the subdivision. After relating area to the number of triangles in this way, the proof concludes by using Euler's polyhedral formula to relate the number of triangles to the number of grid points in the polygon. [ 5 ] Tiling of the plane by copies of a triangle with three integer vertices and no other integer points, as used in the proof of Pick's theorem The first part of this proof shows that a triangle with three integer vertices and no other integer points has area exactly 1 2 {\displaystyle {\tfrac {1}{2}}} , as Pick's formula states. The proof uses the fact that all triangles tile the plane , with adjacent triangles rotated by 180° from each other around their shared edge. [ 9 ] For tilings by a triangle with three integer vertices and no other integer points, each point of the integer grid is a vertex of six tiles. Because the number of triangles per grid point (six) is twice the number of grid points per triangle (three), the triangles are twice as dense in the plane as the grid points. Any scaled region of the plane contains twice as many triangles (in the limit as the scale factor goes to infinity) as the number of grid points it contains. Therefore, each triangle has area 1 2 {\displaystyle {\tfrac {1}{2}}} , as needed for the proof. [ 5 ] A different proof that these triangles have area 1 2 {\displaystyle {\tfrac {1}{2}}} is This allows you to calculate an area of arbitrary polygon with integer vertices by counting points on the boundary of the polygon and on the inside. This area will always be some integer multiple of 1 2 1 2. You are interested in triangles with area of 1 1. The formula for area is A=i+b 2–1 A=i+b 2–1 so when A=1 A=1, we have 1=i+b 2–1 1=i+b 2–1 2=i+b 2 2=i+b 2 i=2−b 2 i=2−b 2 Since b b (points on the boundary) must be at least 3 3 and i i (points in the interior) must be nonnegative, we see that the only combination that works is b=4,i=0 b=4,i=0; if b>4 b>4, 2−2<0 2−2<0 and for b=3 b=3, i i wouldn’t be an integer. For 4 4 points to be on the boundary, you must have one edge of the triangle with three points an the other two with just two. Let’s call the three-point edge the base. Case 1: Base is 2 2. 4 in each 1x2 rectangle, and there’s 4 of those in each of 3 length-5 rows and 2 in each of 5 length-3 column, so 4(12+10)=88 4(12+10)=88 Similar, but since it’s symmetrical, there are just 2 of these in every 1x2 rectangle for total of 44 44 There are 14 1x3 rectangles on the grid, so 4⋅14=56 4⋅14=56 of these triangles. There are 6 1x4 rectangles on the grid, so 4⋅6=24 4⋅6=24 of these triangles. There are just 3 1x5 rectangles, so 4⋅3=12 4⋅3=12 of these triangles. Note that these all have height 1; if they were higher, you would always get another point on the boundary or in the interior. Total number of triangles from case 1 is 88+44+56+24+12=224 88+44+56+24+12=224. Case 2: Base is 2√2 2 2: 8 2x2 squares, 8 of these triangles in every one for total of 64 64. 6+4=10 2x3 rectangles, 4 of these in each for total of 40 40. 2 3x4 rectangles, 4 of these in each for total of 8 8. Total for case 2: 64+40+8=112 64+40+8=112 (All other triangles would have larger area or wouldn’t fit in the 3x5 grid.) Case 3: Base is 2√5 2 5 4 2x4 rectangles, 4 of these in each for total of 16 16. 2 2x5 rectangles, 4 of these in each for a total of 8 8. 4 2x4 rectangles, 4 of these in each for total of 16 16. 1 3x5 rectangle with 4 of these for total of 4 4. Total for case 3: 16+8+16+4=44 16+8+16+4=44 And that is all — all other possibilities for base are too big to fit inside a 3x5 rectangle. So the sum total of the triangles is 224+112+44=380 224+112+44=380. Upvote · 9 1 Sponsored by CDW Corporation What’s the best way to protect your growing infrastructure? Enable an AI-powered defense with converged networking and security solutions from Fortinet and CDW. Learn More 999 119 Aman Roy Intern at Hike Messenger (app) (2018–present) · Author has 74 answers and 270.8K answer views ·8y Related Out of 15 points in a plane, no 3 are in straight line except 8 points which are collinear. How many triangles can be formed by joing them? Ans-504 While studying at VMC we used to solve this problem with two approach I will write both of them. First approach: How can a triangle be formed ? It can be formed in three ways If we take one point from 8 collinear points and 2 from remaining 7 and join them. So this case will give 8c17c2 points which is equal to 821=168. If we take two points from 8 collinear points and 1 from remaining 7 . so this will give 8c27c1=287=196. If we take all three points. From 7 non collinear points . which will give 7c3 = 35 . Hence total number of triangles are 168+196+35=399. Second approach: From 15 points you can Continue Reading While studying at VMC we used to solve this problem with two approach I will write both of them. First approach: How can a triangle be formed ? It can be formed in three ways If we take one point from 8 collinear points and 2 from remaining 7 and join them. So this case will give 8c17c2 points which is equal to 821=168. If we take two points from 8 collinear points and 1 from remaining 7 . so this will give 8c27c1=287=196. If we take all three points. From 7 non collinear points . which will give 7c3 = 35 . Hence total number of triangles are 168+196+35=399. Second approach: From 15 points you can make 15c3 triangles which is equal to 455 . but it will contain triangle made from 8 collinear points which doesn't exists so we have to subtract the number of triangles made from 8 collinear points. No of triangle made from 8 collinear points are 8c3=56. Hence total number of triangles are 455–56=399. PS I don't know why you said answer is 504 in the question. Hope it helps. Upvote · 99 10 Related questions More answers below How many different straight lines can be drawn from 4 points that are not on a straight line? Eight points are marked on a straight line and Nine points are marked on another line which is parallel to the first line. How many straight lines, including the first two, can be formed with those points? How many different straight lines can be formed from 30 points in a plane? (No three points are collinear) Are property lines straight? If there are three non-intersecting straight lines, how many planes can they form together? John Fryer Author has 890 answers and 885.6K answer views ·4y Related There are 6 points in a plane with no 3 points collinear. How many different triangles can be drawn? You have 6 points and if we choose any three we get triangles. So the total will be 6 choose 3 or 6 C 3 Use the full formula or much simpler we use the smaller value to indicate the number of terms top and bottom. Put 1 x 2 x 3 as the denominator Match this on top with a numerator descending from 6 So 6 C 3 = 6 x 5 x 4 / 1 x 2 x 3 = 20 ANSWER With 6 non linear points we can make 20 différent triangles ADDENDUM The Pascal Triangle should be able to be constructed for such problems Note our choice which concerns us is at level 6 We have 1, 6, 15, 20, 15, 6, 1 Suppose the question was 6 points and find the nu Continue Reading You have 6 points and if we choose any three we get triangles. So the total will be 6 choose 3 or 6 C 3 Use the full formula or much simpler we use the smaller value to indicate the number of terms top and bottom. Put 1 x 2 x 3 as the denominator Match this on top with a numerator descending from 6 So 6 C 3 = 6 x 5 x 4 / 1 x 2 x 3 = 20 ANSWER With 6 non linear points we can make 20 différent triangles ADDENDUM The Pascal Triangle should be able to be constructed for such problems Note our choice which concerns us is at level 6 We have 1, 6, 15, 20, 15, 6, 1 Suppose the question was 6 points and find the number of possible hexagons? The answer is 6 choose 6 and comes at the end of our line. Step back one step for the number of pentagons and this is 6 points but choose 5. We know it will be 6 only but we can see the 6 in our line represents 6 C 5 And 20 corresponds logically to 6 C 3 or what our question has asked. Note for similar questions we cannot form polygons with two points and so we ignore 6 C 2, 6 C 1 and 6 C 0 for questions about polygons and work from 6 C 3 onwards. The Pascal Triangle is invaluable for any combination questions Upvote · 9 2 Promoted by Almedia Charlee Anthony Personal Finance @ Almedia \| Gaming Enthusiast ·Sep 17 What are ways to earn 930 within a month? I cashed out $1,200 to PayPal in just 30 days - starting with $0 investment. All I did was try out Freecash, and it actually worked. How I got started At the start of the year, I set myself a simple challenge: make $930 in one month without picking up another job. I was tired of side hustle lists that required selling stuff, investing money I didn’t have, or waiting months for results. That’s when I stumbled on a site called Freecash. Signing up took less than a minute, and I even got a $5 bonus just for creating my account. Honestly, I was skeptical - but since it didn’t cost anything, I gave it Continue Reading I cashed out $1,200 to PayPal in just 30 days - starting with $0 investment. All I did was try out Freecash, and it actually worked. How I got started At the start of the year, I set myself a simple challenge: make $930 in one month without picking up another job. I was tired of side hustle lists that required selling stuff, investing money I didn’t have, or waiting months for results. That’s when I stumbled on a site called Freecash. Signing up took less than a minute, and I even got a $5 bonus just for creating my account. Honestly, I was skeptical - but since it didn’t cost anything, I gave it a try. If you want to test something similar, creating an account is simple and low-risk - you could see if it fits your routine too. What Freecash really is In plain words, Freecash is a platform that rewards you for completing online offers. Some are quick (like surveys), while others are longer but pay more (like reaching levels in mobile games). The big difference I noticed compared to other apps: everything is upfront. You know exactly what you’ll earn before starting, and cashouts are instant - PayPal, Visa, crypto, or gift cards. What really convinced me was seeing other users sharing screenshots of cashouts in the community tab. I wasn’t the only one testing it. If you’re curious, you could just explore Freecash and see what kind of tasks are available. What you can do on Freecash Here are the main ways I earned: Games: I made $65 from one RPG by reaching a certain level. I even spent $10 inside the game to speed up progress, but it was worth it because the payout was much higher. Apps: One finance app paid me $20 just for signing up and completing a quick setup. Surveys: Not glamorous, but consistent. $3 here, $5 there - it added up to $90 that month. Daily bonuses: Logging in and small tasks gave me an extra $70+ I wasn’t expecting. It’s really about finding a mix that fits your time. You could pick whatever seems easiest or most fun for you. You can browse the tasks yourself atFreecash. My earnings on Freecash Breaking down my first month: Game challenge: $350 Finance app signup: $80 Surveys: $90 Another game level-up: $50 Daily bonuses & smaller tasks: $400 Total: $970 I started with the mindset that $930 was a stretch goal, but by week three I already knew I’d hit it. The best part? No budget needed up front - I only chose to make one in-app purchase to speed things up. Step-by-step to start earning If I had to map out exactly how I did it, it would look like this: Signed up atFreecash and claimed the $5 bonus. Picked one high-paying game to focus on. Filled breaks with smaller survey tasks. 3. Made one small in-app purchase to complete an offer faster. 4. Cashed out to PayPal the same day I crossed $50, just to confirm it was real. Rinse and repeat-that’s how I reached $302 in a single month. If you’re looking to make an extra $300, you could try a similar approach and adjust it to your own schedule. Final thoughts For me, Freecash wasn’t about becoming rich - it was about hitting a very specific goal: $930 in a month. I wasn’t expecting it to work, but it turned out to be exactly what I needed at the right time. If you ever want to test something like this for yourself, you can explore Freecash casually and see if it works with your daily routine. Upvote · 99 21 9 4 Henry Burek M.Phil, B.Sc. in Ophthalmic Optics, University of Bradford (MDIS) (Graduated 1977) · Author has 2.2K answers and 3.3M answer views ·Mar 30 Related In a plane, there are 15 points, each pinned with a nail, forming a rectangular grid of 3 rows and 5 columns with 1 cm intervals. Now there are many rubber bands. How many triangles with an area of 1 square centimeter can be formed? In a plane, there are 15 points, each pinned with a nail, forming a rectangular grid of 3 rows and 5 columns with 1 cm intervals. Now there are many rubber bands. How many triangles with an area of 1 square centimeter can be formed? Hopefully I’ve identified all the triangle types with an area of 1 square centimetre. Types A, B, C and D have bases of 1 cm and heights of 2 cm. Types F, G and H have bases of 2 cm and heights of 1 cm. Type E has sides of √2, √10 and √20. Type A can occur 40 times. Type B can occur 12 times. Type C can occur 8 times. Type D can occur 4 times. Type E can occur 4 times. Type Continue Reading In a plane, there are 15 points, each pinned with a nail, forming a rectangular grid of 3 rows and 5 columns with 1 cm intervals. Now there are many rubber bands. How many triangles with an area of 1 square centimeter can be formed? Hopefully I’ve identified all the triangle types with an area of 1 square centimetre. Types A, B, C and D have bases of 1 cm and heights of 2 cm. Types F, G and H have bases of 2 cm and heights of 1 cm. Type E has sides of √2, √10 and √20. Type A can occur 40 times. Type B can occur 12 times. Type C can occur 8 times. Type D can occur 4 times. Type E can occur 4 times. Type F can occur 20 times. Type G can occur 16 times. Type H can occur 8 times. Thus there can be 112 unique occurrences of triangles with an area of 1 square centimetre. Upvote · 9 3 Jitendra Dayma Assistant Audit Officer(AAO) CGL 2017 (2020–present) ·8y Related Five intersecting straight lines are drawn in a plane. What is the maximum number of triangles that can be formed? We can get maximum number of triangles under two assumption: No two lines are parallel to each other. No three lines have common intersection point We will get 10 point under above mentioned assumptions. Number of triangles will be 10 as per my calculations. Way to solve according to my method in figure: Give a number to every closed figure made by lines. And use different combinations to form triangle. Pardon me for ugly figure. Continue Reading We can get maximum number of triangles under two assumption: No two lines are parallel to each other. No three lines have common intersection point We will get 10 point under above mentioned assumptions. Number of triangles will be 10 as per my calculations. Way to solve according to my method in figure: Give a number to every closed figure made by lines. And use different combinations to form triangle. Pardon me for ugly figure. Upvote · 9 6 9 2 Sponsored by Online Shopping Tools Travel More, Spend Less: The Ultimate Free Hack for Travelers. Seniors Can Fly Business Class For Price Of Economy With This. Learn More 99 33 Prachi Sinha Just finished class 12 from DAV Public Schools, India ·Updated 5y Related There are 15 points in a plane out of which 6 are collinear. Find the number of lines that can be formed from 15 points? Any line can be formed by joining two points.Now according to question it is given that out of 15 points 6 are collinear..,like this Then we can choose any 2 points from remaining 9 points for forming line in 9(C)2 ways , again we choose one point from 9 non collinear ponit by 9(c)1 and one point from 6 collinear point by 6(c)1 ways..,and from 6 collinear points we can get a single line,so total no of lines = 9(C)2+9(c)1×6(c)1+1=36+54+1=91 no of different lines Continue Reading Any line can be formed by joining two points.Now according to question it is given that out of 15 points 6 are collinear..,like this Then we can choose any 2 points from remaining 9 points for forming line in 9(C)2 ways , again we choose one point from 9 non collinear ponit by 9(c)1 and one point from 6 collinear point by 6(c)1 ways..,and from 6 collinear points we can get a single line,so total no of lines = 9(C)2+9(c)1×6(c)1+1=36+54+1=91 no of different lines Upvote · 99 10 9 2 Abhishek Thigale 9y Related Ten points are in a plane, no three of them lie on a straight line. How many line segments are formed using these conditions? There are 10 different points such that no three of them don't lie on straight line. This means if you select any 3 points then they won't be collinear. Take any 10 points in a plane and name them as A, B ,C ,D ,E ,F ,G ,H ,I ,J. Now next statement says 4 of these points are joined to 6 of the remaining points. So divide these 10 points into two groups containing 4 and 6 points arbitrarily. Let's assume first group as A, B, C, D and second group contains remaining 6 points. Now each point in first group is joined to each point in second group. So line segments formed are 6+6+6+6=64=24. Because A Continue Reading There are 10 different points such that no three of them don't lie on straight line. This means if you select any 3 points then they won't be collinear. Take any 10 points in a plane and name them as A, B ,C ,D ,E ,F ,G ,H ,I ,J. Now next statement says 4 of these points are joined to 6 of the remaining points. So divide these 10 points into two groups containing 4 and 6 points arbitrarily. Let's assume first group as A, B, C, D and second group contains remaining 6 points. Now each point in first group is joined to each point in second group. So line segments formed are 6+6+6+6=64=24. Because A is joined to 6 points, B is joined to 6 points and so on. Next statement says each point from the group of 6 is joined to 5 other points (these 5 points can be in any group). Now these 6 points are already connected to 4 points in other group, so we need to connect each point to 1 more point from the same group. So divide these group of six into 3 groups of 2 and each group of 2 will form one line segment. For example E and F, G and H, I and J. Here if any point is repeated in group of two, then it will be joined to more than 5 points so we don't repeat. Now the given condition is satisfied. So answer is 24+3=27. Your response is private Was this worth your time? This helps us sort answers on the page. Absolutely not Definitely yes Upvote · 99 13 Navyatha Pandavula Playing with numbers since schooling ·8y Related How do I find the type of a triangle when they have given three straight lines? We can judge the type of triangle by knowing the angles between the straight lines. If the lines are given in the form of ax+by+c =0 , the slope of the line can be found by known formula m=(−a/b)m=(−a/b) Once we know the slopes of three lines , the angle between any two straight lines can be found using the formula t a n A=\|m 1−m 2\|/(1+m 1∗m 2)t a n A=\|m 1−m 2\|/(1+m 1∗m 2) where m1 and m2 are slopes of the two straight lines . By finding out all three angles between pairs of straight lines , we can say that the triangle is Obtuse-angled , if any one of the three angles is greater than 90 degrees. Right angled , if one of the thre Continue Reading We can judge the type of triangle by knowing the angles between the straight lines. If the lines are given in the form of ax+by+c =0 , the slope of the line can be found by known formula m=(−a/b)m=(−a/b) Once we know the slopes of three lines , the angle between any two straight lines can be found using the formula t a n A=\|m 1−m 2\|/(1+m 1∗m 2)t a n A=\|m 1−m 2\|/(1+m 1∗m 2) where m1 and m2 are slopes of the two straight lines . By finding out all three angles between pairs of straight lines , we can say that the triangle is Obtuse-angled , if any one of the three angles is greater than 90 degrees. Right angled , if one of the three angles is 90 degrees. Acute angled , if all three angles are less than 90 degrees. Isosceles triangle , if any two of the three angles are equal. Equilateral triangle , if all three angles are equal to 60 degrees. Hope it helped. Have a nice day :) Upvote · 99 21 9 3 Anil Bapat Have studied Mathematics up to pre-Degree Level · Author has 2.8K answers and 3.8M answer views ·Updated 3y Related There are 6 points, no three of which are collinear. How many straight lines and triangles can be formed by joining the points? There are 6 points, no three of which are collinear. How many straight lines and triangles can be formed by joining the points? There are 6 points and since no 3 points are collinear, for simplicity, we can imagine the points to be vertices of a regular hexagon (A non-regular one also would do, as log as it’s a hexagon). Let’s count the Lines: Firstly there are those 6 lines for forming a hexagon. Th Continue Reading There are 6 points, no three of which are collinear. How many straight lines and triangles can be formed by joining the points? There are 6 points and since no 3 points are collinear, for simplicity, we can imagine the points to be vertices of a regular hexagon (A non-regular one also would do, as log as it’s a hexagon). Let’s count the Lines: Firstly there are those 6 lines for forming a hexagon. Then choose any point and we can draw 3 more lines. Choose the next (adjacent) point, going in anti clockwise direction and we can draw another 3 lines. with the next anti clockwise point, we can draw 2 lines and with the next anti clockwise point we can draw 1 more line. There are two (2) more points but the lines are already drawn (from those points) and thus we have 6 + 3 + 3 + 2 + 1 = 15 Lines, which could be summed up as 6C2 = (65) / (21) = 30 / 2 = 15 Lines. Let’s go for the Triangles: Let’s have 6 distinct points for the vertices of a Regular Hexagon. Let’s call the points as A, B, C, D, E and F in anti clockwise fashion. Let’s take the base as AB and draw Distinct Triangles; we can draw 4 Triangles. Now, let’s shift the base to BC and draw Distinct Triangles; we can draw 3 Triangles. Now, let’s shift the base to CD and draw Distinct Triangles; we can draw 2 Triangles. Now, let’s shift the base to DE and draw Distinct Triangles; we can draw 1 Triangle. With the remaining 2 bases Viz. EF and FA, no more Distinct Triangles are possible. We can draw 2 more Distinct Triangles (1 each) with bases AC and BD and no more Distinct Triangles are possible with the remaining bases as all Triangles are now covered. So, here we have, 4 + 3 + 2 + 1 + 1 + 1 = 12. There must be some formula to arrive at it quickly but I am not able to figure it out, at the moment. Thus... Upvote · 9 3 Related questions Out of 15 points in a plane, no 3 are in straight line except 8 points which are collinear. How many triangles can be formed by joining them? How many soldiers walk in a straight line? Out of four lines, how many triangles can be formed? How many straight lines can be formed from 10 points if no three points are collinear? How many points are there on a line? How many different straight lines can be drawn from 4 points that are not on a straight line? Eight points are marked on a straight line and Nine points are marked on another line which is parallel to the first line. How many straight lines, including the first two, can be formed with those points? How many different straight lines can be formed from 30 points in a plane? (No three points are collinear) Are property lines straight? If there are three non-intersecting straight lines, how many planes can they form together? If there are 10 points on straight line AB and 8 points on straight line AC, with none of them being point A, how many triangles can be formed with these points as vertices? How do you tape straight lines? How many straight lines can be formed from 15 non collinear points? Why are many borders in Africa and America straight lines? How many pairs of coplanar lines are there if three non-parallel straight lines intersect? Related questions Out of 15 points in a plane, no 3 are in straight line except 8 points which are collinear. How many triangles can be formed by joining them? How many soldiers walk in a straight line? Out of four lines, how many triangles can be formed? How many straight lines can be formed from 10 points if no three points are collinear? How many points are there on a line? How many different straight lines can be drawn from 4 points that are not on a straight line? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
15147	https://www.youtube.com/watch?v=FlSYsfwKbkc	Brilliant Puzzle! Andy Math 463000 subscribers 1106 likes Description 17593 views Posted: 30 May 2025 To try everything Brilliant has to offer for free for a full 30 days, visit . You'll also get 20% off an annual premium subscription. 41 comments Transcript: This video is sponsored by Brilliant. Hey guys, this looks like a fun one. We're given two equilateral triangles inside of a square. The area of the larger equilateral triangle is 12 and it wants to know what is the area of the smaller equilateral triangle. If you want to try it on your own, pause it right now cuz I'm going to solve it in 3 2 1. First, let's isolate the two equilateral triangles and let's line them up next to each other. Since these two triangles are similar figures and we're interested in the areas, let's use these notes here. They tell us the relationships of the areas of similar figures. Let's call the side length of this triangle A and the side length of this triangle B. The scale factor between this triangle and this triangle is A to B. And the area ratio is A^2 to B^2. With these notes, we can set up a proportion. It'll be this area 12 over this area question mark equals a^2 B^2. I'm pretty sure we can use this proportion to solve it. In fact, if you're hearing this sentence right now, that means we ended up using this proportion to solve it. So, this must be important. Let's put a box around it. Now, let's bring ourselves back to here. Now, we're back where we started, but we have the larger triangle has a side length of a and the smaller triangle has a side length of b. Since this is a square, this side length is also going to be a since this is an equilateral triangle, this side is also going to be b. Next, let's talk about angles. Inside of a square, all the angles are right angles. And in equilateral triangles, all the angles are equal to 60°. So, this angle here will be 60°. And this angle here will be 60°. Since these were both 90°, that leaves 30° here and 30° here. And on the right hand side here, we have a very nice isoclesles triangle. And since both these angles are equal to 30°, this side is also equal to B. And now we got to figure out a relationship between B and A. Let's draw the altitude here which will be at right angles. And in an isosesles triangle, the altitude is also the perpendicular bis sector. So this base of A is going to be split even between these two sides. They're each going to be 1/2 of A. And now let's focus on this half of the triangle. This is a 30 609 triangle. So let's use the notes for 30 6090 triangles. Anytime you go from the medium side length to the shortest side length, you're going to divide by the of 3. So this side will be a over 2 / roo3, which simplifies to a / 23. And then to go from the shortest side to the longest side, you end up multiplying by two. So this side is going to be 2 a over 23. And this two on top and bottom can cancel each other out. So we end up with b is equal to a over 3. And this is exactly what we needed. We needed a relationship between B and A. So this was really helpful. We are now done with it though. And let's bring up our notes from before. Since b is equal to a over 3 in the place of this b down here, let's plug in a over 3. And in the denominator, this square can distribute to both of these terms. Let's smush everything together. And this a^2 over a^2 can both simplify to 1 over 1. And the of 3^2 is equal to 3. Now we have 1 / 1/3. Anytime you divide by a fraction, that's the same thing as multiplying by the reciprocal. So 1 / 1/3 is the same thing as multiplying by 3. And 1 3 is 3. So now we have 12 over question mark is equal to 3. So there's a couple ways we can do this. We can cross multiply or other things. I think it'll be fun to rewrite this three as a fraction with a 12 on top. Well, that's going to be 124s. So we can rewrite this as 12 over question mark = 12 over 4. And that's only going to be true if the question mark is equal to four. The area of this smaller equilateral triangle is equal to four. There's no units for the 12. So to be consistent, let's not put units here. This is the answer to our question. Let's put a box around it. How exciting. We used a couple different concepts in this video. We used the areas of similar figures and we used the side lengths of 30 60 90 triangles. If you want to have a deeper understanding of either of these two concepts, Brilliant has you covered. Brilliant has thousands of lessons in math, data analysis, programming, and AI. And all of them are interactive, which is the most effective way to learn. In the geometry course, it was easy to find the lesson on similar figures, and they did an amazing job of explaining how the areas are related to each other. I also like the lesson on 30 609 triangles. It was cool how they set it all up, and I like how they explained it. You can tell they put a lot of thought and care into the best way of explaining it. And they have that same level of care and quality in all their lessons. To try Brilliant for free, visit brilliant.org. org/andandymath or scan the QR code on the screen or you can click on the link in the description. And with that link, you can also get 20% off an annual premium plan. How exciting.
15148	https://www.isa-afp.org/browser_info/current/AFP/Lucas_Theorem/document.pdf	Lucas’s Theorem Chelsea Edmonds March 17, 2025 Abstract This work presents a formalisation of a generating function proof for Lucas’s theorem. We ﬁrst outline extensions to the existing For-mal Power Series (FPS) library, including an equivalence relation for coeﬃcients modulo n, an alternate binomial theorem statement, and a formalised proof of the Freshman’s dream (mod p) lemma. The second part of the work presents the formal proof of Lucas’s Theorem. Working backwards, the formalisation ﬁrst proves a well known corollary of the theorem which is easier to formalise and then applies induction to prove the original theorem statement. The proof of the corollary aims to provide a good example of a formalised generating function equivalence proof using the FPS library. The ﬁnal theorem statement is intended to be integrated into the formalised proof of Hilbert’s 10th Problem . Contents 1 Extensions on Formal Power Series (FPS) Library 2 1.1 FPS Equivalence Relation . . . . . . . . . . . . . . . . . . . . 2 1.2 Binomial Coeﬃcients . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Freshman’s Dream Lemma on FPS . . . . . . . . . . . . . . . 4 2 Lucas’s Theorem Proof 5 2.1 Reasoning about Coeﬃcients Helpers . . . . . . . . . . . . . . 6 2.2 Lucas Theorem Proof . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Proof of the Corollary . . . . . . . . . . . . . . . . . . 8 2.2.2 Proof of the Theorem . . . . . . . . . . . . . . . . . . 9 theory Lucas-Theorem imports Main HOL−Computational-Algebra.Computational-Algebra begin notation fps-nth (inﬁxl ‹$› 75) 1 1 Extensions on Formal Power Series (FPS) Li-brary This section presents a few extensions on the Formal Power Series (FPS) library, described in 1.1 FPS Equivalence Relation This proof requires reasoning around the equivalence of coeﬃcients mod some prime number. This section deﬁnes an equivalence relation on FPS using the pattern described by Paulson in , as well as some basic lemmas for reasoning around how the equivalence holds after common operations are applied deﬁnition fpsmodrel p ≡{ (f , g). ∀n. (f $ n) mod p = (g $ n) mod p } lemma fpsrel-iﬀ[simp]: (f , g) ∈fpsmodrel p ← →(∀n. (f $ n) mod p = (g $ n) mod p) by (simp add: fpsmodrel-def ) lemma fps-equiv: equiv UNIV (fpsmodrel p) proof (rule equivI) show reﬂ(fpsmodrel p) by (simp add: reﬂ-on-def fpsmodrel-def ) show sym (fpsmodrel p) by (simp add: sym-def fpsmodrel-def ) show trans (fpsmodrel p) by (intro transI) (simp add: fpsmodrel-def ) qed Equivalence relation over multiplication lemma fps-mult-equiv-coeﬀ: ﬁxes f g :: ( ′a :: {euclidean-ring-cancel}) fps assumes (f , g) ∈fpsmodrel p shows (f ∗h)$n mod p = (g∗h)$n mod p proof − have ((f ∗h) $ n) mod p =(P i=0..n. (f $i mod p ∗h$(n −i) mod p) mod p) mod p using mod-sum-eq mod-mult-left-eq by (simp add: fps-mult-nth mod-sum-eq mod-mult-left-eq) also have ... = (P i=0..n. (g$i mod p ∗h$(n −i) mod p) mod p) mod p using assms by auto also have ... = ((g∗h) $ n) mod p by (simp add: mod-mult-left-eq mod-sum-eq fps-mult-nth) thus ?thesis by (simp add: calculation) qed lemma fps-mult-equiv: ﬁxes f g :: ( ′a :: {euclidean-ring-cancel}) fps assumes (f , g) ∈fpsmodrel p shows (f ∗h, g∗h) ∈fpsmodrel p 2 using fpsmodrel-def fps-mult-equiv-coeﬀassms by blast Equivalence relation over power operator lemma fps-power-equiv: ﬁxes f g :: ( ′a :: {euclidean-ring-cancel}) fps ﬁxes x :: nat assumes (f , g) ∈fpsmodrel p shows (f^x, g^x) ∈fpsmodrel p using assms proof (induct x) case 0 thus ?case by (simp add: fpsmodrel-def ) next case (Suc x) then have hyp: ∀n. f^x $ n mod p = g ^x $ n mod p using fpsrel-iﬀby blast thus ?case proof − have fact: ∀n h. (g ∗h) $ n mod p = (f ∗h) $ n mod p by (metis assms fps-mult-equiv-coeﬀ) have ∀n h. (g ^ x ∗h) $ n mod p = (f ^ x ∗h) $ n mod p by (simp add: fps-mult-equiv-coeﬀhyp) then have ∀n h. (h ∗g ^ x) $ n mod p = (h ∗f ^ x) $ n mod p by (simp add: mult.commute) thus ?thesis using fact by force qed qed 1.2 Binomial Coeﬃcients The fps-binomial deﬁnition in the formal power series uses the n gchoose k operator. It’s deﬁned as being of type ′a fps, however the equivalence relation requires a type ′a that supports the modulo operator. The proof of the binomial theorem based on FPS coeﬃcients below uses the choose operator and does not put bounds on the type of fps-X. lemma binomial-coeﬀs-induct: ﬁxes n k :: nat shows (1 + fps-X)^n $ k = of-nat(n choose k) proof (induct n arbitrary: k) case 0 thus ?case by (metis binomial-eq-0-iﬀbinomial-n-0 fps-nth-of-nat not-gr-zero of-nat-0 of-nat-1 power-0) next case h: (Suc n) have start: (1 + fps-X)^(n + 1) = (1 + fps-X) ∗(1 + fps-X)^n by auto show ?case 3 using One-nat-def Suc-eq-plus1 Suc-pred add.commute binomial-Suc-Suc bino-mial-n-0 fps-mult-fps-X-plus-1-nth h.hyps neq0-conv start by (smt (verit, del-insts) of-nat-add) qed 1.3 Freshman’s Dream Lemma on FPS The Freshman’s dream lemma modulo a prime number p is a well known proof that (1 + xp) ≡(1 + x)p mod p First prove that pn k ≡0 mod p for k ≥1 and k < pn. The eventual proof only ended up requiring this with n = 1 lemma pn-choose-k-modp-0: ﬁxes n k::nat assumes prime p k ≥1 ∧k ≤p^n −1 n > 0 shows (p^n choose k) mod p = 0 proof − have inequality: k ≤p^n using assms (2) by arith have choose-take-1: ((p^n −1) choose ( k −1))= fact (p^n −1) div (fact (k −1) ∗fact (p^n −k)) using binomial-altdef-nat diﬀ-le-mono inequality assms(2) by auto have k ∗(p^n choose k) = k ∗((fact (p^n)) div (fact k ∗fact((p^n) −k))) using assms binomial-fact ′[OF inequality] by auto also have ... = k ∗fact (p^n) div (fact k ∗fact((p^n) −k)) using binomial-fact-lemma div-mult-self-is-m fact-gt-zero inequality mult.assoc mult.commute nat-0-less-mult-iﬀ by (simp add: choose-dvd div-mult-swap) also have ... = k ∗fact (p^n) div (k ∗fact (k −1) ∗fact((p^n) −k)) by (metis assms(2) fact-nonzero fact-num-eq-if le0 le-antisym of-nat-id) also have ... = fact (p^n) div (fact (k −1) ∗fact((p^n) −k)) using assms by auto also have ... = ((p^n) ∗fact (p^n −1)) div (fact (k −1) ∗fact((p^n) −k)) by (metis assms(2) fact-nonzero fact-num-eq-if inequality le0 le-antisym of-nat-id) also have ... = (p^n) ∗(fact (p^n −1) div (fact (k −1) ∗fact((p^n) −k))) by (metis assms(2) calculation choose-take-1 neq0-conv not-one-le-zero times-binomial-minus1-eq) ﬁnally have equality: k ∗(p^n choose k) = p^n ∗((p^n −1) choose (k −1)) using assms(2) times-binomial-minus1-eq by auto then have dvd-result: p^n dvd (k ∗(p^n choose k)) by simp have ¬ (p^n dvd k) using assms (2) binomial-n-0 diﬀ-diﬀ-cancel nat-dvd-not-less neq0-conv by auto then have p dvd (p^n choose k) using mult.commute prime-imp-prime-elem prime-power-dvd-multD assms dvd-result by metis thus ?thesis by simp 4 qed Applying the above lemma to the coeﬃcients of (1 + X)p, it is easy to show that all coeﬃcients other than the 0th and pth will be 0 lemma fps-middle-coeﬀs: assumes prime p n ̸= 0 ∧n ̸= p shows ((1 + fps-X :: int fps) ^p) $ n mod p = 0 mod p proof − let ?f = (1 + fps-X :: int fps)^p have ∀n. n > 0 ∧n < p − →(p choose n) mod p = 0 using pn-choose-k-modp-0 [of p - 1] ‹prime p› by auto then have middle-0: ∀n. n > 0 ∧n < p − →(?f $ n) mod p = 0 using binomial-coeﬀs-induct by (metis of-nat-0 zmod-int) have ∀n. n > p − →?f $ n mod p = 0 using binomial-eq-0-iﬀbinomial-coeﬀs-induct mod-0 by (metis of-nat-eq-0-iﬀ) thus ?thesis using middle-0 assms(2) nat-neq-iﬀby auto qed It follows that (1 + X)p is equivalent to (1 + Xp) under our equivalence relation, as required to prove the freshmans dream lemma. lemma fps-freshmans-dream: assumes prime p shows (((1 + fps-X :: int fps ) ^p), (1 + (fps-X)^(p))) ∈fpsmodrel p proof − let ?f = (1 + fps-X :: int fps)^p let ?g = (1 + (fps-X :: int fps)^p) have all-f-coeﬀs: ∀n. n ̸= 0 ∧n ̸= p − →?f $ n mod p = 0 mod p using fps-middle-coeﬀs assms by blast have ?g $ 0 = 1 using assms by auto then have ?g $ 0 mod p = 1 mod p using int-ops(2) zmod-int assms by presburger then have ?g $ p mod p = 1 mod p using assms by auto then have ∀n . ?f $ n mod p = ?g $ n mod p using all-f-coeﬀs by (simp add: binomial-coeﬀs-induct) thus ?thesis using fpsrel-iﬀby blast qed 2 Lucas’s Theorem Proof A formalisation of Lucas’s theorem based on a generating function proof using the existing formal power series (FPS) Isabelle library 5 2.1 Reasoning about Coeﬃcients Helpers A generating function proof of Lucas’s theorem relies on direct comparison between coeﬃcients of FPS which requires a number of helper lemmas to prove formally. In particular it compares the coeﬃcients of (1+X)n mod p to (1 + Xp)N ∗(1 + X)rn mod p, where N = n/p, and rn = n mod p. This section proves that the kth coeﬃcient of (1 + Xp)N ∗(1 + X)rn = (NchooseK) ∗(rnchooserk) Applying the (oo) operator enables reasoning about the coeﬃcients of (1 + Xp)n using the existing binomial theorem proof with Xp instead of X. lemma fps-binomial-p-compose: assumes p ̸= 0 shows (1 + (fps-X:: ( ′a :: {idom} fps))^p)^n = ((1 + fps-X)^n) oo (fps-X^p) proof − have (1:: ′a fps) + fps-X ^ p = 1 + fps-X oo fps-X ^ p by (simp add: assms fps-compose-add-distrib) then show ?thesis by (simp add: assms fps-compose-power) qed Next the proof determines the value of the kth coeﬃcient of (1 + Xp)N. lemma fps-X-pow-binomial-coeﬀs: assumes prime p shows (1 + (fps-X ::int fps)^p)^N $k = (if p dvd k then (N choose (k div p)) else 0) proof − let ?fx = (fps-X :: int fps) have (1 + ?fx^p)^N $ k = (((1 + ?fx)^N) oo (?fx^p)) $k by (metis assms fps-binomial-p-compose not-prime-0) also have ... = (P i=0..k.((1 + ?fx)^N)$i ∗((?fx^p)^i$k)) by (simp add: fps-compose-nth) ﬁnally have coeﬀs: (1 + ?fx^p)^N $ k = (P i=0..k. (N choose i) ∗((?fx^(p∗i))$k)) using binomial-coeﬀs-induct sum.cong by (metis (no-types, lifting) power-mult) thus ?thesis proof (cases p dvd k) case False — p does not divide k implies the kth term has a coeﬃcient of 0 have ∀i. ¬(p dvd k) − →(?fx^(p∗i)) $ k = 0 by auto thus ?thesis using coeﬀs by (simp add: False) next case True — p divides k implies the kth term has a non-zero coeﬃcient have contained: k div p ∈{0.. k} by simp have ∀i. i ̸= k div p − →(?fx^(p∗i)) $ k = 0 using assms by auto then have notdivpis0: ∀i ∈({0 .. k} −{k div p}). (?fx^(p∗i)) $ k = 0 by simp have (1 + ?fx^p)^N $ k = (N choose (k div p)) ∗(?fx^(p ∗(k div p))) $ k + (P i∈({0..k} −{k div p}). (N choose i) ∗((?fx^(p∗i))$k)) 6 using contained coeﬀs sum.remove by (metis (no-types, lifting) ﬁnite-atLeastAtMost) thus ?thesis using notdivpis0 True by simp qed qed The ﬁnal helper lemma proves the kth coeﬃcient is equivalent to ?N ?K ∗ ?rn ?rk as required. lemma fps-div-rep-coeﬀs: assumes prime p shows ((1 + (fps-X::int fps)^p)^(n div p) ∗(1 + fps-X)^(n mod p)) $ k = ((n div p) choose (k div p)) ∗((n mod p) choose (k mod p)) (is ((1 + (fps-X::int fps)^p)^?N ∗(1 + fps-X)^?rn) $ k = (?N choose ?K) ∗ (?rn choose ?rk)) proof − — Initial facts with results around representation and 0 valued terms let ?fx = fps-X :: int fps have krep: k −?rk = ?K∗p by (simp add: minus-mod-eq-mult-div) have rk-in-range: ?rk ∈{0..k} by simp have ∀i ≥p. (?rn choose i) = 0 using binomial-eq-0-iﬀ by (metis assms(1) leD le-less-trans linorder-cases mod-le-divisor mod-less-divisor prime-gt-0-nat) then have ptok0: ∀i ∈{p..k}. ((?rn choose i) ∗(1 + ?fx^p)^?N $ (k −i)) = 0 by simp then have notrkis0: ∀i ∈{0.. k}. i ̸= ?rk − →(?rn choose i) ∗(1 + ?fx^p)^?N $ (k −i) = 0 proof (cases k < p) case True — When k < p, it presents a side case with regards to range of reasoning then have k-value: k = ?rk by simp then have ∀i < k. ¬ (p dvd (k −i)) using True by (metis diﬀ-diﬀ-cancel diﬀ-is-0-eq dvd-imp-mod-0 less-imp-diﬀ-less less-irreﬂ-nat mod-less) then show ?thesis using fps-X-pow-binomial-coeﬀs assms(1) k-value by simp next case False then have ∀i < p. i ̸= ?rk − →¬(p dvd (k −i)) using mod-nat-eqI by auto then have ∀i ∈{0..<p}. i ̸= ?rk − →(1 + ?fx^p)^?N $ (k −i) = 0 using assms fps-X-pow-binomial-coeﬀs by simp then show ?thesis using ptok0 by auto qed — Main body of the proof, using helper facts above have ((1 + fps-X^p)^?N ∗(1 + fps-X)^?rn) $ k = (((1 + fps-X)^?rn) ∗(1 + fps-X^p)^?N) $ k by (metis (no-types, opaque-lifting) distrib-left distrib-right fps-mult-fps-X-commute fps-one-mult(1) 7 fps-one-mult(2) power-commuting-commutes) also have ... = (P i=0..k.(of-nat(?rn choose i)) ∗((1 + (fps-X)^p)^?N $ (k − i))) by (simp add: fps-mult-nth binomial-coeﬀs-induct) also have ... = ((?rn choose ?rk) ∗(1 + ?fx^p)^?N $ (k −?rk)) + (P i∈({0..k} −{?rk}). (?rn choose i) ∗(1 + ?fx^p)^?N $ (k −i)) using rk-in-range sum.remove by (metis (no-types, lifting) ﬁnite-atLeastAtMost) ﬁnally have ((1 + ?fx^p)^?N ∗(1 + ?fx)^?rn) $ k = ((?rn choose ?rk) ∗(1 + ?fx^p)^?N $ (k −?rk)) using notrkis0 by simp thus ?thesis using fps-X-pow-binomial-coeﬀs assms krep by auto qed 2.2 Lucas Theorem Proof The proof of Lucas’s theorem combines a generating function approach, based oﬀ with induction. For formalisation purposes, it was easier to ﬁrst prove a well known corollary of the main theorem (also often presented as an alternative statement for Lucas’s theorem), which can itself be used to backwards prove the the original statement by induction. This approach was adapted from P. Cameron’s lecture notes on combinatorics 2.2.1 Proof of the Corollary This step makes use of the coeﬃcient equivalence arguments proved in the previous sections corollary lucas-corollary: ﬁxes n k :: nat assumes prime p shows (n choose k) mod p = (((n div p) choose (k div p)) ∗((n mod p) choose (k mod p))) mod p (is (n choose k) mod p = ((?N choose ?K) ∗(?rn choose ?rk)) mod p) proof − let ?fx = fps-X :: int fps have n-rep: n = ?N ∗p + ?rn by simp have k-rep: k =?K ∗p + ?rk by simp have rhs-coeﬀs: ((1 + ?fx^p)^(?N) ∗(1 + ?fx)^(?rn)) $ k = (?N choose ?K) ∗ (?rn choose ?rk) using assms fps-div-rep-coeﬀs k-rep n-rep by blast — Application of coeﬃcient reasoning have ((((1 + ?fx)^p)^(?N) ∗(1 + ?fx)^(?rn)), ((1 + ?fx^p)^(?N) ∗(1 + ?fx)^(?rn))) ∈fpsmodrel p using fps-freshmans-dream assms fps-mult-equiv fps-power-equiv by blast — Application of equivalence facts and freshmans dream lemma then have modrel2: ((1 + ?fx)^n, ((1 + ?fx^p)^(?N) ∗(1 + ?fx)^(?rn))) ∈fpsmodrel p 8 by (metis (mono-tags, opaque-lifting) mult-div-mod-eq power-add power-mult) thus ?thesis using fpsrel-iﬀbinomial-coeﬀs-induct rhs-coeﬀs by (metis of-nat-eq-iﬀzmod-int) qed 2.2.2 Proof of the Theorem The theorem statement requires a formalised way of referring to the base p representation of a number. We use a deﬁnition that speciﬁes the ith digit of the base p representation. This deﬁnition is originally from the Hilbert’s 10th Problem Formalisation project which this work contributes to. deﬁnition nth-digit-general :: nat ⇒nat ⇒nat ⇒nat where nth-digit-general num i base = (num div (base ^ i)) mod base Applying induction on d, where d is the highest power required in either n or k’s base p representation, prime ?p = ⇒(?n choose ?k) mod ?p = (?n div ?p choose ?k div ?p) ∗(?n mod ?p choose ?k mod ?p) mod ?p can be used to prove the original theorem. theorem lucas-theorem: ﬁxes n k d::nat assumes n < p ^ (Suc d) assumes k < p ^ (Suc d) assumes prime p shows (n choose k) mod p = (Q i≤d. ((nth-digit-general n i p) choose (nth-digit-general k i p))) mod p using assms proof (induct d arbitrary: n k) case 0 thus ?case using nth-digit-general-def assms by simp next case (Suc d) — Representation Variables let ?N = n div p let ?K = k div p let ?nr = n mod p let ?kr = k mod p — Required assumption facts have Mlessthan: ?N < p ^ (Suc d) using less-mult-imp-div-less power-Suc2 assms(3) prime-ge-2-nat Suc.prems(1) by metis have Nlessthan: ?K < p ^ (Suc d) using less-mult-imp-div-less power-Suc2 prime-ge-2-nat Suc.prems(2) assms(3) by metis have shift-bounds-fact: (Q i=(Suc 0)..(Suc (d )). ((nth-digit-general n i p) choose (nth-digit-general k i p))) = (Q i=0..(d). (nth-digit-general n (Suc i) p) choose (nth-digit-general k (Suc i) p)) 9 using prod.shift-bounds-cl-Suc-ivl by blast — Product manipulation helper fact have (n choose k ) mod p = ((?N choose ?K) ∗(?nr choose ?kr)) mod p using lucas-corollary assms(3) by blast — Application of corollary also have ...= ((Q i≤d. ((nth-digit-general ?N i p) choose (nth-digit-general ?K i p))) ∗(?nr choose ?kr)) mod p using Mlessthan Nlessthan Suc.hyps mod-mult-cong assms(3) by blast — Using Inductive Hypothesis — Product manipulation steps also have ... = ((Q i=0..(d). (nth-digit-general n (Suc i) p) choose (nth-digit-general k (Suc i) p)) ∗(?nr choose ?kr)) mod p using atMost-atLeast0 nth-digit-general-def div-mult2-eq by auto also have ... = ((Q i=1..(d+1). (nth-digit-general n i p) choose (nth-digit-general k i p)) ∗ ((nth-digit-general n 0 p) choose (nth-digit-general k 0 p))) mod p using nth-digit-general-def shift-bounds-fact by simp ﬁnally have (n choose k ) mod p = ((Q i=0..(d+1). (nth-digit-general n i p) choose (nth-digit-general k i p))) mod p using One-nat-def atMost-atLeast0 mult.commute prod.atLeast1-atMost-eq prod.atMost-shift by (smt (verit, ccfv-threshold)) thus ?case using Suc-eq-plus1 atMost-atLeast0 by presburger qed end References J. Bayer, M. David, A. Pal, B. Stock, and D. Schleicher. The DPRM Theorem in Isabelle (Short Paper). In J. Harrison, J. O’Leary, and A. Tolmach, editors, 10th International Conference on Interactive The-orem Proving (ITP 2019), volume 141 of Leibniz International Proceed-ings in Informatics (LIPIcs), pages 33:1–33:7, Dagstuhl, Germany, 2019. Schloss Dagstuhl–Leibniz-Zentrum für Informatik. A. Chaieb. Formal power series. Journal of Automated Reasoning, 47(3):291–318, Oct. 2011. N. J. Fine. Binomial coeﬃcients modulo a prime. The American Math-ematical Monthly, 54(10):589–592, 1947. L. C. Paulson. Deﬁning Functions on Equivalence Classes. ACM Trans-actions on Computational Logic (TOCL), 7(4):658–675, Oct. 2006. Peter Cameron. Notes on Combinatorics. uk/~pjc/notes/comb.pdf, 2007. 10
15149	https://www.ncbi.nlm.nih.gov/books/NBK6124/	An official website of the United States government 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. Log in Account Logged in as:username Dashboard Publications Account settings Log out Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation Browse Titles Advanced Help Disclaimer NCBI Bookshelf. A service of the National Library of Medicine, National Institutes of Health. Madame Curie Bioscience Database [Internet]. Austin (TX): Landes Bioscience; 2000-2013. Madame Curie Bioscience Database [Internet]. Show details Austin (TX): Landes Bioscience; 2000-2013. < PrevNext > Nuclear Import of Plant Proteins Glenn R. Hicks. Introduction Overview In this Chapter, we will focus primarily on protein import into the nucleus of plants. As in other eukaryotes the partitioning of genetic information into the nucleus necessitates the import and export of macromolecules such as proteins, nucleic acids and protein/nucleic acid complexes across the nuclear envelope. These transport processes are essential and are subject to stringent regulatory controls. In plants, it is clear that in addition to the maintenance of basic cellular processes, the regulated import of proteins plays a vital role in development. As we will discuss, the nuclear import of proteins in a selective manner is essential in responses of plants to light and results in the dramatic morphological changes that occur as plants switch from growth in the dark to growth in the light.1 Protein translocation across the nuclear envelope is also a process that is utilized by pathogenic viruses2 and tumor-inducing bacteria in the genus Agrobacterium3 to transport protein/nucleic acid complexes into the nucleus for replication and even incorporation of pathogen DNA into the host genome. The import pathway for proteins, themselves key factors regulating nuclear transport processes, is the best understood of the nuclear transport processes in plants. Numerous import signals have been characterized, and we are beginning to identify and understand the major components of the import machinery. As expected many of these factors are conserved between plants, animals and fungi, but there are surprising results indicating subtle differences in preferences for nuclear localization signals (NLSs),3,4 the mechanics of import receptor function5, and potential plant-specific import factors.6 As we move forward plants are contributing new knowledge such as potential mechanisms for the targeting of proteins to the nuclear envelope and nuclear pore complex (NPC).7 Recent efforts have begun to focus on other nuclear transport processes in plants such as the export of proteins and nucleic acids.8,9 Protein Import in Animals and Yeast Our knowledge of nuclear transport in vertebrates and yeast is more advanced than our understanding in plants. Among the reasons for this historically are: 1. the ease of recovering nuclei from Xenopus oocytes that are intact for morphological studies, 2. the development of in vitro systems in Xenopus for reconstituting the assembly of nuclei and NPCs, 3. the reconstitution of cytosol-dependent nuclear import in permeabilized mammalian cells, 4. the rapid genetics possible in yeast, 5. the large biomedical research community and potential importance of nuclear processes for medicine. However as we are discovering nuclear translocation is a critical aspect of plant growth and development and thus has broad implications for agriculture in terms of disease and stress resistance and other crop improvements, which are the keys to feeding an expanding worldwide population. To place our knowledge of plant nuclear import in perspective, we will overview processes in animals and yeast throughout the Chapter highlighting important advances and unique aspects in plants. Import from the perspective of non-plant systems is covered in greater detail in other Chapters. There are also excellent recent reviews that are focused on the nuclear/cytoplasmic transport of proteins, nucleic acids and their complexes4,10–22as well as the structure and function of the NPCs in vertebrates and yeast.23–30 Nuclear Translocation in Plants Nuclear Pore Complex The channels through which all transport substrates must pass are the NPCs which are macromolecular complexes embedded in the double-membrane nuclear envelope. The NPCs are estimated to have a mass of 125 MDa in higher eukaryotes and to be composed of 50 to 100 different proteins collectively known as nucleoporins. Morphologically, an NPC is composed of a nucleoplasmic and a cytoplasmic ring. Eight spokes are found within the rings that extend toward a central channel resulting in eight 9 nm channels thought to function in the diffusion of small molecules across the nuclear envelope. A basket-like structure extends from the nucleoplasmic face and fibrils have been observed extending from the cytoplasmic face. Thus far, fewer than 20 nucleoporins have been purified from vertebrates. The NPCs in yeast are less complex (mass about 66 MDa) and are not dissociated and reassembled during mitosis as in higher eukaryotes; nevertheless the development of methods to purify intact complexes31 has permitted the identification and sequencing of all 30 of its nucleoporins. Even with such information available understanding the assembly and function of the NPC will be a daunting task.32 A number of vertebrate NPC proteins have been implicated in nuclear import.30 They include Nup358 which is located on the cytoplasmic filaments. Nup 358 has multiple Ran binding sites and binds to the protein transporter importin β (See Components and Mechanisms of Protein Import) via FXFG (single amino acid code where X represents an amino acid with a small or polar side chain) repeats33 which are characteristic of many nucleoporins. Other nucleoporins reported to bind to importin β include Nup153,34 Nup214,35 Nup116p, Nup100p,36 and p62.37 The p62 protein was one of the first nucleoporins to be purified, and like many nucleoporins from vertebrates it is modified by single O-linked N-acetylglucosamine (O-GlcNAc) residues. While the O-GlcNAc is probably not essential for import, the binding of the lectin wheat germ agglutinin (WGA) inhibits nuclear import in vertebrates and has been used for the identification and purification of nucleoporins from higher eukaryotes (For review see ref. 38). From electron microscopy studies beginning in the 1970s we know that nuclear pores in plants are morphologically similar to those of other organisms.39 However there have been few reports in which plant nucleoporins have been purified (For review see ref. 40). Scofield et al (ref. 41) localized a protein at the NPC and identified a 100 kDa polypeptide in a nuclear matrix fraction from carrots using an antibody to the yeast nucleoporin NSP100; however successful purification was not reported. Other studies indicate that nuclear envelope fractions from maize and tobacco nuclei contain a subset of proteins in the NPC fraction that can bind NLSs specifically.42–44 The binding site is at the NPC indicating a role for plant nucleoporins in protein import.42 This finding is consistent with the unusually tight association of at least one plant importin α NLS receptor with the nuclear envelope in purified nuclei and intact cells.45,46 Using WGA as a probe it is clear that GlcNAc-modifications are present at the periphery of the nucleus,47–49 and electron microscopy has shown that some of these modifications are present at the NPC.47 In fact biochemical characterization of tobacco nuclear fractions has shown that the glycans are attached to proteins via an O-linkage and the moities are longer than five sugar residues in length ending with a terminal GlcNAc residue. This is a novel modification not found in vertebrates, which contain only a single O-GlcNAc residue.47 Although the functional significance of O-GlcNAc modifications is not clear the modification can be used to advantage for purifying plant NPC proteins. Using lectin affinity chromatography four O-GlcNAc proteins were purified from nuclear envelope fractions of cultured tobacco cells. Peptide sequence was obtained from a protein of 40 kDa (gp40) that shares about 30% identity with aldose-1-epimerases (also known as mutarotases) which are involved in aldose sugar metabolism in bacteria.38 Their role in higher eukaryotes is not well defined but they do share structural similarities to the glucose carrier from erythrocytes. Interestingly, it has been reported that glycosylated proteins can be imported in a sugar-specific and NLS-independent manner in mammalian cells in vitro.50 Thus one speculation is that gp40 may be involved in such an import system in plants by binding to glycosylated proteins destined for import.40 Regardless of the role eventually assigned to gp40 a procedure to purify NPC proteins from plants should permit additional proteins to be characterized. The availability of Arabidopsis genome sequence should also contribute to the identification of plant nucleoporins, although it should be noted that beyond several very short repeat motifs such as FXFG there are few similarities even between vertebrate and yeast nucleoporins. While interesting in itself, it suggests that the identification of plant nucleoporins based on protein identity across kingdoms will be only partially successful. Import Signals in Plants Several types of transport can occur through NPCs. Ions and small proteins that are typically less than 20 to 30 kDa can pass by simple diffusion through the 9 nm channel. However even small macromolecules such as histones (14 kDa)51 or tRNAs52 cross the NPC via active processes permitting their translocation to be under cellular control. There is evidence that even calcium ions may be subject to selective concentration within the nucleus;53 this may relate to the suggestion that calcium plays a role in the regulation of active import.54 Nuclear transport processes are mediated by specific import receptors that recognize signals located within their respective substrates. In the case of protein import, NLSs interact with the import machinery that facilitates translocation through the NCP. Unlike most signals for organelles including the chloroplasts, mitochondria, vacuoles, peroxisomes and endoplasmic reticulum, NLSs are not proteolytically removed following import. This permits NLS-containing proteins to shuttle in and out of the nucleus and to be re-imported following post-mitotic nuclear assembly. Many transcription factors and cell cycle-regulatory proteins are able to exert their activities upon the cell based on their relative abundances in the cytosol compared to the nuceloplasm.10,54 Most NLSs are classified as either monopartite or bipartite. The classic monopartite NLS is from the SV40 large T-antigen and is composed of a single short region enriched in the basic residues arginine and lysine, whereas the nucleoplasmin NLS, which first defined the bipartite class, is composed of two basic domains separated by a spacer of variable length and composition. There are also NLSs that are less typical. One unusual NLS class is typified by the signal within the Matα2 protein which requires both basic and hydrophobic residues (for review see ref. 44). As in other organisms nuclear localization signals in plants cannot be defined by a strict consensus sequence and regions of basic amino acids are common within proteins, particularly regions involved in DNA binding. Thus, putative NLSs must be examined for activity in vivo to confirm their function. A number of NLSs have been carefully defined in such a manner in plants. These signals include at least one member for each of the three NLS classes. As examples, the transcriptional activitor R from maize possesses three functional NLSs, two SV40-like (one of them, NLS M, is defined as MSERKRREKL) and one Matα2-like signal (NLS C is defined as MISEALRKAIGKR), each of which are sufficient to target a reporter protein to the nucleus in vivo. Another transcription factor from maize, Opaque-2 possesses two signals, one SV40-like and one bipartite signal (NLS B is defined as RKRKESNRESARRSRYRK), that are sufficient to target a reporter protein to the nucleus. For R and Opaque2, mutations within the intact proteins indicate that multiple NLSs are necessary for efficient targeting in vivo suggesting cooperativity among NLSs for import which is also true in vertebrates.4 It is generally accepted that most NLSs can function across kingdoms pointing to a high degree of functional conservation. For example the SV40 large T-antigen NLS functions in plants (See for example refs. 55, 56) and the single bipartite NLS from the VirD2 protein of the plant pathogen Agrobacterium functions in plant, Xenopus, Drosophila, mammalian, and yeast cells.57–59 Agrobacterium is an opportunistic pathogen that infects a wide variety of plant species.3 In the coarse of pathogenesis Agrobacterium interacts with the host cell and transfers pathogen DNA (T-DNA) into the host cell nucleus through the NPC via the cooperative action of the VirD2 and VirE2 proteins. The T-DNA integrates into the plant genome and utilizes host factors to transcribe pathogen sequences. Agrobacterium has been a valuable system for studying import and will be discussed in later sections. Plant import is not strictly conserved when compared to import in other kingdoms however. The yeast Matα2 NLS targets a β-glucuronidase (GUS) reporter protein to the nucleus in onion epidermal cells which is consistent with the specific association of this class of NLSs with an import receptor from Arabidopsis.44–46 Interestingly, the Mata2 signal does not function in mammals60,61 although other yeast NLSs are known to function in vertebrates.62 Another exception is from Agrobacterium. As mentioned the NLS from VirD2 is broadly functional across kingdoms. Fascinatingly, VirE2 which contains two functional bipartite signals does not function in any of the non-plant organisms described for VirD2.57,3,59 However a single amino acid change which alters one NLS to conform to the animal bipartite consensus permits the NLS to function in Xenopus and Drosophila.57 Overall these results indicate that there may be subsets of import receptors or other components in plants that are not present in animals and fungi. Components and Mechanisms of Protein Import The key components of the classical NLS protein import pathway have been identified within the past decade and are the NLS-receptor importin α, the broad specificity transporter importin β, the GTPase Ran, and the Ran-interacting factor NTF2.21,30,63 In the first event of the protein import pathway, NLSs within nuclear proteins are recognized and bound by the NLS receptor importin α in the cytoplasm (Fig. 1). Another factor, importin β interacts with importin α (via an importin β binding domain within importin α) completing the trimeric import complex. It is importin β that then interacts with specific proteins of the NPC facilitating translocation through the NPC. Thus for protein import importin α functions as an adapter that recognizes the NLS and associates with importin β, whereas importin β functions as the actual transporter. The directionality of import is determined by the binding of the small ras-related GTPase Ran to importin β. Following import of the trimeric complex, the GTP-bound form of Ran (Ran-GTP) binds to importin β in the nucleoplasm resulting in the release of importin α and the NLS-containing cargo from the complex. The importin β/Ran-GTP heterodimer is then exported to the cytoplasm where Ran-GTP is hydrolyzed to Ran-GDP leading to the release of importin β for subsequent rounds of import via reassociation with importin α and NLS-containing cargo. Since monomeric importin α does not typically interact directly with the NPC it is exported back to the cytoplasm via its own export receptor, CAS,64 for subsequent rounds of import. Figure 1 Schematic overview of nuclear protein import. Import of NLS-containing proteins is dependent upon the formation of a trimeric import complex in the cytoplasm. Following translocation, binding of Ran-GTP to importin β releases importin α (more...) The energy and directionality of import are hypothesized to depend on the enrichment of Ran-GTP in the nucleoplasm compared to the cytoplasm which contains mostly Ran-GDP. This is accomplished through the action of the nucleotide exchange factor RCC1 in the nucleus that enhances nucleotide exchange favoring Ran-GTP and the GTPase-activating protein RanGAP1 that favors conversion to Ran-GDP in the cytoplasm. The GTPase activity is further stimulated by Ran-BP1. Import is effectively a process of facilitated diffusion utilizing a gradient of Ran-GTP; a loose analogy would be to envision the import apparatus as an antiporter that causes accumulation of protein against a gradient of Ran-GTP. In the cytoplasm, the factor NTF2 interacts with Ran-GDP and NPC proteins and functions as a receptor/transporter for the re-import of Ran preventing its depletion from the nucleoplasm.65,66 It should be noted that Ran plays essential roles beyond nuclear trafficking such as regulation of cell cycle progression.63 The selectivity and range of cargoes shuttled across the NPC is determined by different isoforms of importin α and importin β within the cell. The importin α receptor for NLS-mediated protein import has been found either as a single gene (SRP1) in yeast or as a small gene family in other organisms. In vertebrates at least six genes encoding importin α isoforms has been reported.21 There is also evidence of distinct but overlapping preferences for different NLSs and possible cell-specific roles for the different importin αs.57,67–71 Conversely, the range of importin β-like proteins, many of which are only distantly related, is far more diverse. This is due to the fact that members of the importin β family participate not only in the import of NLS-containing proteins but also function directly as import (importins) and export (exportins) receptor/transporters for other essential cargoes via their interaction with the NPC. Some examples include transportin 1 (import of hnRNP proteins), CRM1 (exportin 1; export of proteins containing a nuclear export signal), exportin-t (export of tRNAs), importin 7 (import of ribosomal proteins), and snurportin 1 (import of U snRNPs) (For review see ref. 21). The recent understanding of this receptor/transporter family has provided mechanistic details about the transport of diverse cargoes and highlighted potential ways in which the translocation of essential macromolecules is coordinated during the cell cycle. In higher plants we have only recently begun to identify components of the import apparatus. Hicks et al (ref. 48) identified a homologue of the importin α receptor, At-IMP α from Arabidopsis. Immunologically related proteins are found in all organs examined including roots, stems, leaves, and flowers as expected for an essential factor. There is also evidence that At-IMP α is phosphylated in vitro in the presence of cytosolic extracts suggesting a potential mechanism for controlling NLS binding or interaction with other proteins (Hicks and Raikhel unpublished). At the cellular level, At-IMP α is localized in the cytoplasm and nucleoplasm and at the nuclear envelope as expected for a receptor that shuttles between compartments. One unusual aspect of At-IMP α is its tight association with the nuclear envelope even in plant cells that have been treated to permeabilize the plasma membrane and deplete cytosolic contents;48 in animal cells endogenous importin α is mostly cytosolic and easily depleted from permeabilized cells. The plant receptor binds specifically in vitro to monopartite, bipartite and Matα2-like NLSs indicating broad NLS selectivity.5,45 Unlike yeast and mammalian orthologs that require importin β for high-affinity binding to NLSs, At-IMP α binds with high affinity (Kd of 5 to10 nM) in the absence of an importin β subunit, although At-IMP α is capable of binding to mouse importin β. This is consistent with the finding that At-IMP α can mediate association of import substrate at the nuclear envelope in permeabilized animal cells, whereas mouse importin α absolutely requires importin β.5 In fact, At-IMP α can mediate nuclear protein import in permeabilized animal cells in the absence of exogenous importin β indicating that At-IMP α shares some properties with importin β.5 This is a surprising result given that At-IMPa shares significant homology with other importin αs which require importin β and indicates the possibility of an importin β-independent pathway in plants. Although At-IMP α has some unusual properties, other plant importin α homologues are more typical of their animal and yeast counterparts. Using the Agrobacterium protein VirD2 as bait in a yeast two-hybrid screen of an Arabidopsis library, Ballas and Citovsky72 identified an importin α homologue (AtKAP α) that has high identity with At-IMP α except for a 64 amino acid extension at the amino-terminus. AtKAP α interacts specifically with the carboxy-terminal NLS of VirD2 both in vitro and in vivo in the two-hybrid assay, and the AtKAP α gene was found to complement a temperature-sensitive yeast mutant srp1–31 (yeast importin α). Cytosolic extract containing AtKAP α protein was able to restore import to cells of srp1–31 in an in vitro import system using permeabilized yeast cells in which import is dependent upon exogenous cytosol. Interestingly, VirE2 does not interact with AtKAP α suggesting an alternative pathway for its import (For discussion see ref. 72). A two-hybrid screen has identified a candidate for a VirE2 import protein (VIP1) that is related to basic leucine zipper proteins rather than to importin α.3 One possibility is that VirE2 is imported via a "piggy-back" mechanism through association with VIP1 which is a nuclear protein likely having functions other than in import. It is likely that importin α in Arabidopsis is encoded by a small gene family as at least 4 members have been reported.73 Another member of this family has been found in a two-hybrid screen using a WD40 type regulatory protein as a bait.74 PRL1 when disrupted by the insertion of a T-DNA tag results in a pleiotropic phenotype conferring hypersensitivity to glucose, sucrose and hormones. PRL1 was found to interact in vivo and in vitro with ATHKAP2 which has high identity with At-IMP α, but ATHKAP2 has a truncated carboxy terminal end being about 60 amino acids shorter than AtIMP α. As with the other characterized importin αs it possesses motifs required for interaction with importin β as well as the characteristic eight conserved armadillo repeats presumably for protein-protein interactions.45 An importin α homologue (importin α1) from rice has been characterized that is 76% identical to At-IMP α. Expression of the gene in rice is suppressed by light in both etiolated seedlings and in leaves but not in roots or calli which display constitutive expression.75 Binding to NLSs was examined in vitro, and importin α1 was found to bind to the SV40 large T antigen NLS and the bipartite signal from Opaque 2. However no association was observed between importin α1 and either the yeast Matα2 signal or the Matα2-like signal from the R protein.76 This again points to some selectivity in NLS recognition in plants. Rice importin α binds to importin β from mouse, although with an affinity much lower than that of mouse importin α. Nevertheless in vitro import in permeabilized HeLa cells could be made dependent upon the presence exogenous rice importin α1 and mouse importin β indicating that rice importin α1 can function in a manner similar to that of the animal and yeast homologues.76 Importin β in plants has been less studied than importin α, although there should be a significant number of genes for importin βs by analogy with vertebrates. There are a few plant importin βs that are characterized, and the results indicate a high degree of functional conservation between kingdoms. The first importin β homologues to be reported are from rice,77 and these are designated rice importins β1 and β2. Recombinant importin β1 can interact in vitro with rice importin α1 and a second rice importin α homologue (rice importin α2). This was studied by examining mobility shifts of protein complexes on native polyacrylamide gels. Using this approach it is argued that importin β1 specifically interacts with Ran-GTP and not Ran-GDP which is consistent with the ability of Ran-GTP to dissociate the importin α/importin β import complex in animals and yeast. In permeabilized HeLa cells, exogenous rice importin α1 and importin β1 can support import in the presence of mouse Ran.78 This result plus the finding that exogenous importin β1 can bind directly to the nuclear envelope in permeabilized tobacco BY2 cells argues that importin α and β functions are conserved. It will be interesting to examine the importin βs and other components for tissue specific or light regulation as has been observed for rice importin α1 because regulation of development by light is an essential and unique feature of plants. The Ran GTPase has been characterized in plants and homologues have been reported in Arabidopsis,79 tomato80 and tobacco81 among other species.82,83 While a direct role for Ran in nuclear import in plants has not been established, the protein is localized to the nucleus and appears to be encoded by a small gene family of at least 3 members79 that are expressed throughout the plant including tissues that are not actively growing. Furthermore, it has been demonstrated that homologues from tomato or tobacco when expressed in Schizosaccharomyces pombe can suppress the phenotype of the pim46-1 mutant which is defective in cell cycle progression.80,81 Ran is known to be involved in cell cycle control suggesting that the tomato and tobacco Ran homologues have functions analogous to those in other organisms. Both RanGAP1 and RanBP1 can stimulate Ran GTPase activity in the cytoplasm favoring Ran-GDP in that compartment. RanBP1 has been identified functionally from Arabidopsis by using the Ran homologue AtRan1 as bait in a two-hybrid screen in yeast. Haizel et al79 found interaction with two RanBP1-like proteins (At-RanBP1a and At-RanBP1b) having 60% identity with vertebrate Ran BP1 and possessing conserved domains for Ran binding. Further study indicates that both At-RanBP1s are capable of associating in vivo in yeast with the GTP-bound form of each of the three Ran homologues from Arabidopsis (AtRan1, AtRan2, AtRan3). Neither the intracellular location nor the ability of At-RanBP1a or At-RanBP1b to stimulate GTPase activity has been reported. Likewise, neither RanGAP1 nor the nuclear exchange factor RCC1 has been characterized functionally in plants. Two putative RanGAPs have been identified from Arabidopsis, and they appear to have a unique domain not present in Ran GAPs from animals and yeast.84 The motif which has been named a WPP domain is shared with MAF1.85 MAF1 is a recently discovered protein that is composed mostly of WPP repeats and appears to be localized to the nuclear envelope via interaction with another envelope protein, MFP1.86 The WPP domain appears to be specific to plants and is hypothesized to be involved in protein-protein interaction.84 If true, RanGAP may associate with the nuclear envelope and NPCs in plants through this interaction. Table 1 describes components of the nuclear import pathway in plants that have been characterized to date. Table 1 Components of the nuclear protein import pathway in plants. It is increasingly apparent that there is functional conservation of the basic nuclear import pathway in plants, vertebrates and yeast. However many of the interesting biological questions will no doubt reside in the ways that plants have adapted the basic nuclear import pathways to fit their sessile photosynthetic life style. Exceptions have been noted already (importin β independence of At-IMP α, a potential alternative pathway for VirE2 import, differences in NLS selectivity) and others will be discussed below. Systems to Study Import in Plants The essential breakthrough that permitted the biochemical identification and purification of the factors now known to be involved in nuclear translocation of proteins and other substrates was the development of an in vitro import system utilizing permeabilized animal cells.87,88 The method relies on the fact that plasma membranes from animals can be selectively permeabilized with digitonin, a reagent that aggregates to form pores upon binding to cholesterol. This sterol is abundant in the plasma membranes of animal cells but not in other membranes such as the nuclear envelope. The effect of the reagent is to permit the selective depletion of soluble factors from cultured cells while leaving the integrity of the nuclear envelope intact. Nuclear import thus occurs only by authentic facilitated translocation rather than by simple diffusion into damaged nuclei through tears in the nuclear envelope. Import can be directly visualized by microscopy following the addition of fluorescently labeled NLS-containing proteins. Another accepted method to examine import is microinjection into Xenopus oocytes or mammalian cells which is analytical and not amenable to fractionation of components. Other approaches that have been reported include the use of purified nuclei or nuclei mixed with Xenopus cytosol extracts (for a review see ref. 4); however, the ease of the permeabilization assay resulted in broad acceptance. Although digitonin is not the reagent of choice in yeast and plants due to differences in membrane composition, the principle of selective permeabilization has been utilized successfully. A method in yeast was developed in which the plasma membrane of cell wall-less spheroplasts was selectively permeabilized via a simple freeze-thaw technique.89 As in animal cells, import is dependent upon ATP (which can be converted to GTP for import), temperature, and the presence of exogenous cytosol (from yeast or mammalian cells). Again, alternative methods have been reported.90 In plants the development of in vitro import systems was difficult technically due to the fact that plant cells possess a thick cell wall and are highly vacuolated. The first report of in vitro import was an antibody cotranslocation assay in which antibodies to G-box binding factors (GBFs) are translocated into the nucleus (presumably by association with the GBFs) of Triton X-100 permeabilized parsley cells.91 The results indicate that GBFs involved in light-regulated gene expression are imported in response to light and that import is ATP and temperature dependent. The assay is indirect however relying of protease protection of antibody associated with the nucleus. Two groups reported the development of direct import assays using evacuolated protoplasts from tobacco.92 The approaches were similar with Merkle et al49 using Triton X-100 to permeabilize the plasma membrane, whereas Hicks et al48 used an osmotic shift to achieve permeabilization without detergents. In both cases, direct visualization of fluorescent import substrates indicates specific import that is dependent upon GTP hydrolysis and is specific for proteins containing functional NLSs. Some interesting differences are apparent between import in plants and import in animals and yeast. Import in plants is only partially inhibited on ice compared to an almost complete block in yeast and animals (perhaps an adaptation), and import was not blocked by WGA as in animals (perhaps due to the unusual NPC modifications). The most fundamental difference, however, is that import in permeabilized plant cells can occur in the absence of exogenous cytosol. This is not to suggest that cytoplasmic factors are not required in plants as in animals and yeast. However, a significant fraction of specific import factors may be tightly associated with cellular structures such as the cytoskeleton that are not disrupted by the permeabilization techniques used. There is support for this notion. By direct observation in permeabilized protoplasts48 and by fractionation of purified nuclei45 it is clear that in the presence of high concentrations of Triton X-100 significant fractions of At-IMP α remain in the cytoplasm and in association with the nuclear envelope in addition to a soluble pool. Whereas these observations provide an opportunity to investigate potential mechanisms for retention of At-IMP α, unfortunately they limit the utility of the assays for the identification of essential import factors in plants. Alternatives have been used most of which are heterologous systems. As noted Ballas and Cytovsky (ref. 72) have used the yeast permeabilized system to demonstrate the function of AtKAPα in the import of VirD2. Other heterologous systems have been used to examine Agrobacterium Vir proteins including Xenopus, Drosophila, mammalian, and yeast cells.57–59 Other examples previously cited are the functional characterizations of At-IMP α and rice importins α and β in permeabilized HeLa cells. An alternative heterologous approach is to utilize plant cytosol extracts to support import in animal cells. This approach was examined, and it was found that cytosolic extract from petunia could in fact support import in permeabilized HeLa cells. As in animal cells import is temperature dependent, requires GTP hydrolysis and is blocked by WGA.93 The final approach that has proven useful is microinjection of import substrates into the stamen hairs of Tradescantia. This was valuable in characterizing the nuclear import of VirE2-single stranded DNA complexes.94 VirE2 facilitates Agrobacterium infection by associating with pathogen DNA in the plant cell cytoplasm and assisting in its nuclear import. Import of the complexes was found to be dependent upon GTP hydrolysis and was inhibited by WGA. The inconsistency of WGA inhibition upon injection compared to a lack of inhibition in permeabilized cells in unclear. Perhaps the large mass of the VirE2-protein complexes renders them more susceptible to import inhibition than simple protein substrates. Regulated Protein Import in Plant Development Plants lead a sessile life style and thus have evolved sensitive mechanisms to control development and withstand environmental challenges. A large number of gene products are induced in response to light, which is an essential developmental stimulus. Seedlings respond to light by undergoing photomorphogenesis, a process that includes altered morphology (shorter stems, leaf development), the development of chloroplasts and induction of the photosynthetic machinery (greening). Even fully differentiated plants must continually respond to quantitative and qualitative differences in light and other environmental challenges such as temperature fluctuations. For the purposes of illustrating the important roles that nuclear protein import play in such responses, we will discuss several interesting examples in which regulation of import potentiates a response to environmental cues. Photomorphogenesis Genetic screens have identified components in light signaling that include photoreceptors (For a review see ref. 95) and downstream components that couple light signals to gene expression during photomorphogenesis (see for example refs. 96-99). The understanding of photomorphogenesis including the mechanisms of light-modulated gene expression is major area of plant biology beyond the scope of this Chapter but recently reviewed in detail.100–103 In Arabidopsis, screens for defective or inappropriate responses to light have resulted in mutants that are photomorphogenic (ie they possess a light-grown phenotype) in complete darkness and define at least 11 loci known as the COP/DET/FUS loci.1,104,105 One protein that acts as a negative regulator of photomorphogenesis is COP1. Mutations in COP1 result in plants that develop a light-grown phenotype including chloroplast development in complete darkness. When expressed as a GUS fusion protein in Arabidopsis, COP1 localizes primarily to the nucleus in the dark in leaves and shoots but is exclusively nuclear localized in the roots.104 However in the light COP1 partitions between the nucleus and cytoplasm in leaves and shoots. COP1 possesses, in addition to a bipartite NLS, a zinc-binding domain, a coiled-coil domain, and WD-40 repeats. Recent characterization of COP1 domain structure indicates that the amino terminal region containing the zinc-binding domain is essential for basal function, whereas the carboxyl terminal domain is necessary for the repression of photomorphogenesis in the dark.105 Although the precise mechanism by which COP1 targeting is regulated by light is not fully understood, a potential route by which photomorphogenic repression is achieved is via interaction of COP1 with the basic leucine-zipper (bZIP) transcription factor Hy5. Hy5 binds to G-box containing promoters for light regulated genes such as ribulose bisphosphate carboxylase/oxygenase small subunit (RBCS) and chalcone synthase (CHS). It has been identified as a suppressor of the cop1 mutant and has been shown to interact physically with COP1 protein.106 Several studies indicate that Hy5 is an exclusively nuclear protein that is abundant in the light but degraded in the dark. Furthermore, the degradation is clearly dependent upon interaction with COP1 in the nucleus because a truncated Hy5 protein lacking a COP1-interaction domain is no longer degraded in the dark.1 Thus, COP1 functions as a repressor of photomorphogenesis by signaling the selective degradation of downstream affectors including Hy5 which participates gene expression essential for development. Recent evidence suggests that degradation of HY5 is mediated by the ubiquitin pathway by interaction with a COP1-containing proteosome107 and that HY5 degradation can be further modulated by phosphorylation of the COP1 interaction domain.108 Expression of HY5 is interestingly itself under negative regulation in the light by the action of a recently discovered calcium-binding protein, SUB1109 The story is more complicated as COP1 appears to interact via its coiled-coiled domain with yet another protein (CIP1) that is cytoplasmic and is capable of interacting with the cytoskeleton.110 One hypothesis is that CIP1 is involved in the retention of COP1 in the cytoplasm in the light essentially excluding it from the nucleus. The perception of light involves multiple photoreceptors that detect blue light (cryptochrome), UV-B, and red light (phytochrome).95,111 The best characterized family of receptors are the phytochromes which are soluble proteins possessing a tetrapyrrole chromophore for activation by red light. After light absorption, the inactive form of phytochrome (Pr) is converted to the active far-red light absorbing form (Pfr) that participates in signal transduction and expression of genes involved in photomorphogenesis. The morphological consequences of red light perception include characteristic hypocotyl shortening and red light dependence of seed germination. There are five genes encoding phytochromes in Arabidopsis (phy A through phy E). The best characterized are phy A and phy B each of which detects red light in different ways as developmental cues. Phy A is rapidly degraded in the light and much more abundant in dark-grown plants than phy B. PhyA is responsible for the so-called very low fluence responses (VLFR) and for absorption of continuous far-red light known as the high irradiance responses (HIR). Phy B is stable in the light and is responsible for the detection of red light known as the low fluence responses (LFR) which are reversible by far-red light.95 For the induction of gene expression in response to light signals there must be communication between the soluble photoreceptors (that are for the most part cytoplasmic) and the nucleus. Recent experiments indicate that phy A and phy B are transported from the cytoplasm to the nucleus in response to red light. Rice phy A and tobacco phy B were fused to green fluorescent protein (GFP) and overexpressed in tobacco.112 When adapted to growth in the dark, phy A-GFP and phy B-GFP were not detected. However upon exposure to as little as 5 min of far-red light phy A-GFP (i.e., VLFR) localized to the nucleus. In contrast, upon exposure to red light (i.e., LFR), but not far-red light, Phy B-GFP was found in the nucleus and the localization was reversible upon irradiation by far-red light. Furthermore, the nuclear localization of phy B is dependent upon the presence of the chromophore. The kinetics of the light-dependent relocalization have been examine in detail.113,114 These results are consistent with the biological activities of phy A and phy B and indicate that their regulated targeting in response to the quality of the red light is an important step in phytochrome signaling. The mechanism of import inhibition in the dark is hypothesized to involve the masking of putative NLSs within the carboxyl terminus via structural changes that are dependent upon the presence of functional chromophore or perhaps a specific retention of Pr in the cytoplasm in the dark (for discussion see ref. 112). The light-regulated nuclear import of several classes of transcription factors has also been described recently and may provide additional pathways for the control of photomorphogenesis. Nuclear translocation of the common plant regulatory factor (CPRF) proteins in parsley cells has been shown to be under red light control and is far-red reversible.115 The CPRFs are bZIP transcription factors that bind to G-box elements found adjacent to many light-regulated genes. Of the three CPRFs that have been examined (CPRF1, CPRF2, CPRF4), CPRF2 was found in the cytoplasm in the dark and relocated to the nucleus in the light. Immunolocalization indicates that phy A HIR and phy B LFR responses are involved implicating CPRFs in phytochrome signaling and provides regulation in addition to light-modulated nuclear import of the receptors. A deletion analysis of CPRF2 reveals two potential domains involved in cytoplasmic retention in the dark. Neither domain has homology to the COP1 retention factor CIP1110 nor to a retention domain in the bZIP factor G-box binding factor 1 (GBF1; discussed below).116 However, one retention domain of CPRF2 shares 25% identity with an a-helical retention domain from mammalian heat shock factor 2.115 Other examples of light-regulated nuclear import are known. Using the previously discussed parsley in vitro antibody cotranslocation assay, Harter et al91 have found evidence for a cytoplasmic pool of GBF-transcription factors involved in light-regulated gene expression. The GBFs are another class of bZIP transciption factors that bind to G-box elements and participate in light-regulated gene expression. Upon exposure to white light, GBFs were found in the nucleus, presumably due to light stimulated relocalization. More recently, Kircher et al117 have cloned several CPRFs from parsley and using the antibody cotranslocation assay find that parsley CPRF1, CPRF2 and GBF2 are translocated to the nucleus in response to UV light. GBFs have been examined further using three different Arabidopsis GBF genes fused to the reporter GUS and examined by transient expression in soybean protoplasts.116 About 50% of one fusion protein (GUS:GBF2) was found in the nucleus in the dark, whereas this increased to about 80% upon exposure to blue light. Deletion analysis of a different fusion protein (GUS:GBF1) resulted in an increase in nuclear localization from a maximum of 50% to about 90%. This analysis may have identified a region involved in cytoplasmic retention of GBFs in the dark. One caveat is that GUS:GBF1 itself is not under light control being about 50% nuclear under all conditions tested. Given the importance of light in plant development additional examples of modulated nuclear import will no doubt be identified. Other Examples of Regulated Import Plants must continually respond to environmental and pathogen challenges and examples indicating the involvement of regulated import are being discovered. In tomato, the import of several heat shock transcription factors (Hsfs) requires protein-protein interaction. The expression of HsfA1 is constitutive but is accompanied by the expression of several heat shock inducible forms called HsfA2 and HsfB1. HsfA2 has been shown in tomato and tobacco protoplasts to be mostly cytoplasmic upon heat shock, even though related factors such as HsfA1 are translocated to the nucleus under these conditions. If a short region is deleted from the carboxyl terminus HsfA2, the protein is strongly localized to the nucleus.118 Interestingly, when coexpressed with HsfA1 in tomato protoplasts, HsfA2 is efficiently translocated following heat shock. The cotranslocation is dependent upon the physical interaction of HsfA1 and HsfA2 as demonstrated by coimmunoprecipitation and a two-hybrid assay.119 The stress induction of HsfA2 and its interaction with constitutively expressed HfA1 to form a transcriptionally active heterodimer provides a mechanism for dynamic changes in the intracellular distribution of HsfA2. We have already discussed aspects of the Agrobacterium system in which the pathogen utilizes endogenous plant import components to assist is the infection process. Nuclear import in plants can also be under viral control. An interesting example occurs in plants infected with the squash leaf curl virus (SqLCV), a geminivirus (for review see refs. 2, 120). The virus encodes two movement proteins, BR1 and BL1, which cooperativey participate in cell-to-cell spread of the virus. BR1 is an NLS-containing protein that shuttles between the nucleus and cytoplasm and binds to single-stranded DNA. BL1 is localized to peripheral regions of cytoplasm and appears to function in the movement of the BR1:viral DNA complexes across the cell wall to adjacent cells. When expressed transiently in tobacco protoplasts BR1 is strongly localized to the nucleus121,122 However when coexpressed with BL1, BR1 is relocalized to the cell periphery via specific interaction between the proteins122 providing a mechanism for delivery of viral genomes to the cell periphery for cell-to-cell spread. Other examples of viral protein nuclear import and cytoplasmic retention controlling nuclear import in viruses exist.123,124 Besides the specific examples of regulated nuclear import cited above, there is almost certainly broader control of development and environmental responses through modulation of the nuclear import apparatus itself. For example in rice, it is known that light exposure results in the down-regulation of importin α in leaves and dark-grown seedlings.75 Potentially broad control of import in plants could be imparted through phophorylation, which has been clearly implicated in both overall control of the cell cycle and in the specific regulation of imported proteins in animals and yeast (for reviews see refs. 4, 10, 125). Recent Advances in Plant Nuclear Translocation Several recent advances in our understanding of nuclear protein translocation in plants are worthy of mention as they have a potential impact of the field in general. Targeting to the NPC Many of the essential components of the import pathway have been identified animals and yeast, and are beginning to be identified in plants. In addition the NPCs have been the focus of intense investigation in animals and yeast. One fundamental question that has received little or no attention is how proteins in the cytoplasm are targeted to the NPCs for import. The notion that proteins are freely soluble in the cytoplasm where they associate with importins and diffuse to the NPCs for translocation is too simplistic. It is known that organelles and mRNAs can be transported along the cytoskeleton to specific sites.126,127 In fact, animal viruses can be targeted to the nucleus along microtubules,128 and there are strong indications that plant viral movement proteins can associate with the cytoskeleton.2,129 Could the cytoskeleton play a role in transporting complexes to the NPC for nuclear import? Immunolocalization of At-IMP α in tobacco protoplasts is suggestive of cytoskeleton, extending from the nucleus throughout the cytoplasm to the cell periphery.45 In addition as noted previously, At-IMP α like other importin αs contains hydrophobic armadillo repeats implicated in protein-protein interactions including association with the cytoskeleton130. Furthermore, At-IMP α cannot be fully depleted from the cytoplasm of permeabilized cells indicating a tight association with intracelluar components.48 These observations prompted Smith and Raikhel7 to investigate the role of the cytoskeleton in NPC targeting using double-immunofluorescence and confocal microscopy.7,46 importin α was found to colocalize with microfilaments and microtubules in tobacco protoplasts, whereas depolymerization of cytoskeleton results in loss of the cytoskeleton-like staining pattern. Depolymerization of microtubules results in diffuse cytoplasmic staining (Figure 2). Interestingly, depolymerization of microfilaments results in accumulation of receptor in the nucleus, suggesting that microfilaments may be involved in retention of importin α in the cytoplasm.7 An examination of At-IMP α association in vitro in a cytoskeleton-binding assay indicates that association with microtubules and microfilaments requires the presence of a functional NLS. The NLS-dependent association of At-IMP α with cytoskeleton may represent a mechanism for the assembly and transport of import complexes to the NPC. Based upon the data, a working model has been proposed46 in which microfilaments serve as sites for assembly of importin α-NLS protein complexes (Figure 3). Transport to the NPC would likely require the participation of a microtubule motor protein. It is possible that proteins translated from polysomes associated with the cytoskeleton could be assembled into complexes following synthesis. The model is supported by observations of the movement of NLS-containing substrates along neurons toward the nucleus, which is microtubule dependent.131 It is unclear at this time what role importin β would play in the formation of cytoskeletal import complexes. Although At-IMP α appears to function in import in an importin β-independent manner, this is probably not true for other importin αs. Other connections between importins and the cytoskeleton are becoming apparent. For example, it is now known that importin β can inhibit microtubule assembly in Xenopus egg extracts, and it is suggested this serves to suppress aster assembly until interaction with Ran-GTP releases importin β and assemble can proceed during mitosis.132 Figure 2 Colocalization of importin α with microtubules in the cytoplasm of tobacco cells. Fixed protoplasts were double immunolabeled for tubulin (Tubulin panels) and importin α (Importin panels) and examined for coalignment (Overlay panels). (more...) Figure 3 Diagram depicting hypothetical model for intracellular retention and translocation of import complexes prior to nuclear import. Actin microfilaments serve as a site for assembly of import complexes. importin α probably does not bind to microfilaments (more...) Nuclear Export Recently the nuclear shuttling protein BR1 from SqLCV protein has been examined for a nuclear export signal (NES) that would permit its export from the nucleus as has been found in the viral protein HIV Rev133 and others.134,135 This signal, like NLSs, is not strictly conserved, but is a leucine-rich hydrophobic sequence of 10 to 13 amino acids. Such as motif was found within BR1, and when its three leucine residues were mutated to alanines, viral pathogenicity was lost indicating the essential nature of these residues.9 It was further reasoned that only export competent BR1 protein could be relocalized to the cell periphery by association with BL1. In fact, GUS-BR1 fusion proteins when coexpressed in tobacco protoplasts are only relocalized to the cell periphery when the NES is present. Fusion proteins without the NES remain in the nucleus. Using this assay it was demonstrated that the NES from the Xenpous transcription factor IIIA can functionally replace the endogenous NES of BR1 and even restore viral infectivity. A homologue of the NES transporter CRM1 also known as exportin 1135–139 was recently cloned from Arabidopsis (AtXPO1) and characterized functionally in a yeast two-hybrid assay. The AtXPO1 interacts with the functional NES from AtRanBP1a as well as the NES from HIV Rev.140 In another protoplast expression system utilizing an NLS and NES fused to GFP, the NES from HIV Rev functions in export, which is inhibited by leptomycine B as in animal cells. The NES from AtRan BP1a, but not a version in which three leucines were mutated to alanines, also functions in this assay. These are the first examples in plants of characterized NESs and suggest that along with the protein import pathway, there is functional conservation with animals and yeast. Again, the interesting biology will reside in the details of how such pathways are utilized in plants, and the development of a straightforward assay for nuclear export in plants should encourage progress. Conclusions The study of protein import in plants is beginning to yield insight into not only the similarities with other kingdoms, but also the interesting differences that we have described throughout this Chapter. Plants are essential to all life on our planet and are the foundation for our food chain. Protein import processes and their role in development and environmental responses are essential to our understanding of plant biology, an important goal in itself. As our knowledge increases about nuclear protein import in plants, contributions to our general understanding of these processes in all organisms will increase. Some of the important areas to be addressed in the future include: The complete characterization of import components from higher plants as has been under way Answers to the question, is there an importin β-independent pathway in plants? The further investigation of plant NPCs Taking full advantage of pathogens such as Agrobacterium and viruses in understanding pathogenesis and import and export pathways A focus on the regulation of import in essential developmental pathways in plants such as photomorphogensis, stress responses, and perhaps phytohormone signaling The molecular mechanism of import complex targeting to the NPCs for translocation including the development of a system to examine NLS protein movement along microtubules and a search for factors that may mediate complex association with the cytoskeleton There is much to be learned and the future will surely present opportunities for new discovery. References 1. : Hartke CA, Deng X -W. The cell biology of the CP/DET/FUS proteins. Regulating proteolysis in photomorphogenesis and beyond? Plant Physiol. 2000;124:1548–1557. [PMC free article: PMC1539311] [PubMed: 11115873] 2. : Lazarowitz SG, Beachy RN. Viral movement proteins as probes for intracellular and intercellular trafficking in plants. Plant Cell. 1999;11:535–548. [PMC free article: PMC144200] [PubMed: 10213776] 3. : Tzfira T, Rhee Y, Chen M -H. et al. Nucleic acid transport in plant microbe interactions: The molecules that walk through the walls. Annu Rev Microbiol. 2000;54:187–219. [PubMed: 11018128] 4. : Hicks GR, Raikhel NV. Protein import into the nucleus: An integrated view. Annu Rev Cell Dev Biol. 1995;11:155–188. [PubMed: 8689555] 5. : Hubner S, Smith H M S, Hu W. et al. Plant importin α binds nuclear localization sequences with high affinity and can mediate nuclear import independent of importin β J Biol Chem. 1999;274(32):22610–22617. [PubMed: 10428841] 6. : Deng W, Chen L, Wood DW. et al. Agrobacterium VirD2 protein interacts with plant host cyclophilins. Proc Natl Acad Sci USA. 1998;95:7040–7045. [PMC free article: PMC22731] [PubMed: 9618535] 7. : Smith H M S, Raikhel NV. Nuclear localization signal receptor importin α associates with the cytoskeleton. Plant Cell. 1998;10:1791–1799. [PMC free article: PMC143961] [PubMed: 9811789] 8. : Haasen D, Kohler C, Neuhaus G. et al. Nuclear export of proteins in plants: AtXPO1 is the export receptor for leucine-rich nuclear export signals in Arabidopsis thaliana. Plant J. 1999;20(6):695–705. [PubMed: 10652141] 9. : Ward BM, Lazarowitz SG. Nuclear export in plants: Use of geminivirus movement proteins for an in vivo cell based export assay. Plant Cell. 1999;11:1267–1276. [PMC free article: PMC144272] [PubMed: 10402428] 10. : Jans DA, Hubner S. Regulation of protein transport to the nucleus: Central role of phosphorylation. Physiol rev. 1996;76(3):651–685. [PubMed: 8757785] 11. : Boulikas T. Nuclear localization signal peptides for the import of plasmid DNA in gene therapy (Review). Int J Oncol. 1996;10:301–309. [PubMed: 21533376] 12. : Corbett AH, Silver PA. Nucleocytoplasmic transport of macromolecules. Microbiol Mol Biol Rev. 1997;61(2):193–211. [PMC free article: PMC232607] [PubMed: 9184010] 13. : Goldfarb DS. Whose finger is on the switch? Science. 1997;276:1814–1816. [PubMed: 9206840] 14. : Lee MS, Silver PA. RNA movement between the nucleus and the cytoplasm. Curr Opin Genet Dev. 1997;7:212–219. [PubMed: 9115427] 15. : Moroianu J. Molecular mechanisms of nuclear protein import. Crit Rev Eukaryot Gene Expr. 1997;7(1-2):61–72. [PubMed: 9034715] 16. : Nakielny S, Fischer U, Michael WM. et al. RNA transport. Annu Rev Neurosci. 1997;20:269–301. [PubMed: 9056715] 17. : Yoneda Y. How proteins are transported from the cytoplasm to the nucleus. J Biochem. 1997;121:811–817. [PubMed: 9192717] 18. : Mattaj IW, Englmeier L. Nucleocytoplasmic transport: the soluble phase. Annu Rev Biochem. 1998;67:265–306. [PubMed: 9759490] 19. : Weis K. Importins and exportins: How to get in and out of the nucleus. Trends Biochem Sci. 1998;23(5):185–189. [PubMed: 9612083] 20. : Whittaker GR, Helenius A. Nuclear import and export of viruses and virus genomes. Virology. 1998;246(1):1–23. [PubMed: 9656989] 21. : Gorlich D, Kutay U. Transport between the cell nucleus and the cytoplasm. Annu Rev Cell Dev Biol. 1999;15:607–660. [PubMed: 10611974] 22. : Hood JK, Silver PA. In or out? Regulating nuclear transport. Curr Opin Cell Biol. 1999;11(2):241–247. [PubMed: 10209150] 23. : Davis LI. The nuclear pore complex. Annu Rev Biochem. 1995;64:865–896. [PubMed: 7574503] 24. : Pante N, Aebi U. Molecular dissection of the nuclear pore complex. Crit Rev Biochem Mol Biol. 1996;31:153–199. [PubMed: 8740526] 25. : Fabre E, Hurt E. Yeast genetics to dissect the nuclear pore complex and nucleocytoplasmic trafficking. Annu Rev Genet. 1997;31:277–313. [PubMed: 9442897] 26. : Nigg EA. Nucleocytoplasmic transport: Signals, mechanisms and regulation. Nature. 1997;386:779–787. [PubMed: 9126736] 27. : Gant TM, Goldberg MW, Allen TD. Nuclear envelope and nuclear pore assembly: Analysis of assembly intermediates by electron microscopy. Curr Opin Cell Biol. 10:409–415. [PubMed: 9640543] 28. : Yang Q, Rout MP, Akey CW. Three-dimensional architecture of the isolated yeast nuclear pore complex: Function and evolutionary implications. Mol Cell. 1998;1:223–234. [PubMed: 9659919] 29. : Badoor K, Shaikh S, Enarson P. et al. Function and assembly of nuclear pore complex proteins. Biochem Cell Biol. 1999;77(4):321–329. [PubMed: 10546895] 30. : Allen TD, Cronshaw JM, Bagley S. et al. The nuclear pore complex: Mediator of translocation between the nucleus and cytoplasm. J Cell Sci. 2000;113:1651–1659. [PubMed: 10769196] 31. : Rout MP, Aitchison JD, Suprapto A. et al. The yeast nuclear pore complex: Composition, architecture and transport mechanism. J Cell Biol. 148:635–651. [PMC free article: PMC2169373] [PubMed: 10684247] 32. : Rout MP, Aitchison JD, Suprapto A. et al. The yeast nuclear pore complex: Composition, architecture and transport mechanism. J Cell Biol. 2000;148:635–651. [PMC free article: PMC2169373] [PubMed: 10684247] 33. : Delphin C, Guan T, Melchior F. et al. RanGTP targets p97 to RanBP2, a filamentous protein localized at the cytoplasmic periphery of the nuclear pore complex. Mol Biol Cell. 1997;8:2379–2390. [PMC free article: PMC25714] [PubMed: 9398662] 34. : Shah S, Tugendreich S, Forbes DJ. Major binding sites for the nuclear import receptor are the internal nucleoporin Nup153 and the adjacent nuclear filament protein Tpr. J Cell Biol. 1998;141:31–49. [PMC free article: PMC2132719] [PubMed: 9531546] 35. : Fornerod M, van Deursen J, van Baal S. et al. The human homologue of yeast CRM1 is in a dynamic subcomplex with CAN/Nup214 and a novel nuclear pore component Nup88. EMBO J. 1997;16:807–816. [PMC free article: PMC1169681] [PubMed: 9049309] 36. : Iovine MK, Wente SR. A nuclear export signal in Kap95p is required for both recycling the import factor and interaction with the nucleoporin GLFG repeat regions of Nup116p and Nup100p. J Cell Biol. 1997;137:797–811. [PMC free article: PMC2139834] [PubMed: 9151683] 37. : Percipalle P, Clarkson WD, Kent HM. et al. Molecular interactions between the importin αlpha/beta heterodimer and proteins involved in vertebrate nuclear protein import. J Mol Biol. 1997;266:722–732. [PubMed: 9102465] 38. : Heese-Peck A, Raikhel NV. A glycoprotein modified with terminal N-acetylglucosamine and localized at the nuclear rim shows sequence similarity to aldose-1-epimerases. Plant Cell. 1998;10:599–612. [PMC free article: PMC144007] [PubMed: 9548985] 39. : Roberts K, Northcoat DH. Structure of the nuclear pore complex in higher plants. Nature. 1970;228:385–386. [PubMed: 5473989] 40. : Heese-Peck A, Raikhel NV. The nuclear pore complex. Plant Mol Biol. 1998a;38(1-2):145–162. [PubMed: 9738965] 41. : Scofield GN, Beven AF, Shaw PJ. et al. Identification and localization of a nucleoporin-like protein component of the plant nuclear matrix. Plant. 1992;187:414–420. [PubMed: 24178083] 42. : Hicks GR, Raikhel NV. Specific binding of nuclear localization sequences to plant nuclei. Plant Cell. 1993;5:983–994. [PMC free article: PMC160333] [PubMed: 8400874] 43. : Hicks GR, Raikhel NV. Nuclear localization signal binding proteins in higher plant nuclei. Proc Natl Acad Sci USA. 1995;92:734–938. [PMC free article: PMC42694] [PubMed: 7846044] 44. : Hicks GR, Smith H M S, Shieh M. et al. Three classes of nuclear import signals bind to plant nuclei. Plant Physiol. 1995;107:1055–1058. [PMC free article: PMC157236] [PubMed: 7770516] 45. : Smith H M S, Hicks GR, Raikhel NV. importin α from Arabidopsis thaliana is a nuclear import receptor that recognizes three classes of import signals. Plant Physiol. 1997;114:411–417. [PMC free article: PMC158320] [PubMed: 9193081] 46. : Smith H M S, Raikhel NV. Protein targeting to the nuclear pore. What can we learn from plants? Plant Physiol. 1999;119:1157–1163. [PMC free article: PMC1539210] [PubMed: 10198074] 47. : Heese-Peck A, Cole RN, Borkhsenious ON. et al. Plant nuclear pore complex proteins are modified by novel oligosaccharides with terminal N-acetylglucosamine. Plant Cell. 1995;7:1459–1471. [PMC free article: PMC160971] [PubMed: 8589629] 48. : Hicks GR, Smith H M S, Lobreaux S. et al. Nuclear import on permeabilized protoplasts from higher plants has unique features. Plant Cell. 1996;8:1337–1352. [PMC free article: PMC161251] [PubMed: 8776900] 49. : Merkle T, Leclerc D, Marshallsay C. et al. A plant in vitro system for the nuclear import of proteins. Plant J. 1996;10:1177–1186. [PubMed: 9011099] 50. : Duverger E, Pellerin, Mendes C, Mayer R. et al. Nuclear import of glycoconjugates is distinct from the classical NLS pathway. J Cell Sci. 1995;108:1325–1332. [PubMed: 7615655] 51. : Breeuwer M, Goldfarb D. Facilitated nuclear trasnport of histone H1 and other small nucleophilic proteins. Cell. 1990;60:999–1008. [PubMed: 1690602] 52. : Zasloff M. tRNA transport from the nucleus in a eukaryotic cell: Carrier-mediated translocation process. Proc Natl Acad Sci USA. 1983;80:6436–6440. [PMC free article: PMC390128] [PubMed: 6579529] 53. : Al-Mohanna FA, Caddy K W T, Bolsover SR. The nucleus is insulated from large cytosolic calcium ion changes. Nature. 1994;367:745–750. [PubMed: 7993399] 54. : Sweitzer TD, Love DC, Hanover JA. Regulation of nuclear import and export. Curr Top Cell Regul. 2000;36:77–94. [PubMed: 10842747] 55. : Lassner MW, Jones A, Daubert S. et al. Targeting of T7 RNA polymerase to tobacco nuclei mediated by an SV40 nuclear localization signal. Plant Mol Biol. 1991;17:229–234. [PubMed: 1650616] 56. : van der Krol AR, Chua N -H. The basic domain of plant B-ZIP proteins facilitates import of a reporter protein into plant nuclei. Plant Cell. 1991;3:667–675. [PMC free article: PMC160034] [PubMed: 1841723] 57. : Guralnick B, Thomsen G, Citovsky V. Transport of DNA into the nuclei of Xenopus oocytes be a modified VirE2 protein of Agrobacterium. Plant Cell. 1996;8:363–373. [PMC free article: PMC161106] [PubMed: 8721747] 58. : Relic B, Andjelkovic M, Rossi L. et al. Interaction of the DNA modifying proteins VirD1 and VirD2 of Agrobacterium tumefaciens: Analysis by subcellular localization in mammalian cells. Proc Natl Acad Sci USA. 1998;95:9105–9110. [PMC free article: PMC21299] [PubMed: 9689041] 59. : Rhee Y, Gurel F, Gafni Y. et al. A genetic system for detection of protein nuclear import and export. Nat Biotechnol. 2000;18:433–437. [PubMed: 10748526] 60. : Chelsky D, Ralph R, Jonak G. Sequence requirements for synthetic peptide-mediated translocation to the nucleus. Mol Cell Biol. 1989:2487–2492. [PMC free article: PMC362321] [PubMed: 2668735] 61. : Lanford RE, Feldherr CM, White RG. et al. Comparison of diverse transport signals in synthetic peptide-induced nuclear transport. Exp Cell Res. 1990;186:32–38. [PubMed: 2137089] 62. : Wagner P, Hall MN. Nuclear transport is functionally conserved between yeast and higher eukaryotes. FEBS Lett. 1993;321:261–266. [PubMed: 8477860] 63. : Sazer S, Dasso M. The Ran decathlon: Multiple roles of Ran. J Cell Sci. 2000;113:1111–1118. [PubMed: 10704362] 64. : Kutay U, Bischoff FR, Kostka S. et al. Export of importin αlpha from the nucleus is mediated by a specific nuclear transport factor. Cell. 1997;90:1061–0171. [PubMed: 9323134] 65. : Ribbeck K, Lipowsky G, Kent HM. et al. NTF2 mediates nuclear import of Ran. EMBO J. 1998;17:6587–6598. [PMC free article: PMC1171005] [PubMed: 9822603] 66. : Smith A, Brownawell A, Macara IG. Nuclear import of Ran is mediated by the transport factor NTF2. Curr Biol. 1998;8:1403–1406. [PubMed: 9889103] 67. : Kussel P, Frasch M, Pendulin a Drosophila protein with cell cycle-dependent nuclear localization is required for normal cell proliferation. J Cell Biol. 1995;129:1491–1507. [PMC free article: PMC2291176] [PubMed: 7790350] 68. : Torok I, Strand D, Schmitt R. et al. The overgrown hematopoietic organs–31 tumor suppressor gene of Drosophila encodes an importin-like proteins accumulating in the nucleus at the onset of mitosis. J Cell Biol. 1995;129:1473–1489. [PMC free article: PMC2291178] [PubMed: 7790349] 69. : Kohler M, Ansieau S, Prehn S. et al. Cloning of two novel human importin-alpha subunits and analysis of the expression pattern of the importin-alpha protein family. FEBS Lett. 1997;417:104–108. [PubMed: 9395085] 70. : Tsuji L, Takumi T, Imamoto N. et al. Identification of novel homologues of mouse importin αlpha, the alpha subunit of the nuclear pore-targeting complex, and their tissu-specific expression. FEBS Lett. 1997;416:30–34. [PubMed: 9369227] 71. : Kohler M, Speck C, Christiansen M. et al. Evidence for distinct substrate specificities of importin α-family members in nuclear protein import. Mol Cell Biol. 1999;19(11):7782–7791. [PMC free article: PMC84838] [PubMed: 10523667] 72. : Ballas N, Citovsky V. Nuclear localization signal binding protein from Arabidopsis mediates nuclear import of Agrobacterium VirD2 protein. Proc Natl Acad Sci USA. 1997;94:10723–10728. [PMC free article: PMC23464] [PubMed: 9380702] 73. : Schledz M, Leclerc D, Neuhaus G. et al. Characterization of four cDNAs encoding different importin αloha homologs from Arabidopisis thaliana. Plant Physiol. 1998;116:868. 74. : Nemeth K, Sakchert K, Putnoky P. et al. Pleiotropic control of glucose and hormone resposnes by PRL1, a nuclear WD protein, in Arabidopsis. Genes and Dev. 1998;12:3059–3073. [PMC free article: PMC317193] [PubMed: 9765207] 75. : Shoji K, Iwasaki T, Matsuki R. et al. Cloning of an importin-a and down-regulation of the gene by light in rice leaves. Gene. 1998;212:279–286. [PubMed: 9678973] 76. : Jiang C -J, Imamoto N, Matsuki R. et al. Functional characterization of a plant importin α homologue. J Biol Chem. 1998;273(37):24083–24087. [PubMed: 9727027] 77. : Matsuki R, Iwasaki T, Shoji K. et al. Isolation and characterization of two importin-beta genes from rice. Plant Cell Physiol. 1998;39(8):879–884. [PubMed: 9787463] 78. : Jian C -J, Imatoto N, Matsuki R. et al. In vitro characterization of rice importin β1: molecular interaction with nuclear transport factors and mediation of nuclear protein import. FEBS Lett. 1998;437:127–130. [PubMed: 9804185] 79. : Haizel T, Merkle T, Pay A. et al. Characterization of proteins that interact with the GTP-bound form of the regulatory GTPase Ran in Arabidopsis. Plant J. 1997;11:93–103. [PubMed: 9025305] 80. : Ach RA, Gruissem W. A small GTP-binding protein from tomato suppresses a Schizosaccharomyces pombe cell-cycle mutant. Proc Natl Acad Sci USA. 1994;91:5863–5867. [PMC free article: PMC44097] [PubMed: 8016079] 81. : Merkle T, Haizel T, Matsumoto T. et al. Phenotype of the fission yeast cell cycle regulatory mutant pim1–46 is suppressed by a tobacco cDNA encoding a small, Ran-like GTP-binding protein. Plant J. 1994;6:555–565. [PubMed: 7987414] 82. : Saalbach G, Christov V. Sequence of a plant cDNA from Vicia faba encoding a novel Ran-related GTP-binding protein. Plant Mol Biol. 1994;24:969–972. [PubMed: 8204834] 83. : Borg S, Brandstrup B, Jenson TJ. et al. Identification of a new protein species among 33 different small GTP-binding proteins encoded by cDNAs from Lotus japonicus, and expression of corresponding mRNAs in developing root nodules. Plant J. 1997;11:237–250. [PubMed: 9076991] 84. : Meier I. A novel link between Ran signal transduction and nuclear envelope proteins in plants. Plant Physiol. 2000;124:1507–1510. [PMC free article: PMC1539304] [PubMed: 11115866] 85. : Gindullis F, Peffer NJ, Meier I. MAF1, a novel plant protein interacting with matrix attachment region binding protein MFP1, is located at the nuclear envelope. Plant Cell. 1999;11:1755–1767. [PMC free article: PMC144308] [PubMed: 10488241] 86. : Gindullis F, Meier I. Matrix attachment region binding protein MFP1 is localized in discrete domains at the nuclear envelope. Plant Cell. 1999;11:1117–1128. [PMC free article: PMC144256] [PubMed: 10368182] 87. : Adam SA, Sterne-Marr R, Gerace L. Nuclear protein import in permeabilized mammalian cells requires soluble cytoplasmic factors. J Cell Biol. 1990;11:807–816. [PMC free article: PMC2116268] [PubMed: 2391365] 88. : Adam SA, Sterne-Marr R, Gerace L. Nuclear protein import using digitonin-permeabilized cells. Methods Enzymol. 1992;219:97–110. [PubMed: 1488017] 89. : Schlenstedt G, Hurt E, Doye V. et al. Reconstitution of nuclear protein transport with semi-intact yeast cells. J Cell Biol. 1993;123:785–798. [PMC free article: PMC2200159] [PubMed: 8227140] 90. : Kalinich JF, Douglas MG. In vitro translocation through the yeast nuclear envelope. J Biol Chem. 1989;264:17979–17989. [PubMed: 2681187] 91. : Harter K, Kircher S, Frohmeyer H. et al. Light regulated modification and nuclear translocation of cytosolic G-box binding factors in parsley. Plant Cell. 1994;6:545–559. [PMC free article: PMC160457] [PubMed: 8205004] 92. : Griesbach RJ, Sink KC. Evacuolation of mesophyll protoplasts. Plant Sci Lett. 1983;30:297–301. 93. : Broder YC, Stanhill A, Zakai N. et al. Translocation of NLS-BSA conjugates into nuclei of permeabilized mammalian cells can be supported by protoplast extract. An experimental system for studying plant cytosolic factors involved in nuclear imports. FEBS Lett. 1997;412:535–539. [PubMed: 9276462] 94. : Zupan JR, Citovsky V, Zambryski P. Agrobacterium VirE2 protein mediates nuclear uptake of single-stranded DNA in plant cells. Proc Natl Acad Sci USA. 1996;93:2392–2397. [PMC free article: PMC39807] [PubMed: 8637884] 95. : Batschauer A. Photoreceptors of higher plants. Planta. 1998;206:479–492. [PubMed: 9821683] 96. : Buche C, Poppe C, Schafer E. et al. eid1: a new Arabidopsis mutants hypersensitive in phytochrome A-dependent high-irradiance responese. Plant Cell. 2000;12:547–558. [PMC free article: PMC139852] [PubMed: 10760243] 97. : Fairchild CD, Schumaker MA, Quail PH. HFR1 encodes an atypical bHLH protein that acts in phytochrome A signal transduction. Genes Dev. 2000;14:2377–2391. [PMC free article: PMC316929] [PubMed: 10995393] 98. : Fankhauser C, Chory J. RSF1, an Arabidopsis locus implicated in phytochrome A signaling. Plant Physiol. 2000;124:39–46. [PMC free article: PMC59120] [PubMed: 10982420] 99. : Hsieh HL, Okamoto H, Wang A. et al. FIN219, an auxin-regulated gene, defines a link between phytochrome A and the downstream regulator COP1 in light control of Arabidopsis development. Genes Dev. 2000;14:1958–1970. [PMC free article: PMC316819] [PubMed: 10921909] 100. : Casal JJ. Phytochromes, cryptochromes, phototropin: photoreceptor interactions in plants. Photochem Photobiol. 71:1–11. [PubMed: 10649883] 101. : Nagy F, Schafer E. Nuclear and cytosolic events of light-induced, phytochrome-regulated signaling in higher plants. EMBO J. 2000;19:157–163. [PMC free article: PMC305550] [PubMed: 10637220] 102. : Neff MM, Fankhauser C, Chory J. Light An indicator of time and place. Genes Dev. 2000;14:257–271. [PubMed: 10673498] 103. : Wei N, Deng XW. Making sense of the COP9 signalosome: A regulatory protein complex conserved from Arabidopsis to human. Trends Genet. 2000;15:98–103. [PubMed: 10203806] 104. : von Armin AG, Deng XW. Light inactiviation of Arabidopsis photomorphogenic repressor COP1 involves a cell-specific regulation of its nucleocytoplasmic partitioning. Cell. 1994;79:1035–1045. [PubMed: 8001131] 105. : Stacey MG, Kopp OR, Kim T -H. et al. Modular domain structure of Arabidopsis COP1. Reconstitution of activity by fragment complementation and mutational analysis of a nuclear localization signal in planta. Plant Physiol. 2000;124:979–989. [PMC free article: PMC59198] [PubMed: 11080276] 106. : Ang LH, Chattopadhyay S, Wei N. et al. Molecular interaction between COP1 and HY5 defines a regulatory switch for light control of Arabidopsis development. Mol Cell. 1998;1:213–222. [PubMed: 9659918] 107. : Osterlund MT, Hardtke CS, Wei N. et al. Targeted destabilization of HY5 during light-regulated development of Arabidopsis. Nature. 2000;405:462–466. [PubMed: 10839542] 108. : Hardtke CS, Gohda K, Osterlund MT. et al. HY5 stability and activity in Arabidopsis is regulated by a phosphorylation within COP1 binding domain. EMBO J. 2000;19:4997–5006. [PMC free article: PMC314229] [PubMed: 10990463] 109. : Guo H, Mockler T, Duong H. et al. SUB1, an Arabidopsis Ca2+-binding protein involved in cryptochrome and phytochrome coaction. Science. 2001;291:487–490. [PubMed: 11161203] 110. : Matsui M, Stoop CD, von Armin AG. et al. Arabidopsis COP1 protein specifically interactsin vitro with a cytoskeleton-associated protein, CIP1. Proc Natl Acad Sci USA. 1995;92:4239–4243. [PMC free article: PMC41919] [PubMed: 7753789] 111. : Batschauer A. Light perception in higher plants. Cell Mol Life Sci. 1999;55:163–166. [PubMed: 10188582] 112. : Kircher S, Kozma-Bognar L, Kim L. et al. Light quality-dependent import of plant photoreceptors phytochrome A and B. Plant Cell. 1999;11:1445–1456. [PMC free article: PMC144301] [PubMed: 10449579] 113. : Gil P, Kircher S, Adam E. et al. Photocontrol of subcellular partitioning of phytochrome-B:GFP fusion protein in tobacco seedlings. Plant J. 2000;22(2):135–145. [PubMed: 10792829] 114. : Kim L, Kircher S, Toth R. et al. Light-induced nuclear import of phytochrome-A:GFP fusion proteins is differentially regulated in transgenic tobacco and Arabidopsis Plant J 200022(2):125–133. [PubMed: 10792828] 115. : Kircher S, Wellmer F, Nick P. et al. Nuclear import of the parsley bZIP transcription factor CPRF2 is regulated by phytochrome photoreceptors. J Cell Biol. 1999;144(2):201–211. [PMC free article: PMC2132893] [PubMed: 9922448] 116. : Terzaghi WB, Bertekap RL, Cashmore AR. Intracellular localization of BGF proteins and blue light-induced import of GBF2 fusion proteins into the nucleus of cultured Arabidopsis and soybean cells. Plant J. 1997;11(5):967–982. [PubMed: 9193069] 117. : Kircher S, Ledger S, Hayashi H. et al. CPRF4a, a novel plant bZIP protein of the CPRF family: comparative analyses of light-dependent expression, post-transcriptional regulation, nuclear import and heterodimerisation. Mol Gen Genet. 1998;257:595–605. [PubMed: 9604882] 118. : Lyck R, Harmening U, Hohfeld I. et al. Intracellular distribution and identification of the nuclear localization signals of two plant heat-stress transcription factors. Planta. 1997;202:117–125. [PubMed: 9177056] 119. : Scharf K -D, Heider H, Hohfeld I. et al. The tomato Hsf system: HsfA2 needs interaction with HsfA1 for efficient nuclear import and may be localized in cytoplasmic heat shock granules. Mol Cell Biol. 1998;18(4):2240–2251. [PMC free article: PMC121470] [PubMed: 9528795] 120. : Sanderfoot AA, Lazarowitz SG. Getting it together in plant virus movement: Cooperative interactions between bipartite geminivirus movement proteins. Trends Cell Biol. 1996;6:353–358. [PubMed: 15157433] 121. : Pascal E, Sanderfoot AA, Ward BM. et al. The geminivirus BR movement protein binds single-stranded DNA and localizes to the cell nucleus. Plant Cell. 1994;6:995–1006. [PMC free article: PMC160495] [PubMed: 8069108] 122. : Sanderfoot AA, Ingham DJ, Lazarowitz SG. A viral movement protein as a nuclear shuttle: The geminivirus BR1 movement protein contains domains essential for interaction with BL1 and nuclear localization. Plant Physiol. 1996;110:23–33. [PMC free article: PMC157690] [PubMed: 8587985] 123. : Restrepo-Hartwig MA, Carrington JC. The tobacco etch potyvirus 6-kilodalto protein is membrane associated and involved in viral replication. J Virol. 1994;68:2388–2397. [PMC free article: PMC236716] [PubMed: 8139025] 124. : Kunik T, Palanichelvam K, Czosnek H. et al. Nuclear import of the capsid protein of tomato yellow leaf curl virus (TYLCV) in plant and insect cells. Plant J. 1998;13(3):393–399. [PubMed: 9680988] 125. : Nagatani A. Regulated nuclear targeting. Curr Opin Plant Biol. 1998;1:470–474. [PubMed: 10066633] 126. : Bassel G, Singer RH. MRNA and cytoskeletal filaments. Curr Opin Cell Biol. 1997;9:109–115. [PubMed: 9013679] 127. : Hirokawa N. Kinesin and dynein superfamily proteins and the mechanism of organelle transport. Science. 1998;279:519–526. [PubMed: 9438838] 128. : Sodeik B, Ebersold MW, Helenius A. Microtuble-mediated transport of incoming herpes simplex virus 1 capsids to the nucleus. J Cell Biol. 1997;136:1007–1021. [PMC free article: PMC2132479] [PubMed: 9060466] 129. : Heinlein M, Epel BL, Padgett et al. Interaction of tobamovirus movement proteins with the plant cytoskeleton. Science. 1995;270:1983–1985. [PubMed: 8533089] 130. : Barth AI, Nathke IS, Nelson WJ. Cadherins, catenins and APC protein: Interplay between cytoskeletal complexes and signaling pathways. Curr Opin Cell Biol. 1997;9:693–690. [PubMed: 9330872] 131. : Ambron RT, Schmied R, Huang CC. A signal sequence mediates the retrograde transport of proteins from the axon periphery to the cell body and then into the nucleus. J Neurosci. 1992;12:2813–2818. [PMC free article: PMC6575827] [PubMed: 1377237] 132. : Wiese C, Wilde A, Moore MS. et al. Role of importin-b in coupling Ran to downstream targets in microtubule assembly. Science. 2001;291(5504):653–656. [PubMed: 11229403] 133. : Fischer U, Huber J, Boelens WC. et al. The HIV-1 Rev activation domain is a nuclear export signal that accesses an export pathway used by specific cellular RNAs. Cell. 1995;82:475–483. [PubMed: 7543368] 134. : Wen W, Meinkoth JL, Tsien RY. et al. Identification of a signal for rapid export of proteins from the nucleus. Cell. 1995;82:463–473. [PubMed: 7634336] 135. : Fridell RA, Fischer U, Luhrmann R. et al. Amphibian transcription factor IIIA proteins contain a sequence element functionally equivalent to the nuclear export signal of human immunodeficiency virus type 1 Rev. Proc Natl Acad Sci USA. 1996;93:2936–2940. [PMC free article: PMC39738] [PubMed: 8610146] 136. : Fornerod M, Ohno M, Yoshida M. et al. CRM1 is an export receptor for leucine-rich nuclear export signals. Cell. 1997;90:1051–1060. [PubMed: 9323133] 137. : Fukuda M, Asano S, Nakamura T. et al. CRM1 is responsible for intracellular transport mediated by the nuclear export signal. Nature. 1997;390:308–311. [PubMed: 9384386] 138. : Ossareh-Nazari B, Bachlerie F, Dargemont C. Evidence for a role of CRM1 in signal-mediated nuclear protein export. Science. 1997;278:141–144. [PubMed: 9311922] 139. : Stade K, Ford CS, Guthrie C. et al. Exportin 1 (CRM1p) is an essential nuclear export factor. Cell. 1997;90:1041–1050. [PubMed: 9323132] 140. : Haasen D, Kohler C, Neuhaus G. et al. Nuclear export of proteins in plants: AtXPO1 is the export receptor for leucine-rich nuclear export signals in Arabidopsis thaliana. Plant J. 1999;20(6):695–705. [PubMed: 10652141] Copyright © 2000-2013, Landes Bioscience. Bookshelf ID: NBK6124 Contents < PrevNext > PubReader Print View Cite this Page Hicks GR. Nuclear Import of Plant Proteins. In: Madame Curie Bioscience Database [Internet]. Austin (TX): Landes Bioscience; 2000-2013. In this Page Introduction Nuclear Translocation in Plants Regulated Protein Import in Plant Development Recent Advances in Plant Nuclear Translocation Conclusions References Related information PMC PubMed Central citations PubMed Links to PubMed Recent Activity Clear)Turn Off)Turn On) Nuclear Import of Plant Proteins - Madame Curie Bioscience Database Nuclear Import of Plant Proteins - Madame Curie Bioscience Database Your browsing activity is empty. Activity recording is turned off. Turn recording back on) See more... Follow NCBI Connect with NLM National Library of Medicine8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers
15150	https://www.sciencedirect.com/science/article/pii/S1322769624000295	Delirium prevention and management in an adult intensive care unit through evidence-based nonpharmacological interventions: A scoping review - ScienceDirect Skip to main contentSkip to article Journals & Books ViewPDF Download full issue Search ScienceDirect Outline ABSTRACT Keywords Acknowledgements 1. Introduction 2. Background 3. The review 4. Methods 5. Results 6. Discussion 7. Limitations 8. Conclusion Authorship contribution statement Funding Ethical Statement Conflict of interest Appendix A. Supplementary material Data Availability References Show full outline Cited by (6) Figures (1) Tables (3) Table 1 Table 2 Table 3 Extras (1) Supplementary material Collegian Volume 31, Issue 4, August 2024, Pages 232-251 Delirium prevention and management in an adult intensive care unit through evidence-based nonpharmacological interventions: A scoping review Author links open overlay panel Gideon U.Johnson a d e, Amanda Towell-Barnard a b, Christopher McLean c, Beverley Ewens a Show more Outline Add to Mendeley Share Cite rights and content Under a Creative Commons license Open access ABSTRACT Objective To map and review current literature to describe evidence-based nonpharmacological interventions for delirium prevention and management in adult critically ill patients. Introduction Previous research has demonstrated the efficacy of nonpharmacological interventions for intensive care unit (ICU) delirium; however, the heterogeneity and complexity of these interventions make it challenging to disseminate and integrate into clinical practice. Design This scoping review follows the Joanna Briggs Institute (JBI) Protocol Guidelines. Data sources Cumulative Index of Nursing and Allied Health Literature, Medical Literature Analysis and Retrieval System Online, PsycINFO, JBI, ProQuest, and Excerpta Medica databases were searched until August 2023. Review methods Double screening, extraction, and data coding using thematic analysis and frequency counts. Reporting followed the Preferred Reporting Items for Systematic Reviews and Meta-Analysis guidelines using the extension for scoping reviews. Results Thirty-three primary research articles were included; thirty-one were quantitative, and two were qualitative. Four categories of interventions were identified: instrument-based therapeutic interventions (n=10) consisting of the use of music, light, mirror, and occupational therapy; nurse-led interventions (n=5) consisting of interventions directly delivered by the nurses with mobilisation, orientation, and cognitive stimulation being the most common types of intervention. Family-delivered interventions (n=5) are delivered directly by family members, with extended visitation and orientation being the most utilised. Multicomponent interventions (n=13) combine different aspects of single interventions into care bundles and programs. Conclusion This review identified a lack of consistency in applying nonpharmacologic interventions to prevent and manage delirium in adult ICUs. Standardised evidence-based guidelines addressing all aspects of single-component or multicomponent nonpharmacological delirium interventions, along with support for ICU staff utilising these interventions and family member education and support, are required. Without consistent involvement from the healthcare team and patient families, opportunities may have been lost to optimise family-centred care practices in critical care settings. Patient or public contribution No patient or public contribution was necessary for this review. Protocol registration The protocol registration for this review can be accessed via Open Science Framework at Previous article in issue Next article in issue Keywords Delirium Critical care nursing Nonpharmacological interventions Holistic care Summary of relevance Problem or Issue Consistent application of evidence-based nonpharmacological interventions is needed in adult ICUs to address the gap in delirium care. What is already known Nonpharmacological interventions have been identified to be effective in preventing and managing delirium in adult ICUs. What this paper adds This review synthesised and categorised evidence-based nonpharmacological interventions for delirium prevention and management in adult ICUs and provided valuable insights into the interventions. These insights can support their dissemination and integration into patient care, thereby enhancing the quality of care in adult ICUs. Acknowledgements Professor Natalie Pattison and Professor Gavin Leslie’s advice helped shape this review. 1. Introduction Acute delirium is a serious complication that affects hospitalised adults and is associated with increased mortality and morbidity rates (Lauretani et al., 2020). Delirium is an acute brain dysfunction that presents as a collection of signs and symptoms that are characterised by a fluctuating mental status accompanied by inattention, altered level of consciousness, and disorientation in thinking (American Psychiatric Association [APA], 2013; Society of Critical Care Medicine, 2013). Delirium is a reversible condition that can be prevented, yet continues to be under-recognised and under-managed in clinical practice (Lange, Lamanna, Watson, & Maier, 2019;Wijdicks, 2021). All hospitalised adults are at risk of developing delirium (Marquetand et al., 2021), and delirium is estimated to occur in about 53% of the adult patient population during their hospital stay (Inouye, Westendorp, & Saczynski, 2014). However, the incidence of delirium increases to 74% of all patients in adult intensive care units (ICUs) and to 80% in those patients who have been mechanically ventilated (Inouye et al., 2014). 2. Background The cause and pathophysiology of delirium are not fully known but are usually associated with an underlying physiological disturbance (Mooyeon et al., 2018). A range of predisposing factors for delirium have been identified. These include age and gender (older men), single relationship status, living alone, lower educational status, reduced functional status, a previous history of delirium, and clinical factors such as low haemoglobin, previous cognitive decline, depression, and abnormal renal markers (Foroughan et al., 2016). The diversity of predisposing factors indicates that a multifactorial approach may be beneficial to the prevention, treatment, and management of delirium (Foroughan et al., 2016). In ICU, the development of delirium is compounded by other physiologically related factors. These include the severity of critical illness, sleep disturbance, unfamiliarity with the ICU environment, sedatives, and nonreversible factors such as cognitive impairment (McPherson et al., 2013). A recent study by Kim, Jin, Jin, and Sun-Mi (2020) concluded that the development of delirium in adult ICU patients may predominantly be related to sepsis, thrombocytopenia, and the presence of infection. Kim et al. (2020) also identified that sepsis-associated delirium was more likely to occur in critically ill patients aged over 65 years, exhibiting low levels of consciousness, tachypnoea, and high dependency on nursing care. Anticholinergic drug atropine is often used in ICU and has also been linked to delirium due to its effect on the central nervous system, resulting in altered perception, attention, and cognitive function (Maravi, Mishra, Singh, & Niranjan, 2020). Acute alcohol withdrawal was also noted as a predisposing factor to the development of delirium in the ICU (Wijdicks, 2021). Amongst all factors associated with delirium, age, functional dependence, previous cognitive impairment, and critical illness are the most common predisposing factors. These predisposing factors are commonly encountered in ICU, hence contributing to the increased rate of delirium. Delirium also has a significant psychological impact on patients and family members, caregivers, and health professionals. This has been postulated to be because of the complexity of its development, the lack of standardised interventions, and the challenges associated with managing delirium symptoms such as agitation and combativeness (Ewens, Collyer, Kemp, & Arabiat, 2021). Delirium carries a great risk of mortality and morbidity, with a risk ratio of 2.19 (94% confidence interval) for developing delirium in a cohort of ICU patients (Lauretani et al., 2020). Of the 172 patients enroled in a study by Aliberti et al. (2015), the prevalence of delirium symptoms and their impact on one-year mortality in patients with severe pneumonia in a respiratory high-dependency unit, 44% of patients who died during hospitalisation had delirium compared with 27% of those who survived. This is supported by Fiest et al. (2021)who concluded that ICU delirium is associated with increased mortality (hazard ratio, 1.12 [95% confidence interval]) 0–30 days after hospital discharge. The incidence of delirium is also associated with long-term cognitive decline and, in the context of dementia, increasing the severity of it in older patients (Goldberg et al., 2020; Lauretani et al., 2020). The impact of delirium can also result in prolonged hospital length of stay (LoS), with high financial costs to patients, healthcare systems, and the economy (Mattison, 2020). Prolonged LoS and ICU admission have been recognised in patients with delirium-related cognitive impairment compared with those without (Tropea, LoGiudice, Liew, Gorelik, & Brand, 2017). The cost associated with hospital LoS for all cognitively impaired patients, including patients with delirium, was 51% greater than for those without cognitive impairment (Tropea et al., 2017). The management of delirium in the ICU predominantly comprises pharmacological therapies, including antipsychotics. However, there is a paucity of evidence to support the efficacy and safety of these pharmacological interventions in treating delirium (Society of Critical Care Medicine, 2013). The reliance on pharmacological methods to manage delirium has been postulated due to a lack of evidence-based strategies to prevent and manage acute delirium (Ewens et al., 2021; McKenzie & Joy, 2020). Studies have demonstrated that pharmacological interventions are not effective in the context of delirium management; thus, delirium remains highly prevalent in the ICU despite the reliance on pharmacological interventions (Chen et al., 2022; Inouye et al., 2014). The long-term effects of these pharmacological therapies for delirium management are unknown, and implementing strategies with a limited evidence base to manage this significant health issue is not without risk (Chen et al., 2022, McKenzie and Joy, 2020). As a complementary approach, nonpharmacological interventions have been suggested as more effective than the use of drugs (Chen et al., 2022). Nonpharmacological interventions can include environmental reorientation, family involvement, sensory stimulation, early mobilisation, noise minimisation, sleep promotion, music therapy, and other nonpharmacological interventions that have been identified in the literature (Bannon et al., 2018, Collet et al., 2019, Johnson et al., 2018). The involvement of family members in delirium care is evolving and has been identified as acceptable to family members and ICU clinical staff (Bannon et al., 2018; Zamoscik, Godbold, & Freeman, 2017). Family involvement is one of the most utilised nonpharmacological interventions in delirium prevention and management, but how family members are involved in delirium care varies considerably from one clinical setting to another (Bannon et al., 2018; Burton et al., 2021). This emerging area has shown promise, but the evidence base is lacking to be able to be widely implemented into clinical practice. It is therefore timely to explore evidence-based nonpharmacological interventions currently utilised in clinical practice and how these are integrated into patient care. Because of the paucity of literature in this area, it was appropriate to conduct a scoping review of the literature on the topic and assess the quality of the available evidence. This enabled the synthesis and categorisation of the findings and provided insight into the integration of evidence-based nonpharmacological intervention for delirium in the ICU. The significance of this scoping review is that it adds to the body of knowledge around delirium in adult ICU settings by mapping out current evidence up to 2023, which has explored the application of nonpharmacological interventions used in adult ICU to prevent and manage delirium. The interventions identified may provide the foundation for further exploration through systematic review or research study. The review adds to the existing literature by mapping out primary research articles with quantitative and qualitative designs without year limitations and critically appraising the quality of the included articles (Bannon et al., 2019; Burry et al., 2021; Deng, Cao, Zhang, Peng, & Zhang, 2020;Kang et al., 2018; Rivosecchi, Smithburger, Svec, Campbell, & Kane-Gill, 2015; Sahawneh & Boss, 2021). Because of the ambiguity and heterogeneity of the nonpharmacological interventions that exist for delirium in adult ICUs, this comprehensive scoping review will be valuable in enhancing the clarity of these interventions and making the dissemination of these findings more feasible in adult ICUs. 3. The review 3.1. Objectives To identify and review current literature to describe nonpharmacological interventions for delirium prevention and management in critically ill adult patients and to assess and present the published evidence supporting each intervention. We further assessed the quality of individual evidence, which may support the reader to make more informed decisions in the choice of interventions to apply or standardise in clinical practice. 3.2. Review question What is the evidence that underpins nonpharmacological interventions for the prevention and management/minimisation of delirium in the adult ICU? 4. Methods 4.1. Design A scoping review methodology was used to address the aim of the study. The scoping review adhered to the Joanna Briggs Institute (JBI) reporting guidelines for conducting scoping reviews (Peters et al., 2020), with the following nine steps being undertaken: (1) defining and aligning the objectives and question; (2) developing and aligning the inclusion criteria with the objectives and question; (3) describing the planned approach to evidence searching, selection, data extraction, and evidence presentation; (4) searching for the evidence; (5) selecting the evidence; (6) extracting the evidence; (7) analysing the evidence; (8) presentation of the results; and (9) summarising the evidence. Reporting followed the EQUATOR Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines using the extension for scoping reviews (PRISMA-ScR) (Tricco et al., 2018). A protocol was registered on Open Science Framework. 4.2. Search methods The first author (GUJ) drafted the search strategy with the support of an experienced academic librarian and refined it with the co-authors. A preliminary search of Medical Literature Analysis and Retrieval System Online (MEDLINE), the Cochrane Database of Systematic Reviews, and JBI Evidence Synthesis was conducted, and no current systematic reviews or scoping reviews on the topic were identified. An initial limited search of MEDLINE and Cumulative Index of Nursing and Allied Health Literature (CINAHL) was undertaken in June 2022 to identify articles on the topic. The initial limited search of CINAHL and MEDLINE used the Medical Subject Headings (MH) terms “Intensive Care Unit” or “critical ill” and “delirium”. Following this initial search, the text words in the retrieved papers’ titles, abstracts, and index terms were analysed to identify key parameters used to describe the topic area. The authors revised the search strategy, and a systematic search of six databases was commenced, including the CINAHL, MEDLINE, PsycINFO, JBI, ProQuest LLC, and Excerpta Medica databases (Embase). The first author completed the search in December 2022 with no date restriction and rerun the search in August 2023. Table 1 provides the core terms used to develop the search strategy. The Supplementary Material Database Search (Appendices 3–8) includes the combination of keywords used for each database. Table 1. Search terms. \| Participants \| ICU patients \| Intensive Care Units (MeSH) OR intensive care OR critical care OR critical care (MeSH) \| \| Concept \| Nonpharmacological interventions aimed at Delirium prevention/minimisation \| Non-pharmacologic Delirium OR prevention OR intervention OR treatment OR Delirium (MeSH) \| \| Context \| Adult ICU \| Applied as limiter ‘All Adults’ to Intensive Care Units (MeSH) OR intensive care OR critical care OR critical care (MeSH) \| 4.2.1. Inclusion criteria and exclusion criteria The scoping review considered primary research papers of qualitative, quantitative, or mixed-method designs published in English in a peer-reviewed journal and all adult categories until August 2023. The reference list of the identified papers was reviewed for any additional relevant articles. Case reports, systematic reviews, meta-analysis and scoping reviews, opinion papers, and unpublished (grey) articles on the research area were excluded from the review as the review seeks to find and synthesise evidence-based nonpharmacological interventions currently being utilised in practice. The criteria for selecting papers were based on the participants, concept, and context mnemonics recommended by JBI methods (Table 1). 4.2.2. Participants Participants comprised any patient (≥18 years old) admitted to an ICU. 4.2.3. Concept This review’s main concept of interest was the nonpharmacological interventions utilised in the prevention, minimisation, or management of delirium. Articles reporting on nonpharmacological and pharmacological interventions were included if the article’s emphasis was not on pharmacological interventions only; in such articles, data about nonpharmacological interventions were extracted. Papers focusing on other types of brain dysfunction other than delirium were not included. 4.2.4. Context Articles were included if they were conducted in an adult ICU. Nonpharmacological interventions utilised for delirium in the general hospital wards, rehabilitation, or community facilities were not included. 4.2.5. Screening and critical appraisal EndNote (version 20.1; Clarivate Analytics, Boston, Massachusetts) and Rayyan supported the screening of the studies (Ouzzani, Hammady, Fedorowicz, & Elmagarmid, 2016). All search results were uploaded into EndNote, and duplicates were removed. The search results were uploaded into Rayyan to complete the title, abstract, and full-text screening (Ouzzani et al., 2016). One author screened articles at the title level, which another author verified. Two authors reviewed at the abstract and full-text level for eligibility for inclusion. Any conflicts were resolved by discussion with the third author. Study quality was assessed and scored independently by two of the authors (GUJ and BE) using the standardised JBI critical appraisal instruments (Munn et al., 2019). Any conflicts were resolved through discussion with the third and fourth authors (AT-B and CM). Cohort studies required 11 questions to be completed, randomised controlled trials (RCTs) required 13 questions to be completed, the analytical cross-sectional study required 8 questions to be completed, and the quasi-experimental study required 9 questions to be completed, whereas the qualitative research studies required 10 questions to be completed (see Supplementary Material Critical Appraisal, Appendix 1 & 2). 4.2.6. Data abstraction The first author (GUJ) developed a data charting form following JBI recommendations and adapted it to reflect the aims of this review. The data charting form was trialled on a small portion (n=11) of the articles before using it for all the articles. The extracted data included authors, year published, country of origin, settings, study aims specific to delirium, study design/methods and nonpharmacological interventions, sample size, population, inclusion/exclusion criteria, results relevant to the review and limitations. Data extraction was completed by the first author and reviewed by a second author. The authors discussed the results and updated the data charting form wherever applicable (see Table 2). Table 2. Study characteristics of the articles included in the review. \| Code \| Study/ country \| Setting \| Study aim \| Study design/methods \| Study population/criteria Delirium tool/duration \| Results \| Limitations \| --- --- --- --- \| \| 1 \| Álvarez et al. (2017)Chile \| Medical and surgical ICU. \| To determine the impact of occupational therapy intervention in duration, incidence, and severity of delirium in elderly patients in the ICU \| Pilot randomised controlled trial (RCT), nPP (nonpharmacologic prevention),plus early and intensive occupational therapy within the first 24 hrs of admission. Control group (n=70) Experimental group (n=70) \| Nonmechanical ventilated ICU patients aged >60 years, within 24 hrs. 5 days. Confusion Assessment Method for the ICU (CAM-ICU). \| Management of delirium: The density of delirium was significantly lower in the experimental group. Incidence of delirium was 20% (n=14) in the control group and 3% (n=2) in the experimental group. Delirium severity was similar between the two groups (mean score 10 vs 9). Patient days with delirium, the control group (27.5 days of 335)and the experimental group (3.5 days of 339). The control group had higher risk of delirium than the experimental group. \| Exclusion of some predisposing factors of delirium such as mechanical ventilation, cognitive impairment, dementia, and history of delirium. No mention of ethical approval. \| \| 2 \| Contreras et al. (2021) Colombia \| Medical & surgical ICU. \| To determine the efficacy of a multicomponent nursing program to prevent delirium in critically ill patients. \| Parallel, double-blind RCT. Cognitive stimulation nearly 15min: time and space orientation and family support. Control group (n=41) Experimental group (n=40) \| ICU patients >18 and >24 hrs in ICU. Vasoactive drugs, mechanical ventilation, invasive monitoring, RASS −3 to +4, no delirium at admission and 40% chance of delirium per the PRE-DELIRIC model. CAM-ICU. 15 min. \| Delirium incidence was 5% in the intervention group vs 24% in control group. Rate of incidence of delirium was 42.37 per 1000 person/days in the control group vs 7.87 per 1000 person/days in the intervention group. \| Several nonpharmacological interventions were applied together, making it difficult to determine the effectiveness of each single-component intervention. The effectiveness of the interventions to manage delirium severity was not evaluated. \| \| 3 \| Damshens et al. (2018) Iran \| General ICU \| To study the effects of music therapy on the incidence of delirium in ICU patients \| RCT. Music therapy twice a day for 45 min duration. Control group (n=40) Experimental group (n=40) \| ICU patients >15 years with no history of cognitive impairment, depression, taking psychotropic drugs, drug abuse and alcohol abuse, and no visual or hearing loss. CAM-ICU. 45 min. \| 15 patients (37.5%) in the interventional group vs 16 patients (40%) in the control group had delirium. Therefore, no significant difference. \| No mention of ethical approval. The patients were also treated for delirium with pharmacological interventions, and it is unclear if these may have affected the result of the music therapy on the patients. \| \| 4 \| Eghbali-Babadi et al. (2017) Iran \| ICU \| To assess the effect of the relationship between the family and patient on the incidence of delirium. \| RCT. A family member selected by the patient was allowed to visit from the morning after surgery, once a day for 30–40 min and communicated to the patient based on education received by the nurse. Control group (n=34) Experimental group (n=34) \| Cardiovascular ICU patients aged between 18 and 70 years. Availability of family member, no drug addiction history, alcohol, smoking, delirium, cognitive disorder, blindness and deafness, family history of surgery, no intubation. CAM-ICU. 30–40 min. \| Incidence of delirium on postoperative day 2 was 11.76% in the interventional group vs 23.53% in the control group. Postoperative day 3 was 8.83% vs 26.58% in interventional and control groups, respectively. \| The follow-up time was short because a longer follow-up duration might provide greater insights into the effects of the intervention. \| \| 5 \| Fahimi et al. (2020) Iran \| Cardiovascular ICU \| To determine the effect of multimedia education on postoperative delirium in patients undergoing coronary artery bypass graft (CABG) \| Parallel RCT. Multimedia CD with 3 educational videos of 4–6 min duration 5–7 days before surgery. Video 1: cardiologist information about the disease and procedure, Video 2: a nurse information about postoperative care, Video 3: a pre- and post-operative experience of a person who has already undergone CABG. Control group (n=55) Experimental group (n=55) \| CABG patients. First-time CABG, nondevelopment of cardiogenic shock, or myocardial rupture. CAM-ICU. 4–6 min; 5–7 days. \| Highest incidence of delirium at 7.3% in the intervention group on postoperative day 1% and 14.5% in the control group on day 2. No significant difference between the two groups on day 1. Higher incidence of delirium in the control group on the mornings of days 2, 3, and 4. \| Possibility of obtaining information from other sources from the patients. Differences in communication of the nursing care, influence of social support, family, previous levels of anxiety. \| \| 6 \| Faustino et al. (2022) Brazil \| ICU \| To evaluate the effectiveness of combined nonpharmacological interventions in preventing delirium in critically ill patients. \| Parallel RCT. Bundle of 5 combined nonpharmacological interventions: periodic orientation, cognitive stimulation, correction of sensory deficit, environmental management, and sleep promotion. Control group (n=72) Experimental group (n=72) \| ICU patients >18 years and >48 hrs in ICU. E-PRE-DELIRIC score ≥10. Richmond Agitation-Sedation Scale (RASS) and CAM-ICU. 7 months. \| Density of incidence of delirium was significantly lower in the intervention group (1.34×10-2 vs 2.29×10-2 person-days), with a 60% lower risk of developing delirium in the intervention group. Overall incidence of delirium was 22.2% vs 12.5% in the control and intervention groups, respectively. Total delirium cases were 16 in control group and 8 in intervention group. \| Only the research team monitored delirium, and underdiagnoses may have occurred due to the fluctuating nature of delirium. Low adherence to some protocol measures. Patients in coma were excluded who were at risk of delirium. Single-centre study therefore might not be generalised to multicentre due to small population. \| \| 7 \| Giraud et al. (2016) UK \| ICU \| To explore whether the use of an evidence-based mirrors intervention may be effective in reducing delirium and improving postoperative outcomes such as factual memory encoding of the ICU environment in older cardiac surgical patients. \| Pilot time-cluster RCT. Structure mirrors intervention at set times following changes in mental status including standard 23×41 cm unbreakable personal mirror and a standard 160×50 cm mobile posture mirror. Control group (n=108) Experimental group (n=115) \| ICU patients aged≥70 years, admitted after elective or emergency cardiac surgery over a 32-week period. CAM-ICU. 2-weeks. \| No significant difference between groups in delirium incidence, median days with delirium, or total ICU stay with delirium. \| Inability to rule out placebo and other effects to the intervention. No baseline cognitive testing was carried out. \| \| 8 \| Guo et al. (2016) China \| ICU \| To investigate the impact of multicomponent nonpharmacologic interventions (MNI) on perioperative cortisol, melatonin levels, and postoperative delirium in elderly oral cancer patients. \| RCT. MNI include psychological guidance with environment, time, place and character orientation, cognitive stimulation, effective communication, noise reduction, sleep promotion, and music listening. Control group (n=79) Experimental group (n=81) \| Surgical ICU Oral cancer patients age ≥65 years and ≤80 years; stay in ICU ≥3 days. RASS and CAM-ICU. Day before surgery and first 3 days after surgery. \| Higher incidence and longer duration of delirium in the usual care group day and first 3 postoperative days compared with the intervention group. \| No mention of formal ethical approval. There was no clear justification for the age range of participants included in the study. \| \| 9 \| Johnson et al. (2018) USA \| Trauma ICU \| To evaluate the effects of a music listening intervention in preventing delirium through decreasing physiologic variables, such as systolic blood pressure, heart rate, and respiratory rate among older patients. \| Feasibility RCT. Intervention include 60-minute prerecorded and self-selected music listening using iPod and headset, two times per day, 2 pm and 8 pm over a three-day period post admission. Control group (n=20) Experimental group (n=20) \| ICU patients ≥55 years. Oriented to person, time, and place on admission, negative CAM-ICU, ability to hear music played from an iPod, able to consent for the study. CAM-ICU. 60 minutes twice a day for three days. \| CAM-ICU for both intervention and usual care groups were negative. \| Exclusion of mechanically ventilated patients. The researcher was not blinded to the interventions and could generate potential observational bias. \| \| 10 \| Karadas and Ozdemir (2016) Turkey \| Medical ICU \| To determine the effect of range of motion exercises on preventing delirium and reducing the duration of delirium among patients in ICU aged ≥65 years \| RCT. Active, assisted-active or passive range of motion (ROM) exercise once a day until discharge. ROM performed for 4 extremities with 10 repetitions for about 30 min. Control group (n=47) Experimental group (n=47) \| ICU patients ≥65 years, no history of delirium, ICU stay ≥24 hrs, and voluntary participation. RASS and CAM-ICU. Daily until discharged. \| 8.5% of patients in the intervention group experienced delirium compared with 21.3% in the control group. Duration of delirium was shorter in the intervention group; 15 hours vs 38 hours in the control group. All patients in the intervention group experienced delirium at night compared with 70% of patients in the control group. \| Small study population; limited to patients aged ≥65 years only and nonmechanical ventilation. \| \| 11 \| Mailhot et al. (2017) Canada \| Surgical ICU \| To assess the feasibility, acceptability, and preliminary efficacy of a nursing intervention involving family caregivers (FC) in delirium management after cardiac surgery. \| RCT. Mentoring model between nurse and FC. Seven encounters over 3 days between nurse and FC, 30-minute visit at the patients’ bedside. FC used delirium management strategies; reorient the person with delirium. Control group (n=14) Experimental group (n=16) \| Surgical ICU patients with postsurgical delirium, undergoing either CABG or heart valve surgery, having an FC available to visit the patient at the bedside within 24 hrs of delirium onset then twice daily for 3 consecutive days. Intensive Care Delirium Screening Checklist (ICDSC) and CAM-ICU. Two weeks. \| Consent rate of 77% from the FC. Number of patients with clinical complications following delirium onset is similar in the two groups. A more favourable functional recovery in the intervention group. Anxiety scores on days 4, 15, and 30 were better in the intervention group. CAM-ICU scores were positive in 43.8% on day 2 for the intervention group vs 71.4% in the control group. Duration of delirium in the intervention group was shorter (mean days: 1.94 vs 4.14). \| Small sample size resulted in imbalance between group characteristics. Single-centre design limits generalisation. \| \| 12 \| McWilliams et al. (2023) United Kingdom \| ICU \| To conduct a feasibility trial of evening mobilisation to prevent and treat delirium in patients admitted to intensive care \| Mixed-method RCT. Intervention: a planned mobilisation session from day 1 of admission to a maximum of 7 consecutive days between 19:00 and 21:00 hours delivered by ICU physiotherapists and nurses Total patient participants (n=58) Usual care group (n=29) Intervention group (n=29) \| ICU patients with Richmond Agitation Sedation Scale (RASS) ≥−3 and expected to be admitted for ≥24 hrs. CAM-ICU. 7 days. \| Delirium incidence in the intervention group was 5/26 (19%; 95% CI: 6–39%) vs 8/28 (29%; 95% CI: 13–49%) in the control group. Mean delirium duration was two days in the intervention group vs 4.25 days in the control group. \| Short length of stay of the participants recruited. A number of the participants did not complete the maximum 7 days duration of the intervention. \| \| 13 \| Mitchell et al. (2017) Australia \| Medical and surgical ICU. \| To assess the feasibility of design, recruitment, and acceptability for family members and nurses of a family-delivered intervention to reduce delirium in ICU patients \| RCT. Intervention: family members providing orientation and memory clues daily, sensory checks, and cognitive stimulation. Total patient participants (n=90) Pre-randomisation (n=30) Control group (n=32) Experimental group (n=29) Family participants (n=61) Nurses (n=11) \| ICU patients ≥16 years, in ICU ≥4 days, able to be screened for delirium, and had a family member visit. ICU nurses who had provided direct care to at least one ICU patient who received at least an episode of the intervention. RASS and CAM-ICU. Once daily. \| Prevalence of delirium was 59% vs 56% in the intervention and control groups, respectively. For only active participants, delirium prevalence was 50% in intervention vs 54% in control group. The number of days of delirium was similar in both groups. \| Single-centre study that limits generalisability. Only patients expected to remain ≥4 days in ICU were recruited, delirium may occur early on in ICU. There was no regular control over how family members implemented the intervention. \| \| 14 \| Moon and Lee (2015) Republic of Korea \| Medical and surgical ICU \| To examine the effects of applying a tailored delirium preventive protocol to ICU patients by analysing its effects on delirium incidence, in-hospital mortality, ICU readmission, and length of ICU stay in a Korean hospital \| Single-blind RCT. Delirium prevention protocol consists of cognitive assessment and orientation, environmental intervention, and early therapeutic intervention. Applied during the first 7 days of ICU admission and for 10- to 20-minute duration. Control group (n=63) Experimental group (n=60) \| ICU patients age ≥18 years, ability to understand the study purpose, and provide consent independently or via a caregiver, ICU stay ≥48 hrs. CAM-ICU. Daily for 7 days. \| The control group had a higher incidence of delirium (n=21) compared with the intervention group (n=12). \| Insufficient delirium training for the nurses delivering the intervention. Patients were assessed only once daily and delirium fluctuates throughout the day. The severity and duration of delirium was not measured, which made comparison impossible. \| \| 15 \| Munro et al. (2017) USA \| ICU \| To explore the effects of an automated reorientation intervention on ICU delirium in a prospective RCT. \| Three-group, prospective RCT. Group 1: automated reorientation messages in a family member’s voice, Group 2: received the same message in an unfamiliar voice; Group 3: did not receive any automated reorientation messages. During predetermined daytime hours (9:00 am to 4:00 pm) over 3 days. Maximum of 24 recorded messages. Total patient participants (n=30; each group n=10) \| ICU patient ≥18 years, within 24 hrs of ICU admission. CAM-ICU. \| Delirium-free days were 1.9 in the family voice group, 1.6 in the unknown voice group, and 1.6 in the control group. Mean days of delirium were 0.3 in the family voice, 0.6 in the unknown voice, and 0.9 in the control group. \| Small sample size. All subjects did not receive the full intervention. \| \| 16 \| Ono et al. (2011) Japan \| ICU \| To verify the usefulness of bright light therapy (BLT) for patients following oesophagostomy to use the index acquired through physical activity, autonomic activity, incidence of postoperative arrhythmia, and level of acute delirium. \| RCT. Participants assigned to study group and control group. Study group received self-standing exposure device modified to have fluorescent light source. Two-hour of bright light exposure from day 2 from 07:30 am for a total of 4 days. Delirium evaluated twice a day, daytime and night time. Total of patient participants (n=26) \| Adult ICU patients undergoing surgical resection and reconstruction through a right thoracotomy for the treatment of thoracic oesophageal cancer. Patients who can communicate and consent and could undergo extubation the day after surgery. Japanese version of NEECHAM Confusion Scale (J-NCS) Scores. Two hours for four days. \| One out of 10 patients in the study group experienced delirium vs 5 out of 12 patients in the control group. \| Study was not double-blind. Small sample size. No mention of ethical approval. \| \| 17 \| Parry et al. (2014) Australia \| ICU \| To determine safety and feasibility of functional electrical stimulation (FES)-cycling and compare FES-cycling with case-matched controls in terms of functional recovery and delirium outcomes. \| Single-centre interventional observational study with case-matched control. FES-cycling within 96 hrs of ICU admission daily until discharge. FES-cycling was conducted for 20–60 min daily 5 times a week. Control group (n=8) Experimental group (n=8) \| ICU patients age ≥18 years. Diagnosis of sepsis or severe sepsis, mechanical ventilation for ≥48 hrs and remain in ICU for ≥4 days. CAM-ICU. \| Delirium incidence in the control group was 87% (n=7) vs 25% (n=2) in the intervention group. Duration of delirium 6.0 in the control group and 0.0 in the intervention group. \| Small sample size. Case-control design. Restrictive inclusion criteria. No mention of ethical approval. \| \| 18 \| Potharajaroen et al. (2018) Thailand \| Surgical ICU \| To examine the effects of BLT on the incidence of delirium in postoperative patients admitted to a surgical ICU. \| Single-blind RCT. The intervention group was treated with BLT consisting of 5000 lx for 2 hrs from 09:00 to 11:00, placed at a distance of 1.40 m from the patient’s face. Light brightness was checked at 09:00 and at 11:00 with digital illuminance – lux metre. Total participants who remained in the study (n=61) Control group (n=31) Experimental group (n=30) \| Post-surgery ICU patients, aged ≥50 years, understanding of Thai language. APACHE II Score ≥8 and arousal by voice. CAM-ICU. 3 days. \| Significant inverse association of BLT and incidence of delirium. Incidence of delirium: intervention group (2/31) vs control group (11/31). \| Not a double-blinded study. No mention of ethical approval. \| \| 19 \| Rice et al. (2017) USA \| Neurological ICU \| To assess the feasibility of enrolment within the 48-hour window when delirium risk is greatest, measuring cognitive function, delivering interventions 7 days per week and determining delirium incidence in stroke-related cognitive dysfunction. \| RCT. Two groups. Multicomponent intervention included all standardised stroke care, physical function and geriatric outcomes, plus trained nonmedical volunteers administered therapeutic activities twice daily for 7 days and 15 min. Control group (n=67) Experimental group (n=67) \| ICU patients admitted ≤48 hrs with ischaemic and haemorrhagic strokes and ≥50 years, understands English and without delirium on admission. CAM-ICU. \| The rate of delirium incident in the intervention group was 8% (10/125). \| The sample is not representative of those stroke patients with the highest risk for delirium. \| \| 20 \| Rood et al. (2021) The Netherlands \| Medical, surgical, and trauma ICU \| To determine the effects of a multicomponent nursing intervention program on delirium in ICU. \| Multicentre, stepped-wedge cluster-RCT. The UNDERPIN-ICU (Nursing Delirium Preventive Interventions in the Intensive Care Unit): optimising visual and hearing impairment, orientation loss, sleep deprivation, cognitive impairment, and immobility. Total patient participants (n=1749) Total number of ICUs (n=10) \| Medical, surgical and trauma ICU patients aged ≥18 years, high risk of developing delirium (E-PRE-DELIRIC score of ≥35%), free from delirium at admission. CAM-ICU. Two months. \| Delirium-free days in 28 days were 23 days in the intervention period and 23 days in the control period. Delirium incidence was 39% in the intervention period vs 40% in the control period. \| Only patients at high risk of delirium were included; the intervention may be effective in patients who already have delirium. The intervention fidelity was monitored using several proxy measurements instead of direct nursing registration. Overall duration of delirium was lower compared with other recent trials. The large number of interventions of the UNDERPIN-ICU program may have limited the effect determined. There were limited details of the risk factors. \| \| 21 \| Theresa, Fathima, Kayalvizhi, and Puliken (2022) India \| ICU \| To determine the effectiveness of a delirium care bundle on sedation and orientation among ICU-acquired delirium patients with mechanical ventilation. \| RCT. A self-structure scale developed by the investigator. The scale comprise of oriented, mild disorientation, moderate disorientation, and severe disorientation. ABCDEF delirium care bundle. Control group (n=28) Experimental group (n=28) \| ICU patients on mechanical ventilation and diagnosed with delirium. RASS. Daily for 7 days. \| Control group: Day 1: most of the participants (64.28%) were severely disorientated; Day 2: 60.71% were severely disorientated; Day 3: 50% were severely disoriented and 25% were moderately disoriented; Day 4: 42.85% were mild disoriented and 21.42% were oriented; Day 5: 46.42% mild and 28.57% oriented; Day 6: 42.85% mild and 32.14% oriented; Day 7: 42.85% mild and 35.71% oriented. Experimental group: Day 1: 35.71% mild and 39.28% moderate; Day 2: 39.28% mild; Day 3: 42.85% mild and 35.71% oriented; Day 4: 53.57% oriented and 35.71% mild; Day 5: 57.14% oriented and 39.28 mild; Day 6: 67.85% oriented and 25% mild; Day 7: 71.42% oriented and 21.42% mild. \| Non-controlled design. Only ventilated patients with delirium. Small sample size. \| \| 22 \| Van Rompaey et al. (2012) Belgium \| Cardiac-surgical, surgical, and medical ICU \| To determine if the use of earplugs during the night reduces the onset of delirium in the ICU and does the earplugs during the night improve the quality of sleep in the ICU. \| RCT. Patients divided into intervention and control group. Intervention group received earplugs at 22:00 hrs and removed at 06:00 hrs. Control group (n=67) Experimental group (n=69) \| ICU patients ≥18 years. ICU stay ≥24 hrs. Speaks Dutch or English and minimum GCS of 10. Neelon and Champagne Confusion Scale (NEECHAM). 5 nights. \| More cognitively normal patients were found in the intervention group. The control group scored 20% vs 19% in the intervention group. Mild confusion was 15% in the intervention group vs 40% in the control group. Total of 60% in the control group showed delirium and mild confusion vs 35% in the intervention group. \| Specific ICU population therefore the results may not be generalised. Focused only on first 24 hrs of ICU admission. \| \| 23 \| Topcu and Tosun (2022) Turkey \| Medical ICU \| To evaluate the effect of a protocol of nonpharmacological interventions to improve sleep quality in the ICU and the effects on noise levels and delirium rates. \| Pre-test post-test design with a control group. Two stages, in stage one, standard care. Stage two, sleep-promotion practices: light reduction, sound reduction, and disturbance reduction. Control group (n=37) Experimental group (n=37) \| ICU patients with ICU stay ≥24 hrs, GCS ≥11. RASS and CAM-ICU. 7 months. \| Frequency of delirium was 45% in the intervention group and 60% in the control group. \| Single-centre study limited to patients who can self-report, and ICU patient beds were together. Subjective evaluation of the sleep quality. No mention of ethical approval. \| \| 24 \| Patel et al. (2014) UK \| Medical and surgical ICU \| To investigate whether the implementation of a bundle of nonpharmacological interventions, consisting of environment noise and light reduction designed to reduce disturbing patients at night, was associated with improved sleep and a reduced incidence of delirium. \| Cohort study, before and after design. Multicomponent bundle consisted measures taken to reduce noise, light, and iatrogenic sleep disturbance and modify risk factors for delirium over 21-day period. Total number of patients before the bundle (n=167) Total number of patients after the bundle (n=171) \| ICU patients ≥18 years and ≥1 night in ICU. CAM-ICU. 21 days. \| Implementation of the intervention resulted in a reduction in the incidence of delirium (55/167, 33%) compared with before the intervention (24/171, 14%). Decrease in the length of delirium 3.4 days vs before 1.2 days. \| Single-centre design and nonrandomised cohorts. \| \| 25 \| Martinez, Donoso, Marquez, and Labarca (2017) Chile \| Medical and surgical ICU \| To assess the efficacy and describe the implementation strategy of a multicomponent intervention to prevent delirium in an ICU. \| Before and after cohort study. Components included early mobilisation, physical therapy, reorientation, cognitive stimulation, drug reviews, environmental stimulation, avoidance of sensory deprivation, pain control, avoidance of restraint use, and family participation. Delirium was assessed twice daily using CAM-ICU. Total number of patients (n=287) Diagnostic phase (n=60) Intervention phase (n=227) \| ICU patients aged ≥18 years. CAM-ICU. 15 months. \| Before implementation, delirium developed in 23 patients (38%) compared with 55 patients (76%) in the interventional phase. \| Lack of randomisation of the participants. \| \| 26 \| Kruger et al. (2018) South Africa \| Cardiothoracic ICU \| To assess the effect of nonpharmacological interventions on the severity and duration of hypoactive delirium and delirium in ICU patients after cardiothoracic surgery. \| Quasi-experimental nonequivalent control group design. Nonpharmacological interventions consist of visual and hearing aids, familiar objects from home, use of television/radio, nonverbal music, sleep hygiene, noise control, twice a day mobilisation, physical restraint reduction, sedation weaning, timely removal of intravenous lines. Control group (n=30) Experimental group (n=30) \| ICU patients aged ≥18 years. Admitted for cardiothoracic surgery, CABG, and valve replacement. Able to understand English. ICDSC. \| Delirium prevalence was similar in the control and intervention groups. The severity of delirium for both groups improved to 0. Duration of delirium was shorter in the intervention group. \| No blinding or participants. The knowledge of ICU nurses conducting delirium assessment was not tested. \| \| 27 \| Foster and Kelly (2013) USA \| Medical ICU \| To determine the feasibility of and test a multicomponent, nonpharmacologic, nurse-driven intervention for prevention of delirium. \| Prospective, cohort pilot study. Multicomponent protocol consists of sedation cessation, sleep–wake cycles, sensory stimulation, mobility, and music. Two-week education to ICU nurses on CAM-ICU, ICU patients with delirium established over 1-month period. Implementation followed for 2 months. Total number of patients (n=32) \| Medical ICU patients aged ≥18 years, haemodynamically stable and able to hear. CAM-ICU. \| At baseline data collection, 46/164 patients were positive for delirium. In the postintervention phase, 26/84 patients were positive for delirium and 57/84 were negative. \| No informed consent was obtained from patients. \| \| 28 \| Colombo et al. (2012) Italy \| Medical-surgical ICU \| To assess the occurrence of delirium, its risk factors and impact on critically ill outcome, and the efficacy of a reorientation protocol based on mnemonical and environmental stimulation. \| Two-stage prospective observational study. Phase one was standard delirium care, and in phase two, a reorientation strategy was introduced. Reorientation strategy consists of frequently calling patients their first names, giving information on the ward, and mnemonically stimulated remember relatives’ names. Observational phase (n=170) Interventional phase (n=144) \| ICU patients from 7 days to 24 hrs after ICU admission. CAM-ICU. Six months observational phase and six months interventional phase. \| Overall percentage of delirium was 25.5%, with a median onset of 2 days. \| Small population. No mention of ethics. The observational design. \| \| 29 \| Bryczkowski, Lopreiato, Yonclas, Sacca, and Mosenthal (2014) USA \| Surgical ICU \| To evaluate the efficacy of a delirium prevention program and determine whether it decreased the incidence and duration of hospital-acquired delirium in older adults aged ≥50 years admitted to the surgical ICU. \| Prospective pre- and post- cohort study. The intervention involved a pharmacologic protocol to limit the use of medications associated with delirium, decrease sedation, and encourage spontaneous breathing trial. Also a nonpharmacologic sleep enhancement and relaxation protocol and patient and family education. Total number of patients (n=123) Preintervention (n=57) Postintervention (n=66) \| Surgical ICU patients aged ≥50 years, in SICU for ≥24 hrs. CAM-ICU. Phase 1: one month Phase 2: four months Phase 3: nine months. \| Delirium-free days pre-intervention were 24/30 vs 27/30 post-intervention. \| A large number of patients were excluded due to inability to obtain delirium status owing to lack of training of some nurses working in the ICU temporarily. Only older adults included. No randomisation. \| \| 30 \| Balas et al. (2014) USA \| Medical and surgical ICU \| To evaluate the effectiveness and safety of implementing the awakening and breathing coordination, delirium monitoring/management, and early exercise/mobility (ABCDE) bundle into everyday practice. \| Prospective, cohort, before–after study. ABCDE bundle application consist awakening and breathing coordination, delirium monitoring/management, early exercise/mobility. Total number of patients (n=296) Pre-bundle (n=146) Post-bundle (n=150) \| Adult ICU patients aged ≥19 years. RASS and CAM-ICU. 18 months. \| Incidence of delirium was 62.3% pre-intervention and 48.7% post-intervention. Number delirium days decreased by 17% pre- and 50% post-intervention. \| Small sample size. Most of the intervention education occurred pre-intervention. \| \| 31 \| Rosa et al. (2019) Brazil \| Medical surgical ICU \| To evaluate the effect of an extended visitation model compared with a restricted model on the occurrence of delirium among ICU patients. \| Prospective single-centre before and after study. In restricted visitation model, 2 or less family visitors per patient were allowed to visit for up to 4.5 hrs/day over 3 periods. In extended visitation model, at most, 2 family visitors per patient were allowed to visit at daytime and evening and participate in bedside multidisciplinary rounds. Total number of patients (n=286) Restricted visitation model (n=141) Extended visitation model (n=145) \| ICU patients aged ≥18 years. ICU stay ≥24 hrs. CAM-ICU. 90 days. \| Delirium incidence in extended visiting model was 14 of 145 patients vs 29 of 141 patients in restricted visiting model. Median duration of delirium was 1.5 days in extended visiting model vs 3.0 days in restricted visiting model. \| The study design is susceptible to bias due to changes over time. The impact of extended visiting model to ICU staff was not evaluated. Risk factors of delirium was not controlled. \| \| 32 \| Liang (2022) China \| Mixed ICUs \| To identify current implementation of nonpharmacological interventions in ICUs of Mainland China. \| Qualitative design. Individual face-to-face discussion and semistructured, individual interviews. NVivo transcription and thematic analysis. Total number of nurses (n=20) \| Registered nurses who have worked over 3 years in the ICU. ICDSC and CAM-ICU. \| Themes: Lack of a delirium assessment practice, early mobilisation, poor sleep quality of ICU patients, limited duration of ICU visitation, structured sensory stimulation program. \| Small sample size limited to nurses with 3 years of ICU experience. \| \| 33 \| Menza (2022) USA \| Surgical and neurosurgical ICU \| To illuminate the ways in which a diverse group of adults use self-selected recorded music to recover after critical illness and describe patients’ perceptions of the effects of listening to self-selected music on symptom experience during mechanical ventilation after critical injury. \| Qualitative, grounded theory. Semistructured interviews. Total number of patient and family member participants (n=16) \| Current or recent surgical and trauma ICU patient or family aged ≥18 years. Receiving mechanical ventilation and listening to recorded music in the ICU. \| Six novel uses of personally selected music in ICU: Restoring consciousness, maintaining cognition, humanising the hospital experience, providing a source of connection, improving psychological well-being, and resolving the problems of silence. Also for pain and anxiety. \| All participants were male. Single-urban trauma centre. \| CAM-ICU, Confusion Assessment Method for the Intensive Care Unit; CD, Compact Disc; CI, Confidence Interval; GCS, Glasgow Coma Scale; SICU, Surgical Intensive Care Unit. 4.2.7. Data synthesis and analysis Results were synthesised and reported according to the concepts of nonpharmacological interventions aimed at delirium prevention or minimisation. The authors performed a descriptive analysis using the extracted data to index and summarise the included studies. Table 2 contains information related to the description of the included studies. Following this, the authors conducted a thematic analysis to discuss the categories of the key results of the review (Thomas & Harden, 2008). A thematic analysis was used to identify the key characteristics of nonpharmacological interventions utilised in delirium prevention and management to provide further clarification and insights into the interventions. The key results of the study focusing on the nonpharmacological interventions aimed at preventing or minimising the severity of delirium in adult ICU formed the basis of the categories. The authors followed the steps in the inductive thematic analysis described by Thomas and Harden to obtain data (Thomas & Harden, 2008). The analysis consisted of the following steps: (i) Line-by-line codes and related categories were extracted based on the description of the studies’ findings in relation to the overall study aim. (ii) Creating analytical themes from the studies (see Table 3). Table 3. Major themes and subthemes extracted from the studies. \| Study \| Empty Cell \| Alvarez et al. (2017) \| Damshens et al. (2018) \| Giraud et al. (2016) \| Johnson et al. (2018) \| Karadas and Ozdemir (2016) \| Ono et al. (2011) \| Parry et al. (2014) \| Potharajaroen et al. (2018) \| Van Rompaey et al. (2012) \| Menza (2022) \| Total \| --- --- --- --- --- --- \| \| Theme 1: Instrument-based therapeutic interventions \| Occupational therapy \| X \| \| \| \| X \| \| X \| \| \| \| 3 \| \| Empty Cell \| Music therapy \| \| X \| \| X \| \| \| \| \| \| X \| 3 \| \| Empty Cell \| Light therapy \| \| \| \| \| \| X \| \| X \| \| \| 2 \| \| Empty Cell \| Use of ear plugs \| \| \| \| \| \| \| \| \| X \| \| 1 \| \| Empty Cell \| Mirror therapy \| \| \| X \| \| \| \| \| \| \| \| 1 \| \| Study \| Empty Cell \| Contreras et al. (2021) \| Fahimi et al. (2020) \| McWilliams et al. (2023) \| Rood et al. (2021) \| Foster and Kelly (2013) \| Eghbali-Babadi et al. (2017) \| Mailhot et at. (2017) \| Mitchell et al. (2017) \| Munro et al. (2017) \| Rosa et al. (2017) \| Total \| --- --- --- --- --- --- \| \| Theme 2: Nurse-led interventions \| Multimedia education \| \| X \| \| \| \| \| \| \| \| \| 1 \| \| Empty Cell \| Mentorship/family support \| X \| \| \| \| \| \| \| \| \| \| 1 \| \| Empty Cell \| Cognitive stimulation \| X \| \| \| X \| \| \| \| \| \| \| 2 \| \| Empty Cell \| Orientation \| X \| \| \| X \| \| \| \| \| \| \| 2 \| \| Empty Cell \| Optimizing visual & hearing impairment \| \| \| \| X \| \| \| \| \| \| \| 1 \| \| Empty Cell \| Sleep promotion \| \| \| \| X \| X \| \| \| \| \| \| 2 \| \| Empty Cell \| Mobilisation \| \| \| X \| X \| X \| \| \| \| \| \| 3 \| \| Empty Cell \| Sedation cessation \| \| \| \| \| X \| \| \| \| \| \| 1 \| \| Theme 3: Family-led interventions \| Memory cues \| \| \| \| \| \| \| \| X \| \| \| 1 \| \| Empty Cell \| Cognitive stimulation \| \| \| \| \| \| \| \| X \| \| \| 1 \| \| Empty Cell \| Orientation \| \| \| \| \| \| \| X \| X \| \| \| 2 \| \| Empty Cell \| Sensory checks \| \| \| \| \| \| \| \| X \| \| \| 1 \| \| Empty Cell \| Automated reorientation \| \| \| \| \| \| \| \| \| X \| \| 1 \| \| Empty Cell \| Visitation \| \| \| \| \| \| X \| X \| \| \| X \| 3 \| \| Study \| Empty Cell \| Contreras et al. (2021) \| Faustino et al. (2022) \| Guo et al. (2016) \| Moon and Lee (2015) \| Rice et al. (2017) \| Theresa et al. (2022) \| Topcu and Tosun (2022) \| Patel et al. (2014) \| Total \| --- --- --- --- --- \| \| Theme 4: Multicomponent interventions \| Orientation Communication Environment \| X \| X X \| X X \| X X \| \| X \| \| \| 5 1 2 \| \| Empty Cell \| Cognitive stimulation \| X \| X \| X \| X \| \| \| \| \| 4 \| \| Empty Cell \| Sleep promotion Noise reduction Light reduction \| \| X \| X X \| \| \| \| X X X \| X X X \| 4 2 1 2 \| \| Empty Cell \| Early therapy Music therapy \| \| \| X \| X \| X \| \| \| \| 2 1 \| \| Empty Cell \| Optimise visual & hearing impairment Modify delirium risk factors \| \| X \| \| \| \| \| \| X \| 1 1 \| \| Empty Cell \| Avoid sensory deprivation Avoid the use of restraint \| \| \| \| \| \| \| \| \| \| \| Empty Cell \| Family participation Psychological guidance Education to patients & family \| X \| \| X \| \| \| \| \| \| 1 1 \| \| Empty Cell \| Drug reviews Sedation weaning Spontaneous breathing trial Timely removal of lines \| \| \| \| \| \| \| \| \| \| \| Study \| Empty Cell \| Martinez et al. (2017) \| Kruger et al. (2017) \| Foster and Kelly (2013) \| Colombo et al. (2012) \| Bryczkowski et al. (2014) \| Balas et al. (2014) \| Liang (2021) \| Rood et al. (2021) \| Total \| --- --- --- --- --- \| \| Theme 4: Multicomponent Interventions \| Orientation Communication Environment \| X X \| X \| \| X X X \| \| \| \| X \| 4 1 2 \| \| Empty Cell \| Cognitive stimulation \| X \| \| \| X \| \| \| \| X \| 3 \| \| Empty Cell \| Sleep promotion Noise reduction Light reduction \| \| X X \| X \| \| X \| \| X \| X \| 5 1 \| \| Empty Cell \| Early therapy Music therapy \| X \| X X \| X X \| \| \| X \| X \| X \| 6 2 \| \| Empty Cell \| Optimise visual & hearing impairment Modify delirium risk factors \| \| X \| \| \| \| X \| \| X \| 2 1 \| \| Empty Cell \| Avoid sensory deprivation Avoid the use of restraint \| X X \| X X \| X \| \| \| \| X \| \| 4 2 \| \| Empty Cell \| Family participation Psychological guidance Education to patients & family \| X \| \| \| \| X \| \| \| \| 2 \| Empty Cell Drug reviews Pain control Sedation weaning Spontaneous breathing trial Timely removal of lines X X X X X X X X X 2 1 3 2 1 5. Results 5.1. Overview of the review findings A total of 1079 papers were identified through database searches. Of these, the full text of 72 articles was assessed for eligibility in the review (see PRISMA-ScR diagram in Fig. 1). This review yielded 33 published studies dated between 2011 and 2023. Going by the continents, most publications originated from Asia (n=11), followed by North America (n=8), South America (n=5), Europe (n=6), Australia (n=2), and Africa (n=1). ICU was the setting of all included studies. 1. Download: Download high-res image (535KB) 2. Download: Download full-size image Fig. 1. PRISMA 2020 flow diagram. CINAHL = Cumulative Index of Nursing and Allied Health Literature; ICU = Intensive care unit; JBI = Joanna Briggs Institute; MEDLINE =Medical Literature Analysis and Retrieval System Online;PRISMA = Transparent Reporting of Systematic Reviews and Meta-analyses. A heterogeneous range of study designs was utilised, with the most common being quantitative studies with RCT (n = 22), cohort (n = 7), cross-sectional (n = 1), quasi-experimental (n = 1), and qualitative studies (n = 2). Sample sizes were variable, ranging from 16 to 1749. Most studies aimed to study the association of single-component nonpharmacological interventions with delirium (n = 20), whereas others aimed to study the association of multicomponent nonpharmacological interventions with delirium (n = 13). In the assessment of the methodological quality, it was found that the seven cohort studies were of high quality; it was unclear if confounding variables were identified in three of the studies (Balas et al., 2014, Kruger et al., 2018, Rosa et al., 2019), and if strategies to deal with the confounding factors were implemented in four of the studies (Patel, Baldwin, Bunting, & Laha, 2014). In all the cohort studies, it was unclear if there were strategies to address incomplete follow-up with participants, while in six of the studies, it needed to be clarified if follow-up was conducted. Most of the included studies were of high quality, and Supplementary Material JBI Critical Appraisal (Appendix 1) outlines the aspects that needed to be addressed or clarified in the studies. Some of the studies included in this review were of low quality, which compromises the reliability of the results. When considering the application of this evidence to practice, the findings should be considered with caution. 5.2. Main findings The findings of this review have identified that there are many nonpharmacological interventions being utilised for delirium prevention and management in adult ICUs. The first theme was instrument-based therapeutic interventions such as occupational, music, light, and mirror therapy. These interventions were indirectly delivered to patients by the staff by utilising these instruments. The second theme was nurse-led interventions directly delivered by nurses to the patients, consisting of cognitive stimulation, mobilisation, mentorship and family support, multimedia education, and orientation. The third theme was family-led interventions, which are delivered directly to the patients by their family members, and they consist of orientation, memory cues, sensory checks, visitation, cognitive stimulation, and an automated voice reorientation program. The fourth theme is multicomponent interventions, which combine various components of the three categories and bundle-based or protocol-based interventions. Whilst robust nonpharmacological delirium prevention and management protocols exist in non–adult ICU settings, the findings of this scoping review revealed inconsistencies in transferring and disseminating those interventions into practice. 5.2.1. Instrument-based therapeutic interventions Ten of the 32 studies reported using mirrors, earplugs, light devices, music, and occupational therapy involving early mobility as nonpharmacological interventions for delirium prevention and management. All of these studies compared two groups of patients to determine the effect of instrument-based therapeutic interventions on the outcomes of duration, density, incidence, and severity of delirium (Álvarez et al., 2017; Damshens, Sanie, Javadpour, Khaef, & Rastgarian, 2018; Giraud et al., 2016; Johnson et al., 2018; Karadas & Ozdemir, 2016;Menza, 2022; Ono, Taguchi, Kido, Fujino, & Doki, 2011; Parry et al., 2014; Potharajaroen et al. 2018; Van Rompaey, Elseviers, Van Drom, Fromont, & Jorens, 2012). Three studies applied intensive occupational therapy and reported a reduction in the duration, density, incidence, and severity of delirium in the experimental group compared with the control group (Álvarez et al., 2017; Mailhot et al., 2017; Potharajaroen et al., 2018). Three studies implemented music therapy; one of the three studies reported extensive uses of music therapy and its overall outcomes on delirium (Damshens et al., 2018; Johnson et al., 2018; Menza, 2022). However, music therapy was not applied extensively in the other two studies (Damshens et al., 2018; Johnson et al., 2018). Two of the studies applied music therapy to sedated and mechanically ventilated patients (Damshens et al., 2018; Menza, 2022), whereas one of the studies applied music therapy to nonventilated and nonsedated patients (Johnson et al., 2018). Four studies examined the effects of light therapy, earplugs, and mirrors on delirium (Giraud et al., 2016; Parry et al., 2014; Rice et al., 2017; Topcu & Tosun, 2022). Two of these studies found an association between light therapy and delirium, decreasing delirium (Parry et al., 2014; Rice et al., 2017). Earplugs were found to reduce the onset of delirium in one of the studies (Topcu & Tosun, 2022). Occupational, light, and music therapies are the most reported interventions in this category. 5.2.2. Nurse-led interventions Interventions led and delivered directly by nurses were reported in five of the studies (Contreras et al., 2021; Fahimi, Abbasi, Zahedi, Amanpour, & Ebrahimi, 2020; Foster & Kelly, 2013; McWilliams et al., 2023; Rood et al., 2021). Three of these studies were based on multicomponent nonpharmacological protocols (Contreras et al., 2021; Foster & Kelly, 2013; Rood et al., 2021; ), and two were single interventions based on multimedia education and mobilisation (Fahimi et al., 2020, McWilliams et al., 2023). The multimedia education played via videos to patients before cardiac surgery resulted in a reduction in the incidence of delirium on the second, third, and fourth days after surgery (Fahimi et al., 2020). The nurse-led interventions reported in the four studies consisted of cognitive stimulation, reorientation, optimising visual and hearing impairment, sleep promotion, early mobilisation, and family mentorship to provide support. The effects of these interventions were reported in four of the studies (Contreras et al., 2021; Foster & Kelly, 2013; McWilliams et al., 2023; Rood et al., 2021). The interventions reported to be most implemented by nurses are cognitive stimulation, reorientation, sleep promotion, and early mobilisation. 5.2.3. Family-led interventions Five studies reported integrating family members into the care of their loved ones to prevent and manage delirium (Eghbali-Babadi, Shokrollahi, & Mehrabi, 2017; Mailhot et al., 2017; Mitchell et al., 2017; Munro et al. 2017; Rosa et al., 2019). The family-led interventions consist of providing memory cues, orientation, cognitive stimulation, sensory checks, extended visitation, and the use of automated reorientation in the family members’ voices to provide ongoing information and reassurance to critically ill patients. The studies reported the effects of the interventions on the incidence and duration of delirium (Eghbali-Babadi et al., 2017; Mailhot et al., 2017; Mitchell et al., 2017; Munro et al., 2017; Rosa et al., 2019), One of the studies reported 77% (n = 30) participation rate of family members, which encouraged family-led interventions (Mailhot et al., 2017). Visitation was the most utilised intervention reported in three studies that enabled the delivery of the interventions and resulted in better patient outcomes. Orientation is the more implemented intervention in the two studies. 5.2.4. Multicomponent interventions Sixteen of the 32 studies reported using multicomponent delirium prevention and management interventions. These consist of programs combining various nonpharmacological interventions to deliver a care bundle. The UNDERPIN-ICU (Nursing Delirium Preventive Interventions in the Intensive Care Unit) is reported in one study involving the largest cohort of patients (Rood et al., 2021). In two studies, the multicomponent interventions were also nurse-led (Contreras et al., 2021; Rood et al., 2021). Environmental orientation and communication were reported as the highest occurring interventions, with 15 studies reporting the intervention effects on delirium incidence, duration, and severity. Interventions that promote sleep, such as light and noise reduction, were also reported as the highest utilised interventions within the multicomponent programs in 15 of the studies. Early therapy and music were reported as part of a multicomponent intervention in 11 studies. Nine studies reported drug reviews, spontaneous breathing trials, and timely removal of invasive devices. Seven studies also reported cognitive stimulation as one of the interventions, and six studies reported the avoidance of restraint and sensory deprivation. Four studies reported family participation as part of a multicomponent intervention, but family members were not directly delivering the interventions. Multicomponent intervention is the most common intervention for delirium prevention and management out of the four categories of interventions. However, one qualitative study reported some barriers to these interventions, including poor delirium assessment and limited duration of family visitation, which could impact the efficacy of these interventions. 5.3. Outcomes Quantitative delirium outcomes were identified in 31 of the studies and encompassed density (n = 2), incidence (n = 24), severity (n = 2), duration (n = 13), risk (n = 2), acceptability of delirium intervention (n = 2), prevalence (n = 2), and frequency (n = 3). One of the qualitative studies reported six themes of nonpharmacological interventions for delirium (Liang, Chau, Lo, Zhao, & Liu, 2022), and one qualitative study reported six novel uses of personally selected music as a therapeutic intervention for delirium (Menza, 2022). 6. Discussion This scoping review identified the range of evidence-based nonpharmacological interventions used for delirium prevention and management. It highlighted different preventative and management strategies utilised in different countries and regions. This lack of consistency in nonpharmacological interventions is reflected in the paucity of standardised policies and protocols that guide practice (Collet et al., 2019; Ewens et al., 2021; Zamoscik et al., 2017). The revised Delirium Clinical Care Standard, quality statement 2 of the Australian Commission on Safety and Quality in Health Care, provides recommendations for clinicians regarding the interventions to prevent delirium (Australian Commission on Safety and Quality in Health Care, 2021). The standard recommends multicomponent delirium prevention interventions focusing on risk factor management and environmental optimisation. This review found that multicomponent interventions were predominately utilised across the 33 included studies (n = 16). In the Acevedo-Nuevo, González-Gil, Romera-Ortega, Latorre-Marco, and Rodríguez-Huerta (2018) case report, a multimodal approach was utilised to manage delirium in a critically ill patient. The approach involved nursing interventions and nonpharmacological input from multidisciplinary teams, which was effective. A multifactorial, multicomponent approach to delirium prevention and management may benefit ICU patients and clinicians significantly. The evidence supporting the use of multicomponent nonpharmacological interventions for delirium prevention and management is more substantial than a single component (Burton et al., 2021). However, the impact of single-component interventions may be understood if the interventions are standardised and utilised over an extended period. There is a paucity of literature to support the use of single-component nonpharmacological interventions for delirium prevention for an extended period, but the evidence identified suggests that such interventions could be highly effective (McKenzie & Joy, 2020). The integration of family carers of ICU patients as a single-component intervention for delirium management was explored in four studies reported in this review. Mailhot et al. (2017) developed a family carer–nurse intervention that comprised an ICU nurse fostering the efficiency of a family carer to behave in a supportive way towards their loved ones during delirium symptoms, demonstrating improved psychological recovery scores in the intervention group. The intervention was feasible to implement in clinical practice and well received by ICU staff, patients, and families. However, the benefits and impacts of family involvement in delirium management have not been extensively explored in ICU. Studies that include family involvement alongside other nonpharmacological interventions have demonstrated beneficial outcomes for ICU patients and also gained acceptance by the ICU staff and families (Digby et al., 2022; Khan, Digby, Giordano, Hade, & Bucknall, 2022; Liang et al., 2022; McKenzie & Joy, 2020). McKenzie and Joy suggest from their systematic review that family intervention in delirium reduced hospital LoS but were uncertain if it affected the duration of delirium (McKenzie & Joy, 2020). A reduction in delirium has also been observed through the delivery of automated family-recorded voice reorientation messages, showing more delirium-free days in the patients who received the intervention than those who did not (Munro et al., 2017). A significant positive effect of integrating familiar voices into the orientation of critically ill patients to prevent delirium has been demonstrated to be effective in other studies not captured in this review (Nielsen et al., 2020). It is shown that interventions designed around family integration are cost-effective and feasible to implement in delirium care, but it has not been standardised as a nonpharmacological intervention (Nielsen et al., 2020). The evidence by Bannon et al. (2018) suggests that this reluctance to implement family interventions may be due to barriers, including safety concerns, family members’ anxiety, patients’ confidentiality, lack of awareness, and inflexible visiting. These barriers may be managed through staff and relatives’ education about the potential benefits of family involvement in delirium intervention and designing a nurse–family carer intervention, which has been recommended (Ewens et al., 2021, Mailhot et al., 2017). This approach may facilitate understanding and promote family confidence in being included in delirium care. Mailhot et al. (2017) reported no significance in family members’ anxiety in delirium interventions. Interventions involving family presence and participation are nonpharmacological interventions found to have beneficial effects on delirium and are perceived as feasible and acceptable by family members, ICU staff, and ICU survivors (Bannon et al., 2018; Elcokany & Ahmed, 2019; Ewens et al., 2021). Nursing intervention may also have equivalent beneficial effects to family intervention, but this is yet to be explored. This review identified that nursing interventions were used as a nonpharmacological delirium intervention but often integrated as a multicomponent strategy (Acevedo-Nuevo et al., 2018; Collet et al., 2019). Nurses remain the key ICU clinicians who can implement delirium care strategies in the ICU and reported that nonpharmacological interventions were fundamental and natural to nursing care (Collet et al., 2019). However, many factors may influence their decision to implement nonpharmacological interventions. A lack of education and standardisation of nonpharmacological interventions may hinder nurses’ autonomy to initiate and integrate these practices into delirium care (Bannon et al., 2018; Ewens et al., 2021; Zamoscik et al., 2017). Evidence supports environmental optimisation as a nonpharmacological intervention for delirium prevention and management. The environmental interventions evident in the literature include sleep promotion, single room, eyeglasses, noise minimisation, comfort promotion, orientation to the window, clock and television, good sight and hearing, and minimal overnight intervention. Also, nurses are more likely to carry out continuity and routines of care as they serve as the cornerstone of critical care, and their skills are most apt for delirium prevention and management (Zamoscik et al., 2017). Nurses are more likely to influence change in inpatient care due to the depth of engagement and familiarity they establish through a therapeutic relationship with patients and their families (Zamoscik et al., 2017). However, nurses have reported anxiety in leading interventions that involve family participation and nonpharmacological strategies due to their unpreparedness, increased workload, and insufficient resources to implement them (Bannon et al., 2018; Ewens et al., 2021). These anxieties can, however, be mitigated through increased utilisation of nonpharmacological delirium care and the development of nurse-led protocols to guide the clinical decision-making process (McKenzie & Joy, 2020). Forsgren and Eriksson reported that 9% of existing nonpharmacological guidelines for delirium are compared with 26% of pharmacological guidelines (Forsgren & Eriksson, 2010). The limited availability of nonpharmacological guidelines infers that nurses rely on medical practitioners to initiate delirium interventions, especially in urgent circumstances where the nurse does not hold drug prescription authority. Intensive care diaries constructed by family members and ICU staff have been identified in the literature as a memory recall approach to help ICU patients remember events that occurred when they were critically unwell (Nielsen et al., 2020). A total lack of recall of events or delirious recall increases the prevalence of Post-traumatic Stress Disorder (PTSD); thus, family interventions may go beyond reducing delirium to preventing PTSD during recovery. This review found that music therapy, mechanical restraints, and verbal restraints are nonpharmacological interventions utilised in delirium care. Restraints in critical care vary worldwide and depend on the region’s culture and background (Kısacık, Sönmez, & Coşğun, 2020). The knowledge of ICU staff regarding the application of physical restraint is poor (Kısacık et al., 2020). There is a paucity of evidence to substantiate the beneficial effects of this intervention on delirium care. However, Johnson et al. (2016) reported that nurses showed a positive attitude towards physical restraints when protecting patients from falling out of bed or chair. These reasons suggest that restraints may be utilised due to a lack of options available or where other options have been exhausted. Physical restraints incorporated with verbal reassurance and reorientation may help avoid using pharmaceutical products to manage delirium and prevent feelings of an ethical dilemma. However, no evidence supported this combination (Kısacık et al., 2020). Although evidence has been found to show the use of mechanical and verbal restraint for nonpharmacological intervention for delirium, several studies have shown insufficient knowledge of ICU staff in the use of restraints resulting in deferring attitudes and unsafe practices when they use it (Stinson, 2016, Suliman et al., 2017). Physical restraints without formal assessment and diagnosis of delirium have also been reported in the literature (Ankravs, Udy, & Byrne, 2023). Music therapy was shown as one of the common instrument-based therapeutic interventions to prevent and manage delirium (Damshens et al., 2018; Johnson et al., 2018; Menza, 2022). Nature-based sounds significantly reduce anxiety and agitation in critically ill patients and mitigate the physiological variables (Froutan et al., 2020). Instrument-based interventions may be beneficial as a single-component intervention or incorporated with other nonpharmacological delirium interventions. Its benefits stem from the evidence that supports its positive effect on psychological and physiological outcomes for patients (Froutan et al., 2020; Johnson et al., 2018). At least 85% of ICUs utilise nonpharmacological interventions without accompanying guidelines (Forsgren & Eriksson, 2010). More recent studies have reported a high appreciation and acceptability of nonpharmacological interventions for delirium accompanied by a decreased incidence of delirium (Bannon et al., 2018; Collet et al., 2019; Zamoscik et al., 2017). The main problem identified with nonpharmacological delirium interventions is the lack of standardisation of the practice. Collating all the information gathered and designing a protocol focusing on nonpharmacological interventions may increase its effectiveness in preventing and managing delirium. This is worth exploring in future research and clinical practice. An evidence-based nonpharmacological protocol may boost the confidence of nurses and family carers in implementing these strategies to practice. However, it appears uncertain if nurses and ICU staff will utilise a developed protocol. A further area of exploration would be to understand the factors that influence the ICU staff’s choices of nonpharmacological interventions for delirium. An evidence-based protocol may help guide safe delirium care practice and evaluate the effects, resulting in better quality care and overall outcomes for critically ill patients. Evidence-based protocols may vary depending on the ICU culture of the region where it exists. However, places may adopt a standard protocol and make changes to it based on the individual needs of their patient population. A review of resources currently available to ICU staff across different countries and an evaluation of the impact of such resources and their utilisation are also an area of future investigation. 7. Limitations The limitations of this review included that the literature was limited to the peer-reviewed primary research literature published in English, and as such, the relevant literature in other languages or unpublished studies may have been overlooked. The review excluded other reviews and case reports, and these may have papers reported in them or their reference lists, which may have been overlooked. 8. Conclusion This scoping review revealed the heterogeneous and complex nature of current nonpharmacological interventions used in the prevention and management of delirium in adult ICU settings. The variety of interventions may lead to challenges in the evaluation, dissemination, and application of these interventions to guide nonpharmacological management in the prevention and management of delirium. The decision-making process in choosing interventions used for delirium is not always based on the best available evidence, leading to inconsistencies in practice within individual ICUs. There is a significant range of nonpharmacological interventions, and an extensive exploration of each of these categories of interventions may lead to increased utilisation and implementation within ICU clinical practice, along with an increased body of evidence. Increased ICU utilisation of evidence-based nonpharmacological interventions may eliminate pharmacological therapies and foster person-centred delirium care, optimising overall outcomes for ICU patients and family members (Collet et al., 2019; Ewens et al., 2021; McKenzie & Joy, 2020). The development and implementation of nonpharmacological interventions and practices that are flexible but based on best evidence would reduce the risk of inconsistent application of nonpharmacological interventions that are currently evident in practice. The development and implementation of flexible, simplified, and evidence-based nonpharmacological interventions that encourage standardised delirium care practices and the integration of family members and nurses into standardised delirium care practices are recommended. This recommendation could reduce reliance on pharmacological interventions and encourage further exploration of the effects of nonpharmacological delirium interventions using methods such as systematic reviews and meta-analyses (Chen et al., 2022; McKenzie & Joy, 2020). Standardised evidence-based guidelines addressing all aspects of single-component or multicomponent nonpharmacological delirium interventions would support ICU staff in utilising these interventions and further enhance family member education and support. Person-centred care involving family integration as an integral component is an evolving practice in ICUs. Further exploration of the impact of delirium on a critically ill adult patient is yet to be conducted. Hence, as this review identified family-delivered interventions as a category of nonpharmacological interventions for delirium in adult ICU, this is an area that should be explored further. In turn, the authors are conducting a study to develop, implement, and evaluate a digital family-led intervention to prevent and manage adult ICU delirium. Authorship contribution statement Gideon U. Johnson: Conceptualization, Methodology, Data curation, Formal analysis, Writing – original draft, Writing – review & editing, Project administration. Amanda Towell-Barnard: Conceptualization, Methodology, Data curation, Formal analysis, Supervision, Writing – review & editing. Christopher McLean: Methodology, Formal analysis, Supervision, Writing – review & editing. Beverley Ewens: Conceptualization, Methodology, Data curation, Formal analysis, Supervision, Resources, Writing – review & editing. Funding No financial support was provided for the work reported in this manuscript. Ethical Statement The research described in the submitted paper relied on previously published manuscripts and thus does not require ethical approval. Conflict of interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this manuscript. Appendix A. Supplementary material Download: Download Word document (77KB) Supplementary material . Recommended articles Data Availability The authors declare no research data associated with the work in this manuscript. References Acevedo-Nuevo et al., 2018M. Acevedo-Nuevo, M. González-Gil, M. Romera-Ortega, I. Latorre-Marco, M. Rodríguez-Huerta The early diagnosis and management of mixed delirium in a patient placed on ECMO and with difficult sedation: a case report Intensive & Critical Care Nursing, 44 (2018), pp. 110-114, 10.1016/j.iccn.2017.07.013 PMID: 28869145 View PDFView articleView in ScopusGoogle Scholar Aliberti et al., 2015S. Aliberti, M. Belotti, G. Messinesi, A. Pesci, A. Morandi, G. Bellelli, et al. Delirium symptoms during hospitalization predict long-term mortality in patients with severe pneumonia Aging Clinical and Experimental Research, 27 (2015), pp. 523-531, 10.1007/s40520-014-0297-9 View in ScopusGoogle Scholar Álvarez et al., 2017E.A. Álvarez, M.A. Garrido, E.A. Tobar, S.A. Prieto, S.O. Vergara, C.D. Briceño, et al. Occupational therapy for delirium management in elderly patients without mechanical ventilation in an intensive care unit: a pilot randomized clinical trial Journal of Critical Care, 40 (2017), Article 265, 10.1016/j.jcrc.2017.03.016 View PDFView articleView in ScopusGoogle Scholar American Psychiatric Association [APA], 2013American Psychiatric Association [APA] Diagnostic and statistical manual of mental disorders (5th edn), American Psychiatric Publishing, Arlington, VA (2013) Google Scholar Ankravs et al., 2023M.J. Ankravs, A.A. Udy, K. Byrne, S. Knowles, N. Hammond, M.K. Saxena, et al. A multicentre point prevalence study of delirium assessment and management in patients admitted to Australian and New Zealand intensive care units Critical Care and Resuscitation, 22 (2023), pp. 355-360, 10.51893/2020.4.OA8 Google Scholar Australian Commission on Safety and Quality in Health Care, 2021Australian Commission on Safety and Quality in Health Care (2021). Delirium Clinical Care Standard. ACSQHC; Sydney. Retrieved from: 〈 Google Scholar Balas et al., 2014M.C. Balas, E.E. Vasilevskis, K.M. Olsen, K.K. Schmid, V. Shostrom, M.Z. Cohen, et al. Effectiveness and safety of the awakening and breathing coordination, delirium monitoring/management, and early exercise/mobility bundle Critical Care Medicine, 42 (2014), pp. 1024-1036, 10.1097/CCM.0000000000000129 View in ScopusGoogle Scholar Bannon et al., 2018L. Bannon, J. McGaughey, M. Clarke, D.F. McAuley, B. Blackwood Designing a nurse-delivered delirium bundle: what intensive care unit staff, survivors, and their families think? Australian Critical Care, 31 (2018), pp. 174-179, 10.1016/j.aucc.2018.02.007 View PDFView articleView in ScopusGoogle Scholar Bannon et al., 2019L. Bannon, J. McGaughey, R. Verghis, M. Clarke, D.F. McAuley, B. Blackwood The effectiveness of non-pharmacological interventions in reducing the incidence and duration of delirium in critically ill patients: a systematic review and meta-analysis Intensive Care Medicine, 45 (2019), pp. 1-12, 10.1007/s00134-018-5452-x Google Scholar Bryczkowski et al., 2014S.B. Bryczkowski, M.C. Lopreiato, P.P. Yonclas, J.J. Sacca, A.C. Mosenthal Delirium prevention program in the surgical intensive care unit improved the outcomes of older adults The Journal of Surgical Research, 190 (2014), pp. 280-288 View PDFView articleView in ScopusGoogle Scholar Burry et al., 2021L.D. Burry, W. Cheng, D.R. Williamson, N.K. Adhikari, I. Egerod, S. Kanji, et al. Pharmacological and non-pharmacological interventions to prevent delirium in critically ill patients: a systematic review and network meta-analysis Intensive Care Medicine, 47 (2021), pp. 943-960, 10.1007/s00134-021-06490-3 View in ScopusGoogle Scholar Burton et al., 2021J.K. Burton, L.E. Craig, S.Q. Yong, N. Siddiqi, E.A. Teale, R. Woodhouse, et al. Non-pharmacological interventions for preventing delirium in hospitalised non-ICU patients The Cochrane Database of Systematic Reviews, 7 (2021), Article CD013307, 10.1002/14651858.CD013307.pub2 View in ScopusGoogle Scholar Chen et al., 2022T.J. Chen, V. Traynor, A.Y. Wang, C.Y. Shih, M.C. Tu, C.H. Chuang, et al. Comparative effectiveness of non-pharmacological interventions for preventing delirium in critically ill adults: a systematic review and network meta-analysis International Journal of Nursing Studies, 131 (2022), Article 104239, 10.1016/j.ijnurstu.2022.104239 View PDFView articleView in ScopusGoogle Scholar Collet et al., 2019M.O. Collet, T. Thomsen, I. Egerod Nurses' and physicians' approaches to delirium management in the intensive care unit: a focus group investigation Australian Critical Care, 32 (2019), pp. 299-305, 10.1016/j.aucc.2018.07.001 View PDFView articleView in ScopusGoogle Scholar Colombo et al., 2012R. Colombo, A. Corona, F. Praga, C. Minari, C. Giannotti, A. Castelli, et al. A reorientation strategy for reducing delirium in the critically ill. Results of an interventional study Minerva Anestesiologica, 78 (2012), pp. 1026-1033 View in ScopusGoogle Scholar Contreras et al., 2021C.C.T. Contreras, A.N.P. Esteban, M.D. Parra, M.K.R. Romero, C.G.D. Silva, N.P.D. Buitrago Multicomponent nursing program to prevent delirium in critically ill patients: a randomized clinical trial Revista Gaucha de Enfermagem, 42 (2021), Article e20200278, 10.1590/1983-1447.2021.20200278 View in ScopusGoogle Scholar Damshens et al., 2018M. Damshens, M.S. Sanie, S. Javadpour, M. Khaef, A. Rastgarian The role of music on the delirium in traumatic patients: a case study in the ICU of Peymanieh Hospital of Jahrom, Fars Province, Iran Ambient Science, 5 (2018), pp. 1-5, 10.21276/ambi.2018.05.sp1.ra11 Google Scholar Deng et al., 2020L.X. Deng, L. Cao, L.N. Zhang, X.B. Peng, L. Zhang Non-pharmacological interventions to reduce the incidence and duration of delirium in critically ill patients: a systematic review and network meta-analysis Journal of Critical Care, 60 (2020), pp. 241-248, 10.1016/j.jcrc.2020.08.019 View PDFView articleView in ScopusGoogle Scholar Digby et al., 2022R. Digby, E. Manias, K. Haines, J. Orosz, J. Ihle, T. Bucknall Family experiences and perceptions of intensive care unit care and communication during the COVID-19 pandemic Australian Critical Care, 36 (2022), pp. 350-360, 10.1016/j.aucc.2022.03.003 Google Scholar Eghbali-Babadi et al., 2017M. Eghbali-Babadi, N. Shokrollahi, T. Mehrabi Effect of family-patient communication on the incidence of delirium in hospitalized patients in cardiovascular surgery ICU Iranian Journal of Nursing and Midwifery Research, 22 (2017), pp. 327-331, 10.4103/1735-9066.212985 View in ScopusGoogle Scholar Elcokany and Ahmed, 2019N.M. Elcokany, F. Ahmed Effect of family reorientation messages on delirium prevention among critically ill patients Journal of Nursing Education and Practice, 9 (2019), pp. 91-99 Google Scholar Ewens et al., 2021B. Ewens, D. Collyer, V. Kemp, D. Arabiat The enablers and barriers to children visiting their ill parent/carer in intensive care units: a scoping review Australian Critical Care, 34 (2021), pp. 604-619, 10.1016/j.aucc.2020.12.009 View PDFView articleView in ScopusGoogle Scholar Fahimi et al., 2020K. Fahimi, A. Abbasi, M. Zahedi, F. Amanpour, H. Ebrahimi The effects of multimedia education on postoperative delirium in patients undergoing coronary artery bypass graft: a randomized clinical trial Nursing in Critical Care, 25 (2020), pp. 346-352, 10.1111/nicc.12473 View in ScopusGoogle Scholar Faustino et al., 2022T.N. Faustino, N.A. Suzart, R.N.D.S. Rabelo, J.L. Santos, G.S. Batista, Y.S.D. Freitas, et al. Effectiveness of combined non-pharmacological interventions in the prevention of delirium in critically ill patients: a randomized clinical trial Journal of Critical Care, 68 (2022), pp. 114-120, 10.1016/j.jcrc.2021.12.015 View PDFView articleView in ScopusGoogle Scholar Fiest et al., 2021K.M. Fiest, A. Soo, C. Hee Lee, D.J. Niven, E.W. Ely, C.J. Doig, et al. Long-term outcomes in ICU patients with delirium: a population-based cohort study American Journal of Respiratory and Critical Care Medicine, 204 (2021), pp. 412-420 CrossrefView in ScopusGoogle Scholar Foroughan et al., 2016M. Foroughan, A. Delbari, S.E. Said, A. AbariKamrani, V. Rashedi, T. Zandi Risk factors and clinical aspects of delirium in elderly hospitalized patients in Iran Aging Clinical and Experimental Research, 28 (2016), pp. 313-319, 10.1007/s40520-015-0400-x View in ScopusGoogle Scholar Forsgren and Eriksson, 2010L.M. Forsgren, M. Eriksson Delirium — awareness, observation and interventions in intensive care units: a national survey of Swedish ICU head nurses Intensive & Critical Care Nursing, 26 (2010), pp. 296-303, 10.1016/j.iccn.2010.07.003 View PDFView articleView in ScopusGoogle Scholar Foster and Kelly, 2013J. Foster, M. Kelly A pilot study to test the feasibility of a nonpharmacologic intervention for the prevention of delirium in the medical intensive care unit Clinical Nurse Specialist CNS, 27 (2013), pp. 231-238, 10.1097/NUR.0b013e3182a0b9f9 View in ScopusGoogle Scholar Froutan et al., 2020R. Froutan, M. Eghbali, S.H. Hoseini, S.R. Mazloom, M.S. Yekaninejad, R. Boostani The effect of music therapy on physiological parameters of patients with traumatic brain injury: a triple-blind randomized controlled clinical trial Complement Therapies in Clinical Practice, 40 (2020), Article 101216, 10.1016/j.ctcp.2020.101216 View PDFView articleView in ScopusGoogle Scholar Giraud et al., 2016K. Giraud, M. Pontin, L.D. Sharples, P. Fletcher, T. Dalgleish, A. Eden, et al. Use of a structured mirrors intervention does not reduce delirium incidence but may improve factual memory encoding in cardiac surgical ICU patients aged over 70 years: a pilot time-cluster randomized controlled trial Frontiers in Aging Neuroscience, 8 (2016), Article 228, 10.3389/fnagi.2016.00228 View in ScopusGoogle Scholar Goldberg et al., 2020T.E. Goldberg, C. Chen, Y. Wang, E. Jung, A. Swanson, C. Ing, et al. Association of delirium with long-term cognitive decline: a meta-analysis JAMA Neurology, 77 (2020), pp. 1373-1381, 10.1001/jamaneurol.2020.2273 View in ScopusGoogle Scholar Guo et al., 2016Y. Guo, L. Sun, L. Li, P. Jia, J. Zhang, H. Jiang, et al. Impact of multicomponent, nonpharmacologic interventions on perioperative cortisol and melatonin levels and postoperative delirium in elderly oral cancer patients Archives of Gerontology and Geriatrics, 62 (2016), pp. 112-117, 10.1016/j.archger.2015.10.009 View PDFView articleGoogle Scholar Inouye et al., 2014S.K. Inouye, R.G. Westendorp, J.S. Saczynski Delirium in elderly people Lancet, 383 (2014), pp. 911-922, 10.1016/S0140-6736(13)60688-1 View PDFView articleView in ScopusGoogle Scholar Johnson et al., 2016K. Johnson, V. Curry, A. Steubing, S. Diana, A. McCray, A. McFarren, et al. A non-pharmacologic approach to decrease restraint use Intensive & Critical Care Nursing, 34 (2016), pp. 12-19, 10.1016/j.iccn.2015.08.004 Google Scholar Johnson et al., 2018K. Johnson, J. Fleury, D. McClain Music intervention to prevent delirium among older patients admitted to a trauma intensive care unit and a trauma orthopaedic unit Intensive & Critical Care Nursing, 47 (2018), pp. 7-14, 10.1016/j.iccn.2018.03.007 View PDFView articleView in ScopusGoogle Scholar Kang et al., 2018J. Kang, M. Lee, H. Ko, S. Kim, S. Yun, Y. Jeong, et al. Effect of nonpharmacological interventions for the prevention of delirium in the intensive care unit: a systematic review and meta-analysis Journal of Critical Care, 48 (2018), pp. 372-384, 10.1016/j.jcrc.2018.09.032 View PDFView articleView in ScopusGoogle Scholar Karadas and Ozdemir, 2016C. Karadas, L. Ozdemir The effect of range of motion exercises on delirium prevention among patients aged 65 and over in intensive care units Geriatric Nursing, 37 (2016), pp. 180-185, 10.1016/j.gerinurse.2015.12.003 View PDFView articleView in ScopusGoogle Scholar Khan et al, 2021S. Khan, R. Digby, N.A. Giordano, S. Hade, T.K. Bucknall A 6-year retrospective cohort study of family satisfaction with critical care and decision-making in an Australian intensive care unit Australian Critical Care, 35 (2022), pp. 264-272, 10.1016/j.aucc.2021.05.009 View PDFView articleView in ScopusGoogle Scholar Kim et al., 2020Y. Kim, Y. Jin, T. Jin, L. Sun-Mi Risk factors and outcomes of sepsis-associated delirium in intensive care unit patients: a secondary data analysis Intensive & Critical Care Nursing, 59 (2020), Article 102844, 10.1016/j.iccn.2020.102844 View PDFView articleView in ScopusGoogle Scholar Kısacık et al., 2020Ö.G. Kısacık, M. Sönmez, T. Coşğun Use of physical restraints in critical care units: nurses' knowledge, attitudes, and practices Critical Care Nurse, 40 (2020), pp. 37-47, 10.4037/ccn2020856 Google Scholar Kruger et al., 2018A. Kruger, I.M. Coetzee, Z. White The effect of non-pharmacological interventions on the severity and duration of hypoactive delirium and delirium in postoperative cardiothoracic surgery patients Southern African Journal of Critical Care, 34 (2018), Article 30 [Link to Abstract] View in ScopusGoogle Scholar Lange et al., 2019P.W. Lange, M. Lamanna, R. Watson, A.B. Maier Undiagnosed delirium is frequent and difficult to predict: results from a prevalence survey of a tertiary hospital Journal of Clinical Nursing, 28 (2019), pp. 2537-2542 CrossrefView in ScopusGoogle Scholar Lauretani et al., 2020F. Lauretani, G. Bellelli, G. Pelà, S. Morganti, S. Tagliaferri, M. Maggio Treatment of delirium in older persons: what we should not do! International Journal of Molecular Sciences, 21 (2020), p. 2397, 10.3390/ijms21072397 View in ScopusGoogle Scholar Liang et al., 2022S. Liang, J.P.C. Chau, S.H.S. Lo, J. Zhao, W. Liu Non-pharmacological delirium prevention practices among critical care nurses: a qualitative study BMC Nursing, 21 (2022), Article 235, 10.1186/s12912-022-01019-5 View in ScopusGoogle Scholar Mailhot et al., 2017T. Mailhot, S. Cossette, J. Côté, A. Bourbonnais, M.C. Côté, Y. Lamarche, et al. A post cardiac surgery intervention to manage delirium involving families: a randomized pilot study Nursing in Critical Care, 22 (2017), pp. 221-228, 10.1111/nicc.12288 View in ScopusGoogle Scholar Maravi et al., 2020P. Maravi, D.K. Mishra, A. Singh, V. Niranjan Atropine eye-drop-induced acute delirium: a case report General Psychiatry, 33 (2020), Article e100125, 10.1136/gpsych-2019-100125 Google Scholar Marquetand et al., 2021J. Marquetand, L. Bode, S. Fuchs, F. Hildenbrand, J. Ernst, R. von Kaenel, et al. Risk factors for delirium are different in the very old: a comparative one-year prospective cohort study of 5,831 patients Frontiers in Psychiatry, 12 (2021), Article 655087, 10.3389/fpsyt.2021.655087 View in ScopusGoogle Scholar Martínez et al., 2017F. Martínez, A.M. Donoso, C. Marquez, E. Labarca Implementing a multicomponent intervention to prevent delirium among critically ill patients Critical Care Nurse, 37 (2017), pp. 36-46, 10.4037/ccn2017531 Google Scholar Mattison, 2020M.L.P. Mattison Delirium Annals of Internal Medicine, 173 (2020), pp. ITC49-ITC64, 10.7326/AITC202010060 View in ScopusGoogle Scholar McKenzie and Joy, 2020J. McKenzie, A. Joy Family intervention improves outcomes for patients with delirium: systematic review and meta-analysis Australasian Journal on Ageing, 39 (2020), pp. 21-30, 10.1111/ajag.12688 Google Scholar McPherson et al., 2013J.A. McPherson, C.E. Wagner, L.M. Boehm, J.D. Hall, D.C. Johnson, L.R. Miller, et al. Delirium in the cardiovascular ICU: exploring modifiable risk factors Critical Care Medicine, 41 (2013), pp. 405-413, 10.1097/CCM.0b013e31826ab49b View in ScopusGoogle Scholar McWilliams et al., 2023D.J. McWilliams, E.B. King, P. Nydahl, J.L. Darbyshire, L. Gallie, D. Barghouthy, et al. Mobilisation in the EveNing to prevent and TreAt deLirium (MENTAL): a mixed-methods, randomised controlled feasibility trial EClinicalMedicine, 62 (2023), Article 102101, 10.1016/j.eclinm.2023.102101 View PDFView articleView in ScopusGoogle Scholar Menza, R. (2022). Recorded Music Listening Interventions for Symptom Management During Mechanical Ventilation in Critical Care. UCSF. ProQuest ID: Menza_ucsf_0034D_12401. Merritt ID: ark:/13030/m5f263k7. Retrieved from 〈 R (2022): Recorded Music Listening Interventions for Symptom Management During Mechanical Ventilation in Critical Care. UCSF. ProQuest ID: Menza_ucsf_0034D_12401. Merritt ID: ark:/13030/m5f263k7. Retrieved from: 〈 Google Scholar Mitchell et al., 2017M.L. Mitchell, S. Kean, J.E. Rattray, A.M. Hull, C. Davis, J.E. Murfield, et al. A family intervention to reduce delirium in hospitalised ICU patients: a feasibility randomised controlled trial Intensive & Critical Care Nursing, 40 (2017), pp. 77-84, 10.1016/j.iccn.2017.01.001 View PDFView articleView in ScopusGoogle Scholar Moon and Lee, 2015K.-J. Moon, S.-M. Lee The effects of a tailored intensive care unit delirium prevention protocol: a randomized controlled trial International Journal of Nursing Studies, 52 (2015), pp. 1423-1432, 10.1016/j.ijnurstu.2015.04.021 View PDFView articleView in ScopusGoogle Scholar Mooyeon et al., 2018O. Mooyeon, P. Chen, V. Romel-Nichols, K. Hreha, O. Boukrina, A.M. Barrett Delirium screening and management in inpatient rehabilitation facilities American Journal of Physical Medicine & Rehabilitation, 97 (2018), pp. 754-762, 10.1097/PHM.0000000000000962 Google Scholar Munn et al., 2019Z. Munn, E. Aromataris, C. Tufanaru, C. Stern, K. Porritt, J. Farrow, et al. The development of software to support multiple systematic review types: the Joanna Briggs Institute System for the Unified Management, Assessment and Review of Information (JBI SUMARI) International Journal of Evidence-Based Healthcare, 17 (2019), pp. 36-43, 10.1097/XEB.0000000000000152 View in ScopusGoogle Scholar Munro et al., 2017C.L. Munro, P. Cairns, M. Ji, K. Calero, W.M. Anderson, Z. Liang Delirium prevention in critically ill adults through an automated reorientation intervention — a pilot randomized controlled trial Heart & Lung, 46 (2017), pp. 234-238, 10.1016/j.hrtlng.2017.05.002 View PDFView articleView in ScopusGoogle Scholar Nielsen et al., 2020A.H. Nielsen, S. Angel, I. Egerod, T.H. Lund, M. Renberg, T.B. Hansen The effect of family-authored diaries on posttraumatic stress disorder in intensive care unit patients and their relatives: a randomised controlled trial (DRIP-study) Australian Critical Care, 33 (2020), pp. 123-129, 10.1016/j.aucc.2019.01.004 View PDFView articleView in ScopusGoogle Scholar Ono et al., 2011H. Ono, T. Taguchi, Y. Kido, Y. Fujino, Y. Doki The usefulness of bright light therapy for patients after oesophagectomy Intensive & Critical Care Nursing, 27 (2011), pp. 158-166, 10.1016/j.iccn.2011.03.003 View PDFView articleView in ScopusGoogle Scholar Ouzzani et al., 2016M. Ouzzani, H. Hammady, Z. Fedorowicz, A. Elmagarmid Rayyan — a web and mobile app for systematic reviews Systematic Reviews, 5 (2016), Article 210, 10.1186/s13643-016-0384-4 View in ScopusGoogle Scholar Parry et al., 2014S.M. Parry, S. Berney, S. Warrillow, D. El-Ansary, A.L. Bryant, N. Hart, et al. Functional electrical stimulation with cycling in the critically ill: a pilot case-matched control study Journal of Critical Care, 29 (2014), pp. 695.e1-695.e7, 10.1016/j.jcrc.2014.03.017 View PDFView articleGoogle Scholar Patel et al., 2014J. Patel, J. Baldwin, P. Bunting, S. Laha The effect of a multicomponent multidisciplinary bundle of interventions on sleep and delirium in medical and surgical intensive care patients Anaesthesia, 69 (2014), pp. 540-549, 10.1111/anae.12638 View in ScopusGoogle Scholar Peters et al., 2020M. Peters, C. Marnie, A.C. Tricco, D. Pollock, Z. Munn, L. Alexander, et al. Updated methodological guidance for the conduct of scoping reviews JBI Evidence Synthesis, 18 (2020), pp. 2119-2126, 10.11124/JBIES-20-00167 View in ScopusGoogle Scholar Potharajaroen et al., 2018S. Potharajaroen, S. Tangwongchai, T. Tayjasanant, T. Thawitsri, G. Anderson, M. Maes Bright light and oxygen therapies decrease delirium risk in critically ill surgical patients by targeting sleep and acid-base disturbances Psychiatry Research, 261 (2018), pp. 21-27, 10.1016/j.psychres.2017.12.046 View PDFView articleView in ScopusGoogle Scholar Rice et al., 2017K.L. Rice, M.J. Bennett, L. Berger, B. Jennings, L. Eckhardt, N. Fabré-LaCoste, et al. A pilot randomized controlled trial of the feasibility of a multicomponent delirium prevention intervention versus usual care in acute stroke The Journal of Cardiovascular Nursing, 32 (2017), pp. E1-E10, 10.1097/JCN.0000000000000356 View in ScopusGoogle Scholar Rivosecchi et al., 2015R.M. Rivosecchi, P.L. Smithburger, S. Svec, S. Campbell, S.L. Kane-Gill Nonpharmacological interventions to prevent delirium: an evidence-based systematic review Critical Care Nurse, 35 (2015), pp. 39-51, 10.4037/ccn2015423 View in ScopusGoogle Scholar Rood et al., 2021P.J.T. Rood, M. Zegers, D. Ramnarain, M. Koopmans, T. Klarenbeek, E. Ewalds, et al. The impact of nursing delirium preventive interventions in the ICU: a multicenter cluster-randomized controlled clinical trial American Journal of Respiratory and Critical Care Medicine, 204 (2021), pp. 682-691, 10.1164/rccm.202101-0082OC View in ScopusGoogle Scholar Rosa et al., 2019R.G. Rosa, M. Falavigna, D.B. da Silva, D. Sganzerla, M.M.S. Santos, R. Kochhann, et al. Effect of flexible family visitation on delirium among patients in the intensive care unit: the ICU visits randomized clinical trial Journal of American Medical Association, 322 (2019), pp. 216-228 CrossrefView in ScopusGoogle Scholar Sahawneh and Boss, 2021F. Sahawneh, L. Boss Non-pharmacologic interventions for the prevention of delirium in the intensive care unit: an integrative review Nursing in Critical Care, 26 (2021), pp. 166-175, 10.1111/nicc.12594 View in ScopusGoogle Scholar Society of Critical Care Medicine, 2013Society of Critical Care Medicine Clinical practice guidelines for the management of pain agitation and delirium in adult patients in the intensive care unit Critical Care Medicine, 41 (2013), pp. 263-306 Retrieved from Google Scholar Stinson, 2016K.J. Stinson Nurses' attitudes, clinical experience, and practice issues with use of physical restraints in critical care units American Journal of Critical Care, 25 (2016), pp. 21-26, 10.4037/ajcc2016428 View in ScopusGoogle Scholar Suliman et al., 2017M. Suliman, S. Aloush, K. Al-Awamreh Knowledge, attitude and practice of intensive care unit nurses about physical restraint Nursing in Critical Care, 22 (2017), pp. 264-269, 10.1111/nicc.12303 View in ScopusGoogle Scholar Theresa et al., 2022S.J. Theresa, L. Fathima, E. Kayalvizhi, M. Puliken Delirium care bundle on sedation and orientation among ICU-acquired delirium a randomized controlled trial Journal of Pharmaceutical Negative Results, 13 (2022), pp. 1991-1999, 10.47750/pnr.2022.13.S04.243 Google Scholar Thomas and Harden, 2008J. Thomas, A. Harden Methods for the thematic synthesis of qualitative research in systematic reviews BMC Medical Research Methodology, 8 (2008), Article 45, 10.1186/1471-2288-8-45 View in ScopusGoogle Scholar Topcu and Tosun, 2022N. Topcu, Z. Tosun Efforts to improve sleep quality in a medical intensive care unit: effect of a protocol of non-pharmacological interventions Sleep & Breathing = Schlaf & Atmung, 26 (2022), pp. 803-810, 10.1007/s11325-022-02570-w View in ScopusGoogle Scholar Tricco et al., 2018A.C. Tricco, E. Lillie, W. Zarin, K.K. O'Brien, H. Colquhoun, D. Levac, et al. PRISMA Extension for Scoping Reviews (PRISMA-ScR): checklist and explanation Annals of Internal Medicine, 169 (2018), pp. 467-473, 10.7326/M18-0850 View in ScopusGoogle Scholar Tropea et al., 2017J. Tropea, D. LoGiudice, D. Liew, A. Gorelik, C. Brand Poorer outcomes and greater healthcare costs for hospitalized older people with dementia and delirium: a retrospective cohort study International Journal of Geriatric Psychiatry, 32 (2017), pp. 539-547, 10.1002/gps.4491 View in ScopusGoogle Scholar Van Rompaey et al., 2012B. Van Rompaey, M.M. Elseviers, W. Van Drom, V. Fromont, P.G. Jorens The effect of earplugs during the night on the onset of delirium and sleep perception: a randomized controlled trial in intensive care patients Critical Care, 16 (2012), Article R73, 10.1186/cc11330 View in ScopusGoogle Scholar Wijdicks, 2021E.F.M. Wijdicks The discovery of acute alcohol withdrawal as a cause of delirium Neurocritical Care, 37 (2021), pp. 806-809, 10.1007/s12028-021-01196-22 Google Scholar Zamoscik et al., 2017K. Zamoscik, R. Godbold, P. Freeman Intensive care nurses' experiences and perceptions of delirium and delirium care Intensive & Critical Care Nursing, 40 (2017), pp. 94-100, 10.1016/j.iccn.2017.01.003 View PDFView articleView in ScopusGoogle Scholar Cited by (6) Pharmacologic prophylaxis of postoperative delirium in elderly patients: A network meta-analysis of randomized controlled trials 2025, Journal of Psychiatric Research Show abstract The high incidence and mortality rates of postoperative delirium (POD) among elderly patients highlights the pressing need for tailored prophylactic strategies. Despite various pharmacologic prophylactic strategies have been reported effective, their overall benefit and safety remain unclear in the geriatric population. Our network meta-analysis (NMA) aimed to systematically evaluate and rank the effectiveness of various pharmacological interventions in preventing POD in elderly patients. We conducted an extensive search of PubMed, Embase, Cochrane Central Register of Controlled Trials, PsycINFO, and Google Scholar for randomized controlled trials (RCTs) published up to August 1, 2023. We included RCTs examining pharmacological prophylactic effects of POD in elderly patients. To extract data in alignment with predefined areas of interest, we employed the Preferred Reporting Items for Systematic Reviews and Meta-analyses guidelines. The primary outcome was the incidence of POD. For secondary outcomes, we evaluated tolerability through all-cause discontinuation or drop-out rates, as well as all-cause mortality. Our analysis encompassed a total of 44 RCTs involving 11,178 patients. Out of these, 26 RCTs involved comparisons with placebo only. For delirium prevention, the treatment groups receiving atypical antipsychotics (odds ratio (OR) of 0.27 and 95% confidence interval (CI) of 0.12–0.58), haloperidol (OR of 0.42; 95% CI of 0.25–0.71), dexmedetomidine (OR of 0.51 and 95% CI of 0.37–0.71 and melatonergic agents (MMA) (OR of 0.57 and 95% CI of 0.33–0.98) had significantly lower rates of delirium compared to the placebo group. Notably, the atypical antipsychotics ranked as the most effective treatment. For tolerability, no statistically differences in rates of dropout discontinuation and all-cause mortality among groups allocated to the placebo or individual pharmacological treatments. Based on indirect evidence, our network meta-analysis identified atypical antipsychotics, dexmedetomidine, MMA, and haloperidol as effective in preventing POD in the elderly, with atypical antipsychotics ranking highest. However, it is essential to note that these findings should be confirmed through further RCTs. ### The development of a family-led novel intervention for delirium prevention and management in the adult intensive care unit: A co-design qualitative study 2025, Australian Critical Care Citation Excerpt : Family-centred care approaches have evolved in recent years, with an emergence of non-pharmacological interventions involving family members considered effective in delirium prevention and management, with high acceptability by patients, family members, and clinicians.18–20 A scoping review,21 conducted as a foundation for this study, identified that family-delivered interventions including orientation practices, the provision of memory cues, extended visitation, and sensory stimulation were commonly utilised in delirium prevention and management.22–25 However, these approaches are inconsistently applied and evaluated within ICUs, making it challenging to recommend their application in delirium management in this context.26 Show abstract The aim of this study was to codesign a Family Members’ Voice Reorientation Intervention (FAMVR)for delirium prevention and management in critically ill adult patients through collaborative process with previous patients, families, and clinical staff. Delirium is a common consequence of intensive care admission, and there is limited evidence to support family-led interventions to prevent or minimise delirium in intensive care. People with lived experience of intensive care are seldom involved in codesigning delirium prevention and management interventions despite the identified benefits of their involvement in delirium care. Codesign qualitative study. The process of co-designing was undertaken using the four stages of the Double Diamond model. Participants included people with lived experience of the intensive care unit, family members, and intensive care clinicians. The codesign approach was utilised, and data were gathered from a series of focus groups and individual interviews. Data were digitally recorded, transcribed verbatim, and analysed using thematic analysis. Of the 26 people who indicated their interest in participating, 12 (46%) completed the first and second stages, and nine (35%) completed the third and fourth stages of the Family Members’ Voice Reorientation Intervention development. All participant groups were represented in the fourth stage: patients (n=4), family members (n=1), nurses (n=2), and medical staff (n=2). Four themes were identified: message content, wording, reactions, and tone, all of which informed the prototype of the intervention and its associated domains. A codesign approach was important for developing a delirium management intervention. This process enabled participants to provide their feedback in the context of their unique experiences, which in turn enhanced the authenticity and appropriateness of this unique intervention. ### Co-designing a digital family-led intervention for delirium prevention and management in adult critically ill patients: An application of the double diamond design process 2024, International Journal of Nursing Studies Citation Excerpt : Delirium is a common and serious condition in critically ill patients, characterised by acute confusion and cognitive disturbances (Mart et al., 2021). Recent evidence has highlighted the role of familiar voice reorientation in preventing and managing delirium (Johnson et al., 2024b). Involving family members in delirium care and using familiar voices to reorientate and comfort patients leverage the emotional connection to promote cognitive stability. Show abstract Co-designing healthcare interventions is gaining recognition as a novel and collaborative method. Co-design involves end-users from the start, ensuring that an intervention best meets their needs. Despite its potential benefits, this approach is not yet widely used in developing clinical interventions within intensive care units where the perspectives of patients, family members, and clinicians are crucial. To describe the application, benefits and challenges of the Double Diamond model to co-design a digital family-led voice reorientation intervention for delirium prevention and management in critically ill adult patients. The co-design process was guided by the Double Diamond model over a period of 12 months. Development involved patients, family members, and nursing and medical staff as co-designers and decision-makers in the iterative development of the intervention. Data from field notes and group meetings were audio recorded, transcribed verbatim, and content analysed at each phase, which were then presented to the co-designers for verification and refinement. Co-designers included people with lived experience of the ICU as patients (n = 5) and family members (n = 1) and clinical experts (nursing staff n = 3; medical staff n = 3). Co-designers were highly engaged and reported positive experiences and collaboration in the co-design process. Sharing the diversity of their own personal ICU experiences was found to be beneficial as it not only validated individual feelings but also strengthened intervention development. Differences in interpretations and meanings of the voice messages proposed as part of the intervention were challenging. Maintaining sufficient focus on each phase of the Double Diamond was difficult due to the complexity of the context in which the intervention was being co-designed and the resulting challenges of maintaining the engagement of the co-designers throughout the process. There were benefits and challenges of engaging people with lived experience in an intensive care unit as co-designers through the Double Diamond design process to develop a digital family-led intervention for delirium prevention and management. Overall, applying the Double Diamond to co-design a clinical intervention is recommended, whereby the collaboration process benefits patients, family members, and clinical staff. ACTRN12622001568707; ANZCTR — Registration. ### The implementation and evaluation of a family-led novel intervention for delirium prevention and management in adult critically ill patients: A mixed-methods pilot study 2025, Nursing in Critical Care ### Exploring Cognitive Stimulation as a Therapy for the Prevention of Delirium in a Hospital Setting: A Narrative Review 2025, Behavioral Sciences ### The Effect of a Multifactorial Intervention on the Occurrence of Delirium in Patients Hospitalized in Intensive Care Unit: A Clinical Trial Study 2025, Iranian South Medical Journal © 2024 Australian College of Nursing Ltd. Published by Elsevier Ltd. Recommended articles Are inherent requirements a barrier to diversity? An analysis of course entry information Collegian, Volume 31, Issue 4, 2024, pp. 252-257 Joanna Tai, …, Lisa McKenna View PDF ### The family in preventing delirium in the intensive care unit: Scoping review Enfermería Intensiva (English ed.), Volume 33, Issue 1, 2022, pp. 33-43 B.A.Pabón-Martínez, …, A.M.Henao-Castaño ### Decreasing the incidence of delirium via multi-sensory stimulation in patients receiving mechanical ventilation in the intensive care unit: A protocol for a randomized feasibility study Contemporary Clinical Trials Communications, Volume 38, 2024, Article 101263 Bin He, …, Yu Chen View PDF ### Interventions to prevent and treat delirium: An umbrella review of randomized controlled trials Ageing Research Reviews, Volume 97, 2024, Article 102313 Nicola Veronese, …, Marco Solmi View PDF ### The implementation of nUrsiNg DEliRium preventive INterventions in the Intensive Care Unit (UNDERPIN-ICU): A qualitative evaluation International Journal of Nursing Studies, Volume 137, 2023, Article 104386 Bram Tilburgs, …, Mark van den Boogaard View PDF ### The implementation of a nonpharmacologic protocol to prevent intensive care delirium Journal of Critical Care, Volume 31, Issue 1, 2016, pp. 206-211 Ryan M.Rivosecchi, …, Pamela L.Smithburger Show 3 more articles Article Metrics Citations Citation Indexes 6 Captures Mendeley Readers 62 View details About ScienceDirect Remote access Contact and support Terms and conditions Privacy policy Cookies are used by this site.Cookie settings All content on this site: Copyright © 2025 Elsevier B.V., its licensors, and contributors. 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15151	https://www.somersetkey.com/ourpages/auto/2018/2/23/66635553/Geo%20Ch%207%20sec%203.pdf	© Houghton Mifflin Harcourt Publishing Company Name Class Date Explore Exploring Triangle Inequalities A triangle can have sides of different lengths, but are there limits to the lengths of any of the sides? A Consider a △ABC where you know two side lengths, AB = 4 inches and BC = 2 inches. On a separate piece of paper, draw _ AB so that it is 4 inches long. B To determine all possible locations for C with _ BC = 2 inches, set your compass to 2 inches. Draw a circle with center at B. C Choose and label a final vertex point C so it is located on the circle. Using a straightedge, draw the segments to form a triangle. Are there any places on the circle where point C cannot lie? Explain. D Measure and record the lengths of the three sides of your triangle. Resource Locker A B A B C Point C cannot lie on the two points of the circle that intersect ‾→ AB because then the sides will overlap to form a straight line. Possible answer: AB = 4 in., BC = 2 in., AC = 3.2 in. Module 7 341 Lesson 3 7.3 Triangle Inequalities Essential Question: How can you use inequalities to describe the relationships among side lengths and angle measures in a triangle? y ; GE_MNLESE385795_U2M07L3.indd 341 02/04/14 1:36 AM Common Core Math Standards The student is expected to: G-SRT.B.5 Use congruence ... criteria for triangles to solve problems and to prove relationships in geometric figures. Also G-CO.C.10, G-CO.D.12 Mathematical Practices MP.5 Using Tools Language Objective Explain to a partner how to show the three inequalities generated for a triangle with side lengths a, b, and c. COMMON CORE COMMON CORE HARDCOVER PAGES 293302 Turn to these pages to find this lesson in the hardcover student edition. Triangle Inequalities ENGAGE Essential Question: How can you use inequalities to describe the relationships among side lengths and angle measures in a triangle? The sum of any two side lengths of a triangle will be greater than the length of the third side. If the sides of a triangle are not congruent, then the largest angle will be opposite the longest side and the smallest angle will be opposite the shortest side. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo, making sure that students understand the objective of an orienteering competition and the tools used by competing teams. Then preview the Lesson Performance Task. 341 HARDCOVER PAGES 293302 Turn to these pages to find this lesson in the hardcover student edition. © Houghton Mifflin Harcourt Publishing Company Name Class Date Explore Exploring Triangle Inequalities A triangle can have sides of different lengths, but are there limits to the lengths of any of the sides?  Consider a △ABC where you know two side lengths, AB = 4 inches and BC = 2 inches. On a separate piece of paper, draw _ AB so that it is 4 inches long.  To determine all possible locations for C with _ BC = 2 inches, set your compass to 2 inches. Draw a circle with center at B.  Choose and label a final vertex point C so it is located on the circle. Using a straightedge, draw the segments to form a triangle. Are there any places on the circle where point C cannot lie? Explain.  Measure and record the lengths of the three sides of your triangle. Resource Locker G-SRT.B.5 Use congruence ... criteria for triangles to solve problems and to prove relationships in geometric figures. Also G-CO.C.10, G-CO.D.12 COMMON CORE A B A B C Point C cannot lie on the two points of the circle that intersect ‾→ AB because then the sides will overlap to form a straight line. Possible answer: AB = 4 in., BC = 2 in., AC = 3.2 in. Module 7 341 Lesson 3 7 . 3 Triangle Inequalities Essential Question: How can you use inequalities to describe the relationships among side lengths and angle measures in a triangle? DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A GE_MNLESE385795_U2M07L3.indd 341 02/04/14 1:37 AM 341 Lesson 7 . 3 L E S S O N 7 . 3 © Houghton Mifflin Harcourt Publishing Company E The figures below show two other examples of △ABC that could have been formed. What are the values that _ AC approaches when point C approaches _ AB ? Reflect 1. Use the side lengths from your table to make the following comparisons. What do you notice? AB + BC ? AC BC + AC ? AB AC + AB ? BC 2. Measure the angles of some triangles with a protractor. Where is the smallest angle in relation to the shortest side? Where is the largest angle in relation to the longest side? 3. Discussion How does your answer to the previous question relate to isosceles triangles or equilateral triangles? Explain 1 Using the Triangle Inequality Theorem The Explore shows that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. This can be summarized in the following theorem. Triangle Inequality Theorem The sum of any two side lengths of a triangle is greater than the third side length. To be able to form a triangle, each of the three inequalities must be true. So, given three side lengths, you can test to determine if they can be used as segments to form a triangle. To show that three lengths cannot be the side lengths of a triangle, you only need to show that one of the three triangle inequalities is false. A C B A C B AB + BC > AC BC + AC > AB AC + AB > BC B C A AB + BC > AC BC + AC > AB AC + AB > BC B C A AC approaches 2 inches or 6 inches. The sum of any of the two sides is greater than the third side. The smallest angle is opposite the shortest side; the largest angle is opposite the longest side. When angles in a triangle have the same measure, the sides opposite those angles also have the same measure. Module 7 342 Lesson 3 y ; y ; GE_MNLESE385795_U2M07L3.indd 342 23/03/14 2:08 AM Learning Progressions In this lesson, students add to their knowledge of triangles by using a variety of tools to verify the Triangle Inequality Theorem. Students also explore how to order the side lengths given the angle measures of the triangle and how to predict the possible lengths of the third side of a triangle, given the lengths of two of the sides. Triangles have important uses in everyday life and in future mathematics study, including trigonometry. All students should develop fluency with the properties of triangles as they continue their study of geometry. EXPLORE Exploring Triangle Inequalities INTEGRATE TECHNOLOGY Students have the option of completing the triangle inequality activity either in the book or online. QUESTIONING STRATEGIES How do you decide if three lengths can be the side lengths of a triangle? Check the sum of each pair of two sides. The sum must be greater than the third side. EXPLAIN 1 Using the Triangle Inequality Theorem AVOID COMMON ERRORS Some students may have difficulty understanding why all three inequalities must be checked for the Triangle Inequality Theorem. One example may be side lengths of 5 cm, 5 cm, and 10 cm. Straws with these lengths look like they will make a triangle, but they do not. Have them do several examples with different side lengths to test the theorem. QUESTIONING STRATEGIES For side lengths a, b, and c of a triangle, how many inequalities must be true? Write them. 3; a < b + c, b < a + c, c < a + b PROFESSIONAL DEVELOPMENT Triangle Inequalities 342 © Houghton Mifflin Harcourt Publishing Company Example 1 Use the Triangle Inequality Theorem to tell whether a triangle can have sides with the given lengths. Explain.  4, 8, 10 4 + 8 > 10 4 + 10 > 8 8 + 10 > 4 12 > 10 ✓ 14 > 8 ✓ 18 > 4 ✓ Conclusion: The sum of each pair of side lengths is greater than the third length. So, a triangle can have side lengths of 4, 8, and 10.  7, 9, 18 + > + > + > > > > Conclusion: Reflect 4. Can an isosceles triangle have these side lengths? Explain. 5, 5, 10 5. How do you know that the Triangle Inequality Theorem applies to all equilateral triangles? Your Turn Determine if a triangle can be formed with the given side lengths. Explain your reasoning. 6. 12 units, 4 units, 17 units 7. 24 cm, 8 cm, 30 cm 18 18 9 9 9 ? ? ? ? ? ? 7 9 16 25 7 x ✓ 7 7 18 18 27 ✓ Not all three inequalities are true. So, a triangle cannot have these three side lengths. No; 12 + 4 ≯ 17 Yes; 24 + 8 > 30, 8 + 30 > 24, and 24 + 30 > 8 No; These numbers do not result in three true inequalities. 5 + 5 ≯ 10, so no triangle can be drawn with these side lengths. Since all sides are congruent, the sum of any two side lengths will be greater than the third side. Module 7 343 Lesson 3 y ; GE_MNLESE385795_U2M07L3.indd 343 23/03/14 2:08 AM COLLABORATIVE LEARNING Small Group Activity Give all students in groups pieces of raw spaghetti. Have each student break three pieces of spaghetti in different lengths and measure the length of each piece. Then have a group member write the three inequalities for those lengths. Have another group member analyze the inequalities and conjecture if the three lengths will form a triangle. Ask the fourth student to position the pieces to show a triangle or to show no triangle. Switch roles and repeat the activity. 343 Lesson 7 . 3 © Houghton Mifflin Harcourt Publishing Company Explain 2 Finding Possible Side Lengths in a Triangle From the Explore, you have seen that if given two side lengths for a triangle, there are an infinite number of side lengths available for the third side. But the third side is also restricted to values determined by the Triangle Inequality Theorem. Example 2 Find the range of values for x using the Triangle Inequality Theorem.  Find possible values for the length of the third side using the Triangle Inequality Theorem. x + 10 > 12 x + 12 > 10 10 + 12 > x x > 2 x > -2 22 > x 2 < x < 22 Ignore the inequality with a negative value, since a triangle cannot have a negative side length. Combine the other two inequalities to find the possible values for x.  + > + > + > > > > < x < a b 12 10 x 15 15 x x x x x x x 15 15 0 0 30 30 0 15 15 15 15 Module 7 344 Lesson 3 y ; y ; GE_MNLESE385795_U2M07L3.indd 344 23/03/14 2:08 AM EXPLAIN 2 Finding Possible Side Lengths in a Triangle CONNECT VOCABULARY Relate the range of values for a third side length of a triangle given two side lengths by stating that the third side length must be greater than the difference of the other two side lengths and also less than the sum of the other two side lengths. DIFFERENTIATE INSTRUCTION Manipulatives Give students a number of straws and ask them to cut them into various lengths. Then have them measure each length. Ask them to make a table listing the measures of all possible combinations of the three lengths. Then have them manipulate the straws to see if they can form a triangle. Highlight sets of three measurements that do not form a triangle. Triangle Inequalities 344 © Houghton Mifflin Harcourt Publishing Company Reflect 8. Discussion Suppose you know that the length of the base of an isosceles triangle is 10, but you do not know the lengths of its legs. How could you use the Triangle Inequality Theorem to find the range of possible lengths for each leg? Explain. Your Turn Find the range of values for x using the Triangle Inequality Theorem. 9. 10. Explain 3 Ordering a Triangle’s Angle Measures Given Its Side Lengths From the Explore Step D, you can see that changing the length of _ AC also changes the measure of ∠B in a predictable way. Side-Angle Relationships in Triangles If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. 14 21 x 18 9 x As side AC gets shorter, m∠B approaches 0° As side AC gets longer, m∠B approaches 180° A C B A C B As side AC gets shorter, m∠B approaches 0° As side AC gets longer, m∠B approaches 180° A C B A C B AC > BC m∠B > m∠A A B C Possible answer: If x represents the length of one leg, then by the Triangle Inequality Theorem, solve for x + x > 10 and x + 10 > x. The solution of the first inequality is x > 5. The solution of the second inequality is 10 > 0, which is always true. So the range of possible lengths for each leg is x > 5. x + 14 > 21 21 + 14 > x x > 7 35 > x x + 21 > 14 x > -7 7 < x < 35 x + 9 > 18 18 + 9 > x x > 9 27 > x x + 18 > 9 x > -9 9 < x < 27 Module 7 345 Lesson 3 y ; GE_MNLESE385795_U2M07L3.indd 345 5/22/14 5:36 PM LANGUAGE SUPPORT Visual Cues Help students understand how to apply the inequality relationships in this lesson by suggesting they list all the angles and sides before doing an example or exercise. Then, they can list the angles in increasing order and write the side lengths opposite those angles in the same order. QUESTIONING STRATEGIES How do find the range for the length of the third side of a triangle? The range is r < x < s, where r is the difference of the two given side lengths and s is the sum of the two given side lengths. How do you interpret the compound inequality a < x < b as individual inequalities? a < x and x < b EXPLAIN 3 Ordering a Triangle’s Angle Measures Given Its Side Lengths INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Have students use geometry software to create a scalene triangle. Ask them to use the measuring features to measure the side lengths. Then have them measure each angle and verify that the largest angle is opposite the longest side length and that the smallest angle is opposite the shortest side length. Ask them to drag the vertices to vary the side lengths and then observe that the angle measures are ordered in the same way as in the original triangle. 345 Lesson 7 . 3 © Houghton Mifflin Harcourt Publishing Company Example 3 For each triangle, order its angle measures from least to greatest.  Longest side length: AC Greatest angle measure: m∠B Shortest side length: AB Least angle measure: m∠C Order of angle measures from least to greatest: m∠C, m∠A, m∠B  Longest side length: Greatest angle measure: Shortest side length: Least angle measure: Order of angle measures from least to greatest: Your Turn For each triangle, order its angle measures from least to greatest. 11. 12. Explain 4 Ordering a Triangle’s Side Lengths Given Its Angle Measures From the Explore Step D, you can see that changing the the measure of ∠B also changes length of _ AC in a predictable way. Angle-Side Relationships in Triangles If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. 10 12 15 A B C 20 21 22 A B C 15 32 40 A B C 7 24 25 A B C As m∠B approaches 0°, side AC gets shorter As m∠B approaches 180°, side AC gets longer A C B A C B As m∠B approaches 0°, side AC gets shorter As m∠B approaches 180°, side AC gets longer A C B A C B BC AB m∠A m∠C m∠C, m∠B, m∠A Longest side length: AB Shortest side length: CB m∠A, m∠B, m∠C Longest side length: BC Shortest side length: AC m∠B, m∠C, m∠A Module 7 346 Lesson 3 y ; y ; GE_MNLESE385795_U2M07L3.indd 346 24/03/14 12:28 PM QUESTIONING STRATEGIES How do you know the greatest angle is opposite the longest side in a triangle? If one side of a triangle is longer than another, then the angle opposite the longer side is larger than the angle opposite the shorter side. EXPLAIN 4 Ordering a Triangle’s Side Lengths Given Its Angle Measures AVOID COMMON ERRORS Some students may think that they can use angle measures to compare the side lengths of different triangles. Explain that angle measures can be used to order the side lengths only within a single triangle. Give an example of why this must be the case, such as a very small triangle with an obtuse angle and a very large equilateral triangle. The obtuse angle is greater than the 60° angle of the equilateral triangle, but its opposite side may be shorter. QUESTIONING STRATEGIES How do you order the side lengths of a triangle given the angle measures? Explain. The side lengths will be in the same order as the measure of the angles opposite the side lengths. For example, the greatest side length is opposite the greatest angle measure. Use the rule that if one angle of a triangle is larger than another, then the side opposite the larger angle is longer than the side opposite the smaller angle. Triangle Inequalities 346 © Houghton Mifflin Harcourt Publishing Company Example 4 For each triangle, order the side lengths from least to greatest.  Greatest angle measure: m∠B Longest side length: AC Least angle measure: m∠A Shortest side length: BC Order of side lengths from least to greatest: BC, AB, AC  Greatest angle measure: Longest side length: Least angle measure: Shortest side length: Order of side lengths from least to great: Your Turn For each triangle, order the side lengths from least to greatest. 13. 14. Elaborate 15. When two sides of a triangle are congruent, what can you conclude about the angles opposite those sides? 16. What can you conclude about the side opposite the obtuse angle in an obtuse triangle? 17. Essential Question Check-In Suppose you are given three values that could represent the side lengths of a triangle. How can you use one inequality to determine if the triangle exists? 30° 100° 50° A B C 45° 70° 65° A C B 160° 15° 5° A C B 60° 30° A C B m∠A BC m∠C AB AB, AC, BC Greatest angle measure: m∠C Least angle measure: m∠A CB, AC, AB Greatest angle measure: m∠C Least angle measure: m∠B AC, BC, AB They are also congruent. It is the longest side of the triangle. If the sum of the two least values is greater than the remaining value, the triangle exists. Otherwise it does not exist. Module 7 347 Lesson 3 y ; GE_MNLESE385795_U2M07L3.indd 347 6/9/15 6:18 AM ELABORATE AVOID COMMON ERRORS Some students may list angle measures in order and side lengths in order, but not make the connection that the largest side length must be opposite the largest angle measure. Suggest that they list the angles with the corresponding opposite side before they order their measures. QUESTIONING STRATEGIES Can a triangle have side lengths of 7 cm, 12 cm, and 20 cm? Explain. No; the Triangle Inequality Theorem states that the sum of each pair of lengths must be greater than the third length in order for 3 lengths to be side lengths of a triangle. Since 7 + 12 ≯ 20, these lengths cannot be lengths of sides of a triangle. A triangle has angle measures of 30°, 60°, and 90°. Which angle is opposite the longest side of the triangle? Explain. The 90° angle, because the side lengths of a triangle are ordered in the same way as the angle measures. SUMMARIZE THE LESSON How do you know if three segment lengths can be the side lengths of a triangle? Test the sum of each two pairs of segment lengths. By the Triangle Inequality Theorem, if the sum of each pair of segment lengths is greater than the third length, then the segments can be the side lengths of a triangle. 347 Lesson 7 . 3 © Houghton Mifflin Harcourt Publishing Company Use a compass and straightedge to decide whether each set of lengths can form a triangle. 1. 7 cm, 9 cm, 18 cm 2. 2 in., 4 in., 5 in. 3. 1 in., 2 in., 10 in. 4. 9 cm, 10 cm, 11 cm Determine whether a triangle can be formed with the given side lengths. 5. 10 ft, 3 ft, 15 ft 6. 12 in., 4 in., 15 in. 7. 9 in., 12 in., and 18 in. 8. 29 m, 59 m, and 89 m Find the range of possible values for x using the Triangle Inequality Theorem. 9. 10. 11. A triangle with side lengths 22.3, 27.6, and x 12. Analyze Relationships Suppose a triangle has side lengths AB, BC, and x, where AB = 2 · BC. Find the possible range for x in terms of BC. • Online Homework • Hints and Help • Extra Practice Evaluate: Homework and Practice 8 3 x 5 12 x No; if the base is 18 cm compass arcs of lengths 7 cm and 9 cm from each end of the base do not intersect. Yes; if the base is 5 in. compass arcs of lengths 2 in. and 4 in. from each end of the base have two intersections, each forming a triangle. No; if the base is 10 in. compass arcs of lengths 1 in. and 2 in. from each end of the base do not intersect. Yes; if the base is 11 cm compass arcs of lengths 9 cm and 10 cm from each end of the base have two intersections, each forming a triangle. No; 10 + 3 ≯ 15 and 12 + 15 > 4 Yes No; 29 + 59 ≯ 89 Yes; 12 + 4 > 15, 4 + 15 > 12, 5 < x < 11 7 < x < 17 3 + 8 > x 3 + x > 8 8 + x > 3 11 > x x > 5 x > -5 5 + 12 > x 5 + x > 12 12 + x > 5 17 > x x > 7 x > -7 5 .3 < x < 49.9 22.3 + 27.6 > x 22.3 + x > 27.6 27.6 + x > 22.3 49.9 > x x > 5.3 x > −5.3 AB + BC > x BC + x > AB x + AB > BC 2 · BC + BC > x BC + x > 2 · BC x + 2 · BC > BC 3 · BC > x x > BC x > −BC BC < x < 3 · BC Module 7 348 Lesson 3 y ; y ; GE_MNLESE385795_U2M07L3.indd 348 2/26/16 1:16 AM Exercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 1–19 2 Skills/Concepts MP.4 Modeling 20–27 2 Skills/Concepts MP.4 Modeling 28 3 Strategic Thinking MP.4 Modeling 29 3 Strategic Thinking MP.3 Logic 30 3 Strategic Thinking MP.3 Logic EVALUATE ASSIGNMENT GUIDE Concepts and Skills Practice Explore Exploring Triangle Inequalities Exercises 1–4 Example 1 Using the Triangle Inequality Theorem Exercises 5–8 Example 2 Finding Possible Side Lengths in a Triangle Exercises 9–12 Example 3 Ordering a Triangle’s Angle Measures Given Its Side Lengths Exercises 13–15 Example 4 Ordering a Triangle’s Side Lengths Given Its Angle Measures Exercises 16–19 INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Students can check their solutions for correctness by using geometry software to create triangles with the same side lengths or angle measures. When checking solutions, remind students the order of the side lengths gives the order of the opposite angle measures. Triangle Inequalities 348 © Houghton Mifflin Harcourt Publishing Company ⋅ Image Credits: ©Carlos Davila/Photographer's Choice RF/Getty Images For each triangle, write the order of the angle measures from least to greatest. 13. 14. 15. Analyze Relationships Suppose a triangle has side lengths PQ, QR, and PR, where PR = 2PQ = 3QR. Write the angle measures in order from least to greatest. For each triangle, write the side lengths in order from least to greatest. 16. 17. 18. In △ JKL, m∠J = 53°, m∠K = 68°, and m∠L = 59°. 19. In △ PQR, m∠P = 102° and m∠Q = 25°. 20. Represent Real-World Problems Rhonda is traveling from New York City to Paris and is trying to decide whether to fly via Frankfurt or to get a more expensive direct flight. Given that it is 3,857 miles from New York City to Frankfurt and another 278 miles from Frankfurt to Paris, what is the range of possible values for the direct distance from New York City to Paris? A B C 6 14 13 D F E 3.4 3.2 3.7 B C A 45° 65° 70° D F 79° 33° E m∠A, m∠B, m∠C m∠E, m∠F, m∠D AC, BC, AB KL, JK, JL m∠D = 68° DE, EF, DF PR = 2PQ PR = 3QR 1 _ 3 PR < 1 _ 2 PR < PR 1 _ 2 PR = PQ 1 _ 3 PR = QR QR < PQ < PR So m∠ P < m∠R < m∠Q; Order from least to greatest: m∠ P, m∠R, m∠Q m∠R = 53° PR, PQ, QR 3, 857 + 278 > x 3, 857 + x > 278 278 + x > 3, 857 4,135 > x x > −3, 579 x > 3, 57 9 3, 579 < x < 4,135 The direct distance is between 3,579 miles and 4,135 miles. Module 7 349 Lesson 3 y ; GE_MNLESE385795_U2M07L3.indd 349 10/14/14 9:08 PM VISUAL CUES Suggest that students label a side and its corresponding opposite angle in one color and then do the same with other side-angle combinations in different colors. This visual cue can help them to remember the order of the measures of the sides or angles. AVOID COMMON ERRORS If students have trouble with compound inequalities, have them write the inequalities separately and then use a number line to help them combine the inequalities into one. COLLABORATIVE LEARNING Have students work in small groups to make a poster showing a triangle, the side-angle relationships, and the triangle inequality relationship they learned in this lesson. Give each group a different triangle to draw. Then have each group present its poster to the rest of the class, explaining each relationship they listed. 349 Lesson 7 . 3 © Houghton Mifflin Harcourt Publishing Company 21. Represent Real-World Problems A large ship is sailing between three small islands. To do so, the ship must sail between two pairs of islands, avoiding sailing between a third pair. The safest route is to avoid the closest pair of islands. Which is the safest route for the ship? 22. Represent Real-World Problems A hole on a golf course is a dogleg, meaning that it bends in the middle. A golfer will usually start by driving for the bend in the dogleg (from A to B), and then using a second shot to get the ball to the green (from B to C). Sandy believes she may be able to drive the ball far enough to reach the green in one shot, avoiding the bend (from A direct to C). Sandy knows she can accurately drive a distance of 250 yd. Should she attempt to drive for the green on her first shot? Explain. 23. Represent Real-World Problems Three cell phone towers form a triangle, △ PQR. The measure of ∠Q is 10° less than the measure of ∠P. The measure of ∠R is 5° greater than the measure of ∠Q. Which two towers are closest together? 24. Algebra In △ PQR, PQ = 3x + 1, QR = 2x − 2, and PR = x + 7. Determine the range of possible values of x. X Y Z 58° 73° N A 102 yd B 135 yd C 58° + 73° + m∠Z = 180°; m∠Z = 49° m∠Z < m∠X < m∠Y, so XY < YZ < XZ. Therefore, the safest route is to avoid sailing between the islands at X and Y. 102 + 135 > AC 102 + AC > 135 135 + AC > 102 237 > AC AC > 33 AC > -33 33 < AC < 237 Since AC is less than 250 yd, Sandy has a good chance of reaching the green in one shot. Yes; First, each side length must be positive. m∠Q = m∠P − 10° and m∠R = m∠Q + 5° = (m∠P − 10°) + 5° = m∠P − 5° So, m∠Q < m∠R < m∠P, and therefore PR < PQ < QR. The towers at Q and R are closest together. 3x + 1 > 0 2x − 2 > 0 x + 7 > 0 x > − 1 _ 3 x > 1 x > −7 (3x + 1) + (2x − 2) > x + 7 (3x + 1) + (x + 7) > 2x − 2 (2x − 2) + (x + 7) > (3x + 1) x > 2 x > −5 4 > 0 Since the last inequality is always true, x > 2. Module 7 350 Lesson 3 y ; y ; GE_MNLESE385795_U2M07L3.indd 350 6/9/15 6:44 AM INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 When writing side measures or side-angle measure relationships for a triangle, students should remember that there is a range of possible side lengths for making a triangle, and that the side lengths and angle measures must be in the same order. AVOID COMMON ERRORS Some students may think that all three inequalities associated with the Triangle Inequality Theorem must be false. Point out that you need to show only that one of the three triangle inequalities is false to state that the three lengths are not side lengths of a triangle. Triangle Inequalities 350 © Houghton Mifflin Harcourt Publishing Company 25. In any triangle ABC, suppose you know the lengths of _ AB and _ BC , and suppose that AB > BC. If x is the length of the third side, _ AC , use the Triangle Inequality Theorem to prove that AB − BC < x < AB + BC. That is, x must be between the difference and the sum of the other two side lengths. Explain why this result makes sense in terms of the constructions shown in the figure. 26. Given the information in the diagram, prove that m∠DEA < m∠ABC. 27. An isosceles triangle has legs with length 11 units. Which of the following could be the perimeter of the triangle? Choose all that apply. Explain your reasoning. a. 22 units b. 24 units c. 34 units d. 43 units e. 44 units H.O.T. Focus on Higher Order Thinking 28. Communicate Mathematical Ideas Given the information in the diagram, prove that PQ < PS. A C B A C B A E D B C 9 9 2 7 4 P Q S R 35° 100° 37° 63° 45° Since AB > BC, BC - AB < 0, so the second inequality is not relevant. Combining the first and last inequalities gives AB - BC < x < AB + BC. The constructions show that AC approaches but is always greater than AB - BC, and that AC approaches but is always less than AB + BC. AB + BC > x AB + x > BC x > BC − AB BC + x > AB x > AB − BC In , △ADE, DA < DE, so m∠DEA < m∠DAE = m∠BAC. In △ABC , AC = 9 + 2 = 11 (Segment Addition Postulate), so BC < AC, and therefore m∠BAC < m∠ABC. Therefore, m∠DEA < m∠ABC (Transitive Property Of Inequality). B, C, D If x represents the length of the third side of the triangle, then by the Triangle Inequality Theorem, solve for 11 + x > 11 and 11 + 11 > x. The solution of the first inequality is x > 0, which is always true. The solution of the second inequality is 22 > x. So the range of possible lengths for the third side is 0 < x < 22. Use both limits to solve for perimeter. 11 + 11 + 0 = 22 and 11 + 11 + 22 = 44. So, the perimeter for all possible triangles must be greater than 22 units and less than 44 units. So choices A and E are not possible. In, △PQR, m∠PRQ < m∠Q , so PQ < PR. In △PRS, m∠PRS = 180° - 37° - 63° = 80°, so m∠S < m∠PRS, and therefore PR < PS. Therefore, PQ < PS (Transitive Property of Inequality). Module 7 351 Lesson 3 y ; GE_MNLESE385795_U2M07L3.indd 351 23/03/14 2:07 AM JOURNAL Have students use diagrams in their journals to illustrate the angle-side relationships in triangles. 351 Lesson 7 . 3 © Houghton Mifflin Harcourt Publishing Company 29. Justify Reasoning In obtuse △ABC, m∠A < m∠B. The auxiliary line segment _ CD perpendicular to ‾→ AB (extended beyond B) creates right triangles ADC and BDC. Describe how you could use the Pythagorean Theorem to prove that BC < AC. 30. Make a Conjecture In acute △DEF, m∠D < m∠E. The auxiliary line segment _ FG creates △EFG, where EF = FG. What would you need to prove about the points D, G, and E to prove that ∠DGF is obtuse, and therefore that EF < DF? Explain. Lesson Performance Task As captain of your orienteering team, it’s your job to map out the shortest distance from point A to point H on the map. Justify each of your decisions. A B D C D G E F A B C D E F G H I 48° 94° 58° 53° Write two equations, AD 2 + CD 2 = AC 2 and BD 2 + CD 2 = BC 2 . Equating expressions for CD 2 , AC 2 - AD 2 = BC 2 - BD 2 and therefore AC 2 - BC 2 = AD 2 - BD 2 . Since the right side is positive, so is the left side, which leads to BC < AC. You would need to show that G lies on _ DE , i.e. between D and E. In that case, since ∠DGF and ∠EGF are supplementary and ∠EGF is acute, then ∠DGF is obtuse. So ∠DGF is the largest angle in △DGF and FG < DF. Since EF = FG, then by substitution, EF < DF. The shortest route is A-C-D-F-G-H. In △ABC the smallest angle measures 48°, so the shortest side is _ AC . In △BCD the route from C to D is shorter than the route from C to B to D, because _ BD is the longest side of the triangle. In △DEF the route from D to F is shorter than the route from D to E to F by the Triangle Inequality Theorem. In quadrilateral FGHI, draw _ FH . Since FI = HI, △FIH is isosceles, with base angles each measuring 61°. So, _ FH is the shortest side of △FIH. _ FH is opposite the largest angle in △FGH, so _ FH is the longest side in triangle in △FGH by the Triangle Inequality Theorem. So, FI > FH > FG and IH > FH > GH. So, the path from F to G to H is shorter than the path from F to I to H. Module 7 352 Lesson 3 y ; GE_MNLESE385795_U2M07L3.indd 352 5/23/14 12:01 PM 90° 90° 90° 95° 95° 170° EXTENSION ACTIVITY Have students design orienteering courses like the one in the Lesson Performance Task. Courses should consist of at least four stages. At each stage of a course, angle measures or other information should be given that will allow an orienteer to apply the Triangle Inequality Theorem, angle-side relationships, and/or side-angle relationships, in order to gauge the shortest route to follow. Students may wish to work in teams of two or three and tackle routes other students have designed. AVOID COMMON ERRORS Students may attempt to apply the methods discussed in this lesson to figures other than triangles, an approach that is likely to lead to false conclusions. The solution, when confronted with a figure like quadrilateral FGHI in the Lesson Performance Task, is to draw one or more diagonals, dividing the figure into triangles whose inequalities can then be analyzed. INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Define the side opposite an angle in a pentagon as the side that neither forms the angle nor is adjacent to a side that forms the angle. Ask students to draw and label a pentagon in which, unlike a triangle, the side opposite the largest angle is the shortest side of the pentagon. Sample figure: Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain. 0 points: Student does not demonstrate understanding of the problem. Triangle Inequalities 352
15152	https://www.youtube.com/watch?v=bmVrJe0RYfk	Congruent Segments & Midpoints DiagKNOWstics Learning 48900 subscribers 11 likes Description 958 views Posted: 23 Jun 2022 Rebecca Shah explains Congruent Segments and Midpoints, as well as what it means to bisect. Transcript: hey guys today we'll learn three new vocab words congruent midpoint and bisect and these three words they're all related so let's start with congruent so congruent just means equal in size so congruent segments have the same length so in this diagram we can see segment a b is congruent to segment cd because they both have a length of 10 centimeters now the cool thing about the word congruent is it has its own special symbol and that symbol looks like this it's an equal sign which kind of makes sense because it means equal with a little squiggly above it now whenever we see that symbol it actually means the words is congruent to now this is great news if you're like me and you don't like writing everything out every time because it can really save us time when we're writing out and describing what we see in these diagrams for example in this diagram we can write out segment a b is congruent to there's our congruent symbol segment cd now believe it or not what i just wrote out is a complete sentence right if you break it down and look at each of these symbols they just stand for words this symbol set stands for segment a b there's our is congruent to and then last we have segment c d at the end so again we can use these symbols to describe what we see in these geometry diagrams now in a diagram we show that two segments are congruent using these little hash marks so on number one we can see that segment w x is congruent to segment y z because they both have these little hash marks on each segment now look look at this next one we have um segment d g and e f both of those just have one hash mark whereas these other two segments have two hash marks so if a segment has a certain number of hash marks then any other segment in the diagram with the same number of hash marks is congruent to that segment now for number three notice only two of the segments are marked as congruent so that's all we know all right so now that leads us to midpoint now a midpoint is exactly what it sounds like it is a point that is perfectly in the middle of a segment now what that means is it cuts the segment in half which makes those two halves equal but wait we have another word for equal right those two halves are congruent so whenever something is a midpoint it divides the segment into two congruent segments and just be careful just because a point is somewhere in the middle of a segment doesn't make it a midpoint you would have to either be told that it's a midpoint or actually see those hash marks in the diagram now that leads us to our third and final word bisect bisects just means to cut something in half so let's look at this diagram and see what's going on it says line bc okay well line bc is right here bisects segment y z now whenever you're reading this the thing that comes after the word bisect is the thing that's getting cut in half so in this case segment y z that's right here is getting cut into two congruent segments y a and segment a z those two segments are congruent all right so let's see some examples so first off in this diagram we can see that these two segments are marked as congruent which means they're equal so whatever you see for one of the segments you would just set that equal to whatever you see for the other one in this case 3x equals 24. that's it just set up the equation and then solve for x in this case we'll divide both sides by 3 which means that x equals eight all right another example using the word midpoint so let's just read this first off a is the midpoint of c t well that means that it's perfectly in the middle and it cuts c t into two congruent segments so let's see what else we have to solve for x and then find the measures of pretty much all the segments so let's see we've got segment c a which is 5x minus 15 and segment a t those two segments are congruent which means that we can set them equal to each other so we have 5x minus 15 equals 2x plus 3. just setting up that equation is half the battle once you've got it this far then you just have to solve for x so let's see we want we've got x's on both sides so i'm going to subtract 2x from both sides to combine my x's so that'll give me 3x minus 15 equals 3 i'll add 15 to both sides giving me 3x equals 18 and then last we'll divide both sides by 3 giving us x equals 6. now be careful x is 6 but that's not our final answer remember it was asking us to find c a a t and c t so i'm going to take that x value of 6 and plug it back into each of those things so for ca plugging 6 in for x we have 5 times 6 minus 15 and that equals 15 and then same thing for a t we have two times six plus three well two times six is twelve plus three that equals fifteen two so we can say c a equals fifteen and a t equals fifteen that kind of makes sense because we said c a and a t are congruent they're equal and last but not least segment c t is the whole thing so that's just 15 plus 15 so that equals 30.
15153	https://en.wikipedia.org/wiki/Incidence_(epidemiology)	Jump to content Incidence (epidemiology) العربية Български Català Čeština Dansk Deutsch Ελληνικά Español Esperanto فارسی Français Bahasa Indonesia Italiano Қазақша Lietuvių Magyar Nederlands 日本語 Norsk bokmål Norsk nynorsk Polski Português Русский Slovenčina Slovenščina Српски / srpski Srpskohrvatski / српскохрватски Suomi Svenska ไทย Türkçe Українська 中文 Edit links From Wikipedia, the free encyclopedia Chance over time of a medical condition \| \| \| --- \| \| \| This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Incidence" epidemiology – news · newspapers · books · scholar · JSTOR (November 2007) (Learn how and when to remove this message) \| In epidemiology, incidence reflects the number of new cases of a given medical condition in a population within a specified period of time. Incidence proportion [edit] Incidence proportion (IP), also known as cumulative incidence, is defined as the probability that a particular event, such as occurrence of a particular disease, has occurred in a specified period: For example, if a population contains 1,000 persons and 28 develop a condition from the time the disease first occurred until two years later, the cumulative incidence is 28 cases per 1,000 persons, i.e. 2.8%. Incidence rate [edit] The incidence rate can be calculated by dividing the number of subjects developing a disease by the total time at risk from all patients: One of the important advantages of incidence rate is that it doesn't require all subjects to be present for the whole study because it's only interested in the time at risk. Incidence vs. prevalence [edit] See also: Prevalence § Difference between prevalence and incidence Incidence should not be confused with prevalence, which is the proportion of cases in the population at a given time rather than rate of occurrence of new cases. Thus, incidence conveys information about the risk of contracting the disease, whereas prevalence indicates how widespread the disease is. Prevalence is the proportion of the total number of cases to the total population and is more a measure of the burden of the disease on society with no regard to time at risk or when subjects may have been exposed to a possible risk factor. Prevalence can also be measured with respect to a specific subgroup of a population. Incidence is usually more useful than prevalence in understanding the disease etiology: for example, if the incidence rate of a disease in a population increases, then there is a risk factor that promotes the incidence. For example, consider a disease that takes a long time to cure and was widespread in 2002 but dissipated in 2003. This disease will have both high incidence and high prevalence in 2002, but in 2003 it will have a low incidence yet will continue to have a high prevalence (because it takes a long time to cure, so the fraction of individuals that are affected remains high). In contrast, a disease that has a short duration may have a low prevalence and a high incidence. When the incidence is approximately constant for the duration of the disease, prevalence is approximately the product of disease incidence and average disease duration, so prevalence = incidence × duration. The importance of this equation is in the relation between prevalence and incidence; for example, when the incidence increases, then the prevalence must also increase. Note that this relation does not hold for age-specific prevalence and incidence, where the relation becomes more complicated. Example [edit] Consider the following example. Say you are looking at a sample population of 225 people, and want to determine the incidence rate of developing HIV over a 10-year period: At the beginning of the study (t=0) you find 25 cases of existing HIV. These people are not counted as they cannot develop HIV a second time. A follow-up at 5 years (t=5 years) finds 20 new cases of HIV. A second follow-up at the end of the study (t=10 years) finds 30 new cases. If you were to measure prevalence you would simply take the total number of cases (25 + 20 + 30 = 75) and divide by your sample population (225). So prevalence would be 75/225 = 0.33 or 33% (by the end of the study). This tells you how widespread HIV is in your sample population, but little about the actual risk of developing HIV for any person over a coming year. To measure incidence rate you must take into account how many years each person contributed to the study, and when they developed HIV because when a subject develops HIV he stops being at risk. When it is not known exactly when a person develops the disease in question, epidemiologists frequently use the actuarial method, and assume it was developed at a half-way point between follow-ups.[citation needed] In this calculation: At 5 yrs you found 20 new cases, so you assume they developed HIV at 2.5 years, thus contributing (20 2.5) = 50 person-years of disease-free life. At 10 years you found 30 new cases. These people did not have HIV at 5 years, but did at 10, so you assume they were infected at 7.5 years, thus contributing (30 7.5) = 225 person-years of disease-free life. That is a total of (225 + 50) = 275 person years so far. You also want to account for the 150 people who never had or developed HIV over the 10-year period, (150 10) contributing 1500 person-years of disease-free life. That is a total of (1500 + 275) = 1775 person-years of life. Now take the 50 new cases of HIV, and divide by 1775 to get 0.028, or 28 cases of HIV per 1000 population, per year. In other words, if you were to follow 1000 people for one year, you would see 28 new cases of HIV. This is a much more accurate measure of risk than prevalence. See also [edit] Attack rate Attributable risk Rate ratio References [edit] ^ Jump up to: a b c Noordzij, Marlies; Dekker, Friedo W.; Zoccali, Carmine; Jager, Kitty J. (2010-02-19). "Measures of Disease Frequency: Prevalence and Incidence". Nephron Clinical Practice. 115 (1): c17 – c20. doi:10.1159/000286345. PMID 20173345. Retrieved 2024-05-29. ^ Brinks R (2011) "A new method for deriving incidence rates from prevalence data and its application to dementia in Germany", arXiv:1112.2720 External links [edit] Calculation of standardized incidence rate Archived 2015-01-09 at the Wayback Machine PAMCOMP Person-Years Analysis and Computation Programme for calculating standardized incidence rates (SIRs) \| v t e Concepts in infectious disease (Outline) \| \| --- \| \| Determinants \| \| \| \| --- \| \| Agent \| Biofilm Germ theory of disease Infectivity + Infectious dose Pathogenesis Pathogenicity + Attack rate Quorum sensing Virulence + Endotoxin + Exotoxin + Case fatality rate + factors Antimicrobial resistance + Drug resistance + Horizontal gene transfer + Multidrug-resistant bacteria Host tropism \| \| Host \| Burn Comorbidity Diabetes Host–pathogen interaction Immune response + Immunodeficiency + Immunosuppression + Immunopathology + Cytokine storm Microbiome health Opportunistic infection Risk of 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15154	https://www.cathstan.org/faith/the-role-of-a-cardinal-in-the-catholic-church	Catholic Standard El Pregonero Classifieds Buy Photos Local US & World Faith Culture Junior Saints Care for Creation All Sections Search Subscribe The role of a cardinal in the Catholic Church Richard Szczepanowski Nov 28, 2020 Faith Print Cardinal Ignatius Suharyo Hardjoatmodjo of Jakarta, Indonesia, is pictured after receiving his red biretta from Pope Francis during a consistory for the creation of 13 new cardinals in St. Peter's Basilica at the Vatican Oct. 5, 2019. (CNS photo/Paul Haring) When Cardinal-designate Wilton Gregory of Washington is elevated to the College of Cardinals on Nov. 28, 2020, he will be given an honor that is more than 1,700 years old. The rank of cardinal in the Catholic Church dates back to the early fourth century when Pope Sylvester I gave that title to his closest advisors. These advisors were also given a pastoral responsibility for one of the parishes in the Diocese of Rome, of which the pope is bishop. The title “cardinal” comes from the Latin word, “cardo” which means a hinge or a door. It referred to the door – or address – of the church in the Diocese of Rome to which the cardinal was assigned. To this day, cardinals remain as close advisors to the pope and automatically become clergy members of the Diocese of Rome or the surrounding area and are still assigned to one of the churches there. This is called the cardinal’s “titular church” because he oversees that church in name only, and is normally not involved in the day-to-day upkeep and operation of that church. Priests and bishops who are named cardinals are not ordained as such. The Sacrament of Holy Orders includes deacons, priests and bishops. Since cardinals are chosen by the pope, it is he who decrees and bestows that title. Until early last century, it was possible for a layman to be named a cardinal. However, Pope Benedict XV decreed in 1917 that only priests or bishops could be elevated to the College of Cardinals. The Code of Canon Law now requires that a priest who is named a cardinal must first be ordained a bishop prior to the consistory in which he is elevated to the College of Cardinals. The world’s cardinals collectively are known as the College of Cardinals, which is led by a dean. Within the college, there are three ranks of cardinals: • Cardinal Bishops are the senior members of the College of Cardinals who are not only assigned a titular church within the Diocese of Rome, but also assigned a titular diocese near Rome. The group also includes patriarchs from Eastern-rite Catholic Churches. It is from among the Cardinal Bishops that the Dean of the College of Cardinals is chosen. • Cardinal Priests are churchmen who are generally bishops or archbishops of historic or important dioceses around the world or who hold important posts within the Vatican Curia. • Cardinal Deacons are priests who have been named to the College of Cardinals after their 80th birthday – and thus cannot vote in a conclave for a new pope – or who are serving in the Vatican Curia. Until the time of Pope (now Saint) Paul VI and later Pope (now Saint) John Paul II, the number of cardinals was kept at 70 to represent the 70 elders chosen from among the tribes of Israel to assist Moses (Numbers 11:16-30). It was also reminiscent of the 72 men appointed by Jesus to go ahead of Him to every town He was about to go (Luke 10:1). Pope St. Paul VI and Pope St. John Paul II increased the number of cardinals to better reflect the international nature of the Church. In addition to their jobs as bishop or archbishop of a diocese or holding an office within the Vatican Curia or any other post they may hold, cardinals meet with the pope – called a Consistory – to discuss and advise the pope on important matters of the Church. Among the most important duties of a cardinal are those that occur at the death of a pope. When the Holy See is vacant, the College of Cardinals is responsible for the day-to-day administration of the Church. They also gather in a conclave to elect a new pope. In 1971, Paul VI decreed that no more than 120 cardinals – all who must be under the age of 80 – could vote for a new pope. However, both Paul VI and John Paul II exceeded that number during their pontificates. At one time during the pontificate of John Paul II, there were 135 cardinals under the 80. After the Nov. 28 consistory, there will be 128 cardinals under the age of 80 who are eligible to vote in a conclave. A cardinal wears distinctive red or scarlet vestments and a red zucchetto (skullcap) or a biretta (a ceremonial square cap with three flat projections) to symbolize his willingness to die for his faith, since red can represent the blood of martyrs. During the Consistory when the pope elevates new cardinals, he places a red hat – the biretta – on the head of each new cardinal. In the 15th century, the zucchetto skull cap began to be used for ceremonial and liturgical occasions. Using the zucchetto to denote clerical rank came into practice. The color of the zucchetto reflects their episcopal order: bishops wear violet, cardinals wear red and the pope wears white. 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15155	https://www.youtube.com/watch?v=VnDnInldaMM	3.1 Defining the Derivative Ryan Melton 1590 subscribers 42 likes Description 6184 views Posted: 20 Mar 2018 OpenStax Calculus Volume 1 1 comments Transcript: Intro section 3.1 defining the derivative this is all from openstax calculus volume 1 at the bottom of every page is a link to go and you can download this the textbook in its entirety it's a great textbook and these videos are meant to supplement and help you go through that textbook so I would totally suggest that you hat let you look at both so let's go ahead and look at what the derivatives unpack this consider the function that passes through the points 1 5 & 5 13 to find the slope or the rate of change we would calculate M equals y2 minus y1 divided by x2 minus x1 and we would get 2 so the slope between those two points is 2 so let's follow through that same process what if we had an arbitrary set of points what if we had the point a F of a and X f of X well that slope this is actually going to be gonna call this the slope of the secant line that would be f of X minus F of a divided by X minus a ok well what if we only considered we're gonna look at these two graphs here what if we only considered the change from the point a if we did that then we would have a separate formula and this is really thinking of change in Y over change in X and we can use these interchangeably we could think of this as f of a plus h where H is a small change minus F of a over H so the substitution here is x equals a plus h so if you did a plus h minus a you get the H in the denominator and you would get the F of a plus h ok but that is the substitution we're really making there now these two equations as I said can be used interchangeably and there are good times to use each one but this is called the difference quotient the second one there is called the difference quotient f of a plus h minus f of a or H all right so let's there's the grabbes that we want to get that idea from and then let's continue on so as the and X approaches zero we see a more accurate slope okay so if we took a secant line I'm just going to draw it on here if I took the secant line between these two points here well what if I take a point that is closer to a alright well that would be drew on the wrong graph if I put a point right here well the slope of that secant line is a little bit closer to what's happening on the graph itself okay so as the change in X is I get closer and closer to a we have something we're gonna call the instantaneous slope and the formal definition will be forthcoming all right so let's first go ahead and look at the slope of the secant line for the function f of X equals the square root of x with the given point 4 comma 2 and the value of x alright so that means we are going to have the point a F of a that is our that is the 4 comma 2 and our value of x is 5 so this is going to be so for the slope of the secant line f of X minus F of a over X minus a well our value of x is 5 so we'll plug in 5 that is to me the point 5 square root of 5 so this is the square root of 5 minus 2 over our x value 5 minus our point of 4 and that is approximately zero point two three six zero six seven nine okay if we're just rounding out to a bunch of decimal places that's what the slope is okay well let's next consider the value x equals four point five so we're the idea is that we're getting close or closer to four I'm gonna leave that as the square root of four point five so that slope of the secant line is f of X that'll be the square root of four point five minus our y-value that we were given is two okay four point five minus four if you notice the denominator is approaching zero those two numbers are getting closer together well that is approximately zero point two four two six four zero six if we calculate that all right it's all well and good we had point three eight was it 0.236 we have point two four two well let's calculate that once more but we'll use the point four point one square root of four point one well the slope of the secant line will be the square root of four point one minus two over four point one minus four again the denominator is approaching zero those are getting closer together and that is approximately zero point two four eight four five six seven now I want to make this a note all right note that as X as the limit as X approaches four f of X or are actually our slope of our secant line okay is approaching one fourth and here in a couple of questions couple of problems we'll see why that is happening but as we get closer to four the slope of our secant lines is approaching one fourth that is going to be our instantaneous rate of change once we get there all right so here is a definition well f of X be a function Definition defined in an open interval containing a the tangent line to f of X at a is the line passing through the point a F of a having a slope of M equals the limit as X approaches a of f of X minus F of a divided by X minus a provided this limit exists now equivalently we can use the other the difference quotient version of this but this will be the limit as H approaches zero now we will use both of these and in the end you can pretty much pick which one you want Example alright so let's go ahead and dive into this we're going to use the first equation on this one find the equation of the line change to the graph f of x equals x squared at x equals 3 okay so let's go ahead and use that first one so first let's find our slope all right the limit as X approaches a in this case that is as X approaches 3 of f of X well f of X is x squared okay and we have F of a that is going to be 9 because this is the point 3 9 okay divided by X minus a so X minus 3 all right well based on our last sections our last chapter this is going to be the limit as X approaches 3 of X plus 3 that factors and cancels and so that limit is 6 so our slope is 6 all right now let's take our point and we'll say Y minus y1 or y minus F of a if you're looking at this formula right here if F a so Y minus 9 equals slope times X minus 3 y minus 9 is 6x minus 18 and adding 9 to both sides 6x minus 9 okay take a minute and soak that in okay go back at the point of the videos that you can pause and rewind okay so take a minute and pause soak that in see why that makes sense okay so I'm going to go ahead and do the same problem using the second equation so my slope is the limit as H goes to zero goes to zero of f of X plus h or a plus h I'm going to go ahead and write the equation in here f minus F of a over H so if I plug in a plus h okay a plus h squared is a squared plus 2a h plus h squared so evaluating that I'm gonna have the limit as H goes to zero of a squared well in this case a is actually three this is the point three nine so if I plug in three there that is nine plus 6h plus h squared minus f of a which that is nine over H okay well a couple of things cancel first off the nines cancel and then we can divide an H into everything so this is the limit as H goes to zero of 6 plus h as h goes to zero that limit is 6 so our slope is 6 just as we had before so this is y minus 9 equals 6 times X minus 3 and doing all the same distributing and algebra there we get y equals 6x minus 9 all right for our next question find the equation of the line tangent to the graph f of X equals 1 over X at x equals 2 let's go ahead and make a point that is 2 comma 1/2 now it does not matter which of these two we use I'm going to go ahead and use the second equation okay now I'll go ahead and use the first equation so slope is f of X minus F of a okay so f of X is 1 over X F of a is 1/2 divided by X minus a ok now I need to make a note here the limit as X approaches 2 ok well this is a complex fraction so we'll go ahead and clear clear that I'm multiplying by 2 X alright so that'll be 2 minus x over 2 x times X minus 2 okay well the X minus 2 and the 2 minus X cancel which gives us negative 1 over X or over 2 X and so my slope as X approaches 2 is negative 1/4 okay now let's go to our point-slope Y minus 1/2 equals M negative 1/4 times X minus 2 distributing our negative 1/4 negative 1/4 X plus 1/2 adding 1/2 to both sides negative 1/4 X plus 1 now to continue to show that these are interchangeable I'm going to go ahead and use the second equation for this next one so I have find the equation of the line tangent to the graph f of X equal square root of x at x equals 4 so my point is 4 2 and I would like to point out this is going to be very similar in answer at least to that first question that we did alright so that is our slope will be the limit as H approaches 0 of f of a plus h so this would be the square root of a plus h when our value of a is actually 4 so 4 plus h minus our y-value there so - - because we're evaluating at a their over H well multiply by our conjugate would be a good choice here do that in blue square root of 4 plus h plus 2 square root of 4 plus h plus 2 carrying on our limit we cannot forget that that's really important we get 4 plus h minus 4 over H times square root of 4 plus h plus 2 ok a few things cancel out here 1 the fours and the H so this becomes 1 over the limit as H approach 0 1 over square root of 4 plus h + 2 now we actually pulled this conjugate trick previously in chapter 2 so as H goes to 0 this approaches 1/4 so the slope is 1/4 all right now we have y minus y1 equals 1/4 X minus 4 y minus 2 is 1/4 X minus 1 minus 1 and y equals 1/4 X plus 1 and there is the equation of our line that has tangent to that point okay now notice that the slope here was 1/4 is why I made a reference back to another one we noticed that our slope was approaching it was point 2 4 8 something and I said that notice that as X is approaching 4 that the slope of the secant line is approaching 1/4 that's exactly what just happened right there all right so what we've been doing is finding the derivative okay so here is Finding the Derivative the definition of that the tangent line at a given point it's called the derivative and the process of finding our slope that limiting process limits are the most important thing in all of calculus this limiting process the finding we did is called differentiation alright so we now make this definition that f prime of a that is our derivative of f of X at a is equal to that limit that we have just found and both of these are equivalent as necessary alright so let's go ahead and use this alright so for x equals x squared use a table to estimate f prime of 3 now I'm not a big fan of this method because using the limits is a lot more accurate however we'll go ahead and do this so f prime of let's see how do we want to say this we'll go ahead and say the slope of our secant line is f of X minus f of a which in this case would be 9 over X minus 3 so that is the function that we're looking at now we want to know what this approach is so we're really we're really looking at a limit okay so f prime of 3 is the limit as H goes to 0 or as X approaches 3 of x squared minus 9 over X minus 3 it's that limit okay so if we look at a table I'm going to actually make a table of the function x squared minus 9 over X so x squared minus 9 over X minus 3 so as you approach three is the idea okay ask alright so as I approach three so say at a value of two point nine okay this is x squared minus 9 over X minus three at a value of two point nine it is five point nine at a value of two point nine nine we have five point nine nine okay so it appears that it might be getting close to six okay let's go the other direction let's go three point one three point one the value is six point one of that of that Cicotte slope of the secant line 3.01 the slope is six point oh one so it appears from this table that is approaching six now if we evaluate the limit itself that is the limit as X approaches 3 of X plus three because it cancels and our limit is 6 so we can estimate f prime of 3 to say it based on a table that say it is approximately six but we can say from our limit definition F prime of 3 gig is equal to six based on that limit alright number seven find the derivative of f of X equals 3x squared minus 4x plus 1 and we want to find F prime of 2 specifically alright so we'll use our first equation here so that would be the slope okay the slope and we're using a is two so really we have a point two comma 12 minus 8 that's four that's five okay all right yeah okay so f of this would be the limit as X approaches 2 of f of X the definition f of X minus a so f of X so we'll go and just plug that in there 3x squared minus 4x plus 1 minus F of a which I just said is 5 and I'm just gonna go ahead and check that real quick let's see 3 2 squared minus 4 times 2 plus 1 all right 2 5 all right so minus F of 2 which is 5 okay over X minus 2 f of X minus F of a divided by X minus a all right well if we work on that what's C that's gonna get minus 4 it's gonna be an F of okay 3x squared minus 4x minus 4 divided by X minus 2 and I'm gonna go ahead and take a stab at factoring that to be 12 so 2 & 6 yes okay I believe this is going to factor and I'm gonna go ahead and check that of course there's gonna be 3x plus 2 X minus 2 let's see if that works 3x squared that's negative negative 6 X plus 2 okay that factors so then canceling those 2 factors of X minus 2 canceling my two factors there and evaluating at to my slope is 6 that is 8 my slope is 8 so that would mean if I go back and replace M with F prime of 2 F prime of 2 equals 8 right now let's use the second equation just to show these are completely interchangeable all right so this will be my slope or F Prime 2 equals f of a plus h so that is f of that's the a plus h so I'm going to replace everywhere I have I need to limit their limit as H goes to 0 of 3 a plus h squared minus 4a plus h plus 1 minus F of 2 which in this case we said was 2 5 so minus 5 over H okay well our a our value of a is 2 so actually let's replace let's replace that with a - all right now doing some algebra off to the side this is going to be H goes to 0 of 3 H squared + 12 H plus 12 minus 4 H minus 8 plus 1 minus 5 leave that's right we have a minus 8 - 4 H okay and then squaring that that's gonna be a 4 so we get a 12 in there that is a 4 H times 3 so 12 H and then a 3 H squared ok all that divided by H if you notice some terms cancel out we cancel out let's see 12 minus 8 12 minus 8 plus 1 minus 5 so that would be 4 5 that is 0 all that cancels out so we are left with a bunch of H terms all right so I'm gonna go ahead and factor the H out 3h plus 12 minus 4 no that'd just be an 8 just go and do that Plus 8 okay and that's divided by H ok so canceling the H terms and then plugging in 0 we get 8 therefore f prime of 2 equals 8 as we deduced with the other formula as well all right next find the derivative of this and we're going to find F for G prime of 1 so G prime of 1 I'm going to use the second formula so as H goes to 0 of F of a plus h so a plus h in this case a is 1 so I'll replace that so 1 plus h squared plus 3 times 1 plus h plus 2 minus if I evaluate that at 1 that'll be 1 plus 3 so 4 minus 6 over H all right using some algebra here we get a 1 plus 2 h plus h squared plus 3 leave that plus 3 plus 3 h plus 2 minus 6 over h ok combining some terms something should cancel out every time we have 3 that's 5 ok ok there we go the 1 3 all of our constants cancel out to our H terms combine okay so we'll go and factor in H that's gonna be a 5 plus H over H canceling the H terms and letting H go to 0 we get to 5 therefore G prime of 1 equals 5 all right now for some application so recall that s of T is the position function for Application an object's motion so the average velocity is found by using our secant line definition that is the average velocity is s of t minus s fa divided by t minus a for a given point a ok now as we approach the point a the values of the average velocity approach the instantaneous velocity and now we can say the instantaneous velocity at a is the derivative of the position function okay which is that women and the instantaneous rate of change of a function at that point is its derivative at that point okay so a couple of examples we have here a lead weight on spring is oscillating up and down its position at time T with respect to a fixed line horizontal line is s of T equals T determine the value of V of 0 ok so V of 0 is s prime of 0 which is the limit as H goes to 0 of well that would be a plus F of a plus h so in this case s of a plus h ok oh I guess I should go ahead and use that version of it shouldn't I all right so we'll go ahead and do that is the limit as T approaches 0 of s of t minus s of 0 over t minus a so that will just be T well if you evaluate sign at T and sign at 0 this is the same as the limit as T goes to 0 of sine T over tea which we happen to know is one so V of zero is one as we discovered in the previous chapter next question we have a rock dropped from a height of 64 feet so the position is given by this function we have so we want to find the instantaneous velocity one second after it's dropped so that would be V of one which is s prime of one which is the limit as T approaches at this point 1 and if we evaluate that function at T so negative 16t squared actually we're gonna have to evaluate that at T yeah we can just a with that negative 16t squared plus 64 minus evaluating it at 1 gets us 48 divided by t minus 1 so once we combine terms there limit as T goes to 1 okay combining terms I get negative 16t squared plus 16 over t minus 1 now I'm gonna bring over this over the next line if i factor out a negative 16 I'll leave that limit as T approaches 1 i factor at the negative 16 I get T squared minus 1 over t minus 1 and that is going to cancel to be the limit as T approaches 1 of negative 16 T plus 1 now if I evaluate that that is negative 32 therefore V of 1 is negative 32 feet per second all right almost to the end here a homeowner sets the thermostat said the temperature in a house begins drop from 70 degrees Fahrenheit at 9 p.m. reaches a low of 60 during the night and rises back to 70 by 7:00 a.m. the next morning so we have this function here there's the temperature at time T where T is the number of hours past 9:00 p.m. so find the instantaneous rate of change of the temperature at midnight so what we want is the instantaneous rate of change at midnight so that is at T prime of 3 so that is a limit as T approaches 3 of T of T 0.4 T squared minus 4t plus 70 minus if we evaluate that at 3 it's a point 4 3 squared minus 12 plus 70 is 60 1.6 and that is over t minus 3 3 that is right okay so that is 8.4 the next line T is T approaches 3 of 0.4 T squared minus 40 plus 8.4 over t minus 3 all right now I'm gonna go ahead and use the table to evaluate this just because I can see minus four X plus eight point four divided by X minus three all right so I want to get close to three so if I plug in 2.9 that's negative 1.6 for my value of T at two point nine its negative one point six four two point nine nine is negative one point six zero for all right three point one I'm going to three point oh one as well three point one is a negative 1.56 3.16 so it appears based on inspection that the instantaneous rate of change of the temperature at that point is one point six degrees Fahrenheit per hour all right one last question toy company can sell X electronic gaming systems at a price of P equals negative 0.01 X plus four hundred dollars per gaming system the cost of manufacturing that is C of X fun the rate of change of profit okay well first off our revenue function is the price or the amount we make or the premiere right like this the profit is the revenue which is x times negative 0.01 X plus 400 minus our cost in this case 100 X plus $10,000 so our profit function is negative 0.01 x squared plus 300 X minus $10,000 all right so P prime of zero is what we are interested in or know of ten thousand games gaming systems is P of X negative 0.01 x squared plus 300 X minus 10,000 minus if we evaluate that at ten thousand which is going to be let's see how many zeros is that all right 1 million nine hundred and ninety ninety thousand okay so that limit alright so that would be this over and this would be X minus ten thousand and as a limit as X approaches ten thousand okay that limit approaches the value of 100 so since the rate of change is greater than zero the rate of change in profit so they are their profit is increasing okay their profit is increasing and their profit currently is greater than zero they should increase production so that was our question was should they increase or decrease production since their profits are increasing and they're currently positive that is good okay they they should increase their production all right so that is the end of this section probably the most tedious section of this entire course and we'll actually look at some rules and see this as a function in the next section so that is it 3.1
15156	https://cdn.standards.iteh.ai/samples/72505/1d9b745782c54e48b453aa8fc0c61610/ISO-IEC-24570-2018.pdf	Software engineering — NESMA functional size measurement method — Definitions and counting guidelines for the application of function point analysis Ingénierie du logiciel — Méthode de mesure de la taille fonctionnelle NESMA — Définitions et manuel des pratiques de comptage pour l'application de l'analyse des points fonctionnels INTERNATIONAL STANDARD ISO/IEC 24570 Reference number ISO/IEC 24570:2018(E) Second edition 2018-02 © ISO/IEC 2018 iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 ISO/IEC 24570:2018(E) ii © ISO/IEC 2018 – All rights reserved COPYRIGHT PROTECTED DOCUMENT © ISO/IEC 2018 All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below or ISO’s member body in the country of the requester. ISO copyright office CP 401 • Ch. de Blandonnet 8 CH-1214 Vernier, Geneva, Switzerland Tel. +41 22 749 01 11 Fax +41 22 749 09 47 copyright@iso.org www.iso.org Published in Switzerland iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 ISO/IEC 24570:2018(E) Foreword...........................................................................................................................................................................................................................................v Introduction to this Standard....................................................................................................................................................................................vi 1 Scope..................................................................................................................................................................................................................................1 1.1 Purpose. .......................................................................................................................................................................................................... 1 1.2 Conformity. .................................................................................................................................................................................................. 1 1.3 Applicability............................................................................................................................................................................................... 1 1.4 Focus................................................................................................................................................................................................................. 1 2 Introduction to FPA. ............................................................................................................................................................................................2 2.1 Brief description of FPA................................................................................................................................................................... 2 2.1.1 Background, purpose and application of FPA. .........................................................................................2 2.1.2 Rationale behind FPA. ...................................................................................................................................................2 2.2 Use of FPA: application versus project functional size.......................................................................................3 2.3 Types of function point analyses.............................................................................................................................................. 3 2.4 Function point analyses during a project. ......................................................................................................................... 3 2.5 Scope of the analysis and boundary of the application to be analyzed...................................................4 2.6 Users................................................................................................................................................................................................................. 4 2.7 Functions and function types...................................................................................................................................................... 4 2.8 The complexity of a function....................................................................................................................................................... 5 2.9 The valuing of functions. .................................................................................................................................................................. 6 2.10 The functional size. ............................................................................................................................................................................... 6 3 Guidelines to perform an FPA..................................................................................................................................................................7 3.1 Step-by-step plan to perform an FPA. ................................................................................................................................... 7 3.2 Types of function point analyses and their accuracy. .............................................................................................7 3.2.1 Indicative function point analysis...................................................................................................................... 8 3.2.2 High level function point analysis...................................................................................................................... 9 3.2.3 Detailed function point analysis.......................................................................................................................... 9 3.3 Role of the quality of the specifications. ..........................................................................................................................10 3.4 FPA during a project.........................................................................................................................................................................10 3.5 Determining the functional size of an application.................................................................................................11 3.5.1 Determining the application boundary. .....................................................................................................11 3.5.2 Functional size of new applications..............................................................................................................12 3.5.3 Functional size of enhanced applications................................................................................................12 3.5.4 Functional size of re-built applications......................................................................................................12 3.6 Determining the functional size of a project...............................................................................................................13 3.6.1 Determining the scope of a project function point analysis.....................................................13 3.6.2 Functional size of development projects..................................................................................................14 3.6.3 Functional size of enhancement projects.................................................................................................15 3.6.4 The project function point analysis during the replacement of an application......16 3.7 Definition of functional change. ..............................................................................................................................................16 3.7.1 General...................................................................................................................................................................................16 3.7.2 Modification of a transactional function...................................................................................................16 3.7.3 Modification of a data function. .........................................................................................................................16 3.7.4 Modification of a DET................................................................................................................................................17 3.8 FPA in specific situations. .............................................................................................................................................................17 3.8.1 Analyzing on the basis of traditional design..........................................................................................17 3.8.2 Analyzing packaged software.............................................................................................................................17 3.8.3 Analyzing screens or windows..........................................................................................................................19 3.8.4 Analyzing when prototyping...............................................................................................................................20 3.9 Illustration: FPA and the application life cycle..........................................................................................................21 3.9.1 FPA during the requirements phase.............................................................................................................21 3.9.2 FPA during the analysis phase...........................................................................................................................22 3.9.3 FPA during the functional design phase....................................................................................................23 3.9.4 FPA during the construction phase. ...............................................................................................................24 © ISO/IEC 2018 – All rights reserved iii Contents Page iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 ISO/IEC 24570:2018(E) 3.9.5 FPA during the implementation phase.......................................................................................................24 3.9.6 FPA during the operation and maintenance phase..........................................................................24 4 General FPA guidelines................................................................................................................................................................................25 4.1 Analyzing from a logical perspective.................................................................................................................................25 4.2 Applying the rules..............................................................................................................................................................................25 4.3 No double counting...........................................................................................................................................................................25 4.4 Built functionality, non-requested functionality. .....................................................................................................25 4.5 Production of re-usable code. ...................................................................................................................................................26 4.6 Re-use of existing code. ..................................................................................................................................................................26 4.7 Screens, windows and reports................................................................................................................................................26 4.8 Input and output records.............................................................................................................................................................26 4.9 Security and authorization.........................................................................................................................................................26 4.10 Operating systems and utilities..............................................................................................................................................27 4.11 Report generators and query facilities. ............................................................................................................................27 4.12 Graphs. ..........................................................................................................................................................................................................27 4.13 Help facilities..........................................................................................................................................................................................27 4.14 Messages....................................................................................................................................................................................................28 4.15 Menu structures. ..................................................................................................................................................................................28 4.16 List functions..........................................................................................................................................................................................28 4.17 Browse and scroll functions......................................................................................................................................................28 4.18 Cleanup functions. ..............................................................................................................................................................................29 4.19 Completeness check on the function point analysis.............................................................................................29 4.20 FPA tables..................................................................................................................................................................................................29 4.21 Deriving logical files (data functions) from a normalized data model.................................................30 4.21.1 Introduction.......................................................................................................................................................................30 4.21.2 Denormalization rules..............................................................................................................................................30 4.21.3 The nature of the relationship (cardinality and optionality)..................................................31 4.21.4 Independence or dependence of an entity type.................................................................................31 4.21.5 Conversion table: from normalized entity types to logical files...........................................33 4.22 Shared use of data..............................................................................................................................................................................34 4.23 Generic rule for counting data element types. ...........................................................................................................37 5 Internal Logical Files. .....................................................................................................................................................................................37 5.1 Definition of an internal logical file. ....................................................................................................................................38 5.2 Identifying internal logical files. .............................................................................................................................................38 5.3 Determining the complexity of internal logical files............................................................................................39 6 External Logical Files.....................................................................................................................................................................................40 6.1 Definition of an external logical file....................................................................................................................................40 6.2 Identifying external logical files. ............................................................................................................................................41 6.3 Determining the complexity of external logical files...........................................................................................43 7 External Inputs. ....................................................................................................................................................................................................43 7.1 Definition of an external input................................................................................................................................................44 7.2 Identifying external inputs.........................................................................................................................................................45 7.3 Determining the complexity of external inputs. .......................................................................................................46 8 External Outputs. ................................................................................................................................................................................................48 8.1 Definition of an external output. ............................................................................................................................................48 8.2 Identifying external outputs. .....................................................................................................................................................50 8.3 Determining the complexity of external outputs....................................................................................................52 9 External Inquiries.............................................................................................................................................................................................53 9.1 Definition of an external inquiry...........................................................................................................................................54 9.2 Identifying external inquiries..................................................................................................................................................55 9.3 Determining the complexity of external inquiries.................................................................................................56 Annex A (normative) Summary features for valuing function types................................................................................58 Annex B (normative) Function Point Analysis glossary.................................................................................................................63 Annex C (informative) Increase in Functional Size. .............................................................................................................................68 iv © ISO/IEC 2018 – All rights reserved iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 ISO/IEC 24570:2018(E) Foreword ISO (the International Organization for Standardization) and IEC (the International Electrotechnical Commission) form the specialized system for worldwide standardization. National bodies that are members of ISO or IEC participate in the development of International Standards through technical committees established by the respective organization to deal with particular fields of technical activity. ISO and IEC technical committees collaborate in fields of mutual interest. Other international organizations, governmental and non-governmental, in liaison with ISO and IEC, also take part in the work. In the field of information technology, ISO and IEC have established a joint technical committee, ISO/IEC JTC 1. The procedures used to develop this document and those intended for its further maintenance are described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the different types of document should be noted. This document was drafted in accordance with the editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives). Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. ISO and IEC shall not be held responsible for identifying any or all such patent rights. Details of any patent rights identified during the development of the document will be in the Introduction and/or on the ISO list of patent declarations received (see www .iso .org/ patents). Any trade name used in this document is information given for the convenience of users and does not constitute an endorsement. For an explanation on the voluntary nature of standards, the meaning of ISO specific terms and expressions related to conformity assessment, as well as information about ISO's adherence to the World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT) see the following URL: www .iso .org/ iso/ foreword .html. This document was prepared by NESMA and was adopted, under the PAS procedure, by Joint Technical Committee ISO/IEC JTC 1, Information Technology, in parallel with its approval by national bodies of ISO and IEC. This International Standard is the latest release in the continually improving Nesma method. This method is a consistent interpretation of functional size measurement in conformance with ISO/IEC 14143-1:2007. The Nesma functional size measurement method is known as Function Point Analysis (FPA)1) and the unit of functional size is called Function Point. This second edition cancels and replaces the first edition (ISO/IEC 24570:2005), which is now obsolete. Functional size measurements as determined based on this new edition of the standard are identical to those based on the previous edition of the standard. Results obtained in the past do not need to be updated. 1) In this document the abbreviation FPA is used for the term Function Point Analysis. © ISO/IEC 2018 – All rights reserved v iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 ISO/IEC 24570:2018(E) Introduction to this Standard Reason for this International Standard Over the years a number of "dialects" have arisen for function point analysis. These dialects complicate the goal of determining the number of function points and make it almost impossible for organizations to compare results. One insufficiently acknowledged reason for this is that different interpretations of the "Albrecht" method have arisen. This International Standard provides clarity by formulating standards for the definitions and counting guidelines that pertain to FPA. Intended audience This International Standard is meant for everyone who performs function point analyses. It is assumed that the reader has some knowledge of function point analysis. Nevertheless, we have attempted to produce an International Standard that is both complete and includes sufficient introductory material and explanation for the new user. Application of this standard in practice This International Standard is one component in the Nesma publications. It is recommended that it be read in conjunction with the other Nesma publications. These provide guidance to application of the rules specified within this International Standard and background information to aid in understanding the use and applicability of the resulting functional size. Supporting Nesma publications include the following: — Examples to illustrate the use of the Nesma method in specific situations and a fully documented Hotel case. — Nesma website at nesma.org which contains a number of documents that can be used in a specific context, for example guidelines how FPA can be used in a Data Warehouse environment, with UML documentation, or different aspects in contracts. vi © ISO/IEC 2018 – All rights reserved iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 ISO/IEC 24570:2018(E) Organization of this International Standard Clause 1 describes the scope of this International Standard. Clause 2 provides an introduction to FPA and in which the functional aspect of FPA is emphasized. It will also spell out briefly what FPA is and explains the terms that form the basis for the concept of FPA. Matters such as distinguishing between an application function point analysis and a project function point analysis are examined, just as are other various types of function point analyses, the role of FPA during a project, users, and function point analysis. Clause 3 provides an overview of the position of FPA in a project and explains the types of function point analyses that can be carried out during the life cycle of an application. In other words, the clause explains when FPA can be applied and what information is needed minimally in order to count. The clause will also give a step-by-step plan for performing a function point analysis and indicates how projects, applications, and packaged software should be counted. Each of these requires their approach. Clause 4 states general counting guidelines for a function point analysis. Clauses 5, 6, 7, 8 and 9 successively give the definitions and guidelines used to identify function types and to determine the complexity of function types for internal logical files, external logical files, external inputs, external outputs, and external inquiries. The guidelines are broken down per function type for identifying the function type concerned, for determining the number of data element types, and for determining the number of record types or referenced logical files. Annex A is meant to be a short summary of the guidelines and contains the most important features of each function type, as well as the tables for valuing the function types. Annex B contains the definitions of the terms in this International Standard. Annex C describes the mechanisms behind the increase in functional size. This International Standard has been set up in such a way that the reader does not necessarily have to start at Clause 1 before continuing on to Clause 2, then 3 and 4 etc. Instead, the reader can look up what is important to him. For one reader, specific counting guidelines for a particular function type may be important, while someone else may want a more general frame of reference for an initial introduction to FPA. © ISO/IEC 2018 – All rights reserved vii iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 Software engineering — NESMA functional size measurement method — Definitions and counting guidelines for the application of function point analysis 1 Scope 1.1 Purpose This International Standard specifies the set of definitions, rules and guidelines for applying the Nesma Function Point Analysis (FPA) method. 1.2 Conformity This International Standard is conformant with all mandatory provisions of ISO/IEC 14143-1:2007. 1.3 Applicability This International Standard can be applied to all functional domains. 1.4 Focus The International Standard focuses on how the functional size of an application is determined. The International Standard does not go into any of the aspects that play a role when project budgets are established on the basis of this functional size (e.g. productivity standards and productivity attributes). The figure below indicates what this International Standard will and will not cover. Figure 1 — Scope of the International Standard INTERNATIONAL STANDARD ISO/IEC 24570:2018(E) © ISO/IEC 2018 – All rights reserved 1 iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 ISO/IEC 24570:2018(E) 2 Introduction to FPA This clause gives a short description of FPA and explains a number of important concepts related to it. More specifically, subclause 2.1 provides a brief synopsis of FPA. Subclauses 2.2 through 2.4 distinguish between the different types of function point analyses. Subclauses 2.5 through 2.9 discuss each of the following successively within the context of FPA: — The boundaries for an analysis — Users — Function types — The complexity of a function type — The valuing of function types Subclause 2.10 defines the term functional size and describes how it is determined. 2.1 Brief description of FPA 2.1.1 Background, purpose and application of FPA FPA was developed by A.J. Albrecht at IBM between 1974 and 1979 as a result of productivity research into a large number of projects. The first release of FPA was introduced in 1979, followed by adaptations based on practical experiences in 1983 and 1984. FPA introduces a unit, the function point, to help measure the size of an application that is to be developed or maintained. The word "application" within the framework of FPA means "an automated information system". The function point expresses the quantity of information processing that an application provides to a user. This unit of measurement is separate from the way in which the information processing is realized in a technological sense. A function point is an abstract term and can be compared somewhat to so-called "rental points". Rental points are based on the number of rooms in a house, the surface area of these rooms, the number of facilities the house has, and the location of the house. This then serves as a measurement for a residence offered to a potential tenant. FPA was first used to measure the productivity of system development and system maintenance after an application was built. It soon became clear that the technique could also be used to support project budgeting because the data needed for an FPA can be made available early on in a project. 2.1.2 Rationale behind FPA The three separate words that make up the term "Function Point Analysis" can be used to explain the way of thinking behind FPA. Function As mentioned earlier, FPA bases itself on the functionality that an application provides to a user (see subclause 2.6). Because users see only the "outside" or the boundary (see subclause 2.5) of an application, FPA examines the specifications that describe the application's exchange of information with its environment. Functionality is derived from incoming and outgoing information flows (these can be both data or control information), as well as from the logical files that an application contains or uses. The functionality of an application is measured by identifying data functions and transactional functions (see subclause 2.7). Point The complexity of a function type is determined according to certain standard guidelines (see subclause 2.8). Each function is worth a number of points, depending on its complexity (subclause 2.9). The sum of these points yields the functional size (see subclause 2.10). 2 © ISO/IEC 2018 – All rights reserved iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 ISO/IEC 24570:2018(E) Analysis FPA is the analysis of an application or the analysis of the description/specification of an application in order to establish its functional size. The act of establishing the functional size of an application or project is therefore called function point analysis. In order to be able to perform a function point analysis the following must first be determined: — purpose of the function point analysis (subclause 2.2); — scope of the analysis and boundaries of the application or project to be analyzed (subclause 2.5). This concludes a summary of the methodology and a brief description of FPA. The subclauses that follow explain the various terms used in FPA. 2.2 Use of FPA: application versus project functional size Functional size can be linked to applications or to projects. This means that a distinction is made between the following two objectives: — Determining the functional size of an application. — Determining the functional size of a project. Application functional size is the number of function points that is a measure for the amount of functionality that an application is to supply or has already supplied to a user. It also is a measure for the functional size of an application that must be maintained. Project functional size is the number of function points that is a measure for the amount of functionality of a new application or of changes to an existing application. Changes to an existing application pertain to adding, changing, and deleting functions. The project functional size is an essential parameter when determining the effort and schedule required for a project. Determining the application functional size is elaborated on in subclause 3.5. Subclause 3.6 discusses the project functional size further. 2.3 Types of function point analyses One of three types of function point analyses can be chosen, depending on the degree of detail of the specifications available. The following represent the different types of function point analyses. Notice that they are listed by degree of detail, number one having the least detail and number three the most: 1 Indicative function point analysis 2 High level function point analysis (previously known as Estimated) 3 Detailed function point analysis These function point analyses are explained further in subclause 3.2. 2.4 Function point analyses during a project Function point analyses can be carried out at different times during a project. They can therefore be related to the phases of a project (e.g. the planning phase, the execution phase, and the evaluation phase). As a result, the following breakdown of function point analyses arises: the initial function point analysis, the interim function point analysis, and the final function point analysis. These analyses are discussed further in subclause 3.4. © ISO/IEC 2018 – All rights reserved 3 iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 ISO/IEC 24570:2018(E) 2.5 Scope of the analysis and boundary of the application to be analyzed The scope of the analysis is the set of functional requirements/specifications to be included in the function point analysis. When the scope has been determined, the boundary can be defined, the conceptual interface between the application and its users and/or other applications. As indicated earlier in subclause 2.1, the scope of the analysis and the boundary of an application to be counted plays an important role in FPA. Consequently, the boundaries of the application to be counted must first be determined in order to be able to perform a function point analysis. The boundary is necessary in order to be able to determine: — the application that certain data belongs to; — which data crosses the boundary. As mentioned in subclause 2.2, a distinction is made between a function point analysis for an application and a function point analysis for a project. Subclause 3.5.1 provides guidelines for determining the application functional size and subclause 3.6.1 gives guidelines for determining the project functional size. 2.6 Users FPA acknowledges three types of users: — The people and/or organizations that use or are going to use the application to be measured. This category includes, amongst others, the following: end-users, functional managers, and operators. — The owner and/or employee(s) who determine(s) the requirements and wishes included in the specifications. These requirements and wishes are recorded on the basis of the demands of the end-user(s) for example, but also on the basis of requirements that a government or its legislation can impose on the application. — Other applications that use the data or the functions of the application to be analyzed. Because the function point analysis takes place from the perspective of the user(s), it is always necessary to have it done in cooperation with the user or, at the very least, to have the result of the analysis verified by the user. The user, after all, is the only one who can determine whether a certain function is being requested. 2.7 Functions and function types The function point analysis measures the size of the functions of (a part of) an application. The analysis revolves around the what and not the how of the application to be analyzed. Only those components that the user requests, can recognize and considers significant are assessed. These components are called functions or base functional components. A function belongs to a function type. FPA defines function types as follows: The five types of components of which an application exists, as seen from the perspective of FPA. These components determine the amount of functionality an application provides to a user. 4 © ISO/IEC 2018 – All rights reserved iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 ISO/IEC 24570:2018(E) Figure 2 — Functions and function types Function types are categorized into two main groups: — Data function types — Transactional function types A data function is: a logical group of data seen from the perspective of the user. FPA distinguishes between the following data function types: — Internal logical files — External logical files A transactional function is: an elementary process that meets the following criteria: — the function has an autonomous meaning to the user and fully executes one complete processing of information, and — after the function has been executed, the application is in a consistent state. FPA distinguishes between the following transactional function types: — External inputs — External outputs — External inquiries Each function type is discussed in detail in clauses 5 through 9. 2.8 The complexity of a function The complexity of a function is defined as follows: The weight of a function on the basis of which a number of function points are allocated to the function. The complexity of a function is determined by using the appropriate complexity matrix. A separate table has been defined for each function type. Complexity depends on the number of data element types and the number of referenced logical files connected to a given function. Three levels of complexity are distinguished: © ISO/IEC 2018 – All rights reserved 5 iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 ISO/IEC 24570:2018(E) Low: Few data element types and/or referenced logical files are involved with the function. Average: The function is neither low nor high with regards to complexity. High: Many data element types and/or referenced logical files are involved with the function. The complexity tables that determine the levels of complexity are included in Annex A. 2.9 The valuing of functions After the complexity of a function has been determined as described in clauses 5 through 9, the number of function points can be allocated to the function. This shall be done according to the rating in Table 1. Table 1 — Function point table High level specifications are enough to identify functions and their type when performing a high level function point analysis (see subclause 3.2.2), but it will be difficult to determine the complexity of these functions. In such a case, a data function is rated as low, while the rating average is used for a transactional function. 2.10 The functional size The Number of function points: see Functional size functional size is the sum of the number of function points assigned to each of the functions (in the way described above) that lie within the scope of the object to be analyzed, that is the application or the project. The functional size can also serve as a basis for preparing a project budget, by multiplying the number of function points with a productivity rate based on historical data (such as hours per function point). The preparation of a project budget is beyond the focus of this standard. More information on this subject can be found on the Nesma website nesma.org. A Nesma FSM measurement result on the functional user requirements or specifications for a piece of software shall be labeled according to the following convention: F(unction) P(oint) (ISO/IEC 24570:2018) 6 © ISO/IEC 2018 – All rights reserved iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018 ISO/IEC 24570:2018(E) 3 Guidelines to perform an FPA This clause indicates how function point analysis shall be carried out. To this end, subclause 3.1 first presents a generally applicable step-by-step plan. Subclause 3.2 then indicates how to act when dealing with an indicative, high level, and detailed function point analysis. Subclause 3.3 goes into the role of the quality of specifications, while subclause 3.4 explains the use of FPA during a project. Subclauses 3.5 and 3.6 show how an application and a project functional size are determined in the event of development and in the event of enhancement, respectively. Subclause 3.7 introduces the definition of a functional change. Subclause 3.8 states what must be taken into consideration when dealing with the different ways of recording specifications. Subclause 3.9 concludes the clause with an illustration of how the different types of function point analyses can be applied during the life cycle of an application. For illustration purposes, this subclause will assume a generic application life cycle as phasing method. 3.1 Step-by-step plan to perform an FPA Below follows a step-by-step plan to perform a function point analysis Step 1: Collect the available documentation. The documentation that should be present for an in-dicative function point analysis, a high level function point analysis, and a detailed function point analysis is described in subclauses 3.2.1, 3.2.2 and 3.2.3 respectively. Step 2: Determine the users of the application (see subclause 2.6). Step 3: Establish whether an application function point analysis or a project function point analysis must be carried out. If an application function point analysis must be performed, follow the instructions stated in subclause 3.5. If a project function point analysis must be performed, follow the instructions in subclause 3.6. Step 4: Determine from which other application(s) the application to be analyzed receives and/or uses data. Step 5: Identify the functions and determine their type and complexity according to the guidelines described in clauses 5 through 9. When doing so, adhere to the sequence in which the clauses appear. Assign the number of function points using the function point table illustrated in subclause 2.9. This will result in the functional size. Register the structure of the analysis and the number of function points. Particularly record any preconditions that have been used and assumptions that have been made. Step 6: Together with the user(s), verify the result in relation to those aspects where specific in-terpretation of the available specifications was needed. If necessary, make any corrections as a result of that verification. Step 7: Verify the result with an FPA expert in relation to those aspects where specific interpre-tation of the counting guidelines was needed. This may or may not be necessary. Make any corrections that are required as a result of that verification. 3.2 Types of function point analyses and their accuracy Depending on the degree of detail of the specifications available, one of three types of function point analyses can be chosen: an indicative, a high level, or a detailed function point analysis. Each type of function point analysis mentioned in this subclause can be used both for the determination of the project size as for the determination of the application size. The minimum specifications required are different for each of the three types of function point analysis. In the subclauses below, the specifications required to perform each of the three types of analyses are stated. Each subclause, finally, will indicate when a particular type of analysis can be executed in the life cycle of an application. © ISO/IEC 2018 – All rights reserved 7 iTeh STANDARD PREVIEW (standards.iteh.ai) ISO/IEC 24570:2018
15157	https://www.elsevier.com/resources/anatomy/cardiovascular-system/veins/sigmoid-sinus/20952	Sigmoid Sinus \| Complete Anatomy Skip to main content We use cookies that are necessary to make our site work. We may also use additional cookies to analyze, improve, and personalize our content and your digital experience. You can manage your cookie preferences using the “Cookie Settings” link. For more information, see our Cookie Policy. Accept Additional Cookies Cookie settings Unfortunately we don't fully support your browser. If you have the option to, please upgrade to a newer version or use Mozilla Firefox, Microsoft Edge, Google Chrome, or Safari 14 or newer. If you are unable to, and need support, please send us your feedback. Dismiss We'd appreciate your feedback.Tell us what you think! Academic & GovernmentAcademic & Government HealthHealth IndustryIndustry Elsevier ConnectInsights AboutAbout Customer supportSupport Publish with us Open Search Location Selector Show Menu Home Resources Complete Anatomy Articles Cardiovascular System Sigmoid Sinus Cardiovascular System Sigmoid Sinus Sinus sigmoideus Read more Quick Facts Origin Course Tributaries Structures Drained Quick Facts Origin: Transverse sinus. Course: Runs towards the superior jugular bulb of the internal jugular vein inside the jugular foramen. Tributaries: Mastoid and condylar emissary veins, cerebral, inferior cerebellar and diploic veins, superior petrosal sinus. Drainage: From the posterior dural venous sinuses (draining posterior aspect of the skull). Complete Anatomy The world's most advanced 3D anatomy platformTry it for Free See these models in 3D with Complete Anatomy App Related parts of the anatomy Occipital Lateral Lacuna (Right) Parietal Lateral Lacuna Frontal Lateral Lacuna (Left) Frontal Lateral Lacuna (Left) Parietal Lateral Lacuna Occipital Lateral Lacuna (Right) Superior Petrosal Sinus Sigmoid Sinus Marginal Sinus Basilar Venous Plexus Origin The sigmoid sinuses are continuations of the transverse sinuses, beginning where these leave the tentorium cerebellum. Course Each sigmoid sinus curves inferomedially in an S-shaped groove on the mastoid process of the temporal bone. It crosses the jugular process of the occipital bone and turns forwards to end in the superior jugular bulb, lying in the posterior half of the jugular foramen. Tributaries The sigmoid sinus receives the mastoid and condylar emissary veins. It also receives blood from the cerebral, cerebellar, and diploic veins. The superior petrosal sinus joins the sigmoid sinus at its junction with the transverse sinus. Inferiorly, the sigmoid sinus combines with the inferior petrosal sinus to the form the internal jugular vein. Structures Drained The sigmoid sinus receives blood from the transverse sinus, which receives blood from the posterior dural venous sinuses that drain the posterior aspect of the skull. Complete Anatomy The world's most advanced 3D anatomy platform Try it for Free Useful links Submit your paper Shop Books & Journals Open access View all products Elsevier Connect About About Elsevier Careers Global Press Office Advertising, reprints & supplements Modern slavery act statement Support Customer support Resource center Global \| English Copyright © 2025 Elsevier, its licensors, and contributors. All rights are reserved, including those for text and data mining, AI training, and similar technologies. Terms & Conditions Privacy policy Accessibility Cookie settings
15158	http://users.cis.fiu.edu/~iyengar/publication/J-(1993)%20-%20Shape%20from%20perspective%20trihedral%20angle%20constraint%20%20-%20%5BIEEE%5D.pdf	Shape From Perspective Trihedral Angle Constraint Yuyan Wu, S. Sitharama lyengar', Ramesh Jain, Santanu Base' RPL, Dept. of Comp. Sei., Louisiana State Univ., Baton Rouge, LA 70802 Dept. of E.E. and Comp. Sci., Univ. of Michigan, Ann Arbor, MI 48 109 Abstract: This paper defines and investigates a funda- mental problem o f determining the position and orienta- tion o f a 3 0 object using $ingle perspective i m g e view. The technique is based on the interpretation o f trihedral angle constraid informution. A new closed form solution io the problem is proposed. The method also provides a general analytic technique for dealing with a class o f problem o f shape from inverse perspective projection by using "Angle to Angle Correspondence Information ". Simulation experiments show that our method is enective and robust for real application. t This research ispariially supported by ONR NooO14-91-J1306. Key Words : Shape from #angle, Shape j?om perspective projection, Pose Estimation, 3 0 object recognition. 1. Introduction Shape from inverse perspective projection is an essen- tial method for model-based 3D reconstruction. There are many applications of this approach in Robotics, Cartogra- phy and Computer Vision [8-131. The formal definition of shape from inverse perspec- tive projection can be stated as follows : Let perspective projection be the ideal model of a camera, then the cam- era imaging process is given by where, P' = ( n, y, z ) is the description of a 3D point in an object coordinate system and 9 = (U, v ) ~ is the 2D pro- jection of P' on the image plane; rotation R and translation f form the transformation from the object coordinate sys- tem to the camera coordinate system; matrix K describes t h e intrinsic parameters of the camera. The problem of shape from inverse perspective projection i s to determine the unknown rotation matrix R and the translation f from certain 3D geometric features of the object and their 2D image geometric features in a single perspective view. Three types of situations are mostly discussed in the problem of shape from inverse perspective projection : 1 . Perspective ' point to point correspondence problem. This problem is usually called PnP problem , when n pairs of corresponding points are known. 2. Perspective line to line correspondence problem. Like the case in the above, we call the problem as LnL, prob- lem, when n pairs of corresponding lines are specified. 3 . Perspective angle to angle correspondence problem. We name this problem as AnA problem if n pairs of corre- sponding angles are given. A closed form solution is the most desirable result for each of PnP, LnL and AnA problems for its simplicity, stability and speed. In this paper, a closed form solution is presented for the general problem of uihedral angle constraint, which is an A3A problem. In recent years, trihedral angle constraint has been ad- dressed by many authors from different viewpoints. The relevant work can be divided into the following two cate- gories : (a) Direct approach : In this category, angle informa- tion is employed directly. b a d e proposes an analytic solution for the problem under orthographic projection. For perspective projection, algebraic solutions have been given by Kanatani 131, Shakunaga and Kaneko for special cases when two or three space angles are right an- gles; in addition, some constructive algorithms are sug- gested by EIoraud [ 7 ] , Shakunaga and Kaneko [SI for solving the general problem . (b) Indirect approach : Without using angles directly, the configuration of a trihedral angle can also be specified by four space points or by a junction of three 3D lines. In this sense, we can consider triheciral angle constraint as a special case of the P4P o r L3L problem. Therefor, the methods for solving these two types of problems can be applied for t r i h e d r a l angle constraint ( [ 111, [ 121 ). 261 1063-6919193 $03.00 0 1993 IEBE Our new solution for trihedral angle constraint uses the direct approach. This method can be considered as a com- plete closed form solution for the general AnA problems in a minimal condition. The significant advantage of our approach over the methods of P4P [Ill and L3L 1121 is that the angle measure is independent of the coordinate system but the description of a point or a line varies when the related coordinate system is changed. 2. A new mathematical framework 2.1. Preliminary formulation $32i?3= sin72sin73 C O S ( P ~ - P ~ ) + C O S ~ ~ C O S ~ ~ = COS ~ 2 3 2.1.1 Canonical image structure are given. Then, we c a n derive from (1) that When the angles q12, ~13,and 723 are given, we have a system of three equations with three unknowns. So we expect to solve 71, 72, 7 3 and then to determine the orien- tation of the trihedral angle in camera coordinate system. Kanatani 1 3 1 first suggested the formulation for angle constraint and proposed a solution of (5) for the special case where at least two of q12. q13,and ~ ~ 2 3 are right an- gles. we Will now present a Complete Solution for (5). 2.2. Analytical solution for trihedral angle con- straint 23.1 Estimate the orientation Our idea for solving (5) is straightforward. First, assume that 2, can be expressed by 2, and fi2 as W e have Assume that the intrinsic matrix Of a [+-I [?I-( K T I[;] (2) where $ is determined only by the extrinsic parameters of rotation K and translation f. W e call 8 as canonical Image and will consider only this representation in L h e followmg discussions for the development of our method. 2.1.2 View orientation transformation W e define a view orientation transformation as a pure rotation upon a camera coordinate system. Because the corresponding relationship between the image points UII- der such a uansformation is uniquely determined, we can $3 = U $ , + bZ2 + ~ $ 1 x $2 (6) - + - + use this relation for facilitating the problem formulation of trihedral constraint [31. N1N3 = U + b COS 7 1 2 = COS ~ 1 3 N2N3 = U COS ~ 1 2 + b = COS 7 2 3 N3N3 = U COS 913 + b COS q23 + c2 sin2 q 1 2 = 1 (7) + - + + + Among the infinite view orientation uansformations which c a n transform the view axis of a camera coordinate system from an old orientation to a new one, we consider the onc which is formed by a rotation around the y-axis of era. L,et a new view orientation be specified by an image pint (U, v ) ~ , then the rotation matrix is as below : Then, the coefficients a, b and c can be derived a = ( cos qI3 - cos q12 cos ~ 2 3 ) t sin2 7lI2 b = ( cos ~ 2 3 - cos q12 cos 913) t sin q12 c = k l/ ( I - a cos ~ 1 3 - b cos ~ 2 3 ) t sin2 ql;‘ camera, followed by a rotation around the x-axis of cam- 2 (8) (3) 2.1.3 ‘fiihedral angle constraint formulation Let a trihedral angle be formed by si = (xi, yi, zJT. i = 0, . . ,3 with $0 as the angular point, and di = ( u ; , v ~ ) ~ be the perspective projection of gi. without loss of gener- ality, assume that iio is on the view axis. Let pi be tlie an- gle formed by ai and the U-axis; zi=si-?o and be the unit direction vector along ti; and 7i be the angle formed By the values of a, b and c. we c a n rewrite (8) t o get a sin 7 1 cos p1 + b sin y 2 cos 8 2 cos P3 sin yl sin p1 cos 72 - sin 72 sin p 2 cos yll cos 8 3 a sin 7 1 sin PI + b sin yz sin p 2 sin p3 sin 72 cos p2 cos 7 1 - sin 7 1 cos P1 cos 72) sin 8 3 sin73 = + (9) sin 73 = + COS 73 = U COS 7 1 + b COS 72 + c sin 7 1 sin 72siQ2 - Without loss of generality, suppose cos(p1-P2)+0. Then, 262 Substituting ( 1 4 ) into the second equation of (ll), we get 5 i d si cosi 7 1 = 0 Where, ~5 = C2A;” + E2C;” - B2:AlCI ~4 = A2Ai + Fz Ct - B2,AI B1- 8 2 D1 C 1 - DZAlCl+ 2CzAl D1+ 2E2ClBl ~3 = 2AzAIDl- D2AliB1- D2C1D1+ 2EzClE1 + 2C2Al F, + C2Di - BzAIE, - B ~ C I F1 ~2 = 2A2Al F1+ A2@ - D2A1 El - D2 F 1 C l - B2Dl B1 + 2F2C1.B1 + E2Bi - D2D,E1 + 2F2C1 El + F2B: + 2C2D1 F 1 (16) - B2DlEl- B2B1 F 1 + 2E2B1 El S I = 2A2D1 F1- D2Dl El - D2F1 Bl + 2F2BIEi + C2F;”- B2FlEl +. E2E;” SO = A2 F;” + Fz Et - Dz FIE1 By (151, (14) and the third equality of ( 9 ) . we can solve cos 71, cos 72 and cos y3 step by step. Then, a t r i h e d r a l angle can be determined in camera coordinate system by : Pi = Po+ liNj ( i = 1,2,3) (17) where, li > 0 is the length of zi. Because so far only the three unit vectors 13; can be assigned by solving (3, we obtain only the orientation of a trihedral angle. To find its -+ + -+ full position, more information is necessary. 23.2 Determine the full position of a trihedral angle Let a trihedral angle in an object frame be given by i ; : = & + lii$ (i = I, 2,3) ( 1 8 ) Where, the corresponding relationship between (17) and (18) is specified by gi to 8. si to $ and Zi= 1;. We can obtain the rotation R by the relation si=R$. To find the translation f=(tx,fY,fJT, let g and j 3 be a pair of matched object point and image point, R = (rG)3x3, we have ( 1 9 ) If two pairs of matched points are available, the transla- tion f can be obtained by solving ( 1 9 ) . It follows that to get a full solution for trihedral angle constraint, we still need two pairs of matched object point and image point. Alternatively, if one of length li in (17) is known, the go can be simply determined by each of the following two equations, provided the denominator is not zero. rllx + r12y+ r13z + I , - U ( T ~ ~ X + r32y + r33z + t,) = 0 r21 x + r, y + rpz + t, - v( r31 x + r3,y + r 3 3 ~ + 1,) = 0 (20 1) (20) 20 = li(SitI7j cos pi - U, cos 71) I ui zo = li(sin7i sin pi - vi cos x) I vi Then the trihedral angle is completely determined in cam- era frame but does involve any object coordinate system. 23.3 The algorithm The solution procedure for shape from t r i h e d r a l angle constraint is summarized as follows : Prerequisite : Suppose that the intrinsic parameters of the camera are given; then a 2D t r i h e d r a l configuration is picked and the Corresponding 3D angles are specified. Step 1. Use (2) to get the canonical representation for the image features. Step 2. Use the angular vertex of the 2D tnhedral config- uration to compute the matrix R (3); then, convert the original image features to their new coordinates. Step 3. Match the 2D vs. 3D angles and determine the constraint equation system (5). Step 4. Derive the fifth-order equation (15) and solve it to get cos 7 1 ; if there is no solution, go to step 8. Step 5. Calculate cos 72 by ( 1 4 ) ; if there i s no solution, go to step 8. Step 6. Calculate cos y3 by the third equality of (9); if there is no solution, go to step 8; Step 7. Check the solution against the original constraint equation system (5). 263 Constraint Relation I 1 I II 11 Linear I n 2 6 I P4P1111 II Linear Closed Form Solution Solution )I Pi'p . . P3P 1- Nonlinear No In this table, we divide the problems of PnP, LnL and AnA into two categories of linear and nonlinear con- straints. The difference between the two categories is that the 3D features are defined in an object coordinate system for linear constraint, but they are given by a group of scalars for nonlinear constraint. LnL constraint belongs to linear category because it is necessary to refer to some object coordinate system for specifying a 3D line. Con- trarilv, AnA constraint is in nonlinear category because only n scalars are needed for specifying n spatial angles. LnL AIIA An intersting fact is that PnP can be presented in both cat- egories when n > 1 [ 8,9,11]. In mathematics, a linear formulation for the problem LnL or PnP m a y be changed to some nonlinear format of AnA or PnP. But we can not change a nonlinear formula- tion for the problem of AnA or PnP to some linear format of LnL. or PnP because the nonlinear formulation is inde- pendent to object frame. So the nonlinear formulation may be more powerful than the linear formulation for cer- t a i n applications. 3 3 Special configuration C a s e s Some special configurations about trihedral angle con- straint described below are commonly encountered in real applications. For these cases, The general fifth order equation (15) can be simplified to lower order to facilitate the solving procedures. (a) Coplanar configuration a plane, equation ( 15) becomes In this case, the three vectors fil, 6 ' . Z3 are located on s4 cos4 7 1 + s2 cos' yl + so = 0 This is actually a quadratic equation on cos' yl (b) The configuration with two or three right angles In the case that there are at least two right angles in a trihedral angle, we can let $3 be normal to $1 and $2. Then, (15) can be rewritten as s5 cos4 7 1 + s3 cos' yl + s1 = o As in case (a), we obtain a quadratic equation on cos' 7 1 . (c) Special image configurations If one 2D right angle exists, say pI-p2=nf2, we have A,=Ei=O (i=1,2) in (12) and (13), so (15) becomes a cubic s4 cos3 7 1 + s3 cos2 yl + s2 cos yl + sI = 0 If pl-p2=r, we have C1=C2= 0 in (12) and (13), so (15) becomes a quadrinomial as s4 cos4 7 1 + s3 cos3 7 1 + s2 cos2 7 1 + s1 cos yl + s o = o 4. Experimental validation 4.1 Experimental design Regarding the perf0r"ance of the new approach, we are mainly concerned about its effects on the following three aspects : 1. Subject to the following three inherent criteria : C-1 : Each solution obtained by (15), ( 14) and (9) must bein[-1, I]. C-2 : Each group of solutions should satisfy the primi- tive equation system (5). Linear n 2 8 L3L r12.131 Nonlinear No A3A.ThisPaper 264 C-3 : if additional information about the 3D length of the side of a trihedral angle is available, the solution of (20) should be bigger than 1. we will investigate how many solutions can occur for an arbitrary trihedral angle cc4nstraint and whether the true solution is obtainable by our method. 2. We should inspect when a correctly matched trihedral angle constraint is derived, if the real solution is obtained by our method; or when an ill-matched trihedral angle constraint is presented, whether our method can identify the ill-condition. 3. Our next task is to study the presented approach for its sensitivity to noise. To test the three questions in general, we arranged our experimental procedure as fi~llows : Data-1 : Randomly generate a set of ideal trihedral angle constraints in a camera coordinate system. Test-1 : U s e correct angle matching relationship on the ideal data to solve a trihedrill angle constraint and then to investigate the solution pattem. Test-2 : Use incorrect angle matching relationship on the ideal data to solve a trihedral angle constraint and then to check the solution results. Data-2 : For a trihedral angle constraint, the net effects of noises can be simply considered as a noise acted on the PI, pz and p3 of (5). We choose an interval I. -m, m ] as the source of noise. Then, a noise triplet is randomly gen- erated from the noise interval and the trihedral angle con- straint generated b Data-1 is added on the noise triplet to produce a noised &a. Table 4-1 The Solution Distribution of Equation (15) Number of Solutions Frequency ~~~~1 Data FnorMatch 0 11 65 24 0 0 Table 4-2 The Reserved Solution Distribution Test-3 : Do Test-1 for Data-2. Test-4 : Do Test-2 for Data-2. The test results are given in the following paragraphs. 4.2 The solution distribution and patterns According to the procedure depicted in 4.1, 100 groups of d a t a are generated and the tested results are shown in the Table 4-1 and Table 4-2. In the tables, an entry repre- sents the emerging frequency of the case specified by the corresponding row title and column title. For example, the entry 48 in the first row and the third column of Table 4-1 means t h a t , when using a randomly generated ideal trihe- dral angle constraint and supposing that the correct match is employed, we got 48 times of the 2-solution cases in the 100 experiments. By table 4-1, we see that the equation (15) usually has some solutions in the interval [ -1, 1 J no matter what experimental condition is assumed. But there is no signif- icant difference to distinguish the ideal data from noised d a t a or distinguish the correct match from error match by just referring to the solutions of (15). When imposing the criteria C-1, C-2 and C-3 for the fonnal solutions of (15). (14) and (9). Table 4-2 shows that the reserved solutions have a very different distribu- tion compared to Table 4-1 ( where we identify a pair of mirror solutions as one solution ). This time, we see that the overwhelming majority of the error matched trihedral angle constraints have no solution. So they can be effec- tively identified by our method. For each correctly matched case, we always find that true solution is included for ideal d a t a and an approximate solution for the m e value exists for noised d a & and most correctly matched cases have just one or two reserved solutions. More details about the experiments are presented in . 4.3 Noise sensitivity analysis For a trihedral angle constraint (3, the net effect of noise c a n be represented by a disturbance on the 2D angles pi, i=l, 2 , 3. Denote ai as the noised pi; let Y, and fi be the correct solution of (5) corresponding to pi and ai. Then, we consider the covariant relationships for the corresponding pairs ( Afi, APi ) and ( A, fi 1 by following two linear regression models : AY, = ai0 + ailAPi + ~i fi = ai0 + Uily, + Ei where, Afi = Ti - y,, Api = $i - pi, ( i = 1,2,3 ) . (i = 1,2,3) Our intention is to test the statistical hypotheses : Ho: ail = 0 Ho: a:, = 0 by using the analysis of variance (ANOVA) t n check the d a t a fitness for the linear regression models. According 265 to the above procedure, the synthetic test d a t a were gener- ated for regression analysis; where, the noises were se- lected from the noise interval [-5",5"]; and for multiple solution cases, we chose the best approximation to the correct value yi as A. The results of the regression analysis are presented by Table 4-3 ( refer for more details ). We see that there is no definite relationship between Ayi and ASi; but very strong linear relationship exist between yi and A. There- fore, we can conclude that the solution of our method for trihedral angle constraint is stable under a noisy environ- ment So the method is robust in real application situa- tions Models Ayl = alo + allApl A y 2 = a20 + azlAp2 Ay3 = a30 + a31Ap3 Tl = a;,-, + u;171 f2 = aio + uil y 2 f3 = a&, + a31 y3 P-value Acceptance 0.1910 Accept 0.4728 Accept 0.9821 Accept O.OOO1 Reject O . O O O 1 Reject O . O O O 1 Reject 5. Conclusion Methods for solving the orientation and position of an object from a single perspective projection view are important for their wide applications and powers. The method presented in this paper permits us to find an analytic solution of a trihedral angle constraint by directly using angle information. Angle is a very com- mon feature for characterizing a variety of objects. The knowledge about the angles of an object provides a strong clue for estimating the orientation and position of the object. Our method gives the first closed form solution for the problem of trihedral angle constraint in perspective projection. Trihedral angle is the simplest but also the most encountered angle constraint in 3D computer vision. This method also provides a basic approach for dealing with the general AnA problems, provided that the number of constraint equations on AnA problem is greater than or equal to the number of unknowns. The results of simula- tion experiments show that the new method is not only a real time technique of shape from angle constraint, but also powerful enough to cope with noisy environments in real applications. With the new developments, we present a general analysis on the essential characteristics of PnP, LnL and AnA techniques. The combination of the three techniques certainly is a very promising tool to deal with varitns situations of shape from inverse perspective pro- jection. To design a sound algorithm for this unified approach is a topic for our further research. References [l] R. M. Haralick, "Monocular Vision Using Inverse Perspec- tive Projection Geometry : Analytic Relations", Prcc. IEEE P.G. Mulgaonkar, L.G. Shapiro and R.M. Haralick., "Shape from Perspective : A Rule Based Approach" CVGDP Vo1.36, K. Kanatani, "Constraints on Length and Angle", CVGIP, Vo1.41. pp.28-42, 1988. (41 S. T. Bamard, "Choosing a basis for perceptual space", Cod. 00 CVPR, pp.370-378, 1989 pp.298-320, 1989 CVGIP, V01.29, pp.87-99, 1985. T. Shakunaga and H. Kaneko, " Shape from Angles Under Perspective Projection", Proc. IEEE 2nd Int. Conf. on T. Kanade, "Recovery of the Three Dimensional Shape of an Object from a SingleView", AI-17, pp.409-460, 1981. R. Horaud, "New Method for Matching 3-D Object with Single Perspective Views", IEEE Trans. on PAMI, Vo1.9, No.3, pp.401-412, 1987. M.A. Fischler and R.C. Bolles, "Random Sample Consen- sus : A Paradigm for Model Fitting with Application to Image Analysis and Automated Cartography", Communication of ACM., Vo1.24, No.6, pp.381-395, 1981. S. Linnainma, D. Harwood and L.S. Davis, "Pose Determi- nation of a Three-Dimensional Object Using Triangle Pairs", IEEE Trans. on PAMI Vol.PAM1-IO, No.5, pp.634-646, 1988. [ 101 D.G. Lowe, "Three-Dimensional Object Recognilion from Single Two-Dimensional Image", AI-3 1, pp.355-395, 1987. [ll] R. Horaud, B. Coni0 and 0. Leboulleux, "An analytic solu- tion for the perspective 4-point problem". CVGIF', Vo1.47, pp.33-44, 1989. M. Dhome, M. Richetin, J-T. Lapreste and G. Rives, "Determination of the Attitude of 3D Objects from a Single Per- spective View", IEEE Trans. on PAMI, Vol.11, pp.1265-1278, 1989 Homer H. Chen, "Pose Determination from Line-to- Plane Correspondences : Existence Condition and Closed-Form Solution", Proc. IEEE 3rd Int. Conference on CV, pp.374-378. 1990 [ 141 Yuyan Wu, S. Sitharama Iyengar, Ramesh Jain and Santanu Bose, "A New Method Of Finding Object Orientation Using Perspective Trihedral Angle Conshaint", Research Report, Robotics Research Laboratory, Department of Computer Sci- ence, Louisiana State University, Baton Rouge, 1992. CV, pp.671-678, 1988. 266
15159	http://people.umass.edu/bvs/p_q_adiab.pdf	External Parameters. Pressure External Parameters So far, we were characterizing an equilibrium state by only one thermodynamic variable: either T, or E, or S, etc. Fixing one of these variables, say, E, we could (at least in principle) express the rest of them as functions: S(E), T(E), etc., because all the variables were related to each other through the Gibbs distribution. Incidentally, the functions S(E), T(E), E(T), S(T), etc., are called equations of state of a given system. There are, however, other natural parameters which the state of the system can depend on—the system volume, or/and the value of applied magnetic or electric field, inter-particle interaction, and so on. In this case, the Gibbs distribution and, correspondingly, the equations of state will also include these variables. Previously we simply kept all these parameters fixed. Now we are going to let them vary. There is a general relation between small variations of energy, entropy, and any external parameter λ, which follows from the formula E = F + TS . (1) Eq. (1) is valid in any equilibrium state. Hence if we consider an infinitesimal variation of the state— from one equilibrium into another equilibrium—the differentials dE, dF, dT, and dS should be related by dE = dF + TdS + SdT . (2) By its definition, the Helmholtz free energy F is naturally expressed as a function of T and λ: F(T, λ) = −T ln X n e−En(λ)/T ! . (3) [Note that the only place where the dependence on λ comes from is the position of the n-th energy level.] For any function of two variables, from Calculus we have dF = ∂F(T, λ) ∂T dT + ∂F(T, λ) ∂λ dλ . (4) Previously, we have established that ∂F(T, λ) ∂T = −S . (5) [We were differentiating F with respect to T assuming that all the external parameters are fixed, which exactly corresponds to the meaning of this partial derivative.] Hence, dF = −SdT + ∂F(T, λ) ∂λ dλ . (6) Plugging this into (2), we get the desired relation: dE = TdS + ∂F(T, λ) ∂λ dλ . (7) We see that the partial derivative ∂F(T, λ)/∂λ plays an important thermodynamic role. Below we will reveal an explicit physical meaning of this quantity in the case when λ is the system volume. 1 Pressure. Generalized force How do we generically define pressure? We say that the pressure is a force per unit area. This definition needs to be further developed for the case of a quantum system, where the force is not a natural quantum mechanical observable. Consider a piston of the area A squeezing some quantum system. To keep the volume fixed, we need to apply some external force F which in the case of a macroscopic system turns out to be proportional to A: F = P A . (8) The proportionality coefficient P is called pressure. To relate P to the microscopic characteristics of our system, we perform the following gedanken experiment. We shift the piston by some small distance ∆x along the x axis. Then, the work ∆W performed by the external force F is given by ∆W = F ∆x = PA ∆x = −P ∆V , (9) where ∆V is the change of volume. We also assume that our system is thermally isolated and thus the conservation of energy simply implies ∆W = ∆E , (10) where E is the system energy. Hence, for the pressure we have P = −lim ∆V →0 ∆E ∆V . (11) One can introduce a quantum mechanical operator of pressure by noting that energy is the expectation value of the Hamiltonian H. Hence, it is natural to expect that the operator of pressure is ˆ P = −lim ∆V →0 ∆H ∆V = −∂H ∂V . (12) (Below we present a rigorous derivation of this formula.) To calculate the pressure for a thermodynamically equilibrium system we need to find the expec-tation value of this operator with respect to Gibbs distribution: P = X n wn ⟨n\| ˆ P\|n⟩= − X n wn ⟨n\|∂H ∂V \|n⟩. (13) Next steps are generic. If the Hamiltonian H depends on some external parameter λ, then for the quantity ∂H ∂λ ≡ X n wn ⟨n\|∂H ∂λ \|n⟩ (14) we proceed as follows. First, we observe that ⟨n\|∂H ∂λ \|n⟩= ∂ ∂λ ⟨n\|H\|n⟩= ∂En ∂λ . (15) Indeed, ∂ ∂λ ⟨n\|H\|n⟩≡ ∂ ∂λ ⟨ψn\|H\|ψn⟩= ⟨∂ψn ∂λ \|H\|ψn⟩+ ⟨ψn\|∂H ∂λ \|ψn⟩+ ⟨ψn\|H\|∂ψn ∂λ ⟩= En⟨∂ψn ∂λ \|ψn⟩+ ⟨ψn\|∂H ∂λ \|ψn⟩+ En⟨ψn\|∂ψn ∂λ ⟩= En ∂ ∂λ ⟨ψn\|ψn⟩+ ⟨ψn\|∂H ∂λ \|ψn⟩. (16) But ⟨ψn\|ψn⟩≡1 by normalization, and any derivative from it is identically equal to zero. 2 Hence, ∂H ∂λ = X n wn ∂En ∂λ . (17) Then we differentiate the partition function with respect to λ, temperature being fixed: ∂Z ∂λ T = ∂ ∂λ X n e−En/T = −1 T X n e−En/T ∂En ∂λ = −Z T X n wn ∂En ∂λ , (18) and thus find that ∂H ∂λ = −T Z ∂Z ∂λ T = ∂F ∂λ T . (19) Replacing now λ →V , we get the formula P = −∂F(T, V ) ∂V . (20) With this result, the relations (6) and (7), yield dF = −SdT −P dV , (21) dE = TdS −P dV . (22) Note also that in a general case, Eq. (19) allows us to write Eqs. (6)-(7) as dF = −SdT + ∂H ∂λ dλ , (23) dE = TdS + ∂H ∂λ dλ . (24) There is a subtlety about the operator of pressure. Namely, we were taking for granted that ∆E was independent of the protocol of producing corresponding small change in volume. Now we elaborate on this issue. Let the Hamiltonian depend on time through some time-dependent parameter λ(t). [In the case of pressure, λ ≡V .] Consider the (time-dependent) quantum mechanical expectation of energy E(t) = ⟨ψ(t)\|H(t)\|ψ(t)⟩ (25) and differentiate it with respect to time: ˙ E = ⟨∂ψ ∂t \|H\|ψ⟩+ ⟨ψ\|∂H ∂t \|ψ⟩+ ⟨ψ\|H\|∂ψ ∂t ⟩. (26) Taking into account Schr¨ odinger equation i¯ h ∂ψ ∂t = Hψ , (27) we notice that ⟨∂ψ ∂t \|H\|ψ⟩+ ⟨ψ\|H\|∂ψ ∂t ⟩= (i/¯ h) ⟨ψ\|H2\|ψ⟩−(i/¯ h) ⟨ψ\|H2\|ψ⟩= 0 , (28) and thus ˙ E = ⟨ψ\|∂H ∂t \|ψ⟩. (29) 3 The dependence of H on t comes through λ(t). That is ∂H ∂t = ˙ λ ∂H ∂λ . (30) Plugging this into Eq. (29), we conclude that ˙ E = ⟨ψ\|∂H ∂λ \|ψ⟩˙ λ , (31) or, equivalently, dE = ⟨ψ\|∂H ∂λ \|ψ⟩dλ . (32) This is the crucial result saying that if the variation of λ is small enough, then the corresponding variation of energy is defined only by the state ψ and the variation of λ, the time-dependence of λ being irrelevant. Moreover, variation of energy is proportional to the variation of λ, the proportionality coefficient—generalized force—being given by the expectation value of the following operator ˆ F = ∂H ∂λ , (33) which we refer to as the operator of generalized force. Work and Heat The First Law of Thermodynamics Suppose the system energy has been changed by a certain amount ∆E. Since the total energy is conserved, ∆E should be equal to the sum of energies taken from other systems either in the form of heat (the energy associated with chaotic microscopic motions) or in the form of mechanical work, or both. Hence, for a thermodynamic processes with a given system the conservation of energy can be written as ∆E = ∆Q + ∆W , (34) where ∆Q is the portion of energy transferred to the system in the form of heat, while ∆W is the energy transferred in the form of mechanical work. Eq. (34) is known as the first law of thermody-namics. Heat and Entropy Eq. (34) is always true, since it is just the law of energy conservation. Consider now a quasi-equilibrium process, that is a process when temporal variations of parameters are slow enough so that the evolution is just a chain of equilibrium states. In this case, we can use the equilibrium relations established in the previous section. In particular, we have dE = TdS −P dV , (35) which implies that ∆E ≈T ∆S −P ∆V , (36) 4 if all ∆’s are small enough. Comparing (36) and (34), with (9) taken into account, we come to a fundamental relation between the heat transfer, entropy change, and temperature: ∆Q = T∆S (∆Q →0) . (37) Introducing a pseudo-differential dQ standing for an infinitesimal amount of transferred heat, we write (37) as dS = dQ T . (38) One can employ this relation for an experimentally measuring the entropy. Indeed, starting from some very low temperature T0, where the entropy is almost zero, and adding energy to the system of interest in the form of heat by small enough portions, and keeping track of the temperature, one gets the entropy in the form of the following integral (sum) over the experimental data: S(Tn) = Z dQ T ≈∆Q0 T0 + ∆Q1 T1 + . . . + ∆Qn Tn (n ≫1) . (39) In an essentially non-equilibrium process there is an extra increase of entropy due to relaxation processes. (We have established this fact previously by considering two systems with different tem-peratures relaxing towards some common temperature.) In this case instead of (37) we have ∆S > ∆Q/T (non-equilibrium process) . (40) As we will show later, this thermodynamic inequality leads to an upper bound of the efficiency of heat machines. Adiabatic Process Suppose our system is isolated from any heat baths and we slowly change some external parameter λ (say, volume or magnetic field) so that the process is quasi-equilibrium. Such a process is called adiabatic. The word ‘adiabatic’ is just a term for ‘quasi-equilibrium and thermally isolated’. Because of the quasi-equilibrium character of the process, we can use Eq. (38), while the fact that the system is thermally isolated means that there is no heat transfer: dQ = 0. We conclude that in adiabatic process entropy remains constant. And this allows us to relate the temperature to the particular value of the parameter λ by an implicit function S(T, λ) = S(T0, λ0) , (41) where T0 and λ0 are the initial values of T and λ. The curve T(λ) specified by Eq. (41) is called adiabat. Problem 22. A macroscopic system of almost non-interacting spins-1/2 (of one and the same magneton) is in an external magnetic field. The system is thermally isolated and is in equilibrium. The external magnetic field is adiabatically decreased by a factor of 100. What happens to the temperature? Heat Capacities Most generally, by heat capacity we mean the heat required for a unit change in the temperature of a body. Hence, to obtain the heat capacity we need to find the limit of the ratio ∆Q/∆T as ∆T →0. In case when the state of a system depends not only on the temperature, but also on some other quantities, like volume or pressure, this definition of heat capacity is ambiguous. For example, if we change the temperature of water in a glass (open surface), then the volume changes as well due to 5 the expansion of the liquid, while the pressure remains constant. Alternatively, if we heat up water in a closed container with rigid walls, the volume remains constant. Hence, the value of heat capacity may depend on the character of a process, and to fix the definition, we need to specify the parameter (set of parameters), ξ, that is (are) kept constant when we add heat to the system. We thus write Cξ = lim ∆T →0 ξ = const ∆Q ∆T . (42) Eq. (37) states that this is equivalent to Cξ = T lim ∆T →0 ξ = const ∆S ∆T ≡T ∂S ∂T ξ . (43) For the two most typical heat capacities, CV and CP , we have: CV = T ∂S ∂T V , CP = T ∂S ∂T P . (44) 6
15160	https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/52%3A_Fluid_Dynamics/52.09%3A_Fluid_Flow_through_a_Pipe	52.9: Fluid Flow through a Pipe - Physics LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode 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"65:_Further_Reading" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "66:_Appendices" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "67:_References" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "zz:_Back_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } Thu, 08 Aug 2024 03:59:31 GMT 52.9: Fluid Flow through a Pipe 92335 92335 Joshua Halpern { } Anonymous Anonymous 2 false false [ "article:topic", "license:ccbyncsa", "showtoc:no", "licenseversion:40", "Churchill\'s equation", "L.L. Simpson", "Moody friction factor", "Weisbach friction factor", "Darcy friction factor", "Fanning friction factor", "Stanton friction factor" ] [ "article:topic", "license:ccbyncsa", "showtoc:no", "licenseversion:40", "Churchill\'s equation", "L.L. Simpson", "Moody friction factor", "Weisbach friction factor", "Darcy friction factor", "Fanning friction factor", "Stanton friction factor" ] Search site Search Search Go back to previous article Sign in Username Password Sign in Sign in Sign in Forgot password Contents 1. Home 2. Campus Bookshelves 3. Prince George's Community College 4. General Physics I: Classical Mechanics 5. 52: Fluid Dynamics 6. 52.9: Fluid Flow through a Pipe Expand/collapse global location General Physics I: Classical Mechanics Front Matter 1: What is Physics? 2: The Greek Alphabet 3: Quotations from the Classical Greek 4: Deductive Logic 5: Units 6: Problem-Solving Strategies 7: Density 8: Kinematics in One Dimension 9: Vectors 10: The Dot Product 11: Kinematics in Two or Three Dimensions 12: Projectile Motion 13: Newton’s Method 14: Mass 15: Force 16: Newton’s Laws of Motion 17: The Inclined Plane 18: Atwood’s Machine 19: Statics 20: Friction 21: Blocks and Pulleys 22: Resistive Forces in Fluids 23: Circular Motion 24: Work 25: Simple Machines 26: Energy 27: Conservative Forces 28: Power 29: Linear Momentum 30: Impulse 31: Collisions 32: The Ballistic Pendulum 33: Rockets 34: Center of Mass 35: The Cross Product 36: Rotational Motion 37: Moment of Inertia 38: Torque 39: Measuring the Moment of Inertia 40: Newton’s Laws of Motion- Rotational Versions 41: The Pendulum 42: Simple Harmonic Motion 43: Rocking Bodies 44: Rolling Bodies 45: Galileo’s Law 46: The Coriolis Force 47: Angular Momentum 48: Conservation Laws 49: The Gyroscope 50: Elasticity 51: Fluid Statics 52: Fluid Dynamics 53: Hydraulics and Pneumatics 54: Gravity 55: Earth Rotation 56: Geodesy 57: Celestial Mechanics 58: Astrodynamics 59: Partial Derivatives 60: Lagrangian Mechanics 61: Hamiltonian Mechanics 62: Special Relativity 63: Quantum Mechanics 64: The Standard Model 65: Further Reading 66: Appendices 67: References Back Matter 52.9: Fluid Flow through a Pipe Last updated Aug 8, 2024 Save as PDF 52.8: Stokes’s Law 52.10: Gases picture_as_pdf Full Book Page Downloads Full PDF Import into LMS Individual ZIP Buy Print Copy Print Book Files Buy Print CopyReview / Adopt Submit Adoption Report View on CommonsDonate Page ID 92335 ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents No headers If a viscous fluid is flowing through a pipe, then there is an additional term called the friction head introduced into Bernoulli’s equation: (52.9.1)P ρ⁢g+f⁡L D⁢v 2 2⁢g+v 2 2⁢g+y=constant where the second term on the left is the friction head; f is a dimensionless constant called the friction factor, 1 L is the pipe length, D is the pipe diameter, and v is the average fluid velocity (the fluid will flow faster at the center of the pipe than near the edges). For laminar flow , the friction factor f is given simply by (52.9.2)f=64 Re( laminar flow ), where Re is the Reynolds number . For a nonviscous fluid, the viscosity μ=0, the Reynolds number Re=∞, and so f=0, so that Eq. 52.9.1 reduces to the previous form of Bernoulli's equation , Eq. 52.3.1. For turbulent flow , the analysis to find the friction factor is more complicated and depends on the Reynolds number and the ratio of the pipe surface roughness to pipe diameter. There is a general formula due to S.W. Churchill that gives the friction factor f for all values of Reynolds numbers and all types of flow (laminar, transitional, and turbulent) through both rough and smooth pipes. Churchill's equation (as modified by L.L. Simpson to produce accurate results for turbulent flow ) is (52.9.3)f=\|(64 Re)12+{[2⁢log 10⁡(ε 3.7⁢D−5.02 Re⁢log 10⁡(ε 3.7⁢D+(7 Re)0.9))]16+(13269 Re)16}−3/2\|1/12 where ε is the pipe roughness and D is the pipe diameter. The friction factor vs. Reynolds number is shown in Figure 52.9.1 Figure 52.9.1: Friction factor as a function of Reynolds number, for both laminar and turbulent flow. (Ref. ) 1 Sometimes f is called the Moody friction factor, Weisbach friction factor, or Darcy friction factor. One sometimes also encounters the Fanning friction factor equal to f⁡/4, and the Stanton friction factor equal to f⁡/8. The Moody friction factor used here is the most common. 52.9: Fluid Flow through a Pipe is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts. Back to top 52.8: Stokes’s Law 52.10: Gases Was this article helpful? Yes No Recommended articles 52.2: The Continuity Equation 52.3: Bernoulli’s Equation 52.4: Torricelli’s Theorem 52.5: The Siphon 52.6: Viscosity Article typeSection or PageLicenseCC BY-NC-SALicense Version4.0Show TOCno Tags Churchill's equation Darcy friction factor Fanning friction factor L.L. Simpson Moody friction factor Stanton friction factor Weisbach friction factor © Copyright 2025 Physics LibreTexts Powered by CXone Expert ® ? The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org. Support Center How can we help? Contact Support Search the Insight Knowledge Base Check System Status× contents readability resources tools ☰ 52.8: Stokes’s Law 52.10: Gases Complete your gift to make an impact
15161	https://en.wikipedia.org/wiki/American_toad	Jump to content Search Contents (Top) 1 Eggs 2 Tadpoles 3 Biogeography 4 Subspecies 4.1 Eastern American toad 4.2 Dwarf American toad 4.3 Hudson Bay toad 5 Inbreeding avoidance 6 Gallery 7 See also 8 References 9 External links American toad Atikamekw Azərbaycanca Български Cebuano Diné bizaad Español Esperanto Euskara فارسی Français Bahasa Indonesia Italiano Magyar مصرى Nederlands 日本語 Português Русский Simple English Svenska Українська Tiếng Việt Winaray 中文 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 Wikimedia Commons Wikispecies Wikidata item Appearance From Wikipedia, the free encyclopedia Species of amphibian Not to be confused with the cane toad. \| American toad \| \| \| \| Specimen from Jacques-Cartier National Park, Quebec, Canada \| \| Conservation status \| \| Least Concern (IUCN 3.1) \| \| Secure (NatureServe) \| \| Scientific classification \| \| Kingdom: \| Animalia \| \| Phylum: \| Chordata \| \| Class: \| Amphibia \| \| Order: \| Anura \| \| Family: \| Bufonidae \| \| Genus: \| Anaxyrus \| \| Species: \| A. americanus \| \| Binomial name \| \| Anaxyrus americanus (Holbrook, 1836) \| \| Subspecies \| \| A. a. americanus A. a. charlesmithi A. a. copei \| \| \| \| Range of A. americanus \| \| Synonyms \| \| Bufo americanus Holbrook, 1836 \| The American toad (Anaxyrus americanus) is a common species of toad found throughout the eastern half of Canada and the United States. It is divided into three subspecies: the eastern American toad (A. a. americanus), the dwarf American toad (A. a. charlesmithi) and the rare Hudson Bay toad (A. a. copei). Recent taxonomic treatments place this species in the genus Anaxyrus instead of Bufo. Eggs [edit] A. americanus eggs are bicolored. They are often a roughly equal mixture of a black or brown with white or cream. The eggs are deposited in two long ropes that have been recorded to transcend 60 m in length. Single egg diameter ranges from 0.1 cm to 0.2 cm.[failed verification] Tadpoles [edit] The American toad lays between 2,000-20,000 eggs in two strings which hatch in 3-12 days. The hatched tadpoles, while very small, are recognizable by their skinny tails in relation to the size of their round black bodies. They reach adulthood in 50–65 days. When metamorphosis is complete, the "toadlets" may stay in the water for a short period of time before transitioning to be mostly land based. Often times, groups of tadpoles reach the toadlet stage at once and a begin a mass migration to higher ground. Typically, toadlets migrate to shaded areas in the mid-range and upland forests that border the marshes where they were bred. Toadlets can be observed eating microscopic bugs in their roaming ground between various vegetation; they are also known to eat ants, spiders, slugs and worms. Studies have shown that they have a mutualistic relationship with Chlorogonium algae, which makes tadpoles develop faster than normal. Leaf litter from invasive plants like autumn olive or purple loosestrife in the larval nursery can increase the burden of trematode parasite infestation among tadpoles. This is possibly mediated by faster development of tadpoles in these aquatic environments than in nurseries with leaves of native black huckleberry or swamp loosestrife. Tadpoles have several mechanisms to reduce predation. They avoid predators by swimming in very shallow water often with thick grass vegetation and by swimming close together in schools during the day. Tadpoles also produce toxic chemicals in their skin that discourage some predation. Fish have been reported to die after consuming one tadpole; however, most fish quickly learn to avoid eating American toad tadpoles. Biogeography [edit] Based on DNA sequence comparisons, Anaxyrus americanus and other North American species of Anaxyrus are thought to be descended from an invasion of toads from South America prior to the formation of the Isthmus of Panama land bridge, presumably by means of rafting. Subspecies [edit] Races tend to hybridize with Anaxyrus woodhousii in their overlapping ranges. Eastern American toad [edit] The eastern American toad (A. a. americanus) is a medium-sized toad usually ranging in size from 5–9 cm (2.0–3.5 in); the record length for an eastern American toad is 11.1 cm (4.4 in). The color and pattern is somewhat variable, especially for the females. Skin color can change depending on habitat colors, humidity, stress, and temperature. Color changes range from yellow to brown to black, from solid colors to speckled. Their breeding habits are very similar to Anaxyrus fowleri. The call or voice of a breeding male is a high trill that lasts between 6–30 seconds and sounds similar to a ringing telephone. Males call for an average of 6-7 nights during their breeding period. Females show preference for call efforts (rate × duration), but not call frequency. They hibernate during the winter. The eastern American toad has spots that contain only one to two warts. It also has enlarged warts on the tibia or lower leg below the knee. While the belly is usually spotted, in some areas many are, and it is generally more so on the forward half (in some rare individuals there may be few or no spots). This subspecies of the American toad has no or very little markings on it. The spades on the back legs are blackish. Some toads of this subspecies have a more pervasive red and deep brown color, many with red warts on their bodies. Also eastern American toads have parotoid glands that are the same color as the surrounding skin. The glands don't usually have any patterning on them. Other species that may be confused with the eastern American toad are Fowler's toad, which has three or more warts in the largest dark spots, and in the far west of its range woodhouse's toad. Fowler's toad can be especially difficult to identify in comparison to the eastern American toad but one difference is that it never has a spotted belly and both cranial crests touch the parotoid glands. Also, Fowler's toads are very fast hoppers (bursts of 5–10 fast hops) in comparison to Eastern toads lethargic, casual hopping and walking locomotion. In the eastern American toad these crests almost never touch the parotoid glands, which secrete bufotoxin, a poisonous substance meant to make the toad unpalatable to potential predators. Bufotoxin is a mild poison in comparison to that of other poisonous toads and frogs, but it can irritate human eyes and mucous membranes and is dangerous to smaller animals (such as dogs) when ingested. American toads require a semi-permanent freshwater pond or pool with shallow water in which to breed, to gather their water supplies in times of drought or as a routine, and for their early development. They also require dense patches of vegetation, for cover and hunting grounds. Given these two things and a supply of insects for food, American toads can live almost everywhere, ranging from forests to flat grassland. Females when caught are silent and easily tamed, adapting to terrarium life readily, while the smaller males are readily communicative. The smaller males do not adapt well to terrarium life and should be released after a few days of observation.[original research?] Adult toads are mostly nocturnal, although juveniles are often abroad by day. When it rains, these toads will become active and can be observed eating robustly worms and insects leaving their burrows and walking in front of an opportunist toad. These toads are 'creatures of habit' once they have a certain area they prefer to live within... an acre of wooded forest with water in proximity for soaking, a home with cool ledges and window wells; they commonly seek cover in burrows, under boardwalks, flat stones, boards, logs, wood piles, or other cover. When cold weather comes, these toads dig backwards and bury themselves in the dirt of their summer homes, or they may choose another site in which to hibernate. Their diet includes crickets, mealworms, earthworms, ants, spiders, slugs, centipedes, moths, and other small invertebrates. Some of these toads have been known to live over 30 years and currently a female specimen (over 13 centimeters long) is living healthily into her late 30s. Another female toad of 17 centimeters is known to have existed in Wisconsin from Washington Island on Lake Michigan. The eastern American toad may be confused with the Canadian toad in the area where they overlap, but the cranial crests in the American toad do not join to form a raised "boss" (bump) like they do in the Canadian toad. Its range also overlaps with the southern toad's, but in this species the cranial crests form two unique knobs. Dwarf American toad [edit] The dwarf American toad (A. a. charlesmithi), is a smaller version of the American toad, which reaches lengths of about 6 cm (2+1⁄4 in), is generally a dark reddish color ranging to light red in some specimens in isolated populations. The spots on the back are reduced or absent, and when present they contain a few small red warts and a black ring around it like in the normal American toad. The warts are always darker than the skin of the toad. Some specimens have a white dorsal line in the middle of their backs. The ventral surface or belly is usually cream colored with a few dark spots in the breast area. This subspecies can be distinguished from the above-mentioned species in the same manner as for the eastern American toad. The southwestern portion of the Dwarf American toad's range overlaps with that of the Gulf Coast toad. The latter species is distinguished by the presence of a dark lateral stripe as well as a deep "valley" between its prominent cranial crests. It eats mainly spiders, worms and small insects. Hudson Bay toad [edit] The Hudson Bay toad (A. a. copei) is a rare Canadian subspecies of A. americanus. This subspecies of the American toad has been seen in the northern parts of Ontario where there are a few isolated populations. These northern dwarf toads mostly have the red coloring on the sides of their bodies and have an unusually high number of warts for the subspecies. Interbreeding with eastern American toads caused this subspecies to lose the red coloring on their backs. Inbreeding avoidance [edit] Toads display breeding site fidelity, as do many other amphibians. Individuals that return to natal ponds to breed will likely encounter siblings as potential mates. Although incest is possible, Anaxyrus americanus siblings rarely mate. These toads likely recognize and actively avoid close kins as mates. Advertisement vocalizations by males appear to serve as cues by which females recognize their kin. Gallery [edit] Newly metamorphosed American Toad (~5mm) at confluence of Keay Brook, Berwick, ME and the Salmon Falls River. Approximately one week into adult cycle, shown in comparison to an adult female human hand Eastern American toad, seen from behind, shows characteristic markings and "warts" Closeup of the Eastern American toad Young American toad Front view of Eastern American toad Side view of camouflaged A. americanus Eastern American toad (A. a. americanus), North Dumfries, Ontario American toad eating its skin as it sheds American toad chirp - mating call American toad feeding time 5 year old female American toad. A closeup of an Eastern American toad seen in Tennessee. See also [edit] European toad European green toad Japanese toad References [edit] ^ IUCN SSC Amphibian Specialist Group (2015). "Anaxyrus americanus". IUCN Red List of Threatened Species. 2015 e.T54570A56843565. doi:10.2305/IUCN.UK.2015-4.RLTS.T54570A56843565.en. Retrieved 19 November 2021.{{cite iucn}}: uses deprecated \|page= identifier (help) ^ "Anaxyrus americanus". NatureServe Explorer. Retrieved 17 April 2024. ^ a b Frost, Darrel R. (2015). "Anaxyrus americanus (Holbrook, 1836)". Amphibian Species of the World: an Online Reference. Version 6.0. American Museum of Natural History. Retrieved 23 December 2015. ^ Review: The Amphibian Tree of Life, by Frost, D.R. et al. Archived 2013-04-04 at the Wayback Machine, Amphibiatree ^ "Bufonidae". AmphibiaWeb: Information on amphibian biology and conservation. [web application]. Berkeley, California: AmphibiaWeb. 2015. Retrieved 23 December 2015. ^ Kapfer, Joshua M. (2010-03-01). "A Survey of Gravid Snakes at Several Sites in Southern Wisconsin". Reptiles & Amphibians. 17 (1): 22–25. doi:10.17161/randa.v17i1.16058. ISSN 2332-4961. ^ "Species Profile: American Toad (Bufo [Anaxyrus] americanus) \| SREL Herpetology". srelherp.uga.edu. Retrieved 2024-04-24. ^ Tumlison, Renn; Trauth, Stanley E. (July 2006). "A novel facultative mutualistic relationship between bufonid tadpoles and flagellated green algae" (PDF). Herpetological Conservation and Biology. 1: 51–55. ^ Devin G. DiGiacopo; Jessica Hua (2022). "The effects of novel leaf litter deposition on competitive, predator-prey and host-parasite interactions of American toad larvae". Aquatic Ecology. 56 (1): 59–73. Bibcode:2022AqEco..56...59D. doi:10.1007/s10452-021-09893-y. ^ "ADW: Bufo americanus: Information". Animaldiversity.ummz.umich.edu. Retrieved 2011-04-01. ^ a b "University of Notre Dame: Yellow perch predation on tadpoles" (PDF). Retrieved 2011-04-01. ^ Pauly, G. B.; Hillis, D. M.; Cannatella, D. C. (November 2004). "The History of a Nearctic Colonization: Molecular Phylogenetics and Biogeography of the Nearctic Toads (Bufo)". Evolution. 58 (11): 2517–2535. doi:10.1554/04-208. PMID 15612295. S2CID 198155461. ^ American toad (Bufo americanus) Archived June 18, 2009, at the Wayback Machine, Natural Resources Canada ^ a b c Conant, Roger (1975). A Field Guide to Reptiles and Amphibians of Eastern and Central North America. Boston: Houghton Mifflin. ISBN 0-395-19979-4. ^ Sullivan, Brian K. (1992). "Sexual Selection and Calling Behavior in the American Toad (Bufo americanus)". Copeia. 1992 (1): 1–7. doi:10.2307/1446530. ISSN 0045-8511. JSTOR 1446530. ^ Sullivan, Brian K. (1992). "Sexual Selection and Calling Behavior in the American Toad (Bufo americanus)". Copeia. 1992 (1): 1–7. doi:10.2307/1446530. ISSN 0045-8511. JSTOR 1446530. ^ Lannoo, Michael. "Amphibian Declines: The Conservation Status of United States Species". Amphibiaweb. Regents of the University of California. Retrieved 8 May 2019. ^ Vigil, Stacey; Mengak, Michael (October 2006). "American Toad (Bufo americanus)". WSFS Natural History Series. 7: 2–3. ^ "American toad". Frog Toad Newt and Salamander Species of Canada. amphibians.ca. Retrieved 17 August 2013. ^ a b Waldman, B; Rice, JE; Honeycutt, RL (1992). "Kin recognition and incest avoidance in toads". Am. Zool. 32: 18–30. doi:10.1093/icb/32.1.18. External links [edit] Wikispecies has information related to Anaxyrus americanus charlesmithi. Wikimedia Commons has media related to American toad (Bufo americanus — Anaxyrus americanus). Wikispecies has information related to American toad. B. americanus at the United States Geological Survey site USDA – Bufo americanus – United States Department of Agriculture Integrated Taxonomic Information System. Animal Diversity Web: Bufo americanus Photograph and audio recording of male American toad \| Taxon identifiers \| \| Anaxyrus americanus \| Wikidata: Q694496 Wikispecies: Anaxyrus americanus ADW: Anaxyrus_americanus AmphibiaWeb: 100 ASW: Anaxyrus-americanus BOLD: 759868 CoL: 66LTT EoL: 1019159 EPPO: BUFOAM GBIF: 2422872 GISD: 1947 iNaturalist: 64968 ITIS: 773511 IUCN: 54570 NAS: 44 NatureServe: 2.102753 NCBI: 8389 Observation.org: 201611 ODNR: american-toad Open Tree of Life: 889326 Paleobiology Database: 133747 Xeno-canto: Anaxyrus-americanus \| \| Bufo americanus \| Wikidata: Q24248632 Wikispecies: Bufo americanus CoL: NNYD GBIF: 5217021 IRMNG: 11037189 ITIS: 173473 Open Tree of Life: 889326 Paleobiology Database: 133747 \| Retrieved from " Categories: IUCN Red List least concern species NatureServe secure species Anaxyrus Amphibians of the United States Amphibians of Canada Fauna of the Eastern United States Fauna of the Great Lakes region (North America) Amphibians described in 1836 Taxa named by John Edwards Holbrook Least concern biota of the United States Hidden categories: Cite IUCN maint Webarchive template wayback links Articles with short description Short description matches Wikidata Articles with 'species' microformats All articles with failed verification Articles with failed verification from August 2024 All articles that may contain original research Articles that may contain original research from September 2020 Commons category link is on Wikidata Taxonbars with automatically added original combinations Taxonbars with 20–24 taxon IDs American toad Add topic
15162	https://pages.uoregon.edu/cashman/GEO311/311pages/L5_exercise.htm	Geology 311 HOMEWORK I’d like you to try to work through the process of “normalizing” a mineral analysis (that is, converting a chemical analysis to a mineral formula) between now and Thursday’s class. I will not require you to hand your work in – instead, bring it with you and I will go over it at the start of class. Analytical instruments can be used to obtain chemical analyses. The resulting data are generally reported in weight percent of the major oxides in the mineral. Use the following chemical analysis and the instructions below to determine the specific mineral formula and the identity of the unknown mineral. Be sure to show all your work! Using a spreadsheet program like Excel makes these calculations easier, although this particular analysis can be done pretty quickly by hand as well. A similar set of instructions appears in Box 1.5 (p. 22) of your text. Analysis SiO 2 38.05% FeO 20.38% MgO 41.57% Total 100 % (Assume the known total number of oxygen atoms per formula unit is 4) Mineral formula_________ What mineral is this?________ Steps for determining a specific mineral formula: 1)Convert oxide wt.% into molecular proportion of each oxide. This is done by dividing the wt. % of each oxide by the molecular weight of the oxide. This gives the molecular proportion of each oxide. (The molecular weight for each is calculated from their atomic weights.) 2)Multiply the molecular proportion for each oxide by the # of oxygen atoms present in each oxide. This gives the O atomic proportion. 3)Sum the O atomic proportion column. 4)Divide the known total # of Oxygen atoms per unit cell in the mineral by the sum of the O atomic proportion. (e.g. Olivine is know to have 4 oxygen atoms in its mineral formula, Feldspars have 8). This operation gives you a normalization factor. 5)Next, normalize the O atomic proportions from each oxide by multiplying each entry by this normalization factor. This gives the number of anions based on the known number in the mineral formula. 6)Determine the number of cations associated with the oxygens by dividing the number of anions determined in step 5 by the number of oxygens in the reported oxide. (e.g. SiO2 has 2 O per 1 Si, Al2O3 has 1.5 O per 1 Al). 7)The number you obtain after doing step six is the number of cations that are in the final mineral formula. EXAMPLE: Olivine: step 1step 2step 4step 5step 6step 7 Element Oxide wt. %Molec. Wt.Molec. Prop.# Oxygen O atomic Normaliza.# of Anions Oxygen Cations of oxides proportion factor per cation SiO2 31.85 60.074 0.530179 2 1.060368 1.90543 2.02043 2 1.0102 FeO 58.64 71.841 0.816247 1 0.816247 1.90543 1.5553 1 1.5553 MnO 0.85 70.937 0.011982 1 0.011982 1.90543 0.02283 1 0.02283 MgO 8.49 40.299 0.210675 1 0.210675 1.09543 0.04142 1 0.40142 (step 3)Total 2.09926 Normalization factor=4Oxygens/2.09926 Formula: (Fe 1.555 Mg 0.401 Mn 0.023) Si 1.01 O 4
15163	https://annamalaiuniversity.ac.in/studport/download/engg/civil/resources/FlowInOpenChannels.pdf	Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 1 FLOW IN OPEN CHANNELS (UNIFORM FLOW) GEOMETRICAL PROPERTIES OF CHANNEL SECTION The geometrical properties of a channel section can be defined by the shape of the section and the depth of flow. 1. Depth of flow, y: It is the vertical distance of the lowest point of a channel section from the free surface of water. 2. Top width, T: It is the width of the channel section at the free surface of water. 3. Wetted area, A: It is the cross-sectional area of the flow of the channel section. 4. Wetted perimeter, P: It is the length of the channel boundary in contact with the flowing water at any section. 5. Hydraulic radius, R (or Hydraulic mean depth): It is the ratio of wetted area, A and wetted perimeter, P. R = P A 6. Hydraulic depth, D: It is the ratio of wetted area, A and the top width, T D = T A 7. Section factor, Z, for critical flow computation: It is the product of wetted area and the square root of the hydraulic depth, D Z = 2 / 1 3 2 / 1 2 / 3            T A T A T A A D A 8. Section factor, Z, for uniform flow computation: It is the product of the wetted area and the hydraulic radius raised to the two-thirds power. Z = AR2/3 Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 2 VELOCITY DISTRIBUTION IN A CHANNEL SECTION The velocity of flow at any channel section is not uniformly distributed. Why? This is due to the presence of free surface and the frictional resistance offered to free flow of water by the boundary of the channel. The velocity distribution in a channel section can be measured either by a pitot tube or a currentmeter. The typical patterns of velocity distribution in rectangular, trapezoidal, triangular and circular channel sections are represented in Figure 1 below. The pattern of velocity distribution in a channel section is represented by lines of equal velocity. For typical velocity distribution curve along a vertical line of a channel section, refer to a standard text book/reference on Fluid Mechanics/Hydraulics/Open Channel Flow. In a straight reach of a channel, the maximum velocity generally occurs at a distance of 0.05 to 0.15 depth of flow from the free surface of flow. The velocity distribution in a channel section depends upon various factors such as the shape of the section, roughness of the boundary of the channel and the alignment of the channel. The average velocity of flow in a channel section can be computed from the vertical velocity distribution curve obtained for that section from actual measurements. From measurements, it is observed that the velocity measured at 0.6 depth of flow from the free surface is near to the average velocity of flow in the vertical section. A still better approximation for the average velocity of flow can be obtained by taking the average of velocities measured at 0.2 depth of flow and 0.8 depth of flow measured from the free surface. As the velocity distribution in a channel section is non-uniform, correction factors have to be applied while computing the kinetic energy and momentum. The kinetic energy correction factor, also called Coriolis coefficient, is denoted by the Greek symbol . The momentum correction factor, also called Boussinesq coefficient, is denoted by them Greek symbol . The values of  and  can be obtained from the actual velocity distribution profile for a channel section. From experiments, it is found that the value of kinetic energy correction factor  varies from 1.03 to 1.36 for turbulent flow in fairly straight prismatic channels. Similarly, it is found from experiments that the value of momentum correction factor Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 3  varies from 1.01 to 1.12 for fairly straight prismatic channels. However, for the sake of simplicity, the values of  and  are assumed to be unity in the present analysis. UNIFROM FLOW IN OPEN CHANNELS When water flows in an open channel, it experiences resistance offered by the boundary of the channel. This causes loss of energy of the flowing water in the direction of flow. This resistance is overcome by the flowing water by the components of gravity forces acting on the body of water in the direction of flow. Problem : An open channel is V – shaped with each side being inclined at 45 to the vertical. If it carries a discharge of 0.04 m3/s, when the depth of flow at the centre is 225 mm, calculate the slope of the channel assuming that Chezy’s C = 50. Solution. Data given: Shape of section of open channel: Triangular Slope of each side of the channel section = 45 to the vertical.  L V AL sin W = AL o F2 F1 y V2/2g  Free surface TEL 2 1 A C B D Channel bottom Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 4 Discharge, Q = 0.04 m3/s Depth of flow at the centre of channel section, y = 225 mm = 0.225 m Bed slope of the channel, S = ? Chezy’s C = 50 Wetted area of channel section, A = Zy2 = 1 x (0.225)2 = 0.050625 m2 As per the continuity principle, we have, Q = AV where V = mean velocity of flow in channel Therefore, V = 050625 . 0 04 . 0  A Q = 0.79 m/s Wetted perimeter, P = 1 2 2  Z y = 1 1 ) 225 . 0 ( 2 2 = 0.6364 m Hydraulic radius, R =   6364 . 0 050625 . 0 P A 0.07955 m Chezy’s formula: RS C V   0.79 = 50 x S ) 07955 . 0 (  S = 07955 . 0 x ) 50 ( ) 79 . 0 ( 2 2 = 0.00314 = 319 1 Problem: A rectangular channel is 2.5 m wide and has a uniform bed slope of 1 in 500. If the depth of flow is constant at 1.7 m calculate (a) the hydraulic mean depth, (b) the velocity of flow, (c) the volume rate of flow. Assume that the value of the coefficient C in Chezy’s formula is 50. Solution. y = 225 mm 1 Z 1 Z 45 tan 45 = 1 Z  Z = tan 45 = 1 Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 5 (a) Hydraulic mean depth, R = P A where, A = wetted area = By = 2.5 x 1.7 = 4.25 m2 P = wetted perimeter = B + 2y = 2.5 + (2 x 1.7) = 2.5 + 3.4 = 5.9 m R = 72 . 0 9 . 5 25 . 4   P A m (b) Velocity of flow, RS C V  = 50         500 1 72 . 0 = 1.897 ms-1 (Note: For uniform steady flow, energy gradient, S = bed slope, So = 1 / 500) © Volume rate of flow, Q = AV = 4.25 x 1.897 = 8.064 m3/s Problem: An open channel has a vee-shaped cross section with sides inclined at an angle of 60 to the vertical. If the rate of flow is 80 dm3 s-1 when the depth at the centre is 0.25 m, what must be the slope of the channel assuming C = 45. Solution. Q = 80 dm3 s-1 = 80 x (10-1)3 = 80 x 10-3 m3 s-1 (Note: 1 dm = 10 cm = 0.1 m = 10-1 m) Wetted area of channel section, A = Zy2 = 1.734 x (0.25)2 = 0.108 m2 As per the continuity principle, we have, y = 0.25 m 1 z 1 z 60 tan 60 = 1 z  z = tan 60 = 1.734 B = 2.5 m y = 1.7 m Data: Bottom width of channel, B = 2.5 m Depth of flow, y = 1.7 m Bed slope, So = 1 in 500 Chezy’s constant, C = 50 Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 6 Q = AV where V = mean velocity of flow in channel Therefore, V = 108 . 0 08 . 0  A Q = 0.738 m/s Wetted perimeter, P = 1 2 2  Z y =   1 734 . 1 ) 25 . 0 ( 2 2 = 1.000 m Hydraulic radius, R =   000 . 1 108 . 0 P A 0.108 m Chezy’s formula: RS C V   0.738 = 45 x S ) 108 . 0 (  S = x 108 . 0 ) 45 ( ) 738 . 0 ( 2 2 = 0.00249 = 401 1 Problem: A channel 5 m wide at the top and 2 m deep has sides sloping 2 vertically in 1 horizontally. The slope of the channel is 1 in 1000. Find the volume rate of flow when the depth of water is constant at 1 m. Take C as 53. What would be the depth of water if the flow rate were to be doubled? Solution. Data: Top width of channel, Tw = 5 m Depth of channel, d = 2 m Let Bottom width of channel be B Side slope = 2 vertical : 1 horizontal = 1 vertical : 0.5 horizontal = 1 : z Bottom slope of the channel, So = 1 in 1000 Depth of flow in the channel, y = 1 m Tw = B + 2zd  5 = B + 2 x 0.5 x 2 = B + 2  B = 5 – 2 = 3 m For uniform steady flow, 2 1 B 2 m 5 m 1 m Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 7 Energy gradient, S = Bed slope of channel, So = 001 . 0 1000 1  Mean velocity of flow, V = C RS Wetted area, A =     1 1 5 . 0 3 ) (    y zy B = 3.5 m2 Wetted perimeter, P =    1 2 2 z y B   1 5 . 0 1 2 3 2   = 5.236 m Hydraulic radius, R = P A = 668 . 0 472 . 7 8  m V =    001 . 0 668 . 0 53 = 1.37 ms-1 Q = AV = 3.5 x 1.37 = 4.8 m3 s-1 Depth of flow when the flow is doubled = ? Now, Q = 2 (4.8) = 9.6 m3 s-1 Q = 9.6 = AV Wetted area A =     2 5 . 0 3 5 . 0 3 ) ( y y y y y zy B      V = C RS Wetted perimeter, P =    1 2 2 z y B   1 5 . 0 2 3 2  y = 3 + 2.236y R = P A = y y y 236 . 2 3 5 . 0 3 2   Hence, Q = 9.6 = ( 2 5 . 0 3 y y  ) (53)   001 . 0 236 . 2 3 5 . 0 3 2           y y y Solving by trial and error, y = 1.6 m Problem: Water is conveyed in a channel of semi-circular cross-section with a slope of 1 in 2500. The Chezy coefficient C has a value of 56. If the radius of the channel is 0.55 m, what will be the volume flowing per second when the depth of flow is equal to the radius? If the channel had been rectangular in form with the same width of 1.1 m and depth of flow of 0.55 m, what would be the discharge for the same slope and value of C? Solution. Bed slope of channel, So = 1 in 2500 y = 0.55 m r Data: Radius of semicircular channel, r = 0.55 m Depth of flow, y = r = 0.55 m Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 8 Chezy coefficient, C = 56 Q = AV Where A = wetted area =    2 55 . 0 2 2 2  r 0.4754 m2 V = mean velocity of flow = C RS R = hydraulic mean depth = P A  P = wetted perimeter = r =  (0.55) = 1.729 m R =  729 . 1 4754 . 0 0.275 m Hence, Q = AC RS = 0.4754 x 56 x         2500 1 275 . 0 = 0.279 m3 s-1 Wetted area, A = By = 1.1 x 0.55 = 0.605 m2 Wetted perimeter, P = B + 2y = 1.1 + (2 x 0.55) = 1.1 + 1.1 = 2.2 m Hydraulic radius, R =   2 . 2 605 . 0 P A 0.275 m Mean velocity of flow, V = C RS = 56         2500 1 275 . 0 = 0.587 m s-1 Volume rate of flow, Q = AV = 0.605 x 0.587 = 0.355 m3 s-1 Problem: An open channel has a cross-section in the form of trapezium as shown in Figure below. Assuming that the roughness coefficient n is 0.025, the bed slope is 1 in 1800 and the depth of flow is 1.2 m, find the volume rate of flow Q using (a) Chezy’s formula with C determined from the Kutter’s formula, and (b) the Manning’s formula. B = 1.1 m y = 0.55 m Data: Bottom width of channel, B = 1.1 m Depth of flow, y = 0.55 m Bed slope, So = 1 in 2500 Chezy’s constant, C = 56 Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 9 Solution. Data: Bottom width of channel, B = 4 m Side slope = 1 vertical: z horizontal = 1 : 1.5 Depth of flow, y = 1.2 m Manning’s roughness coefficient, n = 0.025 Bed slope, So = 1 / 1800 Required: Volume flow rate, Q (a) Using Chezy’s formula: Wetted area, A =       2 . 1 2 . 1 5 . 1 4 ) (    y zy B = 6.96 m2 Wetted perimeter, P =    1 2 2 z y B    1 5 . 1 2 . 1 2 4 2   = 8.327 m Hydraulic radius, R = P A = 836 . 0 327 . 8 96 . 6  m For uniform steady flow, energy gradient, S = bed slope, So = 1800 1 Chezy’s C is computed from Kutter’s formula as: C = R n S n S           00155 . 0 23 1 1 00155 . 0 23 = 836 . 0 025 . 0 1800 1 00155 . 0 23 1 025 . 0 1 1800 1 00155 . 0 23                             = 38.6 Mean velocity of flow, V = C RS Q = AV = AC RS = 6.96 x 38.6         1800 1 836 . 0 = 5.79 m3s-1 1 1.5 4 m 1.2 m Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 1 0 (b) Using Manning’s formula: Manning’s formula for mean velocity of flow is given by 2 / 1 3 / 2 1 S R n V  Q = AV = A 2 / 1 3 / 2 1 S R n = (6.96)   2 / 1 3 / 2 1800 1 836 . 0 025 . 0 1             = 5.82 m3s-1 Problem: An earth channel is trapezoidal in cross-section with a bottom width of 1.8 m and side slopes of 1 vertical to 2 horizontal. Taking the friction coefficient in the Bazin formula as 1.3 and the slope of the bed as 0.57 m per kilometre, find the discharge in cubic metres per second when the depth of flow is 1.5 m. Solution. Data: Bottom width, B = 1.8 m Side slope = 1 vertical : 2 horizontal = 1 vertical : z horizontal Bazin’s friction coefficient, K = 1.3 Bed slope, So = 0.57 m per km = 0.57 m / 1000 m = 0.00057 Depth of flow, y = 1.5 m Required: Discharge, Q = ? Wetted area, A =      5 . 1 5 . 1 2 8 . 1 ) (    y zy B = 7.2 m2 Wetted perimeter, P =    1 2 2 z y B   1 2 5 . 1 2 8 . 1 2   = 8.508 m Hydraulic radius, R = P A = 846 . 0 508 . 8 2 . 7  m 1 2 1.8 m 1.5 m Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 1 1 Bazin’s formula for evaluating Chezy’s constant C: R K C   1 9 . 86 = 846 . 0 3 . 1 1 9 . 86  = 36 Mean velocity of flow, V = C RS = 36    00057 . 0 846 . 0 = 0.791 m s-1 Discharge, Q = AV = 7.2 x 0.791 = 5.69 m3s-1 Problem: An open channel is to be constructed of trapezoidal section and with side slopes 1 vertical to 1.5 horizontal. Find the proportions, that is, the relation between bottom width and depth of flow) for minimum excavation (that is, best hydraulic section). If the flow is to be 2.7 m3/s, calculate the bottom width and the depth of flow assuming Chezy’s C as 44.5 and the bed slope as 1 in 4000. Solution. Side slope of channel section = 1 vertical : z horizontal = 1 : 1.5 Let the bottom width of channel section be B and the depth of flow be y. Q = 2.7 m3/s C = 44.5 Bed slope S = 1 in 4000 = 00025 . 0 4000 1  B = ? y = ? Required: To find the relation between bottom width B and the depth of flow y for most economical section of channel 1 Z B y Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 1 2 For most economical trapezoidal section, we have, 2 1 2 2 z y zy B    2 ) 5 . 1 ( 1 2 ) 5 . 1 ( 2    y y B  y y y y B 6056 . 3 25 . 3 2 25 . 2 1 2 3       y y y B 6056 . 0 3 6056 . 3      y B 0.6056 Wetted area, A = 2 1056 . 2 ) 5 . 1 6056 . 0 ( ) ( y y y y y zy B     Wetted perimeter, P =    1 2 2 z y B 1 ) 5 . 1 ( 2 6056 . 0 2   y y = 0.6056y + 3.6056y = 4.2112y Hydraulic radius, R = P A = y y y 5121 . 0 2112 . 4 1056 . 2 2  As per continuity principle, we have, Q = A.V Hence, V = A Q  V = 2 2 2823 . 1 1056 . 2 7 . 2 y y  As per Chezy’s formula, we have, RS C V   2 2823 . 1 y =44.5 ) 00025 . 0 )( 5121 . 0 ( y  ) 00025 . 0 )( 5121 . 0 )( 25 . 1980 ( 64428 . 1 4 y y   1.64428 = 0.253522 y5  y5 = 6.485762 253522 . 0 64428 . 1   y = 1.453 m Hence, bottom width, B = 0.6056 y = 0.6056 x 1.453 = 0.880 m Problem: A trapezoidal channel has side slopes of 3 horizontal to 4 vertical and the slope of its bed is 1 in 2000. Determine the optimum dimensions of the channel if it is to carry water at 0.5 m3 s-1. Use the Chezy’s formula, assuming that C = 80 m1/2 s-1. Solution. Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 1 3 Data: Side slope of channel = 3 horizontal : 4 vertical = z horizontal : 1 vertical = 0.75 : 1 Bottom slope, So = 1 in 2000 Discharge, Q = 0.5 m3 s-1 Chezy’s constant C = 80 m1/2 s-1 Required: To find the optimum dimensions of the channel. For most economical trapezoidal section, we have, 2 1 2 2 z y zy B    2 ) 75 . 0 ( 1 2 ) 75 . 0 ( 2    y y B  y y y y B 5 . 2 5625 . 1 2 5625 . 0 1 2 5 . 1       y y y B    5 . 1 5 . 2   y B 1 Wetted area, A = 2 75 . 1 ) 75 . 0 ( ) ( y y y y y zy B     Wetted perimeter, P =    1 2 2 z y B 1 ) 75 . 0 ( 2 2   y y = y + 2.5y = 3.5y Hydraulic radius, R = P A = y y y 5 . 0 5 . 3 75 . 1 2  As per continuity principle, we have, Q = A.V Hence, V = A Q  V = 2 2 285714 . 0 75 . 1 5 . 0 y y  As per Chezy’s formula, we have, RS C V  4 3 B y Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 1 4  2 285714 . 0 y =80       2000 1 ) 5 . 0 ( y  ) 0005 . 0 )( 5 . 0 )( 6400 ( 081633 . 0 4 y y   0.081633 = 1.6 y5  y5 = 051 . 0 6 . 1 081633 . 0   y = 0.552 m Hence, bottom width, B = y = 0.552 m Problem: It is required to excavate a canal of rectangular section out of rock to bring 15 m3 of water per second from a distance of 6.4 km with a velocity of 2.25 m/s. Determine the most suitable section for the canal and its gradient. Take Manning’s n = 0.02. Solution. Discharge, Q = 15 m3/s Velocity of flow in channel, V = 2.25 m/s Manning’s n = 0.02 As per continuity principle, we have, Q = AV where A = wetted area A =   25 . 2 15 V Q 6.667 m2 For a rectangular channel section, we have, A = By where B = bottom width y = depth of flow For most economical rectangular channel section, we have, y = 2 B (or) B = 2y Hence, A = 6.667 = (2y)y = 2y2  y =  2 667 . 6 1.826 m B = 2y = 2 x 1.826 = 3.651 m Wetted perimeter, P = B + 2y = 3.651 + 2(1.826) = 3.651 + 3.651 = 7.302 m Hydraulic radius, R =   302 . 7 667 . 6 P A 0.913 m As per Manning’s formula, we have, 2 / 1 3 / 2 1 S R n V  Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 1 5  S1/2 = 3 / 2 ) 913 . 0 ( 0.02 x 25 . 2 =  941 . 0 045 . 0 0.047815  S = 0.00229 Problem: The water supply for a turbine passes through a conduit which for convenience has its cross-section in the form of a square with one diagonal vertical. If the conduit is required to convey 8.5 x 10-3 m3 s-1 under conditions of maximum discharge at atmospheric pressure when the slope of the bed is 1 in 4900, determine its size assuming that the velocity of flow is given by . 80 2 / 1 3 / 2 S R V  Solution. When the same square section is placed with one of its diagonals vertical as shown in Figure, for maximum discharge, the depth of flow y becomes equal to half the height of the vertical diagonal and the free surface of flow coincides with the other diagonal that is horizontal. Length of diagonal = B 2 Depth of flow, y = 2 2B Hydraulic radius, R = 4 2 2 2 2 2 B B y           Wetted area, A =    2 2 1 2 1 2 B xBxB EH EF   . 80 2 / 1 3 / 2 S R V  For most economical square section (that is, for square section to carry the maximum discharge), we have, Depth of flow = half the bottom width i.e., y = B/2 E G B B 45 45 y F H Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 1 6 = 2 / 1 3 / 2 4900 1 4 2 80               B Q = AV =         2 2 B 2 / 1 3 / 2 4900 1 4 2 80               B  8.5 x 10-3 =         2 2 B 2 / 1 3 / 2 4900 1 4 2 80               B = 0.285714 B8/3  B8/3 = 02975 . 0 285714 . 0 0085 . 0   B = (0.02975)3/8 = 0.268 m Problem: Determine the most efficient section of a trapezoidal channel with side slopes 1 vertical to 2 horizontal. The channel carries a discharge of 11.25 m3/s with a velocity of 0.75 m/s. What should be the bed slope of the channel? Take Manning’s n = 0.025. Solution. Side slope of channel section = 1 vertical : z horizontal = 1 : 2 Discharge, Q = 11.25 m3/s Velocity of flow in channel, V = 0.75 m/s Manning’s n = 0.025 The channel section is considered to be most economical. Bed slope of channel, S = ? For most economical trapezoidal channel section, we have, 2 1 2 2 z y zy B    2 2 1 2 ) 2 ( 2    y y B  y y y y B 472 . 4 5 2 4 1 2 4       y y y B 472 . 0 4 472 . 4      y B 0.472 As per continuity principle, we have, Q = AV where A = wetted area A =   75 . 0 25 . 11 V Q 15 m2 = 2 472 . 2 ) 2 472 . 0 ( ) ( y y y y y zy B     Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 1 7  y2 = 472 . 2 15  y = 2.463 m Hence, B = 0.472 y = 0.472 x 2.463 = 1.163 m Wetted perimeter, P =    1 2 2 z y B 1 2 2 472 . 0 2  y y = 0.472 y + 4.472 y = 4.944 y = 4.944 x 2.463 = 12.177 m Hydraulic radius, R = 177 . 12 15  P A = 1.232 m As per Manning’s formula, we have, 2 / 1 3 / 2 1 S R n V   0.75 = 2 / 1 3 / 2 ) 232 . 1 ( 025 . 0 1 S  S1/2 = 3 / 2 ) 232 . 1 ( 0.025 x 75 . 0 = 0.0163  S = 0.000266 Problem: A canal is to have a trapezoidal section with one side vertical and the other side sloping at 45. It has to carry a discharge of 30 m3/s with an average velocity of 1 m/s. Compute the dimensions of the section which will require the minimum lining. Solution. Discharge, Q = 30 m3/s Average velocity of flow, V = 1 m/s The trapezoidal section ABCD is made up of a rectangular portion ABCE of dimensions (B x y) and a triangular portion AED right – angled at E. From triangle AED, tan 45 = 1   y ED AE ED Hence, ED = y 45 B y A B C D E Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 1 8 Hence, area of triangular portion = AE x ED x 2 1 = y x y x 2 1 = 2 2 y Area of rectangular portion = By Hence, area of trapezoidal section, A = By + 2 2 y By = A - 2 2 y B = 2 y y A  Wetted perimeter, P = DA + AB + BC = y B ED AE    2 2 ) ( ) ( = y B y y    2 2 = y B y   2 2 = y B y   2 = 2.414 y + B Putting B = 2 y y A  in P = 2.414 y + B, we have, P = 2.414 y + 2 y y A  Assuming area A to be constant, the above equation can be differentiated with respect to y and equated to zero for obtaining the condition for minimum P. Hence, 0 914 . 1 2 1 414 . 2 2 2       y A y A dy dP Putting A = By + 2 2 y , we have, 1.914 - 0 2 2 2   y y By  1.914 y2 – By - 0 2 2  y  1.414 y2 – By = 0  By = 1.414 y2  B = 1.414 y = y 2 This is the condition for most economical trapezoidal section of channel defined in the problem. From continuity principle, we have, Wetted area of flow, A = V Q = 30 1 30  m2 Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 1 9 We have, A = By + 2 2 y Putting B = 1.414 y, we have, A = 30 m2 = (1.414 y) y + 2 2 y  1.414 y2 + 2 2 y = 30  1.914 y2 = 30  y2 = 674 . 15 914 . 1 30   y = 3.959 m Hence, B = 1.414 y = 1.414 x 3.959 = 5.598 m Problem: An open channel laid at a constant slope is required to carry a maximum discharge of 5 m3/s and a minimum discharge of 1 m3/s at a constant velocity of 1 m/s at all depths of flow. Compute the top width at the free surface and the depths of flow corresponding to minimum and maximum discharges. For minimum discharge, a rectangular channel section of the most economical type may be designed. Solution. Maximum discharge, Qmax = 5 m3/s Minimum discharge, Qmin = 1 m3/s Constant velocity of flow, V = 1 m/s For minimum discharge, a rectangular section that is most economical is to be designed. Case (i) Discharge is minimum Qmin = 1 m3/s As per continuity principle, we have, Qmin = AV where A = wetted area of rectangular channel section when the discharge is minimum A = 1 1 1 min   V Q m2 For rectangular channel section, A = By where B = bottom width of channel section y = depth of flow corresponding to minimum discharge Further, for most economical rectangular channel section, we have, Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 2 0 y = 2 B Putting y = 2 B in A = By, we have, 1 = B x 2 B = 2 2 B  B2 = 2  B =1.414 m Hence, y = 1.414 / 2 = 0.707 m Wetted perimeter of most economical rectangular section, P = B + 2y = 1.414 + 2 x 0.707 = 2.828 m Hydraulic radius, R = A/P = 1 / 2.828 = 0.3536 m Note: the velocity of flow is constant at all depths of flow if the hydraulic radius is constant at all depths of flow. When the discharge is maximum equal to 5 m3/s, A = Q / V = 5 / 1 = 5 m2 R = 0.375 m = A / P = 5 / P  P = 5 / 0.375 = 13.333 m = width of bottom rectangular section + 2 (depth of rectangular section) + 2 (length of each side of the constant velocity section) i.e., 13.333 = 1.414 + 2(0.707) + 2(length of each side of the constant velocity section) Length of each side of the constant velocity section = [13.333 – 1.414 – 2(0.707)] / 2 = 5.2525 m The cross-section of a channel with constant velocity at all depths of flow is defined by the equation y =  C R x x R e    2 2 log For x = 707 . 0 2 414 . 1  m; y = 0 Hence, C = -   2 2 log R x x R e   = -   2 2 ) 707 . 0 ( 707 . 0 log R R e   Thus, y =   2 2 log R x x R e   -   2 2 ) 707 . 0 ( 707 . 0 log R R e   =                2 2 2 2 ) 707 . 0 ( 707 . 0 log R R x x R e Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 2 1 Solution incomplete Problem: A trapezoidal channel with side slopes of 2 horizontal to 1 vertical has to carry a discharge of 20 m3/s. if the bottom width is 4 m, calculate the bottom slope required to maintain a uniform flow at a depth of 1.5 m. Take Manning’s n = 0.015. What would be the normal depth of flow for the above channel to carry a discharge of 27 m3/s? Solution. Wetted area of channel section, A = (B + zy) y = [4 + (2) (1.5)] (1.5) = 10.5 m2 Wetted Perimeter, P =    1 2 2 z y B 4 + 2(1.5) 1 22  = 10.708 m Hydraulic radius, R = A/P = 10.5 / 10.708 = 0.981 m Manning’s formula: 2 / 1 3 / 2 1 S R n V  As Q = AV, we have, Q = A 2 / 1 3 / 2 1 S R n = 2 / 1 3 / 2 1 S AR n  20 =   2 / 1 3 / 2 981 . 0 ) 5 . 10 ( 015 . 0 1 S  S1/2 = 3 / 2 ) 981 . 0 )( 5 . 10 ( ) 015 . 0 )( 20 ( = 0.0289  S = 0.000837 = 1 in 1194 (i.e., 1 vertical to 1194 horizontal) Let y be the normal depth of flow Q = 27 m3/s A = (B + zy) y = (4 + 2 y) y P =    1 2 2 z y B 4 + 2y 1 22  = 4 + 2y 5 R = A / P = ) 5 2 4 ( ) 2 4 ( y y y   Q = AV = A 2 / 1 3 / 2 1 S R n  27 = [(4 + 2 y) y] n 1   2 / 1 3 / 2 000837 . 0 ) 5 2 4 ( ) 2 4 (         y y y  27 =       015 . 0 1       2 / 1 3 / 2 3 / 5 000837 . 0 ) 5 2 4 ( ) 2 4 ( y y y   Topic: Flow in Open Channels (Uniform Flow) DR.A.MURUGAPPAN, PROF. OF CIVIL ENGG., ANNAMALAI UNIVERSITY, ANNAMALAI NAGAR 2 2 Solving for y by trial and error, we have, y = 1.745 m
15164	https://www.metalsales.us.com/blog/thermal-expansion/	Learn about thermal expansion and contraction with metal panels Skip to content PRODUCT SEARCH PRODUCT TYPESMenu Toggle ROOF PANELS WALL PANELS IMPACT STEEL BUILDINGSMenu Toggle 3D BUILDING DESIGNER BUILDING QUOTE IMPACT DEALER BECOME AN APPROVED INSTALLER STEEL BUILDING GLOSSARY SELF STORAGE SECONDARY FRAMING RETROFITMenu Toggle RETRO-MASTER ROOF HUGGER FLAT SHEETS – SLIT COIL RESOURCESMenu Toggle BRANCH PAGESMenu Toggle ANCHORAGE, AK BAY CITY, MI DEER LAKE, PA DENVER, CO DETROIT LAKES, MN FONTANA, CA FORT SMITH, AR INDEPENDENCE, MO JACKSONVILLE, FL JEFFERSON, OH MOCKSVILLE, NC NASHVILLE, TN NEW ALBANY, IN ROCK ISLAND, IL ROGERS, MN SELLERSBURG, IN SIOUX FALLS, SD SPOKANE, WA TEMPLE, TX WOODLAND, CA COLORSMenu Toggle COLOR CHARTS & WARRANTIES COLOR VISUALIZER CONDENSED TECHNICAL REFERENCES INSTALL GUIDES LITERATURE SPECIFICATIONSMenu Toggle 3-PART SPECS SPECIFICATION DATA SHEETS MASTERSPEC ARCHITECTSMenu Toggle ARCHITECT RESOURCES AIA CONTINUING EDUCATION PRODUCT SUBMITTAL COLOR GUIDES & TRENDS CASE STUDIES TESTING & APPROVALSMenu Toggle FLORIDA APPROVED PRODUCTS MIAMI-DADE APPROVED PRODUCTS ICC-ES EVALUATION REPORTS CONTRACTORSMenu Toggle CONTRACTOR RESOURCES COLOR GUIDES PRODUCT SUBMITTAL CONDENSED TECHNICAL REFERENCES STANDING SEAM ROOF SYSTEMS FLAT SHEETS – SLIT COIL VIDEOS SUSTAINABILITY TESTING & APPROVALSMenu Toggle FLORIDA APPROVED PRODUCTS MIAMI-DADE APPROVED PRODUCTS ICC-ES EVALUATION REPORTS ARTICLES AUTOCAD & BIM GLOSSARY PROJECT GALLERY Main Menu Main Menu Generic selectors [x] Exact matches only [x] Search in title [x] Search in content [x] Post Type Selectors [x] [x] Go Back March 29, 2021 THERMAL EXPANSION AND CONTRACTION When metal is subjected to a change in temperature, it responds by changing size, known as thermal expansion and contraction. An increase in temperature makes the member bigger and a decrease in temperature makes the member smaller. For metal panels, change in length is the significant concern. The change in length is proportional to both the original length, the change in temperature and the response characteristic of the metal. Δ Length (in inches) = Length (in inches) Δ Temperature (in °F) ε (in / °F). Each metal has its own response characteristic, ε, known as the coefficient of thermal expansion. ε for steel is 0.0000065 / °F. For aluminum, ε is 0.0000128 / °F. A 100’ long steel panel experiencing a 100°F temperature change will increase in length by 0.78”; while an aluminum panel of the same length experiencing the same temperature change will increase in length by 1.56”. For metal panels, an endlap is considered to be a rigid attachment, so that the total considered length of a panel is the sum of the lengths of the panels attached by endlap. The lowest temperature that a panel will experience is approximately equal to the lowest air temperature that the panel is exposed to. The highest temperature that a panel will experience is significantly greater than the highest air temperature that the panel is exposed to. This greater temperature is due to the continual absorption of solar energy through daylight hours. The amount of energy absorbed is a function many factors including: climate, building geometry, shading and reflectivity of the panel surface. Direct fastened panel systems are commonly limited to 60’ to 80’ runs. The change in panel length can affect the fasteners by working them loose or by slotting the fastener holes. The stiffness of the support system can also affect the fastener / panel interaction. Many standing seam panel systems can accommodate 150’ runs or greater. Standing seam systems have a clip attachment mechanism that accommodates the thermal movement of the panel without damage. Standing seam roof systems must be attached to the building with a fixed point, a point where the roof does not move relative to the support framing. This fixed point is commonly located near an end of the panels – either at the eave or ridge. The other, free end of the panels will experience the maximum amount of thermal movement. About Metal Sales Metal Sales Manufacturing Corporation is the premier nationwide provider of metal panels for the construction industry. Metal Sales works with architectural specifiers and commercial construction professionals to create inspirational design solutions. With the industry’s largest and most knowledgeable sales and technical support team, Metal Sales has the expertise to address today’s challenges in high-performance, sustainable and Net-Zero building. Metal Sales has outreach around the world. Delivering outstanding roof, wall and fascia metal panels from its 21 facilities throughout the U.S. For more information, visitwww.metalsales.us.com To find the nearest branch near you, please visithere. ← Previous Post Next Post → Related Posts Benefits and Cost of a Metal Roof What’s the Difference Between Galvanized Steel and Galvalume? 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15165	https://www.mashupmath.com/blog/how-to-add-fractions	How to Add Fractions in 3 Easy Steps March 9, 2023 by Anthony Persico How to Add Fractions in 3 Easy Steps Math Skills: How to add fractions with the same denominator and how to add fractions with different denominators Knowing how to add fractions is an important and fundamental math skill. Since fractions are a critically important math topic, understanding how to add fractions is a fundamental building block for mastering more complex math concepts that you will encounter in the future. (Looking to learn how to subtract fractions? Click here to access our free guide) Luckily, learning how to add fractions with like and unlike (different) denominators is a relatively simple process. The free How to Add Fractions Step-by-Step Guide will teach you how to add fractions when the denominators are the same and how to add fractions with different denominators using a simple and easy 3-step process. This guide will teach you the following skills (examples included): What is the difference between the numerator and denominator of a fraction? How to add fractions with the same denominator? How to add fractions with different denominators? But, before you learn how to add fractions, let’s do a quick review of some key characteristics and vocabulary terms related to fractions before we move onto a few step-by-step examples of how to add fractions. Are you ready to get started? How to Add Fractions: Definitions and Vocabulary In order to learn how to add fractions, it is imperative that you understand the difference between a numerator and a denominator. Definition: The numerator of a fraction is the top number in the fraction. For example, in the fraction 3/4, the numerator is 3. Definition: The denominator of a fraction is the bottom number in the fraction. For example, in the fraction 3/4, the denominator is 4. Pretty simple, right? These terms are visually represented in Figure 01 below. Make sure that you understand the difference between the numerator and the denominator of a fraction before moving forward in this guide. If you mix them up, you will not learn how to add fractions correctly. Figure 01: The numerator is the top number of a fraction, and the denominator is the bottom number of a fraction. Now that you know the difference between the numerator and the denominator of a fraction, you are ready to learn how to identify whether or not a given problem involving adding fractions falls into which of the following categories: Like Denominators (the denominators are the same) Unlike Denominators (the denominators are different) Fractions with like denominators have bottom numbers that equal the same value. For example, in the case of 1/5 + 3/5, you would be adding fractions with like denominators since both fractions have a bottom number of 5. Conversely, fractions with different (or unlike) denominators have bottom numbers that do not equal the same value. For example, in the case of 1/2 + 3/7, you would be adding fractions with different denominators since the fractions do not share a common denominator (one has a denominator of 2 and the other has a denominator of 7). These examples are featured in Figure 02 below. Figure 02: In order to learn how to add fractions, you must be able to identify when the fractions have denominators that are the same and when they have different denominators. Again, this concept should be simple, but a quick review was required because you will need to be able to identify whether or not a fractions addition problem involves like or unlike denominators in order to solve it correctly. Now, let’s move onto a few examples. How to Add Fractions with Like Denominators How to Add Fractions with Like Denominators: Example #1 Example #1: 1/4 + 2/4 Our first example is rather simple, but it is perfect for learning how to use our easy 3-step process, which you can use to solve any problem that involves adding fractions: Step One: Identify whether the denominators are the same or different. Step Two: If they are the same, move onto Step Three. If they are different, find a common denominator. Step Three: Add the numerators and find the sum. Okay, let’s take our first attempt at using these steps to solve the first example: 1/4 + 2/4 = ? Step One: Identify whether the denominators are the same or different. Clearly, the denominators are the same since they both equal 4. Step Two: If they are the same, move onto Step Three. If they are different, find a common denominator. Since the denominators are the same, you can move onto Step Three. Step Three: Add the numerators and find the sum. To complete this first example, simply add the numerators together and express the result as one single fraction with the same denominator as follows: 1/4 + 2/4 = (1+2)/4 = 3/4 Since 3/4 can not be simplified further, you can conclude that… Final Answer: 3/4 This process is summarized in Figure 03 below. Figure 03: How to Add Fractions: The process is relatively simple when the denominators are the same. As you can see from this first example, learning how to add fractions when the denominators are the same is very simple. To add fractions with the same denominator, simply add the numerators and keep the same denominator. Let’s take a look at one more example of adding fractions when the denominators are the same before you learn how to add fractions with different denominators. How to Add Fractions with Like Denominators: Example #2 Example #2: 2/9 + 4/9 To solve this second example, let’s apply the 3-step process like we did in the previous example as follows: Step One: Identify whether the denominators are the same or different. The denominators in this example are the same since they both equal 9. Step Two: If they are the same, move onto Step Three. If they are different, find a common denominator. Again, you can skip the second step because the denominators are the same. Step Three: Add the numerators and find the sum. The final step is to add the numerators and keep the denominator the same: 2/9 + 4/9 = (2+4)/9 = 6/9 In this case, 6/9 is the correct answer, but this fraction can actually be reduced. Since both 6 and 9 are divisible by 3, 6/9 can be reduced to 2/3. Final Answer: 2/3 This process is summarized in Figure 04 below. Figure 04: How to Add Fractions: 6/9 can be reduced to 2/3 Next, let’s learn how to add fractions with different denominators. How to Add Fractions with Different Denominators How to Add Fractions with Different Denominators: Example #1 Example #1: 1/3 + 1/4 Step One: Identify whether the denominators are the same or different. In this case, the denominators are different (one is 3 and the other is 4) Step Two: If they are the same, move onto Step Three. If they are different, find a common denominator. For this example, you can not skip the second step. Before you can continue on, you will need to find a common denominator—a number that both denominators can divide into evenly. The easier way to do this is to multiply the denominator of the first fraction by the second fraction and the denominator of the second fraction by the first fraction (i.e. multiply the denominators together). 1/3 + 1/4 (4x1)/(4x3) + (3x1)/(3x4) = 4/12 + 3/12 This process is shown in Figure 05 below. Figure 05: How to Add Fractions with Different Denominators: Get a common denominator by multiplying the denominators together. (If you need some help with multiplying fractions, click here to access our free guide). Now, we have transformed the original question into a scenario involving adding two fractions that do have common denominators, which means that the hard work is over and we can solve by adding the numerators and keep the same denominator: 4/12 + 3/12 = (4+3)/12 = 7/12 Since 7/12 can not be simplified further, you can conclude that… Final Answer: 7/12 Figure 06: Once you have common denominators, you can simply add the numerators together and keep the same denominator. Now, let’s work through one final example of adding fractions with unlike denominators. How to Add Fractions with Different Denominators: Example #2 Example #1: 3/5 + 4/11 For this last example, let’s again apply the 3-step process: Step One: Identify whether the denominators are the same or different. The denominators are clearly different (one is 5 and the other is 11) Step Two: If they are the same, move onto Step Three. If they are different, find a common denominator. Just like the last example, the second step is to find a common denominator by multiplying the denominators together as follows: 3/5 + 4/11 (11x3)/(11x5) + (5x4)/(5x11) = 33/55 + 20/55 = 53/55 This process is shown in Figure 07 below. Figure 07: How to Add Fractions with Different Denominators: Get a common denominator by multiplying the denominators together. Finally, now that you have common denominators, you can solve the problem as follows: 33/55 + 20/55 = (33+20)/55 = 53/55 Since there is no value that divides evenly into both 53 and 55, you can not simplify the fraction further. Final Answer: 53/55 Figure 08: How to Add Fractions with Different Denominators: 53/55 can not be simplified further. Conclusion: How to Add Fractions To add fractions with the same denominator, simply add the numerators (top values) and keep the same denominator (bottom value). To add fractions with different denominators, you need to find a common denominator. A common denominator is a number that both denominators can divide into evenly. You can solve problems involving adding fractions for either scenario by applying the following 3-step process: Step One: Identify whether the denominators are the same or different. Step Two: If they are the same, move onto Step Three. If they are different, find a common denominator. Step Three: Add the numerators and find the sum. Keep Learning: How to Subtract Fractions in 3 Easy Steps Search Tags: how to add fractions, how to add fractions with different denominators, how to add fraction , how to add fractions with unlike denominators, how to. add fractions, how to add fractions with, how to add a fraction Comment Mashup Math About Us Contact Us Accessibility More Info Membership Purchase Orders FAQ Terms of Use Privacy Policy Copyright Info Terms of Service © 2025 Mashup Math LLC. All rights reserved.
15166	http://www.royalhydraulics.com/newrsg/support/2-uncategorised/29-fluid-power-formulas	Fluid Power Formulas Monday, 29 September 2025 Toll-Free:888.333.7692 Home Power Generation Products/Services Systems Integration Systems Products Test Stands Power Generation Industrial Presses Electronics Controls Military Coalescer Skydrol Systems Bosch Rexroth Repairs Systems Design Request Support About Us News Career Inquiry Employment application Royal Hydraulics Newsletter HomeSupportSupportFluid Power Formulas Fluid Power Formulas Basic Fluid Power Formulas / Hydraulics / Pneumatics VariableWord Formula w/ UnitsSimplified Formula Fluid Pressure - P(PSI) = Force (Pounds) / Area ( Sq. In.)P = F / A Fluid Flow Rate - QGPM= Flow (Gallons) / Unit Time (Minutes)Q = V / T Fluid Power in Horsepower - HPHorsepower = Pressure (PSIG) × Flow (GPM)/ 1714 HP = PQ / 1714 Actuator Formulas VariableWord Formula w/ UnitsSimplified Formula Cylinder Area - A( Sq. In.) = ? × Radius (inch)2 A = ? × R 2 (Sq. In.) = ? × Diameter (inch)2/ 4 A = ? × D 2/ 4 Cylinder Force - F(Pounds) = Pressure (psi) × Area (sq. in.)F = P × A Cylinder Speed - v(Feet / sec.) = (231 × Flow Rate (gpm))/ (12 × 60 × Area)v = (0.3208 × gpm) / A Cylinder Volume Capacity - VVolume = ? × Radius 2× Stroke (In.) / 231 V = ? × R 2× L / 231 (L = length of stroke) Cylinder Flow Rate - QVolume = 12 × 60 × Velocity (Ft./Sec.) × Net Area(In.)2/ 231 Q = 3.11688 × v × A Fluid Motor Torque - TTorque (in. lbs.) = Pressure (psi) × disp. (in.3/ rev.) / 6.2822 T = P × d / 6.2822 Torque = HP × 63025 / RPM T = HP × 63025 / n Torque = Flow Rate (GPM) × Pressure × 36.77 / RPM T = 36.77 × Q × P / n Fluid Motor Speed- nSpeed (RPM) = (231 × GPM) / Disp. (in.)3 n = (231 × GPM) / d Fluid Motor Horsepower - HPHP = Torque (in. lbs.) × rpm / 63025 HP = T × n / 63025 Pump Formulas VariableWord Formula w/ UnitsSimplified Formula Pump Output Flow - GPMGPM= (Speed (rpm) × disp. (cu. in.)) / 231 GPM = (n ×d) / 231 Pump Input Horsepower - HPHP = GPM × Pressure (psi) / 1714 × Efficiency HP = (Q ×P) / 1714 × E Pump Efficiency - EOverall Efficiency = Output HP / Input HP E Overall= HP Out/ HP In X 100 Overall Efficiency = Volumetric Eff. × Mechanical Eff.E Overall= Eff Vol.× Eff Mech. Pump Volumetric Efficiency - EVolumetric Efficiency = Actual Flow Rate Output (GPM) / Theoretical Flow Rate Output (GPM) × 100 Eff Vol.= Q Act./ Q Theo.X 100 Pump Mechanical Efficiency - EMechanical Efficiency = Theoretical Torque to Drive / Actual Torque to Drive × 100 Eff Mech= T Theo./ T Act.× 100 Pump Displacement - CIPRDisplacement (In.3/ rev.) = Flow Rate (GPM) × 231 / Pump RPM CIPR = GPM × 231 / RPM Pump Torque - TTorque = Horsepower × 63025 / RPM T = 63025 × HP / RPM Torque = Pressure (PSIG) × Pump Displacement (CIPR) / 2?T = P × CIPR / 6.28 Contact Info Royal Systems Group Inc 18301 Napa St. Northridge, CA 91325 Toll Free: 888.333.7692 Local: 818.717.5010 Fax: 818.885.3940 Contact us Home Contact Us Terms and Conditions Privacy Policy Site Map
15167	https://dart.deloitte.com/USDART/home/codification/broad-transactions/asc842-10/roadmap-leasing/appendix-c-differences-between-asc-842/appendix-c-differences-between-asc-842	Home Accounting FASB FASB Accounting Standards Codification Manual Codification Broad Transactions 842 Leases 10 Overall Deloitte's Roadmap: Leases Appendix C — Differences Between ASC 842 and Previous Guidance Under ASC 840 Appendix C — Differences Between ASC 842 and Previous Guidance Under ASC 840 ... Appendix C — Differences Between ASC 842 and Previous Guidance Under ASC 840 Appendix C — Differences Between ASC 842 and Previous Guidance Under ASC 840 Appendix C — Differences Between ASC 842 and Previous Guidance Under ASC 840 The table below illustrates the key differences between ASC 842 and ASC 840. \| Key Provision \| ASC 842 \| ASC 840 \| \| Substantive substitution rights \| For a substitution right to be substantive and thus preclude lease accounting, the supplier must both (1) have the practical ability to substitute the asset and (2) economically benefit from the substitution. The concept of economically benefitting from the substitution is a new concept under ASC 842. \| A lease does not exist if the supplier has the right and ability to substitute other PP&E to fulfill the arrangement. The supplier does not need to economically benefit from the substitution for lease accounting to be precluded (the substitution must be practicable and economically feasible). \| \| Right to control the use of the asset \| To control the use of the asset, the customer must have the right to (1) obtain substantially all of the economic benefits from using the asset and (2) direct the use of the asset (i.e., determine HAFWP the asset will be used throughout the period of use). \| A customer does not need to both obtain substantially all of the economic benefits and direct the use of the asset to control the use of the asset. For example, the customer can control the use of the asset if (1) it obtains substantially all of the output or other utility generated by the asset and (2) the price it pays is neither contractually fixed per unit of output nor equal to the current market price of the output. \| \| Separating land and other lease components \| A lessee should account for land and buildings as separate lease components unless the accounting effect of doing so would be insignificant (e.g., there would be no impact on lease classification or the amount recognized for the land component would be insignificant). \| When the lease meets either the transfer-of-ownership or bargain-purchase-price classification criteria, the lessee should account for the land and other assets separately. Otherwise, when the fair value of the land is 25 percent or more of the total fair value of the leased property at lease inception, the lessee should classify the land and other assets separately. \| \| Separating lease and nonlease components \| As a practical expedient,1 a lessee may elect not to separate lease components and nonlease components associated with those lease components in the contract. If this practical expedient is elected, the lessee must account for the combined components as a single lease component. \| A similar practical expedient does not exist under ASC 840. \| \| Maintenance \| Maintenance services represent a nonlease component that must be separated from the lease component(s) in the contract (if the practical expedient described above is not elected). \| Amounts paid by the lessee to the lessor for maintenance are generally considered executory costs. \| \| Classification criteria \| Lessee — There are two accounting models for leases, and the model will dictate the pattern of expense recognition associated with the lease. Therefore, a lessee must perform a lease classification assessment as of the commencement date. Under ASC 842-10-25-2, a lessee must consider the following five criteria when assessing lease classification: Transfer of ownership. Bargain purchase option. Lease term is for a major part of the estimated economic life of the leased property. Present value of the lease payments is substantially all of the fair value of the leased property. The leased asset is so specialized that it would have no alternative use to the lessor at the end of the lease term. If any one of the five criteria is met, the lease is classified as a finance lease; otherwise, the lease is classified as an operating lease. Although the lease classification criteria under ASC 842 differ from those under ASC 840, the FASB has stated that an entity may use the bright lines established under ASC 840 when evaluating the more principles-based criteria in ASC 842. Lessor — A lessor must perform a lease classification assessment as of the commencement date. The criteria governing when a lessor must classify a lease as a sales-type lease are the same as those that govern when a lessee must classify a lease as a finance lease (noted above). If none of these criteria are met, the lessor would classify the lease as a direct financing lease in accordance with ASC 842-10-25-3 if (1) the sum of the lease payments and any third-party guarantee of the residual value “equals or exceeds substantially all of the fair value of the underlying asset” and (2) “[i]t is probable that the lessor will collect the lease payments plus any amount necessary to satisfy a residual value guarantee.” Otherwise, the lease would be classified as an operating lease. In addition, ASC 842-10-25-3A requires a lessor to classify a lease with variable lease payments that do not depend on an index or rate as an operating lease at lease commencement if: (1) the lease would have been classified as a sales-type or direct financing lease in accordance with the classification criteria in ASC 842-10-25-2 and 25-3, respectively, and (2) the lessor would have recognized a selling loss on the lease commencement date. \| Lessee — There are two accounting models for leases, and the model will dictate the pattern of expense recognition associated with the lease. Therefore, a lessee must perform a lease classification assessment as of the inception date. Under ASC 840-10-25-1, a lessee must consider the following four criteria when assessing lease classification: Transfer of ownership. Bargain purchase option. Lease term is equal to 75 percent or more of the estimated economic life of the leased property. Present value of the minimum lease payments equals or exceeds 90 percent of the fair value of the leased property. If any one of the four criteria is met, the lease is classified as a capital lease; otherwise, the lease is classified as an operating lease. ASC 840 includes special considerations related to situations in which the sole asset being leased is land. A lease of land is classified as an operating lease unless the transfer-of-ownership criterion or bargain purchase option criterion is met. Lessor — A lessor must perform a lease classification assessment as of the inception date. A lessor, when assessing the classification of the lease, must consider the four lease classification criteria described above and both of the following additional criteria in ASC 840-10-25-42: Collectibility of the minimum lease payments is reasonably predictable. No uncertainties are associated with the amount of unreimbursable costs yet to be incurred by the lessor under the lease. Important uncertainties might include commitments by the lessor to guarantee performance of the leased property in a manner more extensive than the typical product warranty or to effectively protect the lessee from obsolescence of the leased property. If, as of the lease inception date, the lease meets any of the four lease classification criteria and both of the additional criteria, it will be classified as a sales-type lease, direct financing lease, leveraged lease, or operating lease as follows: Sales-type lease — If the lease gives rise to manufacturer’s or dealer’s profit (or loss) and either (1) involves real estate for which ownership is transferred to the lessee by the end of the lease term and the two additional criteria in ASC 840-10-25-2 are met or (2) does not involve real estate and any of the criteria in ASC 840-10-25-1 are met and both of the criteria in ASC 840-10-25-42 are met. Direct financing lease — If the lease (1) meets any of the criteria in ASC 840-10-25-1 and both of the criteria for sales-type lease classification, (2) does not result in manufacturer’s or dealer’s profit for the lessor, and (3) does not meet the definition of a leveraged lease. Leveraged lease — If the (1) lease meets the criteria for a direct financing lease; (2) lease involves at least three parties, including a lessee, long-term creditor, and lessor; (3) financing provided by the long-term creditor is nonrecourse to the lessor’s general credit; and (4) lessor’s net investment declines in the early years once the investment has been completed and rises during the later years of the lease. Operating lease — If the lease does not qualify as a sales-type, direct financing, or leveraged lease. Lessors should apply the guidance in ASC 840-10-25-55 through 25-59 when the sole asset being leased is land. \| \| Executory costs \| Executory costs (e.g., reimbursement for a lessor’s property taxes and insurance) are allocated to both lease and nonlease components in the contract on the same basis as the other consideration in the contract. The portion of executory costs allocated to the lease component(s) in the contract is considered part of the lease payments (to the extent that the payments are fixed). \| All executory costs are excluded from the determination of minimum lease payments for classification and measurement purposes. \| \| Residual value guarantees \| A lessee should include in the lease payments only the amount of the residual value guarantee that it is probable the lessee will owe at the end of the lease term. \| A lessee should include in the minimum lease payments the full amount of the residual value guaranteed by the lessee. \| \| Discount rate \| A lessee should use the rate implicit in the lease if it is readily determinable, regardless of whether it is higher than the lessee’s incremental borrowing rate. The rate implicit in the lease takes into account the lessor’s initial direct costs. If the rate implicit in the lease is not readily determinable, a lessee uses its incremental borrowing rate. The incremental borrowing rate is the rate that the lessee would pay to borrow an amount equal to the lease payments on a collateralized basis over a similar term. In addition, the lessee’s incremental borrowing rate is the rate of interest that a lessee would have to pay to borrow on a collateralized basis and must reflect a secured borrowing rate. Non-PBEs can elect a practical expedient to use a risk-free discount rate in lieu of an incremental borrowing rate. \| A lessee should use the rate implicit in the lease if it is readily determinable, unless that rate exceeds the lessee’s incremental borrowing rate. The rate implicit in the lease does not incorporate the lessor’s initial direct costs. The incremental borrowing rate is the rate the lessee could obtain to borrow the funds necessary to purchase the underlying asset. The lessee’s incremental borrowing rate may be unsecured if it is consistent with the financing that would have been obtained to purchase the underlying asset (and not leased). The lessee is not required to use a rate that takes collateral into account but should use a rate that is “determinable, reasonable, and consistent with the financing that would have been used in the particular circumstances,” regardless of whether the rate is secured. \| \| Inception date vs. commencement date \| A lease is classified and initially measured on the lease commencement date. \| A lease is classified and initially measured on the lease inception date. \| \| Lessee accounting \| All leases (finance and operating) other than those that qualify for the short-term recognition exception must be recognized on the lessee’s balance sheet. A lessee recognizes a liability for its lease obligation under the lease and a corresponding asset representing its right to use the underlying asset over the lease term. \| A lessee only recognizes capital leases on its balance sheet. Leases classified as operating leases are not recognized on a lessee’s balance sheet. \| \| Sales-type vs. direct financing lease \| The distinction between a sales-type lease and a direct financing lease is based on whether the lessee obtains control of the underlying asset. This assessment is not affected by the relationship of the fair value to the carrying amount of the underlying asset. If the lessee obtains control of the underlying asset, the lease is classified as a sales-type lease. If the lessee does not obtain control of the underlying asset (but the lessor relinquishes control), the lease is classified as a direct financing lease. \| The distinction between a sales-type lease and a direct financing lease is based on whether there is a difference between the fair value and carrying amount of the underlying asset. If the fair value equals the carrying amount of the underlying asset, the lease is classified as a direct financing lease. Otherwise, the lease is classified as a sales-type lease. \| \| Collectibility \| Lessors should consider collectibility in accounting for their leases, and such consideration differs depending on whether the lease is classified as a sales-type, direct financing, or operating lease. A lease can still be classified as a sales-type lease if there are collectibility concerns. For a sales-type lease to be recognized, collectibility of the lease payments must be probable (in a manner consistent with ASC 606). The collectibility guidance for direct financing and operating leases is aligned with ASC 840 (i.e., if the lease is not a sales-type lease, the lease should be classified as an operating lease if collectibility of the lease payments and any residual value guarantee is not probable at lease commencement). \| A lessor cannot recognize a capital lease unless collectibility of the minimum lease payments is reasonably predictable. \| \| Lessor’s accounting for direct financing leases \| A lessor must defer selling profit for a direct financing lease and recognize the deferred amount over the lease term. \| Because a direct financing lease can only arise if the fair value equals the carrying amount of the underlying asset, no profit or loss arises under a direct financing lease. \| \| Leases involving real estate \| There is no unique guidance on classifying and accounting for leases involving real estate. Leases involving real estate are subject to the same general classification and measurement guidance as leases involving other PP&E. \| Leases involving real estate are subject to specific guidance that is unique to real estate (e.g., the lessor will only classify a lease involving real estate as a sales-type lease if it meets the transfer-of-ownership criterion in ASC 840-10-25-1(a)). \| \| Sale-and-leaseback arrangements \| All assets are subject to the same sale-and-leaseback guidance (i.e., there is no unique guidance on sale-and-leaseback arrangements involving real estate). An entity should assess the criteria in ASC 606 to determine whether a sale has occurred. A repurchase option precludes sale accounting unless (1) the option is priced at the fair value of the asset on the date of exercise and (2) alternative assets exist that are substantially the same as the transferred asset and are readily available in the marketplace. The existence of a leaseback by itself would not indicate that a sale has not occurred unless the leaseback is classified as a finance lease. \| There is specific guidance on sale-and-leaseback arrangements involving real estate (including integral equipment). A sale-and-leaseback arrangement involving nonintegral equipment that includes a repurchase option may not result in a failed sale if the option does not economically compel the seller-lessee to repurchase the equipment. \| \| Leveraged lease accounting \| ASC 842 does not include guidance on leveraged leases. Entities are not permitted to account for any new lease arrangements as leveraged leases after the effective date of ASC 842. Leveraged leases existing as of the effective date of ASC 842 would be subject to the guidance in ASC 842-50, which is generally consistent with the legacy accounting requirements for leveraged leases and effectively grandfathers in that guidance. If a leveraged lease is modified after the effective date of ASC 842, it would be accounted for as a new lease. \| A lease is a leveraged lease if it has all of the following characteristics: It meets the criteria to be classified as a direct financing lease. It involves at least three parties, including a long-term creditor. The financing provided by the creditor is nonrecourse with respect to the lessor’s general credit. The lessor’s net investment declines during the early years and rises during the later years. \| \| Build-to-suit lease arrangements \| ASC 842 supersedes the guidance in ASC 840 on build-to-suit arrangements. Under ASC 842, the accounting for a build-to-suit arrangement depends on whether the lessee controls the underlying asset during the construction period. \| A lessee is considered the owner of an asset during the construction period if the lessee has substantially all of the construction period risks. There are certain automatic indicators of ownership, which have historically caused a number of lessees to be deemed owners of assets during the construction period. \| \| Related-party leases \| Entities should account for related-party leasing arrangements on the basis of the legally enforceable terms and conditions of the lease rather than the substance of the arrangement. Common-Control Arrangements ASU 2023-01 allows private companies, as well as not-for-profit entities that are not conduit bond obligors, to elect a practical expedient in which the written terms and conditions of the arrangement are used to determine whether a lease exists and the subsequent accounting for the lease. \| Entities should account for related-party leasing arrangements on the basis of the substance of the contract. \| \| Reassessment (identifying a lease) \| An entity should only reassess whether the contract is or contains a lease if the terms and conditions of the contract are changed. \| An entity should only reassess whether an arrangement is or contains a lease if any of the following occur: Change in contractual terms. Renewal or extension (excluding a modification). Dependency on specific PP&E. Physical change to PP&E. \| \| Reassessment (lessee measurement) \| Upon a reassessment event, a lessee should remeasure its ROU asset and lease liability on its balance sheet. A lessee should use the discount rate that applies as of the date of the reassessment event to remeasure its ROU asset and lease liability. \| A lessee should not remeasure a capital lease liability during the lease term unless the lease is modified. If the lease is modified and remeasured, the remeasurement should be based on the discount rate that was used at lease inception (i.e., the discount rate should not be updated). \| \| Lease modifications \| The lease modification guidance is more extensive under ASC 842, and the two-step model from ASC 840 is not carried forward. The changes are primarily related to aligning the modification guidance with the guidance in ASC 606. \| A modification of a lease should be accounted for as a new lease if the modification would have resulted in a different lease classification had the changed terms been in effect at lease inception. A lease modification does not include renewals or extensions of the lease if such renewals or extensions were already included in the lease term. \| \| Initial direct costs \| Initial direct costs include only those costs that are incremental to the arrangement and that would not have been incurred if the lease had not been obtained. \| In addition to costs that are incremental to the arrangement and that would not have been incurred if the lease had not been obtained, initial direct costs include costs directly related to the following activities: Evaluating the prospective lessee’s financial condition. Evaluating and recording guarantees, collateral, and other security arrangements. Negotiating lease terms. Preparing and processing lease documents. Closing the transaction. \| \| \| Operating leases are subject to the impairment guidance in ASC 360. \| Operating leases are subject to the guidance in ASC 420 on exit and disposal activities. \| \| Statement of cash flows (lessor) \| A lessor must classify cash received from leases in the operating activities section of its statement of cash flows.2 \| ASC 840 does not include guidance on the classification of cash received from leases in a lessor’s statement of cash flows. \| \| \| An entity must disclose significantly more quantitative and qualitative information under ASC 842. \| Lessees and lessors are subject to relatively limited disclosure requirements. \| Footnotes 1 In July 2018, the FASB issued ASU 2018-11, which includes a practical expedient that allows lessors, when certain conditions are met, not to separate lease and nonlease components. Under ASU 2018-11, lessors availing themselves of this practical expedient would not account for affected nonlease components separately. See Section E.3.1.4.2 for further discussion. 2 In March 2019, the FASB issued ASU 2019-01, which amended the presentation of the statement of cash flows for entities within the scope of ASC 942. Such entities are required to classify principal payments received from sales-type and direct financing leases in the investing activities section of their statement of cash flows. This requirement is associated with an illustrative example in ASC 942 that existed before the adoption of, and was not amended by, ASC 842. Confidential and Proprietary — for Use Solely by Authorized Personnel This publication provides comprehensive guidance; however, it does not address all possible fact patterns, and the guidance is subject to change. Consult a Deloitte & Touche LLP professional regarding your specific issues and questions. Your feedback will help us improve the FASB Accounting Standards Codification Manual. Please let us know what you think. Copyright © 2025 Deloitte Development LLC. All rights reserved. FASB Accounting Standards Codification: Copyright © 2005 - 2025 by Financial Accounting Foundation, Norwalk, Connecticut. Here's how we use cookies We use preference and analytical cookies to better understand your preferences so that we can bring you the best, most personalized experience possible. You can learn more by visiting our Cookie Notice.
15168	https://www.youtube.com/watch?v=Mw9P4S-uvfQ	Pythagorean Theorem Solving for Hypotenuse or a Leg MooMooMath and Science 584000 subscribers 202 likes Description 74079 views Posted: 20 Dec 2020 How to use the Pythagorean Theorem to find the hypotenuse or a leg length. Learn how to use the Pythagorean theorem. The Pythagorean theorem expresses the relation among the three sides of a right triangle. A right triangle is a triangle with one angle of 90 degrees. This theorem can be used to calculate the missing length of a right triangle. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the two sides equals the square of the hypotenuse. The hypotenuse is the side opposite the right angle. Example problems Calculate the hypotenuse of a right triangle given the side lengths. Calculate the length of the missing side. Additional Resources The Pythagorean theorem Skittles demonstration of the Pythagorean theorem 16 comments Transcript: Intro [Music] hey guys in this video we are going to be using pythagorean theorem to solve for a missing side of a right triangle so that side could be the hypotenuse or that side could be a leg Formula so recall that the pythagorean theorem formula is a squared plus b squared equals c squared where a and b are the legs and the two legs a and b are the sides of the right triangle that form the right angle so a and b intersect at 90 degrees and then c is the hypotenuse and that's the longest side of the right triangle and it's also going to be the side that is across from the directly across from the right angle so we're going to look at the steps for the steps for this process or to write the formula a squared plus b squared equals c squared substitute the numbers or the values into the formula and the correct with the correct variables so a and b are the legs and c is the hypotenuse solve for the missing side using inverse operations and use the height and remember that the hypotenuse must be the longest side so example number one we're going to Example calculate the missing length of the hypotenuse and this problem tells you to solve for hypotenuse but remember the hypotenuse is always the side that is opposite the right angle so we're going to write our formula a squared plus b squared equals c squared we're going to substitute the values into the formula so 9 and 12 are a and b and those could be swapped a and b can be swapped but c has to be the hypotenuse value so 9 squared plus 12 squared equals c squared square the 9 which gives you 81 square the 12 which is 144 equals c squared 81 plus 144 is 225 and then the final step this is your inverse operation take the square root to unsquare c squared to give you c equals 15 and that would be the length of the longest side of the right triangle so keep in mind that value has to the hypotenuse has to be the longest the largest number the longest side all right example number two calculate the missing length of the leg so this one we're solving for a leg if we locate that right angle and draw the align to the opposite side that side has to be the hypotenuse so 24 will have to be substituted in for c 8 can be subbed in for a and we can solve for b so let's write our formula a squared plus b squared equals c squared a is going to be 8 squared we don't know what b is so we're going to leave b squared there and then 24 squared let's go ahead and square the numbers we know so 8 squared is 64 plus we don't know b squared then 24 squared is 576. now we're going to use inverse operations to isolate b squared so let's subtract 64 from both sides of the equation so we are going to get that b squared is equal to 512 and then finally we take the square root of both sides of the equation and we are going to get that b is equal to 22.6 and let's just double check that that value cannot be greater than the hypotenuse the hypotenuse is 24 so we have calculated this correctly you
15169	https://brainly.com/question/21230989	[FREE] Directions: Using the digits 1 to 6, at most one time each, fill in the boxes so that top two equations - brainly.com 5 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +81k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +10,7k Ace exams faster, with practice that adapts to you Practice Worksheets +5,6k Guided help for every grade, topic or textbook Complete See more / Mathematics Textbook & Expert-Verified Textbook & Expert-Verified Directions: Using the digits 1 to 6, at most one time each, fill in the boxes so that top two equations are equal and the bottom equation has the greatest value. You can move the numbers around on the graph to help. First Attempt: column 1 column 2 2 See answers Explain with Learning Companion NEW Asked by jadelynwhite12 • 02/04/2021 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 7022536 people 7M 0.0 0 Upload your school material for a more relevant answer Given a number placement puzzle, organise the digits to make the top two equations false and the bottom one yielding the maximum value. Suitable placement could be Column 1: 6, 5, 4 and Column 2: 3, 2, 1. Explanation This problem is a number placement puzzle where you're looking to arrange digits to create equations. Since having higher numbers is better for maximising the value, place the bigger numbers in spots having more value. A possible solution could be: Column 1: 6, 5, 4 Column 2: 3, 2, 1 The top equations (6 = 3 and 5 = 2) will be false, but the bottom equation (4 = 1) yields the greatest value that you can get from these digit placements. Learn more about Number Placements here: brainly.com/question/34140875 SPJ11 Answered by AngelinaGermanotta •13.7K answers•7M people helped Thanks 0 0.0 (0 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 7022536 people 7M 0.0 0 Psychology 2e - Rose M. Spielman, William J. Jenkins, Marilyn D. Lovett Supplemental Modules (Modern Physics) Principles of Chemistry I - Michael Nelson Upload your school material for a more relevant answer To fill the boxes using digits from 1 to 6 while ensuring the top equations are false and maximizing the bottom equation, an effective arrangement could be Column 1: 6, 5, 4 and Column 2: 3, 2, 1. This ensures that the top equations do not equal each other, while allowing for a maximum bottom value. This strategy prioritizes larger numbers in the lower positions to achieve greater sums. Explanation To solve the problem of filling the boxes with the digits from 1 to 6 so that the top two equations are equal, while maximizing the value of the bottom equation, we need to think strategically about how numbers can interact in equations. Let's assume the equations are structured as follows: Column 1 can have numbers that form one or more equations with Column 2. The goal is to have meaningful equations that are false (i.e., not equal) on the top while maximizing the number in the bottom equation. Steps to Approach the Problem: Choose Largest Values for the Bottom Equation: Since we want the bottom equation to be as high as possible, we should place larger numbers there. Arrange Smaller Values on the Top: To make sure that the top equations are false, we can arrange smaller numbers mismatched between the two columns. For practical placement: Column 1: 6, 5, 4 Column 2: 3, 2, 1 Analyzing the Results: Equations Created: 6 (Column 1) does not equal 3 (Column 2) → FALSE 5 (Column 1) does not equal 2 (Column 2) → FALSE For the bottom we can take 4, which will equate to whatever we want (imagining an operation here, like addition or multiplication, to maintain its maximum value). Conclusion: Thus, placing the digits in this manner allows us to meet the requirements of having the top equations false and having a higher resultant value for the bottom equation. Examples & Evidence For instance, using 6 in Column 1 and 3 in Column 2 guarantees that 6 does not equal 3, maximizing the lowest result by having higher numbers positioned in Column 1. This pattern can be applied generally to maintain false relationships while maximizing values. Such arrangement and strategies are based on the foundational rules of equality and the maximization principle in arithmetic, where larger numbers yield higher results, commonly utilized in math puzzles. Thanks 0 0.0 (0 votes) Advertisement Community Answer This answer helped 11390 people 11K 4.3 7 -6+3=-3 -2-1=-3 -4-(-5)=1 Answered by MrNarwhalWasTaken •32 answers•11.4K people helped Thanks 7 4.3 (3 votes) 1 Advertisement ### Free Mathematics solutions and answers Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD Community Answer 4.1 17 How many positive integers between 100 and 999 inclusive are divisible by three or four? Community Answer 4.0 9 N a bike race: julie came in ahead of roger. julie finished after james. david beat james but finished after sarah. in what place did david finish? Community Answer 4.1 8 Carly, sandi, cyrus and pedro have multiple pets. carly and sandi have dogs, while the other two have cats. sandi and pedro have chickens. everyone except carly has a rabbit. who only has a cat and a rabbit? Community Answer 4.1 14 richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points. Community Answer 4 Click an Item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers. Express In simplified exponential notation. 18a^3b^2/ 2ab New questions in Mathematics 24 b sec 5×4 Choose an equation that would be used to solve 0=−x 2+10 x−8. Solve the equation to find where the trestle meets ground level. Enter your answers from the nearest tenth. The approximate antler length L (in inches) of a deer buck can be modeled by L=9 3 t+15 where t is the age in years of the buck. If a buck has an antler length of 36 inches, what is its age? Cholesterol levels (in m g/d L) were collected from a random sample of 24 patients two days after they had a heart attack. \| Cholesterol Levels (in mg/dL) \| \| 186 \| \| 224 \| \| 280 \| \| 236 \| \| 142 \| \| 226 \| \| 244 \| \| 282 \| \| 236 \| \| 220 \| \| 278 \| \| 272 \| \| 234 \| \| 276 \| \| 310 \| \| 288 \| \| 242 \| \| 280 \| \| 294 \| \| 282 \| \| 160 \| \| 288 \| \| 206 \| \| 266 \| For the data shown above, find the following. Do not round any of your answers. a) Find the 5-number summary: Evaluate 1 F 4+9 C 2 in base Sixteen. 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15170	https://www.quora.com/How-can-I-determine-parallelity-of-two-2d-vectors	Something went wrong. Wait a moment and try again. 2D Vector Basic Linear Algebra Vectors (mathematics) Geometric Mathematics 2D GEOMETRY Vectors (2-Dimensional) 5 How can I determine parallelity of two 2d vectors? · To determine if two 2D vectors are parallel, you can use the following methods: Method 1: Cross Product For two vectors a=(ax,ay) and b=(bx,by), you can compute the cross product in 2D. In 2D, the cross product can be represented as a scalar: Cross Product=ax⋅by−ay⋅bx If the result is zero, then the vectors are parallel. Method 2: Scalar Multiplication Two vectors a and b are parallel if one is a scalar multiple of the other. This means: a=k⋅b for some scalar k. In component form, this means: \frac{a_x}{b To determine if two 2D vectors are parallel, you can use the following methods: Method 1: Cross Product For two vectors a=(ax,ay) and b=(bx,by), you can compute the cross product in 2D. In 2D, the cross product can be represented as a scalar: Cross Product=ax⋅by−ay⋅bx If the result is zero, then the vectors are parallel. Method 2: Scalar Multiplication Two vectors a and b are parallel if one is a scalar multiple of the other. This means: a=k⋅b for some scalar k. In component form, this means: axbx=ayby This holds as long as bx and by are not both zero. Let’s say you have two vectors: - a=(2,4) - b=(1,2) Using Cross Product: Cross Product=2⋅2−4⋅1=4−4=0 Since the result is zero, a and b are parallel. Using Scalar Multiplication: 21=2and42=2 Both ratios are equal, confirming that a is a scalar multiple of b. You can use either method to determine the parallelity of two 2D vectors effectively. Related questions How do you prove vectors are parallel? How do I prove that two vectors are parallel or not? Explain with an example. Can two non-coincident parallel vectors define a plane? How do I find the components of a vector parallel to other two vectors? How do you know if two vectors are parallel? Isfan B. Gunawan 9y Ok, you should also view other answers and double check the internet because I’m 2 months out of highschool and vectors isn’t my brightest area. But what you researched was right. If the cross-product of the two vectors equal to zero, it means the angle between the two vectors are either 0 ot 180 degrees. But this method is used for 3D vectors (and an easier one is there) since e.g vector (a, b, c). While 2D vectors doesn’t have the third value (c), comparing the slope of the two vectors can be done to determine whether or not they’re parallel. The other method that I’m referring to is checking Ok, you should also view other answers and double check the internet because I’m 2 months out of highschool and vectors isn’t my brightest area. But what you researched was right. If the cross-product of the two vectors equal to zero, it means the angle between the two vectors are either 0 ot 180 degrees. But this method is used for 3D vectors (and an easier one is there) since e.g vector (a, b, c). While 2D vectors doesn’t have the third value (c), comparing the slope of the two vectors can be done to determine whether or not they’re parallel. The other method that I’m referring to is checking whether vector a1,a2 can be multiplied by a scalar value that would make it equal to vector b1,b2. If a1 = k b1, and a2 = k b2, you’ve proven that both vectora are parallel to each other. 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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Sc. in mathematics. · Author has 353 answers and 529K answer views · 9y The problem with the cross product method is, that it only works in three dimensions. So you can’t use it here. You can however find the determinant corresponding to the vectors to check the same thing: Determinant [ ]. If the determinant is zero, the vectors are parallel (some texts do refer to this as the 2d cross product, so your adv The problem with the cross product method is, that it only works in three dimensions. So you can’t use it here. You can however find the determinant corresponding to the vectors to check the same thing: Determinant [ ]. If the determinant is zero, the vectors are parallel (some texts do refer to this as the 2d cross product, so your advice may be OK after all). Howeve... Sanidhya Gupta Love it · Author has 136 answers and 892.9K answer views · 11y Related How do I prove that two vectors are parallel or not? Explain with an example. THE more mathematically rigorous method - there is an operation on vectors defined as a U x V where U and V are your vectors, this operation is called the cross product. lets define this is equivalent to U = (u1,u2,u3) this is equivalent to V = (v1,v2,v3) if you dont already know what these i,j,k are, then these are simply the x,y,z components of the vector respectively :) now their cross product is defined as so now if the 2 vectors are parallel you will get UxV =0 that will be a zero vector = (0,0,0) lets take two vectors (1,1,1) and (2,2,2) now if you calculate its cross product it come THE more mathematically rigorous method - there is an operation on vectors defined as a UxV where U and V are your vectors, this operation is called the cross product. lets define this is equivalent to U = (u1,u2,u3) this is equivalent to V = (v1,v2,v3) if you dont already know what these i,j,k are, then these are simply the x,y,z components of the vector respectively :) now their cross product is defined as so now if the 2 vectors are parallel you will get UxV =0 that will be a zero vector = (0,0,0) lets take two vectors (1,1,1) and (2,2,2) now if you calculate its cross product it comes out to be 0i +0j+0k so they are parallel Now the Easier way ;) let your vectors be U = (u1,u2,u3) and V = (v1,v2,v3) now if (u1/v1)=(u2/v2)=(u3/v3) then the vectors are parallel you can check it your self for the case (1,1,1) and (2,2,2) :) Related questions If two vectors are parallel, what is the condition? How can we determine a unit vector perpendicular to two vectors? Is the product of two vectors also a vector? How do I find a [2D] vector which is perpendicular to a line and points to a specific half-plane? How can you determine the dot products of parallel vectors? Oliver Jennrich Physicist turned rocket scientist, in some sense of the word. · Author has 766 answers and 1.2M answer views · 9y There is more than one way to skin a cat. As two vectors are parallel if →a=k→b, you can also try a1/b1=a2/b2, which is of course equivalent to a1/a2=b1/b2. The cross-product is anti-symmetric, i.e.→a×→b=−→b×→a, the cross-product of two parallel vectors →a and k→a gives zero. So that works as well. Promoted by The Penny Hoarder Lisa Dawson Finance Writer at The Penny Hoarder · Updated Sep 16 What's some brutally honest advice that everyone should know? Here’s the thing: I wish I had known these money secrets sooner. 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Shahriar Shishir 4y Originally Answered: How do you know if two vectors are parallel? · Find the cross products of the two vectors, if the cross product is equal to zero then the given 2 vectors are parallel otherwise not Timothy Johnson B.S. in Mathematics, California Institute of Technology (Caltech) (Graduated 2013) · Author has 1.5K answers and 7.7M answer views · Updated 11y Related How do I prove that two vectors are parallel or not? Explain with an example. The answers about using the cross product are correct, but needlessly complicated. If two vectors are parallel, then one of them will be a multiple of the other. So divide each one by its magnitude to get a unit vector. If they're parallel, the two unit vectors will be the same. Edit: Someone pointed out in the comments that two vectors are still parallel if they point in opposite directions. So the two unit vectors might have different signs. Sponsored by CDW Corporation How can AI help your teams make faster decisions? CDW’s AI solutions offer retrieval-augmented generation (RAG) to expedite info with stronger insights. Omer Hertz Updated 8y Related How do I prove that two vectors are parallel or not? Explain with an example. This was answered here before me, but here's an easy example: Let's assume we are dealing with a vector in the R3 space. Thus, any two vectors V1 and V2 will look something like this: (X1,Y1,Z1) and (X2,Y2,Z2) respectively. Now, since vectors have a direction and length but no fixed location we can displace them, if they are parallel, one on top of the other. To do this all we need to do is solve three simple equations in one of two ways: (1) If there exists a single constant, A, for which these three are all true, then V1and V2 are paralleled: X1=A∗X2;Y1=A∗Y2;Z1=A∗Z2; (2) Else, V1 a This was answered here before me, but here's an easy example: Let's assume we are dealing with a vector in the R3 space. Thus, any two vectors V1 and V2 will look something like this: (X1,Y1,Z1) and (X2,Y2,Z2) respectively. Now, since vectors have a direction and length but no fixed location we can displace them, if they are parallel, one on top of the other. To do this all we need to do is solve three simple equations in one of two ways: (1) If there exists a single constant, A, for which these three are all true, then V1and V2 are paralleled: X1=A∗X2;Y1=A∗Y2;Z1=A∗Z2; (2) Else, V1 and V2 are paralleled if the following is true: X1X2=Y1Y2=Z1Z2. Hope this helps. By the way, this question was posted with a Calculus tag - however, vectors are more accurately a part of Linear Algebra, a not-dissimilar but distinct field. _______________ Edit(s): Trying to get the math notations right :-) By the way, if anyone can comment on how I can write actual fractions (not this A / B hodge-podge), I would be grateful. Ritchie Brannan Game developer. · Author has 163 answers and 305.1K answer views · Updated 9y Related How do I prove that two vectors are parallel or not? Explain with an example. There are several answers already but they apply to limited numbers of dimensions or require matrices. Here's my attempt at a more general answer: The dot product of 2 vectors is defined for any number of dimensions and is equal to the length of the first vector multiplied by the length of the second vector multiplied by the cosine of the angle between the vectors. We can calculate the length of a vector by taking the square root of the dot product of a vector with itself. 2 vectors are parallel if the angle between them is 0 degrees or 180 degrees. At 0 degrees or 180 degrees, the cosine will be 1 There are several answers already but they apply to limited numbers of dimensions or require matrices. Here's my attempt at a more general answer: The dot product of 2 vectors is defined for any number of dimensions and is equal to the length of the first vector multiplied by the length of the second vector multiplied by the cosine of the angle between the vectors. We can calculate the length of a vector by taking the square root of the dot product of a vector with itself. 2 vectors are parallel if the angle between them is 0 degrees or 180 degrees. At 0 degrees or 180 degrees, the cosine will be 1 or -1 and the cosine squared will be 1. At all other angles, the cosine squared will be less than 1. Given the above, when two vectors are parallel: length( A )length( B )cos(0) = abs(dot( A, B )) Squaring both sides: dot( A, A )dot( B, B ) = dot( A, B ) dot( A, B ) Rearranging: dot( A, A )dot( B, B ) - dot( A, B ) dot( A, B ) = 0 So any 2 vectors are parallel if the above formula is true (equal to zero). Sponsored by CDW Corporation Want document workflows to be more productive? The new Acrobat Studio turns documents into dynamic workspaces. Adobe and CDW deliver AI for business. Anshul Gupta Worked at Mathematics · Author has 473 answers and 682.8K answer views · 11y Related How do I prove that two vectors are parallel or not? Explain with an example. By taking their cross product. If value comes to be zero then either they are parallel or overlap each other (3î + 4j + 5k) x (6î + 8j + 10k) = 0 Chris Manning B. Math. (Hons.) in Mathematics, The University of Newcastle (Australia) (Graduated 1981) · Author has 1.8K answers and 4.2M answer views · 5y Related If two vectors are parallel, what is the condition? Other answerers have mentioned dot products, cross products and even Banach spaces. I don’t know what a Banach space is (I have heard of them) and I have a degree in maths, so they are not exactly mainstream maths. Two vectors are parallel if and only if one is a multiple of the other. That is, (x1, y1) and (x2, y2) are parallel if and only if there is a number k such that (x1, y1) = k(x2, y2). For example: (1, - 2) and (- 3, 6) are parallel because (- 3, 6) = - 3(1, - 2). (4, - 2, 7) and (8, - 4, 14) are parallel because (8, - 4, 14) = 2(4, - 2, 7). (5, 3) and (4, 6) are not parallel because ther Other answerers have mentioned dot products, cross products and even Banach spaces. I don’t know what a Banach space is (I have heard of them) and I have a degree in maths, so they are not exactly mainstream maths. Two vectors are parallel if and only if one is a multiple of the other. That is, (x1, y1) and (x2, y2) are parallel if and only if there is a number k such that (x1, y1) = k(x2, y2). For example: (1, - 2) and (- 3, 6) are parallel because (- 3, 6) = - 3(1, - 2). (4, - 2, 7) and (8, - 4, 14) are parallel because (8, - 4, 14) = 2(4, - 2, 7). (5, 3) and (4, 6) are not parallel because there is no number k such that (5, 3) = k(4, 6). Proof: k(4, 6) = (4k, 6k). So if (5, 3) = k(4, 6) then 5 = 4k or k = 5/4. But also, 3 = 6k or k = 3/6 = 1/2. But we already showed that k = 5/4. This is a contradiction. So there is no possible value of k and (5, 3) and (4, 6) are not parallel. Note that it doesn’t matter which way around you do it. For example I wrote above that: (1, - 2) and (- 3, 6) are parallel because (- 3, 6) = - 3(1, - 2). You could just as well write: (1, - 2) and (- 3, 6) are parallel because (1, - 2) = - (1/3)(- 3, 6). James M. Volo Physics Professor PhD (1969–present) · Author has 28.4K answers and 54M answer views · 3y Related What is a parallel vector? Two vectors can be described in terms of direction and orietation. They can be parallel or collinear. All collinear vectors (on the same line) are also parallel, but not all parallel vectors are collinear. Parallel vectors may create rotation. The placement of parallel vectors above may represent a force–system of moments of rotation(s) about x, y, z in vector form. The parallel vectors above can be resolved into a single resultant force and shown as an equivalent force-moment system. Two vectors can be described in terms of direction and orietation. They can be parallel or collinear. All collinear vectors (on the same line) are also parallel, but not all parallel vectors are collinear. Parallel vectors may create rotation. The placement of parallel vectors above may represent a force–system of moments of rotation(s) about x, y, z in vector form. The parallel vectors above can be resolved into a single resultant force and shown as an equivalent force-moment system. Khenan Mak Engineering mathematics · Author has 2.6K answers and 1.2M answer views · 4y Related How do I tell if 2 vectors are parallel? Two vectors are parallel if one can be written as a multiple of the other. In the words, when they are pointing in the same, or opposite, direction. Related questions How do you prove vectors are parallel? How do I prove that two vectors are parallel or not? Explain with an example. Can two non-coincident parallel vectors define a plane? How do I find the components of a vector parallel to other two vectors? How do you know if two vectors are parallel? If two vectors are parallel, what is the condition? How can we determine a unit vector perpendicular to two vectors? Is the product of two vectors also a vector? How do I find a [2D] vector which is perpendicular to a line and points to a specific half-plane? How can you determine the dot products of parallel vectors? Given the components of two three dimensional vectors, how do you determine if the two are parallel to each other? How do I add two parallel vector using triangle method? How can we multiply two vectors? How do we determine whether two vectors are linearly independent or not in a vector space? Can we calculate the sum of two vectors that are not parallel? 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15171	https://link.springer.com/article/10.1007/BF00339601	Cloning and expression of the SalI restriction-modification genes of Streptomyces albus G \| Molecular Genetics and Genomics Your privacy, your choice We use essential cookies to make sure the site can function. We also use optional cookies for advertising, personalisation of content, usage analysis, and social media. By accepting optional cookies, you consent to the processing of your personal data - including transfers to third parties. Some third parties are outside of the European Economic Area, with varying standards of data protection. See our privacy policy for more information on the use of your personal data. Manage preferences for further information and to change your choices. Accept all cookies Skip to main content Log in Menu Find a journalPublish with usTrack your research Search Cart Search Search by keyword or author Search Navigation Find a journal Publish with us Track your research Home Molecular and General Genetics MGG Article Cloning and expression of the Sal I restriction-modification genes of Streptomyces albus G Published: August 1988 Volume 213,pages 346–353, (1988) Cite this article Molecular and General Genetics MGGAims and scopeSubmit manuscript M. R. Rodicio1,2nAff3& Keith F. Chater1,2 107 Accesses 14 Citations 9 Altmetric Explore all metrics Summary The Streptomyces albus G genes (salR and salM) for the class II restriction enzyme Sal I (Sal GI) and its cognate modification enzyme were cloned in Streptomyces lividans 66. Selection was initially for the salR gene. From a library of S. albus G DNA in the high copy number plasmid pIJ486 several clones of S. lividans were obtained that were resistant to phage ϕC31 unmodified at the many Sal I sites in its DNA, but were sensitive to modified phages last propagated on a restriction-deficient, modification-proficient mutant of S. albus G. Sal I activity was detected in cell-free extracts of the clones, though only at levels comparable with that in S. albus G. Five different recombinant plasmids were isolated, with inserts of 5.6, 5.7, 8.9, 10 and 18.9 kb that contained a common region of 4.5 kb. These plasmids could not be digested by Sal I, although the vector has four recognition sites for this enzyme, indicating that the salM gene was also cloned and expressed. Subcloning experiments in S. lividans indicated the approximate location of salR and salM, and in Escherichia coli led to detectable expression of salM but not of salR. A variety of previously isolated S. albus G mutants affected in aspects of Sal I-specific restriction and modification were complemented by the cloned DNA; they included a mutant temperature-sensitive for growth apparently because of a mutation in salM. Southern blotting showed that DNA homologous to the cloned sal genes was present in Xanthomonas and Rhodococcus strains, but not detectably in Herpetosiphon strains, all of which produce Sal I isoschizomers. This is a preview of subscription content, log in via an institution to check access. Access this article Log in via an institution Subscribe and save Springer+ from $39.99 /Month Starting from 10 chapters or articles per month Access and download chapters and articles from more than 300k books and 2,500 journals Cancel anytime View plans Buy Now Buy article PDF USD 39.95 Price excludes VAT (USA) Tax calculation will be finalised during checkout. Instant access to the full article PDF. Institutional subscriptions Similar content being viewed by others SlnR is a positive pathway-specific regulator for salinomycin biosynthesis in Streptomyces albus Article 09 November 2016 Emergence of a clinical Salmonella enterica serovar 1,4,, 12: i:-isolate, ST3606, in China with susceptibility decrease to ceftazidime-avibactam carrying a novel bla CTX-M-261 variant and a bla NDM-5 Article Open access 22 February 2024 Development of the first gene expression system for Salinicoccus strains with potential application in bioremediation of hypersaline wastewaters Article 09 August 2017 Explore related subjects Discover the latest articles, books and news in related subjects, suggested using machine learning. Archaeal Genetics Archaeal Genes Bacterial Genetics Bacterial Genes DNA restriction-modification enzymes Bacillus subtilis References Arrand JR, Myers PA, Roberts RJ (1978) A new restriction endonuclease from Streptomyces albus G. J Mol Biol 118:127–135 Google Scholar Bibb MJ, Cohen SN (1982) Gene expression in Streptomyces: construction and application of promoter-probe plasmid vectors in Streptomyces lividans. Mol Gen Genet 187:265–277 Google Scholar Bibb MJ, Chater KF, Hopwood DA (1983) Developments in Streptomyces cloning. In: Inouye M (ed) Experimental manipulation of gene expression. Academic Press, New York, London, pp 53–82 Google Scholar Bolivar F, Rodriguez RI, Greene PJ, Betlach MCV, Heyneker HL, Boyer HW, Crosa JH, Falkow S (1977) Construction and characterisation of new cloning vehicles, II A multipurpose cloning system. Gene 2:95–113 Google Scholar Chater KF (1982) Streptomyces in the ascendant. Nature 299:10–11 Google Scholar Chater KF, Carter AT (1978) Restriction of a bacteriophage in Streptomyces albus P (CMI 52766) by endonuclease Sal P1. J Gen Microbiol 109:181–185 Google Scholar Chater KF, Carter AT (1979) A new wide host-range temperate bacteriophage, R4, of Streptomyces and its interaction with some restriction modification systems. J Gen Microbiol 115:431–442 Google Scholar Chater KF, Wilde LC (1976) Restriction of a bacteriophage of Streptomyces albus G involving endonuclease Sal I. J Bacteriol 128:644–650 Google Scholar Chater KF, Wilde LC (1980) Streptomyces albus G mutants defective in the Sal GI restriction-modification system. J Gen Microbiol 116:323–334 Google Scholar Chater KF, Bruton CJ, Suarez JE (1981a) Restriction mapping of the DNA of the Streptomyces temperate phage ϕC31 and its derivatives. Gene 14:183–194 Google Scholar Chater KF, Bruton CJ, Springer W, Suarez JE (1981b) Dispensable sequences and packaging constraints of DNA from the Streptomyces temperate phage ϕC31. Gene 15:249–256 Google Scholar Chater KF, Hopwood DA, Kieser T, Thompson CJ (1982) Gene cloning in Streptomyces. Curr Top Microbiol Immunol 96:69–95 Google Scholar Daniels MJ, Barber CE, Turner PC, Sawczyc MK, Byrde RJW, Fielding AH (1984) Cloning of genes involved in pathogenicity of Xanthomonas campestris pv campestris using the broad host range cosmid pLAFR1. EMBO J 3:3323–3328 Google Scholar Hintermann G, Crameri R, Kieser T, Hütter R (1981) Restriction analysis of the Streptomyces genome by agarose gel electrophoresis. Arch Mikrobiol 130:218–222 Google Scholar Holmes DS, Quigley M (1981) A rapid boiling method for the preparation of bacterial plasmids. Anal Biochem 114:193–197 Google Scholar Hopwood DA, Bibb MJ, Chater KF, Kieser T, Bruton CJ, Kieser HM, Lydiate DJ, Smith CP, Ward JM, Schrempf H (1985) Genetic manipulation of Streptomyces—A laboratory manual. The John Innes Foundation, Norwich Google Scholar Jaurin B, Cohen SN (1984) Streptomyces lividans RNA polymerase recognizes and uses Escherichia coli transcription signals. Gene 28:83–91 Google Scholar Katz E, Thompson CJ, Hopwood DA (1983) Cloning and expression of the tyrosinase gene from Streptomyces antibioticus in Streptomyces lividans. J Gen Microbiol 129:2703–2714 Google Scholar Kieser T (1984) Factors affecting the isolation of CCC DNA from Streptomyces lividans and Escherichia coli. Plasmid 12:19–36 Google Scholar Kieser T, Moss MT, Dale JW, Hopwood DA (1986) Cloning and expression of Mycobacterium bovis BCG DNA in Streptomyces lividans. J Bacteriol 168:72–80 Google Scholar Kröger M, Hobom G, Schütte H, Mayer H (1984) Eight new restriction endonucleases from Herpetosiphon giganteus—divergent evolution in a family of enzymes. Nucleic Acids Res 12:3127–3141 Google Scholar Lomovskaya ND, Mkrtumian NM, Gostimskaya NL, Danilenko VN (1972) Characterisation of temperate actinophage ϕC31 isolated from Streptomyces coelicolor A3(2). J Virol 9:258–262 Google Scholar Lomovskaya ND, Chater KF, Mkrtumian NM (1980) Genetics and molecular biology of Streptomyces bacteriophages. Microbiol Rev 44:206–229 Google Scholar Maniatis T, Fritsch EF, Sambrook J (1982) Molecular cloning: a laboratory manual. Cold Spring Harbor Laboratory Press, New York Google Scholar Maxwell A, Halford SE (1982a) The Sal GI restriction endonuclease. Purification and properties. Biochem J 203:77–84 Google Scholar Maxwell A, Halford SE (1982b) The Sal GI restriction endonuclease. Mechanism of DNA cleavage. Biochem J 203:85–92 Google Scholar Maxwell A, Halford SE (1982c) The Sal GI restriction endonuclease. Enzyme specificity. Biochem J 203:93–98 Google Scholar Messing J (1979) Recombinant DNA Tech Bull 2:43–48 Google Scholar Murray NE, Brammar WJ, Murray K (1977) Lambdoid phages that simplify the recovery of in vitro recombinants. Mol Gen Genet 150:53–61 Google Scholar Norrander J, Kempe T, Messing J (1983) Construction of improved M13 vectors using oligonucleotide-directed mutagenesis. Gene 26:101–106 Google Scholar Reichenbach H, Dworkin M (1981) The order Cytophagales (with addenda on the genera Herpetosiphon, Saprospira and Flexithrix). In: Starr MP, Stolp H, Trüper HG, Balows A, Schlegel HG (eds) The prokaryotes. Vol I. Springer-Verlag, Berlin Heidelberg New York, pp 356–379 Google Scholar Roberts RJ (1987) Restriction enzymes and their isoschizomers. Nucleic Acids Res 15:r189-r217 Google Scholar Suarez JE, Chater KF (1980) DNA cloning in Streptomyces: a bifunctional replicon comprising pBR322 inserted into a Streptomyces phage. Nature 286:527–529 Google Scholar Ward JM, Janssen GR, Kieser T, Bibb MJ, Buttner MJ, Bibb MJ (1986) Construction and characterisation of a series of multi-copy promoter-probe plasmid vectors for Streptomyces using the aminoglycoside phosphotransferase gene from Tn5 as indicator. Mol Gen Genet 203:468–478 Google Scholar Welsch M (1936) Propriétés bactérioliques du Streptothrix et sporulation. C R Soc Biol 123:1013–1016 Google Scholar Wespheling J, Ranes M, Losick R (1985) RNA polymerase heterogeneity in Streptomyces coelicolor. Nature 313:22–27 Google Scholar Download references Author information Author notes 1. M. R. Rodicio Present address: Departamento de Microbiologia, Universidad de Oviedo, 33006, Oviedo, Spain Authors and Affiliations John Innes Institute, Colney Lane, NR4 7UH, Norwich, England M. R. Rodicio&Keith F. Chater AFRC Institute of Plant Science Research, Colney Lane, NR4 7UH, Norwich, England M. R. Rodicio&Keith F. Chater Authors 1. M. R. RodicioView author publications Search author on:PubMedGoogle Scholar 2. Keith F. ChaterView author publications Search author on:PubMedGoogle Scholar Additional information Communicated by W. Arber Rights and permissions Reprints and permissions About this article Cite this article Rodicio, M.R., Chater, K.F. Cloning and expression of the Sal I restriction-modification genes of Streptomyces albus G . Mol Gen Genet213, 346–353 (1988). Download citation Received: 24 March 1988 Issue Date: August 1988 DOI: Share this article Anyone you share the following link with will be able to read this content: Get shareable link Sorry, a shareable link is not currently available for this article. Copy shareable link to clipboard Provided by the Springer Nature SharedIt content-sharing initiative Key words Bacteriophage ϕC31 Streptomyces genetics Xanthomonas amarinthicola Rhodococcus rhodochrous Herpetosiphon giganteus Access this article Log in via an institution Subscribe and save Springer+ from $39.99 /Month Starting from 10 chapters or articles per month Access and download chapters and articles from more than 300k books and 2,500 journals Cancel anytime View plans Buy Now Buy article PDF USD 39.95 Price excludes VAT (USA) Tax calculation will be finalised during checkout. Instant access to the full article PDF. Institutional subscriptions Sections References Summary References Author information Additional information Rights and permissions About this article Advertisement Arrand JR, Myers PA, Roberts RJ (1978) A new restriction endonuclease from Streptomyces albus G. J Mol Biol 118:127–135 Google Scholar Bibb MJ, Cohen SN (1982) Gene expression in Streptomyces: construction and application of promoter-probe plasmid vectors in Streptomyces lividans. Mol Gen Genet 187:265–277 Google Scholar Bibb MJ, Chater KF, Hopwood DA (1983) Developments in Streptomyces cloning. In: Inouye M (ed) Experimental manipulation of gene expression. Academic Press, New York, London, pp 53–82 Google Scholar Bolivar F, Rodriguez RI, Greene PJ, Betlach MCV, Heyneker HL, Boyer HW, Crosa JH, Falkow S (1977) Construction and characterisation of new cloning vehicles, II A multipurpose cloning system. Gene 2:95–113 Google Scholar Chater KF (1982) Streptomyces in the ascendant. Nature 299:10–11 Google Scholar Chater KF, Carter AT (1978) Restriction of a bacteriophage in Streptomyces albus P (CMI 52766) by endonuclease Sal P1. J Gen Microbiol 109:181–185 Google Scholar Chater KF, Carter AT (1979) A new wide host-range temperate bacteriophage, R4, of Streptomyces and its interaction with some restriction modification systems. J Gen Microbiol 115:431–442 Google Scholar Chater KF, Wilde LC (1976) Restriction of a bacteriophage of Streptomyces albus G involving endonuclease Sal I. J Bacteriol 128:644–650 Google Scholar Chater KF, Wilde LC (1980) Streptomyces albus G mutants defective in the Sal GI restriction-modification system. J Gen Microbiol 116:323–334 Google Scholar Chater KF, Bruton CJ, Suarez JE (1981a) Restriction mapping of the DNA of the Streptomyces temperate phage ϕC31 and its derivatives. Gene 14:183–194 Google Scholar Chater KF, Bruton CJ, Springer W, Suarez JE (1981b) Dispensable sequences and packaging constraints of DNA from the Streptomyces temperate phage ϕC31. Gene 15:249–256 Google Scholar Chater KF, Hopwood DA, Kieser T, Thompson CJ (1982) Gene cloning in Streptomyces. Curr Top Microbiol Immunol 96:69–95 Google Scholar Daniels MJ, Barber CE, Turner PC, Sawczyc MK, Byrde RJW, Fielding AH (1984) Cloning of genes involved in pathogenicity of Xanthomonas campestris pv campestris using the broad host range cosmid pLAFR1. EMBO J 3:3323–3328 Google Scholar Hintermann G, Crameri R, Kieser T, Hütter R (1981) Restriction analysis of the Streptomyces genome by agarose gel electrophoresis. Arch Mikrobiol 130:218–222 Google Scholar Holmes DS, Quigley M (1981) A rapid boiling method for the preparation of bacterial plasmids. Anal Biochem 114:193–197 Google Scholar Hopwood DA, Bibb MJ, Chater KF, Kieser T, Bruton CJ, Kieser HM, Lydiate DJ, Smith CP, Ward JM, Schrempf H (1985) Genetic manipulation of Streptomyces—A laboratory manual. The John Innes Foundation, Norwich Google Scholar Jaurin B, Cohen SN (1984) Streptomyces lividans RNA polymerase recognizes and uses Escherichia coli transcription signals. Gene 28:83–91 Google Scholar Katz E, Thompson CJ, Hopwood DA (1983) Cloning and expression of the tyrosinase gene from Streptomyces antibioticus in Streptomyces lividans. J Gen Microbiol 129:2703–2714 Google Scholar Kieser T (1984) Factors affecting the isolation of CCC DNA from Streptomyces lividans and Escherichia coli. Plasmid 12:19–36 Google Scholar Kieser T, Moss MT, Dale JW, Hopwood DA (1986) Cloning and expression of Mycobacterium bovis BCG DNA in Streptomyces lividans. J Bacteriol 168:72–80 Google Scholar Kröger M, Hobom G, Schütte H, Mayer H (1984) Eight new restriction endonucleases from Herpetosiphon giganteus—divergent evolution in a family of enzymes. Nucleic Acids Res 12:3127–3141 Google Scholar Lomovskaya ND, Mkrtumian NM, Gostimskaya NL, Danilenko VN (1972) Characterisation of temperate actinophage ϕC31 isolated from Streptomyces coelicolor A3(2). J Virol 9:258–262 Google Scholar Lomovskaya ND, Chater KF, Mkrtumian NM (1980) Genetics and molecular biology of Streptomyces bacteriophages. Microbiol Rev 44:206–229 Google Scholar Maniatis T, Fritsch EF, Sambrook J (1982) Molecular cloning: a laboratory manual. Cold Spring Harbor Laboratory Press, New York Google Scholar Maxwell A, Halford SE (1982a) The Sal GI restriction endonuclease. Purification and properties. Biochem J 203:77–84 Google Scholar Maxwell A, Halford SE (1982b) The Sal GI restriction endonuclease. Mechanism of DNA cleavage. Biochem J 203:85–92 Google Scholar Maxwell A, Halford SE (1982c) The Sal GI restriction endonuclease. Enzyme specificity. Biochem J 203:93–98 Google Scholar Messing J (1979) Recombinant DNA Tech Bull 2:43–48 Google Scholar Murray NE, Brammar WJ, Murray K (1977) Lambdoid phages that simplify the recovery of in vitro recombinants. Mol Gen Genet 150:53–61 Google Scholar Norrander J, Kempe T, Messing J (1983) Construction of improved M13 vectors using oligonucleotide-directed mutagenesis. Gene 26:101–106 Google Scholar Reichenbach H, Dworkin M (1981) The order Cytophagales (with addenda on the genera Herpetosiphon, Saprospira and Flexithrix). In: Starr MP, Stolp H, Trüper HG, Balows A, Schlegel HG (eds) The prokaryotes. Vol I. Springer-Verlag, Berlin Heidelberg New York, pp 356–379 Google Scholar Roberts RJ (1987) Restriction enzymes and their isoschizomers. Nucleic Acids Res 15:r189-r217 Google Scholar Suarez JE, Chater KF (1980) DNA cloning in Streptomyces: a bifunctional replicon comprising pBR322 inserted into a Streptomyces phage. Nature 286:527–529 Google Scholar Ward JM, Janssen GR, Kieser T, Bibb MJ, Buttner MJ, Bibb MJ (1986) Construction and characterisation of a series of multi-copy promoter-probe plasmid vectors for Streptomyces using the aminoglycoside phosphotransferase gene from Tn5 as indicator. Mol Gen Genet 203:468–478 Google Scholar Welsch M (1936) Propriétés bactérioliques du Streptothrix et sporulation. C R Soc Biol 123:1013–1016 Google Scholar Wespheling J, Ranes M, Losick R (1985) RNA polymerase heterogeneity in Streptomyces coelicolor. Nature 313:22–27 Google Scholar Discover content Journals A-Z Books A-Z Publish with us Journal finder Publish your research Language editing Open access publishing Products and services Our products Librarians Societies Partners and advertisers Our brands Springer Nature Portfolio BMC Palgrave Macmillan Apress Discover Your privacy choices/Manage cookies Your US state privacy rights Accessibility statement Terms and conditions Privacy policy Help and support Legal notice Cancel contracts here 34.96.49.56 Not affiliated © 2025 Springer Nature
15172	https://www.ifo.de/DocDL/cesifo1_wp1035.pdf	TAX POLICY AND LABOR MARKET PERFORMANCE A. LANS BOVENBERG CESIFO WORKING PAPER NO. 1035 CATEGORY 1: PUBLIC FINANCE SEPTEMBER 2003 PRESENTED AT VENICE SUMMER INSTITUTE, WORKSHOP ON TAX POLICY AND LABOUR MARKET PERFORMANCE, JULY 2003 An electronic version of the paper may be downloaded • from the SSRN website: www.SSRN.com • from the CESifo website: www.CESifo.de CESifo Working Paper No. 1035 TAX POLICY AND LABOR MARKET PERFORMANCE Abstract In exploring the impact of tax policy on labor-market performance, the paper first investigates how tax reform impacts labor supply and equilibrium unemployment in representative agent models. The impact of tax policy on labor market performance depends importantly on various other labor-market institutions, such as minimum wage laws, wage bargaining, and unemployment benefits. In non-competitive labor markets, employment declines if a higher tax burden makes the outside option (i.e. unemployment) relatively more attractive. Marginal tax rates typically differ substantially across individuals. To explore the impact of specific tax policies, therefore, the paper relies on an applied general equilibrium model to investigate the consequences of tax reform with heterogeneous households. The model simulations reveal several trade-offs between various objectives, such as cutting unemployment, stimulating the participation of secondary workers into the labor force, raising the quality and quantity of labor supply, and establishing an equitable income distribution. The paper also analyses how efficiency considerations affect the optimal progressiveness of labor income taxes. Finally, the optimal progression of the labor income tax is investigated in the presence of search unemployment, heterogeneous households and distributional concerns. JEL Code: H2, H5, I2, J2. A. Lans Bovenberg Tilburg University Algemene Economie B-511 P. O. Box 90153 5000 LE, Tilburg The Netherlands A.L.Bovenberg@uvt.nl The author thanks Jan Boone, Peter Sørensen,and participants of the CESifo workshop on Tax Policy and Labour Market Performance in Venice, July 21-23, 2003 for helpful comments on an earlier draft. 2 1 Introduction This paper explores the link between tax policy and labor-market performance. As far as labor-performance is concerned, we focus on labor supply, employment and the diﬀerence between these two variables: unemployment. As regards tax policy, we consider a number of elements: ﬁrst, the level of taxation as measured by average tax burdens for major groups of workers; second, the composition of the tax burden over payroll taxes, personal labor income taxes, capital income taxes, and consumption taxes; third, the progressiveness of the tax system (as measured by the speed with which average tax rates rise with income levels) and — related to this — the magnitude of marginal tax rates for workers with middle- and high incomes; and fourth, marginal tax rates faced by low-income earners and non-participating or unemployed individuals as a result of means-tested safety-net provisions and retirement beneﬁts. The analysis is mainly theoretical. Nevertheless, at several stages we survey evidence on the empirical importance of various theoretical mechanisms. The paper is structured as follows. After section 2 provides information about the tax and beneﬁt systems in various OECD countries, the paper turns to the labor-market eﬀects of taxation in representative agents models. Section 3 explores how tax reform impacts employment through the channels of labor supply and equilibrium unemployment. Section 4 investigates how eﬃciency considerations aﬀect the optimal progressiveness of labor income taxes. Sections 5 and 6 account for distributional con-siderations by allowing for heterogeneous workers. In particular, Section 5 employs an applied general equilibrium model with heterogeneous agents to investigate the conse-quences of tax reform for not only the labor market but also the income distribution. Section 6 constructs a framework for exploring the optimal progression of the labor income tax in the presence of search unemployment, heterogeneous households and distributional concerns. Section 7, ﬁnally, summarizes the main policy conclusions. 2 Taxes and labor-market performance Labor-market performance can be assessed in several ways. This paper focuses on unemployment and labor supply, which can be measured as the participation rate (i.e. the labor force as percentage of the working age population (15-65)) and average number of hours worked by employees (see Table 1). Together, unemployment and labor supply yield employment. The ﬁfth column of Table 1 gives the average hours worked per member of the working-age population as a percentage of a full-time work-week (of 40 hours). This can be considered as the best available aggregate measure of labor-market performance. It in fact measures the share of the potential labor-market endowment that is actually used. Table 1 ranks countries in decreasing order of this utilization rate of labor resources. On this measure, the United States, Canada and Japan perform better than Europe and continental Europe in particular. Within Eu-rope, the largest continental European countries (Germany, France and Italy) do worse than most other European countries. Also tax policy can be assessed in diﬀerent ways. Table 2, which ranks countries according to their aggregate labor-market performance, contains average tax rates and marginal tax rates at diﬀerent income levels for a single-person household.1 The tax wedges used here include personal income taxes, employers’ and employees’ social 1The data are for an average production worker who is 40 years of age. For more details, see OECD (2002b). Table 1. Indicators of labor market performance, 2001 Labour force Unemployment Employment/population Average hours worked Employment indicator (d) Employment/population ratio Long-term unemployment participation rate (%) (c) rate (%) (c) ratio (%) (c) per person in employment for older workers (55-64) share (e) UK 74,9 4,8 71,3 1711 58,7 52,2 27,7 Sweden 79,3 5,1 75,3 1603 58,0 67,0 22,3 Finland 74,6 9,2 67,7 1694 55,1 45,9 26,2 Denmark 79,2 4,2 75,9 1482 54,1 56,6 22,2 Ireland 67,5 3,7 65,0 1674 52,3 46,6 55,3 (b) Spain 65,8 10,5 58,8 1816 51,3 39,2 44,0 Greece 62,1 10,4 55,6 1921 51,3 38,0 52,8 Netherlands 75,7 2,1 74,1 1346 48,0 39,3 43,5 (b) Germany 71,6 8,0 65,9 1467 46,5 36,8 51,5 (a) France 68,0 8,8 62,0 1532 45,7 36,5 37,6 Belgium 63,6 6,2 59,7 1528 43,9 25,2 51,7 Italy 60,7 9,6 54,9 1606 42,4 18,6 63,4 Portugal 71,8 4,3 68,7 n.a. n.a. 50,3 38,1 Austria 70,7 4,0 67,8 n.a. n.a. 27,4 23,5 Luxembourg 64,2 1,9 63,0 n.a. n.a. 24,8 27,6 US 76,8 4,8 73,1 1821 64,0 58,4 6,1 Canada 76,5 7,3 70,9 1801 (a) 61,4 48,3 9,5 Japan 72,6 5,2 68,8 1821 (a) 60,2 62,0 26,6 OECD Europe 66,8 8,6 61,1 n.a. 37,9 40,4 Total OECD 69,8 6,4 65,3 n.a. 48,4 27,5 Source: OECD (2002a) (a) Data refer to 2000 (b) Data refer to 1999 (c) Persons aged 15-64 years (d) (column 3 x column 4)/2080 (e) The share of long-term (12 months and over) in employment in total unemployment n.a. Not available Table 2. Effective tax (a) rates on labor Income, 1999 Total average tax rate (ta) Total marginal tax rate (tm) for a single-person household % of gross labor costs (b) % of gross labor costs (c) 0,67APW (d) APW (d) 1,67APW (d) 0,67APW (d) APW (d) 1,67APW (d) UK 37,2 41,4 43,7 49,4 49,4 41,9 Sweden 59,9 61,3 64,9 64,6 62,8 71 Finland 53,9 57,7 63,3 63,6 67,2 70,7 Denmark 54,9 57,4 63 62,4 62,4 71,9 Ireland 37,1 45,8 53,8 47,2 66 63,4 Spain 42,7 46,9 50,1 52,3 53,8 56,4 Greece 45,2 46,4 50 48 53,4 61,7 Netherlands 50 53,4 53,3 60,6 65 58,1 Germany 55 59,2 62,5 66,1 68,9 68,9 France 52,1 58,4 59,8 78,7 62,5 60,9 Belgium 59,3 64,2 68,4 71,9 72,8 75,7 Italy 54,5 57 59,8 59,8 63,6 63,6 Portugal 43,1 45,7 50,1 50,6 51,3 57,9 Austria 52,7 56,2 60,1 60,9 64,5 68,6 Luxembourg 43,9 48 54,6 53,3 60 67,1 US 35,2 37 42,2 40,4 40,4 51,5 Source: Cnossen (2001) (a) Taxes include direct taxes, i.e. personal income taxes, employers' and employees' social security contributions, and payroll taxes (if levied), plus indirect taxes, i.e. VAT and excises. (b) ta = (a+c)/(1+c), where a is the total direct average tax rate and c is the average effective indirect tax rate. (c) tm = (m+c)/(1+c), where m is the total direct marginal tax rate. (d) APW stands for the income level of an average productive worker. For a definition, see OECD (2002b) 3 security contributions,2 payroll taxes, and indirect consumption taxes (such as VAT and excises). The marginal tax wedge drives a wedge between marginal labor costs (which a competitive, proﬁt-maximizing employer equates to the marginal productivity of labor (i.e. the social beneﬁt of labor)) and after-tax disposable income from work (which a utility-maximizing household sets equal to the monetary value of the marginal disutility of labor (i.e. the reservation wage or the social costs of labor)). Tax rates are quite high. Marginal tax rates for the average production worker exceed 60 % in most countries on the European continent. Although marginal rates of personal income tax generally rise with income, overall marginal tax rates do not rise substantially with income. This is because social security contributions are typically due only on incomes below a ceiling. The United Kingdom and the United States combine high employment rates with relatively low marginal and average tax burdens. Within continental Europe, however, labor-market performance and tax rates do not show a clear correlation. Also other taxes may harm the reward to labor, even though they are not assessed on labor income. To illustrate, by reducing labor productivity, source-based taxes on capital may be shifted unto labor in a small open economy with internationally mobile capital. These implicit taxes on labor are not included in Table 2. The same holds true for labor-market regulations that give rise to implicit taxes on employment. Minimum wage policies, for example, in eﬀect levy implicit taxes on employers hiring low-skilled workers, with the revenues being transferred to these workers.3 Other implicit taxes on employers are employment regulations that constrain the ability of employers to reduce the labor force in response to weak business conditions. By reducing labor demand, the implicit taxes associated with these regulations harm employment. For low income levels, the average and marginal tax burdens as contained in Table 2 become a less reliable indicator of the incentives to supply labor because social insurance beneﬁts and means-tested welfare beneﬁts (including other safety-net beneﬁts, such as housing allowances) imply signiﬁcant implicit tax burdens on work. Indeed, many of these beneﬁts are withdrawn when a worker ﬁnds work or works more hours. To provide some information about the magnitude of these implicit tax rates, Table 3 contains the replacement rates (in after-tax terms) for both short-term and long-run unemployed. The desire to protect households with young children from poverty implies that lone-parent families and two-parent families feature the highest replacement rates. These replacement rates are closely related to the eﬀective marginal tax rate on ﬁnding a full-time job. In particular, to arrive at the overall implicit rate on work ¯ t, one should perform the following calculation: ¯ t = t+(1−t)r, where t is the (average) tax wedge (as given in the ﬁrst three columns of Table 2, but then for the relevant household types) and r is the net replacement rate (as provided in Table 3). 2The marginal tax rates assume that social security contributions are not linked to insurance beneﬁts on an individual level. 3Neary and Roberts (1980) show how rigid wages and prices can be modelled as implicit tax rates. Table 3. Net replacement Rates for three family types at two earnings levels, 1999 After tax and including unemployment benefits, family and housing benefits in the first month of benefit receipt APW (a) 66,7% of APW (a) Single Couple, 2 children Lone parent, 2 children Single Couple, 2 children Lone parent, 2 children UK 46 49 49 66 54 55 Sweden 71 78 85 82 90 93 Finland 65 83 87 79 88 92 Denmark 63 73 78 89 95 96 Ireland 31 57 52 42 67 59 Spain 74 73 76 76 76 77 Greece 47 44 47 48 46 50 Netherlands 82 89 81 88 85 80 Germany 60 70 71 67 75 76 France 71 72 72 78 82 83 Belgium 64 64 65 85 79 81 Italy 42 53 50 39 49 47 Portugal 79 79 80 88 87 87 Luxembourg 82 87 87 82 88 88 Austria 60 76 73 61 82 78 US 58 57 58 59 49 49 Canada 62 91 91 62 97 97 Japan 67 64 70 82 77 82 After tax and including unemployment benefits, family and housing benefits in the sixtiest month of benefit receipt APW (a) 66,7% of APW (a) Single Couple, 2 children Lone parent, 2 children Single Couple, 2 children Lone parent, 2 children UK 46 80 71 66 88 81 Sweden 54 85 59 79 110 70 Finland 53 89 62 73 100 69 Denmark 60 80 79 85 102 97 Ireland 31 56 56 41 66 64 Spain 23 39 37 32 57 51 Greece 8 10 11 8 11 12 Netherlands 60 71 61 74 85 76 Germany 54 65 63 63 71 71 France 30 42 43 43 59 60 Belgium 45 68 69 60 84 86 Italy 0 18 14 0 21 17 Portugal 49 63 64 70 87 87 Luxembourg 50 75 59 70 93 82 Austria 55 72 69 58 78 74 US 7 46 38 10 59 48 Canada 24 62 60 35 81 80 Japan 33 68 61 49 87 84 Source: OECD (2002b) (a) APW stands for the income level of an average productive worker. For a definition, see OECD (2002b) Table 4. Average effective tax rates on transitions for two-earner couples, 1999 (a) From unemployed breadwinner and non-employed partner to From full-time employed breadwinner and non-employed partner to Part-time (40%) employed Full-time employed breadwinner/ Full-time employed breadwinner/ Full-time employed breadwinner/ breadwinner/non-employed partner non-employed partner part-time (40%) employed partner full-time employed partner UK 18 56 15 26 Sweden 87 84 29 34 Finland 86 87 21 34 Denmark 86 81 51 51 Ireland 101 61 21 34 Spain 166 76 18 19 Greece 103 43 16 18 Netherlands 84 92 50 46 Germany 52 75 51 53 France 80 72 32 37 Belgium 107 74 46 51 Italy 80 60 33 41 Portugal 174 81 15 20 Austria 142 79 21 30 Luxembourg 17 87 13 26 US 99 63 19 25 Canada 73 92 30 34 Japan 151 68 14 14 Source: OECD (2002b) (a) If employed, the earnings are as a percentage of the full-time employed salary of an average production worker. 4 Table 4 contains the implicit tax rates for several transitions.4 In particular, the ﬁrst two columns consider the transition of an unemployed average production worker, with a non-employed spouse and two children, to part-time employment (40 %) and full-time employment, respectively. The third and fourth columns present the eﬀective tax rates for a secondary earner previously out of the labor force who starts working part-time or full-time, while the principal earner within the same household continues to work full-time. These data reveal that eﬀective tax rates on principal earners typically substantially exceed those on secondary earners. The main reason is the beneﬁt system. In particular, means-tested beneﬁts and unemployment beneﬁts are withdrawn if the principal earner ﬁnds work. Secondary earners do not have access to welfare beneﬁts if the primary worker (i.e. the breadwinner) is employed. Moreover, if they do not have an employment history, these workers are also ineligible for beneﬁts from unemployment insurance. Table 4 reveals that eﬀective tax rates diﬀer substantially across various households, even within the same country. The ﬁrst two columns of Table 4 suggest that marginal eﬀective tax rates are relatively high for primary workers at the bottom of the labor market, especially since many OECD countries have cut top marginal tax rates over the last two decennia. Another important reason why marginal tax rates are highest at the bottom of the labor market is that unemployment insurance beneﬁts for high-income earners are typically only limited in duration. For high-income earners, therefore, the short-run replacement rates on which the data in Table 4 are based may overstate disincentives to seek work. For low-income earners, in contrast, safety-net provisions, which are typically unlimited in duration, imply high replacement rates for longer periods of time. A comparison between the short-run and long-run replacement rates in Table 3 does indeed reveal that beneﬁts tend to drop less over time for low-income earners, thus producing higher long-run replacement rates than for those earning higher incomes. This implies substantial disincentives for low-income earners. Indeed, the duration rather than the magnitude of unemployment beneﬁts may be the main determinant of disincentives to work and maintain human capital. Older workers often face very high marginal tax rates on continuing to work because early retirement beneﬁts are withdrawn if workers continue to work instead of retire. In any case, pension beneﬁts are typically not increased in an actuarially fair manner if older workers delay retirement. Gruber and Wise (1999) show that marginal tax rates for older workers may sometimes exceed 100 %. Their analysis reveals that high marginal tax rates on older workers are strongly correlated with labor-force participation of older workers, which is in fact quite low in most European countries (see the next-to-last column in Table 1). In addition to workers facing these high explicit tax rates, employers of older workers may be subject to implicit tax rates as a result of downward rigid wages. The government could oﬀset these implicit taxes by explicit job subsidies for employers who employ older workers. These job subsidies, however, need to be ﬁnanced through distortionary taxation. A more direct way to protect employment of older workers is to make wages of older workers more ﬂexible. In this way, wages can be more in line with individual productivity. To achieve this, age-related pay schemes have to be reconsidered. For example, occupational pension systems that link pension beneﬁts to ﬁnal pay discourage gradual retirement through occupational downgrading with lower rates of pay. Ljungqvist and Sargent (1998) show how generous unemployment and disability beneﬁts that are based on previous earnings prevent the labor market from easily adjusting to adverse shocks. In particular, in the face of generous insurance beneﬁts that exceed their labor productivity, older skilled 4In contrast to the ﬁgures in Table 2, these ﬁgures abstract from indirect taxes on consumption. 5 workers who suﬀer a substantial capital loss on their human capital (e.g. as a result of being laid oﬀ) are discouraged from searching for new jobs and from reducing their reservation wage in line with their reduced productivity. In this way, social insurance sets in motion a vicious circle of high unemployment and skill loss. This explains the high incidence of long-term unemployment and disability among European workers (see the last column of Table 1). As the work force ages, these moral hazard problems associated with social insurance beneﬁts based on previous earnings become more serious. Indeed, social insurance beneﬁts based on ﬁnal pay discourage workers from maintaining their human capital, since workers can rely on generous social beneﬁts when their human capital becomes obsolete. Private insurance policies supplementing public disability and unemployment insurances worsen these moral hazard problems (see Pauly (1974)).5 Welfare, unemployment and early retirement beneﬁts are typically conditional beneﬁts. In particular, unemployment beneﬁts are paid only if one has left one’s job involuntarily and if one is actively looking for work. Furthermore, many countries are enforcing obligations on welfare recipients, sometimes in the form of welfare-to-work programs or workfare programs. Early retirement beneﬁts may be similar to disability beneﬁts in that they are conditional on failing health. Countries may diﬀer substan-tially in the eligibility criteria for categorical beneﬁts like unemployment and disability beneﬁts and in how strictly they enforce these criteria.6 In interpreting the replace-ment rates and implicit tax rates in Tables 3 and 4, one needs to be aware of these considerations. Indeed, the duration of the beneﬁts and the obligations associated with social beneﬁts (workfare, training, work tests) are key aspects of the design of unemployment insurance.7 3 Tax reform and employment This section uses representative agent models to explore the impacts of tax reform on employment. It ﬁrst considers the channel of labor supply before it turns to the channel of equilibrium unemployment. The analysis in this section is positive rather than normative. Normative aspects of labor taxation in representative agent models are explored in section 4. 3.1 Labor supply A representative household derives utility from consumption of goods (C) and leisure (V ). The utility function U(C, V ) is concave and homothetic. Total time available to each household is normalized to one, which can be used to enjoy leisure V or to work Ls = 1 −V . The only source of income is labor income. The household budget constraint is thus given by P cC = (1 −T a)WLs, where P c stands for the consumer price and W denotes the market wage. T a ≡T(WLs)/WLs represents the average personal income tax rate on labor, where the income tax paid by the household T(WLs) is a function of the market value of labor supply. Households determine labor supply from the condition that the marginal rate of substitution between leisure and 5The data in Tables 3 and 4 only account for publicly provided unemployment beneﬁts. Supplemen-tary, private beneﬁts, which may be provided by the previous employer, sometimes raise replacement rates further. 6To illustrate, many countries do not strictly enforce on older unemployed persons the obligation to look for work in order to be eligible for unemployment or welfare beneﬁts. 7For a recent overview of the literature on the optimal design of these important elements for unemployment insurance, see Frederiksson and Holmlund (2003) . 6 consumption should equal the marginal consumer wage, i.e. Uv/Uc = (1 −T m)W/P c, where subscripts stand for partial derivatives. T m ≡dT(WLs)/d(WLs) denotes the marginal tax rate on labor income. Both the marginal and the average tax rates depend on the market value of wage income WLs.8 A measure of the progressivity of the income tax is the elasticity of after-tax labor income with respect to pre-tax labor income, i.e. S ≡d log(WLs−T(WLs))/d log(WLs) = (1 −T m)/(1 −T a). This coeﬃcient is also known as the coeﬃcient of residual income progression (Musgrave and Musgrave, 1976). In a proportional tax system, the aver-age and the marginal tax rates coincide so that S = 1. In a progressive tax system, in contrast, the average tax rate T a ≡T(WLs)/WLs rises with pre-tax labor income WLs, so that the marginal tax rate exceeds the average tax rate (i.e. T m > T a ) and thus S < 1. We use lower-case variables to denote loglinear deviations from an initial equilibrium (e.g., c ≡dC/C), except for the tax rates where we deﬁne ti ≡dT i/(1 − T i), i = a, m. The logarithmic change in the degree of progressivity is thus given by s = ta −tm. The household budget constraint in relative changes is given by pc + c = S(w + ls) −ta. Together with the loglinearised optimality condition, i.e. c−v = σ(w −tm −pc) where σ ≡−d log(C/V )d log(Uc/Uv) represents the elasticity of substitution between leisure and consumption goods in utility, we obtain the relative change in labor supply:9 ls = ϵu(w −pc) −ϵctm −ϵita = ϵu(w −ta −pc) + ϵcs. (1) Here, ϵi ≡−V < 0, ϵc ≡V σ > 0, and ϵu ≡V (σ −1) = ϵc + ϵi stand for the income, compensated wage, and uncompensated wage elasticities of labor supply, respectively. Ceteris paribus the average tax rate T a and the market wage W, a higher marginal tax rate tm > 0 reduces the opportunity cost of leisure at the margin. Hence, households substitute leisure for consumption and thus reduce labor supply. The com-pensated elasticity of labor supply ϵc reﬂects the strength of this substitution eﬀect on account of a more progressive tax system (i.e. s < 0 with ta = 0). For a given marginal tax rate T m, a higher average tax ta > 0 makes workers poorer and thus increases the incentive to work. The magnitude of this income eﬀect is reﬂected in the income elasticity of labor supply ϵi. The average and marginal tax rates thus exert opposite eﬀects on labor supply: whereas higher marginal tax rates harm labor supply through the substitution eﬀect, higher average tax rates raise it through the income eﬀect. If both the marginal and average tax rates are increased in tandem such that the progression of the tax system is unaﬀected (i.e. tm = ta so that s = 0), the uncompensated wage elasticity of labor supply (i.e. ϵu = ϵc+ϵi ) captures the combined labor-supply impact of the substitution eﬀect of a higher marginal tax rate and the income eﬀect of a higher average tax rate. The negative substitution eﬀect dominates the positive income eﬀect on labor supply if the elasticity of substitution between leisure and consumption goods exceeds unity (i.e. σ > 1) so that the uncompensated wage elasticity of labor supply is positive. The previous analysis has assumed that the market wage W is constant. To explore the impact on the market wage, we model the demand side of the labor 8These tax rates may also depend on the personal characteristics of the individual. Moreover, since the tax authorities observe individual labor incomes, the tax schedule may be non linear in individual labor income WLs. 9Here we have assumed that the initial coeﬃcient of residual income progression is unity. If this assumption is not met, labor supply is given by [(S −1)V + 1]ls = ϵc(w −tm −pc) + ϵi(Sw −ta −pc). 7 market. A representative ﬁrm maximizes proﬁts, taking wages as given. It may have some market power on the commodity market in that the (absolute value of) price elasticity of demand for its output ε remains ﬁnite. Proﬁts are given by Π ≡ Py(AF(Ld))AF(Ld) −(1 + T l)WLd, where T l denotes the payroll tax rate and Ld represents labor demand. AF(Ld), F ′ > 0, F” < 0 stands for a production function with diminishing returns to labor. These diminishing returns are due to a second pro-duction factor (e.g. capital), which is taken as ﬁxed in the short run. Hence, proﬁts originate in not only market power on commodity markets but also this second produc-tion factor. Exogenous technology shocks are captured by changes in the productivity parameter A. Firms hire labor until the marginal revenue from the last worker equals the producer wage, i.e. Py(1 −1 ε)AF ′(Ld) = (1 + T l)W. Using tl ≡dT l/(1 + T l), loglinearizing the marginal productivity condition for ﬁrms, and taking the producer price Py as numeraire, we obtain the relative change in the demand for labor:10 ld = −ϵd(w + tl −a), (2) where ϵd stands for the wage elasticity of labor demand.11 Expression (2) reveals that an adverse productivity shock (i.e. a < 0) acts like a payroll tax. Indeed, we investigate not only changes in explicit taxes on labor income but also exogenous changes in labor productivity. This enables us to explore the impact of implicit labor taxes that reduce the productivity of labor. To illustrate, in a small open economy, source-based taxes on capital act like implicit taxes on labor by reducing the productivity of labor if capital is perfectly mobile internationally. In the same fashion, if world prices of energy are ﬁxed, a tax on the intermediate use of energy into production exerts similar adverse eﬀects on labor productivity.12 In a competitive labor market, the market wage ensures that aggregate labor supply equals labor demand. Ignoring open economy considerations, we deﬁne the consumer price as P c ≡1 + T c, where T c denotes the consumer tax rate. Using tc ≡dT c/(1 + T c) = pc and imposing equilibrium on the labor market (i.e. NLs = Ld where N denotes the ﬁxed number of households), we can solve for employment, wage costs per unit of output, and the consumer wage: l = −ϵd/ ϵu + ϵd ϵctm + ϵita + ϵu tl −a + tc = ϵd/ ϵu + ϵd [ϵcs + ϵu (a −t)] (3) 10We assume here that ﬁrms face a constant price elasticity of demand for their output ε. An increase in market power, reﬂected in a decrease in ε, amounts to an implicit tax on labor. Indeed, a relative change in ε would enter (2) in the same way as tl (but with the opposite sign). If other production factors besides labor are ﬁxed, the labor demand elasticity is given by ϵd ≡1/ 1−α σf + α ε , where σf is the substitution elasticity between labor and the other produc-tion factor(s), and α ≡F ′(L)L/F(L). With a Cobb-Douglas production function (i.e. σf = 1) and a constant price elasticity ε, the labor demand elasticity is constant, i.e. ϵd ≡1/ 1 −α(1 −1 ε ) . In that case, a smaller share of ﬁxed factors (i.e. a higher value for α) raises ϵd. 11Two important aspects of the labor-demand elasticity are the time horizon and the aggregation level to which the elasticity applies. As regards the time horizon, other production factors may respond to changes in wage costs, especially in the longer run. The long-run wage elasticity of labor demand is therefore likely to exceed the corresponding short-run elasticity. As regards the aggregation level, the labor-demand elasticity on a macroeconomic level is likely to be smaller than on a sectoral or microeconomic level. 12Another reason for investigating the impact of changes in productivity is that modern economies experience steady growth in labor productivity while the rate of unemployment remains more or less stationary. In the face of this empirical observation, most models impose conditions that ensure that changes in labour productivity do not impact unemployment. Sub-section 3.2 develops models in which, in contrast to productivity, the tax burden does aﬀect the structural rate of unemployment. 8 w + tl −a = ϵu(t −a) −ϵcs)/(ϵu + ϵd) (4) w −ta −pc = ϵd(a −t) −ϵcs /(ϵu + ϵd) (5) where s = ta −tm, t ≡ta + tl + tc, and the labor-demand elasticity ϵd applies to the macroeconomic level. 3.1.1 tax progression and overall tax burden We consider the impact of three exogenous policy shocks, namely (i) a higher marginal tax rate ceteris paribus the average tax wedge (i.e. tm = −s > 0; t = 0); (ii) a higher average tax wedge ceteris paribus the marginal tax rate (i.e. t = ta = s > 0; tm = 0); and (iii) a higher average tax wedge ceteris paribus the coeﬃcient of residual income progression (i.e. t = tm > 0; s = 0). The average tax wedge between the producer wage and the consumer wage consists of the sum of the employees’ tax rate, employers’ tax rate and consumer tax rate.13 The ﬁrst two shocks are driven by labor-supply eﬀects (compare the impact of tm and ta in (1) and (3)). As described above, the marginal average and average tax rates shift the labor supply curve down and up, respectively. How the impact is distributed over employment and wage responses depends on the labor-demand elasticity. If this latter elasticity is large (in absolute value), employment moves substantially while wages do not change much. In any case, a higher marginal tax rate associated with a more progressive tax system harms employment. Under a linear tax system, therefore, cutting the marginal tax rate, while at the same time lowering tax allowances or the tax credit in order to keep the average tax rate on labor income unaﬀected, enhances labor market participation. The incentive mechanism operates entirely through the substitution eﬀect in labor supply. Raising the average tax rate on labor for a given marginal tax rate (e.g., by reducing tax allowances) depresses wage costs and boosts employment as ϵi < 0. This shock is transmitted entirely through the income eﬀect in labor supply. We now turn to the impact of a higher average tax wedge (i.e. t > 0) while leaving the tax structure in terms of the degree of progressivity unaﬀected (i.e. s = 0). If the uncompensated wage elasticity of labor supply is positive, a larger tax burden on labor income (t > 0) is partially shifted onto ﬁrms by raising the producer wage. Workers are particularly successful in doing so if labor supply is rather elastic and labor demand relatively inelastic. A higher average tax wedge between the producer and consumer wage lowers employment and raises the producer wage if the substitution eﬀect dominates the income eﬀect in labor supply (i.e. σ > 1 so that ϵc > −ϵi), i.e. if the uncompensated wage elasticity of labor supply is positive (i.e. ϵu > 0).14 If the uncompensated labor-supply curve bends backwards, however, employment rises as the producer wage declines. In this case, workers bear more than 100 percent of the tax burden. Unemployment beneﬁts do not have a natural place in this equilibrium model of the labor market. However, if one is willing to interpret unemployment as leisure, one can model unemployment beneﬁts as a subsidy to leisure (see Pissarides (1998) and van der Ploeg (2003)). Unemployment beneﬁts hurt labor supply through both 13In analyzing these policy changes, we do not explicitly consider the government budget constraint. The analysis implicitly assumes that changes in government revenues are transmitted into correspond-ing changes in government spending that is separable from other arguments in the utility function in households. Hence, the changes in government spending produced by public revenue eﬀects do not aﬀect private decisions. 14Empirical evidence suggests that this elasticity is indeed positive, being quite small for men (see also section 5). 9 substitution and income eﬀects. Indeed, unemployment beneﬁts raise the eﬀective marginal tax on work while reducing the overall average tax rate.15 3.1.2 composition of tax burden All components of the average tax wedge exert the same impact on employment. The reason is that ﬂexible wages ensure that ﬁrms can partially shift a higher payroll tax (i.e. tl > 0) onto workers through lower wages, while higher wages allow workers to shift higher income taxes and consumption taxes (i.e. ta, tc > 0) onto employers. This equivalence of various taxes depends on ﬂexible wages. If market wages W were rigid, in contrast to income and consumption taxes, payroll taxes would hurt employment. In the presence of a ﬁxed binding statutory minimum wage, therefore, replacing payroll taxes by income taxes boosts employment. Such a tax reform in eﬀect undoes some of the implicit labor tax that workers impose on employers as a result of the minimum wage. Why the same employment eﬀect cannot be achieved by simply lowering the implicit tax by reducing the minimum wage directly is unclear. Indeed, if the statutory minimum wage is raised to protect the purchasing power of workers after raising the income tax to replace the payroll tax, the tax reform does not succeed in raising employment. An adverse productivity shock (a < 0) yields exactly the same eﬀects as a rise in the payroll tax rate, namely a drop in after-tax wages and a fall in (boost to) employment if the substitution eﬀect dominates (is outweighed by) the income eﬀect in labor supply.16 Hence, an adverse supply shock amounts to an implicit labor tax. To keep employment constant in the face of a steady increase in productivity, one needs to impose a unitary elasticity of substitution between leisure and consumption of goods (King, Plosser and Rebelo, 1998). In that case, not only productivity shocks but also changes in the average tax rate leave employment unaﬀected.17 3.1.3 human capital Labor taxes impact not only the quantity of labor (i.e. hours worked) but also the quality of labor (i.e. eﬀort and human capital).18 With exogenous labor supply, pro-portional labor taxes do not aﬀect human capital accumulation if all costs of training are deductible against the proportional tax rate. Intuitively, just as cash-ﬂow taxes leave capital investment unaﬀected,19 proportional labor taxes aﬀect the costs and beneﬁts of investments in human capital in the same way. The neutrality of pro-portional labor taxes no longer holds if hours worked is endogenous. At lower hours worked, human capital accumulation becomes less attractive because human capital 15The latter eﬀect on the average tax rate drops out if one imposes government budget balance with exogenous public spending on other purposes besides unemployment beneﬁts. In that case, higher tax rates required to ﬁnance the additional beneﬁts may further raise eﬀective marginal tax rates. 16If other production factors besides labor enter the production function, labor productivity may de-cline as a result of higher taxes on either these factors or (depending on the degree of complementarity between labor and these factors) labor itself. 17Alternatively, one could assume that A increases productivity not only of labor in the formal sector but also the productivity of leisure time (see, e.g. Heckman (1976)). In that case, one does not have to impose σ = 1 to reconcile productivity growth with a constant employment level. Hence, in contrast to productivity changes, proportional taxes may aﬀect employment. 18Human capital is another channel through which tax policy may aﬀect long-run productivity growth. In fact, in endogenous growth models in which human capital drives growth (see e.g. Lucas (1988)), labor taxes may exert permanent eﬀects on growth. 19In the presence of uncertain returns, proportional taxes may boost investments in human capital by risk-averse agents. Indeed, in the presence of such a tax, the government in eﬀect shares in the risk of the investment. For the role of taxes as an insurance device, see Eaton and Rosen (1980). 10 is utilized less intensively. This so-called utilization eﬀect makes schooling and labor supply complementary activities.20 This indirect complementarity is further strengthened in learning-by-doing mod-els (see Heckman, Lochner, and Cossa (2002)). In these models, learning and working are directly complementary, whereas in the traditional learning-or-doing model work and schooling compete for a worker’s time. Nevertheless, if the labor market ap-propriately prices the beneﬁts from learning-by-doing in wages21, the implications of learning-by-doing turn out to be equivalent to those of learning or doing. In any case, with endogenous leisure, both models predict that permanent tax policies that stimu-late labor supply also boost human capital accumulation. At the same time, policies that encourage schooling increase long-run labor supply. Labor supply and human capital thus exert positive feedback eﬀects on each other. Apart from the utilization and learning-by-doing eﬀects, labor taxes may harm human capital if not all costs of schooling are tax deductible (see, e.g. Trostel (1993)). Similarly, if marginal taxes rise with income, beneﬁts of schooling may be taxed at higher rates than the rates against which costs are tax deductible, thereby discourag-ing human capital accumulation (see Bovenberg and van Ewijk (1997)). Residence-based taxes on capital income, in contrast, stimulate schooling because they encourage agents to substitute human capital for ﬁnancial capital. These taxes can therefore help to oﬀset the human-capital distortions due to non-deductible training costs or rising marginal tax rates (see Nielsen and Sørensen (1998)). Alternatively, training subsidies or compulsory schooling may be employed to alleviate the adverse eﬀect of progressive labor taxes on human capital accumulation (see Bovenberg and Jacobs (2002)). 3.2 Equilibrium unemployment Turning to the analysis of labor taxes in imperfect labor markets, we enter the realm of second-best economics. Distortionary labor taxes may either alleviate or exacerbate non-tax distortions in the labor market that give rise to involuntary unemployment. Whereas the previous sub-section considered only a representative agent and did not address distributional issues at all, this sub-section investigates the separate impacts of taxes on employed and unemployed agents. Workers, however, are still homogeneous.22 3.2.1 right-to-manage model To illustrate the impact of taxes in imperfect labor markets, we formulate a right-to-manage model of the labor market.23 Many symmetric decentralized unions exert market power in a labor-market segment but are too small to internalize the eﬀects of higher wages on prices, proﬁts and the government budget constraint. Unions and employers bargain about wages, after which ﬁrms set employment. 20Jacobs (2002) demonstrates that positive feedback eﬀects between human capital and labor supply raise the long-run wage elasticity of eﬀective labor supply above the corresponding standard elastic-ities that assume exogenous levels of human capital. If the government optimally employs schooling subsidies to undo the eﬀect of taxation on schooling, eﬀective elasticities again correspond to the standard elasticities. The utililization eﬀects depends on human capital being more productive in work time than leisure time. Heckman (1976), in contrast, assumes that human capital is equally productive in leisure and work. 21Heckman, Lochner, and Cossa (2002), however, seriously doubt whether ﬁrms can diﬀerentiate wages on the basis expected future human capital beneﬁts of current work. Indeed, they cite empirical evidence for the learning-by-doing model (without suﬃcient wage discrimination). 22For a more complete distributional analysis, see section 5, which introduces heterogeneous workers. 23The eﬀects of labor taxes on wages and unemployment are very similar in eﬃciency wage models and search models, although the normative implications for welfare may be diﬀerent. See Bovenberg and van der Ploeg (1994), Sørensen (1997), and Pissarides (1998). 11 Union preferences are characterized by the following objective function24 Lv(W a) + (N −L) WU r ; v′(W a) > 0, v”(W a) ≤0, (6) where N denotes the number of trade union members of which L are employed. W a ≡W −T(W) represents after-tax wages earned in the industry, where W and T(W) represent the market wage and the personal income tax function respectively. A concave felicity function v(W a) implies risk-averse workers. In several cases, we will assume that felicity is isoelastic, i.e. v(W a) = (W a)1−ρ/(1 −ρ), where ρ stands for the (constant) coeﬃcient of relative risk aversion. In wage bargaining, the union takes the outside option (i.e. expected utility outside the industry Ur) as given and accounts for labor-demand behavior by ﬁrms which is modelled along the lines of sub-section 3.1. The perceived wage elasticity of labor demand ϵd depends on the price elasticity of demand for the bargaining unit as a whole (i.e. the industry). This price elasticity is likely to be smaller than the price elasticity facing an individual ﬁrm. Nash bargaining maximizes [L(v(W a)−U r)]βΠ1−β with respect to W, where proﬁts Π are given by Py(AF(L))AF(L)−(1+T l)WL. This yields v(W a) −Ur v(W a) = Sm, (7) where m ≡ v′(Wa)Wa v(Wa) [ϵd+ 1−β β (1+T l)WL Π ]. The union sets utility in work as a mark up on outside utility U r.25 A more progressive tax system, which implies a lower coeﬃcient of residual income progression S, moderates wages. Intuitively, a high marginal tax rate implies that higher wages accrue mainly to the government rather than union members. Hence, from the union’s point of view, the pay oﬀfrom higher wage costs (and the associated loss in employment) in terms of higher net incomes for working members is only low so that the union moderates wages. By aﬀecting wage setting behavior, a more progressive tax system for workers thus combats inequality between employed union members enjoying utility v(W a) and union members who only obtain outside utility U r. Apart from S, however, tax policy cannot aﬀect this inequity as measured by the ratio v(W a)/U r. In particular, unions undo the eﬀect on v(W a)/U r of a higher tax on union members employed in the sector by raising the before-tax reward to work, W, so that v(W a)/Ur does not decline. In this way, union behavior limits the scope for redistribution between workers and the unemployed.26 Various non-tax factors aﬀect the mark-up m. To explore these factors, we assume that other production factors besides labor are ﬁxed. In that case, the mark-up can be written as v′(W a)W a v(W a) / 1 ( 1−α σf + α ε ) + 1−β β α(1−1 ε ) [1−α(1−1 ε )] . Wages are thus moderated if bargaining ﬁrms as a unit do not yield much market power on the commodity market (i.e. 1 ε is small), labor is a good substitute for the ﬁxed factor (i.e. σf is large), proﬁts account only for a small share of value added (i.e. α is large), and unions do not exert 24The objective function can be interpreted as an expected utility function of union members. (6) assumes that hours worked in a full-time job are exogenously ﬁxed. For a model in which unions also set hours worked in a full-time job, see Sørensen (1997). Alternatively, employed individual workers can determine working time after wages have been set. For this formulation of endogenous individual labor supply within a bargaining framework, see Kilponen and Sinko (2003) for a monopoly union model, and Holmlund (2000) for individual bargaining with home production. 25An eﬃciency wage model in which eﬀort by workers, e, is given by e = (W a −W r)ζ and ﬁrms set wages to maximize proﬁts yields a similar expression for the wage mark-up, except that the mark-up m is replaced by the exponent ζ in the eﬀort function. 26The same holds true in an eﬃciency wage model; see Stiglitz (1999). 12 much bargaining power (i.e. β is small).27 The ﬁrst two conditions ensure that the (quasi-)rents parties bargain over are only small. In order to solve for wages in general equilibrium in which outside utility Ur is endogenous, we use the following expression for outside utility U r = g(u)[v(B) + δ] + (1 −g(u))v( ¯ W a), (8) where B and ¯ W a denote the unemployment beneﬁt and after-tax wages of workers em-ployed in other sectors, respectively, and δ ≥0 is the utility of leisure if unemployed.28 (1 −g(u)), measures the probability of ﬁnding a job outside the current sector. This probability is decreasing with the aggregate unemployment rate (i.e. dg/du > 0). In a symmetric equilibrium, all unions set the same wages, so that ¯ W a = W a. Using this equilibrium condition in (8) to eliminate ¯ W a and using the result in (7) to eliminate Ur, we arrive at the following expression for equilibrium unemployment u : g(u) = Sm [1 −(v(B) + δ)/v(W a)). (9) This expression can be interpreted as the wage-setting curve. Together with the labor-demand function (2), the curve determines labor-market equilibrium. Thus, compared to the competitive equilibrium analyzed in sub-section 3.1, the wage-setting curve rather than the labor-supply curve aﬀects actual wages. The labor-supply curve implicit in the reservation wage is given by v(B) + δ. It is thus horizontal in (L, w) space, reﬂecting an inﬁnite elasticity of labor supply associated with constant utility of leisure δ. 3.2.2 net replacement rate ﬁxed We ﬁrst look at the special case in which the eﬀective net (i.e. after-tax) replacement rate R ≡B/W a is ﬁxed, utility is isoelastic, utility of leisure is absent (i.e. δ = 0), and S and the mark-up m are ﬁxed. In these circumstances, the wage curve determines equilibrium unemployment g(u) = Sm [1 −R1−ρ]. The wage curve is vertical in (L, W) space. Intuitively, an increase in the wage rate does not make work more attractive because the ﬁxed replacement implies that such a wage increase is accompanied by an equivalent increase in the unemployment beneﬁt B. tax progression The tax system aﬀects employment through the coeﬃcient of resid-ual income progression S. The employment impacts of a higher marginal tax rate (ce-teris paribus the average tax wedge, i.e. tm = −s > 0) and a higher average tax wedge (ceteris paribus the marginal tax rate, i.e. t = ta = s > 0) have the opposite sign as the employment impacts established in sub-section 3.1. In particular, whereas sub-section 3.1 showed that a higher marginal tax rate hurts employment by harming 27In case of a Cobb-Douglas production function, a ﬁxed price elasticity ε, an isoelastic utility function v(C) = C1−ρ, and a ﬁxed S, this mark-up is constant and given by S(1−ρ)[1−α(1−1 ε )] 1+ 1−β β α(1−1 ε ) . Hence, more risk aversion (i.e. a higher value for ρ) moderates wages. 28With an isoelastic utility function v(C) = C1−ρ, one can interpret the coeﬃcient of risk aversion ρ also as the reciprocal of the substitution elasticity between consumption and leisure. A lower substitution elasticity makes unemployment less attractive and thus moderates wages. In principle, δ can be negative if the unemployed are stigmatized. 13 labor supply, we now ﬁnd that it actually boosts employment by moderating wages.29 Indeed, in our second-best setting, a high marginal tax rate alleviates the distortions implied by the market power of unions. overall tax burden Given a ﬁxed eﬀective replacement rate R = B/W a, a higher average tax wedge (ceteris paribus the coeﬃcient of residual income progression, i.e. t = tm > 0; s = 0) leaves equilibrium unemployment unaﬀected. Raising the tax burden while maintaining the structure of taxation (as measured by the coeﬃcient of residual income progression) thus does not impact labor-market transactions. Workers completely accommodate the higher tax burden in terms of lower after-tax wages so that wage costs (and hence labor demand) remain constant. This is known in the literature as the complete absence of real wage resistance. The intuition behind this lack of real wage resistance is that the unemployed are in eﬀect subject to the higher tax burden. With the outside option thus eﬀectively being taxed, the bargaining position of the union weakens so that wages are moderated. Indeed, the key eﬀective tax rate in this model is the eﬀective (after-tax) replacement rate. As long as the tax system does not aﬀect this key variable, it leaves unemployment unaﬀected. With a vertical wage-setting curve, also payroll taxes or capital taxes harming labor productivity (thereby shifting the labor-demand curve) do not aﬀect employment but are transmitted fully as changes in market wages. This result of lack of real wage resistance has been quite popular for both the-oretical and empirical reasons. Regarding the theoretical reasons, one would like to clearly separate the unemployment impact of the tax burden and that of the social insurance system (and the replacement rate). A higher tax burden aﬀects equilibrium unemployment only through the channel of the eﬀective (after-tax) replacement rate. Another reason for the popularity of this result is that most labor-market models im-pose conditions that ensure that productivity growth does not impact unemployment. Fixing the replacement rate ensures that the models replicate this stylized fact. As regards empirical reasons, several cross-country studies could not establish signiﬁcant empirical correlation between average tax rates and unemployment (see e.g. Layard, Nickell and Jackman (1991)). Indeed, as long as one measures the after-tax replace-ment rate R = B/W a and the coeﬃcient of residual income progression S correctly, one would not expect to ﬁnd an additional separate eﬀect of the tax burden. composition of the tax burden Changes in the tax structure, i.e. replacing payroll taxes by consumption taxes, do not aﬀect equilibrium unemployment. A upward shift in the labor-demand curve as a consequence of lower payroll taxes results in higher wages. This protects the purchasing power of workers and beneﬁt recipients after the increase in consumption taxes. 3.2.3 gross replacement rate ﬁxed Daveri and Tabellini (2000) have challenged the result that a higher tax burden does not aﬀect unemployment, which has been supported by empirical studies that could not ﬁnd any signiﬁcant correlation between cross-section variations in unemployment and tax rates. They argue that labor-market institutions diﬀer signiﬁcantly across countries and that ﬁxed eﬀects thus dominate cross-sectional variations in unemployment rates. Accordingly, they rely on time-series instead of cross-section evidence to establish the link between the labor tax burden and unemployment. They ﬁnd that time variation 29The wage moderating eﬀects of high marginal tax rates have been established empirically by Lockwood and Manning (1993), Tyrvainen (1995), and Graaﬂand and Huizinga (1999). 14 in labor taxes tends to be strongly correlated with unemployment changes in highly unionized countries of continental Europe. The correlation is substantially less strong, however, in the Scandinavian countries with centralized trade unions. Hence, the unemployment impact of labor taxes depends importantly on the non-tax institutions of a country. To establish real wage resistance theoretically, Daveri and Tabellini (2000) as-sume that the replacement rate is ﬁxed in before-tax terms and that unemployment beneﬁts are not subject to income tax.30 With isoelastic utility but without leisure (i.e. δ = 0), equilibrium employment amounts to g(u) = Sm [1 −(Rg/(1 −T a))1−ρ], (10) where Rg = B/W is the ﬁxed gross (i.e. before-personal tax) replacement rate. overall tax burden At ﬁxed S, a higher average tax burden T a raises unemploy-ment by in eﬀect increasing the net replacement rate Rg/(1 −T a), thereby making unemployment relatively more attractive. Union members pay for the higher tax bur-den less in terms of lower after-tax wages, and more in terms of a higher probability of becoming unemployed. The intuition behind the higher unemployment rate can be understood as follows. (7) implies that the utility of non-employed union members is proportional to employed union members. Hence, a higher tax burden raising the net replacement rate does not make non-employed union members better oﬀcompared to employed members. The eﬀect of a higher net replacement rate is oﬀset by a higher unemployment rate increasing the expected duration of unemployment. Indeed, a change in the net replacement rate is powerless to aﬀect the relative position of the unemployed compared to the employed. Whether the higher tax burden makes the employed and the unemployed worse oﬀdepends on the elasticity of labor demand.31 With inelastic labor demand, a higher tax burden may even raise after-tax wages so that workers can shift more than 100 % of the tax burden unto employers. The intuition for the overshifting is that higher wages increase unemployment beneﬁts, thereby improving the outside option and thus increasing wage pressure. If at the same time labor demand is inelastic, the higher wages do not result in much additional unemployment, so that wage pressures remain.32 Indeed, with inelastic labor demand, workers are able to shift the tax burden onto proﬁts, consumers and other taxpayers. With overshifting, despite the higher unemployment rate increasing the expected duration of unemployment, also the unemployed in eﬀect gain because higher wages raise income not only in employment but also in unemployment (since unemployment beneﬁts are linked to wages). One way to justify the separate unemployment eﬀect of T a in empirical wage equations (even if they directly measure S and the after-tax replacement rates from 30Similar results would be found if the unemployment beneﬁts were taxed but at a lower average rate ¯ T a than wages and if the coeﬃcient (1−¯ T a)/(1−T a) would increase with the tax burden. If the tax schedule features a constant coeﬃcient of residual income progression S and unemployment beneﬁts would be subject to the same income tax schedule, the coeﬃcient (1 −¯ T a)/(1 −T a) would not vary with the overall tax burden but would depend only on S. Indeed, as shown below in (11), equilibrium unemployment is given by g(u) = Sm 1−(Rg)S , where Rg = B/W stands for the ﬁxed replacement rate in before-tax terms. 31Similar conditions determine the distributional eﬀects of a higher replacement rate Rg. Note that the (absolute value of the) elasticity of labor demand is likely to increase with the time horizon considered. In particular, the long-term labor elasticity is likely to exceed the labor-demand elasticity that the union employs to estimate the employment impact of higher wage costs. 32With higher wages raising unemployment beneﬁts and proﬁts, the negative external eﬀects on the government budget and proﬁts become more substantial. 15 social insurance beneﬁts) is that, in addition to taxed unemployment beneﬁts, un-employed may derive untaxed incomes from the informal sector (see Bovenberg and van der Ploeg (1998) and Holmlund (2000)) or enjoy utility of untaxed leisure (see Sørensen (1997)). The oﬃcial replacement rates, which include only public unemploy-ment beneﬁts, thus do not correctly measure the eﬀective replacement rates. Indeed, with a ﬁxed net replacement rate from unemployment insurance R, isoelastic utility and positive non-taxed other income in unemployment δ, the wage-setting curve is given by (from (9)) g(u) = Sm [1 −R1−ρ −δ/(W(1 −T a)1−ρ]. With non-taxed sources of unemployment income, productivity growth is consistent with stationary unemployment if non-taxed incomes rise with productivity in the for-mal economy so that δ∗≡δ/W is ﬁxed. In that case, the wage-setting curve is vertical again. With a ﬁxed net oﬃcial replacement rate R, in contrast to the situation without non-tax sources of unemployment income, an increase in the average tax burden T a moves this vertical curve to the right.33 Hence, conditions that ensure that changes in labor productivity do not impact unemployment do not necessarily imply that a higher tax burden leaves unemployment unaﬀected. The result that the tax burden raises the equilibrium unemployment rate is an important, and controversial policy conclusion. In the Netherlands, for example, the empirical result that — even if one controls for the oﬃcial replacement rate — the tax wedge signiﬁcantly aﬀects equilibrium unemployment has been quite robust (see, e.g. Graaﬂand and Huizinga (1999)). It has played an important role in supporting policies to contain the tax burden. Indeed, the numerical impacts of a lower tax wedge can be substantial. To illustrate, Daveri and Tabellini (2000) ﬁnd that the rise of 10 percentage points in the rate of eﬀective labor tax in continental Europe in the seventies and eighties can explain about 3 percentage points of the increase in European unemployment during this period. Nickell and Layard (1999) estimate an unemployment eﬀect of about 2 percentage points of such a tax increase.34 tax progression Tax progression raises the net replacement rate if unemployment beneﬁts are subject to the same tax schedule as wage income, while the replacement rate is ﬁxed in before-tax terms. This is in fact the case in many OECD countries (see OECD (2002b)). In that case, more progression exerts two oﬀsetting eﬀects in equi-librium unemployment. In addition to moderating wages, it boosts wages by raising the net replacement rate. To illustrate these two eﬀects, we assume a tax schedule featuring a constant coeﬃcient of residual income progression S. In particular, the tax schedule is given by T(Y ) = Y −gY S, where g and S are positive constants and Y is gross (labor or unemployment) income. This tax schedule implies that the coeﬃ-cient of residual income progression is ﬁxed at S. With this tax schedule, equilibrium unemployment amounts to g(u) = Sm 1 −(Rg)S , (11) 33Also an increase in the consumption tax moves this curve to the right is the price of output in the formal sector is proportional to consumer prices. 34The macro-economic model of CPB Netherlands Bureau of Economic Policy Analysis implies that rise in the tax wedge of one percentage point reduces equilibrium unemployment eﬀect by about 1/4 percentage point. 16 where Rg denotes the ﬁxed replacement rate (in before-tax terms).35 In that case, the wage moderation eﬀect of more progression still dominates the replacement rate eﬀect. Accordingly, more progression alleviates unemployment, even if high gross replacement rates imply that more progression raises eﬀective net replacement sub-stantially.36 Replacing a proportional consumption tax by a progressive income tax thus boosts employment. This result is modiﬁed if we allow for positive income from leisure δ. In that case, a more progressive tax schedule may actually raise equilibrium unemployment. Intu-itively, with high leisure, the replacement rate eﬀect becomes relatively more important compared to the wage-moderation eﬀect. At high replacement rates, a progressive tax system thus becomes less powerful in boosting employment. Indeed, a more progres-sive tax schedule raises unemployment if non-taxable income δ∗= δ/W(1 −T a) and the gross replacement rate Rg are large while the tax system is quite progressive to start with (i.e. S is small). All these three factors contribute to a high net replacement rate. Starting from a proportional income tax system, the introduction of some pro-gression may help to ﬁght unemployment. At higher levels of progression, however, further increasing progression may be counterproductive in terms of the objective of reducing unemployment. There is thus a level of progression that minimizes unem-ployment deﬁned by the following implicit equation for S 37 1 −δ∗= (Rg)α(1 −α log Rg). The optimal tax system is progressive (i.e. α < 1) as long as the gross replacement rate and utility from leisure δ∗are not very large, so that (Rg)(1 −log Rg) < 1 −δ∗. This example suggests that the unemployment impacts of taxes and unemploy-ment beneﬁts are related. If tax systems are progressive, a given gross replacement rate implies a higher after-tax replacement rate, so that this replacement rate wors-ens unemployment more. At the same time, at higher replacement rates, changes in progression are less likely to reduce unemployment, as the replacement rate eﬀect of more progression becomes stronger compared to the wage-moderating eﬀect. composition of the tax burden The diﬀerential impacts of productivity and ex-plicit labor taxes in case the gross compensation ratio is ﬁxed generate scope for tax 35Note that a higher average tax burden raising the parameter g does not impact unemployment. Hence, a higher tax burden does not aﬀect unemployment even if, as documented by Daveri and Tabellini (2000), the unemployed pay less taxes than the employed do. Hence, just as in the case with a ﬁxed after-tax replacement rate (see sub-section 3.2.2), the average tax rate does not aﬀect equilibrium unemployment. 36The unemployment eﬀect of more progression (with a ﬁxed gross eﬀective replacement rate) is very similar to that of more risk aversion (with a ﬁxed net eﬀective replacement rate). To determine these eﬀects, one takes the derivative of the function f(a) = a/(1 −ba) with respect a. The sign of this derivative is determined by 1 −ba(1 −a log b). This expression is non-negative because g(a, b) ≡ ba(1−a log b) ≤1. This inequity can be established by showing that δg/δa and δg/δb are both positive for a > 0 and 0 < b < 1 so that g(a, b) reaches a maximum of 1 at a = 1 and b = 1 (we impose the restriction a ≤1, b ≤1). 37Here we assume that δ is proportional to gross wages W, while T a is constant. We thus vary progression at a constant average tax burden on workers. The expression is found by taking the ﬁrst derivative of g(u) = α/(1 −δ∗−(Rg)α) with respect to α. The sign of this derivative depends on h(Rg, α) ≡1 −δ∗−(Rg)α(1 −α log Rg). This function is decreasing in Rg and increasing in α. Since we have h(Rg, 0) = −δ, h(Rg, 1) > 0 (i.e. (Rg)(1 −log Rg) < 1 −δ) is a suﬃcient condition for the existence of a unique optimal value for 0 < α < 1 if δ∗> 0. At this unique value of α, we have 1 −δ∗−(Rg)α = −α(Rg)α log Rg > 0 if Rg < 1. Hence, at the optimal value of α, g(u) = α/(1 −δ∗−(Rg)α) is well deﬁned (as the denominator is positive). Note that the optimal marginal tax rate is 100 % (i.e. α = 0) if δ∗= 0 or Rg = 0. 17 policies to boost employment. In particular, replacing explicit labor taxes by implicit labor taxes boosts employment. Intuitively, such a tax switch in eﬀect reduces the net replacement rate and shifts the tax burden onto non-labor income. With their non-taxed incomes being tied to before-tax wages, the unemployed bear the burden of the implicit labor taxes (since these taxes reduce before-tax wages), yet escape the burden of explicit labor taxes. Hence, this tax reform succeeds in shifting the tax burden towards the unemployed so that the outside option becomes less attractive. This stimulates wage moderation and thus employment. Implicit labor taxes can take various forms. Source-based capital income taxes in small open economies are one example. With mobile capital, these taxes are shifted onto labor (and with unemployment beneﬁts and other non-labor income δ being linked to wages also onto the unemployed) in the form of lower labor productivity. Environ-mental taxes on tradable intermediate inputs such as energy are another example. Indeed, Bovenberg and van der Ploeg (1998) and Koskela and Schob (1999) show that green tax reforms may boost employment if they succeed in shifting the tax burden to non-labor income and income of the unemployed, in particular. If they succeed in increasing employment, these reforms are thus an indirect way to cut the eﬀective net replacement rate. These conclusions are consistent with Bovenberg (1995), who argues that a green tax reform boosts employment only if the tax burden is shifted away from workers to people outside the active labor force (e.g. pensioners, owners of natural resources, transfer recipients). A change in the tax structure thus succeeds in alleviating unemployment if it replaces a tax that is borne by workers only by a tax that is also paid by the unemployed.38 The role of the bargaining level The employment eﬀect of labor taxes depends crucially on wage-setting institutions (see also Daveri and Tabellini (2000)). Up to now, we have assumed decentralized wage setting. Some countries, however, feature more centralized wage setting. Centralized unions may internalize the adverse impacts of high unemployment on the government budget constraint, thereby moderating wages (see Calmfors and Driﬃll (1988) and Summers, Gruber and Vergara (1993)).39 In fact, taxes may no longer aﬀect unemployment at all. Intuitively, unions see through the veil of the government budget constraint and oﬀset changes in the tax rate through transfers that exactly oﬀset the real eﬀects of taxes. This is a mixed blessing. The good news is that a higher average tax burden is less harmful for employment if (part of) income in unemployment is untaxed.40 The bad news is, however, that a high marginal tax rate becomes less eﬀective in reducing the monopoly distortion. This latter distortionis absent, however, if unions also internalize the impact of wages 38If unemployment beneﬁts are indexed to producer prices, for example, replacing the payroll tax, the income tax or implicit labor taxes by an indirect tax on consumption raises employment by shifting the tax burden towards the unemployed (i.e. imposing a larger burden on the outside option of unions). Whereas replacing consumption taxes by implicit taxes thus boosts employment if unemployment beneﬁts are linked to gross wages, such a tax reform hurts employment if these unemployment beneﬁts are linked to producer prices. Accordingly, whether replacing implicit taxes by consumption taxes raises or reduces employment thus crucially depends on how unemployment beneﬁts respond to prices and wages and how unemployment beneﬁts are taxed. 39This explains why smaller European countries feature lower unemployment rates than larger ones. An alternative explanation, however, is that small countries features less market power on commodity markets (i.e. εd and ε are larger) so that the union mark-up m in (9) is smaller. 40Indeed, Nickell and Layard (1999, pp. 3059) ﬁnd empirical evidence that coordination in wage bargaining reduces the impact of taxes on equilibrium unemployment. If unions also set working hours (as in Sørensen (1997) for example), taxes also leave labor supply unaﬀected if unions internalize the government budget constraint. Hence, in contrast to the competitive model with endogenous labor supply (see sub-section 3.1), taxes are non distortionary. 18 on proﬁts (e.g. because union members derive their pensions from shareholdings in the ﬁrms) and consumer prices (because union members consume the commodities produced at home). Indeed, unions may internalize the eﬀect of high wages on not only the government budget constraint but also on proﬁts and prices.41 In addition to centralization, another important aspect of wage-setting insti-tutions is the time horizon unions consider in setting wages. In particular, if they have a short-term horizon, unions can take the capital stock as given. Hence, the labor-demand elasticity in wage bargaining is rather low, implying a relatively high wage mark-up m and thus a high equilibrium unemployment rate. If reputational con-siderations allow unions to commit, in contrast, unions use a longer time horizon in considering the eﬀects of high wages. Hence, they in eﬀect employ larger labor-demand elasticity in setting wages. With a smaller union mark-up, changes in labor taxes exert a smaller impact on unemployment.42 unemployment beneﬁts linked to consumer prices Unemployment beneﬁts may be linked to prices rather than wages, especially in the short run.43 Indeed, these beneﬁts may be associated with an exogenous minimum income a country wants to maintain to keep liquidity-constrained households above the poverty line. In this case, the wage-setting curve is no longer vertical but slopes upward. Intuitively, higher wages are no longer transmitted into higher income during unemployment. This makes a higher wage more eﬀective in equilibrating the labor market. In particular, a higher wage makes work more attractive compared to unemployment, thereby boosting em-ployment. If unemployment beneﬁts are linked to consumer prices and are thus ﬁxed in real terms, the wage-setting curve describing the target real wage (in terms of producer prices) is given by (from (9)) v(W(1 −T a)/(1 + T c)) = [v(B) + δ] 1 −Sm g(u) −1 . In this case, taxes are paid only by workers, while the unemployed are pro-tected. Since higher taxes make work less attractive compared to unemployment, a higher overall tax burden raises unemployment, just as in the case in which the gross replacement rate is ﬁxed. Moreover, there is real wage resistance in that after-tax wages do not fully absorb a higher tax burden. How much wage costs increase and after-tax wages decline in response to a higher tax burden depends on the slopes of the labor-demand and wage-setting curves. Workers shift most of the tax burden onto ﬁrms if labor demand is inelastic (i.e. hori-zontal in (L, W) space) and unemployment does not aﬀect the target real wage much, so that the wage-setting curve is vertical in (L, W) space. In that case, employed union members and non-employed union members do not experience much of a loss in utility. Utility of non-employed union members, U r, does not decline much, as un-employment does not increase substantially. The initial decrease in inequity between workers and unemployed is thus restored mainly through an increase in W a. The other extreme case involves elastic labor demand and elastic wage setting (with respect to 41Indeed, if all budget constraints are linked through one representative agent, policy becomes completely neutral (see Bernheim and Bagwell (1988)). 42For commitment problems facing unions, see van der Ploeg (1987). In a small open economy with mobile capital, the long-run wage elasticity of labor demand may become very large. This explains why smaller European countries feature lower unemployment rates than larger European countries. 43If this link would be maintained in the long run, productivity growth reduces the replacement rate so that unemployment declines over time. 19 unemployment). In this case, the initial gap between utilities of workers and unem-ployed agents is re-established through higher unemployment reducing U r rather than through higher market wages increasing W a. 4 Optimal taxes: eﬃciency 4.1 Progressive taxation and eﬃciency In a second-best world, progressive taxes may help to alleviate various imperfec-tions. The previous section considered one speciﬁc labor-market imperfection, namely monopsony power of unions. In the particular model we explored, a 100 % marginal tax rate (i.e. S = 0) would be optimal to eliminate the mark-up of unions on the reservation wage. High marginal tax rates may also help to combat leapfrogging of employers when they set eﬃciency wages. To illustrate, if eﬀort depends on relative wages (i.e. the wage paid in a ﬁrm compared to the average wage level in the economy), employers impose adverse externalities on other ﬁrms if they raise wages in order to stimulate eﬀort of their own workers. High marginal taxes on wage increases may help to internalize these externalities. Marginal labor taxes may internalize adverse externalities also if utility from consumption depends in part on one’s relative position in society (see Layard (1980)). In that case, an individual raising his consumption by working harder reduces the utility of others. Marginal taxes help to combat these negative external eﬀects of additional consumption. In addition to labor-market imperfections, progressive tax systems may also help to alleviate distortions on capital markets. In particular, poor agents and young agents may suﬀer from liquidity constraints when they want to borrow in order to invest in human capital or to smooth consumption over their lifecycle. Progressive taxes redistribute income towards the poor and the young (since young workers tend to earn less than older workers) and thus help to alleviate these capital-market imperfections (see Hubbard and Judd (1986)). Another relevant non-tax market failure concerns insurance markets. Parents can not sign contracts insuring their children against career risks. Moreover, insur-ance against various human capital risks suﬀers from adverse selection. Hence, private insurance contracts are not available or are excessively expensive. Agents may thus demand a high risk premium on their investments, which inhibits risk taking and en-trepreneurship (see Sinn (1995)). By helping to pool human capital risks, a progressive labor tax helps to create the missing market for insurance of human capital (see Eaton and Rosen (1980)). Indeed, from an ex-ante point of view (i.e. behind the veil of igno-rance), a redistributive labor tax can be viewed as insurance of human capital; what is insurance ex ante (before the uncertainty has been realized) becomes redistribution ex post (after one knows the outcome). Two considerations are important when considering whether or not to employ high marginal tax rates on labor income as an instrument to address these imper-fections. As a ﬁrst consideration, beneﬁts must be weighed against costs. Indeed, high marginal tax rates impose various costs. Section 3.1 focussed on labor-leisure distortions. However, other potential costs are diminished work eﬀort and human capital accumulation, thereby harming not only the quantity but also the quality of labor supply. Moreover, high marginal taxes may hamper labor mobility, stimulate tax avoidance and tax evasion, and redistribute activities from the formal sector into the informal sector or the black economy. Finally, they may encourage jobs with 20 substantial nontaxable non-pecuniary beneﬁts. The second, related, consideration is whether alternative instruments are avail-able to combat market imperfections. Often more direct instruments are available to address the market imperfections. 4.2 Optimal progressivity from an eﬃciency point of view This sub-section explores the optimal progressivity of the labor income tax in a stan-dard search model of the labor market. With homogeneous households, a progressive income tax does not generate any distributional beneﬁts. Hence, optimal progressivity is explored from a pure eﬃciency point of view. In emphasizing the pure eﬃciency case for tax progressivity in imperfect labor markets, the approach is similar to that of Sørensen (1999), who investigates the optimal progressivity of the labor income tax in various labor-market models. Whereas Sørensen (1999) relies on numerical simula-tions, we derive an explicit analytical solution for the optimal labor income tax. This allows us to gain more insight into the determinants of optimal progression. To investigate optimal progression in imperfect labor markets, we simplify the workhorse of modern labor economics — the search model developed by Mortensen and Pissarides (see, e.g., Pissarides (1990) and Mortensen and Pissarides (1999)) — by for-mulating a one-shot, static version of the model. While facilitating the interpretation of the results considerably, the simpliﬁed model still contains the main determinants of the optimal system. Most importantly, it retains the major market failure of the standard search model: search activities, which amount to speciﬁc investments in a labor-market relationship, are non-contractible and may thus be held up.44 We explore how the tax system, by acting as a commitment device, can avoid hold-up of search activities. By eﬃciently allocating property rights, the tax system in eﬀect acts as a substitute for complete contracts in protecting the appropriate incentives for search activities. In this way, the tax system internalizes both positive and negative search externalities. The crucial element here is that wages are negotiated after search eﬀorts on both sides of the labor market have been sunk. The quasi rents from the search ac-tivities are thus distributed on the basis of ex-post bargaining power rather than the marginal eﬀectiveness of search in generating matches. Accordingly, if the marginal productivity of search activities exceeds the ex-post bargaining power, speciﬁc invest-ments in the match are held up. This hold-up problem arises because the party with excessive bargaining power cannot credibly commit to reward his partner according to her contribution to concluding the match. Indeed, parties can bargain only after they have met. Since contracts can thus be signed only after the contracting parties have sunk their search activities, the market for search is missing. The missing market for speciﬁc investments in the match is the key non-tax distortion in the model. 4.2.1 model The sequencing of decisions is as follows. In the ﬁrst stage of the one-shot game, tax policy is set. In the second stage, ﬁrms enter. In the third stage, workers and ﬁrms (or entrepreneurs or employers), which are unmatched, search for a partner on the labor market. At the supply side of the labor market, workers i ∈[0, 1] select their search intensities 0 ≤Xi ≤1 at a cost γ(Xi) ≥0, with γ′′(.) > 0 and limXi↑1 γ′(Xi) → +∞. At the demand side, entrepreneurs simultaneously decide how many vacancies 44Also Hosios (1990) and Acemoglu and Shimer (1998) formulate static versions of the search model. See also Boone and Bovenberg (2002). 21 to create. Vacancy costs are linear, so that economy-wide vacancy costs amount to cD, where D denotes the economy-wide number of vacancies and c are the per-unit vacancy costs. In the fourth stage of the game, workers and entrepreneurs are matched; the number of matches equals m(X, D), where X = 1 0 Xidi. The matching function m(., .) is increasing in its two arguments. Moreover, it exhibits constant returns in both arguments together, but decreasing returns in each of the arguments separately. Since a Cobb Douglas matching function ﬁts the data rather well,45 we assume that the matching function is of the Cobb Douglas form where the exponent of the vacancies is given by η. After they have been matched, workers and entrepreneurs bargain about the (after-tax) wage rate W in the ﬁfth stage of the game. Entrepreneurs and workers who do not ﬁnd a match receive a payoﬀof, respectively, zero and the (after-tax) unemployment (or welfare) beneﬁt B. The unemployment beneﬁt B can be interpreted as the minimum standard of living that the government guarantees.46 Finally, output is produced, taxes are collected and tax revenues G are spent on a public good. The model is solved backwards. Accordingly, before determining search inten-sities X and labor-market tightness θ ≡D X , we solve for (after-tax) wages. production Each matched ﬁrm-worker combination produces Y units of output. Output is the numeraire. Output net of search costs, Ω, is given by Ω= m(X, D)Y −γ(X) −cD, (12) where m(X, D)Y represents total output and γ(X) + cD stands for total search costs. The exogenous public good G and the exogenous unemployment beneﬁt B are ﬁnanced by a linear tax on wages: G + B = m(X, D)(τ(W −B) + τ a + B). (13) Here, τ represents the proportional (or ad valorem) tax on wages (net of the unem-ployment beneﬁt B). The other component of the linear wage tax, τ a, is a ﬁxed (or speciﬁc) tax on the match. This tax depends only on the existence of a match and is not conditioned on how the quasi-rents from the match are shared between ﬁrms and workers. Wage taxation is progressive (i.e. the average tax burden rises with the wage) if the speciﬁc tax τ a is negative. wage setting Wages are determined by Nash bargaining after a match has been found. The bargaining is about the (after-tax) quasi rent (or surplus) from the match, Y −τ(W−B) −τ a−B. The after-tax wage W that maximizes the Nash Bargaining function (W−B) β(Y −W −τ(W−B) −τa)1−β is given by W = β(Y −τa + τB) + (1 −β)(1 + τ)B 1 + τ . (14) This is the value of the match for the worker. The value of a match for the entrepreneur, Π, amounts to Π ≡Y −W(1 + τ) + τB −τ a = (1 −β)(Y −τ a −B). (15) 45See, e.g., Broersma and Van Ours (1999). 46This is in fact optimal if the utility function of individuals is given by u(Y ) = −∞ if Y < B Y if Y ≥B . Note that the unemployment beneﬁt is not subject to the labor income tax. Boone and Bovenberg (2002) explore the model developed here with B = 0. 22 The burden of the ﬁxed tax component τa is shared between the worker (i.e. the supply side of the labor market) and the ﬁrm (i.e. the demand side of the labor market) in proportion to their respective bargaining powers β and (1 −β); after-tax wages W decline and before-tax wages (i.e. wage costs) W(1 + τ) + τ a rise with τ a. The proportional tax rate τ, in contrast, reduces only the worker’s value of a match (14); before-tax wages W + τ( W−B) + τ a and the ﬁrm’s value of the match (15) are not aﬀected by τ. The proportional tax rate thus bears on the supply side rather than the demand side of the labor market. Intuitively, by taxing the quasi rents that accrue to workers (i.e. the after-tax wage W), the proportional tax not only reduces the (after-tax) surplus from the match but also raises the eﬀective bargaining strength of employers. In the presence of a higher proportional tax, employers bargain more aggressively because a given increase in the after-tax wage W results in a larger increase in wage costs W(1 + τ) + τ a. search intensity and vacancies The wage agreed upon in ex-post bargaining (i.e. after the match has been concluded) aﬀects the incentives facing workers and ﬁrms to search for a partner in the preceding stage of the game. In selecting their search intensity, workers trade oﬀadditional search costs against the higher probability of ﬁnding a job. With a constant-returns-to-scale matching function, the probability that a worker with search intensity Xi is matched with a ﬁrm can be written as a function of labor-market tightness only: Xi X m(X, D) = Xim(θ) where m(θ) ≡m(1, θ). The risk-neutral worker selects search intensity Xi so as to maximize the expected surplus from search max Xi≥0{Xim(θ)(W −B) −γ(Xi)}. With homogeneous individuals, all households feature the same search intensity γ′(X) = m(θ)(W −B), (16) where the left-hand side represents the marginal costs from higher search intensity and the right-hand side the corresponding expected marginal beneﬁt in terms of raising the probability of ﬁnding a job. The net expected surplus for the worker, Xm(θ)( W− B) −γ( X) = Xγ′( X) −γ( X), is assumed to be positive. The expression for optimal search intensity (16) can be interpreted as the im-plicit labor-supply equation. With the aid of (14), labor supply can alternatively be written as γ′(X) = m(θ)β(Y −τ a −B) 1 + τ . (17) Demand for labor is determined by ﬁrms. The probability that a ﬁrm is matched with a worker equals m(X,D) D = m(θ) θ . With free entry of ﬁrms, expected proﬁts from posting an additional vacancy are zero c = m(θ) θ Π = m(θ) θ (1 −β)(Y −τ a −B). (18) Here the left-hand side represents the costs for a ﬁrm entering the labor market, while the right-hand side stands for the ﬁrm’s expected beneﬁts of doing so. By reducing the probability of ﬁlling a vacancy m(θ) θ , a tighter labor market decreases the expected beneﬁts from posting a vacancy. Since labor-market tightness θ is the only endogenous variable in (18), the free-entry condition determines tightness as a function of τa. As 23 in most non-competitive models of the labor market, a more progressive tax system (i.e. a smaller value for τ a) raises the employment rate (i.e. the number of matches per unit of labor supply) m(X, D)/X = m(θ). Welfare Substituting the government budget constraint, Xm(θ)Y = Xm(θ) [W + Π] + G + (1−Xm(θ)) B into the expression for Ωin (12), we ﬁnd Ω = Xm(θ) [W −B + Π] + G + B −γ(X) −cθX = Xγ′(X) −γ(X) + G + B. With the free-entry condition ensuring a zero expected return for entrepreneurs, welfare consists of the ex-ante return to workers Xm(θ)( W−B)−γ( X)+ B and the resources allocated to the government G. The second equality follows from (16) (to eliminate W) and (18) (to eliminate Π). Since G and B are exogenously given and Xγ′(X) −γ(X) is rising in X, maximizing welfare is equivalent to maximizing search X. Optimal progressiveness If the government employs τ and τ a to maximize welfare and search, it sets τ a according to (see the Appendix) τ a ≡1 −β −η 1 −β Y −B. (19) The optimal degree of progressiveness of the income tax ensures that the match-ing process is eﬃcient by establishing an eﬃcient distribution of property rights over the fruits from non-contractible speciﬁc investments in search. In particular, the party that carries out the most important non-contractible investments should be able to reap most of the quasi rents from the relationship. In this way, property rights act as a substitute for complete contracts in protecting the incentives for speciﬁc investments. By moderating wages, a progressive tax system in eﬀect allows ﬁrms to increase their share in the quasi rents from search. A progressive tax system is thus optimal if ﬁrms can not reap the full social beneﬁts of their search eﬀort in a laissez-faire equilibrium. This is the case if vacancies are important in generating matches (as reﬂected in a high value for η) and workers can appropriate a large share of the surplus from the match because of substantial bargaining power β and a good outside option B.47 In that case, workers in eﬀect levy an implicit tax on the speciﬁc investments of employers (i.e. the posting of vacancies) by expropriating part of the marginal social beneﬁts of these investments.48 With employers being held up by workers, labor demand (i.e. the posting of vacancies) is too low from a social point of view. A progressive tax un-does the implicit ’hold-up’ tax levied by workers on employers so that employers face adequate incentives to enter the labor market. By in eﬀect subsidizing labor demand and taxing labor supply, tax policy restores the socially optimal mix of labor demand and supply. 47A proportional tax is optimal if unemployment beneﬁts are absent (i.e. B = 0) and the so-called Hosios condition holds. The latter condition (see Hosios (1990)), which reads 1−β = η, states that the bargaining power of ﬁrms 1 −β should correspond to the eﬀectiveness of ﬁrms in producing matches as measured by η. 48Unions can aﬀect hold-up problems. On the one hand, they may worsen these problems if, by monopolizing labor supply in an industry, they are able to hold up ﬁrms’ investments that are speciﬁc to this industry. On the other hand, unions tend to feature a longer time horizon than individual workers do. The reputational mechanism may thus induce them to keep their commitments to mod-erate wages, thereby alleviating the hold-up problem. These opposite eﬀects of unions resemble the opposite eﬀects of industry unions worsening monopoly distortions and national unions internalizing externalities (as discussed in sub-section 3.2.3). 24 Tax policy, which is set before search activities are determined, allows workers to commit to not expropriate ﬁrms. In this way, tax policy eﬀectively creates the market for search that is missing in the laissez-faire equilibrium. Before workers and ﬁrms meet each other after they match, tax policy in eﬀect allows them to conclude a contract stipulating that their search activities will be rewarded according to the marginal contribution to the match. Indeed, if workers would vote on the tax rate in the ﬁrst stage of the game (i.e. when they are still unmatched and in eﬀect face inﬁnitely elastic labor demand), they would vote for the optimal social contract (i.e. the optimal allocation of property rights) implicit in the optimal tax structure. The results can be interpreted also in terms of the distortions due to imperfect competition. If workers exercise too much power ex post (i.e. 1−β−η 1−β Y −B < 0 so that θ is too low), the market can be characterized as being monopolized. Tax policy corrects the associated monopoly distortions by levying a tax on the excessive wages. In this way, tax policy oﬀsets the implicit taxes imposed by the party with excessive market power. An eﬃcient matching process maximizes the incentives of workers to participate in this matching process through labor supply. If the bargaining power of workers is too strong (i.e. τa −1−β−η 1−β Y + B > 0), workers are discouraged from looking for a job by a low probability of ﬁnding a job on account of a lax labor market (as reﬂected in a low value for tightness θ).49 The unemployment rate is too high in that case. If workers’ bargaining power is too weak (i.e. τ a −1−β−η 1−β Y + B < 0), workers’ search is depressed by excessively low wages. Accordingly, beyond the point at which τa ≡ 1−β−η 1−β Y −B a more progressive tax system harms the eﬃciency of the matching process by reducing labor supply and in eﬀect giving the supply side of the labor market insuﬃcient bargaining power compared to labor demand.50 As a direct consequence, labor supply is too low compared to labor demand. The resulting excessively tight labor market implies that the unemployment rate 1 −m(X, D)/X is too low from a pure eﬃciency point of view. The optimal τa does not depend on revenue requirements. The intuition behind this result is the following. The linear vacancy costs imply that demand for labor is inﬁnitely elastic. Hence, ﬁrms are able to shift the entire tax burden required to ﬁnance government spending to workers. Thus, whereas a tax on labor supply, τ, taxes the supply side directly through a lower after-tax wage W, a tax on labor demand τ a is also borne by labor supply — albeit indirectly (namely, through the general equilibrium eﬀect of fewer ﬁrms entering the labor market, which reduces the probability of ﬁnding a job by producing a less tight labor market). It is more eﬃcient to tax workers directly through τ than indirectly through the general equilibrium eﬀect on θ; both ways distort search intensity, but the second way distorts also labor-market tightness.51 This result is closely related to the celebrated Diamond-Mirrlees (1971) result on the optimality of production eﬃciency. With constant-returns-to-scale production (or tax instruments to tax away rents due to decreasing returns) and suﬃcient tax 49Thus, progressive taxation and wage moderation may raise labor supply if workers have excessive bargaining power in the laissez-fair equilibrium. This contrasts with models (explored in the section 3.1) that focus on the intensive rather than the extensive margin of labor supply and in which labor supply is set after workers have found a job. In these models, the discouraged-worker eﬀect is absent and the unemployment rate thus does not depress labor supply. 50This contrasts with the union model explored in section 3.2. In that model, which abstracts from endogenous labor supply, the optimal marginal tax rate is 100 %, resulting in the elimination of involuntary unemployment. 51This strong result no longer holds if labor demand is not inﬁnitely elastic with respect to wage costs because ﬁrms cannot freely enter the labor market and a lump-sum proﬁt tax is not feasible (see Boone and Bovenberg (2002)). In that case, not only τ but also τa rises with the government revenue requirement. 25 instruments to tax consumers directly, the government should ensure production eﬃ-ciency. The government ﬁnds it optimal to tax consumers directly through consumer taxes rather than indirectly through taxes that violate production eﬃciency. Similarly, in the current context, the government should not distort labor-market tightness, θ, by raising revenues through τ a. Indeed, keeping labor-market tightness at its ﬁrst-best level can be viewed as maintaining eﬃciency in the production of matches. A higher unemployment beneﬁt is translated into an equivalent increase in in-work beneﬁts τ a (i.e. −dτ a = dB). A higher unemployment beneﬁt thus results in a more progressive tax system, as a higher in-work beneﬁt in eﬀect oﬀsets the adverse impact of the unemployment beneﬁt on job creation. The beneﬁt system thus determines the optimal progressiveness of the labor tax. The combination of an employment beneﬁt and a job subsidy can be interpreted as a basic income. Indeed, if the Hosios condition is met, the overall tax on job creation is zero (i.e. τ a + B = 0). The higher marginal tax rates depress labor supply but it is more eﬃcient to depress labor supply through lower after-tax wages than to depress labor supply indirectly through violating production eﬃciency.52 A progressive tax (i.e. τ a < 0) thus oﬀsets the distortions of the welfare system. 5 Employment and distribution: applied general equi-librium analysis This section explores the impact of labor tax reform with the help of an applied general equilibrium model for the Netherlands, the so-called MIMIC model developed at CPB Netherlands Bureau for Economic Policy Analysis. The model combines a rich theoretical framework based on modern economic theories, a ﬁrm empirical foundation, and an elaborate description of the actual tax and social insurance systems in the Netherlands. The model considers the two main transmission channels through which tax policy impacts the labor market, namely, labor supply and wage determination. In addition, it considers various other ways through which taxes and beneﬁts aﬀect the labor market namely, the black economy, human capital accumulation, eﬃciency wages, costly job matching, and search behavior of the unemployed. Hence, in addition to wages, unemployment and the quantity of labor supply, taxes aﬀect the quality of labor supply. Through the replacement rate, the beneﬁt system aﬀects not only wage setting but also search intensity and the reservation wages of the unemployed. 5.1 MIMIC model This sub-section provides a bird-eye’s view of MIMIC (Graaﬂand e.a. (2001) provides a more detailed overview of the model). MIMIC allows for considerable heterogeneity among households. In particular, the model accounts for heterogeneity in household composition (including the number of children), educational level, age, ability, pref-erences for leisure, and labor-market status. Incorporating this heterogeneity allows one to explore the income distribution and hence various trade-oﬀs between equity and eﬃciency. Moreover, Tables 3 and 4 document on fact that replacement rates and marginal tax rates vary considerably across various individuals, depending on household composition and income level. The same holds true for labor-supply elas-52With less than inﬁnitely elastic labor demand, higher unemployment beneﬁts would in part be ﬁnanced through lower job subsidies τa so that τa + B > 0. A similar result holds in a model in which labor supply is endogenous on not only the extensive margin but also the intensive margin (see section 6). 26 ticities.53 Whereas particular policies may have little impact on a representative in-dividual, they may signiﬁcantly aﬀect the behavior of particular types of individuals, such as secondary part-time workers, low-skilled agents and older employees close to retirement (see also Disney (2000)). A careful analysis of the labor-supply eﬀects of tax policy therefore requires substantial disaggregation. Indeed, representative agent models conceal this variation in eﬀective tax rates and labor-supply behavior. MIMIC embeds a standard microsimulation model in a general equilibrium set-ting. As an applied general equilibrium model, MIMIC draws on microeconomic the-ory to derive supply and demand from optimizing behavior by decentralized agents. This allows one to interpret the model results in terms of microeconomic behavior of households and ﬁrms. In modelling equilibrium on the labor market, the model de-parts from the traditional assumption of market clearing in most general equilibrium models. In modelling labor-market imperfections that give rise to involuntary unem-ployment, MIMIC employs modern labor-market theories. In particular, in addition to legal minimum wages, it includes elements of wage bargaining, eﬃciency wages and costly job matching. In this way the model describes equilibrium unemployment in terms of the structure of the tax system, minimum wages and the features of social insurance and assistance. MIMIC has a ﬁrm empirical basis. Various crucial relationships in the model, in-cluding contractual wage formation and the production function, have been estimated from time series data. Furthermore, microeconometric estimates on Dutch labor sup-ply helped to calibrate the labor-supply model. Moreover, income distributions are based on micro data. MIMIC describes the institutional features of taxation and social insurance in much detail. This institutional detail makes the model especially relevant for policy making because actual policy proposals typically involve particular details of the tax and social insurance systems. Moreover, as section 3.2 documents, the impact of tax policies depend crucially on how unemployment and welfare beneﬁts respond to changes in wages and taxes. Incorporating the main transmission channels of tax policy in an empirically based model with substantial household heterogeneity is not without costs. In partic-ular, the various sub-models are not fully consistent with each other. To illustrate, the wage setting model does not take into account endogenous labor supply. Moreover, the labor-supply model assumes that households are not rationed on the labor market. At the same time, the models describing search behavior of the unemployed and training and schooling are not part of the household model describing labor supply. 5.1.1 households and labor supply MIMIC distinguishes 40 types of households in order to adequately describe labor sup-ply and explore the income distribution. In particular, MIMIC distinguishes couples, single persons, single parents, pensioners and students. To model the speciﬁc labor-supply behavior of those close to retirement, people aged between 55 and 65 years are represented by a separate household type. Couples consist of a so-called breadwinner (i.e. the individual with the highest personal income) and a partner (i.e. the adult with the lowest personal income). Couples are subdivided into families with children and families without children. Individuals within each household may diﬀer with re-spect to their skill level (high skilled, low skilled or unskilled) and their job status (i.e 53This is shown by microeconometric evidence. Econometric work on labor-supply behavior is increasingly exploiting this microeconomic variation and is hence moving away from macroeconomic estimation. 27 holding a job in the formal sector, unemployed and collecting a social beneﬁt, or not participating in the labor force). For each household type, MIMIC employs class-frequency income distributions based on micro data to describe the distribution of gross incomes. These income distributions are important determinants of the eﬃciency costs of high marginal tax rates: the more people are concentrated in a particular income range, the higher become the eﬃciency costs of high marginal rates in this income range. By applying the corresponding statutory tax and premium rates to gross incomes, MIMIC determines net incomes and the average and marginal tax rates that aﬀect labor-supply decisions. The labor-supply model has been calibrated so that the model reproduces labor-supply elasticities estimated in the empirical literature for the Netherlands. In partic-ular, the uncompensated wage elasticity of labor supply by partners is set at 1.0, single persons feature a corresponding elasticity of 0.25 and most breadwinners of around 0.1. Older breadwinners, who may change their retirement decisions in response to changes in wages, feature a somewhat higher elasticity of 0.15. The income elasticities of labor supply are smaller than the corresponding wage elasticities — namely, 0.2 for partners, 0.05 for single persons and almost zero for breadwinners. In addition to supplying labor to the formal labor market, households can supply labor to the black labor mar-ket. The model has been calibrated to reproduce the size of the black economy in the Netherlands, which is estimated at about 3% of GDP, and an uncompensated wage elasticity of black labor supply of 0.75. A separate training model endogenizes the distribution of the labor force over unskilled, low-skilled and high-skilled workers. By engaging in training activities, workers can increase the transition rates to higher skill levels with higher wages. In setting their training level, workers trade oﬀ(non-taxable) eﬀort costs and the beneﬁts of training (in terms of a higher probability of moving towards a higher skill level earning a higher wage). Based on Groot and Oosterbeek (1995), the model is calibrated such that a 10 % increase in after-tax wage diﬀerentials raises the share of workers participating in training by 8 %. 5.1.2 wage formation The black labor market is modelled as the competitive labor market in section 3.1; for each of the three skill categories, the wage clears this market. Firms in the sheltered sector and the construction sector54 demand labor from the black market. The elas-ticity of substitution between black and formal labor in the production function is set at 2, which is based on empirical evidence in Baartmans et al. (1986). Furthermore, ﬁrms may pay formal labor in part informally, i.e. without reporting the wages to the tax authorities. Firms determine this informal labor by trading oﬀlower taxes against a potential penalty for fraud. The formal labor markets for the three skill categories do not clear. The imper-fections of this market originate in market power of unions, eﬃciency wages and costly job matching. To describe wage formation in these markets, MIMIC distinguishes between contractual wages, which are determined in collective bargaining between employers and unions, and incidental wages, which are set by individual employers based on the tightness of the skill-speciﬁc labor markets. Social beneﬁts are linked to contractual, rather than incidental, wages. 54In addition to these two sectors, MIMIC includes a mining sector, a residential sector and an exposed sector, which consists not only of capital-intensive manufacturing industries subject to intense foreign competition but also of agriculture and transport. The sheltered sector includes trade, banking and insurances, and other private services. 28 Contractual wages are determined by a right-to-manage model in which em-ployers and unions bargain over wages at the industry level. The rents that unions and employers bargain over originate in the market power of ﬁrms on product mar-kets. In particular, each industry produces a good that is an imperfect substitute for goods produced by other domestic industries or by foreign ﬁrms. The unions are small compared to the labor market as a whole and therefore do not internalize the impact of their bargain on the government budget constraint, proﬁts and prices. The resulting wage equation is calibrated on the basis of estimates by Graaﬂand and Huizinga (1999). Using macro data, they found that the positive elasticity of the average tax rate is six times as large in absolute value as the negative elasticity of the marginal tax rate (-0.1). The elasticity of the consumer price equals the sum of the elasticities of the marginal and average tax rates, which is 0.5. Accordingly, at constant unemployment and replacement rates, the incidence of a higher tax wedge (by simultaneously increasing average and marginal tax rates) is split equally between employers and employees through higher gross wages and lower after-tax wages. On average, the wage elasticity of the replacement rate is about 0.2.55 The wage structure among skills is further modiﬁed by a skill-speciﬁc, so-called incidental, wage component. The employer uses this incidental wage component to minimize search costs. The incidental wage can thus be interpreted as an eﬃciency wage associated with hiring costs. It is set as a mark-up on the contractual wage. This mark up rises with the tightness of the labor market. 5.1.3 job matching To model labor-market tightness and mismatch, MIMIC incorporates costly job match-ing. Heterogeneity in the matching process allows MIMIC to model also the adverse impact of high minimum wages and high reservation wages on the eﬃciency of the matching process. In particular, low-productivity matches may fail because they do not meet the minimum productivity standard of the employer (determined by the minimum wage) or the reservation wage of the unemployed. In the matching model, the behavior of the unemployed is described in terms of the reservation wage and search intensity. In particular, in setting search intensity, the unemployed trade oﬀthe loss of leisure against the increased probability of moving into the employed state. The optimal search intensity increases in the average transition rate into employment (because it raises the marginal return on search) and decreases in the replacement rate (which decreases the diﬀerence in life-time utility between the employed and unemployed states). The second variable describing the behavior of the unemployed is the reservation wage, which is the wage at which an unemployed job seeker is indiﬀerent between the employed and the unemployed states. The reserva-tion wage rises with both the unemployment beneﬁt and the average transition rate into employment. Together with the lognormal wage distribution of job oﬀers, the reservation wage determines the acceptance rate of the unemployed (i.e. the share of contacts that is acceptable to unemployed job seekers). A higher replacement rate thus exacerbates the mismatch on the labor market by lowering search intensity and raising the reservation wage. This pushes up incidental wages, thereby raising unemployment in equilibrium. 55Since both skill-speciﬁc and macroeconomic factors play a role in determining skill-speciﬁc wages, skill-speciﬁc wages are determined by both a macroeconomic wage equation, which adopts macro-aggregates for the average tax rate, the marginal tax rate, the replacement rate and unemployment, and a corresponding skill-speciﬁc wage equation, which employs skill-speciﬁc explanatory variables. Based on Graaﬂand and Lever (1996), the macro and skill-speciﬁc wage equations carry equal weights in determining the contractual wage for a speciﬁc skill. 29 The long-term unemployed typically diﬀer from the short-term unemployed in their search behavior, reservation wage and productivity. MIMIC therefore distin-guishes between short- and long-term unemployment by using a steady-state ﬂow model for job matches akin to Holmlund and Linden (1993).56 In particular, the long-term unemployed are less productive than the short-term unemployed because they lost some human capital during their prolonged period of unemployment. If they ﬁnd a job, the long-term unemployed face some (exogenous) probability to restore their human capital. The long-term unemployed take into account this beneﬁt of entering work and hence feature a relatively low reservation wage. Accordingly, rather than the reservation wage, the minimum eﬀective productivity standard of the employer, which is determined mainly by the minimum wage, mainly restricts the number of successful matches for the long-term unemployed. For the short-term unemployed, in contrast, a relatively high reservation wage is the most important barrier to successful job matches. As a relatively large number of long-term unemployed are unskilled, the minimum eﬀective productivity standard (and hence the minimum wage) is the most important barrier in the job-matching process of the unskilled. The model is calibrated so as to conform closely to the observed transition rates between the various states and to the main empirical ﬁndings on search intensities and reservation wages. 5.1.4 public institutions MIMIC contains several public institutions, including the Dutch personal income tax system in 1998. The personal income tax features a tax-free allowance and three tax brackets. A partner whose labor income remains below the tax-free allowance can transfer the tax-free allowance to the breadwinner. The rate in the ﬁrst tax bracket is about 36% in 1998. The tax rate in the second bracket is 50% and has to be paid on incomes above about 25,000 euro. The marginal rate in the third tax bracket, which amounts to 60%, is paid on incomes above about 50,000 euro. Workers beneﬁt from a special earned-income tax deduction, which amounts to 12% of labor income with a maximum of around 1500 euro. Unemployment beneﬁts are subject to the progressive personal income tax. VAT in the Netherlands imposes a low rate on necessary goods (6%) and a high rate for other goods (17%). Other public institutions in MIMIC include employee and national social insurance schemes,57 the employers’ and employ-ees’ contributions to employee social insurances, premiums for health insurance, the statutory minimum wage (which is linked to the average contractual wage rate), social assistance (which is linked to the statutory minimum wage), and a number of policy instruments targeted at speciﬁc groups, such as the long-term unemployed and the unskilled. Households with incomes just above the minimum wage face overall eﬀec-tive marginal tax rates close to 80 % on account of employee insurance premiums, income-dependent public health care premiums, and means-tested housing allowances. Indeed, overall marginal taxes in this income range exceed marginal tax rates (of 60 %) facing high-income earners. 56A detailed description of this model can be found in Jongen and Graaﬂand (1998). 57Employee insurances apply only to working people and cover employment risks — namely, un-employment, disability, and sickness. Beneﬁts depend on previously earned wages. All residents are entitled to national social insurance, which involves family allowances, disability beneﬁts for the hand-icapped, special health costs, and a basic pension. In contrast to beneﬁts from employee insurances, beneﬁts from national social insurance are not related to previously earned wages. 30 5.2 Cutting taxes in MIMIC This section employs the MIMIC model to investigate the long-run eﬀects of a number of tax cuts. In all experiments, the ex-ante (i.e. before behavioral responses have been taken into account) reduction in tax revenues amounts to 0.25 % of GDP. A cut in public consumption balances the government budget ex post (i.e. after the eﬀects of the behavioral responses on the public budget have been taken into account). Hence, the required cut in public consumption reﬂects the impact of behavioral responses on the public budget. In particular, if the reduction in public consumption is less than the ex-ante cut in revenues of 0.25 % of GDP, behavioral responses help to mitigate budgetary costs. This section consists of three parts. The ﬁrst part explores cuts in personal income taxes. The second part turns to cuts in social security contributions (i.e. payroll taxes) paid by employers. Finally, the third part investigates various forms of an Earned Income Tax Credit (EITC) aimed at increasing the reward of work in general and of low-skilled work in particular. 5.2.1 personal income taxation Cutting marginal tax rates The detailed modelling of the personal income tax system allows MIMIC to explore the labor-market eﬀects of various parameters of the Dutch tax system. The ﬁrst three columns of Table 5 contain the long-run eﬀects of cuts in each of the three tax brackets of the Dutch personal income tax (of respectively 1.2, 6.9 and 24.5 % points). These tax cuts reduce both marginal and average tax rates. However, the tax cut in the ﬁrst bracket is inframarginal for many workers whose incomes reach into the second and third tax brackets. Hence, this particular tax cut reduces the average marginal tax rate (i.e. the marginal tax rate averaged over the various workers) substantially less than tax cuts in the higher brackets do (see the third next-to-last row of Table 5). Indeed, in contrast to a reduction in the tax rates in the upper brackets, a tax cut in the lowest bracket makes the tax system somewhat more progressive (as measured by the coeﬃcient of residual income progression). First income Second income Third income Basic income General cut Targeted cut tax bracket tax bracket tax bracket tax allowance in payrol tax in payrol tax Private consumption 0,51 0,64 0,62 0,39 0,48 0,43 Exports 0,36 0,53 0,53 0,12 0,29 0,28 Imports 0,21 0,30 0,30 0,09 0,17 0,17 Formal Production 0,37 0,57 0,57 0,09 0,29 0,27 Black production -0,02 -0,20 -0,63 0,06 -0,02 1,42 Employment 0,39 0,47 0,44 0,07 0,29 0,47 - unskilled 0,46 0,20 0,20 0,18 0,40 3,92 - low-skilled 0,50 0,17 0,07 0,10 0,36 0,05 - high-skilled 0,35 0,60 0,60 0,04 0,25 0,04 Labour supply (pers.) 0,39 0,05 -0,02 -0,07 0,05 0,10 Labour supply (hours) 0,18 0,26 0,30 -0,04 0,10 0,10 - breadwinners 0,03 0,28 0,54 -0,02 0,03 0,02 - partners 0,39 0,02 -0,20 -0,16 0,19 0,31 - single persons 0,26 0,31 0,08 -0,06 0,13 0,19 - 55+ 0,09 0,42 0,77 0,03 0,06 0,06 Black labour (hours) -0,03 -0,14 -0,33 0,07 -0,02 1,45 Training - unskilled and low-skilled 0,04 0,10 0,03 -0,05 0,00 -0,83 - high-skilled -0,09 0,65 0,89 -0,06 -0,02 -0,02 Unemployment rate -0,13 -0,12 -0,08 -0,08 -0,12 -0,23 - unskilled -0,21 -0,37 -0,27 -0,07 -0,20 -1,41 - low-skilled -0,18 -0,12 -0,08 -0,13 -0,17 -0,24 - high-skilled -0,10 -0,08 -0,04 -0,07 -0,10 -0,08 Share long term unemployment -1,29 -0,74 -0,47 -0,48 -0,75 -1,61 Replacement rate (a) -0,03 -0,29 -0,18 0,15 -0,01 -0,15 Average tax burden (a) -0,35 -0,36 -0,33 -0,25 -0,32 -0,29 Marginal tax burden (a) -0,32 -0,91 -1,07 -0,07 -0,20 0,38 Government consumption (b) -0,17 -0,15 -0,14 -0,21 -0,18 -0,16 Source: Graafland et. al. (2001) (a) Average over all households (b) In percentage of GDP percentage deviations absolute deviations Table 5. Macroeconomic effects of cuts in income and payroll taxes 31 Labor supply All three tax cuts boost aggregate labor supply (in hours) because the substi-tution eﬀect associated with a lower marginal tax rate dominates the income eﬀect on account of a lower average tax rate. The composition of additional labor supply, however, diﬀers. In particular, a lower tax rate in the ﬁrst bracket raises especially the labor supply of partners (i.e. secondary earners). This is because partners tend to work in part-time jobs with relatively low (annual) labor incomes. Hence, their marginal labor income is typically subject to the tax rate in the ﬁrst bracket. A cut in this tax rate therefore encourages partners to work longer hours, especially in view of the relatively large uncompensated wage elasticity of partner’s labor supply. Breadwinners and older workers generally earn higher labor incomes than part-ners do. Indeed, the incomes of many of these workers reach into the second or third tax brackets. For these workers, a lower tax rate in the ﬁrst bracket reduces the aver-age tax rate without aﬀecting the marginal tax rate. The inframarginal character of the tax cut in the ﬁrst bracket for many breadwinners explains why such a cut barely aﬀects aggregate labor supply of breadwinners and older workers; the income eﬀect is relevant for all breadwinners and older workers, while the substitution eﬀect applies only to those workers whose marginal labor income falls in the ﬁrst bracket. In contrast to tax cuts in the ﬁrst bracket, tax cuts in the second and third brackets are eﬀective in stimulating labor supply of breadwinners and older workers. Although these groups feature relatively low labor-supply elasticities, the relatively large cuts in marginal tax rates produce signiﬁcant labor-supply responses. The im-pact on aggregate labor supply (in hours) is substantial because breadwinners, single persons and elderly account for a large share of aggregate labor supply (in hours). Tax cuts in the highest bracket discourage partners from supplying labor, because the income eﬀect rather than the substitution eﬀect mainly impacts the labor supply of partners. In particular, by raising the incomes of breadwinners, a tax cut in the high-est bracket reduces partners’ labor supply through the channel of higher household incomes. At the same time, the substitution eﬀect is not important because only few partners earn incomes that are suﬃciently high to be marginally taxed in the third bracket. These simulations illustrate the added value of the extensive labor-supply model of MIMIC, which accounts for heterogeneity in preferences and wages, incorporates the actual Dutch tax system, and explicitly models labor supply of partners. The incorpo-ration of the actual income distribution and the institutional detail of the Dutch tax system allows MIMIC to determine to what extent cuts in particular tax brackets are (infra)marginal. Furthermore, the explicit modelling of labor-supply behavior of part-ners and breadwinners modiﬁes the predictions from aggregate models. To illustrate, tax cuts in the ﬁrst brackets are more inframarginal and thus reduce marginal tax rates, on average, only a third as much as tax cuts in the higher brackets do (see the next-to-last row of Table 5). Despite the relatively small decline in average marginal tax rates, tax cuts in the ﬁrst bracket are still quite eﬀective in stimulating aggregate labor supply. The reason is that these tax cuts reduce marginal tax rates of partners — the group featuring the most elastic labor supply. Indeed, these tax cuts are most eﬀective in raising labor supply in persons, as more partners are encouraged to enter the labor force. Black labor supply and training All three tax cuts reduce the size of the black economy. Supply of black-market labor declines because lower marginal income taxes make formal labor supply more attractive. Firm demand for black labor decreases because formal wage costs decline 32 on account of a lower average tax burden. This encourages ﬁrms to hire formal rather than informal labor. Tax cuts in the higher brackets are most eﬀective in combatting the black economy because these tax cuts reduce marginal tax rates most. The lower marginal tax rates in the upper tax brackets raise the marginal return on training activities. Accordingly, human capital and labor productivity increase and the expansion of production exceeds the rise in employment. Unemployment The income tax cuts reduce equilibrium unemployment for two main reasons.58 The ﬁrst is the drop in the average tax burden, which moderates contractual wages. The lower marginal tax wedge produces upward wage pressure, but the positive elas-ticity of the average tax burden in the contractual wage equation (of 0.6) substantially exceeds the absolute value of the negative elasticity of the marginal tax burden (of 0.1). Hence, the overall eﬀect of the tax cut is to moderate wages, thereby reduc-ing equilibrium unemployment. Cutting taxes in the ﬁrst bracket is most eﬀective in reducing unemployment through this channel because it combines the decline in the average tax burden (the magnitude of which is similar for tax cuts in each of the three brackets) with the smallest decline in the marginal tax rate. The second factor explaining the decline in unemployment is the lower replace-ment rate; workers tend to beneﬁt more from lower marginal rates of personal income tax than transfer recipients do because the incomes of workers tend to exceed those of transfer recipients. This is especially so for tax reductions in the second bracket of the income tax.59 Subtracting the results of the second column from those of the ﬁrst column, one ﬁnds the impact of increasing tax progression by using revenues from a higher tax rate in the second tax bracket to cut the ﬁrst tax bracket. Increasing progression in this way leaves the unemployment rate more or less unaﬀected. The replacement rate eﬀect thus oﬀsets the direct wage moderation eﬀect associated with a higher marginal tax rate.60 Employment The three tax cuts raise aggregate employment through the channels of both lower unemployment and higher labor supply. In fact, all tax cuts generate a similar increase in aggregate employment. However, the composition of the employment gains diﬀers. A tax cut in the ﬁrst bracket is most eﬀective in raising employment for the unskilled, low skilled and partners. The other tax cuts are somewhat more eﬀective in boosting aggregate labor supply (in hours) and high-skilled employment and in combatting the black economy. raising the tax allowance We now turn to the eﬀects of raising the general tax allowance (see the fourth column in Table 5). Partners who do not earn suﬃcient labor income to fully use the tax allowance can transfer the allowance to the breadwinner. The tax credit is thus in fact refundable for households with non-participating partners. Hence, this tax credit reduces the average tax burden but leaves the marginal tax 58According to these simulations, a cut of one percentage point in the average tax burden reduces the unemployment rate by about 0.3 percentage point. This is somewhat higher than the estimates of Nickell and Layard (1999) and similar to the estimates of Daveri and Tabellini (2000). 59The tax rate in the third bracket exerts a smaller eﬀect on the replacement rate because this income range is less relevant for unemployed persons. 60Sub-section 3.2.3 identiﬁes these two eﬀects in an analytical right-to-manage model in which unemployment beneﬁts are subject to the progressive labor income tax. 33 burden unaﬀected, even for partners with small part-time jobs.61 The tax allowance applies to both transfer recipients and workers. Formal labor supply falls because the tax credit exerts only income eﬀects on labor supply. Unemployment declines despite an increase in the average replacement rate. The unemployed beneﬁt relatively more from a tax credit than those in work because the unemployed typically collect lower incomes than the employed. The main reason for lower equilibrium unemployment is that the lower average tax burden to-gether with the constant marginal tax burden moderates contractual wages. To summarize, a lower average tax rate at a constant marginal tax rate reduces both labor supply and unemployment. On balance, aggregate employment expands somewhat. The main diﬀerence with the cuts in tax brackets is thus that labor supply falls. 5.2.2 payroll taxes This sub-section explores two alternative ways to reduce social security contributions (SSC) that are imposed on employers, namely an across-the-board reduction in the rate of SSC and a targeted reduction of SSC for unskilled workers (see the last two columns of Table 5). Across-the-board reductions of employers’ SSC The ﬁfth column of Table 5 shows the eﬀects of an across-the-board cut in payroll tax paid by employers. Cuts in the rate of SSC reduce the average tax rate more than the marginal tax rate, thereby raising the coeﬃcient of progression. This is because the contributions are paid only on labor incomes up to 36,000 euro. Indeed, the impact of the cut in the SSC rate on the marginal tax rate and hence on the labor market is quite similar to a weighted average of a reduction in the tax rate in the ﬁrst bracket of the personal income tax and an increase in the general tax allowance (which are explored in sub-section 5.2.1). The lower SSC burden directly reduces labor costs. Accordingly, employment for all types of labor expands, while unemployment falls. Workers succeed in collecting part of the SSC cut in the form of higher net wages. In particular, employees increase their wage claims in contractual wage formation as the higher proﬁt margin raises the rents that are bargained over. Moreover, incidental wages rise as ﬁrms try to attract more applicants to ﬁll the increasing number of vacancies. Also recipients of social security beneﬁts gain because of the institutional link between beneﬁts and contractual wages. Higher wages mildly stimulate labor supply because substitution eﬀects dominate the income eﬀects. Targeted SSC cut In order to enhance the employability of low productivity work-ers, the SSC cut can be targeted at unskilled labor. We investigate a targeted SSC cut for low-skilled labor, which amounts to 1500 Euro for full-time workers earning the statutory minimum wage. It is phased out between hourly wages of 100% and 130% of the statutory minimum wage.62 The phasing out of the cut raises the marginal tax rates on higher hourly wages in this range. However, it does not raise the marginal tax rate on hours worked because the SSC cut is based on hourly wages and hence increases proportionally for workers who work longer hours. 61For students with low annual incomes, however, the tax allowance reduces the marginal tax rate. This explains the minor decline in the average marginal tax rate in the next-to-last row of the fourth column of Table 6. 62The Dutch government has introduced a reduction in employer’s SSC that is structured similarly: the so-called SPAK (SPeciale AfdrachtsKorting). 34 A comparison between the ﬁfth and sixth columns of Table 5 reveals that a targeted SSC cut is more eﬀective in raising employment than an across-the-board SSC cut is, especially as far as unskilled employment is concerned. The cut in SSC for unskilled workers boosts the demand for these workers through substitution towards unskilled labor. Moreover, lower labor costs at the minimum wage level facilitate job matching. In particular, the lower wage costs reduce the minimum productivity stan-dards due to minimum wage scales. Accordingly, an increasing number of unskilled unemployed, which often feature rather low productivities, meet the minimum produc-tivity standards of employers. Indeed, as described in sub-section 3.1.2, the minimum wage can be viewed as a tax on employers. With a cut in SSC paid by employers, the overall tax rate on labor demand (consisting of the implicit tax rate implied by minimum wages and explicit taxes) is reduced. The targeted SSC cut suﬀers from a number of drawbacks. First, gradual re-ductions in the tax allowance cause the marginal tax rate on increases in hourly wages to rise. Accordingly, increasing the net hourly wage is rather expensive because it substantially raises SSC. The high marginal tax burden on higher hourly wages harms the incentives for training by unskilled employees. The productivity level of workers therefore drops. Indeed, the last column of Table 5 reveals that production rises less than employment, which reﬂects the loss in human capital of the low skilled. Another disadvantage of a high marginal tax burden for employers is that it stimulates sub-stitution between formal labor and informal labor. In particular, a high marginal tax burden encourages ﬁrms to pay additional wage income above the formal minimum wage in an informal fashion. fixed annual hourly 80 hourly 50 hourly 30 Private consumption 0,56 0,30 0,51 0,51 0,51 Investment 0,47 0,12 0,46 0,48 0,45 Export 0,52 0,17 0,50 0,51 0,48 Imports 0,29 0,09 0,28 0,29 0,28 Formal production 0,55 0,13 0,53 0,55 0,52 Black production 0,09 -0,17 0,55 0,64 0,67 Employment 0,61 0,17 0,67 0,74 0,71 - unskilled 1,00 1,50 2,89 3,81 4,07 - low-skilled 0,72 0,14 0,47 0,28 0,08 - high-skilled 0,51 -0,03 0,39 0,42 0,39 Labour supply (pers.) 0,11 0,58 0,11 0,13 0,16 Labour supply (hours) 0,19 -0,22 0,12 0,14 0,13 - breadwinners 0,01 -0,33 -0,13 -0,12 -0,17 - partners 0,44 0,54 0,42 0,50 0,59 - single persons 0,28 -0,55 0,26 0,30 0,31 - 55+ 0,10 -0,29 0,06 0,05 0,03 Black labour (hours) 0,09 -0,19 0,55 0,69 0,78 Training -0,08 -0,49 -0,94 -1,29 -1,42 - unskilled and low-skilled -0,14 -0,14 -0,26 -0,37 -0,28 - high-skilled -0,03 -0,07 -0,17 -0,24 -0,20 Unemployment rate -0,26 -0,28 -0,36 -0,39 -0,37 - unskilled -0,40 -0,16 -0,23 -0,22 -0,32 - low-skilled -0,37 -0,47 -0,62 -0,70 -0,66 - high-skilled -0,21 -0,25 -0,32 -0,35 -0,33 Share long term unemployment -1,64 -1,75 -2,30 -2,54 -2,45 Replacement rate (a) -0,42 -0,28 -0,60 -0,78 -0,74 Average burden (a) -0,50 -0,43 -0,51 -0,52 -0,51 Marginal burden (a) -0,14 0,63 0,80 0,67 0,38 Government consumption (b) -0,12 -0,16 -0,09 -0,08 -0,08 Source: Graafland et. al. (2001) (a) Average over all households (b) In percentage of GDP Tabel 6. Macroeconomic effects of an earned income tax credit percentage deviations absolute deviations 35 5.2.3 earned income tax credit Table 6 contains the long-term eﬀects of introducing various forms of a tax credit that applies only to workers — the so-called Earned Income Tax Credit (EITC). In several EU countries, this instrument is increasingly perceived as an attractive instrument to combat unemployment by raising the return to low-skilled work. This policy in eﬀect directly reduces the net replacement rate, as the unemployed do not beneﬁt from the EITC. Hence, an EITC corresponds to the case in which unemployment beneﬁts are not subject to tax (see sub-section 3.2.3).63 Flat EITC The ﬁrst column of Table 6 contains the impact of a ﬂat EITC of 140 euro per year (corresponding to about 0.5% of the median gross wage). This non-refundable EITC reduces the marginal tax rate on small part-time jobs so that partners ﬁnd it more attractive to enter the labor force. Accordingly, the participation rate (i.e. labor supply in persons) increases. Unemployment declines substantially. The reason is that the EITC accrues only to those in work and hence reduces the replacement rate. The lower replacement rate enhances job matching by reducing the reservation wage and by encouraging the unemployed to search more intensively for a job. Moreover, it moderates contractual wages. This wage moderation reduces the current incomes from transfer recipients because social beneﬁts are linked to gross wages. Targeted EITC based on annual labor incomes The second column of Table 6 explores the impact of an EITC that focuses on raising the reward to low-skilled work. The EITC analyzed here depends on annual labor income of an individual.64 It amounts to 3 % of annual labor income of the individual in a phase-in range up to the statutory minimum wage (13,500 euro) and stays at 340 euro in a ﬂat range up to 115 % of the minimum wage. Subsequently, the EITC is phased out linearly up to 180 % of the minimum wage. The EITC reduces the marginal tax burden on small part-time jobs, thereby encouraging partners to join the labor force. Accordingly, the participation rate in-creases. However, aggregate labor supply measured in hours drops. Only partners raise their average labor supply (in hours) because many partners fall in the phase-in range of the EITC. Breadwinners and single persons, in contrast, reduce their labor supply because of a positive income eﬀect and, to the extent that they fall in the phase-out range, a negative substitution eﬀect associated with a higher marginal tax rate. On balance, for labor supply in hours, the reduction in labor supply on account of the substitution eﬀect in the phase-out range and the income eﬀect dominates the positive eﬀect on the participation rate. The high marginal tax rate in the phase-out range harms the incentives for training. This reduces the transition rates into higher skill levels. Hence, unskilled labor supply rises at the expense of low-skilled labor supply. The changing composition of labor supply aﬀects the distribution of employment and unemployment over skill levels. Whereas the training eﬀect mitigates the decline in unskilled unemployment, it raises unskilled employment. Since unskilled workers face a higher replacement ratio than low skilled workers do, this tends to contain the decline in the average replacement ratio, thereby moderating the employment gains. 63In case of a cut in SSC, in contrast, not only workers enjoy higher net wages but also beneﬁt recipients gain because of the institutional link between wages and beneﬁts. 64Hence, this EITC diﬀers from the EITC implemented in the US, which depends on family income and the number of children in a family. 36 Targeted EITC based on hourly wages If the objective is to reduce the num-ber of unskilled who collect unemployment and welfare beneﬁts, the targeted EITC explored above suﬀers from the disadvantage that it accrues also to part-time workers with high hourly wages but low annual incomes. This is relevant especially in the Netherlands, which features the highest share of part-time work of all OECD coun-tries. Hence, in the Dutch policy discussion, a targeted EITC has been proposed that depends on hourly wages rather than annual incomes. Workers who earn the hourly minimum wage and hold a full-time job are eligible for the full EITC. Just as the targeted SSC cut considered in sub-section 5.2.2, the credit is reduced proportionally for workers who work less than a full-time job. It gradually drops also with the level of the hourly wage rate. By reducing the credit for part-time workers with high wages, the EITC for full-time workers who earn an hourly wage up to 115 % of the statutory minimum wage can almost be doubled to 625 euro. The phase-out range runs up to an hourly wage of 180% of the minimum wage. This EITC reduces the marginal tax burden only on part-time jobs with low hourly wages. Hence, the eﬀect on the participation rate is smaller than in the previous experiment. The higher marginal tax rate in the phase-out range applies only to higher hourly wages and not to higher labor incomes on account of more hours worked. In fact, additional hours worked raise the credit for unskilled workers. This explains why, in contrast to the case with an EITC based on annual labor income, labor supply (in hours) increases slightly. The marginal tax rate on higher hourly wages in the phase-out range is higher than in the previous experiment because the maximum credit is about twice as large. This harms the incentives to accumulate human capital. Hence, compared to an EITC that depends on annual incomes, an EITC that depends on hourly wages does less harm to the quantity of labor supply but does more harm to the quality of labor supply. Another drawback of this variant of the EITC is that it relies on additional information (namely the number of hours worked in the formal sector) that is vulnerable to fraud. Indeed, the black economy expands substantially. Unemployment This EITC reduces the replacement rate for low-skilled workers more substan-tially than the other EITCs explored above. Through skill-speciﬁc wage formation, this decline in the replacement rate reduces low-skilled wages, thereby boosting demand for low-skilled labor. Moreover, the lower replacement rate stimulates search and lowers the reservation wage, thereby facilitating the matching process for low-skilled labor. Accordingly, the unemployment rate for the low skilled drops more substantially than under the EITCs analyzed above. Trade-oﬀs The comparison between an EITC that depends on annual incomes and an EITC that depends on hourly wages reveals a trade-oﬀbetween two objectives: increasing the participation rate of partners and reducing the unemployment rate for the low skilled. An EITC that depends on annual incomes advances the ﬁrst objective, while an EITC that depends on hourly wages is more eﬀective in cutting low-skilled unemployment. This trade-oﬀis similar to that uncovered in studies for the U.S. and the U.K. (see, e.g., Blundell e.a. (2000)). In these countries, the EITC depends on household rather than individual incomes. The advantage is that the tax incentives can be better targeted at low-income households who often face high replacement rates, thereby stimulating employment of primary wage earners and single mothers. The disadvantage, however, is that income eﬀects and higher marginal tax rates in the phase-out range harm labor-37 supply incentives facing secondary earners with working partners. This illustrates how reducing one obstacle to employment may increase another obstacle. Another trade-oﬀinvolves the quality versus the quantity of labor supply. Com-pared to an EITC that depends on annual incomes, an EITC that depends on hourly wages enhances the quantity of labor supply (in hours) but harms its quality (in terms of human capital). Targeting the EITC The last two columns of Table 6 show the eﬀects of two EITC’s (based on hourly wages) that are phased out more rapidly than the previous experiment, namely, at 150% (the fourth column) and 130% (the ﬁfth column) of the minimum wage. Fewer people fall in the phase-out range, but those who remain in the phase out range face even higher marginal tax rates. The advantage of more targeting is that the maximum credit for people who earn the minimum wage rate can be larger, thereby cutting the replacement rate of the unskilled more substantially. The disadvantage is that the marginal tax rate in the phase-out range increases more sharply and the (larger) decline in the replacement rate applies to fewer persons. A moderately targeted version of the EITC (in the fourth column of Table 6) is slightly more eﬀective in reducing the aggregate unemployment rate than the most targeted EITC (in the ﬁfth column of Table 6). Also, compared to the less targeted EITC (in the third column of Table 6), the moderately targeted EITC is more eﬀective in reducing the aggregate unemployment rate. This suggests that an inverse U-shaped curve describes how the eﬀectiveness of the EITC in cutting unemployment varies with the degree of targeting. Hence, moderately targeting the EITC seems the most eﬀective way to reduce the overall unemployment rate. At the same time, these simulations illustrate the drawbacks of targeting: more targeting implies that more workers remain unskilled. Indeed, the adverse shift in the skill composition boosts unskilled employment at the expense of low-skilled employment and limits the decline in unskilled unemployment. 5.2.4 targeted SSC cut versus targeted EITC A comparison between the targeted cut in SSC paid by employers (see the last column in Table 5) with a similar targeted EITC (see the last column in Table 6) reveals that the SSC cut is more eﬀective in ﬁghting unemployment among the unskilled but less eﬀective in reducing aggregate unemployment. The SSC cut enhances the eﬃciency of the matching process primarily through lower minimum wage costs. This substantially reduces unskilled unemployment because the minimum productivity standard is the most restrictive factor in the matching process for the unskilled. Indeed, with rigid wages, cuts in payroll taxes boost employment more than cuts in income taxes do (see sub-section 3.1) The EITC improves the matching process primarily through a lower replace-ment rate reducing the reservation rate of the unemployed. A lower reservation wage is less important for the matching process of the unskilled than a lower minimum productivity standard. However, a lower replacement rate also moderates wages in collective bargaining. This makes the targeted EITC more eﬀective in reducing aggre-gate unemployment. The substantial decline in the replacement rate produced by the EITC is as-sociated with a decline in the current incomes of transfer recipients. In case of a targeted SSC, in contrast, beneﬁt recipients are better oﬀbecause wages (to which beneﬁts are linked) rise rather than fall. Tables 7 and 8 illustrate these income ef-fects. These tables use MIMIC’s search model to compute the welfare eﬀects of the 38 various policies considered here. The temporal welfare gains correspond to the static welfare eﬀects (i.e. abstracting from changes in employment prospects). For the un-employed, the measure corresponds to after-tax income in unemployment (analogous to B in sub-section 3.2). The intertemporal welfare eﬀects incorporate changes in fu-ture employment prospects (for the unemployed analogous to U r in sub-section 3.2). For the unemployed, the contrast between the temporal and intertemporal measures can be striking. In particular, whereas the EITC produces temporal losses for the un-employed, the intertemporal gains for the unemployed are large because employment prospects improve substantially as a result of lower unemployment rates. The scope for Pareto-improving policies increases substantially if one considers the impact on life-time rather than static incomes.65 65Note, however, that even an EITC is not without costs, as public spending must be cut. First income Second income Third income Basic income General cut in Targeted cut in tax bracket tax bracket tax bracket tax allowance payroll tax payroll tax Temporal welfare Unemployed (long-term) - unskilled 0,3 0,1 0,2 1,0 0,4 1,3 - low-skilled 0,4 0,2 0,2 0,9 0,4 0,2 - high-skilled 0,4 0,3 0,1 0,8 0,4 0,1 Employed - unskilled 0,5 0,3 0,2 0,6 0,5 2,6 - low-skilled 0,5 0,5 0,3 0,5 0,5 0,4 - high-skilled 0,5 1,1 0,6 0,4 0,4 0,2 Intertemporal welfare Unemployed (long-term) - unskilled 0,6 0,7 0,5 0,9 0,7 3,8 - low-skilled 0,8 0,6 0,4 0,9 0,8 0,7 - high-skilled 0,9 1,1 0,6 0,8 0,8 0,4 Employed - unskilled 0,5 0,4 0,3 0,7 0,5 3,0 - low-skilled 0,6 0,6 0,3 0,5 0,5 0,5 - high-skilled 0,6 1,1 0,6 0,5 0,5 0,2 Source: Graafland et. al. (2001) fixed annual hourly 80 hourly 50 hourly 30 Temporal welfare Unemployed (long-term) - unskilled -0,3 -0,8 -1,2 -1,6 -1,8 - low-skilled -0,2 -0,2 -0,1 - high-skilled -0,1 -0,1 -0,1 -0,1 -0,1 Employed - unskilled 0,9 1,6 3,2 4,2 4,2 - low-skilled 0,7 0,2 0,5 0,6 0,3 - high-skilled 0,6 0,2 0,4 0,4 0,4 Intertemporal welfare Unemployed (long-term) - unskilled 0,9 0,8 1,5 1,9 2,0 - low-skilled 1,1 1,0 1,5 1,7 1,5 - high-skilled 1,1 1,0 1,4 1,5 1,4 Employed - unskilled 0,9 1,4 2,8 3,7 3,8 - low-skilled 0,8 0,4 0,7 0,8 0,6 - high-skilled 0,7 0,4 0,6 0,6 0,5 Source: Graafland et. al. (2001) Table 7. Welfare effects of cuts in income and payroll taxes percentages of base path current income Table 8. Welfare effects of an earned income tax credit percentages of base path current income 39 6 Optimal redistribution We now turn to a model of risk-averse agents with heterogeneous abilities. Accordingly, progressive taxes not only aﬀect the eﬃciency of the labor market but also insure agents against the risk of being born with heterogenous abilities. Moreover, in addition to an extensive labor-supply margin, we allow taxes to impact labor supply on the intensive margin. Hence, whereas progressive taxes may stimulate job search, they may harm the incentives of workers to exert eﬀort and work long hours after workers have found a job. Compared to the analysis in section 4, we thus include both an additional beneﬁt of progressive taxation (namely income redistribution) and an additional cost (namely lower labor supply on the intensive margin). As another extension, welfare beneﬁts may be set optimally. Moreover, the income tax does not have to be linear, but may be non linear as the government can observe individual labor incomes. Furthermore, the government can imperfectly monitor the search eﬀort of agents. This allows us to investigate how the monitoring technology aﬀects the optimal welfare beneﬁt and the optimal tax system. In order to incorporate the additional complications of heterogenous, risk-averse agents with endogenous work eﬀort, we simplify the model of section 4 in two ways. First, we abstract from wage bargaining and search (and the associated externalities) at the demand side of the labor market: workers are paid their marginal product.66 Second, we simplify the formulation of search at the supply side so that the search mar-gin is relevant for low-skilled workers only. Our formulation of labor-market matching allows for two types of unemployment: ﬁrst, involuntary unemployment of high-skilled agents and, second, voluntary unemployment of low-skilled agents who do not face suﬃcient incentives to search. As regards this last type of unemployment, optimal un-employment beneﬁts in eﬀect set a wage ﬂoor below which agents no longer search for work. Hence, the desire to protect the involuntary unemployed produces an optimal rate of voluntary unemployment. 6.1 The model The economy is populated by agents featuring homogeneous preferences but hetero-geneous skills. A worker of ability (or skill or eﬃciency level) n working E hours (or providing E units of work eﬀort) supplies nE eﬃciency units of homogeneous labor. With constant unitary labor productivity, these eﬃciency units are transformed in the same number of units of output. With output as the numeraire, the before-tax wage per hour is thus given by exogenous skill n. Hence, overall gross output produced by a worker of skill n, Z(n), amounts to Z(n) = nE(n). Since workers collect only labor income, this gross output Z(n) corresponds to total gross (i.e. before-tax) income collected by a worker of that skill n. The density of agents of ability n is denoted by f(n), and F(n) represents the corresponding cumulative distribution function. The support of the distribution of abilities is given by [n0, n1], while f (.) is diﬀerentiable. Workers share the following quasi-linear utility function over consumption C and hours worked (or work eﬀort) E u(C, E) = v (C) −E, where v(C) is increasing and strictly concave: v′ (C) > 0, v′′ (C) < 0 for all C ≥0. The speciﬁc cardinalization of the utility function aﬀects the distributional preferences of a utilitarian government. In particular, the concavity of v(.) implies that a utilitarian 66In terms of the model of section 4, β = 1 and η = 0 so that the Hosios condition is met. 40 government aims to ﬁght poverty. In other words, such a government wants to insure agents against the risk of a low consumption level. As in Lollivier and Rochet (1983), Weymark (1987), Ebert (1992), and Boadway, Cuﬀand Marchand (2000), utility is linear in work eﬀort E and separable in work eﬀort and consumption C. This has three important consequences. First, consumption C is not aﬀected by income eﬀects. A higher average tax rate thus induces households to raise work eﬀort E rather than to cut consumption C. Second, the speciﬁc quasi-linear utility function allows for a closed-form solution of the standard optimal income tax problem. Third, a utilitarian government cares only about aggregate work eﬀort in the economy. Such a government thus aims at an equal distribution of consumption (i.e. the alleviation of poverty) rather than an equal distribution of work eﬀort over the various agents. In line with the optimal income tax literature, the government is assumed not to be able to observe skills n but to know the distribution function f(n) and before-tax income of each individual Z(n). We depart from the standard optimal tax literature by incorporating job search: agents have to search for a job and the government can only imperfectly monitor agents’ search eﬀort (see below). In particular, we allow agents to adjust their labor supply not only on the intensive margin (i.e. by varying hours of work) but also on the extensive margin (i.e. by deciding whether or not to look for a job). In particular, by searching with intensity X ∈[0, 1], agents ﬁnd a job with probability X. Agents’ search costs γ(X) are given by γ(X) = γX if X ∈[0, ¯ X] +∞ otherwise, where γ ≥0 is a parameter representing the magnitude of the search costs. ¯ X < 1 captures the idea that agents may fail to ﬁnd a job, even if they search at full capacity. By modelling the costs and eﬀectiveness of search, the parameters γ and (1−¯ X) represent labor-market imperfections that give rise to unemployment. Agents thus diﬀer in both ability n and employment status and face two types of risks: being born with low ability n and being involuntarily unemployed. If an agent does not succeed in ﬁnding a job, (s)he receives a welfare (or social assistance) beneﬁt B ≥0.67 An agent who does not search for a job, while (s)he is expected to look for a job by the government has a probability pc ∈⟨0, 1⟩of receiving a penalty π ≥0. This penalty is in the form of lost leisure time. An agent of ability n who is expected to search by the government searches at full capacity if and only if −γ ¯ X + ¯ XU (n) + 1 −¯ X v (B) ≥v (B) −pcπ (20) The linear speciﬁcation of the search cost function implies that a worker either does not search at all (and is voluntarily unemployed) or searches at the level ¯ X (and faces a probability of (1 −¯ X) of involuntary unemployment). After a worker has found a job, (s)he has to determine her work eﬀort. Ex-post utility of a type n agent who ﬁnds a job is determined by type n’s choice of gross income Z: U (n) = max Z v Z −˜ T (Z) −Z n , (21) 67An alternative interpretation of B is a categorial unemployment insurance beneﬁt. Indeed, the beneﬁt is paid only to those who have not found a job. In most countries, however, unemployment beneﬁts depend on the previously earned wage and are thus likely to increase with ability n. This is the main reason why we interpret B as a social assistance beneﬁt, i.e. the minimum income level provided by the government. Another interpretation of B is an early retirement or disability beneﬁt that is paid if an agent does not work. 41 where ˜ T (Z) denotes the tax schedule as a function of gross income Z. We can write ˜ T (n) = T (Z (n)) , since type n chooses gross income Z (n) in equilibrium. The enve-lope theorem yields the ﬁrst-order incentive compatibility constraint68 U ′ (n) = Z (n) n2 . (22) The utilitarian government maximizes ex-ante expected utility (i.e. expected utility before ability and labor market status have been revealed) Ω≡ n1 n0 −γX (n) + X(n)[U(n) + ξ] + (1 −X (n)) [v (B) −κe(n)] f(n)dn, where κe(n) represents the expected penalty for type n with dκe(n)/dn ≥0. We allow for positive employment externalities ξ > 0. If these externalities are positive, the government attaches more value to work than individual agents. The government faces the following budget constraint n1 n0 f (n) X (n) [B + T (n)]dn = G + B, (23) where G represents exogenously given exhaustive government expenditure, and T (n) ≡ Z (n) −C (n) denotes the tax paid by type n. The government employs the non-linear income tax and welfare beneﬁts to optimize social welfare and takes public spending G and the search monitoring and penalty system as given. 6.2 The optimal tax problem In optimizing social welfare, the government faces three constraints: the incentive compatibility constraint (22), the participation constraint (20), and the government budget constraint (23). Since Z (n) ≥0, incentive compatibility (22) implies that utilities do not decline with skill (i.e. U ′ (n) ≥0). Accordingly, if the participation constraint U (n) ≥γ + v (B) −κe(n) is met for skill ¯ n, it is met also for higher skills n > ¯ n. Deﬁning nw as the lowest skill that looks for work, we thus have X (n) = 0 for n < nw and X (n) = ¯ X for n ≥nw. The agents with skill n < nw can be viewed as being voluntarily unemployed. The higher skills n > nw look for work but may be involuntarily unemployed (if ¯ X < 1). The productivity level nw is called the minimum productivity level. It is in fact the minimum gross wage implied by the welfare and tax systems. These observations allow us to formulate the social planner’s problem as69 max nw,U(.),Z(.), B F (nw) v (B) + [1 −F (nw)] −γX ¯ X + 1 −¯ X v (B) + n1 nw ¯ XU(n)f(n) −λU(n) U ′(n) −Z(n) n2 + λE f(n) ¯ XT(n) dn −λE B F (nw) + (1 −F (nw)) 1 −¯ X + G −ηw γ −U(nw) + v(B) −pc ¯ X π , (24) 68The second-order condition for the agents’ optimal choice of consumption and gross income implies that consumption and gross income are non-decreasing in type n. Boone and Bovenberg (2003a) analyze these constraints (and the associated bunching implications) in depth and argue that they are not relevant for understanding optimal taxation and welfare beneﬁts at the bottom of the labor market. We therefore ignore these constraints here and refer the interested reader to Boone and Bovenberg (2003a). 69Instead of C (n) , we employ U (n) as a control variable in order to facilitate the inclusion of ﬁrst-order incentive compatibility (22) into the optimization problem. 42 where T (n) ≡Z (n) −C (n) = Z (n) −v−1 U (n) + Z(n) n . λU (n) represents the Lagrange multiplier of the incentive compatibility constraint, and λE stands for the multiplier of the government budget constraint. ηw denotes the Lagrange multiplier on the participation constraint for type nw. It measures the social value of increasing employment by forcing more people to search, and can therefore be interpreted as the value of a work test (and the required information on search intensity) inducing more skills to look for work. Boone and Bovenberg (2003b) derive the ﬁrst-order conditions for the optimal tax problem and establish the following proposition. Proposition 1 If γ ¯ X > pcπ, employed agents of type n > n0 face positive marginal tax rates. If in addition ¯ X < 1 −n0f (n0) , there is voluntary unemployment (i.e. nw > n0), marginal taxes are positive at the bottom (i.e. τ(nw) > 0), and the following relationship holds at the minimum productivity level70 τ(nw)Z(nw) = (B + T (nw)) + ξ −π pc ¯ X λE . (25) The inequality γ ¯ X > pcπ implies that search costs are so high that agents can be induced to search only if they can expect higher consumption levels in work than in unemployment. Since the unemployed enjoy less consumption than workers, the government wants to redistribute resources away from the employed skills n > nw to the unemployed skills n < nw. This desire to redistribute towards the unemployed results in positive marginal tax rates for all workers, including the marginal workers with skill nw. The left-hand side of inequality 1 −¯ X > n0f (n0) stands for involuntary un-employment among the skills that are actively searching for a job. Hence, if these labor-market imperfections as measured by this involuntary unemployment 1 −¯ X are substantial, voluntary unemployment (i.e. nw > n0 so that the least skilled do not look for a job) becomes optimal. Intuitively, to avoid poverty among the substantial numbers of involuntarily unemployed, the welfare level B is set at such high levels that the participation constraint becomes binding and the least skilled workers no longer search for work, especially if these workers feature only low labor productivity (i.e. n0 is small). The desire to combat poverty among the low skilled and the involun-tarily unemployed agents without imposing excessive distortions on the work eﬀort of high-skilled agents thus optimally creates additional, voluntary unemployment. To interpret expression (25), we ﬁrst consider the case without employment externalities and penalties (i.e. ξ = π = 0). In that case, the right-hand side of (25) represents the direct budgetary implications of raising employment by reducing nw: by bringing a marginal worker into work, the government saves a welfare beneﬁt B and collects additional tax revenue T(nw). The indirect implications, namely the eﬀects on other workers, are captured by the left-hand side of (25). Bringing a marginal type nw into work encourages workers who are marginally more skilled to work less hard — as they can now mimic type nw. An optimal tax system balances the welfare implications of this latter behavioral response on the intensive margin of the more productive workers (represented by the left-hand side of (25)) with the budgetary implications of the behavioral response on the extensive margin of the marginal workers. The government thus faces a trade-oﬀbetween obtaining revenues from either inducing more agents to search or encouraging a smaller group of agents to work harder. As a result, the distortion on the extensive margin (i.e. the right-hand side of (25)) should equal the distortion on the intensive margin (i.e. the left-hand side of (25)). 70This expression holds also if the welfare beneﬁt is ﬁxed exogenously. 43 In this particular case (i.e. ξ = π = 0), the progressiveness of the income tax is directly related to the level of welfare beneﬁts. To see this, we rewrite (25) as τ(nw) −T(nw) Z(nw) = B Z(nw) = B C(nw) 1 −T (nw) Z(nw) . For the least-skilled worker, the marginal tax rate τ(nw) minus the average tax rate T (nw) Z(nw) is directly related to the replacement rate B C(nw). In particular, the tax system is progressive at the minimum productivity level nw if and only if the welfare beneﬁt is positive. This result resembles the corresponding result on optimal progression in section 4. In particular, if the Hosios condition holds, this latter section establishes also that the income tax is progressive if and only if the welfare beneﬁt is positive. An important diﬀerence is, however, that in section 4 the search margin is not distorted in the optimum (i.e. τ a + B = 0 so that at a wage level of B, workers collect a tax subsidy of −τ a = B). In the presence of an extensive labor-supply margin, in contrast, the distortions on the search margin have to be traded oﬀagainst the distortions in work eﬀort. Hence, the search margin remains distorted in equilibrium as the tax on search B + T (nw) is positive. We now turn to the case with positive employment externalities ξ > 0. In that case, ceteris paribus the gross replacement rate B Z(nw), 71 the gap between the marginal tax and average tax rates widens at the minimum productivity level. In-tuitively, with positive employment externalities, the government wants to subsidize search of unskilled workers in order to have these workers internalize the positive ex-ternalities from search. These subsidies are ﬁnanced by higher skilled agents so that marginal tax rates increase and the tax system becomes more progressive. An alter-native way to understand why employment externalities tend to make the tax system more progressive and therefore reduce work eﬀort of high-skilled workers is that posi-tive employment externalities can be viewed as implicit taxes on search. With a larger overall tax on search, the optimal trade-oﬀbetween distortions on the intensive and extensive margins demands that the explicit tax on search (i.e. the extensive margin) is reduced and that the tax on eﬀort (i.e. the intensive margin) is raised. If the em-ployment externalities are large enough and the revenue requirements are only small (so that the marginal tax rate τ(nw) can be small), search may even be subsidized in equilibrium (i.e. B + T(nw) < 0). With penalties on inadequate search (i.e. πpc > 0), the government reduces the search distortions originating in the welfare system and can rely less on explicit search subsidies for low-skilled labor. With less need to subsidize the low skilled, the income tax system has to be less progressive. Hence, compared to positive employment externalities, the penalty system exerts exactly the opposite eﬀect on marginal tax rates facing workers. Indeed, whereas positive employment externalities can be viewed as an implicit tax on search, the penalty system works as an implicit subsidy on search. If the penalties are strong enough to internalize the employment externalities (i.e. π pc ¯ X > ξ), the penalty eﬀect dominates. Hence, (25) together with τ(nw) > 0 (see Proposition 1) implies that search is necessarily taxed (i.e. B + T(nw) > 0). Intuitively, the net tax on search helps to redistribute resources away from workers to the unemployed, who are poorer than the workers. The monitoring system allows the government to alleviate the distortions im-posed by the welfare system on the search margin.72 It thus alleviates the distortions 71Equation (25) holds also with an exogenous beneﬁt level. If B is not optimally set, however, τ(nw) may be negative (see Boone and Bovenberg (2003a)). With positive search externalities, the optimal welfare beneﬁt B will typically be lower than without positive search externalities. Indeed, the search externalities will be internalized by reducing both B and T(nw). Indeed, redistribution away from workers towards the poorer unemployed becomes more problematic. 72Whereas we thus model the beneﬁts of monitoring, we do not specify the costs of monitoring. We thus can not compute optimal monitoring levels. 44 from redistribution resources away from workers to poorer unemployed. In particular, the agents who are required to search collect B −π pc ¯ X /λE rather than B in unemploy-ment. As long as B −[π pc ¯ X /λE] > 0, the welfare system distorts search and the tax system is progressive at the minimum productivity level. 7 Conclusions The link between taxes and labor-market performance depends crucially on non-tax institutions. In particular, the impact of taxes on wages and unemployment depend on how wages are set and on welfare and unemployment beneﬁts. Indeed, a key channel through which taxes aﬀect unemployment is the eﬀective replacement rate. The highest eﬀective tax rates on work typically originate in welfare and unemployment beneﬁts that are withdrawn if work is found. Changes in the tax structure can cut unemployment if they succeed in shifting the tax burden unto the unemployed, thereby reducing the eﬀective replacement rate. Moreover, whether a higher tax burden raises unemployment depends crucially on whether the unemployed share in the higher tax burden or not. These two insights explain how revenue neutral environmental tax reforms can create a double dividend by producing not only a cleaner environment but also a lower level of unemployment. How unemployment beneﬁts are indexed is crucial in determining whether a change in the tax structure can aﬀect the eﬀective after-tax replacement rate and whether a higher tax burden is shared by the unemployed.73 In particular, an environmental tax reform can shift the tax burden onto the unemployed by taxes on dirty consumption replacing labor income taxes if unemployment beneﬁts are linked to producer prices and not subject to personal income tax. Alternatively, this can be accomplished by taxes on dirty inputs into production replacing consumption taxes if unemployment beneﬁts are linked to wages. In all these cases, environmental tax reform in eﬀect succeeds in cutting the eﬀective after-tax replacement rate.74 With revenue-neutral reforms, the employment impact also depends on the additional implicit tax burden associated with a better quality of the public good of the environment. A double dividend is feasible only if the beneﬁt recipients pay a more than proportional share of the larger supply of the environmental public good (see Bovenberg and van der Ploeg (1998)). The tax system may impact unemployment also through rigid market wages. If statutory minimum wages prevent market wages from falling, tax policy can then in eﬀect oﬀsets the implicit tax on employers imposed by workers by reducing payroll taxes paid by employers. The question also applies here why these non-tax institutions cannot be reformed directly but have to be changed indirectly through tax policy. Another channel through which the tax system impacts unemployment is the progressiveness of the tax system. In particular, by taxing wage rises, progressive taxes moderate wages, thereby reducing unemployment. However, progression may also increase the eﬀective net replacement rate if unemployment beneﬁts are subject to tax. 73We demonstrated that higher income or payroll tax rates can raise unemployment even though productivity growth does not aﬀect the unemployment rate. This may happen if the ’outside wage’ is indexed to labour productivity in the formal sector. This case seems particularly relevant if the outside option is employment in the untaxed, informal economy. 74This raises the question why the government cannot cut the replacement rate directly but has to rely on an environmental tax reform to do so. One reason may be that beneﬁt recipients reap the largest gains from the improvement in environmental quality. With environmental beneﬁts oﬀsetting the decline in after-tax unemployment beneﬁts, beneﬁt recipients may favor an environmental tax reform that cuts their after-tax incomes while they would not support a direct cut in the replacement rate (see Bovenberg (1999)). 45 This latter eﬀect may in fact be stronger than the ﬁrst one if gross replacement rates and non-taxable incomes in unemployment are substantial. Moreover, even though progressive taxes combat unemployment, they typically imply other costs, for example reducing labor supply, work eﬀort, human capital accumulation and labor mobility while stimulating tax avoidance, tax evasion, jobs with substantial nontaxable non-pecuniary beneﬁts, and the informal and black economies. Tax policy impacts the labor market not only through wage setting but also through labor supply. The labor-supply eﬀects of tax policy require microeconomic analysis of speciﬁc, disaggregated groups (such as secondary part-time workers, low-skilled agents and older employees close to retirement) in order to do justice to sub-stantial variation in eﬀective marginal tax rates and labor supply elasticities. The general equilibrium model MIMIC incorporates a disaggregated household model in a general equilibrium setting. In addition to labor supply and wage determination, various other ways through which taxes and beneﬁts aﬀect the labor market are in-corporated, namely the black economy, human capital accumulation, eﬃciency wages, costly job matching, and search behavior of the unemployed. The simulations with MIMIC reveal several trade-oﬀs between various objec-tives. These objectives include cutting unemployment in general and low-skilled un-employment in particular, stimulating the participation of women in the labor force, raising the quality and quantity of labor supply (both in hours and in persons), and establishing an equitable income distribution, including a reasonable income level for those dependent on social beneﬁts. Indeed, these objectives imply diﬀerent prior-ities for how tax cuts should be structured. In particular, cutting unemployment primarily requires widening the gap between labor incomes and transfer incomes in unemployment. Stimulating labor-force participation of women calls for widening the gap between, on the one hand, after-tax incomes of households with two partners who are active on the formal labor market and, on the other hand, after-tax incomes of households with a non-participating partner. Such a larger income gap encourages partners to start participating in the labor force so that the latter households turn into the former households. Raising the quantity and quality of labor supply in the formal economy calls for widening the income diﬀerentials between low formal labor incomes and high formal labor incomes. The most eﬀective way to ﬁght economy-wide unemployment is through in-work beneﬁts. These beneﬁts widen the gap between after-tax income from work and net transfer income, thereby raising the reward to work compared to relying on social beneﬁts. This moderates wage costs, reduces reservation wages and encourages job search. Wage moderation reduces social beneﬁts if these beneﬁts are linked to market wages. Targeting in-work beneﬁts at the low skilled is most eﬀective in cutting economy-wide unemployment. This is because the gap between labor income and transfer income is smallest for low-skilled workers. Hence, widening this small gap produces the largest pay-oﬀin terms of reducing unemployment. However, by decreasing the gap between low and high labor incomes through a more progressive tax system for workers, a targeted EITC reduces the hours of labor supplied. The cost of higher marginal tax rates in the phase-out range is particularly high in European countries, where marginal tax rates are already quite high. The trade-oﬀbetween cutting unemployment and raising labor supply (in hours) can be mitigated by linking the EITC to hourly wages rather than annual incomes and by reducing the EITC proportionally for small part-time jobs. Doing so, however, raises the marginal tax burden on hourly wage increases, thereby discouraging the accumulation of human capital and stimulating the black 46 economy. Moreover, whereas the tax cuts are better targeted at beneﬁt recipients, the lower beneﬁts to small part-time jobs do not help to raise the labor-force participation of women. This points to a trade-oﬀbetween targeting tax cuts at small part-time jobs of partners or at full-time jobs of breadwinners and singles earning low hourly wages. Tax cuts in the higher tax brackets are most eﬀective in raising the quantity and quality of formal labor supply (in hours). Indeed, these policies widen the after-tax income diﬀerentials between low and high labor incomes by reducing marginal tax rates. However, cuts in higher tax brackets are less eﬀective in reducing unemployment (by widening the income gap between being in work and collecting unemployment beneﬁts), raising low-skilled employment, and stimulating female labor supply. Indeed, the contrast between cuts in the highest tax brackets and a targeted EITC reveals a trade-oﬀbetween raising the quality and quantity of labor supply and combatting unemployment. We formalized the trade-oﬀbetween high levels of labor supply and low unem-ployment rates in a model of optimal taxation with involuntary unemployment. In a model with homogeneous households without an intensive margin of labor supply, a progressive labor tax eliminates non-tax distortions on wage setting. In particular, a progressive tax allows workers to commit not to expropriate speciﬁc investments of ﬁrms. This is especially relevant if unions feature a short time horizon and thus set wages on the basis of low short-run labor-demand elasticities rather than higher long-run labor-supply elasticities. In other words, progressive taxes restore the eﬃ-cient balance of power between workers and employers. A progressive tax also corrects for the impact of the welfare system on wage setting, thereby alleviating the adverse impact of the welfare beneﬁt on job creation and the unemployment rate. The beneﬁt system thus determines the optimal progressiveness of the labor tax: a higher welfare beneﬁt is accompanied by a higher in-work beneﬁt so that the improved outside op-tion of workers as a result of the higher welfare beneﬁt does not raise unemployment. The more progressive labor tax depresses labor supply, but it is more eﬃcient to re-duce labor supply through lower after-tax wages than through the discouraged-worker eﬀect. In the presence of an intensive labor-supply margin, the government faces a trade-oﬀbetween obtaining revenues from either inducing more agents to search or encouraging a smaller group of agents to work harder. In that case, therefore, the government does not completely eliminate the impact of the welfare beneﬁt on the unemployment rate. In particular, the government balances the distortions on job creation and the unemployment rate against those on hours worked and work eﬀort. The government can improve this trade oﬀin various ways. First of all, agents may insure themselves against the risk of involuntary unemployment through precau-tionary saving and compulsory saving schemes so that the unemployment insurance beneﬁts paid to the involuntarily unemployed can be cut.75 Self insurance seems a particularly attractive instrument for high-skilled agents who face relatively short un-employment spells during their careers. In this connection, the government may want to relieve liquidity constraints by oﬀering loans to the unemployed.76 This combats the 75Indeed, the desire to provide income to involuntarily unemployed agents creates distortions on the extensive margin in the model laid out in section 7. If the government does not need to provide income to the involuntarily unemployed, even the least skilled workers can be oﬀered suﬃcient incentives to look for jobs. 76Table 10 suggest that lowering unemployment beneﬁts may actually beneﬁt the unemployed if the unemployed do not face liquidity constraints. The reason is that lower unemployment beneﬁts enhance the probability of ﬁnding a job, thereby improving expected incomes in the future. 47 capital-market distortions that may give rise to labor-market distortions. Compulsory saving schemes with liquidity insurance in eﬀect provide a stronger link between con-tributions and insurance beneﬁts on a micro level (see Sørensen (2003)). This protects incentives to search for work, work hard and moderate wages. For agents with low life-time incomes, self insurance does not work well. To protect these agents against poverty, the government needs to transfer resources to these agents. For these agents, other ways need to be found to improve the trade-oﬀ between the extensive and the intensive margins. In particular, the government may collect more information by monitoring job search and imposing penalties on less active job search.77 In this connection, workfare may also play a role, even though it may to some extent crowd out private employment. In particular, the mere threat of being put on workfare is likely to boost job search of able individuals and prevent nonworkers who highly value leisure78 from claiming unemployment beneﬁts (see Fredriksson and Holmlund (2003)). Workfare can thus be seen as a way to redistribute resources to low-skilled agents who are involuntarily unemployed. 8 References Acemoglu, D., and R. Shimer, 1998, ’Eﬃcient Unemployment Insurance,’ Journal of Political Economy, Vol. 107, pp. 893-928. Baartmans, K., F. Meyer, and A. van Schaik, 1986, ’Houserepair and the Infor-mal Sector,’ mimeo, University of Delft, the Netherlands. Bernheim, B.D., and K. Bagwell, 1988, ’Is Everything Neutral?’ Journal of Political Economy, Vol. 96, pp. 308-338. Blanchard, O.J. and P.A. Diamond, 1989, ’The Beveridge curve’, Brookings Papers on Economic Activity 1, 1-60. Blanchard, O. and L.H. Summers, 1986, ’Hysteresis and the European unem-ployment problem,’ in S. Fischer (ed.), NBER Macroeconomics Annual, MIT Press, Cambridge, Mass. Blundell, R., A. Duncan, J. McCrae, and C. Mehir, 2000, ’The Labor Market Impact of the Working Families’ Tax Credit,’ Fiscal Studies, Vol. 21, pp. 75-104. Boadway, R., K. Cuﬀ, and M. Marchand, 2000, Optimal Income Taxation with Quasi-Linear Preferences Revisited, Journal of Public Economic Theory, Vol. 2, pp. 435-460. Boone, J., and A.L. Bovenberg, 2002, ’Optimal Taxation and Search,’ Journal of Public Economics, Vol. 85, No. 1, pp. 53-98. Boone J., and A.L. Bovenberg, 2003a, ’The Optimal Taxation of Unskilled Labor with Job Search and Social Assistance,’ NBER Discussion Paper No. 9785. Boone J., and A.L. Bovenberg, 2003b, ’Optimal Welfare Beneﬁts and Non-Linear Income Taxation with Unemployment,’ mimeo, CentER, Tilburg University, the Netherlands. 77The government can also collect more information regarding why workers lost their jobs. If they were laid oﬀbecause of misconduct or if they quit voluntarily, the government may refuse unemployment beneﬁts. This may reduce the wage pressure from higher unemployment beneﬁts (see van der Ploeg (2003)). Higher categorical beneﬁts for veriﬁable disabilities (so-called ’tagging’) may also provide valuable insurance without inducing moral hazard. Indeed, the trade-oﬀbetween eﬃciency and equity originates in assymetric information about the skills and behavior of agents. The agencies paying welfare and unemployment beneﬁts typically collect much more information about the skills and health of beneﬁt recipients than the tax oﬃce does about taxpayers. 78In terms of the model developed in section 3.2, workfare in eﬀect reduces δ (i.e. the value of leisure in unemployment). 48 Bovenberg, A.L., and F. van der Ploeg, 1994, ’Eﬀects of the Tax and Beneﬁt System on Wage Formation and Unemployment,’ Mimeo Tilburg University. Bovenberg, A.L., 1995, ’Environmental Taxation and Employment,’ De Economist, Vol. 143, No. 2, pp. 111-140. Bovenberg, A.L., and C. van Ewijk, 1997, ’Progressive Taxes, Equity, and Hu-man Capital Accumulation in an Endogenous Growth Model with Overlapping Gen-erations, Journal of Public Economics, Vol. 64, pp. 154-179. Bovenberg, A.L., and F. van der Ploeg, 1998, ’Tax Reform, Structural Unem-ployment and the Environment,’ Scandinavian Journal of Economics, Vol. 100, pp. 593-610. Bovenberg, A.L., ’Green Tax reforms and the Double Dividend,: An Updated Reader’s Guide,’ International Tax and Public Finance, Vol. 6, pp. 421-443. Bovenberg, A.L., and B. Jacobs, 2002, ’Redistribution and Education Subsidies are Siamese Twins,’ CEPR Discussion Paper No. 3099. Broersma, L., and J.C. van Ours, 1999, ’Job Searchers, Job Matches and the Elasticity of Matching’, Labor Economics, 6, 77-93. Calmfors, L., and E.J. Driﬃll, 1988, ’Bargaining Structure, Corporatism and Macroeconomic Performance,’ Economic Policy, Vol. 6, pp 13-62. Cnossen, S., 2001, ’Tax Policy in the European Union. A Review of Issues and Options,’ Studies in Economic Policy No. 5, Ocfeb, Erasmus University Rotterdam, the Netherlands. Daveri, F., and G. Tabellini, 2000, ’Unemployment and Taxes: Do Taxes Aﬀect the Rate of Unemployment?’ Economic Policy, pp. 49-104. Diamond P.A., and J.A. Mirrlees 1971, ’Optimal Taxation and Public Produc-tion 1: Production Eﬃciency and 2: Tax Rules,’ American Economic Review, Vol. 61, pp. 8-27 and pp. 261-278. Disney, R., 2000, ’The Impact of Tax and Welfare Policies on Employment and Unemployment in OECD Countries,’ IMF Working Paper No. 2000/64, Washington D.C. Eaton, J., and H.S. Rosen, 1980, ’Taxation, Human Capital, and Uncertainty,’ American Economic Review, Vol. 70, pp. 705-715. Ebert, U., 1992, ’A Reexamination of the Optimal Nonlinear Income Tax,’ Jour-nal of Public Economics, Vol. 49, pp. 47-73. Frederiksson, P., and B. Holmlund, 2003, ’Improving Incentives in Unemploy-ment Insurance: A Review of Recent Research,’ Institute for labor Market Policy Evaluation, Uppsala, Working Paper 2003:5. Goulder, L.H., 1995, ’Environmental Taxation and the Double Dividend: A Reader’s Guide,’ International Tax and Public Finance, Vol. 2, pp. 157-183. Graaﬂand, J.J., and M H. C. Lever, 1996, ’Internal and External Forces in Sec-toral Wage Formation: Evidence from the Netherlands,’ Oxford Bulletin of Economics and Statistics, Vol. 58, pp. 241-252. Graaﬂand, J.J., and F.H. Huizinga, 1999, ’Taxes and Beneﬁts in a Non-Linear Wage Equation,’ De Economist, Vol. 147, pp. 39-54. Graaﬂand, J.J., R.A. de Mooij, A.G.H. Nibbelink, and A. Nieuwenhuis, 2001, Mimicing Tax Policies and the Labor Market (North-Holland, Amsterdam). Groot, W., and H. Oosterbeek, 1995, ’Determinants and Wage Eﬀects of Partici-pation in On- and Oﬀ-the-Job Training,’ Tinbergen Institute Research Memonrandum TI 95-122. Gruber, J., and D. Wise, 1999, Social Security and Retirement Around the World (Unversity of Chicago Press, Chicago). 49 Jacobs, B., 2002, Public Finance and Human Capital, Tinbergen Institute The-sis (Thela Thesis, Amsterdam). Heckman, J.J., 1976, ’A Life-Cycle Model of Earnings, Learning, and Consump-tion,’ Journal of Political Economy, Vol. 4, S11-S44. Heckman, J.J., L. Lochner, and R, Cossa, 2002, ’Learning By Doing Versus On-the-Job Training: Using Variantion Induced by the EITC to Distinguish between Models of Skill Formation,’ NBER Working Paper No. 9083. Hersoug, T., 1984, Union wage responses to tax changes, Oxford Economic Papers, 36, 37-51. Hoel, M., 1990, Eﬃciency wages and income taxes, Journal of Economics, 51, 1, 89-99. Holmlund, B., and J. Linden, ’Job Matching, Temporary Public Employment, and Equilibrium Unemployment,’ Journal of Public Economics, Vol. 51, pp. 329-343. Holmlund, B., 2000, ’Labor Taxation in Search Equilibrium with Home Produc-tion,’ Mimeo, Department of Economics, Uppsala, Sweden. Hosios, A.J., 1990, ’On the Eﬃciency of Matching and Related Models of Search and Unemployment,’ Review of Economic Studies, Vol. 57, pp. 279-298. Hubbard, R.G., and K.L. Judd, 1986, ’Liquidity Constraints, Fiscal Policy, and Consumption,’ Brookings Papers on Economic Activity, pp. 1-50. King, R., C. Plosser and S. Rebelo, 1988, Production, growth and business cycles, I: The basis neoclassical model, Journal of Monetary Economics, 21, 195-232. Koskela, amd Schob, 1999, ’Alleviating Unemployment. The Role of Unemploy-ment Beneﬁts and Tax Structure,’ European Economic Review, Vol. 43, pp. 1723-1746. Layard, R., 1980, ’Human Satisfactions and Public Policy,’ Economic Journal, Vol. 90, pp. 737-750. Layard, R., S. Nickell and R. Jackman, 1991, Unemployment. Macroeconomic Performance and the Labour Market, Oxford University Press, Oxford. Lockwood, B. and A. Manning, 1993, Wage setting and the tax system. Theory and evidence for the United Kingdom, Journal of Public Economics, 52, 1-29. Lollivier, S., and J. Rochet, 1983, Bunching and Second-Order Conditions: A Note on Optimal Tax Theory, Journal of Economic Theory, Vol. 31, pp. 392-400. Lucas, R.E., 1988, ’On the Mechanisms of Economic Development,’ Journal of Monetary Economics, Vol. 22, pp. 3-42. Lungqvist, L, and T.J. Sargent, 1998, ’The European Unemployment Dilemma,’ Journal of Political Economy, Vol. 106, pp. 514-550. Malcolmson, J.M. and N. Sator, 1987, Tax push inﬂation in a universal labour market, European Economics Review, 31, 1581-1596. Mortensen D.T., and C.A. Pissarides, 1999, ’New Developments in Models of Search in the labor Market.’ in O. Ashenfelter and D. Card (eds.), Handbook of Labor Economics (North Holland). Musgrave, R. and P.B. Musgrave, 1976, Public Finance in Theory and Practice, second edition, McGraw-Hill, New York. Neary, J.P., and K.W.S. Roberts, ’The Theory of Household Behaviour and Rationing,’ European Economic Review, Vol. 13, pp. 25-42. Nickell, S., and R. Layard, 1999, Labor Market Institutions and Economic Per-formance, in O. Ashenfelter and D. Card (ed.), Handbook of Labor Economics, Vol. 3 (North Holland, Amsterdam). Nielsen, S.B., and P.B. Sørensen, ’On the Optimality of the Nordic System of Dual income Taxation,’ Journal of Public Economics, Vol. 63, pp. 311-329. OECD, 2002a, Employment Outlook, OECD, Paris. 50 OECD, 2002b, Beneﬁts and Wages, OECD Indicators, OECD, Paris. Pauly, M.V., 1974, ’Over Insurance and Public Provision of Insurance. The Roles of Moral Hazard and Adverse Selection,’ Quarterly Journal of Economics, Vol. 88, pp. 44-62. Pissarides, C.A., 1990, Equilibrium Unemployment Theory (Blackwell, Oxford). Pissarides, C.A., 1998, ’The Impact of Employment Tax Cuts on Unemployment and Wages: the Role of Unemployment Beneﬁts and the Tax Structure,’ European Economic Review, Vol. 42, pp. 155-183. Ploeg, F. van der, 1987, ’Trade Unions, Employment and Investment: A Non-Cooperative Approach,’ European Economic Review. Ploeg, F. van der, 2003, ’Do Social Policies Harm Employment and Growth? Eﬀects of Taxes and Beneﬁts on Non-Competitive Labour Markets,’ Mimeo, European University Institute, Florence, Italy. Sandmo, A., 1975, ’Optimal Taxation in the Presence of Externalities,’ Swedish Journal of Economics, Vol. 77, pp. 86-98. Shapiro, C. and J.E. Stiglitz, 1984, ’Equilibrium unemployment as a worker discipline device,’ American Economic Review, 74, 3, 433-444. Sinn, 1995, ’A Theory of the Welfare State,’ Scandinavian Journal of Economics, Vol. 97, pp. 495-526. Sørensen, P.B., 1997, ’Public Finance Solutions to the European Unemployment Problem?’ Economic Policy, Vol. 25, pp. 223-264. Sørensen, P.B., 1999, ’Optimal Tax Progressivity in Imperfect Labor Markets,’ Labour Economics, Vol. 6, pp. 435-452. Sørensen, P.B., 2003, ’Social Insurance Based on Individual Savings Accounts,’ Forthcoming as Chapter 10 in S. Cnossen and H.-W. Sinn (eds.), Public Finances and Public Policy in the New Century, MIT Press. Stiglitz, J.E., 1999, ’Taxation, Public Policy and the Dynamics of Unemploy-ment,’ International Tax and Public Finance, Vol. 6, pp. 239-262. Summers, L., J. Gruber, and K. Vergara, 1993, ’Taxation and the Structure of Labor Markets: The Case of Corporatism,’ Quarterly Journal of Economics, Vol. 108, pp. 385. Trostel, P.A., 1993, ’The Eﬀect of Taxation on Human Capital,’ Journal of Political Economy, Vol. 101, pp. 327-350. Tyrvainen, T., 1995, ’Wage Setting, Taxes and Demand for labour: Multivariate Analysis of Cointegrating Relations,’ Empirical Economics, Vol. 20, pp. 271-297. Weymark, J.A., 1987, Comparative Static Properties of Optimal Nonlinear Taxes, Econometrica, Vol. 55, pp. 1165-1185. 9 Appendix To establish (19), we linearize the following equations characterizing the decentralized equilibrium γ′(X) = m(θ) Y −Π −G + B m(θ) , (26) cθ m(θ) = Π. (27) To derive the ﬁrst equation, we have substituted the government budget constraint Xm(θ)Y = Xm(θ) [W + Π] + (1−Xm(θ))B+ G to eliminate the after-tax wage W from (16). The second equation follows from (15) and (18). 51 Loglinearization yields ˜ X ˜ θ = 1 cθ(1−η) m(θ) Xγ′′(X) −G+B X cθ(1−η) m(θ) m(θ)η (Y −Π) 0 Xγ′′(X) −G+B X −m(θ)Π˜ Π −G X ˜ G −B X ˜ B ΠΠ . (28) We thus have (the second equality follows from (27)) ˜ X ˜ Π = Πm(θ) cθ(1−η) m(θ) Xγ′′(X) −G+B X −cθ m(θ)(1 −η) + η(Y −Π) = Πm(θ) cθ(1−η) m(θ) Xγ′′(X) −G+B X [−Π + ηY ] = ≥0 if Π ≤ηY < 0 if Π > ηY . (29) Hence, X and thus welfare is maximized if Π = ηY. Using (15) to eliminate Π, we arrive at the expression for the optimal tax τ a in (19). QED CESifo Working Paper Series (for full list see www.cesifo.de) ____________ 970 Hartmut Egger and Josef Falkinger, The Role of Public Infrastructure for Firm Location and International Outsourcing, June 2003 971 Dag Morten Dalen and Trond E. 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15173	https://www.quora.com/How-do-you-solve-for-grams-from-molality	How to solve for grams from molality - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Chemistry Molality Problems Measurement Units Grams Chemical Solutions Scientific Calculations Molal Concentration Solution Thermodynamics Molarity and Molality 5 How do you solve for grams from molality? All related (37) Sort Recommended Alexander Fleming Games to play and math to do ·11y We know that molality is m o l e s s o l u t e k i l o g r a m s s o l v e n t m o l e s s o l u t e k i l o g r a m s s o l v e n t. We also know that we have 0.5kg of water: 500 m l∗1.000 g 1 m l=500 g=0.5 k g 500 m l∗1.000 g 1 m l=500 g=0.5 k g. Set up an equation to solve for moles of glucose: m o l s g l u c o s e 0.5 k g w a t e r=0.450 m o l s g l u c o s e 0.5 k g w a t e r=0.450. Solving for moles of glucose yields that the required number of moles is 0.225 moles. Then convert moles of glucose to grams of glucose 0.225 m o l s g l u c o s e∗180.1559 g g l u c o s e 1 m o l g l u c o s e=40.535 g g l u c o s e 0.225 m o l s g l u c o s e∗180.1559 g g l u c o s e 1 m o l g l u c o s e=40.535 g g l u c o s e. Upvote · 9 2 9 1 Sponsored by Grammarly 92% of professionals who use Grammarly say it has saved them time Work faster with AI, while ensuring your writing always makes the right impression. Download 999 207 Related questions More answers below How do you calculate the molality of the solution? How can the formula for molality be solved? How can the formula to solve for molality be determined? What is the difference between a gram and a kilogram? Why is it important to use the correct unit when calculating the molality of a solution? What is the equation for molality? Saroj Bhatia 8y Molality = weight of solute/( molecular weight x weight of solvent in Kg) Weight of solute =Molality x mol.wt. x wt. of solvent(Kg) solution- wt. of solute (glucose)= ? gram molecular weight of glucose (C6H12O6)=12x6 +12x1+16x6=180 molality = 0.450 m volume =500 ml mass or weight= volume x density=500 x 1 =500 gm. =500/1000 kg=0,5 kg Weight of solute =Molality x mol.wt. x wt. of solvent(Kg) w= 0.450 x 180 x 0,5=40.5 gm mass of glucose =40.5 gm. Read more articles on chemistryonline.guru Your response is private Was this worth your time? This helps us sort answers on the page. Absolutely not Definitely yes Upvote · 9 2 Guy Clentsmith Chemistry tutor... at Self-Employment (2018–present) · Author has 26.5K answers and 19.7M answer views ·Updated 4y Originally Answered: How do we find grams from molarity? · Well, molarity is a measure of concentration, and is given by the quotient… Molarity=Moles of solute Volume of solution Molarity=Moles of solute Volume of solution …accordingly, molarity molarity has units of m o l∙L−1 m o l•L−1. And of course (i) the Moles of solute Moles of solute is itself defined by a quotient… Moles of solute=Mass of solute Molar mass of solute Moles of solute=Mass of solute Molar mass of solute And of course (ii) we need to specify a Continue Reading Well, molarity is a measure of concentration, and is given by the quotient… Molarity=Moles of solute Volume of solution Molarity=Moles of solute Volume of solution …accordingly, molarity molarity has units of m o l∙L−1 m o l•L−1. And of course (i) the Moles of solute Moles of solute is itself defined by a quotient… Moles of solute=Mass of solute Molar mass of solute Moles of solute=Mass of solute Molar mass of solute And of course (ii) we need to specify a given molarity, and a given volume of solution. And I will g... Upvote · 9 8 Roderick Mobley Former AP/IB Chemistry Teacher ·6y Originally Answered: How do we find grams from molarity? · Molarity is the concentration of a solution in units of moles of solute per liter of solution. If you know the volume of the solution in liters, you can multiply the molarity by the volume in liters to determine the number of moles of solute. This can be converted to grams by multiplying the number of moles by the molar mass of the solute. Upvote · 9 2 Sponsored by All Out Kill Dengue, Malaria and Chikungunya with New 30% Faster All Out. Chance Mat Lo, Naya All Out Lo - Recommended by Indian Medical Association. Shop Now 999 621 Related questions More answers below What is the molality of a 1.24 M solution of KI that has a density of 1.15 g/mL? How do I solve this? Can you explain how to calculate molality from moles and grams? 4 grams of NaOH is dissolved in enough water to prepare a 500ml PF solution. What is its molality? What is the concentration measurement for molality? Why would we need to know the molality of a solution? Hcbiochem Former Professor of Biology and Chemistry (1982–2019) · Author has 4.6K answers and 1.9M answer views ·3y Originally Answered: How do you calculate molar mass from grams? · I wish you had provided more information, since it will depend on the specific problem you are working. So, for example, if you know the mass, temperature, pressure and volume of a sample of a gas, you can use the ideal gas law to calculate the moles of gas present. Dividing the mass of the sample by the moles of gas present gives you the molar mass of the gas. You might want to submit a new question with more information so that someone can help you out. Upvote · Ishika Khurana Studied at Indus Public School ·6y Originally Answered: How do we find grams from molarity? · Molarity=no.of moles/Volume of solution(in L) Also, no.of moles=mass in grams/molecular mass So, wt in grams=(molarity)(molecular mass)(volume in L) Upvote · 9 2 9 1 Sponsored by Book Geists If you're a Kiwi, this could be the best day of your life! Available to Kiwis only. Read today. Learn More 1.3K 1.3K Daniel Iyamuremye Former Senior Lecturer (Retired) (2000–2018) · Author has 12.1K answers and 2M answer views ·11mo Originally Answered: How do we find grams from molarity? · Molarity is: number of moles per liter of solution. Number of moles of solute in a solution = (molarity/1L) Volume of solution in L = X moles Then you convert X moles into grams Upvote · Sri Sandhya 7y Related How do you relate mole fraction and molality? The mole fraction is the ratio of moles of solute to moles of solvent. Molality is the number of moles of solute per mass of solvent. Continue Reading The mole fraction is the ratio of moles of solute to moles of solvent. Molality is the number of moles of solute per mass of solvent. Upvote · 999 201 99 10 9 4 Sponsored by RedHat Know what your AI knows, with open source models. Your AI should keep records, not secrets. Learn More 99 36 Gaurav Kumar Influencer, Perfectionist, researcher · Author has 204 answers and 1.4M answer views ·7y Related How do I calculate density if molality and molarity of a solution is given? By separately arranging them gives a way to find density by the help of molarity and molality. If you like the answer, then please upvote it. Continue Reading By separately arranging them gives a way to find density by the help of molarity and molality. If you like the answer, then please upvote it. Upvote · 99 72 9 2 9 2 Ajithkumar N Btech chemicalengineering from Sri Venkateswara College of Engineering (Graduated 2021) · Upvoted by Malcolm Sargeant , Degree level applied chemistry + 20yr experience in corrosion prevention and water treatment ·6y Related What is the molality of the solution containing 36 grams of glucose into 50 grams of water? As we know molality is defined as no of moles of solute in 1 kg of solvent. So it has SI unit as mol/kg. And it is denoted as letter ‘m’. Here the solute is Glucose and we need to find its moles. No. Of moles = given mass / molecular mass Given mass is 36 grams. And molecular mass of the glucose is 180 grams(C6H12O6). No. Of moles = 36/180 = 0.2 moles. Here solvent is water and we can take 50 grams of water as 0.05kg of water to get SI Unit precisely. Molality = no of moles / kg of solvent = 0.2 / 0.05 (mol/kg) =4 m. Thanks for reading ❤️ Upvote · 9 7 Alexander Mathey Former Chemical Engineer, retired, lives in Athens, GR · Author has 5.6K answers and 10.9M answer views ·8y Related How do you convert mols to grams? By multiplying by the molecular weight of the substance. Example: Atomic weight of Oxygen = 16 Atomic weight of Hydrogen = 1 Molecular weight of water = 116 + 21 = 18 Therefore: 1 mol of water = 18 grams. (Notification: We still use the expression ‘Atomic and Molecular Weight’ just because we are accustomed to it, well knowing that what we mean is ‘Atomic and Molecular Mass’. It is as, when we say ‘I weigh 80 Kilos’, we don’t refer to the force by which our feet press the scale, but to our mass.) Upvote · Aidan Cooney PhD in Chemistry&Physical Organic Chemistry, Durham University (Graduated 1987) · Author has 553 answers and 1.4M answer views ·6y Related How do you find molarity without grams? If you know molar mass in grams per mole and some other amount parameter like parts per million or weight percent or even volume percent with density and the total volume then you could back calculate to get the number of moles and calculate molairity in moles per litre. You can get molar mass from molecular weight or formula weight or relative atomic mass Upvote · 9 1 Howard Neilly Studied at University of the Bahamas (Graduated 1983) · Author has 3.1K answers and 1.5M answer views ·3y Related How do you calculate molality from molarity and density? Assuming that I am given a 1M NaCl solution with a density of 3g/cm^3. The formula for finding the molality of a solution is molality= moles of solute/kg of solvent. A 1M solution means that I have 1mol of NaCl in 1liter of solution so moles of solute=1mol kg of solvent=kg of water=unknown formula for finding molarity is Molarity= moles of solute/liters of solution where mol of solute= 1mol volume of solution=1L So in order to find the molality of this solution I have to subtract the mass of solvent from the mass of the solvent and the solute which constitute the solution. The mass of the combined solute Continue Reading Assuming that I am given a 1M NaCl solution with a density of 3g/cm^3. The formula for finding the molality of a solution is molality= moles of solute/kg of solvent. A 1M solution means that I have 1mol of NaCl in 1liter of solution so moles of solute=1mol kg of solvent=kg of water=unknown formula for finding molarity is Molarity= moles of solute/liters of solution where mol of solute= 1mol volume of solution=1L So in order to find the molality of this solution I have to subtract the mass of solvent from the mass of the solvent and the solute which constitute the solution. The mass of the combined solute and solvent (the solution) is found by using the formula for density density=mass/volume or D=m/V so solving for mass by multiplying both sides of the equation by V DV=(m/V)V both V's on the right side cancels each other out leaving DV=m where D=density=3g/cm^3 V=volume=1L which must be converted to cm^3 (1L=1000cm3) so V=1000cm^3 plugging these values into the formula 31000=m so the mass of the solution=3000g mass of the solute = mass of 1mol of NaCl 1mol of NaCl=58g mass of solvent=mass of solution -mass of solute mass of solvent in grams=3000g-58g=3942g this must be converted to kg (1kg=1000g) (3942g)(1kg)/(1000g)=3.9421kg=3.942kg plugging this value into the formula for finding molality the molality=1/3.942 the molality=0.25molal Upvote · 9 5 9 1 9 1 Related questions How do you calculate the molality of the solution? How can the formula for molality be solved? How can the formula to solve for molality be determined? What is the difference between a gram and a kilogram? Why is it important to use the correct unit when calculating the molality of a solution? What is the equation for molality? What is the molality of a 1.24 M solution of KI that has a density of 1.15 g/mL? How do I solve this? Can you explain how to calculate molality from moles and grams? 4 grams of NaOH is dissolved in enough water to prepare a 500ml PF solution. What is its molality? What is the concentration measurement for molality? Why would we need to know the molality of a solution? What will be the molality of the solution made by dissolving 10g of O? How many grams are there in an ml? How do I find molality? What is the molality of a substance of 3.42 grams and molecular weight is 34. There is 250 grams of water? Why is the molality of a solution always larger than the molarity? Related questions How do you calculate the molality of the solution? How can the formula for molality be solved? How can the formula to solve for molality be determined? What is the difference between a gram and a kilogram? Why is it important to use the correct unit when calculating the molality of a solution? What is the equation for molality? What is the molality of a 1.24 M solution of KI that has a density of 1.15 g/mL? How do I solve this? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
15174	https://cl.desmos.com/t/modelling-equations-with-scales/1732	Modelling Equations with Scales - Questions - Computation Layer Support Forum Skip to main content Sign Up Log In Welcome to the Desmos Computation Layer support forum. Computation Layer is the underlying technology in which lets us connect different components and different representations when authoring activities. The goal of this forum is to help anyone that is authoring activities with Computation Layer be successful. This is a great place to: ask questions about how to use Activity Builder features or Computation Layer learn about new Activity Builder and Computation Layer features show off your new activities and ideas, and get feedback on them meet other teachers that are creating things with Activity Builder You might want to check out the official Computation Layer documentation or read the Desmos Guide to Building Great (Digital) Math Activities Modelling Equations with Scales Questions You have selected 0 posts. select all cancel selecting Oct 2020 1 / 2 Oct 2020 Oct 2020 Spencer_Wenzel 1 Oct 2020 Hi everyone. I am looking for any examples people have made where they model equations using a scale. There is a similar balance one in the desmos made activities but it would be much too difficult for my grade 6 class. Any help would be great! 389 views 1 link JAC9186Javier CabezasFellow Oct 2020 I remember seeing this on Twitter a while back: [Public version of] Equation Pan Balance â¢ Activity Builder by Desmos I haven’t looked at anything beyond the screens shown, but it might be the start of what you’re hoping for. Reply Related topics Topic list, column headers with buttons are sortable.\| Topic \| Replies \| Views \| Activity \| --- --- \| \| A few “game” ideas for creating equations Discussion \| 0 \| 609 \| Apr 2019 \| \| Racing Dots! Classroom report Discussion \| 1 \| 620 \| Jun 2018 \| \| Can anybody Help with this activity Questions \| 56 \| 2.4k \| Mar 2021 \| \| Visual feedback on equations (math input + table) Examples \| 0 \| 549 \| Mar 2021 \| \| Graphing more than one Equation Questions \| 6 \| 4.8k \| May 2018 \| Invalid date Invalid date
15175	https://www.chegg.com/homework-help/questions-and-answers/fundamental-period-smallest-positive-period-find-cosnx-sinnx-cos-2-pi-x-k-sin-2p-x-k-cos-2-q18969962	Solved The fundamental period is the smallest positive \| Chegg.com Skip to main content Books Rent/Buy Read Return Sell Study Tasks Homework help Understand a topic Writing & citations Tools Expert Q&A Math Solver Citations Plagiarism checker Grammar checker Expert proofreading Career For educators Help Sign in Paste Copy Cut Options Upload Image Math Mode ÷ ≤ ≥ o π ∞ ∩ ∪           √  ∫              Math Math Geometry Physics Greek Alphabet Engineering Mechanical Engineering Mechanical Engineering questions and answers The fundamental period is the smallest positive period. Find it for cosnx, sinnx, cos (2(pi)x)/k, sin (2p(i)x)/k, cos (2(pi)nx)/k, sin (2(pi)nx)/k Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on. See Answer See Answer See Answer done loading Question: The fundamental period is the smallest positive period. Find it for cosnx, sinnx, cos (2(pi)x)/k, sin (2p(i)x)/k, cos (2(pi)nx)/k, sin (2(pi)nx)/k The fundamental period is the smallest positive period. Find it for cosnx, sinnx, cos (2(pi)x)/k, sin (2p(i)x)/k, cos (2(pi)nx)/k, sin (2(pi)nx)/k There are 2 steps to solve this one.Solution 100%(1 rating) Share Share Share done loading Copy link Step 1 Time period of different functions are View the full answer Step 2 UnlockAnswer Unlock Previous questionNext question Not the question you’re looking for? Post any question and get expert help quickly. 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15176	https://www.youtube.com/watch?v=TNqP5EDqGOU	GeoGebra - Construct segments tangent to two circles Nick Hershman 5 subscribers Description 411 views Posted: 26 Oct 2017 Transcript:
15177	https://online.stat.psu.edu/stat500/Lesson09	9 Linear Regression Foundations – STAT 500 \| Applied Statistics We use cookies We use cookies and other tracking technologies to improve your browsing experience on our website, to show you personalized content and targeted ads, to analyze our website traffic, and to understand where our visitors are coming from. I agree I decline Change my preferences STAT 500 \| Applied Statistics Department of Statistics Lessons 9 Linear Regression Foundations About this course Lessons 0: Overview 1 Collecting and Summarizing Data 2 Probability 3 Probability Distributions 4 Sampling Distributions 5 Confidence Intervals 6 Hypothesis Testing 7 Comparing Two Population Parameters 8 Chi-Square Test for Independence 9 Linear Regression Foundations 10 Introduction to ANOVA 11 Introduction to Nonparametric Tests and Bootstrap 12 Summary and Review In this lesson 9.1 Linear Relationships 9.1.1 Scatterplots 9.1.2 Correlation 9.2 Simple Linear Regression 9.2.1 The SLR Model 9.2.2 Interpreting the Coefficients 9.2.3 Assumptions for the SLR Model 9.2.4 Inferences about the Population Slope 9.2.5 Other Inferences and Considerations 9.2.6 Examples 9.3 Coefficient of Determination 9.4 Inference for Correlation 9.4.1 Hypothesis Testing for the Population Correlation 9.4.2 Comparing Correlation and Slope 9.5 Multiple Regression Model 9.6 Lesson Summary Lessons 9 Linear Regression Foundations 9 Linear Regression Foundations Code Overview In this Lesson, we will first introduce the Simple Linear Regression (SLR) Model and the Correlation Coefficient. Inferences for the simple linear regression model will be discussed, and the critical distinction between inference for mean response and inference for the outcome will be clarified. We will also introduce a basic understanding of the multiple regression model. Regression analysis is a tool to investigate how two or more variables are related. Quite often we want to see how a specific variable of interest is affected by one or more variables. For example, one may wish to use a person’s height, gender, race, etc. to predict a person’s weight. Let us first consider the simplest case: using a person’s height to predict the person’s weight. Example If you are asked to estimate the weight of a STAT 500 student, what will you use as a point estimate? If I tell you that the height of the student is 70 inches, can you give a better estimate of the person’s weight? Answer For the first part, the point estimate would be the average weight (or the median weight) of a STAT 500 student. If you know the student is 70 inches, then, yes, you can give a better estimate of the person’s weight, but only if you have some idea about how height and weight are related. It is important to distinguish between the variable of interest and the variable(s) we will use to predict the variable of interest. 9.1 (Response Variable) Denoted, Y, is also called the variable of interest or dependent variable. In the example, weight is the response variable. 9.2 (Predictor Variable) Denoted, X, is also called the explanatory variable or independent variable. In the example, height is the predictor. When there is only one predictor variable, we refer to the regression model as a simple linear regression model. To use known information to provide a better estimate, we need to understand how the dependent and independent variables are related. In statistics, we can describe how variables are related using a mathematical function. The function along with other assumptions is called a model. There are many models we can consider. In this class, we will focus on linear models, particularly, when there is only one predictor variable. We refer to this model as the simple linear regression model. Objectives Upon completion of this lesson, you should be able to: Use plots and summary statistics to describe the relationship between the response variable and the predictor variable. Perform a hypothesis test for the population correlation. Find the regression equation and interpret the results. Apply the regression model and know the limitations. Find an interval estimate for the population slope and interpret the interval. 9.1 Linear Relationships To define a useful model, we must investigate the relationship between the response and the predictor variables. As mentioned before, the focus of this Lesson is linear relationships. For a brief review of linear functions, recall that the equation of a line has the following form: y=m x+b where m is the slope and b is the y-intercept. Given two points on a line, (x 1,y 1) and (x 2,y 2), the slope is calculated by: m=y 2−y 1 x 2−x 1=change in y change in x=rise run The slope of a line describes a lot about the linear relationship between two variables. If the slope is positive, then there is a positive linear relationship, i.e., as one increases, the other increases. If the slope is negative, then there is a negative linear relationship, i.e., as one increases the other variable decreases. If the slope is 0, then as one increases, the other remains constant. When we look for linear relationships between two variables, it is rarely the case where the coordinates fall exactly on a straight line; there will be some error. In the next sections, we will show how to examine the data for a linear relationship (i.e., the scatterplot) and how to find a measure to describe the linear relationship (i.e., correlation). 9.1.1 Scatterplots If the interest is to investigate the relationship between two quantitative variables, one valuable tool is the scatterplot. 9.3 (Scatterplot) A graphical representation of two quantitative variables where the explanatory variable is on the x-axis and the response variable is on the y-axis. When we look at the scatterplot, keep in mind the following questions: What is the direction of the relationship? Is the relationship linear or nonlinear? Is the relationship weak, moderate, or strong? Are there any outliers or extreme values? We describe the direction of the relationship as positive or negative. A positive relationship means that as the value of the explanatory variable increases, the value of the response variable increases, in general. A negative relationship implies that as the value of the explanatory variable increases, the value of the response variable tends to decrease. Example 9.1 (Student height and weight (Scatterplots)) Suppose we took a random sample from students at a large university and asked them about their height and weight. The data can be found here university_ht_wt.csv. The first three observations are: \| Height (inches) \| Weight (pounds) \| --- \| \| 72 \| 200 \| \| 68 \| 165 \| \| 69 \| 160 \| We let X denote the height and Y denote the weight of the student. The observations are then considered as coordinates (x,y). For example, student 1 has coordinates (72,200). These coordinates are plotted on the x-y plane. We can use Minitab to create the scatterplot. We can create our scatterplot in Minitab following these steps. Choose Graph>Scatterplot>Simple For the Y-variable: select ‘weight’ and for the X-variable: select ‘height’. Choose OK. Fig 9.1: Scatterplot of Weight vs Height The scatterplot shows that, in general, as height increases, weight increases. We say “in general” because it is not always the case. For example, the observation with a height of 66 inches and a weight of 200 pounds does not seem to follow the trend of the data. The two variables seem to have a positive relationship. As the height increases, weight tends to increase as well. The relationship does not seem to be perfectly linear, i.e., the points do not fall on a straight line, but it does seem to follow a straight line moderately, with some variability. Try It An elementary school teacher gives her students two spelling tests a year. Each test contains 24 words, and the score is the number of words spelled correctly. The teacher is interested in the relationship between the score on the first test and the score on the second test. Using the scatterplot, comment on the relationship between the two variables. Answer There seems to be a weak positive linear relationship between the two test scores. In the next section, we will introduce correlation. Correlation is a measure that gives us an idea of the strength and direction of the linear relationship between two quantitative variables. 9.1.2 Correlation If we want to provide a measure of the strength of the linear relationship between two quantitative variables, a good way is to report the correlation coefficient between them. The sample correlation coefficient is typically denoted as r. It is also known as Pearson’s r. The population correlation coefficient is generally denoted as ρ, pronounced “rho.” Sample Correlation Coefficient The sample correlation coefficient, r, is calculated using the following formula: r=∑(x i−x¯)(y i−y¯)∑(x i−x¯)2∑(y i−y¯)2 Properties of the correlation coefficient, r: −1≤r≤1, i.e.r takes values between -1 and +1, inclusive. The sign of the correlation provides the direction of the linear relationship. The sign indicates whether the two variables are positively or negatively related. A correlation of 0 means there is no linear relationship. There are no units attached to r. As the magnitude of r approaches 1, the stronger the linear relationship. As the magnitude of r approaches 0, the weaker the linear relationship. If we fit the simple linear regression model between Y and X, then r has the same sign as β 1, which is the coefficient of X in the linear regression equation. – more on this later. The correlation value would be the same regardless of which variable we defined as X and Y. Note! The correlation is unit free. We can see this easier using the equation above. Consider, for example, that we are interested in the correlation between X = height (inches) and Y = weight (pounds). In the equation above, the numerator would have the units of pounds∗inches. The denominator would include taking the square root of pounds squared and inches squared, leaving us again with units of pounds∗inches. Therefore the units would cancel out. Fig 9.2: r>0 Fig 9.3: r<0 Fig 9.4: r≈0 Fig 9.5: r≈0 Example 9.2 (Sales and Advertising (Correlation)) We have collected five months of sales and advertising dollars for a small company we own. Sales units are in thousands of dollars, and advertising units are in hundreds of dollars. Our interest is determining if a linear relationship exists between sales and advertising. The data is as follows: \| Sales (Y) \| Advertising (X) \| --- \| \| 1 \| 1 \| \| 1 \| 2 \| \| 2 \| 3 \| \| 2 \| 4 \| \| 4 \| 5 \| Find the sample correlation and interpret the value. Answer By Hand Minitab The mean of Sales (Y) is y¯=2 and the mean of advertising (X) is x¯=3. We can calculate the sample correlation in steps. \| y i−y¯ \| x i−x¯ \| (x i−x¯)(y i−y¯) \| --- \| 1−2=−1 \| 1−3=−2 \| (−1)(−2)=2 \| \| 1−2=−1 \| 2−3=−1 \| (−1)(−1)=1 \| \| 2−2=0 \| 3−3=0 \| (0)(0)=0 \| \| 2−2=0 \| 4−3=1 \| (0)(1)=0 \| \| 4−2=2 \| 5−3=2 \| (2)(2)=4 \| From the table, we can calculate the following sums… ∑(y i−y¯)2=(−1)2+(−1)2+0+0+2 2=6(sum of first column) ∑(x i−x¯)2=(−2)2+(−1)2+0+1 2+2 2=10(sum of second column) ∑(x i−x¯)(y i−y¯)=2+1+0+0+4=7(sum of third column) Using these numbers in the formula for r… r=∑(x i−x¯)(y i−y¯)∑(x i−x¯)2∑(y i−y¯)2=7 10 6=0.9037 Using Minitab to calculate r To calculate r using Minitab: Open Minitab and upload the data (for this example type the Y data into a column (e.g., C1) and the X data into a column (e.g., C2)) Choose Stat>Basic Statistics>Correlation Specify the variables in the dialog box (X and Y in this example). Minitab output for this example: Correlation: Sales (Y), Advertising (X) Method Correlation type Pearson Number of rows used 5 Correlation \| \| Sales (Y) \| --- \| \| Advertising (X) \| 0.904 \| The sample correlation is 0.904. This value indicates a strong positive linear relationship between sales and advertising. Try It Using the following data, calculate the correlation and interpret the value. \| X \| Y \| --- \| \| 2 \| 7 \| \| 4 \| 11 \| \| 14 \| 29 \| \| 13 \| 28 \| \| 15 \| 32 \| Answer By Hand Minitab The mean of X is 9.6 and the mean of Y is 21.4. The sums are… ∑(x i−x¯)2=149.2 ∑(y i−y¯)2=529.2 ∑(x i−x¯)(y i−y¯)=280.8 Using these sums in the formula for r… r=∑(x i−x¯)(y i−y¯)∑(x i−x¯)2∑(y i−y¯)2=0.9993 Following the steps for finding correlation with Minitab you should get the following output: Method Correlation type Pearson Number of rows used 5 Correlation \| \| X \| --- \| \| Y \| 0.999 \| 9.2 Simple Linear Regression Statisticians use models as a mathematical formula to describe the relationship between variables. Even with models, we never know the true relationship in practice. In this section, we will introduce the Simple Linear Regression (SLR) Model. In simple linear regression, there is one quantitative response and one quantitative predictor variable, and we describe the relationship using a linear model. In the linear regression model view, we want to see what happens to the response variable when we change the predictor variable. If the value of the predictor variable increases, does the response tend to increase, decrease, or stay constant? We use the slope to address whether or not there is a linear relationship between the two variables. If the average response variable does not change when we change the predictor variable, then the relationship is not a predictive one using a linear model. In other words, if the population slope is 0, then there is no linear relationship. In this section, we present the model, hypotheses, and the assumptions for this test. 9.2.1 The SLR Model Before we set up the model, we should clearly define our notation. The variable Y is the response variable and y 1,y 2,…,y n are the observed values of the response, Y. The variable X is the predictor variable and x 1,x 2,...x n are observed values of the predictor, X. The observations are considered as coordinates, (x i,y i), for i=1,…,n. As we saw before, the points, (x 1,y 1),…,(x n,y n), may not fall exactly on a line, (like the weight and height example). There is some error we must consider. We combine the linear relationship along with the error in the simple linear regression model. Simple Linear Regression Model The general form of the simple linear regression model is… Y=β 0+β 1 X+ϵ For an individual observation, y i=β 0+β 1 x i+ϵ i where, β 0 is the population y-intercept, β 1 is the population slope, and ϵ i is the error or deviation of y i from the line, β 0+β 1 x i. To make inferences about these unknown population parameters, we must find an estimate for them. There are different ways to estimate the parameters from the sample. In this class, we will present the least squares method. 9.4 (Least Squares Line) The least squares line is the line for which the sum of squared errors of predictions for all sample points is the least. Note! When writing the least squares regression line, one must put the “hat” on top of y to distinguish predicted response from the observed response. Using the least squares method, we can find estimates for the two parameters. The formulas to calculate least squares estimates are: Sample Slope β^1=∑(x i−x¯)(y i−y¯)∑(x i−x¯)2 Sample Intercept β^0=y¯−β^1 x¯ Note! You will not be expected to memorize these formulas or to find the estimates by hand. We will use Minitab to find these estimates for you. We estimate the population slope, β 1, with the sample slope denoted β 1^. The population intercept, β 0, is estimated with the sample intercept denoted β 0^. The intercept is often referred to as the constant or the constant term. Once the parameters are estimated, we have the least square regression equation line (or the estimated regression line). Least Squares Regression Equation y^=β^0+β^1 x We can also use the least squares regression line to estimate the errors, called residuals. 9.5 (Residual)ϵ^i=y i−y^i is the observed error, typically called the residual. The graph below summarizes the least squares regression for the height and weight data. Select the icons to view the explanations of the different parts of the scatterplot and the least squares regression line. We will go through this example in more detail later in the lesson. 9.2.2 Interpreting the Coefficients Once we have the estimates for the slope and intercept, we need to interpret them. Recall from the beginning of the Lesson what the slope of a line means algebraically. If the slope is denoted as m, then m=change in y change in x In other words, the slope of a line is the change in the y variable over the change in the x variable. If the change in the x variable is one, then the slope is: m=change in y 1 The slope is interpreted as the change of y for a one unit increase in x. This is the same idea for the interpretation of the slope of the regression line. Interpreting the slope of the regression equation,β^1 β^1 represents the estimated increase in Y per unit increase in X. Note that the increase may be negative which is reflected when β^1 is negative. Again going back to algebra, the intercept is the value of y when x=0. It has the same interpretation in statistics. Interpreting the intercept of the regression equation,β^0 β^0 is the Y-intercept of the regression line. When X=0 is within the scope of observation, β^0 is the estimated value of Y when X=0. Note, however, when X=0 is not within the scope of the observation, the Y-intercept is usually not of interest. Example 9.3 (Student height and weight (Interpreting the coefficients)) Suppose we found the following regression equation for weight vs.height. weight^=−222.5+5.49 height Interpret the slope of the regression equation. Does the intercept have a meaningful interpretation? If so, interpret the value. Answer Answer A slope of 5.49 represents the estimated change in weight (in pounds) for every increase of one inch of height. A height of zero, or X=0 is not within the scope of the observation since no one has a height of 0. The value β^0 by itself is not of much interest other than being the constant term for the regression line. If the slope of the line is positive, then there is a positive linear relationship, i.e., as one increases, the other increases. If the slope is negative, then there is a negative linear relationship, i.e., as one increases the other variable decreases. If the slope is 0, then as one increases, the other remains constant, i.e., no predictive relationship. Therefore, we are interested in testing the following hypotheses: H 0:β 1=0 H a:β 1≠0 There are some assumptions we need to check (other than the general form) to make inferences for the population parameters based on the sample values. We will discuss these topics in the next section. 9.2.3 Assumptions for the SLR Model In this section, we will present the assumptions needed to perform the hypothesis test for the population slope: H 0:β 1=0 H a:β 1≠0 We will also demonstrate how to verify if they are satisfied. To verify the assumptions, you must run the analysis in Minitab first. Assumptions for Simple Linear Regression Linearity: The relationship between X and Y must be linear. Check this assumption by examining a scatterplot of x and y. Independence of errors: The errors ϵ 1,ϵ 2,...,ϵ n are independent. To check the validity of this assumption, see how the data was collected. Based on the method of data collection, think if one observation has anything to do with another. For example, in a situation where the response was observed over time, independence of errors may not be a reasonable assumption.” Normality of errors: The residuals must be approximately normally distributed. Check this assumption by examining a normal probability plot; the observations should be near the line. You can also examine a histogram of the residuals; it should be approximately normally distributed. Equal variances: The variance of the residuals is the same for all values of X. Check this assumption by examining the scatterplot of “residuals versus fits”; the variance of the residuals should be the same across all values of the x-axis. If the plot shows a pattern (e.g., bowtie or megaphone shape), then variances are not consistent, and this assumption has not been met. Example 9.4 (Student height and weight (SLR Assumptions)) Recall that we would like to see if height is a significant linear predictor of weight. Check the assumptions required for simple linear regression. The data can be found here university_ht_wt.txt. The first three observations are: \| Height (inches) \| Weight (pounds) \| --- \| \| 72 \| 200 \| \| 68 \| 165 \| \| 69 \| 160 \| To check the assumptions, we need to run the model in Minitab. Using Minitab to Fit a Regression Model To find the regression model using Minitab… To check linearity create the fitted line plot by choosing Stat>Regression>Fitted Line Plot. For the other assumptions run the regression model. Select Stat>Regression>Regression>Fit Regression Model In the ‘Responses’ box, specify the desired response variable. In the ‘Continuous Predictors’ box, specify the desired predictor variable. Choose Graphs. In ‘Residuals plots’, choose ‘Four in one.’ Select OK and OK. Note! Of the ‘four in one’ graphs, you will only need the Normal Probability Plot, and the Versus Fits graphs to check the assumptions 3-4. The basic regression analysis output is displayed in the session window. But we will only focus on the graphs at this point. The graphs produced allow us to check our assumptions. Assumption 1: Linearity - The relationship between height and weight must be linear. Fig 9.6 The scatterplot shows that, in general, as height increases, weight increases. There does not appear to be any clear violation that the relationship is not linear. Assumption 2: Independence of errors - There is not a relationship between the residuals and weight. It is a random sample of students. So, one person’s weight has nothing to do with another person. Therefore, independence of the errors is a reasonable assumption. Remove the plot and comments about the plot. Assumption 3: Normality of errors - The residuals must be approximately normally distributed. Fig 9.7 Most of the data points fall close to the line, but there does appear to be a slight curving. There is one data point that stands out. Assumption 4: Equal Variances - The variance of the residuals is the same for all values of X. Fig 9.8 In this plot, there does not seem to be a pattern. All of the assumptions except for the normal assumption seem valid. 9.2.4 Inferences about the Population Slope In this section, we will present the hypothesis test and the confidence interval for the population slope. A similar test for the population intercept, β 0, is not discussed in this class because it is not typically of interest. Hypothesis Test for the Population Slope \| Research Question \| Is there a linear relationship? \| Is there a positive linear relationship? \| Is there a negative linear relationship? \| --- --- \| \| Null Hypothesis \| β 1=0 \| β 1=0 \| β 1=0 \| \| Alternative Hypothesis \| β 1≠0 \| β 1>0 \| β 1<0 \| \| Type of Test \| Two-tailed, non-directional \| Right-tailed, directional \| Left-tailed, directional \| The test statistic for the test of population slope is: t∗=β^1 S E^(β^1) where S E^(β^1) is the estimated standard error of the sample slope (found in Minitab output). Under the null hypothesis and with the assumptions shown in the previous section, t∗ follows a t-distribution with n−2 degrees of freedom. Note! In this class, we will have Minitab perform the calculations for this test. Minitab’s output gives the result for two-tailed tests for β 1 and β 0. If you wish to perform a one-sided test, you would have to adjust the p-value Minitab provides. Confidence Interval for the Population Slope (1−α)100% Confidence Interval for the Population Slope The (1−α)100% confidence interval for β 1 is: β^1±t α/2(S E^(β^1)) where t has n−2 degrees of freedom. Note! The degrees of freedom of t depend on the number of independent variables. The degrees of freedom is n−2 when there is only one independent variable. 9.2.5 Other Inferences and Considerations Inferences about Mean Response for New Observation Let’s go back to the height and weight example: Example If you are asked to estimate the weight of a STAT 500 student, what will you use as a point estimate? If I tell you that the height of the student is 70 inches, can you give a better estimate of the person’s weight? Now that we have our regression equation, we can use height to provide a better estimate of weight. We would want to report a mean response value for the provided height, i.e.70 inches. The mean response at a given X value is given by: E(Y)=β 0+β 1 X This is an unknown but fixed value. The point estimate for mean response at X=x is given by: β^0+β^1 x The example for finding this mean response for height and weight is shown later in the lesson. Inferences about Outcome for New Observation The point estimate for the outcome at X=x is provided above. The interval to estimate the mean response is called the confidence interval. Minitab calculates this for us. The interval used to estimate (or predict) an outcome is called prediction interval. For a given x value, the prediction interval and confidence interval have the same center, but the width of the prediction interval is wider than the width of the confidence interval. That makes good sense since it is harder to estimate a value for a single subject (say predict your weight based on your height) than it would be to estimate the average for subjects (say predict the mean weight of people who are your height). Again, Minitab will calculate this interval as well. Minitab: Find the Condidence and Prediction Intervals Steps: Fit the regression model (if not already done). Generate the prediction and intervals: Go to Stat>Regression>Regression>Predict. In the dialog box: Under Predictors, enter the value of the predictor variable (e.g., 70 if height is the X variable). Select OK. The output provides the point estimate obtained by plugging 70 into the fitted model along with confidence and prediction intervals when the height is 70 inches. Cautions with Linear Regression First, use extrapolation with caution. Extrapolation is applying a regression model to X-values outside the range of sample X-values to predict values of the response variable Y. For example, you would not want to use your age (in months) to predict your weight using a regression model that used the age of infants (in months) to predict their weight. Second, the fact that there is no linear relationship (i.e.correlation is zero) does not imply there is no relationship altogether. The scatter plot will reveal whether other possible relationships may exist. The figure below gives an example where X and Y are related, but not linearly related i.e.the correlation is zero. Outliers and Influential Observations Influential observations are points whose removal causes the regression equation to change considerably. It is flagged by Minitab in the unusual observation list and denoted as X. Outliers are points that lie outside the overall pattern of the data. Potential outliers are flagged by Minitab in the unusual observation list and denoted as R. The following is the Minitab output for the unusual observations within the height and weight example: Fits and Diagnostics for Unusual Observations \| Obs \| weight \| Fit \| Resid \| Std Resid \| \| --- --- --- \| \| 24 \| 200.00 \| 139.74 \| 60.26 \| 3.23 \| R \| R Large residual Some observations may be both outliers and influential, and these are flagged by R and X (R X). Those observational points will merit particular attention. In our height and weight example, we have an R (potential outlier) observation, but it is not an influential point (RX observation). Estimating the standard deviation of the error term We can estimate the standard deviation of the error by finding the standard deviation of the residuals, e p s i l o n^i=y i−y^i. Our simple linear regression model is: Y=β 0+β 1 X+ϵ The errors for the n observations are denoted as ϵ i, for i=1,…,n. One of our assumptions is that the errors have equal variance (or equal standard deviation). We can estimate the standard deviation of the error by finding the standard deviation of the residuals, ϵ i=y i−y^i. Minitab also provides the estimate for us, denoted as S, under the Model Summary. We can also calculate it by: s=MSE Find the MSE in the ANOVA table, under the MS column and the Error row. 9.2.6 Examples In this section, we present an example and review what we have covered so far in the context of the example. Example 9.5 (Student height and weight (SLR)) We will continue with our height and weight example. Answer the following questions. Is height a significant linear predictor of weight? Conduct the test at a significance level of 5%. State the regression equation. Does β 0 have a meaningful interpretation? Find the confidence interval for the population slope and interpret it in the context of the problem. If a student is 70 inches, what weight could we expect? What is the estimated standard deviation of the error? Answer The model for this problem is: weight=β 0+β 1 height+ϵ The hypotheses we are testing are: H 0:β 1=0 H a:β 1≠0 Recall that we previously examined the assumptions. Here is a summary of what we presented before: Assumptions 1. Linearity: The relationship between height and weight must be linear. The scatterplot shows that, in general, as height increases, weight increases. There does not appear to be any clear violation that the relationship is not linear. 2. Independence of errors: Since it’s a random sample, one person’s weight has nothing to do with another person. 3. Normality of errors: The residuals must be approximately normally distributed. Most of the data points fall close to the line, but there does appear to be a slight curving. There is one data point that clearly stands out. In the histogram, we can see that, with that one observation, the shape seems slightly right-skewed. 4. Equal variances: The variance of the residuals is the same for all values of X. In this plot, there does not seem to be a pattern. All of the assumptions except for the normal assumption seem valid. Minitab output for the height and weight data: Regression Equation weight = -222.5 + 5.49 height Coefficients \| Term \| Coef \| SE Coef \| T-Value \| P-Value \| VIF \| --- --- --- \| \| Constant \| -222.5 \| 72.4 \| -3.07 \| 0.005 \| \| \| height \| 5.49 \| 1.06 \| 5.16 \| 0.000 \| 1.00 \| Model Summary \| S \| R-sq \| R-sq(adj) \| R-sq(pred) \| --- --- \| \| 19.1108 \| 50.57% \| 48.67% \| 44.09% \| The regression equation is: weight^=−222.5+5.49 height The slope is 5.49, and the intercept is -222.5. The test for the slope has a p-value of less than 0.001. Therefore, with a significance level of 5%, we can conclude that there is enough evidence to suggest that height is a significant linear predictor of weight. We should make this conclusion with caution, however, since one of our assumptions might not be valid. The intercept is -222.5. Therefore, when height is equal to 0, then a person’s weight is predicted to be -222.5 pounds. It is also not possible for someone to have a height of 0 inches. Therefore, the intercept does not have a valid meaning. The confidence interval for the population slope is: β^1±t α/2 S E^(β^1) The estimate for the slope is 5.49 and the standard error for the estimate (SE Coef in the output) is 1.06. There are n=28 observations so the degrees of freedom are 28−2=26. Using Minitab, we find the t-value to be 2.056. Putting the pieces together, the interval is: 5.49±2.056(1.06) (3.31,7.67) We are 95% confident that the population slope is between 3.31 and 7.67. In other words, we are 95% confident that, as height increases by one inch, that weight increases by between 3.31 and 7.67 pounds, on average. Using the regression formula with a height equal to 70 inches, we get: weight^=−222.5+5.49(70)=161.8 For a student with a height of 70 inches, we would expect a weight of 162.3 pounds. If we wanted, we could have Minitab produce a confidence interval for this estimate. We will leave this out for this example. Using the output, under the model summary: s=19.1108 Try It The No-Lemon used car dealership in a college town records the age (in years) and price of cars it sold in the last year (cars_sold.txt). The table is a preview of this data. \| age \| price \| --- \| \| 4 \| 6200 \| \| 4 \| 5700 \| \| 4 \| 6800 \| \| 5 \| 5600 \| \| … \| … \| \| 11 \| 2600 \| Using the data above, answer the following questions. Is age a significant negative linear predictor of price? Conduct the test at a significance level of 5%. Does β 0 have a meaningful interpretation? Find the confidence interval for the population slope and interpret it in the context of the problem. If a car is seven years old, what price could we expect? What is the estimate of the standard deviation of the errors? Answer a. b. c. d. e. The linear regression model is: price=β 0+β 1 age+ϵ To test whether age is a statistically significant negative linear predictor of price, we can set up the following hypotheses:. H 0:β 1=0 H a:β 1<0 We need to verify that our assumptions are satisfied. Let’s do this in Minitab. Remember, we have to run the linear regression analysis to check the assumptions. Assumption 1: Linearity The scatterplot below shows that the relationship between age and price scores is linear. There appears to be a strong negative linear relationship and no obvious outliers. Assumption 2: Independence of errors In this setting the price of one car in the probably has nothing to do with the price of another car. Therefore, the independence of the errors seems to be a reasonable assumption. Assumption 3: Normality of errors On the normal probability plot, we are looking to see if our observations follow the given line. This graph does not indicate that there is a violation of the assumption that the errors are normal. If a probability plot is not an option we can refer back to one of our first lessons on graphing quantitative data and use a histogram or boxplot to examine if the residuals appear to follow a bell shape. Assumption 4: Equal Variances Again, we will use the plot of residuals versus fits. Now we are checking that the variance of the residuals is consistent across all fitted values. This assumption seems valid. Regression Equation price = 7850 - 485 age Coefficients \| Term \| Coef \| SE Coef \| T-Value \| P-Value \| VIF \| --- --- --- \| \| Constant \| 7850 \| 362 \| 21.70 \| 0.000 \| \| \| age \| -485.0 \| 43.9 \| -11.04 \| 0.000 \| 1.00 \| Model Summary \| S \| R-sq \| R-sq(adj) \| R-sq(pred) \| --- --- \| \| 503.146 \| 88.39% \| 87.67% \| 84.41% \| From the output above we can see that the p-value of the coefficient of age is 0.000 which is less than 0.001. The Minitab output is for a two-tailed test and we are dealing with a left-tailed test. Therefore, the p-value for the left-tailed test is less than 0.001 2 or less than 0.0005. We can thus conclude that age (in years) is a statistically significant negative linear predictor of price for any reasonable α value. β 0 is the y-intercept, which means it is the value of price when age is equal to 0. It is possible for a vehicle to have the number of years equal to 0. Therefore, it does have an interpretable meaning. We should use caution if we use this model to predict the price of a car with age equal to 0 because it is outside the range of values used to estimate the model. The 95% confidence interval for the population slope is: β^1±t α/2 SE(β^1) Using the output, β^1=−485 and the SE(β^1)=43.9. We need to have t α/2 with n−2 degrees of freedom. In this case, there are 18 observations so the degrees of freedom are 18−2=16. Using software, we find t α/2=2.12. The 95% confidence interval is: −485±2.12(43.9) (−578.068,−391.932) We are 95% confident that the population slope for the regression model is between -578.068 and -391.932. In other words, we are 95% confident that, for every one year increase in age, the price of a vehicle will decrease between 391.932 and 578.068 dollars. We can use the regression equation with age=7: price^=7850−485(7)=4455 We can expect the price to be $4455. The residual standard error is estimated by s, which is calculated as: s=MSE=253156=503.146 Note! The MSE is found in the ANOVA table which is part of the regression output in Minitab. It is also shown as s under the model summary in the output. 9.3 Coefficient of Determination Now that we know how to estimate the coefficients and perform the hypothesis test, is there any way to tell how useful the model is? One measure is the coefficient of determination, denoted R 2. 9.6 (Coefficient of Determination R 2) The coefficient of determination measures the percentage of variability within the y-values that can be explained by the regression model. Therefore, a value close to 100% means that the model is useful and a value close to zero indicates that the model is not useful. It can be shown by mathematical manipulation that: SST=SSR+SSE ∑(y i−y¯)2=∑(y^i−y¯)2+∑(y i−y^i)2 Total variability in the y value = Variability explained by the model + Unexplained variability To get the total, explained, and unexplained variability, first, we need to calculate corresponding deviances. Drag the slider on the image below to see how the total deviance (y i−y¯) is split into explained (y^i−y¯) and unexplained deviances (y i−y^i). The breakdown of variability in the above equation holds for the multiple regression model also. Coefficient of Determination R 2 Formula R 2=variability explained by the model total variability in the y values R 2 represents the proportion of total variability of the y-value that is accounted for by the independent variable x. For the specific case when there is only one independent variable X (i.e., simple linear regression), one can show that R 2=r 2, where r is the correlation coefficient between X and Y. Example 9.6 (Student height and weight (R 2)) Let’s take a look at Minitab’s output from the height and weight example (university_ht_wt.txt) that we have been working within this lesson. Regression Equation weight = -222.5 + 5.49 height Coefficients \| Term \| Coef \| SE Coef \| T-Value \| P-Value \| VIF \| --- --- --- \| \| Constant \| -222.5 \| 72.4 \| -3.07 \| 0.005 \| \| \| height \| 5.49 \| 1.06 \| 5.16 \| 0.000 \| 1.00 \| Model Summary \| S \| R-sq \| R-sq(adj) \| R-sq(pred) \| --- --- \| \| 19.1108 \| 50.57% \| 48.67% \| 44.09% \| Find the coefficient of determination and interpret the value. Answer The coefficient of determination, R 2 is 0.5057 or 50.57%. This value means that 50.57% of the variation in weight can be explained by height. Remember, for this example, we found the correlation value, r, to be 0.711. So, we can now see that r 2=(0.711)2=.506 which is the same reported for R-sq in the Minitab output. Try It! Used car sales continued… For the age and price of the car example (cars_sold.txt), what is the value of the coefficient of determination and interpret the value in the context of the problem? Answer Regression Equation price = 7850 - 485 age Coefficients \| Term \| Coef \| SE Coef \| T-Value \| P-Value \| VIF \| --- --- --- \| \| Constant \| 7850 \| 362 \| 21.70 \| 0.000 \| \| \| age \| -485.0 \| 43.9 \| -11.04 \| 0.000 \| 1.00 \| Model Summary \| S \| R-sq \| R-sq(adj) \| R-sq(pred) \| --- --- \| \| 503.146 \| 88.39% \| 87.67% \| 84.41% \| From the Minitab output, we see an R-sq value of 88.39%. We want to report this in terms of the study, so here we would say that 88.39% of the variation in vehicle price is explained by the age of the vehicle. Note! The two other references to R-sq, (adj) and (pred), are used for model comparisons. These two metrics do not provide any interpretive value to the model in regards to X and Y. 9.4 Inference for Correlation Let’s review the notation for correlation. r: The sample correlation (Pearson’s correlation) ρ: “rho” is the population correlation The sample correlation is found by: r=∑(x i−x¯)(y i−y¯)∑(x i−x¯)2∑(y i−y¯)2 In this section, we will present a hypothesis test for the population correlation. Then, we will compare the tests and interpretations for the slope and correlation. 9.4.1 Hypothesis Testing for the Population Correlation In this section, we present the test for the population correlation using a test statistic based on the sample correlation. Assumptions As with all hypothesis test, there are underlying assumptions. The assumptions for the test for correlation are: The are no outliers in either of the two quantitative variables. The two variables should follow a normal distribution Hypotheses If there is no linear relationship in the population, then the population correlation would be equal to zero. H 0:ρ=0 (X and Y are linearly independent, or X and Y have no linear relationship) H a:ρ≠0 (X and Y are linearly dependent) \| Research Question \| Is there a linear relationship? \| Is there a positive linear relationship? \| Is there a negative linear relationship? \| --- --- \| \| Null Hypothesis \| ρ=0 \| ρ=0 \| ρ=0 \| \| Alternative Hypothesis \| ρ≠0 \| ρ>0 \| ρ<0 \| \| Type of Test \| Two-tailed, non-directional \| Right-tailed, directional \| Left-tailed, directional \| Test Statistic Under the null hypothesis and with the above assumptions, the test statistic, t∗, found by: t∗=r n−2 1−r 2 which follows a t-distribution with n−2 degrees of freedom. As mentioned before, we will use Minitab for the calculations. The output from Minitab previously used to find the sample correlation also provides a p-value. This p-value is for the two-sided test. If the alternative is one-sided, the p-value from the output needs to be adjusted. Example 9.7 (Student height and weight (Tests for ρ)) For the height and weight example, university_ht_wt.txt, conduct a test for correlation with a significance level of 5%. Answer The output from Minitab is: Correlations \| \| height \| --- \| \| weight \| 0.711 \| For the sake of this example, we will find the test statistic and the p-value rather than just using the Minitab output. There are 28 observations. The test statistic is: t∗=r n−2 1−r 2=(0.711)28−2 1−0.711 2=5.1556 Next, we need to find the p-value. The p-value for the two-sided test is: p-value=2 P(T>5.1556)<0.0001 Therefore, for any reasonable α level, we can reject the hypothesis that the population correlation coefficient is 0 and conclude that it is nonzero. There is evidence at the 5% level that Height and Weight are linearly dependent. Try It For the sales and advertising example, conduct a test for correlation with a significance level of 5% with Minitab. Sales units are in thousands of dollars, and advertising units are in hundreds of dollars. \| Sales (Y) \| Advertising (X) \| --- \| \| 1 \| 1 \| \| 1 \| 2 \| \| 2 \| 3 \| \| 2 \| 4 \| \| 4 \| 5 \| Answer The Minitab output gives: Correlations \| \| height \| --- \| \| weight \| 0.904 \| The sample correlation is 0.904. This value indicates a strong positive linear relationship between sales and advertising. For the Sales (Y) and Advertising (X) data, the test statistic is… t∗=(0.904)5−2 1−(0.904)2=3.66 …with df of 3, we arrive at a p-value = 0.035. For α=0.05, we can reject the hypothesis that the population correlation coefficient is 0 and conclude that it is nonzero, i.e., that sales and advertising are linearly dependent. 9.4.2 Comparing Correlation and Slope Some of you may have noticed that the hypothesis tests for correlation and slope are very similar. Also, the test statistic for both tests follows the same distribution with the same degrees of freedom, n−2. This similarity is because the two values are mathematically related. In fact, β^1=r∑(y i−y¯)2∑(x i−x¯)2 Here is a summary of some of the similarities and differences between the sample correlation and the sample slope. Similarities The test for correlation will lead to the same conclusion as the test for slope. The sign of the slope (i.e.negative or positive) will be the same for the correlation. In other words, both values indicate the direction of the relationship Differences The value of the correlation indicates the strength of the linear relationship. The value of the slope does not. The slope interpretation tells you the expected change in the response for a one-unit increase in the predictor. Correlation does not have this kind of interpretation. 9.5 Multiple Regression Model In this section, we (very) briefly discuss Multiple Linear Regression. In practice, it is not usual that there is only one predictor variable. In Multiple Linear Regression, there is one quantitative response and more than one predictor or independent variable. The model will contain the constant or intercept term, β 0, and more than one coefficient, denoted β 1,…,β k, where k is the number of predictors. Multiple Linear Regression Model Y=β 0+β 1 X 1+...+β k X k+ϵ Where Y is the response variable and X 1,…,X k are independent variables. β 0,β 1,…,β k are fixed (but unknown) parameters and ϵ is a random variable that is normally distributed with mean 0 and having a variance σ ϵ 2. F-Test for Overall Significance There is a statistical test we can use to determine the overall significance of the regression model. The F-test in Multiple Linear Regression test the following hypotheses: H 0:β 1=...=β k=0 H a:At least one β i is not equal to zero The test statistic for this test, denoted F∗, follows an F distribution. We will not expect you to understand Multiple Linear Regression. We included it here to show you what you may see in practice. If you are interested in learning more, you can take STAT 501: Regression Methods. 9.6 Lesson Summary In this Lesson, we examined the relationship between a quantitative response (or dependent variable) and a quantitative explanatory variable (or independent variable). We are interested in whether or not the independent variable is a significant linear predictor of the response. In the next Lesson, we consider a different situation. We look back on the example where we have a quantitative response and an explanatory variable with more than two levels. In other words, we look at the situation where we are comparing means from more than two groups. 8 Chi-Square Test for Independence 10 Introduction to ANOVA Source Code ``` categories: [] image: /assets/L9card.png Linear Regression Foundations Overview {.unnumbered .unlisted} In this Lesson, we will first introduce the Simple Linear Regression (SLR) Model and the Correlation Coefficient. Inferences for the simple linear regression model will be discussed, and the critical distinction between inference for mean response and inference for the outcome will be clarified. We will also introduce a basic understanding of the multiple regression model. Regression analysis is a tool to investigate how two or more variables are related. Quite often we want to see how a specific variable of interest is affected by one or more variables. For example, one may wish to use a person's height, gender, race, etc. to predict a person's weight. Let us first consider the simplest case: using a person's height to predict the person's weight. :::{.ms-3 .border-start .border-success .rounded-2 .ps-2} [Example]{.fs-4} \ If you are asked to estimate the weight of a STAT 500 student, what will you use as a point estimate? If I tell you that the height of the student is 70 inches, can you give a better estimate of the person's weight? ::: {.card .card-body .bg-light .ms-3 .mb-3 .pt-0} Answer For the first part, the point estimate would be the average weight (or the median weight) of a STAT 500 student. If you know the student is 70 inches, then, yes, you can give a better estimate of the person’s weight, but only if you have some idea about how height and weight are related. ::: ::: It is important to distinguish between the variable of interest and the variable(s) we will use to predict the variable of interest. ::: {#def-ResponseVariable .ms-3} Response Variable Denoted, Y, is also called the variable of interest or dependent variable. In the example, weight is the response variable. ::: ::: {#def-PredictorVariable .ms-3} Predictor Variable Denoted, X, is also called the explanatory variable or independent variable. In the example, height is the predictor. ::: When there is only one predictor variable, we refer to the regression model as a simple linear regression model. To use known information to provide a better estimate, we need to understand how the dependent and independent variables are related. In statistics, we can describe how variables are related using a mathematical function. The function along with other assumptions is called a model. There are many models we can consider. In this class, we will focus on linear models, particularly, when there is only one predictor variable. We refer to this model as the simple linear regression model. ::: objectiveblock [Objectives]{.callout-header} Upon completion of this lesson, you should be able to: Use plots and summary statistics to describe the relationship between the response variable and the predictor variable. Perform a hypothesis test for the population correlation. Find the regression equation and interpret the results. Apply the regression model and know the limitations. Find an interval estimate for the population slope and interpret the interval. ::: Linear Relationships To define a useful model, we must investigate the relationship between the response and the predictor variables. As mentioned before, the focus of this Lesson is linear relationships. For a brief review of linear functions, recall that the equation of a line has the following form: $$y=mx+b$$ where m is the slope and b is the y-intercept. Given two points on a line, $(x_1, y_1)$ and $(x_2, y_2)$, the slope is calculated by: $$\begin{align} m&=\dfrac{y_2-y_1}{x_2-x_1}\&=\dfrac{\text{change in y}}{\text{change in x}}\&=\frac{\text{rise}}{\text{run}} \end{align}$$ The slope of a line describes a lot about the linear relationship between two variables. If the slope is positive, then there is a positive linear relationship, i.e., as one increases, the other increases. If the slope is negative, then there is a negative linear relationship, i.e., as one increases the other variable decreases. If the slope is 0, then as one increases, the other remains constant. When we look for linear relationships between two variables, it is rarely the case where the coordinates fall exactly on a straight line; there will be some error. In the next sections, we will show how to examine the data for a linear relationship (i.e., the scatterplot) and how to find a measure to describe the linear relationship (i.e., correlation). Scatterplots If the interest is to investigate the relationship between two quantitative variables, one valuable tool is the scatterplot. ::: {#def-Scatterplot .ms-3} Scatterplot A graphical representation of two quantitative variables where the explanatory variable is on the x-axis and the response variable is on the y-axis. ::: When we look at the scatterplot, keep in mind the following questions: What is the direction of the relationship? Is the relationship linear or nonlinear? Is the relationship weak, moderate, or strong? Are there any outliers or extreme values? We describe the direction of the relationship as positive or negative. A positive relationship means that as the value of the explanatory variable increases, the value of the response variable increases, in general. A negative relationship implies that as the value of the explanatory variable increases, the value of the response variable tends to decrease. ::::: {#exm-studenthtwt} Student height and weight (Scatterplots) {.unnumbered .unlisted} \ Suppose we took a random sample from students at a large university and asked them about their height and weight. The data can be found here university_ht_wt.csv{download="" target="_blank"}. The first three observations are: \| Height (inches) \| Weight (pounds) \| :---------------:\| \| 72 \| 200 \| \| 68 \| 165 \| \| 69 \| 160 \| : {.w-auto .table-sm .table-responsive .mx-auto} We let $X$ denote the height and $Y$ denote the weight of the student. The observations are then considered as coordinates $(x,y)$. For example, student 1 has coordinates (72,200). These coordinates are plotted on the x-y plane. We can use Minitab to create the scatterplot. ::: {.mticon .float-end .d-flex} ::: ::: {.ms-3 .border-start .border-secondary-subtle .ps-3} We can create our scatterplot in Minitab following these steps. Choose Graph > Scatterplot > Simple For the Y-variable: select 'weight' and for the X-variable: select 'height'. Choose OK. {#fig-htwtscatter .mx-auto .d-block .lightbox fig-alt="Scatterplot with height on x-axis and weight on y-axis of the student data" width="60%"} ::: The scatterplot shows that, in general, as height increases, weight increases. We say “in general” because it is not always the case. For example, the observation with a height of 66 inches and a weight of 200 pounds does not seem to follow the trend of the data. The two variables seem to have a positive relationship. As the height increases, weight tends to increase as well. The relationship does not seem to be perfectly linear, i.e., the points do not fall on a straight line, but it does seem to follow a straight line moderately, with some variability. ::::: Try It! {.unnumbered .unlisted} ::::: tryit An elementary school teacher gives her students two spelling tests a year. Each test contains 24 words, and the score is the number of words spelled correctly. The teacher is interested in the relationship between the score on the first test and the score on the second test. Using the scatterplot, comment on the relationship between the two variables. {#fig-500-l9-try-it-spelling-tests .mx-auto .d-block .lightbox fig-alt="Scatter plot showing the score of the first tests on the x-axis and the second on the y-axis" width="60%"} Answer :::: {#collapsespelling .collapse} ::: {.card .card-body .bg-light} There seems to be a weak positive linear relationship between the two test scores. ::: :::: ::::: In the next section, we will introduce correlation. Correlation is a measure that gives us an idea of the strength and direction of the linear relationship between two quantitative variables. Correlation If we want to provide a measure of the strength of the linear relationship between two quantitative variables, a good way is to report the correlation coefficient between them. The sample correlation coefficient is typically denoted as $r$. It is also known as Pearson’s $r$. The population correlation coefficient is generally denoted as $\rho$, pronounced “rho.” ::: {.callout-note appearance="minimal"} [Sample Correlation Coefficient]{.lead} The sample correlation coefficient, $r$, is calculated using the following formula: $$r=\dfrac{\sum (x_i-\bar{x})(y_i-\bar{y}) }{\sqrt{\sum (x_i-\bar{x})^2}\sqrt{\sum (y_i-\bar{y})^2}}$$ ::: Properties of the correlation coefficient, $r$: {.unnumbered .unlisted} $-1\le r\le 1$, i.e. $r$ takes values between -1 and +1, inclusive. The sign of the correlation provides the direction of the linear relationship. The sign indicates whether the two variables are positively or negatively related. A correlation of 0 means there is no linear relationship. There are no units attached to $r$. As the magnitude of $r$ approaches 1, the stronger the linear relationship. As the magnitude of $r$ approaches 0, the weaker the linear relationship. If we fit the simple linear regression model between Y and X, then $r$ has the same sign as $\beta_1$, which is the coefficient of X in the linear regression equation. -- more on this later. The correlation value would be the same regardless of which variable we defined as X and Y. ::: {.callout-caution appearance="minimal"} Note!\ The correlation is unit free. We can see this easier using the equation above. Consider, for example, that we are interested in the correlation between X = height (inches) and Y = weight (pounds). In the equation above, the numerator would have the units of $\text{pounds}^\text{inches}$. The denominator would include taking the square root of pounds squared and inches squared, leaving us again with units of $\text{pounds}^\text{inches}$. Therefore the units would cancel out. ::: ::::: grid ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} {#fig- .mx-auto .d-block .lightbox fig-alt="Scatter plot with a positive linear correlation (increasing from left to right)" width="80%"} ::: ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} {#fig-scatternegative .mx-auto .d-block .lightbox fig-alt="Scatterplot with a negative linear association (decreaseing from left to right)" width="80%"} ::: ::::: ::::: grid ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} {#fig-scatternone .mx-auto .d-block .lightbox fig-alt="Scatterplot with no linear association" width="80%"} ::: ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} {#fig-scatterparabola .mx-auto .d-block .lightbox fig-alt="Scatter plot with a downward facing parabolic trend" width="80%"} ::: ::::: :::::::::: {#exm-salesads} Sales and Advertising (Correlation) {.unnumbered .unlisted} \ We have collected five months of sales and advertising dollars for a small company we own. Sales units are in thousands of dollars, and advertising units are in hundreds of dollars. Our interest is determining if a linear relationship exists between sales and advertising. The data is as follows: \| Sales (Y) \| Advertising (X) \| :---------------:\| \| 1 \| 1 \| \| 1 \| 2 \| \| 2 \| 3 \| \| 2 \| 4 \| \| 4 \| 5 \| : {.w-auto .table-sm .table-responsive .mx-auto} Find the sample correlation and interpret the value. ::::::::: {.card .card-body .bg-light .ms-3 .mb-3 .pt-0} Answer :::::::: panel-tabset By Hand The mean of Sales ($Y$) is $\bar{y}=2$ and the mean of advertising ($X$) is $\bar{x}=3$. We can calculate the sample correlation in steps. \| $y_i-\bar{y}$ \| $x_i-\bar{x}$ \| $(x_i-\bar{x})(y_i-\bar{y})$ \| :-------------: \| $1-2=-1$ \| $1-3=-2$ \| $(-1)(-2)=2$ \| \| $1-2=-1$ \| $2-3=-1$ \| $(-1)(-1)=1$ \| \| $2-2=0$ \| $3-3=0$ \| $(0)(0)=0$ \| \| $2-2=0$ \| $4-3=1$ \| $(0)(1)=0$ \| \| $4-2=2$ \| $5-3=2$ \| $(2)(2)=4$ \| : {.w-auto .table-sm .table-responsive .mx-auto} From the table, we can calculate the following sums... ::: {style="overflow-x:auto;overflow-y:hidden;"} $$\sum(y_i-\bar{y})^2=(-1)^2+(-1)^2+0+0+2^2=6 \;\text{(sum of first column)}$$ $$\sum(x_i-\bar{x})^2=(-2)^2+(-1)^2+0+1^2+2^2=10 \;\text{(sum of second column)}$$ $$\sum(x_i-\bar{x})(y_i-\bar{y})=2+1+0+0+4=7 \;\text{(sum of third column)}$$ ::: Using these numbers in the formula for r... ::: {style="overflow-x:auto;overflow-y:hidden;"} $$r=\dfrac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum(x_i-\bar{x})^2}\sqrt{\sum(y_i-\bar{y})^2}}=\dfrac{7}{\sqrt{10}\sqrt{6}}=0.9037$$ ::: Minitab Using Minitab to calculate $r$ {.unnumbered .unlisted} ::: {.mticon .float-end .d-flex} ::: ::: {.ms-3 .border-start .border-secondary-subtle .ps-3} To calculate r using Minitab: Open Minitab and upload the data (for this example type the Y data into a column (e.g., C1) and the X data into a column (e.g., C2)) Choose Stat > Basic Statistics > Correlation Specify the variables in the dialog box (X and Y in this example). Minitab output for this example: ::: ::: minitab_output Correlation: Sales (Y), Advertising (X) {.unnumbered .unlisted} {.d-block .lightbox fig-alt="Matrix plot of advertising vs sales with the correlation of 0.904" width="60%"} Method \| Correlation type Pearson \| Number of rows used 5 Correlation ```{=html} Sales (Y) Advertising (X) 0.904 ``` The sample correlation is 0.904. This value indicates a strong positive linear relationship between sales and advertising. ::: :::::::: ::::::::: :::::::::: Try It! {.unnumbered .unlisted} ::: {.tryit} Using the following data, calculate the correlation and interpret the value. \| X \| Y \| :---------------:\| \| 2 \| 7 \| \| 4 \| 11 \| \| 14 \| 29 \| \| 13 \| 28 \| \| 15 \| 32 \| : {.w-auto .table-sm .table-responsive .mx-auto} {=html} <button class="btn btn-outline-success collapsed" type="submit" data-bs-toggle="collapse" data-bs-target="#collapsecorrel" aria-expanded="false" aria-controls="collapsecorrel"> Answer </button> :::: {#collapsecorrel .collapse} ::: {.card .card-body .bg-light} :::::::: panel-tabset By Hand The mean of $X$ is 9.6 and the mean of $Y$ is 21.4. The sums are... :::{.ms-3} $\sum (x_i-\bar{x})^2=149.2$ $\sum (y_i-\bar{y})^2=529.2$ $\sum (x_i-\bar{x})(y_i-\bar{y})=280.8$ ::: Using these sums in the formula for r... ::: {style="overflow-x:auto;overflow-y:hidden;"} $$r=\dfrac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum(x_i-\bar{x})^2}\sqrt{\sum(y_i-\bar{y})^2}}=0.9993$$ ::: Minitab Following the steps for finding correlation with Minitab you should get the following output: ::: {.minitab_output} Method \| Correlation type Pearson \| Number of rows used 5 Correlation ```{=html} X Y 0.999 ``` ::: :::::::: ::: :::: ::: Simple Linear Regression Statisticians use models as a mathematical formula to describe the relationship between variables. Even with models, we never know the true relationship in practice. In this section, we will introduce the Simple Linear Regression (SLR) Model. In simple linear regression, there is one quantitative response and one quantitative predictor variable, and we describe the relationship using a linear model. In the linear regression model view, we want to see what happens to the response variable when we change the predictor variable. If the value of the predictor variable increases, does the response tend to increase, decrease, or stay constant? We use the slope to address whether or not there is a linear relationship between the two variables. If the average response variable does not change when we change the predictor variable, then the relationship is not a predictive one using a linear model. In other words, if the population slope is 0, then there is no linear relationship. In this section, we present the model, hypotheses, and the assumptions for this test. The SLR Model Before we set up the model, we should clearly define our notation. The variable $Y$ is the response variable and $y_1, y_2, …, y_n$ are the observed values of the response, $Y$. The variable $X$ is the predictor variable and $x_1, x_2, ...x_n$ are observed values of the predictor, $X$. The observations are considered as coordinates, $(x_i, y_i)$, for $i=1, …, n$. As we saw before, the points, $(x_1,y_1), …,(x_n,y_n)$, may not fall exactly on a line, (like the weight and height example). There is some error we must consider. We combine the linear relationship along with the error in the simple linear regression model. ::: {.callout-note appearance="minimal"} [Simple Linear Regression Model]{.lead} The general form of the simple linear regression model is... $$Y=\beta_0+\beta_1X+\epsilon$$ For an individual observation, $$y_i=\beta_0+\beta_1x_i+\epsilon_i$$ where, $\beta_0$ is the population y-intercept, $\beta_1$ is the population slope, and $\epsilon_i$ is the error or deviation of $y_i$ from the line, $\beta_0+\beta_1x_i$. ::: To make inferences about these unknown population parameters, we must find an estimate for them. There are different ways to estimate the parameters from the sample. In this class, we will present the least squares method. ::: {#def-LeastSquaresLine .ms-3} Least Squares Line The least squares line is the line for which the sum of squared errors of predictions for all sample points is the least. ::: ::: {.callout-caution appearance="minimal"} Note! \ When writing the least squares regression line, one must put the “hat” on top of y to distinguish predicted response from the observed response. ::: Using the least squares method, we can find estimates for the two parameters. The formulas to calculate least squares estimates are: ::::: grid ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12 .text-center} Sample Slope $$\hat{\beta}_1=\dfrac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2}$$ ::: ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12 .text-center} Sample Intercept $$\hat{\beta}_0=\bar{y}-\hat{\beta}_1\bar{x}$$ ::: ::::: ::: {.callout-caution appearance="minimal"} Note!\ You will not be expected to memorize these formulas or to find the estimates by hand. We will use Minitab to find these estimates for you. ::: We estimate the population slope, $\beta_1$, with the sample slope denoted $\hat{\beta_1}$. The population intercept, $\beta_0$, is estimated with the sample intercept denoted $\hat{\beta_0}$. The intercept is often referred to as the constant or the constant term. Once the parameters are estimated, we have the least square regression equation line (or the estimated regression line). ::: {.callout-note appearance="minimal"} [Least Squares Regression Equation]{.lead} $$\hat{y}=\hat{\beta}_0+\hat{\beta}_1x$$ ::: We can also use the least squares regression line to estimate the errors, called residuals. ::: {#def-Residual .ms-3} Residual $\hat{\epsilon}_i=y_i-\hat{y}_i$ is the observed error, typically called the residual. ::: The graph below summarizes the least squares regression for the height and weight data. Select the icons to view the explanations of the different parts of the scatterplot and the least squares regression line. We will go through this example in more detail later in the lesson. ```{=html} ``` Interpreting the Coefficients Once we have the estimates for the slope and intercept, we need to interpret them. Recall from the beginning of the Lesson what the slope of a line means algebraically. If the slope is denoted as $m$, then $$m=\dfrac{\text{change in y}}{\text{change in x}}$$ In other words, the slope of a line is the change in the y variable over the change in the x variable. If the change in the x variable is one, then the slope is: $$m=\dfrac{\text{change in y}}{1}$$ The slope is interpreted as the change of y for a one unit increase in x. This is the same idea for the interpretation of the slope of the regression line. ::: {.callout-tip appearance="minimal"} Interpreting the slope of the regression equation, $\hat{\beta}_1$ $\hat{\beta}_1$ represents the estimated increase in Y per unit increase in X. Note that the increase may be negative which is reflected when $\hat{\beta}_1$ is negative. ::: Again going back to algebra, the intercept is the value of y when $x = 0$. It has the same interpretation in statistics. ::: {.callout-tip appearance="minimal"} Interpreting the intercept of the regression equation, $\hat{\beta}_0$ $\hat{\beta}_0$ is the $Y$-intercept of the regression line. When $X = 0$ is within the scope of observation, $\hat{\beta}_0$ is the estimated value of Y when $X = 0$. ::: Note, however, when $X = 0$ is not within the scope of the observation, the Y-intercept is usually not of interest. :::::: {#exm-studenthtwtinterpret} Student height and weight (Interpreting the coefficients) \ Suppose we found the following regression equation for weight vs. height. $$\hat{\text{weight }}=-222.5 +5.49\text{ height }$$ a. Interpret the slope of the regression equation. b. Does the intercept have a meaningful interpretation? If so, interpret the value. Answer ::::: {#collapsestudentsint .collapse} :::: {.card .card-body} ::: {.card .card-body .bg-light .ms-3 .mb-3 .pt-0} Answer a. A slope of 5.49 represents the estimated change in weight (in pounds) for every increase of one inch of height. b. A height of zero, or $X = 0$ is not within the scope of the observation since no one has a height of 0. The value $\hat{\beta}_0$ by itself is not of much interest other than being the constant term for the regression line. ::: :::: ::::: :::::: \ If the slope of the line is positive, then there is a positive linear relationship, i.e., as one increases, the other increases. If the slope is negative, then there is a negative linear relationship, i.e., as one increases the other variable decreases. If the slope is 0, then as one increases, the other remains constant, i.e., no predictive relationship. Therefore, we are interested in testing the following hypotheses: \| $H_0\colon \beta_1=0$ \| $H_a\colon \beta_1\ne0$ \ There are some assumptions we need to check (other than the general form) to make inferences for the population parameters based on the sample values. We will discuss these topics in the next section. Assumptions for the SLR Model In this section, we will present the assumptions needed to perform the hypothesis test for the population slope: \| $H_0\colon \beta_1=0$ \| $H_a\colon \beta_1\ne0$ \ We will also demonstrate how to verify if they are satisfied. To verify the assumptions, you must run the analysis in Minitab first. Assumptions for Simple Linear Regression {.unnumbered .unlisted} Linearity: The relationship between $X$ and $Y$ must be linear. Check this assumption by examining a scatterplot of x and y. Independence of errors: The errors $\epsilon_1, \epsilon_2,..., \epsilon_n$ are independent. To check the validity of this assumption, see how the data was collected. Based on the method of data collection, think if one observation has anything to do with another. For example, in a situation where the response was observed over time, independence of errors may not be a reasonable assumption.” Normality of errors: The residuals must be approximately normally distributed. Check this assumption by examining a normal probability plot; the observations should be near the line. You can also examine a histogram of the residuals; it should be approximately normally distributed. Equal variances: The variance of the residuals is the same for all values of $X$. Check this assumption by examining the scatterplot of “residuals versus fits”; the variance of the residuals should be the same across all values of the x-axis. If the plot shows a pattern (e.g., bowtie or megaphone shape), then variances are not consistent, and this assumption has not been met. ::: {#exm-studenthtwtslrassumption} Student height and weight (SLR Assumptions) \ Recall that we would like to see if height is a significant linear predictor of weight. Check the assumptions required for simple linear regression. The data can be found here university_ht_wt.txt{download="" target="_blank"}. The first three observations are: \| Height (inches) \| Weight (pounds) \| :---------------:\| \| 72 \| 200 \| \| 68 \| 165 \| \| 69 \| 160 \| : {.w-auto .table-sm .table-responsive .mx-auto} To check the assumptions, we need to run the model in Minitab. ::: {.card .card-body .bg-light .ms-3 .mb-3 .pt-0} Using Minitab to Fit a Regression Model {.unnumbered .unlisted} ::: {.ms-3 .border-start .border-secondary-subtle .ps-3} To find the regression model using Minitab... ::: {.mticon .float-end .d-flex} ::: To check linearity create the fitted line plot by choosing Stat > Regression > Fitted Line Plot. For the other assumptions run the regression model. Select Stat > Regression > Regression > Fit Regression Model In the 'Responses' box, specify the desired response variable. In the 'Continuous Predictors' box, specify the desired predictor variable. Choose Graphs. In 'Residuals plots', choose 'Four in one.' Select OK and OK. ::: ::: {.callout-caution appearance="minimal"} Note! \ Of the 'four in one' graphs, you will only need the Normal Probability Plot, and the Versus Fits graphs to check the assumptions 3-4. ::: The basic regression analysis output is displayed in the session window. But we will only focus on the graphs at this point. The graphs produced allow us to check our assumptions. ::: {.grid} ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} Assumption 1: Linearity - The relationship between height and weight must be linear. {#fig-fittedlinestudenthtwt fig-alt="Scatter plot with the fitted line for weight vs height in the university student data" .mx-auto .d-block width="80%" .lightbox} The scatterplot shows that, in general, as height increases, weight increases. There does not appear to be any clear violation that the relationship is not linear. ::: ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} Assumption 2: Independence of errors - There is not a relationship between the residuals and weight. It is a random sample of students. So, one person's weight has nothing to do with another person. Therefore, independence of the errors is a reasonable assumption. Remove the plot and comments about the plot. ::: ::: ::: {.grid} ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} Assumption 3: Normality of errors - The residuals must be approximately normally distributed. {#fig-normpropplotstudenthtwt fig-alt="Normal probability plot for weight vs height" .mx-auto .d-block width="80%" .lightbox} Most of the data points fall close to the line, but there does appear to be a slight curving. There is one data point that stands out. ::: ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} Assumption 4: Equal Variances - The variance of the residuals is the same for all values of $X$. {#fig-vsfitsstudenthtwt fig-alt="Versus Fits graph for student weight vs height data" .mx-auto .d-block width="80%" .lightbox} In this plot, there does not seem to be a pattern. ::: ::: All of the assumptions except for the normal assumption seem valid. ::: ::: Inferences about the Population Slope In this section, we will present the hypothesis test and the confidence interval for the population slope. A similar test for the population intercept, $\beta_0$, is not discussed in this class because it is not typically of interest. Hypothesis Test for the Population Slope {.unnumbered .unlisted} ```{=html} Research Question Is there a linear relationship? Is there a positive linear relationship? Is there a negative linear relationship? Null Hypothesis $\beta_1=0$ $\beta_1=0$ $\beta_1=0$ Alternative Hypothesis $\beta_1\ne0$ $\beta_1>0$ $\beta_1<0$ Type of Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional ``` The test statistic for the test of population slope is: $$t^=\dfrac{\hat{\beta}_1}{\hat{SE}(\hat{\beta}_1)}$$ where $\hat{SE}(\hat{\beta}_1)$ is the estimated standard error of the sample slope (found in Minitab output). Under the null hypothesis and with the assumptions shown in the previous section, $t^$ follows a $t$-distribution with $n-2$ degrees of freedom. ::: {.callout-caution appearance="minimal"} Note! \ In this class, we will have Minitab perform the calculations for this test. Minitab's output gives the result for two-tailed tests for $\beta_1$ and $\beta_0$. If you wish to perform a one-sided test, you would have to adjust the p-value Minitab provides. ::: Confidence Interval for the Population Slope {.unnumbered .unlisted} ::: {.callout-note appearance="minimal"} [$(1-\alpha)100$% Confidence Interval for the Population Slope]{.lead} The $(1-\alpha)100$% confidence interval for $\beta_1$ is: $$\hat{\beta}1\pm t{\alpha/2}\left(\hat{SE}(\hat{\beta}_1)\right)$$ where $t$ has $n-2$ degrees of freedom. ::: ::: {.callout-caution appearance="minimal"} Note! \ The degrees of freedom of t depend on the number of independent variables. The degrees of freedom is $n - 2$ when there is only one independent variable. ::: Other Inferences and Considerations Inferences about Mean Response for New Observation {.unnumbered .unlisted} Let’s go back to the height and weight example: :::{.ms-3 .border-start .border-success .rounded-2 .ps-2} [Example]{.fs-4} If you are asked to estimate the weight of a STAT 500 student, what will you use as a point estimate? If I tell you that the height of the student is 70 inches, can you give a better estimate of the person's weight? Now that we have our regression equation, we can use height to provide a better estimate of weight. We would want to report a mean response value for the provided height, i.e. 70 inches. ::: The mean response at a given X value is given by: $$E(Y)=\beta_0+\beta_1X$$ This is an unknown but fixed value. The point estimate for mean response at $X=x$ is given by: $$\hat{\beta}_0+\hat{\beta}_1x$$ The example for finding this mean response for height and weight is shown later in the lesson. Inferences about Outcome for New Observation {.unnumbered .unlisted} :::{.ms-3} The point estimate for the outcome at $X=x$ is provided above. The interval to estimate the mean response is called the confidence interval. Minitab calculates this for us. The interval used to estimate (or predict) an outcome is called prediction interval. For a given x value, the prediction interval and confidence interval have the same center, but the width of the prediction interval is wider than the width of the confidence interval. That makes good sense since it is harder to estimate a value for a single subject (say predict your weight based on your height) than it would be to estimate the average for subjects (say predict the mean weight of people who are your height). Again, Minitab will calculate this interval as well. Minitab: Find the Condidence and Prediction Intervals {.unnumbered .unlisted} ::: {.mticon .float-end .d-flex .m-2 .rounded} ::: ::: {.ms-3 .border-start .border-secondary-subtle .ps-3} Steps: Fit the regression model (if not already done). Generate the prediction and intervals: Go to Stat > Regression > Regression > Predict. In the dialog box: Under Predictors, enter the value of the predictor variable (e.g., 70 if height is the X variable). Select OK. The output provides the point estimate obtained by plugging 70 into the fitted model along with confidence and prediction intervals when the height is 70 inches. ::: ::: Cautions with Linear Regression {.unnumbered .unlisted} :::{.ms-3} {#fig-scatterparabola .img-fluid .float-end .lightbox fig-alt="Scatter plot with a downward facing parabolic trend" width="30%"}First, use extrapolation with caution. Extrapolation is applying a regression model to X-values outside the range of sample X-values to predict values of the response variable $Y$. For example, you would not want to use your age (in months) to predict your weight using a regression model that used the age of infants (in months) to predict their weight. Second, the fact that there is no linear relationship (i.e. correlation is zero) does not imply there is no relationship altogether. The scatter plot will reveal whether other possible relationships may exist. The figure below gives an example where X and Y are related, but not linearly related i.e. the correlation is zero. ::: Outliers and Influential Observations {.unnumbered .unlisted} :::{.ms-3} Influential observations are points whose removal causes the regression equation to change considerably. It is flagged by Minitab in the unusual observation list and denoted as X. Outliers are points that lie outside the overall pattern of the data. Potential outliers are flagged by Minitab in the unusual observation list and denoted as R. The following is the Minitab output for the unusual observations within the height and weight example: ::: {.minitab_output} Fits and Diagnostics for Unusual Observations {.unnumbered .unlisted} \| Obs \| weight \| Fit \| Resid \| Std Resid \| \| :------::-----:---\| \| 24 \| 200.00 \| 139.74 \| 60.26 \| 3.23 \| R \| : {.w-auto .table-sm .table-responsive} [R Large residual]{.small } ::: Some observations may be both outliers and influential, and these are flagged by R and X (R X). Those observational points will merit particular attention. In our height and weight example, we have an R (potential outlier) observation, but it is not an influential point (RX observation). ::: Estimating the standard deviation of the error term {.unnumbered .unlisted} We can estimate the standard deviation of the error by finding the standard deviation of the residuals, $\hat{epsilon}_i=y_i-\hat{y}_i$. :::{.ms-3} Our simple linear regression model is: $$Y=\beta_0+\beta_1X+\epsilon$$ The errors for the $n$ observations are denoted as $\epsilon_i$, for $i=1, …, n$. One of our assumptions is that the errors have equal variance (or equal standard deviation). We can estimate the standard deviation of the error by finding the standard deviation of the residuals, $\epsilon_i=y_i-\hat{y}_i$. Minitab also provides the estimate for us, denoted as $S$, under the Model Summary. We can also calculate it by: $$s=\sqrt{\text{MSE}}$$ Find the MSE in the ANOVA table, under the MS column and the Error row. ::: Examples In this section, we present an example and review what we have covered so far in the context of the example. ::: {#exm-studenthtwtslr} Student height and weight (SLR) {.unnumbered .unlisted} \ We will continue with our height and weight example. Answer the following questions. a. Is height a significant linear predictor of weight? Conduct the test at a significance level of 5%. State the regression equation. b. Does $\beta_0$ have a meaningful interpretation? c. Find the confidence interval for the population slope and interpret it in the context of the problem. d. If a student is 70 inches, what weight could we expect? e. What is the estimated standard deviation of the error? ::: {.card .card-body .bg-light .ms-3 .mb-3 .pt-0} Answer a. The model for this problem is: $$\text{weight}=\beta_0+\beta_1\text{height}+\epsilon$$ The hypotheses we are testing are: $H_0\colon \beta_1=0$ $H_a\colon \beta_1\ne 0$ Recall that we previously examined the assumptions. Here is a summary of what we presented before: [Assumptions]{.fs-4} 1. Linearity: The relationship between height and weight must be linear. The scatterplot shows that, in general, as height increases, weight increases. There does not appear to be any clear violation that the relationship is not linear. 2. Independence of errors: Since it’s a random sample, one person's weight has nothing to do with another person. 3. Normality of errors: The residuals must be approximately normally distributed. Most of the data points fall close to the line, but there does appear to be a slight curving. There is one data point that clearly stands out. In the histogram, we can see that, with that one observation, the shape seems slightly right-skewed. 4. Equal variances: The variance of the residuals is the same for all values of $X$. In this plot, there does not seem to be a pattern. All of the assumptions except for the normal assumption seem valid. Minitab output for the height and weight data: ::: {.minitab_output} #### Regression Equation weight = -222.5 + 5.49 height #### Coefficients \| Term \| Coef \| SE Coef \| T-Value \| P-Value \| VIF \| :------::-------::----:\| \| Constant \| -222.5 \| 72.4 \| -3.07 \| 0.005 \| \| \| height \| 5.49 \| 1.06 \| 5.16 \| 0.000 \| 1.00 \| : {.w-auto .table-sm .table-responsive} #### Model Summary \| S \| R-sq \| R-sq(adj) \| R-sq(pred) \| :------::----------:\| \| 19.1108 \| 50.57% \| 48.67% \| 44.09% \| : {.w-auto .table-sm .table-responsive} ::: The regression equation is: $$\hat{\text{weight}}=-222.5+5.49\text{height}$$ The slope is 5.49, and the intercept is -222.5. The test for the slope has a p-value of less than 0.001. Therefore, with a significance level of 5%, we can conclude that there is enough evidence to suggest that height is a significant linear predictor of weight. We should make this conclusion with caution, however, since one of our assumptions might not be valid. b. The intercept is -222.5. Therefore, when height is equal to 0, then a person’s weight is predicted to be -222.5 pounds. It is also not possible for someone to have a height of 0 inches. Therefore, the intercept does not have a valid meaning. c. The confidence interval for the population slope is: $$\hat{\beta}_1\pm t_{\alpha/2}\hat{SE}(\hat{\beta}_1)$$ The estimate for the slope is 5.49 and the standard error for the estimate (SE Coef in the output) is 1.06. There are $n=28$ observations so the degrees of freedom are $28-2=26$. Using Minitab, we find the t-value to be 2.056. Putting the pieces together, the interval is: $$5.49\pm 2.056(1.06)$$ $$(3.31, 7.67)$$ We are 95% confident that the population slope is between 3.31 and 7.67. In other words, we are 95% confident that, as height increases by one inch, that weight increases by between 3.31 and 7.67 pounds, on average. d. Using the regression formula with a height equal to 70 inches, we get: $$\hat{\text{weight}}=-222.5+5.49(70)=161.8$$ For a student with a height of 70 inches, we would expect a weight of 162.3 pounds. If we wanted, we could have Minitab produce a confidence interval for this estimate. We will leave this out for this example. e. Using the output, under the model summary: $$s=19.1108$$ ::: ::: Try It! {.unnumbered .unlisted} ::: {.tryit} ::: {.grid} ::: {.g-col-lg-8 .g-col-md-6 .g-col-sm-12} The No-Lemon used car dealership in a college town records the age (in years) and price of cars it sold in the last year (cars_sold.txt{download="" target="_blank"}). The table is a preview of this data. ::: ::: {.g-col-lg-4 .g-col-md-6 .g-col-sm-12} \| age \| price \| -------\| \| 4 \| 6200 \| \| 4 \| 5700 \| \| 4 \| 6800 \| \| 5 \| 5600 \| \| ... \| ... \| \| 11 \| 2600 \| : {.w-auto .table-sm .table-responsive .mx-auto} ::: ::: Using the data above, answer the following questions. a. Is age a significant negative linear predictor of price? Conduct the test at a significance level of 5%. b. Does $\beta_0$ have a meaningful interpretation? c. Find the confidence interval for the population slope and interpret it in the context of the problem. d. If a car is seven years old, what price could we expect? e. What is the estimate of the standard deviation of the errors? {=html} <button class="btn btn-outline-success collapsed" type="submit" data-bs-toggle="collapse" data-bs-target="#collapselemon" aria-expanded="false" aria-controls="collapselemon"> Answer </button> :::: {#collapselemon .collapse} ::: {.card .card-body .bg-light} ::: {.panel-tabset} a. The linear regression model is: $$\text{price}=\beta_0+\beta_1\text{age}+\epsilon$$ To test whether age is a statistically significant negative linear predictor of price, we can set up the following hypotheses:. \| $H_0\colon \beta_1=0$ \| $H_a\colon \beta_1< 0$ We need to verify that our assumptions are satisfied. Let's do this in Minitab. Remember, we have to run the linear regression analysis to check the assumptions. ::: {.grid} ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} Assumption 1: Linearity {#fig-fittedlineplotlemon fig-alt="Scatter plot with fitted line for price vs age data" .mx-auto .d-block width="80%" .lightbox} The scatterplot below shows that the relationship between age and price scores is linear. There appears to be a strong negative linear relationship and no obvious outliers. ::: ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} Assumption 2: Independence of errors In this setting the price of one car in the probably has nothing to do with the price of another car. Therefore, the independence of the errors seems to be a reasonable assumption. ::: ::: ::: {.grid} ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} Assumption 3: Normality of errors {#fig-normprobplotlemon fig-alt="Normal probability plot of price vs age data" .mx-auto .d-block width="80%" .lightbox} On the normal probability plot, we are looking to see if our observations follow the given line. This graph does not indicate that there is a violation of the assumption that the errors are normal. If a probability plot is not an option we can refer back to one of our first lessons on graphing quantitative data and use a histogram or boxplot to examine if the residuals appear to follow a bell shape. ::: ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} Assumption 4: Equal Variances {#fig-vsfitslemon fig-alt="Versus fits graphs for the price vs age data" .mx-auto .d-block width="80%" .lightbox} Again, we will use the plot of residuals versus fits. Now we are checking that the variance of the residuals is consistent across all fitted values. This assumption seems valid. ::: ::: ::: {.minitab_output} Regression Equation price = 7850 - 485 age Coefficients \| Term \| Coef \| SE Coef \| T-Value \| P-Value \| VIF \| :------::-------::----:\| \| Constant \| 7850 \| 362 \| 21.70 \| 0.000 \| \| \| age \| -485.0 \| 43.9 \| -11.04 \| 0.000 \| 1.00 \| : {.w-auto .table-sm .table-responsive} Model Summary \| S \| R-sq \| R-sq(adj) \| R-sq(pred) \| :------::----------:\| \| 503.146 \| 88.39% \| 87.67% \| 84.41% \| : {.w-auto .table-sm .table-responsive} ::: From the output above we can see that the p-value of the coefficient of age is 0.000 which is less than 0.001. The Minitab output is for a two-tailed test and we are dealing with a left-tailed test. Therefore, the p-value for the left-tailed test is less than $\frac{0.001}{2}$ or less than 0.0005. We can thus conclude that age (in years) is a statistically significant negative linear predictor of price for any reasonable $\alpha$ value. b. $\beta_0$ is the y-intercept, which means it is the value of price when age is equal to 0. It is possible for a vehicle to have the number of years equal to 0. Therefore, it does have an interpretable meaning. We should use caution if we use this model to predict the price of a car with age equal to 0 because it is outside the range of values used to estimate the model. c. The 95% confidence interval for the population slope is: $$\hat{\beta}1\pm t{\alpha/2}\text{SE}(\hat{\beta}_1)$$ Using the output, $\hat{\beta}1=-485$ and the $\text{SE}(\hat{\beta}_1)=43.9$. We need to have $t{\alpha/2}$ with $n-2$ degrees of freedom. In this case, there are 18 observations so the degrees of freedom are $18-2=16$. Using software, we find $t_{\alpha/2}=2.12$. The 95% confidence interval is: $$-485\pm 2.12(43.9)$$ $$(-578.068, -391.932)$$ We are 95% confident that the population slope for the regression model is between -578.068 and -391.932. In other words, we are 95% confident that, for every one year increase in age, the price of a vehicle will decrease between 391.932 and 578.068 dollars. d. We can use the regression equation with $\text{age}=7$: $$\hat{\text{price}}=7850-485(7)=4455$$ We can expect the price to be $4455. e. The residual standard error is estimated by s, which is calculated as: $$s=\sqrt{\text{MSE}}=\sqrt{253156}=503.146$$ ::: {.callout-caution appearance="minimal"} Note! \ The MSE is found in the ANOVA table which is part of the regression output in Minitab. ::: It is also shown as $s$ under the model summary in the output. ::: ::: ::: :::: Coefficient of Determination Now that we know how to estimate the coefficients and perform the hypothesis test, is there any way to tell how useful the model is? One measure is the coefficient of determination, denoted $R^2$. ::: {#def-rsquared .ms-3} Coefficient of Determination $R^2$ The coefficient of determination measures the percentage of variability within the $y$-values that can be explained by the regression model. ::: Therefore, a value close to 100% means that the model is useful and a value close to zero indicates that the model is not useful. It can be shown by mathematical manipulation that: $$\text{SST }=\text{ SSR }+\text{ SSE}$$ $$\sum (y_i-\bar{y})^2=\sum (\hat{y}_i-\bar{y})^2+\sum (y_i-\hat{y}_i)^2$$ Total variability in the y value = Variability explained by the model + Unexplained variability To get the total, explained, and unexplained variability, first, we need to calculate corresponding deviances. Drag the slider on the image below to see how the total deviance $(y_i-\bar{y})$ is split into explained $(\hat{y}_i-\bar{y})$ and unexplained deviances $(y_i-\hat{y}_i)$. :::{.mx-auto .w-75} ```{=html} ``` ::: The breakdown of variability in the above equation holds for the multiple regression model also. ::: {.callout-caution appearance="minimal"} Coefficient of Determination $R^2$ Formula \ $$R^2=\dfrac{\text{variability explained by the model}}{\text{total variability in the y values}}$$ $R^2$ represents the proportion of total variability of the $y$-value that is accounted for by the independent variable $x$. ::: For the specific case when there is only one independent variable $X$ (i.e., simple linear regression), one can show that $R^2 =r^2$, where $r$ is the correlation coefficient between $X$ and $Y$. ::: {#exm-studenthtwtrsq} Student height and weight ($R^2$) \ Let's take a look at Minitab's output from the height and weight example (university_ht_wt.txt{download="" target="_blank"}) that we have been working within this lesson. ::: {.minitab_output} Regression Equation weight = -222.5 + 5.49 height Coefficients \| Term \| Coef \| SE Coef \| T-Value \| P-Value \| VIF \| :------::-------::----:\| \| Constant \| -222.5 \| 72.4 \| -3.07 \| 0.005 \| \| \| height \| 5.49 \| 1.06 \| 5.16 \| 0.000 \| 1.00 \| : {.w-auto .table-sm .table-responsive} Model Summary \| S \| R-sq \| R-sq(adj) \| R-sq(pred) \| :------::----------:\| \| 19.1108 \| 50.57% \| 48.67% \| 44.09% \| : {.w-auto .table-sm .table-responsive} ::: Find the coefficient of determination and interpret the value. ::: {.card .card-body .bg-light .ms-3 .mb-3 .pt-0} Answer The coefficient of determination, $R^2$ is 0.5057 or 50.57%. This value means that 50.57% of the variation in weight can be explained by height. Remember, for this example, we found the correlation value, $r$, to be 0.711. So, we can now see that $r^2 = (0.711)^2 = .506$ which is the same reported for R-sq in the Minitab output. ::: ::: Try It! Used car sales continued... {.unnumbered .unlisted} ::: {.tryit} For the age and price of the car example (cars_sold.txt{download="" target="_blank"}), what is the value of the coefficient of determination and interpret the value in the context of the problem? {=html} <button class="btn btn-outline-success collapsed" type="submit" data-bs-toggle="collapse" data-bs-target="#collapselemon2" aria-expanded="false" aria-controls="collapselemon2"> Answer </button> :::: {#collapselemon2 .collapse} ::: {.card .card-body .bg-light} ::: {.minitab_output} Regression Equation price = 7850 - 485 age Coefficients \| Term \| Coef \| SE Coef \| T-Value \| P-Value \| VIF \| :------::-------::----:\| \| Constant \| 7850 \| 362 \| 21.70 \| 0.000 \| \| \| age \| -485.0 \| 43.9 \| -11.04 \| 0.000 \| 1.00 \| : {.w-auto .table-sm .table-responsive} Model Summary \| S \| R-sq \| R-sq(adj) \| R-sq(pred) \| :------::----------:\| \| 503.146 \| 88.39% \| 87.67% \| 84.41% \| : {.w-auto .table-sm .table-responsive} ::: From the Minitab output, we see an R-sq value of 88.39%. We want to report this in terms of the study, so here we would say that 88.39% of the variation in vehicle price is explained by the age of the vehicle. ::: {.callout-caution appearance="minimal"} Note! \ The two other references to R-sq, (adj) and (pred), are used for model comparisons. These two metrics do not provide any interpretive value to the model in regards to $X$ and $Y$. ::: ::: :::: ::: Inference for Correlation Let’s review the notation for correlation. $r$: The sample correlation (Pearson’s correlation) $\rho$: “rho” is the population correlation The sample correlation is found by: $$r=\dfrac{\sum (x_i-\bar{x})(y_i-\bar{y}) }{\sqrt{\sum (x_i-\bar{x})^2}\sqrt{\sum (y_i-\bar{y})^2}}$$ In this section, we will present a hypothesis test for the population correlation. Then, we will compare the tests and interpretations for the slope and correlation. Hypothesis Testing for the Population Correlation In this section, we present the test for the population correlation using a test statistic based on the sample correlation. Assumptions {.unnumbered .unlisted} :::{.ms-3} As with all hypothesis test, there are underlying assumptions. The assumptions for the test for correlation are: The are no outliers in either of the two quantitative variables. The two variables should follow a normal distribution ::: Hypotheses {.unnumbered .unlisted} :::{.ms-3} If there is no linear relationship in the population, then the population correlation would be equal to zero. :::{.ms-3} $H_0\colon \rho=0$ ($X$ and $Y$ are linearly independent, or X and Y have no linear relationship) $H_a\colon \rho\ne0$ ($X$ and $Y$ are linearly dependent) ::: ```{=html} Research Question Is there a linear relationship? Is there a positive linear relationship? Is there a negative linear relationship? Null Hypothesis $\rho=0$ $\rho=0$ $\rho=0$ Alternative Hypothesis $\rho\ne0$ $\rho>0$ $\rho<0$ Type of Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional ``` ::: Test Statistic {.unnumbered .unlisted} :::{.ms-3} Under the null hypothesis and with the above assumptions, the test statistic, $t^$, found by: $$t^=\dfrac{r\sqrt{n-2}}{\sqrt{1-r^2}}$$ which follows a $t$-distribution with $n-2$ degrees of freedom. ::: As mentioned before, we will use Minitab for the calculations. The output from Minitab previously used to find the sample correlation also provides a p-value. This p-value is for the two-sided test. If the alternative is one-sided, the p-value from the output needs to be adjusted. ::: {#exm-studenthtwtrho} Student height and weight (Tests for $\rho$) \ For the height and weight example, university_ht_wt.txt{download="" target="_blank"}, conduct a test for correlation with a significance level of 5%. ::: {.card .card-body .bg-light .ms-3 .mb-3 .pt-0} Answer The output from Minitab is: ::: {.minitab_output} Correlations {.unnumbered .unlisted} \| \| height \| :------:\| \| weight \| 0.711 \| : {.w-auto .table-sm .table-responsive} ::: For the sake of this example, we will find the test statistic and the p-value rather than just using the Minitab output. There are 28 observations. The test statistic is: $$\begin{align} t^&=\dfrac{r\sqrt{n-2}}{\sqrt{1-r^2}}\&=\dfrac{(0.711)\sqrt{28-2}}{\sqrt{1-0.711^2}}\&=5.1556 \end{align}$$ Next, we need to find the p-value. The p-value for the two-sided test is: $$\text{p-value}=2P(T>5.1556)<0.0001$$ Therefore, for any reasonable $\alpha$ level, we can reject the hypothesis that the population correlation coefficient is 0 and conclude that it is nonzero. There is evidence at the 5% level that Height and Weight are linearly dependent. ::: ::: Try It! {.unnumbered .unlisted} ::: {.tryit} For the sales and advertising example, conduct a test for correlation with a significance level of 5% with Minitab. Sales units are in thousands of dollars, and advertising units are in hundreds of dollars. \| Sales (Y) \| Advertising (X) \| :---------------:\| \| 1 \| 1 \| \| 1 \| 2 \| \| 2 \| 3 \| \| 2 \| 4 \| \| 4 \| 5 \| : {.w-auto .table-sm .table-responsive .mx-auto} {=html} <button class="btn btn-outline-success collapsed" type="submit" data-bs-toggle="collapse" data-bs-target="#collapseads" aria-expanded="false" aria-controls="collapseads"> Answer </button> :::: {#collapseads .collapse} ::: {.card .card-body .bg-light} The Minitab output gives: ::: {.minitab_output} Correlations {.unnumbered .unlisted} \| \| height \| :------:\| \| weight \| 0.904 \| : {.w-auto .table-sm .table-responsive} ::: The sample correlation is 0.904. This value indicates a strong positive linear relationship between sales and advertising. For the Sales (Y) and Advertising (X) data, the test statistic is... $$t^=\dfrac{(0.904)\sqrt{5-2}}{\sqrt{1-(0.904)^2}}=3.66$$ ...with df of 3, we arrive at a p-value = 0.035. For $\alpha=0.05$, we can reject the hypothesis that the population correlation coefficient is 0 and conclude that it is nonzero, i.e., that sales and advertising are linearly dependent. :::: ::: ::: Comparing Correlation and Slope Some of you may have noticed that the hypothesis tests for correlation and slope are very similar. Also, the test statistic for both tests follows the same distribution with the same degrees of freedom, $n-2$. This similarity is because the two values are mathematically related. In fact, $$\hat{\beta}_1=r\dfrac{\sqrt{\sum (y_i-\bar{y})^2}}{\sqrt{\sum(x_i-\bar{x})^2}}$$ Here is a summary of some of the similarities and differences between the sample correlation and the sample slope. ::: {.grid} ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} Similarities {.unnumbered .unlisted} The test for correlation will lead to the same conclusion as the test for slope. The sign of the slope (i.e. negative or positive) will be the same for the correlation. In other words, both values indicate the direction of the relationship ::: ::: {.g-col-lg-6 .g-col-md-6 .g-col-sm-12} Differences {.unnumbered .unlisted} The value of the correlation indicates the strength of the linear relationship. The value of the slope does not. The slope interpretation tells you the expected change in the response for a one-unit increase in the predictor. Correlation does not have this kind of interpretation. ::: ::: Multiple Regression Model In this section, we (very) briefly discuss Multiple Linear Regression. In practice, it is not usual that there is only one predictor variable. In Multiple Linear Regression, there is one quantitative response and more than one predictor or independent variable. The model will contain the constant or intercept term, $\beta_0$, and more than one coefficient, denoted $\beta_1, …, \beta_k$, where $k$ is the number of predictors. ::: {.callout-caution appearance="minimal"} [Multiple Linear Regression Model]{.fs-4} \ $$Y=\beta_0+\beta_1X_1+...+\beta_kX_k+\epsilon$$ Where $Y$ is the response variable and $X_1, …, X_k$ are independent variables. $\beta_0, \beta_1, …, \beta_k$ are fixed (but unknown) parameters and $\epsilon$ is a random variable that is normally distributed with mean 0 and having a variance $\sigma^2_\epsilon$. ::: F-Test for Overall Significance {.unnumbered .unlisted} There is a statistical test we can use to determine the overall significance of the regression model. The F-test in Multiple Linear Regression test the following hypotheses: :::{.ms-3} $H_0\colon \beta_1=...=\beta_k=0$ $H_a\colon \text{ At least one }\beta_i\text{ is not equal to zero}$ ::: The test statistic for this test, denoted $F^$, follows an $F$ distribution. We will not expect you to understand Multiple Linear Regression. We included it here to show you what you may see in practice. If you are interested in learning more, you can take STAT 501: Regression Methods. Lesson Summary In this Lesson, we examined the relationship between a quantitative response (or dependent variable) and a quantitative explanatory variable (or independent variable). We are interested in whether or not the independent variable is a significant linear predictor of the response. In the next Lesson, we consider a different situation. We look back on the example where we have a quantitative response and an explanatory variable with more than two levels. In other words, we look at the situation where we are comparing means from more than two groups. ``` Except where otherwise noted, content on this site is licensed under a CC BY-NC 4.0 license. 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15178	https://www.studypug.com/trigonometry-help/law-of-sines	Sign InTry Free Home Trigonometry Trigonometric Ratios and Angle Measures Law of sines Mastering the Law of Sines: Your Key to Solving Triangle Problems Unlock the power of the Law of Sines to solve complex triangle problems. Learn how to apply this fundamental trigonometric principle to real-world scenarios in navigation, surveying, and more. Get the most by viewing this topic in your current grade. Pick your course now. Now Playing:Law of sines– Example 0 Intros The Law of Sines Examples Given the following triangle △ABC, Solve for∠C Solve for a Solve for side x Ambiguous case: SSA triangles In △DEF, DE=21cm, ∠F=45°, and EF=24cm; find DF. View All Practice Now Practicing:Law Of Sines 3 Free to Join! StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun — with achievements, customizable avatars, and awards to keep you motivated. Students Parents Try Free Easily See Your Progress We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details. #### Make Use of Our Learning Aids ###### Last Viewed ###### Practice Accuracy ###### Suggested Tasks Get quick access to the topic you're currently learning. See how well your practice sessions are going over time. Stay on track with our daily recommendations. Try Free #### Earn Achievements as You Learn Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service. #### Create and Customize Your Avatar Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug. Try Free Law of sines Jump to:NotesConceptExampleFAQsPrerequisitesRelated Notes In this section, we will learn about the Law of Sines, also known as the Sines Rule. The Law of Sines is a formula that models the relationship between the sides and the angles of any triangle, be it a right-angled triangle, an obtuse triangle, or an acute triangle. In order to use the Law of Sines, we need to satisfy the "one pair, one additional information" condition (i.e. Angle-Angle-Side abbreviated as AAS, and Angle-Side-Angle abbreviated as ASA). We will also explore the concept of the Ambiguous Case of the Law of Sines. Law of Sine For any △ ABC, sin(A)a =sin(B)b =sin(C)c and, asin(A) =bsin(B) =csin(C) Use the Law of Sine when given a pair! Ambiguous case Ambiguous case of the Law of Sine arises when given SSA (side-side-angle) Step 1) Use the given angle to find the height of the triangle: h=bsin(A) Step 2) Check if, Sidea < h, then no triangles Sidea=h, then 1 triangle Sidea > h, then 1 triangle h < Sidea < Sideb, then 2 triangles Step 3) Solve the triangle(s)! Concept Introduction to the Law of Sines The Law of Sines is a fundamental concept in trigonometry that plays a crucial role in solving triangles. This powerful mathematical tool allows us to find unknown sides or angles in a triangle when we have limited information. Our introduction video provides a clear and concise explanation of this important trigonometric principle, making it easier for students to grasp its significance. By understanding the Law of Sines, learners can tackle a wide range of problems in geometry, physics, and engineering. The video demonstrates how to apply this law in various scenarios, from simple right-angled triangles to more complex oblique triangles. Mastering the Law of Sines is essential for anyone studying trigonometry or pursuing fields that involve spatial reasoning and calculations. With this knowledge, students can confidently approach triangle-related problems and develop a deeper understanding of trigonometric relationships. The Law of Sines is particularly useful when dealing with non-right triangles. It provides a way to find unknown sides or angles using the ratios of the sides and the sines of their opposite angles. This method is not only applicable in theoretical mathematics but also in practical fields such as physics and engineering, where understanding the properties of triangles is essential. By mastering the Law of Sines, students can enhance their problem-solving skills and gain a deeper appreciation for the beauty and utility of trigonometric relationships. Example Given the following triangle △ABC, Solve for ∠C Step 1: Understanding the Law of Sines In this question, we are going to solve for the angle C. To solve for angle C, we need to use the Law of Sines. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, it is expressed as: sinAa=sinBb=sinCc where a, b, and c are the lengths of the sides opposite to angles A, B, and C respectively. Step 2: Identifying the Given Information To use the Law of Sines, we need at least one pair of a side length and its opposite angle. In the given triangle △ABC, we need to check if we have this information. We are given: Angle B=62∘ Side b=8.4 (opposite to angle B) Side c=6.3 (opposite to angle C) We have a pair: angle B and side b. Additionally, we have the length of side c. Step 3: Setting Up the Law of Sines Since we are solving for angle C, it is more efficient to use the version of the Law of Sines with the angles on top: bsinB=csinC Substituting the known values, we get: 8.4sin62∘=6.3sinC Step 4: Solving for sinC To isolate sinC, we multiply both sides of the equation by 6.3: sinC=8.46.3×sin62∘ Using a calculator, we find: sin62∘≈0.8829 Therefore: sinC=8.46.3×0.8829≈0.6615 Step 5: Finding Angle C To find angle C, we take the inverse sine (arc sine) of 0.6615: C=sin−1(0.6615) Using a calculator, we find: C≈41.47∘ Therefore, angle C is approximately 41.47∘. FAQs Here are some frequently asked questions about the Law of Sines: 1. How do I use the law of sines to solve problems? To use the Law of Sines, follow these steps: Identify the known sides and angles in your triangle. Set up the Law of Sines equation: a/sin(A) = b/sin(B) = c/sin(C). Plug in the known values and solve for the unknown side or angle. Use your calculator to compute the final answer. 2. What is an example of the sine law? Here's a simple example: In triangle ABC, if angle A = 30°, angle B = 45°, and side a = 10 cm, we can find side b using the Law of Sines. The equation would be: 10/sin(30°) = b/sin(45°). Solving this, we get b 12.47 cm. 3. What is the problem with the sine rule? The main issue with the Law of Sines is the ambiguous case. This occurs when solving an SSA (Side-Side-Angle) triangle, where two sides and a non-included angle are known. In this scenario, there may be zero, one, or two possible solutions, requiring careful analysis to determine the correct answer. 4. What is a real life example of the law of sines? A real-life application of the Law of Sines is in navigation. For instance, a ship's captain can use it to determine the distance to a lighthouse. By measuring two angles from different positions on the ship and knowing the distance between these positions, the captain can calculate the distance to the lighthouse using the Law of Sines. 5. How does the Law of Sines differ from the Law of Cosines? While both laws are used to solve triangles, they have different applications. The Law of Sines is typically used when we know two angles and one side (AAS) or two sides and a non-included angle (SSA). The Law of Cosines is used when we know three sides (SSS) or two sides and the included angle (SAS). The Law of Cosines involves squaring side lengths, while the Law of Sines uses only ratios. Prerequisites Mastering the Law of Sines requires a solid foundation in several key trigonometric concepts. Understanding these prerequisite topics is crucial for students to grasp the full scope and application of this important trigonometric law. One of the fundamental skills needed is the ability to use sine ratio to calculate angles and sides in right triangles. This forms the basis for understanding how the Law of Sines extends these principles to non-right triangles. Similarly, familiarity with the cosine ratio is essential, as it complements the sine ratio in trigonometric calculations. Students should also be comfortable with various combinations of SohCahToa questions, which reinforce the relationships between sides and angles in triangles. This knowledge is directly applicable when using the Law of Sines to solve more complex triangular problems. A strong grasp of trigonometric identities is also beneficial. While not directly used in the Law of Sines, understanding these identities enhances overall trigonometric proficiency and problem-solving skills. For practical applications, students should be familiar with word problems relating to angles in trigonometry. This helps in recognizing real-world scenarios where the Law of Sines can be applied. Additionally, understanding angles of elevation and depression is crucial for many practical applications of the Law of Sines, especially in surveying and navigation problems. While more advanced, knowledge of inverse trigonometric functions can provide a deeper understanding of the relationships between angles and sides in triangles, which is at the core of the Law of Sines. By mastering these prerequisite topics, students will be well-prepared to tackle the Law of Sines. This law is a powerful tool in trigonometry, allowing for the solution of non-right triangles and opening up a wide range of applications in physics, engineering, and navigation. The strong foundation provided by these prerequisites ensures that students can not only understand the Law of Sines but also appreciate its significance and versatility in solving complex trigonometric problems. Use sine ratio to calculate angles and sides (Sin = ho ) Use cosine ratio to calculate angles and sides (Cos = ha ) Use tangent ratio to calculate angles and sides (Tan = ao ) Quotient identities and reciprocal identities Pythagorean identities Sum and difference identities Become a member to get more! Try FreeLearn More
15179	https://www.youtube.com/watch?v=XLDJS7GsCFs	Tangents of circles problems : Khan Academy Khan Man Math 1200 subscribers 194 likes Description 10924 views Posted: 6 Jun 2022 9 comments Transcript: this will be for the con assignment tangents of circles problems all sides of a b c d are tangent to circle p what is the perimeter of the quadrilateral below so for this problem here you want to remember um if two tangent segments meet at the same endpoint outside the circle then those tangent segments are congruent all right so this line segment here is eight which means this line segment here is also going to be eight and the reason that is is because both of these are tangent to the same circle and they meet at the same point so you want to go around the circle and do that whole do the whole thing that we just did there so at this point here this section here is 3.1 so this section right here is also going to be 3.1 all right and at this end point here this line segment right here is eight therefore this set line segment right here is also going to be eight all right and the whole thing down here the whole segment is 11.1 and this segment is eight so this one right here is going to be 11.1 minus eight all right because the whole thing is 11.1 and this one is eight so that just equals 3.1 so this segment right here is going to be 3.1 all right and at this end point right here if this segment is 3.1 this one right here is also going to be 3.1 all right so it looks like all the sides of the quadrilateral are equal it's not always going to work out that way but we can see each side is 8 plus 3.1 which is 11.1 and we have that on the bottom on the right and the left okay so since each side is 11.1 and there's four sides in this case we can just do 11.1 times four for the total perimeter and that's going to be 44.4 final answer all right like i said it's not always gonna be that all four sides are congruent it just worked out that way in this case so click in the box and just type 44.4 okay ac is tangent to circle o at point c what is the length of ac all right so remember the tangent line the tangent line is a line that hits the circle or intersects the circle at just one point all right it has to be a continuous line you can't draw something like this right there and say that's a tangent line that's just not true that's not a tangent line all right and the other thing you want to remember if you draw a radius okay they said circle oh so so this has to be the center and that would make this a radius okay and this line right here ac is a tangent line so if you draw a radius to the point of tangency it's going to make a right angle all right so the radius will make a right angle with the tangent line all right so this is perpendicular right here that's a right hand conception so the other thing what you want to remember the radius of a given circle is the same regardless of where you draw okay so this right here is the radius all right so if we drew another radius over here that would also be the same length that would be five five units all right so this is a radius and from here to here is also a radius so this distance right here is five all right and we can see this line segment right here is going to be eight plus five so this whole side right here is just going to be 13. all right since we have a right triangle and we have two sides we can use pythagorean theorem a squared plus b squared equals c squared okay c is always the hypotenuse across from the right angle so if you go across from the right angle this whole side right here is the hypotenuse so that is 13 and that's what we're going to use for c so for c squared i'm going to use 13 squared and a and b are the legs uh this segment here i'll call that a and this side length right here i'm going to call that b so a is what we're looking for so a squared we're gonna leave alone and b is five so b squared is going to be five squared all right so we have a squared five times five is 25 13 times 13 is 169. subtracting 25 from both sides 25 minus 25 cancels bring down the a squared 169 minus 25 equals 144. the opposite of squared is square root so in order to solve for a we take the square root of both sides and a is just 12. okay of course we're not going to use negative 12 because a side length cannot be negative so this side ac right here is 12 units and that's what they ask they ask for side length ac so you just click in the box and type 12. all right it says angle a is circumscribed about about circle o what is the measure of angle a all right so since angle a here is circumscribed we know that point c and point b are points of tangency all right and it said circle o so we know that o is the center so these lines right here and this one right here they're both radiuses or radii okay so when you draw a line um when you draw a radius to the point of tangency it makes a right angle so this is going to be a right angle and this one down here is going to be a right angle all right now we can see angle d as an inscribed angle all right the vertex is on the circle itself and we know that an inscribed angle is half the measure of the arc it intercepts all right so if this is 65 that's going to be half of this arc right here so in other words this arc is going to be 2 times 65. if we do 2 times 65 we get 130. all right so the measure of this arc right here is 130 degrees and the central angle the measure of the central angle matches the measure of the arc so if this arc is 130 the central angle right here is also going to be 130. all right so in other words this central angle is just two times the inscribed angle and now we can solve for angle a [Music] all right so here we have a quadrilateral this is a side this is a side this is a side and this is a side all right we know the angles the interior angles of a quadrilateral add to 360. so i'm going to call this one x alright so we have angle x and going around we have 90. right here we have 90 and right here we have 130 okay those four angles of a quadrilateral equal 360. all right i don't have enough well just write it right here they equal 360. all right so combining like terms nothing to combine with x 90 plus 90 is 180 plus 130 that's going to be 310. all right so i just combined 90 90 and 130 and that whole thing equals 360. all right subtracting 310 from both sides the three tens cancel and x is 360 minus 310 that's just 50. final answer so click inside the box and just type 50. okay angle a is circumscribed about circle o what is the measure of angle o all right since angle a is circumscribed that means these are points of tangency point c and point b okay they said circle oh we know this is the center and we know these two segments right here are radiuses or radii if you draw a radius to the point of tangency it makes a right angle so there's one here and there's a right angle down here and they want us to find the measure of angle o all right we're just going to call that x and we said a minute ago the four angles interior angles of a quadrilateral add to 360. so we have 46 plus 90. plus 90 plus x and that equals 360. combining like terms 90 plus 90 is 180 220 226 that's plus x and that equals 360. all right so i just combine these three to get 226. solving for x subtract 226 on both sides cancels right here bring down your x and let's see ten minus six is four five minus two is three and 3 minus 2 is 1. so x is going to be 134. all right so that's this angle right here and the answer makes sense we can see it looks like it's about 134 and you just click inside the box and you type 134 all right last one it says ac is tangent to circle o at point c what is the measure of angle oac all right so this is let me see here this is the point of tangency right here at point c and they said circle o so we know that o is the center and we know that this is a radius when you draw a radius to the point of tangency it makes a right angle all right and we want to find angle a i'll just call that x and since we have the two angles here we can solve for x we know the sum of the interior angles of a triangle is 180 so we have x plus 63 plus 90. and that equals 180. combining like terms 63 plus 90 is going to be 153 and that equals 180 all right so i just combine those two and subtracting 153 on both sides right here bring down your x 180 minus 153 is 27. final answer okay and the answer makes sense this looks like an acute angle that would be about 27. so you just click inside the box and type 27. you
15180	https://www.wyzant.com/resources/answers/902837/discuss-the-difference-between-a-rational-and-radical-exponent	Discuss the difference between a rational and radical exponent \| Wyzant Ask An Expert Log inSign up Find A Tutor Search For Tutors Request A Tutor Online Tutoring How It Works For Students FAQ What Customers Say Resources Ask An Expert Search Questions Ask a Question Wyzant Blog Start Tutoring Apply Now About Tutors Jobs Find Tutoring Jobs How It Works For Tutors FAQ About Us About Us Careers Contact Us All Questions Search for a Question Find an Online Tutor Now Ask a Question for Free Login WYZANT TUTORING Log in Sign up Find A Tutor Search For Tutors Request A Tutor Online Tutoring How It Works For Students FAQ What Customers Say Resources Ask An Expert Search Questions Ask a Question Wyzant Blog Start Tutoring Apply Now About Tutors Jobs Find Tutoring Jobs How It Works For Tutors FAQ About Us About Us Careers Contact Us Subject ZIP Search SearchFind an Online Tutor NowAsk Ask a Question For Free Login Math Sunshine R. asked • 08/23/22 Discuss the difference between a rational and radical exponent Discuss the difference between a rational and radical exponent Follow •1 Add comment More Report 1 Expert Answer Best Newest Oldest By: Raymond B.answered • 08/23/22 Tutor 5(2) Math, microeconomics or criminal justice See tutors like this See tutors like this a rational number = any number that can be written in the form a/b where both a and b are integers square root of 2 = sqr(2) is irrational. It can't be writen as a/b with a and b as integers. a radical number has a number inside a radical. which could make the number rational or irrational or imaginary. "Mostly"imaginary and irrational. although there is an infinite number of each. sqr(2) is irrational. sqr(4) is rational. sqr(-2) is imaginary Upvote • 0Downvote Add comment More Report Still looking for help? Get the right answer, fast. Ask a question for free Get a free answer to a quick problem. Most questions answered within 4 hours. OR Find an Online Tutor Now Choose an expert and meet online. 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15181	https://artofproblemsolving.com/wiki/index.php/Pigeonhole_Principle?srsltid=AfmBOopg4h-RURzKlnXrF8puNDzKtEbnMmdza-pTVy4Eydr2MTlFcjCb	Art of Problem Solving Pigeonhole Principle - 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 Pigeonhole Principle Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Learn more about the Pigeonhole Principle and other powerful techniques for combinatorics problems in our Intermediate Counting & Probability textbook by USA Math Olympiad winner (and MIT PhD) David Patrick. LEARN MORE Pigeonhole Principle In combinatorics, the pigeonhole principle states that if or more pigeons are placed into holes, one hole must contain two or more pigeons. This seemingly trivial statement may be used with remarkable creativity to generate striking counting arguments, especially in Olympiad settings. In older texts, the principle may be referred to as the Dirichlet box principle. A common phrasing of the principle uses balls and boxes and is that if balls are to be placed in boxes and , then at least one box must contain more than one ball. Contents [hide] 1 Proof 2 Problems 3 Introductory 3.1 Example 1 3.2 Example 2 3.3 Example 3 3.4 Example 4 4 Intermediate 4.1 Example 1: Rational Approximation Theorem 5 Olympiad Problems 6 See Also Proof An intuitive proof of the pigeonhole principle is as follows: suppose for contradiction that there exists a way to place balls into boxes where such that all boxes contain at most one ball. Let how many balls each box contains. Our condition that all boxes contain at most one ball implies that for all , so However, we know that there are a total of balls across all our boxes, so this sum must equal : Therefore, . This contradicts our definition that . Therefore, our assumption must be incorrect; at least one box must contain two or more balls. In formal terms, the pigeonhole principle is a consequence of how one set is defined to be larger than another set. Let be a set of balls and be a set of boxes such that . The definition that (as in our problem) is that there exists a surjective mapping from to , but not an injection. In other words, there exists a way to map every ball of to every box of , but it does not hold that if the boxes of two balls are the same, then the balls must be the same. That is to say, there must be two or more balls in the same box—which is the pigeonhole principle. Problems Introductory Example 1 If a Martian has an infinite number of red, blue, yellow, and black socks in a drawer, what is the minimum number of socks that the Martian must pull out of the drawer to guarantee they have a pair?` Solution: Intuitively, you might realize that after we select four socks of different colors (one red, one blue, one yellow, and one black), the Martian can't select a fifth sock without creating a pair. We may use this to prove the problem: Note that the Martian may select socks without a pair: one red, one blue, one yellow, and one black sock. However, if the Martian selects socks from colors, the pigeonhole principle guarantees that socks must have the same color (where the socks are "pigeons" and the colors are "holes"). Therefore, is the minimum number of socks they must draw to guarantee a pair. Example 2 Let . Show that if we choose numbers from , then there exist two numbers such that one is a multiple of the other. Solution: We write each number as for some integers and , where is nonnegative and is odd ( can also be ). Because we select integers and there are possible values of , Pigeonhole Principle guarantees that two numbers will share the same value. These numbers are multiples—if we define the two numbers and where , we may multiply by to get , as desired. Example 3 Suppose is a set of integers. Prove that there exists distinct in such that is a multiple of . Solution: Note that for any such and , We may rewrite this in modular arithmetic as , or Therefore, we wish to show that there exist and with the same remainder modulo n. Note that there are integers in and possible remainders (namely, ) modulo . Then by the pigeonhole principle, there exist two integers with the same remainder modulo . As shown earlier, this implies that their difference is a multiple of , as required. Example 4 Show that in any group of people, there are two who have an identical number of friends within the group. Solution: Note that for any person from the group, the minimum number of in-group friends they can have is (nobody) and the maximum is (everybody but themselves). Hence, there are possible values for an individual's number of in-group friends. However, note that if a person has in-group friends, nobody else can be friends with all other individuals; vice-versa if an individual has friends. There cannot exist two individuals with and friends. Therefore, there are only possible values of an individual's number of in-group friends. Then because there are individuals and possible values of in-group friends, the Pigeonhole Principle guarantees that two individuals have the same number of friends within the group. Intermediate Example 1: Rational Approximation Theorem Show that for any irrational number and positive integer , there exists a rational number with such that . Solution: Take a moment to digest the question; in short, our task is to prove the existence of a rational number close to our irrational . In particular, is almost like a "confidence level"—where a higher increases the denominator and decreases the distance between and . To simplify the problem, we multiply both sides of the inequality by to get Note that we wish to be less than one; hence, we might think to define such that is , the fractional part of . In formal terms, we let such that . Now, we wish to demonstrate that out of , there exists a positive integer such that lies in the interval ![Image 125: $0,1/n)$. We can view the intervals ![Image 126: $[0, 1/n), [1/n,2/n), \ldots, (n-1)/n, 1)$ as windows that contain all our multiples of . Note that if ![Image 128: $0, 1/n)$ and ![Image 129: $(n-1)/n, 1)$ contain a multiple, then we are done. Suppose every interval is filled; then one value of must lie in ![Image 131: $0, 1/n)$. If there exists an interval that is not filled, we have at most filled intervals and multiples. The Pigeonhole Principle thus guarantees that there exists an interval with two values of . Letting these values be and , we note that is either in the first or last interval. Hence, in either case, there exists a such that lies in ![Image 140: $0, 1/n)$, which is the theorem. We may assemble this into a formal proof: Proof: Let be an integer from to inclusive. Note that for all of , we can write , where is an integer and . Consider the intervals from ![Image 149: $0, 1)$ of size . We have total ; hence, the Pigeonhole Principle guarantees the existence of some and such that and are in the same interval. Without loss of generality, let . Then . We have that Note that the fact allows us to divide both sides of this inequality by to obtain Therefore, is a rational such that , which completes the proof. Olympiad Problems Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle. (Solution) (Manhattan Mathematical Olympiad 2004) Prove that having 100 whole numbers, one can choose 15 of them so that the difference of any two is divisible by 7. (Solution) (Manhattan Mathematical Olympiad 2005) Prove that from any set of one hundred whole numbers, one can choose either one number which is divisible by 100, or several numbers whose sum is divisible by 100. (Solution) (Manhattan Mathematical Olympiad 2003) Prove that among any ten points located inside a circle with diameter 5, there exist at least two at a distance less than 2 from each other. (Solution) (Japan 1997) Every point in a plane is either red, green, or blue. Prove that there exists a rectangle in the plane such that all of its vertices are the same color. (Solution) (USAMTS Year 18 - Round 1 - Problem 4) There are 51 senators in a senate. The senate needs to be divided into committees such that each senator is on exactly one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does 'not' necessarily hate senator A.) Find the smallest such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee. (Solution) (Red MOP lecture 2006) Given a real number , we define a sequence by , , and for . Prove that if for some , then the sequence is periodic. (Solution) (2018 Putnam B) See Also Combinatorics This article is a stub. Help us out by expanding it. 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15182	https://mathoverflow.net/questions/437561/asymptotics-of-int-0-infty-fracx2z-gamma1z-dz-for-large-x	real analysis - Asymptotics of $\int_0^\infty \frac{x^{2z}}{\Gamma(1+z)}\,dz$ for large $x$ - 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 Asymptotics of ∫∞0 x 2 z Γ(1+z)d z∫0∞x 2 z Γ(1+z)d z for large x x Ask Question Asked 2 years, 9 months ago Modified2 years, 9 months ago Viewed 625 times This question shows research effort; it is useful and clear 11 Save this question. Show activity on this post. I'm interested in the asymptotics of ∫∞0 x 2 z Γ(1+z)d z∫0∞x 2 z Γ(1+z)d z as x→∞x→∞. I expect the results to behave similarly to e x 2=∑k≥0 x 2 k k!e x 2=∑k≥0 x 2 k k!. However, I'm not quite sure how to develop the leading asymptotics of the integrals. I first thought that for large x x, the integral should be dominated by its integral over a small ball around the maximum of the integrand e 2 z log x Γ(1+z)e 2 z log⁡x Γ(1+z). To find this maximum, I computed f′(z)=(2 log(x)−ψ(0)(1+z))f(z)f′(z)=(2 log⁡(x)−ψ(0)(1+z))f(z) with ψ(0)(z)ψ(0)(z) the logarithmic derivative of Γ(z)Γ(z). Since f f never vanishes, the maximum must occur at z∈(0,∞)z∈(0,∞) such that ψ(0)(1+z)=2 log x ψ(0)(1+z)=2 log⁡x. Assuming that we will take x→∞x→∞ as well as the asymptotics ψ(z)=log z+O(1/z)ψ(z)=log⁡z+O(1/z) for large and positive z z, we seek to solve 2 log x=log(1+z)+O(1/z)2 log⁡x=log⁡(1+z)+O(1/z). Exponentiating, we find that x 2=1+z+O(1)x 2=1+z+O(1). Thus, we have that arg max f(z)∼x 2 arg max⁡f(z)∼x 2 is asymptotically correct for large x x. Let z 0=x 2 z 0=x 2. We can now rewrite the integral as dominated by e 2 x 2 log x Γ(1+x 2)∫z 0+ϵ z 0−ϵ e 2(z−z 0)log x Γ(1+z 0)Γ(1+z)d z.e 2 x 2 log⁡x Γ(1+x 2)∫z 0−ϵ z 0+ϵ e 2(z−z 0)log⁡x Γ(1+z 0)Γ(1+z)d z. However, I'm not sure what size to take ϵ ϵ as a function of x x. I do know that the expression outside of the integral is asymptotic to (2 π)−1/2 e x 2 x.(2 π)−1/2 e x 2 x. I'm not sure how to deal with the actual integral though. Input is much appreciated. Edit: Math Stack Exchange cross-post real-analysis ca.classical-analysis-and-odes asymptotics 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 Dec 31, 2022 at 22:18 DispersionDispersion asked Dec 31, 2022 at 6:22 DispersionDispersion 936 1 1 gold badge 7 7 silver badges 15 15 bronze badges 1 Consider g(x)=∫∞0 x t Γ(t+1)d t g(x)=∫0∞x t Γ(t+1)d t. You differentiate and then split the integral at 1, you find that g(x)/e x g(x)/e x has bounded derivative.user473423 –user473423 2022-12-31 08:06:58 +00:00 Commented Dec 31, 2022 at 8:06 Add a comment\| 2 Answers 2 Sorted by: Reset to default This answer is useful 13 Save this answer. Show activity on this post. Let g(x)g(x) denote your integral. Then g(x)∼e x 2(1)(1)g(x)∼e x 2 as x→∞x→∞. Proof: g(x)=∫∞0 d z x 2 z Γ(1+z).(1.5)(1.5)g(x)=∫0∞d z x 2 z Γ(1+z). So, for x>0 x>0, g′(x)=∫∞0 d z 2 z x 2 z−1 Γ(1+z)=∫∞0 d z 2 x 2 z−1 Γ(z)=2∫∞−1 d t x 2 t+1 Γ(1+t)=2 x∫∞−1 d z x 2 z Γ(1+z).g′(x)=∫0∞d z 2 z x 2 z−1 Γ(1+z)=∫0∞d z 2 x 2 z−1 Γ(z)=2∫−1∞d t x 2 t+1 Γ(1+t)=2 x∫−1∞d z x 2 z Γ(1+z). So (and this is the key), g g is a solution of the ODE g′(x)=2 x g(x)+h(x),g′(x)=2 x g(x)+h(x), where h(x):=2 x∫0−1 d z x 2 z Γ(1+z).h(x):=2 x∫−1 0 d z x 2 z Γ(1+z). Also, g(0)=0 g(0)=0. So, g(x)=e x 2 G(x),(2)(2)g(x)=e x 2 G(x), where G(x):=∫x 0 d u e−u 2 h(u).(3)(3)G(x):=∫0 x d u e−u 2 h(u). As x→∞x→∞, G(x)→∫∞0 d u e−u 2 h(u)=∫∞0 d u e−u 2 2 u∫0−1 d z u 2 z Γ(1+z)=∫∞0 d v e−v∫0−1 d z v z Γ(1+z)=∫0−1 d z Γ(1+z)∫∞0 d v v z e−v=∫0−1 d z Γ(1+z)Γ(1+z)=1.G(x)→∫0∞d u e−u 2 h(u)=∫0∞d u e−u 2 2 u∫−1 0 d z u 2 z Γ(1+z)=∫0∞d v e−v∫−1 0 d z v z Γ(1+z)=∫−1 0 d z Γ(1+z)∫0∞d v v z e−v=∫−1 0 d z Γ(1+z)Γ(1+z)=1. Now (1)(1) follows from (2)(2). □◻ Remark: Note that −Γ′(1)−Γ′(1) is Euler's γ>0 γ>0. Therefore and because Γ Γ is log convex on (0,∞)(0,∞), we see that Γ Γ is decreasing on (0,1](0,1]. So, for x≥1 x≥1 we have h(x)<2 x Γ(1)∫0−1 d z x 2 z<2 x.h(x)<2 x Γ(1)∫−1 0 d z x 2 z<2 x. So, by (2)(2)--(3)(3), for x≥1 x≥1 the relative error of the asymptotic approximation (1)(1) is 0<R(x):=∫∞x d u e−u 2 h(u)<∫∞x d u e−u 2 2 u=e−x 2,0<R(x):=∫x∞d u e−u 2 h(u)<∫x∞d u e−u 2 2 u=e−x 2, which goes to 0 0 very fast as x→∞x→∞. This bound on the relative error seems hard (if at all possible) to get by the Laplace method/Watson lemma applied to the original integral expression (1.5)(1.5) of g(x)g(x), which gives an asymptotic expansion in integral powers of x x. However, one can apply the Watson lemma to the expression (2)(2) of g(x)g(x) to get the following asymptotic expansion of the absolute error of approximation (1)(1), in integral powers of ln x ln⁡x: E(x):=e x 2−g(x)=e x 2∫∞x d u e−u 2 h(u)∼∑k≥0(−1)k c k(2 ln x)k+1(4)(4)E(x):=e x 2−g(x)=e x 2∫x∞d u e−u 2 h(u)∼∑k≥0(−1)k c k(2 ln⁡x)k+1 as x→∞x→∞, where c k:=d k d z k 1 Γ(1+z)∣∣z=0 c k:=d k d z k 1 Γ(1+z)\|z=0, so that c 0=1 c 0=1. □◻ For an illustration, below are the graphs {(x,g(x)/e x 2):0≤x≤4}{(x,g(x)/e x 2):0≤x≤4} (solid black), {(x,E(x)/E 0(x)):5≤x≤100}{(x,E(x)/E 0(x)):5≤x≤100} (solid red), {(x,E(x)/E 1(x)):5≤x≤100}{(x,E(x)/E 1(x)):5≤x≤100} (solid green), and {(x,E(x)/E 2(x)):5≤x≤100}{(x,E(x)/E 2(x)):5≤x≤100} (solid blue), where (cf. (4)(4)) E m(x):=∑m k=0(−1)k c k(2 ln x)k+1 E m(x):=∑k=0 m(−1)k c k(2 ln⁡x)k+1: Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications edited Jan 2, 2023 at 4:00 answered Jan 1, 2023 at 0:13 Iosif PinelisIosif Pinelis 141k 9 9 gold badges 120 120 silver badges 251 251 bronze badges Add a comment\| This answer is useful 11 Save this answer. Show activity on this post. The saddle point of the integrand is at z=x 2 z=x 2 for large x x, this gives the saddle point approximation ∫∞0 x 2 z Γ(1+z)d z→x 2 x 2 Γ(1+x 2).∫0∞x 2 z Γ(1+z)d z→x 2 x 2 Γ(1+x 2). You may then obtain higher order terms by performing the gaussian integral around the saddle point,∗∗ which gives a pre-exponential factor 2 π−−√x 2 π x. Substituting also the large-x x asymptotics of the Γ Γ function, I arrive at ∫∞0 x 2 z Γ(1+z)d z→e x 2.∫0∞x 2 z Γ(1+z)d z→e x 2. The e x 2 e x 2 approximation is already quite good for relatively small values of x x; see the plot (blue curve = exact, orange curve = approximation): ∗∗The integrand e F(z)e F(z) with F((z)=2 z ln x−ln Γ(1+z)F((z)=2 z ln⁡x−ln⁡Γ(1+z) is expanded to second order about the saddle point z 0 z 0, where the first derivative vanishes, F′(z 0)=0 F′(z 0)=0. For x≫1 x≫1 one has z 0=x 2 z 0=x 2 and F(z)≈2 z 0 ln x−ln Γ(1+z 0)−1 2 x−2(z−z 0)2.F(z)≈2 z 0 ln⁡x−ln⁡Γ(1+z 0)−1 2 x−2(z−z 0)2. The integral ∫∞−∞e F(z)d z∫−∞∞e F(z)d z then gives the asymptotic approximation e x 2 e x 2. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications edited Jan 2, 2023 at 14:26 answered Dec 31, 2022 at 8:04 Carlo BeenakkerCarlo Beenakker 199k 19 19 gold badges 479 479 silver badges 698 698 bronze badges 8 1 Re: The discriminatory power of graphical comparisons, have you tried plotting e x 2 e x 2 or x−1 e x 2 x−1 e x 2 just for fun as well?Michael Engelhardt –Michael Engelhardt 2022-12-31 14:54:52 +00:00 Commented Dec 31, 2022 at 14:54 1 Incidentally, I cross posted the question to math.stack.exchange (with the link to the post now in my mathoverflow post), and an answer there finds the asymptotics to be purely e x 2 e x 2.Dispersion –Dispersion 2022-12-31 22:19:23 +00:00 Commented Dec 31, 2022 at 22:19 What tool do you use for plotting?Sidharth Ghoshal –Sidharth Ghoshal 2022-12-31 22:29:38 +00:00 Commented Dec 31, 2022 at 22:29 the plots are Mathematica output...Carlo Beenakker –Carlo Beenakker 2022-12-31 22:53:53 +00:00 Commented Dec 31, 2022 at 22:53 3 Based on your Taylor expansion I would expect a prefactor of 2 π−−√x 2 π x rather than 2 π x−−−√2 π x, which should explain the discrepancy with the other answers.Terry Tao –Terry Tao 2023-01-01 02:29:24 +00:00 Commented Jan 1, 2023 at 2:29 \|Show 3 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 ca.classical-analysis-and-odes asymptotics See similar questions with these tags. Featured on Meta Spevacus has joined us as a Community Manager Introducing a new proactive anti-spam measure Linked 2When is it true that ∑k≥0 x k Γ(1+a(k))∼∫∞0 x t Γ(1+a(t))d t∑k≥0 x k Γ(1+a(k))∼∫0∞x t Γ(1+a(t))d t as x→∞x→∞? 1Necessary and sufficient conditions so that ∑k≥0 x k Γ(1+a(k))∼∫∞0 x t Γ(1+a(t))d t∑k≥0 x k Γ(1+a(k))∼∫0∞x t Γ(1+a(t))d t as x→∞x→∞? Related 3Asymptotic behaviour/upper bound for ∫∞0 exp(−c x a+K x b)d x∫0∞exp⁡(−c x a+K x b)d x for a>b>0 a>b>0 as K→∞K→∞? 8Asymptotics of a special function 0Solving or bounding the real part of the integral ∫2 π i m 0 e−t t−a d t∫0 2 π i m e−t t−a d t 0Stirling's approximation for normalized Γ Γ 3Asymptotic bound for ∑∞x=0∑∞y=0(x+y)m e−x 2 2 i−y 2 2 j∑x=0∞∑y=0∞(x+y)m e−x 2 2 i−y 2 2 j for i i and j j large 2When is it true that ∑k≥0 x k Γ(1+a(k))∼∫∞0 x t Γ(1+a(t))d t∑k≥0 x k Γ(1+a(k))∼∫0∞x t Γ(1+a(t))d t as x→∞x→∞? 6Bessel function J ν(x)J ν(x) asymptotics for ν≈x ν≈x Question feed Subscribe to RSS Question feed To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Why are you flagging this comment? It contains harassment, bigotry or abuse. This comment attacks a person or group. Learn more in our Code of Conduct. It's unfriendly or unkind. This comment is rude or condescending. Learn more in our Code of Conduct. Not needed. 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15183	https://books.google.com/books/about/Concepts_of_Modern_Physics.html?id=LP0eAAAACAAJ	Concepts of Modern Physics - Arthur Beiser - Google Books Sign in Hidden fields Try the new Google Books Books Add to my library Try the new Google Books Check out the new look and enjoy easier access to your favorite features Try it now No thanks Try the new Google Books My library Help Advanced Book Search Get print book No eBook available Amazon.com Barnes&Noble.com Books-A-Million IndieBound Find in a library All sellers» ### Get Textbooks on Google Play Rent and save from the world's largest eBookstore. Read, highlight, and take notes, across web, tablet, and phone. Go to Google Play Now » My library My History Concepts of Modern Physics Arthur Beiser McGraw-Hill, 1967 - Matter - 405 pages From inside the book Contents THE SPECIAL THEORY OF RELATIVITY 3 THE PARTICLE PROPERTIES OF WAVES 40 THE WAVE PROPERTIES OF PARTICLES 61 Copyright 14 other sections not shown Other editions - View all Concepts of Modern Physics Arthur Beiser Snippet view - 1967 Common terms and phrases alpha decayalpha particleangleangular momentumatomic numberaveragebaryonbeambeta decaybinding energyBohrBroglie wavescellcenter of massclassicalcollisionconservationconstantcorrespondingcross sectioncrystaldeuterondirectiondistancedistributionelecelectromagneticelectrostaticelementary particlesemissionemittedenergy levelsexclusion principleexperimentFIGUREfissionforceformulaframe of referencefrequencyHencehydrogen atomhyperoninteractionionskinetic energym/secm₁magnetic fieldmagnetic quantummass numbermesonmetalmoleculemomentamotionmovingn₁neutrinoneutronnucleonsnucleusobservedoscillatorparityphotoelectric effectphysicspositionpositronpotential energyprobability densityprotonquantum mechanicsradiationradiusraysrelativeresultrotationalscatteredSchrödinger's equationshellssodiumspacespectraspectral linesspectrumsubshelltargettemperaturetiontotal energytransitiontronvectorvelocityvibrationswave functionwavelengthX-rayzero References to this book Mechanical Behaviour of Engineering Materials: Metals, Ceramics, Polymers ... Joachim Roesler,Harald Harders,Martin Baeker Limited preview - 2007 Medical Infra-Red Thermography Woodrough No preview available - 1982 All Book Search results » Bibliographic information Title Concepts of Modern Physics International student edition MacGraw-Hill series in fundamentals of physics : an undergraduate textbook program McGraw-Hill Series in Fundamentals of Physics. Upper-Division Texts McGraw-Hill series in fundamentals of physics AuthorArthur Beiser Edition 2, revised Publisher McGraw-Hill, 1967 Original from the University of Michigan Digitized Nov 20, 2007 ISBN 0070043485, 9780070043480 Length 405 pages Export CitationBiBTeXEndNoteRefMan About Google Books - Privacy Policy - Terms of Service - Information for Publishers - Report an issue - Help - Google Home
15184	https://www.mathbootcamps.com/binomial-probabilities-examples/	Binomial probabilities – examples (calculator) Once you have determined that an experiment is a binomial experiment, then you can apply either the formula or technology (like a TI calculator) to find any related probabilities. In this lesson, we will work through an example using the TI 83/84 calculator. If you aren’t sure how to use this to find binomial probabilities, please check here: Details on how to use a calculator to find binomial probabilities. [adsenseWide] Example A student is taking a multiple choice quiz but forgot to study and so he will randomly guess the answer to each question. There are a total of 12 questions, each with 4 answer choices. Only one answer is correct for each question. Verifying the experiment is binomial We know that this experiment is binomial since we have (n = 12) trials of the mini-experiment “guess the answer on a question”. There are two outcomes: “guess correctly”, “guess incorrectly”. If we treat a success as guessing a question correctly, then since there are 4 answer choices and only 1 is correct, the probability of success is: (p = \dfrac{1}{4} = 0.25) Finally, since the guessing is random, it is reasonable to assume that each guess is independent of the other guesses. Calculating probabilities We will let (X) represent the number of questions guessed correctly. Let’s now use this binomial experiment to answer a few questions. (a) Find the probability that he answers 6 of the questions correctly. This is asking for the probability of 6 successes, or (P(X = 6)). For finding an exact number of successes like this, we should use binompdf from the calculator. For this problem, (n = 12) and (p = 0.25). Therefore: (\begin{align} P(X=6) &= \text{binompdf(12,0.25,6)} \ &\approx \boxed{0.0401}\end{align}) (b) Find the probability that he correctly answers 3 or fewer of the questions. The probability of 3 or fewer successes is represented by (P(X < 3)). Anytime you are counting down from some possible value of (X), you will use binomcdf. (\begin{align}P(X < 3) &= \text{binomcdf(12, 0.25, 3)} \ &\approx \boxed{0.6488}\end{align}) It isn’t looking good. This is a pretty high chance that the student only answers 3 or fewer correctly! (c) Find the probability that he correctly answers more than 8 questions. This probability is represented by (P(X > 8)). To understand how to find this probability using binomcdf, it is helpful to look at the following diagram. This shows all possible values of (X) with the values which would represent “more than 8 successes” highlighted in red. To calculate this, we could do the binompdf of 9, the binompdf of 10, the binompdf of 11, and the binompdf of 12 and add them all together. But, this would take quite a while. Instead, we could use the complementary event. Recall that (P(A)) is (1 – P(A \text{ complement})). So, we can write: (\begin{align} P(X > 8) &= 1 – P( X < 8) \ &= 1 - \text{binomcdf(12, 0.25, 8)}\ &\approx \boxed{3.9 \times 10^{-4}}\end{align}) This is a very small probability. Looks like the random guessing probably won’t pay off too much. (d) Find the probability that he correctly answers 5 or more questions. This probability is represented by (P(X \geq 5)). Using our diagram: Again, since this is asking for a probability of > or (\geq);, and the CDF only counts down, we will use the complement. Notice that the complementary event starts with 4 and counts down. So, we will use 4 in the CDF. (\begin{align}P(X \geq 5) &= 1 – P(X < 5)\ &= 1 - \text{binomcdf(12, 0.25, 4)}\ &\approx \boxed{0.1576}\end{align}) (e) Find the probability that he correctly answers fewer than 2 questions. This is asking for (P(X < 2)). Since this is counting down, we can use binomcdf. But, the event “fewer than 2” does not include 2. So, we will put 1 into the cdf function. (\begin{align} P(X < 2) &= \text{binomcdf(12, 0.25, 1)}\ &\approx \boxed{0.1584}\end{align}) (e) Find the probability that he correctly answers between 5 and 10 questions (inclusive) correctly. Since this is inclusive, we are including the values of 5 and 10. That is, we are finding (P(5 \leq X \leq 10)). Remember, you can always find the PDF of each value and add them up to get the probability. Here however, we can creatively use the CDF. Recall that the CDF takes whatever value you put in and adds the PDFs for each value starting with that number all the way down to zero. If we find the CDF of 10, it will add the PDFs of 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, and 0. This will include all the values below 5, which we don’t want. So, we will subtract them out! This will leave exactly the values we want: (\begin{align}P(5 \leq X \leq 10) &= \text{binomcdf(12,0.25,10)} – \text{binomcdf(12,0.25,4)}\ &\approx \boxed{0.1576}\end{align}) Interesting observation Did you notice that two of our answers were really similar? We found that: (\begin{align}P(X \geq 5) &= 1 – P(X < 5)\ &= 1 - \text{binomcdf(12, 0.25, 4)}\ &\approx \boxed{0.1576}\end{align}) and (\begin{align}P(5 \leq X \leq 10) &= \text{binomcdf(12,0.25,10)} – \text{binomcdf(12,0.25,4)}\ &\approx \boxed{0.1576}\end{align}) What is going on here? Well, these probabilities aren’t exactly the same. The first is actually 0.1576436761 while the second is 0.1576414707. These are certainly very close though! The tiny difference is because (P(X \geq 5)) includes (P(X = 11)) and (P(X = 12)), while (P(5 \leq X \leq 10)) does not. Further, (P(X = 11)) represents the probability that he correctly answers 11 of the questions correctly and latex (P(X = 12)) represents the probability that he answers all 12 of the questions correctly. Both events are very unlikely since he is guessing! In fact: (\begin{align}P(X = 11) &= \text{binompdf(12,0.25,11)} \ &\approx \boxed{2.14 \times 10^{-6}}\end{align}) and (\begin{align} P(X = 12) &= \text{binompdf(12,0.25,12)} \ &\approx \boxed{5.96 \times 10^{-8}}\end{align}) Since these are so tiny, including them in the first probability only increases the probability a little bit. Rounding to 4 decimal places, we didn’t even catch the difference. [adsenseLargeRectangle] Summary The only reason we were able to calculate these probabilities is because we recognized that this was a binomial experiment. This fact allowed us to use binompdf for exact probabilities and binomcdf for probabilities that included multiple values. Just remember – binomcdf is cumulative. It adds up PDFs for the value you put in, all the way down to zero. Using this, you can find pretty much any binomial probability as long as you use something like the diagrams we drew above to keep track of the needed values. Share this: Click to share on Twitter (Opens in new window) Click to share on Facebook (Opens in new window) Loading Comments...
15185	https://sciendo.com/article/10.2478/chilat-2020-0015	Diabetic Retinopathy. Association with Metabolic... Skip to main content Enable accessibility for low vision Open the accessibility menu Accessibility Menu Skip to content Publish & DistributePublishing SolutionsDistribution SolutionsLibrary Services SubjectsArchitecture and DesignArtsBusiness and EconomicsChemistryClassical and Ancient Near Eastern StudiesComputer SciencesCultural StudiesEngineeringGeneral InterestGeosciencesHistoryIndustrial ChemistryJewish StudiesLawLibrary and Information Science, Book StudiesLife SciencesLinguistics and SemioticsLiterary StudiesMaterials SciencesMathematicsMedicineMusicPharmacyPhilosophyPhysicsSocial SciencesSports and RecreationTheology and Religion PublicationsJournalsBooksProceedingsPublishersJournal Matcher Blog Contact EnglishEnglishDeutschPolskiEspañolFrançaisItaliano Home Journals Acta Chirurgica Latviensis Volume 18 (2020): Issue 1 (June 2020) Open Access Diabetic Retinopathy. Association with Metabolic Compensation, Duration of Diabetes and Other Micro and Macrovascular Complications in Patients with Type 1 Diabetes Mellitus Lelde Ullase Lelde Ullase Resident in Ophthalmology, Faculty of Medicine, University of Latvia Riga Latvia Department of Ophthalmology, Pauls Stradins Clinical University Hospital Riga, Latvia Search for this author on Sciendo\|Google Scholar Ullase, Lelde , Kristīne Ducena Kristīne Ducena SIA University of Latvia Medical postgraduate education institute Riga, Latvia Search for this author on Sciendo\|Google Scholar Ducena, Kristīne , Dace Markevica Dace Markevica Department of Ophthalmology, Pauls Stradins Clinical University Hospital Riga, Latvia Search for this author on Sciendo\|Google Scholar Markevica, Dace and Guna Laganovska Guna Laganovska Department of Ophthalmology, Pauls Stradins Clinical University Hospital Riga, Latvia Search for this author on Sciendo\|Google Scholar Laganovska, Guna Nov 18, 2020 Acta Chirurgica Latviensis Volume 18 (2020): Issue 1 (June 2020) About this article Previous Article Next Article Abstract References Authors Articles in this Issue Preview PDF Cite Share Download Cover Published Online: Nov 18, 2020 Page range:56 - 62 DOI: Keywords diabetic retinopathy, metabolic compensation, glycated hemoglobin, type 1 diabetes mellitus, nephropathy, neuropathy, atherosclerosis, arterial hypertension, coronary heart disease, dyslipidemia © 2020 Dr. Lelde Ullase et al., published by Sciendo This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. Introduction Diabetic retinopathy (DR) is a severe complication that can lead to complete vision loss and still is one of the main blindness-causing reasons among patients with type 1 diabetes mellitus (T1DM). DR as a complication can cause vision loss to people at their working age. More than 90% of patients with type 1 diabetes will develop DR by 20 years post diagnosis (Leslie R. Dye, 2018). DR is more likely to develop in patients with T1DM (Kanski's Clinical Ophthalmology, 2016). This complication can be very serious speaking of the ability to see. Sometimes vitrectomy plays a vital role in the management of severe complications of DR at its end-stage (Myron Yanoff et al., 2020). Aim of the study To prove the development severity of DR that depends on the duration of diabetes and metabolic compensation. Additionally, to determine retinopathy's association with other micro and macrovascular diabetes mellitus complications for a better understanding of what are the contributory factors for these complications to develop and which of those may coexist. Materials and Methods A retrospective study was held at the Pauls Stradins Clinical University Hospital (Riga, Latvia). From January 2016 to March 2018, 79 (158 eyes) patient histories were analyzed who have type 1 diabetes mellitus. To obtain more precise research results, almost all patients have done check-up visits to one certain ophthalmologist. The IBM SPSS Statistics version 25.0 was used to process data. Tables were made in SPSS and Microsoft Excel 2016 programs. Statistically significant value (p) was set at < 0,05. Results No statistically significant difference is seen in the mean duration of the disease: in the group of proliferative diabetic retinopathy (PDR): 25.23 (median = 22.0) years and non-proliferative group: 24.68 (median = 23.50) years. Results show that the duration of diabetes mellitus is considerably smaller in a group without DR 11.24 (median = 8.50) years. Metabolic compensation (%) in diabetes mellitus is not statistically different between patients with diverse forms of DR; no association found either. No statistically significant difference in best corrected visual acuity (BCVA) was detected among patients with various forms of DR. Three groups were compared: both types of DR and no DR. It was detected that BCVA in patients without DR was higher in both eyes: 0.83 ± 0.27 dioptres. No statistically significant difference (pχ > 0.05) was detected between the groups of DR and therefore no association was made between the form/existence of DR and arterial hypertension. There is a strong association between DR and microvascular complications (V = 0.40) with the existence of DR and there is an even stronger association (V = 0.61) with the forms of DR. There is no statistically reliable difference (pχ > 0.05) between the groups of DR; therefore, no association with the existence of microvascular complications and also risk factors. Conclusions More than two-thirds of patients included in the research have some signs of DR. Because of the strong association of DR and other microvascular complications, patients with diabetes should be screened regularly for retinopathy, nephropathy, and neuropathy. And likewise, if a patient has at least one microvascular complication, he or she should be tested for the rest possible complications as well. According to data, most of the patients in this study have poor metabolic compensation; consequently, the metabolic compensation screening should be done certainly every three months. Preview Recommended articles from TrendMD We recommend Assessment of Prevalence and Risk Factors for Diabetic Retinopathy in Patients with Type 1 and Type 2 Diabetes Examined at a Tertiary CareBrankica Krstevska, PRILOZI, 2023 Correlation between diabetic nephropathy and diabetic retinopathy as a long term complications of diabetes mellitusMuamer Dervišević, Acta Medica Marisiensis, 2023 Retinal Complications in Diabetes Mellitus: Importance of Screening and ManagementMilena Golubovik, PRILOZI, 2015 The association between insulin resistance and proliferative retinopathy in type 1 diabetesIrina Duţă, Romanian Journal of Internal Medicine, 2015 The Corneal Changes in Diabetic PatientsSuncica Sreckovic, Serbian Journal of Experimental and Clinical Research, 2021 Multi-Scale Class Attention Network for Diabetes Retinopathy GradingHongyu Chen, WU Rong-hua, Tao Chen, et al., International Journal of Network Dynamics and Intelligence, 2023 Identification of glutamine as a potential therapeutic target in dry eye diseaseXiaoniao Chen, Signal Transduction and Targeted Therapy, 2025 Macrophage immunometabolism in diabetes-associated atherosclerosisGoedeke, Leigh, Immunometabolism, 2023 Prognostic impact of MALs and potential immunotherapy targets in uveal melanomaJing Yang, Zhou Fu, Qin Xiang, Pediatric Discovery, 2024 Regulatory T cells and bioenergetics of peripheral blood mononuclear cells linked to pediatric obesityRose, Shannon, Immunometabolism, 2024 Powered by Targeting settings Do not sell my personal information Language:English Publication timeframe:1 times per year Journal Subjects:Medicine, Clinical Medicine, Surgery, Surgery, other Journal RSS Feed The Sciendo Team is part of Paradigm Publishing Services Blog Career Team Contact Terms Privacy Cookie Policy Publishing and Ethical Policies Worldwide De Gruyter Brill Sp. z o.o Nowogrodzka 4/3 00-513 Warsaw, Poland US & Canada De Gruyter Brill, Inc. 121 High Street, 3rd Fl. 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15186	https://www.sciencedirect.com/science/article/abs/pii/S1756231711000132	Pitfalls in the diagnosis of endometriosis: a condition characterized by a plethora of unusual histological features - ScienceDirect Skip to main contentSkip to article Journals & Books Access throughyour organization Purchase PDF Patient Access Other access options Search ScienceDirect Article preview Abstract Introduction Section snippets References (63) Cited by (7) Diagnostic Histopathology Volume 17, Issue 4, April 2011, Pages 193-202 Mini-symposium: ovarian pathology Pitfalls in the diagnosis of endometriosis: a condition characterized by a plethora of unusual histological features Author links open overlay panel Oisin Houghton, W. Glenn McCluggage Show more Add to Mendeley Share Cite rights and content Abstract The histological diagnosis of endometriosis is usually straightforward. However, for a large number of reasons, there may be diagnostic difficulties. These difficulties may be related to both unusual morphological features in endometriosis and the occurrence of endometriosis at unusual sites. In this review, 10 problematic areas in the histological diagnosis of endometriosis are discussed. Introduction Endometriosis, defined as the presence of endometrial tissue in locations outside the uterine corpus, is an extremely common condition, especially in women in the reproductive years.1 The most common locations are the ovary, fallopian tube, pouch of Douglas and pelvic peritoneum but a variety of sites may be affected. In most cases, the histological diagnosis is straightforward with endometrioid type glands and stroma often associated with pigment-laden macrophages. However, in a not insignificant number of cases, problems may occur in establishing a diagnosis and this is obviously important from a therapeutic point of view. We discuss 10 problematic areas related to the diagnosis of endometriosis. In some of the scenarios discussed, the identification of the secondary changes described is a clue to the presence of underlying endometriosis and further sampling may assist the pathologist by revealing more diagnostic foci.2 In other cases, the presence of endometriosis may explain unusual and worrisome morphological findings, an example being florid reactive mesothelial proliferations which may occur in association with endometriosis. In this review, we do not discuss in detail malignant transformation of endometriosis but note that there is a firm association between endometriosis and the development of certain Mullerian neoplasms, especially endometrioid and clear cell carcinoma and Mullerian type mucinous borderline tumours. However, neoplastic transformation of endometriosis is uncommon and it would be wrong to consider this as a premalignant disease; rather endometriosis can be considered as a disease with the potential to develop malignancy in a small number of cases. Access through your organization Check access to the full text by signing in through your organization. Access through your organization Section snippets Stromal endometriosis In our opinion, this is one of the commonest pitfalls associated with the diagnosis of endometriosis. Stromal endometriosis is characterized by the presence of endometrioid type stroma in the absence of glands and has been described in the peritoneum, cervix and ovary.3, 4, 5 We have also seen this occasionally on the omentum and at other sites. Stromal endometriosis is relatively common, one study identifying it in 44.9% of peritoneal biopsies containing endometriosis, usually, but not always, Stromal metaplasias and other alterations in the stromal component of endometriosis The stromal component of endometriosis may undergo a number of morphological alterations which if not recognized may result in underdiagnosis or in misinterpretation as an alternative pathological process. These are discussed in the next paragraphs. Stromal elastosis is manifested by focal or total replacement of the endometrial stromal cells by fibrillary, pink to blue-grey elastic tissue which can be highlighted with an elastic stain.2, 4 It has been speculated that the elastic tissue may be Endometriosis composed of glands only The stromal component of endometriosis may be atrophic and barely discernible especially in postmenopausal women. In this situation, the finding of isolated endometrioid glands may result in misinterpretation as endosalpingiosis. A relative paucity of cilia (cilia may be present focally in endometrioid type epithelium) and the identification of typical endometriosis elsewhere assist in diagnosis. In some cases, there is subtle mild hypercellularity of the stroma surrounding the glands and this Nuclear atypia in endometriosis (atypical endometriosis) Nuclear atypia is a common finding involving the epithelial lining of ovarian endometriotic cysts. Usually the atypia is mild but on occasions it is quite striking. The atypia may be focal, multifocal or widespread.23, 24, 25, 26, 27, 28 The atypical cells often have a polygonal appearance with enlarged nuclei, prominent nucleoli and sometimes a smudged chromatin (Figure 4). They are often infiltrated by polymorphs. There is characteristically abundant eosinophilic cytoplasm and hobnail cells Epithelial metaplasias in endometriosis The range of epithelial metaplasias that potentially occur within the eutopic endometrium may be seen in endometriosis, including ciliated, eosinophilic, hobnail, clear cell, squamous and mucinous types.29, 30 The Arias-Stella effect may occur in pregnancy or rarely in association with hormonal medications.2, 31 Epithelial metaplasias are a very common finding in endometriosis with a prevalence in one study of 63%29 and they usually pose no diagnostic problem, especially if a focal finding. Polypoid endometriosis Polypoid endometriosis is a term used for a mass forming polypoid endometriotic lesion, sometimes with microscopic features resembling an endometrial polyp.35 As well as endometrioid type glands and stroma, there is often abundant fibrous stroma with thick walled blood vessels. The most common sites of involvement are the colon and ovary, followed by the uterine serosa, cervix, vagina, ureter, fallopian tube, omentum, bladder, paraurethral and paravaginal soft tissues and retroperitoneum.35 In Vascular, perineural and lymph node involvement in endometriosis Endometrial tissue within myometrial vascular channels is a common finding in adenomyosis and is of no clinical significance.39 Although much less common, endometriotic tissue is occasionally seen within vascular spaces. This has been used to support the embolic theory of endometriosis.40, 41, 42 It is usually minor in degree. Vascular involvement in endometriosis has been reported in the ureter, ovarian hilar and mesovarian vessels and vessels within the colonic wall.40, 41, 42 Involvement of Mesothelial hyperplasia associated with endometriosis Mesothelial hyperplasia is a common response to peritoneal inflammation or some other insult and is usually minor in degree and poses no diagnostic difficulty. There may be a wide variety of underlying lesions including endometriosis, pelvic inflammatory disease, tubo-ovarian abscess, ovarian torsion or an ovarian tumour or the mesothelial reaction may be a response to mechanical irritation following surgery.49, 50, 51 Mesothelial hyperplasia, usually involving the surface of the ovary, is Superficial cervical endometriosis There are two distinct forms of cervical endometriosis with a different underlying pathogenesis. Deep (secondary) cervical endometriosis, occurring in the outer aspects of the cervical stroma, is usually associated with pelvic endometriosis or endometriosis of the rectovaginal septum. Diagnosis is usually straightforward. Superficial (primary) cervical endometriosis is more common and is located superficially in the cervix, usually close to the transformation zone.53 It is thought to occur Intestinal endometriosis Endometriosis of the gastrointestinal tract has been estimated to occur in up to 37% of patients with pelvic endometriosis.57, 58 The most common site of involvement is the rectosigmoid followed by the proximal colon, small intestine and caecum. It is also not uncommon to identify endometriosis as an incidental finding in the appendix. All layers of the intestinal wall may be involved with the subserosa and muscularis propria being the predominant layers affected. When the mucosa and submucosa Recommended articles References (63) S.A. Missmer et al. The epidemiology of endometriosis Obstet Gynecol Clin North Am (2003) A. LaGrenade et al. Ovarian tumors associated with atypical endometriosis Hum Pathol (1988) F. Prefumo et al. Epithelial abnormalities in cystic ovarian endometriosis Gynecol Oncol (2002) R.E. Jimenez et al. Unilateral hydronephrosis resulting from intraluminal obstruction of the ureter by adenosquamous endometrioid carcinoma arising from disseminated endometriosis Urology (2000) K. Ooi et al. Intravascular endometrial tissue in an ovary of a patient with abnormal endometrial histology Pathology (1994) L. Insabato et al. Endometriosis of the bowel with lymph node involvement. A report of three cases and review of the literature Pathol Res Pract (1996) P.B. Clement The pathology of endometriosis: a survey of the many faces of a common disease emphasizing diagnostic pitfalls and unusual and newly appreciated aspects Adv Anat Pathol (2007) D.P. Boyle et al. Peritoneal stromal endometriosis: a detailed morphological analysis of a large series of cases of a common and under-recognised form of endometriosis J Clin Pathol (2009) P.B. Clement et al. Two previously unemphasized features of endometriosis: micronodular stromal endometriosis and endometriosis with stromal elastosis Int J Surg Pathol (2000) P.B. Clement et al. Stromal endometriosis of the uterine cervix. A variant of endometriosis that may simulate a sarcoma Am J Surg Pathol (1990) V.P. Sumathi et al. CD10 is useful in demonstrating endometrial stroma at ectopic sites and in confirming a diagnosis of endometriosis J Clin Pathol (2002) G.M. Groisman et al. CD10 is helpful in detecting occult or inconspicuous endometrial stromal cells in cases of presumptive endometriosis Arch Pathol Lab Med (2003) M.M. Kennedy et al. Cyclin D1 expression and HHV8 in Kaposi sarcoma J Clin Pathol (1999) R.H. Young et al. Endometrioid stromal sarcomas of the ovary. A clinicopathologic analysis of 23 cases Cancer (1984) M. Fukunaga Smooth muscle metaplasia in ovarian endometriosis Histopathology (2000) M. Fukunaga Uterus-like mass in the uterine cervix: superficial cervical endometriosis with smooth muscle metaplasia? Virchows Arch (2001) S.A. Pai et al. Uterus-like masses of the ovary associated with breast cancer and raised serum CA125 Am J Surg Pathol (1998) P.B. Clement et al. Endometriosis with myxoid change. A case simulating pseudomyxoma peritonei Am J Surg Pathol (1994) F.F. Nogales et al. Myxoid change in decidualized scar endometriosis mimicking malignancy J Cutan Pathol (1993) W.G. McCluggage et al. Pregnancy associated endometriosis with pronounced stromal myxoid change J Clin Pathol (2000) L.R. Bégin Florid soft-tissue decidual reaction: a potential mimic of neoplasia Am J Surg Pathol (1997) P.L. Perrotta et al. Liesegang rings and endometriosis Int J Gynecol Pathol (1998) D.V. Luna Liesegang rings in paratubal cysts: case report and literature review Int J Gynecol Pathol (2010) P.B. Clement et al. Necrotic pseudoxanthomatous nodules of ovary and peritoneum in endometriosis Am J Surg Pathol (1988) S.C. Chou et al. Malacoplakia of the ovary, fallopian tube and uterus: a case associated with diabetes mellitus Pathol Int (2002) M. Furuya et al. Pseudoxanthomatous and xanthogranulomatous salpingitis of the fallopian tube: a report of four cases Int J Gynecol Pathol (2002) R. Colella et al. Endometriosis-associated skeletal muscle regeneration: a hitherto undescribed entity and a potential diagnostic pitfall Am J Surg Pathol (2010) J.D. Seidman Prognostic importance of hyperplasia and atypia in endometriosis Int J Gynecol Pathol (1996) B. Czernobilsky et al. A histologic study of ovarian endometriosis with emphasis on hyperplastic and atypical changes Obstet Gynecol (1979) M. Fukunaga et al. Ovarian atypical endometriosis: its close association with malignant epithelial tumours Histopathology (1997) F. Ballouk et al. Ovarian endometriotic cysts. An analysis of cytologic atypia and DNA ploidy patterns Am J Clin Pathol (1994) View more references Cited by (7) Glypican3 and serglycin as potential biomarkers involved in the pathogenesis of ovarian endometriosis 2025, Tissue and Cell Show abstract Endometriosis, a non-malignant gynecological disorder characterized by debilitating symptoms, displays several cancer-like characteristics, including metastatic behavior and extracellular matrix (ECM) remodeling. The dynamics of ECM are largely influenced by proteoglycans (PGs), a family of glycosaminoglycan (GAG)-decorated proteins known for their regulatory impact on cellular behavior through ECM modulation. This study aimed to investigate the dysregulated expression of 20 PG genes in ovarian endometrioma (n = 24) in comparison to eutopic endometrial tissue samples (n = 16) from patients diagnosed with ovarian endometriosis, employing quantitative real-time PCR (qPCR) and immunohistochemistry (IHC). qPCR screening identified four upregulated PG genes—glypican 3 (GPC3), decorin (DCN), serglycin (SRGN), and glypican 5 (GPC5)—whereas 16 PG genes were found to be downregulated. In ovarian endometrioma, relative to eutopic endometrial tissue, GPC3 and SRGN expression were further verified to be significantly overexpressed by 18.6-fold (P< 0.05) and 6.7-fold (P< 0.01), respectively, whereas brevican (BCAN) and syndecan 4 (SDC4) were markedly downregulated by approximately 90 % and 86 %, respectively (both P< 0.001). IHC staining further validated the significant overexpression of GPC3 protein in ovarian endometrioma compared to eutopic and control endometrial tissues (P< 0.0001). In-silico analysis using the Enrichr database identified enriched functional pathways associated with the top overexpressed genes, such as hypoxia, glycolysis, and WNT signaling, known to be implicated in endometriosis. These findings suggest that the overexpression of GPC3 and SRGN may contribute to the pathogenesis of ovarian endometrioma, highlighting their potential as biomarkers and therapeutic targets for this disease. ### Histotyping and grading of endometriosis and its association with clinico-pathological parameters 2022, Journal of Obstetrics and Gynaecology ### A Clinical and Pathologic Exploration of Suspected Peritoneal Endometriotic Lesions 2021, International Journal of Gynecological Pathology ### Endometriosis-related pathology: a discussion of selected uncommon benign, premalignant and malignant lesions 2020, Histopathology ### Menstrual endometrial supernatant may induce stromal endometriosis in baboons 2014, Frontiers in Bioscience Scholar ### Abdominal wall endometriosis associated with ventriculoperitoneal and lumboperitoneal shunts: A report of 2 cases of an extremely rare phenomenon 2012, International Journal of Surgical Pathology View all citing articles on Scopus View full text Copyright © 2011 Elsevier Ltd. All rights reserved. Recommended articles Nonlinear low-velocity impact analysis of matrix cracked hybrid laminated plates containing CNTRC layers resting on visco-Pasternak foundation Composites Part B: Engineering, Volume 117, 2017, pp. 9-19 Yin Fan, Hai Wang ### A novel use of surgicel® as a spacer for intraoperative contour defect JPRAS Open, Volume 13, 2017, pp. 46-48 Ammar Allouni, David Dujon ### Vingt ans après… Histoire d’un textilome intra-thoracique Revue des Maladies Respiratoires, Volume 36, Issue 2, 2019, pp. 214-218 L.Lebas, …, A.Didier ### Fibrous hamartoma of infancy in the scrotum – Report of a case Journal of Pediatric Surgery Case Reports, Volume 11, 2016, pp. 25-27 Naoki Hashizume, …, Minoru Yagi ### Obesity does not alter endometrial gene expression in women with endometriosis Reproductive BioMedicine Online, Volume 41, Issue 1, 2020, pp. 113-118 Sarah J Holdsworth-Carson, …, Jane E Girling ### Cytomorphology of follicular dendritic cell sarcoma: a report of 7 cases with an emphasis on the diagnostic challenges Journal of the American Society of Cytopathology, Volume 12, Issue 3, 2023, pp. 229-238 Cody Weimholt, …, Cedric Bailey Show 3 more articles About ScienceDirect Remote access Contact and support Terms and conditions Privacy policy Cookies are used by this site.Cookie settings All content on this site: Copyright © 2025 Elsevier B.V., its licensors, and contributors. 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15187	https://bookdown.org/egarpor/PM-UC3M/lm-iii-constr.html	4.2 Constrained linear models \| Notes for Predictive Modeling Type to search Predictive Modeling Preface Welcome Main references and credits Contributions License Citation 1 Introduction 1.1 Course overview 1.2 What is Predictive Modeling? 1.3 General notation and background 1.4 Scripts and datasets 2 Linear models I: multiple linear model 2.1 Case study: The Bordeaux equation 2.2 Model formulation and least squares 2.2.1 Simple linear model 2.2.2 Case study application 2.2.3 Multiple linear model 2.2.4 Case study application 2.3 Assumptions of the model 2.4 Inference for model parameters 2.4.1 Distributions of the fitted coefficients 2.4.2 Confidence intervals for the coefficients 2.4.3 Testing on the coefficients 2.4.4 Case study application 2.5 Prediction 2.5.1 Case study application 2.6 ANOVA 2.6.1 Case study application 2.7 Model fit 2.7.1 The R 2 R 2 2.7.2 The R 2 Adj R 2 Adj 2.7.3 Case study application 3 Linear models II: model selection, extensions, and diagnostics 3.1 Case study: Housing values in Boston 3.2 Model selection 3.2.1 Case study application 3.2.2 Consistency in model selection 3.3 Use of qualitative predictors 3.3.1 Case study application 3.4 Nonlinear relationships 3.4.1 Transformations in the simple linear model 3.4.2 Polynomial transformations 3.4.3 Interactions 3.4.4 Case study application 3.5 Model diagnostics 3.5.1 Linearity 3.5.2 Normality 3.5.3 Homoscedasticity 3.5.4 Independence 3.5.5 Multicollinearity 3.5.6 Outliers and high-leverage points 3.5.7 Case study application 3.6 Dimension reduction techniques 3.6.1 Review on principal component analysis 3.6.2 Principal components regression 3.6.3 Partial least squares regression 4 Linear models III: shrinkage, multivariate response, and big data 4.1 Shrinkage 4.1.1 Ridge regression 4.1.2 Lasso 4.1.3 Variable selection with lasso 4.2 Constrained linear models 4.3 Multivariate multiple linear model 4.3.1 Model formulation and least squares 4.3.2 Assumptions and inference 4.3.3 Shrinkage 4.4 Big data considerations 5 Generalized linear models 5.1 Case study: The Challenger disaster 5.2 Model formulation and estimation 5.2.1 Logistic regression 5.2.2 General case 5.3 Inference for model parameters 5.3.1 Distributions of the fitted coefficients 5.3.2 Confidence intervals for the coefficients 5.3.3 Testing on the coefficients 5.3.4 Case study application 5.4 Prediction 5.4.1 Case study application 5.5 Deviance 5.6 Model selection 5.7 Model diagnostics 5.7.1 Linearity 5.7.2 Response distribution 5.7.3 Independence 5.7.4 Multicollinearity 5.8 Shrinkage 5.9 Big data considerations 6 Nonparametric regression 6.1 Nonparametric density estimation 6.1.1 Histogram and moving histogram 6.1.2 Kernel density estimation 6.1.3 Bandwidth selection 6.1.4 Multivariate extension 6.2 Kernel regression estimation 6.2.1 Nadaraya–Watson estimator 6.2.2 Local polynomial regression 6.2.3 Asymptotic properties 6.2.4 Bandwidth selection 6.3 Kernel regression with mixed multivariate data 6.4 Prediction and confidence intervals 6.5 Local likelihood Appendix A Further topics A.1 Informal review on hypothesis testing A.2 Least squares and maximum likelihood estimation A.3 Multinomial logistic regression A.4 Dealing with missing data A.5 A note of caution with inference after model-selection B Software B.1 Installation of R and RStudio B.2 Introduction to RStudio B.3 Introduction to R Simple computations Variables and assignment Vectors Some functions Matrices, data frames, and lists More on data frames Vector-related functions Logical conditions and subsetting Plotting functions Distributions Functions Control structures References ISBN 978-84-09-29679-8 Licensed under Published with bookdown A A Serif Sans White Sepia Night Notes for Predictive Modeling 4.2 Constrained linear models As outlined in the previous section, after doing variable selection with lasso,126 two possibilities are: (i) fit a linear model on the lasso-selected predictors; (ii) run a stepwise selection starting from the lasso-selected model to try to further improve the model.127 Let’s explore the intuitive idea behind (i) in more detail. For the sake of exposition, assume that among p p predictors, lasso zeroed out the first q q of them.128 Then, once q q is known, we would seek to fit the model Y=β 0+β 1 X 1+⋯+β p X p+ε,subject to β 1=…=β q=0.Y=β 0+β 1 X 1+⋯+β p X p+ε,subject to β 1=…=β q=0. This is a very simple constraint that we know how to solve: just include the p−q p−q remaining predictors in the model and fit it. It is however a specific case of a linear constraint on β,β, since β 1=…=β q=0 β 1=…=β q=0 is expressible as (I q 0 q×(p−q))q×p β−1=0 q,(4.11)(4.11)(I q 0 q×(p−q))q×p β−1=0 q, where I q I q is an q×q q×q identity matrix and β−1=(β 1,…,β p)′.β−1=(β 1,…,β p)′. The constraint in (4.11) can be generalized as A β−1=c,A β−1=c, which results in the (linearly) constrained linear model Y=β 0+β 1 X 1+⋯+β p X p+ε,subject to A β−1=c,(4.12)(4.12)Y=β 0+β 1 X 1+⋯+β p X p+ε,subject to A β−1=c, where A A is an q×p q×p matrix129 of rank q q and c∈R q.c∈R q. The constrained linear model (4.12) is useful when there is prior information available about a linear relation that the coefficients of the linear model must satisfy (e.g., in piecewise polynomial fitting). Before fitting the model (4.12), let’s assume from now on that the variables Y Y and X 1,…,X p,X 1,…,X p, as well the sample {(X i,Y i)}n i=1,{(X i,Y i)}i=1 n, are centered (see the tip at the end of Section 2.4.4). This means that ¯Y=0 Y¯=0 and that ¯X:=(¯X 1,…,¯X p)′X¯:=(X¯1,…,X¯p)′ is zero. More importantly, it also means that β 0 β 0 and ^β 0 β^0 are null, hence they are not included in the model. That is, that the model Y=β 1 X 1+⋯+β p X p+ε(4.13)(4.13)Y=β 1 X 1+⋯+β p X p+ε is considered. In this setting, β=(β 1,…,β p)′β=(β 1,…,β p)′130 and ^β=(X′X)−1 X′Y β^=(X′X)−1 X′Y is the least squares estimator, with the design matrix X X now omitting the first column of ones. Now, the estimator of β β in (4.13) from a sample {(X i,Y i)}n i=1{(X i,Y i)}i=1 n under the linear constraint A β=c A β=c is defined as ^β A:=arg min β∈R p A β=c RSS 0(β),RSS 0(β):=n∑i=1(Y i−β 1 X i 1−⋯−β p X i p)2.(4.14)(4.14)β^A:=arg⁡min β∈R p A β=c RSS 0(β),RSS 0(β):=∑i=1 n(Y i−β 1 X i 1−⋯−β p X i p)2. Solving (4.14) analytically is possible using Lagrange multipliers, and the explicit solution to (4.14) can be seen to be ^β A=^β+(X′X)−1 A′[A(X′X)−1 A′]−1(c−A^β).(4.15)(4.15)β^A=β^+(X′X)−1 A′[A(X′X)−1 A′]−1(c−A β^). For the general case given in (4.12), in which neither Y Y and X X nor the sample are centered, the estimator of β β in (4.12) is unaltered for the slopes and equals (4.15). The intercept is given by ^β A,0=¯Y−¯X′^β A.β^A,0=Y¯−X¯′β^A. The next code illustrates how to fit a linear model with constraints in practice. ``` Simulate data set.seed(123456) n <- 50 p <- 3 x1 <- rnorm(n, mean = 1) x2 <- rnorm(n, mean = 2) x3 <- rnorm(n, mean = 3) eps <- rnorm(n, sd = 0.5) y <- 1 + 2 x1 - 3 x2 + x3 + eps Center the data and compute design matrix x1Cen <- x1 - mean(x1) x2Cen <- x2 - mean(x2) x3Cen <- x3 - mean(x3) yCen <- y - mean(y) X <- cbind(x1Cen, x2Cen, x3Cen) Linear restriction: use that beta_1 + beta_2 + beta_3 = 0 beta_2 = -3 In this case q = 2. The restriction is codified as A <- rbind(c(1, 1, 1), c(0, 1, 0)) c <- c(0, -3) Fit model without intercept S <- solve(crossprod(X)) beta_hat <- S %% t(X) %% yCen beta_hat [,1] x1Cen 1.9873776 x2Cen -3.1449015 x3Cen 0.9828062 Restricted fit enforcing A beta = c beta_hat_A <- beta_hat + S %% t(A) %% solve(A %% S %% t(A)) %% (c - A %% beta_hat) beta_hat_A [,1] x1Cen 2.0154729 x2Cen -3.0000000 x3Cen 0.9845271 Intercept of the constrained fit beta_hat_A_0 <- mean(y) - c(mean(x1), mean(x2), mean(x3)) %% beta_hat_A beta_hat_A_0 [,1] [1,] 1.02824 ``` What about inference? In principle, it can be obtained analogously to how the inference for the unconstrained linear model was obtained in Section 2.4, since the distribution of ^β A β^A under the assumptions of the linear model is straightforward to obtain. We keep assuming that the model is centered. Then, recall that (4.15) can be expressed as ^β A=(X′X)−1 A′[A(X′X)−1 A′]−1 c+(I−(X′X)−1 A′[A(X′X)−1 A′]−1 A)^β.β^A=(X′X)−1 A′[A(X′X)−1 A′]−1 c+(I−(X′X)−1 A′[A(X′X)−1 A′]−1 A)β^. Therefore, using (1.4) and proceeding similarly to (2.11), ^β A∼N p(β+b(β,A,c,X),σ 2(X′X)−1−v(σ 2,A,X)),(4.16)(4.16)β^A∼N p(β+b(β,A,c,X),σ 2(X′X)−1−v(σ 2,A,X)), where b(β,A,c,X):=(X′X)−1 A′[A(X′X)−1 A′]−1(c−A β),v(σ 2,A,X):=σ 2(X′X)−1 A′[A(X′X)−1 A′]−1 A(X′X)−1.b(β,A,c,X):=(X′X)−1 A′[A(X′X)−1 A′]−1(c−A β),v(σ 2,A,X):=σ 2(X′X)−1 A′[A(X′X)−1 A′]−1 A(X′X)−1. The inference for constrained linear models is not built within base R. Therefore, we just give a couple of insights about (4.16) and do not pursue inference further. Note that: The variances of ^β A,j,β^A,j,j=1,…,p,j=1,…,p,decrease with respect to the variances of ^β j,β^j, given by the diagonal elements of σ 2(X′X)−1.σ 2(X′X)−1. This is perfectly coherent, after all we are constraining the possible values that the estimator of β β can take in order to accommodate A β=c.A β=c. More importantly, these variances remain the same irrespective of whether A β=c A β=c holds or not.131 The bias of ^β A β^A depends on the veracity of A β=c A β=c. If the restriction is verified, then b(β,A,c,X)=0 b(β,A,c,X)=0 and ^β A β^A is still unbiased. However, if A β≠c,A β≠c, then ^β A β^A is severely biased in estimating β.β. Verify by Monte Carlo that the covariance matrix in (4.16) is correct. To do so: Choose β,β,A,A, and c c at your convenience. Sample n=50 n=50 observations for the predictors. Sample n=50 n=50 observations for the responses from a linear model based on β.β. Use the same n n observations for the predictors from Step 2. Compute ^β A.β^A. Repeat Steps 3–4 M=500 M=500 times, saving each time ^β A.β^A. Compute the sample covariance matrix of the ^β A β^A’s. Compare it with the covariance matrix in (4.16). Do the same study for checking the expectation in (4.16), for the cases in which A β=c A β=c and A β≠c.A β≠c. For example, based on the data-driven penalization parameters ^λ k-CV λ^k-CV or ^λ k-1SE.λ^k-1SE.↩︎ Note that with this approach we assign to the more computationally efficient lasso the “hard work” of coming up with a set of relevant predictors from the whole dataset, whereas the betterment of that model is done with the more demanding stepwise regression (if the number of predictors is smaller than n n).↩︎ Note that this is a random quantity, but we ignore this fact for the sake of exposition.↩︎ Therefore, usually not invertible.↩︎ We do not require the previous notation β−1 β−1 They do not depend on c c
15188	https://stacks.math.columbia.edu/tag/005N	Lemma 5.17.1 (005N): Tube lemma—The Stacks project Typesetting math: 100% The Stacks project bibliography blog Table of contents Part 1: Preliminaries Chapter 5: Topology Section 5.17: Characterizing proper maps Lemma 5.17.1: Tube lemma (cite) [x] next definition [x] numbers tags history statistics 2 tags refer to this tag Lemma 5.17.1(Tube lemma). Let X and Y be topological spaces. Let A⊂X and B⊂Y be quasi-compact subsets. Let A×B⊂W⊂X×Y with W open in X×Y. Then there exists opens A⊂U⊂X and B⊂V⊂Y such that U×V⊂W. Proof. For every a∈A and b∈B there exist opens U(a,b) of X and V(a,b) of Y such that (a,b)∈U(a,b)×V(a,b)⊂W. Fix b and we see there exist a finite number a 1,…,a n such that A⊂U(a 1,b)∪…∪U(a n,b). Hence A×{b}⊂(U(a 1,b)∪…∪U(a n,b))×(V(a 1,b)∩…∩V(a n,b))⊂W. Thus for every b∈B there exists opens U b⊂X and V b⊂Y such that A×{b}⊂U b×V b⊂W. As above there exist a finite number b 1,…,b m such that B⊂V b 1∪…∪V b m. Then we win because A×B⊂(U b 1∩…∩U b m)×(V b 1∪…∪V b m). □ Comments (0) There are also: 2 comment(s) on Section 5.17: Characterizing proper maps Post a comment Your email address will not be published. Required fields are marked. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). All contributions are licensed under the GNU Free Documentation License. Name: E-mail: Site: Comment: \| \| \| You can type your comment here, use the preview option to see what it will look like. If your comment is on a lemma, please leave your comment on the page of the lemma. Same with remarks, propositions, theorems, etc. In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 005N. Beware of the difference between the letter'O' and the digit'0'. Tag: Post comment The tag you filled in for the captcha is wrong. You need to write 005N, in case you are confused. next definition [x] numbers tags View Lemma 5.17.1 as pdf history statistics 2 tags refer to this tag
15189	https://www.cuemath.com/ncert-solutions/if-a-and-b-are-acute-angles-such-that-cos-a-cos-b-then-show-that-a-b/	If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B We use cookies to improve your experience. Learn more OK Sign up LearnPracticeDownload If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B Solution: Using the basic trigonometric ratios, we can solve this problem. In the right-angled triangle ABC as shown below, ∠A and ∠B areacute angles and ∠C is right angle. cos A = side adjacent to ∠A /hypotenuse= AC/AB cosB = side adjacent to ∠B / hypotenuse = BC/AB Given that cos A = cos B Therefore, AC/AB = BC/AB AC = BC Hence, ∠A = ∠B(angles opposite to equal sides of a triangleare equal.) Let's look into an alternative approach to solve the question. Let us consider a triangle ABC in which CO ⊥ AB. It is given that cos A = cos B AO/AC = BO/BC AO/BO = AC/BC Let AO/BO = AC/BC = k AO = k × BO ...(i) AC = k × BC ...(ii) By applyingPythagoras theorem in ΔCAO and ΔCBO, we get AC 2 = AO 2 + CO 2 (from ΔCAO) CO 2 = AC 2 - AO 2...(iii) BC 2 = BO 2 + CO 2 (from ΔCBO) CO 2 = BC 2 - BO 2...(iv) From equation (iii) and equation (iv), we get AC 2 - AO 2 = BC 2 - BO 2 (kBC)2 - (kBO)2 = BC 2 - BO 2[From equation (i) and (ii)] k 2 BC 2 - k 2 BO 2 = BC 2 - BO 2 k 2 (BC 2 - BO 2) = BC 2 - BO 2 k 2= (BC 2 - BO 2)/(BC 2 - BO 2) = 1 k = 1 Putting this value in equation (ii) we obtain, AC = BC Thus, ∠A = ∠B (angles opposite to equal sides of triangle are equal.) ☛ Check:NCERT Solutions Class 10 Maths Chapter 8 Video Solution: If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B Maths NCERT Solutions Class 10 Chapter 8 Exercise 8.1 Question 6 Summary: If ∠A and ∠B are acute angles such that cos A = cos B we have proved that ∠A = ∠B since AC = BC and angles opposite to equal sides of a triangle are equal. ☛ Related Questions: If cotθ = 7/8, evaluate: (i) (1 + sinθ)(1 - sinθ) / (1 + cosθ)(1 - cosθ), (ii) cot2θ If 3 cot A = 4, check whether (1 - tan2 A) / (1 + tan2 A) = cos2 A - sin2 A or not. In the triangle ABC right-angled at B, if tan A = 1/√3 find the value of:(i) sin A cos C + cos A sin C(ii) cos A cos C - sin A sin C In ΔPQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P. Explore math program Math worksheets and visual curriculum Sign up FOLLOW CUEMATH Facebook Youtube Instagram Twitter LinkedIn Tiktok MATH PROGRAM Online math classes Online Math Courses online math tutoring Online Math Program After School Tutoring Private math tutor Summer Math Programs Math Tutors Near Me Math Tuition Homeschool Math Online Solve Math Online Curriculum NEW OFFERINGS Coding SAT Science English MATH ONLINE CLASSES 1st Grade Math 2nd Grade Math 3rd Grade Math 4th Grade Math 5th Grade Math 6th Grade Math 7th Grade Math 8th Grade Math ABOUT US Our Mission Our Journey Our Team QUICK LINKS Maths Games Maths Puzzles Our Pricing Math Questions Blogs Events FAQs MATH TOPICS Algebra 1 Algebra 2 Geometry Calculus math Pre-calculus math Math olympiad Numbers Measurement MATH TEST CAASPP CogAT STAAR NJSLA SBAC Math Kangaroo AMC 8 MATH CURRICULUM 1st Grade Math 2nd Grade Math 3rd Grade Math 4th Grade Math 5th Grade Math 6th Grade Math 7th Grade Math 8th Grade Math FOLLOW CUEMATH Facebook Youtube Instagram Twitter LinkedIn Tiktok MATH PROGRAM Online math classes Online Math Courses online math tutoring Online Math Program After School Tutoring Private math tutor Summer Math Programs Math Tutors Near Me Math Tuition Homeschool Math Online Solve Math Online Curriculum NEW OFFERINGS Coding SAT Science English MATH CURRICULUM 1st Grade Math 2nd Grade Math 3rd Grade Math 4th Grade Math 5th Grade Math 6th Grade Math 7th Grade Math 8th Grade Math MATH TEST CAASPP CogAT STAAR NJSLA SBAC Math Kangaroo AMC 8 ABOUT US Our Mission Our Journey Our Team MATH TOPICS Algebra 1 Algebra 2 Geometry Calculus math Pre-calculus math Math olympiad Numbers Measurement QUICK LINKS Maths Games Maths Puzzles Our Pricing Math Questions Blogs Events FAQs MATH ONLINE CLASSES 1st Grade Math 2nd Grade Math 3rd Grade Math 4th Grade Math 5th Grade Math 6th Grade Math 7th Grade Math 8th Grade Math Terms and ConditionsPrivacy Policy
15190	https://pmc.ncbi.nlm.nih.gov/articles/PMC11444988/	Nuclear structures and their emerging roles in cell differentiation and development - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer \| PMC Copyright Notice BMB Rep . 2024 Aug 22;57(9):381–387. doi: 10.5483/BMBRep.2024-0101 Search in PMC Search in PubMed View in NLM Catalog Add to search Nuclear structures and their emerging roles in cell differentiation and development Hye Ji Cha Hye Ji Cha 1 Department of Biomedical Science & Engineering, Dankook University, Cheonan 31116, Korea Find articles by Hye Ji Cha 1, Author information Article notes Copyright and License information 1 Department of Biomedical Science & Engineering, Dankook University, Cheonan 31116, Korea Corresponding author. Tel: +82-41-550-3671; Fax: +82-41-559-7881; E-mail: hyejicha@dankook.ac.kr Received 2024 Jun 21; Revised 2024 Jul 16; Accepted 2024 Jul 31; Issue date 2024 Sep 30. Copyright © 2024 by the The Korean Society for Biochemistry and Molecular Biology This is an open-access article distributed under the terms of the Creative Commons Attribution Non-Commercial License ( which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. PMC Copyright notice PMCID: PMC11444988 PMID: 39219044 Abstract The nucleus, a highly organized and dynamic organelle, plays a crucial role in regulating cellular processes. During cell differentiation, profound changes occur in gene expression, chromatin organization, and nuclear morphology. This review explores the intricate relationship between nuclear architecture and cellular function, focusing on the roles of the nuclear lamina, nuclear pore complexes (NPCs), sub-nuclear bodies, and the nuclear scaffold. These components collectively maintain nuclear integrity, organize chromatin, and interact with key regulatory factors. The dynamic remodeling of chromatin, its interactions with nuclear structures, and epigenetic modifications work in concert to modulate gene accessibility and ensure precise spatiotemporal control of gene expression. The nuclear lamina stabilizes nuclear shape and is associated with inactive chromatin regions, while NPCs facilitate selective transport. Sub-nuclear bodies contribute to genome organization and gene regulation, often by influencing RNA processing. The nuclear scaffold provides structural support, impacting 3D genome organization, which is crucial for proper gene expression during differentiation. This review underscores the significance of nuclear architecture in regulating gene expression and guiding cell differentiation. Further investigation into nuclear structure and 3D genome organization will deepen our understanding of the mechanisms governing cell fate determination. Keywords: Cell differentiation, Development, Nuclear architecture, Nuclear structure INTRODUCTION The nucleus is a highly organized and dynamic organelle that plays a crucial role in maintaining cellular integrity and function. Its complex architecture, comprising chromatin and the nuclear structure, provides a tightly regulated environment for essential processes such as gene expression, DNA replication, and RNA processing. The nuclear envelope, consisting of a double lipid bilayer and nuclear pore complexes, separates the nucleus from the cytoplasm, allowing for the selective exchange of molecules and creating a unique nuclear microenvironment (1-3). Within the nucleus, chromatin is organized into distinct territories, with active and inactive regions that are dynamically regulated by epigenetic modifications and chromatin remodeling complexes. The nuclear lamina, a dense fibrillar network beneath the inner nuclear membrane, provides structural support and helps maintain nuclear shape and mechanical stability (4, 5). Additionally, sub-nuclear structures, such as the nucleolus, nuclear speckles, and Cajal bodies, serve as specialized hubs for various nuclear processes, further contributing to the functional organization of the nucleus (6-10). Recent advances in our understanding of nuclear architecture have begun to reveal the intricate interplay between nuclear structure and cellular differentiation. As cells differentiate into specific lineages, they undergo profound changes in gene expression, chromatin organization, and nuclear morphology. The dynamic remodeling of chromatin and its interactions with various nuclear structures, such as the nuclear lamina, nuclear pore complexes, and sub-nuclear bodies, are considered to play essential roles in fine-tuning gene expression patterns during differentiation (11-15). Epigenetic modifications, including DNA methylation and histone modifications, work in concert with chromatin remodeling complexes to modulate gene accessibility and ensure precise spatiotemporal control of gene expression (16-18). Furthermore, the spatial organization of chromatin within the nucleus, as seen in particular in its interactions with the nuclear periphery and inner nuclear scaffold, has emerged as an important factor in regulating cell type-specific gene expression programs (19, 20). As our knowledge of the complex relationship between nuclear structure and cellular differentiation continues to expand, it becomes increasingly clear that the nucleus is not merely a passive container for genetic material, but rather an active and dynamic organelle that plays a vital role in determining cell fate and function. NUCLEAR STRUCTURE AND FUNCTION The nucleus, the commend center of the cell, is a highly structured organelle that is essential to maintain cell integrity and function (Fig. 1). The nuclear envelope, composed of a double lipid bilayer, encloses the nucleus, separating the nucleoplasm from the cytoplasm and creating a unique environment. This separation is crucial for regulating processes, such as gene expression and DNA replication, in a controlled manner. Within the nuclear envelope are large multiprotein structures called nuclear pore complexes (NPCs), which are composed of various proteins known as nucleoporins. NPCs are well-known for their crucial role in regulating the selective exchange of molecules between the nucleus and the cytoplasm, thereby facilitating nucleocytoplasmic transport (3). Recent research has expanded our understanding of NPCs by identifying their role in regulating gene expression independently of their transport functions (14). Of the two membranes of the nuclear envelope, the outer membrane is continuous with the endoplasmic reticulum, while the inner membrane is associated with the nuclear lamina, which is composed of lamin proteins (5). This structure provides mechanical support, maintaining the shape of the nucleus and organizing chromatin. Mutations in lamin proteins can lead to genetic disorders known as laminopathies, highlighting their crucial role in nuclear function (21). Fig. 1. Open in a new tab Nuclear structure of a mammalian cell. The mammalian cell nucleus is composed of chromatin and functional nuclear structures. The nuclear pore complex is embedded within the nuclear envelope, while the nuclear lamina lies beneath it. Inside the nucleus, inner nuclear scaffold proteins and various sub-nuclear bodies can be found, including the nucleolus, nuclear speckles, Cajal bodies, and promyelocytic leukemia (PML) bodies. Within the nucleus, DNA is packaged into chromatin, a complex of DNA and histone proteins. Chromatin exists in two forms: euchromatin, which is less condensed and actively transcribed, and heterochromatin, which is highly condensed, and in general, transcriptionally inactive. The dynamic organization of chromatin is crucial to regulate gene expression in response to developmental cues and environmental changes. Additionally, the nuclear scaffold, or nuclear matrix, is a framework of fibrous proteins that provides structural support to the nucleus, organizing its three-dimensional architecture and facilitating processes such as gene transcription and RNA processing (22, 23). This structural support is essential to maintain the shape and integrity of the nucleus, ensuring the proper functional organization of chromatin and the regulation of gene expression. Sub-nuclear structures, or nuclear bodies, are distinct membraneless regions within the nucleus that specialize in various nuclear processes. These include the nucleolus, nuclear speckles, Cajal bodies, and promyelocytic leukemia (PML) bodies. The nucleolus is primarily involved in ribosome biogenesis, while Cajal bodies function as hubs for ribonucleoprotein particle formation and RNA metabolism (8, 10, 24). Nuclear speckles play a crucial role in the storage and modification of pre-mRNA splicing factors (7, 25). PML bodies, though their precise role remains unclear, are implicated in processes such as post-translational regulation and stress response (26, 27). These sub-nuclear bodies primarily occupy the interchromatin space and are associated with factors involved in specific functions (6), contributing to the complex and dynamic organization of the nucleus. INTERPLAY BETWEEN NUCLEAR STRUCTURE AND CELL DIFFERENTIATION Cellular differentiation is a highly orchestrated process that involves the precise control of gene expression to drive the specialization of cells into distinct lineages. At the core of this process lies the complex interplay between chromatin structure, nuclear architecture, and gene regulation. Chromatin undergoes extensive remodeling during differentiation, resulting in altered accessibility of genes to transcriptional machinery. Studies indicate that the three-dimensional organization of chromatin and its interactions with various nuclear structures, such as the nuclear lamina, nuclear pore complexes, sub-nuclear bodies, and nuclear scaffold, contribute to the modulation of gene expression patterns. As cells differentiate, they experience profound changes in their epigenetic landscape, chromatin conformation, and nuclear morphology that work in concert to establish and maintain cell type-specific gene expression programs. This section explores the intricate relationship between chromatin structure, nuclear architecture, and gene expression during cellular differentiation. Epigenetic regulation and chromatin reorganization play a significant role in regulating gene expression during cell differentiation. Epigenetic modifications, such as DNA methylation and histone modifications, serve as molecular markers that can activate or silence genes, without altering the underlying DNA sequence. These modifications are critical to determining cell fate, as they enable the dynamic and reversible regulation of gene activity in response to developmental signals and environmental factors. For example, DNA methylation is typically associated with gene silencing influencing the binding of various regulatory proteins (16). Histone modifications encompass a diverse range of changes, including acetylation, methylation, and phosphorylation, which affect the chromatin compaction, thereby modulating accessibility to transcriptional regulators. Chromatin remodeling complexes further facilitate these processes by repositioning nucleosomes, thereby altering the chromatin landscape to either expose or shield regulatory regions of the genome (17, 18). The complex coordination between epigenetic modifications and chromatin structure ensures that specific sets of genes are precisely turned on or off during the various stages of development, allowing progenitor cells to develop into specialized cell types with distinct functions. The spatial organization of chromatin within the nucleus influences gene expression by affecting the proximity of genes to transcriptional regulators (28-30). Actively transcribed genes are typically located in the euchromatin, which is less densely packed and more accessible, whereas inactive genes reside in the heterochromatin, which is tightly packed and less accessible. The positioning of genes relative to nuclear structures also impacts their expression. Genes near the nuclear periphery tend to be transcriptionally silent, while those in the nuclear interior are often transcriptionally active. Recent studies have shown that the role of nuclear structures in the regulation of gene expression is also critical during cell differentiation and development. The 3D organization of chromatin changes dynamically during differentiation, leading to the rearrangement of gene accessibility and the activation of lineage-specific genes. Thus, the highly organized and dynamic structure of chromatin within the nucleus is essential for precise gene regulation and cell differentiation, allowing genes to be expressed at the right time and space. To further elucidate the intricate relationship between nuclear structure and cellular differentiation, it is crucial to examine specific components of nuclear architecture. The nuclear lamina, nuclear pore complexes, sub-nuclear bodies, and the inner nuclear scaffold all play vital roles in shaping the nuclear environment and influencing gene expression during differentiation. These components contribute to the establishment and maintenance of cell type-specific gene expression programs through their interactions with chromatin and other nuclear factors. By exploring each of these elements in detail, we can gain a more comprehensive understanding of how nuclear architecture orchestrates cellular differentiation. Nuclear lamina structure and its significance in cellular differentiation The nuclear lamina, a dense fibrillar network that lies beneath the inner membrane of the nuclear envelope, is primarily composed of intermediate filament proteins, called lamins, which are classified into two types: A-type and B-type lamins (21, 31). Lamins and associated proteins form a filamentous meshwork, providing structural support to the nucleus and maintaining its shape and mechanical stability (5). This structural framework stabilizes the integrity of the nucleus, while also playing a crucial role in organizing chromatin, influencing gene expression, and participating in various nuclear processes, such as DNA replication, transcription, and cell cycle progression (32-35). By anchoring chromatin to the nuclear envelope, the lamina establishes distinct nuclear compartments that segregate active and inactive regions of the genome. This spatial organization is essential to maintaining gene expression patterns that are specific to cell type and function. For instance, genes that need to be silenced are often located in peripheral heterochromatin regions, where they are tightly packed and less accessible to transcriptional machinery. Conversely, actively transcribed genes are found in euchromatin regions, which are more centrally located and loosely packed, facilitating access by transcription factors and RNA polymerase. During cell differentiation, the nuclear lamina undergoes significant remodeling to facilitate the necessary changes in the regulation of gene expression (11). Recent studies have also identified that Lamina-Associated Domains (LADs), regions of the genome that interact with the nuclear lamina, dynamically restructure during differentiation (4). This discovery has led to various investigations into the correlation between the relative positioning of genes at the nuclear lamina and the regulation of their expression. For example, nuclear lamina– genome interactions are involved in lineage commitment to cardiac and neural cells, and the relative positioning of genes to the nuclear lamina changes during the differentiation process of myoblasts (36-38). Genes located within LADs often exhibit specific epigenetic marks, such as histone H3K9 methylation, which are associated with transcriptional repression, and epigenetic modifications are involved in facilitating this process (36). In fact, experiments involving the artificial tethering of genes to the nuclear lamina have demonstrated transcriptional repression mediated by the repositioning of genes to the nuclear lamina in mouse fibroblasts (39). These findings underscore the role of the nuclear lamina and LADs in regulating gene expression through spatial organization within the nucleus, highlighting their importance in the context of cellular differentiation. Nuclear pore complexes and their impact on differentiation processes Nuclear Pore Complexes (NPCs) are large protein assemblies that are embedded in the nuclear envelope, serving as gateways that regulate molecular transport between the nucleus and cytoplasm. These structures are crucial for maintaining cellular homeostasis by controlling the exchange of RNA, proteins, and other macromolecules (40). Composed of multiple proteins called nucleoporins, NPCs form a cylindrical structure with a central channel that facilitates selective molecular passage. Small molecules diffuse freely through this channel, while larger molecules, such as RNA and proteins, require active transport mediated by specific receptors. Interestingly, studies have demonstrated that NPC components influence the regulation of gene expression across various organisms, independent of their primary transport function (1, 41-43). For example, Nup153, a nucleoporin, is essential for maintaining pluripotency in mouse embryonic stem cells; its depletion leads to the derepression of developmental genes by affecting polycomb-repressive complex 1 (PRC1) recruitment to chromatin. Similarly, Nup98 is involved in epigenetic memory, and has been shown to associate with MBD-R2/NSL and Trx/MLL histone-modifying complexes and regulate Hox gene expression in developing flies. By interacting with transcription factors, other NPC proteins, and epigenetic factors, these components can either activate or repress the expression of a wide range of genes, including those involved in developmental processes. The spatial arrangement of chromatin near NPCs also plays a role in regulating gene expression. Protein networks can anchor a subset of genes and their associated protein complexes to NPCs in the nuclear envelope, creating a microenvironment that allows for the distinct regulation of gene expression, such as transcriptional activation (2, 44-47). Intriguingly, the expression levels of various nucleoporins and transport proteins have been observed to vary during cell differentiation, and the expression of certain NPC components is required for cell fate determination and the differentiation process (12, 14). For instance, changes in expression levels of various nucleoporins were observed during mesenchymal stem cell differentiation (48). Moreover, during the neural differentiation of mouse embryonic stem cells, the expression of importin-α subtype transport proteins undergoes precise regulation, with shifts in importin-α subtype expression playing a crucial role in facilitating neural differentiation (49). The influence of these nucleocytoplasmic transport proteins on cellular processes, particularly cell differentiation, can be primarily attributed to the involvement of NPCs in the transport of diverse transcription factors, differentiation regulators, and pluripotency factors. As these factors are shuttled between the nucleus and cytoplasm, their spatial and temporal distribution can significantly impact gene expression patterns, and ultimately, cell fate decisions. To fully elucidate the complex roles of NPCs in differentiation, further research focusing on various stages of the process is necessary. Sub-nuclear bodies and their associations with cell differentiation processes Sub-nuclear bodies are specialized, membraneless structures within the nucleus that play diverse roles in regulating various nuclear functions. These structures include the nucleolus, Cajal bodies, nuclear speckles, and PML bodies, each with distinct functions. The nucleolus, one of the most noticeable structures observed in cells under phase contrast microscopy, is primarily involved in ribosome biogenesis, synthesizing ribosomal RNA (rRNA) and assembling it with ribosomal proteins to form ribosomes, which are essential for protein synthesis (10). Cajal bodies, which are distinguished by the presence of coiled threads of the marker protein coilin (50), serve as preassembly sites for transcriptosomes and contain protein components essential for the transcription and processing of nuclear RNAs, effectively functioning as hubs for ribonucleoprotein (RNP) particle formation and RNA metabolism (8, 24). Nuclear speckles, rich in pre-mRNA splicing factors and commonly identified by the marker SC35 (9), play a crucial role in the storage and modification of these factors, ensuring efficient splicing and processing of pre-mRNAs (7, 25). The precise role of other sub-nuclear bodies, such as PML bodies, remains elusive (26, 27, 51, 52). PML bodies form around the PML protein, a tumor suppressor that polymerizes into punctate structures and recruits many seemingly unrelated partner proteins. While these structures are implicated in a broad spectrum of biological processes, including post-translational control and stress response, a unifying biochemical function has yet to be clearly defined. The composition and dynamics of sub-nuclear bodies are notably associated with cellular differentiation and development. As cells differentiate, they experience alterations in gene expression and nuclear organization, which are accompanied by changes in sub-nuclear structures. For example, when cells differentiate from embryonic stem cells to neural progenitor cell, the nucleolus undergoes dramatic changes in size and number as cells differentiate, with a general trend towards a greater number of smaller nucleoli in the differentiated cells (53-55). This change appears to reflect a decrease in ribosome biogenesis and a shift toward more specialized cellular functions. Similarly, the number of Cajal bodies varies, but tends to be higher in the early stages of embryo development and to decrease upon differentiation, which somewhat correlates with the transcriptional and metabolic activity of the cells (13, 24). In contrast, the number and size of PML bodies have been shown to vary during differentiation, as observed in hematopoietic and prostate cells, possibly reflecting their dispensable role during development (51, 56). Emerging studies demonstrate that the proximity of chromatin to nuclear speckles is associated with gene expression that can regulate cell differentiation (9, 57). For example, the chromatin architectural protein CTCF forms stress-sensitive complexes localizing to nuclear speckles during specific stages of neuronal commitment but not in differentiated neurons. The mechanism linking nuclear speckles and gene expression levels is mediated by the dynamic properties of the 3D genome architecture, where genes located closer to nuclear speckles tend to exhibit higher levels of transcription and splicing. This spatial organization is likely due to the high concentration of splicing factors within nuclear speckles, offering insights into how the nuclear environment influences gene regulation and contributes to the process of cellular differentiation. Inner nuclear scaffold architecture and its impact on cellular differentiation The nuclear scaffold, also known as the nuclear matrix, is a complex network of fibrogranular proteins and RNA that forms a structural framework within the nucleus of eukaryotic cells. This matrix structure was identified through a series of sequential salt extractions, detergent, and DNase treatments (58, 59). The nuclear matrix has been implicated in organizing chromatin structure by providing attachment sites for matrix attachment regions (MARs) to MAR-binding proteins, thereby forming chromatin loops (60). Although the interaction between chromatin and the nuclear matrix was predicted to influence many biological functions, it has only recently been thoroughly investigated at the molecular level (61). Efforts to identify components of nuclear matrix proteins have led to the identification of nuclear lamins, nucleolar proteins, and inner nuclear proteins, such as heterogeneous nuclear ribonucleoproteins and nuclear matrins, which were named after their discovery as major nuclear matrix components (62). Among these, Matrin-3 has gained attention due to its abundance in the nucleus and recent studies suggesting its specific functions, emphasizing the role of inner nuclear proteins (63). Matrin-3 associates with other structural and regulatory factors in the nucleus, controls RNA processing, and coding mutations in its gene have been linked to rare genetic disorders (64-66). Very recently, the regulatory role of Matrin-3 at the chromatin level and in transcriptional control has been proposed, highlighting its more direct involvement in the regulation of gene expression, further expanding our understanding of the roles of nuclear matrix components in maintaining nuclear structure and regulating cellular processes (19, 23, 67). While the role of nuclear membrane proteins, such as lamins, in regulating gene accessibility during differentiation has been well-established, the contribution of nucleoplasmic proteins, which constitute a large component of the inner nucleus, to chromatin remodeling during transcription and differentiation remains less explored. Studies have shown that hnRNP components, such as SAF-A/hnRNP U and SAF-B, contribute to the organization and regulation of chromatin structure (68, 69). In the context of development, Matrin-3, another inner nuclear protein, has been suggested to maintain the undifferentiated state of neural stem cells, albeit limited to morphological observations (70). A more direct relationship between inner nuclear scaffold proteins and differentiation was recently examined using Matrin-3. In erythroid cells, Matrin-3 was found to interact with architectural proteins, such as CTCF and cohesin, stabilizing chromatin structure and negatively regulating differentiation (19). This association was also observed in mouse embryonic stem cells and muscle cells, suggesting that the role of Matrin-3 in coordinating chromatin organization and gene expression during cellular differentiation may be more general (19, 20). These findings provide new insights into the complex interplay between nuclear architecture and cell fate determination, underscoring the importance of inner nuclear scaffold proteins in this process. CONCLUSION The intricate relationship between nuclear structure and cellular differentiation is an intriguing area of research that has gained notable attention in recent years. As evidenced by the studies discussed in this review, the nucleus is a highly organized and dynamic organelle that undergoes profound changes during the process of cellular differentiation. The complex interplay between chromatin structure, epigenetic modifications, and nuclear architecture plays a crucial role in regulating gene expression and driving cell fate decisions. The nuclear lamina, nuclear pore complexes, sub-nuclear bodies, and inner nuclear scaffold all contribute to the functional organization of the nucleus, and have been implicated in various aspects of cellular differentiation. As cells differentiate, they undergo significant remodeling of chromatin and alterations in the spatial arrangement of genes, which are mediated by interactions with these nuclear structures. Epigenetic modifications and chromatin remodeling complexes further fine-tune gene expression patterns, ensuring precise control over cell type-specific transcriptional programs. Despite the significant progress made in understanding the relationship between nuclear structure and cellular differentiation, many questions remain unanswered. Future research could concentrate on further elucidating the molecular mechanisms underlying the dynamic changes in nuclear architecture during differentiation, as well as the specific roles of individual nuclear components in regulating gene expression and cell fate. Emerging approaches for future investigation include the use of degrader molecules for dynamic regulation of protein expression and the application of high-resolution chromatin capture techniques. These advanced approaches could provide unprecedented insights into the temporal dynamics of nuclear reorganization and the fine-scale chromatin interactions that occur during cellular differentiation. As our knowledge of the nucleus continues to expand, it will deepen our understanding of the fundamental principles governing cellular differentiation, while also potentially leading to the development of novel therapeutic strategies for diseases associated with aberrant nuclear structure and function. The study of nuclear structure and its role in cellular differentiation therefore represents an exciting and rapidly evolving field that promises to provide new insights into the complex mechanisms underlying cell fate determination and organismal development. ACKNOWLEDGEMENTS The present research was supported by the research fund of Dankook University in 2023. Footnotes CONFLICTS OF INTEREST The author has no conflicting interests. References 1.Raices M, D’Angelo MA. Nuclear pore complexes and regulation of gene expression. Curr Opin Cell Biol. 2017;46:26–32. doi: 10.1016/j.ceb.2016.12.006. [DOI] [PMC free article] [PubMed] [Google Scholar] 2.Strambio-De-Castillia C, Niepel M, Rout MP. The nuclear pore complex: bridging nuclear transport and gene regulation. Nat Rev Mol Cell Biol. 2010;11:490–501. doi: 10.1038/nrm2928. [DOI] [PubMed] [Google Scholar] 3.Kabachinski G, Schwartz TU. The nuclear pore complex - structure and function at a glance. J Cell Sci. 2015;128:423–429. doi: 10.1242/jcs.083246. [DOI] [PMC free article] [PubMed] [Google Scholar] 4.Luperchio TR, Wong X, Reddy KL. Genome regulation at the peripheral zone: lamina associated domains in development and disease. Curr Opin Genet Dev. 2014;25:50–61. doi: 10.1016/j.gde.2013.11.021. [DOI] [PubMed] [Google Scholar] 5.Wong X, Melendez-Perez AJ, Reddy KL. The nuclear lamina. Cold Spring Harb Perspect Biol. 2022;14:a040113. doi: 10.1101/cshperspect.a040113. [DOI] [PMC free article] [PubMed] [Google Scholar] 6.Sleeman JE, Trinkle-Mulcahy L. Nuclear bodies: new insights into assembly/dynamics and disease relevance. Curr Opin Cell Biol. 2014;28:76–83. doi: 10.1016/j.ceb.2014.03.004. [DOI] [PubMed] [Google Scholar] 7.Spector DL, Lamond AI. Nuclear speckles. Cold Spring Harb Perspect Biol. 2011;3:1–12. doi: 10.1101/cshperspect.a000646. [DOI] [PMC free article] [PubMed] [Google Scholar] 8.Gall JG. Cajal bodies: the first 100 years. Annu Rev Cell Dev Biol. 2000;16:273–300. doi: 10.1146/annurev.cellbio.16.1.273. [DOI] [PubMed] [Google Scholar] 9.Faber GP, Nadav-Eliyahu S, Shav-Tal Y. Nuclear speckles - a driving force in gene expression. J Cell Sci. 2022;135:jcs259594. doi: 10.1242/jcs.259594. [DOI] [PMC free article] [PubMed] [Google Scholar] 10.Pederson T. The nucleolus. Cold Spring Harb Perspect Biol. 2011;3:1–15. doi: 10.1101/cshperspect.a000638. [DOI] [PMC free article] [PubMed] [Google Scholar] 11.Hampoelz B, Lecuit T. Nuclear mechanics in differentiation and development. Curr Opin Cell Biol. 2011;23:668–675. doi: 10.1016/j.ceb.2011.10.001. [DOI] [PubMed] [Google Scholar] 12.Khan AU, Qu R, Ouyang J, Dai J. Role of nucleoporins and transport receptors in cell differentiation. Front Physiol. 2020;11:1–12. doi: 10.3389/fphys.2020.00239.428edfd6c0c54aaeb7ecf9abce4321fd [DOI] [PMC free article] [PubMed] [Google Scholar] 13.Arias Escayola D, Neugebauer KM. Dynamics and function of nuclear bodies during embryogenesis. Biochemistry. 2018;57:2462–2469. doi: 10.1021/acs.biochem.7b01262. [DOI] [PubMed] [Google Scholar] 14.D’Angelo MA, Gomez-Cavazos JS, Mei A, Lackner DH, Hetzer MW. A change in nuclear pore complex composition regulates cell differentiation. Dev Cell. 2012;22:446–458. doi: 10.1016/j.devcel.2011.11.021. [DOI] [PMC free article] [PubMed] [Google Scholar] 15.Grosch M, Ittermann S, Shaposhnikov D, Drukker M. Chromatin-associated membraneless organelles in regulation of cellular differentiation. Stem Cell Reports. 2020;15:1220–1232. doi: 10.1016/j.stemcr.2020.10.011. [DOI] [PMC free article] [PubMed] [Google Scholar] 16.Li E, Zhang Y. DNA methylation in mammals. Cold Spring Harb Perspect Biol. 2014;6:a019133. doi: 10.1101/cshperspect.a019133. [DOI] [PMC free article] [PubMed] [Google Scholar] 17.Torchy MP, Hamiche A, Klaholz BP. Structure and function insights into the NuRD chromatin remodeling complex. Cell Mol Life Sci. 2015;72:2491–2507. doi: 10.1007/s00018-015-1880-8. [DOI] [PMC free article] [PubMed] [Google Scholar] 18.Centore RC, Sandoval GJ, Soares LMM, Kadoch C, Chan HM. Mammalian SWI/SNF chromatin remodeling complexes: emerging mechanisms and therapeutic strategies. Trends Genet. 2020;36:936–950. doi: 10.1016/j.tig.2020.07.011. [DOI] [PubMed] [Google Scholar] 19.Cha HJ, Uyan Ö, Kai Y, et al. Inner nuclear protein Matrin-3 coordinates cell differentiation by stabilizing chromatin architecture. Nat Commun. 2021;12:6241. doi: 10.1038/s41467-021-26574-4.0375ca998e104d8ca0b29f84e0c5f8f6 [DOI] [PMC free article] [PubMed] [Google Scholar] 20.Liu T, Zhu Q, Kai Y, et al. Matrin3 mediates differentiation through stabilizing chromatin loop-domain interactions and YY1 mediated enhancer-promoter interactions. Nat Commun. 2024;15:1–18. doi: 10.1038/s41467-024-45386-w.23d68e42d7da4813946ed12c450bf64b [DOI] [PMC free article] [PubMed] [Google Scholar] 21.Schreiber KH, Kennedy BK. When lamins go bad: nuclear structure and disease. Cell. 2013;152:1365–1375. doi: 10.1016/j.cell.2013.02.015. [DOI] [PMC free article] [PubMed] [Google Scholar] 22.Malik AM, Miguez RA, Li X, Ho YS, Feldman EL, Barmada SJ. Matrin 3-dependent neurotoxicity is modified by nucleic acid binding and nucleocytoplasmic localization. Elife. 2018;7:1–30. doi: 10.7554/eLife.35977.bd94e4720db945cab0da54b4986177c0 [DOI] [PMC free article] [PubMed] [Google Scholar] 23.Skowronska-Krawczyk D, Ma Q, Schwartz M, et al. Required enhancer-matrin-3 network interactions for a homeodomain transcription program. Nature. 2014;514:257–261. doi: 10.1038/nature13573. [DOI] [PMC free article] [PubMed] [Google Scholar] 24.Machyna M, Heyn P, Neugebauer KM. Cajal bodies: where form meets function. Wiley Interdiscip Rev RNA. 2013;4:17–34. doi: 10.1002/wrna.1139. [DOI] [PubMed] [Google Scholar] 25.Galganski L, Urbanek MO, Krzyzosiak WJ. Nuclear speckles: molecular organization, biological function and role in disease. Nucleic Acids Res. 2017;45:10350–10368. doi: 10.1093/nar/gkx759. [DOI] [PMC free article] [PubMed] [Google Scholar] 26.Corpet A, Kleijwegt C, Roubille S, et al. Survey and summary PML nuclear bodies and chromatin dynamics: catch me if you can! Nucleic Acids Res. 2020;48:11890–11912. doi: 10.1093/nar/gkaa828. [DOI] [PMC free article] [PubMed] [Google Scholar] 27.Lallemand-Breitenbach V, de Thé H. PML nuclear bodies: from architecture to function. Curr Opin Cell Biol. 2018;52:154–161. doi: 10.1016/j.ceb.2018.03.011. [DOI] [PubMed] [Google Scholar] 28.Pombo A, Dillon N. Three-dimensional genome architecture: players and mechanisms. Nat Rev Mol Cell Biol. 2015;16:245–257. doi: 10.1038/nrm3965. [DOI] [PubMed] [Google Scholar] 29.Ghosh RP, Meyer BJ. Spatial organization of chromatin: emergence of chromatin structure during development. Annu Rev Cell Dev Biol. 2021;37:199–232. doi: 10.1146/annurev-cellbio-032321-035734. [DOI] [PMC free article] [PubMed] [Google Scholar] 30.Schneider R, Grosschedl R. Dynamics and interplay of nuclear architecture, genome organization, and gene expression. Genes Dev. 2007;21:3027–3043. doi: 10.1101/gad.1604607. [DOI] [PubMed] [Google Scholar] 31.Smith ER, Meng Y, Moore R, Tse JD, Xu AG, Xu XX. Nuclear envelope structural proteins facilitate nuclear shape changes accompanying embryonic differentiation and fidelity of gene expression. BMC Cell Biol. 2017;18:1–14. doi: 10.1186/s12860-017-0125-0. [DOI] [PMC free article] [PubMed] [Google Scholar] 32.van Steensel B, Belmont AS. Lamina-associated domains: links with chromosome architecture, heterochromatin, and gene repression. Cell. 2017;169:780–791. doi: 10.1016/j.cell.2017.04.022. [DOI] [PMC free article] [PubMed] [Google Scholar] 33.Andrés V, González JM. Role of A-type lamins in signaling, transcription, and chromatin organization. J Cell Biol. 2009;187:945–957. doi: 10.1083/jcb.200904124. [DOI] [PMC free article] [PubMed] [Google Scholar] 34.Pope BD, Ryba T, Dileep V, et al. Topologically associating domains are stable units of replication-timing regulation. Nature. 2014;515:402–405. doi: 10.1038/nature13986. [DOI] [PMC free article] [PubMed] [Google Scholar] 35.Tsai MY, Wang S, Heidinger JM, et al. A mitotic lamin B matrix induced by RanGTP required for spindle assembly. Science. 2006;311:1887–1893. doi: 10.1126/science.1122771. [DOI] [PubMed] [Google Scholar] 36.Poleshko A, Shah PP, Gupta M, et al. Genome-nuclear lamina interactions regulate cardiac stem cell lineage restriction. Cell. 2017;171:573–587.e14. doi: 10.1016/j.cell.2017.09.018. [DOI] [PMC free article] [PubMed] [Google Scholar] 37.Peric-Hupkes D, Meuleman W, Pagie L, et al. Molecular maps of the reorganization of genome-nuclear lamina interactions during differentiation. Mol Cell. 2010;38:603–613. doi: 10.1016/j.molcel.2010.03.016. [DOI] [PMC free article] [PubMed] [Google Scholar] 38.Yao J, Fetter RD, Hu P, Betzig E, Tjian R. Subnuclear segregation of genes and core promoter factors in myogenesis. Genes Dev. 2011;25:569–580. doi: 10.1101/gad.2021411. [DOI] [PMC free article] [PubMed] [Google Scholar] 39.Reddy KL, Zullo JM, Bertolino E, Singh H. Transcriptional repression mediated by repositioning of genes to the nuclear lamina. Nature. 2008;452:243–247. doi: 10.1038/nature06727. [DOI] [PubMed] [Google Scholar] 40.Beck M, Hurt E. The nuclear pore complex: understanding its function through structural insight. Nat Rev Mol Cell Biol. 2017;18:73–89. doi: 10.1038/nrm.2016.147. [DOI] [PubMed] [Google Scholar] 41.Light WH, Freaney J, Sood V, et al. A conserved role for human nup98 in altering chromatin structure and promoting epigenetic transcriptional memory. PLoS Biol. 2013;11:e1001524. doi: 10.1371/journal.pbio.1001524.601700e520d3430998c17d5c6c48a45e [DOI] [PMC free article] [PubMed] [Google Scholar] 42.Jacinto FV, Benner C, Hetzer MW. The nucleoporin Nup153 regulates embryonic stem cell pluripotency through gene silencing. Genes Dev. 2015;29:1224–1238. doi: 10.1101/gad.260919.115. [DOI] [PMC free article] [PubMed] [Google Scholar] 43.Pascual-Garcia P, Jeong J, Capelson M. Nucleoporin Nup98 associates with Trx/MLL and NSL histone-modifying complexes and regulates Hox gene expression. Cell Rep. 2014;9:433–442. doi: 10.1016/j.celrep.2014.09.002.99106ec6e52345a28ecc1abaf1a4deed [DOI] [PubMed] [Google Scholar] 44.Brickner JH. Transcriptional memory at the nuclear periphery. Curr Opin Cell Biol. 2009:21127–21133. doi: 10.1016/j.ceb.2009.01.007. [DOI] [PMC free article] [PubMed] [Google Scholar] 45.Casolari JM, Brown CR, Komili S, West J, Hieronymus H, Silver PA. Genome-wide localization of the nuclear transport machinery couples transcriptional status and nuclear organization. Cell. 2004;117:427–439. doi: 10.1016/s0092-8674(04)00448-9. [DOI] [PubMed] [Google Scholar] 46.Taddei A, Van Houwe G, Hediger F, et al. Nuclear pore association confers optimal expression levels for an inducible yeast gene. Nature. 2006;441:774–778. doi: 10.1038/nature04845. [DOI] [PubMed] [Google Scholar] 47.Brickner DG, Cajigas I, Fondufe-Mittendorf Y, et al. H2A.Z-mediated localization of genes at the nuclear periphery confers epigenetic memory of previous transcriptional state. PLoS Biol. 2007;5:704–716. doi: 10.3410/f.1086141.539124. [DOI] [PMC free article] [PubMed] [Google Scholar] 48.Liu E, Gordonov S, Treiser MD, Moghe PV. Parsing the early cytoskeletal and nuclear organizational cues that demarcate stem cell lineages. Cell Cycle. 2010;9:2108–2117. doi: 10.4161/cc.9.11.11864. [DOI] [PubMed] [Google Scholar] 49.Yasuhara N, Shibazaki N, Tanaka S, et al. Triggering neural differentiation of ES cells by subtype switching of importin-α. Nat Cell Biol. 2007;9:72–79. doi: 10.1038/ncb1521. [DOI] [PubMed] [Google Scholar] 50.Morris GE. The cajal body. Biochim Biophys Acta - Mol Cell Res. 2008;1783:2108–2115. doi: 10.1016/j.bbamcr.2008.07.016. [DOI] [PubMed] [Google Scholar] 51.Bernardi R, Pandolfi PP. Structure, dynamics and functions of promyelocytic leukaemia nuclear bodies. Nat Rev Mol Cell Biol. 2007;8:1006–1016. doi: 10.1038/nrm2277. [DOI] [PubMed] [Google Scholar] 52.Lallemand-Breitenbach V, de Thé H. PML nuclear bodies. Cold Spring Harb Perspect Biol. 2010;2:a000661. doi: 10.1101/cshperspect.a000661. [DOI] [PMC free article] [PubMed] [Google Scholar] 53.Gupta S, Santoro R. Regulation and roles of the nucleolus in embryonic stem cells: from ribosome biogenesis to genome organization. Stem Cell Reports. 2020;15:1206–1219. doi: 10.1016/j.stemcr.2020.08.012. [DOI] [PMC free article] [PubMed] [Google Scholar] 54.Jevtic P, Edens LJ, Vukovic LD, Ley DL. Sizing and shaping the nucleus: mechanisms and significanc. Curr Opin Cell Biol. 2014;28:16–27. doi: 10.1016/j.ceb.2014.01.003. [DOI] [PMC free article] [PubMed] [Google Scholar] 55.Meshorer E, Misteli T. Chromatin in pluripotent embryonic stem cells and differentiation. Nat Rev Mol Cell Biol. 2006;7:540–546. doi: 10.1038/nrm1938. [DOI] [PubMed] [Google Scholar] 56.Flenghi L, Fagioli M, Tomassoni L, et al. Characterization of a new monoclonal antibody (PG-M3) directed against the aminoterminal portion of the PML gene product: immunocytochemical evidence for high expression of PML proteins on activated macrophages, endothelial cells, and epithelia. Blood. 1995;85:1871–1880. doi: 10.1182/blood.v85.7.1871.bloodjournal8571871. [DOI] [PubMed] [Google Scholar] 57.Lehman BJ, Lopez-Diaz FJ, Santisakultarm TP, et al. Dynamic regulation of CTCF stability and subnuclear localization in response to stress. PLoS Genetics. 2021;17:1–34. doi: 10.1371/journal.pgen.1009277.885202f5c4b9446984f6463da2b4e510 [DOI] [PMC free article] [PubMed] [Google Scholar] 58.Nelson WG, Pienta KJ, Barrack ER, Coffey DS. The role of the nuclear matrix in the organization and function of DNA. Annu Rev Biophys Biophys Chem. 1986;15:457–475. doi: 10.1146/annurev.bb.15.060186.002325. [DOI] [PubMed] [Google Scholar] 59.Verheijen R, van Venrooij W, Ramaekers F. The nuclear matrix: structure and composition. J Cell Sci. 1988;90:11–36. doi: 10.1242/jcs.90.1.11. [DOI] [PubMed] [Google Scholar] 60.Boulikas T. Nature of DNA sequences at the attachment regions of genes to the nuclear matrix. J Cell Biochem. 1993;52:14–22. doi: 10.1002/jcb.240520104. [DOI] [PubMed] [Google Scholar] 61.Albrethsen J, Knol JC, Jimenez CR. Unravelling the nuclear matrix proteome. J Proteomics. 2009;72:71–81. doi: 10.1016/j.jprot.2008.09.005. [DOI] [PubMed] [Google Scholar] 62.Nakayasu H, Berezney R. Nuclear matrins: identification of the major nuclear matrix proteins. Proc Natl Acad Sci U S A. 1991;88:10312–10316. doi: 10.1073/pnas.88.22.10312. [DOI] [PMC free article] [PubMed] [Google Scholar] 63.Fenelon KD, Hopyan S. Structural components of nuclear integrity with gene regulatory potential. Curr Opin Cell Biol. 2017;48:63–71. doi: 10.1016/j.ceb.2017.06.001. [DOI] [PubMed] [Google Scholar] 64.Coelho MB, Attig J, Bellora N, et al. Nuclear matrix protein Matrin3 regulates alternative splicing and forms overlapping regulatory networks with PTB. EMBO J. 2015;34:653–668. doi: 10.15252/embj.201489852. [DOI] [PMC free article] [PubMed] [Google Scholar] 65.Johnson JO, Pioro EP, Boehringer A, et al. Mutations in the Matrin 3 gene cause familial amyotrophic lateral sclerosis. Nat Neurosci. 2014;17:664–666. doi: 10.1038/nn.3688. [DOI] [PMC free article] [PubMed] [Google Scholar] 66.Malyavantham KS, Bhattacharya S, Barbeitos, et al. Identifying functional neighborhoods within the cell nucleus: proximity analysis of early S-phase replicating chromatin domains to sites of transcription, RNA polymerase II, HP1γ, Matrin 3 and SAF-A. J Cell Biochem. 2008;105:391–403. doi: 10.1002/jcb.21834. [DOI] [PMC free article] [PubMed] [Google Scholar] 67.Pandya-Jones A, Markaki Y, Serizay J, et al. A protein assembly mediates Xist localization and gene silencing. Nature. 2020;587:145–151. doi: 10.1038/s41586-020-2703-0. [DOI] [PMC free article] [PubMed] [Google Scholar] 68.Marenda M, Lazarova E, Gilbert N. The role of SAF-A/hnRNP U in regulating chromatin structure. Curr Opin Genet Dev. 2022;72:38–44. doi: 10.1016/j.gde.2021.10.008. [DOI] [PubMed] [Google Scholar] 69.Huo X, Ji L, Zhang Y, et al. The nuclear matrix protein safb cooperates with major satellite rnas to stabilize heterochromatin architecture partially through phase separation. Mol Cell. 2020;77:368–383.e7. doi: 10.1016/j.molcel.2019.10.001. [DOI] [PubMed] [Google Scholar] 70.Niimori-Kita K, Tamamaki N, Koizumi D, Niimori D. Matrin-3 is essential for fibroblast growth factor 2-dependent maintenance of neural stem cells. Sci Rep. 2018;8:1–10. doi: 10.1038/s41598-018-31597-x. 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15191	https://artofproblemsolving.com/wiki/index.php/Telescoping_series?srsltid=AfmBOoo7FqfeCK3eb3dOcy9ZQLmv_cR1Ox3L7gUJGXbAZa4kAIweA5tA	Art of Problem Solving Telescoping series - 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 Telescoping series Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Telescoping series In mathematics, a telescoping series is a series whose partial sums eventually only have a finite number of terms after cancellation. This is often done by using a form of for some expression . Contents [hide] 1 Example 1 2 Solution 1 3 Example 2 4 Solution 2 5 Problems 5.1 Introductory 5.2 Intermediate 5.3 Olympiad 6 See Also Example 1 Derive the formula for the sum of the first counting numbers. Solution 1 We wish to write for some expression . This expression is as . We then telescope the expression: . (Notice how the sum telescopes— contains a positive and a negative of every value of from to , so those terms cancel. We are then left with , the only terms which did not cancel.) Example 2 Find a general formula for , where . Solution 2 We wish to write for some expression . This can be easily achieved with as by simple computation. We then telescope the expression: . Problems Introductory When simplified the product becomes: (Source) The sum can be expressed as , where and are positive integers. What is ? (Source) Which of the following is equivalent to (Hint: difference of squares!) (Source) Intermediate Let denote the value of the sum can be expressed as , where and are positive integers and is not divisible by the square of any prime. Determine . (Source) Olympiad Find the value of , where is the Riemann zeta function See Also Algebra Summation Retrieved from " Category: Algebra 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.
15192	https://artofproblemsolving.com/wiki/index.php/AM-GM_Inequality?srsltid=AfmBOoogReJvt6R9NRqJxaZJ3Ax7UC6ePJI8ZHKCjCV0fnDqkRE2oRpF	Art of Problem Solving AM-GM Inequality - 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 AM-GM Inequality Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search AM-GM Inequality In algebra, the AM-GM Inequality, also known formally as the Inequality of Arithmetic and Geometric Means or informally as AM-GM, is an inequality that states that any list of nonnegative reals' arithmetic mean is greater than or equal to its geometric mean. Furthermore, the two means are equal if and only if every number in the list is the same. In symbols, the inequality states that for any real numbers , with equality if and only if . The AM-GM Inequality is among the most famous inequalities in algebra and has cemented itself as ubiquitous across almost all competitions. Applications exist at introductory, intermediate, and olympiad level problems, with AM-GM being particularly crucial in proof-based contests. Contents 1 Proofs 2 Generalizations 2.1 Weighted AM-GM Inequality 2.2 Mean Inequality Chain 2.3 Power Mean Inequality 3 Problems 3.1 Introductory 3.2 Intermediate 3.3 Olympiad 4 See Also Proofs Main article: Proofs of AM-GM All known proofs of AM-GM use induction or other, more advanced inequalities. Furthermore, they are all more complex than their usage in introductory and most intermediate competitions. AM-GM's most elementary proof utilizes Cauchy Induction, a variant of induction where one proves a result for , uses induction to extend this to all powers of , and then shows that assuming the result for implies it holds for . Generalizations The AM-GM Inequality has been generalized into several other inequalities. In addition to those listed, the Minkowski Inequality and Muirhead's Inequality are also generalizations of AM-GM. Weighted AM-GM Inequality The Weighted AM-GM Inequality relates the weighted arithmetic and geometric means. It states that for any list of weights such that , with equality if and only if . When , the weighted form is reduced to the AM-GM Inequality. Several proofs of the Weighted AM-GM Inequality can be found in the proofs of AM-GM article. Mean Inequality Chain Main article: Mean Inequality Chain The Mean Inequality Chain, also called the RMS-AM-GM-HM Inequality, relates the root mean square, arithmetic mean, geometric mean, and harmonic mean of a list of nonnegative reals. In particular, it states that with equality if and only if . As with AM-GM, there also exists a weighted version of the Mean Inequality Chain. Power Mean Inequality Main article: Power Mean Inequality The Power Mean Inequality relates all the different power means of a list of nonnegative reals. The power mean is defined as follows: The Power Mean inequality then states that if , then , with equality holding if and only if Plugging into this inequality reduces it to AM-GM, and gives the Mean Inequality Chain. As with AM-GM, there also exists a weighted version of the Power Mean Inequality. Problems Introductory For nonnegative real numbers , demonstrate that if then . (Solution) Find the maximum of for all positive . (Solution) Intermediate Find the minimum value of for . (Source) Olympiad Let , , and be positive real numbers. Prove that (Source) See Also Proofs of AM-GM Mean Inequality Chain Power Mean Inequality Cauchy-Schwarz Inequality Inequality Retrieved from " Categories: Algebra Inequalities Definition Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
15193	https://www.chegg.com/homework-help/questions-and-answers/products-combustion-flowing-heat-exchanger-12-co2-13-h2o-75-n2-volume-basis-rate-01-kg-s-1-q8556907	Solved The products of combustion are flowing through a heat \| Chegg.com Skip to main content Books Rent/Buy Read Return Sell Study Tasks Homework help Understand a topic Writing & citations Tools Expert Q&A Math Solver Citations Plagiarism checker Grammar checker Expert proofreading Career For educators Help Sign in Paste Copy Cut Options Upload Image Math Mode ÷ ≤ ≥ o π ∞ ∩ ∪           √  ∫              Math Math Geometry Physics Greek Alphabet Engineering Mechanical Engineering Mechanical Engineering questions and answers The products of combustion are flowing through a heat exchanger with 12% CO_2, 13% H_2O and 75% N_2 on a volume basis at the rate 0.1 kg/s and 100 kPa. What is the dew-point temperature? If the mixture is cooled 10 degree C below the dew-point temperature, how long will it take to collect 10 kg of liquid water? Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on. See Answer See Answer See Answer done loading Question: The products of combustion are flowing through a heat exchanger with 12% CO_2, 13% H_2O and 75% N_2 on a volume basis at the rate 0.1 kg/s and 100 kPa. What is the dew-point temperature? If the mixture is cooled 10 degree C below the dew-point temperature, how long will it take to collect 10 kg of liquid water? Show transcribed image text There are 4 steps to solve this one.Solution Share Share Share done loading Copy link Step 1 Introduction The problem involves analyzing a gas mixture of combustion products (CO₂, H₂O, and N₂) f... 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Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Dealing with ties in quantcut() Ask Question Asked 13 years, 2 months ago Modified13 years, 2 months ago Viewed 6k times Part of R Language Collective This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. I'm trying to use the R function quantcut() to recode a numeric variable as a factor with levels corresponding to quantiles. For example: ```r X 6 4 9 6 1 2 5 3 5 7 10 7 2 7 7 5 6 6 3 4 6 4 2 7 6 7 4 3 5 3 7 6 8 12 4 4 0 1 7 6 7 4 7 1 1 1 2 3 3 1 1 6 5 3 1 1 1 3 3 3 1 1 3 1 1 1 3 3 0 1 3 1 8 5 3 0 0 2 1 3 8 0 1 4 1 1 1 1 1 1 3 2 1 4 1 5 5 12 7 2 6 6 2 6 0 1 4 1 4 0 7 3 2 1 1 8 5 5 3 0 5 6 2 4 2 2 2 6 4 2 2 2 2 6 8 5 1 2 8 3 2 7 4 6 6 6 7 5 1 5 5 6 1 4 4 5 6 2 4 7 2 4 10 6 3 5 2 2 6 6 2 4 5 7 4 5 11 6 6 8 2 4 4 6 12 16 9 7 14 13 11 5 5 2 2 7 7 6 4 3 4 3 5 4 5 7 9 4 3 12 4 4 4 8 7 6 1 3 6 7 5 5 6 9 6 4 7 8 5 6 3 6 4 7 3 3 4 7 5 7 5 9 5 8 3 4 3 2 5 2 4 3 8 4 2 2 1 5 3 5 8 5 6 4 5 1 1 2 6 2 7 2 4 4 3 3 4 10 5 6 10 2 5 5 0 1 6 2 5 4 6 6 9 5 5 6 3 8 1 5 1 8 5 2 5 2 4 2 4 4 bins=10 labels = 1:bins library(gtools) x2 = quantcut(X, q = seq(0, 1, by=1/bins), labels=labels) ``` I get the error: "Error in cut.default(x[!flag], breaks = newquant, include.lowest = TRUE, : 'breaks' are not unique". I thought this was because there are ties in the quantiles, but the documentation for quantcut specifically shows an example of how the function can handle ties by using fewer intervals. The error occurs regardless of whether I specify the labels argument. Any advice would be greatly appreciated. EDIT: Here is code to enter the variable X: r X = c(6L, 4L, 9L, 6L, 1L, 2L, 5L, 3L, 5L, 7L, 10L, 7L, 2L, 7L, 7L, 5L, 6L, 6L, 3L, 4L, 6L, 4L, 2L, 7L, 6L, 7L, 4L, 3L, 5L, 3L, 7L, 6L, 8L, 12L, 4L, 4L, 0L, 1L, 7L, 6L, 7L, 4L, 7L, 1L, 1L, 1L, 2L, 3L, 3L, 1L, 1L, 6L, 5L, 3L, 1L, 1L, 1L, 3L, 3L, 3L, 1L, 1L, 3L, 1L, 1L, 1L, 3L, 3L, 0L, 1L, 3L, 1L, 8L, 5L, 3L, 0L, 0L, 2L, 1L, 3L, 8L, 0L, 1L, 4L, 1L, 1L, 1L, 1L, 1L, 1L, 3L, 2L, 1L, 4L, 1L, 5L, 5L, 12L, 7L, 2L, 6L, 6L, 2L, 6L, 0L, 1L, 4L, 1L, 4L, 0L, 7L, 3L, 2L, 1L, 1L, 8L, 5L, 5L, 3L, 0L, 5L, 6L, 2L, 4L, 2L, 2L, 2L, 6L, 4L, 2L, 2L, 2L, 2L, 6L, 8L, 5L, 1L, 2L, 8L, 3L, 2L, 7L, 4L, 6L, 6L, 6L, 7L, 5L, 1L, 5L, 5L, 6L, 1L, 4L, 4L, 5L, 6L, 2L, 4L, 7L, 2L, 4L, 10L, 6L, 3L, 5L, 2L, 2L, 6L, 6L, 2L, 4L, 5L, 7L, 4L, 5L, 11L, 6L, 6L, 8L, 2L, 4L, 4L, 6L, 12L, 16L, 9L, 7L, 14L, 13L, 11L, 5L, 5L, 2L, 2L, 7L, 7L, 6L, 4L, 3L, 4L, 3L, 5L, 4L, 5L, 7L, 9L, 4L, 3L, 12L, 4L, 4L, 4L, 8L, 7L, 6L, 1L, 3L, 6L, 7L, 5L, 5L, 6L, 9L, 6L, 4L, 7L, 8L, 5L, 6L, 3L, 6L, 4L, 7L, 3L, 3L, 4L, 7L, 5L, 7L, 5L, 9L, 5L, 8L, 3L, 4L, 3L, 2L, 5L, 2L, 4L, 3L, 8L, 4L, 2L, 2L, 1L, 5L, 3L, 5L, 8L, 5L, 6L, 4L, 5L, 1L, 1L, 2L, 6L, 2L, 7L, 2L, 4L, 4L, 3L, 3L, 4L, 10L, 5L, 6L, 10L, 2L, 5L, 5L, 0L, 1L, 6L, 2L, 5L, 4L, 6L, 6L, 9L, 5L, 5L, 6L, 3L, 8L, 1L, 5L, 1L, 8L, 5L, 2L, 5L, 2L, 4L, 2L, 4L, 4L) R Language Collective r grouping quantile Share Share a link to this question Copy linkCC BY-SA 3.0 Improve this question Follow Follow this question to receive notifications edited Jul 6, 2012 at 22:50 thelatemail 94.3k 12 12 gold badges 139 139 silver badges 197 197 bronze badges asked Jul 6, 2012 at 20:48 half-passhalf-pass 1,941 4 4 gold badges 23 23 silver badges 34 34 bronze badges Add a comment\| 1 Answer 1 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. Okay, the issue can be traced to here, where as you say, the 70% and 80% quantiles are the same. quantile is used internally by quantcut r quantile(X,probs=seq(0,1,0.1)) 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 0.0 1.0 2.0 3.0 3.6 4.0 5.0 6.0 6.0 8.0 16.0 I can't see how to address this issue using quantcut itself, but you could always just use cut and quantile and unique in combination to sort it out. From what I can tell, this is what quantcut does internally when there are ties anyway. ```r result <- cut(X,unique(quantile(X,probs=seq(0,1,0.1))),include.lowest=TRUE) result[2:10] (3.6,4] (8,16] (5,6] [0,1] (1,2] (4,5] (2,3] (4,5] (6,8] Levels: [0,1] (1,2] (2,3] (3,3.6] (3.6,4] (4,5] (5,6] (6,8] (8,16] X[2:10] 4 9 6 1 2 5 3 5 7 ``` Share Share a link to this answer Copy linkCC BY-SA 3.0 Improve this answer Follow Follow this answer to receive notifications answered Jul 6, 2012 at 21:39 thelatemailthelatemail 94.3k 12 12 gold badges 139 139 silver badges 197 197 bronze badges 3 Comments Add a comment half-pass half-passOver a year ago Thanks. I've been playing with this, and I don't understand why it is level 4 that gets omitted rather than level 8 or 9 (since those levels represent the tied quantiles). If I call levels(result), it lists 1:9. But if I call table(result), there are 0 observations of level 4. 2012-07-06T23:53:59.437Z+00:00 0 Reply Copy link thelatemail thelatemailOver a year ago @half-pass - I'm guessing this is because the quantile grouping is >3 to 3.6 in which it is impossible for an integer to fall into. You might need to floor the break points of the cut function to get some values to fall into that grouping: result <- cut(X,unique(floor(quantile(X,probs=seq(0,1,0.1)))),include.lowest=TRUE) 2012-07-07T00:17:35.043Z+00:00 1 Reply Copy link rafa.pereira rafa.pereiraOver a year ago Is it possible to use this solution and still keep quantile labels? I've tried your suggestion including labels like this: result <- cut(X, labels=c(1:10), breaks= unique(quantile(X,probs=seq(0,1,0.1))),include.lowest=TRUE) but I get an Error: lengths of 'breaks' and 'labels' differ 2015-09-07T16:30:27.66Z+00:00 0 Reply Copy link Add a comment Your Answer Thanks for contributing an answer to Stack Overflow! 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15195	https://www.teacherspayteachers.com/Product/Two-Digit-Addition-without-Regrouping-Adding-Two-Digit-Numbers-Tens-and-Ones-1453470	Log InSign Up Cart is empty Total: View Wish ListView Cart Two Digit Addition without Regrouping - Adding Two Digit Numbers Tens and Ones Rated 4.93 out of 5, based on 173 reviews 4.9 (173 ratings) $5.00 DescriptionReviews173Q&AStandards2 Share What others say "This was a great resource. Easy for my students to engage with and it makes planning super easy. Thank you!" Maeghan R. "I used these as a free choice game for students who were finished with their daily problems and they loved it! I like the scaffolds in place to differentiate for those ready to regroup and those who aren't." Abby B. Description These 2 digit addition without regrouping games give your students the valuable practice they need to master double digit addition without regrouping. More motivating than a worksheet, your class will love this hands-on approach to practicing double digit addition by adding tens and ones without regrouping. Each game requires players to add 2 digit numbers and record the equation they've made before they can move on the game board. The self-correcting element of the games ensures that students get the answer correct or find their mistake if they've made one. Just print each game with its illustrated rules (there's no cutting involved) and place them in your math centers. Include the recording sheets and collect them at the end of the session to include in student portfolios. You'll be provided with valuable data on who requires more support with 2 digit addition without regrouping. Students can play in small groups independently or with minimal adult support. The non-seasonal themes make these games perfect for your class to play all year long. Addition without Regrouping Games contents: Fish Bowl (2 - 4 players) Players must spin and record their equation and then place a counter on the answer. The first player to get three numbers in a row is the winner. Pirate Gold (2 players) Players travel around the game board, record their equation and place their counter on the answer. To win a player must make a trail from the pirate to the gold. Taste Test (2-4 players) Players must spin an equation and record it. They may only move to the next food item if the answer is there. The first player to reach the finish is the winner. Who stole the Cookie? (2 - 4 players) Players must spin and record their equation. They then move their counter either forward or backward to the closest answer. Gumball Gallery (2 - 4 players) Players must spin and record their equation. Each player who can match the answer to the code shown on their section of the game board can take a counter. You may prefer the bundled pack which includes these games and 5 more games with regrouping. Addition Games - 2 digit addition with and without regrouping Also available: 2 digit addition without regrouping - addition riddle worksheets and an addition game Interesting in regrouping games? View the whole addition with regrouping range here: addition with regrouping games If you are interested in subtraction with regrouping games please click here: subtraction with regrouping games ____________________________________________________________________________________ Become a follower and take advantage of my Early Bird Specials. All new products, excluding bundles, are 50% off for the first day of listing. Look for the green star near the top of any page within my store and click it to become a follower. You will then receive customized email updates about this store. Report this resource to TPT Reported resources will be reviewed by our team. Report this resource to let us know if this resource violates TPT's content guidelines. Two Digit Addition without Regrouping - Adding Two Digit Numbers Tens and Ones Rated 4.93 out of 5, based on 173 reviews 4.9 (173 ratings) Teaching Trove 14.5kFollowers $5.00 Grade 1st - 2nd Mostly used with 1st and 2nd Subject Basic Operations, Math Standards CCSS1.NBT.C.4 CCSS2.NBT.B.5 Tags Centers, Games Save even more with bundles Two Digit Addition with Regrouping + Adding 2 Digit Numbers without Regrouping You’ve taught your students adding two digit numbers without regrouping and two digit addition with regrouping, now they need practice. This bundled two digit addition with and without regrouping game pack gives your students the valuable practice they need to master 2 digit addition.More motivating $7.97Price $7.97$10.00Original Price $10.00Save $2.03 2 2 and 3 digit Addition and Subtraction with Regrouping and without Regrouping Looking for an engaging way to help your students master 2- and 3-digit addition and subtraction with and without regrouping? These hands-on math games make learning fun while reinforcing essential math skills that are critical to building fluency with multi-digit operations.Instead of traditional w $20.00Price $20.00$30.00Original Price $30.00Save $10.00 6 Reviews 4.9 Rated 4.93 out of 5, based on 173 reviews 173ratings 5 162 4 12 3 0 2 0 1 0 Mostly used with 1st and 2nd grades Reviews 15 19 4 2 2 1st 2nd 3rd 4th 5th All verified TPT purchases 5 stars 4 stars 1st grade 2nd grade 3rd grade 4th grade 5th grade Learning difficulties Mild to severe disabilities Autism Emerging bilingual Most relevant Most recent Highest rating Lowest rating Great Resource Rated 5 out of 5 August 19, 2025 My students really benefited from this resource while practicing double-digit addition. Verna Lynn G. 7,295reviews• Outside the United States Grades taught: 1st, 2nd Rated 5 out of 5 July 14, 2025 This was great to use during small group practice! Miss Kadilak's Corner (TPT Seller) 438reviews Grades taught: 4th Student populations: Autism, Learning difficulties, Mild to severe disabilities Rated 5 out of 5 March 25, 2025 These are fun centres and my students loved them! Easy to print and go. Sabrina M. 29reviews Grades taught: 2nd Rated 5 out of 5 December 4, 2024 This was a great resource. Easy for my students to engage with and it makes planning super easy. Thank you! Maeghan Robertson (TPT Seller) 918reviews Grades taught: 2nd Rated 5 out of 5 August 26, 2024 This is a fantastic resource for helping the students practice double digit addition. Jennifer Weales (TPT Seller) 2,798reviews Grades taught: 1st Rated 5 out of 5 August 26, 2024 I used these as a free choice game for students who were finished with their daily problems and they loved it! I like the scaffolds in place to differentiate for those ready to regroup and those who aren't. Abby Boruff (TPT Seller) 743reviews Grades taught: 1st Rated 5 out of 5 July 17, 2024 These games were so much fun for my students! Most of the time they hate practicing this skill, but the games made it much more engaging! Thank you! Stephanie Nicholson (TPT Seller) Grades taught: 3rd, 4th, 5th Student populations: Autism, Emerging bilinguals, Learning difficulties, Mild to severe disabilities Rated 5 out of 5 July 3, 2024 I used this with my summer camp students. They absolutely loved it because it was very engaging. They preferred this instead of sitting at their desk doing a basic worksheet. Cynthia A. Questions & Answers Standards Log in to see state-specific standards (only available in the US). CCSS1.NBT.C.4 Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. CCSS2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Loading TPT is the largest marketplace for PreK-12 resources, powered by a community of educators. Who we are We're hiring Press Blog Gift Cards Help & FAQ Security Privacy policy Student privacy Terms of service Tell us what you think Get our weekly newsletter with free resources, updates, and special offers. 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15196	https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Mechanics/06._One_Dimensional_Harmonic_Oscillator/Kinetic_Isotope_Effects	Skip to main content Kinetic Isotope Effects Last updated : May 4, 2024 Save as PDF Harmonic Oscillator 7: Angular Momentum Page ID : 1685 ( \newcommand{\kernel}{\mathrm{null}\,}) The kinetic isotope effect (KIE) is a phenomenon associated with isotopically substituted molecules exhibiting different reaction rates. Isotope effects such as KIEs are invaluable tools in both physical and biological sciences and are used to aid in the understanding of reaction kinetics, mechanisms, and solvent effects. Introduction Research was first introduced on this topic over 50 years ago and has grown into an enormous field. The scientists behind much of the understanding and development of kinetic isotope effects were Jacob Bigeleisen and Maria Goeppert Mayer who published the first paper on isotope effects [J. Chem. Phys., 15, 261 (1947)]. Kinetic isotope effects specifically explore the change in rate of a reaction due to isotopic substitution. An element is identified by its symbol, mass number, and atomic number. The atomic number is the number of protons in the nucleus while the mass number is the total number of protons and neutrons in the nucleus. Isotopes are two atoms of the same element that have the same number of protons but different numbers of neutrons. Isotopes are specified by the mass number. As an example consider the two isotopes of chlorine, you can see that their mass numbers vary, with 35Cl being the most abundant isotope, while their atomic numbers remain the same at 17. The most common isotope used in light atom isotope effects is hydrogen () commonly replaced by its isotope deuterium (). Note: Hydrogen also has a third isotope, tritium (). Isotopes commonly used in heavy atom isotope effects include carbon (, , nitrogen (, ), oxygen, sulfur, and bromine. Not all elements exhibit reasonably stable isotopes (i.e. Fluorine, ), but those that due serve as powerful tools in isotope effects. Potential Energy Surfaces Understanding potential energy surfaces is important in order to be able to understand why and how isotope effects occur as they do. The harmonic oscillator approximation is used to explain the vibrations of a diatomic molecule. The energies resulting from the quantum mechanic solution for the harmonic oscillator help to define the internuclear potential energy of a diatomic molecule and are where n is a positive integer (n=1,2,3...), h is Planck's constant and is the frequency of vibration. The Morse potential is an analytic expression that is used as an approximation to the intermolecular potential energy curves: where is the potential energy, is the dissociation energy of the molecule, is the measure of the curvature of the potential at its minimum, is displacement, and is the equilibrium bond length. The , , and variables can be looked up in a textbook or CRC handbook. Below is an example of a Morse potential curve with the zero point vibrational energies of two isotopic molecules (for example R-H and R-D where R is a group/atom that is much heavier than H or D). The y-axis is potential energy and the x axis is internuclear distance. In this figure EDo and EHo correspond to the zero point energies of deuterium and hydrogen. The zero point energy is the lowest possible energy of a system and equates to the ground state energy. Zero point energy is dependent upon the reduced mass of the molecule as will be shown in the next section. The heavier the molecule or atom, the lower the frequency of vibration and the smaller the zero point energy. Lighter molecules or atoms have a greater frequency of vibration and a higher zero point energy. We see this is the figure below where deuterium is heavier than hydrogen and therefore has the lower zero point energy. This results in different bond dissociation energies for R-D and R-H. The bond dissociation energy for R-D (ED) is greater than the bond dissociation energy of R-H (EH). This difference in energy due to isotopic replacement results in differing rates of reaction, the effect that is measured in kinetic isotope effects. The reaction rate for the conversion of R-D is slower than the reaction rate for the conversion of R-H. It is important to note that isotope replacement does not change the electronic structure of the molecule or the potential energy surfaces of the reactions the molecule may undergo. Only the rate of the reaction is affected. Activation Energies The energy of the vibrational levels of a vibration (i.e., a bond) in a molecule is given by where we assume that the molecule is in its ground state and we can compare zero-point vibrational energies, Using the harmonic oscillator approximation the fundamental vibrational frequency is where is the force constant of the bond and is the reduced mass The Arrhenius equation is used to determine reaction rates and activation energies and since we are interested in the change in rate of reactions with different isotopes, this equation is very important, where is the reaction rate, is the activation energy, and is the Arrhenius constant. The Arrhenius equation can be used to compare the rates of a reaction with R-H and R-D, where kH and kD are the rates of reaction associated with R-H and the isotope substituted R-D. We will then assume the Arrhenius constants are equal ( ). The ratio of the rates of reaction gives an approximation for the isotope effect resulting in: By using the relationship that for both R-H and R-D a substitution can be made resulting in The vibrational frequency (Equation 5) can then be substituted for R-H and R-D and the value of the expected isotope effect can be calculated. The same general procedure can be followed for any isotope substitution. In summary, the greater the mass the more energy is needed to break bonds. A heavier isotope forms a stronger bond. The resulting molecule has less of a tendency to dissociate. The increase in energy needed to break the bond results in a slower reaction rate and the observed isotope effect. Kinetic Isotope Effects Kinetic Isotope Effects (KIEs) are used to determine reaction mechanisms by determining rate limiting steps and transition states and are commonly measured using NMR to detect isotope location or GC/MS to detect mass changes. In a KIE experiment an atom is replaced by its isotope and the change in rate of the reaction is observed. A very common isotope substitution is when hydrogen is replaced by deuterium. This is known as a deuterium effect and is expressed by the ratio kH/kD (as explained above). Normal KIEs for the deuterium effect are around 1 to 7 or 8. Large effects are seen because the percentage mass change between hydrogen and deuterium is great. Heavy atom isotope effects involve the substitution of carbon, oxygen, nitrogen, sulfur, and bromine, with effects that are much smaller and are usually between 1.02 and 1.10. The difference in KIE magnitude is directly related to the percentage change in mass. Large effects are seen when hydrogen is replaced with deuterium because the percentage mass change is very large (mass is being doubled) while smaller percent mass changes are present when an atom like sulfur is replaced with its isotope (increased by two mass units). Primary KIEs Primary kinetic isotope effects are rate changes due to isotopic substitution at a site of bond breaking in the rate determining step of a reaction. Example Consider the bromination of acetone: kinetic studies have been performed that show the rate of this reaction is independent of the concentration of bromine. To determine the rate determining step and mechanism of this reaction the substitution of a deuterium for a hydrogen can be made. When hydrogen was replaced with deuterium in this reaction a of 7 was found. Therefore the rate determining step is the tautomerization of acetone and involves the breaking of a C-H bond. Since the breaking of a C-H bond is involved, a substantial isotope effect is expected. Heavy Atom Isotope Effects A rule of thumb for heavy atom isotope effects is that the maximum isotopic rate ratio is proportional to the square root of the inverse ratio of isotopic masses. Expected: Experimental: Secondary KIEs Secondary kinetic isotope effects are rate changes due to isotopic substitutions at a site other than the bond breaking site in the rate determining step of the reaction. These come in three forms: , , and effects. secondary isotope effects occur when the isotope is substituted at a position next to the bond being broken. This is thought to be due to hyperconjugation in the transition state. Hyperconjugation involves a transfer of electron density from a sigma bond to an empty p orbital (for more on hyperconjugation see outside links). Solvent Effects in Reactions Reactions may be affected by the type of solvent used (for example H2O to D2O or ROH to ROD). There are three main ways solvents effect reactions: The solvent can act as a reactant resulting in a primary isotope effect. Rapid hydrogen exchange can occur between substrate molecules labeled with deuterium and hydrogen atoms in the solvent. Deuterium may change positions in the molecule resulting in a new molecule that is then reacted in the rate determining step of the reaction. The nature of solvent and solute interactions may also change with differing solvents. This could change the energy of the transition state and result in a secondary isotope effects. References Baldwin, J.E., Gallagher, S.S., Leber, P.A., Raghavan, A.S., Shukla, R.; J. Org. Chem. 2004, 69, 7212-7219 (This is a great paper using kinetic isotope effects to determine a reaction mechanism. It will interest the organic chemistry oriented reader.) Bigeleisen, J., Goeppert, M., J. Chem. Phys. 1947, 15, 261. Chang, R.; Physical Chemistry for the Chemical and Biological Sciences; University Science Books: Sausalito, CA, 2000, pp 480-483. Isaacs, N.; Physical Organic Chemistry; John Wiley & Sons Inc.: New York, NY; 1995, 2nd ed, pp 287-313. March, J., Smith, M.B.; March’s Advanced Organic Chemistry; John Wiley & Sons, Inc.: Hoboken, NJ, 2007; 6th ed. McMurry, J.; Organic Chemistry; Brooks & Cole: Belmont, CA; 2004, 6th ed. McQuarrie, D.; Quantum Chemistry; University Science Books: Sausalito, CA, 2008, 2nd ed. Rouhi, A.; C&EN. 1997, 38-42. Problems Describe the difference between primary and secondary kinetic isotope effects. Estimate the kN-H/kN-D for a deuterium substitution on nitrogen given that vH=9.3x1013 Hz and the activation energy is equal to 5.31 kJ/mol. Using the 'rule of thumb' for heavy isotope effects, calculate the expected effect for a bromine isotope substitution, 79Br and 81Br. Explain some of the main ways kinetic isotope effects are used. As discussed, the rate-limiting step in the bromination of acetone is the breaking of a carbon-hydrogen bond. Estimate kC-H/KC-D for this reaction at 285 K. (Given: vtildeC-H=3000 cm-1 and vtildeC-D=2100 cm-1) Solutions Primary isotope effects involve isotopic substitution at the bond being broken in a reaction, while secondary isotope effects involve isotopic Substitution on bonds adjacent to the bond being broken. 8.5 1.0126 To determine reaction mechanisms, to determine rate limiting steps in reactions, to determine transition states in reactions. 9.685 Harmonic Oscillator 7: Angular Momentum
15197	https://www.chemguide.co.uk/physical/entropy/moreadvanced.html	Taking entropy changes further TAKING ENTROPY CHANGES FURTHER This page considers various entropy changes: of the system, of the surroundings, and the total change. It goes on to look at how you can use the total entropy change to decide whether or not a reaction is feasible. This is going to be quite a long page. Don't rush it. Note:If your syllabus doesn't specifically mention entropy change terms like system, surroundings and total, you could safely ignore this page. You will still probably have to be able to work out the feasibility of reactions, but that will be done by the rather less confusing use of an equation based on Gibbs free energy. If you do need to read this page, make sure you have read the page explaining how you calculate the entropy change of the system first. Introducing total entropy changes If you only calculate the entropy change of the reaction (the entropy change of the system), you are leaving out an important factor. Suppose your reaction is exothermic. Heat is given off to the surroundings, and that extra heat increases the entropy of the surroundings. If you add more energy to the surroundings, the number of different possibilities for arranging the energy over the molecules increases. And so increasing the temperature increases the entropy of the surroundings. The reverse is true for an endothermic change. An endothermic reaction will cool the surroundings, and so the entropy of the surroundings decreases. What matters is the total entropy change, which is the sum of the entropy changes of the system and the surroundings. ΔS total = ΔS surroundings + ΔS system Note:I have deliberately left out the "standard" symbols in this equation. That is because we shall be using this equation under non-standard conditions. You should be able to judge what is acceptable to your examiners by looking at how this is presented in your syllabus. Calculating the entropy change of the surroundings. So far, you know how to work out the entropy change of the system for a given reaction if you are told the entropies of all the substances involved in the reaction. There is a simple equation for the entropy change of the surroundings. ΔH is the enthalpy change for the reaction. T is the temperature. That seems easy, but there is a major trap to fall in here, and if you manage to get through your course without falling into it at least once, you will have done really well! There is a mismatch between the units of enthalpy change and entropy change. When you quote figures for enthalpy change they will have energy units of kJ. But entropy change is quoted in energy units of J. That means that if you are calculating entropy change, you must multiply the enthalpy change value by 1000. So if, say, you have an enthalpy change of -92.2 kJ mol-1, the value you must put into the equation is -92200 J mol-1. If the temperature was 298 K . . . Notice that the negative sign in the equation converts the negative exothermic enthalpy change into a positive entropy change. An exothermic change heats the surroundings, and increases the entropy of the surroundings. Working out the total entropy change If, for example, the entropy change of the reaction (the system) was +112 J K-1 mol-1, then the total entropy change would be The importance of total entropy change For a reaction to be feasible, the total entropy has to increase - in other words the sign of the total entropy change must be positive. So what does "feasible" mean in reaction terms? A feasible reaction is one that is possible in terms of energy, but it doesn't mean that it will necessarily happen. Although energetically it might be feasible, it may have a large activation energy barrier that will slow it down, or even prevent it from happening altogether at a particular temperature. If the total entropy change is negative (if entropy decreases) then the reaction isn't feasible. Feasible or spontaneous? The word spontaneous is often used in place of feasible. In everyday life, something is spontaneous if it happens of its own accord, without any input from outside. The same thing is true in chemistry, but there is one major difference which defies everyday common sense. If you drop marble chips (calcium carbonate) into dilute hydrochloric acid, there is an immediate fizzing. You don't need to do anything else - the reaction happens entirely of its own accord. It is a spontaneous change. But in chemistry, a spontaneous change doesn't have to be rapid; in fact, it can be very, very, very slow indeed - even infinitely slow! For example, carbon burns in oxygen to make carbon dioxide, but a piece of carbon will stay totally unchanged however long you keep it unless you first heat it. The energetics are right for a reaction to happen, but there is a huge activation energy. Chemistry counts the reaction between carbon and oxygen as spontaneous! Personally, I think that is daft, and I prefer the word "feasible", which is often used in this topic. However, If your examiners use the word spontaneous, then you will be expected to as well. Some examples Example 1: Dissolving ammonium nitrate in water This is a simple example of an endothermic change which nevertheless happens because there is a large increase in disorder when the crystal breaks up into its separate ions and mixes with the water. The entropy change to the surroundings will be negative because of the cooling caused by the ammonium nitrate dissolving, but this is more than made up for by the large increase in the entropy of the system. So the total entropy change is positive, and the change is feasible - actually also literally spontaneous in this instance. Example 2: The reaction between concentrated ethanoic acid and solid ammonium carbonate This is another endothermic change which becomes feasible because the increase in entropy due to the gaseous carbon dioxide formed outweighs the fall in entropy of the surroundings. The total entropy increases and so the reaction is feasible (and again literally spontaneous). Example 3: The reaction between magnesium ribbon and oxygen This is simple to do as a calculation. We will work out the total entropy change if a reaction happened at room temperature (say 293 K). The enthalpy change for the reaction is -1204 kJ mol-1, and the entropy change of the system is -216 J K-1 mol-1. The overall calculation looks like this, with the answer quoted to 3 signficant figures: Don't forget to convert the enthalpy change into joules! There has been a very large increase in entropy overall, so is the reaction feasible? Yes! Is it spontaneous (in the usual meaning of the word) at 293 K (room temperature)? No! Magnesium may tarnish forming magnesium oxide or hydroxide very, very slowly in air, but you need to supply a lot of activation energy to get it to burn. You mustn't assume that all feasible reactions actually happen quickly (or even at all) in the lab. Example 4: The reaction between solid hydrated barium hydroxide and solid ammonium chloride I am including this rather obscure reaction because it is mentioned specifically by one of the UK's A level Exam Boards. The enthalpy change for the reaction is +164 kJ mol-1 assuming the barium chloride is as shown in the equation, or somewhat less if it is formed as BaCl 2,2H 2 O. Assuming it is produced as BaCl 2, the entropy change for the system is +591J K-1 mol-1. That is a high value because of the gas and liquid molecules being formed from two more ordered solids. If you use these figures to calculate the total entropy change, and assuming a temperature of 293 K, you should find that it comes to +31.3 J K-1 mol-1. It is positive, and the reaction is feasible. Note:You will find this reaction described and explained in more detail in this page from the RSC's Practical Chemistry project. That page also contains all the raw data for the calculations involved. Work the whole lot out for yourself! If you find this link doesn't work, please contact me via the address on the about this site page. Example 5: Making benzene from its elements The enthalpy change of the reaction is +49.0 kJ mol-1, and the entropy change of the system is -254 J K-1 mol-1. So it is an endothermic reaction with a decrease in entropy of the system. We don't need to do a calculation with this. It is quicker just to give it a bit of thought! The entropy change of the surroundings is going to be negative because of the minus sign in the equation. Look back at the equation further up the page if you aren't sure. Since the entropy change of the system is also negative, the total entropy change is bound to be negative whatever the temperature you choose. The reaction isn't feasible at any temperature. A quick introduction to Gibbs free energy The next page in this sequence of pages looks at Gibbs free energy and how you can predict the feasibility of reactions using that concept. All I want to do for now is to see how this new term comes from what we have already discussed on this page. You know that: ΔS total = ΔS surroundings + ΔS system You also know how the entropy change of the surroundings is related to the enthalpy change of the reaction: Putting these together, and rearranging slightly by multiplying everything by T to get rid of the fraction gives: The term on the left-hand side is known as the Gibbs free energy, and is given the symbol ΔG. That means that you can write the fairly simple looking equation: ΔG = ΔH - TΔS The ΔS in this version is always just the entropy change of the system. You will see how this equation is used in the next page of this sequence. Questions to test your understanding Because this is all covered in detail in my calculations book I shan't be setting any questions throughout this section on entropy and free energy Where would you like to go now? To the entropy and free energy menu . . . To the Physical Chemistry menu . . . To Main Menu . . . © Jim Clark 2017 (modified May 2025)
15198	https://artofproblemsolving.com/wiki/index.php/Absolute_value?srsltid=AfmBOoqEa2JCkxynqJ6OnekZ0I8gcrxBfcYrzXyuAJlcNhLjMUhe-syJ	Art of Problem Solving Absolute value - 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 Absolute value Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Absolute value The absolute value of a real number, denoted , is the unsigned portion of . Geometrically, is the distance between and zero on the real number line. The absolute value function exists among other contexts as well, including complex numbers. Contents [hide] 1 Real numbers 2 Complex numbers 3 Examples 4 Problems 5 See Also Real numbers When is real, is defined as For all real numbers and , we have the following properties: (Alternative definition) (Non-negativity) (Positive-definiteness) (Multiplicativeness) (Triangle Inequality) (Symmetry) Note that and Complex numbers For complex numbers, the absolute value is defined as , where and are the real and imaginary parts of , respectively. It is equivalent to the distance between and the origin, and is usually called the complex modulus. Note that , where is the complex conjugate of . Examples If , for some real number , then or . If , for some real numbers , , then or , and therefore or . Problems Find all real values of if . Find all real values of if . (AMC 12 2000) If , where , then find . See Also Magnitude Norm Valuation Retrieved from " Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
15199	https://www.nagwa.com/en/lessons/987167464608/	Lesson: Perimeter of a Composite Figure \| Nagwa Lesson: Perimeter of a Composite Figure \| Nagwa Sign Up Sign In English English العربية English English العربية My Wallet Sign Up Sign In My Classes My Messages My Reports My Wallet My Classes My Messages My Reports Lesson: Perimeter of a Composite Figure Mathematics Join Nagwa Classes Attend live Mathematics sessions on Nagwa Classes to learn more about this topic from an expert teacher! Check Available Classes Next Session: Tuesday 30 September 2025 • 2:00pm Number of Seats: One-to-One Class Try This In this lesson, we will learn how to find the perimeters of composite figures. Lesson Plan Students will be able to calculate missing side lengths on diagrams of composite figures, calculate the perimeter of composite figures by summing the side lengths, calculate the area of composite figures by dividing them into squares or rectangles. Lesson Video 17:53 Lesson Playlist 02:20 04:51 +1002:34 Lesson Menu Lesson Lesson Plan Lesson Video Lesson Playlist Join Nagwa Classes Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher! Interactive Sessions Chat & Messaging Realistic Exam Questions Nagwa is an educational technology startup aiming to help teachers teach and students learn. Company About Us Contact Us Privacy Policy Terms and Conditions Careers Tutors Content Lessons Lesson Plans Presentations Videos Explainers Playlists Copyright © 2025 Nagwa All Rights Reserved Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy Accept

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